Thermodynamics, constitutive equations and kineticssestak/yyx/Thermodyn-kinetics.pdf ·...

Thermodynamics, constitutive equations and kinetics RATIONAL APPROACH TO THERMODYNAMIC PROCESSES AND CONSTITUTIVE EQUATIONS IN ISOTHERMAL AND NON-ISOTHERMAL KINETICS J. Sestak, J. Kratochvil Journal of Thermal Analysis and Calorimetry 5 (1973) 193 KLINETICS WITH REGARD TO (NON) EQUILIBRIUM PROCESSES STUDIED BY (NON) ISOTHERMAL TECHNIQUES P. Holba, J. Sestak Zeitzeschrift fur Fyzikalishe Chemie Neue Folge 80 (1072) 1 THERMODYNAMIC BASIS FOR THE THEORETICAL DESCRIPTION AND CORRECT INTERPRETATION OF THERMOANALYTICAL EXPERIMENTS J. Sestak, Thermochimica Acta 28 (1979) 197 IRREVERSIBLE THERMODYNAMICS AND TRUE THERMAL STATE DYNAMICS IN VIEW OF GENERALIZED SOLID-STATE REACTION KINETICS J. Sestak, Z. Chvoj Thermochimica Acta 388 (1002) 427 THERMODYNAMICS AND THERMOCHEMISTRY OF KINETIC (REAL) PHASE DIAGRAMS INVOLVING SOLIDS J. Sestak, Z. Chvoj Journal of Thermal Analysis and Calorimetry 32 (1987) 1645

Journal o f Thermal Analysis, Vol. 5 (1973) 193--201

R A T I O N A L A P P R O A C H TO T H E R M O D Y N A M I C RROCESSES

A N D C O N S T I T U T I V E E Q U A T I O N S

I N I S O T H E R M A L A N D N O N - I S O T H E R M A L KINETICS

J. gEST~K and J. KRATOCHVfL

Institute o f Solid State Physics o f the Czech Academy o f Sciences, Prague, Czechoslovakia

Isothermal and non-isothermal kinetics are classified according to the viewpoint of rational approach. The appropriate selection of basic quantities and constitutive equations is stressed. The extensive discussion recently focused to the meaning of the partial derivatives is reinvestigated and clarified considering the origin of following equation

. = f(T, t)

where ~ is the extent of reaction, T and t are the temperature and time respectively, and f represents a function. The meaning of partial derivatives is demonstrated in details. The disagreement sometimes claimed between the data evaluated by means of isothermal and non-isothermal kinetics is also reviewed, but no fundamental differences are established.

A recent article by MacCallum and Tanne~ [1 ] on a non-isothermal rate equation has created an extensive discussion as to the applicability of the isothermal mathematical treatment to non-isothermal conditions for which it is assumed that the concentration, C, under non-isothermal conditions in a homogeneous system is a function of both the temperature, T, and the time, t. This assumption leads to the following equation, based upon a normal mathematical procedure for partial differentiation

dC ( ~ C j ( a C ) dT - + ~ - : d / (1) dt 0 t - T t

Some authors [1, 2] have claimed that the term (gC/Ot)r is appropriate to describe the isothermal rate of a process only. Hill [3] first argued that the term (~ C/~ T)t is effectively zero, comparing the situation to that of the arrow in flight. Although the arrow is in motion, at any instance it is at rest, similarly to the state of a chemically reacting system defined at any instant of time without reference to change. Felder and Stehel [4] later pointed out that such an instantaneous rate of Eq. (1) would depend not only on the present state of the system (e.g. the fre- quency of collisions, relative energies and the orientation of the molecules of the system), but also on previous and future states. They emphasized that Eq. (1)is invalid because C is a path function rather than a state function of t•e variables t and T. Holba and Sestfik [5 ] made a mathematical attempt to calculate possible consequences of Eq. (1) upon the analytical form of the non-isothermal rate equa-

2* 3". Thermal Anal 5, 1973

194 ~EST~.K, KRATOCHV[L: CONSTITUTIVE EQUATIONS IN KINETICS

tion. By assuming that the degree of reaction*, c~, is a state function of T and t, they arrived at a relationship yielding a non-isothermal rate about ten times faster than the isothermal one. Hrma and Satava [6] discussed the actual meaning of partial derivatives using the established form and significance of the specific rate constant. They concluded that the non-isothermal a of Eq. (1) would then depend only on the total duration of the process and its final temperature.

In his defence of the validity of Eq. (1), MacCallum [7] assumed that the temperature is only one of the physical parameters which may be varied during a kinetic reaction; the volume of the reactant solution, the pressure of volatile products, etc., could also be changed during an isothermal decomposition, just as in non-isothermal experiments the temperature is linearly raised holding other variables constant. He considered the case of adding an inert diluent at a constant rate during a process, resulting in a mathematical description analogous to Eq. (1), for the volume, V:

d C { t? C } (c3C d t - ~ - v + ~-V , t -d -t (2)

where the partial terms were further analytically expressed. Another more experimental approach to this discussion was based on a long

kinetic practice which has exhibited that tke reaction rate, dc~/dt, is proportional to the product of two separate functions; the first, k(T), is temperature-dependent, and the second, f(e), is related to c~ only:

dT/dt = k(T) f(c 0 . (3)

The validity of Eq. (3) has also been confirmed in non-isothermal kinetics, e.g. by Sestfik [8], who showed a simple accumulation procedure based on infinitesimal changes in ~ scanned along a non-isotherm, and by Simmons and Wendlandt [9], who made a similar stepwise calculation of the instantaneous rate constant under isothermal conditions with linearly or hyperbolically elevated temperature. Gilles and Tompa [10] stressed the fact that the value of ~, as the solution of a special form of the differential equation (3), dc~/dt = k(T)c~, depends at time t on the functional relationship between T and t, and in general therefore is not a function of two independent variables T and t.

Because the situation is still not completely clear, the aim of this article is to review the problem employing a well-defined concept.

* C o n c e n t r a t i o n C, as well as any o the r phys ica l p rope r ty wh ich is chosen to represen t the s y s t e m inves t iga ted , can be n o r m a l i s e d in the f o r m of a so-ca l led f rac t ion conver s ion c~, by the equation

c~(t) = [ C ( t ) - C o ] / [ C c e - - C0]

where subscr ip t s indicate the va lue o f C at the initial t ime (t = 0) and at t--+ oo. The case where Coo is n o t c o n s t a n t a n d varies wi th the t empe ra tu r e [5], is no t cons idered here.

J. T h e r m a l A n a l . 5 . 1973

gESTAK, KRATOCHV1L: CONSTITUTIVE EQUATIONS IN KINETICS 195

Rational approach to the kinetics

In our opinion the misunderstanding arose from a vague definition of the concepts which appear in the considerations. A convenient framework in which al necessary concepts can be exactly defined is the rational thermodynamics devel- oped by Coleman and Noll [11] (for applications in chemical kinetics see e.g. [12]). This method does not introduce new physical principles but its progres- siveness is based on the precise establishment of logical connections between the thermodynamic notions. In contrast to classical thermodynamics which is suitable for the description of a reversible equilibration, rational thermodynamics may also be used for processes distant from equilibrium and hence may cover the field of kinetics as well. Only some general features of the rational thermodynamic approach which are relevant to the present discussion will be utilized here, namely to clear the definition of a thermodynamic process and a constitutive equation and their mutual relations [13].

Any theory which attempts to describe a physical phenomenon requires drastic assumptions as to what is to be included and what can reasonably be neglected. These assumptions are set up in a list of basic quantities which unambiguously describe the given thermodynamic system (e.g. volume V, pressure P, heat exchange Q, temperature T, fraction conversion ~, etc.). We say that the thermodynamic process (i.e. continuous sequences of the states of the system), or just a process, is completely described if the basic quantities are specified as a function of the time*, t [e.g. V = V(t), P = P(t), Q = Q(t), T = T(t), ~ = cfft) etc.]. We neglect here all quantities except a kinetic variable c~, and the temperature T. Hence a process in our system is represented by the pair of functions c~ = a(t), and T = T(t), denoted [ct(t), T(t)].

A special class of processes in which T(t) = K = constant is called the class of isothermal processes [a(t), K]. Similarly we can have the class of linear processes [a(t), K't] where Tis given as a linear function of t, and K' is a constant; the class of quadratic processes [a(t), K"t2], etc.

The basic quantities are not independent. They must satisfy constitutive equations which are characteristic for a given system (in general the basic quantities must satisfy both the constitutive equations and balance laws. The balance laws express properties common to all systems covered by the theory, and the constitutive equations formalised diversities in the system allowed by the balance laws**).

A process which satisfies the constitutive equations is called admissible. From this point of view isothermal kinetics concerns the class of admissible isothermal processes; non-isothermal kinetics concerns the class of admissible linear, quadratic

* The dependence of basic quantities on the position can generally be considered. A process independent of the position is called homogeneous .

** The balance laws, e.g. the conservation of mass and energy, are not considered here as they do not appear in the present discussion.

J. Thermal Anal 5, 1973

196 SESTA_K, KRATOCHVIL: CONSTITUTIVE EQUATIONS IN KINETICS

or other dynamic processes. According to the class of processes involved, we have linear non-isothermal kinetics, quadratic non-isothermal kinetics, hyperbolic non-isothermal kinetics, etc.

Const i tut ive equations in chemica l k inet ics

The entire form of constitutive equations for a given system can be deduced from suitably designed experiments or derived from a microscopic theory. In our case the choice of the constitutive equation should be found through the available kinetic relations justified by both experimental practice and the statistical approach based on the microscopic theory, as for example the Arrhenius rate equation, the collision theory, the activated complex theory, etc. [14].

For the system characterised by the basic quantities ~(t) and T(t) it seems to be fairly well established that Eq. (3) holds as the constitutive equation; this will be used in the following discussion in a slightly generalised form, i.e.

d~/dt = F(~, T) (4)

where F denotes a function (as well as G later). a) In isothermal kinetics we have from Eq. (4)

de/dt = F(cq K) . (5)

We denote the solution of Eq. (5), assuming the admissible isothermal process, characterised by K as

~(t) = ~(t, K)*. (6)

We may write formally

- + - -~ ~ - = ~ - . (7)

Eq. (7)follows from dK/dt = 0 as K is a constant in an isothermal process. The derivative (a&/g K)t may in general be non-zero. Physically this derivative measures the change of ~ at t, if we consider instead of the process with T = K the process with T = K + dK (see also Fig. 1 but replace K' with K).

b) In linear non-isothermal kinetics we have from Eq. (4)

(de/d0 - F(t, K't). (8)

We denote the solution of Eq. (8) in the admissible linear process characterised by K' as

c,(t) = K' ) and we may proceed

' - - - - b 7 t K'

(9)

(10)

* Note tha t the superposed care t in ~ serves to d i s t ingu i sh this func t ion f r o m its value.

3. Thermal Anal. 5, 1973

SESTAK, KRATOCHVIL: CONSTITUTIVE EQUATIONS IN KINETICS 197

(the illustration of which is given in Fig. 1). Similarly to Eq. (7) dK'/dt = 0 according to the assumption that K' is a constant for a linear process and (8~/8K')~ indicates the change in ~ in two infinitesimally close processes differing by dK' (see Fig. 1).

/

S

d ' t ~at j ~z,

~ (t.K')

I l ' , d t ,, ,

t F I [ ] i

il//~dK' K~

K~

~ / / t lm"

Fig. l . Diagrammatic representation o f the system ~ = ~(t, K')

Fig. 2. D iagrammat ic representation of the system ~ = ~(t, T)

.L Thermal Anal. 5, 1973

198 ~ESTAK, KRATOCHV~L: CONSTITUTIVE EQUATIONS IN KINETICS

However, the solution of Eq. (8) may also be obtained in the form

e(t) = 8(t, T/t) def. = 8(t, T) (11)

using K ' = Tit. The function 8 for given values of t and T yields the value of at t in the linear process characterised by K ' = Tit. In terms of ~ we obtain

d--T = ~ - T + ~ < ~-=1, a t ) T 8 T t

(as graphically demonstrated in Fig. 2). In Eq. (12) both partial derivatives are generally non-zero, and (8 8/a t)T means the change of ~ if we replace the process K ' at t with the process K ' + dK ' ( = T/(t + d t ) ) at t + dt (see Fig. 3b). Accordingly, (~8/ST)t reflects the change of ~ at t if we replace the process K ' at Tit with the process K' + dK' (where dK ' = dT/t) (see Fig. 3a).

/ 8~ \ ~(t +dr ~ K'+dK')

o) ~ K'

I ~ ( t *d t , K" dK')

I 1 I , /

I /I V /1 / [ / l1

b) ~K'

Fig. 3. Diagrammatic representation of the partial derivatives in the system ~ = ~(t, T)

It is important to point out that the partial derivatives (O&/St)K in Eq. (7) and (OS/St)T in Eq. (12) are in general not equal as is indicated by their different physical meanings.* Hence, comparison of isothermal and non-isothermal kinetics is not possible to reduce to the question of the meaning and value of the derivatives (ac~/0T)t, but the difference in the values of (S&/Ot)T=K and (88/OtT) is also significant.

A similar analysis can be applied for any type of process, e.g. quadratic, hyperbolic, etc.

* We can demonstrate a simple example for the special form of Eq. (4) i.e. d~/dt = -- aT [where f(~) = -- e and k(T) ~- T, compare Eq. (3)]. It is easy to find that &(t, K) = exp (-- Kt); (~t, K') = exp (-- K't 2/2) and ~(t, T) ---- exp (-- T t/2). Hence (~ ~! ~ t)K = -- K exp (-- Kt ) an d (t?~/t~t)r=K =- --K/2 exp ( - Kt/2).

J. Thermal Anal. 5, 1973


Discussion

We may now consider once more the meaning of Eq. (1) in the light of the preceding rational approach. We may ascribe to it two different meanings:

a) Eq. (1) rewritten as

(13)

This is a consequence of the constitutive equation of the type

= G(T, t) (14)

which, in fact, is implicitly involved in [1, 7]. However, this constitutive equation would physically describe a material controlled by an internal clock, which is not the case for an ordinary chemical reaction.

b) Eq. (1) as understood in the sense of Eqs (7), (10) and (12). It is evident that its meaning then depends critically on the precise definition of the symbol C on the right-hand side of Eq. (1). This is rather an extension of the idea expressed by Felder and Stehel [4], Hrma and Satava [6], and Gilles and Tompa [10] that the interpretation of Eq. (1) is necessarily related to the process itself.

As we can see, there is no essential discrepancy between the isothermal and non- isothermal kinetics. In usual practice the function k(T) is expressed by an exponen- tial type equation [14] and instead off(a) a particular model relation is used, based upon a physico-geometrical hypothesis [8]. In such a special case the kinetic constants (i.e. parameters defining the analytical form of a particular differential equation [3]) can be determined by either kinetics.* Disagreements sometimes re- ported [1 ] between the kinetic data observed under isothermal and non-isothermal conditions are therefore not fundamental and may be caused by:

a) experimental reasons, e.g. inaccurate determination of basic quantities and/or not exact satisfaction of the required and predetermined conditions for a given process;

b) oversimplified separation of c~ and Tfunctions in Eq. (4) [as given in Eq. (3)] and/or inaccurate formulation of the particular functions k(T) and f(c0;

c) a more complex constitutive equation. Considering here only point c), we can continue using the same method as above.

Assuming a constitutive equation (4) which involves the higher derivatives of temperature T (e.g. T) we have for instance

de/dt = F(c~, T, ~i'). (15)

It then follows for the isothermal kinetics that

(16) da/dt = F(~, K, 0) ~ ~(t) = &(c~, K)

* For the integration of Eq. (8) [solution Eq. (9)] the temperature-dependence of k ( T ) must be kept in mind (for the analytical solution see [15]).

J Thermal Anal. 5, 1973

200 SESTAK, KRATOCHV]~L: CONSTITUTIVE EQUATIONS IN KINETICS

and for the l inear non- iso thermal kinetics

d~/ dt = F(~, K ' t, K ' ) ---* ~( t ) = ~( t, K ' ) (17)

and /o r

~(t) = ~(t, ~r).

I t can be seen that the result ing isothermal kinetics is eqmvalent to that described in Eqs (5), (6) and (7), bu t this l inear non- i so thermal kinetics is not of the same na ture as that discussed in p. 196, Eqs (5 ) - (12 ) . I t gives more in fo rmat ion about the process bu t would be appropria te only for systems exhibit ing very fast changes with temperature increase. This is no t the case for an ord inary chemical react ion either.

The authors are obliged to Dr. P. Holba of the Institute of Solid State Physics and Dr. P. Hrma of the Joint Laboratory for Silicate Research, The Czechoslovak Academy of Sciences, Prague, for their helpful suggestions and friendly discussions on this subject.

References

1. J. R. MACCALLUM and J. TANNER, Nature, 225 (1970) 1127. 2. A. L. DRAPER, Proceedings of the 3rd Toronto Symposium on Thermal Analysis

(ed. H. G. MacAddie), 1970, p. 63. 3. R. A. W. HILL, Nature, 227 (1970) 703. 4. R. M. FEEDER and E. P. STEHEE, Nature, 228 (1970) 1085. 5. P. HOEBA and J. SEST$.K, Z. Phys. Chem., Neue Folge, 80 (1972) 1. 6. P. HRMA and V. ~ATAVA, unpublished results. 7. J. R. MACCALLUM, Nature Phys. Sci., 232 (1971) 41. 8. J, ~ESrA, K, Plenary lecture "Non-isothermal Kinetics" Proceedings of the 3rd ICTA,

Davos, Switzerland Vol. 2., p. 3., Birkh/iuser, Basel-Stuttgart, 1972. 9. E. L. SIMMONS and W. W. WENDLANDT, Thermochim. Acta, 3 (1972) 498.

10. J. M. GILLES and H. TOMI'A, Nature Phys. Sci., 229 (1971) 57. 11. B. D. COLEMAN and W. NOEL, Arch. Rational Mech. Anal., 13 (1963) 167. 12. C. TRUESDELL, Rational Thermodynamics, McGraw-Hill Book Co., New York, 1969. 13. J, KRATOCRVIL, Rational Thermodynamics, J. Czech. Phys., A, 23 (1973) 1. 14. C. H. BAMFORD and C. F. H. TIPPER (eds), Series of Comprehensive Chemical Kinetics,

Elsevier Publ. Co., Amsterdam (The Theory of Kinetics, 1970). 15. J, SEST/~K, Therrnochim. Acta, 3 (1971) 150.

R~SUM~ -- Dans l'optique d'une approche rationnelle, on proc6de ~t un classement de la cin6- tique en r6gimes isotherme et non-isotherme. On met en 4vidence le choix judicieux de valeurs fondamentales et d'6quations d'6tat. On 6tudie la discussion r6cente concernant la signification des d6riv6es partielles et l'on 6claircit la question en consid6rant l'6quation

= f(T, t)

oh cr est le degr6 d'avancement de la r6action, Tet t la temp4rature et le temps, fune fonction. On discute les causes du d6saccord quelquefois observ6 entre les donn6es 6valu4es en r6gimes isotherme ou non-isotherme.



ZUSAMMENFASSUNG - - I so the rme und nicht- isotherme Kinetik wurden entsprechend einer rationellen Ann~iherung klassifiziert. Die geeignete Auswahl der fundamenta len Mengen und konst i tut iven Gleichungen wurde betont. Die neuerliche lebhafte Diskuss ion fiber die Be- deutung der partiellen Derivativen wurde tiberprtift und durch die Gleichung gekl/irt:

= f ( T , t )

wobei ~ den Reakt ionsgrad, T und t die Tempera tu r und die Zeit, f eine Funk t ion bedeuten. Es wurde auch die in einigen F~illen beobachtete Nichtf ibereinst immung der durch isotherme und nicht- isotherme Kinetik erhal tenen Daten behandelt .

Pe3ioMe - - C TO~lI(rI 3peI-IH~ palIrlOHaJIbnOrO IIO~XO~Ia KJiaccri~OrlttrlpoBanbi i, i30TepMrlqecKa~ rr Be~i30TepMri~iecKa~ K~IHeTHKa. IIO~IqepKHyTa Heo6xo~II, IMOCTb COCTBeTCTBylOIILeFo BbI6opa OCHOBHI,IX Be:II, Iq~IH rI nprlMeHSeMblX ypaBHei-m~. YIpeiIMeT IItHpOKO~ ~HCKyCCrlrI, rlanpaB~IaeMo~t B llocJie~Hee BpeMa Ha 3Ha~IeHHe qaCTHBIX IIpOH3BO~H]bIX, paCCMOTpeH CHOBa H ~aH~I pa3"b.qCHeHH~I o,rlOCrlTeabao ypaBrlenrIn

= f ( T , t)

r~Ie ~ -- Mepa peaKl/i,Iyi, T i.i t - - TeMnepaTypa ri BpeM~[, COOTBeTCTBeHHO, rI f -- ~OyHKIII, I~l. Pacc- MOTpeltO TO)Ke pacxox~ei-iHe, O6rlapy)K~iBaeMoe rlrlOr~Ia MeeKly IlaI-IHblMH, pacc'-I/,ITaHHblM~ IIocpe~ICTBOM I, I3oTeplvLr,I'-IeCKO~ H HeH3oTepMHKeCKO~ KI, I/-IeTI, IKIiL

d. Thermal Anal. 5, 1973

Thernlochimica Acta, 28 ( 1979) 197-227 @I Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

Review

Thermodynamic basis for the theoretical description and correct interpretation of thermoanalytical experiments*

J. SESTAK

Institute of Solid State ?hysics, Czechoslovak Academy of Science, 16253 Prague (Czechoslovakia)

(Received 13 October 1977)

ABSTRACT

The basic principles of the description and processing of thermal analysis (TA) curves are examined. A rational approach is used to investigate the limits of thevalidity of phenomenological thermodynamics under non-isothermal conditions. The necessary thermodynamic relations and response functions are derived for thermophysical measurements of thermal and non-thermal (dielectric, magnetic and mechanic) property. Sixteen basic thermal coefficients are listed. Simple phase transformations are analyzed (including generalized Clausius-Clapeyron and Ehrnfest equations for the first- and second-order processes) and their theoretical courses are related to the experimental TA curves. Variant and invariant processes are distinguished with regard to the thermal development of their equilibrium background. Actual conditions of dissociation processes are also discussed.

INTRODUCTION

For the correct interpretation of an arbitrary physical measurement, where a particular physical property of the sample is measured as a function of an externally controlled parameter, it is necessary to consider the conditions under which the experiment is conducted and, consequently, the effect of these conditions upon the resulting data ‘_ The term conditions (or experimental conditions) means the way by which the studied material is transferred into the form of a sample and the properties of the environment which surround the sample during the whole course of measurement. From the definition of thermal analysis (usually abbreviated TA) as recently proposed by us elsewhere’ it follows that dynamic TA covers the group of selected thermophysical measurements where the state of the sample is investigated on the basis of its interaction with the heat of the surroundings if the temperature of the

* See author’s note, p. 226.

surroundings is intentionally programmed, preferably as a linear function of time. The complexity of relations between the assumingly dynamic experimental conditions and the resultiilg TA record is the major obstacle in utilizing TA methods exactly in physical chemistry, particularly if theoretical thermal analysis is still diverse and yet disproportiona13.

With regard to the dynamic character of TA methods the generalized quantitative description must be looked for through the flux relations, as introduced in our previous works4- 6, i.e. the flux formulation of energy conservation law, principles of heat and mass transfer, chemical kinetics, etc. The application of these laws is not easy and is not unambiguous because many of them were originally derived for the conditions which are not always fulfilled during TA experiments_ We face the greatest difficulty, however, when coordinating the ordinary equilibrium thermodynamics with the dynamic character of TA measurements. Hence, the aim of this comprehensive work is the analysis of the validity of some generally used relations in non- isothermal conditions, the systematization of these relations within the logistic of TA and, last but not least, the correlation between thermodynamic processes and the types of resulting curves.

Chapter 1

ANALYSIS QF THERMOPHYSICAL MEASUREMENTS

The unifying element of all thermophysical measurements to be looked after is the investigated sample itself and the way in which the sample is thermally treated. The detected physical property of the material is understood to represent the instantaneous state of the sample and it is up to the investigator to which temperature it is ascribed and in what manner it is analyzed_ Qualitative applications, as common in DTA, are a frequent subject of most TA books’. Quantitative measurements at an equilibrated (constant) temperature (static methods) are the most general methods of extracting thermodynamic information in solid state chemistry and physics. From the moment when temperature becomes time-dependent (dynamic methods) we face all possible problems connected with temperature gradients, hysteresis, supercooling and/or superheating, etc. Utility of dynamic measurements is thus not general and strongly depends upon the sort of material investigated and its capability to equilibrate fast enough to follow changing temperature; it must, of course, be tested for each experimental case. The comparison of mutual advantages of the equilibrium but time- consuming static methods with the comparatively fast but non-equilibrium dynamic methods provides the basis for the appropriate set up of our experiment. For the study of reversible processes the dynamic methods were found satisfactory.

Nowadays the TA measurements are often understood in their broad sense and may cover almost all thermophysical measurements if the temperature is continuously varied and if the physical property is also continuously registered. Although this all-covering approach provokes assumingly numerous physicists (because many

199

thermophysical measurements have their origin in experimental physics) we, in fact, have frequent reports on such TA techniques ’ as thermospectrometry, thermo- luminiscence, thermorefractometry, thermqacoustimetry, thermomicroscopy or even high-temperature (oscillation) X-ray diffraction and/or spectroscopy. These methods evidently belong among the measrrremzezzts of stmctur.al properties. These methods can be contrasted with the more classical methods of TA based on the measrueuzents

of tJzermody~zanzic properties as temperature (Direct TA, DTA), heat content (DSC), volume, density, weight (TG), content of volatile products (EGD), as well as less common magnetization (MTA), polarization (ETA), deformation (TMA) and pressure (DPA). The theoretical description of later methods lies within the proposed scope of our introductory thermodynamics and hence is dealt: with in detail later.

Almost all TA measurements yield merely single valued data of a given physical property although its local value within the mass of the sample may vary. Averaging such space inhomogeneities” (as well as most important temperature distribution) is in accordance with the theory of phenomenological thermodynamics where all quantities are assumed and characterized by their mean values. The neglect of gradients, however, is a most serious simplification particularly in determining the true state of solids but, on the other hand, is adequate to the present level of TA instrumentation. The use of gradient theory (particularly assuming most effective temperature gradients) is thus not actual unless a more sophisticated instrumentation is introduced as, for example, space multidetection devices.

The framework of phenomenological thermodynamics’ seems to be the most useful tool in finding the unifying groundwork of TA. For the sake of simplicity,

we start by treating a simple one-component system to illustrate the principles of

caloric and volume TA measurements, see Chapter 3 (In Chapter 2 we attempt to investigate the validity of basic thermodynamic relations in a more general sense of non-equilibrium condition of non-isothermal studies.) Consequently, the system is complicated by assuming the externally applied fields necessary for the description of dielectric, magnetic and thermomechanic TA and finally generalized for multicomponent materials to describe TG, EGD, etc., based on the detection of volatile products, see Chapter 6. Although all these TA methods never take place at the same time, even as multisimultaneous techniques, their joint description well demonstrates the principal features of a complex thermodynamic approach. To achieve a uniform link and to ease our understanding we commence each of our system descriptions by listing the so-called corzstitrttive eqrratiozzs (material relations), see Chapter 2, e.g., Y = @x1, X,, ..) h w ere the state quantity Y is a function Pof variables X,, X,, etc.

* We are not concerned with the magnitude of the system3* I-I when considering its influence on the extent of, for example, temperature gradients (macro- versus micro-methods of investigation and their accuracy of measurements), neither with the description of the microscopic state of the system3. * as ordering of crystallographic sites, distribution of species (and vacancies) on, for example, cation regular and interstitial sites, nor with their thermodynamic potentials which were dealt with in detail elsewhere”.

200

TYPE OF TA RECOR[

a

DTG, DMTA. DETA.DTMA

b igqJ EGD

IDTA, DDTA, DDSC, DEGD

scillation

i ‘D

E

R I

” A T

; E

C

Fig. I. Graph of typical TA curves as recorded for the individua1 TA techniques, where a corresponding physical property 2 of the sample is plotted versus temperature T-

Z

It should be noted that we intentionally delete from this approach the descrip-

tion of electric conductivity (amperometric TA) because this method is directed to

investigate the Aux property of materiaIs, i.e. the detection of electric current passing

through the sample layer which is evidently a case similar to the purely kinetic studies

of mass diffusion and/or heat conductivity bearing their own values of the energy of activation. Neither have we dealt with the description of the energetics of surfaces

and interfaces which is discussed in the forthcoming text at a minimum level just to

give the basis of heterogeneous new phase formation_

Let us now -turn our attention to the possible kinds of TA records” as shown

schematically for the above selected methods of TA in Fig. l_ Every record can be

divided into smooth lines called base Zincs and their sudden changes called e&cts.

The upper part of Fig. 1 exhibits two sorts of effects: the change of the base line

slope called break and the stepwise displacement of base lines called step (wave).

This is typical for thermogravimetry, thermodilatometry, thermomechanical and

electromagnetic measurements. Another effect called peak arises from a sudden

increase and decay of the measured property 2 and occurs at direct TA (heating and

201

cooling curves), DTA, DSC and methods associated with evolved gas detection, see middle part of Fig_ l(b)_ A similar effect, however, can be obtained through an electronic derivation of upper curve (a) sometimes presented as an independent measuring technique as DTG, similarly DDTA, DDSC, IDTA and DEGD shown in the lower part of Fig. 1. Here also an additional effect called oscillation appears.

It follows that each TA technique provides only that kind of record appropriate for further analyzing_ Additional electronic, analogic as well as numerical treatment merely provides a derived record which can serve as complementary information only I O or for an advanced characterization when using computers.

To extract the desired data, we must identify mathematically individual base lines and effects and then relate them to a given thermodynamic and/or kinetic description. Base line can be analytically expressed in the form of a series, most conveniently as Z = ni= I + a2T + a3T2 - a4T3, where T is the temperature and ai are constants. In most cases, a linear approximation (first two terms of the series) is satisfactory (linear materials). The thermodynamic meaning of thermal coefficients for individual TA methods in question is thus analyzed in Chapter 3

where a simple development of the thermal state of the sample is described_ Chemical reactions and structural transformations are indicated by a base line discontinuity and the resulting effects should be analyzed with regard to their positiotz (characteristic temperatures), size (integral change of the measured property), see Chapter 4, and drape (time-development of the measured property), see Chapter 5. The last two phenomena, however, may exhibit a mutual interference of reaction kinetics and thermal development of equilibrium (thermodynamics), compare Chapter 5. The mathematical description of individual effects falls into two categories: determination of characteristic poit2ts1 I, e.g. the beginning and end of the break, extrapolated point of base lines intersection; beginning and end of the step inflection point, step width and height 12; beginning and end of the peak, extrapolated onset and offset, front and rear inflection point, peak width and height, extrapolated peak width, actual and linearly interpolated peak background, etc., and determination of itwtarttarleous

values (compare Chapter 5) which may be eased by fitting the curve with a suitable function’ 3, e.g. higher-order polynomials Z = p,(T)/p,(T), exponentials Z =

w_1 t a1 exp (a,T)], logarithm Z = a, + [a,/ln(a2T)]a3 and hyperbolic tangent Z = tanh(TO). To establish the total change of the measured property, Z, the step is the most appropriate curve because the peak must be gradually integrated as well as the break derived_ Evidentiy a satisfactorily readable record is the essential requirement for a successful interpretation of TA data*.

* The discussion of partial or whole curve fitting and smoothing with regard to further computer treatment which is usually accomplished by applying more complex, e.g. orthogonal functions possibly under curvature tension (i.e. spline-functions to avoid the creation of inevitable inflections), is not the aim of this review. Neither do we assume the backwards consequence of certain mathematical operations such as derivation or integration which result in changing the scatter and/or smoothness of the originally recorded trace. A detailed mathematical approach will be dealt with elsewhereld.

202

THE SURROUNDIN~GS

MATERlAL

UNDER STUDY

S 7 T

V

P U

HEAT l%

INTERACTION

/

THE SAMPLE

T,

-ii

Fig. 2. Simple closed thermodynamic system, where the state of the system under TA study is represented by the entropy S, temperature T, volume V, pressure P and internal energy ZJ- Externally controiled parameters of the surroundings are the temperature T,, pressure P, and the time progress of temperature i,.

Chapter 2

RATIONAL DESCRIPTION OF A THERMODYNAMIC SYSTEM WHILE EXTERNALLY HEATED

The basic need of a rational approach is to choose the minimum number of

variables necessary to describe a system satisfactorily. For the sake of simplicity, we assume a simple (homogeneous) system with constant (one component) composi- tioL where no chemical reactions occur. In order to give a mathematical description of such a system, we must define it as a physical object in which we can specify certain

basic quantities- They may be temperature, T, volume, V, pressure, P, entropy, S, internal energy, U, and heat exchange, 0 (= dQ/dt), between the sample and its surroundings (*)_ Such a system is shown schematically in Fig. 2 and exhibits all

the basic features of a simple thermoanalytical arrangement. The temperature, r, pressure, P, and the rate of heating, ?! (= dT/dr), can be

externally controlled and are thus independent while the remaining quantities behave

as the dependent variables. It is assumed that the system is not in its equilibrium state and the possibility of obtainin g its description by means of ordinary thermodynamics must be first analyzed.

The main idea of how to make a more flexible framework for the thermodynamic description of the system can be obtained on the basis of rational thernzodyna- micsl ‘- 1 ’ as recently summarized by Kratochvil ’ 8 To this effect, let us define our .

thermodynamic process in question as continuous sequences of the state of the

203

system. Simply, the process is fully described when the basic quantities are given as functions (superscript A ) of time, t, or

T = p(t) V = P(t)

P = P(t)

s = S(t) u = U(t)

0 = Q(t) 1

process (1)

The term e is the typical phenomenon accompanying any thermoanalytical experiment and can always be obtained from the energy conservation law

fT=Q-pv

where superposed dotts mean the time derivative.

(2)

Specific properties of the sample can be characterized by three material relations, G, V and S, expressed as functions of the state of the sample. The state is now identified with the instantaneous values given for the three externally varied parameters, say pressure, P, temperature, 7-, and its time change, i; namely

G = &P, T, p)

V = v(P, T, T)

S = f?(P, T, r) (3)

The entropy principle requires that for all processes of this system the rate of heat exchange, (i, be limited by the maximum value of entropy change, s, or

s r o/T (4)

Employing this requirement, we can draw important conclusions_ Introducing a state function in the form

G=U-TS+PV (5)

which is conveniently caIIed the Gibbs free energy, we have from eqns. (2) and (4)

OzG++P-VP (6)

Using eqn. (3) we proceed by introducing the partial derivatives into the inequality (6) to obtain

02 [I %P.T3.) I a- 1P.f

+ S(P, T, T) ] T +

(7)

Equation (7) must be fulfilled for any process so that the values of T, P and their

204

time derivatives 5!‘, P can be chosen independently and arbitrarily. Hence for F = 0 and 9 = 0, eqn. (7) is reduced to (dG/Zi‘)T 9 0, which can be identified for all possible values of ZC only if the term in the parentheses is equal to zero. From this it follows that the function G cannot be dependent on p; thus G in eqns. (3) is only G = G(P, T), i.e. Gibbs free energy obtains the form known from ordinary thermo-

dynamics. SimilarIy, if we choose i, we have eqn. (7) in the form [aG/M + VJP Q 0

being already aware that o’G/a?- = 0. Because the term in square brackets is independent of P, and P can be chosen arbitrarily, we obtain

aGWJ-, I I ap T.T

= V(P, T) (8)

The analysis of the last term of eqn. (7) is more complicated. Let us divide the entropy, .S, into its equilibrium part, SC, = sC,(P, T) 3 s(P, T’, p = 0), and the complementary part [S - Ses]_ What remains from the inequality (7) can now be rewritten

as

P.i + Seq(P, T) 1 ?’ + [S(P, T, T) - S,,(P, T)] i-

For fixed P and T, this inequality expresses the variable i-in the analytical form of 0 2 aYf + b(f)?, where b(p) approaches zero if ?-, 0. Such inequality can be satis- fied for arbitrary Ponly if a = 0 and [b(?)P] d 0, or

%PJ-, I I aT p,i_ = - SCCJR T)

cw, T, T) - &q(P, Tll T I 0 (11)

Equation (11) represents here the so-called dissipatiorz izzeqztality, i.e. the non-ideality of our material under study. If the term b(T) is negligible or small enough, we come to the so-called quasistatic processes, where an ordinary description by means of classical thermodynamic relations [see eqns. (8) and (lo)] is satisfactory_

The application of our approach now depends on the kind of material investigated and on the rate of temperature change. It is evident that, for example, a perfect gas will always behave ideally regardless of the conditions externally applied. However, in thermal analysis we often encounter rather non-ideal materials such as solids and, hence, the heating rate becomes decisive. Nevertheless, for ordinary TA runs (p > 0, r = 0), where the heating rates are of the order of magnitude of lo- l K set- l, the state functions of G, V and S depend predominantly on P and T and the effect of ri-is negligible. This, in fact, is in accordance with the well-known

result following from thermodynamics of irreversible processes which says that for the systems which are not too far from their equilibrium state and where the processes proceed fast enough, the ordinary thermodynamics can be utilizedlg- ‘l.

205

It is evident that this “classical” thermodynamics or "therrnostatics" forms a limiting case of a general rational approach and thus its applicability must be care- fully investigated for each experimental case. For example, if we start to deal with greater and non-uniform heating rates (7? $ 0, acceleration T # 0), the system of eqns. (3) may not be adequate because it does not include the systems possible dependence upon the second (or even higher) derivatives in T, e.g. G = G(p, T, i, r). This holds true for some extreme conditions when, for example, some explosive reactions are studied and/or for such a non-ideal material which can remember its thermal history. The ordinary Gibbs free energy then alters by an additional termI’, dG/a? # 0 exp ressing higher dissipation. The discussion of such a system, however, is beyond the scope of this review and also beyond an ordinary thermo-

analytical experiment.

Chapter 3

BASIC THERMODYNAMIC RELATIONS AND MEASURABLE QUANTITIES

When investigating quasistatical transformations of the energy [see eqn. (2)] of our studied macro-system into its particular forms, the so-called phenome~zologicaf

t17ern70dyi7an7ics3* ' is of great help to a better understanding of the general principles of a TA experiment. It aids our interpretation of how to construct mutual inter- connections between the thermal and non-thermal properties of the system. The most ready-to-use result of such a description is the set of relations, conveniently called respoizse fur7ctions, correlating thermodynamic quantities with those which can be detected by means of a direct thermophysical measurement. For the simplest case of a closed system discussed above we readily transform3 internal energy U = o(S, V)anddU= TdS - Pd V into the form of Gibbs free energy G = @T, P) and dG = -SdT + VdP by replacing the extensive parameters S and V by the intensive ones, T and P, which do not depend upon the quantity of the system and which can be more easily externally controlled. On the other hand, we should bear in mind that the experimentally measurable state of the system is best reflected by

the instantaneous values of the extensive parameters V and S as functions of the intensiveparan7eters P and T [see eqns. (3)] with regard to the size of the system*.

By the use of thermoanalytical convention the state of this system can be investigated in two different ways: by volume measurements V = @‘(P, T) and/or

* The greater the magnitude of the system investigated, the better the sensibility and resolution achieved for the detection of an extensive parameter. On the other hand, the accuracy of determination of an intensive parameter improves for smaller systems (mainly due to decreasing gradients). There, however, arises the controversy of how to measure experimentally intensive parameters through which the representative extensive quantity is estimated. The typical case is that of temperature10 (thermometric measurements as DTA, spontaneous heat flux. measurements as DCC) or pressure (non-isobaric measurements, isochoric measurements as DPA), etc.

206

enthalpy measurements H = &P, S) = l?[P, s(P, T)]. As both functions are the

state functions, we can express their total differential as

dV= (g)=dP + (g)pdT

and

(12)

dH = ($$U’ + (g), [(g)=dP + ($r),dT]

= VdP + T E-(g) dP + (g) dT] (13)

where the partial derivatives can be identified with the following experimentally

attainable coefficients

thermal compressibility - /? = ?._

thermal expansion CL, = t(%)P = ( ::T),,$

and

heat capacity c, = (s), = - T($)p = T($)p

Finally, it yields

dV= a,VdT- j?VdP

dH = V(I - a,T)dP + c,dT (15)

It can be seen that the change of the thermal state of our macro-system is

accompanied by the changes of all the macroscopic properties in question and,

conversely, the change of any macroscopic property results in changing the thermal

property. This fact documents the major importance of thermodynamics in describing

a TA experiment in its broad sense of a general thermophysical measurement. In

addition, the change of temperature will not only affect the above-listed thermal

properties but will also change mechanical, electromagnetic22- 25 and optical

properties, and will affect the rate of chemical reactions3, heat and mass transfer, etc.

Consequently, let us consider a more complex system assuming the exchange of a volatile component between the sample and its surroundings (a partly open

system) as well as new externally applied fields: electromagnetic and mechanical, as schematically shown in Fig. 3. New intensive parameters which are to be externally

207

EXTERNALLY APPLIED FIELDS

T, S

p,v

c&l E, U’ F, Z

@

(HI C I

lNTERAC7lON 1 OF ENERGY i AND MASS 1

L ---

Fig. 3. Partly open, quasistationar thermodynamic system suitable for the portrayal of TA methods as calloric, thermodilatometric, weight, evolved gas, magnetic, dielectric and thermomechanical measurements (symbols see text).

controlled are p, S, E and Fcalled the chemical potential, magnetic and electric fields

and mechanical tension, respectively. Corresponding extensive parameters represent-

ing the state of the sample are n, M, 9 and T, known as the mole number, magnetization, polarization and deformation. We certainly cannot control directly the chemical potential of the surroundings but we choose the partial pressure p. (assuming the validity 11~ = ,Q, + RT In po). It should also be noted that the previously used term VdP retains its real meaning only if the material investigated does not become anisotropic under the action of the external fields (i.e. homogeneity condition)*. The chemical potential term izd/c is expressed in the form of a summation according to the number of phases in the system.

Let us imagine a generalized state function @ depending exclusively on the intensive parameters as independent parameters of our TA experiment

@ = @(T, P, /1,X, E, F) (If9

The differential form of this thermodynamic potential is analogical to Gibbs free energy G but more comprehensive

d@ = -SdT + VdP - Ndp - MdS’ - BdE - +rdF (17)

l For a more rigorous analysis of electromagnetic and mechanic measurements it is more convenient to replace the term Pd V by the term pdg, which indicates the energy change in a unit volume of the sample as a result of a mass change for given E, A? and F_ The symbol p is density and q means here a chemical potential of a unit amount of the mass of the sample.

TA

BL

E t

Ttj

ER

MA

L

CO

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FIC

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TS

DE

RIV

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ON

RA

SJS

OF

A G

EN

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TH

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MO

DY

NA

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P

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EN

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L r

l, =

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Exp

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nerr

tally

co

rrtr

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l iii

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sive

pa

wm

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X

Co w

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av

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av

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as

$(l,

as

ax

-=-

-z-z

- ex

terr

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par

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ax2

ax

axar

aT

ax

-..

-_--

= -=

- aT

aY

ay

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to b

e d&

?ctc

d Y

(=

aqax

)

Pre

ssur

e P

Che

mic

al

pote

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l jl

Mol

e num

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I1

Mag

netic

fkld

,X?

Mag

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n h4

Ele

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c fie

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Mec

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c ten

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F

Def

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n r

Tem

pera

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T

Vol

ume

Y

Pola

riza

tion 9

Ent

ropy

(ent

halp

y)

S (W

1 av

-_

- v

aP

= /

I the

rmal

co

mpr

essi

bilit

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all

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(lo

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thm

aP

of

act

ivity

)

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= 1

mag

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su

scep

tibili

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- =

XI>

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su

scep

tibili

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- =

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tic

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stif

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s coc

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-=

a~ th

erm

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expa

nsio

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- =

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= u

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cf

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uv

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=

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pjns

ivity

‘

ail

- =

Sp

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l m

olar

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en

trop

y

a8

m

-=- 3T

x

ther

mal

aE

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w

ccpt

ibili

ties

-=- aT

%

r

at

- =

ut

ther

mal

stra

ins

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coef

fici

ent

8F

- =

KIV

the

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cocf

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C

,Y th

erm

al ca

paci

ty, e

ssen

tial te

rm in

TA

209

By making partial derivatives we obtain an extended number of thermal coefficients as listed in Table 1. Here we are merely concerned with the derivatives between the intensive (X) and the corresponding extensive (Y) parameters and the mixed derivatives with temperature as the essential coefficients for theoretical thermal analysis. The parameters which are kept constant during the derivation and which are usually marked as the subscripts of individual partial derivatives are, for the sake of simplicity, omitted. Some coefficients are evidently simplified when we assume real materials.

The electromagnetic field and particularly the mechanical tension must be understood as tensors which may yield for an anisotropic material as many as six coefficients for

each c,, cc, and KFT. Some interesting coefficients can also be obtained on the basis of interactions between non-thermal properties only, as e.g. electromagnetic field with mechanical tension (piezoelectric coefficient dr = 8 g/aFor = &/dE 2 2, magneto-

elastic coefficient C,, = BM/dF = &l&Z) and/or pressure with mole number (molar volume V = 8~(/8P or = aV/&z, change of concentration with pressure V Knp z

anlap or = avlapc), etc. Following Table 1 we can derive a set of equations in analogy with the proce-

dure given for volume by eqns. (12) and (14) suitable to describe thermodilatometry or differential pressure analysis and/or experiments carried out in sealed ampoules (dV = 0). Th ese relations describing the change of the selected extensive property Y as a function of the corresponding intensive parameter X and the temperature T

then hold the form

II = 2(/l, T) dtz = (d/z/a&- d/r + K,,,- dT (18)

M = Ici(H, T) dM = xdx i- CL,, dT (19)

9 = &(E, T) dg = x,,dE f pp dT (20)

Z = ?(F, T) dt = C,dF+ a,dT (21)

These equations become essential for the respective thermogravimetric, thermo-

magnetic, thermodielectric and thermomechanical measurements under given

experimental conditions. In analogy with the well known form of entropy equation suitable to describe

caloric measurements

dS = c+VdP - cp/TdZ- (22)

we can distinguish four particular forms of investigations common in physical

chemistry and physics

dY=O isolated (thermally closed) system (e-g. dS = 0 adiabatic or for eqn. (14) dV = 0 isochoric)

dX=O system under constant external field (e.g. dP = 0 isobaric) dT=O system at constant temperature (e.g. dT = 0 isothermal) dT = const. system under constant heating (TA) (e.g. dT = 4 non-iso-

thermal)

210

dY, dX, dT # 0 system out of control (e.g. undefined experimental condition)

Equations (14) and (19-o-(1) express the thermal development of non-thermal (non-caloric) property of the sample. In basic TA methods, however, we are usually concerned with the thermal development of thermal (caloric) property of the system which is essential to any calorimetric work. Thus we can derive the relations of the enthalpy-like term H upon a given intensive parameter X [compare eqns. (18)-(21)] and the temperature T in analo,T with eqns. (13) and (15).

H = H(p, %, T)) dH = (n - T K,,,)dp + cp dT (23)

H = @(A@, s(Z’, T)) dH = (M - Tcc,)dc% + c, dT (24)

H = fi(E, f?(E, T)) dH=(Y- Tp,)dE + CE dT (25)

H = fi(F, s(F, T)) dH = (r - TccJdF + c, dT (26)

Chapter 4

THERMODYNAMICS OF SIMPLE PHASE TRANSFORMATIONS

In the preceding part we shown the thermodynamic picture of a heated system where no chemical reactions and structural transformations occur which, in fact, corresponds to the base line of respective TA records. Moreover, it demonstrates well the necessity of a proper definition of the set of intensive parameters to be chosen in accordance with the externally applied, so-called experimental conditions to achieve a successful description and classification of individual TA methods’* 6p 12. In our experiments with rising temperature, however, we are more concerned with the characterization of thermal effects which change the smooth course of the base line of a TA record. Such turns are usually caused by the process of phase transforma- tionZ6’ 27, where a stable (or metastable) phase A is transformed into the other, thermally more stable phase B; or A with @(TA, PA, /(A, S’A, EA, FA) + B with @(TR, PB, pB, sB, I!&, FB)_ There arise two basic problems: (1) how to find the conditions of the two equilibrium phases and, (2) where and how one phase transforms into the other.

The principal condition of equilibrium is given by the minimum of the potential @ (d@ = 0) associated with the secondary conditions valid for all the extensive parameters, dYA -j- dY, = 0. It follows that such a system must be in thermal as well as mechanic and electromagnetic equilibrium, where XA = X, = X& In other words, with a continuous change of temperature, the other intensive parameters must also change continuously [which is particularly important for the chemical potential p as classically assumed for multicomponent systems]. With regard to the given couples of intensive variables T and X chosen in eqns. (18)-(22), we obtain for a single component system the equality

@WA, TA) = @(G,, TB) (28)

211

Fig. 4. Three-dimensional representation of the state of two phases (A and B) mutual relation. The equilibrium (eq) is described by the double solid line, which holds for the general thermodynamic potential @A(TA, XA) = @B(TB, XB), see text-

(se& Fig. 4) the solution of which are the curves X = X(T) usually represented in the form of diagrams, P-T for liquids, p-T for solid solutions, S--T for ferromagnetics, E-T for dielectrics and F-T for mechanically stressed systems2 ‘.

The conditions of a thermodynamic equilibrium do not put any limitation on the change of the derivative of the general potential with an intensive variable, &D/3X, which may have in the different phases different values. As each phase represents here a homogeneous system and the coexistence of two phases, A and B, becomes discontinuous (i.e. a heterogeneous system defined as the sum Y = pi Yi), the two phases must differ by the value of at least one property, as, for instance, the density, specific heat, magnetization, etc. The discontinuity in &P/6X is thus most suitable for the classification of phase transformations and the characteristic value of X, at which the transformation occurs, is the equilibrium value of the given intensive

Fig. 5. Graphical illustration of possible types of phase transformation for the non-isothermal degree of transformation, L, and the general thermodynamic potential, @, as a function of temperature. The underscripts in, var, corn, eq, o and F mean invariant, variant, composed, equilibrium, initial and final, respectively and dashed line indicates actual “response” curves.

property, X,,. From the viewpoint of TA the most interesting are the derivatives with temperature, see Fig. 5, as the caloric term -SdT always takes part in any of the so far used forms of our potential CD. This yields the discontinuity in entropy AS, which means that during this so called$rst-order transformation a certain amount of latent heat (TdS = LIEI) is always absorbed or generated. If the first derivative in @ is continuous, at least the second derivative is stepwise showing the so-called second- order transformations (as also illustrated in Fi g. 5) which are always accompanied by the change in the value of thermal capacity AC,. It follows that the thermophysical measurement of the enthalpy content is the principal and most general method of thermal analysis because it can detect any physico-chemical process. The most important result may be found in generalized Clausius-Clapeyron and Ehrnfest relations which are listed in Table 2’ ’ * 2 ‘* 27. Th e set of these relations is experimentally very useful as they reflect the alternation of externally applied parameters

213

s A

Fi ri Ek s I I I I

II II II II II

214

in terms of the stepwise changes of experimentally’measurable extensive quantities. During a TA experiment, the temperature is gradually raised so that the whole

transformation does not have time to proceed instantly at a single temperature, T_,.

Thus, it is of great importance to determine the temperature dependence of AH

inr.luding~the simultaneou, effect_ of the second intensive parameter X, i.e. AH =

AH(X, AS(X, T)). Making partial derivatives we can proceed according to eqn. (13)

to obtain

dAH -= dT

AY-Tg g >

-I- AC,

By the substitution for B’,,aX from Table 2 we have’

dAH

-=Acx+-T- dT AH-- aln AYAH

aT

(2%

(30)

where AY equals respective AV, An, AM, A 9 and Ar, according to the type of transformation listed in Table 2. In the sense of Fig. 4, eqn. (24) expresses the Shift of AH along the transformation boundary shown as the double-solid line. This equation is most often used for the .simplified description -of subIimation and/or melting

(X = P, A Y = A VB), where the second and the last term of the right-hand side of eqn. (30) cancel each other yielding dAH/dT z AC,. If multipIe effects of axl/Zr, GX,/aT - - - are assumed, the last two terms in eqn. (24) reappear for each new A Y.

According to the preceding scheme, we can derive the required increase of any non-thermal property dX, to balance the equilibrium of transformation if the other non-thermal property is changed by dX,, i.e. dX,/dX, = X,,A Y,/AY, (compare

Table 2), as well as to find out the non-thermal dependence of any d Y’in analogy with eqns. (29) and (30) [AF = A$((x,, &Xl, X2))] which, of course, falls beyond

the normal TA practice. It should be emphasized that all precedin g equations-are correct only if the

experimental conditions are well defined and restricted to given values. If one of the externally applied parameters is out of control, the process of transformation becomes undetermined within our measures which can be graphically illustrated by the dotted line in Fig. 4. This is particularly important for the so-called self-gezzer-ating corzditiozzs

which are often applied in terms of non-constant pressure P and/or partial pressure p.

(2 p) usually found when the dissociating sample is placed in a partly sealed crucible or when applying undefined vacuum. Similar effects may result from a free motion of

the sample, which is suspended in a non-homogeneous electromagnetic field.

Chap#er 5

DESCRIPTION OF THEORETICAL AND EXPERIMENTAL CURVES

The object of the majority of TA measurements is to find out and to describe the effects which occur during the heating of the material under investigation. The

215

analyses -of TA data are thus directed to establishing the relation between the ex-

perimentally obtained curves and the actual course of our thermodynamic process, compare eqns. (1). Let us focus our attention to the simple one-component trans-

formation,.see eqn. (28), ?zA + nB- For the sake of simplicity the set of eqns. (1) may

be reduced to

(or -nB = n&t))

where, however, we intentionally neglect the energetics of A-B interface formation_ As the transformation cannot take place at temperature TCq infinitesimally fast, it is necessary to define the progress of transformation by a dimensionless parameter 5,

called the true degree of transformation28 (or generally conversion)

< = iIA/(nA f %) or (1 - 5) = %&B + nA) (31)

where i2A and tlg are the instantaneous mole numbers of the respective phases. The

choice of non-dimensional parameters normalized within the interval (0,I) is in accordance with ordinary kinetic convenience.

The practice of TA measurements is to coIlect al1 information on the in-

stantaneous state of the sample and on the time progress of the change of its state on the basis of a certain physical property of the sample, which is experimentalty

measured and chosen to represent the state of the sample. This property must be evidently dependent on the quantity of the sample, i.e. it must be an extensive parameter characterizing the material investigated such as enthalpy content, density, weight, length, volume, magnetization, polarization, mechanic deformation, possibly

weight loss and/or amount of volatilized component, as will also be shown Iater on. The experimentally detected course of the process is thus best described by the

effective degree of transformation, A, defined on the basis of an experimentally measured property 2 by the equation28P 2g

]=z--O ., zF-zO

(32)

where 2, is the initial and 2, is the final (ultimate) value of Z. One of the principal premises of TA is the identity between 5 and /, which is usualiy accepted without

proofs as proportionality

5 = KJ (33)

Hence it is interesting to compare the thermal development of both these degrees. We can assume that the measured property is proportional to an extensive parameter of the system, 2 = KY Y. For the constant mole number of the system (nA + nB) =

const. we have by eqns. (31)-(33)

216

For the total change (,4 Y = YF - Yo) the proportionality coefficient KA keeps a

certain value in accordance with the corresponding thermal coefficients in Table 1, e.g.

KJ_ = (35)

which is approximately constant for a narrow temperature interval of the process

duration. The experimental task of TA measurements is the transformation of experi-

mentally recorded signal Z,,,,,,. to the true value of measured physical property 2. For most TA apparatus there exists a direct or almost direct proportionality in the form

z = zne,s_ Kpp. (36)

where K&_ is known as apparatus constant to be established by calibration_ This is

adequate, for example, for weight measurement of magnetization, compensation heat fiux measurements in DSC or length measurements in thermodilatometry. For some special instrumentation, however, the proportionality (36) reaches the form of a more complicated function as3 O - 32

Z = 2(Z,,,,., imeas.7 T, ri: P, 9, Kapp., - - -1 (37)

the typical example of which is the case of DTA’ 3, DCC or DPA. Having determined Z we can evaluate the so-called reaction kinetics from the

time-dependence of our process using L = L(t) and T = F(t). The kinetics of the

-

I T

LINEAR NONISOTtiERMAL T= 0

ISOTCrERMAL T= conrc

i=o

AC_TUAL ~ONISOTUERMAL T=O,T’O

Fig. 6. ExempliScation of the most common methods of thermal treatments. Solid lines show the idealized course of temperature while dashed lines express the actual course of temperature due to the heat absorbed (assuming an endothermic process). Dashed-and-dott lines specify the characteristic temperatures, compare Fig. 5.

simple first-order transformation tions33* 34

217

can be expressed by two basic constitutive equa-

(38)

They are practically evaluated in three different ways, see Fig. 6, (i) isothermal kinetics assuming

T = constant (and F = 0) x = f(l) k(T)

(39)

where f(3,) and k(T) are functions called the model relation and the rate constant respectively being dependent on the separable variables 3. and T only.

(ii) litwar non-isothermal kirletics assuming3 ’

T= 4 = constant (i.e. T = T,=, + q5 t)

j, = f’(J, T)k’(T, @) z f(il)k(T)@ (W

where 4 is the constant heating rate. (iii) actual (nonlinear) kinetics, where eqns. (38) are to be applied without

simplifications_ Here we are not able to separate individual parameters as in eqns. (39) and (40) 2g* 3 6_ Simple numerical determinations of ordinary kinetic parameters, e.g. activation energy which is popular in isothermal and linear non-isothermal kinetics, are not possible in this complex case (iii).

Let us investigate in greater detail the speciality of a thermoanalytical description of an actual “kinetic” curve, particularly if the process nA --f 11~ is not thermo- dynamically to take place at a point temperature T,, but within a certain temperature interval To - TF, where phases A and B can coexist. The terminal state of the system may thus become temperature dependent due to the change of equilibrium with temperature. This effect is important in all non-isothermal studies and ought to be incorporated into the calculation. Considering eqn. (32), where the value 2, reaches evidently its maximum value, we can imagine that this equation is composed of two parts: kinetics and equilibrium. Accordingly, we may introduce a new term28 called the advancement of equilibrium of the process ,I,, defined as

3 _ Zr - =0 -=q - 2 - zo TF

(41)

where Z-r is the terminal equilibrium value of Z for the given temperature T, compare Fig. 6, while Zr is its ultimate value reached at the end of whole process evidently independent of temperature. This term, in fact, describes the propagation of equilibrium under the conditions of an infinitesimally slow temperature increase. Combining eqns. (41) and (32), we obtain=’

z--o _ /, = 3.,, z - I,,, a T

_ z 0

(42)

218

where o! is the isothermal degree of transformation defined in accordance with ordinary

isothermal studies. It follows that 1. can be understood as the non-isothermal degree

of transformation containing all complex information about the process normally

provided by a set of cc determined for a series of T(Z,,) within the interval T&Z,)

to T&Z,) (compare Fig. 6).

From the point of view of the equilibrium background of the process we can

classify the first order process into invariant (single temperature T,,), variant (temper-

ature interval TO-T,) and combined (where the process proceeds partly as invariant

and partly as variant), as it is graphically demonstrated in Fig. 5, where dashed lines

show the actual course of the process owing to kinetic retardation.

For TA practice there follow important considerations2’.

(i) For invariant processes, where 2- = CL, it is not necessary to take any pre-

caution for the curves interpretation.

(ii) For variant and combined processes we should either use the truly non-

isothermal degree of transformation, which is not common as yet, or to employ

enough high heating rates which enable us to evaluate the major part of TA curve

above T,, i.e. above two-phase region where again I. = cc.

(iii) For the description of the second-order processes it is only convenient

to use the derivative of the experimentally obtained curve which can then be treated

in the same way as first-order processes (compare Fig. 5) last column, i.e. the trans-

formation of the break with no inflections to the step with one inflection point.

Let us concentrate our attention on Fig. 5, which is instructive enough to find

analogy between the graphical demonstration of increasing derivatives in ~25 with the

kind of experimental curves and their derivatives (compare Fig. 1). With the gradual

transition from lower to higher derivatives, the curve changes its character; it be-

INTERNAL I EXTERNAL ,_ FLUXES FLUXES

Fig. 7. ActuaI conditions of a heterogeneous system during a TA experiment, where externally applied fields (compare Fig. 3) are absent but where we consider internal fluxes across the phase A and B interface I.

219

comes mathematically more distinguishable as the number of inflection points and extremes increase99 “9 37. From a certain stage, however, the effect cannot be quantitatively evaluated as, for example, oscillation (see Fig. l), defined by three inflection points. Therefore, it is important to know the proper form of any experimental curve for a given TA instrumentation and to distinguish its electronically derived analogue to make possible a correct determination of a true (“kinetic”) curve.

Chapter 6

ACTUAL CONDITIONS OF DISSOCIATION PROCESSES

So far we have not considered the content of a volatile component (v) in our sample, i.e. the transformation well known as dissociation process and often en- countered in TA practice. Let us imagine our system in the form illustrated in Fig. 7. Two basic external fluxes take place between the sample and the surroundings: 0 (heat flux) and ri’ (mass flux). The overall behaviour of the system can be expressed on the basis of the energy and mass conservation law, compare eqn. (2), as shown in our previous works4- 6

already

(43)

which, in fact, is the flux formulation of the first law of thermodynamics. As the enthalpy is a more convenient parameter in TA measurements, then

A=& vP+/l*iY (44)

where H is the system enthalpy which for our heterogeneous system composed of phases A and B holds as the sum of HA and HB. In accordance with eqn. (13), we have to assume H as the state function of S, P and n, i.e. H = l?(P, n, f?(P, n, T)), so that

(45)

where the superposed strip describes molar values for the given extensive property (F). By combining with fi, we get

if(V,+ VB)= V,R,-TsA=GA= pA and P, = P = const., i-e_ P = 0, we have

by eqn. (44)

0 = (CPA + CPu)p + PASTA + p~fi~ - /.&fiv (47)

which can be considered as the basic relation of our system. However, there arises the problem of how to treat the terms pi in an understandable enough way. Therefore, let us use the molar values4 but related to the sum of conservative components

220

(superscript c) as introduced by Holba as a convenient mean for the description of non-stoichiometry3 ‘. Hence assuming the transformation of the type

ijv(A) + ij(B) + vt (*) (48)

where i and j are two conservative components of phases A and B and v is the volatile component (superscript v). For nc = Zi + c

c,,/rz& Ni = nA/n& N’ = tziJrzc, ?zi = ni + nj we have for example c$ =

etc. These parameters are stable regardless of the mass lost during dissociation except for the volatile part. Introducing the degree of transformation e which expresses the portion of phase A converted to phase B related again to the sum of conservative components for any extensive parameter, we have49 ’

Y” = Y”,(l - 0 + YE r (49)

where

5 = ni[nc = 1 - nilno (50)

Assuming, for example, the fluxes in the form of hi = ZLiQz~ + ZN$; (e.g. II’; = ii@“, + Nirii) together with eqns. (49) and (50), we obtain after some algebraic manipulation4

The first three lines of this equation describe the thermal development of the states of the individual phases A and B and (*) while the last Iine expresses the entire change occurring due to the transformation of A into B. For a better explanation of its physical meaning, let us rewrite eqn. (51) in the following form4.

[K,‘“(T, - T) + Kzd(T,4 - T4)]lrz" =

+

(a)

U-9

(cl

(4

(4

(52)

where the individual lines describe the heat consumption due to: (a) specific heat flux between the surroundings and the sample, where K,‘*

and Kf” are the sample heat transfer coefficients for conduction and radiation; (b) the change of the system temperature, where AC, is the difference between

the specific heats of the initial phase A and product phase B;

221

(c) the change of the content of the volatile component in respective phases (called stoichiornetry);

(d) the formation of product phase B (where AH,,_ is the effective enthalpy change) including the specific loss of the volatile component (AN”) by dissociation taking place during A + B transformation;

(e) the redistribution of conservative components between the phases often called phase separation, where Api = pi - pi and 6 is a new parameter called extent of phase separation (normalized 0-6-O) by equation <(N’ - Nk) = (1 - 5)

(N: - N’) = 6. This analysis shows that this system must also be described by help of 6 besides

the previously introduced 5. Furthermore, it is worth noting that each mass flux is diffusion controlled3 ‘, i.e. is carried out across the interface (I) the area of which must also be introduced to our consideration as a necessary parameter describing, in fact, the energetics of phase A and B discontinuity. From this viewpoint, the set of constitutive equations (38) for this heterogeneous system takes up the form3g’ 4o

E = f(C, 6, r, I-) 6 = is<<, 6, r, T)

1 = H(& 6, r, T)

i- = +(S, S, I, T),

where f is bounded with 5, 6 and T by nucleation-growth indicated in the form2gW 369 3g 1 = &,uc,_(L 6 T) &,,(<, 6, T)

(53)

equation which is usually

(54) Equation (54) is well known in the simplified shape of Kolgomorov-Johnson-Mehl- Avrami-Yerofeev equation3* 3 6, where the parameter 6 is not considered and the course of T is idealized (F = 0 or F = 4).

Equation (52), although havin g a rather small practical applicability, well demonstrates the complex behaviour of a heterogeneous system and possible inter- connections between the individual parameters. It can be seen that the specific properties of such a system can be investigated by, for example, weight loss (TG) and/or evolved gas (EGD) measurements (AN”); compensation calorimetry (DSC) and enthalpiometry (AH,,_, 5f = q5 = const.), Calvet microcalorimetry (DCC) and other spontaneous heat flux measurements (Q, T # const.)32* 37, direct TA (T) and its derived techniques32* 3 7 as DTA measuring the temperature difference

AT = (T - Lrerence) or even possibly high-temperature X-ray diffraction (6). This approach also yields the virtually new parameter 6 as an entire property of a multicomponent heterogeneous system3’* 4o which, for advanced kinetic studies, ought to be incorporated into the calculation_ Consequently, for variant processes proceeding in two-phase region the values of < and 6 again contain the equilibrium parts4’, teq and Seq, compare eqn. (41).

For the actual interpretation of TA curves, Holba6 thoroughly analyzed the relations between the change of a measured extensive property 2 and the state of the sample. For the transformation of the type eqn. (48) he assumes the state of the

222

sample to be dependent upon the degree of conversion < [eqn. (50)], temperature T, pressure P, partial pressure of volatile components p,‘, total chemical composition N’ = lzi/ltc and composition of product condensed phase B, NA, i.e. Z = Z(T, P, p;, N’, NL). Using eqn. (49) for N’ and the derived derivative for fib we can write for the rate of the change of a measured extensive property Z6

+ az= Ft azcp + azc az= A_ nc

aT ap -@B’;.+--+ -

ad ax I (53

where Zc = Z/ lzc and X mean, in accordance with the above symbolic, any additional intensive parameter varied from the surroundings. The term AZ’ = 2; - Z; is the integral change of Z’ due to the transformation. The following term which is also multiplied by f is of similar meaning as the extent of phase separation in eqn. (52e). The primitive proportionality &’ = II~A~“~~, often used in TG, holds, however, in the case of daltonides only, i.e. the constant stoichiometry of phase A and B. For a practical use of eqn. (55) it is necessary to assume that the values AZ’ and aZ/i3X are not constant but dependent upon the state of the sample. To express their instantaneous values, it is possible to employ Taylor’s expansion in the vicinity of equilibrium points of the transformation6, e.g. T = T, + AT, p,’ = pt + Ap’, etc., where AT and Ap’ have the meaning of the deviation of the sample temperature or partial pressure from those in the surroundings (*).

Chapter 7

DISCUSSION

The importance of thermodynamic relations as elementary rules to be applied in theoretical TA has already been stressed in an earlier review3. The present extension with the heating rate as an independent parameter and, particularly, the inclusion of electromagnetic and mechanoelastic measurements comes within the theoretical scope of TA description and is also required with respect to the individual methods classification although, for illustrative purposes, it is still idealized.

It is the matter of mathematical manipulation to extend the above two-parameter description, see Chapter 4, by additional parameters, which is most often P alike always accounted T. For instance, by a mere combination of individual terms in eqns. (23)-(26), we can achieve a generalized relation for N in the form of N = W, X WY p, 0.

dH = V(1 - TaJdP + (Y - TaJdX + C,dT (56)

223

This equation clearly illustrates the possible interdependence of externally controlled parameters. It can be seen that the originally used symbols of heat capacities, as exhibited by eqns. (23)-(26) and Table 2, and unusually defined under either constant p, X, E or F, may now be identified with the classical C,, compare eqn. (13). An analogous result can be reached for the generalized Clausius-Clapeyron equation (Table 2) using the plausible extension to three- or multi-variable relations.

Attention should now be paid to the possibility of practical utilization of the previously listed equations. Above all, it may be the direct numerical extraction or tabulation of the standard enthalpy and/or free energy changes. By approximating the material coefficients in eqn. (56) as, for example, LX, = a, + b,T and the heat capacity as Cp = a, + b,T + cc/T’, and after substitution and integration we obtain

AH = AH, + AV(l - Aa,T2/2 - Ab,T3/3)(P - PO) t

+ (AY - Aa,T2/2 - Ab,T3/3)(X - X0) +

+ [Aa,T - Ab,T2/2 - AcJT];,, (57)

or, for A@ = AH - TAS, the last term of the right-hand side of eqn. (57) being altered to

[Aa,T(l - In T) + Ab,T - Ac,/2T2]~,.

These functions are not in a sufficiently convenient form for

nor does the second term on the right-hand side of eqn. (57) have standardization* any applicability

except in the special determination of boundary curves in eIectromagnetic or mechanoelastic system, cf. eqn. (28). Moreover, the partial derivative of @ with respect to IZ does not provide the convenient parameter ,v, so useful in the standard description of equilibria**. Hence, let us restrict our attention to the ordinary AH and AG functions and their most frequent use when constructing phase diagrams by the direct determination of either characteristic temperatures or the heats of fusion for a given

composition X. For instance, modifying the Clausius-Clapeyron equation into the form of the Le Chatelier-Schrederer equation

* The choice of standard state is solely a matter of convenience for ease of calculation and should not affect the resulW J2. So it may be To = 298 K, PO = 1, Xo = 0, etc. The proper selection of the consistent units should be noted, e.g. AH (cal mole-l), ACT (cal K-l mole-l), AS (e-u.), A V (cal bar-l = cm3/41,84), etc.

** This, of course, can be helped by introducing an additional parameter II into eqn. (16) thus yielding the desired term pdtz (and a@/&~ = cc). as the additional term of eqn. (17). Presumably this would be more convenient in everyday thermodynamic practice 25 but it is not consistent with our simplified approach to use exclusively the intensive parameters as variables in &-

224

A+- TA-- T++-)]

AC, 2 [ (

1

T,T

(58

where Ti and T are the temperature of fusion and of the system, respectively, and

AH, is the change of standard enthalpy at TA_ Practical applicability and common

simplifications were surveyed by, for example, Adams and Cohen43. Assuming ideal

behaviour, the most popular simplification employs only the first term of eqn. (58),

plotting In -VA versus l/T to obtain a straight line the slope of which yields the enthalpy

of fusion AH,. On the other hand, if known, this equation may be used to

give the ratio of the activities instead of XA We also should not forget the possibility

of using the Hess and Kirchhof additive laws as convenient means in all cases where

experimental difficulties in the direct determination of the state functions occur.

Let us now consider a typical TA recording in Fig. 8 to demonstrate the type of information that can be extracted. First of all, we may look for thermodynamic quantities. From the base line we can read the thermal development of the measured property 2 as well as non-thermal progress if an additional external parameter is also

z T d-0

heating(+) ____---.

l

supCheated METASrABLE A

Kinetics

I undercooled

Fig. 8. Graphical illustration of a TA record with regard to possible ways of data selection. The rectanguular solid line reflects the development of equilibrium (background) of the phase transformation (A+B) of invariant type assuming an infinitesimal temperature change (Cf. Fig. 5, Ii,)- The possible distortion of its rectangularity (I-shaped and/or diffuse-like phase transformatiorS6* 2;) due to fluctuations of concentration, magnetic moments or temperature etc. is not accounted for here (see ref. 14). The horizontal liner-like parts correspond to the base lines representing the change of system state if no reaction occurs. The actual (S-shaped) course of the TA curves for different heating rates 41, 42, $3, is caused by the time relaxation process necessary to reach equilibrium, called kinetics.

225

time-temperature dependent. The displacement of base lines gives the integral change of the measured property but its equilibrium value can only be achieved by extrapolation to zero heating rate or by recalculation using eqns. (29) and (30) when dealing with the enthalpy change, or by using analogous forms for any non-thermal property. IIowever, one should be careful over the correct interpretation of DTA

measurements3” 31 ccf. eqn. (37)J in ordinary dynamic calorimetry32* 37. Equi- librium temperatures can also be obtained by extrapolation to zero heating rate. In determining its equilibrium values, there always remains a certain error proportional to the temperature gradient44-46. This was quantitatively estimated by Proks4’ and reduced in practice by, for example, using thin layers of investigated materials spread over the large surface of a well conducting sample holder shaped, for example, in the form of multiplate crucible46, already convenient for TG measurements. Beside this error due to quasi-stationary gradients, there can arise an additional (kinetic) delay caused by the impingement of interface energetics of new phase formation which is commonly associated with the effect of superheating and/or, most probably, supercooling. This is already connected with the second type of data to be evaluated from TA records: kinetics and mechanism of the process. The logistics of this procedure were briefly touched on in eqns. (38)(40) and its entire mathematics have been presented with full details elsewhere3. Some notoriously discussed and yet unclear points of view, however, are discussed in the form of questions and answers in a subsequent critical review47.

In conclusion, it should be noted that truly equilibrium thermodynamics cannot be completely sufficient to describe correctly the real dynamic features of TA experiments because even steady temperature increments may give rise to non- equilibrium states, cf. Fig. 6. In other words, the state functions, similar to those used above, must be considered as functions of the space coordinates and time. Kluge4* has produced a nice approach for generally solving and interpreting the basic differential equations for the independent state fields (e.g., temperature, density, concentration, etc.). This has already received attention in kinetics in the determination of the decisive dimensionless parameters such as diffusion coefficient, rate constant, activation energy, etc. 4g Such new, actually non-isothermal approaches to non- .

isothermal kinetics3 ‘* 4g* So are a good guarantee of a promising theoretical future. It can be anticipated that the development in theoretical TA will also be affected by flux methods as pioneered by Sestgk et a1.4* ’ and matured in stating the fundamental equation of TA in the form (cf. Chapter 7)” 6.

e =ri+PV--lli--~-E~-F~+fyA+~oC (59) heat interaction with = response change of the state of TA system the surroundings (according to type of TA method)

The additional work terms, such as rA and cpC, can be chosen according to the type of TA in question, e.g. emanation TA, where A, y, C and cp are the surface area, interfacial tension, surface curvature and curvature coefficient, respectively (which would bring desired attention to the surface chemistry, in this article intentionally

226

underestimated). A definite 3ut not yet easy aid for the near future can be sought in a more concise framework of rational thermodynamics’ 5 - * ‘.

AUTHOR’S NOTE

In some aspects, the nomenclature employed throughout this article slightly but intentionally deviates from the nomenclature recommended by ICTA (see four reports of ICTA nomenclature committee published, for example, in the Proceedings of ICTA Conferences). It was found necessary from the point of view of theoretical TA but it should be stressed that, so far, it is restricted to this article, having no general validity. The notation of the most important, and thus recommendable, basic terms are denoted in the text by the italic Iettering. In our attempt to present a concise list of symbols suitable in theoretical TA, some replacement of individual symbols may also be found more convenient as, for example, heating rate a or p, compressibility 2, defdrmation e, stress cr or r, force F, etc.

ACKNOWLEDGEMENTS

The wishes to Prof. V. (University of Technoloa in Dr. J. and Dr. Loos (Institute Solid State and Dr. I-PoIba (Institute Inorganic Chemistry Re2) for kind discussion preparing the and Dr. Kluge in Jena) critically reading manuscript.

REFERENCES

1 2

5

6 7

8

9

10 11 12 13

14

J. SestBk, J. Thermal AtIaI., 7 (1977) 257. J. Se&k and P. Holba, 7th Czech. COJI$ OJI TA in High Tatras 1975; Proc. Termanal ‘76 by SVST Bratislava, 1976, pp. 7-13 (in Czech.). J. SestBk, V. Satava and W. W. Wendland& Thermochim. Acta, 5 (1973) 333. J. Se&k, P. Holba and J. Kratochvil, 2nd Nordic Symposium on TA in Rim, 1973, Proc. by Nordforsk, Stockholm, 1974, p. 7. J. &stBk, 4111 ICTA, Budapest 1974, Proc. Thermal AtlaIysis, Akademiai Kiado, Budapest, 1975, Vol. 1, p- 3. P. Holba, Silikaty, 20 (1976) 20 (in Czech.). R- C. Mackenzie (Ed.), Or I erential ThermaI Analysis, Vols. 1 and 2. Academic Press, London, 1972. V. Satava, PrirlcipIes of PhenomenoIogicaI and Statistical Thermodynamics, Scriptum VSCHT by SNTL, Prague, 1977 (in Czech.). J. SestBk, 5th Czech. Conf. on PhysicaI Methods in Ceramic Industry, Horni BXza, 1976, Proc. by (SVTS DT Plzefi, 1976, p. 5 (in Czech.). P. Holba and J. &&ik, Thermochim. Acta, 13 (1975) 471. P. Holba, Silikaty, 21 (1977) 19 (in Czech.). P. Holba, M. NevFiva and J. &stBk, Thermochim. Acta, 23 (1977), 223. I=. Skv&ra, J. .%st5k and V. &stBkovB, in Thermal Analysis, 4th ICJYA, Akademiai Kiado, Budapest, 1975, Vol. 1, p. 105. J. %stBk, Thermophysical Measurements of Solid-State Properties, Academia, Prague, 1980 (in course of preparation) (in Czech.).

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15 16

17 18 19

20 21 22 23 24

25

26

27 28 29

30

31 32 33 34 35 36 37 38 39

40 41 42

43 44 45 46 47 48

49 50

C. Truesdel, Rational Thermodynamics, McGraw-Hill, New York, 1969. G. Astarita, Itltrodaction to Non-Linear Contimutm Thermodynamics, Societa Editrice di Chimica, Milano, 1975. I. Samoh and P. Hrma, Rational Thermodynamics, Academia, Prague, 1980 (in Czech.). J. Kratochvil, Czech. J. Phys., A23 (1973) 1 (in Czech.). A. Sanfeld, Irreversible Processes, in the series Advanced Treutcs of Physical Chemistry, Academic Press, New York, 1971, Vol. 1. p. 217. H. B. Cullen, Thermodynamics, Willey, New York, 1960. J. Kvasnica, Thermodynamics, SNTL, Prague, 1965 (in Czech.). B. Biezina and P. Glogar, Ferroelectrics, Academia, Prague, 1973 (in Czech.). W. Kiinziga, Ferroelectrics in Solid-State Physics, New York 1957, Vol. 4. J. Loos, seminary lecture Critical phenomena and thermodynamics of magnetics, preprints by Inst. Solid Stare Phys., Prague, 1975 (in Czech.). P. Holba, seminary lecture Phase transitions and theory of chemical phase, preprints by Inst. Solid Srate Phys.. Prague, 1975 (in Czech.). H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Clarendon Press, Oxford, 1971. B. N. Rolov, Di@se Phase Transformations, Zinatne U.S.S.R., Riga, 1972 (in Russian). P. Holba and J. Sestlk, Z. Phys. Chem., Neae Folge, 80 (1971) 1. P. Holba and J. %-stBk, in Termanal ‘73, Proc. 6th Czech. CoujI TA in High Tart-as, SV&I’ Bratislava, 1973, p_ Pl (in Czech.). M. Nevfiva, P. Holba and J. Se&k, in Thermal Analysis, 4th IOTA, Akademiai Kiado, Budapest, 1975, Vol. 3, p. 981. M. Nevziva, P. Holba and J. SestBk, Sifikary, 20 (1976) 33 (in Czech.). J. Se&k, P. Holba and R. Barta, Silikaty, 20 (1976) 83 (in Czech.). J. Se&k and J. Kratochvil, J. Therm. Anal., 5 (1973) 193. J. Kratochvil and J. SestBk, Thermochim. Acla, 7 (1973) 330. J. &st5k, Thermochim. Acta, 3 (1971) 150. J. Sest&k and G. Berggren, Arbetsrapport Srrtdsvik AB, AE-BP-275, Sweden, 1977. J. SestBk, Ana. Chim. (Rome), 67 (1977) 73. P. Holba, Thermochim. Acra, 3 (1972) 475. J. Sestrik, P. Holba and J. Kratochvil, in M. Pavlyuchenko and E. Prodana (Eds.), Hetero- geneous Chemical Reacrions and Reacting Capability, Nauka i technika, Minsk, U.S.S.R., 1975, p. 57 (in Russian). P. Holba, in Tizermai Analysis, 4th TCTA, Akademiai Kiado, Budapest, 1975, Vol. 1, p. 33. R. A. Swalin, Thermodynamics of Solids, Wiley, New York, 1962. B. J. Wood and D. G. Fraser, Elementary Thermodynamics for Geologists, Oxford University Press, London, 1976. L. H. Adams and L. H. Cohen: Am. J. Sci., 264 (1966) 543. I. Proks, Silikaty, 5 (1961) 115 (in Czech)_ I. Proks, Silikaty, 14 (1970) 287. J. Sestrik, Silikaty, 5 (1961) 68 (in Czech). J. Sestrik, Philosophy of non-isothermal kinetics, in course of preparation. G. Kluge, in J. Sestak (Ed.), Proc. Treatments of TA cwves by Compztters, DT CVTS, Prague, 1978, p. 170. G. Kluge and K. Heide, Thermochim. Acra, 21 (1977) 423. A. G. Merzhanov, V. V. Barzykin, A. S. Shteinberg and V. T. Gontkovskaya, Thermochim. Acta, 21 (1977) 301.

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Journal of Thermal Analysis, Vol. 32 (1987) 1645-1650

T H E R M O D Y N A M I C S AND T H E R M O C H E M I S T R Y O F K I N E T I C ( R E A L ) P H A S E D I A G R A M S I N V O L V I N G S O L I D S

J. ~estdk and Z. Chvoj

INSTITUTE OF PHYSICS, CZECHOSLOVAK ACADEMY OF SCIENCES, NA SLOVANCE 2, 180 40 PRAGUE 8, CZECHOSLOVAKIA

A theoretical analysis of the formation of materials with metastable microstructures under non-ideal and highly non-equilibrium conditions is presented.

The most important processes in material engineering include crystal growth or dissolution from multicomponent (usually high-temperature) solutions. Selection of the most suitable conditions for obtaining a controlled type of crystalline (or even non-crystalline) materials requires a good knowledge of the heterogeneous equilibria between the solid and liquid phases, conveniently collected graphically in the form of phase diagrams. The most widespread are those diagrams describing the equilibrium dependence of the composition upon the temperature, but this is not easy to achieve experimentally [1]. In technology we can practically control or present (or often leave undefined) many parameters as follows:

- - Chemical and physical properties of the sample (composition, compactness, homogeneity, impurity, diffusivity, viscosity or even prenuclei sites or mechanical tensions).

- - External forces (temperature, pressure, partial pressure, or electromagnetic, hydrostatic or gravitational force fields) and their changes (heating or cooling, convection, atmospheric composition and hydrodynamics and associated flows).

- - Sample geometry, including the factors affecting the heat exchange, sink and generation, the extent of volatile component exchange; the interactions between interfaces and/or with the surroundings or with the measuring head.

Furthermore, the classical approach to the study 'of phase equilibria by long annealing at increased temperature is generally found to be tedious and experimentally demanding. Therefore, dynamic techniques are often preferred as more convenient, in which the tempered sample is investigated in the freeze--in state after a suitable rapid quenching and/or during the entire programmed cooling. A detailed corisideration of all the peculiarities involved in such a non-equilibrium procedure is definitely required. It can be verified that for fast enough processes,

John Wiley & Sons, Limited, Chichester Akad~miai Kiadr, Budapest

1646 gESTAK, CHVOJ: THERMODYNAMICS AND THERMOCHEMISTRY

occurring, for instance, in metals and alloys, near-equilibrium conditions can be assumed even at the relatively high cooling rate of 102 degree s- a, whereas for the very slow diffusional or viscous processes in silicate systems the cooling rate m~ast be as low as 10 4 degree s -a, and to avoid undercooling (or even limiting glass- formation) equilibration of the order of weeks may be required.

To look for materials with desired (specific) properties, we certainly have to use phase diagrams which correspond to non-equilibrium conditions of their preparation, which provide us with the necessary information of possible thermodynamic stability, metastability andeven instability under the assumed standard (equilibrium) conditions. There is increased interest in the mathematical analysis and synthesis of such "kinetic" phase diagrams. Besides the classical requirements if the minima and equality of the Gibbs energies of the individual components of the coexisting stable phases, we have to assume the same for the metastable phases occurring at higher energy levels (and different compositions) than for the stable phases, the latter not being achieved due to the kinetic hindrance of new phase formation (nucleation). For the sake of practical use, metastable boundaries can be predicted by a simple extrapolation of the coexistence lines for the stable states into non-equilibrium regions, usually down to lower temperatures. Alternatively, the preliminary shapes of metastable lines can be estimated from a superposition of two corresponding (e.g. simple eutectic) phase diagrams [2]. This is of considerable importance for all dynamic methods (from thermal analysis to sample quenching) for the correct interpretation of the phases observed~

For a system which cannot follow the experimentally enforced (strong) changes, even by establishing the previously discussed metastable equilibria, the boundary lines shift freely along both the concentration and temperature axes, forming thereby the regions of unstable phases to be described in terms of the kinetics of the physico-chemical processes. Such a truly kinetic phase diagram is fully dependent upon the experimental conditions applied (cooling rate, sample geometry and measuring conditions) and can be best treated mathematically in the case of a stationary process conditioning. Paper [6] summarizes the results attained in this field by using the soRttion of the kinetic equations (Focker-Planck equation) or the Monte-Carlo method. The new method for the description of kinetic phase diagrams based on the stochastic process theory [7, 10] enables us to describe non- stationary processes too. These include such phenomena as "coring" and "surrounding" (see (1)), known long ago in metallurgical engineering, explaining the concentration-dependences across the grains precipitated in the vicinity of hypoeutectic points, respectively.

The kinetic phase diagrams can be used for the description of the processes connected with the preparation of metallic glasses (103-107 degree s- 1) or with the photon, electron or ion beam processing of the surfaces of many materials. The


~ESTAK, CHVOJ: THERMODYNAMICS AND THERMOCHEMISTRY 1647

cooling rates reached during, for example the laser glassing of metals, can be as fast as of the order of 10 l~ degree s- 1, yielding associated processes of phase formation that are very highly non-equilibrium.

In the case of the non-stationary processes, the composition of the solid phase changes with time as well as the local cooling rate and degree of undercooling at the phase interfaces. Mathematical treatment is then extremely difficult, requiring a joint solution of the equations for heat and mass transfer under given boundary conditions, and the kinetic equations describing the kinetics of the phase transformation on the solidification front. Evaluations under the stochastic theory yield interesting dependences between the undercooling, concentration, linear growth rate and cooling rate (see [5, 8, 10]). The results for Cu--Ni alloy and similar systems are demonstrated in Table 1. For instance, if the cooling rate R increases, the difference in the concentrations of the solid and the liquid phase decreases and tends to zero. This result can simulate the creation of the amorphous phase during rapid cooling.

Not less important in the theoretical treatment is the thermochemistry of mixtures and solid solutions. For example, with regular solutions ( A H ex ~> TASeX),

interactions decisive for the system behaviour take place between spherical species

Table 1 Qualitative dependences of the growth parameters on the kinetic properties of solidification

ange of lues

Cooling rate, R (10 s -,106 Ks-l )

Diffusion coefficient, D (i0-7 __, 10-3 m 2 S -l)

Kinetic coefficient,* k (10o ~10 ' - s- l )

Concentration Temperature Concentration

difference on Growth rate on the Undercooling, of the

the solidifi- of solid phase solidification d T liquid phase,

cation front, formation, G front, 7', Cli q

C . q - Col

intensive increases decreases decreases nonlinear decreases increases

decreases increases increases decreases increases

increases to decreases equilibrium to zero

value

decreases increases increases

* i.e. rate constant of the front solidification including thermal vibration of molecules, probability of molecules incorporation into solid phase dependent on temperature, interface energy barrier and the difference of Gibbs energy of both phases.


1648 ~EST,~K, CHVOJ: THERMODYNAMICS AND THERMOCHEM1STRY

Table 2 Gibbs energy AGm~x (= A H i~ + T A S ~ + A H e~ + T A S ex) for binary mixture (A + B) assuming different models of the system nonideality

Name zl H e~ AS ex

ideal 0 0 athermal xA In (x ~ + Qnx n) + XB In (XB + Q ~X A)

regular QR 0 subregular Q R " Qs 0

pseudoregular Q R / Qs 0

quaziregular Q R " Q r }

quazisubregular QR " QK " Qs ~ x AxaOr

q uazipseudosubregular QR'QK/Qs

Where T, AH, AS, .(2 and x are temperature, enthalpy and entropy change, interaction parameter and mole fraction, respectively, for ideal (id) and excess (ex) quantities. Read A H id = O, A S id = x A. In xA + x n l n x n , Qs = OXAX~, Qs = (1 +OsxB), and Qr = ( 1 - t ~ r T ).

of(molecular) mixtures of metals (see Table 2), whereas for the decisive effect of the mutual arrangement of usually geometrically complicated species of polymers and silicates we have to take up athermal solutions ( A H ex ~ TASeX). A specific case of ionic (Temkin) solutions requires the incorporation of energetically unequal sites in the solid-state network into the above concept of regular solutions. This is usually true for oxides where, for example, energetically unequal (disparate) tetrahedral (i) and octahedral (j) sites are occupied on different levels of the species A and B, so that the ordinary term x A In x A must be read as the term ~ x m j In xmj , etc.

For the sake of practical applicability, a graphical form of the above effects is shown for the case of a hypothetical phase diagram of a binary mixture of A and B. Under the conditions of ultrafast cooling, the phase boundary is depressed to lower temperatures (see Fig. lb), to be observed experimentally later in a position not associated.with that normally called the equilibrium phase boundary (Fig. la). If the chemical phases of the system (in Fig. la) change in such a way that the mixing of components A and B is accompanied by a large enthalpy'change due to strong interactions between A and B solutions (see Table 2), a eutectic point [1] emerges. For still intensive interactions, a more complicated form may occur, exhibiting peritectic compound C. Such a type of phase diagram (see Fig. lc) is illustrative to show possible consequences of phase metastability represented by dashed lines. If the solidified compositions of this system are reheated, phase boundaries corresponding to metastable phases are often observed, making correct interpretation of the experimental data difficult; this is particularly true for thermoanalytical measurements [9] (e.g. DTA).


Fig. 1

~ESTA.K, CHVOJ: THERMODYNAMICS AND THERMOCHEMISTRY 1649

A C B

•

Possible shifts and extrapolations in a hypothetical phase diagram of binary mixture A and B. a near-ideal behaviour of solid-liquid curves with solid-solid phase separation at bottom; b - shift of original (solid) phase boundaries to nonequilibrium (dashed) ones due to the effect of ultrafast cooling (cf. Table l )exhibi t ing the consequences of preparation technology; c - possible change of the structure of phase boundaries when accounting chemical effect of strong interactions (cf. Table 2) between two components A and B to exhibit eutectic and peritectic points and incongruently melting compound C. It results from the system nonideality; d inclusion of metastable phase boundaries (dashed lines) by extrapolation of equilibrium (solid) lines. It shows all possible stable and metastable states available at the system to occur as combined effect of system nonideality and experimental conditions of its treatment; e - possible experimental result while studying such a system (d) by thermal analysis upon heating. By fitting hypothetical experimental points (dots) the lines can assume a form of two separated, eutectic- type diagrams of the A component with B superposed to that of A with C. It is worth noting that the both experimentally detected horizontal lines must not necessarily represent stable phase boundary being often a source of false interpretation. Whatmore any of metastable boundary lines may appear as a potential source of an effect detectable by a given thermoanalytical technique in a more or less pronounced (or negligible) form depending on the sample history

References

1 J. ~estfik, Thermophysical Properties of Solids; Their Measurement and Theoretical Thermal Analysis, Elsevier, Amsterdam 1984.

2 M. Nevfiva and J. ~estk, k, Thermodynamic Approach to Study Solid-Liquid Phase Equili- bria in Uninvestigated Oxide Systems; in Thermal Analysis (Z. D. ~ivkovic, ed.), Collection of Papers of Technical University, Bor 1984 (Yugoslavia), p. 3.

3 L. H. Bennet, T. B. Massalski and B. C.

Giassen (eds), Alloy Phase Diagrams, North Holland, Amsterdam 1983. Moffat et al. (eds), The Structure and Prop- erties of Materials, Vol. I., J. Wiley and Sons Inc., New York 1967. J. gest~ik, Z. Strnad and A. T~'iska (eds), Modern Crystalline and Non-crystalline Ma- terials and Their Technologies, Academia, Prague 1988; Elsevier, Amsterdam 1988 (in print).


1650 ~ESTAK, CHVOJ: THERMODYNAMIC S AND THERMOCHEMISTRY

6 T. A. Cherepanova, J. Crystal Growth, 52 (1981) 319.

7 Z. Chvoj, Cryst. Res. Technol., 21 (1986) 1003.

8 Z. Chvoj, Czech. J. Phys., 1987 (in print).

9 M. Nevfiva and J. ~estfik, Thermochim. Acta 92 (1985) 623.

t0 Z. Chvoj, A. Tfiska and J. ~estak, Kinetic Phase Diagrams (in course of preparation) Academia, Prague 1989; Elsevier, Amsterdam 1989.

Zusammenfassung - - Es wird eine theoretische Untersuchung der Bildung von Stoffen mit metastabilen Mikrostrukturen unter nichtidealen Bedingungen weitab vom Gleichgewicht dargelegt.

Pe31oMe - - l-lpe~CTaB.rleH TeOpeTl, lqecgHfi aHaYlH3 o6pa30Banxa BelIIeCTB C MeTaCTa6H.rlbHblMit MngpOCTpyKTypaMrl a HeH,/IeaJlbHblX I.I CH.qbHO HepaBHOBeCHblX yCJIOBtt~IX.

T~ermal Anal. 32, 1987

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