Computer simulation of fast ion transport in fluorites

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Computer simulation of fast ion transport in fluorites

M. Dixon, M. Gillan

To cite this version:

M. Dixon, M. Gillan. Computer simulation of fast ion transport in fluorites. Journal de Physique Colloques, 1980, 41 (C6), pp.C6-24-C6-27. �10.1051/jphyscol:1980606�. �jpa-00220001�

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JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 7 , Tome 41, Juillet 1980, page C6-24

Computer simulation of fast ion transport in fluorites

iV Dixon (*! and IM. J. Gillan

Theoretical Physics Division, AERE Harwell, Oxfordshire OX11 ORA, UK

RksumC. - Nous reportons des nouvelles ttudes de simulation du fluorure de calcium i l'ttat supraconducteur ionique par la technique de la Dynamique Molkculaire. Les simulations sont effectuCes d'une part avec un modele d'ion rigide a differentes temperatures en utilisant des systemes de deux tailles differentes et d'autre part avec un moditle d'ion polarisable i une seule tempkrature. L'Ctude faite avec les ions polarisables, qui se sert du shell model, nous permet de trouver comment la presence de la polarisabilite modifie les proprietes du systitme. Dans chaque cas nous avons CvaluC le coefficient d'auto-diffusion des anions et nous avons analyse leur repartition spatiale ainsi que leur comportement dynamique. L'ordre de grandeur du coefficient de diffusion ainsi que sa dkpendance en fonction de la temperature sont en accord qualitatif avec 11exp6rience. Nos rksultats donnent pour le CaF, a 1'6tat supraconducteur ionique l'image qualitative suivante : a) malgrt: leur grande mobilitC, les anions passent la plupart du temps A vibrer autour de leurs sites de rCseau et ne restent pas localisCs de facon bien dCfinie sur les sites inters- ticiels ; b) le mode de diffusion des anions peut &tre interprtte en termes de sauts discrets, et dts lors il se distingue du processus de diffusion i l'ttat liquide ; c ) la diffusion peut s'interpreter en termes de mouvement de dtfauts ; d) le temps de vol des ions est d'un ordre de grandeur plus petit que leur temps de residence ; e) la presence de la polarisabiliti: ne modifie pas qualitativement les caracteristiques du systeme.

Abstract. - We report on new simulation studies of fluorite crystals in the superionic state using the Molecular Dynamics technique. The simulations were performed on a rigid ion model of CaF, at a series of temperatures using two sizes of simulated system and on a polarizable-ion model of CaF, at a single temperature. Our polarizable-ion study, which uses shell-model potentials, allows us to assess the ways in which the properties of the system are modified by the introduction of polarizability. In each case we have calculated the anion self-diffusion constant and the spatial distribution of the anions and we have made a detailed analysis of the dynamics of the diffusing ions.

We show that the magnitude and temperature dependence of the diffusion constant are in qualitative accord with experiment. Our results yield the following qualitative picture of the superionic state in CaF, : a) in spite of their high mobility, the anions spend most of their time vibrating about the regular sites and do not reside in a well- defined manner on the cube-centreinterstitial sites ; b) anion diffusion can be analysed in terms of discrete hops and does not resemble liquid-state diffusion; c ) the diffusive motion can be interpreted in terms of the motion of discrete defects ; d) the flight time of the ions is an order of magnitude less than the residence time in the superionic state ; e ) the introduction of polarizability does not qualitatively alter the characteristics of the system.

1 . Introduction. - This is a preliminary report o f new simulation calculations o n superionic CaF, using the Molecular Dynamics technique. CaF,, like other materials having t h e fluorite structure, possesses a liquid-like ionic conductivity o in t h e temperature range from a few hundred degrees below the melting point u p to the melting point itself [I, 21.

T h e high value o f o is caused b y disordering of the anion sublattice [ l , 31, which also gives rise t o a pronounced peak in the specific heat a t the temperature of the diffuse transition t o the superionic state [4]. The value o f Molecular Dynamics simulation

in investigating these phenomena was first demonstrat- e d by R a h m a n [5-71 in his work o n CaF,. M o r e recently, we [8] have performed a n extensive study o f SrCl,, i n which we showed that the simulated temperature dependent diffusion constant and ^{t h e}

(*) Theoretical Physics Department, University of Oxford, Oxford, UK.

specific heat anomaly were in qualitative agreement with experiment.

In t h e present investigation, we have extended o u r work o n CaF, t o cover a range of temperatures, using improved rigid-ion potentials. W e have also studied t h e effects o f varying the size o f the simulated system, a n d o f introducing electronic polarizability o f the ions, t h e latter being included through the use of shell-model potentials. W e will show that the simulations yield a good account of t h e experimental results a n d allow o n e t o obtain valuable information o n the distribution of t h e ions a n d the dynamics of the diffusion process. W e will also demonstrate that neither system size variation n o r the introduction of polarizability produces a large change in the results.

2. Interionic potentials. - I n all our calculations, t h e interionic potentials have the form :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980606

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COMPUTER SIMULATION OF FAST ION TRANSPORT IN FLUORITES C6-25

in the usual notation [9]. The short-range cation- cation potential is set equal to zero ( A ⁺⁺= C+ ⁺= 0) as is the cation-anion dispersion potential ( C + ^-= 0) [9].

For the anion-anion potential, the values A _ - ⁼1 8 0 8 e V , p - - = 0.293A and

C - - = 109.1 eV.W6

of Catlow and Norgett [9] are taken. For the rigid-ion simulations, the cation-anion parameters have the values A ^{+ -}= 674.3 eV and p + - = 0.336

A

obtained by fitting to the experimental values of the lattice constant a, = 5.444

A

and the anion Frenkel energy E , = 2.71 eV [lo], the latter being calculated using the HADES program of Norgett [Ill. We have verified that the normal cube-centre interstitial is stable. For the shell-model simulations, we take the values A + - = 446.6 eV, p + - = 0.360 and shell charges Y + = 5.24, Y - = - 2.38 and spring constants k+ = 390.9, k- = 101.2 eV.

k 2

given by Catlow and Norgett [9]. Table I gives values of dielectric and elastic constants, cohesive energy, anion Frenkel energy and anion vacancy and interstitial migration energies calculated using the rigid-ion and shell-model potentials compared with their experimental values. The results suggest the rigid-ion potentials are not much inferior to the shell- model potentials.

Table I. - Perfect-crystal and anion-defect quantities calculated froni rigid-ion and shell-niodel potentials conipared with the experiniental values given in refs. [9]

and [lo] for perfect-crystal and defect quantities respectively.

Shell-model Expt.

- -

6.42 6.47

2.01 2.05

16.9 17.12

4.80 4.68

3.23 3.62

- 28.06 - 26.76

2.63 2.71

0.20

-

^0.42

0.69 0.79

3. Simulation runs. - The Molecular Dynamics technique is fully described in the review by Sangster and Dixon [12]. We have performed rigid-ion runs at a series of temperatures and a lattice constant a, ⁼5.712

13

using systems of 96 and 324 ions.

Since shell-model Molecular Dynamics is very demanding on computer time, we have performed only a single shell-model run with a system of 96 ions at a lattice constant a , = 5.707

13

and temperature T = 1 639 K (the experimental melting temperature is 1 691 K [4]). For the rigid-ion simulations we used time steps of 0.007 5 ps (96 ions) and 0.008 0 ps

(324 ions). For the shell-model simulation, a conside- rably shorter step of 0.003 5 ps was found necessary to ensure energy conservation. Even with this time step, a slight energy drift occurred. In order to eli- minate this, we adopted the device of continually rescaling the velocities by an amount calculated from the average value of the energy over successive sections of 100 steps. In each run we allowed

-

1 000 (1 500) steps for equilibration in rigid-ion (shell-model) simulations. The subsequent 1 000 (1 800) steps were used for analysis.

4. Diffusion constant and enthalpy anomaly. - The self-diffusion constants can be calculated either from the time-dependent mean square displacement of the two ionic species or by directly counting hops of particles between sites [8] (see hopping analysis below) and using the Einstein relation. In all cases, the cation diffusion constant was zero, which shows that the system was in the solid state. The two methods yield values for the anion diffusion constant D- which agree to within about 20

%.

We give results obtained by the hopping method.

T I K

Fig. 1.'- Simulation results for anion diffusion constant (0 and 96- and 324-part~cle rigid-ion. V : 96-particle shell-model) compared with values derived from conductlvlty measurements [l]

(solid line).

Figure 1 shows the simulation results for D - as a function of temperature T, together with values obtained from experimental measurement [I] of the electrical conductivity ^ovia the Nernst-Einstein relation :

where k , and e have their usual. meanings ; the corre- lation factor is taken to be unity. No experimental error is quoted for the conductivity measurements, but they are known not to be of high accuracy [I].

Three points should be noted : a) the simulation results are in qualitative agreement with experiment

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C6-26 M. DIXON AND M. J. GILLAN

regarding both the magnitude of D - and the temperature region of the diffuse transition : b) the 96-par- ticle rigid-ion results are in qualitative agreement with those for 324 particles, though they overestimate

D -

at low temperatures and underestimate it at high temperatures ; c) the shell-model value of D- is close to the rigid-ion values.

The peak in the experimental specific heat C, is associated with a diffuse step in the enthalpy as a function of temperature, the size of the step being of order 10-20 kJ. mol -

'

in CaF, [4]. We would expect to see a diffuse step of similar magnitude in the internal energy U calculated at constant a, in our simulations.

Our results do indeed reveal such a step, whose size is 10

+

2 kJ.mol-'.

We conclude from the above results that the simulated systems, both for rigid-ion and shell-model potentials, give a qualitatively faithful representation of real CaF,.

5. Spatial distribution of ions. - We have made a detailed investigation of the spatial distribution of the ions in the unit cell, but we confine ourselves here to the distribution of ions with respect to the cube- centre site; this is the site occupied by anion interstitials at low defect concentrations. In figure 2 we show results for go(r), the spherically averaged mean density of anions at a distance r from the cube-centre site [5] (the mean number of anions between distances r and r

+

dr from the site is 4 xr2 g,(r) dr). The results are for the 96- and 324-particle rigid-ion systems and the 96-particle shell-model system at temperatures 1 651, 1 585 and

Fig. 2. - Anion density distribution function g,(r). Chain, solid.

dotted and broken curves : 96-particle rigid-ion T = 1651 K, 324-particle rigid-ion T = 1 585 K, 96-particle shell-model T = 1 639 K and 324-particle rigid-ion T = 1 277 K respectively.

Chain and dotted curves omitted where same as solid curve.

1 639 K respectively. For comparison, we show the distribution for the 324-particle rigid-ion system at T = 1 277 K, which is below the superionic regime.

The following points are noteworthy : a) neither the introduction of polarizability, nor variation of the size of the system has a marked effect on the results;

b) the main qualitative changes in g,(r) on going to the superionic state are a lowering of the peaks and an increase of anion density near the cube-centre site ; c) the extra density has a rather flat distribution and has at most a feeble peak at the origin (if each cube-centre site were occupied by 0.1 anion with a thermal distribution similar to that of the anions on regular sites, a peak of height

-

3.0 would be produced). This confirms our earlier conclusion 17, 8, 131 that, in the superionic state, the disordered anions do not reside stably on the cube-centre positions, but are dynamically distributed over a rather large volume of the unit cell.

6. Hopping analysis. - We have analysed the diffusive motion of the anions into discrete hops between regular lattice sites. using the method described previously [7. 81 : a sphere of radius a,/6 is centred on each regular anion site, and hopping events are identified as those in which an anion leaves one sphere and enters another; the flight time T , is the mean time between entering and leaving, and the residence time ^{T ,}is the mean time between hops of a given anion. For the 324-particles system at T = 1 5 8 5 K we find T , = 0.32ps and T, = 5.1 ps;

the residence time is therefore at least an order of magnitude greater than the flight time, which means that the anions spend the majority of their time on the regular sites - a conclusion which is also supported by an analysis of their spatial distribution. The correlations between hops of anions can be analysed in terms of the motion of vacancy and interstitial defects [6-81, though we find that the interstitials do not reside on identifiable sites. The results of this analysis will be published later.

7. Conclusions. - The results of our simulations of superionic CaF, lead to the following conclusions : a) the temperature-dependent diffusion constant and internal energy are in qualitative accord with experiment; b) the simulated results are not crucially affected either by change of system size or by the introduction of polarizability ; c ) the anions displaced from their regular sites in the superionic state do not reside in a well-defined manner on the cube-centre sites; d ) anion diffusion occurs by discrete hops between regular sites; e ) in the superionic state, the residence time of anions is at least an order of magnitude greater than the flight time.

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COMPUTER SIMULATION O F FAST ION TRANSPORT IN FLUORITES

DISCUSSION Question. ^-A. B. LIDIARD.

From the insight you obtain by these calculations could you comment on the different inferences made for PbF, by Dr. Nolting and by the neutron diffraction workers as reported by Dr. Hayes ? Do you expect PbF, to be different from the other fluorite compounds ?

Reply. - M . J . GILLAN.

In all our calculations, both on CaF, and on SrCl,, we find defect concentrations of only a few percent.

I cannot a t present see why PbF, should be so radically different. One should note that in interpreting the

diffraction results there is some ambiguity in the definition of a defect, and I would think that the difference between Dr. Nolting and Dr. Hayes must stem from this.

Question. - R. J . FRIAUF.

What typical values do you find for residence times and flight times in CaFz ?

Reply. - M . J . GILLAN.

The exact values are given in the paper. The essential result is that there is a difference of over an order of magnitude between the two times.

References

[I] DERRINGTON, C. E., LINDNER, A. and O'KEEFFE, M., J. Solid State Chem. 15 (1975) 171.

[2] HAYES, W., Contemp. Phys. 19 (1978) 469.

[3] DICKENS, M. H., HAYES, W., HUTCHINGS, M. T. and SMITH, C., J . Phys. C . 12 (1979) L-97.

[4] DWORKIN, A. S. and BREDIG, M. A,, J. Phys. Chem. 72 (1968) 1277.

[5] RAHMAN, A,, J. Chem. Phys. 65 (1976) 4845.

[6] JACUCCI, G. and RAHMAN, A., J. Chem. Phys. 69 (1978) 41 17.

[7] DIXON, M. and GILLAN, M. J., J. Phys. C . 11 (1978) L-165.

[8] GILLAN, M. J. and DIXON, M., AERE report TP. 794 (1979).

[9] CATLOW, C. R. A. and NORGETT, M. J., J. Phys. C . 6 (1973) 1325.

[lo] JACOBS, P. W. M. and ONG, S. H., J. Physique CoNoq. 37 (1976) C7-331.

[I I] NORGETT, M. J., UKAEA report AERE-R7650 (1974).

[I21 SANGSTER, M. J. and DIXON, M., Adv. Phys. 25 (1976) 247.

[13] GILLAN, M. J. and RICHARDSON, D. D., J. P h p . C . 12 (1979) L61.