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Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse percolation of triangular tiles of side k (k-tiles) on triangular lattices. In the case of standard percolation, the lattice is initially empty. Then, k-tiles are randomly and sequentially deposited on the lattice. In the case of inverse percolation, the process starts with an initial configuration where all lattice sites are occupied by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then, the system is diluted by randomly removing k-tiles [composed by k(k+1)/2 monomers] from the lattice. Two schemes are used for the depositing and removing process: the isotropic scheme, where the deposition (removal) of the objects occurs with the same probability in any lattice direction; and the anisotropic (perfectly oriented or nematic) scheme, where one lattice direction is privileged for depositing (removing) the tiles. The study is conducted by following the behavior of four critical concentrations with the size k: (i) [(ii)] standard isotropic (oriented) percolation threshold θ_{c,k} (Ï_{c,k}), which represents the minimum concentration of occupied sites at which an infinite cluster of occupied nearest-neighbor sites extends from one side of the system to the other. θ_{c,k} (Ï_{c,k}) is reached by isotropic (oriented) deposition of k-tiles on an initially empty lattice; and (iii) [(iv)] inverse isotropic (oriented) percolation threshold θ_{c,k}^{i} (Ï_{c,k}^{i}), which corresponds to the maximum concentration of occupied sites for which connectivity disappears. θ_{c,k}^{i} (Ï_{c,k}^{i}) is reached after removing isotropic (completely aligned) k-tiles from an initially fully occupied lattice. The obtained results indicate that (1)θ_{c,k} (θ_{c,k}^{i}) is an increasing (decreasing) function of k in the range 1≤k≤6. For k≥7, all jammed configurations are nonpercolating (percolating) states and, consequently, the percolation phase transition disappears. (2)Ï_{c,k} (Ï_{c,k}^{i}) show a behavior qualitatively similar to that observed for isotropic deposition. In this case, the minimum value of k at which the phase transition disappears is k=5. (3) For both isotropic and perfectly oriented models, the curves of standard and inverse percolation thresholds are symmetric to each other with respect to the line θ(Ï)=0.5. Thus, a complementary property is found θ_{c,k}+θ_{c,k}^{i}=1 (and Ï_{c,k}+Ï_{c,k}^{i}=1), which has not been observed in other regular lattices. (4) Finally, in all cases, the jamming exponent ν_{j} was measured, being ν_{j}=1 regardless of the orientation (isotropic or nematic) or 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 five (three) for isotropic (nematic) deposition scheme, has the same universality class as the standard percolation problem.
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Combining Monte Carlo simulations and thermodynamic integration method, we study the configurational entropy per site of straight rigid rods of length k (k-mers) adsorbed on three-dimensional (3D) simple cubic lattices. The process is monitored by following the dependence of the lattice coverage θ on the chemical potential µ (adsorption isotherm). Then, we perform the integration of µ(θ) over θ to calculate the configurational entropy per site of the adsorbed phase s(k,θ) as a function of the coverage. Based on the behavior of the function s(k,θ), different phase diagrams are obtained according to the k values: k≤4, disordered phase; k=5,6, disordered and layered-disordered phases; and k≥7, disordered, nematic and layered-disordered phases. In the limit of θâ1 (full coverage), the configurational entropy per site is determined for values of k ranging between 2 and 8. For k≥6, MC data coincide (within the statistical uncertainty) with recent analytical predictions [D. Dhar and R. Rajesh, Phys. Rev. E 103, 042130 (2021)2470-004510.1103/PhysRevE.103.042130] for very large rods. This finding represents the first numerical validation of the expression obtained by Dhar and Rajesh for d-dimensional lattices with d>2. In addition, for k≥5, the values of s(k,θâ1) for simple cubic lattices are coincident with those values reported in [P. M. Pasinetti et al., Phys. Rev. E 104, 054136 (2021)2470-004510.1103/PhysRevE.104.054136] for two-dimensional (2D) square lattices. This is consistent with the picture that at high densities and k≥5, the layered-disordered phase is formed on the lattice. Under these conditions, the system breaks to 2D layers, and the adsorbed phase becomes essentially 2D. The 2D behavior of the fully covered lattice reinforces the conjecture that the large-k behavior of entropy per site is superuniversal, and holds on d-dimensional hypercubical lattices for all d≥2.
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Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse percolation by removing semirigid rods from a L×L square lattice that contains two layers (and M=L×L×2 sites). The process starts with an initial configuration where all lattice sites are occupied by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then the system is diluted by removing groups of k consecutive monomers according to a generalized random sequential adsorption mechanism. The study is conducted by following the behavior of two critical concentrations with size k: (1) jamming coverage θ_{j,k}, which represents the concentration of occupied sites at which the jamming state is reached, and (2) inverse percolation threshold θ_{c,k}, which corresponds to the maximum concentration of occupied sites for which connectivity disappears. The obtained results indicate that (1) the jamming coverage exhibits an increasing dependence on the size k-it rapidly increases for small values of k and asymptotically converges towards a definite value for infinitely large k sizes θ_{j,kâ∞}≈0.2701-and (2) the inverse percolation threshold is a decreasing function of k in the range 1≤k≤17. For k≥18, all jammed configurations are percolating states (the lattice remains connected even when the highest allowed concentration of removed sites is reached) and, consequently, there is no nonpercolating phase. This finding contrasts with the results obtained in literature for a complementary problem, where straight rigid k-mers are randomly and irreversibly deposited on a square lattice forming two layers. In this case, percolating and nonpercolating phases extend to infinity in the space of the parameter k and the model presents percolation transition for the whole range of k. The results obtained in the present study were also compared with those reported for the case of inverse percolation by removal of rigid linear k-mers from a square monolayer. The differences observed between monolayer and bilayer problems were discussed in terms of vulnerability and network robustness. Finally, the accurate determination of the critical exponents ν, ß, and γ reveals that the percolation phase transition involved in the system has the same universality class as the standard percolation problem.
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Monte Carlo simulations and finite-size scaling theory have been carried out to study the critical behavior and universality for the isotropic-nematic (IN) phase transition in a system of straight rigid pentamers adsorbed on a triangular lattice with polarized nonhomogeneous intermolecular interactions. The model was inspired by the deposition of 2-thiophene molecules over the Au(111) surface, which was previously characterized by experimental techniques and density functional theory. A nematic phase, observed experimentally by the formation of a self-assembled monolayer of parallel molecules, is separated from the isotropic state by a continuous transition occurring at a finite density. The precise determination of the critical exponents indicates that the transition belongs to the three-state Potts universality class. The finite-size scaling analysis includes the study of mutability and diversity. These two quantities are derived from information theory and they have not previously been considered as part of the conventional treatment of critical phenomena for the determination of critical exponents. The results obtained here contribute to the understanding of formation processes of self-assembled monolayers, phase transitions, and critical phenomena from novel compression algorithms for studying mutual information in sequences of data.
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Polymers are frequently deposited on different surfaces, which has attracted the attention of scientists from different viewpoints. In the present approach polymers are represented by rigid rods of length k (k-mers), and the substrate takes the form of an L×L square lattice whose lattice constant matches exactly the interspacing between consecutive elements of the k-mer chain. We briefly review the classical description of the nematic transition presented by this system for k≥7 observing that the high-coverage (θ) transition deserves a more careful analysis from the entropy point of view. We present a possible viewpoint for this analysis that justifies the phase transitions. Moreover, we perform Monte Carlo (MC) simulations in the grand canonical ensemble, supplemented by thermodynamic integration, to first calculate the configurational entropy of the adsorbed phase as a function of the coverage, and then to explore the different phases (and orientational transitions) that appear on the surface with increasing the density of adsorbed k-mers. In the limit of θâ1 (full coverage) the configurational entropy is obtained for values of k ranging between 2 and 10. MC data are discussed in comparison with recent analytical results [D. Dhar and R. Rajesh, Phys. Rev. E 103, 042130 (2021)2470-004510.1103/PhysRevE.103.042130]. The comparative study allows us to establish the applicability range of the theoretical predictions. Finally, the structure of the high-coverage phase is characterized in terms of the statistics of k×l domains (domains of l parallel k-mers adsorbed on the surface). A distribution of finite values of l (lâªL) is found with a predominance of k×1 (single k-mers) and k×k domains. The distribution is the same in each lattice direction, confirming that at high density the adsorbed phase goes to a state with mixed orientations and no orientational preference. An order parameter measuring the number of k×k domains in the adsorbed layer is introduced.
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In this work we study the deposition phenomena on a modified electrode in the framework of the mean field theory. The electrode surface is modified by irreversible deposition of impurities which can block a fraction of the adsorption sites. Then, an electroactive species is allowed to adsorb on the accessible sites, transferring electric charge and generating a current that can be calculated and measured. Nearest-neighbor lateral interactions are considered both between electroactive particles and between particles and impurities. A modified Bragg-Williams theoretical approach considers both the blocking effects of impurities and the lateral interactions, through different concentrations of impurities and particles. The analysis is based on the study of adsorption isotherms and voltammograms, considering different interaction energies and impurity concentrations. The potentialities and limitations of the analytical approximation are discussed by comparing theoretical predictions with Monte Carlo simulations and experimental measurements in which artificial clay represents the impurity and a [Fe(CN)6]4 redox probe is the species that transfers the charge.
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Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse percolation of straight rigid rods on triangular lattices. In the case of standard percolation, the lattice is initially empty. Then, linear k-mers (particles occupying k consecutive sites along one of the lattice directions) are randomly and sequentially deposited on the lattice. In the case of inverse percolation, the process starts with an initial configuration where all lattice sites are occupied by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then the system is diluted by randomly removing sets of k consecutive monomers (linear k-mers) from the lattice. Two schemes are used for the depositing/removing process: an isotropic scheme, where the deposition (removal) of the linear objects occurs with the same probability in any lattice direction, and an anisotropic (perfectly oriented) scheme, where one lattice direction is privileged for depositing (removing) the particles. The study is conducted by following the behavior of four critical concentrations with size k: (i) [(ii)] standard isotropic[oriented] percolation threshold θ_{c,k}[Ï_{c,k}], which represents the minimum concentration of occupied sites at which an infinite cluster of occupied nearest-neighbor sites extends from one side of the system to the other. θ_{c,k}[Ï_{c,k}] is reached by isotropic[oriented] deposition of straight rigid k-mers on an initially empty lattice; and (iii) [(iv)] inverse isotropic[oriented] percolation threshold θ_{c,k}^{i}[Ï_{c,k}^{i}], which corresponds to the maximum concentration of occupied sites for which connectivity disappears. θ_{c,k}^{i}[Ï_{c,k}^{i}] is reached after removing isotropic [completely aligned] straight rigid k-mers from an initially fully occupied lattice. θ_{c,k}, Ï_{c,k}, θ_{c,k}^{i}, and Ï_{c,k}^{i} are determined for a wide range of k (2≤k≤512). The obtained results indicate that (1)θ_{c,k}[θ_{c,k}^{i}] exhibits a nonmonotonous dependence on the size k. It decreases[increases] for small particle sizes, goes through a minimum[maximum] at around k=11, and finally increases and asymptotically converges towards a definite value for large segments θ_{c,kâ∞}=0.500(2) [θ_{c,kâ∞}^{i}=0.500(1)]; (2)Ï_{c,k}[Ï_{c,k}^{i}] depicts a monotonous behavior in terms of k. It rapidly increases[decreases] for small particle sizes and asymptotically converges towards a definite value for infinitely long k-mers Ï_{c,kâ∞}=0.5334(6) [Ï_{c,kâ∞}^{i}=0.4666(6)]; (3) for both isotropic and perfectly oriented models, the curves of standard and inverse percolation thresholds are symmetric to each other with respect to the line θ(Ï)=0.5. Thus a complementary property is found θ_{c,k}+θ_{c,k}^{i}=1 (and Ï_{c,k}+Ï_{c,k}^{i}=1) which has not been observed in other regular lattices. This condition is analytically validated by using exact enumeration of configurations for small systems, and (4) in all cases, the critical concentration curves divide the θ space in a percolating region and a nonpercolating region. These phases extend to infinity in the space of the parameter k so that the model presents percolation transition for the whole range of k.
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The site-bond percolation problem in two-dimensional kagome lattices has been studied by means of theoretical modeling and numerical simulations. Motivated by considerations of cluster connectivity, two distinct schemes (denoted as Sâ©B and SâªB) have been considered. In Sâ©B (SâªB), two points are connected if a sequence of occupied sites and (or) bonds joins them. Analytical and simulation approaches, supplemented by analysis using finite-size scaling theory, were used to calculate the phase boundaries between the percolating and nonpercolating regions, thus determining the complete phase diagram of the system in the (p_{s},p_{b}) space. In the case of the Sâ©B model, the obtained results are in excellent agreement with previous theoretical and numerical predictions. In the case of the SâªB model, the limiting curve separating percolating and nonpercolating regions is reported here.
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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) θ_{j,k} is a decreasing function with increasing k, being θ_{j,kâ∞}=0.6007(6) the limit value for infinitely long k-mers; and (ii) θ_{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.
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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 Lâ∞. 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.