## Diffusion Coefficients

### Quick Start

It can be difficult to get hold of a diffusion coefficient for a specific molecule in a specific polymer at a specific temperature. This app gives you an "industry standard" value from your choice of MWt, temperature and one of the standard polymers.

For those who use nanoparticles you can get an estimate of their diffusion coefficients, though this is hardly necessary as they are all "very, very small".

### Diffusion Coefficients

_{P}

_{mP}°C

_{m}

_{g}°C

_{NP}nm

The chief difficulty with any diffusion scenario is finding the diffusion coefficient, D. One approach is to follow a method (effectively) approved by US and European regulatory bodies (see T. Begley, L. Castle, A. Feigenbaum, R. Franz, K. Hinrichs, T. Lickly, P. Mercea, M. Milana, A. O’Brien,S. Rebre, R. Rijk & O. Piringer (2005): *Evaluation of migration models that might be used in support of regulations for food contact plastics*, Food Additives and Contaminants, 22:1, 73-90) which provides a set of standard polymers and estimates D from the MWt of the molecule and the temperature, T (°K), for the specific scenario. The formula is:

D=10^{4}exp(A-C_{1}MWt^{0.666}+C_{2}MWt-C_{3}/T) cm²/s

C_{1}=0.135, C_{2}=0.003, C_{3}=10454.

A in turn depends on T and two polymer specific terms which are part of the dataset, A* and τ.

A=A*-τ/T

When you choose a polymer the A* and τ values are shown and from your given MWt and T, D is calculated

Some alternative results, including for nanoparticles, are shown and the details are explained below.

Although this formula is often used in food packaging applications, be aware that this app carries no guarantees that the calculated D values are correct. For example, the linear, branched and cyclic versions of a molecule with the same molecular weight will each have different D values; branched and cyclic molecules have lower D values than linear ones as they are less able to wiggle into spare free volume within the polymer.

### A revised formula

A 2008 formula for polyolefins from Pringer and Bauer in the book *Plastic Packaging - Interactions with food and pharmaceuticals* uses four input parameters: T and MWt as before, adding T_{mP} and MWt_{P} as the melting point and molecular weight of the polymer. The formula is much too complex to be reproduced here: the result is simply provided as the D (2008) value. If you change the selected polymer the T_{mP} slider is changed to a representative value, but as these vary strongly in the literature, feel free to move the slider to your preferred value. For PET a more specific formula from Welle^{1}at the Fraunhofer IVV is used. For PEN I have made an ad hoc variant of the Welle formula. For COP (Cyclo Olefin Polymer) there are no published values for the original formula and the formula from another Welle^{2} paper is used.

### Brandsch

A formula from Brandsch gives a dependency on the Tg of the polymer, the Tm of the small molecule, the MVols of the polymer and the small molecule, here approximated by their MWts. Each of these different techniques is intended to provide rational options for judging diffusion coefficients - there is no substitute for measuring them if the application is of great importance.

### Nanoparticles

Following another paper from the Fraunhofer IVV^{3}, by using a pseudo MWt calculated from the diameter of a nanoparticle it is possible to create an estimate of the diffusion coefficient based on the 2008 formula. With diameters above 5nm the diffusion coefficients are very, very small! The Welle PET/COP formula is also used for these values.

The inspiration to add the nanoparticle calculations (and the 2008 formula) came from reading the Franz and Welle Chapter 15 *Mathematic Modelling of Migration of Nanoparticles from Food Contact Polymers* in the marvellous book The Use of Nanomaterials in Food Contact Materials edited by Dr Rob Veraart and published in 2017 by DEStech..

^{1}Frank Welle,

*A New Method for the Prediction of Diffusion Coefficients in Poly(ethylene terephthalate)*, J. Appl. Polym. Sci. 129,2013,1845–1851

^{2}Frank Welle,

*Activation energies of diffusion of organic migrants in cyclo olefin polymer*, Int. J. Pharmaceutics, 473, (2014), 510–517

^{3}J. Bott, A. Störmer, R. Franz,

*A model study into the migration potential of nanoparticles from plastics nanocomposites for food contacts*, Food Packaging and Shelf Life, 2, 2014, 73-80