Calculation of Critical Points

Miscibility gaps appear in many alloy systems and, consequently, spinodals and critical points. Three methods are presented for determining a critical point in a multicomponent system. The first method is Gibbs thermodynamic conditions for spinodal and critical point. The spinodal is where the determinant of Gibbs energy Hessian equals zero. The critical point is on the spinodal and where the determinant of another matrix, the Gibbs energy Hessian with any one row replaced by the gradient of the determinant of Gibbs energy Hessian, also equals zero. The second method for a critical point uses the fact that the third directional derivative of the Gibbs energy along the eigenvector direction of Gibbs energy Hessian must be zero at a critical point. The third method uses the geometric features of binodal, spinodal, and eigenvector of Gibbs energy Hessian to determine the position of a critical point on a spinodal curve. The conditions for a spinodal and a critical point in a multicomponent system can be expressed in terms of directional derivatives of Gibbs energy, which are the natural extension of the binary conditions. The ternary Al-Cu-Sn alloy system is used as an example to demonstrate how to apply these conditions to determine the critical points and critical lines. A proof is given in Appendix for the gauge invariance of the determinant of the Gibbs energy Hessian.