On the chain length dependence of local correlations in polymer melts and a perturbation theory of symmetric polymer blends

The self-consistent field (SCF) approach to the thermodynamics of dense polymer liquids is based on the idea that short-range correlations in a polymer liquid are almost independent of how monomers are connected into polymers over larger scales. Some limits of this idea are explored in the context of a perturbation theory for symmetric polymer blends. We consider mixtures of two structurally identical polymers, A and B, in which the AB monomer pair interaction differs slightly from the AA and BB interactions by an amount proportional to a parameter α. An expansion of the free energy to first order in α yields an excess free energy of mixing per monomer of the form αz (N) φA φB in both lattice and continuum models, where z (N) is a measure of the number of intermolecular near neighbors per monomer in a one-component (α=0) reference liquid with chains of length N. The quantity z (N) decreases slightly with increasing N because the concentration of intramolecular near neighbors is slightly higher for longer chains, creating a slightly deeper intermolecular correlation hole. We predict that z (N) =z (∞) [1+Β N ^-1/2], where N- is an invariant degree of polymerization and Β= (6/π) 3/2 is a universal coefficient. This and related predictions about the slight N dependence of local correlations are confirmed by comparison to simulations of a continuum bead-spring model and to published lattice Monte Carlo simulations. We show that a renormalized one-loop theory for blends correctly describes this N dependence of local liquid structure. We also propose a way to estimate the effective interaction parameter appropriate for comparisons of simulation data to SCF theory and to coarse-grained theories of corrections to SCF theory, which is based on an extrapolation of perturbation theory to the limit N→∞.