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Workshop on thermodynamic models and data for pure elements and other endmembers of solutions
Calphad Vol. 19, NO. 4, pp. 449-480, 1995 Copyright (0 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0384-5918195 $9.50 + 0.00 PII SO364-5916(96)
00003-X
Worlkshop on
THERMODYNAMIC MODELS AND DATA FOR PURE ELEMENTS AND OTHER ENDMEMBERS OF SOLUTIONS Schlol3 Ringberg, Febr. 21, to March 3, 1995
GROUP 2: Extrapolation Of The Heat Capacity In Liquid And Amorphous Phases
Group members: John &g-en Dept. Mater. Sci. & Eng., Royal Institute of Technology,
Stockholm, Sweden
Bertrand Cheynet Thermodata, Grenoble, France
Maria Teresa Clavaguera-Mora Dept. Fisica Universitet de Barcelona, Barcelona, Spain
Klaus Hack GlT Technologies, Herzogenrath, Germany
Lab. de Thermodynamique,
Jean Hertz Univ. Henri Poincan5, Nancy, France
Ferdinand Sommer Inst. Wekstoffwissenschaft,
Max-Planck Institut fur Eisenforschung,
Stuttgart, Germany
Ursula Kattner Metallurgy division, National Institute of Standards and Techn., Gaithersburg,
USA
Abstract Various methods of extrapolating the thermodynamic properties of the liquid will be discussed. A phanomenological method based on polynomials as well as a more physical method based on an extended two-state Schottky formalism will be presented and their application to some substances will be demonstrated.
449
J. AGREN
et al.
1. Introduction In many practical applications it is essential to know the thermodynamic properties of a liquid below its stable solidification temperature, e.g. when analysing crystallisation of highly undercooled liquids during rapid solidification processing or when calculating the phase equilibria between a high and a low-melting substance by means of the CALPHAD method. In the present report some aspects of the representation of thermodynamic data for the liquid will be discussed. 2. Experimental information All calculations of phase equilibria are based experimental information and in particular on heat capacity data. These data sets are a&able for most substances in their stable solid state (s), e.g. 91Din, 76Bar, 85Cha. However, the heat capacity of the liquid state has usually been measured only in a limited temperature range above the melting temperature T, /91Din, 765ar, 85Cha/. The c,” of high melting metals withl’, > 9OOK exhibit constant cf values because the temperature range is too narrow to evaluate any variation of c,” with temperature. hand, the results for several low melting-point
On the other
metals show a temperature dependent
above T,. It has been further observed that upon supercooling
cb values ci increases. e.g. as for In, Sn
and Bi l84Perl or Te l84Tsul. For other substances like Se or SiO, cf is constant or increases as the temperatures is increased from below T, to above T,, /78Ste, 76Ang/. For several inorganic phases such as Al,O, (corundum) or SiO, (tridymite) c; data above T, may be obtained for temperature ranges covering more than 1000 K aboveT,. Such data may be obtained by means of high-pressure experiments 193Saxl. The data extrapolated to atmospheric pressure show a monotonous increase of ci from the stable solid state into the metastable superheated regime /93Sax/. In addition, other pure substances and solution phases exhibit a monotonous increase of the experimental cf and ci as the melting point is passed. One may thus conclude that it seems as a general behaviour, inherit in the liquid state, that the entropy decreases upon cooling and that the heat capacity increases upon heating if the temperature is sufficiently high.
3. Problems arising from extrapolation 3.1 Common extrapolation formulae Using a simple mathematical representation of the experimental information discussed in the previous section should enable the reader to extrapolate the heat capacity of the solid phase above the melting point and that of the liquid below the melting point. The simplest method is to fit the experimental data with a polynomial of T such as the well known Meyer-Kelley expression: cp =a+bT+cT-‘+dT’
3.1
Of course, such an equation can be used outside the temperature range where the phase is thermodynamically stable. This method is quite acceptable in many cases when the extrapolation does not extend too far. However, when applying it to practical cases two problems may occur, namely artificial phase stability and the so-called Kauzmann paradox. We shall now turn to discuss these phenomena.
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
451
3.2 Artificial phase stabiliv When. the thermodynamic properties are extrapolated beyond the stability range of liquid and crystalline solid one may found that the solid phase may become stable at temperatures far above the melting point while in turn the liquid phase may become stable at low temperatures below the melting point. This situation is depicted in a schematic way in Fig. 1. Some examples are the occurrence of solid Al very close to the boiling point and the occurrence of liquid W near room temperature. 3.3 The Kauzmann paradox and inverse Kauzmann paradox Kauzmann 148Kaul has pointed out that if the supercooled liquid could be cooled to very low temperatures and crystallisation is avoided a paradox would occur because the entropy of the undercooled liquid would become smaller than that of the stable crystalline state. However, in real systems a glass transition occurs at temperatures above the Kauzmann temperature and the paradox thus is avoided because the glass has almost the same entropy as the solid crystalline phase. A similar paradox may occur when considering a superheated crystal. When extrapolating the thermodynamic functions towards high temperatures one may find that the entropy of such a crystal can be larger than that of the stable liquid phase above the “inverse” Kauzmann temperature /88Fec/. A typical situation of the two isentropic temperatures both below and above the melting point is given in Fig.2. Of course one may wonder if there is any physical significance to these problems. However, it is easy to conclude that they are just a consequence of unsuitable extrapolation procedures and could be avoided if the extrapolations are performed differently. 3.4 The SGTE extrapolation In order to avoid the above problems SGTE /91Din/ et al. have suggested
to extrapolate
c,”
and ci respectively such that the difference between these goes to zero approximately at OX,, and l.ST, respectively. This method ensures that the Gibbs energy curves intersect only once, i.e. at the melting point, see Fig. 3. In addition, the Kauzmann paradox also disappears. However, the resulting behaviour of the cp function for Cu is shown in Fig. 4. It is obvious that the curves do not describe correctly the cP for undercooled liquid and superheated solid, see section 2. It should also be pointed out that this method results in unphysical cP curves of the reference state of solution phases if the melting points of the components differ strongly. 4. Options for the description of the metastable liquid and solid state 4.1 Extrapolation based on Debye function for the heat capacity A possible description for c,, has been suggested by Hoch /69Hoc, 76Hoci who introduced a series expansion of the Debye function and two additional physically based terms covering explicit electronic (a) and anharmonic (b) contributions assuming the same Debye temperature for solid and liquid. The respective equations for solid and liquid are :
ci = f(BIT)+aT+bT’
4.1
cf =f(fl/T)+aT+bT-*
4.2
f(~~T)a3R[1-(9/T)*/20+(9/T)‘/560]
4.3
with
J. AGREN er al.
452
The “anharmonic” term has been chosen to fit the experimental information. The above two equations have been used to describe the experimental information for Sn and Cu /95Som/. Sn is one of the above mentioned substances for which experimentalcj data are available both in the stable and undercooled regime. Cu belongs to the group of substances were the experimental information suggests constant cf (31.38 Jlmo1.K). Table 1 shows the coefficients used in the above equations for c6 and c,“. Table 1: Coeffirients in expansion of Debye function for Sn and Cu
Element Sn Sn CU
Phase BCT Liq
0 (K) 200 200
a b (1 lo-’ J mol-’ K-*2) (110” J mol-’ Kd) 7.3826 1.81520 2.6855 9.12957 2.9132
0.1247
Using the data in Table 1 together with the values for OS,,, and T, as well as AH,,,in, for Sn and Cu from SGTE /91Din/ the entropy curves for the solid and liquid state have been calculated in a temperature range covering both the Kauzmann and inverse Kauzmann temperatures (isentropic temperatures). The results are shown in Figs.5 and 6 respectively. It can be reasonably well assumed that at the Kauzmann temperature the undercooled liquid transforms into an “ideal” glassy state. The entropy of the “ideal” glass follows the entropy of the solid crystal for all temperatures below the Kauzmann temperature. For the superheated crystal a similar argument holds for its transformation into the liquid phase at the “inverse“ Kauzmann temperature. The entropy of the hypothetical solid phase follows therefore the entropy of the stable liquid above the “inverse” Kauzmann temperature. In consequence the heat capacity curve of the supercooled liquid is only defined down to the Kauzmann temperature and that of the superheated crystal up to the “inverse” Kauzmann temperature. In order to be able to extrapolate below and above these temperatures a discontinuous change of their heat capacities occurs. For Sn and Cu this behaviour is shown in Figs.7 and 8. 5. Amorphous soIldlcation by two-states model 5.1 Development of model The aim of the present section is to develop a phenomenological model capable of representing the thermodynamic properties of the liquid in a physically sound way. The model should allow reasonable predictions, extended to low temperatures as well as high temperatures. The present approach will be based on a model suggested by Agren /88Agr/. As temperature is decreased the thermodynamic properties of the liquid approach gradually those of a crystalline solid. As already mentioned, there is a gradual decrease in entropy, i.e. the liquid undergoes a gradual ordering transformation. This ordering would probably have different origins in different classes of liquids but we shall call it amorphous solidification, since the liquid becomes more solid like. It should be emphasised though, that since only pure substances are considered here we would have no contribution from chemical ordering. On the other hand pure substances may exhibit polymerisation, as is the case, e.g. with boron and sul-
453
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
phur. However, discussed here.
models for polymerisation
are already well established and will not be further
In a simple liquid substance, like a pure metal, one may give the following physical picture of the amorphous solidification. In the crystalline state the atoms or molecules only have vibrational degrees of freedom whereas in the liquid state one may imagine that a large fraction of the atoms or molecules have translational degrees of freedom. The difference in entropy between a state where all the atoms are free to move all over the system and a state where each atom. can only move inside its cell is R, i.e. the gas constant. This is sometimes referred to as the communal entropy /54Hir/. It so:ms reasonable to represent the gradual loss in translational degrees of freedom with a classical two-state model. In this model it is suggested that the atoms could be either in a solidlike state or in a liquid-like state. For simple liquids one may imagine that the form?r state is connected with vibrational motion whereas the latter with translational motion. An internal ordering parameter 5 is introduced as the fraction of atoms being in the liquid-like state. For the molar Gibbs energy one may thus write G,” =(1-c)
“G~‘+5”G~+RT{~ln~+(l-t)ln(l-5))
5.1
where “Gf’ and “G? are the molar Gibbs energy if all the atoms were in the solid-like state and liquid-like state respectively. The equilibrium value of 5 at a given temperature is the value that minimises the above expression. exp(-AC,
5= l+exp(-AC,
I RT)
5.2
I RT)
where AGd=uG~-oG~‘.
5.3
Inserting the value of 5 in the equation for Gk one finds G,” =“Gz’ - RT ln[ I+ exp(-AC,
/ RT)]
5.4
It is c:onvenient to define an excess Gibbs energy as ~,E=~,L-OG~I=~AG~+RT(~I~~+(~-~)~(A-~)}
5.5
For a liquid in internal equilibrium one thus have G,” =-RTln[l+exp(-AG,/RT)]
5.6
If AC;, is temperature the above model is identical to the classical Schottky two-level model. Sometimes a degeneracy is introduced into the Schottky model and this is formally equivalent with having a linear temperature dependency of AC,, provided that the temperature dependency is chosen in such a way that exp(AS / R) is an integer. In general we would expect AGd to have any temperature dependence. In 1ac.kof experimental
information one may take as a first approximation
J. AGREN et al.
454
where A and B are adjustable parameters. Assuming that the ideal amorphous state obeys the third law one would have to accept B=O. In any case one would expect B to be small and may therefore take
If experimental information is available for both the undercooled and stable liquid one may simply write AGd =~+bT+cTlnT+dT~+...
5.9
where a,& and d are adjustable parameters to fit the experimental information. However, in most cases, except for low-melting substances, not much information is available. For example, if the Gibbs energy of the crystalline phase is known, which may often be the case, but only the melting point and the heat of melting are known for the liquid only two parameters can be evaluated for the liquid. It is reasonable to assume that b would be close to R, the communal entropy. Setting c and d to zero it may thus be possible to evaluate A and a. 5.2 General behaviour of two-statemodel We shall now investigate the general behaviour of the model. The results of a series of calculations are shown in Fig. 9 where the excess heat capacity normal&d to R has been plotted as a function of RTla. The excess heat capacity is defined as the derivative -Ta=G,E/aT’
5.10
where G,” is given by eq 5.6. All the curves are calculated with a/R=1000 K and c=O. Only b and dare varied. As can be seen, the peak will become higher as the magnitude of b increases (b < 0). By introducing also a finite negative d value there will be a continuous increase in excess heat capacity with temperature, approaching asymptotically a linear behaviour. 5.3. Example of an application where heat capacity data are available - Liquid Sn The heat capacity of liquid Sn has been measured almost 200 K below the melting point (505 K) and up to 1000 K /84Per/. Sn is thus chosen in order to test if the model is capable of representing the thermodynamic properties of a substance that is well characterised. The first step in the evaluation is to chose a suitable “Gz’. In order to test if one may apply eq. 5.8 one may plot the heat capacity of pure solid white Sn according to the SGTE database, see Fig. 10. It should be emphasised that the expression valid below the melting point has simply been applied also above the melting point, i.e. the so-called SGTE extrapolation has not been used. As can be seen, the heat capacity first increases and reaches a maximum at around 1000 K. This behaviour does not seem suitable as a reference for”Gz’ because the experimental information indicates that the liquid heat capacity approaches a constant value or even increases above 800 K. Thus we cannot use eq. 5.8. Instead one may apply a description of the ideal amorphous phase based on the Einstein model. For the 8 temperature we take 155 K as the Einstein temperature of white Sn. We apply the following polynomial approximation of the F(WT) F(WT)~33RT[ln(01T)+(BIT)2/24]
5.11
HEAT CAPACtTY
IN LtQUtD AND AMORPHOUS
455
PHASES
Thus~the following expression was chosen “G;‘=A+BT-3RTlnT+(3RB2/24)T-’
where A and B zue to be adjusted to frt the experimental AG, =a+bT+dT’
5.12
data. For AC, we choose 5.13
The 5 coefficients A, B. a, b and d were now adjusted to tit the following experimental mation: AHmclang= 7029 J mol-’ , T,c,,i,g= 505.1 K AlDin/ as well as the heat-capacity
inforvalues
from 84Per. The following values were obtained A = -2864.6+
HinER J mol-’
B=:112.537
J mol-’ K-’
a = 5662.3
J mol-’
b = -11.3298
J mol-’ K-’
d = --1.4328
1O-3
J mol-’ Kb2
where SER denotes the so called stable element reference, i.e. the element in its stable state at 298.15 K. The calculated heat capacity is shown as the solid line in Fig. 11 and the symbols are the experimental values. As can be seen the model yields an excellent representation of the experimental data. Fig. 12 shows the calculated entropy for the liquid (solid line) and for the stable crystalline phase (dashed line). The entropy of liquid and solid comes very close below 200 :K. Fig. 13. shows an extrapolation of the heat capacity to 3000 K. Finally Fig.14 shows the 5 at equilibrium as a function of temperature. It is interesting to notice that 5 is around 0.5 at the: melting point. 5.4 Example of an application where a constant heat capacity value is available - Liquid Cu For Cu the experimental information is not extensive enough to reveal any temperature dependence of the heat capacity but only a single value at the melting point is available. On the other hand it was found that the Gibbs energy of the stable FCC phase, given by the SGTE database, behaved reasonably well upon extrapolation to high temperatures and it was thus chosen to use 5.14 and for AG, we first choose AG, = AH,,,i,,p - RT
5.15
The optimisation of A and d was now performed adjusting to the following infonnationbJSm,,i~, = 13263 J rn~l-‘.T,,,~,, = 1358 K, ck(T,) = 31.4 J mol-’ K’ I9lDinl. The result is
J. AGREN et al.
456
A= 6681.5+ H;; d = 1.095 lo4
J mol-’ J mol-’ KJ
It was thus possible to fit the melting point as well as the enthalpy of melting almost exactly but the heat capacity came out somewhat less well, namely 32.4 rather than 3 1.4 J mol-’ K-’ , see Fig. 15. The entropy of the liquid and solid phases is shown in Fig. 16. As can be seen they come very close around 250 K. We may now try to improve the fit by taking instead AG, =a-RT
5.16
treating a as an adjustable parameter. We thus have three parameters which can be derived exactly from the 3 data points. The result is A = 7985
J mol -’
a = 10609.7
J mole’
d = 1.0135 lo4
J mole’ K”
It is encouraging to see that the values of the parameters do not change very much compared to when a = AHmehh8.
6. Extensions into multicomponent systems For multicomponent solutions the reference state is the mechanical mixture of the pure components in the same state as the solution. The reference temperature of the databases used at present is 298.15 K. As a consequence, the heat capacity data for both the stable crystalline state and the undercooled liquid are only given for temperatures above 298.15 K. Based on the approach given in paragraph 4.1 it is only possible to define the reference of the liquid above the highest Kauzmann temperature of the components while the reference state of the solid can only be defined below the lowest “inverse” Kauzrnann temperature of the components. The Kauzmann temperatures are generally of the order of 0.3T, and the “inverse” Kauzmann temperatures of the order of 3T, /92Hoc/. Calculations of phase equilibria using this approach are therefore possible only in a limited temperature range. For a binary alloy with the melting temperatures T,” and T,” of the components, with T,” c Tf, the solid phase can be described in the temperature range 298.15 K I T I3T/ and the liquid phase above 0.3Ti or 298.15 K. Fig.17 shows the calculated heat capacity of solid equimolar Cu-Sn alloy up to 4000 K using the extrapolation based on isentropic temperature approach. It can be directly seen that the extrapolation beyond 3T: is artificial. The SGTE extrapolation given in paragraph 3.4 has also its limitations. The Figs. 18 and 19 show the heat capacity of the solid FCC-Al phase for Cu and Sn. the heat capacity of the reference state for one mol of the FCC-Al Cu,,Sn,, phase in Fig. 20 obtained from the values shown in Figs. 18 and 19 exhibit also artificial values below about 700 K. Both approaches to describe the metastable liquid and solid state can be used therefore only in a limited temperature range.
HEAT CAPACITY
IN LIQUID AND AMORPHOUS
PHASES
On the other hand, when using the approach based on the two-state model it is natural to accept eq. 5.1 also for the alloy and expand ‘Gz’ and AGd as functions of composition in the usual regular solution formalism, i.e.
AG, = x,“AG,”+ x;AG; + x;x;AG;
6.2
Higher-order terms as well as temperature dependencies are easily included whenever experimental data indicate a more complex behaviour. Applied to the equimolar Cu-Sn alloy we would, as a first approximation, take the parameters evaluated in sections 5.3 and 5.4 and in lack more detailed data set AG,AB=0 The heat capacity calculated from the two-state model for the equimolar Cu-Sn melt is shown in Fig. 2 1. As can be seen, the curve has now a reasonable behaviour over the whole temperature range.
Ack.nowledgements
The authors wish to thank Prof. Bo Sundman for organising a wonderful workshop and one of the authors (JA) would like to thank Dr. Nigel Saunders for animated discussions. Finally we thank the sponsors of the “Ringberg Workshop” for their moral and financial support. References
48K.au WKauzmann, Chem.Rev. 43, 1948,219 54Hir J.O. Hirschfelder, CF. Curtiss and R.B. Bird, Molecular Theory of Liquids and Gases, John Wiley & Sons Inc, New York 1954, 274 69Hoc M.Hoch, High Temp. High Pres. 1, 1969,531 76Ang C.A.Angell, W.Sichina, Ann.N.Y.Acad.Sci., 279, 1976,53 76Bar LBarin, O.Knacke, O.Kubaschewski, Thermochemical Tables, Springer 76Hoc M.Hoch, T.Vemadakis, Ber. Bunsenges. 80, 1976,770 78Ste R.B.Stephens, J.Appl.Phys., 49, 1978.5855 84P’er J.H.Perepezko, J.S.Paik, J.Non-Cryst.Solids, 61, 1984, 113 84Tsu Y.Tsuchiya, J.Phys.Cond.Mat., 3, 1984,125 85Cha M.V.Chase et al, JANAF Thermochemical Tables, Third Edition, J.Phys. Chem.Ref.Data, 14, suppl.1, 1985 88Fsec H.J.Fecht, W.L.Johnson, Nature 334, 1988,50 88A.gr J. Agren, Phys. Chem. Liq., 18, 1988, 123 91Din A.T.Dinsdale, CALPHAD 15,1991,317 92H:oc M.Hoch, Met.Trans.B 23B, 1992,309 93%~ S.K. Saxena, N. Chatterjee, Y. Fei and G. Shen: Thermodynamic data on oxides and silicates, Springer Verlag, Berlin,-Heidelberg-New York 1993, p 428 95Som F.Sommer, private corn.
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Figure captions: Fig. 1: Schematic phases. Solid part extrapolation. The ing artificial phase
diagram of Gibbs energy as a function of temperature for liquid and solid of curves represent experimental data whereas dashed parts are obtained by curves intersect at the normal melting point and at two temperatures yieldstability.
Fig. 2: The entropy of liquid and crystalline Al as a function of temperature in the stable and metastable region. AS, denotes the entropy of fusion. Reproduced from /88Fec/. Fig. 3: The Gibbs energy of liquid and crystalline Cu as a function of temperature and metastable region calculated from the SGTE /91Din/ database.
in the stable
Fig. 4: The heat capacity of liquid and crystalline Cu as a function of temperature and metastable region calculated from the SGTE /91Din/ database.
in the stable
Fig. 5: The entropy of liquid and crystalline Sn as a function of temperature metastable region calculated using the coefficients in Table 1.
in the stable and
Fig. 6: The entropy of liquid and crystalline Cu as a function of temperature metastable region calculated using the coefficients in Table 1.
in the stable and
Fig. 7: The heat capacity of liquid and crystalline Sn as a function of temperature in the stable and metastable region calculated using the coefficients in Table 1 and taking AC, = 0 below the Kauzmann temperature and above the inverse Kauzmann temperature. Fig. 8: The heat capacity of liquid and crystalline Cu as a function of temperature in the stable and metastable region calculated using the coefficients in Table 1 and taking AC, = 0 below the Kauzmann temperature and above the inverse Kauzmann temperature. Fig. 9: The excess heat capacity of a liquid as function of temperature the coefficients in Eq. 5.9.
for different values of
Fig. 10: The heat capacity of white Sn as a function of temperature in the stable and metastable region calculated from the SGTE data without the SGTE extrapolation above the melting point. Fig. 11: The heat capacity of liquid Sn as a function of temperature in the stable and metastable region calculated using the two-state model. Symbols denote experimental values /84Per/ Fig. 12: The entropy of liquid Sn as a function of temperature in the stable and metastable region calculated using the two-state model, solid line. The entropy of solid white Sn (BCT) is given by the dashed line. Fig. 13: The heat capacity of liquid Sn as a function of temperature in the stable and metastable region calculated using the two-state model. Extrapolation to high temperatures. Fig. 14: Fraction of liquid-like atoms as a function of temperature for Sn.
HEAT CAPACITY IN LIQUID AND AMORPHOUS
459
PHASES
Fig. 15: The heat capacity of liquid Cu as a function of temperature in the stable and metastable region calculated using the two-state model. The symbol denotes the experimental at the normal melting point. Fig. 16: The entropy of liquid Cu as a function of temperature in the stable and metastable regicln calculated using the two-state model, solid line. The entropy of crystalline Cu (FCC) is given by the dashed line. Fig. 17: The heat capacity of a liquid CuO,,SnO,, alloy as a function of temperature in the stable and metastable region calculated using the approach based on isentropic temperatures. Note that there are two different isentropic temperatures. Fig. 18: The heat capacity of crystalline Cu as a function of temperature tastable region calculated from the SGTE /91Din/ database.
in the stable and me-
Fig. 19: The heat capacity of hypothetic crystalline FCC Sn as a function of temperature stable and metastable region calculated from the SGTE /91Din/ database.
in the
Fig. 20: The heat capacity of hypothetical metastable Cu,,Sn,, ahoy with FCC structure as a function of temperature calculated from the SGTE /91Din/ database. Fig. 121:The heat capacity of liquid Cu, Cu,,Sn,, alloy and Sn as functions of temperature the stable and metastable region calculated using the two-state model.
in
J. AGREN
460
Fig. 1:Schematic phases. Solid part extrapolation. The ing artificial phase
et at.
diagram of Gibbs energy as a function of temperature for liquid and solid of curves represent experimental data whereas dashed parts are obtained by curves intersect at the normal melting point and at two temperatures yieldstability.
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
60
40
L I 20 z E 7 W-J 0
-20
0
200 400 600 800 1000 1200 1400 T(K)
Fig. 2: The entropy of liquid and crystalline Al as a function of temperature in the stable and metastable region. AS, denotes the entropy of fusion. Reproduced from /88Fec/.
461
J. AGREN et a/.
462
2 0 -2 -4 -6 -8 -10 -12 E4 -14L 0
500
1000
1500
2000
T (K) Fig. 3: The Gibbs energy of liquid and crystalline Cu as a function of temperature in the stable and metastable region calculated from the SGTE /91Din/ database.
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
463
383634L
32 -
z: 30 , 2 a: LI ii
28 262422 -
I *OO
I
500
I
1000
I
1500
2000
T (Kl Fig.4: The
heat capacity of liquid and crystalline Cu as a function of temperature in the stable and metastablc region calculated from the SGTE /9 lDin/ database.
J. k3REN
464
I
I
I
et al.
I
I
I
f
1630
T (to Fig. 5: The entropy of liquid and crystalline Sn as a function of temperature in the stable and met&able region calculated using the coefficients in Table 1.
465
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
I
I
1
I
I
I
I
I
I
5( 30
T (K) Fig. 6: The entropy of liquid and crystalline Cu as a function of temperature in the stable and metastable region calculated using the coefficients in Table 1.
466
16 30
Fig. 7: The heat capacity of liquid and crystalline Sn as a function of temperature in the stable and metastable region calculated using the coefficients in Table 1 and taking AcP= 0 below the Kauzmann temperature and above the inverse Kauzmann temperature.
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
70
Fig. 8: The heat capacity of liquid and crystalline Cu as a function of temperature in the stable and mctastable region calculated using the coefficients in Table 1 and taking AC,,= 0 below the Kauzmann temperature and above the inverse Kauzmann temperature.
467
1
468
J. AGREN et a/.
RT/a Fig. 9: The excess heat capacity of a liquid as function of temperature for different values of the coefficients in Eq. 5.9.
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
469
50 45 40 35 30 25 20 15 IO sI
OO
I
I
200
400
I
600
I
800
I
1000
1:
-i-(K) Fig. 10: The heat capacity of white Sn as a function of temperature in the stable and metastable region calculated from the SGTE data without the SGTE extrapofntion above the melting poinl..
J.AGREN et a/.
470
50 45 40 35 30
;
p_@__
25 20 15 10 5 OO
I
200
I
400
I
600
I
800
I
1000 Ii10
Fig. 11: The heat capacity of liquid Sn as a function of temperature in the stable and metastable region calculnted using the two-state model. Symbols denote experimental values /84Per/
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
200 180 160 140 'i- 120 LL c $ 100 :
80
5; 60 40 20 0
.* 5’ :
I
200
1
400
I
600
I
800
I
1000 Ii30
T(K)
Fig. 12: The entropy of liquid Sn as a function of temperature in the stable and metastable rcgion c,alculatcd using the two-state inodcl, solid line. The entropy of solid white Sn (BCT) is given by the dashed line.
471
J. AGREN et al.
472
-i
5oL 45
kz 40 2 7
35
25:: I
20
IS10 51
OO
!
500
!
I
I
I
1000 1500 2000 2500 31 10
1 (K)
Fip. 13: The heat capacity of liquid Sn as a function of temperature in the stable and mctnstable region calculated using the two-state model. Extrapolation to high temperatures.
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
0
a”’
I
500
I
I
I
I
1000 1500 2000 2500 3000
TIKI !:ig. i3: Fraction of liqtlid-like xoms as 2 function of tempernture for Sn.
473
J.
474
50 h
c
40
7
35
7
et al.
r-
45
x
zE
kXEN
30 25 20 15 IO 5 t
0
500
1
1000
1
1500
2000
T (IO
Fig. 15: The heat capacity of liquid Cu as a function of temperature in the stable and metastable region calculated using the two-state model. The symbol denotes the experimental at the normal me1 ting point.
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
100 90 80 70 60 50 40 30
20 IO OL 0
500
1000
1500
2000
T WI
Fig. 16: The entropy of liquid Cu as a function of temperature in the stable and metastable region calculated using the two-state model, solid line. The entropy of crystalline Cu (FCC) is given by the dashed line.
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65
I
75
I
I
I
Ld
0
T 0
I
I
I
woo
rn,
Fig. 17: The heat capacity of a liquid CuO,,,Sn,,, altoy ns a function of temperature iu the st;lble and metastable region calculated usin g the approach based on isentropic temperatures. Note that there are two different isentropic temperatures.
r-I I I
HEAT CAPACITY I
I I
(J rnol-'K-') w iQ
J. AGREN et al.
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I
I
I
I
I
I
I
I
I
5000 T Ml Fig. 19:The heat capacity of hypothetic crystalline FCC Sn as a function of temperature stable and metastable region calculated from [IICSGTE /91Din/ database.
in the
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
25
0
I
1
I
I
I
I
I
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479
I
5( 30 T (K)
Pig. 20: The heat capacity of hypothetic mctastable Cu,&&, alloy with FCC structure as a function of temperature calculated from tilt: SGTE /91Din/ database.
J. AGREN et
480
al.
50 45 Sn
40 35 30/ 2s I-
20 15 IO 5 OO8.
200 I
400 1
600 I
800 I
1000 I
1200 1
T MI Fig. 21: The heat capacity of liquid Cu, Cu,,,Sn,,, alloy and Sn as functions of temperature in the stable and metastable region calculated using the two-state model.
11
00003-X
Worlkshop on
THERMODYNAMIC MODELS AND DATA FOR PURE ELEMENTS AND OTHER ENDMEMBERS OF SOLUTIONS Schlol3 Ringberg, Febr. 21, to March 3, 1995
GROUP 2: Extrapolation Of The Heat Capacity In Liquid And Amorphous Phases
Group members: John &g-en Dept. Mater. Sci. & Eng., Royal Institute of Technology,
Stockholm, Sweden
Bertrand Cheynet Thermodata, Grenoble, France
Maria Teresa Clavaguera-Mora Dept. Fisica Universitet de Barcelona, Barcelona, Spain
Klaus Hack GlT Technologies, Herzogenrath, Germany
Lab. de Thermodynamique,
Jean Hertz Univ. Henri Poincan5, Nancy, France
Ferdinand Sommer Inst. Wekstoffwissenschaft,
Max-Planck Institut fur Eisenforschung,
Stuttgart, Germany
Ursula Kattner Metallurgy division, National Institute of Standards and Techn., Gaithersburg,
USA
Abstract Various methods of extrapolating the thermodynamic properties of the liquid will be discussed. A phanomenological method based on polynomials as well as a more physical method based on an extended two-state Schottky formalism will be presented and their application to some substances will be demonstrated.
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1. Introduction In many practical applications it is essential to know the thermodynamic properties of a liquid below its stable solidification temperature, e.g. when analysing crystallisation of highly undercooled liquids during rapid solidification processing or when calculating the phase equilibria between a high and a low-melting substance by means of the CALPHAD method. In the present report some aspects of the representation of thermodynamic data for the liquid will be discussed. 2. Experimental information All calculations of phase equilibria are based experimental information and in particular on heat capacity data. These data sets are a&able for most substances in their stable solid state (s), e.g. 91Din, 76Bar, 85Cha. However, the heat capacity of the liquid state has usually been measured only in a limited temperature range above the melting temperature T, /91Din, 765ar, 85Cha/. The c,” of high melting metals withl’, > 9OOK exhibit constant cf values because the temperature range is too narrow to evaluate any variation of c,” with temperature. hand, the results for several low melting-point
On the other
metals show a temperature dependent
above T,. It has been further observed that upon supercooling
cb values ci increases. e.g. as for In, Sn
and Bi l84Perl or Te l84Tsul. For other substances like Se or SiO, cf is constant or increases as the temperatures is increased from below T, to above T,, /78Ste, 76Ang/. For several inorganic phases such as Al,O, (corundum) or SiO, (tridymite) c; data above T, may be obtained for temperature ranges covering more than 1000 K aboveT,. Such data may be obtained by means of high-pressure experiments 193Saxl. The data extrapolated to atmospheric pressure show a monotonous increase of ci from the stable solid state into the metastable superheated regime /93Sax/. In addition, other pure substances and solution phases exhibit a monotonous increase of the experimental cf and ci as the melting point is passed. One may thus conclude that it seems as a general behaviour, inherit in the liquid state, that the entropy decreases upon cooling and that the heat capacity increases upon heating if the temperature is sufficiently high.
3. Problems arising from extrapolation 3.1 Common extrapolation formulae Using a simple mathematical representation of the experimental information discussed in the previous section should enable the reader to extrapolate the heat capacity of the solid phase above the melting point and that of the liquid below the melting point. The simplest method is to fit the experimental data with a polynomial of T such as the well known Meyer-Kelley expression: cp =a+bT+cT-‘+dT’
3.1
Of course, such an equation can be used outside the temperature range where the phase is thermodynamically stable. This method is quite acceptable in many cases when the extrapolation does not extend too far. However, when applying it to practical cases two problems may occur, namely artificial phase stability and the so-called Kauzmann paradox. We shall now turn to discuss these phenomena.
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
451
3.2 Artificial phase stabiliv When. the thermodynamic properties are extrapolated beyond the stability range of liquid and crystalline solid one may found that the solid phase may become stable at temperatures far above the melting point while in turn the liquid phase may become stable at low temperatures below the melting point. This situation is depicted in a schematic way in Fig. 1. Some examples are the occurrence of solid Al very close to the boiling point and the occurrence of liquid W near room temperature. 3.3 The Kauzmann paradox and inverse Kauzmann paradox Kauzmann 148Kaul has pointed out that if the supercooled liquid could be cooled to very low temperatures and crystallisation is avoided a paradox would occur because the entropy of the undercooled liquid would become smaller than that of the stable crystalline state. However, in real systems a glass transition occurs at temperatures above the Kauzmann temperature and the paradox thus is avoided because the glass has almost the same entropy as the solid crystalline phase. A similar paradox may occur when considering a superheated crystal. When extrapolating the thermodynamic functions towards high temperatures one may find that the entropy of such a crystal can be larger than that of the stable liquid phase above the “inverse” Kauzmann temperature /88Fec/. A typical situation of the two isentropic temperatures both below and above the melting point is given in Fig.2. Of course one may wonder if there is any physical significance to these problems. However, it is easy to conclude that they are just a consequence of unsuitable extrapolation procedures and could be avoided if the extrapolations are performed differently. 3.4 The SGTE extrapolation In order to avoid the above problems SGTE /91Din/ et al. have suggested
to extrapolate
c,”
and ci respectively such that the difference between these goes to zero approximately at OX,, and l.ST, respectively. This method ensures that the Gibbs energy curves intersect only once, i.e. at the melting point, see Fig. 3. In addition, the Kauzmann paradox also disappears. However, the resulting behaviour of the cp function for Cu is shown in Fig. 4. It is obvious that the curves do not describe correctly the cP for undercooled liquid and superheated solid, see section 2. It should also be pointed out that this method results in unphysical cP curves of the reference state of solution phases if the melting points of the components differ strongly. 4. Options for the description of the metastable liquid and solid state 4.1 Extrapolation based on Debye function for the heat capacity A possible description for c,, has been suggested by Hoch /69Hoc, 76Hoci who introduced a series expansion of the Debye function and two additional physically based terms covering explicit electronic (a) and anharmonic (b) contributions assuming the same Debye temperature for solid and liquid. The respective equations for solid and liquid are :
ci = f(BIT)+aT+bT’
4.1
cf =f(fl/T)+aT+bT-*
4.2
f(~~T)a3R[1-(9/T)*/20+(9/T)‘/560]
4.3
with
J. AGREN er al.
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The “anharmonic” term has been chosen to fit the experimental information. The above two equations have been used to describe the experimental information for Sn and Cu /95Som/. Sn is one of the above mentioned substances for which experimentalcj data are available both in the stable and undercooled regime. Cu belongs to the group of substances were the experimental information suggests constant cf (31.38 Jlmo1.K). Table 1 shows the coefficients used in the above equations for c6 and c,“. Table 1: Coeffirients in expansion of Debye function for Sn and Cu
Element Sn Sn CU
Phase BCT Liq
0 (K) 200 200
a b (1 lo-’ J mol-’ K-*2) (110” J mol-’ Kd) 7.3826 1.81520 2.6855 9.12957 2.9132
0.1247
Using the data in Table 1 together with the values for OS,,, and T, as well as AH,,,in, for Sn and Cu from SGTE /91Din/ the entropy curves for the solid and liquid state have been calculated in a temperature range covering both the Kauzmann and inverse Kauzmann temperatures (isentropic temperatures). The results are shown in Figs.5 and 6 respectively. It can be reasonably well assumed that at the Kauzmann temperature the undercooled liquid transforms into an “ideal” glassy state. The entropy of the “ideal” glass follows the entropy of the solid crystal for all temperatures below the Kauzmann temperature. For the superheated crystal a similar argument holds for its transformation into the liquid phase at the “inverse“ Kauzmann temperature. The entropy of the hypothetical solid phase follows therefore the entropy of the stable liquid above the “inverse” Kauzmann temperature. In consequence the heat capacity curve of the supercooled liquid is only defined down to the Kauzmann temperature and that of the superheated crystal up to the “inverse” Kauzmann temperature. In order to be able to extrapolate below and above these temperatures a discontinuous change of their heat capacities occurs. For Sn and Cu this behaviour is shown in Figs.7 and 8. 5. Amorphous soIldlcation by two-states model 5.1 Development of model The aim of the present section is to develop a phenomenological model capable of representing the thermodynamic properties of the liquid in a physically sound way. The model should allow reasonable predictions, extended to low temperatures as well as high temperatures. The present approach will be based on a model suggested by Agren /88Agr/. As temperature is decreased the thermodynamic properties of the liquid approach gradually those of a crystalline solid. As already mentioned, there is a gradual decrease in entropy, i.e. the liquid undergoes a gradual ordering transformation. This ordering would probably have different origins in different classes of liquids but we shall call it amorphous solidification, since the liquid becomes more solid like. It should be emphasised though, that since only pure substances are considered here we would have no contribution from chemical ordering. On the other hand pure substances may exhibit polymerisation, as is the case, e.g. with boron and sul-
453
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
phur. However, discussed here.
models for polymerisation
are already well established and will not be further
In a simple liquid substance, like a pure metal, one may give the following physical picture of the amorphous solidification. In the crystalline state the atoms or molecules only have vibrational degrees of freedom whereas in the liquid state one may imagine that a large fraction of the atoms or molecules have translational degrees of freedom. The difference in entropy between a state where all the atoms are free to move all over the system and a state where each atom. can only move inside its cell is R, i.e. the gas constant. This is sometimes referred to as the communal entropy /54Hir/. It so:ms reasonable to represent the gradual loss in translational degrees of freedom with a classical two-state model. In this model it is suggested that the atoms could be either in a solidlike state or in a liquid-like state. For simple liquids one may imagine that the form?r state is connected with vibrational motion whereas the latter with translational motion. An internal ordering parameter 5 is introduced as the fraction of atoms being in the liquid-like state. For the molar Gibbs energy one may thus write G,” =(1-c)
“G~‘+5”G~+RT{~ln~+(l-t)ln(l-5))
5.1
where “Gf’ and “G? are the molar Gibbs energy if all the atoms were in the solid-like state and liquid-like state respectively. The equilibrium value of 5 at a given temperature is the value that minimises the above expression. exp(-AC,
5= l+exp(-AC,
I RT)
5.2
I RT)
where AGd=uG~-oG~‘.
5.3
Inserting the value of 5 in the equation for Gk one finds G,” =“Gz’ - RT ln[ I+ exp(-AC,
/ RT)]
5.4
It is c:onvenient to define an excess Gibbs energy as ~,E=~,L-OG~I=~AG~+RT(~I~~+(~-~)~(A-~)}
5.5
For a liquid in internal equilibrium one thus have G,” =-RTln[l+exp(-AG,/RT)]
5.6
If AC;, is temperature the above model is identical to the classical Schottky two-level model. Sometimes a degeneracy is introduced into the Schottky model and this is formally equivalent with having a linear temperature dependency of AC,, provided that the temperature dependency is chosen in such a way that exp(AS / R) is an integer. In general we would expect AGd to have any temperature dependence. In 1ac.kof experimental
information one may take as a first approximation
J. AGREN et al.
454
where A and B are adjustable parameters. Assuming that the ideal amorphous state obeys the third law one would have to accept B=O. In any case one would expect B to be small and may therefore take
If experimental information is available for both the undercooled and stable liquid one may simply write AGd =~+bT+cTlnT+dT~+...
5.9
where a,& and d are adjustable parameters to fit the experimental information. However, in most cases, except for low-melting substances, not much information is available. For example, if the Gibbs energy of the crystalline phase is known, which may often be the case, but only the melting point and the heat of melting are known for the liquid only two parameters can be evaluated for the liquid. It is reasonable to assume that b would be close to R, the communal entropy. Setting c and d to zero it may thus be possible to evaluate A and a. 5.2 General behaviour of two-statemodel We shall now investigate the general behaviour of the model. The results of a series of calculations are shown in Fig. 9 where the excess heat capacity normal&d to R has been plotted as a function of RTla. The excess heat capacity is defined as the derivative -Ta=G,E/aT’
5.10
where G,” is given by eq 5.6. All the curves are calculated with a/R=1000 K and c=O. Only b and dare varied. As can be seen, the peak will become higher as the magnitude of b increases (b < 0). By introducing also a finite negative d value there will be a continuous increase in excess heat capacity with temperature, approaching asymptotically a linear behaviour. 5.3. Example of an application where heat capacity data are available - Liquid Sn The heat capacity of liquid Sn has been measured almost 200 K below the melting point (505 K) and up to 1000 K /84Per/. Sn is thus chosen in order to test if the model is capable of representing the thermodynamic properties of a substance that is well characterised. The first step in the evaluation is to chose a suitable “Gz’. In order to test if one may apply eq. 5.8 one may plot the heat capacity of pure solid white Sn according to the SGTE database, see Fig. 10. It should be emphasised that the expression valid below the melting point has simply been applied also above the melting point, i.e. the so-called SGTE extrapolation has not been used. As can be seen, the heat capacity first increases and reaches a maximum at around 1000 K. This behaviour does not seem suitable as a reference for”Gz’ because the experimental information indicates that the liquid heat capacity approaches a constant value or even increases above 800 K. Thus we cannot use eq. 5.8. Instead one may apply a description of the ideal amorphous phase based on the Einstein model. For the 8 temperature we take 155 K as the Einstein temperature of white Sn. We apply the following polynomial approximation of the F(WT) F(WT)~33RT[ln(01T)+(BIT)2/24]
5.11
HEAT CAPACtTY
IN LtQUtD AND AMORPHOUS
455
PHASES
Thus~the following expression was chosen “G;‘=A+BT-3RTlnT+(3RB2/24)T-’
where A and B zue to be adjusted to frt the experimental AG, =a+bT+dT’
5.12
data. For AC, we choose 5.13
The 5 coefficients A, B. a, b and d were now adjusted to tit the following experimental mation: AHmclang= 7029 J mol-’ , T,c,,i,g= 505.1 K AlDin/ as well as the heat-capacity
inforvalues
from 84Per. The following values were obtained A = -2864.6+
HinER J mol-’
B=:112.537
J mol-’ K-’
a = 5662.3
J mol-’
b = -11.3298
J mol-’ K-’
d = --1.4328
1O-3
J mol-’ Kb2
where SER denotes the so called stable element reference, i.e. the element in its stable state at 298.15 K. The calculated heat capacity is shown as the solid line in Fig. 11 and the symbols are the experimental values. As can be seen the model yields an excellent representation of the experimental data. Fig. 12 shows the calculated entropy for the liquid (solid line) and for the stable crystalline phase (dashed line). The entropy of liquid and solid comes very close below 200 :K. Fig. 13. shows an extrapolation of the heat capacity to 3000 K. Finally Fig.14 shows the 5 at equilibrium as a function of temperature. It is interesting to notice that 5 is around 0.5 at the: melting point. 5.4 Example of an application where a constant heat capacity value is available - Liquid Cu For Cu the experimental information is not extensive enough to reveal any temperature dependence of the heat capacity but only a single value at the melting point is available. On the other hand it was found that the Gibbs energy of the stable FCC phase, given by the SGTE database, behaved reasonably well upon extrapolation to high temperatures and it was thus chosen to use 5.14 and for AG, we first choose AG, = AH,,,i,,p - RT
5.15
The optimisation of A and d was now performed adjusting to the following infonnationbJSm,,i~, = 13263 J rn~l-‘.T,,,~,, = 1358 K, ck(T,) = 31.4 J mol-’ K’ I9lDinl. The result is
J. AGREN et al.
456
A= 6681.5+ H;; d = 1.095 lo4
J mol-’ J mol-’ KJ
It was thus possible to fit the melting point as well as the enthalpy of melting almost exactly but the heat capacity came out somewhat less well, namely 32.4 rather than 3 1.4 J mol-’ K-’ , see Fig. 15. The entropy of the liquid and solid phases is shown in Fig. 16. As can be seen they come very close around 250 K. We may now try to improve the fit by taking instead AG, =a-RT
5.16
treating a as an adjustable parameter. We thus have three parameters which can be derived exactly from the 3 data points. The result is A = 7985
J mol -’
a = 10609.7
J mole’
d = 1.0135 lo4
J mole’ K”
It is encouraging to see that the values of the parameters do not change very much compared to when a = AHmehh8.
6. Extensions into multicomponent systems For multicomponent solutions the reference state is the mechanical mixture of the pure components in the same state as the solution. The reference temperature of the databases used at present is 298.15 K. As a consequence, the heat capacity data for both the stable crystalline state and the undercooled liquid are only given for temperatures above 298.15 K. Based on the approach given in paragraph 4.1 it is only possible to define the reference of the liquid above the highest Kauzmann temperature of the components while the reference state of the solid can only be defined below the lowest “inverse” Kauzrnann temperature of the components. The Kauzmann temperatures are generally of the order of 0.3T, and the “inverse” Kauzmann temperatures of the order of 3T, /92Hoc/. Calculations of phase equilibria using this approach are therefore possible only in a limited temperature range. For a binary alloy with the melting temperatures T,” and T,” of the components, with T,” c Tf, the solid phase can be described in the temperature range 298.15 K I T I3T/ and the liquid phase above 0.3Ti or 298.15 K. Fig.17 shows the calculated heat capacity of solid equimolar Cu-Sn alloy up to 4000 K using the extrapolation based on isentropic temperature approach. It can be directly seen that the extrapolation beyond 3T: is artificial. The SGTE extrapolation given in paragraph 3.4 has also its limitations. The Figs. 18 and 19 show the heat capacity of the solid FCC-Al phase for Cu and Sn. the heat capacity of the reference state for one mol of the FCC-Al Cu,,Sn,, phase in Fig. 20 obtained from the values shown in Figs. 18 and 19 exhibit also artificial values below about 700 K. Both approaches to describe the metastable liquid and solid state can be used therefore only in a limited temperature range.
HEAT CAPACITY
IN LIQUID AND AMORPHOUS
PHASES
On the other hand, when using the approach based on the two-state model it is natural to accept eq. 5.1 also for the alloy and expand ‘Gz’ and AGd as functions of composition in the usual regular solution formalism, i.e.
AG, = x,“AG,”+ x;AG; + x;x;AG;
6.2
Higher-order terms as well as temperature dependencies are easily included whenever experimental data indicate a more complex behaviour. Applied to the equimolar Cu-Sn alloy we would, as a first approximation, take the parameters evaluated in sections 5.3 and 5.4 and in lack more detailed data set AG,AB=0 The heat capacity calculated from the two-state model for the equimolar Cu-Sn melt is shown in Fig. 2 1. As can be seen, the curve has now a reasonable behaviour over the whole temperature range.
Ack.nowledgements
The authors wish to thank Prof. Bo Sundman for organising a wonderful workshop and one of the authors (JA) would like to thank Dr. Nigel Saunders for animated discussions. Finally we thank the sponsors of the “Ringberg Workshop” for their moral and financial support. References
48K.au WKauzmann, Chem.Rev. 43, 1948,219 54Hir J.O. Hirschfelder, CF. Curtiss and R.B. Bird, Molecular Theory of Liquids and Gases, John Wiley & Sons Inc, New York 1954, 274 69Hoc M.Hoch, High Temp. High Pres. 1, 1969,531 76Ang C.A.Angell, W.Sichina, Ann.N.Y.Acad.Sci., 279, 1976,53 76Bar LBarin, O.Knacke, O.Kubaschewski, Thermochemical Tables, Springer 76Hoc M.Hoch, T.Vemadakis, Ber. Bunsenges. 80, 1976,770 78Ste R.B.Stephens, J.Appl.Phys., 49, 1978.5855 84P’er J.H.Perepezko, J.S.Paik, J.Non-Cryst.Solids, 61, 1984, 113 84Tsu Y.Tsuchiya, J.Phys.Cond.Mat., 3, 1984,125 85Cha M.V.Chase et al, JANAF Thermochemical Tables, Third Edition, J.Phys. Chem.Ref.Data, 14, suppl.1, 1985 88Fsec H.J.Fecht, W.L.Johnson, Nature 334, 1988,50 88A.gr J. Agren, Phys. Chem. Liq., 18, 1988, 123 91Din A.T.Dinsdale, CALPHAD 15,1991,317 92H:oc M.Hoch, Met.Trans.B 23B, 1992,309 93%~ S.K. Saxena, N. Chatterjee, Y. Fei and G. Shen: Thermodynamic data on oxides and silicates, Springer Verlag, Berlin,-Heidelberg-New York 1993, p 428 95Som F.Sommer, private corn.
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Figure captions: Fig. 1: Schematic phases. Solid part extrapolation. The ing artificial phase
diagram of Gibbs energy as a function of temperature for liquid and solid of curves represent experimental data whereas dashed parts are obtained by curves intersect at the normal melting point and at two temperatures yieldstability.
Fig. 2: The entropy of liquid and crystalline Al as a function of temperature in the stable and metastable region. AS, denotes the entropy of fusion. Reproduced from /88Fec/. Fig. 3: The Gibbs energy of liquid and crystalline Cu as a function of temperature and metastable region calculated from the SGTE /91Din/ database.
in the stable
Fig. 4: The heat capacity of liquid and crystalline Cu as a function of temperature and metastable region calculated from the SGTE /91Din/ database.
in the stable
Fig. 5: The entropy of liquid and crystalline Sn as a function of temperature metastable region calculated using the coefficients in Table 1.
in the stable and
Fig. 6: The entropy of liquid and crystalline Cu as a function of temperature metastable region calculated using the coefficients in Table 1.
in the stable and
Fig. 7: The heat capacity of liquid and crystalline Sn as a function of temperature in the stable and metastable region calculated using the coefficients in Table 1 and taking AC, = 0 below the Kauzmann temperature and above the inverse Kauzmann temperature. Fig. 8: The heat capacity of liquid and crystalline Cu as a function of temperature in the stable and metastable region calculated using the coefficients in Table 1 and taking AC, = 0 below the Kauzmann temperature and above the inverse Kauzmann temperature. Fig. 9: The excess heat capacity of a liquid as function of temperature the coefficients in Eq. 5.9.
for different values of
Fig. 10: The heat capacity of white Sn as a function of temperature in the stable and metastable region calculated from the SGTE data without the SGTE extrapolation above the melting point. Fig. 11: The heat capacity of liquid Sn as a function of temperature in the stable and metastable region calculated using the two-state model. Symbols denote experimental values /84Per/ Fig. 12: The entropy of liquid Sn as a function of temperature in the stable and metastable region calculated using the two-state model, solid line. The entropy of solid white Sn (BCT) is given by the dashed line. Fig. 13: The heat capacity of liquid Sn as a function of temperature in the stable and metastable region calculated using the two-state model. Extrapolation to high temperatures. Fig. 14: Fraction of liquid-like atoms as a function of temperature for Sn.
HEAT CAPACITY IN LIQUID AND AMORPHOUS
459
PHASES
Fig. 15: The heat capacity of liquid Cu as a function of temperature in the stable and metastable region calculated using the two-state model. The symbol denotes the experimental at the normal melting point. Fig. 16: The entropy of liquid Cu as a function of temperature in the stable and metastable regicln calculated using the two-state model, solid line. The entropy of crystalline Cu (FCC) is given by the dashed line. Fig. 17: The heat capacity of a liquid CuO,,SnO,, alloy as a function of temperature in the stable and metastable region calculated using the approach based on isentropic temperatures. Note that there are two different isentropic temperatures. Fig. 18: The heat capacity of crystalline Cu as a function of temperature tastable region calculated from the SGTE /91Din/ database.
in the stable and me-
Fig. 19: The heat capacity of hypothetic crystalline FCC Sn as a function of temperature stable and metastable region calculated from the SGTE /91Din/ database.
in the
Fig. 20: The heat capacity of hypothetical metastable Cu,,Sn,, ahoy with FCC structure as a function of temperature calculated from the SGTE /91Din/ database. Fig. 121:The heat capacity of liquid Cu, Cu,,Sn,, alloy and Sn as functions of temperature the stable and metastable region calculated using the two-state model.
in
J. AGREN
460
Fig. 1:Schematic phases. Solid part extrapolation. The ing artificial phase
et at.
diagram of Gibbs energy as a function of temperature for liquid and solid of curves represent experimental data whereas dashed parts are obtained by curves intersect at the normal melting point and at two temperatures yieldstability.
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
60
40
L I 20 z E 7 W-J 0
-20
0
200 400 600 800 1000 1200 1400 T(K)
Fig. 2: The entropy of liquid and crystalline Al as a function of temperature in the stable and metastable region. AS, denotes the entropy of fusion. Reproduced from /88Fec/.
461
J. AGREN et a/.
462
2 0 -2 -4 -6 -8 -10 -12 E4 -14L 0
500
1000
1500
2000
T (K) Fig. 3: The Gibbs energy of liquid and crystalline Cu as a function of temperature in the stable and metastable region calculated from the SGTE /91Din/ database.
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
463
383634L
32 -
z: 30 , 2 a: LI ii
28 262422 -
I *OO
I
500
I
1000
I
1500
2000
T (Kl Fig.4: The
heat capacity of liquid and crystalline Cu as a function of temperature in the stable and metastablc region calculated from the SGTE /9 lDin/ database.
J. k3REN
464
I
I
I
et al.
I
I
I
f
1630
T (to Fig. 5: The entropy of liquid and crystalline Sn as a function of temperature in the stable and met&able region calculated using the coefficients in Table 1.
465
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
I
I
1
I
I
I
I
I
I
5( 30
T (K) Fig. 6: The entropy of liquid and crystalline Cu as a function of temperature in the stable and metastable region calculated using the coefficients in Table 1.
466
16 30
Fig. 7: The heat capacity of liquid and crystalline Sn as a function of temperature in the stable and metastable region calculated using the coefficients in Table 1 and taking AcP= 0 below the Kauzmann temperature and above the inverse Kauzmann temperature.
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
70
Fig. 8: The heat capacity of liquid and crystalline Cu as a function of temperature in the stable and mctastable region calculated using the coefficients in Table 1 and taking AC,,= 0 below the Kauzmann temperature and above the inverse Kauzmann temperature.
467
1
468
J. AGREN et a/.
RT/a Fig. 9: The excess heat capacity of a liquid as function of temperature for different values of the coefficients in Eq. 5.9.
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
469
50 45 40 35 30 25 20 15 IO sI
OO
I
I
200
400
I
600
I
800
I
1000
1:
-i-(K) Fig. 10: The heat capacity of white Sn as a function of temperature in the stable and metastable region calculated from the SGTE data without the SGTE extrapofntion above the melting poinl..
J.AGREN et a/.
470
50 45 40 35 30
;
p_@__
25 20 15 10 5 OO
I
200
I
400
I
600
I
800
I
1000 Ii10
Fig. 11: The heat capacity of liquid Sn as a function of temperature in the stable and metastable region calculnted using the two-state model. Symbols denote experimental values /84Per/
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
200 180 160 140 'i- 120 LL c $ 100 :
80
5; 60 40 20 0
.* 5’ :
I
200
1
400
I
600
I
800
I
1000 Ii30
T(K)
Fig. 12: The entropy of liquid Sn as a function of temperature in the stable and metastable rcgion c,alculatcd using the two-state inodcl, solid line. The entropy of solid white Sn (BCT) is given by the dashed line.
471
J. AGREN et al.
472
-i
5oL 45
kz 40 2 7
35
25:: I
20
IS10 51
OO
!
500
!
I
I
I
1000 1500 2000 2500 31 10
1 (K)
Fip. 13: The heat capacity of liquid Sn as a function of temperature in the stable and mctnstable region calculated using the two-state model. Extrapolation to high temperatures.
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
0
a”’
I
500
I
I
I
I
1000 1500 2000 2500 3000
TIKI !:ig. i3: Fraction of liqtlid-like xoms as 2 function of tempernture for Sn.
473
J.
474
50 h
c
40
7
35
7
et al.
r-
45
x
zE
kXEN
30 25 20 15 IO 5 t
0
500
1
1000
1
1500
2000
T (IO
Fig. 15: The heat capacity of liquid Cu as a function of temperature in the stable and metastable region calculated using the two-state model. The symbol denotes the experimental at the normal me1 ting point.
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
100 90 80 70 60 50 40 30
20 IO OL 0
500
1000
1500
2000
T WI
Fig. 16: The entropy of liquid Cu as a function of temperature in the stable and metastable region calculated using the two-state model, solid line. The entropy of crystalline Cu (FCC) is given by the dashed line.
475
J. AGREN et al.
476
65
I
75
I
I
I
Ld
0
T 0
I
I
I
woo
rn,
Fig. 17: The heat capacity of a liquid CuO,,,Sn,,, altoy ns a function of temperature iu the st;lble and metastable region calculated usin g the approach based on isentropic temperatures. Note that there are two different isentropic temperatures.
r-I I I
HEAT CAPACITY I
I I
(J rnol-'K-') w iQ
J. AGREN et al.
478
I
I
I
I
I
I
I
I
I
5000 T Ml Fig. 19:The heat capacity of hypothetic crystalline FCC Sn as a function of temperature stable and metastable region calculated from [IICSGTE /91Din/ database.
in the
HEAT CAPACITY IN LIQUID AND AMORPHOUS PHASES
25
0
I
1
I
I
I
I
I
I
479
I
5( 30 T (K)
Pig. 20: The heat capacity of hypothetic mctastable Cu,&&, alloy with FCC structure as a function of temperature calculated from tilt: SGTE /91Din/ database.
J. AGREN et
480
al.
50 45 Sn
40 35 30/ 2s I-
20 15 IO 5 OO8.
200 I
400 1
600 I
800 I
1000 I
1200 1
T MI Fig. 21: The heat capacity of liquid Cu, Cu,,,Sn,,, alloy and Sn as functions of temperature in the stable and metastable region calculated using the two-state model.
11