Diagrammatic analysis of correlations in polymer fluids: Cluster diagrams via Edwards' field theory

TL;DR

This paper develops a diagrammatic cluster expansion for polymer fluids based on Edwards' field theory, enabling systematic calculation of correlation functions and direct correlation functions, with applications to coarse-grained models.

Contribution

It introduces a new diagrammatic formalism for polymer fluids that relates Edwards' functional integral to Mayer cluster expansion, including renormalized and loop expansions.

Findings

Derived a new expression for the Ornstein-Zernicke direct correlation function.

Recovered known intramolecular correlation results to first order.

Demonstrated convergence of loop expansion for coarse-grained models.

Abstract

A straightforward expansion of Edwards' functional integral representation of the grand partition function for a polymer liquid as an infinite set of Feynman diagrams is shown to yield a cluster expansion that is closely related to the corresponding Mayer cluster expansion developed for flexible molecules by Chandler and coworkers. The procedure initially yields a perturbative cluster expansion in which all free energies and correlation functions are expressed diagrammatically as functionals of single-molecule correlation functions of non-interacting molecules. Topological reduction yields several renormalized expansions for collective correlation functions of all orders as functionals of single-molecule correlation functions in the interacting fluid. Renormalized expansions are also obtained for a generalized Ornstein-Zernicke (OZ) direct correlation function and for intramolecular correlation functions. The application of the formalism to coarse-grained models of polymer fluids is discussed, and a loop expansion about self-consistent field theory is shown to converge for sufficiently coarse-grained models, in which monomers are strongly overlapping. The formalism is used to derive a new expression for the OZ direct correlation function and to recover known results for the 2-point intramolecular correlation function to first order in a loop expansion, for binary blends and diblock copolymer melts.