Structural and transport properties of β″-Al2O3

Solid State Ionics 13 (1984) 33-38 North-Holland, Amsterdam

STRUCTURAL AND TRANSPORT PROPERTIES OF p"-AI203 M.L. WOLF, J.R. WALKER * and C.R.A. CATLOW

Department of Chemistry, University College London, 20 Gordon Street, London WCIH OAJ, UK Received 8 April 1983

We report results of a molecular dynamics simulation of sodium fl"-Al203. We confirm that the superionic properties of this material require deviation from the stoichiometric composition. Our results yield values for the transport coefficients of the material that are in good agreement with experiment. Thermal parameters, measured by diffraction studies are well reproduced by the calculations. The simulations suggest a change in the diffusion mechanism from hopping, at low temperatures, to liquid-like at higher temperatures. The change in migration mechanism is aeeompanied by corresponding structural changes.

1. Introduction

0

0

o

ff'-A1203 is a layer structured fast ion conductor closely related to the widely studied ~-A1203 class o f compounds. Like the latter it consists o f spinel-structured alumina blocks between which the mobile Na + ions are located in "conduction planes" [ 1 ]. The structure differs in that the spinel blocks are smaller, and contain Mg2+ in addition to A13+. (The formula o f "stoichiometric" ~" .A1203 is Na 20.MgO.5AI 203 compared with Na20.11A1203 for stoichiometric /3-A1203. ) The higher Na + content of/~"-A1203 is reflected in the structure of the conduction plane (see fig. 1) which comprises a hexagonal net o f Na + ions in the stoichiometric compound, in contrast to/3-A1203 in which points on the hexagonal net are alternately occupied and unoccupied. Nonstoichiometry is, however, an important feature o f the structural chemistry o f both classes o f compound. In/3-A1203 excess Na + is accommodated, together with oxygen interstitials, in the conduction plane. In contrast, fft-Al203 shows Na + deficiency which is caused by the replacement of Mg 2+ by A13+ in the spinel structured blocks. In both systems, the conductivity appears to be greatly enhanced by the deviation from stoichiometry [2]. The conductivities of/~- and/~"-aluminas are of * Present address: Atomic Energy of Canada, Pinawa, Manitoba, Canada.

0

O

o

o

O

o

e.

0 • o

"'""O""'"Q'"'"O "''''''• o

•

o

i

.e,

"" "'"o'""

o""""

o

o

0

0

0

0

o

o

O

""'o

o

o @

0

0 • 0

0

• 0

~"

IS)/

0 •

F

b/

0

• +&

( ~

+0

o Fig. 1. Plan of conduction plane in stoichiometric ff'-Al203 : O 0(5) ion, o Na+ ion above 0(5) plane, • Na+ ion below 0(5) plane, + Na(2) site, zxmid-oxygen site. similar magnitudes (for nonstoichiometric ~A1203 at 550 K, o is ~0.3 f l - 1 c m - 1 [3]); for nonstoichiometric ff'-A1203, o is ~0.8 ~2-1 cm -1 [4]). In both systems there are, however, important fundamental problems; the first concerning structural properties, and in particular, the effect o f temperature and composition on the distribution o f Na + ions in the conduction plane;the second concerning migration mechanisms, whose nature is uncertain, as is the effect on the mechanism of temperature and the deviation from stoichiometry. Insight into structural and transport properties of superionics may be obtained by application of the

34

M.L. Wolfet aL/Structural and transport properties of 1~""Al203

static simulation techniques [6,7] which yield defect formation and migration energies. Such studies have been undertaken for/~- and ff'-Al203 [8] ; they have yielded activation energies which appear to accord well with experimental data. However, detailed dynamical properties may not be predicted by these methods, and the methods are unsuitable for describing systems in which there is "liquid-like" rather than hopping conductivity. For this reason we have undertaken a molecular dynamical simulation study of these systems. We report here results of our study of ff'-A1203 . Our results predict interesting variations with temperature in the structural and dynamical properties of the conduction plane ions. They demonstrate, moreover, the viability of dynamical simulation techniques for these complex layer structured superionics.

2. Techniques and potentials The molecular dynamics technique solves numerically the classical equations of motion for an ensemble of particles, to which periodic boundary conditions are applied, and for which interatomic forces, generally of a pairwise nature, are specified. Extensive discussions of the method are available [9,10]. Special features regarding the application of the technique to solids are discussed by Walker [ 11 ] ; and applications to superionics are reported by Dixon and Gillan [12,13], Walker and Catlow [14], and Vashishta and Rahman [15]. The simulations discussed in this paper were all performed using the FUNGUS program [11 ] - a molecular dynamics code written specifically for solid state simulations which also optimises use of the parallel processor facilities available on CRAY computer systems; the calculations were performed on the CRAY-1S at the Daresbury Laboratory, UK. The following special features of our simulation of ~"-A1203 should be noted.

(i) "Box size" The hexagonal unit cell of stoichiometric ff'-Al203 contains 90 ions. In this study this cell was quadrupled normal to the c axis to yield a " b o x " (i.e. the unit to which the periodic boundary conditions are applied) of 360 ions. These include 24 Na + ions distributed, as sets of 8, over 3 independent planes. Calculations with

a larger box size of 720 ions did not yield significantly different results. We note that a full description of the spinel blocks was included in our calculations, in contrast to an earlier study by de Leeuw and Perram [5].

(ii) Description of the nonstoichiometric compound Here we removed one Na ÷ ion from each conduction plane (and replaced an Mg2+ by an AI3+ in a cation site.in the spinel block). The resulting composition (Nal.75Mgo.75 Al10.25017 ) is as close to that of the commonly observed material (Nal.66Mgo.66AI10.33017) as can be achieved with a "box" of the dimensions described above. (iii) The time step used in the simulations (i.e. the time lapse between successive configurations in the simulation) was taken as 4 X 10 -15 s, which is at least an order of magnitude smaller than the period of the highest atomic vibrational frequency in the material. (iv) The "start~p" of the simulations involved placing the ions at their lattice sites and supplying velocities (whose directions were random) according to the temperature required for the simulation. The system was then allowed to equilibrate, i.e. attain an equilibrium distribution of velocities appropriate to the temperature of the simulation; three temperatures were studied between 300 K and 700 K. About a thousand time steps were necessary to achieve equilibration. The equilibrated system was then allowed to evolve for ~1000 time steps, and the results stored for subsequent analysis. (v) lnteratomic potentials. In common with most molecular dynamics simulations, the effects of electronic polarisabilities were omitted from this study, owing to the excessive computational requirements of any models which incorporate polarisation. Ionic model potentials were used with the short range potentials specified for Na + ...Na+, Na+...O 2- , A13+...O 2- , Mg2+...O 2- and 0 2 - ...02- interactions; the parameters used here were those reported in our earlier static simulation studies of/3- and/~".AI203 [8]. We note that these potentials correctly reproduced the observed crystal structure of room temperature stoichiometric ~-AI203 . It is one of the aims of this study to test their success in the prediction of high temperature structural and dynamical properties.

M.L. Wolf et al./Structural and transport properties of ff'-Al20a

3. Results

3.1. Structural properties Fig. 2 a - d shows Na...Na, radial distribution functions, g(r), for both stoichiometric and nonstoichiometric systems at various temperatures. For the stoichiometric compounds (fig. 2a, b) the results indicate the expected structure for the conduction plane with Na + ions well localised into the regular cation sites in the conduction plane. The positions of the peaks correspond to the expected Na ÷...Na ÷ spacings in the conduction plane. The increase with temperature of the peak width follows from the increase in the amplitude of the thermal vibrations; the extent to which our predictions of these thermal parameters are compatible with crystallographic data is discussed below. A marked contrast is, however, noted between

9"0

35

the g(r) for stoichiometric and nonstoichiometric compounds. The latter differ, first, in that they have less structure at higher r - an observation indicative of much greater disorder in the nonstoichiometric system. In addition, however, the first peak is displaced to larger distances, and at the two lower temperatures shows a shoulder at higher r, which disappears at the highest temperature (fig. 2c, d). We interpret these differences as follows. At low temperatures, the nonstoichiometric system contains well defined vacancies; inward relaxation of neighbouring Na + ions occurs around the vacancies as illustrated in fig. 3. Indeed we identify the relaxed Na + ions with the "Na(2)" sites detected in crystallographic studies [ 16,17] of nonstoichiometric/~"-A1203 . The only feasible assignment of the shoulder discussed above would seem to be an Na(2)...Na(2) correlation; the main peak is then taken as being an average over the Na(1)...Na(1) and

b)

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I

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1.0 0

0"5

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0

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0.5

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Fig. 2. (a) N a - N a radial distribution function: ( - - ) stoichiometric 300 K, (....) nonstoichiometric 300 K; (b) N a - N a radial distribution function: ( - - ) stoichiometric 600 K, (....) nonstoichiomctric 550 K; (c) N a - N a radial distribution function: nonstoichiornetric 300 K. (d) N a - N a radial distribution function: nonstoichiometric 700 K.

M.L. Wolfet al./Structuraland transport properties of ff'-Al203

36 o

o

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o

0

0

0

o

~ o

o

0

0

0

[]

•

Table 1 Calculated and experimental conductivities for nonstoichiometfic ~¢'-A12O3 T (K)

Calc. D (cm2 s-1)

Calc. o (S2-1 cm-1 )

Obs. o a) (S2-1 cm- t )

0.94 X 10-s 1A9 x 10-s 3.85 X 10-s

0.89 0.72 1.45

0.014-0.160 0.80 1.24

0 •

0

O*

0

300 550 700

o

o

o

a) Refs. [4,20].

o

o

~o.

o

Fig. 3. Schematic plan of conduction plane around an isolated Na+ vacancy: O 0(5) ion, o Na+ ion above 0(5) plane, • Na+ ion below 0(5) plane,l'VINa+ vacancy, × displacement of Na(2) site from Na(1) site. Na(1)...Na(2) correlations (where Na(1) refers to unrelaxed lattice ions). Given this assignment we may calculate the relaxation of the Na (2) ions towards the vacancy as 0.81 A. This calculated relaxation is in excellent agreement with that deduced from the position observed for the Na(2) ion in the neutron diffraction studies [16]. A relaxation of this magnitude is unusual; but in view o f the flatness of the potential surface for conduction plane ions, it is quite plausible. The most interesting feature of the results is, however, the disappearance at high temperature of the "shoulder", assigned to Na(2)...Na(2) correlations. This suggests that well-defined vacancies are no longer present at this temperature, although the persistence of sharp well-defined peaks in the g(r) Na +...Na + and the Na +...O correlations (which were calculated but are not discussed further here) suggests that at least structurally, the conduction plane is more solid than liquid-like. A simple physical interpretation of the results is that although we may describe the structure in terms of vacancies at Na + sites, the lifetime of these vacancies is too short to permit relaxation o f the surrounding lattice ions. We note that such a description has important implications for the transport mechanism discussed in the next section.

cients and conductivities for the nonstoichiometric Na fl"-A1203; the diffusion coefficients for the stoichiometric compound were found to be negligible. The values of D were obtained via calculated rms displacements as discussed in refs. [10] and [12]. The conductivities were obtained from the calculated diffusion coefficients via the Nernst-Einstein relationship assuming a correlation coefficient,f, of 0.4 which is appropriate for a vacancy mechanism in this structure

[181. It is clear from these results that the simulations are correctly reproducing the superionic properties of this material. Indeed the calculated conductivities at 550 K and 700 K are in excellent agreement with experiment. The agreement is much poorer at the lower temperature of 300 K although we should note that the accuracy of calculated transport coefficients, obtained via mean square displacements, is reduced for these lower values of D. However, the discrepancy has, we consider, a physical cause in the formation of a vacancy superlattice which has been observed in/3"-A1203 at 300 K [ 17] and which will reduce the rate of vacancy transport. Our simulated system is, however, too small to permit formation of the observed superlattice. The simulations would therefore be expected to yield a higher value of D at this temperature.

3.2.2. Transport mechanism As discussed by Rahman [19] and by Dixon and Gillan [ 12], useful information on migration mechanisms can be obtained from calculation of the "moment"

3.2. Dynamics

P(t) = 3
3.2.1. Transport coefficients

A strong deviation of this quantity from unity indicates a non-gaussian Van Hove self correlation func-

In table 1 we report the calculated diffusion coeffi-

M.L. Wolf et al./Structural and transport properties of ~"-Al203

3.0

Table 2 Thermal parameters for nonstoichiometrie ff'-A12Oa .

300 K

O (5) ions

=.oi

Obs. a)

Calc.

Obs. b)

(0~3) 1/2

0.056

0.068

0.170

0.166

(U~I) 1/2

0.125

0.218

0.218

0.309

7

0.45

0.31

0.78

0.54

1.0

I.O

:~I0

3"0

Na+ ions

Calc.

Pit)

0

37

a) Ref. [21].

b) Ref. [17].

t los Fig 4. Moment ratio for Na+ ions in nonstoichiometrie systems.

tion, Gs(r, t), which is expected for hopping conductivity. Fig. 4 illustrates the results of calculations of P(t) for the three temperatures. The results at 300 K are indicative of hopping conductivity in line with the calculated Na+...Na + g(r) which, we recall, suggested well defined vacancies stabllised by relaxation of surrounding lattice ions. At 700 K, however, the mechanism has clearly changed: the hopping description appears no longer to be appropriate, and a liquid like or "continuous" diffusion mechanism operates; the absence at this temperature of lattice relaxation around vacant sites, would be expected for a continuous rather than a hopping mechanism. The prediction of a change in the mechanism of Na + migration is supported by the experimental conductivity data, the Arrhenius plot of which changes slope between ~-420 K and 570 K. A change in the mechanism of Na + migration would be expected to yield a change in the activation energy for the conductivity.

3.2.3. Thermal parameters Molecular dynamics simulations contain the necessary information to calculate the thermal parameters due to lattice vibrations, which are obtained experimentally in ref'mements of diffraction data. The methods used to calculate the rms displacement, (~2)1/2, of ions with respect to lattice sites are discussed elsewhere [22]. The anisotropy of/~"-A120 3 structure introduces an added complication. Two distinct values can be calculated - (021)1/2 which refers to vibrational motion - 2 1/2 in which parallel to the conduction plane, and (U~3)

the motion is perpendicular to the plane. Table 2 reports the results for these parameters for Na+ and conduction plane oxygen ions ( 0 ( 5 ) ions) in the nonstoichiometric compound at room temperature; the anisotropy ratio, 3' = (023)1/2/(~111)1/2 is also reported. The results demonstrate a pronounced anisotropy in the thermal motion. The agreement with experiment is excellent for the Z direction; in the X direction calculated values are lower than experimental values. This is expected due to the lower degree of nonstoichiometry. These results encourage confidence in the quantitative reliability of our simulations, and shows the value of molecular dynamics simulations in determination of thermal parameters, which are commonly a considerable source of uncertainty in structural refinements. This application of simulation studies is discussed in greater detail elsewhere [22].

4. Conclusions

Two main points have been established by this paper. First, we have shown that simulation techniques and the interatomic potentials used in our work are capable of yielding accurate results for structural and dynamical properties of fl-A1203-like superionics; the application of the techniques to other systems in this class, especially sodium/3-A1203, is clearly encouraged. Secondly for ff'-Al203 we have predicted an interesting transformation with temperature in structural and dynamical properties, with a low temperature hopping transport mechanism changing to a higher temperature liquid-like transport mechanism. It would be of interest to test this prediction experimentally.

38

M.L. Wolf et al./Structural and transport properties of ff'.Al 2 03

Acknowledgements We wish to thank Dr. M.J. Gillan for helpful discussions and SERC for research grants supporting this work.

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[ 10] M.J. Sangster and M. Dixon, Adv. Phys. 25 (1976) 247. [11] J.R. Walker, in: Computer simulation of solids, eds. C.R.A. Catlow and W.C. Mackrodt, Vol. 166, Lecture Notes in Physics (Springer, Berlin, 1982). [12] M.J. Gillan and M. Dixon, J. Phys. C13 (1980) 1901. [13] M. Dixon and M.J. Gfllan, J. Phys. C13 (1980) 1919. [14] J.R. Walker and C.R.A. Catlow, J. Phys. C14 (1981) L979. [15] P. Vashishta and A. Rahman, in: Fast ion transport in solids, eds. P. Vashishta, J.N. Mundy and G.K. Shenoy (North-Holland, Amsterdam, 1979) p. 527. [16] W.L'Roth, M. Anne, D. Tranqui and A. Heidemann, in: Fast ion transport in solids, eds. P. Vashishta, J.N. Mundy and G.K. Shenoy (North-Holland, Amsterdam, 1979) p. 267. [17] G. Coliin, Ph. Colomban, J.P. Boilot and R. Comes, in: Fast ion transport in solids, eds. P. Vashishta, J.N. Mundy and G.K. Shenoy (North-Holland, Amsterdam, 1979) p. 309; J.P. Boilot, Phys. Rev. B22 (1980) 5912. [18] D. Wolf, in: Fast ion transport in solids, eds. P. Vashishta, J.N. Mundy and G.K, Shenoy (North-Holland, Amsterdam, 1979) p. 341. [19] A. Rahman, J. Chem. Phys. 65 (1976) 4845. [20] J.D. Jorgenaen, F.J. Rotella and W.L. Roth, in: Fast ion transport in solids, eds. J.B. Bates and G.C. Farrington (North-Holland, Amsterdam, 1981). [21 ] W.L. Roth, F. Reidinger and S. La Placa, in: Superionic conductors, eds. G.D. Mahan and W.L. Roth (Plenum Press, New York, 1976). [22] L. Moroney, M.L. Wolf and C.R.A. Catlow, to be published.