Statistical mechanics of frustrated assemblies and incompatible graphs

TL;DR

This paper develops a graph theory framework to understand the statistical mechanics of geometrically frustrated assemblies, revealing new phenomena like dual percolation transitions and stress responses.

Contribution

It introduces a novel graph theory approach to model frustration in assemblies, linking structural connectivity with frustration phenomena.

Findings

Identification of two distinct percolation transitions

Discovery of a crossover between frustration regimes

Divergent length scale in assembly response

Abstract

Geometrically frustrated assemblies where building blocks misfit have been shown to generate intriguing phenomena from self-limited growth, fiber formation, to structural complexity. We introduce a graph theory formulation of geometrically frustrated assemblies, capturing frustrated interactions through the concept of incompatible flows, providing a direct link between structural connectivity and frustration. This theory offers a minimal yet comprehensive framework for the fundamental statistical mechanics of frustrated assemblies. Through numerical simulations, the theory reveals new characteristics of frustrated assemblies, including two distinct percolation transitions for structure and stress, a crossover between cumulative and non-cumulative frustration controlled by disorder, and a divergent length scale in their response.