Introduction
Polymeric materials (Plastics, Coatings, Rubbers, etc.) transfer their components (low molecular weight substances like monomers, additives, etc.) to contacting media to more or less extent, due to thermodynamic driving forces. This phenomenon is called mass transfer and the common wording is migration and/or emission depending on the type of contacting medium, i.e. liquid or gaseous. Due to transfer of substances (chemicals) from the polymeric material to the contact medium the substance amount in the polymeric material decreases and a concentration profile is established. The amount of substance is depleted first at the material/medium interface. The release of the substances from polymeric materials to contacting media obeys in most cases the law of diffusion because the diffusion process is the rate determining step. In case of gaseous contact media the evaporation process may become under certain circumstances the rate determining step. It is obvious that mass transfer can occur as well from the contacting medium to the polymeric material.
The release of substances from the polymeric material was historically assessed by experimental testing under conventional test conditions. Due to advanced understanding of the mass transfer processes and translation into science based computational tools simulation became the method of choice.

Simulation & diffusion models
The complex migration process (mass transfer) is reduced to the rate determining step, i.e. the diffusion process. Correspondingly migration modelling is based on diffusion models. The migration process may be simulated according to real use or according to conventional test conditions. Modelling migration according to conventional test conditions makes a one to one comparison between simulation and experimental results possible
From mathematical point of view the system polymeric material in contact with a medium is

Figure 1 Law of diffusion

To simplify the mathematical problem it is assumed that the polymeric material and its individual layers as well as the contacting medium are ordered parallel to each other. The diffusion equation is a partial differential equation which can be solved with recognised numerical algorithms. For more details regarding the numerical algorithms employed the reader is referred to the help menu of the SML software.
With the numerical solution of the diffusion equation the migration kinetic, i.e. migration of substances from the polymeric material to the contacting medium with time and the concentration profile of the substance in the system can be calculated.
One has to distinguish between monolayer and multilayer materials. If monolayer materials are in contact with a liquid medium the system is described by two mass transfer constants, the diffusion coefficient of the substance in the polymeric material and the partition coefficient of the substance between polymeric material and contacting medium.
If multilayer materials with n layers are in contact with a liquid medium, all mass transfer constants describing the system must be considered, i.e. n diffusion coefficients of the substance in each layer
As long as the contacting medium is a liquid or a gas the diffusion process in the medium is neglected because the diffusion rates are much lower compared to solids like polymeric materials. If the contacting medium is a solid like some foods or another polymeric material, the diffusion process in the medium must be considered accordingly.

Figure 2 Symbols for diffusion models (n - number of layers)

The solution of the diffusion equation requires variables for the simulation of migration kinetics and concentration profiles:

• geometry related variables (thicknesses, contact area, volumes) as well as time and temperature, e.g. according to real use or conventional test conditions
• initial or residual concentration of the migrant in each layer including the contacting medium) e.g. residual amount of monomers or initial concentration of additives.
• mass transfer constants (diffusion- and partition coefficients). These are not available from the literature and must be estimated according to generally recognised and validated estimation procedures based on scientific evidence.

A model is valid if it describes precisely enough the behaviour of the real system. The comparison between time dependent experimental migration data and simulated data is suitable for the validation of diffusion models. This procedure was chosen in the frame of several EU-projects as well as many publications in the scientific literature. It was shown that migration processes in the system polymeric material in contact with a medium can be well described by the solution of the diffusion equation.

Estimation of mass transfer coefficients
Diffusion coefficients
The diffusion coefficient is a time related mass transfer constant which specifies how fast a substance is released from a polymeric material to a contacting medium by diffusion.

Figure 3 Diffusion coefficient (Arrhenius)

partition coefficients
The partition coefficient is a thermodynamic mass transfer constant which considers the equilibrium concentrations of a substance in the polymeric material and the contacting medium. The partition coefficient defines the maximum amount of substance which can be transferred from the polymeric material to the contacting medium.

Figure 4 Partition coefficient

For the estimation of diffusion- and partition coefficients there are several scientific approaches:

Figure 5 Estimation procedures for estimation of mass transfer constants

For the estimation procedures of diffusion coefficients according to Arrhenius and Piringer experimental testing is required to determine material specific parameters like the pre-exponential factor, ; the activation energy, ; the polymer specific constant, and the polymer specific temperature constant, tau (corresponds to the deviation from the reference activation energy, with R the gas constant). The influence of the migrant size to the diffusion rate may be considered by its molecular weight. The -value concept of Piringer was validate in the frame of an EU-project, refined in further EU activities and validated for migration modelling from plastics into food simulants (polymeric material in contact with liquids).

The estimation of diffusion coefficients according to Brandsch is an ab initio technique based on well known thermodynamic parameters like the glass temperature of polymeric materials. Validation of the new procedure is possible against the existing -value approach as well as against experimental data.

The estimation of partition coefficients is less advanced. Nevertheless some scientific approaches exist based on a group contribution method developed by Piringer or an empirical approach developed by Brandsch in the frame of the Foodmigrosure EU-project. The empirical approach by Brandsch considers the polarity of the migrant by its octanol/water partition coefficients and sets it in relation to the partition coefficient between polymeric materials and real foods or food simulants. Knowledge of the water solubility may allow for the estimation of worst case partition coefficients between polymeric materials and aqueous media.

-value concept (Piringer)
The estimation procedure for diffusion coefficients according to Piringer, the so called -value concept, requires the experimental determination of the polymer specific constant, -value and the polymer specific temperature constant, tau once for each polymer type. For each polymeric material a minimum of time dependent migration experiments must be run at different temperatures. From the experimental data diffusion coefficients are determined. These are translated into -values according to the relationship developed by Piringer. From all available -values an upper limit value for one polymeric material is calculated as the 95-percentile of that data set. Provided the data set available is big enough, the -value for that polymeric material is considered to be validated.
Experimental time dependent migration data, diffusion coefficients and -values derived there from are collected and evaluated by the Modelling Task Force at the Joint Research Centre of the EU-Commission.
For most of the polymeric materials used in food contact materials upper limit *-values and mean tau-values are listed in the Practical Guide supporting the Plastics Directive 2002/72/EC of the EU-Commission. Upper limit *-values result in overestimated migration values in support of consumer safety.

Overestimation and consumer protection
It is in support of consumer safety to develop procedures for estimation of mass transfer constants which systematically overestimate the migration behaviour of real polymeric materials. Due to systematic overestimation the risk of underestimation compared to the real system is minimized.
Due to simulation of one single migration cycle it is easy for food contact materials to implement overestimation. The use of upper limit -values in the estimation procedure of diffusion coefficients and lower limit partition coefficients results in estimated migration value which are higher compared to the real ones.

Repeated use
Polymeric materials with repeated use or dynamic flow behaviour of the contacting medium need specific considerations. The experimental conventional test conditions as well as migration simulations account for the repeated use or dynamic flow behaviour by implementation of several migration test cycles. Overestimating the migration in the first migration cycle may cause underestimation in later migration cycles. This can happen only if the overestimation based on diffusion and partitioning is very high and more than 50% of the substance migrates out of the polymeric material in the first migration cycle. In these special cases, which can be easily identified during simulation, only the first migration cycle can be used for comparison with a specific migration limit.