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

[+] Author and Article Information
L. B. Freund

Division of Engineering,  Brown University, Providence, RI 02912-9104freund@brown.edu

J. Appl. Mech 74(3), 421-426 (Mar 31, 2006) (6 pages) doi:10.1115/1.2201888 History: Received March 27, 2006; Revised March 31, 2006

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.

Copyright © 2007 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

The dependence of the unknown functions φ(n) and ψ(n) appearing in 26 as determined by means of the variational principle. The deviations from their initial values shows the influence of the excluded volume effect for the case when v=b3

Grahic Jump Location
Figure 2

The influence of the excluded volume of fact on the shape of the generating function for an isolated molecule. The curve shows the generating function g(r,50) for the case in which the excluded volume parameter is v=b3. For purposes of comparison, the corresponding function for the case in which the excluded volume effect is neglected is shown as a dashed curve plotted against the right hand scale, the range of which is approximately seven times the range of the left hand scale

Grahic Jump Location
Figure 3

Estimate of the expected value of the end to end distance ⟨r⟩ and the square root of the expected value of the square of the distance ⟨r2⟩1∕2 for an isolated molecule as determined by 28 for excluded volume v=b3. Also shown (dashed) for comparison is the expected value of radius when v=0



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