Lattice Monte Carlo dynamics

Hoffmann, D., Knapp, E.W. Polypeptide folding with off-lattice Monte Carlo dynamics the method. Eur. Biophys. J. 1996, 24, 387-403. 

A. Rey and J. Skolnick, Comparison of lattice Monte Carlo Dynamics and Brownian... 

A. Rey and J. Skolnick, Comparison of lattice Monte Carlo Dynamics and Brownian Dynamics Folding Pathways of -Helical Hairpins, Chem. Phys. 158,199-219 (1991). 

Milik, M., and Skolnick, J. 1993. Insertion of peptide chains into lipid membranes an off-lattice Monte Carlo dynamics model. Proteins. 15 10. 

A. Milchev, K. Binder. Polymer solutions confined in slit-hke pores with attractive walls An off-lattice Monte Carlo study of static properties and chain dynamics. J Computer-Aided Mater Des 2 167-181, 1995. 

I. Gerroff, A. Milchev, W. Paul, K. Binder. A new off-lattice Monte Carlo model for polymers A comparison of static and dynamic properties with the bond fluctuation model and application to random media. J Chem Phys 95 6526-6539, 1993. 

Dynamics of Crystal Growth hi the preceding section we illustrated the use of a lattice Monte Carlo method related to the study of equilibrium properties. The KMC and DMC method discussed above was applied to the study of dynamic electrochemical nucleation and growth phenomena, where two types of processes were considered adsorption of an adatom on the surface and its diffusion in different environments. 

Lattice Monte Carlo Model for Polymers A Comparison of Static and Dynamic Properties with the Bond-Fluctuation Model and Application to Random Media. 

Gerroff, A. Milchev, K. Binder, and W. Paul, /. Chem. Phys., 98, 6526 (1993). A New Off-Lattice Monte Carlo Model for Polymers A Comparison of Static and Dynamic Properties with the Bond-Fluctuation Model and Application to Random Media. 

Graf, P., Nitzan, A., Kurnikova, M.G., Coalson, R.D. A dynamic lattice Monte Carlo model of ion transport in inhomogeneous dielectric environments Method and implementation. J. Phys. Chem. B 2000,104,12324-38. 

This chapter will focus on a simpler version of such a spatially coarse-grained model applied to micellization in binary (surfactant-solvent) systems and to phase behavior in three-component solutions containing an oil phase. The use of simulations for studying solubilization and phase separation in surfactant-oil-water systems is relatively recent, and only limited results are available in the literature. We consider a few major studies from among those available. Although the bulk of this chapter focuses on lattice Monte Carlo (MC) simulations, we begin with some observations based on molecular dynamics (MD) simulations of micellization. In the case of MC simulations, studies of both micellization and microemulsion phase behavior are presented. (Readers unfamiliar with details of Monte Carlo and molecular dynamics methods may consult standard references such as Refs. 5-8 for background.)... 

These results show that simplified molecular dynamics simulations can qualitatively account for micellization quite well. However, the computation time necessary for even such simple models is too great to allow the model to be useful for the calculation of other micellar properties or phase behavior or for an in-depth study of solubilization. Stochastic dynamics simulations, in which the solvent effects are accounted for through a mean-field stochastic term in the equations of motion, can also be used to study surfactant self-assembly [22], but the most efficient approach to date is still the one based on lattice Monte Carlo simulations, which are discussed next. 

Finally, Mattice and coworkers have used lattice Monte Carlo simulations for various studies of micellization of block copolymers in a solvent, including micellization of triblock copolymers [43], steric stabilization of polymer colloids by diblock copolymers [44], and the dynamics of chain interchange between micelles [45]. Their studies of the self-assembly of diblock copolymers [46-48] are roughly equivalent to those of surfactant micellization, as a surfactant can in essence be considered a short-chain diblock copolymer and vice versa. In fact, Wijmans and Linse [49,50] have also studied nonionic surfactant micelles using the same model that Mattice and coworkers used for a diblock copolymer. Thus, it is interesting to compare whether the micellization properties and theories of long-chain diblock copolymers also hold true for surfactants. 

In this chapter we have reviewed several studies of micellar and microemulsion systems based on molecular dynamics and lattice Monte Carlo simulations. The major observations based on these studies can be summarized as follows. 

A Dynamic Lattice Monte Carlo Model of Ion Transport in Inhomogeneous Dielectric Environments Method and Implementation. 

Comparison of Dynamic Lattice Monte Carlo Simulations and the Dielectric Self-Energy Poisson-Nernst-Planck Continuum Theory for Model Ion Channels. 

Graf, P, Kurnikova, M. G., Coalson, R. D., and Nitzan, A. 2004. Comparison of dynamic lattice Monte Carlo simulations and the dielectric self-energy Poisson-Nernst-Planck continuum theory for model ion channels, 108(6), 2006-2015. 

The corresponding rate in the lattice model, with its Monte Carlo dynamics, is the acceptance rate for the moves. Here we also drop the arbitrary time constant of the Monte Carlo process and define a mean barrier as... 

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. 

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. 

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). 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

Another special case of weak heterogeneity is found in the systems with stepped surfaces [97,142-145], shown schematically in Fig. 3. Assuming that each terrace has the lattice structure of the exposed crystal plane, the potential field experienced by the adsorbate atom changes periodically across the terrace but exhibits nonuniformities close to the terrace edges [146,147]. Thus, we have here another example of geometrically induced energetical heterogeneity. Adsorption on stepped surfaces has been studied experimentally [95,97,148] as well as with the help of both Monte Carlo [92-94,98,99,149-152] and molecular dynamics [153,154] computer simulation methods. 

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. 

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