Flowchart of null-event CGMC algorithm. The cell coverage and cell CG transition rate will only be used if CG cell homogenization is applied.
(a) Schematic of top (T) and bridge (B) sites of a (100) surface and (b) layout of CG cells over the microscopic lattice in example A. CG cells are uniform squares with a side length of four lattice constants. Periodic boundary conditions are used. (c) Coverage evolution against simulated time.
(a) Schematic of the CG mesh. Each CG cell encompasses microscopic sites. Periodic boundary conditions are used. (b) Coverage of vacancies at increasing times (, , and ). CGMC is compared against traditional KMC. The slight difference is attributed to the well-mixed assumption along the -axis.
Schematic of 1D diffusion through window and intercage sites. Traditional KMC will store and simulate all potential events shown in (a). The CGMC method with assumes neighboring and sites are in equilibrium with each other (b). The two diffusion rates between neighbors are lumped into one coarse process via homogenization (c).
(a) Steady state loading profiles in a 1D zeolite membrane with Dirichlet boundary conditions. CGMC using a stochastic LMF closure with varying levels coarse graining compares well with traditional KMC. Steady state solution based on the deterministic LMF closure ( only is shown for clarity; not shown for clarity). (b) Zeolite loading vs time. (c) CPU relative to traditional KMC of simulating the same amount of time at steady state with varying levels of spatial coarse graining on two lattice sizes.
Examples of MF coverages within CG cells using the deterministic and stochastic LMF approximation. The error in the deterministic LMF solution relative to the accurate stochastic LMF solution increases as the CG cell size, , decreases: (a) , (b) .
Summary of transition probability rates for CGMC (without homogenization).
Summary of processes and transition probability rates for example A.
Summary of processes and transition probability rates for example B.
Summary of transition probability rates for CGMC (with homogenization).
Summary of processes and transition probability rates for Example C.
Transition probability rate constants for example C.
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