A new semi-analytical method for phase transformations in binary alloys

In the present paper, a new semi-analytical method is developed to cover a wide range of phase transformation problems and their practical applications. The solution procedure consists of two parts: first, determination of the position of the moving boundary named the homogenous part and second, determination of the concentration named the non-homogenous part. The homogenous part leads to a system of homogenous linear equations, based on the mathematical fact that a homogenous system has a non-trivial solution if the determinant of the coefficient matrix equals zero. This determinant leads to an ordinary differential equation for the moving boundary, and its solution leads to a closed form formula for the position of the moving boundary. The non-homogenous part transforms the governing equations to a non-homogenous linear system of equations, having three unknowns that appear in the concentration profile assumed in the beginning of the proposed method. Solution of the non-homogenous system leads to a value of these unknowns. Once these unknowns are computed, the concentration at any time and at any point can be found easily.