  The self-consistent field (SCF) theory of dense polymer liquids assumes that
short-range correlations are almost independent of how monomers are connected
into polymers. Some limits of this idea are explored in the context of a
perturbation theory for mixtures of structurally identical polymer species, A
and B, in which the AB pair interaction differs slightly from the AA and BB
interaction, and the difference is controlled by a parameter alpha Expanding
the 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 a
measure of the number of inter-molecular near neighbors of each monomer in a
one-component liquid. This quantity decreases slightly with increasing N
because the self-concentration of monomers from the same chain is slightly
higher 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 of
polymerization, and $\beta=(6/\pi)^{3/2}$. This and other predictions are
confirmed by comparison to simulations. 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 coefficients in this perturbation theory
to the limit $N \to \infty$. We show that a renormalized one-loop theory
contains a quantitatively correct description of the $N$-dependence of local
structure studied here.
