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Simulation Details

Figure 1: 1 dimensional representation of the potential energy surface. $E_{ij}$ is the energy barrier between the two states $i$ and $j$.
\includegraphics[height=.15\textheight, width=.5\textwidth]{surface.bmp}

Our approach follows the same methodology provided by D. N. Theodorou et al. [6] and K. A. Fichthorn et al. [9]. We take a similar approach in solving for the rate constants $(k_{i\rightarrow j})$ in the Markovian Master equation,

\begin{displaymath} \frac{\partial{p_{i}}}{\partial{t}} = \sum_{j} k_{j\rightarrow i} p_{j} - \sum_{j} k_{i\rightarrow j} p_{i} \end{displaymath} (4)


where the rate constants shown in equation(5) below

\begin{displaymath} k_{i\rightarrow j} = \nu_{i} \ e^{\left[-\frac{E_{i j} - E_{i}}{k_{b} T_{i}}\right]} \end{displaymath} (5)


are obtained via the application of the Arrhenius Law in Transition State Theory (TST). Of most importance is the energy barrier ($E_{ij}$) between the vacancy state (j) and the diffusive atoms state (i). Figure(1) above displays a fictitious potential energy for a diffusing atom in 1 dimension. In a majority of the past simulations that apply (TST) to KMC methods the frequency factor $\nu_{i}$ is assumed to be a constant. In general $\nu_{i}$ is of the form,

\begin{displaymath} \nu_{i} \cong \frac{k_{b} T_{i} Q^{+}}{h Q} \end{displaymath} (6)


where $Q^{+}$ is the partition function of the transition state and $Q$ is the partition function of the reactant state. A major concern in our simulation was whether or not the local rate constants could be considered a function of the local temperature, and if so is this assumption sufficient to capture the actual coupling mechanism between the thermal and concentration gradients.

Figure 2: The simulation lattice above displays the imposed perodic boundrary conditions, vacancy locations, atom species, and the temperature field.
\includegraphics[height=.35\textheight, width=.75\textwidth]{Lattice.bmp}


As shown in figure(2), our 2D simulation lattice is subject to periodic boundary conditions and a linear temperature gradient in the x direction. The transitions or diffusive jumps are limited to only movements between the vacancy site and nearest neighbor atoms. Also, the temperature $(T_{i})$ in the rate constant equation(5) is a function of the local lattice temperture (vacancy position). For example, given that the temperature in the positive x direction is decreasing, $(T_{i})$ in equation(5) would be less for transitions to the right than for transitions to the left.
next up previous
Next: Driven KMC Algorithm Up: Methods Previous: Methods
Shawn A. Putnam
2001-12-17