Pair correlation function in random sequential adsorption processes

Random sequential adsorption of aligned regular polygons and rounded squares: Transition in the kinetics of packing growth

We study two-dimensional random sequential adsorption (RSA) of flat polygons and rounded squares aligned in parallel to find a transition in the asymptotic behavior of the kinetics of packing growth. Differences in the kinetics for RSA of disks and parallel squares were confirmed in previous analytical and numerical reports. Here, by analyzing the two classes of shapes in question we can precisely control the shape of the packed figures and thus localize the transition. Additionally, we study how the asymptotic properties of the kinetics depend on the packing size. We also provide accurate estimations of saturated packing fractions. The microstructural properties of generated packings are analyzed in terms of the density autocorrelation function.

Treating random sequential addition via the replica method

While many physical processes are non-equilibrium in nature, the theory and modeling of such phenomena lag behind theoretical treatments of equilibrium systems. The diversity of powerful theoretical tools available to describe equilibrium systems has inspired strategies that map non-equilibrium systems onto equivalent equilibrium analogs so that interrogation with standard statistical mechanical approaches is possible. In this work, we revisit the mapping from the non-equilibrium random sequential addition process onto an equilibrium multi-component mixture via the replica method, allowing for theoretical predictions of non-equilibrium structural quantities. We validate the above approach by comparing the theoretical predictions to numerical simulations of random sequential addition.

Random sequential adsorption of oriented rectangles with random aspect ratio

We studied random sequential adsorption (RSA) of parallel rectangles with random aspect ratio but fixed area using a newly developed algorithm that allows to generate strictly saturated packing of this kind. We determined saturated packing fraction for several different distributions of a random variable used for selecting side length ratio of deposited rectangles. It was also shown that the anisotropy of deposited rectangles changes during packing generation. Additionally, we examined the kinetics of packing growth, which near saturation obeys the power law with the exponent 1/d≈1/3, typical for the RSA of unoriented anisotropic shapes on a two-dimensional surface. Kinetics in the low coverage limit is determined using the concept of the available surface function. The microstructural properties of obtained random packings are evaluated in terms of two-point density correlation function.

Paris car parking problem for partially oriented discorectangles on a line

The random sequential adsorption (RSA) of identical elongated particles (discorectangles) on a line ("Paris car parking problem") was studied numerically. An off-lattice model with continuous positional and orientational degrees of freedom was considered. The possible orientations of the discorectanles were restricted between θ∈[-θ_{m};θ_{m}] while the aspect ratio (length-to-width ratio) for the discorectangles was varied within the range ɛ∈[1;100]. Additionally, the limiting case ɛ=∞ (i.e., widthless sticks) was considered. We observed that the RSA deposition for the problem under consideration was governed by the formation of rarefied holes (containing particles oriented along a line) surrounded by comparatively dense stacks (filled with almost parallel particles oriented in the vertical direction). The kinetics of the changes of the order parameter and the packing density are discussed. Partial ordering of the discorectangles significantly affected the packing density at the jamming state, φ_{j}, and shifted the cusps in the φ_{j}(ɛ) dependencies. This can be explained by the effects on the competition between the particles' orientational degrees of freedom and the excluded volume effects.

Kinetics of random sequential adsorption of two-dimensional shapes on a one-dimensional line

Saturated random sequential adsorption packings built of two-dimensional ellipses, spherocylinders, rectangles, and dimers placed on a one-dimensional line are studied to check analytical prediction concerning packing growth kinetics [A. Baule, Phys. Rev. Lett. 119, 028003 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.028003]. The results show that the kinetics is governed by the power law with the exponent d=1.5 and 2.0 for packings built of ellipses and rectangles, respectively, which is consistent with analytical predictions. However, for spherocylinders and dimers of moderate width-to-height ratio, a transition between these two values is observed. We argue that this transition is a finite-size effect that arises for spherocylinders due to the properties of the contact function. In general, it appears that the kinetics of packing growth can depend on packing size even for very large packings.

Saturated random packing built of arbitrary polygons under random sequential adsorption protocol

Random packings and their properties are a popular and active field of research. Numerical algorithms that can efficiently generate them are useful tools in their study. This paper focuses on random packings produced according to the random sequential adsorption (RSA) protocol. Developing the idea presented by G. Zhang [Phys. Rev. E 97, 043311 (2018)2470-004510.1103/PhysRevE.97.043311], where saturated random packings built of regular polygons were studied, we create an algorithm that generates strictly saturated packings built of any polygons. Then, the algorithm was used to determine the packing fractions for arbitrary triangles. The highest mean packing density, 0.552814±0.000063, was observed for triangles of side lengths 0.63:1:1. Additionally, microstructural properties of such packings, kinetics of their growth, as well as distributions of saturated packing fractions and the number of RSA iterations needed to reach saturation were analyzed.

Random sequential adsorption of particles with tetrahedral symmetry

We study random sequential adsorption (RSA) of a class of solids that can be obtained from a cube by specific cutting of its vertices, in order to find out how the transition from tetrahedral to octahedral symmetry affects the densities of the resulting jammed packings. We find that in general solids of octahedral symmetry form less dense packing; however, the lowest density was obtained for the packing built of tetrahedra. The densest packing is formed by a solid close to a tetrahedron but with vertices and edges slightly cut. Its density is θ_{max}=0.41278±0.00059 and is higher than the mean packing fraction of spheres or cuboids but is lower than that for the densest RSA packings built of ellipsoids or spherocylinders. The density autocorrelation function of the studied packings is typical for random media and vanishes very quickly with distance.

Random sequential adsorption of Platonic and Archimedean solids

The aim of this study is the analysis of packings generated according to random sequential adsorption protocol consisting of identical Platonic and Archimedean solids. The computer simulations performed showed that the highest saturated packing fraction θ=0.40210(68) is reached by packings build of truncated tetrahedra and the smallest one θ=0.35635(67) by packings composed of regular tetrahedra. The propagation of translational and orientational order exhibited microstructural properties typically seen in random sequential adsorption packings and the kinetics of three-dimensional packings growth were again observed not to be strictly connected with the dimension of the configuration space. Moreover, a fast overlap criterion for Platonic and Archimedean solids based on separating axis theorem has been described. The criterion, together with other optimizations, allowed us to generate significantly larger packings, which translated directly to a lower statistical error of the results obtained. Additionally, the polyhedral order parameters provided can be utilized in other studies regarding particles of polyhedral symmetry.

Random sequential adsorption of cuboids

The subject of this study was random sequential adsorption of cuboids of axes length ratio of a : 1 : b for a ∈ [0.3, 1.0] and b ∈ [1.0, 2.0], and the aim of this study was to find a shape that provides the highest packing fraction. The obtained results show that the densest packing fraction is 0.401 87 ± 0.000 97 and is reached for axes ratios near cuboids of 0.75:1:1.30. Kinetics of packing growth was also studied, and it was observed that its power-law character seems not to be governed by the number of cuboid degrees of freedom. The microstructural properties of obtained packings were studied in terms of density correlation function and propagation of orientational ordering.

Precise algorithm to generate random sequential adsorption of hard polygons at saturation

Random sequential adsorption (RSA) is a time-dependent packing process, in which particles of certain shapes are randomly and sequentially placed into an empty space without overlap. In the infinite-time limit, the density approaches a "saturation" limit. Although this limit has attracted particular research interest, the majority of past studies could only probe this limit by extrapolation. We have previously found an algorithm to reach this limit using finite computational time for spherical particles and could thus determine the saturation density of spheres with high accuracy. In this paper, we generalize this algorithm to generate saturated RSA packings of two-dimensional polygons. We also calculate the saturation density for regular polygons of three to ten sides and obtain results that are consistent with previous, extrapolation-based studies.

Surface fine structure influence on saturated random packings

Random packings of disks on a mesh are studied numerically using random sequential adsorption algorithm. The mesh is built of straight horizontal and vertical one-dimensional lines of a given distance between them. The packing fraction and structure as well as the kinetics of packing growth dependence on mesh size are analyzed to provide information, whether surface inhomogeneity will affect the properties of random packings. It has been shown that the number of disks in a packing slightly decreases with growing distance between mesh lines while the kinetics may change significantly even for very dense meshes. As packings obtained in random sequential adsorption resemble monolayers produced by irreversible adsorption processes, results of this study show that by measuring properties of a random packing it may be possible to determine fine structure of an underlying surface.

Random sequential adsorption of starlike particles

Random packing of surfaceless starlike particles built of 3 to 50 line segments was studied using random sequential adsorption algorithm. Numerical simulations allow us to determine saturated packing densities as well as the first two virial expansion coefficients for such objects. Measured kinetics of the packing growth supports the power law known to be valid for particles with a finite surface; however, the dependence of the exponent in this law on the number of star arms is unexpected. The density autocorrelation function shows fast superexponential decay as for disks, but the typical distance between closest stars is much smaller than between disks of the similar size, especially for a small number of arms.

Random packing of regular polygons and star polygons on a flat two-dimensional surface

Random packing of unoriented regular polygons and star polygons on a two-dimensional flat continuous surface is studied numerically using random sequential adsorption algorithm. Obtained results are analyzed to determine the saturated random packing ratio as well as its density autocorrelation function. Additionally, the kinetics of packing growth and available surface function are measured. In general, stars give lower packing ratios than polygons, but when the number of vertexes is large enough, both shapes approach disks and, therefore, properties of their packing reproduce already known results for disks.

Properties of random sequential adsorption of generalized dimers

Saturated random packing of particles built of two identical, relatively shifted spheres in two- and three-dimensional flat and homogeneous space was studied numerically using a random sequential adsorption algorithm. The shift between centers of spheres varied from 0.0 to 8.0 sphere diameters. Numerical simulations allowed the determination of random sequential adsorption kinetics, the saturated random coverage ratio, as well as the available surface function and density autocorrelation function.

Studying nanostructure gradients in injection-molded polypropylene/montmorillonite composites by microbeam small-angle x-ray scattering

The core-shell structure in oriented cylindrical rods of polypropylene (PP) and nanoclay composites (NCs) from PP and montmorillonite (MMT) is studied by microbeam small-angle x-ray scattering (SAXS). The structure of neat PP is almost homogeneous across the rod showing regular semicrystalline stacks. In the NCs the discrete SAXS of arranged crystalline PP domains is limited to a skin zone of 300 μm thickness. Even there only frozen-in primary lamellae are detected. The core of the NCs is dominated by diffuse scattering from crystalline domains placed at random. The SAXS of the MMT flakes exhibits a complex skin-core gradient. Both the direction of the symmetry axis and the apparent perfection of flake-orientation are varying. Thus there is no local fiber symmetry, and the structure gradient cannot be reconstructed from a scan across the full rod. To overcome the problem the rods are machined. Scans across the residual webs are performed. For the first time webs have been carved out in two principal directions. Comparison of the corresponding two sets of SAXS patterns demonstrates the complexity of the MMT orientation. Close to the surface (< 1 mm) the flakes cling to the wall. The variation of the orientation distribution widths indicates the presence of both MMT flakes and grains. The grains have not been oriented in the flowing melt. An empirical equation is presented which describes the variation from skin to core of one component of the inclination angle of flake-shaped phyllosilicate filler particles.

Random sequential adsorption of trimers and hexamers

Adsorption of trimers and hexamers built of identical spheres was studied numerically using the random sequential adsorption (RSA) algorithm. Particles were adsorbed on a two-dimensional, flat and homogeneous surface. Numerical simulations allowed us to determine the maximal random coverage ratio, RSA kinetics as well as the available surface function (ASF), which is crucial for determining the kinetics of the adsorption process obtained experimentally. Additionally, the density autocorrelation function was measured. All the results were compared with previous results obtained for spheres, dimers and tetramers.

Continuum random sequential adsorption of polymer on a flat and homogeneous surface

Random sequential adsorption (RSA) of polymer, modeled as a chain of identical spheres, is systematically studied. In order to control precisely anisotropy and number of degrees of freedom, two different kinds of polymers are used. In the first one, monomers are placed along a straight line, whereas in the second, relative orientations of particles are random. Such polymers fill a flat homogeneous surface randomly. The paper focuses on maximal random coverage ratio and adsorption kinetics dependence on polymer size, shape anisotropy, and numbers of degrees of freedom. Obtained results were discussed and compared with other numerical experiments and theoretical predictions.

Correlation effects in sequential energy branching: an exactly solvable model of Fano statistics

Correlation effects in the fluctuation of the number of particles in the process of energy branching by sequential impact ionizations are studied using an exactly soluble model of random parking on a line. The Fano factor F calculated in an uncorrelated final-state "shot-glass" model does not give an accurate answer even with the exact gap-distribution statistics. Allowing for the nearest-neighbor correlation effects gives a correction to F that brings F very close to its exact value. We discuss the implications of our results for energy resolution of semiconductor gamma detectors, where the value of F is of the essence. We argue that F is controlled by correlations in the cascade energy branching process and hence the widely used final-state model estimates are not reliable--especially in the practically relevant cases when the energy branching is terminated by competition between impact ionization and phonon emission.

Random sequential addition of hard spheres in high Euclidean dimensions

Sphere packings in high dimensions have been the subject of recent theoretical interest. Employing numerical and theoretical methods, we investigate the structural characteristics of random sequential addition (RSA) of congruent spheres in d -dimensional Euclidean space R{d} in the infinite-time or saturation limit for the first six space dimensions (1< or =d < or =6) . Specifically, we determine the saturation density, pair correlation function, cumulative coordination number and the structure factor in each of these dimensions. We find that for 2< or =d <or =6 , the saturation density phi{s} scales with dimension as phi{s}=c{1}/2{d}+c{2}d/2{d} , where c{1}=0.202048 and c{2}=0.973872 . We also show analytically that the same density scaling is expected to persist in the high-dimensional limit, albeit with different coefficients. A byproduct of this high-dimensional analysis is a relatively sharp lower bound on the saturation density for any d given by phi{s}> or =(d+2)(1-S{0})2;{d+1} , where S{0}[0,1] is the structure factor at k=0 (i.e., infinite-wavelength number variance) in the high-dimensional limit. We demonstrate that a Palàsti-type conjecture (the saturation density in R{d} is equal to that of the one-dimensional problem raised to the d th power) cannot be true for RSA hyperspheres. We show that the structure factor S(k) must be analytic at k=0 and that RSA packings for 1< or =d< or =6 are nearly "hyperuniform." Consistent with the recent "decorrelation principle," we find that pair correlations markedly diminish as the space dimension increases up to six. We also obtain kissing (contact) number statistics for saturated RSA configurations on the surface of a d -dimensional sphere for dimensions 2< or =d< or =5 and compare to the maximal kissing numbers in these dimensions. We determine the structure factor exactly for the related "ghost" RSA packing in R{d} and demonstrate that its distance from "hyperuniformity" increases as the space dimension increases, approaching a constant asymptotic value of 12 . Our work has implications for the possible existence of disordered classical ground states for some continuous potentials in sufficiently high dimensions.

Early oriented isothermal crystallization of polyethylene studied by high-time-resolution SAXS/WAXS

During cooling from the quiescent melt of a highly oriented polyethylene rod, highly oriented proto-lamellae are formed first, which are not crystalline. This is shown in scattering data which are recorded on two-dimensional detectors with a cycle time of 1 s and an exposure of 0.1 s. In the experiments small-angle X-ray scattering (SAXS) and wide-angle X-ray scattering (WAXS) are registered simultaneously during the first 3 min after quenching to a crystallization temperature. A non-uniform thickness between 20 and 100 nm is characteristic for the ensemble of proto-lamellae. During the first minute of isothermal treatment the number of proto-lamellae slowly increases without a change of the thickness distribution. As crystallization starts, the crystallites are not oriented in contrast to the proto-lamellae. During crystallization the layer thickness distribution narrows. The number of lamellae rapidly increases during the following 2 min of isothermal treatment (at 128 degrees C and 124 degrees C). The results are obtained by interpretation of the WAXS and of the multidimensional chord distribution function (CDF), a model-free real-space visualization of the nanostructure information contained in the SAXS data.

DNA charge neutralization by linear polymers. II. Reversible binding

We model the way in which polymers bind to DNA and neutralize its charged backbone by analyzing the dynamics of the distribution of gaps along the DNA. We generalize existing theory for irreversible binding to construct deterministic models which include polymer removal, movement along the DNA, and allow for binding with overlaps. We show that reversible binding alters the capacity of the DNA for polymers by allowing the rearrangement of polymer positions over a longer time scale than when binding is irreversible. When the polymers do not overlap, allowing reversible binding increases the number of polymers adhered and hence the charge that the DNA can accommodate; in contrast, when overlaps occur, reversible binding reduces the amount of charge neutralized by the polymers.

DNA charge neutralization by linear polymers: irreversible binding

We develop a deterministic mathematical model to describe the way in which polymers bind to DNA by considering the dynamics of the gap distribution that forms when polymers bind to a DNA plasmid. In so doing, we generalize existing theory to account for overlaps and binding cooperativity whereby the polymer binding rate depends on the size of the overlap. The proposed mean-field models are then solved using a combination of numerical and asymptotic methods. We find that overlaps lead to higher coverage and hence higher charge neutralizations, results which are more in line with recent experimental observations. Our work has applications to gene therapy where polymers are used to neutralize the negative charges of the DNA phosphate backbone, allowing condensation prior to delivery into the nucleus of an abnormal cell.

Statistical mechanical description of the parking-lot model for vibrated granular materials

We apply the statistical mechanical approach proposed by Edwards and co-workers to the parking-lot model, a model that reproduces the main features of the phenomenology of vibrated granular materials. We first build the compactivity-based measure for the case of vanishingly small tapping strength and then generalize the approach to finite tapping strengths by introducing a "thermodynamic" parameter, the available volume for particle insertion, in addition to the particle density. This description is able to take into account the various memory effects observed in vibrated granular media. Although not exact, the approach gives a good description of the behavior of the parking-lot model in the regime of slow compaction.

Adsorption-desorption model and its application to vibrated granular materials

We investigate both analytically and by numerical simulation the kinetics of a microscopic model of hard rods adsorbing on a linear substrate, a model that is relevant for compaction of granular materials. The computer simulations use an event-driven algorithm that is particularly efficient at very long times. For a small, but finite desorption rate, the system reaches an equilibrium state very slowly, and the long-time kinetics display three successive regimes: an algebraic one where the density varies as 1/t, a logarithmic one where the density varies as 1/ln(t), followed by a terminal exponential approach. The characteristic relaxation time of the final regime, though incorrectly predicted by mean field arguments, can be obtained with a systematic gap-distribution approach. The density fluctuations at equilibrium are also investigated, and the associated time-dependent correlation function exhibits a power law regime followed by a final exponential decay. Finally, we show that denser particle packings can be obtained by varying the desorption rate during the process.