Continuum approximation boundary conditions

If the Brownian particles were macroscopic in size, the solvent could be treated as a viscous continuum, and the particles would couple to the continuum solvent through appropriate boundary conditions. Then the two-particle friction may be calculated by solving the Navier-Stokes equations in the presence of the two fixed particles. The simplest approximation for hydrodynamic interactions is through the Oseen tensor [54],... 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

In the previous chapter, we have seen how Born s simple and successful idea of a dielectric continuum approximation for the description of solvation effects has been developed to a considerable degree of perfection. Almost all workers in this area have been trying to obtain more efficient and more precise methods for the solution of dielectric boundary conditions combined with molecular electrostatics, but the question of the validity of Born s basic assumption has rarely been discussed. This will be done in the following sections, with a surprising result. 

The rationale of using hybrid simulation here is that a classic diffusion-adsorption type of model, Eq. (2), can efficiently handle large distances between steps by a finite difference coarse discretization in space. As often happens in hybrid simulations, an explicit, forward discretization in time was employed. On the other hand, KMC can properly handle thermal fluctuations at the steps, i.e., provide suitable boundary conditions to the continuum model. Initial simulations were done in (1 + 1) dimensions [a pseudo-2D KMC and a ID version of Eq. (2)] and subsequently extended to (2 + 1) dimensions [a pseudo-3D KMC and a 2D version of Eq. (2)] (Schulze, 2004 Schulze et al., 2003). Again, the term pseudo is used as above to imply the SOS approximation. Speedup up to a factor of 5 was reported in comparison with KMC (Schulze, 2004), which while important, is not as dramatic, at least for the conditions studied. As pointed out by Schulze, one would expect improved speedup, as the separation between steps increases while the KMC region remains relatively fixed in size. At the same time, implementation is definitely complex because it involves swapping a microscopic KMC cell with continuum model cells as the steps move on the surface of a growing film. 

The failure diameter is deduced from Fig. 8, and is approximately equal to 100 A. All the results presented so far the existence of a failure diameter, the curvature of the front and the variation of the detonation velocity with the cylinder radius, are consistent with continuum theory. Nevertheless, the necessary condition to get a steady detonation wave is the existence of a sonic point that isolates the reaction zone from the rarefaction waves coming from the back. This is investigated in the case of a planar detonation wave (using periodic boundary conditions in 3D). 

The continuum model, in which solvent is regarded as a dielectric continuum, has been used for a long time to study solvent effects [2]. Solvation energies can be primarily approximated by a reaction field owed to polarization of the dielectric continuum as solvent, and other contributions such as dispersion interactions, which must be explicitly considered for non-polar solvent systems, have usually been treated with an empirical quantity such as the macroscopic surface tension of the solvent. An obvious advantage of the method is its handiness, whilst its disadvantage is an artifact introduced at the boundary between the solute and solvent. Agreement between experiment and theory is considerably governed by the boundary conditions. 

To overcome problems arising fi-om the finite system size used in MC or MD simulation, boundary conditions are imposed using periodic-stochastic approximations or continuum models. In particular, in stochastic boundary conditions the finite system is not duplicated but a boundary force is applied to interact with atoms of the system. This force is set as to reproduce the solvent regions that have been neglected. Anyway, in general any of the methods used to impose boundary conditions in MC or MD can be used in the QM/MM approach. 

---