The Gaussian stiffness of graphene deduced from a continuum model based on Molecular Dynamics potentials

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.

Determining the Gaussian Modulus and Edge Properties of 2D Materials: From Graphene to Lipid Bilayers

“…This is in sharp contrast to the negative edge tension that is believed to exist for graphene edges. For Gaussian modulus also, we obtained κ G ∼ 1kT ∼ 25meV, which is positive and much larger than its predicted bare value at zero Kelvin ∼ −1.52 to −1.62 eV [25,26]. Our results bring values of Gaussian modulus [25] and edge properties [45] in the literature are used.…”

“…As alluded to earlier, two notable (and congruent) estimates exist for the zero Kelvin value [25,26]. Accordingly, we have also explored the temperature dependency of the Gaussian modulus and edge properties.…”

“…al. [26] used a rather interesting approach where they derived a continuum model linked with the 2nd-generation reactive empirical bond-order (REBO) interatomic potential [27] to extract the Gaussian modulus at zero Kelvin and find its value to be around −1.62eV . These approaches exclude entropic contributions.…”

An atomistic-based Föppl–von Kármán model for graphene

“…The out-of-plane deformations have been considered by various authors. In particular, by exploiting a formal analysis we deduced a continuum model of a graphene sheet [5], and provided explicit expressions for the bending and Gaussian stiffness by starting from the study of the lattice kinematics and assuming the reactive empirical bond-order potential (REBO) of 2nd generation by Brenner et al [3]. The approach takes into account the role of self-stress and provides a quantitative estimate of the self-stress contribution to the overall bending and Gaussian stiffness.…”

“…We consider an array of Catoms sitting at the nodes of a hexagonal lattice, and assume that the lattice interactions are governed by the Brenner's REBO potential of 2nd generation and that self-stress is present. Thus, the starting point is the same as in [5], but the changes of edge lengths, wedge angles and dihedral angles are calculated by keeping the quadratic term in the out-of-plane displacements, according to the form of the in-plane Green-Lagrange strain used in Föppl-von Kármán plate theory.…”