To address the thermodynamics of solid solutions we need an efficient
technique to sample the configuration space spanned by the possible
arrangements of soluble atoms over the available sites. Solute and solvent
in Mg/Fe-silicates or alumino-silicates are chemically dissimilar. In this
situation, the method of choice is the Cluster Expansion (CE) method. The
alloy is treated as a lattice problem on N sites. For each configuration
one assigns a set of spin variables Si (i=1,2,
, N) to each site, with Si =
1 according to whether site i is occupied by atom A or B. Then the energy
of various configurations are used to parameterize an Ising Hamiltonian to
be used in the estimation of the free energy of any configuration. It is a
well-tested approach that has successfully addressed isovalent
size-matched alloys. However, for systems with large strain relaxations or
heterovalent substitutions, long-range interactions lead to slow
convergence rates. An alternative approach that can deal easily and
exactly with long range interactions mediated by the structural relaxation
is Computational Alchemy (CA) that has been successfully applied to
lattice mismatched semiconductor alloys. Another possibility is to fit in
reciprocal space the long-range part of the interaction. A combination of
standard CE for the short-range chemical interactions and CA or reciprocal
space fit for long range interactions will allow to correctly predict the
energies of arbitrary configurations for our lattice mismatched
substitutional alloys. (Wentzcovitch, de Gironcoli)
