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Diffusivity and mobility of lattice gases in lattices with randomly blocked sites
Solid State Ionics 40/41 North-Holland
(1990)
167-170
DIFFUSIVITY AND MOBILITY OF LAlTICE WITH RANDOMLY BLOCKED SITES K.W. KEHR and M. BRAUN’ ftir Festkiirperforschung der Kernforschungsanlage
Institut
GASES IN LATTICES
Jiilich, D-51 70 Jiilich, Federal Republic of Germany
The relations between the single-particle, the collective, the tagged-particle diffusion coefficients, and the mobility were studied by numerical simulations for the model of lattice gases on lattices with randomly blocked sites. The coefftcients of single-particle and collective diffusion are identical and closely related to the mobility. The tagged-particle diffusion coefficient cannot be simply related to the former coefftcients.
1.
Introduction and model
The transport of particles in heterogeneous materials is of great practical and theoretical interest. Such materials may be composed of a mixture of one practically immobile and one mobile component. It is evident that inhomogeneous ionic conductors provide examples of such systems. Other examples are fluids in porous rocks. On the theoretical side, much interest has been concentrated on the percolation case [ 1 ] where the immobile particles provide a structure for incipient transport of the mobile particles. In that case the well-known properties of critical phenomena appear. This investigation has been restricted to the case where the mobile particles can perform longrange diffusion. The aim of this contribution is the discussion of the relations between the different diffusion and transport coefficients that can be introduced for this model. These are the single-particle, the collective, and the tagged-particle diffusion coefficients, as well as the mobility. More explicit detinitions will be given in the following sections. The details of the investigations will be published elsewhere [ 21, hence this contribution will try to give an overview of the results, together with an attempt of putting them into perspective with related work. The model to be investigated is a lattice where the sites are randomly occupied with particles with con’ Present address: Institut fur Theoretische ersitlt Ttibingen, 0167-2738/90/$03.50 ( North-Holland )
D-7400 Tubingen
Astrophysik, I, FRG.
0 Elsevier Science Publishers
B.V.
Univ-
centration p. These particles are considered as immobile and their concentration will be smaller than the critical concentration pC, where the empty sites form a percolating cluster. In the numerical simulations three dimensional simple-cubic lattices with up to 120’ sites were used. The empty sites of the randomly blocked lattices are then randomy tilled with “mobile” particles with concentration c where c is defined as the quotient of the particle number to the number of lattice sites. Clearly c+p< 1, and the difference gives the vacancy concentration. The mobile particles may perform transitions to empty neighboring sites with transition rate I7 Apart from the exclusion of double occupancy no further interactions between the particles are considered.
2. Single-particle diffusion
The single-particle diffusion coefficient D,, is obtained from the mean-square displacement of single, mobile particles on the empty, accessible, sites of the lattice with randomly blocked sites. Averages over different random walks and over different realizations of the random lattice are implied. The diffusion coefficient of single particles may also be interpreted as the diffusion coefficient of lattice gases, in the lattice with blocked sites, in the limit c-0. A complementary way of introducing D,, is to consider
168
K. W Kehr, M. Braun / Difusivity and mobility of latticegases
a lattice gas with c approaching 1 -p, and creating a single vacancy in this lattice gas. The exchange rate of the vacancy is identical to the exchange rate of the mobile particles with empty sites. Hence the diffusion coefficient of the vacancy is identical, in this model, to the diffusion coefficient of a single particle 131. One may introduce a formal correlation factor for the single-particle diffusion by dividing D, by the mean-field expression D,,( 1 -p) where Do is the diffusion coefficient on the completely empty lattice (Do=ra2 for simple cubic lattices with the lattice constant a). This correlation factor will be designated by JK(p) following [4], and it is identical to the tracer correlation factorfv which was introduced by Manning [ 5 ] and discussed, e.g., in [ 3 1. The diffusion coefficient of single particles in the lattice with randomly blocked sites was investigated repeatedly. One of the earliest simulations was made by Brandt [6] for fee lattices. This work was critically examined by Murch and Rothman [ 31 who observed that the value of the correlation factor& depended on the length of the time interval employed to estimate the diffusion coefficient. Kutner and Kehr [ 41 investigated the diffusion of a low concentration of tagged particles with transition rates that were different from the background particles and obtained &(p) for fee lattices in the limit of immobile latticegas particles. In our present work we employed an ensemble of many independent particles, averaged over several realizations, and took the corrections to the asymptotic behavior into account in a systematic way. Moreover, we determined also the fluctuations of the diffusion coefficients between different samples, and their dependence on sample size and length of the time intervals used. The theory of the single-particle diffusion coefficient is nontrivial for arbitrary p. For smaller p, Tahir-Kheli [6] deduced from a self-consistent improvement of a theory based on truncations of the master equations the relation D,,=&(l
-p/f+...)
,
ments of the blocking defects. Of course, after division with 1 -p eq. ( 1) yields a theoretical prediction for&(p) or fv that is valid for small p. Our computer simulations [ 21 confirm ( 1) for small and moderate defect concentrations up to about ~~0.2. Even for large p the deviations between ( 1) and the simulations are not really large. Eq. ( 1) predicts a percolation threshold p,=f: This value is 0.653... whereas the correct value is p,=O.689... .
3. Collective diffusion The coefficient of collective diffusion is defined by Fick’s law and it describes the decay of density fluctuations in the lattice gas. In the noninteracting lattice gas in ideal lattices Dcoll= D,,. An elegant derivation was given by Kutner [9]; the point is the cancellation of site exclusion terms in the master equation for the probability of site occupation by untagged particles. The same argument also applies [ 41 to the model of randomly blocked sites. The equivalent result for the conductivity was established by Harder et al. [ 10 1. In contrast to the ideal lattice, in the model with randomly blocked sites D,, is a nontrivial quantity. Note that the equality Dcoll= D,, implies the absence of any concentration dependence. In our work [2] we estimated the collective diffusion by setting up cosine-like density variations in the lattice gas and monitoring their decay, which is governed by Fick’s law. For a detailed description of the technique see [ 111. The simulations verify the equality Dmu= Dspfor different values of p within the numerical accuracy.
4. Mobility The mobility of the particles is related to the collective diffusion coefficient by the generalized Nemst-Einstein relation [ 12 ]
(1)
wherefis the correlation factor for tracer diffusion in lattice gases on ideal lattices, in the limit c+ 1. The coefficient beyond the linear term has been derived only in d= 2 [ 7 1. Koiwa and Ishioka [ 8 ] noted that eq. ( 1) can already be deduced for regular arrange-
(2) The derivative ap/ac is easily calculated for the lattice gas model with randomly blocked sites and we have
K. W. Kehr, M. Braun / Difiivity
kBTB=D,,,,(l-c-P)/(l-P)
*
(3)
The “physical correlation factor” may be introduced [ 13 ] in this context by factoring out the mean-field mobility of the lattice gas particles, kr,TB= (1 -c-P)&~(P) In view of the identity
.
(4)
DColl=DsP we obtain
x(P)=GJ[&(1--P)l~
(5)
If we compare this relation with the definition of the formal single-particle correlation factor, we observe f;(P) =fR(P)
(6)
and this correlation factor is also identical to the vacancy correlation factor, fv. The identity f,(p) =fv was already found by Murch and Rothman [ 3 1. Our derivation establishes its validity for arbitrary concentrations of the lattice gas. It should be emphasized that (6) has been derived for the model of a noninteracting lattice gas in the lattice with randomly blocked sites. It is an open question whether and to what extent (6) is modified for interacting lattice gases. For further discussions of the interpretation of the physical correlation factor, see Murch [ 121, Murch and Dyre [ 141, and Dyre and Murch [ 15 1. We also checked the validity of (3) by direct numerical simulations [ 21. For this purpose we imposed a bias on the transition rates in one direction, e.g., T,=bT and T_,=b-‘r with b> 1. From the mean velocity induced in the particles the Onsager coefficient and the product of mobility and kgT can be obtained. In the linear regime (3) is satisfied within the accuracy of the numerical data.
5. Tagged-particle diffusion The asymptotic mean-square displacement of tagged particles in the lattice gas allows to define the tagged-particle or tracer diffusion coefficient 0,. This quantity is especially amenable to simulations where each particle can be treated as tagged. It is usual to define a tracer correlation factor f( c,p) by writing 0, =Do( 1 --c--P)~(c,P)
.
We note
cases f(c,p=O)
the limiting
(7) =f(c)
and
169
and mobility of lattice gases
f( c+O,p) =f;(p). The simulations were done for a fixed vacancy concentration cv~O.04, i.e., c+p was fixed. The results for various p are given in [ 2 1. Manning [ 5 ] gave the relation f( c,p) =ffv where f=f( c+ 1) and fv has been introduced in section 2. Since we employed a fixed cv CK 1, the behavior of f(c= 0.96 -p,p) should be similar to that off,. Our simulations are at variance with this prediction. This points to the necessity to develop better theories for f (GP). Qualitatively, we point out that there are two sources of correlations present: first, the backward correlations induced by the blocked sites, second, the dynamic backward correlations caused by the other lattice gas particles. The two correlation effects are coupled together. It is instructive to consider the representation of the tracer diffusion coefftcient in terms of Onsager coefficients [ 16 1, D, =AM/ca -A,./c,.
,
(8)
where A refers to the untagged particles and A* to the tagged ones. The coefficient AMIcA has been probed by the simulations of the mobility. Since D, has been determined directly, one can obtain information on the cross coefficients. As a particular example we consider p=O.48. Here A,Jc,=O.OlSS and A,./ c,. = 0.0113. The cross coefficient contains the backward correlations of the tagged-particle diffusion and the example demonstrates that these are appreciable. The Haven ratio HR has been frequently discussed in solid state ionics, see, e.g., ref. [ 12 1. It is given by
Dt
f(c,p)
HR=gFl=fr(p)
(9)
We observed [ 2 ] an approximately linear decrease of HR with p, for fixed c+p. A theory of Manning [ 5,17 ] (see also [ 18 ] ) that expresses Onsager coefficients in terms of tracer diffusion coefficients would predict that for the model with randomly blocked sites HR is independent of p, for fixed vacancy concentration. Hence our simulation results are in disagreement with this prediction.
6. Concluding remarks In this simple model of noninteracting lattice gases on lattices with randomly blocked sites the relation
170
K. W. Kehr, h4. Braun /Diffuvity
betweenQp, Roll, and B is now well established. Since the particles of real materials do have interactions, it would be of great interest to extend the investigations to interacting lattice gases. The tracer diffusion needs more detailed investigations for different concentrations of the lattice gases. Also an adequate theoretical description of this quantity is desired. A special point of interest is the percolation threshold. Some studies were made by Heupel [ 191 who found at the percolation threshold the same critical exponents for the anomalous mean-square displacement of tagged particles of lattice gases as for single particles. Also the critical behavior of the tracer diffusion coeffkient on approach to the percolation threshold has been studied in a modified percolation model [ 10 1. The authors found that it is governed by the same exponent as the critical behavior of the single-particle diffusion coefficient.
References [ 1 ] D. Stauffer, Introduction to percolation Francis, London, 1985 )
theory (Taylor and
and mobility of latticegases [2] M. Braun and K.W. Kehr, Philos. Mag. A, to be published. Philos. Mag. A 43 ( 1981) 229. [4] R. Kutnerand K.W. Kehr, Philos. Mag. A 48 ( 1983) 199. [ 5 ] J.R. Manning, Acta Metall. 15 ( 1967) 8 17; Phys. Rev. B 4 (1971) 1111. [ 61 R.A. Tahir-Kheli, Phys. Rev. B 28 ( 1983) 3049. [7] M.H. Ernst, Th.M. Nieuwenhuizen and P.F.J. van Velthoven, J. Phys. A 20 (1987) 5335. [8] M. Koiwa and S. Ishioka, in: Solute-Defect Interaction, Theory and Experiment, Proc. Intemat. Seminar, Kingston, Canada,‘eds. S. Saimoto, G.R. Purdy and G.V. Kidson (Pergamon Press, New York, 1986) p.232. [9] R. Kutner, Phys. Letters 81A (1981) 239. [lo] H. Harder, A. Bunde and W. Dieterich, J. Chem. Phys. 85 (1986) 4123. [ 111 K.W. Kehr, SM. Reulein and K. Binder, Phys. Rev. B 39 (1989) 4891. [ 121 G.E. Murch, Philos. Mag. A 45 (1982) 685; Solid State Ionics 7 (1982) 177. [ 131 H. Sato and R. Kikuchi, J. Chem. Phys. 55 (1971) 677. [ 141 G.E. Murch and J.C. Dyre, Solid State Ionics 20 (1986) 203. [ 151 J.C. Dyre and G.E. Murch, Solid State Ionics 21 (1986) 139. [ 161 R.E. Howard and A.B. Lidiard, Rep. Prog. Phys. 27 (1964) 161. [ 171 J.R. Manning, Metal. Trans. 1 (1970) 499. [ 181 A.R. Allnatt and A.B. Lidiard, Rep. Prog. Phys. 50 (1987) 373. [ 191 L. Heupel, J. Stat. Phys. 42 (1986) 541.
[ 31 G.E. Murch and S.J. Rothman,
(1990)
167-170
DIFFUSIVITY AND MOBILITY OF LAlTICE WITH RANDOMLY BLOCKED SITES K.W. KEHR and M. BRAUN’ ftir Festkiirperforschung der Kernforschungsanlage
Institut
GASES IN LATTICES
Jiilich, D-51 70 Jiilich, Federal Republic of Germany
The relations between the single-particle, the collective, the tagged-particle diffusion coefficients, and the mobility were studied by numerical simulations for the model of lattice gases on lattices with randomly blocked sites. The coefftcients of single-particle and collective diffusion are identical and closely related to the mobility. The tagged-particle diffusion coefficient cannot be simply related to the former coefftcients.
1.
Introduction and model
The transport of particles in heterogeneous materials is of great practical and theoretical interest. Such materials may be composed of a mixture of one practically immobile and one mobile component. It is evident that inhomogeneous ionic conductors provide examples of such systems. Other examples are fluids in porous rocks. On the theoretical side, much interest has been concentrated on the percolation case [ 1 ] where the immobile particles provide a structure for incipient transport of the mobile particles. In that case the well-known properties of critical phenomena appear. This investigation has been restricted to the case where the mobile particles can perform longrange diffusion. The aim of this contribution is the discussion of the relations between the different diffusion and transport coefficients that can be introduced for this model. These are the single-particle, the collective, and the tagged-particle diffusion coefficients, as well as the mobility. More explicit detinitions will be given in the following sections. The details of the investigations will be published elsewhere [ 21, hence this contribution will try to give an overview of the results, together with an attempt of putting them into perspective with related work. The model to be investigated is a lattice where the sites are randomly occupied with particles with con’ Present address: Institut fur Theoretische ersitlt Ttibingen, 0167-2738/90/$03.50 ( North-Holland )
D-7400 Tubingen
Astrophysik, I, FRG.
0 Elsevier Science Publishers
B.V.
Univ-
centration p. These particles are considered as immobile and their concentration will be smaller than the critical concentration pC, where the empty sites form a percolating cluster. In the numerical simulations three dimensional simple-cubic lattices with up to 120’ sites were used. The empty sites of the randomly blocked lattices are then randomy tilled with “mobile” particles with concentration c where c is defined as the quotient of the particle number to the number of lattice sites. Clearly c+p< 1, and the difference gives the vacancy concentration. The mobile particles may perform transitions to empty neighboring sites with transition rate I7 Apart from the exclusion of double occupancy no further interactions between the particles are considered.
2. Single-particle diffusion
The single-particle diffusion coefficient D,, is obtained from the mean-square displacement of single, mobile particles on the empty, accessible, sites of the lattice with randomly blocked sites. Averages over different random walks and over different realizations of the random lattice are implied. The diffusion coefficient of single particles may also be interpreted as the diffusion coefficient of lattice gases, in the lattice with blocked sites, in the limit c-0. A complementary way of introducing D,, is to consider
168
K. W Kehr, M. Braun / Difusivity and mobility of latticegases
a lattice gas with c approaching 1 -p, and creating a single vacancy in this lattice gas. The exchange rate of the vacancy is identical to the exchange rate of the mobile particles with empty sites. Hence the diffusion coefficient of the vacancy is identical, in this model, to the diffusion coefficient of a single particle 131. One may introduce a formal correlation factor for the single-particle diffusion by dividing D, by the mean-field expression D,,( 1 -p) where Do is the diffusion coefficient on the completely empty lattice (Do=ra2 for simple cubic lattices with the lattice constant a). This correlation factor will be designated by JK(p) following [4], and it is identical to the tracer correlation factorfv which was introduced by Manning [ 5 ] and discussed, e.g., in [ 3 1. The diffusion coefficient of single particles in the lattice with randomly blocked sites was investigated repeatedly. One of the earliest simulations was made by Brandt [6] for fee lattices. This work was critically examined by Murch and Rothman [ 31 who observed that the value of the correlation factor& depended on the length of the time interval employed to estimate the diffusion coefficient. Kutner and Kehr [ 41 investigated the diffusion of a low concentration of tagged particles with transition rates that were different from the background particles and obtained &(p) for fee lattices in the limit of immobile latticegas particles. In our present work we employed an ensemble of many independent particles, averaged over several realizations, and took the corrections to the asymptotic behavior into account in a systematic way. Moreover, we determined also the fluctuations of the diffusion coefficients between different samples, and their dependence on sample size and length of the time intervals used. The theory of the single-particle diffusion coefficient is nontrivial for arbitrary p. For smaller p, Tahir-Kheli [6] deduced from a self-consistent improvement of a theory based on truncations of the master equations the relation D,,=&(l
-p/f+...)
,
ments of the blocking defects. Of course, after division with 1 -p eq. ( 1) yields a theoretical prediction for&(p) or fv that is valid for small p. Our computer simulations [ 21 confirm ( 1) for small and moderate defect concentrations up to about ~~0.2. Even for large p the deviations between ( 1) and the simulations are not really large. Eq. ( 1) predicts a percolation threshold p,=f: This value is 0.653... whereas the correct value is p,=O.689... .
3. Collective diffusion The coefficient of collective diffusion is defined by Fick’s law and it describes the decay of density fluctuations in the lattice gas. In the noninteracting lattice gas in ideal lattices Dcoll= D,,. An elegant derivation was given by Kutner [9]; the point is the cancellation of site exclusion terms in the master equation for the probability of site occupation by untagged particles. The same argument also applies [ 41 to the model of randomly blocked sites. The equivalent result for the conductivity was established by Harder et al. [ 10 1. In contrast to the ideal lattice, in the model with randomly blocked sites D,, is a nontrivial quantity. Note that the equality Dcoll= D,, implies the absence of any concentration dependence. In our work [2] we estimated the collective diffusion by setting up cosine-like density variations in the lattice gas and monitoring their decay, which is governed by Fick’s law. For a detailed description of the technique see [ 111. The simulations verify the equality Dmu= Dspfor different values of p within the numerical accuracy.
4. Mobility The mobility of the particles is related to the collective diffusion coefficient by the generalized Nemst-Einstein relation [ 12 ]
(1)
wherefis the correlation factor for tracer diffusion in lattice gases on ideal lattices, in the limit c+ 1. The coefficient beyond the linear term has been derived only in d= 2 [ 7 1. Koiwa and Ishioka [ 8 ] noted that eq. ( 1) can already be deduced for regular arrange-
(2) The derivative ap/ac is easily calculated for the lattice gas model with randomly blocked sites and we have
K. W. Kehr, M. Braun / Difiivity
kBTB=D,,,,(l-c-P)/(l-P)
*
(3)
The “physical correlation factor” may be introduced [ 13 ] in this context by factoring out the mean-field mobility of the lattice gas particles, kr,TB= (1 -c-P)&~(P) In view of the identity
.
(4)
DColl=DsP we obtain
x(P)=GJ[&(1--P)l~
(5)
If we compare this relation with the definition of the formal single-particle correlation factor, we observe f;(P) =fR(P)
(6)
and this correlation factor is also identical to the vacancy correlation factor, fv. The identity f,(p) =fv was already found by Murch and Rothman [ 3 1. Our derivation establishes its validity for arbitrary concentrations of the lattice gas. It should be emphasized that (6) has been derived for the model of a noninteracting lattice gas in the lattice with randomly blocked sites. It is an open question whether and to what extent (6) is modified for interacting lattice gases. For further discussions of the interpretation of the physical correlation factor, see Murch [ 121, Murch and Dyre [ 141, and Dyre and Murch [ 15 1. We also checked the validity of (3) by direct numerical simulations [ 21. For this purpose we imposed a bias on the transition rates in one direction, e.g., T,=bT and T_,=b-‘r with b> 1. From the mean velocity induced in the particles the Onsager coefficient and the product of mobility and kgT can be obtained. In the linear regime (3) is satisfied within the accuracy of the numerical data.
5. Tagged-particle diffusion The asymptotic mean-square displacement of tagged particles in the lattice gas allows to define the tagged-particle or tracer diffusion coefficient 0,. This quantity is especially amenable to simulations where each particle can be treated as tagged. It is usual to define a tracer correlation factor f( c,p) by writing 0, =Do( 1 --c--P)~(c,P)
.
We note
cases f(c,p=O)
the limiting
(7) =f(c)
and
169
and mobility of lattice gases
f( c+O,p) =f;(p). The simulations were done for a fixed vacancy concentration cv~O.04, i.e., c+p was fixed. The results for various p are given in [ 2 1. Manning [ 5 ] gave the relation f( c,p) =ffv where f=f( c+ 1) and fv has been introduced in section 2. Since we employed a fixed cv CK 1, the behavior of f(c= 0.96 -p,p) should be similar to that off,. Our simulations are at variance with this prediction. This points to the necessity to develop better theories for f (GP). Qualitatively, we point out that there are two sources of correlations present: first, the backward correlations induced by the blocked sites, second, the dynamic backward correlations caused by the other lattice gas particles. The two correlation effects are coupled together. It is instructive to consider the representation of the tracer diffusion coefftcient in terms of Onsager coefficients [ 16 1, D, =AM/ca -A,./c,.
,
(8)
where A refers to the untagged particles and A* to the tagged ones. The coefficient AMIcA has been probed by the simulations of the mobility. Since D, has been determined directly, one can obtain information on the cross coefficients. As a particular example we consider p=O.48. Here A,Jc,=O.OlSS and A,./ c,. = 0.0113. The cross coefficient contains the backward correlations of the tagged-particle diffusion and the example demonstrates that these are appreciable. The Haven ratio HR has been frequently discussed in solid state ionics, see, e.g., ref. [ 12 1. It is given by
Dt
f(c,p)
HR=gFl=fr(p)
(9)
We observed [ 2 ] an approximately linear decrease of HR with p, for fixed c+p. A theory of Manning [ 5,17 ] (see also [ 18 ] ) that expresses Onsager coefficients in terms of tracer diffusion coefficients would predict that for the model with randomly blocked sites HR is independent of p, for fixed vacancy concentration. Hence our simulation results are in disagreement with this prediction.
6. Concluding remarks In this simple model of noninteracting lattice gases on lattices with randomly blocked sites the relation
170
K. W. Kehr, h4. Braun /Diffuvity
betweenQp, Roll, and B is now well established. Since the particles of real materials do have interactions, it would be of great interest to extend the investigations to interacting lattice gases. The tracer diffusion needs more detailed investigations for different concentrations of the lattice gases. Also an adequate theoretical description of this quantity is desired. A special point of interest is the percolation threshold. Some studies were made by Heupel [ 191 who found at the percolation threshold the same critical exponents for the anomalous mean-square displacement of tagged particles of lattice gases as for single particles. Also the critical behavior of the tracer diffusion coeffkient on approach to the percolation threshold has been studied in a modified percolation model [ 10 1. The authors found that it is governed by the same exponent as the critical behavior of the single-particle diffusion coefficient.
References [ 1 ] D. Stauffer, Introduction to percolation Francis, London, 1985 )
theory (Taylor and
and mobility of latticegases [2] M. Braun and K.W. Kehr, Philos. Mag. A, to be published. Philos. Mag. A 43 ( 1981) 229. [4] R. Kutnerand K.W. Kehr, Philos. Mag. A 48 ( 1983) 199. [ 5 ] J.R. Manning, Acta Metall. 15 ( 1967) 8 17; Phys. Rev. B 4 (1971) 1111. [ 61 R.A. Tahir-Kheli, Phys. Rev. B 28 ( 1983) 3049. [7] M.H. Ernst, Th.M. Nieuwenhuizen and P.F.J. van Velthoven, J. Phys. A 20 (1987) 5335. [8] M. Koiwa and S. Ishioka, in: Solute-Defect Interaction, Theory and Experiment, Proc. Intemat. Seminar, Kingston, Canada,‘eds. S. Saimoto, G.R. Purdy and G.V. Kidson (Pergamon Press, New York, 1986) p.232. [9] R. Kutner, Phys. Letters 81A (1981) 239. [lo] H. Harder, A. Bunde and W. Dieterich, J. Chem. Phys. 85 (1986) 4123. [ 111 K.W. Kehr, SM. Reulein and K. Binder, Phys. Rev. B 39 (1989) 4891. [ 121 G.E. Murch, Philos. Mag. A 45 (1982) 685; Solid State Ionics 7 (1982) 177. [ 131 H. Sato and R. Kikuchi, J. Chem. Phys. 55 (1971) 677. [ 141 G.E. Murch and J.C. Dyre, Solid State Ionics 20 (1986) 203. [ 151 J.C. Dyre and G.E. Murch, Solid State Ionics 21 (1986) 139. [ 161 R.E. Howard and A.B. Lidiard, Rep. Prog. Phys. 27 (1964) 161. [ 171 J.R. Manning, Metal. Trans. 1 (1970) 499. [ 181 A.R. Allnatt and A.B. Lidiard, Rep. Prog. Phys. 50 (1987) 373. [ 191 L. Heupel, J. Stat. Phys. 42 (1986) 541.
[ 31 G.E. Murch and S.J. Rothman,