Editing Kirkendall effect (section)

===Darken's equations===
{{main|Darken's equations}}
Shortly after the publication of Kirkendall's paper, L.&nbsp;S.&nbsp;Darken published an analysis of diffusion in binary systems much like the one studied by Smigelskas and Kirkendall. By separating the actual diffusive flux of the materials from the movement of the interface relative to the markers, Darken found the marker velocity <math>v</math> to be<ref name=Darken>{{cite journal |last=Darken |first=L. S. |title=Diffusion, Mobility, and Their Interrelation through Free Energy in Binary Metallic Systems |journal=Trans. AIME |date=February 1948 |volume=175 |page=194}}</ref>
<math display="block">
 v = (D_1 - D_2) \frac{dN_1}{dx},
</math>
where <math>D_1</math> and <math>D_2</math> are the diffusion coefficients of the two materials, and <math>N_1</math> is an atomic fraction.
One consequence of this equation is that the movement of an interface varies linearly with the square root of time, which is exactly the experimental relationship discovered by Smigelskas and Kirkendall.<ref name="original journal" />

Darken also developed a second equation that defines a combined chemical diffusion coefficient <math>D</math> in terms of the diffusion coefficients of the two interfacing materials:<ref name=Darken />
<math display="block">
 D = N_1 D_2 + N_2 D_1.
</math>
This chemical diffusion coefficient can be used to mathematically analyze Kirkendall effect diffusion via the [[Boltzmann–Matano analysis|Boltzmann–Matano method]].