The concept of thermodynamic excess at interfaces in multi-component systems was generalized to crystalline materials in such a way that it became useful for practical atomistic calculations. It was necessary to terminate the region around a defect at the atomic scale in a manner which was consistent with macroscopic thermodynamics, and this problem was solved by introducing a tapered termination. This was equivalent to averaging over an ensemble of terminations which was distributed broadly at the atomic scale. This approach led to a unique definition of excess which generalized the concept of a charged or polar interface. Five cases were considered. These were grain boundaries, an heterogeneous interface with more than 2 components, an heterogeneous interface in a 2-component system, free surfaces, and dislocations. General formula were also derived for the excess free energies of extended planar defects, and their dependence upon the chemical potentials and excesses of the components.

Accessing the Excess - an Atomistic Approach to Excesses at Planar Defects and Dislocations in Ordered Compounds. M.W.Finnis: Physica Status Solidi A, 1998, 166[1], 397-416