A new method for the prediction of diffusion coefficients in poly(ethylene terephthalate)—Validation data

Prediction of the migration is a useful tool in compliance evaluation of food contact materials. In our previous work, such a prediction model had been established for polyethylene terephthalate (PET), which is widely used as packaging material for beverages as well as for meat and cheese. Within the actual study, 263 diffusion coefficients in PET for 66 substances at temperatures between 40°C and 120°C were determined from permeation kinetic experiments. The diffusion coefficients DP were compared with the predicted values by use of a log/log plot as well as in direct comparison of the diffusion coefficients. When applying the migration prediction model for compliance evaluation of food packaging materials, it is mandatory that the migration prediction is over‐estimative in any case. As a result, the predicted values are in good agreement with the experimental results. The prediction model slightly over‐estimates the real migration by a factor of 1.3 in average. Reduction of the molecular volume of 20% results in an average over‐estimation of 3 of the migration (worst‐case). Another finding of this study is that the diffusion behaviour at the glass transition temperature not significantly change. The prediction model is applicable below and above the glass transition temperature.

Prediction of the migration is a useful tool in compliance evaluation of food contact materials. In our previous work, such a prediction model had been established for polyethylene terephthalate (PET), which is widely used as packaging material for beverages as well as for meat and cheese. Within the actual study, 263 diffusion coefficients in PET for 66 substances at temperatures between 40 C and 120 C were determined from permeation kinetic experiments. The diffusion coefficients D P were compared with the predicted values by use of a log/log plot as well as in direct comparison of the diffusion coefficients. When applying the migration prediction model for compliance evaluation of food packaging materials, it is mandatory that the migration prediction is over-estimative in any case. As a result, the predicted values are in good agreement with the experimental results. The prediction model slightly overestimates the real migration by a factor of 1.3 in average. Reduction of the molecular volume of 20% results in an average over-estimation of 3 of the migration (worstcase). Another finding of this study is that the diffusion behaviour at the glass transition temperature not significantly change. The prediction model is applicable below and above the glass transition temperature.

K E Y W O R D S
diffusion coefficients, diffusion modelling, permeation kinetics, polyethylene terephthalate

| INTRODUCTION
The prediction of the migration is a useful tool in compliance evaluation of food contact materials, especially for packaging polymers with a low diffusion. For low diffusive polymers, the migration process is very time consuming and result at the end in low concentrations in food that challenges the analytical approaches. Increasing the temperature is one of the approaches to increase the speed of the mass transfer. However, higher temperatures like 60 C are far away from realistic storage conditions, which is in most cases room temperature.
One of these low-diffusive polymers for which migration prediction is important in food law compliance evaluation is polyethylene terephthalate PET. PET is mostly used for beverage packaging, but also for trays for meat and cheese.
The most important factors influencing the mass transfer from the packaging material into food is the concentration of the migrant in the polymer, the storage time, the storage temperature, the diffusion coefficients D P , and the partition coefficient K P/F . Other factors are the surface volume ratio and the film thickness of the packaging material. Most of the abovementioned factors are known or analytically available. Only the diffusion coefficients D P and the partition coefficients K P/F are rarely available from the scientific literature. However, for low-diffusive polymers like PET, the partition coefficients are negligible because the equilibrium between the polymer and the food will not be reached under normal storage conditions of packed foods. Therefore, the diffusion coefficient D P is the most important factor in the prediction of the migration for low-diffusive polymers like PET.
Predictive models for the migration from food packaging materials have been developed with the last 25 years. 1-5 A comprehensive review on the different approaches is given in the scientific literature. 6 These prediction models should be over-estimative to show the food regulatory compliance with a safety factor. 7 Regulation 10/2011 stated that "To screen for specific migration the migration potential can be calculated on the residual content of the substance in the material or article applying generally recognized diffusion models based on scientific evidence that are constructed in a way that must never underestimate real migration levels." 8 However, the prediction should be not too over-estimative; otherwise, compliance with current food law cannot be shown in some cases. Therefore, realistic but still over-estimative diffusion coefficients D P should be available or should be predicted.
In our previous work, a prediction model for the diffusion coefficients in PET was developed. 9 This prediction model is based on the molecular volume V of the substances and the temperature in Kelvin (Equation 1). The prediction model was derived from two correlations: (i) a correlation between the activation energy of diffusion E A and the molecular volume V of the substance and (ii) a correlation between the activation energy E A and the pre-exponential factor D 0 of the Arrhenius equation. 9 The activation energy of diffusion E A and the pre-exponential factor D 0 were experimentally determined mainly from desorption kinetics of spiked PET sheets into the gas phase. 10 The parameters a and b are the slope and the intercept of the correlation between the activation energy E A and the pre-exponential factor D 0 . The parameters c and d are the intercept and the slope of the correlation between the activation energy E A and the molecular volume V. The parameters a to d for the prediction of diffusion coefficients D P in PET are given Table 1.
Due to the fact that PET is a low-diffusive polymer, high temperatures have to be applied for the desorption kinetics in our previous study. Therefore, most of the activation energies of diffusion E A were derived from diffusion coefficients D P above the glass transition temperature T g.
10 Also the activation energies of an alternative prediction model are determined mainly above T g. 3 On the other hand, the low diffusion of PET is also responsible that at temperatures below the glass transition temperature only the diffusion coefficients for small molecules are available. 11

| RESULTS
Within this study, a permeation method was applied to determine the diffusion coefficients D P for several substances in PET. From the experimental permeation data, the diffusion coefficients D P were derived from the so-called lag time according to Equation 2, where l is the thickness of the PET film. This method was applied in previous studies for the determination diffusion coefficients in PET, 14 polyamide PA6, 15 polyethylene naphthalate PEN, 16 and general purpose polystyrene GPPS. 17 However, in these studies, the permeants were limited to homologous rows of n-alkanes and 1-alcohols. In the actual study, we determined also other substances than n-alkanes and 1-alcohols towards their permeation through a thin PET film and derived diffusion coefficients for a broad range of substances and functional groups.
The permeants used within this study are summarized in Table 2.
The substances cover various molecular volumes V, functional groups, and therefore also polarities. Overall, 66 substances were tested at 14 temperatures between 40 C and 120 C. In addition, the diffusion coefficients were determined also at different gas phase with the parameters given in Table 1. This comparison is visualized in Figure 1 for each of the tested temperatures between 40 C and T A B L E 1 Parameters for the prediction of diffusion coefficients in PET from Equation 1 9 Parameter Value    are not available or cannot be handled because they are too thin. The molecular weight range of the permeants was tried to expand the molecular weight range as much as possible. However, the bounders are very narrow. Therefore, diffusion coefficients below T g are only available for low molecular weight substances.
It is important to note that all diffusion coefficients D P are derived from permeation kinetics into the gas phase. Interactions between the PET polymer and food (simulants) like swelling are therefore excluded.
The diffusion coefficients derived from this method can be considered as the pure diffusion coefficients in PET. Swelling effects of food simulants on the PET surface are well-known especially at high temperatures of 60 C, which is mostly used for PET beverage bottle testing. 12,18,19 These swelling effects significantly increases the migration into the simulants and lead in some cases to migration levels, that exceed the specific migration limits. On the other hand, real foods did not significantly swell PET and lead to much lower migration limits.
Diffusion modelling is therefore more suitable to realistically predict the migration into food at the end of shelf life compared to experimental migration tests using high ethanolic food simulants. 19 It is important to note, that also moisture can swell the PET polymer especially at high temperatures. In order to reduce these swelling effects, the diffusion coefficients in this study were determined at virtually zero levels. Therefore, the pure diffusion coefficients in the PET polymer were determined without any swelling effects. Regarding real application, foods in contact with PET include high moisture conditions, but only at low temperatures. At low temperatures swelling is negligible for moisture but are significant for high ethanolic simulants. 12,18,19 As mentioned above, most of the diffusion coefficients were determined above the glass transition temperature T g of 81 C. At the glass transition temperature T g , the diffusion behaviour might change.
In order to investigate this effect, the diffusion coefficients derived from this study were compared with previous studies. For four permeants, diffusion coefficients were available in the scientific literature from desorption experiments at temperatures between 120 C and 180 C 20 and from migration kinetics into mineral water and 10% ethanol at temperatures between 23 C and 50 C. 12,13 The permeation experiments of this study are in between the temperature intervals.
The results are visualized in Figure 3 with the correlation of the reciprocal temperature (in Kelvin) and the logarithm of the diffusion coefficient D P (Arrhenius plot). The results for these four substances show, that at the glass transition temperature T g no significant change of the diffusion behaviour of PET was detectable. In addition, diffusion coefficients predicted from three independent methods (desorption kinetics, 20 migration kinetics 12,13 and permeation kinetics [this study] for the determination of diffusion coefficients are in good agreement with each other.

| PET film sample and model chemicals
For the permeation tests, a commercial biaxially oriented PET film was used. The thickness was determined to 11.9 ± 0.1 μm. The glass transition temperature of the investigated PET film was determined by F I G U R E 2 Comparison between the predicted diffusion coefficients and the experimental diffusion coefficients (log/log plot). Dotted lines 95% confidence interval, solid dots: diffusion coefficients below T g , open dots, diffusion coefficients above T g differential scanning calorimetry (DSC) to 81 C. The melting range was also determined by DSC to 243-260 C (peak at 255 C). The permeation was tested with 66 different substance ( Table 2). The substances were purchased with a purity of 99% and used without further purification. The concentration applied in the lower space of the permeation cell are given in the supporting information together with the corresponding diffusion coefficients.

| Molecular volume
The molecular volume V of the molecules was calculated with the free internet program molinspiration. 21

| CONCLUSIONS
The modelling parameters derived from our previous study 9 were validated with 263 diffusion coefficients from 66 different organic Correlation between the diffusion coefficients and reciprocal temperature (Arrhenius plots) for (a) benzene, (b) toluene, (c) chlorobenzene, and (d) tetrahydrofuran. Desorption data from Ewender and Welle, 20 migration data into water and 10% ethanol from Franz and Welle 12 and Welle and Franz, 13 and permeation data from this study substances with various functional groups, volatility, and polarity. The parameters given in Table 1 in combination with Equation 1 predict the diffusion coefficients D P of organic substances in PET with a slight average over-estimative factor of 2.6. When using the molecular volume V of the substances minus 20% the modelling parameters are in any case over-estimative which is mandatory for compliance evaluation of food packaging materials according to European Regulation 10/2011. This worst-case prediction results in an average overestimative factor of 9.2 in the diffusion coefficients. Increasing the diffusion coefficient D P by a one order of magnitude (factor 10) results in an increase of the migration by a factor of 3.16 (square root of 10). An average over-estimation factor of 9.2 results in a factor of 3.0 higher migration, which seems to be suitable for food law compliance evaluation purposes. The average over-estimation without virtual volume reduction is 1.6, which results in an over-estimation of the migration compared to the predicted migration of only 1.3. The parameters in Table 1 therefore realistically describe the migration from PET. Such a realistic migration prediction is useful for the evaluation of non-intentionally added substances [22][23][24] as well as in the evaluation of post-consumer PET recyclates in direct food contact applications. 25 Another important finding of this study is, that the diffusion behaviour at the glass transition temperature do not significantly change. Even if the most of the diffusion coefficients were derived above the glass transition temperature the diffusion coefficients D P can be predicted below the glass transition temperature. On the other hand, assuming that there is a slight change, the diffusion behaviour will be probably higher above the glass transition temperature as below. Therefore, the modelling parameters given in Table 1 can be considered as worst case for temperatures below the glass transition temperature T g of PET.