Diffusion in Oxides. Department of Geological Sciences Case Western Reserve University Euclid Avenue Cleveland, Ohio U.S.A.

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1 Reviews in Mineralogy & Geochemistry Vol. 72 pp , 2010 Copyright Mineralogical Society of America 17 Diffusion in Oxides James A. Van Orman 1,2 and Katherine L. Crispin 1 1 Department of Geological Sciences Case Western Reserve University Euclid Avenue Cleveland, Ohio U.S.A. james.vanorman@case.edu 2 Department of Chemical Engineering Case Western Reserve University Euclid Avenue Cleveland, Ohio U.S.A. INTRODUCTION Non-silicate oxide minerals are minor but important constituents of many igneous and metamorphic rocks, a major component of Earth s lower mantle, and are well represented in planetary and meteoritic materials. Oxide minerals also have important roles in technology, for example as semiconductors, thermal and electrical insulators, fuel cell components, substrates for thin films, photovoltaic materials, and as products of metal oxidation. Because of their technological applications, and their fundamental interest to geosciences, materials science, physics and chemistry, the diffusion properties of many oxide minerals have been studied intensively, using a wide range of experimental, analytical and computational approaches. In many cases, particular attention has been devoted to deciphering the atomic-level mechanisms involved in diffusion. With the possible exceptions of metals and halides, oxides have probably been studied in more detail with regard to their point defect and diffusion properties than any other group of minerals. The oxide minerals considered in this chapter are relatively simple in terms of their structure and chemistry, but nonetheless exhibit quite complicated diffusion behavior in many cases. Due to the simplicity of the minerals, and to the amount and quality of data available, the origin of many of these complicated behaviors is fairly well understood. In magnetite, for example, cation diffusion rates have a complex dependence on oxygen fugacity. This dependence is due to internal redox reactions, and to a transition from an interstitial diffusion mechanism at low oxygen fugacities to a vacancy mechanism under more oxidizing conditions. Oxygen and titanium diffusion rates in rutile also vary strongly with f O2, due to internal reduction of titanium and the associated production of oxygen vacancies and titanium interstitials. Some cations in rutile exhibit strong diffusional anisotropy, which is thought to result from rapid diffusion along interstitial channels that extend along the c direction in the rutile structure. In periclase, trivalent cations diffuse rapidly compared to divalent cations, opposite to the trend observed in most silicate minerals. This behavior arises from the Coulombic attraction between trivalent cations and cation vacancies. Similar interactions between oxygen vacancies and cation vacancies appear to be responsible for lattice diffusion of oxygen in periclase. In this chapter, we focus on four oxide minerals periclase, spinel, magnetite and rutile that are important in high-temperature geochemistry and geophysics, and for which a significant volume of diffusion data are available. A large body of experimental data also exists /10/ $10.00 DOI: /rmg

2 758 Van Orman & Crispin on diffusion in corundum, and this has been reviewed recently by Doremus (2006). Wüstite is addressed here as a solid solution with periclase; for information on cation self-diffusion in the pure phase see McKee (1983) and references therein. Diffusion in silica minerals is reviewed by Cherniak (2010). PERICLASE Periclase is an important constituent of Earth s lower mantle, thought to comprise roughly 15 to 20% of its mass, and is also an important technological material. Its mass transport properties are important for many applications, and a large body of experimental diffusion data has been acquired over the past several decades. Periclase has also been the focus of many theoretical studies that have helped establish the diffusion mechanisms and have provided quantitative predictions of point defect formation and migration energies, as well as absolute diffusion rates. These predictions are, in general, in good agreement with the experimental results. The convergence of theory and experiment in the study of diffusion in periclase illustrates the power of a combined approach for understanding the mechanisms and controls on diffusion in minerals, and provides a benchmark for multi-disciplinary studies of diffusion in more complicated minerals. General considerations Periclase has the same simple structure as halite (NaCl) and is stable over the entire range of temperature and pressure relevant to Earth s interior. Its broad range of stability has made it possible to study diffusion over a large range of temperatures (to 2500 C) and pressures (to 35 GPa). The simple structure and stoichiometry of periclase make it somewhat easier, compared to most minerals, to infer the point defect mechanisms involved in diffusion. Periclase has a single cation site and a single anion site, each with octahedral coordination, as well as interstitial sites with tetrahedral coordination. Theoretical calculations indicate that interstitial magnesium and oxygen in periclase are energetically unfavorable. The enthalpy of formation for a Frenkel defect, which involves either Mg or O moving from its regular lattice site to an interstitial site, leaving behind a vacancy, is predicted to be very large, on the order of 1200 kj/mol for Mg Frenkel defects and perhaps even greater for O Frenkel defects (Hirsch and Shankland 1991, and references therein). The concentrations of Mg and O interstitials are thus expected to be very low. A Schottky defect, formed when both an Mg atom and an O atom leave their lattice sites to create vacancies, is predicted to have a formation enthalpy on the order of 650 kj/mol (Alfè and Gillan 2005; Karki and Khanduja 2006), only about half as large as that of a Frenkel defect. Cation and anion vacancies are thus predicted to be much more favorable than interstitials, and vacancy mechanisms for diffusion are therefore be expected to predominate in periclase. The experimental data for oxygen and most cations are indeed consistent with diffusion by a vacancy mechanism. However, some small cations like Be 2+ may diffuse by an interstitial mechanism. Although the Schottky defect formation enthalpy is smaller than the formation enthalpy for Frenkel defects, it is still quite large and the concentrations of intrinsic cation and anion vacancies are therefore expected to be small. At equilibrium, the concentration of oxygen and magnesium vacancies in pure MgO is given by: f = = G S VMg VO exp ( 1) 2RT f where G S is the free energy of formation of the Schottky pair, R is the gas constant, T is the absolute temperature, and the brackets refer to concentrations. In this equation, the identity of the point defect is expressed using Kröger-Vink notation; V Mg represents a vacant Mg site with

3 Diffusion in Oxides 759 an effective charge (relative to that of the occupied site) of 2, and V O represents a vacant O site with an effective charge of +2. For a free energy of formation of 650 kj/mol, the equilibrium concentrations of intrinsic vacancies are less than a few ppm at any temperature up to the melting point (~2850 C). Even for a formation energy as low as 500 kj/mol, intrinsic vacancy concentrations are <70 ppm at the melting point, and much smaller at lower temperatures. In addition to intrinsic vacancies, extrinsic vacancies are present in periclase to balance the charge of aliovalent solutes. Even synthetic MgO crystals of ultra-high purity contain extrinsic cation vacancies to compensate trivalent cation impurities. Extrinsic anion vacancies are much less abundant in most samples than cation vacancies, because positively charged cation solutes (mainly trivalent cations such as Fe 3+, Al 3+ and Cr 3+ ) tend to be present at higher concentrations than solutes with effective negative charges (monovalent cations such as Li + and Na + ). The commercial nominally pure MgO crystals used in most experimental diffusion studies are calculated to contain cation vacancies at the level of ~ ppm (e.g., Wuensch 1975; Oishi et al. 1983), which would likely predominate over intrinsic cation vacancies at all temperatures up to the melting point. As in other highly ionic crystals, oppositely charged point defects in periclase may bind to form pairs and other associates. These defect associates can have a strong influence on diffusion rates. Cation and anion vacancies may bind to form electrically neutral pairs, which have been inferred by some authors to be the defect primarily responsible for oxygen diffusion (Ando et al. 1983; Yang and Flynn 1994). Cation vacancies also may bind to positively charged trivalent cations to form either negatively charged pairs, neutral dimers consisting of two trivalent ions and a vacancy on adjacent cation sites, or larger clusters. In general, the larger defect associates (dimers and clusters) are expected to be significant only at relatively high concentrations and low temperatures (e.g., Carroll et al. 1988), but cation-vacancy pairs may be present at significant concentrations over a broad range of conditions. The formation of these pairs impedes the motion of cation vacancies, and thus reduces the diffusivity of unbound cations that diffuse by a vacancy mechanism. On the other hand, diffusion of the trivalent cation is enhanced by up to several orders of magnitude when a vacancy is bound to it on an adjacent cation site (Perkins and Rapp 1973; Van Orman et al. 2009). Oxygen More than a dozen experimental studies on oxygen self-diffusion in periclase have been performed over the last five decades (See Appendix Table 1). Most of the early studies determined diffusion coefficients by measuring the rate of exchange of 18 O with a gas phase, and relied on bulk measurements of the isotopic composition. Later studies, beginning in the 1980s, employed secondary ion mass spectrometry (SIMS) and proton activation techniques to measure 18 O diffusion profiles in periclase samples that had undergone diffusive exchange with an isotopically enriched gas or solid reservoir. One study included in Table 1 (see Appendix), by Narayan and Washburn (1973), determined oxygen diffusion coefficients indirectly, based on measurements of the shrinkage rate of dislocation loops, where oxygen self-diffusion was interpreted to be the rate-limiting step in dislocation climb. The diffusion coefficients determined from these studies span three orders of magnitude at a particular temperature (Fig. 1a), neglecting two studies that used MgO doped with lithium (Shirasaki et al. 1973; Oishi et al. 1987). Diffusion coefficients determined in the studies on Li-doped samples are more than an order of magnitude larger than from any of the other studies, consistent with an enhancement of oxygen diffusivity by extrinsic vacancies created to compensate Li + on cation sites. The activation enthalpies determined from these studies, 186 kj/mol (Shirasaki et al. 1973) and 279 kj/mol (Oishi et al. 1987) have relatively large uncertainties, due to the small temperature range covered in each study and to experimental problems including Li volatilization, but agree reasonably well with recent theoretical calculations of the migration energy for oxygen vacancies, 261 kj/mol (Ita and Cohen 1997)

4 760 Van Orman & Crispin O83 O60 S73 (a) a. -18 Y94 Y96-19 R83 S73ws O87 log D (m 2 /s) Y84,02 N73 S73ls H72-23 R66se R66n -24 O in periclase /T (K) (b) Calculated bulk diffusion coefficients, for different dislocation densities b. log D (m 2 /s) m m m O in periclase m /T (K) Figure 1. Summary of experimental diffusion data for oxygen in periclase at atmospheric pressure. In (a), dashed lines refer to data from studies using Li + -doped periclase; solid curves show data from studies on nominally pure MgO, or MgO doped with trivalent cations. In (b), the experimental data (solid curves) are compared with calculated bulk diffusion coefficients for crystals with different dislocation densities (dashed curves, calculated using Eqn. 2). Abbreviations: N73 Narayan and Washburn (1973); O60 Oishi and Kingery (1960); O83 Oishi et al. (1983); O87 Oishi et al. (1987); R66n Rovner (1966) Norton crystals; R66se Rovner (1966) Semi-Elements crystals; R83 Reddy and Cooper (1983); S73 Shirasaki et al. (1973) 3.5 at.% Li-doped; S73ls Shirasaki and Hama (1973) loosely sintered ; S73ws Shirasaki and Hama (1973) well sintered ; Y Yoo et al. (1984, 2002); Y94 Yang and Flynn (1994); Y96 Yang and Flynn (1996).

5 Diffusion in Oxides 761 and 234 kj/mol (Karki and Khanduja 2006). However, as pointed out by Oishi et al. (1987), the interpretation of the activation enthalpy in these experiments is subject to considerable uncertainty. The measured activation enthalpy would be equivalent to the migration enthalpy for oxygen if Li resided solely on cation sites and did not bind significantly with oxygen vacancies to form defect associates, but it is not clear whether these conditions were met in the experiments. Further measurements of oxygen diffusivity in samples with a broad range of Li concentration, over a greater temperature interval, would undoubtedly shed more light on this issue. In general, the use of monovalent cation dopants in diffusion studies on periclase appears to be a promising avenue for further exploration of oxygen diffusion mechanisms. Oxygen diffusion coefficients in periclase crystals that contain mainly trivalent rather than monovalent cation impurities the type used in all studies except the two discussed above that used Li-doped samples do not appear to vary significantly with the concentration of trivalent cations. Ando et al. (1983) studied both pure MgO and samples doped with between 310 and 12,900 ppm Fe, in experiments performed under oxidizing conditions where a substantial proportion of the iron must have been present as Fe 3+, and found no significant variation in oxygen diffusivity with the level of doping. Similarly, Henriksen et al. (1983) found no significant variation in oxygen diffusion coefficients among samples that were nominally pure, or that were doped with Sc 3+ at 550 ppm and 1400 ppm. These experimental observations indicate that free oxygen vacancies are not the species responsible for oxygen diffusion in periclase containing predominantly trivalent impurities, because while trivalent solutes enhance the concentration of cation vacancies, they decrease the concentration of free oxygen vacancies according to the law of mass action. Hence, if oxygen diffusion were to occur by means of free vacancies, a negative correlation would be expected between the trivalent cation concentration in the periclase sample and the oxygen diffusivity. Ando et al. (1983) proposed that in samples doped with trivalent cations oxygen diffuses by means of a neutral species whose concentration is independent of the level of doping, and suggested that this neutral species was a bound pair consisting of a cation vacancy and an anion vacancy. Cationanion vacancy pairs have similarly been inferred to contribute to anion diffusion in alkali halides (e.g., Fuller 1966) and later studies reinforced the inference that vacancy pairs were largely responsible for oxygen diffusion in MgO. Yang and Flynn (1994) found quantitative agreement between their experimental results on oxygen diffusion in the high-temperature intrinsic regime and theoretical calculations on the diffusivity of ( V Mg V O ) pairs, and in later experiments (Yang and Flynn 1996) observed no significant variation in oxygen diffusivity between samples with ~30 and ~3 ppm trivalent impurities. Van Orman et al. (2003), in a highpressure experimental study, found no variation in oxygen diffusivity along an Al 3+ gradient in either single crystals or grain boundaries, suggesting that vacancy pairs (or another neutral species) are responsible for oxygen diffusion through periclase grain boundaries as well as single crystals, and at high as well as low pressures. Yang and Flynn (1994, 1996) employed thin-film diffusion couples of exceptionally high quality grown by molecular beam epitaxy, and determined oxygen diffusion coefficients that were smaller than in any other study. The samples were grown on a polished MgO substrate and consisted of a nm layer of epitaxial single-crystal Mg 16 O deposited at the base, followed by a 5 nm Mg 18 O tracer layer and nm top layer of Mg 16 O. Two of these samples were clamped face to face and packed in MgO powder to prevent evaporation during the hightemperature diffusion anneals. Oxygen isotope profiles across the sample were measured before and after each diffusion experiment using SIMS depth profiling. The activation enthalpy for oxygen diffusion at high temperatures was found to be 667 kj/mol, much higher than calculated values for the migration enthalpy of oxygen but consistent with the activation enthalpy for intrinsic diffusion, which includes half the formation enthalpy of the Schottky pair. Further, Yang and Flynn (1994) showed that their experimental oxygen diffusion coefficients in the hightemperature regime fell between the narrow band of values for intrinsic diffusion with no pairing

6 762 Van Orman & Crispin and with complete pairing of oxygen vacancies to cation vacancies. A later theoretical study of defect formation and migration energies in MgO by Ita and Cohen (1997) also yields intrinsic oxygen diffusion coefficients in close agreement with the Yang and Flynn (1994) values. At lower temperatures, Yang and Flynn (1994) found a deviation from intrinsic behavior, with lower activation enthalpy (257 kj/mol). The lower temperature results have an Arrhenius slope similar to those found in many other experimental studies of oxygen diffusion, but with lower diffusivity values. A similar low-temperature regime, with diffusion coefficients an order of magnitude higher, was found in a study of epitaxially-grown MgO with even higher chemical purity but a higher density of unidentified structural defects (Yang and Flynn 1994, 1996). Evidently, diffusion in this low temperature regime is enhanced by these structural defects. The activation enthalpy for oxygen diffusion in the low-temperature regime observed by Yang and Flynn (1994, 1996) and in many studies of oxygen diffusion using bulk gas-exchange methods, is similar to the measured activation enthalpy of 252 kj/mol for oxygen diffusion along dislocation cores (Narayan and Washburn 1973). This suggests that dislocations may have enhanced oxygen diffusion in these studies, and that variations in dislocation density among the samples used might explain at least some of the wide scatter in oxygen diffusion coefficients determined in different experimental studies. Other experimental observations support this interpretation. Oishi et al. (1983) found that oxygen diffusion coefficients decreased by one to two orders of magnitude when samples were chemically polished to remove surface damage produced by crushing or cleaving the samples in preparation for gas-exchange experiments. Also, Henriksen et al. (1983) found that oxygen diffusivity increased by a factor of ~5 in a single crystal that had been deformed prior to the diffusion anneal. In general, the influence of dislocations and other extended defects becomes more important when lattice diffusion is slow, as is the case for intrinsic oxygen diffusion in periclase, especially at lower temperatures. Figure 1b shows calculated values for the effective diffusivity of oxygen in periclase single crystals with four different dislocation densities, along with the experimental data for periclase single crystals (excluding those that were doped with Li). The effective diffusivity is calculated as: D eff = D lat + ρad disl (2) where D lat is the lattice diffusion coefficient, assumed to follow the Arrhenian dependence determined by Yang and Flynn (1994) at high temperatures, D disl is the diffusion coefficient in a dislocation core assumed to follow the Arrhenian dependence determined by Narayan and Washburn (1973), ρ is the dislocation density, and a is the cross-sectional area of a dislocation. In writing Equation (2), diffusion along subgrain boundaries is not considered, although subgrains would be expected to contribute to oxygen transport in deformed periclase crystals. Thus the calculated curves can be taken to represent the lower limit on effective diffusivity at a given dislocation density, for the lattice and dislocation diffusion coefficients used in the calculations. The lowest dislocation density shown in the figure, m 2, is typical of wellannealed crystals, yet even in this case dislocations are predicted to contribute significantly to diffusion at temperatures below 1400 C. The highest dislocation density shown, m 2, is relevant to a highly deformed crystal, subjected to shear stress on the order of 1 GPa according to the shear stress/dislocation-density relationship reported for MgO by Takeuchi and Argon (1976). In this case the intrinsic regime is not seen; diffusion along dislocations dominates oxygen transport at all temperatures. The influence of pressure on oxygen diffusion in periclase has been addressed in one experimental (Van Orman et al. 2003) and several theoretical studies (Ita and Cohen 1997, 1998; Karki and Khanduja 2006; Ito and Toriumi 2007). Van Orman et al. (2003) determined oxygen diffusion coefficients in single crystals and grain boundaries at 2273 K and pressures of GPa. Oxygen diffusion coefficients in the single crystals were found to decrease significantly with increasing pressure between 15 and 25 GPa, with an activation volume of ~3.3 cm 3 /mol.

7 Diffusion in Oxides 763 However, the diffusion coefficients obtained from these high-pressure experiments are larger than those obtained from most studies at atmospheric pressure. It was suggested (Van Orman et al. 2003) that oxygen transport in the high-pressure experiments may have been enhanced by dislocations generated during non-hydrostatic compression of the samples. In this case, the activation volume determined in this study would be that for diffusion along dislocations which, as discussed above, may be the relevant activation volume for oxygen diffusion in periclase in many cases. The oxygen grain boundary diffusion coefficients determined by Van Orman et al. (2003) were seven orders of magnitude higher than the volume diffusion coefficients measured in the same study, and had a similar pressure dependence. Because the high-pressure experiments on oxygen diffusion in MgO appear to have been conducted in an extrinsic regime, the influence of pressure on the intrinsic diffusion of oxygen has been addressed only by theoretical calculations. Ita and Cohen (1997) used a Gordon-Kim method to determine the Gibbs free energy of formation of Schottky pairs in MgO, and the Gibbs free energy of oxygen migration at pressures up to 140 GPa and temperatures of 1000 to 5000 K. The activation volume for intrinsic oxygen diffusion derived from these results at 0 to 20 GPa is 9 cm 3 /mol, and decreases at higher pressures. Karki and Khanduja (2006) studied defect formation and migration energies in MgO using calculations based on density functional theory. The activation volume for oxygen intrinsic diffusion derived from their results between 0 and 20 GPa is 7.6 cm 3 /mol, and also decreases at higher pressures. The Schottky pair formation energy found in these two studies is in good agreement with the value determined by Alfè and Gillan (2005) using a quantum Monte Carlo method. Ito and Toriumi (2007) studied diffusion in MgO over a wide range of temperatures and pressures using molecular dynamics simulations with empirically fitted force fields. The activation volume for intrinsic diffusion of oxygen derived from their results at 0 to 20 GPa is 13.5 cm 3 /mol. In general, the Schottky formation energy found by Ito and Toriumi (2007) is higher than those found by Ita and Cohen (1997) or Karki and Khanduja (2006), especially at high pressures. The migration energies determined in all three studies are similar at low pressures, but at very high pressures the migration energies derived by Ito and Toriumi (2007) are significantly smaller than those found in the other two studies. Magnesium Magnesium self-diffusion in periclase has been studied extensively using both radioactive and stable isotope tracers (See Appendix Table 2). Important information on Mg self-diffusion is also available from studies of ionic conductivity in periclase samples doped with trivalent cation impurities. Sempolinski and Kingery (1980) studied periclase doped with various levels of trivalent cations and found that the data were consistent with a simple model wherein cation vacancies are the charge-carrying species and their concentration is controlled by the trivalent cation substituents through charge balance. Conductivity data for samples doped with 65 to 1500 ppm trivalent cations yield consistent data on the diffusivity of the cation vacancies, with an Arrhenius relation (Sempolinski and Kingery 1980): D VMg 5 Hm = exp RT () 3 where is the diffusion coefficient for cation vacancies in m 2 /s, H m is the migration enthalpy for cation vacancies (220 kj mol 1 ), R is the gas constant ( J mol 1 K 1 ), and T is the temperature in Kelvin. The migration enthalpy determined by Sempolinski and Kingery (1980) is in reasonable agreement with recent theoretical calculations of 241 kj/mol (Ita and Cohen 1997) and 218 kj/mol (Karki and Khanduja 2006). Information on the rate of diffusion of cation vacancies derived from these ionic conductivity measurements is extremely useful because it can be used to calculate the coefficient for Mg self-diffusion as a function of the concentration of cation vacancies and the temperature, according to the relation:

8 764 Van Orman & Crispin D = x D f ( 4 ) Mg VMg VMg where x VMg is the fraction of cation sites that are vacant and f is a correlation factor with numerical value of for the face-centered cubic cation sublattice in periclase (Shewmon 1989, p. 111). Magnesium diffusion coefficients calculated by combining Equations (3) and (4) are compared with experimental data on Mg self-diffusion in Figure 2. The data are reasonably consistent with the calculations, for cation vacancy concentrations between ~25 and 2000 ppm. Early experimental studies, published prior to 1973, are consistent with high vacancy concentrations, while those published more recently are generally consistent with a vacancy concentration of ~50 ppm, within the range expected for high-purity MgO single crystals that are presently available from commercial suppliers based on the trace impurity contents reported by the manufacturers. When the Mg diffusion coefficients determined in the various experimental studies are corrected to a common cation vacancy concentration, they are in good agreement (Fig. 2). Although it was suggested in some early studies (Harding et al. 1971; Harding and Price 1972) that intrinsic Mg self-diffusion was measured at high temperatures, this interpretation of the data now seems unlikely. The inference of intrinsic diffusion was made on the basis of data covering a short high-temperature interval on the Arrhenius plots, with an apparent activation enthalpy somewhat higher than at lower temperatures. It is not clear, however, that a distinct high temperature segment is resolvable; re-fitting each of the Harding et al. (1971) data sets, obtained using Ventron and Monocrystal MgO crystals, with a single Arrhenius line yields activation energies of 242 and 257 kj/mol, respectively, with r 2 values of 0.98 and Similarly, fitting the Harding and Price (1972) data set with a single Arrhenius line yields an activation energy of 245 kj/mol with an r 2 value of On the whole, the Mg diffusion data for periclase appear to be consistent with an extrinsic vacancy mechanism, with an activation energy similar to values for the migration energy determined in theoretical and ionic conductivity studies. This interpretation is consistent with recent first-principles calculations of the formation energy for Schottky defects in periclase (Alfè and Gillan 2005; Karki and Khanduja 2006), which show (a) that the concentrations of intrinsic cation vacancies will be far below the extrinsic concentrations in real crystals at all temperatures up to the melting point, as noted above, and (b) that the activation energy in the intrinsic regime should be ~600 kj/mol, far higher than observed in any experimental study. The influence of pressure on Mg self-diffusion has been determined in one experimental study (Van Orman et al. 2003) and has also been addressed by several theoretical studies. Van Orman et al. (2003) measured Mg self-diffusion coefficients in single crystals and fine-grained polycrystals at 2273 K and pressures of 15 to 25 GPa. The single crystal data were found to be consistent with data from studies at atmospheric pressure using samples with comparable purity. Magnesium diffusion in the high-pressure experiments appears not to have been affected significantly by the presence of dislocations or other extended defects, in contrast to oxygen diffusion in the same experiments (Van Orman et al. 2003). Oxygen is much more prone to the influence of extended defects because its intrinsic diffusivity in the lattice is so slow; lattice diffusion of Mg is much faster due to the relatively high concentration of extrinsic cation vacancies. In combination with the atmospheric pressure data, the high-pressure data for Mg self-diffusion yield an activation volume of 3.0 cm 3 /mol, which is in reasonable agreement with the migration volumes determined over a similar pressure range using various theoretical methods (Ita and Cohen 1997; Karki and Khanduja 2006; Ito and Toriumi 2007). At higher pressures the theoretical calculations diverge significantly, although all predict a decreasing activation volume at higher pressures (Fig. 3). Clarifying the pressure dependence of cation diffusion in MgO is an important goal for the future, as it has important implications for rates of

9 Diffusion in Oxides 765 H72 H71 25 ppm L57 W73 S ppm 500 ppm 100 ppm Mg in periclase Calculated diffusion coefficients, for different cation vacancy concentrations Harding & Price, 1972 Mg in periclase X v = 50 ppm Lindner & Parfitt, 1957 M85 Harding et al., 1971 Wuensch & Vasilos, 1973 Sakaguchi et al., /T (K) log D (m 2 /s) log D (m 2 /s) Martinelli et al., /T (K) Figure 2. Summary of experimental diffusion data for magnesium in periclase at atmospheric pressure. Solid lines are based on the experimental data, and dashed lines show Mg diffusion coefficients calculated at different vacancy concentrations (indicated by the labels) from measurements of the cation vacancy diffusivity (Sempolinski and Kingery 1980), using Equations (3) and (4). In the inset, the experimentally determined Mg diffusion coefficients are corrected to a common cation vacancy concentration of 50 ppm, according to D corr = D exp (50 ppm/x V,exp ), where x V,exp is the cation vacancy concentration in the periclase samples used for the diffusion experiments. The experimental vacancy concentrations were estimated from reported values of the aliovalent impurity concentrations, as follows: Lindner and Parfit (1957) 500 ppm; Harding et al. (1971) 500 ppm; Harding and Price (1972) 500 ppm; Wuensch et al. (1973) 100 ppm; Martinelli et al. (1985) 50 ppm; Sakaguchi et al. (1992) 50 ppm.

10 766 Van Orman & Crispin Mg pressure dependence Ita & Cohen, 1997 H m = E m + PV m (kj/mol) Van Orman et al., 2003 Karki & Khanduja, 2006 Ito & Toriumi, Pressure (GPa) Figure 3. Experimental and theoretical results on the pressure dependence of Mg diffusion in MgO. Theoretical values of the migration enthalpy from Karki and Khanduja (2006) and Ito and Toriumi (2007) are those reported in the original reference. The theoretical values of Ita and Cohen (1997) were calculated from the reported values of the Gibbs free energy of migration (G m ) at different pressures according to the relation H m = E m + P(dG m /dp) T, with the reference value of E m (220 kj/mol) taken to be the migration energy determined by Sempolinski and Kingery (1980). The experimental values of the migration enthalpy, shown as squares, were similarly calculated as H m = E m + PV m, where E m is the migration energy determined by Sempolinski and Kingery (1980) and V m (3.0 cm 3 /mol) is the activation volume determined by Van Orman et al. (2003). diffusion in Earth s deep mantle. At the pressure of Earth s core-mantle boundary (~140 GPa) the migration enthalpy extrapolated from experimental data assuming a constant activation volume of 3.0 cm 3 /mol is 640 kj/mol, whereas theoretical values are 445 kj/mol (Ita and Cohen 1997), 390 kj/mol (Karki and Khanduja 2006) and 205 kj/mol (Ito and Toriumi 2007). These differences in migration enthalpy correspond to differences in the Mg self diffusion coefficient of six orders of magnitude at 4000 K (and even larger differences at lower temperatures). Other group IIA divalent cations Diffusion data have been acquired for all five stable elements in group IIA of the periodic table. A compilation of the experimental results is presented in Table 3 (see Appendix), and a summary is shown on an Arrhenius plot in Figure 4. Be 2+. Diffusion of divalent beryllium in nominally pure MgO has been investigated by Harding and Mortlock (1966) at temperatures of 1000 to 1700 C and by Harding (1973a) from 636 to 2341 C. Both studies used 7 Be as a radioactive tracer, and employed serial sectioning techniques to measure the diffusion profiles. The two studies yielded consistent results, despite using MgO crystals from two different sources (Monocrystals and Ventron), and no difference was observed between experiments performed in air vs. an argon atmosphere (Harding 1973a). In each case the diffusion coefficients were well described by a single Arrhenius equation over the entire temperature range studied, with activation energy of 154 kj/mol (Harding and Mortlock 1966) and 162 kj/mol (Harding 1973a). The small activation energy, rapid diffusion rates relative to other divalent cations, and low degree of scatter in the diffusion coefficients (perhaps implying that variations in purity from sample to sample are not important) led Harding (1973a) to tentatively suggest an interstitial diffusion mechanism.

11 Diffusion in Oxides Mg 2+ ; W73 Ca 2+ ; H73b Mg 2+ ; HP72 Mg 2+ ; H71 Be 2+ ; H66 Group IIA cations in periclase x v = 50 ppm -15 Mg 2+ ; L57 log D (m 2 /s) Ba 2+ ; H72 Sr 2+ ; M73 Be 2+ ; H73a Ca 2+ ; R66 Mg 2+ ; M Ba 2+ ; H67 Ca 2+ ; W68 Ca 2+ ; Y96 Mg 2+ ; S92 Ca 2+ ; Y /T (K) Figure 4. Summary of experimental diffusion data for group IIa cations in periclase at atmospheric pressure. Data for each cation have been corrected to a common cation vacancy concentration of 50 ppm (as in Fig. 2b), with the exception of Be which may diffuse by an interstitial mechanism. Abbreviations and estimated experimental cation concentrations are as follows: H66 Harding and Mortlock (1966); H73a Harding (1973a); L57 Lindner and Parfitt (1957), 500 ppm; H71 Harding et al (1971), 500 ppm; HP72 Harding and Price (1972), 500 ppm; W73 Wuensch et al. (1973), 100 ppm; M85 Martinelli et al. (1985), 50 ppm; S92 Sakaguchi et al. (1992), 50 ppm; R66 Rungis and Mortlock (1966), 500 ppm; W68 Wuensch and Vasilos (1968), 70 ppm; H73b Harding (1973b), 500 ppm; Y94 Yang and Flynn (1994), 30 ppm; Y96 Yang and Flynn (1996), 3 ppm; M73 Mortlock and Price (1973), 500 ppm; H67 Harding (1967), 500 ppm; H72 Harding (1972), 500 ppm. Ca 2+. Calcium diffusion in nominally pure MgO has been studied by Rungis and Mortlock (1966) and Harding (1973b) using a thin surface deposit of 45 Ca radiotracer with diffusion profiles determined by serial sectioning; by Wuensch and Vasilos (1968) using thin film and vapor exchange techniques with electron microprobe analysis of the diffusion profiles; and by Yang and Flynn (1994, 1996) using high-purity MgO single crystals grown by molecular beam epitaxy with an embedded Ca-doped tracer layer, with diffusion profiles characterized by SIMS depth profiling. As with Mg there is considerable scatter among the different studies in the diffusion coefficients determined at a particular temperature, but much of this scatter is removed by correcting the data to a common cation vacancy concentration. Yang and Flynn (1994) found quantitative agreement between their experimental results and theoretical predictions for diffusion by a vacancy mechanism, at the impurity levels relevant to their samples (10-50 ppm cation vacancies). Yang and Flynn (1996) used samples of exceptionally high purity, with ~3 ppm extrinsic cation vacancies, and found diffusion coefficients an order of magnitude smaller than in their 1994 study, consistent with diffusion by an extrinsic vacancy mechanism. Sr 2+. Strontium diffusion in nominally pure MgO was studied between 1000 and 1600 C by Mortlock and Price (1973), using 85 Sr radiotracer thin films and serial sectioning. Two distinct zones were identified in the diffusion profiles near the surface where the radiotracer

12 768 Van Orman & Crispin was deposited the concentration gradient was relatively steep, and a segment with much shallower gradient extended much deeper into the crystal. This second, deeper zone, which yielded diffusion coefficients two orders of magnitude higher than in the near-surface region, was found to disappear if the samples were pre-annealed at 1700 C, and was attributed to dislocation-enhanced transport. Enhancement of Sr mobility by dislocations is not surprising given the low solubility of the large Sr 2+ cation in periclase; cations with low solubility in the crystal lattice are often found to have high relative concentrations in extended crystal defects (e.g., Hiraga et al. 2004). Ba 2+. Diffusion of Ba 2+ in nominally pure MgO was studied by Harding (1967) at C and by Harding (1972) over an extended temperature range from 1100 to 2500 C. Both studies utilized 133 Ba radiotracer thin films and serial sectioning. As with Sr 2+, the diffusion profiles from these studies have relatively steep gradients near the surface and tails extending much deeper into the crystal. The atmosphere in which the experiments were conducted (air vs. argon) was found not to have a significant influence on the diffusion rates. Harding (1972) suggested that three regimes could be identified on an Arrhenius plot, corresponding to different diffusion mechanisms with different activation energies. These three regimes are not obviously distinguishable, and the data are actually represented quite well by a single line. A non-weighted least squares fit of the entire data set to a single Arrhenius equation yields an activation energy of 292 kj/mol with r 2 of 0.98, which is in reasonable agreement with the value of 326 kj/mol determined by Harding (1967) over a smaller temperature interval. Influence of ionic radius. Among the elements of group IIA the activation energy increases and the diffusion coefficient decreases with increasing cation radius (Fig. 5). The small activation energy and large diffusivity for Be 2+, the smallest of the group IIA cations, may result from diffusion by an interstitial rather than a vacancy mechanism, but for the larger cations an extrinsic vacancy mechanism appears to be well established. The observed correlations are broadly consistent with expectations based on elastic strain considerations (e.g., Mullen 1966); a larger cation is expected to encounter a larger energy barrier as it passes through a constriction during a jump from one site to the next, and thus would have a larger activation energy and, all else being equal, slower diffusivity. However, this general expectation is not observed for diffusion of the transition metals, where there is no obvious correlation between ionic radius and diffusivity. For the group IIa cations, which share similar valence electron shell configurations, the size of the ion appears to play a primary role in governing its Mg Be Ca Sr Ba Group IIA cations in periclase at 1573K Q (kj/mol) Ionic Radius ( ) log D (m 2 /s) Be Mg -17 Ca Group IIA cations Sr a. in periclase Ba b Ionic Radius ( ) Figure 5. Variation with ionic radius of the activation energy (a) and diffusion coefficient at 1573 K (b) for group IIa cations in periclase. In (b), the diffusion coefficients for all cations except Be have been corrected to a cation vacancy concentration of 50 ppm. Ionic radii are from Shannon (1976).

13 Diffusion in Oxides 769 diffusivity, while for the transition metals other factors, including the crystal field effect and the dipole polarizability, may come into play. Group IIIA and IIIB trivalent cations Diffusion of trivalent cations differs fundamentally from diffusion of divalent cations in that their substitution into periclase must be coupled with that of a charge-balancing species. In most cases the charge-balancing species is a cation vacancy, which is the point defect that controls diffusion of the trivalent cations, as well as most other cations. The coupled substitution of cation vacancies and trivalent cations leads in general to a strong concentration dependence of the diffusivity; the concentration of cation vacancies increases with the concentration of the trivalent cations, and thus the diffusivity increases. In the simplest case the diffusion coefficient is a linear function of the concentration. However, as has been observed in the alkali halides and other strongly ionic crystals, positively charged solutes and negatively charged cation vacancies are attracted to each other in periclase, and tend to form strongly bound M 3+ - vacancy pairs (and larger defect complexes, at low temperatures and high concentrations). The formation of bound pairs enhances the mobility of the trivalent cation, because it increases the fraction of trivalent cations that have vacant nearest-neighbor cation sites. On the other hand, the formation of bound pairs steals vacancies from other cations, and thus reduces their mobility. These effects are most important at high trivalent cation concentrations, and at relatively low temperatures. A summary of diffusion data for Group IIIA and IIIB cations, at low concentration, is shown in Figure 6, and a compilation of experimental results is given in Table 4 (see Appendix). Al 3+. Diffusion of Al 3+ in periclase has been studied using MgO-Al 2 O 3 (Whitney and Stubican 1971a) and MgO-MgAl 2 O 4 diffusion couples (Whitney and Stubican 1971b; Van Orman et al. 2009), with electron microprobe analyses of the experimental diffusion profiles. Whitney and Stubican (1971a,b) calculated Al diffusion coefficients in MgO as a function of concentration by graphical analysis of the diffusion profiles using an equation presented by Wagner (1969). Interdiffusion between MgO and Al 2 O 3 (Whitney and Stubican 1971a) is complicated by the formation of MgAl 2 O 4 spinel between them, resulting in a continuously moving interface between the periclase and spinel; despite this complication, the results of these experiments are in reasonable agreement with those using spinel as the source of diffusant (Fig. 7). In both studies the diffusion coefficient was found to be a strong function of concentration. Van Orman et al. (2009) performed experiments similar in design to those of Whitney and Stubican (1971b) but instead of performing a graphical analysis to determine diffusion coefficients, the Al diffusion coefficient as a function of concentration was derived by numerically fitting the diffusion profiles to a theoretical model for diffusion in the presence of bound Al 3+ - vacancy pairs. The theoretical model used was based on the model presented by Lidiard (1955) for diffusion of a divalent cation in an alkali halide crystal, and extended to a trivalent cation in a simple metal oxide by Perkins and Rapp (1973). The equation derived by Van Orman et al. (2009) for the Al diffusion coefficient as a function of the Al cation fraction, x Al, is: D Al = D x x 1 Al Al exp( G / RT) 576exp( 2G / RT ) 2 2 where G 2 is the Gibbs free energy of the point defect association reaction: V + ( ) 1 2 x Al exp( G / RT ) 2 () 5 Al V Al ( 6) Mg Mg Mg Mg and D 2 is the diffusion coefficient of the bound (V Mg Al Mg ) pair. Synthetic diffusion profiles

14 770 Van Orman & Crispin log D (m 2 /s) Ge 4+ ; Harding, 1973b Groups IIIA, IIIB, IVA cations in periclase Ga 3+ (100 ppm); Crispin & Van Orman, 2010 Y 3+ ; Berard, 1971 Sc 3+ ; Solaga & Mortlock, /T (K) Al 3+ (100 ppm) Van Orman et al., 2009 Figure 6. Summary of experimental diffusion data for group IIIa, IIIb and IVa cations in periclase. Data for Y, Sc and Ge are from radiotracer studies, and data for Ga and Al are from interdiffusion studies that characterized the concentration dependence of the diffusion coefficient. Diffusion coefficients for Ga and Al are shown at cation concentrations of 100 ppm, similar to the aliovalent impurity concentrations in the periclase samples used in the tracer diffusion studies MgO-Al 2 O 3 ; Whitney & Stubican, 1971a Figure 7. Summary of experimental data for Al-Mg interdiffusion in periclase. All data are shown at an Al cation fraction of 2%. log D (m 2 /s) MgO-MgAl 2 O 4 ; Whitney & Stubican, 1971b MgO-MgAl 2 O 4 Van Orman et al., Al-Mg interdiffusion in periclase; x Al = /T (K) based on Equation (5) with the appropriate boundary conditions were found to provide a good description of the experimental diffusion profiles at all conditions, from K and 1 atm to 25 GPa. The average binding energy of the Al-vacancy pair derived from fitting the experimental diffusion profiles, 50 kj/mol, is in good agreement with the temperaturecorrected theoretical values derived from shell model calculations (Gourdin and Kingery 1979; Carroll et al. 1988). The activation energy and activation volume for diffusion of the (V Mg Al Mg ) pair were found to be similar to those for Mg self-diffusion. Analysis of the frequencies of Al-vacancy and Mg-vacancy exchange both of which are involved in the motion of the Al-vacancy pair shows that the Al jump frequency is only about one third the Mg jump frequency. Despite this, diffusion of Al is about an order of magnitude faster than Mg at the same conditions, due to the attraction of cation vacancies to Al 3+. Ga 3+. The concentration-dependent diffusion of Ga 3+ in MgO was studied by Crispin and Van Orman (2010) using an experimental design similar to that used by Van Orman et al.

15 Diffusion in Oxides 771 (2009) for Al 3+. These authors found that cation vacancies bind more tightly to Ga 3+ than they do to Al 3+, with a binding energy of 83 kj/mol, which appears to be the primary reason Ga diffuses more rapidly than Al. Sc 3+. A series of measurements of Sc 3+ diffusion was made by Solaga and Mortlock (1970) using radiotracer ( 46 Sc) and sectioning methods. Each experiment was performed at 1773 K for a duration of 50 hours, and the amount of radiotracer applied to the surface of the MgO crystal was varied between 0.5 and 10 5 ppm. For experiments performed with surface tracer concentrations less than about 50 ppm the diffusion coefficients were found to be constant. Beyond 50 ppm the diffusive penetration distance increased with the tracer concentration, and the diffusion profiles showed clear evidence for concentration-dependent diffusivity. Y 3+. Yttrium diffusion in nominally pure MgO was studied by Berard (1971) using a radiotracer ( 91 Y) and sectioning method. No mention is made of the concentration of radiotracer used, and the diffusion profiles were fit to a thin film solution with the diffusion coefficient independent of concentration. Tetravalent cations Tetravalent cations, like trivalent cations, are expected in most cases to be chargebalanced by cation vacancies in periclase. Cation vacancies should also bind more strongly to tetravalent cations than to trivalent cations, due to the greater Coulombic attraction. To the authors knowledge, only one study of tetravalent cation diffusion in periclase has been published, on diffusion of Ge 4+ (Harding 1973b). A small concentration of tracer was used in these experiments, which the author suggested would not increase the concentration of cation vacancies in the MgO crystal significantly. Unfortunately no information on the binding energy between Ge 4+ and cation vacancies can be extracted from the Harding (1973b) experiments. However, the diffusion coefficients for Ge 4+ were found to be higher than the Ca 2+ diffusion coefficients reported in the same paper, despite the higher activation energy of Ge 4+. Ordinarily an ion with larger activation energy is expected to diffuse more slowly. It seems likely that the enhanced diffusivity of Ge 4+ results from strong binding to cation vacancies. Transition metals An extensive body of experimental data exists on the diffusion of 10 transition metals in periclase, including Sc and Y which were discussed above in the section Group IIIA and IIIB trivalent cations. A compilation of the published experimental data is presented in Tables 4-7 (see Appendix). The transition metals differ in several ways from the elements considered above, in that many of them (a) are stable in multiple valence states, (b) have extensive solid solution with MgO, and (c) have partially filled d-orbitals resulting in non-spherical valence electron shells. Cr. Diffusion of chromium has been studied experimentally by several groups, under relatively oxidizing conditions where it is thought to have been present primarily as Cr 3+. As with the trivalent cations discussed in section Group IIIA and IIIB trivalent cations, diffusion of Cr 3+ is strongly concentration-dependent and its attraction to cation vacancies appears to have a significant influence on its diffusion. Tagai et al. (1965) studied diffusion of Cr 3+ in air by depositing a thin oxide film and analyzing diffusion profiles with an electron microprobe. In contrast to later studies, no concentration dependence of the diffusion coefficient was noted. Greskovich and Stubican (1969) examined interdiffusion in the MgO-Cr 2 O 3 system (which, as in the MgO-Al 2 O 3 system discussed above, is complicated by the formation of a spinel phase, MgCr 2 O 4 ) and found that diffusion of Cr 3+ in MgO was a strong and quasi-linear function of concentration. Osenbach et al. (1981) obtained similar results using MgO crystals doped with 0.05 and 0.56 cation% Cr 3+ as the diffusion source, rather than Cr 2 O 3. Crispin and Van Orman (2010) examined interdiffusion between nominally pure MgO single crystals and polycrystalline (Mg 0.99 Cr 0.01 )O over a large range of temperature at 2 GPa, and numerically fitted the dif-