Safe Design of Packaging with SFPPy

from Virgin and Recycled Materials

Olivier Vitrac, PhD | Generative Simulation Initiative 🌱🌿🌳🍃🍂

Abstract

This document introduces a tiered framework (M0–M3) for the safe design of food, cosmetic, and pharmaceutical packaging. The framework progresses from simple conservative balances to complex scenarios integrating diffusion, functional barriers, and recycled materials. It relies on: (i) explicit indicators (severity, criticity, release potential); (ii) constraint-based optimization rules; (iii) open databases (PubChem, ToxTree); and (iv) Python tools (SFPPy) to automate calculations and scenarios, including coupling with generative AI (RAG) for reasoning and traceability. The goal is design-for-compliance: build intrinsically safe, conformant packaging upstream, not only verify ex-post.

See the overview of tiers in Section 6, thermodynamic principles in Section 3, and SFPPy examples in Section 9.

This document is released "AS IS".
The content of this article is intended to be published in French and English in several peer-reviewed journals..

Read this 📄file in PDF with this link | Access all 📂files with this link

Note

A glossary of key terms and expressions, as well as a table of acronyms, notations, and symbols used throughout the article, is provided at the end.

Introduction

Motivation

The safe design of food packaging consists in integrating health and safety constraints from the very beginning. The objective is to ensure that migrating substances from materials remain below regulatory toxicological thresholds, regardless of the complexity of structures or the origin of materials, including recycled ones. This proactive approach is decisive, as it steers formulation and design before market release, in a context where European, American, and Chinese regulations strictly frame compliance requirements.

The proposed method rests on a clear doctrine: translate the safety problem into explainable and verifiable design constraints that can be tested by humans, AI agents and both. It is structured around four complementary principles.

Formalization of the regulatory constraint, linking the specific migration of each substance to a toxicological reference value.
Conservative modeling of migration: a generic thermodynamic model provides robust upper bounds, applicable to any material and any contact condition.
Introduction of risk indicators, condensing information on substances and components into severities (individual) and criticities (cumulative), facilitating ranking, optimization, and justification of technical choices.
A tiered approximation method: a system of “tiers” ranging from the most conservative scenario to refined evaluations by substance and material.

This approach is designed to be operational at multiple scales. It can be implemented simply in a spreadsheet for punctual evaluations, but it is also compatible with automated environments and reasoning chains driven by Artificial Intelligence (AI). It is integrated in the open-source tools FMECAengine (Vitrac 2025) and SFPPy / SFPPyLite (Vitrac, 2025a; Vitrac, 2025b). These tools utilize PubChem (Kim et al., 2024) as a molecular structure engine and ToxTree (Patlewicz et al., 2008) for toxicological classification in a fully automated manner. This dual usage — manual or automated — is deliberate: it aims to foster transparency and auditability, both essential in a regulated domain, while anticipating the industrialization of repetitive calculations and multicriteria explorations, beyond the substances explicitly listed in regulations. In this sense, this article illustrates a philosophy of “design for compliance”: the objective is no longer to demonstrate a posteriori that a packaging is safe, but to build intrinsically compliant and safe solutions from the outset.

This article focuses on the objective construction of health safety indicators and their integration in a design logic (optimization of components, formulation, and uses), treated as an engineering mathematical problem. Readers interested in the physics of transfers and their resolution techniques are referred to Zhu et al.( 2019). Finally, the approaches presented here constitute an important update compared to the last European guide (Brandsch et al., 2015), to which one of us contributed.

Key takeaways

Because this is a design approach, it takes place before experimental testing. The proposed indicators can accommodate calculated or partial data (composition, decontamination rates, intermediate migration results). They cover:

All materials: while tiers 2 and 3 are particularly suited to thermoplastics and elastomers, an extension is foreseen for paper and board.
Virgin and recycled materials, without distinction of recycling technology.
Single-use and repeated-use contexts, accounting for forming steps that modify substance distribution.
All sensitive contact applications: primarily food, but also pharmaceuticals, cosmetics, medical devices, and household products.

In contrast, this article does not provide a ranking of materials or recommendations on processes/materials. The application of indicators depends on the intended use, without privileging virgin versus recycled plastics or paper. The indicators are generic: they can be combined, if necessary, with other criteria such as economic costs or environmental impacts.

Tip

Readers wishing to explore the concepts while maintaining control over calculation steps may use the SFPP3 platform (Vitrac, 2025c), which relies on the same computational engine (Nguyen et al., 2013) as SFPPy / SFPPyLite. Prerequisites for a good understanding of this article can be acquired through the FitNESS platform (Vitrac et al., 2024).

1 | General Problem

“It would be ideal if the migration of each additive and monomer into the packed foodstuff could be determined when the package has been filled and stored under normal conditions of practice. This would ensure that no physiologically objectionable plastics material would be admitted and, on the other hand, that no suitable plastics material would be rejected because of a hypercritical assessment.”
— Figge (1980) (Figge 1980)

1.1 | Its evolution

Designing safe packaging remains a challenge because the distinction between hazard and risk is not always intuitive for designers. Hazard refers to the intrinsic toxicity of substances, whereas risk depends on the quantities actually ingested by the consumer. Two emblematic publications, published more than half a century apart in the prestigious journals Science and Nature, illustrate the persistence of this problem: Jaeger and Rubin (1970), which highlights migration of major constituents such as monomers and plasticizers, and Monclús et al. (2025), which reveals the migration of hundreds of minor substances.

The difficulty is therefore not removed but shifted: on the one hand, from controlling a few well-characterized additives to assessing multiple contaminants with uncertain profiles; on the other, from an analytical approach focused on known substances to an integrated approach capable of setting priorities and ranking risks.

1.2 | The European regulatory perspective

In Europe, assessment rests on a clear distinction between hazard and risk. Substances are assessed individually when their structures differ sufficiently. For homogeneous families (isomers, substituted molecules), they are grouped by aggregation. This logic extends to substances sharing a common toxicological mode of action such as phthalates or benzophenones.

Certain effects, however, are excluded from the regulatory scope. Synergistic effects, often called “cocktail effects,” are not considered. Cumulative exposures to genotoxic or carcinogenic substances are likewise not considered. Thus, the presence of several carcinogens at low levels may be deemed acceptable as long as none individually exceeds its alert threshold.

Note

These rules may evolve, but they do not challenge the safe design method presented in this article. The core of the approach—translating the toxicological constraint into a design constraint—remains valid regardless of the criteria adopted.

1.3 | Exposure calculation in Europe and the “safety condition”

conservative, simplified hypothesis $DI=1\,\text{kg of food}\cdot\text{day}^{-1}$ $BW=60\,\text{kg}$ $TDI$ $\text{mg}\cdot\text{kg body weight}^{-1}\cdot\text{day}^{-1}$ maximum allowable concentration in food $SML$ ).

$i$ :

\begin{matrix} (SML) & S M L_{i} = B W \times D I \times T D I_{i} \end{matrix}

Note

$SML_i$ corresponds to the maximum acceptable exposure for an adult consuming 1 kg of food daily packaged in the same packaging, excluding other sources of exposure.

safe $i$ $C_{i,F}$ results in an exposure below the maximum acceptable exposure:

\begin{matrix} (Safety) & \frac{C_{i, F}}{ρ_{F}} \leq S M L_{i} \end{matrix}

$\rho_F$ is the apparent density of the food, i.e., the mass of food occupying the food volume. For a granular product such as breakfast cereal, this is the mass of food in the bag divided by the internal volume of the bag.

This approach ignores cumulative exposure but requires aggregation for substances with a common toxicological targetrecycled material $\ref{eq:SafetyCondition}$ $i$ it contains. These substances may be identified and quantified, only identified, or unknown.

Note

$C_{i,F}$ volumetric concentration $\text{kg}\cdot\text{m}^{-3}$ ) consistent with the physical description of transfers: contaminants fill a volume, not a mass. This choice is also consistent with the design problem, in which volumes are known via the geometry of layers and components.
Conversely, exposure limitsingested mass of food $\tfrac{C_{i,F}}{\rho_F}$ mass concentration $\text{kg}\cdot\text{kg}^{-1}$ ).

1.4 | Toxicological reference values

When a substance appears on a positive listregulatory value $SML_i$ as a priority $TDI$ value is published, it should also be preferred over other estimates.

$TDI$ is used in a generic sense to denote any toxicological reference value. For certain substances, notably those present in recycled materials, other notions may apply, such as the threshold of toxicological concern (TTC).

TTC thresholds (Munro et al., 2008) to be applied to non-evaluated substances—and thus usable for unknown substances—were set by the European Food Safety Authority (EFSA) (EFSA, 2016) and are recalled in Table 1.

Table 1 — TTC thresholds by toxicological class (EFSA) (Source: EFSA, 2016)

Toxicological class	TTC $\mu\text{g}\cdot\text{kg}^{-1}\cdot\text{bw}^{-1}\cdot\text{day}^{-1}$ )
Possible genotoxicity (conservative)	0.0025
Cramer class III (high concern)	1.5
Cramer class II (intermediate toxicity)	9
Cramer class I (low toxicity)	30

Tip

TTC values can be predicted for arbitrary molecules using ToxTree (Patlewicz 2008) or its ports in open-source projects such as SFPPy (Vitrac 2025).

$BW$ $DI$ 2 $SML_i$ values can be adjusted for packaging intended for sensitive populations.

Table 2 — Recommended BW and DI values by age group

Age group	$BW$ (kg)	$DI$ $^{-1}$ )	Source
Newborns (0–16 weeks)	5	0.75	EFSA 2017
Infants (0–6 months)	5–6	0.75–1.0	EFSA 2017
Young children (1–3 years)	12	1.0	EFSA 2012
Children (4–9 years)	20	1.0–1.5	EFSA 2012
Adults (≥ 15 years)	60–70	1.0	EU 10/2011

1.5 | Equivalent mathematical problem

$C_{i,F}$ for hundreds of substances and several componentsover-estimator $\hat{C}_{i,F}$ satisfies two properties:

(i) Robust overestimation property:

\begin{matrix} (Equiv-1) & \begin{matrix} \begin{aligned} {\hat{C}}_{i, F} & \geq C_{i, F} \\ \Pr ({\hat{C}}_{i, F} > ρ_{F} \cdot S M L_{i}) & \leq \Pr (C_{i, F} > ρ_{F} \cdot S M L_{i}) \end{aligned} \end{matrix} \end{matrix}

An explainable and fast calculation $\hat{C}_{i,F}$ .

$\ref{eq:PropMathEquiv1}$ $\tfrac{\hat{C}_{i,F}}{\rho_F}\le SML_i$ translates into actual safetyreal, experimentally verifiable concentration $\tfrac{C_{i,F}}{\rho_F}\le SML_i$ $\ref{eq:PropMathEquiv1}$ bounds on the initial concentrations $C_{i,j}^0$ $j=1,\dots,m$ $j=0$ ):

\begin{matrix} (1) & C_{i, j}^{0} \leq {\hat{C}}_{i, j}^{0} \end{matrix}

$\hat{C}_{i,j}^0$ maximum acceptable concentration $j$ $i$ $\hat{C}_{i,F}=SML_i$ all possible sources $\mathbf{C}_i^0=(C_{i,1}^0,\dots,C_{i,m}^0)$ to an admissible polytope defined by these inequalities.

Note

Initial concentrations $C_{i,j}^0$ volumetric concentrations $\text{kg}\cdot\text{m}^{-3}$ mass concentrations $\tfrac{C_{i,j}^0}{\rho_j}$ $\rho_j$ density $j$ .
For porous materials $\rho_j$ is an apparent density. For paperboard, it is defined by the basis weight divided by the average thickness.

1.6 | A tiered approach

$\ref{eq:PropMathEquiv1}$ can be satisfied at several levels of approximation that balance uncertainty, compute time, and explainability. We structure these approximations into tiers along two independent axes:

Toxicology axis (T) $SML_i$ $T0$ worst case $0.0025\,\mu\text{g}\cdot\text{kg}^{-1}\cdot\text{bw}^{-1}\cdot\text{day}^{-1}$ $T1, T2, \dots$ use TTC by Cramer classespublished TDI/SML $T$ , the more informedless conservative $SML_i$ becomes.
Transfer axis (M) $\hat{C}_{i,F}$ $M0$ total transfer $\bar v_i^*\!\to 1$ $M1$ unit partitioning $K\simeq 1$ $M2$ realistic partitions $M3$ diffusion dynamics $M$ , the more realistic and thus less conservative the estimate.

$T \times M$ decomposition, reproduced in Table 3, enables tailoring the effort: $T$ when toxicology progresses, or $M$ when material/use data become available—without changing the decision logic.

Table 3 — Tier matrix for design (𝐓×𝐌)

	T0 — worst case — genotoxic TTC	T1 — TTC — Cramer classes	T2 — published TDI/SML
M0 — total transfer	$(T0,M0)$	$(T1,M0)$	$(T2,M0)$
M1 — unit partition	$(T0,M1)$	$(T1,M1)$	$(T2,M1)$
M2 — realistic partitions (solubility, polarity, crystallinity)	$(T0,M2)$	$(T1,M2)$	$(T2,M2)$
M3 — dynamic diffusion (time, temperature)	$(T0,M3)$	$(T1,M3)$	$(T2,M3)$

$T$ $M$ $(T,M)$ $\hat{C}_{i,j}^0$ .

Estimators derived from the M-tiers must be constructed to satisfy monotonicity properties on concentration scales in food and in materials:

\begin{matrix} (Monotony) & \begin{aligned} {{\hat{C}}_{i, F}}_{M = 0} & \geq {{\hat{C}}_{i, F}}_{M = 1} \geq {{\hat{C}}_{i, F}}_{M = 2} \geq {{\hat{C}}_{i, F}}_{M = 3} \geq C_{i, F}, \\ {{\hat{C}}_{i, j}^{0}}_{M = 0} & \leq {{\hat{C}}_{i, j}^{0}}_{M = 1} \leq {{\hat{C}}_{i, j}^{0}}_{M = 2} \leq {{\hat{C}}_{i, j}^{0}}_{M = 3} \leq C_{i, j}^{0} \end{aligned} \end{matrix}

Note

each component can be a source $\hat{C}_{i,F}$ adds these contributions conservatively, including when no-contact steps redistribute masses before contact.
For unknown substancessubstitution rules $DL_{i,j}$ $(1-\Delta_{i,j})$ , or dimensioning molecules (e.g., toluene) depending on the chosen tier.

1.7 | Operational formulation of the design problem

given tier pair $(T,M)$ $\mathbf{C}_i^0\in\mathbb{R}_+^m$ $\hat{C}_{i,F}^{(T,M)}(\mathbf{C}_i^0)\ \le\ \rho_F\cdot SML_i^{(T)}$ $i$ . In practice, we prefer to compute per-component thresholds via the sizing equality

\begin{matrix} (FormulationProblem) & \begin{matrix} \begin{aligned} {\hat{C}}_{i, F}^{(T, M)} ({\hat{C}}_{i}^{0}) & = ρ_{F} \cdot S M L_{i}^{(T)} \\ with C_{i}^{0} & = (C_{i, 1}^{0}, \dots, C_{i, m}^{0}) \end{aligned} \end{matrix} \end{matrix}

simple constraints $C_{i,j}^0\le \hat{C}_{i,j}^0$ usable in a spreadsheet, in a formulation database, or in a tooled calculation chain (scripts, RAG).

Note

script $(T,M)$ and applies them to a large number of substances and components (e.g., in Python or Matlab).
RAG (Retrieval-Augmented Generation) denotes an AI approach that combines knowledge-base retrieval with the automatic generation of scripts or explanations, used here to document and secure sizing calculations. Finally, the article is written so that it can be directly exploited by large language models (LLMs).

Key takeaways for Section 1

$C_{i,F}$ explainable over-estimator $\hat{C}_{i,F}$ design bounds $C_{i,j}^0\le \hat{C}_{i,j}^0$ .

The tierstoxicology $T$ $SML_i$ transfer physics $M$ : partitioning then diffusion). Monotonicitysafety by construction $T$ $M$ enlarges the admissible solution space without weakening it.

Tip

$C_{i,F}$ $\hat{C}_{i,F}$ $\hat{C}_{i,F}$ $PR$ $M2$ $M3$ .

2 | General model for the final concentration in food

2.1 | Hierarchical construction and conservative assumptions

$\{\hat{C}_{i,F}\}_{M=0..3}$ are all built conservatively: they aim to maximize the final contamination of food. Each tier progressively complicates a common parent model, in order to reduce part of the uncertainty when justified. The representation remains deliberately general to cover any type of material or substance, at least in the lower tiers. Further refinements may then introduce assumptions specific to a family of materials.

All models share a common set of assumptions:

the total system volume (food + packaging) is additive;
the system is closed to the outside: no losses, no chemical decomposition, no secondary production;
use conditions are maximized to reflect the longest contact durations and highest foreseeable temperatures.

2.2 | Isolating the time contribution

The temporal component is the most complex to handle because it depends on:

geometry (thickness, contact surface),
use conditions (duration, temperature, type of contact),
transport and thermodynamic properties of materials.

thermodynamic equilibrium concentration $C_{i,F}^{eq}$ dynamic term $\bar{v}_i^*(t)$ that reflects the relative transfer rate (Vitrac and Hayert, 2005; Vitrac and Hayert, 2006):

\begin{matrix} (2) & C_{i, F}^{(t)} = C_{i, F}^{(0)} + {\bar{v}}_{i}^{*} (t) (C_{i, F}^{e q} - C_{i, F}^{(0)}) \end{matrix}

Note

$\bar{v}_i^*(t)$ increases monotonically from 0 to 1. For a multilayer material, it may temporarily exceed 1 due to internal redistributions, but never departs by more than an order of magnitude from unity.

2.3 | Equilibrium concentration

$p_{i,j}$ $i$ $j$ .
Henry’s law gives:

\begin{matrix} (HenryIsotherm) & p_{i, j} = k_{i, j}^{H} C_{i, j}, j = 0. . m \end{matrix}

$k^H_{i,j}$ $j$ .

Mass conservation imposes:

\begin{matrix} (MassBalance) & \sum_{j = 0}^{m} V_{j} C_{i, j}^{0} = \sum_{j = 0}^{m} V_{j} C_{i, j}^{e q} = p_{i}^{e q} \sum_{j = 0}^{m} \frac{V_{j}}{k_{i, j}^{H}} \end{matrix}

Thus, the equilibrium concentration in food is:

\begin{matrix} (3) & C_{i, F}^{e q} = \frac{p_{i}^{e q}}{k_{i, 0}^{H}} = \frac{\sum_{j = 0}^{m} V_{j} C_{i, j}^{0}}{\sum_{j = 0}^{m} \frac{k_{i, 0}^{H}}{k_{i, j}^{H}} V_{j}} \end{matrix}

2.3.1 | Multiple sources

In the general case, contaminants may come from all packaging components (plastic, adhesive, ink, paper/board), independently of regulations that usually handle substances by material type. It is useful to distinguish the contribution from any initial food contamination (before contact) and that from non-food components.

$j=0$ $j\ge1$ ):

\begin{matrix} (4) & C_{i, F}^{e q} = C_{i, F}^{0} \frac{V_{0}}{\sum_{j = 0}^{m} \frac{k_{i, 0}^{H}}{k_{i, j}^{H}} V_{j}} + \frac{\sum_{j = 1}^{m} V_{j} C_{i, j}^{0}}{\sum_{j = 0}^{m} \frac{k_{i, 0}^{H}}{k_{i, j}^{H}} V_{j}} \end{matrix}

conservative approximation $\hat{C}_{i,F}^{0}$ which ignores dilutionnot recontaminate $j>0$ ):

\begin{matrix} (5) & {\hat{C}}_{i, F}^{e q} = {\hat{C}}_{i, F}^{0} + \frac{\sum_{j = 1}^{m} V_{j} {\hat{C}}_{i, j}^{0}}{\sum_{j = 0}^{m} \frac{k_{i, 0}^{H}}{k_{i, j}^{H}} V_{j}} with {\hat{C}}_{i, F}^{0} \geq C_{i, F}^{0}, {\hat{C}}_{i, j}^{0} \geq C_{i, j}^{0} \end{matrix}

no bias $C_{i,F}^{eq}=\hat{C}_{i,F}^{eq}$ .

Specializations by tier (axis M):

M0 (total transfer).
All substances present in packaging fully migrate to food:
$\begin{matrix} (6) & {\hat{C}}_{i, F}^{e q} = {\hat{C}}_{i, F}^{(0)} + \frac{{\hat{m}}_{i}^{0}}{V_{0}} \end{matrix}$
M1 (unit partition).
$k_{i,j}^H = k_{i,0}^H$ $j$ .
The expression simplifies to a volume-weighted mean:
$\begin{matrix} (7) & {\hat{C}}_{i, F}^{e q} = \frac{\sum_{j = 0}^{m} V_{j} {\hat{C}}_{i, j}^{0}}{\sum_{j = 0}^{m} V_{j}} \end{matrix}$
M2 (realistic partitions).
$k_{i,0}^H/k_{i,j}^H$ are accounted for to reflect solubility, polarity, and crystallinity of materials.
minimum plausible values $k_{i,0}^H/k_{i,j}^H$ $\hat{C}_{i,F}^{eq}$ .

Note

$\hat{C}_{i,F}^{(0)}$ , it is convenient to write:

\begin{matrix} (8) & {\hat{C}}_{i, F}^{e q} = {\hat{C}}_{i, F}^{(0)} + Δ {\hat{C}}_{i, F}^{e q} \end{matrix}

$\Delta\hat{C}_{i,F}^{eq}$ $j\ge1$ . This form simplifies spreadsheet use and contribution traceability.

2.3.2 | Single source and static migration rate

$s$ , a simplified expression is obtained:

\begin{matrix} (9) & {\hat{C}}_{i, F}^{e q} = {\hat{C}}_{i, F}^{0} + \frac{{\hat{C}}_{i, s}^{0}}{\sum_{j = 0}^{m} \frac{k_{i, 0}^{H}}{k_{i, j}^{H}} \frac{V_{j}}{V_{s}}} = {\hat{C}}_{i, F}^{0} + {\hat{m}}_{i}^{*, e q} \frac{{\hat{m}}_{i}^{0}}{V_{0}} \end{matrix}

where the static migration rate

\begin{matrix} (10) & {\hat{m}}_{i}^{*, e q} = \frac{1}{\sum_{j = 0}^{m} \frac{k_{i, 0}^{H}}{k_{i, j}^{H}} \frac{V_{j}}{V_{0}}} \end{matrix}

$0 \le \hat{m}_i^{*,eq} \le 1$ $\hat{m}_i^0$ that is transferred to food at equilibrium.

Note

cautious assumption $i$ $\hat{m}_i^0$ .

2.4 | Practical expression of dynamic concentration in food

$t$ is:

\begin{matrix} (11) & {\hat{C}}_{i, F}^{(t)} = {\hat{C}}_{i, F}^{(0)} + {\bar{v}}_{i}^{*} (t) {\hat{m}}_{i}^{*, e q} \frac{{\hat{m}}_{i}^{0}}{V_{0}} \end{matrix}

This formalism introduces two release potentials:

dynamic $PR_i^T = \bar{v}_i^*(t)$ ,
static $PR_i^E = \hat{m}_i^{*,eq}$ ,

$\boxed{PR_i = PR_i^T \times PR_i^E}$ represents the total released fraction.

2.5 | Case of porous foods

$V_0$ $V_F$ , corresponds to the apparent volumeexperimental volume $\hat{C}_{i,F}$ must be considered averaged at the volume scale.

In porous foods (e.g. bread), since contamination accumulates in condensed phases, the calculated concentration in food will be lower than the concentration measured in the solid phase alone (crumb and crust) by extraction. The two measures reconcile if we note that the concentration in the solid phase is:

\begin{matrix} (12) & {\hat{C}}_{i, F, solid} = \frac{{\hat{C}}_{i, F}}{1 - ϵ_{F}} \end{matrix}

$\epsilon_F$ is the porosity of food, i.e. the void volume fraction.

Key takeaways for Section 2

The general model expresses the final concentration in food as the sum of four components:

initial contamination,
a time factor,
a static migration rate,
and the total mobilizable amount.

This formulation preserves thermodynamic rigor while remaining usable in a spreadsheet. It prepares the introduction of release potentials and ranking metrics.

Tip

The next section develops the concept of partial release potentials, essential for analyzing process sequences and introducing the concept of a functional barrier.

3 | Partial release potentials and step sequences

3.1 | Intuition and objective

Within a sequence of steps in a package’s life, we aim to assign an explainabletotal release $i$ to each step. Contrary to common intuition, a partial potential may be non-zero even if no direct transfer to food occurs during the step considered. It is enough that the step mobilizes the substance (internal redistribution, surface approach, interface creation, geometry change) so as to increase what will be released in a later step.

Examples of “mobilizing” mechanisms without food contact:

thermoforming that thins a layer and increases the specific surface area;
thermoforming or any step in which temperature rise changes the internal distribution, notably by bringing substances closer to the exchange surface;
bringing internal and external surfaces into contact by stacking or roll storage (e.g., contact with printed surfaces);
folding that puts the packaging into contact with an overwrap;
sealing or printing that creates active interfaces;
contamination of contact layers by transfer during storage of components (e.g., cross-contamination with or without physical contact).

Tip

Key point. The concept of a partial release potential is independent of the detailed transfer physics and relies on a material balance between a reservoir of contaminants and a receiving phase (e.g., food or simulant). It is useful to trace the causes and sources of food contamination.

3.2 | Formal definition for independent steps

partial potential $k$ by an “with vs without” comparison:

\begin{matrix} (13) & P R_{i} |_{k} \equiv 1 - \frac{1 - P R_{i}^{(N)}}{1 - P R_{i}^{(N ∖ k)}} \in [0, 1] \end{matrix}

$PR_i^{(N)}$ $N$ $PR_i^{(N\setminus k)}$ $k$ is omitted (all else equal). This definition entails the multiplicative composition of complementssurvival factors $\sigma_{i,k}=1-PR_i\!\big|_k$ $k$ :

\begin{matrix} (14) & P R_{i}^{(N)} = 1 - \prod_{k = 1}^{N} (1 - P R_{i} |_{k}) = 1 - \prod_{k = 1}^{N} σ_{i, k} . \end{matrix}

$k$ involves no food contact $PR_i\!\big|_k>0$ $PR_i^{(N\setminus k)}=1$ $k$ $\frac{1-PR_i^{(N)}}{1-PR_i^{(N\setminus k)}}$ $PR_i\!\big|_k=PR_i^{k}$ $PR_i^{k}$ is the release potential $k$ is considered.

3.3 | Calculation procedure

Partial release potentials are computed iteratively using the omission rule.

Define the actual sequence of steps and their conditions (time, temperature, geometry).
$PR_i^{(N)}$ for the full sequence.
$k$ $PR_i^{(N\setminus k)}$ by $k$ (same conditions for the others).
$PR_i\!\big|_k$ $\ref{eq:PRw-w/o}$ .
independent $PR_i^{(N)}\stackrel{?}{=}1-\prod_k(1-PR_i\!\big|_k)$ (numerical equality within tolerance).

Table 4 — Calculation procedure for the estimation of partial potential release associated with indpendent steps

$k$	$PR_i^{(N\setminus k)}$	$PR_i\lvert_k$	$1-\prod(1-PR_i\lvert_k)$	Comment (mobilizing/contact/purge)
1
2
…
N

Mechanisms associated with each step are identified as:
"mobilizing": step which redistributes or put closer the susbtances (e.g. heating, thinning, storage);
"contact": step where the packaging material is in contact with food or a food simulant and transfer mobilized subsances;
"purge": step contribution to the removal os substances from the packaging system (e.g. desorption, washing, outgasing).

Note

Repeated use $N$ $PR_i^\text{cycle}$ :

\begin{matrix} (15) & P R_{i}^{(N)} = 1 - (1 - P R_{i}^{cycle})^{N} . \end{matrix}

3.4 | Example: storage followed by food contact (2 steps)

Consider a simple situation with a package undergoing two distinct steps; the corresponding release potentials can be obtained experimentally or by simulation:

Step 1: storage of the package at 40 °C for 30 days;
Step 2: an “accelerated” food contact of 10 days at 40 °C.

$1\!\to\!2$ $PR_{i}^{(2)}=50\%$ food-contact $PR_{i\lvert_ 2}=30\%$ . Step 1 has no contact and corresponds to stacked storage of a multilayer article. The contribution of step 1 to total transfer is:

\begin{matrix} (16) & P R_{i |_{1}} = 1 - \frac{1 - P R_{i}^{(2)}}{1 - P R_{i |_{2}}} = 1 - \frac{1 - 0.5}{1 - 0.3} \approx 0.29 . \end{matrix}

This elementary example confirms that non-contact storage can have a significant effect on the final transfer potential and, by extension, on the risk-assessment outcome.

one additional month $1\!\to\!1'\!\to\!2$ $1'$ $PR_{i\lvert_ 2}=30\%$ $PR_{i}^{(2)}=50\%$ $PR_{i}^{(2’)}=75\%$ $PR_{i\lvert_ 1'}$ then follows the “with vs without” relation:

\begin{matrix} (17) & P R_{i |_{1}^{'}} = 1 - \frac{1 - P R_{i}^{(2^{'})}}{(1 - P R_{i |_{1}}) (1 - P R_{i |_{2}})} = 1 - \frac{1 - 0.75}{(1 - 0.29) (1 - 0.30)} \approx 49.7 % . \end{matrix}

Tip

Interpretation.

$PR_{i\lvert_ 1'}\!\approx\!50\%$ means that transfer of half of the initially present substances $1\!\to\!1'\!\to\!2$ sequence.”
$PR_{i\lvert_ 1'} > PR_{i\lvert_ 2}$ initiated $1'$ $C_F$ only at the endreconstructs causality $PR$ at the end of the chain. These variations stem from dynamic effects of internal redistribution measured by the temporal potential (mobilization toward interfaces, shorter diffusion paths, increased diffusivity).

Calculation check. One extra month induces a relative increase:

\begin{matrix} (18) & \frac{P R_{i}^{(2^{'})} - P R_{i}^{(2)}}{P R_{i}^{(2)}} = \frac{P R_{i |_{1}^{'}} (1 - P R_{i |_{1}}) (1 - P R_{i |_{2}})}{P R_{i}^{(2)}} = P R_{i |_{1}^{'}} (\frac{1 - P R_{i}^{(2)}}{P R_{i}^{(2)}}) \approx 0.497 \times \frac{1 - 0.5}{0.5} \approx 50 % . \end{matrix}

3.5 | Metrics to rank and prioritize steps

$PR_i\!\big|_k$ provide an explainable decomposition of total release. To steer design choices, it is useful to transform them into metrics that compare and prioritize steps in a sequence.

Different, complementary metrics are possible:

Absolute indicators. $A_{i,k}$ $PR_i\!\big|_k$ :
$\begin{matrix} (19) & A_{i, k} \equiv - \log (1 - P R_{i} |_{k}) (\geq 0) . \end{matrix}$
Relative scores and elasticities.relative score $PR_i\!\big|_k$ $PR_i^{(N)}$ :
$\begin{matrix} (20) & {Score}_{i, k} \equiv \frac{P R_{i} |_{k}}{P R_{i}^{(N)}} . \end{matrix}$
Elasticities identify the most sensitive parameters for mastering release:
$\begin{matrix} (21) & E_{i, k} \equiv \frac{\partial P R_{i}^{(N)}}{\partial θ_{k}} \cdot \frac{θ_{k}}{P R_{i}^{(N)}}, \end{matrix}$
$\theta_k$ $k$ (duration, temperature, thickness).
Order-invariant indicators.strongly coupled $PR_i\!\big|_k$ may depend on step order. Shapley values provide an “equitable” attribution:
$\begin{matrix} (22) & {Shapley}_{i, k} \equiv \frac{1}{N!} \sum_{π} [P R_{i}^{(previous + {k})} - P R_{i}^{(previous)}], \end{matrix}$
$\pi$ of step order. Each step receives a share proportional to its marginal contribution in all possible configurations.

These metrics (absolute, relative, elasticities, Shapley) can be combined with health, economic, or environmental weights to obtain a multi-criteria ranking of steps—directly usable to guide safe design and justify decisions to authorities.

Note

Key point. Ranking metrics translate partial potentials into decision tools. Absolute metrics support quick and intuitive analysis; relative metrics and elasticities give a comparative/parametric view; Shapley values and multi-criteria indicators enable robust analysis in complex or strongly coupled systems.

3.6 | Dependent steps

When steps are independent, their total contribution to release does not depend on order, even though each step’s specific contribution may vary with its position. In contrast, introducing dependent steps breaks commutativity: steps occurring before or after a dependent step see their contributions modified.

Dependent steps correspond to situations where contaminants are redistributed, consumed, or produced inside the material, not necessarily transferred to food. This symmetry breaking is the very origin of dependence.

A sequence’s degree of dependence can be quantified by the deviation:

\begin{matrix} (23) & dependence = \frac{\prod_{k} (1 - P R_{i} |_{k})}{1 - P R_{i}^{(N)}} - 1. \end{matrix}

Any value significantly different from zero indicates a history effect:

Positive dependence: the theoretical transfer (estimated from individual partial potentials) is higher than observed for the full sequence. This reflects an effective decrease in transfers to food, e.g., back-flux, crust formation (plasticizer depletion or induced crystallization), or consumption of substances via chemical reactions.
Negative dependence: the effective transfer is higher than anticipated from individual contributions. This corresponds to bringing substances closer to the food interface, accelerating transfers (plasticization or thermal activation), or generating new substances by chemical reactions.

3.7 | Continuous extension (limit 𝐍→∞)

3.7.1 | Principles in a nutshell

instantaneous release rate $\zeta_i(t)$ $PR_i(t)$ follows a mass-action-type differential form:

\begin{matrix} (24) & \frac{d P R_{i} (t)}{d t} = (1 - P R_{i} (t)) ζ_{i} (t), P R_{i} (0) = 0. \end{matrix}

$\zeta_i(t)$ $\boxed{\zeta_i(t)\sim t^{-1/2}}$ short times $PR_i\!\big|_k$ to cumulative risk in an extended scenario.

Tip

Piecewise approximation. $\zeta_i(t)\simeq\zeta_{i,k}$ $t\in(t_{k-1},t_k]$ $PR_i\!\big|_k$ additive decomposition $\zeta_i(t)$ also applies to multiple sources, as will be shown at tier M3.

3.7.2 | Demonstration

From discrete to continuous. $S_i^{(N)} \equiv 1-PR_i^{(N)}=\prod_{k=1}^N \sigma_{i,k}$ be the survivalcumulative hazard $H_i^{(N)} \equiv -\ln S_i^{(N)}=\sum_{k=1}^N h_{i,k}$ $h_{i,k}\equiv -\ln\sigma_{i,k}$
$\tau$ $\tau_0<\tau_1<\cdots<\tau_N$ $\Delta\tau_k=\tau_k-\tau_{k-1}$ $h_{i,k}=O(\Delta\tau_k)$ instantaneous hazard $\lambda_i(\tau)$ by

\begin{matrix} (25) & h_{i, k} = \int_{τ_{k - 1}}^{τ_{k}} ζ_{i} (u) d u = ζ_{i} (ζ_{k}^{*}) Δ ζ_{k} + o (Δ τ_{k}) . \end{matrix}

$N\to\infty$ ,

\begin{matrix} (26) & H_{i} (τ) = \int_{0}^{τ} ζ_{i} (u) d u, S_{i} (τ) = \exp (- H_{i} (τ)), P R_{i} (τ) = 1 - \exp (- \int_{0}^{τ} ζ_{i} (u) d u) . \end{matrix}

Compact differential form.

\begin{matrix} (27) & \frac{d P R_{i}}{d τ} = (1 - P R_{i} (τ)) ζ_{i} (τ), P R_{i} (0) = 0, \end{matrix}

$S_i(\tau)=1-PR_i(\tau)$ $\dot S_i=-\zeta_i S_i$ .

Exact link to discrete partial potentials. $[\tau_{k-1},\tau_k]$ ,

\begin{matrix} (28) & P R_{i} |_{k} = 1 - \exp (- \int_{τ_{k - 1}}^{τ_{k}} ζ_{i} (u) d u), \int_{τ_{k - 1}}^{τ_{k}} ζ_{i} (u) d u = - \ln (1 - P R_{i} |_{k}) . \end{matrix}

Tip

$\zeta_i$ $k$ $\zeta_{i,k} = -\ln(1-PR_i\!\big|_k)/\Delta\tau_k$ .

3.7.3 | Extensions

Useful parameterizations. $\zeta_i(\tau)$ encodes the physics/chemistry:

Independent pieces (parallel mechanisms). $\zeta_i(\tau)=\sum_m \zeta_{i,m}(\tau)$ (additivity of hazards).
Dependence on process conditions. $\zeta_i(\tau)=\Zeta_i\!\bigl(T(\tau),\mathrm{RH}(\tau),\dot\zeta(\tau),\ldots\bigr)$ $\zeta_i(\tau)=\zeta_{0,i}\exp\!\{-E_{a,i}/(R\,T(\tau))\}\,g(\tau)$ .
Piecewise-constant (calibrated on step data). $\zeta_i(\tau)=\zeta_{i,k}$ $\tau\in(\tau_{k-1},\tau_k]$ $\lambda_{i,k}$ $PR_i\!\big|_k$ .

Link with mass-transfer models.effective first-order $\dot M_\mathrm{rel}(t)=k_\mathrm{eff}(t)\,\bigl(M_\infty-M_\mathrm{rel}(t)\bigr)$ $PR_i(t)=M_\mathrm{rel}(t)/M_\infty$ $\tau=t$ $\zeta_i(t)=k_\mathrm{eff}(t)$ .

For Fickian diffusion (planar slab, perfect sink), the non-released fraction admits the classical series

\begin{matrix} (29) & S_{i} (t) = 1 - P R_{i} (t) = \frac{8}{π^{2}} \sum_{n = 0}^{\infty} \frac{1}{(2 n + 1)^{2}} \exp (- a_{n} t), a_{n} = \frac{(2 n + 1)^{2} π^{2} D_{i}}{4 L^{2}} . \end{matrix}

The corresponding instantaneous hazard is

\begin{matrix} (30) & ζ_{i} (t) = - \frac{d}{d t} \ln S_{i} (t) = \frac{\sum_{n \geq 0} \frac{a_{n}}{(2 n + 1)^{2}} e^{- a_{n} t}}{\sum_{n \geq 0} \frac{1}{(2 n + 1)^{2}} e^{- a_{n} t}}, \end{matrix}

$t$ $\propto t^{-1/2}$ $K$ series-resistance $\zeta_i(t)$ $Bi$ $K$ $PR_i$ $\zeta_i=-\dot S/S$ ).

Key takeaways for Section 3

Partial potentials decompose total release into step contributions. They can be non-zero without food contactmobilization $PR_i^{(N)}=1-\prod_k(1-PR_i\!\big|_k)$ , enabling straightforward spreadsheet implementation.

$PR_i\!\big|_k$ feed into metrics (absolute impact, elasticity, weighted scores, Shapley value) to prioritize actionscompatible $M$ $PR$ can be estimated in M0–M2 (equilibrium/partition) or in M3 (explicit diffusion).

Tip

The next section introduces the functional barrier concept, which formalizes how a component reduces release. It integrates naturally with partial potentials (barriers in series, cumulative effects, prioritization) and their metrics.

4 | Functional barrier concept

4.1 | Towards a robust definition

The functional barrier (FB) concept is central in U.S. (FDA) and EU (Regulation (EU) No 10/2011) frameworks. It enables derogations: a recycled material or a debated additive may be used provided that a demonstrated functional barrier keeps the effective release below regulatory thresholds.

There is no formal, official definition beyond the idea of “a layer preventing migration”. We propose two operationaltotal $\mathrm{PR}$ partial $\ref{eq:FBdef1}$ $\ref{eq:FBdef2}$ ):

\begin{matrix} (4.1) & (Total PR) P R_{i}^{f b} = P R_{i}^{T, f b} \times P R_{i}^{E, f b} < P R_{i} = P R_{i}^{T} \times P R_{i}^{E}, \end{matrix}

\begin{matrix} (4.2) & (Partial PRs) P R_{i}^{f b} (S) = 1 - \prod_{ℓ \in S} (1 - P R_{i}^{f b} |_{ℓ}) < P R_{i} (S) = 1 - \prod_{ℓ \in S} (1 - P R_{i} |_{ℓ}), for a significant time t \leq t_{f b} and \forall i, \end{matrix}

$^{fb}$ with $\ref{eq:FBdef2}$ $k$ effective $[0,t_{fb}]$ $S$ $k$ , a reduction in transfer potential is observed.

These conditions allow the effect to be limited to a single step. The barrier is neither absolute (perfectly tight layer) nor permanent: its effect is temporary and of variable intensity.

Traditionally seen as a pure diffusion barrier, a functional barrier may actually combine several mechanisms:

Reservoir effect $PR_i^{T,fb}<PR_i^T$ , temporary accumulation upstream (backflow effect).
Retention effect $PR_i^{E,fb}<PR_i^E$ , chemical or physical trapping.
Diffusive effectlag time $t_\text{lag}$ and flux reduction.

Note

$PR$ ), pure dilution effects (e.g., blending recycled with virgin) are excluded from the FB concept.

Warning

Key point. Poor FB sizing can have opposite consequences:

Undersized ⇒ potential health risk due to uncontrolled transfer from recycled constituents.
Oversized ⇒ unnecessary drag on circularity (lower recycled content, needless decontamination, poorer recyclability of FB-containing materials).

4.2 | Derived performance indices

additive gains $G_{i,k}$ :

\begin{matrix} (4.3) & G_{i, k} \equiv \ln \frac{1 - P R_{i}^{f b} |_{k}}{1 - P R_{i} |_{k}} . \end{matrix}

$G_{i,k}>0$ partial $PR$ $k$ .
Choice of dimensioning molecules is critical (e.g., toluene as a reference).
For homologous series, lag times scale with molar mass squared (rubbery polymers) or follow power laws (glassy polymers).

Note

Worked example. $PR_i\lvert_2$ $PR^{fb}(1\!\to\!1'\!\to\!2)$ $1-(1-0.2857)(1-0.50)(1-0.12)\approx 57\%$ $G_{i,2}=\ln\!\frac{1-0.12}{1-0.30}\approx 0.229$ .

4.3 | Functional barrier typology

The FB concept covers interlayers between content and contaminants, but may also include additives, micro-/nanoparticles modifying chemical affinity and/or mobility of potential migrants. Purge stepsFB assembly steps $PR_i^{fb}$ .

Classical diffusive barriers
- Principle: slow the flux via low diffusivity or large thickness.
- Example: virgin PET interlayers between a recycled layer and the food.
- Indicator $PR_i\lvert_k$ ; never total.
- Limit $PR_i^{fb}\to 0$ only if $\to 0$ or infinite thickness (unrealistic).
Trapping barriers
- Principle: chemical/physical fixation of contaminants.
- Example: scavenger additives, sorptive mineral fillers.
- Indicatorselective $PR_i\lvert_k$ .
- Limit: efficiency limited to substances with affinity for the trap.
Dilution / redistribution barriers
- Principle: disperse the initial contamination within a mass of virgin material.
- Example: a virgin polyethylene ply between recycled board and food.
- Indicator $\hat{C}_{i,F}^{eq}$ without removing substances.
- Limit: reversible; contaminants remain and may migrate later.
Dynamic or temporal barriers
- Principle: introduce a delay shifting transfers to longer times.
- Example: saturable absorbent layers, slowly swelling polymers.
- Indicator $PR_i^{fb}$ short times $PR_i$ at long times.
- Limit: efficiency tied to use duration (single-use vs. long storage).
Purge steps
- Principle: remove part of contaminants before use.
- Example: thermal degassing of a recycled film; aqueous washing of fibers.
- Indicatorirreversible $PR_i^{fb}$ $C_{i,j}^0$ .
- Limit: depends on industrial reproducibility.

4.4 | Application to recycled material

$t_\text{contact}$ , one may theoretically use less-decontaminated recycled material if the FB compensates, provided:

\begin{matrix} (4.4) & \frac{1 - Δ_{L}}{1 - Δ_{H}} \leq \frac{P R_{i}^{T} \cdot P R_{i}^{E}}{P R_{i}^{T, f b} \cdot P R_{i}^{E, f b}} for t \leq t_{contact}, \end{matrix}

$\Delta_L<\Delta_H$ $95\%$ $99\%$ ).
minimum reduction factor $99\%$ $95\%$ $\frac{1-0.95}{1-0.99}=5$ $PR_i^T\cdot PR_i^E$ 5 $t_\text{contact}$ $i$ .

Tip

Key point.Misestimation $(PR_i^T, PR_i^E$ or their FB versions) can yield higher contamination than without the FB, due to the use of more contaminated feedstockappropriate safety factors $PR$ values.

4.5 | Points of attention and recommendations

Points of attention

Link to recycling. FBs are often invoked to include recyclates. Poor sizing leads to either (i) under-protection (underestimated release, health risk) or (ii) over-protection (unrealistic requirements, blocked streams).
Measurability. Effectiveness must be shown via measurable indicators (tests, validated simulations), not material assumptions alone.
Selectivity. An FB protects some substances better than others—analyze per substance or via dimensioning migrants.
Identification. FB-equipped materials should be identifiable to route recycling streams; otherwise, risk of cross-contamination if non-authorized recyclates are reintroduced into direct-contact loops.

Recommendations

Favor FBs implemented cold and as close to the content as possible to avoid pre-contamination.
An FB may be removable (sachet, inner film, gel) provided it is not confused with the content.
Typology: diffusive barriers, reservoirs, trapping, or hybrids.
An FB can justify less-decontaminated recyclates if it ensures sufficient reductionstatic × dynamic $(PR_i^E \times PR_i^T)$ .
FBs are ephemeral: sizing must include all prior steps (forming, storage, elevated temperatures) that may erode their real effectiveness.

Key takeways for section 4

Functional barriers are a core instrumenteffective reductions $PR_i$ static $PR_i^E$ dynamic $PR_i^T$ $G_{i,k}$ apply at step level and at sequence level. FBs have finite service lives and must be optimized for the intended application. Use of FBs should not result in introducing materials unsuitable for food contact into recycling loops—FB-equipped packaging should be identified.

Tip

tier M3 (dynamic tiers) $PR$ predictions confirm or challenge expected FB performance and support regulatory validation.

5 | Severity and criticality scales

The notions of severity and criticality, rooted in FMECA (Failure Mode Effects and Criticality Analysis) and generalized to mass transfer (see Nguyen 2013), yield intensive quantities (severities) and their cumulants (criticalities) to compare substances, components, and packages—pinpointing what carries the highest risk for consumers.

5.1 | Severity scale

$i$ $\hat{C}_{i,F}/(\rho_0\,SML_i)$ :

\begin{matrix} (5.1) & \begin{aligned} {\hat{s}}_{i}^{(t)} & = 100 \times \frac{{\hat{C}}_{i, F}}{ρ_{0} S M L_{i}} \\ = {\hat{s}}_{i}^{0} + 100 \times {\bar{v}}_{i}^{*} (t) \cdot {\hat{m}}_{i}^{*, e q} \cdot \frac{{\hat{m}}_{i}^{0}}{ρ_{0} V_{0} S M L_{i}} \\ = {\hat{s}}_{i}^{0} + 100 \times P R_{i}^{T} P R_{i}^{E} \times \frac{{\hat{m}}_{i}^{0}}{ρ_{0} V_{0} S M L_{i}} . \end{aligned} \end{matrix}

$\hat{s}_i^{(t)}$ is dimensionless and defined on an open scale (can exceed 100).
Factor 100 flags severecatastrophic $\gg 100$ ).
It is an estimate (hats) and depends on the tier (approximation level).
$PR_i^T=\bar{v}_i^*(t)$ dynamic $PR_i^E=\hat{m}_i^{*,eq}$ (static potential).

5.1.1 | Partial severity by packaging component

$\hat{m}_i^0=\sum_{j=1}^m \hat{m}_i^0\lvert_j$ $\hat{m}_i^{*,eq}\lvert_j$ $j$ $\bar{v}_i^*\lvert_j(t)$ its dynamic factor:

\begin{matrix} (5.2) & {\hat{s}}_{i}^{(t)} |_{j} = {\hat{s}}_{i}^{0} |_{j} + 100 \times {\bar{v}}_{i}^{*} (t) |_{j} \cdot {\hat{m}}_{i}^{*, e q} |_{j} \cdot \frac{{\hat{m}}_{i}^{0} |_{j}}{ρ_{0} V_{0} S M L_{i}}, {\hat{s}}_{i}^{(t)} = \sum_{j = 1}^{m} {\hat{s}}_{i}^{(t)} |_{j} . \end{matrix}

Note

(additivity—tier M3).additivity $\hat{m}_i^{*,eq}\lvert_j$ $\bar{v}_i^*\lvert_j(t)$ from tier M2–M3 simulations; reuse across similar uses/substances.

5.1.2 | Suspected substances (below LOD)

$i$ is suspectedlimit of detection $DL_{i,j}$ $j$ :

\begin{matrix} (5.3) & {\hat{s}}_{i}^{(t)} |_{j} = {\hat{s}}_{i}^{0} |_{j} + 100 \times {\bar{v}}_{i}^{*} (t) |_{j} \cdot {\hat{m}}_{i}^{*, e q} |_{j} \cdot \frac{ρ_{j} V_{j}}{ρ_{0} V_{0}} \cdot \frac{D L_{i, j}^{mass}}{S M L_{i}} . \end{matrix}

5.1.3 | Mechanically recycled materials

$j$ maximum contamination $C_{i,j}^g$ minimum $\Delta_{i,j}$ :

\begin{matrix} (5.4) & {\hat{s}}_{i}^{(t)} |_{j} = {\hat{s}}_{i}^{0} |_{j} + 100 \times {\bar{v}}_{i}^{*} (t) |_{j} \cdot {\hat{m}}_{i}^{*, e q} |_{j} \cdot (1 - Δ_{i, j}) \cdot \frac{ρ_{j} V_{j}}{ρ_{0} V_{0}} \cdot \frac{C_{i, j}^{g, mass}}{S M L_{i}} . \end{matrix}

Note

Example (mechanical rPET). $\Delta_{i,j}\ge 0.90$ $C_{i,j}^{g,\text{mass}}=3$ $\rho_{PET}\simeq 1380$ $^3$ $\hat{C}_{i,j=\mathrm{PET}}^0 \approx (1-0.9)\times \times10^{-6}\,\text{kg/kg}\times 1380\,\text{kg/m}^3= 4.14\times 10^{-4}\,\text{kg/m}^3.$

5.2 | Criticality scale

Criticality aggregates severities across substances (and optionally components). It is not intensive; it is a cumulant used to rank components, packages, or designs.

5.2.1 | Component criticality

From conditional severities:

\begin{matrix} (5.5) & {\hat{c}}^{(t)} |_{j} = \sum_{i = 1}^{n} pr (i \in j) {\hat{s}}_{i}^{(t)} |_{j}, j = 1, \dots, m, \end{matrix}

$\operatorname{pr}(i\in j)$ is the probability of presence (intentional) or likelihood inferred from identification databases (retention times and/or mass spectra).

Note

Inheritance. $\hat{s}_i^{(t)}\lvert_j$ $PR_i^T\,PR_i^E$ , criticality inherits dynamic and equilibrium effects.

5.2.2 | Package criticality

Double sum over substances × components:

\begin{matrix} (5.6) & {\hat{c}}^{(t)} |_{packaging} = \sum_{j = 1}^{m} {\hat{c}}^{(t)} |_{j} = \sum_{j = 1}^{m} \sum_{i = 1}^{n} pr (i \in j) {\hat{s}}_{i}^{(t)} |_{j} . \end{matrix}

$\hat{c}^{(t)}$ $\ref{eq:CriticityPack}$ is well-suited to compare design options: virgin, mechanical/chemical recycling, with/without FB.

Note

Comparability. Always compare criticalities at the same tier. If a refinement changes criticality significantly, moving up a tier is justified; otherwise it may be deferred.

5.2.3 | Step criticality (preview)

To link FMECA with process sequencesmarginal step criticality $k$ marginal release gain $\Delta_{i,k}=PR_i^{(N)}-PR_i^{(N\setminus k)}$ :

\begin{matrix} (5.7) & {\hat{c}}^{(t)} |_{k} = \sum_{j = 1}^{m} \sum_{i = 1}^{n} pr (i \in j) {\hat{s}}_{i}^{(t)} |_{j} \cdot Δ_{i, k} . \end{matrix}

Note

coupled $\Delta_{i,k}$ with the corresponding Shapley value (fair attribution).
A full development is given at tier 3 (boundary conditions, kinetics).

Warning

Points of attention

Count detected substances to estimate criticality for non-identified ones at/above LOD.
Reasonable working assumption: same distribution of Cramer classes for identified and non-identified substances ⇒ a $SML_i$ usable for severities.
$\hat{s}_i^{(t)}\gg 100$ should trigger a higher-tier analysis or be backed by analytical/technical evidence. Exceeding 100 is an objective alert, not per se non-compliance.
$\hat{s}_i^{(t)}$ to the tiers used (M0–M3 and T0–T2).
It is generally acceptable to aggregate severities from different toxicology tiers (T0–T2) provided the same transfer tiers (M0–M3) were used.
Document methods and maintain prudence regarding criticality calculations.
$\hat{c}^{(t)}$ with prioritization metrics and FB effectiveness indicators.

Key Takeaways for Section 5

Severity is a dimensionless, intensiveacceptable exposure $PR_i^T$ $PR_i^E$ when available. A severity of 100 corresponds to contamination reaching the acceptable exposure threshold under standardized conditions (e.g., 1 kg food/day for a 60-kg adult).
Criticalities are cumulants to rank components, packages, and designs by exposure risk. They may include both identified and suspected substances and decompose along two axes: components and use cycles.

6 | Principles for constructing tiers M0–M3

6.1 | Objective

Up to this point, the proposed approach has not introduced any simplifying assumptions about package geometry, composition, transfer mechanisms, or interactions among substances.
Only the thermodynamic description of transfers has been linearized so that concentrations can be used as transfer potentials instead of chemical potentials.

In addition, the equilibrium description was adjusted in the case of pre-existing food contamination: contaminations are strictly additive across steps—put simply, the package never acts to decontaminate the food.

$\bar{v}_i^*(t)$ $\hat{m}_i^{*,eq}$ $\hat{m}_i^0/V_0$ $SML_i$ nevertheless requires further assumptions—often delicate when many components and substances are involved.

The tiers (levels of approximation) therefore aim to identify the substances and components that most influence the final safety outcome, before optimizing them via design, formulation, or use conditions.

The construction of tiers for transfer physics (M0 to M3) is artificial and follows a logic of decreasing severities and criticalities: tier 0 provides a very conservative upper bound; subsequent tiers progressively introduce refinements.

6.2 | Overview of tiers

Figure 1 presents the available Tx/Mx options based on what is known about migrants:

T0–M0 if the substance is unknown/not detected,
T1–M0 or T1–M1 if the substance is identified but no material data are available,
T2–M1 or T2–M2 if material properties are available,
T2–M3 if a dynamic description is required.

The decision tree illustrates the progression from conservative scenarios toward detailed kinetic models by systematically combining a toxicology tier (Tx) with a transfer-modeling tier (Mx).

Table 5 provides a concise, progressive view of how assumptions evolve across Mx scenarios. The four transfer-modeling levels (M0–M3) are compared by their required inputs and expected outputs.

Figure 1 — Decision tree for tier combinations Tx/Mx.
Flow diagram guiding the choice among T0–T2 / M0–M3 according to data availability on substances and materials.

Table 5 — Operational summary of calculations at tiers M0–M3

Tier	Minimal inputs	Condensed calculation procedure	Equations (overview)	Outputs	Notes & conservatism
M0 — Total transfer	$BW$ $DI$ $V_0$ $V_j$	(1) $TDI$ $^{-1}$ $^{-1}$ (2) $SML_i=BW\cdot DI\cdot TDI_i$ (3) $(\bar{v}_i^(t),\hat m_i^{,eq}\to 1)$ . — (4) Deduce material thresholds.	$\{\hat C^0_{i,j}\}_\mathrm{M0}=\dfrac{\rho_F\,V_0}{V_j}SML_i$	$\hat C^{(0)}_{i,j}$ $\hat{s}_i$ $\hat{c}\lvert_j$	Very $k_j^H\gg k_0^H$ ).
M1 — Unit partition	$BW$ $DI$ $V_0$ $V_j$	(1) $SML_i$ (2) $k_0^H\approx k_j^H$ . — (3) Volume-weighted distribution.	$\{\hat C^0_{i,j}\}_\mathrm{M1}=\dfrac{V_0+\sum_{k=1}^m V_k}{V_j}\rho_F\,SML_i$ $=\Bigl(1+\sum_{k=1}^m\dfrac{V_k}{V_0}\Bigr)\{\hat C^0_{i,j}\}_\mathrm{M0}$	$\hat C^{(1)}_{i,j}$ $\hat{s}_i$ $\hat{c}$	$V_0$ is not dominant; does not discriminate materials.
M2 — Realistic partitions	$V_0$ $V_j$ $C^\text{sat}_{i,j}$ $P_{i,\text{sat}}$ $logP_i$ $V_{i,\text{mol}}$ $c_j$ $\epsilon_j$ $\theta_{i,j}$	(1) $k_{i,j}^H$ (2) $K_{i,j_1/j_2}$ (3) $\hat m_i^0=\sum_j V_j(1-\theta_{i,j})\hat C^0_{i,j}$ (4) $\hat m_i^{eq}$ $C^{eq}_{i,F}$ (5) $\hat C^{(2)}_{i,j}$ $C^{eq}_{i,F}=SML_i$ .	$\displaystyle k_{i,j}^H=\frac{P_{i,\text{sat}}^{(T_K)}V_{i,\text{mol}}}{(1-c_j)(1-\epsilon_j)}\,\gamma_{i,j}^v$ $\displaystyle K_{i,j_1/j_2}=\frac{k_{i,j_2}^H}{k_{i,j_1}^H}$ $\displaystyle \hat m_i^0=\sum_j V_j(1-\theta_{i,j})\hat C^0_{i,j}$ $\displaystyle \hat C^{eq}_{i,F}=\hat C^0_{i,F}+ \hat m_i^{eq}\frac{\hat m_i^0}{V_0}$ .	$\hat C^{(2)}_{i,j}$ $\hat{s}_i(t)$ $\hat{c}(t)$ ; barrier-layer diagnostics	Realistic; supports optimization of material choice and multilayer architecture.
M3 — Dynamic transfers	$D_{i,j}$ $k_{i,j}^H$ $T(t)$ $t_\text{contact}$	(1) Solve Fick’s equations per layer. — (2)(3) $\bar v_i(t)$ $\hat C^{(t)}_{i,F}$ .	$\displaystyle J_{i,j}(t)=-D_{i,j}\frac{\partial C_{i,j}}{\partial x}$ $\displaystyle \hat C^{(t)}_{i,F}=\hat C^0_{i,F}+\bar v_i(t)\hat m_i^{eq}\frac{\hat m_i^0}{V_0}$ .	$\hat C^{(t)}_{i,F}$ $\hat s_i^{(t)}$ $\hat c^{(t)}$	Richest tier: introduces time scale and true use conditions (heating, storage). Sensitive to diffusion parameters; requires experimental validation.

6.3 | Tier T0–M0 — Total transfer

most conservative $TDI$ most stringent toxicology scenario $(0.0025\,\mu\text{g}\,\text{kg}^{-1}\,\text{bw}^{-1}\,\text{day}^{-1})$ total transfer $(\bar{v}_i^*(t)\cdot\hat{m}_i^{*,eq}\to 1)$ .

$SML_i$ does not depend on the substance but on the consumer age group. One obtains a factor 16 between adults and newborns. Values are compared in Table 6.

Table 6 — BW, DI, and SML values by age group

Age group	BW (kg)	DI (kg/day)	SML $^{-3}$ $^9$
Newborns (0–16 weeks)	5	0.75	9.375
Infants (0–6 months)	5	1	12.5
Young children (1–3 y)	12	1	30
Children (4–9 y)	20	1.5	75
Adults (≥15 y)	60	1	150

The interest of the coupled T0 and M0 tiers is to deliver a per-material concentration threshold that applies to any substance, detected or not, evaluated or not:

\begin{matrix} (6.1) & {{\hat{C}}_{i, j}^{0}}_{tier=M0} = \frac{V_{0}}{V_{j}} S M L_{i} . \end{matrix}

Note

Tier T0–M0 in practice. A per-material threshold valid for any substance under total-transfer and maximalist toxicology assumptions. Simple, fast, and safe—but it ignores real material properties and fine geometry.

Strengths

Per-material threshold, applicable to any substance (known or unknown).
Total transfer + maximal toxicology scenario.
Simple and fast computation.
Useful for screening substances/projects without data.

Tip

Moving to tier M1 incorporates volume partitioning between food and packaging to obtain more realistic yet broadly conservative thresholds.

6.4 | Tier M1 — Unit partition coefficients

equilibrium $(\bar{v}_i^*(t)\to 1)$ $k_j^H\gg k_0^H$ equivalence $k_0^H\approx k_j^H$ .

This amounts to assuming a homogeneous partition proportional to volumes:

\begin{matrix} (6.2) & {{\hat{C}}_{i, j}^{0}}_{tier=M1} = \frac{1}{V_{j}} \sum_{k = 0}^{m} V_{k} \cdot S M L_{i} = (1 + \sum_{j = 1}^{m} \frac{V_{j}}{V_{0}}) \cdot {{\hat{C}}_{i, j}^{0}}_{tier=0} . \end{matrix}

Tiers 0 and 1 yield similar results as long as the food volume dominates over the packaging components.

Note

Tier M1 in practice. Introduces global geometry (relative volumes) and a unit partition between phases, refining thresholds according to the actual configuration. Still conservative because it does not discriminate materials by physicochemical properties.

Strengths

Introduces global geometry (relative volumes).
Still conservative, but less severe than M0.
Does not yet discriminate materials by chemistry.
Prepares tier M2, which introduces polarity, solubility, and barrier effects.

Tip

Tier M2 will enrich the calculation with realistic partition coefficients derived from solubilities, polarities, and material structures—opening the way to optimization via retention layers or barriers.

6.5 | Additional data required for tiers M2 and M3

Tiers M2 (see § M2) and M3 (see § M3) explicitly introduce sorption and diffusion properties into the transfer-risk evaluation. These properties are directly tied to the identity of the considered substances (migrants/contaminants, polymers, simulants or food components).

To keep the assessment exhaustive, property-estimation methods should apply equally to:

identified contaminants,
probable substances inferred from mass-spectral and/or retention-time matches,
unknown compounds represented by typical, well-characterized physicochemical proxies.

This framework justifies using coarse estimatorsmolar mass $logP$ , molar volume, van der Waals volume. Links between these descriptors and the target properties used in transport equations are summarized in Table 7.

Table 7 — Minimal data for computing thermodynamic and transport properties of an arbitrary solute

Required data	Use	Target properties
$M_i$	$V_{i,\mathrm{mol}}$	$k_{i,j}^H$ $\gamma_{i,j}^v$ $K_{i,F/P}$ $D_{i,j}^{\text{Piringer}}$
Optimized 3D geometry (`.sdf`,`.mol`,`.pdb`)	$V_{i,\mathrm{vdW}}^\dagger$ $\xi_i$	$D_{i,j}^{\text{Welle}}$ $D_{i,j}^{\text{hFV}}$
$logP$ (octanol/water)	$P'$	$k_{i,j}^H$ $\gamma_{i,j}^v$ $K_{i,F/P}$

$^\dagger$ $V_{i,\mathrm{vdW}}$ $\tfrac{4}{3}\pi r_{\mathrm{vdW}}^3$ with overlap corrections. Robust implementations exist in RDKit (Landrum 2022), OpenBabel (O’Boyle 2011), and in SFPPy (Vitrac 2025a, Vitrac 2025b).

Important

The approach is not limited to known monomers and additives; it extends to emerging contaminants and potential migrants from recycled inputs. Descriptors are purposely simplified to allow rapid, automatable calculations across hundreds of substances. Macroscopic effects (porosity, crystallinity, plasticization, trapping) must also be considered.

Key takeaways (Section 6)

M0 and M1 are simple tiers, easy to implement in a spreadsheet. Coupled with T0, they allow decisions on risks from identified and non-evaluated substances.
For identified substances, they help shortlist those that must be analyzed with detailed transfer scenarios, accounting for material nature, substance identity, and use conditions.
M2 and M3 progressively add realistic partitions and dynamic diffusion, enabling design optimization (materials, multilayers, barriers) and supporting regulatory justification with quantified severities and criticalities.

7 | Advanced Tier M2: realistic partition coefficients, retention layers, and solubility barriers

7.1 | Objective

The goal of Tier M2 is to provide a realistic yet conservative estimate of equilibrium contamination by introducing physicochemical rules on affinities (Henry-type notations) and immobile fractions, without requiring the partial differential equations of Tier M3.
static release potential $PR_i^E=\hat{m}_i^{*,eq}$ dynamic release potential $PR_i^T=\bar{v}_i^*(t)$ is kept bounded (often set to 1 in M2) or approximated by simple time tiers, and will be fully developed in Section 8.

7.2 | Positioning and assumptions

The assumptions are aligned with those already used in the literature (Nguyen et al., 2016; Zhu et al., 2019a; Vitrac et al., 2022) to describe transfers between packaging/polymer materials and foods/liquids. See Zhu et al., 2019a for a comprehensive review.

Framework: multi-source, multi-layer, multi-material systems (plastics, adhesives, inks, papers/boards, elastomers), virgin and recycled, single-use and repeated-use.
Purpose of M2: discriminate food ↔ material affinitiesweight the mobile mass $\hat{C}_{i,F}^{eq}$ that is both explainable and conservative.
Principle of conservatism: select minimally plausible affinity ratios (Henry ratios) from the perspective of transfer toward food so as to maximize the final concentration in food.
Separation of scales: microstructural/supramolecular effects (crystallinity, trapping) are treated by homogenization separately from molecular mean-field effects.
Simplified molecular framework $k_{i,j}^H$ values from chemical databases, usable in automated calculation workflows.

Notations (recall):

$k_{i,j}^H$ $i$ $j$ $j=0$ $j=1..m$ ).
$V_j$ $j$ $\hat{C}_{i,j}^0$ : initial (estimated) concentration.
$K_{i,j/F} \equiv \tfrac{k_{i,0}^H}{k_{i,j}^H}$ $j$ and the contacting food (F).
$\theta_{i,j}\in[0,1]$ : immobile fraction (effective crystallinity, trapping, occlusion, inaccessibility).
Linearized equilibrium with multiple sources and immobile fractions:

\begin{matrix} (31) & {\hat{C}}_{i, F}^{e q} = {\hat{C}}_{i, F}^{(0)} + \frac{\sum_{j = 1}^{m} V_{j} (1 - θ_{i, j}) {\hat{C}}_{i, j}^{0}}{V_{0} + \sum_{j = 1}^{m} {\hat{K}}_{i, j / F} V_{j} (1 - θ_{i, j})} \end{matrix}

7.3 | Macroscopic scale: apparent partition coefficients

7.3.1 | Definitions

$j_1$ $j_2$ ,

\begin{matrix} (32) & K_{i, j_{1} / j_{2}} = \frac{C_{i, j_{1}}^{e q}}{C_{i, j_{2}}^{e q}}, \end{matrix}

$j_1$ $j_2$ :

\begin{matrix} (33) & K_{i, j_{1} / j_{2}} = \frac{k_{i, j_{2}}^{H}}{k_{i, j_{1}}^{H}}, j_{1} \neq j_{2}, j_{1}, j_{2} \geq 0 \end{matrix}

Extrapolating Henry’s law up to saturation:

\begin{matrix} (34) & k_{i, j}^{H} = \frac{P_{i, v}^{sat} (T_{K})}{C_{i, j}^{sat}}, j \geq 0 \end{matrix}

$P_{i,v}^\text{sat}(T_K)$ $T_K$ $C_{i,j}^\text{sat}$ $j$ .

$j$ is:

\begin{matrix} (35) & K_{i, F / j} = \frac{k_{i, j}^{H}}{k_{i, 0}^{H}} = \frac{C_{i, F}^{sat}}{C_{i, j}^{sat}} \end{matrix}

Food-favoring migration occurs if solubility in food is greater than in the material. With volume ratios included:

\begin{matrix} (36) & {\hat{C}}_{i, F}^{e q} = \frac{{\hat{C}}_{i, j}^{0}}{(V_{0} / V_{j}) + (C_{i, j}^{sat} / C_{i, F}^{sat})} \end{matrix}

and is significantly constrained only if:

\begin{matrix} (37) & C_{i, j}^{sat} ≫ \frac{V_{0}}{V_{j}} C_{i, F}^{sat} \end{matrix}

7.3.2 | Link to solubility barriers

Unit partition coefficient (M1) does not allow optimizing designs with retention layers or solubility barriers. Table 8 contrasts the two strategies.

Table 8 — Differences between a retention layer and a solubility barrier

$s>0$ )	Condition	Interpretation	Limitation	Example
$s$ as retention layer	$K_{i,F/s}\ll1$	$s$ retains its own substances	$K_{i,F/s}\ll V_j/V_0$	Hydrophobic polymer for hydrophobic contaminants
$j>s$ as reservoir	$K_{i,j/s}\gg K_{i,F/s}$	Substances accumulate opposite to food side	Temporary (depends on diffusion time)	Waxy external coating
$j<s$ as solubility barrier	$K_{i,j/s}\gg1$	Interposed layer reduces driving force	$s$ )	Dense polar film, or air gap

7.3.3 | Non-partition cases — example of inks

Partition coefficients apply only to exchangeable fractions. Substances co-crystallizedimmobile fraction $\theta_{i,j}$ .
Transferable mass is:

\begin{matrix} (38) & {\hat{m}}_{i}^{0} = \sum_{j = 1}^{m} V_{j} (1 - θ_{i, j}) {\hat{C}}_{i, j}^{0} \end{matrix}

$\theta>0$ ) must be experimental (extraction, DSC). Pigments often qualify; soluble dyes typically do not.

$\theta_{i,j}^{(T_K)}$ may be estimated from DSC melting enthalpy:

\begin{matrix} (39) & θ_{i, j}^{(T_{K})} = \frac{A_{i, j}^{T_{K} \to}}{ϕ_{i, j} H_{i}^{ref}} \end{matrix}

$\phi_{i,j}$ $i$ $j$ .

7.3.4 | Ready-to-use spreadsheet procedure

9 $\hat{C}_{i,F}^{eq}$ across layers and food.

Table 9 — Spreadsheet template for Tier M2 macroscopic effects

Col.	Content	Example formula
A	$j$ (0=food, 1..m)	0,1,2,…
B	$V_j$	thickness × area
C	$\hat{C}_{i,j}^0$	input
D	$\theta_{i,j}$	0–1
E	$\hat{m}_{i,j}^{0,\text{mob}}$	`=B(1-D)C`
F	$k_{i,j}^H$	proxy/score
G	$\hat{K}_{i,j/F}$	`=k0H/kjH`
H	Num. term	`=B(1-D)C`
I	Den. term	`=IF(j=0;B;GB(1-D))`
J	Sum num.	`=SUM(H[j≥1])`
K	Sum den.	`=V0+SUM(GB(1-D)[j≥1])`
L	$\hat{C}_{i,F}^{eq}$	`=C_F0+J/K`

7.4 | Parameterization of molecular effects

This section explains how molecular effects are translated into Henry-type partition coefficients ki,jH \,k_{i,j}^H\,ki,jH from physico-chemical data available in public databases such as PubChem (see Kim et al., 2024) or its aggregated version PubChemLite (see Schymanski et al., 2025). The approach relies on well-known techniques from the scientific literature (see Gillet et al., 2009, Gillet et al., 2010, Vitrac and Gillet, 2010, Nguyen et al., 2016, Nguyen et al., 2017, Vitrac et al., 2022), but is deliberately simplified and automatable, so it can be applied to hundreds or even thousands of substances using a spreadsheet or an online query.

The method has been implemented in SFPPy (see Vitrac 2025a) and SFPPyLite (see Vitrac 2025b) and integrated into AI agents built on top of them. It provides an operational framework (for risk assessment and regulatory modeling) while retaining a solid theoretical foundation that supports updates, extensions to new families of compounds, and generalization to other migration and material–content interaction domains.

7.4.1 | Link to activity coefficients

$k_{i,j}^H$ $j$ $k_{i,j}^H$ $j$ . In condensed phases, attractive or repulsive interactions between neighbors modify heats of mixing: exothermic (attraction, spontaneous dissolution) or endothermic (dissolution requiring heat input). Even without specific interactions, any solute whose size differs from the polymer’s rigid unit (the monomer) imposes a local reorganization, changing the entropy of mixing compared with the pure components.

$i{+}j$ $i$ $j$ $i$ $j$ . The compressible Flory–Huggins theory at the atomic scale properly describes these attraction/repulsion and compaction/excess-free-volume effects (see Gillet et al., 2010, Vitrac and Gillet, 2010). However, it requires heavy molecular modeling and is not suitable for evaluating hundreds of substances within seconds.

$k_{i,j}^H$ $\gamma_{i,j}^v$ is:

\begin{matrix} (40) & k_{i, j}^{H} = \frac{P_{i, sat}^{(T_{K})} \cdot V_{i, mol}}{(1 - c_{j}) (1 - ϵ_{j})} \cdot γ_{i, j}^{v} \end{matrix}

$V_{i,\text{mol}}$ $i$ $c_j$ $\epsilon_j$ $j$ $P_{i,\text{sat}}^{(T_K)}$ $P_{i,\text{sat}}^{(T_K)}=1$ .

7.4.2 | Link to compressible Flory–Huggins theory

$\gamma_{i,j}^v$ $\chi_{i+j}$ $r_{i,j}\simeq V_{i,\text{mol}}/V_{j,\text{mol}}$ $i$ $j$ $n(r_{i,j})$ $j$ within the solute’s interaction sphere. We obtain:

\begin{matrix} (41) & \ln γ_{i, j} \approx χ_{i + j} + 1 - [r_{i, j} - n (r_{i, j})] \end{matrix}

Parameterization of entropic effects (volume and reorganization) $r_{i,j}=n(r_{i,j})\to 0$ .

For solutes, food simulants, or solid ink solvents, we use Miller’s empirical approximation for molar volume:

\begin{matrix} (42) & V_{k, mol} = 0.997 \cdot M_{k, mol}^{1.03} (k = i, j) \end{matrix}

$M_{k,\text{mol}}$ is the molar mass from the chemical formula.

$r_{i,j}\ge 1$ ), solvent molecules can reorganize and reduce the effective number of molecules displaced as the solute is introduced. A compressibility correction is then applied:

\begin{matrix} (43) & n (r_{i, j}) = \frac{r_{i, j}}{3} - 1 for r_{i, j} \geq 3 else 0 \end{matrix}

This correction reflects that bulky solutes occupy an effective space smaller than their geometric volume thanks to flexibility and possible rearrangements in the surrounding matrix.

Parameterization of enthalpic effects (interactions) $i$ $j$ cover both van der Waals and electrostatic interactions, including potential hydrogen bonding. As contact energies and coordination are not available without a 3D molecular representation, we resort to the quadratic regular-solution (Hildebrand–Scatchard) approximation as in Gillet et al. (2009):

\begin{matrix} (44) & χ_{i + j} = \frac{V_{ref}}{R T_{K}} (δ_{i} - δ_{j})^{2} \end{matrix}

$V_{\text{ref}}$ $j$ $i$ $\delta_k=\sqrt{\tfrac{\Delta H_{k,\text{mol}}^{\text{vap}}}{V_{k,\text{mol}}}}$ $k$ (at zero pressure) normalized by its molar volume.

$V_{\text{ref}}$ $i$ $j$ $P'$ $i$ between a solvent and a gas phase.

$\hat{P'}$ $(\log P)$ , widely available in databases:

\begin{matrix} (45) & χ_{i + j} \approx α [\hat{P^{'}} (\log P_{i}, V_{i, mol}) - \hat{P^{'}} (\log P_{j}, V_{j, mol})]^{2} \end{matrix}

7.4.3 | Building a polarity index scale inspired by Snyder

Snyder, 1974 $P'$ from solvent/air partition coefficients of three probe solutes (ethanol, dioxane, nitromethane) (see Rohrschneider ,1973). It is adapted here because (i) it is familiar in analytical chemistry, (ii) it reflects dipole–dipole interactions, hydrogen bonding, and other polar interactions, and (iii) it covers the main contaminants of recycled plastics.

Reference values used for calibration are given in Table 10. The lower bound (0) corresponds to n-hexane, and the upper bound (10.2) to water (neutral molecules). Ionic species or metal solvates can exceed this bound.

Table 10 — Reference values to calibrate a P' polarity scale, based on logP and Miller molar volumes.

Solvent	$P'$	logP	$(\text{cm}^3\!\cdot\!\text{mol}^{-1})$
n-hexane	0.0	3.90	98.2
toluene	1.8	2.73	105.2
dichloromethane	3.1	1.25	96.7
acetone	5.6	−0.21	65.4
ethanol	6.0	−0.24	51.5
acetonitrile	6.8	−0.22	45.8
methanol	8.1	−0.77	35.4
water	10.2	−1.38	19.6

$\gamma_{i,j}$ $k_{i,j}^H$ $P'$ uses the same Flory–Huggins theory applied to the octanol/water partition coefficient:

\begin{matrix} (46) & \ln 10 \cdot \log P_{k} + (r_{k, w} - r_{k, o}) \approx χ_{k + w} - χ_{k + o} = Δ χ_{k, o w} = α [(P_{k}^{'} - P_{w}^{'})^{2} - (P_{k}^{'} - P_{o}^{'})^{2}] \end{matrix}

which expands to:

\begin{matrix} (47) & Δ χ_{k, o w} = α [- 2 P_{k}^{'} (P_{w}^{'} - P_{o}^{'}) + (P_{w}^{'} + P_{o}^{'}) (P_{w}^{'} - P_{o}^{'})] = α (P_{w}^{'} - P_{o}^{'}) (P_{w}^{'} + P_{o}^{'} - 2 P_{k}^{'}) = C_{1} - C_{2} P_{k}^{'} \end{matrix}

$C_1$ $C_2$ $\alpha=\tfrac{C_2}{2(P'_w-P'_o)}$ .

Table 10 $C_1\approx 11.7$ $C_2\approx 1.49$ $P'_o=2\tfrac{C_1}{C_2}-P'_w\approx 5.5$ $P'=3.1$ $P'=5.6$ $\boxed{\alpha\approx 0.163}$ .

$P'\to 0$ ):

\begin{matrix} (48) & Δ χ_{k, o w} = A P^{' 2} + B P^{'} + C \end{matrix}

$\alpha$ $\boxed{A=0.181613}$ $\boxed{B=-3.412679}$ $\boxed{C=13.079540}$ . The polarity estimator is then:

\begin{matrix} (49) & \begin{matrix} {\hat{P}}_{k}^{'} (\log P_{k}, V_{k, mol}) = {\begin{cases} 10.2, & if Δ χ_{k, o w} \leq Δ χ_{k, o w}^{min} \approx - 4.11 \\ \frac{- B - \sqrt{B^{2} - 4 A (C - Δ χ_{k, o w})}}{2 A}, & if Δ χ_{k, o w}^{min} \leq Δ χ_{k, o w} \leq Δ χ_{k, o w}^{max} \approx 13.08 \\ 0, & if Δ χ_{k, o w} > Δ χ_{k, o w}^{max} \end{cases} \end{matrix} \end{matrix}

Note

Calibration update. $\alpha$ $\alpha=0.162$ ).

7.4.4 | Polarity indices for polymers

$\log P$ values. A robust estimate can nevertheless be obtained from their monomer or, more faithfully, from a saturated analogue/proxy. This molecular-modeling approach replaces the polymerizable function by hydrogens to reproduce the local steric environment of the repeating unit.

Table 11— Molecular homologues/proxies to estimate the effective polarity P' of common polymer repeat units.

Polymer	Monomer	$P'$	Substitute / analogue	$P'$	Remark
LDPE / HDPE / LLDPE	ethylene	4.7	n-hexane (LD/HD), iso-pentane (LLD)	0.2 / 1.8	Saturated, apolar chains (linear vs. slightly branched)
PP (isotactic)	propylene	3.9	iso-pentane (tert-butyl-like branching)	1.8	Reflects branching of PP
PS	styrene	0.9	ethylbenzene (saturated styrylic motif)	0.7	Faithful proxy of phenyl-alkyl group
HIPS	styrene	0.9	ethylbenzene (matrix) + butadiene (rubber)	0.7 / 2.8	Bicontinuous (PS + PB)
SBS	styrene	0.9	ethylbenzene (PS) + iso-pentane (PB)	0.7 / 1.8	PS/PB blocks
PMMA	methyl methacrylate	2.8	isobutyl acetate (compact ester)	2.0	Represents pendant ester function
PVC	vinyl chloride	3.3	1-chlorobutane (chlorinated alkane)	1.5	Increased polarity due to chlorine
PVAc	vinyl acetate	3.9	ethyl acetate (acetate ester)	3.9	Same functional family
PET	ethylene terephthalate	0.0	dimethyl terephthalate (DMT)	0.4	Aromatic ester proxy (DMT)
PBT	butylene terephthalate	0.1	dibutyl terephthalate	0.0	DMT usable as lower bound
PEN	ethylene naphthalate	0.0	dimethyl 2,6-naphthalene dicarboxylate	0.0	Faithful proxy of naphthalene-ester motif
PA6	caprolactam	4.5	n-hexanamide	2.6	Linear amide, realistic local proxy
PA66	hexamethylenediamine + adipic acid	0.7–3.6	adipamide (amide motif)	5.5	Di-functional amide, highly polar

Note

Usage notes

Proxies are notlocal solvation homologues/proxies $P'$ at Tier M2.
$P'$ $\log P$ reported in PubChem (XLOGP3), consistent with the simplified Flory–Huggins approach embedded in SFPPy.
Using proxies better discriminates apolar polymers (PE, PP) from polar ones (PA, PVAc, PMMA), especially in multilayer systems.

7.4.5 | Validation and prediction-error estimates

deliberately overestimate $\Delta\chi_{k,ow}$ $\chi_{i,w} > \chi_{i,o}$ $\tfrac{d\hat{P}'}{d\Delta\chi_{k,ow}} < 0$ . This is conservative for chemical risk assessment since it promotes migration into lipophilic phases.

Figure 2 $\pm 20\%$ $P'$ Figure 3 $R^2=0.713$ .

Figure 2 — Comparison of estimated χ_k,w-χ_k,o with tabulated P' values. Calibration data in red; validation data (35 substances) in blue. The model curve and its theoretical asymptotes at the bounds of P' are shown. The octanol estimate is highlighted in orange.

Figure 3 — Comparison of estimated and calculated P'. The dashed line is the 1:1 line.

Note

$\log P$ (2025 version). PubChem values (see Kim et al., 2024) use the XLOGP3 algorithm (see Cheng et al., 2007). For water an inconsistency appears: PubChem value: $\log P=-0.5$ (XLOGP3, 87 atom/group types). Expected value: $[H_2O]_{\text{octanol}} \approx 2.3\ \text{mol·L}^{-1}$ $\log_{10}\!\big(\tfrac{2.3}{55}\big)\approx -1.38$ , confirmed by AlogPS and ChemAxon. Consequence: $\log P=-0.5$ $P'=8.00$ $P'\approx 7.4$ ) instead of the theoretical upper bound 10.2. Impact: $0.1663 \cdot (10.2-8.00)^2 \simeq 14.7$ . This limitation remains acceptable as it yields a more conservative scenario (increased transfer to aqueous phases).

7.4.6 | Ready-to-use procedure to parameterize molecular effects

Table 12 — Calculation template for molecular effects (ready-to-use in a spreadsheet).

Col.	Content	Example cell (pseudo-Excel/Calc)
A	$i$	`"limonene"`
B	$\log P_i$	input (PubChem)
C	$V_{i,\text{mol}}$ $^3$ $^{-1}$ )	input (Miller or DB)
D	$j$	`"amorphous PET"`, `"PP (c=0.30)"`, `"ethanol"`, `"water"`
E	$V_{j,\text{mol}}$ (if condensed)	input (monomer/analogue proxy)
F	$r_{i,j}=V_{i,\text{mol}}/V_{j,\text{mol}}$	`=C/E`
G	$n(r)$	`=IF(F>=3, F/3-1, 0)`
H	$\Delta\chi_{i,ow}$	`=LN(10)*B + (r_i,w - r_i,o)`
I	$P'_i$ (positive root)	`=(-B - SQRT(B^2 - 4A(C - H)))/(2*A)` bounded to [0; 10.2]
J	$P'_j$	$j$ )
K	$\chi_{i+j}=\alpha (P'_i - P'_j)^2$	`=0.163*(I-J)^2`
L	$\ln\gamma_{i,j}\approx \chi_{i+j}+1-[r_{i,j}-n(r)]$	`=K+1-(F-G)`
M	$\gamma_{i,j}=\exp(\ln\gamma)$	`=EXP(L)`
N	$c_j$ $\epsilon_j$ (porosity)	input (0–1)
O	$k_{i,j}^H$ (condensed phase)	`=MV_i/((1-c_j)(1-epsilon_j))` $P_{\text{sat}}=1$ )
P	$k_{i,0}^H$ $F$ )	$j=0$
Q	$K_{i,F/j}=k_{i,j}^H/k_{i,0}^H$	`=O/P`
R	$K_{i,j/F}=k_{i,0}^H/k_{i,j}^H$	`=P/O`
S	Justification	source, version, assumptions

Computation steps
$P'$ $(\log P, M_i \text{ or } V_{i,\text{mol}})$ via the octanol/water calibration (constant parameters provided).
$\gamma_{i,j}$ (simplified Flory–Huggins, infinite dilution).
$k_{i,j}^H$ $c_j$ $\epsilon_j$ .
$K_{i,F/j}=k_{i,j}^H/k_{i,0}^H$ .

Calibration constants
$\alpha = 0.163$
$\Delta\chi_{k,ow}$ $A=0.181613$ $B=-3.412679$ $C=13.079540$
$\Delta\chi_{\min}\approx -4.11$ $P'=10.2$ $\Delta\chi_{\max}=C$ $P'=0$ )

Implementation notes
Gas phases: $k_{i,\text{air}}^H = R\,T_K$ .
Condensed phases only: $P_{\text{sat}}=1$ is consistent (thermodynamic equilibrium without a vapor phase).
Semi-crystalline polymersporous media $c_j$ $\epsilon_j$ reduce the available phaseamorphous/solid fraction $k_{i,j}^H$ $\sim \dfrac{1}{(1-c_j)(1-\epsilon_j)}$ .
Ionizable substances: $K_{i,F/j}\to\infty$ for aqueous contact); document separately.

7.4.7 | Worked example

Table 13 — Examples of Henry and partition coefficients (K_i,F/j) for limonene under typical conditions (SFPPy outputs).

Case	Pair (j/F)	$k_{i,j}^H$	$k_{i,F}^H$	$K_{i,F/j}=k_{i,j}^H/k_{i,F}^H$
1	$c=35\%$ ) / Ethanol	0.845	3.612	0.234
2	$c=35\%$ ) / Water	0.845	5.805	0.146
3	$c=50\%$ ) / Ethanol	1.351	3.612	0.374
4	$c=50\%$ ) / Water	1.351	5.805	0.233

Note

M2 reading. $K<1$ ). Water is the least favorable solvent, confirming the molecule’s hydrophobic nature. PP and PET behave similarly here and illustrate that fruit-juice terpenes sorb readily into most plastics.

7.5 | Comment on the temperature effect on partitioning

$j_1$ $j_2$ $\chi_{i,j_1}-\chi_{i,j_2}$ $j_1$ $j_2$ . The main cases are summarized in Table 14.

Table 14 — Temperature effect on partition coefficients.

Situation	Expected effect
Two endothermic mixtures (common case)	$K_{j_1/j_2}$ overall stable.
Specific interaction with one phase only	$T_K$ other $T$ has the opposite effect.
Presence of a gas layer	$T_K$ dominates: activation proportional to latent heat of vaporization. Volatiles migrate uniformly; low-volatiles show a marked threshold.

7.6 | Identified limitations

Tier M2 enables:

Incorporation of solubility, polarity, crystallinity, porosity, immobile fractions.
Differentiation of partition coefficients between layers.
Evaluation of barrier/retention layers, supporting recycled/multilayer design.
Routine application via spreadsheets or automated agents.

But: limitations include handling of ions/zwitterions, non-linear sorption, poorly characterized immobile fractions, multiphase tie-layers, food matrix effects, multi-solute competition, and recycled material variability.
Table M2-4 summarizes the main limitations and best practices.

Fast, explainable partition values come with limits that must be documented. They are summarized in Table 1.

Tip

Operational recommendations

$k_{i,j}^H$ .
$K_{i,F/j}$ from the “food ← material” perspective.
$\theta_{i,j}$ only if supported by experimental data (extraction, DSC).
$\hat{c}(t)$ $PR_i^E$ .

Table 15 — Main limits of Tier M2 and practical consequences.

Aspect	Limit	Consequence / Good practice
$P'$ indices, logP	Valid for neutral migrants. Out of scope for ions, zwitterions, siloxanes, heavy halogenated species, etc.	Document the source (PubChem/XLOGP3). If anomalies arise (e.g., water), retain the more solvating phase.
Henry’s law	Assumes a linear isotherm. Ignored: dual-mode sorption, plasticization, swelling.	Keep the unfavorable slope. Move to M3 when strong non-linearities are present.
$\theta_{i,j}$	Hard to estimate (DSC, extraction, NMR). Strong dependence on thermal history.	$\theta>0$ only with analytical evidence. Otherwise, use a lower bound.
Multiphase systems & interphases	Tie-layers, micro-domains, actual porosity not explicit.	Add an interphase compartment or bias toward the more solvating phase.
Real matrices vs simulants	pH, proteins, co-solvency not captured.	$P'$ to the matrix or choose the unfavorable scenario.
Multi-solute competition	Additivity simplified. Ignored: competition, co-solvency, complexation.	Use the most mobile migrant for screening. Switch to M3/experiments for critical cases.
Recycled materials	Variability, NIAS, pre-contamination.	$C_{i,j}^0$ (P90–P95). Include NIAS present at the LOD\textsuperscript{†}.
Temperature and time	$K$ $\theta$ $C^{\text{sat}}$ .	Move to M3 for long storage, pasteurization, or temperature cycling.
Adhesives, inks, varnishes	Oligomers, residual solvents, pigments.	$\theta$ minimal unless strongly evidenced.
Extensive quantities	Risk of mass/volume confusion, A/V, heterogeneous thicknesses.	$V_0$ ; document the calculation chain.
Uncertainties	Poorly quantified. Risk of over- or under-estimation.	Report ranges (low, central, high). Separate screening (M0–M2) from validation (M3).
Switch to M3	$K$ $T/t$ , plasticization, critical interphases, high health stakes.	Couple M2+M3 for higher accuracy.

^† $0.3\ \mathrm{mg}\,\mathrm{kg}^{-1}$ is generally a good approximation.

Key takeaways (Section 7)

Tier M2 enriches M1 by incorporating material/solute-specific properties (solubility, polarity, crystallinity, porosity, immobile fractions).
Partition coefficients become differentiated per layer and included in a filtered mass balance, enabling the evaluation of barrier or retention layers.
complementary $\bar v_i^*(t)$ ).
M2 is suitable for screening and design justification, while M3 refines results under real usage conditions (time, temperature, cycling).

8 | Advanced Tier M3: Diffusion dynamics, barrier layers, and functional barriers

Note

Micro-recap of what changes at M3

M3 adds time $\bar v_i^*(t)$ $D_{i,j}$ , and leverages two powerful properties: temporal commutativity (sum of Fourier numbers) and spatial additivity (superposition by layer). Use M3 together with M1/M2.

8.1 | Introduction

Tier M3 extends M1 and M2. While M1 provides a global mass balance and M2 refines it with thermodynamic partitioning at equilibrium, M3 explicitly introduces transfer kinetics. It couples equilibrium quantities with diffusive phenomena and ties real contact conditions (duration, temperature, sequences) to the effective migration.

M3 is never used alone: its results only make sense when combined with M1/M2, which supply thermodynamic framing and mass-balance coherence. It is indicated when:

contact time is short or variable (transport, forming, thermal treatments);
temperature varies strongly (processing, storage, use);
the effectiveness of one or more barrier layers must be demonstrated;
contamination is governed by diffusion kinetics rather than equilibria;
polymer plasticization, swelling, or aging affects diffusion;
recycled/aged materials make kinetics critical.

$\bar v_i^*(t)$ $D_{i,j}$ . Two fundamental properties structure the approach: temporal commutativity $Fo$ ), (ii) spatial additivity (superpose per-layer contributions).

$\hat D_{i,j}$ , then a summary emphasizing complementarity with M2. For overviews see Fang 2015, Zhu 2019; for molecular descriptors used in learning-based models see Vitrac 2006. Mathematical properties for mono- and multilayers appear in Vitrac 2005 and Nguyen 2013.

Important

Key idea.
Because diffusion laws are linear in time, two key properties follow: (i) temporal commutativitycumulative Fourier number $Fo^{(N)}$ ; (ii) spatial additivity, allowing you to decompose the final contamination into distinct contributions from each source layer. These greatly simplify complex scenarios (multiple barriers, storage/processing sequences) and partial severities by step or layer.

Note

A diffusive transport description is also assumed applicable to paper and board in a mean-field sense, though the microscopic picture involves diffusion plus sorption/desorption at fiber surfaces.

8.2 | Diffusive phenomena

$j_1$ $j_2$ balance; off-equilibrium, a net flux appears from the source layer to others.

$i$ through free-volume pockets created by thermal agitation of rigid polymer segments. Jump length and frequency depend on:

the ratio of molecular volumes (solute vs rigid segment),
$T-T_g$ ,
the fraction of free volume, linked to polymer density.

Important

$T>T_g$ $T<T_g$ ) it is moderate.

8.3 | Invariance and temporal commutativity

$D$
$m$ $l_j = V_j/A$ ), kinetics is controlled by the limiting layer:

\begin{matrix} (50) & j_{min} = \underset{1 \leq j \leq m}{\arg \min} {\frac{D_{i, j}}{k_{i, j}^{H} l_{j}}} . \end{matrix}

Define the cumulative Fourier number:

\begin{matrix} (51) & F o^{(N)} = \sum_{k = 1}^{N} Δ F o_{k}, Δ F o_{k} = \frac{D_{j_{min}} (T_{k})}{l_{j_{min}}^{2}} t_{k} . \end{matrix}

Note. $j_{\min}$ $i$ .

$N$ steps writes:

\begin{matrix} (52) & {\hat{C}}_{i, F}^{(N)} = {\hat{C}}_{i, F} (F o^{(N)}) . \end{matrix}

relative severity $\ell$ is:

\begin{matrix} (53) & {\hat{s}}_{i} |_{ℓ} = 100 \times \frac{max ({\hat{C}}_{i, F}^{(N)} - {\hat{C}}_{i, F}^{(N ∖ ℓ)}, {\hat{C}}_{i, F} |_{ℓ})}{ρ_{0} \cdot {SML}_{i}} . \end{matrix}

8.4 | Spatial additivity (superposition of contributions)

$s$ :

\begin{matrix} (54) & {\hat{C}}_{i, F}^{(N)} = \sum_{j = 1}^{m} {\hat{C}}_{i, j \to F}^{(N)}, {\hat{s}}_{i, s} = 100 \times \frac{{\hat{C}}_{i, s \to F}^{(N)}}{ρ_{0} \cdot {SML}_{i}} \end{matrix}

$\hat{C}_{i,j\rightarrow F}$ $j$ $F$ $\hat{s}_{i,j}$ its associated severity.

$F$ $G_{j\to F}(t)$ $C_{i,j}^0$ :

\begin{matrix} (55) & C_{i, F} (t) = \sum_{j = 1}^{m} (G_{j \to F} \otimes C_{i, j}^{0}) (t), \end{matrix}

$\otimes$ the convolution operator.

Each layer/component behaves as an independent source; contributions add without artificial interaction as long as the tracer diffusion assumption holds (properties independent of concentration).

Important

Practical note.
Superposition is valid only if all else is unchanged (number of layers, properties, boundary conditions). Only the initial contamination conditions may vary. This property applies to any source arrangement—1D, 2D or 3D; in series, in parallel, or any combination.

8.5 | Diffusivity prediction models

8.5.1 | Physical definitions

$D_{i,j}$ $i$ $j$ $\mathrm{m}^2\mathrm{s}^{-1}$ $\mathbf{r}(t)$ , averaged over directions (Fang 2015):

\begin{matrix} (56) & D_{i, j} = lim_{t \to \infty} \frac{⟨ ‖ r (t) - r (0) ‖^{2} ⟩}{6 t} . \end{matrix}

$\langle \cdot \rangle$ $t=0$ . Displacements are relative to the system’s center of mass to remove drift.

In condensed phases, three regimes are distinguished:
tracer diffusion $D$ constant),
mutual diffusion $D$ depends on composition),
self-diffusion $D$ constant).

$D_{i,j}$ : hydrodynamic Stokes–Einstein, free-volume theory (solute translation + polymer relaxation), and WLF (frequency vs temperature).

Important

concentration-independent $D$ hFV model $T_g$ .

8.5.2 | Model taxonomy

Three operational models are proposed—Piringer, Welle, hFV (hole Free-Volume)—with the following conservative hierarchy:

\begin{matrix} (57) & {\hat{D}}_{i, j}^{Piringer} \geq {\hat{D}}_{i, j}^{Welle} \geq {\hat{D}}_{i, j}^{hFV} \approx D_{i, j} . \end{matrix}

A usage comparison is given in Table 16.

Note

Model precedence (as in SFPPy; see Vitrac 2025a, Vitrac 2025b):
– hFV whenever the toluene/polymer couple is covered;
– Welle where polymer-specific parameters exist;
– Piringer as the universal envelope elsewhere.
Using hFV with toluene is preferred to qualify functional barrier effectiveness, notably for recycled/aged materials.

Table 16 — Compared properties of diffusivity prediction models

Model / effect	$\hat{D}_{i,j}^\text{Piringer}$	$\hat{D}_{i,j}^\text{Welle}$	$\hat{D}_{i,j}^\text{hFV}$
Principle	$M_i$ $T$ )	$V_{\text{vdW}}, T$	$T-T_g$ )
Parameters	$M, T, (A_{pp},\tau)$	$V_{\text{vdW}}, T, (a,b,c,d)$	$T,T_g,(D_0,\xi,K_a,K_b,r)$
$T_g$	No	Via parameter sets	Yes (continuous)
Plasticization	No	No	$\to$ self-diffusion)
Polymers covered	Broad (universal)	PET, PS/HIPS	PE, polyesters, PA, PV, etc.
Solutes covered	$M>40$ $^{-1}$	$M>40$ $^{-1}$	Rigid/linear solutes
Pros	Simplicity, upper bound	Tuned for glassy polymers	$T$ effects
Limits	Variable over-prediction; not conservative for very small solutes in glassy polymers	Spherical solutes; no plasticization	More complex; needs fine parametrization
SFPPy status	complete	complete	partial (toluene)

8.5.3 | Piringer model: a near-universal envelope

Begley 2005 $M_i$ $A_{pp}$ $\tau$ :

\begin{matrix} (58) & D_{i, j}^{Piringer} (M, T) = \exp [(A_{p p} - \frac{τ}{T_{K}}) - 0.135 M_{i}^{2 / 3} + 0.003 M_{i} - \frac{10454}{T_{K}}] \end{matrix}

$T_K = T\,[^\circ\mathrm{C}] + 273.15$ $M$ $^{-1}$
$(A_{pp},\tau)$ material-specific $(11.5,0)$ $(3.1,1577)$ $(6.4,1577)$ $(13.1,1577)$ $T_g$ nor plasticization.

Note

Use vs limits.
Use: broad coverage, robust upper envelope when specifics are missing.
Limit: may be non-conservative for very small molecules in glassy polymers. Applicability to paper/board is limited.

Table 17 — Polymer parameters for the Piringer model

Key†	className†	material†	$A_{pp}$	$\tau$
HDPE	HDPE	high-density polyethylene	14.5	1577
LDPE	LDPE	low-density polyethylene	11.5	0
LLDPE	LLDPE	linear low-density polyethylene	11.5	0
PP	PP	isotactic polypropylene	13.1	1577
aPP	PPrubber	atactic polypropylene	11.5	0
oPP	oPP	bioriented polypropylene	13.1	1577
pPVC	plasticizedPVC	plasticized PVC	14.6	0
PVC	rigidPVC	rigid PVC	–1.0	0
HIPS	HIPS	high-impact polystyrene	1.0	0
PBS	PBS	styrene-based polymer PBS	10.5	0
PS	PS	polystyrene	–1.0	0
PBT	PBT	polybutylene terephthalate	6.5	1577
PEN	PEN	polyethylene naphthalate	5.0	1577
PET	gPET	$T_g$ )	3.1	1577
rPET	rPET	$T_g$ )	6.4	1577
wPET	wPET	(duplicate of rPET)	6.4	1577
PA6	PA6	polyamide 6	0.0	0
PA6,6	PA66	polyamide 6,6	2.0	0
Acryl	AdhesiveAcrylate	acrylate adhesive	4.5	83
EVA	AdhesiveEVA	EVA adhesive	6.6	–1270
rubber	AdhesiveNaturalRubber	natural rubber adhesive	11.3	–421
PU	AdhesivePU	polyurethane adhesive	4.0	250
PVAc	AdhesivePVAC	PVAc adhesive	6.6	–1270
sRubber	AdhesiveSyntheticRubber	synthetic rubber adhesive	11.3	–421
VAE	AdhesiveVAE	VAE adhesive	6.6	–1270
board_polar	Cardboard	cardboard (polar migrants)	4.0	–1511
board_apol	Cardboard	cardboard (apolar variant)	7.4	–1511
paper	Paper	paper	6.6	–1900
gas	air	ideal gas	–	–

^† SFPPy nomenclature: material key, class name, material.

8.5.4 | Welle model: specific for PS/HIPS (and PET)

Ewender and Welle, 2022 $V_{i,\mathrm{vdW}}$ :

\begin{matrix} (59) & D_{i, j}^{Welle} (V_{vdW}, T) = 10^{- 4} b {(\frac{V_{i, vdW}}{c})}^{\frac{a - 1 / T_{K}}{d}} \end{matrix}

$a,b,c,d$ $a=1.93\cdot10^{-3}$ $b=2.27\cdot10^{-6}$ $c=11.1$ $d=1.50\cdot10^{-4}$ $T_g$ effects are implicit through distinct parameter sets.

Note

Use vs limits.
Use: validated accuracy for PET and PS/HIPS; uses true 3D size.
Limit: $V_{\mathrm{vdW}}$ (atomic connectivity). Parameters established for dry polymers, not equilibrated with ambient humidity.

Table 18 — Polymer parameters for the Welle model

Polymer key†	$a$ (1/K)	$b$ (cm²/s)	$c$ (Å³)	$d$ (1/K)
gPET	$1.93\times10^{-3}$	$2.27\times10^{-6}$	11.1	$1.50\times10^{-4}$
PS	$2.59\times10^{-3}$	$7.38\times10^{-9}$	55.71	$2.73\times10^{-5}$
rPS	$2.44\times10^{-3}$	$6.46\times10^{-8}$	25.51	$7.55\times10^{-5}$
HIPS	$2.55\times10^{-3}$	$9.21\times10^{-9}$	73.28	$2.04\times10^{-5}$
rHIPS	$2.46\times10^{-3}$	$2.07\times10^{-7}$	45.00	$2.07\times10^{-7}$

^† SFPPy nomenclature: polymer key.

8.5.5 | hFV model: rigid blob in free-volume theory

The reformulated free-volume theory (Fang et al., 2013; Zhu et al., 2019b) includes a smoothed glass transition and a scaling law for linear probes (rigid blobs). It is parameterized here for toluene on several polymers.

$T_g$ :

\begin{matrix} (60) & H (T) = \frac{1}{2} (1 + \tanh \frac{4}{Δ T} [T_{K} - T_{g}]), Δ T ≃ 2 K \end{matrix}

Thermal dilations:

\begin{matrix} (61) & α (T) = 1 + \frac{K_{a}}{T_{K} - T_{g} + K_{b}}, α_{g} (T) = 1 + \frac{K_{a}}{r (T_{K} - T_{g}) + K_{b}} \end{matrix}

Mixture:

\begin{matrix} (62) & α_{T} (T) = (1 - H) α_{g} (T) + H α (T) \end{matrix}

Free-volume measure:

\begin{matrix} (63) & P_{j, like} (T) = \frac{α_{T} (T) + β_{lin}}{0.24}, β_{lin} = 1 \end{matrix}

Diffusivity:

\begin{matrix} (64) & D_{i, j}^{hFV} (T) = D_{i, 0} \exp (- \frac{E_{i, j}}{R T_{K}}) \exp (- ξ_{i} P_{j, like} (T)) \end{matrix}

Note

Use vs limits.
Use: good accuracy over many conditions (polar/apolar solutes; glassy/rubbery polymers).
Limit: more complex; needs careful parameterization for arbitrary solutes.

Table 19 — Polymer parameters for the blob model applied to toluene

Polymer key†	$T_g$ (K)	$D_{i,0}$ (m²/s)	$\xi_i$	$K_a$ (K)	$K_b$ (K)	$r$
LDPE	148.15	$1.87\times10^{-8}$	0.6150	144	40	0.5000
PMMA	381.15	$1.87\times10^{-8}$	0.5600	252	65	0.5000
PS	373.15	$4.80\times10^{-8}$	0.5840	144	40	0.5000
PVAc	305.15	$1.87\times10^{-8}$	0.8600	142	40	0.5000
gPET	349.15	$1.02\times10^{-8}$	0.6761	252	65	0.6153
wPET	316.15	$1.02\times10^{-8}$	0.6761	252	65	0.2777

^† SFPPy nomenclature: polymer key.

8.6 | When to use Tier M3?

$\hat{D}_{i,j}$ $\hat{k}_{i,j}^H$ .
Thanks to tools such as FMECAengineSFPPy $\bar v^*(t)$ can be computed in < 0.1 s, enabling analysis of dozens/hundreds of substances across designs.

not $\hat{s}_i \le 100$ $\bar v^*(t)\to 1$ would not change your decision. Switch to M3 if:

$\bar v^*(t)$ or on its optimization;
transfer mechanisms require explicit kinetics;
multiple layers function as barriers and need dynamic coupling.

Figure 4 — Decision guide for choosing M2 alone vs M2+M3

Key takeaways for Section 8

Blue box — What to remember.
M3 enriches M2 by introducing diffusion kinetics, indispensable when time, temperature, or barrier performance governs migration. Three model families are available: Piringer (universal envelope, conservative), Welle (PET/PS/HIPS-specific correlations, more realistic), and hFV (free-volume physics, accurate but more complex).

Tip

Their hierarchical combination, already implemented in SFPPy (Vitrac 2025a, Vitrac 2025b), yields explainable, consistent predictions for both screening and in-depth evaluation of recycled materials, with uncertainties appropriately framed.

9 | Risk reduction by optimization

9.1 | Objective and stakes

Risks arise from formulation (substances, concentrations), usedesign $i$ $\hat{s}_{i,\max}\le 100$ $\hat{C}_{i,F}\le \hat{C}_{i,F}^{\max}$ (maximum admissible concentration in food).

Contamination from multiple sources is common (layers of the same material, multiple printings, multi-function additives such as benzophenones, use of recyclates). It requires linked decisions across components.

Note

Operational note. Optimization is applied substance by substance, targeting the most critical compounds due to quantity, toxicity, or propensity to migrate. An equivalentsurrogate/proxy molecules $D$ $k$ ).
An article/package is rarely designed for a single product/use. Conception should therefore remain robust across several use cases.

A large-scale example (multi-criteria, multiple constraints and parameters) for a bottle-type packaging can be found in Zhu 2019.

9.2 | Useful properties and identities

Two indicators help rank layers/components to target first, assuming the food is not pre-contaminated:

current relative contribution $s$ :

\begin{matrix} (65) & u_{i, s} = \frac{{\hat{C}}_{i, s \to F}^{(N)}}{{\hat{C}}_{i, F}^{(N)}}, 0 \leq u_{i, s} \leq 1, \sum_{s = 1}^{m} u_{i, s} = 1 \end{matrix}

propensity to transfer $s$ , i.e., its marginal leverage:

\begin{matrix} (66) & w_{i, s} = \frac{V_{0}}{V_{s}} \cdot \frac{{\hat{C}}_{i, s \to F}^{(N)}}{{\hat{C}}_{i, s}^{0}}, 0 \leq w_{i, s} \leq 1 \end{matrix}

Total contamination and the associated severity are:

\begin{matrix} (67) & {\hat{C}}_{i, F}^{(N)} = \sum_{s = 1}^{m} w_{i, s} \cdot \frac{V_{s}}{V_{0}} \cdot {\hat{C}}_{i, s}^{0}, {\hat{s}}_{i}^{(N)} = \frac{100}{S M L_{i}} {\hat{C}}_{i, F}^{(N)} \end{matrix}

The general optimization constraint is:

\begin{matrix} (68) & \sum_{s = 1}^{m} \frac{V_{s}}{V_{0}} w_{i, s} {\hat{C}}_{i, s}^{0, new} \leq C_{i, F}^{max} \end{matrix}

Two levers (marginal sensitivities) follow:

\begin{matrix} (69) & η_{i, s} = \frac{\partial {\hat{C}}_{i, F}^{(N)}}{\partial {\hat{C}}_{i, s}^{0}} = \frac{V_{s}}{V_{0}} w_{i, s}, \frac{\partial {\hat{C}}_{i, F}^{(N)}}{\partial w_{i, s}} = \frac{V_{s}}{V_{0}} {\hat{C}}_{i, s}^{0} . \end{matrix}

Note

Reading the levers. $\eta_{i,s}$ is the marginal sensitivity of total contamination to a change in initial content. The second lever quantifies sensitivity to a change in transfer rate via design or use. Its effect scales with the initial contaminant load, confirming that such modifications should first target the most formulated or most contaminated parts/components.

9.3 | Mitigation rules (content and transfer)

9.3.1 | General principles

For a single source, options are straightforward:

avoidance: substitute the substance or remove the source;
mitigation: reduce content, add a barrier, adapt use conditions.

With multiple sources, it is essential to distinguish:

$u_{i,s}$ : current contribution,
$w_{i,s}$ : latent risk of future transfer.

The joint reading is illustrated in Table 20.

Table 20 — Joint reading of indicators u_i,s (current contribution) and w_i,s (future propensity).

Configuration	Interpretation	Comment
$u$ $w$	Priority layer	Already contributes and transfers strongly
$u$ $w$	Contribution due to high content	Content reduction is effective
$u$ $w$	“Hot” layer with latent risk	$Fo$ ) before rise
$u$ $w$	Low priority	Marginal contribution

Note

On the indicators. $u_{i,s}$ is composite: it includes mass-action (transfer ∝ amount present) plus design effects. $w_{i,s}$ is more specific to causes (physico-chemistry and design), independent of the present quantities.

9.3.2 | Uniform content reduction

$\lambda$ the reduction factor:

\begin{matrix} (70) & {\hat{C}}_{i, s}^{0, new} = λ {\hat{C}}_{i, s}^{0}, λ = min (1, \frac{C_{i, F}^{max}}{\sum_{r} \frac{V_{r}}{V_{0}} w_{i, r} {\hat{C}}_{i, r}^{0}}) . \end{matrix}

Practical note (recyclates). $\Delta_2$ is:
$\begin{matrix} (71) & Δ_{2} = 1 - λ (1 - Δ_{1}), \end{matrix}$
$\Delta_1$ is the initial decontamination rate.

9.3.3 | Selective prioritization by marginal impact (step-by-step)

$\eta_{i,s}$ $\eta_{i,s}$ is largest.

$\kappa_s$ $\eta_{i,s}$ by a cost-effectiveness index

\begin{matrix} (72) & ν_{i, s} = \frac{η_{i, s}}{κ_{s}} . \end{matrix}

Iterate mitigation until

\begin{matrix} (73) & \sum_{s} \frac{V_{s}}{V_{0}} w_{i, s} {\hat{C}}_{i, s}^{0, new} \leq C_{i, F}^{max} . \end{matrix}

$u$ $w$

$\eta_{i,s}=\tfrac{V_s}{V_0}\,w_{i,s}$ $u_{i,s}$ . Table 21 introduces composite indices covering both aspects.

Table 21 — Prioritization indices combining u_i,s and w_i,s.

Indicator	Interpretation	Properties
$R^{(\times)}_{i,s}=u_{i,s}\,\eta_{i,s}$	Priority to layers that already contribute and transfer strongly	$u$ $w$ (i.e., never mis-prioritizes)
$R^{(\alpha)}_{i,s}=u_{i,s}^{\beta}\,\eta_{i,s}^{1-\beta}$	$\beta$ )	$\beta\uparrow$ $\beta\downarrow$ : emphasize marginal leverage
$R^{(\times,Euros)}_{i,s}=\dfrac{u_{i,s}\,\eta_{i,s}}{\kappa_s}$	Cost-effectiveness index	Impact per euro invested

Note

Coupling content & design. $\kappa_s$ content reduction $\mathbf{C}$ barrier improvement $\mathbf{w}$ ). This couples optimization of content and design in one framework.

$w$

$w_{i,s}$ :

$Fo$ (time/temperature),
$K_{F/s}$ (material choice),
$j_{\min}$ ,
$\theta_{i,s}$ (immobilization, crystallinity).

Caution

Focusing on $u_{i,s}$ alone (“who contributes today”) can miss a high $w_{i,s}$ tomorrow $\eta_{i,s}=(V_s/V_0)\,w_{i,s}$ .

9.4 | Safe design as a linear programming problem

$\kappa_s$ not only prices design modifications on an existing package, it also prioritizes solutions that are technically/economically feasible (available and effective).

Safe design becomes a cost-minimization problem with a health-safety constraint. The cost may include raw material, package mass, recycling, etc. Compactly:

\begin{matrix} (74) & min_{{Δ C_{s}}} \sum_{s = 1}^{m} κ_{s} Δ C_{s} s.t. \sum_{s = 1}^{m} \frac{V_{s}}{V_{0}} w_{i, s} ({\hat{C}}_{i, s}^{0} - Δ C_{s}) \leq C_{i, F}^{max} . \end{matrix}

$w_{i,s}$ sequential $w$ $C$ , then vice-versa). The computation chain is summarized in Figure 5.

Figure 5 — Optimization chain driven by the joint values of u_i and w_i.

Key takeaways (Section 9)

The optimization framework formalizes risk reduction through targeted actions: content reduction, barriersuse conditions $u_{i,s}$ $w_{i,s}$ $\eta_{i,s}$ $\kappa_s$ , safe design becomes a linear programming problem that reconciles safety and feasibility—either substance-wise or via homologues/proxies representative of critical migrants.

10 | Application perspectives with SFPPy and Python

This section illustrates how to operationalize the concepts of tiers M1–M3 with SFPPy (Vitrac 2025a, 2025b) and Python, through five progressive examples. Code snippets follow the idioms in example1.py–example5.py and the project README. They cover the full chain: (i) defining migrants and layers, (ii) M2 partitioning, (iii) M3 dynamics (time/temperature sequences), (iv) visualization, and (v) automation (fitting, optimization, AI integration).

Note

The SFPPy project (Safe Food Packaging Portal in Python) is an open-source initiative that implements the principles presented in this article as Python scripts and notebooks. Based on a finite-volume solver (Patankar method, see Nguyen et al., 2013) and interfaced with public databases (PubChem Kim et al., 2024, ToxTree Patlewicz et al., 2008, EU/US/CN regulations), SFPPy bridges theory and operational simulation in a few lines of code — or without local installation via online notebooks (e.g., Google Collab) or browser-based WebAssembly builds (Vitrac 2025b).

10.1 | Example 1 — Transfer simulation through a monolayer

Objective of example 1
Estimate partitioning and potential migration of an additive (e.g., Irganox 1076) from a thin LDPE film wrapping a sandwich, then quickly evaluate the effect of duration/temperature (simplified M3).

Table 21 summarizes the studied configurations.

Table 21 — Parameters of Example 1.

Properties	Values
Film	$l=100\,\mu$ m
Food (simulant)	Cylindrical shape (Ø=19 cm, L=19 cm), ethanol 95%
Migrant	Irganox 1076 (CAS 2082-79-3), properties from public DBs
Outputs	$K_{F/\text{LDPE}}$ $\hat{C}_{i,F}$ $\hat{s}_i$ after 10 days at 7 °C, plus 4 h at 25 °C

Listing 1 — Simulation of transfers between LDPE film and a sandwich


xxxxxxxxxx
92
1
# Listing 1 — Simulation of transfers between LDPE film and a sandwich
2

3
# ==========================================================================
4
#      ////////////////////// Minimalist version //////////////////////
5
# ==========================================================================
6
# Inspired by example1.py (SFPPy distribution)
7
from patankar.layer import LDPE
8
from patankar.loadpubchem import migrant
9
from patankar.geometry import Packaging3D
10
import patankar.food as food
11

12
# Geometry: cylinder (h=19 cm, r=30 mm)
13
internal_volume, surface_area = Packaging3D(
14
    "Cylinder", length=(19, "cm"), radius=(30, "mm")
15
).get_volume_and_area()
16

17
# Migrant + film
18
m = migrant("irganox 1076")
19
film = LDPE(migrant=m, l=(100, "um"))
20

21
# Food proxy: fatty semi-solid sandwich
22
class Sandwich(food.realfood, food.semisolid, food.fat): name = "sandwich"
23
F = Sandwich(volume=internal_volume, surfacearea=surface_area,
24
    contacttime=(10,"days"), contacttemperature=(7,"C"),
25
    substance=m, simulant="ethanol")
26

27
# Tier M2 — partition
28
K_F_over_P = film.k / F.k0
29

30
# Tier M3 — kinetics
31
kin = F.migration(film)
32
CF = kin.CFtarget
33
s = 100 * CF / m.SML
34

35
# ==========================================================================
36
#       ////////////////////// Detailed version //////////////////////
37
# ==========================================================================
38
# --- Imports -------------------------------------------------------------
39
from patankar.layer import LDPE          # LDPE model
40
from patankar.loadpubchem import migrant # PubChem + props
41
from patankar.geometry import Packaging3D# 3D geometry
42
import patankar.food as food             # food/simulants
43

44
# --- 1) Geometry: cylinder (height x radius) -----------------------------
45
# Units as tuples (value, "unit"), handled by SFPPy.
46
internal_volume, surface_area = Packaging3D(
47
   "Cylinder",length=(19, "cm"),  # height = 19 cm
48
    radius=(30, "mm")   # radius = 30 mm (diam = 60 mm)
49
    ).get_volume_and_area()
50
print(f"[Geometry] V = {internal_volume.item():.6g} m^3")
51
print(f"[Geometry] A = {surface_area.item():.6g} m^2")
52

53
# --- 2) Migrant + LDPE film ---------------------------------------------
54
# Example: Irganox 1076 (phenolic AO) from public DBs.
55
m = migrant("irganox 1076")
56
film = LDPE(migrant=m, l=(100, "um"))  # 100 um
57
print(f"[Migrant] {m.compound} | CAS={m.CAS} | "
58
      f"M={m.M} g/mol | logP={m.logP}")
59

60
# --- 3) Food proxy: semi-solid, fatty sandwich ---------------------------
61
class Sandwich(food.realfood,food.semisolid,food.fat):name = "sandwich"
62
F = Sandwich(volume=internal_volume, surfacearea=surface_area,
63
    contacttime=(10, "days"), contacttemperature=(7, "C"),
64
    substance=m, simulant="ethanol")  # fatty matrix proxy
65

66
# --- 4) Equilibrium partition (Tier M2) ----------------------------------
67
# K_{F/P} = k_P / k_F
68
K_F_over_P = film.k / F.k0
69
print(f"[M2] K_(F/LDPE) = {K_F_over_P.item():.4g} "
70
       "(<1: stronger LDPE affinity)")
71

72
# --- 5) Kinetics (Tier M3): 10 d @ 7 C -----------------------------------
73
# SFPPy couples partition + diffusion with T,t.
74
kin = F.migration(film)
75
CF = kin.CFtarget
76
s = 100 * CF / m.SML
77
print(
78
 f"[M3] 10 d @ 7 C: C_F = {CF.item():.6g} mg/kg, "
79
 f"severity = {s.item():.2f}"
80
 )
81
# Optional plots: time course + layer profiles
82
kin.plotCF() # kinetics
83
kin.plotCx() # concentration profiles
84

85
# --- 6) Step 2: +4 h @ 25 C (warming before use) -------------------------
86
F2 = F.copy().update( contacttime=(4, "hours"),contacttemperature=(25, "C"))
87
kin2 = kin >> F2 # simulation chaining with ">>"
88
s2 = 100 * kin2.CFtarget / m.SML # updated severity
89
print(f"[M3] +4 h @ 25 C: C_F = {kin2.CFtarget.item():.6g} mg/kg, "
90
      f"severity = {s2.item():.2f}")
91
# Cumulative kinetics over both steps
92
(kin + kin2).plotCF()

Typical outputs (Listing 2):

Output 1 — outputs of listing 1


xxxxxxxxxx
6
1
[Geometry] V = 0.000537212 m^3
2
[Geometry] A = 0.041469 m^2
3
[Migrant] irganox 1076 | CAS=['2082-79-3'] | M=530.9 g/mol | logP=[13.8]
4
[M2] K_(F/LDPE) = 6.741 (<1: stronger LDPE affinity)
5
[M3] 10 d @ 7 C: C_F = 5.22691 mg/kg, severity = 87.12
6
[M3] +4 h @ 25 C: C_F = 5.56426 mg/kg, severity = 92.74

Takeaway. This example shows the SFPPy syntax and basic M2/M3 calculations from just the compound’s name. Severities are computed automatically. While 4 h at 25 °C breaks cold chain for microbiological risk, it has minor impact on chemical risk.

10.2 | Example 2 — Functional barrier in recycled materials

Objective of example 2
Assess migration of *toluene* from a 300 µm recycled polypropylene (rPP) bottle wall into a fatty liquid food, then evaluate the efficiency of a PET functional barrier (FB) of various thicknesses).

Table 22 gives the setup; Listing 3 shows the script.

Table 22 — Parameters of Example 2.

Properties	Values
Geometry	$V\approx 1$ L (cylinder + neck)
Migrant	Toluene (CAS 108-88-3), typical NIAS proxy
Wall	$l=300\,\mu$ $C_0=10$ mg/kg
Barrier	PET FB, base case 30 µm, variants 2–60 µm
Food	Fatty liquid, 450 d @ 20 °C
Outputs	$\hat{C}_{i,F}(t)$ , profiles, severity, FB efficiency

Listing 2 — Optimizing FB thickness for a PP bottle (SFPPy, example2.py)


xxxxxxxxxx
84
1
# Listing 2 — Optimizing FB thickness for a PP bottle (SFPPy, example2.py)
2
# ==========================================================================
3
#      ////////////////////// Minimalist version //////////////////////
4
# ==========================================================================
5
from patankar.loadpubchem import migrant
6
from patankar.geometry import Packaging3D
7
import patankar.food as food
8
import patankar.layer as polymer
9
from patankar.migration import senspatankar, CFSimulationContainer
10

11
# Bottle geometry (1 L)
12
bottle = Packaging3D("bottle",
13
    body_radius=(40,"mm"), body_height=(0.2,"m"),
14
    neck_radius=(1.8,"cm"), neck_height=0.05)
15
Vint, Acontact = bottle.get_volume_and_area()
16

17
# Migrant + rPP wall
18
tol = migrant("toluene")
19
PP = polymer.PP(l=(300,"um"), substance=tol, C0=10, T=(20,"C"))
20

21
# Food proxy
22
class FattyLiquid(food.realfood, food.liquid, food.fat): name="fattyFood"
23
F = FattyLiquid(volume=Vint, surfacearea=Acontact,
24
    contacttime=(450,"days"), contacttemperature=(20,"C"))
25

26
# Simulation without FB
27
ref = senspatankar(PP, F)
28

29
# Add 30 µm PET FB
30
PETfb = polymer.wPET(l=(30,"um"), substance=tol, C0=0, T=(20,"C"))
31
walls_FB = PETfb + PP
32
fb = senspatankar(walls_FB, F)
33

34
# ==========================================================================
35
#       ////////////////////// Detailed version //////////////////////
36
# ==========================================================================
37
# --- Imports -------------------------------------------------------------
38
from patankar.loadpubchem import migrant      # migrants (PubChem)
39
from patankar.geometry import Packaging3D     # 3D bottle geometry
40
import patankar.food as food                  # food/simulants
41
import patankar.layer as polymer              # polymer layers
42
from patankar.migration import senspatankar   # solver
43
from patankar.migration import CFSimulationContainer as store
44
from patankar.layer import _toSI
45

46
# --- 1) Bottle geometry (1 L) -------------------------------------------
47
bottle = Packaging3D("bottle",
48
    body_radius=(40,"mm"), body_height=(0.2,"m"),
49
    neck_radius=(1.8,"cm"), neck_height=0.05)
50
Vint, Acontact = bottle.get_volume_and_area()
51
print(f"[Geometry] V={Vint.item():.3e} m3, A={Acontact.item():.3e} m2")
52

53
# --- 2) Migrant + rPP wall ----------------------------------------------
54
tol = migrant("toluene")
55
PP = polymer.PP(l=(300,"um"), substance=tol, C0=10, T=(20,"C"))
56

57
# --- 3) Food proxy -------------------------------------------------------
58
class FattyLiquid(food.realfood,food.liquid,food.fat): name="fattyFood"
59
F = FattyLiquid(volume=Vint, surfacearea=Acontact,
60
    contacttime=(450,"days"), contacttemperature=(20,"C"))
61

62
# --- 4) Simulation without FB -------------------------------------------
63
ref = senspatankar(PP, F, name="rPP_bottle")
64
ref.plotCF(); ref.plotCx()
65

66
# --- 5) Add PET functional barrier (30 um) -------------------------------
67
PETfb = polymer.wPET(l=(30,"um"), substance=tol, C0=0, T=(20,"C"))
68
walls_FB = PETfb + PP
69
fb = senspatankar(walls_FB, F, name="rPP_bottle_FB")
70
fb.plotCF(); fb.plotCx()
71

72
# --- 6) Compare with/without FB ------------------------------------------
73
comp = store(name="bottles")
74
comp.add(ref, "without FB","b"); comp.add(fb,"with PET FB","m")
75
comp.plotCF()
76

77
# --- 7) Systematic study: FB thickness (2-60 um) ------------------------
78
study = store(name="FBthickness")
79
study.add(ref,"without FB","k")
80
for lfb in range(2,61,4):
81
    walls_FB.l[0] = _toSI((lfb,"um")).item()
82
    sim = senspatankar(walls_FB,F,name=f"FB{lfb}um")
83
    study.add(sim, f"PET {lfb} um")
84
study.plotCF()

Results. Figure 6 shows kinetic profiles for varying PET FB thickness. Key points:

rapid contamination without FB (reference),
drastic reduction with 30 µm PET (even plasticized “wPET”),
only moderate pure lag effect (weeks).

Figure 6 — Simulated kinetics for various PET FB thicknesses (Example 2).

Tip

Takeaway. SFPPy models functional barriers (FB) without extra equations: same M2/M3 solvers apply. Thickness sweeps directly size the FB needed for compliance. Full script: example2.py.

10.3 | Example 3 — Multilayer structure and sequential scenarios

Objective of example 3
Simulate limonene migration from a 500 µm rPP core through a PET/PP/PET (ABA) tray, across 3 steps: (1) 4 mo @ 20 °C storage, (2) hot fill @ 90 °C with fatty food, (3) 6 mo @ 30 °C storage.

Variants: (v1) substitute with toluene, (v2) halve PET layers (20→10 µm), (v3) combine v1+v2.

Table 23 summarizes the studied configurations.

Table 23 — Parameters of Example 3.

Properties	Values
Geometry	Open box 19×10×8 cm (rectangular prism)
Migrant	$C_0=200$ mg/kg; variant: toluene
Structure	ABA: PET 20 µm / rPP 500 µm / PET 20 µm
Steps	(1) 4 mo @ 20 °C, (2) hot fill 90 °C, (3) 6 mo @ 30 °C
Outputs	Concentration profiles, cumulative kinetics, variant comparisons

Key findings (Figure 6):

PET 20 µm stabilizes contamination (steady permeability regime),
toluene (v1) migrates more (higher mobility),
thinner PET (v2) worsens migration,
v3 is worst case.

Listing 3 — Multilayer structure and sequential scenarios (SFPPy, example3.py)


xxxxxxxxxx
50
1
# Listing 3 — Multilayer structure and sequential scenarios
2
# Imports
3
from patankar.loadpubchem import migrant
4
from patankar.layer import gPET,wPET,PP
5
from patankar.geometry import Packaging3D
6
from patankar.food import setoff,ambient,hotfilled,realfood,liquid,fat
7
from patankar.migration import CFSimulationContainer as store
8

9
# --- 1) Geometry ----------------------------------------------------------
10
box = Packaging3D("box_container", length=(19,"cm"),
11
                  width=(10,"cm"), height=(8,"cm"))
12
Vint, Acontact = box.get_volume_and_area()
13

14
# --- 2) Migrant and layers ------------------------------------------------
15
m = migrant("limonene")
16
A1 = wPET(l=(30,"um"), migrant=m, C0=0)
17
B  = PP(l=(0.5,"mm"), migrant=m, CP0=200) # CP0 or C0 are aliases
18
A2 = gPET(l=(30,"um"), migrant=m, C0=0)
19
ABA = A1 + B + A2  # trilayer structure
20

21
# --- 3) Contact conditions ------------------------------------------------
22
class Step1(setoff,ambient): contacttime=(4,"months"); contacttemperature=(20,"C")
23
class Step2(hotfilled,realfood,liquid,fat): pass
24
class Step3(ambient,realfood,liquid,fat): contacttime=(6,"months"); contacttemperature=(30,"C")
25
F1,F2,F3 = Step1(), Step2(), Step3()
26
box >> F1 >> F2 >> F3  # propagate geometry
27

28
# --- 4) Chained simulation (>> operator) ----------------------------------
29
F1 >> ABA >> F1 >> F2 >> F3
30
sol_ref = F1.lastsimulation + F2.lastsimulation + F3.lastsimulation
31

32
# --- 5) Variants ----------------------------------------------------------
33
# operator % is used to inject substances
34
m2 = migrant("toluene")
35
m2 % F1 >> ABA >> F1 >> F2 >> F3
36
sol_v1 = F1.lastsimulation + F2.lastsimulation + F3.lastsimulation
37
m % F1 >> ABA.copy(l=[10e-6,0.5e-3,10e-6],migrant=m) >> F1>>F2>>F3
38
sol_v2 = F1.lastsimulation + F2.lastsimulation + F3.lastsimulation
39
m2 % F1 @ ABA.copy(l=[10e-6,0.5e-3,10e-6],migrant=m2) >> F1>>F2>>F3
40
sol_v3 = F1.lastsimulation + F2.lastsimulation + F3.lastsimulation
41

42

43
# --- 6) Compare all cases -------------------------------------------------
44
comp = store(name="ABA study")
45
comp.add(sol_ref, "Limonene with 20 um thick FB (ref)", "Teal", linewidth=3, linestyle='--')
46
comp.add(sol_v1, "Toluene with 20 um thick FB (v1)", "Crimson", linewidth=3)
47
comp.add(sol_v2, "Limonene with 10 um thick FB (v2)", "deepskyblue", linewidth=2, linestyle='--')
48
comp.add(sol_v3, "Toluene-FB with 10 um thick FB (v3)", "tomato", linewidth=2)
49
hfig = comp.plotCF()
50
hfig.print('cmp_pltCF_container_with_FB',destinationfolder="tmp/",overwrite=True)

Figure 6 — Simulated kinetics for ABA tray and variants (Example 3).

!Note]
Takeaway. SFPPy’s symbolic operators: + (layer assembly/sum), >> (step chaining), % (migrant substitution), @ (thermal reinit). Applicable at M3 for kinetics and release potentials.

10.3.1 | Intrinsic Release Potentials

The extension proposed in Listing \ref{lst:Ex3multilayerPR}intrinsic potential releases per step $PR_i|_k$ intrinsic potentials per layer $PR_{i,j}|_k$ $j$ in turn. Comparison with the full sequence allows the specific contribution of each step or layer to be inferred (see Eq. \ref{eq:PRdependence}).

Listing 4 — Calculate potential release corresponding to listing 3


xxxxxxxxxx
57
1
# %% Example 3 - Extension (1/2) - Potential Release (PR)
2
# ---------------------------------------------------------
3
"""
4
Analysis at the scale of the full packaging (the initial distribution of contaminants is preserved)
5
    PR = PRE * PRT
6

7
PR : ndarray of shape (n_steps,)
8
    Array of intrinsic potential release values for each step in the
9
    current sequence of transfer (e.g. storage, hot fill, long-term storage).
10
"""
11
import numpy as np
12

13
# effective potential release of steps 1+2+3, 2+3, 1+2
14
# Fj @ ABA or Fj >> ABA = thermalize
15
# ABA >> Fj >> Fj+1 propagates migration from ABA to Fj and then Fj+1
16
sim123 = m2 % F1 >> ABA >> F1 >> F2 >> F3
17
PRall123 = np.array([m.lastsimulation.PR.PRtarget_effective for m in [F1,F2,F3]])
18
F2 @ ABA >> F2 >> F3  # step 1 omitted
19
PRall23 = np.array([m.lastsimulation.PR.PRtarget_effective for m in [F2,F3]])
20
F1 @ ABA >> F1 >> F3  # step 2 omitted
21
PRall13 = np.array([m.lastsimulation.PR.PRtarget_effective for m in [F1,F3]])
22

23
# intrinsic potential release for steps: 1=1+2+3\2+3, 2=1+2+3\1+3, 3=1+2+3\1+2
24
PR = np.zeros((len(PRall123),), dtype=float)
25
PRN = PRall123[-1,0]  # potential release with all steps
26
PR[0] = 1.0 - (1.0-PRN)/(1.0-PRall23[1,0])
27
PR[1] = 1.0 - (1.0-PRN)/(1.0-PRall13[1,0])
28
PR[2] = 1.0 - (1.0-PRN)/(1.0-PRall123[1,0])
29
print("PR =", PR)
30
print("PRT =", PR/F3.lastsimulation.PR.PRE_effective)
31
print("score = PR/PRN =", PR/PRN)
32
# compute dependency error on intrinsic PR
33
PRNcontrol = (1-np.prod(1-PR))
34
print(f"PR dependency ~ {(PRNcontrol/PRN - 1).item()*100:0.3f} %")
35

36

37
# %% Example 3 - Extension (2/2) - layer-based Potential Release (PR)
38
# --------------------------------------------------------------------
39
"""
40
Analysis at the scale of each layer (j=1,2,3).
41

42
PR_layers : ndarray of shape (n_steps, n_layers)
43
    Array of intrinsic potential release values per step and per layer
44
    in a multilayer system, reconstructed by comparison of full and
45
    step-omitted sequences.
46
"""
47
R123 = F1.potentialRelease(ABA) >> F2 >> F3
48
R23 = F2.potentialRelease(ABA) >> F3
49
R13 = F1.potentialRelease(ABA) >> F3
50

51
# intrinsic layer-based potential release
52
PR_layers = np.zeros((len(R123), len(ABA)), dtype=float)  # nsteps x nlayers
53
PRN_layers = R123.PR[2,:]
54
PR_layers[0,:] = 1.0 - (1.0-PRN_layers)/(1.0-R23.PR[1,:])
55
PR_layers[1,:] = 1.0 - (1.0-PRN_layers)/(1.0-R13.PR[1,:])
56
PR_layers[2,:] = 1.0 - (1.0-PRN_layers)/(1.0-R123.PR[1,:])
57
PRN_layers_control = 1 - np.prod(1-PR_layers, axis=0)

10.3.2 | Results and Interpretation (Package Scale)

The computed release potentials for the ABA tray are summarized in Table 24. PRN ≈ 0.338 $(F_1 + F_2 + F_3)$ $F_3$ alone accounts for about 97 %score[2] $F_2$ score[1] $F_1$ (storage without contact) still contributes about 11 %, moving the contaminant closer to the contact interface.

4.29 % $PR_i|_k$ $PR_i^{N=F_1+F_2+F_3}$ $F_1$ composite $A_1$ partial backflow $A_3$ $F_1$ durations significantly alter these contributions.

Table 24 — Release potentials per step (script extension 1/2).

↓ Variable / index `o` →	0	1	2	Interpretation
`PRall123[o]`	0	0.0154	PRN = 0.3375	$PR_i^{(N = F_1, F_1+F_2, F_1+F_2+F_3)}$
`PRall23[o]`	2.887 × 10⁻⁶	0.3134	–	$PR_i^{(N = F_2, F_2+F_3)}$
`PRall13[o]`	0	0.3361	–	$PR_i^{(N = F_1, F_1+F_3)}$
`PR[o]`	0.0350	0.0020	0.3271	$PR_i\lvert_{k=F_1,F_2,F_3}$
`PRT[o]`	0.0362	0.0020	0.3381	$PR_i^T\lvert_{k=F_1,F_2,F_3}$
`score[o]`	0.1039	0.0059	0.9692	$PR_i\lvert_k/PR_i^{(F_1+F_2+F_3)}$

† In Python, the index o = k – 1 (step index). Variables PRall123, PRall23, and PRall13 capture cumulative effects up to step k. PR, PRT, and score are intrinsic to each step k.

10.3.3 | Results and Interpretation (Layer Scale)

$PR_{i,j}|_k$ (variable PR_layersm2 $A_1$ $j = 1$ $B$ $j = 2$ $A_3$ $j = 3$ ). Results (Table \ref{tab:Ex3Ext2}PR_layers[:,1] $j = 2$ ) reproduces PR $A_1$ $F_1$ PR_layers[0,0] $B_2$ $A_3$ confirm that contaminants initially located on the external side have a low transfer risk toward the food.

Note

Practical note: this analysis yields a component-wise release indicator, highly useful for recycled materials. It identifies the parts that are structurally prone to releasing the most (for a given content) and thus where to install or optimize barriers. When a layer is protected, the barrier imposes its resistance on the global transfer; conversely, a reservoir layer behaves almost neutral and contributes little to the total release.

Table 25 — PR_layers[o,p] values per step (o) and per layer (p) (script extension 2/2).

↓ Index `o` / Index `p` →	A₁ (p = 0)	B₂ (p = 1)	A₃ (p = 2)
F₁ (o = 0)	−0.89269	0.03505	0.00293
F₂ (o = 1)	0.00195	0.00198	0.00699
F₃ (o = 2)	0.32807	0.32706	0.02125

† Python indices start at 0 → o = k – 1 (step) and p = j – 1 (layer). The smallest index corresponds to the layer in contact with the food.

10.4 | Example 4 — Parametric identification

Objective of example 4
Estimate diffusivity D and partition coefficient k of a monolayer directly from kinetic data (real or noisy simulated). Generate pseudo-experimental kinetics, then recover true values by non-linear fit.

Table 6 details the case; Listing 4 shows the script.

Table 26 — Parameters of Example 4.

Properties	Values
Layer	$P$ $l=100\,\mu$ $D=1{\times}10^{-10}\,\text{cm}^2/s$ $C_0=1000$ $k_P=0.1$
Medium	$F$ $V=1$ $A=6$ $h=10^{-6}$ $k_F=1$
Data	$C_F(t)$ $\sigma$ =1%)
Outputs	$D,k$ ; compare fitted vs. true

Note

Result Overview. $D,k$ $D\simeq9.9\times10^{-15}\,$ $1.0\times10^{-14}$ $k\simeq0.108$ vs. 0.10).

Listing 5 — Script to identify D and k from experimental data (adapted from example4.py)


xxxxxxxxxx
38
1
# Listing — Script to identify D and k from experimental data (adapted from example4.py)
2

3
# --- Imports --------------------------------------------------------------
4
from patankar.layer import layer, layerLink
5
from patankar.food import foodlayer
6

7
# --- 1) Define layer P and food F ----------------------------------------
8
P = layer(l=(100,"um"), D=(1e-10,"cm**2/s"), C0=1000, k=0.1)
9
F = foodlayer(contacttime=(10,"days"), volume=(1,"L"),
10
              surfacearea=(6,"dm**2"), h=(1e-6,"m/s"), CF0=0, k=1)
11

12
# --- 2) Link parameters (D,k) for external control -----------------------
13
Dref, kref = P.D, P.k
14
P.Dlink = layerLink("D", indices=0, values=Dref)
15
P.klink = layerLink("k", indices=0, values=kref)
16

17
# --- 3) Baseline simulation + pseudo-experiment --------------------------
18
R = F.migration(P)            # reference kinetics
19
E = R.pseudoexperiment(npoints=30, std_relative=0.01)  # noisy data
20

21
# --- 4) Sensitivity scan (optional visual check) -------------------------
22
R.comparison.update(0, label="**Reference**", color="black")
23
for _ in range(10):
24
    P.Dlink[0] /= 1.1          # decrease D
25
    P.klink[0] *= 1.1          # increase k
26
    R.rerun(name=f"{P.Dlatex()[0]}, {P.klatex()[0]}",
27
            color=R.comparison.jet(10)[_])
28
R.comparison.add(E, label="Pseudo Experiment", discrete=True)
29
R.comparison.plotCF()
30

31
# --- 5) Fit D and k on the experimental kinetics -------------------------
32
d2 = R - E                       # squared-distance callable
33
print("Before fit:", d2())
34
res = R.fit(E)                   # non-linear optimization of D and k
35
print("After  fit:", d2())
36
print("Fitted D:", P.Dlink.values, "| True:", Dref)
37
print("Fitted k:", P.klink.values, "| True:", kref)
38
R.comparison.plotCF()            # show fitted curve vs. data

Key result. $D$ $k$ $D \simeq 9.89\times 10^{-15}$ $^2$ $1\times 10^{-14}$ $k \simeq 0.108$ $0.10$ Figure 7 $D/1.1^\ell$ $k\times 1.1^\ell$ $\ell = 1 \dots 10$ ).

Figure 7 — Joint fit of D and k from a noisy kinetic (“pseudo-experiment”) using Listing 5. The optimized curve (orange) overlays both the simulated experimental points and the noise-free reference. Other curves illustrate sensitivity to D and k.

Tip

Real data. The procedure works identically with experimental series (E = R.from_csv(...) or by creating an equivalent “experiment” object). The layerLinkbinds $D$ $k$ $C_F(t)$ while preserving the M2/M3 physics. The only constraint is to remain within the linear (tracer) regime presented in Section 8 (properties independent of concentration).

Note

Takeaway.layerLink $D,k$ $C_F(t)$ , while keeping M2/M3 physics intact. Applies equally to real datasets (from_csv). Domain validity: tracer regime (concentration-independent properties, see §8).

10.5 | Example 5 — AI integration with local knowledge base (RAG)

Automating calculations and verifying regulatory thresholds with AI becomes possible by integrating local document bases (Retrieval-Augmented Generation, RAG) and Python scripts. The whole stack can run offline, with confidentiality and auditability—key conditions for safe-by-design workflows.

Objective of example 5
Couple SFPPy with a local regulatory Q/A engine by indexing a Markdown knowledge base (regulations, internal guides, Python scripts). Documents are vectorized using HuggingFace embeddings, then queried via LlamaIndex/Ollama with a mistral model.

A minimal setup can be built from open tools and modest hardware (an 8–16 GB VRAM GPU is recommended). Components are summarized in Table 27.

Table 7 shows minimal setup; Listing 5 demonstrates.

Table 27 — Components of RAG system coupled to SFPPy.

Properties	Values
KB (source)	Markdown files under `./docs/KB`
Index	Persistent `VectorStoreIndex` (Chroma)
Embeddings	`BAAI/bge-small-en-v1.5` (HuggingFace)
LLM	`mistral` via `ollama`
I/O	User question → synthetic answer + cited sources

Outputs.8 $D$ , thickness, partitioning.

Listing 6 — Minimal RAG configuration (adapted from example5.py)


xxxxxxxxxx
22
1
# Listing — Minimal RAG configuration (adapted from example5.py)
2

3
from llama_index.core import SimpleDirectoryReader, VectorStoreIndex
4
from llama_index.embeddings.huggingface import HuggingFaceEmbedding
5
from llama_index.llms.ollama import Ollama
6

7
# 1) Load the Markdown/Python KB (recursive)
8
documents = SimpleDirectoryReader(input_dir="./docs/KB", recursive=True).load_data()
9

10
# 2) Embeddings (HuggingFace) then vector index
11
embed_model = HuggingFaceEmbedding(model_name="BAAI/bge-small-en-v1.5")
12
index = VectorStoreIndex.from_documents(documents, embed_model=embed_model)
13

14
# 3) Declare the local LLM (Ollama)
15
llm = Ollama(model="mistral")  # other models may be installed
16

17
# 4) Query (RAG) with source citations
18
query_engine = index.as_query_engine(llm=llm, similarity_top_k=3)
19
response = query_engine.query("What is the definition of a functional barrier in EU 10/2011?")
20
print(response.response)
21
for src in response.source_nodes:
22
    print(" -", src.metadata.get("file_path","unknown"))

Note

Good practices. Store regulatory excerpts and Python scripts in ./docs/KB, rebuild the persistent index, and preserve source citations. The SFPPy–RAG coupling enables hybrid agents where the LLM’s answer directly steers physical simulation and design choices.

10.5.1 | A Practical Illustration: Legal Definition of Functional Barriers in the EU and US

Table 28 — Comparison of “functional barrier” definitions in the EU and US, as returned by a RAG system built on the SFPPy KB.

Question (EU)	Question (US)
What is the definition of a functional barrier in EU 10/2011?	What is the definition of a functional barrier in the US?
RAG answer	RAG answer
Definition in EU Regulation 10/2011 (Art. 3): “Functional barrier means a layer which prevents migration of substances from behind that barrier into food.”Implications: $D$ ; (ii) very low/negligible permeability; (iii) use `nofood()` or `setoff()` boundaries in SFPPy.	In the US, “functional barrier” is not legally defined, but FDA recognizes the concept in FCS Notifications. Informal definition:Key points: $< 0.5$ $0.5\,\mu$ $D$ , high thickness, and partitioning conditions. Also see FDA ToR policy, FCNs, and CFR 21 Parts 174–179.

In SFPPy. $D$ ; (ii) sufficiently high thickness; and (iii) partitioning such that downstream migration remains < 0.5 µg/kg.

10.5.2 | Advanced Use: Asking a RAG System to Build Predictions

Extension request (to the RAG agent). “Generate a simulation example with a functional barrier applied in a multilayer system, justify your choice.”

The RAG system can combine example strategies and known documentation to generate new simulations. Listing 7 is auto-generated and can be executed in the SFPPy environment to produce Figure 8 and the interpretation summarized in Table 28.

The selections (migrant = toluene; recycled PP as the source layer; glassy PET as the barrier; simulant = ethanol) are consistent with SFPPy recommendations. Ethanol is intentionally chosen as a fatty-food simulant: it provides a severe and reproducible stress, although other options like oliveoil are available. Keep in mind these are numerical calculations: ethanol does not interact with glassy PET (“gPET”) as it might experimentally.

Value for SMEs. Such auto-generations provide concrete technical support—both executable scripts and expert interpretations. They do not replace regulatory evaluation, but they speed up preparation of credible, well-argued scenarios. Limitations remain: for instance, the agent proposes an un-plasticized PET barrier without discussing plasticization risk (see Example 2, Section 10.2) or polymer-compatibility issues.

Listing 7 — Multilayer with a Functional Barrier


xxxxxxxxxx
51
1
"""
2
SFPPy demo - Multilayer with a Functional Barrier
3
-------------------------------------------------
4
Scenario: Toluene in a recycled PP core, with/without a PET functional barrier
5
Contact: ethanol, 60 days @ 40oC
6
Outputs: CF(t) curves, snapshots, and a small thickness study for the FB
7
"""
8
# --- Imports ---
9
from patankar.loadpubchem import migrant
10
from patankar.layer import PP, gPET
11
from patankar.food import ethanol
12
from patankar.migration import senspatankar, CFSimulationContainer
13
import matplotlib.pyplot as plt
14

15
# --- Define Migrant (Toluene) ---
16
tol = migrant("toluene")
17

18
# --- Define Layers ---
19
PPcore = PP(l=500e-6, C0=200, substance=tol)    # 500 um PP with 200 mg/kg toluene
20
PETbarrier = gPET(l=30e-6, C0=0, substance=tol) # 30 um PET barrier, initially no migrant
21

22
# --- Build Multilayer Systems ---
23
multilayer_with_FB = PETbarrier + PPcore    # PET on food side
24
multilayer_no_FB = PPcore                   # no barrier
25

26
# --- Define Food (Ethanol, 40oC, 10 days) ---
27
food_contact = ethanol(contacttime=(60, "days"), contacttemperature=(40, "degC"))
28

29
# --- Run Simulations ---
30
result_no_FB = senspatankar(multilayer_no_FB, food_contact, name="No Barrier")
31
result_with_FB = senspatankar(multilayer_with_FB, food_contact, name="With 30 um PET FB")
32

33
# --- Plot & Compare ---
34
result_no_FB.plotCF()
35
result_with_FB.plotCF()
36
comparison = CFSimulationContainer(
37
                name="Toluene Migration into Fatty Food (Ethanol)",
38
                SML=0.15e-3, SMLunit="mg/kg")
39
comparison.add(result_no_FB, label="No Barrier", color='red')
40
comparison.add(result_with_FB, label="With 30 um PET FB", color='blue')
41
comparison.plotCF()
42
plt.yscale("log")                 # log-scale for better visibility
43
plt.ylim(1e-10,10)                # concentration range (mg/kg)
44
plt.axhline(0.5e-3, color='black', linestyle='--', label="FDA ToR (0.5 ug/kg)")
45
plt.title("Toluene Migration into Fatty Food")
46
plt.ylabel("C_F (mg/kg)")
47
plt.xlabel("Time (days)")
48
plt.legend()
49
plt.grid(True)
50
plt.tight_layout()
51
plt.show()

Figure 8. Output figure produced by the auto-generated script in Listing 7.

Table 29 — RAG system’s interpretation of the results from Listing 7 and Figure 8.

Question	Answer
Why these inputs.	$D$ $T$ $C_0=200$ mg/kg to stress the barrier; (vi) SML line at 0.15 µg/kg; (vii) FDA ToR line at 0.5 µg/kg.
How to read the figure.	The blue curve (30 µm PET FB) lies orders of magnitude below red (no barrier): clear functional-barrier effect (lag + reduced flux). On the log scale: staying below both SML and ToR at 60 days meets both criteria; between the lines meets ToR but fails SML; above ToR fails both.
Is 60 days sufficient?	Depends on intended use. Simulate the full profile (hot-fill → cooling → storage). If life exceeds 60 days, extend duration or increase PET thickness. Align simulant choice to the actual food (e.g., `ethanol95()` or a real-fat model as needed).

10.6 | Integrating Other Capabilities: Analytical Fingerprints and Neo-formed Compounds

SFPPy Vitrac 2025a is an AI-ready open-source project already integrating molecular-modeling modules, a dynamic PubChem client (Kim et al., 2024), EU/US/CN regulatory bases, and a ToxTree port (Patlewicz et al., 2008). It is compatible with Vitrac 2025b, dedicated to predicting oxidation products and their kinetics under controlled thermo-oxidative conditions—thus enabling realistic migrant scenarios from degradation processes (oxidative scission, more polar low-MW products).

The Vitrac 2025c project—aimed at recycled-material management—leverages chemical fingerprints to recognize/classify contaminants from chromatograms or spectroscopic signals. These tools pave the way to an integrated approach that combines modeling, automated recognition, and release scenarios.

More broadly, SFPPy sits within a rich ecosystem of complementary tools (Table 30) orchestrable in Python. This ecosystem is now usable by large language models (LLMs), which can directly operate these software bricks to answer targeted questions. Coupled with RAG techniques, these capabilities enable repeatable, traceable answers grounded in vetted internal guides and examples.

Table 30 — Ecosystem of tools combinable with SFPPy for extended computation, analysis, and decision support.

Tool / Library	Primary role	Use with SFPPy
sig2dna	Analytical-signal interpretation; chemical fingerprints	Map GC/LC–MS peaks to likely chemical families; feed SFPPy with “migrant/NIAS” candidates.
radigen	Neo-formed product generation (oxidation, thermolysis)	$T$ ; test worst-cases in SFPPy.
OpenChrom*	GC/LC(/MS) data management & processing	Import, filter, integrate chromatograms; export peak lists to SFPPy (CSV).
pymzML, pyteomics	MS data I/O (mzML, mzXML)	Python pipeline for intensities and m/z; couple to `sig2dna`/`matchms`, then SFPPy.
matchms	Spectral matching	Rapid candidate annotation (public libs); prioritize migrants to simulate.
RDKit	Cheminformatics (structures, 3D, descriptors)	$V_{\mathrm{vdW}}$ , computed logP; feed M2/M3 modules.
Open Babel	Chemical-format conversion/optimization	Normalize `.sdf/.mol/.pdb`; complement geometric properties for diffusion (Welle/hFV).
PubChemPy	Programmatic PubChem access	Retrieve formulas, masses, IDs, logP; auto-complete SFPPy “migrant” sheets.
SciPy (optimize)	Deterministic optimization (LS, simple constraints)	$D$ $k$ from kinetics (cf. Ex. 4); small design optimizations.
Pyomo, cvxpy	Mathematical programming (LP, QP, MIP)	$\kappa_s$ $\hat{C}_{i,F}^{\max}$ ).
scikit-optimize, DEAP/pymoo	Bayesian / evolutionary optimization	Multi-objective search (migration, cost, mass); explore complex, gradient-free surfaces.
SALib	Global sensitivity analysis (Sobol, Morris)	$D$ $k$ $Fo$ $K$ ) and target measurement/qualification effort.
PyMC	Bayesian inference & uncertainty	$D$ $k$ $\hat{s}_i$ .
pandas, DuckDB	Tables, joins, analytics	Aggregate M2/M3 outputs; build dashboards across trials/variants; fast local queries.
Matplotlib, Plotly/Dash, Streamlit	Visualization / light apps	$C(x,t)$ profiles, parametric studies; mini-apps for quality engineers.
Dask, joblib	Parallelization, batch	Sweep hundreds of substances/designs; accelerate Monte-Carlo/sensitivity.
Prefect, Snakemake	Workflow orchestration	Reproducible studies (data ETL → M2/M3 → optimization → report).
pint	Robust units handling	$\mu$ $^2$ , mg/kg); align with SFPPy units.
LlamaIndex, LangChain + Ollama	Local RAG/LLMs	Regulatory Q/A, study generation; hybrid agents (code + regulation) around SFPPy (cf. Ex. 5).
unstructured, pdfplumber	PDF/HTML ingestion	Extract tables/normative excerpts for RAG; build parameter sets.

* OpenChrom (like ToxTree) is a Java application; integration is via CSV/Mass-List export and command-line calls.

Key takeaways (Section 10)

automated and coupled to AI agents $\hat{D}_{i,j}, \hat{k}_{i,j}^H$ ) are extracted from substance/material names. Literature data and scripts can be indexed and exploited by LLMs. The Python ecosystem integrates diverse libraries — from data analysis to analytical chemistry — building pipelines from experimental fingerprints to design scenarios. This capability enables safe designs, adaptable to recyclate variability and interoperable among industry and regulators.

11 | Conclusion

This article proposes the main ingredients of a safe-by-design approach, progressing from the simplest indicators to complex scenarios where formulation, design, and use interact. These two levels are not opposed: one helps prioritize when not everything can be controlled, the other integrates all real constraints (geometry, format, presence of contaminants).

$\mathbb{R}^n$ $\mathbb{R}^n$ $n$ $n>100$ , diverse polymers and contact conditions), scripted automation becomes essential.

The proposed formulation is thus suited to managing recycled feedstock and usage diversity. It paves the way for environments where the paper itself can serve as a RAG (Retrieval-Augmented Generation) knowledge base, building scenarios and optimizations in natural language. Relying on open and interoperable tools and databases, the proposed ecosystem (SFPPy, Python, PubChem, ToxTree…) enables rapid adoption without prior modeling expertise and lends itself to the development of hybrid agents combining physical simulation, regulatory bases, and artificial intelligence.

Hence, this approach provides a robust and extensible foundation for research, industry, and regulation—combining scientific rigor, operational requirements, and openness to digital innovation.

12 | Acronyms, notations, and symbols

12.1 | Notations and symbols

Table 31 — Notations and symbols used.

Symbol	*Definition (FR; EN)*	Units
$\mathsf{i}$	$i$ ; solute index.	—
$\mathsf{j}$	$F$ $s$ : couche); phase index.	—
$\mathsf{s}$	Index d’une couche / composant d’emballage; layer index.	—
$\mathsf{A}$	Surface de contact; contact area.	$^2$
$\mathsf{l_s}$	$s$ ; layer thickness.	m
$\mathsf{V_F}$ $\mathsf{V_s}$	Volume de l’aliment / d’une couche; food/layer volume.	$^3$
$\mathsf{\rho_j}$	$j$ ; density.	$^{-3}$
$\mathsf{C_{i,j}}$ $\mathsf{C_{i,j}^0}$	$i$ $j$ $^{0}$ ); concentration (initial).	$^{-3}$ $^{-1}$
$\mathsf{\hat{C}_{i,F}^{(N)}}$	$N$ ); predicted food concentration.	$^{-3}$ $^{-1}$
$\mathsf{w_{i,s}}$ $\mathsf{\theta_{i,j}}$	$s$ $j$ ; weight / availability fraction.	—
$\mathsf{D_{i,j}(T)}$	$i$ $j$ ; diffusivity.	$^2$ $^{-1}$
$\mathsf{D_{0,i,j}}$ $\mathsf{E_{a,i,j}}$	Pré-exponentielle et énergie d’activation (Arrhenius); pre-exponential, activation energy.	$^2$ $^{-1}$ $^{-1}$
$\mathsf{k_{i,j}^H}$	$j{=}0$ ); apparent Henry coefficient.	—
$\mathsf{K_{i,F/j}}$	$j$ $=k_{i,j}^H/k_{i,0}^H$ ); partition coefficient.	—
$\mathsf{\gamma_{i,j}}$ $\mathsf{a_{i,j}}$	Coefficient d’activité et activité; activity coefficient & activity.	—
$\mathsf{\chi_{i+j}}$	Paramètre d’interaction de Flory–Huggins; Flory–Huggins interaction.	—
$\mathsf{\delta_j}$	Paramètre de solubilité (Hildebrand/Hansen); solubility parameter.	$^{1/2}$
$\mathsf{P'}$	$logP$ ; polarity index.	—
$\mathsf{logP}$	Partage octanol/eau (base 10); octanol/water partition.	—
$\mathsf{Fo}$	$\displaystyle \int D\,dt / l^2$ ; Fourier number.	—
$\mathsf{Bi_m}$ $\mathsf{h_m}$	Nombre de Biot de masse; coeff. de transfert convectif; mass Biot number; mass transfer coeff.	$^{-1}$
$\mathsf{Sh}$ $\mathsf{Pe}$	Nombres de Sherwood et de Péclet; Sherwood, Péclet.	—
$\mathsf{M_{i,\mathrm{mig}}(t)}$	Masse cumulée migrée dans l’aliment; cumulative migrated mass.	mol or mg
$\mathsf{m_{i,0}}$	Masse initiale disponible dans la source; initial available mass.	mol or mg
$\mathsf{s_{i,\max}}$	Sévérité maximale visée (critère d’optimisation); target maximal severity.	—
$\mathsf{SML_i}$ $\mathsf{OML}$	Limites de migration spécifique / globale; SML / OML.	$^{-1}$ $^{-2}$
$\mathsf{Q_M}$	Teneur autorisée dans le polymère; quantity in material.	$^{-1}$ (polymer)
$\mathsf{R}$ $\mathsf{T}$	Constante des gaz et température absolue; gas constant, temperature.	$^{-1}$ $^{-1}$ ; K
$\mathsf{t}$	Temps de contact; contact time.	s, h, d
$\mathsf{A_{pp}}$ $\mathsf{\tau}$	Paramètres polymère (corrélation de Piringer : diffusivité et facteur thermique); Piringer polymer/thermal parameters.	—; K
$\mathsf{V_i}$ $\mathsf{V_{i,\mathrm{mol}}}$	$i$ ; molecular (molar) volume.	$^3$ $^3$ $^{-1}$
$\mathsf{\varepsilon}$ $\mathsf{\tau_{por}}$	Porosité et tortuosité effectives; porosity, tortuosity.	—
$\mathsf{PR_i^T}$ $\mathsf{PR_i^E}$ $\mathsf{t_{fb}}$	Potentiels de relargage dynamique/statique; durée de service de la barrière fonctionnelle; dynamic/static release potentials; FB service life.	—; —; s

12.2 | Indices and exponents

Table 32 — Indices and exponents used.

Index/Exponent	*Meaning (FR; EN)*
$i$	Substance/soluté; solute.
$j$	$F$ $s$ : couche polymère); phase (reference/food/layer).
$s$	Couche/composant d’emballage; layer/component.
$F$	Aliment (ou simulant); food (or simulant).
$0$ (zero)	Référence thermodynamique (air/eau ou phase de référence); reference phase.
$^{0}$ (superscript)	Valeur initiale; initial value.
$(N)$	Itération / ordre d’approximation; iteration/order.
$fb$	Barrière fonctionnelle; functional barrier.
$eq$	Équilibre thermodynamique; thermodynamic equilibrium.
$ref$	Référence (condition/état de référence); reference.

13 | Literature cited

[1] Vitrac, O.. FMECAengine: FMECA software developed in the framework of the project SafeFoodPack Design (version v0.45). GitHub repository. Last commit: 2019 (commit 3a9f417); available at: https://github.com/ovitrac/Fmecaengine. (2019). URL. [2] Vitrac, O.. SFPPy: Python Framework for Food Contact Compliance and Risk Assessment (version v1.42). GitHub repository. Last commit circa mid-2025; available at https://github.com/ovitrac/SFPPy (accessed on July 27, 2025). (2025). URL. [3] Vitrac, O.. SFPPyLite: lightweight browser-based version of SFPPy (version v1.42). GitHub repository. Lightweight browser-based version of SFPPy. Last commit circa mid-2025; available at https://github.com/ovitrac/SFPPylite (accessed on July 27, 2025).. (2025). URL [4] Kim, S.; Chen J.; Cheng, T.; Gindulyte, A.; He, J.; He, S.; Li, Q.; Shoemaker, B.; Thiessen, P.; Yu, B.; Zaslavsky, L.; Zhang, J.; Bolton, E.. – Pubchem 2025 update. Nucleic Acids Research, 53(D1) :D1516–D1525 (Nov 2024). doi [5] Patlewicz, G.; Jeliazkova, N.; Safford, R.; Worth, A.; Aleksiev, B. An evaluation of the implementation of the cramer classification scheme in the toxtree software. SAR and QSAR in Environmental Research, 19(5–6) :495–524 July 2008 ;doi, available at URL (accessed on July 27, 2025). [6] Zhu, Y.; Nguyen, P.; Vitrac, O.. Risk assessment of migration from packaging materials into food. Elsevier. (2019). doi. [7] Brandsch, Rainer; Dequatre, Claude; Mercea, Peter; Milana, Maria Rosaria; Stoermer, Angela; Trier, Xenia; Vitrac, Olivier; Schaefer, Annette; Simoneau, C. Practical guidelines on the application of migration modelling for the estimation of specific migration.. JRC Report JRC98028, Publications Office, LU. (2015). doi. [8] Vitrac, O.. SFPP3: Safe Food Packaging Portal – migration simulation and decision framework (version 1.28). Web-based platform. Access: https://sfpp3-simulation.contactalimentaire.fr/; default login credentials: user: demouser, password: inramig. Platform hosted by LNE (accessed on July 27, 2025).. (2016). URL. [9] Nguyen, P.; Goujon, A.; Sauvegrain, P.; Vitrac, O.. A computer‐aided methodology to design safe food packaging and related systems. AIChE Journal, 59(4): 1183–1212. (mar 2013). doi. [10] Vitrac, O.; Ouadi, S.; Hayert, M.; Domenek, S.; Cornuau, G.; Nguyen, P. FitNESS an E-learning platform to design safe and responsible food packaging. Journal of Chemical Education, 101(8): 3179–3192. (aug 2024). doi. URL. [11] Figge, K.. Migration of components from plastics-packaging materials into packed goods — test methods and diffusion models. Progress in Polymer Science, 6(4): 187–252. (jan 1980). doi. [12] Jaeger, R. J.; Rubin, R. J.. Plasticizers from plastic devices: extraction, metabolism, and accumulation by biological systems. Science, 170(3956): 460–462. (oct 1970). doi. [13] Monclús, L.; Arp, H. P. H.; Groh, K. J.; Faltynkova, A.; Løseth, M. E.; Muncke, J.; Wang, Z.; Wolf, R.; Zimmermann, L.; Wagner, M.. Mapping the chemical complexity of plastics. Nature, 643(8071): 349–355. (jul 2025). doi. [14] Munro, I.; Renwick, A.; Danielewska-Nikiel, B.. The Threshold of Toxicological Concern (TTC) in risk assessment. Toxicology Letters, 180(2): 151–156. (aug 2008). doi. [15] Authority, E. F. S.; Organization, W. H.. Review of the Threshold of Toxicological Concern (TTC) approach and development of new TTC decision tree. EFSA Supporting Publications, 13(3). (Mar 2016). doi. [16] Patlewicz, G.; Jeliazkova, N.; Safford, R.; Worth, A.; Aleksiev, B.. An evaluation of the implementation of the Cramer classification scheme in the Toxtree software. SAR and QSAR in Environmental Research, 19(5–6): 495–524. (jul 2008). doi. URL. [17] Committee, E. S.; Hardy, A.; Benford, D.; Halldorsson, T.; Jeger, M. J.; Knutsen, H. K.; More, S.; Naegeli, H.; Noteborn, H.; Ockleford, C.; Ricci, A.; Rychen, G.; Schlatter, J. R.; Silano, V.; Solecki, R.; Turck, D.; Bresson, J.; Dusemund, B.; Gundert‐Remy, U.; Kersting, M.; Lambré, C.; Penninks, A.; Tritscher, A.; Waalkens‐Berendsen, I.; Woutersen, R.; Arcella, D.; Court Marques, D.; Dorne, J.; Kass, G. E.; Mortensen, A.. Guidance on the risk assessment of substances present in food intended for infants below 16 weeks of age. EFSA Journal, 15(5). (May 2017). doi. [18] Committee, E. S.. Guidance on selected default values to be used by the EFSA Scientific Committee, Scientific Panels and Units in the absence of actual measured data. EFSA Journal, 10(3). (Mar 2012). doi. [19] Commission. Commission Regulation (EU) No 10/2011 of 14 January 2011 on plastic materials and articles intended to come into contact with food. Official Journal of the European Union, L 12: 1–89. (2011). URL. [20] Vitrac, O.; Hayert, M.. Risk assessment of migration from packaging materials into foodstuffs. AIChE Journal, 51(4): 1080–1095. (feb 2005). doi. [21] Vitrac, O.; Hayert, M.. Identification of Diffusion Transport Properties from Desorption/Sorption Kinetics: An Analysis Based on a New Approximation of Fick Equation during Solid-Liquid Contact. Industrial & Engineering Chemistry Research, 45(23): 7941–7956. (oct 2006). doi. [22] Landrum, G.; Tosco, P.; Kelley, B.; Ric; Sriniker; Gedeck; Vianello, R.; NadineSchneider; Kawashima, E.; Dalke, A.; N, D.; Cosgrove, D.; Cole, B.; Swain, M.; Turk, S.; AlexanderSavelyev; Jones, G.; Vaucher, A.; Wójcikowski, M.; Take; Probst, D.; Ujihara, K.; Scalfani, V. F.; Godin, G.; Pahl, A.; Berenger; JLVarjo; Strets123; JP; DoliathGavid. rdkit/rdkit: 2022_03_1b1 (Q1 2022) Release. Zenodo. ; available at https://www.rdkit.org/ (accessed July 27, 2025).. (2022). doi. URL. [23] O'Boyle, N. M.; Banck, M.; James, C. A.; Morley, C.; Vandermeersch, T.; Hutchison, G. R.. Open Babel: An open chemical toolbox. Journal of Cheminformatics, 3(1). (oct 2011). doi. URL. [24] Nguyen, P.; Guiga, W.; Vitrac, O.. Molecular thermodynamics for food science and engineering. Food Research International, 88: 91–104. (oct 2016). doi. [25] Vitrac, O.; Nguyen, P.; Hayert, M.. In silico prediction of food properties: a multiscale perspective. Frontiers in Chemical Engineering, 3. (jan 2022). doi. [26] Kim, S.; Chen, J.; Cheng, T.; Gindulyte, A.; He, J.; He, S.; Li, Q.; Shoemaker, B. A.; Thiessen, P. A.; Yu, B.; Zaslavsky, L.; Zhang, J.; Bolton, E. E.. PubChem 2025 update. Nucleic Acids Research, 53(D1): D1516–D1525. (nov 2024). doi. URL. [27] Schymanski, E.; Kondic, T.; Elapavalore, A.; Bolton, E.; Thiessen, P.; Zhang, J.; Kim, S.; Krinsky, A. M.; Ross, D. H.; Xu, L.. PubChemLite for Exposomics + predicted CCS from CCSbase - 2 August 2025. Zenodo. (2025). doi. [28] Gillet, G.; Vitrac, O.; Desobry, S.. Prediction of solute partition coefficients between polyolefins and alcohols using a generalized Flory-Huggins approach. Industrial & Engineering Chemistry Research, 48(11): 5285–5301. (may 2009). doi. [29] Gillet, G.; Vitrac, O.; Desobry, S.. Prediction of partition coefficients of plastic additives between packaging materials and food simulants. Industrial & Engineering Chemistry Research, 49(16): 7263–7280. (jul 2010). doi. [30] Vitrac, O.; Gillet, G.. An off-lattice flory-huggins approach of the partitioning of bulky solutes between polymers and interacting liquids. International Journal of Chemical Reactor Engineering, 8(1). (jan 2010). doi. [31] Nguyen, P.; Guiga, W.; Dkhissi, A.; Vitrac, O.. Off-lattice Flory–Huggins approximations for the tailored calculation of activity coefficients of organic solutes in random and block copolymers. Industrial & Engineering Chemistry Research, 56(3): 774–787. (jan 2017). doi. [32] Snyder, L.. Classification of the solvent properties of common liquids. Journal of Chromatography A, 92(2): 223–230. (may 1974). doi. [33] Rohrschneider, L.. Solvent characterization by gas-liquid partition coefficients of selected solutes. Analytical Chemistry, 45(7): 1241–1247. (jun 1973). doi. [34] Cheng, T.; Zhao, Y.; Li, X.; Lin, F.; Xu, Y.; Zhang, X.; Li, Y.; Wang, R.; Lai, L.. Computation of octanol-water partition coefficients by guiding an additive model with knowledge. Journal of Chemical Information and Modeling, 47(6): 2140–2148. (nov 2007). doi. [35] Fang, X.; Vitrac, O.. Predicting diffusion coefficients of chemicals in and through packaging materials. Critical Reviews in Food Science and Nutrition, 57(2): 275–312. (apr 2015). doi. [36] Begley, T.; Castle, L.; Feigenbaum, A.; Franz, R.; Hinrichs, K.; Lickly, T.; Mercea, P.; Milana, M.; O’Brien, A.; Rebre, S.; Rijk, R.; Piringer, O.. Evaluation of migration models that might be used in support of regulations for food-contact plastics. Food Additives and Contaminants, 22(1): 73–90. (jan 2005). doi. [37] Ewender, J.; Welle, F.. A new method for the prediction of diffusion coefficients in poly(ethylene terephthalate)—Validation data. Packaging Technology and Science, 35(5): 405–413. (jan 2022). doi. [38] Fang, X.; Domenek, S.; Ducruet, V.; Réfrégiers, M.; Vitrac, O.. Diffusion of aromatic solutes in aliphatic polymers above glass transition temperature. Macromolecules, 46(3): 874–888. (jan 2013). doi. [39] Zhu, Y.; Welle, F.; Vitrac, O.. A blob model to parameterize polymer hole free volumes and solute diffusion. Soft Matter, 15(43): 8912–8932. (2019). doi. [40] Zhu, Y.; Guillemat, B.; Vitrac, O.. Rational design of packaging: toward safer and ecodesigned food packaging systems. Frontiers in Chemistry, 7. (may 2019). doi. [41] Vitrac, O.. radigen: Python-based chemical kernel for simulating oxidation reactions from prompts. GitHub repository. Available at https://github.com/ovitrac/radigen (accessed on on July 27, 2025). (2025). URL.

Generative Simulation Initiative 🌱 | Dr Olivier Vitrac | Last edition on October 11^th, 2025