Kinetics and thermodynamics of cation site-exchange reaction in olivines

Solid State Ionics 181 (2010) 473–478

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Solid State Ionics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / s s i

Kinetics and thermodynamics of cation site-exchange reaction in olivines Jianmin Shi ⁎, Klaus Dieter Becker ⁎ Institute of Physical and Theoretical Chemistry, Technische Universität Braunschweig, Germany Hans-Sommer-Strasse 10, D-38106 Braunschweig, Germany

a r t i c l e

i n f o

Article history: Received 23 November 2009 Received in revised form 3 February 2010 Accepted 3 February 2010 Keywords: Kinetics Thermodynamics Cation distribution Site exchange Optical spectroscopy Olivine

a b s t r a c t In many complex oxides cations are found to occupy structurally non-equivalent sites. The internal distribution equilibria determine the structure, as well as physical and chemical properties. Olivine is one of such complex oxides where divalent cations occupy two non-equivalent octahedral sites. The kinetics of cation exchange between non-equivalent sites in cobalt-containing (CoxMg1-x)2SiO4 olivines have been studied by means of temperature-jump optical relaxation spectroscopy. Results derived from the modeling of the experimental absorbance relaxation curves show that the kinetics of cation site-exchange are strongly temperature- and composition-dependent with activation energies ranging from 196 to 221 kJ/mol in (CoxMg1-x)2SiO4 mixed crystals for x between 0.6 and 0.1. The dependence of kinetics on oxygen activity supports a local defect mechanism for cation site-exchange involving vacancies. Finally, literature data reported for cation distributions in quenched samples are analyzed in respect to the temperature dependence of the cation distribution in cobalt-containing olivines yielding relevant thermodynamic parameters. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The most commonly occurring chemical solid state processes are of heterogeneous nature involving at least two phases. Formation reactions provide prominent examples, but also phase separation, decomposition and precipitation processes. Especially, however, redox processes are to be mentioned. In view of their relevance and ubiqueness, an immense body of quantitative and empirical knowledge has been accumulated concerning equilibrium properties but also in respect to the kinetic aspects of heterogeneous solid state reactions. For monographs on the subject the reader is referred to Refs. [1,2]. In comparison, knowledge about homogeneous solid state processes is very limited despite their fundamental importance in establishing thermodynamic equilibrium of materials. In the first place, thermal intrinsic point defect disorder is to be mentioned in this context, i.e. Schottky and Frenkel disorder. However, in complex solids possessing two or more non-equivalent sites – mostly for two or more different types of cations – , atomic disorder can also be created due to internal equilibria causing cations to exchange between these non-equivalent structural sites. Spinels, with cations on sites of tetrahedral and octahedral coordination by oxygen, represent well-known examples for this type of internal disordering processes and their cation distributions often are found to be temperature dependent, see e.g. [3–12]. In this case, the attainment of thermody⁎ Corresponding authors. E-mail addresses: [email protected] (J. Shi), [email protected] (K.D. Becker). 0167-2738/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2010.02.003

namic equilibrium after temperature changes requires the exchange of cations between the non-equivalent sites. In spinels, the kinetics of the cation distribution have been studied in a few cases [8–12]. Apart from spinels, however, knowledge about the temperature dependence of cation distributions in complex oxides is scarce and virtually nothing is known about the kinetics with which these internal equilibria are attained. In the present contribution, we will report on a kinetic study of cobalt-containing olivines, (CoxMg1-x)2SiO4, where the divalent cations are distributed over two octahedrally coordinated sites (M1 and M2) of different local symmetry, Fig. 1. Recently, olivine-type complex oxides, (MxMg1-x)2SiO4, with M denoting a divalent cation, like for example Mn, Fe, Co, and Ni, have been studied with increasing interest because of their chemical reactivity and catalytic activity [13,14]. On the other hand, especially LiFePO4 and other members of the olivine family like LiMPO4 and Li2MSiO4 have attracted considerable recent interest in the context of electrode materials for lithium ion batteries, see e.g. [15–17]. The exchange of cations in (CoxMg1-x)2SiO4 between the two octahedral sites M1 and M2 in olivines can be formulated as kf

CoM2 þMgM1 ⇌ CoM1 þMgM2 kb

ð1Þ

where CoM1, MgM1, CoM2, and MgM2 denote cations in M1 and M2 sites, respectively. kf and kb are the respective rate constants for the cation site-exchange and their ratio defines the equilibrium constant for the internal cation distribution equilibrium. At thermodynamic

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Fig. 1. Crystal structure of olivine (Pbnm) looking along a-axis. Large dark (blue) spheres are cations in M1 sites and large light (green) spheres are cations in M2 sites. Si atoms are not shown. Fig. 2. Sketch of the experimental setup used in this study.

equilibrium, the distribution coefficient, KD, in (CoxMg1-x)2SiO4 is given by Eq. (2) following the mass action law KD = =

kf ½Co∞M1 ½Mg∞M2  = kb ½Co∞M2 ½Mg∞M1  ð2x−½Co∞M2 Þð1−½Co∞M2 Þ : ½Co∞M2 ð1−2x + ½Co∞M2 Þ

ð2Þ

Earlier studies on the cation distribution in cobalt-containing olivines were mainly concerned with its composition dependence using X-ray diffraction on quenched samples [18–21]. These studies showed that the Co2+ ions prefer the smaller M1 site, (see also Section 3.5), but provided no information on the temperature dependence of the cation distribution in olivines. A recent study using in-situ synchrotron X-ray and neutron powder diffraction has shown that the cation distribution is temperature dependent with cobalt ions increasingly populating the M2 site with rising temperatures [22]. This result is consistent with our previous in-situ high temperature optical spectroscopy studies on cobalt-containing olivines [23–26]. In this paper, we report on the temperature- and compositiondependence of the kinetics of cation exchange between M1 and M2 sites in cobalt-containing olivines. The study is performed by means of in-situ optical spectroscopy following the evolution of site populations after sudden temperature changes induced by laser heating. From experiments made at different oxygen activities insight is obtained into the atomic mechanism for cation exchange. Thermodynamic parameters of the cation site-exchange reaction have been estimated for the temperature dependence of cation distribution in olivines. 2. Experimental Single crystals of (CoxMg1-x)2SiO4 olivines with x = 0.1, x = 0.21, and x = 0.6 were grown using floating zone, Czochralski, and Bridgman techniques, respectively, as described elsewhere [23–25]. For absorbers in the optical experiments thin sections were used of thicknesses ranging from 200 μm to 400 μm, depending on cobalt concentration, which were cut perpendicular to the [001] direction and polished on both sides. Optical absorption spectra of (CoxMg1-x)2SiO4 at room temperature and high temperatures were collected with a modified optical spectrometer (Perkin Elmer, Lambda 9) together with an external optical setup including a custom-built furnace and a CO2 laser (Synrad, Model 48-2) to introduce temperature jumps by local laser heating, Fig. 2. The experimental quantity determined is the absorbance A A = lgðI0 = IÞ = ε⋅c⋅d

ð3Þ

where I0 and I denote the intensity of the incoming and transmitted light, respectively. ε is the (decadic) molar absorption coefficient, c the concentration of absorbing species, and d the thickness of the absorber. For the relaxation experiments, samples are equilibrated in the furnace at a given temperature T1. This temperature is measured by a type S thermocouple placed in the vicinity of the sample. The sample then is heated by means of the CO2 laser (beam size ϕ ≤ 3 mm) to a higher temperature, T2 = T1 + ΔT, where ΔT depends on the heating power of the laser. Temperatures T2 are determined from the temperature dependent optical spectra. Sudden changes in temperature can then be induced by switching the laser on and off. Additional information on the experimental setup and temperature-jump relaxation experiments may be found in Refs. [23–26]. 3. Results and discussion 3.1. Point defect structure and mechanism of cation site-exchange in olivines Following Schmalzried and coworkers [27,28], it will be assumed that the majority point defects in (CoxMg1-x)2SiO4 are trivalent cobalt ions, Co•M, and cation vacancies, V”M, where M indicates a divalent cation site. This assumption is also fully compatible with the well-studied analogous case of Fe-containing olivine, (FexMg1-x)2SiO4 [29–32]. Some of the defect reactions that have been proposed in the literature [27–32] for the incorporation of oxygen into olivine are listed below together with the respective conditions of electroneutrality and the predicted dependence of the concentration of cation vacancies, [V”M], on oxygen activity, ao2: Reaction 1:  q þ 4Co& þ Co SiO ; 2½V q  ¼ ½Co&  6CoM þ SiO2 þ O2 ðgÞ ¼ 2VM M M 2 4 M 1=6

q  ¼ K a ½VM 1 O2 :

Reaction 2: 0   q þ 7Co& þ Co þ Co SiO ; 10CoM þ Sisi þ 2O2 ðgÞ ¼ 3VM M 2 4 si 0

q  þ ½Co  ¼ ½Co  2½VM M si 1=5:5

q ≈K a ½VM 2 O2 :

ð4Þ

Reaction 3:   q þ 6Co& þ ðCo0 Co& Þ þ Co SiO ; 10CoM þ Sisi þ2O2 ðgÞ ¼ 3VM M M 2 4 si 0

&

q  þ ½Co  ¼ ½Co  2½VM M si 1=5

q ≈K a ½VM 3 O2 :

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The above-shown defect reactions predict an oxygen activity dependent cation vacancy concentration following [V"M] ∝ao2m with the m-parameter, m = ∂lg[V"M]/∂lgao2, taking values between about 1/5 and 1/6 depending on details of the point defect structure of the material assumed in the reactions. In the present work, the absorber was kept in contact with a piece of quartz glass and therefore, Reaction (1) may apply, see Section 3.4. In case of a vacancy mechanism of cation site-exchange, the overall exchange reaction of Eq. (1) proceeds via the following two independent sublattice exchange reactions which both involve the participation of vacancies, V"M1 and V"M2, on the two sites k1

q

q

CoM2 þVM1 ⇌ CoM1 þ VM2

ð5aÞ

k2

q

k3

q

MgM1 þVM2 ⇌ MgM2 þVM1 :

ð5bÞ

k4

In the framework of this model of local cation-vacancy exchange involving nearest neighbor M1 and M2 sites, the exchange kinetics can be described by a system of two coupled differential equations, see [25,26]. The solution of this system leads to double-exponential time dependencies for species concentrations on the sublattices after sudden perturbations of site populations. According to tracer diffusion studies in isostructural (FexMg1-x)2SiO4 [31,33,34], Mg2+ ions are less mobile than the transition metal cations. It appears reasonable to assume that this also holds true for the present case which renders Mg2+ ions the rate determining species for the exchange kinetics. As a consequence, time dependent concentrations of Co2+ ions, e.g., in the M2 sites, [CoM2](t), will follow a single exponential behavior 0

∞

∞

½CoM2 ðtÞ¼ð½CoM2 −½CoM2 Þ expð−t = τÞþ½CoM2 :

ð6Þ

In the frame work of the model indicated by Eqs. (5a,b), the rate constant k = τ−1 depends on the cation distribution and is proportional to the vacancy concentrations on the sublattices [25]. This implies that the kinetics of cation site-exchange in olivine at given temperature and composition will possess the same oxygen activity dependence as the vacancy concentration if cation site-exchange in olivine is due to a vacancy mechanism. According to Eq. (6) and the Beer-Lambert law, Eq. (3), the timedependent absorbance in a relaxation experiment after a temperature jump is given by AðtÞ = εðtÞ⋅cðtÞ⋅d = ½ε∞ + ðε0 −ε∞ Þexpð−t=σÞ  ½c∞ + ðc0 −c∞ Þexpðt=τÞ⋅d:

ð7Þ

Here, σ is the relaxation time of the absorption coefficient ε after the temperature jump, which is a direct reflection of the temperature change of the absorber; τ is the relaxation time of the concentration of Co2+ ions on the particular site under investigation after a perturbation of site populations, e.g., by a temperature jump. ε0, c0 and ε∞, c∞ represent absorption coefficients and concentrations of Co2+ ions on a specific site in the initial (at time t = 0) and final state (at time t = ∞), respectively. 3.2. Optical spectra of cobalt-containing olivine Fig. 3 shows representative optical absorption spectra of single crystalline (CoxMg1-x)2SiO4 olivine (x = 0.21) in the range of 10,000– 30,000 cm−1 at room temperature and at elevated temperatures between 773 and 1173 K. With increasing temperatures, the absorption bands are characterized by a broadening in width and a shift to lower energies. The change in cation distribution in the two octahedral sites is not clearly reflected by the high temperature equilibrium spectra

Fig. 3. Optical absorption spectra of (Co0.21 Mg0.79)2SiO4 in air at room temperature and at temperatures up to 1173 K. Arrows M1 and M2 indicate the energies selected for monitoring temperature-jump relaxation experiments on the M1 and M2 sites, respectively.

due to vibronic coupling on both sites which is especially strong for the centrosymmetric M1 position. According to previous work [23,35], the observed transitions are well understood as being due to ligand field transitions of the Co2+ ions in M1- and M2-sites. In particular, the absorption band centered at about 13,300 cm−1 at room temperature is well separated from others and has been solely attributed to the 4T1g → 4A2g electronic transition of Co2+ ions in the M2 site [23,35]. Obviously, this band offers a good choice to perform relaxation experiments on the kinetic behavior of Co2+ ions on M2 sites. Because of the band shift at high temperatures, the cation redistribution process on this site is monitored at 11,900 cm−1 (840 nm). The absorption band centered at about 21,500 cm−1 at room temperature has been assigned to one of the electronic transitions of Co2+ ions located on M1-sites [23]. However, due to the strong increase of absorbance with temperature, the temperature-jump relaxation measurements on the M1 site have been performed at 23,300 cm−1 (430 nm), which is located in the M1dominated high-energy wing of the afore mentioned band [24–26]. 3.3. Temperature-jump optical relaxation experiments on M2 sites Fig. 4 shows the time dependent absorbance of Co2+ ions on M2 sites in relaxation experiments in air with temperature jumps between T1 = 873 K and various temperatures T2 for (CoxMg1-x)2SiO4 with x = 0.1, x = 0.21, and x = 0.6, respectively. In the relaxation experiments, the time evolution of absorbance after a perturbing temperature jump is composed of a rapid change followed by a slow relaxation process. The sudden changes in absorbance are caused by the optical absorption coefficient ε, Eq. (7), which instantly reflects the sudden temperature change. The subsequent slow changes of absorbance are attributed to the concentration change of Co2+ ions in the M2 site, i.e., to the relaxation of the cation distribution to its new equilibrium state. From the relaxation curves for 873 K, recorded in temperature jumps from T2 to 873 K, one obtains relaxation times τ1 for the cation site-exchange process of 5598 s, 176 s, and 95 s for olivines with x = 0.1, x = 0.21, and x = 0.6, respectively. From the relaxation data obtained for temperature jumps from 873 K to different T2, relaxation times τ2 are found to be 268 s at about 973 K for x = 0.1, 19 s at about 948 K for x = 0.21, and 2.3 s at 1053 K for x = 0.6 respectively. It is also noted that the relaxation times, σ1 and σ2 for the absorption coefficient ε are less than 3 s confirming that the thermal equilibration of the sample is reached within a few seconds in the present temperature-jump experiments. The experimental time

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high temperature while it decreases isothermally upon a temperature drop. Generally, good agreement is obtained between relaxation times for M2 sites and those observed for M1 sites [24–26]. In the present paper, however, only results from relaxation experiments performed on the M2 sites are presented due to the better experimental conditions for monitoring the cation site-exchange on this site. 3.4. Kinetic parameters of cation site-exchange in (CoxMg1-x)2SiO4 The kinetics of cation site-exchange in cobalt-containing olivines (CoxMg1-x)2SiO4 was studied in the temperature range of 873 to 1073 K for x = 0.1, of 773 to 973 K for x = 0.21, and of 753 to 953 K for x = 0.6. For each case, the upper temperature limit is constrained by the condition that the time for reaching thermal equilibrium in the temperature jumps – which in the present experimental setup takes several seconds – is short in comparison to the chemical relaxation, i.e. τ N σ. In Fig. 5, the exchange rate constant, k, i.e., the inverse of relaxation time, k=

1 = k0 exp½−Ea = RT τ

ð8Þ

is shown in an Arrhenius plot for the three samples studied. As seen, the kinetics of cation site-exchange processes in cobalt-containing olivines exhibit a considerable temperature- and compositiondependence. Activation energies for cation site-exchange in (CoxMg1-x)2SiO4 of about 221 kJ/mol, 213 kJ/mol, and 196 kJ/mol are found for x = 0.1, x = 0.21, and x = 0.6, respectively. Table 1 reports the kinetic parameters obtained from linear fits to the kinetic data of Fig. 5. By extrapolation, the relaxation time for cation site-exchange in cobalt-containing olivines at 1273 K is found to be less than 0.5 s. At 1473 K, relaxation times of even less than 0.02 s are estimated from the present data, Table 1. As a consequence, it will be very difficult, if not impossible, to freeze-in cation distributions at and above 1273 K in quench experiments. Apart from the variation of activation energies, the inverse relaxation times of the three samples show a systematic variation with composition. Because the inverse relaxation time is directly proportional to the cation vacancy concentrations, the increase of 1/τ observed with increasing cobalt content indicates an increasing vacancy concentration. This observation can easily be rationalized by the fact that vacancies on the M1- and M2-sublattice may be formed due to the oxidation of Co2+

Fig. 4. Results of temperature-jump optical relaxation experiments on M2 sites together with fits according to the kinetic model for (CoxMg1-x)2SiO4, Eq. (7). (a) temperature jump between 873 K and 973 K for x = 0.1, (b) temperature jump between 873 K and 948 K for x = 0.21, and (c) temperature jump between 873 K and 1053 K for x = 0.6. Relaxation times for cation site-change at 873 K are 5598 s, 176 s, and 95 s for x = 0.1, x = 0.21, x = 0.6, respectively. All experiments were performed in air at 11,900 cm−1.

dependences unambigously demonstrate that M2 sites are increasingly populated with rising temperatures by Co2+ ions: the absorbance contributed by the changing concentration of Co2+ ions on the M2 sites increases isothermally upon a temperature jump from low to

Fig. 5. Arrhenius plots of the rate constants k = 1/τ for cation site-exchange reactions in (CoxMg1-x)2SiO4 with x = 0.1, x = 0.21, and x = 0.6, obtained from relaxation experiments on M2 sites. Linear fits of the data points yield apparent activation energies for the cation exchange reaction of 221 ± 4 kJ/mol, 213 ± 4 kJ/mol and 196 ± 16 kJ/mol for x = 0.1, x = 0.21, and x = 0.6, respectively.

J. Shi, K.D. Becker / Solid State Ionics 181 (2010) 473–478 Table 1 Kinetic parameters for cation site-exchange in (CoxMg1-x)2SiO4 determined from relaxation experiments in air on M2 sites and extrapolated relaxation times at 1273 K. x in (CoxMg1-x)2SiO4

Ea/kJmol k0/s−1 τ1273 K/s

−1

Table 2 Gibbs energies, enthalpies and entropy for the site-exchange reaction, Eq. (1) in (CoxMg1-x)2SiO4. x

0.1

0.21

0.6

221 ± 4 (2.8 ± 1.6) × 109 0.4

213 ± 4 (2.1 ± 1.3) × 1010 0.02

196 ± 16 (0.5 ± 1.1) × 1010 0.02

ions and, hence, the vacancy concentration can be expected to increase with increasing molar fraction of Co2SiO4 in (CoxMg1-x)2SiO4 olivines. Thus, the compositional dependence of the kinetics provides some preliminary evidence for a vacancy mechanism of cation site-exchange in cobalt-containing olivines. This evidence is confirmed and substantiated by the experimental oxygen activity dependence of the site exchange kinetics reported in Fig. 6 for a (CoxMg1-x)2SiO4 mixed crystal with x = 0.6. It is noted that the rate constant at given temperature in N2 atmosphere is approximately one order of magnitude smaller than in air. This demonstrates the dependence of the kinetic process on the oxygen activity (in the external atmosphere) and, thus, on the concentration of cation vacancies, see Eqs. (4). From the kinetic data obtained in air and N2 one finds an average m-value of 0.21 ± 0.07, which is in good agreement with the predictions made on the basis of the defect reactions (1)–(3), Eqs. (4), predicting m-values between about 1/5 and 1/6. 3.5. Thermodynamics of the cation distribution in (CoxMg1-x)2SiO4 In previous work [26], we have used the experimentally determined the temperature dependence of the rate constant for cation site-exchange to estimate the effective “freezing-in” temperature in quench experiments. As a result, it is estimated that the cation distribution of cobalt-containing olivine in a realistic quench experiment is frozen-in at about 973 K, see Ref. [26]. Hence, the cation distributions obtained in the literature for quenched (CoxMg1-x)2SiO4 samples may reflect the cation distribution state at around this temperature. This estimate is in fair agreement with the work on (CoxMg1-x)2SiO4 performed by Müller-Sommer et al. [21]. These authors examined the cation distribution using designed quench experiments and concluded that their results from Rietveld refinement of room temperature X-ray data correspond to the cation distribution state at about 1073 K. We have therefore collected the cation

Fig. 6. Arrhenius plots of the rate constants cation site-exchange reactions from experiments on the M2 site in air (logaO2 =−0.7) and in N2 (logaO2 =−4.5) for (CoxMg1-x)2SiO4 with x=0.6.

477

0.10 0.21 0.60

ΔG0ex(1000 K)

ΔS0ex

ΔH0ex

kJ/mol

J/mol K

kJ/mol

−12.50 −12.92 −13.20

+ 5.76 + 5.76 + 5.76

−6.74 −7.16 −7.43

distribution coefficients KD, Eq. (2), reported in the literature for quenched samples of similar compositions [18–21] and obtained the following averaged values of KD = 4.50, 4.73, and 4.89 for x = 0.1, 0.21, and 0.6, respectively, at an estimated freezing-in temperature of 1000 K. From these values the standard Gibbs energy of the cation siteexchange reaction in (CoxMg1-x)2SiO4 can be calculated according to 0

ΔGex = −RT ln KD

ð9Þ

where 0

0

0

ΔGex = ΔHex − TΔSex :

ð10Þ

Here, the entropy change, ΔS0ex is composed of a vibrational and el electronic contribution, ΔSvib ex and ΔSex. The vibronic contribution can be assumed to be negligible due to identical coordination numbers of M1 and M2-sites. Also, no electronic contribution can be expected from the closed shell 2p6 Mg2+ ions and therefore 0

vib

el

el

el

2+

el

2+

ΔSex = ΔSex + ΔSex ≅ΔSex ≅S ðCoM1 Þ−S ðCoM2 Þ:

ð11Þ

From statistical mechanics the electronic entropy is given by [5,36] el

S = R ln w

ð12Þ

where w is the product of spin and orbital degeneracy of the electronic state under consideration. For Co2+ in M1 and M2 sites, the electronic ground states are approximated by 4Eg and 4A2 according to the local symmetry of M1 (D4h) and M2 (C2v) sites, respectively [23]. Thus, w equals 8 for Co2+ on M1 sites, and 16 for Co2+ on M2 sites and the (electronic) exchange entropy is given by ΔS0ex≈Rln2 = 5.76 J/mol K. The thermodynamic parameters obtained from the experimental KD (1000 K) values and the theoretical ΔS0ex are shown in Table 2. The temperature dependence of the cation distribution coefficient, KD,

Fig. 7. Temperature dependence of the cation distribution coefficient, KD, in (CoxMg1-x)2SiO4 for x= 0.1, x= 0.21, and x= 0.6, respectively, see text.

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Eq. (2), derived on the basis of these thermodynamic parameters is displayed in Fig. 7 for (CoxMg1-x)2SiO4 olivines with x = 0.1, 0.21, and 0.6. As seen, KD, deviates clearly from the random distribution with KD = 1. Indeed, it predicts an increasing population of Co2+ ions on M2 sites with increasing temperatures. The expected relative concen∞ tration changes, Δ[Co∞ M2]/[CoM2] of about 2–3% for a temperature jump of about 100 °C are fully compatible with the experimentally observed isothermal absorbance changes.

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

4. Conclusions The kinetics of the homogeneous cation site-exchange reaction in cobalt-containing olivines (CoxMg1-x)2SiO4 have been studied by means of temperature-jump optical relaxation spectroscopy. The kinetics of cation site-change exhibit a strong temperature- and compositiondependence. The dependence of kinetics on oxygen activity gives evidence of a local defect mechanism for cation site-exchange involving cation vacancies. Thermodynamic parameters of equilibrium cation distributions in cobalt-containing olivines were estimated from experimental data on cation distributions of quenched samples. Acknowledgements This work was financially supported by the German Research Foundation (DFG). We also thank K. Ullrich, S.G. Ebbinghaus, and S. Ganschow for growing single crystals. References [1] H. Schmalzried, Solid State Reactions, Weinheim, Verlag Chemie, 1975. [2] H. Schmalzried, Chemical Kinetics of Solids, VCH, Weinheim, New York, 1995. [3] H. Schmalzried, Z. Phys. Chem. NF 28 (1961) 203.

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