Monte Carlo transport simulations

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. 

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. 

Molecular Simulations Molecular simulations are useful for predicting properties of bulk fluids and solids. Molecular dynamics (MD) simulations solve Newton s equations of motion for a small number (on the order of 10 ) of molecules to obtain the time evolution of the system. MD methods can be used for equilibrium and transport properties. Monte Carlo (MC) simulations use a model for the potential energy between molecules to simulate configurations of the molecules in proportion to their probability of occurrence. Statistical averages of MC configurations are useful for equilibrium properties, particularly for saturated densities, vapor pressures, etc. Property estimations using molecular simulation techniques are not illustrated in the remainder of this section as commercial software implementations are not generally available at this time. 

Systematic transport studies of Li ions in oxide and chalcogenide glasses of the general formula, xLi2X (l-x)SiX2 (X = O, S or Se) by Pradel and Ribes (1994), revealed that conductivities in oxide glasses are significantly low. The d.c. conductivities were found to vary as yfx" and this behaviour found support in Monte Carlo computer simulation. Since... 

In this chapter I will review random walk Monte Carlo, MC, simulation models of electron transport in DSSC. In Sect. 2,1 will place these studies in the context of DSSC transport models. MC methods and results are covered in Sect. 3 The concluding section. Sect. 4, looks at future applications of these methods and the related MC models for polymer blend cells that are covered in [7]. 

A standard Grand Canonical Monte Carlo (GCMC) simulation is employed in the equilibrium study, while MD simulation is more preferable to study the nonequilibrium transport properties [104]. 

In this paper we report on the first results of a calculation by Monte Carlo computer-simulation techniques of the mass transport coefficient in a lattice gas model. We calculate in a rectangular geometry the mass flux, Q(T), and the mass density profile, p(x, r), for an open system which is subject to an externally imposed chemical potential gradient, x is a spatial coordinate. From these quantities, the mass transport coefficient, D(T), can be derived using the equivalent of Ohm s law. 

Ensemble Monte Carlo (MC) simulations have been the popular tools to investigate the steady-state and transient electron transport in semiconductors theoretically. In particular, the steady-state velocity field characteristics have been determined using the Monte Carlo method for electric field strengths up to 350kVcm in bulk wurtzite structure ZnO at lattice temperatures of 300, 450, and 600 K [158]. The conduction bands ofwurtzite-phase ZnO structure were calculated using FP-LMTO-LDA method. For the MC transport simulations, the lowest F valley (Fi symmetry) and the satellite valleys located at F (F3 symmetry) and at U point, f/rmn (Efi symmetry), which is located two-thirds of the way between the M and L symmetry points on the edge of the Brillouin zone, have been considered. 

By running Monte Carlo (MC) simulations of carriers within such a landscape and by assuming that the probability of a carrier hopping from one site to the other follows the MA expression this model provides us with semi-empirical expressions for the full-temperature and field-dependence of the carrier mobility. Improvements and developments upon this model in common use include the correlated disorder model (CDM) of Novikov and co-workers [28]. Goto and co-workers have also carried out MC simulations using a modified Gay-Berne potential specifically for application to anomalous charge transport in nematic liquid crystals [29]. 

The alternative simulation approaches are based on molecular dynamics calculations. This is conceptually simpler that the Monte Carlo method the equations of motion are solved for a system of A molecules, and periodic boundary conditions are again imposed. This method pennits both the equilibrium and transport properties of the system to be evaluated, essentially by numerically solvmg the equations of motion... 

Barzykin A V, Barzykina N S and Fox M A 1992 Electronic excitation transport and trapping in micellar systems— Monte-Carlo simulations and density expansion approximation Chem. Rhys. 163 1-12... 

One application of the grand canonical Monte Carlo simulation method is in the study ol adsorption and transport of fluids through porous solids. Mixtures of gases or liquids ca separated by the selective adsorption of one component in an appropriate porous mate The efficacy of the separation depends to a large extent upon the ability of the materit adsorb one component in the mixture much more strongly than the other component, separation may be performed over a range of temperatures and so it is useful to be to predict the adsorption isotherms of the mixtures. 

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). 

In the next section we describe the basic models that have been used in simulations so far and summarize the Monte Carlo and molecular dynamics techniques that are used. Some principal results from the scaling analysis of EP are given in Sec. 3, and in Sec. 4 we focus on simulational results concerning various aspects of static properties the MWD of EP, the conformational properties of the chain molecules, and their behavior in constrained geometries. The fifth section concentrates on the specific properties of relaxation towards equilibrium in GM and LP as well as on the first numerical simulations of transport properties in such systems. The final section then concludes with summary and outlook on open problems. 

Whenever the polymer crystal assumes a loosely packed hexagonal structure at high pressure, the ECC structure is found to be realized. Hikosaka [165] then proposed the sliding diffusion of a polymer chain as dominant transport process. Molecular dynamics simulations will be helpful for the understanding of this shding diffusion. Folding phenomena of chains are also studied intensively by Monte Carlo methods and generalizations [166,167]. 

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