# The Gaussian Stiffness of Graphene deduced from a Continuum Model based on Molecular Dynamics Potentials

## TL;DR

This paper derives an analytical expression for the Gaussian stiffness of graphene using a continuum model based on molecular dynamics potentials, clarifying its atomic-scale origins and resolving previous conflicting values.

## Contribution

It introduces the first analytical formula for graphene's Gaussian stiffness derived from a continuum limit of a molecular dynamics model.

## Key findings

- Analytical expression for Gaussian stiffness of graphene.
- Quantitative evaluation of atomic-scale sources of stiffness.
- Clarification of previously conflicting values in literature.

## Abstract

We consider a discrete model of a graphene sheet with atomic interactions governed by a harmonic approximation of the 2nd-generation Brenner potential that depends on bond lengths, bond angles, and two types of dihedral angles. A continuum limit is then deduced that fully describes the bending behavior. In particular, we deduce for the first time an analytical expression of the Gaussian stiffness, a scarcely investigated parameter ruling the rippling of graphene, for which contradictory values have been proposed in the literature. We disclose the atomic-scale sources of both bending and Gaussian stiffnesses and provide for them quantitative evaluations.