Periodic boundary conditions effects

If the simulated system uses periodic boundary conditions, the logical long-range interaction includes a lattice sum over all particles with all their images. Apart from some obvious and resolvable corrections for self-energy and for image interaction between excluded pairs, the question has been raised if one really wishes to enhance the effect of the artificial boundary conditions by including lattice sums. The effect of the periodic conditions should at least be evaluated by simulation with different box sizes or by continuum corrections, if applicable (see below). 

Fig. 1. Periodic boundary conditions protect the inner simulation cell from disturbing effects of having all its particles close to the surface. With PBCs in force, as a particle moves out of the box on one side, one of its images will move back into the box on the opposite side.

It is sometimes desirable to include the effect of the rest of the system, outside of the QM and MM regions. One way to do this is using periodic boundary conditions, as is done in liquid-state simulations. Some researchers have defined a potential that is intended to reproduce the effect of the bulk solvent. This solvent potential may be defined just for this type of calculation, or it may be a continuum solvation model as described in the next chapter. For solids, a set of point charges, called a Madelung potential, is often used. 

The tests in the two previous paragraphs are often used because they are easy to perform. They are, however, limited due to their neglect of intermolecular interactions. Testing the effect of intennolecular interactions requires much more intensive simulations. These would be simulations of the bulk materials, which include many polymer strands and often periodic boundary conditions. Such a bulk system can then be simulated with molecular dynamics, Monte Carlo, or simulated annealing methods to examine the tendency to form crystalline phases. 

HyperChem uses the TIP3P water model for solvation.You can place the solute in a box of TIP3P water molecules and impose periodic boundary conditions. You may then turn off the boundary conditions for specific geometry optimization or molecular dynamics calculations. However, this produces undesirable edge effects at the solvent-vacuum interface. 

With the Monte Carlo method, the sample is taken to be a cubic lattice consisting of 70 x 70 x 70 sites with intersite distance of 0.6 nm. By applying a periodic boundary condition, an effective sample size up to 8000 sites (equivalent to 4.8-p.m long) can be generated in the field direction (37,39). Carrier transport is simulated by a random walk in the test system under the action of a bias field. The simulation results successfully explain many of the experimental findings, notably the field and temperature dependence of hole mobilities (37,39). 

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. 

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... 

A realistic model of a solution requires at least several hundred solvent molecules. To prevent the outer solvent molecules from boiling off into space, and minimizing surface effects, periodic boundary conditions are normally employed. The solvent molecules are placed in a suitable box, often (but not necessarily) having a cubic geometry (it has been shown that simulation results using any of the five types of space filling polyhedra are equivalent ). This box is then duplicated in all directions, i.e. the central box is suiTounded by 26 identical cubes, which again is surrounded by 98 boxes etc. If a... 

The orientational relationships between the martensite and austenite lattice which we observe are partially in accordance with experimental results In experiments a Nishiyama-Wasserman relationship is found for those systems which we have simulated. We think that the additional rotation of the (lll)f< c planes in the simulations is an effect of boundary conditions. Experimentally bcc and fee structure coexist and the plane of contact, the habit plane, is undistorted. In our simulations we have no coexistence of these structures. But the periodic boundary conditions play a similar role like the habit plane in the real crystals. Under these considerations the fact that we find the same invariant direction as it is observed experimentally shows, that our calculations simulate the same transition process as it takes place in experiments. The same is true for the inhomogeneous shear system which we see in our simulations. 

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]. 

Fig. 5. 12. Snapshot pictures of configurations of the effective bonds for PE melts at T = 2000 K (top) and at T = 210 K (bottom). Effective monomers are not shown, and effective bonds are shown as cylinders of different brightness. The edges of the cubic simulation box are also highlighted. Note that periodic boundary conditions are applied throughout. From [184]... 

Both a uniform bulk fluid and an inhomogeneous fluid were simulated. The latter was in the form of a slit pore, terminated in the -direction by uniform Lennard-Jones walls. The distance between the walls for a given number of atoms was chosen so that the uniform density in the center of the cell was equal to the nominal bulk density. The effective width of the slit pore used to calculate the volume of the subsystem was taken as the region where the density was nonzero. For the bulk fluid in all directions, and for the slit pore in the lateral directions, periodic boundary conditions and the minimum image convention were used. 

Figure 7 also shows results for the thermal conductivity obtained for the slit pore, where the simulation cell was terminated by uniform Lennard-Jones walls. The results are consistent with those obtained for a bulk system using periodic boundary conditions. This indicates that the density inhomogeneity induced by the walls has little effect on the thermal conductivity. 

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