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

¹ Institut für Mineralogie, Westfälische Wilhelms-Universität, Corrensstr. 24, 48149 Münster, Germany
² Mineralogisch-Petrologisches Institut, Poppelsdorfer Schloss, 53115 Bonn, Germany

^* Corresponding author, e-mail: kroll{at}nwz.uni-muenster.de

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,

G⁰_exch = H⁰_exch – TS⁰_exch relates to the exchange reaction Fe^M2 + Mg^M1 Fe^M1 + Mg^M2. Since for the investigated olivine both H⁰_exch and S⁰_exch are positive (H⁰_exch = 1.2 kJ/mol, S⁰_exch = 3.7 J/mol K), an ordered Fe²⁺,Mg configuration is favoured by the enthalpic part of G⁰_exch whereas the vibrational entropic part favours anti-ordering. As a result, at low temperatures, where H⁰_exch > TS⁰_exch, Fe²⁺ prefers M2. Since, however, the energy TS⁰_exch steadily increases with increasing temperature it promotes Fe²⁺ into M1 and full disorder is attained at a crossover temperature T_co where H⁰_exch = T_co S⁰_exch. Above T_co, TS⁰_exch becomes progressively larger than H⁰_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 H⁰_exch being relatively small in FeMg olivine compared to the temperature proportional energy TS⁰_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 G⁰_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 () 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 H⁰_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 H⁰_exch. In MnMg olivine, it is the radius ratio that dominates, contrary to CoMg and NiMg olivine where the electronegativity ratios control H⁰_exch. Consequently, Mn prefers M2, and Co and Ni segregate into M1.

S⁰_exch can be split into vibrational, S^0,vib_exch, and electronic exchange entropies, S^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, S⁰_exch can be satisfactorily modelled by

The resulting lnK_D = –(H⁰_exch – T S⁰_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.

Key-words: olivine, order, anti-order, thermodynamic analysis, exchange enthalpy, nonconfigurational entropy, electronic entropy, topological analysis, ionic radius, electronegativity.

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