Emanuel A. Lazar

Topological framework for local structure analysis in condensed matter

Significance Richard Feynman famously described the hypothesis “All things are made of atoms” as among the most significant of all scientific knowledge. How atoms are arranged in “things” is an interesting and natural question. However, aside from perfect crystals and ideal gases, understanding these arrangements in an insightful yet tractable manner is challenging. We introduce a unified mathematical framework for classifying and identifying local structure in imperfect condensed matter systems using Voronoi topology. This versatile approach enables visualization and analysis of a wide range of complex atomic systems, including highly defected solids and glass-forming liquids. The proposed framework presents a new perspective into the structure of discrete systems of particles, ordered and disordered alike. Physical systems are frequently modeled as sets of points in space, each representing the position of an atom, molecule, or mesoscale particle. As many properties of such systems depend on the underlying ordering of their constituent particles, understanding that structure is a primary objective of condensed matter research. Although perfect crystals are fully described by a set of translation and basis vectors, real-world materials are never perfect, as thermal vibrations and defects introduce significant deviation from ideal order. Meanwhile, liquids and glasses present yet more complexity. A complete understanding of structure thus remains a central, open problem. Here we propose a unified mathematical framework, based on the topology of the Voronoi cell of a particle, for classifying local structure in ordered and disordered systems that is powerful and practical. We explain the underlying reason why this topological description of local structure is better suited for structural analysis than continuous descriptions. We demonstrate the connection of this approach to the behavior of physical systems and explore how crystalline structure is compromised at elevated temperatures. We also illustrate potential applications to identifying defects in plastically deformed polycrystals at high temperatures, automating analysis of complex structures, and characterizing general disordered systems.

Statistical topology of three-dimensional Poisson-Voronoi cells and cell boundary networks.

Voronoi tessellations of Poisson point processes are widely used for modeling many types of physical and biological systems. In this paper, we analyze simulated Poisson-Voronoi structures containing a total of 250000000 cells to provide topological and geometrical statistics of this important class of networks. We also report correlations between some of these topological and geometrical measures. Using these results, we are able to corroborate several conjectures regarding the properties of three-dimensional Poisson-Voronoi networks and refute others. In many cases, we provide accurate fits to these data to aid further analysis. We also demonstrate that topological measures represent powerful tools for describing cellular networks and for distinguishing among different types of networks.

A geometric formulation of the law of Aboav?Weaire in two and three dimensions

The law of Aboav–Weaire is a simple mathematical expression deriving from empirical observations that the number of sides of a grain is related to the average number of sides of the neighboring grains, and is usually restricted to natural two-dimensional microstructures. Numerous attempts have been made to justify this relationship theoretically, or to derive an analogous relation in three dimensions. This paper provides several exact geometric results with expressions similar to that of the usual law of Aboav–Weaire, though with additional terms that may be used to establish when the law of Abaov–Weaire is a suitable approximation. Specifically, we derive several local relations that apply to individual grain clusters, and a corresponding global relation that is identical in two and three dimensions except for a single parameter ζ. The derivation requires the definition and investigation of the average excess curvature, a previously unconsidered physical quantity. An approximation to our exact result is compared to the results of extensive simulations in two and three dimensions, and we provide a compact expression that strikes a balance between complexity and accuracy.

On Fiber Diameters of Continuous Maps

Abstract We present a surprisingly short proof that for any continuous map f: ℝn → ℝ;m, if n > m, then there exists no bound on the diameter of fibers of f. Moreover, we show that when m = 1, the union of small fibers of f is bounded; when m > 1, the union of small fibers need not be bounded. Applications to data analysis are considered.

VoroTop: Voronoi cell topology visualization and analysis toolkit

This paper introduces a new open-source software program called VoroTop, which uses Voronoi topology to analyze local structure in atomic systems. Strengths of this approach include its abilities to analyze high-temperature systems and to characterize complex structure such as grain boundaries. This approach enables the automated analysis of systems and mechanisms previously not possible.

Statistical topology of perturbed two-dimensional lattices

The Voronoi cell of any atom in a lattice is identical. If atoms are perturbed from their lattice coordinates, then the topologies of the Voronoi cells of the atoms will change. We consider the distribution of Voronoi cell topologies in two-dimensional perturbed systems. These systems can be thought of as simple models of finite-temperature crystals. We give analytical results for the distribution of Voronoi topologies of points in two-dimensional Bravais lattices under infinitesimal perturbations and present a discussion with numerical results for finite perturbations.

Many-faced cells and many-edged faces in 3D Poisson–Voronoi tessellations

Motivated by recent new Monte Carlo data we investigate a heuristic asymptotic theory that applies to n-faced 3D Poisson–Voronoi cells in the limit of large n. We show how this theory may be extended to n-edged cell faces. It predicts the leading order large-n behavior of the average volume and surface area of the n-faced cell, and of the average area and perimeter of the n-edged face. Such a face is shown to be surrounded by a toroidal region of volume n/λ (with λ the seed density) that is void of seeds. Two neighboring cells sharing an n-edged face are found to have their seeds at a typical distance that scales as n−1/6 and whose probability law we determine. We present a new data set of 4 × 109 Monte Carlo generated 3D Poisson–Voronoi cells, larger than any before. Full compatibility is found between the Monte Carlo data and the theory. Deviations from the asymptotic predictions are explained in terms of subleading corrections whose powers in n we estimate from the data.