Derivation of darken equation
1. Find the net flux of atoms
From Fick’s first law we can state that:
\[{J_A} = - {D_A}\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\}\] \[{J_B} = - {D_B}\left\{ {\frac{{\partial {C_B}}}{{\partial x}}} \right\}\]
We can assume that the total number of atoms is constant, therefore the two concentration gradients are equal and opposite:
\[\frac{{\partial {C_A}}}{{\partial x}} = - \frac{{\partial {C_B}}}{{\partial x}}\]
Therefore:
\[{J_B} = {D_B}\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\}\]
The net flux of atoms is given by JB − JA, and must be equal to the net flux of vacancies, Jv.
\[{J_v} = \left( {{D_A} - {D_B}} \right)\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\}\]
2. Use this to find the velocity of the lattice drift
The velocity of the lattice drift can be found by relating it to the flux of vacancies, Jv.
In a time increment δt a lattice plane of area A will sweep out a volume Avδt. If there are C0 atoms per unit volume, this volume contains Avδt C0 atoms.
This must be the total number of vacancies passing through the plane, given by Jv Aδt
Thus,
\[{J_v}A\partial t = Av\partial t{C_0}\] \[{J_v} = v{C_0}\]
We now employ the expression for Jv found in section 1:
\[\left( {{D_A} - {D_B}} \right)\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\} = v{C_0}\]
\[v = \frac{1}{{{C_0}}}\left( {{D_A} - {D_B}} \right)\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\}\]
3. Find the effect of this lattice drift on the diffusivity species A and B
From Fick’s first law:
\[{J_A} = - {D_A}\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\}\]
This flux is opposed by the flux due to lattice drift:
Jlatt = ν CA
If JA0 is the net flux:
\[{J_A}' = - {D_A}\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\} + v{C_A}\]
We know v from section 2:
\[{J_A}' = - {D_A}\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\} + \frac{1}{{{C_0}}}\left( {{D_A} - {D_B}} \right)\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\}{C_A}\] \[{J_A}' = - {D_A}\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\} + {X_A}\left( {{D_A} - {D_B}} \right)\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\}\] \[{J_A}' = \left( {{X_A}{D_A} - {X_A}{D_B} - {D_A}} \right)\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\}\] \[{J_A}' = \left( {\left( {1 - {X_B}} \right){D_A} - {X_A}{D_B} - {D_A}} \right)\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\}\] \[{J_A}' = - \left( {{X_A}{D_B} + {X_B}{D_A} + {D_A} - {D_A}} \right)\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\}\] \[{J_A}' = - \left( {{X_A}{D_B} + {X_B}{D_A}} \right)\left\{ {\frac{{\partial {C_A}}}{{\partial x}}} \right\}\] \[{J_A}' = - \tilde D\left\{{\frac{{\partial {C_A}}}{{\partial x}}} \right\}\]
Where \(\tilde D\) is the interdiffusion coefficient, such that
\[\tilde D = {X_A}{D_B} + {X_B}{D_A}\]