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Fick's Laws of Diffusion

By Paul A. Steward.

Diffusion is the mechanism by which components of a mixture are transported around the mixture by means of random molecular (Brownian) motion (cf. permeation: the ability of a diffusant to pass through a body - dependent on both the diffusion coefficient, D, and the solubility coefficient, S, ie, permeability coefficient, P = D.S). Flynn et al. cite Berthalot as postulating, at the beginning of the nineteenth century, that the flow of mass by diffusion (ie, the flux), across a plane, was proportional to the concentration gradient of the diffusant across that plane.

In the mid-1800's, Fick introduced two differential equations that quantified the above statement for the case of transport through thin membranes. Fick's First Law states that the flux, J, of a component of concentration, C, across a membrane of unit area, in a predefined plane, is proportional to the concentration differential across that plane (see note), and is expressed by:

Equation for Fick's First Law

Fick's Second Law states that the rate of change of concentration in a volume element of a membrane, within the diffusional field, is proportional to the rate of change of concentration gradient at that point in the field, as given by:

Equation for Fick's Second Law

where t = time.

Adapted from: Modification of the Permeability of Polymer Latex Films., Nottingham Trent University PhD Thesis, 1995.
Copyright © Paul Steward, 1995.

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References.

  1. Flynn G.L. Yalkowsky S.H., Roseman T.J., Mass Transport Phenomena and Models: Theoretical Concepts., J. Pharm. Sci., 63 (4), pp479-510, 1974. Return to text.
  2. Berthalot C.L., Eassai de Statique Chimique., Paris, France, 1803. Return to text.
  3. Fick A., Ann. Physik, Leipzig, 170, pp59, 1855. Return to text.

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Copyright © Paul A. Steward 1995.
Created by Paul A. Steward.
Last Revised: Saturday, June 06, 1998 07:44 PM
Email: paul.steward@initium.demon.co.uk

Note:

Diffusion typically means the interdiffusion of two species, and requires two equations for its description. However, in the absence of a net volume change across the plane of reference, the rates are equal but opposite. Therefore, one equation only need be considered. This is especially true if one species is large (eg, polymer) and dictates a low diffusion rate for both species. A membrane is usually considered as fixed, and considered to act as a reference to the flux of mobile permeant.
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