Monte Carlo lattice

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

A theoretical description of all the factors controlling the effect of processing conditions on polymer structure and properties is extremely complex because of the need to consider a wide variety of different factors such as polymer concentration, molecular weight, as well as external variables (time, temperature, pressin-e). In view of this complexity, this work has focused on the development of kinetic Monte-Carlo lattice models. These models are mesoscopic in the sense that the imit lattice length is of the order of the statistical segment length for the polymer chain and atomic level details are omitted. The objective of this review is to describe these models in some detail and show their value in getting a better... 

Spin-polarized mobile hydrogen and deuterium atoms were detected in silica glass containing 1200 ppm of OH or OD groups by the time-resolved pulsed EPR technique.The EPR spectra of the H/D atoms indicate the occurrence of CIDEP in reactions of H/D atoms with radiolytically induced metastable spin centres. The CIDEP effects in silica glass exhibit unusual polarization patterns, temperature dependence and dependence on hfs constants. A Monte Carlo lattice model is introduced to simulate the RPM spin polarization in amorphous silica. Although the model does not explain all the features observed, it indicates where peculiarities of CIDEP in disordered solids may originate. 

J. Skolnick and A. Kolinski, Monte Carlo lattice dynamics and the prediction of protein folds, in Computer simulations of biomolecular systems. Theoretical and experimental studies., W.F. van Gunsteren, P.K. Weiner, and A.J. Wilkinson, Editors. 1996, ESCOM Science Publ. 

Monte Carlo Lattice Model for Chain Diffusion in Dense Polymer Systems and its Interlocking with Molecular Dynamics Simulations. 

We regard these dynamic Monte Carlo lattice chain simulations as computer "experiments" which complement analytical calculations. A lattice chain model has been widely used in both analytical and numerical calculations to describe equilibrivim properties of polymers (l6). Numerical calculations have been paurticularly helpful in treating equilibrium properties which are difficult to handle analytically (3J, 12 ). Similarly, the Monte Carlo... 

Salsburg Z W, Jacobson J D, Fickett W and Wood W W 1959 Application of the Monte Carlo method to the lattice gas model. Two dimensional triangular lattice J. Chem. Phys. 30 65-72... 

Chesnut D A and Salsburg Z W 1963 Monte Carlo procedure for statistical mechanical calculation in a grand canonical ensemble of lattice systems J. Chem. Phys. 38 2861-75... 

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... 

Fabbri U and Zannoni C 1986 A Monte Carlo investigation of the Lebwohl-Lasher lattice model in the vicinity of its orientational phase transition Mol. Phys. 58 763-88... 

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... 

Bruce A D, Wilding N B and Ackland G J 1997 Free energies of crystalline solids a lattice-switch Monte-Carlo method Phys. Rev. Lett. 79 3002-5... 

Allan N L, G D Barrera, J A Purton, C E Sims and M B Taylor 2000. Ionic Solids at High Temperatures and Pressures Ah initio, Lattice Dynamics and Monte Carlo Studies. Physical Chemistry Chemical Physics 2 1099-1111. 

With the Monte Carlo method, the sample is taken to be a cubic lattice consisting of 70 x 70 x 70 sites with intersite distance of 0.6 nm. By applying a periodic boundary condition, an effective sample size up to 8000 sites (equivalent to 4.8-p.m long) can be generated in the field direction (37,39). Carrier transport is simulated by a random walk in the test system under the action of a bias field. The simulation results successfully explain many of the experimental findings, notably the field and temperature dependence of hole mobilities (37,39). 

Various equations of state have been developed to treat association ia supercritical fluids. Two of the most often used are the statistical association fluid theory (SAET) (60,61) and the lattice fluid hydrogen bonding model (LEHB) (62). These models iaclude parameters that describe the enthalpy and entropy of association. The most detailed description of association ia supercritical water has been obtained usiag molecular dynamics and Monte Carlo computer simulations (63), but this requires much larger amounts of computer time (64—66). 

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

El Shakhnovich, G Earztdmov, AM Gutm, M Karplus. Pi otem folding bottlenecks A lattice Monte Carlo simulation. Phys Rev Lett 67 1665-1668, 1991. 

Figure 12.5. (a) Lattice model showing a polymer chain of 200 beads , originally in a random configuration, after 10,000 Monte Carlo steps. The full model has 90% of lattice sites occupied by chains and 10% vacant, (b) Half of a lattice model eontaining two similar chain populations placed in contact. The left-hand side population is shown after 50,0000 Monte Carlo steps the short lines show the loeation of the original polymer interface (courtesy K. Anderson). 

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