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Derivation of darken equation

1. Find the net flux of atoms 

From Fick’s first law we can state that:

Equation

We can assume that the total number of atoms is constant, therefore the two concentration gradients are equal and opposite:

Equation

Therefore:

Equation

The net flux of atoms is given by JB − JA, and must be equal to the net flux of vacancies, Jv.

Equation

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,

Equation

We now employ the expression for Jv found in section 1:

Equation

Equation

3. Find the effect of this lattice drift on the diffusivity species A and B 

From Fick’s first law:

Equation

This flux is opposed by the flux due to lattice drift:

Jlatt = ν CA

If  JA0 is the net flux:

Equation

We know v from section 2:

Equation
Equation

Where Equation is the interdiffusion coefficient, such that

Equation


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