Simulation of Systems

Simulating the dynamical properties of single molecules is all very interesting, but molecular dynamics (MD) was originally developed to study systems of particles 

Studies in Molecular Dynamics 1 General Method B. J. Alder and T. E. Wainwright The Journal of Chemical Physics 31 (1959) 459-466 

A method is outlined by which it is possible to calculate exactly the behavior of several hundred interacting classical particles. The study of this many-body problem is carried out by an electronic computer which solves numerically the simultaneous equations of motion. The limitations of this numerical scheme are enumerated and the important steps in making the program efficient on the computer are indicated. The applicability of this method to the solution of many problems in both equilibrium and nonequilibrium statistical mechanics is discussed. 

Alder and Wainwright gave MD treatments of particles whose pair potential was very simple, typically the square well potential and the hard sphere potential. Rahman (1964) simulated liquid argon in 1964, and the subject has shown exponential growth since then. The 1970s saw a transition from atomic systems 

A number of intermolecular potentials have been developed over the years that treat molecules as collections of point charges. The intermolecular electrostatic potential is taken as a sum of the mutual electrostatic interaction of these point charges, summed over interacting pairs of molecules. Occasionally, extra van der Waals terms are added to the potential. 

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A way of looking at the points raised in the previous section is to compare energy distributions in two systems whose free energies we wish to relate. In particular, consider measuring, in a simulation of system 0, the fiinction Pq(AE), i.e., the probability density per unit AE of configurations for which and differ by the... 

As has probably become obvious already, the study of micelles has been one of the big topics in simulations of systems with surfactants. We have cited many of the related publications in the previous sections. Here, we shall discuss some special aspects of micelle simulations in order to illustrate the use of idealized chain models for this type of problem. 

The function / incorporates the screening effect of the surfactant, and is the surfactant density. The exponent x can be derived from the observation that the total interface area at late times should be proportional to p. In two dimensions, this implies R t) oc 1/ps and hence x = /n. The scaling form (20) was found to describe consistently data from Langevin simulations of systems with conserved order parameter (with n = 1/3) [217], systems which evolve according to hydrodynamic equations (with n = 1/2) [218], and also data from molecular dynamics of a microscopic off-lattice model (with n= 1/2) [155]. The data collapse has not been quite as good in Langevin simulations which include thermal noise [218]. 

Couette Flow Simulation. MD typically simulate systems at thermodynamic equilibrium. For the simulation of systems undergoing flow various methods of nonequilibrium MD have been developed (Ifl iZ.). In all of these methods the viscosity Is calculated directly from the constitutive equation. 

As shown in this chapter, in the simulation of systems described by partial differential equations, the differential terms involving variations with respect to length are replaeed by their finite-differenced equivalents. These finite-differenced forms of the model equations are shown to evolve as a natural eonsequence of the balance equations, according to the manner of Franks (1967). The approximation of the gradients involved may be improved, if necessary, by using higher order approximations. Forward and end sections can... 

Consideration of the various methods points to certain issues that could arise in MEHMC simulations of systems other than those studied here. One source for concern is that the efficiency of sampling clearly depends on the choice of averaging period. A similar criticism has been put forward with regard to the essential... 

An important aspect of the methods described in the preceding section is that Lx and Ly can be time dependent. As we will show in this section, this flexibility allows the simulation cell to fluctuate independently along different spatial dimensions during the simulation. This capability is useful in simulations of systems such as self-assembled monolayers under shear. However, care must be taken when allowing for this additional flexibility because, for some systems, e.g., simple fluids under shear, there is no particular reason why Lx and Ly should be chosen to be independent of one another. In this... 

First-order instabilities may not only involve the translational motion of atoms confined within contacts, but they may also involve chemical reactions within the confined fluid itself. This has been demonstrated recently in first-principles studies of zinc phosphates, which are found in protective films formed in automobile engines.19,83 Here, we focus on simulations of systems containing phosphate molecules in which pressure-induced chemical reactions lead to hysteresis and energy dissipation. The reactions involving zinc phosphates are discussed below along with other tribochemical reactions. 

There are problems with using both of these methods in the simulation of inhomogeneous systems. Because the periodicity of the system is lost in the direction normal to the interface (unless one uses image charges with the flat wall model, which effectively results in a 3D periodic system implementation of the ES method is not straightforward for certain type of systems. Hautman and Klein have presented a modified Ewald sum method for the simulation of systems that are periodic in two... 

As discussed in Section 6.5.3, coarse-grained molecular modeling approaches offer the most viable route to the molecular modeling of hydrated ionomer membranes. The coarse-grained treatment implies simplification in interactions, which can be systematically improved with advanced forcematching procedures, but allows simulations of systems with sufficient size and sufficient statistical sampling. Structural correlations, thermodynamic properties, and transport parameters can be studied. 

The effect of the surface of the box on the solute is of major importance in the simulation of systems such as the one described here. The sudden cut-off of long-range nonbonded potentials at the box surface (beyond which is vacuum) would have an unnatural effect on the dynamics of the simulation. Only an extremely large system size could ensure a small influence of this surface effect on the solute. The computational cost of such a large system would be prohibitive. For this reason, periodic boundary conditions are used. The image of the simulation box is translated repeatedly to form an infinite lattice. When a particle in the simulation box moves, the image in all other translated boxes moves correspond-... 

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