A Variational Principle Governing the Generating Function for Conformations of Flexible Molecules

The generation of a random walk path under the action of an external potential field has been of interest for decades. The motivation derives largely from the prospect of incorporating the nonlocal excluded volume effect through such a potential in characterizing the statistical behavior of a long flexible polymer molecule. In working toward a continuum mean-field model, a central feature is a partial differential equation incorporating the influence of the potential and governing the generating function for the dependence of end to end separation distance of the molecule on its pathlength. The purpose here is to describe an approach in which the differential equation is recast as a global minimization of a functional. The variational approach is illustrated by an application to familiar configurations, the first of which is a molecule attached at one end to a noninteracting plane barrier in the presence of a uniform potential field. As a second illustration, the generating function is sought for a free molecule for the case in which conformations must be consistent with the excluded volume condition. This is accomplished by adapting a local form of the Flory approach to the phenomenon and extracting estimates of the expected end to end separation distance, the entropy and other statistical features of behavior. By means of the variational principle, the problem is recast into a form that admits a direct, noniterative analysis of conformations within the context of the self-consistent field theory.