Diffusion Coefficient of Gas, Liquid & Vapour in Polymer |

The diffusion coefficient or so called diffusivity has the dimensions of [length^{2} time^{-1}], [m^{2} s^{-1}]. These dimensions result from the underlying kinetic theory. The diffusion 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:

(1)

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):

(2)

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:

(3)

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

(4)

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

(5)

**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:

(6)

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

(7)

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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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:

(1)

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):

(2)

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

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:

(3)

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

(4)

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

(5)

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:

(6)

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

(7)

From the experiment, the time and distance is known. Hence, the diffusion coefficient can be calculated. The reader is

[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)

-Watch the video on Brownian movement.

-More on temperature dependence of diffusion, solubility & permeability.

-More on temperature dependence of diffusion, solubility & permeability.

i need diffusion coeficients of Mg , Ba and Ta. please let me know.

Hello, I want to add some analytical information concerning the diffusion coefficient of for example Carbon Dioxide (CO2), Carbonic Acid or Water Vapour in membrane (i.e. Polyimide, PDMS or Sulfonated PEEK), coating or fibre composite applications.

There are number of analytical- experimental techniques which can be used to determine the diffusion coefficient through polymer interfaces. They are summarized as follows:

1. Scanning infrared microscopy

- infrared microdensitometry

- scanning infrared microscopy

2. light scattering

- optical Schlieren technique

- spectroscopic ellipsometry

- dynamic light scattering

3. Neutron scattering

- small-angle neutron scattering (SANS)

- neutron reflection spectroscopy (NRS)

4.Raman scattering

-surface-enhanced Raman scattering (SERS)

5. Infrared spectroscopy

- external reflection infrared spectroscopy

- attenuated total reflectance spectroscopy

- transmission FTIR

- reflection absorption spectroscopy

- attenuated total reflection microspectrometry.

6 Other methods

- Photon correlation spectroscopy

- Donor-acceptor fuorescence method

- Small-angle x-ray scattering (SAXS)

- Electron microprobe analysis

- Nuclear reaction analysis (NRA)

- Ellipsometry

- Electrical (Electro Chemical) Impedance Spectroscopy (EIS)

- Electrochemical Noise Spectroscopy / Measurement (ENM)

There are number of analytical- experimental techniques which can be used to determine the diffusion coefficient through polymer interfaces. They are summarized as follows:

1. Scanning infrared microscopy

- infrared microdensitometry

- scanning infrared microscopy

2. light scattering

- optical Schlieren technique

- spectroscopic ellipsometry

- dynamic light scattering

3. Neutron scattering

- small-angle neutron scattering (SANS)

- neutron reflection spectroscopy (NRS)

4.Raman scattering

-surface-enhanced Raman scattering (SERS)

5. Infrared spectroscopy

- external reflection infrared spectroscopy

- attenuated total reflectance spectroscopy

- transmission FTIR

- reflection absorption spectroscopy

- attenuated total reflection microspectrometry.

6 Other methods

- Photon correlation spectroscopy

- Donor-acceptor fuorescence method

- Small-angle x-ray scattering (SAXS)

- Electron microprobe analysis

- Nuclear reaction analysis (NRA)

- Ellipsometry

- Electrical (Electro Chemical) Impedance Spectroscopy (EIS)

- Electrochemical Noise Spectroscopy / Measurement (ENM)

I read that Electrical Impedance Spectroscopy (EIS), Dielectric Sorption Analysis (DSA) or Electrochemical Noise Measurement(ENM) is particularly useful for service life prediction of coatings. With this regard I have the following questions:

1. Are these electrochemical methods also the preferred methods for measuring diffusion coefficients of a polymer or polymer based material on a substrate?

2. Or are they only suitable for measuring a general degradation state of the coating or polymer without detailed information on primarly diffusion rate, thermodynamic information, chemical corrosion rates and interfacial issues?

3. Does anyone have experience with a continuous Electrical Impedance Spectroscopy (EIS) coating measurement in the field (oil platforms, vessels, bridge, etc.) and the forthcoming results regarding the diffusion coefficient?

Thanks Ralph

p.s I am mainly interested in polyurethane / epoxy based coatings on metals (Zinc, Aluminum, Stainless Steel) and exposure to weathering conditions

1. Are these electrochemical methods also the preferred methods for measuring diffusion coefficients of a polymer or polymer based material on a substrate?

2. Or are they only suitable for measuring a general degradation state of the coating or polymer without detailed information on primarly diffusion rate, thermodynamic information, chemical corrosion rates and interfacial issues?

3. Does anyone have experience with a continuous Electrical Impedance Spectroscopy (EIS) coating measurement in the field (oil platforms, vessels, bridge, etc.) and the forthcoming results regarding the diffusion coefficient?

Thanks Ralph

p.s I am mainly interested in polyurethane / epoxy based coatings on metals (Zinc, Aluminum, Stainless Steel) and exposure to weathering conditions