Monte Carlo simulation lattice models

Key words Micelle - lattice model -Monte Carlo simulation - entropy... 

Van de Walle, A. and Asta, M. (2002) Self-driven lattice-model Monte Carlo simulations of alloy thermodynamic properties and phase diagrams. Model Simul. Mater. ScL Eng., 10 (5), 521-538. 

Figure 5.11 Order-disorder transition (ODT) of a symmetric diblock copolymer studied by a soft, coarse-grained, off-lattice model. Monte Carlo simulations are performed in the npT ensemble and the pressure is kept constant at pb /kgT = 18 (with b = Reo/VW-l)-Xo = 1.5625. The invariant degree of polymerization, and the chain discretization are indicated in the key. The figure presents the excess thermodynamic potential, per...

Mapping Atomistically Detailed Models of Flexible Polymer Chains in Melts to Coarse-Grained Lattice Descriptions Monte Carlo Simulation of the Bond Fluctuation Model... 

Abstract. The interlamellar domain of semicrystalline polyethylene is studied by means of off-lattice Metropolis Monte Carlo simulations using a realistic united atom force field with inclusion of torsional contributions. Both structural as well as thermal and mechanical properties are discussed for systems with the 201 crystal plane parallel to the interface. In so doing, important data is obtained which is useful for modeling semicrystalline polyethylene in terms of multiphase models. Here, we review the main results published previously by us (P.J. in t Veld, M. Hiitter, G.C. Rutledge Macromolecules 39, 439 (2006) M. Hiitter, P.J. in t Veld, G.C. Rutledge Polymer (in press), (2006)]. 

Harris J and Rice S A 1988 A lattice model of a supported monolayer of amphiphile molecules—Monte Carlo simulations J. Ohem. Phys. 88 1298-306... 

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

Kremer K and Binder K 1988 Monte Carlo simulations of lattice models for macromolecules Comp. Phys. Rep. 7 259... 

A Kolinski, J Skolmck. Monte Carlo simulations of protein folding. I. Lattice model and interaction scheme. Pi-otem 18 338-352, 1994. 

The effects due to the finite size of crystallites (in both lateral directions) and the resulting effects due to boundary fields have been studied by Patrykiejew [57], with help of Monte Carlo simulation. A solid surface has been modeled as a collection of finite, two-dimensional, homogeneous regions and each region has been assumed to be a square lattice of the size Lx L (measured in lattice constants). Patches of different size contribute to the total surface with different weights described by a certain size distribution function C L). Following the basic assumption of the patchwise model of surface heterogeneity [6], the patches have been assumed to be independent one of another. 

In this review we put less emphasis on the physics and chemistry of surface processes, for which we refer the reader to recent reviews of adsorption-desorption kinetics which are contained in two books [2,3] with chapters by the present authors where further references to earher work can be found. These articles also discuss relevant experimental techniques employed in the study of surface kinetics and appropriate methods of data analysis. Here we give details of how to set up models under basically two different kinetic conditions, namely (/) when the adsorbate remains in quasi-equihbrium during the relevant processes, in which case nonequilibrium thermodynamics provides the needed framework, and (n) when surface nonequilibrium effects become important and nonequilibrium statistical mechanics becomes the appropriate vehicle. For both approaches we will restrict ourselves to systems for which appropriate lattice gas models can be set up. Further associated theoretical reviews are by Lombardo and Bell [4] with emphasis on Monte Carlo simulations, by Brivio and Grimley [5] on dynamics, and by Persson [6] on the lattice gas model. 

The bond fluctuation model (BFM) [51] has proved to be a very efficient computational method for Monte Carlo simulations of linear polymers during the last decade. This is a coarse-grained model of polymer chains, in which an effective monomer consists of an elementary cube whose eight sites on a hypothetical cubic lattice are blocked for further occupation (see... 

K. Kremer, K. Binder. Monte Carlo simulation of lattice models for macromolecules. Comp Phys Rep 7 259-310, 1988. 

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. 

Monte Carlo simulation shows [8] that at a given instance the interface is rough on a molecular scale (see Fig. 2) this agrees well with results from molecular-dynamics studies performed with more realistic models [2,3]. When the particle densities are averaged parallel to the interface, i.e., over n and m, and over time, one obtains one-dimensional particle profiles/](/) and/2(l) = 1 — /](/) for the two solvents Si and S2, which are conveniently normalized to unity for a lattice that is completely filled with one species. Figure 3 shows two examples for such profiles. Note that the two solvents are to some extent soluble in each other, so that there is always a finite concentration of solvent 1 in the phase... 

In general, percolation is one of the principal tools to analyze disordered media. It has been used extensively to study, for example, random electrical networks, diffusion in disordered media, or phase transitions. Percolation models usually require approximate solution methods such as Monte Carlo simulations, series expansions, and phenomenological renormalization [16]. While some exact results are known (for the Bethe lattice, for instance), they are very rare because of the complexity of the problem. Monte Carlo simulations are very versatile but lack the accuracy of the other methods. The above solution methods were employed in determining the critical exponents given in the following section. 

Dimensional Lattice Model for the Glass Transition of Polymer Melts. A Monte Carlo Simulation. 

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