Random Sequential Adsorption Process Research Articles

Density and Correlation in a Random Sequential Adsorption Model

We consider the random sequential adsorption process on the one- dimensional lattice with nearest-neighbor exclusion. In this model, each site s 2 Z starts empty and a particle will be deposited in it at time ts, where (ts)s2Z is a sequence of independent random variables uniformly distributed on the interval [0; 1]. The site will be occupied if both of its neighbors are vacant. Analytical expressions for the density of occupied sites and the pair correlation function, for all time t, are well-established and have been obtained through methods such as generating functions and differential equations. In this study, we present a method based on probabilistic arguments for the calculation of these expressions.

Random adsorption process of linear k-mers on square lattices under the Achlioptas process.

We study the explosive percolation with k-mer random sequential adsorption (RSA) process. We consider both the Achlioptas process (AP) and the inverse Achlioptas process (IAP), in which giant cluster formation is prohibited and accelerated, respectively. By employing finite-size scaling analysis, we confirm that the percolation transitions are continuous, and thus we calculate the percolation threshold and critical exponents. This allows us to determine the universality class of the k-mer explosive percolation transition. Interestingly, the numerical simulation suggests that the universality class of the explosive percolation transition with the AP alters when the k-mer size changes. In contrast, the universality class of the transition with the IAP is independent of k, but it differs from that of the RSA without the IAP.

Multilayer Random Sequential Adsorption

In this work, we present a variant of the multilayer random sequential adsorption (RSA) process that is inspired by orthogonal resource sharing in wireless communication networks. In the one-dimensional (1D) version of this variant, the deposition of overlapping rods is allowed only if they are assigned two different colors, where colors are symbolic of orthogonal resources, such as frequency bands, in communication networks. Owing to a strong spatial coupling among the deposited rods of different colors, finding an exact solution for the density of deposited rods of a given color as a function of time seems intractable. Hence, we propose two useful approximations to obtain the time-varying density of rods of a given color. The first approximation is based on the recursive use of the known monolayer RSA result for the indirect estimation of the density of rods for the multilayer version. The second approximation, which is more accurate but computationally intensive, involves accurate characterization of the time evolution of the gap density function. This gap density function is subsequently used to estimate the density of rods of a given color. We also consider the two-dimensional (2D) version of this problem, where we estimate the time-varying density of deposited circles of a given color as a function of time by extending the first approximation approach developed for the 1D case. The accuracy of all the results is validated through extensive Monte Carlo simulations.

A hard-sphere model of protein corona formation on spherical and cylindrical nanoparticles

A nanoparticle (NP) immersed in biological media rapidly forms a corona of adsorbed proteins, which later controls the eventual fate of the particle and the route through which adverse outcomes may occur. The composition and timescale for the formation of this corona are both highly dependent on both the NP and its environment. The deposition of proteins on the surface of the NP can be imitated by a process of random sequential adsorption, and, based on this model, we develop a rate-equation treatment for the formation of a corona represented by hard spheres on spherical and cylindrical NPs. We find that the geometry of the NP significantly alters the composition of the corona through a process independent of the rate constants assumed for adsorption and desorption of proteins, with the radius and shape of the NP both influencing the corona. We further investigate the roles of protein mobility on the surface of the NP and changes in the concentration of proteins.

Jamming densities of random sequential adsorption on d-dimensional cubic lattices.

The rate of convergence of the jamming densities to their asymptotic high-dimensional tree approximation is studied, for two types of random sequential adsorption (RSA) processes on a d-dimensional cubic lattice. The first RSA process has an exclusion shell around a particle of nearest neighbors in all d dimensions (N1 model). In the second process the exclusion shell consists of a d-dimensional hypercube with length k=2 around a particle (N2 model). For the N1 model the deviation of the jamming density ρ_{r}(d) from its asymptotic high d value ρ_{asy}(d)=ln(1+2d)/2d vanishes as [ln(1+2d)/2d]^{3.41}. In addition, it has been shown that the coefficients a_{n}(d) of the short-time expansion of the occupation density of this model (at least up to n=6) are given for all d by a finite correction sum of order (n-2) in 1/d to their asymptotic high d limit. The convergence rate of the jamming densities of the N2 model to their high d limits ρ_{asy}(d)=dln3/3^{d} is slow. For 2≤d≤4 the generalized Palasti approximation provides by far a better approximation. For higher d values the jamming densities converge monotonically to the above asymptotic limits, and their decay with d is clearly faster than the decay as (0.432332...)^{d} predicted by the generalized Palasti approximation.

Modeling Rydberg gases using random sequential adsorption on random graphs

The statistics of strongly interacting, ultracold Rydberg gases are governed by the interplay of two factors: geometrical restrictions induced by blockade effects, and quantum mechanical effects. To shed light on their relative roles in the statistics of Rydberg gases, we compare three models in this paper: a quantum mechanical model describing the excitation dynamics within a Rydberg gas, a Random Sequential Adsorption (RSA) process on a Random Geometric Graph (RGG), and a RSA process on a Decomposed Random Intersection Graph (DRIG). The latter model is new, and refers to choosing a particular subgraph of a mixture of two other random graphs. Contrary to the former two models, it lends itself for a rigorous mathematical analysis; and it is built specifically to have particular structural properties of a RGG. We establish for it a fluid limit describing the time-evolution of number of Rydberg atoms, and show numerically that the expression remains accurate across a wider range of particle densities than an earlier approach based on an RSA process on an Erdos-Renyi Random Graph (ERRG). Finally, we also come up with a new heuristic using random graphs that gives a recursion to describe a normalized pair-correlation function of a Rydberg gas. Our results suggest that even without dissipation, on long time scales the statistics are affected most by the geometrical restrictions induced by blockade effects, while on short time scales the statistics are affected most by quantum mechanical effects.

Nonlinear dynamical modeling of adsorption and desorption processes with power-law kinetics: Application to CO_{2} capture.

Modeling of random sequential adsorption (RSA) process is studied in this paper as this kind of process is close to the surface adsorption phenomenon that is, for instance, exploited in gas sensors or for liquid or gas purification. Analysis and simulation of the RSA process is first performed to highlight a power-law kinetic behavior. Such behaviors are often modeled in the literature with fractional models. The paper, however, shows that fractional models are not able to capture some important properties of the RSA process. A nonlinear model and the associated parameters tuning method are, thus, proposed. A discussion on the ability of the proposed model to capture the power-law kinetics without exhibiting some of the drawbacks of fractional models is proposed. This nonlinear model is then modified to take into account the reverse desorption process. The proposed modeling approach is applied to experimental data of CO_{2} capture.

Density distribution functions of random sequential adsorption on a [formula omitted] lattice

Random sequential adsorption (RSA) is an irreversible random deposition process with short range exclusion interaction. It is shown that the distribution function of the occupation densities at the jamming limit of a RSA process on a 2×n lattice converges for n→∞ to a Gaussian around an average density. The second moment decreases like 1∕n.

Jammed systems of oriented dimers always percolate on hypercubic lattices

Random sequential adsorption (RSA) is a standard method of modeling adsorption of large molecules at the liquid-solid interface. Here we consider jammed states of the RSA process of nonoverlapping dimers(objects occupying two nearest-neighbor lattice sites) in a hypercubic lattice of arbitrary space dimension D ⩾ 2. We show that each dimer in such a state belongs to a percolating cluster.

A novel FastRSA algorithm: Statistical properties and evolution of microstructure

While the Random Sequential Adsorption (RSA) process for the generation of 2D geometries containing discrete entities has been extensively studied, both in terms of numerical simulations and in terms of its statistical properties, all the mechanisms involved are not fully understood, especially in dense configurations of elongated particles. This is mainly due to the very slow asymptotic approach to high packing configurations, especially when highly elongated particles are involved, which makes the creation of such configurations a time consuming task. For the estimation of the statistical properties of such configurations we therefore have to resort to extrapolations that do not always give accurate results. In this work we reveal the interaction of the mechanisms that come into play in the RSA process. We specifically show that the overall result of an RSA process is the summary outcome of these interwoven mechanisms, namely those involving the formation and destruction of Particle Area, Overlap Area and Influence Area – terms which we introduce and define in this work – resulting in a behavior that often appears counter-intuitive. We also show the shift of their importance as the particle aspect ratio α varies and explain how nematic structures are created when high aspect ratio particles are involved as well as the mechanisms behind their appearance. Following this, we propose a new algorithm for the process of random sequential adsorption (named FastRSA) which is capable of creating very high count configurations through all the range of particle aspect ratios and which follows Feder’s law with a θ∼τ−1∕2 behavior instead of the θ∼τ−1∕3 of the classic approach, where τ is the number of attempts to place a particle and θ is the degree of packing. We show how the new algorithm can be coupled with the classic RSA approach and point out the benefits of such a coupling. Use of the FastRSA algorithm has enabled us to study the evolution of the extent of packing using actual geometries, without the need to resolve to extrapolations and assumptions. For the case of highly elongated particles, this is the first time in our knowledge that estimations of maximum packing from actual configurations near the jamming limit have been obtained.

Effects of Brownian Motions on Electrical Conductivity and Optical Transparency of Two-Dimensional Films Filled by Needle-Like Particles

The effects of Brownian motions on the electrical conductivity and optical transparency of two-dimensional films filled with needle-like particles (needles) have been investigated, using the Monte-Carlo method. The initial state of the system was produced with the use of the random-sequential adsorption process. In the subsequent evolution (aging) of the system, the translation and rotation diffusion motions are taken into account. The intersections between needles are forbidden. The interaction potential between needles is short-range (i.e., it is nonzero at distances less than Rc) and is dependent on the angle between needles ф(∝ cos2 ф). The aging results in the formation of island, net-like, and hole-like (with significant cavities) structures depending on parameters of the interaction potential. The relations between the electrical conductivity and the optical transparency during the aging are discussed.

Random sequential adsorption and its long-time limit

We discuss random sequential adsorption (RSA), a model of an irreversible adsorption process of large noninteracting molecules on a solid surface, and discuss an algorithm that efficiently simulates the RSA process in the long-time limit.

A Computer‐Aided Model for the Simulation of Railway Ballast by Random Sequential Adsorption Process

AbstractThis article presents a computer‐aided multistage methodology for the simulation of railway ballasts using the Random Sequential Adsorption (RSA – 2D domain) paradigm. The primary stage in this endeavor is the numerical generation of a synthetic sample by a “particle sizing and positioning” process followed by a “compaction” process. The synthetic samples of ballast are then visualized in the Computer‐Aided Design (CAD) environment. The outcomes of the simulation are analyzed by comparison with the results of an experimental investigation carried out using a methacrylate container in which real samples of railway ballast are formed. A test of model reliability is carried out between the aggregates number and the grading curves of the synthetic sample and the real one. A validation is therefore performed using the ground‐penetrating radar (GPR) nondestructive testing (NDT) method and the finite‐difference time‐domain (FDTD) simulation developed in a computer‐aided environment. The results prove the viability and the applicability of the proposed modeling for the assessment of railway ballast conditions.

Electrical conductivity of a monolayer produced by random sequential adsorption of linear k-mers onto a square lattice.

The electrical conductivity of a monolayer produced by the random sequential adsorption (RSA) of linear k-mers (particles occupying k adjacent adsorption sites) onto a square lattice was studied by means of computer simulation. Overlapping with predeposited k-mers and detachment from the surface were forbidden. The RSA process continued until the saturation jamming limit, p_{j}. The isotropic (equiprobable orientations of k-mers along x and y axes) and anisotropic (all k-mers aligned along the y axis) depositions for two different models-of an insulating substrate and conducting k-mers (C model) and of a conducting substrate and insulating k-mers (I model)-were examined. The Frank-Lobb algorithm was applied to calculate the electrical conductivity in both the x and y directions for different lengths (k=1 - 128) and concentrations (p=0 - p_{j}) of the k-mers. The "intrinsic electrical conductivity" and concentration dependence of the relative electrical conductivity Σ(p) (Σ=σ/σ_{m} for the C model and Σ=σ_{m}/σ for the I model, where σ_{m} is the electrical conductivity of substrate) in different directions were analyzed. At large values of k the Σ(p) curves became very similar and they almost coincided at k=128. Moreover, for both models the greater the length of the k-mers the smoother the functions Σ_{xy}(p),Σ_{x}(p) and Σ_{y}(p). For the more practically important C model, the other interesting findings are (i)for large values of k (k=64,128), the values of Σ_{xy} and Σ_{y} increase rapidly with the initial increase of p from 0 to 0.1; (ii) for k≥16, all the Σ_{xy}(p) and Σ_{x}(p) curves intersect with each other at the same isoconductivity points; (iii) for anisotropic deposition, the percolation concentrations are the same in the x and y directions, whereas, at the percolation point the greater the length of the k-mers the larger the anisotropy of the electrical conductivity, i.e., the ratio σ_{y}/σ_{x} (>1).

Three-dimensional distribution of cortical synapses: a replicated point pattern-based analysis.

The biggest problem when analyzing the brain is that its synaptic connections are extremely complex. Generally, the billions of neurons making up the brain exchange information through two types of highly specialized structures: chemical synapses (the vast majority) and so-called gap junctions (a substrate of one class of electrical synapse). Here we are interested in exploring the three-dimensional spatial distribution of chemical synapses in the cerebral cortex. Recent research has showed that the three-dimensional spatial distribution of synapses in layer III of the neocortex can be modeled by a random sequential adsorption (RSA) point process, i.e., synapses are distributed in space almost randomly, with the only constraint that they cannot overlap. In this study we hypothesize that RSA processes can also explain the distribution of synapses in all cortical layers. We also investigate whether there are differences in both the synaptic density and spatial distribution of synapses between layers. Using combined focused ion beam milling and scanning electron microscopy (FIB/SEM), we obtained three-dimensional samples from the six layers of the rat somatosensory cortex and identified and reconstructed the synaptic junctions. A total volume of tissue of approximately 4500μm3 and around 4000 synapses from three different animals were analyzed. Different samples, layers and/or animals were aggregated and compared using RSA replicated spatial point processes. The results showed no significant differences in the synaptic distribution across the different rats used in the study. We found that RSA processes described the spatial distribution of synapses in all samples of each layer. We also found that the synaptic distribution in layers II to VI conforms to a common underlying RSA process with different densities per layer. Interestingly, the results showed that synapses in layer I had a slightly different spatial distribution from the other layers.

Kinetics of Preferential Adsorption of Amphiphilic Star Block Copolymers that Tether by Their Corona Blocks at the Solid/Fluid Interface

The kinetics of preferential adsorption of a series of amphiphilic star block copolymers having cores made of polystyrene, PS, and coronas made of poly(2-vinylpyridine), PVP, have been studied using in situ phase modulated ellipsometry. The stars have, on average, either 26 or 40 PS-b-PVP diblock copolymer arms with PS/PVP molecular weight ratios (S/V) of 1/1, 5/1, or 9/1. Preferential adsorption from dilute toluene solutions onto silicon surfaces tethers these topologically complex star copolymers through their PVP blocks, yielding an interfacial structure resembling a dome. The adsorption kinetics are concentration dependent and show two distinct regimes: an initial rapid adsorption is followed by a slow approach toward equilibrium that is dominated by relaxations and rearrangements. For all PS/PVP ratios studied, star copolymers having 40 arms exhibit greater final adsorbed amounts compared to those having 26 arms. A kinetics model that considers diffusion of molecules at the solid/fluid interface as well as surface relaxation/reorganization events is used to characterize the adsorption process. In general, while both processes are necessary to describe the kinetics of preferential adsorption of these materials, surface rearrangements dominate for the two stars having the largest size and number of arms. It is also found that star block copolymers having symmetric arms (PS/PVP = 1/1) approach adsorption equilibrium in agreement with random sequential adsorption (RSA) model kinetics, attaining final adsorbed amounts that are close to those predicted for RSA processes. In contrast, the asymmetric stars having PS/PVP = 9/1 and 5/1 are able to rearrange on the surface, reaching final adsorbed amounts that are greater than those predicted for RSA processes, suggesting surface relaxation/reorganization and possible stretching due to confinement occurring at the solid/fluid interface. The results reported provide key insights as to how composition and connectivity of highly branched soft materials impact their self-assembly and interfacial structure.

Mechanics of random discontinuous long-fiber thermoplastics. Part II: Direct simulation of uniaxial compression

The mechanical interactions between fibers in a dense random-fiber network transmit stress, cause fiber curvature, and influence fiber orientation in the processing of many types of composites. A few theories describe the mechanics of fiber networks, but almost no simulation results are available. Here, we report a direct numerical simulation of the mechanical behavior of random-fiber networks. The finite element method is used, and each fiber is represented by a small number of 3D beam elements. The calculations assume a periodic structure to avoid boundary effects, but within the unit cell, the fibers are placed randomly. A special algorithm that uses the random sequential adsorption process creates an initial structure of straight, random, nonintersecting fibers from which a unit cell with periodic boundary conditions is built automatically [A. I. Abd El-Rahman and C. L. Tucker III, “Mechanics of random discontinuous long-fiber thermoplastics. Part I: Generation and characterization of initial geometry...

Mechanics of Random Discontinuous Long-Fiber Thermoplastics—Part I: Generation and Characterization of Initial Geometry

The deformation mechanics of dry networks of large-aspect-ratio fibers with random orientation controls the processing of long-fiber thermoplastics (LFTs) and greatly affects the mechanical properties of the final composites. Here, we generate initial geometries of fiber networks in a cubic unit cell with a fiber aspect ratio of l/d = 100 and fully periodic boundary conditions for later numerical simulation. The irreversible random sequential adsorption (RSA) process is first used to generate a quasi-random structure due to the excluded-volume requirements. In order to investigate the nonequilibrium character of the RSA, a second method, which is similar to the mechanical contraction method (MCM) (Williams and Philipse, 2003, “Random Packings of Spheres and Spherocylinders Simulated by Mechanical Contraction,” Phys. Rev. E, 67, pp. 1–9) and based on a simplified Metropolis Monte Carlo (MC) simulation is then developed to produce quasi-equilibrium fiber geometries. The RSA packing results (ϕ ≈ 4.423% when using a fiber aspect ratio of 100) are in good agreement with the maximum unforced random packing limits (Evans and Gibson, 1986, “Prediction of the Maximum Packing Fraction Achievable in Randomly Oriented Short-Fibre Composites,” Compos. Sci. Technol., 25, pp. 149–162). The fiber structures were characterized by several distribution functions, including pair-spatial and pair-orientation distributions, based on either the center-to-center distance or the shortest distance between the particles. The results show that the structures generated by the RSA have an easily-detectable long-range spatial correlation but very little orientational correlation. In contrast, the quasi-equilibrium structures have reduced spatial correlation but increased short-range orientational correlation.

Study of random sequential adsorption by means of the gradient method

By using the gradient method (GM) we study random sequential adsorption (RSA) processes in two dimensions under a gradient constraint that is imposed on the adsorption probability along one axis of the sample. The GM has previously been applied successfully to absorbing phase transitions (both first and second order), and also to the percolation transition. Now, we show that by using the GM the two transitions involved in RSA processes, namely percolation and jamming, can be studied simultaneously by means of the same set of simulations and by using the same theoretical background. For this purpose we theoretically derive the relevant scaling relationships for the RSA of monomers and we tested our analytical results by means of numerical simulations performed upon RSA of both monomers and dimers. We also show that two differently defined interfaces, which run in the direction perpendicular to the axis where the adsorption probability gradient is applied and separate the high-density (large-adsorption probability) and the low-density (low-adsorption probability) regimes, capture the main features of the jamming and percolation transitions, respectively. According to the GM, the scaling behaviour of those interfaces is governed by the roughness exponent α = 1/(1 + ν), where ν is the suitable correlation length exponent. Besides, we present and discuss in a brief overview some achievements of the GM as applied to different physical situations, including a comparison of the critical exponents determined in the present paper with those already published in the literature.

Statistical analysis and stochastic modelling of fibre composites

The fibre component of many fibre reinforced composites can be modelled by a system of non-overlapping straight cylinders. In this paper we discuss a model based on a random sequential adsorption (RSA) process. Geometric characteristics of the fibre system such as the fibre volume fraction or the fibre direction distribution are estimated from tomographic images of composite samples. Using this information, we fit RSA models to samples of a glass fibre reinforced polymer and a fibre reinforced ultra high performance concrete.