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2D crystals

The Wulff construction goes low-symmetry

Nature Materials (2022)Cite this article

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An apparent quirk of mathematics draws on a symmetry and resolves the issue of how to determine the equilibrium shape of crystals of two-dimensional materials with asymmetric terminations.

The striking geometrical simplicity of naturally occurring minerals has inspired human imagination for millennia, suggesting the emergence of order in an otherwise seemingly amorphous world. The appearance of spinel, quartz or topaz crystals that exhibit faceting visible to the naked eye is the direct result of the arrangement of the atoms inside these materials on a (crystal) lattice. The atom arrangement can be terminated in countless distinct ways, yet each surface formed by such a cut is associated with an energy cost. As shown already by Gibbs1 and Curie2, the equilibrium shape of a crystallite is the one that minimizes the total surface energy. A formal geometric procedure to determine the shape given by this condition was introduced by Wulff3 and later proved, for example by Laue4. The Wulff construction has been extremely successful for understanding the shapes of both two-dimensional (2D) and 3D materials, all the way from the macro to the nanoscale5, as one has to approach the sub-10-nm size regime in order to observe deviations6.

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Fig. 1: A simple example of a Wulff construction for an asymmetric crystal.

References

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Acknowledgements

This work was funded by the Swedish Research Council (grant numbers 2018-06482, 2020-04935 and 2021-05072) and the Excellence Initiative Nano at Chalmers.

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  1. Department of Physics, Chalmers University of Technology, Gothenburg, Sweden

    Paul Erhart & J. Magnus Rahm

  2. Compular AB, Gothenburg, Sweden

    J. Magnus Rahm

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  1. Paul Erhart
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  2. J. Magnus Rahm
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Correspondence to Paul Erhart.

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Erhart, P., Rahm, J.M. The Wulff construction goes low-symmetry. Nat. Mater. (2022). https://doi.org/10.1038/s41563-022-01407-x

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