Simulation techniques periodic boundary conditions

The basis of these very widely used techniques is again conceptually simple. We simulate the system studied by an ensemble of particles (usually representing atoms, but occasionally groups of atoms) which are contained in a simulation box. Periodic boundary conditions (pbcs) are commonly applied, i.e. the basic simulation box is replicated infinitely in three directions (although 2-D pbcs may be used in surface simulations) in simulations of crystals, the simulation box will comprise the unit cell or, more commonly, a supercell. As the simulations are dynamical , the atoms are given velocities that are chosen with... 

The Gibbs Ensemble MC simulation methodology [17-19] enables direct simulations of phase equilibria in fluids. A schematic diagram of the technique is shown in Fig. 10.1. Let us consider a macroscopic system with two phases coexisting at equilibrium. Gibbs ensemble simulations are performed in two separate microscopic regions, each within periodic boundary conditions (denoted by the dashed lines in Fig. 10.1). The thermodynamic requirements for phase coexistence are that each... 

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). 

Once the boundary conditions have been implemented, the calculation of solution molecular dynamics proceeds in essentially the same manner as do vacuum calculations. While the total energy and volume in a microcanonical ensemble calculation remain constant, the temperature and pressure need not remain fixed. A variant of the periodic boundary condition calculation method keeps the system pressure constant by adjusting the box length of the primary box at each step by the amount necessary to keep the pressure calculated from the system second virial at a fixed value (46). Such a procedure may be necessary in simulations of processes which involve large volume changes or fluctuations. Techniques are also available, by coupling the system to a Brownian heat bath, for performing simulations directly in the canonical, or constant T,N, and V, ensemble (2,46). 

The computer simulations employed the molecular dynamics technique, in which particles are moved deterministically by integrating their equations of motion. The system size was 864 Lennard-Jones atoms, of which one was the solute (see Table II for potential parameters). There were no solute-solute interactions. Periodic boundary conditions and the minimum image criterion were used (76). The cutoff radius for binary interactions was 3.5 G (see Table II). Potentials were truncated beyond the cutoff. 

This general technique was employed to simulate the motion of a mono-layer of identical spherical particles subjected to a simple shear. Confinement to a monolayer represents tremendous economies of computer time compared with a three-dimensional simulation. Hopefully, these highly specialized monolayer results will provide comparable insights into the physics of three-dimensional suspensions. Periodic boundary conditions were used in the simulation, and the method of Evans (1979) was incorporated to reproduce the imposed shear. 

In practice, one must compromise the number of molecules in the simulation and/or the number of configurations calculated to conserve computer cycles. Two essential techniques that are utilized are periodic boundary conditions and sampling algorithms, which we discuss separately. 

Recent years have seen the extensive application of computer simulation techniques to the study of condensed phases of matter. The two techniques of major importance are the Monte Carlo method and the method of molecular dynamics. Monte Carlo methods are ways of evaluating the partition function of a many-particle system through sampling the multidimensional integral that defines it, and can be used only for the study of equilibrium quantities such as thermodynamic properties and average local structure. Molecular dynamics methods solve Newton s classical equations of motion for a system of particles placed in a box with periodic boundary conditions, and can be used to study both equilibrium and nonequilibrium properties such as time correlation functions. 

Fig. 13. Illustration of the grand-canonical simulation technique for temperature UbT/e = 1.68 and p = pcoex- A cuboidal system geometry 13.8cr x 13.8cr x 27.6cr is used with periodic boundary conditions in all three directions. The solid line corresponds to the negative logarithm of the probability distribution, P p), in the grand canonical ensemble. The two minima correspond to the coexisting phases and the arrows on the p axis mark their densities. The height of the plateau yields an accurate estimate for the interfacial tension, yLV- The dashed line is a parabohc fit in the vicinity of the liquid phase employed to determine the compressibihty. Representative system configurations are sketched schematically. From [62]...

A computer simulation of the surface of the amorphous Si02 has been reported in Ref. [16]. It was accomplished in two steps. First, the bulk amorphous atomic structure was simulated by the usual MD melt-quench technique described above. Then a free surface was created by removing the periodic boundary condition in one dimension (Z) and freezing the bottom layer of atoms. After that the system was annealed at 1000 K and then cooled gradually to 300 K. 

Our Monte Carlo (MC) simulation uses the Metropolis sampling technique and periodic boundary conditions with image method in a cubic box(21). The NVT ensemble is favored when our interest is in solvent effects as in this paper. A total of 344 molecules are included in the simulation with one solute molecule and 343 solvent molecules. The volume of the cube is determined by the density of the solvent and in all cases used here the temperature is T = 298K. The molecules are rigid in the equilibrium structure and the intermolecular interaction is the Lennard-Jones potential plus the Coulombic term... 

Dynamics simulations employing conventional periodic boundary conditions are usually carried out in the microcanonical (constant N, constant volume, constant energy) ensemble. However, techniques have been introduced that allow NVT (constant N, constant volume, constant temperature), NPH... 

For an understanding of protein-solvent interactions it is necessary to explore the modifications of the dynamics and structure of the surrounding water induced by the presence of the biopolymer. The theoretical methods best suited for this purpose are conventional molecular dynamics with periodic boundary conditions and stochastic boundary molecular dynamics techniques, both of which treat the solvent explicitly (Chapt. IV.B and C). We focus on the results of simulations concerned with the dynamics and structure of water in the vicinity of a protein both on a global level (i.e., averages over all solvation sites) and on a local level (i.e., the solvent dynamics and structure in the neighborhood of specific protein atoms). The methods of analysis are analogous to those commonly employed in the determination of the structure and dynamics of water around small solute molecules.163 In particular, we make use of the conditional protein solute -water radial distribution function,... 

In all simulations of clay mineral systems we apply periodic boundary conditions at constant pressure and temperature (constant NPT), This allows the system volume to change freely at 100 kPa (1 bar) external pressure and 298 K. Furthermore we employ Ewald summation to compute both electrostatic potentials and dispersive van der Waals interactions, and the simulations are fully dynamic, using the Discover module and Insight II graphical user interface of the MSI molecular modeling suite (MSI, 1997). The free energy perturbation technique is not implemented in this software per se so that many of the aforementioned calculations have to be performed with spreadsheet software (e.g., Microsoft Excel). 

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