PART II: THERMOCHEMICAL PROPERTIES 144 CHAPTER 7 ... · Materials Science Fall, 2008 Page 139 PART...

Materials Science Fall, 2008

Page 139

PART II: THERMOCHEMICAL PROPERTIES 144

CHAPTER 7: THERMODYNAMICS 146

7.1 Introduction 146

7.2 Entropy 147 7.2.1 Entropy and time 147 7.2.2 Entropy and heat 149 7.2.3 Entropy and randomness 151

7.3 The Conditions of Equilibrium 154 7.3.1 The equilibrium of an isolated system 154 7.3.2 Internal equilibrium 156 7.3.3 Non-equilibrium states; constrained equilibria 158

7.4 The Thermodynamic Potentials 160 7.4.1 The Helmholtz free energy 161 7.4.2 The Gibbs free energy 163 7.4.3 The work function 165

7.5 The Fundamental Equation 168 7.5.1 The entropy function 168 7.5.2 The energy function 169 7.5.3 Alternate forms of the fundamental equation 172 7.5.4 The integrated form of the fundamental equation 174 7.5.5 The statistical form of the fundamental equation 175

7.6 The Thermodynamics of Surfaces 176 7.6.1 The Gibbs construction 177 7.6.2 The fundamental equation of an interface 178 7.6.3 The conditions of equilibrium at an interface 179

CHAPTER 8: SIMPLE SOLIDS 185


8.2 The Perfect Crystal 186 8.2.1 The internal energy 186 8.2.2 Lattice vibrations 187 8.2.3 The vibrational energy 196 8.2.4 The vibrational contribution to the specific heat 197 8.2.5 A qualitative version of the Debye model 203 8.2.6 The electronic contribution to the specific heat 205 8.2.7 The Helmholtz free energy 209


Page 140

8.2.8 Thermodynamic properties 211

8.3 The Random Solid Solution 213 8.3.1 The Bragg-Williams model 213 8.3.2 The internal energy 214 8.3.3 The configurational entropy 215 8.3.4 The free energy and thermodynamic behavior 216

8.4 Equilibrium Defect Concentrations 218 8.4.1 The equilibrium vacancy concentration 218 8.4.2 Dislocations and grain boundaries 220

CHAPTER 9: PHASES AND PHASE EQUILIBRIUM 222


9.2 Phase Equilibria in a One-Component System 223 9.2.1 Phase equilibria and equilibrium phase transformations 224 9.2.2 Metastability 226 9.2.3 First-order phase transitions: latent heat 226 9.2.4 Transformation from a metastable state 228

9.3 Mutations 228 9.3.1 The Nature of a Mutation 228 9.3.2 Common Transitions that are mutations 229

9.4 Phase Equilibria in Two-Component Systems 232 9.4.1 The free energy function 233 9.4.2 The common tangent rule 236 9.4.3 The phases present at given T, P 239 9.4.4 Equilibrium at a congruent point 241 9.4.5 Equilibrium at the critical point of a miscibility gap 242

9.5 Binary Phase Diagrams 243

9.6 The Solid Solution Diagram 244 9.6.1 The thermodynamics of the solid solution diagram 245 9.6.2 Equilibrium information contained in the phase diagram 247 9.6.3 Equilibrium phase changes 247

9.7 The Eutectic Phase Diagram 249 9.7.1 Thermodynamics of the eutectic phase diagram 249 9.7.2 Equilibrium phase changes 251 9.7.3 Precipitation from the α phase 251 9.7.4 The eutectic microstructure 252 9.7.5 Mixed microstructures in a eutectic system 253 9.7.6 Phase diagrams that include a eutectoid reaction 254


Page 141

9.8 Common Binary Phase Diagrams 254 9.8.1 Solid solution diagrams 255 9.8.2 Low-temperature behavior of a solid solution 256 9.8.3 Phase diagrams with eutectic or peritectic reactions 259 9.8.4 Structural transformations in the solid state 263 9.8.5 Systems that form compounds 265 9.8.6 Mutation lines in binary phase diagrams 271 9.8.7 Miscibility gap in the liquid 272

CHAPTER 10: KINETICS 274


10.2 Local Equilibrium 275

10.3 The Conduction of Heat 277 10.3.1 Heat conduction in one dimension: Fourier's Law 277 10.3.2 Heat conduction in three dimensions 279 10.3.3 Heat sources 281

10.4 Mechanisms of Heat Conduction 282 10.4.1 Heat conduction by a gas of colliding particles 283 10.4.2 Heat conduction by mobile electrons 285 10.4.3 Heat conduction by phonons 287 10.4.4 Heat conduction by photons 292

10.5 Non-equilibrium Thermodynamics 293 10.5.1 The thermodynamic forces 293 10.5.2 The non-equilibrium fluxes and the kinetic equations 294 10.5.3 Simplification of the kinetic equations 295

10.6 Diffusion 298 10.6.1 Fick's First Law for the diffusion flux 298 10.6.2 Fick's Second Law for the composition change 299 10.6.3 Solutions of the diffusion equation 301

10.7 The Mechanism of Diffusion in the Solid State 304 10.7.1 The mobility of interstitial species 304 10.7.2 The mobility of substitutional species 305 10.7.3 Random-walk diffusion; Fick's First Law 307 10.7.4 The mean diffusion distance in random walk diffusion 309 10.7.5 Uses of the mean diffusion distance 311

10.8 Microstructural Effects in Diffusion 313 10.8.1 The vacancy concentration 313 10.8.2 Grain boundary diffusion 315 10.8.3 Diffusion through dislocation cores 317


Page 142

CHAPTER 11: PHASE TRANSFORMATIONS 318

11.1 Common Types of Phase Transformations 318

11.2 The Basic Transformation Mechanisms 319 11.2.1 Nucleated transformations and instabilities 319 11.2.2 First-order transitions and mutations 321

11.3 Homogeneous Nucleation 321 11.3.1 Nucleation as a thermally activated process 321 11.3.2 The activation energy for homogeneous nucleation 323 11.3.3 The nucleation rate 324 11.3.4 The initiation time 326

11.4 Heterogeneous Transformations 327 11.4.1 Nucleation at a grain boundary 327 11.4.2 Other heterogeneous nucleation sites 328 11.4.3 Implications for materials processing 329

11.5 The Thermodynamics of Nucleation 330 11.5.1 The thermodynamic driving force for nucleation 331 11.5.2 Nucleation in a one-component system 332 11.5.3 Nucleation from a supersaturated solid solution 334

11.6 Nucleation of Non-Equilibrium States 336 11.6.1 Congruent nucleation 337 11.6.2 Coherent nuclei 341 11.6.3 Nucleation of metastable phases: the Ostwald rule 343 11.6.4 Sequential Nucleation in a Eutectic 344

11.7 Recrystallization 347

11.8 Growth 349 11.8.1 Primary growth and coarsening 349 11.8.2 Time-temperature-transformation (TTT) curves 350

11.9 Interface-Controlled Growth 351 11.9.1 Isotropic growth of a congruent phase 351 11.9.2 Interface control at stable surfaces 353

11.10 Diffusion-Controlled Growth 357 11.10.1 Growth controlled by thermal diffusion 357 11.10.2 Growth controlled by chemical diffusion 359

11.11 Chemical Segregation During Growth 364

11.12 Grain Growth and Coarsening 366


Page 143

11.13 Instabilities 367

11.14 Martensitic Transformations 368

11.15 Spinodal Decomposition 372 11.15.1 Spinodal instability within a miscibility gap 372 11.15.2 Spinodal decomposition 374 11.15.3 Spinodal decomposition to a metastable phase 377 11.15.4 Use of spinodal decomposition in materials processing 378

11.16 Ordering Reactions 378 11.16.1 Ordering reactions that are mutations 379 11.16.2 First-order transitions that change the state of order 380 11.16.3 Ordering at the stoichiometric composition 382 11.16.4 Ordering at an off-stoichiometric composition 382 11.16.5 Implications for materials processing 384

CHAPTER 12: ENVIRONMENTAL INTERACTIONS 385


12.2 Chemical Changes near the Surface 385 12.2.1 Thermal treatment 385 12.2.2 Diffusion Across the interface 387 12.2.3 Ion implantation 390

12.3 Chemical Reactions at the Surface: Oxidation 392 12.3.1 Thermodynamics of oxidation 392 12.3.2 The kinetics of oxidation 394 12.3.3 Protecting against oxidation 399

12.4 Electrochemical Reactions 402 12.4.1 The galvanic cell 403 12.4.2 The electromotive series and the galvanic series 405 12.4.3 The influence of concentration: concentration cells 407 12.4.4 Reactions at the cathode 408 12.4.5 The influence of an impressed voltage 411 12.4.6 Thermodynamics of the galvanic cell 411

12.5 The Kinetics of Electrochemical Reactions 418 12.5.1 The current in an electrochemical cell 418 12.5.2 The current-voltage characteristic 420

12.6 Aqueous Corrosion in Engineering Systems 427 12.6.1 Corrosion cells in engineering systems 427 12.6.2 The corrosion rate 430 12.6.3 Corrosion protection 431


Page 144

P a r t I I : T h e r m o c h e m i c a l P r o p e r t i e sP a r t I I : T h e r m o c h e m i c a l P r o p e r t i e s

...In war the victorious strategist seeks battle after the victory has been won, while he who is destined for defeat fights first and looks for victory afterwards - Sun Tzu, The Art of War

The thermochemical properties of materials govern two kinds of behavior: internal reactions that occur within the material and determine its microstructure, and environmental interactions that alter the chemistry of the material or its environment. The most important of these are the reactions that are manipulated to control microstructure and fix engineering properties. The management of thermochemical processes within the material is the essence of materials processing. Thermochemical properties also affect behavior in service. If the microstructure evolves as a material is used its engineering properties change accordingly. Such changes must be anticipated and, if necessary, prevented. To understand thermochemical processes it is necessary to recognize the interplay of two central themes: thermodynamics and kinetics. Thermodynamics governs whether a given process is possible and fixes the magnitude of the forces that drive it. Kinetics determines how quickly the process can happen, given the thermodynamic driving force. In many cases, kinetic constraints are used to prevent thermodynamically favorable processes from happening at all. Virtually all useful materials have microstructures that would spontaneously change if they were free to do so. They are, nonetheless, useful because the rate of change is negligible. Silica glasses and amorphous polymers are used routinely despite the fact that the crystalline state is preferred, and crystallization will eventually occur. Almost all metallic structures are polygranular despite the fact that grain boundaries are unfavorable, high-energy defects that will eventually be removed by grain growth. Electronic devices employ silicon chips that are doped to create chemical heterogeneities that will eventually homogenize and disappear. These are all examples of useful microstructures that are maintained by kinetic constraints. Because the kinetics of change are slow, "eventually" is a time much longer than the expected life of the device in which the engineer uses it (and, often, much longer than the expected life of the engineer). Materials are made to have useful microstructures by combining thermodynamics and kinetics in a constructive way. The material is brought into a condition in which the microstructure that is wanted is both thermodynamically possible and kinetically achiev-able. The conditions are then altered, usually by cooling, to prevent further micro-structural changes. For example, glasses are manufactured by melting, then cooling


Page 145

quickly to a temperature at which both the amorphous and the crystalline solid states are thermodynamically preferred to the liquid. Since the amorphous structure is easier to achieve, it forms first. If the composition is such that crystallization is difficult and the glass is kept reasonably cool, it will remain amorphous for as long as desired. Chemically heterogeneous silicon chips are manufactured by allowing chemical species to diffuse into particular regions of the silicon crystal at high temperature, then cooling to a temperature at which the rate of subsequent diffusion is negligible. Almost all other useful engineering materials are made in a conceptually similar way. To appreciate how this is done it is necessary to understand both thermodynamics and kinetics. In the following chapter we discuss thermodynamics. Thermodynamics de-fines the conditions of equilibrium, which lead to the notions of homogeneous phases, which are volumes of material that have uniform structure and composition, and equilib-rium phase diagrams, which are maps that show the phases that are preferred in a material as a function of its temperature and composition. These concepts identify the two ways in which the microstructures of materials evolve: heterogeneities in single-phase regions diminish and vanish with time, and distinct phases appear or disappear as suggested by the phase diagram. The rate at which these processes occur is governed by the kinetics of diffusion in the former case, and the kinetics of structural change in the latter. The kinetics of thermochemical changes will be treated in general, and then specifically applied to the problems of chemical diffusion and structural change in the solid state. The environmental interactions we shall specifically consider are those that govern the deterioration of materials at high temperature or in aqueous environments: high-temperature oxidation and aqueous corrosion. Both are important problems that concern almost all branches of engineering. Finally, we shall consider two types of catalytic behavior: wetting, in which a solid determines the morphology of a second phase that may coat it, and chemical catalysis, in which a solid promotes a chemical reaction in which it does not directly participate. Wetting phenomena are important in coating, bonding and the catalysis of structural change. Chemical catalysis is particularly relevant to chemical engineering, and has given rise to a particular branch of materials science, the science of catalytic materials.


Page 146

C h a p t e r 7 : T h e r m o d y n a m i c sC h a p t e r 7 : T h e r m o d y n a m i c s

The great German physicist, Boltzmann, spent a lifetime deciphering the laws of thermodynamics, and died by his own hand. His intellectual successor, Paul Ehrenfest, also killed himself. We shall now study thermodynamics ... - I can't recall who wrote this. However, per Oscar Wilde , a good quote should not be held responsible for its source.

7.1 INTRODUCTION While the origins of materials engineering are lost in the mists of prehistory, it is arguably possible to date the beginning of materials science. The year was 1876, and the occasion was the publication of a paper by a then obscure Professor of Mathematical Physics at Yale University named Josiah Willard Gibbs. The paper appeared in an even more obscure scientific journal, the Transactions of the Connecticut Academy. It was entitled "On the Equilibrium of Heterogeneous Substances", and it was the first part of a two-part paper (the second part appeared in the same journal in 1878) that discussed how the recently formulated laws of thermodynamics might be used to understand the microstructures of materials. For centuries before Gibbs, perceptive engineers had recognized that the properties of materials were determined by their nature, or, as we would say today, by their composition and microstructure. They also recognized that the nature of a material could be intentionally modified or controlled by processing it in an appropriate way. Even in the ancient world engineers achieved a degree of control over the materials they used that is impressive today. But the rules that governed processing were almost entirely empirical. The competent materials engineer was an artisan who resembled a master chef more than a modern technologist. Recipes were passed from master to apprentice, to be memorized rather than understood. Progress almost invariably reflected accidental discovery or observations derived from casual experimentation. The development of the thermodynamics of materials in the hands of Gibbs and his successors revolutionized materials engineering and turned it into materials science. The keys lay in Gibbs derivation of the fundamental equation that governs the properties of a material, his demonstration that the fundamental equation could be written in alternative forms to define convenient thermodynamic potentials that rule behavior in common experimental situations, and his formulation of the conditions of equilibrium and stability that determine when microstructural changes can occur. Much of this information can be presented in the form of equilibrium phase diagrams that are plots of the equilibrium states of materials under given conditions. The equilibrium phase diagrams identify the state a material will seek under a given set of conditions, for


Page 147

example, a given composition, temperature and pressure, and can be used to infer the microstructural changes that will happen when these conditions are altered. Thermodynamics is a field in its own right that is the subject of a number of courses in various disciplines. I assume that you have some familiarity with it from your general background in chemistry and physics, and will give only a short overview here. The content of thermodynamics that is essential to understand the thermochemical behavior of materials includes the fundamental equation and the conditions of equilibrium. Both of these are consequences of the Second Law of Thermodynamics, which defines the entropy of a system. I shall assume that you are generally familiar with the concept of energy and its conservation, which is the content of the First Law of Thermodynamics, and move immediately to the Second Law. 7.2 ENTROPY The concept of entropy is one of the most fundamental in science, and is also one of the more difficult for the typical student to grasp. A major reason is that entropy is not a thing that is easily seen or measured, and hence does not call up a concrete physical picture of the type that often helps illustrate other fundamental principles of physics. To understand entropy it is useful to think of it in connection with one of the more familiar concepts with which it is intimately associated: time, heat, equilibrium, and probability. 7.2.1 Entropy and time The classical definition of entropy likens it to time, in the sense of real, or evolu-tionary time. To illustrate how and why we consider the simple system shown in Fig. 7.1. The system contains a uniform gas in a container whose walls insulate it from thermal or chemical interaction with its environment. Such a container is called adiabatic, and establishes a situation in which the state of the gas within the container can only be changed by doing mechanical work on it. In the example shown mechanical work can be done by displacing the piston at one end of the container or by rotating the paddle-wheel in its interior. The state of the gas is defined by its fixed chemical composition ({N}, the set of atom or mole numbers of the components it contains), by its volume (V), which can be changed by displacing the piston, and by its internal energy (E), which can be changed in the positive sense by compressing the piston or turning the paddle wheel to do mechanical work on the system, and in the negative sense by displacing the piston to expand the gas so that it does work on its environment. It is found experimentally that all of the possible states of a gas in an adiabatic container can be divided into three sets on the basis of their accessibility from the initial state. The first set (1) includes states that can be reached in a reversible way from the initial state. These states are achieved by displacing the piston so slowly and smoothly that friction and turbulence are negligible. The initial state can be recovered by slowly returning the piston to its original position. The second set (2) includes all other states that can be reached from the initial state without changing the nature of the system or its


Page 148

adiabatic container. The transitions that lead to these states are irreversible: the system can never return to its original state as long as it remains adiabatic. All states that are reached by processes that stir the gas by turning the paddle wheel fall into this category. One can introduce an arbitrary amount of energy into the gas by turning the paddle wheel, but once this is done and the gas has become quiescent the wheel will never spontaneously turn itself to do work on the environment. The third set (3) includes all states that are inaccessible in the sense that they cannot be obtained from the initial state by any adiabatic process at all. These states have the property that they could only be reached if the energy could be decreased at constant volume, but this could only happen if gas spontaneously turned the paddle wheel.

Fig. 7.1: A gas in an adiabatic container fitted with a piston and a stirrer. Each of the three sets of states that are defined above includes an infinite number of distinct states that have a common property: their accessibility from the initial state by adiabatic processes (adiabatic accessibility). Since adiabatic accessibility can be measured experimentally, it is possible to give this property a name and define it in an objective way. We call this property the entropy, S, and define it so that states that are mutually accessible have the same entropy, those that are irreversibly accessible have greater entropy, and those that are inaccessible have lower entropy. By changing the initial state it is possible to define the entropy of all the states of the system. In any adiabatic process ÎS ≥ 0. If ÎS = 0 the process is reversible; if ÎS > 0 it is not. The definition of the entropy and its monotonic behavior in adiabatic processes form the content of the Second Law of Thermodynamics. The concept of entropy may appear to be restricted by the fact that it was defined for a particular type of system, the adiabatic system. In fact, this is not a restriction. An example of an adiabatic system is an isolated system, which does not interact with its environment at all. Any system can be made into an isolated system by joining it to the environment with which it interacts. The composite system is adiabatic and, hence, can only evolve in such a way that its entropy increases or remains the same. The end of its evolution, its preferred or ultimate equilibrium state, is that which has the greatest possible entropy. In a very real sense the entropy is the thermodynamic measure of time. The entropy of an adiabatic system orders its states in time. Those whose entropies are higher must necessarily come after those whose entropies are lower. Since the entropy can only


Page 149

increase, time moves forward and cannot be reversed. The association between entropy and time is not casual, but fundamental; the Second Law of Thermodynamics is the only fundamental principle of theoretical physics in which time has a direction. 7.2.2 Entropy and heat A body of material that is homogeneous in its composition and structure is called a phase. The thermodynamic state of a homogeneous phase of a simple material, such as the gas of Fig. 7.1, is fixed by its energy, its volume and its chemical content, that is, by the variables E, V, and {N}. Since each state of the system has an entropy, S, the entropy can be written as a function S = ¡S(E,V,{N}) 7.1 of the variables that characterize the state. It is possible to choose the entropy so that the entropy function, eq. 7.1, is a continuous function of its variables that is additive in the sense that the joint entropy of a composite of two or more systems is the sum of their individual entropies. Moreover, S can be defined so that the partial derivatives of the function ¡S(E,V,{N}) have the following simple physical values:

∆¡S

∆E = 1T 7.2

where T is the absolute temperature,

∆¡S

∆V = PT 7.3

where P is the mechanical pressure, and

∆¡S

∆Nk = -

µkT 7.4

where Nk is the mole number of the kth component and µk is its chemical potential. This definition of S is called the metrical entropy. As we shall discuss below, the entropy function, eq. 7.1, that expresses the entropy as a function of the thermodynamic content of the system is Gibbs' fundamental equation that governs the thermodynamic behavior of the system. When the entropy is given by equation 7.1 its change in an infinitesimal change of state, that is, a change of state that involves an infinitesimal change in one or more of the variables, E, V, and {N}, is


Page 150

dS =

∆¡S

∆E dE +

∆¡S

∆V dV + ∑k

∆¡S

∆Nk dNk

= 1T (dE + PdV - ∑

k µkdNk) 7.5

Solving equation 7.5 for the incremental change in the energy, dE, yields the relation dE = TdS - PdV + ∑

k µkdNk 7.6

Eq. 7.6 has a simple physical interpretation. If the state is changed quasi-statically, that is, slowly enough that friction can be ignored, then the second term on the right-hand side, -PdV, is the mechanical work done. The third term on the right is the en-ergy due to the chemical change, {dN}, and is hence the chemical work. The energy change that is unaccounted for is that due to the thermal interaction, the thermal work, or heat, dQ. Hence in a quasi-static change of state dQ = TdS 7.7 and there is a direct association between the entropy change and the heat, or thermal work. The association between the entropy change and the quasi-static heat that is ex-pressed by eq. 7.7 makes it relatively easy to measure the entropy of a phase as a function of its energy and composition. If the system is heated incrementally and quasi-statically at constant volume and composition then the change in entropy is given by the change in energy, which can be found independently, divided by the absolute temperature. It is also possible to evaluate the second and third terms on the right-hand side of equation 7.6 independently, so the entropies of states that differ in volume and composition can be re-lated to one another. Such measurements evaluate the entropy with respect to its value in a fixed reference state. It is, however, possible to proceed further to define the absolute value of the metrical entropy. The Third Law of Thermodynamics states that in the limit of zero temperature the most stable state of a system is a state of perfect order whose entropy may be set equal to zero. If this state is used as a reference the metrical entropy is fixed to within a multiplicative constant that sets the scale of the temperature. When the process that changes the state of the system is not quasi-static, for example, when friction cannot be neglected, then the simple association between the heat added and the change in the entropy is lost. Since energy is conserved and the change in energy is equal (by convention) to the sum of the work done on the system and the heat added to it then dE = dQ + dW 7.8


Page 151

for any infinitesimal change of state whatever. (In the context of the First Law of Thermodynamics eq. 7.8 is the definition of the heat.) Since non-static phenomena such as friction or turbulence increase the energy of the system (work is done on the system by the frictional forces or the stirring that introduces turbulence), it follows from eqs. 7.6 and 7.8 that dW ≥ - PdV + ∑

k µkdNk 7.9

and hence that dQ ≤ TdS 7.10 The entropy change is always greater than or equal to the heat added divided by the tem-perature. The additional entropy is due to internal changes in the system of the sort that are caused by frictional forces or turbulence. Since these processes are inherently irreversible, the inequality (7.10) measures the irreversibility of a change of state. 7.2.3 Entropy and randomness The metrical entropy of a homogeneous phase of an isolated simple system can be calculated statistically from the relation S(E,V,{N}) = k ln„(E,V,{N}) 7.11 where „(E,V,{N}) is the degeneracy of the phase. If the atoms contained in the system can be treated as classical particles then the degeneracy is the total number of distinct ways of assigning positions and momenta to the particles that are consistent with the definition of the phase and yield the right value of the total energy. In the context of quantum mechanics the degeneracy is the total number of distinct quantum states of the system that are consistent with its energy and phase. We shall not prove eq. 7.11 here, but will show that the entropy defined by 7.11 is consistent with the nature of the entropy as we have described it. Properties of the statistical entropy First, eq. 7.11 predicts that the preferred phase of an isolated system is the one that maximizes its entropy. Suppose that there are two possible phases of an isolated system with parameters E, V, and {N}. Let the phases be designated å and ∫, and let them have, respectively, degeneracies „å and „∫. If there is no constraint on the system that prevents it from taking on the configurations of either phase then its instantaneous state is equally likely to be in any one of the possible states of either phase. The probability that an instantaneous measurement of the state of the system will find it in phase å is


Page 152

På = „å

„å + „∫ 7.12

which is greater than P∫ if „å > „∫, or, by eq. 7.11, if Så > S∫. In the usual case the system is virtually certain to be in phase å if its entropy is greater; the statistical degen-eracy of a state, „, increases exponentially with the number of particles it contains, and „å >> „∫ if it is greater at all. It follows that an isolated system evolves into the phase (or mixture of phases) that has the highest statistical entropy. Second, the statistical entropy is additive. Let system 1 have energy E1, volume V1 and chemical content {N1} while system 2 has E2, V2 and {N2}. When they are sepa-rated, system 1 has degeneracy „1(E1,V1,{N1}) and system 2 has degeneracy „2(E2,V2,{N2}). The degeneracy of the two separated systems taken together is the product „0 = „1„2 7.13 since each individual state in „1 can coexist with any state in „2, and conversely. Hence the joint entropy of the separated systems is S0 = k ln(„1„2) = k ln(„1) + k ln(„2) = S1 + S2 7.14 and the statistical entropy is additive. Third, if two isolated systems are joined so that they interact with one another the interaction can only increase the statistical entropy. When the two systems are joined and interact all of the states in „1 and „2 are still possible. Since the total energy, volume and chemical content are E = E1+ E2, V = V1 + V2 and {N} = {N1 + N2}, it is still possible for the volume V1 to have the chemical content {N1} and the energy E1, in which case V2 contains {N2} and E2. The total system has at least the degeneracy „1„2 and the entropy S1 + S2. But the interaction between the two systems creates the possibility of new states that, depending on the nature of the interaction, correspond to new distributions of E, V or {N}. If the interaction does result in a redistribution of E, V or {N} the states associated with the new distribution add to the degeneracy. Hence, after the interaction, „ ≥ „0 = „1„2 7.15 and S ≥ S0 = S1 + S2 7.16 It follows that the statistical entropy of an isolated system can only increase.


Page 153

The statistical entropy of a system that is not isolated can also be calculated. The method is given in standard texts on Statistical Mechanics. We limit this discussion to the simpler, isolated system. The statistical entropy of a solid of given energy The degeneracy in the energy states of a typical crystalline solid has three principle sources. First, the valence electrons in the solid can be distributed in many different ways over the quantum states available to them, which produces the electronic entropy, Se. The electronic entropy is much higher in a metal than in a semiconductor or insulator. There are many empty electron states that have energies comparable to those that are occupied by the valence electrons, and hence many distinguishable ways in which the electrons can be distributed without changing the energy or violating the Pauli Exclusion Principle. The electronic entropy of a semiconductor or insulator is much smaller since almost all of the valence electrons are confined to particular atomic or bonding states that are very nearly full. Second, a crystalline solid has a vibrational entropy, Sv, that is due to the thermal oscillations of its atoms about their equilibrium positions on the crystal lattice. The small displacements associated with the lattice vibrations significantly increase the degeneracy of the crystalline phase (the degeneracy remains finite because the motions of the individual atoms are correlated in quantized vibrational states called phonons). The vibrational entropy decreases with the strength and directionality of the crystal bonds, which inhibit atom displacements, and increases with the openness of the crystal structure. Hence very stable, high-melting solids tend to have relatively low vibrational entropies, and metals with the more open BCC structure tend to have higher vibrational entropies than the same metals in the close-packed structures. Third, a multi-component solid has a configurational entropy, Sc, which is due to the many different ways in which the various chemical species can be distributed over the different atom sites in its structure. The number of distinct configurations is relatively easy to calculate, and it is useful to do so. Assume that NA atoms of type A and NB of type B are distributed over N = NA + NB atom sites. There are N! different ways of distributing N particles over N sites (the number of permutations of an ordered list of N objects). However, distributions that differ only through the interchange of A atoms with one another or B atoms with one another are physically indistinguishable. Since there are NA! ways of redistributing the A atoms over the sites occupied by A atoms in a given configuration, and NB! ways of redistributing the B atoms, the total number of distinguishable configurations is

„ = N!

NA!NB! 7.17

If the energy of the crystal is approximately independent of the way in which the atoms are distributed then its configurational entropy is


Page 154

Sc = k ln(„) = k [ln(N!) - ln(NA!) - ln(NB!)] 7.18 This expression can be simplified considerably by using Sterling's Approximation, ln(N!) = N ln(N) - N 7.19 which is a very good approximation when N is greater than about 10. The configurational entropy can then be written Sc = k [N ln(N) - NA ln(NA) - NB ln(NB)] = - kN [x ln(x) + (1-x) ln(1-x)] 7.20 where x is the atom fraction of component A, x = NA/N. An amorphous solid or glass has an additional entropy due to the irregularity of its spatial configuration; the atom positions are not confined to a crystal lattice but are distri-buted in a less regular way. It follows that the entropy of an amorphous configuration of atoms or molecules is always greater than that of a crystalline state of the same material. Low-temperature behavior: the Third Law The Third Law of Thermodynamics states that the entropy of the preferred, or ultimate equilibrium phase of a system vanishes in the limit of zero absolute temperature. This is equivalent to the statement that the ground state of the system, the state of lowest energy, is unique (non-degenerate). The electronic and vibrational entropies vanish naturally at zero temperature. The electrons are reduced to their ground state. While the atoms continue to vibrate in the zero-temperature limit, these ground state vibrations are quantum phenomena that have no entropy since they are associated with a single quantum state. However, the configurational entropy is independent of the temperature. It vanishes at zero temperature only if the solid takes on a perfectly ordered state that has no configurational entropy. This leads to the important conclusion that the preferred state of a solid at low temperature is a perfectly ordered crystal or macromolecule. 7.3 THE CONDITIONS OF EQUILIBRIUM 7.3.1 The equilibrium of an isolated system If the entropy of an isolated system can be increased by an infinitesimal change in its state, then that change will inevitably occur. The reason is that any real system is in constant thermal agitation on a microscopic scale. Through local fluctuations it constantly samples thermodynamic states that are incrementally close to whatever


Page 155

macroscopic state it is currently in. If any of these nearby states have higher entropy the system reaches them, but then cannot return. It necessarily evolves in the direction of increasing entropy. Its evolution continues until it finds itself in a state that provides a local extremum in the entropy such that no infinitesimally adjacent state increases the entropy further. The states that correspond to local extrema in the entropy can be maintained, at least momentarily, and are hence possible equilibrium states. They satisfy a condition of equilibrium that can be written (∂S)E,V,{N} ≤ 0 7.21 where ∂S is the infinitesimal change in the entropy that would result if the state of the system were changed infinitesimally at the given values of E, V and {N}, and the inequality holds for every possible infinitesimal change. There are four kinds of equilibria that satisfy the condition expressed in the inequality 7.21, of which only two are of any real interest. The four are illustrated in Fig. 7.2, where we have assumed that a single parameter, x, describes the path between states in order to make a two-dimensional plot (since the possible states differ in microstructure their differences are described by many independent variables; the entropy should be plotted in a multi-dimensional space).

S

X

metastable

unstable

stable

Fig. 7.2: Illustration of four kinds of equilibrium: metastable, unstable (inflection point), unstable (minimum), and stable equilibrium.

A local minimum or saddle point in the entropy creates a state of unstable equilib-rium. Such states satisfy the mathematical condition for equilibrium but cannot be pre-served in practice since they are unstable under small, finite perturbations. A local maxi-mum in the entropy defines a state that is stable with respect to small changes and can, in principle, be preserved for a very long time. However, if the locally stable state does not provide the maximum entropy for all possible states of the system then it will transform if it is given a sufficiently large perturbation in the right direction. Such states are called metastable states. They must eventually evolve to the state of maximum entropy , or


Page 156

stable equilibrium state, since the natural fluctuations of the system will eventually cause an appropriate perturbation, however large that perturbation must be. How long a metastable state can be preserved is a kinetic issue that depends on the size of the perturbation that is required to change it and the frequency with which perturbations occur. Many engineering materials are used in metastable states that are preserved almost indefinitely, including all amorphous materials and glasses and many crystalline solids. 7.3.2 Internal equilibrium The general condition of equilibrium, equation 7.21, leads to specific conditions of thermal, mechanical and chemical equilibrium that are satisfied in all equilibrium states. To find the conditions of internal equilibrium we consider two subsystems of the system that have volumes V1 and V2, energies E1 and E2 and chemical contents {N1} and {N2}, and are in contact with one another, as illustrated in Fig. 7.3. If the system is to be in equilibrium its entropy must not increase if the energy, volume or chemical content of the two subsystems is redistributed between them.

1 2

Fig. 7.3: Two subsystems within an isolated system that interact only with one another.

Thermal equilibrium First let the two subsystems exchange an infinitesimal amount of energy without changing the volume or chemical content of either. This exchange describes a thermal interaction. Since energy is conserved the energy gained by subsystem 1 must be lost by subsystem 2. Hence dE2 = - dE1 7.22 By eq. 7.5 the total entropy change in the process is

dS = dS1 + dS2 = dE1T1

+ dE2T2

= dE1

1

T1 -

1T2 7.23


Page 157

where T1 and T2 are the temperatures of the two subsystems. Eq. 7.23 shows that the system can only be in equilibrium if T1 = T2. If this is not the case then dS is positive for a transfer of energy from the high-temperature subsystem to the low-temperature one. Hence the two subvolumes are not in equilibrium with one another unless their tempera-tures are the same. The same reasoning applies to any choice of subvolumes within the system. It also applies when the system is not isolated, since a system cannot be in equilibrium if it is out of equilibrium with respect to internal changes that do not affect its environment. We are therefore led to the condition of thermal equilibrium, which holds in general: a system in equilibrium has a uniform temperature. T = const. 7.24 Mechanical equilibrium A second possible interaction between two subsystems is a mechanical interaction in which they distort one another. Mechanical interactions in a solid can be rather complicated; they may be elastic in the sense that they stretch the bonds without rearranging the atoms, or plastic, in the sense that the atoms are permanently displaced. We will discuss these deformations at a later point in the course. However, if the system is in mechanical equilibrium then it must at least be in equilibrium with respect to the simple, fluid-like mechanical interaction that exchanges volume between the two subsystems by displacing the boundary between them, and must satisfy the condition of mechanical equilibrium that governs that interaction. Even then the mechanical interaction is complicated by the presence of external fields, such as the gravitational field, that impose mechanical forces directly on the material. However, external fields, including the gravitational field, have a negligible effect in most situations and we shall ignore them (the usual negligibility of gravity is one reason why materials processing in space has turned out to be less attractive than many had hoped, although there are potential applications that may prove important). Assuming fluid-like deformation and neglecting external fields, the mechanical interaction displaces the boundary between the two subvolumes so that dV2 = - dV1 7.25 Since energy is conserved (dE2 = - dE1) and temperature is constant (thermal equilibrium), the associated change in the entropy is

dS = 1T [P1dV1 + P2dV2] =

dV1T [P1 - P2] 7.26


Page 158

If P1 ≠ P2, dS is positive when the volume of the high-pressure subsystem increases at the expense of that of the low-pressure one. We are hence led to the condition of mechanical equilibrium in the absence of external fields: a system in mechanical equilibrium in the absence of external fields has a uniform pressure. P = const. 7.27 Chemical equilibrium The third type of interaction between the two subsystems is a chemical interaction in which they exchange matter. Since any one of the independent chemical components may be exchanged there is a separate condition of chemical equilibrium for each. As in the case of mechanical equilibrium, the conditions of chemical equilibrium are affected by external fields since external fields such as the gravitational field apply forces to the individual atoms that do work when they are moved. We assume that external fields can be neglected. Since the total amount of the kth component is conserved, the quantity added to V2 is equal to that lost from V1, dN2

k = - dN1k 7.28

Letting the exchange occur at constant volume (we have already found the condition of mechanical equilibrium) the entropy change at constant total energy and uniform temperature is

dS = - 1T µ1

kdN1k + µ2

kdN2k = -

1T µ1

k - µ2k dN1

k 7.29

where µi

k is the chemical potential of the kth component in the ith subsystem. If µ1k ≠

µ2k the entropy increases if a quantity of the kth component is transferred from the

system of higher chemical potential to that in which the potential is lower. We are therefore led to the condition of chemical equilibrium in the absence of external fields: a system in chemical equilibrium in the absence of external fields has a uniform value of the chemical potential of each of its components. µk = const. 7.30 7.3.3 Non-equilibrium states; constrained equilibria The materials that are used in engineering are not usually in equilibrium states, even in the sense of metastable equilibrium. They are in non-equilibrium states that evolve continuously. The materials can be used as if their properties were constant only because the rate of evolution is so slow that it can be neglected.


Page 159

A familiar example of a non-equilibrium material is a semiconductor that has been chemically doped to create islands that contain local concentrations of electrically active solutes. The chemical potentials of the solute species are not uniform; the material evolves toward an equilibrium state in which the solutes are spread uniformly through the semiconductor crystal. However, non-equilibrium behavior can ordinarily be neglected because the materials are used at relatively low temperature where the rate of solute diffusion is so slow that the solute distribution changes very little over the useful life of the device. A second example is a polygranular crystalline solid. The crystal grains never have equilibrium shapes. Their surfaces tend to become slightly curved to achieve local mechanical equilibrium. It can be shown that the chemical potential is slightly higher on the convex side than on the concave side of the grain boundary. Atoms move across the grain boundary in response to the chemical potential gradient so that the grain boundary migrates toward its center of curvature. The smaller grains disappear with time and the average grain size grows. However, at the normal temperatures at which polygranular materials are used the rate of grain growth is negligible. A third example is a crystalline solid that contains dislocations. The dislocations are non-equilibrium defects. They exert forces on one another and, given sufficient time, move by glide and climb to annihilate by interacting with one another or with free surfaces. Again, this process is very slow at ordinary temperatures, so the dislocation distribution is nearly fixed in the absence of mechanical forces that are large enough to force dislocation motion. To apply thermodynamics to the behavior of non-equilibrium systems like these, which include almost all real materials, we treat them as idealized systems in constrained equilibria, that is, we analyze the behavior of a hypothetical system that is physically similar to the material of interest, but includes physical constraints that maintain it in thermodynamic equilibrium. Formally, we replace the kinetic constraints that maintain the non-equilibrium state by imaginary physical constraints that accomplish the same purpose in a static way. For example, an idealized polygranular solid whose grain boundaries are rigid, impermeable membranes behaves in many ways like a real polygranular solid, but its grains cannot grow. One can gain insight into the thermochemical behavior of polygranular solids by considering the behavior of solids with impermeable grain boundaries, which can reach thermodynamic equilibrium in the polygranular state. A silicon crystal in which certain regions have been doped with active solutes behaves at low temperature very much like a hypothetical crystal in which the doped regions are surrounded by impermeable membranes. The low-temperature thermochemical behavior of a dislocated solid is very much like that of a hypothetical solid whose dislocations are artificially pinned in space. We implicitly use constrained equilibrium models of this sort almost whenever we apply thermodynamics to real systems. The constraints fix those non-equilibrium features of the microstructure that remain nearly constant with time. The conditions of equilibrium govern those features of the microstructure that are kinetically capable of


Page 160

reconfiguration during the time of experimental interest. The microstructural evolution of a real solid can be incorporated in this kind of model by periodically relaxing the hypothetical constraints so that the grains can grow, the solutes migrate or the dislocations move. 7.4 THE THERMODYNAMIC POTENTIALS The condition of maximum entropy is sufficient to analyze the equilibrium states of any system; it applies to an isolated system, and any system can be made part of an isolated system by joining it to its environment. However, it is usually inconvenient to do this. An engineering material is almost never used in a condition that can be regarded as isolated. It interacts with its environment. The relevant isolated system is a composite system that contains both the material and the environment that interacts with it. But the physical nature of the environment is rarely of interest. In engineering practice the "environment" may be nothing more than a heat source, such as an oven or furnace, that is used to maintain temperature, or a mechanical linkage that maintains the load on a material that is used as a structural member, or an electrical condenser that sets the electric field in a material that is used as a capacitor. Its only function is to establish the conditions under which the material is used; its detailed internal state is uninteresting. If the environment is regarded as part of the system to which the conditions of equilibrium are applied then one has to worry about it. It is far preferable to find an alternate way of phrasing the conditions of equilibrium so that they can be applied to a system that is not isolated, and involve only the state of the material itself. We accomplish this by representing the environment as a thermodynamic reservoir. A thermodynamic reservoir is a system that is so large compared to the system of interest that it can interact by exchanging energy, volume or chemical species without its own state being affected in any sensible way. The reservoir is assumed to be in equilibrium to the extent that it has a well-defined temperature, pressure and set of chemical potentials. Since its state is unaffected by its interaction with the system the values of these intensities remain constant, and the reservoir serves to fix their values within the system. When the environment can be approximated as a reservoir it is always possible to treat the behavior of the system in terms of its own properties and thermodynamic states without reference to the physical nature of the environment. However, when the environment is treated as a reservoir the thermodynamic quantity that governs the equilibrium of the system is not its entropy, but an appropriate thermodynamic potential whose identity depends on the nature of the interaction between the system and environment. The thermodynamic potentials that are most often useful are the Helmholtz free energy, which governs the equilibrium of a system with fixed volume and chemical content that interacts with a thermal reservoir, and the Gibbs free energy, which governs the equilibrium of a system with fixed composition that interacts with a reservoir that sets both its temperature and its pressure. Other common potentials include the enthalpy, which governs a system of fixed entropy and composition that interacts with a pressure reservoir (the enthalpy is frequently used in fluid dynamics, but


Page 161

rarely in materials science), and the work function, which governs the behavior of an open system, a system that is enclosed by imaginary boundaries that fix its volume, and interacts with a reservoir that sets its temperature and chemical potentials. The various thermodynamic potentials are readily defined by considering the vari-ous experimental situations to which they naturally apply. Solids are ordinarily used in one of three experimental situations. First, a solid of given composition and volume has its temperature controlled by a reservoir. An example is a material that is heated in a furnace, when we can neglect the thermal expansion of the solid and the possibility of chemical interaction with the atmo-sphere in the furnace. In this case the controlled variables are V, {N} and T, and, as we shall see, the relevant thermodynamic potential is the Helmholtz free energy, F = E - TS. Second, and more realistically, the solid has a constant composition, but has both its temperature and pressure controlled by the environment. The temperature may be controlled by a furnace, or, simply, by ambient air, and the pressure may be fixed by the ambient, or by some mechanical device that loads the solid. In this case controlled variables are T, P and {N}, and the relevant thermodynamic potential is the Gibbs free energy, G = E - TS + PV. Third, the solid has a fixed size and shape, and its temperature and chemical potentials controlled by its environment. The common example is a small subvolume within a larger body, which we define by simply drawing an imaginary boundary around it. This open system has fixed volume, but can freely exchange energy and matter with its environment. In this case the controlled variables are T, V and {µ} (the set of chemical potentials), and the relevant thermodynamic potential is the work function, „ = E - TS - ÍµkNk. We consider each case in turn. 7.4.1 The Helmholtz free energy Consider a system that has fixed volume, V, and chemical content, {N}, and is in contact with a reservoir with fixed temperature, T. The system and reservoir together form an isolated system as shown in Fig. 7.4. The system can exchange energy with the reservoir through a thermal interaction across its boundary, but the boundary is rigid and impermeable. The system (1) and reservoir (2) together make up an isolated system. By eq. 7.21 the system is in equilibrium only if every possible infinitesimal change of state leads to a decrease in the entropy, that is, if (eq. 7.21) (∂S)E,V,{N} ≤ 0 7.21 Assume an infinitesimal transfer of energy (∂E) between the system and the reser-voir. The reservoir acquires the energy increment ∂E2 = - ∂E1. The total entropy change in the system is ∂S1, which may include an entropy increment due to internal changes as


Page 162

well as that associated with the energy transfer. The entropy change in the reservoir may also include internal changes, but these must be positive, and hence cannot affect the equilibrium of the system. We can, therefore, neglect them and use eq. 7.5 to write

∂S2 = ∂E2T = -

∂E1T 7.31

{N},V

T

dE

Fig. 7.4: A system of fixed volume and composition that is in contact with a thermal reservoir.

The total entropy change is

∂S = ∂S1 - ∂E1T = -

1T[ ]∂E1 - T∂S1

= - 1T ∂[E1 - TS1] ≤ 0 7.32

where the last form of the right-hand side follows since T is constant. Defining the quan-tity F1 = E1 - TS1 7.33 and using the fact that the temperature, T, is constant and the same in the system and the reservoir, the condition of equilibrium becomes (∂F1)T,V,{N} ≥ 0 7.34 The quantity F = E -TS is called the Helmholtz free energy. Eq. 7.34 states that if a system with fixed volume and chemical composition interacts with a thermal reservoir, its behavior is governed by its Helmholtz free energy, F, which has a minimum value for an equilibrium state. It is useful to examine the functional dependence of the Helmholtz free energy. Using eq. 7.6, the total differential of F is


Page 163

dF = dE - d(TS) = TdS - PdV + ∑k

µkdNk - TdS - SdT

= - SdT - PdV + ∑

k µkdNk 7.35

If the Helmholtz free energy is evaluated as a function of the temperature, T, volume, V, and composition, {N}, as F = ¡F(T,V,{N}) 7.36 then

dF =

∆¡F

∆T dT +

∆¡F

∆V dV + ∑k

∆¡F

∆Nk dNk 7.37

It follows by comparison with equation 7.35 that

∆¡F

∆T = - S 7.38

∆¡F

∆V = - P 7.39

∆¡F

∆Nk = µk 7.40

which show the dependence of the Helmholtz free energy on its natural variables, T, V and {N}. 7.4.2 The Gibbs free energy

{N}

P, T

dE

dV

Fig. 7.5: A system with fixed composition, {N}, that is in mechanical and thermal contact with a reservoir with fixed P and T.


Page 164

In a second common situation (which is, ordinarily, more realistic) the system has a given chemical content, but interacts with a reservoir that fixes both its temperature and pressure. This case is diagrammed in Fig. 7.5. The system can exchange volume and en-ergy with the reservoir through mechanical and thermal interactions at its boundary, but the boundary is impermeable. As in the previous example, the system (1) and reservoir (2) together make up an isolated system, which is in equilibrium only if every possible infinitesimal change of state leads to a decrease in the entropy. Assume an infinitesimal transfer of energy (∂E) between the system and the reservoir that is accompanied by an infinitesimal displacement of the boundary so that the volume of the system changes by ∂V. The reservoir acquires the energy increment ∂E2 = - ∂E1 and the volume increment ∂V2 = - ∂V1. The total entropy change in the system is ∂S1. The relevant entropy change in the reservoir can be written, according to eq. 7.5,

∂S2 = 1T[ ]∂E2 + P∂V2 = -

1T[ ]∂E1 + P∂V1 7.41

Hence the total entropy change is

∂S = ∂S1 - 1T[ ]∂E1 + P∂V1 = -

1T[ ]∂E1 - T∂S1 + P∂V1

= - 1T ∂[E1 - TS1 + PV1] ≤ 0 7.42

where the last form of the right-hand side follows since T and P are constant. Defining the quantity G1 = E1 - TS1 + PV1 7.43 and using the fact that T and P are constant, the condition of equilibrium becomes (∂G1)T,P,{N} ≥ 0 7.44 The quantity G = E -TS + PV is called the Gibbs free energy. Eq. 7.44 states that if a system with fixed chemical content interacts both thermally and mechanically with a reservoir that fixes its pressure and temperature, its behavior is governed by its Gibbs free energy, G, which has a minimum value for an equilibrium state. The functional dependence of the Gibbs free energy can also be found with the help of eq. 7.6. The total differential of G is dG = dE - d(TS) + d(PV)


Page 165

= - SdT + VdP + ∑k

µkdNk 7.45

If the Gibbs free energy is written as a function of the temperature, T, pressure, P, and composition, {N}, as G = ¡G(T,P,{N}) 7.46 then

dG =

∆ ¡G

∆T dT +

∆¡G

∆P dP + ∑k

∆¡G

∆Nk dNk 7.47

It follows by comparison with eq. 7.45 that

∆¡G

∆T = - S 7.48

∆¡G

∆P = V 7.49

∆¡G

∆Nk = µk 7.50

which show the dependence of the Gibbs free energy on its natural variables, T, P and {N}. In real systems the environment usually fixes the pressure and temperature, so the Gibbs free energy is the pertinent thermodynamic potential. However, when the system is a solid at atmospheric pressure, which is the case that is most frequently of interest, the PV term is almost always negligible compared to F and there is virtually no numerical difference between the Gibbs and Helmholtz free energies. 7.4.3 The work function

T, {µ}

VdEdN


Page 166

Fig. 7.6: An open system, with fixed volume, in contact with a reser-voir that fixes T and {µ}.

We are often interested in behavior in the interior of a material. Since regions in the interior of a material can, ordinarily, exchange both energy and chemical content with one another, the best way to study their behavior is usually by defining an open system, of the kind diagrammed in Fig. 7.6. A volume within the material is surrounded by an imaginary boundary. Since the boundary is fixed in space, its volume is given. However, its energy and chemical content are not. The material beyond the boundary acts as a thermal and chemical reservoir that fixes the temperature and chemical potentials within the system. Again, the combination of system (1) and reservoir (2) make up an isolated system, which is in equilibrium only if every possible infinitesimal change of state leads to a decrease in the entropy. Assume an infinitesimal transfer of energy (∂E) that may be accompanied by infinitesimal transfers (∂Nk, k = 1,...,n) of each of the n chemical species present. The reservoir acquires the energy increment ∂E2 = - ∂E1 and the chemical additions ∂N2

k = - ∂N1k , k = 1,...,n. The total entropy change in the system is ∂S1. The

entropy change in the reservoir is (eq. 7.5),

∂S2 = 1T

∂E2 - ∑

k µk∂N2

k = - 1T

∂E1 - ∑

k µk∂N1

k 7.51

Hence the total entropy change is

∂S = ∂S1 - 1T

∂E1 - ∑

k µk∂N1

k = - 1T

∂E1 - T∂S1 - ∑

k µk∂N1

k

= - 1T ∂

E1 - TS1 - ∑

k µkN1

k ≤ 0 7.52

where the last form of the right-hand side follows since T and the chemical potentials, µk, are held constant by the reservoir. Defining the work function, „ = E - TS - ∑

k µkNk 7.53

the condition of equilibrium becomes (∂„)T,V,{µ} ≥ 0 7.54 Eq. 7.54 states that if a system with fixed boundaries interacts both thermally and chemi-cally with a reservoir that fixes the temperature and chemical potentials its behavior is governed by its work function, „, which has a minimum value for an equilibrium state.


Page 167

Using eq. 7.6, the total differential of „ is d„ = dE - d(TS) - d

∑

k µkNk

= - SdT - PdV - ∑

k

Nkd µk 7.55

If the work function is written as a function of the temperature, T, volume, V, and chemical potentials, {µ}, as „ = ¡„(T,V,{µ}) 7.56 then

d„ =

∆ ¡„

∆T dT +

∆¡„

∆V dV + ∑k

∆¡„

∆µk dµk 7.57

It follows by comparison with eq. 7.55 that

∆¡„

∆T = - S 7.58

∆¡„

∆V = - P 7.59

∆¡G

∆µk = Nk 7.60

which show the dependence of the work function on its natural variables. A fourth thermodynamic potential, the enthalpy, H = E + PV 7.61 governs the behavior of systems whose entropy, pressure, and chemical content are con-trolled. This is seldom the situation in Materials Science, where the enthalpy is rarely used for any purpose other than as a shorthand notation for the sum E + PV. However, the enthalpy is often the preferred potential in fluid mechanics, since entropy is locally conserved in many types of fluid flow.


Page 168

7.5 THE FUNDAMENTAL EQUATION The various thermodynamic functions defined in the previous section are different and equivalent forms of the fundamental equation of the system. The fundamental equation is a concept introduced by Gibbs, who recognized that there is a single thermodynamic function that contains a complete description of the thermodynamic state of a material, provides values for all of its thermodynamic properties, and governs equilibrium. 7.5.1 The entropy function The entropy function, S = ¡S(E,V,{N}) 7.62 is the form of the fundamental equation that is provided directly by the Second Law of thermodynamics. It is the most convenient form of the fundamental equation when the system of interest is isolated (or adiabatic). It then has the following features. First, the natural variables of the entropy function, E, V and {N}, are precisely the variables that can be controlled experimentally when the system is isolated. Their values are fixed by setting the content of the system at the time it is isolated, and cannot be altered afterwards. Second, the values of E, V, {N} are sufficient to fix the equilibrium state of an isolated system. The Second Law asserts that the equilibrium state has a maximum value of the entropy, S, with respect to all other ways of configuring the system with the given values of E, V and {N}. The entropy function, 7.62, is just the function that gives the en-tropy of the equilibrium state as E, V and {N} are varied. Of course, there may be physical or kinetic constraints on the system that limit the configurations it can take on, and set it in a metastable equilibrium state or a constrained equilibrium state. In that case the entropy function incorporates the constraints; the entropy has the largest value it can have for given E, V, and {N} when those constraints are imposed. Third, the conjugate forces, T, P and {µ}, are determined by the first partial derivatives of the entropy function

∆¡S

∆E = 1T 7.63

∆¡S

∆V = PT 7.64

∆¡S

∆Nk = -

µkT 7.65


Page 169

That is, each thermodynamic force is specified by the partial derivative of the entropy function with respect to the thermodynamic quantity that is conjugate to it. Fourth, the thermodynamic properties of the system are specified by the second and higher derivatives of the fundamental equation. The thermodynamic properties are material properties that govern the changes in the thermodynamic contents, E, V, and {N}, with the values of the thermodynamic forces. The most commonly used are the specific heat, which governs the change of energy with temperature, the compressibility, which governs the change of volume with pressure, and the coefficient of thermal expansion , which governs the change of volume with temperature. The thermodynamic properties are discussed in more detail in the following section. Finally, the fundamental equation can be transformed so that its independent vari-ables are changed from the set E, V and {N} to some other set that is more natural for a particular experimental situation. The simplest transformation is accomplished by solving the entropy function for the energy. The result is the energy function, E = ¡E(S,V,{N}) 7.66 which, as we shall see, contains precisely the same information as the entropy function, but with the different set of independent variables {S,V,{N}} replacing {E,V,{N}}. Since the energy function contains the same information, it is also a form of the fundamental equation. Other useful forms of the fundamental equation are obtained by applying a technique known as the Legendre transform to the energy function. The various Legendre transforms of the energy function are the thermodynamic potentials we discussed in the previous section. 7.5.2 The energy function The independent variables that appear in the energy function (7.66) are the set S, V and {N}. While we shall not prove it here, it is a consequence of the Second Law that the energy of an equilibrium state has the minimum value that it can have for any way of reconfiguring the system at given values of S, V and {N}. This is the energy minimum principle, which you have probably encountered fairly frequently in elementary physics. It is the complement of the entropy maximum principle that the entropy has the largest possible value for given E, V and {N}. The thermodynamic quantities S, V and {N} that appear as independent variables in the energy function (7.66) characterize the system; the equilibrium state of the system is that which has the least value of the energy for the given values of S, V and {N}. The function ¡E(S,V,{N}) gives the minimum (equilibrium) value of the energy. While it is not easy to construct experimental systems in which the entropy is con-trolled, there are hypothetical situations in which the variables S, V and {N} are fixed, and these are important in theoretical analysis. First, in the limit of low temperature the Third Law of thermodynamics asserts that the entropy of a system approaches a least


Page 170

value that is independent of V and {N}; in this limit the value of the entropy is fixed. Since it is much easier to compute the energy of a solid than its entropy, most of the available theoretical analysis of the solid state concerns the identification and properties of minimum energy states. Since many stable crystalline solids assume their minimum energy structures at temperatures well above room temperature, this approach is often fruitful. In many other cases we can usefully discuss the properties of solids at finite temperature by identifying the properties of the minimum-energy state and inferring the changes that should occur as the temperature is raised. We have already used this approach to discuss the electronic configurations of atoms and solids in terms of their minimum-energy, ground states. The change in the energy in an infinitesimal change in the state is dE = TdS - PdV + ∑

k µkdNk 7.67

where the successive terms on the right-hand side correspond to the thermal, mechanical and chemical work done if the change of state is accomplished without turbulence or fric-tion. Note that each term in this expression has the form of a thermodynamic force, T, P or µk, multiplied by the differential change in one of the thermodynamic quantities, S, V or Nk. Each of the forces, f = T, P or µk, is said to be conjugate to the corresponding quantity, x = S, V or Nk, in that the work done by that force in an infinitesimal change of state is obtained by multiplying it by the differential change in its conjugate quantity. To complete the specification of the state of a system that has given values of S, V and {N} we need to know the values of the conjugate forces, T, P and {µ}. These are determined from the fundamental equation by the first partial derivatives of the energy function

∆¡E

∆S = T 7.68

∆¡E

∆V = - P 7.69

∆¡E

∆Nk = µk 7.70

That is, each thermodynamic force is given by the partial derivative of the function ¡E(S,V,{N}) with respect to its conjugate quantity. The thermodynamic properties of the system are the materials properties that govern the changes in the values of the thermodynamic forces with a change of state. For example, the isometric specific heat, CV, governs the change in the temperature with a


Page 171

differential change in the energy at constant volume and composition according to the relation

dT = dECv

7.71

while the isentropic compressibility, ˚S, gives the change in pressure on a differential change in volume at constant entropy and composition according to the relation

dP = dVV˚S

7.72

These and the other first-order thermodynamic properties of the system are given by the second partial derivatives of the energy function. For example,

∆2¡E

∆S2 =

∆T

∆S V{N} =

TCV

7.73

where the subscripts on the partial derivative indicate that it is to be taken at constant vol-ume and composition, and the final form of the right-hand side follows from the fact that if a sample is heated reversibly at constant V and {N} the change in entropy is

dS = dQT =

dET =

CVdTT 7.74

The isentropic compressibility is given by the second derivative of the energy function with respect to the volume,

∆2¡E

∆V2 = -

∆P

∆V S{N} =

1V˚S

7.75

where the isentropic compressibility is defined as

˚S = - 1V

∆V∆P S{N}

7.76

If the system is an n-component fluid there is a total of (n+2)2 second partial derivatives, of which only 12(n+2)(n+3) are independent since the value of a second partial derivative is independent of the order of differentiation. The only other first-order thermodynamic property that is commonly given a name is the isentropic coefficient of thermal expansion,

åS = 1V

∆V∆T S{N}

7.77


Page 172

This property is related to the cross-derivative of the energy with respect to entropy and volume:

∆2¡E

∆S∆V =

∆T

∆V S{N} =

1VåS

7.78

The third and higher derivatives of the energy function are higher order thermody-namic properties that determine how the first-order thermodynamic properties vary with a change in state. To recapitulate, the energy function is a form of the fundamental equation. Its in-dependent variables are a sufficient set of thermodynamic quantities to determine the thermodynamic state. Its first partial derivatives with respect to these quantities give the thermodynamic forces. Its second and higher partial derivatives give the thermodynamic properties of the system. Moreover, the fundamental equation determines the conditions of equilibrium for a system in which the variables S, V and {N} are experimentally controlled; by the minimum energy principle the energy has a minimum value for all possible changes in the state of the system that maintain the total values of these quantities. 7.5.3 Alternate forms of the fundamental equation Ordinarily the variables that are controlled when a material is processed, used, or experimented on are some mixture of thermodynamic forces and quantities. For example, when the system is in a rigid, impermeable, diathermal container, T, V and {N} are controlled. Note, however, that one can never control a thermodynamic force and its conjugate thermodynamic quantity at the same time; to control the temperature of a system it must interact thermally with a reservoir, and hence its entropy cannot be controlled, to control the pressure it must interact mechanically with a reservoir, and hence its volume cannot be controlled, to control its chemical potential it must interact chemically, and hence the composition cannot be controlled. More generally, if we divide the thermodynamic variables into the conjugate pairs (S,T), (V,P), (Nk,µk) then we can control only one variable within each conjugate pair during an experiment. As we have already seen, it is useful to express the condition of equilibrium for a given experimental situation in terms of the minima of that thermodynamic potential whose natural variables are the variables that are controlled experimentally (maxima in case of the entropy of an isolated system). We can now see that when these thermodynamic potentials are written as functions of their natural variables they are alternate forms of the fundamental equation.


Page 173

Consider, for example, the Gibbs free energy, which governs the equilibrium of a fluid system whose temperature, pressure and composition are controlled. In terms of its natural variables, the Gibbs free energy is given by the function G = E - TS + PV = ¡G(T,P,{N}) 7.79 We have already found that the first partial derivatives of the Gibbs free energy are

∆¡G

∆T = - S 7.48

∆¡G

∆P = V 7.49

∆¡G

∆Nk = µk 7.50

Hence the thermodynamic quantities and forces that are not fixed by the experimental situation are determined by the first derivatives of the Gibbs free energy function. The second derivatives of the Gibbs free energy function given the first-order thermodynamic properties. These appear in a slightly different form from those derived from the energy function. For example, the second partial derivative of the Gibbs free energy with respect to the temperature is

∆2¡G

∆T2 = ∆

∆T

∆¡G∆T = -

∆S

∆T P{N} = -

CPT 7.80

which defines the isobaric specific heat, CP, rather than the isometric specific heat, CV. The second derivative of ¡G(T,P,{N}) with respect to the pressure is

∆2¡G

∆P2 = ∆

∆P

∆¡G∆P =

∆V

∆P T{N} = - V˚T 7.81

where ˚T is the isothermal compressibility,

˚T = - 1V

∆V∆P T{N}

7.82

The cross-derivative of ¡G(T,P,{N}) with respect to temperature and pressure is

∆2¡G

∆T∆P = ∆

∆T

∆¡G∆P =

∆V

∆T P{N} = VåT 7.83


Page 174

where åT is the coefficient of thermal expansion,

åT = 1V

∆V∆T P{N}

7.84

While the thermodynamic properties defined by second derivatives of the Gibbs free energy function differ slightly from those derived from the energy function, it can be shown that these (and the properties defined by the second derivatives of any other thermodynamic potential) can be derived from those defined by the second derivatives of the energy function. There is a general mathematical method for doing this, called the Jacobian method, which we shall not review here. The same results follow for any of the other thermodynamic potentials defined in the previous section. When they are written as function of their natural variables their first derivatives give the values of the thermodynamic forces and quantities that are conjugate to the set of independent variables, and the second and higher partial derivatives give the thermodynamic properties. When the natural variables of a given thermodynamic potential are the variables that are experimentally controlled, that potential governs the equilibrium of the system. Hence, as Gibbs recognized, the problem of determining the macroscopic equilib-rium behavior of a material is reduced to the problem of finding its fundamental equation. This can be done experimentally, or, in principle, theoretically using the methods of statistical thermodynamics. 7.5.4 The integrated form of the fundamental equation The energy of a system can be written in the integrated form E = TS - PV + ∑

k µkNk 7.85

To prove this equation, we use the fact that E, S, V and N are all additive quantities. If we double the size of the system without changing its internal state, we simply double the value of each. However, the energy is a function of S, V and {N}, E = ⁄E(S,V,{N}). This function must have the mathematical property that, if we multiply each of the independent variables, S, V, and {N}, by the same constant, å, the energy is also multiplied by å. That is, ⁄E(åS,åV,{åN}) = å⁄E(S,V,N) = åE 7.86 Functions that have this property are called homogeneous functions of the first order. If we now differentiate both sides of eq. 7.86 with respect to å (which we can do, since å can be any number), the result is


Page 175

∆ ¡E

∆(åS) S +

∆¡E

∆(åV) V + ∑k

∆¡E

∆(åNk) Nk = E 7.87

Since eq. 7.87 holds for any value of å, it holds if å = 1. Setting å = 1 and using eqs. 7.68-70 yields eq. 7.85. Given eq. 7.85, the integrated forms of the other common thermodynamic poten-tials are: F = E - TS = - PV + ∑

k µkNk 7.88

G = E - TS + PV = ∑

k µkNk 7.89

„ = E - TS - ∑

k µkNk = - PV 7.90

Eq. 7.90 reveals why „ is called the work function of the system; it is given by the product of pressure and volume. Eq. 7.89 shows that the Gibbs free energy is associated with the total chemical energy of the material. 7.5.5 The statistical form of the fundamental equation One can, in principle, calculate the fundamental equation of a material by applying the techniques of Statistical Thermodynamics. In the formal sense, these are straight-forward, and were introduced in Section 7.2.3 when we discussed the relation between the entropy and the degeneracy, or randomness of the system. In that case we considered a material that was isolated from its environment, so that its energy, E, volume, V, and composition. {N} were fixed. Suppose that we are able to identify every state that this material can possibly have, consistent with the rules of quantum mechanics. The possible states include the various ways of configuring the ion cores into static equilibrium configurations, the various ways of distributing the electrons among the allowable electron states for a given distribution of the ion cores, and the possible vibrational states of the ion cores about their equilibrium positions, under the restriction that all of these states have the same energy, E. If the number of these states (the degeneracy of the system) is „(E,V,{N}), then the entropy is given by S(E,V,{N}) = k ln„(E,V,{N}) 7.11 and we have evaluated the fundamental equation of the material. The problem, of course, is counting all the states. This is always a difficult exercise, and is made particularly


Page 176

difficult by the restriction that the states must have the same energy. When the configuration of atoms or electrons is changed, the energy ordinarily changes as well. It is almost always easier to evaluate the Helmholtz free energy, F(T,V,{N}). To do this we let the material have a given volume and composition, but let it exchange energy with a reservoir that fixes its temperature. In this case the state of the material can have any energy, provided that it has the given volume and composition. Let an arbitrary state be denoted by the index, n, and let the material have energy, En, when it is in its nth state. To compute the fundamental equation of the solid we form the canonical partition function,

Z(T,V,N) = ∑n

e- ∫En 7.91

where the coefficient, ∫ = 1/kT and the sum is taken over all admissible states. The parti-tion function, Z, is a function of T, V and N since the energy of the nth state depends on the volume of the system and the number of particles, while the coefficient, ∫, is the reciprocal temperature. The Helmholtz free energy, ¡F(T,V,N), is obtained directly from the partition function by the relation ¡F(T,V,N) = - kT ln[Z(T,V,N)] 7.92 Once the Helmholtz free energy has been found all other thermodynamic potentials and properties can be computed from it. The statistical relations that evaluate the fundamental equation, eqs. 7.91-92, are formally simple. The problem is to identify and compute the energies of all the possible states, which is not simple at all. While a number of interesting problems have been solved, at least approximately, with the techniques of statistical thermodynamics, in almost all cases the thermodynamic properties of real materials must be measured experimentally. 7.6 THE THERMODYNAMICS OF SURFACES External surfaces and internal interfaces in solids influence their behavior in many important ways. Their energies influence the shapes of solids, including grains and phases within solids, and, as we shall see in a later chapter, have a strong influence on whether and where a new phase will form when it is thermodynamically favorable for it to do so. Their permeability controls the exchange of material from phase to phase. Interfaces often have chemical content of their own. Adsorbed species influence the reactivity of a solid, and may strongly affect its mechanical properties. For example, superficially minor concentrations of metalloid impurities in structural metals, such as sulfur or phosphorous in steel, can cause catastrophic embrittlement when they are adsorbed on grain boundaries.


Page 177

The thermodynamics of interfaces is made difficult by their complex structure. The interfaces that separate phases in contact are not strict discontinuities. They are rather thin transition shells across which the materials properties and thermodynamic densities change from the values appropriate to one phase to those appropriate to the other. The reason for the thickness of the interface is relatively straightforward: the two phases perturb one another over a distance that is at least equal to the effective range of atomic interaction. Even in the simplest case, a crystalline solid that presents a close-packed surface to a vacuum, the atomic packing of the first few layers below the surface is distorted; there is an asymmetry in bonding since there are no atoms beyond the surface. The interface structure is difficult to predict, and often impossible to observe. Even the best modern characterization tools reveal very little about the internal structure of real interfaces. We are hence faced with the problem of describing the behavior of an inhomogeneous material whose internal structure we know very little about. The thermodynamics of surfaces was developed by Gibbs (Equilibrium of Hetero-geneous Substances), who devised a mathematical technique that acknowledges the finite thickness of the interface while avoiding it in a simple formal way. The method uses a geometric representation of the interface, the Gibbs construction, together with the as-sumption that while the internal state of the interfacial shell may be unknown it is fixed by equilibrium with the bulk phases on either side. 7.6.1 The Gibbs construction

å

∫

transition shelldividing surface

... Fig. 7.7: The Gibbs model of an interface. The three-dimensional

transition shell is replaced by a two-dimensional surface. The model that is used for the thermodynamics of surfaces is illustrated in Fig. 7.7. The figure shows two homogeneous phases separated by an interface. The transition shell between the two phases is shaded, and includes the whole volume of material that is perturbed by the interface. In the Gibbs construction the three-dimensional transition shell is replaced by a hypothetical, two-dimensional dividing surface, which is placed so that it is roughly coincident with the physical interface. The bounding phases are imagined to extend homogeneously up to the dividing surface from either side. The model system is then given a thermodynamic content that duplicates that of the actual


Page 178

transition shell by defining surface excess quantities of energy, entropy and mass that are imputed to the dividing surface itself. Let EI, SI and NI

k be the actual quantities of energy, entropy and mole number of the kth chemical species contained in a segment of the interfacial shell. Let the dividing surface that replaces that segment have area, A, and let it divide the volume, V, of the transition shell into subvolumes Vå and V∫, which lie on the å and ∫ sides of the interface, respectively. If the thermodynamic content of the model of the interfacial shell is to reproduce the content of the actual shell, then the dividing surface must have surface excesses of energy, ES, entropy, SS, and chemical content, NI

k , such that ES = EI - Eå

V Vå - E∫V V∫ 7.93

SS = SI - Så

V Vå - S∫V V∫ 7.94

NS

k = NSk - nå

k Vå - n∫k V∫ (k = 1,...,m) 7.95

where EV and SV are the energy and entropy per unit volume, nk is the molar density of the kth component, and there are m chemical components in the system. The surface excess quantities are said to be adsorbed on the surface. Their densities are two-dimensional (quantity per unit area), and are given by

ES = ES

S 7.96

SS = SS

S 7.97

©k = N

Sk

S (k = 1,...,m) 7.98

7.6.2 The fundamental equation of an interface Since the thermodynamic state of the transition region is given by its entropy, chemical content and volume, the state of the surface can be characterized by the adsorbed entropy and chemical species and the surface area. (As Gibbs showed, this is true even when the surface is curved, if the dividing surface is placed properly at the interface.) It follows that the fundamental equation for the interface can be written ES = ⁄ES(SS, {NS}, A) 7.99


Page 179

By analogy to the energy function for a bulk phase, the partial derivatives of this function with respect to entropy and chemical content define the temperature and chemical potentials at the interface:

∆¡ES

∆SS = T 7.100

∆¡ES

∆NSk

= µk 7.101

The change of energy with area defines the interfacial tension, ß:

∆¡ES

∆A = ß 7.102

The interfacial, or surface tension is the two-dimensional analog of the pressure, and is the force that resists the extension of the surface. It is necessarily positive: ß > 0 7.103 since, if it were negative, the surface would grow spontaneously. Since the adsorbed energy, entropy and chemical content of a homogeneous inter-face increase linearly with its area, the fundamental equation of the surface has the inte-grated form (cf. Sec. 7.5.4 above) ES = TSS + ßA + ∑

k µkNS

k 7.104

7.6.3 The conditions of equilibrium at an interface Since the interface can exchange energy and chemical species with either of the adjacent phases, it can be easily shown that, when the system is in equilibrium, its temperature, T, and chemical potentials, µk (k = 1,...,m), are the same as those in the surrounding phases. It follows that the bulk phases act as a thermochemical reservoir for the interface, which is, effectively, an open system. The controllable variables are T, {µ} and A, where the temperature and chemical potentials are controlled by setting their values in the bulk phases, and A is the area of the interface. As discussed in Sec. 7.4.3, the equilibrium of an open system requires that its work function, „ = E - TS - ∑

k µkNS

k 7.105


Page 180

have a minimum value for given values of T, V and the set {µ}. The work function of the interface is „S = E - TSS - ∑

k µkNS

k = ßA 7.106

Equation 7.54 shows that the surface tension is just the surface excess of the work function per unit area, just as the pressure is the (negative) work function per unit volume of the bulk. The general condition of equilibrium is, by analogy to eq. 7.54, (∂„S)T,{µ} = ∂(ßA)T,{µ} ≥ 0 7.107 that is, the interface is in equilibrium only if every possible change in the state of the interface that does not alter its temperature or chemical potentials increases „S. The possible changes in the interface are of two types: those that change the state of the interface at fixed area (that is, those that change the internal state of the transition shell), and those that change the area. Consider each of these in turn. Changes in the internal state of the interface If we fix the area, eq. 7.105 becomes (∂ß)T,{µ} ≥ 0 7.108 Hence, the equilibrium state of the interface is that which has the least value of ß for given values of T and {µ}. If we were trying to compute the internal structure of the interface, we would do that by comparing various possible structures and accepting that which leads to the lowest value of ß. It is useful to recognize that the condition 7.108 can be read in reverse: any spontaneous change in the state of the interface must decrease its interfacial tension. Many important interfacial phenomena can be easily understood on the basis of this simple rule. For example, an impurity that is adsorbed onto a solid surface from the atmosphere will remain on the surface if its presence decreases ß, but will be drawn into the interior, or repelled back into the atmosphere, if it increases ß. An impurity in a polygranular solid, such as sulfur in steel, will segregate to grain boundaries if it lowers the interfacial tension of the boundary, but will otherwise remain in the bulk. Two solids will spontaneously bond together only if their interfacial tension is less that the sum of their surface tensions in air. We will encounter other examples later in the course. Equilibrium at a curved interface Next, consider changes in the shape of the interface at constant interfacial tension, for example, when a curved interface is deformed or displaced. In the simplest example,


Page 181

the interface is spherical, and separates phases (call them å and ∫) that have different pressures (På and P∫), as illustrated in Fig. 7.8. If the radius, R, of the interface changes, it also changes the volumes of the bounding phases. The work function for the system shown in the figure is „ = - PåVå - P∫V∫ + ßA 7.109 When R is changed, infinitesimally, to R + ∂R the change in „ is ∂„ = - (P∫ - På)4πR2∂R + ß(8πR)∂R

= - 4πR2

(P∫ - På) - 2ßR ∂R 7.110

The change in „ can always be made negative, by choosing the sign of ∂R, unless the bracketed term vanishes. Hence the sphere is only in equilibrium with respect to expansion or contraction of the interface if

(P∫ - På) = 2ßR 7.111

Note that equilibrium is only possible if the pressure inside the sphere, P∫, is greater than that outside, På. In the limit of a plane interface, R “ ∞, the pressures are the same.

R

∂R

På P ∫

... Fig. 7.8: A sphere of phase ∫, with pressure, P∫, and radius, R, enclosed

by a surface and embedded in phase å with pressure På. Equation 7.111 is a special case of a condition of mechanical equilibrium that ap-plies to curved surfaces in general. Whatever the shape of a curved surface, it is always possible to characterize its local curvature by measuring its radius of curvature along two perpendicular axes that lie in the surface. Its mean curvature, –K is related to these two radii by the equation


Page 182

–K = 1

R1 +

1R2

7.112

The condition for mechanical equilibrium across the interface is, then, (P∫ - På) = ß–K 7.113 where P∫ is the pressure in the interior, which is defined as the side of the interface that contains the shortest radius of curvature. In the case of a sphere, –K = 2/R, which regener-ates eq. 7.111. In the case of a cylinder, –K = 1/R, where R is the cylinder radius. Equilibrium at a three-phase junction line A two-phase interface cannot simply end; it either closes on itself, as it does in the case of the sphere shown in Fig. 7.8, or terminates along a line where three interfaces meet, as illustrated in Figs. 7.9 and 7.10. We distinguish two types of three-phase junc-tion lines. In the first type, which is exemplified by the line around the periphery of a drop of oil floating on water and by a three-grain junction line in a polygranular solid, all three phases can change shape by growth or mechanical deformation. In this case the three interfaces met at angles that are, ordinarily, not all that far from 120º, as illustrated in Fig. 7.9.

å

∫

©

ßå∫

ßå©

ß∫©

... Fig. 7.9: View perpendicular to a three-phase junction line where phases

å, ∫ and © meet. In the second case, which is exemplified by a liquid droplet sitting on a solid sur-face, one of the phase is much more rigid than the other two, and the surface of that phase continues straight through the junction line (Fig. 7.10). To find the equilibrium configuration at a three-phase junction line it is simplest to use a force balance, in which each interface is imagined to exert a perpendicular pull on the junction line with a force per unit length equal to its surface tension, ß. The justification for this approach is illustrated in Fig. 7.11, which shows an element of surface that terminates in a junction line. If a unit length of the junction line is displaced normal to itself, as shown in the figure, the work done is W = ∂„S = ß∂A 7.114


Page 183

which is exactly equal to the work done by a force, ß, per unit length of junction line that acts perpendicular to it. It follows that equilibrium can only be obtained at a junction line if the three interfacial tensions are in balance.

å

∫

©

ß

ß ß

å∫

å©∫©

R

Fig. 7.10: A liquid-like droplet (∫) sitting on a rigid surface (©). The pe-

riphery of the droplet is a three-phase junction line where the å∫ and ∫© interfaces form the contact angle, œ.

In the case illustrated in Fig. 7.9, the force balance leads to an equation that is known as the Neumann triangle of forces. If we define the vector force, ßij, as a force that has a magnitude equal to the interfacial tension of the ij interface, and a direction that lies in the plane of the ij interface perpendicular to the three-phase junction line, then the condition of equilibrium at the line illustrated in Fig. 7.9 is that ßå∫ + ß∫© + ßå© = 0 7.115 When the three tensions are identical, as they are, for example, when the interfaces are grain boundaries in a solid that is isotropic in its surface properties, then the equilibrium condition is that the three interfaces make angles of 120º to one another.

dAß

Fig. 7.11: Illustration of the action of the surface tension on a three-phase

junction line. In the case illustrated in Fig. 7.10, the force balance leads to a relation that is known as the Young equation. Since the substrate solid is rigid, we need only balance forces parallel to the substrate surface. These forces are in balance if the å∫ interface meets the © surface at an angle, œ, the contact angle, that satisfies the relation

cos(œ) = ßå© - ß∫©

ßå∫ 7.116


Page 184

Of course, cos(œ) must have a value that lies between -1 and 1. If the right-hand side of eq. 7.116 is greater than 1 or less than -1, the equation has no solution. In the former case the ∫ phase spreads over the © surface to form a continuous film, and is said to wet the surface. In the latter case, the ∫ phase is repelled by ©, and is separated from it by a thin film of å. The macroscopic shape of a crystal If a crystal grows in free contact with the atmosphere then it can take on any shape it chooses. The preferred shape is that which minimizes the total surface energy „S = ∑

k ßkAk 7.117

where ßk is the surface tension of the kth element of the external surface of the crystal, Ak is its area, and the sum is taken over the whole external surface. Every crystal is at least slightly anisotropic in its surface tension; close-packed or, in ionic crystals, electrically neutral surface planes invariably have lower surface tension. On the other hand, the total surface area is minimized if the crystal is a sphere, which uses all planes, and increases as the crystal develops facets with preferred orientations. The macroscopic shape of a crystal reflects the competition between the drive to minimize area, which favors a spherical shape, and the drive to present low-tension sur-faces, which favors a faceted shape. In most metallic and covalently bonded solids the anisotropy in the surface tension is not great enough to drive a faceted shape. The faceted "crystals" that form in nature are almost always strongly ionic materials whose interfacial anisotropy is due to the fact that only certain planes are electrically neutral.


Page 185

C h a p t e r 8 : S i m p l e S o l i d sC h a p t e r 8 : S i m p l e S o l i d s

Once out of nature I shall never take My bodily form from any natural thing But such a form as Grecian goldsmiths make Of hammered gold and gold enamelling - William Butler Yeats, "Sailing to Byzantium"

8.1 INTRODUCTION While it is difficult to calculate the thermodynamic properties of solids with numerical accuracy, it is relatively easy to develop a qualitative understanding of them. For this purpose, it is useful to consider the three model solids: a perfect crystal, a ran-dom solid solution with near-neighbor bonding, and a slightly imperfect crystal. First, we shall consider the perfect crystal. Its energy can be written as the sum of three terms: the net binding energy at zero temperature, which is the binding energy when the atoms are located at their equilibrium positions on the crystal lattice, the vibrational energy, which is the energy of atom motion about the equilibrium positions, and the electronic energy, which is the energy due to thermal excitations of the electrons. As we have defined it, the binding energy depends only on the volume of the crystal. The vibra-tional and electronic energies increase with temperature. That increase, which is mea-sured by the specific heat, CV, is primarily due to the thermal excitation of lattice vibra-tions. In most solids, the vibrational specific heat is governed by a material property, the Debye temperature, ŒD, which provides a rough measure of the energy required to excite all possible lattice vibrations. In the low-temperature limit (T < ŒD/4), the specific heat increases with the cube of the temperature, Cv fi T3, essentially because more and more lattice vibrational modes are excited as the temperature is raised. At high temperature, T > ŒD, the specific heat approaches a constant, Cv « 3Nk, essentially because all possible lattice vibrations are excited. The electrons also contribute to the specific heat since they are also excited to higher energy levels as the temperature is raised. However, because of the Pauli exclusion principle, only those electrons that are very close to the Fermi energy can be excited. Since the number of such electrons is always small, their contribution to Cv is negligible at ordinary temperatures. Given the specific heat, it is possible to find the thermal contribution to the Helmholtz free energy of a crystal, which establishes the form of the fundamental equa-tion. Differentiating this equation leads to the entropy, and differentiating it again pro-duces the compressibility and the coefficient of thermal expansion (as well as regenerat-ing Cv). The entropy depends on the Debye temperature. The compressibility and coef-ficient of thermal expansion depend both on ŒD and on its derivative with respect to volume, which is specified by a material property known as the Grªuneisen parameter, ©.


Page 186

Second, we shall consider a random solid solution. When the solid is a solution with more than one chemical component, as are most of the materials of interest to us, its free energy is affected by the configurational entropy that arises from the many different ways in which distinct kinds of atoms can be arranged over the lattice sites. When all configurations are equally likely, the configurational energy can be calculated as de-scribed in the previous chapter. We use it to compute the fundamental equation and ex-plore the behavior of a simple solution, in which each atom is assumed to bond to its nearest neighbors only. The model illustrates how the configurational entropy dominates behavior when the temperature is sufficiently high, and produces mutual solubility be-tween chemical species that would segregate apart (or form ordered compounds) at lower temperature. Finally, we shall consider an imperfect crystal and calculate the equilibrium den-sity of defects as a function of temperature. The results show that the crystal always con-tains vacancies, whose concentration increases exponentially with the temperature. However, the other, high-energy defects, such as dislocations and grain boundaries, are non-equilibrium features that would seldom be found if an unconstrained equilibrium were easily attained. 8.2 THE PERFECT CRYSTAL 8.2.1 The internal energy Consider a perfect, crystalline solid that contains a given number of atoms (N) in an essentially fixed volume (V), and whose temperature, T, is fixed by its environment. As discussed in the previous chapter, the Helmholtz free energy of this solid can be found by identifying all of its possible states, computing their energies, and using the results to find the canonical partition function (eq. 7.91). This method is difficult, but it works for any system. When the system is a single, perfect crystal. however, there is an easier way. We need only calculate the mean value of the internal energy, E, of the material as a function of temperature. The equation that relates the internal energy of a system to its temperature, volume and particle (or mole) number is called the caloric equation of state: E = ™E(T,V,N) 8.1 In the general case, the caloric equation of state is not equivalent to the fundamen-tal equation. From the definition of the Helmholtz free energy,

E = F + TS = F - T

∆¡F

∆T

= - T2

∆

∆T¡F

T 8.2


Page 187

Eq. 8.2 shows that the caloric equation of state is related to the partial derivative of ¡F(T,V,N) with respect to T. If there is a part of ¡F that is linear in T (that is, an additive term that has the form Tg(V,N), where g is a function of V and N only) this part is not de-termined by the caloric equation of state. To completely determine ¡F we also need the thermal equation of state,

P = ™P(T,V,N) = -

∆¡F

∆V 8.3

to fix the volume dependence of ¡F. However, in the specific case of a perfect crystal, the fundamental equation does not contain volume-dependent terms that divide out of 8.2. It follows that the fundamen-tal equation of a perfect crystal, ¡F(T,V,N), can be calculated from the caloric equation of state. To compute the internal energy of a perfect crystal we write it in the form E = E0(V) + ED(V,T) 8.4 where we have left the dependence on the particle number, N, implicit (N is fixed). E0(V) is the energy of the solid in the limit T = 0. It is essentially equal to the binding energy of the solid when all atoms are in their static equilibrium positions and the elec-trons are in the lowest-energy electron states. (There is also a small zero-point vibra-tional energy which is a quantum-mechanical effect that makes a negligible contribution to the energy at finite temperature). We discussed the binding energy in Chapter 3. In this chapter we are concerned with the temperature-dependent part of the energy, ED(V,T), the thermal energy. There are two potentially significant contributions to the thermal energy: the vi-bration of atoms about their equilibrium positions, which become increasingly violent as the temperature is raised, and the excitation of electrons, particularly the valence elec-trons that have energies near the Fermi level and can be excited to relatively high-energy states. We consider these in turn, and will find that, except in the case of a metal at very low temperature, almost all of the thermal energy is due to the atomic vibrations. To appreciate the vibrational energy we need to understand the nature of lattice vibrations in a solid. 8.2.2 Lattice vibrations At finite temperature the atoms in a solid are in continuous thermal motion about their equilibrium positions. To describe the vibration of an atom it is necessary to specify its motion along each of the three perpendicular directions in space. Since an atom can move at different velocities along the three directions in space, each atom has three vi-


Page 188

brational degrees of freedom. It follows that a solid that contains N atoms has 3N vibra-tional degrees of freedom, three for each atom. However, the motions of the atoms in a solid are not independent of one another. The atoms are coupled together by strong bonding forces. If a particular atom is dis-placed from its equilibrium position the lengths and angles of the bonds it makes with its neighbors are changed. The resulting forces act both to restore the displaced atom to its equilibrium position and to displace its neighbors to accommodate its displaced position. The displaced neighbors, in turn, exert forces that tend to displace their neighbors. In this way the displacement of an atom generates a vibrational wave that propagates through the solid. These waves are called the normal modes of vibration of the solid. Since there are a total of 3N vibrational degrees of freedom of the atoms, there are 3N normal modes. Transverse vibration of a linear chain of atoms A particular example of a displacement wave in a linear chain of atoms is shown in Fig. 8.1. Each atom displaces its neighbor with the result that the vertical displace-ments of the atoms are described by a sinusoidal wave with wavelength, ¬. The normal vibrations include waves with all possible values of the wavelength, ¬.

¬/2

a

Fig. 8.1: A transverse displacement wave of wavelength, ¬, in a linear

chain of atoms. To find all of the possible vibrational waves in a linear chain like that shown in Fig. 8.1, let the chain have a total of N atoms, located at positions x = na, where n is an integer. A transverse wave like that illustrated can be described mathematically by the relation

u(x) = u0eikx

8.5 where u(x) is the vertical displacement of the atom at x = na and k is the wave vector,

k = 2π¬ 8.6

Since eik(x+¬) = eikx eik¬ = eikx e2πi = eikx 8.7


Page 189

the displacement is periodic with period, ¬. The displacement waves that actually can appear in a finite chain of N atoms are affected by end effects at the terminations of the chain. If N is large, these are surface ef-fects that make a negligible contribution to the energy. To remove them we adopt what are called periodic boundary conditions. We imagine the chain of N atoms to be embed-ded in an infinite chain that has the property that every N-atom segment is like every other, in the sense that identically situated atoms in every segment have the same dis-placement. The displacement is then periodic with the macroscopic period, Na: u(x + Na) = eik(x+Na) = u(x) = eikx 8.8 By an analysis like that used in eq. 8.7, equation 8.8 holds only if kNa = 2πm 8.9 where m is an integer. Hence the allowed values of the wave vector, k, are

k = 2πmNa 8.10

There are exactly N independent values of the integer, m. To see this, let m = N + p, where p is an integer. Then

eikx = eikna = exp

i2πn + 2πinp

N

= exp

2πinp

N = eik'x 8.11

where k' = 2πp/Na. Hence the waves that are generated by wave vectors for which m > N simply repeat those for which m ≤ N. It is most convenient to chose the N values of m to be the set

- N2 ≤ m ≤

N2 8.12

(there are only N independent values since - N/2 gives the same wave as N/2). With this choice the N independent values of k are

- πa < k ≤

πa 8.13

and are symmetric about k = 0. The range given in 8.13 is called the first Brillouin zone of the linear chain.


Page 190

The vibrational waves in a solid are not static. The displacement due to the kth wave at any given atom oscillates with angular frequency, ∑, so that the displacement of the atom at position, x = na, at time, t, is given by the time-dependent function u(x,t) = u0eikx e-i∑t 8.14 The frequency, ∑, increases with the magnitude of the wave vector, k. To see why this should be so refer back to Fig. 8.1. The larger the value of k, the smaller the wavelength, ¬, and, hence, the larger the relative displacement between adjacent atoms in the chain. The larger the displacement, the larger the force that acts on the displaced atom and, hence, the greater the frequency of vibration. For this reason ∑ is an increasing function of k. The function ∑ = ∑(k) 8.15 is called the dispersion relation for the lattice waves. The dispersion relation for a trans-verse vibration in a simple one-dimensional chain of atoms is plotted schematically in Fig. 8.2.

∑

kπ/a- π/a 0

Fig. 8.2: The dispersion relation, ∑(k), between the wave vector and fre-quency for the transverse vibration of a one-dimensional chain of atoms.

The ratio

c(k) = ∑(k)

k 8.16

is the velocity of the wave. Using equation 8.16, the displacement imparted by the wave can be written u(x,t) = u0eik(x-ct) 8.17


Page 191

and hence propagates with the velocity, c. When the wave vector, k, is small the disper-sion relation is linear: ∑ = ck 8.18 The waves in a solid that have very long wave-lengths (small k) are sound waves, of the type that might be made, for example, by striking the solid with a hammer. Hence the constant velocity, c, is the speed of sound in the linear chain. The transverse displacement of an atom, ∂u(x,t), at a point, x = na, in the linear chain at time, t, is the sum of the instantaneous displacements imposed by all of the vibra-tional waves that are excited. Hence, ∂u(x,t) = ∑

k u0(k)ei[kx - ∑(k)t] 8.19

While the behavior of the individual vibrational waves is simple and sinusoidal, the oscil-lation of an atom is the sum of many such waves, and may appear quite chaotic. It is for that reason that it is simplest to visualize the thermal motion of the atom in terms of the lattice waves, or normal modes of vibration. Displacement waves in three dimensions The extension of these concepts into the three-dimensional motion of atoms in real crystals is relatively straightforward, but more difficult to visualize.

∑

kπ/a- π/a 0

longitudinal

transverse (2)

Fig. 8.3: Schematic drawing of the dispersion relation for a linear chain of atoms that can move in three dimensions. The two transverse vibrational modes have the same dispersion relation. Note the higher sound speed for the longitudinal mode.

If the atoms of the linear chain vibrate in three dimensions we need two more sets of waves to describe their motion. These can be taken to be a second set of transverse waves that displace the atoms perpendicular to the plane of Fig. 8.1, and a set of longitu-


Page 192

dinal waves that displace the atoms toward one another along the line. The displacements associated with each of these sets of waves are described by equations like 8.8, with wave numbers given by 8.10. The two transverse modes ordinarily have the same dispersion relation. However, the longitudinal mode has a different dispersion relation. The frequency of a longitudinal wave of given k is ordinarily greater than that of a transverse wave since the atoms are displaced toward one another and, hence, experience larger restoring forces. You may already be familiar with the fact that longitudinal and transverse sound waves travel with different velocities through solids. Fig. 8.3 is a schematic drawing of the dispersion relation of a linear chain of identical atoms that move in three dimensions. Lattice vibrations in a simple cubic crystal The changes that result when the one-dimensional chain of atoms is replaced by a three-dimensional crystal are easiest to see when the crystal has a primitive cell with a simple cubic structure. A simple cubic crystal has linear chains of {100} planes of atoms in three perpendicular directions. If we refer the crystal to a Cartesian coordinate system with axes along the three edges of the cubic unit cell, then the wave numbers of the vibrational waves are described by the wave vector k = kxex + kyey + kzez 8.20 where kx, ky, and kz are the wavenumbers of one-dimensional vibrational waves in the coordinate directions specified by the perpendicular unit vectors, ex, ey and ez. If the crystal contains N atoms (N unit cells since there is one atom/cell), then the length of the crystal in each of the coordinate directions is N1/3a, where a is the edge length of the unit cell. If periodic boundary conditions are imposed along the x, y and z-axes, the allowed values of kx, ky and kz are

ki = 2πmiN1/3a 8.21

where i = x, y or z. There are N1/3 independent values of mi for each value of i (we can always choose N such that N1/3 is an integer). If we let these be the values - N1/3/2 ≤ mi ≤ N1/3/2, as in the one-dimensional case, then the independent values of ki are given by

- πa ≤ ki ≤

πa (i = x, y, z) 8.22

The independent wave vectors, k, are obtained by selecting distinguishable sets of one-dimensional wave vectors {kx, ky, kz} in all of the ways that are permitted by eq. 8.22. Hence the total number of independent wave vectors is the product of the numbers of values of kx, ky and kz, which is N, the total number of atoms. Since there are three polarizations of the wave for every value of k (two perpendicular transverse waves and


Page 193

one longitudinal wave in the simplest case) the total number of independent waves is 3N, equal to the total number of vibrational degrees of freedom of the N atoms.

πa

πa

πa

πa-

πa-

πa-

k

Fig. 8.4: The first Brillouin zone of a simple cubic crystal, showing an arbitrary wave vector, k, in the Brillouin zone.

The independent wave vectors, k, are confined to the three-dimensional Brillouin zone shown in Fig. 8.4. The volume of the Brillouin zone is

„ = (2π)3

a3 = (2π)3

vc 8.23

where vc is the volume of the unit cell. The terminal points of the wave vectors, k, that is, the points {kx, ky, kz}, are uniformly distributed through the Brillouin zone. Since there are N wave vectors, and 3 vibrational states per vector, the independent vibrational states (which are identified by the points {kx, ky, kz}) are homogeneously distributed through the volume of the first Brillouin zone with the density

n0 = 3N„ =

3Nvc(2π)3 =

3V(2π)3 8.24

where V is the volume of the crystal. The dispersion relation for the solid assigns a frequency, ∑, to each wave vector, k: ∑ = ∑(k) 8.25 The dispersion relation along any given line through the Brillouin zone qualitatively re-sembles that shown in Fig. 8.3, although, in the case of a three-dimensional crystal it may be much more complicated, with peaks and valleys at intermediate values of k. Moreover, the value of ∑(k) varies with direction in the crystal. The vibrations may not


Page 194

be strictly longitudinal and transverse if the direction is not along one of the symmetry axes, [100]. Vibrational modes in primitive crystal structures When the solid is not a simple cubic crystal its vibrational behavior is more complicated still. The simplest case is a one-component solid or random solid solution with a primitive crystal lattice (atoms on the points of a Bravais lattice) that is not simple cubic. The FCC and BCC metals and alloys are examples (HCP and diamond cubic ma-terials are not since they have a minimum of two atoms per unit cell). In these solids the repeat vectors for the atoms are the translation vectors of the primitive cell. As described in Chapter 3, these vectors are not orthogonal. The consequence is that, while the wave vector, k, can always be referred to cubic axes and written in the form given in eq. 8.20, its components, kx, ky and kz are not given by the simple relation, 8.22, but by more complicated relations that reflect the fact that the periodic boundary conditions are im-posed along axes (the crystal axes of the primitive cell) that are not parallel to x, y and z and are not orthogonal to one another. The result is that the first Brillouin zone is not a cube, but a figure with a more complicated shape that depends on the particular lattice. However, whatever the shape of the first Brillouin zone, it always contains ex-actly 3N vibrational states, and these are always homogeneously distributed with the density n0 given by equation 8.24. Moreover, if the dispersion relations are measured along a line through the Brillouin zone that passes through the origin, in the usual case the curves are qualitatively similar to those drawn in Fig. 8.3. Lattice vibrations in non-primitive structures A different kind of complication intrudes when the crystal has more than one atom in its smallest unit cell (the primitive cell of its Bravais lattice). Two kinds of crys-talline solids have this property: one-component solids and solid solutions with non-primitive lattices, such as the HCP and diamond cubic structure, and compound structures in which each lattice site is occupied by a group of atoms, including interstitial com-pounds like NaCl and ∫-ZnS, and ordered substitutional compounds like Cu3Au, where the ordering of substitutional species on the atom sites changes the unit cell (in the case of Cu3Au, from FCC to simple cubic with four atoms per simple cubic unit cell).

a

Fig. 8.5: A linear diatomic chain of atoms vibrating in the optical mode that corresponds to k = 0. The center of mass of each unit cell remains fixed.


Page 195

In these solids a group of atoms sits at every site of the Bravais lattice. As a con-sequence, the vibrational states include two distinct kinds of vibrations: acoustic modes, in which the atoms in the group associated with a given lattice point are displaced to-gether, and optical modes, in which the different atoms in the atom group are displaced with respect to one another. The optical modes are essentially vibrational states in which the whole molecule is displaced according to a lattice wave with vector, k, while at the same time the atoms in the molecular group vibrate with respect to one another. As an example, the optical mode that corresponds to k = 0 in a diatomic linear chain is shown in Fig. 8.5. The center of mass of the unit cell remains stationary (k = 0) while the two dif-ferent atoms vibrate with respect to one another.

∑

kπ/a- π/a 0

acoustic

optical

Fig. 8.6: Schematic drawing of the dispersion relation for a diatomic linear chain like that shown in Fig. 8.5. There are 3 acoustic modes and 3 optical modes: 2 transverse, with the same dis-persion relations, and 1 longitudinal.

The dispersion relation of a diatomic linear chain like that shown in Fig. 8.5 is drawn schematically in Fig. 8.6. The optical modes introduce three new dispersion rela-tions, two for the transverse optical modes and one for the longitudinal optical mode, in which the atoms in the molecules are displaced toward one another. The frequencies of these modes are significantly higher than those of the acoustic modes, since nearest-neighbor atoms are displaced. The number of vibrational states in each mode, acoustic or optical, is equal to 3Nc, where Nc is the number of unit cells rather than the number of atoms. There are three acoustic modes, and there are three optical modes for each addi-tional atom in the unit cell. Hence the total number of vibrational states is 3N, as it should be. The angular frequency, ∑, of the optical vibration varies much less strongly with the wave vector, k, than in the acoustic mode. The variation is often so weak that it is a good approximation to assign a single, constant frequency, ∑opt , to each optical mode.


Page 196

8.2.3 The vibrational energy Quantized energy of vibration; phonons The quantum theory of the harmonic oscillator applies to lattice vibrations, since each normal mode of vibration behaves like a harmonic oscillator with angular frequency, ∑. The quantum theory asserts that the amplitude and, hence, the vibrational energy of a harmonic oscillator is quantized, and can only increase in discrete steps. The energy of a harmonic oscillator takes the discrete values E = (n + 1/2)Ó∑ 8.26 where n is an integer, Ó = h/2π, where h is Planck's constant, and the added factor 1/2 cor-responds to the zero-point energy of the oscillator, a purely quantum effect. Equation 8.26 has a form that suggests that the harmonic oscillator defines a quantum state with an energy Ó∑ that has an inherent energy Ó∑/2 and can hold an arbitrary number of particles, n. These particles are called phonons, and are the elementary quantized excitations of the lattice vibrations. They define the number of elementary excitations that have been used in making the wave. The fact that phonons behave like particles should not be surprising. The lattice vibrations are waves that propagate through the lattice, and it is a common feature of the quantum theory that waves behave like particles, and vice versa. The phonons of lattice vibration are very much like photons, which are the particles of light and other electro-magnetic waves. Thermal effects: Bose-Einstein statistics The phonons of lattice vibration do not obey the Pauli exclusion principle. A vi-brational wave can have an arbitrary amplitude and energy, which requires that an arbi-trary number of phonons can be in the same state. Quantum particles that have this prop-erty are called Bosons, and they have a thermal behavior that is described by Bose-Einstein statistics. In particular, the expected level of excitation, that is, the expected number of phonons, ń(∑,T)¨, in a vibrational state with frequency, ∑, at temperature, T, is given by the relation

ń(∑,T)¨ = 1

eÓ∑/kT

- 1 8.27

This complex relation can be simplified in two limiting cases. First, when kT >> Ó∑, that is, when the temperature is very high or the frequency very low, we can use the expan-sion

eÓ∑/kT

~ 1 + Ó∑kT 8.28


Page 197

to derive the approximate result

ń(∑,T)¨ ~ kTÓ∑ (kT >> Ó∑) 8.29

Second, when Ó∑ >> kT, that is, when the temperature is low or the frequency is very high, eÓ∑/kT >> 1 and

ń(∑,T)¨ ~ e- Ó∑/kT

8.30 The vibrational energy The expected value of the total energy of the lattice vibrations in a solid can be written Ev(T,V,N) = ∑

∑ (ń(∑,T)¨ + 1/2) Ó∑ 8.31

where the summation is taken over the allowed frequencies of the lattice vibrations. The energy depends on N because the total number of lattice vibrations increases with N. It depends on V because the dispersion relation, and hence the spectrum of frequencies, ∑, depends on the volume of the unit cell (the edge length, a, in the case of a linear chain or a simple cubic crystal). It depends on the temperature, T, through the Bose-Einstein for-mula that gives the expected number of phonons, ń(∑,T)¨. The general solution of equation 8.31 is very difficult, and requires a detailed knowledge of the Brillouin zone and the dispersion relations. However, there are two cases in which the equation can be solved almost exactly without a detailed knowledge of the dispersion relation: the high-temperature limit, where kT >> Ó∑ for every vibration, and the low-temperature limit, where ń(∑,T)¨ « 0 for all but the lowest-frequency acous-tic modes. In most cases, the two limiting solutions can be fit together to provide a rela-tively simple picture of the vibrational energy, and, hence, the vibrational contribution to the specific heat, at all temperatures. We focus on the specific heat, which is easily mea-sured experimentally. Its prototypic behavior is characterized by a material property, called the Debye temperature, ŒD. 8.2.4 The vibrational contribution to the specific heat The high-temperature limit: the Dulong-Petit Law The vibrational energy in the high-temperature limit is found by substituting the high-temperature expression for ń(∑,T)¨, eq. 8.29, into eq. 8.31. Since kT/Ó∑ >> 1/2, we can neglect the zero-point vibrations. The result is


Page 198

E(T,V,N) ~ ∑∑

kT = 3NkT 8.32

Differentiating eq. 8.32 with respect to the temperature gives the vibrational contribution to the specific heat in the high-temperature limit: (CV)v = 3Nk 8.33 Equation 8.33 is called the Dulong-Petit Law. It asserts that the specific heat of a solid approaches a constant, 3Nk, at high temperature, where 3N is the total number of vibra-tional degrees of freedom. Low temperature: the Debye interpolation formula The vibrational energy can also be calculated in the low-temperature limit where only the lowest-energy vibrational modes are activated. For solids that have primitive unit cells the result can be extended to provide a reasonable approximation for the spe-cific heat at all temperatures. This result was originally obtained by Peter Debye early in this century, and is known as the Debye interpolation formula for the specific heat. To formulate the model used by Debye note that the low-energy vibrational modes are the long-wavelength acoustic modes for which ∑ = ck, where k = |k| is the magnitude of the wave vector. For simplicity we ignore the difference between the sound speed, c, for the longitudinal and transverse vibrations, or use an appropriate weighted average of the two speeds. The low-energy waves that have a particular energy, Ó∑, lie on the surface of a sphere with radius k = ∑/c in the first Brillouin zone (for example, in Fig. 8.4). Since there are n0 states per unit volume in the Brillouin zone (eq. 8.24), the number of vibrational states, g(k)dk, that have wave vectors, k, with magnitude between k and k+dk, is equal to the number of states per unit volume, n0, times the differential volume of the Brillouin zone between spheres of radii k and k+dk: g(k)dk = n0(4πk2)dk

= 4πn0

∑2d∑

c3 = g(∑)d∑ 8.34

where g(∑)d∑ is the number of states with energies between Ó∑ and Ó(∑+d∑). To solve for the vibrational energy in the low-temperature limit we use the fact that the summation over frequencies in eq. 8.32 can be replaced by an integral according to the rule

∑k

“ ⌡⌠0

km

dkg(k)“ ⌡⌠0

∑m

d∑g(∑) 8.35


Page 199

where km is the maximum value of k, and ∑m is the maximum value of ∑. Equation 8.32 then becomes

Ev(V,N,T) = ⌡⌠0

∑m

(ń(∑,T)¨ + 12) Ó∑g(∑)d∑

= Eº + ⌡⌠0

∑m

ń(∑,T)¨Ó∑g(∑)d∑ 8.36

where Eº is the energy of the zero-point vibrations. It is independent of temperature, Eº = Eº(V,N), and so does not affect the specific heat.

∑

kπ/a- π/a 0

∑D

...

Fig. 8.7: The Debye approximation for a linear chain. The dispersion relation is replaced by its long-wavelength form.

To solve this integral Debye approximated the dispersion relation of the solid by assuming that all waves propagate with sound speed, that is, that ∑ = ck 8.37 for all values of k. The approximation is illustrated in Fig. 8.7 for the case of a linear chain of atoms. It is a good approximation for the low-energy modes, and hence should provide a good estimate of the vibrational energy at low temperature, where only these modes are activated, but becomes less accurate for the more energetic vibrations near the edge of the Brillouin zone. If all of the vibrations obeyed the dispersion relation given by eq. 8.37 then the vibrational states would be located in a sphere centered around k = 0; the Brillouin zone would be spherical. Since the total number of vibrational modes is 3N, the frequency of the states at the outer boundary of the sphere, ∑D, would satisfy the relation


Page 200

3N = ⌡⌠0

∑D

g(∑)d∑ = ⌡⌠0

∑D

4πn0

c3 ∑2d∑

=

4πn0

3c3 ∑D3 8.38

This equation defines the Debye cut-off frequency,

∑D = c

6π2

vc1/3

8.39

where we have used equation 8.24 for n0; vc is the volume of the primitive unit cell. The cut-off frequency, ∑D, is used to define the Debye temperature,

ŒD = Ó∑D/k = Óck

6π2

v1/3

8.40

where k is Boltzmann's constant. The Debye temperature is the temperature at which the thermal energy, kT, becomes equal to the energy of a phonon with frequency, ∑D, and is, hence, a measure of the temperature at which all the vibrational states become activated. When T > ŒD, the high-temperature limit has certainly been reached. If we substitute these approximations and results into equation 8.36, the vibra-tional energy is given by the integral

Ev(T,V,N) = Eº + ⌡⌠0

∑D

ń(∑,T)¨Ó∑g(∑)d∑

= Eº +

4πn0

c3 ⌡⌠0

∑D

Ó∑3

eÓ∑/kT - 1 d∑ 8.41

Now defining the integration variable, x = Ó∑/kT 8.42 and substituting the value of n0, we obtain, after a bit of algebra,


Page 201

Ev(T,V,N) = Eº +

3V(kT)4

2π2(Óc)3 ⌡⌠0

xD

x3

ex - 1 dx

= Eº + 9NkT

T

ŒD3 ⌡⌠

0

xD

x3

ex - 1 dx 8.43

where the limit of integration is xD = ŒD/T 8.44 Since the value of the integral depends on its upper limit, xD, equation 8.43 can be written in the simpler form Ev(T,V,N) = Eº(V,N) + 3NkTf(ŒD/T) 8.45 where Eº is the zero-point vibrational energy and f(ŒD/T) is a universal function (the Debye function) of the variable ŒD/T:

f(ŒD/T) = 3

T

ŒD3 ⌡⌠

0

xD

x3

ex - 1 dx 8.46

The Debye function estimates the correction to the Dulong-Petit relation for the vibra-tional energy when the temperature is below ŒD. Its value is always less than 1. The specific heat at low temperature In the limit of low temperature xD “ ∞, and the Debye function can be found analytically. The definite integral,

⌡⌠

0

∞ x3

ex - 1 dx = π4

15 8.47

so the vibrational energy is

Ev(T,V,N) = Eº(V,N) + 3NkT

π4

5

TŒD

3 8.48

The vibrational specific heat in the low-temperature limit is obtained by differentiating equation 8.48 with respect to T, and is

CV = 3Nk

4π4

5

T

ŒD3

8.49


Page 202

which differs from the Dulong-Petit law by the factor in braces. Equation 8.49 is exact in the limit of low temperature, and shows that the vibrational specific heat of a solid varies as T3 in the limit of low temperature.

T

CV

Dœ

Fig. 8.8: The vibrational specific heat in the Debye approximation.

For T > ŒD/2 the specific heat is approximately constant, and given by the Dulong-Petit Law.

The specific heat at finite temperature in the Debye model The Debye function can be solved numerically to estimate the vibrational energy and the specific heat for all temperatures. The result is important because it is universal. In the Debye approximation the vibrational energy depends on only a single material pa-rameter, the Debye temperature, ŒD. In particular, the vibrational specific heat depends only on the dimensionless combination, T/ŒD. To see this we differentiate equation 8.43:

CV = ∆

∆T [Ev(T,V,N)]

= 3Nk{f(ŒD/T) -

ŒD

T f'(ŒD/T)} 8.50

where f'(ŒD/T) is the derivative of f(ŒD/T) with respect to its argument:

f'(ŒD,T) = df(ŒD/T)d(ŒD/T) 8.51

Eq. 8.50 shows that, to within the accuracy of the Debye approximation, the vibrational specific heats of all simple solids should have the same form when expressed as a func-tion of the dimensionless temperature, T/ŒD. This universal function is shown in Fig. 8.8. It is reasonably well obeyed by all solids that have primitive unit cells.


Page 203

Solids with non-primitive unit cells Strictly speaking, the Debye model is restricted to solids that have only acoustic vibrational modes, that is, to solids that have primitive unit cells such as FCC and BCC solids. It can be extended to solids with non-primitive cells by accounting for optical modes of vibration. It is often possible to do this to a reasonable approximation by as-signing a single frequency to all of the optical modes. In any case, the Debye T3 law governs the vibrational specific heat at very low temperature, since only the acoustic modes are activated when T is small. We can simply define a value for ŒD for solids with non-primitive structures, such as diamond and the HCP metals, by fitting equation 7.126 to the low-temperature specific heat. With this approximation the Debye model can be applied to crystals with non-primitive structures, and works reasonably well for elemental solids, solid solutions and ordered substitutional compounds. The model is less useful for interstitial compounds and molecular solids, since the optical modes in these solids often have high frequencies. The same approximation can be used to define the Debye temperatures of amorphous solids, and works reasonably well for amorphous metals, semiconductors and simple glasses. It is less useful for polymeric and other molecular glasses. The Debye temperature as a material property Given that the Debye temperature characterizes the vibrational specific heat of a solid it is useful to see how it can be related to other material properties. According to equation 7.117, the only material property on which it depends is the mean speed of sound, the mean velocity of propagation of waves of long wavelength. The speed of sound can be calculated from the elastic constants of the material, which we shall discuss later, but can also be related to the strength of bonding in the solid. The stronger the bonds, the stronger the force that tends to restore a displaced atom to its equilibrium posi-tion, and, hence, the higher the frequency and velocity of a sound wave. Hence strongly bonded solids have high Debye temperatures (diamond has ŒD « 2000K), while weakly bonded solids have low values (Pb has ŒD « 95K). For most metals, ŒD lies in the range 200-500K. 8.2.5 A qualitative version of the Debye model As the theoretical models of solid state physics go, the Debye model is a relatively simple one. Nonetheless, it involves some rather messy integrals whose form obscures the qualitative features of the physics that govern the lattice contribution to the specific heat. For that reason, it is useful to describe an even simpler model that may help to clarify the physics of the problem. While it is seriously deficient in mathematical rigor, it does give the right functional dependence of the specific heat. Following Debye, let us assume that the frequency of a lattice vibration is linearly related to its wave number, ∑ = ck, as in eq. 8.37. At temperature, T, the lattice vibra-tions can be divided into two sets: those for which the phonon energy, Ó∑, is small com-


Page 204

pared to kT, and those for which it is large compared to kT. The former, low-energy vi-brational modes are strongly excited, while the latter are not. Let ∑T be the cut-off fre-quency that divides strongly excited vibrations from weakly excited ones, as illustrated in Fig. 8.9. A crude representation of the Bose-Einstein function suggests that Ó∑T « 2kT is a reasonable choice.

∑

kπ/a- π/a 0

∑D

∑T

... Fig. 8.9: Schematic drawing illustrating the division between excited

modes (∑ < ∑T) and quiescent modes at temperature T. Now make the approximation that all vibrations with ∑ < ∑T are excited into the high-temperature limit, with energy kT, while those with ∑ > ∑T remain quiet. Since Ó∑D = kŒD, and since the 3N vibrational modes are spherically distributed through the Brillouin zone, the number of excited modes is

N(T) ~ 3N

2T

ŒD3

T<ŒD/2

3N T>ŒD/2 8.52

the vibrational energy is

E(T) ~ 3NkT

2T

ŒD3

T<ŒD/2

3NKT T>ŒD/2 8.53

and the specific heat is

Cv(T) ~ 12Nk

2T

ŒD3

T<ŒD/2

3Nk T>ŒD/2 8.54


Page 205

The vibrational energy and specific heat given by eqs. 8.53 and 8.54 are in reasonable numerical agreement with those predicted by the Debye model, and have the right tem-perature dependence in both the high-T and low-T limits. A useful qualitative feature of this model is that it makes it clear why the specific heat changes with temperature. At high temperature all of the available vibrational modes are strongly activated. Increasing the temperature further raises the level of acti-vation of each of the 3N modes by the same amount, so the specific heat is constant. At low temperature, however, only some of the modes are strongly activated. Increasing the temperature raises the energy in two ways: the energy of the activated modes is raised, and vibrational modes that were previously quiet are activated. In this case the tempera-ture increases both the energy per mode and the effective number of modes. As a conse-quence, the specific heat is a string function of the temperature. 8.2.6 The electronic contribution to the specific heat The second contribution to the thermal energy of a perfect crystal is made by the valence electrons, which reconfigure into excited states above the Fermi energy, EF, as the temperature rises. There are Nz electrons, where N is the number of atoms and z is the valence. Since these are in rapid motion through the crystal lattice one might expect them to make a large contribution to the specific heat. However, they do not. The elec-tronic specific heat is very small at all realistic temperatures, even in metals with high electrical conductivity. The reason is that only a few of the valence electrons are excited by an increase in temperature.

T

C V

T

A

2

B

Fig. 8.10: Plot of CV/T against T2. The slope gives the coefficient of the vibrational contribution; the intercept is the coefficient of the electronic term.

On the other hand, the electronic contribution to the thermal energy is responsible for an anomalous feature of the specific heats of metals at very low temperature. In the limit of low temperature the specific heat of a metal obeys an equation of the form CV = AT + BT3 8.55


Page 206

where A and B are constants. The T3 term is due to lattice vibrations. While A is much smaller than B, at temperatures so low that T >> T3 the linear term dominates. If one plots the specific heat of a metal in the manner shown in Fig. 7.14, as CV/T against T2, then the result is a straight line with slope, B, and asymptote, A. The behavior of the electronic specific heat has its source in the Pauli exclusion principle, which requires that there be only one electron in each permissible state. Quantum particles that obey the Pauli exclusion principle are called Fermions. At finite temperature they are distributed over the set of states available to them according to the Fermi-Dirac distribution function, which states that the expected number of particles, ń(E)¨, in a state with energy, E, at temperature, T is

ń(E)¨ = 1

e(E-EF)/kT

+ 1 8.56

where EF is the Fermi energy. Note that n(E) is always less than 1, as required by the Pauli exclusion principle. The Fermi energy is the energy at which the probability that a state is occupied is exactly 1/2:

ń(EF)¨ = 12 8.57

In the limit T “ 0 the exponential in the denominator of eq. 8.57 vanishes if E < EF, and is arbitrarily large if E > EF. It follows that

limT“0[ń(E)¨] = 1 E < EF

0 E > EF 8.58

As we assumed in Chapter 2, all states below EF are filled in the ground state of the solid (T « 0) while all states above EF are empty. The Fermi level, EF, itself lies half-way be-tween the highest filled and lowest empty states. In a metal the allowed electron states are so close together at EF that the Fermi energy is essentially equal to the energy of the highest filled state. In an intrinsic semiconductor or insulator the highest filled state lies at the top of the valence band while the lowest empty state is at the bottom of the conduc-tion band; EF is in the center of the band gap. The Fermi-Dirac distribution function is plotted for some finite T in Fig. 8.11. In the limit T “ 0 all states below EF are filled, all states above EF are empty. At finite temperature the distribution differs significantly from the low temperature limit only for states whose energies differ from EF by no more than the thermal energy, kT. When E-EF > kT,

ń(E)¨ « e- (E-EF)/kT

(E-EF > kT) 8.59


Page 207

and rapidly asymptotes to 0 as the energy increases. When EF-E > kT,

ń(E)¨ « 1 - e(E-EF)/kT

(EF -E > kT) 8.60 and asymptotes rapidly to 1 as the energy falls below EF. [To derive equation 8.60 note that (1 + x)-1 « 1 - x when x is small.]

E

n(E)

FE1

0

.5

kT

...

Fig. 8.11: The Fermi-Dirac distribution function for electrons at finite temperature. Note that the difference from the ground state is essentially confined to the shaded band of energies within kT of EF.

The electronic specific heat of a metal In a metal the Fermi energy lies within a band of allowed electron states. As the temperature increases the electron distribution is smeared over the states within a dis-tance, kT, of EF as shown in Fig. 8.11. Since the widths of bands of electron states are of the order of several electron volts (eV), and 1 eV is the value of the thermal energy at nearly 104K, only a very small fraction of the electrons within the valence band of the metal are affected. The thermal contribution to the electron energy can be roughly estimated in the following way. Let Ng(EF)dE be the electron density of states at the Fermi level, the number of electron states with energies between EF and EF+dE. The factor g(EF) is the density of states per atom, and is a small number since there are only 1-4 valence elec-trons per atom. Assuming that g(EF) is approximately constant for a range of width kT on either side of EF, then the total number of electrons that are excited at temperature, T, is approximately

Ne(T) « NkT

2 [g(EF)] 8.61


Page 208

where we have also assumed that 1/2 of the available electrons are excited. Each of these electrons acquires a thermal energy of the order of kT. Hence the thermal contribution to the electron energy is, approximately,

Ee(T) « N[g(EF)]

2 (kT) 2 8.62

E

n(E)

FE

kT

valenceband

conductionband

GE

1

0

.5

...

Fig. 8.12: Fermi-Dirac distribution for a narrow-gap semiconductor at moderate temperature.

It follows from eq. 8.62 that the electronic specific heat is of the form suggested by eq. 8.55: (CV)e = AT 8.63 where the coefficient, A, is A « Nk2[g(EF)] 8.64 A much more rigorous theoretical calculation [J. Ziman, Principles of the Theory of Solids, p. 144] gives the result

A =

π2

3 Nk2[g(EF)] 8.65

which shows that the simple model is not so bad. The coefficient, A, is a small number. At ordinary temperatures the electronic term accounts for only about 1% of the value of the specific heat. The electronic contri-bution is only observable at very low temperatures, where T >> T3. The electronic con-tribution to the thermal energy is otherwise negligible, and is not ordinarily included in calculations of the fundamental equation of solids.


Page 209

Semiconductors and insulators The electronic contribution to the specific heat of a semiconductor or insulator is significantly smaller than in a metal. The reason is shown schematically in Fig. 8.12, in which the Fermi-Dirac distribution is plotted against the distribution of allowed states in an intrinsic semiconductor or insulator. The Fermi energy is located within the energy gap. The magnitude of kT is less than .1eV at normal temperatures. The band gaps of typical semiconductors are of the order of an electron volt while the band gaps of insula-tors are several eV. Hence very few electrons are excited across the band gap at finite temperature, and the electronic specific heat is negligible. 8.2.7 The Helmholtz free energy We can find the Helmholtz free energy of the perfect crystal by integrating the differential expression given in eq. 7.82:

E = - T2

∆

∆T¡F

T 8.2

Since eq. 8.2 is a partial differential equation in which the partial differential is taken at constant volume, it actually determines ¡F/T only to within an additive function of vol-ume. However, in the case of the simple vibrating solid we are considering here no such additive function appears (a fact that can be proved by invoking the Third Law of ther-modynamics). The low-temperature form of the fundamental equation Equation 8.2 is easily solved in the low-temperature limit. Using eq. 8.48,

∆

∆T¡F

T = - EºT2 -

3π2Nk

5ŒD3 T2 8.66

whose solution is

¡F(T,V,N) = Eº(V,N) - π4

5 NkT

T

ŒD3 8.67

The Debye temperature, ŒD, is a function of the volume per atom, since expanding the solid changes the strength of binding between the atoms and, hence, changes the vibra-tional frequencies. Hence the right-hand side of eq. 8.67 is a function of T, V and N. Since these are the natural variables for the Helmholtz free energy, eq. 8.67 is the low-temperature form of the fundamental equation of a simple solid. The entropy of the solid at low temperature is the vibrational entropy


Page 210

S = -

∆¡F

∆T = 4π4

5 Nk

T

ŒD3 8.68

The pressure; the Grªuneisen equation of state The pressure of the solid is given as a function of (T,V,N) by the partial derivative

P = -

∆¡F

∆V = - ∆Eº∆V -

3π4

5 NkT4

1

ŒD4

dŒD

dV

= - ∆Eº∆V -

3π4

5 NkT

T

ŒD3

1

V

dln(ŒD)

dln(V) 8.69

where dln(x) = dx/x. The Debye temperature depends on the volume through the atomic volume, v = V/N; the vibrational spectrum of the solid does not change if we simply in-crease the size of the solid by adding atoms at constant atomic volume. Since the partial derivative in eq. 8.69 is taken at constant N, dln(V) = dln(v). If we now define the Grªuneisen parameter

© = - dln(ŒD)dln(v) = -

VŒD

∆ŒD

∆V 8.70

which specifies the volume dependence of the Debye temperature, the pressure is given by

P = P(N,V,T) = - ∆Eº∆V -

3π4

5 NkT

T

ŒD3

1

V

dln(ŒD)

dln(v)

= - ∆Eº∆V +

©EDV 8.71

where ED is the vibrational energy. It can be shown that equation 8.71 holds at all temperatures for solids whose spe-cific heats are well represented by the Debye interpolation formula. Equation 8.71 is called the Grªuneisen equation of state of a solid, and is obeyed reasonably well by all simple solids, including simple compounds. The Grªuneisen parameter, ©, is a dimension-less constant whose value lies in the range 1-3 for elemental solids, solutions and com-pounds that have simple crystal structures. The two terms that contribute to the pressure represent two different physical pro-cesses. The first term contains the effect of the interatomic bonding. The binding energy has a minimum at V0, the volume of the solid in the limit of zero pressure and tempera-ture. If the solid is compressed to V < V0 the bonding term exerts a positive pressure


Page 211

that tends to expand it to restore V0. If the solid is expanded to V > V0 the solid tends to contract, and the bonding contribution to the pressure is negative. The second term in-cludes the effect of the lattice vibrations. It is always positive. The lattice vibrations tend to push the atoms apart. 8.2.8 Thermodynamic properties Given the fundamental equation, we can also find the thermodynamic properties of the solid. We have already discussed the specific heat, CV. The isothermal compress-ibility was defined in Chapter 7, and is:

˚T = - 1V

∆V∆P T,{N}

8.72

Its reciprocal (the bulk modulus) is obtained by differentiating the Grªuneisen equation of state with respect to V. Assuming that © is constant, the low-temperature result is

(˚T)-1 = - V

∆P

∆V T,{N}

= V

∆2Eº

∆V2 {N} +

©EDV - ©

∆ED

∆V T,{N}

= V

∆2Eº

∆V2 {N} -

©EDV [3© - 1] 8.73

rr 0Ï

Ï(r)

vibrationalenergy states

change of equilibriumradius with vibration

. Fig. 8.13: The binding potential of a diatomic molecule, showing the

vibrational energy states and the increase of the equilibrium radius as the vibrational energy increases.

The first term on the right in 8.73 is the bulk modulus at zero T, which is deter-mined by the interatomic binding. The second term is much smaller in magnitude and is due to the lattice vibrations. Since 3© > 1, the vibrational term decreases (˚T)-1, and hence decreases the bulk modulus. The physical reason for this behavior is that, since ©


Page 212

is positive, an increase in volume decreases the Debye temperature, ŒD, and hence increases the vibrational energy, which is proportional to [ŒD]-3. In effect, the binding strength, which provides the spring constant for the lattice vibrations, goes down. This has the result that the solid contracts more for a given increase in the pressure than it would at T = 0. Hence the compressibility increases with the temperature. The coefficient of thermal expansion is (Chapter 7):

åT = 1V

∆V∆T P,{N}

8.74

It can be shown that

1V

∆V∆T P,{N}

= - 1V

∆V∆P V,{N}

∆P

∆T V,{N}

= ˚T

∆P

∆T V,{N} 8.75

By differentiating the Grªuneisen equation of state (8.71) with respect to the temperature we find

∆P

∆T V,{N} =

©CVV 8.76

and, hence,

åT = ©CV˚T

V 8.77

Equation 8.77 is known as the Grªuneisen relation. Its validity is not confined to low tem-perature; it applies whenever the Grªuneisen equation of state is valid. The physical source of the coefficient of thermal expansion lies in the anhar-monicity of the atom vibrations in the solid. To visualize this consider a diatomic molecule, as we did in Chapter 2. The binding potential of the molecule, Ï(r), has the form shown in Fig. 8.13. The vibrations of the molecule have quantized values that are indicated by the horizontal lines within the potential well. Because of the anharmonic shape of the binding potential, the restoring force on the atoms is smaller when r > r0 than it is when r < r0, and, hence, the atoms spend a greater fraction of the time at r > r0. This has the consequence that the time average interatomic separation, r0(T), increases as the vibrational energy increases, as shown in the figure. The situation in a vibrating solid is qualitatively the same. The total binding po-tential between atoms is anharmonic, roughly as shown in the figure. As the vibrational


Page 213

energy increases the time average of the interatomic separation increases, so the solid ex-pands. 8.3 THE RANDOM SOLID SOLUTION 8.3.1 The Bragg-Williams model The previous section showed how the fundamental equation of a perfect crystal could be found by considering its binding energy at zero temperature together with the changes in its energy due to vibrational and electronic excitations as the temperature is raised. In this section we formulate the fundamental equation of a random solid solution, so that we can gain some insight into the influence of composition on the thermochemical properties of solid solutions. To do this, we shall use a model called the Bragg-Williams model, which is the simplest model of a solid solution that includes compositional effects. The Bragg-Williams model treats a binary solution of two kinds of atoms, A and B, that are distributed over the sites of the fixed crystal lattice.. It assumes that the inter-nal energy of the solution is equal to its binding energy, and assumes that E is indepen-dent of temperature (we discussed the thermal contribution to the energy in the previous Section, and can add it to the model later, if we wish). To evaluate the binding energy, it assumes that each atom interacts only with the atoms that are closest to it on the crystal lattice. Since the energy is independent of temperature, the entropy is just the configura-tional entropy of the solution (Section 7.2.3), and is also independent of temperature. The free energy of the solution is, then, F(T,{N}) = Eº({N}) - TS({N}) 8.78 where Eº is the binding energy at the equilibrium volume, V0, and S is the configura-tional entropy of the distribution of atoms over the atom sites of the solid solution. Both Eº and S depend on the composition.

VAB AAV

BBV

= A

= B

...

Fig. 8.14: Solution of A and B atoms, illustrating AA, BB and AB bonds. To evaluate Eº, let each atom interact only with those atoms that are nearest neighbors to it, as illustrated in Fig. 8.14, and assume that the bonding interaction is inde-


Page 214

pendent of temperature and concentration. The atom fraction of component B is x =NB/N, where NB is the number of B-atoms and N is the total number of atoms (equal to the total number of lattice sites). The atom fraction of A is (1-x). 8.3.2 The internal energy If VAA is the energy of a bond between A atoms, VBB the energy of a B-B bond, and VAB the energy of an A-B bond then the energy of the solution is E = VAANAA + VBBNBB + VABNAB 8.79 where Nij is the number of nearest neighbor bonds of type ij (= AA, BB or AB). Let each atom have z nearest neighbors. If the solution is random, the probability that an atom site is occupied by a B-atom is x, the atom fraction of B, while the probability that it contains an A-atom is (1-x). Hence each A atom has, in the mean, z(1-x) AA bonds and zx AB bonds, while each B atoms has z(1-x) AB bonds and zx BB bonds. The total energy is, then

E = 12 {NAz[(1-x)VAA + xVAB] + NBz[(1-x)VAB + xVBB]} 8.80

where NA is the number of A atoms, NB is the number of B atoms, and the factor 1/2 is included because each bond is counted twice in the expression in braces. Since the num-ber of A-atoms, NA = (1-x)N and NB = xN, equation 8.80 can be re-written:

E = Nz2 {(1-x)2VAA + x2 VBB + 2x(1-x)VAB}

= Nz2

(1-x)VAA + xVBB + 2x(1-x)

VAB - 12(VAA + VBB)

= Nz2 { }(1-x)VAA + xVBB + 2x(1-x)V 8.81

where V is the relative binding energy

V = VAB - 12(VAA + VBB) 8.82

The relative binding energy, V, has a simple physical meaning: it is the difference between the energy of an AB bond and the average of the energies of AA and BB bonds. If V is positive the energy of the system is decreased if 2 AB bonds are replaced by an AA bond and a BB bond, that is, if A atoms and B atoms associate preferentially with one another. In this case the energy of the system is lowered if it decomposes into A-rich and B-rich solutions. If V is negative, on the other hand, the energy is decreased if AA and BB bonds are replaced by AB bonds, and we should expect the system to prefer a


Page 215

solid solution that eventually orders into an AB-type compound. The energy of the random solution is graphed in Fig. 8.15 for the cases V > 0, V < 0 and V = 0.

E

xA B

VAA

BBV

V<0

V>0

V=0

...

Fig. 8.15: The binding energy of a random solution with near neighbor interactions for three values of the relative binding energy.

8.3.3 The configurational entropy To complete the calculation of the free energy we require the value of the entropy. In this model we assume that only the configurational entropy is important. The configu-rational entropy of the random solution is given as a function of the composition, x, by equation 7.20 of the previous chapter: Sc = - kN[x ln(x) + (1-x) ln(1-x)] 8.83 The configurational entropy is plotted as a function of x in Fig. 8.16.

S

xA B

0

...

Fig. 8.16: The configurational entropy of a random solution as a function of composition.

Note that the configurational entropy is well behaved in the limits x “ 0 and x “ 1; the function x ln(x) “ 0 in the limits x “ 0 or 1. However, the compositional deriva-tive of the configurational entropy


Page 216

dSdx = - kN[ln(x) - ln(1-x)] 8.84

is singular at both limits. It explodes to ∞ as x “ 0 and to - ∞ as x “ 1. 8.3.4 The free energy and thermodynamic behavior The free energy of the random solution is obtained by summing the energy and entropy: F = ⁄F(T,V,{N}) = E - TS

= Nz2 { }(1-x)VAA + xVBB + 2x(1-x)V + NkT[x ln(x) + (1-x) ln(1-x)]} 8.85

Both the configurational energy and entropy are independent of the temperature. The temperature enters the expression through its multiplication of the configurational en-tropy. Unlike-atom attraction; low-temperature order The thermodynamic behavior that is predicted by eq. 8.85 depends on the sign of the relative binding energy, V. If V ≤ 0 then the free energy is negative and concave (curved upward) at all values of x and T. As we shall see in the next chapter, the concav-ity of the fundamental equation has the consequence that the random solid solution is at least metastable at all compositions and temperatures, and, therefore, can exist at all x and T. However, it does not follow that the random solid solution is the preferred config-uration for all x and T. If V < 0 and T is low, the solution can lower its free energy by forming an ordered arrangement that maximizes the number of AB bonds. As a simple example, let x = 0.5, and assume that the crystal has a BCC structure (z = 8), so that it is possible to arrange the atoms in a CsCl pattern with A atoms at the cell centers and B atoms at the corners. In the ordered state, all bonds are AB bonds, and the energy is Eorder = 4NVAB 8.86 However, the configurational entropy (Sord) is zero. In the disordered state the energy is

Edis = 2N

VAB + 12(VAA + VBB) 8.87

while the configurational entropy is


Page 217

Sdis = Nkln(2) 8.88 Since Sdis > Sord, the random solution has the lower free energy when T is large, and is thermodynamically preferred. However, since Eord < Edis, there is a temperature low enough that the ordered state has the lower free energy. It follows that the random solid solution becomes at least metastable with respect to the ordered phase as the temperature is lowered. This is precisely the behavior observed in ∫-brass (CuZn). It is a solid solu-tion with a BCC structure at high T, but an ordered compound with the CsCl structure at low T. While eqs. 8.86-8.88 are written for a specific case, the qualitative result is gen-eral. When V< 0 the random solid solution is always at least metastable with respect to transformation into an ordered phase (or mixture of ordered phases) at sufficiently low temperature. This result is in keeping with the Third Law of thermodynamics; when the temperature is very small the equilibrium state of a multicomponent solid is always an ordered structure or mixture of ordered structures. Like-atom attraction; low-temperature decomposition If V > 0, then the energy is a convex function of x while the entropy term, -Ts, is concave. The behavior of the free energy per atom f(x) (= F(x)/N) is plotted in Fig. 8.17 for three values of the temperature. At high temperature the entropy dominates and the function f(x) is concave; the system is a random solid solution at all compositions. At low temperature the energy dominates the behavior of f(x) at intermediate values of x but, because of the singularity in its compositional derivative, the entropy still dominates in the limits x “ 0 or 1. Hence f(x) is a concave function near x = 0 and x = 1, but is convex (curved downward) at intermediate values of the composition.

f

xA B

VAA

BBVT3

2T

1T

...

Fig. 8.17: Free energy curves for a random solution with V > 0 at three temperatures, T1 > T2 > T3.

As we shall see in the following chapter, a solid solution cannot exist when its free energy function, f(x), is a concave function of x. If we try to mix equal concentra-tions of two components so that x = 0.5 at temperature T2 or T3 in Fig. 8.17, we will dis-cover that we cannot do so; the solution will spontaneously decompose into a mixture of


Page 218

two solutions with different concentrations, one A-rich (x < 0.5) and one B-rich (x > 0.5). We say that the A-B solution has a miscibility gap. We can form a solid solution at all compositions at high temperature where the configurational entropy dominates the free energy, but, if V > 0, cannot make a solid solution at intermediate compositions at low temperature where the energy dominates. Note, however, that the singularity in the slope of the free energy at x = 0 and 1, which is due to the singularity in the slope of the configurational entropy, has the conse-quence that the free energy function is always concave when x is close to 0 or 1. This is true no matter how high V may be, that is, no matter how much the species A and B dis-like one another chemically. As we shall see in the following Chapter, this has the con-sequence that there is always some solubility for A in B and B in A. If the solution de-composes, it does not decompose into pure A and pure B, but into two solid solutions, one of which is predominantly A with some B in solution, and the other predominantly B with some A in solution. This is one example of the general thermodynamic principle that everything is at least slightly soluble in everything else. 8.4 EQUILIBRIUM DEFECT CONCENTRATIONS All real crystals are defective. They not only contain impurities in solution, but are filled with internal defects, including vacancies and interstitialcies, dislocations, and grain boundaries. It is important to recognize that vacancies, and, to a much lesser extent, interstitialcies, are inherent defects that are present in finite concentration in the equilibrium state of the crystal. They can be created or destroyed by processes that occur spontaneously within the solid. Moreover, there are active mechanisms in the solid that maintain the vacancy concentration at near-equilibrium values. Other crystal defects, such as dislocations and grain boundaries, have very high formation energies and essen-tially zero concentrations in the equilibrium state. They are non-equilibrium defects that are created during processing, and preserved because of the slow kinetics of the mecha-nisms that eliminate them. To establish this, we compute the equilibrium concentration of defects in an oth-erwise perfect crystal. 8.4.1 The equilibrium vacancy concentration When a crystalline solid is held at fixed temperature and pressure its equilibria are controlled by the Gibbs free energy. The addition of a single vacancy to a crystalline solid increases its Gibbs free energy by the amount Îgv = Îev + PÎvv - TÎsv 8.89 where Îev is the formation energy of the vacancy, Îvv is the associated volume increase, and Îsv is the entropy change which, in the case of an isolated vacancy, is due to the change in the vibrational and electronic entropies of the atoms in the neighborhood of the


Page 219

vacancy. In most crystalline solids it is reasonable to neglect the volume and entropy in-crement for the addition of a single vacancy, in which case Îgv « Îev 8.90 Let there be n vacancies in the solid, where n is sufficiently small compared to the number of atoms, N, that the vacancies do not interact significantly with one another. The free energy increment due to the vacancies is, then, ÎG = nÎgv - kT ln(„) « nÎev - kT[(N+n) ln(N+n) - n ln(n) - N ln(N)] 8.91 where the second term on the right-hand side is the configurational entropy, which we have evaluated with the help of eq. 7.20. N+n is the total number of lattice sites; since the number of atoms is conserved, the number of lattice sites in the solid changes when vacancies are added. The equilibrium number of vacancies (ne) minimizes the Gibbs free energy. Differentiating eq. 8.91 and setting the derivative equal to zero yields the condi-tion:

d(ÎG)

dn n = ne = 0 = Îev - kT[ln(N+n) - ln(n)] 8.92

The solution is

ne

N+ne = xe

v = e-

ÎevkT 8.93

where xe

v is the equilibrium concentration of vacancies, the fraction of lattice sties that are vacant in the equilibrium state. [Note that the left-hand side of eq. 8.92 is the definition of the chemical potential of a vacancy, regarded as a chemical species that occupies atomic positions in the crystal. Eq. 8.92 shows that, in equilibrium, the chemical potential of a lattice vacancy is zero. This is true because the number of vacancies is not conserved. Vacancies can be created and destroyed, so their concentration adjusts to minimize G.]

...

Fig. 8.18: Illustration of the creation or annihilation of vacancies by ex-change with adsorbed atoms at an interface.


Page 220

While the vacancy concentration in a typical crystalline solid at moderate temper-ature is not large, it can be significant. The energy to form a vacancy, Îev, is of the order of 1 eV (electron volt) in a typical solid. An electron volt corresponds to the value of kT at about 104 K. Hence the vacancy concentration in a typical solid at 1000 K (« 700 ºC) is on the order 10-4, giving about 1018 vacancies per cm3 of crystal. The equilibrium va-cancy concentration drops exponentially as the temperature decreases. At moderate temperature most solids maintain vacancy concentrations that are close to the equilibrium value. The reason is that defects within the solid act as sources and sinks for vacancies, which are reasonably mobile at moderate temperature. Efficient sources and sinks for vacancies include free surfaces, which can add or subtract vacancies by adjusting the number of adsorbed atoms (as illustrated in Fig. 8.18), grain boundaries, which can absorb or emit vacancies by reconfiguring the atoms in the boundary, and dis-locations, which absorb or emit vacancies during climb, as discussed in Chapter 4. At low temperature the vacancy mobility decreases dramatically. When a solid is quenched from high temperature the vacancy concentration is frozen in for some time after the quench. This behavior affects atomic diffusion in the solid, as we shall discuss in the next chapter. It is also very important in the processing of semiconductor crystals such as silicon. Vacancies are electrically active defects in a semiconductor and must be held to a low concentration so that they do not alter its electrical properties. Hence, after silicon crystals are processed at high temperature they are cooled slowly to achieve an equilibrium vacancy concentration at a temperature where the vacancy concentration has an acceptably small value before cooling to room temperature where Si vacancies are nearly immobile. 8.4.2 Dislocations and grain boundaries It is possible to compute the equilibrium concentrations of dislocations and grain boundaries just as we calculated the concentration of vacancies. However, the formation energies of these defects is so high that their equilibrium concentrations are negligible at all normal temperatures. The energy of formation of a dislocation is of the order of 1 eV for each plane through which the dislocation threads. The energy of formation of a grain boundary is a fraction of an electron volt for each atom it contains (as one can see by rec-ognizing that a low-angle, relatively low-energy grain boundary is often an array of dislo-cations). These defects are non-equilibrium defects in normal solids at normal tempera-tures. Dislocations form and multiply not only during plastic deformation, but also dur-ing crystal growth, processing, structural phase transformations, and heating or cooling, particularly when the material contains more than one phase. The principle source of dis-locations in the latter cases are the mechanical stresses that are introduced when different grains or phases do not quite fit together in the solid, or contract at different rates when the solid is cooled. Since the dislocations are non-equilibrium defects there is a thermo-dynamic driving force to eliminate them. However, a dislocation can only disappear at a position in the crystal where the crystal discontinuity (the extra half-plane in the case of


Page 221

an edge dislocation) can be eliminated, such as a boundary or matching dislocation of op-posite sense. Dislocations are eliminated at a finite rate at moderate temperature. The process is called recovery. However, since recovery requires solid state diffusion its rate is relatively slow at all temperatures and is nearly zero at low temperature. Grain boundaries are formed during the original solidification of the solid or dur-ing recrystallization or structural transformations that reconstitute the grain structure. The grain boundaries are always non-equilibrium, high-energy defects. There is, hence, a thermodynamic driving force to eliminate them. The smaller the grain the higher its sur-face-to-volume ratio; hence larger grains grow at the expense of smaller ones to decrease the grain boundary area, causing gradual grain growth. In order for a grain to grow atoms must move across its boundary into its interior. This diffusional process is slow at ordinary temperatures, so grain growth only occurs at an appreciable rate at high tem-perature. Moreover, the decrease in boundary area per incremental increase in volume becomes smaller as the grain grows. As a consequence the rate of grain growth decreases as the grain size increases. In many cases the rate of grain growth is such that the grain size increases with the square root of the aging time. The equilibrium state of a macroscopic solid is a single crystal, and in some cases it is possible to make single crystals by growing grains until the individual crystallites become large enough to separate and use individually. However, in most cases it is not practical to make single crystals by simply growing grains since the rate of growth is slow, and the driving force for growth becomes vanishingly small as the grain size becomes large.


Page 222

C h a p t e r 9 : P h a s e s a n d P h a s e E q u i l i b r i u mC h a p t e r 9 : P h a s e s a n d P h a s e E q u i l i b r i u m

Phases and stages Circles and cycles Scenes that we've all seen before... - Willie Nelson, "Phases and Stages"

9.1 INTRODUCTION All pure (one-component) materials take on several different structures, or phases, as the temperature is varied at atmospheric pressure. At high temperature the material is a vapor, at lower temperature it condenses into a liquid phase, and at still lower temperature it freezes into a solid. Moreover, many common solids can be found in several different crystal structures. In many cases, the multiple crystal structures of the solid are different equilibrium phases, as in iron, which transforms from FCC to BCC as the temperature is lowered, and in quartz, which takes on several different crystal structures as the temperature is lowered. In other cases the multiple structures are due to metastable equilibria. Common examples include diamond, which forms at high temperature and pressure but can be retained indefinitely under atmospheric conditions because of the difficulty of its structural transformation into graphite, which is the more stable form, and silica glasses, which form from the liquid and can be retained indefinitely because of the difficulty of the structural transformation to crystalline quartz. When a system contains more than one component, it is often a multiphase mixture in which volumes that have two or more distinct structures or compositions are intermingled on a very fine scale. For example, if water that contains a small amount of salt is held at a temperature slightly below Oº C it becomes a mixture of solid ice that contains only a small concentration of salt and liquid water that is relatively salty. This two-phase mixture persists with time, and hence must represent an equilibrium state. If iron that contains a bit of carbon in solution is cooled from a high temperature at which the FCC structure is preferred to a lower temperature, the iron not only transforms in structure from FCC to BCC, but also segregates into a two-phase mixture in which small precipitates of carbide, Fe3C, appear in a parent matrix of BCC iron that has a very small carbon content in solid solution. These and many other phenomena are examples of phase equilibria and equilibrium phase transformations in solids. They can be understood and predicted from the thermodynamic behavior of materials, and can be represented graphically in equilibrium phase diagrams, which show the equilibrium phases that appear at given values of the thermodynamic variables. We treat phase equilibria and phase diagrams in this chapter.


Page 223

The principles of phase equilibria help to understand why materials have the microstructures they do, and why these microstructures are often complex mixtures of two or more distinct phases. However, the equilibrium behavior gives only part of the story. To understand microstructure we must also appreciate why the equilibrium distribution of phases is often not observed, and why the phases that do appear have the shapes and distributions that they do. These phenomena reflect the kinetics of transformations in materials. Even though they are influenced by kinetics, many of the important features of real microstructures can be inferred from the equilibrium phase diagrams, and will be discussed in this chapter. Other important features can only be understood after a more detailed study of the kinetics of changes in solids, which will be presented in the following chapter. The second law of thermodynamics provides a general criterion that governs the equilibrium of all systems: the equilibrium state of an isolated system maximizes the en-tropy of the system for given values of its energy, volume, and chemical content. The material systems in which we are interested usually are not isolated, but interact with the environment. As we discussed in Chapter 7, it is possible to rephrase the condition of equilibrium so that it applies to a system under the experimental conditions that pertain to it. Often the temperature and pressure are controlled by the environment, and the chemical content is fixed. In this situation the condition of equilibrium is that the Gibbs free energy of the material has a minimum value with respect to all changes that may occur at the given T, P and {N}. Mathematically, if (ÎG)T,P,{N} ≥ 0 9.1 for all possible changes in the way the atoms are configured in the material at the given T, P, and {N}, the system is at equilibrium. Since this is the most common experimental situation, we ordinarily describe the equilibrium states of materials in terms of their Gibbs free energy. We saw in the previous chapter that the Helmholtz free energy (F) of a material is somewhat easier to compute theoretically than the Gibbs free energy. In fact, when the material is solid it does not matter very much whether we choose the Gibbs or the Helmholtz free energy to describe its equilibrium states. The two differ by the factor PV: G = E - TS + PV = F + PV 9.2 At atmospheric pressure the PV product is very small, and the Gibbs and Helmholtz free energies are numerically almost equal. While we shall use the Gibbs free energy in the following, we can often speak of equilibrium in the solid state as representing the minimum free energy without being specific about which free energy is meant. 9.2 PHASE EQUILIBRIA IN A ONE-COMPONENT SYSTEM


Page 224

The phases of one-component systems (including molecular solids and compounds with fixed composition) are distinguished by their structures. In the vapor and liquid phases the atoms or molecules are randomly distributed, but differ in the fact that the liquid is condensed while the vapor is not. Crystalline solid phases differ in crystal structure. Amorphous solids (glasses) are essentially continuations of the liquid state that differ only in properties. Phase relations like that between a liquid and a glass are examples of what we shall call mutations, in which one phase simply becomes another at a particular value of T and P. The phase behavior of one-component systems can be understood by comparing the Gibbs free energies of the various phases. In particular, we can understand why the equilibrium phase changes, usually discontinuously, as the temperature is lowered at constant pressure.

G

T

- S

E + PV

Fig. 9.1: The Gibbs free energy of a phase as a function of T. Fig. 9.1 shows a plot of the Gibbs free energy of a phase as a function of tempera-ture at constant pressure. The slope of the curve at any temperature is given by equation 7.48, and is equal to - S. Since S is positive the curve slopes down. However, as T “ 0 the entropy of an ordered phase vanishes according to the Third Law; even when the phase is a non-equilibrium one that retains some entropy, the entropy becomes small. Hence the Gibbs free energy approaches the constant value, E + PV, as the temperature approaches zero (in the usual case PV is negligible compared to E). 9.2.1 Phase equilibria and equilibrium phase transformations A one-component material has at least three possible phases, solid (S), liquid (L) and vapor (V). The internal energy of these phases increases in the order S “ L “ V, since the energy is determined by the bonding. The entropy increases in the same sequence since the system is increasingly disordered as it changes from solid to liquid to gas. As a consequence the Gibbs free energy curves for the three phases appear roughly as shown in Fig. 9.2. At low temperature the solid has the least free energy and is the sta-ble phase. Because of the higher entropy of the liquid, however, the free energy of the liquid drops below that of the solid at the melting temperature, Tm, and the liquid is the stable phase at higher temperature. The vapor has still higher entropy, so its free energy


Page 225

eventually falls below that of the liquid, at Tv, and the vapor phase is preferred at all higher temperatures. The free energy relations that lead to multiple solid structures are essentially identical to those shown in Fig. 9.2. Suppose that a particular one-component solid has three possible crystal structures, as shown in Fig. 9.3 and labeled å, ∫ and ©. If the ∫ structure has the lowest energy then it will be the equilibrium phase at sufficiently low temperature. If another structure (å) has greater entropy then its free energy may drop below that of the ∫ phase as the temperature rises. The å-phase is then the equilibrium structure. The range of preference of å and ∫ phases in a hypothetical material is shown in Fig. 9.3. The figure also includes the free energy curve for a third phase, ©, which does not appear at equilibrium because its free energy is always above that of the preferred phase.

G

T

TTm v

S

L

V

solid liquid vapor

Fig. 9.2: Free energies and equilibrium temperatures for solid, liquid, and

vapor phases of a hypothetical material.

G

T

Tå∫Tå©

∫

©

å

∫ å

©

Fig. 9.3: Free energy relations in a material that has three phases. Note that while phase © does not appear at equilibrium, it can still form as a metastable phase.


Page 226

9.2.2 Metastability In addition to revealing the equilibrium state of a material, the free energy curves also suggest how metastable phases can be formed. Suppose, for example, that the mate-rial of Fig. 9.3 is made in the å phase and then cooled. The equilibrium diagram suggests that it should transform to the phase ∫ at the temperature Tå∫. However, if the transformation is kinetically difficult, as structural transformations generally are, then phase å may be retained at temperatures below Tå∫. When this happens the free energy varies along the continuation of the å free energy curve as the temperature drops. As the system is cooled further the thermodynamic driving force for the å “ ∫ transformation, the free energy difference between the two phases, increases monotonically. If the driving force becomes sufficient to force the transformation then å will transform to ∫ at a lower temperature. If it does not, then å will be retained as a metastable structure. The possible retention of phase å in a metastable state may also make it possible to form phase ©, which does not appear in the equilibrium sequence at all. When phase å is cooled below the temperature Tå© then there is a net thermodynamic driving force for the transformation å “ ©, and this transformation will occur if it is kinetically possible. The phase © may then transform to the equilibrium phase, ∫, but if the © “ ∫ transformation is kinetically difficult phase © may be retained indefinitely as a metastable phase. It is possible to form © from å at any temperature in the hatched region of the figure. Metastable behavior of the type exhibited by the å, ∫ and © phases in Fig. 9.3 is common, and greatly increases the variety of structures that can be realized in engineering solids. A classic example of the use of this behavior is the formation of an amorphous film from a vapor phase. In this case the phase å is the vapor, ∫ is the crys-talline solid, and © is the amorphous solid. By condensing the vapor onto a cold substrate it is brought to below the temperature Tå© at which it can transform to the amorphous phase. Since the amorphous phase is kinetically easier the film takes on an amorphous structure and is trapped in it. Metastability also increases the variety of transformation paths that can be used in the processing of a material. By using the metastability of the å phase one can often con-trol the temperature at which it transforms to ∫, and can also sometimes force it to take an indirect path in which it first forms an intermediate structure like ©. Since the defect type and concentration in the microstructure of the final product is sensitive to the path of the transformation, metastable states are often utilized in materials processing to control the ultimate microstructure. 9.2.3 First-order phase transitions: latent heat The phase changes that we considered above are all first-order phase transitions. One phase transforms into another because the free energy of the product phase is lower than that of the parent phase. The phases are distinct at the transition temperature, and will continue to exist as metastable phases at temperatures beyond the transition


Page 227

temperature unless there is an easy kinetic path that facilitates the transformation between them. Transitions of this type are called first-order phase transitions because they involve discontinuous changes in the first derivatives of the free energy function: the entropy and the volume. The substantial majority of the phase transitions that are important in engineering are of this type, including normal vapor-liquid and liquid-solid transitions, and transitions between solid phases that differ in crystal structure. Since the entropy changes discontinuously in a first-order phase transition the transition always involves a latent heat. Heat is released or absorbed when the transformation happens. To find the latent heat, note that the equilibrium transformation occurs when the Gibbs free energies of the two phases are the same. When this is true, ÎG = 0 = ÎE + PÎV - TÎS = ÎH - TÎS 9.3 where H is the enthalpy, H = E + PV. Hence the entropy change is

ÎS = ÎHT ~

ÎET

= - QT 9.4

Where Q = - ÎH ~ - ÎE 9.5 is the latent heat of the transition, the heat released to the environment when the transfor-mation happens. It follows from eq. 9.5 that the latent heat is positive when the transformation is from a relatively high-energy phase to a relatively low-energy one. Heat is released, for example, when a vapor condenses, a liquid solidifies, or a high-temperature solid phase transforms to a phase that is preferred at low temperature. This has the consequence, for example, that the temperature remains almost constant while a system is undergoing a phase transformation that happens on cooling, such as solidification. If the system is cooled, the rate of transformation increases, and the heat released causes the temperature to rise to the transformation temperature. The temperature cannot increase further since the transformation would then reverse itself. Conversely, the latent heat is negative in a transition from a low-energy phase to a phase of higher energy, as, for example, when a liquid vaporizes or a solid melts or vaporizes. This phenomenon is used in a number of practical ways. An intelligent but


Page 228

overheated undergraduate pours water on his head. The evaporation of water absorbs heat and cools him. In an identical, but more high-tech example, the exposed surfaces of missiles and spacecraft are often coated with ablative materials that vaporize at high temperature with very high latent heat. These materials are used to cool the exposed surfaces of a spacecraft during its re-entry into the atmosphere, since atmospheric friction would otherwise raise the surface temperature to values that might destroy it. 9.2.4 Transformation from a metastable state The latent heat of a transformation is only slightly changed by heating or cooling beyond the transformation point, which reflects the fact that ÎS and ÎH are approximately constant. Hence the thermodynamic driving force for transformation from a metastable phase at a temperature T that is not too far from the equilibrium transformation temperature, T0, is

ÎG(T) = ÎH - TÎS = Q

T

T0 - 1

= QT0

[T - T0] 9.6

Equation 9.6 is reasonably accurate even when at temperatures well below or well above the transformation point. 9.3 MUTATIONS 9.3.1 The Nature of a Mutation The phase changes we have considered thus far are first-order transitions that con-nect states that are physically different from one another. Since both of the phases in a first-order transition are stable at the equilibrium transformation point, metastability is always possible for at least a small range of conditions around the transformation point. There is, however, a second type of phase transformation, in which two phases that are otherwise distinct become identical at the transformation point. Rather than transforming discontinuously, the two phases simply mutate into one another at the transformation point, although they have quite different properties when they are in states some distance away. We therefore call them mutations.


Page 229

T

g

Tc

C

TTc

åå' åå'

...

Fig. 9.4: (a) The free energy curve of a system that passes through a mutation at Tc. (b) Possible behavior of the specific heat near the mutation.

Fig. 9.4 shows the free energy curve of a system that mutates at the temperature Tc. The free energy curve is uninteresting and uninformative unless it is examined in minute detail. The free energy and its first derivatives, the entropy and volume, have well-defined values at the mutation, so the curve is continuous there to first order. A mutation affects the second and higher derivatives of the free energy. The specific heat is ordinarily singular at Tc, and usually has a behavior near Tc like that shown in Fig. 9.4b. The singularities in the thermodynamic properties at Tc have the consequence that both phases are unstable there. Metastability is impossible; the two phases become one another at Tc. The kind of transition we have called a mutation here is called by several other names in the literature. The most common is second-order transition, since the transition affects the second or higher derivatives of the free energy. Mutations whose free energy functions resemble Fig. 9.4b near the transition point (most do) are sometimes called ¬-transitions because of the shape of the curve. 9.3.2 Common Transitions that are mutations The simplest mutation is the glass transition we discussed in Chapter 5. At the glass transition point, Tg, the liquid simply becomes solid. The free volume available to the molecules in the liquid is no longer sufficient for them to move around one another, so they are frozen into position. In contrast to the case of solidification into a crystalline solid, there is no discontinuity in the volume at a glass transition. The discontinuities at the glass transition occur in physical properties, for example, the coefficient of thermal expansion changes discontinuously there.

...


Page 230

Fig. 9.5: Examples of mutations: (a) ferromagnetic crystal; (b) ordered crystal structure of ∫'-CuZn; (c) ferroelectric displacement in BaTiO3.

There are also important examples of mutations in crystalline solids. However, the requirement that two otherwise distinct phases become identical at a mutation restricts severely restricts the kinds of phases that can be connected by a mutation. Their crystallographic or physical symmetries must satisfy a relation called the Landau symmetry rule. Four classes of crystalline phase transitions that are important in materials science satisfy this rule and are always or often mutations. They are magnetic transitions from paramagnetic to ferromagnetic or antiferromagnetic order, some chemical ordering reactions, ferroelectric transitions, and superconducting transitions. In each case the mutation introduces a type of order that is not present above the mutation temperature. The first three classes of mutation are illustrated in Fig. 9.5. Ferromagnetism and antiferromagnetism The mutation that is probably most familiar is the ferromagnetic transition that oc-curs in many metals, alloys and ceramics. We shall discuss this transition in some detail when we come to consider magnetic properties. For the moment it is sufficient to recog-nize that magnetic materials contain transition metals, such as Fe and Ni, or rare earth elements that have partly filled d- or f-shells in the electron configurations of the atoms. The electrons in the partly filled shells have aligned spins so that the ion core of the atom has a net magnetic moment. At high temperature the magnetic moments of adjacent atoms are uncorrelated (to maximize the entropy) so the crystal has no net magnetic moment. In this state it is said to be paramagnetic. At a critical temperature, called the Curie temperature, Tc, the magnetic moments spontaneously align as shown in Fig. 9.5a. At the Curie point itself the degree of alignment is zero. However, at a temperature very slightly below Tc there is a measurable degree of magnetic order. The degree of alignment in the magnetic moments can be measured by the value of a long-range order parameter, ˙, that is zero when the spin directions are random, but takes a finite value when the spins are aligned and approaches 1 in the limit of complete alignment. The long-range order parameter is zero at and above Tc, but finite at any lower temperature, as illustrated in Fig. 9.6a. At Tc the ferromagnetic and paramagnetic states become identical; the transition is a mutation at Tc.

T

˙

Tc

å' å

T

˙

Tc

å' å

...


Page 231

Fig. 9.6: The behavior of the long-range order parameter (˙) near (a) a mutation; (b) a first-order transition.

All transition metals and rare earth elements with unpaired inner shell electrons have magnetic moments. It is a consequence of the Third Law of thermodynamics that these moments must align at sufficiently low temperature. However, alignment does not always lead to ferromagnetism. Adjacent magnetic moments can also align anti-parallel, in which case they cancel one another and the crystal has no net magnetic moment. This type of order is called antiferromagnetism, and appears through a mutation at a critical temperature called the Neel point. There are even materials that are ferromagnetic and antiferromagnetic at the same time. These are materials like the ferrites that have antiferromagnetic order, but are chemically ordered so that species with different magnetic moments appear on adjacent sites, and the net magnetic moment is non-zero. The Curie and Neel points of these materials are the same. Chemical order A second class of mutations includes chemical ordering reactions like that which leads to the ∫'-CuZn structure (CsCl) that was discussed in Chapter 3 and is drawn in Fig. 9.5b. Above the transition point the crystal is disordered; all sites have the same probability of being filled by either chemical specie. At the transition the lattice sites divide into two sets, each of which is preferentially occupied by one specie. In the example given, ∫'-CuZn, the corner sites of the basic BCC cell are preferentially filled by one specie while the center sites are filled by the other. Let x be the composition of the solute in CuZn, chosen so that x ≤ .5, and let the solute fill the body-centered sites. Then the long-range order parameter (˙) is defined by the equation x1 = x(1 + ˙) 9.7 where x1 is the fraction of body-center sites filled by solute atoms. The atom fraction on the corner sites is x2 = 1 - x1. As T“0, ˙“1 and all the solute atoms are in center sites. As T “ Tc there are two possibilities. If ˙ “ 0 as in Fig. 9.6a the ordering reaction at Tc is a mutation; if ˙ is discontinuous at Tc, as in Fig. 9.6b, the ordering reaction is a first-order transition. Only certain ordering reactions can be mutations since only these have sym-metries that satisfy the Landau rule. An ordering reaction that can be a mutation may be a first-order transition for some systems or under some conditions. Theory tells us which reactions can be mutations; we must rely on more precise analysis or experiment to know which of these are mutations, and under what conditions. For example, the order that cre-ates ∫'-CuZn from the disordered BCC solution in the Cu-Zn system is a mutation, the order reaction that creates the Ni3Al structure from disordered FCC is always first-order, and the order reaction that creates FeAl3 in the Fe-Al system is a mutation for one range of composition and temperature and a first-order transition for another. Ferroelectric transitions


Page 232

The third class of mutations that is illustrated in Fig. 9.5 is the ferroelectric transi-tion. The classic examples of materials with ferroelectric behavior are perovskite crystals like barium titanate (BaTiO3), which is drawn in Fig. 9.5c. The cations in BaTiO3 have a ∫'-CuZn configuration, which is drawn with Ti+4 in the center of the cell and Ba+2 at the corners. At high temperature the Ti ion has a average position precisely in the center of the cube. However, at low temperature the Ti ion is slightly displaced from the cube center so that the cell has a permanent dipole moment. The displacement is shown, though it is greatly exaggerated, by the arrow in Fig. 9.5c. The displacements in neighboring cells are correlated so that the crystal has a net dipole moment and a very high dielectric constant. This is the phenomenon of ferroelectricity, which has technological applications in many electronic devices. The ferroelectric transition is a mutation. Letting the order parameter, ˙, measure the average displacement of the central ion as a fraction of its displacement in the limit T “ 0, the order parameter behaves as shown in Fig. 9.6a. It vanishes at Tc, the Curie point for the transition. Superconducting transitions The final class of mutations that is of great interest in materials science is the superconducting transition, which is discussed in some detail in Chapter 19. While the mechanism of superconductivity is not fully understood in all cases (the mechanism in the high-Tc oxides is still under dispute) the normal source of superconductivity is the formation of coupled pairs of electrons (Cooper pairs) that also couple to lattice vibrations (phonons) so that they can move through the crystal without being scattered. The ordering reaction is the formation of Cooper pairs, and it occurs through a mutation at a specific temperature, the superconducting critical temperature, Tc. 9.4 PHASE EQUILIBRIA IN TWO-COMPONENT SYSTEMS When a system contains more than one component it is not always meaningful to speak of the phase that minimizes its free energy. The equilibrium state can also be a mixture of two or more phases with different compositions. While the conditions of equilibrium require that the temperature and chemical potentials be the same everywhere in the system, it is possible to have the same values of the chemical potentials in two or more phases that have very different compositions. These phases can then coexist at equilibrium. We shall confine this discussion to binary (two-component) systems since binary systems are relatively easy to understand and since they exhibit most of the new phenom-ena that appear in multicomponent systems, such as the coexistence of phases of different composition. Moreover, systems of engineering interest can often be treated as binary systems, either because they are essentially binary or because the phenomena of interest involve changing the content of one component while the remainder of the system is


Page 233

fixed in its chemistry. Phase equilibria in two-component systems are often presented in binary phase diagrams, which are simply maps that show which phases appear at given values of the temperature and composition. To understand the equilibrium relations that give phase diagrams their characteristic forms it is necessary to understand the concept of free energy curves and the common tangent rule. 9.4.1 The free energy function Consider a two-component system for which the temperature, pressure and com-position are controlled. Let the two components be labeled A and B. Let component A be taken as the solvent, or reference component. If the mole fraction of the solute, B, is given by the variable, x, the mole fraction of A is (1-x). The equilibrium of the system is governed by the Gibbs free energy. The Gibbs free energy per mole is given by the function g = ¡g(T,P,x) 9.8 To find the partial derivatives of the molar free energy, begin from equation 7.45 of the previous chapter: dG = - SdT + VdP + µA dNA + µBdNB 9.9 if we fix the value of N and divide through by it, the result is dg = - sdT + vdP + µAd(1-x) + µBdx = - sdT + vdP + (µB - µA)dx 9.10 Defining the relative chemical potential, ¡µ = µB - µA 9.11 the partial derivatives of the function ¡g are

∆¡g

∆T = - s 9.12

∆¡g

∆P = v 9.13

∆¡g

∆x = ¡µ = µB - µA 9.14


Page 234

We have already studied the behavior of the free energy function as T and P are varied. We now study the behavior of the function ¡g(x) when the temperature and pressure are fixed. Three useful general statements can be made about the free energy function, ¡g(x), that governs a particular phase at given T and P. First, the free energy curve of a given phase of the system must be concave, as drawn in Fig. 9.7. The slope of the curve at a given point (say, x1) is the relative chemical potential, ¡µ, defined by equation 9.14. The condition of concavity is that

∆2¡g

∆x2 =

∆¡µ

∆x PT > 0 9.15

which is the condition of local stability. If 9.15 is not satisfied for a state of the system then, as we shall demonstrate below, that state is unstable with respect to decomposition into regions of slightly different composition and can only exist as a transient state in an on-going transformation.

g

xA

å

x1

¡µ( )x1

Fig. 9.7: The free energy curve of a phase of a binary system. There is some subtlety in the definition of a phase since the free energy function of a continuous set of states of a system, for example, a random solution of two atoms on the sites of a FCC lattice, may include unstable states whose free energies can be computed or even measured by achieving them in a transient sense. One example is the simple binary solution whose free energy was computed in the previous chapter. When the temperature is low and the relative bonding potential is positive, V > 0, its free energy curve is like that shown in Fig. 9.8, with two minima and an intervening maximum as x is varied from zero to one. All states in the region shown shaded in the figure, where ∆¡µ/∆x < 0, are unstable and cannot persist. Hence these states cannot meaningfully be ascribed to any phase. The stable states, where ∆¡µ/∆x > 0, are divided into two sets that fall in different ranges of composition. The corresponding portions of the free energy curve are marked å1 and å2 in the figure. Not only are these states stable, we shall see below that states of å1 can coexist in equilibrium with states of å2 to form two-phase mixtures. Hence a free energy curve like that shown in Fig. 9.8 contains two distinct phases; each separate concave segment of the free energy curve represents a different phase.


Page 235

g

xA B

å2å1 unstable

Fig. 9.8: Free energy curve of a system that has two distinct phases sepa-rated by a continuous set of unstable states.

An important theorem follows from this definition of a phase of a binary system:

two distinct examples of the same phase can never be in equilibrium with one another.

This important result is a consequence of the conditions of equilibrium. If two states of the same phase were in equilibrium then they would have to have the same values of T and ¡µ. But since ∆¡µ/∆x > 0 for a given phase the chemical potential increases monotonically with composition. Two states of the same phase cannot have the same chemical potential, ¡µ, and hence cannot be in equilibrium with one another. When a system contains more than one distinguishable state at equilibrium, these states must be different phases. The second universal property of the free energy curve is its behavior in the limit of zero concentration (x “ 0). It can be shown that as x “ 0 the relative chemical potential takes the form

¡µ = ¡µ0(T,P) + RTln

x

1-x 9.16

where ¡µ0(T,P) is a function of the temperature and pressure only. We shall not prove this relation, which is due to Gibbs, but note that it is obeyed by the model solid solution studied in Chapter 8. Eq. 9.16 has the consequence that the free energy curve asymptotes to become parallel to the g-axis as x“0, as drawn in Figs. 9.7 and 9.8. Since eq. 9.16 holds whatever the chemical nature of the solvent and solute, it has the consequence that everything is at least slightly soluble in everything else.


Page 236

g

xA

T1T2 <

T1

g

xA

T1T2 <

T1

x0

Fig. 9.9: The behavior of the free energy curve as T decreases toward

zero: (a) solution; (b) compound stoichiometric at x0. The third universal property of the free energy curve concerns its behavior in the limit of zero temperature. The Third Law of thermodynamics requires that the entropy of an equilibrium phase vanishes (or reaches a small constant value in the case of a metastable equilibrium) in the limit of zero temperature. This has the consequence that the free energy curve develops a sharper and sharper trough as T “ 0, as illustrated in Fig. 9.9. When the phase is a random solution the free energy curve gradually folds onto the axis, as shown in Fig. 9.9a. When the phase is a compound with a stoichiometric composition, x0, the free energy curve becomes more and more sharply centered about the equilibrium composition, as shown in Fig. 9.9b. This behavior has the consequence that solubility disappears as T approaches zero; the only possible phases are pure compo-nents or stoichiometric compounds. 9.4.2 The common tangent rule In a binary system two-phase, three-phase and four-phase equilibria are possible. However, when the pressure is fixed it is extremely unlikely that more than two phases will appear at equilibrium. Four-phase equilibrium requires that the pressure, temperature and relative chemical potential have specific values. It is extremely unlikely that the pressure of interest for a particular system would be the four-phase equilibrium pressure. At any other pressure the maximum number of possible phases is three. But when the pressure is fixed, three-phase equilibria occur at specific values of T and ¡µ. Since ¡µ = ¡µ(T,x) at given P, three-phase equilibrium requires that the temperature and composition have mathematically precise values. But it is impossible to control these variables to mathematically precise values in a real system. Hence only two-phase equilibria are ordinarily observed. Two-phase equilibria occur when the Gibbs free energy of a two-phase mixture is lower than that of either phase alone. We can find a general mathematical condition that must be satisfied for two-phase equilibrium directly from the conditions of equilibrium and stability. However, there is an alternative, geometric method that is less formal and somewhat easier to visualize.


Page 237

g

xA B

å

∫

xå x∫x

xåx∫ -

xåx -

Fig. 9.10: Free energy curves for a binary system. The overall composi-tion is x. A possible two-phase state would include å at com-position xå and ∫ at x∫. The free energy of this state is marked by the dot in the figure.

Consider a binary system that has two possible phase, å and ∫, at given T, P. Possible free energy curves for the two phase are drawn in Fig. 9.10. Let the overall composition of the system be x, as indicated by the vertical dashed line in the figure. Since x is beyond the limit of the ∫ free energy curve, the system cannot be in a single-phase state of phase ∫. However, it can be either in a single-phase state, phase å of composition, x, or in a two-phase state that is a mixture of å and ∫ phase with different composition. The equilibrium state is that which minimizes the Gibbs free energy. To determine whether a two-phase state minimizes the free energy we compute the free energy of an arbitrary two-phase mixture with net composition x. Since states of phase å cannot be in equilibrium with one another we need only consider equilibria be-tween states of phase å and states of phase ∫. The states must be on opposite sides of the composition, x, so that the average composition can be x. The free energy of a two-phase mixture that has a mole fraction få of phase å in composition xå and a fraction f∫ of ∫ at composition x∫ is g = få¡gå(xå) + f∫ ¡g∫(x∫) = ¡gå(xå) + f∫[¡g∫(x∫) - ¡gå(xå)] 9.17 where we have used the condition få + f∫ = 1. The mole fraction of the ∫ phase can be found from the condition that the overall composition is x: x = fåxå + x∫f∫ = xå + f∫(x∫ - xå) 9.18 from which it follows that

f∫ = x - xå

x∫ - xå 9.19


Page 238

Equation 9.19 is called the lever rule, since the fraction of a phase, say f∫, is equal to the length on the concentration axis, x - xå, on the far side of the fulcrum at x divided by the total length, x∫ - xå. It follows from equation 9.17 and the lever rule that

the free energy of a two-phase mixture of å with composition xå and ∫ with composition x∫ is given by the intersection of a straight line connecting the free energies of the two states on a g-x plot with a vertical line at the average composition, x.

This relation is illustrated in Fig. 9.10. The equilibrium state of a binary system of overall composition, x, is given by the phase or two-phase mixture that intersects a vertical line at x at the lowest value of g. There are two cases. First, let the composition be such that a tangent line to the free energy curve of the phase with least free energy never intersects the free energy curve of the other. This situation is diagrammed in Fig. 9.12 for a case in which phase å has the least free energy. As can be seen by inspecting the figure, in this case there is no two-phase combination of å and ∫ that leads to a free energy below that of the å phase alone. This result is an alternative proof of the general theorem that was established in the previous section. When the tangent to the free energy curve of the å phase at x never touches the free energy curve of the ∫ phase, phase å is stable with respect to the formation of ∫ and the equilibrium state is single-phase.

g

xA B

å

∫

xå x∫x

... Fig. 9.12: Free energy relations for a binary system at a composition

where the å phase is stable. Second, let the composition be such that a tangent line to the phase with the lower free energy at x cuts the free energy curve of the other phase, as illustrated in Fig. 9.13. In this case it is simple to find two-phase states that lead to free energies lower than that of the homogeneous å phase. It can be seen by inspection that the two-phase mixture that provides the least free energy at composition x is a mixture of phases å and ∫ with the compositions xå and x∫ indicated in the figure. These are the states of å and ∫ that


Page 239

are connected by a common tangent, a straight line that just touches the free energy curves of both phases.

g

xA B

å

∫

xå x∫x

... Fig. 9.13: Free energy relations in a binary system at a composition

where a two-phase equilibrium is stable. Note that the same result holds for every composition between xå and x∫, that is, every composition of the system that is internal to the common tangent. Throughout this range of composition the least free energy is obtained when the system is a two-phase mixture of phase å with composition xå and phase ∫ with composition x∫. The states of the phases in the two-phase mixture are the same for all x; however, their proportions change according to the lever law, eq. 9.19. The rule that governs this behavior is called the common tangent rule:

if the composition of a binary system is internal to a common tangent be-tween the free energy curves of two phases, and if the common tangent line lies below the free energy curve of any other possible phase, then the equilibrium state of the system is a two-phase mixture in which the states of the two phases are the states at the terminal points of the common tangent and the fractions of the two phases are given by the lever rule.

9.4.3 The phases present at given T, P We are now in a position to construct the minimum free energy curve for a system and find the equilibrium phases as a function of composition. To do this we construct the common tangents between the free energy curves of the possible phases. The states of least free energy lie on a curve that can be drawn by connecting segments of the free energy curves of the individual phases with their common tangents.


Page 240

g

xA B

å∫

xå ∫xå ∫å + ∫

...

Fig. 9.14: The minimum free energy curve and equilibrium phases for a binary system at given temperature and pressure.

The minimum free energy curve for an example binary system that contains two stable phases is drawn in Fig. 9.14. When the composition of this system is x < xå, the equilibrium state at the given values of T and P is homogeneous phase å. When the com-position is x > x∫ the equilibrium state is homogeneous phase ∫. When the composition is in the range xå < x < x∫ the equilibrium state is a two-phase mixture of å and ∫ in which phase å has composition xå and phase ∫ has composition x∫. The relative fractions of the two phases are given by the lever rule. It is common to find three or more phases in a binary system as the composition is varied at given temperature and pressure. Such a case is diagrammed in Fig. 9.15. Its analysis is straightforward. To find the minimum free energy curve we construct the common tangents between the free energy curves of each of the phases. The common tangent lines join segments of the free energy curves of the individual phases. If a segment of a single-phase curve has the lowest free energy for given composition, x, then the equilibrium state at that composition is a homogeneous state of the preferred phase. If a common tangent has the lowest free energy at x then the equilibrium state is a two-phase mixture. The compositions of the two phases are the compositions of the states that are connected by the common tangent. The mole fractions of the two phases are determined by the lever rule.

g

å∫

xA B

å ∫

©

å + © © ∫ + ©

x1 x2 x3 x4


Page 241

Fig. 9.15: Minimum free energy surface and phase relations in a binary system in which three phases appear at T, P.

In the case illustrated the equilibrium state of the system is a homogeneous å solution when 0 ≤ x < x1, a two phase mixture of å (composition x1) and © (composition x2) when x1 < x < x2, a homogeneous © phase when x2 < x < x3, a two phase mixture of © (composition x3) and ∫ (composition x4) when x3 < x < x4, and a homogeneous ∫ solution when x4 < x ≤ 1. More complex situations are treated in the same way. Note, however, that it is possible for three phases, say å, © and ∫ in Fig. 9.15, to touch the same common tangent line. For given pressure, this happens at a unique value of the temperature; since the phases have different entropies their free energy curves shift by different amounts as the temperature changes. Since it is impossible to control an exact temperature, three-phase equilibrium is rarely important in the engineering sense. If three phases touch the common tangent all three can coexist at equilibrium. The compositions of the three phases are given by the compositions at which their free energy curves touch the tangent, just as in the case of two-phase equilibrium. However, the lever rule does not apply and their relative fractions cannot be determined from the information contained in a plot like Fig. 9.15. 9.4.4 Equilibrium at a congruent point In a one-component system two phases are in equilibrium when their free energies are equal. Their equilibrium is said to be congruent because they have the same composi-tion and can transform into one another without changing composition. A similar situation often occurs in two-component systems; the free energy curves of two different phases touch at particular values of T and x so that the two phases are in equilibrium there. An example of the situation that leads to a congruent point is drawn in Fig. 9.16b. In order that the two-phase equilibrium at the congruent point be an equilibrium state the free energy curves of the two phases must touch without crossing; hence they have the same common tangent there (¡µå = ¡µ∫). However, since the phases are distinct, ∆¡µå/∆x ≠ ∆¡µ∫/∆x, so they touch only at the congruent point. It follows that if å and ∫ have a congruent point at a given temperature, one of the phases (say, ∫) is present in the equilibrium state only at the congruent composition; å is the equilibrium phase at all surrounding compositions.


Page 242

g

xA x0

xA x0

åå+∫∫å+∫

å

∫å

xA x0

å ∫å ∫

...

Fig. 9.16: Possible relation between the free energy curves of phases å and ∫ that have a congruent point at (T0,x0). (a) T > T0; phase å is stable. (b) T = T0; phase å is stable, two-phase equilib-rium at x0. (c) T < T0; å, ∫ and å+∫ phase fields appear.

The usual reason for the appearance of a congruent point is that the system has two phases of different entropy whose free energy curves pass through one another as the temperature is varied, and first touch at some non-zero value of x. The situation is illustrated in Fig. 9.16 for the case in which å is a high-temperature phase whose free energy curve is penetrated by that of the low-temperature phase ∫ as the temperature decreases. Above the congruent point, that is, when T > T0, phase å is stable at all compositions near x0 (Fig. 9.16a). Just below the congruent point the å phase appears on both sides of the ∫ phase, with two-phase regions separating them (Fig. 9.16c). At the congruent point the two free energy curves just touch. 9.4.5 Equilibrium at the critical point of a miscibility gap Sometimes a phase that is stable develops an instability at an interior point of its composition range as the temperature changes so that it decomposes into two phases. This behavior is exhibited by many systems that are solutions at high temperature, but decompose on cooling. An example is the model solution discussed in Chapter 8. There are also unusual systems (long-chain polymers or complex organic systems) that decompose on heating. The temperature and composition at which the instability that leads to decomposition first appears is called the critical point of a miscibility gap, since a single-phase solution divides into two phases of distinct composition (i.e., the components become immiscible) as the state of the system passes through the critical point. An example of the behavior of the free energy function near the critical point of a miscibility gap is shown in Fig. 9.17. In this case a high-temperature solution decomposes on cooling. When the temperature is above the critical point, T > Tc, a homogeneous å solution is stable everywhere near the critical composition, xc; that is, ∆¡µå/∆x > 0 in this composition range. As the critical point is approached, ∆¡µå/∆x decreases near xc, and vanishes at xc when the critical temperature, Tc, is reached. At Tc the å phase is stable everywhere with the possible exception of the isolated point at xc. When the temperature is below Tc there is a range of compositions about xc for which


Page 243

∆¡µ/∆x < 0, so the homogeneous state is unstable. The single, å-phase has decomposed into two distinct phases, å' and å", which produce a two-phase equilibrium at compositions near xc. Above Tc the equilibrium state is a single-phase solution; the two components are miscible. Below Tc the equilibrium state near xc is a two-phase mixture; the components are immiscible in this composition range. The temperature, Tc, and composition, xc, define the critical state at the miscibility gap.

g

xA xc

xA xc

xA

åå

xc

å' å"å'+å"

å' å"

...

Fig. 9.17: Behavior of the free energy curve of a phase that decomposes at a miscibility critical point. (a) T > Tc. (b) T = Tc; å is sta-ble except at the point xc. (c) T < Tc; two distinct phases, å' and å" have a two-phase equilibrium at xc.

9.5 BINARY PHASE DIAGRAMS Equilibrium phase relations are represented by phase diagrams, which are maps that present the equilibrium phases as a function of the thermodynamic variables. The simplest phase diagrams are binary phase diagrams that show the equilibrium phases of a two-component system as a function of the temperature, T, and composition, x. While the Gibbs free energy of a two-component system also depends on the pressure, the pressure is ordinarily fixed by the atmosphere. Binary phase diagrams have the advantage that they can be drawn in two dimensions. While most engineering materials are multicomponent systems, binary phase dia-grams are often useful for analyzing their equilibria. Many important materials have only two dominant components, and can hence be treated as binary systems that are perturbed by the addition of other components as minor solutes or impurities. Examples include most alloy steels, which are solutions of Fe with C, Ni, Mn or Cr, with third, fourth, and often fifth, sixth and seventh species included as minor alloying additions. Other important engineering materials can be approximated as two-component systems in which the components are compounds. Examples include many of the oxide ceramics, which are approximately binary systems of oxide compounds. Still other engineering materials can be approximated as two-component systems in which one element is added against a fixed background provided by the other elements. An example that is of great current interest is the high-temperature superconductor, YBa2Cu3O6+∂, whose variable


Page 244

oxygen content is expressed by the inclusion of the variable, ∂, in the chemical formula, where ∂ varies between 0 and 1. The solid phases that appear when two elements are joined are of two types. The first type is a primary solution, in which the basic structure of the phase is the structure of the pure component, for example, phase å of component A, and the second element, B, is added to form a substitutional or interstitial solution. Since every element has at least a small solubility in every possible phase, a primary solid solution is always formed when a sufficiently small quantity of B is added to a phase å of the pure component, A, or when a sufficiently small quantity of A is added to phase ∫ of the pure component, B. The second type of solid phase is an ordered compound, which is an ordered arrangement of A and B over the sites of a crystal lattice in stoichiometric proportions that are expressed by the chemical formula AxBy. Ordered compounds also give rise to solid solutions. It is always possible to add some excess of A or B to the compound AxBy to create a non-stoichiometric compound which is essentially a solid solution of A or B in AxBy. In the previous section we discussed how the equilibrium phase fields at any given temperature and pressure are determined by the free energy curves. The phase diagram in the T-x plane is a plot of the equilibrium phase fields as the temperature is varied at fixed pressure. The phase fields, or regions of T and x over which the primary solid solutions or compounds exist at equilibrium, are separated by two-phase fields in which two phases coexist. As discussed in the previous section, the two-phase fields are a consequence of the common tangent rule. It is also possible for three phases to be in equilibrium at a particular value of the temperature, although three-phase equilibrium is almost never observed in practice since the temperature cannot be precisely controlled. Because of the possibility of forming ordered compounds at intermediate compositions, the phase diagrams of most binary systems include a number of distinct phases separated by two-phase regions and are very complicated in their appearance. Whatever its complexity, however, a binary phase diagram is always just a map of the equilibrium phases of the system A+B that contains three pieces of information: (1) the phase or phases present at equilibrium at given temperature and composition (T,x); (2) the compositions of the phases if the point (T,x) lies in a two-phase region; (3) the fractions of the phases at a point in a two-phase region. To establish this and illustrate how phase diagrams are generated from the free energy curves we shall discuss two classic types of binary phase diagram in some detail: the solid solution, or phase diagram of a system that forms solutions at all compositions, and the simple eutectic, or phase diagram of a system that has two distinct phases in the solid state. This discussion is followed by a systematic enumeration of the possible shapes of the phase diagrams of simple systems. 9.6 THE SOLID SOLUTION DIAGRAM


Page 245

Let a binary system of elements A and B have two phases: a liquid solution (L) that is preferred at high temperature and a solid solution (å), preferred at low temperature, in which the atoms are distributed over the sites of a particular crystal lattice. A possible phase diagram for the system is shown in Fig. 9.18. It contains a liquid phase at high T, a solid solution phase, å, at all compositions at lower temperature, and a two-phase å+L region that separates the two. (Here, as in the following, we shall ignore the vapor phase that appears at still higher temperature.)

L

L + å

å

A B

T

x

TA

T B

...

Fig. 9.18: Phase diagram of a binary system that forms solutions at all compositions.

The most interesting feature of the diagram is the two-phase å + L region. At the limits of the phase diagram, x = 0 and x = 1, the binary system contains only one compo-nent. The å “ L transition in a one-component system occurs at a particular temperature, the melting point of the pure component in phase å. Hence the two-phase region terminates at the melting points of the two pure components, TA and TB, respectively. Between these limits it opens out into a region in which solid and liquid phases coexist. 9.6.1 The thermodynamics of the solid solution diagram The two-phase region is a consequence of the difference between the shapes of the free energy curves of the liquid and solid phases. The relevant thermodynamics are illustrated in Fig. 9.19, which shows hypothetical forms of the free energy curves at three temperatures. When T > TB, the higher of the two melting points, then the free energy curve of the liquid lies below that of the å phase at all compositions, and the liquid is preferred at all compositions. When T < TA, the lower of the two melting points, the free energy curve of the å phase lies below that of the liquid at all compositions, and å is preferred. When TA < T < TB, however, the two curves intersect, as shown in Fig. 9.19b. The common tangent rule has the consequence that when the composition lies in the range xL < x < xå both å and L are present at equilibrium. The liquid has composition xL, the solid has composition xå, and the fractions of the two phases are given by the lever rule.


Page 246

g

xA B

åL

Bx

A

å

L

xA B

å

L

xåxL

L åå+L

(a) (b) (c) ...

Fig. 9.19: Possible free energy curves of the liquid and solid phases at three temperatures: (a) T > TB, (b) TA < T < TB, (c) T < TA.

L

å

L + å

xåxL

x

T

x

...

Fig. 9.20: A section of the phase diagram including an isotherm (tie-line) that connects the equilibrium concentration xL and xå.

The free energy relations drawn in Fig. 9.19 occur sequentially as the temperature drops because the entropy of the liquid phase is higher than that of the solid. Since, for any composition,

∆ ¡g

∆T = - s 9.20

where s is the molar entropy, the free energy curve of the relatively low-entropy solid phase is displaced downward with respect to that of the liquid as the temperature is low-ered. In the example drawn, it first touches the liquid free energy curve at TB, and moves through it as the temperature is lowered to TA, producing a continuous sequence of rela-tions like that drawn in Fig. 9.19b that generate the two-phase region of the phase diagram. At any given temperature between TA and TB the free energy curves overlap, and the extent of the two-phase region is governed by their common tangent. Hence the boundaries of the L+å region in the phase diagram at a temperature T such that TA < T < TB are just the compositions xL and xå that are in equilibrium by the common tangent rule, as shown in Fig. 9.20.


Page 247

9.6.2 Equilibrium information contained in the phase diagram Once the relation between the binary phase diagram (Fig. 9.20) and the free energy curves (Fig. 9.15) is understood, it becomes clear that three types of information can be found from the binary phase diagram. First, given a temperature and composition (T,x), the equilibrium phase or phases that appear can be read off the phase diagram by simply identifying the phase field in which the point (T,x) lies. If the point (T,x) is in a one-phase field the equilibrium state is a homogeneous state of that phase (å or L in the above example). If the point (T,x) falls in a two-phase field then the equilibrium state is a two-phase mixture of the phases that label the field (å and L in the above example). Second, the compositions of the phases can be determined from the phase diagram. If only one phase is present then its composition is the overall composition, x, of the system. If two phases are present their compositions can be found by drawing an isothermal line through the point (T,x). The compositions of the two phases are given by the intersections of that isothermal line with the boundaries of the two-phase region, as illustrated in Fig. 9.20. The isothermal line connecting the boundaries of the two-phase region is called a tie-line. Note that the compositions of the two phases that are in equilibrium depend on the temperature only, and are the same for every composition of the system that falls within the two-phase region at that temperature. If the overall composition of the system is changed at given T so that it remains within the two-phase region, the fractions of the two phases change, but their compositions remain the same. Third, the fractions of the phases present at equilibrium can be determined from the phase diagram. If the point (T,x) falls in a single-phase region then the system contains that phase alone. If (T,x) lies in a two-phase region then the fractions can be computed from the equilibrium compositions at that temperature by applying the lever rule. In the present example, for a point within the two-phase å + L region,

få = x - xL

xå - xL 9.21

and

fL = 1 - få = xå - x

xå - xL 9.22

9.6.3 Equilibrium phase changes In addition to showing the phases, compositions and phase fractions that are pre-sent, the phase diagram also permits an analysis of the phase changes that occur when a system is cooled or heated slowly enough to preserve equilibrium. The most obvious dif-ference between a one-component system and a binary system is in its solidification


Page 248

behavior. While a one-component system freezes at a particular temperature, a two-component system freezes over a range of temperature through a gradual increase in the fraction of solid as the temperature is lowered through the two-phase region.

L

å

A B

T

x

TA

TB

x

... Fig. 9.21: Phase diagram indicating cooling of sample with composition x

= 0.8. For example, Fig. 9.21 depicts the cooling of a sample of the system diagrammed in Fig. 9.18 at x = 0.8. Following the vertical line downward shows that the system re-mains liquid until the temperature drops to slightly below TB, at which point it enters the two-phase field. According to the tie-line at the top of the two-phase field, the first solid to form within the two-phase region is rich in B, with x « 0.98. As the temperature is lowered, more and more solid appears, while the composition of the solid adjusts along the å boundary of the two-phase field, becoming progressively less rich in B. At the same time the composition of the residual liquid evolves along the L boundary of the two-phase field and becomes richer in A. When the temperature reaches the bottom of the two-phase field for x = 0.8 the solidification is completed and the system becomes homogeneous in phase å at a composition of x = 0.8. Note that two phases that are in equilibrium in the two-phase field each have uni-form composition. If the system solidifies in equilibrium then on each increment of cooling two changes occur in the solid phase: the fraction of solid increases, and the composition of the solid evolves to the new equilibrium value. This latter change requires that the composition of the sold that has already formed adjust to the new equilibrium value. The composition change requires diffusion in the solid state. As we shall see, this is a slow process in most solids, so the system must ordinarily be cooled very slowly to maintain equilibrium during solidification. The microstructure that results from equilibrium solidification will ordinarily be a polygranular å phase in which the grains are equiaxed and fairly large since they remain at temperatures near the melting point for a significant period of time.


Page 249

9.7 THE EUTECTIC PHASE DIAGRAM The second classic type of binary phase diagram applies to a system that has two distinct equilibrium phases in the solid state, one an A-rich terminal solution, å, and the other a B-rich solution, ∫. There must be A- and B-rich terminal solutions with different structures when the components A and B have different crystal structures in their pure forms. It is also possible to have A- and B-rich terminal solutions that are distinct phases with the same structure. This happens, for example, when the system has a miscibility gap like that shown in Fig. 9.17. Assuming a liquid solution at all compositions at high temperature, the phase diagram of the system appears as shown in Fig. 9.22.

T

A Bx

å ∫

L

å+L ∫+L

å + ∫

Fig. 9.22: A eutectic phase diagram of a binary system. The eutectic phase diagram derives its name from the reaction that occurs where the liquid phase field touches the two-phase å+∫ region. If the system is cooled slowly through this point the reaction is L “ å + ∫, which is called a eutectic reaction. 9.7.1 Thermodynamics of the eutectic phase diagram The free energy relations that lead to the eutectic phase diagram are shown in Fig. 9.23. The central feature is the eutectic reaction. The thermodynamic reason for this reaction is contained in the behavior of the free energy curves near the eutectic point. Just above the eutectic point all three phases appear in an isothermal section through the phase diagram. The liquid phase has common tangents with both the å and ∫ phases. However, the liquid phase also has higher entropy, so its free energy curve rises with respect to those of the å and ∫ phases when the temperature is lowered. At the eutectic point the free energy curve for the liquid just touches the common tangent between the å and ∫ curves to establish a three-phase equilibrium. Just below the eutectic temperature the å-∫ common tangent passes below the free energy curve for the liquid phase, which no longer appears in the phase diagram. For any combination of the temperature and composition (T,x), the phases present at equilibrium, their compositions and their phase fractions can be found from the phase diagram. The label of the phase field in which the point (T,x) falls identifies the phases that are present. If there are two phases, their compositions can be found by drawing an


Page 250

isotherm across the two -phase region that passes through the state (T,x), the tie-line shown in Fig. 9.24. The two compositions are given by the intersections of the tie-line with the boundaries of the two-phase region. The fractions of the two phases are given by the lever rule.

g

å∫

xA B

å ∫

L

L

x1 x2 x3 x4

å∫

xA B

å ∫

L

x1 x2

å + L ∫ + L å + ∫

(a) (b)

... Fig. 9.23: (a) Free energy curves for the å, ∫ and L phases just above the

eutectic temperature. (b) Free energy curves just below the eu-tectic temperature.

T

A Bx

xx xå ∫

å ∫

L

å+L ∫+L

å + ∫

...

Fig. 9.24: The compositions of the å and ∫ phases in equilibrium in the state shown by the dot.

It is also possible to have three-phase equilibrium in a system with a eutectic phase diagram, but only when the system is held at precisely the eutectic temperature. The compositions of the three phases in equilibrium at the eutectic temperature are given by the points at which the isotherm at the eutectic temperature touches the å, L and ∫ single-phase fields. However, the lever rule cannot be applied to determine the quantities of the three phases. This is not an important limitation; since it is not physically possible to control the temperature of a system to a precise point, three-phase equilibrium is not an important case in the practical applications of binary systems.


Page 251

9.7.2 Equilibrium phase changes Characteristic phase transformations occur in a system that has a eutectic phase diagram when it is cooled slowly enough that equilibrium phase relations are preserved. The behavior of the system and the resulting microstructure depend on where its overall composition falls in the phase diagram. Three distinct cases are indicated in Fig. 9.25.

T

A Bx

å ∫

L

å+L ∫+L

å + ∫

1x 2x 3x

Fig. 9.25: Compositions leading to three distinct microstructures in a eu-tectic system.

9.7.3 Precipitation from the α phase

(a) (b)

... Fig. 9.26: (a) Equiaxed grain structure of the primary å solid solution.

(b) Precipitates of ∫ in grain interiors and on boundaries. First consider an alloy of composition x1, which is less than the å-phase solubility limit at the eutectic temperature. Let this alloy be cooled slowly enough to maintain equilibrium, beginning from the temperature at the top of the dotted line shown in Fig. 9.25. The system remains liquid until it reaches the temperature at which the dotted line drops into the two-phase, å + L field. It then solidifies over a range of temperature as it is cooled through the two-phase field to become a homogeneous solid in the å phase. It remains homogeneous until its temperature drops into the two-phase, å + ∫ field. At that point a small amount of B-rich ∫-phase precipitates out of the å. The ∫-phase increases in volume and B-content as the temperature is decreased further.


Page 252

The probable microstructure of the system can be inferred from the equilibrium phase diagram. If the system solidifies slowly enough to remain close to equilibrium then the microstructure of the primary å solid solution is normally a polygranular aggregate of equiaxed å grains, as shown in Fig. 9.26a. The ∫ phase typically forms as small precipitates either in the interiors or along the boundaries of the å grains (Fig. 9.26b), depending on how rapidly the system is cooled. 9.7.4 The eutectic microstructure When the system has composition x3, the composition of the eutectic point on the phase diagram in Fig. 9.25, its cooling behavior is qualitatively different. On cooling, the system remains liquid until the eutectic temperature is reached. At this temperature the system freezes completely, just as if it had a single component, but freezes into a two-phase mixture of å and ∫ phases whose compositions and phase fractions are given by the tie-line and lever rule just below the eutectic temperature. If the system is cooled further the compositions and phase fractions adjust according to the tie-lines across the two-phase å+∫ region. The eutectic reaction, L “ å+∫, ordinarily produces a characteristic microstructure, called the eutectic microstructure, which is illustrated in Fig. 9.27. The elementary constituent of eutectic microstructure is a very fine-scale mixture of the two phases in which thin plates of one phase alternate with thin plates of the other or aligned rods of one phase sit in a continuous matrix of the other. The plates or rods are ordinarily single crystals, but may be polycrystalline on a fine scale. The eutectic microstructure is made up of grain-like colonies within which the plates or rods have a common orientation.

... Fig. 9.27: Schematic drawing of a eutectic microstructure. The grain-like

features are eutectic colonies containing aligned plates or rods. The eutectic microstructure forms for kinetic reasons. Consider a stacking of parallel plates of å and ∫, and let the stack grow into the liquid phase along the long axis of the plates, as shown in Fig. 9.28. In order for the A-rich plates of å phase to grow, A atoms must diffuse from the front of the ∫ plates. If the å and ∫ plates are immediately adjacent to one another the distance the A atoms must travel is very small, and lies entirely in the liquid phase, where the atom mobility is high. The B atoms counterflow


Page 253

through the liquid from the front of the growing å plates to the ∫ plates. Hence the eutectic microstructure grows with relative ease.

∫å∫

∫

∫

å

å

L

Fig. 9.28: Illustration of a growing eutectic colony.

9.7.5 Mixed microstructures in a eutectic system The composition x2 in Fig. 9.25 lies between the solubility limit of the å phase and the eutectic composition. If a system of this composition is cooled from a temperature at the top of the dotted line in the figure then it begins to solidify, by forming islands of å phase, when the temperature drops into the two-phase region, and solidifies continuously as the temperature is lowered to the eutectic temperature. However, the system is only partly solidified when the temperature reaches the eutectic line; according to the lever rule only about 60% of the system is solid at a temperature incrementally above the eutectic temperature. The residual liquid has precisely the eutectic composition and solidifies by the eutectic reaction, L “ å + ∫. Hence the product ordinarily has a mixed microstructure that includes islands of "proeutectic" å phase that formed during cooling through the two-phase region. These are embedded in a eutectic constituent that formed through solidification of the residual liquid at the eutectic temperature, as shown in Fig. 9.29.

Proeutectic å

Eutectic colony

...

Fig. 9.29: Probable microstructure of a solidified sample having the com-position x2 in Fig. 9.21.

Note that å phase is present both in the proeutectic å that formed during cooling through the two-phase region, and in the å-phase plates within the eutectic constituent. According to the lever rule the system is about 85% å just below the eutectic temperature. This total is the sum of « 60% proeutectic å phase and « 25% å that is contained in a eutectic constituent that is « 60% å and has « 40% molar fraction in the microstructure.


Page 254

9.7.6 Phase diagrams that include a eutectoid reaction In strict metallurgical terminology the term "eutectic reaction" is reserved for a reaction of the type L “ å+∫ in which a liquid solidifies to a mixture of two solid phases. There are also many systems whose phase diagrams include points at which a high-temperature solid solution transforms into a mixture of two other solid solutions. The reaction is of the type © “ å+∫, where å, ∫ and © are solid solutions with different structures. An important example occurs in the Fe-C phase diagram that governs the behavior of carbon steels. A section of the Fe-C diagram is shown in Fig. 9.30. The high-temperature phase is the © phase, which is the FCC phase of iron, and is an FCC solution of carbon in iron in this case. The phase diagram has a eutectic-like shape and undergoes a reaction of the form © “ å + carbide at the bottom of the © phase field.

1000

800

600

0.5 1.0 1.5 2.0

©

å + ©© + carbide

å + carbide

å

T (ºC)

weight percent carbon ...

Fig. 9.30: A section of the Fe-C diagram that includes the eutectoid reac-tion © “ å + carbide(Fe3C)

For historical reasons a reaction of the type © “ å+∫, where all three phases are solid, is called a eutectoid reaction. The shape of the diagram and the behavior of the system are essentially identical to those in the eutectic diagram. The eutectoid reaction yields a eutectic-like microstructure (Fig. 9.28) for precisely the same kinetic reasons that govern the eutectic case. If a system that has a composition below the eutectoid composition is cooled from the © field, the microstructure contains a proeutectoid constituent as in Fig. 9.29. There is, in fact, no reason to distinguish between the eutectic and eutectoid reactions, and we shall rarely do so in the following. 9.8 COMMON BINARY PHASE DIAGRAMS Many binary systems contain several solid phases, and, hence, have rather complicated phase diagrams. However, most of these diagrams can be simplified and understood by breaking them into parts that involve the equilibrium of only a few phases.


Page 255

In this section we consider possible binary phase diagrams for systems that contain one, two or three solid phases, and also describe one common example of a phase diagram with two liquid phases. Almost all binary phase diagrams can be divided into segments whose behavior is like that of one of the diagrams listed below. 9.8.1 Solid solution diagrams The systems that form solid solutions at all compositions (at least at intermediate temperature) have one of three phase diagrams: the simple solution diagram discussed in Section 9.5, or a slight modification of it that has a congruent point either at the top or the bottom of the two-phase (å+L) region. Of course, solid solutions are only possible when the two components have the same crystal structure in the solid state.

L

å

A B

T

x ...

Fig. 9.31: The simplest phase diagram for the solid solution. The simplest phase diagram for the solid solution is re-drawn in Fig. 9.31. This diagram appears when the free energy curve of the solid solution first cuts the liquid free energy curve at the higher of the two melting points of the pure components, and cuts it last at the lower melting point, so there is no congruent point. Many binary systems have this simple phase diagram, including Ag-Au, Ag-Pd, Au-Pd, Bi-Sb, Nb-Ti, Nb-W, Cd-Mg, Cr-W, Cu-Ni, Cu-Pt, Cu-Pd, Hf-Zr, Mo-Ta, Mo-Ti, Mo-V, Mo-W, Ge-Si, Pd-Rh, Ta-Ti, Ta-V, Ta-Zr, U-Zr, and V-W.

L

å

A B

T

x

TA

TB

...

Fig. 9.32: Solid solution with a high-temperature congruent point.


Page 256

However, almost as many binary solutions have congruent points in their phase diagrams, which shows that the free energy curves of the liquid and the solid solution touch before they cross at x = 0 or x = 1. If the first contact that happens between the liq-uid and solid free energy curves on cooling falls at an intermediate composition then the system has a elevated congruent point, as in Fig. 9.32. If the last contact on cooling falls at an intermediate composition the system has a depressed congruent point, as in Fig. 9.33.

L

å

A B

T

x

TA

TB

...

Fig. 9.33: Solid solution with a low-temperature congruent point. There are very few binary systems with elevated congruent points; the Pb-rich solution in the Pb-Tl system is one of the few examples. On the other hand, depressed congruent points are common. Au-Cu, Au-Ni, Nb-Mo, Nb-Ni, Nb-V, Co-Pd, Cs-K, Fe-Ni, Fe-Pd, Hf-Ta, Mn-Ni, Mn-Fe, Pu-U, Ti-V and Ti-Zr, show this behavior, among oth-ers. Perhaps the strangest example is the behavior of the Ti-Zr system. The high temperature solid structures of both components are BCC, and both transform to HCP on cooling. The phase diagram contains two solid solutions, ∫(BCC) at high temperature and å(HCP) at lower temperature. Both the liquid-å equilibrium and the å-∫ equilibrium are separated by two-phase regions with depressed congruent points like that shown in Fig. 9.33. There is a simple thermodynamic reason for the preference for a low-temperature congruent point. The molar free energy is g = h - Ts 9.23 where h is the molar enthalpy, e + Pv, and s is the molar entropy. The high-temperature phase is the more disordered one, and generally has a higher entropy of mixing. As a consequence its free energy curve tends to have a deeper trough at intermediate composition, so that the liquid and solid free energy curves contact at intermediate composition at a temperature below the melting points of the pure components. 9.8.2 Low-temperature behavior of a solid solution


Page 257

One of the fundamental laws of thermodynamics is the Third Law, which asserts that the entropy of an equilibrium phase vanishes in the limit T “ 0. The Third Law has the consequence that a solid solution cannot be the equilibrium state in the limit of zero temperature. At sufficiently low temperature, the equilibrium state must be a perfectly ordered phase or a simple mixture of perfectly ordered phases. This criterion can be satisfied in two simple ways: the solid solution can decompose into two terminal solutions at low temperature or the system can rearrange itself into an ordered compound or mixture of ordered compounds. The two possibilities are illustrated by the Bragg-Williams model of the solid solution that is described in Chapter 8. The Bragg-Williams solution decomposes via a miscibility gap if its components prefer like bonds, and orders if they prefer unlike bonds. In many real systems this low-temperature behavior intrudes at temperatures so low that it is never observed; such systems are solid solutions for all practical purposes. However, in other cases complete solubility is lost at moderate temperature through the formation of either a miscibility gap or an ordered phase. We consider the two possibilities in turn. Phase diagrams containing a miscibility gap A possible binary phase diagram that contains a miscibility gap is shown in Fig. 9.34. The system freezes into a solid solution (å) at all compositions. However, at lower temperature the solid solution spontaneously decomposes into two solid solutions, å' and å'', that have the same structure but different compositions. The two-phase, å' + å'' re-gion within the miscibility gap contains the same information as any other two-phase region in a binary phase diagram. The compositions of the two phases, å' and å'', are determined as a function of temperature by the isothermal tie-lines. The phase fractions are determined from the tie-line by the lever rule.

L

å

A B

T

x

å' å''

...

Fig. 9.34: The phase diagram of a binary system that contains a miscibil-ity gap in a homogeneous solid solution. The two-phase re-gions are shown shaded; the horizontal lines are the tie-lines.

A miscibility gap is caused by an instability in the free energy curve that develops as the temperature decreases. The sequence of free energy curves that lead to a miscibility gap like that in Fig. 9.34 were presented in Fig. 9.17. Let Tc be the


Page 258

temperature at the top of the miscibility gap. Well above Tc the å free energy curve is concave and well behaved. As Tc it approached the free energy curve flattens, until just below Tc it develops a small convex region, as shown in Fig. 9.17. The convex region has the consequence that two points on the free energy curve are connected by a common tangent. Hence the system decomposes into two solutions with different compositions, but the same structure. As the temperature decreases further the convex region becomes more pronounced and the miscibility gap broadens. As T “ 0 the compositions of the terminal solid solutions approach x = 0 and x = 1 to satisfy the Third Law. As suggested by the Bragg-Williams model (Chapter 8), a miscibility gap is due to a preference for bonds between atoms of like kind, with the consequence that the energy is lowered when the system segregates into A-rich and B-rich solutions. At higher temperature the energetic preference for decomposition is outweighed by the entropic preference for the solid solution. The binary systems whose components are mutually soluble at intermediate tem-perature, but become immiscible at lower temperature include Au-Ni, Au-Pt, Cr-Mo, Cr-W, Cu-Ni, Cu-Rh, Ir-Pa, Ir-Pt and Ta-Zr. Ceramic systems such as NiO-CaO also form solid solutions with low-temperature miscibility gaps. According to the Third Law, the miscibility gap must extend to the pure component lines at T = 0, as drawn in Fig. 9.34. Not all of the diagrams that appear in compilations of binary phase diagrams are drawn this way since decomposition is kinetically slow and difficult to observe at low tem-perature. Phase diagrams with low-temperature ordered phases

L

å

A B

T

x

©

...

Fig. 9.35: Phase diagram of a binary system that has an ordered phase (©) at low temperature. The two-phase regions are shaded with horizontal tie-lines.

The phase diagram of a binary system that forms an ordered phase at low temperature is shown in Fig. 9.35. The single-phase region of the ordered phase is closed at a congruent point at its top (T0) and asymptotes to a point in the limit T “ 0, to satisfy the Third Law. The conditions at the two limiting temperatures have the consequence that the ordered phase field has a shape something like that of an inverted teardrop. At finite temperatures the ordered phase has at least a slight solubility for the species A and


Page 259

B, and is in equilibrium over a range of compositions about its stoichiometric value. The single-phase © field is bounded by two-phase (å+©) fields that separate it from the single-phase å field on either side. If only one ordered phase is present, then the two-phase regions that bound it must spread across the phase diagram in the limit T “ 0 so that the equilibrium phases at T = 0 are the stoichiometric © ordered phase and the å phase at x = 0 or x = 1, in agreement with the Third Law. In many systems that order, several ordered phases are present. The free energy relations that give rise to a phase diagram like that in Fig. 9.35 are illustrated in Fig. 9.36. The free energy curve of the ordered compound lies above that of the å solid solution when T > T0 and passes through it to create a congruent point at T = T0. The free energy curve of the ordered © phase has a strong minimum at its stoichiometric composition. When T < T0 the free energy curve of the © phase lies below that of the å solid solution only at compositions near the stoichiometric value. Hence there are common tangents between the © and å free energy curves on both sides of the © curve. The © phase is the equilibrium phase at compositions near the stoichiometric value; the å phase is at equilibrium at compositions that deviate significantly from the stoichiometric value to either side.

g

xA B

å

©

g

xA B

å

©

åå+©

©å+©

å

...

Fig. 9.36: Free energy relations leading to the appearance of an ordered compound: (a) T > Tc; (b) T < Tc.

Many binary systems whose components have complete or extensive solid solubility at intermediate temperature are known to order into one or more stoichiometric compounds at lower temperature. Examples include Au-Cu, Cu-Pt, Cd-Mg, Co-Pt, Cu-Pd, Fe-Ni, Fe-Pt, Fe-V, Mn-Ni, Ni-Pt, and Ta-V. 9.8.3 Phase diagrams with eutectic or peritectic reactions Binary systems that have two distinct phases in the solid state often have phase diagrams of the simple eutectic or peritectic (inverted eutectic) form. The eutectic diagram


Page 260

The simple eutectic diagram was discussed in some detail in Section 9.5, and is re-drawn in Fig. 9.37. It takes its name from the eutectic reaction, L “ å + ∫ 9.24 which occurs at the minimum point of the liquid phase field. A system that has a eutectic phase diagram is usually one whose components have different crystal structures in the pure form. Since components with different structures cannot form a continuous range of solid solutions, there are always at least two phases in the solid state and the eutectic diagram is one of the simplest the system can have. Among the systems with simple eutectic diagrams are Ag-Bi, Al-Ge, Al-Si, Al-Sn, Au-Co, Au-Si, Bi-Cu, Bi-Cd, Bi-Sn, Cd-Pb, Cu-Li, In-Zn, Pb-Sb, Pb-Sn, Si-Zn and Sn-Zn. Ceramic systems with simple eutectic diagrams include MgO-CaO, among others. All of these systems have components with different crystal structures, and are sufficiently different chemically that it is plausible that they form no stable compounds.

T

A Bx

å ∫

L

...

Fig. 9.37: A binary system with a simple eutectic diagram. The two-phase regions are shown shaded with horizontal tie-lines.

However, there are also systems that have simple eutectic phase diagrams even though their components have the same crystal structure. Examples include Ag-Cu, in which both components are FCC, Cd-Zn, both components HCP, and Na-Rb, both com-ponents BCC.

g

xA

å ∫

B

L

xA

å ∫

B

Lå ∫å+L ∫+L

L

xA

å ∫

B

å ∫

L

å+∫

...


Page 261

Fig. 9.38: Free energy relations leading to a eutectic diagram for a system whose components have a miscibility gap at high temperature. (a) Liquid phase stable; (b) three phases appear at lower T; (c) two solid phase appear below the eutectic point.

The most plausible interpretation of the eutectic behavior in this case is that the two terminal solutions have a miscibility gap at a temperature so high that the liquid phase is retained to temperatures well below Tc. The relations between the free energy curves that lead to a eutectic diagram in a system whose solid phases have a miscibility gap is diagrammed in Fig. 9.38. Fig. 9.38a diagrams a situation in which a solid phase has decomposed into two solutions with the same structure, å and ∫, at a temperature at which the liquid is still stable. As the temperature decreases the solid free energy curve drops with respect to the liquid, and leads to a eutectic diagram. Fig. 9.38b shows the situation just above the eutectic point where all three phases appear. Fig. 9.38c shows the situation just below the eutectic point where only å and ∫ solid solutions appear at equilibrium. The peritectic diagram The classic peritectic phase diagram is drawn in Fig. 9.39. It is characterized by the appearance of a peritectic reaction of the form ∫ + L “ å 9.25 that appears at the top of the å field.

A B

x

å

∫

L

T

...

Fig. 9.39: A simple peritectic phase diagram in a binary system. The peritectic reaction is, essentially, an inverse eutectic. The classic eutectic reaction occurs when the free energy curve of the liquid cuts through a common tangent to the curves of two solid phases on heating. The peritectic occurs when the free energy curve of a solid phase cuts through a common tangent to the curves of liquid and solid phases on cooling. The relations between the free energy curves just above and just below the peritectic point are illustrated in Fig. 9.40. Just above the peritectic the common tangent to the L and ∫ phases lies below the å free energy curve, as in Fig.


Page 262

9.40a. At the peritectic the å free energy curve contacts that common tangent, and drops below it as the system is cooled further to create the configuration shown in Fig. 9.40b. Simple peritectic diagrams are much less common that simple eutectic ones. The thermodynamic reason is apparent from Fig. 9.40. To create a peritectic point the free energy curve of the solid must cut that of the liquid at finite x, that is, at a composition away from the axis. For that to happen the free energy of the solid phase must decrease more quickly than that of the liquid at small x. Since the liquid has higher entropy, this is only likely to happen when the enthalpy of the solid phase drops rapidly with the solute content, that is, when there is a strong preferential bonding between the two components in the å phase. But this is precisely the situation that leads to the formation of stable compounds between the two components. A system that has a simple peritectic diagram is, therefore, likely to be one that almost forms stable compounds.

g

xA

å

∫ L

L L+∫ ∫

B

g

xA

å ∫

L

LL+åå å+∫ ∫

B

... Fig. 9.40: Free energy relations leading to a peritectic phase diagram. (a)

T just above the peritectic temperature; (b) T just below. Examples of metal systems that have peritectic diagrams are Cu-Nb, Cu-Co, Ni-Mo, Ni-Re, Ni-Ru, Os-Ir, Os-Pd, Re-Rh, W-Pd and W-Pt. Not surprisingly, some of these systems, such as Ni-Mo, form intermetallic compounds at lower temperature. There are also several examples of systems that have peritectic reactions in which the high-temperature solid phase is an intermetallic compound. Examples include Al-Al3Ti, Al-Al3Zr and Al-AlSb.

g

xA

å

∫

B

L

xA

å

∫

B

Lå å+L

L

xA

å

∫

B

å ∫

L

å+∫

... Fig. 9.41: A peritectic reaction in a system with a high-temperature mis-

cibility gap. (a) The liquid phase is stable at a high tempera-ture below the miscibility gap in the solid; (b) two-phase equi-


Page 263

librium just above the peritectic; (c) two-phase equilibrium below the peritectic temperature.

There is at least one example, Ag-Pt (FCC), in which two components with the same crystal structure have a peritectic phase diagram. As in the case of a eutectic diagram between elements with the same structure, this suggests that the components have a high-temperature miscibility gap, leading to free energy relations like those shown in Fig. 9.41. The condition is that the second solid phase that intersects the liquid curve (∫ in the case shown) makes its first appearance as a stable phase by cutting the tie-line between the liquid and the å solid solution. 9.8.4 Structural transformations in the solid state When one of the components of a binary system undergoes a structural transformation on cooling, not only is a new structure introduced into the binary phase diagram, but new two-phase equilibria appear. Since the phases that are connected by the structural transformation are different, they respond differently to the introduction of the solute. The result is a two-phase equilibrium field between them. Figs. 9.42 and 9.43 show the common forms of the binary phase diagram of a system in which one component (A) undergoes a structural transformation (© “ å) as the temperature is lowered. The configurations at the solid-solid transformation are geometrically identical to those at the eutectic or peritectic points of the liquid-solid transformation.

T

A Bx

å∫

L

∫+L

å + ∫

©©+L

©+∫

...

Fig. 9.42: A binary system with a eutectic reaction at the bottom of the liquid phase field and a eutectoid reaction at the bottom of the phase field of the high-temperature (©) phase. The two-phase fields are shaded with isothermal tie-lines.

If å is the low-temperature phase of a component (A) that also has a high-temperature phase, ©, then the free energy of å falls below that of © as the temperature is lowered. On the x=0 axis (where the system has only one component) the two free energies cross at a particular transition temperature. However, since the two phases are distinct they respond differently to the solute, and hence have different free energy curves at finite x. As these curves pass through one another they generate a two-phase region, just as in the liquid-solid case. The shape of the two-phase (å+©) region depends on


Page 264

where the å free energy curve first contacts the © curve as the temperature is lowered. If the first contact is between the å and © curves rather than between the å curve and the ©-∫ common tangent then the behavior is just like that near a eutectic point in the liquid-solid case; the high-temperature phase field (©) extends to a temperature minimum at finite x, as shown in Fig. 9.42. The reaction at the bottom of the © field is © “ å + ∫ 9.26 This reaction is called a eutectoid reaction since it is eutectic-like, but involves only solid phases. In the writer's opinion the separate terminology is redundant. In a system that has a eutectoid reaction the first contact between å and © is ordinarily at x = 0, in which case the phase diagram near the reaction looks like that shown in Fig. 9.42. However, it is also possible that the first contact occurs slightly off the x-axis at finite composition. In this case the å field has a maximum at a congruent point between © and å, while the © field has a minimum at a eutectic point at slightly higher composition. We shall not illustrate this case. Eutectoid reactions are common in binary systems that include a component that transforms in the solid state. We discussed the case of the ©“å transformation in Fe-Fe3C in Section 9.5. Other examples include the ∫“å transformation of Ti in Ti-Cr and Ti-W, the å“∫ transformation in Mn in Ni-Mn, and the å“∫ transformation in Th in Th-U and Th-Zr. Eutectoid reactions also occur in the transformations of many intermetallic and oxide compounds. The second possibility is that, on cooling, the å free energy curve cuts the ©-∫ common tangent before it contacts the © free energy curve. Then the situation near the transition temperature is like that illustrated for the peritectic transition in the liquid-solid case. The å phase field extends to a temperature maximum at finite x, and the configura-tion near the transition has a shape like that drawn in Fig. 9.43. The reaction at the maxi-mum point of the å field is © + ∫ “ å 9.27 and is called a peritectoid reaction.

T

A Bx

å

∫

L

©


Page 265

... Fig. 9.43: A binary system with a eutectic reaction at the bottom of the

liquid phase field and a peritectoid reaction at the top of the phase field of the low-temperature (å) phase. The two-phase fields are shaded with isothermal tie-lines.

Peritectoid reactions are reasonably common. Examples include the ©“å transition of Fe in Fe-Nb and Fe-Ta, the å“‰ transition of Co in Co-Cr and Co-W, and the å“∫ transition of Mn in Mn-Cr. Peritectic reactions are also found in a number of intermetallic and oxide compounds. 9.8.5 Systems that form compounds A substantial fraction of all binary systems form ordered compounds in the solid state. In fact, it is common for several compounds to appear in the phase diagram. To explore the influence of ordered compounds on the shape of the phase diagram we con-sider systems that contain a single one. Fig. 9.35 illustrates the phase field of a compound that emerges directly from a solid solution. We now consider compounds in systems that contain two terminal solid solutions. Four cases are reasonably common: (1) a compound first appears at a congruent point in the liquid; (2) a compound first appears at a peritectic point in a two-phase region (å+L); (3) a high-temperature compound disappears at a eutectoid; (4) a low-temperature first appears at a peritectoid. Finally, we consider the equilibrium phase fields near a structural transformation of an ordered compound. Compounds that form directly from the liquid Many binary systems have stable compounds that can be formed directly from the liquid at a congruent point. The simplest phase diagram for a system of this type is shown in Fig. 9.44. The compound essentially divides the phase diagram into two eutectic diagrams between the compound and the terminal solid solutions.

T

A Bx

å∫

L

©

...

Fig. 9.44: The phase diagram of a system that forms a stable compound at an intermediate composition.


Page 266

Phase diagrams like that shown in Fig. 9.44 govern a large number of binary sys-tems, including Al-Sb, Al-Ca, Al-Au, As-In, As-Pb, Ca-Mg, Nb-Cr, Cd-Sb, Cd-Te, Cr-Ta, Cr-Zr, Ga-Sb, Hf-V, In-Sb, Mg-Pb, Mg-Si, Mg-Sn, Mo-Pt, Pb-Te, Sn-Te, and Zn-Te, among others. Phase diagrams of this type are particularly common in the III-V and II-VI systems. In these cases the stable compounds are the semiconducting III-V and II-VI compounds. The reaction at the congruent point in Fig. 9.44 is L “ ©, where © is the com-pound. Compounds that form congruently are particularly easy to make since they can be gotten by direct solidification (casting or crystal growth) from a liquid of appropriate composition. Phase diagrams of this type are basic to a number of technologically important processes. Perhaps the most important is the growth of large crystals of III-V and II-VI semiconducting compounds from the melt, which is only possible when the compound has a congruent point with the liquid. Compounds that form through a peritectic reaction If a binary system contains a single compound (©) whose free energy curve is such that its first appearance breaks a solid-liquid (å+L) tie-line then the compound is derived from a peritectic reaction (å + L “ ©) and the simplest phase diagram is like that shown in Fig. 9.45.

T

A Bx

∫

L

å

©

...

Fig. 9.45: The phase diagram of a binary system with an intermediate compound that forms by a peritectic reaction.

Several binary systems that have phase diagrams that closely resemble Fig. 9.45, including Bi-Pb, Cd-Sn, Hg-Pb, In-Pb, Hf-W, Mo-Zr, Ru-W and Sn-Tl. The phase dia-grams of Mo-Hf and Sb-Sn differ from Fig. 9.45 only in that the ∫ phase at the far end of the phase diagram has a peritectic rather than a eutectic relation to ©. The phase diagrams of many binary systems that form multiple compounds are such that some of these com-pounds form through peritectic reactions and have local phase relationships like those in the left-hand side of Fig. 9.45. Because the © compound in Fig. 9.45 is the product of a peritectic reaction it cannot be cast or grown directly from the melt. Moreover, many of the more useful


Page 267

compounds of this type include elements that diffuse slowly in the solid state so that it is difficult to make the compound by holding the system at a point within equilibrium phase field. Technologically important compounds that have this behavior include the A15 superconducting compounds such as Nb3Sn and Nb3Al, high-temperature intermetallic structural materials like Ni3Al, and low-density intermetallics with potential high-temperature structural applications such as the Al-Ti intermetallics. Complex processing techniques such as reaction from a ternary solution, vapor deposition, or powder processing are required to synthesize these compounds. Finally, note that a phase diagram like that shown in Fig. 9.45 has a eutectic reac-tion, but the phases that border the eutectic include intermediate compounds (© in the fig-ure). Nonetheless, a system that has the eutectic composition will solidify into a eutectic microstructure like that illustrated in Fig. 9.27. One or both of the interleaved phases are intermetallics rather than terminal solid solutions. Compounds that disappear at a eutectoid Many binary phase diagrams contain ordered compounds that only appear at intermediate temperature. They are stable at high temperature, but eventually disappear if the system is cooled. For this to happen in a simple system that contains only one ordered compound the common tangent to the free energy curves of the terminal solid solutions must fall beneath the free energy curve of the compound at sufficiently low temperature. This is more likely to happen if the terminal solution is more stable than the compound, and is hence most often observed in systems whose compounds result from a peritectic reaction like that shown in Fig. 9.45. Fig. 9.46 contains a sketch of a simple phase diagram containing a single ordered compound that is confined to intermediate temperature. The top of the phase field of the compound is a peritectic point, å+L “ ©. The phase field terminates in a eutectoid reac-tion, © “ å+∫.

T

A Bx

∫

L

å

©

...

Fig. 9.46: Phase diagram of a simple binary system that forms a com-pound at intermediate temperature.


Page 268

Several binary systems have phase diagrams that resemble Fig. 9.46 very closely, including Bi-Pb, Cd-Sn and Ru-W. Many other systems contain compounds whose ther-mal stability is limited by the intrusion of other ordered compounds. Compounds that form at a peritectoid If an intermetallic compound first appears in the solid state then it intrudes either into a single-phase region or a two-phase region. In the former case the maximum tem-perature of the equilibrium field of the compound is a congruent point, as in Fig. 9.35, where the compound forms by a reaction of the type å “ ©. In the latter case the maximum temperature is the temperature at which the free energy curve of the compound cuts a two-phase tangent line. The top of the field is a peritectoid point, and the compound forms by a reaction of the type å+∫ “ ©. A simple phase diagram for a binary system with a compound that forms by a peritectoid reaction is shown in Fig. 9.47. Several binary systems have phase diagrams that resemble this one, including Ru-Mo, Ru-Nb and Pd-V. In binary systems that contain several compounds it is common that one or more appear at low temperature through peritectic reactions.

L

å

A B

T

x

©

∫

...

Fig. 9.47: Phase diagram of a simple system in which a compound ap-pears through a low-temperature peritectoid reaction.

Structural transformation of a compound Compounds may undergo structural transformations just as pure phases do. In fact, there is a richer set of possibilities for transformations in compounds, since com-pounds can change in chemical order as well as in basic lattice structure. A compounds transforms when there are two separate phases of essentially the same compound (nearly the same stoichiometric composition) whose free energies become equal at some temperature and composition. If the high-temperature phase of the compound were cooled to that temperature and composition it would transform ho-mogeneously to the low-temperature phase. If the two phases are related by a first-order transformation, that is, if they are distinct phases at the transformation point, then they


Page 269

are represented by different free energy curves and their first contact on lowering the temperature is at an isolated point. A compound differs from a pure component in that its composition can deviate from stoichiometry in either the positive or the negative sense. At finite temperature the free energy curve of a compound is continuous through its stoichiometric composition (as illustrated, for example, in Fig. 9.25) and its chemical potential is not singular there. This has the consequence that the free energy curves of two phases of essentially the same compound (that is, compounds that have the same stoichiometric composition in the limit T “ 0) may first touch one another on cooling at a composition that is off-stoichiometric and possibly outside the equilibrium phase field of the high-temperature phase. The two possibilities are illustrated in Fig. 9.48, which shows free energy curves at a temperature just above that at which a compound, ©, transforms to a second compound, ∂, in a system whose phase diagram is like that in Fig. 9.44 at temperatures above that shown. In the left-hand figure the free energy curve of the ∂ phase contacts that of the © phase within its region of stability. In the right-hand figure the ∂ free energy curve contacts the common tangent between the © phase and the primary ∫ solution.

g

å

∫

xA B

å ∫

©

©

∂

å+© ∫+©

å

∫

xA B

å ∫

©

©

∂

å+© ∫+©

...

Fig. 9.48: Possible shapes of the free energy curves near the transforma-tion ©“∂. (a) The ∂ free energy curve contacts within the © stability range. (b) The ∂ free energy curve contacts the ©-∫ common tangent.


Page 270

T

A Bx

å∫

L

©

∂

...

Fig. 9.49: Phase diagram of a binary system in which a high-temperature compound, ©, transforms to a low-temperature compound, ∂, at a congruent point, as in Fig. 9.48a.

The situation shown in Fig. 9.48a leads to a phase diagram like that shown in Fig. 9.49. The contact of the © and ∂ free energy curves gives rise to a congruent point in the © phase field at which © “ ∂ without change of composition. The congruent point is en-closed by two-phase (©+∂) fields that terminate at eutectic points for the reactions © “ å+∂ and © “ ∫+∂. Structural transformations of compounds that lead to a phase relationship like that drawn in Fig. 9.49 occur in a number of binary systems, including W-C, Ag-Ga, Ag-Li, Au-Zn, Cu-In, Cu-Sn, Mo-C, Mn-Zn, Ni-S, Ni-Sn and Ni-Sb. When the configuration of the free energy curves near the structural transformation resembles that in Fig. 9.48b, the low-temperature phase, ∂, first appears as the product of a peritectic reaction, ∫+© “ ∂. The given fact that ∂ becomes stable means that the ∂ curve is displaced downward relative to the © free energy curve as the temperature decreases. The form of the phase diagram near the peritectic can be approximated by translating the ∂ free energy curve in 9.48b downwards as T decreases and constructing the successive common tangents. The resulting phase diagram is drawn in Fig. 9.50. The © and ∂ stability fields never touch; they are separated by a narrow two-phase region that terminates at a eutectic point where the reaction is © “ å + ∂.

T

A Bx

å∫

L

©

∂

...


Page 271

Fig. 9.50: Phase diagram for a system in which a compound transforms through a peritectic reaction, as in Fig. 9.48b.

Compound structural transformations of the type that appears in Fig. 9.50 are found in many binary phase diagrams. Among the systems that have reactions of this type are Ag-Cd, Ag-In, Bi-Mg, Co-Cr, Ge-Cu, Cu-Sn, Hf-Ir, Mn-Ni, Mn-Pt, Mn-Zn, Mo-Pt, Ni-V, and Zn-Sb. As this extensive list suggests, the geometry of the transformation in Fig. 9.50 is, in fact, more common than that the congruent geometry shown in Fig. 9.49. Its prevalence reflects the narrow width of the equilibrium phase fields of most solid compounds; a small difference in the relative composition dependence of the free energies of the two phases can then shift the first intersection of the two curves out of the stability field of the high-temperature phase. 9.8.6 Mutation lines in binary phase diagrams The kind of phase transition known as a mutation, or second-order phase transition, was discussed in Section 9.3. In a mutation, one phase simply becomes another. There is no two-phase equilibrium and, hence, there are no two-phase regions associated with mutations. However, in a binary system the critical temperature for a mutation can be a func-tion of composition, and almost always is. Hence the mutation appears as a simple curve in a pseudo-single phase region that contains both of the phases that are related by the mutation. There is also no discontinuity in the boundary of the pseudo-single phase re-gion where the mutation line contacts it. The composition of the phase in equilibrium in a two-phase region is fixed by temperature. Hence a mutation line is a horizontal isotherm through a two-phase region that gives the temperature at which one phase mutates.

T

A Bx

å ∫

L

å'

...

Fig. 9.51: A eutectic system with a mutation in the å terminal solid solu-tion, indicated by the dashed line.


Page 272

T

A Bx

å∫

L

©

©'

...

Fig. 9.52: A binary system with an intermediate compound that under-goes a mutation, indicated by the dashed line.

Fig. 9.51 illustrates the appearance of a eutectic phase diagram with a mutation in the å-rich solid solution. The ferromagnetic transition in Fe and Ni and the rare earths leads to phase relationships like those shown in Fig. 9.51. Fig. 9.52 illustrates the phase relationships in a simple system with an intermediate ordered phase that mutates. Many intermediate compounds undergo ordering reactions that are mutations. The classic example is the ∫ “ ∫' transition in Cu-Zn. 9.8.7 Miscibility gap in the liquid As a final example we consider a binary system in which a miscibility gap intrudes in the liquid, as it does in many real systems. The simplest system of this type has only the two terminal solid solutions in the solid state. The phase diagram is drawn in Fig. 9.53.

T

A Bx

∫

å

L1 L2

...

Fig. 9.53: A possible phase diagram for a binary system with a miscibility gap in the liquid. The shaded region is an equilibrium between two liquid phases.

A sequence of free energy relations that lead to a phase diagram like that shown in Fig. 9.53 is drawn in Fig. 9.54. Fig. 9.54a pertains to a temperature just below the


Page 273

melting point of the å phase. The miscibility gap in the liquid is due to the inflection in its free energy curve, which divides it into two stable phases with a common tangent. The three phases å, L1 and L2 appear in the section. Fig. 9.54b is drawn at a lower temperature at which the liquid phase L1 no longer appears. As shown in the diagram the free energy curve of the å phase has dropped with respect to that of the liquid, with the consequence that the lowest common tangent connects the å and L2 free energy curves directly. The ∫ free energy curve is everywhere above that of L2. As a consequence there are two phases in the section, å and L2. Fig. 9.54c illustrates behavior at a still lower temperature where both the å and ∫ free energy curves are well below that of the liquid. The lowest common tangent in this case connects å and ∫ directly; only these phases appear in an isothermal section through the phase diagram.

g

xA

å

L' L"

∫

åå+L'L' L'+L" L"

B xA

åL' L"

∫

å

B

å+L" L"

xA

å L' L" ∫

å

B

å+∫ ∫

...

Fig. 9.54: Free energy relations at three temperatures in a system with the phase diagram shown in Fig. 9.53: (a) just below the melting point of å; (b) at a T where only å and L2 appear; (c) at a T where only å and ∫ appear.

The binary systems that have phase diagrams that resemble Fig. 9.53 include Al-Bi, Al-In, Bi-Zn, Cu-Cr, Cu-Pb, Cu-Tl, Ni-Pb, Pb-Zn and Th-U. These systems contain species that are very different in their chemical behavior, which leads to the miscibility gap in the liquid. A like behavior is seen on the silica-rich side of the SiO2-MgO diagram. Similar phase relations are found at low temperature in the solid state in a number of systems that form extensive solid solutions, including Al-Zn, Nb-Zr and Hf-Ta. Phase fields like those in Fig. 9.53 result from a miscibility gap in the solid solution. In the case of Al-Zn the miscibility gap has a bottom because of its interaction with the terminal Zn solid solution. In Nb-Zr and Hf-Ta the bottom of the miscibility gap is due to interference by the low-temperature phase of one of the components; both Hf and Zr have structural transformations at low temperature.


Page 274

C h a p t e r 1 0 : K i n e t i c sC h a p t e r 1 0 : K i n e t i c s

Many years ago, on a visit to Korea I was introduced to a famous palmist. After examining my palm, he noted that I have an exceptionally long "life line" "Great", I said, "does that mean I can expect a long, long life?" "Perhaps." he replied, "It is not just the length of the line, but how fast you are moving along it."

10.1 INTRODUCTION The science of kinetics is concerned with the rate of change. Thermodynamics sets the driving forces that impel changes of state; kinetics determines how quickly those changes happen. When the process can happen in more than one way, the relative kinetics also determine the path taken in a change of state; the system evolves along the path that permits the most rapid rate of change. The path that is preferred kinetically does not necessarily lead to the preferred equilibrium state. It may, instead, terminate in a metastable state where the system is trapped and cannot evolve further. This fact is central to materials processing, which is the art of manipulating the microstructure to control engineering properties. To achieve a desired microstructure one brings the system into a non-equilibrium state that is chosen so that the desired microstructure is both thermodynamically possible and kinetically achievable. Once the desired mi-crostructure is established, one changes the forces or constraints on the system to "freeze" the microstructure so that it is retained for at least the intended service life of the material. It is useful to divide changes of state into two types that are based, respectively, on internal and global equilibrium. Internal equilibrium requires that the temperature (T), the pressure (P) and the chemical potentials {µ} have uniform values. If any of these conditions is violated, the material evolves to re-establish it. A local gradient in the temperature causes a flow of heat, a local gradient in the pressure causes a mechanical relaxation that redistributes the volume, and a local gradient in the chemical potential induces a flow of the associated chemical species. These are, ordinarily, continuous changes of state that do not change the basic structure, or phase of the material. The stronger condition of thermodynamic equilibrium is the global condition of equilibrium, which asserts that the total entropy of the system and its environment should be as large as possible or, equivalently, that the appropriate thermodynamic potential of the system should be as small as possible. Changes of state that result from violations of the global conditions of equilibrium are called phase transformations. They are discon-


Page 275

tinuous transitions in the sense that the microstructure is physically different after the transformation has happened. We shall defer the discussion of phase transformations to the following chapter, and consider the kinetics of continuous transitions here. We shall specifically describe the conduction of heat in solids and the diffusion of chemical species through solids. While the microstructural mechanisms of heat and mass flow are very different, they are governed by a differential equation, called the diffusion equation, that has the same form for both. Heat is transported by particles whose motion is controlled by collisions with the lattice and with one another. Kinetic processes of this sort are called frictional processes. A familiar example is the fall of a solid body through the air; its terminal (steady state) velocity is determined by the balance between the force of gravity, which tends to accelerate it, and the frictional force of the air, which is due to collisions with air molecules that tend to slow it down. Mass diffusion through solids ordinarily occurs by discrete atom jumps from one equilibrium site to another. In moving from site to site the atom must pass through inter-mediate positions in which it has a relatively high energy. The energy required to pass through these intermediate configurations is supplied by thermal fluctuations. Kinetic processes of this type are called thermally activated processes. Diffusion in the solid state is a prototypic example. This chapter begins with a discussion of heat transfer as an example of a continuous transition. This example is followed by a brief introduction to the general theory of continuous transitions, called non-equilibrium thermodynamics. The chapter concludes with the analysis of diffusion in solids. 10.2 LOCAL EQUILIBRIUM We ordinarily describe the states of systems that are not in equilibrium by giving the distribution of the thermodynamic forces within them, with the understanding that, if the thermodynamic forces are not uniform, the system will evolve in a direction that makes them so. For example, we describe a material that is heated on one side as having a non-uniform temperature. Strictly speaking, it is incorrect to do this. Temperature is a characteristic of thermodynamic equilibrium, and is not well defined when the conditions of equilibrium are violated. (As an example, those of you who are studying chemical or mechanical engineering may have already encountered the practical problem of measuring the temperature of a flowing fluid, and learned that the temperature you measure may depend on how the measurement is done.) However, provided that the system is not too far from equilibrium we can construct a perfectly useful theory that uses equilibrium properties like the local temperature and entropy and chemical potential. It is worth taking a moment to consider how this can be done.


Page 276

To apply equilibrium thermodynamics to non-equilibrium states we use the as-sumption of local equilibrium. In most of the continuous transitions that are of interest in materials science, the deviation from equilibrium is imperceptible on the atomic scale. The variation in temperature or composition is significant on a scale of millimeters, and possibly micrometers (microns), but even a micron corresponds to several thousand interatomic spacings. Moreover, the rate of change of thermodynamic quantities is usually very slow compared to the processes that establish equilibrium on the atomic level. The electron cloud around an atom adjusts itself at a natural frequency that is called the plasmon frequency, and is about 1015 cycles per second, and the atom as a whole vibrates with a mean frequency on the order of 1013 cycles per second. In contrast, sensible changes in the temperature ordinarily require at least a fraction of a second and significant chemical redistributions require minutes or hours. In the usual situation, an atom comes to an essentially complete equilibrium with its immediate en-vironment long before it senses any macroscopic violation of the conditions of equilib-rium. It is, therefore, usually reasonable to assume that any microscopic subvolume of a material is locally in equilibrium at any instant of time.

x1

x2

R

dV(R)

Fig. 10.1: Macroscopic system divided into differential subvolumes. We can construct a non-equilibrium thermodynamics of continuous processes in the following formal way. Let the system have total volume, V, and let it be divided into subvolumes, dV, that are microscopic, but still large on the atomic scale (Fig. 10.1). Each differential volume is instantaneously in equilibrium so that it has well-defined values of the thermodynamic variables. The position of a particular volume element, dV, can be designated by its vector position, R, in a coordinate system fixed in space. It has the instantaneous temperature T(R), pressure P(R), and chemical potentials {µ(R)}. When the thermodynamic state is non-uniform these are continuous functions of the position variable, R. When two adjacent differential volumes are not in equilibrium with one another their interaction changes the state of the system. They exchange heat and mass across their fixed boundaries. The process by which this happens can be modeled by replacing the system by an idealized one that remains in equilibrium at all times. Let the differential volume elements be separated by partitions that isolate them from one another, but periodically become transparent so that heat or mass can move across them.


Page 277

While the partitions are impermeable each differential volume element comes to equilibrium. When the partitions are transparent each element exchanges incremental quantities of heat and mass with its neighbors. The time interval between periods of transparency is taken to be long enough to establish internal equilibrium, but short enough that the system seems to evolve continuously on the time scale that is pertinent to macroscopic changes. Each differential volume element of the ideal system is in a well-defined equilibrium thermodynamic state at any instant of time. However, the system as a whole is in a non-equilibrium state that changes continuously. We use this idealized system to model and predict the behavior of real systems in non-equilibrium situations. The simplest case of non-equilibrium behavior is the conduction of heat through a material whose pressure and composition are fixed. We consider this case first, and then return to a discussion of the general equations of non-equilibrium thermodynamics. 10.3 THE CONDUCTION OF HEAT 10.3.1 Heat conduction in one dimension: Fourier's Law Consider a solid whose atoms are immobile, so that atomic diffusion can be ne-glected, and assume that the temperature varies in the x-direction, T = T(x). This means that two adjacent subvolumes have different temperatures, as shown in Fig. 10.2. According to the Second Law, the temperature difference induces a flow of heat across the boundary, as shown in the figure. The heat flow (°Q) is in the direction of decreasing temperature and vanishes when the temperature difference vanishes. The simplest equation that has these properties is °Q = - C(ÎT) 10.1 where C is a positive constant.

T1T2JQ

Fig. 10.2: Heat flow from one differential volume to another in response to a temperature difference, T2 > T1.

The constant, C, in equation 10.1 cannot be a material property since both the temperature difference, ÎT, and the heat flow, °Q, depend on the specific geometry of the differential volume elements. When the local temperature variation is small the temperature difference can be written


Page 278

ÎT = dTdx Îx 10.2

where x is a coordinate in the direction that connects the centers of the volume elements, Îx is the distance between the centers and dT/dx is the temperature gradient in the direction, x. The flow of heat per unit time from one volume element to the other can be written °Q = JQ dA 10.3 where dA is the area of the interface and JQ is the heat flux perpendicular to the interface, the heat flow across the interface per unit area per unit time. Substituting equations 10.2 and 10.3 into 10.1 yields the relation

JQ = - k dTdx 10.4

Equation 10.4 relates the heat flux to the temperature gradient. The equation holds independent of the geometry of the differential volume elements, and the coeffi-cient, k, is a material property, which is called the thermal conductivity. Equation 10.4 is the one-dimensional form of Fourier's Law of Heat Conduction. According to equation 10.4 the local deviation from thermal equilibrium produces a driving force, the temperature gradient, dT/dx. The response to this force is a thermodynamic flux, JQ, in a direction that would relieve the driving force. The magnitude of the flux induced by a given force is determined by the thermal conductivity, which is a property of the material. When heat flows in accordance with Fourier's law it may or may not change the temperature of the volume element that receives it. For the temperature to change there must be an accumulation of energy in the volume element, which only happens if there is a net flux of heat across its boundary. To find the equation that governs the temperature change consider the one-dimensional case shown in Fig. 10.3.

T1T2T3

JQ12JQ23

Fig. 10.3: One-dimensional flow of heat. Heat flows into the central volume element in Fig. 10.3 from the left, and flows out across the interface to the right. The net heat added per unit time, °QT, is °QT = JQ

23 - JQ12 dA 10.5


Page 279

where dA is the interface area. If the volume element is small and the flux is continuous then the flux at the 12 interface is related to that at the 23 interface by

JQ12 = JQ

23 +

dJQ

dx dx 10.6

where dx is the length of the volume element. It follows that

°QT = -

dJQ

dx dV =

d

dx

kdTdx dV 10.7

where we have used Fourier's Law (equation 10.4). When the temperature of a differential volume element is changed by dT at constant volume (dV) and composition the change in energy is equal to the heat added, and is proportional to the change in temperature: dE = dQ = CvdTdV 10.8 where Cv is the isometric specific heat. Dividing equation 10.8 by the time differential, dt, substituting equation 10.7, and dividing through by the volume, dV, we obtain the partial differential equation that determines the change in the temperature as a function of position and time, that is, the partial differential equation that determines the function, T(x,t),

∆∆t T(x,t) =

1Cv

∆

∆x

k

∆∆xT(x,t) 10.9

When the thermal conductivity is constant this simplifies to

∆T∆t =

kCv

∆2T

∆x2 10.10

The partial differential equation 10.10 is an example of the diffusion equation, whose solutions are tabulated in standard texts on differential equations. 10.3.2 Heat conduction in three dimensions In a more typical case the temperature is a function of position in three-dimensional space, and is described by the function, T(R,t), where R = x1e1 + x2e2 + x3e3 10.11 is the position vector, and x1, x2, x3 are the coordinates of the point, R, in a Cartesian co-ordinate system with the unit vectors e1, e2, e3. In the three-dimensional case the local variation in the temperature is described by the gradient,


Page 280

ÂT =

∆T

∆x1 e1 +

∆T

∆x2 e2 +

∆T

∆x3 e3 10.12

which is a vector that has both magnitude and direction. The heat flux that is induced by the temperature gradient is also a vector, JQ. The equation that relates JQ to ÂT (the three-dimensional form of Fourier's Law) is complicated by the fact that the heat flux is not necessarily parallel to the temperature gradient. The component of the heat flux in the direction e1 depends on all three com-ponents of the temperature gradient:

JQ1 = - k11

∆T

∆x1 - k12

∆T

∆x2 - k13

∆T

∆x3 10.13

where k11, k12 and k13 may be different material constants. The other two components of the heat flux obey similar equations. Hence there are a total of nine thermal conductivities of the form kij (i,j = 1,2,3). These nine coefficients obey the symmetry relation, kij = kji (for example, k12 = k21) and, hence, only six of them are independent. However, the most general material still has six independent thermal conductivities. Fortunately, equation 10.13 has a much simpler form in most of the cases of inter-est to us. Equations like 10.13 that describe the behavior of materials are called constitutive equations. A constitutive equation must predict a behavior that is compatible with the symmetry of the material; if the direction of the temperature gradient is changed to a symmetrically equivalent direction, as from one <100> direction to another in a cubic crystal, then the direction of the induced flux must change in exactly the same way. A material with an amorphous structure is the same in every direction, and is said to have isotropic symmetry. A material that has a cubic crystal structure is said to have cubic symmetry. It can be shown that the symmetry of an isotropic or cubic material has the consequence that the off-diagonal coefficients, kij (i ≠ j), vanish, while the diagonal components, kij (i = j), are all equal. Hence when the material is isotropic or cubic the heat flux is parallel to the temperature gradient JQ = - kÂT 10.14 where the thermal conductivity, k, has the same value in all directions. In a cubic or isotropic material the heat flux in each of the three coordinate directions obeys an equation of the form 10.4, for example,

JQ1 = - k

∆T

∆x1 10.15

Most materials of engineering interest are isotropic or cubic to a good approxima-tion. Amorphous solids and glasses are isotropic, materials with cubic crystal structures


Page 281

are cubic, and polygranular materials that are aggregates of many crystalline grains with random orientations (random polycrystals) are isotropic in the macroscopic sense whatever their crystal structures. Hence the vector form of the Fourier Law of Heat Conduction given in equation 10.14 is usually adequate. When the material is cubic or isotropic the three-dimensional equation that governs the change in the temperature with time can be derived by considering the total rate of accumulation of heat in a cubic volume element. The heat flow along each of the three coordinate directions obeys an equation of the form 10.15, and the total heat accumulated per unit time is obtained by summing the heat contributed by flow in the three coordinate directions. It follows from equation 10.7 that

°QT = -

∆JQ

1∆x1

+ ∆JQ

2∆x2

+ ∆JQ

3∆x3

dV

= - (Â^JQ)dV 10.16 where the symbol Â^JQ is the vector notation for the quantity in brackets, and is called the divergence of the vector JQ. Using equation 10.8 and assuming that the thermal conductivity is constant, it follows by a derivation analogous to the one that leads to equation 10.10 that the temperature distribution through the body, T(R,t), is governed by the partial differential equation

∆T∆t =

kCv

∆2T

∆x12 + ∆2T∆x22 +

∆2T∆x32

= k

Cv Â2T 10.17

where symbol Â2T is a vector notation for the quantity in brackets, which is called is called the Laplacian of T. 10.3.3 Heat sources The temperature change in a differential volume of the solid may also be affected by the presence of heat sources, where heat is created, or heat sinks, where heat is lost. If there are heat sources or sinks distributed through the volume that produce a net heat, •q, per unit volume per unit time then equation 10.16 must be changed to •QT = - [Â^JQ + •q] dV 10.18 so that the evolution of the temperature is governed by the partial differential equation

Cv

∆T

∆t = kÂ2T + •q 10.19


Page 282

The most common heat source is a chemical reaction or first-order phase transition within the volume that produces a latent heat, QL, per mole of product. The latent heat was discussed in Chapter 7. It may be positive or negative. A phase transition to a high-temperature phase is endothermic; heat is absorbed and QL is negative. The rate at which heat is evolved per unit volume of material, •q, is the product of the latent heat per mole and the number of moles created per unit volume per unit time, •n: •q = QL•n 10.20 A second source of heat added to a volume of material is the absorption or emission of radiation. The most common type of radiation is the thermal radiation between bodies of different temperature, which gives rise to a heat flux that is proportional to the fourth power of the temperature. Thermal radiation provides a heat source or sink at the external surface of an opaque body. The net flux to a surface at temperature T from a medium of temperature T0 is JQ = aßB(T04 - T4) 10.21 where (a) is the absorption coefficient for the material and ßB is the Stefan-Boltzmann constant.. The absorption coefficient is a material property that depends both on the nature of the material and on the nature of its surface. The more reflective the surface, the lower the value of the absorption coefficient. There is also a net radiative heat transfer from point to point within a material whose temperature is non-uniform. However, this internal radiative transfer is included in the thermal conductivity. According to equation 10.21, radiant heat transfer in an opaque body gives a contribution to the thermal conductivity that is proportional to T3. This contribution is negligible except in semi-transparent materials at very high temperature. The absorption of radiation creates a heat source when the radiant photons are ab-sorbed at discrete sites within the body. Examples include x-rays and energetic ions, which penetrate well into a solid before being absorbed, and light photons in a transparent body, which are often absorbed at discrete impurity atom sites (color centers) that are distributed through the body. Radiation can also create an effective heat sink, for example, when energy is liberated by the radioactive decay of atoms within the material. 10.4 MECHANISMS OF HEAT CONDUCTION Heat is transported through a solid by three carriers: lattice vibrations, conduction electrons, and optical photons. Since elementary lattice vibrations can be treated as parti-cles, called phonons, as we discussed in Chapter 7, all three cases can be visualized as heat conduction by energetic particles that move through the solid. In the course of their


Page 283

motion they occasionally collide with one another or with the crystal lattice itself. These collisions produce an effective friction that opposes their motion. The type of carrier that is mainly responsible for the transport of heat depends on the type of material and the temperature, and is also sensitive to the microstructure. Phonons are the predominant carriers in all materials at very low temperature. Electrons become the predominant carriers in metals at moderate temperature. Photons are relatively unimportant except in insulators at very high temperature. 10.4.1 Heat conduction by a gas of colliding particles The basic equation that governs thermal conductivity can be derived by considering thermal conduction in a dilute gas of particles that collide with one another. The theory of this process was originally developed to treat the thermal conductivity of a gas of atoms, but also provides a reasonable description of thermal conduction by "gases" of electrons, phonons or photons in solids. Assume a gas of particles that move at an average velocity, v, that is randomly oriented in space. The particles occasionally collide with one another. Assume that they travel an average distance, ´l¨, the mean free path, between collisions, where ´l¨ = v† and † is the mean time between collisions. Let the gas have a temperature, T(x), that varies in the x-direction, and assume that the particles come to thermal equilibrium at the local value of the temperature when they collide.

v

J = (1/2)nev

...

Fig. 10.4: Flux of particles in one dimension. The particles in the region shown that travel in the positive direction cross the shaded sur-face in unit time.

Now let an imaginary plane be placed perpendicular to the x-axis, as shown in Fig. 10.4. The number of particles that cross the plane in the positive direction per unit time is 12 nvx, where vx is the average speed of travel in the x-direction and the factor 1/2 appears because half of the particles are moving in the positive x-direction at any given time. If these particles carry an average energy, ‰, then the net flux of energy across the plane in the positive x-direction is

J+ = 12 n‰vx =

12 EV(T)vx 10.22


Page 284

where EV(T) (= n‰) is the energy of the mobile particles per unit volume, which depends on the temperature. The typical particle that crosses the plane had its most recent collision a distance of the order ´lx¨ away, where ´lx¨ is the mean free path of travel in the x-direction between collisions. Hence

J+(x) = 12 EV[T(x-´lx¨)]vx 10.23

where J+(x) is the flux in the positive direction across a plane at x, and EV[T(x-´lx¨)] is the energy density at the position, x - ´lx¨, where the previous collisions occurred. The energy density depends on the position, x, through the local value of T at that position. Since particles move in both directions along the x-axis, there is also a flux of en-ergy across the plane in the negative x-direction, equal to

J-(x) = 12 EV[T(x+´lx¨)]vx 10.24

Hence the net flux in the positive x-direction across a plane at x is

J(x) = 12 vx{EV[T(x-´lx¨)] - EV[T(x+´lx)]} 10.25

Let the temperature gradient be constant over the mean free path, ´lx¨, as it is, at least ap-proximately, when ´lx¨ is a microscopic distance and the temperature variation is macro-scopic. Then

EV[T(x-´lx¨)] = EV[T(x)] - dEVdT

dT

dx ´lx¨

= EV[T(x)] - Cv´lx¨

dT

dx 10.26

where Cv is the isometric specific heat. If we write a similar relation for the energy density at x + ´lx¨, and substitute the two relations into equation 10.25, the result is

J(x) = - Cvvx´lx¨

dT

dx 10.27

which has the form of Fourier's Law for heat conduction, with the thermal conductivity k = Cvvx´lx¨ = Cvvx2† 10.28 where † is the mean time between collisions. Since the particles are equally likely to move in any direction,


Page 285

vx2 = vy2 = vz2 = 13 v2 10.29

where v is the average particle speed, and

k = 13 Cvv2† =

13 Cvv´l¨ 10.30

where ´l¨ is the mean free path. Equation 10.30 is valid whatever the nature of the conducting particles, provided that they travel with velocity, v, and equilibrate by collisions with the mean free path, ´l¨. The electrons, phonons and photons that carry heat through a solid behave in this way. 10.4.2 Heat conduction by mobile electrons Conduction electrons are the primary carriers of heat in metals at normal tempera-tures. This may seem surprising, since we found in Chapter 8 that the specific heat is pri-marily due to lattice vibrations (phonons). However, while the phonon specific heat of a metal at room temperature is roughly an order of magnitude greater than the electron spe-cific heat, electrons travel at speeds approaching the speed of light, « 108 cm/sec, while phonons travel at the speed of sound, « 105 cm/sec. Hence the electrons make a greater contribution to the thermal conductivity. In a pure metal at room temperature the electron conductivity is « 30 times the phonon conductivity. In a disordered alloy it is roughly 3 times greater. The mean free path of the electrons is determined by their collisions with the crystal lattice. These collisions transfer energy to the lattice and cause the electron distribution to equilibrate at the local value of the lattice temperature. The electron-lattice collisions are also responsible for the electrical resistance of the metal, and we shall discuss them further when we treat electronic properties. The collisions have two sources: collisions with phonons and collisions with defects, primarily solute or impurity atoms. Both types of collisions are due to the interaction of the electron with a local disturbance of the crystal pattern of the ion cores. If the solid were a perfectly ordered crystal in which all ion cores were sited in their equilibrium positions, the valence electrons would not interact with the lattice at all. The reason is that the electrons are in quantum states that are determined by the ion configuration. The interaction with the ions is incorporated into the wave function of the electron itself. The similar situation in the free atom is, perhaps, more familiar. The electrons in an atom orbit around the nucleus. Their interaction with the nucleus determines the wave functions that describe their states. They do not collide with the nucleus in any other way as they move about in the ion core. In a real crystal the perfect pattern of the ion cores is disturbed in two ways. Because of thermal oscillations the ions are not instantaneously situated on their


Page 286

equilibrium sites, and in the neighborhood of defects or solutes the charge distribution is locally disturbed by the different charge distribution of the solute and the distortion of the positions of the lattice atoms around it. Hence a traveling electron "sees" lattice vibrations and defects as disturbances in the perfectly periodic charge distribution its wave function is designed to expect. Its motion is perturbed by these disturbances. We describe the perturbation by saying that the electron periodically "collides" with them and is "scattered" by them. The mean free path between collisions of conduction electrons and phonons in a pure metal at room temperature is 100-600‹. The mean free path between collisions with solute atoms is of the order of the mean separation of the solutes, and is, hence, proportional to n-1/3, where n is the number of impurity atoms per unit volume. The collisions between the electrons and the lattice determine the electrical conductivity as well as the thermal conductivity, so we might expect the two to be related. The relation between them is known as the Wiedemann-Franz Law,

k = LßT = LT® 10.31

where L is a constant, called the Lorenz number, ß is the electrical conductivity, and ® = 1/ß is the electrical resistivity. The Wiedemann-Franz Law can be understood by referring back to equation 10.30. The electrical resistivity, ®, is inversely proportional to the mean free path, ´l¨, since the motion of an electron through the lattice is resisted by its collisions. The electronic specific heat, CV, is proportional to the temperature, T, as we found in Chapter 8. Hence the product, CV´l¨, yields a relation of the form given in equation 10.31. The resistivity of a typical metal varies linearly with the temperature: ® = ®0 + bT 10.32 where ®0 is the residual resistivity, due to solutes, impurities and lattice defects, and the linear increase in ® with T is due to electron collisions with lattice phonons. This simple equation leads to the following rough approximation for the thermal conductivity:

k ~ LT

®0 + bT 10.33

At high temperature the thermal contribution to the resistivity dominates, and

k ~ Lb 10.34

so the thermal conductivity approaches a constant. At low temperature the residual resis-tivity, ®0, dominates the denominator, and


Page 287

k ~ LT®0

10.35

so thermal conduction by electrons increases approximately linearly with T. The low-temperature limit, equation 10.35, is only reached in alloys and defective metals with high values of ®0, since, as we shall see, phonon conductivity becomes dominant at sufficiently low temperatures. The thermal conductivity of a metallic solid solution (alloy) is always much lower than that of the pure metal because of its higher residual resistivity, ®0. Highly alloyed materials, such as stainless steels, have relatively low thermal conductivities. A familiar example of the engineering application of this result is in the design of stainless steel skil-lets, which are sometimes plated with high-conductivity copper to distribute heat and pre-vent the development of hot spots in the skillet. For the same reason aluminum kitchen foil is made of aluminum that is alloyed as little as possible. 10.4.3 Heat conduction by phonons Lattice vibrations, or phonons, are the principle carriers of heat in all materials at very low temperature, and dominate the thermal conductivity of insulating materials at all but very high temperatures. When a solid is heated, for example, on one of its surfaces, the local increase in temperature excites lattice vibrations that propagate through the lattice. Since the vibrations are harmonic waves, we can picture them as particles (phonons) that move through the lattice. The phonons experience two distinct types of collisions: they collide with one another, and they collide with defects in the lattice, such as solute atoms. Phonon-phonon collisions Phonon-phonon collisions occur because the interatomic forces are slightly anhar-monic, as discussed in Chapter 8. Because of this anharmonicity, an atom that is initially given a simple harmonic vibration gradually changes its frequency. We can model this process by saying that lattice phonons excite one another by collisions. As we discussed in Chapter 8, a one-dimensional lattice vibration is described by a wave function of the form u(x,t) = u0 exp[ ]i(kx - ∑t) 10.36 where u(x,t) is the displacement of an atom at position, x, at time, t, k is the wave vector of the vibrational wave, and ∑ is its frequency, which depends on k through the dispersion relation. The wave vector, k, lies in the range - π/a ≤ k ≤ π/a that defines the first Brillouin zone, where a is the atom spacing in the x-direction.


Page 288

The momentum of the lattice wave is given, in quantum mechanics, by the relation

px = - iÓ

∆u

∆x = Ók 10.37

Hence conservation of momentum in phonon-phonon collisions requires that the total wave vector be conserved. If phonons with wave vectors k1 and k2 collide to produce phonons with wave vectors k3 and k4, we must have k1 + k2 = k3 + k4 10.38 so the net direction of propagation of the lattice waves is not changed. However, it is possible to have a phonon-phonon collision that is consistent with equation 10.38 in which one of the product vectors exceeds the value, π/a. We saw in Chapter 8 that a phonon with k ≥ π/a is indistinguishable from one with

k' = k - 2πa 10.39

Hence when a collision produces a phonon with k ≥ π/a, it is indistinguishable from one with k' = k - 2π/a; a large fraction of its momentum is apparently lost. This momentum is not actually lost from the system. It is transferred to the crystal lattice. However, it is lost from the phonon spectrum. Since the net phonon momentum decreases, the net transport of energy by phonons goes down, and the phonon thermal conductivity is diminished. Inelastic phonon collisions of this sort are called Umklapp processes, and they are responsible for the finite value of the phonon thermal conductivity in a solid. A more elaborate analysis shows that an equation of the form 10.30,

k = 13 Cvv´l¨ 10.30

applies to the phonon conductivity, k, where Cv is the vibrational specific heat and ´l¨ is the mean free path of the phonons between inelastic collisions. The mean free path between inelastic collisions is strongly dependent on the tem-perature. A simple relation can be found from the following one-dimensional argument. Since inelastic collisions require a product phonon with k ≥ π/a, and since momentum is conserved, these collisions will involve at least one phonon with an initial wave vector, k ≥ π/2a. In the Debye model these phonons have energies, E ≥ kŒD/2, where ŒD is the Debye temperature. The mean free path between such collisions should decrease with the density of these high-energy phonons, giving


Page 289

´l¨ fi 1

n(kŒD/2) 10.40

where n(kŒD/2) is the equilibrium density of high-energy phonons. We found in Chapter 8 that the equilibrium density of phonons with energy, Ó∑, has a simple form in the high- and low-temperature limits:

n(Ó∑) ~ e- Ó∑/kT Ó∑>> kT

kT Ó∑<<kT 10.41

Using this relation in eq. 10.40, the mean free path for inelastic phonon-phonon collisions obeys the relation

´l¨ fi eŒD/2T T << ŒD

1/kT T>>ŒD 10.42

Equations 10.42 are obeyed by most pure crystalline materials. For temperatures above ŒD, the phonon mean free path varies inversely with temperature. Hence the ther-mal conductivity of an insulator decreases as T increases above ŒD until photon conductivity intrudes at very high temperature. As T decreases below ŒD, on the other hand, the mean free path between phonon-phonon collisions becomes very large. As a consequence, all pure crystalline materials have high thermal conductivities at temperatures well below ŒD. The low-temperature value of the thermal conductivity is limited by the defect density. The phonon mean free path is determined by defect scattering rather than by phonon-phonon interactions when the temperature is sufficiently low. Phonon-defect collisions The second kind of collision that limits the phonon mean free path involves colli-sions with lattice defects, such as solute atoms, grain boundaries, dislocations, precipitates, or, in the case of a glass, irregular atom configurations. The mean free path for phonon collisions with defects depends on the type of the defects as well as on their separation. Defects that dramatically disturb the crystal lattice, such as free surfaces, pores, precipitate particles and grain boundaries, interact strongly with phonons. When these are the dominant scattering centers the mean free path is equal to the mean defect spacing. The distorted atom configurations in glasses scatter phonons strongly. The effective mean free path in a silica glass is 3-5 ‹, which is of the order of the size of a basic silica tetrahedron. Solute atoms are less effective scattering centers. The phonon mean free path in a solid solution is proportional to the mean solute separation, but is greater, since, effec-tively, only a fraction of the phonon-solute collisions cause phonon scattering. The


Page 290

phonon mean free path for collisions with solutes decreases as the temperature rises, and becomes nearly constant when T ≥ ŒD/2. Phonon thermal conductivity When several different phonon scattering processes are active, the total number of collisions per unit time is the sum of the collisions due to each scattering process. Since the frequency of collisions is inversely proportional to the mean free path, the net value of mean free path is determined by the reciprocals of the mean free paths for each independent mechanism:

1

´l¨ = 1

´l¨1 +

1´l¨2

+ ... 10.43

where ´l¨1, ´l¨2, refer to the different collision mechanisms. It follows that the mean free path at a given temperature, and, hence, the phonon conductivity, is dominated by that process whose mean free path is the smallest. The result is a phonon conductivity that varies with temperature as drawn schematically in Fig. 10.5. Two curves are drawn in the figure to illustrate the characteristically different behaviors of a pure, crystalline solid and a highly defective solid such as a glass.

k

TŒ

Crystal

Glass

...

Fig. 10.5: The variation of thermal conductivity with temperature for a crystal and a glass.

In a pure crystalline solid the concentration of defects is small, so the associated mean free path is large. At normal temperatures the mean free path is determined by phonon-phonon collisions. It is useful to distinguish three temperature regimes. (1) T > ŒD. When T is above the Debye temperature, ŒD, the specific heat is constant and the phonon mean free path is inversely proportional to the temperature (eq. 10.42). It follows that the phonon conductivity, k, is given by a relation of the form

k = 13 Cvv´l¨ ~

AT (T ≥ ŒD) 10.44


Page 291

where A is a constant. (2) 0 << T < ŒD. At lower temperatures the mean free path for phonon-phonon collisions increases exponentially (eq. 10.42). When T is low and phonon-phonon colli-sions dominate thermal conductivity, k « A'eŒD /2T (0<<T << ŒD) 10.45 where A' is a pre-exponential factor. (3) T “ 0. The mean free path for phonon-phonon collisions increases with de-creasing T, and, eventually, exceeds the mean free path for phonon-defect collisions with whatever defect dominates scattering in the material (usually grain boundaries). When this happens, ´l¨ becomes fixed. Since Cv varies as T3 at low temperature, the thermal conductivity obeys a relation of the form k = A"T3 (T “ 0) 10.46 Note that the thermal conductivity vanishes in the limit of zero temperature. Equations 10.44-46 are obeyed rather well by crystalline materials that are electrical insulators, except at very high temperatures where photon conductivity becomes important. Pure metals also conduct heat primarily via phonons at low temperatures. The thermal conductivities of pure metals have the characteristic shape given in Fig. 10.5 for temperatures less than about half the Debye temperature (T < ŒD/2). At higher temperatures the electronic contribution dominates the thermal conductivity. For this reason the thermal conductivities of metals are much higher than those of insulators when they are compared at the same value of the dimensionless temperature, T/ŒD, for T greater than about ŒD/2. At temperatures near room temperature almost all the best thermal conductors are metals, and they are the materials that are the best electrical conductors, such as Cu, Ag, Au and Al. However, there are a few simple insulators that have very high Debye temperatures. These materials are also excellent thermal conductors at normal temperatures, since their phonon thermal conductivities are unusually high. Diamond, which has a Debye temperature near 2000 K, is the best thermal conductor known at temperatures near room temperature. Glasses and highly defective crystalline solids are relatively poor thermal conduc-tors at all temperatures. The phonon mean free path in such a material is necessarily small because of the high concentration of defects. Except at high temperature, the mean free path is nearly constant, so the temperature dependence of the thermal conductivity is determined by the specific heat. Hence the thermal conductivity decreases with the temperature, and obeys equation 10.43 as the temperature approaches zero. At normal temperatures the thermal conductivity of a glass is small relative to that of the same material in crystalline form. However, as the temperature increases the mean free path


Page 292

for phonon-phonon collisions in the crystal decreases, eventually becoming comparable to that in the glass. The crystal and glass have similar thermal conductivities at high temperature. Amorphous polymers that are electrical insulators (as are almost all of them) behave like glasses. Their thermal conductivities are small and decrease with temperature. 10.4.4 Heat conduction by photons A body that has the temperature, T, radiates energy in the form of photons with an intensity that is proportional to T4. There is, hence a radiant heat transfer between bodies of different temperature that is proportional to the difference between the fourth powers of their temperatures (see equation 10.21): JQ = K(T14 - T24) 10.47 where the constant, K, depends on the thermal emissivities of the bodies. Equation 10.47 suggests that there is a radiant heat transfer between regions of different temperature in a solid. This transfer is negligible at all temperatures unless the solid is transparent to photons at frequencies near the peak of the thermal spectrum, which shifts from the infrared at low temperatures into the visible spectrum at high temperature (the spectral peak in the thermal emission of photons is responsible for the characteristic color of the glow of a hot metal, and is used to measure the temperature by instruments known as optical pyrometers). In a transparent solid the radiant heat transfer contributes an additive term to the thermal conductivity, kr = Ar´l¨T3 10.48 where Ar is a constant and ´l¨ is the mean free path for photons. Even in these materials the radiant contribution to the thermal conductivity is negligible until the temperature is very high. It is responsible for a gradual increase in the thermal conductivities of transparent insulators that becomes significant at temperatures above about 1000 K. The principle engineering interest in the radiant thermal conductivity lies in reducing it. It is a source of increased thermal conductivity in the ceramic materials that are used as thermal insulators at high temperature. Radiant conduction is particularly important in transparent glasses which conduct heat poorly by other mechanisms. To decrease the radiant contribution to the thermal conductivity one decreases the photon mean free path by introducing microstructural defects that scatter light. The most effective defects are grain boundaries in polycrystalline ceramics and small voids or pores in both crystalline ceramics and glasses. Both grain boundaries and pores are ef-


Page 293

ficient reflectors (you can easily verify, for example, that typical opaque sand consists of small grains of transparent material). The photon mean free path, ´l¨, is equal to the distance between such scattering centers. Hence the radiant conductivity decreases significantly if the grain size of a polycrystalline ceramic is refined, or if a high density of fine pores is introduced into a ceramic or glass. For this reason thermal insulators that are intended for use at high temperature tend to be porous compacts of polycrystalline or glassy ceramics. The walls of thermally insulating containers usually have a sandwich design whose center is an open space that is evacuated (hence the term vacuum bottle). Thermal radiation is the only mechanism available for heat conduction across an evacuated space. To minimize it, the walls of the evacuated space are silvered to make them highly reflective, so that the incident thermal energy is reflected rather than absorbed on the cold side of the evacuated space. 10.5 NON-EQUILIBRIUM THERMODYNAMICS Now that we have learned a bit about the simplest continuous transition, the con-duction of heat, it is worthwhile to consider the general theory of continuous changes be-fore turning to our second specific example, the diffusion of atoms through solids. 10.5.1 The thermodynamic forces The conditions of local equilibrium in a simple solid require that the temperature, pressure, and chemical potentials of all components be constant. When these conditions are violated they give rise to thermodynamic forces whose local values are given by the gradients, ÂT, ÂP, and {Âµ}, where {Âµ} is the set of all chemical potential gradients. This list contains (n+2) thermodynamic driving forces, where n is the number of chemical components. However, only n+1 of the thermodynamic forces are independent. This is a consequence of a general relation in thermodynamics that is known as the Gibbs-Duhem equation, which states that the changes in T, P and {µ} are related by the differential equation:

SdT - VdP + ∑k=1

n Nkdµk = 0 10.49

Equation 10.49 can be solved for one of the differentials dT, dP, or any one of the dµk. It follows that the gradients of the thermodynamic intensities are related by the equation

SÂT - VÂP + ∑k=1

n NkÂµk = 0 10.50


Page 294

and, hence, that only n+1 of the gradients are independent. It is usually best to choose the independent forces so that the conditions of thermal, mechanical and chemical equilibrium are most clearly separated. To do this we divide equation 10.50 through by the total mole number, N, to obtain

sÂT - vÂP + ∑k=1

n ckÂµk = 0 10.52

where s is the molar entropy, v is the molar volume, and ck = Nk/N is the mole fraction of the kth component (we shall use the symbol ck rather than xk to avoid confusion with the coordinate, x). Since the mole fractions sum to 1, the mole fraction of the nth component (the solvent) is given by the equation

cn = 1 - ∑k=1

n-1 ck 10.53

When this equation is substituted into equation 10.52 the result can be cast in the form

Âµn = - sÂT + vÂP - ∑k=1

n-1 ckÂ –µk 10.54

where –µk is the relative chemical potential of the kth component,

–µk = µk - µn =

∆

∆ck¡g(T,P,{c}) 10.55

It follows that the (n+1) independent thermodynamic forces can be taken to be the set ÂT, which measures the local deviation from thermal equilibrium, ÂP, which measures the deviation from mechanical equilibrium, and the n-1 gradients Â –µk, which measure that part of the deviation from chemical equilibrium that is independent of thermal and mechanical equilibrium. 10.5.2 The non-equilibrium fluxes and the kinetic equations The differential volume elements within a non-equilibrium solid can exchange heat and atomic species. Hence the evolution toward equilibrium is characterized by the n+1 fluxes: JQ, the heat flux, and Jk (k = 1,...,n), the molar fluxes of the n chemical components. In the most general case the values of the non-equilibrium fluxes depend on all of the non-equilibrium forces. The reason is that the flows tend to couple to one another.


Page 295

For example, atoms have vibrational energy, and hence transport thermal energy, or heat, when they diffuse down a temperature gradient. Therefore the heat flow is affected by gradients in the chemical potential. Since atoms are more mobile at high temperature, atom diffusion is enhanced if the temperature decreases in the direction of atom flow since there is less counterflow from random movements of atoms in the colder region; hence chemical diffusion is affected by a temperature gradient. The diffusional flows of different chemical species influence one another, particularly when the chemical species are substitutional solutes that occupy the same crystal lattice, or when one of the species is so highly mobile that it can transfer momentum to another by collision and help it to move. The most general form of the kinetic equations for a cubic or isotropic material can, therefore, be written

JQ = - LQQÂT - LQPÂP - ∑k=1

n-1 LQkÂ–µk 10.56

Jk = - LkQÂT - LkPÂP - ∑k'=1

n-1 Lkk'Â –µk' 10.57

where there is one equation of the form 10.57 for each independent chemical component. The coefficients, Lij , are numbers because of the material symmetry, and are material properties. 10.5.3 Simplification of the kinetic equations It is, of course, extremely difficult to analyze a process that involves all of the coupled flows that appear in equations 10.56-57. Fortunately the kinetic equations can be simplified substantially in most cases of interest. A major reason is the kinetic hierarchy of the processes that lead to equilibrium. The kinetic hierarchy is such that mechanical equilibrium is ordinarily established very quickly, thermal equilibrium is achieved in a reasonably short time, and chemical evolution is relatively slow. Mechanical equilibrium A pressure gradient in a solid that is otherwise unconstrained is relaxed by small displacements of the lattice that happen at the speed of sound. Hence a disturbance in the mechanical equilibrium of a solid is ordinarily removed in a time that is short compared to the time needed to reach thermal or chemical equilibrium. However, the pressure is not always constant when a material is in mechanical equilibrium.. A more complete analysis of mechanical equilibrium in solids shows that if balanced, non-uniform forces are applied to the boundary of a solid, it develops an inho-mogeneous elastic stress in which the pressure can vary from point to point. The inhomogeneous stress causes stress-induced diffusion of the chemical species that is important in many material processes.


Page 296

The analysis of stress-induced diffusion is beyond the scope of this course. We shall confine the following discussion to thermochemical processes at constant pressure. In this case the governing equations reduce to the set

JQ = - LQQÂT - ∑k=1

n-1 LQkÂ–µk 10.58

Jk = - LkQÂT - ∑k'=1

n-1 Lkk'Â –µk' (k = 1,...,n) 10.59

Equations 10.58-59 are a set of n+1 equations for the fluxes JQ and Jk (k = 1,...,n) that involve only the n independent variables ÂT and Â –µk (k=1,...,n-1). However, when the pressure is constant in a material process the fluxes JQ and Jk are related to one another by the requirement that the net flow leave the pressure unchanged. The exact nature of the mathematical relationship depends on the process and the boundary conditions that are imposed on the surface of the material, but has the consequence that one of the fluxes can be found from the others. Using this result to eliminate the nth chemical flux, Jn, equations 10.59 reduce to the set

Jk = - LkQÂT - ∑k'=1

n-1 Lkk'Â –µk' (k = 1,...,n-1) 10.60

Thermal equilibrium; heat conduction Since heat can be conducted much more rapidly than atoms can move, thermal gradients in a solid ordinarily disappear in a time that is short compared to the time required for any significant chemical change. In this case thermal equilibrium is estab-lished by heat conduction alone, and chemical equilibrium is reached by chemical diffusion at essentially constant temperature. The kinetic equations 10.58-60 then decouple to read JQ = - kÂT 10.61

Jk = - ∑k'=1

n-1 Lkk'Â –µk' (k = 1,...,n-1) 10.62

Equation 10.61 is just Fourier's Law of heat conduction, and we have replaced the kinetic coefficient LQQ by the normal symbol, k, for the thermal conductivity. It should be noted, however, that the heat and chemical fluxes cannot always be decoupled. The most important exception is when a temperature gradient is imposed


Page 297

externally, as when the opposing surfaces of a solid are held at different temperatures. Even in this case it is usually possible to neglect the chemical contribution to heat flow, which is small compared to that due to thermal conductivity. However, the temperature gradient ordinarily has a significant effect on the chemical flux; its contribution is known as thermal diffusion. In this case equations 10.60 govern chemical diffusion. Chemical equilibrium; diffusion The slowest of the processes involved in the evolution toward internal equilibrium are the chemical redistributions that lead to chemical equilibrium. Given thermal equilib-rium, mass diffusion is described by the kinetic equations gathered in equation 10.62, where the number of independent equations is one less than the number of components. In a binary solution only two species are present. There is only one relative chemical potential, –µB, which refers to the specie, B, chosen as the solute, and only one independent diffusion flux, JB, which, still assuming that the solid is isotropic or cubic, is governed by the constitutive equation JB = - LBÂ–µB 10.63 The kinetic coefficient, LB, is called the mobility. Like any other material property of a binary solution the mobility depends on the nature of the solution, including the identity of the solvent and the phase, and on the thermodynamic state, which is specified by the temperature, pressure and composition, LB = ¡LB(T,P,c). For reasons we shall discuss below the dependence on the temperature is exponential

LB = LB0 e

- QkT 10.64

where Q is the activation energy for diffusion. We shall restrict our discussion of diffusion in the solid state to the binary case described by equation 10.63, both because it is the simplest case and because it provides an adequate description of most of the chemical redistributions that occur in materials. Even when a solid has several mobile components it is often possible to treat its chemical evolution as a sequence of binary diffusion processes. The reason is that the different solute species usually have very different mobilities. The mobility of the fastest-moving specie is often so much greater than that of the next-fastest that the former can be assumed to evolve into an equilibrium distribution before the latter redistributes at all. When this is true the redistribution of the most mobile component occurs against the background of an almost fixed distribution of the other species, and is, hence, binary diffusion. As the slower component diffuses, the distribution of the more mobile component simply evolves with it to preserve equilibrium, so the diffusion of the slower component can be treated as a binary diffusion problem as well.


Page 298

10.6 DIFFUSION 10.6.1 Fick's First Law for the diffusion flux The equations that govern chemical diffusion in a binary solution at constant tem-perature and pressure can be written so that they are mathematically identical to those which govern the conduction of heat (Section 10.2). As shown in Chapter 7, the relative chemical potential of the solute (specie B in the A-B solution) is the compositional derivative of the molar Gibbs free energy,

–µB = ∆∆c [¡g(T,P,c)] = –µB(T,P,c) 10.65

where we have used the symbol c for the concentration of B, and noted that the value of –µB is determined by the variables (T,P,c) if g is. The gradient of the relative chemical potential at constant T, P is

Â –µB =

∆ –µB

∆c Âc 10.66

that is, it is simply proportional to the concentration gradient. When equation 10.66 is substituted into equation 10.63, the result is JB = - nDB Âc 10.67 where n is the total number of atoms (or moles) per unit volume and the coefficient, D, is a material property that is called the diffusivity, and is related to the mobility by the equation

DB = 1n

∆–µB∆c LB 10.68

and it follows from equation 10.64 that its temperature dependence is governed by an exponential relation of the form

DB = DB0 e

- QkT 10.69

Equation 10.69 is known as Fick's First Law. It states that the diffusion flux of the solute is proportional to the negative of its concentration gradient. Note that Fick's First Law is valid under three assumptions: (1) the material is a binary solution (or behaves like a binary solution in the sense discussed in the previous section), (2) the temperature and pressure are constant, (3) the material has isotropic or cubic symmetry. If the material does not behave as a binary solution then the diffusion fluxes interact and equations 10.62 must be used. If the temperature and pressure are inhomogeneous then terms in ÂT and ÂP must be included in the kinetic equations. When the symmetry of


Page 299

the material is not isotropic or cubic then the diffusivity depends on the direction in the material and the diffusion flux is not generally parallel to the concentration gradient. 10.6.2 Fick's Second Law for the composition change The change in composition with time can be found by applying the same reasoning that was used to obtain equations 10.4 and 10.10 for the change in the temperature with time. First consider a one-dimensional diffusion flow as shown in Fig. 10.6. The rate of change in the quantity of component B in the central cell in Fig. 10.6 is

∆NB∆t = JB

12 - JB23 dA 10.70

where dA is the interface area. If dx is the length of the volume element, dV2, of the cen-tral cell then

JB23 = JB

12 +

∆JB

∆x dx 10.71

So that

∆NB∆t = -

∆JB

∆x dV 10.72

Since the volume element, dV, is constant, equation 10.72 can be re-written

∆nB∆t = -

∆JB

∆x 10.73

Substituting in the one-dimensional form of Fick's First Law, equation 10.67, yields the equation

∆nB∆t =

∆∆x

nDB

∆c

∆x 10.74

JB23JB12

c c c1 2 3

Fig. 10.6: Binary diffusion in one dimension.


Page 300

If the total atom or molar density, n, is constant during diffusion then both sides of equation 10.74 can be divided by n to obtain the simpler diffusion equation

∆c∆t =

∆∆x

DB

∆c

∆x 10.75

By its definition the diffusivity is a material property that is a function of T, P and c. Since T and P are constant by assumption,

∆DB

∆x =

∆DB

∆c∆c∆x 10.76

and equation 10.75 can be written

∆c∆t = DB

∆2c

∆x2 +

∆DB

∆c

∆c

∆x2 10.77

which is a partial differential equation for the concentration field, c(x,t). When the dependence of the diffusivity on the composition can be ignored the composition is determined by the simple equation

∆c∆t = DB

∆2c

∆x2 10.78

which is an example of the diffusion equation like the governing equation for the temperature, eq. 10.10. Equation 10.78 is called Fick's Second Law. Fick's Second Law is a valid equation for the concentration, c(x,t), when Fick's First Law (eq. 10.67) applies, and when, in addition, the total atom density is constant and the diffusivity is independent of concentration. These conditions are often satisfied to a reasonable approximation. The total atom density is essentially equal to the total number of atom sites per unit volume; when the solution is substitutional the number of vacancies is always small, and when the solution is interstitial the number of interstitial atoms is almost always small. Since the pressure is constant the number of atom sites is nearly constant unless the volume per site is a strong function of the composition. This is the case when the two components have very different atomic volumes. An equation like 10.78 holds even then, but with a multiplying constant on the right-hand side. The diffusivity becomes a strong function of composition when the solute concentration is high and the two components have very different atomic volumes, since the lattice is then distorted to an extent that depends on the concentration. In this case equation 10.78 governs the evolution of the concentration and the mathematical analysis of diffusion is much more difficult.


Page 301

10.6.3 Solutions of the diffusion equation A thorough discussion of the solutions to the diffusion equation is beyond the scope of this course. However, it is useful to know the solution for a few simple situations that be used as reference cases to help understand material behavior. The one-dimensional diffusion equation, 10.78, has simple solutions in each of the cases shown in Fig. 10.7.

(a) (b) (c)

Fig. 10.7: Three examples of diffusion in one dimension. (a) A thin slab

(point source) of B is embedded in an infinite bar of A and al-lowed to diffuse. (b) Semi-infinite bars of A and B are joined together and allowed to interdiffuse. (c) B is plated on the sur-face of A and allowed to interdiffuse.

First, let a thin slab of element B be located in the interior of an essentially infinite bar of material A, as shown in Fig. 10.7(a). If the total number of moles of B is NB, and the slab is located at x = 0, the concentration of B as a function of position and time is given by the equation

c(x,t) = NB

2 πDt exp

- x2

4Dt 10.79

One can verify by direct substitution that eq. 10.79 is a solution of the differential equation, 10.78, and satisfies the constraint

⌡⌠

-∞

∞ c(x,t)dx = NB 10.80

which insures that the total quantity of B is the same at all times. The concentration field has a simple Gaussian form, and develops with time as shown in Fig. 10.8. Second, let a semi-infinite bar that has composition, c0, of B in A be brought into contact with a semi-infinite bar of pure A, as shown in Fig. 10.7(b). Let the interface be at x = 0. The total quantity of B per unit area perpendicular to the axis of the bar at position x' < 0 is c0dx'. According to eq. 10.79, this increment of material makes a contribution


Page 302

∂c(x,t) = c0dx'

2 πDt exp

- (x-x')2

4Dt 10.81

to the concentration at position x at time t. If we change the variable on the right-hand side of eq. 10.81 from x' to ç = (x-x')/2 Dt , and integrate over the range of ç that corresponds to -∞ < x' < 0, the result is

c(x,t) = c02 erfc

x

2 Dt 10.82

where the symbol erfc(˙) stands for the "complementary error function", which is defined by the integral

erfc(z) = 1 - 2π

⌡⌠0

z exp[ ]- ç2 dç 10.83

The complementary error function is tabulated in standard math tables. It has the limiting properties erfc(0) = 1 erfc(∞) = 0 erfc(-z) = 2 - erfc(z) 10.84

distance

concentration

t1

t 2t 3

Fig. 10.8: The development of the concentration profile with time for

one-dimensional diffusion from a point source (Fig. 10.7(a)). The development of the concentration field as a function of diffusion time is illus-trated in Fig. 10.9. Note that the concentration at x = 0 is fixed at the value c0/2.


Page 303

concentration

distance

t1t 2

t 3

Fig. 10.9: The development of the concentration profile with time for one-dimensional diffusion in the geometry shown in Fig. 10.7(b).

Third, consider the diffusion of B from the surface into a semi-infinite body of A. There are two relevant cases. In the first, a quantity, NB, of B is plated onto the surface and allowed to diffuse in. In the second, the environment is controlled so that the concentration of B at the surface is maintained at a fixed value. For a fixed quantity of B, eq. 10.79 applies. We can solve the problem by, effectively, slicing Fig. 10.7(a) in half. The result is

c(x,t) = NB

πDt exp

- x2

4Dt (x ≥ 0) 10.85

The concentration field is shown as a function of time in Fig. 10.10(a). For a fixed concentration, c0, of B, eq. 10.82 applies. For positive x this equation yields a solution to the diffusion equation that has a fixed, constant concentration on the boundary at x = 0. Setting this concentration at c0, we have

c(x,t) = c0erfc

x

2 Dt 10.86

the development of the concentration field is illustrated in Fig. 10.10(b).

concentration

distance

t1t 2 t 3

distance

concentration

t1

t 2t 3

(a) (b) Fig. 10.10: Concentration profiles for diffusion from a surface with: (a)

fixed quantity of deposit; (b) fixed concentration.


Page 304

10.7 THE MECHANISM OF DIFFUSION IN THE SOLID STATE 10.7.1 The mobility of interstitial species With very few exceptions atoms in solids diffuse by discrete jumps from one atom site to another. The jumping process is easiest to visualize when the diffusing atom is interstitial. Fig. 10.11 is a two-dimensional drawing of an interstitial atom in a square lattice. The atom moves by jumping to one of the closest interstitial sites. However, the atom is held in its initial position by the surrounding lattice atoms, and can only move if these are displaced as shown in the upper right-hand figure. Any such displacement increases the free energy of the system. The change in free energy as the interstitial atom is displaced from one equilibrium position to another is diagrammed in the figure. The maximum free energy increment during the displacement is ÎGm, as shown in the figure. ÎGm is called the activation free energy for atom motion. If the atoms were quiescent then the interstitial atom could only move if the activation energy were supplied from some external source. However, the atoms in a solid are in constant thermal agitation. At normal temperatures they oscillate about their equilibrium positions with a frequency of about 1013 per second (the Einstein frequency ). With each oscillation the interstitial atom approaches the barrier of lattice atoms that separates it from a neighboring interstitial position. If the vibrations of the lattice atoms are such that their instantaneous configuration is like that shown in the upper right in Fig. 10.11 then the atom can pass through to the neighbor site.

ÎGmG

x

Fig. 10.11: An interstitial atom in a square lattice moves from site to site through intermediate configurations like that shown at upper right. The associated free energy change is diagrammed at lower right.

If a system whose temperature, pressure and composition are fixed has Gibbs free energy, G, then the probability that it will sample an excited state of free energy G + ÎG in a single trial is


Page 305

P(ÎG) « e - ÎG

kT = eÎS/k

e - ÎH

kT 10.87 where we have used the relation ÎG = ÎH - TÎS. The thermal dependence of the proba-bility is governed by the enthalpy change, ÎH. The number of jumps per unit time (∑) that the interstitial atom of Fig. 10.7 makes to nearest neighbor sites is, therefore, equal to the number of trials multiplied by the probability of success per trial, or

∑ = ê -

QmkT 10.88

where the effective frequency, ˆ, is ˆ = zˆ0e

ÎSm/k 10.89

In this last expression z is the number of nearest neighbor interstitial sites (which are as-sumed separated by identical activation barriers), ˆ0 is the frequency of atom vibration, ÎGm = ÎHm - TÎSm, and Qm = ÎHm is the activation energy for migration. The effective frequency, ˆ, is about 1014 per second in a typical cubic solid. The activation energy for migration of an interstitial depends on the nature of the interstitial and the lattice, but is usually of the order of 1 eV, which is the value of kT at about 104 K. It follows that at 500 K (223 ºC) an interstitial atom makes about 105 site-to-site jumps per second. 10.7.2 The mobility of substitutional species While a lattice vacancy moves from site to site very much as an interstitial atom does, a substitutional atom can only jump to a neighboring site if that site is vacant. Hence the rate of diffusion of a substitutional specie depends strongly on the concentration of vacancies. Even when a vacant site is available for a substitutional atom jump, the jump is still complicated by two effects. First, there is an activation barrier (ÎGm) opposing the jump. One source of the activation barrier is illustrated in Fig. 10.12, which shows a two-dimensional close-packed group of atoms including a vacancy. As shown in the figure, the central atom can only move into the vacancy if its neighbors "open up" and allow it through. The open configuration is a non-equilibrium configuration with a free energy ÎGm above that of the ground state.


Page 306

(a) (b) ...

Fig. 10.12: Mechanism of diffusion of a substitutional atom: (a) substitu-tional atom with a neighboring vacancy (shaded); (b) neighboring atoms "open up" to let the atom move into the vacancy.

Second, only some of the atom-vacancy exchanges actually contribute to diffusion. Immediately after a substitutional atom has exchanged positions with a vacancy, as in Fig. 10.12, the vacancy is in the particular neighbor site that would reverse the atom jump that had just been performed. There is, therefore, a higher than random probability that the next atom jump will simply cancel the previous one so that both jumps are ineffective. This phenomenon is known as the correlation effect in the diffusion of substitutional species. When the lattice atoms have roughly equal probabilities of exchanging with the vacancy the correlation effect is not large. For example, an atom site in an FCC structure has twelve nearest neighbors. If the vacancy is equally likely to exchange with any one of its neighbors then the probability that an atom jump will be reversed by the succeeding jump is only 1/12. Even if we include the possibility that a jump is reversed after two or more atom exchanges with the vacancy the correlation effect remains small. Hence. we ordinarily ignore the correlation effect when we are concerned with the self-diffusion of the lattice specie in a one-component solid or the diffusion of a substitutional solute that is similar in size to the solvent specie. Correlation effects can be important when the different components of a substitutional solution have significantly different activation energies for exchanging with a vacancy. We shall not consider such cases here. The jump frequency of a substitutional atom is the product of two terms: the jump frequency into vacant neighbor sites, and the probability that a neighbor site is vacant to accept the jump. If the vacancies are randomly distributed the probability that a neighbor site is vacant is just cv, the atom fraction of vacancies, and

∑ = ˆcv e -

QmkT 10.90

where ˆ is the effective frequency (eq. 10.89) and Qm is the activation energy for migra-tion. As shown in Chapter 8, the equilibrium vacancy concentration, cv, is

cv = Ae -

QVkT 10.91


Page 307

where A is a pre-exponential factor and Qv is the energy of formation of a vacancy. When the vacancy concentration has its equilibrium value the jump frequency is

∑ = Aê - Q

kT 10.92

where Q, the activation energy for an atom jump, is Q = Qv + Qm 10.93 the sum of the activation energy to form a vacancy and the activation energy for atom ex-change with a vacancy. These activation energies vary significantly from material to material, but both are of the order of an electron volt. Hence the effective activation energy that governs the jump of a substitutional atom is of the order of twice that for an interstitial atom. Equation 10.92 is adequate to understand the mechanism of diffusion of a substitutional specie, but should be modified in at least two respects for quantitative accuracy. First, because of the correlation effect a larger than random fraction of the atom jumps of a substitutional specie are reversed and do not contribute to diffusion. The effective frequency of diffusional jumps is, therefore, ∑eff = f∑ 10.94 where the factor f accounts for correlation. Second, the vacancy concentration that appears in the jump frequency is, strictly, the vacancy concentration at a site adjacent to the atom that performs the jump. If the atom has a strong interaction with vacancies then the activation energy, Qv, for forming a vacancy may change significantly. Even when these changes are important, however, the effective jump frequency is given by an equation of the form of equation 10.92 with an activation energy of the form 10.93. 10.7.3 Random-walk diffusion; Fick's First Law The individual jumps of the atoms within a uniform solid are random in direction. Over time an atom experiences a sequence of such jumps and meanders through the solid in an aimless path known as a random walk. But even though the individual atoms wander aimlessly through the solid there is a net diffusional transport of matter when the composition is non-uniform. The reason is that a random walk of the atomic species creates a more random, or uniform overall distribution and gradually smoothes compositional heterogeneities. An analysis of random-walk diffusion reveals the connection between the jumps of the individual atoms and the macroscopic diffusivity that is defined by Fick's First Law. For example, consider a solid with a simple cubic structure that is uniform except for a concentration variation in the [100], or x-direction. The diffusional flux in the x-


Page 308

direction is the net flow of atoms per unit area across a plane perpendicular to x, which is the net flow between adjacent (100) planes.

J21

c1 c2

J12

a

Fig. 10.13: Diffusion between adjacent (100) planes in a simple cubic

crystal. Fig. 10.13 shows three adjacent unit cells of a simple cubic crystal that abut in the x-direction. The net flow of solute atoms per unit area between adjacent (100) planes (which are labeled 1 and 2 and shaded in the figure) is the net flux across a plane midway between them. Hence J = J12 - J21 10.95 where J12 is the flow per unit area from plane 1 to plane 2 and J21 is the counterflow from 2 to 1. The flow is due to atom jumps in the x-direction. Let the frequency with which a solute atom jumps in the positive x-direction be ∑x. Then J12 = n1∑x 10.96 where n1 is the number of solute atoms per unit area of plane 1. If the solute concentration on plane 1 is c1 then n1 = nc1a 10.97 where n is the number of atom sites per unit volume and the volume per unit area of plane is equal to the interplanar spacing, a. If the solute concentration is a continuous function of x that varies only slightly between planes 1 and 2,

c2 = c1 +

∆c

∆x a 10.98

If an atom is equally likely to exchange with each of its six neighbor sites in the simple cubic structure,

∑x = ∑6 10.99


Page 309

where ∑ is the effective frequency of solute atom jumps per unit time. Writing an equation like 10.96 for the counterflow, J21, and using eqs. 10.97=99, the net flux is

J = 16 ∑a(c1 - c2) = -

16 ∑a2

∆c

∆x 10.100

Equation 10.100 has the one-dimensional form of Fick's First Law,

J = - nD

∆c

∆x 10.101

with the diffusivity,

D = 16 ∑a2 10.102

It can be shown that equation 10.102 holds in general for materials with cubic structures. The parameter, a, is the length of the elementary atom jump. It is the inter-atomic distance for substitutional diffusion, and the distance between neighboring interstitial sites for interstitial diffusion. Equation 10.102 provides the connection between the frequency of atom jumps and the diffusivity. It follows that the diffusivity can be written

D = D0 exp

- QDkT 10.102

where the activation energy for diffusion, QD, is equal to Qm for the diffusion of vacancies or interstitial atoms, and is equal to Qm+Qv for the diffusion of substitutional species in a solid that has an equilibrium vacancy concentration. 10.7.4 The mean diffusion distance in random walk diffusion Further useful insight into the mechanism and consequences of random-walk dif-fusion can be garnered from an analysis of the mean diffusion distance. If an atom diffuses by performing a random walk in a homogeneous solid then its position after time t is displaced by the vector X from its initial position, as illustrated by the example in Fig. 10.14. Since the net displacement, X, is equally likely to lie in any direction its expected value, ´X¨, is zero at all times. However, it is very unlikely that the atom will be found in its initial position after it has performed many jumps. The expected value of the mean diffusion distance, XD, should not be zero, and should increase with the number of atom jumps.


Page 310

X

...

Fig. 10.14: The random walk of an interstitial atom causes a net displacement, X, in time t.

The mean diffusion distance is the expected value of the magnitude, |X|, of the displacement, X, independent of its direction. Hence XD = ´|X|¨ = ´X^X¨ 10.103 where ´X^X¨ is the mean square displacement, the expected value of the scalar product of the displacement vector with itself. The mean square displacement is relatively easy to compute for a cubic solid with random atom jumps. Let ai be the ith jump in the sequence of the n jumps an atom makes in the time, t. Assuming that all jumps are to nearest neighbor positions, all jump vectors have the same magnitude, a, where a is the distance to the nearest neighbor site. They differ only in direction. The total displacement vector, X, is the vector sum of the vector displacements in the n elementary jumps:

X = ∑i=1

n ai 10.104

Hence

X^X = ∑i=1

n ∑j=1

n aiâj

= ∑k=1

n aiâi + ∑

i=1

n ∑j≠i=1

n aiâj 10.105

The first summation on the right in this last expression includes the terms in which the two jumps are the same while the second includes all terms that involve the scalar product of the ith jump vector and the jth when j ≠ i. If there are many jumps in the se-quence the expected value of the second term on the right in 10.105 is zero; the jump aj is equally likely to be in the negative as the positive direction, so the terms in the series cancel one another in pairs. Since aiâi = a2 the value of the first sum is just na2, so


Page 311

´X^X¨ = na2 = ∑a2t 10.106 where ∑ is the effective number of atom jumps per unit time. The mean square displace-ment is the sum of the mean square displacements in the three coordinate directions: ´X^X¨ = ´(xex + yey + zez)^(xex + yey + zez)¨ = ´x2 ¨ + ý2¨ + ´z2¨ = 3´x2¨ 10.107 where x, y and z are the displacements in the three coordinate directions. In a cubic solid the mean square displacements in the three coordinate directions are equal, so

´x2¨ = 13 ∑a2t

= 2Dt 10.108 where D is the diffusivity, D = ∑a2/6. Since all directions are the same for diffusion in a cubic solid equation 10.108 gives the mean linear diffusion distance xD = 2Dt 10.109 which measures the expected linear displacement of an atom that has diffusivity, D, after a random walk for time t. 10.7.5 Uses of the mean diffusion distance Equation 10.109 is a very useful relation. The two most common situations in the practice of materials science in which it is desirable to know the kinetics of diffusion are in the processing of materials, where it is important to know the time required to complete a diffusional process, and in the use of chemically heterogeneous materials, where it is important to know that diffusion will not significantly disturb the chemical heterogeneity during the expected service life of the material. In both cases it is often un-necessary to know the detailed evolution of the concentration; what is needed is a rough estimate of the time required for the relevant diffusion process to occur. This estimate can often be made directly from equation 10.109, as illustrated by the following two examples. Time required to homogenize a heterogeneous material The first example is the homogenization of a freshly solidified material with an inhomogeneous distribution of solute. For reasons we shall discuss below, when a multi-component solid is cast from the liquid its composition is usually inhomogeneous even


Page 312

when its microstructure contains only a single phase. The compositional heterogeneity develops because solidification occurs gradually on cooling through the two-phase region, and is a roughly periodic variation about the mean concentration, –c, as illustrated in Fig. 10.15. The compositional heterogeneity is eliminated by giving the material a homogenization treatment in which it is heated to the highest practical temperature and held for a time long enough for the concentration to become uniform.

c

x

–c¬

Fig. 10.15: Inhomogeneous concentration profile after solidification. To estimate the time required to homogenize a material we note that the peaks and valleys that appear in the composition profile will certainly disappear if the material is held at temperature for a time long enough that the linear diffusion distance, xD, is large compared to the peak-to-valley distance, ¬/2, where ¬ is the effective wavelength of the composition. The material should homogenize in a time of the order

t > th = ¬2

8D 10.110

Since the diffusivity varies exponentially with the temperature, equation 10.110 can be used to select a temperature at which homogenization is completed in a practical time. Eq. 10.110 also reveals when th is impractically long at all reasonable temperatures (as it is, for example, for many alloys that are potentially useful in high-temperature structures) in which case some alternative to conventional homogenization must be found. Service life of a chemically heterogeneous material A second important example occurs in semiconducting devices. As we shall dis-cuss at a later point in the course, these devices usually consist of silicon crystals, or chips, whose surfaces are "doped" locally with selected solutes to control their electrical characteristics, as illustrated in Fig. 10.16. Silicon chips are, hence, non-equilibrium structures with implanted compositional heterogeneities. The moment the doping is completed the heterogeneity begins to diffuse away.

¬Si

P


Page 313

Fig. 10.16: An Si crystal locally doped with P to the depth ¬. The maximum possible service life of the chip is limited by the diffusion time of the dopant. Assuming a device depth, ¬, the service time, ts , and the service temperature must be such that

ts << ¬2

8D 10.111

This requirement is relatively easy to satisfy when the chip is kept cool, but can become a limiting problem when ¬ is very small and the device is heated by the environment or by the current developed during operation. 10.8 MICROSTRUCTURAL EFFECTS IN DIFFUSION The diffusivity of an atom is strongly affected by the composition and structure of the material in which it appears. The diffusivity is also affected by other aspects of the microstructure, such as grain boundaries, dislocations, and point defects. The microstruc-tural effects are most pronounced when the diffusing specie is substitutional. 10.8.1 The vacancy concentration The diffusivity of a substitutional component is proportional to the vacancy concentration, cv. The vacancy concentration can be significantly changed, at least in a transient sense, by cooling the material from high temperature, by deforming it, by bombarding it with radiation, or by adding solutes that promote vacancy formation. Quench-enhanced diffusivity The equilibrium vacancy concentration is an exponential function of the tempera-ture. If a material is brought into equilibrium at high temperature and then cooled quickly, the high-temperature vacancy concentration is preserved for some time; the excess vacancies must diffuse to boundaries, dislocations or other sinks before they can be eliminated. Consequently the diffusivity of a substitutional species is anomalously high immediately after a quench. It decreases gradually to the equilibrium value as illustrated in Fig. 10.17. The plot given in Fig. 10.17 is on a semi-logarithmic scale since the initial vacancy concentration, and hence the initial diffusivity, is raised by several orders of magnitude if the material is quenched from high temperature.


Page 314

ln(D)

t

Deq

...

Fig. 10.17: Schematic drawing of the variation of the diffusivity of a substitutional specie with time after a quench. Deq is the equilibrium value of D at the final temperature.

Quench-enhanced solute diffusivity is often useful in materials processing. Processes such as precipitation in the solid state are sluggish at the low temperatures at which it is often desirable to make them happen. Their rates can be increased by quenching from high temperature. Deformation-enhanced diffusivity The diffusivity of a substitutional component is increased by concurrent plastic deformation. The motion of dislocations in plastic deformation always involves at least some dislocation climb, which produces vacancies. The diffusivity is raised accordingly. Concurrent plastic deformation may also increase the diffusivity of very mobile interstitial species such as hydrogen. Interstitial species tend to accumulate near dislocations that have some edge component because the lattice beneath the extra half-plane of an edge dislocation is expanded and can better accommodate interstitial defects. The excess solute is said to form a solute atmosphere about the dislocation. If the interstitials are sufficiently mobile they remain with the dislocation when it moves. This process, which is known as dislocation sweeping, is a significant source of hydrogen diffusion in solids that are deformed at low temperature. Irradiation-enhanced diffusivity Irradiation substantially enhances diffusivity by increasing the density of point de-fects. Intense radiation knocks atoms off of their lattice sites to create vacancy-interstitialcy pairs. The vacancies increase diffusivity on the lattice while the interstitials are mobile through the interstitial sites. The enhanced diffusivity of irradiated solids is sometimes useful in materials processing, but accelerates the degradation of materials used in nuclear devices. Solute-enhanced diffusivity Solute atoms may increase the concentration of vacancies by lowering their energy. Important examples include solutes of relatively large atomic volume, which


Page 315

promote the formation of vacancies on adjacent sites to minimize the net lattice strain, and solutes of different ionic charge in ionic solids, which promote the formation of vacancies to reestablish charge neutrality. An example of the latter is Mg in NaCl; Mg++ ions are balanced by Na+ vacancies, which raise the diffusivity of Na. 10.8.2 Grain boundary diffusion If the material is polycrystalline, substitutional solutes can diffuse along the continuous network of grain boundaries as well as through the bulk of the crystal. A grain boundary is a region of disorder where adjacent crystals do not fit perfectly together. To a first approximation the grain boundary can be pictured as a region of distorted crystal that contains a very high density of vacant sites. The grain boundary diffusivity obeys an equation that is of the form

DB = DB0 exp

-

QB

kT 10.112

where QB is of the order of one-half the activation energy for diffusion through the bulk, QD, reflecting the fact that it is not usually necessary to form a vacancy to accomplish an atom jump. The pre-exponential factor, D

B0 , for grain boundary diffusion is also much

smaller than D0 for diffusion through the bulk. To understand why, recall that the diffusivity governs the net flux of atoms across a cross-sectional plane through the solid. The flux due to grain boundary diffusion is restricted by the small areal fraction of grain boundary on a cross-section through the material. Consider a solid with cubic grains of size, d, whose grain boundaries have an ef-fective width, ∂, as drawn in Fig. 10.18. The cross-sectional area per grain is d2 and the cross-sectional area of grain boundary per grain is 4d(∂/2); the fractional area of grain boundary in a cross-section is proportional to (∂/d). If the activation energy for diffusion in both parts of the microstructure were the same the flux would be proportional to the relative cross-section. Hence

DB0 « D0

∂

d 10.113

Since ∂ is of the order of a few ‹, while d is typically of the order 104-106 ‹, D

B0 << D0 10.114

Atoms diffuse simultaneously through the bulk and along the grain boundaries, so these provide parallel diffusion paths. The net diffusion flux is the sum of the flux each contributes. The ratio of the two diffusivities is


Page 316

DBD =

D

B0

D0 exp

( )Q-QB

kT 10.115

When the temperature is very large the exponential term approaches 1 and DB << D. Hence at high temperature almost all of the flux is due to bulk diffusion. As the temperature approaches zero, however, the exponential term becomes arbitrarily large since Q > QB. At sufficiently low temperature, DB >> D, and the flux is almost exclusively along the grain boundaries.

d

∂

Fig. 10.18: Section through a microstructure of square grains whose grain boundaries have effective width, ∂.

The variation of the effective diffusivity with temperature is sketched qualitatively in Figure 10.19. Since

ln(D) = ln(D0) - QkT 10.116

a plot of ln(D) against (1/kT) is a straight line with slope -Q that approaches ln(D0) in the limit T “ ∞ (T-1 “ 0). The total diffusivity is determined by the greater of the bulk and grain boundary diffusivities. The effective diffusivity is given by two straight lines that join at the temperature at which the dominant mechanism of diffusion changes, as shown in the figure. The value of DB

0 increases as the grain size becomes smaller. Hence the grain boundary diffusivity and the temperature at which it becomes dominant increase as the grain size decreases as shown in the figure. A particular case in which grain boundary diffusion is important is in the selection of diffusion barriers to separate reactive species at low temperature. This is a particularly important problem in microelectronics. Microelectronic devices usually contain dissimilar materials in intimate contact, such as metallic conductors in contact with silicon or oxide insulators. Because of the extremely small dimensions of the individual material elements and the sensitivity of their electrical properties, chemical reactions can be a problem even at very low temperature. Thin films of very stable materials are, therefore, often inserted as diffusion barriers to separate reactive materials from one another. An effective diffusion barrier must be such that both reactive species


Page 317

diffuse very slowly through it. However, the diffusion barriers are deposited films and are almost always polycrystalline. At the low temperature of operation of microelectronic devices grain boundary diffusion usually predominates, so diffusion barriers must be selected on the basis of the grain boundary diffusivities rather than the bulk diffusivities of the species to be separated.

ln(D)

1/kT

D0

D0B

QB-

QD- decreasing grain size

... Fig. 10.19: Semi-logarithmic plot of diffusivity against reciprocal

temperature for a polygranular material. The grain boundary diffusivity increases as grain size decreases as shown.

10.8.3 Diffusion through dislocation cores Diffusion is also easier along the core of a dislocation that has at least some edge component. Interstitial species move more easily through the expanded lattice below the dislocation (that is, below the termination of the extra half-plane), and substitutional species are more mobile in the compressed lattice above the dislocation since vacancies have lower energy there. However, excepting the case in which planar arrays of dislocations form low-angle grain boundaries, dislocation cores fill a negligible fraction of the cross-section of a crystal. Diffusion along dislocation cores is rarely important except at very low temperature in materials that have been severely deformed to create a very high dislocation density.


Page 318

C h a p t e r 1 1 : P h a s e T r a n s f o r m a t i o n sC h a p t e r 1 1 : P h a s e T r a n s f o r m a t i o n s

After conquering Sardis, capital of Lydia and wealthiest city in the world, the Persian King Cyrus the Great had the captured Lydian King, Croesus, brought to him ... For a while Croesus was deep in his thoughts and did not speak. Then he turned and, seeing that the Persians were sacking the town, said, "Should I tell you, my lord, what I have in my mind, or must I now keep silent?" Cyrus replied that he might say what he pleased without fear, so Croesus put another question: "What is it," he asked, "that all those men of yours are so intent upon doing?" "They are plundering your city and carrying off your treasures." "Not my city or my treasures," Croesus answered. "Nothing here any longer belongs to me. It is you they are robbing." -Herodotus, "The Histories"

11.1 COMMON TYPES OF PHASE TRANSFORMATIONS A phase transformation is a discontinuous change in the microstructure of a mate-rial. Phase transformations happen when it is both thermodynamically and kinetically possible to reconfigure the material so that its thermodynamic potential is lowered. In the most common case the temperature, pressure and composition are fixed, so the thermody-namic potential that governs equilibrium is the Gibbs free energy. A material can change its phase if there is an alternate phase with a lower free energy, but it will do so only if there is a transformation mechanism that can act at a measurable rate. There are several physically distinct types of phase transformations, including the following: 1. Structural transformations alter the basic arrangement of atoms in the material. Familiar examples include boiling and freezing, crystal structure changes such as the ©(FCC) to å(BCC) phase transition that occurs when iron is cooled, and glass transitions that "freeze" a rapidly cooled liquid into an amorphous solid state. 2. Recrystallization reactions occur in severely deformed metals, and replace the deformed grains with new ones that have the same basic crystal structure, but are relatively free of defects. 3. Ordering reactions change the distribution of atoms over a fixed framework of atom positions to create an ordered compound. An example is the ∫ (disordered BCC) to ∫' (CsCl) transition in ∫-brass (50Cu-50Zn). 4. Precipitation reactions create second-phase particles within a parent solution to make a two-phase mixture. The chemical composition of the precipitate is always different from that of the parent phase, but the basic crystal structure (the arrangement of


Page 319

atom sites, irrespective of atom type) may be the same. If it is the same, then the precipitation reaction may be coherent in the sense that the parent lattice is continuous across the precipitate interface. An example of a coherent precipitation reaction is the precipitation of Ni3Al from an FCC solid solution of Al in Ni. If the basic crystal structure of the precipitate is different from that of the parent then the precipitation reaction is necessarily incoherent, and the lattice is discontinuous at the precipitate interface. An example of an incoherent precipitation reaction is the precipitation of Fe3C from a supersaturated solution of C in Fe. 5. Ferroelectric transitions occur in some ionic solids, and involve spontaneous ion displacements that create permanent dipole moments. The most frequently cited ex-ample is the ferroelectric transition in BaTiO3, which was described in Chapter 9. When the material is cooled to a temperature below a critical temperature called the Curie temperature the Ti+4 ions are slightly and spontaneously displaced from the center of the BaTiO3 unit cell to create a permanent electric dipole. 6. Magnetic transitions occur in materials whose atoms have permanent magnetic moments. At sufficiently low temperature the magnetic moments spontaneously align with one another. If the alignment is parallel the transition is called ferromagnetic and the transition temperature is called the Curie temperature (as described in Chapter 9). If the alignment is anti-parallel the transition is called antiferromagnetic and the transition temperature is called the Neel temperature. 7. Electronic transitions are changes in the distribution of electrons that alter the nature of electrical conduction. Ordinarily electronic transitions change the nature of bonding in a way that is so fundamental that the crystal structure changes at the same time. An example is the transition from gray tin, a semiconductor with a diamond cubic structure, to white tin, a metal with a tetragonal structure, that occurs on heating. In cases like this the electronic transition can usually be treated as an incidental feature of a structural transition. However, there are cases in which the electronic structure changes without altering the crystal lattice. The most spectacular example is the superconducting transition in which charge carriers spontaneously pair in a nominal metal or superconductor so that all electrical resistance is lost. In this chapter we shall concentrate on the phase transformations that change the structure or chemical distribution. The phase transformation that change the electromag-netic properties of materials will be treated in the discussion of electromagnetic properties later in the course. 11.2 THE BASIC TRANSFORMATION MECHANISMS 11.2.1 Nucleated transformations and instabilities While the driving force for a phase transformation at given T and P is a decrease in the Gibbs free energy, the kinetics of the transformation depend on the mechanism by


Page 320

which it is accomplished. It is useful to divide the mechanisms of phase transitions into two characteristic types: nucleated transitions and instabilities. The distinction is easiest to see in the case of a monolithic structural transformation in a one-component system. If the equilibrium of a one-component system places it in the å structure at high temperature and in the ∫ structure at low temperature then the free energy curves for the two phases will ordinarily appear as shown in Fig. 11.1. The free energy curves intersect at the temperature T0. However, it is usually possible to preserve the å structure in a metastable state at temperatures below T0, as indicated by the extension of its free energy curve. The driving force, ÎG, for the å“∫ transformation ordinarily increases as the temperature is decreased below T0, but, so long as the å phase is metastable, the system must be perturbed in some finite way to cause the transformation to take place.

G

TT0

∫

∫

nucleatedinstability

å

å

=> equilibrium å

=> metastable å

Fig. 11.1: Free energy curves for hypothetical phases å and ∫ showing the range of metastability of å, and situations in which nucleated and instability transitions occur.

The most favorable transformation mechanism is that which requires the smallest disturbance of the metastable å phase. This is usually the formation of a small seed, or nucleus, of the ∫ phase within the body of the å. Once a stable nucleus of ∫ has formed then the free energy decreases continuously as the nucleus grows to consume the å and complete the transformation. The nucleation-and-growth process is the usual mechanism by which a metastable phase transforms. Since nucleation requires a significant (though local) change in the parent phase, a nucleated transformation can often be suppressed or delayed by cooling the material at a sufficiently rapid rate. However, it often happens that the å phase cannot be cooled indefinitely. At some temperature below its equilibrium transformation temperature the å phase may become thermodynamically unstable, either because of internal changes or because the driving force for the transformation becomes so large that a significant volume of å can transform spontaneously. The limit of stability of the å phase is indicated in Fig. 11.1 by a termination of the å free energy curve. The termination defines an instability at which the å phase must transform; a transformation that is due to an instability cannot be suppressed.


Page 321

11.2.2 First-order transitions and mutations As we discussed in Chapter 9, a phase transition of the type diagrammed in Fig. 11.1 is called a first-order phase transition. A first-order transition connects structures that are physically different and involves discontinuous changes in at least some of the thermodynamic quantities. For example, even when the transformation diagrammed in Fig. 11.1 happens at the equilibrium temperature the entropy changes discontinuously. Structural transformations are ordinarily first-order as are precipitation reactions and many of the reactions that create chemical order. When two phases are related by a first-order transition it is always possible to pre-serve them in metastable states beyond the equilibrium transformation temperature. Hence first-order transitions that occur sufficiently near the equilibrium transformation temperature must be nucleated. On the other hand, the range of metastability of a phase that participates in a first-order transition may or may not be limited by an instability. In many cases the metastable phase can be preserved at temperatures as far below the transition temperature as desired. There is always a nucleation-and-growth mechanism for a first-order transition. There may also be an instability mechanism, but there need not necessarily be. There is a second class of phase transitions, which we have called mutations, in which two phases that have distinct structures away from the transition temperature become identical to one another at the transition point. In this case the free energy curves merge at the transition temperature and metastability is impossible. Effectively, the stability limit and the equilibrium transition temperature are the same. Such transitions do not require nucleation; the transformation mechanism is an instability that cannot be suppressed. These transitions are often called second-order transitions, since the discontinuity at the transition temperature is not in the thermodynamic quantities, such as the entropy, but in their derivatives, such as the isobaric specific heat. Important examples of mutations include glass transitions, in which the liquid and solid states merge together, ferroelectric transitions, where the ion displacement and, hence, the induced dipole become imperceptibly small at the Curie point, magnetic transi-tions, in which the degree of magnetic alignment vanishes at the Curie or Neel points, and superconducting transitions. As we discussed in Chapter 9, chemical ordering reactions are sometimes mutations and sometimes first-order phase transitions, depending on whether the degree of order vanishes continuously at the transition temperature or changes discontinuously there (the ordering reactions that can be transmutations are identified theoretically by the Landau symmetry rules, which are beyond the scope of this course). 11.3 HOMOGENEOUS NUCLEATION 11.3.1 Nucleation as a thermally activated process


Page 322

When the parent phase is stable with respect to infinitesimal changes in its local state (i.e., when it is metastable), the phase transformation must be nucleated if it is to happen at all. Nucleation is a finite fluctuation in which a small element of the parent spontaneously takes on a state that is very close to that of the product phase. Since the spontaneous reconfiguration of a significant volume of stable material is extremely unlikely, the fluctuations that lead to nucleated transformations are local, and produce candidate nuclei that are very small. When nuclei form at more or less random positions within the metastable parent phase the process is known as homogeneous nucleation. When the nuclei form preferentially at catalytic sites in the parent the process is called heterogeneous nucleation. We shall consider homogeneous nucleation first. The thermodynamics of nucleation can be understood by writing the free energy of the nucleus as the sum of two separate contributions (Fig. 11.2). First, the nucleus introduces a small volume of product phase. Since this phase is more stable than the parent phase, the associated free energy change is negative. The formation of the nucleus also creates an interface between the parent and product phases. The interfacial energy is necessarily positive, and raises the free energy. The total free energy change is the sum of the two effects.

∫

å

Fig. 11.2: A spherical nucleus of phase ∫, surrounded by an interface,

forms within homogeneous phase å. However, the volume and surface contributions to the free energy depend differ-ently on the nucleus size; the surface energy is proportional to the surface area, and hence to r2, where r is the effective radius of the nucleus, while the volume energy is proportional to r3. When the nucleus is very small, r2 > r3 and the surface term domi-nates; the formation of a nucleus raises the free energy. When the nucleus is larger, r3 > r2 and the volume term dominates; the growth of the nucleus lowers the free energy of the system if the product phase is thermodynamically stable. This relationship is illustrated in Fig. 11.3. The free energy change on forming the nucleus is a function of its size, and passes through a maximum, ÎG*, as the effective radius increases. The maximum, ÎG*, is the activation energy that must be overcome to form the nucleus. Once the critical nucleus size, rc, is exceeded, the free energy decreases monotonically as the nucleus grows to complete the phase transition.


Page 323

Fig. 11.3 shows that the nucleation of a new phase is a thermally activated process, like the diffusional processes we discussed in Chapter 9. In order for the trans-formation to proceed spontaneously it must first overcome the activation barrier, ÎG*. To determine the rate at which nuclei of the new phase form under particular conditions, it is necessary to evaluate the activation energy, ÎG*.

ÎGr

ÎG*

rc

Fig. 11.3: The free energy to form a nucleus as a function of its effective radius, r.

11.3.2 The activation energy for homogeneous nucleation The free energy change on forming the ∫ nucleus is the sum of two terms. First, a volume, V∫ of the ∫ phase is created from the parent å. The associated change in the free energy can be written ÎG = V∫ÎGv 11.1 where ÎGv is the free energy change per unit volume of the nucleus. Since ÎGv relates to the free energy change of the whole system, rather than that of the nucleus alone, there is some subtlety in its evaluation. We discuss this matter in more detail below. However, when å and ∫ are incompressible phases of a one-component system, ÎGv is just the free energy change per unit volume for the transformation å “ ∫ at the given temperature. Secondly, an interface appears between the ∫ nucleus and the å phase. The interface adds the free energy ÎG = ßå∫S∫ 11.2 where ßå∫ > 0 is the interfacial tension of the å∫ interface and S∫ is its area. It follows that the free energy change on forming a spherical nucleus of radius, r, is

ÎG = 43 πr3ÎGv + 4πr2ß 11.3

The critical nucleus size and the activation energy for homogeneous nucleation can be found by differentiating equation 11.3. The maximum is reached when


Page 324

d(ÎG)

dr = 4πr2ÎGv + 8πrß = 0 11.4

Hence the critical radius is

rc = - 2ß

ÎGv 11.5

and the activation energy for homogeneous nucleation is the value of ÎG when r = rc. We use the symbol ÎGH to designate the activation energy to create a spherical, homogeneous nucleus:

ÎGH = ÎG(rc) = 16πß3

3(ÎGv)2 11.6

11.3.3 The nucleation rate The rate of homogeneous nucleation is proportional to the frequency with which free energy fluctuations of magnitude ÎG* occur in the system. Hence

•n fi Nexp

- ÎGHkT 11.7

where •n is the number of critical nuclei that form per unit area per unit time and N is the number of atom sites per unit volume, any one of which can be the locus of the center of the critical nucleus. However, there is a second kinetic term that influences the nucleation rate. To create a critical nucleus it is necessary to assemble the material that it contains. Even in a one-component system the growth of the nucleus requires transport of material across the interface from å to ∫. When å and ∫ are solid solutions the composition of the nucleus ordinarily differs from that of the parent phase, so species must diffuse to the nucleus in order for it to grow. When å and ∫ are condensed phases these are diffusional processes that require thermal activation. Let QD be the relevant activation energy. Then the net nucleation rate for a transformation between condensed phases is given by an equation of the form

•n = ANexp

- (ÎGH+QD)

kT 11.8

where A is a pre-exponential factor. Equation 11.8 reveals two important qualitative features of the kinetics of nucleated transformations in condensed phases. The first is that a nucleated transformation never happens precisely at the transformation temperature; the system


Page 325

always overshoots the transformation temperature at least slightly to provide the positive driving force that is needed to trigger the transformation. Let T0 be the transformation temperature. When T = T0, ÎGv = 0 and ÎGH is infinite. The nucleation rate, •n, is zero when T = T0. A nucleated transformation can only occur when the temperature passes T0 to create a finite driving force, ÎGv.

ÎT

ln(•n)

nucleation limited

growth limited

Fig. 11.4: The variation of the nucleation rate with temperature. The second important kinetic feature applies only to transformations that occur in condensed systems on cooling. In these systems the nucleation rate has a maximum at a finite temperature below T0, and the transformation can be suppressed by quenching the system at a sufficiently rapid rate. To see this note that near T0 the nucleation rate is dominated by the activation energy for nucleation, ÎGH, since ÎGH >> QD. We refer to this regime as nucleation-dominated. In the nucleation-dominated regime the nucleation rate is given approximately by

•n ~ ANexp

- ÎGHkT 11.9

Since the magnitude of ÎGv ordinarily increases rapidly at temperatures slightly below T0, ÎGH decreases rapidly and •n increases exponentially.. However, the increase in ÎGv eventually has the consequence that ÎGH falls below QD. When this happened the nucleation rate is given approximately by

•n ~ ANexp

- QDkT 11.10

We refer this regime as growth-dominated since the kinetics of growth limit the nucleation rate. Since QD is approximately constant with temperature, the nucleation rate decreases exponentially as the temperature decreases in the growth-dominated regime. The balance of these two effects causes the nucleation rate to vary with temperature as shown in Fig. 11.4. It increases at high temperature, decreases at low temperature, and has a maximum at an intermediate temperature that roughly corresponds to the change from nucleation to growth control.


Page 326

11.3.4 The initiation time The consequence of the maximum in the nucleation rate is perhaps more clear when we plot the time required to initiate the transformation as a function of temperature below T0. Defining the initiation time (†) for the transformation as the time required to form at least one nucleus per unit volume then † = (•n)-1 11.11 and varies with the temperature as shown in Fig. 11.5.

ln(†)

nucleation limited

growth limitedÎT

Fig. 11.5: The initiation time (†) as a function of temperature for a nucle-ated transformation that occurs on cooling.

T ln(†)

∫ “ å

å “ ∫

Fig. 11.6: Contrast between the c-curve behavior of a transformation that happens on cooling and the monotonically decreasing initiation time of a transformation that happens on heating.

Fig. 11.5 shows the characteristic c-curve behavior of a nucleated transformation in a condensed phases on cooling. The initiation time decreases rapidly when the material is cooled below the transformation temperature, T0, but then reaches a minimum (the nose of the transformation curve) and increases exponentially if the system is cooled further. The transformation initiates at approximately the temperature at which the cooling curve intersects the c-curve. Hence the undercooling that precedes the start of the transformation increases with the rate of cooling. The transformation can be sup-pressed entirely by cooling the system so rapidly that it passes the nose of the initiation


Page 327

curve, provided that initiation at low temperature takes an arbitrarily long time. It is also often possible to control the temperature at which a nucleated transformation proceeds by quenching to suppress the transformation and then re-heating to the desired reaction temperature. As we shall discuss below, this freedom facilitates control of the microstructure of the product phase. Note that the kinetic features that are illustrated in Figs. 11.4-5 apply only to transformations that happen on cooling. Consider the reverse transformation, ∫“å, that happens when the temperature is increased to above T0. Both nucleation and growth be-come easier the further the temperature rises above T0, so the initiation time decreases monotonically. The kinetic behavior of nucleated transformations of the type å“∫ and ∫“å are contrasted in Fig. 11.6. 11.4 HETEROGENEOUS TRANSFORMATIONS Homogeneous nucleation is relatively easy to describe, but is often difficult to achieve in practice. The reason is that there are usually catalytic sites where nuclei can form with relative ease. The common heterogeneous nucleation sites include free surfaces, grain boundaries, pre-existing precipitates, inclusions and dislocation tangles. Each of these sites includes a defect whose excess energy is at least partly recovered when the nucleus forms, lowering the total energy of the nucleation event. 11.4.1 Nucleation at a grain boundary As an example, consider the formation of a nucleus on a grain boundary, as illus-trated in Fig. 11.7. If the spherical nucleus is simply placed on the boundary it eliminates a small area of boundary. Since the boundary has a positive tension, ßåå, the energy ßååÎSåå is recovered in nucleation, which decreases the free energy required to form the nucleus.

ßåå

ßå∫ßå∫

(a) (b)

Fig. 11.7: (a) A nucleus formed at a grain boundary showing the elimina-tion of a section of the boundary. (b) The nucleus deforms into a lens shape to destroy more boundary area.

A more complete analysis shows that a nucleus that would be spherical in the bulk takes a lens shape on the boundary, as shown in Fig. 11.7b. This is the shape that minimizes the total free energy change


Page 328

ÎG = ÎGvV∫ + ßå∫S∫ + ßååÎSåå 11.12 The nucleation barrier is set by the maximum of ÎG, which is ÎG*

b = ÎGHf(ßåå/ßå∫) 11.13 where f(ßåå/ßå∫) is a catalytic factor that is less than 1 and decreases as the ratio ßåå/ßå∫ increases. The rate of nucleation on the grain boundary is given by an equation of the form 11.8 with the difference that ÎG* is the activation energy for grain boundary nucleation and the pre-factor is proportional to the number of grain boundary sites per unit volume (Nb):

•nb = ANbexp

- ÎG*

b + QD

kT 11.14

When the undercooling is small, the transformation is nucleation-limited. Since ÎG*

b < ÎGH the rate of grain boundary nucleation is usually much greater than the rate of homogeneous nucleation . However, at large values of the undercooling the transforma-tion is growth-limited, and homogeneous nucleation dominates since Nv >> Nb. 11.4.2 Other heterogeneous nucleation sites

T

ln(†)

grain boundaries

homogeneous sitesvolume heterogeneities

Fig. 11.8: Diagram showing a possible sequence of dominant nucleation sites in a polygranular solid that contains heterogeneous nucle-ation sites.

Nuclei also form at other heterogeneous sites such as free surfaces, junction lines and nodes in the polygranular network of grain boundaries, and internal heterogeneous sites such as precipitates, inclusions or dislocation tangles. The nucleation rate at the pth heterogeneous site is given by an equation of the form

•np = ANpexp

- ÎG*

p + QD

kT 11.15


Page 329

where ÎG*

p is the activation energy for nucleation on a site of type p and Np is the number density of such sites. The value of ÎG*

p depends on the nature of the site, and is generally less than the activation energy for homogeneous nucleation. However, Np << Nv, so nucleation at heterogeneous sites is most important when the undercooling is small and the transformation is nucleation-limited. In a typical nucleated transformation there are several active nucleation sites. The nucleation rate is the sum of the rates of all of them. Grain boundary nucleation tends to dominate in transformations of polygranular materials at small values of the undercooling. Heterogeneous sites that are distributed through the volume, such as pre-existing precipitates and dislocation tangles, become important at larger values of the undercooling, and homogeneous nucleation dominates when the undercooling is very large. This situation is illustrated in Fig. 11.8. 11.4.3 Implications for materials processing When there are several different sets of heterogeneous nucleation sites it is often possible to control the product microstructure by selecting the dominant one. For example, by cooling a polygranular solid slowly through the transformation temperature, the transformation may be made to initiate at a low value of the undercooling where grain boundary nucleation is dominant. The microstructure of the transformed product will then be set by crystallites that nucleated and grew from grain boundaries. If the solid is quenched past the nose of the initiation curve and then re-heated to a temperature well below the nose of the curve the transformation may be made to nucleate homogeneously through the volume.

ln(†)

grain boundary precipitatesprecipitates in grain interiorsÎT

...

Fig. 11.9: Alternative heat treatments for a precipitation-hardening alloy. Slow cooling leads to precipitation on grain boundaries. Quenching followed by re-heating causes precipitation through the volume.

The possibility of controlling the nucleation site is critical to the processing of many materials. An important particular example is the precipitation hardening of structural alloys. The mechanical strengths of many structural alloys are enhanced and controlled by introducing a dense distribution of fine precipitates through the volume. If these precipitates appear at the grain boundaries instead, they not only fail to strengthen


Page 330

the alloy, but embrittle the alloy by introducing preferential paths for intergranular fracture. The precipitates are forced into the grain interiors by quenching the alloy to suppress nucleation on the grain boundaries, and then re-heating to induce precipitation at a low temperature where nucleation happens predominantly in the interiors of the grains. The alternative heat treatments are diagrammed in Fig. 11.9. 11.5 THE THERMODYNAMICS OF NUCLEATION Assuming that the parent phase in which a nucleus forms has a given temperature, pressure, and chemical content then its equilibrium is governed by the Gibbs free energy. The change in Gibbs free energy on forming a homogeneous nucleus of volume, V∫, and surface area, S∫, is ÎG = ÎGvV∫ + ßå∫S∫ 11.16 where ÎGv is the free energy change per unit volume of the nucleus, the thermodynamic driving force for the transformation. However, equation 11.16 is not as straightforward as it might appear. ÎGv is the free energy change of the whole system (å + nucleus) when the nucleus forms. If we are to understand the driving force for nucleation it is necessary to evaluate ÎGv in terms of the state of the nucleus itself. Both the pressure and the composition of the material within the ∫ nucleus will, generally, differ from those within the bulk ∫ phase. The conditions of mechanical equilibrium for material enclosed in a spherical surface with radius, r, and tension, ßå∫, require that its pressure satisfy the equation

P∫ - På = 2ßå∫

r 11.17

The pressure difference is very large when r is small. Moreover, when å and ∫ are solu-tions then the composition of the ∫ nucleus is a variable, and will adjust itself to minimize its thermodynamic potential. To understand these, and related phenomena that influence the nature of the material that actually nucleates in a phase transition we must consider the thermodynamics of nucleation.


Page 331

11.5.1 The thermodynamic driving force for nucleation To find the thermodynamic potential that governs the behavior of the nucleus itself we use the construction shown in Fig. 11.10. Let the volume within which the ∫ nucleus forms be separated from the remainder of the å phase by an imaginary boundary, as shown in the figure. Since the nucleus is very small compared to the å phase, the temperature, pressure and chemical potentials in the å phase are not perceptibly changed by the formation of the nucleus. The ambient å phase acts as a thermodynamic reservoir that fixes T, P and the chemical potentials {µ} within the small subvolume that contains the nucleus. The region within the imaginary boundary shown in Fig. 11.10 is an open system. As shown in Chapter 7 (Sec. 7.4.3) the thermodynamic potential that governs its equilib-rium is the work function „ = E - PV - ∑

k µkNk 11.18

which must have a minimum value when the open system is at equilibrium. The change in „ on the formation of the ∫ nucleus is the sum of volume and surface terms: ∂„ = Î„vV∫ + ßå∫S∫ 11.19 where Î„v is the change in work function per unit volume of the ∫ nucleus. Since

„v = - PV

V = - P 11.20

It follows that the change in the work function per unit volume of the ∫ nucleus is just the negative of the change in pressure: Î„v = - (P∫ - På) 11.21

∫

å

...


Page 332

Fig. 11.10: A small ∫ nucleus forms within the outlined volume in a parent phase of homogeneous å, which acts as a thermodynamic reservoir.

Comparing equations 11.21 and 11.16 shows that the thermodynamic driving force for nucleation, the change in the Gibbs free energy of the system (å+∫) as a whole, is equal to the change in the value of the work function of the ∫ nucleus alone: ÎGv = Î„v = - (P∫ - På) 11.22 Hence the activation barrier that opposes homogeneous nucleation is (eq. 11.6)

ÎGH = 16πß3

3(ÎP)2 11.23

and the critical radius is

rc = 2ßÎP 11.24

Comparison with equation 11.17 shows that the critical nucleus satisfies the condition of mechanical equilibrium for the pressure drop across the interface. To calculate the activation energy to form a critical nucleus it is necessary to have an estimate for the pressure difference, ÎP = (P∫ - På), between the two phases at the ambient values of T and {µ}. Since thermodynamic measurements are usually made in vacuum or at atmospheric pressure the function, ¡P(T,{µ}), that gives the pressure of a phase as a function of its temperature and chemical potentials is rarely known. The quantity that is known or easily measured is the change in the Gibbs free energy on transformation at constant temperature and pressure, Îgå∫ = ¡g∫(T,P,{x∫}) - ¡gå(T,P,{xå}) 11.25 where g is the molar density of the Gibbs free energy. When the molar volume of the product phase, v∫, is independent of pressure and composition it is possible to relate ÎP to this quantity. The relation is simplest for a one-component system. 11.5.2 Nucleation in a one-component system When a phase ∫ nucleates from å the condition of chemical equilibrium requires that the chemical potential be the same in both phases. Since the ∫ nucleus has pressure, P∫, while the å phase has pressure, På, we must have µå(På) = µ∫(P∫) 11.26 The Gibbs free energy of a one-component system is


Page 333

G = E - TS + PV = µN 11.27 where µ is the chemical potential. Hence the molar free energy is

g = GN = µ 11.28

The free energy change per mole for a transformation in which a homogeneous å phase is replaced by ∫ at the ambient pressure, På, can hence be written Îgå∫(På) = Îµå∫(På) = µ∫(På) - µå(På) = µ∫(På) - µ∫(P∫) 11.29 where µ∫(P∫ ) is the chemical potential in the nucleus of the ∫ that forms in å at På, and we have used eq. 11.26 to obtain the second form of the right-hand side. The change in chemical potential with pressure at constant temperature is just the molar volume, v:

∆µ

∆P T =

VN = v 11.30

... Fig. 11.11: Illustration of the driving force for nucleation at temperature,

T, when the product phase, ∫, is an incompressible phase in a one-component system.

Hence equation 11.29 can be re-written

Îgå∫(På) = - ⌡⌠

På

P∫

∆µ∫

∆P dP = - ⌡⌠

På

P∫

v∫ dP

= - v∫(P∫ - På) 11.31

T T 0

∫ ∫

å å

=> equilibrium å => metastable å

G v

v ÎG = -ÎP


Page 334

where v∫ is the molar volume of the ∫ phase, and we have assumed that the ∫ phase is in-compressible, as most condensed phases are to a good approximation. It follows that

- ÎP « Îgå∫(På)

v∫ = ÎGå∫v 11.32

where ÎGå∫

v is the change in free energy per unit volume when phase å is replaced by ∫ at pressure På, as illustrated in Fig. 11.11. 11.5.3 Nucleation from a supersaturated solid solution An important structural transformation in a multicomponent solid is precipitation from a supersaturated solid solution, which happens, for example, when the system is cooled from a one-phase into a two-phase field. A prototypic example is illustrated in Fig. 11.12. A binary system that is initially in the one-phase å field is cooled into the two-phase å+∫ field to initiate precipitation of ∫. If the supersaturated å solution is metastable the precipitation reaction is a nucleated structural transformation. If the volume of the ∫ phase is nearly independent of its pressure and composition the free energy change on forming a nucleus of it in the body of å can again be written ÎG = Î„vV∫ + ßå∫S∫ ~ ÎGå∫

v V∫ + ßå∫S∫ 11.33 Where the driving force, ÎGå∫

v , is the free energy change per unit volume for the formation of ∫ from å at ambient temperature and pressure. However, the free energy change, ÎGå∫

v , depends on the precise composition of the ∫ phase in the nucleus.

T

A Bx

å ∫

L

å+L ∫+L

å + ∫

Fig. 11.12: A binary system with a eutectic phase diagram. When an å solution is cooled along the dashed line it becomes supersatu-rated with respect to precipitation of ∫.


Page 335

To find ÎGå∫v we begin by recalling that the free energy of a binary solution is

G = E - TS + PV = µANA + µBNB 11.34 where the symbols A and B label the two components. To calculate the free energy change for precipitation of ∫ from å at constant T and P, let the precipitate take in NA atoms of component A and NB atoms of component B. The free energy change is ÎG = ÎG∫ + ÎGå = NA

µ∫A - µå

A + NB µ∫

B - µåB 11.35

If the total number of atoms in ∫ is N then NB = x∫N, NA = (1-x∫)N, and ÎG∫ = Ng∫, where g∫ is the free energy per atom in the ∫ phase and x∫ is the mole fraction of B in the ∫ phase. Equation 11.35 can then be re-written ÎG = N

g∫ - x∫ µå

B - µåA - µå

A = N{ }g∫ - [ ]gå - (x∫-xå)–µå 11.36 where –µ is the relative chemical potential, –µ = µB - µA, and xå is the solute concentration in the å phase. It follows that ÎGå∫

v = n∫{ }g∫ - [ ]gå + (x∫-xå)–µå 11.37 where n∫ is the number of atoms per unit volume of ∫.

g

xA B

å

∫

xå ∫xexåe

- Îg

∫x

Fig. 11.13: Free energy curves of å and ∫ phases. A supersaturated solu-

tion of composition xå is metastable with respect to precipitation of ∫ at any composition in the shaded region.


Page 336

The å phase is metastable with respect to the formation of a precipitate of ∫ that has any composition for which ÎGå∫

v is negative. The range of composition for which ÎGå∫

v < 0 has a simple graphical representation, which is shown in Fig. 11.13. Since

–µå =

∆gå

∆x xå 1.38

the value of the relative chemical potential, –µå, in the metastable å solution is the slope of the line that is tangent to the free energy curve gå(x) at concentration xå. Hence the term in brackets on the right-hand side of equation 11.37 is the equation of the tangent line. The value of the term in braces in equation 11.37 at a given value of x∫ [Îg(x∫)] is the negative of the difference between the value of g on the tangent line at x∫ and the value of g∫(x∫) on the ∫ free energy curve, as shown in the figure. The supersaturated solution is metastable with respect to the formation of a ∫ nucleus with any composition, x∫, at which the tangent line to gå at xå lies above the ∫ free energy curve. There is, however, a preferred composition for the ∫ nucleus; the composition that maximizes ÎGå∫

v minimizes ÎG*, the activation energy for nucleation. The composition that maximizes ÎGå∫

v can be found from the condition

d(ÎGå∫

v )dx∫ = 0 = n∫[–µ∫ - –µå] 11.39

Hence the composition, x∫, that maximizes the driving force for nucleation is the composition at which the relative chemical potential, –µ∫, in the ∫ phase is equal to the potential, –µå, in the parent å phase. Graphically, it is the composition, x∫, where the tangent line to the free energy curve of the ∫ phase, g∫, parallels the tangent line to gå at xå. This composition is indicated in Fig. 11.13; note that it is greater than the equilibrium composition, x∫

e , of the ∫ phase at the precipitation temperature. The kinetic equations that govern a nucleated transformation in a binary system are formally the same as in the case of the one-component system, though now the activation energy, QD, is the activation energy for interdiffusion of the B-component. The c-curve that governs the initiation of a precipitation reaction has the same qualitative form and interpretation as that which governs a structural transformation in a one-component system, and heterogeneous nucleation has the same qualitative effects on the kinetics and the morphology of the precipitation reaction. 11.6 NUCLEATION OF NON-EQUILIBRIUM STATES The phase that nucleates from a metastable parent phase need not be in the state that is in equilibrium at the given temperature and pressure. The nucleated phase may be a non-equilibrium state of the preferred product phase, or it may be a different phase of


Page 337

the system entirely. Examples of both phenomena can be inferred from the results of the previous section. First, the initial composition of a nucleus that forms in a binary solution is different from its equilibrium composition, as shown in Fig. 11.13. Second, the pressure within the nucleus is ordinarily much higher than that in the parent phase, as indicated by equations 11.23 and 11.26. If the most stable phase of the system depends on the pressure then the phase that nucleates from the metastable parent may be a high pressure phase rather than the phase that is in equilibrium under the ambient conditions. Kinetic factors also affect the state of the product phase. Generalizing equation 11.15, the rate of formation of nuclei whose state is designated by the index, i, is

•ni = ANiexp

- ÎG*

i + Qi

kT 11.40

where ΔG*i is the activation energy to form a critical nucleus in state (i), Qi is the activation energy for growth of the nucleus, and Ni is the density of sites at which the nucleus can form. The actual state of the nucleus is the one that maximizes the nucleation rate. We already saw an example of this in heterogeneous nucleation; when a heterogeneous site decreases ÎG* by an amount that is great enough to provide a large value of •n, the nuclei appear predominantly at heterogeneous sites. By the same reasoning, if the structure or composition of the nucleus can be changed so that •n is increased, the nuclei tend to have that structure or composition. This general principle is responsible for many important (and, sometimes, unanticipated) phenomena that influence nucleated phase transformations. We will specifically consider three in this section: congruent nucleation, in which the phase that initially forms in a solid solution has the same composition as the parent, coherent nucleation, in which the phase that nucleates shares the crystal lattice of the parent, and Ostwald nucleation, in which the phase that nucleates is a high-temperature, metastable phase. 11.6.1 Congruent nucleation It is relatively difficult to form a nucleus that has a chemical composition very different from that of the parent phase because the excess solute must be gathered by diffusion. It is, therefore, often easy to suppress precipitation from a multicomponent parent phase by cooling it rapidly. However, when the undercooling is sufficiently large an alternative transformation path is available; the parent phase loses equilibrium with re-spect to the nucleation of the product phase at the same composition. A transformation from a multicomponent parent phase to a product phase with the same composition is called a congruent transformation. The nucleation of a congruent phase is called congruent nucleation. Since congruent nucleation requires no long-range diffusion it is often kinetically preferred to incongruent nucleation when both are possible.


Page 338

The thermodynamic relations that lead to congruent nucleation are illustrated in Fig. 11.14, which shows the free energy curves for hypothetical å and ∫ phases at a given temperature. Let the system be in the å phase, and let its composition (x) be increased slowly. The å phase is preferred until the system enters the two-phase (å+∫) region at the composition xå

e . For compositions between xαe and x0 the system is out of equilibrium, but, since the å-phase is thermodynamically stable (its free energy curve is concave) and its free energy is below that of the ∫ phase at the same composition, it can only approach equilibrium by nucleating a B-rich ∫ phase. This situation continues until the composition passes the value x0 at which the å and ∫ free energy curves intersect. For compositions above x0, however, the ∫ free energy curve is lower and the ∫ phase can nucleate congruently.

g

xA B

å∫

∫xexåe x0

= å “ å+∫

= å “ ∫ “ å+∫

Fig. 11.14: Free energy curves for a binary system at given temperature showing composition ranges for incongruent and congruent nucleation.

A system that undergoes a congruent transformation may be left in a homo-geneous, metastable state, as in the transformation å “ ∫ in Fig. 11.14, or the transformation may continue to equilibrium through subsequent decomposition, as in the path å “ ∫ “ å+∫ shown in the figure.

g

xA B

å

LT2

1T

< 1T

å

L

–x

Fig. 11.15: Free energy curves of å and L phases at temperatures above

(T1) and below (T2) the congruent transformation temperature.


Page 339

One of the most common examples of congruent nucleation is in the rapid solidification of liquid solutions. When a liquid solution first enters a two-phase region on cooling, its free energy curve is related to that of the solid phase (call it å) as shown in the lower part of Fig. 11.15: the composition of the liquid solution is to the right of the intersection of the å and L free energy curves, and solidification can only occur through nucleation of A-rich å phase. However, as the temperature is lowered the increasing relative stability of the å solid has the consequence that the intersection of the å and L free energy curves is displaced to higher concentrations. Eventually the liquid becomes metastable with respect to a congruent transformation to å, as illustrated by the upper free energy curves in Fig. 11.15.

T

A x

å

L

å+L

å + ∫

L “ å

Fig. 11.16: A possible locus for the L“å congruent transformation line in a binary phase diagram.

The locus of the congruent L“å transition can be plotted on the phase diagram, and has a form like that shown in Fig. 11.16. For all compositions and temperatures to the left of the L“å line shown in the figure the free energy of the liquid solution is greater than that of an å solid solution of the same composition, and a direct L“å transformation can occur through congruent nucleation of the å phase. In the system illustrated in this figure a L“å congruent transformation can occur even at temperatures within the two-phase (å+∫) region if the liquid solution is cooled rapidly enough to suppress nucleation of the eutectic constituent. Because there is no need for long-range diffusion the activation energy for growth of a congruent nucleus, QD, is the activation energy for a direct jump across the interface, which is more like the activation energy for grain boundary diffusion than the much higher activation energy for bulk diffusion. Hence the nucleation of congruent particles is ordinarily predominant at temperatures where a congruent nucleus can form. A possible form for the kinetic transformation diagram of a system that can form congruent nuclei is shown in Fig. 11.17. The congruent transformation is not possible until the temperature is well below the equilibrium transformation temperature, but is then nucleated at a much more rapid rate.


Page 340

Congruent solidification is particularly desirable in the processing of alloys of materials that have very high melting points or very low diffusivities, since it is difficult to remove chemical heterogeneities that may develop during the equilibrium solidification of these materials. To achieve it, materials scientists have developed the technique of rapid solidification, in which the liquid is quenched very rapidly to suppress nucleation of the solid phase until the undercooling is high enough for congruent nuclei to form. It is usually only possible to achieve such high rates of cooling when the liquid has a very high surface-to-volume ratio so that heat can be extracted quickly. The conventional techniques of rapid solidification include pouring the liquid onto a chilled substrate that is rapidly moving, so that the liquid solidifies rapidly into a thin ribbon, or spraying the liquid in the form of small droplets into a chilled gas, so the droplets solidify rapidly into a fine powder.

T

ln(†)

incongruentnucleation

congruentnucleation

Fig. 11.17: A possible kinetic diagram for the transformation of a multi-component system that can transform congruently.

Another important application of rapid solidification is the formation of metallic glasses. As discussed in Chapter 5, to force a metallic material into a glassy structure it is necessary to cool it rapidly enough to suppress nucleation of the crystalline phase so that the lower, glass transition temperature, Tg, can be reached. Given the high rate of congruent nucleation it is very difficult to suppress nucleation by this mechanism. Hence the systems that are used to form metallic glasses tend to be solutions whose compositions are chosen so that the congruent nucleation temperature is very low.

T

A Bx

å ∫

L

å+L ∫+L

å + ∫

glasses


Page 341

Fig. 11.18: Illustration of the formation of glasses in a eutectic system. The congruent nucleation lines are shown. Glasses form in the low-temperature bay between where congruent nucleation is not possible.

A simple example is shown in Fig. 11.18. In order to form a glass in a system with a simple eutectic phase diagram it must be possible to cool the system to a temperature well below the crystallization temperature without inducing congruent nucleation. If the congruent nucleation lines for the two terminal solid solutions extrapolate to a temperature below Tg without crossing, as they do in the example shown, glasses can be formed by quenching into the low-temperature "bay" between the congruent nucleation lines. This is generally only possible when the two species have very low solubilities in one another. Otherwise the congruent nucleation lines intersect above Tg. Hence metallic glasses tend to be concentrated solutions of species that have low solubility, such as solutions between metals and metalloids like B, P or C. 11.6.2 Coherent nuclei When the parent phase is a solid the work to form the nucleus is also affected by its coherency, or crystallographic registry with the parent lattice. A phase that can nucleate coherently has a very low interfacial tension, ß, because its atom planes are in registry with those of the surrounding matrix, but has a significant strain energy distributed through its volume since it is necessarily distorted to maintain registry with the matrix. The coherent nucleus can only form when the undercooling is sufficiently high that ÎGå∫

v + Îeel < 0 11.41 where Îeel is the (positive) elastic strain energy per unit volume and ÎGå∫

v is the (negative) free energy change for transformation to the ∫ phase in its relaxed (unstrained) state. Assuming spherical shape, the work to form the coherent nucleus is, then,

ÎGH = 16πßc3

3(ÎGå∫v + Îeel)2

11.42

where ßc is the interfacial tension of the coherent nucleus. Since ßc is usually much less than ß, the interfacial tension of an incoherent nucleus, the work to form the coherent nu-cleus is less than that to form an incoherent nucleus when the undercooling is appreciable.


Page 342

T

ln(†)

incoherent nucleation

coherent nucleation

Fig. 11.19: Example transformation kinetics for a solid-solid transforma-tion in which the product phase can form coherently.

The kinetics of a solid-solid transformation that can be nucleated coherently are illustrated in Fig. 11.19, and are qualitatively similar to those in Fig. 11.17. If the transformation occurs at small undercooling the nucleus must be incoherent, since only the unstrained state of the product phase is stable. However, the large interfacial tension of the incoherent phase usually has the consequence that it is difficult to nucleate, and then only forms at heterogeneous sites such as grain boundaries. Coherent nucleation is not possible until the undercooling is appreciable, but is then kinetically preferred. In fact, most solid-solid transformations that occur in the bulk are initiated by coherent nuclei. The coherency of the product phase is ordinarily lost when the nucleus grows to significant size. The reason is that the strain energy due to the coherent strain increases with the volume of the nucleus, while the surface energy increases with the surface area. There is always a volume large enough that it is energetically favorable for the growing particle to replace its coherent interface with an incoherent one that relaxes the transformation strain. The larger the strain energy, that is, the larger the crystallographic mismatch between the parent and product phases, the smaller the size at which the incoherent interface is preferred. The mechanism by which coherency is lost is usually the accumulation of dislocations at the interface to destroy the registry between the two phases. Coherency strains may also affect the shape of the nucleus. It often happens that the strain energy of a coherent particle is minimized when the particle takes the shape of a thin plate that lies on a particular crystallographic plane of the matrix (called the habit plane). The tendency to assume a plate-like shape is, however, opposed by the surface energy since a platelet has a much higher surface area per unit volume than a more equiaxed figure. For this reason solid nuclei are found in a variety of shapes. Those that have very high coherency strains tend to nucleate as platelets with specific habit planes. Those that fit better in the parent matrix tend to nucleate as spheres, but even then the shape tends to evolve into a cubic and, eventually, a platelet morphology as the particle grows. Since many of the important properties of solid precipitates, such as their resistance to cutting by dislocations, are affected by their shapes, the prediction and


Page 343

control of precipitate shape is an important engineering consideration in designing and processing structural materials. 11.6.3 Nucleation of metastable phases: the Ostwald rule When the parent phase in a phase transformation is far away from equilibrium it is often the case that several product phases are possible. For example, when a condensed phase is deposited from the vapor on a chilled substrate the nucleated phase can be a liquid or a solid, and can often have any one of several solid structures. The phase that actually forms will be the one whose nucleation rate is the highest. In the context of classical nucleation theory this is the phase that minimizes the total activation energy, ÎG* + Q. Neglecting differences in the activation energy for growth, the preferred phase minimizes the nucleation barrier, ÎG*. In the case of homogeneous nucleation,

ÎGH = 16πß3

3(ÎGå∫v + Îeel)2

11.43

where we have included the elastic strain energy, Îeel, to allow for coherency. This ex-pression suggest two factors that may promote the formation of a metastable phase. The first factor is the interfacial tension. Phases that have low surface tension are strongly preferred. Since phases that have high entropy tend to have low interfacial ten-sions, there is a strong tendency for high temperature phases to nucleate preferentially at temperatures well below their equilibrium temperature range. The initial deposits on solid surfaces are often liquid at temperatures well below the melting point. The solid phases that form during vapor deposition on a cold substrate are often high temperature or amorphous structures (glassy phases have particularly low interfacial tensions). The second factor is the strain energy, Îeel. This factor is important when the product phase is coherent, as it often is when a solid nucleates in the bulk or when a solid is deposited onto a crystalline substrate. If there is a metastable phase that provides a better lattice match than the equilibrium phase, it is often preferred. There are many examples of precipitation reactions in the solid state that lead to metastable precipitate phases that are preferred because of their relatively small strain with the solid matrix. The hardening precipitates in most important aluminum alloys are examples. The various carbide precipitates in steel are even more common examples; the equilibrium precipitate in carbon steel is graphite, which has a very poor match with the iron lattice. The structures of deposited films are also often determined by lattice compatibility with the substrate. A particular product phase can also be formed catalytically by ensuring that there are heterogeneous nucleation sites that prefer that phase to any other. For example, if it is desirable to deposit a crystalline thin film whose surface is a particular crystallographic


Page 344

plane, this can often be done by depositing it on a substrate that provides a particularly good crystallographic match to that plane (the substrate is said to be epitaxial to the desired film). The good interfacial match lowers both ß and Îeel for a film with the desired orientation, and, hence, promotes its nucleation. These considerations suggest that the nucleation of metastable phases should be a common phenomenon. It is. The prevalence of metastable products was so well docu-mented by the early years of the present century that W. Ostwald was led to formulate it as a law of science. His Ostwald Rule states essentially that if a metastable phase can form in preference to the equilibrium one, it will. The Ostwald Rule is something of an overstatement, but it contains more than a grain of truth. 11.6.4 Sequential Nucleation in a Eutectic Some phase transformations involve two or more products that nucleate sequen-tially. A simple and important example is the solidification of a binary system that has a eutectic phase diagram. Three qualitatively different cases occur. They are illustrated by the three compositions shown in Fig. 11.20 First consider solidification at the composition x1. Assuming that the solid reaches equilibrium, it transforms in two discrete steps. First, the solid solution, å, nucleates out of the liquid, and grows until it consumes the liquid. The result will ordinarily be a polygranular å solid solution. At equilibrium, the å solution is homogeneous in composition. However, on further cooling the å solution enters the two-phase (å+∫) field, and becomes metastable with respect to precipitation of the ∫ phase. The final, two-phase state is obtained through a sequence of two independent reactions that involve independent nucleation events: å nucleates and grows at the expense of the liquid, and then, at a lower temperature, ∫ nucleates and grows out of å. The kinetics of these sequential reactions can be analyzed separately.

T

A Bx

å ∫

L

å+L ∫+L

å + ∫

x1 x2 x3

...

Fig. 11.20: Eutectic phase diagram showing solidification at three com-positions.


Page 345

A qualitatively different behavior occurs if the liquid phase has the eutectic composition, x3. In this case the liquid first becomes metastable with respect to a two-phase mixture of å and ∫. In order to form the eutectic constituent, both must nucleate.

...

Fig. 11.21: Schematic illustration of the nucleation of the eutectic con-stituent in a eutectic microstructure.

A mechanism for the nucleation of the eutectic constituent is illustrated in Fig. 11.21. Let one of the two solid phases, say å, nucleate first. Since the å nucleus is lean in solute (B), it rejects solute, so that a solute-rich "halo" forms around it, as shown in the Fig. 11.21a. The nucleation of the B-rich ∫-phase is relatively easy in this solute-rich halo, so a nucleus of ∫ forms on å. As the ∫ nucleus grows it rejects solvent (A), and develops an A-rich halo on the side away from the growing å nucleus. A second å-nucleus forms in this A-rich halo, creating a B-rich region to the far side of it. The sequential nucleation eventually leads to a composite nucleus of the eutectic constituent: a laminate of the å and ∫ phases like that shown in Fig. 11.21d. The eutectic nucleus shown in Fig. 11.21d can grow vertically into the liquid by the process described in Chapter 9; å and ∫ lamellae grow side-by-side at essentially constant composition as solute is redistributed over the short distance between the head of the å lamella and the heads of the ∫ lamellae that border it. The eutectic nucleus extends itself sideways by alternate nucleation of the å and ∫ phases, creating the colonies that appear in the eutectic microstructure described in Chapter 9. While the nucleation of a eutectic constituent is a bit more complicated than the nucleation of a single product phase, it is thermally activated and is described by kinetic equations that have the same form as those for the simpler å“∫ transformation. Hence the initiation time for a eutectic transformation has a typical c-curve dependence on the undercooling. The solidification of a liquid solution that has an intermediate composition, such as the composition x2 in Fig. 11.20, involves two sequential reactions. When this solution is cooled into the two-phase (å+L) region, at Tå, it becomes metastable with respect to the nucleation of å solid solution. However, the nucleation of å cannot complete the reaction. Even if the liquid solidifies to the extent needed for equilibrium, there is residual liquid when the eutectic temperature, TE, is reached. Hence the eutectic constituent must also nucleate to complete the transformation. The nucleation of the eutectic is facilitated in this case, however, because islands of å-phase are already present. The eutectic reaction simply continues the growth of the


Page 346

å particles that are already present by nucleating ∫-phase heterogeneously on the pre-existing (proeutectic) å. Hence the nucleation of the eutectic constituent is relatively easy.

T

ln(†)åT

ET å

å + ∫

...

Fig. 11.22: Initiation kinetics for nucleation on cooling a liquid of com-position x2 in Fig. 11.21. Proeutectic å nucleates below Tå. The eutectic constituent (å+∫) nucleates below TE.

The initiation times of the sequential nucleation reactions behave as shown in Fig. 11.22. When the temperature falls below Tå the liquid is metastable with respect to the nucleation and growth of proeutectic å. The nucleation of å obeys the usual kinetics of L“å transformations. When the system falls below TE it is metastable with respect to the nucleation and growth of the eutectic constituent. The eutectic constituent nucleates heterogeneously on the proeutectic å, and is, hence, relatively easy to form. However, some undercooling is still required to initiate the heterogeneous nucleation of ∫ and form the eutectic.

weight percent carbon

1000

800

600

0.5 1.0 1.5 2.0

©

å + ©© + carbide

å + carbide

åT (ºC)

...

Fig. 11.23: A portion of the Fe-C phase diagram. If a sample of the composition shown by the dashed line is cooled, it first precipitates å, then the eutectoid constituent, å + Fe3C.

The behavior of a system that undergoes a eutectoid reaction (solid solution trans-forms to a mixture of two solid phases, © “ å+∫) is entirely similar; the only difference is that the high-temperature phase is a solid solution rather than a liquid. For example,


Page 347

the Fe-C phase diagram appears as shown in Fig. 11.23. If a sample of ©-phase with the composition shown by the dashed line is cooled to low temperature, the initiation kinetics for the proeutectoid, © “ å reaction and the eutectoid reaction, © “ å + Fe3C, combine as shown in Fig. 11.22. 11.7 RECRYSTALLIZATION In the nucleated reactions we have discussed to this point the product phase of the transformation is a different phase from the parent. However, there is one important case in which the parent and the product are examples of the same phase: the recrystallization reaction that occurs when a deformed metal is heated to a sufficiently high temperature. Recrystallization is a common phenomenon, and is the reaction that is ordinarily used to control the grain size of a structural alloy. The thermodynamic driving force for recrystallization is the free energy of non-equilibrium defects in the deformed material. As we shall discuss when we come to me-chanical properties, the plastic deformation of a material introduces a high density of dislocations and other defects. These defects have energy, and hence add to the free energy of the material. While there are mechanisms, called recovery processes, that gradually eliminate them, the rate of recovery of non-equilibrium defects is very slow at ordinary temperatures. The alternative mechanism is recrystallization: the nucleation and growth of new, relatively defect-free grains. The actual mechanism of nucleation of a defect-free grain in a deformed material can be subtle. The dislocation network in a deformed material is inhomogeneous and evolves with time at high temperature, and there are always small volumes of essentially defect-free material that may act as pre-existing embryos of defect-free phase. However, we can obtain a qualitative understanding of recrystallization by assuming that the nucleation of recrystallized grains is mechanistically the same as the nucleation of grains of a new phase. Since the material is solid and relatively incompressible, the driving force for nucleation is approximately, ÎGåå

v , the free energy per unit volume of the defects in the deformed phase. The surface tension is ßåå, the interfacial tension of a grain boundary in the material. The activation energy for homogeneous nucleation of defect-free grains is, hence,

ÎGH = 16π(ßåå)3

3(ÎGååv )2

11.44

Heterogeneous nucleation of recrystallized grains is both possible and likely, since the deformed solid is polygranular and also has a dense distribution of internal clusters of defects that may act as heterogeneous nucleation sites. Hence the activation barrier, ÎG*, that opposes the nucleation of a recrystallized grain should be less than ÎGH by some factor:


Page 348

ÎG*n = fpÎGH (f < 1) 11.45

where fp is the catalytic factor of the dominant heterogeneous sites, which are denoted by the index, p. The elementary growth step for a nucleus of recrystallized grain is the migration of an atom across a grain boundary. The nucleation rate for recrystallization process should, hence, obey a rate equation of the familiar form

•n = ANpexp

- ÎG*

p + QD

kT 11.46

where Np is the number of heterogeneous sites per unit volume. While the rate equation that governs the nucleation of recrystallized grains in this model, eq. 11.46, is formally the same as that which governs the other transformation processes we have studied, it is qualitatively different in two respects. First, the activation barrier that opposes the nucleation of recrystallized grains, ÎG*

n , decreases with the degree of prior deformation, but is nearly independent of the temperature, since the free energy of a dislocation changes very little with temperature. The second im-portant difference is that recrystallization is induced by heating rather than cooling, and, hence, cannot be suppressed. From equation 11.46, the initiation time for recrystallization can be written in the simple form

ln(†) = - ln(•n) = - ln(ANp) + QRkT 11.47

where QR, the activation energy for recrystallization, is QR = ÎG*

p + QD 11.48 It is relatively insensitive to the temperature, but decreases with the degree of prior defor-mation. If the logarithm of the initiation time is plotted as a function of 1/kT, as in Fig. 11.24, the result is a straight line of slope QR. If the initiation rate is plotted for various degrees of prior deformation, the result is a series of straight lines whose slope decreases with prior deformation.


Page 349

decreasing prior deformation

ln(†)

1/kT

QR

...

Fig. 11.24: The initiation time for recrystallization as a function of tem-perature and degree of prior deformation.

The time to initiate a recrystallization reaction in a typical material is prohibitively long unless the temperature is a significant fraction of the melting point (T greater than about 0.4 Tm), and unless the prior deformation is substantial. The temperature at which recrystallization initiates in a given, reasonably short time is the recrystallization temperature. The recrystallization temperature decreases with the extent of prior deformation. 11.8 GROWTH 11.8.1 Primary growth and coarsening After nucleation, grains of the product phase grow to complete the transformation and establish the final microstructure. This happens in two distinct stages. First, the product phase grows at the expense of the parent. In this stage, primary growth, the prod-uct phase grows relatively quickly until it consumes the parent or, in the case of a precipitation reaction, reaches a volume fraction close to its equilibrium value. At the end of primary growth, however, the product is usually in a fine-grained microstructure that may also be chemically inhomogeneous. In the second stage, secondary growth, or coarsening, the grains or particles of product phase grow at the expense of one another to reconfigure themselves into a more stable microstructure. Two things must happen for the product phase to grow. The material that is to be incorporated into the growing particle must be delivered to the interface, and it must cross the interface to enter the product phase. Either step may govern the kinetics of growth. We therefore distinguish two types of growth on the basis of the slower, or rate-limiting step: diffusion-controlled growth and interface-controlled growth. The diffusional processes that may control growth are the diffusion of mass and heat to or from the growing interface. A first-order phase transformation generates latent heat. If the heat of the reaction is not transported away from the interface it accumulates there and raises the temperature (if it is positive) until the reaction stops. Unless the


Page 350

transformation is congruent, chemical species must also diffuse to (or from) the growing interface to produce the composition of the product phase. Since mass diffuses more slowly than heat, mass diffusion is usually the rate-limiting step for primary growth in multicomponent systems. The material that diffuses to the surface of the growing phase must also be incorporated into it. When the growing phase is congruent and the interface is rough and irregular on the atomic scale this step requires no more than a diffusional jump across the interface. The jump across the interface is, nonetheless, often the rate-controlling step for congruent transformations in solids where the latent heat of the transformation is small and the thermal conductivity is high. Even when the growing particle has a composition very different from that of the matrix, the interface still controls the rate of growth if its nature is such that the addition of atoms is difficult. The mechanisms of growth control the grain morphology of the product phase, and may also control the chemical distributions within them. Chemical heterogeneities are particularly common in as-solidified materials. In the usual case, solidification initiates when the liquid is cooled into a two-phase (solid + liquid) region, and proceeds through the growth of a solid whose composition is initially very different from that of the liquid. Solute is rejected into the liquid and accumulates there. Since the rate of solidification is much more rapid that the rate of diffusion in the solid state, the composition of the solid cannot adjust itself during solidification, so the final product is a chemically heterogeneous solid in which the last material to solidify is very rich in solute. In the following sections we discuss the features of growth in more detail. We shall first consider primary growth, and first examine the case of interface control, because it is the simplest. We then discuss diffusion-controlled primary growth, chemical segregation during primary growth, and secondary growth (coarsening). 11.8.2 Time-temperature-transformation (TTT) curves The net rate of a phase transformation reflects the kinetics of growth as well as the kinetics of nucleation. It is customary to represent the overall rate graphically by plotting the time required to achieve various fractions of transformation as a logarithmic function of the temperature. A hypothetical example is given in Fig. 11.25, which shows lines for 1%, 50% and 99% transformation as a function of the undercooling. When such a diagram refers to transformation under isothermal conditions, which can ordinarily be achieved by quenching the sample and then re-heating it to the appropriate temperature, it is called a TTT-diagram (time-temperature-transformation). In practice the first curve on a TTT diagram is drawn at some small fraction, since the very first transformation is difficult to detect. For the same reason the last curve is drawn at a fraction near, but less than 100% transformation.


Page 351

ln(t)

ÎT

1%

50%

99%

...

Fig. 11.25: A TTT-curve for a hypothetical nucleated transformation that occurs on cooling.

The extent of transformation in a sample that is cooled continuously through the phase transformation is also of interest. A diagram like that in Fig. 11.25 that plots the time to achieve a given fractional reaction on continuous cooling is called a CCT diagram (continuous cooling transformation). The two kinds of diagrams have a qualitatively similar appearance, but differ in detail since the incremental progress of a nucleated transformation as it is cooled through a temperature, T, depends on events that happened at higher temperature as well as on those that transpire at T. 11.9 INTERFACE-CONTROLLED GROWTH 11.9.1 Isotropic growth of a congruent phase The simplest example of interface-controlled growth is the growth of a solid phase that has the same composition as the matrix, has an incoherent interface, and produces a latent heat of transformation that is small enough that it is conducted away to preserve an essentially constant temperature. While this combination of conditions may seem stringent, it pertains, at least roughly, to a large number of solid-solid and liquid-solid transformations, including many structural transformations in one-component solids, recrystallization reactions, massive (congruent) transformations in solid solutions, and eutectic solidification. In the simplest case the temperature and composition are constant and the rate-limiting step for growth is the motion of an atom across the interface from the ambient to the growing phase, as illustrated in Fig. 11.26. If the interface is incoherent and the sur-face of the growing phase is rough on the atomic scale, then the interface has essentially the same structure over its whole boundary with the fluid, and the rate of growth is nearly isotropic (the same in all directions).


Page 352

Îg*

Îgå∫

x

g

å

∫

...

Fig. 11.26: The free energy change during the motion of an atom across the interface of a growing particle.

The rate of growth can be understood from the following simple model. Let an atom cross the interface. Its free energy changes along a line connecting its initial and final positions roughly as shown in Fig. 11.26. This curve governs atom jumps across the interface in both directions. To cross the boundary from the å ambient to the ∫ particle an atom must overcome the activation barrier, Îg*; to cross from the ∫ particle to the å ambient an atom must overcome the higher barrier, Îg* + Îgå∫, where Îgå∫ is the difference in the free energy per atom between the two phases. If there are n atoms per unit area of interface, the net flux of atoms across the interface into the ∫ particle is J = Jå∫ - J∫å

= nˆ

exp

- Îg*kT - exp

- Îg* + Îgå∫

kT

= nˆ exp

- Îg*kT

1 - exp

- Îgå∫

kT 11.49

where ˆ is the frequency of jumps attempts. Assuming that Îgå∫ is small compared to kT, we can use the approximation ex « 1 - x to evaluate the bracket in eq. 11.49. The result is

J = nˆ

Îgå∫

kT e-

Îg*kT 11.50

The velocity of the interface (the linear growth rate) is

Ë = Jv1/3

n = nˆv1/3

Îgå∫

kT e-

Îg*kT 11.51

where v is the atomic volume.


Page 353

Equation 11.51 suggests that the congruent growth of a ∫ particle is thermally activated, with an activation energy equal to Îg*, the activation energy for migration from å to ∫, and a pre-exponential factor that increases with the thermodynamic driving force for the transformation, Îgå∫. If the transformation happens on cooling, then Îgå∫ vanishes at the transformation temperature, T0, and increases with the undercooling. Since the exponential term decreases monotonically as T decreases, the growth rate reaches a maximum at some moderate undercooling, and vanishes when the temperature is very small. If the transformation happens on heating, for example, the recrystallization reaction, then the rate of growth increases monotonically with the superheating. The microstructures that result from isotropic growth depend on the type of nucle-ation that predominates. This is true even if the nuclei are distributed through the volume. If nuclei form throughout the volume at a roughly constant rate, then the microstructure at the end of primary growth is obtained by the impingement of grains that initiated at different times. The microstructure is an equiaxed, polygranular microstructure that contains grains with a variety of sizes, with curved grain boundaries. Such a microstructure is found, for example, in recrystallized iron. In a second common case nuclei form throughout the volume, but at discrete, heterogeneous sites that are exhausted very early in the transformation. In this case the microstructure contains equiaxed grains with almost straight grain boundaries. Such a microstructure is sometimes seen in recrystallized FCC metals, such as high-Ni stainless steels. If the nuclei form heterogeneously on a surface then the microstructure contains parallel, columnar grains that grow out from the surface. Such a microstructure is often found in solid films, in metal ingots that were cast by pouring into a chilled mold, and in weldments in metals. 11.9.2 Interface control at stable surfaces A second class of growth processes that are interface-controlled includes the growth of crystals whose surfaces are relatively immobile because of the difficulty of adding atoms onto them. The processes that behave in this way generally have two characteristics: the growing crystal has a highly anisotropic surface tension, and the connection between the crystal and its ambient is such that the crystal can develop a form in which it is bounded by low-energy planes. The reactions that satisfy these constraints include the growth of compounds or crystals with complex structures from the liquid or vapor, the precipitation of crystalline compounds from aqueous solution, and, at least in some cases, the growth of coherent precipitates in solids. The structural reasons for interface control The reason that these reactions are interface-controlled is the structure of the growing surface. As we discussed in Chapter 4, when a solid has an anisotropic surface tension its surfaces tend to be made up of segments of low-energy surface (ledges) that are terminated by steps. Such surfaces grow by the addition of atoms at the steps, as illustrated in Fig. 11.27a. This mechanism causes the ledges to grow relatively rapidly


Page 354

over the surface, as shown in Fig. 11.27b, until the steps disappear at the boundary and a low-energy surface is presented to the ambient.

(111)(hkl)

...

Fig. 11.27: The structure of a stepped surface, and its growth to achieve a low-energy interface.

If the growing crystal has a highly anisotropic surface tension, the growth of a low-energy surface is relatively difficult. The low-energy surface moves in the direction normal to itself through the lateral growth of ledges, as shown in the figure. Its growth terminates when all of the existing ledges have exhausted themselves at the boundary, and can only continue if new ledges form. However, in the absence of a catalytic mechanism, the formation of a new ledge requires the nucleation of a disc of material on the plane crystal surface. The nucleation of a disc of new plane is opposed by the line tension of the step on the periphery of the disc, and is, hence, a thermally activated nucleation process that is difficult to accomplish unless the thermodynamic driving force is high. Hence materials with strongly anisotropic surface tension tend to develop polygonal shapes that are bounded by low-energy surface planes. When they have achieved these shapes they grow at the rate at which new ledges can nucleate on the low-energy surfaces. Unless the undercooling is high, the normal growth rate is slow. This phenomenon is responsible for the fact that oxide minerals and other materials of complex crystal structure often appear "crystalline" in their natural form, while normal metals and elemental semiconductors do not. Materials with complex crystal structures tend to have anisotropic surface tension, since the atom density in the surface plane depends strongly on the indices of the plane. The anisotropy is particularly strong in compounds with partly ionic bonding, a class that includes most oxide minerals, since only certain crystal planes are electrically neutral (contain equal numbers of positive and negative charges). Hence many minerals grow from solution as polygonal bodies whose macroscopic shapes reflect their atomic structure, and their growth is interface-controlled unless the driving force for the transformation is very high. On the other hand, metals tend to have close-packed or nearly close-packed structures whose interfacial tensions are not strongly dependent on orientation. They normally form in irregular shapes that do not reflect their inherent crystallinity. (We are so accustomed to


Page 355

associate crystallinity with macroscopic form that most undergraduates are as surprised as I was to learn that metals are crystalline at all.) However, there are at least two cases in which metallic phases exhibit a crystalline appearance and an interface-controlled growth. One is when the metal has a complex crystal structure. An example is antimony (Sb), which has a rhombohedral structure in which pairs of Sb atoms are closely bound to one another. An Sb-rich phase can be crystallized out of a Pb-Sb solution (as it is in one of the E45 labs) by choosing a composition on the Sb-rich side of the eutectic. The solidified Sb crystals are nearly cubes. A second example is found in the shapes of coherent precipitates in solids. These often appear as discs or plates with flat interfaces whose habit plane parallels the plane that provides the best match between the crystal structures of the precipitate and the matrix. Moreover, these coherent interfaces are relatively immobile during growth. The platelet shapes of these precipitates reflect two factors, minimization of the interfacial tension between the precipitate and matrix and simultaneous minimization of the elastic energy due to coherency, that ordinarily promote the same shape and growth behavior. The screw-dislocation mechanism of growth In order for a faceted crystal to grow under small supersaturation there must be a mechanism that continually produces mobile ledges on its low-energy surfaces. Several such mechanisms have been proposed; all of them involve dislocations that penetrate the free surface. The simplest and best documented is the screw-dislocation mechanism that is diagrammed in Fig. 11.28.

...

Fig. 11.28: The screw-dislocation mechanism of crystal growth. The left figure shows a growth spiral around an emerging screw dislo-cation. The right is a side view showing growth steps.

Let a screw dislocation intersect the free surface of a growing crystal, as shown in Fig. 11.28a. Recalling the discussion of screw dislocations in Chapter 4, the screw dislo-cation is one whose Burgers' vector, b, parallels the dislocation line. The displacement introduced by a screw dislocation has the consequence that a given plane of atoms spirals around the dislocation; since b parallels the dislocation, a Burgers' circuit around the simplest screw dislocation finishes one atom plane above its starting point. It follows that a screw dislocation introduces a vertical step when it emerges at a free surface. If


Page 356

this step grows laterally across the surface, as drawn in Fig. 11.28a, it develops a spiral step around the dislocation core that is self-perpetuating; each time the growth of the step causes the spiral to complete a revolution around the dislocation core, the dislocation advances one atom plane perpendicular to the interface so that the step is maintained. The surface near the emerging screw dislocation has the shape shown in cross-section in Fig. 11.28b. The spiral step is preserved as the growing ledge rotates around the dislocation, and the interface grows by one atom plane with each revolution. The screw-dislocation mechanism of crystal growth permits a low-energy surface to grow normal to itself without successive nucleation events on the plane, and hence permits the growth of a faceted crystal under a relatively small thermodynamic driving force. The rate of growth is relatively slow, so almost isothermal conditions are maintained. An examination of the faceted surface of a growing crystal ordinarily reveals a number of emerging screw dislocations whose growth spirals overlap to maintain the growth rate. Instabilities in interface-controlled growth: whiskers Under certain conditions of interface control a growing crystal develops an anomalously high growth rate in one direction leading to the formation of a long, thin single-crystal "whisker". Whiskers are most commonly observed during deposition from a vapor or precipitation from a liquid under conditions such that nucleation is confined to a surface. The surface fixes the base of the whisker, which then grows out from it into the ambient. Whiskers originated as a scientific curiosity. They are often free of dislocations, and hence have exceptionally high resistance to plastic deformation. More recently, whiskers have become important in materials technology, in both the positive and negative sense. In the positive sense, high-strength whiskers of materials like SiC are grown commercially and harvested for use as reinforcements in metal-matrix composite materials (SiC-reinforced aluminum). In the negative sense, whiskers sometimes develop spontaneously on aluminum conducting lines in microelectronic devices, and cause electrical shorts by bridging conductors. Current understanding suggests that several different mechanisms can lead to the formation of whiskers. All are interface controlled. Two of the best documented are illustrated in Fig. 11.29. One mechanism of whisker growth is the screw-dislocation mechanism shown on the left in Fig. 11.29. Let a small crystallite contain a single screw dislocation that is ori-ented perpendicular to one face. The face that contains the screw dislocation can grow. The perpendicular faces cannot grow easily. If the screw dislocation is oriented out of the plane of the substrate a thin whisker may form and grow by the addition of atoms at its stepped tip. Whiskers that grow in this way are nearly dislocation-free, and can actually become dislocation-free if the central screw dislocation eventually glides to the free surface. This is, therefore, a useful mechanism for creating very strong, almost perfect whiskers.


Page 357

strained solid

screw dislocation

surface diffusionto pinned base

Fig. 11.29: Two mechanisms of whisker growth: growth at the tip by a screw dislocation and growth at the base by surface diffusion to a pinned droplet on a strained surface.

A second mechanism is shown on the left in Fig. 11.29, and appears to be pertinent to the growth of whiskers from conducting lines in microelectronic devices. Let the substrate be strained mechanically, for example, let it be a thin film that is strained by a lattice mismatch with the body on which it rests. It can be shown that the strained solid is thermodynamically unstable with respect to the nucleation and growth of an unstrained mound of material ("hillock") on its surface. However, suppose that this hillock cannot spread over the surface, most likely because of pinning by surface debris. Then the only way the nucleated crystal can grow is out of the surface. Since the surface crystal is unstrained, the free energy is lowered if atoms of the strained substrate migrate to it, causing a whisker to be pushed out from the bottom as shown in the figure. 11.10 DIFFUSION-CONTROLLED GROWTH In many important phase transformations the growth of the product phase is con-trolled by the rate at which the heat of reaction can diffuse away from the interface or the rate at which the chemical species that preferentially congregate in the new phase can dif-fuse to it. Common and important examples include the solidification of typical metals or semiconductors from the melt and the growth of common precipitates in the solid state. The simplest example is thermal control during congruent solidification from a liquid, so we shall consider this case first. 11.10.1 Growth controlled by thermal diffusion Consider the freezing of a solid at constant composition, and assume that the interface is sufficiently mobile that the solid can grow as rapidly as other conditions permit. The velocity of the interface is limited by the need to eliminate the latent heat of solidification. The temperature at the growing interface The positive latent heat of solidification has the consequence that, at least shortly after the transformation begins, the temperature at the growing interface is Tm, the


Page 358

melting point of the material. To see this note that if the interface temperature were less than Tm the solid could grow more rapidly by using its own heat capacity to absorb the latent heat. Hence the rate of growth increases until T = Tm. But the interfacial temperature cannot exceed Tm. If it were to do so the solid would melt. The rate of growth is hence set by the requirement that T = Tm at the growing interface. To maintain T = Tm the latent heat must be removed from the interface as fast as it is generated. The primary mechanism is thermal conduction down a temperature gradient that leads away from the interface. In most cases of solidification, whether the casting of metal ingots or the growing of semiconductor crystals, the growing solid is attached to an external wall or fixture that can serve as a heat sink. Hence there is a negative temperature gradient from the interface into the solid. On the other hand, the temperature gradient into the liquid may be either positive or negative. It is positive, for example, when hot, molten metal is poured into a chilled mold to cast and ingot. In this case the liquid is a source of heat, and the solid can only grow as fast as the sum of the latent heat of the transformation and the heat that flows in from the liquid can be conducted away. The temperature gradient into the liquid is negative, for example, when the liquid is supercooled to below its melting point to induce nucleation. In this case the latent heat is conducted away by both the solid and the liquid. Thermal instability: dendrites Solidification is much more rapid when the liquid is supercooled. However, the solidification front is also unstable. To see this, consider the example shown in Fig. 11.30, where the liquid temperature is below the melting point.

T

x

å L

å

...

Fig. 11.30: Initiation of a dendrite during solidification of a supercooled liquid.

The temperature profile through the interface of a solid that is growing into a supercooled liquid is illustrated on the left in Fig. 11.30. The maximum temperature is Tm, at the solid-liquid interface. Now suppose that a small protrusion appears on the solid surface. It penetrates into the liquid. The liquid that is sampled by the material at the tip of this protrusion is cooler than that sensed by the trailing surface of the solidifying solid, so the temperature gradient into the liquid from the tip of the protrusion is greater than that at the trailing solid surface. Moreover, heat is conducted radially away from the protrusion into the liquid, while the heat flow from the trailing surface is one-dimensional. Both factors cause a higher growth rate at the head of the protrusion,


Page 359

so the protrusion becomes more and more pronounced. The interface is unstable with respect to the formation of long, thin dendrites that reach out into the liquid. As dendrites develop during unstable solidification they extend and branch until the solidification front comes to resemble a forest of Christmas trees, as illustrated in Fig. 11.31. The dendrites branch for the same reason they formed in the first place. The pe-ripheral surface of a dendrite is an essentially plane surface that thickens by planar growth. It is unstable with respect to formation of a secondary dendrite, just as the original interface was. Hence dendrites branch and branch again. (You may be most familiar with dendrites from the shapes of snowflakes, which are dendritic structures of ice that form by deposition of ice from cold air supersaturated with water. The old folk saying that "no two snowflakes are alike" is a semi-accurate description of the statistical nature of dendrite growth and the almost infinite variety of the branching patterns that may develop. Is the folklore true? I don't know, and I am not anxious to examine all the snowflakes that there have ever been to find out.)

...

Fig. 11.31: Dendrites growing into a supercooled liquid. It is important to recognize that the growth of thermal dendrites, like those dis-cussed above, is peculiar to solidification into a supercooled liquid. If the liquid is above its melting point then a nascient dendrite that protrudes from the solid surface encounters temperatures above the melting point of the solid, and remelts. When the temperature of the liquid is above Tm, a congruent solid can form a stable, planar growth front. This fact is used, for example, in the growth of large single crystals from the melt. The melt must be held at a temperature above its melting point and heat extracted through the solidifying solid; otherwise the solidifying surface is unstable with respect to the formation of dendrites. 11.10.2 Growth controlled by chemical diffusion When the product phase has a different composition from the parent and the inter-face is mobile the rate of growth is limited by the rate of atom transport to the solid surface. As a simple example, consider the growth of a solid precipitate of phase ∫, of composition, c∫, in a matrix of phase, å, whose mean composition is cå. Assume that the interface is plane and that growth occurs at essentially constant temperature (as it ordinarily does when å is solid since the latent heat of precipitation in a solid is small).


Page 360

c

∫ å

cå

ecå

c∫

x ...

Fig. 11.31: Composition profile near the plane surface of a precipitate growing under diffusion control.

The composition profile across the growing interface is illustrated in Fig. 11.31. The solute concentration in the ∫ precipitate is c∫. The concentration in the å phase well away from the interface is cå. Equilibrium at the interface requires that the solute concentration in the å phase there be cå

e , the concentration of solute in the å phase when it is in equilibrium with ∫ at the interface temperature, T. The concentration gradient in the å phase provides the driving force for the diffusion of solute to the solid surface. The kinetics of diffusional growth We can find an approximate relation for the diffusional growth of ∫ from the fol-lowing simple argument. If the solute diffusivity is D, the mean diffusion distance for the solute in time, t, is, by the arguments of Chapter 10, –x = ´x2¨ = 2Dt 11.52 To approximate the width of the precipitate, xp, at time, t, assume that all of the excess solute (B) atoms that are located within a mean diffusion distance of the center of the pre-cipitate have incorporated into it, leaving behind an layer of solution that has the equilib-rium composition, cå

e , while the remainder of the solution is unaffected by the growing interface and has he composition cå. The total number of B-atoms in a slab of solution of width 2Dt and unit cross-sectional area is NB = ncå 2Dt 11.53 where n is the number of atoms per unit volume. The number of B-atoms in a composite of a precipitate of width, xp, and an equilibrium solution of width, xp - 2Dt is NB = nc∫xp + ncå

e [ ]2Dt - xp 11.54 Equating the two expressions gives


Page 361

xp = cå - cå

e

c∫ - cåe

2Dt 11.55

which suggests that the growth rate, dxp/dt, is given by a relation of the form

dxpdt = A(cå - cå

e )Dt 11.56

The linear growth rate increases with the supersaturation of the å solid solution, (cå - cå

e ), and decreases with the square root of the time. While this derivation is based on an oversimplified model, its features are the same as those obtained by more accurate theoretical analyses of the linear growth rate: the interface velocity increases with the supersaturation (though not always linearly), and decreases with the time as D/t . Constitutional supercooling It is apparent from Fig. 11.31 that the planar interface of the growing precipitate treated here is unstable with respect to the formation of dendrites, even when the temperature is constant. If a small protrusion forms on the surface of the growing precipitate its tip encounters solution that is richer in solute than the equilibrium value. Hence the concentration gradient is larger at the tip of the protrusion. Moreover, the composition gradient at the tip is radial, so the total flux to the protrusion is enhanced still further over the flux to the plane interface. Consequently, the protrusion extends into a dendrite, and the planar growth front breaks up into dendritic elements. A transformation product that differs in composition from its parent phase is even less stable with respect to dendrite formation than is a congruent product. Despite this instability, precipitate particles in solids do not ordinarily produce dendrites until they grow to moderate size. The reason is that there is a minimum lateral dimension to a growing protrusion, just as there is a minimum size for a stable nucleus. Small particles are not large enough to generate dendrites; large particles often do. When the growing crystal is large it is still possible to stabilize its interface by ad-justing the temperature gradient to compensate for the concentration-induced instability. To see how this is done in the particular case of the solidification of a solid solution, consider the segment of phase diagram shown in Fig. 11.32. Assume that ∫ is solidified out of the liquid at the temperature, T, that is indicated by the dashed line in the phase diagram. The equilibrium concentration of B in the liquid is cL

e , given by the tie-line that is drawn in the phase diagram. Assume that the liquid is heated with respect to the solid so that its temperature increases with distance from the interface. If the solid were congruent, a positive temperature gradient would stabilize its surface against the formation of dendrites.


Page 362

However, when the solid is incongruent the temperature gradient into the liquid must not only be positive, it must also be fairly steep.

∫

å + ∫

∫ + L

L cLe(T)

T

c B ...

Fig. 11.32: B-rich portion of the eutectic phase diagram of a hypothetical binary system showing a ∫-L tie lie across the two-phase re-gion and the increased solubility of B in the liquid as the tem-perature is increased further.

To appreciate this, return to the phase diagram shown in Fig. 11.32. According to the phase diagram, increasing the temperature raises the solubility of B in the liquid phase. The solubility is given as a function of T by the equation cL

e = cLe (T) 11.57

that defines the boundary of the L+∫ region on the liquid side. Now suppose that the temperature of the liquid varies as T(x) near the growing solid surface. Substituting this function into equation 11.57 yields the solubility of B in the liquid phase as a function of distance from the interface: cL

e (x) = cLe [T(x)] 11.58

The actual composition of the liquid as a function of distance from the interface is cL(x), and is determined by the diffusion profile. If the temperature rise near the interface is sufficiently rapid that cL

e (x) ≥ cL(x) 11.59 everywhere, then a protrusion from the solid surface encounters liquid that is unsaturated with solute, and dissolves. Hence if the temperature and composition profiles are such that the inequality 11.59 is satisfied, the interface is stable with respect to the formation of protrusions. On the other hand, if, for x near the interface, cL

e (x) < cL(x) 11.60


Page 363

a protrusion penetrates a supersaturated solution and grows spontaneously, at least for the distance over which the inequality 11.60 is satisfied. When 11.60 is true the solution is said to be constitutionally supercooled.

c

∫ å

cå

ecå

c∫

x

(ÂT )1ÂT2

ÂT3

( )

( )

cc

c

...

Fig. 11.33: Composition profile near a growing interface with the solubility profiles associated with three temperature gradients. The regions of constitutional supercooling for the gradients ÂT2 and ÂT3 are shaded.

The relations that govern the geometric stability of the growing interface are illus-trated in Fig. 11.33. Three values of the temperature gradient into the liquid are included: a steep gradient that stabilizes the interface, and two more shallow gradients that lead to various degrees of constitutional supercooling. Equation 11.59 and Fig. 11.33 show that a growing interface is stabilized by a steep temperature gradient. However, it is difficult to maintain such a gradient more than temporarily during normal solidification. Since the thermal conductivity of a condensed phase is much larger than its diffusivity, a steep temperature gradient is quickly dissipated into a shallow one. For example, we can solidify a solution by pouring it into a chilled mold, so that heat is extracted through the mold wall. A steep temperature gradient is established immediately after pouring, and the solidification front from the wall is initially plane and stable. However, the temperature gradient quickly softens, generating the succession of conditions indicated by the gradients ÂT1 to ÂT3 in Fig. 11.33. After some growth the sample is constitutionally supercooled for a distance ahead of the growing interface that increases with time. Hence the growing interface becomes increasingly unstable as it grows. Dendrites almost inevitably appear. However, dendrite formation can be suppressed in carefully controlled solidifica-tion reactions that use special furnaces to maintain a temperature gradient at the interface. A technologically important example is the growth of large crystals for microelectronic devices. These must be both structurally and chemically uniform, and hence cannot tolerate dendrite formation at the interface. In the usual methods for crystal growth from the melt the growing interface is stabilized by heating the melt in a controlled manner to


Page 364

establish a sharp temperature gradient at the interface that is maintained during growth. Special gradient furnaces have been designed for this purpose. 11.11 CHEMICAL SEGREGATION DURING GROWTH One of the major practical problems encountered in the processing of multicomponent systems is the development of undesirable chemical segregations during diffusional transformations. The clearest example occurs in the solidification of a solid solution. As a specific example, consider the solidification of the solution indicated by the dashed line in the phase diagram drawn in Fig. 11.34.

T

A Bx

å ∫

L

å+L ∫+L

å + ∫

...

Fig. 11.34: A eutectic phase diagram showing a composition that would, ideally, solidify into a homogeneous å solid solution.

If the solution indicated by the dashed line were solidified under equilibrium conditions it would become completely solid at the temperature at which the dashed line crosses into the single-phase, å field. However, a real system that is cast or solidified at a practical cooling rate is more likely to remain at least partly liquid until the eutectic temperature, TE, is reached, and solidify into a microstructure that has a chemically inhomogeneous å phase in the interiors of grains or dendrites, while the boundaries of these are decorated by high-solute material, including some of the eutectic constituent. To understand why this should happen, assume that solidification starts when the system is in the two-phase å+L region, and assume that solid state diffusion can be ig-nored, as it usually can be when the å phase is a substitutional solution and the system is cooled at a moderate rate. The first solid to form is rich in A, and causes the liquid to be-come slightly richer in B. Ideally, when the solution is cooled to lower temperature the composition of this first solid should adjust to the equilibrium value by incorporating more of the B-specie. But this chemical redistribution requires diffusion in the solid å-phase, and cannot happen if solid state diffusion is negligible. The physical consequence of forming a solid whose composition cannot be adjusted on the time scale of the solidification process is the same as if the solid were removed from the system entirely, leaving a solution slightly richer in B. At a slightly


Page 365

lower temperature the liquid solution deposits an A-rich solid, though it is slightly less rich in A than the material that solidified previously, and becomes richer still in B. A continuation of this process causes the composition of the residual liquid to follow the liquidus curve (the boundary between the liquid and the two-phase region) all the way down to the eutectic point, as shown in Fig. 11.35. The last bit of liquid to solidify has the eutectic composition. It is trapped between the grains or dendrites of å since these are formed by the material that solidifies earlier in the reaction.

åT

L

å+L

å+∫

... Fig. 11.35: Schematic drawing of a section of the eutectic phase diagram

illustrating the evolution of the composition of the solid and liquid when an A-rich solution is solidified rapidly. The last material to solidify has the eutectic composition.

The chemical segregation that develops during solidification is disadvantageous for at least three reasons. First, it degrades the properties of the solidified material, particularly its mechanical properties. In most systems the eutectic constituent has poor mechanical properties; it is either very soft or very brittle. In the structure discussed above this weak material is located in the worst possible place, in a semi-continuous network along the grain boundaries. This morphology maximizes any deleterious effect on the mechanical properties. Second, segregation interferes with further processing. For example, in structural materials the å solid solution in a system that has a phase diagram like that shown in Fig. 11.34 is precipitation hardened by introducing a fine dispersion of ∫ precipitates into the interiors of the å grains. It is not possible to achieve a homogeneous precipitation of the ∫ phase if the parent å is chemically heterogeneous; the precipitation times and temperatures vary from point to point with the composition. Third, the presence of the eutectic constituent makes it difficult to homogenize the material to remove segregation. Homogenization, which was discussed in Chapter 10, requires solid state diffusion, and hence proceeds more rapidly the higher the temperature at which it is carried out. But an alloy with the inhomogeneous microstructure described here must be homogenized at a temperature below the eutectic temperature, TE. The eutectic constituent melts at TE, and the alloy will fall apart if heated to a higher temperature. This problem has the consequence that alloys that have components with high melting points and eutectics with low melting points often cannot be homogenized at all, and require elaborate treatments, such as rapid solidification processing, before their potential can be realized.


Page 366

In many engineering applications it is desirable or necessary to use multicomponent systems in the as-solidified condition. The metal alloys used for welding and joining usually must be used in the as-solidified condition. There is often a large financial benefit to making parts of complex shape by direct casting. Special casting or welding alloys have been created for these purposes. Their compositions are carefully engineered to avoid the most serious problems encountered in the non-equilibrium solidification of more conventional alloys. 11.12 GRAIN GROWTH AND COARSENING When a phase transformation has gone essentially to completion the material is usually left in an unstable microstructure that evolves further. The as-solidified material contains residual chemical heterogeneities of the type discussed in the previous section, has a polygranular microstructure, and often contains small second-phase precipitates. If the material is maintained at high temperature the chemical heterogeneities tend to homogenize. At the same time the grains grow and the precipitates coarsen. The principal driving force for grain growth and precipitate coarsening is the surface energy. These processes lower the total surface area, and hence the total surface energy. Both grain growth and coarsening are thermally activated diffusional processes. In order for a grain to grow material must diffuse across its boundary from a neighboring grain. On balance, large grains grow at the expense of small ones. Since the large grain has less surface area per unit volume, the free energy per atom (chemical potential) de-creases with grain size. There is, hence, a thermodynamic driving force for diffusion across the boundary from the small grain to the larger one. Large grains grow, small ones disappear. The mechanism of precipitate coarsening is slightly different. Large precipitates grow at the expense of smaller ones by diffusion through the matrix. Let the precipitates be spherical particles in a binary solution at constant temperature and let the precipitate phase have a composition that is richer in solute that the matrix. The larger the precipitate particle, the smaller its free energy per atom, and, hence, the smaller its relative chemical potential. Since –µ = µB - µA = ∆¡g/∆x increases with x for both the matrix and the precipitate, the solute concentration that is in equilibrium with the precipitate decreases as the precipitate grows. Hence the matrix is relatively depleted in solute near the large precipitates, and relatively enriched near the smaller ones. The concentration gradient from the small precipitates to the large ones induces solute diffusion to the large precipitates, which causes the matrix composition to exceed the local equilibrium value, so the large precipitate grows. The smaller precipitate shrinks as matrix diffusion removes solute, which is restored by the dissolution of the precipitate. The rate of grain growth or coarsening increases with the temperature in a pre-dictable way, with the result that either process can be controlled, and used to fix the mean grain size or mean precipitate size of a material to a reasonable degree of accuracy.


Page 367

11.13 INSTABILITIES At the beginning of this chapter we discussed the fact that there are two basic phase transformation mechanisms: nucleated transformations and instabilities. When the parent phase is stable the transformation must be nucleated. The probability of nucleation depends on the thermodynamics of both the parent and product phases. When the parent phase becomes unstable it must transform. As diagrammed in Fig. 11.1, instabilities define the limits of metastability of the parent phase, and depend on its properties alone. Solid phases are distinguished from one another by differences in one or more of three characteristics: composition, chemical or electromagnetic order, and basic crystal structure. If we identify the basic crystal structure with the distribution of atom sites, without regard to the specific types of atoms that occupy them, and define the state of order (segregation of atom types onto specific sets of lattice positions) so that perfect order corresponds to the most complete possible segregation for the given composition, then these three characteristics are independent. For a solid phase to be thermodynam-ically stable it must be stable with respect to small changes in any one of the three characteristics: a change in the state of order at fixed composition and structure, a change in the structure at fixed composition and state of order, and a decomposition into regions of different composition on a fixed, ordered lattice. When a solid phase becomes unstable then, barring fortuitous degeneracy, the initial instability concerns only one of these three characteristics. While other characteristics may change as the transformation proceeds, the instability that triggers the transformation inevitably involves the composi-tion, the structure, or the state of order alone. There are common phase transformations that are initiated by instability with respect to each of the three characteristics of structure. First, and most familiar, because of its importance in the heat treatment of steel, are the transitions in which the crystal structure changes spontaneously at constant composition. For historical reasons, these are called martensitic transformations. Second, there are instabilities in which a random solid solution becomes unstable with respect to decomposition into two solutions with the same structure, but different compositions, and spontaneously develops a fine scale chemical heterogeneity. There is always an instability of this type when a miscibility gap appears in the phase diagram. The transformation that is induced by a compositional instability is called a spinodal de-composition. (The name derives from the characteristic shape of the surface that results when the relative chemical potential of a system that undergoes a spinodal composition is plotted as a function of temperature and pressure, which we shall not illustrate here.) Third, it is both possible and fairly common for disordered solid solutions to order spontaneously when the ordering reaction is first-order and the system is cooled to well below the equilibrium ordering temperature. These reactions are called ordering in-stabilities. Such instabilities affect disordered solutions in systems that have ordered


Page 368

phases. An ordered phase may also become unstable, and be replaced by another that is very like it. Another group of instabilities includes the mutations that were discussed in Chapter 9 and in Section 11.2. These are simpler in the sense that the equilibrium transformation temperature and the stability limit of the parent phase are the same; the parent phase simply becomes the product at the phase transformation. The glass transition is a mutation, as are almost all of the electromagnetic transformations in solids, and some chemical ordering reactions. We have already described the common mutations. In the following sections we discuss the martensitic, spinodal and ordering transitions. 11.14 MARTENSITIC TRANSFORMATIONS The martensitic transformation takes its name from the spontaneous structural change that occurs when carbon steel is rapidly quenched from an elevated temperature at which the ©(FCC) phase is stable. The microstructural features of the transformation were studied at an early date by a German scientist named Martens, so the product is called "martensite". The martensitic transformation is central to the processing of a number of important structural steels and has, consequently, received a great deal of attention. However, similar transformations occur in many other metal alloys and oxide ceramics. Transformations that have martensitic characteristics have even been observed in biological systems.

Fig. 11.36: The body-centered tetragonal cell within an FCC structure that transforms to BCC when given the tetragonal distortion shown at right.

The simplest model of the martensitic transformation (which is oversimplified, but sufficient for our purposes) begins from the observation that the driving force for a structural transformation, ÎGv, increases monotonically as the temperature is lowered below the equilibrium transformation temperature. If the material is cooled so rapidly that nucleation is suppressed then the driving force may become large enough to force a structural transformation through a spontaneous mechanical distortion of the crystal lattice.


Page 369

To see how this might happen in the specific case of the FCC“BCC transformation in iron, consider two adjacent cells of the FCC structure, as shown in Fig. 11.36. It is possible to define a body-centered tetragonal cell within the FCC crystal structure, as shown in the figure. This cell has height, a, while its other edges have length a/ 2 . It can be transformed to BCC by giving it the net tetragonal distortion shown in the figure to make its edge lengths equal. This distortion preserves the atomic volume if the length and width are expanded slightly as the height is compressed. At finite temperature the atoms in a solid undergo thermal vibrations about their equilibrium positions. When the basic structure is stable the free energy increases as an atom moves away from its equilibrium position, so the amplitude of the vibration is small and centered about the equilibrium site. However, some of the vibrational modes of the atoms in the FCC structure carry them in the direction of the tetragonal distortion that would generate BCC. If the driving force for the FCC“BCC transformation is great enough it can enhance the amplitude of the vibrational modes that carry the system toward BCC. There are regions within the FCC crystal in which the atoms are already distorted toward BCC, for example, near properly oriented grain boundaries or clusters of dislocations. When the thermodynamic driving force becomes sufficiently large, small volumes of FCC material in these regions spontaneously distort into the BCC structure, initiating a mechanical lattice transformation from FCC to BCC. When these small volumes are large enough to grow into the FCC matrix their boundaries propagate as a mechanical disturbances at the speed of sound, transforming the structure as they move. In fact, one can often hear a martensitic transformation in steel.

T

ln(†)

Ms

Mf

martensitictransformation

nucleated transformation

Fig. 11.37: Plot of the transformation initiation time (†) against T for a system that can transform by a martensitic mechanism.

Since a pure martensitic transformation is caused by a thermodynamic instability, it cannot be suppressed; when the driving force is great enough the transformation happens spontaneously. Such transformations are called athermal. If an athermal martensitic transformation occurs in a given system, it only happens if the system is quenched rapidly enough to suppress transformation by nucleation and growth, and then initiates at a particular value of the undercooling, at a temperature called the martensite start temperature, Ms. The martensite start temperature is represented by a horizontal line on the transformation initiation plot shown in Fig. 11.37. If the system is cooled slowly,


Page 370

as illustrated by the upper arrow in the figure, the transformation is nucleated. If the system is quenched rapidly, as indicated by the lower arrow, the transformation is martensitic. The martensite transformation starts at the temperature, Ms, but in most cases it does not go to completion at that temperature. To sustain the transformation the tempera-ture must be lowered further. The martensite fraction is a function of the temperature, and approaches completion at the temperature, Mf, the martensite finish temperature. The reason for this behavior is the strain induced in the structure by the martensite transformation. When the structure is locally changed as, for example, in an FCC “ BCC transformation like that shown in Fig. 11.36, the new phase does not fit properly in the crystalline matrix of the parent phase, and hence distorts the region of parent phase around it. For this reason the individual particles of the martensite tend to appear in the form of thin plates, or laths, that are internally defective, and have a very small volume fraction at the martensite start temperature, Ms. To sustain the transformation the thermodynamic driving force must be sufficient to form additional product phase in a matrix that is strained by the prior transformation; hence the undercooling, ÎT, must be increased continuously as the transformation proceeds. The progress of a martensitic transformation through the successive formation of product phase is illustrated in Fig. 11.38.

(a) (b)

Fig. 11.38: Successive stages in the martensite transformation on

cooling: (a) near Ms; (b) on further cooling below Ms. The pattern of an athermal martensite transformation often looks like that shown in Fig. 11.38. The new phase first appears as thin plates that cross grains and terminate at grain boundaries. The subsequent transformation produces smaller plates that tend to subdivide the untransformed regions until the transformation is complete. Because the martensite transformation involves a distortion of the lattice it is pro-moted if the solid is subjected to mechanical stresses (forces) that act to deform the solid in the direction of the crystallographic transformation. Hence Ms increases when a suitable stress is applied, and the enhanced martensite is called stress-induced martensite. If the solid is plastically strained at a temperature below T0, high local internal stresses are caused by interaction of the dislocations that cause plastic deformation. These stresses can be very effective in promoting the martensite transformation. The


Page 371

temperature at which martensite forms on deforming the material is called Md, and can be a hundred degrees or more above Ms in steels. This deformation-induced martensite has important effects on the mechanical properties of many austenitic (FCC) stainless steels, which have kinetically stabilized FCC structures and are used at temperatures above Ms, but below Md.

T

ln(†)

Ms

Mf

martensitictransformation

nucleated transformation

Fig. 11.39: Two cooling paths that lead to a mixed microstructure in which the transformation is accomplished partly by nucleation and partly by the martensitic mechanism.

It is also possible to produce mixed microstructures in which the product phase is achieved partly through a martensitic transformation and partly through a nucleated trans-formation. Two methods for doing this are diagrammed in Fig. 11.39. In the first the sample is cooled slowly to initiate the transformation by a nucleated mechanism, but then quenched before the nucleated transformation is complete. The residual material transforms by a martensitic mechanism. In the second the material is quenched below Ms to accomplish a partial transformation to martensite, but is then held isothermally (or heated slightly) so that the residual material transforms by the nucleated mechanism. Finally, since the martensite transformation is caused by a structural instability it does not permit chemical redistribution. If the transformation is one that would normally create a two-phase microstructure, as, for example, the transformation of carbon steel on cooling which creates a mixture of the BCC å-phase and a carbide precipitate phase (Fe3C), the martensitic transformation creates a single-phase microstructure that is super-saturated with solute. In the particular case of carbon steel the martensitic product is supersaturated with carbon, and is usually distorted into a body-centered tetragonal (BCT) structure by the ordering of excess carbon interstitials in asymmetric octahedral interstitial sites, as discussed in Chapter 4. The resulting structure is mechanically very hard, but tends to be brittle (easily fractured). Hence martensitic carbon steels are often heated to a moderate temperature and held for some time to permit precipitation of the excess carbon in Fe3C precipitates. This process is called tempering, and leads to a product that is still very hard, but is much more difficult to fracture (that is, it is tougher). If the martensitic transformation of carbon steel happens at relatively high temperature (that is, when Ms is a relatively high temperature) or if the steel is cooled relatively slowly between Ms and Mf then carbides may precipitate out of the


Page 372

supersaturated martensite while the transformation is going on. The carbides form from carbon that is rejected from the supersaturated martensite as it grows. This autotempered martensite is a mixture of martensitic plates and carbide precipitates. It is called Bainite after the American metallurgist, E.C. Bain, who was among the first to study it. The microstructure of Bainite depends on the kinetic balance between the growth of martensite and the rate of carbide precipitation, and many complex microstructures are possible. 11.15 SPINODAL DECOMPOSITION Spinodal decomposition describes a phase transformation in which a solid solution spontaneously decomposes into a mixture of two solutions that have the same crystal structure and degree of order. Spinodal instabilities appear near miscibility gaps in the phase diagram, where a parent solid solution decomposes into two solutions of different composition. 11.15.1 Spinodal instability within a miscibility gap

L

å

A B

T

x

å' å''å'+å"

...

Fig. 11.40: The phase diagram of a binary system that contains a miscibility gap in a homogeneous solid solution.

An example of a miscibility gap is shown in Fig. 11.40. This figure shows the phase diagram of a binary system that forms a solid solution (å) at all compositions at high temperature, but divides into two solid solutions, å' and å", at lower temperature. The thermodynamic reasons for the appearance of a miscibility gap were discussed in Chapters 8 and 9. A miscibility gap appears when the when the free energy of the solution has the behavior shown in Fig. 11.41. At T > Tc, where Tc is the temperature at the top of the miscibility gap, the free energy depends on composition as shown in the left-hand figure in 11.41; g is a concave function of x, so the equilibrium phase is a homogeneous solution at all compositions shown. At T <Tc, on the other hand, the function, g(x), has double minima as shown in the right-hand figure in Fig. 11.41. The two concave branches of the curve represent different phases, with the same crystal structure, that share a common tangent. When the composition falls within the region of


Page 373

the common tangent the solution divides into a mixture of two solutions with different compositions, which generates the two-phase region in the equilibrium phase diagram.

xA

å

xA

å' å"å'+å"

å' å"

T > T T < Tcc

g

...

Fig. 11.41: Free energy curves for a solid solution in a system that ex-hibits a miscibility gap (a) above and (b) below the tempera-ture, Tc, at the top of the gap.

It is often possible to preserve a homogeneous solution in a metastable state within a miscibility gap by cooling it quickly enough to suppress nucleation. However, the range of metastability is limited. The parent solution becomes unstable with respect to spontaneous, spinodal decomposition once it reaches a temperature such that its composition is significantly inside the miscibility gap. The thermodynamic source of the instability is illustrated in Fig, 11.42. The solution is unstable when the free energy becomes a convex function of x, that is, when ∆2¡g/∂x2 ≤ 0.

A B

g

x

å''å'

x1 x2

...

Fig. 11.42: The free energy curve of a solution that has a miscibility gap. The solution is stable when g(x) is concave, for example, at x1, but unstable when g(x) is convex, as at x2.

To see this consider the free energy curve shown in Fig. 11.42, which is drawn at a temperature for which the solution has a miscibility gap. As we discussed in Chapter 9, if the system is a mixture of two distinct states, its free energy is given by the intersection of a straight line connecting these two states with a vertical line at the overall composition of the system, x. This graphical relation is illustrated in Fig. 11.42 for two


Page 374

compositions, x1, where g(x) is concave, and x2, where it is convex. A homogeneous solution of composition x1 is at least metastable. Because g(x) is concave, decomposition into two solutions with compositions very close to x1 leads to an increase in the free energy. The solution is, hence, stable with respect to spontaneous fluctuations in its local composition. On the other hand, a homogeneous solution of composition x2 is unstable. If it is separated into two solutions with slightly different compositions, the free energy decreases. In a real solution atoms continually move by diffusional jumps, so that the composition of any small volume within the solution fluctuates constantly. When the overall composition is such that ∆2¡g/∆x2 ≤ 0, these local fluctuations decrease the free energy, and grow spontaneously to decompose the solution. The stability limit for the homogeneous solution is the locus of points for which ∆2¡g/∆x2 = 0, that is, the locus of inflection points in the free energy curve, ¡g(x), as the temperature is lowered. These points generate spinodal lines that can be plotted in the phase diagram, as in the example given in Fig. 11.43. The spinodal lines define a spinodal gap in the phase diagram within which a homogeneous solution is unstable.

å

A B

T

x

å' å''

spinodal gap

xx1 2

T1

T2

Fig. 11.43: A solid solution containing a miscibility gap, with the curves that define the stability limits of the solid solution drawn in-side.

11.15.2 Spinodal decomposition To illustrate how a homogeneous solution behaves when it is cooled into a misci-bility gap, we consider the two compositions, x1 and x2, that are labeled in Fig. 11.43. Symmetric spinodal reactions The composition, x2, is the composition at the top of the miscibility gap. A solu-tion of this composition is unstable as soon as it is cooled below Tc. As a consequence it inevitably decomposes by a spinodal mechanism. Solutions with composition near x2 develop the symmetric spinodal that is illustrated in Fig. 11.44.


Page 375

Spinodal decomposition develops in a characteristic pattern like that illustrated in Fig. 11.44. Spontaneous diffusion of the solute causes the development of alternate re-gions that are rich and poor in solute. However, the interfaces between these regions can-not be arbitrarily sharp without introducing a high interfacial tension that would oppose the transformation. As a consequence the composition varies in a roughly sinusoidal pattern like that shown in Fig. 11.44, in which the composition varies gradually with distance.. An analysis of the kinetics of spinodal decomposition shows that there is a particular wavelength of the spinodal wave that can grow more rapidly than any other under a given set of conditions. Hence solutions that decompose by the spinodal mechanism tend to have sinusoidal composition profiles with a particular dominant wavelength.

r r

´x¨´x¨x x

Fig. 11.44: Development of a symmetric spinodal wave in a solution with x « x2 in Fig. 11.45.

Note that during spinodal decomposition the solute diffuses uphill, from regions that are poor in solute to regions that are rich. This behavior violates Fick's First Law, but should not be surprising. As we discussed in Chapter 10, the actual thermodynamic driving force for diffusion is the chemical potential gradient rather than the composition gradient. When an unstable solution undergoes spinodal decomposition the chemical potential gradient is in the opposite direction to the composition gradient; the free energy is reduced as the solution divides into regions that are alternately rich and poor in solute. Rather different microstructures may result from spinodal decomposition, depend-ing on the number of dimensions in which the spinodal waves develop. In some materials the spinodal wave is essentially a one-dimensional wave; the formation of a wave in one dimension suppresses the development of similar waves in the perpendicular directions. In this case the final microstructure contains alternate lamellae of å' and å". In other materials waves develop simultaneously in two or three perpendicular directions. Two-dimensional decomposition leads to a microstructure in which one phase (that with the lower volume fraction) appears in rod-shaped grains in the matrix of the other. In three-dimensional decomposition the minority phase ultimately forms equiaxed grains in the matrix of the other. Asymmetric spinodal reactions


Page 376

The transformation behavior of the solution is qualitatively different when its overall composition is either much less or much greater than the composition at the top of the miscibility gap, that is, when it produces an asymmetric spinodal. An example is the composition, x1, that is indicated in Fig. 11.43. There are two principle differences. First, when the composition is asymmetric the spinodal line is inside the miscibility gap. When a solution of composition, x1, is quenched, it passes through a range of temperatures in which it is metastable before penetrating the spinodal gap. One such temperature is indicated by the upper dot on the vertical line at x1 in the figure. The transformation mechanism in this range is the nucleation and growth of the solute-rich å" phase. Spinodal decomposition is only observed if the solution is cooled to a temperature inside the miscibility gap at a rate fast enough to suppress nucleation until the spinodal line is reached. The initiation curve for the transformation is like that drawn schematically in Fig. 11.45. The nucleated transformation has a c-curve behavior, while the spinodal transformation initiates as soon as the solution penetrates the two-phase region.

ln(†)

ÎT nucleated transformation

spinodal decomposition

Fig. 11.45: Initiation kinetics for a solution like that with composition x1 Fig. 11.43. Spinodal decomposition is observed if the solu-tion is quenched to suppress nucleation.

Second, the ultimate microstructure that results from spinodal decomposition is different when the composition differs significantly from the symmetric value. The development of the composition profile in an asymmetric decomposition is illustrated in Fig. 11.46. The decomposition initially develops as a sinusoidal decomposition wave that grows in amplitude (Fig. 11.46a). But since the average composition is close to one boundary of the miscibility gap, the decomposition cannot be completed by the symmetric growth of this wave. The result is shown in Fig. 11.46b; narrow, precipitate-like regions develop that are rich in solute concentration, and these grow to complete the reaction. As a spinodal like that in Fig. 11.46 develops in three dimensions it produces a homogeneous distribution of precipitates through the bulk that is nearly indistinguishable from that which would form by homogeneous nucleation.


Page 377

x

r

x

r

´x¨ ´x¨

Fig. 11.46: Development of the composition profile in a asymmetric spinodal decomposition. A low-amplitude spinodal wave develops into discrete precipitates.

11.15.3 Spinodal decomposition to a metastable phase Spinodal decomposition may cause the formation of a metastable precipitate phase rather than an equilibrium transformation product. Fig. 11.47 illustrates a set of thermodynamic relationships that cause this to happen. The free energy curves in the figure are drawn for a temperature at which there are two equilibrium phases, å and ∫. If a homogeneous å solution that has a composition within the two-phase (å+∫) field is quenched to this temperature it is metastable with respect to nucleation of the ∫ phase. However, the å free energy curve contains a spinodal instability, marked by the shaded region in the figure. If the composition of the quenched å solution lies in the shaded region it is unstable with respect to a spinodal decomposition that produces a metastable phase, å'. Since the kinetics of decomposition are more rapid than the kinetics of nucleation, the metastable, å' phase forms preferentially. The å' phase has the same crystal structure as the å parent, and hence appears in the form of a dense distribution of coherent precipitates.

g

xA

å∫

B

∫å å+∫

å'å“å'

Fig. 11.47: Free energy curves leading to formation of a metastable, co-herent precipitate. If å has a composition within the shaded region, it is unstable with respect to å', as well as being metastable with respect to ∫.


Page 378

ln(†)

ÎTå “ ∫

å “ å'

Fig. 11.48: Initiation kinetics for a transformation that may proceed by nucleating an equilibrium phase, ∫, or by spinodal decompo-sition to a coherent, metastable phase, å'.

When a solution that has the behavior shown in Fig. 11.47 is cooled, the transfor-mation kinetics tend to obey a relation like that shown in Fig. 11.48. Phase ∫ can nucleate at small undercooling, but a quench suppresses this nucleation, and leads to the spontaneous formation of a coherent, metastable phase, å'. 11.15.4 Use of spinodal decomposition in materials processing Spinodal decomposition has been intentionally used in materials processing to ac-complish two different objectives. First, solutions with compositions well inside the mis-cibility gap are decomposed to create a microstructure that contains a fine admixture of two phases of different composition. One particular example is in the processing of magnetic materials in which it is desirable to have fine particles of magnetic phase distributed through a non-magnetic, metallic matrix. If the solute-rich constituent of the spinodal is magnetic while the solute-poor constituent is not, then the desired microstructure can be created by spinodally decomposing a solution that is slightly rich in solvent, so that the non-magnetic phase becomes the matrix. Another important example is in the processing of ceramics. While nucleated transformations are often very difficult to initiate in oxide ceramics, fine-grained two-phase microstructures can be made by spinodal decomposition. This technique is used in the processing of a number of modern ceramics. The second use of the spinodal decomposition is to create a homogeneous distribution of fine precipitates. In systems whose transformations resemble those shown in Fig. 11.45 or 11.48, quenching into the spinodal gap produces a distribution of precipitates in the interior of grains and, hence, provides an alternative to low-temperature heat treatment as a method for achieving this microstructure. 11.16 ORDERING REACTIONS The third important class of instabilities includes transitions that change the state of order of the material while the crystal structure and overall composition remain the same. The ordering transition was discussed in Section 8.4. The simplest ordering


Page 379

reaction is a transition from a disordered to an ordered state in a binary solution. In the high-temperature, disordered state the atoms are randomly distributed over the sites of a background crystal lattice. In the ordered state the sites of the crystal lattice are divided into two sets, one of which is preferentially occupied by atoms of type A, while the other is preferentially occupied by atoms of type B. Examples of substitutionally ordered phases we have studied include the CsCl structure, which is a BCC-based structure with one kind of atom at the corner sites and a second at the body-centered sites (see Fig. 11.49 below), and the Cu3Au structure, which is an FCC-based structure with one kind of atom at the corners and a second at the face-centered sites of the unit cell (Fig. 11.51). It is only possible to have perfect order when the composition of the solution is the stoichiometric composition of the ordered phase (AB for the CsCl structure, A3B for the Cu3Au structure), and when the temperature is so low that the fraction of anti-site defects (A atoms on B sites) is negligible. In other cases we measure the degree of order by the long-range order parameter, ˙, which was defined in Section 9.4. To phrase the most general definition of the order parameter in a binary solution let x be the atom fraction of the component whose composition exceeds the stoichiometric value, xs, for the ordered phase, and let x1 be the atom fraction of that component on the sites it occupies in the ordered phase (x1 = 1 when all of these sites are filled). Then

˙ = x1 - x1 - x 11.65

When ˙ < 1 both types of sites contain a mixture of the two kinds of atoms. As ˙ increases sites of type 1 become increasingly filled with atoms of type 1. However, even when ˙ = 1 the compound is not perfectly ordered unless its composition is stoichiometric. Even when all sites of type 1 are filled, if x ≥ xs there are additional atoms of type 1 that must sit on sites that are appropriate to the other specie. That is, there is a residual concentration of anti-site defects in the crystal. 11.16.1 Ordering reactions that are mutations As we discussed in Section 8.4, there are two different kinds of ordering reactions. In the first kind the structures of the ordered and disordered phases are so closely related to one another that the reaction is a mutation; order happens spontaneously when the disordered solution is cooled to a critical temperature, Tc. An example of an ordering reaction that is a mutation is the transition from a disordered BCC solution to a CsCl-ordered solution that occurs in brass (CuZn), which is shown in Fig. 11.49. At the critical temperature, Tc, order spontaneously appears.


Page 380

Fig. 11.49: Transition from disordered BCC to CsCl in CuZn. The

shading in BCC indicates random occupancy by Cu and Zn.

T

˙

Tc

∫' ∫

Fig. 11.50: Variation of the long-range order parameter with T in a muta-tion, like the disorder-order transition in ∫-brass.

A phase transition that is a mutation is never nucleated since the parent phase is never metastable. The transformation occurs spontaneously at the critical temperature, Tc. The order parameter becomes finite at a temperature incrementally below Tc, and increases in magnitude as the temperature is lowered further, as illustrated in Fig. 11.50. 11.16.2 First-order transitions that change the state of order

...

Fig. 11.51: The first-order transition from a disordered, FCC-solution to the Cu3Au structure in Cu-Au.

The second kind of ordering transition includes transitions in which the structures of the two phases are sufficiently different from one another that the transition is first-or-der. There is a well-developed theory, the Landau symmetry rules, that tell us how to distinguish these transformations, but the development of this theory is beyond the scope of this course. An example is the transition from a disordered solution to a Cu3Au structure in Cu-Au, which is illustrated in Fig. 11.51. When the ordering reaction is first-order there is a discontinuous change in the equilibrium value of the order parameter at the equilibrium transformation temperature, as described in Section 9.4. Moreover, it is possible to cool the disordered solution, or heat the ordered solution, into metastable states beyond the equilibrium transformation temperature. This behavior is illustrated in Fig. 11.52, which shows the variation of the equilibrium value of the order parameter near the transition temperature, T0.


Page 381

˙

T0

å' å

= metastable ordered phase

= metastable disordered phase

Fig. 11.52: Variation of the order parameter for a first-order ordering reaction, showing the ranges of metastability near the equilibrium transition temperature, T0.

As indicated in Fig. 11.52 the range of metastability is limited. If the disordered phase is cooled below the transition temperature at a rate that is rapid enough to suppress nucleation, the material becomes unstable with respect to the spontaneous appearance of long-range order when the undercooling becomes great enough. The instability temperature for the ordering reaction varies with composition, and can be plotted on the phase diagram as is done in Fig. 11.53. This figure shows a simple prototype reaction in which a disordered solution, å, has a first-order transition to an or-dered phase, ©. The phase diagram is like that shown in Fig. 8.35; the equilibrium phase field of the ordered phase has a teardrop shape whose top defines a congruent (constant composition) ordering transformation from the parent phase. The congruent point is at, or very close to, the stoichiometric composition for the ordered phase (Cu3Au in the copper-gold case). The line in the phase diagram indicates the onset of instability. If a disordered å solution is cooled to the instability line, it spontaneously orders, insofar as its composition allows, to produce a homogeneous ordered phase.

A B

T

x

x1 x2

©©+å©+å

å å

Fig. 11.53: A section of the equilibrium phase diagram of a system that contains an ordered phase. The disordered solution is unstable when the temperature falls into the lightly shaded region.

The transformation behavior of a system like that shown in Fig. 11.53 changes with the composition of the å phase. In particular, the transformation behavior at the stoichiometric composition (x2 in the figure) is qualitatively different from that at a significantly off-stoichiometric composition, such as the composition x1 in the figure.


Page 382

11.16.3 Ordering at the stoichiometric composition

ln(†)

ÎT nucleated

spontaneous

T0

Ti

Fig. 11.54: Kinetic diagram for ordering at the stoichiometric composition. The transformation is nucleated at T > Ti, but becomes spontaneous at the instability temperature, Ti.

First consider transformation at the stoichiometric composition, indicated by x2 in Fig. 11.53. In this case the å solution becomes metastable with respect to congruent nu-cleation of the © phase as soon as its temperature falls into the © region. It remains metastable until the temperature drops below the instability line shown in the figure. If nucleation has not happened when this point is reached the system spontaneously becomes ordered; A-atoms spontaneously move to A-sites and B-atoms to B-sites. The kinetic diagram for the stoichiometric reaction is like that shown in Fig. 11.54. The nucleated transformation becomes possible when the temperature drops below T0. If the transformation has not happened when the temperature reaches Ti, the instability temperature, then a homogeneous ordering reaction initiates spontaneously. 11.16.4 Ordering at an off-stoichiometric composition The transformation behavior is qualitatively different when the composition is off-stoichiometric, for example, the composition x1 in Fig. 11.53. To understand the transformation behavior in this case it is useful to consider the free energy curves of the disordered, å, and ordered, ©, phases, which change with temperature roughly as shown in Fig. 11.55. As the temperature is lowered the free energy curve for the © phase drops relative to that of the å phase. In addition, the composition at which the å-phase becomes unstable (the solid curve in Fig. 11.53) decreases. This composition is indicated by the termination of the å free energy curve in Fig. 11.55.


Page 383

g

x x xx1 x1 x1

å å å

©©

©

T1 T1T2 < Ti

Fig. 11.55: Possible form of the free energy curves of the å and © phases in Fig. 11.53 as the temperature decreases. The dot marks a disordered solution with composition, x1.

When a solution of composition, x1, is cooled to a temperature just inside the two-phase region the free energy curves appear like those shown in Fig. 11.55a. The system is metastable, but can only transform by nucleating a solute-rich © phase. If the sample is cooled to a lower temperature at a rate that is fast enough to suppress nucleation of the equilibrium phase then å becomes metastable with respect to the formation of a non-stoi-chiometric ordered phase, ©, that has the same composition, as illustrated in Fig. 11.55b. Hence congruent nucleation becomes possible. However, as the temperature drops the composition at which the disordered solution becomes thermodynamically unstable de-creases, so the termination of the free energy curve of the å phase moves closer to x1. Finally, at the instability temperature, Ti, an å phase of composition, x1, becomes unstable with respect to spontaneous order into the © phase, as illustrated in Fig. 11.55c. If the solution is cooled to Ti quickly enough to suppress congruent nucleation, the system orders spontaneously at Ti.

ÎT

ln(†)

incongruentnucleation

congruentnucleation

spontaneous orderTi

Fig. 11.56: Kinetic diagram for the ordering of an off-stoichiometric dis-ordered solution.

The kinetic diagram for a transformation at composition, x1, appears as shown in Fig. 11.56. Slow cooling leads to nucleation of the equilibrium ordered phase, more rapid cooling produces congruent nucleation of an off-stoichiometric ordered phase, and quenching leads to spontaneous order.


Page 384

Note that when an off-stoichiometric solution of composition like x1 transforms through congruent nucleation or spontaneous ordering the product of the transformation is a non-equilibrium ordered phase. To achieve equilibrium the homogeneous reaction, å “ ©, must be followed by a second reaction, © “ å + ©, in which islands of disordered solution reform out of the ordered phase. 11.16.5 Implications for materials processing Many of the materials that are coming into use in advanced engineering systems are ceramics or intermetallic compounds with complex, ordered structures. These often have multiple ordered structures that are, in practice, connected to one another by congruent ordering reactions. Largely for this reason, ordering reactions and ordering instabilities are becoming increasingly important in the processing of modern materials. Ordering reactions can also be exploited for the control of microstructure. This is particular true in solids with very low diffusivities. If diffusion is very slow, as it is in many ceramic and intermetallic phases, then decomposition is easily suppressed and non-stoichiometric ordered phases can be made and retained almost indefinitely. Hence the basic crystal structure of the compound can be controlled by adjusting the cooling rate. When the diffusivity is higher, it is possible to engineer the microstructure of a two-phase mixture of ordered and disordered phases by controlling the reaction sequence that leads to the equilibrium state. A solution like that shown in Fig. 11.56 can be brought to equilibrium by two very different paths that may lead to very different final microstructures. If the disordered parent solution is cooled slowly, it decomposes by nucleation of the equilibrium ordered phase. The reaction is å “ å + ©. If the sample is cooled quickly it orders before decomposing. The reaction is a two-step one in which å “ © “ å + ©.


Page 385

C h a p t e r 1 2 : E n v i r o n m e n t a l I n t e r a c t i o n sC h a p t e r 1 2 : E n v i r o n m e n t a l I n t e r a c t i o n s

In lodgings frail as dew upon the reeds I left you, and the four winds tear at me - Murasaki Shikibu, "The Tale of Genji"

12.1 INTRODUCTION In the last few chapters we have considered the changes that occur within a material when we alter its temperature or chemical composition. In the present chapter we consider the changes that are brought about in a material when it reacts with its environment. The important environmental interactions are of three types. The first type includes reactions that change the microstructure of the material near the surface by adding heat or chemical species from the environment. These include the surface hardening reactions that are widely applied to structural materials and the doping reactions that control the properties of semiconductors. The second type includes chemical reactions between the metal and its environment that form compounds at the interface. There are many important reactions of this type. We shall specifically consider the oxidation reaction, since it is both important in engineering and illustrative of the mechanisms that govern interfacial reactions. The third type includes electrochemical reactions, in which the chemical reaction at the interface is assisted by electric currents so that it has an appreciable rate at low temperature. The most important electrochemical reaction is aqueous corrosion, which is responsible for the gradual deterioration of metallic structures that are exposed to water, soil or moist air. 12.2 CHEMICAL CHANGES NEAR THE SURFACE The first class of environmental interactions we shall consider are those that change the microstructure of the solid immediately beneath its surface. It is often desirable to do that to control the mechanical, chemical or electrical properties of the material near the surface. Rapid thermal treatments are used to change the phase, grain structure or chemical distribution near the interface. Chemical diffusion or ion implantation is used to change the chemical composition near the surface. 12.2.1 Thermal treatment To control the microstructure of the layer of material just beneath a solid surface it is useful to have a way of heating that particular material without significantly heating the interior. It is difficult to do this because heat is conducted so rapidly in most materials that heating the surface quickly raises the temperature in the interior. The methods that are useful for the thermal treatment of surfaces create intense, local heat


Page 386

sources in the material immediately beneath the surface. The possible heat sources include electrical currents and locally absorbed beams of light or high-energy electrons. These can cause local heating at the surface that is so rapid that a high temperature is reached before heat can be conducted away. The thermal conductivity of the solid then becomes an advantage, since the hot surface layer is rapidly quenched by heat flow into the bulk, often freezing it in a non-equilibrium microstructure that has desirable properties. The common methods of rapid thermal treatment are induction heating, laser surface processing, and electron beam processing. Induction hardening If a high-frequency alternating current is passed through a coil located near the surface of a metal it induces electrical currents. These raise the temperature by Joule heating. If the coil is small and close to the surface the induced current does not penetrate very far into the interior. If the induction current is shut off after a short time, the heated layer cools rapidly as heat flows into the interior.

hardness

x

casedepth

Fig. 12.1: The variation of hardness with depth below the surface for a

case-hardened part. Local induction heating is a common method for producing a hard, wear-resistant surface on steel parts, such as gears, shafts and bearing races. A layer of material adjacent to the surface is heated to a temperature above that at which the low-temperature BCC (ferrite) phase reverts to the high-temperature FCC (austenite) phase, and then quenched to form a layer of fresh martensite. The fresh martensite layer is very hard because of its high defect density. The induction hardening of a typical carbon steel produces a hardness profile like that shown in Fig. 12.1. The rapid drop in hardness defines the limit of the martensite layer, and measures what is called the "case depth" of the hardened part. The manufacturing specifications for hardened gears set both the level of surface hardness and the case depth over which some minimum hardness must be maintained. Laser surface processing Laser surface processing is an even more efficient method of thermal treatment. In this method the surface is subjected to an intense pulse of coherent light from a high-power laser. The frequency of the light is chosen so that a high fraction of the incident


Page 387

energy is absorbed within a short distance of the surface. The almost instantaneous absorption of the optical energy of the laser beam by a small volume of solid raises its temperature dramatically. Commonly a layer of material parallel to the surface is actually melted by the beam. However, since the melted layer is small in thickness, it is very rapidly quenched back into the solid state as heat flows into the bulk. A significant advantage of laser surface treatment is the thin layer that is processed; properly designed laser treatments can create treated layers whose thickness is on the order of microns. Laser surface processing is sometimes used simply to modify the microstructure of the surface layer. Passing a laser beam over the surface of a metal may create a thin surface layer with a highly defective microstructure that has exceptional hardness, and, hence, good wear resistance. Laser surface treatment of a semiconductor, such as silicon, can produce a thin amorphous layer that has useful electrical properties. Lasers are also used to make surface layers with combinations of composition and microstructure that are not easily obtained in other ways. For example, elements such as carbon, phosphorous and boron have very limited solubility in solid steel, but are much more soluble in the liquid. If a thin layer of carbon or phosphorous is deposited onto the surface of a steel and then treated with a laser pulse, the carbon or phosphorous dissolves into the surface layer while it is molten, and is trapped there by rapid solidification. The result is a very highly doped surface layer with very high hardness. It may also be amor-phous in its structure. Among other potentially useful properties, amorphous surface films on metals often lead to exceptional corrosion resistance. The uniformity of an amorphous coating eliminates the local galvanic couples that promote corrosion of normal metal surfaces. Electron beam treatments High-energy electron beams are also used for surface treatments. When an electron beam strikes a metallic conductor, its energy is absorbed in the layer immediately beneath the metal surface, so the beam has nearly the same effect as a laser pulse. Electron beams are advantageous in that they easier and cheaper to create than intense laser pulses, and provide a more efficient energy transfer to the metal. However, it is difficult to produce electron beams with the short, intense pulses that are possible with lasers, so it is more difficult to confine the heating to the immediate surface region. High-energy electrons have complex interactions with semiconductors; laser processing is usually more controllable and useful for these materials. 12.2.2 Diffusion Across the interface The simplest method for changing the composition near the solid surface is by diffusion from the environment. Diffusion can be induced by heating the solid in a vapor that contains the solute or by plating the solute onto the surface and maintaining a high temperature until it diffuses in. The two processes are roughly equivalent. In the former case the surface concentration is fixed by adsorption of solute from the vapor phase. In the latter, it is fixed by the composition of the deposited surface film.


Page 388

The concentration profile The solute concentration at a distance, x, below the surface at a time, t, can be found by solving the diffusion equation (Fick's Second Law; Chapter 9):

∆∆t c(x,t) = D

∆2

∆x2 c(x,t) 12.1

where D is the diffusion coefficient for the solute. If the solute concentration at the surface is fixed (c = c0 at x = 0), the initial concentration in the solid is negligible, and the solid is very large compared to the effective diffusion distance, then the solution to equation 12.1 is

c(x,t) = c0

1 - erf

x

2 Dt 12.2

where erf(∫) is a function that is called the error function, and has the value

erf(∫) = 2π

⌡⌠0

∫ e- u2

du 12.3

The error function is tabulated in most compilations of mathematical functions.

c0

c

x

increasingtime

Fig. 12.2: Successive profiles for diffusion from the interface.

The concentration profile is plotted for several values of the time in Fig. 12.2. The equation x = 2∫ Dt 12.4 gives the value of x at which the concentration is c = c0[1 - erf(∫)] at time t, so the thick-ness of the layer that has been doped to a given concentration increases with the square root of the time. Since erf(∫) « ∫ to within a few percent when ∫ is less than about 0.75, the concentration profile satisfies the simple equation


Page 389

cc0

« 1 - x

2 Dt 12.5

for c/c0 greater than about 0.25. Equation 12.5 can be used to set the temperature and time for a process that is intended to insure a minimum dopant concentration within a pre-selected depth. At the mean diffusion distance, –x = 2Dt , the solute concentration is approximately 0.3c0. Surface diffusion has the advantage of being a relatively simple and inexpensive processing step. Its principle disadvantages are that it produces a steep concentration gra-dient rather than the flat concentration profile that is often desirable, and requires a rela-tively high temperature so that diffusion is reasonable fast. Surface hardening: carburizing and nitriding A common use of diffusion from the vapor is in the hardening of metal surfaces through carburizing and nitriding. The most common method of carburizing a steel part is to pass the part through a furnace that contains an atmosphere from which carbon diffuses into the metal. The carbon hardens the steel in one of two ways, depending on the temperature at which the carbon is added. If the carburizing temperature is below the temperature at which the steel reverts from the BCC to the FCC phase (the austenitization temperature), then the carbon is added to a microstructure that is more or less fixed. Carbon hardens the steel by an amount that is roughly proportional to the carbon concentration, but is limited by the very low solubility of carbon in the ferritic phase; if the carbon concentration exceeds the solubility limit it precipitates into carbides that form primarily on the grain boundaries and do not increase the hardness very much further. In the usual case carbon is added at high temperature where the steel has the FCC austenite phase. The FCC phase has a relatively high carbon solubility. On subsequent cooling, the austenite transforms to martensite with carbon trapped at high concentration. This martensite is very hard because of its high density of crystal defects, and forms a hardened surface layer. The hardness of the carburized layer is roughly proportional to the fraction of martensite and varies with distance beneath the surface roughly as shown in Fig. 12.1. Nitrogen is an alternative to carbon for surface hardening. Nitrogen also has a very low solubility in ferritic steel, but has an advantage over carbon in that excess nitrogen forms hardening precipitates within the ferrite grains, particularly in steels that are alloyed with Mo, Cr or Al. For this reason steels can be nitrided at relatively low temperature where they retain the ferritic structure. In addition, nitrogen has a high solubility in the FCC phase of iron (austenite) and is, therefore, particularly useful in hardening high-alloy stainless steels that retain the FCC structure at room temperature.


Page 390

Semiconductor doping As we shall discuss below, the electrical properties of semiconductors are con-trolled by adding small concentrations of electrically active impurities that act as donors or acceptors of electrons. Almost all semiconducting devices are based on semiconductor junctions, which are surfaces across which the nature of the conductivity changes from n-type conduction based on excess electrons in the conduction band from donor impurities to p-type conduction based on the holes in the valence band from acceptor impurities. A simple example is diagrammed in Fig. 12.3. To create these devices it is necessary to in-troduce controlled distributions of selected impurities.

p nn

Fig. 12.3: A semiconductor with an n-p-n junction near its surface. The

n- and p-type regions are doped with donor and acceptor im-purities, respectively.

A typical method for doing this is diagrammed in Fig. 12.4, where we have used silicon as a specific example. A thin layer of oxide is plated over the silicon surface by reacting with oxygen. Using techniques that will be described in a later chapter, the oxide is etched away over the region that is to be doped. The crystal is then exposed to an atmosphere that contains the dopant impurity, which adsorbs or deposits onto its surface. The sample is held at a temperature high enough to stimulate diffusion into the silicon, but low enough that diffusion through the oxide layer is negligible. The result is a distribution of dopant through the surface region beneath the cut in the oxide layer, as illustrated in Fig. 12.4b. If the cut in the oxide layer is large compared to the mean diffusion distance the concentration profile well inside the cut is the one-dimensional profile given by eq. 12.2. In any case, the diffusion profile varies with distance into the solid roughly as illustrated in Fig. 12.2.

(a) (b) Fig. 12.4: Locally doping a semiconductor. (a) An oxide layer is etched

to expose surface and the dopant is deposited. (b) Dopant is diffused in to create a diffuse doped region and the oxide re-moved.

12.2.3 Ion implantation Ion implantation provides a more direct means for controlling the composition be-neath the surface. In this technique the dopant species is ionized and accelerated toward the solid surface by an electric field. The kinetic energy of ions causes them to penetrate


Page 391

a short distance into the solid before decelerating and embedding. The mean penetration distance is fixed by the kinetic energy, which can be controlled by setting the potential through which the ions are accelerated. Ion implantation ordinarily leads to a composition profile like that shown in Fig. 12.5; the profile has a Gaussian form that is centered about the mean penetration distance, ∂. Ion implantation has three advantages over diffusional doping. First, virtually any species can be implanted by ionization, including those whose low diffusivities or low solubilities make them difficult to add by diffusional doping. Second, it is not necessary to heat the material before implantation, so dopants can be implanted into crystals that would degrade if they were heated to the temperatures required for diffusional doping. Third, ion implantation creates a very different initial composition profile than diffusional doping. If the material is heated slightly to diffuse the ion after implantation, the Gaussian profile of implanted ions spreads to become relatively flat, as illustrated in Fig. 12.5. The result is a much more uniform composition over the doped region. This is a particularly desirable feature in the manufacture of semiconducting devices, which should have relatively constant electrical properties within each separately doped region.

diffusion

ion implantation

c

x Fig. 12.5: Typical dopant profiles produced by ion implantation, and by

impantation followed by diffusion. The disadvantages of ion implantation include the relative complexity of the equipment required, which makes it expensive to use ion implantation for the surface treatment of large metal parts, and the physical damage that is done by the ions as they burrow into the solid at high energy, which causes defects that degrade the electrical properties of semiconductors. The latter problem requires that ion implantation processes for semiconductors be carefully designed, by adjusting the kinetic energy of the ions, their implantation rate, and subsequent annealing treatments, so that residual defects are held within tolerable levels.


Page 392

12.3 CHEMICAL REACTIONS AT THE SURFACE: OXIDATION The second class of reactions that occur between a solid and its environment are chemical reactions that cause the formation and growth of new compounds at the interface. The most familiar of these is oxidation. If a typical metal or semiconductor is heated to high temperature in air an oxide forms on its surface that grows until the material is consumed. We shall focus on the oxidation reaction because of its familiarity, its importance in engineering, and its value as a prototypic example of compound formation at an interface. However, oxidation is only one of many interfacial reactions that are important in the processing and use of engineering materials, and is a particular example of a reaction at a solid-vapor interface. Reactions that occur at solid-solid interfaces have distinct features that should also be appreciated. 12.3.1 Thermodynamics of oxidation When a metal or elemental semiconductor is exposed to an atmosphere that contains strong oxiding agents such as oxygen, sulfur or chlorine, the solid and gas react at the interface to form a scale of oxide, sulfide or chloride. If the reactant is oxygen the chemical reaction is

xM + y2 O2 = MxOy 12.6

where M is the solid MxOy is the stoichiometric formula for the oxide. The free energy change per mole of oxygen consumed is

Îg = 2y µ(MxOy) -

2xy µ(M) - µ(O2) 12.7

The chemical potential is usually written in the form µ(T,P,c) = µ0(T) + RTln(a) 12.8 where µ0 is a function of T (we assume atmospheric pressure), and a, which is called the activity of the species, gives its dependence on the composition, c. Eq. 12.8 is, in fact, the definition of the activity. The activity is a simple function of the composition in four limiting cases: (1) if the specie is pure (c = 1), then a = 1 and µ0 is its molar Gibbs free energy in the pure state; (2) if the specie is the solvent of a dilute solution (c « 1), then a = c and µ0 is the molar free energy in the pure state (this is called Raoult's Law); (3) if the specie is a gas in an approximately ideal gas mixture then µ0 is its free energy in the pure state and a = p, its partial pressure in the gas; (4) if the specie is a solute in a dilute liquid or solid solution (c << 1) then a = ©c, where © is called the activity coefficient, and has a constant value when c is small (this is known as Henry's Law). If the oxidizing solid, M, is nearly pure and the oxide, MxOy, is formed in the solid state, equation 12.7 can be rewritten


Page 393

Îg = Îg0 - RT ln(pO2

) 12.9

where Îg0 is called the standard free energy of formation of the oxide,

Îg0 = 2y µ0(MxOy) -

2xy µ0(M) - µ0(O2) 12.10

and pO2

is the partial pressure of oxygen in the gas. When the reaction is in equilibrium the oxygen partial pressure is given by Îg0 = RT ln(pO2

) 12.11

when the partial pressure exceeds this value the sample oxidizes.

-1000

-200

-800

-600

-400

0

200 400 600 800 1000

Al O2 3

SiO2

Cr O2 3

NiOFeO

Cu O2

Îgº (J/mole)

T (ºC)

Fig. 12.6: Standard free energies of formation of selected oxides. The standard free energy of formation of several common oxides are plotted as a function of temperature in Fig. 12.6. Îg0 is negative for all common structural metals and elemental semiconductors. The only notable exceptions are the noble metals, Au, Ag and Pt, whose oxidation resistance is largely responsible for their economic value. Two features of the data shown in Fig. 12.6 may be surprising. First, the common metals have very negative values of Îg0 at low temperatures, which suggests that they should oxidize rapidly in air at room temperature. Second, Îg0 increases (becomes less negative) as the temperature rises; the thermodynamic driving force for oxidation decreases with temperature. The engineering behavior of structural metals and alloys is the opposite of this. Metals do not oxidize appreciably at room temperature, but do oxidize at high temperature at a rate that increases with the temperature. This behavior shows that the oxidation that is of practical concern is controlled by the kinetics of the process.


Page 394

12.3.2 The kinetics of oxidation Oxidation occurs by one of three mechanisms that lead to the three distinct kinetic relations illustrated in Fig. 12.7. In the most basic reaction the oxide grows as a continuous film that thickens through the reaction of the metal and the oxide at the oxide-vapor interface or, less commonly, at the metal-oxide interface. The rate of this reaction is limited by the rate at which the reactants can diffuse through the film, and obeys the parabolic relation Îm = k t 12.12 where Îm is the mass of metal oxidized in time t. The second mechanism, which is re-sponsible for rapid oxidation at high temperature, occurs when the stresses that are devel-oped in the growing film cause it to crack or spall off of the surface as it grows. The re-curring damage to the film continuously exposes fresh surface to the oxidizing environ-ment, and the rate of oxidation is linear: Îm = k't 12.13 The third mechanism applies to very thin films, particularly those that form at low temperature. It remains poorly understood. However, it is clear that when a film is only a few nanometers thick its growth is strongly affected by inhomogeneous charge distributions that develop as the metal and oxygen ionize at the film surfaces. The electric field set up by these charges promotes the diffusion of ionic species. The result is a logarithmic behavior in which the oxidation is given by an expression of the approximate form Îm = k"ln(Bt + 1) 12.14

Îm

Ôt t ln(t) (a) (b) (c)

Fig. 12.7: The three common kinetic relations for interfacial reactions: (a) parabolic growth; (b) linear growth; (c) logarithmic growth.

Parabolic oxidation To understand parabolic oxidation, assume that a thin oxide film has formed, as shown in Fig. 12.8, and consider the sequence of events that must happen at the atomic level in order for it to grow. As shown in the figure, oxygen molecules (O2) are absorbed


Page 395

onto the surface and dissociate into adsorbed oxygen atoms. To grow the film, these must ionize and react with ionized atoms from the underlying metal. In most cases the reaction occurs at the outer surface of the oxide layer. Metal atoms ionize at the inner surface of the oxide. The metal ions (M++) diffuse into the oxide film, creating a diffusion current of ions to the surface. At the same time, electrons are transported to the surface as an electrical current; most metal oxides are at least semiconducting and permit some current flow. Both electrons and ions participate in the oxidation reaction at the outer surface: M++ + 2e- + O “ MO 12.15

metal

oxide2e-

OO2 O= MO

M++

Fig. 12.8: The typical mechanism of oxide film growth. Oxygen atoms

adsorb, dissosciate, ionize by accepting electrons from the metal, and grow oxide by combining with ions from the metal.

The rate-limiting step in the reaction described above is the diffusion of metal ions through the oxide film. Since the mean diffusion distance of the metal ions is proportional to Dt , where D is the ion diffusivity, the film thickness, ∂, varies with time as ∂ = A Dt 12.16 where A is a constant. The total mass of metal consumed is simply proportional to the film thickness; hence Îm = k t 12.12 which is the parabolic growth law. The rate of oxidation, the mass of metal consumed per unit time, is

dmdt =

k2 t

12.15

which decreases monotonically with the time.


Page 396

The temperature dependence of the oxidation rate is primarily due to its dependence on the diffusivity. The rate constant, k, varies with temperature as

k = Aexp

- QD2kT 12.16

where QD is the activation energy for diffusion of the rate-limiting species, usually the metal ion. The rate of oxidation is very low at low temperatures where the diffusivity is negligible. The rate increases exponentially with (1/T), and becomes significant when the temperature is high enough for solid state diffusion to proceed at a reasonable rate. To understand the rate of film growth in a particular case one must identify the rate-controlling specie and understand its diffusion mechanism. In most cases the fast-diffusing species is the metal cation. It is useful to divide the oxides that are controlled by cation diffusion into two groups: metal-deficit oxides (called p-type oxides after their semiconducting characteristics) and metal-excess oxides (n-type oxides). The two types of oxide are illustrated in Fig. 12.9. Metal-deficit oxides contain point defects of the Shottky type (described in Chapter 4) in which paired vacancies appear on the cation and anion lattices. Metal ions diffuse by cation-vacancy exchange on the cation lattice, and, since the cations are more mobile, govern the rate of oxide growth. Given the presence and mobility of cation vacancies, the metal content in these oxides is, ordinarily, a bit below the stoichiometric value, giving them a p-type electrical character. Common oxides that fall into this class include Cr2O3, MnO, FeO, NiO and Cu2O.

= cation (+)

= anion (-)

= vacancy

(a) (b)

... Fig. 12.9: Defects in oxides. (a) Shottky defect: paired vacancies on

anion and cation sites. (b) Frenkel defect: cation interstitial and vacancy.

Metal-excess oxides contain point defects of the Frenkel type (Chapter 4), in which a cation interstitial is paired with a cation vacancy. Metal ions diffuse as highly mobile cation interstitials. In these oxides there is ordinarily a slight excess of in-terstitials that are injected by ionization at the oxide-metal interface, hence these oxides tend to be n-type semiconductors. Common metal-excess oxides include BeO, MgO, CaO, ZnO, and CdO. A third group of oxides form at a rate that is governed by the diffusion of oxygen. Most of these are oxides that have an excess of oxygen vacancies due to multiple valance


Page 397

of the cation. The classic example is ZrO2. Zr can have valence 2 or 4, and ZrO2 films have high concentrations of oxygen vacancies with the missing charge balanced by Zr cations with valence 2 rather than 4. The high concentration of oxygen vacancies has the consequence that oxygen diffusion controls the rate of film growth, and the oxide thickens at the metal-oxide interface rather than at the oxide-vapor interface. Whatever the dominant ion, almost all oxide films that form at moderate to high temperature obey parabolic growth kinetics during the initial stages of their growth. Since the rate of oxidation decreases monotonically with time, these films are protective; they seal the surface from catastrophic oxidation. Linear oxidation A parabolic oxidation rate is maintained as long as the oxide film remains intact and well-bonded to the metal surface. However, in most cases of high temperature oxidation the film does not remain intact. The reason is that the oxidation reaction involves a volume change. The volume change per mole of oxide produced is described by the Piling-Bedworth ratio

P = vO

nvM 12.17

where vO is the molar volume of the oxide, vM is the molar volume of the metal, and n is the number of moles of metal consumed to create a mole of oxide.

(a) (b) ... Fig. 12.10: Modes of film failure. (a) Film cracks due to a negative vol-

ume change. (b) Film spalls due to a positive volume change.

If P < 1, as it is, for example, for the oxides of most alkali metals, then the volume of the oxide formed is less than that of the metal consumed. The oxide can only be fit onto the metal surface by stretching it in tension. It can be shown that the tension in the film increases with its thickness. When the film is very thin the tension is sustainable, but when it grows to a critical thickness, the tension exceeds the breaking strength of the film and it cracks, exposing fresh surface to the oxidizing atmosphere. The film also fails if P is much greater than 1, as it is in the case of iron oxides. When P > 1 the stress in the film is compressive. As the film grows, the compressive stress increases. Some oxides are sufficiently deformable that the compressive stresses


Page 398

can be relieved by plastic flow. Others are not. If the compressive stress cannot be relieved by plastic flow, it accumulates until the oxide fails, usually by losing cohesion at the interface and spalling away from the surface, as shown in Fig. 12.10(b). This process is responsible for the flakes of oxide that appear on the surface of oxidized iron. The failure of the oxide film exposes fresh surface to the oxidizing atmosphere, and increases the oxidation rate. In either case the local failure of the oxide film periodically exposes fresh surface to the oxidizing medium. The fresh surface oxidizes through the formation of a film that grows until it also fails. The net result is an almost linear corrosion rate, like that shown in Fig. 12.9(b). Several other mechanisms can also lead to failure of the protective film, and to an almost linear oxidation rate. The most spectacular case is called catastrophic oxidation, and occurs when the oxide film melts. Relatively low-melting oxides are formed on V, Mo, Bi and Pb. A molten oxide layer permits the diffusion of oxygen to the metal surface, producing a very rapid, linear oxidation. In other systems, such as Al, Si and W, the oxide is volatile at high temperature. These metals develop a steady-state oxide film whose thickness is such that the rate of oxide formation by ion diffusion to the surface is just equal to the rate at which oxide is removed by evaporation. Once steady state is achieved, the oxidation is linear. A final case includes metals that form porous oxide films, as Ni does at high temperature. These are metals that form "metal-deficit" oxides in which diffusion occurs by a vacancy mechanism on the cation sublattice. The net flux of cations (M++) to the oxide surface produces a counterflow of cation vacancies toward the metal surface. At high temperature the vacancy flux may becomes sufficient to create local vacancy supersaturations at imperfections in the film that nucleate voids. This process can lead to a porous oxide film. The surfaces of the pores provide easy diffusion paths that increase the oxidation rate, and may result in linear oxidation if the porosity is severe. Linear oxidation is responsible for the relatively rapid oxidation of most metals at high temperature. To control high-temperature oxidation it is critically important to prevent linear oxidation by ensuring the retention of a coherent oxide film. Logarithmic oxidation Logarithmic oxidation, like that shown in Fig. 12.9(c), governs the very thin oxide films that form on metals that are exposed to air at low temperature. This mechanism governs behavior under conditions where the oxidation rate is almost negligible. The mechanism of logarithmic oxidation remains controversial. There are several competing theories, any one of which may pertain to a particular oxide system. All are based on the fact that film growth requires the transport of both ions and electrons. Since the ions and electrons move independently, charge inhomogeneities build up in the film and govern the rate of oxidation.


Page 399

The simplest and, arguably, most plausible theory of logarithmic growth is due to Mott and Cabrera. It is based on the observation that electrons are much more mobile than ions. It is, therefore, possible to develop a layer of ionized oxygen at the oxide-va-por interface. This produces an excess negative charge at the oxide surface that is balanced by excess positive charge at the metal surface (Fig. 12.11) The charge separation creates an electric field within the film, which can be very intense if the film is only a few nanometers thick. The electric field imposes a force on the cations and increases their diffusion rate (the enhancement of diffusion of a charged species in the presence of an electric field is known as electromigration). However, the field strength decreases with film thickness, and the effective diffusivity of the cations falls back to the field-free value when the film thickness exceeds about 100 nm. The result is a logarithmic rate of film growth.

O

metal

oxide2e-

O=

M++

O=O=O= O= O= O=

M++

M++

M++

M++

M++ E

... Fig. 12.11: Mechanism of logarithmic growth. Oxygen and metal ions

are created at the two interfaces. The resulting electric field, E, raises the rate of cation diffusion through the film.

Since the cation diffusivity in most oxides is negligible at room temperature, when a metal is exposed to ambient air logarithmic growth leads to the formation of a very thin film that essentially ceases to grow after it has reached a few nanometers in thickness. At higher temperature, logarithmic growth governs the initial kinetics of film formation, but growth becomes parabolic as soon as the film thickness exceeds a few nanometers. 12.3.3 Protecting against oxidation Protective films The principal requirement for oxidation resistance is the presence of an intact, co-herent oxide film. Some materials, such as Al and Cr, form such films naturally and suc-cessfully resist oxidation until the temperature becomes so high that the oxide loses stability. Most other materials form protective films at low temperature, where low cation diffusivity limits film thickness, but lose the protective layer at higher temperatures where the film cracks, spalls or becomes porous. It is often possible to increase the oxidation resistance of these materials by alloying with species that are incorporated into the oxide film and help preserve its integrity.


Page 400

An alloy addition will be effective if it changes film growth in either of two ways. First, an alloy addition that lowers the diffusion rate through the film increases the time required for the film to grow to the thickness at which film stress causes failure. Second, an alloy addition that improves the geometric fit between the film and the substrate reduces stresses and permits significantly greater film growth before failure. The most common example of a beneficial alloy addition is Cr in Fe. Fig. 12.12 illustrates the variation of the parabolic rate constant for corrosion of Fe-Cr alloys at 1000ºC as a function of Cr content. The addition of 10% Cr lowers the rate constant by more than four orders of magnitude. The reason is the preferential incorporation of Cr into the oxide film, which produces a mixed oxide with a high content of Cr2O3. The low cation diffusivity through Cr2O3 decreases the parabolic rate constant. The parabolic rate constant decreases monotonically with Cr content up to about 18% Cr, by which point the initial oxide film is essentially pure Cr2O3. At the same time, the Cr-bearing film has improved coherence with the substrate. Steels that contain more than 10.5% Cr are called stainless steels because of their excellent oxidation and corrosion resistance. Al alloy additions have a similar beneficial effect; in this case the protective film is Al2O3.

ln (k)

Cr (wt%)5 10 15 20

Fig. 12.12: The influence of Cr on the parabolic rate constant for

oxidation of steel. While significant additions of Al and Cr replace the oxide film with one that has a significantly lower growth rate, other alloy additions influence the oxidation rate by changing the cation diffusivity in the original oxide film. The alloy additions that have this effect decrease the concentration of the defects that are responsible for diffusion. In metal-excess (n-type) oxides the diffusing species are interstitial cations, such as Zn++ interstitials in ZnO (Fig. 12.9(b)). Since these interstitials produce a local excess of positive charge, their formation energy is reduced by the addition of cations that have lower valence, and is increased by the addition of cations with higher valence. It follows that the concentration of interstitials increases with the concentration of low-valence solutes, and decreases with the concentration of high-valence solutes. Hence a small alloy addition of Li, which ionizes to Li+, increases the parabolic rate constant for the oxidation of Zn, while a small addition of Cr, which ionizes to Cr+3, decreases it.


Page 401

ln (k)

Cr (wt%)5 10 15 20

Fig. 12.13: The influence of Cr on the parabolic rate constant for

oxidation of Ni. These species have the opposite effect on metal-deficit (p-type) oxides, such as NiO. Cations diffuse through these oxides by exchanging with vacancies. Since cation vacancies cause an excess of negative charge, their concentration is increased by substitu-tional cations that have higher valence, and decreased by cations of lower valence. Hence a small alloy addition of Li decreases the rate constant for oxidation of Ni, while a small addition of Cr increases it. This effect explains the anomalous oxidation behavior of Ni-Cr alloys, which is illustrated in Fig. 12.13. The parabolic rate constant for oxidation of Ni at 1000ºC increases with the addition of Cr up to 3 wt.%, but decreases if the Cr content is increased to 10 wt.%. When the Cr content is small the Cr is incorporated substitutionally into the NiO film, and increases Ni++ diffusion by raising the vacancy concentration. A higher chromium content produces a mixed oxide, and, eventually, a Cr2O3 protective layer, which significantly decreases the rate of oxidation. Protective coatings While one can increase oxidation resistance by alloying, there are limits to how effectively this can be done, and there are often associated disadvantages, since the alloy additions affect other properties as well. For this reason structural alloys that are intended for use in extreme environments are often processed to optimize their mechanical properties, and then coated with a second alloy that has exceptional oxidation resistance. An effective coating must satisfy at least two criteria. It must form a stable protective oxide film, and it must remain attached to the metal during service, which often includes repeated cycles to elevated temperature. The first criterion suggests the use of elements like Cr and Al that are known to form coherent oxides with low diffusivity. The second criterion requires that the coating remain well bonded to the substrate through severe thermal cycles. To accomplish this, the coating and its protective oxide layer must have thermal expansion coefficients that do not differ greatly from one another or from that of the protected metal. Otherwise the coating will develop thermal stresses that cause it to spall off of the underlying metal.


Page 402

The most successful of the coatings now in use are alloys of cobalt, chromium, aluminum and yttrium (called CoCrAlY coatings). Cr and Al are the oxidizing species that form the protective film. Their concentration is balanced with that of Co to minimize the thermal expansion mismatch with the underlying metal. A small addition of yttrium substantially improves the retention of the protective layer during thermal cycling. There are several possible explanation for the yttrium effect. One that has attracted recent attention is the apparent action of Y in "pegging" the oxide to the coating; Y2O3 forms at the coating-oxide interface in the form of spikes that penetrate into the coating and act as pegs to prevent its decohesion during thermal cycling. 12.4 ELECTROCHEMICAL REACTIONS Most metals are protected from oxidation at room temperature by the slow kinetics of the process. If this were not true much of engineering, and a good part of civilization, would not exist. Unfortunately, there is an alternative reaction mechanism that can oxidize metal at an appreciable rate at ambient temperature. It is a familiar fact that iron rusts. It converts to its oxide at a disturbingly rapid rate if it immersed in water, embedded in the soil, or simply left out in the rain and fog. The corrosion of iron and other structural metals is a costly engineering problem. The mechanism that permits oxidation at ambient temperature is the electrochemical reaction that dissolves metal in ionic solution. While most metals ionize rather easily when they are in contact with an aqueous solution at ambient temperature, the reaction is impeded by the fact that the release of a positive ion leaves electrons behind in the metal. The accumulation of negative charge quickly stops the release of positive ions. If, however, the metal is joined with another in an electrical circuit through which electrons can flow, the metal can be kept electrically neutral as it ionizes so that dissolution proceeds at a significant rate. Two metals that are joined electrically so that electrons can pass between them, and are in contact with an ionic solution through which ions can migrate to complete an electrical circuit are said to form an electrochemical cell. One of the metals, the anode in the cell, gradually dissolves. The ions that are released into the solution as it dissolves ordinarily combine with hydroxyl ions from the water to form oxide; the net effect is the oxidation of the metal. The electrochemical reactions that cause corrosion are harmful. However, other electrochemical reactions are useful, and are used in many important engineering devices. A familiar example is a battery, in which an electrochemical reaction generates electric power. A second important example is electroplating, in which a controlled electrochemical reaction is used to deposit one metal as a coating on the surface of another. In this section we shall first discuss the general principles that govern the behavior of electrochemical, or galvanic, cells, and then focus on the important issue of aqueous corrosion and its prevention. Here we will consider only metal removal by


Page 403

corrosion. As we shall see later, corrosion can also affect the mechanical properties of a material through such mechanisms as stress corrosion cracking, corrosion fatigue, and hydrogen embrittlement. 12.4.1 The galvanic cell Suppose we place plates of two dissimilar metals, say Zn and Cu, in an solution in a configuration like that shown in Fig. 12.14. Each metal is in equilibrium with an ionic solution (electrolyte) that contains its own ion, Zn++ in the case of Zn, Cu++ in the case of Cu. For simplicity, let both of the solutions be sulfate: ZnSO4 and CuSO4, and assume that the ion concentrations in the solutions, [Cu++] and [Zn++] are fixed at the same values (this can be done by setting the concentration of the solution and making it large compared to the metal). The two metals are in equilibrium with their respective solutions; nothing further happens. Now suppose that we change the experiment in two ways. We connect the Zn and Cu plates with a thin conducting wire, and we connect the solutions with a salt bridge, which may be just a permeable membrane through which ions such as SO4= can flow to preserve charge neutrality. This simple act triggers a spontaneous chemical reaction. The Zn plate begins to dissolve, while the Cu plate grows by the addition of Cu from the solution. This experiment illustrates two important processes: the galvanic corrosion of one metal (Zn) by another (Cu) and the electroplating of a metal (Cu) onto the surface of an electrode. The experimental arrangement in which this happens is called a galvanic cell.

Zn CuZn

++Cu+

SO4= SO4

=

V

e-Zn “ Zn++

+ 2 Cu+ e-+ “ Cu

... Fig. 12.14: A galvanic cell between Cu and Zn, connected by a sulfate

solution and a wire and voltmeter. The anode (Zn) and cath-ode (Cu) reactions are indicated.

A galvanic cell is formed because the two metals differ in their electrical potential (ƒ). The ionization reaction at the Zn-solution interface, Zn ¿ Zn++ + 2e- 12.18


Page 404

produces an electrical potential within the Zn, ƒZn, whose magnitude is given by the Nernst equation

ƒZn = ƒ0Zn +

RT2F ln[Zn++] 12.19

where ƒ0

Zn is the standard potential for the half-cell reaction given in eq. 12.18, F is the Faraday constant , the charge in a mole of electrons, F = N0e 12.20 where N0 is Avogadro's number and e is the magnitude of the electron charge, the factor 2 appears because there are two electrons produced in the reaction, and [Zn++] is the concentration of Zn++ ions in the solution. The electrical potential in the Cu is fixed by the ionization reaction at the Cu interface, Cu ¿ Cu++ + 2e- 12.21 and is

ƒCu = ƒ0Cu +

RT2F ln[Cu++] 12.22

When the two metals are joined electrically their potential difference is

Îƒ = ƒZn - ƒCu = Îƒ0 + RT2F ln

[Zn++]

[Cu++] 12.23

Since the initial concentrations in the two solutions are the same, the potential difference is Îƒ = Îƒ0 = ƒ0

Zn - ƒ0Cu 12.24

where Îƒ0 is the standard potential for a Zn/Cu galvanic couple through a sulfate solution. This is the potential that would be measured by a voltmeter placed as shown in Fig. 12.14. The standard potential, Îƒ0, is negative for this particular case; ƒ0

Cu > ƒ0Zn .

Since electrons spontaneously flow from regions of low potential to regions of high potential, there is a net flux of electrons from Zn to Cu. The electrons are produced by the ionization of Zn at the Zn-solution interface according to eq. 12.18, and are consumed by the formation of Cu from Cu++ ions at the Cu-solution interface by reversing eq. 12.21. The circuit is completed by the flow of SO4= ions toward the Zn across the membrane separating the solutions. The net reaction is


Page 405

Zn + Cu++ “ Cu + Zn++ 12.25 The Zn electrode, where ions and electrons are produced, is called the anode of the cell. The Cu electrode, where ions and electrons combine to form neutral atoms, is called the cathode of the cell. In this cell, Zn is corroded by Cu; the Zn anode gradually disappears. Cu is electroplated by Zn; the Cu anode grows through the deposition of Cu. The example discussed above uses the metals Zn and Cu. A similar galvanic cell is established if any two metals are brought into electrical contact with one another while placed in an electrolyte. Let one electrode be formed of metal, A, which ionizes to an ion of valence, a, while the other is made of metal, B, which ionizes to an ion of valence, b. Assume that A is the anode. The two electrode reactions are A ¿ A+a + ae- 12.26 B ¿ B+b + be- The half-cell potentials are

ƒA = ƒ0A +

RTaF ln[A+a]

12.27

ƒB = ƒ0B +

RTbF ln[B+b]

When the two electrodes are joined the overall reaction is, per mole of A consumed,

A + ab B+b “ A+a +

ab B 12.28

and the potential is

Îƒ = Îƒ0 + RTaF ln

[A+a]

[B+b]a/b 12.29

where

Îƒ0 = ƒ0A -

ab ƒ0

B 12.30

12.4.2 The electromotive series and the galvanic series If we ignore the influence of the ion concentration, the identity of the anode and cathode and the basic cell potential are determined by the standard potential, Îƒ0. The standard potential is the difference between the standard half-cell potentials, ƒ0, of the anode and cathode. The value of the standard half-cell potential is affected by the nature


Page 406

of the solution. As we shall discuss in more detail below, the standard potential decreases if the solution contains species that interact favorably with the metal ion and decrease its chemical potential. The half-cell potential is also affected by the temperature. Hence any tabulation of half-cell potentials is specific to a particular solution and temperature. A table of half-cell potentials is, nonetheless, useful. It helps to identify which electrode will be the anode and which the cathode in solutions like those for which the tabulation is made, and suggests the magnitude of the galvanic potential when the concentrations are uniform. Two tabulations of standard half-cell potentials are ordinarily included in standard texts on electrochemistry and corrosion. The first is the electromotive force series (EMF), which is a tabulation of the standard half-cell potentials for the solution of metals in aqueous solutions of their own simple salts at 25ºC. Since the zero of potential is arbitrary, the potentials are given relative to that of the formation of H2, H+ + e- ¿ H2 12.31 whose standard potential is arbitrarily set equal to zero. The electromotive force series for a number of important reactions is given in Table 1. The reactions are listed in descending order of ƒ0. It follows that when a galvanic cell includes two reactions in the series, then, in the absence of concentration effects, the one that appears higher in the series is the cathode reaction while the one that appears lower is the anode reaction. That is, the metal that appears higher in the series will corrode that which is listed lower down. The standard potential of the cell is the difference between the two half-cell potentials.

Table 12.1: The Electromotive Force Series (EMF)

Electrode Reaction Îƒº (25ºC)

Au “ Au+3 + 3e- 1.50 volts Hg “ Hg++ + 2e- 0.854 Ag “ Ag+ + e- 0.800 Cu “ Cu+ + e- 0.521 H2 “ 2H+ + 2e- 0.00 Pb “ Pb++ + 2e- -0.126 Sn “ Sn++ + 2e- -0.136 Ni “ Ni++ + 2e- -0.250 Cd “ Cd++ + 2e- -0.403 Fe “ Fe++ + 2e- -0.440 Cr “ Cr+3 + 3e- -0.74 Zn “ Zn++ + 2e- -0.763 Mn “ Mn++ + 2e- -1.18


Page 407

Zr “ Zr+4 + 4e- -1.53 Ti “ Ti++ + 2e- -1.63 Al “ Al+3 + 3e- -1.66 Mg “ Mg++ + 2e- -2.37 Ca “ Ca++ + 2e- -2.87 Li “ Li+ + e- -3.05

Table 12.2: The Galvanic Series in Sea Water

The second tabulation that is widely given is the galvanic series in sea water, which is reproduced as Table 2. This tabulation is qualitative. It is simply a columnar listing of metals in decreasing order of potential. In the absence of concentration effects, the metal that appears higher in table is cathodic with respect to one that listed lower in a galvanic couple in sea water. The more cathodic metal will ordinarily corrode the more anodic one when they are joined in a galvanic couple. The galvanic series incorporates two changes from the electromotive force series. The solution is different, and the series lists engineering alloys as well as pure metals. Since engineering alloys are multicomponent and, often, multiphase materials with complex microstructures, their standard potentials cover a range of values. The galvanic series indicates the relative position of this range of values with respect to the standard potentials of other metals and alloys. 12.4.3 The influence of concentration: concentration cells The potential of the galvanic cell shown in Fig. 12.14 depends on the concentra-tions of Zn++ and Cu++ ions as well as on the standard potential for the cell. The govern-ing relation is given by the Nernst equation, eq. 12.23. A decrease in the Cu++ concentra-tion relative to that of Zn++ reduces the potential difference between the two electrodes and lowers the driving force for the electrochemical reaction. It is even possible, in theory, to reverse the polarity of the cell, that is, to make the Cu electrode the anode in the cell by adjusting the ion concentrations.

Increasingly anodic ‘ Tin Magnesium Nickel Magnesium alloys Brasses (Cu-Zn) Zinc Copper Aluminum Bronzes (Cu-Sn) Al-Cu alloys Silver solders Mild steel Nickel (passive) Wrought iron Monel (70Ni-30Cu) Cast iron Titanium 18Cr-8Ni stainless steel (non-passivated) 18Cr-8Ni stainless steel (passive) 50Pb-50Sn solder Gold Lead Increasing cathodic ’


Page 408

Zn Zn++

SO4= SO4

=

V

e-Zn “ Zn++

+ 2

Zn

e-Zn++

+ 2 “ Zn

low

Zn++

high

Fig. 12.15: An example of a concentration cell between Zn++ solutions

of different concentrations. The low-Zn++ solution is the anode; the high-Zn++ solution is the cathode.

The influence of concentration is most obvious in a type of galvanic cell known as the concentration cell, an example of which is shown in Fig. 12.15. Let both electrodes be made of Zn, but let the concentrations of the two solutions be [Zn++]1 and [Zn++]2. Let the two electrodes be connected by a wire and the two solutions by a salt bridge that neutralizes the solution by passing negative ions. According to eq. 12.23 the potential difference in this case is

Îƒ = RT2F ln

[Zn++]1

[Zn++]2 12.32

which is negative if [Zn++]1 < [Zn++]2. It follows that the electrode that is in contact with the solution of lower concentration is the anode. It will be corroded, while the electrode at the cathode grows by adding Zn from the solution. The behavior of concentration cells simply illustrates how electrochemical reactions lead to thermodynamic equilibrium. The equilibrium condition of the solution in Fig. 12.15 is one in which the Zn++ concentration is uniform throughout. This equilibrium cannot be achieved by Zn++ diffusion through the solution, since the membrane that separates the two solutions is (by assumption) impermeable to Zn++. However, it can be achieved by an electrochemical reaction that adds Zn++ ions to the solution at the anode, where the concentration is relatively low, and removes them at the cathode, where the concentration is relatively high. 12.4.4 Reactions at the cathode In addition to Cu++ ions, the solution near the Cu cathode in Fig. 12.14 contains H+ and OH- ions from the ionization of water. Ordinarily, the solution also contains dis-solved O2 gas that has diffused in from the atmosphere. The equilibrium of H+, OH-, and


Page 409

O2 leads to two common reactions that compete with electrodeposition to consume elec-trons at the cathode: the evolution of hydrogen is governed by the reaction H2 ¿ 2H+ + 2e- 12.33 while the consumption of oxygen is governed by the reaction

OH- ¿ 14 O2 +

12 H20 + e- 12.34

The half-cell potential for the ionization of hydrogen gas is

ƒH2 = ƒ

0H2 +

RT2F ln

[H+]2

[H2] 12.35

Equation 12.35 can be written in a form that is easier to use. By convention, for aqueous solutions at ambient temperature, ƒ0

H2 = 0 12.36

The zero of energy, and, hence, the zero of potential is arbitrary, and electrochemists have agreed to set the zero so that equation 12.36 holds. The concentration of hydrogen gas in solution is approximately equal to the partial pressure of hydrogen in the gas above the solution, in atmospheres pressure. Hence

ƒH2 =

RTF ln[H+] -

RT2F ln(PH2

) 12.37

The concentration of hydrogen ion in solution, [H+], defines the pH of the solution by the relation pH = - log[H+] = - 2.303 ln[H+] 12.38 In these equations ƒH2

is the potential to which an electron is effectively raised when it joins with H+ to form H2, and is, hence, the effective cathode potential when hydrogen evolution is the governing reaction at the cathode. The half-cell potential for the consumption of oxygen in a basic solution is,

ƒOH- = ƒ0OH- +

RTF ln

[O2]1/4

[OH-] 12.39

When the oxygen concentration in solution is determined by equilibrium with a vapor in which the partial pressure of oxygen is PO2

(measured in atmospheres) , the half-cell potential is


Page 410

ƒOH- = ƒ0OH- +

RTF ln

(PO2

)1/4

[OH-] 12.40

As before, ƒOH- is the potential to which an electron is raised when it joins with dissolved oxygen and water to produce OH- ions at the cathode. In most cases the cathode reaction in a galvanic cell is hydrogen evolution or oxy-gen consumption rather than metal deposition. As we shall discuss further below, the dominant cathode reaction is the one that leads to the highest cell current. This is usually the reaction with the greatest half-cell potential, ƒ, since this reaction causes the highest potential difference across the galvanic couple. Unless the solution at the cathode is con-strained to have a high concentration of the cathodic metal species, the metal ion concentration at the cathode is very low, or quickly becomes so by cathodic deposition. As the metal ion concentration falls to a low value at the cathode, the cathode potential becomes strongly negative, as shown by eq. 12.27 (the logarithm of a number that is much less than 1 is large and negative). Hence the cathode reaction ordinarily produces H2 or OH- rather than deposited metal. In strongly acidic solutions or in solutions that have low oxygen concentrations, hydrogen is evolved at the cathode. In intermediate to basic solutions with moderate oxygen concentrations, oxygen is consumed. The oxygen concentration cell A galvanic cell that is very important in practice is the cell that is set up by a significant difference in the oxygen concentration at two electrodes that are otherwise the same. Consider the cell shown in Fig. 12.16, both of whose electrodes are made of Fe. Assume that the solutions in contact with these electrodes are approximately neutral (pH « 7),. but differ significantly in their oxygen concentrations. According to eq. 12.40, the solution with the higher oxygen concentration provides the higher electrode potential, so the electrode that is embedded in this solution should be the cathode of a concentration cell. The anode is the electrode in the solution that is relatively poor in oxygen.

V

e-Fe “ Fe++

+ 2

Fe++

Fe Fe

(OH)-

anode:O2low high

O2

cathode:O2 + 2H O + 2e “ 42

- (OH)-

... Fig. 12.16: An oxygen concentration cell between two Fe electrodes.

Fe++ is produced at the low-O2 anode, (OH)- at the high-O2 cathode.


Page 411

If we consider only the oxygen solution then the reaction at the anode would appear to be the evolution of oxygen by the reaction of OH- ions, according to eq. 12.34. However, there is ordinarily a much more favorable anodic reaction; the electrode potential for the ionization of iron Fe “ Fe++ + 2e- 12.41 is much lower than that for the evolution of oxygen. It follows that Fe++ ions are produced at the anode, and the overall reaction is

Fe + 12 O2 + H2O “ Fe++ + 2OH- 12.42

Equation 12.42 is the reaction that usually governs the atmospheric corrosion of iron. The Fe++ ions and the (OH)- ions react to form iron oxide (rust). This simple example shows how a gradient in the concentration of oxygen can cause corrosion even in the absence of dissimilar metal contact. 12.4.5 The influence of an impressed voltage Since the galvanic cell is an electrical circuit, it is possible to impose a voltage to control the direction or rate of the reaction. External voltages are commonly used for three purposes. First, an impressed voltage can halt the corrosion of a natural anode by reversing the galvanic cell. If the natural potential of the cell is Îƒ, a battery that imposes an opposite potential that is greater than Îƒ reverses the anode and cathode, and stops corrosion of the anode. This is one to accomplish the cathodic protection of iron, a method of corrosion prevention that we shall discuss further below. Second, an impressed voltage can be used to electroplate a metal coating onto a metal that is naturally anodic to it. For example, gold coatings are sometimes electroplated onto metal surfaces to beautify them or protect them from corrosion. Since Au is cathodic to all metals, it is ordinarily impossible to design a natural galvanic cell that will accomplish this. However, an impressed voltage can be used to reverse the cell and electroplate Au onto almost any metal. Third the rate of corrosion of the anode is governed by the cell potential. Using an external potential, it is possible to control the voltage to a value that sets the rate of corrosion. This technique provides a method of corrosion protection in that it can be used to minimize the corrosion rate. It is also used in electroplating, to control the rate of metal deposition on the cathode. 12.4.6 Thermodynamics of the galvanic cell In discussing the galvanic cell, we have taken the standard half-cell potentials as experimental fact. To understand how these potentials arise, and why they have the values they do, it is necessary to consider the thermodynamics of the galvanic cell more closely. There are three species that appear in a standard half-cell reaction that governs


Page 412

the ionization of a metal: the atoms in the metal, the ions that go into solution, and the excess electrons that are left behind in the metal. Let the reaction have the generic form M ¿ M+ + e- 12.42 The change in Gibbs free energy per mole of metal consumed is Îg = µ(M+) + µ(e-) - µ(M) 12.43 where µ(A) is the chemical potential of the species, A. To solve equation 12.43 we need to know the chemical potentials of each of the species involved. The work function and the contact potential First consider the electrons in a metal that is not in contact with an ionic solution. As we discussed in Chapter 2, the electrons in a metal fill its available energy states up to the Fermi energy, EF. The chemical potential of a species in a system is defined as the work done, or energy increment, when an infinitesimal amount of that species is added. Since the addition of an electron to a metal adds the energy, EF, the chemical potential of an electron in a neutral metal is equal to EF. To compare the electron energies in different metals it is necessary to set the zero level of the energy in a way that guarantees that the energies are measured with respect to the same reference. This is done by converting the electron energy into a potential according to the relation E = - eƒ 12.44 where e is the electron charge, and setting the zero of the potential at the same reference for every material. The chemical potential per mole of electrons at potential, ƒ, is, then, µe- = N0E = - Fƒ 12.45 where N0 is Avogadro's number and F is the Faraday constant, the magnitude of the charge of a mole of electrons. The Fermi energy of a neutral metal in free space is negative, since the electrons are bound in the solid, as illustrated in Fig. 12.17. The potential at the Fermi level is ƒF = - EF/e 12.45 and is positive. The potential, ƒF, is called the work function of the solid, since the work, eƒF, must be done to raise the energy of the electron to the level at which it can flow out of the solid into free space.


Page 413

Free space

filled electron states

EFFeƒ

E

x Fig. 12.17: Schematic diagram of electron energies in a simple metal,

illustrating the Fermi level, the potential increase to the free space value at the metal surfaces, and the work function.

Now suppose that two different metals, A and B, are joined together by a conduct-ing wire, as shown in Fig. 12.18. The two metals have different Fermi energies, and, hence, have different internal potentials. Let these be ƒA and ƒB, where ƒB > ƒA. There is, hence, a potential difference between the two metals of magnitude Îƒ = ƒB - ƒA 12.47 This is called the contact potential, since it is developed spontaneously when the two metals are joined. Since electrons have lower energy in regions of higher potential, electrons flow through the wire from A to B. The accumulation of electrons (excess negative charge) in metal B lowers its potential (raises the Fermi energy) while the loss of electrons (excess positive charge) in metal A raises its potential (lowers the Fermi energy). A current continues to flow until the potentials in the two metals become the same. At this point the Fermi level, EF, is the same in both metals, and, hence, the electron chemical potential is also the same. Equilibrium is established.

EFAeƒ

EF

BeƒeÎƒ

Fig. 12.18: Energy diagram illustrating the source of the contact

potential when two metals are joined together. Electrons will flow from A to B until EF has the same value in both metals.

The contact potential is a principle source of the galvanic potential that develops when the two metals are joined in a galvanic cell, although, as we shall see, the ionic part of the cell also plays an important role.


Page 414

In addition to its importance in galvanic cells, the contact potential between dissimilar conductors is also responsible for the important junction properties of semiconducting devices, which we shall discuss in Chapter 14. It is also used in the temperature measuring device known as a thermocouple. It is worth making a small digression to describe how a thermocouple works. A thermocouple consists of wires of two different metals that are joined at one end to establish an electrical contact and are connected at the other end to a potentiometer that measures the contact potential between them. Not only are the Fermi energies of the two metals different, but they also have different thermal derivatives. The contact potential is a function of temperature, that can be measured once to calibrate the thermocouple. Since the contact potential is established at the dissimilar metal junction, it measures the temperature at the place where the junction is located. One can, for example, measure the temperature within a furnace by placing the junction of the thermocouple wires at the point where the temperature is to be measured, and reading the temperature from a potentiometer located well outside the furnace. The chemical potential of an atom in solution Now consider the chemical potential of the metal, M, in the reaction given in eq. 12.43. For generality, assume that the metal is a solid solution that contains metal M along with other species. Let the concentration of M be denoted [M]. The chemical potential of a species, M, in a solid solution can always be written µM = µ*

M + RT ln(aM) 12.48 where µ*

M is the potential of the species in its pure state (µoM is a function of T and P,

but we assume these are fixed at ambient values), and aM is the activity of M in its solution. In fact, eq. 12.48 defines the activity, aM. The activity is a function of the composition of the solution, but has well-defined values in two limits. First, when M is the solvent and its concentration is nearly 1, the activity is equal to the concentration: aM = [M] ([M] « 1) 12.49 This relation is one form of Raoult's Law. Second, when M is a solute and its concentra-tion is very small, aM is proportional to the concentration: aM = ©M[M] ([M] << 1) 12.50 where ©M is a constant known as the activity coefficient. In either of these two cases we can replace eq. 12.48 by the simpler relation µM = µ0

M + RT ln[M] 12.51


Page 415

where µ0

M = µ*M ([M] « 1)

12.52 µ0

M = µ*M + RT ln(©M) ([M] <<1)

These two limits are sufficient to cover most of the cases we will be interested in, so we shall assume that eq. 12.51 holds. The physical reason for the difference between these two expressions lies ultimately in the environment of an M atom in the solution. When [M] « 1 an M atom is surrounded almost entirely by atoms of its own kind. Since its environment is almost identical to that it should have in pure M, its chemical potential is almost the same. On the other hand, when [M] << 1 an M atom is surrounded almost entirely by solvent atoms. The chemical potential of an isolated atom of M is determined almost entirely by its interaction with the solvent atoms, so µ0

M ordinarily has a very different value than it would have in pure M. The activity coefficient, ©M, has no fundamental significance; it is simply a number that measures the difference between µ0

M in dilute solution and that in the pure metal. If the interaction between M and the solvent species is energetically favorable, µ0

M for the dilute solution is less than µ*M , so ©M < 1. When the interaction

is energetically unfavorable, ©M > 1. The chemical potential of an ion in a dilute solution Now consider the potential of the ion, M+, in a dilute aqueous solution of the sort we are interested in here. Since the ion is charged, its chemical potential is affected both by its concentration and by the electrical potential of the solution. The electrical potential of the solution that contains M+, ƒM+ , must necessarily be considered if the solution contains an excess of positive or negative ions. The electrical contribution to the chemical potential of a positive ion of charge, z, in a potential, ƒ, is zFƒ, since this is the electrical work that would have to be done to move a mole of ions to this potential from a place where the potential was zero. Using equation 12.51, the chemical potential of a singly charged ion, like M+, is

µM+ = µ0M+ + FƒM+ + RT ln[M+] 12.53

where µ

0M+ is the reference chemical potential for the dilute solution.

The potential at a metal-solution interface We can now combine these relations to find the half-cell potential for the reaction M ¿ M+ + e- 12.42


Page 416

When equilibrium is established at the interface, Îg = 0 in eq. 12.43. Hence µM - µM+ - µe- = 0 12.54 Substituting equations 12.45, 12.51, and 12.53, and letting [M] = 1 for the pure metal, we have

µ0M - µ

0M+ - FƒM+ + FƒM - RT ln[M+] = 0 12.55

This equation can be solved for the potential difference between the metal and the solution. The result is

ÎƒM = ƒM - ƒM+ = ƒ0M +

RTF ln[M+] 12.56

where

ƒ0M =

µ0M+ - µ0

MF 12.57

This equation is identical in form to the basic equation (12.27) for the half-cell potential of an electrode in a galvanic cell. Note, however, that the standard potential (eq. 12.57) is not quite the same as that which appears in the electromotive series (Table 1). Since electrochemists measure standard potential relative to that for ionization of hydrogen, the potential in the electromotive series differs from that in equation 12.56 by a constant, which subtracts out when two half-cell reactions are combined in a galvanic cell. It is useful to consider the physical meaning of equation 12.56. This equation de-scribes a situation that leads to a dynamic equilibrium in which the rate of creation of M+ ions and electrons is precisely equal to the rate of their recombination to generate pure M. Consider four prototypic situations. First, suppose that we place a sheet of M in a neutral solution that contains no M+ ions. Since, initially, ƒM+ is zero and ƒM is positive, the left-hand side of the equation is positive, while the right-hand side is negative since [M+] is arbitrarily small. Therefore, some M ionizes to M+ to establish equilibrium. As it does so, the right-hand side of the equation shifts toward positive values as [M+] increases. However, ionization causes the accumulation of excess electrons in the metal and excess ions in the solution. The excess electrons decrease ƒM, while the excess ions increase ƒM+ , so the left-hand side of the equation decreases. The two sides of the equation move toward one another until they become equal and equilibrium is established. Second, suppose that we place a small sheet of M in a large body of neutral solution that contains M+. If the solution is very large, [M+] is essentially fixed, and ƒM+ remains very close to zero. The right-hand side of the equation then has a fixed value. M is either ionized or plated at the interface until the electron concentration within


Page 417

the metal reaches a value such that ƒM is equal to this value. It follows that we can use equilibrium with an ionic solution to fix the value of the potential within the metal. Third, suppose that the solvent of the solution is replaced by one that interacts strongly and favorably with M+. In this case µ

0M+ decreases, so ƒ0

M decreases and more M+ is produced by ionization. If the solution is so large that [M+] is fixed, electrons accumulate in the metal until ƒM decreases to its new equilibrium value. Fourth, suppose that the nature of the metal is changed, for example, by making it polygranular or deforming it to introduce dislocations or other crystallographic defects. These microstructural changes raise the chemical potential of the metal in its pure state, µ0

M . As a consequence, they lower ƒ0M and make the metal more anodic. This has the

important consequence that one can form a galvanic couple between two samples of the same metal by processing one of them to introduce defects. The defective sample is the anode in the galvanic couple, and is corroded by the more perfect one. The opposite effect is achieved by alloying the metal with solute to form a solution. If the solute is dilute, then, whatever its nature, the chemical potential is lowered according to eq. 12.51. It follows that one can form a galvanic couple between a pure metal and a similar metal that is lightly alloyed. The alloy is the cathode, and corrodes the pure metal.

Metal Solution

e-

e-

e-

e-

e-

e-

+M+M+M+M+M+M

... Fig. 12.19: The double-layer of charge at the interface between a metal

and an electrolyte. The figure assumes excess electrons in the metal and excess positive ions in the solution.

Finally, consider where the excess electrons and ions will normally be found. Since the electrons are negatively charged and the ions are positively charged, the two species attract one another. The excess electrons and ions form layers of excess charge along the interface, with the electrons just inside the solid and the ions just inside the solution, as illustrated in Fig. 12.19. If there is a deficit of charge in the metal, the holes (missing electrons) are concentrated along the interface of the metal while the excess negative ions lie along the solution side of the interface. This configuration of parallel layers of charge is called the double-layer of charge. The double-layer contains essentially all of the excess charge, and, since the total excess electron charge equals the total excess ion charge, it acts as a thin capacitor with balanced charge. Excepting the region within the double-layer, the metal and the solution are electrically neutral; hence the potential is constant in the interiors of both the metal and the solution. The potential gradient that changes the value of ƒ from ƒM to ƒM+ lies across the double-layer. A


Page 418

double-layer of charge is formed at every surface where the potential changes; hence there is a double-layer at virtually every interface between a metal and an electrolyte. The galvanic cell Now assume that M is joined with another metal, P, to form a galvanic cell. For simplicity, let P also form a monovalent ion P ¿ P+ + e- 12.58 and let M and P be in contact with solutions of their ions that have equal concentrations. If P is the cathode, the cell reaction is M + P+ “ P + M+ 12.59 and the potential of the cell is Îƒ = ÎƒP - ÎƒM = ƒ0

P - ƒ0M = (ƒP - ƒM) - (ƒP+ - ƒM+ ) 12.60

12.5 THE KINETICS OF ELECTROCHEMICAL REACTIONS The engineering significance of an electrochemical reaction is usually determined by its kinetics, that is, by the rate at which the reaction proceeds. To sustain the reaction each of the electrons that is liberated at the anode must travel to the cathode and be consumed there. Since each ion that is liberated at the anode produces a given number of electrons, the reaction rate is simply proportional to the current that circulates through the electrochemical cell. 12.5.1 The current in an electrochemical cell Since the electrochemical cell is a simple electrical circuit, the electrical current, I, must be the same for every cross-section through the circuit, and must obey Ohm's Law: ÎV = IR 12.61 where ÎV is the total voltage drop between the electrodes and R is the total resistance. In a simple galvanic cell, that is, one that does not contain a battery or other voltage source, ÎV = Îƒ 12.62 the galvanic potential between the electrodes. The total resistance, R, is the sum of the resistance of the metallic circuit through which the electrons pass and the effective resistance of the ionic solution through which the ions pass. The current that is given by eq. 12.61 is the electric current. Recall that, by the strange historical convention of


Page 419

electrical engineers, the electric current is the flow of positive charge, and is equal and opposite to the electron flow. While electrons flow from anode to cathode through the metallic conductor, and negative ions flow from cathode to anode through the solution, the electric current travels from cathode to anode through metal, and from anode to cathode through the electrolyte. You might guess that we could find the current in a simple electrochemical cell by inserting the value of Îƒ that is determined by the thermodynamics of the cell according to eq. 12.60. But this is not the case. The potentials that appear in eq. 12.60 apply when both electrodes are in equilibrium. When the electrodes are in equilibrium there is no net ionization and no net production or consumption of electrons. To induce a current we must perturb the anode and cathode potentials so that equilibrium is violated and the reactions proceed at a finite rate. For example, let the anode reaction be M “ M+ + e- 12.63 and let its potential be ƒa = ƒM - ƒM+ 12.64 where ƒM is the potential in the metal and ƒM+ is the potential in the solution at the anode. If ƒa = ƒºa = ÎƒM 12.65 where ƒºa is the equilibrium potential at the anode, then the anode is in equilibrium and no net reaction happens. To drive the anodic reaction we must increase ƒa so that electrons are attracted into the metal and ions are attracted into the solution. When this is done the electrode is said to be polarized. The rate of reaction, and, hence, the current at the anode, ordinarily increases with the degree of polarization as measured by the difference, ƒa - ƒºa. Similar reasoning applies to the cathode. If the cathode potential, ƒc, is equal to its equilibrium value, ƒºc, no reaction occurs. To drive the cathodic reaction we must de-crease ƒc so that electrons are attracted to the solution. The current at the cathode ordinarily increases with the potential difference ƒºc - ƒc. In an active galvanic cell both electrodes must be polarized so that both pass cur-rent. The electrode potentials and the resulting current are determined by two conditions. First, the total current is the same at the anode and the cathode: Ia = Ic = I 12.66


Page 420

Second, the potential difference between the anode and the cathode satisfies Ohm's Law, which, in the absence of an impressed voltage, reads V = (ƒc - ƒa) = IR 12.67 where R is the sum of the resistances of the metallic and ionic paths. If R is small, ƒc ~ ƒa = ƒ 12.68 where ƒ is the potential that leads to the same overall current at the two electrodes. Eq. 12.68 is usually a good approximation in natural corrosion cells where the anode and cathode are physically close to one another. In this case ƒ is called the cell potential, or, in the specific case of aqueous corrosion, the corrosion potential. 12.5.2 The current-voltage characteristic To solve the simultaneous equations 12.66 and 12.67 we must know the current-voltage characteristics for the two electrodes. The current-voltage characteristic of an electrode relates the current density, j, at the electrode to the overpotential, ƒ - ƒº. The current, I, at the electrode is the product of its current density and its area, A: I = jA 12.69 If the resistance, R, is negligible, the corrosion potential, ƒ, is the potential at which jaAa = jcAc 12.70 where the subscripts a and c refer to the anode and cathode, respectively. If the current-voltage characteristics of the two electrodes are known, the cell po-tential and current can be found by simple graphical methods, as illustrated in Fig. 12.20. Fig. 12.20(a) applies when the electrodes have the same area and the circuit resistance, R, is negligible. The cell potential, ƒ, is determined by the intersection of the two current-voltage characteristic curves. Fig. 12.20(b) applies when the electrode areas are different. In the case shown, the anode is much smaller than the cathode. The equality of the overall current requires that the current density at the anode be higher than that at the cathode by the factor Ac/Aa (eq. 12.70). As the area ratio becomes larger, the anode current density increases (the rate of corrosion of the anode increases) and the cell potential rises toward the equilibrium potential of the cathode. Fig. 12.20(c) applies when the circuit resistance, R, is appreciable. As illustrated in the figure, the cell potentials at the anode and cathode are separated by the factor, IR, and the current density is lower than its value in the absence of the resistance.


Page 421

ƒ

ln (j) ln (j) ln (j)

cathode

anode

ƒ

ln (j)

AaAc jajc =ƒ

cathode

anode

ln ( ) ln ( )j c ja

Îƒ = IR

aƒ

cƒ

ln (j)

(a) (b) (c)

Fig. 12.20: Graphical solution for the electrode potentials and currents

under three conditions: (a) Ac = Aa, R = 0; (b) Ac >> Aa, R = 0; (c) Ac = Aa, R ≠ 0.

To go further we need to understand the current-voltage characteristics of the anode and the cathode. First consider the anode. The current density at the anode (ja) is equal to the rate at which positive ions are liberated per unit area of anode, multiplied by the ionic charge:

ja = zF

dN

dt 12.71

where z is the charge per ion, zF is the charge per mole of ions, and dN/dt is the mole number of ions produced per unit area per unit time. The reaction at the anode occurs in two sequential steps. First, positive ions are produced at the anode and liberated into the solution. Once in the solution they migrate toward the cathode. The rate of ion production is determined by the slower of these two processes. If the ionization reaction is slower, the current is said to be activation controlled. If ion migration away from the interface is slower, the reaction is mass transport controlled. However, mass transport control at the anode almost invariably leads to an even more restrictive reaction. If ions accumulate at the cathode, they tend to react to form oxide or hydroxide films. If these are insoluble, they coat the interface, severely restricting the reaction rate. In this case the anode is said to be passivated. The reaction steps that occur at the cathode are similar, with the difference that the cathode reaction may either consume positive ions, as in the case of metal deposition and hydrogen evolution, or consume oxygen to produce negative ions. In either case the reaction involves an ionization step that consumes or produces ions, and a diffusion step that delivers the reacting species to the surface. Hence the cathode reaction may be activation controlled or mass transport controlled. Passivation does not normally occur at the cathode. The three possible rate-limiting steps, activation, mass transport, and passivation, lead to different reaction kinetics.


Page 422

Activation control When the current is low it is invariably controlled by the rate of ionization, since there is ample time for the ions to migrate away from the interface. Ionization is a thermally activated process, whose rate is ordinarily given by an equation of the form

dNdt = Aexp

Îg

2RT 12.72

where A is a constant, Îg is the free energy change per mole of ions, and the factor 2 ap-pears in the exponent because both ions and electrons are produced in the reaction. The free energy change, Îg, for ionization at an electrode is related to the electrode potential according to Îg = ± zF(ƒ - ƒº) 12.73 where the (+) sign applies at the anode (positive ions produced) and the (-) sign applies at the cathode (positive ions consumed or negative ions produced). Combining equations 12.71, 12.72 and 12.73, the current-voltage characteristic at the anode can be written in the form

(ƒa - ƒºa) = ka ln

ja

jºa 12.74

where ka and jºa are constants. (This equation applies when ƒa is significantly greater than ƒºa. When ƒa « ƒºa we must also consider the reverse reaction that creates neutral atoms from ions. When ƒa = ƒºa the forward and reverse reactions happen at the same rate, and the net current is zero.) When the cathode is activation-controlled, a similar analysis shows that its current-voltage characteristic is given by

(ƒºc - ƒc) = kc ln

jc

jºc 12.75

where kc and jºc are constants. The cell potential, ƒ, under activation-controlled conditions are determined as shown graphically in Fig. 12.20. In the example used in Fig. 12.20 it is assumed that k and jº have comparable values at the two electrodes. This is not necessarily the case. The constant, k, is inversely proportional the ion charge, and the constant, jº, is proportional to the logarithm of the rate constant, A. The intersection of the two curves, which determines the cell potential, ƒ, always falls between the equilibrium potentials, ƒºc and ƒºa. However, ƒ may lie anywhere in this range.


Page 423

Finally, consider the case in which two or more independent reactions can occur at an electrode. The most common case is the competition between metal deposition, hydrogen evolution, and oxygen consumption at the cathode. The dominant cathode reaction is the one that produces the highest cell current. This is ordinarily the reaction that has the largest equilibrium cathode potential, ƒºc. However, exceptions are possible. If a cathode reaction has a smaller value of ƒº, but has a smaller value of k or a larger value of jº, it may produce a higher cell current. An example is given in Fig. 12.21.

ƒ

ln (j)

anode

cathode

... Fig. 12.21: Graphical illustration of a case in which a cathodic reaction

with a relatively low equilibrium potential, ƒº, can lead to a higher cell current (j), and, hence, a higher reaction rate.

Mass transport control Mass transport may control either the anode or cathode current. It is most often encountered as the limiting factor in the cathode current when the cathode reaction consumes oxygen. If the oxygen content is relatively low, the rate of oxygen consumption by ionization at the cathode may exceed the rate at which oxygen can be delivered to the cathode so that the current is limited by the rate of oxygen diffusion to the surface.

c

x

cathode

solution

cO∂

... Fig. 12.22: Mass transport control due to oxygen starvation at the

cathode.


Page 424

The maximum current that can be produced by oxygen consumption at the cathode is equal to the rate at which oxygen can be delivered to the cathode by diffusion from the solution. We can estimate this current from the following simple argument. Let the concentration of oxygen molecules (O2) in the solution be cO, let the oxygen concentration at the electrode surface be zero (all oxygen is consumed), and approximate the oxygen profile by assuming that c increases linearly to cO over an effective diffusion distance, ∂, as shown in Fig. 12.22. The flux of oxygen to the interface is, then,

J = - D

dc

dx = DcO

∂ 12.78

where D is the diffusivity of oxygen molecules in the solution. The oxygen consumption reaction at the cathode can be written O2 + 2H2O + 4e- = 4(OH)- 12.79 Hence each oxygen molecule produces four negative charges, so the maximum possible current density is

(jc)max = 4FJ = 4FDcO

∂ 12.80

When the cathode potential is such that the cathode current is much less than this value, the current is determined by the ionization rate (eq. 12.77). However, as the current ap-proaches the limiting value given by eq. 12.80 the current-voltage curve drops to an asymptote at (jc)max, as diagrammed in Fig. 12.23. The cell current cannot exceed (jc)max, and reaches this value if the current characteristic of the anode does not intersect that of the cathode at j < (jc)max, as illustrated in Fig. 12.23.

ƒ

ln (j)

anode

cathode

ln jmax)(

... Fig. 12.23: Decrease of cell current and potential due to mass transport

control at the cathode.


Page 425

In a large, quiescent solution the diffusion distance, ∂, is proportional to the mean diffusion distance, 2Dt , and, hence, increases with time. It follows that (jc)max decreases with time. The cell current is eventually controlled by mass transport, and decreases with time. However, real solutions are almost always stirred by convective currents or by fluid flow. As the flow rate increases, ∂ decreases. The limiting current (jc)max increases with the flow rate, and the cell current returns to the activation-controlled value when the flow rate is very high. This effect is responsible for the fact that the rate of corrosion of a metal in flowing water is often much higher than the rate in still water, and increases with the flow rate. Passivation When mass transport begins to limit the reaction at the anode, metal ions accumulate there. If the metal is one that can form a coherent oxide or hydroxide at the interface by reacting with the solution, then a film forms and seals the anode. The film drives the cell current to a very low value since ions must diffuse through the film to reach the solution. This phenomenon is called passivation. The passivating film forms when the ion concentration at the anode exceeds a critical value. In the unpassivated condition, the ion concentration is determined by a balance between the rate at which ions are produced by ionization and the rate at which they migrate into the solution. It follows that the ion concentration increases with the anode potential, ƒa, and that passivation occurs when the anode potential reaches a critical value, called the passivation potential, ƒp, which depends on the nature of the anode and the composition of the electrolyte.

ƒ

pH

Al

Al+32Al 3O

solu

ble

oxid

es

Fig. 12.24: Simplified form of the Pourbaix diagram for aluminum. The

passivating oxide, Al2O3, is stable at intermediate values of the pH.

Assuming that the electrolyte is an aqueous solution, its most important characteristic is its acidity, as measured by its pH. Oxide films are formed when metal ions react with (OH)- ions. For a given concentration of metal ions, oxides only form if the (OH)- concentration, and, hence, the pH, exceeds a critical value. Moreover, most metals can form any one of several oxides or hydroxides, and the nature of the film that is


Page 426

formed also depends on the pH. Since the type of oxide is determined by the metal ion concentration and the pH, and since the metal ion concentration is determined by the anode potential, it is possible to construct a type of "phase diagram" that identifies the presence and type of oxide as a function of the pH and the anode potential, ƒa. Such a diagram is called a Pourbaix diagram after its inventor. A simple example is given in Fig. 12.24. The passivation potential can be estimated from the Pourbaix diagram of the anode. If the pH of the solution is such that a stable oxide forms, the passivation potential is given approximately by the lowest potential at which it appears in the Pourbaix diagram. If the pH is such that no stable oxide forms, the anode does not passivate. However, a Pourbaix diagram must be used with some caution. Not all oxides make coherent films; some are soluble or form loose, porous aggregates. Even those oxides that do form coherent films do not necessarily do so at the lowest potential at which they are stable.

ƒ

ln (j)

anode

cathode

active

passive

transpassive

cathode

1

2ƒ p

Fig. 12.25: Current-voltage characteristic for a metal that passivates at

intermediate potential. Current-voltage characteristics are drawn for two cathodes. Note that the cathode that has the lower ƒ produces the higher cell current.

When a metal does passivate its current-voltage characteristic takes the rather strange form shown in Fig. 12.25. The current increases logarithmically with the potential until the passivation potential is reached. It then decreases dramatically, and remains very small until the potential becomes so high that new anodic reactions, such as the decomposition of water to oxygen and hydrogen ions, intrude and attack the oxide film. The effect of passivation on the cell current is illustrated in Fig. 12.25. Surprisingly, a metal that passivates is corroded by a cathode that has a relatively small equilibrium potential, since the current-voltage characteristics intersect below the passivation potential, but is not significantly corroded by a cathode with a relatively high equilibrium potential, since the current-voltage characteristics intersect at ƒ > ƒp.


Page 427

12.6 AQUEOUS CORROSION IN ENGINEERING SYSTEMS Engineering systems are most often corroded by aqueous solutions that are com-mon in the environment. Common problems include atmospheric corrosion by water va-por, rain or salt-water spray, corrosion by immersion in fresh or salt water, and corrosion by ground water in the soil. Aqueous corrosion only happens when the corroded metal is an integral piece of a complete galvanic cell. This requires that the two materials that act as anode and cathode are in both electrical contact through a conductor and ionic contact through an electrolyte. Aqueous corrosion is only an engineering problem if the rate of corrosion leads to significant damage or failure in a time less than the expected lifetime of the device. Since the rate of corrosion is determined by the current density at the anode, corrosion is only a problem if the current density is reasonably high. Where aqueous corrosion is a potential problem, it can be addressed by using corrosion protection schemes. These employ one of two basic tactics. The first is to break the electrical circuit or alter it so that the rate of corrosion is negligible. The second is to provide an alternate circuit so that corrosion occurs in unimportant places within the engineering system. 12.6.1 Corrosion cells in engineering systems The three most common sources of galvanic cells in engineering systems are dis-similar metal contact, microstructural surface couples, and oxygen concentration cells. Dissimilar metal contact Whenever two different conductors (metals or semiconductors) are brought into electrical contact in the presence of an ionic solution a galvanic cell develops. The anode in the cell is corroded unless some protective mechanism intrudes. Dissimilar metal contact usually establishes a high potential difference, and often causes rapid corrosion, particularly if the anode in the cell has a much smaller surface area than the cathode. Most of the spectacular corrosion failures that color the history of engineering have this cause. One can identify the cathode in the cell that is formed by contact between metals A and B by finding the sign of the potential for the reaction

A + ab B+b “ A+a +

ab B 12.79

The potential is given by eq. 12.29. If the potential is negative, A is the anode and the reaction proceeds in the direction of the arrow. However, the reaction at the cathode is ordinarily not the condensation of B, but the hydrogen evolution or oxygen consumption reactions discussed in Section 12.4. The dominant reaction at the cathode is the one that


Page 428

yields the highest current which is, ordinarily, the one that provides the highest potential difference. The implications of dissimilar metal cells for engineering design is straight-forward. Whenever two different conductors are used in proximity to one another, rapid corrosion is a possibility that must be taken seriously. Dissimilar metal couples are likely to be found in bolted or riveted structures, where the bolts or rivets are often made of a very different alloy than the structural metal, in piping systems, where different metals are used for various sections of pipe (for example, municipal water systems usually use steel pipe while individual homes use copper pipe), and in microelectronic devices, where many different conductors are used in intimate contact with one another. Microstructural couples Galvanic couples are also formed when nominally uniform metals are brought into contact with an electrolyte. The reason is that the metal surface is never perfectly uniform. Nonuniformities produce galvanic cells in which local areas that have relatively high free energy act as anodes while those that have relatively low free energy behave as cathodes. For this reason, grain boundaries are anodic to grain interiors, deformed (and, hence, defective) regions are anodic to those that are less heavily deformed, and regions that are relatively low in solute are normally anodic to those that are higher. When a metal, A, is in contact with a uniform electrolyte, two different areas of its surface, A1 and A2, may react according to the relation A1 + A+a “ A2 + A+a 12.80 The anode can be identified by comparing the free energies of A1 and A2; the region with the higher free energy density is the anode. Again, however, the cathode reaction in the actual cell is usually not the deposition of metal, but the evolution of hydrogen or con-sumption of oxygen, whichever provides the higher potential. The corrosion of a nominally homogeneous metal that is placed in a homogeneous electrolyte is often relatively slow and uniform over the surface. The reason is that the locally active regions of the surface corrode away. Other regions become active, and the site of corrosion gradually shifts over the surface. This uniform corrosion may not be very important unless the metal part is intended for long-time service, as, for example, a steel beam in a building, or unless the environment is particularly severe, as in the corrosion of unprotected automobile bodies on the salted roads of the Northeastern United States. When the metal is inhomogeneous on a macroscopic scale its corrosion may be more serious. Macroscopic heterogeneities are often introduced during processing. For example, the weld in a welded plate differs in microstructure and, often, in composition form the plate itself, and the small radius or corner of a severely formed metal part has a different microstructure from the relatively undeformed areas of the part. These large-


Page 429

scale inhomogeneities produce permanent galvanic couples that concentrate corrosion in particular places on the part. Oxygen concentration cells The final common source of galvanic couples is the oxygen concentration cell that was discussed in Section 12.4. As shown by eq. 12.40, the cathode in an oxygen concen-tration cell is located where the solution that contacts the metal is relatively rich in oxygen; the anode is located where the solution is relatively poor. However, if the metal is one that is susceptible to corrosion, the reaction that occurs at the anode is metallic corrosion, since this provides a higher potential, and the overall reaction is

A + a4 O2 +

a2 H2O “ A+a + aOH- 12.81

The metal, A, is corroded at the cathode. Oxygen concentration cells may be produced by either the electrolyte or the metal. The electrolyte creates an oxygen concentration cell when it has an inhomogeneous oxygen concentration. The usual reason is oxygen enrichment at the water-vapor interface, where oxygen enters from the atmosphere. A simple example is a drop of water on a metal surface. Oxygen enters the drop along its periphery. Hence the solution near the periphery is cathodic with respect to the solution in the interior of the drop, and the metal inside the drop is corroded. A second example is when a metal is partly immersed in water, a situation that applies to bridge pilings and ship hulls, among many other engineering structures. The layer of water very near the surface is oxygen-rich because of its contact with the atmosphere. The metal-water interface just beneath the water line is, therefore, cathodic to the interface below it. Corrosion occurs predominantly near, but just beneath the water line, where the gradient in the oxygen concentration is greatest. The metal produces an oxygen concentration cell when it contains pits or crevices. When the metal is exposed to electrolyte, the oxygen inside the crevice is partly consumed by local corrosion. The oxygen cannot be replenished nearly as quickly as that on the free surface because of the tortuous diffusion path that must be followed to access the bottom of the crevice. Hence an oxygen concentration cell is established between the bottom of the crevice and the surface of the metal outside the crevice. This tends to deepen the crevice, driving it into the material. Pitting or crevice corrosion is a damaging form of attack because it is localized. It is a particularly serious problem in thin-walled structures, such as tank walls and automo-bile fenders, since it can lead to rapid perforation of the wall. Crevice corrosion happens almost whenever crevices are present on a surface. Common sites are at bolts and rivets, and at places where metal plates are joined so that they overlap one another. Pitting corrosion develops spontaneously when corrosion is localized to initiate a pit. Pitting is particularly likely when a painted or coated layer is locally broken. Pitting also occurs on


Page 430

bare metal surfaces. In this case it is initiated by rapid corrosion at local anodes. Rapid, local corrosion ordinarily requires a relatively high corrosion potential that is sustained for some reasonable period of time. Corrosion engineers define a "pitting potential" that is required to initiate pitting on a bare surface. To develop the pitting potential there must be strong local couples, such as those produces by dissimilar metal contact, weldments, or deformation gradients at bends or corners. Once pitting has initiated, it is usually self-sustaining and eventually causes perforation of the metal. 12.6.2 The corrosion rate The rate of aqueous corrosion is governed by the kinetic relations developed in Section 12.5. The local rate of corrosion at the anode is proportional to the current density at the anode, which is related to the current density at the cathode by eq. 12.68. The corrosion potential, ƒa, is determined by the current-voltage characteristics of the anode and cathode and the resistance of the cell according to eq. 12.75. The current-voltage characteristics of the anode and cathode are determined by the rate-limiting steps in the ionization reactions at the two electrodes as described in Section 12.5.2. In practice, the corrosion rate usually increases with the equilibrium potential difference, ƒºc - ƒºa, and with the area ratio, Ac/Aa, unless passivation occurs at the anode or mass transport rules at the cathode. The kinetic features that are important in engineering design include the following. The increase in corrosion rate with the equilibrium potential difference has the consequence that high corrosion rates must be expected when there is dissimilar metal contact, a substantial difference in oxygen concentration, or a significant change in the composition or microstructure of the material. Hence the corrosion engineer is anxious to avoid dissimilar metal cells, crevice corrosion, or microstructural cells that develop at weldments or regions of high local deformation in formed parts. The increase in corrosion rate with the area ratio, Ac/Aa, has the consequence that one must particularly avoid situations in which a small anode is in ionic contact with a large cathodic area. For this reason, bolts, rivets and other fasteners should, if possible, be cathodic to the metal they fasten and crevices should be avoided. If a metal surface is protected with a cathodic coating, such as a gold coating, one must be very careful to avoid breaks in the coating since these may lead to intense corrosion. Even passivation may not protect a small anode against corrosion by dissimilar metal contact. The corrosion potential increases with the area ratio, and may reach trans-passive values if the equilibrium potentials are sufficiently different. The rate of aqueous corrosion in non-acidic solutions is often limited by oxygen diffusion at the cathode. The corrosion rate is increased by any mechanism that decreases the diffusion distance or brings fresh solution into contact with the corroding metal. Hence corrosion is ordinarily faster in flowing water than in still water. Waterline corrosion of pilings and ship hulls is faster in rough water than in smooth, since waves constantly wet the surface with fresh solution. The same mechanism tends to make the corrosion rate in a water spray higher than that when the metal is fully immersed.


Page 431

Passivation can be very effective in decreasing the corrosion rate. However, it may lead to an unexpectedly high rate of corrosion in relatively mild conditions, where the corrosion potential does not reach the passivation value. It may also lead to an unexpected and dramatic increase in the corrosion rate when the pH of the solution falls outside the range of stability of the passive layer. Engineering structures that are ordinarily safe against corrosion may deteriorate rapidly in an environment in which they are subjected to an acidic (or also, in the case of aluminum, highly basic) effluent or when they experience acid rain. Passivation is also ineffective when the corrosion potential is very high (trans-passive) and may not protect against dissimilar metal corrosion, particularly when the area ratio is high. 12.6.3 Corrosion protection Since dissimilar metal contact often cannot be avoided in engineering systems, and since nominally homogeneous metals contain microstructural couples and are subject to oxygen potential cells, most engineering systems must be protected against corrosion. There is an entire industry devoted to this subject. Very many different schemes are used. However, almost all effective corrosion protective schemes are based on one of two basic ideas. First, corrosion cannot occur unless a galvanic cell is established. If the electrical circuit is broken corrosion stops. The circuit can be broken by insulating the metal from the ionic solution or by insulating the anode and cathode from one another. Second, corrosion occurs at the anode in the galvanic cell. One can protect a given metal part by making it the cathode, either by using an external voltage to reverse the cell, or by introducing a metal into the circuit that is anodic to the metal of interest and corrodes preferentially. Insulation at the metal-solution interface The common methods that are used to break the corrosion circuit at the metal-solution interface include organic coatings such as paint, inorganic coatings such as natural oxides or passivation layers, and noble metal coatings such as gold. The most common protective coating is paint. Insulating organic paints provide excellent corrosion resistance so long as they are intact. However, they must be well-bonded to the metal surface, and even then may break or spall off the surface because of mechanical damage. If the paint layer is perforated so that the solution penetrates to the surface, then metal corrosion may spread under the paint, causing it to bubble and peel from the surface. Under normal conditions the corrosion under the paint layer is a kind of crevice corrosion; the solution that penetrates along the paint-metal interface is depleted in oxygen. For this reason corrosion may spread rapidly after the paint is penetrated, so it is important to inspect painted structures often and repair any damage. Some of the best protective paints incorporate chemicals, such as PbCrO4, that act as corrosion inhibitors. If a PbCrO4 paint is perforated to expose the metal surface,


Page 432

chromate ions go into solution and react at the interface to produce thin, coherent oxide coatings that inhibit corrosion. These paints tend to be red-orange to red in color and are sometimes called "red-lead" paints. Thicker and more elaborate organic coatings are used to seal microelectronic devices from moisture. Since these devices contain many dissimilar conductors that are in intimate contact, they are liable to catastrophic corrosion and must be hermetically sealed from all moisture. The sealing of electronic devices is made difficult by the fact that conducting wires or contacts must be provided to supply current to the device. These must be well bonded to the sealant to prevent the influx of moisture along the wire. Oxide insulation coatings are of two types: those that are coated on the surface and those that are formed by natural oxidation processes. A common example of an applied oxide coating is the porcelain enamel that is often used on carbon steel, for example, in sinks and bathtubs in the home. Natural oxide protective coatings form spontaneously on aluminum, chromium and many stainless steels when they are exposed to air. These metals are naturally passive in mildly corrosive environments, and are widely used for their corrosion resistance. However, it must be kept in mind that these metals do not have good inherent corrosion resistance. To the contrary, they are highly reactive. It is, therefore, important that they not be used in conditions where the protective oxide may be lost. Aluminum and chromium are liable to corrosion in acidic media since the protective oxides, Al2O3 and Cr2O3, become unstable al low pH values. Aluminum is also corroded in highly basic solutions since the nature of the oxide changes at high pH into a phase that is soluble. Chlorine ions attack both Al2O3 and Cr2O3; the range of pH over which these oxides are protective decreases in the presence of chlorine. Materials that passivate at high corrosion potentials can often be protected by exposing them to a high corrosion potential before service to establish a semi-permanent passivation layer. This technique is also used with aluminum alloys to thicken and stabilize the natural oxide coating; aluminum processed in this way is called anodized aluminum. Other materials passivate, but form a protective layer that is not retained after the potential has been removed. These materials can be protected in service by introducing a strongly cathodic material into the circuit, or imposing an external voltage, so that the anode potential is maintained above the passivation value. Finally, a structural metal can be protected from corrosion by coating it with a second metal that seals the surface and does not itself corrode. A common example is gold coating, which is widely used to protect Ni and Cu contact pads in microelectronic devices from corroding during the period between their manufacture and their final assembly. Au is useful because it does not corrode, and can be easily electroplated into a coherent film on Ni and Cu surfaces. The problem with Au is that it is strongly cathodic to the metal it is intended to protect. If the Au coating is imperfectly applied, or if it is perforated for any reason, the result is a strong corrosion couple with a very large area ratio.


Page 433

A second approach that has been developed very recently is to protect the metal with a thin deposited coating of amorphous metal. The coating can be made in situ by using ion implantation or laser surface processing to implant a high concentration of a glass-forming solute (such as P or C in Fe) and rapidly heating, with a laser or electron beam, to drive the coating into the amorphous structure. Amorphous coatings of Fe highly doped with P or C have excellent corrosion resistance. They resist corrosion for a unique reason; since their structures are statistically uniform on the atomic scale, their surfaces contain virtually no microstructural couples that can form galvanic cells. The amorphous structure has the additional advantage that it has a relatively high free energy. If the coating is perforated for any reason, the coating is anodic to the base metal and corrodes preferentially. Insulation at the metal-metal interface It is often desirable to use dissimilar metals in structures that are exposed to the environment. Examples include bolted or riveted structures, where the bolt or rivet is made of a different material than the bolted plate, and piping systems, where it is often desirable to use pipe of one alloy for part of the system and pipe of a second metal or alloy for another. These systems can be protected from galvanic corrosion by the simple device of using a sleeve or coupling that is electrically insulating and separates the metals from one another. Since the metals are not in electrical contact, there is no galvanic corrosion. Cathodic protection by a sacrificial anode The final method of corrosion protection we shall consider is cathodic protection; the metal of interest is protected by making it the cathode in the galvanic cell. There are several ways to do this. One of the simplest is to introduce a sacrificial anode into the system. The sacrificial anode is a metal, usually Zn or Mg, that is anodic to the metal of interest. If the metal and sacrificial anode are brought into electrical contact in an elec-trolyte, the sacrificial anode corrodes. The metal of interest is the cathode in the cell, and does not corrode until the sacrificial anode has disappeared from the circuit. Sacrificial anodes are attached to many structures, such as ship hulls, buried tanks and others, to prevent corrosion. It should be kept in mind that a sacrificial anode is only effective if it is in metallic contact with the metal it is to protect, and in ionic contact with the electrolyte. A Mg anode that is attached to the hull of a ship will protect the hull from corrosion by sea water when it is immersed. However, it will not protect the exposed portions of the hull from corrosion by salt spray, since a droplet that attaches to the hull does not ordinarily contact the Mg anode. A particularly good corrosion protection scheme uses a sacrificial anode as a coating over the surface of the metal. If the coating completely covers the exposed metal surface then it corrodes at a very slow rate since it is only subjected to its own microstructural galvanic couples. Moreover, if the coating is perforated by corrosion or


Page 434

mechanical attack, then the exposed metal is cathodically protected by the remainder of the coating and does not corrode. Two important examples of anodic coatings are "alclad" aluminum and galvanized steel. "Alclad" aluminum is widely used in aircraft. It consists of a coating of essentially pure aluminum on a high-strength aluminum alloy. The pure aluminum layer is anodic to the alloy. So long as the pure aluminum coating is intact, it is highly resistant to corrosion in its own right, and offers very good long-term corrosion protection. Galvanized steel is used in many applications for its corrosion resistance, and has recently become the material of choice for external panels in automobile bodies. Galvanized steel has a superficial zinc or zinc alloy coating. Zinc is anodic to steel. It resists corrosion so long as it remains intact on the steel surface, and provides cathodic protection to the steel if it is perforated. Cathodic protection by an impressed potential A second common method of cathodic protection uses an auxiliary electrode of any type along with a battery or other external voltage source. The imposed voltage ensures that the auxiliary electrode is the anode, while the metal that is to be protected is the cathode. This method has the advantage that the anode need not corrode at a significant rate. The external potential can be chosen so that the anode is passivated, or it can be made of a relatively noble material so that the anode reaction is an anodic reaction in the electrolyte (for example, the decomposition of water: H2O “ O2 + 2H+ + 2e-). A cathodic protection scheme of this type can function almost indefinitely. The cathodic protection methods that are used for pipelines and other safety-critical underground structures are usually variations of this method.

Date post:	04-Apr-2018
Category:	Documents
Upload:	phamnhan
View:	219 times
Download:	6 times

PART II: THERMOCHEMICAL PROPERTIES 144 CHAPTER 7 ... · Materials Science Fall, 2008 Page 139 PART...

Documents