We theoretically study dense polymer solutions under open (capillary and slit) and closed (box) confinement. The theory is formulated for grand-canonical polymers and corrections to the self-consistent mean-field results are discussed. In contrast to the mean-field prediction, we found that the partition function of a labeled chain is affected by confinement even under neutral von Neumann boundary conditions and the chain length distribution is biased to short chains. As the container size increases, the contribution of the transverse excited states to the free energy of a labeled chain is found to approach its bulk value nonmonotonically (through an extremum) for the box and the capillary confinement but not for the slit. So does the confinement free energy of a labeled chain. The confinement energy of the solution is well behaved for open confinement but formally diverges for a closed box in the limit that the average chain length goes to infinity. Counted per chain, the confinement energy of the dense solution is qualitatively weaker than for a single ideal chain under similarly strong confinement (by one power in transverse container size). The container boundary contributes a surface tension to the free energy, which makes the effective monomer-wall affinity more repulsive. This correction increases with the average chain length. If present, edge or vertex singularities also contribute to the grand potential of the solution.
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There are also systems where the bare excluded volume is larger than at the fixed-point in which case short polymer sections, and likely the small scale ξ, are overswollen.14 This happens for flexible chains with strong excluded volume, so-called strong coupling limit, like polystyrene in toluene15 and is typical for standard algorithms in silico.16 In this case, MF never applies within scale ξ. In contrast, polystyrene in hexane17 is in the weak coupling regime.
As was also found for Casimir effects by others.26
