Order and anti-order in olivine II: Thermodynamic analysis and crystal-chemical modelling

Abstract

The equilibrium order/anti-order behaviour in olivine Fe 0.48 Mg 0.52 [SiO 4 ] is analysed in terms of the Thompson (1969, 1970) model for the Gibbs energy due to ordering, G ord , \[G^{ord} \frac{1}{2}{\Delta}G^{0}_{exch} Q {-} TS^{ord}_{conf}.\] ΔG 0 exch = ΔH 0 exch − TΔS 0 exch relates to the exchange reaction Fe M2 Mg M1 ↔ Fe M1 Mg M2 . Since for the investigated olivine both ΔH 0 exch and ΔS 0 exch are positive (ΔH 0 exch = 1.2 kJ/mol, ΔS 0 exch = 3.7 J/mol K), an ordered Fe 2+ ,Mg configuration is favoured by the enthalpic part of ΔG 0 exch whereas the vibrational entropic part favours anti-ordering. As a result, at low temperatures, where ΔH 0 exch > TΔS 0 exch , Fe 2+ prefers M2. Since, however, the energy TΔS 0 exch steadily increases with increasing temperature it promotes Fe 2+ into M1 and full disorder is attained at a crossover temperature T co where ΔH 0 exch = T co ΔS 0 exch . Above T co , TΔS 0 exch becomes progressively larger than ΔH 0 exch and stimulates further fractionating of Fe 2+ into M1 corresponding to increasing anti-order. The unusual phenomenon of anti-order increasing at increasing temperatures is due to ΔH 0 exch being relatively small in FeMg olivine compared to the temperature proportional energy TΔS 0 exch . 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 ΔG 0 exch 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 M 2 [SiO 4 ] 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 ΔH 0 exch 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 ΔH 0 exch . In MnMg olivine, it is the radius ratio that dominates, contrary to CoMg and NiMg olivine where the electronegativity ratios control ΔH 0 exch . Consequently, Mn prefers M2, and Co and Ni segregate into M1. ΔS 0 exch can be split into vibrational, ΔS 0,vib exch , and electronic exchange entropies, ΔS 0,el exch . Describing the first in terms of a new octahedral distortion parameter, D f , and estimating the second from the Boltzmann distribution of the 3d-electrons, ΔS 0 exch 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 lnK D = −(ΔH 0 exch − T ΔS 0 exch )/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 lnK D allows also for predicting the temperature variation of equilibrium cation distributions in AB olivines not investigated so far.