We consider the non-overlapping irreversible random sequential adsorption (RSA) process on one-dimensional finite line, which is known also as the car parking process. The probability of each coverage in saturating states is analytically and exactly obtained. In the derivation, a new representation of states in RSA process is introduced, which effectively works to make the calculation clear and simple.
A notorious problem in mathematics and physics is to create a solvable model for random sequential adsorption of non-overlapping congruent spheres in the $d$-dimensional Euclidean space with $d\geq 2$. Spheres arrive sequentially at uniformly chosen locations in space and are accepted only when there is no overlap with previously deposited spheres. Due to spatial correlations, characterizing the fraction of accepted spheres remains largely intractable. We study this fraction by taking a novel approach that compares random sequential adsorption in Euclidean space to the nearest-neighbor blockin…
We study the random sequential adsorption of $k$-mers on the fully-connected lattice with $N=kn$ sites. The probability distribution $T_n(s,t)$ of the time $t$ needed to cover the lattice with $s$ $k$-mers is obtained using a generating function approach. In the low coverage scaling limit where $s,n,t\to\infty$ with $y=s/n^{1/2}={\mathrm O}(1)$ the random variable $t-s$ follows a Poisson distribution with mean $ky^2/2$. In the intermediate coverage scaling limit, when both $s$ and $n-s$ are ${\mathrm O}(n)$, the mean value and the variance of the covering time are growing as $n$ and the fluctu…
The influence of different boundary conditions on the density of random packings of disks is studied. Packings are generated using the random sequential adsorption algorithm with three different types of boundary conditions: periodic, open, and wall. It is found that the finite size effects are smallest for periodic boundary conditions, as expected. On the other hand, in the case of open and wall boundaries it is possible to introduce an effective packing size and a constant correction term to significantly improve the packing densities.
A novel approach for deriving the equation of state for a 2D lattice gas is proposed, based on arguments similar to those used in the derivation of the Langmuir-Szyszkowski equation of state for localized adsorption. The relationship between surface coverage and excluded area is first extracted from Random Sequential Adsorption simulations incorporating surface diffusion (RSAD). The adsorption isotherm is then obtained using kinetic arguments and the Gibbs equation gives the relation between surface pressure and coverage. Provided surface diffusion is fast enough to ensure internal equilibrium…
Random sequential adsorption of spheres on a wavy surface was studied. It was determined how surface structure influences random packing properties such as the packing fraction, the kinetics of packing growth, and the two-particle density correlation function. Until the substrate varies within the range one order of magnitude smaller than the particle's diameter, the properties of the packings obtained do not differ significantly from those on a flat surface. On the other hand, for the higher amplitude of unevenness, the packing fraction, low-density growth kinetics, and the density autocorrel…
The properties of the number of iterations in random sequential adsorption protocol needed to generate finite saturated random packing of spherically symmetric shapes were studied. Numerical results obtained for one, two, and three dimensional packings were supported by analytical calculations valid for any dimension $d$. It has been shown that the number of iterations needed to generate finite saturated packing is subject to Pareto distribution with exponent $-1-1/d$ and the median of this distribution scales with packing size according to the power-law characterized by exponent $d$. Obtained…
The random sequential adsorption (RSA) model is a classical model in Statistical Physics for adsorption on two-dimensional surfaces. Objects are deposited sequentially at random and adsorb irreversibly on the landing site, provided that they do not overlap any previously adsorbed object. The kinetics of adsorption ceases when no more objects can be adsorbed (jamming state). Here, we investigate the role of post-relaxation on the jamming state and percolation properties of RSA of dimers on a two-dimensional lattice. We consider that, if the deposited dimer partially overlaps with a previously a…
Restricted-valence random sequential adsorption~(RSA) is studied in its pure and disordered versions, on the square and triangular lattices. For the simplest case~(pure on the square lattice) we prove the absence of percolation for maximum valence $V_{\rm max}=2$. In other cases, Monte Carlo simulations are used to investigate the percolation threshold, universality class, and jamming limit. Our results reveal a continuous transition for the majority of the cases studied. The percolation threshold is computed through finite-size scaling analysis of seven properties; its value increases with th…
We report a surprising result, established by numerical simulations and analytical arguments for a one-dimensional lattice model of random sequential adsorption, that even an arbitrarily small imprecision in the lattice-site localization changes the convergence to jamming from fast, exponential, to slow, power-law, with, for some parameter values, a discontinuous jump in the jamming coverage value. This finding has implications for irreversible deposition on patterned substrates with pre-made landing sites for particle attachment. We also consider a general problem of the particle (depositing …
Random sequential adsorption (RSA) of various two dimensional objects is studied in order to find a shape which maximizes the saturated packing fraction. This investigation was begun in our previous paper [Cieśla et al., Phys. Chem. Chem. Phys. 17, 24376 (2015)], where the densest packing was studied for smoothed dimers. Here this shape is compared with a smoothed $n$-mers, spherocylinders and ellipses. It is found that the highest packing fraction out of the studied shapes is $0.58405 \pm 0.0001$ and is obtained for ellipses having long-to-short axis ratio of $1.85$, which is also the largest…
In random sequential adsorption (RSA), objects are deposited randomly, irreversibly, and sequentially; attempts leading to an overlap with previously deposited objects are discarded. The process continues until the system reaches a jammed state when no further additions are possible. We analyze a class of lattice RSA models in which landing on an empty site in a segment is allowed when at least $b$ neighboring sites on the left and the right are unoccupied. For the minimal model ($b=1$), we compute the full counting statistics of the occupation number. We reduce the determination of the full c…
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. Let. 119, 028003 (2017)]. 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 va…
We present the algorithm for generating strictly saturated random sequential adsorption packings built of rounded polygons. It can be used to study various properties of such packings built of a wide variety of different shapes and in modelling monolayers obtained during the irreversible adsorption processes of complex molecules. Here, we apply the algorithm to study the densities of packings built of rounded regular polygons. Contrary to packings built of regular polygons, where packing fraction grows with an increasing number of polygon sides, the packing fraction reaches its maximum for pac…
An extension of the Random Sequential Adsorption (RSA) model has been proposed recently, motivated by the coverage of oil droplets by DNA-functionalized colloidal particles. Particles arrive to a flat substrate with a uniform flux F but they can only adsorb on patches. Patches diffuse on the substrate with a diffusion coefficient D if they are free and they remain immobile when attached to an adsorbed particle. The adsorption is considered irreversible and particles cannot adsorb on top of each other. Thus, the system reaches a jammed state, consisting of a monolayer where no more particles ca…
Computer simulation was used to study the random sequential adsorption of identical discorectangles onto a continuous plane . The problem was analyzed for a wide range of discorectangle aspect ratios ($\varepsilon \in [1;100]$). We studied anisotropic deposition, i.e., the orientations of the deposited particles were uniformly distributed within some interval such that the particles were preferentially aligned along a given direction. The kinetics of the changes in the packing density $φ$ found at different values of $S_0$ are discussed. Partial ordering of the discorectangles significantly af…
The behavior of the percolation threshold and the jamming coverage for isotropic random sequential adsorption samples has been studied by means of numerical simulations. A parallel algorithm that is very efficient in terms of its speed and memory usage has been developed and applied to the model involving large linear $k$-mers on a square lattice with periodic boundary conditions. We have obtained the percolation thresholds and jamming concentrations for lengths of $k$-mers up to $2^{17}$. New large $k$ regime of the percolation threshold behavior has been identified. The structure of the perc…
Irreversible adsorption of objects of different shapes and sizes on Euclidean, fractal and random lattices is studied. The adsorption process is modeled by using random sequential adsorption (RSA) algorithm. Objects are adsorbed on one-, two-, and three-dimensional Euclidean lattices, on Sierpinski carpets having dimension $d$ between 1 and 2, and on Erdos-Renyi random graphs. The number of sites is $M=L^d$ for Euclidean and fractal lattices, where $L$ is a characteristic length of the system. In the case of random graphs it does not exist such characteristic length, and the substrate can be c…
Using computer simulation, we have studied the random sequential adsorption of stiff linear $k$-mers onto a square lattice. Each such particle occupies $k$ adjacent lattice sites. During deposition, the two mutually perpendicular orientations of the particles are equiprobable, hence, a macroscopically isotropic monolayer is formed. However, this monolayer is locally anisotropic, since the deposited particles tend to form domains of particles with the same orientation. Using the "excluded area" concept, we have classified lattice sites into several types and examined how the fraction of each ty…
Superballs represent a class of particles whose shapes are defined by ${|x|}^{2p}+{|y|}^{2p}+{|z|}^{2p} \le R^{2p}$, with $p\in(0,\infty)$ being the "deformation parameter". $0<p<0.5$ represents a family of hexapodlike (concave octahedrallike) particles, while for $0.5\leq p<1$ and $p>1$ one has, respectively, families of convex octahedrallike and cubelike particles, with $p=1,\;0.5$ and $\infty$ representing spheres, octahedra, and cubes. Colloidal zeolite suspensions, catalysis, and adsorption, as well as biomedical magnetic nanoparticles are but a few of the applications of packing of super…
Relaxation of the packing of elongated particles (discorectangles) aligned on a line was studied numerically. The aspect ratio (length-to-width ratio) for the discorectangles was varied within the range $\varepsilon \in [1;50]$. The initial jamming (saturated) state was produced using the basic variant of the random sequential adsorption (RSA) model with random positions and orientations of particles. The relaxation was performed by allowing rotational and translational diffusion motions of the particles wile their centers remained located on the line. The effects of the aspect ratio $\varepsi…
We consider two-species random sequential adsorption (RSA) in which species A and B adsorb randomly on a lattice with the restriction that opposite species cannot occupy nearest-neighbor sites. When the probability $x_A$ of choosing an A particle for an adsorption trial reaches a critical value $0.626441(1)$, the A species percolates and/or the blocked sites X (those with at least one A and one B nearest neighbor) percolate. Analysis of the size-distribution exponent $τ$, the wrapping probabilities, and the excess cluster number shows that the percolation transition is consistent with that of …
We study random sequential adsorption of particles from pool onto a one dimensional substrate following ballistic deposition rules, with separate nucleation and growth processes occurring simultaneously. Nucleation describes the formation of point-sized seeds, and after a seed is sown, it acts as an attractor and grows in size by the addition of grains of a fixed-sized. At each time step either an already-nucleated seed can increase in size, or a new seed may be nucleated. We incorporate a parameter $m$, to describe the relative rates of growth to nucleation. We solve the model analytically to…
The jamming and percolation for two generalized models of random sequential adsorption (RSA) of linear $k$-mers (particles occupying $k$ adjacent sites) on a square lattice are studied by means of Monte Carlo simulation. The classical random sequential adsorption (RSA) model assumes the absence of overlapping of the new incoming particle with the previously deposited ones. The first model LK$_d$ is a generalized variant of the RSA model for both $k$-mers and a lattice with defects. Some of the occupying $k$ adjacent sites are considered as insulating and some of the lattice sites are occupied …
We study approach to the large-time jammed state of the deposited particles in the model of random sequential adsorption. The convergence laws are usually derived from the argument of Pomeau which includes the assumption of the dominance, at large enough times, of small landing regions into each of which only a single particle can be deposited without overlapping earlier deposited particles and which, after a certain time are no longer created by depositions in larger gaps. The second assumption has been that the size distribution of gaps open for particle-center landing in this large-time sma…
Random sequential adsorption of straight rigid rods of length $k$ ($k$-mers) on a simple cubic lattice has been studied by numerical simulations and finite-size scaling analysis. The calculations were performed by using a new theoretical scheme, whose accuracy was verified by comparison with rigorous analytical data. The results, obtained for \textit{k} ranging from 2 to 64, revealed that (i) in the case of dimers ($k=2$), the jamming coverage is $θ_j=0.918388(16)$. Our estimate of $θ_j$ differs significantly from the previously reported value of $θ_j=0.799(2)$ [Y. Y. Tarasevich and V. A. Cher…
The behavior of a system of two-dimensional elongated particles (discorectangles) packed into a slit between two parallel walls was analyzed using a simulation approach. The packings were produced using the random sequential adsorption model with continuous positional and orientational degrees of freedom. The aspect ratio (length-to-width ratio, $\varepsilon=l/d$) of the particles was varied within the range $\varepsilon \in [1;32]$ while the distance between the walls was varied within the range $h/d\in [1;80]$. The properties of the deposits when in the jammed state (the coverage, the order …
Jamming and percolation transitions in the standard random sequential adsorption of particles on regular lattices are characterized by a universal set of critical exponents. The universality class is preserved even in the presence of randomly distributed defective sites that are forbidden for particle deposition. However, using large-scale Monte Carlo simulations by depositing dimers on the square lattice and employing finite-size scaling, we provide evidence that the system does not exhibit such well-known universal features when the defects have spatial long-range (power-law) correlations. T…
Monte Carlo simulations and finite-size scaling analysis have been performed to study the jamming and percolation behavior of linear $k$-mers (also known as rods or needles) on the two-dimensional triangular lattice, considering an isotropic RSA process on a lattice of linear dimension $L$ and periodic boundary conditions. Extensive numerical work has been done to extend previous studies to larger system sizes and longer $k$-mers, which enables the confirmation of a nonmonotonic size dependence of the percolation threshold and the estimation of a maximum value of $k$ from which percolation wou…
A Generalized Model for Disordered Random Sequential Adsorption with Charge-Dependent Deposition
Centro de Ciências Exatas e Sociais Aplicadas, Universidade Estadual da Paraíba（帕拉伊巴州立大学精确科学与应用科学学院）；Departamento de Física, Universidade Federal de Pernambuco（伯南布哥联邦大学物理系）
Connectedness percolation phenomena in two-dimensional packings of elongated particles (discorectangles) were studied numerically. The packings were produced using random sequential adsorption (RSA) off-lattice model with preferential orientations of particles along a given direction. The partial ordering was characterized by order parameter $S$, with $S=0$ for completely disordered films (random orientation of particles) and $S=1$ for completely aligned particles along the horizontal direction $x$. The aspect ratio (length-to-width ratio) for the particles was varied within the range $\vareps…
