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 d escription 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 latt ice 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 > T-c) usin g a general arbitrary-range pair potential decaying exponentially with dist ance. The solution describes the segregation profile (decay lengths and amp litudes) around a grain boundary or free surface. The method reveals severa l more length scales (compared with an analysis based on a simple nearest-n eighbour 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 interactio ns between planes on the same side and on either side of the boundary. Anal ytical results are demonstrated for a free surface and Sigma = 5 grain boun dary in Cu3Au. Monte Carlo simulations are performed and compared with anal ytical 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 Sigma = 5 boundary in Cu3Au, which, at the same temperature, ga ve qualitatively completely different results for a rigid and a relaxed lat tice.