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. An expansion of the free energy to first order in alpha yields an excess free energy of mixing per monomer of the form alpha z(N)phi(A)phi(B) 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 (alpha=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(infinity)[1+beta(N) over bar (-1/2)], where (N) over bar is an invariant degree of polymerization and beta= (6/pi)(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 -> infinity. (C) 2009 American Institute of Physics. [DOI: 10.1063/1.3108460]