2.6 Discrete / atomistic models

In case of continuum models, it is easy to consider the diffusion asymmetry. We have just to suppose the diffusion coefficients to be composition dependent: $D=D(c)$ , e.g. $D=D_0\exp(mc)$ . However, in case of atomistic models, the composition dependence of the jump frequencies or jump probabilities needs more considerations.

2.6.1 One dimensional kinetic mean filed model (KMF)

Our model to calculate the time evolution of the composition on a one-dimensional lattice is based on Martin's equations.[3] However, we use our own composition dependent activation barriers (diffusion asymmetry) in the exchange frequencies, which unify the advantages of other barriers used in the literature as was shown in Ref. [4]. The time derivatives of atomic fractions of $A$ atoms in the $i$ -th atomic layer perpendicular to the $x$ -axis can be given by:

Here $J_{i,i+1}$ is the net flux of $A$ atoms from plane $i$ to $(i+1)$ :

$$J_{i,i+1}=z_v \left[ c_i(1-c_{i+1}) \Gamma_{i,i+1}- c_{i+1}(1-c_i) \Gamma_{i+1,i} \right],$$

where $c_i$ is the atomic fraction of A atoms in plane $i$ , $\Gamma_{i,i+1}$ is the frequency with which an A atom in plane $i$ exchanges with a B atom in plane $i+1$ and $z_v$ is the vertical coordination number. It is usually assumed that the exchange frequencies have an Arrhenius type temperature dependence:

$$\Gamma_{i,i+1}=\Gamma_i \gamma_i \Gamma_{i+1,i}=\Gamma_i/ \gamma_i$$

with

$$\Gamma_i=\Gamma_0\exp[\alpha_i/kT] \gamma_i=\exp[-\varepsilon_i/kT],$$

were $\Gamma_0=\nu\exp[-\hat E_0/kT]$ , $\nu$ denotes the attempt frequency, $\hat E_0$ is composition independent and contains the saddle point energy, $k$ is the Boltzmann constant, $T$ is the absolute temperature, and

$$\alpha_{i}=\left[z_{v}(c_{i-1}+c_{i+1}+c_{i}+c_{i+2})+z_{l}(c_{i}+c_{i+1})\right]M$$

as well as

$$\varepsilon_{i}=\left[z_{v}(c_{i-1}+c_{i+1}-c_{i}-c_{i+2})+z_{l}(c_{i}-c_{i+1})\right]V.$$

Here $z_l$ is the lateral coordination number, $V$ is the same regular solid solution parameter as in KMC, and also similarly to the KMC model, $M$ measures the diffusion asymmetry. Here $m'=-2ZM/kT\log_{10}e$ , where $e$ is the base of natural logarithm and $(Z= 2z_v + z_l)$ .[4]

The input parameters in the model are, therefore:[4] $V$ regular solid solution parameter; $T$ temperature; $z_v$ , $z_l$ vertical and lateral coordination numbers [$z_v=4$ , $z_l=0$ for a BCC structure and $z_v=4$ , $z_l=4$ for an FCC structure in (100) direction]; and $m'$ diffusion asymmetry parameter.

2.6.2 Three dimensional kinetic Monte Carlo model (KMC)

Monte Carlo simulations of the kinetic process can be performed using for instance the residence-time algorithm.[5] It is quite common to use the so-called one-vacancy model, when there is always one vacancy in the simulation box. In this case, for comparison to real time scale, the simulation time should be rescaled to corresponding real concentrations.

If only atom-vacancy exchange is allowed, the exchange probability of a vacancy-atom pair can be calculated from the binding energy of the atom. This energy can be calculated easily in a simple nearest-neighbour interaction approximation either for an $A$ ($E_A$ ) or $B$ ($E_B$ ) atom:

$$E_A = n_A V_{AA} + n_B V_{AB}, \notag$$ (2.29)

$$E_B = n_A V_{AB} + n_B V_{BB},$$ (2.30)

where $n_A$ and $n_B$ are the number of $A$ and $B$ atoms in the vicinity of the given atom, $V_{ij}$ ($i,j=A$ or $B$ ) is the interaction energy between an $ij$ atom pair. Introducing $V = V_{AB} - \frac{V_{AA}+V_{BB}}{2}$ and $M = \frac{V_{AA} - V_{BB}}{2}$ , Eqs. (2.29) can be written as:

$$E_A = -n_A (V-M) + n V_{AB}, \notag$$

$$E_B = -(n-n_A) (V+M) + n V_{AB}.$$

where $n=n_A+n_B$ is the number of neighbors of an atom, $V$ , is the regular solid solution parameter [proportional to the mixing energy and measures the phase separating $(V>0)$ or ordering $(V<0)$ tendency] and $M$ measures the diffusion asymmetry ( $m'=-2nM/kT\log_{10}e$ , where $e$ is the base of natural logarithm).[4] Using the usual Arrhenius- relationship between the activation energy ( $Q_i=E^0-E_i$ , where $E^{0}$ is the saddle point energy, and $i=A,B$ ) and the probability [ $\Gamma_{iV} = \nu \exp(-Q_i/kT)$ ], the exchange probabilities of a vacancy-$i$ atom pair are:

$$\Gamma_{VA} = \nu \exp \left[-\frac{\hat{E}^{0}+n_A (V-M)}{kT}\right], \notag$$

$$\Gamma_{VB} = \nu \exp \left[-\frac{\hat{E}^{0}+(n-n_A) (V+M)}{kT}\right],$$

where $k$ and $T$ are the Botzmann's constant and the absolute temperature, $\nu$ is the attempt frequency, and $\hat{E}^0 = -E^0 + nV_{AB}$ . Note that $\hat{E}^0$ is taken to be constant for a given system, and thus it may be set to zero during calculations (but can be considered in the time scaling if needed).