Impact of defects on percolation in random sequential adsorption of linear k-mers on square lattices

Percolation and jamming in random sequential adsorption of straight k-mers on square, triangular, and cubic lattices

Random sequential adsorption (RSA) is one of the most widely used theoretical models of macromolecule and particle adsorption. Here we consider jammed (saturated) states of the RSA process of aligned, nonoverlapping k-mers (linear objects occupying k consecutive lattice sites) in square, triangular, and cubic lattices and show that each k-mer in such a state belongs to a percolating cluster. We also discuss several generalizations of our approach, including a hypercubic lattice and models with several different k-mer lengths.

Random sequential adsorption with correlated defects : A series expansion approach

The random sequential adsorption (RSA) problem holds crucial theoretical and practical significance, serving as a pivotal framework for understanding and optimizing particle packing in various scientific and technological applications. Here the problem of the one-dimensional RSA of k-mers onto a substrate with correlated defects controlled by uniform and power-law distributions is theoretically investigated: the coverage fraction is obtained as a function of the density of defects and several scaling laws are examined. The results are compared with extensive Monte Carlo simulations and more traditional methods based on master equations. Emphasis is given in elucidating the scaling behavior of the fluctuations of the coverage fraction. The phenomenon of universality breaking and the issues of conventional Gaussian fluctuations and the Lévy type fluctuations from a simple perspective, relying on the central limit theorem, are also addressed.

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.

Breaking universality in random sequential adsorption on a square lattice with long-range correlated defects

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. The critical exponents ν_{j} and ν associated with the jamming and percolation transitions, respectively, are found to be nonuniversal for strong spatial correlations and approach systematically their own universal values as the correlation strength is decreased. More crucially, we have found a difference in the values of the percolation correlation length exponent ν for a small but finite density of defects with strong spatial correlations. Furthermore, for a fixed defect density, it is found that the percolation threshold of the system, at which the largest cluster of absorbed dimers first establishes the global connectivity, gets reduced with increasing the strength of the spatial correlation.

Adsorption kinetics and thermodynamic properties of a binary mixture of hard-core particles on a square lattice

The adsorption kinetics and thermodynamic properties of a binary mixture on a square lattice are studied using the random sequential adsorption with surface diffusion (RSAD). We compare the adsorption of binary species with different equilibrium rate constants and effective rates of adsorption to a surface and find that the temporal evolution of surface coverages of both species can be obtained through the use of the blocking function of a system with irreversible adsorption of highly diffusive particles. Binary mixtures, when one of the components follows the random sequential adsorption (RSA) without surface diffusion and the other follows the RSAD model, display competitive adsorption in addition to cooperative phenomena. Specifically, (i) species replacement occurs over a long period of time, while the total coverage remains unchanged after a short time, (ii) the presence of the RSAD component shifts the jamming coverage to the higher values, and (iii) the maximum jamming coverage is obtained when the effective adsorption of the RSA type components is lower than the other adsorbing particles.

Percolation of aligned rigid rods on two-dimensional triangular lattices

The percolation behavior of aligned rigid rods of length k (k-mers) on two-dimensional triangular lattices has been studied by numerical simulations and finite-size scaling analysis. The k-mers, containing k identical units (each one occupying a lattice site), were irreversibly deposited along one of the directions of the lattice. The connectivity analysis was carried out by following the probability R_{L,k}(p) that a lattice composed of L×L sites percolates at a concentration p of sites occupied by particles of size k. The results, obtained for k ranging from 2 to 80, showed that the percolation threshold p_{c}(k) exhibits a increasing function when it is plotted as a function of the k-mer size. The dependence of p_{c}(k) was determined, being p_{c}(k)=A+B/(C+sqrt[k]), where A=p_{c}(k→∞)=0.582(9) is the value of the percolation threshold by infinitely long k-mers, B=-0.47(0.21), and C=5.79(2.18). This behavior is completely different from that observed for square lattices, where the percolation threshold decreases with k. In addition, the effect of the anisotropy on the properties of the percolating phase was investigated. The results revealed that, while for finite systems the anisotropy of the deposited layer favors the percolation along the parallel direction to the alignment axis, in the thermodynamic limit, the value of the percolation threshold is the same in both parallel and transversal directions. Finally, an exhaustive study of critical exponents and universality was carried out, showing that the phase transition occurring in the system belongs to the standard random percolation universality class regardless of the value of k considered.

Morphology and kinetics of random sequential adsorption of superballs: From hexapods to cubes

Superballs represent a class of particles whose shapes are defined by the domain |x|^{2p}+|y|^{2p}+|z|^{2p}≤R^{2p}, with p∈(0,∞) being the deformation parameter. 0<p<0.5 represents a family of hexapodlike (concave octahedral-like) particles, 0.5≤p<1 and p>1 represent, respectively, families of convex octahedral-like and cubelike particles, with p=1,0.5, and ∞ 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 superballs. We introduce a universal method for simulating random sequential adsorption of superballs, which we refer to as the low-entropy algorithm, which is about two orders of magnitude faster than the conventional algorithms that represent high-entropy methods. The two algorithms yield, respectively, precise estimates of the jamming fraction ϕ_{∞}(p) and ν(p), the exponent that characterizes the kinetics of adsorption at long times t, ϕ_{∞}(p)-ϕ(p,t)∼t^{-ν(p)}. Precise estimates of ϕ_{∞}(p) and ν(p) are obtained and shown to be in agreement with the existing analytical and numerical results for certain types of superballs.

Tensor renormalization group study of hard-disk models on a triangular lattice

High accuracy and performance of the tensor renormalization group (TRG) method have been demonstrated for the model of hard disks on a triangular lattice. We considered a sequence of models with disk diameter ranging from a to 2sqrt[3]a, where a is the lattice constant. Practically, these models are good for approximate description of thermodynamics properties of molecular layers on crystal surfaces. Theoretically, it is interesting to analyze if and how this sequence converges to the continuous model of hard disks. The dependencies of the density and heat capacity on the chemical potential were calculated with TRG and transfer-matrix (TM) methods. We benchmarked accuracy and performance of the TRG method comparing it with TM method and with exact result for the model with nearest-neighbor exclusions (1NN). The TRG method demonstrates good convergence and turns out to be superior over TM with regard to considered models. Critical values of chemical potential (μ_{c}) have been computed for all models. For the model with next-nearest-neighbor exclusions (2NN) the TRG and TM produce consistent results (μ_{c}=1.75587 and μ_{c}=1.75398 correspondingly) that are also close to earlier Monte Carlo estimation by Zhang and Deng. We found that 3NN and 5NN models shows the first-order phase transition, with close values of μ_{c} (μ_{c}=4.4488 for 3NN and 4.4<μ_{c}<4.5 for 5NN). The 4NN model demonstrates continuous yet rapid phase transition with 2.65<μ_{c}<2.7.

Site-bond percolation of heteronuclear dimers irreversibly deposited on square lattices

A generalization of the site-bond percolation problem was studied, in which pairs of neighboring sites (site dimers) and bonds are occupied irreversibly, randomly, and independently on homogeneous square surfaces. A dimer is composed of two segments and occupies two adjacent sites. Each segment can be either a conductive segment (segment type A) or a nonconductive segment (segment type B). Two types of dimers are considered, AA and AB, and the connectivity analysis is carried out by accounting only for the conductive segments (segments type A) in combination with bonds. For the combination of dimers and bonds, two different criteria were analyzed: the union or the intersection between the adsorbed percolating particles and the bonds. By means of numerical simulations and finite-size scaling analysis, the complete phase diagram separating a percolating from a non-percolating region was determined.

Jammed systems of oriented needles always percolate on square lattices

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, percolation is impossible if the needles are sufficiently long (k of order of several thousand). We refute these claims and present rigorous proof that in any jammed configuration of nonoverlapping, fixed-length, horizontal, or vertical needles on a square lattice, all clusters are percolating clusters.

Jamming for nematic deposition in the presence of impurities

The deposition of one-dimensional objects (such as polymers) on a one-dimensional lattice with the presence of impurities is studied in order to find saturation conditions in what is known as jamming. Over a critical concentration of k-mers (polymers of length k), no further depositions are possible. Five different nematic (directional) depositions are considered: baseline, irreversible, configurational, loose-packing, and close-packing. Correspondingly, five jamming functions are found, and their dependencies on the length of the lattice, L, the concentration of impurities, p=M/L (where M is the number of one-dimensional impurities), and the length of the k-mer (k) are established. In parallel, numeric simulations are performed to compare with the theoretical results. The emphasis is on trimers (k=3) and p in the range [0.01,0.15], however other related cases are also considered and reported.

Percolation of heteronuclear dimers irreversibly deposited on square lattices

The percolation problem of irreversibly deposited heteronuclear dimers on square lattices is studied. A dimer is composed of two segments, and it occupies two adjacent adsorption sites. Each segment can be either a conductive segment (segment type A) or a nonconductive segment (segment type B). Three types of dimers are considered: AA, BB, and AB. The connectivity analysis is carried out by accounting only for the conductive segments (segments type A). The model offers a simplified representation of the problem of percolation of defective (nonideal) particles, where the presence of defects in the system is simulated by introducing a mixture of conductive and nonconductive segments. Different cases were investigated, according to the sequence of deposition of the particles, the types of dimers involved in the process, and the degree of alignment of the deposited objects. By means of numerical simulations and finite-size scaling analysis, the complete phase diagram separating a percolating from a nonpercolating region was determined for each case. Finally, the consistency of our results was examined by comparing with previous data in the literature for linear k-mers (particles occupying k adjacent sites) with defects.

Percolation and jamming of linear k-mers on a square lattice with defects: Effect of anisotropy

Using the Monte Carlo simulation, we study the percolation and jamming of oriented linear k-mers on a square lattice that contains defects. The point defects with a concentration d are placed randomly and uniformly on the substrate before deposition of the k-mers. The general case of unequal probabilities for orientation of depositing of k-mers along different directions of the lattice is analyzed. Two different relaxation models of deposition that preserve the predetermined order parameter s are used. In the relaxation random sequential adsorption (RRSA) model, the deposition of k-mers is distributed over different sites on the substrate. In the single-cluster relaxation (RSC) model, the single cluster grows by the random accumulation of k-mers on the boundary of the cluster (Eden-like model). For both models, a suppression of growth of the infinite (percolation) cluster at some critical concentration of defects d(c) is observed. In the zero-defect lattices, the jamming concentration p(j) (RRSA model) and the density of single clusters p(s) (RSC model) decrease with increasing length k-mers and with a decrease in the order parameter. For the RRSA model, the value of d(c) decreases for short k-mers (k<16) as the value of s increases. For k=16 and 32, the value of d(c) is almost independent of s. Moreover, for short k-mers, the percolation threshold is almost insensitive to the defect concentration for all values of s. For the RSC model, the growth of clusters with ellipselike shapes is observed for nonzero values of s. The density of the clusters p(s) at the critical concentration of defects d(c) depends in a complex manner on the values of s and k. An interesting finding for disordered systems (s=0) is that the value of p(s) tends towards zero in the limits of the very long k-mers, k→∞, and very small critical concentrations d(c)→0. In this case, the introduction of defects results in a suppression of k-mer stacking and in the formation of empty or loose clusters with very low density. On the other hand, denser clusters are formed for ordered systems with p(s)≈0.065 at s=0.5 and p(s)≈0.38 at s=1.0.

Jamming and percolation in generalized models of random sequential adsorption of linear k-mers on a square lattice

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 RSA model assumes the absence of overlapping of the new incoming particle with the previously deposited ones. The first model 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 by defects (impurities). For this model even a small concentration of defects can inhibit percolation for relatively long k-mers. The second model is the cooperative sequential adsorption one where, for each new k-mer, only a restricted number of lateral contacts z with previously deposited k-mers is allowed. Deposition occurs in the case when z≤(1-d)z(m) where z(m)=2(k+1) is the maximum numbers of the contacts of k-mer, and d is the fraction of forbidden contacts. Percolation is observed only at some interval k(min)≤k≤k(max) where the values k(min) and k(max) depend upon the fraction of forbidden contacts d. The value k(max) decreases as d increases. A logarithmic dependence of the type log(10)(k(max))=a+bd, where a=4.04±0.22,b=-4.93±0.57, is obtained.