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

Malena I. Español, Dmitry Golovaty, J. Patrick Wilber

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


In this paper, we derive a continuum variational model for a two-dimensional deformable lattice of atoms interacting with a two-dimensional rigid lattice. The starting point is a discrete atomistic model for the two lattices which are assumed to have slightly different lattice parameters and, possibly, a small relative rotation. This is a prototypical example of a three-dimensional system consisting of a graphene sheet suspended over a substrate. We use a discrete-to-continuum procedure to obtain the continuum model which recovers both qualitatively and quantitatively the behaviour observed in the corresponding discrete model. The continuum model predicts that the deformable lattice develops a network of domain walls characterized by large shearing, stretching and bending deformation that accommodates the misalignment and/or mismatch between the deformable and rigid lattices. Two integer-valued parameters, which can be identified with the components of a Burgers vector, describe the mismatch between the lattices and determine the geometry and the details of the deformation associated with the domain walls.

Original languageEnglish (US)
Article number20170612
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2209
StatePublished - 2018
Externally publishedYes


  • Bilayer graphene
  • Discrete-to-continuum modelling
  • Domain wall
  • Ginzburg–Landau energy
  • Heterostructure
  • Moiré pattern

ASJC Scopus subject areas

  • General Mathematics
  • General Engineering
  • General Physics and Astronomy


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