The ideal polymer chain near planar hard wall beyond the Dirichlet boundary conditions

摘要

We present a new ab initio approach to describe the statistical behavior of long ideal polymer chains near a plane hard wall. Forbidding the solid half-space to the polymer explicitly (by the use of Mayer functions) without any other requirement, we derive and solve an exact integral equation for the partition function G(D)(r, r', N) of the ideal chain consisting of N bonds with the ends fixed at the points r and r'. The expression for G(r, r', s) is found to be the sum of the commonly accepted Dirichlet result G(D)(r, r', N) = G(0)(r, r', N) - G(0)(r, r '', N), where r '' is the mirror image of r', and a correction. Even though the correction is small for long chains, it provides a non-zero value of the monomer density at the very wall for finite chains, which is consistent with the pressure balance through the depletion layer (so-called wall or contact theorem). A significant correction to the density pro. le (of magnitude 1/root N) is obtained away from the wall within one coil radius. Implications of the presented approach for other polymer-colloid problems are discussed.