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Geometrical frustration: A study of four-dimensional hard spheres

J. A. van Meel, D. Frenkel, and P. Charbonneau
Phys. Rev. E 79, 030201(R) – Published 3 March 2009

Abstract

The smallest maximum-kissing-number Voronoi polyhedron of three-dimensional (3D) Euclidean spheres is the icosahedron, and the tetrahedron is the smallest volume that can show up in Delaunay tessellation. No periodic lattice is consistent with either, and hence these dense packings are geometrically frustrated. Because icosahedra can be assembled from almost perfect tetrahedra, the terms “icosahedral” and “polytetrahedral” packing are often used interchangeably, which leaves the true origin of geometric frustration unclear. Here we report a computational study of freezing of 4D Euclidean hard spheres, where the densest Voronoi cluster is compatible with the symmetry of the densest crystal, while polytetrahedral order is not. We observe that, under otherwise comparable conditions, crystal nucleation in four dimensions is less facile than in three dimensions, which is consistent with earlier observations [M. Skoge et al., Phys. Rev. E 74, 041127 (2006)]. We conclude that it is the geometrical frustration of polytetrahedral structures that inhibits crystallization.

    • Received 26 November 2008

    DOI:https://doi.org/10.1103/PhysRevE.79.030201

    ©2009 American Physical Society

    Authors & Affiliations

    J. A. van Meel

    • FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands

    D. Frenkel

    • Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom and FOM Institute for Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands

    P. Charbonneau

    • Department of Chemistry, Duke University, Durham, North Carolina 27708, USA

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    Issue

    Vol. 79, Iss. 3 — March 2009

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