SKMF

Short description of the model

[for further details see: Erdélyi Z, Pasichnyy M, Bezpalchuk V, Tomán J, B. Gajdics, Gusak A, Stochastic Kinetic Mean Field Model, Computer Physics Communications 204: pp. 31-37. (2016) (available online: http://dx.doi.org/10.1016/j.cpc.2016.03.003)]

Rate of change of concentration in each site of a three-dimensional grid is defined according to conservation of matter and the corresponding local flux balance at each site:

\begin{equation}
\frac{dc_i}{dt}=-\sum_{j=1}^Z\left[c_i(1-c_j)\left(\Gamma_{i,j}^{mean-field} + \delta\Gamma_{i,j}^{Lang}\right) \left. \\
\right. - c_j(1-c_i)\left(\Gamma_{j,i}^{mean-field} + \delta\Gamma_{j,i}^{Lang}\right)\right]. \label{eq:master}
\end{equation}

where is the atomic fraction of A atoms at site , is the atomic fraction on a neighbouring site , and the total number of nearest neighbours is . is in fact the probability that the site is occupied by an A atom and a neighbouring site by a B atom; i.e. an A-B exchange is possible. is the probability of such an exchange per unit time in mean-field approximation, i.e. the jump rate of A atoms from site to a neighbouring site and backward jumps of B atoms ( is for an exchange of an A and a B atoms being on site and , respectively):

\begin{equation}
\Gamma_{i,j}^{mean-field}= \Gamma_{0} \exp\left(-\frac{\hat{E}_{i,j}}{kT}\right) \label{eq:Gamma_mean_field}
\end{equation}

with

\begin{equation}
\hat{E}_{i,j} = \left(M-V\right) \sum_{l=1}^Z c_l + \left(M+V\right) \sum_{n=1}^Z c_n \label{eq:E_ij}
\end{equation}

where ( A, B) are the pair interaction energies, , and ( is the attempt frequency and is the saddle point energy).

Note that is the regular solid solution parameter – proportional to the heat of mixing – and determines the strength of the composition dependence of the tracer diffusion coefficient (diffusion asymmetry). Last but not least, in eq. \ref{eq:master} are the noise terms, which are random additions to the mean-field exchange rates:

\begin{equation}
\delta\Gamma_{i,j}^{Lang} = \frac{A_n}{\sqrt{dt}}\sqrt{3}\left(2 random - 1\right) \label{eq:Gamma_Lang}
\end{equation}

where is a uniform random number between and . It is easy to check that the random expression has the mean squared value equal to . Here is the noise of the amplitude and is the time time step.

Actually, eqs. (\ref{eq:master}-\ref{eq:Gamma_Lang}) have to be used to calculate the time evolution of the composition at each site of a 3D lattice. (see open source code: Downloads) With , we perform a fully mean-field calculation, whereas with increasing the calculation becomes more and more stochastic, that is the dispersion of composition becomes higher.