- Email: [email protected]
Soret effect in solid—II
J. Phys. Chem. Solids Vol. 48, No. 6, pp. 579-586, Printed in Great Britain.
0022.3697187 $3.00+0.00 0 1987PergamonJournals Ltd.
1987
SORET EFFECT IN SOLID-II S. A. AKBAR, M. KABURAGI~ and H. SATO School of Materials Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. (Received
3 September
1986; accepted 21 November
1986)
Abstract--Atomistic features of the Soret effect in multi-component systems, especially those in binary systems, are treated by the path probability method of irreversible statistical mechanics based on a rigid lattice model as an extension of our earlier treatment of the effect in one-component systems. While in one-component systems, particles tend to accumulate toward the low temperature side in the present model, in two-component systems, the constituent components can accumulate on opposite sides as long as the vacancy concentration is low. Explanations of the features in the multi-component Soret effect which do not appear in one-component systems are specifically given. The difference between the Soret effect and the demixing effect under a temperature gradient is also discussed. Keywords: Soret effect, thermal diffusion, thermotransport,
1. INTRODUCTION
When a multi-component system is placed under the influence of a temperature gradient, concentration gradients of constituent components are built up as a response to the temperature gradient. This phenomenon is known as the Soret effect. The phenomenon falls in the category of thermal diffusion in general and involves coupling of atomic diffusion and heat conduction. The theoretical description of the Soret effect has been based on irreversible thermodynamics [l-l 01, so that it is difficult to understand the atomistic features of the phenomenon. Recently, Kikuchi et al. [l 1] treated the problem of thermal diffusion in binary systems by utilizing the path probability method (PPM) [12] of irreversible statistical mechanics on an atomistic basis. They could thus derive a set of linear equations corresponding to the Onsager equations for thermal diffusion analytically. Based on those equations, Wada et al. [13] treated the Soret effect in one-component systems in detail; this work will be referred to as I hereafter. By a one-component system we mean that only one component of the system can change its distribution. Here, we try to extend the treatment to many-component systems. As far as the Soret effect is concerned, binary systems (i.e. two components of the system can change their distribution) represent general many-component systems. Therefore, the present paper is devoted to treating the effect in binary systems. The Soret effect represents zero macroscopic atomic flow under a temperature gradient. In other words, the balance of flow under the temperature gradient is established by the flows under the chemical potential gradients of the constituent components created by thermal diffusion. The concentration gra7 On leave of absence from College of Liberal Arts, Kobe University, Kobe, Japan.
rigid lattice model, multi-component systems.
dients of the constituent species thus established represent the Soret effect. The relation can thus be determined by the Onsager equation for thermal diffusion under the condition of zero macroscopic flow of constituents, which leads to the relation between the concentration gradients and the temperature gradient. In I, it was shown that, in a one-component system, the rigidity (or the inverse of the compressibility), which is defined as the resistance of the system to change the vacancy concentration [13] (the change of the chemical potential caused by changing the density of the particle) was the major quantity which determines the magnitude of the Soret effect. Therefore, in the low vacancy concentration limit, the effect becomes extremely small, because in the rigid lattice model the compressibility tends to zero in the low vacancy limit. Also, it was concluded that the Soret effect in a one-component system can have only one sign (atoms accumulate toward the low temperature side) based on the model adopted [13]. This conclusion, however, does not explain experimental observations that, in some interstitial solid solutions such as C or N in Fe, the solute atoms tend to accumulate toward the high temperature side [20,21], or that in superionic conductors, the thermoelectromotive force, which can be explained parallel to the Soret effect, can either take a plus or minus sign [22,23] depending on the material. In order to find possible sources of discrepancy within the rigid lattice model, we tried to extend the treatment to twocomponent systems to see if any further factor should appear in determining the sign of these effects. Here, we will treat the Soret effect in binary systems in the same fashion as that of I. Therefore, we will emphasize the important features of binary systems which are different from those of onecomponent systems. At the same time, we try to clarify the differences between the Soret effect and the 579
580
S.
A.
AKBAR et al.
demixing effect in binary oxides under a temperature gradient [14, 18, 191. 2. THERMAL DIFFUSION BY THE PATH PROBABILITY METHOD In this section, we briefly review the model and summarize formulae necessary for deriving the Soret effect from our previous treatment of thermal diffusion in binary systems by the PPM [ll]. 2.1. Model As in Ref. [11], the thermal diffusion problem, is here, also treated based on the rigid lattice model. In this model, a system of a fixed number of idealized crystal lattice points is taken and constituent atoms and vacancies are distributed on these lattice points. Interactions among constituent atoms, generally interactions only between nearest neighboring atoms, are then introduced. Based on these interactions, the distribution of the constituent components is calculated under the assumption of local equilibrium by the Cluster Variation method CVM, a static version of the PPM. Atomic migration via vacancies is treated by the PPM by introducing kinetic parameters such as jump frequencies for constituent atoms. For diffusion under a temperature gradient in the present model, the effect of the temperature gradient appears through the activation energy in the jump frequency of the form tIeeuikT. Both fl and u, here, are assumed to be unaffected by temperature. Therefore, the model implicitly assumes that atoms move from the high to the low temperature side under the influence of the temperature gradient, unless there are other opposing terms. 2.2. Onsager equation Based on the model explained in Section 2.1. a set of linear equations corresponding to the Onsager equations for thermal diffusion for two-component systems is derived under a steady-state condition by the pair approximation of the PPM in the following form [ll]:
here to show the existence of the energy carried by other than atomic diffusion such as by phonons and electrons which essentially establish the temperature gradient [ll]. However, because we are interested in only terms by atomic flow here, the existence of this term, although this is a major term in the energy flow in actual systems, will be neglected in the following discussion. Eventually, a concentration gradient of vacancies will be set up due to the change of distribution of atoms under the influence of the temperature gradient. In the rigid lattice model, the concentration gradient of vacancies can be specified by the existence of the internal pressure gradient in the system. Therefore, at a reference plane, the form of the Onsager equation is unchanged as long as the extra boundary condition which determines the distribution of vacancies exists [14]. The Onsager matrix elements thus derived are shown to satisfy the reciprocal relations, i.e. 42
=
(2.2a)
L21
and L,E = L,
(i = 1 and 2).
(2.2b)
The Onsager matrix elements L,‘s and L$s are expressed analytically in terms of static and kinetic parameters such as pair interaction energies eii and effective jump frequencies. The equilibrium distribution is specified in the pair approximation by xi (probability of finding an atom i on a lattice point; in a disordered system this represents the concentration of the ith species) and yii (probability of finding an i-j pair) in the CVM. The pair variables yij are conveniently expressed in terms of site variables qi and qi as [16] Y, = qiqjK,T’,
(2.3)
Kij = e-p%
(2.4)
where
and the qis are determined by a set of simultaneous equations Jz = -L&,
- L,,ci + L&j
pi = xi = qi i qjK,
(2.5)
j=l
JE = -LE,ci, - L,d
+ (L, - K*T/B)j.
(2.1)
Here, C,X~ = &( pi being the chemical potential for the ith component and fi = I//CT). The dots represent the derivatives along the length of the specimen (the x-direction). The terms J, and J2 represent atomic flows of two species 1 and 2, respectively and JE represents the energy flow. Equation (2.1) is defined at a reference plane perpendicular to the direction of the gradients, or with respect to the laboratory frame fixed at a lattice plane. The term (K*T/fl)d is added
using the concentration pi or xi in the disordered state. Here, 3 is used to denote a vacancy as one of the constituents. By utilizing the atomistic quantities described so far, the Onsager’s matrix elements are derived as [ 111 L, = Yie(l - 2Q,$/H),
L, = 2YieQjGj/H,
(2.6)
581
Soret effect in solid L,r = (ui + E,)LB + (Uj+ q)L, + ALiz, Lzz = i Yk[(Ui+ L’i)’ + am i-1
where 6, = exp[(o - l)F, - In F2],
- ZQj(t+iEf
(2.11a)
Fi =
B[Qt~~+ Q+ti+ Q.x(G + tu)l,(2.11b)
F2 =
Qieh
- b+jEiEj)/H], + Qjeh
+ Q,ek+,).
(2.1 lc)
where AL,E = - 7Yie(Qj/H)[(c,
- ct)$
- (ci/ - 6f)Gj],
In other words, eqn (2.10) gives an interpretation of the effective jump frequency in a binary interacting system. The quantity
(Ae)’ = c;iQ, + &Q2 - (c;)~, LJ,?= ui - (2~ - l)(Qieii + Q,e,) 6: =
(2.12)
ci,Q, + GZQZ> can be termed the effective activation energy and 6 the effective attempt frequency.
ff = 7(*1+
G2) - 5(G,Q2 + G2Q,),
Y,= AtGiy,T.
(2.7)
Here, Y, represents the actual jump probability of the ith species in the time span of At in the equilibrium state. It should be noticed that y,, or the probability of finding a vacancy at the nearest neighbor of atoms of the ith species, appears in addition to the commonly defined jump frequency Gi. Furthermore, the expression for tli is derived by the CVM as [17] ai=BPi=2w
ln(dq3)-(2~
- l)ln(PilL%h
(2.8)
and, hence, d, can be evaluated analytically. The meaning of these terms for a one-component system is explained in detail in I. Unless specifically required, explanations of individual terms are not attempted. The expression Gi in eqns (2.6) and (2.7) represents the jump frequency of the ith species including the effect of bond breaking (the effect of the immediate surroundings on the jump frequency). A physical image of bond breaking has been illustrated in I. The jump frequency, Gi, is derived as [ 161 CYijKtT’ 1 = Wi I &vi Pi i. i
*u-l ’
(2.9a)
2.3. Energy of transport and heat of transport We give the expressions of the energy of transport and the heat of transport for binary systems in a similar fashion to that in I. These quantities in irreversible thermodynamics are introduced as phenomenological quantities and, hence, their contents are not known. The advantage of the present treatment is that an atomistic description of these quantities is possible. The energy flux JE is expressed in terms of atomic fluxes as [14] JE = E:J, + E: J2 + &fi, where ET = (LiELii-- Li,Lij)/&
B = L,,L,,
- Lf,.
(2.14)
Equation (2.13) gives a physical interpretation of the quantities ET as the energy carried by unit Ilow of species i (i.e. for J, = 1) at uniform temperature (b = 0). These quantities are called the energies of transport. In eqn (2.13), these quantities are only phenomenological parameters in irreversible thermodynamics but, in this treatment, analytical expressions can be given via expressions L,‘s and L,‘s. Utilizing eqn (2.6) in eqn (2.14), the energy of transport can be expressed as
where
E: = ui + 6: + Ai,
wi = 0, e-“ilkT.
(2.9b)
Here 20 represents the coordination number of the lattice (2~ = 8, for bee lattice). The jump frequency ti, (eqn (2.9a)) thus gives a local jump frequency. The expression G, can be expressed in an analogous fashion (as was done in I) as Gi=eexp[-B{ai-(2Cc
-l)(Q+ii+Q,+)}],
(2.10)
(2.13)
(2.15)
where Ai = (LjAL,
- LijALjE)@.
(2.16)
The term Ai represents a feedback effect [l l] due to the presence of other components. In a onecomponent system, this term, therefore, does not appear.
S. A. AKBAR et al.
582
By changing the variables, the heat of transport can be defined in a similar fashion based on irreversible thermodynamics. The entropy production dS/dt is written in terms of the driving forces dl and fl and their conjugate flows as dS - = J,j dt
-J&q
-J,&,.
(2.17)
If the driving force di is expressed as di =
(BrCi)
=
B/A
+
P+B>
3. DERIVATION
The Soret effect is calculated from the Onsager equation under the condition that the macroscopic atomic flows J, and J2 are zero. Zero macroscopic flow is attained when the flow under the temperature gradient is counterbalanced by the flows under the concentration gradients created by thermal diffusion. Utilizing the expressions given in section 2, the expression for the Soret effect is readily obtained. Using the conditions
(2.18)
/3& and fl can be taken as driving forces. Therefore, utilizing eqn (2.18) in eqn (2.17), we can rewrite entropy production in terms of driving forces &ii and b and their conjugate flows as
OF THE SORJCT EFFECT
and
J,=O
J,=O
in eqn (2.22), the following equations j.i, +
F
T = 0,
(3.2a)
ii,++0 where
Jo= JE-PIJ, -PzJ,.
(2.20)
(3.1)
(3.2b)
are obtained. Because the chemical potentials p, and pZ are functions of concentrations and temperature, fir and fi2 are given as follows:
By definition Je represents the heat Row and can be expressed in terms of atomic flows as 1143 Je=Q:Ji+QfJ,-&‘.
(2.21)
Equation (2.21) gives physical interpretations of the quantities Q: and k The quantity QT is interpreted as being the heat carried by unit flow of atom i at uniform temperature (d = 0) and is called the heat of transport. The notation for the heat of transport Qt should not be confused with the quantity Qi defined in eqn (2.7). The term a is the thermal conductivity due to atomic flow. Atomic flows (eqn (2.1)) can thus be expressed in terms of the heats of transport as [14]
J,=
By utilizing eqn (3.3) in eqn (3.2) and solving for
we obtain
dp,
(3.4a)
--_-
dT
-$,+$f)-+(r2+$$), dp, -=dT
(3.4b)
or in a matrix form In this fashion, the heats of transport are specifically identified in terms of atomistic quantities. For a binary system, based on the rigid lattice model, heats of transport are specifically expressed as [14] Q:=u~+E:+A~--~~~=E:-~~.
@I*-H*)
(2.23)
The quantities on the right hand side of eqn (2.23) have been introduced earlier (eqns (2.6), (2.7) and (2.16)).
=--
i
T
-ap -1 (IS* - H*).
0ap
Equation (3.4) thus represents the concentration
(3.4c) dis-
Soret effect in solid tributions of the two components for the Soret effect. The inverse matrix in eqn (3.4~) represents the 90s called ~m~re~ibility matrix. Here, HF is the partial molar enthatpy [13] defined as
583
Here, ui is a part of the activation energyof motion defined in eqn (2.9b) and is positive. Substituting eqn (3.8) in eqn (3.4), we obtain (3.9a)
and would ap~rop~a~ly be called the enthalpy of transport. For the sake of simplicity, the inverse matrix in a many~omponent system is written hereafter simply as (~~/~~)-I. In vector form, the expression for man~component systems (Appendix) can be written in the same form as eqn (3.4~). As discussed in I, the signs of Et - H: and the determinant l&/+ I are all positive and hence the relative magnitudes af EF - Ht and each component of the compressibility matrix determine the sign of distribution of the two components as shown in eqns (3.4a) and (3.4b). Again, the concentration of vacancies plays a decisive role, especiatly when the cormentration of vacancies is low. Equation (3.4) represents the Soret effect in a two-component system in the general case. Equation (3.4) is often represented as
and the sign of the Soret effect or the thermoelectric power in superionic conductors is said to be determined by the sign of Q*. This derivation, however, ignores the temperature dependence of ~1, and the appearance of (E* -H*) instead of Q* is a mare exact expression as pointed out in I. In addition, the value of Q* evaluated by the present model can vary from -co to + 00 due to the variation of ,u and the sign of Q* cannot be determined reliably. On the other hand, the value E* - H* can have a finite value. The corresponding equation of eqn (3.4), in a one-component system was derived in I as
(3.9b) The Co~pon~ng equation for a one-component Soret effect eqn (3.7) can be expressed as
dp, -=----_ dT
4
(3.10)
kT2 _!_ 0 P3
It is clearly seen that the concentration of vacancies is the determining term in the comp~ssibi~ity matrix as seen in eqn (3.10). Because, there does not exist any cross term, eqn (3.7) or (3.10) suggests that in one-component systems, the Soret effect has only one sign (even in a non-ideal solution, Ef - Hf was found to be positive 119) and the ~~rnp~sibiI~ty (represented by (l/p,)-‘) is the major term determining the magnitude of the Soret effect, esp&aIly in the low-vacancy con~n~a~on limit. This is due to the fact that in the lattice gas model given here, the internal pressure P is the conjugate variable of the vacancy concentration, or p3. This situation is the same for many-component systems. Let us examine binary systems or eqn (3.9) with this fact in mind. The term dp,/dT in eqn (3.9b) is essentially a change of the vacancy concentration along the specimen (along the temperature gradient) and is related to the com-
‘.Oj
(3.7)
Here, the cross-terms (off-diagonal terms in the compressibility matrix) do not enter and the physical meaning of the competition of the compressibility matrix (a~/&)-~ and (E+ - H*) is clearly understood. To understand the qualitative features of the problem, let us discuss the most simple case of ideal solution 1141 (eil = 0). Under this assumption, the chemical potential (eqn (2.8)) can be simplified as (3.8a) and Eif-If:=&.
(3.8b)
Fig. I, Tw0-~mponent Soret effect for low vacancy conizntration. b<: total Iength of the specimen; pt. p2: e0ncentration of the two ~rn~~n~ tiX, y : activation energy of motion for ~o~pou~~ I and 2, respectively; T,, T,: temperatures on the two surCa_ p3: vacancy
584
S. A. AKBARet al. demixing effect under a temperature gradient only. The demixing effect under a temperature gradient only should be an open case when a vacancy concentration gradient inside the specimen does not exist [14]. In Ref. [14], we carefully discussed the difference between the open case and the closed case under such idealized boundary conditions. 4. DISCUSSION
The treatment of the Soret effect has been extended to binary and many-component systems in the rigid lattice model. The extension is straightforward. However, one distinct difference exists. In the one-component system treated earlier, the effect is determined by the -0.5 -0.1 0.1 0.3 -0.3 0.5 competition of (E* -H*), the energy of transport minus the enthalpy of transport, and the comFig. 2. Two-component Soret effect for high vacancy pressibility (~~~~~)-I as given in eqn (3.7). Because concentration. both quantities are positive, the present model predicts that particle tends to segregate towards the low tem~rature side. In the rigid lattice model, pressibility of the system. In the low vacancy concentration limit, the ~ompre~ibility is very low (i.e. (~~/a~)-’ tends to zero in the Iow vacancy concentration limit and the change of distribution of vadp, fdT = 0) as in the one-component case. Hence, cancies does not occur in practice, and the Soret effect depending on the activation energy difference, the is negligible in such a case. In many-component two components redistribute in different directions relation is systems, this replaced by without changing the vacancy concentration. In other words, if one component preferentially segregates to (Z$/i?p)-‘(E* - H*) as defined in eqn (3.k). Alone side, the other segregates to the opposite side though det]+/ap 1 and (E: - H:) are positive, the (illustrated in Fig. 1). This is due to the difference in matrix multiplication (ap/ap)-‘(E* - H*) creates the magnitude of (JP - H*) of each component and the combination of (E* -H*) and the compressibility component having the larger value of (E* - H*) matrix elements in the numerator in eqn (3.4) and, segregates towards the low temperature side, As the depending on the relative magnitude of (E* - H*) for each component, the segregation of components can vacancy concentration increases, it can change and, occur in either direction. An essential factor in the at a certain stage, the segregation of both components can occur to the lower temperature side compressibility matrix is again the concentration of (Fig. 2). In general, therefore, the Soret effect is vacancies of the system. This is because the vacancy concentration and the internal pressure are conjugate determined by the relative ma~itude of (E* -H*) variables and one determines the other. As in a and the con~ntration of vacancies. system, the deviation from a hoBoth the demixing effect and the Soret effect one-com~n~t mogeneous distribution of vacancies does not occur discussed so far belong to general demixing phenomin practice when the concentration of vacancies is ena, but are with two idealized boundary conditions. low. However, redistribution of constituent atoms In this respect, it is worthwhile to point out the can occur without changing the vacancy distribution difference between the Soret effect and the demixing in many-component systems. Figure 1 represents effect under a temperature gradient. Some demixing experiments have been carried out for binary oxides such a case for binary systems. If the vacancy is large enough, redistribution of under a temperature gradient under a variety of boundary conditions [18, 191. Among these, experi- vacancies readily occurs. Because the thermal diffusion process in the present model tends to drive ments carried out in the presence of an oxygen atoms toward the low temperature side, if the reatmosphere in contact with the specimen can allow sistance of the redistribution of vacancies is low the flow of vacancies in and out of the specimen, and this is called the open case [14]. This corresponds to enough, all components can accumulate toward the low temperature side. Figure 2 for the binary system the idealized demixing effect. In the experiments under a temperature gradient on encapsulated sam- represents this case. In many-component systems, the distribution of components would be a combination ples, on the other hand, specimens are not in contact with an oxygen atmosphere and do not allow the flow of these two cases. Another major conclusion obtained in the treatof vacancies. This case is called the closed case. The ment is that the Soret effect should be corte~tly latter type of experiment corresponds to the Soret represented as in eqn (3.k) but not as in eqn (3.6) effect [14]. In other words, according to our where the sign ofQ* determines the sign of the effect. definition, the closed case does not represent the
o.o-
Soret effect in solid This difference apparently comes from the difference in handling eqn (3.2) in which p is treated as a function of p and Tin our treatment, while in many other treatments, p is treated only as a function of p, assuming an isothermal condition in deriving the Soret concentration distribution [18-211. It is worth mentioning here that, in the original treatment of the Soret effect by de Groot [l], the same conclusion as ours was drawn. The notion that the sign of the effect is determined by the sign of Q* may also come from expressions like eqn (3.2) where
dPi --. QT
dT=
T
As pointed out in I, it was not possible to explain in interstitial solid solutions that interstitial atoms segregate in either direction of the temperature gradient by our model for a one-component system. Although in many-component systems, each component can assemble in either direction of the temperature gradient, it does not seem proper to utilize the result of many-component systems for this explanation. Although the present rigid lattice model is generally utilized to discuss the phase diagram of alloy systems, it seems that it is necessary to utilize a different model for the assembly of interstitial atoms in order to explain the effect properly. As we discussed earlier, if solid solutions are placed under a temperature gradient, the composition can change due to diffusion [14]. This effect is generally called the demixing effect. A steady state demixing process which can occur in oxides was discussed in detail in another paper [14]. This effect and the Soret effect are cases under two idealized boundary conditions which can be treated analytically. These two idealized cases have been classified as the open case and the closed case, respectively. However, it should be mentioned that a variety of demixing processes can occur depending on the boundary conditions.
585
7. Callen H. B., Thermodynamics.
Wiley, New York (1963). 8. Peterson N. L., Solid Stare Physics (Edited by F. Seitz, D. Tumbull and H. Ehrenreich), Vol. 12. Academic Press, New York (1968). 9. Gillan M. J., Mass Transport in Solids (Edited by F. Beniera and C. R. A. Catlow). Plenum Press, New York (1981). 10. Howard R. E. and Manning J. R., J. them. Phys. 36, 910 (1962). 11. Kikuchi R., Ishikawa T. and Sato H., Physics a 123A, ?27 (J984).
12. Kikuchi R., Prog. rheor. Phys. (Kyoto), Suppl. No. 35, 1 (1966). 13. Wada K., Suzuki A., Sato H. and Kikuchi R., 1. Phys. Chem. Solia!s 46. 11951202 (1985). This uaner is referred to as I in’the text. . ’ _ _ 14. Akbar S. A., Kaburagi M., Sato H. and Kikuchi R., J. Am. Ceram. SOL, in print. 15. Kikuchi R., Phys. Rev. 81, 988 (1951). 16. Sato H., Noniraditional Methodr in Diffusion (Edited by G. E. Murch, H. Bimbaum and J. R. Cost), TMSAIME Conference Proceedings, Vol. 1984, pp. 203-235. 17. Kikuchi’ R. and Sato H., J. them. Phys. 51(l), 161 (1969). 18. Petuskey W. T. and Bowen H. K., J. Am. Ceram. Sot. 64(10), 611 (1981). 19. Stephenson G. B., M.S. Thesis, MIT (1981). 20. Shewmon P. G., Acta melall. 8, 605 (1960). 21. Oriani R. A., J. them. Phys. 34, 1773 (1961). 22. Kuwamoto H. and Sato H., Solid St. Ionics 5, 187 (1981). 23. Schiraldi A., Baldini P. and Pezzati E., Solid St. Ionics 9 and 10, 1187 (1983).
APPENDIX Sore1 effect in many-component systems The matrix representation, eqn (3.4c), for
binary systems
for the Soret effect holds for an n component system and can thus be written as
dp
dT=-T
’ f! -I@*
0
_H*).
ap
C-4.1)
Here, p represents the density and is defined as a column matrix
64.2) Acknowledgements-Discussions with Professor R. Kikuchi, Dr. A. Suzuki, Dr. T. Ishii and Dr. T. Ishikawa are gratefully acknowledged. The work was supported by the U.S. Department of Energy under the grant number DEEGO2-84ER45133.
1p.J while (ay/ap)-’
is the compressibility matrix given by
r%...F?!Q -’ ap,
REFERENCES 1. de Groot S. R., Thermodynamics of Irreversible Process. North-Holland, Amsterdam (195 1). 2. de Groot S. R. and Mazur P.. Non-eauilibrium Thermodynamics. North-Holland, Amsterdam (1962). 3. Shewmon P. G., Diffusion in Solids. McGraw Hill. New York (1963). 4. Howard R. E. and Lidiard A. B., Repl. Prog. Phys. 27, 161 (1964). 5. Howard R. E. and Lidiard A. B., Acia metail. 13, 443 (1965). 6. Allnatt A. R. and Chadwick A. V., Chem. Rev. 67,681 (1967).
ak
G Quantities E* and H* are the energy of transport ethalpy of transport represented by column matrices
E* =
lE.*J
and
586
S. A. AKBARet al.
and
(A.3
The nature determined (&I/@-‘. and it can
of the multi-component Soret effect is essentially by the nature of the compressibility matrix For a stable phase, the rigidity must be positive, generally be shown that det 1ap/ap I> 0. As is
discussed for binary systems the magnitude of the vacancy concentration plays a decisive role in determining the nature of the Soret effect. If the vacancy concentration is small, the vacancy distribution does not change in practice along the specimen and constituent components segregate to either sides of specimens according to the magnitude of (E* - H*) in such a way as to cancel out the change of vacancy concentrationWOn the other hand, if the number of vacancies is large, all constituent components can segregate to the low temperature side only. In other words, the qualitative feature of the Soret effect in multi-component systems is essentially the same as in the case of binary systems.
0022.3697187 $3.00+0.00 0 1987PergamonJournals Ltd.
1987
SORET EFFECT IN SOLID-II S. A. AKBAR, M. KABURAGI~ and H. SATO School of Materials Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. (Received
3 September
1986; accepted 21 November
1986)
Abstract--Atomistic features of the Soret effect in multi-component systems, especially those in binary systems, are treated by the path probability method of irreversible statistical mechanics based on a rigid lattice model as an extension of our earlier treatment of the effect in one-component systems. While in one-component systems, particles tend to accumulate toward the low temperature side in the present model, in two-component systems, the constituent components can accumulate on opposite sides as long as the vacancy concentration is low. Explanations of the features in the multi-component Soret effect which do not appear in one-component systems are specifically given. The difference between the Soret effect and the demixing effect under a temperature gradient is also discussed. Keywords: Soret effect, thermal diffusion, thermotransport,
1. INTRODUCTION
When a multi-component system is placed under the influence of a temperature gradient, concentration gradients of constituent components are built up as a response to the temperature gradient. This phenomenon is known as the Soret effect. The phenomenon falls in the category of thermal diffusion in general and involves coupling of atomic diffusion and heat conduction. The theoretical description of the Soret effect has been based on irreversible thermodynamics [l-l 01, so that it is difficult to understand the atomistic features of the phenomenon. Recently, Kikuchi et al. [l 1] treated the problem of thermal diffusion in binary systems by utilizing the path probability method (PPM) [12] of irreversible statistical mechanics on an atomistic basis. They could thus derive a set of linear equations corresponding to the Onsager equations for thermal diffusion analytically. Based on those equations, Wada et al. [13] treated the Soret effect in one-component systems in detail; this work will be referred to as I hereafter. By a one-component system we mean that only one component of the system can change its distribution. Here, we try to extend the treatment to many-component systems. As far as the Soret effect is concerned, binary systems (i.e. two components of the system can change their distribution) represent general many-component systems. Therefore, the present paper is devoted to treating the effect in binary systems. The Soret effect represents zero macroscopic atomic flow under a temperature gradient. In other words, the balance of flow under the temperature gradient is established by the flows under the chemical potential gradients of the constituent components created by thermal diffusion. The concentration gra7 On leave of absence from College of Liberal Arts, Kobe University, Kobe, Japan.
rigid lattice model, multi-component systems.
dients of the constituent species thus established represent the Soret effect. The relation can thus be determined by the Onsager equation for thermal diffusion under the condition of zero macroscopic flow of constituents, which leads to the relation between the concentration gradients and the temperature gradient. In I, it was shown that, in a one-component system, the rigidity (or the inverse of the compressibility), which is defined as the resistance of the system to change the vacancy concentration [13] (the change of the chemical potential caused by changing the density of the particle) was the major quantity which determines the magnitude of the Soret effect. Therefore, in the low vacancy concentration limit, the effect becomes extremely small, because in the rigid lattice model the compressibility tends to zero in the low vacancy limit. Also, it was concluded that the Soret effect in a one-component system can have only one sign (atoms accumulate toward the low temperature side) based on the model adopted [13]. This conclusion, however, does not explain experimental observations that, in some interstitial solid solutions such as C or N in Fe, the solute atoms tend to accumulate toward the high temperature side [20,21], or that in superionic conductors, the thermoelectromotive force, which can be explained parallel to the Soret effect, can either take a plus or minus sign [22,23] depending on the material. In order to find possible sources of discrepancy within the rigid lattice model, we tried to extend the treatment to twocomponent systems to see if any further factor should appear in determining the sign of these effects. Here, we will treat the Soret effect in binary systems in the same fashion as that of I. Therefore, we will emphasize the important features of binary systems which are different from those of onecomponent systems. At the same time, we try to clarify the differences between the Soret effect and the 579
580
S.
A.
AKBAR et al.
demixing effect in binary oxides under a temperature gradient [14, 18, 191. 2. THERMAL DIFFUSION BY THE PATH PROBABILITY METHOD In this section, we briefly review the model and summarize formulae necessary for deriving the Soret effect from our previous treatment of thermal diffusion in binary systems by the PPM [ll]. 2.1. Model As in Ref. [11], the thermal diffusion problem, is here, also treated based on the rigid lattice model. In this model, a system of a fixed number of idealized crystal lattice points is taken and constituent atoms and vacancies are distributed on these lattice points. Interactions among constituent atoms, generally interactions only between nearest neighboring atoms, are then introduced. Based on these interactions, the distribution of the constituent components is calculated under the assumption of local equilibrium by the Cluster Variation method CVM, a static version of the PPM. Atomic migration via vacancies is treated by the PPM by introducing kinetic parameters such as jump frequencies for constituent atoms. For diffusion under a temperature gradient in the present model, the effect of the temperature gradient appears through the activation energy in the jump frequency of the form tIeeuikT. Both fl and u, here, are assumed to be unaffected by temperature. Therefore, the model implicitly assumes that atoms move from the high to the low temperature side under the influence of the temperature gradient, unless there are other opposing terms. 2.2. Onsager equation Based on the model explained in Section 2.1. a set of linear equations corresponding to the Onsager equations for thermal diffusion for two-component systems is derived under a steady-state condition by the pair approximation of the PPM in the following form [ll]:
here to show the existence of the energy carried by other than atomic diffusion such as by phonons and electrons which essentially establish the temperature gradient [ll]. However, because we are interested in only terms by atomic flow here, the existence of this term, although this is a major term in the energy flow in actual systems, will be neglected in the following discussion. Eventually, a concentration gradient of vacancies will be set up due to the change of distribution of atoms under the influence of the temperature gradient. In the rigid lattice model, the concentration gradient of vacancies can be specified by the existence of the internal pressure gradient in the system. Therefore, at a reference plane, the form of the Onsager equation is unchanged as long as the extra boundary condition which determines the distribution of vacancies exists [14]. The Onsager matrix elements thus derived are shown to satisfy the reciprocal relations, i.e. 42
=
(2.2a)
L21
and L,E = L,
(i = 1 and 2).
(2.2b)
The Onsager matrix elements L,‘s and L$s are expressed analytically in terms of static and kinetic parameters such as pair interaction energies eii and effective jump frequencies. The equilibrium distribution is specified in the pair approximation by xi (probability of finding an atom i on a lattice point; in a disordered system this represents the concentration of the ith species) and yii (probability of finding an i-j pair) in the CVM. The pair variables yij are conveniently expressed in terms of site variables qi and qi as [16] Y, = qiqjK,T’,
(2.3)
Kij = e-p%
(2.4)
where
and the qis are determined by a set of simultaneous equations Jz = -L&,
- L,,ci + L&j
pi = xi = qi i qjK,
(2.5)
j=l
JE = -LE,ci, - L,d
+ (L, - K*T/B)j.
(2.1)
Here, C,X~ = &( pi being the chemical potential for the ith component and fi = I//CT). The dots represent the derivatives along the length of the specimen (the x-direction). The terms J, and J2 represent atomic flows of two species 1 and 2, respectively and JE represents the energy flow. Equation (2.1) is defined at a reference plane perpendicular to the direction of the gradients, or with respect to the laboratory frame fixed at a lattice plane. The term (K*T/fl)d is added
using the concentration pi or xi in the disordered state. Here, 3 is used to denote a vacancy as one of the constituents. By utilizing the atomistic quantities described so far, the Onsager’s matrix elements are derived as [ 111 L, = Yie(l - 2Q,$/H),
L, = 2YieQjGj/H,
(2.6)
581
Soret effect in solid L,r = (ui + E,)LB + (Uj+ q)L, + ALiz, Lzz = i Yk[(Ui+ L’i)’ + am i-1
where 6, = exp[(o - l)F, - In F2],
- ZQj(t+iEf
(2.11a)
Fi =
B[Qt~~+ Q+ti+ Q.x(G + tu)l,(2.11b)
F2 =
Qieh
- b+jEiEj)/H], + Qjeh
+ Q,ek+,).
(2.1 lc)
where AL,E = - 7Yie(Qj/H)[(c,
- ct)$
- (ci/ - 6f)Gj],
In other words, eqn (2.10) gives an interpretation of the effective jump frequency in a binary interacting system. The quantity
(Ae)’ = c;iQ, + &Q2 - (c;)~, LJ,?= ui - (2~ - l)(Qieii + Q,e,) 6: =
(2.12)
ci,Q, + GZQZ> can be termed the effective activation energy and 6 the effective attempt frequency.
ff = 7(*1+
G2) - 5(G,Q2 + G2Q,),
Y,= AtGiy,T.
(2.7)
Here, Y, represents the actual jump probability of the ith species in the time span of At in the equilibrium state. It should be noticed that y,, or the probability of finding a vacancy at the nearest neighbor of atoms of the ith species, appears in addition to the commonly defined jump frequency Gi. Furthermore, the expression for tli is derived by the CVM as [17] ai=BPi=2w
ln(dq3)-(2~
- l)ln(PilL%h
(2.8)
and, hence, d, can be evaluated analytically. The meaning of these terms for a one-component system is explained in detail in I. Unless specifically required, explanations of individual terms are not attempted. The expression Gi in eqns (2.6) and (2.7) represents the jump frequency of the ith species including the effect of bond breaking (the effect of the immediate surroundings on the jump frequency). A physical image of bond breaking has been illustrated in I. The jump frequency, Gi, is derived as [ 161 CYijKtT’ 1 = Wi I &vi Pi i. i
*u-l ’
(2.9a)
2.3. Energy of transport and heat of transport We give the expressions of the energy of transport and the heat of transport for binary systems in a similar fashion to that in I. These quantities in irreversible thermodynamics are introduced as phenomenological quantities and, hence, their contents are not known. The advantage of the present treatment is that an atomistic description of these quantities is possible. The energy flux JE is expressed in terms of atomic fluxes as [14] JE = E:J, + E: J2 + &fi, where ET = (LiELii-- Li,Lij)/&
B = L,,L,,
- Lf,.
(2.14)
Equation (2.13) gives a physical interpretation of the quantities ET as the energy carried by unit Ilow of species i (i.e. for J, = 1) at uniform temperature (b = 0). These quantities are called the energies of transport. In eqn (2.13), these quantities are only phenomenological parameters in irreversible thermodynamics but, in this treatment, analytical expressions can be given via expressions L,‘s and L,‘s. Utilizing eqn (2.6) in eqn (2.14), the energy of transport can be expressed as
where
E: = ui + 6: + Ai,
wi = 0, e-“ilkT.
(2.9b)
Here 20 represents the coordination number of the lattice (2~ = 8, for bee lattice). The jump frequency ti, (eqn (2.9a)) thus gives a local jump frequency. The expression G, can be expressed in an analogous fashion (as was done in I) as Gi=eexp[-B{ai-(2Cc
-l)(Q+ii+Q,+)}],
(2.10)
(2.13)
(2.15)
where Ai = (LjAL,
- LijALjE)@.
(2.16)
The term Ai represents a feedback effect [l l] due to the presence of other components. In a onecomponent system, this term, therefore, does not appear.
S. A. AKBAR et al.
582
By changing the variables, the heat of transport can be defined in a similar fashion based on irreversible thermodynamics. The entropy production dS/dt is written in terms of the driving forces dl and fl and their conjugate flows as dS - = J,j dt
-J&q
-J,&,.
(2.17)
If the driving force di is expressed as di =
(BrCi)
=
B/A
+
P+B>
3. DERIVATION
The Soret effect is calculated from the Onsager equation under the condition that the macroscopic atomic flows J, and J2 are zero. Zero macroscopic flow is attained when the flow under the temperature gradient is counterbalanced by the flows under the concentration gradients created by thermal diffusion. Utilizing the expressions given in section 2, the expression for the Soret effect is readily obtained. Using the conditions
(2.18)
/3& and fl can be taken as driving forces. Therefore, utilizing eqn (2.18) in eqn (2.17), we can rewrite entropy production in terms of driving forces &ii and b and their conjugate flows as
OF THE SORJCT EFFECT
and
J,=O
J,=O
in eqn (2.22), the following equations j.i, +
F
T = 0,
(3.2a)
ii,++0 where
Jo= JE-PIJ, -PzJ,.
(2.20)
(3.1)
(3.2b)
are obtained. Because the chemical potentials p, and pZ are functions of concentrations and temperature, fir and fi2 are given as follows:
By definition Je represents the heat Row and can be expressed in terms of atomic flows as 1143 Je=Q:Ji+QfJ,-&‘.
(2.21)
Equation (2.21) gives physical interpretations of the quantities Q: and k The quantity QT is interpreted as being the heat carried by unit flow of atom i at uniform temperature (d = 0) and is called the heat of transport. The notation for the heat of transport Qt should not be confused with the quantity Qi defined in eqn (2.7). The term a is the thermal conductivity due to atomic flow. Atomic flows (eqn (2.1)) can thus be expressed in terms of the heats of transport as [14]
J,=
By utilizing eqn (3.3) in eqn (3.2) and solving for
we obtain
dp,
(3.4a)
--_-
dT
-$,+$f)-+(r2+$$), dp, -=dT
(3.4b)
or in a matrix form In this fashion, the heats of transport are specifically identified in terms of atomistic quantities. For a binary system, based on the rigid lattice model, heats of transport are specifically expressed as [14] Q:=u~+E:+A~--~~~=E:-~~.
@I*-H*)
(2.23)
The quantities on the right hand side of eqn (2.23) have been introduced earlier (eqns (2.6), (2.7) and (2.16)).
=--
i
T
-ap -1 (IS* - H*).
0ap
Equation (3.4) thus represents the concentration
(3.4c) dis-
Soret effect in solid tributions of the two components for the Soret effect. The inverse matrix in eqn (3.4~) represents the 90s called ~m~re~ibility matrix. Here, HF is the partial molar enthatpy [13] defined as
583
Here, ui is a part of the activation energyof motion defined in eqn (2.9b) and is positive. Substituting eqn (3.8) in eqn (3.4), we obtain (3.9a)
and would ap~rop~a~ly be called the enthalpy of transport. For the sake of simplicity, the inverse matrix in a many~omponent system is written hereafter simply as (~~/~~)-I. In vector form, the expression for man~component systems (Appendix) can be written in the same form as eqn (3.4~). As discussed in I, the signs of Et - H: and the determinant l&/+ I are all positive and hence the relative magnitudes af EF - Ht and each component of the compressibility matrix determine the sign of distribution of the two components as shown in eqns (3.4a) and (3.4b). Again, the concentration of vacancies plays a decisive role, especiatly when the cormentration of vacancies is low. Equation (3.4) represents the Soret effect in a two-component system in the general case. Equation (3.4) is often represented as
and the sign of the Soret effect or the thermoelectric power in superionic conductors is said to be determined by the sign of Q*. This derivation, however, ignores the temperature dependence of ~1, and the appearance of (E* -H*) instead of Q* is a mare exact expression as pointed out in I. In addition, the value of Q* evaluated by the present model can vary from -co to + 00 due to the variation of ,u and the sign of Q* cannot be determined reliably. On the other hand, the value E* - H* can have a finite value. The corresponding equation of eqn (3.4), in a one-component system was derived in I as
(3.9b) The Co~pon~ng equation for a one-component Soret effect eqn (3.7) can be expressed as
dp, -=----_ dT
4
(3.10)
kT2 _!_ 0 P3
It is clearly seen that the concentration of vacancies is the determining term in the comp~ssibi~ity matrix as seen in eqn (3.10). Because, there does not exist any cross term, eqn (3.7) or (3.10) suggests that in one-component systems, the Soret effect has only one sign (even in a non-ideal solution, Ef - Hf was found to be positive 119) and the ~~rnp~sibiI~ty (represented by (l/p,)-‘) is the major term determining the magnitude of the Soret effect, esp&aIly in the low-vacancy con~n~a~on limit. This is due to the fact that in the lattice gas model given here, the internal pressure P is the conjugate variable of the vacancy concentration, or p3. This situation is the same for many-component systems. Let us examine binary systems or eqn (3.9) with this fact in mind. The term dp,/dT in eqn (3.9b) is essentially a change of the vacancy concentration along the specimen (along the temperature gradient) and is related to the com-
‘.Oj
(3.7)
Here, the cross-terms (off-diagonal terms in the compressibility matrix) do not enter and the physical meaning of the competition of the compressibility matrix (a~/&)-~ and (E+ - H*) is clearly understood. To understand the qualitative features of the problem, let us discuss the most simple case of ideal solution 1141 (eil = 0). Under this assumption, the chemical potential (eqn (2.8)) can be simplified as (3.8a) and Eif-If:=&.
(3.8b)
Fig. I, Tw0-~mponent Soret effect for low vacancy conizntration. b<: total Iength of the specimen; pt. p2: e0ncentration of the two ~rn~~n~ tiX, y : activation energy of motion for ~o~pou~~ I and 2, respectively; T,, T,: temperatures on the two surCa_ p3: vacancy
584
S. A. AKBARet al. demixing effect under a temperature gradient only. The demixing effect under a temperature gradient only should be an open case when a vacancy concentration gradient inside the specimen does not exist [14]. In Ref. [14], we carefully discussed the difference between the open case and the closed case under such idealized boundary conditions. 4. DISCUSSION
The treatment of the Soret effect has been extended to binary and many-component systems in the rigid lattice model. The extension is straightforward. However, one distinct difference exists. In the one-component system treated earlier, the effect is determined by the -0.5 -0.1 0.1 0.3 -0.3 0.5 competition of (E* -H*), the energy of transport minus the enthalpy of transport, and the comFig. 2. Two-component Soret effect for high vacancy pressibility (~~~~~)-I as given in eqn (3.7). Because concentration. both quantities are positive, the present model predicts that particle tends to segregate towards the low tem~rature side. In the rigid lattice model, pressibility of the system. In the low vacancy concentration limit, the ~ompre~ibility is very low (i.e. (~~/a~)-’ tends to zero in the Iow vacancy concentration limit and the change of distribution of vadp, fdT = 0) as in the one-component case. Hence, cancies does not occur in practice, and the Soret effect depending on the activation energy difference, the is negligible in such a case. In many-component two components redistribute in different directions relation is systems, this replaced by without changing the vacancy concentration. In other words, if one component preferentially segregates to (Z$/i?p)-‘(E* - H*) as defined in eqn (3.k). Alone side, the other segregates to the opposite side though det]+/ap 1 and (E: - H:) are positive, the (illustrated in Fig. 1). This is due to the difference in matrix multiplication (ap/ap)-‘(E* - H*) creates the magnitude of (JP - H*) of each component and the combination of (E* -H*) and the compressibility component having the larger value of (E* - H*) matrix elements in the numerator in eqn (3.4) and, segregates towards the low temperature side, As the depending on the relative magnitude of (E* - H*) for each component, the segregation of components can vacancy concentration increases, it can change and, occur in either direction. An essential factor in the at a certain stage, the segregation of both components can occur to the lower temperature side compressibility matrix is again the concentration of (Fig. 2). In general, therefore, the Soret effect is vacancies of the system. This is because the vacancy concentration and the internal pressure are conjugate determined by the relative ma~itude of (E* -H*) variables and one determines the other. As in a and the con~ntration of vacancies. system, the deviation from a hoBoth the demixing effect and the Soret effect one-com~n~t mogeneous distribution of vacancies does not occur discussed so far belong to general demixing phenomin practice when the concentration of vacancies is ena, but are with two idealized boundary conditions. low. However, redistribution of constituent atoms In this respect, it is worthwhile to point out the can occur without changing the vacancy distribution difference between the Soret effect and the demixing in many-component systems. Figure 1 represents effect under a temperature gradient. Some demixing experiments have been carried out for binary oxides such a case for binary systems. If the vacancy is large enough, redistribution of under a temperature gradient under a variety of boundary conditions [18, 191. Among these, experi- vacancies readily occurs. Because the thermal diffusion process in the present model tends to drive ments carried out in the presence of an oxygen atoms toward the low temperature side, if the reatmosphere in contact with the specimen can allow sistance of the redistribution of vacancies is low the flow of vacancies in and out of the specimen, and this is called the open case [14]. This corresponds to enough, all components can accumulate toward the low temperature side. Figure 2 for the binary system the idealized demixing effect. In the experiments under a temperature gradient on encapsulated sam- represents this case. In many-component systems, the distribution of components would be a combination ples, on the other hand, specimens are not in contact with an oxygen atmosphere and do not allow the flow of these two cases. Another major conclusion obtained in the treatof vacancies. This case is called the closed case. The ment is that the Soret effect should be corte~tly latter type of experiment corresponds to the Soret represented as in eqn (3.k) but not as in eqn (3.6) effect [14]. In other words, according to our where the sign ofQ* determines the sign of the effect. definition, the closed case does not represent the
o.o-
Soret effect in solid This difference apparently comes from the difference in handling eqn (3.2) in which p is treated as a function of p and Tin our treatment, while in many other treatments, p is treated only as a function of p, assuming an isothermal condition in deriving the Soret concentration distribution [18-211. It is worth mentioning here that, in the original treatment of the Soret effect by de Groot [l], the same conclusion as ours was drawn. The notion that the sign of the effect is determined by the sign of Q* may also come from expressions like eqn (3.2) where
dPi --. QT
dT=
T
As pointed out in I, it was not possible to explain in interstitial solid solutions that interstitial atoms segregate in either direction of the temperature gradient by our model for a one-component system. Although in many-component systems, each component can assemble in either direction of the temperature gradient, it does not seem proper to utilize the result of many-component systems for this explanation. Although the present rigid lattice model is generally utilized to discuss the phase diagram of alloy systems, it seems that it is necessary to utilize a different model for the assembly of interstitial atoms in order to explain the effect properly. As we discussed earlier, if solid solutions are placed under a temperature gradient, the composition can change due to diffusion [14]. This effect is generally called the demixing effect. A steady state demixing process which can occur in oxides was discussed in detail in another paper [14]. This effect and the Soret effect are cases under two idealized boundary conditions which can be treated analytically. These two idealized cases have been classified as the open case and the closed case, respectively. However, it should be mentioned that a variety of demixing processes can occur depending on the boundary conditions.
585
7. Callen H. B., Thermodynamics.
Wiley, New York (1963). 8. Peterson N. L., Solid Stare Physics (Edited by F. Seitz, D. Tumbull and H. Ehrenreich), Vol. 12. Academic Press, New York (1968). 9. Gillan M. J., Mass Transport in Solids (Edited by F. Beniera and C. R. A. Catlow). Plenum Press, New York (1981). 10. Howard R. E. and Manning J. R., J. them. Phys. 36, 910 (1962). 11. Kikuchi R., Ishikawa T. and Sato H., Physics a 123A, ?27 (J984).
12. Kikuchi R., Prog. rheor. Phys. (Kyoto), Suppl. No. 35, 1 (1966). 13. Wada K., Suzuki A., Sato H. and Kikuchi R., 1. Phys. Chem. Solia!s 46. 11951202 (1985). This uaner is referred to as I in’the text. . ’ _ _ 14. Akbar S. A., Kaburagi M., Sato H. and Kikuchi R., J. Am. Ceram. SOL, in print. 15. Kikuchi R., Phys. Rev. 81, 988 (1951). 16. Sato H., Noniraditional Methodr in Diffusion (Edited by G. E. Murch, H. Bimbaum and J. R. Cost), TMSAIME Conference Proceedings, Vol. 1984, pp. 203-235. 17. Kikuchi’ R. and Sato H., J. them. Phys. 51(l), 161 (1969). 18. Petuskey W. T. and Bowen H. K., J. Am. Ceram. Sot. 64(10), 611 (1981). 19. Stephenson G. B., M.S. Thesis, MIT (1981). 20. Shewmon P. G., Acta melall. 8, 605 (1960). 21. Oriani R. A., J. them. Phys. 34, 1773 (1961). 22. Kuwamoto H. and Sato H., Solid St. Ionics 5, 187 (1981). 23. Schiraldi A., Baldini P. and Pezzati E., Solid St. Ionics 9 and 10, 1187 (1983).
APPENDIX Sore1 effect in many-component systems The matrix representation, eqn (3.4c), for
binary systems
for the Soret effect holds for an n component system and can thus be written as
dp
dT=-T
’ f! -I@*
0
_H*).
ap
C-4.1)
Here, p represents the density and is defined as a column matrix
64.2) Acknowledgements-Discussions with Professor R. Kikuchi, Dr. A. Suzuki, Dr. T. Ishii and Dr. T. Ishikawa are gratefully acknowledged. The work was supported by the U.S. Department of Energy under the grant number DEEGO2-84ER45133.
1p.J while (ay/ap)-’
is the compressibility matrix given by
r%...F?!Q -’ ap,
REFERENCES 1. de Groot S. R., Thermodynamics of Irreversible Process. North-Holland, Amsterdam (195 1). 2. de Groot S. R. and Mazur P.. Non-eauilibrium Thermodynamics. North-Holland, Amsterdam (1962). 3. Shewmon P. G., Diffusion in Solids. McGraw Hill. New York (1963). 4. Howard R. E. and Lidiard A. B., Repl. Prog. Phys. 27, 161 (1964). 5. Howard R. E. and Lidiard A. B., Acia metail. 13, 443 (1965). 6. Allnatt A. R. and Chadwick A. V., Chem. Rev. 67,681 (1967).
ak
G Quantities E* and H* are the energy of transport ethalpy of transport represented by column matrices
E* =
lE.*J
and
586
S. A. AKBARet al.
and
(A.3
The nature determined (&I/@-‘. and it can
of the multi-component Soret effect is essentially by the nature of the compressibility matrix For a stable phase, the rigidity must be positive, generally be shown that det 1ap/ap I> 0. As is
discussed for binary systems the magnitude of the vacancy concentration plays a decisive role in determining the nature of the Soret effect. If the vacancy concentration is small, the vacancy distribution does not change in practice along the specimen and constituent components segregate to either sides of specimens according to the magnitude of (E* - H*) in such a way as to cancel out the change of vacancy concentrationWOn the other hand, if the number of vacancies is large, all constituent components can segregate to the low temperature side only. In other words, the qualitative feature of the Soret effect in multi-component systems is essentially the same as in the case of binary systems.