Mean-field analytical calculation of the segregation profile around grain boundaries and free surfaces in binary alloys

We derive a mean field equation for a [001] twist grain boundary. For the description of the one-site probability function we use the coincident-site lattice symmetry and the displacement shift complete lattice with two order parameters: one corresponding to the direction of the coincident-site lattice supercell lattice vector and one to the [001] direction (normal to the boundary). We then present a new method to obtain an analytical solution to the linearized mean-field equation for the disordered state (T>Tc) using a general arbitrary-range pair potential decaying exponentially with distance. The solution describes the segregation profile (decay lengths and amplitudes) around a grain boundary or free surface. The method reveals several more length scales (compared with an analysis based on a simple nearest-neighbour or next-nearest-neighbour interaction). Their relative amplitudes vary differently with temperature, giving a richer description of the shape of the segregation profile depending on the strength of average interactions between planes on the same side and on either side of the boundary. Analytical results are demonstrated for a free surface and &#135 = 5 grain boundary in Cu3Au. Monte Carlo simulations are performed and compared with analytical results. The profiles are found to be very sensitive to interatomic interaction, a feature also captured by a Monte Carlo simulation performed for a twist &#135 = 5 boundary in Cu3Au, which, at the same temperature, gave qualitatively completely different results for a rigid and a relaxed lattice.