Mean-field analytical calculation of the segregation profile around grain boundaries 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 CSL symmetry and the DSC lattice with two order parameters one corresponding to the direction of the CSL supercell lattice vector and one to direction [001] (normal to boundary). We then present a new analytical method to obtain an asymptotic analytical solution to the mean-field equation for the disordered state (T>T-c) for a general pair potential decaying exponentially for arbitrary range interactions. The solution describes the segregation profile around the grain boundary (decay lengths and amplitudes) and can also be applied to free surfaces. The method reveals several more pertinent length scales (compared to a simple NN or NNN interaction) whose relative amplitudes vary differently with temperature giving a richer description for 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. Either no fit to experimental data is required (if one knows the potential) or a fit to a known profile at a specific temperature can provide the potential parameters and then profiles can be calculated for all other temperatures. The profile obtained is found to be very sensitive to interatomic interaction, a feature also captured by a MC simulation we performed for a twist Sigma 5 boundary in Cu3Au which at the same temperature gave qualitatively completely different results for a rigid and a relaxed lattice.