The onset of self-assembly

TL;DR

This paper develops a gauge-invariance-based theory of self-assembly at the mesoscale, explaining the onset of structure formation in diblock copolymers through critical phenomena and correlation length divergence.

Contribution

It introduces a novel gauge-invariance framework to describe mesoscale self-assembly and derives critical exponents for correlation length near the transition.

Findings

Correlation length diverges as (c - c*)^-2/3

Renormalized diffusion constant approaches zero at criticality

Correlation function remains finite at the transition

Abstract

We have formulated a theory of self-assembly based on the notion of local gauge invariance at the mesoscale. Local gauge invariance at the mesoscale generates the required long- range entropic forces responsible for self-assembly in binary systems. Our theory was applied to study the onset of mesostructure formation above a critical temperature in estane, a diblock copolymer. We used diagrammatic methods to transcend the Gaussian approximation and obtain a correlation length zeta ~ (c-c*)^-gamma, where c* is the minimum concentration below which self-assembly is impossible, c is the current concentration, and gamma was found numerically to be close to 2/3. The renormalized diffusion constant vanishes as the c* is approached, indicating the occurrence of critical slowing down, while the correlation function remains finite at the transition point.