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1 Thermodynamics and Phase Diagrams of Materials

Arthur D. Pelton

Centre de Recherche en Calcul Thermochimique, cole Polytechnique,Montral, Qubec, Canada

List of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Gibbs Energy and Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Gibbs Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Chemical Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Predominance Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 Calculation of Predominance Diagrams . . . . . . . . . . . . . . . . . . . 71.3.2 Ellingham Diagrams as Predominance Diagrams . . . . . . . . . . . . . . 81.3.3 Discussion of Predominance Diagrams . . . . . . . . . . . . . . . . . . . 91.4 Thermodynamics of Solutions . . . . . . . . . . . . . . . . . . . . . . . 91.4.1 Gibbs Energy of Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.2 Chemical Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4.3 Tangent Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.4 Gibbs-Duhem Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.5 Relative Partial Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.6 Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.7 Ideal Raoultian Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.8 Excess Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4.9 Activity Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4.10 Multicomponent Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Binary Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5.1 Systems with Complete Solid and Liquid Miscibility . . . . . . . . . . . . 141.5.2 Thermodynamic Origin of Phase Diagrams . . . . . . . . . . . . . . . . . 161.5.3 Pressure-Composition Phase Diagrams . . . . . . . . . . . . . . . . . . . 191.5.4 Minima and Maxima in Two-Phase Regions . . . . . . . . . . . . . . . . . 201.5.5 Miscibility Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.5.6 Simple Eutectic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5.7 Regular Solution Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.5.8 Thermodynamic Origin of Simple Phase Diagrams Illustrated by Regular

Solution Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5.9 Immiscibility Monotectics . . . . . . . . . . . . . . . . . . . . . . . . . 261.5.10 Intermediate Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.5.11 Limited Mutual Solubility Ideal Henrian Solutions . . . . . . . . . . . . 291.5.12 Geometry of Binary Phase Diagrams . . . . . . . . . . . . . . . . . . . . . 31

Phase Transformations in Materials. Edited by Gernot KostorzCopyright 2001 WILEY-VCH Verlag GmbH, WeinheimISBN: 3-527-30256-5


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1.6 Application of Thermodynamics to Phase Diagram Analysis . . . . . . 341.6.1 Thermodynamic/Phase Diagram Optimization . . . . . . . . . . . . . . . . 341.6.2 Polynomial Representation of Excess Properties . . . . . . . . . . . . . . . 341.6.3 Least-Squares Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.6.4 Calculation of Metastable Phase Boundaries . . . . . . . . . . . . . . . . . 391.7 Ternary and Multicomponent Phase Diagrams . . . . . . . . . . . . . . 391.7.1 The Ternary Composition Triangle . . . . . . . . . . . . . . . . . . . . . . 391.7.2 Ternary Space Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.7.3 Polythermal Projections of Liquidus Surfaces . . . . . . . . . . . . . . . . 411.7.4 Ternary Isothermal Sections . . . . . . . . . . . . . . . . . . . . . . . . . 431.7.4.1 Topology of Ternary Isothermal Sections . . . . . . . . . . . . . . . . . . 451.7.5 Ternary Isopleths (Constant Composition Sections) . . . . . . . . . . . . . 461.7.5.1 Quasi-Binary Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 471.7.6 Multicomponent Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . 471.7.7 Nomenclature for Invariant Reactions . . . . . . . . . . . . . . . . . . . . 491.7.8 Reciprocal Ternary Phase Diagrams . . . . . . . . . . . . . . . . . . . . . 491.8 Phase Diagrams with Potentials as Axes . . . . . . . . . . . . . . . . . . 511.9 General Phase Diagram Geometry . . . . . . . . . . . . . . . . . . . . . 561.9.1 General Geometrical Rules for All True Phase Diagram Sections . . . . . . 561.9.1.1 Zero Phase Fraction Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 581.9.2 Choice of Axes and Constants of True Phase Diagrams . . . . . . . . . . . 581.9.2.1 Tie-lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601.9.2.2 Corresponding Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . 60

1.9.2.3 Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 601.9.2.4 Other Sets of Conjugate Pairs . . . . . . . . . . . . . . . . . . . . . . . . 611.10 Solution Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621.10.1 Sublattice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621.10.1.1 All Sublattices Except One Occupied by Only One Species . . . . . . . . . 621.10.1.2 Ionic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621.10.1.3 Interstitial Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641.10.1.4 Ceramic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641.10.1.5 The Compound Energy Formalism . . . . . . . . . . . . . . . . . . . . . . 65

1.10.1.6 Non-Stoichiometric Compounds . . . . . . . . . . . . . . . . . . . . . . . 651.10.2 Polymer Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661.10.3 Calculation of Limiting Slopes of Phase Boundaries . . . . . . . . . . . . 661.10.4 Short-Range Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681.10.5 Long-Range Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711.11 Calculation of Ternary Phase Diagrams From Binary Data . . . . . . . 721.12 Minimization of Gibbs Energy . . . . . . . . . . . . . . . . . . . . . . . 741.12.1 Phase Diagram Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 761.13 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

1.13.1 Phase Diagram Compilations . . . . . . . . . . . . . . . . . . . . . . . . . 761.13.2 Thermodynamic Compilations . . . . . . . . . . . . . . . . . . . . . . . . 771.13.3 General Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781.14 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2 1 Thermodynamics and Phase Diagrams of Materials


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List of Symbols and Abbreviations 3

List of Symbols and Abbreviations

Symbol Designation

ai activity of component iC number of componentscp molar heat capacity

E electrical potential of a galvanic cellF degrees of freedom/varianceG Gibbs energy in Jg molar Gibbs energy in J/molgi partial molar Gibbs energy ofiGi

0 standard Gibbs energy ofigi

0 standard molar Gibbs energy ofiDgi relative partial Gibbs energy ig

E excess molar Gibbs energygi

E excess partial Gibbs energy ofiDG Gibbs energy changeDG0 standard Gibbs energy changeDgm molar Gibbs energy of mixingDgf

0 standard molar Gibbs energy of fusionDgv

0 standard molar Gibbs energy of vaporizationH enthalpy in Jh molar enthalpy in J/molhi partial enthalpy ofi

Hi0 standard enthalpy ofi

hi0 standard molar enthalpy ofi

Dhi relative partial enthalpy ofihE excess molar enthalpyhi

E excess partial enthalpy ofiDH enthalpy changeDH0 standard enthalpy change

Dhm molar enthalpy of mixingDhf0 standard molar enthalpy of fusion

Dhv0 standard molar enthalpy of vaporization

K equilibrium constantkB Boltzmann constantn number of molesni number of moles of constituent i

Ni number of particles ofiN0 Avogadros number

pi partial pressure ofiP total pressureP number of phasesqi general extensive variable


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R gas constantS entropy in J/Ks molar entropy in J/mol Ksi partial entropy ofi

Si0

standard entropy ofisi0 standard molar entropy ofi

Dsi0 relative partial entropy ofi

s E excess molar entropysi

E excess partial entropy ofiDS entropy changeDS0 standard entropy changeDsm molar entropy of mixingDsf

0 standard molar entropy of fusionDsv

0 standard molar entropy of vaporizationT temperatureTf temperature of fusionTc critical temperatureTE eutectic temperatureU internal energyvi molar volume ofivi

0 standard molar volume ofiXi mole fraction ofiZ coordination number

gi activity coefficient ofie bond energyh empirical entropy parametermi chemical potential ofin number of moles of foreign particles contributed by a mole of solutex molar metal ratios vibrational bond entropyfi generalized thermodynamic potential

w empirical enthalpy parameter

b.c.c. body-centered cubicf.c.c. face-centered cubich.c.p. hexagonal close-packed



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1.2 Gibbs Energy and Equilibrium 5

1.1 Introduction

An understanding of thermodynamicsand phase diagrams is fundamental and es-

sential to the study of materials science. Aknowledge of the equilibrium state under agiven set of conditions is the starting pointin the description of any phenomenon orprocess.

The theme of this chapter is the relation-ship between phase diagrams and thermo-dynamics. A phase diagram is a graphicalrepresentation of the values of thermody-namic variables when equilibrium is estab-lished among the phases of a system. Mate-rials scientists are used to thinking of phasediagrams as plots of temperature versus com-position. However, many other variablessuch as total pressure and partial pressuresmay be plotted on phase diagrams. In Sec.1.3, for example, predominance diagramswill be discussed, and in Sec. 1.8 chemicalpotentialcomposition phase diagrams will

be presented. General rules regarding phasediagram geometry are given in Sec. 1.9.

In recent years, a quantitative couplingof thermodynamics and phase diagramshas become possible. With the use of com-puters, simultaneous optimizations of ther-modynamic and phase equilibrium data canbe applied to the critical evaluation of bi-nary and ternary systems as shown in Sec.

1.6. This approach often enables good esti-mations to be made of the thermodynamicproperties and phase diagrams of multi-component systems as discussed in Sec.1.11. These estimates are based on structu-ral models of solutions. Various modelssuch as the regular solution model, the sub-lattice model, and models for interstitialsolutions, polymeric solutions, solutions of

defects, ordered solutions, etc. are dis-cussed in Secs. 1.5 and 1.10.The equilibrium diagram is always cal-

culated by minimization of the Gibbs en-

ergy. General computer programs are avail-able for the minimization of the Gibbs en-ergy in systems of any number of phases,components and species as outlined in Sec.

1.12. When coupled to extensive databasesof the thermodynamic properties of com-pounds and multicomponent solutions,these provide a powerful tool in the studyof materials science.

1.1.1 Notation

Extensive thermodynamic properties arerepresented by upper case symbols. For ex-

ample, G = Gibbs energy in J. Molar prop-erties are represented by lower case sym-bols. For example, g = G/n = molar Gibbsenergy in J/mol where n is the total numberof moles in the system.

1.2 Gibbs Energy and Equilibrium

1.2.1 Gibbs Energy

The Gibbs energy of a system is definedin terms of its enthalpy,H, entropy, S, andtemperature, T:

G =H T S (1-1)

A system at constant temperature and pres-sure will approach an equilibrium state thatminimizes G.

As an example, consider the question ofwhether silica fibers in an aluminum ma-trix at 500 C will react to form mullite,Al6Si2O13

If the reaction proceeds with the formationof dn moles of mullite then, from the stoi-

chiometry of the reaction, dnSi =(9/2) dn,dnAl = 6 dn, and dnSiO2 =13/2dn. Sincethe four substances are essentially immis-cible at 500C, we need consider only the

132

692

SiO Al = Si Al Si O (1-2)2 6 2 13+ +


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standard molar Gibbs energies, Gi0. The

Gibbs energy of the system then varies as:

where DG0 is the standard Gibbs energychange of reaction, Eq. (1-2), at 500C.

Since DG0 0, then the reaction, Eq. (1-5),will proceed to the left in order to minimizeG until the equilibrium condition of Eq.(1-7) is attained.

As a further example, we consider thepossible precipitation of graphite from agaseous mixture of CO and CO2 . The reac-tion is:

2 CO = C + CO2 (1-8)

Proceeding as above, we can write:

dG/dn = gC + gCO2 2 gCO= ( g0C + g

0CO2

2 g0CO) +RT ln (pCO2/p2CO)

= DG0 +RT ln (pCO2/p2CO) (1-9)

= DG = RT ln K+RT ln (pCO2/p2CO)

If (pCO2

/p2

CO) is less than the equilibriumconstant K, then precipitation of graphitewill occur in order to decrease G.

Real situations are, of course, generallymore complex. To treat the deposition ofsolid Si from a vapour of SiI4 , for example,we must consider the formation of gaseousI2 , I and SiI2 so that three reaction equa-tions must be written:

SiI4(g) = Si(sol) + 2I2(g) (1-10)SiI4(g) = SiI2(g) + I2(g) (1-11)

I2(g) = 2 I (g) (1-12)



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1.3 Predominance Diagrams 7

The equilibrium state, however, is still thatwhich minimizes the total Gibbs energy ofthe system. This is equivalent to satisfyingsimultaneously the equilibrium constants

of the reactions, Eqs. (1-10) to (1-12), aswill be shown in Section 1.12 where thisexample is discussed further.

1.3 Predominance Diagrams

1.3.1 Calculation of Predominance

Diagrams

Predominance diagrams are a particu-larly simple type of phase diagram whichhave many applications in the fields of hotcorrosion, chemical vapor deposition, etc.Furthermore, their construction clearly il-lustrates the principles of Gibbs energyminimization and the Gibbs Phase Rule.

A predominance diagram for the CuSO system at 1000 K is shown in Fig.1-1. The axes are the logarithms of thepartial pressures of SO2 and O2 in thegas phase. The diagram is divided intoareas or domains of stability of the various

solid compounds of Cu, S and O. For ex-ample, at point Z, where pSO2= 10

2 andpO2= 10

7 bar, the stable phase is Cu2O.The conditions for coexistence of two and

three solid phases are indicated respectivelyby the lines and triple points on the diagram.For example, along the univariant line

(phase boundary) separating the Cu2O andCuSO4 domains the equilibrium constantK=p2SO2pO2

3/2 of the following reaction issatisfied:

Cu2O + 2 SO2 + 3

2O2 = 2CuSO4 (1-13)

Hence, along this line: (1-14)log K= 2 logpSO2

3

2logpO2 = constant

This boundary is thus a straight line with aslope of ( 3/2)/2 = 3/4.

In constructing predominance diagrams,we define a base element, in this case Cu,which must be present in all the condensedphases. Let us further assume that there is

no mutual solubility among the condensedphases.Following the procedure of Bale et al.

(1986), we formulate a reaction for the for-

Figure 1-1. Predomi-

nance diagram. logpSO2versus logpO2 (bar) at1000 K for the Cu S Osystem (Bale et al., 1986).


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mation of each solid phase, always fromone mole of the base element Cu, and in-volving the gaseous species whose pres-sures are used as the axes (SO2 and O2 in

this example):Cu + 1

2O2 = CuO

DG = DG0 +RT lnpO21/2 (1-15)

Cu + 14

O2 = 1

2Cu2O

DG = DG0 +RT lnpO21/4 (1-16)

Cu + SO2 = CuS + O2DG = DG0 +RT ln (pO2p

1SO2

) (1-17)

Cu + SO2 + O2 = CuSO4DG = DG0 +RT ln (p1SO2p

1O2

) (1-18)

and similarly for the formation of Cu2S,Cu2SO4 and Cu2SO5 .

The values ofDG0 are obtained from ta-bles of thermodynamic properties. For anygiven values ofpSO2 and pO2, DG for eachformation reaction can then be calculated.

The stable compound is simply the onewith the most negative DG. If all the DGvalues are positive, then pure Cu is thestable compound.

By reformulating Eqs. (1-15) to (1-18) interms of, for example, S2 and O2 ratherthan SO2 and O2 , a predominance diagramwith ln pS2 and ln pO2 as axes can be con-structed. Logarithms of ratios or productsof partial pressures can also be used asaxes.

1.3.2 Ellingham Diagrams

as Predominance Diagrams

Rather than keeping the temperatureconstant, we can use it as an axis. Figure1-2 shows a diagram for the FeSO sys-tem in whichRT lnpO2 is plotted versus T

at constant pSO2

= 1 bar. The diagram is ofthe same topological type as Fig. 1-1.A similar phase diagram ofRT lnpO2

versus T for the CuO system is shown in

Fig. 1-3. For the formation reaction:

4 Cu + O2 = 2 Cu2O (1-19)

we can write:

DG0 = RT ln K=RT ln (pO2)equilibrium= DH0 TDS0 (1-20)

The diagonal line in Fig. 1-3 is thus a plotof the standard Gibbs energy of formationof Cu2O versus T. The temperatures indi-cated by the symbol M andM are the melt-ing points of Cu and Cu2O respectively.This line is thus simply a line taken from

the well-known Ellingham Diagram orDG0 vs. T diagram for the formation ofoxides. However, by drawing vertical linesat the melting points of Cu and Cu2O asshown in Fig. 1-3, we convert the plot to atrue phase diagram. Stability domains forCu(sol), Cu(l), Cu2O(sol), and Cu2O(l)are shown as functions ofTand of imposed

pO2. The lines and triple points indicate


Figure 1-2. Predominance diagram. RT ln pO2 ver-sus Tat pSO2 = 1.0 bar for the Fe SO system.


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1.4 Thermodynamics of Solutions 9

conditions of two- and three-phase equilib-rium.

1.3.3 Discussion of Predominance

Diagrams

In this section discussion is limited tothe assumption that there is no mutual sol-ubility among the condensed phases. The

calculation of predominance phase dia-grams in which mutual solubility is takeninto account is treated in Sec. 1.9, wherethe general geometrical rules governingpredominance diagrams and their relation-ship to other types of phase diagrams arediscussed.

We frequently encounter predominancediagrams with domains for solid, liquid,

and even gaseous compounds which havebeen calculated as if the compounds wereimmiscible, even though they may actuallybe partially or even totally miscible. Theboundary lines are then no longer phaseboundaries, but are lines separating regionsin which one species predominates. Thewell knownEpHor Pourbaix diagrams ofaqueous chemistry are examples of such

predominance diagrams.Predominance diagrams may also be con-structed when there are two or more baseelements, as discussed by Bale (1990).

Predominance diagrams have foundmany applications in the fields of hot cor-rosion, roasting of ores, chemical vapordeposition, etc. A partial bibliography ontheir construction and applications includesYokokowa (1999), Bale (1990), Bale et al.(1986), Kellogg and Basu (1960), Ingra-ham and Kellogg (1963), Pehlke (1973),

Garrels and Christ (1965), Ingraham andKerby (1967), Pilgrim and Ingraham(1967), Gulbransen and Jansson (1970),Pelton and Thompson (1975), Shatynski(1977), Stringer and Whittle (1975), Spen-cer and Barin (1979), Chu and Rahmel(1979), and Harshe and Venkatachalam(1984).

1.4 Thermodynamics of Solutions

1.4.1 Gibbs Energy of Mixing

Liquid gold and copper are completelymiscible at all compositions. The Gibbs en-ergy of one mole of liquid solution, gl, at1400 K is drawn in Fig. 1-4 as a function ofcomposition expressed as mole fraction,

XCu , of copper. Note thatXAu = 1 XCu . Thecurve ofgl varies between the standard mo-lar Gibbs energies of pure liquid Au andCu, g0Au and g

0Cu .

Figure 1-3. Predominance diagram(also known as a Gibbs energy-tem-perature diagram or Ellingham dia-gram) for the CuO system. PointsM andM represent the meltingpoints of the metal and oxide re-spectively.


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The function Dglm shown on Fig. 1-4 iscalled the molar Gibbs energy of mixing ofthe liquid solution. It is defined as:

Dglm = gl (XAu g

0Au +XCu g

0Cu) (1-21)

It can be seen that Dglm is the Gibbs energychange associated with the isothermal mix-ing ofXAu moles of pure liquid Au andXCumoles of pure liquid Cu to form one moleof solution:

XAu Au(l) +XCu Cu(l)= 1 mole liquid solution (1-22)

Note that for the solution to be stable it isnecessary that Dglm be negative.

1.4.2 Chemical Potential

Thepartial molar Gibbs energy of com-ponent i, gi , also known as the chemical

potential, mi , is defined as:

gi = mi = (G/ni)T,P, nj (1-23)where G is the Gibbs energy of the solu-tion, ni is the number of moles of compo-

nent i, and the derivative is taken with allnj (j9i) constant.

In the example of the AuCu binary liq-uid solution, gCu = (Gl/nCu)T,P, nAu, where

Gl

= (nCu + nAu) gl

. That is, gCu , which hasunits of J/mol, is the rate of change of theGibbs energy of a solution as Cu is added.It can be seen that gCu is an intensive prop-erty of the solution which depends uponthe composition and temperature but notupon the total amount of solution. That is,adding dnCu moles of copper to a solutionof given composition will (in the limit asdnCu 0) result in a change in Gibbs en-ergy, dG, which is independent of the totalmass of the solution.

The reason that this property is called achemical potential is illustrated by the fol-lowing thought experiment. Imagine twosystems, I and II, at the same temperatureand separated by a membrane that permitsonly the passage of copper. The chemicalpotentials of copper in systems I and II

are gICu = GI/nICu and gIICu = GII/nIICu .Copper is transferred across the membrane,with dnI = dnII. The change in the totalGibbs energy accompanying this transferis then: (1-24)dG = d (GI + GII) = (gICu g

IICu) dn

IICu

If gICu > gIICu , then d(G

I + GII) is negativewhen dnIICu is positive. That is, the total

Gibbs energy will be decreased by a trans-fer of Cu from system I to system II.Hence, Cu will be transferred spontane-ously from a system of higher gCu to a sys-tem of lower gCu . Therefore gCu is calledthe chemical potential of copper.

An important principle of phase equilib-rium can now be stated. When two or more

phases are in equilibrium, the chemical po-

tential of any component is the same in allphases.


Figure 1-4. Molar Gibbs energy, g l, of liquid AuCu alloys at constant temperature (1400 K) illustrat-ing the tangent construction.


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1.4.3 Tangent Construction

An important construction is illustratedin Fig. 1-4. If a tangent is drawn to thecurve of gl at a certain composition

(XCu = 0.6 in Fig. 1-4), then the interceptsof this tangent on the axes at XAu =1 and

XCu =1 are equal to gAu and gCu respec-tively at this composition.

To prove this, we first consider that theGibbs energy of the solution at constant Tand P is a function ofnAu and nCu . Hence:

Eq. (1-25) can be integrated as follows:

where the integration is performed at con-stant composition so that the intensiveproperties gAu and gCu are constant. Thisintegration can be thought of as describinga process in which a pre-mixed solution ofconstant composition is added to the sys-tem, which initially contains no material.

Dividing Eqs. (1-26) and (1-25) by(nAu + nCu) we obtain expressions for themolar Gibbs energy and its derivative:

gl =XAu gAu +XCu gCu (1-27)

and

dgl = gAu dXAu + gCu dXCu (1-28)

Since dXAu = dXCu , it can be seen thatEqs. (1-27) and (1-28) are equivalent to thetangent construction shown in Fig. 1-4.

These equations may also be rearranged

to give the following useful expression fora binary system:

gi = g + (1Xi) dg/dXi (1-29)

0

l

0Au Au

0Cu Cu

lAu Au Cu Cu

lAu Cu

d = d d

= (1-2 )

G n n

G n n

G n n

+

+

g g

g g 6

d = d d

= d d (1-2 )

ll

Au

Au

l

Cu

Cu

Au Au Cu Cu

GG

n

nG

n

n

n n

T P,

+

+g g 5

1.4.4 GibbsDuhem Equation

Differentiation of Eq. (1-27) yields:

dgl = (XAu dgAu +XCu dgCu)

+ (gAu dXAu + gCu dXCu) (1-30)

Comparison with Eq. (1-28) then gives theGibbsDuhem equation at constant Tand P:

XAu dgAu +XCu dgCu = 0 (1-31)

1.4.5 Relative Partial Properties

The difference between the partial Gibbs

energy gi of a component in solution andthe partial Gibbs energy gi

0 of the samecomponent in a standard state is called therelative partial Gibbs energy (or relativechemical potential ), Dgi . It is most usual tochoose as standard state the pure compo-nent in the same phase at the same temper-ature. The activity ai of the component rel-ative to the chosen standard state is then

defined in terms of Dgi by the followingequation, as illustrated in Fig. 1-4.

Dgi = gi gi0 = mi mi

0 =RT ln ai (1-32)

Note that gi and mi are equivalent symbols,as are gi

0 and mi0, see Eq. (1-23).

From Fig. 1-4, it can be seen that:

Dgm =XAu DgAu +XCu DgCu

=RT(XAu ln aAu +XCu ln aCu) (1-33)The Gibbs energy of mixing can be di-

vided into enthalpy and entropy terms, ascan the relative partial Gibbs energies:

Dgm = Dhm TDsm (1-34)

Dgi = Dhi TDsi (1-35)

Hence, the enthalpy and entropy of mixing

may be expressed as:Dhm =XAu DhAu +XCu DhCu (1-36)

Dsm =XAu DsAu +XCu DsCu (1-37)


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Furthermore, and for the same reason, theAu and Cu atoms will be randomly distrib-uted over the lattice sites. (In the case of aliquid solution we can think of the lattice

sites as the instantaneous atomic positions.)For a random distribution of NAugold atoms and NCu copper atoms over(NAu +NCu) sites, Boltzmanns equationcan be used to calculate the configurationalentropy of the solution. This is the entropyassociated with the spatial distribution ofthe particles:

(1-42)Sconfig = kB ln (NAu +NCu) ! /NAu !NCu !

where kB is Boltzmanns constant. The con-figurational entropies of pure Au and Cuare zero. Hence the configurational entropyof mixing, DSconfig, will be equal to Sconfig.Furthermore, because of the assumed closesimilarity of Au and Cu, there will be nonon-configurational contribution to the en-tropy of mixing. Hence, the entropy ofmixing will be equal to Sconfig.

Applying Stirlings approximation, whichstates that lnN! = [ (NlnN) N] if N islarge, yields:

For one mole of solution, (NAu +NCu) =N0,

where N0 = Avogadros number. We alsonote that (kBN

0) is equal to the ideal gasconstantR. Hence:

(1-44)DSm

ideal = R (XAu lnXAu +XCu lnXCu)

Therefore, since the ideal enthalpy of mix-ing is zero:

(1-45)Dgm

ideal =R T(XAu lnXAu +XCu lnXCu)

By comparing Eqs. (1-33) and (1-45) weobtain:

Dgiideal =R Tln ai

ideal =R TlnXi (1-46)

DS S k N N

NN

N NN

N

N N

mideal config

B Au Cu

AuAu

Au CuCu

Cu

Au Cu

= = (1- ) +

+

++

( )

ln ln

43

Hence Eq. (1-40) has been demonstratedfor an ideal substitutional solution.

1.4.8 Excess Properties

In reality, Au and Cu atoms are not iden-tical, and so AuCu solutions are not per-fectly ideal. The difference between a solu-tion property and its value in an ideal solu-tion is called an excess property. The ex-cess Gibbs energy, for example, is definedas:

gE = Dgm Dgm

ideal (1-47)

Since the ideal enthalpy of mixing is zero,the excess enthalpy is equal to the enthalpyof mixing:

hE = Dhm Dhmideal = Dhm (1-48)

Hence:

gE = hE T sE

= Dhm T sE (1-49)

Excess partial properties are defined simi-larly:

giE = Dgi Dgi

ideal

=R Tln ai R TlnXi (1-50)

siE = Dsi Dsi

ideal = Dsi +R lnXi (1-51)

Also:

gi

E

= hiE

T siE

= Dhi T siE (1-52)

Equations analogous to Eqs. (1-33),(1-36) and (1-37) relate the integral andpartial excess properties. For example, inAuCu solutions:

gE =XAu g

EAu +XCu g

ECu (1-53)

sE =XAu sEAu +XCu s

ECu (1-54)

Tangent constructions similar to that ofFig. 1-4 can thus also be employed for ex-cess properties, and an equation analogous


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to Eq. (1-29) can be written:

giE = gE + (1Xi) dg

E/dXi (1-55)

The Gibbs Duhem equation, Eq. (1-31),

also applies to excess properties:XAu dg

EAu +XCu dg

ECu = 0 (1-56)

In Au Cu alloys, gE is negative. That is,Dgm is more negative than Dgm

ideal and sothe solution is thermodynamically morestable than an ideal solution. We say thatAuCu solutions exhibit negative devia-tions from ideality. IfgE>0, then the solu-

tion is less stable than an ideal solution andis said to exhibitpositive deviations.

1.4.9 Activity Coefficient

The activity coefficient of a componentin a solution is defined as:

gi = ai/Xi (1-57)

From Eq. (1-50):gi

E =R Tln gi (1-58)

In an ideal solution gi =1 and giE =0 for

all components. Ifgi1, then gi

E >0 and the

driving force for the component to enterinto solution is less than in the case of anideal solution.

1.4.10 Multicomponent Solutions

The equations of this section were de-rived with a binary solution as an example.However, the equations apply equally tosystems of any number of components. For

instance, in a solution of componentsA BCD , Eq. (1-33) becomes:

Dgm =XA DgA +XB DgB +XC DgC+XD DgD + (1-59)

1.5 Binary Phase Diagrams

1.5.1 Systems with Complete Solid

and Liquid Miscibility

The temperaturecomposition (TX)phase diagram of the CaOMnO system isshown in Fig. 1-5 (Schenck et al., 1964;Wu, 1990). The abscissa is the composi-


Figure 1-5. Phase dia-

gram of the CaO MnOsystem at P =1bar(after Schenck et al.,1964, and Wu, 1990).


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1.5 Binary Phase Diagrams 15

tion, expressed as mole fraction of MnO,XMnO . Note thatXMnO = 1 XCaO . Phase di-agrams are also often drawn with the com-position axis expressed as weight percent.

At all compositions and temperatures inthe area above the line labelled liquidus, asingle-phase liquid solution will be ob-served, while at all compositions and tem-peratures below the line labelled solidus,there will be a single-phase solid solution.A sample at equilibrium at a temperatureand overall composition between these twocurves will consist of a mixture of solidand liquid phases, the compositions ofwhich are given by the liquidus and soliduscompositions at that temperature. For ex-ample, a sample of overall composition

XMnO = 0.60 at T= 2200C (at point R inFig. 1-5) will consist, at equilibrium, of amixture of liquid of composition XMnO =0.70 (point Q) and solid of composition

XMnO = 0.35 (point P).The line PQ is called a tie-line or co-

node. As the overall composition is variedat 2200 C between points P and Q, thecompositions of the solid and liquid phasesremain fixed at P and Q, and only the rela-tive proportions of the two phases change.From a simple mass balance, we can derivethe lever rule for binary systems: (moles ofliquid)/(moles of solid)= PR/RQ. Hence,at 2200 C a sample with overall composi-

tionXMnO = 0.60 consists of liquid and solidphases in the molar ratio (0.600.35)/(0.700.60)=2.5. Were the compositionaxis expressed as weight percent, then thelever rule would give the weight ratio ofthe two phases.

Suppose that a liquid CaOMnO solu-tion with composition XMnO = 0.60 iscooled very slowly from an initial tempera-

ture of about 2500 C. When the tempera-ture has decreased to the liquidus tempera-ture 2270C (point B), the first solidappears, with a composition at point A

(XMnO = 0.28). As the temperature is de-creased further, solid continues to precipi-tate with the compositions of the twophases at any temperature being given by

the liquidus and solidus compositions atthat temperature and with their relativeproportions being given by the lever rule.Solidification is complete at 2030 C, thelast liquid to solidify having composition

XMnO = 0.60 (point C).The process just described is known as

equilibrium cooling. At any temperatureduring equilibrium cooling the solid phasehas a uniform (homogeneous) composition.In the preceding example, the compositionof the solid phase during cooling variesalong the lineAPC. Hence, in order for thesolid grains to have a uniform compositionat any temperature, diffusion of CaO fromthe center to the surface of the growinggrains must occur. Since solid-state dif-fusion is a relatively slow process, equi-librium cooling conditions are only ap-

proached if the temperature is decreasedvery slowly. If a sample of composition

XMnO = 0.60 is cooled very rapidly from theliquid, concentration gradients will be ob-served in the solid grains, with the concen-tration of MnO increasing towards the sur-face from a minimum ofXMnO = 0.28 (point

A) at the center. Furthermore, in this casesolidification will not be complete at

2030 C since at 2030C the average con-centration of MnO in the solid particleswill be less than XMnO = 0.60. These con-siderations are discussed more fully inChapter 2 of this volume (Mller-Krumb-haar et al., 2001).

At XMnO = 0 andXMnO =1 in Fig. 1-5 theliquidus and solidus curves meet at theequilibrium melting points, or tempera-

tures of fusion of CaO and MnO, which areT0f(CaO) =2572C, T0f(MnO) =1842C.

The phase diagram is influenced by thetotal pressure, P. Unless otherwise stated,


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TX diagrams are usually presented forP = const. = 1 bar. For equilibria involvingonly solid and liquid phases, the phaseboundaries are typically shifted only by the

order of a few hundredths of a degree perbar change in P. Hence, the effect of pres-sure upon the phase diagram is generallynegligible unless the pressure is of the or-der of hundreds of bars. On the other hand,if gaseous phases are involved then the ef-fect of pressure is very important. The ef-fect of pressure will be discussed in Sec.1.5.3.

1.5.2 Thermodynamic Origin

of Phase Diagrams

In this section we first consider the ther-modynamic origin of simple lens-shapedphase diagrams in binary systems withcomplete liquid and solid miscibility.

An example of such a diagram was givenin Fig. 1-5. Another example is the GeSi

phase diagram in the lowest panel of Fig.1-6 (Hansen, 1958). In the upper three pan-els of Fig. 1-6, the molar Gibbs energies ofthe solid and liquid phases, gs and gl, atthree temperatures are shown to scale. Asillustrated in the top panel, gs varies withcomposition between the standard molarGibbs energies of pure solid Ge and of puresolid Si, gGe

0(s) and gSi0(s), while gl varies

between the standard molar Gibbs energiesof the pure liquid components gGe0(l) and

gSi0(l).The difference between gGe

0(l) and gSi0(s) is

equal to the standard molar Gibbs energyof fusion (melting) of pure Si, Dg0f(Si)=(gSi

0(l) gSi0(s)). Similarly, for Ge, Dg0f(Ge)=

(gGe0(l) gGe

0(s)). The Gibbs energy of fusion ofa pure component may be written as:

Dgf0 = Dhf0 TDsf0 (1-60)where Dhf

0 and Dsf0 are the standard molar

enthalpy and entropy of fusion.

Since, to a first approximation, Dhf0 and

Dsf0 are independent ofT, Dgf

0 is approxi-mately a linear function of T. If T> Tf

0,then Dgf

0 is negative. IfT< Tf0, then Dgf

0 is

positive. Hence, as seen in Fig. 1-6, as Tdecreases, the gs curve descends relative tog

l. At 1500C, gl < gs at all compositions.Therefore, by the principle that a systemalways seeks the state of minimum Gibbsenergy at constant Tand P, the liquid phaseis stable at all compositions at 1500 C.

At 1300 C, the curves ofgs and gl cross.The line P1Q1, which is the common tan-gentto the two curves, divides the compo-sition range into three sections. For compo-sitions between pure Ge and P1, a single-phase liquid is the state of minimum Gibbsenergy. For compositions between Q1 andpure Si, a single-phase solid solution is thestable state. Between P1 and Q1, a totalGibbs energy lying on the tangent lineP1Q1 may be realized if the system adoptsa state consisting of two phases with com-

positions at P1 and Q1 and with relativeproportions given by the lever rule. Sincethe tangent line P1 Q1 lies below both g

s andg

l, this two-phase state is more stable thaneither phase alone. Furthermore, no otherline joining any point on gl to any point ong

s lies below the line P1 Q1. Hence, this linerepresents the true equilibrium state of thesystem, and the compositions P1 and Q1 are

the liquidus and solidus compositions at1300C.As T is decreased to 1100C, the points

of common tangency are displaced tohigher concentrations of Ge. For T


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Ge and Si are equal in the solid and liquidphases at equilibrium. That is:

glGe = g

sGe (1-61)

glSi = g

sSi (1-62)

This equality of chemical potentials wasshown in Sec. 1.4.2 to be the criterion forphase equilibrium. That is, the commontangent construction simultaneously mini-

mizes the total Gibbs energy and ensuresthe equality of the chemical potentials,thereby showing that these are equivalentcriteria for equilibrium between phases.

If we rearrange Eq. (1-61), subtractingthe Gibbs energy of fusion of pure Ge,Dg0f(Ge)= (gGe

0(l) gGe0(s)), from each side, we

get:

(glGe gGe0(l)) (gsGe gGe

0(s))

= (gGe0(l) gGe

0(s)) (1-63)

Using Eq. (1-32), we can write Eq. (1-63)as:

DglGe Dg

sGe = Dg

0f(Ge) (1-64)

or

RTln alGe RTln asGe = Dg

0f(Ge) (1-65)

Figure 1-6. GeSi phase diagram atP =1 bar (after Hansen, 1958) andGibbs energy composition curves atthree temperatures, illustrating the com-mon tangent construction (reprintedfrom Pelton, 1983).


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where alGe is the activity of Ge (with re-spect to pure liquid Ge as standard state) inthe liquid solution on the liquidus, and asGeis the activity of Ge (with respect to pure

solid Ge as standard state) in the solid solu-tion on the solidus. Starting with Eq. (1-62), we can derive a similar expression forthe other component:

RTln alSi RTln asSi = Dg

0f(Si) (1-66)

Eqs. (1-65) and (1-66) are equivalent to thecommon tangent construction.

It should be noted that absolute values ofGibbs energies cannot be defined. Hence,the relative positions of gGe

0(l) and gSi0(l) in

Fig. 1-6 are completely arbitrary. However,this is immaterial for the preceding discus-sion, since displacing both gSi

0(l) and gSi0(s) by

the same arbitrary amount relative to gGe0(l)

and gGe0(s) will not alter the compositions of

the points of common tangency.It should also be noted that in the present

discussion of equilibrium phase diagrams

we are assuming that the physical dimen-sions of the single-phase regions in thesystem are sufficiently large that surface(interfacial) energy contributions to theGibbs energy can be neglected. For veryfine grain sizes in the sub-micron range,however, surface energy effects can notice-ably influence the phase boundaries.

The shape of the two-phase (solid + liq-

uid) lens on the phase diagram is deter-mined by the Gibbs energies of fusion,Dgf

0, of the components and by the mixingterms, Dgs and Dgl. In order to observehow the shape is influenced by varyingDgf

0, let us consider a hypothetical systemA B in which Dgs and Dgl are ideal Raoul-tian (Eq. (1-45)). Let T0f(A) = 800 K andT0f(B) = 1200 K. Furthermore, assume that

the entropies of fusion of A and B are equaland temperature-independent. The enthalp-ies of fusion are then given from Eq. (1-60)by the expression Dhf

0= Tf0Dsf

0 since

Dgf0 = 0 when T= Tf

0. Calculated phase dia-grams for Dsf

0 = 3, 10 and 30 J/mol K areshown in Fig. 1-7. A value ofDsf

0 10 istypical of most metals. However, when the

components are ionic compounds such asionic oxides, halides, etc., Dsf0 can be sig-

nificantly larger since there are severalions per formula unit. Hence, two-phaselenses in binary ionic salt or oxide phasediagrams tend to be fatter than thoseencountered in alloy systems. If we areconsidering vaporliquid equilibria ratherthan solidliquid equilibria, then theshape is determined by the entropy ofvaporization, Dsv

0. Since Dsv0 is usually an

order of magnitude larger than Dsf0, two-

phase (liquid + vapor) lenses tend to be


Figure 1-7. Phase diagram of a system AB with

ideal solid and liquid solutions. Melting points of Aand B are 800 and 1200 K, respectively. Diagramsare calculated for entropies of fusion S0f(A)= S0f(B) =3, 10 and 30 J/mol K.


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very wide. For equilibria between two solidsolutions of different crystal structure, theshape is determined by the entropy ofsolidsolid transformation, which is usu-

ally smaller than the entropy of fusion byapproximately an order of magnitude.Therefore two-phase (solid + solid) lensestend to be very narrow.

1.5.3 PressureComposition Phase

Diagrams

Let us consider liquidvapor equilib-rium with complete miscibility, using as an

example the ZnMg system. Curves ofgv

and gl can be drawn at any given T and P,as in the upper panel of Fig. 1-8, and the

common tangent construction then givesthe equilibrium vapor and liquid composi-tions. The phase diagram depends upon theGibbs energies of vaporization of the com-

ponents Dgv(Zn) and Dgv(Mg) as shown inFig. 1-8.To generate the isothermal pressure

composition (PX) phase diagram in thelower panel of Fig. 1-8 we require theGibbs energies of vaporization as functionsof P. Assuming monatomic ideal vaporsand assuming that pressure has negligibleeffect upon the Gibbs energy of the liquid,we can write:

Dgv(i) = Dg0v(i) +RTln P (1-67)

where Dgv(i) is the standard Gibbs energyof vaporization (when P =1 bar), which isgiven by:

Dg0v(i) = Dh0v(i) TDs

0v(i) (1-68)

For example, the enthalpy of vaporization

of Zn is Dh0v(Zn)= 115 300 J/mol at its nor-mal boiling point of 1180 K (Barin et al.,

1977). Assuming that Dh0v is independentof T, we calculate from Eq. (1-68) thatDs0v(Zn) = 115 300 /1180 = 97.71 J/mol K.From Eq. (1-67), Dgv(Zn) at any T and P isthus given gy:

(1-69)Dgv(Zn) = (115 300 97.71 T) +RTln P

A similar expression can be derived for theother component Mg.At constant temperature, then, the curve

of gv in Fig. 1-8 descends relative to gl asthe pressure is lowered, and the PXphasediagram is generated by the common tan-gent construction. The diagram at 1250 Kin Fig. 1-8 was calculated under the as-sumption of ideal liquid and vapor mixing

(gE(l)

= 0, gE(v)

=0).PX phase diagrams involving liquidsolid or solidsolid equilibria can be cal-culated in a similar fashion through the fol-

Figure 1-8. Pressurecomposition phase diagramof the ZnMg system at 1250 K calculated for idealvapor and liquid solutions. Upper panel illustratescommon tangent construction at a constant pressure.


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lowing general equation, which gives theeffect of pressure upon the Gibbs energychange for the transformation of one moleof pure component i from an a-phase to a

b-phase:

where Dg0ab is the standard (P =1 bar)Gibbs energy of transformation, and vi

b andvia are the molar volumes.

1.5.4 Minima and Maxima

in Two-Phase Regions

As discussed in Sec. 1.4.8, the Gibbs en-ergy of mixing Dgm may be expressed asthe sum of an ideal term Dgm

ideal and an ex-cess term gE. As has just been shown inSec. 1.5.2, ifDg sm and Dg

lm for the solid

and liquid phases are both ideal, then alens-shaped two-phase region always re-sults. However in most systems even ap-

proximately ideal behavior is the exceptionrather than the rule.

Curves of gs and gl for a hypotheticalsystem AB are shown schematically inFig. 1-9 at a constant temperature (belowthe melting points of pure A and B) such

D Da b a bb ag g v v + = d (1- )

=

0

1

70P

P

i i P( )

that the solid state is the stable state forboth pure components. However, in thissystem gE(l) < gE(s), so that gs presents aflatter curve than does gl and there exists a

central composition region in which gl

< gs

.Hence, there are two common tangentlines, P1 Q1 and P2 Q2 . Such a situationgives rise to a phase diagram with a mini-mum in the two-phase region, as observedin the Na2CO3K2CO3 system (Dessu-reault et al., 1990) shown in Fig. 1-10. At acomposition and temperature correspond-ing to the minimum point, liquid and solidof the same composition exist in equilib-rium.

A two-phase region with a minimumpoint as in Fig. 1-10 may be thought of as atwo-phase lens which has been pusheddown by virtue of the fact that the liquid isrelatively more stable than the solid. Ther-modynamically, this relative stability is ex-pressed as gE(l) < gE(s).

Conversely, ifgE(l) > gE(s) to a sufficient

extent, then a two-phase region with amaximum will result. Such maxima in (liq-uid+solid) or (solid+solid) two-phase re-gions are nearly always associated with theexistence of an intermediate phase, as willbe discussed in Sec. 1.5.10.


Figure 1-9. Isothermal Gibbs energy-compositioncurves for solid and liquid phases in a system AB inwhich g E(l) > g E(s). A phase diagram of the type ofFig. 1-10 results.

Figure 1-10. Phase diagram of the K2CO3Na2CO3system at P =1 bar (Dessureault et al., 1990).


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1.5.5 Miscibility Gaps

If gE > 0, then the solution is thermody-namically less stable than an ideal solution.

This can result from a large difference insize of the component atoms, ions or mole-cules, which will lead to a (positive) latticestrain energy, or from differences in elec-tronic structure, or from other factors.

In the AuNi system, gE is positive inthe solid phase. In the top panel of Fig. 1-11,g

E(s) is plotted at 1200 K (Hultgren et al.,1973) and the ideal Gibbs energy ofmixing, Dgm

ideal, is also plotted at 1200 K.The sum of these two terms is the Gibbsenergy of mixing of the solid solution,Dgm

s , which is plotted at 1200 K as wellas at other temperatures in the central panelof Fig. 1-11. Now, from Eq. (1-45), Dgm

ideal

is always negative and varies directlywith T, whereas gE varies less rapidly withtemperature. As a result, the sum Dgm

s =Dgm

ideal + gE becomes less negative as T de-

creases. However, the limiting slopes to theDgm

ideal curve atXAu =1 andXNi =1 are bothinfinite, whereas the limiting slopes ofgE

are always finite (Henrys Law). Hence,Dgsm will always be negative as XAu 1andXNi 1 no matter how low the temper-ature. As a result, below a certain tempera-ture the curve ofDgsm will exhibit two neg-ative humps. Common tangent lines

P1Q1, P2 Q2 , P3 Q3 to the two humps at dif-ferent temperatures define the ends of tie-lines of a two-phase solidsolid miscibilitygap in the AuNi phase diagram, which isshown in the lower panel in Fig. 1-11(Hultgren et al., 1973). The peak of the gapoccurs at the critical or consolute tempera-ture and composition, Tc andXc .

When gE(s) is positive for the solid phase

in a system it is usually also the case thatgE(l) < gE(s) since the unfavorable factors(such as a difference in atomic dimensions)which are causing gE(s) to be positive will

have less of an effect upon gE(l) in the liq-uid phase owing to the greater flexibility ofthe liquid structure to accommodate differ-ent atomic sizes, valencies, etc. Hence, a

solidsolid miscibility gap is often asso-ciated with a minimum in the two-phase(solid + liquid) region, as is the case in theAuNi system.

Figure 1-11. Phase diagram (after Hultgren et al.,1973) and Gibbs energy composition curves of solidsolutions for the AuNi system at P =1 bar. Letterss indicate spinodal points (Reprinted from Pelton,1983).


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Below the critical temperature the curveofDgsm exhibits two inflection points, indi-cated by the letter s in Fig. 1-11. Theseare known as the spinodal points. On the

phase diagram their locus traces out thespinodal curve (Fig. 1-11). The spinodalcurve is not part of the equilibrium phasediagram, but it is important in the kineticsof phase separation, as discussed in Chap-ter 6 (Binder and Fratzl, 2001).

1.5.6 Simple Eutectic Systems

The more positive gE is in a system, thehigher is Tc and the wider is the miscibility

gap at any temperature. Suppose that gE(s)

is so positve that Tc is higher than the min-imum in the (solid + liquid) region. The re-sult will be a phase diagram such as that of

the MgOCaO system shown in Fig. 1-12(Doman et al., 1963; Wu, 1990).The lower panel of Fig. 1-12 shows the

Gibbs energy curves at 2450C. The twocommon tangents define two two-phase re-gions. As the temperature is decreased be-low 2450 C, the gs curve descends relativeto gl and the two points of tangency P1and P2 approach each other until, at T=2374 C, P1 and P2 become coincident atthe composition E. That is, at T=2374C


Figure 1-12. Phase diagramat P =1 bar (after Doman etal., 1963, and Wu, 1990) andGibbs energy composition

curves at 2450C for theMgO CaO system. SolidMgO and CaO have thesame crystal structure.


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there is just one common tangent line con-tacting the two portions of the gs curve atcompositions A and B and contacting the gl

curve at E. This temperature is known as

the eutectic temperature, TE , and the com-position E is the eutectic composition. Fortemperatures below TE , g

l lies completelyabove the common tangent to the two por-tions of the gs curve and so for T< TE asolidsolid miscibility gap is observed.The phase boundaries of this two-phase re-gion are called the solvus lines. The wordeutectic is from the Greek for to meltwell since the system has its lowest melt-ing point at the eutectic composition E.

This description of the thermodynamicorigin of simple eutectic phase diagrams isstrictly correct only if the pure solid com-ponents A and B have the same crystalstructure. Otherwise, a curve for gs whichis continuous at all compositions cannot bedrawn.

Suppose a liquid MgOCaO solution of

composition XCaO =0.52 (composition P1)is cooled from the liquid state very slowlyunder equilibrium conditions. At 2450 Cthe first solid appears with composition Q1.As T decreases further, solidification con-tinues with the liquid composition follow-ing the liquidus curve from P1 to E and thecomposition of the solid phase followingthe solidus curve from Q1 to A. The rela-

tive proportions of the two phases at any Tare given by the lever rule. At a tempera-ture just above TE , two phases are ob-served: a solid of composition A and a liq-uid of composition E. At a temperature justbelow TE , two solids with compositions Aand B are observed. Therefore, at TE , dur-ing cooling, the following binary eutecticreaction occurs:

liquid solid1 + solid2 (1-71)Under equilibrium conditions the tempera-ture will remain constant at T= TE until all

the liquid has solidified, and during the re-action the compositions of the three phaseswill remain fixed at A, B and E. For thisreason the eutectic reaction is called an in-

variant reaction. More details on eutecticsolidification may be found in Chapter 2(Mller-Krumbhaar et al., 2001).

1.5.7 Regular Solution Theory

Many years ago Van Laar (1908) showedthat the thermodynamic origin of a greatmany of the observed features of binaryphase diagrams can be illustrated at leastqualitatively by simple regular solutiontheory. A simple regular solution is one forwhich:

gE =XAXB (w hT) (1-72)

where wand h are parameters independentof temperature and composition. Substitut-ing Eq. (1-72) into Eq. (1-29) yields, forthe partial properties:

(1-73)g

EA =XB

2 (w hT) , gEB =XA2 (w hT)

Several liquid and solid solutions con-form approximately to regular solution be-havior, particularly ifgE is small. Examplesmay be found for alloys, molecular solu-tions, and ionic solutions such as moltensalts and oxides, among others. (The verylow values ofgE observed for gaseous solu-

tions generally conform very closely to Eq.(1-72).)

To understand why this should be so, weonly need a very simple model. Supposethat the atoms or molecules of the compo-nents A and B mix substitutionally. If theatomic (or molecular) sizes and electronicstructures of A and B are similar, then thedistribution will be nearly random, and the

configurational entropy will be nearlyideal. That is:

gE Dhm T SE(non-config) (1-74)


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More will be said on this point in Sec.1.10.5.

We now assume that the bond energieseAA, eBB and eAB of nearest-neighbor pairs

are independent of temperature and com-position and that the average nearest-neighbor coordination number, Z, is alsoconstant. Finally, we assume that the en-thalpy of mixing results mainly from thechange in the total energy of nearest-neigh-bor pair bonds.

In one mole of solution there are (N0

Z/2) neareast-neighbor pair bonds, whereN0 is Avogadros number. Since the distri-bution is assumed random, the probabilitythat a given bond is an AA bond is equalto X2A. The probabilities of BB and ABbonds are, respectively, X2B and 2XAXB .The molar enthalpy of mixing is then equalto the sum of the energies of the nearest-neighbor bonds in one mole of solution,minus the energy of the AA bonds inXAmoles of pure A and the energy of the BB

bonds inXB moles of pure B:Dhm = (N

0Z/2) (X2A eAA +X

2B eBB + 2XAB eAB)

(N0Z/2) (XA eAA) (N0Z/2) (XB eBB)

= (N0Z) [eAB (eAA + eBB)/2]XAXB= wXAXB (1-75)

We now define sAB , sAA and sBB as the

vibrational entropies of nearest-neighborpair bonds. Following an identical argu-ment to that just presented for the bondenergies we obtain:

sE(non-config) (1-76)= (N0Z) [sAB (sAA + sBB)/2] = hXAXB

Eq. (1-72) has thus been derived. If ABbonds are stronger than AA and BB

bonds, then (eAB hABT) 0 and gE> 0.

Simple non-polar molecular solutionsand ionic solutions such as molten salts of-

ten exhibit approximately regular behavior.The assumption of additivity of the energyof pair bonds is probably reasonably realis-tic for van der Waals or coulombic forces.For alloys, the concept of a pair bond is, atbest, vague, and metallic solutions tend toexhibit larger deviations from regular be-havior.

In several solutions it is found that|hT| < |w| in Eq. (1-72). That is, gE Dhm= wXAXB , and to a first approximationg

E is independent ofT. This is more oftenthe case in non-metallic solutions than inmetallic solutions.

1.5.8 Thermodynamic Origin

of Simple Phase Diagrams Illustrated

by Regular Solution Theory

Figure 1-13 shows several phase dia-grams, calculated for a hypothetical systemAB containing a solid and a liquid phasewith melting points of T0f(A) = 800 K andT0f(B) = 1200 K and with entropies of fusionof both A and B set to 10 J/mol K, which isa typical value for metals. The solid andliquid phases are both regular with temper-ature-independent excess Gibbs energies

gE(s) = wsXAXB and gE(l) = wlXAXB

The parameters ws and wl have been variedsystematically to generate the various pan-els of Fig. 1-13.

In panel (n) both phases are ideal. Panels(l) to (r) exhibit minima or maxima de-pending upon the sign and magnitude of(gE(l) gE(s)), as has been discussed in Sec.

1.5.4. In panel (h) the liquid is ideal butpositive deviations in the solid give rise toa solidsolid miscibility gap as discussedin Sec. 1.5.6. On passing from panel (h) to



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panel (c), an increase in gE(s) results in awidening of the miscibility gap so that the

solubility of A in solid B and of B in solidA decreases. Panels (a) to (c) illustrate thatnegative deviations in the liquid cause arelative stabilization of the liquid with re-

sultant lowering of the eutectic tempera-ture.

Eutectic phase diagrams are often drawnwith the maximum solid solubility occur-ring at the eutectic temperature (as in Fig.1-12). However, panel (d) of Fig. 1-13, in

Figure 1-13. Topological changes in the phase diagram for a system AB with regular solid and liquid phases,brought about by systematic changes in the regular solution parameters s and l. Melting points of pure A andB are 800 K and 1200 K. Entropies of fusion of both A and B are 10.0 J/mol K (Pelton and Thompson, 1975).The dashed curve in panel (d) is the metastable liquid miscibility gap (Reprinted from Pelton, 1983).


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which the maximum solubility of A in theB-rich solid solution occurs at approxi-mately T= 950 K, illustrates that this neednot be the case even for simple regular so-

lutions.

1.5.9 Immiscibility Monotectics

In Fig. 1-13(e), positive deviations inthe liquid have given rise to a liquidliquidmiscibility gap. The CaOSiO2 system(Wu, 1990), shown in Fig. 1-14, exhibitssuch a feature. Suppose that a liquid ofcomposition XSiO

2

=0.8 is cooled slowlyfrom high temperatures. At T= 1815 C themiscibility gap boundary is crossed and asecond liquid layer appears with a compo-sition ofXSiO2= 0.97. As the temperature islowered further, the composition of eachliquid phase follows its respective phaseboundary until, at 1692 C, the SiO2-richliquid has a composition of XSiO2=0.99(point B), and in the CaO-rich liquid

XSiO2= 0.74 (point A). At any temperature,the relative amounts of the two phases aregiven by the lever rule.

At 1692 C the following invariant bi-nary monotectic reaction occurs upon cool-ing:

Liquid B Liquid A + SiO2 (solid) (1-77)

The temperature remains constant at

1692 C and the compositions of the phasesremain constant until all of liquid B is con-sumed. Cooling then continues with pre-cipitation of solid SiO2 with the equilib-rium liquid composition following the liq-uidus from point A to the eutectic E.

Returning to Fig. 1-13, we see in panel(d) that the positive deviations in the liquidin this case are not large enough to produce

immiscibility, but they do result in a flat-tening of the liquidus, which indicates atendency to immiscibility. If the nuclea-tion of the solid phases can be suppressed

by sufficiently rapid cooling, then a meta-stable liquidliquid miscibility gap is ob-served as shown in Fig. 1-13(d). For exam-ple, in the Na2OSiO2 system the flattened

(or S-shaped) SiO2 liquidus heralds theexistence of a metastable miscibility gap ofimportance in glass technology.

1.5.10 Intermediate Phases

The phase diagram of the AgMgsystem (Hultgren et al., 1973) is shown inFig. 1-15(d). An intermetallic phase, b, isseen centered approximately about thecomposition XMg =0.5. The Gibbs energycurve at 1050 K for such an intermetallicphase has the form shown schematically inFig. 1-15(a). The curve gb rises quite rap-idly on either side of its minimum, whichoccurs near XMg= 0.5. As a result, the bphase appears on the phase diagram onlyover a limited composition range. Thisform of the curve gb results from the fact

that when XAg XMg a particularly stablecrystal structure exists in which Ag and Mgatoms preferentially occupy different sites.The two common tangents P1Q1 and P2 Q2give rise to a maximum in the two-phase(b + liquid) region of the phase diagram.(Although the maximum is observed verynear XMg= 0.5, there is no thermodynamicreason for the maximum to occur exactly at

this composition.)Another intermetallic phase, the e phase,is also observed in the AgMg system,Fig. 1-15. The phase is associated with a

peritectic invariant ABC at 744 K. TheGibbs energy curves are shown schemati-cally at the peritectic temperature in Fig.1-15(c). One common tangent line can bedrawn to gl, gb and ge.

Suppose that a liquid alloy of composi-tion XMg =0.7 is cooled very slowly fromthe liquid state. At a temperature just above744 K a liquid phase of composition C and



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Figure 1-14. CaOSiO2 phase diagram at P= 1 bar (after Wu, 1990) and Gibbs energy curves at 1500 C illus-trating Gibbs energies of fusion and formation of the stoichiometric compound CaSiO3 .


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a b phase of composition A are observedat equilibrium. At a temperature just below744 K the two phases at equilibrium are bof composition A and e of composition B.

The following invariant binary peritecticreaction thus occurs upon cooling:

Liquid + b (solid) e (solid) (1-78)

This reaction occurs isothermally at 744 Kwith all three phases at fixed compositions(at points A, B and C). For an alloy withoverall composition between points A and

B the reaction proceeds until all the liquidhas been consumed. In the case of an alloywith overall composition between B and C,the b phase will be the first to be com-pletely consumed.

Peritectic reactions occur upon coolingwith formation of the product solid (e inthis example) on the surface of the reactantsolid (b), thereby forming a coating whichcan prevent further contact between the re-actant solid and liquid. Further reactionmay thus be greatly retarded so that equi-librium conditions can only be achieved byextremely slow cooling.

The Gibbs energy curve for the e phase,ge, in Fig. 1-15(c) rises more rapidly on ei-

ther side of its minimum than does theGibbs energy gb for the b phase in Fig. 1-15(a). As a result, the width of the single-

phase region over which the e phase exists(sometimes called its range of stoichiome-try or homogeneity range) is narrower thanfor the b phase.

In the upper panel of Fig. 1-14 for theCaOSiO2 system, Gibbs energy curves at1500 C for the liquid and CaSiO3 phasesare shown schematically. g0.5(CaSiO3) risesextremely rapidly on either side of its min-

imum. (We write g0.5(CaSiO3) for 0.5 molesof the compound in order to normalize to abasis of one mole of components CaO andSiO2 .) As a result, the points of tangencyQ1 and Q2 of the common tangents P1Q1and P2 Q2 nearly (but not exactly) coincide.Hence, the range of stoichiometry of theCaSiO3 phase is very narrow (but neverzero). The two-phase regions labelled

(CaSiO3 + liquid) in Fig. 1-14 are the twosides of a two-phase region that passesthrough a maximum at 1540 C just as the(b + liquid) region passes through a maxi-


Figure 1-15. AgMg phase diagram at P =1 bar (af-ter Hultgren at al., 1973) and Gibbs energy curves atthree temperatures.


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mum in Fig. 1-15(d). Because the CaSiO3single-phase region is so narrow, we referto CaSiO3 as a stoichiometric compound.Any deviation in composition from the

stoichiometric 1:1 ratio of CaO to SiO2results in a very large increase in Gibbsenergy.

The e phase in Fig. 1-15 is based on thestiochiometry AgMg3 . The Gibbs energycurve, Fig. 1-15(c), rises extremely rapidlyon the Ag side of the minimum, but some-what less steeply on the Mg side. As a re-sult, Ag is virtually insoluble in AgMg3,while Mg is sparingly soluble. Such aphase with a narrow range of homogeneityis often called a non-stoichiometric com-

pound. At low temperatures the b phaseexhibits a relatively narrow range of stoi-chiometry about the 1:1 AgMg composi-tion and can properly be called a com-pound. However, at higher temperatures itis debatable whether a phase with such awide range of composition should be called

a compound.From Fig. 1-14 it can be seen that if stoi-

chiometric CaSiO3 is heated it will meltisothermally at 1540 C to form a liquid ofthe same composition. Such a compound iscalled congruently melting or simply a con-gruent compound. The compound Ca2SiO4in Fig. 1-14 is congruently melting. The bphase in Fig. 1-15 is also congruently melt-

ing at the composition of the liquidus/sol-idus maximum.It should be noted with regard to the con-

gruent melting of CaSiO3 in Fig. 1-14 thatthe limiting slopes dT/dXof both branchesof the liquidus at the congruent meltingpoint (1540C) are zero since we are reallydealing with a maximum in a two-phase re-gion.

The AgMg3 (e) compound in Fig. 1-15 issaid to melt incongruently. If solid AgMg3is heated it will melt isothermally at 744 Kby the reverse of the peritectic reaction,

Eq. (1-78), to form a liquid of compositionC and another solid phase, b, of composi-tion A.

Another example of an incongruent

compoundis Ca3Si2O7 in Fig. 1-14, whichmelts incongruently (or peritectically) toform liquid and Ca2SiO4 at the peritectictemperature of 1469 C.

An incongruent compound is always as-sociated with a peritectic. However, theconverse is not necessarily true. A peritec-tic is not always associated with an inter-mediate phase. See, for example, Fig. 1-13(i).

For purposes of phase diagram calcula-tions involving stoichiometric compoundssuch as CaSiO3 , we may, to a good approx-imation, consider the Gibbs energy curve,g0.5(CaSiO3) , to have zero width. All thatis then required is the value ofg0.5(CaSiO3)at the minimum. This value is usuallyexpressed in terms of the Gibbs energyof fusion of the compound, Dg0f (0.5 CaSiO3)or the Gibbs energy of formationDg0form(0.5 CaSiO3) of the compound fromthe pure solid components CaO and SiO2according to the reaction: 0.5 CaO(sol) +0.5SiO2(sol)=0.5CaSiO3(sol). Both thesequantities are interpreted graphically inFig. 1-14.

1.5.11 Limited Mutual Solubility Ideal Henrian Solutions

In Sec. 1.5.6, the region of two solids inthe MgOCaO phase diagram of Fig. 1-12was described as a miscibility gap. That is,only one continuous gs curve was assumed.If, somehow, the appearance of the liquidphase could be suppressed, then the two

solvus lines in Fig. 1-12, when projectedupwards, would meet at a critical pointabove which one continuous solid solutionwould exist at all compositions.


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Such a description is justifiable only ifthe pure solid components have the samecrystal structure, as is the case for MgOand CaO. However, consider the AgMg

system, Fig. 1-15, in which the terminal(Ag) solid solution is face-centered-cubicand the terminal (Mg) solid solution is hex-agonal-close-packed. In this case, one con-tinuous curve for gs cannot be drawn. Eachsolid phase must have its own separateGibbs energy curve, as shown schemati-cally in Fig. 1-15(b) for the h.c.p. (Mg)phase at 800 K. In this figure, gMg

0(h.c.p.) andgAg

0(f.c.c.) are the standard molar Gibbs ener-gies of pure h.c.p. Mg and pure f.c.c. Ag,while gAg

0(h.c.p.-Mg) is the standard molarGibbs energy of pure (hypothetical) h.c.p.Ag in the h.c.p. (Mg) phase.

Since the solubility of Ag in the h.c.p.(Mg) phase is limited we can, to a good ap-proximation, describe it as aHenrian idealsolution. That is, when a solution is suffi-ciently dilute in one component, we can ap-

proximate gEsolute=RTln gsolute by its valuein an infinitely dilute solution. That is, if

Xsolute is small we set gsolute= g0solute where

g0solute is theHenrian activity coefficientatXsolute = 0. Thus, for sufficiently dilute solu-tions we assume that gsolute is independentof composition. Physically, this means thatin a very dilute solution there is negligibleinteraction among solute particles because

they are so far apart. Hence, each addi-tional solute particle added to the solutionproduces the same contribution to the ex-cess Gibbs energy of the solution and so gE-solute= dG

E/dnsolute=constant.From the GibbsDuhem equation, Eq.

(1-56), if dgEsolute = 0, then dgEsolvent = 0.

Hence, in a Henrian solution gsolute is alsoconstant and equal to its value in an infi-

nitely dilute solution. That is, gsolute =1 andthe solvent behaves ideally. In summarythen, for dilute solutions (Xsolvent 1)

Henrys Law applies:

gsolvent 1gsolute g0solute = constant (1-79)

(Care must be exercised for solutions otherthan simple substitutional solutions. HenrysLaw applies only if the ideal activity is definedcorrectly, as will be discussed in Sec. 1.10).

Treating, then, the h.c.p. (Mg) phasein the AgMg system (Fig. 1-15(b)) as aHenrian solution we write:

gh.c.p. = (XAg gAg

0(f.c.c.) +XMg gMg0(h.c.p.))

+RT(XAg ln aAg +XMg ln aMg)

= (XAg

gAg

0(f.c.c.) +XMg

gMg

0(h.c.p.)) (1-80)

+RT(XAg ln (g0AgXAg) +XMg lnXMg)

where aAg and g0Ag are the activity and ac-

tivity coefficient of silver with respect topure f.c.c. silver as standard state. Let usnow combine terms as follows:

gh.c.p. = [XAg (gAg

0(f.c.c.) +RTln g0Ag)

+XMg gMg0(h.c.p.)] (1-81)

+RT(XAg lnXAg +XMg lnXMg)

Since g0Ag is independent of composition,let us define:

gAg0(h.c.p.-Mg) = (gAg

0(f.c.c.) +RTln g0Ag) (1-82)

From Eqs. (1-81) and (1-82) it can be seenthat, relative to gMg

0(h.c.p.) and to the hypothet-ical standard state gAg

0(h.c.p.-Mg) defined in

this way, the h.c.p. solution is ideal. Eqs.(1-81) and (1-82) are illustrated in Fig. 1-15(b). It can be seen that as g0Ag becomeslarger, the point of tangency N movesto higher Mg concentrations. That is, as(gAg

0(h.c.p.-Mg) gAg0(f.c.c.)) becomes more posi-

tive, the solubility of Ag in h.c.p. (Mg) de-creases.

It must be stressed that gAg0(h.c.p.-Mg) as de-

fined by Eq. (1-82) is solvent-dependent.That is, gAg0(h.c.p.-Mg) is not the same as, say,

gAg0(h.c.p.-Cd) for Ag in dilute h.c.p. (Cd) solid

solutions.



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Henrian activity coefficients can usuallybe expressed as functions of temperature:

RTln gi0 = a b T (1-83)

where a and b are constants. If data are lim-ited, it can further be assumed that b 0 sothatRTln gi

0 constant.

1.5.12 Geometry of Binary Phase

Diagrams

The geometry of all types of phase dia-grams of any number of components is

governed by the Gibbs Phase Rule.Consider a system with Ccomponents inwhich P phases are in equilibrium. Thesystem is described by the temperature, thetotal pressure and the composition of eachphase. In a C-component system, (C1) in-dependent mole fractions are required todescribe the composition of each phase(because SXi=1). Hence, the total numberof variables required to describe the systemis [P (C1) + 2]. However, as shown in Sec.1.4.2, the chemical potential of any compo-nent is the same in all phases (a, b, g, )since the phases are in equilibrium. That is:

gia (T, P,X1

a,X2a,X3

a, )

= gib(T, P,X1

b,X2b,X3

b, )

= gig(T, P,X1

g,X2g,X3

g, ) = (1-84)

where gia(T, P, X1a, X2a, X3a, ) is a func-tion of temperature, of total pressure, andof the mole fractions X1

a, X2a, X3

a, inthe a phase; and similarly for the otherphases. Thus there are C(P 1) indepen-dent equations in Eq. (1-84) relating thevariables.

Let F be the differences between thenumber of variables and the number of

equations relating them:F= P (C 1) + 2 C(P 1)F= C P + 2 (1-85)

This is the Gibbs Phase Rule. F is calledthe number ofdegrees of freedom or vari-ance of the system and is the number of pa-rameters which can and must be specified

in order to completely specify the state ofthe system.Binary temperaturecomposition phase

diagrams are plotted at a fixed pressure,usually 1 bar. This then eliminates one de-gree of freedom. In a binary system, C= 2.Hence, for binary isobaric TX diagramsthe phase rule reduces to:

F= 3 P (1-86)

Binary TX diagrams contain single-phase areas and two-phase areas. In the sin-gle-phase areas, F= 3 1 = 2. That is, tem-perature and composition can be specifiedindependently. These regions are thuscalled bivariant. In two-phase regions,F= 3 2 = 1. If, say, T is specified, then thecompositions of both phases are deter-mined by the ends of the tie-lines. Two-

phase regions are thus termed univariant.Note that the overall composition can bevaried within a two-phase region at con-stant T, but the overall composition is not aparameter in the sense of the phase rule.Rather, it is the compositions of the indi-vidual phases at equilibrium that are theparameters to be considered in counting thenumber of degrees of freedom.

When three phases are at equilibrium ina binary system at constant pressure,F= 3 3 = 0. Hence, the compositions ofall three phases, as well as T, are fixed.There are two general types of three-phaseinvariants in binary phase diagrams. Theseare the eutectic-type and peritectic-typeinvariants as illustrated in Fig. 1-16. Letthe three phases concerned be called a, b

and g, with b as the central phase as shownin Fig. 1-16. The phases a, b and g can besolid, liquid or gaseous. At the eutectic-type invariant, the following invariant re-


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action occurs isothermally as the system iscooled:

b a + g (1-87)

whereas at the peritectic-type invariant theinvariant reaction upon cooling is:

a + g b (1-88)

Some examples of eutectic-type invari-

ants are: (i) eutectics (Fig. 1-12) in whicha =solid1, b =liquid, g =solid2; the eutecticreaction is l s1 + s2; (ii) monotectics (Fig.1-14) in which a =liquid1, b =liquid2 , g = -solid; the monotectic reaction is l2 l1 + s;(iii) eutectoids in which a =solid1, b = solid2 , g = solid3; the eutectoidreaction is s2 s1+ s3; (iv) catatectics inwhich a =liquid, b =solid1, g =solid2; the

catatectic reaction is s1 l + s2 .Some examples of peritectic-type invari-ants are: (i)peritectics (Fig. 1-15) in whicha =liquid, b =solid1, g =solid2. The peri-

tectic reaction is l+s2 s1; (ii) syntectics(Fig. 1-13(k)) in which a =liquid1, b =solid, g = liquid2 . The syntectic reaction isl1+ l2 s; (iii) peritectoids in which a =

solid1, b = solid2, g = solid3 . The peritec-toid reaction is s1+ s3 s2 .An important rule of construction which

applies to invariants in binary phase dia-grams is illustrated in Fig. 1-16. This ex-tension rule states that at an invariant theextension of a boundary of a two-phase re-gion must pass into the adjacent two-phaseregion and not into a single-phase region.Examples of both correct and incorrectconstructions are given in Fig. 1-16. Tounderstand why the incorrect extensionsshown are not right consider that the (a + g)phase boundary line indicates the composi-tion of the g-phase in equilibrium with thea-phase, as determined by the commontangent to the Gibbs energy curves. Sincethere is no reason for the Gibbs energycurves or their derivatives to change dis-

continuously at the invariant temperature,the extension of the (a + g) phase boundaryalso represents the stable phase boundaryunder equilibrium conditions. Hence, forthis line to extend into a region labeled assingle-phase g is incorrect.

Two-phase regions in binary phase dia-grams can terminate: (i) on the pure com-ponent axes (atXA= 1 orXB =1) at a trans-

formation point of pure A or B; (ii) at acritical point of a miscibility gap; (iii) at aninvariant. Two-phase regions can also ex-hibit maxima or minima. In this case, bothphase boundaries must pass through theirmaximum or minimum at the same point asshown in Fig. 1-16.

All the geometrical units of constructionof binary phase diagrams have now been

discussed. The phase diagram of a binaryalloy system will usually exhibit several ofthese units. As an example, the FeMophase diagram (Kubaschewski, 1982) is


Figure 1-16. Some geometrical units of binary phasediagrams, illustrating rules of construction.


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shown in Fig. 1-17. The invariants in thissystem are peritectics at 1540, 1488 and1450 C; eutectoids at 1235 and 1200 C;peritectoids at 1370 and 950C. The two-phase (liquid+ g) region passes through a

minimum atXMo=0.2.Between 910C and 1390 C is a two-phase (a + g) g-loop. Pure Fe adopts thef.c.c. g structure between these two temper-atures but exists as the b.c.c. a phase athigher and lower temperatures. Mo, how-ever, is more soluble in the b.c.c. thanin the f.c.c. structure. That is, gMo

0(b.c.c.-Fe)

< gMo0(f.c.c.-Fe) as discussed in Sec. 1.5.11.

Therefore, small additions of Mo stabilizethe b.c.c. structure.In the CaOSiO2 phase diagram, Fig.

1-14, we observe eutectics at 1439, 1466

and 2051C; a monotectic at 1692 C; anda peritectic at 1469 C. The compoundCa3SiO5 dissociates upon heating to CaOand Ca2SiO4 by a peritectoid reaction at1789 C and dissociates upon cooling to

CaO and Ca2SiO4 by a eutectoid reaction at1250 C. Maxima are observed at 2130 and1540 C. At 1470 C there is an invariantassociated with the tridymite cristobalitetransition of SiO2 . This is either a peritec-tic or a catatectic depending upon the rela-tive solubility of CaO in tridymite and cris-tobalite. However, these solubilities arevery small and unknown.

Figure 1-17. FeMophase diagram at P =1 bar(Kubaschewski, 1982).


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1.6 Application of

Thermodynamics to Phase

Diagram Analysis

1.6.1 Thermodynamic/Phase DiagramOptimization

In recent years the development ofsolution models, numerical methods andcomputer software has permitted a quanti-tative application of thermodynamics tophase diagram analysis. For a great manysystems it is now possible to perform asimultaneous critical evaluation of avail-able phase diagram measurements and ofavailable thermodynamic data (calorimet-ric data, measurements of activities, etc.)with a view to obtaining optimized equa-tions for the Gibbs energies of each phasewhich best represent all the data. Theseequations are consistent with thermody-namic principles and with theories of solu-tion behavior.

The phase diagram can be calculatedfrom these thermodynamic equations, andso one set of self-consistent equations de-scribes all the thermodynamic propertiesand the phase diagram. This technique ofanalysis greatly reduces the amount of ex-perimental data needed to fully character-ize a system. All data can be tested forinternal consistency. The data can be inter-

polated and extrapolated more accuratelyand metastable phase boundaries can becalculated. All the thermodynamic proper-ties and the phase diagram can be repre-sented and stored by means of a small setof coefficients.

Finally, and most importantly, it is oftenpossible to estimate the thermodynamicproperties and phase diagrams of ternary

and higher-order systems from the assessedparameters for their binary sub-systems, aswill be discussed in Sec. 1.11. The analysisof binary systems is thus the first and most

important step in the development of data-bases for multicomponent systems.

1.6.2 Polynomial Representation

of Excess Properties

Empirical equations are required to ex-press the excess thermodynamic propertiesof the solution phases as functions of com-position and temperature. For many simplebinary substitutional solutions, a good rep-resentation is obtained by expanding theexcess enthalpy and entropy as polynomi-als in the mole fractionsXA and XB of thecomponents:

hE =XAXB [h0 + h1 (XB XA) (1-89)

+ h2 (XB XA)2 + h3 (XB XA)

3 + ]

sE =XAXB [s0 + s1 (XB XA) (1-90)

+ s2 (XB XA)2 + s3 (XB XA)

3 + ]

where the hi and si are empirical coeffi-cients. As many coefficients are used as

are required to represent the data in agiven system. For most systems it is a goodapproximation to assume that the coeffi-cients hi and si are independent of tempera-ture.

If the series are truncated after the firstterm, then:

gE = hE T sE =XAXB (h0 T s0) (1-91)

This is the equation for a regular solutiondiscussed in Sec. 1.5.7. Hence, the polyno-mial representation can be considered to bean extension of regular solution theory.When the expansions are written in termsof the composition variable (XBXA), as inEqs. (1-89) and (1-90), they are said to bein Redlich Kister form. Other equivalentpolynomial expansions such as orthogonal

Legendre series have been discussed byPelton and Bale (1986).Differentiation of Eqs. (1-89) and (1-90)

and substitution into Eq. (1-55) yields the



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1.6 Application of Thermodynamics to Phase Diagram Analysis 35

following expansions for the partial excessenthalpies and entropies:

hAE =XB

2 Si=0

hi [(XB XA)i

2 i XA

(XB

XA

)i 1] (1-92)

hBE =XA

2Si=0

hi [(XB XA)i

+ 2 i XB (XB XA)i 1] (1-93)

sAE =XB

2 Si=0

si [(XB XA)i

2 i XA (XBXA)i 1] (1-94)

sBE =XA

2 Si=0

si [(XB XA)i

+ 2 i XB (XB XA)i 1] (1-95)

Partial excess Gibbs energies, giE

, arethen given by Eq. (1-52).Eqs. (1-89) and (1-90), being based upon

regular solution theory, give an adequaterepresentation for most simple substitu-tional solutions in which deviations fromideal behavior are not too great. In othercases, more sophisticated models are re-quired, as discussed in Sec. 1.10.

1.6.3 Least-Squares Optimization

Eqs. (1-89), (1-90) and (1-92) to (1-95)are linear in terms of the coefficients.Through the use of these equations, allintegral and partial excess properties (gE,hE, sE, gi

E, hiE, si

E) can be expressed bylinear equations in terms of the one set ofcoefficients {hi , si}. It is thus possible to

include all available experimental data fora binary phase in one simultaneous linearleast-squares optimization. Details havebeen discussed by Bale and Pelton (1983),Lukas et al. (1977) and Drner et al.(1980).

The technique of coupled thermody-namic/phase diagram analysis is best illus-trated by examples.

The phase diagram of the LiFNaFsystem is shown in Fig. 1-18. Data pointsmeasured by Holm (1965) are shown onthe diagram. The Gibbs energy of fusion of

each pure component at temperature T isgiven by:

where Dh0f (Tf) is the enthalpy of fusion atthe melting point Tf, and c

lp and c

sp are the

heat capacities of the pure liquid and solid.The following values are taken from Barinet al. (1977):

Dg0f (LiF) = 14.518 + 128.435T+ 8.709 103 T2 21.494 Tln T

2.65 105 T1 J/mol (1-97)Dg0f (NaF) = 10.847 + 156.584T

+ 4.950 103 T2 23.978 Tln T 1.07 105 T1 J/mol (1-98)

Thermodynamic properties along the liq-uidus and solidus are related by equationslike Eqs. (1-64) and (1-65). Taking theideal activities to be equal to the mole frac-

tions:R TlnXi

l R TlnXis + gi

E(l) giE(s)

= Dg0f (i) (1-99)

D Dgf0

f f

pl

ps

=

d (1- )

f

f

h T T

c c T T

T

T

T

( ) ( / )

( ) ( / )

0 1

1 1 96

+

Figure 1-18. LiFNaF phase diagram at P=1 barcalculated from optimized thermodynamic parame-ters (Sangster and Pelton, 1987). Points are experi-

mental from Holm (1965). Dashed line is theoreticallimiting liquidus slope for negligible solid solubility.


36/80

where i = LiF or NaF. Along the LiF-richliquidus, the liquid is in equilibrium withessentially pure solid LiF. Hence, XsLiF = 1and gLif

E(s) =0. Eq. (1-99) then reduces to:

R TlnXlLiF + gLifE(l) = Dg0f (LiF) (1-100)

From experimental values ofXlLiF on theliquidus and with Eq. (1-97) for Dg0f (LiF),values of gLif

E(l) at the measured liquiduspoints can be calculated from Eq. (1-100).

Along the NaF-rich solidus the solid so-lution is sufficiently concentrated in NaFthat Henrian behavior (Sec. 1.5.11) can be

assumed. That is, for the solvent, gNaFE(s)

= 0.Hence, Eq. (1-99) becomes:

R TlnX(l)NaF R TlnX(s)NaF + gNaF

E(l)

= Dg0f (NaF) (1-101)

Thus, from the experimental liquidus andsolidus compositions and with the Gibbsenergy of fusion from Eq. (1-98), values ofgNaF

E(l) can be calculated at the measured liq-

uidus points from Eq. (1-101).Finally, enthalpies of mixing, hE, in theliquid have been measured by calorimetryby Hong and Kleppa (1976).

Combining all these data in a least-squares optimization, the following expres-sions for the liquid were obtained by Sang-ster and Pelton (1987):

hE(l) =XLiFXNaF (1-102)

[ 7381 + 184(XNaF XLiF)] J/mol

sE(l) =XLiFXNaF (1-103)

[ 2.169 0.562(XNaF XLiF)] J/mol

Eqs. (1-102) and (1-103) then permit allother integral and partial properties of theliquid to be calculated.

For the NaF-rich Henrian solid solution,

the solubility of LiF has been measured byHolm (1965) at the eutectic temperaturewhere the NaF-rich solid solution is inequilibrium with pure solid LiF. That is,

aLiF =1 with respect to pure solid LiF asstandard state. In the Henrian solution atsaturation,

aLiF = g0LiF

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