text: Shortly after the publication of Kirkendall's paper, L.S. 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  to be 
  where  and  are the diffusion coefficients of the two materials and  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.
  Darken also developed a second equation that defines a combined chemical diffusion coefficient  in terms of the diffusion coefficients of the two interfacing materials:
  This chemical diffusion coefficient can be used to mathematically analyze Kirkendall effect diffusion via the Boltzmann-Matano method.