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 fluctuations are Gaussian. When full coverage is approached the scaling functions diverge, which is the signal of a new scaling behaviour. Indeed, when $u=n-s={\mathrm O}(1)$, the mean value of the covering time grows as $n^k$ and the variance as $n^{2k}$, thus $t$ is strongly fluctuating and no longer self-averaging. In this scaling regime the fluctuations are governed, for each value of $k$, by a different extreme value distribution, indexed by $u$. Explicit results are obtained for monomers (generalized Gumbel distribution) and dimers.
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…
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 …
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…
The effect of defects on the percolation of linear $k$-mers (particles occupying $k$ adjacent sites) on a square lattice is studied by means of Monte Carlo simulation. The $k$-mers are deposited using a random sequential adsorption mechanism. Two models, $L_d$ and $K_d$, are analyzed. In the $L_d$ model, it is assumed that the initial square lattice is non-ideal and some fraction of sites, $d$, is occupied by non-conducting point defects (impurities). In the $K_d$ model, the initial square lattice is perfect. However, it is assumed that some fraction of the sites in the $k$-mers, $d$, consists…
We have performed extensive simulations of random sequential adsorption and diffusion of $k$-mers, up to $k=5$ in two dimensions with particular attention to the case $k=2$. We focus on the behavior of the coverage and of vacancy dynamics as a function of time. We observe that for $k=2,3$ a complete coverage of the lattice is never reached, because of the existence of frozen configurations that prevent isolated vacancies in the lattice to join. From this result we argue that complete coverage is never attained for any value of $k$. The long time behavior of the coverage is not mean field and n…
The electrical conductivity of a monolayer produced by the random sequential adsorption (RSA) of linear $k$-mers onto a square lattice was studied by means of computer simulation. Overlapping with pre-deposited $k$-mers and detachment from the surface were forbidden. The RSA continued until the saturation jamming limit, $p_j$. The isotropic and anisotropic 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 electrica…
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 simple statistical mechanical approach for studying multilayer adsorption of interacting polyatomic adsorbates (k-mers) has been presented. The new theoretical framework has been developed on a generalization in the spirit of the lattice-gas model and the classical Bragg-Williams (BWA) and quasi-chemical (QCA) approximations. The derivation of the equilibrium equations allows the extension of the well-known Brunauer-Emmet-Teller (BET) isotherm to more complex systems. The formalism reproduces the classical theory for monomers, leads to the exact statistical thermodynamics of interacting k-me…
Numerical simulations by means of Monte Carlo method and finite-size scaling analysis have been performed to study the percolation behavior of linear $k$-mers (also denoted in the literature as rigid rods, needles, sticks) on two-dimensional square lattices $L \times L$ with periodic boundary conditions. Percolation phenomena are investigated for anisotropic relaxation random sequential adsorption of linear $k$-mers. Especially, effect of anisotropic placement of the objects on the percolation threshold has been investigated. Moreover, the behavior of percolation probability $R_L(p)$ that a la…
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…
The diffusion-driven self-assembly of rod-like particles was studied by means of Monte Carlo simulation. The rods were represented as linear $k$-mers (i.e., particles occupying $k$ adjacent sites). In the initial state, they were deposited onto a two-dimensional square lattice of size $L\times L$ up to the jamming concentration using a random sequential adsorption algorithm. The size of the lattice, $L$, was varied from $128$ to $2048$, and periodic boundary conditions were applied along both $x$ and $y$ axes, while the length of the $k$-mers (determining the aspect ratio) was varied from $2$ …
Pattern formation in a two-dimensional system of rod-like particles has been simulated using a lattice approach. Rod-like particles were modelled as linear $k$-mers of two mutually perpendicular orientations ($k_x$- and $k_y$-mers) on a square lattice with periodic boundary conditions (torus). Two kinds of random sequential adsorption model were used to produce the initial homogeneous and isotropic distribution of $k$-mers with different values of packing fraction. By means of the Monte Carlo technique, translational diffusion of the $k$-mers was simulated as a random walk, while rotational di…
Random sequential adsorption (RSA) is a standard method of modeling adsorption of large molecules at the liquid-solid interface. Several studies have recently conjectured that in the RSA of rectangular needles, or $k$-mers, on a square lattice the percolation is impossible if the needles are sufficiently long ($k$ of order of several thousand). We refute these claims and present a strict proof that in any jammed configuration of nonoverlapping, fixed-length, horizontal or vertical needles on a square lattice, all clusters are percolating clusters.
Diffusion in a two-species 2D system has been simulated using a lattice approach. Rodlike particles were considered as linear $k$-mers of two mutually perpendicular orientations ($k_x$- and $k_y$-mers) on a square lattice. These $k_x$- and $k_y$-mers were treated as species of two kinds. A random sequential adsorption model was used to produce an initial homogeneous distribution of $k$-mers. The concentration of $k$-mers, $p$, was varied in the range from 0.1 to the jamming concentration, $p_j$. By means of the Monte Carlo technique, translational diffusion of the $k$-mers was simulated as a r…
Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse site percolation by the removal of $k \times k$ square tiles ($k^{2}$-mers) from square lattices. The process starts with an initial configuration, where all lattice sites are occupied and, obviously, the opposite sides of the lattice are connected by occupied sites. Then, the system is diluted by removing $k^{2}$-mers of occupied sites from the lattice following a random sequential adsorption mechanism. The process finishes when the jamming state is reached and no more objects can be r…
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…
We study the percolation and jamming of rods ($k$-mers) on a square lattice that contains defects. The point defects are placed randomly and uniformly on the substrate before deposition of the rods. The general case of unequal probabilities for orientation of depositing of rods along different directions of the lattice is analyzed. Two different models of deposition are used. In the relaxation random sequential adsorption model (RRSA), the deposition of rods is distributed over different sites on the substrate. In the single cluster relaxation model (RSC), the single cluster grows by the rando…
We consider the large-time dynamics of one-dimensional processes involving adsorption and desorption of extended hard-core particles (dimers, trimers,\,$\cdots,k$-mers), while interacting through their constituent monomers. Desorption can occur whether or not these latter adsorbed together, which leads to reconstitution of $k$-mers and the appearance of sectors of motion with nonlocal conservation laws for $k \ge 3$. Dynamic exponents of the sector including the empty chain are evaluated by finite-size scaling analyses of the relaxation times embodied in the spectral gaps of evolution operator…
Regular structures of an intractable enumeration problem: a diagonal recurrence relation of monomer-polymer coverings on two-dimensional rectangular lattices
