Fundamentals
Food packaging or drinking water contact materials consists of one or multiple layers of different materials, e. g. polymers or paper. The system of all these layers and the food simulant or drinking water builds the matrix. The migration of a substance from a matrix into food simulant or drinking water is controlled
 by the diffusion of the substance through the matrix and
 by the solubility / partitioning of the substance in the matrix and the food simulant or drinking water.
If the diffusion is according to Fick's second law of diffusion, the migration can be simulated by solving the diffusion equation. Therefore a diffusion coefficient at the current temperature is needed for each layer. The diffusion coefficient and the partitioning coefficient are hard to determine experimentally for each substance, so they need to be estimated by a given method.
Estimation of diffusion coefficients
Diffusion coefficients of migrants in polymers can be estimated based on established estimation procedures or can be a user defined numerical input based on existing knowledge. Well known estimation procedures for diffusion coefficients of migrants in polymers were published by Pringer (2005), Welle (2013) or Brandsch (2017) (nonexhaustive list).
Piringer estimation method
The estimation procedures for diffusion coefficients of migrants in polymers according to Piringer is based on the parameters:
 polymer specific constant of polymer (Ap),
 molecular mass of migrant, and
 temperature.
Polymer specific constants of polymers are determined experimentally from time, temperature and migrant dependent migration kinetics. To some extend experimental results for a single combination of time, temperature and migrant were used as well. For each polymer type (e.g. LDPE, HDPE, PP, PS, PET, etc.) a whole set of experimentally determined Ap values exists. The distribution of these values is assumed to be log normal from which a mean value (= "best case") and a right side 95percentile value (= "upper limit") for the data set results. It is assumed that the "best case" value will generate a migration result which is close to an experimental migration result determined under the same conditions. The "upperlimit" value will generate a migration result which is in \(95\,\%\) of all cases above an experimental migration result determined under the same conditions.
If the \(A_p\) value for the polymer is not known, the \(A_p\) value can be derived from the glass transition temperature of the polymer ^{1}.
Welle estimation method
The estimation procedures for diffusion coefficients of migrants in polymers according to Welle is based on the parameters:
 polymer specific parameter set a, b, c, and d,
 molecular volume of migrant, and
 temperature.
Polymer specific parameter sets of polymers are determined experimentally from time, temperature, and migrant dependent migration kinetics. For each polymer type (e.g. PET, PS, HIPS, ABS, etc.) one experimentally determined parameter set a, b, c, and d is considered to represent the "best case" which will generate a migration result which is close to an experimental migration result determined under the same conditions. Reducing the molecular volume of the migrant by \(20\,\%\) will generate an "upper limit" diffusion coefficient. It is expected that the "upperlimit" value will generate a migration result which is in \(95\,\%\) of all cases above an experimental migration result determined under the same conditions.
Brandsch estimation method
The estimation procedures for diffusion coefficients of migrants in polymers according to Brandsch is an "insilico" approach based on the parameters:
 glasstransition temperature, density, and molar volume of polymer
 melting point, density, and molar volume of migrant, and
 temperature.
All parameters are most likely available from literature or can be determined experimentally in case of need for specific cases. Based on the parameters from literature the Brandsch estimation procedure will generate a theoretic diffusion coefficient which so far is of no practical relevance. Under consideration of a minimum migrant density of \(700\,\frac{kg}{m^3}\) for all migrants and a minimum value \(R \cdot (T_{g,P} + T_{m,S})\) of \(4100\,\frac{J}{mol}\) for all polymer migrant combinations an estimated best value (unwisely denoted as "upper limit" diffusion coefficients in the original publication) result. Setup of an "upperlimit" concept to fulfill the legal requirement of always overestimating an experimental migration result determined under the same conditions is in preparation.
For a comparision of the estimation methods see case study: diffusion coefficients estimation methods
Initial mass fraction
Beside the properties of the matrix, the initial mass fraction of the migrant(s) are needed for each layer. In the model, at the beginning the migrant is homogenously distributed in the layer. The initial mass fraction can be taken from the formulation of the polymer or, if needed, by a measurement in a laboratory.
Determination of the residual amount of a peroxide used as polymerisation initiator
If the migration of a peroxide needs to be modelled, it is hard to determine the residual amount of the peroxide after the production process of the polymer from the used amount. In case the peroxide is a polymerisation initiator the decay kinetics (download file) can be computed.
Reproduction of the examples of the modelling guideline
The Guideline for the Mathematical Estimate of the Migration of Individual Substances from Organic Material in Drinking Water (Modelling Guidline, 7 October 2008) of the German Umweltbundesamt (Federal Environment Agency) is currently revised. The draft of the new guideline contains a set of examples. These examples can be reproduced by migraSIM. A summary can be downloaded here (download file).

R. Brandsch. Probabilistic migration modelling focused on functional barrier efficiency and low migration concepts in support of risk assessment. Food Additives & Contaminants: Part A, 34:10, 17431766; (2017);, 2017. doi:10.1080/19440049.2017.1339235. ↩