Theory of the Order-Disorder Phase Transition in Cross-Linked Polymer Blends

We consider the system of a network formed by crosslinking a polymer blend. In such a system there is a competition between monomer-monomer interactions and elastic energy which may result in a microphase separation. We first treat the problem by adapting a model due to de Gennes to include the effect of concentration fluctuations `frozen in' by the crosslinking process. This yields an expression for the scattering structure factor which gives finite scattering in the limit of low wavenumber (this has been observed in experiments on crosslinked blends). The expression also shows that the frozen-in fluctuations `seed' the phase transition, so that the structure factor diverges as $(\chi{\rm s} -\chi)^{-2}$ on approaching the spinodal. We then reconsider the problem at a molecular level. We model the network as a blend of interacting chains anchored at either end to fixed points in space. The system is treated using a variant of the Random Phase Approximation (RPA) which deals with the quenched chain-end variables but which does not resort to replica methods. The resulting structure factor has an identical form to that obtained by modifying the de Gennes model, but allows us to investigate the effect of varying composition, crosslink density, and applied strain. We find that the characteristic lengthscale for the early stages of microphase separation is controlled by the least concentrated component or the one with the shortest chain length between crosslinks, depending on which parameter shows the strongest difference between the two chain types. We also find that strain produces a phase separation with wavevectors in the direction(s) of least stretching (or greatest compression)