Phase-Boundaries near Critical End points: Applications to Cross-linked Copolymers

TL;DR

This paper investigates the phase boundaries near critical end points in cross-linked copolymer blends, revealing nonanalytic behaviors and providing explicit expressions for phase boundaries close to end points.

Contribution

It offers a detailed analysis of phase boundaries near critical end points in cross-linked copolymer systems, including explicit formulas and singularity checks.

Findings

Identification of continuous and first-order phase boundaries

Explicit expressions for phase boundaries near end points

Detection of nonanalyticities at critical end points

Abstract

The phase behavior of a cross-linked polymer blend made of two incompatible species, $A$ and $B$, of different chemical nature is analyzed. Besides a homogeneous phase, this system also exhibits two microphases and a phase of total segregation. The transition between the homogeneous and the microphase is continuous along a $\lambda$-line; a first-order phase boundaries separate the microphase and the disordered phase from the phase of complete segregation. The critical line meets the first-order phase boundaries at an end point. Scaling arguments indicate that, close to any end point, the equations for the first-order phase boundaries exhibit nonanaliticities associated with the singularities present at the thermodynamic functions near the critical line. Explicit expressions for the phase boundaries near the end point for a cross-linked polymer mixture are obtained and checked for singularities.