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A new method of describing lattice stabilities
0364-5916/87 $3.00 t .OO fc) 1987 Pergamon Journals Ltd.
CALPHAD Vol. 11, No. 1, pp. 93-98, 1987 Printed in the USA.
A NEW METHOD OF DESCRIBING
LATTICE
STABILITIES
Jan-Olof Andersson, Armando Fernandez Guillermet, Per Gustafson, Bo Sundman and John Agren. Bo J&s&on, Div. of Physical Metallurgy Royal Institute of Technology, S-10044 Stockholm,
(Presented
at
CALPHAD XV,
Mats
Hillert,
Bo Jansson,
Sweden.
London, England July 1986
stability is defined as the Gibbs energy of an element in a certain ABSLTRACT The lattice structure relative to its value in another structure. It is often assumed to vary an assumption which is either limited to a narrow range 1 inearly with temperature, of temperature or implies that CP of the two phases are the same. A better method would be to use the information available for Cl, and derive a more realistic extrait is demonstrated that even the best assessment available today polation. However, may give absurd results for the stability of a liquid phase relative to a solid. SGTE has recommended another method which makes use of the experimental difference in Cl, at the melting point. It is a simple method and guarantees that the results will be reasonable. The method should not be considered as ideal but it provides a workable framework until reliable physical models are available to represent thermodynamic data for phases outside their range of stability.
In the thermodynamic calculations of alloy phase diagrams one often needs information on the stability of an element in a given structure relative to another structure. This quantity is defined as the difference in the molar Gibbs energy between the two structures (phases) and is often called lattice stability (I). Close to the transformation point between the two phases one may estimate the lattice stability by assuming that the enthalpy and entropy of transformation are effectively constant,
with
AG = AH0 - TASo
(1)
AS0 = AHo/To
(2)
T here denotes u$!e this simple
the transformation expression even
over
temperature. Lacking further a wide range of temperatures.
information See for
it is common to instance ref. 2.
As more information becomes available and the assessments of alloy systems are made it seems justified to consider experimental information for the heat with greater ambition, for each one within the temperature range where it is stable, capacity, CP, of the phases, 1 is exact only if C of the two phases are equal. By applying eq. 1 together However, eq. in their stable regions it is with the direct experimental information for C of the phases thus implied that C of each phase has a disco I! tinuity at the transformation temperature and This is not a very attractive consequence and it should then follows CB of e he other phase. be better to give up eq. 1. In fact, there is no need for eq. 1 if the experimental information for CB of each phase is extrapolated across the transformation temperature. However, with the llmited information available today that method may yield unacceptable results, As an example we may examine a recent assessment of Mo (3). Fig. 1 shows the experimental information for CP of the bee and liquid phases and the curve obtained by fitting the data to an analytical expression. Below the melting point the following expression was obtained for the bee phase at zero pressure.
Received 28 January 1986 final form 5 March 1986
93
J. -0.
94
PULSE
HEATING
ANDERSSON et al.
+
: +
CEZAIRLIYANCiQ83> RIGHINIC 1 Q83> TAYLOR<
+
1964)
MODULATION
91
t
:
NAKARENKOClQ78> LOWENTHALClQ831
I000
1400
1800 TEMPERATURE,
FIG.
1
CP for Ho in liquid states ref. 3.
2288
2600
the bee and according to
c 3888
K
FIG.
2
The difference in molar Gibbs energy of MO of the relative to the bee state, liquid. Curve 1 is the usual, 1 inear representation, curve 2 is obtained by an extrapolation using equations in ref. 3, obtained in the stable ranges of the phases and curve 3 is the extrapolation actually proposed in ref. 3. It was obtained by the SGTE method of extrapolation.
0
1280
2480
TEMPERATURE-KELVIN
3808
4880
6888
NEW METHOD OF DESCRIBING LATTICE STABILITIES
bee GMo
_ hSER = -7747.247 MO
+ 5.662834.10-7T3
-
+ 131.9197T
-
1 . 309265.10-‘“T4
23.56414TlnT
-
95
0.003443396T2
+ 65812.39T-’
J/mol
(3)
SER
is the molar enthalpy at 298.15 K and 1 bar of the most stable phase of MO The quantity H Above the melting point the information for the liquid phase is too under these co%itions. and the result is meagre to allow the temperature dependence of CR to be evaluated I iq GMo
_ HSER = 3538.018 MO
+ 271.6697
-
42.63829TlnT
bcc_(;l iq Curve 1 is the lattice stability of liquid MO expressed as G 1 and curve 2 from eqs. 3 and 4. It is evident t F1” at t R”e method of extracalculated from eq. polation, yielding curve 2 is not good enough to give a realistic estimate of the lattice stability far away from the melting temperature. For instance, the prediction that the lattice stability of liquid MO turns negative above about 5000 K implies that the bee phase becomes stable there. This absurd result may be eliminated by assuming that the strong increase of CR of the bee phase below the melting point should not continue very far above. Unthere is no information or model available today by which we can make a better fortunately, prediction. Fig.
2 shows
It is also possible to eliminate the absurd result by assuming that CR of the liquid phase should increase drastically at high temperatures. However, the meagre information available for C of liquid metals rather indicates that C is approximately constant above the melting poiKt and should increase on cooling below thg melting point down to a glass transition below which CR of the liquid or amorphous phase quickly changes to become similar to that of the solid. A better estimate of the lattice stability below the melting point could certainly be obtained if we had a reliable model for the thermodynamics of the transition from liquid to glass. Such a model would undoubtedly predict that the lattice stability decreases less rapidly below the melting point than according to curve 2 in Fig. 2. As an example, we may examine the results recently obtained for Mg (4). Curve 1 in Fig. 3 has the same definition as in Fig. 2 but curve 2 was here obtained after a rather ambitious assessment of the properties of the liquid phase below the melting temperature. It was performed under the assumption that the entropy of solid and liquid Mg are equal at a third of the melting temperature. This isentropic condition is sometimes regarded as a reasonable definition of the thermodynamic glass transition and below that temperature it is assumed that the entropies of the two phases will stay equal. Thus curve 2 in Fig. 3 is horizontal below the glass transition. Curve 2(L) in Fig. 4 shows the very drastic increase in the heat capacity of the liquid when cooled towards the glass transition temperature. Before discussing further the extrapolation of the liquid phase below the melting point it should be emphasized that an important practical use of the extrapolated liquid form of an element is as a basis for the description of liquid alloys which may be stable at such temperatures. In order to describe the Gibbs energy of mixing of such a solution phase with a reasonably simple expression it is important to choose the correct reference states of the components. This is today a common procedure for ferromagnetic elements. Solutions which are not ferromagnetic at a temperature below the Curie temperature of one of the component elements are described with reference to a hypothetical para-magnetic state of the ferromagnetic element (5). Rather than referring to the actual state of the pure element one thus refers to a high-temperature state. In fact, the general method is to subtract the ferro-magnetic effect from the properties of the pure elements, to describe the “chemical” part of the Gibbs energy of mixing with a simple polynomial and then to add the ferromagnetic effect for the alloy. The treatment of liquid alloys will not be satisfactory until a similar procedure can be applied to the glass transition. Even though the glass transition, as defined by changes in mechanical properties, occurs in a rather limited temperature range its thermodynamic effect is spread over a very wide temperature range. It has even been suggested that it is not completed at the melting point (6). As a consequence, with the present lack of understanding of the glass transition it is difficult to give any recommendation on the extrapolation of the properties of a liquid element below the melting point. In order to promote collaboration on the assessment work on necessary to define some method of extrapolation across the melting in an unambiguous way. Such a recommendation was recently made by ration called SGTE (Scientific Group Thermodata Europe). The main was to define a simple method of representing the lattice stabilities phase to become stable at high temperatures, or a liquid phase to
alloy systems it is point, which can be applied a European group for collabopurpose of the recommendation without risking a solid become stable at low tempera-
J.-O. ANDERSSON et al.
96
FIG. 3 The difference in molar Gibbs energy of Mg in the hcp state, relative to the liquid, Curve 1 is the usual i inear representation. Curve 2 is obtained by extrapolation according to the assessment in ref. 4 using an isentropic point at To/3.
70 c
X
clE a
-100! 0
t *
.
400
888
#, I 280
t 880
2008
TEHPERATURE-KELVXN
6% 55
FIG.
4
The heat capacity for hcp and liquid Mg according to the assessment in ref. 4.
15 18 0
200
400
600
TEMPERATURE-KELVIN
800
1000
1208
NEW METHOD OF DESCRIBING LATTICE STABILITIES
tures. point
A second purpose which is suitable
was to obtain as a reference
portant to notice that eq. 1 fulfils a discontinuity in C at the melting simple way of retainyng the advantage
a description of a liquid for the liquid phase in
97
element below its low-melting alloys.
both requirements but has the point. The SGTE recommendation of eq. 1 but eliminating its
disadvantage of may be regarded disadvantage.
melting It is
im-
producing as a
The SGTE method forces C of the extrapolated phase to approach C of the stable phase It makes use of thePexperimental difference in Cp at the me1 f. Ing point and accepts gradually. a kink in Cp for both phases at the melting point but not a discontinuity. This is accompthe melting point with the following expressions lished by extrapolating Cp across
T is p!?ase
T > To
: C;(T)
= C;(T)
+ @To)
-
C,e(To)l
(T/To)-”
T < To
:
= C;(T)
+ [CF(To)
-
CF(To)l
(T/To)’
the stable stable at
CF(il
melting
298.15
point independent K and 1 bar. a is
By the use of the exponents -10 and 2% of the difference at the meltinq point, ting point. Fig. 5 shows how the difference with the temperature.
(5)
(6)
of which solid any solid phase.
phase
is
considered
and
c1 is
the
6,
the remaining difference in C will be less than when the temperature is 50% abode or below the melin heat capacity between c1 and the liquid varies
The result of the SGTE method, applied to MO, is shown by curve 3 in Fig. 2. It is very similar to the straight-line behaviour of eq. 1 (curve 1) which is a consequence of the rapid decrease of the difference in C according to the SGTE method. It is interesting to note that the SGTE method yields a correc !. Ion to the straight-line behaviour of eq. 1 which is of the same sign on both sides of the melting point. This correction can be calculated directly from 5 and 6. When there is only one solid phase one obtains eqs. T > To
: A(G; - G:) = [c~(T~) - c~(T~)I
. t-0
T < To
: A(G; - G;) = [C;(To) - C;(To)l
. {-(To - T)/6 + [l
- T~)/IO
+ [I
-
. To/901(7)
(~/-r~)-~l (T/T~)~I
*
To/421
(8)
Fig. 6 shows how this correction varies with the temperature. For Mg the SGTE method would yield a curve practically indistinguishable from the straight line obtained from eq. 1 because Cp of the liquid and solid phases happen to be practically equal at the melting point, as demonstrated by Fig. 4. As a consequence, no curve 3 was included in Fig. 3. It should be emphasized that the SGTE method is not based on any physical model but is rather a substitute for such a model. It simply forces the lattice-stability expressions to obey certain rules, such that extrapolations will yield reasonable results. If good physical models were available they could be used to perform more realistic extrapolations and there would be no need for the SGTE method. It should also be emphasized that the SGTE method is intended to be used for the solid/ liquid transformations only. This is illustrated for Zr in Fig. 7, where the data were taken from a recent assessment (7). That element has two stable, solid phases, hcp below 1139 K and bee above. Hcp has a metastable melting point at 1999 K but the SGTE method is not applied until the stable melting point of Zr, which is 2128 K. It should further be noticed that the C P curve for the I iquid phase is made to approach the extrapolated curve for the low temperature phase, hcp, and not bee. See curve 3(L). In Fig. 7 it is well illustrated that the SGTE method produces a kink in Cp at the melting point but not a discontinuity as eq. 1 would require. Even the kink in Cp could be eliminated by applying a more complicated method of decreasing the difference in Cp at high and low unless one uses a real extrapolation of Cp such methods will always retemperatures. However, sult in a kink in Cp or some of its derivatives. For most practical purposes it seems sufficient to have removed the discontinuity in CP itself and the SGTE method is thus appropriate for many purposes. Finally, we should discuss the consequences for the description of intermediary phases. The molar Gibbs energy of phases sfkth a fixed composition can be ef2cribed with an expression similar to eq. 3. The quantity H would then be defined as Cx.H. and the method of treating Cp for the pure elements would not necessarily affect the descrfption of the intermediary comIf an intermediary phase exists over a range of compositions, it should be described as pound. a solution phase between the pure component elements at the temperature under consideration. For such.a phase the SGTE method would always give a kink in C, at the temperatures of melting of the component elements. This is not more serious than the kink in Cp of the liquid phase.in low-melting alloys at the melting points of the component elements.
J.-O. ANDERSSON et al.
6 RELATIVE
FIG.
TE”PERINRE.
1.66
6.48 RELATlVE
T/To
5
FIG.
The temperature dependence of the difference in CP between the solid and liquid phases according to the SGTE method of extrapolation across the melting point, To.
The
P.00
T/lo
6 correction
the straight-line culated by the polat ion.
FIG.
TMPERATUFZ,
to
G”-G~
relative
aFpr%imation, SGTE method of
to
calextra-
7
The heat capacity of hcp, bee and liquid Zr. Curves 2 describe extrapolations outside the range of stability of thephases,obtained by using equations from ref. 7, derived inside the range of stability. Curves 3 were obtained by the SGTE method of extrapolation and were actually proposed in ref. 7. 25 98%
1180
I708
2388
2900
3588
Acknowledgement The authors gratefully with the colleagues in SGTE. Technical Development.
acknowledge many discussions on the This work was financially supported
lattice by the
stability Swedish
problem Board for
References 1. 2. 3. 4. 5. 6. 7.
L. Kaufman and H. Bernstein, “Computer Calculation of Phase Diagrams”, Academic Press, New York 1970. L. Kaufman and H. Nesor, Calphad 2 (1978) 55-80, 81-108, 117-146, 295-318, 325-348. A. Fernandez Guillermet, Intern. JT Thermophys. 6 (1985) 395-409. B. JBnsson and J. Agren, Hetall. Trans. (in press). S. Hertzman and B. Sundman, Calphad 6 (1982) 67-80. M. Hillert, “The thermodynamics of tKe glass transition”, in “Rapidly Sol idif ied Amorphous and Crystalline alloys”, (Eds. B.H. Kear. B.C. Giessen and N. Cohen), Elsevier 1982. TRITA-MAC 271, (revised version), Royal Inst. Technology, StockA. Fernandez Guillermet, holm 1985.
CALPHAD Vol. 11, No. 1, pp. 93-98, 1987 Printed in the USA.
A NEW METHOD OF DESCRIBING
LATTICE
STABILITIES
Jan-Olof Andersson, Armando Fernandez Guillermet, Per Gustafson, Bo Sundman and John Agren. Bo J&s&on, Div. of Physical Metallurgy Royal Institute of Technology, S-10044 Stockholm,
(Presented
at
CALPHAD XV,
Mats
Hillert,
Bo Jansson,
Sweden.
London, England July 1986
stability is defined as the Gibbs energy of an element in a certain ABSLTRACT The lattice structure relative to its value in another structure. It is often assumed to vary an assumption which is either limited to a narrow range 1 inearly with temperature, of temperature or implies that CP of the two phases are the same. A better method would be to use the information available for Cl, and derive a more realistic extrait is demonstrated that even the best assessment available today polation. However, may give absurd results for the stability of a liquid phase relative to a solid. SGTE has recommended another method which makes use of the experimental difference in Cl, at the melting point. It is a simple method and guarantees that the results will be reasonable. The method should not be considered as ideal but it provides a workable framework until reliable physical models are available to represent thermodynamic data for phases outside their range of stability.
In the thermodynamic calculations of alloy phase diagrams one often needs information on the stability of an element in a given structure relative to another structure. This quantity is defined as the difference in the molar Gibbs energy between the two structures (phases) and is often called lattice stability (I). Close to the transformation point between the two phases one may estimate the lattice stability by assuming that the enthalpy and entropy of transformation are effectively constant,
with
AG = AH0 - TASo
(1)
AS0 = AHo/To
(2)
T here denotes u$!e this simple
the transformation expression even
over
temperature. Lacking further a wide range of temperatures.
information See for
it is common to instance ref. 2.
As more information becomes available and the assessments of alloy systems are made it seems justified to consider experimental information for the heat with greater ambition, for each one within the temperature range where it is stable, capacity, CP, of the phases, 1 is exact only if C of the two phases are equal. By applying eq. 1 together However, eq. in their stable regions it is with the direct experimental information for C of the phases thus implied that C of each phase has a disco I! tinuity at the transformation temperature and This is not a very attractive consequence and it should then follows CB of e he other phase. be better to give up eq. 1. In fact, there is no need for eq. 1 if the experimental information for CB of each phase is extrapolated across the transformation temperature. However, with the llmited information available today that method may yield unacceptable results, As an example we may examine a recent assessment of Mo (3). Fig. 1 shows the experimental information for CP of the bee and liquid phases and the curve obtained by fitting the data to an analytical expression. Below the melting point the following expression was obtained for the bee phase at zero pressure.
Received 28 January 1986 final form 5 March 1986
93
J. -0.
94
PULSE
HEATING
ANDERSSON et al.
+
: +
CEZAIRLIYANCiQ83> RIGHINIC 1 Q83> TAYLOR<
+
1964)
MODULATION
91
t
:
NAKARENKOClQ78> LOWENTHALClQ831
I000
1400
1800 TEMPERATURE,
FIG.
1
CP for Ho in liquid states ref. 3.
2288
2600
the bee and according to
c 3888
K
FIG.
2
The difference in molar Gibbs energy of MO of the relative to the bee state, liquid. Curve 1 is the usual, 1 inear representation, curve 2 is obtained by an extrapolation using equations in ref. 3, obtained in the stable ranges of the phases and curve 3 is the extrapolation actually proposed in ref. 3. It was obtained by the SGTE method of extrapolation.
0
1280
2480
TEMPERATURE-KELVIN
3808
4880
6888
NEW METHOD OF DESCRIBING LATTICE STABILITIES
bee GMo
_ hSER = -7747.247 MO
+ 5.662834.10-7T3
-
+ 131.9197T
-
1 . 309265.10-‘“T4
23.56414TlnT
-
95
0.003443396T2
+ 65812.39T-’
J/mol
(3)
SER
is the molar enthalpy at 298.15 K and 1 bar of the most stable phase of MO The quantity H Above the melting point the information for the liquid phase is too under these co%itions. and the result is meagre to allow the temperature dependence of CR to be evaluated I iq GMo
_ HSER = 3538.018 MO
+ 271.6697
-
42.63829TlnT
bcc_(;l iq Curve 1 is the lattice stability of liquid MO expressed as G 1 and curve 2 from eqs. 3 and 4. It is evident t F1” at t R”e method of extracalculated from eq. polation, yielding curve 2 is not good enough to give a realistic estimate of the lattice stability far away from the melting temperature. For instance, the prediction that the lattice stability of liquid MO turns negative above about 5000 K implies that the bee phase becomes stable there. This absurd result may be eliminated by assuming that the strong increase of CR of the bee phase below the melting point should not continue very far above. Unthere is no information or model available today by which we can make a better fortunately, prediction. Fig.
2 shows
It is also possible to eliminate the absurd result by assuming that CR of the liquid phase should increase drastically at high temperatures. However, the meagre information available for C of liquid metals rather indicates that C is approximately constant above the melting poiKt and should increase on cooling below thg melting point down to a glass transition below which CR of the liquid or amorphous phase quickly changes to become similar to that of the solid. A better estimate of the lattice stability below the melting point could certainly be obtained if we had a reliable model for the thermodynamics of the transition from liquid to glass. Such a model would undoubtedly predict that the lattice stability decreases less rapidly below the melting point than according to curve 2 in Fig. 2. As an example, we may examine the results recently obtained for Mg (4). Curve 1 in Fig. 3 has the same definition as in Fig. 2 but curve 2 was here obtained after a rather ambitious assessment of the properties of the liquid phase below the melting temperature. It was performed under the assumption that the entropy of solid and liquid Mg are equal at a third of the melting temperature. This isentropic condition is sometimes regarded as a reasonable definition of the thermodynamic glass transition and below that temperature it is assumed that the entropies of the two phases will stay equal. Thus curve 2 in Fig. 3 is horizontal below the glass transition. Curve 2(L) in Fig. 4 shows the very drastic increase in the heat capacity of the liquid when cooled towards the glass transition temperature. Before discussing further the extrapolation of the liquid phase below the melting point it should be emphasized that an important practical use of the extrapolated liquid form of an element is as a basis for the description of liquid alloys which may be stable at such temperatures. In order to describe the Gibbs energy of mixing of such a solution phase with a reasonably simple expression it is important to choose the correct reference states of the components. This is today a common procedure for ferromagnetic elements. Solutions which are not ferromagnetic at a temperature below the Curie temperature of one of the component elements are described with reference to a hypothetical para-magnetic state of the ferromagnetic element (5). Rather than referring to the actual state of the pure element one thus refers to a high-temperature state. In fact, the general method is to subtract the ferro-magnetic effect from the properties of the pure elements, to describe the “chemical” part of the Gibbs energy of mixing with a simple polynomial and then to add the ferromagnetic effect for the alloy. The treatment of liquid alloys will not be satisfactory until a similar procedure can be applied to the glass transition. Even though the glass transition, as defined by changes in mechanical properties, occurs in a rather limited temperature range its thermodynamic effect is spread over a very wide temperature range. It has even been suggested that it is not completed at the melting point (6). As a consequence, with the present lack of understanding of the glass transition it is difficult to give any recommendation on the extrapolation of the properties of a liquid element below the melting point. In order to promote collaboration on the assessment work on necessary to define some method of extrapolation across the melting in an unambiguous way. Such a recommendation was recently made by ration called SGTE (Scientific Group Thermodata Europe). The main was to define a simple method of representing the lattice stabilities phase to become stable at high temperatures, or a liquid phase to
alloy systems it is point, which can be applied a European group for collabopurpose of the recommendation without risking a solid become stable at low tempera-
J.-O. ANDERSSON et al.
96
FIG. 3 The difference in molar Gibbs energy of Mg in the hcp state, relative to the liquid, Curve 1 is the usual i inear representation. Curve 2 is obtained by extrapolation according to the assessment in ref. 4 using an isentropic point at To/3.
70 c
X
clE a
-100! 0
t *
.
400
888
#, I 280
t 880
2008
TEHPERATURE-KELVXN
6% 55
FIG.
4
The heat capacity for hcp and liquid Mg according to the assessment in ref. 4.
15 18 0
200
400
600
TEMPERATURE-KELVIN
800
1000
1208
NEW METHOD OF DESCRIBING LATTICE STABILITIES
tures. point
A second purpose which is suitable
was to obtain as a reference
portant to notice that eq. 1 fulfils a discontinuity in C at the melting simple way of retainyng the advantage
a description of a liquid for the liquid phase in
97
element below its low-melting alloys.
both requirements but has the point. The SGTE recommendation of eq. 1 but eliminating its
disadvantage of may be regarded disadvantage.
melting It is
im-
producing as a
The SGTE method forces C of the extrapolated phase to approach C of the stable phase It makes use of thePexperimental difference in Cp at the me1 f. Ing point and accepts gradually. a kink in Cp for both phases at the melting point but not a discontinuity. This is accompthe melting point with the following expressions lished by extrapolating Cp across
T is p!?ase
T > To
: C;(T)
= C;(T)
+ @To)
-
C,e(To)l
(T/To)-”
T < To
:
= C;(T)
+ [CF(To)
-
CF(To)l
(T/To)’
the stable stable at
CF(il
melting
298.15
point independent K and 1 bar. a is
By the use of the exponents -10 and 2% of the difference at the meltinq point, ting point. Fig. 5 shows how the difference with the temperature.
(5)
(6)
of which solid any solid phase.
phase
is
considered
and
c1 is
the
6,
the remaining difference in C will be less than when the temperature is 50% abode or below the melin heat capacity between c1 and the liquid varies
The result of the SGTE method, applied to MO, is shown by curve 3 in Fig. 2. It is very similar to the straight-line behaviour of eq. 1 (curve 1) which is a consequence of the rapid decrease of the difference in C according to the SGTE method. It is interesting to note that the SGTE method yields a correc !. Ion to the straight-line behaviour of eq. 1 which is of the same sign on both sides of the melting point. This correction can be calculated directly from 5 and 6. When there is only one solid phase one obtains eqs. T > To
: A(G; - G:) = [c~(T~) - c~(T~)I
. t-0
T < To
: A(G; - G;) = [C;(To) - C;(To)l
. {-(To - T)/6 + [l
- T~)/IO
+ [I
-
. To/901(7)
(~/-r~)-~l (T/T~)~I
*
To/421
(8)
Fig. 6 shows how this correction varies with the temperature. For Mg the SGTE method would yield a curve practically indistinguishable from the straight line obtained from eq. 1 because Cp of the liquid and solid phases happen to be practically equal at the melting point, as demonstrated by Fig. 4. As a consequence, no curve 3 was included in Fig. 3. It should be emphasized that the SGTE method is not based on any physical model but is rather a substitute for such a model. It simply forces the lattice-stability expressions to obey certain rules, such that extrapolations will yield reasonable results. If good physical models were available they could be used to perform more realistic extrapolations and there would be no need for the SGTE method. It should also be emphasized that the SGTE method is intended to be used for the solid/ liquid transformations only. This is illustrated for Zr in Fig. 7, where the data were taken from a recent assessment (7). That element has two stable, solid phases, hcp below 1139 K and bee above. Hcp has a metastable melting point at 1999 K but the SGTE method is not applied until the stable melting point of Zr, which is 2128 K. It should further be noticed that the C P curve for the I iquid phase is made to approach the extrapolated curve for the low temperature phase, hcp, and not bee. See curve 3(L). In Fig. 7 it is well illustrated that the SGTE method produces a kink in Cp at the melting point but not a discontinuity as eq. 1 would require. Even the kink in Cp could be eliminated by applying a more complicated method of decreasing the difference in Cp at high and low unless one uses a real extrapolation of Cp such methods will always retemperatures. However, sult in a kink in Cp or some of its derivatives. For most practical purposes it seems sufficient to have removed the discontinuity in CP itself and the SGTE method is thus appropriate for many purposes. Finally, we should discuss the consequences for the description of intermediary phases. The molar Gibbs energy of phases sfkth a fixed composition can be ef2cribed with an expression similar to eq. 3. The quantity H would then be defined as Cx.H. and the method of treating Cp for the pure elements would not necessarily affect the descrfption of the intermediary comIf an intermediary phase exists over a range of compositions, it should be described as pound. a solution phase between the pure component elements at the temperature under consideration. For such.a phase the SGTE method would always give a kink in C, at the temperatures of melting of the component elements. This is not more serious than the kink in Cp of the liquid phase.in low-melting alloys at the melting points of the component elements.
J.-O. ANDERSSON et al.
6 RELATIVE
FIG.
TE”PERINRE.
1.66
6.48 RELATlVE
T/To
5
FIG.
The temperature dependence of the difference in CP between the solid and liquid phases according to the SGTE method of extrapolation across the melting point, To.
The
P.00
T/lo
6 correction
the straight-line culated by the polat ion.
FIG.
TMPERATUFZ,
to
G”-G~
relative
aFpr%imation, SGTE method of
to
calextra-
7
The heat capacity of hcp, bee and liquid Zr. Curves 2 describe extrapolations outside the range of stability of thephases,obtained by using equations from ref. 7, derived inside the range of stability. Curves 3 were obtained by the SGTE method of extrapolation and were actually proposed in ref. 7. 25 98%
1180
I708
2388
2900
3588
Acknowledgement The authors gratefully with the colleagues in SGTE. Technical Development.
acknowledge many discussions on the This work was financially supported
lattice by the
stability Swedish
problem Board for
References 1. 2. 3. 4. 5. 6. 7.
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