Calculation of Spinodal and Critical Point in Multicomponent Systems

Miscibility gaps appear in many binary alloy systems and extend into multicomponent systems. Spinodal exists within a miscibility gap. Critical points are where a miscibility gap and its spinodal are tangent to each other. Both spinodal and critical points can be determined from the Gibbs energy Hessian as Gibbs presented in 1870’s. Each Gibbs energy Hessian has its eigenvalues and eigenvectors. Spinodal is where the smallest eigenvalue is zero, or the second directional derivative of Gibbs energy along the eigenvector is zero. A critical point is on a spinodal with a constraint that the third directional derivative of Gibbs energy along the eigenvector is also zero. These conditions will be used to calculate spinodal and critical points in multicomponent Systems. Both the second and third directional derivatives of Gibbs energy along the eigenvector of its Hessian have been integrated with Pandat software so that the spinodal and critical points can be calculated for multi-component systems. An example of Al-Bi-Cu-Sn system will be presented to demonstrate the calculation of liquidus spinodal and critical line.