Integral equation theory of the structure and thermodynamics of polymer blends

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Abstract
Our recently developed RISM integral equation theory of the structure and thermodynamics of homopolymer melts is generalized to polymer mixtures. The mean spherical approximation (MSA) closure to the generalized Ornstein–Zernike equations is employed, in conjunction with the neglect of explicit chain end effects and the assumption of ideality of intramolecular structure. The theory is developed in detail for binary blends, and the random phase approximation (RPA) form for concentration fluctuation scattering is rigorously obtained by enforcing incompressibility. A microscopic, wave vector‐dependent expression for the effective chi parameter measured in small angle neutron scattering (SANS) experiments is derived in terms of the species‐dependent direct correlation functions of the blend. The effective chi parameter is found to depend, in general, on thermodynamic state, intermolecular forces, intramolecular structure, degree of polymerization, and global architecture. The relationship between the mean field Flory–Huggins expression for the free energy of mixing and our RISM‐MSA theory is determined, along with general analytical connections between the chi parameter and intermolecular pair correlations in the liquid. Detailed numerical applications to athermal and isotopic chain polymer blend models are presented for both the chi parameter and the structure. For athermal blends a negative, concentration‐dependent chi parameter is found which decreases with density, structural asymmetry, and increases with molecular weight. For isotopic blends, the effective (positive) chi parameter is found to be strongly renormalized downward from its mean field enthalpic value by long range fluctuations in monomer concentration induced by polymeric connectivity and excluded volume. Both the renormalization and composition dependence of the chi parameter increase with chain length and proximity to the spinodal instability. The critical temperature is found to be proportional to the square root of the degree of polymerization in stark contrast to the classical mean field prediction of a linear dependence. Comparison of the theoretical predictions with SANS measurements and computer simulations is presented, along with brief discussions of nonideal effects and lower critical solution temperature phenomena.