Discrete-to-continuum modeling of weakly interacting incommensurate chains

Abstract

In this paper we use a formal discrete-to-continuum procedure to derive a continuum variational model for two chains of atoms with slightly incommensurate lattices. The chains represent a cross-section of a three-dimensional system consisting of a graphene sheet suspended over a substrate. The continuum model recovers both qualitatively and quantitatively the behavior observed in the corresponding discrete model. The numerical solutions for both models demonstrate the presence of large commensurate regions separated by localized incommensurate domain walls.

Discrete-to-continuum modelling of weakly interacting incommensurate two-dimensional lattices

“…4.5 as functions of χ 1 while holding χ 2 = 0.25 fixed. These graphs are very similar to what is observed for an analogous one-dimensional system considered in [7]. In particular, the small dips next to the cross-section of a wall on the right inset in Fig.…”

“…The corresponding non-dimensional displacement components εξ 1 and εη are shown in figure 14 as functions of χ 1 while holding χ 2 = 0.25 fixed. These graphs are very similar to what is observed for an analogous one-dimensional system considered in [7]. In particular, the small dips next to the cross-section of a wall in figure 14b result from the presence of the bending stiffness term and the linear stiffness of the van der Waals interaction near equilibrium.…”

“…Then, an appropriate truncation of the resulting series yields a continuum elastic energy, which is close to the energy of the classical von Föppl-Kármán shell theory [6]. Our recent work on a similar one-dimensional problem [7] indicates that this choice provides a reasonable generic approximation of the mesoscopic elastic energy.…”

“…The discrete-to-continuum modelling in this paper generalizes to two dimensions the work in [7], in which a continuum theory for weakly interacting chains of atoms is derived. The atomistic model includes stretching and bending energies for strong covalent bonds between atoms in the same chain and an interaction energy between atoms in adjacent chains.…”

Moiré of Moiré: Modeling Mechanical Relaxation in Incommensurate Trilayer van der Waals Heterostructures

“…Crystal relaxation in vdW heterostructures has been studied in the continuum limit by energy minimization [12][13][14][15][16][17][18] . In twisted bilayer graphene, for instance, relaxation enlarges the AB/BA stacking regions (the equilibrium stacking) and forms a thin domain line in between adjacent AB/BA stacking regions, while the AA stacking regions (high energy stacking order) shrink to localized spots.…”