Static Scaling Behavior of High-Molecular-Weight Polymers in Dilute Solution: A Reexamination

TL;DR

This paper reexamines the scaling behavior of high-molecular-weight polymers in dilute solutions, clarifying the applicability of the continuum Edwards model to different long-chain limit scenarios and connecting crossover scaling functions with renormalization group theory.

Contribution

It demonstrates that the continuum Edwards model applies specifically to the limit where temperature approaches the theta point, not the fixed-temperature limit, and links crossover functions with renormalization group methods.

Findings

The Edwards model accurately describes the crossover scaling for $x \,\geq\, 0$.

A Wilson-type renormalization group elucidates the connection between crossover functions and field theories.

Clarifies the distinction between two different long-chain limit approaches.

Abstract

Previous theories of dilute polymer solutions have failed to distinguish clearly between two very different ways of taking the long-chain limit: (I) $N \to\infty$ at fixed temperature $T$, and (II) $N \to\infty$, $T \to T_\theta$ with $x \equiv N^\phi (T-T_\theta)$ fixed. I argue that the modern two-parameter theory (continuum Edwards model) applies to case II --- not case I --- and in fact gives exactly the crossover scaling functions for $x \ge 0$ modulo two nonuniversal scale factors. A Wilson-type renormalization group clarifies the connection between crossover scaling functions and continuum field theories. [Also contains a general discussion of the connection between the Wilson and field-theoretic renormalization groups. Comments solicited.]