The simulation method

Abstract Monte Carlo simulations of lattice spin models are a powerful method for the investigation of confined nematic liquid crystals and allow for a study of the molecular organization and thermod3mamics of these systems. Investigations of models of polymer-dispersed liquid cr3rstals are reviewed devoting particular attention to the calculation of deuterium NMR spectra from the simulation data. 

Pasini et al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 1-25. 2005 Kluwer Academic Publishers. Printed in the Netherlands. 

---

Ihe allure of methods for calculating free energies and their associated thermod)mai values such as equilibrium constants has resulted in considerable interest in free ene calculations. A number of decisions must be made about the way that the calculatior performed. One obvious choice concerns the simulation method. In principle, eit Monte Carlo or molecular dynamics can be used in practice, molecular dynamics almost always used for systems where there is a significant degree of conformatio flexibility, whereas Monte Carlo can give very good results for small molecules which either rigid or have limited conformational freedom. 

We close these introductory remarks with a few comments on the methods which are actually used to study these models. They will for the most part be mentioned only very briefly. In the rest of this chapter, we shall focus mainly on computer simulations. Even those will not be explained in detail, for the simple reason that the models are too different and the simulation methods too many. Rather, we refer the reader to the available textbooks on simulation methods, e.g.. Ref. 32-35, and discuss only a few technical aspects here. In the case of atomistically realistic models, simulations are indeed the only possible way to approach these systems. Idealized microscopic models have usually been explored extensively by mean field methods. Even those can become quite involved for complex models, especially for chain models. One particularly popular and successful method to deal with chain molecules has been the self-consistent field theory. In a nutshell, it treats chains as random walks in a position-dependent chemical potential, which depends in turn on the conformational distributions of the chains in... 

A rather crude, but nevertheless efficient and successful, approach is the bond fluctuation model with potentials constructed from atomistic input (Sect. 5). Despite the lattice structure, it has been demonstrated that a rather reasonable description of many static and dynamic properties of dense polymer melts (polyethylene, polycarbonate) can be obtained. If the effective potentials are known, the implementation of the simulation method is rather straightforward, and also the simulation data analysis presents no particular problems. Indeed, a wealth of results has already been obtained, as briefly reviewed in this section. However, even this conceptually rather simple approach of coarse-graining (which historically was also the first to be tried out among the methods described in this article) suffers from severe bottlenecks - the construction of the effective potential is neither unique nor easy, and still suffers from the important defect that it lacks an intermolecular part, thus allowing only simulations at a given constant density. 

To perform simulations of relatively large systems for relatively long times, it is essential to optimize the computational strategy of discrete particle simulations. Obviously, the larger the time step 5t, the more efficient the simulation method. For the soft-sphere model, the maximum value for 5t is dictated by the duration of a contact. Since there are two different spring-dashpot systems in our current model, it is essential to assume that tcontact>n — tcontacUU so that... 

In this type of apparatus, the two phases do not come to equilibrium, at any point in the contactor and the simulation method is based, therefore, not on a number of equilibrium stages, but rather on a consideration of the relative rates of transport of material through the contactor by flow and the rate of interfacial mass transfer between the phases. For this, a consideration of mass transfer rate theory becomes necessary. 

An important advantage of the simulation methods consists in their explicit treatment of solvent effects as the effects of a set of individual... 

The simulation method proceeds as follows a number of particles is placed randomly on the sites of the cylinder, according to the initial concentration, avoiding double occupancy. The diffusion process is simulated by selecting a particle at random and moving it to a randomly selected nearest-neighbor site. 

There remains one problem, that of the values of Fi needed for the approximations. Their determination depends on the simulation method used, but at this point, it can be said that the major term, DC/dT, always present, can be approximated simply as... 

Fig. 9 Schematic drawing illustrating the simulation method based on the competition technique...

Fig. 20 MWD of emulsion-polymerized polyethylene. The experimental conditions are discussed in detail in [307], and the simulation method is described in [310]...

To understand the relationships between the simulation methods and the desired thermodynamic quantities, a short review of the major concepts of statistical mechanics may be in order. This is not meant to be comprehensive, but rather to remind the reader of the relevant ideas. 

The simulation method used by Ziff et al. is a fixed time step variant of RSM. Their simulations show that the system has three steady states. For values of the control parameter Y < Y = 0.389 0.005, the lattice is completely covered by B particles and for F > Y2 = 0.525 0.001, the surface is completely covered by A. For Yi < F < the system is in the reactive steady state [43]. At F = Fi, the adsorbate coverages change continuously, which means that a second-order phase transition occurs at that point. At F = F2, a first order phase transition occurs, as is clear from the stepwise change in adsorbate coverages at that point. 

Computation was carried out using a leap-frog algorithm proposed by Fincham with a time step of 1 fs. The total run was 20 ns for the CH4 system and 14 ns for the Xe system. Temperature and pressure were kept constant at 298 K and 100 MPa, respectively, using a method proposed by Berendsen et al Details of the simulation method is given in Ref. 4. 

The simulation method developed in IMPROVE (cf. Sect. 5.2) is based on correlations between selected influencing factors which have been determined by literature survey and expert discussions in a theoretical rather than empirical way. However, these are just a small excerpt of the success factors relevant for actual design processes. If additional influencing factors are integrated in the simulation for their practical relevance, the models must be validated again. 

In many cases, simulation methods are used in a complementary manner to experimental studies, with the validity of the calculations assessed by comparing simulated properties (e.g., crystal structure and activation energies) with those determined experimentally. The major factor in determining the reliability of all the simulation methods is the accuracy of the description of the interaction between the ions. The majority of studies of ionically conducting systems have utilized parameterized potentials containing explicit expressions for the various interactions (short-range repulsion. Coulomb, etc.), although recent advances in available computer power have enabled the application of ab initio methods (see Chapter 7). 

Numerical simulations of styrene free-radical polymerization in micro-flow systems have been reported. The simulations were carried out for three model devices, namely, an interdigital multilamination micromixer, a Superfocus interdigital micromixer, and a simple T-junction. The simulation method used allows the simultaneous solving of partial differential equations resulting from the hydrodynamics, and thermal and mass transfer (convection, diffusion and chemical reaction). 

---