In a material composed of two or more chemical species in which there are spatial inhomogeneities of the composition, there is a driving force for inter-diffusion of the various molecular species so as to render the composition of the material uniform. In a mixture of just two species, the diffusive flux of each molecular species is proportional to the gradient of its composition. This proportionality is known as Fick's Law of Diffusion and is, to a limited extent, the mass transfer analogue of Newton's Law of Viscosity and Fourier's Law of Heat Conduction [Bird et al. (1960)].
The mathematical formulation of Fick's Law must be performed with great care, because there are a number of ways of expressing the composition of a material and because it is necessary to prescribe the frame of reference with respect to which the flux of a particular species is measured. For this reason, it is first necessary to identify the molar average velocity in a mixture of n species as
where vi is the mean velocity of molecules of species i with respect to a fixed coordinate axis. Here, c is the amount-of-substance density of species i and c, the total amount-of-substance density for the mixture. This definition means that cv* is the total amount of substance that flows through unit area perpendicular to v* in unit time.
According to Fick's Law for a binary system, the diffusive amount-of-substance flux of species 1 relative to a reference frame moving with the molar average velocity of the two species is given by
where x1 is the amount-of-substance fraction of species 1 and D12 is the inter-diffusion coefficient of the two species, sometimes known as the mutual diffusion coefficient.
It is rather difficult to arrange in practice to study diffusion relative to the molar average velocity of the two species so it is usually studied relative to a reference frame fixed in the laboratory. As a consequence, the total amount-of-substance flux observed in this reference frame differs from that given by Fick's Law. This can arise whenever there is a net flow of a species from one point to another, for example, when a liquid evaporates into a gas and diffuses through it or in a case of adsorption of a gas on to a solid surface. It may also arise when there is a change of volume of the mixture upon mixing since, then, the molar average velocity is not constant.
In multicomponent mixtures the process of diffusion is much more complicated since there can be coupling of the gradients of some species to the fluxes of others [Bird et al. (1960)].
Bird, R. B., Stewart, W. E., and Lightfoot, E. N. (1960) Transport Phenomena, John Wiley, New York.
Tyrrell, H. J. V. and Harris, K. (1984) Diffusion in Liquids, Butterworths, London.
Cussler, E. L. (1984) Diffusion: Mass Transfer in Fluid Systems, Cambridge University Press, Cambridge.