J. Phys. Chem. C 2009, 113, 6267–6274
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Octahedral Cation Exchange in (Co0.21Mg0.79)2SiO4 Olivine at High Temperatures: Kinetics, Point Defect Chemistry, and Cation Diffusion Jianmin Shi,*,† Steffen Ganschow,‡ Detlef Klimm,‡ Klaus Simon,§ Rainer Bertram,‡ and Klaus-Dieter Becker*,† Institut fu¨r Physikalische and Theoretische Chemie, Technische UniVersita¨t Braunschweig, Hans-Sommer-Strasse 10, D-38106 Braunschweig, Germany, Leibniz-Institut fu¨r Kristallzu¨chtung, Max-Born-Strasse 2, D-12489 Berlin, Germany, and Go¨ttinger Zentrum der Geowissenschaften, UniVersity of Go¨ttingen, Goldschmidtstrasse 1, D-37077 Go¨ttingen, Germany ReceiVed: December 12, 2008; ReVised Manuscript ReceiVed: February 12, 2009
The potential applications of transition metal-containing olivines, (M,Mg)2SiO4, in environmental sustainability and renewable energy rely on a better understanding of their internal structure, defect chemistry, and sublattice processes at high temperatures. In the olivine crystal structure, divalent cations occupy two nonequivalent octahedral sites. The kinetics of octahedral cation exchange between the two sites in a (Co0.21Mg0.79)2SiO4 single crystal has been studied from 500 to 700 °C by means of time-resolved optical relaxation spectroscopy upon rapid temperature-jumps. Our experiments show that the cation distribution in the two octahedral sites changes toward a random distribution with increasing temperature and that the exchange kinetics is strongly temperature dependent. Modeling of experimental relaxation data using a kinetic equation of cation exchange yields relaxation times of about 12 300 s at 500 °C and about 6 s at 700 °C, respectively, and an activation energy of about 230 ( 12 kJ/mol for the cation exchange reaction in the (Co0.21Mg0.79)2SiO4 olivine. Calculations of vacancy concentrations based on a defect model and impurity levels in the (Co0.21Mg0.79)2SiO4 single crystal have been used to interpret the experimentally observed dependence on oxygen activity. We developed as well a formula to correlate experimental relaxation times to the cation diffusion coefficient along the b-axis in olivines. Such a relation allows one to estimate Mg self-diffusion coefficients DMg b as well as Co-Mg interdiffusion coefficients Db along the b-axis, especially at low temperatures, for example, DMg b ) 4.78 × 10-23 m2/s and Db ) 9.33 × 10-23 m2/s at 600 °C. Cation interdiffusion coefficients from the extrapolation of our diffusion data to high temperatures are in agreement with available literature data from Co-Mg interdiffusion experiments. 1. Introduction Olivine-type complex ionic solids, (MxMg1-x)2SiO4, where M denotes a transition metal cation, for example, Mn, Fe, Co, and Ni, have been studied recently with increasing interest because of their technical applications in the reduction of greenhouse effect.1-5 Natural olivine (Fe0.1Mg0.9)2SiO4 has been tested for CO2 sequestration by mineral transformations.2 Olivines have also shown catalytic activity in syngas production from the biomass gasification process.3-5 The chemical reactivity and catalytic activity of olivines are related not only to their physical properties but also to their internal structure, for example, cation distribution and point defect chemistry at high temperatures. A better understanding of internal structure and sublattice processes in olivines will contribute in part to the improvement of their chemical reactivity and to the development of catalysts with enhanced properties. The olivine structure belongs to the orthorhombic space group (Pbnm), and the unit cell contains 4 formula units. In the crystal structure, the oxygen ions form an approximately hexagonally close-packed array where one-eighth of the tetrahedral sites are * Corresponding authors. E-mail:
[email protected] (J.S.);
[email protected] (K.-D.B.). † Technische Universita¨t Braunschweig. ‡ Leibniz-Institut fu¨r Kristallzu¨chtung. § University of Go¨ttingen.
filled with Si atoms and one-half of octahedral sites are occupied by divalent cations, M and Mg. The octahedral positions are divided into two crystallographically distinct sites: M1, with a center of symmetry, and M2, on a mirror plane. The smaller M1 sites are interconnected by common edges to form straight chains parallel to the c-axis ([001] direction in Pbnm). Chains of alternating M1 and M2 sites are arranged along the b-axis ([010] direction) as shown in Figure 1. In some transition metal containing mixed olivines, for example, (FexMg1-x)2SiO4, (CoxMg1-x)2SiO4, and (NixMg1-x)2SiO4, the divalent cations are not randomly distributed on the two nonequivalent sites mainly due to the crystal field stabilization energy and the bonding nature of cations in the sites.6 Earlier studies on the cation distribution in cobalt-containing olivines, (CoxMg1-x)2SiO4, were mainly concerned with the composition dependence using X-ray diffraction on quenched samples.7-10 These studies showed that the Co2+ ions strongly prefer the smaller M1 site, but, however, provided no information on the temperature dependence of the cation distribution in olivines. A recent study by Sutanto et al.11 using in situ neutron and synchrotron powder diffraction has revealed that with increasing temperature from about 500 °C the Co2+ ions move from M1 to the M2 sites in (Co0.51Mg0.49)2SiO4, that is, toward a random distribution, but that the M1 preference holds up to the melting point temperature. Another study using in situ neutron powder diffraction on
10.1021/jp810968q CCC: $40.75 2009 American Chemical Society Published on Web 03/24/2009
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CoM2 + MgM1 y\z CoM1 + MgM2
(1)
kb
where CoM2, MgM1 and CoM1, MgM2 denote cations in M2 and M1 sites, respectively. kf and kb are the respective rate constants for the cation exchange reaction in olivine. At thermodynamic equilibrium, the cation distribution coefficient, KD, is given by eq 2 following the mass action law:
Figure 1. Crystal structure of olivine (Pbnm) looking along the a-axis. Small (red) spheres indicate oxygen, large dark (blue) spheres are divalent cations, and Si atoms are not shown. The smaller octahedra (gray) are M1 sites, and the larger ones (cyan) are M2 sites. Note the alternating arrangement of M1 and M2 layers along the b-axis and the M1 chains parallel to the c-axis.
polycrystalline olivine of nearly the same composition shows a different temperature dependence of cation distribution in ref 12. Thanks to its site and chemical selectivity, optical spectroscopy is an important and useful experimental method for the study of cation distributions and the investigation of kinetic processes in transition metal-containing complex oxides as demonstrated by Becker and co-workers.13,14 Olivine is one of such complex oxide systems. Studies on the electronic transitions in cobalt-containing olivines provided information on the cation distribution and the assignments of absorption bands to Co2+ ions in the M1 or M2 sites in the olivine crystal structure.15,16 This knowledge makes it feasible to monitor site-selective cation exchange processes in olivines. In-situ optical spectroscopy can probe the cation distribution not only in thermal equilibrium but also during nonequilibrium conditions introduced, for example, by sudden temperature-jumps. By means of the combination of optical spectroscopy and the temperature-jump technique, our recent work17-19 on the internal cation kinetics in olivines indirectly confirms the temperature dependence of cation distribution in cobalt containing olivines as observed by Sutanto et al.11 In this Article, we present a study of another cobalt-containing olivine, (Co0.21Mg0.79)2SiO4, by using temperature-jump relaxation optical spectroscopy. We report on the effect of temperature and oxygen activity on the kinetics of cation exchange between octahedral M1 and M2 sites and discuss the observed phenomena within the framework of a point defect model allowing for impurities. A correlation established between kinetic parameters and divalent cation diffusion allows one to estimate diffusion coefficients at low temperatures. 2. Theory of Octahedral Cation Exchange in Olivine 2.1. Kinetics of Cation Exchange Reaction. In recent publications,17-19 we have provided in detail the theoretical analysis and mathematical treatment of the kinetics of octahedral cation intersite exchange in olivines by a phenomenological description as well as within the framework of a vacancy mechanism. Here, we present only equations essential in the context of this Article. For (CoxMg1-x)2SiO4 olivines, the cation exchange reaction between the two octahedral sites M1 and M2 can be written as
KD )
kf [Co∞M1][Mg∞M2] ) kb [Co∞M2][Mg∞M1]
(2)
∞ ∞ ∞ ∞ ], [CoM2 ], [MgM1 ], and [MgM2 ] denote the equilibHere, [CoM1 rium concentrations of each of the cations in the indicated site. In view of the earlier studies on point defects and interdif′′ , and electron fusion in (CoxMg1-x)2SiO4, cation vacancies, VM • , are considered to be the majority of point defects holes, CoM in this system.20,21 Therefore, the exchange of cations between sublattices by means of a vacancy mechanism appears to be a very likely process. Accordingly, the overall exchange reaction 1 is described in detail by the two reactions:
k1
CoM2 + V′′M1 y\z CoM1 + V′′M2
(3a)
k2
and k3
MgM1 + V′′M2 y\z MgM2 + V′′M1
(3b)
k4
From these elementary reactions, one obtains a system of two differential equations, which are coupled due to the fact that ′′ ′′ the two types of vacancies, VM1 and VM2 , are involved in both exchange reactions:
d[CoM2] ) -k1[CoM2][V′′M1] + dt d[CoM1] dt
(4a)
d[MgM1] dt
(4b)
k2[CoM1][V′′M2] ) d[MgM2] ) -k3[MgM2][V′′M1] + dt k4[MgM1][V′′M2] ) -
The differential equations can be linearized on the condition that the deviations of cation concentrations from their equilibrium values are sufficiently small. To make the solution of eq 4a feasible, it is further assumed that the total concentration of ′′ ], is constant in the course of a temperaturevacancies, [VM jump relaxation experiment (conservation of sites). Together with the assumption that the sublattice exchange of Mg2+ ions is much slower than that of Co2+ ions as has been experimentally confirmed in the isostructural (FexMg1-x)2SiO4 olivines using tracer diffusion,22,23 the cation exchange kinetics in (CoxMg1-x)2SiO4 olivines is determined by the sublattice
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TABLE 1: Concentrations of Cationic Trace Impurities in the (Co0.21Mg0.79)2SiO4 Olivine (×10-6 per formula unit) Ni
Ca
Cr
Cd
Ti
Al
Sn
Ba
334
104
12
4.0
6.0
4.0
2.0
2.0
exchange of the slower Mg2+ ions. As a consequence of the kinetic coupling, the time-dependent concentration of Co2+ ions, for example, in the M2 sites, [CoM2](t), follows a single exponential behavior:
[CoM2](t) ) ([Co0M2] - [Co∞M2]) · exp(-t/τ) + [Co∞M2]
(5) 2+
and similarly for the concentration of Co ions in the M1 sites, where τ is the relaxation time for the cation concentration relaxation upon a change in temperature. To a good approximation, the inverse relaxation time, 1/τ, is finally given, for example, in jumps from a high temperature to a lower temperature, by
(
∞ 1/τ = k3[V′′∞ M1][MgM2]
1 + [Mg∞M1] 1 1 1 + + ∞ ∞ [MgM2] [CoM1] [Co∞M2]
)
(6)
2.2. Modeling of Optical Relaxation Experiments. On the basis of the above indicated theoretical analysis of cation exchange between M1 and M2 sites, the concentration change of Co2+ ions on both sites can be monitored and analyzed using time-resolved optical spectroscopy. Upon a rapid temperaturejump, it is assumed that the temperature of the olivine sample changes exponentially from the initial to the final value following Newton’s law of cooling and heating. The temperature change of the olivine sample is instantly reflected by the absorption coefficient ε due to its electronic nature. To a first approximation, ε is linearly dependent on temperature for d-d electronic transitions in a centrosymmetric site.24 Finally, according to the Beer-Lambert law, see section 3.2, the timedependent absorbance in a relaxation experiment after a temperature-jump is given by
A(t) ) ε(t) · c(t) · d )
[ε
∞
( σt )] · [c
+ (ε0 - ε∞) exp -
∞
( τt )] · d
+ (c0 - c∞) exp -
(7) where σ is the relaxation time of the absorption coefficient ε after the temperature-jump, which is a direct reflection of the temperature change; τ is the relaxation time of the concentration of Co2+ ions in M1 or M2 sites. ε0, c0 and ε∞, c∞ are 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. d is the thickness of the sample. 3. Experimental Section 3.1. Crystal Growth and Characterization. The cobaltcontaining olivine single crystal investigated in this study was grown by an automated Czochralski technique in an atmosphere of 2 vol % CO in CO2. As starting materials, the 4N and 5N pure metal oxides were dried, mixed in the stoichiometric
Figure 2. Schematic illustration of the experimental setup for the temperature-jump relaxation experiments, including a modified optical spectrometer, an external optics including lenses, mirrors, a chopper, and a furnace with controllable atmosphere. A CO2 laser is used for introducing temperature-jumps.
composition (Co0.35Mg0.65)2SiO4, and melted in an inductively heated iridium crucible of 38 mm inner diameter. Pure single crystalline forsterite, Mg2SiO4, oriented along [100] was used as a seed, and the crystal was rotated at 10 rpm and pulled with a rate relative to the melt level of 1.0 mm/h. Chemical element analysis of the grown crystal was carried out using an inductively coupled plasma spectrometer IRIS Intrepid HR Duo (Thermo Electron). The measured element concentrations yield the chemical composition of the grown crystal as (Co0.213Mg0.787)2SiO4. The crystal was oriented using the Laue backscattering technique, and the unit cell parameters are a ) 4.7627 Å, b ) 10.2178 Å, and c ) 5.9838 Å. Thin sections of 4 × 4 × 0.4 mm cut perpendicular to the [001] direction were polished on both sides and used as absorbers for the optical experiments. The concentrations of trace impurities in the single crystal (Co0.21Mg0.79)2SiO4 were determined using laser ablationinductively coupled plasma mass spectrometry (LA-ICPMS). The LA is a GeoLas system (MicroLas) working with a ComPex 110 Excimer Laser (Lambda Physik) at 193 nm (ArF) and 120 µm pit size at about 3 J/cm2. The ICPMS is a DRC II (PerkinElmer). Table 1 lists the concentration of major cationic trace impurities in the sample. 3.2. High Temperature Optical Spectroscopy and Relaxation Experiments. The experimental quantity determined in the optical experiments is the absorbance A
()
A ) log
I0 I
(8)
of the sample, where I0 and I denote the intensities of incoming and transmitted light, respectively. Allowing for absorption according to Beer-Lambert’s law and for a contribution due to the reflectivity of the sample surfaces, AR, the experimental absorbance A is given by A ) εcd + AR with AR ) -2 log(1 R) and R ) ((n - 1)/(n + 1))2, where n is the refractive index. Measurements have been performed using an experimental setup as shown in Figure 2, which consists of a modified commercial UV/vis/NIR double beam spectrometer (Perkin-Elmer, Lambda 9), an external optics to avoid the influence on the absorbance of unmodulated thermal radiation emitted by the sample and by the furnace at high temperatures, a CO2 laser (Synrad, Model 48-2) for generating temperature-jumps, and a computer for data acquisition. For the relaxation experiments, the sample was
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Figure 3. Optical absorption spectra of (Co0.21Mg0.79)2SiO4 in air between 10 000 and 30 000 cm-1 at 25 °C and at temperatures up to 900 °C, showing the temperature evolution of the equilibrium spectra characterized by band broadening and shifts to lower energies at high temperatures. Arrows M1 and M2 indicate the energies selected for performing temperature-jump relaxation experiments on the M1 and M2 sites, respectively.
heated in the homemade furnace to the temperature of interest, T1, which is measured using a type S thermocouple placed in the vicinity of the sample. The sample can then be heated using the CO2 laser (beam size φ e 3 mm) to a higher temperature, T2 ) T1 + ∆T. ∆T depends on the heating power of the laser and was determined spectroscopically to be about 80 °C throughout this study. Temperature-jumps between T1 and T2 can be introduced by opening and closing a shutter in front of the laser. In this way, the sample is heated and cooled rapidly, and thermal equilibrium of the sample after temperature-jumps is reached within a few seconds as confirmed in our experiments. To define the silica activity of the sample, the sample was kept in contact with a piece of quartz glass in the sample holder during all of the experiments. The area of the sample over which optical spectra are collected is constrained by the hole of the sample holder (φ ) 3 mm). Temperature-jump relaxation experiments were performed with T1 ranging from 500 to 700 °C and in three different atmospheres within the stability region of olivine, that is, air, nitrogen, and a gas mixture of 0.3 vol % CO in CO2. In each atmosphere, the sample was equilibrated for about 2 days before measurements. The oxygen activity was measured using an external oxygen sensor (Metrotech, type A15N) in the outlet of the furnace. 4. Results 4.1. Optical Spectra at High Temperatures. The optical absorption spectra of the (Co0.21Mg0.79)2SiO4 single crystal at room temperature and between 500 and 900 °C are shown in Figure 3. 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 seen in the high temperature equilibrium spectra due to vibronic coupling on both sites, which is especially strong for the centrosymmetric M1 position. According to the work of Ullrich et al.,15 who studied polarized electronic absorption spectra of cobalt-containing olivines between 20 and 1000 °C, the observed transitions are well understood as being due to ligand field transitions of the Co2+ ions on M1 as well as on M2 sites. The assignment of the absorption bands to the two different sites provides the basis for the choice of transitions or spectral regions dominated by Co2+ either on M1 sites or on 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 f 4A2g electronic transition of Co2+ ions
Figure 4. Time-dependent absorbance relaxation curves and fits according to eq 7 for the M1 site (a) and M2 site (b) upon temperaturejumps between 520 and about 600 °C in air. The time-dependent evolution of absorbance is composed of a rapid change and a slow relaxation process after temperature-jumps. Relaxation times of about 4400 s at 520 °C and 130 s at about 600 °C are obtained for the cation concentration relaxation process for both sites.
in the M2 site.15,16 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 complex band system between about 14 000 and about 24 000 cm-1 at high temperatures, which is not resolved in detail in Figure 3, was found to be formed by the superposition of absorption bands of Co2+ ions on M1 and M2 sites.15 The absorption band centered at about 21 400 cm-1 at room temperature has been assigned to an electronic transition of Co2+ ions located on M1 sites. However, due to the high absorbance at this energy, measurements on the M1 site are performed at 23 300 cm-1 (430 nm), which is located in the M1-dominated high-energy wing of the band centered at 21 400 cm-1 at room temperature. 4.2. Temperature-Jump Relaxation Experiments. Figure 4a and b shows the time-dependent absorbance of Co2+ ions on M1 and M2 sites in relaxation experiments in air with temperature-jumps between 520 and about 600 °C, respectively. In both cases, the time-dependent variation of absorbance, A, is composed of a rapid change and a slow relaxation process after temperature-jumps. The sudden changes in absorbance are caused by the optical absorption coefficient ε, eq 7. The subsequent slow relaxation of absorbance is attributed to the concentration changes of Co2+ ions in the respective sites. For the M1 site, Figure 4a, with a temperature-jump from 520 to 600 °C, the absorbance of Co2+ ions in the M1 site almost instantaneously jumps from 0.872 to 0.930 and then slowly decreases with time until a final equilibrium value of 0.923 is reached. On the other hand, in the temperature drop from 600 to 520 °C, a slow increase in absorbance is observed with time from 0.865 to 0.872, following the rapid decrease in absorbance from 0.923 to 0.865. For the M2 site, Figure 4b, the time
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Figure 5. Time-dependent absorbance relaxation data from experiments on the M2 site in air at (a) T1 ) 550 °C and (b) T1 ) 600 °C, showing the temperature dependence of cation exchange processes. Fits of eq 7 to the data yield relaxation times of about 1420 s at 550 °C and of about 180 s at 600 °C for the concentration relaxation process.
Figure 6. Time-dependent absorbance relaxation curves and fits according to eq 7 for the M1 site (a) and M2 site (b) upon temperaturejumps between 700 and 780 °C in air. Note that the relaxation time of cation exchange processes at 700 °C is about 6 s, and at temperatures above 700 °C it approaches that of the absorption coefficient, ε.
evolution of absorbance due to the concentration changes shows a trend opposite of that observed for the M1 site in the corresponding temperature-jumps between 520 and 600 °C. Altogether, the time evolution of absorbance of Co2+ ions in both sites upon temperature-jumps unambiguously indicates that the concentration of Co2+ ions in the M2 sites increases with temperature, which is paralleled by the corresponding decrease in concentration of Co2+ ions in the M1 sites. Fitting the kinetic model eq 7 to the experimental relaxation data measured on the M1 sites, one obtains relaxation times τ (τ2 in Figure 4a) at 600 °C of about 120 s and σ (σ2 in Figure 4a) of about 1 s. At 520 °C, on the other hand, relaxation times (τ1 in Figure 4a) of 4400 s for the concentration relaxation and of about 1 s for the absorption coefficient σ (σ1 in Figure 4a) are determined from modeling the experimental data of the temperature-jump from 600 to 520 °C. Similarly, the relaxation times of cation exchange processes recorded for the M2 site, Figure 4b, are 144 s at 600 °C and 4370 s at 520 °C. The relaxation times for the cation distribution obtained from both sites are in fair agreement, yielding average values of about 130 s at 600 °C and of about 4400 s at 520 °C. Figure 5 displays experimental relaxation curves at T1 ) 550 and 600 °C on the M2 sites and fits according to eq 7. At T1 ) 700 °C, we found that the characteristic time, τ, of the concentration relaxation is only about 6 s, which approaches the relaxation time σ of about 3 s, Figure 6. This then determines the high temperature limit for the kinetic studies on this sample using the present experimental setup. The fast kinetics at temperatures above 700 °C found in this study is consistent with a theoretical study on Fe-Mg olivine25 and our previous studies on Co-Mg olivines of other compositions.17-19 Figure 7 shows experimental relaxation curves collected on the M2 sites at T1 ) 620 °C in various atmospheres of different
oxygen activity. As can be seen, the time constants for the cation concentration relaxation do not vary with the oxygen activity as expected. They are all about 100 s, although the atmosphere varies from air (log aO2 ) -0.68) to N2 (log aO2 ) -3.84) and to a more reducing one of CO/CO2 (log aO2 ) -18.7). This observation of an independence of relaxation kinetics on oxygen activity implies that the concentration of cation vacancies in the sample does not significantly change, eq 6, in the temperature range studied. 5. Discussion 5.1. Activation Energy of the Octahedral Cation Exchange Reaction. The temperature-jump-induced cation exchange kinetics in air was studied in the temperature range from T1 ) 500 to 700 °C for both M1 and M2 sites. The relaxation time, τ, is found to span a range from about 12 300 s at 500 °C to about 6 s at 700 °C, thus showing significant temperature dependence. Figure 8 displays the temperature dependence of the inverse relaxation time for the cation exchange reaction in an Arrhenius plot. The measurements on the M2 and M1 site yield activation energies, Ea, of 212 ( 4 and 250 ( 11 kJ/mol, respectively, with a mean value of 230 ( 12 kJ/mol and a pre-exponential factor in the order of 1010 s-1. At 800 and 1000 °C, the relaxation time of cation exchange in air obtained by extrapolation is found to be about 1.0 and 0.02 s, respectively. 5.2. Point Defect Model. The most abundant point defects in Co-containing olivines are cation vacancies, V′′M, and electron • , which are produced by oxidation of Co2+ ions in holes, CoM the cation lattice.20 There may be additional minority defects, • ′′ ′ • such as Co′Si, V′′′′ Si and associates of type (CoMVM)′ and (CoSiCoM). If the concentration of minority defects is negligible, the charge ′′ • ] ) [CoM ]. In our neutrality in the crystal is given by 2[VM experiments, the single crystal sample was in contact with SiO2
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glass. Thus, the formation reaction of the majority point defects in olivine (CoxMg1-x)2SiO4 is given by
6Co×M + SiO2 + O2(g) ) 2V′′M + 4Co•M + Co2SiO4 (9) The corresponding mass law for reaction 9 is
K9 )
[V′′M]2[Co•M]4aCo2SiO4 [Co•M]6aSiO2aO2
(10)
By using the electrical neutrality condition, one obtains
[V′′M] ) (4K9)1/6x5/6aO1/62
(11)
As seen, at a given temperature and composition x, the vacancy concentration [V′′M] is proportional to the power of 1/6 of the oxygen
Figure 8. Arrhenius plot of the inverse relaxation time, τ-1, between 500 and 700 °C for (Co0.21Mg0.79)2SiO4 in air. The “4” and “9” are data from relaxation experiments in air on M1 and M2 sites, respectively. The line shown is the linear fit of the data points for M2 sites, which yields an apparent activation energy for the cation exchange process of 213 ( 4 kJ/mol and a prefactor in the order of 1010 s-1. The linear fit of the data points from both M1 and M2 sites gives an activation energy of 230 ( 12 kJ/mol.
activity in the atmosphere. Although computational studies show ′′ that vacancies on M1 sites, VM1 , have a smaller formation energy ′′ than the ones on M2 sites, VM2, in olivines, there is no information on the distribution of vacancies on the two nonequivalent M1 and M1 sites.26,27 Assuming that the overall concentration of vacancies remains constant in the course of a temperature-jump relaxation experiment, the inverse relaxation times of the cation exchange reaction, eq 6 is expected to vary with aO2 as the equilibrium vacancy concentration, eq 11. Schwier et al.20 experimentally determined the cation vacancy concentration in (CoxMg1-x)O and (CoxMg1-x)2SiO4 using highpurity powders. For (Co0.2Mg0.8)2SiO4, for example, at 1100 °C in ′′ ] ) 1.8 × 10-5 per formula unit. Using this air, they obtained [VM value together with the temperature dependence of the equilibrium constant K9 derived for the (CoxMg1-x)O system, the vacancy concentration in a pure (Co0.21Mg0.79)2SiO4 olivine at 600 °C in air can be estimated to be about 2.56 × 10-6. This vacancy concentration in the temperature range of our optical relaxation experiments is comparable to or even smaller than the impurity concentrations in the sample. Therefore, the effect of impurity ions on the point defect chemistry has to be taken into consideration. The alivolent cations Cr3+, Al3+, Ti4+, and Sn4+, however, may occupy octahedral sites, and they are the most likely trace elements to influence the vacancy concentration in the single crystal (Co0.2Mg0.8)2SiO4 olivine. Because little is known about the substitution on the sites, we simply assume an effective impurity (EIP) possessing 3+ valence. Under this condition, the charge neutrality equation is now given as
2[V′′M] ) [Co•M] + [EIP•M]
(12)
and eq 10 is to be written as
K9 )
Figure 7. Time-dependent absorbance relaxation curves and fits according to eq 7 for the M2 site at T1 ) 620 °C in (a) air, log aO2 ) -0.68, (b) N2, log aO2) -3.84, and (c) CO/CO2, log aO2 ) -18.70. Note that the relaxation times of cation exchange processes at 620 °C in the three atmospheres are about 100 s and that no oxygen activity dependence is observed.
[V′′M]2(2[V′′M] - [EIP•M])4aCo2SiO4 [Co•M]6aSiO2aO2
(13)
For [EIP3+] ) 15 × 10-6, the vacancy concentration according to eq 13 on the octahedral cation sites as a function of oxygen activity is shown in Figure 9 at 1100 and 600 °C. As can be seen, the ′′ becomes less aO2 dependent at 600 °C than concentration of VM at 1100 °C in the oxygen activity range from air to N2 and further
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′′ Figure 9. Cation vacancy concentration, [VM ], in the (Co0.21Mg0.79)2SiO4 single crystal as a function of oxygen activity, log aO2, at 1100 and 600 °C calculated using data from ref 20 and impurity levels in the sample, Table 1. Note that the change in [V′′M] at 600 °C from -0.68 (air) to -3.84 (N2) or a more reducing atmosphere is less than 10%.
′′ to a more reducing environment. The concentration of VM varies less than 10% at 600 °C when the atmosphere is changed from air to N2, and then remains constant in more reducing atmospheres. In our relaxation experiments, the kinetic process is reflecting the equilibrium vacancy concentration on the respective sublattices, ′′∞ ′′∞ or VM2 , eq 6. In conclusion, calculations including impurities VM1 can explain the experimental observations that the cation exchange kinetics becomes virtually independent of aO2 in the temperature range studied due to the fact that the change in the concentration ′′ ′′ or VM2 is too small to be reflected in the relaxation of VM1 experiments. 5.3. Estimation of Cation Diffusion Coefficient. A site exchange of divalent cations in the olivine crystal structure represents also a fundamental step for cation diffusion. Diffusion along the c-axis in olivine via cation jumps between neighboring M1 sites is predicted to have the lowest migration energy and the fastest rate.27 From lattice geometry, it is obvious that cation diffusion in the b-direction is due to the cation site-exchange between M1 and M2. Thus, the intersite exchange process is associated with cation diffusivity along this axis. In the following, we will correlate the experimental relaxation times, τ, with the magnesium self-diffusion coefficient in the b-direction, DMg b , derive an estimate for the interdiffusion coefficient along this direction, Db, and compare it with the literature data on Co-Mg interdiffusion in (CoxMg1-x)2SiO4 for x ) 0.2. The Mg self-diffusion coefficient, DbMg along the b-axis in olvine is given by
DMg b )
1 Mg 2 ·Γ ·l 2 b
(14)
where l ) b/4 and ΓMg b are the jump distance and the average jump frequency of Mg2+ ions along the b-axis, that is, the frequency of sublattice exchange, respectively. Denoting the frequency of jumps originating from M1 (M2) sites by ΓM1 (ΓM2), the average jump frequency of Mg2+ ions along the b-axis is ΓbMg ) ΓM1 + ΓM2. To extract relations between ΓbMg and τ from our kinetic data, eq 4b is rewritten in terms of the average sublattice exchange frequencies, ΓM1 and ΓM2:
d[MgM2] ) -k3[MgM2][V′′M1] + k4[MgM1][V′′M2] dt ) -ΓM2[MgM2] + ΓM1[MgM1]
(15)
Figure 10. Interdiffusion coefficients, Db, for diffusion along the b-axis in air calculated from relaxation times, τ (Figure 8), and extrapolation to high temperatures in comparison to interdiffusion data, Dc, for diffusion along the c-axis in air in (Co0.2Mg0.8)2SiO4 from ref 21.
Taking advantage of the equilibrium conditions, which is implicitly assumed in eq 14:
ΓM2[Mg∞M2] ) ΓM1[Mg∞M1] and ΓM2 ) k3[V′′M1∞], etc. one obtains ′′∞ ΓMg b ) k3 · [VM1] · [Mg] ·
1 [Mg∞M1]
(16)
where [Mg] denotes the total concentration of Mg2+ ions. By making use of eqs 6, 14, and 16, the self-diffusion coefficient of Mg2+ along the b-axis is
DMg b )
(
1 1 1 1 2 [Mg] · · + + ·l · ∞ ∞ ∞ 2 τ [MgM1][MgM2] [MgM1] [Mg∞M2] -1 1 1 + (17) ∞ ∞ [CoM1] [CoM2]
)
For the calculation of DMg b with eq 17, the equilibrium cation distribution at about 800 °C reported by Mu¨ller-Sommer et al.10 for a similar composition of cobalt-containing olivine was used for our temperature range. Using the extrapolated relaxation time at 800 °C, τ ) 1.0 s, and l ) b/4 ) 2.55 × 10-10 m, DbMg at 800 °C is estimated to be of the order of 10-21 m2/s. The interdiffusion coefficient along the b-axis, Db, in (CoxMg1-x)2SiO4 olivines can be expressed in terms of the tracer diffusion coefficients, DbCo* and DbMg*, assuming nearly ideal behavior of the olivine solid solution.22
Db )
Mg* DCo* b · Db
x · DCo* + (1 - x) · DMg* b b
(18)
There is, however, no information on the ratio of tracer diffusion coefficients, DbCo* and DbMg*, in olivine. Using a ratio of about 10 as observed in the case of the (Mg, Co)O system,22 one obtains from eq 18 that Db is about 2.6 times of DbMg*. Using DbMg ) 4.78 × 10-23 m2/s at 600 °C from eq 17 and Figure 10, one arrives at Db ) 9.33 × 10-23 m2/s, where a correlation factor fb of 0.745 has been used to approximately account for geometrical correlation in the olivine structure.22 Figure 10
6274 J. Phys. Chem. C, Vol. 113, No. 15, 2009 shows the calculated Co-Mg interdiffusion coefficient Db from the present work and the one along the c-axis, Dc, as determined by Morioka21 using a couple annealing method. As can be seen, in the temperature range from 1150 to 1450 °C, the interdiffusion value Dc is about 6 times larger than Db obtained by extrapolation of our low temperature data. This difference of Db and Dc agrees to a remarkable degree with the behavior shown by the experimental diffusion data in Fe-Mg olivines along the b- and c-axes.28-30 6. Conclusions We have reported a kinetic study on octahedral cation sublattice exchange in a (Co0.21Mg0.79)2SiO4 olivine single crystal by monitoring the time evolution of optical absorbance on the two nonequivalent octahedral M1 and M2 sites upon temperature-jumps. The temperature-jump-induced relaxation on both sites shows conclusive evidence that the concentration of Co2+ ions on the M2 site increases and that on M1 sites decreases with increasing temperature. The absorbance relaxation curves from experiments on both octahedral sites can be modeled using a kinetic equation based on the vacancy mechanism for cation exchange predicting that the concentration change in the octahedral sites follows a single exponential behavior. Temperature-jump relaxation experiments were performed from 500 to 700 °C and in three different atmospheres. The relaxation times of cation exchange processes were found to span a range from about 6 s at 700 °C to about 12 300 s at 500 °C, thus showing a strong temperature dependence with an activation energy of about 230 ( 12 kJ/mol. No oxygen activity dependence was observed in our experiments. This observation can well be explained by the quantitative evaluation of the concentration of cation vacancies in the octahedral sites using literature data and trace impurity levels in the single crystal, which resulted in an almost constant vacancy concentration in the temperature range studied. A quantitative relation between the relaxation time and the Mg self-diffusion coefficient, DMg b , along the b-axis in the olivine structure has been derived in this work. This allows one to estimate the Co-Mg interdiffusion coefficient in the b-direction, Db, for example, Db ) 9.33 × 10-23 m2/s at 600 °C. Thus, the optical relaxation technique provides a new set of diffusion data in cobalt-containing olivines in the low temperature range. Acknowledgment. Financial support from the German Research Foundation (DFG) is greatly acknowledged. Albert
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