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The self-consistent field (SCF) theory of dense polymer liquids assumes thatshort-range correlations are almost independent of how monomers are connectedinto polymers. Some limits of this idea are explored in the context of aperturbation theory for mixtures of structurally identical polymer species, Aand B, in which the AB pair interaction differs slightly from the AA and BBinteraction, and the difference is controlled by a parameter alpha Expandingthe free energy to O(\alpha) yields an excess free energy of the form alpha$z(N)\phi_{A}\phi_{B}$, in both lattice and continuum models, where z(N) is ameasure of the number of inter-molecular near neighbors of each monomer in aone-component liquid. This quantity decreases slightly with increasing Nbecause the self-concentration of monomers from the same chain is slightlyhigher for longer chains, creating a deeper correlation hole for longer chains.We analyze the resulting $N$-dependence, and predict that $z(N) = z^{\infty}[1+ \beta \bar{N}^{-1/2}]$, where $\bar{N}$ is an invariant degree ofpolymerization, and $\beta=(6/\pi)^{3/2}$. This and other predictions areconfirmed by comparison to simulations. We also propose a way to estimate theeffective interaction parameter appropriate for comparisons of simulation datato SCF theory and to coarse-grained theories of corrections to SCF theory,which is based on an extrapolation of coefficients in this perturbation theoryto the limit $N \to \infty$. We show that a renormalized one-loop theorycontains a quantitatively correct description of the $N$-dependence of localstructure studied here.

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