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Ehrenfest relations in crystalline media
Viswanathan, R. Rajagopal, E. S. 1961
Physica 27 765-767
LETTER Ehrenfest
TO THE
relations
EDITOR
in crystalline
media
The Ehrenfest relations for the second order phase transitions are well-known and a clear discussion of the various considerations obeyed at these transitions is given by Temperley 1) and Ubbelohde 2). However, most of the discussions are confined to the case of fluids, where P and V are sufficient to describe the mechanical state. In the case of solids, one has to consider the six stress and strain components 74 and E(. A general treatment of phase transitions is given by T i s z a 3) and Semen c h e n ko 4) who discuss the stability conditions in solids. Other thermodynamic relations in solids have been considered by Verhoogen 5), MacDonald 6, and Li and co-workers 7)*)e). It is of interest to see whether a generalization of the Ehrenfest relations can be obtained in the case of crystals. Considering the second order transition equilibrium between two phases to be represented by a curve, one can write down G(Te + AT, 7( + ATE) in terms of G(To,,%) by a Taylor series, as G(T” + AT, T”, + Aq)
= G(T”, $)
+
+ + I(s),,
(dT)2 + 2 (z),,[dT
+ (+$)T
(dn d,))
+
dn) +
...
where i, j, k = 1, 2, . . . . 6 and k # i, j. At TO,,! both the phases are in equilibrium. In this connection Bridgman lo), Oldenburger ii), Fisher 12) and various others have shown how the equilibrium curve rather than the equilibrium point occurs more often in practice. The condition that B/BT(Gl - Ge) (represented as a/aT Gi-s henceforth) vanishes, gives
The conditions that a/&r(Gi -
Gs) = 0, give six equations of the type (2b)
Now the increase in the internal energy is dU = TdS-
VX,td.e(
(3)
where C 76 da4 is the strain energy per unit volume. The Gibb’s function is G=
Vc,.,,,
U--S+
(4)
From eqns. (3) and (4) one gets z,=
From
(5) one gets
(5)
-
(a2G/aT2)1, = -
-
C,/T,
the specific
765 -
heat
at constant
stress,
R. VISWANATHAN
766
AND
E. S. RAJAGOPAL
(a2G/aTaTi)., = Vjlrthe thermal expansion coefficient and (@G/&&~)r,., the isothermal compliance constant. Eliminating dT and dTi from the eqns. (2a) and (2b) one gets
- C,1-2/T V
a:-2
ai-
Sll
Ai-2
l-2 s21
ai-2
.
.
.
ng-2
. .
. . . .
. . . .
. . . .
l-2 ‘16 l-2 S26
..
..
..
l-2 s66
l-2
..
1*--z S61
Expanding
zzz
0
= VQ
(6)
(6) and introducing the stiffness constants:
C;-2
z
-
T’V
;
C&-2
Ail-2
(7)
Ii’-2
i,i=l
The axes of reference has been chosen quite arbitrarily and by a suitable chaise of them one can make d74 = d7s = d76 = 0, that is one has a triaxial stress system. Then, eqn. (7) becomes C)-2
= _ TV
;
C&s Al-2 A;-”
(8)
i,j=l
For the particular case of a cubic crystal, with xi as the axes eqn. (8) reduces to C,14/TV
_ _
(9)
(Al-2)2Jj31-2
where A = tr + 1s + As is the volume coefficient of expansion and B=
3/(&l
+
2G2)
=
3(Sll +
2s12)
is the cubical compressibility. Experimental verification of the eqns. (7) and (8) is not possible because of the difficulty in determining the specific heat at constant stress. On the other hand, the usual experiments are conducted in an atmosphere of air or other gases which impresses on the materials a uniform hydrostatic pressure. In such a case, dri = dra = drs = d$. d74 = d7s = d7e = 0, C7 = C+ and eqns. (1) and (3) will involve only dTr, dT2 and dTs. Hence from the eqns. (2a) and (2b) one gets dT
- fC;-2)
+ V d+ (A#-” + A;-2 + A;-“)
V dT A;-2 + V dp ;
= 0
s&T” = 0.
j=l
(10)
Eliminating dT/dp from eqns. (10) one has C;-g/TV
(A1-2)2
=
p-2
(11)
The eqns. (9) and (11) are nothing but the usual Ehrenfest relation. Moreover, from the four eqns. of (10)one can also get C;-s/TV =
_/p-2~;-2//4-2
=
-Al-21:~2//j;-2 =
ai-2
-111-2B1--2 3
(14
EHRENFEST
RELATIONS
IN CRYSTALLINE
MEDIA
767
where
are the linear compressibilities related to cubical compressibility by /7 = /Ii + /?Q+ Bs. The expression (12) can be easily verified for crystals, since the quantities involved are measurable. Some of the common crystals with which the verification can be done are rochelle salt (T, N 26”)rs, potassium dihydrogen phosphate (T, - 123’K) i4) and arsenate ( Te - 97°K) r4), potassium dideuterium phosphate (T, .- 213°K) 14) am15), barium titanate (T, 120°K. 5°K) is), ammonium monium chloride (T, -243’K) chromate (T, - 128”K, 155”K, 268°K) r7), quartz (T, -573OC) 1s) and sodium nitrate (T, N 290°C) IQ). Thus one finds that even in the crystalline media the usual Ehrefest relations are valid. In this case, eqn. (12) is a more general relation which does not appear to be known earlier as in the case with eqn. (8). One also notices that purely thermodynamic relations do not yield further information. In order to estimate the magnitudes of the jumps in the second derivatives of G one must use other statistical mechanical considerations, see e.g. Dzialoshinskii and Lifshitz QQ),Slater 21) and others. Acknowledgement. The authors thank Professor R. S. Krishnan for his guidance and encouragement. One of them (R.V.) thanks the Indian Institute of Science for the award of a Research scholarship and the other (E.S.R.) thanks the Council of Scientific & Industrial Research for the award of a Senior Research Fellowship. R. VISWANATHAN E. S. RAJAGOPAL Department of Physics, Indian Institute of Science, Bangalore, India. Received l-4-6 1 REFERENCES
1) Temperley, 2) IJbbelohde, 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
H. N. V., Changes of State, Cleaver-Hume, London, (1956). A. R., Quart. Rev. (London) 11 (1957) 246. Tisza, L., Phase Transformations in Solids, Eds. R. Smouluchowski, J. E. Mayer A. Weyl, Wiley, New York (1954). Semenchenko, V. K., Zh. Fiz. Khim. 21 (1945) 1461; 25 (1950) 121. Verhoogen, J., Amer. geophys. UnionTrans. 32 (1951) 251. MacDonald, G. J., Amer. J. Sci. 255 )1957) 266. Ting, T. W. and Li, J. C. M., Phys. Rev., 106 (1957) 1165. Ting, T. W. and Li, .J. C. K., J. them. Phys. 27 (1957) 693. Li, J. C. M. and Kiang, H. S., J. them. Phys. 32 (1960) 1644. Bridgman, Oldenburger,
P. W., Phys. Rev. 70 (1946) 425. R., Phys. Rev. 70 (1946) 433.
Fisher, G., Am. J. Phys. 2S (1957) 100. Rusterholz, A., Helv. phys. Acta 8 (1935) Bantle, Lawson,
and W.
W., Helv. phys. Acta A. W., Phys.
39.
12 (1939) 279;
15 (1942) 373.
Rev. 57 (1940) 417.
KLnzig, W., Blattner, H. and Merz, W., Helv. phys. Acta 22 (1949) 35. Jaffray, J., C. R. Acad. Sci. (Paris) 241 (1955) 1114. Silverman, S. M., J. them. Phys. 25 (1956) 1081.
Kornfel’d, M. I. and Chudinov, A. A., Zh. eksper. teor. Fiz. (USSR) 33 (1957) 33; Soviet Physics, JETP R (1958) 26. I. E. and Lifshitz, E. M., Zh. eksper. teor. Fiz. 33 (1957) 299; Soviet 20) Dzialoshinskii, Physics, JETP 6 (1958) 233. J. C., J. them. Phys. 2 (1943) 16. 21) Slater,
Physica 27 765-767
LETTER Ehrenfest
TO THE
relations
EDITOR
in crystalline
media
The Ehrenfest relations for the second order phase transitions are well-known and a clear discussion of the various considerations obeyed at these transitions is given by Temperley 1) and Ubbelohde 2). However, most of the discussions are confined to the case of fluids, where P and V are sufficient to describe the mechanical state. In the case of solids, one has to consider the six stress and strain components 74 and E(. A general treatment of phase transitions is given by T i s z a 3) and Semen c h e n ko 4) who discuss the stability conditions in solids. Other thermodynamic relations in solids have been considered by Verhoogen 5), MacDonald 6, and Li and co-workers 7)*)e). It is of interest to see whether a generalization of the Ehrenfest relations can be obtained in the case of crystals. Considering the second order transition equilibrium between two phases to be represented by a curve, one can write down G(Te + AT, 7( + ATE) in terms of G(To,,%) by a Taylor series, as G(T” + AT, T”, + Aq)
= G(T”, $)
+
+ + I(s),,
(dT)2 + 2 (z),,[dT
+ (+$)T
(dn d,))
+
dn) +
...
where i, j, k = 1, 2, . . . . 6 and k # i, j. At TO,,! both the phases are in equilibrium. In this connection Bridgman lo), Oldenburger ii), Fisher 12) and various others have shown how the equilibrium curve rather than the equilibrium point occurs more often in practice. The condition that B/BT(Gl - Ge) (represented as a/aT Gi-s henceforth) vanishes, gives
The conditions that a/&r(Gi -
Gs) = 0, give six equations of the type (2b)
Now the increase in the internal energy is dU = TdS-
VX,td.e(
(3)
where C 76 da4 is the strain energy per unit volume. The Gibb’s function is G=
Vc,.,,,
U--S+
(4)
From eqns. (3) and (4) one gets z,=
From
(5) one gets
(5)
-
(a2G/aT2)1, = -
-
C,/T,
the specific
765 -
heat
at constant
stress,
R. VISWANATHAN
766
AND
E. S. RAJAGOPAL
(a2G/aTaTi)., = Vjlrthe thermal expansion coefficient and (@G/&&~)r,., the isothermal compliance constant. Eliminating dT and dTi from the eqns. (2a) and (2b) one gets
- C,1-2/T V
a:-2
ai-
Sll
Ai-2
l-2 s21
ai-2
.
.
.
ng-2
. .
. . . .
. . . .
. . . .
l-2 ‘16 l-2 S26
..
..
..
l-2 s66
l-2
..
1*--z S61
Expanding
zzz
0
= VQ
(6)
(6) and introducing the stiffness constants:
C;-2
z
-
T’V
;
C&-2
Ail-2
(7)
Ii’-2
i,i=l
The axes of reference has been chosen quite arbitrarily and by a suitable chaise of them one can make d74 = d7s = d76 = 0, that is one has a triaxial stress system. Then, eqn. (7) becomes C)-2
= _ TV
;
C&s Al-2 A;-”
(8)
i,j=l
For the particular case of a cubic crystal, with xi as the axes eqn. (8) reduces to C,14/TV
_ _
(9)
(Al-2)2Jj31-2
where A = tr + 1s + As is the volume coefficient of expansion and B=
3/(&l
+
2G2)
=
3(Sll +
2s12)
is the cubical compressibility. Experimental verification of the eqns. (7) and (8) is not possible because of the difficulty in determining the specific heat at constant stress. On the other hand, the usual experiments are conducted in an atmosphere of air or other gases which impresses on the materials a uniform hydrostatic pressure. In such a case, dri = dra = drs = d$. d74 = d7s = d7e = 0, C7 = C+ and eqns. (1) and (3) will involve only dTr, dT2 and dTs. Hence from the eqns. (2a) and (2b) one gets dT
- fC;-2)
+ V d+ (A#-” + A;-2 + A;-“)
V dT A;-2 + V dp ;
= 0
s&T” = 0.
j=l
(10)
Eliminating dT/dp from eqns. (10) one has C;-g/TV
(A1-2)2
=
p-2
(11)
The eqns. (9) and (11) are nothing but the usual Ehrenfest relation. Moreover, from the four eqns. of (10)one can also get C;-s/TV =
_/p-2~;-2//4-2
=
-Al-21:~2//j;-2 =
ai-2
-111-2B1--2 3
(14
EHRENFEST
RELATIONS
IN CRYSTALLINE
MEDIA
767
where
are the linear compressibilities related to cubical compressibility by /7 = /Ii + /?Q+ Bs. The expression (12) can be easily verified for crystals, since the quantities involved are measurable. Some of the common crystals with which the verification can be done are rochelle salt (T, N 26”)rs, potassium dihydrogen phosphate (T, - 123’K) i4) and arsenate ( Te - 97°K) r4), potassium dideuterium phosphate (T, .- 213°K) 14) am15), barium titanate (T, 120°K. 5°K) is), ammonium monium chloride (T, -243’K) chromate (T, - 128”K, 155”K, 268°K) r7), quartz (T, -573OC) 1s) and sodium nitrate (T, N 290°C) IQ). Thus one finds that even in the crystalline media the usual Ehrefest relations are valid. In this case, eqn. (12) is a more general relation which does not appear to be known earlier as in the case with eqn. (8). One also notices that purely thermodynamic relations do not yield further information. In order to estimate the magnitudes of the jumps in the second derivatives of G one must use other statistical mechanical considerations, see e.g. Dzialoshinskii and Lifshitz QQ),Slater 21) and others. Acknowledgement. The authors thank Professor R. S. Krishnan for his guidance and encouragement. One of them (R.V.) thanks the Indian Institute of Science for the award of a Research scholarship and the other (E.S.R.) thanks the Council of Scientific & Industrial Research for the award of a Senior Research Fellowship. R. VISWANATHAN E. S. RAJAGOPAL Department of Physics, Indian Institute of Science, Bangalore, India. Received l-4-6 1 REFERENCES
1) Temperley, 2) IJbbelohde, 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
H. N. V., Changes of State, Cleaver-Hume, London, (1956). A. R., Quart. Rev. (London) 11 (1957) 246. Tisza, L., Phase Transformations in Solids, Eds. R. Smouluchowski, J. E. Mayer A. Weyl, Wiley, New York (1954). Semenchenko, V. K., Zh. Fiz. Khim. 21 (1945) 1461; 25 (1950) 121. Verhoogen, J., Amer. geophys. UnionTrans. 32 (1951) 251. MacDonald, G. J., Amer. J. Sci. 255 )1957) 266. Ting, T. W. and Li, J. C. M., Phys. Rev., 106 (1957) 1165. Ting, T. W. and Li, .J. C. K., J. them. Phys. 27 (1957) 693. Li, J. C. M. and Kiang, H. S., J. them. Phys. 32 (1960) 1644. Bridgman, Oldenburger,
P. W., Phys. Rev. 70 (1946) 425. R., Phys. Rev. 70 (1946) 433.
Fisher, G., Am. J. Phys. 2S (1957) 100. Rusterholz, A., Helv. phys. Acta 8 (1935) Bantle, Lawson,
and W.
W., Helv. phys. Acta A. W., Phys.
39.
12 (1939) 279;
15 (1942) 373.
Rev. 57 (1940) 417.
KLnzig, W., Blattner, H. and Merz, W., Helv. phys. Acta 22 (1949) 35. Jaffray, J., C. R. Acad. Sci. (Paris) 241 (1955) 1114. Silverman, S. M., J. them. Phys. 25 (1956) 1081.
Kornfel’d, M. I. and Chudinov, A. A., Zh. eksper. teor. Fiz. (USSR) 33 (1957) 33; Soviet Physics, JETP R (1958) 26. I. E. and Lifshitz, E. M., Zh. eksper. teor. Fiz. 33 (1957) 299; Soviet 20) Dzialoshinskii, Physics, JETP 6 (1958) 233. J. C., J. them. Phys. 2 (1943) 16. 21) Slater,