Simple cubic periodic boundary conditions

The hexagonal prism is the space-filling object closest to a cylinder and thus a natural choice for an approximate implementation of the cell model. Still, it can be implemented in simple cubic periodic boundary conditions. 

FIGURE 16-3 The grid of wave numbers allowed by periodic boundary conditions, as in Fig. 15-2. Also shown are several lattice wave numbers (neavier dots). The relative spacings and geometry correspond to a simple cubic lattice of 6859 ions (see Problem 16-1). A sample wave number k is shown. 

N = 1056 simple cubic lattice subject to periodic boundary conditions ... 

Fig. 7a. Plot of light scattering intensity, ocS(q, t) vs wavenumbers q for different times after the quench from To = 75 °C to T — 49 °C, for the same mixture as in Fig. 6a). From Bates and Wilzius [36], b Structure factor S(q, t) vs wavenumber q for different times after a quench from infinite temperature (i.e. the Initial state is an ather-mal blend) to kBT/t = 1.0, for N = 32, <[>, = 0.6, eAB = e, eAA = cBB = 0, and averages over 40 quenches in a simple cubic lattice of size 40 x 40 x 40 (with periodic boundary conditions) are taken. Arrow shows the prediction for qm of the linearized theory of spinodal decomposition. Time t after the quench is measured in attempted moves per monomer. From Sariban and Binder [155]...

Most of the above simulations are performed on three-dimensional simple cubic lattices with periodic boundary conditions in all directions. (Some of the early studies were based on two-dimensional square lattices but have since been updated.) Additionally, all of the works discussed in this section (except where noted otherwise) use the standard Metropolis Monte Carlo algorithm discussed in detail in Sec. III. B, but the major difference lies in the selection of which of the components contribute to the total energy of the system. Other differences include the lattice rearrangement methodology and parameters such as surfactant structure, temperature, composition, lattice size, and dimensionality. The specifics of each model are summarized below. 

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.

Choosing a simple cubic lattice of linear dimensions L with periodic boundary conditions, obviously only discrete values of q are accessible, namely... 

We simulate a simple cubic lattice with size 15 x 15 x 20 and periodic boundary conditions in all directions. Because the electric potential is different at the z = 0 and z = 20 electrodes we cannot take the periodic boundary conditions literally. We have to imagine that the system is infinite in the z-direction and consists of an infinite repetition of identical configurations. 

As we will consider above all the highly filled polymers, we have chosen to use numerical models which are best suited for this case. One has considered periodic distribution of particles (radius a) in the matrix with a simple cubic arrangement and a distance 2b between particles. Figure 3 shows the elementary 3D cell with at the boundary, prescribed temperatures ti, zi) in two parallel planes along the main heat flux direction (z) and adiabatic conditions on the 4 other sides [7],... 

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