Lees-Edwards boundary conditions

As an example of such applications, consider the dynamics of a hexible polymer under a shear how [88]. A shear how may be imposed by using Lees-Edwards boundary conditions to produce a steady shear how y = u/Ly, where Ly is the length of the system along v and u is the magnitude of the velocities of the boundary planes along the x-direction. An important parameter in these studies is the Weissenberg number, Wi = Ti y, the product of the longest... 

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. 

The Lagrangian L for Lees-Edwards boundary conditions combined with Parrinello-Rahman fluctuations for the cell geometry now reads as follows ... 

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 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. 

Figure 1 Lees-Edwards boundary conditions for planar Couette flow. On the left, the sliding brick form of the boundary conditions are shown, while the deforming cube version is shown on the right. In both cases, the displacement of the centers of the image cells above and below the central simulation is yLt where y is the strain rate, L is the length of the side of the cubic simulation cell and t is time...

Figure 3.10 Illustration of the implementation of simple shear in Lees-Edwards boundary conditions for a packing of bidisperse disks. The particles in the main cell are given the affine deformation in Equation 3.9. The image cells above and below in the central cell in the shear gradient direction are also shifted by LAy in the shear flow direction.

A simple shear strain can he applied to each particle in increments of Ay = 10 (coupled with Lees-Edwards boundary conditions) followed by potential energy minimization from total strain yj j = 0 to yjot = 10 . The shear stress is... 

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 earliest efforts to develop non-equilibrium molecular dynamics (NEMD) methods used special boundary conditions and/or external fields to induce non-equilibrium behavior in the system. Important contributions to this development include those of Lees and Edwards [72], Gosling et al. [73], Hoover and Ashurst [74] and Ciccotti and Jacucci... 

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... 

Multiple replica simulations can be extended to driven systems (e.g., S5 tems with time-dependent boundary conditions, such as Lees-Edwards or Lagrangian-rhomboid boundary conditions). Each of P processors simulates a replica at a driving rate that is P times faster than the desired... 

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. 

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