Non-Hookean statistical mechanics of clamped graphene ribbons

TL;DR

This paper investigates how thermal fluctuations affect the bending rigidity of clamped graphene ribbons, revealing size-dependent violations of Hooke's Law through combined theoretical and numerical analysis.

Contribution

It introduces a combined statistical mechanics and simulation approach to study thermal renormalization of graphene ribbon bending rigidity, highlighting size-dependent effects.

Findings

Bending rigidity independent of width for W<ell_th

Rigidity increases with width W for W>ell_th

Observation of large-scale random walk behavior when L>ell_p

Abstract

Thermally fluctuating sheets and ribbons provide an intriguing forum in which to investigate strong violations of Hooke's Law: large distance elastic parameters are in fact not constant, but instead depend on the macroscopic dimensions. Inspired by recent experiments on free-standing graphene cantilevers, we combine the statistical mechanics of thin elastic plates and large-scale numerical simulations to investigate the thermal renormalization of the bending rigidity of graphene ribbons clamped at one end. For ribbons of dimensions $W\times L$ (with $L\geq W$), the macroscopic bending rigidity $\kappa_R$ determined from cantilever deformations is independent of the width when $W<\ell_\textrm{th}$, where $\ell_\textrm{th}$ is a thermal length scale, as expected. When $W>\ell_\textrm{th}$, however, this thermally renormalized bending rigidity begins to systematically increase, in agreement with the scaling theory, although in our simulations we were not quite able to reach the system sizes necessary to determine the fully developed power law dependence on $W$. When the ribbon length $L > \ell_p$, where $\ell_p$ is the $W$-dependent thermally renormalized ribbon persistence length, we observe a scaling collapse and the beginnings of large scale random walk behavior.