Polymer Chain Size from Geodesic Path and Geometrical Bolyai–Lobachevskij Partition Function. Application to Swelling of Macromolecules in Solution and Micellar Growth

Abstract

The Bolyai–Lobachevskij (BL) formula, relating parallelism angle and distance in a Non-Euclidean space, is used to introduce a geometrical partition function. Employing a correspondence between Boltzmann factor and BL characteristic length, allows us to get a simple relation for average size and space curvature, which is the analogy to the equation for the mean energy derived from the ordinary partition function. Due to the equivalence, recently proposed, between a chain molecule in a liquid and a geodesic path in a relativistic space, the equation obtained is expected to be suitable for describing geometrical phenomena in polymer-like networks. Simple applications to swelling of polymer solutions and micellar growth are presented and discussed.