Abstract
Oxygen diffusion in the oxygen ionic conductor yttria-stabilized zirconia is investigated by means of the molecular dynamics simulation technique. Oxygen ions migrate by means of a discrete hopping process, mainly between neighbouring tetrahedral sites. Diffusion appears to occur in a short time and a long time regime. Only when the oxygen ions have moved over distances much larger than the characteristic distances of the underlying crystal structure, a linear relation is found between the mean square displacement and time. The oxygen tracer diffusion coefficient, obtained from this long time regime, is 1.86 × 10 -6 and 3.23 × 10 -6 cm2/s at 1759 and 2057 K, respectively. The ionic conductivity, calculated from the tracer diffusion coefficient, agrees well with experimental values.
Introduction
The molecular dynamics (MD) technique is used to study the diffusion of oxygen ions in yttria-stabilized zirconia (YSZ). This material is well known as an oxygen ionic conductor at elevated temperatures and is applied in oxygen sensors and in solid oxide fuel cells. At room temperature undoped zirconia (ZrO 2) has a monoclinic structure; doping with more than 17 mol% YOi. 5 (or about 9 mol% Y203) stabilizes the cubic fluorite structure at room temperature. The Zr 4+ and y3+ ions form an FCC lattice in which most of the tetrahedral positions are occupied by 0 2-ions. The substitution of Zr 4÷ by y3+ causes the formation of oxygen vacancies (equal to half the amount of y3+ ions) to maintain electrical neutrality. These oxygen vacancies are mobile at high temperatures and give rise to oxygen ionic conductivity. The octahedral sites are not occupied.
Introduction
Although YSZ is a material widely applied in inorganic materials science, only one research group [1, 2] has performed MD simulations with this material. Three compositions (ZrO2)l_x(Y203) x with x = 0.0485, 0.102 and 0.227 at 1800°C were investigated. The oxygen tracer diffusion coefficient has a maximum at x = 0.102, in accordance with experimental measurements. In this Letter an MD investigation of oxygen diffusion in (ZrO2)0.92(Y203)0.08 at 1759 and 2057 K is discussed. Compared to Shimojo et al.'s results [ 1 ] , our simulation time is much longer, which turns out to be crucial if one wants to calculate diffusion coefficients. An extended investigation is published elsewhere [3] .
Details of the simulation
Constant (N, V, E) MD simulations have been performed using the GROMOS package [4] on a 0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD10009-2614(95)01231-1 Table 1 Interionic potential parameters i-j
Details of the simulation
Zr-Zr Challenge L computer. The GROMOS code has been adjusted to calculate Coulombic interactions using the Ewald transformation. The equations of motion were integrated by a leap-frog algorithm; periodic boundary conditions were applied. Short range interactions between the ions were represented by Born-Mayer-Buckingham pair potentials,
Details of the simulation
Parameters were obtained from Refs. [5, 6] and are given in Table 1 . For the cation-cation interactions only Coulombic interactions were assumed. It was assumed that the YSZ system is fully ionic; the ionic charges were chosen to be 4 +, 3 + and 2 -for Zr, Y and O, respectively. A cubic MD simulation cell was chosen, consisting of 3 X 3 X 3 crystallographic unit cells (in the x, y and z directions) containing a maximum of 324 ions. For the system (ZrO2)0.92(Y203)0.08 the MD cell contained 92 Zr 4+ ions, 16 y3+ ions and 208 O 2-ions. According to experiments, the lattice parameter for the cubic YSZ unit cell increases slightly with temperature and yttria content. Therefore, the length of an MD cell was chosen to be 1.5633 nm at a temperature of 1759 K and 1.5678 nm at 2057 K. A cubic (FCC) ZrO 2 cell was generated by positioning 108 Zr 4+ ions and 216 0 2-ions at their regular lattice sites. Next, 16 of the Zr 4+ ions were randomly substituted by y3+ ions; 8 of the 0 2-ions were randomly substituted by oxygen vacancies.
Details of the simulation
Velocities were attributed to the ions, and were re-scaled to an initial temperature of 1 K. In the equilibration stage of the simulation, the system was brought from 1 K to the desired simulation temperature using 5000 steps of 2 fs by weak coupling to an external temperature bath [7] . The systems appeared to be equilibrated already after 4 ps. The real simula-tion run consisted of 500000 steps of 2 fs (total simulation time 1000 ps). Every 125 steps (every 0.25 ps) the positions of all ions in the system were recorded for 1000 ps. Due to periodic boundary conditions the positions of the ions were recorded with x, y and z coordinates between 0 and the MD cell length. Every 2500 steps (every 5 ps) the kinetic, potential and total energies of the system were recorded for 1000 ps. For each run, the variation in total energy was not larger than 0.007%. During these runs the temperatures varied by approximately 3.2%.
Results and discussion

Migration mechanism
The ion density distribution (i.e. the probability of finding ions in certain regions of the cell) for (ZrO2)0.92(Y203)0.08 at 1759 K over a 12.5 ps simulation time in the (110) face (slab width 0.01 nm) of a crystallographic unit cell is shown in Fig. 1 . This face covers all box positions of interest: cube corners, face centers, tetrahedral and octahedral sites, Fig. 1 shows that the cations vibrate around their regular sites, that on average the anions tend to dwell in the region of their regular tetrahedral sites and that some of the anions migrate from one regular lattice site to another; so the migration process is a discrete hopping process and not a continuous liquid-like process. The octahedral sites remain vacant during almost the entire simulation. This indicates that octahedral sites hardly play a role in the diffusion process of oxygen ions in YSZ. In Fig. 1 can be seen that anions migrate mainly in the [001] direction between regular lattice sites and not much in other directions. This is investigated further by looking at the numbers of jumps in the (100) directions (to a neighbouring tetrahedral site), in the (110) directions (to a tetrahedral site on '~/2 distance') and in the (111 ) directions (to an octahedral site). In Table 2 the number of jumps are shown in each of these directions for (ZrO2)0.92(Y203)0.08 at 1759 K for a 200 ps simulation time. It is seen that almost all jumps occur in (100) directions. The fractions of jumps in the [100], [010] and [001] directions are given separately in Table 2 and are almost equal, about 30-35%; this indicates that diffusion in the (100) directions is rather isotropic. Similar results have been found by Shimojo et al. [2] for the system (ZrO2)0.898(Y203)0.102 at 2073 K.
Migration mechanism
It is observed during analysis of the simulation data that some of the anions that migrate to a neighbouring tetrahedral site, migrate back to the original tetrahedral site a few time steps later. These migrations will not contribute to the diffusion process and are further referred to as non-diffusive migrations. In Ref. [3] we have found that within the range of 3.85 to 16.13 mol% Y203 dopant at 1759 K, the system with 8.00 mol% Y203 has the largest number of diffusive jumps. This is in agreement with literature where a maximum in ionic conductivity is observed for YSZ with about 8-10 mol% yttria. 
Diffusion coefficient
The brackets ( ) are to be understood as an average over all time origins t o within the simulation run; summation occurs over all N oxygen ions to obtain a statistical accuracy as good as possible. This relation is known as the Einstein relation; the (bulk) tracer diffusion coefficient Db, Jk is obtained when time goes to infinity.
Diffusion coefficient
The horizontal arrows in Figs. 3 and 4 at MSD values of 0.068 and 0.136 nm 2 correspond to the (squared) nearest-and next-nearest distance, respectively, between two tetrahedral sites. It is seen that only when the ions have moved over distances much larger than these characteristic distances, a linear relation between MSD and time is found. The bulk oxygen tracer diffusion coefficients are calculated from the MSD versus time curves according to Eq.
Diffusion coefficient
(2). Both MSD versus time curves are fitted to a straight line between 150 and 400 ps. For the simulation at 1759 K, Db,~k is 1.86 X 10 -6 cm2//s; at 2057 K Obulk is 3.23 X 10 -6 cm2//s. In Fig. 2 it is seen that anions move faster with rising temperature. At the lower temperature it takes longer for the system to reach the 'bulk' regime than at the higher temperature; in Fig. 3 (1759 K) the 'transition time' is around 60 ps whereas in Fig. 4 (2057 K) this time is around 35 ps.
Diffusion coefficient
For comparison with Shimojo et al. [1] , estimates of a 'short time diffusion constant' are given by fitting the MSD versus time curves between 3 and 15 ps to a straight line. These fits are given as dotted lines in Figs. 3 and 4 . For the simulation at 1759 K, Dsho~ t is 2.38 × 10 -6 cmE/s; at 2057 K Dshor t is 3.80 X 10 -6 cm2/s. Notice that these 'short time diffusion constants' are 18-28% larger than the 'bulk' diffusion constants. In their investigation, Shimojo et al. [1] only observe the first, short time regime. Their total correlation time is 30 ps, which is much too short to observe a transition to 'bulk' behaviour. From Fig. 6 in Ref. [1] an oxygen tracer diffusion coefficient of 3.6 X 10 -6 cmE//s is found at 1800°C, which compares well to our value for Dsho~ t at 2057 K.
Diffusion coefficient
The oxygen tracer diffusion coefficient D can be transformed to the dc oxygen ionic conductivity (tr) via the Nernst-Einstein relation (see e.g. Ref. [9] ),
Diffusion coefficient
kaT Here H R is the Haven ratio. For vacancy diffusion in an FCC structure, H R is equal to the tracer correlation factor f. At low defect concentrations (in our system the oxygen vacancy concentration is 3.7% of the total number of anion sites) f is 0.7815 [9] . N is 1) -1 cm -1 at 1759 K and 1.9 f~-i cm-1 at 2057 K financial aid from the Netherlands Organization for [10] ), these values differ by a factor of 3-4.
Diffusion coefficient
Scientific Research (NWO).
Conclusions
The results presented here show that oxygen diffusion in (ZrO2)0.92(Y203)0.08 occurs by means of a discrete hopping mechanism, mainly in (100) directions. Our results clearly show a transition from a short time to a long time diffusion regime. This transition occurs when the displacements of the oxygen ions are roughly equal to the characteristic distances of the underlying crystal structure. In the past this has not been realized well enough. The 'bulk' oxygen tracer diffusion coefficient, obtained from the long time regime, is 1.86 × 10 -6 and 3.23 × 10 -6 cm2//s at 1759 and 2057 K, respectively. The ionic conductivities, calculated from these diffusion coefficients, differ only by a factor of 3-4 from experimental values.

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