Complexions in a modified Langmuir-McLean model of grain boundary segregation

The Langmuir-McLean isotherm is often interpreted as providing an approximation to the most probable grain boundary segregation as a function of the bulk mole solute fraction $x_B$, even though $x_B$ is not an independant parameter in the free energy minimization on which it is based. In this paper it is shown that the most probable segregation for a specified $x_B$ differs from the standard Langmuir-McLean relation. Numerical solution of the derived equation suggests that two potentially stable interface compositions are associated with most bulk compositions. One solution represents a state with an excess of solute along the boundary relative to the bulk, while the other represents a deficit. The vacancy content ratio between the interface and the bulk plays a large role in determining the shape of the derived isotherm.