Maximally random jamming of one-component and binary hard-disk fluids in two dimensions

Critical pore radius and transport properties of disordered hard- and overlapping-sphere models

Transport properties of porous media are intimately linked to their pore-space microstructures. We quantify geometrical and topological descriptors of the pore space of certain disordered and ordered distributions of spheres, including pore-size functions and the critical pore radius δ_{c}. We focus on models of porous media derived from maximally random jammed sphere packings, overlapping spheres, equilibrium hard spheres, quantizer sphere packings, and crystalline sphere packings. For precise estimates of the percolation thresholds, we use a strict relation of the void percolation around sphere configurations to weighted bond percolation on the corresponding Voronoi networks. We use the Newman-Ziff algorithm to determine the percolation threshold using universal properties of the cluster size distribution. The critical pore radius δ_{c} is often used as the key characteristic length scale that determines the fluid permeability k. A recent study [Torquato, Adv. Wat. Resour. 140, 103565 (2020)10.1016/j.advwatres.2020.103565] suggested for porous media with a well-connected pore space an alternative estimate of k based on the second moment of the pore size 〈δ^{2}〉, which is easier to determine than δ_{c}. Here, we compare δ_{c} to the second moment of the pore size 〈δ^{2}〉, and indeed confirm that, for all porosities and all models considered, δ_{c}^{2} is to a good approximation proportional to 〈δ^{2}〉. However, unlike 〈δ^{2}〉, the permeability estimate based on δ_{c}^{2} does not predict the correct ranking of k for our models. Thus, we confirm 〈δ^{2}〉 to be a promising candidate for convenient and reliable estimates of the fluid permeability for porous media with a well-connected pore space. Moreover, we compare the fluid permeability of our models with varying degrees of order, as measured by the τ order metric. We find that (effectively) hyperuniform models tend to have lower values of k than their nonhyperuniform counterparts. Our findings could facilitate the design of porous media with desirable transport properties via targeted pore statistics.

Bidisperse Nanospheres Jammed on a Liquid Surface

Jammed packings of bidisperse nanospheres were assembled on a nonvolatile liquid surface and visualized to the single-particle scale by using an in situ scanning electron microscopy method. The PEGylated silica nanospheres, mixed at different number fractions and size ratios, had large enough in-plane mobilities prior to jamming to form uniform monolayers reproducibly. From the collected nanometer-resolution images, local order and degree of mixing were assessed by standard metrics. For equimolar mixtures, a large-to-small size ratio of about 1.5 minimized correlated metrics for local orientational and positional order, as previously predicted in simulations of bidisperse disk jamming. Despite monolayer uniformity, structural and depletion interactions caused spheres of a similar size to cluster, a feature evident at size ratios above 2. Uniform nanoparticle monolayers of high packing disorder are sought in many liquid interface technologies, and these experiments outlined key design principles, buttressing extensive theory/simulation literature on the topic.

Tiled Monolayer Films of 2D Molybdenum Disulfide Nanoflakes Assembled at Liquid/Liquid Interfaces

Thin films of MoS₂ bilayer nanoflakes, which are predominantly a single flake thick and with flakes in edge-to-edge contact, have been produced via self-assembled tiling at the planar interface between two immiscible liquids. Films of several square centimeters extent can be produced with a total covered area approaching 90% and over 70% of the film covered by single flakes without overlap. Films produced through liquid/liquid assembly are shown to produce a lower uncovered area fraction and more uniform thickness when compared with films of similar areal coverage produced by the "top-down" techniques of spin coating and spray coating. Statistical analysis of flake coverage data, measured by atomic force microscopy (AFM), shows that liquid/liquid assembly produces a distinctly different variation in film thickness than conventional top-down deposition. This supports the hypothesis that the two-dimensional (2D) confinement of liquid/liquid assembly produces more uniform films. Demonstrator field-effect transistors (FETs) manufactured from the films exhibit mobility and on/off current ratios of 0.73 cm² V^-1 s^-1 and 10⁵, respectively, comparable to FETs of similar layout and chemical vapor deposition (CVD)-grown or mechanically cleaved single-crystal MoS₂ channel material. This work demonstrates the use of liquid/liquid interfaces as a useful tool for the self-assembly of high-performance thin-film devices made from dispersions of 2D materials.

Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions

In the first two papers of this series, we characterized the structure of maximally random jammed (MRJ) sphere packings across length scales by computing a variety of different correlation functions, spectral functions, hole probabilities, and local density fluctuations. From the remarkable structural features of the MRJ packings, especially its disordered hyperuniformity, exceptional physical properties can be expected. Here we employ these structural descriptors to estimate effective transport and electromagnetic properties via rigorous bounds, exact expansions, and accurate analytical approximation formulas. These property formulas include interfacial bounds as well as universal scaling laws for the mean survival time and the fluid permeability. We also estimate the principal relaxation time associated with Brownian motion among perfectly absorbing traps. For the propagation of electromagnetic waves in the long-wavelength limit, we show that a dispersion of dielectric MRJ spheres within a matrix of another dielectric material forms, to a very good approximation, a dissipationless disordered and isotropic two-phase medium for any phase dielectric contrast ratio. We compare the effective properties of the MRJ sphere packings to those of overlapping spheres, equilibrium hard-sphere packings, and lattices of hard spheres. Moreover, we generalize results to micro- and macroscopically anisotropic packings of spheroids with tensorial effective properties. The analytic bounds predict the qualitative trend in the physical properties associated with these structures, which provides guidance to more time-consuming simulations and experiments. They especially provide impetus for experiments to design materials with unique bulk properties resulting from hyperuniformity, including structural-color and color-sensing applications.

Equation of state of polydisperse hard-disk mixtures in the high-density regime

A proposal to link the equation of state of a monocomponent hard-disk fluid to the equation of state of a polydisperse hard-disk mixture is presented. Event-driven molecular dynamics simulations are performed to obtain data for the compressibility factor of the monocomponent fluid and of 26 polydisperse mixtures with different size distributions. Those data are used to assess the proposal and to infer the values of the compressibility factor of the monocomponent hard-disk fluid in the metastable region from those of mixtures in the high-density region. The collapse of the curves for the different mixtures is excellent in the stable region. In the metastable regime, except for two mixtures in which crystallization is present, the outcome of the approach exhibits a rather good performance. The simulation results indicate that a (reduced) variance of the size distribution larger than about 0.01 is sufficient to avoid crystallization and explore the metastable fluid branch.

Analytic solution of the Ornstein-Zernike relation for inhomogeneous liquids

The properties of a classical simple liquid are strongly affected by the application of an external potential that supports inhomogeneity. To understand the nature of these property changes, the equilibrium particle distribution functions of the liquid have, typically, been calculated directly using either integral equation or density functional based analyses. In this study, we develop a different approach with a focus on two distribution functions that characterize the inhomogeneous liquid: the pair direct correlation function c(r₁,r₂) and the pair correlation function g(r₁,r₂). With g(r₁,r₂) considered to be an experimental observable, we solve the Ornstein-Zernike equation for the inhomogeneous liquid to obtain c(r₁,r₂), using information about the well studied and resolved g⁽⁰⁾(r₁,r₂) and c⁽⁰⁾(r₁,r₂) for the parent homogeneous (⁽⁰⁾) system. In practical cases, where g(r₁,r₂) is available from experimental data in a discrete form, the resulting c(r₁,r₂) is expressed as an explicit function of g(r₁,r₂) in a discrete form. A weaker continuous form of solution is also obtained, in the form of an integral equation with finite integration limits. The result obtained with our formulation is tested against the exact solutions for the correlation and distribution functions of a one-dimensional inhomogeneous hard rod liquid. Following the success of that test, the formalism is extended to higher dimensional systems with explicit consideration of the two-dimensional liquid.

Characterization of maximally random jammed sphere packings. II. Correlation functions and density fluctuations

In the first paper of this series, we introduced Voronoi correlation functions to characterize the structure of maximally random jammed (MRJ) sphere packings across length scales. In the present paper, we determine a variety of different correlation functions that arise in rigorous expressions for the effective physical properties of MRJ sphere packings and compare them to the corresponding statistical descriptors for overlapping spheres and equilibrium hard-sphere systems. Such structural descriptors arise in rigorous bounds and formulas for effective transport properties, diffusion and reactions constants, elastic moduli, and electromagnetic characteristics. First, we calculate the two-point, surface-void, and surface-surface correlation functions, for which we derive explicit analytical formulas for finite hard-sphere packings. We show analytically how the contact Dirac delta function contribution to the pair correlation function g_{2}(r) for MRJ packings translates into distinct functional behaviors of these two-point correlation functions that do not arise in the other two models examined here. Then we show how the spectral density distinguishes the MRJ packings from the other disordered systems in that the spectral density vanishes in the limit of infinite wavelengths; i.e., these packings are hyperuniform, which means that density fluctuations on large length scales are anomalously suppressed. Moreover, for all model systems, we study and compute exclusion probabilities and pore size distributions, as well as local density fluctuations. We conjecture that for general disordered hard-sphere packings, a central limit theorem holds for the number of points within an spherical observation window. Our analysis links problems of interest in material science, chemistry, physics, and mathematics. In the third paper of this series, we will evaluate bounds and estimates of a host of different physical properties of the MRJ sphere packings that are based on the structural characteristics analyzed in this paper.

Lateral diffusion in a discrete fluid membrane with immobile particles

Due to the coupling between the plasma membrane and the actin cytoskeleton, membrane molecules such as receptor proteins can become immobilized by binding to cytoskeletal structures. We investigate the effect of immobile membrane molecules on the diffusion of mobile ones by modeling the membrane as a two-dimensional (2D) fluid composed of hard particles and performing event-driven molecular dynamics simulations at a particle density where the system is in an isotropic liquid state. We show that the diffusion coefficient sharply decreases with increasing immobile fraction, dropping by a factor of ∼ 3 as the fraction of immobile particles increases from 0 to 0.1, in a system-size dependent manner. By combining our results with earlier calculations, we estimate that a factor-of-∼ 20 reduction in diffusion coefficients in live cell membranes, a puzzling finding in cell biology, can be accounted for when less than ∼ 22% of the particles in our model system is immobilized. Furthermore, we investigate the effects of confinement induced by a correlated distribution of immobile particles by calculating the distribution of the time it takes for particles to escape from a corral. In the regime where the particles can always escape from the corral, it is found that the escape times follow an exponential distribution, and the mean escape time grows exponentially with the density of obstacles at the corral boundary, increasing by a factor of 3-5 when immobile particles cover 50% of the boundary, and is approximately proportional to the area of the corral. We believe that our findings will be useful in interpreting (1) single molecule observations of membrane molecules and (2) results of particle based simulations that explore the effect of fluid dynamics on molecular transport in a 2D fluid.

Shape-induced chiral ordering in two-dimensional packing of snowmanlike dimeric particles

Understanding the distinctive phase behaviors in random packing due to particle shapes is an important issue in condensed matter physics. In this paper, we investigate the random packing structure of two-dimensional (2D) snowmen via wax-snowman packing experiments and Brownian dynamics simulations. Both experiments and simulations reveal that neighboring snowmen have a strong short-range orientational correlation and consequently locally form particular conformations. A chiral conformation is dominant for high area fractions near the jamming condition (φ>0.8), and the proportion of the chiral conformation increases with γ. We also found that the attractive interaction does not have a significant impact on the results. The geometry of chirally ordered snowmen causes a mismatch with 2D crystalline symmetries and thus inhibits the development of long-range spatial order, despite the strong orientational correlation between neighbors.

Solid-liquid transition in polydisperse Lennard-Jones systems

We study melting of a face-centered crystalline solid consisting of polydisperse Lennard-Jones spheres with Gaussian polydispersity in size. The phase diagram reproduces the existence of a nearly temperature invariant terminal polydispersity (δ(t) =/~ 0.11), with no signature of reentrant melting. The absence of reentrant melting can be attributed to the influence of the attractive part of the potential upon melting. We find that at terminal polydispersity the fractional density change approaches zero, which seems to arise from vanishingly small compressibility of the disordered phase. At constant temperature and volume fraction the system undergoes a sharp transition from crystalline solid to the disordered amorphous or fluid state with increasing polydispersity. This has been quantified by second- and third-order rotational invariant bond orientational order, as well as by the average inherent structure energy. The translational order parameter also indicates a similar sharp structural change at δ =/~ 0.09 in case of T(*) = 1.0, φ = 0.58. The free energy calculation further supports the sharp nature of the transition. The third-order rotationally invariant bond order shows that with increasing polydispersity, the local cluster favors a more icosahedral arrangement and the system loses its local crystalline symmetry. Interestingly, the value of structure factor S(k) of the amorphous phase at δ =/~ 0.10 (just beyond the solid-liquid transition density at T(*) = 1) becomes 2.75, which is below the value of 2.85 required for freezing given by the empirical Hansen-Verlet rule of crystallization, well known in the theory of freezing.

Phase transitions in four-dimensional binary hard hypersphere mixtures

Previous Monte Carlo investigations of binary hard hyperspheres in four-dimensional mixtures are extended to higher densities where the systems may solidify. The ratios of the diameters of the hyperspheres examined were 0.4, 0.5, and 0.6. Only the 0.4 system shows a clear two phase, solid-liquid transition and the larger component solidifies into a D4 crystal state. Its pair correlation function agrees with that of a one component fluid at an appropriately scaled density. The 0.5 systems exhibit states that are a mix of D4 and A4 regions. The 0.6 systems behave similarly to a jammed state rather than solidifying into a crystal. No demixing into two distinct fluid phases was observed for any of the simulations.

Monte Carlo study of four dimensional binary hard hypersphere mixtures

A multithreaded Monte Carlo code was used to study the properties of binary mixtures of hard hyperspheres in four dimensions. The ratios of the diameters of the hyperspheres examined were 0.4, 0.5, 0.6, and 0.8. Many total densities of the binary mixtures were investigated. The pair correlation functions and the equations of state were determined and compared with other simulation results and theoretical predictions. At lower diameter ratios the pair correlation functions of the mixture agree with the pair correlation function of a one component fluid at an appropriately scaled density. The theoretical results for the equation of state compare well to the Monte Carlo calculations for all but the highest densities studied.