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J. Phys. Chem. B 2006, 110, 15302-15311
Ionic Motion in Crystalline Cryolite Lindsay Foy and Paul A. Madden* School of Chemistry, Edinburgh UniVersity, The King's Building, West Mains Road Edinburgh EH9 3JJ, U.K. ReceiVed: April 26, 2006; In Final Form: June 8, 2006
The character of the ion dynamics in crystalline cryolite, Na3AlF6, a model double perovskite-structured mineral, has been examined in computer simulations using a polarizable ionic potential obtained by forcefitting to ab initio electronic structure calculations. NMR studies, and conductivity measurements, have indicated a high degree of mobility, in both Na+ ion diffusion and reorientation of the AlF6 octahedral units. The simulations reproduce the low-temperature (tilted) crystal structure and the existence of a transition to a cubic structure at elevated temperatures, in agreement with diffraction measurements, though the calculated transition temperature is too low. The reorientational dynamics of the AlF6 octahedra is shown to consist of a hopping motion between the various tilted positions of the low-temperature form, even above the transition temperature. The rate of reorientation estimated by extrapolation to the temperature re´gime of the NMR measurements is consistent with the experimental data. In addition, we report a novel cooperative “tilt-swapping” motion of the differently tilted sublattices, just below the transition temperature. The perfect crystals show no Na+ diffusion, in apparent disagreement with observation. We argue, following previous analyses of the cryolite phase diagram, that the diffusion observed in the experimental studies is a consequence of defects that are intrinsic to the thermodynamically stable form of cryolite. By introducing defects into the simulation cell, we obtain diffusion rates that are consistent with the NMR and conductivity measurements. Finally, we demonstrate a link between diffusion of the Na+ ions and the reorientation of AlF6 units, though the correlation between the two is not very strong.
I. Introduction Cryolite, Na3AlF6, is a naturally occurring mineral that has the elpasolite crystal structure, a double perovskite (A2BB′X6) with Na+ ions occupying both the A and B′ sites.1 As indicated in Figure 1 this structure may be viewed as a set of cornersharing AlF6 and NaF6 octahedra (the B′ site) with the remaining Na+ ions occupying eight-coordinate holes between these octahedra (the A sites). Below 885 K, the AlF6 (and NaF6) octahedra adopt two tilt orientations and the overall crystal structure is monoclinic (but only slightly distorted from orthorhombic).1-3 At this temperature, a transition to a cubic structure occurs, with the octahedra aligned, on average, with the crystallographic axes.3 NMR observations4-6 and conductivity measurements7 show that ionic diffusion in cryolite is surprisingly facile considering that the crystal structure does not appear to show the kind of empty sites characteristic of typical fast-ion conducting crystal, the fluorites for example.8 In the fluorites, such sites allow the formation of thermally activated Frenkel defects and thus mobile vacancies and interstitial ions. In view of the interest in novel material properties of transition-metal oxides with the double perovskite structure,9 it is of interest to examine the mechanisms that permit ionic motion in this structure because this process could prevent the formation of substitutionally ordered phases in target materials. Conductivity measurements show diffusion of the Na+ ions persisting down to room temperature;7 because the Al3 +-Finteraction is much stronger than that of the Na+-F- interaction, the AlF6 octahedral units remain stable species throughout the solid state. All three of the nuclei have good NMR properties * Corresponding author. E-mail:
[email protected].
Figure 1. Low-temperature crystal structure of cryolite. In the lefthand figure, where the c axis is vertical, the red and blue octahedra indicate the two tilt domains of the AlF6 octahedra and the NaF6 octahedra are shown in yellow. The A-site Na+ ions are shown as small spheres. In the right-hand figure, the view down the c axis is shown and the Na-centered octahedra have been omitted for clarity.
(which is not the case for the more technologically relevant double-perovskite transition-metal oxides), though 27Al and 23Na are quadrupolar, and there have been several NMR studies of the ionic motion.4-6 These studies confirm the exchange of Na+ ions between A and B′ sites; a rate constant of ∼200 s-1 at 473 K is quoted in Lacassagne et al.,6 far below the phase transition temperature. The NMR studies also demonstrate appreciable reorientation of the AlF6 octahedra4 well below the phase transition; this results in a narrowing of the dipole and quadrupole broadening. There has been a good deal of speculation that these effects are connected, with the Na+ motion facilitated by a paddle-wheel-type effect involving the AlF6 octahedra, by analogy with arguments used to explain the Li+ diffusion in Li2SO4 inter alia.8 Above the phase transition at
10.1021/jp062563o CCC: $33.50 © 2006 American Chemical Society Published on Web 07/08/2006
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885 K, the AlF6 octahedra rotate with relative freedom and the Na+ diffusion is sufficiently rapid as to be observable in quasielastic neutron scattering.10 Finally, at 1290 K, cryolite melts. Several considerations motivate our interest in performing computer simulation studies of the ionic motion in solid cryolite and comparing calculated quantities with NMR data. The NMR studies indicate the existence of a considerable degree of ionic motion but, as yet, they have not been able to present a consistent picture of the ion dynamics. As indicated above, Na3AlF6 should be an ideal material for refining NMR techniques to examine the motion of quadrupolar nuclei:5 however, a notable feature of the NMR results is that there are inconsistencies at a quantitative level between the results obtained by different groups and even between the results obtained by the same group on different samples (for example, natural and synthetic cryolite4). Furthermore, there are qualitative inconsistencies between the NMR picture of the solid as one sustaining a high degree of ionic motion even at about 450 K and that which emerges from the diffraction data, which indicates that the ions occupy well-defined crystallographic lattice sites with relatively small thermal ellipsoids even above the phase transition at 885 K. Simulations should, in principle, give insight into the character of possible motions and may indicate the source of these inconsistencies. Apart from the difficulty of obtaining a sufficiently good description of the ionic interactions to represent the real material, the different time scales of NMR observations and computer simulations present a substantial challenge to the use of the method to enhance the information coming from the experiments. The time scale of the NMR experiment is set by the frequency differences between different lines in the spectrum, and is on the order of 10-5 seconds or longer, whereas the total duration of an MD run is unlikely to exceed a nanosecond. Although we may make quantitative comparisons in either the static limit, where all motions are slow compared to the NMR time scale, or the opposite extreme narrowing re´gime, we cannot follow the evolution of the spectra between these limits. We can only follow the sequence of dynamical events that influence the NMR observations at elevated temperatures, where they occur on the MD time scale, and attempt to extrapolate the information back to the temperatures at which the observations are actually made. This has proven successful in a number of studies of ionically conducting solids.11,12 Apart from the interest in studying the solid state, molten cryolite is the electrolyte in the production of aluminum from alumina and thus of enormous importance in its own right,13 so the development of good, transferable models of the ionic interactions is a significant goal. We have developed empirically parametrized simulation models on several occasions14,15 and have shown how these potentials can reproduce the crystal structures though we have not successfully connected their dynamical properties to observation. In this work, we will use a potential derived from ab initio considerations, using techniques used previously to model oxides.16 We begin with an account of the development of this potential and we will contrast its predictions with those of the empirical potentials we have used previously in the body of the paper.
model for stabilizing the tilting of the Al-centered octahedra in cryolite,14 as illustrated in Figure 1, and in pure AlF3,15 thereby giving a good account of the crystal structure and phase behavior. These interaction potentials, which comprise formally charged Born-Mayer pair potentials supplemented by an account of anion polarization,17 were also used in the first phase of the present work. Motivated by a general need to extend the modeling of fluoride materials onto a predictiVe level18 and also by a desire to improve quantitative aspects of the cryolite dynamical behavior obtained with the empirical potentials, revealed in the early stages of this work, we also derived a potential of the same type from purely ab initio considerations that was used for the simulations yielding the results described below. Detailed descriptions of the ab initio “force-fitting” procedure, used to obtain the parameters in a potential of a chosen form from condensed phase, planewave density functional, and electronic structure calculations have been given recently16,18,19 and will not be repeated here. It involves varying the parameters in the potential to minimize the difference between the ab initio forces, dipoles, and quadrupoles on each ion and those obtained from the potential for a set of condensed-phase configurations. To generate a cryolite potential, we used configurations for pure AlF3 and NaF. For AlF3, we generated configurations from a ReO320 structure by running short MD simulations using an empirical potential at a temperature of 800 K. In this structure, the Al3 + ions are coordinated by six F- ions, as they are in cryolite, and the F- coordination number is two. For NaF, the configurations were derived from 1000 K MD simulations of the rocksalt, zinc blende, and CsCl structures, in which the Fions have coordination numbers of 6, 4, and 8, respectively, and liquid configurations were also generated. The configurations therefore sample as wide a range of thermally distorted coordination environments for the F- ions as are likely to be found in cryolite. Excellent fits to the ab initio forces for these configurations were obtained with a potential that allowed for dipole polarization of the fluoride ions as the only many-body effect; the details of this “fitted potential” are given in the Appendix. The fitted potential was obtained by fitting the ab initio information for pure AlF3 and NaF, but we can examine its ability to predict good properties for cryolite by performing further ab initio calculations on cryolite itself and comparing the forces and multipoles with those predicted by the interaction model. Such a comparison is illustrated in Figure 2, bottom panel. The abcissa in these plots represent the index of an ion in a configuration of the structure indicated and the ordinate gives values for the chosen quantity (force, dipole, etc.) for that ion with the point giving the ab initio value and the line the value predicted by the fitted potential. As Figure 1 shows, in cryolite the fluoride ions are sandwiched between triply charged Al3 + ions and singly charged Na+ so that they may be strongly polarized in a coordination environment not found in either of the pure materials; consequently, the transferability of the potential to this mixed system is not guaranteed. Nevertheless, the figure shows that without further optimization the fitted potential predicts the induced dipoles very well, and the forces quite well.
II. Interaction Potentials for Cryolite
III. Temperature Dependence of the Crystal Structure 15
In previous work, ionic interaction potentials for AlF3 and its mixtures with NaF14 have been proposed on the basis of empirical considerations. It has been shown how anion dipole polarization is the key physical ingredient in an ionic interaction
Crystallographic studies show that,1-3 in both high- and lowtemperature phases, cryolite can be described as an alternating three-dimensional network of corner-sharing AlF6 and NaF6 octahedra. In the high-temperature structure, all octahedra are
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Foy and Madden
Figure 2. Dipoles and forces on the ions from ab initio calculations on a high-temperature configuration for solid cryolite (solid line) are compared with those predicted by the fitted potential (points). The horizontal axis is the index of an ion within the configuration of 80 ions, the vertical axes give the values of the dipole and force on the ion; unpolarizable Al3 + ions are omitted from the dipole plots.
Figure 3. Angles θ and φ used to define the AlF6 tilt structure.
TABLE 1: Unit Cell Lengths and the Angles Characterizing the AlF6 Tilt Domains source
a/Å
b/Å
c/Å
θ°
φ°
simulation X-ray diffraction
5.5121 5.4139
5.5884 5.6012
7.8385 7.7769
19° 19.5°
(30 ° (23.5 °
on average aligned with the cell axes in the same manner: two axial bonds along the c or z direction and four orthogonal lying in the ab or xy plane, giving an overall cubic arrangement. However, in the low-temperature structure the AlF6 octahedra tilt slightly from their common axes, with alternating layers adopting different orientations that are related by reflection in the yz plane. The AlF6 tilting necessarily causes the interconnecting NaF6 to adopt a similar alternating tilt-pattern. The tilting results in a monoclinic space group, though the departure of the non-90 ° angle, β, is only 0.185 ° at 295 K and decreases steadily toward the phase transition at 885 K. In our simulations, at finite temperature, we are insensitive to such a small departure from an orthorhombic structure (with our choice of simulation cell, a * b * c, R ) β ) γ ) 90°) and we will discuss the comparison with the experimental structure on this basis. The fitted potential was found to reproduce the lowtemperature crystal structure well. The average cell lengths from NPT (“constant stress”, but with the above constraint on cell angles) simulations21 are compared with those from X-ray diffraction at 295 K1 in Table 1. The simulation produces a slightly larger unit cell, which is at least in part attributable to ignoring dispersion effects in the fitted potential.16,18,19 Angles characterizing the tilts of the AlF6 octahedra are shown in Figure 3. The octahedra may be described as belonging to two tilt domains whose spatial arrangement is shown in Figure 3. They possess the same tilt of the axial Al-F bonds from the z axis (angle θ) and oppositely signed angles (φ) between the x axis and the projection of the axial Al-F bond onto the xy plane. Table 1 gives the average values of θ and φ from
simulation and experiment at 295 K,1 and these compare well, particularly θ. Similar results for the low-temperature structure are obtained with the empirically constructed potentials. Experiment shows that the enthalpy and volume differences between the low- and high-temperature phases are weak and the transition is almost continuous.2,7 We have not been able to locate the transition by looking for sharp changes in structural parameters (such as cell lengths) or thermodynamic quantities, like the mean potential energy, in a series of NPT simulations as the temperature is raised, in the manner of the previous work on AlF3.15 Instead we see a progressive change in the mean tilt angles of the AlF6 units and in the anisometry of the simulation cell. The evolution of the distribution of tilt angles with increasing temperature is illustrated in Figure 4. In Figure 4a the probability distribution of the axial tilt θ is shown; it is peaked about a nonzero angle of about 19° at the two lower temperatures and then broadens and re-centers to about a zero tilt at the two highest temperatures illustrated (recall that the probability of seeing perfect alignment with the axis is zero). We will associate the temperature, Ttrans, at which the distribution broadens and collapses with the phase transition temperature predicted by the simulation. The assignment of a transition temperature by reference to the distribution of tilts is supported by the evolution of average cell lengths, which fit the orthorhombic pattern below Ttrans. As the temperature is raised, they coalesce fairly suddenly to give a ) b ) c/x2, consistent with cubic symmetry, at the same temperature as the tilt distribution re-centers about 0°. As we shall see in detail below, the dynamical properties of the AlF6 units are also consistent with this assignment. Figure 4b-d illustrates the corresponding behavior of the angle φ; at the lowest temperature illustrated the distribution is peaked about a tilt of 30° and at the highest temperature (above Ttrans) the distribution is symmetrical and peaked about zero tilt and has a flat background. Just below Ttrans, as assigned above, in Figure 4c the distribution becomes symmetrical about zero but shows peaks at finite tilts of about 20°, we will discuss this effect as a consequence of “tilt-swapping” and examine the associated dynamics more closely below. The empirical and fitted interaction potentials produce a similar pattern of behavior as the temperature is raised, and this extends to the dynamical information we will discuss below as well as the average structure. This suggests that this pattern is robust with respect to changes in the interaction model; it appears that the behavior predicted by each of the potentials is very similar if they are compared at the scaled temperature T/Ttrans where Ttrans is particular to each potential. However, the actual transition temperature, Ttrans, does differ from one potential to another. The ab initio parametrized fitted potential gives best agreement with the experimental value (885 K) with Ttrans ) 550 K, with the transition temperature predicted by the two empirical potentials significantly lower. It appears that the value is particularly sensitive to the Na-F interaction parameters because these are the parts of the potential that vary most significantly between the different models. This makes physical sense because for an AlF6 octahedron to reorient, at least four (V.i.) neighboring (B′)Na-F bonds must be broken. The actual value of Ttrans is somewhat disappointing; however, as remarked above, we are dealing with a very weak transition and its location is sensitive to tiny energy differences between the two phases. Further work is in progress to discover if more sophisticated models for the interactions give better results. The fact that dynamical behavior is robust to changes in the potential and scales with Ttrans suggests that we should compare
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Figure 4. Changes in the tilt structure formed by the AlF6 octahedra in cryolite. Panel a shows the distribution of the angle θ at several different temperatures; panels b-d show the distribution of φ for the two tilt domains at a series of temperatures from just below to just above the assigned transition temperature.
the predicted and experimental observed behavior at corresponding scaled temperatures. IV. AlF6 Reorientation In this section we will investigate the mechanism for the reorientation of the AlF6 units, suggested by the structural measures discussed above. An immediate picture of this motion is obtained from the mean-squared displacement (MSD) of the F- ions across a range of temperatures. At low temperatures (T/Ttrans < 0.8), a constant amplitude of
