The partitioning of cations between coexisting single- and multi-site phases with application to the assemblages: orthopyroxene-clinopyroxene and orthopyroxene-olivine

Abstract

The partitioning of cations between coexisting phases is a function of the number of energetically distinct cation sites available within each phase and the exchange free energies between the intracrystalline sites, as well as an inter-phase exchange energy, temperature and composition. The form of the function describing partitioning between a single-site phase and a phase containing two or more energetically distinct sites can be deduced from the cation exchange free energies between pairs of sites. In some cases the cation exchange free energies between pairs of sites can be deduced explicitly from the form of the partitioning curve. If a phase, φ 2, containing equal numbers of two energetically distinct sites, α and β, with ideal solution of cations A and B on both sites, coexists in equilibrium with an ideal one-site phase, φ 1, the standard state free energy of intercrystalline exchange between φ 1 and φ 2, ΔG T 0, is ▪, where X A 1 is the mol fraction of species A in 1 measured at X A φ 2 = 1 2 . The standard state free energy of intracrystalline exchange between α and β in φ 2, ΔG E 0, is ▪ for ΔG T 0 ≠ 0, where ( ∂X A φ 1 ∂X A φ 2 )′ is the slope of the observed partition function at X A φ 2 = 1 2 . For ΔG T 0 = 0, Δ E 0 = 2 RT arc cosh [2( ∂X A φ 1 ∂X A φ 2 )′ − 1]. If φ 1 is non-ideal, the expressions for ΔG T 0 and Δ E 0 are somewhat more complex. In the double-site phase, an enthalpy of mixing term arises when intracrystalline partitioning is imposed upon ideal mixing on the individual sites. This enthalpy is directly related to the free energy of exchange between the sites by the expression, H mix = (x A β−X A α) 2 ΔG E 0 , where X A α and X A β are the sublattice mol fractions. A single site-double site model is proposed to explain the Fe-Mg partitioning observed at different temperatures in the natural and experimental assemblages orthopyroxene-clino-pyroxene and orthopyroxene-olivine.