Flory theory revisited

TL;DR

This paper revisits Flory theory for polymers, deriving it via cumulant expansion, recovering the full free energy including the logarithmic term, and discusses extensions and systematic corrections to the classical approach.

Contribution

It presents a derivation of Flory theory using cumulant expansion, including the full free energy and systematic corrections, applicable to various polymer systems.

Findings

Full Flory free energy including logarithmic term recovered.

Method applicable to different monomer interactions and polymer solutions.

Systematic expansion around Flory theory with controlled corrections.

Abstract

The Flory theory for a single polymer chain is derived as the lowest order of a cumulant expansion. In this approach, the full original Flory free energy (including the logarithmic term), is recovered. %This term does not change the wandering exponent $ \nu $ but turns out to %be responsible for the crossover from Brownian $ (d>4) $ to swollen $ %(d\leq4) $ %regime. The prefactors of the elastic and repulsive energy are calculated from the microscopic parameters. The method can be applied to other types of monomer-monomer interactions, and the case of a single chain in a bad solvent is discussed . The method is easily generalized to many chain systems (polymers in solutions), yielding the usual crossovers with chain concentration. Finally, this method is suitable for a systematic expansion around the Flory theory. The corrections to Flory theory consist of extensive terms (proportional to the number $N$ of monomers) and powers of $N^{2-\nu d}$ . These last terms diverge in the thermodynamic limit, but less rapidly than the usual Fixman expansion in $N^{2- d/2}$.