BEYOND THE MARGULES-EQUATION: A TREATMENT FOR MULTI-COMPONENT SOLID SOLUTIONS
In the simplest case in which the enthalpy of an atom on a certain site is not influenced by any neighbors occupying same site, the formalism reduces trivially to that of a mechanical mixture. If the enthalpy of that atom is influenced by one neighbor then the presented formulation for the excess enthalpy reduces to that of a regular solution. In the case of two influencing neighbors the excess enthalpy comes in the form of the Margules-equation. One can now proceed with more complex environments with more neighbors on the same site or influencing neighbors on other sites. In any case the resulting equations are rather simple, besides they contain multiple summations. They are always of the same type and can be readily set up for any configuration. It is important to note that the formalism is not pairwise additive.
To calculate the entropy the occupancy of each mixing site is considered to be totally random without any clustering. If clustering occurs, the site can be simply split into two.
The same procedure allows the description of excess mixing volumes necessary for calculating the pressure dependence of equilibria and is able to handle excess zero point energies and heat capacities.
The approach is applied to the system calcite-magnesite-siderite. In this layered structure each cation is surrounded by six neighboring positions of the same site on the same layer. In addition these cations have 3 nearest neighbors in the layer above and 3 nearest neighbors in the layer below. So each cation interacts with 12 neighbors.
This example also shows that the formalism has its price. For a carbonate binary 96 constants have to be considered, for a ternary 2349. The necessary parameters can clearly not be calculated using experimental data. Instead frequent force-field or DFT calculations can be used to provide the required input parameters.