- Email: [email protected]
Can the mechanisms of ion transport in SSIs be determined by computer modelling?
SOLID STATE
Solid State Ionics 53-56 (1992) 955-963 North-Holland
IONICS Can the mechanisms of ion transport in SSIs be determined by computer modelling? C.R.A. Catlow The Royal Institution, 21 Albemarle Street, London W1X 4BS, UK
Simulation methods are now used routinely to examine structural propenies and transport mechanisms in solid state ionic materials. This paper will review recent applications of the techniques with a strong emphasis on their use in the elucidation of mechanism and of the factors controlling ionic mobility. We will consider crystalline, amorphous and polymeric materials. The developing field of electronic structure calculations in solid state ionics will also be briefly discussed.
1. Introduction C o m p u t e r simulation techniques are now a stand a r d tool in the study o f solid state ionic materials. A n d there is ample evidence o f their ability to model accurately the structures and defect properties o f ionic and semi ionic solids [ 1-8 ]. In this article we concentrate on their role in understanding the mechanisms o f ionic transport in solids, where we aim to show that the methods now have a predictive capacity which will be o f special value as their use is extended to highly complex systems such as polymers, glasses and proton conductors. Very recent applications to these latter systems will be described after a review o f work undertaken during the last few years on crystalline materials which has clearly established the value of simulations in mechanistic studies. We preface this by a short s u m m a r y o f the principal techniques used in current studies.
2. Methodologies These are now well established and have been reviewed in the last few years by Gillan [3 ], Catlow [6], Vashishta [4] and H a r d i n g [8 ]. We shall be concerned mainly with simulation techniques, which rest upon the specification o f an interatomic potential which expresses the total energy o f the system in some readily usable way as a function o f the nuclear
coordinates. F o r ionic and semi-ionic systems, Born model potentials are c o m m o n l y employed; these partition the total energy into long-range Coulomb and short-range pair and 3-body potentials. Detailed discussions are given in refs. [6 ] and [8 ]. Three classes of simulation technique have been employed in the study of solid state ionics: first, static lattice simulations in which energy-minimisation methods are used to generate the lowest energy configuration o f a perfect periodic crystal structure of a region o f crystal a r o u n d a defect or defect cluster. In the latter case, methods based on the M o t t - L i t t l e t o n a p p r o x i m a t i o n [9] have proved to be particularly effective and have for several systems yielded quantitative agreement with experimentally d e t e r m i n e d defect parameters [ 1,3,8 ]. Secondly, molecular dynamics ( M D ) methods have been successfully employed in the study o f transport in solid state ionics. This widely applied c o m p u t a t i o n a l technique entails numerical solution o f classical equations o f m o t i o n o f an ensemble o f p a n i c l e s (to which periodic b o u n d a r y conditions are applied in the case o f crystalline materials); as such, kinetic energy is included explicitly in the simulation. Unlike the case o f the static methods, dynamical details are available directly from M D simulations. There is, however, as we have argued elsewhere [10], a natural complem e n t a r i t y between static and dynamical methods with the latter being most a p p r o p r i a t e for superionic systems while the former are the natural technique
0167-2738/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
956
C.R.A. Catlow/Mechanisms of ion transport in SSIs
for systems with more slowly diffusing ions (in practice for systems where D < 10 -8 cm 2 s-1 ). Thirdly, Monte Carlo (MC) methods may be employed in systems with high levels of disorder, especially when a variety of j u m p mechanisms are involved. MC is essentially a technique of computational statistical mechanics, in which an ensemble of configurations is generated by a succession of random moves. The "acceptance criterion" (i.e. the procedure which decides which configurations are to be included in the ensemble) leads to an ensemble which weights the probability of inclusion of a configuration according to its Boltzmann factor. Detailed discussions are given in refs. [ 11,12 ]. The same procedure may be applied to studying atomic transport, with atoms being moved at random and the probability of accepting a jump being weighted in proportion to its frequency. Successful MC simulation studies of several alloy systems have been reported by Murch and coworkers [ 13,14] and the same techniques have been applied to the solid state oxygen ion conductor Y/CeO2 [ 15]. We should stress that the accuracy and reliability of all simulation methods depends critically on the quality of the interatomic potentials employed. Considerable experience has been built up during the last 15 years in deriving and refining interatomic potentials for ionic and semi-ionic systems; and there is now abundant evidence that for many classes o f inorganic solid it is possible to obtain quantitative agreement between experimental and calculated parameters relating to atomic transport processes in solid state ionic materials. In this context we should note an important current limitation of the MD method is that it has not, except in a very limited number of cases, been possible to include a treatment of ionic polarisation (normally described using the Shell model [ 16 ] ). Work is, however, in progress to remove this limitation. Finally, we should note that there is an important and expanding role for quantum mechanical studies of properties of solids. In particular there has recently been major progress in the development of embedded quantum mechanical cluster techniques for studying defects in solids [ 17,18 ] ; and an illustration of their application to proton transport processes will be given later in this article.
3. Mechanistic studies of crystalline solid state ionic materials To demonstrate the power of simulation studies in the determination of mechanisms we choose four systems which have been extensively investigated by both theoretical and experimental techniques in recent years. Each of them will highlight different types of mechanistic complexity that can be unravelled by simulation methods. 3.1. Oxygen ion transport in complex oxides
Here we take two topical oxide materials where recent applications of simulations have led to valuable mechanistic information. 3. I. 1. Pyrochlore structured oxides Anion deficient fluorite structured oxides e.g. Y / ZrO2 and rare earth doped CeO2 are amongst the most widely studied oxygen ion conducting materials. Simulation studies have made a major contribution to our knowledge of these systems, and in conjunction with experimental results they have led to a good basic understanding of the mechanism of oxygen transport. Oxygen migration is effected by anion vacancy jumps for which both calculations and experiment give an activation energy of ~0.5 eV. Vacancies are trapped by dopants with the trapping energy being strongly dependent on the dopant ionic radius, in particular the extent of the mismatch between the radius of the dopant ion and the host lattice ion [19,20]. At higher dopant concentrations ( > ~ 8 tool%) there is a dramatic reduction in the conductivity [21 ] due to more complex dopant-vacancy interactions - a process that was successfully simulated by Monte Carlo methods [ 15 ]. The pyrochlore structure shown in fig. 1 is an intriguing variant of the parent fluorite structure. It is of general composition A2BzOv (where A is normally a trivalent and B a tetravalent ion) with ordering of the A and B ions and of the oxygen "vacancies". Early work of van Dijk et al. [22] showed that the pyrochlore structured materials are promising oxygen ion conductors. More detailed studies have been reported by Moon et al. [ 23,24 ], who have undertaken a comprehensive study of systems with A=Gd or Y and B=Zr or Ti; mixed Z r / T i systems were included
957
C.R.A. Catlow/Mechanisms of ion transport in SSIs
Y2Ti207 and YzZr207 - are summarised in table 1. They show first that rapid vacancy migration between 4 8 ( f ) sites may occur; indeed the activation energy for the titanates is rather low compared with experiment. The mechanism is a direct linear oxygen j u m p between the two 48 ( f ) sites. The question of the origin of the 48 ( f ) vacancies is more intriguing. Excitation of the oxygen ions from the 4 8 ( f ) into the vacant 8 (b) sites requires a large amount of energy - ~ 5 eV. In a perfectly ordered pyrochlore the "constitutional vacancies" are therefore locked into their sites, and low oxygen ion conductivity must follow. However, preliminary calculations o f Wilde show that introduction of disorder on the cation sublattices dramatically reduces (by more than 50%) the energy to excite oxygen ions from the 48 (f) into the 8 (b) sites. Further work is needed to provide a satisfactory understanding of this system. But both experiment and theory have already revealed a fascinating dependence of the anion transport processes on the degree of cation disorder.
Structural"'"i~Vacancy
Q
A3+ in 16c3m000
•
B 4÷ in 16 d 3m 1/2 1/2 1/2
(~
0(1) in 48 f mm x 1/8 1/8 0(2) in 8 a ~,3rn 1/8 1/8
+ Vo in8b~,3m 3/8 3/8 3/8 Fig. 1. Projection of the contents of the pyrochlore supercell for 0 < z < ¼.The displacement of the oxygen ions (0.46/~.) from their ideal tetrahedral locations in the fluorite structure corresponds to that found for Y2Ti207 in the present work.
3.1.2. High temperature superconductors The celebrated YBCO(YBa2Ca3Oy_x) superconductor is, in addition to being a high Tc material, a good oxygen conductor; and since it is now clear that disorder on the anion sub-lattice has an important influence on the superconducting properties of the material, there have been a number of studies of oxygen diffusion in the material, for example, the work o f X i e et al. [26] and Rothman et al. [27], the results of which are summarised in table 2. The well known structure of the materials is shown in fig. 2. It comprises two types of copper: Cu( 1 ) (copper chains) and Cu (2) (copper planes), and five types of oxygen site of which the O ( 5 ) is not
in this study. Their work reveals low oxygen migration energies ( ~ 0.8 eV) with an interesting correlation with the degree o f A / B cation order: increase in disorder leads to a reduction in the activation energy. The basic mechanistic problems posed by these studies concern first the nature of the mobile species, and secondly, their migration mechanisms. In particular, what is the role of the "built in" vacancies on oxygen ion transport? The problem has been investigated using simulation techniques by Wilde [25], whose results for four systems - Gd2Ti2Ov, GdzZr207, Table 1 Calculated anion defect energies (eV) in pyrochlore structured oxides. 0 2 - defect
Gd2Ti207
Gd2Zr207
YETi207
Y2Zr207
8a vacancy 48f vacancy 8b interstitial Frenkel pair saddle point activation energy (48(f)-~48(f))
20.47 16.96 - 11.64 5.32 17.13 0.18
20.10 17.96 - 13.75 4.21 18.70 0.73
20.38 17.02 - 12.42 4.60 17.19 0.18
19.88 17.91 - 14.64 3.27 18.76 0.86
CR.A. Catlow/Mechanisms of ion transport in SSIs
958
Table 2 Arrhenius parameters for oxygen diffusion in YBCO: D = Doexp ( - Ea/kT). Do (crn2 so-~ )
Ea (eV)
Ref.
3.5×10 -~ 1.4× 10-4 3.7× 10-4
1.03 0.97 0.99
[261 [27] this work, [28]
Y
y
I.... I
e (3-
)
c
to run at higher temperatures (1400-1800 K) than those used in the experimental diffusion studies. However, from the analysis of the temperature dependence of the simulated diffusion coefficients (which showed Arrhenius behaviour) we are able to obtain pre-exponential factors and activation energies that are in good agreement with experiment as is evident from the results given in table 2. Our main interest is, however, in the mechanisms of ionic transport revealed by the simulations. Analysis of the trajectories of migrating oxygen ions shows that ions move predominantly between O ( 1 ), O (4) and 0 ( 5 ) sites; and in the time scale of our simulation no jumps to 0 ( 2 ) and 0 ( 3 ) sites were observed - a result that is clearly consistent with the much higher measured diffusivity in the ah plane. We found that migration takes place between O( 1 ) sites, but not directly; rather jumps take place via 0 ( 4 ) and 0 ( 5 ) sites. Complex sequences may occur, e.g. O(1 ) - ~ O ( 4 ) - * O ( 5 ) - ~ O ( 1 ) , but no direct O ( 5 ) - ~ O ( 5 ) jumps are observed. The speculative mechanism of Rothman et al. [271 is not therefore supported by these calculations. The results accord with static lattice simulations of Islam [291 and Baetzold [30], and demonstrate the ability of molecular dynamics to reveal the mechanism directly. 3.2. Systems with complex correlated motions
¥
Fig. 2. The slructure of YBa:Cu307. normally occupied. The material is nonstoichiometric, with x ranging from 0 to I. The diffusion studies all relate to substoichiometric materials, for example the sample in the work of Rothman et al. had x ~ 0.1. The resulting vacancies are predominantly in the O( 1 ) site. The oxygen diffusivity is found to be far higher in the (ab) plane than in the c direction. There have been several speculative discussions of the mechanisms of oxygen ion transport in these materials. For example, Rothman et al. [27 ] suggested that oxygen ions jumped into the vacant O ( 5 ) sites between which they then migrated until returning to a vacant O ( 1 ) site. To investigate this problem further we have recently undertaken a detailed MD study of YBa2Cu3069 [28]. In order to obtain sufficiently high diffusion coefficients it was necessary
In contrast to the materials discussed above in which diffusion is effected by single discrete vacancy jumps, there are many superionics in which correlated motions of ions provide the key to the high ionic conductivity. We give here two examples in which simulations have played an important r6le in elucidating details of migration mechanisms. 3.2. I. Li~N This material has one of the highest known Li + ion conductivities. Its unique layer structure shown in fig. 3 also results in high anisotropy of the conductivity which is much greater parallel than perpendicular to the layers [31 ]. A detailed MD study reported by Wolf [ 32,33 ] revealed nicely the origins of the high Li conductivity. Analysis of his results showed that Li ions are readily excited from layer sites into interlayer interstitial positions. Following the creation of the vacancies within the layers there
959
C.R.A. Catlow/Mechanisms of ion transport in SSIs 3.2.2. R b B i F 4
perspective
i i I i I I I
o--:--o
I I
rl
~
....
I
,' ?
?/, ',/ #
. . . t.
D
~/ ;,'
/
o-
•
. . . .
/
/. i.A-iT../
001 plane I
ii
ii
I
o"--di'--d: :--o' O
N3
•
Li ~' ~on
Jan
Fig. 3. The structure of Li3N.
O
O
6
5 4
O
O 3
•
O
•
O
N ~
•
U + ion
ion
O
Li + v a c a n c y
I
Fig. 4. Schematic illustration of the mechanism of correlated Li migration in Li3N. is a cascade of migration events involving several ions moving simultaneously. A typical example is shown in fig. 4 where we see that the concerted motion of several ions effects transport of the vacancy from position 7 to position 1 in one step. This remains one of the best examples of correlated motion in superionic conductors; and it was revealed naturally and straightforwardly by application of the MD technique. Wolf's work also revealed fascinating mechanisms effecting transport perpendicular to the layers, which also effect interchange of "layer" and "bridging" lithium ions. Indeed the MD simulations have led to a very detailed understanding of the rich mechanistic variety in this system.
The high conductivity of this cation disordered fluorite structured material was first shown by Reau and coworkers [34]. The material has the fluorite structure with a disordered distribution of Rb and Bi over the cation sites. An extensive study of the local structure around both Rb and Bi, carried out using the EXAFS technique, has shown that disorder is generated preferentially around the Rb ions [ 35,37 ]. Indeed we have postulated that at anion sites around which there is a local excess of Rb, it will be energetically preferable to create vacancies; the neighbouring Rb ions then relax away from the vacancy as shown in fig. 5. The process would lead to a net shortening of the R b - F bond; such a reduction in the R b - F bond length on raising the temperature is demonstrated by our EXAFS studies. Of greater interest to our present discussion is the question of the migration mechanism for the interstitials that are created by the excitation of F ions from their regular lattice sites. It has long been known that interstitials in the fluorite structure migrate by the "interstitialcy" mechanism in which the migrating F - ion displaces neighbouring lattice ions into interstitial sites. A molecular dynamics study of RbBiF4 undertaken by Cox [35,36] has shown that such mechanisms occur in RbBiF4; fig. 6 illustrates the trajectory of one F - ion which is seen moving between lattice and interstitial sites, the characteristic behaviour in a system in which interstitialcy motion is occurring. However, the MD study revealed far more complex mechanisms. Several F ions are found to move in a correlated manner, the motion of each ion being of the interstitialcy type. As
F - a t corners of cube []
F-vacancy
Fig. 5. Proposed model for Rb relaxation awayfrom vacancyin RbBiF4.
960
C.R.A. Catlow/Mechanisms of ion transport in SSIs
/ Fig. 6. Trajectoryof migrating F interstitial in RbBiF4. with Li3N, ease of creation of the defects is vital for the high conductivity; but correlated migration mechanisms are again clearly of central importance.
ions resulting in a material with high conductivity for both cations (and anions); second, doped electronically conducting polymers (e.g. Li doped polyacetylene) in which the dopant by acting as a donor or acceptor state greatly enhances the conductivity of the polymer. In both cases it is most important to obtain a detailed understanding of the mechanisms of ion transport in the polymer matrix. Simulation provides a unique opportunity for deriving this information. Work is currently in progress on both types of system; but in the discussion which follows we concentrate on the doped polyacetylenes. Simulations of pristine and crystalline Li doped trans-polyacetylene at a temperature of 300 K have recently been undertaken by Ses6 [38]. Following earlier static lattice studies [39] a standard molecular mechanics force field was used to model the polymer chain; no bond length or bond angle constraints were employed. The crystal structure of the material (see fig. 7) was well reproduced. Analysis of the dynamics of the undoped system revealed vibrational frequencies in reasonable accord with experiment. Lithium ions were then introduced in low concentrations (Li/C 0.01 ) and dynamical simulations performed for up to 100 ps. The results dem-
3.3. Mechanisms in crystalline superionics: Summary
a) G~
From the examples given above it should be clear that simulations have a proven success in the elucidation of transport mechanisms in a wide variety of systems. MD is able to yield mechanisms directly, while static methods enable us to study the energetics of the migration process. In the remainder of this article we will concentrate on recent applications of the methods to more complex systems.
C~ G, @.
= 7.32A 4.24A c -
llliii
,,i2 ,G ,,G ,,G
4.96A
b)
4. Simulation of ion transport in polymers
Two types of doped polymer system have attracted considerable attention in recent years: first the salt doped polymer systems (e.g. NaI:PEO) in which metal salts dissolve in polymers containing polar atoms, which are thought to "solvate" the cat-
b
½
5
b
Fig. 7. Crystal structure of trans polyacetylene,viewed (i) perpendicular (ii) parallel to the c axis.
961
C.R.A. Catlow/Mechanismsof ion transport in SSIs
onstrate that Li diffuses very rapidly within the crystalline polymer matrix as shown by the diffusion coefficients collected in table 3. Note that the diffusivity is higher along the z axis, i.e. down the channels, between the polymers, than it is in the x y direction, i.e. perpendicular to the polymer chains. Analysis of the results yielded the following valuable mechanistic information: (i) The Li ÷ ions are not localised at particular sites but move freely along the interchain channels. (ii) The motions of the different Li + ions are strongly correlated. The Li+...Li + Coulomb repulsions are the strongest forces in the system and act so as to maximise the separation between the Li + ions. (iii) Li ÷ ions can move between different channels but this is a slower process than diffusion down the channels (hence Dz>Dxy as evident from table 3). Simulations on systems with high Li concentration showed instabilities. However, it has proved possible to simulate K doped systems at high concentrations. Very different results are obtained with low dopant diffusivity and "crystallisation" o f the K into well defined sites being observed in the simulation. More detailed accounts of these studies (and of related work on NaI doped PEO) will be reported shortly. The work summarised above demonstrates the viability and value of M D studies o f cation migration in doped polymers.
al. [42] has shown that it is possible by simulated quenching from the melt to generate structures that are in excellent agreement with experimental RDFs, as shown by fig. 8, which compares R D F s calculated by Vessal et al. [42] with the recent results from neutron data of Wright [43]. These techniques are beginning to be applied to systems with network modifying cations e.g. alkali silicate glasses, which have been studied recently by both Cormack et al. [44] and by Vessal [45]. Their
28
'
I
I
i
J
26 24. 22 II II II II II II
20 18
~
/ ~ f ~ / / ~
!
\
16
I
I
12
I
10 5. S i m u l a t i o n
;
I
. . . . . . . . .
\
/
/ / \
/
d
8
of glasses
Since the pioneering work o f Woodcock et al. [ 40 ] on vitreous silica there have been several successful M D simulations o f glassy materials. Recent work of both Feuston and Garafolini [41 ] and of Vessal et Table 3 Calculated diffusion coefficients at 300 K in lightly Li doped polyacetylene. Axis
D (cm2 S-1 )
(x, y) z overall
2× 10-6 4X10 -5 4.2 X 10- 5
6 ,-,4
0 0
I
I
1
2
3
1
I
I
I
.4 rCA)
5
a
7
8
Fig. 8. Measured and simulated (dotted line) RDFs for SiO2; three upper curves indicate respectively Si...Si, Si...O, and O...O calculated partial RDFs.
962
C.R.A. Catlow/Mechanisms of ion transport in SSIs
results support the modified random network models advanced by Greaves [46], as they show that the presence of the cations which have reasonably well defined coordination extensively modifies the silicate network. Evidence for aggregation of the cations is also found. In mixed cation systems (e.g. R b / N a glasses) intimate mixing of the two types of cation occurs - a result which is of clear relevance to the long standing puzzle of the mixed alkali effect [47 ]. There is a major future role for M D methods for elucidating the mechanistic details of cation transport in these materials.
6. Proton transport in oxides
One of the most interesting types of proton conductor is based on the perovskite or closely related structures with low valence dopants introduced on the B site; examples are Yb/SrCeO3 and Fe/KTaO3. Doping on the B site would be expected to lead to oxygen vacancy formation. However, the following reaction with water can cause the introduction of OH groups: Oo + V~ + H2 O--, 2 O H o . The protons so introduced give rise to the conductivity. What, however, is their migration mechanism? To investigate this problem we [48] have undertaken quantum mechanical calculations on the transfer of protons between two O 2- ions, i.e. we have calculated the activation energy for the process OH-+O2-~O2+OH-, assuming that the maxim u m corresponds to the proton equidistant between the oxygen ions. The barrier was calculated for two O...O spacings: the larger of 3.02 A corresponds to that in SrCeO3 - a perovskite with a relatively large O...O distance; the smaller of 2.75 to that in LaA103 - a material with a comparatively short O...O spacing. The calculations were performed on oxygen ions embedded in an array of point charges to simulate the remainder of the lattice. Calculations were performed both at the Hartree-Fock level and including correlation effets via the MP2 technique. The results are summarised in table 4. They demonstrate an appreciable dependence of the calculated energies on bond lengths. The values calculated for the barrier height are, of
Table 4 Calculated barrier heights for proton migration. O...O distance (A)
Barrier height (eV) A a~
Bbl
2.75
1.40
0.74
3.02
1.27
1.38
a) Hartree-Fock; ~ Hartree-Fock correlation. course, an upper limit on the activation energy which will be appreciably reduced by the ability of the proton to tunnel through the barrier. Furthermore, inclusion of a more detailed treatment of lattice relaxation may modify the barrier. The values are, however, sufficiently low to suggest that the process we have proposed will provide the dominant proton transfer mechanism. Further studies of these systems using both simulation and quantum mechanical techniques are in progress.
7. Summary and conclusions
We hope that the work summarised above has demonstrated that simulations can indeed provide valuable guidance as to transport mechanisms in solid state ionics. The use of the techniques in studying crystalline materials is now routine. The future will see much more extensive applications of the methods to glassy and polymeric materials and an increasing use of electronic structure techniques.
Acknowledgement
I am grateful to Drs. G. Ses6, B. Vessal and J.D. Gale for permission to refer to their unpublished work.
References
[ 1]C.R.A. Catlow and W.C. Mackrodt, eds., in: Computer Simulation of Solids: Lecture Notes in Physics vol. 166 (Springer, Berlin, 1982). [2] M.J. Gillan, Solid State Ionics 9/10 (1983) 735. [3] W.C. Mackrodt, Solid State lonics 12 (1984) 175. [4] P. Vashishta, Solid State Ionics 18/19 (1986) 3.
C.R.A. Catlow/Mechanisms of ion transport in SSls
[5] M.J. Gillan, Physica B 13 (1985) 157. [6] C.R.A. Catlow, Ann. Rev. Mater. Sci. 16 (1986) 517. [7] P.W.M. Jacobs, J. Chem. Soc. Faraday Trans. 86 (1989) 415. [8] J.H. Harding, Rep. Prog. Phys. 53 (1990) 1403. [ 9 ] N.F. Mort and M.J. Littleton, Trans. Faraday Soc. 34 ( 1938 ) 485. [10]C.R.A. Catlow, in: Proc. European Mat. Res. Soc. (Strasbourg, 1991 ), to be published. [11] M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids (Oxford University Press, Oxford, 1987). [12]C.R.A. Catlow, S.C. Parker and M.P. Alien, eds., in: Computer Simulation of Fluids, Polymers and Solids, NATO ASI Series (Kluwer, Dordrecht, 1989). [ 13] G.E. Murch, in: Diffusion in Crystalline Solids, eds. G.E. Murch and A.S. Nowick (Academic Press, New York, 1984). [ 14] G.E. Murch, in: Non Traditional Methods in Diffusion, eds. G.E. Murch, H.K. Birnbaum and J.R. Cost (Warrendale, Mich., AIME, 1984). [15] G.E. Murch, A.D. Murray and C.R.A. Catlow, Solid State Ionics 18/19 (1986) 196. [ 16] B.G. Dick and A.W. Overhauser, Phys. Rev. 112 (1958) 90. [ 17] A.L. Shluger, N. Itoh, R.W. Grimes and C.R.A. Catlow, to be published. [18] R.W. Grimes, C.R.A. Catlow and A.L. Shluger, eds., in: Quantum Mechanical Calculations in Solid State Studies (World Scientific, Singapore, 1991 ). [19] G.R. Gerhardt-Anderson and A.S. Nowick, Solid State Ionics 5 (1981) 547. [20] V. Butler, C.R.A. Catlow, B.E.F. Fender and J.H. Harding, Solid State Ionics 8 (1983) 109. [21 ] D.Y. Wang, D.S. Park, J. Griffiths and A.S. Nowick, Solid State Ionics 2 (1981) 95. [22] M.P. van Dijk, A.J. Burgraaf, A.N. Cormack and C.R.A. Catlow, Solid State lonics 17 (1985) 159. [23] P.K. Moon and H.L. Tuller, Solid State Ionics 28-30 (1988) 470.
963
[24] P.K. Moon and H.L. Tuller, Solid State lonics (Mater. Res. Soc. Symp. Proc., 1989), eds. G. Nazri, R.A. Huggins and B.F. Shriver, Vol. 135 (1989) p. 149. [25 ] P.J. Wilde and C.R.A. Catlow, to be published. [ 26 ] X.M. Xie, T.G. Chen and Z.L. Wu, Phys. Rev. B 40 ( 1989 ) 4549. [27] S.J. Rothman, J.L. Routbart and J.E. Baker, Phys. Rev. B 40 (1989) 8852. [28] X. Zhang and C.R.A. Catlow, Phys. Rev. B, in press. [29] M.S. Islam, Supercord. Sci. Technol. 3 (1990) 531. [30] R. Baetzold, Phys. Rev. B 38 (1988) 11304. [31 ] A. Rabenau, Solid State Ionics 6 (1982) 277. [32] M.L. Wolf, J.R. Walker and C.R.A. Catlow, J. Phys. C 17 (1984) 6623. [33] M.L. Wolf and C.R.A. Catlow, J. Phys. C 17 (i984) 6635. [34] S.M. Matar and J.M. R6au, C.R. Sci. Acad. Sci. Paris 294 (1982) 649. [35] P.A. Cox, PhD Thesis (University of Keele, UK, 1989). [36] P.A. Cox and C.R.A. Catlow, to be published. [37]P.A. Cox, A.V. Chadwick and C.R.A. Catlow, to be published. [ 38 ] G. Ses6 and C.R.A. Catlow, to be published. [ 39] J. Corish, V.C.A. Hanratty, D.A. Morton-Blake, F. B6ni~re andA. Morin, J. Mol. Struct. 207 (1990) 41. [40] L.V. Woodcock, C.A. Angell and P. Cheeseman, J. Chem. Phys. 65 (1976) 1565. [41 ] B.P. Feuston and S.H. Garafolini, J. Chem. Phys. 89 (1988) 5818. [42 ] B. Vessal, M. Amini, D. Fincham and C.R.A. Catlow, Philos. Mag. 60 (1989) 753. [43 ] A.C. Wright, private communication. [44] C. Huang and A.N. Cormack, J. Chem. Phys. 93 (1990) 8180. [45] B. Vessal, M. Amini and C.R.A. Catlow, J. Phys. Chem., in press. [46] G.N. Greaves, J. Non-Cryst. Solids 71 (1985) 203. [47] J.M. Stevels, Handbuch der Physik, Vol. 20, ed. S. Fl/igge (Springer, Berlin, 1957) p. 246. [48 ] J.G. Gale and C.R.A. Catlow, to be published.
Solid State Ionics 53-56 (1992) 955-963 North-Holland
IONICS Can the mechanisms of ion transport in SSIs be determined by computer modelling? C.R.A. Catlow The Royal Institution, 21 Albemarle Street, London W1X 4BS, UK
Simulation methods are now used routinely to examine structural propenies and transport mechanisms in solid state ionic materials. This paper will review recent applications of the techniques with a strong emphasis on their use in the elucidation of mechanism and of the factors controlling ionic mobility. We will consider crystalline, amorphous and polymeric materials. The developing field of electronic structure calculations in solid state ionics will also be briefly discussed.
1. Introduction C o m p u t e r simulation techniques are now a stand a r d tool in the study o f solid state ionic materials. A n d there is ample evidence o f their ability to model accurately the structures and defect properties o f ionic and semi ionic solids [ 1-8 ]. In this article we concentrate on their role in understanding the mechanisms o f ionic transport in solids, where we aim to show that the methods now have a predictive capacity which will be o f special value as their use is extended to highly complex systems such as polymers, glasses and proton conductors. Very recent applications to these latter systems will be described after a review o f work undertaken during the last few years on crystalline materials which has clearly established the value of simulations in mechanistic studies. We preface this by a short s u m m a r y o f the principal techniques used in current studies.
2. Methodologies These are now well established and have been reviewed in the last few years by Gillan [3 ], Catlow [6], Vashishta [4] and H a r d i n g [8 ]. We shall be concerned mainly with simulation techniques, which rest upon the specification o f an interatomic potential which expresses the total energy o f the system in some readily usable way as a function o f the nuclear
coordinates. F o r ionic and semi-ionic systems, Born model potentials are c o m m o n l y employed; these partition the total energy into long-range Coulomb and short-range pair and 3-body potentials. Detailed discussions are given in refs. [6 ] and [8 ]. Three classes of simulation technique have been employed in the study of solid state ionics: first, static lattice simulations in which energy-minimisation methods are used to generate the lowest energy configuration o f a perfect periodic crystal structure of a region o f crystal a r o u n d a defect or defect cluster. In the latter case, methods based on the M o t t - L i t t l e t o n a p p r o x i m a t i o n [9] have proved to be particularly effective and have for several systems yielded quantitative agreement with experimentally d e t e r m i n e d defect parameters [ 1,3,8 ]. Secondly, molecular dynamics ( M D ) methods have been successfully employed in the study o f transport in solid state ionics. This widely applied c o m p u t a t i o n a l technique entails numerical solution o f classical equations o f m o t i o n o f an ensemble o f p a n i c l e s (to which periodic b o u n d a r y conditions are applied in the case o f crystalline materials); as such, kinetic energy is included explicitly in the simulation. Unlike the case o f the static methods, dynamical details are available directly from M D simulations. There is, however, as we have argued elsewhere [10], a natural complem e n t a r i t y between static and dynamical methods with the latter being most a p p r o p r i a t e for superionic systems while the former are the natural technique
0167-2738/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
956
C.R.A. Catlow/Mechanisms of ion transport in SSIs
for systems with more slowly diffusing ions (in practice for systems where D < 10 -8 cm 2 s-1 ). Thirdly, Monte Carlo (MC) methods may be employed in systems with high levels of disorder, especially when a variety of j u m p mechanisms are involved. MC is essentially a technique of computational statistical mechanics, in which an ensemble of configurations is generated by a succession of random moves. The "acceptance criterion" (i.e. the procedure which decides which configurations are to be included in the ensemble) leads to an ensemble which weights the probability of inclusion of a configuration according to its Boltzmann factor. Detailed discussions are given in refs. [ 11,12 ]. The same procedure may be applied to studying atomic transport, with atoms being moved at random and the probability of accepting a jump being weighted in proportion to its frequency. Successful MC simulation studies of several alloy systems have been reported by Murch and coworkers [ 13,14] and the same techniques have been applied to the solid state oxygen ion conductor Y/CeO2 [ 15]. We should stress that the accuracy and reliability of all simulation methods depends critically on the quality of the interatomic potentials employed. Considerable experience has been built up during the last 15 years in deriving and refining interatomic potentials for ionic and semi-ionic systems; and there is now abundant evidence that for many classes o f inorganic solid it is possible to obtain quantitative agreement between experimental and calculated parameters relating to atomic transport processes in solid state ionic materials. In this context we should note an important current limitation of the MD method is that it has not, except in a very limited number of cases, been possible to include a treatment of ionic polarisation (normally described using the Shell model [ 16 ] ). Work is, however, in progress to remove this limitation. Finally, we should note that there is an important and expanding role for quantum mechanical studies of properties of solids. In particular there has recently been major progress in the development of embedded quantum mechanical cluster techniques for studying defects in solids [ 17,18 ] ; and an illustration of their application to proton transport processes will be given later in this article.
3. Mechanistic studies of crystalline solid state ionic materials To demonstrate the power of simulation studies in the determination of mechanisms we choose four systems which have been extensively investigated by both theoretical and experimental techniques in recent years. Each of them will highlight different types of mechanistic complexity that can be unravelled by simulation methods. 3.1. Oxygen ion transport in complex oxides
Here we take two topical oxide materials where recent applications of simulations have led to valuable mechanistic information. 3. I. 1. Pyrochlore structured oxides Anion deficient fluorite structured oxides e.g. Y / ZrO2 and rare earth doped CeO2 are amongst the most widely studied oxygen ion conducting materials. Simulation studies have made a major contribution to our knowledge of these systems, and in conjunction with experimental results they have led to a good basic understanding of the mechanism of oxygen transport. Oxygen migration is effected by anion vacancy jumps for which both calculations and experiment give an activation energy of ~0.5 eV. Vacancies are trapped by dopants with the trapping energy being strongly dependent on the dopant ionic radius, in particular the extent of the mismatch between the radius of the dopant ion and the host lattice ion [19,20]. At higher dopant concentrations ( > ~ 8 tool%) there is a dramatic reduction in the conductivity [21 ] due to more complex dopant-vacancy interactions - a process that was successfully simulated by Monte Carlo methods [ 15 ]. The pyrochlore structure shown in fig. 1 is an intriguing variant of the parent fluorite structure. It is of general composition A2BzOv (where A is normally a trivalent and B a tetravalent ion) with ordering of the A and B ions and of the oxygen "vacancies". Early work of van Dijk et al. [22] showed that the pyrochlore structured materials are promising oxygen ion conductors. More detailed studies have been reported by Moon et al. [ 23,24 ], who have undertaken a comprehensive study of systems with A=Gd or Y and B=Zr or Ti; mixed Z r / T i systems were included
957
C.R.A. Catlow/Mechanisms of ion transport in SSIs
Y2Ti207 and YzZr207 - are summarised in table 1. They show first that rapid vacancy migration between 4 8 ( f ) sites may occur; indeed the activation energy for the titanates is rather low compared with experiment. The mechanism is a direct linear oxygen j u m p between the two 48 ( f ) sites. The question of the origin of the 48 ( f ) vacancies is more intriguing. Excitation of the oxygen ions from the 4 8 ( f ) into the vacant 8 (b) sites requires a large amount of energy - ~ 5 eV. In a perfectly ordered pyrochlore the "constitutional vacancies" are therefore locked into their sites, and low oxygen ion conductivity must follow. However, preliminary calculations o f Wilde show that introduction of disorder on the cation sublattices dramatically reduces (by more than 50%) the energy to excite oxygen ions from the 48 (f) into the 8 (b) sites. Further work is needed to provide a satisfactory understanding of this system. But both experiment and theory have already revealed a fascinating dependence of the anion transport processes on the degree of cation disorder.
Structural"'"i~Vacancy
Q
A3+ in 16c3m000
•
B 4÷ in 16 d 3m 1/2 1/2 1/2
(~
0(1) in 48 f mm x 1/8 1/8 0(2) in 8 a ~,3rn 1/8 1/8
+ Vo in8b~,3m 3/8 3/8 3/8 Fig. 1. Projection of the contents of the pyrochlore supercell for 0 < z < ¼.The displacement of the oxygen ions (0.46/~.) from their ideal tetrahedral locations in the fluorite structure corresponds to that found for Y2Ti207 in the present work.
3.1.2. High temperature superconductors The celebrated YBCO(YBa2Ca3Oy_x) superconductor is, in addition to being a high Tc material, a good oxygen conductor; and since it is now clear that disorder on the anion sub-lattice has an important influence on the superconducting properties of the material, there have been a number of studies of oxygen diffusion in the material, for example, the work o f X i e et al. [26] and Rothman et al. [27], the results of which are summarised in table 2. The well known structure of the materials is shown in fig. 2. It comprises two types of copper: Cu( 1 ) (copper chains) and Cu (2) (copper planes), and five types of oxygen site of which the O ( 5 ) is not
in this study. Their work reveals low oxygen migration energies ( ~ 0.8 eV) with an interesting correlation with the degree o f A / B cation order: increase in disorder leads to a reduction in the activation energy. The basic mechanistic problems posed by these studies concern first the nature of the mobile species, and secondly, their migration mechanisms. In particular, what is the role of the "built in" vacancies on oxygen ion transport? The problem has been investigated using simulation techniques by Wilde [25], whose results for four systems - Gd2Ti2Ov, GdzZr207, Table 1 Calculated anion defect energies (eV) in pyrochlore structured oxides. 0 2 - defect
Gd2Ti207
Gd2Zr207
YETi207
Y2Zr207
8a vacancy 48f vacancy 8b interstitial Frenkel pair saddle point activation energy (48(f)-~48(f))
20.47 16.96 - 11.64 5.32 17.13 0.18
20.10 17.96 - 13.75 4.21 18.70 0.73
20.38 17.02 - 12.42 4.60 17.19 0.18
19.88 17.91 - 14.64 3.27 18.76 0.86
CR.A. Catlow/Mechanisms of ion transport in SSIs
958
Table 2 Arrhenius parameters for oxygen diffusion in YBCO: D = Doexp ( - Ea/kT). Do (crn2 so-~ )
Ea (eV)
Ref.
3.5×10 -~ 1.4× 10-4 3.7× 10-4
1.03 0.97 0.99
[261 [27] this work, [28]
Y
y
I.... I
e (3-
)
c
to run at higher temperatures (1400-1800 K) than those used in the experimental diffusion studies. However, from the analysis of the temperature dependence of the simulated diffusion coefficients (which showed Arrhenius behaviour) we are able to obtain pre-exponential factors and activation energies that are in good agreement with experiment as is evident from the results given in table 2. Our main interest is, however, in the mechanisms of ionic transport revealed by the simulations. Analysis of the trajectories of migrating oxygen ions shows that ions move predominantly between O ( 1 ), O (4) and 0 ( 5 ) sites; and in the time scale of our simulation no jumps to 0 ( 2 ) and 0 ( 3 ) sites were observed - a result that is clearly consistent with the much higher measured diffusivity in the ah plane. We found that migration takes place between O( 1 ) sites, but not directly; rather jumps take place via 0 ( 4 ) and 0 ( 5 ) sites. Complex sequences may occur, e.g. O(1 ) - ~ O ( 4 ) - * O ( 5 ) - ~ O ( 1 ) , but no direct O ( 5 ) - ~ O ( 5 ) jumps are observed. The speculative mechanism of Rothman et al. [271 is not therefore supported by these calculations. The results accord with static lattice simulations of Islam [291 and Baetzold [30], and demonstrate the ability of molecular dynamics to reveal the mechanism directly. 3.2. Systems with complex correlated motions
¥
Fig. 2. The slructure of YBa:Cu307. normally occupied. The material is nonstoichiometric, with x ranging from 0 to I. The diffusion studies all relate to substoichiometric materials, for example the sample in the work of Rothman et al. had x ~ 0.1. The resulting vacancies are predominantly in the O( 1 ) site. The oxygen diffusivity is found to be far higher in the (ab) plane than in the c direction. There have been several speculative discussions of the mechanisms of oxygen ion transport in these materials. For example, Rothman et al. [27 ] suggested that oxygen ions jumped into the vacant O ( 5 ) sites between which they then migrated until returning to a vacant O ( 1 ) site. To investigate this problem further we have recently undertaken a detailed MD study of YBa2Cu3069 [28]. In order to obtain sufficiently high diffusion coefficients it was necessary
In contrast to the materials discussed above in which diffusion is effected by single discrete vacancy jumps, there are many superionics in which correlated motions of ions provide the key to the high ionic conductivity. We give here two examples in which simulations have played an important r6le in elucidating details of migration mechanisms. 3.2. I. Li~N This material has one of the highest known Li + ion conductivities. Its unique layer structure shown in fig. 3 also results in high anisotropy of the conductivity which is much greater parallel than perpendicular to the layers [31 ]. A detailed MD study reported by Wolf [ 32,33 ] revealed nicely the origins of the high Li conductivity. Analysis of his results showed that Li ions are readily excited from layer sites into interlayer interstitial positions. Following the creation of the vacancies within the layers there
959
C.R.A. Catlow/Mechanisms of ion transport in SSIs 3.2.2. R b B i F 4
perspective
i i I i I I I
o--:--o
I I
rl
~
....
I
,' ?
?/, ',/ #
. . . t.
D
~/ ;,'
/
o-
•
. . . .
/
/. i.A-iT../
001 plane I
ii
ii
I
o"--di'--d: :--o' O
N3
•
Li ~' ~on
Jan
Fig. 3. The structure of Li3N.
O
O
6
5 4
O
O 3
•
O
•
O
N ~
•
U + ion
ion
O
Li + v a c a n c y
I
Fig. 4. Schematic illustration of the mechanism of correlated Li migration in Li3N. is a cascade of migration events involving several ions moving simultaneously. A typical example is shown in fig. 4 where we see that the concerted motion of several ions effects transport of the vacancy from position 7 to position 1 in one step. This remains one of the best examples of correlated motion in superionic conductors; and it was revealed naturally and straightforwardly by application of the MD technique. Wolf's work also revealed fascinating mechanisms effecting transport perpendicular to the layers, which also effect interchange of "layer" and "bridging" lithium ions. Indeed the MD simulations have led to a very detailed understanding of the rich mechanistic variety in this system.
The high conductivity of this cation disordered fluorite structured material was first shown by Reau and coworkers [34]. The material has the fluorite structure with a disordered distribution of Rb and Bi over the cation sites. An extensive study of the local structure around both Rb and Bi, carried out using the EXAFS technique, has shown that disorder is generated preferentially around the Rb ions [ 35,37 ]. Indeed we have postulated that at anion sites around which there is a local excess of Rb, it will be energetically preferable to create vacancies; the neighbouring Rb ions then relax away from the vacancy as shown in fig. 5. The process would lead to a net shortening of the R b - F bond; such a reduction in the R b - F bond length on raising the temperature is demonstrated by our EXAFS studies. Of greater interest to our present discussion is the question of the migration mechanism for the interstitials that are created by the excitation of F ions from their regular lattice sites. It has long been known that interstitials in the fluorite structure migrate by the "interstitialcy" mechanism in which the migrating F - ion displaces neighbouring lattice ions into interstitial sites. A molecular dynamics study of RbBiF4 undertaken by Cox [35,36] has shown that such mechanisms occur in RbBiF4; fig. 6 illustrates the trajectory of one F - ion which is seen moving between lattice and interstitial sites, the characteristic behaviour in a system in which interstitialcy motion is occurring. However, the MD study revealed far more complex mechanisms. Several F ions are found to move in a correlated manner, the motion of each ion being of the interstitialcy type. As
F - a t corners of cube []
F-vacancy
Fig. 5. Proposed model for Rb relaxation awayfrom vacancyin RbBiF4.
960
C.R.A. Catlow/Mechanisms of ion transport in SSIs
/ Fig. 6. Trajectoryof migrating F interstitial in RbBiF4. with Li3N, ease of creation of the defects is vital for the high conductivity; but correlated migration mechanisms are again clearly of central importance.
ions resulting in a material with high conductivity for both cations (and anions); second, doped electronically conducting polymers (e.g. Li doped polyacetylene) in which the dopant by acting as a donor or acceptor state greatly enhances the conductivity of the polymer. In both cases it is most important to obtain a detailed understanding of the mechanisms of ion transport in the polymer matrix. Simulation provides a unique opportunity for deriving this information. Work is currently in progress on both types of system; but in the discussion which follows we concentrate on the doped polyacetylenes. Simulations of pristine and crystalline Li doped trans-polyacetylene at a temperature of 300 K have recently been undertaken by Ses6 [38]. Following earlier static lattice studies [39] a standard molecular mechanics force field was used to model the polymer chain; no bond length or bond angle constraints were employed. The crystal structure of the material (see fig. 7) was well reproduced. Analysis of the dynamics of the undoped system revealed vibrational frequencies in reasonable accord with experiment. Lithium ions were then introduced in low concentrations (Li/C 0.01 ) and dynamical simulations performed for up to 100 ps. The results dem-
3.3. Mechanisms in crystalline superionics: Summary
a) G~
From the examples given above it should be clear that simulations have a proven success in the elucidation of transport mechanisms in a wide variety of systems. MD is able to yield mechanisms directly, while static methods enable us to study the energetics of the migration process. In the remainder of this article we will concentrate on recent applications of the methods to more complex systems.
C~ G, @.
= 7.32A 4.24A c -
llliii
,,i2 ,G ,,G ,,G
4.96A
b)
4. Simulation of ion transport in polymers
Two types of doped polymer system have attracted considerable attention in recent years: first the salt doped polymer systems (e.g. NaI:PEO) in which metal salts dissolve in polymers containing polar atoms, which are thought to "solvate" the cat-
b
½
5
b
Fig. 7. Crystal structure of trans polyacetylene,viewed (i) perpendicular (ii) parallel to the c axis.
961
C.R.A. Catlow/Mechanismsof ion transport in SSIs
onstrate that Li diffuses very rapidly within the crystalline polymer matrix as shown by the diffusion coefficients collected in table 3. Note that the diffusivity is higher along the z axis, i.e. down the channels, between the polymers, than it is in the x y direction, i.e. perpendicular to the polymer chains. Analysis of the results yielded the following valuable mechanistic information: (i) The Li ÷ ions are not localised at particular sites but move freely along the interchain channels. (ii) The motions of the different Li + ions are strongly correlated. The Li+...Li + Coulomb repulsions are the strongest forces in the system and act so as to maximise the separation between the Li + ions. (iii) Li ÷ ions can move between different channels but this is a slower process than diffusion down the channels (hence Dz>Dxy as evident from table 3). Simulations on systems with high Li concentration showed instabilities. However, it has proved possible to simulate K doped systems at high concentrations. Very different results are obtained with low dopant diffusivity and "crystallisation" o f the K into well defined sites being observed in the simulation. More detailed accounts of these studies (and of related work on NaI doped PEO) will be reported shortly. The work summarised above demonstrates the viability and value of M D studies o f cation migration in doped polymers.
al. [42] has shown that it is possible by simulated quenching from the melt to generate structures that are in excellent agreement with experimental RDFs, as shown by fig. 8, which compares R D F s calculated by Vessal et al. [42] with the recent results from neutron data of Wright [43]. These techniques are beginning to be applied to systems with network modifying cations e.g. alkali silicate glasses, which have been studied recently by both Cormack et al. [44] and by Vessal [45]. Their
28
'
I
I
i
J
26 24. 22 II II II II II II
20 18
~
/ ~ f ~ / / ~
!
\
16
I
I
12
I
10 5. S i m u l a t i o n
;
I
. . . . . . . . .
\
/
/ / \
/
d
8
of glasses
Since the pioneering work o f Woodcock et al. [ 40 ] on vitreous silica there have been several successful M D simulations o f glassy materials. Recent work of both Feuston and Garafolini [41 ] and of Vessal et Table 3 Calculated diffusion coefficients at 300 K in lightly Li doped polyacetylene. Axis
D (cm2 S-1 )
(x, y) z overall
2× 10-6 4X10 -5 4.2 X 10- 5
6 ,-,4
0 0
I
I
1
2
3
1
I
I
I
.4 rCA)
5
a
7
8
Fig. 8. Measured and simulated (dotted line) RDFs for SiO2; three upper curves indicate respectively Si...Si, Si...O, and O...O calculated partial RDFs.
962
C.R.A. Catlow/Mechanisms of ion transport in SSIs
results support the modified random network models advanced by Greaves [46], as they show that the presence of the cations which have reasonably well defined coordination extensively modifies the silicate network. Evidence for aggregation of the cations is also found. In mixed cation systems (e.g. R b / N a glasses) intimate mixing of the two types of cation occurs - a result which is of clear relevance to the long standing puzzle of the mixed alkali effect [47 ]. There is a major future role for M D methods for elucidating the mechanistic details of cation transport in these materials.
6. Proton transport in oxides
One of the most interesting types of proton conductor is based on the perovskite or closely related structures with low valence dopants introduced on the B site; examples are Yb/SrCeO3 and Fe/KTaO3. Doping on the B site would be expected to lead to oxygen vacancy formation. However, the following reaction with water can cause the introduction of OH groups: Oo + V~ + H2 O--, 2 O H o . The protons so introduced give rise to the conductivity. What, however, is their migration mechanism? To investigate this problem we [48] have undertaken quantum mechanical calculations on the transfer of protons between two O 2- ions, i.e. we have calculated the activation energy for the process OH-+O2-~O2+OH-, assuming that the maxim u m corresponds to the proton equidistant between the oxygen ions. The barrier was calculated for two O...O spacings: the larger of 3.02 A corresponds to that in SrCeO3 - a perovskite with a relatively large O...O distance; the smaller of 2.75 to that in LaA103 - a material with a comparatively short O...O spacing. The calculations were performed on oxygen ions embedded in an array of point charges to simulate the remainder of the lattice. Calculations were performed both at the Hartree-Fock level and including correlation effets via the MP2 technique. The results are summarised in table 4. They demonstrate an appreciable dependence of the calculated energies on bond lengths. The values calculated for the barrier height are, of
Table 4 Calculated barrier heights for proton migration. O...O distance (A)
Barrier height (eV) A a~
Bbl
2.75
1.40
0.74
3.02
1.27
1.38
a) Hartree-Fock; ~ Hartree-Fock correlation. course, an upper limit on the activation energy which will be appreciably reduced by the ability of the proton to tunnel through the barrier. Furthermore, inclusion of a more detailed treatment of lattice relaxation may modify the barrier. The values are, however, sufficiently low to suggest that the process we have proposed will provide the dominant proton transfer mechanism. Further studies of these systems using both simulation and quantum mechanical techniques are in progress.
7. Summary and conclusions
We hope that the work summarised above has demonstrated that simulations can indeed provide valuable guidance as to transport mechanisms in solid state ionics. The use of the techniques in studying crystalline materials is now routine. The future will see much more extensive applications of the methods to glassy and polymeric materials and an increasing use of electronic structure techniques.
Acknowledgement
I am grateful to Drs. G. Ses6, B. Vessal and J.D. Gale for permission to refer to their unpublished work.
References
[ 1]C.R.A. Catlow and W.C. Mackrodt, eds., in: Computer Simulation of Solids: Lecture Notes in Physics vol. 166 (Springer, Berlin, 1982). [2] M.J. Gillan, Solid State Ionics 9/10 (1983) 735. [3] W.C. Mackrodt, Solid State lonics 12 (1984) 175. [4] P. Vashishta, Solid State Ionics 18/19 (1986) 3.
C.R.A. Catlow/Mechanisms of ion transport in SSls
[5] M.J. Gillan, Physica B 13 (1985) 157. [6] C.R.A. Catlow, Ann. Rev. Mater. Sci. 16 (1986) 517. [7] P.W.M. Jacobs, J. Chem. Soc. Faraday Trans. 86 (1989) 415. [8] J.H. Harding, Rep. Prog. Phys. 53 (1990) 1403. [ 9 ] N.F. Mort and M.J. Littleton, Trans. Faraday Soc. 34 ( 1938 ) 485. [10]C.R.A. Catlow, in: Proc. European Mat. Res. Soc. (Strasbourg, 1991 ), to be published. [11] M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids (Oxford University Press, Oxford, 1987). [12]C.R.A. Catlow, S.C. Parker and M.P. Alien, eds., in: Computer Simulation of Fluids, Polymers and Solids, NATO ASI Series (Kluwer, Dordrecht, 1989). [ 13] G.E. Murch, in: Diffusion in Crystalline Solids, eds. G.E. Murch and A.S. Nowick (Academic Press, New York, 1984). [ 14] G.E. Murch, in: Non Traditional Methods in Diffusion, eds. G.E. Murch, H.K. Birnbaum and J.R. Cost (Warrendale, Mich., AIME, 1984). [15] G.E. Murch, A.D. Murray and C.R.A. Catlow, Solid State Ionics 18/19 (1986) 196. [ 16] B.G. Dick and A.W. Overhauser, Phys. Rev. 112 (1958) 90. [ 17] A.L. Shluger, N. Itoh, R.W. Grimes and C.R.A. Catlow, to be published. [18] R.W. Grimes, C.R.A. Catlow and A.L. Shluger, eds., in: Quantum Mechanical Calculations in Solid State Studies (World Scientific, Singapore, 1991 ). [19] G.R. Gerhardt-Anderson and A.S. Nowick, Solid State Ionics 5 (1981) 547. [20] V. Butler, C.R.A. Catlow, B.E.F. Fender and J.H. Harding, Solid State Ionics 8 (1983) 109. [21 ] D.Y. Wang, D.S. Park, J. Griffiths and A.S. Nowick, Solid State Ionics 2 (1981) 95. [22] M.P. van Dijk, A.J. Burgraaf, A.N. Cormack and C.R.A. Catlow, Solid State lonics 17 (1985) 159. [23] P.K. Moon and H.L. Tuller, Solid State Ionics 28-30 (1988) 470.
963
[24] P.K. Moon and H.L. Tuller, Solid State lonics (Mater. Res. Soc. Symp. Proc., 1989), eds. G. Nazri, R.A. Huggins and B.F. Shriver, Vol. 135 (1989) p. 149. [25 ] P.J. Wilde and C.R.A. Catlow, to be published. [ 26 ] X.M. Xie, T.G. Chen and Z.L. Wu, Phys. Rev. B 40 ( 1989 ) 4549. [27] S.J. Rothman, J.L. Routbart and J.E. Baker, Phys. Rev. B 40 (1989) 8852. [28] X. Zhang and C.R.A. Catlow, Phys. Rev. B, in press. [29] M.S. Islam, Supercord. Sci. Technol. 3 (1990) 531. [30] R. Baetzold, Phys. Rev. B 38 (1988) 11304. [31 ] A. Rabenau, Solid State Ionics 6 (1982) 277. [32] M.L. Wolf, J.R. Walker and C.R.A. Catlow, J. Phys. C 17 (1984) 6623. [33] M.L. Wolf and C.R.A. Catlow, J. Phys. C 17 (i984) 6635. [34] S.M. Matar and J.M. R6au, C.R. Sci. Acad. Sci. Paris 294 (1982) 649. [35] P.A. Cox, PhD Thesis (University of Keele, UK, 1989). [36] P.A. Cox and C.R.A. Catlow, to be published. [37]P.A. Cox, A.V. Chadwick and C.R.A. Catlow, to be published. [ 38 ] G. Ses6 and C.R.A. Catlow, to be published. [ 39] J. Corish, V.C.A. Hanratty, D.A. Morton-Blake, F. B6ni~re andA. Morin, J. Mol. Struct. 207 (1990) 41. [40] L.V. Woodcock, C.A. Angell and P. Cheeseman, J. Chem. Phys. 65 (1976) 1565. [41 ] B.P. Feuston and S.H. Garafolini, J. Chem. Phys. 89 (1988) 5818. [42 ] B. Vessal, M. Amini, D. Fincham and C.R.A. Catlow, Philos. Mag. 60 (1989) 753. [43 ] A.C. Wright, private communication. [44] C. Huang and A.N. Cormack, J. Chem. Phys. 93 (1990) 8180. [45] B. Vessal, M. Amini and C.R.A. Catlow, J. Phys. Chem., in press. [46] G.N. Greaves, J. Non-Cryst. Solids 71 (1985) 203. [47] J.M. Stevels, Handbuch der Physik, Vol. 20, ed. S. Fl/igge (Springer, Berlin, 1957) p. 246. [48 ] J.G. Gale and C.R.A. Catlow, to be published.