Regarding that the excess volume in grain boundaries was released as the vacancies which were accommodated by the crystal bulk during grain growth, a free-energy function for binary nanocrystalline solid solution was proposed, based upon the pair-wise nearest-neighbour interactions. The model, for the given composition and temperature, predicts an equilibrium grain size, subjected to a mixed effect due to solute segregation and due to excess vacancies. Furthermore, excess-vacancy-inhibited grain coarsening could be attained, which played a minor role in holding the thermal stability of nanocrystalline alloys, as compared to the effect of solute segregation.

Thermodynamic Stability of Binary Nanocrystalline Alloys: Analysis of Solute and Excess Vacancy. M.M.Gong, F.Liu, K.Zhang: Applied Physics A, 2011, 105[4], 927-34