The diffusion coefficient or so called diffusivity has the dimensions of [length² time^-1], [m² s^-1]. These dimensions result from the underlying kinetic theory. The theory states that chemicals move with a certain molecular velocity, [m/s], depending on particle size and temperature, along a free path, [m]. The free path length is determined by the amount of matter per cubic meter. The less matter is available, the longer the available path length. Hence, self diffusion rates of gases are much higher than liquids.

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):

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 a solvent into a polymer or 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 Diffusivities

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)!

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Diffusion coefficient according free volume theory

Probably still the best theory for calculation of an unknown diffusion coefficient of a liquid or gas in a polymer, is the free volume theory for diffusion. The theory is developed by Cohen and Turnbull (1959) who considered transport in a liquid of hard spheres. Molecules reside, most of the time, in cages bound by their neighbours. Occasionaly a fluctuation in density opens up a hole within a cage large enough to permit considerable displacement of the molecule contained by it. Succesful diffusive transport occurs if another molecule jumps into the hole before the first can return to its original position. In the model of Cohen and Turnbull, diffusion is treated as translation of a molecule across the void within its cage.

Diffusion occurs not only as a result of an activation in the ordinary sense, but rather as a result of redistribution of the free volume within the liquid or frozen liquid: a polymer.

Now, the diffusion activation related theories aren't very useful: some models are too simple (Arrhenius / Van't Hoff) and others are too complex, in the sense that they require too much parameters to fit the eventually experimentally obtained diffusivity figures.

On the contrary, the free volume approach requires a limited amount of parameters, which can be calculated rather well by using the work from Guggenheim (for the free volume of the liquid) and Positron Lifetime Spectroscopic information (for the free volume in the polymer and resins).

Below we will present an example of application of the free volume theory, in line with previous posting on the diffusion coefficient. Hence we will first discuss the calculation of the molecular velocity and secondly the path length, determined by the free volume theory.

1:Molecular Velocity

The molecular velocity of chemical i [v in m/s] is a function of the molecular mass [μi in kg / mol], temperature [T in Kelvin] and the gas constant [R: 8.314 Joule / mol K]:

vi = SQRT (3 R T / μi)

2:Path Length by Free Volume Theory

Little more complicated is the determination of the free path length in polymers and inclusion of the change of free path length when passing the glass transition (a sudden increase in path length for the amorphous fraction) and swelling of the polymer (usually causing plasticizing, increasing the path length).

We will determine the path length using the statistical model from Cohen and Turnbull for spherical particles. It is based on the fact that the chance of finding a hole larger than a certain volume is a simple function of the free volume per particle. If we multiply this chance with the diameter of the molecule, we acquire a good estimation of the free path available for diffusion.

Below the path equation for a relatively small gas in a polymer is shown.

lamdai = di x e-[vi/vpfree]

With di [m] is the diameter of the molecule, [m3 / mol] the compressed molecular volume of i, and [m3 / mol] the available free volume in the polymer. The compressed volume is for all chemicals a function of the molar volume at the critical temperature:

vi = 0.286 x vicritical

So, if we now the molar volume at the critical temperature (available in most reference books), we can calculate the compressed molar volume (i.e. the molar volume at 0 Kelvin) rather easily. The free volume in a polymer, PE, is at a temperature of 298 Kelvin and a atmospheric pressure of 1 bar, equal to a fraction of 9 to 15% of the compressed molar volume for Polyethylene with a respective crystallinity of 56 to 0%.

3: the result

The diffusion coefficient for gases and liquids that have a low solubility (smaller than 0.1 mass percent) can now be calculated by multiplication of the thermal velocity with the path length:

Di = vi x di x e-[vi/vpfree]

Mass balance for (multicomponent) diffusion

For studying permeation of one or more gas or liquid components diffusing alone (binary diffuson) or simultaneously (multicomponent) in a polymer or composite material, a mass balance is essential.

The mass balance is used for interpretation of the diffusion experiment, according to gravimetric methods and gas chromatography measurements and subsequently for application of the diffusive mass transfer in the real life application, such as mass transport through a membrane, containment, pipeline or package.

Usually Fick's first and second laws are used to balance the driving force (the concentration gradient) with the friction force (the diffusivity).

This goes well as long as we have binary diffusion and the concentration gradients acting as the only driving force. If we have to consider multicomponent diffusion, Fick's laws hardly make any sense.

The Maxwell-Stefan equation uses the chemical potential gradient as the driving force for diffusing chemicals. The motion with respect to other chemicals (especially the polymer matrix) causes friction. The driving force is equal to the sum of these friction forces, because acceleration effects are negligible in diffusion.

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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