HETEROGENEOUS AND HOMOGENEOUS CRITICAL-POINTS OF POLYMER DISTRIBUTIONS

General equations for the convergence of more than two phases (heterogeneous multiple critical points) and the merging of critical points associated with phase regions (homogeneous multiple critical points) are developed in the first half of this paper and are applied to three polymer distributions in the second half. At a heterogeneous point the spinodal surface is locally translation invariant along a local zero eigenvector field of the stability matrix and at a homogeneous point the gradient of the spinodal surface is null. Local invariance is extended to (1) field parameters (as temperature or pressure) along the streamlines of the zero eigenvector field on the spinodal surface and to (2) the classical approach with Legendre transform free energy of one composition variable. The heterogeneous critical conditions of Solc for a mixture of two homopolymers with molecular weight distribution are confirmed. For a copolymer with mer distribution, or equivalently a mixture of regularly interacting components, these equations are just successive moment expressions of a Gaussian distribution. The last example concerns the local destabilization of a homopolymer/copolymer mixture by copolymer mer distribution. Solc's formalism can be extended to match these results also. Homogeneous multiple critical points are forbidden by molecular weight distribution in the first example and by a mer distribution that represents more than two components in the last two examples. A simple Landau expansion describing the local convergence of phases and critical points about a ternary tricritical point emphasizes the invariance of the critical point hierarchy with eigenvector scaling that was employed in developing all examples.