Theoretical Investigation of the Effect of Intramolecular Interactions on the Configuration of Polymeric Chains

Abstract

A theoretical investigation of the effect of intramolecular interactions on the configurational statistics of a polymer molecule is presented. This has been studied by many authors and is known as the volume problem in the literature. A statistical mechanical approach is used. Many of the similarities between the theory of classical fluids and the excluded volume are exploited. The configurational statistics of 2 and 3 segment chains are computed exactly for the sphere The integrations were performed by introducing bipolar and tripolar coordinate systems. It was found that the mean square end-to-end distance for these cases was n1.33 where n is the number of segments. These results are of no practical use in predicting the properties of real polymer chains which are much longer. It is instructive, however, to compare these exact results with approximate theories in the limit of short chain length. A is written for the partition function of a polymer chain with the ends of the chain fixed. This is analogous to the cluster expansion for the partition function of an imperfect gas. The first-order term in this expansion is evaluated for the hard core potential. In the limit of small hard core diameters, the first-order term leads to the wellknown first-order perturbation theory for the mean square end-to-end distance. The exact results of this first-order correction term are used to construct higher-order terms of a specified isolated topology. If only these terms are used in the cluster expansion, incorrect results are obtained for the mean square end-to-end distance. This indicates that higher-order terms of complicated topology are significant for longer chain length. Various approximate integral equations for the restricted partition function of a polymer chain are presented. The most promising of these equations is the analog of the well-known Percus-Yevick equation in the theory of liquids. In deriving this equation two topologically distinct types of graphs are defined. These are the nodal and elementary graphs. An exact equation relating these types of graphs is presented. The analog of the Percus-Yevick approximation is made which leads to an integro-difference equation. This equation is solved exactly using the hard core potential for the special case of the hard core diameter equal to the polymer segment length. Results of numerical calculations are given for other intermediate values of this diameter ranging from zero to the segment length (the pearl necklace model). This leads to values of γ ranging correspondingly from 1.0 to 2.0 where with