Abstract
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.
Highlights
Self-avoiding random walks (SAWs) have long been used in polymer science as one of the simplest and most useful descriptions of polymeric chains
It must be emphasized that the goal of this work is not to achieve high-accuracy values [27,85,86,89,99,100] in the calculation of the critical exponents or the leading or sub-leading correction-to-scaling exponents, but to obtain correlations for c N for chains of moderate length to be used in the understanding of the entropic mechanisms of phase transitions observed in the off-lattice simulations of confined and densely-packed polymers
The values of c N for self-avoiding random walks (SAWs) on lattices restricted to a tube of cross section L × L oriented along the h100i direction are presented in Tables A1–A3 for the simple cubic (SC) lattice, together with their average squared end-to-end distance
Summary
Self-avoiding random walks (SAWs) have long been used in polymer science as one of the simplest and most useful descriptions of polymeric chains. The relative simplicity of SAWs has made them an ideal tool to investigate static and dynamic properties of polymers both analytically and computationally [1,2,3,4,5,6,7]. They have proved useful in the determination of universal behavior and scaling laws for polymer systems ranging from individual chains to melts. In spite of the very simple idea underlying SAWs, comparatively few results have been rigorously solved in a mathematical sense [19]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
