Abstract
We analyse in a common framework the properties of the Voronoi tessellations resulting from regular 2D and 3D crystals and those of tessellations generated by Poisson distributions of points, thus joining on symmetry breaking processes and the approach to uniform random distributions of seeds. We perturb crystalline structures in 2D and 3D with a spatial Gaussian noise whose adimensional strength is α and analyse the statistical properties of the cells of the resulting Voronoi tessellations using an ensemble approach. In 2D we consider triangular, square and hexagonal regular lattices, resulting into hexagonal, square and triangular tessellations, respectively. In 3D we consider the simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) crystals, whose corresponding Voronoi cells are the cube, the truncated octahedron, and the rhombic dodecahedron, respectively. In 2D, for all values α>0, hexagons constitute the most common class of cells. Noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α=0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise with α0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α>2, results converge to those of Poisson-Voronoi tessellations. In 2D, while the isoperimetric ratio increases with noise for the perturbed hexagonal tessellation, for the perturbed triangular and square tessellations it is optimised for specific value of noise intensity. The same applies in 3D, where noise degrades the isoperimetric ratio for perturbed FCC and BCC lattices, whereas the opposite holds for perturbed SCC lattices. This allows for formulating a weaker form of the Kelvin conjecture. By analysing jointly the statistical properties of the area and of the volume of the cells, we discover that also the cells shape heavily fluctuates when noise is introduced in the system. In 2D, the geometrical properties of n-sided cells change with α until the Poisson-Voronoi limit is reached for α>2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established, which agrees with exact asymptotic results. Anomalous scaling relations are observed between the perimeter and the area in the 2D and between the areas and the volumes of the cells in 3D: except for the hexagonal (2D) and FCC structure (3D), this applies also for infinitesimal noise. In the Poisson-Voronoi limit, the anomalous exponent is about 0.17 in both the 2D and 3D case. A positive anomaly in the scaling indicates that large cells preferentially feature large isoperimetric quotients. As the number of faces is strongly correlated with the sphericity (cells with more faces are bulkier), in 3D it is shown that the anomalous scaling is heavily reduced when we perform power law fits separately on cells with a specific number of faces.
Highlights
A Voronoi tessellation [1, 2] is a partitioning of an Euclidean N-dimensional space defined in terms of a given discrete set of points X
By analysing jointly the statistical properties of the area and of the volume of the cells, we discover that the cells shape heavily fluctuates when noise is introduced in the system
We study how the statistical properties of the resulting Voronoi cells change with α, covering the whole range going from the symmetry breaking, occurring when α becomes positive, up to the progressively more and more uniform distribution of xi, obtained when 1, so that the distributions of nearby points xi overlap more and more significantly
Summary
Since Voronoi tessellations boil up to being optimal partitionings of the space resulting from a set of generating points, they define a natural discrete mathematical measure, and have long been considered for applications in several research areas, such as telecommunications [5], biology [6], astronomy [7], forestry [8], atomic physics [9], metallurgy [10], polymer science [11], materials science [12, 13], and biophysics [14]. A great deal of theoretical and computational work has focused on the more specific and tractable problem of studying the statistical properties of Poisson-Voronoi tessellations These are Voronoi tessellations obtained starting from a random set of points X generated as output of a homogeneous. Examples of BCC structures include Fe, Cr, W, and Nb. thanks to its extremal properties in terms of packing fraction and the resulting high density, FCC crystals are fairly common and specific examples include Pb, Al, Cu, Au and Ag. The extremal properties in terms of surface–to-area ratio of the hexagonal (in 2D) and truncated octahedral (in 3D) tessellations imply that they may serve as optimal tool for achieving data compression [57].
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