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In this work two theoretical approximations, the so-called theoretical approach of energy levels and an extension of the modified mean field approach (TAEL and MMFA, respectively) are applied to the study of surface decoration of modified nanostructures like crystalline nanoparticles. The surface of the nanoparticles is modified by the irreversible random deposition of defects consisting in isolated atoms. Such deposition is carried out until a certain surface density is reached, leaving the rest of the sites available for a second species to adsorb. Through the formulation of the integral equation, the theoretical approaches permit obtaining the adsorption isotherms and the compressibility of the adlayer. The main difference between the two approaches is the degree of details considered in their mathematical formulations: TAEL takes in account all the energy levels meanwhile MMFA only an average. The degree of precision and usefulness of both theories were evaluated in comparison with Monte Carlo simulations in the grand canonical assembly. Several cases were studied: attractive and repulsive lateral interactions and different fraction of defects. The effects of the nanoscale were considered for different types and sizes of nanoparticles. By calculating an integral error, we are able to affirm that TAEL reproduces all the properties of the analyzed quantities from the reference simulated curves. On the other hand, the MMFA performance is good only for a certain limited range of the parameters, however the strength is in the mathematical simplicity compared to TAEL.

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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 θ→1 (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,θ→1) 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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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 θ→1 (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 (l≪L) 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.

RESUMO

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}[ϑ_{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}[ϑ_{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}[ϑ_{c,k}^{i}], which corresponds to the maximum concentration of occupied sites for which connectivity disappears. θ_{c,k}^{i}[ϑ_{c,k}^{i}] is reached after removing isotropic [completely aligned] straight rigid k-mers from an initially fully occupied lattice. θ_{c,k}, ϑ_{c,k}, θ_{c,k}^{i}, and ϑ_{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,k→∞}=0.500(2) [θ_{c,k→∞}^{i}=0.500(1)]; (2)ϑ_{c,k}[ϑ_{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 ϑ_{c,k→∞}=0.5334(6) [ϑ_{c,k→∞}^{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(ϑ)=0.5. Thus a complementary property is found θ_{c,k}+θ_{c,k}^{i}=1 (and ϑ_{c,k}+ϑ_{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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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.

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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.

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The statistical thermodynamics of binary mixtures of polyatomic species was developed based on a generalization in the spirit of the lattice-gas model and the quasi-chemical approximation (QCA). The new theoretical framework is obtained by combining: (i) the exact analytical expression for the partition function of non-interacting mixtures of linear k-mers and l-mers (species occupying k sites and l sites, respectively) adsorbed in one dimension, and its extension to higher dimensions; and (ii) a generalization of the classical QCA for multicomponent adsorbates and multisite-occupancy adsorption. This process is analyzed using the partial adsorption isotherms corresponding to both species of the mixture. Comparisons with analytical data from Bragg-Williams approximation (BWA) and Monte Carlo simulations are performed in order to test the validity of the theoretical model. Even though a good fitting is obtained from BWA, it is found that QCA provides a more accurate description of the phenomenon of adsorption of interacting polyatomic mixtures.

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The statistical thermodynamics of polyatomic species mixtures adsorbed on two-dimensional lattices was developed based on generalization of the semiempirical approximation for the adsorption of single components [Romá, F. et al., Langmuir, 2006, 22, 3192-3197]. In this scheme, the partial adsorption isotherms are obtained using a correction function C[combining tilde], which relates to the conditional probability of finding the ith empty site to a lattice with i- 1 already vacant sites. This approximation allows us to write a new theoretical model using a combination of the correction functions corresponding to exact 1-D calculations and the Guggenheim-DiMarzio approach. Finally, comparisons with MC simulations and experimental data of methane-ethane and ethane-propylene mixtures on activated carbon are used to test the accuracy and reliability of the proposed model. The obtained results indicate that the new thermodynamic description is significantly better than the existing theoretical models developed to treat adsorption of interacting binary mixtures of polyatomics.

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In a previous paper [F. Romá, A. J. Ramirez-Pastor, and J. L. Riccardo, Phys. Rev. B 72, 035444 (2005)], the critical behavior of repulsive rigid rods of length k (k-mers) on a square lattice at half coverage has been studied by using Monte Carlo (MC) simulations. The obtained results indicated that (1) the phase transition occurring in the system is a second-order phase transition for all adsorbate sizes k; and (2) the universality class of the transition changes from 2D Ising-type for monomers (k = 1) to an unknown universality class for k ≥ 2. In the present work, we revisit our previous results together with further numerical evidences, resulting from new extensive MC simulations based on an efficient exchange algorithm and using high-performance computational capabilities. In contrast to our previous conclusions (1) and (2), the new numerical calculations clearly support the occurrence of a first-order phase transition for k ≥ 2. In addition, a similar scenario was found for k-mers adsorbed on the triangular lattice at coverage k/(2k+1).

Assuntos

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In the present paper, the adsorption thermodynamics of a lattice-gas model which mimics a nanoporous environment is studied by considering nonadditive interactions between the adsorbed particles. It is assumed that the energy linking a certain atom with any of its nearest neighbors strongly depends on the state of occupancy in the first coordination sphere of such an adatom. By means of Monte Carlo (MC) simulations in the grand canonical ensemble, adsorption isotherms and differential heats of adsorption were calculated. Their striking behaviors were analyzed and discussed in terms of the low temperature phases formed in the system. Finally, the results obtained from MC simulations were compared with the corresponding ones from Bragg-Williams approximation.

Assuntos

RESUMO

Monte Carlo simulations and finite-size scaling analysis have been carried out to study the critical behavior in a submonolayer lattice-gas which mimics a nanoporous environment. In this model, the adsorbent is modeled as one-dimensional channels of equivalent adsorption sites arranged in a triangular cross-sectional structure. Two kinds of lateral interaction energies have been considered: (1) w(L) interaction energy between nearest-neighbor particles adsorbed along a single channel and (2) w(T) interaction energy between particles adsorbed across nearest-neighbor channels. We focus on the case of repulsive transverse (w(T)>0) and attractive longitudinal (w(L)<0) lateral interactions, where a rich variety of structural orderings are observed in the adlayer depending on the value of the parameters k(B)T/w(T) (being k(B) the Boltzmann constant) and w(L)/w(T). The results reveal the existence of a first-order phase transition in the adlayer between a low-temperature "condensed" phase and a high-temperature "disordered" phase.

RESUMO

Monte Carlo simulations and finite-size scaling analysis have been carried out to study the critical behavior in a submonolayer lattice-gas of interacting monomers adsorbed on one-dimensional channels arranged in a triangular cross-sectional structure. Two kinds of lateral interaction energies have been considered: (1) w(L), interaction energy between nearest-neighbor particles adsorbed along a single channel and (2) w(T), interaction energy between particles adsorbed across nearest-neighbor channels. We focus on the case of repulsive transverse interactions (w(T)>0), where a rich variety of structural orderings are observed in the adlayer, depending on the value of the parameters k(B)Tw(T) (being k(B) the Boltzmann constant) and w(L)w(T). For w(L)w(T)=0, successive planes are uncorrelated, the system is equivalent to the triangular lattice, and the well-known ([square root] 3 x [square root] 3) [([square root] 3 x ([square root] 3)(*)] ordered phase is found at low temperatures and a coverage, theta, of 13. In the more general case (w(L)/w(T) not equal 0), a competition between interactions along a single channel and a transverse coupling between sites in neighboring channels leads to a three-dimensional adsorbed layer. Consequently, the ([square root] 3 x ([square root] 3) and (([square root] 3 x ([square root] 3)(*) structures "propagate" along the channels and new ordered phases appear in the adlayer. Each ordered phase is separated from the disordered state by a continuous order-disorder phase transition occurring at a critical temperature, T(c), which presents an interesting dependence with w(L)/w(T). The Monte Carlo technique was combined with the recently reported free energy minimization criterion approach (FEMCA) [F. Roma et al., Phys. Rev. B 68, 205407 (2003)] to predict the critical temperatures of the order-disorder transformation. The excellent qualitative agreement between simulated data and FEMCA results allows us to interpret the physical meaning of the mechanisms underlying the observed transitions.

Assuntos

RESUMO

The configurational entropy of interacting particles adsorbed on one-dimensional channels arranged in a triangular cross-sectional structure is studied by combining Monte Carlo simulation and thermodynamic integration method. Three different energies have been considered in the adsorption process: (1) epsilono, constant interaction energy between a monomer and an adsorption site; (2) wL, interaction energy between nearest-neighbor particles adsorbed along a single channel, and (3) wT, interaction energy between particles adsorbed across nearest-neighbor channels. Special attention is devoted to the case of repulsive transversal interactions (wT>0), for which a rich variety of ordered phases are observed in the adlayer, depending on the value of the parameters kBT/wT (being kB the Boltzmann constant) and wL/wT. The influence of each ordered structure on the configurational entropy of the adlayer has been analyzed and discussed in the context of the lattice-gas model.