Path Integral Approach to a Single Polymer Chain with Excluded Volume Effect

The effect of a polymer chain with excluded volume representing the long-range interaction between segments along the chain is studied using the Feynman path integral method. The main problem is to calculate the mean square distance, < R² >, that is <R^2> = AN^v where N is the degree of polymerization, v is a scaling exponent varying from 1-2 for free chain and stiff chain, respectively; and A is a coefficient that depends on the details of the polymer. In the case that the excluded volume interactions are present, v = 6/5.

The model proposed by Edwards and Singh [1] and Muthukumar and Nickel [2] are employed. Instead of using the idea of an effective step-length technique and the perturbation technique, the idea of Feynman [3] in relation to the polaron problem is used and developed by handling a disordered system. The idea is to model the polymer action as a model of quardatic trial action and consider the differences between the polymer action and the trial action as the first cumulant approximation in one parameter. The variational principle is used to find the optimal values of the variational parameters and the mean square distance is obtained. A comparison between these approaches and effective step-length and perturbation approach will be discussed.