Diffusion in binary dilute systems is governed by Fick\'s first law, linking mass flux \(J_i\) to the concentration gradient:

\[J_i = -D_{ij} \nabla C_i\]

Limitations of Fick\'s Model

Fickian diffusion assumes that solute molecules interact only with background solvent molecules. In concentrated multi-component mixtures, solute-solute drag forces cannot be ignored. The **Maxwell-Stefan** formulation relates forces to concentration profiles:

\[\nabla x_i = \sum_{j \neq i} \frac{x_i N_j - x_j N_i}{C D_{ij}}\]

Where \(x_i\) is the mole fraction, \(N_i\) is the molar flux, \(C\) is total concentration, and \(D_{ij}\) represents Maxwell-Stefan friction coefficients.

References

  • Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2002). Transport Phenomena. Wiley.
  • Taylor, R., & Krishna, R. (1993). Multicomponent Mass Transfer. Wiley.