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In this message I will explain the use of the Arrhenius Law for calculating species diffusivity, solubility and permeability in unreinforced and reinforced polymers as a function of temperature. Most important are the physical considerations, in the second paragraph. Here, I will demonstrate some important solubility and permeation highlights and anomalies with regard to the use of the Arrhenius equation in this context. I end with a practical example on Carbon Dioxide (CO2) diffusion, solubiltiy and permeation in Polyethylene (PE).

The Arrhenius Equation

D(T)=D₀ e ^{-[E / R x T]} or S(T)=S₀ e ^{-[H / R x T]}

D(T) / S(T): diffusion coefficient / solubility as function of temperature [m2/s] / [gram / m3].

D₀ / S₀: the diffusion coefficient / solubility when the temperature goes to infinitity [m²/s] / [gram/m³].

E / H: the activation energy for diffusion / the mixture enthalpy [Joule/ mole].

R: Universal gas constant (8.314 Joule / Mole x Kelvin).

T: Temperature (Kelvin).


Physical Considerations

The temperature dependence of the diffusion coefficient (D) of, for example, carbon dioxide in high density polyethylene can be obtained by using the formula based on the Van 't Hoff Equation or Arrhenius Law. Realize that the temperature dependence of the solubility (S) of substance in materials is often also calculated by this Arrhenius Law. Moreover, as the steady state permeability (P), follows from D times S, the permeation rate also follows an Arrhenius type of temperature behaviour. Below we list some important consistencies with regard to the above:

1] From the general accepted picture of the mechanism of the activated diffusion process, it is known that larger holes need to be formed in the polymer for the diffusion of larger molecules. These will require a larger energy for their formation and hence the activation energy will be larger for the diffusion of larger molecules, and the diffusivity will be smaller.

2] Hence, polymers and coatings with a small molecular free volume because of high degree of crystallinity, will have a high activation energy for diffusion.

3] When passing the Glass Transition temperature heading for a higher temperature, the Activation Energy stays constant or increases. Concerning polymers that have a low free volume, such as partly crystalline polyamide (PA) or polyimide (PI), the change of activation energy is significant. For amorphous polymers, like polymethamethacrylate (PMMA) there is no change of activation energy. Whether the activation enery increases or decreases most probably depends on the degree of plasticizing by the permeant and the size of permeant relative to the free volume in the polymer.

4] Although the diffusion coefficient increases as function of temperature, the solublity of chemical in polymers does not increase by definition. In general:

-Heat of Solution for small gases is usually positive, hence an increase of solubility with temperature. Heat of Solutions of large gases are negative, so a decrease of solubility with temperature. Helium and Hydrogen are per definition small gases. The solubility temperature dependence of larger chemicals, like Oxygen, Nitrogen, Carbon Monoxide, Argon and Carbon Dioxide depends on the polymer under consideration.
-Solubility of liquids increases as a function of temperature.
Read more on solubility of gases and liquids in polymers in the case stories section (check the case on Hildebrand solubility parameters).

5] Effect of crosslinking
Increasing the degree of crosslinking in thermosetting resins such as polyester, epoxy and vinyl ester or for example sulphur in natural rubber, increases the activation energy.

6] One may question what happens with permeability P as a function of temperature. In case of liquid and solvent permeation in polymer and composite materials this is obvious: permeability increases as a function of temperature. When dealing with a gas, usually the acitvation energy for diffusion is higher than the possible negative enthalpy of solution, hence the permeability also increases as a function of temperature.

Example
We measured the diffusivity of carbon dioxide in high density polyethylene with a crystalline fraction of 0.77, at a temperature of 40 and 50 degrees Celsius (313 and 323 Kelvin). We want to know the diffusion coefficient at 60 degrees Celsius.

D (313 Kelvin) = 4.0 E-11 m²/s
D (323 Kelvin) = 6.1 E-11 m²/s

Applying the Arrhenius equation on the two data points we obtained an activation energy of:
37 Kilo Joule (37 KJ).
Then the diffusion coefficient at 60 degrees Celsius is:

D (333 Kelvin) = 9 E-11 m²/s
The diffusion coefficient is strongly dependent on temperature.

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