Local structure characterization in particle systems

TL;DR

This work surveys a broad set of local-structure metrics for particulate systems, spanning coordination-based, radial- and angular-order descriptors, environment matching, structure identification, and machine-learned approaches. It ties these tools to concrete calculations and practical considerations, illustrating crystallization, stacking faults, and phase transitions with representative soft-matter examples and software recommendations. A key contribution is organizing methods from simple to advanced, highlighting how combined metrics yield robust structural insights and practical guidance for analyzing real data. The paper's significance lies in providing a practical roadmap for researchers to diagnose and track local order in simulations and experiments using widely available software and well-grounded theory.

Abstract

Many tools and techniques measure local structure in materials in contexts ranging from biology to geology. We provide a survey of those tools and metrics that are especially useful for analyzing particulate soft matter. The metrics we discuss can all be computed from the positions of particles, and are thus most useful when there is access to this information, either from simulation or experimental imaging. For each metric, we provide derivations, intuition regarding its implications, example uses, and references to software packages that compute the metric. Our survey encompasses characterization techniques ranging from the simplest to the most complex, and will be useful for students getting started in the structural characterization of particle systems.

Figures (14)

• Figure 1: The extraction of an environment vector set from a local neighborhood. (a) For any set of particle neighbors near a central particle, (b) a set of vectors can be defined pointing from the center of the central particle $i$ to the centers of its neighbors, indexed by $j$. (c) This vector set, denoted as $\{ \mathbf{r}_{ij} \}$, can be viewed as the abstract representation of any local environment.
• Figure 2: Examples of coordination environments which may require different definitions of the local neighborhood. (a) A square lattice has a single simple nearest-neighbor distance, (b) a rectangular lattice may be defined as having two nearest neighbor distances, and (c) a lattice of ellipsoids may have a complex definition of the distance to a neighbor.
• Figure 3: A schematic representation for calculating $g(r)$ for a a square lattice with noise. The red particle indicates the central reference particle. $g(r)$ is constructed by calculating the number density of particles within each shell moving outward, and comparing to the overall density. This process is repeated taking each particle as the central reference to calculate an average.
• Figure 4: Plots of $g(r)$ for (a) liquid, (b) fcc crystal, and (c) bcc crystal structures. The liquid is generated from 500 particles interacting via the Lennard-Jones potential discussed in Sec. \ref{['section:detectingcrystals']}, with parameters $\varepsilon = 1$ and $\sigma = 1$. The system was simulated via molecular dynamicsAnderson2020 in the NVT ensemble at $k_B T = 1.2$ and number density $\rho = 0.8$; $g(r)$ was calculated from a single system snapshot after equilibration. Each crystal structure consists of 10 replicated unit cells using the UnitCell class within freud, with Gaussian noise of standard deviation 0.03 added to each particle position. Unit cells were scaled so that the nearest-neighbor distance is unity in both structures.
• Figure 5: Local environments may differ primarily in their rotational symmetry. (a) The fcc local environment and (b) the icosahedral environment both consist of 12 particles packed tightly around a central particle, but have significantly different rotational symmetry.