Lattice gas computers

Y. Hashimoto, Y. Chen, and H. Ohashi, Immiscible real-coded lattice gas, Comput. Phys. Commun, 129, 56 (2000). 

In a lattice gas, computational particles are allowed to move on a lattice according to certain rules. The lattice may be either two or three dimensional. By considering a hexagonal lattice, Frisch et al. [109] showed that one could obtain solutions of the two-dimensional Navier-Stokes equation by suitable averaging of the lattice gas solutions for appropriate collision rules. Their work was extended to three dimensions by d Humieres et al [110] who considered a projection of the four-dimensional face-centered-hyper-cubic (FCHC) lattice. 

Wetting phenomena on irregularly rough surfaces have not been studied so far. It seems quite reasonable to use computer simulation methods for this purpose. Of course, such computer simulation would be very expensive as the finite size of the simulation cells would require appropriate averaging over different spatial distributions of surface inhomogeneities. Nevertheless, with modern fast computers and using multispin coding techniques such calculations can be efficiently carried out for lattice gas systems. 

For adsorbates out of local equilibrium, an analytic approach to the kinetic lattice gas model is a powerful theoretical tool by which, in addition to numerical results, explicit formulas can be obtained to elucidate the underlying physics. This allows one to extract simplified pictures of and approximations to complicated processes, as shown above with precursor-mediated adsorption as an example. This task of theory is increasingly overlooked with the trend to using cheaper computer power for numerical simulations. Unfortunately, many of the simulations of adsorbate kinetics are based on unnecessarily oversimplified assumptions (for example, constant sticking coefficients, constant prefactors etc.) which rarely are spelled out because the physics has been introduced in terms of a set of computational instructions rather than formulating the theory rigorously, e.g., based on a master equation. 

The lattice gas has been used as a model for a variety of physical and chemical systems. Its application to simple mixtures is routinely treated in textbooks on statistical mechanics, so it is natural to use it as a starting point for the modeling of liquid-liquid interfaces. In the simplest case the system contains two kinds of solvent particles that occupy positions on a lattice, and with an appropriate choice of the interaction parameters it separates into two phases. This simple version is mainly of didactical value [1], since molecular dynamics allows the study of much more realistic models of the interface between two pure liquids [2,3]. However, even with the fastest computers available today, molecular dynamics is limited to comparatively small ensembles, too small to contain more than a few ions, so that the space-charge regions cannot be included. In contrast, Monte Carlo simulations for the lattice gas can be performed with 10 to 10 particles, so that modeling of the space charge poses no problem. In addition, analytical methods such as the quasichemical approximation allow the treatment of infinite ensembles. 

Both the lattice-gas model and computer simulations predict that the interface is rough on a molecular scale. If this roughness is not too large, the interface can be represented by a... 

Perhaps the most extensive computational study of the kinetics of NO reactions on Rh and Pd surfaces has been provided by the group of Zgrablich. Their initial simulations of the NO + CO reaction on Rh(lll) corroborated the fact that the formation of N-NO intermediate is necessary for molecular nitrogen production [83], They also concluded that an Eley-Rideal mechanism is necessary to sustain a steady-state catalytic regime. Further simulations based on a lattice-gas model tested the role of the formation of... 

What is next Several examples were given of modem experimental electrochemical techniques used to characterize electrode-electrolyte interactions. However, we did not mention theoretical methods used for the same purpose. Computer simulations of the dynamic processes occurring in the double layer are found abundantly in the literature of electrochemistry. Examples of topics explored in this area are investigation of lateral adsorbate-adsorbate interactions by the formulation of lattice-gas models and their solution by analytical and numerical techniques (Monte Carlo simulations) [Fig. 6.107(a)] determination of potential-energy curves for metal-ion and lateral-lateral interaction by quantum-chemical studies [Fig. 6.107(b)] and calculation of the electrostatic field and potential drop across an electric double layer by molecular dynamic simulations [Fig. 6.107(c)]. 

A promising study of the lattice gas model is the computer statistical tests (by the Monte Carlo method). Such calculations have been carried out since the mid-1960s (see, for example, refs. 66 and 105). For calculations of gas adsorption on metals, see refs. 106-110. However, no systematic application of the Monte Carlo method to heterogeneous reactions has been carried out it is to be done in the future. 

Except the kinetic equations, now various numerical techniques are used to study the dynamics of surfaces and gas-solid interface processes. The cellular automata and MC techniques are briefly discussed. Both techniques can be directly connected with the lattice-gas model, as they operate with discrete distribution of the molecules. Using the distribution functions in a kinetic theory a priori assumes the existence of the total distribution function for molecules of the whole system, while all numerical methods have to generate this function during computations. A success of such generation defines an accuracy of simulations. Also, the well-known molecular dynamics technique is used for interface study. Nevertheless this topic is omitted from our consideration as it requires an analysis of a physical background for construction of the transition probabilities. This analysis is connected with an oscillation dynamics of all species in the system that is absent in the discussed kinetic equations (Section 3). 

The aim of this chapter is to provide the reader with an overview of the potential of modern computational chemistry in studying catalytic and electro-catalytic reactions. This will take us from state-of-the-art electronic structure calculations of metal-adsorbate interactions, through (ab initio) molecular dynamics simulations of solvent effects in electrode reactions, to lattice-gas-based Monte Carlo simulations of surface reactions taking place on catalyst surfaces. Rather than extensively discussing all the different types of studies that have been carried out, we focus on what we believe to be a few representative examples. We also point out the more general theory principles to be drawn from these studies, as well as refer to some of the relevant experimental literature that supports these conclusions. Examples are primarily taken from our own work other recent review papers, mainly focused on gas-phase catalysis, can be found in [1-3]. 

Results of recent theoretical and computer simulation studies of phase transitions in monolayer films of Lennard-Jones particles deposited on crystalline solids are discussed. DiflFerent approaches based on lattice gas and continuous space models of adsorbed films are considered. Some new results of Monte Carlo simulation study for melting and ordering in monolayer films formed on the (100) face of an fee crystal are presented and confronted with theoretical predictions. In particular, it is demonstrated that the inner structure of solid films and the mechanism of melting transition depend strongly on the effects due to the periodic variation of the gas - solid potential. 

In this chapter we have summarized the fundamentals and recent advances in thermodynamic and kinetic approaches to lithium intercalation into, and deintercalation from, transition metals oxides and carbonaceous materials, and have also provided an overview of the major experimental techniques. First, the thermodynamics oflithium intercalation/deintercalation based on the lattice gas model with various approximations was analyzed. Lithium intercalation/deintercalation involving phase transformations, such as order-disorder transition or two-phase coexistence caused by strong interaction oflithium ions in the solid electrode, was clearly explained based on the lattice gas model, with the aid of computational methods. 

Stockman (1997) considered dispersion in fractures using a lattice gas simulation incorporating sorption kinetics and buoyancy factors. He advocated the use of solute slugs over step changes in concentration because moment calculations are simplified and noise in the computation of second moments is reduced. 

Soil structure, antecedent soil moisture and input flow rate control rapid flow along preferential pathways in well-structured soils. The amount of preferential flow may be significant for high input rates, mainly in the intermediate to high ranges of moisture. We use a three-dimensional lattice-gas model to simulate infiltration in a cracked porous medium as a function of rainfall intensity. We compute flow velocities and water contents during infiltration. The dispersion mechanisms of the rapid front in the crack are analyzed as a function of rainfall intensity. The numerical lattice-gas solutions for flow are compared with the analytical solution of the kinematic wave approach. The process is better described by the kinematic wave approach for high input flow intensities, but fails to adequately predict the front attenuation showed by the lattice-gas solution. 

We recall now that at p = p, (4.68) is exact for a lattice gas hence, on the basis of widely held notions of universality, p can also be assumed in a continuum-fluid computation without doing violence to the structure of the dominant singularities that emerge at the fluid critical point as t— 0, p = p. Since (4.71) follows without further assumptions from (4.64) and (4.68), (4.71) appears to be an appropriate expression for the study of critical behavior of S2 in the constant-polarizability model. It is especially useful if we note that there is gross similarity between the function O2 in (4.71) and the attractive part of a typical pair potential. To exploit this, consider the potential... 

LBM was originally proposed by McNamara and Zanetti [3] to circumvent the limitations of statistical noise that plagued lattice gas automata (EGA). LBM is a simplified kinetic (mesoscopic) and discretized approximation of the continuous Boltzmarui equation. LBM is mesoscopic in nature because the particles are not directly related to the number of molecules like in DSMC or MD but representative of a collection of molecules. Hence, the computational cost is less demanding compared with DSMC and MD. Typical LBM consists of the lattice Boltzmann equation (LBE), lattice stmcture, transformation of lattice units to physical units, and boundary conditions. 

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