Grain boundaries with impurities in a two-dimensional lattice-gas model

Abstract

The two-dimensional lattice-gas model used previously to study grain-boundary structure and thermodynamic properties of a single-component system is extended to two components. In addition to the pair-wise potential interactions ${\epsilon }_{4A}$ and ${\epsilon }_{5A}$ used previously for two different interaction distances in a pure system, two interaction parameters Δ${\epsilon }_4$ and Δ${\epsilon }_5$ are introduced; they are sufficient to display a rich set of grain-boundary phenomena, while maintaining ideal solutions in the crystals away from the boundary. The ratio of the parameters Δ${\epsilon }_4$ and Δ${\epsilon }_5$ permits the selection of which of the observed sites in the grain boundary are occupied at low temperatures. Keeping the magnitude of Δ${\epsilon }_4$ and Δ${\epsilon }_5$ small compared to ${\epsilon }_{4A}$ and ${\epsilon }_{5A}$ permits the approximate uncoupling of chemical change (i.e., exchange of atomic species) and structural change (similar to the changes in the single-component system) in the boundary. As a result the low-temperature structure and the high-temperature structure can be clearly distinguished. In the low-temperature regime the structure remains unchanged, but exchange of atoms with the bulk changes the adsorption. On any given site the magnitude of adsorption decreases with increasing temperature, but with positive and negative adsorption on different sites the total adsorption can change sign. At high temperature, adsorption increases logarithmically as the melted layer (which is enriched by segregation) increases in thickness. The concept of the grain-boundary melting transition, introduced in our previous paper, remains and now is defined as the transition from the low-temperature structure to the high-temperature structure. The transition region is gradual as before but can be clearly identified based on the impurity adsorption behavior.