Avoiding Surfaces Periodic Boundary Conditions

The most severe problem concerns surface effects. If the small number of particles is supposed to be confined to an isolated finite box, it is clear that all of them will be close to the surface of the box, and they will not be in an environment appropriate to particles within a macroscopic sample. In an attempt to overcome this problem, Metropolis etal introduced the so-called periodic boundary conditions. Under those conditions the N particles of the 

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Since these microscopic simulations typically can only treat short times and small samples, it is important to avoid surface effects. It is common to employ periodic boundary conditions. A special trick often used for these kinds of simulation is, instead of simulating a melt of many chains, to simulate one very long chain which falls back again and again into the box. In this way, the effect of the chain ends, which introduces artificially high free volume can be reduced. However, one should keep in mind that this chain interacts with its own periodic images. It is known that this may... 

The general setup of an MD run comprehends the system size (dimensions of the simulation box), the temperature and the pressure which affect the molecule(s) and the treatment of the system boundaries. In almost all cases it is recommended to simulate with periodic boundary conditions in order to avoid unwanted surface effects due to the artificial box walls. Periodic boundary conditions are implemented in such a way that the box is completely encircled by identical copies of itself [140, 141]. 

Therefore, whenever we introduce symmetries into our systems, we risk observing behavior that is inconsistent with that observed when these symmetries are absent. Because opposing surfaces are almost always incommensurate unless they are prepared specifically, it will be important to avoid symmetries in simulations as much as possible. Unfortunately, it can be difficult to make two surfaces incommensurate in simulations, particularly when the interface is composed of two identical crystalline surfaces. These difficulties arise from the fact that only a limited number of geometries conform to the periodic boundary conditions in the lateral direction. Each geometry needs to be analyzed separately... 

In practice the equation of motion is solved first without considering the constraint force and in the next step the constraint forces are obtained by correcting the positions such that the molecule conserves its minimum structure, i.e., such that the constraints are fiilfilled. For small molecules direct inversion is possible, for large molecules iterative procedures are applied (60). This means that each constraint is corrected after the other until a certain convergence is reached. This algorithm is called Shake (65). Another important aspect of simulations concerns periodic boundary conditions. A virtual replication of the central box at each of its planes is carried out in order to avoid surface effects. A detailed description can be found in the excellent textbooks of Allen and Tildesley (60) and Frenkel and Smit (61). 

Thanks to their speed and relatively low computational cost, M D and MC simulations can be used for studying the physical properties of large systems. This is extremely useful in heterogeneous catalysis, e.g., for modeling the structure and the properties of the bulk and the surface of a solid catalyst, or the properties of the bulk and interface of liquid/liquid biphasic systems. However, since the number of particles modeled is still very small compared to real materials, the models are susceptible to wall effects. One neat trick for avoiding this problem is to apply periodic boundary conditions The volume containing the model is treated as the primitive cell of an... 

Equation 1.3 represents a system of usually several thousand coupled differential equations of second order. It can be solved only numerically in small time steps At via finite-difference methods [16]. There always the situation at t + At is calculated from the situation at t. Considering the very fast oscillations of covalent bonds, At must not be longer than about 1 fs to avoid numerical breakdown connected with problems with energy conservation. This condition imposes a limit of the typical maximum simulation time that for the above-mentioned system sizes is of the order of several ns. The limited possible size of atomistic polymer packing models (cf. above) together with this simulation time limitation also set certain limits for the structures and processes that can be reasonably simulated. Furthermore, the limited model size demands the application of periodic boundary conditions to avoid extreme surface effects. 

Low-energy boron bombardment of silicon has been simulated at room temperature by MD. Tersoff potential T3 was used in the simulation and smoothly linked up with the universal potential. The boron-silicon interaction was simulated according to Tersoff potential for SiC but modified to account for the B-Si interaction. Silicon crystal (Si-c) in the (001) direction, with (2x1) surface reconstruction, was bombarded with boron at 200 and 500 eV. Reasonably good statistics are obtained with 1000 impact points uniformly distributed over a representative surface area. The simulation size was 16x 16x 14 unit cells. Periodic boundary conditions were applied laterally. The temperature was kept at 300 K with a thermal bath applied to the more external cells in the crystal except the top surface. In these conditions the crystal was relaxed during 19 ps. In order to avoid direct channeling, the incidence was inclined 7° out of the normal, as usual in experiments, with random azimuthal direction. 

This gas model has periodic boundary conditions. This is a common trick to avoid the surface problem. We consider a box having a property such that if something goes out through one wall, it enters through the opposite wall (cf. p. 524). 

The only serious problem lies in the choice of boundary conditions. The use of the usual periodic boundary conditions (for samples of realistic size) would correspond to a not very dilute lattice of ions, and the mean forces in this situation are not of immediate interest. What is wanted instead is the effective force between only two ions immersed in a solvent. There have been two independent attempts to choose boundary conditions appropriate to that situation. Both retained basic periodicity of the boundary conditions in order to avoid drastic surface effects, but they suppressed certain interaction terms to make the ionic environment more appropriate. 

In order to simulate more closely the behavior of an infinite system, periodic boundary conditions are imposed on the solution to the equations of motion. If a molecule labeled i is located at position (xj, yt, z,) at time t, we imagine that there are 26 additional images of i located at (x, L, 0 yj L, 0 L, 0). The particle and its 26 images have the same orientation and velocity. Another molecule j may interact with any i within its interaction range. If molecule i should cross a face of the box, it is reinserted at the opposite face. Constant density is thus maintained. These periodic boundary conditions avoid the strong surface effects that would result from a box with reflecting walls. 

Here hence denotes the position of monomer with label i (i= 1,..., N) in the feth chain molecule (fe = 1,..., N ). For simplicity, we have specialized here to a monodisp>erse system of linear homopolymers hut the generalization to polydisperse systems or to heteropolymers or to branched architecture is straightforward, as well as to multicomponent systems (including solvent molecule coordinates, for instance). Typically, the volume in which the S3 tem is considered is a cubic LxLxL box (in d = 3 dimensions, or a square LxL box in d = 2 dimensions), and one chooses periodic boundary conditions to avoid surface effects but if the latter are of interest, the corresponding change of boundary conditions is straightforward. All of what has been said so far applies to lattice models as well as to models in the continuum. 

Unlike the bulk morphology, block copolymer thin films are often characterized by thickness-dependent highly oriented domains, as a result of surface and interfacial energy minimization [115,116]. For example, in the simplest composition-symmetric (ID lamellae) coil-coil thin films, the overall trend when t>Lo is for the lamellae to be oriented parallel to the plane of the film [115]. Under symmetric boundary conditions, frustration cannot be avoided if t is not commensurate with L0 in a confined film and the lamellar period deviates from the bulk value by compressing the chain conformation [117]. Under asymmetric boundary conditions, an incomplete top layer composed of islands and holes of height Lo forms as in the incommensurate case [118]. However, it has also been observed that microdomains can reorient such that they are perpendicular to the surface [ 119], or they can take mixed orientations to relieve the constraint [66]. 

Block copolymers comprise chemically distinct polymer chains bonded together to form a single macromolecule. The repulsion between chemically dissimilar blocks enables the formation of periodic microdomains and, as a result of the covalent bond between blocks, macroscopic phase segregation is avoided. The presence and type of equilibrium microstructure (e.g., lamella, cylinders, and spheres) depends on temperature, composition, the extent of repulsion between blocks, surface boundary conditions, and imposed fields. 

Boundary conditions where only one dimension is periodic and the other two are open or finite appear in physical situations such as electronic structures of supported structures on metal surfaces, e.g., steps and atomic chains in one dimension, or in systems which can model a charged stiff polymer or DNA piece where, for avoiding end effects, the rod is made infinitely long [64-66]. It was only very recently that a ID Ewald method (EWID) for these systems was developed [67,68], although the Lekner method [11] de-... 

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