Renormalization of the one-loop theory of fluctuations in polymer blends and diblock copolymer melts

TL;DR

This paper analyzes the ultraviolet divergence in one-loop fluctuation theories of polymer blends and copolymer melts, showing how to renormalize parameters to achieve a UV-convergent correction to self-consistent field theory.

Contribution

It demonstrates that UV divergences in one-loop corrections can be absorbed into parameter renormalizations, enabling a UV-convergent theory for polymer mixture correlations.

Findings

UV divergences can be renormalized into phenomenological parameters.

The divergent contributions include renormalizations of the $hi$ parameter, segment lengths, and gradient terms.

A framework for UV-convergent corrections to SCFT is established.

Abstract

Attempts to use coarse-grained molecular theories to calculate corrections to the random-phase approximation (RPA) for correlations in polymer mixtures have been plagued by an unwanted sensitivity to the value of an arbitrary cutoff length, {\it i.e.}, by an ultraviolet (UV) divergence. We analyze the UV divergence of the inverse structure factor $S^{-1}(k)$ predicted by a `one-loop' approximation similar to that used in several previous studies. We consider both miscible homopolymer blends and disordered diblock copolymer melts. We show, in both cases, that all UV divergent contributions can be absorbed into a renormalization of the values of the phenomenological parameters of a generalized self-consistent field theory (SCFT). This observation allows the construction of a UV convergent theory of corrections to SCFT phenomenology. The UV-divergent one-loop contribution to $S^{-1}(k)$ are shown to be the sum of: (i) a $k$-independent contribution that arises from a renormalization of the effective $\chi$ parameter, (ii) a $k$-dependent contribution that arises from a renormalization of monomer statistical segment lengths, (iii) a contribution proportional to $k^{2}$ that arises from a square-gradient contribution to the one-loop fluctuation free energy, and (iv) a $k$-dependent contribution that is inversely proportional to the degree of polymerization, which arises from local perturbations in fluid structure near chain ends and near junctions between blocks in block copolymers.