Branched Polymers with Excluded Volume Effects/ Relationship between Polymer Dimensions and Generation Number

TL;DR

This paper extends empirical models of branched polymers to include excluded volume effects, deriving relationships between polymer dimensions, architecture, and generation number, and discusses inequalities and potential violations in specific cases.

Contribution

It introduces a generalized scaling relation for polymer architectures and explores the restrictions imposed by excluded volume effects on polymer dimensions.

Findings

Derived the exponent or various architectures

Established inequalities relating , nd or polymers in good solvents

Discussed potential violations of these inequalities in specific examples

Abstract

We discuss the extension of the empirical equation: $\left\langle s_{N}^{2}\right\rangle_{0}\propto g\,l^{2}$, where the subscript 0 denotes the ideal value with no excluded volume and $g$ the generation number from the root to the youngest (outermost) generation. By analogy with the linear chain problem, we introduce the assumption that the scaling relation, $\left\langle s_{N}^{2}\right\rangle_{0}\propto g^{2\lambda}\,l^{2}$, exists for arbitrary polymeric architectures, where $\lambda$ is an exponent for the backbone structure. Then, making use of the relationship between $g$ and $N$ (monomer number), we can deduce the exponent, $\nu$, for polymers with various architectures. The theory of the excluded volume effects impose the severe restriction on the quantities: $\nu_{0}$, $\nu$, and $\lambda$; for instance, the inequality, $\nu_{0}\ge\frac{1}{d+1}$, must be satisfied for isolated polymers in good solvents. An intriguing question is whether or not there exists an actual molecule that violates this inequality. We take up two examples having $\nu_{0}=1/4$ for $d=2$ and $\nu_{0}=1/6$ for $d=3$, and discuss this question.