Abstract

The equilibrium order/anti-order behaviour in olivine Fe0.48Mg0.52[SiO4] is analysed in terms of the Thompson (1969, 1970) model for the Gibbs energy due to ordering, Gord,

 
\[G^{ord}\ =\ \frac{1}{2}{\Delta}G^{0}_{exch}\ Q\ {-}\ TS^{ord}_{conf}.\]

ΔG0exch = ΔH0exch − TΔS0exch relates to the exchange reaction FeM2 + MgM1 ↔ FeM1 + MgM2. Since for the investigated olivine both ΔH0exch and ΔS0exch are positive (ΔH0exch = 1.2 kJ/mol, ΔS0exch = 3.7 J/mol K), an ordered Fe2+,Mg configuration is favoured by the enthalpic part of ΔG0exch whereas the vibrational entropic part favours anti-ordering. As a result, at low temperatures, where ΔH0exch > TΔS0exch, Fe2+ prefers M2. Since, however, the energy TΔS0exch steadily increases with increasing temperature it promotes Fe2+ into M1 and full disorder is attained at a crossover temperature Tco where ΔH0exch = Tco ΔS0exch. Above Tco, TΔS0exch becomes progressively larger than ΔH0exch and stimulates further fractionating of Fe2+ into M1 corresponding to increasing anti-order. The unusual phenomenon of anti-order increasing at increasing temperatures is due to ΔH0exch being relatively small in FeMg olivine compared to the temperature proportional energy TΔS0exch. In other AB olivines (A, B = Mn, Fe, Co, Ni, Mg) the exchange enthalpies are much larger, between 9 and 20 kJ/mol, so that they dominate ΔG0exch to a degree that precludes a crossover from ordered to anti-ordered states up to the melting point.

The exchange enthalpies reported for MnMg, FeMg, CoMg, NiMg and MnFe olivines can be rationalized in terms of cation radius (r) and electronegativity (χ) ratios of the A and B cations. In a novel approach, both radii and electronegativities have been derived from topological analyses of the procrystal electron density distributions of pure M2[SiO4] olivines (M = Mn, Fe, Co, Ni, Mg) yielding a very satisfactory description by

 
\[{\Delta}H^{0}_{exch}\ =\ 252.6({\pm}6.1)\ [r(A)/r(B)\ {-}\ 1]\ {-}\ 75.8({\pm}1.9)\ [{\chi}(A)/{\chi}(B)\ {-}\ 1].\]

Accordingly, the small value of ΔH0exch found for FeMg olivine is a consequence of opposite radius and electronegativity contributions which almost cancel. In MnFe olivine, although both contributions are small, they cooperate resulting in a moderate value of ΔH0exch. In MnMg olivine, it is the radius ratio that dominates, contrary to CoMg and NiMg olivine where the electronegativity ratios control ΔH0exch. Consequently, Mn prefers M2, and Co and Ni segregate into M1.

ΔS0exch can be split into vibrational, ΔS0,vibexch, and electronic exchange entropies, ΔS0,elexch. Describing the first in terms of a new octahedral distortion parameter, Df, and estimating the second from the Boltzmann distribution of the 3d-electrons, ΔS0exch can be satisfactorily modelled by

 
\[{\Delta}S^{0}_{exch}\ =\ 35.76({\pm}\ 0.34)\ {\{}[D_{f}\ (A)^{M1}\ +\ D_{f}\ (B)^{M2}]/[D_{f}\ (B)^{M1}\ +\ D_{f}\ (A)^{M2}]\ {-}\ 1{\}}\ +\ {\Delta}S^{0,el}_{exch}\]

The resulting lnKD = −(ΔH0exch − T ΔS0exch)/RT allows, for the first time and to the best of our knowledge, an exclusively electron density based description of the experimentally observed temperature variations of the site occupancies in AB olivines. This modelling of lnKD allows also for predicting the temperature variation of equilibrium cation distributions in AB olivines not investigated so far.

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