Lees-Edwards periodic boundary conditions

The first method we will discuss was proposed by Lees and Edwards,51 and is outlined qualitatively in Figure 13. In this technique, periodic boundary conditions are employed in all three spatial directions however, although the... 

The treatment of applying periodic boundary conditions discussed here is markedly different from that traditionally employed in simulations of planar Couette flow. The PBC method that is commonly used is called the Lees-Edwards boundary condition. In its simplified form applied to cubic boxes, it represents a translation of the image boxes in the y direction, at a rate equal to y. Further details on this method can be found elsewhere. In contrast to the method involving the dynamical evolution of h presented here, the Lees-Edwards method is much harder to develop and implement for noncubic simulation cells. Also, in simulations involving charged particles, the Coulom-bic interaction is handled in both real and recipro l spaces. The reciprocal space vectors k of the simulation cell represented by h can be written " " as follows ... 

In the method of nonequilibrium molecular dynamics (NEMD), transport processes are usually driven by boundary conditions. For example, the calculation of shear viscosity is based on the Lees-Edwards flow-adapted sliding brick periodic boundary conditions (PBCs) (Panel 4 or their equivalent Lagrangian-rhomboid... 

As in equilibrium molecular dynamics, the equations of motion have to be solved for a system with periodic boundaries. For shear, the boundaries are modified to become the Lees-Edwards sliding brick conditions (Lees Edwards 1972), in which periodic images of the simulation cell above and below the unit cell are moved in opposite directions at a velocity determined by the imposed shear rate (see Fig. 9.9). The properties of the system follow firom the appropriate time averages, <. . >, usually (but not necessarily) after the system has reached the steady state. Given, for example, a system at a number density, n = N/V, under an applied shear rate, the kinetic temperature is constrained with an appropriate thermostat Different properties can then be evaluated, for example, the internal energy. 

An alternative to Lees-Edwards boundary conditions is the formalism put forth by Parrinello and Rahman for the simulation of solids under constant stress.52,53 They described the positions of particles by reduced, dimensionless coordinates ra, where the ra can take the value 0 < ra < 1 in the central image. Periodic images of a given particle are generated by adding or subtracting integers from the individual components of r. 

In order for this approach to be effective, it is necessary to ensure that the compressive and/or shear strains to which the simulation cell is subjected are transmitted to the atoms within the cell. Indeed, it is possible in principle to deform the cell in arbitrary ways without altering the positions of the atoms at all and stiU obtain a perfectly suitable periodically repeated system. To ensure the atoms in the cell move in conjunction with the lattice vectors, it is common practice to represent the atomic positions in fractional coordinates. This approach also ensures that the Lees-Edwards boundary conditions [124] are satisfied to ensure that artificial slip planes are not introduced at the interface between each periodically repeated cell. 

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