Lattice Gas Systems Research Articles

Hydrodynamic fluctuations, diffusion and correlation functions in systems of particles adsorbed on solid surfaces

We present the detailed derivation of the equations describing the evolution of the hydrodynamic fluctuations of the coverage of particles adsorbed on homogeneous lattices. Using the method of the non-equilibrium statistical operator, we reduce the balance equation governing the behavior of the individual particles to the diffusion equation. On a macroscopic level, this equation describes the approach to equilibrium of the hydrodynamic coverage fluctuations. We obtain the analytical expressions for the Fickian diffusivity and Onsager phenomenological coefficient. These expressions are derived with account of the lateral interaction between the particles. They are simple functions of the thermodynamic quantities — derivatives of the thermodynamic potential over its arguments. The transport coefficients accurately describe the development of fluctuations in the entire coverage region and in the wide range of lateral interactions. We presented an elementary introduction to the theory of fluctuations in the lattice gas systems. For calculations of the correlation function and spectral density of fluctuations, we use the Langevin approach and the method of moments. The exact coincidence of the analytical expressions for the diffusion coefficients obtained by the two independent calculations is the direct proof of the accuracy of the approach developed in Chumak and Tarasenko (1980).

Diffusion in a binary mixture of molecules adsorbed on a multisite two-dimensional lattice

The diffusion in a binary mixture of species coadsorbed on a multisite square lattice is investigated by a theoretical approach (Chumak and Tarasenko, 1980) based on the non-equilibrium statistical operator method proposed by Zubarev (1961). The investigated lattice gas system is rather complex. There is a mixture of two types of particles (atoms, and/or molecules) adsorbed on a multisite lattice which consists of the three types of adsorption sites. This lattice can be subdivided onto three homogeneous square sublattices composed of the sites of the same type. As the binding energies for the molecules of distinct types adsorbed on the nonidentical sublattices are different, there are six average occupancies (coverages) describing the molecule distribution over the sublattices. A system of the balance equations, which controls the exchange of the molecules between the sublattices on the atomistic level is reduced to the diffusion equations describing the evolution of small hydrodynamic fluctuations of these coverages on the macroscopic level. The diffusion equations are written in the Onsager representation, when the driving forces are gradients of the chemical potentials and in the Fickian representation, when the driving forces are the gradients of coverages. The derivation of these equations results in the analytical expressions for the Fickian diffusivities and Onsager phenomenological coefficients. Despite the complex process of derivation, the final results are simple. The kinetic coefficients, which describe the non-equilibrium state of the molecule system, are expressed via the thermodynamic quantities calculated in the equilibrium state of the system: the chemical potentials (or activities), pair correlation functions and the isothermal susceptibilities. These expressions are derived with account of the lateral interactions between the nearest neighbor molecules. The matrix of the Fickian diffusivities has the ordinary classical form. The diffusivity is a non-symmetric 2 × 2 matrix. The Onsager coefficients are represented by a diagonal 2 × 2 matrix, where the cross terms are absent.

Condensation and Crystal Nucleation in a Lattice Gas with a Realistic Phase Diagram.

We reconsider model II of Orban et al. (J. Chem. Phys. 1968, 49, 1778–1783), a two-dimensional lattice-gas system featuring a crystalline phase and two distinct fluid phases (liquid and vapor). In this system, a particle prevents other particles from occupying sites up to third neighbors on the square lattice, while attracting (with decreasing strength) particles sitting at fourth- or fifth-neighbor sites. To make the model more realistic, we assume a finite repulsion at third-neighbor distance, with the result that a second crystalline phase appears at higher pressures. However, the similarity with real-world substances is only partial: Upon closer inspection, the alleged liquid–vapor transition turns out to be a continuous (albeit sharp) crossover, even near the putative triple point. Closer to the standard picture is instead the freezing transition, as we show by computing the free-energy barrier relative to crystal nucleation from the “liquid”.

Phase transitions in hard-core lattice gases on the honeycomb lattice.

We study lattice gas systems on the honeycomb lattice where particles exclude neighboring sites up to order k (k=1,...,5) from being occupied by another particle. Monte Carlo simulations were used to obtain phase diagrams and characterize phase transitions as the system orders at high packing fractions. For systems with first-neighbors exclusion (1NN), we confirm previous results suggesting a continuous transition in the two-dimensional Ising universality class. Exclusion up to second neighbors (2NN) lead the system to a two-step melting process where, first, a high-density columnar phase undergoes a first-order phase transition with nonstandard scaling to a solidlike phase with short-range ordered domains and, then, to fluidlike configurations with no sign of a second phase transition. 3NN exclusion, surprisingly, shows no phase transition to an ordered phase as density is increased, staying disordered even to packing fractions up to 0.98. The 4NN model undergoes a continuous phase transition with critical exponents close to the three-state Potts model. The 5NN system undergoes two first-order phase transitions, both with nonstandard scaling. We, also, propose a conjecture concerning the possibility of more than one phase transition for systems with exclusion regions further than 5NN based on geometrical aspects of symmetries.

The general theory of diffusion in a mixture of molecules coadsorbed on a homogeneous two-dimensional lattice

The diffusion in the multicomponent mixture of molecules coadsorbed on a surface with a square symmetry of the adsorption sites is investigated in the framework of a lattice gas model by a theoretical approach and the Monte Carlo simulations. Using the approach based on the theory of the non-equilibrium statistical operator I have derived the systems of equations describing the diffusion of the molecules in the Fickian and Onsager forms and obtained the analytical expressions for the Fickian diffusivities. These expressions are derived taking into account the lateral interactions between the molecules and the interactions of the activated molecules in the saddle points of the lattice potential relief with their surroundings. The diffusion equations in the Fickian representation exactly coincide with the corresponding classical counterparts, but the Onsager-like transport equations differ from their standard lore. In the new Onsager representation the migration of the diffusants is controlled only by the gradients of their own chemical potentials and the cross terms are absent. The Monte Carlo simulations have been used to test the correctness of the new expressions. There is a very good coincidence between the analytical and numerical data in the whole coverage region and in the wide range of the lateral interactions. The lateral interactions substantially influence on the behavior of the thermodynamic quantities and the Fickian diffusivities. These results demonstrate that the classical description in the Onsager presentation is not universal and cannot be applied for some lattice gas systems.

The theory of diffusion in a binary mixture of molecules coadsorbed on a two-dimensional lattice

The diffusion in the binary mixture of molecules coadsorbed on a square lattice is investigated by a theoretical approach and the Monte Carlo simulations. Using a lattice gas model I have derived a system of equations describing the diffusion of the molecules in the Fickian and Onsager representations. Also, I have obtained the analytical expressions for the Fickian diffusion coefficients. These expressions are derived taking into account the lateral interactions between the molecules and the interactions of the activated molecules in the saddle points of the lattice potential relief with their surroundings. The diffusion equations in the Fickian representation coincide with the corresponding classical counterparts, but the Onsager-like equations differ significantly from their standard lore – the cross terms are absent. It occures that the new transport coefficients describing the diffusion in the new Onsager representation are expressed in a rather nontrivial way via the phenomenological coefficients entering the classical equations. The Monte Carlo simulations have been used to test the correctness of the new expressions. There is a very good coincidence between the thermodynamic and kinetic data in the whole coverage region and in the wide range of the lateral interactions. The interactions substantially influence on the behavior of the thermodynamic quantities and the tracer and Fick diffusivities. These results demonstrate that the classical description of the molecule diffusion in the Onsager representation is not universal and may be not applied for some lattice gas systems.

Diagrammatic approach to gas-liquid phase transition in statistical theory

The lattice gas system is studied. In frameworks of statistical mechanics, it is proposed the analytic algorithm of approximations construction when pressure calculating. Algorithm is built on the basis of virial expansion using special small parameter ν. The parameter is introduced by the prescription of the corresponding ν-power weight to each Mayer diagram without contact vertexes. The obtained formulas describe the gas-liquid phase transition. The density dependence on pressure has the jump if the temperature is less than the critical one. The algorithm permits to calculate the phase diagrams contained the critical point. The formula of pressure obtained at null approximation corresponds to the self-consistent field approximation. Corrections to the critical temperature, the density jump value and the curves of phase diagrams are calculated in the approximation with the accuracy to first order on ν. It is manifested the forbiddance of phase transition in one-dimensional system.

Soft-particle lattice gas in one dimension: One- and two-component cases

The object of the present article is a 1d lattice-gas system comprised of soft-particles, wherein particles interact only if they occupy the same or a neighboring site, as a simple representation of penetrable particles of soft condensed matter. To represent different scenarios, two different realizations of the lattice model are considered, a one-component and a two-component system, where in the two-component case particles of the same species repel and those of opposite species attract each other. The systems are analyzed entirely within the transfer matrix framework. Special attention is paid to the criterion devised in Ref. [Phys. Rev. E 63, 031206 (2001)], which serves to separate two classes of behavior encountered in a one-component penetrable particle systems. In addition to confirm the existence of a similar criterion for the one-component lattice-gas model, we find that the same criterion can be applied to the two-component system to provide conditions for thermodynamic catastrophe.

Diffusion on a one-dimensional sawtooth lattice with the nearest and the next nearest neighbor interactions

The diffusion of particles adsorbed on a homogeneous one-dimensional sawtooth lattice with the nearest neighbor and the next nearest neighbor interactions between the particles is investigated using a theoretical approach and the kinetic Monte Carlo simulations. I have derived the analytical expressions for the diffusion coefficients. The calculated concentration dependencies of the coefficients have been compared with the numerical data generated by the simulations. The comparison reveals an interesting peculiarity of the particle diffusion. The lateral interactions induce correlations in the successive jumps of the particles. The thorough investigations of the dependencies clearly demonstrate how with the strengthening of the interactions the classical uncorrelated jumps are gradually replaced by the sequences of the correlated jumps. This behavior principally differs from the usual diffusion of particles in the homogeneous one-dimensional systems. It is an interesting and unexpected novelty of this system. Another uncommon feature of this model is a rich variety of the ‘diffusion coefficients’. There are four combinations of the interaction parameter signs, and every combination has its own specific behavior of the particle diffusion, qualitatively opposite from the other combinations. Also, it should be noted a quite unusual result. In the homogeneous lattice gas system (all sites have an equal depth) with the strong, repulsive nearest neighbor and attractive next nearest neighbor interactions the diffusion is perfectly described by the diffusion coefficients derived for the heterogeneous lattices with deep and shallow sites. It is a direct evidence of the specific mode of the particle migration when the correlated pairs of jumps give the main contribution to the diffusion coefficients.

Local Equilibrium in Inhomogeneous Stochastic Models of Heat Transport

We extend the duality of Kipnis Marchioro and Presutti to inhomogeneous lattice gas systems where either the components have different degrees of freedom or the rate of interaction depends on the spatial location. Then the dual process is applied to prove local equilibrium in the hydrodynamic limit for some inhomogeneous high dimensional systems andin the nonequilibrium steady state for one dimensional systems with arbitrary inhomogeneity.

Analytical approach for collective diffusion: One-dimensional heterogeneous lattice.

Diffusion of particles adsorbed on the heterogeneous chain was investigated using a theoretical approach and Monte Carlo method. I present the derivation of the analytical expressions for the diffusion coefficients. This approach is based on the theory of the non-equilibrium statistical operator developed by Zubarev. The concentration dependencies of the center-of-mass and Fickian diffusion coefficients have been calculated for some representative values of lateral interactions between the adsorbed particles. The analytical dependencies are compared with the numerical data generated by the kinetic Monte Carlo simulation. The almost perfect coincidence of the data obtained by the two quite different methods clearly demonstrates the applicability of the approach for the investigations of the particle migration in the lattice gas systems.

Microscopic reversibility and macroscopic irreversibility: A lattice gas model

We present coarse-grained descriptions and computations of the time evolution of a lattice gas system of indistinguishable particles, whose microscopic laws of motion are exactly reversible, in order to investigate how or what kind of macroscopically irreversible behavior may eventually arise. With increasing coarse-graining and number of particles, relative fluctuations of entropy rapidly decrease and apparently irreversible behavior unfolds. Although that behavior becomes typical in those limits and within a certain range, it is never absolutely irreversible for any individual system with specific initial conditions. Irreversible behavior may arise in various ways. We illustrate one possibility by replacing detailed integer occupation numbers at lattice sites with particle probability densities that evolve diffusively.

Dynamics of a lattice gas system of three species

This paper considers a mutualism system of three species in which each species provides resource for the next one in a one-directional loop, while there exists spatial competition among them. The system is characterized by a lattice gas model and the cases of obligate mutualisms, obligate–facultative mutualisms and facultative mutualisms are considered. Using dynamical systems theory, it is shown that (i) the mutualisms can lead to coexistence of species; (ii) A weak mutualism or an extremely strong mutualism will result in extinction of species, while even the superior facultative species will be driven into extinction by its over-strong mutualism on the next one; (iii) Initial population density plays a role in the coexistence of species. It is also shown that when there exists weak mutualism, an obligate species can survive by providing more benefit to the next one, and the inferior facultative species will not be driven into extinction if it can strengthen its mutualism on the next species. Moreover, Hopf bifurcation, saddle-node bifurcation and bifurcation of heteroclinic cycles are shown in the system. Projection method is extended to exhibit bistability in the three-dimensional model: when saddle-node bifurcation occurs, stable manifold of the saddle-node point divides intR+3 into two basins of attraction of two equilibria. Furthermore, Lyapunov method is applied to exhibit unstability of heteroclinic cycles. Numerical simulations confirm and extend our results.

The studies of particle diffusion on a heterogeneous one-dimensional lattice

Abstract Diffusion of particles adsorbed on a heterogeneous one-dimensional chain is investigated using the theoretical approach and Kinetic Monte Carlo simulations. The concentration dependencies of the center-of-mass and Fickian diffusion coefficients are calculated for some representative values of the lateral interaction between the adsorbed particles. We thoroughly compared the analytical dependencies with the numerical data. The perfect coincidence of the data obtained by the two different methods clearly demonstrates that the analytical expressions obtained for the diffusion coefficients exactly describe the particle migration in the heterogeneous lattice gas systems.

Population dynamics of intraguild predation in a lattice gas system

In the system of intraguild predation (IGP) we are concerned with, species that are in a predator–prey relationship, also compete for shared resources (space or food). While several models have been established to characterize IGP, mechanisms by which IG prey and IG predator can coexist in IGP systems with spatial competition, have not been shown. This paper considers an IGP model, which is derived from reactions on lattice and has a form similar to that of Lotka–Volterra equations. Dynamics of the model demonstrate properties of IGP and mechanisms by which the IGP leads to coexistence of species and occurrence of alternative states. Intermediate predation is shown to lead to persistence of the predator, while extremely big predation can lead to extinction of one/both species and extremely small predation can lead to extinction of the predator. Numerical computations confirm and extend our results. While empirical observations typically exhibit coexistence of IG predator and IG prey, theoretical analysis in this work demonstrates exact conditions under which this coexistence can occur.

Analytical approach for collective diffusion: one-dimensional homogeneous lattice.

Diffusion of particles adsorbed on the homogeneous one-dimensional chain was investigated using a theoretical approach and kinetic Monte Carlo simulations. The concentration dependencies of the center-of-mass and Fickian diffusion coefficients have been calculated for some representative values of lateral interactions between adsorbed particles. The analytical dependencies have been compared with the numerical data. The perfect coincidence of the data obtained by the two quite different methods clearly demonstrates that the analytical expressions for the diffusion coefficients derived in the framework of the approach based on the non-equilibrium statistical operator exactly describe the particle migration in the lattice gas systems.

The role of available sites in the activity of lattice gases with geometric constraints

The activity in lattice-gas systems with geometric constraints is shown to be the ratio of the number of particles to the number of available sites. The key role of sites available for occupation is emphasized. Available sites may be different for different species and are not necessarily just unoccupied sites. Location-specific or non-local constraints are allowed. An analytical expression for the number of available sites is given for the hard-hexagon model. The utility of an expression for available sites is illustrated for the non-trivial case of a mixed Langmuir/hard-hexagon adsorption system, where the influence of the Langmuir adsorbates on the hard-hexagon phase transition is investigated. The dependence on available sites indicates how to extend these results to the kinetic regime and simulations of kinetic voltammograms for the hard-hexagon model are given as an example.

Kinetics of hydrogen adsorption and desorption on Si(100) surfaces

The kinetics of molecular hydrogen reactions at the Si (100) surface has been studied by simulation to extract the physics underlying two unexpected experimental observations: apparently first-order desorption kinetics and an increase in sticking probability with hydrogen coverage. At a partially H-terminated Si(100) surface, each Si dimer assumes an unoccupied dimer (UOD), singly occupied dimer (SOD), or doubly occupied dimer (DOD) structure. In our hydrogen reaction model based on an inter-dimer mechanism, a site consisting of an adjacent pair of a DOD and a UOD (DOD/UOD) is a key component for the desorption and adsorption kinetics of hydrogen at the Si(100) surface. To simulate reaction kinetics of both reactions, DU (D: DOD, U: UOD) and SS (S: SOD) pathways are proposed: DU pathway claims that the adsorption as well as desorption of hydrogen takes place at common sites having a cis-configured SOD/SOD pair that is transformed transiently from a DOD/UOD pair by H(D) diffusion. Thus the adsorption obeys the so-called 4H mechanism, but the desorption obeys the 2H mechanism. SS pathway claims that the adsorption occurs at sites having a UOD/UOD pair, and the desorption occurs at sites having a cis-configured SOD/SOD pair that is generated by diffusion of isolated SODs. To simulate temperature-programmed-desorption spectra and sticking probability vs coverage curves, thermo-statistics for a lattice-gas system characterized with parameters for hydrogen pairing and dimer clustering is used to evaluate equilibrium populations of DOD/UOD pairs and isolated SODs. The model simulation based on the above reaction model successfully reproduces all of the complicated, coverage dependent adsorption and desorption reactions of hydrogen at Si(100) surfaces. Specifically, at high coverage above 0.1 ML majority of the adsorption and desorption proceed along the DU pathway. Hence, it is suggested that the adsorption and desorption in the high coverage regime are not microscopically reversible. On the other hand, at low coverages below 0.1 ML, the simulation shows up that the majority of adsorption proceeds along the SS pathway, and the desorption by the DU pathway. Since both reactions obey the 2H mechanism, it is suggested that the desorption and adsorption in the low coverage regime are microscopically reversible.

Chemical potential evaluation in NVT lattice-gas simulations

The discrete nature of the partition function of a lattice-gas system can be exploited to build an efficient strategy for the evaluation of the chemical potential of a periodic lattice-gas with arbitrarily ranged interactions during a simulation in the canonical ensemble, with the need of no additional sampling as it were required instead by the Widom insertion/deletion approach. The present method is based on the main concepts of the small system grand ensemble [for details, see G. Soto-Campos, D. S. Corti, and H. Reiss, J. Chem. Phys. 108, 2563 (1998)], whose key idea is to study the properties of a sublattice (called small system) and of its complementary (the reservoir sublattice) as they were two separated subsystems. The accuracy of the measured chemical potential can be further improved by artificially "restoring" the missing connections among the reservoir sublattice sites located at the boundaries with the small system. We first illustrate the theory and then we compare μVT with NVT simulation results on several test systems.

Surface diffusion near first-order phase transitions for a model on a triangularlattice

We rigorously study the thermodynamic factor of surface diffusion for a two-dimensionalone-component lattice-gas system on a regular triangular lattice. First, we prove thatfour low-temperature phases exist in the system, their surface coverages being 0,1/3,2/3 and 1, and that, depending on the interparticle interactions, a phase transition between anytwo of them can occur. Three of these transitions are shown to be of first order at lowtemperatures, while in the other three cases there is an infinite number of ground states,yielding in fact a residual entropy. Then, for the three first-order phase transitions, weprovide explicit finite-size formulae for the free energy, surface coverage and isothermalsusceptibility, showing their low-temperature dependence on the chemical potential and thetotal number of adsorption sites. Using these formulae, we numerically obtain thecoverage dependence of the thermodynamic factor and compare it with analyticalresults.