Spatial Upscaling of Distributed Lattice KMC Simulation

While CG-KMC can reach large scales at reasonable computational cost, it can lead to substantial errors at boundaries and interfaces where large gradients exist, and the local mean field assumption is not as accurate. Recent 

an example of CG-KMC from pattern formation on surfaces is presented. Another application to relatively thick membranes was given in Snyder et al. (2004). In the example considered here, atoms adsorb from a fluid reservoir on a flat surface. Subsequently, they may desorb back to the fluid, diffuse on the surface, or be annihilated by a first-order surface reaction, as shown in Fig. 11a. Attractive interactions between atoms trigger a phase transition from a dilute phase (a low coverage) to a dense phase (a high coverage) (Vlachos et al., 1991), analogous to van der Waals loops of fluid vapor coexistence. Surface reactions limit the extent of phase separation the competition between microphase separation and reaction leads to nanoscopic patterns by self-organization under certain conditions (Hildebrand et al., 1998). 

A major challenge in simulating such problems is that nucleation occurs at the nanometer scale whereas self-organization entails competition between numerous pattern blocks for reagents over microns to millimeters. These problems do not exhibit an obvious separation of length scales. From a different point of view, the stochasticity is built within the PDE as a source or sink term (if one were able to write such a PDE). Furthermore, surface diffusion is faster than the other microscopic processes by many orders of magnitude, but PE cannot be applied since the actual value of diffusion dictates the presence or absence of patterns. 

(a) Schematic of microscopic processes for fluid-surface interacting systems, (b) Spa-tiotemporal evolution of ID concentration patterns (coarse graining of two sites into each coarse cell is used). Bifurcation splittings and mergings occur as time evolves. The fast diffusion necessary for pattern formation (five to six orders of magnitude faster than the rest of the processes) renders microscopic KMC unsuitable even for small domains. 

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