Monte Carlo MC Simulation Method

Polymerization rate represents the instantaneous status of reaction locus, but the whole history of polymerization is engraved within the molecular weight distribution (MWD). Recently, a new simulation tool that uses the Monte Carlo (MC) method to estimate the whole reaction history, for both hnear [263-265] and nonlinear polymerization [266-273], has been proposed. So far, this technique has been applied to investigate the kinetic behavior after the nucleation period, where the overall picture of the kinetics is well imderstood. However, the versatility of the MC method could be used to solve the complex problems of nucleation kinetics. 

The MC method is a powerful technique for investigating complicated phenomena that are difficult to solve by the conventional differential equation approach. In the MC approach, all one needs are the individual probabilities of various kinetic events. It is easy to understand the advantages of applying the MC method to emulsion polymerization if we note that it is possible to simulate the formation processes of all polymer molecules in each polymer particle directly because the volume of the reaction locus is very small. One 

In this section we discuss unique MWDs formed via Hnear emulsion polymerization, while the kinetics of branched and crossUnked polymer formation are considered in Sect. 4.2. 

In MC simulation, any kinetic event can be accounted for, as long as the probability of each kinetic event is represented exphcitly. Chain length dependent kinetics can be accounted for in a straightforward manner if the functional form is provided. In conventional MC simulations of molecular build-up processes, the monomeric units are added to each growing polymer molecule one-by-one therefore, a multitude of random numbers and calculations are required to simulate the formation of each polymer molecule. To get around this problem, a new concept, the competition techniqueyV/as proposed in order to drastically reduce the amount of calculation required for the simulation [263,264]. 

In this technique, the imaginary time (or equivalently, the imaginary chain length, given by P=kp[M]pt) for a certain event to occur is calculated by using the appropriate probability distribution for each type of event. If the given process 

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Monte Carlo search methods are stochastic techniques based on the use of random numbers and probability statistics to sample conformational space. The name Monte Carlo was originally coined by Metropolis and Ulam [4] during the Manhattan Project of World War II because of the similarity of this simulation technique to games of chance. Today a variety of Monte Carlo (MC) simulation methods are routinely used in diverse fields such as atmospheric studies, nuclear physics, traffic flow, and, of course, biochemistry and biophysics. In this section we focus on the application of the Monte Carlo method for... 

Other methods which are applied to conformational analysis and to generating multiple conformations and which can be regarded as random or stochastic techniques, since they explore the conformational space in a non-deterministic fashion, arc genetic algorithms (GA) [137, 1381 simulation methods, such as molecular dynamics (MD) and Monte Carlo (MC) simulations 1139], as well as simulated annealing [140], All of those approaches and their application to generate ensembles of conformations arc discussed in Chapter II, Section 7.2 in the Handbook. 

In Chapter 2, a brief discussion of statistical mechanics was presented. Statistical mechanics provides, in theory, a means for determining physical properties that are associated with not one molecule at one geometry, but rather, a macroscopic sample of the bulk liquid, solid, and so on. This is the net result of the properties of many molecules in many conformations, energy states, and the like. In practice, the difficult part of this process is not the statistical mechanics, but obtaining all the information about possible energy levels, conformations, and so on. Molecular dynamics (MD) and Monte Carlo (MC) simulations are two methods for obtaining this information... 

For reasons of space and because of their prime importance, we focus here on free energy calculations based on detailed molecular dynamics (MD) or Monte Carlo (MC) simulations. However, several other computational approaches exist to calculate free energies, including continuum dielectric models and integral equation methods [4,14]. 

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. 

The Monte Carlo (MC) simulation is performed using standard procedures [33] for the Metropolis sampling technique in the isothermal-isobaiic ensemble, where the number of molecules N, the pressure P and the temperature T are fixed. As usual, we used the periodic boundary conditions and image method in a cubic box of size L. In our simulation, we use one F embedded in 1000 molecules of water in normal conditions (T—29S K and P= 1 atm). The F and the water molecules interact by the Lennard-Jones plus Coulomb potential with three parameters for each interacting site i (e, o, - and qi). 

The need to reliably describe liquid systems for practical purposes as condensed matter with high mobility at a given finite temperature initiated attempts, therefore, to make use of statistical mechanical procedures in combination with molecular models taking into account structure and reactivity of all species present in a liquid and a solution, respectively. The two approaches to such a description, namely Monte Carlo (MC) simulations and molecular dynamics (MD), are still the basis for all common theoretical methods to deal with liquid systems. While MC simulations can provide mainly structural and thermodynamical data, MD simulations give also access to time-dependent processes, such as reaction dynamics and vibrational spectra, thus supplying — connected with a higher computational effort — much more insight into the properties of liquids and solutions. 

There are four contributions based on Monte Carlo (MC) simulations of different systems. These contributions include lucid discussion of the fundamentals of MC methods used in electronic structure calculations by Lester, the MC simulation, and... 

DFT and Monte Carlo (MC) simulations are modern methods for extracting information on PSDs. Both methods are used for obtaining the local isotherms as functions of pore width and geometry. Knowing the set of local isotherms, the PSDs are obtained by an inverse process solving the GAI equation (Equation 4.28). 

A great deal of efforts were made to understand the systematics of this new method. Fig. 5 and Table 1 give some such examples. The detail of the analysis can be found in Refs. [29,30]. The resulting DS fusion time spectrum and its comparison with Monte Carlo (MC) simulations are shown in Fig. 6, which clearly establishes the resonance structure. From the time-of-fiight analysis of 2036 116 DS fusion events, a formation rate consistent with 0.7.3 (0.16)meas (0.09)rrto( e times the theoretical prediction of Faifman et al. [9] was obtained (the first error... 

Solvation of cyclen 36 was studied using Monte Carlo (MC) simulations <1996JPC17655, 1997JCF3045>. Potentials for cyclen were calculated by an ab initio method. It was found that the water hydration sphere is composed of three layers two water molecules are strongly bound in close vicinity, six molecules form the inner hydration sphere, and... 

Comparison is attempted between the results of density distribution calculated by the DF method and those from Monte Carlo(MC) simulation. The present. MG simulations have been... 

We treat, in this chapter, mainly solid composed of water molecules such as ices and clathrate hydrates, and show recent significant contribution of simulation studies to our understanding of thermodynamic stability of those crystals in conjunction with structural morphology. Simulation technique adopted here is not limited to molecular dynamics (MD) and Monte Carlo (MC) simulations[l] but does include other method such as lattice dynamics. Electronic state as well as nucleus motion can be solved by the density functional theory[2]. Here we focus, however, our attention on the ambient condition where electronic state and character of the chemical bonds of individual molecules remain intact. Thus, we restrict ourselves to the usual simulation with intermolecular interactions given a priori. 

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