Surface growth during random and irreversible multilayer deposition of straight semirigid rods.

Surface growth properties during irreversible multilayer deposition of straight semirigid rods on linear and square lattices have been studied by Monte Carlo simulations and analytical considerations. The filling of the lattice is carried out following a generalized random sequential adsorption mechanism where the depositing objects can be adsorbed on the surface forming multilayers. The results of our simulations show that the roughness evolves in time following two different behaviors: an "homogeneous growth regime" at initial times, where the heights of the columns homogeneously increase, and a "segmented growth regime" at long times, where the adsorbed phase is segmented in actively growing columns and inactive nongrowing sites. Under these conditions, the surface growth generated by the deposition of particles of different sizes is studied. At long times, the roughness of the systems increases linearly with time, with growth exponent ß=1, at variance with a random deposition of monomers which presents a sublinear behavior (ß=1/2). The linear behavior is due to the segmented growth process, as we show using a simple analytical model.

Critical behavior of repulsively interacting particles adsorbed on disordered triangular lattices.

A simple model for amorphous solids, consisting of a triangular lattice with a fraction of attenuated bonds randomly distributed (which simulate the presence of defects in the surface), is used here to find out, by using grand canonical Monte Carlo simulations, how the adsorption thermodynamics of repulsively interacting monomers is modified with respect to the same process in the regular lattice. The degree of disorder of the surface is tunable by selecting the values of (1) the fraction of attenuated bonds ρ (0 ≤ρ≤ 1) and (2) the attenuation factor r (0 ≤r≤ 1), where r is defined as the ratio between the value of the lateral interaction associated to an attenuated bond and that corresponding to a regular bond. Adsorption isotherm and differential heat of adsorption calculations have been carried out showing and interpreting the effects of the disorder. A rich variety of behavior has been observed for different values of ρ and r, varying between two limit cases: bond-diluted lattices (r = 0 and ρ≠ 0) and regular lattices (r = 1 and any value of ρ). In addition, the critical behavior of the system was studied, showing that the order-disorder phase transition observed for the regular lattice survives, though with modifications, above a critical curve (ρ-r-temperature).

Jamming and percolation of linear k-mers on honeycomb lattices.

Numerical simulations and finite-size scaling analysis have been performed to study the jamming and percolation behavior of elongated objects deposited on two-dimensional honeycomb lattices. The depositing particle is modeled as a linear array of length k (so-called k-mer), maximizing the distance between first and last monomers in the chain. The separation between k-mer units is equal to the lattice constant. Hence, k sites are occupied by a k-mer when adsorbed onto the surface. The adsorption process starts with an initial configuration, where all lattice sites are empty. Then, the sites are occupied following a random sequential adsorption mechanism. The process finishes when the jamming state is reached and no more objects can be deposited due to the absence of empty site clusters of appropriate size and shape. Jamming coverage θ_{j,k} and percolation threshold θ_{c,k} were determined for a wide range of values of k (2≤k≤128). The obtained results shows that (i) Î¸_{j,k} is a decreasing function with increasing k, being θ_{j,k→∞}=0.6007(6) the limit value for infinitely long k-mers; and (ii) Î¸_{c,k} has a strong dependence on k. It decreases in the range 2≤k<48, goes through a minimum around k=48, and increases smoothly from k=48 up to the largest studied value of k=128. Finally, the precise determination of the critical exponents ν, ß, and γ indicates that the model belongs to the same universality class as 2D standard percolation regardless of the value of k considered.

Irreversible bilayer adsorption of straight semirigid rods on two-dimensional square lattices: Jamming and percolation properties.

Numerical simulations and finite-size scaling analysis have been performed to study the jamming and percolation behavior of straight semirigid rods adsorbed on two-dimensional square lattices. The depositing objects can be adsorbed on the surface forming two layers. The filling of the lattice is carried out following a generalized random sequential adsorption (RSA) mechanism. In each elementary step, (i) a set of k consecutive nearest-neighbor sites (aligned along one of two lattice axes) is randomly chosen and (ii) if each selected site is either empty or occupied by a k-mer unit in the first layer, then a new k-mer is then deposited onto the lattice. Otherwise, the attempt is rejected. The process starts with an initially empty lattice and continues until the jamming state is reached and no more objects can be deposited due to the absence of empty site clusters of appropriate size and shape. A wide range of values of k (2≤k≤64) is investigated. The study of the kinetic properties of the system shows that (1) the jamming coverage θ_{j,k} is a decreasing function with increasing k, with θ_{j,k→∞}=0.7299(21) the limit value for infinitely long k-mers and (2) the jamming exponent ν_{j} remains close to 1, regardless of the size k considered. These findings are discussed in terms of the lattice dimensionality and number of sites available for adsorption. The dependence of the percolation threshold θ_{c,k} as a function of k is also determined, with θ_{c,k}=A+Bexp(-k/C), where A=θ_{c,k→∞}=0.0457(68) is the value of the percolation threshold by infinitely long k-mers, B=0.276(25), and C=14(2). This monotonic decreasing behavior is completely different from that observed for the standard problem of straight rods on square lattices, where the percolation threshold shows a nonmonotonic k-mer size dependence. The differences between the results obtained from bilayer and monolayer phases are explained on the basis of the transversal overlaps between rods occurring in the bilayer problem. This effect (which we call a "cross-linking effect"), its consequences on the filling kinetics, and its implications in the field of conductivity of composites filled with elongated particles (or fibers) are discussed in detail. Finally, the precise determination of the critical exponents ν, ß, and γ indicates that, although the increasing in the width of the deposited layer drastically affects the behavior of the percolation threshold with k and other critical properties (such as the crossing points of the percolation probability functions), it does not alter the nature of the percolation transition occurring in the system. Accordingly, the bilayer model belongs to the same universality class as two-dimensional standard percolation model.

Irreversible multilayer adsorption of semirigid k-mers deposited on one-dimensional lattices.

Irreversible multilayer adsorption of semirigid k-mers on one-dimensional lattices of size L is studied by numerical simulations complemented by exhaustive enumeration of configurations for small lattices. The deposition process is modeled by using a random sequential adsorption algorithm, generalized to the case of multilayer adsorption. The paper concentrates on measuring the jamming coverage for different values of k-mer size and number of layers n. The bilayer problem (n≤2) is exhaustively analyzed, and the resulting tendencies are validated by the exact enumeration techniques. Then, the study is extended to an increasing number of layers, which is one of the noteworthy parts of this work. The obtained results allow the following: (i) to characterize the structure of the adsorbed phase for the multilayer problem. As n increases, the (1+1)-dimensional adsorbed phase tends to be a "partial wall" consisting of "towers" (or columns) of width k, separated by valleys of empty sites. The length of these valleys diminishes with increasing k; (ii) to establish that this is an in-registry adsorption process, where each incoming k-mer is likely to be adsorbed exactly onto an already adsorbed one. With respect to percolation, our calculations show that the percolation probability vanishes as L increases, being zero in the limit L→∞. Finally, the value of the jamming critical exponent ν_{j} is reported here for multilayer adsorption: ν_{j} remains close to 2 regardless of the considered values of k and n. This finding is discussed in terms of the lattice dimensionality.

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.

Percolation phase transition by removal of k

Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse site percolation by the removal of k×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 removed due to the absence of occupied sites clusters of appropriate size and shape. The central idea of this paper is based on finding the maximum concentration of occupied sites, p_{c,k}, for which the connectivity disappears. This particular value of the concentration is called the inverse percolation threshold and determines a well-defined geometrical phase transition in the system. The results obtained for p_{c,k} show that the inverse percolation threshold is a decreasing function of k in the range 1≤k≤4. For k≥5, all jammed configurations are percolating states, and consequently, there is no nonpercolating phase. In other words, the lattice remains connected even when the highest allowed concentration of removed sites is reached. The jamming exponent ν_{j} was measured, being ν_{j}=1 regardless of the size k considered. In addition, the accurate determination of the critical exponents ν, ß, and γ reveals that the percolation phase transition involved in the system, which occurs for k varying between one and four, has the same universality class as the standard percolation problem.

Jamming and percolation for deposition of k

Percolation and jamming of k×k square tiles (k^{2}-mers) deposited on square lattices have been studied by numerical simulations complemented with finite-size scaling theory and exact enumeration of configurations for small systems. The k^{2}-mers were irreversibly deposited into square lattices of sizes L×L with L/k ranging between 128 and 448 (64 and 224) for jamming (percolation) calculations. Jamming coverage θ_{j,k} was determined for a wide range of k values (2≤k≤100 with many intermediate k values to allow a fine scaling analysis). θ_{j,k} exhibits a decreasing behavior with increasing k, being θ_{j,k=∞}=0.5623(3) the limit value for large k^{2}-mer sizes. In addition, a finite-size scaling analysis of the jamming transition was carried out, and the corresponding spatial correlation length critical exponent ν_{j} was measured, being ν_{j}≈1. On the other hand, the obtained results for the percolation threshold θ_{c,k} showed that θ_{c,k} is an increasing function of k in the range 1≤k≤3. For k≥4, all jammed configurations are nonpercolating states and, consequently, the percolation phase transition disappears. An explanation for this phenomenon is offered in terms of the rapid increase with k of the number of surrounding occupied sites needed to reach percolation, which gets larger than the sufficient number of occupied sites to define jamming. In the case of k=2 and 3, the percolation thresholds are θ_{c,k=2}(∞)=0.60355(8) and θ_{c,k=3}=0.63110(9). Our results significantly improve the previously reported values of θ_{c,k=2}^{Naka}=0.601(7) and θ_{c,k=3}^{Naka}=0.621(6) [Nakamura, Phys. Rev. A 36, 2384 (1987)0556-279110.1103/PhysRevA.36.2384]. In parallel, a comparison with previous results for jamming on these systems is also done. Finally, a complete analysis of critical exponents and universality has been done, showing that the percolation phase transition involved in the system has the same universality class as the ordinary random percolation, regardless of the size k considered.

Jamming and percolation of k

Jamming and percolation of three-dimensional (3D) k×k×k cubic objects (k^{3}-mers) deposited on simple cubic lattices have been studied by numerical simulations complemented with finite-size scaling theory. The k^{3}-mers were irreversibly deposited into the lattice. Jamming coverage θ_{j,k} was determined for a wide range of k (2≤k≤40). θ_{j,k} exhibits a decreasing behavior with increasing k, being θ_{j,k=∞}=0.4204(9) the limit value for large k^{3}-mer sizes. In addition, a finite-size scaling analysis of the jamming transition was carried out, and the corresponding spatial correlation length critical exponent ν_{j} was measured, being ν_{j}≈3/2. However, the obtained results for the percolation threshold θ_{p,k} showed that θ_{p,k} is an increasing function of k in the range 2≤k≤16. For k≥17, all jammed configurations are nonpercolating states, and consequently, the percolation phase transition disappears. The interplay between the percolation and the jamming effects is responsible for the existence of a maximum value of k (in this case, k=16) from which the percolation phase transition no longer occurs. Finally, a complete analysis of critical exponents and universality has been done, showing that the percolation phase transition involved in the system has the same universality class as the 3D random percolation, regardless of the size k considered.

Random sequential adsorption on Euclidean, fractal, and random lattices.

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 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-Rényi 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, such a characteristic length does not exist, and the substrate can be characterized by a fixed set of M vertices (sites) and an average connectivity (or degree) g. This paper concentrates on measuring (i) the probability W_{L(M)}(θ) that a lattice composed of L^{d}(M) elements reaches a coverage θ and (ii) the exponent ν_{j} characterizing the so-called jamming transition. The results obtained for Euclidean, fractal, and random lattices indicate that the quantities derived from the jamming probability W_{L(M)}(θ), such as (dW_{L}/dθ)_{max} and the inverse of the standard deviation Δ_{L}, behave asymptotically as M^{1/2}. In the case of Euclidean and fractal lattices, where L and d can be defined, the asymptotic behavior can be written as M^{1/2}=L^{d/2}=L^{1/ν_{j}}, with ν_{j}=2/d.

Standard and inverse bond percolation of straight rigid rods on square lattices.

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse bond percolation of straight rigid rods on square lattices. In the case of standard percolation, the lattice is initially empty. Then, linear bond k-mers (sets of k linear nearest-neighbor bonds) are randomly and sequentially deposited on the lattice. Jamming coverage p_{j,k} and percolation threshold p_{c,k} are determined for a wide range of k (1≤k≤120). p_{j,k} and p_{c,k} exhibit a decreasing behavior with increasing k, p_{j,k→∞}=0.7476(1) and p_{c,k→∞}=0.0033(9) being the limit values for large k-mer sizes. p_{j,k} is always greater than p_{c,k}, and consequently, the percolation phase transition occurs for all values of k. In the case of inverse percolation, the process starts with an initial configuration where all lattice bonds are occupied and, given that periodic boundary conditions are used, the opposite sides of the lattice are connected by nearest-neighbor occupied bonds. Then, the system is diluted by randomly removing linear bond k-mers from the lattice. The central idea here is based on finding the maximum concentration of occupied bonds (minimum concentration of empty bonds) for which connectivity disappears. This particular value of concentration is called the inverse percolation threshold p_{c,k}^{i}, and determines a geometrical phase transition in the system. On the other hand, the inverse jamming coverage p_{j,k}^{i} is the coverage of the limit state, in which no more objects can be removed from the lattice due to the absence of linear clusters of nearest-neighbor bonds of appropriate size. It is easy to understand that p_{j,k}^{i}=1-p_{j,k}. The obtained results for p_{c,k}^{i} show that the inverse percolation threshold is a decreasing function of k in the range 1≤k≤18. For k>18, all jammed configurations are percolating states, and consequently, there is no nonpercolating phase. In other words, the lattice remains connected even when the highest allowed concentration of removed bonds p_{j,k}^{i} is reached. In terms of network attacks, this striking behavior indicates that random attacks on single nodes (k=1) are much more effective than correlated attacks on groups of close nodes (large k's). Finally, the accurate determination of critical exponents reveals that standard and inverse bond percolation models on square lattices belong to the same universality class as the random percolation, regardless of the size k considered.

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.

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.

From single-file diffusion to two-dimensional cage diffusion and generalization of the totally asymmetric simple exclusion process to higher dimensions.

A two-dimensional constrained diffusion model is presented and characterized by numerical simulations. The model generalizes the one-dimensional single-file diffusion model by considering a cage diffusion constraint induced by neighboring particles, which is a more stringent condition than volume exclusion. Using numerical simulations we characterize the diffusion process and we particularly show that asymmetric transition probabilities lead to the two-dimensional Kardar-Parisi-Zhang universality class. Therefore, this very simple model effectively generalizes the one-dimensional totally asymmetric simple exclusion process to higher dimensions.

Site trimer percolation on square lattices.

Percolation of site trimers (k-mers with k=3) is investigated in a detailed way making use of an analytical model based on renormalization techniques in this problem. Results are compared to those obtained here by means of extensive computer simulations. Five different deposition possibilities for site trimers are included according to shape and orientation of the depositing objects. Analytical results for the percolation threshold p(c) are all close to 0.55, while numerical results show a slight dispersion around this value. A comparison with p(c) values previously reported for monomers and dimers establishes the tendency of p(c) to decrease as k increases. Critical exponent ν was also obtained both by analytical and numerical methods. Results for the latter give values very close to the expected value 4/3 showing that this percolation case corresponds to the universality class of random percolation.

Fractional statistical theory of adsorption applied to protein adsorption.

Experimental adsorption isotherms of bovine serum albumin (BSA) adsorbed on sulfonated microspheres were described by means of two analytical models: the first is the well-known Langmuir-Freundlich model (LF), and the second, called fractional statistical theory of adsorption (FSTA), is a statistical thermodynamics model developed recently by Ramirez-Pastor et al. [Phys. Rev. Lett. 93 (2004) 186101]. The experimental data, obtained by Hu et al. [Biochem. Eng. J. 23 (2005) 259] for different concentrations of sulfonate group on the surface of the microspheres, were correlated by using a fitting algorithm based on least-squares statistics. The combination of LF and FSTA models, along with the choice of an adequate fitting procedure, allowed us to obtain several conclusions: (i) as previously reported in the literature, the maximum amount adsorbed increases as the amount of sulfonate group increases; (ii) the equilibrium constant does not appear as a sensitive parameter to the amount of sulfonate group on the surface of the microspheres; and (iii) the values of the fitting parameters obtained from FSTA may be indicative of a mismatch between the equilibrium separation of the intermolecular interaction and the distance between the adsorption sites. The exhaustive study presented here has shown that FSTA model is a good one considering the complexity of the physical situation, which is intended to be described and could be more useful in interpreting experimental data of adsorption of molecules with different sizes and shapes.

Percolation of aligned rigid rods on two-dimensional square lattices.

The percolation behavior of aligned rigid rods of length k (kmers) on two-dimensional square lattices has been studied by numerical simulations and finite-size scaling analysis. The kmers, containing k identical units (each one occupying a lattice site), were irreversibly deposited along one of the directions of the lattice. The process was monitored 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 1 to 14, show that (i) the percolation threshold exhibits a decreasing function when it is plotted as a function of the kmer size; (ii) for any value of k (k>1), the percolation threshold is higher for aligned rods than for rods isotropically deposited; (iii) the phase transition occurring in the system belongs to the standard random percolation universality class regardless of the value of k considered; and (iv) in the case of aligned kmers, the intersection points of the curves of R(L,k)(p) for different system sizes exhibit nonuniversal critical behavior, varying continuously with changes in the kmer size. This behavior is completely different to that observed for the isotropic case, where the crossing point of the curves of R(L,k)(p) do not modify their numerical value as k is increased.

Relaxation of surface steps after thermal quenches: a numerical study within the terrace-step-kink model.

We study the out-of-equilibrium relaxation of surface steps after thermal quenches using numerical simulations of the terrace-step-kink model for a vicinal surface. We analyze both single and interacting steps in a situation where the temperature is suddenly changed at a given quench time. We focus on a physically relevant range of temperatures and show that the relaxation of the roughness is compatible with a power-law behavior with an effective relaxation exponent close to γ = 1/2 in all cases. This value is consistent with a one-dimensional Edwards-Wilkinson equation. In particular, this means that, although the case of interacting steps is effectively a two-dimensional system, its relaxation is dominated by short length-scale fluctuations, where steps are not interacting.

Effective Edwards-Wilkinson equation for single-file diffusion.

In this work, we present an effective discrete Edwards-Wilkinson equation aimed to describe the single-file diffusion process. The key physical properties of the system are captured defining an effective elasticity, which is proportional to the single particle diffusion coefficient and to the inverse squared mean separation between particles. The effective equation gives a description of single-file diffusion using the global roughness of the system of particles, which presents three characteristic regimes, namely, normal diffusion, subdiffusion, and saturation, separated by two crossover times. We show how these regimes scale with the parameters of the original system. Additional repulsive interaction terms are also considered and we analyze how the crossover times depend on the intensity of the additional terms. Finally, we show that the roughness distribution can be well characterized by the Edwards-Wilkinson universal form for the different single-file diffusion processes studied here.