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The periodic boundary conditions

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The correct description of the finite size effects is a crucial point in numerical simulation methods. Thus it allows to evaluate macroscopic quantities using small size systems. 

Indeed even considering the calculation power of the actual computers it is not possible to run molecular dynamics simulations on systems with a number of atoms close to the Avogadro number ( $ \simeq$ 1023).

The importance of the surface effects can be easily illustrated using a 1 dm3 edged cube of water (1 L) at room temperature. This cube contains approximately  3.3×1025water molecules, each of them can be considered as a sphere having a diameter of 2.8 Å. Following this scheme surface interactions can affect up to 10 layers of spheres (water molecules) far from the surface of the cube. In such conditions the number of water molecules in interaction with the surface is about 2×1019, which represents a very small proportion of 1/15×106 molecules influenced by the surface of the cube.

In a molecular dynamics simulation with 1 thousand of waler molecules in the sames conditions almost all the molecules (if not all of them) will be influenced by finite size effects of the simulation box.
To overcome these size effects involved by the small size of the simulated samples  it is usual applied the Periodic Boundary Conditions "PBC" which means to repeat infinitely the simulation box in the 3 directions of space.

Image pbc-seb
Figure 1: Schematic representation of the idea of periodic boundary conditions.

Figure [Fig. 1] illustrates the principle of the periodic boundary conditions: a particle which goes out from the simulation box by one side is reintroduced in the box by the opposite side (in the 3 dimension of space).
The maximum inter-atomic distance rcut which can be taken into account in the calculations is therefore equal to the half of the edge of the simulation box:

rcut = L/2 (1)

The surface effects would therefore be small in the calculation, furthermore the bigger the simulation box and the number of atoms will be, the smaller the surface effects will be.
Last Updated on Friday, 27 February 2009 18:35  

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