Universal hidden order in amorphous cellular geometries

Klatt, M.A; Lovrić, Jakov; Chen, Duyu; Kapfer C., Schaller; Schönhöfer W. A., Phillipp; Gardiner, S. Bruce; Smith, Ana-Sunčana; Schröder-Turk, E. Gerd; Torguato, Salvatore (2019) Universal hidden order in amorphous cellular geometries. Nature Communications, 10 . ISSN 2041-1723

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Official URL: https://www.nature.com/articles/s41467-019-08360-5

Abstract

Partitioning space into cells with certain extreme geometrical properties is a central problem in many fields of science and technology. Here we investigate the Quantizer problem, defined as the optimisation of the moment of inertia of Voronoi cells, i.e., similarly-sized ‘sphere-like’ polyhedra that tile space are preferred. We employ Lloyd’s centroidal Voronoi diagram algorithm to solve this problem and find that it converges to disordered states associated with deep local minima. These states are universal in the sense that their structure factors are characterised by a complete independence of a wide class of initial conditions they evolved from. They moreover exhibit an anomalous suppression of long-wavelength density fluctuations and quickly become effectively hyperuniform. Our findings warrant the search for novel amorphous hyperuniform phases and cellular materials with unique physical properties.

Item Type:

Article

Additional Information:

We thank Ge Zhang for supplying us stealthy point patterns in two dimensions and Steven Atkinson for providing MRJ sphere packings. We thank Adil Mughal for comments, Sara Kaliman for preliminary work on Lloyd's algorithm and Thomas Pigeon for the 3D visualisations in Fig. 2. We thank the German Academic Exchange Service and Universities Australia for travel funding through a collaborative grant scheme. We acknowledge the support of the European Research Council (ERC) under grant ERC StG Membranes Act 2013-33728 and of the German Science Foundation (DFG) through the research group 'Geometry and Physics of Spatial Random Systems' (GPSRS) under grants number SCHR-1148/3-2, HU1874/3-2, and LA965/6-2. S.K., M.K., and G.S.T. are grateful to Klaus Mecke for years of support, moral and financial, and for scientific guidance and inspiration without which this article would have never materialised.

Uncontrolled Keywords:

geometrical properties; cells

Subjects:

NATURAL SCIENCES > Mathematics

Divisions:

Division of Physical Chemistry

Projects: