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Kinetics of relaxation process of hopping ionic conduction in lattice gas systems
$
J O U R N A L OF
ELSEVIER
Journal of Non-Crystalline Solids 203 (1996) 306-311
Kinetics of relaxation process of hopping ionic conduction in lattice gas systems Hiroshi Sato *, Anuradha Datta, Takuma Ishikawa 1 School of Materials Engineering, Purdue University, IV. Lafayette, IN 47907-1289, USA
Abstract An analytical, microscopic treatment which can correctly describe the relaxation process of hopping ionic conduction from the adiabatic region to the isothermal region with equal rigor has been developed in interacting lattice gas systems. It is demonstrated that the relaxation motion of a single particle in the assembly is intrinsically non-Debye and follows the Kohlrausch-Williams-Watts (KWW) relaxation behavior even without interactions among mobile ions. This dependence is an apparent contrast to the current view that the anomaly is 'created' by interaction and disorder. Some conclusions with respect to the universal dynamic response observed in solid electrolytes are drawn from results of the analytical treatment.
1. Introduction
A major interest of this treatment is to understand the 'universal dynamic response' observed in the ionic conductivity in solid electrolytes by an analytical treatment from first principles. The phenomenon 'universal dynamic response' is succinctly represented by the following two empirical relations [1,2]; the first indicates that the frequency dependent ionic conductivity is represented by Jonscher's power law ~(~o) = ~o~
(O
l),
(1)
with p limited between 0 and 1, and second, the relaxation process in the induced current when a constant external field is switched on at t = 0 in such system is non-Debye (non-exponential) with the be-
* Corresponding author. Tel.: + 1-317 494 4099; fax: + 1-317 494 1204; e-mail: [email protected]. Present address: Faculty of Engineering, Tokyo Institute of Polytechnics, Atsugi, Kanagawa 243-02, Japan.
havior to reach the stationary state represented by the Kohlrausch-Williams-Watts (KWW) stretched exponential function, _=(t) [2], ~(t)=exp
-
,
1-p=/3,0
(2) Eq. (2) indicates that, in the expression of the form e x p [ - (t/'r)], -r increases with time. In the hopping process, a particle which jumps out to a nearest neighboring site leaves behind a vacancy in the wake and there is always a high probability for that particle to jump back into the vacancy. This jump back process thus constitutes the relaxation process and, hence, the above mentioned problem is reduced to a question of whether this jump back process is exponential with time or not. The task involved here is to derive the relaxation curve for the induced current upon the application of a constant external field at t = 0 in the lattice gas model. Here, we limit ourselves to the derivation of
0022-3093/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PH S 0 0 2 2 - 3 0 9 3 ( 9 6 ) 0 0 3 6 3 - 8
H. Sato et al. / Journal of Non-Crystalline Solids 203 (1996) 306-31 l
flow of a single particle in an assembly rather than that of the assembly, because the treatment of the former is far simpler. The relation of the two is now well known [3], but, as far as the mechanism of the relaxation process is concerned, the former is satisfactory as well. Typically, the relaxation curve can be divided into three time regions: adiabatic, intermediate and isothermal [4]. If the relaxation curve is normalized, the curve represents the time dependent correlation factor f(t) as introduced by Funke [2] and the relaxed value in the isothermal region for flow of a single particle is called the tracer correlation factor. The difficulty in deriving the relaxation behavior theoretically is to deal with all the time regions with equal rigor. Although the adiabatic region and isothermal region are theoretically manageable, it is necessary to rely on kinetics in the intermediate state, because the reference state is not the equilibrium state. If the same equilibrium state persists from the adiabatic range through the isothermal range, the jump back process should be exponential with time, because this reduces the relaxation process to the Markovian process [5]. Prior theoretical efforts emphasize that the KWW effect should arise from interactions among mobile particles and their disordered arrangements [6]. This means that the origin of the nonlinear relaxation process is ascribed to a coupling with the relaxation process of the surroundings through interactions with neighbors [2,7,8]. Also, highly sophisticated simulation methods in disordered systems such as glass are being developed to account for the universal dynamic response. However, in order to understand the real origin of the universal dynamic response, it is necessary to develop an analytical treatment based on first principles rather than resorting to phenomenological models. We have confirmed that the combination of the pair approximation of the path probability method [9] and the Bethe method [10] can describe this jump back process for the whole time range properly in the pair approximation. This confirmation is based on the fact that the pair approximation (not the higher degree of approximation) of the PPM represents the adiabatic state correctly and the Bethe approximation leads this adiabatic state to the isothermal state with t also in the pair approximation
307
so that the whole relaxation process can be derived consistently with the same rigor by the combination. By applying this treatment to the jump back process of a single particle in the assembly of particles of the same species (without any interaction) with a single vacant site, it is shown that the existence probability of the vacancy at the site left behind increases with time from the value expected at t = 0 due to the return of the vacancy after its escape. This leads the jump back process to exhibit the KWW behavior.
2. The derivation of the K W W behavior The examination of the formalism of the PPM reveals that, when applied to transport problems, the same equilibrium state is assumed for the reference state in the PPM for all time regions from the adiabatic state (t = 0) to the isothermal state (t ~ ~). This also means that, the equilibrium state the PPM represents is that of the adiabatic state. Our intention here is to derive the change of the relaxation behavior with t to that of t ~ ~ by means of the Bethe method based on the correct knowledge at t = 0 obtained by the PPM and show that this leads to the KWW relaxation behavior. Here we start with the calculation of the relaxation of flow when a constant driving force for the tagged particle, &i, is switched on at t = 0 in a binary system consisting of A and B particles (and of a small number of tracer particles of B or B * to follow its motion) indicated by i and j which specify A, B and B* as 1, 2 and 3, respectively in a lattice gas with pairwise, nearest neighbor interactions, eAA, e:AB and ~3AB= 8BA among mobile particles, A and B, by means of the pair approximation of the PPM. A detailed derivation of the tracer correlation factor in the range t - ~ zc has been carried out and is shown in the appendix of Ref. [11]. The flow equation for a small number of tracer ions in the disordered distribution in the linear approximation at a time instant t is derived, according to [11], as 3
~ i ( t ) = - & i + y" Qj~ji(t) j=1 ×(i=
1,2and3, j=l,2and3).
(3)
H. Sato et al. / Journal of Non-Crystalline Solids 203 (1996) 306-311
308
If i is meant to be the tagged species here, ~i indicates the normalized flow of the tagged species at t, Qj is the distribution of particles around the vacancy, and ~j~(t) indicates the gradient in the distribution of particles created by the driving force for the ith species, &~. ~ ( t ) is evaluated in a short time period between t and t + At at a reference plane under the value of Qj at any t. Based on Eq. (3), we can rewrite =
-
1 -
The jump back probability of the tagged particle with time, -ri*(t) (the same as f(t) defined in Eq. (4)), in competition with other particles around the vacancy can then be easily calculated by referring to Fig. 1:
zi*(t ) = "ri(1 - e
1 -- "ri*(t ) L* (t) -
(4) This defines the time dependent correlation factor, fi(t). In this expression, the first term represents the jumping out of the tagged particle (at t = 0) and the second term represents the jumping back of the tagged particle into the vacancy. Fig. 1 indicates the jumping back of the tagged particle (with asterisk) into the vacancy at any time instant in competition with particles Qj around the vacancy represented by the second term in Eq. (3) [11]. Since the nearest neighbor distant jump of particles is assumed for the kinetics of flow, at the adiabatic stage, only particles at the nearest neighbor of the vacancy can contribute to the change of state. Therefore, the pair approximation of the PPM shown in Fig. 1 represents kinetics at the adiabatic state well. If one considers the assembly of particles of the same species and no interactions among particles is taken into account, the pair approximation shown in Fig. 1 represents the adiabatic condition exactly as long as the concentration of vacancies is negligibly small.
I./ ®
S j'"o ®
P~ =z, Pr= 1-z
2if)--1
Fig. 1. The jump back process of the tagged particle (represented by an asterisk) in a disordered binary alloy by means of the pair approximation of the PPM.
(5)
and, based on the PPM, the time dependent correlation factor f*(t), Eq. (4), is represented in terms of "ri*(t) as [11]
a,= -f,.(t)a,.
E
j=l
(~"/~')')
(t)
1+
(6)
'
where
(1
--
Zi)
Ti= ( 1 - Z i ) + ( 2 ~ - I ) Z
i
(7)
is defined as the competition or the normalized jump back probability under the equilibrium condition and
z,= E
+
(s)
J
Therefore, the quantity 1 - Z i , represents the jump back (return) probability Pr of the particle whereas Z~ represents the escape probability of the vacancy Pe (the jumping in probability of a particle around the vacancy). Here 2 & represents the coordination number and v~ etc. are defined as the jump frequency of the ith species including the bond breaking factor due to the interaction of the ith particle with the neighbors. The function "ri*(t) given in Eq. (5) changes exponentially with t in the PPM because in the PPM, the reference state remains the same with t and, hence, "r remains unchanged with t. If T increases with t, -r* changes in a KWW fashion and so does f * (t). The examination of the behavior of -r with time is the major concern of our work here. The original Bethe method [10] is devised to deal with the order-disorder transformation in alloys and is known to correspond to the pair approximation of the cluster variation method (CVM) on which the PPM is based [9]. Here, the consistency relation [10] in the Bethe method which connects the thermodynamical properties of a small cluster to that of the outside, gives a change from the local order to the long range order with distance in the form of the
H. Sato et al. / Journal of Non-Crystalline Solids 203 (1996) 306-311
propagation of order [12]. This concept can be extended in t-space. By means of Fig. 1, we can define "r in the adiabatic condition by the pair approximation of the PPM through Eqs. (7) and (8). The change of -r with time can then be calculated by the Bethe method in the form of the propagation of order as t changes from 0 to ~ consistently in the pair approximation. Now we show the calculation of the relaxation curve for the motion of a single particle in the assembly of particles of the same species without any mutual interaction with only a single vacant site in a lattice gas with the combination of the pair approximation of the PPM represents the exact adiabatic condition. In the pair approximation of the PPM in the adiabatic state, Z in Eq. (7) is given as z = w / ( w + w),
(9)
where w represents the jump frequency of the particle (no need to include the bond breaking factor). From this, the adiabatic value of r is obtained using Eq. (7) as 7 = w / { w + ( 2 t S 1)w}. For t > 0 , the continuity condition requires that w(1-1-) z=
w+w(1-'r)
1-'r -
2-r'
(10)
The consistency relation in "r is then obtained with the use of Eq. (7) as 1
~'=
(11)
in the form of an iteration formula. As the number of iterations, N, increases, "r approaches the isothermal value T ~ 1 / ( 2 ~ 5 - 1) ( N ~ 2 ) from the adiabatic value -r = 1/(2~5) ( N = 1) in Eq. (11). Therefore, if N is related to time t, we obtain the propagation of order, d"r/dt, in t-space. The number of iterations N represents the distance of propagation of information, and, hence, represents the number of jumps n which effectively covers N unit distance by the vacancy which makes a random walk. If so, N and the number of jumps n are related as N = ~ - and t = n/w. Because N = 1 corresponds to t = 0, N - 1 = f ~ [13]. This is the relation between N and t. The propagation of order in Eq. (11) means that information at the center of
1.00
309
,
2~=2 0.80
0.60
2~ = 3
0.40
26 = 4
0.20 ~_
26,) = 8
0.00
0.0
i
i
,
10.0
20.0
30.0
40.0
50.0
wt Fig. 2. " r - t relations for structures with different coordination numbers, 2 ~5.
the group in Fig. 1 is accumulated over a distance with time by hopping of particles [14]. Here, specifically, the increase of "r indicated in Fig. 2 represents the return of the escaped vacancy with time. The presence of interactions among conduction ions, especially those which tend to create an ordering of conduction ions enhances the K W W behavior. In Fig. 2, -r-t relations are plotted for different values of 2 & with 2 ~ = 2 for the one dimensional lattice, 3 for the two dimensional honeycomb lattice, 4 for the two dimensional square net and 8 for the body-centered cubic lattice. The increase of -r with t means that "r * (t) and, hence, f * (t) also exhibits the K W W behavior. In other words, the non-Debye behavior is inherent even in this single mode case although the effect is almost negligible except for the case 2 ~5 = 2. Based on Eq. (11), the increase of "r as t ~ zc for cases with 2 & > 2 is exponential, but for the one dimensional case (2& = 2), where the percolation limit "r = 1 is reached as t - ~ oo, the decay becomes proportional to t - i / 2 . The change of the rate of approach from exponential to t t/2 is observed in Fig. 3 which shows d(logf*(t))/d(log t) at the value f * ( t ) around 0.1. The extensive linearity in the relation log f(t) - log t such as that observed for the case 2 ~5 = 2 is also observed in systems where f(t) decreases with time to a small enough value such as in mixed alkali systems [15]. This is due to the existence of a scaling law for the decrease of f(t) with time near the percolation limit. The value of p is obtained from the slope of this curve as d(logf*(t))/d(log t) = - p and are 0.12, 0.22, 0.32 and 0.5 for 2 ~5 = 6, 4, 3 and 2. The value 0.5 for the
310
H. Sato et al./ Journal of Non-Crystalline Solids 203 (1996) 306-311 I
*~
0.1
o
'"'~. . . . . . . . . . . . . . . . . . . . . . . . . .
0.01
0.0010.01 0.I
1
I0
100 1000
log wt 1
the assembly. In the case of the assembly of particles of the same species with a negligible amount of vacancies as treated here, the correlation effect for the assembly vanishes in the completely homogeneous case due to the indistinguishability among particles. In order to regain the correlation effect for the assembly of particles, it is necessary to deal with more general cases with mutual interactions among particles and a fluctuation in the distribution is necessary.
3. Universal dynamic response
o simulation 0.1 0.01
..... ~ ' '"'~ ...... ~ ....... ' ...... 0.I 1 l0 I 0 0 1000
log wt Fig. 3. Comparison of Monte Carlo simulation results with the analytical theory for 2 & = 2 and 2 ~7~= 3.
one dimensional lattice is proven to be the exact value for this case. The relation l o g f * ( t ) - l o g t can be converted to log f * ( e 0 ) - log eo by simply converting t into 2w/e0. The Jonscher's power law exponent, p can be obtained from d(log f * ( e o ) ) / d ( log o~) = p, too. Monte Carlo simulation of the relaxation process can be made in a straightforward fashion. The simulation calculates f * ( t ) = - f ( ~ ( c ( 0 ) c ( t ' ) ) / ( v Z ( 0 ) ) d t ' directly, assuming a single vacancy in the system which makes the random walk starting from an instant when the vacancy has exchanged places with the tagged particle. The agreement of the analytical calculation of f * ( t ) obtained from "r * ( t ) with the Monte Carlo simulation of f * ( t ) is excellent. In the case 2 & = 3, where the tracer correlation factor obtained by the pair approximation coincides with the exact value of 1 / 3 , an almost exact agreement of the results is observed (Fig. 3). This shows that the pair approximation is quite acceptable for the calculation of f * ( t ) . In the one dimensional case, a change in the slope in the log f * ( t ) - log t curve is observed at the value of f * ( t ) - - ~ 0.1 (Fig. 3) in agreement with the theory. It is to be mentioned here that universal response is a property of the total assembly of charged particles and not that of the motion of a single particle in
We have shown an analytical derivation of the relaxation motion of a single particle in an assembly of particles of the same species with only one vacant site in a lattice gas. The treatment of more general models with interactions among mobile particles and with disordered distribution can be made analytically in a similar fashion. These analytical results provide us with information with respect to qualitative features of the universal dynamic response. These are listed in the following. (1) The K W W behavior becomes pronounced in systems where the tracer correlation factor is small.
0.60
,
i
;
r
r
,
(a)
0.50
~'",,..... 0.40
'"',.,,
..
p ~,-,. ,.
0.30
',.. ,, t.
0.20 0.10 0.0
i
I
0.1
0.2
i
i
0.3
i
0.4
i
0.5
0.6
" 0.7
f
1.0o
0.50
\
0.00
• 0.0
0,2
0.4
tI 0.6
f
i 0.8
concentration
1,0
Fig. 4. The relation between p and f'~. (a) Single mode case: p and f * for 2~ = 2.3, 4 and 6. (b) Two mode case: p and f* for a large concentration of vacant sites.
H. Sato et al. / Journal of Non-Crystalline Solids 203 (1996) 306-311
(2) A general scaling factor exists which restricts Jonscher's exponent p within a certain limit, but p is not necessarily restricted within 0 and 1. (3) Generally, the value of p is larger when the tracer correlation factor f is smaller. In other words, p is more or less proportional to l / f . (4) When f becomes extremely small such as in cases of mixed alkali glasses, the value of p is found to be around 0.5. Here, 0.5 is the exact value of p in the one dimensional lattice where the percolation limit ( f * ( t ) ~ 0 as t ~ ~) exists. More details will be given elsewhere. In order to support the above statements, Fig. 4 is shown. Fig. 4(a) indicates the relation between p and f in the case of the assembly of particles of the same species (without interactions) with a negligible amount of vacant sites whereas Fig. 4b shows the relation between p and f in a system with a large concentration of vacant sites with a nearest neighbor repulsive pair interaction between conducting particles.
Acknowledgements This work is a result of cooperation with K. Funke of the University of Miinster, Germany. The
31 I
authors also owe valuable discussions to R. Kikuchi of UCLA and to G. Liedl of Purdue University. The work was supported by the US Department of Energy under grant No. DE-EG02-84ER45133.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
A.K. Jonscher, Nature 267 (1977) 673. K. Funke, Prog. Solid State Chem. 22 (1993) 1. G.E. Murch and J.C. Dyre, CRC Crit. Rev. 15 (1989) 348. C. Zener, Elasticity and Anelasticity of Metals (University of Chicago, Chicago, 1948). J.C. Dyre, J. Appl. Phys. 64 (1988) 2456. J.P. Bouchard and A. George, Phys. Rep. 195 (1990) 127. K.L. Ngai, A.K. Rajagopal and S. Teitler, J. Chem. Phys. 88 (1988) 5086. S.R. Elliott and A.P. Owens, Phys. Rev. B44 (1991) 47. R. Kikuchi, Prog. Theor. Phys. (Kyoto) Suppl. No. 35 (1966) I. H.A. Bethe, Proc. R. Soc. 150 (1935) 552. H. Sato and R. Kikuchi, Phys. Rev. B28 (1983) 648. F. Zernike, Physica 1 (1940) 565. H. Sato and A. Datta, Solid State Ionics 72 (1994) 19. H. Sato, T. Ishikawa and K. Funke, Solid State Ionics 53-56 (1992) 907. H. Sato, in: Theoretical Aspects of Mixed Alkali Effect, Ceramic Transactions, Vol. 20, Glasses for Electronic Applications, ed. K.M. Nair (American Ceramic Society, Westerville, OH, 1991) p. 19.
J O U R N A L OF
ELSEVIER
Journal of Non-Crystalline Solids 203 (1996) 306-311
Kinetics of relaxation process of hopping ionic conduction in lattice gas systems Hiroshi Sato *, Anuradha Datta, Takuma Ishikawa 1 School of Materials Engineering, Purdue University, IV. Lafayette, IN 47907-1289, USA
Abstract An analytical, microscopic treatment which can correctly describe the relaxation process of hopping ionic conduction from the adiabatic region to the isothermal region with equal rigor has been developed in interacting lattice gas systems. It is demonstrated that the relaxation motion of a single particle in the assembly is intrinsically non-Debye and follows the Kohlrausch-Williams-Watts (KWW) relaxation behavior even without interactions among mobile ions. This dependence is an apparent contrast to the current view that the anomaly is 'created' by interaction and disorder. Some conclusions with respect to the universal dynamic response observed in solid electrolytes are drawn from results of the analytical treatment.
1. Introduction
A major interest of this treatment is to understand the 'universal dynamic response' observed in the ionic conductivity in solid electrolytes by an analytical treatment from first principles. The phenomenon 'universal dynamic response' is succinctly represented by the following two empirical relations [1,2]; the first indicates that the frequency dependent ionic conductivity is represented by Jonscher's power law ~(~o) = ~o~
(O
l),
(1)
with p limited between 0 and 1, and second, the relaxation process in the induced current when a constant external field is switched on at t = 0 in such system is non-Debye (non-exponential) with the be-
* Corresponding author. Tel.: + 1-317 494 4099; fax: + 1-317 494 1204; e-mail: [email protected]. Present address: Faculty of Engineering, Tokyo Institute of Polytechnics, Atsugi, Kanagawa 243-02, Japan.
havior to reach the stationary state represented by the Kohlrausch-Williams-Watts (KWW) stretched exponential function, _=(t) [2], ~(t)=exp
-
,
1-p=/3,0
(2) Eq. (2) indicates that, in the expression of the form e x p [ - (t/'r)], -r increases with time. In the hopping process, a particle which jumps out to a nearest neighboring site leaves behind a vacancy in the wake and there is always a high probability for that particle to jump back into the vacancy. This jump back process thus constitutes the relaxation process and, hence, the above mentioned problem is reduced to a question of whether this jump back process is exponential with time or not. The task involved here is to derive the relaxation curve for the induced current upon the application of a constant external field at t = 0 in the lattice gas model. Here, we limit ourselves to the derivation of
0022-3093/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PH S 0 0 2 2 - 3 0 9 3 ( 9 6 ) 0 0 3 6 3 - 8
H. Sato et al. / Journal of Non-Crystalline Solids 203 (1996) 306-31 l
flow of a single particle in an assembly rather than that of the assembly, because the treatment of the former is far simpler. The relation of the two is now well known [3], but, as far as the mechanism of the relaxation process is concerned, the former is satisfactory as well. Typically, the relaxation curve can be divided into three time regions: adiabatic, intermediate and isothermal [4]. If the relaxation curve is normalized, the curve represents the time dependent correlation factor f(t) as introduced by Funke [2] and the relaxed value in the isothermal region for flow of a single particle is called the tracer correlation factor. The difficulty in deriving the relaxation behavior theoretically is to deal with all the time regions with equal rigor. Although the adiabatic region and isothermal region are theoretically manageable, it is necessary to rely on kinetics in the intermediate state, because the reference state is not the equilibrium state. If the same equilibrium state persists from the adiabatic range through the isothermal range, the jump back process should be exponential with time, because this reduces the relaxation process to the Markovian process [5]. Prior theoretical efforts emphasize that the KWW effect should arise from interactions among mobile particles and their disordered arrangements [6]. This means that the origin of the nonlinear relaxation process is ascribed to a coupling with the relaxation process of the surroundings through interactions with neighbors [2,7,8]. Also, highly sophisticated simulation methods in disordered systems such as glass are being developed to account for the universal dynamic response. However, in order to understand the real origin of the universal dynamic response, it is necessary to develop an analytical treatment based on first principles rather than resorting to phenomenological models. We have confirmed that the combination of the pair approximation of the path probability method [9] and the Bethe method [10] can describe this jump back process for the whole time range properly in the pair approximation. This confirmation is based on the fact that the pair approximation (not the higher degree of approximation) of the PPM represents the adiabatic state correctly and the Bethe approximation leads this adiabatic state to the isothermal state with t also in the pair approximation
307
so that the whole relaxation process can be derived consistently with the same rigor by the combination. By applying this treatment to the jump back process of a single particle in the assembly of particles of the same species (without any interaction) with a single vacant site, it is shown that the existence probability of the vacancy at the site left behind increases with time from the value expected at t = 0 due to the return of the vacancy after its escape. This leads the jump back process to exhibit the KWW behavior.
2. The derivation of the K W W behavior The examination of the formalism of the PPM reveals that, when applied to transport problems, the same equilibrium state is assumed for the reference state in the PPM for all time regions from the adiabatic state (t = 0) to the isothermal state (t ~ ~). This also means that, the equilibrium state the PPM represents is that of the adiabatic state. Our intention here is to derive the change of the relaxation behavior with t to that of t ~ ~ by means of the Bethe method based on the correct knowledge at t = 0 obtained by the PPM and show that this leads to the KWW relaxation behavior. Here we start with the calculation of the relaxation of flow when a constant driving force for the tagged particle, &i, is switched on at t = 0 in a binary system consisting of A and B particles (and of a small number of tracer particles of B or B * to follow its motion) indicated by i and j which specify A, B and B* as 1, 2 and 3, respectively in a lattice gas with pairwise, nearest neighbor interactions, eAA, e:AB and ~3AB= 8BA among mobile particles, A and B, by means of the pair approximation of the PPM. A detailed derivation of the tracer correlation factor in the range t - ~ zc has been carried out and is shown in the appendix of Ref. [11]. The flow equation for a small number of tracer ions in the disordered distribution in the linear approximation at a time instant t is derived, according to [11], as 3
~ i ( t ) = - & i + y" Qj~ji(t) j=1 ×(i=
1,2and3, j=l,2and3).
(3)
H. Sato et al. / Journal of Non-Crystalline Solids 203 (1996) 306-311
308
If i is meant to be the tagged species here, ~i indicates the normalized flow of the tagged species at t, Qj is the distribution of particles around the vacancy, and ~j~(t) indicates the gradient in the distribution of particles created by the driving force for the ith species, &~. ~ ( t ) is evaluated in a short time period between t and t + At at a reference plane under the value of Qj at any t. Based on Eq. (3), we can rewrite =
-
1 -
The jump back probability of the tagged particle with time, -ri*(t) (the same as f(t) defined in Eq. (4)), in competition with other particles around the vacancy can then be easily calculated by referring to Fig. 1:
zi*(t ) = "ri(1 - e
1 -- "ri*(t ) L* (t) -
(4) This defines the time dependent correlation factor, fi(t). In this expression, the first term represents the jumping out of the tagged particle (at t = 0) and the second term represents the jumping back of the tagged particle into the vacancy. Fig. 1 indicates the jumping back of the tagged particle (with asterisk) into the vacancy at any time instant in competition with particles Qj around the vacancy represented by the second term in Eq. (3) [11]. Since the nearest neighbor distant jump of particles is assumed for the kinetics of flow, at the adiabatic stage, only particles at the nearest neighbor of the vacancy can contribute to the change of state. Therefore, the pair approximation of the PPM shown in Fig. 1 represents kinetics at the adiabatic state well. If one considers the assembly of particles of the same species and no interactions among particles is taken into account, the pair approximation shown in Fig. 1 represents the adiabatic condition exactly as long as the concentration of vacancies is negligibly small.
I./ ®
S j'"o ®
P~ =z, Pr= 1-z
2if)--1
Fig. 1. The jump back process of the tagged particle (represented by an asterisk) in a disordered binary alloy by means of the pair approximation of the PPM.
(5)
and, based on the PPM, the time dependent correlation factor f*(t), Eq. (4), is represented in terms of "ri*(t) as [11]
a,= -f,.(t)a,.
E
j=l
(~"/~')')
(t)
1+
(6)
'
where
(1
--
Zi)
Ti= ( 1 - Z i ) + ( 2 ~ - I ) Z
i
(7)
is defined as the competition or the normalized jump back probability under the equilibrium condition and
z,= E
+
(s)
J
Therefore, the quantity 1 - Z i , represents the jump back (return) probability Pr of the particle whereas Z~ represents the escape probability of the vacancy Pe (the jumping in probability of a particle around the vacancy). Here 2 & represents the coordination number and v~ etc. are defined as the jump frequency of the ith species including the bond breaking factor due to the interaction of the ith particle with the neighbors. The function "ri*(t) given in Eq. (5) changes exponentially with t in the PPM because in the PPM, the reference state remains the same with t and, hence, "r remains unchanged with t. If T increases with t, -r* changes in a KWW fashion and so does f * (t). The examination of the behavior of -r with time is the major concern of our work here. The original Bethe method [10] is devised to deal with the order-disorder transformation in alloys and is known to correspond to the pair approximation of the cluster variation method (CVM) on which the PPM is based [9]. Here, the consistency relation [10] in the Bethe method which connects the thermodynamical properties of a small cluster to that of the outside, gives a change from the local order to the long range order with distance in the form of the
H. Sato et al. / Journal of Non-Crystalline Solids 203 (1996) 306-311
propagation of order [12]. This concept can be extended in t-space. By means of Fig. 1, we can define "r in the adiabatic condition by the pair approximation of the PPM through Eqs. (7) and (8). The change of -r with time can then be calculated by the Bethe method in the form of the propagation of order as t changes from 0 to ~ consistently in the pair approximation. Now we show the calculation of the relaxation curve for the motion of a single particle in the assembly of particles of the same species without any mutual interaction with only a single vacant site in a lattice gas with the combination of the pair approximation of the PPM represents the exact adiabatic condition. In the pair approximation of the PPM in the adiabatic state, Z in Eq. (7) is given as z = w / ( w + w),
(9)
where w represents the jump frequency of the particle (no need to include the bond breaking factor). From this, the adiabatic value of r is obtained using Eq. (7) as 7 = w / { w + ( 2 t S 1)w}. For t > 0 , the continuity condition requires that w(1-1-) z=
w+w(1-'r)
1-'r -
2-r'
(10)
The consistency relation in "r is then obtained with the use of Eq. (7) as 1
~'=
(11)
in the form of an iteration formula. As the number of iterations, N, increases, "r approaches the isothermal value T ~ 1 / ( 2 ~ 5 - 1) ( N ~ 2 ) from the adiabatic value -r = 1/(2~5) ( N = 1) in Eq. (11). Therefore, if N is related to time t, we obtain the propagation of order, d"r/dt, in t-space. The number of iterations N represents the distance of propagation of information, and, hence, represents the number of jumps n which effectively covers N unit distance by the vacancy which makes a random walk. If so, N and the number of jumps n are related as N = ~ - and t = n/w. Because N = 1 corresponds to t = 0, N - 1 = f ~ [13]. This is the relation between N and t. The propagation of order in Eq. (11) means that information at the center of
1.00
309
,
2~=2 0.80
0.60
2~ = 3
0.40
26 = 4
0.20 ~_
26,) = 8
0.00
0.0
i
i
,
10.0
20.0
30.0
40.0
50.0
wt Fig. 2. " r - t relations for structures with different coordination numbers, 2 ~5.
the group in Fig. 1 is accumulated over a distance with time by hopping of particles [14]. Here, specifically, the increase of "r indicated in Fig. 2 represents the return of the escaped vacancy with time. The presence of interactions among conduction ions, especially those which tend to create an ordering of conduction ions enhances the K W W behavior. In Fig. 2, -r-t relations are plotted for different values of 2 & with 2 ~ = 2 for the one dimensional lattice, 3 for the two dimensional honeycomb lattice, 4 for the two dimensional square net and 8 for the body-centered cubic lattice. The increase of -r with t means that "r * (t) and, hence, f * (t) also exhibits the K W W behavior. In other words, the non-Debye behavior is inherent even in this single mode case although the effect is almost negligible except for the case 2 ~5 = 2. Based on Eq. (11), the increase of "r as t ~ zc for cases with 2 & > 2 is exponential, but for the one dimensional case (2& = 2), where the percolation limit "r = 1 is reached as t - ~ oo, the decay becomes proportional to t - i / 2 . The change of the rate of approach from exponential to t t/2 is observed in Fig. 3 which shows d(logf*(t))/d(log t) at the value f * ( t ) around 0.1. The extensive linearity in the relation log f(t) - log t such as that observed for the case 2 ~5 = 2 is also observed in systems where f(t) decreases with time to a small enough value such as in mixed alkali systems [15]. This is due to the existence of a scaling law for the decrease of f(t) with time near the percolation limit. The value of p is obtained from the slope of this curve as d(logf*(t))/d(log t) = - p and are 0.12, 0.22, 0.32 and 0.5 for 2 ~5 = 6, 4, 3 and 2. The value 0.5 for the
310
H. Sato et al./ Journal of Non-Crystalline Solids 203 (1996) 306-311 I
*~
0.1
o
'"'~. . . . . . . . . . . . . . . . . . . . . . . . . .
0.01
0.0010.01 0.I
1
I0
100 1000
log wt 1
the assembly. In the case of the assembly of particles of the same species with a negligible amount of vacancies as treated here, the correlation effect for the assembly vanishes in the completely homogeneous case due to the indistinguishability among particles. In order to regain the correlation effect for the assembly of particles, it is necessary to deal with more general cases with mutual interactions among particles and a fluctuation in the distribution is necessary.
3. Universal dynamic response
o simulation 0.1 0.01
..... ~ ' '"'~ ...... ~ ....... ' ...... 0.I 1 l0 I 0 0 1000
log wt Fig. 3. Comparison of Monte Carlo simulation results with the analytical theory for 2 & = 2 and 2 ~7~= 3.
one dimensional lattice is proven to be the exact value for this case. The relation l o g f * ( t ) - l o g t can be converted to log f * ( e 0 ) - log eo by simply converting t into 2w/e0. The Jonscher's power law exponent, p can be obtained from d(log f * ( e o ) ) / d ( log o~) = p, too. Monte Carlo simulation of the relaxation process can be made in a straightforward fashion. The simulation calculates f * ( t ) = - f ( ~ ( c ( 0 ) c ( t ' ) ) / ( v Z ( 0 ) ) d t ' directly, assuming a single vacancy in the system which makes the random walk starting from an instant when the vacancy has exchanged places with the tagged particle. The agreement of the analytical calculation of f * ( t ) obtained from "r * ( t ) with the Monte Carlo simulation of f * ( t ) is excellent. In the case 2 & = 3, where the tracer correlation factor obtained by the pair approximation coincides with the exact value of 1 / 3 , an almost exact agreement of the results is observed (Fig. 3). This shows that the pair approximation is quite acceptable for the calculation of f * ( t ) . In the one dimensional case, a change in the slope in the log f * ( t ) - log t curve is observed at the value of f * ( t ) - - ~ 0.1 (Fig. 3) in agreement with the theory. It is to be mentioned here that universal response is a property of the total assembly of charged particles and not that of the motion of a single particle in
We have shown an analytical derivation of the relaxation motion of a single particle in an assembly of particles of the same species with only one vacant site in a lattice gas. The treatment of more general models with interactions among mobile particles and with disordered distribution can be made analytically in a similar fashion. These analytical results provide us with information with respect to qualitative features of the universal dynamic response. These are listed in the following. (1) The K W W behavior becomes pronounced in systems where the tracer correlation factor is small.
0.60
,
i
;
r
r
,
(a)
0.50
~'",,..... 0.40
'"',.,,
..
p ~,-,. ,.
0.30
',.. ,, t.
0.20 0.10 0.0
i
I
0.1
0.2
i
i
0.3
i
0.4
i
0.5
0.6
" 0.7
f
1.0o
0.50
\
0.00
• 0.0
0,2
0.4
tI 0.6
f
i 0.8
concentration
1,0
Fig. 4. The relation between p and f'~. (a) Single mode case: p and f * for 2~ = 2.3, 4 and 6. (b) Two mode case: p and f* for a large concentration of vacant sites.
H. Sato et al. / Journal of Non-Crystalline Solids 203 (1996) 306-311
(2) A general scaling factor exists which restricts Jonscher's exponent p within a certain limit, but p is not necessarily restricted within 0 and 1. (3) Generally, the value of p is larger when the tracer correlation factor f is smaller. In other words, p is more or less proportional to l / f . (4) When f becomes extremely small such as in cases of mixed alkali glasses, the value of p is found to be around 0.5. Here, 0.5 is the exact value of p in the one dimensional lattice where the percolation limit ( f * ( t ) ~ 0 as t ~ ~) exists. More details will be given elsewhere. In order to support the above statements, Fig. 4 is shown. Fig. 4(a) indicates the relation between p and f in the case of the assembly of particles of the same species (without interactions) with a negligible amount of vacant sites whereas Fig. 4b shows the relation between p and f in a system with a large concentration of vacant sites with a nearest neighbor repulsive pair interaction between conducting particles.
Acknowledgements This work is a result of cooperation with K. Funke of the University of Miinster, Germany. The
31 I
authors also owe valuable discussions to R. Kikuchi of UCLA and to G. Liedl of Purdue University. The work was supported by the US Department of Energy under grant No. DE-EG02-84ER45133.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
A.K. Jonscher, Nature 267 (1977) 673. K. Funke, Prog. Solid State Chem. 22 (1993) 1. G.E. Murch and J.C. Dyre, CRC Crit. Rev. 15 (1989) 348. C. Zener, Elasticity and Anelasticity of Metals (University of Chicago, Chicago, 1948). J.C. Dyre, J. Appl. Phys. 64 (1988) 2456. J.P. Bouchard and A. George, Phys. Rep. 195 (1990) 127. K.L. Ngai, A.K. Rajagopal and S. Teitler, J. Chem. Phys. 88 (1988) 5086. S.R. Elliott and A.P. Owens, Phys. Rev. B44 (1991) 47. R. Kikuchi, Prog. Theor. Phys. (Kyoto) Suppl. No. 35 (1966) I. H.A. Bethe, Proc. R. Soc. 150 (1935) 552. H. Sato and R. Kikuchi, Phys. Rev. B28 (1983) 648. F. Zernike, Physica 1 (1940) 565. H. Sato and A. Datta, Solid State Ionics 72 (1994) 19. H. Sato, T. Ishikawa and K. Funke, Solid State Ionics 53-56 (1992) 907. H. Sato, in: Theoretical Aspects of Mixed Alkali Effect, Ceramic Transactions, Vol. 20, Glasses for Electronic Applications, ed. K.M. Nair (American Ceramic Society, Westerville, OH, 1991) p. 19.