Polymers in highly confined geometries can display complex morphologies including ordered phases. A basic component of a theoretical analysis of their phase behavior in confined geometries is the knowledge of the number of possible single-chain conformations compatible with the geometrical restrictions and the established crystalline morphology. While the statistical properties of unrestricted self-avoiding random walks (SAWs) both on and off-lattice are very well known, the same is not true for SAWs in confined geometries. The purpose of this contribution is a) to enumerate the number of SAWs on the simple cubic (SC) and face-centered cubic (FCC) lattices under confinement for moderate SAW lengths, and b) to obtain an approximate expression for their behavior as a function of chain length, type of lattice, and degree of confinement. This information is an essential requirement for the understanding and prediction of entropy-driven phase transitions of model polymer chains under confinement. In addition, a simple geometric argument is presented that explains, to first order, the dependence of the number of restricted SAWs on the type of SAW origin.

Recent simulation studies have revealed a wealth of distinct crystal polymorphs encountered in the self-organization of polymer systems driven by entropy or free energy. The present analysis, based on the concept of self-avoiding random walks on crystal lattices, is useful to calculate upper bounds for the entropy difference of the crystals that are formed during polymer crystallization and thus provide predictions on polymorph thermodynamic stability. Here, we compare two pairs of crystals sharing the same coordination number, ncoord: hexagonal close packed (HCP) and face centered cubic (FCC), both having ncoord = 12 and the same packing density, and the less dense hexagonal (HEX) and body centered cubic (BCC) lattices, with ncoord = 8. In both cases, once a critical step length is reached, one of the crystals shows a higher number of SAWs compatible with the crystal. We explain the observed trends in terms of the bending and torsion angles corresponding to the different chain geometry as imposed by the geometric constraints of the crystal lattice.