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

The equilibrium order/anti-order behaviour in olivine Fe_0.48Mg_0.52[SiO₄] is analysed in terms of the Thompson (1969, 1970) model for the Gibbs energy due to ordering, G^ord,

ΔG⁰_exch = ΔH⁰_exch − TΔS⁰_exch relates to the exchange reaction Fe^M2 + Mg^M1 ↔ Fe^M1 + Mg^M2. Since for the investigated olivine both ΔH⁰_exch and ΔS⁰_exch are positive (ΔH⁰_exch = 1.2 kJ/mol, ΔS⁰_exch = 3.7 J/mol K), an ordered Fe²⁺,Mg configuration is favoured by the enthalpic part of ΔG⁰_exch whereas the vibrational entropic part favours anti-ordering. As a result, at low temperatures, where ΔH⁰_exch > TΔS⁰_exch, Fe²⁺ prefers M2. Since, however, the energy TΔS⁰_exch steadily increases with increasing temperature it promotes Fe²⁺ into M1 and full disorder is attained at a crossover temperature T_co where ΔH⁰_exch = T_co ΔS⁰_exch. Above T_co, TΔS⁰_exch becomes progressively larger than ΔH⁰_exch 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 ΔH⁰_exch being relatively small in FeMg olivine compared to the temperature proportional energy TΔS⁰_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 ΔG⁰_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 M₂[SiO₄] olivines (M = Mn, Fe, Co, Ni, Mg) yielding a very satisfactory description by

Accordingly, the small value of ΔH⁰_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 ΔH⁰_exch. In MnMg olivine, it is the radius ratio that dominates, contrary to CoMg and NiMg olivine where the electronegativity ratios control ΔH⁰_exch. Consequently, Mn prefers M2, and Co and Ni segregate into M1.

ΔS⁰_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, ΔS⁰_exch can be satisfactorily modelled by

The resulting lnK_D = −(ΔH⁰_exch − T ΔS⁰_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.