General Solution for Elastic Networks on Arbitrary Curved Surfaces in the Absence of Rotational Symmetry

TL;DR

This paper develops a numerical framework to analyze elastic networks on arbitrary curved surfaces without relying on symmetry, revealing how domain shape influences defect formation and stability.

Contribution

It introduces a novel numerical method for solving non-linear elasticity equations on arbitrary surfaces lacking rotational symmetry, applicable to complex biological and material systems.

Findings

Transition from defect-free to defected structures depends on domain shape

Edge geometry significantly influences defect stability

Method applicable to virus capsid assembly and complex crystal growth

Abstract

Understanding crystal growth over arbitrary curved surfaces with arbitrary boundaries is a formidable challenge, stemming from the complexity of formulating non-linear elasticity using geometric invariant quantities. Solutions are generally confined to systems exhibiting rotational symmetry. In this paper, we introduce a framework to address these challenges by numerically solving these equations without relying on inherent symmetries. We illustrate our approach by computing the minimum energy required for an elastic network containing a disclination at any point and by investigating surfaces that lack rotational symmetry. Our findings reveal that the transition from a defect-free structure to a stable state with a single 5-fold or 7-fold disclination strongly depends on the shape of the domain, emphasizing the profound influence of edge geometry. We discuss the implications of our results for general experimental systems, particularly in elucidating the assembly pathways of virus capsids. This research enhances our understanding of crystal growth on complex surfaces and expands its applications across diverse scientific domains.