Darkens Equation

The Darkens Equation relates the atomic mobility's of the binary alloy species \(M_A, M_B\), to the total difussion mobility \(M\). As we have defined \(c_A = c\), then \(M\) is the diffusion mobility of species \(A\) and is estimated by the Darkens equation:

$$ M = c_A c_B \left[c_A M_B + c_B M_A\right] = c (1-c)\left[c M_B + (1-c) M_A\right] $$ .

In SpinDec2, the user can set which definition of mobility they would like to use using the problem string input.

• "constant" - Mobility is taken to be constant everywhere at all times
• "nontemp" - Mobility is dependent on the order paramter c over time
• "temp" - The atomic mobilities are defined in terms of local tempeartures

Constant Mobility

If the user sets the problem string to "Constant", the diffusive mobility field is constant everywhere and is defined via the Darkens equation for \(c=c_0\):

$$ M = c_0 (1-c_0)\left[c_0 M_B + (1-c_0) M_A\right],$$

where \(c_0\) is the average value of the initial \(c\) grid. This simplfies the Cahn-Hilliard equation as the mobility can be taken out of the divergence operation:

$$\frac{\partial c}{\partial t} = M\nabla^2 Q$$,

where \(Q = \frac{\delta F}{\delta c}\).

Non-Constant Mobility

The Darkens equation shows how the mobility is dependent on the concentration field. This then suggests that the diffusive mobility should also evolve with time inline with the concentration field. Due to this dependcey on \(c\), the Cahn-Hilliard becomes slightlu more complex with an additional chain rule term:

$$ \frac{\partial c}{\partial t} = \nabla \cdot (M \nabla Q) = M \left( \frac{\partial^2 Q}{\partial x^2} + \frac{\partial^2 Q}{\partial y^2}\right) + \left( \frac{\partial M}{\partial x}\frac{\partial Q}{\partial x} + \frac{\partial M}{\partial y}\frac{\partial Q}{\partial y} \right).$$

The model can further evolved by considering how to define the atomic mobilities. In SpinDec2, the user can chose wheter they want to use consatnt values for \(M_A\) and \(M_B\), by initialsing the problem string to "nontemp", or to make the atomic mobilities dependent on a local value of tempertaure which remains constant through out (i.e doesnt evolve via conction of difussion equation).

Temperature dependcey

If the user sets the problem string to "temp", the atomic mobilities \(M_A\) and \(M_B\) are no longer constant and are dependent on a temperature initialised from the input.txt file. This is done by setting the minimum and maximum temperature, from which the grid points are selected from samples of a uniform distribution defined by these bounds. The formula used to define the atomic mobilities, in this case, is taken from Y. H. Wang et al., J. Appl. 12 85102 (2019):

$$ M_{\alpha} = \frac{1}{R T} \exp \left(\frac{E_{\alpha}}{R T} \right) = \frac{1}{R T} D_{\alpha}, \,\,\, \alpha = A, B ,$$

where \(R\) is the molar gas constant, \(T\) is the absolute temperature and \(E_{\alpha}\) is the activation energy for the select species. To simplify notation in the next part, we rename the exponential factor to be \(D_{\alpha}\) and shall be known as the diffusion coefficient for species \(\alpha\). Therefore the diffusion mobility is redefined as:

$$ M = \frac{c (1-c)}{RT} \left[D_B c + D_A (1-c) \right].$$