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THEORETICAL ASPECTS OF DIFFUSION AND
IONIC CONDUCTION IN
SUPER-IONIC-CONDUCTORS IN CONNECTION
WITH ORDERING TRANSITIONS
H. Sato, R. Kikuchi
To cite this version:
THEORETICAL ASPECTS OF DIFFUSION AND IONIC CONDUCTION
IN SUPER-IONIC-CONDUCTORS IN CONNECTION
WITH ORDERING TRANSITIONS
(*I
H. SAT0
School of Materials Engineering, Purdue University, West Lafayette, Indiana 47907, U.S.A. and R. KTKUCHI
Hughes Kesearch Laboratories, Mahbu, Cal~l'orn~a 90265, b .S. A.
RCsumC. - La caractkristique fondamentale des superconducteurs ioniques est que leur rkseau cristallin contient un grand nombre de lacunes permettant aux ions de diffuser. Dans de tels systkmes, il existe plusieurs possibilitks d'ordre, ce qui influence la conductivitk. Dans cet article la mCthode path probability (qui permet de traiter thkoriquement les phknomknes coopkratifs irrkversibles) est utiliske pour synthktiser trois types d'interactions, apparemment diffkrentes, entre la mise en ordre et la superconductivitk ionique. Des transitions ordre-dksordre peuvent avoir lieu (a) parmi les ions conducteurs, (b) dans les sous-reseaux rigides ou (c) sous forme de dtplacements cooperatifs d ~ , ions conducteurs des sites normaux vers les sites interstitiels. Dans chaque cas, l'on observe une diminution trks nette de la conductivitk ionique dans l'ktat ordonnk. On montre que cette diminution est due principalement z i la dkcroissance du facteur de corrtlation.
Abstract. - The fundamental characteristic of the super-ionic-conductors is that the lattice contains many vacancies among which the ions can migrate. In such systems, possibilities of different kinds of ordering exist and, furthermore, the ordering is reflected in ionic conductivity. In this paper, the path probability method (of theoretically treating irreversible cooperative phenomena) is used to summarize, from a unified point of view, three apparently differenbkinds of interplay between ordering and super-ionic-conductivity. Order-disorder transitions can occur (a) among conducting ions, (b) in rigid sublattices or (c) in the form of cooperative excitation of conducting ions from their normal lattice sites to interstitial sites. In each case, a sharp decrease in the ionic conductivity in the ordered state is observed. The decrease is found to be mainly due to the decrease in the correlation factor.
1. Introduction. - The high ionic conductivity of super-ionic-conductors is closely related to its pecu- liar structural characteristic known as the Sublattice Disorder [I]. This term indicates that the number of available sites for conducting ions is greater than the number of conducting ions and hence the occu- pancy rate of these sites by ions is much less than unity, depending on their site occupancy energies: In other words, the number of vacancies available for conducting ions is comparable to the number of conducting ions. Since conducting ions are interacting among themselves (directly or indirectly), the diffu- sion of-ions in such a large number of available sites is a many-body problem, and only an appropriate treat- ment based on a time-dependent statistical mechanics can handle such problems properly. The conventional theories of diffusion which are designed for calculating the isotope diffusion coefficient or the impurity diffu-
(*) This work was supported in part by the National Science Foundation Grant DMR 7502959.
sion coefficient are based on the assumption that the number of vacancies is negligible and, therefore, these theories are not capable of calculating super-ionic- conductor problems. As a method which is qualified for our present purposes, we choose the pair appro- ximation of the Path Probability method (PPM) [l, 21.
The details of PPM, as applied to diffusion and ionic conduction, can be found in our previous publica- tions [l-31.
The PPM describes the isotope diffusion coefficient D and the ionic conductivity o in such disordered systems as follows [3] :
D = a2 8 e-s"
VWf
(1)Here, a is a numerical factor related to the jump distance and the dimensionality of the lattice,
C7-160 H. SAT0 AND R. KIKUCHI 8 is a vibrational frequency of an ion at a lattice point,
u is the activation energy for a jump of an ion, and n and e are the density and the electric charge of the conducting ions, respectively. The factors V, Wand f
indicate the many-body effect; V is related to the vacancy distribution with respect to a conducting ion and is called the vacancy availability factor; W is related to the effect of interaction of neighboring ions on the conducting ion, and is called the effective jump frequency factor ; and f is a generalized correlation factor which becomes essentially the correlation factor in the limit of self-diffusion. The factor
f,
in Eq. (2) represents the correlation factor for ionic conduction, and this deviates from unity only when there are physical origins to cause correlation effects [3, 41.While the PPM could account for many important features of diffusion and ionic conduction in super- ionic-conductors, it has been found that numerical values for correlation factor f are not as exact as hoped for when compared to those obtained by the random walk theory for self-diffusion for which exact solutions can be obtained [5, 61. Monte Car10 computer simulation on equivalent cases shows that its results agree very well with those obtained by the PPM for V and W, but not so well for f [7, 81. The cause of the discrepancy in f has since been located to a somewhat inappropriate time-averaging process in connection with the PPM treatment of the flow [9], which reduced the degree of approximation. However, since even the results for f are still qualitatively correct in most cases, we utilize the results of our previous calculations in this paper.
An important result of the PPM as applied to the disordered system is that the temperature dependence of the many-body terms VWf as a whole is such that it
gives an additional activation energy for diffusion and ionic conduction [3], additional to the ordinary activa- tion energy u in Eq. (1). As shown in figure 1, the additional activation energy due to the correlation among migrating ions is negative when negative (repulsive) interactions exist among conducting ions, and its magnitude is approximately the same as the magnitude of the interaction energy
1
E1.
The effect of VWf becomes dominant [3] when there exists a sizable number of vacancies and .the repulsive inter- actions further tend to enhance'the magnitude of V . In other words, in disordered systems, a cooperative behavior in diffusion and ionic conduction exists, and this reduces the overall activation energy a great deal and, at the same time, makes the pre- exponential factor small compared to those of one ion motion. In view of this situation, it is legitimate to say that the existence of the sublattice disorder is a necessary condition for a high ionic conductivity of super-ionic-conductors.2. Effect of Order-Disorder Transitions on Diffusion and Ionic Conduction. - Order-disorder transitions in super-ionic-conductors affect the diffusion and the ionic conduction a great deal. In all known cases, ordering decreases the ionic conductivity drastically. Categorically, we can classify the order-disorder transitions in three types. The PPM, being a treatment of time-dependent cooperative processes [2] (like the time-dependent Ising model), is very suitable to deal with the effects of order-disorder transitions on the diffusion and ionic conductivity. As specific examples, we explain on idealized models which were used earlier for p"-alumina, calcia-stabilized zirconia, and
P-
alumina. However, it should be borne in mind that these examples are chosen only for demonstration purposes, and that the theories can apply to similar but different cases of the respective classes.2.1 The first type of ordering is the one occurring among conducting ions. When the number of avai- lable sites is much larger than the number of ions, and if there exist repulsive interactions among ionsj the system is equivalent to binary alloys (consisting of conducting ions and vacancies). The simplest case would then be the case in which all the available sites are equivalent. As a typical example of such a case, we treated the p"-alumina model in reference [3]. Here the two-dimensional honeycomb lattice is assum- ed for conducting Na+ ions, and the occupancy rate p
of these sites (density) is assumed to vary from 0 to 1. If a repulsive interaction of magnitude
1
e1
existsU.U U.3 1.U 1.3 L V L.3
I / T among conducting ions, an orderedregion of conduct-
ing ions and vacancieswill be established in the p-T FIG. 1. - Diffusion coefficient of Na' calculated for the p-alu- diagram near = 112. figure 2, such a region mina model. The D plotted here is the VWf part of Eq. (1) only,
the exp(- pu) factor being excluded. Note that the
o
case calculated by the pair approximation of the Cluster- contributes a negative value to the activation energy. The tempe- Variation method [lo] (the static version of PPM) is0.3 0.4 0 5 0.6 0.7 Na CONCENTRATION p
FIG. 2. -The phase diagram which separates the ordered region (inside) from the disordered (outside) among Naf ions in the /?"-alumina model when Na' ions are repulsive ( 8 i 0). The curve is calculated for a honeycomb lattice using the pair approxi-
mation of the CV method [lo].
The order-disorder transition is most typical at
p = 112. In figure 3, the result of calculation of the p = 112 case for the specific heat, the change of appa- rent activation energy of diffusion and the diffusion coefficient D with respect to 1/T are plotted. The conductivity o (times temperature
7')
shows prac- tically the same behavior as D. [See Eq. (2).] Figure 3shows a sharp break in the log D us. 1/T curve with a discontinuous jump in the overall activation energy at the critical point of the order-disorder transition. At the same time, a lambda-type specific heat peak characteristic of the second-order phase transition is observed at exactly the point of discontinuity of the
€/kT
FIG. 3. -Specific, heat c, the diffusion coefficient D and the overall activation energy [the derivative - d In D/d(l/T)], which accompany the second-order phase transition in the conducting
-ions for the /?"-alumina model with p = 0.5.
lag
D
vs. l/~c;rve. The change effected in the log D vs.1/T curve by ordering is the effect of VWf in Eq. (1). The analysis of the results [3] shows that the decrease in the diffusion coefficient by ordering is due partly to the increase in the activation energy of ionic motion caused by ordering as emphasized by Pardee and Mahan [ll], who used a lattice gas model for the calculation of the phase change. However, the major part of the decrease in D is found to be' due to the decrease in f upon ordering. This means that the limitation imposed by the ordering upon the paths which are available to ions for long-range diffusion is mostly responsible for the decrease in the conduc- tivity and for the increase in the overall activation energy.
The situation shown in figure 3 is very similar to that
observed in RbAg,I, at 209 K [12, 131. Geller investi- gated in detail the change in the structure due to this phase change and found a distortion of the crystal which created a further subdivision of sublattices for Ag ions [14]. Since both phases are characterized as sublattice-disordered phases, he called this phase change the disorder-disorder transition. Salamon,
et al., measured change of the specific heat and the
ionic conductivity at the transformation in detail and confirmed that this was a typical second-order trans- formation [15, 161. He then ascribed the change in structure to the Jahn-Teller distortion which accompa- nies the change in the correlation among Ag ions [17].
Aside from a semantic question of whether this can be called an order-disorder transition or not, it may now seem that this transition is due to an apparent inter- action among Ag ions as discussed above.
2.2 In mixed (alloyed) compounds, an order- disorder transition can occur in the distribution of two kinds of ions in the rigid sublattice. It is possible to show that a decrease in the conductivity similar to the above case can occur in the ordered state even if no long-range order among conducting ions exists. A
C7-162 H. SAT0 AND R. KIKUCHI
tice [18]. We will show in this section that the flow of oxygen ions in their own (s.c.) sublattice can be affected if the ordering of Ca and Zr ions exists in the cation (f.c.c.) sublattice.
If there should be a cation ordering caused by the nearest-neighbor interactions in the f.c.c. sublat- tice around 15
%
Ca, the ordered structure should correspond to that of Cu,Au type [lo, 201. The phase diagram of the ordered structure in the f.c.c. lattice with nearest-neighbor interactions has been calculated with the tetrahedron approximation of the Cluster Variation method [20] and is shown in figure 4. This indicates the onset of ordering around 15 % Ca. Figure 5 shows the ordered cation sublattice (circles) and the anion sublattice (crosses). Filled circles indicate Ca ions and open circles Zr ions. In both the ordered and the disordered state (of the cation sublat-0.15 0.20 0.25 0 30 0 35 0.40 0.45 0.50
DENSITY OF T H E B COMPONENT
FIG. 4. - Phase diagram of ordered structures in the Cu-Au type alloy in f.c.c. lattice. The parameter e is defined as.
4 E = EAA
+
EBB - 2 EAB.0 AND
.
: CATIONSX : OXYGENS AND VACANCIES
5. - Two possible types of jump of 0-- in calciastabilized zirconia at 25 % Ca composition.
tice), a unit cube (or more strictly a tetrahedron) composed of cations in figure 5 include one Ca ion each on -the average (at 25
%
Ca) and, hence, the occupation probability of 0-- ions on the anion sublattice is not affected by the order-disorder trans- formation in the cation sublattice. Since the concen- tration of vacancies of anion sites is low, it is not likely to have a long-range order among 0-- ions through their repulsive interactions [3] (of figure 4). Although there are no preferential sites for 0-- ions, two types of paths (a- and b- in figure 5) can be differentiated for oxygen ion migration in the ordered state. If the Ca-0 repulsion is stronger than the Zr-0 repulsibn, the b-type jump in figure 5 is easier than the a-type jump and hence the a-type jump tends to be avoided. This affects the efficiency of jumps which contribute to the conductivity, and the conductivity in the ordered state is lowered. This is the correlation effect represented by f and f , in Eqs. (1) and (2), respectively. The result of the calculation is compared with experiments [18] in figure 6 . The agreement is quite favorable. Therefore, it is reasonable to expect a similar situation in reality.I I I
-q
- I I IP
;
- I \ I 1-
A' \ \ - - - \ - - \ - - - 4 --
.
- -n - - - -&--, TIEN, EXPERIMENT 800°C
-
-
THIS PAPER
I I I
FIG. 6 . - Dependence of ionic conductivity on composition, showing the effect of ordering in the cation sublattice on the anion
conduction.
The parameters adopted for the calculation are the ordering energy 8 , in'the cation sublattice, E~ which indicates the difference in the interaction energies of oxygen ions and two kinds of cations, respectively,
40
104
MOLE % CaO
and c3 which indicates the pair interaction energy among oxygen ions. These are defined as follows :
E O p O = &3 > 0
with relative magnitudes
It is clear that the limitation in the path for diffusion or the decrease in the correlation factor
f
by ordering is responsible for the decrease in the conductivity and for the increase in the activation energy in the ordered state.2.3 Recently, it has become apparent that there are a number of ionic compounds in which the ionic conductivity passes smoothly from values typical of normal solid to values typical of super-ionic- conductors. Such solids have been classified as Class 111 solids, and the transformation as the Faraday transition, by O'Keeffe [21]. Such transition to the super-ionic-conducting state has been shown to have a characteristic similar to melting, and the term sublat- tice melting is sometimes used [21]. It is, however, easily shown that a transition with such characteristics can be created by a cooperative excitation of ions from the normal sites to interstitial sites to form the sublat- tice disordered state.
The Faraday transition is well-documented for crystals. with the fluorite structure. The normal sites for anions in the fluorite structure are the tetrahedral sites of the f.c.c. cation lattice. These anions are known to be easily excited into the octahedral sites, thus creating vacancies in the tetrahedral sites and the partially occupied octahedral sites ; this process,is the formation of Frenkel defects' The two sites combined constitute the sublattice disordered phase. If any interaction among excited ions exists, the excitation becomes more abrupt than the ordinary Frenkel defect formation. Let us call such excitation as coope- rative excitation. Such a cooperative excitation of Frenkel defects has been discussed theoretically by Rice, et al. [22], and.Huberman [23], based on simple models. If the excitation of ions occurs in a way to promote a further excitation, ,the excitation becomes more enhanced and, if the interaction becomes strong enough, the excitation becomes eventually a first order transition [22, 231.
Even without using such strongly cooperative model, it is easy theoretically to reproduce a result similar to that of the Faraday transition. Actually, we have calculated the relation between the excitation and the change in the ionic conductivity similar to the above based on a simple model taken for P-alu- mina [3]. Here, the two-dimensional honeycomb lattice is divided intb. two sublattices, A and B
(figure 7) ; the A-sites are assigned to the normal sites and the B-sites to the interstitial sites. If the density p
of~the conducting ions is 112, the case corresponds to the stoichiometric compound, and then ions exclusively occupy the A-sites at low temperatures. By assigning a site occupancy energy w to B-sites and a nearest- neighbor interaction energy
-
E (repulsion) to conducting ions, the distribution change of ions on A- and B-sites with temperature and hence the specific heat involved in the excitation can -be calculated as shorn in figure 8. The calculated results [3] are forw = 5 1 8 1 .
The calculation of the ionic conductivity is, how- ever, very similar to the case of the ordered state of
FIG. 7. - TWO sublattices A and B of the two-dimensional honey- comb lattice.
FIG. 8. - Change'of aT, specific heat and the overall activation energy of ionic conduction which accompany a cooperative exci-
C7-164 H. SAT0 AND R. KIKUCHI
the p"-alumina model treated earlier, the only diffe- rence being that in the /?-case two built-in sublattices are involved, whereas two sublattices are created in the p"-case as a consequence of ordering. In other words, as the occupancy rate of the A-sites is increased at low temperatures, a sharp decrease in the correlation factor and hence in the ionic conductivity is observed. Therefore, the change in o from the conducting state
to the normal state becomes enhanced. In figure 8,
log oT(o
=
VWfI p), specific heat, and the change in the apparent activation energy are plotted against 1/T.As is clear from the figure, a smooth but a sharp bend in the log oT us. 1/T curve is observed. At the
same time, a specific heat curve with a rounded peak around the bendpoint of the log oT us. 1/T curve is
well observed. The curve is reminiscent of the Class I11 solids or solids with the Faraday transitions which O'Keeffe classified 1211. The calculation can be extend-
ed to the case of fluorite structure with the aid of PPM. It might be worthwhile to mention the signifi- cance o f f , in the product o of Eq. (2). The appearance
o f f , is due to the existence of a real correlation effect in flow (physical correlation effect) like that in the ordered state. In general cases of diffusion, the corre- lation factor f also includes the effect of physical
correlation, while in ordinary self-diffusion only the geometrical correlation factor is involved for the isotope diffusion [4]. Therefore, if the Haven ratio, the ratio of the isotope diffusion coefficient D and the charge diffusion coefficient
D,
(= o k ~ l n e ' ) is taken,the ratio gives the ratio o f f and
f,,
and the effect of physical correlation can be cancelled out 131. The physical correlation effect always appears if the preferential sites exist in sublattice disordered phases because there is always a tendency for ions to jump into preferential sites in such a case. Such an effect should appear even in the short-range ordered statewhere no distinct sublattices are created. In our pre- vious treatment on the model of p"-alumina, we failed to show the appearance o f f , in the absence of the long- range order [3]. This is due to an inappropriate averag- ing process introduced in the calculation mentioned earlier [9]. In the computer simulation of the ionic conduction for a p"-alumina model carried out b y Murch, the appearance o f f , in the short-rangq ordered state is clearly shown (*). A correction will be made to our previous calculations by -PPM in order to show this effect analytically.
3. Summary. - Utilizing the Path Probability method, it is shown that the existence of the sublattice disorder is a necessary condition for the appearance of
the super-ionic-conduction and that the cooperative features of ion transport in disordered systems are responsible for the reduction in the overall activation energy of ionic conduction from that of one ion motion. The effects of ordering on the conductivity are discussed in three different categories :
a) ordering among conducting ions,
b) ordering of ions in the rigid sublattice and
c) a cooperative excitation of ions from their
normal sites to interstitial sites.
In each case, a sharp decrease in the ionic conduc- tivity is expected. The decrease in the ionic conducti- vity in the ordered state is mainly due to the decrease in the correlation factor rather than to the increase in
the activation energy of ionic motion. Since the correlation effect is a characteristic of the dc conduc- tivity, a high-frequency conductivity is expected to show a less significant decrease in the ordered .state.
(*) Murch, G. E.., private communication.
References
[I] SATO, H. and KIKUCHI, R., Superionic Conductors, eds.
G. D. Mahan and W. L. Roth (Plenum Press, New York)
1976, p. 135.
[2] KIKUCHI, R., Prog. Th. Phys. Suppl. No. 35 (1966) p. 1. [3] SATO, H. and KIKUCHI, R., J. Chem. Phys. 55 (1971) 677. [4] SATO, H. and KIKUCHI, R., Mass Transport Phenomena in
Ceramics, eds. A. R. Cooper and A. H. Heuer (Plenum Press, New York) 1975, p. 149.
[5] MANNING, J. R., Dzffusion Kinetics for Atoms in Crystals (Van Nostrand, Princeton) 1968.
[6] PETERSON, N. L., Solid State Fhysics, eds, F. Seitz, D. Turn- bull and H. Ehrenreich (Academic Press) 1968, Vol. 22. p. 409.
[7] BAKKAR, H., STOLWIJK, N. A., VAN DER MEIJ, L. and ZUUREN- DONK, T. J., Proceedings of Computer Simulation for
r, Material Application, AIME 20, part 1 (1976) p. 96. [8] MURCH, G. E. and THORN, R. J., Phil. Mag. 35 (1977) 493. [9] GSCHWEND, K., SATO, H. and KIKUCHI, R., BUN. Am. Phys.
Soc. 22 (1977) 442.
[lo] KIKUCHI, R., The Cluster Variation Method, J. Physique
Colloq. 38 (1977) C7.
[ll] PARDEE, W. J. and MAHAN, G. D., J. Solid State Chem. 15 (1975) 310.
[12] OWENS, B. B., Advances in Electrochemistry and Electro- chemical Engineering 8, ed. C . W. Tobias (John Wiley and Sons, New York) 1971, p. 1.
[13] JOHNSTON, W. V., WIEDERSICH, H. and LINDBERG, G. D.,
J. Chem. Phys. 51 (1969) 3739. [14] GELLER, S . , Science 157 (1967) 310.)
[I51 LEDERMAN, F. L., SALAMON, M. B. and PEISL, H., Solid State
Commun. 19 (1976) 147.
[16] VARGAS, R., SALAMON, M. .B. and FLYNN, C. P., Phys. Rev.
Lett. 37 (1977) 1550.
[17] SALAMON, M. B., Phys. Rev. B 15 (1977) 2236.
[18] TIEN, T. Y., J. Am. Cer. Soc. 47 (1964) 430.
[19] CARTER, R. E. and ROTH, W. L., Electromotive Force Measure- ments in High Temperature Systems, ed. C. B. Alcock (Institute of Minerals and Metals, London) 1968, p. 125. [20] VAN BAAL, C . M., 'Physica 64 (1973) 571.
I211 O'KEEFPE, M., Superionic Conductors, eds. G. D. Mahan and W. L. Roth (Plenum Publishing Co., New York) 1976, p. 101.
[22] RICE, M. J., STRASSLER, S. and TOOMBS, G. A., Phys. Rev. Lett. 32 (1974) 596.
