New Cascade Theory of Branched Polymers and Its Application to Size Exclusion Chromatography

Abstract

A generalized cascade theory is developed and applied to study the size distribution of branched polymers. The configuration of a polymer is expressed by a rooted tree. The rooted tree is described by the “S-expression” of Lisp language. Thus, the distribution of polymers in a system can be expressed by the polynomial W = ∑αkTk, where Tk denotes the k-th S-expression corresponding to a k-th type of polymer and αk is the probability that a randomly chosen unit in a system belongs to the polymer of k-th type. Terms of the polynomial are classified into equivalent classes according to the types of characterization of polymers, and the polynomial in these classes can be calculated from recursive coupled equations on the basis of the Markov branching process. Solving these equations enables us to calculate the parameters for configurations of polymers as well as to obtain the results of the conventional cascade theory. As a special case of this generalized theory, a recursive equation for size distribution is formulated. Using this formulation, the distributions of the mean-square radius of gyration and the shrink factor of branched polymers are calculated on the basis of the Gaussian chain statistics. The substitution effects of neighboring groups are reasonably found to give significant effects on these distributions.