Local anisotropy of fluids using Minkowski tensors

Abstract

Statistics of the free volume available to individual particles have previously been studied for simple and complex fluids, granular matter, amorphous solids, and structural glasses. Minkowski tensors provide a set of shape measures that are based on strong mathematical theorems and easily computed for polygonal and polyhedral bodies such as free volume cells (Voronoi cells). They characterize the local structure beyond the two-point correlation function and are suitable to define indices 0 ≤ β a,b ν ≤ 1 of local anisotropy. Here, we analyze the statistics of Minkowski tensors for configurations of simple liquid models, including the ideal gas (Poisson point process), the hard disks and hard spheres ensemble, and the Lennard-Jones fluid. We show that Minkowski tensors provide a robust characterization of local anisotropy, which ranges from β a,b ν ≈ 0.3 for vapor phases to β a,b ν → 1 for ordered solids. We find that for fluids, local anisotropy decreases monotonously with increasing free volume and randomness of particle positions. Furthermore, the local anisotropy indices β a,b ν are sensitive to structural transitions in these simple fluids, as has been previously shown in granular systems for the transition from loose to jammed bead packs.

Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter

“…The morphometric neighborhood has previously been characterized using Minkowski tensors, 38,53,61 which measure the distribution of normal vectors of the Voronoi cells. The Minkowski structure metrics presented here can be interpreted as the rotational invariants of a multipole expansion of the same distribution of normal vectors; indeed, the approaches of higher-rank Minkowski tensors and Minkowski structure metrics turn out to be mathematically equivalent ways to cure the shortcomings of bond-orientational order parameters.…”

“…Using non-equilibrium molecular dynamics (MD) simulations, [52][53][54] super-cooled configurations are generated that represent entirely disordered states with densities larger than the fluid-crystal coexistence density of hard spheres (HS) of φ ≈ 0.494. , and q D 6 differ significantly, which is important when comparing these values to that of a specific crystalline phase such as fcc. Second, and of greater concern for the use of q 6 as a structure metric, the behavior of q r c =1.…”

“…The moment tensors W 0,l 1 are special types of Minkowski tensors. 53,59 These versatile shape metrics have been studied in the field of integral geometry 60 and successfully applied to analyze structure in jammed bead packs, 38,61 bi-phasic assemblies, 62,63 foams, 64 and other cellular structures. 59,64 There is a one-to-one correspondence between this class of Minkowski tensors and the multipole expansion of the surface normal vector distribution ρ(n) of a convex Voronoi polytope F(a).…”

“…Figure 5 shows q 2 , q 4 , and q 6 of hard-sphere systems in a wide range of packing fractions. The plot includes data from Monte Carlo (MC) simulations of the thermal equilibrium fluid/solid, 53 from fully disordered and partially crystalline jammed Lubachevsky-Stillinger (jLS), 61,69 and also from unjammed non-equilibrium simulations (uLS) from LS simulations before jamming, 70 and the data from Fig. 2 (Matsumoto algorithm (MA-MD)).…”

Tensorial Minkowski functionals of triply periodic minimal surfaces

“…The sensitivity of the technique has been studied in Arns et al [85]. With these algorithms, we discovered an intrinsic local anisotropy in granular bead packs [58], in fluids [86] and porous solid foams [57], and (by an analysis of higher-rank tensors) onset of crystallization in jammed spherical bead packs [87]. Minkowski functionals have been used to derive a density functional theory for fluids of non-spherical particles [88,89], possibly applicable to nematic fluids confined in pores.…”