Spinodal curves and critical points in mixtures containing polydisperse polymers with many components

Abstract

In this paper, we report on a new method for computing stability limits (i.e., spinodal curves) and critical points in mixtures containing polydisperse polymers. At points on the spinodal curve, the determinant of a matrix is zero. When the model is an equation of state in temperature, volume and the mole numbers of C components, the matrix is of dimension C by C. Polymer characterization may entail use of very many pseudo-components, thereby complicating evaluation of the necessary determinant. The new analysis in this paper shows that the same result can be obtained by zeroing a determinant with only two dimensions required for a polymer of a given type, regardless of the number of pseudo-components used to characterize the polymer. The analysis does not rely on algebraic operations seeking closed-form evaluation of the determinant. It applies to mixtures with any number of non-polymer species and with more than one type of polymer; features not possible with earlier methods.Critical points are on the stability limit and satisfy a second condition. In this paper, the second criticality criterion is evaluated through an easily obtained third-order numerical derivative of the Helmholtz free energy.