Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem

Abstract

We show that the emerging field of discrete differential geometry can be usefully brought to bear on crystallization problems. In particular, we give a simplified proof of the Heitmann–Radin crystallization theorem (Heitmann and Radin in J Stat Phys 22(3):281–287, 1980), which concerns a system of N identical atoms in two dimensions interacting via the idealized pair potential \(V(r)=+\infty \) if \(r<1\), \(-1\) if \(r=1\), 0 if \(r>1\). This is done by endowing the bond graph of a general particle configuration with a suitable notion of discrete curvature, and appealing to a discrete Gauss–Bonnet theorem (Knill in Elem Math 67:1–7, 2012) which, as its continuous cousins, relates the sum/integral of the curvature to topological invariants. This leads to an exact geometric decomposition of the Heitmann–Radin energy into (i) a combinatorial bulk term, (ii) a combinatorial perimeter, (iii) a multiple of the Euler characteristic, and (iv) a natural topological energy contribution due to defects. An analogous exact geometric decomposition is also established for soft potentials such as the Lennard–Jones potential \(V(r)=r^{-6}-2r^{-12}\), where two additional contributions arise, (v) elastic energy and (vi) energy due to non-bonded interactions.

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Acknowledgements

This work was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”. GF thanks Sasha Bobenko and John Sullivan for their most helpful advice at an early stage of this project that discrete Gauss–Bonnet theorems might be relevant to the goal of relating defect-induced curvature to atomistic energies. Also, LDL and GF thank Oliver Knill for sharing valuable intuition and insights related to Refs. Knill (2012) and Knill (2011).

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Correspondence to L. De Luca.

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De Luca, L., Friesecke, G. Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem. J Nonlinear Sci 28, 69–90 (2018). https://doi.org/10.1007/s00332-017-9401-6

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Keywords

  • Crystallization
  • Interaction potential
  • Discrete differential geometry
  • Energy minimization
  • Gauss–Bonnet theorem

Mathematics Subject Classification

  • Primary 82D25
  • Secondary 05C10
  • 74E15