Analytic reference interaction site model-mean spherical approximation theory of flexible polymer blends: Effects of spatial and fractal dimensions

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Abstract
Recently developed methods for obtaining exact and approximate analytical solutions of the reference interaction site model-mean spherical approximation (RISM-MSA) integral equations for liquid mixtures composed of long, flexible polymers are applied to study the critical temperature Tc for phase separation of symmetric isotopic binary blends as a function of degree of polymerization N, spatial dimension D, and fractal dimension df of the individual macromolecules. For ideal random walk coils, the theory predicts a nonclassical behavior given by Tc∝N(D−2)/2 in two and three dimensions, and the classical Flory–Huggins mean field Tc∝N law is recovered in four and higher dimensions. For arbitrary interpenetrating polymeric fractals, the theory predicts Tc∝N(D−df)/df for spatial dimensions below 2df and Flory–Huggins behavior for D>2df. These novel scaling laws for isotopic mixtures are a consequence of a consistent treatment of chain connectivity on all length scales, intermolecular excluded volume, and a short range unfavorable interaction between hydrogenated and deuterated polymers. A general, closure-independent physical argument based on a renormalization of the bare chi parameter by relatively long range correlated fluctuations in the blend is proposed which reproduces all the qualitative predictions of the RISM-MSA integral equation theory. Analogies with nonclassical critical fluctuation effects are established. Application of the analytical approach to purely athermal blends is also presented. The magnitude and composition dependence of the effective chi parameter is found to be a sensitive function of both spatial and fractal dimensions, and also local nonuniversal features. The various theoretical predictions are favorably compared with recent small angle neutron scattering measurements on binary polymer alloys.