## 3 Results and Discussion
### Prediction of Diffusion Coefficients
The correlation above the glass transition temperature therefore is associated with a high degree of uncertainty and the values should be used with caution. The parameters for HIPS were derived from ten activation energies of diffusion below and above the glass transition temperature. The parameter \(a\) is the slope of the correlation between the activation energies of diffusion \(\mathrm{E_{A}}\) and the pre-exponential factor \(\mathrm{D_{0}}\) of the Arrhenius equation. When this reciprocal temperature \(a\) is reached, the diffusion coefficients reaches its maximum value, which is the factor \(b\). The parameter \(b\) is therefore the highest diffusion coefficient, which can be reached in the investigated temperature interval. When the molecular volume reaches the parameter \(c\), the activation energy of diffusion reaches zero and the diffusion  
Figure 4: Correlation between the molecular volume V and the activation energy of diffusion \(\mathrm{E_{A}}\) for (**a**) GPPS and (**b**) HIPS below \(\mathrm{T_{g}}\) (solid dots) and above \(\mathrm{T_{g}}\) (non-solid dots). Figure 5: Correlation between the pre-exponential factor \(\mathrm{D_{0}}\) and the activation energy of diffusion \(\mathrm{E_{A}}\) below \(\mathrm{T_{g}}\) (solid dots, solid line) and above \(\mathrm{T_{g}}\) (non-solid dots, dashed line): black dots GPPS, grey dots HIPS. coefficients are the same at any temperatures, which is again the parameter \(b\) but at the same time also \(\mathrm{D_{0}}\) when the molecular volume V is the parameter \(c\). The parameter \(d\) is a measure of the extent to which the activation energy of diffusion is influenced by the molecular volume V of the migrant in different polymers. \[\mathrm{D_{P}}\ =\ b\ \left(\frac{\mathrm{V}}{c}\right)^{\frac{\epsilon-\frac{1}{ 2}}{d}} \tag{3}\]  
For migration modelling used for food law compliance evaluation, the applied prediction models should be over-estimative, because the predicted migration is in any case higher as the experimentally determined migration [31]. As a consequence, the predicted diffusion coefficients \(\mathrm{D_{P}}\) needs to be higher as the measured diffusion coefficients for each molecule in any case. An artificial reduction, e.g., of 20% of the molecular volume seems to sufficient to overestimate the experimentally determined migration in case of PET [25]. These diffusion coefficients might be considered as worst-case diffusion coefficients for food law compliance evaluation. Systematic studies for polystyrene are not available to date, so that the amount of volume reduction needs to part of further investigations. On the other hand, when using migration modelling for the prediction of the decontamination efficiency in PS recycling processes, over-estimation will be the best-case. In this case, the molecular volume should be increased in order to get slightly lower diffusion coefficients \(\mathrm{D_{P}}\) for the prediction of the worst-case cleaning efficiencies of a recycling process. In conclusion, by use of Equation (3), the diffusion coefficients of other non-tested substances in GPPS and HIPS can be predicted for any temperature between 0 \({}^{\circ}\)C and 115 \({}^{\circ}\)C if the molecular volume V is known. Polarity of the substances seems to play a minor role on the correlations in Figure 4. The polarity and the functional groups of the molecules influences the partition coefficient K between the polymer and the gas phase. This is discussed in previous studies [24; 26]. It is interesting to note that the very low molecular weight alcohols like methanol and ethanol have molecular volumes V which are smaller or near the volume of parameter \(c\) for GPPS. According to Equation (3), molecules with a molecular volume of parameter \(c\) result in an activation energy of diffusion of 0 kJ/mol and the diffusion coefficient is equivalent to factor \(b\). For methanol, an activation energy of diffusion of 4.9 kJ/mol was determined which is within the analytical uncertainties in good agreement with Equation (3).