Periodic boundary conditions in two dimensions

FIGURE 39.1 Periodic boundary conditions in two dimensions. The molecules that appear to be around the center box are actually copies of the center box. 

Computer simulations of bulk liquids are usually performed by employing periodic boundary conditions in all three directions of space, in order to eliminate artificial surface effects due to the small number of molecules. Most simulations of interfaces employ parallel planar interfaces. In such simulations, periodic boundary conditions in three dimensions can still be used. The two phases of interest occupy different parts of the simulation cell and two equivalent interfaces are formed. The simulation cell consists of an infinite stack of alternating phases. Care needs to be taken that the two phases are thick enough to allow the neglect of interaction between an interface and its images. An alternative is to use periodic boundary conditions in two dimensions only. The first approach allows the use of readily available programs for three-dimensional lattice sums if, for typical systems, the distance between equivalent interfaces is at least equal to three to five times the width of the cell parallel to the interfaces. The second approach prevents possible interactions between interfaces and their periodic images. 

If our goal is to study a surface, our ideal model would be a slice of material that is infinite in two dimensions, but finite along the surface normal. In order to accomplish this, it may seem natural to take advantage of periodic boundary conditions in two dimensions, but not the third. There are codes in which this technique is implemented, but it is more common to study a surface using a code that applies periodic boundary conditions in all three dimensions, and it is this approach we will discuss. The basic idea is illustrated in Fig. 4.1, where the supercell contains atoms along only a fraction of the vertical... 

Fig. 1. Periodic boundary conditions in two dimensions, for N = 4 particles. The central box is marked by a heavy boundary. In calculating the energy of the dark particle by the Ewald method, the other partides in the central box are induded, but also all the other images of all the particles (including itself) in the array of boxes. In the MI method the potential function is truncated at the dotted redangle, so that the nearest image of each of the N— 1 other particles is induded. In the cutoff method the truncation is at the dotted dr-cle, and fewer interactions are induded.

Figure 4. (a) Periodic boundary conditions (PBC). Heavy solid lines indicate location of PBC that surround Box I, which is the central cell where atoms are located. Assume the third dimension, Z, is similarly drawn, (b) Periodic boundary conditions (see text), (c) Periodic boundary conditions in two dimensions (assuming the PBC in the rd dimension is unchanged). 

FIGURE 5.5 Periodic boundary conditions in two dimensions. Twelve molecules neighbor list boundary shown for molecule A. The molecule entering the box just above A is leaving at the bottom. 

Preliminary results have successfully shown the initial repulsion and de-stabilisation of CO adsorbed on the (3u (100) surface. Skin depth effects are limited to a two-layer copper slab, with periodic boundary conditions in two dimensions. The CO molecule transfers electronic charge to the surface and subsequently presents a partial positive charge on the carbon atom, near the copper surface. This has then be shown to provide a site for nucleophilic attack by the hydride ion (H-). This adsorbed reaction has been compared to its gas-phase counterpart and shown to have a lower activation barrier. All these systems require a DMC fixed node calculation to obtain sufficiently low QMC variance energies to argue reliably the case for a surface catalyst effect. Other nucleophiles are now being considered. 

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. 

Boundary Conditions although CA are a.ssumed to live on infinitely large lattices, computer simulations must necessarily be run on finite sets. For a one dimensional lattice with N cells, it is common to use periodic boundary conditions, in which ctn + i is identified with ai. Alternatively, all cells to the left and right of a finite block of N cells may be arbitrarily defined to possess value 0 for all time, so that their dynamics remains uncoupled with that taking place within the block. Similarly, in two dimensions, it is usual to have the dynamics take place on a torus, in which o m+i = <7, 2 and = cTi,j- As we will see later it turns one... 

Lastly, we would like to mention here results of the two kinds of large-scale computer simulations of diffusion-controlled bimolecular reactions [33, 48], In the former paper [48] reactions were simulated using random walks on a d-dimensional (1 to 4) hypercubic lattice with the imposed periodic boundary conditions. In the particular case of the A + B - 0 reaction, D = Dq and nA(0) = nB(0), the critical exponents 0.26 0.01 0.50 0.02 and 0.89 0.02 were obtained for d = 1 to 3 respectively. The theoretical value of a = 0.75 expected for d = 3 was not achieved due to cluster size effects. The result for d = 4, a = 1.02 0.02, confirms that this is a marginal dimension. However, in the case of the A + B — B reaction with DB = 0, the asymptotic longtime behaviour, equation (2.1.106), was not achieved at all - even at very long reaction times of 105 Monte Carlo steps, which were sufficient for all other kinds of bimolecular reactions simulated. It was concluded that in practice this theoretically derived asymptotics is hardly accessible. 

The basic cell employed in a canonical ensemble MC simulations has been chosen in form shown in Fig. 4. The cell has the shape of a rectangular box of dimensions xxlyx z = 30x10x30 (in the units of macroion diameter, D) and contains of both bulk and confined regions. There are two (left and right) bulk regions which are the parts of the same reservoir connected by means of periodic boundary conditions in x direction, i.e., particles that left the cell from the left side enter on the right side and vice versa. 

However, not all systems require periodic boundary conditions in all spatial directions.For membranes, for example, only two dimensions are periodic, while the third one is finite. In that case, the Ewald method is computationally highly inefficient and wouid not aiiow to treat more than a few hundred charged particles. We present two alternative approaches, the MMM2D and ELC methods, which allow for computational efficiency similar to the bulk case. It is also simple to adapt the MMM2D method for systems with only one periodic dimension. 

Fig. 7.18 Profiles of the order parameter p z) - psiz) for the symmetric (/= 1/2) block copolymer model with N = 16,= 0.2, and Lx Lx D simple cubic lattices with periodic boundary conditions in the x and y directions, while at the two repulsive walls of size Lx L a repulsive energy of strength eAs = ab/2 acts on A-monomers, with units such that As/ka = 1- Linear dimension L in parallel directions is always L = 16, while the linear dimension in z-direction is D = 10 (a), 14 (b), and 18 (c). Normalized inverse temperatures are indicated by different symbols l/T = 0.3 (circles), 0.4 (diamonds), 0.5 (triangles), and 0.6 (squares). The critical temperature in the bulk is estimated as l/Tc 0.52 0.05. (From Kikuchi and Binder.

The simulations described above were performed at constant density, i.e., a volume was imposed on the system irrespective of the resulting pressure or chemical potential. MD simulations performed at constant chemical potential, where the confined liquid is in equilibrium with a vapor or bulk liquid phase, have also been performed. Simulations with free surfaces, i.e., with vapor/polymer interfaces, allow for the study of the equilibrium liquid-vapor interface structure and the calculation of the surface tension, a thermodynamic property fundamental to the understanding of the behavior of a material at interfaces. An MD study of the equilibrium liquid-vapor interface structure and surface tension of thin films of n-decane and n-eicosane (C20H42) has been performed in Ref. 26. The system studied consisted of a box with periodic boundary conditions in all directions. The liquid polymer, however, while fully occupying the x and y dimensions, occupied only a fraction of the system in the z direction, resulting in two liquid-vapor interfaces. The liquid phase ranged from about 4.0 to 7.0 nm in thickness. Simulations were performed at 400 K for both decane and eicosane, with additional decane simulations at 300 K. A similar system of tridecane molecules, using a well calibrated EA force field, has been studied at 400 K and 300 K in Ref 32. 

Fig. 1.Two-dimensional lattice gas model on a square lattice with open boundaries A and B in one dimension and periodic boundary conditions in the other dimension. Chemical potentials, /i, are applied at the two ends. Black dots denote particles on the lattice. 

Here, ry is the separation between the molecules resolved along the helix axis and is the angle between an appropriate molecular axis in the two chiral molecules. For this system the C axis closest to the symmetry axes of the constituent Gay-Berne molecules is used. In the chiral nematic phase G2(r ) is periodic with a periodicity equal to half the pitch of the helix. For this system, like that with a point chiral centre, the pitch of the helix is approximately twice the dimensions of the simulation box. This clearly shows the influence of the periodic boundary conditions on the structure of the phase formed [74]. As we would expect simulations using the atropisomer with the opposite helicity simply reverses the sense of the helix. 

The heptane water and toluene water interfaces were simulated by the use of the DREIDING force field on the software of Cerius2 Dynamics and Minimizer modules (MSI, San Diego) [6]. The two-phase systems were constructed from 62 heptane molecules and 500 water molecules or 100 toluene molecules and 500 water molecules in a quadratic prism cell. Each bulk phase was optimized for 500 ps at 300 K under NET ensemble in advance. The periodic boundary conditions were applied along all three directions. The calculations of the two-phase system were run under NVT ensemble. The dimensions of the cells in the final calculations were 23.5 A x 22.6 Ax 52.4 A for the heptane-water system and 24.5 A x 24.3 A x 55.2 A for the toluene-water system. The timestep was 1 fs in all cases and the simulation almost reached equilibrium after 50 ps. The density vs. distance profile showed a clear interface with a thickness of ca. 10 A in both systems. The result in the heptane-water system is shown in Fig. 3. Interfacial adsorption of an extractant can be simulated by a similar procedure after the introduction of the extractant molecule at the position from where the dynamics will be started. 

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