RISM theory of polymer liquids: Analytical results for continuum models of melts and alloys

Exact and approximate analytical solutions to the polymer RISM integral equation theory for melts and alloys are derived for long, flexible Gaussian chains. Comparisons of predictions for the isothermal compressibility and site-site intermolecular pair correlation function with exact numerical results reveals that the simplifications invoked to achieve analytic solutions are surprisingly accurate. A detailed study of the effective chi-parameter and critical temperature of binary isotopic blends of linear chains and copolymers of various microstructures is presented. In three dimensions, novel non-mean field predictions for the scaling of the critical temperature with degree of polymerization and copolymer architecture are found, and a rich dependence of the corresponding prefactor on system-specific features is determined. The breakdown of Flory-Huggins theory is due to a relatively long range, but weak, concentration fluctuation process which is a consequence of the combined influences of chain connectivity, intermolecular excluded volume and dispersion-like interactions. Physical analogies with critical phenomena ideas are developed, and comparisons are made of the theoretical predictions with recent small-angle neutron scattering observations.