Then, it is hypothesized that the chance of travelling one times the free path distance in the positive x-direction is equal to 1; the chance of travelling two times the free path in the positive x-direction is 1/2; the chance of travelling four times the free path in the positive x-direction is 1/4, and so on. One can imagine that this yields the following expression for the diffusivity:

Since we live in a three dimensional world, we have to add the chance of going into the positive x-direction (we could also have gone into the - x, +y, -y, +z, -z direction):

Watch species diffusie in real life (including a calculation of the diffusion coefficient): the video on Brownian movement.

Diffusion Distance in Fick's Laws

Fick's First Law adds a driving force - the concentration gradient - to the diffusion coefficient. This enables one to calculate the diffusion flux or mass transport in a preferential direction of for example water, a solvent or natural gas into a coating, packaging polymer, multilayer plastic , fibre metal laminate or glass fibre composite. Moreover, by solving the partial differential equation of Fick's Second Law we can define a function c(x,t) that gives the concentration of the diffusing species as a function of time and place in unsteady conditions. As long as the medium is semi-infinite (penetration from one side), the diffusion coefficient of species in polymer, multilayer and composite materials can be calculated from the weighted average distance travelled. This distance is calculated from the concentration function as follows:

With delta c the concentration gradient. The diffusion coefficient now follows from:

If Fick's First Law applies, the penetration depth for small times (Fourier Mass number << 0.1) follows from:

Determination of Diffusion Coefficients

Often the so called time lag method is used to determine the diffusion coefficient of molecules through plastic materials. In this method the polymer or composite sample is exposed on one side to the gas, liquid, solvent or vapour of interest. On the opposite side, the concentration of molecules is continuously measured by use of analytical equipment. At the same time, species are removed on this side to prevent concentration build-up. Then, after a certain time a steady state diffusion flux is obtained. This time relates to a weighted average diffusion distance. If the diffusion coefficient is constant, this steady state distance is calculated as follows:

With delta x is the thickness of the polymer based material. The related time lag formula is then:

From the experiment, the time and distance is known. Hence, the diffusion coefficient can be calculated. The reader is warned that this formula for time lag can applies when (i) diffusion is governed by Fick's First and Second Law (no pressure gradients or other driving forces than concentration gradients involved), (ii) when the diffusion coefficient is constant and not a function of concentration (when the polymer or composite material swells) or distance (such as the case in multilayer and fibre reinforced composite materials)!

References

[1] Einstein, A., Investigations on the theory of the Brownian movement, Dover Publ. (1956)
[2] Frisch, H.L., Time lag in transport theory, Journal of Chemical Physics, 36, 2(1962)
[3] Cranck. J, The Mathematics of Diffusion, Oxford Clarendon Press (1956)
[4] Cranck, J.; Park G.S., Diffusion in Polymers, Academic Press London
[5] Dlubek, G.; et al., Free Volume Variation in Polyethylenes of Different Crystallinities: Positron Lifetime, Density and X-Ray Studies, J. of Pol. Sci., Part B, 40, 65-81 (2001)
[6] Wesselingh J.A.; Krishna R., Mass Transfer in Multicomponent Mixtures, Delft University Press (2000)

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