The Entropic Penalty of Confining a Chain Polymer into a Very Small Space

Abstract

The confinement of a flexible polymer is thermodynamically unfavorable, because of the reduction in the number of conformational states. The determination of the entropic penalty of confinement into a very small space is an important unsolved problem in polymer statistical mechanics. We present a method for calculating TΔS for the confinement of an elastic polymer of persistence length P when volume exclusion effects are ignored, considering three geometries: (1) parallel planes separated by a distance d; (2) a circular tube of diameter d; and (3) a sphere of diameter d. As d/P drops from 100 to 0.01, TΔS rises from about 0.01kT to about 30kT for both cases (1) and (2), with the cost in the latter case being consistently about twice that for confinement between parallel planes. The entropic penalty for confinement to a sphere is ∼5kT per persistence length, when d = P, in the absence of excluded volume effects. TΔS can be determined fairly easily when chains of finite diameter are confined into thin tubes, or into spheres with diameters on the order of the persistence length. We also show how volume exclusion effects can be determined in other cases. Excluded volume effects can be very large, especially for confinement to spheres.