Risk Assessment of Migration From Packaging Materials Into Food

Yan Zhu

, Phuong-Mai Nguyen

, and Olivier Vitrac

INRA, French National Institute of Agricultural Research, UMR 1145

Ingénierie Procédés Aliments, AgroParisTech, INRA, Université Paris-Saclay, Massy, France; and

LNE, French National Laboratory

of Metrology and Testing, Trappes Cedex, France

Introduction 2

What Is Migration? 2

Our Approach to Risk Assessment 2

The Scope of Migration Modeling 2

Lessons From Crises 3

A Short History 3

Why Thermodynamics Counts 4

Modern Crises 5

Risk Assessment for Decision Making 6

Overview of Migration Modeling 6

What Is Currently Permitted? 6

Modeling Using a Tiered Approach: From Worst-Case Scenarios to Detailed Conservative Ones 9

Key Steps in Migration Modeling and Risk Assessment Approaches 12

Migration Modeling for Compliance Testing and Beyond 12

Principles of Migration Modeling 12

Governing Equations for Monolayer Materials 16

Governing Equations for Multilayers 18

Strategies and Equations to Simulate Multiple Steps and Conditions 24

Discussion on the Choice of Accelerated Conditions and the Identiﬁcation of Critical Steps 28

Diffusion Properties in Polymers 30

Deﬁnitions of Diffusion Coefﬁcients 30

Self- and Trace-Diffusion Coefﬁ cients 30

Mutual Diffusion Coefﬁcients 31

Effect of the Geometry of Migrants on D

Values 32

Effect of the Polymer 34

Activation of Diffusion by Temperature 34

Sorption Properties and Partition Coefﬁcients 34

Some Deﬁnitions 36

Chemical Potentials, Fugacities, and Activities 36

Effective Partition Coefﬁcients Between P and F 38

Sorption Isotherms 38

Linear Isotherms 38

Binary Flory Isotherms 38

Ternary Flory Isotherms 39

Binary Flory-Huggins Coefﬁcients in a Copolymer AB 40

High Throughput Calculations of Flory-Huggins Coefﬁcients at Atomistic Scale 40

Justiﬁcation and Limitations 40

Principles 40

Probabilistic Modeling of the Migration 41

Beyond Intuition 41

Epistemic Uncertainty 42

Sensitivity Analysis of Migration Models 43

Local Sensitivity Analysis 43

Global Sensitivity Analysis via Stochastic Simulation 44

Principles of the Probabilistic Interpretation of Mass Transfer 44

Input Distributions 44

Estimation of Probabilities via Monte-Carlo Sampling 44

Estimation of Joint Probabilities via the Composition Theorem 45

Some Illustrations 45

Typical Probabilistic Migration Kinetics 46

Effect of Bi and s

What Will Be the Future? 47

Extending the Legal Scope of Migration Modeling 47

Reference Module in Food Sciences https://doi.org/10.1016/B978-0-08-100596-5.22501-8 1

Extending the Capacity of Migration Modeling 48

Bridging Migration Modeling, Safe-by-Design, and Eco-Design Approaches 49

Online Resources Ease Risk Assessment 49

Lower Bounds of Toxicological Thresholds for Non-evaluated Substances 49

Migration Modeling Tools 49

References 51

Introduction

What Is Migration?

Migration is a general term for spontaneous mass transfer of chemical substances, and, in the context of food packaging, it indicates

an extraction of packaging constituents and their transfer to the food. Industry and authorities both recognized the fate of cross mass

transfer between materials and the resulting contamination of the food. The term migration was consequently preferred to the

contamination one in the scientiﬁc literature and legal documents. During the last few decades, the concern about the safety of

food contact materials (FCM) has increased with the public’s appetite for transformed food and ready-to-eat (RTE) meals, and

with their ever-expanding needs for disposable packaging. Nowadays, FCM is identiﬁed as the prevalent source of exogenous chem-

ical contaminants in food, ahead of pesticides, veterinary drug residues and other environmental contaminants.

The ubiquitous contamination issue was thought to be conﬁned initially to contact layers and materials, but this is far from

being the case. Modern food packaging systems are, indeed, printed, coated, laminated and associated with other materials. The

whole history of these materials must be considered, as they may have been subjected to repeated use, brought to a second life

via a recycling process, stored in reels or stacks contacting internal and external surfaces; or shipped with other materials during

long and warm periods. The whole contamination problem can be envisioned as a cross mass transfer of several substances between

Matryoshka or nesting dolls. The pre-weighted food feeds the smallest doll and is surrounded by many layers including the rigid

walls of the primary packaging, a sleeve, the transport cardboard box containing several sale units, the treated wood pallet,

a wrap cling ﬁlm, etc. until the freight container. In RTE meals and convenience foods, additional components may be present inter-

nally such as a bag preventing direct contact with walls, individual packages or wraps for portion control, separators and speciﬁc

holders, sachets for seasonings, active elements to increase food shelf life, etc. The whole picture is not complete without citing

the many tie layers, glued labels, and printed and coated surfaces. The ultravacuum and aerospace industry would have regarded

similar materials and combinations as a substantial reservoir of organic compounds without exception. As an illustration, NASA

compiled more than 35,000 outgassing data (ESA; NASA) for almost any material which could enter in a spacecraft, including

many commodity materials such as thermoplastics, coatings, and tapes.

Our Approach to Risk Assessment

This article provides a comprehensive description of molecular processes responsible for the migration of packaging constituents

and their pathways to the food with or without direct contact. The recurring structure of contamination scena rios leading to the

past major crises is discussed and analyzed regardless of the modalities of the regulations. The distance between the facts and prac-

tices is maintained throughout this article, as the regulations and the good manufacturing practices follow the crises and rarely

precede them. Major regulations (the US, European and Chinese) are referenced to highlight their convergence on the use of

modeling to demonstrate compliance and to evaluate the safety of recycled materials. The miscalculation of the connection between

chemical structure and physicochemical properties (volatility, solubility, diffusivity) has been the foremost cause of past crises.

Computer and proper simulation procedures can assist efﬁciently small and intermediate industries in overcoming internal knowl-

edge limitations on materials, mass transfer, and physical chemistry. By comparison with acceptable thresholds, migration

modeling can be extended at low cost to non-evaluated and non-intentionally added substances. In the foreseeable future, similar

techniques might be used to tackle the diffuse risks raised by endocrine-disrupting chemicals (Kortenkamp et al., 2009; Lee, 2018)

alone or in mixed cocktails (Kortenkamp, 2007). More globally, the extension of predictive tools and approaches will beneﬁt not

only the evaluation of the contribution of FCM to the global exposome (Wild, 2005), but it will also facilitate the adoption of

preventive approaches all along the supply chain. Safe-by-design approaches, including additive redesign, optimization of the

formulations (choice of substances and amounts), new packaging design and good manufacturing practices, will reduce the risk

of unintended food packaging interactions (Eicher et al., 2015; Muhamad et al., 2016). Improving the way food ingredients are

stored and processed will bring additional risk reduction, beyond the reduction of the migration in the ﬁnished product.

The Scope of Migration Modeling

The scope of migration modeling has been underestimated in the past and limited to compliance testing under worst-case scenarios.

The uncertainty and the pioneering methods were too coarse thirty years ago. The scope can be broadly extended today as shown in

2 Risk Assessment of Migration From Packaging Materials Into Food

Fig. 1. Earlier models could cover only single materials, and simple geometries without any dynamic change of conditions. The most

advanced models can today incorporate information at molecular scale and cover an entire supply chain. The article focuses on the

key details and features required to get robust modeling of migration at the scale of a material, component (label, cap .), an entire

food packaging, or industrial practices. The methodology to get estimates of consumer exposure are not covered because they are

directly related to the design of the packaging itself.

Lessons From Crises

Highlights crises associated with food contact materials

•

The contamination of food by materials in contact is never fortuitous, but it may be not avoidable.

•

Only the nature of the migrating substances, and the extent of the migration can be controlled by our choices.

•

Most, if not all, crises could have been predicted with relatively simple descriptions of mass transfer phenomena and

thermodynamics.

•

As a corollary, consumer exposure to substances from food contact materials could be reduced.

A Short History

Crises tend to predate regulations. A crisis is a situation where food safety is seriously questioned due to systematic contamination

by one or several FCM. The contamination was usually unexpected but not unforeseeable due to its anthropogenic nature.

Compared to food infection and food intoxication, the possibility of crises by FCM substances has been recognized lately in Europe.

The obligation of traceability of all packaging components to organize the recall of packaged food products were implemented only

in 2004 through the framework regulation 2035/2004/EC.

Thirty years ago, western governments endorsed enthusiastically an early idea hypothesized by Jerome Nriagu (Nriagu, 1983)

and popularized by Clair Patterson (Patterson et al., 1987), whereby Roman civilization collapsed as a result of lead poisoning.

“Although today lead is no longer seen as the prime culprit of Rome’s demise” (quoting Delile et al., 2014), lead poisoning

from leaded pottery and earthenware was known for centuries. All possible sources of lead were individually tracked by authorities

in the 19th century, in particular, after the development of wrought-iron canisters. The French ordinance of March 21st, 1879 (see p

231 of Doumerc and de Leymarie, 1895) prohibited, for instance, the use of alloys of tin and lead for all inner parts including

Figure 1 Evolution of the scope of migration modeling during the last decades.

Risk Assessment of Migration From Packaging Materials Into Food 3

welding. The French regulation of 1908 was even more explicit “no food substance should contain any harmful product or chemical

substance” (Hamel, 1910). In modern times, the editorial of the New England Journal of Medicine was headed in 1972 “The invis-

ible pollution” ( Rall, 1972) after the discovery of plasticizers in human blood stored in polyvinyl chloride (PVC) bags (Jaeger and

Rubin, 1970, 1972). Similar contaminations were associated with tubing used for culture tissues (Lawrence et al., 1969). The exact

nature of the contamination mechanism was not fully established at the time, and an analogy with the corrosion of metallic mate-

rials was falsely suggested (Kestel’man et al., 1972). The ﬁrst mechanistic review of what is called “extractivity” or “migration

propensity” appeared only in 1980 (Figge, 1980; Giacin, 1980). Three mechanisms of contamination were listed:

•

contamination only from the extreme surface of the material in contact;

•

the diffusion-controlled release of the material in contact;

•

penetration of the polymer matrix by the contacting phase (liquid) and subsequent extraction of material constituents.

Early descriptions were strongly inﬂuenced by the behavior and the dominance of PVC in the seventies and eighties, and by the lack

of sensitivity of contemporaneous analytical techniques. In spite of erratic results and difﬁculties in getting reproducible kinetics, the

corollary reasoning supported the conﬁdence of stakeholders in the apparent inertia of thermoplastics and thermosets. It was falsely

thought that:

•

the absence of direct or permanent contact,

•

aqueous contact,

•

high molecular weight additive or residues,

•

low temperatures

would prevent any signiﬁcant migration and did not need proper attention. At the time, only a study using radio-labeled additives

(Till et al., 1982a, 1982b; Schwope et al., 1987a, 1987b) carried out under contract for the US Food and Drug Administration and,

subsequently, interpreted in detail during the PhD thesis of Thomas P. Gandek at MIT (Gandek, 1986) highlighted several abnor-

malities, which anticipated future crises. In poor barrier polymers, the hydrodynamic conditions in the contacting liquid were

shown to control the release; but neglecting it was not underestimating migration, on the contrary. In aqueous food simulants

and presumably in any aqueous-type food, the decomposition of additives displaces the apparent thermodynamic equilibrium

between the material and the liquid in contact. Contrary to previous descriptions, the contamination was found to be unbounded

(Gandek et al., 1989a, 1989b).

Similar conclusions were found for large additives migrating to dry food simulants for short periods at temperatures elevated but

sufﬁciently close to those met during transportation (Schwope et al., 1987b). The most outstanding ﬁnding was that the migration

rate could not be overestimated by experiments using corn oil. Accelerated testing using food-simulating liquids is a common prac-

tice to evaluate the risk of migration, but it was emphasized that more rationale was required for evaluating with sufﬁcient conﬁ-

dence the risk of migration for new polymers. Simulating liquids should be chosen with respect to the nature of both the migrating

substance and the original polymer. Fatty food simulants offer worst-case migration and extraction capabilities only for hydro-

phobic substances in apolar polymers. Aqueous simulants are more aggressive for polar or charged substances, and polar polymers

(Fornasiero et al., 2002). Since most foods are multicomponent and multiphasic (e.g., emulsions, gels, cake with chocolate, etc.),

they cannot be reduced easily to a single contact phase when different classes of migrants are involved.

Why Thermodynamics Counts

Thermodynamics has been pra ised by the whole packaging community, including the chemic al, compounding, processing, recy-

cling and food industries, as well as authorities and safety agencies. It has been regularly used as the primary argument to justify the

condi tions of compliance testing (choice of simulan t, test temperature and contact time, extrapolation rules and migration calcu-

latio ns) and to authorize the recycling processes of polyme rs, active packaging, etc. A naïve reasoning may, however, lead to severe

consequences, which should not be underestimated. A common mistake is to assume that mass transfer sto ps after some long time.

The transferred amoun t is assumed to reach a maximum controlled by the partition coefﬁcient between the packaging and the

food. Statistical mechanics teaches that this macroscopic description is oversimpliﬁ ed and proceeds with an analogy between

a mechanical equilibrium and a chemical equilibrium. At molecular scale, all the substances continue to move freely at equilib-

rium as t hey were moving before the whole packaging-food system reaches a macroscopic equilibrium. In a closed system, the

equilibrium is associated with a zero net mass balance across the packaging-food interface: the number of mo lecules of type A

entering in the food is exactly compensated by the number of molecules of type A leaving the food. It mig ht be thought that

because the food has a larger volume than the packaging, the return of mo lecules is unlikely. It is however not correct because

the substa nces are transported faster in food than in dense polymer matrices. Only when the food-to-packaging volume ratio

becomes very large, does the probability of return approach zero and a total extraction of A is expected regardless of its afﬁnity

for the food.

When a chemical reaction transforms species A into species B in the food, the previous balance is profoundly modiﬁed, and more

substances A are invading the food than substances A are returning to the packaging. Similarly, when substance A cleaves into two

breakdown products a1 and a2 in the packaging, parts of substances A that migrated in the food are partly reabsorbed by the pack-

aging itself. When the reaction is almost complete, only traces can be identiﬁed in the food. As well, substances a1 and a2 can

migrate (faster than A) and may remain undetected if not speciﬁcally targeted.

4 Risk Assessment of Migration From Packaging Materials Into Food

Adding more packaging components (several layers, cap, label, etc.), variable temperatures and complex contact conditions

complicates the mass transfer description, but thermodynamics always provides the relationships to encompass all possible

exchanges including in multiphasic foods. The thermodynamics needed is not equilibrium thermodynamics, as it ignores the

time-course of the migration processes, but a local version, where equilibrium is reached only at the interface between each phase

and component. The classical non-equilibrium thermodynamic concept of local thermodynamic equilibrium is robust enough to

integrate non-linear sorption isotherms, coupled mass transfer and plasticizing effects. When the relaxation of polymer systems is

longer than the timescale of mass transfer (e.g., in glassy materials or hysteresis effects), constitutive equations need to be modiﬁed

to integrate the mechanical behavior of the materials (swelling or densiﬁcation).

No food packaging system is isolated enough from the rest of the world so that a true thermodynamic equilibrium cannot be

ﬁnally observed literally in real-life systems. Preventing the loss with surroundings in tests and calculations maximizes the amount

transferred to the food. Conversely, not considering the possibility of redistribution of migrants between materials during their life-

times hampers the proper management of non-intentionally added substances.

In shorts, most of the experimental evidence supporting regulation and rules was obtained on simple materials, preferably

monolayer and apolar ones in a rubber state. Equilibrium was reached rapidly in conditions accelerated comparatively to the

real shelf life of foods. In this case and only in this case, the concentration in the simulating liquid increases monotonously

with time.

Thermodynamics offer a robust framework to address all previous issues at the molecule scale, where the interactions and the

macroscopic properties emerge from the vibration of atoms. Theories or robust inference rules have been developed to relate the

chemical structure of the migrants and the polymers to the diffusion and partition coefﬁcients without requiring an explicit descrip-

tion of molecular interactions. The principles of quantitative structure-relationships detailed in this article are shown in Fig. 2. They

should not be considered as deﬁnitive rules, but as an ongoing process, where updates are regularly obtained.

Modern Crises

“Early civilizations adopted laws that punished sellers of tainted food” as quoted by Merill (Merrill, 1997), In 1958, the US Food,

Drug and Cosmetic Act introduced the Delauney clause, which prohibited the use of any substance as a food additive “if it is found

to induce cancer when ingested by man or animal” (National Research Council (US), 1982). In the US regulatory system, the

concept of additives is broad and comprises substances which may become a component of food or otherwise may affect the

food characteristics (Roberts, 1976). They include therefore any substance released by food packaging, regardless of the nature

of the materials. As a result, it is the responsibility of the industry to submit a dossier to the FDA to get a new substance, a new

polymer and new application of packaging approved. Similar rules were enforced successively in the EU via directive 1990/128/

EEC (EC. Commission, 1990), the regulations 2002/72/EC (EC. Commission, 2002) and 10/2011/EC (EC, 2011). The dossiers

apply, however, only to initial substances of plastic materials monomers and additives. China recently adopted legal requirements

close to the European system for plastics with a premarket approval for both the plastic resins and the additives (CFDA, 2016). The

inventories of substances are compared in Table 1.

A crisis occurs when an un authorized substance is found in the packaging material or when the levels in food raise concern due

to its ubiquitous distribution or due to signiﬁcant exposure to speciﬁc or global populations. As an illustration, the substances from

Figure 2 Principles of quantitative structure relationships to predict common transport and thermodynamic properties needed for migration

modeling.

Risk Assessment of Migration From Packaging Materials Into Food 5

materials intended to be in contact with food and exceeding EU migration tolerances during the period 2002–18 are listed in Fig. 3

in decreasing order of occurrence. Half of the 1956 cases are associated with imported tableware and with contamination by heavy

metals. Packaging alone represents less than 6% of the total with lids involved in 45% of cases. The main crises associated with

packaging materials and their likely causes and consequences are reviewed in Table 2.

The inset of Fig. 3 and Table 2 would suggest that the number of contamination cases would increase with time. The better effort

of authorities to identify, track and contain the contamination by FCM led to more frequent reporting. Regardless of the real risks,

the successive crises with can coatings, bisphenol A and mineral oils increased the awareness of the general population and, in

return, promoted different management strategies and better integration along the supply chain.

Risk Assessment for Decision Making

Highlights on modeling, risk assessment, and decision making

•

Migration modeling is a cognitive process aiming to capture the essential mass transfer phenomena responsible for the

contamination of food by substances originating from materials.

•

Calculations are carried out in a way that guarantees they are more severe than real conditions.

•

Only compliance can be demonstrated by calculations and simulations.

•

Calculations are in essence different than accelerated tests and should be used to reproduce real but conservative contact

conditions (time, temperature, interactions with food).

•

Sophistications in calculations required to be introduced progressively and in a comprehensive manner starting from the most

conservative (severe) conditions.

•

Variability and uncertainty must be characterized and documented.

•

The whole process can be integrated into preventive approaches (safe-by-design) or stochastic calculations (see Probabilistic

Modeling of the Migration section)

Overview of Migration Modeling

What Is Currently Permitted?

Notwithstanding the introduction of accelerated tests for compliance testing, simpliﬁed alternatives to traditional migration testing

were sought in the nineties. The common interests of the industry and the authorities were summarized by Begley (Begley, 1997):

“Traditionally, migration tests are performed by using food-simulating liquids such as water, edible oils, ethanol/water solutions

Table 1 Comparison of the Inventories of food contact substances in US, EU and China. Sources (EC, 2011; CFDA, 2016; FDA, 2018)

Materials

Regulations, resolutions, provisions & recommendations

US ( FDA, 2018) EU related China ( CFDA, 2016)

Plastics, resins, additives,

polymerization aids

1340 food contact notiﬁcations þ GRAS

and prior sanctioned substances

843 in the EU positive list (EC, 2011)

including

339 in positive list

Rubber 428 with SML or SML group, and 587

additives

86 with SML/QM or SML

group

176 substances in the French positive list

(DGCCRF, 1995)

88 substances

23 with SML/QM

Printing inks (Title 21 CFR Parts 175, 177, 178) 5104 substances (IAS) in EuPIA guidance

(EuPIA, 2016a) and Swiss Ordinance

(EDI, 2016). For NIAS, see (EuPIA,

2016b)

97 substances

33 with SML/QM

Paper and board (including

recycled)

482 substances 1556 substances additives in Council of

Europe resolutions (COE, 2009) and

good manufacturing practices (CEPI,

2012). For recycled materials, see (BfR,

2017; EFSA, 2012a)

277 substances

(Title 21 CFR Part 176) 77 with SML/QM

Compliance testing by modeling Yes (plastics) Yes (restricted to plastics) No restriction on applicable

materials

Risk Assessment including

migration modeling

Broad range of applications in petitions

in relation with consumer exposure

determination

Broad range of applications in petitions in

relation with an upper estimation of

consumer exposure

Not applicable in petitions

GRAS ¼ generally recognized as safe; SML: speciﬁc migration limit (maximum concentration in food); QM: quantity maximum (maximum amount in the material before contact); IAS:

intentionally added substances; NIAS: non-intentionally added substances.

6 Risk Assessment of Migration From Packaging Materials Into Food

and sometimes food. These tests are time-consuming in two ways; generally, the accelerated tests run for at least ten days, and the

analysis of the migrants at low concentrations in the simulants or food is generally difﬁcult. These analyses are also expensive and

generate hazardous laboratory waste”. Migration modeling was proposed both in the US (earlier works Limm and Holliﬁeld, 1996)

and in the EU (earlier works Baner et al., 1996; Baner et al., 1994a; Hinrichs and Piringer, 2002a) to tailor the process of migration

assessment. Migration modeling is nowadays mostly extended in the EU via a speciﬁc task force TF-MATHMOD publishing updated

guidance (Hoekstra et al., 2015a). Migration modeling is broadly accepted for compliance testing ( EC, 2011) and risk assessment

(see Schwope et al., 1990) of food contact materials with the following strict restrictions:

1) estimated values – whatever the calculation procedure and underlying assumptions, it must be at least as severe as the real test

and overestimate the migration;

2) calculations cannot be used to demonstrate non-compliance.

Similar methodologies are used by the industry to calculate the level of decontamination (cleaning) of materials in mechanical recy-

cling processes. But in this case, the amount released by the material should be estimated accurately and not overestimated. Addi-

tionally, it is worth noting that the state of polymers is very different between the conditions met by the consumer (moderated

temperatures) and the industrial conditions of recycling (high temperature, solvent-swollen). Only mechanistic modeling can cover

both cases. Modeling is not restricted to any material (thermoplastic, thermoset, paper, and board), but it has been tested chieﬂy for

thermoplastics and with non-uniform coverage.

There is no limit to the scope of modeling in current regulations and practices. Compliance testing can be seen as the simplest use

of modeling and validating closed loop recycling as the most complex application (see Fig. 1). Repeated use, composite materials

and safe-by-design approaches can be envisioned of intermediate complexity. All applications (small and large) obey in fact the

Figure 3 List of contaminants from food contact materials reported in the European Rapid Alert System for Food and Feed (EC) (extracted on

December 31st, 2018). The inset shows the evolution with time of the number of alerts issued by all member states, corresponding to the period

2002–18 to 955 border rejections, 459 alerts, 288 information for attention, 284 information and 216 information for follow-up.

Risk Assessment of Migration From Packaging Materials Into Food 7

Table 2 Analysis of the major crisis involving food contact materials

Crisis (class) Period (key references) Cause Consequence

Levels of plasticizers from PVC cling ﬁlms above

100 mg/kg

198x (Bradley et al., 2013a, 2013b; Maff, 1995,

1996a, 1996b, 1998)

Widely used as additives (plasticizer, solvent) with

large amounts, not covalently bonded to the

material backbone

Restrictions or ban of many phthalates in FCM

applications

High levels BADGE and of its reaction products of

BADGE from can coatings

1997-today (EFSA, 2004) Starting substance for the manufacture of epoxy

resins, used as an additive, functioning as

a stabilizer and as a plasticizer / reaction with

medium in contact

TDI 0.15 mg/kg b.w. (BADGE)

SML 1 mg/kg of food (reaction products)

Primary aromatic amines (PAA) from

agglomerated cork stoppers

2000-today (Six et al., 2002; Six and Feigenbaum,

2003)

Surface treatment, adhesive, lubricant / reaction

products

SML 0.01 mg/kg of food applied to the sum of PAA

released (annex II of EC, 2011)

Primary aromatic amines from laminates 2000-today (Campanella et al., 2015; Mutsuga

et al., 2014; Athenstädt et al., 2012)

polyurethane adhesive / reaction due to thermal

treatment

SML 0.01 mg/kg of food applied to the sum of PAA

released (annex II of (EC, 2011))

High levels of epoxidized soybean oils 1988 – today (Castle et al., 1988) Large amounts in plasticized PVC, high lipid

solubility, long time storage

Lowering of SML to 30 mg/kg for infant food

High levels Nonyl-phenols 2004-today (EC, 2002) Breakdown product of tris(nonylphenyl)phosphite

(authorized in EU 10/2011)

strictly limited by Directive 2003/53/EC (EC, 2003)

restriction REACH annex XVII (ECHA, 2016)

Semicarbazide leached from the thermal

decomposition of azodicarboamide from gaskets

used in baby food jar closure technology (press

twist and twist-off lids)

2003–05 (EFSA, 2005a; Stadler et al., 2004) Breakdown product of azodicarbonamide during

the heat treatment

ban (EC, 2004)

ITX 2005-today (

EFSA, 2005b) Poor identiﬁcation of transfer/contamination paths

from ink to foods

Not regulated in plastic material regulation

Poor information exchange between stakeholders

in supply chain

REACH study

Benzophenones from printing inks 1995-today (Johns et al., 1995) Poor identiﬁcation of transfer/contamination paths SML: 0.6 mg/kg

Poor information exchange between stakeholders

in supply chain

Bisphenol A leached from Baby Bottles 2004-today (EFSA, 2006; Gundert-Remy et al.,

2017)

Widely used in infant products Lowering of SML to 0.05 mg/kg

Replaced by other substances

New risk assessment ongoing by EFSA

Ubiquitous contamination by mineral oils from

recycled papers and boards

1993-today (Castle et al., 1993; EFSA, 2012b) Recycled paperboard, difﬁculties in analytical

analysis

Proposition of SML of 0.5 mg/kg

8 Risk Assessment of Migration From Packaging Materials Into Food

same global scheme, where the result obtained at the scale immediately below is used at the upper scale, as shown in Fig. 4. The

complexity of modeling relies on the number of scales considered and not on the system (ﬁlm, bottle, etc.) subjected to the

modeling activity. The keys to identifying the number of scales, components and steps are discussed later.

Modeling Using a Tiered Approach: From Worst-Case Scenarios to Detailed Conservative Ones

The exact value of the contamination of the food is never achievable because the conditions of contact are variable (time, temper-

ature) between comparable products and because knowledge of molecular mechanisms is not perfect. As a result, the practice seeks

successive approximations of the migration in a tiered approach, as shown in Fig. 5. At the ﬁrst tier, the estimation is very coarse and

connected with the highest overestimation factor. If the determined concentration at tier n is higher than the threshold of concern,

the next tier is triggered by introducing substantial reﬁnements and details, and so on. The process stops when no additional infor-

mation can be introduced (experiments need to be preferred) or when the calculated concentration is lower than the threshold of

concern. The lowest tier within the threshold of concern deﬁnes the proper level of knowledge required to demonstrate compliance

or to guarantee the safe use of a material, substance, or process. There is no systematic procedure to identify the minimum tier to

reach the goal, and only the needed information can be listed.

The possibilities and prerequisites for using modeling in compliance testing are reviewed in Table 3. The mentioned tiers R1, R2

and R3 are indicative and correspond to a modeling level more sophisticated than the ﬁrst tier. The ﬁrst tier is usually coined “total

migration” and corresponds to the total transfer of substances into food (see Fig. 13). The corresponding concentration in food,

tier 1

, is determined by a “dilution” model:

tier 1

¼ L

(1)

where C

is the initial concentration in the material (regardless of its distribution) expressed in mass per volume (preferred in this

work) or in mass per mass (industrial practice). L

P/F

is the material-to-food volume or mass ratio. When one-dimensional repre-

sentations are used, L

P/F

is also the ratio between the thickness of packaging walls, denoted l

, and the characteristic dimension of the

Figure 4 Overview of the nested modeling strategy to predict migration for all applications depicted in Fig. 1.

Figure 5 Principle of the tiered approach to demonstrate compliance for food contact materials. Compliance is demonstrated as soon as the esti-

mated concentration is greater than the threshold of concern. Tier 1 is usually associated with total migration (see Eq. 1).

Risk Assessment of Migration From Packaging Materials Into Food 9

Table 3 Prerequisites to use calculations as an alternative to migration testing

Prerequisites Type of estimate Examples of works Tier

Migration modeling or related calculations • Lectures • Lectures on migration (Vitraca; Vitracb; Vitracc)R1

• Reference text books (speciﬁc or general) • Text books on migration (Vergnaud, 1991; Piringer and Baner, 2000, 2008; Barnes et al.,

2006; Vergnaud and Rosca, 2006; Singh et al., 2017)

• Reviews and case studies on migration modeling (Piringer and Baner, 2000, 2008; Poças

et al., 2008; Helmroth et al., 2002; Arvanitoyannis and Bosnea, 2004; Lau and Wong, 2000;

Gillet et al., 2009a)(Vitrac et al., 2007a, 2007b; Vitrac and Leblanc, 2007)

• Reference text books on packaging (Singh et al., 2017; Robertson, 2016)R3

• Reference text books on mass transfer (Cussler, 2009; Vieth, 1991; Crank, 1975; Stastna and

De Kee, 1995; Mehrer, 2010; Neogi, 1996; Ben-Naim, 2013; Vrentas and Vrentas, 2013)

Identity of material

• technical speciﬁcation • supplier R1

• recycling code • regulation, standard (EC, 1997; GB, 2008)R2

• measurement • FTIR spectra R3

Characteristics of the polymer • density • supplier R1

• glass transition temperature • handbooks (van Krevelen and te Nijenhuis, 2009; Chemical Retrieval)R2

• crystallinity • measurements R3

Identity of the substance • real substance • supplier R1

• chemical structure • deformulation (Gillet et al., 2009b) and/or spectroscopy (Gillet et al., 2011; Nguyen et al.,

2015)

• chemical descriptors • analogous substance R3

Packaging geometry • 1 kg packed in 6 dm

• regulation R1

• 1D approximation of real geometry • supplier, end-user R2

• 3D real geometry • Research work (Chemical Retrieval; Masood and KeshavaMurthy, 2005; Huang et al., 2018;

Hu et al., 2012; Demirel and Daver, 2009)

Contact conditions (time, temperature, phase in

contact .)

• standard test conditions • end-user R1

• accelerated conditions • regulation R2

• real conditions R3

Initial concentration

• real values • supplier R1

• overestimates • guidance, orientation formula R2

• brute force deformulation (Gillet et al., 2011; Nguyen et al., 2015)R3

Diffusion coefﬁcients (see Diffusion Properties in

Polymers section)

• real values • measurements, literature (Roe et al., 1974; Moisan, 1980; Ju et al., 1981; Ehlich and Sillescu,

1990; Mauritz et al., 1990; Arnould and Laurence, 1992; Grifﬁths et al., 1998; Hall et al., 1999;

Vitrac and Hayert, 2006; von Meerwall et al., 2007; Grabowski and Mukhopadhyay, 2014;

Vagias et al., 2015)

• overestimates • Piringer model (Begley et al., 2005) or equivalent (Cussler, 2009)R2

• molecular theory • free-volume and its extensions (Mauritz et al., 1990; Arnould and Laurence, 1992; Hall et al.,

1999; Hong, 1996; Vrentas and Vrentas, 1998; Vrentas et al., 1996; Zielinski and Duda, 1992;

Fang and Vitrac, 2017)

• molecular modeling • Molecular Dynamics simulation (Karlsson et al., 2002; Vitrac et al., 2006; Harmandaris et al.,

2007; Durand et al., 2010; Gautieri et al., 2010; Lin et al., 2017)

10 Risk Assessment of Migration From Packaging Materials Into Food

Partition coefﬁcients or sorption isotherms (see

Sorption Properties and Partition Coefﬁcients

section)

• real values • measurements, literature R1

• overestimates • guidance (Hoekstra et al., 2015b)R2

• molecular theory • Kirkwood-Buff theory (Kirkwood and Buff, 1951)R3

• molecular modeling (Flory Approximation) • Flory -Huggins theory (Flory, 1942, 1949, 1953; Kadam et al., 2014)R3

• molecular modeling • solubility parameters (van Krevelen and te Nijenhuis, 2009; Gillet et al., 2009b; Hansen, 2007;

Baner and Piringer, 1991)

• Theory of interacting liquids (Kontogeorgis and Folas, 2009)R3

• Thermodynamic integration, insertion method (De Angelis et al., 2010; Boulougouris, 2011;

Boulougouris, 2010; Özal et al., 2008; Hess and van der Vegt, 2008; Hess et al., 2008)at

atomistic scale

• Atomistic calculations within the Flory approximation (Gillet et al., 2009b, 2010, 2011; Vitrac

and Gillet, 2010; Nguyen et al., 2017a, 2017b; Tribble, 2000; Lee, 1989; Weissler and Carlson,

1979)

Mass transfer resistance in the contacting phase

(see Limiting Mass Transfer Resistance section)

• none • worst-case R1

• correlation • liquid or gas model (Cussler, 2009; Vieth, 1991; Stastna and De Kee, 1995; Mehrer, 2010;

Neogi, 1996; Alexander Stern, 1994; Klopffer and Flaconneche, 2001; Masaro and Zhu, 1999)

• diffusion coefﬁcient • Graetz type problem, full simulation coupling (Pigeonneau et al., 2016)R2

• ﬂow R3

Acceptable thresholds (see Lower Bounds of

Toxicological Thresholds for Non-evaluated

Substances section)

• QM (EC, 2011; CFDA, 2016; FDA, 2018) • Regulations R1

• SML (EC, 2011; CFDA, 2016; Gao et al., 2014;

Aparicio and Elizalde, 2015; Salafranca et al.,

2015; Szendi et al., 2018; Careghini et al., 2015;

Spack et al., 2017)

• regulation policy (Begley, 1997)R2

• threshold of regulation (Food, 1995) • recommendations (GMP), literature R3

• TTC (Kroes et al., 2004)

Recommended for compliance testing at the ﬁrst (R1), second (R2) and third (R3) tier. If the value of the migration is larger at the second tier than at the ﬁrst one, use the second tier. If the value of the migration is larger at the third tier than at the second

one, use the third one. SML ¼ speciﬁc migration limit; QM ¼ maximum amounts; TTC ¼ threshold of toxicological concern; GMP ¼ good manufacturing practices. QM and maximum concentration in food

max

are related as C

max



F =P



1

QM, with V

and V

the food and packaging volume respectively, r

the density of the polymer and K

F/P

the partition coefﬁcient.

Risk Assessment of Migration From Packaging Materials Into Food 11

food, denoted l

, where A is the effective surface area in contact, counting usually the total surface area in contact with the food

and the headspace. By contrast, the food volume V

is restricted to the condensed part of the food (solid or liquid).

Key Steps in Migration Modeling and Risk Assessment Approaches

Any migration modeling for compliance testing, risk assessment, and safe-by-design approaches should be initiated by the review of

ﬁve important sections with an intent of providing an inventory on:

1. the formulation of materials (intentionally-added or not substances),

2. the components included in the design;

3. the steps followed by the material, the ﬁnished packaging, and the packaged food;

4. the information obtained from suppliers, regulations, industrial recommendations;

5. the described mechanisms of contamination.

For each section, the items need to be ranked and prioritized according to their suspected or foreseen importance on the contam-

ination of the packaged food. The principles and illustrations described in this article can be used to extend the systems, steps,

substances, etc. under scrutiny beyond primary food packaging and contact layers Fig. 6.

Migration Modeling for Compliance Testing and Beyond

In agreement with EU recommendations (Hoekstra et al., 2015a), this section details the assumptions and conditions suitable for

compliance testing at high tiers using a comprehensive description of mass transfer. The levels of description correspond to tiers R2

and R3 in Table 3.

Principles of Migration Modeling

Highlights on the principles of migration modeling.

•

Migration obeys (simply speaking) the well-known laws of diffusion.

Figure 6 Generic steps to review in migration modeling and safe-by design approaches. The depicted example corresponds to the review for a new

aseptic carton packaging for milk to be consumed by infants.

12 Risk Assessment of Migration From Packaging Materials Into Food

•

The mass transfer from one material to another, or the food requires speciﬁc treatment and attention as it is not implemented by

default generic numerical software (commercial or not).

•

One-dimensional mass transfer calculations are sufﬁcient for most of the applications if mass balances are well preserved.

•

The cost of modeling is dramatically reduced by simulating the contacting phase implicitly with proper boundary conditions. It

is important to note that the substances within any eventual mass transfer boundary layer are not included in the food mass

balance when implicit models are used.

•

Beyond its obvious numerical advantages (abacuses, master curves, pre-tabulated results), dimensionless formulations based on

similitude principles facilitate the review of model assumptions and results.

Underlying Microscopic Assumptions

Substances non-covalently bound to the polymer are subjected to thermal agitation, which causes in return a random translation of

their center-of-mass. Each additive, monomer, or residue jumps or walks randomly in the polymer matrix from one accessible void

to the next. The substances eventually reach the interface with the food, where the same hopping process is repeated, usually at

a higher pace. When volatile substances meet a gas phase, their skew trajectories are governed by the collision with gas molecules.

In all cases, random walks occur in three-dimensions, but a concentration gradient develops only at leaching interfaces, in a perpen-

dicular direction. As the walls of the packaging are thin compared to the characteristic food dimension, migration can be approx-

imated as a one-dimensional mass transfer problem as shown in Fig. 7. The migrating substances are depicted either as individual

molecules or solutes (i.e., scales are not respected) showing microscopic concentration ﬂuctuations or as smooth macroscopic

concentration proﬁles (continuous lines). Different symbols are used depending on whether the solutes are in the polymer (posi-

tion 0  x/l

< 1); in a small boundary layer in the food and of thickness x

, where a concentration gradient exists (position 1 < x/

 1 þ x

); or in the food bulk (x/l

> 1 þ x

) without any signiﬁcant concentration gradient (e.g., assumption of a perfectly

mixed medium). Choosing x

as a free parameter enables covering almost all contacting phases (gas, liquid food or simulant,

solid and semi-solid food) with reasonable complexity.

Fig. 7 plots simulation results using the concepts of statistical physics (i.e., the molecules jump randomly vertically and hori-

zontally without “knowing” where the interface is located) and by using the concept of continuum mechanics (i.e., balance on pop-

ulations and macroscopic ﬂuxes on elementary volumes). The stochastic and deterministic point of views are equivalent and

highlight that the observed macroscopic gradients are the consequence of the evolution of the distributions of solutes with time

and not its cause. In the upward direction, the random displacements are compensated by the same and opposite microscopic

ﬂux in the downward direction. The net balance is zero, and no concentration gradient can develop. The substances translate at

the same frequency in the horizontal direction (i.e., isotropic diffusion), but since no solute comes to compensate the ﬂux from

left to right at the beginning of the contact, a net ﬂux develops from left to right, resulting in the spreading of a concentration

gradient from the source (the polymer: P) to the food (the food: F). Statistical physics and continuum mechanics counts molecules

in a very similar fashion, via the concepts of probability density, r ¼

and of volume concentration C ¼ rN

, respectively. N

the total number of migrating substances in the whole system and N the number of molecules in an elemental volume V. The

concentration at the interface between F and P (denoted P-F) requires speciﬁc treatment and analysis. Since the principle zero of

thermodynamics does not hold for mass transfer, both density and concentration are discontinuous at the P-F interface. If no reac-

tion occurs at the interface, the mass balance is kept across the interface (i.e., no substance is lost). Additionally, the principle of

microscopic reversibility entails that the amount of substances crossing the interface per time unit from left to right (denoted

P/F) is exactly matched by the amount of substances crossing the interface in the opposite direction (denoted F/P). In other

words, any substance located exactly at the interface x ¼ l

has the same probability to go in P and F, irrespective of its origin.

This principle of microscopic mass balance reads:

P/F

¼ r

F/P

(2)

where f

A/B

is the frequency of translating from the compartment A to the compartment B. The concept of local chemical equi-

librium developed among others by Henry Eyring provides a robust framework to express the frequencies of passage from one state

to another without justifying the details on how the change in conformations and local velocities affect the passage from A to B. The

frequency of passage is written as:

A/B

¼ k exp





 G



(3)

with G

the free energy associated with the transition state and G

the free energy of sorption of the solute in A. The preexponential

factor k is independent of temperature; its expression depends on the statistical ensemble used to express probabilities. By

combining Eqs. (2) and (3), the molecules distribute across the interface with a ratio of probabilities equal to:

F=P

Cðx ¼ l

þ ε; tÞ

Cðx ¼ l

 ε; tÞ

¼ exp





 G



with ε/0 and t > 0 (4)

where x and t indicate the position and time; the food-packaging interface is located at x ¼ l

The concentration proﬁles are, therefore, continuous across the P-F interface only when the free sorption energies are similar in

both compartments. In the general case, the concentration proﬁle is discontinuous across the interface. Fig. 7 plots cases for an

Risk Assessment of Migration From Packaging Materials Into Food 13

apparent partition coefﬁcient K

F/P

of 0.5. In other words, the substance has double the chemical afﬁnity for P than for F, as expected

for plastic additives and monomers. For conservative migration modeling, values of K

F/P

higher than unity are preferred to maxi-

mize the gradient and consequently the amount transferred to the food. When the release of a substance does not modify the prop-

erties of the polymer (e.g. polymer densiﬁcation, or a shift in the glass transition temperature T

), the ratio of concentrations is likely

to be constant at the interface.

Migration Modeling and Similitude Properties

Under the assumption of uniform and constant transport and thermodynamic properties in each compartment (polymer: P,

boundary layer: BL, bulk contacting phase or food: F), the mass transfer problem is similar according to a small number of

Figure 7 One-dimensional description of solute diffusion (e.g., additive, monomer) from the packaging wall (position:0  x/l

 1, individual

solutes identiﬁed as ) to the contacting phase (individual solutes identiﬁed as ) via the food boundary layer (individual solutes identiﬁed as ):

(A) random distribution of solutes and corresponding concentration proﬁle at Fo ¼ 0.1 and (B) after contact times up to Fo ¼ 2. The percentages in

the top part represent the residual amount in each compartment.

14 Risk Assessment of Migration From Packaging Materials Into Food

dimensionless parameters or ratios. According to the principle of similitude, a real problem can be compared to a theoretical case

without dimensions if all independent dimensionless quantities are similar. The key dimensionless quantities are reviewed in

Table 4.

Explicit Versus Implicit Food Representation

It would be logical that migration models describe explicitly how the migrants distribute in the food. Solid foods, such as a chicken

or a pizza, are not expected to have all parts contaminated similarly. For risk assessment, all the parts that are intended to be

ingested, including the most contaminated sauce in contact with the packaging are considered. As a result, only a global estimate

of the food contamination is required, as measured with a liquid simulating the entire food. Replacing a solid by a liquid or vice-

versa has a consequence on the rates of mass transfer. This section discusses the differences between explicit and implicit represen-

tations of the food and the risk of underestimating the real migration.

Fig. 7 represents explicitly mass transfer in the food, that is the concentration proﬁles in the food are also calculated. The depicted

cases correspond to a characteristic food length l

of 12l

(i.e., thin food to make the boundary layer visible). The total domain

has a length of 13l

. When the food is represented explicitly, the amount transferred to the food is deﬁned as:

ðtÞ¼

x¼l

þl

x¼l

Cðx; tÞdx (5)

Eq. (5) accumulates substances in the boundary layer (round symbols) and in the food bulk (square symbols). Implicit food repre-

sentation will describe mass transfer only in the packaging and apply a proper boundary condition between the food and the pack-

aging at the position x ¼ l

. With the help of Eq. (4),theﬂux at the interface, denoted j, can be expressed only with concentrations

taken inside the packaging or in the food far from the interface, Cðx/N;tÞ. This assumption opens the way for an implicit represen-

tation of the food via a boundary condition relating the diffusion at the packaging-food interface with the ﬂux entering into the food:

jðtÞ¼D



x¼lpε;t

¼ hðCðx þ ε; tÞCðx/N; tÞÞ

¼ h



F=P

Cðx  ε; tÞCðx/N; tÞ



with ε/0 and t > 0

(6)

Eq. (6) offers a good approximation of the explicit representation when the contact time is sufﬁcient to reach a fully developed

concentration proﬁle (linear, so-called Prandtl approximation) inside the boundary layer. The critical Fourier number is given by

critical

¼ðx

=ð6D

Þ¼ðx

Þ=ð6BiÞ. The proﬁle plotted in Fig. 7A for Bi ¼ 1 and Fo ¼ 0.1 deviates from the assumption

above. The critical Fourier number is 1.67 and the value of C ðx /N; tÞ is close to zero, whereas the food is already contaminated via

its boundary layer.

Implicit food representation approximates the concentration in the food, C

(t), by its concentration far from the interface. By

noting that the ﬂux j(t) is taken after the boundary layer, one gets:

ðtÞ¼Cðx/N; tÞzC

ðt ¼ 0Þþ

jðtÞdt for t > Fo

critical

(7)

Table 4 Key dimensionless quantities of the migration from monomaterials. Contact is assumed to be initiated at t ¼ 0

Dimensionless quantity Meaning Justiﬁcation

uðx;tÞ¼

Cðx ;tÞ

ref

Dimensionless concentration C

ref

is a reference concentration, usually the initial concentration in the

polymer C

t ¼0

F/P

it is deﬁned from Eq. (4).

Partition coefﬁcient

At macroscopic equilibrium, it is also deﬁned as K

F =P

ðt /NÞ

with C

(t )

and C

(t ) the volume-averaged concentrations in F and P, respectively.

q ¼

Dimensionless position l

is characteristic food dimension, usually the thickness or the ratio

if the

geometry is complex, with V

the volume of the material and A the surface

area in contact with F. It is recommended to maximize A by also including

the headspace in the calculations of A.

Fo ¼

ðtÞdt

Dimensionless time D

is the diffusion coefﬁcient in the polymer at the temperature of contact.

The integral form is preferred if the diffusion coefﬁcient is variable with time

(temperature change)

Bi or Sh ¼

mass Biot or Sherwood number where x

is the thickness of the boundary layer and D

the diffusion

coefﬁcient in F, such that h ¼

is the effective mass transfer coefﬁcient

across the boundary.

P=F

Dilution ratio This number is deﬁned as the ratio of characteristic lengths and controls how

the substances are diluted in the food, usually much larger than the

packaging.

Risk Assessment of Migration From Packaging Materials Into Food 15

For Bi > 50 and if the threshold of concern is not too low, the amount present in the boundary layer can be neglected (less

than 1% i n Fig. 7AwhenBi ¼ 50) and an impli cit representation can be used. Its main advantage is the dramatic reduction i n the

problem complexity and t he computat ional t ime. In very t hin or low barrier ﬁlms and in solid foods, the implicit fo od repres en-

tation may severely underestimat e the contamination of the food. The amounts in the boundary layer reported in Fig. 7Areach

9%, 18% and 22% for Bi ¼ 10, 5, 1 respectively. When an implicit representat ion is used, calculating the concentration in the

food from the mass balance in the packaging between t ¼ 0andt does not so lve the issue as the closure equalities are enforced

at any time:

Cðx; tÞdx

¼jðtÞ¼

dCðx/N; tÞ

for t > 0 (8)

The amount present in the boundary layer is always neglected in migration representing the food implicitly. Only by choosing

artiﬁcially Bi/N as a worst-case scenario makes this amount negligible at the price of migration much faster than that expected in

the real conditions.

Other Assumptions

In this section the constitutive equations to describe mass transfer from monolayer and multilayer materials, when the food is rep-

resented implicitly via the boundary condition (6) and the food mass balance (8) are presented. The total packaging thickness is

denoted l

i¼1

for a packaging (e.g. laminate) consisting of m layers. At the position x ¼ 0 (usually the surface exposed to

the ambient environment), an impervious boundary layer is assumed so that the amount transferred to the food is maximized.

All substances are assumed to be well dispersed where they have been incorporated (e.g., no blooming effect) and below their

concentration at saturation (i.e., no supersaturation).

Governing Equations for Monolayer Materials

Overview

The full set of equations for monolayer materials including the initial condition (IC), the transport equation (TE), the boundary

conditions (BC) and the mass balance on the food compartment (MB) are:

IC : C

ðx;t¼0Þ

¼ C

t¼0

for 0  x  l

TE :

ðx;tÞ





ðx;tÞ



ðx;tÞ



for 0  x  l

; t > 0

BC : jðtÞ¼







x¼l

ε;t

¼ h

F=P

x¼l

ε;t

 C

ðtÞ

;



x¼0;t

¼ 0 for t > 0 and ε/0

MB : C

¼ C

t ¼ 0

ds for t  0

(9)

where n is an e xponent controlling the system of coordinates (n ¼ 0: Cartesian, n ¼ 1: cylindrical; n ¼ 2: spherical); D

(x,t)

) is the diffusion coefﬁcient that possibly varies with concentration (e.g., in the case of plast icizers used at high

concentrations).

When the diffusion coefﬁcient in the packaging is considered constant along with C

(t ¼ 0) ¼0, Table 4 and system (9) yield the

following dimensionless formulation for Cartesian coordinates:

IC : u

ðq;Fo¼0Þ

¼ 1 for 0  q  1

TE :

ðq;FoÞ

vFo

ðq;FoÞ

for 0  q  1; Fo > 0

BC : j



ðFoÞ¼

ðq;FoÞ



q¼1ε

¼ Bi

ðq¼1ε;FoÞ



ðtÞ

t¼0

;

ðq;FoÞ



q¼0

¼ 0 for t > 0 and ε/0

MB :

ðtÞ

t¼0

¼ L



ðsÞds for t  0

(10)

16 Risk Assessment of Migration From Packaging Materials Into Food

When Bi/N, the third type boundary condition at q ¼ 1 (Robin boundary condition) can be replaced by a simple coupling

with the mass balance. Differentiating BM with respect to Fo, yields:



ðFoÞ¼ 

ðq;FoÞ



q¼1ε

t¼0

ðtÞ

dFo

F=P

ðq¼1;FoÞ

vFo

(11)

The worst-case scenario when Bi/N and K

F=P

/N corresponds to the Dirichlet boundary condition u

(q¼1,Fo)

¼ 0.

Concentration in the Contact Phase at Thermodynamic Equilibrium

According to Eq. (10), the maximum concentration in food is obtained at thermodynamic equilibrium (j* / 0):

t¼0

ðFo/NÞ

t¼0

þ 1

(12)

In practice it is convenient to express the kinetics of desorption in food as a function of the fraction of the equilibrium value:



Fo; Bi; K

F=P

; L

P=F



t¼0

P=F

þ 1

F=P



Fo; Bi; K

F=P

; L

P=F



¼ C



Fo; Bi; K

; L



(13)

Dimensionless Migration Kinetics and Their Analytical Approximations

The dimensionless migration kinetics



ðFo; Bi; K

F=P

; L

P=F

Þ are plotted in Fig. 8. The analytical solution associated with the Dirichlet

condition C(x ¼ l

, t) ¼ 0 is given by Eq. (4.18) in Crank’s book (Crank, 1975) and reads:



ðFoÞ¼

ðFoÞ

¼ 1 

lim

n/N

ðFoÞ with S

ðFoÞ¼

i¼0

exp





ð2i þ 1Þ



ð2i þ 1Þ

(14)

For small Fo values, the approximation (14) requires n to be very large (10

or 10

) and a more efﬁcient approximation can be

obtained by combining an approximation of Eq. (4.20) in Crank’s book (Crank, 1975) with Eq. (14) for i ¼ 0 as:



ðFoÞ¼min



ﬃﬃﬃﬃﬃ

ﬃﬃﬃﬃ

; 1 

exp







 min



ﬃﬃﬃﬃﬃ

ﬃﬃﬃﬃ

; 1



(15)

Approximations (14) and (15) are plotted along with the results of numerical simulations in Fig. 8. The common assumption of

the linearity of



with

ﬃﬃﬃﬃﬃ

is well veriﬁed while Fo  0.8 and when the equivalent length of the contacting phase ‘

¼ K

F=P

P=F

and Bi are large. At intermediate Bi values, the Dirichlet condition offers a conservative approximation (i.e., v



is overestimated) and

Eq. (15) can be used safely for compliance testing. However, when ‘

 100 (e.g., small food volume, low chemical afﬁnity for the

food), Eq. (15) must be avoided due to a signiﬁcant risk of underestimation of low Fo values. The equilibrium is, indeed, reached

much faster due to a much smaller amount to transfer and because the concentration at the F-P interface never vanishes. An accurate

estimation of migration requires ad-hoc numerical solutions or special analytical solutions. In mathematical terms, ﬁnite volume

effects cause the propagation of shockwaves between the polymer and the contacting phase: the addition of substances to F modiﬁes

instantaneously the capacity of P to transfer additional substances, and so on for the next substances creating positive and destruc-

tive interference across the mass transfer boundary layer when it exists. One practical consequence is that analytical solutions are

without closed-forms and therefore more complex than Eq. (15).

•

For Bi/N and L

P/F

< 1, the analytical solution verifying ﬁnite volume effects and boundary condition (8) is given by Eq. 4.37 in

Crank’s book (Crank, 1975) and is very accurate at a reasonable cost for Fo > 10

–4

•

For arbitrary Bi and ‘

values, new solutions verifying the general boundary condition (6) (see Vitrac and Hayert, 2006; Sagiv,

2001, 2002, Goujot and Vitrac, 2013). Short-time and long-time solutions were optimized for efﬁciency and to integrate more

complex physics such as non-linear sorption isotherms or for boundary conditions variable in time.

The general solutions are not detailed here as their expressions exceed the scope of the article. When Bi/N, the Eq. 4.37 in Crank’s

book (Crank, 1975) reads:



ðFoÞ¼1 

n¼1

‘

ð1 þ‘

1 þ ‘

þ ‘

exp



q



where are zeros of tan q

¼‘

(16)

The zeros of the transcendental equation, q

, increase with n and also when ‘

decreases. This behavior demonstrates that v



ðFoÞ

converges exponentially and more rapidly to equilibrium when ‘

values are low. The linearization with

ﬃﬃﬃﬃﬃ

ceases to be correct

earlier and is associated to slopes varying with ‘

. As the values of q

are usually not tabulated beyond n ¼ 6in reference text books

(see Table 4.1, p. 379 in Crank, 1975), it is recommended to restrict the use of Eq. (16) to Fo > 10

–4

Risk Assessment of Migration From Packaging Materials Into Food 17

Governing Equations for Multilayers

Highlights for multilayers

•

Multilayers can be seen as generalizations of monolayer materials but with additional features such as functional barriers

(delayed migration) and reservoir layers (accumulation inside one or several internal virgin layers).

•

When the partition coefﬁcients between layers and their diffusion coefﬁcients are constant with time, the total migration is the

sum of the migrations associated with the contribution of individual layers.

•

Uncertainty in partitioning and the initial distribution of migrants can be overcome by moving “artiﬁcially” the content on one

layer to a layer closer to the contacting phase.

•

The calculation procedures presented in this section have been introduced in the guidance to EU Regulation 10/2011/EC.

Thermodynamic Assumptions

The case of materials consisting of m materials or layers (m > 1) can be seen as a generalization of the monolayer case (m ¼ 1) at the

expense of a few additional assumptions and conventions. Because monolayer systems were dominating in the 20th century, the

reasoning supporting US and EU regulations was, therefore, devised based on an assumption of migration without delay and

obeying a scaling proportional to the square root of time. The conventional condition of 10 day at 40



C was thus thought to

be equivalent to a test of one hundred days with a factor comprised between unity (equilibrium) and

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

100 days

10 days

z3:16 (Fo  1).

Figure 8 Dimensionless desorption kinetics v



ðFoÞ¼

ðFoÞ

for various values of L

P/F

, K

F/P

and Bi with C

1=L

P=F

þ1=K

F =P

. Approximations (1) and

(2) are given by Eqs. (14) and (15), respectively.

18 Risk Assessment of Migration From Packaging Materials Into Food

Moving the substances away from the P-F interface delays substantially the mass transfer to the contacting phase. This behavior is

central to the concept of functional barrier (Feigenbaum et al., 2005); it was initially explored to promote the incorporation of

recycled material – possibly contaminated due to post-consumer misuse or mixing with non-food grade materials – in co-

extrudates. The recycled polymer is sandwiched within two layers of virgin polymers. Similar problems can be resolved only by adapt-

ing the initial condition C (x, t ¼ 0) in the equation system (9) to the need instead of using a uniform distribution.

When the materials are different in nature, a contact condition similar to Eq. (4) needs to be considered. The proposed descrip-

tion relies on an assumption of a linear and reversible sorption of substances in each layer so that a linear sorption isotherm is

assumed in any layer, including in the food. By denoting p(x, t) the partial pressure of the migrating substance, the isotherm asso-

ciated with the Henri constant

j¼0::m

for each layer is:

pðx; tÞ¼k

Cðx; tÞ for 0  L

j1

 x < L

and t > 0 ; with L

i¼1

and 0  j  m (17)

For the sake of generalization, the layers including the food are indexed from j ¼ 0 (food: F) to j ¼ m (layer the most distant to F),

with j ¼ 1 being the contact layer, as shown in Fig. 9. The residual concentration in each layer is C

ðx; tÞ¼

j1

Cðx; tÞdt for 1  j 

m and C

(t) ¼ C

(t).

Concentration in the Contact Phase at Thermodynamic Equilibrium

The Henri isotherm deﬁned in Eq. (17) offers a robust but simpli ﬁed thermodynamic representation of the variation of the chem-

ical potential with the local composition in the system. The validity of the model and its generalization are discussed in Sorption

Isotherms section. Partial pressure p(x, t) is a continuous potential, and thermodynamic equilibrium is achieved when its value

is uniform across the structure. By neglecting mass losses, Eq. (17) and the mass balance between t ¼ 0 and t/N

j¼0

t/N

j¼0

t¼0

enables generalization of Eq. (12):

¼C

t/N

j¼0

t¼0

1 þ

j¼1

(18)

where k

¼ C

t/N

t/ N

is the partition coefﬁcient between the layer j and the food.

Transport Equations

Transport equations are unchanged and are deﬁned as:

vCðx; tÞ

¼D

Cðx; tÞ

for 0  L

j1

 x < L

and 1  j  m (19)

and connected at internal interfaces via the double conditions of mass conservation and local thermodynamic equilibrium:

vCðx; tÞ



x¼L

ε

¼ D

jþ1

vCðx; tÞ



x¼L

þε

for 1  j < m



x ¼ L

 ε; t



¼ k

jþ1



x ¼ L

 ε; t



for 0  j < m

and ε/0 ; t > 0

(20)

Figure 9 Indexing rule of a material including m layers (total thickness l

) in contact with a food indexed 0. The left and right external boundaries

are considered impervious (no mass loss). The concepts of “functional barrier” and “reservoir” assume that layer j is the source (with non-zero initial

concentration). It is used in Typologies of Migration Behaviors section and Superposition Principles and Conservative Scenarios for Multilayer and

Multicomponent Systems section.

Risk Assessment of Migration From Packaging Materials Into Food 19

Limiting Mass Transfer Resistance

A dimensionless formulation of Eqs. (6), (19) and (20) is achievable at the expense of choosing a reference layer, 1  j

ref

 m,

and setting the reference time scale as

ref

. In numerical algorithms, where stability and convergence are very stringent, it is conve-

nient to choose as a ref erence layer, the layer associated with the highest mass transfer resistance (lowest permeabilit y) is the best

choice:

ref

such that

ref

¼ max





j¼1::m

(21)

If several conditions need to be compared, a natural choice is to choose the contact layer as the reference layer (j

ref

¼ 1).

Typologies of Migration Behaviors

The main behaviors, which can be observed with multila yers, are illustrate d in simple conﬁgurations corresponding to

a bi layer structure (each layer has a thickness l

/2) in contact with a liquid phase with a characteristic thickness 50l

and asso-

ciated with k

¼ 1. The ﬁve considered cases are summarized in Table 5 and were a ssociated with a similar initial amount and

comparable ﬁnal concentration in the contacting phase ca.2.0 10

–2

,(C

þ C

)/2. The other parameters were Bi/N,

¼ l

, D

¼ D

and the time scale

. The calcula ted proﬁles and k inetics correspo nding to the ﬁve scenario s are plotted

in Fig. 10.

Monolayers and functional barriers lead to uniformly decreasing concentration proﬁles. The corresponding desorption kinetics

in F are respectively proportional to the square-root of time and proportional to time after some lag time equal to

. For the same

initial amount in the structure and after the lag time, the functional barrier ceases to operate and leads to a migration proportional

. Only a functional barrier combining a diffusion barrier







and a solubility one



[



can slow down the desorp-

tion durably. Reservoirs behave very differently; they are associated with non-monotonous concentration proﬁles, accelerated

desorption kinetics while converging to a very similar concentration at equilibrium. For the same initial content and when the

barrier on the right is higher than the barrier on the left

[

, the “reservoir” conﬁguration overestimates the migration kinetics

associated with all other conﬁgurations.

Superposition Principles and Conservative Scenarios for Multilayer and Multicomponent Systems

Mathematical Principles

Multilayer structures offer a broad range of behaviors. In the simplest cases, as shown in Fig. 10, desorption kinetics are monoto-

nous with time. It may not be the case if the functional barrier and reservoir effects are combined. The calculations for complex

multilayers are complicated by the difﬁculty to associate the uncertainty in diffusion

j¼1::m

and sorption properties

j¼1::m

with the concentration in food. For monolayer materials, a conservative scenario is achieved by overestimating simultaneously

and Bi. For multilayer materials, no similar rule holds. Intuitively based on the illustrated conﬁgurations in Fig. 10, it can

be stated that for the capacity of the layer j to transfer its content, M

(t) (amount transferred to F at the time t), is maximized

(denoted





ðtÞ) if the following properties are fulﬁlled:

is replaced by dD

e > D

for 1  i  j and k

is replaced by bk

c < k

for 0  i  j

is replaced by bD

c < D

and k

is replaced by dk

e > k

for j < i  m

(22)

Table 5 Illustration of the main behaviors associated with multilayer structures. The concepts of functional barrier and

reservoir are illustrated in Fig. 9

Interpretation Codej ¼ 1 j ¼ 2 j ¼ 1 j ¼ 2

1 1 1 1 uniform distribution (equivalent to a monolayer) [1,1][1,1]

0 2 1 1 functional barrier (barrier to diffusion only) [0,2][1,1]

2 0 1 1 reservoir layer (same capacity) [2,0][1,1]

0 2 2 1 functional barrier (barrier to diffusion and of solubility) [0,2][2,1]

2 0 1 2 reservoir layer (with reduced capacity by half) [2,0][1,2]

20 Risk Assessment of Migration From Packaging Materials Into Food

Conditions (22) have a mathematical justiﬁcation in the linear properties of the Eqs. (10,19–20). The solution of any linear

decomposition of the initial concentration proﬁle is equal to the sum of the individual solutions:

Cðt ¼ 0; xÞ¼

j¼1

ðt ¼ 0; xÞ where p  1 is the number of profiles; 0  x  l

ðt; Cðt ¼ 0; xÞÞ ¼

Mðt; Cðt ¼ 0; xÞÞ

j¼1



t; C

ðt ¼ 0; xÞ





j¼1





t; C

ðt ¼ 0; xÞ



eð

t; Cðt ¼ 0; xÞÞ

(23)

Example for a Trilayer Material ABC

Eq. (23) is particularly signiﬁcant as it is valid for any partitioning of the source terms in the material, irrespective of the positions of

the layers. An application of the additivity of M

values (concentration proﬁles and kinetics) is shown in Fig. 11 for a trilayer struc-

ture ABC detailed in Table 6.

The simulation of each layer individually underlines the different mechanisms controlling the contribution of each layer: reser-

voir effect for source A (scaling of desorption kinetics with the square root of time) and functional barriers for B and C (desorption

kinetics linear with time after signiﬁcant lag-times). The depicted proﬁles are assumed to the “likely” or “true” ones. They are consid-

ered inaccessible to simulation and should be approximated at some tier in a way that the concentration in F is always overesti-

mated (see Fig. 13).

Figure 10 Concentration proﬁles (top) and migration kinetics (bottom) for the bilayer structure and scenarios detailed in Table 5.

Risk Assessment of Migration From Packaging Materials Into Food 21

Eq. (22) provides a numerical procedure to devise conservative scenarios for multilayer structures. A similar procedure has been

detailed in Migration Modeling for Compliance Testing and Beyond section of the European guidance document (Hoekstra et al.,

2015b). Here the procedure for the sole overestimation of the chemical afﬁnity effects is repeated by keeping the diffusion coefﬁ-

cients to their “likely” values. The core idea is to prevent or hinder the diffusion of the substances in the jth layer to the right (by

assuming that the food is on the left) and to facilitate their desorption to the left, to bring the contaminants closer to the food. The

“conservative scenario” of Table 6 applies a factor ten to the Henry-like coefﬁcient(s) k

of the source and behind. The likely k

and D

Figure 11 Illustration of the additivity of the sources (see Eq. 23) for a trilayer structure ABC associated with the case study detailed in Table 6:

concentration proﬁles (top), kinetics (bottom). The case “sources ABC” is obtained by simulating the whole structure. The result A þ B þ C corre-

sponds to the mathematical addition of the contributions of the three sources.

Figure 12 Illustration of the conservative scenario of Table 6 based on the overestimation of the contribution of each source. The reference corre-

sponds to the initial case-study conﬁguration also depicted in Fig. 11.

22 Risk Assessment of Migration From Packaging Materials Into Food

values are kept for the layers between the food and the jth layer. To prevent back diffusion in the reservoir layers the diffusion coef-

ﬁcients were divided arbitrarily by a factor 10

. The corresponding kinetics are shown in Fig. 12, with their parameters listed in the

section “conservative scenario” of Table 6. The diffusion coefﬁcients towards the food are overestimated by a factor of 10. At inter-

mediate Fourier numbers below 0.4, the kinetics is not signiﬁcantly overestimated, but a signiﬁcant conservatism is achieved at

larger Fourier numbers when the overall migration is controlled by the second and third layers as sources. This case study demon-

strates how migration scenarios can be ﬁnely tuned to decrease the uncertainty related to the behavior of internal layers. The contact

layer as a source can always be considered as a single monolayer.

In practice, any uncertainty on the internal partitioning between layers can be converted into a conservative scenario by forcing

mass transfer to the food and by relocating “artiﬁcially” the content from layer j to layer j  1. The iterative procedure is illustrated in

Fig. 13 and can be applied to decrease a m-layer problem into a m  1, m  2, etc. layer problem, until reaching tier 1 (total migra-

tion). The ﬁrst iteration applied to the case-study is denoted “worst-case” scenario in Table 6. The procedure may overestimate

Table 6 Parameters used to construct realistic and conservative migration scenarios depicted in Figs. 11 and 12. Quantities are expressed

respectively to the likely values

for the ﬁrst layer (the three layers ABC are indexed 1,2,3). They are scalar when the contribution of each

layer as a source is considered in combination with others (the three sources are considered at once). The contributions of individual

sources are indicated by 3  1 vectors mentioning the properties of all layers considered as a source or not

Property

Case-study (likely)

Contribution of the jth layer

Conservative scenario (high tier)

Contribution of the jth layer

Worst-case scenario (low tier)

Contribution of the jth layer

j ¼ 1 j ¼ 2 j ¼ 3 j ¼ 1 j ¼ 2 j ¼ 3 j ¼ 1 j ¼ 2 j ¼ 3

likely

þl

likely

þl

likely

0.2 0.5 0.3 0.2 0.5 0.3 0.2 0.5 none

0;likely

1 1 1 [1,0,0] [0,1,0] [0,0,1] 3.5 0.6 none

likely

1 1 1 [1,10

3

,10

3

] [1,1,10

3

] [1,10,10] 1 1 none

likely

0.3 0.8 2.0 [10,10,10] [1,10,10] [1,1,10] 10 10 none

Likely value ¼ true value or close to the true value in the considered scenario.

Figure 13 Principles of the simpliﬁcation of a m-layer problem (here m ¼ 3) into a problem with a lower number of layers and, therefore, easier

parameterization. The represented distributions in the packaging correspond to initial conditions at various tiers.

Risk Assessment of Migration From Packaging Materials Into Food 23

dramatically the real migration, but it keeps the applicability of conservative calculations to demonstrate safety at low cost. For risk

assessment, the procedure may be applied with caution as it may lead to unrealistic consumer exposure.

Strategies and Equations to Simulate Multiple Steps and Conditions

This part discusses the invariance of migration estimates, namely C

(t), with the order of contact and the thermodynamic conditions

met by the components of the packaging before and during food contact.

Highlights on the strategies to simulate multiple steps and variable conditions.

•

The initial dispersion of the substances between materials and the subsequent migration into the contact phase requires at least

a two-step modeling and simulation.

•

Temperature and relative humidity are variable during storage, transportation, etc. and subjected to uncontrolled diurnal and

seasonal variations. Their variations can be integrated in chained simulations (one condition ¼ one step), where the outputs

calculated at the previous step are used as inputs for the next steps.

•

Chained scenarios can be factorized under certain conditions in a shorter series of conditions without respecting the original

order of steps or variations.

•

Factorization keeps unchanged migration results only if the contact conditions are unchanged and if apparent partition coef-

ﬁcients are independent of temperature.

•

Factorization should be avoided if the dispersion in food plays a signiﬁcant role and in the presence of polar migrants.

•

Factorized scenarios offer practical estimates for repeated uses.

•

Probabilistic migration modeling (see Probabilistic Modeling of the Migration section) can be used to analyze the effects of

known and unknown variations.

Problem Formulation

Mass transfer between components and materials occur insidiously along the supply chain. Fig. 14 illustrates conditions triggering

or altering migration from printed materials. Many uncontrolled factors may affect the extent of mass transfer: i) variable contact or

exposure times, ii) random combinations of storage and transportation steps for intermediate, ﬁnished packaging materials and

packaged foods, iii) changes in temperature and relative humidity (e.g., seasonal, diurnal, international transportation), iv) modi-

ﬁcations of boundary conditions during any stage of materials lifetime and product shelf life. The redistribution of migrants

between materials, layers, and components deserve special attention as it usually remains ignored by end-users. In practice,

cross-contamination occurrences can also be considered indirectly (i.e., without causality) as impurities and non-intentionally

added substances. Without being exhaustive, cross-contamination is highly likely from cured adhesives and printing inks, recycled

materials and any material with rich volatile organic compounds. Packaging and materials stored in stacks and reels ease cross

contaminations by contacting internal and external layers, regardless of the presence of a functional barrier (absolute such as

a metallic layer or relative such as a barrier polymer) in the structure. Due to periodic conditions, the inner layer can act as a reservoir

of contaminants before the food is put in contact.

Figure 14 Illustration of the redistribution of the migrants from UV-cured printing ink and their subsequent migration in food for long shelf life

products. The depicted cases cover hot-ﬁlled or aseptically-ﬁlled products (e.g., soups, pasteurized juices, sterilized dairy products), and dry or

ready-to-eat products stored in cardboard boxes.

24 Risk Assessment of Migration From Packaging Materials Into Food

From a mathematical viewpoint, the succession of steps and temperature variations can be seen as a sequence of constant condi-

tions occurring in variable order. In the presence of n

steps

conditions occurring at time t ¼ 0; t

; :::; t

steps

, the composite solution is

obtained by integrating the coupled system (9) via the Chasles’ relation:



x; t

steps



nsteps

Cðx; tÞ



Cðx;t¼0Þ

dt ¼

steps

i¼0

iþ1

Cðx; tÞ



Cðx;t ¼t



steps

; C

t¼0

; C

t¼0



nsteps

ðt; C ðx ; tÞÞ



t¼0

dt ¼

steps

i¼0

iþ1

ðt; Cðx; tÞÞ



t¼t

(24)

Determining the duration of each step

i¼t

iþ1

t

and their corresponding temperatures are critical. Representing all diurnal

and seasonal temperature variations shown on the timeline of Fig. 14 would require 2  450 ¼ 900 successive simulations (one

every 12 hours). On the one hand, a rigorous approach suggests that the congruence of all the steps should be strictly preserved

to get reliable conclusions. In this case, how to identify the worst-case combination of conditions or steps? If the temperature vari-

ations are uncontrolled, how to build a conservative sequence? On the other hand, a naïve approach would suggest that the times

series in Eq. (24) could be built from mutually independent steps assembled as the sum of a series of standard and well-controlled

steps (e.g., cold, moderate and warm days), and one series with stochastic contributions, representing an extra safety margin.

A First Intuitive Approach

The variations of C

between steps are not factorizable but the dimensionless times are. Their effects are additive and commutative

Þ¼C

Þ (see Eq. 10 for the demonstration). If the diffusion coefﬁcient is the only quantity varying

with temperature and the plasticizer content (i.e., no change in the partitioning, and the packaging dimensions), the effects of n

steps

is captured via the generalized Fourier number:



nsteps



nsteps

ref

ðTðtÞ; plasticizerðtÞÞdt

ref

(25)

where D

ref

is the diffusion coefﬁcient of the most limiting material, component or layer, and l

ref

the one-dimensional equivalent

thickness (see Eq. 21 to identify j

ref

Eq. (25) is equival ent to the low of the composition of velocities along a curvilinear coordinate system tangent to the

trajectory going from A (



ðt ¼ 0Þ¼0) to B (v



ðt/NÞ¼1). The analogy between spatial translation alo ng a winding

road and the translation along the curve



vs. Fo is illustrated in Fig. 18 by comparing the travel via three modes of transport

(each of duration

i¼1;2;3

at a speed:

i¼1;2;3

) with the cumulative contamination aft er three s teps (of duration

i¼1;2;3

at temperature

i¼1;2;3

. The total distance is v

þ v

, and the cumulati ve migration is



ððDðT

þ DðT

Þ=l

Þ independent of the order of the steps. Replacing the physical time by cumula tively

measuring the total arc-length of the curve or the road makes it possible to use end point estimates (one simulation) instead

of chained simulations (n

steps

simulations). In this new space, the reside nce times – represented by the density of markers on

the curves – are not uniform. They are distributed more densely at departure and destinati on, but more sparsely in the middle

region where the studied system follows different routes Fig. 15.

Eq. (25) suffers, however, from a lack of generality as it applies only to the limiting mass transfer resistance and not to all layers.

As a rule of thumb, it offers an acceptable solution if the function

v is monotone with Fo (i.e., v



ðFo

Þv



ðFo

Þ when Fo

 Fo

simple counter-example can be, however, constructed by noting that the concentration at equilibrium C

depends only on the

initial and ﬁnal states, but not on intermediate steps.

The conditions of exchangeability of steps (which is more generic than Eq. 25) is discussed hereafter in more general terms. Two

conditions are analyzed: i) when the effect of the mass transfer resistance is considered (explicit representation) and the number of

molecules does not change, and ii) when an implicit food representation is used (i.e., Eqs. 9,10,20). The distinction between explicit

and implicit food representation is relevant, as the boundary layer delays the effects of perturbations and may contain a signiﬁcant

amount of contaminants, which are ignored at low Bi values in implicit representations (see Explicit Versus Implicit Food Repre-

sentation section).

Strict Conditions of Exchangeability With Explicit Food Models

Microscopic Description of the Random Walk of Molecules Between P and F

The visited distance by a deterministic system is ‘ ¼

‘

þ / and is invariant with the order of the visits. At a micro-

scopic scale (i.e., at a scale where they can be separated), the trajectories of migrants in materials and the food verify this property

(distances are additives), but with a different relationship with time (displacements are proportional with time). The random

displacements of additives and residues shares, instead, notable features with random walks and continuous stochastic paths. In

a loose sense, substances jump randomly at discrete times (random walks) or as continuous events (stochastic paths). Their skewed

trajectories are nowhere differentiable, and velocities cannot be deﬁned in a classical sense. Under a hypothesis of stationarity of the

Risk Assessment of Migration From Packaging Materials Into Food 25

microscopic random process (the quantity X(t) has the same statistics as Xðt þ εÞfor any ε), a law of composition can be justiﬁed for

the mean-square-distances, denoted h‘

i, visited by the molecules: h‘

i¼D

þ D

þ /. h ‘

i is mathematically

deﬁned as hx

(t)ihx(t)i

, where hx(t)i is the average distance (ﬁrst moment) and hx

(t)ithe square distance to the food-

packaging interface located at x ¼ 0:

Þi

þN

N

x; t



ðtÞ



þN

N

rðx; tÞdx

(26)

The one-dimensional space approximation with explicit food representation is deﬁned on the domain l

 x  l

so that the

probability density rðx; tÞ veriﬁes:

l

rðx; tÞ dx ¼ 1 and rðx; tÞ¼0 for x < l

or x > l

. A differential form of the growth of hx

(t)i

with time is inferred from a special case of the Fokker-Planck equation

vrðx;tÞ

¼ D

rðx;tÞ

(similar to Eq. 9). By multiplying both sides

by x

and by integrating over the entire domain, gives:

þN

N

vrðx; tÞ

dx ¼

þN

N

rðx; tÞ

dx with D ¼ D

when x  0 and D ¼ D

otherwise (27)

with D

and D

being the diffusion coefﬁcients in P and F, respectively. The left-hand side is equivalent to

ðtÞi. The right-hand

side requires two successive integration by parts along with the impervious boundary conditions at x ¼l

and x ¼ l

, and the

conservation of the ﬂux at x ¼ 0. The simpliﬁcations associated with the compact support of rðx; tÞ leads to:



ðtÞ



þN

N

rðx; tÞ

dx ¼



vrðx; tÞ



N



vrðx; tÞ



þN

2D

N

vrðx; tÞ

dx 2D

þN

vrðx; tÞ

¼ 0 þ 0  2D

0 

N

rðx; tÞ dx

 2D

0 

þN

rðx; tÞ dx

¼ 2D

l

rðx; tÞ dx þ 2D

þl

rðx; tÞdx ¼ 2D

þ 2ðD

 D

l

rðx; tÞdx

¼ 2D

eff

ðtÞ

(28)

Figure 15 Illustration of the composition rules (A) for distances and (B) for the migration from a monolayer material, and of their invariance with

the order of the steps (see Eqs. 25 and 32).

26 Risk Assessment of Migration From Packaging Materials Into Food

For any initial distribution of substances rðx; t ¼ 0Þ (single molecule or a collection of molecules), Eq. (28) describes the evolu-

tion of the mean-square distance hx

(t)i to the FP interface. Since equations do not include any thermodynamic consideration,

(Eq. (4) was not enforced, Eq. (28) is valid while hx

ðtÞi <



þl



(beyond this length scale, ﬁnite size effects dominate and

eff

(t) / 0). It shows that hx

(t)iincreases as 2D

t, when the concentration in F

l

rðx; tÞ dx

is low. When the amount

of substances in F becomes larger and because the molecules diffuse faster in F than in P, hx

(t)i increases more rapidly. D

eff

(t) is the

effective diffusion coefﬁcient between P and F when both materials are replaced by an equivalent medium. A version similar to Eq.

(28) for a homogeneous medium and known as the Einstein equation is presented in Self- and Trace-Diffusion Coefﬁcients

section (see Eq. 37).

Condition of Invariance of the Dispersion of Solutes With the Properties of the Contacting Phase

Eq. (28) shows that the composition hx

(t)i is independent of D

and of the amount already transferred to the food only if the

inequality

N

rðx; tÞ dx 

þN

rðx; tÞdx is veriﬁed. The dispersion of contaminants is, hence, independent of the order of vari-

ations of D

with time when:

ðtÞ

þ D

P=F

(29)

By noting that L

P=F

is the total migration (see Eq. 1) and that the diffusion in polymers is 1:100 or less lower than in the food,

it can be seen that the invariance with the order of the steps hold only at the beginning of the migration process or when the chem-

ical afﬁnity for the food is very low. Eq. (29) could be justiﬁed with the example of a large food volume submitted to a cold condi-

tion during

, denoted ð

; T

Þ followed by a warm condition during the same time, denoted (t

, T

) with T

> T

. The normal

order ð

; T

Þð

; T

Þ would lead to a small mass transfer during ð

; T

Þ and a very strong one during ð

Þ. Performing

the transfer in the reverse order ð

; T

Þð

; T

Þ will cause an even higher mass transfer during the ﬁrst step. If the food is large,

the extra number of molecules transferred to the food during the ﬁrst step ð

; T

Þ will not have enough time to be transferred back

to the packaging. As the food-packaging contact is not symmetrical D

[ D

and l

[ l

, the two orders might not lead to similar

irreversible behaviors: the food is contaminated in both cases but not to the same extent.

Discussion on the Limits Introduced by Implicit Models

The fundamental results exposed here rely on an explicit representation of the food, where molecules displace at a ﬁnite velocity.

This subtle detail is not reproduced in implicit representations, which assume a perfect mixing outside the mass transfer boundary

layer (the velocity of molecules). Only a delay is considered in the boundary assuming a linear proﬁle instead of a parabolic one (see

the distribution of molecules depicted by green symbols in Fig. 7). The next paragraph reviews the conditions of commutativity of

implicit models under variable conditions. The condition of commutativity is less severe as the back ﬂux from the contacted phase is

immediately compensated in the numerical scheme. But as shown here, the condition of commutativity is expected to be veriﬁed in

real life, only when condition (29) is met. The equivalence between time and temperature is acceptable only far from the equilib-

rium (



/0), but it is unacceptable closer to equilibrium (v



/1 ), where the effective mass transfer is governed by an effective

partition coefﬁcient across the P-F interface. From Eq. (4), partitioning coefﬁcients are independent of the temperature only if

the difference of free sorption energies between P and F are kept constant (G

¼ G

) when the temperature is raised.

Conditions of Exchangeability in Food Implicit Models

Overview of Implicit Numerical Models and Their Solutions

Food implicit models are by far the most used. They are more ﬂexible to accommodate variable conditions and chained conditions.

They have been implemented with various numerical models using different spatial discretization schemes. The ﬁnite difference

method is the dominant approach in one-dimension problems, but it loses accuracy at interfaces when large jumps in concentra-

tions and diffusion coefﬁcients occur. The ﬁnite element method is the standard in the industry as it enables integration of any

partial differential equation system on arbitrary geometrical domains, using a grid approximation (consisting of triangles, quadran-

gles, and curvilinear polygons). The ﬁnite volume method is in essence similar (values are calculated on a meshed geometry), but

the equations are integrated on small, but not inﬁnitesimal, volumes. By positioning the interface between volumes at the exact

location where thermodynamic constraints such as Eq. (4) need to be strictly veriﬁed, the method enables maintenance of the exact

mass balances and the continuity of chemical potentials between materials. The pros and cons of each method are discussed in

Nguyen et al. (2013). The three methods can be put in a matrix form, with coding for a system of ordinary differential equation:

vCðtÞ

M CðtÞ (30)

The common practice is to include the concentration in P discretized n

nodes

and the concentration in F in CðtÞ. Since the food is

represented implicitly, one node is sufﬁcient for the food and

CðtÞ is a (n

nodes

þ 1)  1 vector mapping its continuous version C(x, t).

M is a triband matrix (n

nodes

þ 1)  (n

nodes

þ 1) for a discretization scheme at order 1 and pentaband matrix for quadratic ﬁnite

elements techniques.

Risk Assessment of Migration From Packaging Materials Into Food 27

When the transport and thermodynamic properties are constant, the solution of Eq. (30) with respect to the initial condition

CðtÞ¼0 is expð MtÞCðt ¼ 0Þ, with expð MtÞ¼

k¼0

ðtÞ

Composition Rules When Chained Simulations Are Used (Example With Three Steps)

The solution of the mass transfer associated with three conditions: ð

;

ÞðM

;

Þ



;



, with





i¼1::3

the stiffness

matrix for the ith step (e.g., coding the effect of temperature on diffusion and partition coefﬁcients) is:

Cð

Þ¼exp



 M



expð

ÞexpðM

ÞCðt ¼ 0Þ (31)

The steps are exchangeable if the equality (31) satisﬁes also:

¼ exp





 M



t ¼ 0

(32)

which is veriﬁed only if





i¼1::3;isj

and

j¼1::3;isj

commutes (meaning that M

¼ M

for isj).

Conditions of Exchangeability Imposed by the Physical Chemistry

Nguyen et al. (2013) demonstrated that a necessary and sufﬁcient condition to have solution (32) applicable is that the ratios of the

Henry coefﬁcients



ðtÞ



u;v¼0:::m;usv

(see their deﬁnitions in Eq. 17) remain constant with time between all considered steps. As

a result, adding or removing a material/layer/food (i.e., changing k

from N to 0 or the reverse) breaks the condition of exchange-

ability of steps. The condition of exchangeability is also likely to be lost for polar solutes dispersed between polar and apolar phases.

As shown in Binary Flory Isotherms section (see Eq. 49 at inﬁnite dilution when f

/0 ), the excess enthalpies of mixing are nega-

tive for the ﬁrsts and positive for the lasts. In this particular case, the apparent activation energy of the partition coefﬁcient between u

and v is maximal, and the effect of temperature needs to be simulated by respecting the order of the temperature variations (i.e., by

using Eq. (31) of Eq. (32)). For apolar solutes, the activation of k

is, conversely, almost compensated by a symmetrical variation of

, when the temperature is changed.

Discussion on the Choice of Accelerated Conditions and the Identiﬁcation of Critical Steps

Highlights for multiple steps and variable conditions.

•

Accelerated conditions and time-temperature relationships need to be veriﬁed according to the rules of factorization mentioned

in Strategies and Equations to Simulate Multiple Steps and Conditions section.

•

The contribution of any step or component is obtained by difference if the comparison includes all steps and components.

•

Probabilistic migration modeling (see Probabilistic Modeling of the Migration section) can be used to evaluate the safety

margin associated with accelerated and equivalent conditions.

Factors of Acceleration and Possible Biases

The different theoretical developments presented in this part highlight the complications to reach exact modeling of migration for

a given product. The real contact conditions and couple timetemperature need to be considered and incorporated in a proper

simulation scenario. Subtle variations of temperature and contact times may modify the estimates. The reader will recall that a tiered

approach, as shown in Fig. 5 is always preferable. A robust approach to assess the effects of various sources of variability (e.g.,

temperature variations) and uncertainty (e.g., effects of temperature on diffusion and partition coefﬁcients) is presented in Prob-

abilistic Modeling of the Migration section. Probabilistic modeling applied to Eq. (9) can be used to derive conservative estimates

either with a prescribed risk or with a prescribed safety margin (see. Fig. 23).

Accelerated tests are used experimentally to reproduce the level of contamination after the end of the food shelf life while using

much shorter contact times. By assuming that migration is controlled by one single mass transfer resistance (index j

ref

), the principle

relies on replacing the real condition ðt

shelf

life

; T

shelf

life

Þ by the accelerated condition ðt

shelf

life

; T

accelerated

conditions

Þ. When the amount transferred at

equilibrium is not affected by temperature (see Binary Flory Isotherms section for discussion), the ratio of C

values between accel-

erated and real conditions is:



accelerated

conditions

; T

accelerated

conditions





shelf

life

; T

shelf

life





ref



accelerated

conditions



ref



shelf

life



accelerated

conditions

shelf

life



exp



ref

accelerated

conditions



shelf

life

accelerated

conditions

shelf

life



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

exp



ref

accelerated

conditions



shelf

life

accelerated

conditions

shelf

life

for monolayer materials

(33)

28 Risk Assessment of Migration From Packaging Materials Into Food

where E

ref

is the apparent activation energy of the diffusion. This parameter is critical and will be discussed in Activation of

Diffusion by Temperature section. For monolayer materials, an activation energy of 60 kJ mol

1

yields an activation factor of ca. 5

when the temperature is raised from 20



Cto40



C. Testing food packaging at 40



C during ten days is equivalent to a test of

50 days at 20



C. If the shelf life is 100 days, the risk of underestimation of the true migration can reach up to factor

ﬃﬃﬃ

=2z 0:7. In

the presence of a functional barrier, the underestimation factor can be even larger as shown in Fig. 10.

Causality, Critical Steps, and Crtical Components

In the presence of multiple steps, it is recommended to perform a test which mimics the most critical steps for the ﬁnal value of C

Due to the lack of generality of Eqs. (25) and (32), the factorization strategies should be avoided to identify critical steps. Nguyen

et al. (2013) demonstrated that the contribution of one single step, denoted

ðstep iÞ , or of one component (label, cap, ﬁlm,

bottle, etc.), denoted

ðcomponent iÞ , could always be determined by difference as:

ðstep iÞ¼max



1/2///n

steps

 C

1/2///n

steps

; C

step i



ðcomponent iÞ¼max



assembly

1þ2þ/þn

components

 C

assembly

1þ2þ/þn

components

i

; C

component i



(34)

where 1 / 2 / . / n

steps

/i represents the sequence including all steps except the step i;1þ 2 þ / þ n

components

i including all

components except the component i.

The contributions of the sum of all steps and components are larger than the real contamination. This behavior is expected as

several steps can lead to similar effects. as a corollary, Eq. (34) enables the identiﬁcation of steps, which occur without contact with

food (see Fig. 14) or of any component, even if there is no direct or intended contact with food. When the contributions

are

normalized by a threshold of concern, SML, they can be used to seek the likely causes (industrial practices or components) which

lead or which could lead C

to exceeding tolerances. The effects can be cumulative over several substances to get a global criticality of

the step or the component of interest. This procedure has been used as the foundation of a preventive approach of mass transfer

reusing the principles of the methodology FMECA (Failure Mode Effects and Criticality Analysis) (Nguyen et al., 2013). The

approach can be employed to analyze an entire supply chain or industrial practices, as illustrated in Fig. 13. The assessment of

the risk of cross-contamination of the constituents of UV-cured printing ink during the long-term storage of pots in stacks is shown

in Fig. 16. The stacking step is indexed 1. The redistribution of contaminants triggers contamination immediately during and after

ﬁlling the pot at 80



C. Migrants accumulate continuously in the food during storage until ﬁnal consumption. By difference, the

contribution of the ﬁrst step can be evaluated. After normalization by SML, it is demonstrated that the ﬁrst step provides the largest

contribution to the risk of exceeding the legal limit.

Figure 16 Contribution of the stacking of pots before ﬁlling them at hot temperatures with a Chinese soup along the supply chain: (A) steps of

supply chain; (B) assessment of a severity of a single step; (C) comparison of several steps. After Nguyen et al. (2013).

Risk Assessment of Migration From Packaging Materials Into Food 29

Diffusion Properties in Polymers

Highlights on the diffusion coefﬁcients in polymers (thermoplastics, thermosets, elastomers)

•

Diffusion coefﬁcients are essential properties to evaluate the performances of functional barriers and to assess the kinetics of

migration in food.

•

Diffusion coefﬁcients are strongly affected by the size of migrants: volume effects dominate for bulk and rigid solutes (e.g.,

aromatic solutes), while the number of rigid sub-units dominates in ﬂexible solutes.

•

The activation of diffusion follows an Arrhenius relationship only far from the glass transition temperature (T

). Near T

, free-

volume effects dominate, and apparent activation energies depend on both temperature and the size of diffusants.

•

For compliance testing, it is recommended to use overestimates of diffusion coefﬁcients to reduce the risk of underestimation of

migration.

•

When modeling is used to validate mechanical recycling processes of polymers, it is recommended to use either realistic esti-

mates or probabilistic modeling (see Probabilistic Modeling of the Migration section).

Deﬁnitions of Diffusion Coefﬁcients

Diffusion coefﬁcients in the polymer, D

, are essential properties to calculate migration according to Eqs. (9) and (19). Reference

textbooks [see Chapter 11 of Crank, 1975 and Vrentas and Vrentas, 2013) have proposed several de ﬁnitions of diffusion coefﬁcients

including overestimations (see chapters in Piringer and Baner, 2008). In this section the deﬁnitions which are relevant in macro-

scopic migration models and which are capable of incorporating the effects of the molecular structure of the migrants and the poly-

mer are reviewed.

Self- and Trace-Diffusion Coefﬁcients

As a starting point, it is worth noticing that the random walk of migrants (without gradient) does not occur in an empty space

but among other molecules (polymer segments and other solutes, such as plasticizers). By reusing the illustrations of Fig. 7, the

net ﬂux density perpendicular to a cross section located at a position x is proportional to the difference of the net velocities

between the solutes (i.e., molecules of the migrant of interest), u(x, t) and the Stokesian velocity of the surrounding molecules

(x, t):

Jðx; tÞ¼Cðx; tÞ½uðx; tÞu

ðx; tÞ ¼ D

vCðx; tÞ



(35)

In the limiting case where the migrant is of the same nature as the surrounding molecules (e.g., pure liquid plasticizers, pure

solvents), u(x, t) ¼ u

(x, t ), there is no net ﬂux (J(x, t) ¼ 0) and no macroscopic gradient

vCðx;tÞ



¼ 0. The diffusion coefﬁcient

is still deﬁned, and is called the self-diffusion coefﬁcient but is not correlated to any macroscopic gradients.

The opposite limiting case corresponding to inﬁnite dilut ion (C(x, t) / 0) a nd a solid behavior (u

(x, t) / 0) also leads

to an extremely low net ﬂux (J(x, t) / 0), but with a concentration gradient. The corresponding diffusion coefﬁcient is called

the trace diffusion coefﬁcient. Experimentally, the velocity u(x, t) can be determined from the lag-time, t

lag

, associated with

the migration across a functional barrier of thickness l

.Notingthatt

lag

(see Typologies of Migration Behaviors

section) leads to:



x ¼ 0; t ¼ t

lag





x ¼ 0; t ¼ t

lag





x ¼ 0; t ¼ t

lag



lag

¼ 6

(36)

Eq. (36) is correct but suffers from a lack of generality for an arbitrary initial distribution of solutes. The approach presented in

Strict Conditions of Exchangeability With Explicit Food Models section is more general. Eq. (28) associated toD

¼D

shows that:

migrants

lim

t/N

migrants

i¼1



ðtÞr

ð0Þ



lim

t/ N

ðtÞz

ðtÞ

(37)

Factor 6 appears in Eqs. 36 and 37 instead of factor 2 shown in Eq. (28), because the random walks are considered in three

dimensions and not anymore in one dimension. The mean-square-displacement g

ðtÞ¼h

migrants

i¼1



ðtÞ

ðt ¼ 0Þ



i=N

migrants

is calculated accordingly from the 3D positions of the center-of-mass of all migrants, fr

ðtÞg

i¼1::N

migrants

and averaged over all possible initial positions

ðt ¼ 0Þg

i¼1::N

migrants

. The deﬁnition (37) is used to calculate diffusion coefﬁcients

by molecular dynamics simulations in polymers (Durand et al., 2010; Vitrac and Hayert, 2007). In practice and to remove any

uncontrolled drift (i.e., ju

j > 0) due to the displacement of surrounding molecules, the positions are not absolute but taken respec-

tively to the center-of-mass of the whole system.

30 Risk Assessment of Migration From Packaging Materials Into Food

Mutual Diffusion Coefﬁcients

Self- and Trace-Diffusion Coefﬁcients section describes diffusion either in a stream or when there is only one diffusing species.

Diffusion in a medium with variable composition generates ﬂuctuations in u

(see Eq. 35) and changes the nature of the interactions

with the surrounding molecules. The picture is complete if consideration is given to the point of view of the surrounding molecules

whose diffusion is also affected. An example could be the diffusion of an antioxidant (or any large molecule) in a heterogeneously

plasticized polymer (i.e., see Courgneau et al., 2013), where the local mobility of the antioxidant is strongly enhanced by the local

amount of plasticizer in the ternary system (migrant þ plasticizer þ polymer). Similarly, the diffusion of the plasticizer increases the

mobility of polymer segments. Strictly speaking, all these effects cannot be described by the sole shift of the glass transition temper-

ature and the increase of free volumes presented in Effect of the Polymer section (see also discussion in Chapters 5 and 6 of the

reference textbook (Vrentas and Vrentas, 2013)). As the diffusion of one species affects the diffusion of all other species and recip-

rocally, self- and trace diffusion coefﬁcients cannot be used. The correct description requires the use of Onsager’s theory and gener-

alized forces. The entire system (polymer þ solutes) reorganizes to minimize its total free-energy. A cationic surfactant will, for

example, accumulate at the surface of the material to minimize its interaction with the polymer. A poorly soluble colorant or

pigment will do the same at high concentration. The resulting concentration proﬁles therefore evolve spontaneously from an initial

uniform distribution after processing to a highly heterogeneous one. Such evolutions with matter moving from low to high concen-

trated regions cannot be predicted with concentration gradients as effective driving forces. The gradients of chemical potentials need

to be used as an effective driving force. For the sake of simplicity and to avoid a tensor deﬁnition of the diffusion coefﬁcient, only the

effect of the chemical potential of the migrant, denoted mðx; tÞ , as the driving force is considered. Rigorously, a linear relationship

should be considered between the ﬂux of the migrating substance and the driving forces associated with all species in the mixture.

The generalized driving force, f, induced by the local variation of the chemical potential of the migrant in the mixture is:

f ¼

vmðx; tÞ



¼ z

mutual

½uðx; tÞu

ðx; tÞ ¼ z

mutual

Jðx; tÞ

x; t

(38)

where z

is a friction coefﬁcient; and mðx; tÞ¼ m

þ RT lnðg

fÞ is the chemical potential of the migrant deﬁned respectively to its

volume fraction f and activity coefﬁcient g

. The relationship between the gradient of chemical potential and the concentration

gradient is given by:

vmðx; tÞ



vmðx; tÞ



vfðx; tÞ



¼ RT

vlnðg

fðx; tÞÞ



vfðx; tÞ



Cðx; tÞ



vlnðg

vlnðfðx; tÞÞ



þ 1



vCðx; tÞ



Cðx; tÞ

vCðx; tÞ



(39)

with

ðfÞ¼1 þ

vlnðg

vlnðfðx;tÞÞ



being the thermodynamic factor depending on the composition, whose dependence with concentration

can be determined from the expression of the sorption isotherm see Sorption Isotherms section). An effective diffusion coefﬁcient,

mutual

, related to the ﬂux Jðx; tÞ¼D

mutual

vCðx;tÞ



is identiﬁed from Eqs. (38) and (39) as:

mutual

ðfÞ¼

ðfÞ

mutual

ðfÞ

trace

mutual

ðfÞ

trace

(40)

where subscripts trace and mutual refer to the value of the property at inﬁnite dilution and in mixture, respectively. For binary

solvent–polymer mixtures, Vrentas and Vrentas (Vrentas and Vrentas, 1993) proposed the following evolution of the friction

coefﬁcient:

mutual

ðfÞ

trace

¼af

þð1  fÞð1 þ 2fÞ (41)

with a ¼

pure

solvent

dissolved

polymer

self

solvent

trace

dissolved

polymer

and V

the molar volume of X. Eq. (41) is not thought to be general for non-solvent or not plasticizing

molecules such as hindered phenols or aromatic amines. The fundamental reason is that such migrants are crystalline (solids) at

high concentration.

It is worth noting that D

trace

can contain additional concentration dependence due to the extra free-volumes brought by the

solute itself and the possible shift in the glass transition temperature. For most applications and in the absence of a uniﬁed theory,

the equation of Fujita can be applied to describe the concentration dependence:

mutual

ðCÞ

mutual



ref



¼b

1 

ref

(42)

with b a concentration to be determined experimentally.

Risk Assessment of Migration From Packaging Materials Into Food 31

Effect of the Geometry of Migrants on D

Values

Migrants from polymers are not gas molecules such as water, oxygen, and carbon dioxide, but heavy molecules larger than voids in

the polymer. The smallest additives are monomer residues and solvents commensurable to one or several monomers. At the concen-

tration of use of most of additives and residues (i.e., except plasticizers which are used at concentration ranges from 10 w% to 50 w

%), the many pair contacts between the segments of the polymer and the migrant control the rate of the translation of the most

mobile species (the migrant). This conﬁguration is very different from the situation in food or liquid food simulants, where the

food constituents (water, oil, ethanol, etc.) are much smaller than the polymer chains and are usually packed less densely. The

correct picture is to consider that the trace diffusion of the migrant in the polymer is smaller than the self-diffusion of food constit-

uents (typically 10

9

–10

10

m (NASA) s

1

), but much larger than the self-diffusion of the polymer itself (<10

22

m (NASA) s

1

The exact mechanism of translation of molecules larger than voids in solid polymers has not been fully elucidated yet. They have

been investigated independently by two communities: the community of free-volume-theory was interested in the mutual-diffusion

of polymer solvents whereas the food packaging community was focused on the development of practical overestimate models for

compliance testing. Due to the different nature of the considered substances and the type of considered diffusion coefﬁcients

(mutual diffusion measured in

H spin-echo nuclear magnetic resonance and trace diffusion coefﬁcients measured via desorption

kinetics), the interactions between the two communities have been limited. Based on the work (Fang and Vitrac, 2017; Fang et al.,

2013), a uniﬁed approach has been sketched and is summarized in Fig. 17. Early works (Vitrac et al., 2006; Ewender and Welle,

2013, 2014, 2018; Welle, 2013) suggested that the volume of the entire molecule or its rough estimate the molecular mass (see

the discussion in Zhao et al., 2003) offered a proper scaling parameter of diffusivities, at least, at the ﬁrst tier. The use of the entire

mass or volume of the migrant is misleading as it covers very different realities for rigid and ﬂexible solutes. Flexible migrants,

possibly with large masses, can beneﬁt from the translation of smaller rigid units; whereas a large rigid migrant rarely requires

free-volumes matching the shape and size of a larger rigid block. Without paying attention, both the effects of block sizes and their

numbers combine, giving an apparent correlation of the logarithm of D

with the logarithm of the molecular mass M(concerted

displacements of rigid units or blocks) and with the total volume of the migrant n

rigid

block

vdW

rigid

block

. In this description, short n-alkanes

are partly rigid since the constraints of torsion prevent blocks from translating independently.

Fig 18 presents the diffusion coefﬁcients of 49 substances (105 D

values) in low-density polyethylene (LDPE) at 23



C collected

by the National Institute of Standards (NIST) (NIST) and the European Commission (EC) (Hinrichs and Piringer, 2002b) from the

literature. The data were selected to reproduce the main features of Fig. 17: the decrease of D

values with the number of rigid blocks

and the almost invariance of D

with the exact shape and ﬂexibility of the solute. The concept of invariance considers here that there

is no visible effect if the induced variation is lower than the uncertainty associated with the measurement of D

. As an illustration,

Figure 17 Scaling of diffusion coefﬁcients between rigid and connected blocks with molar mass and van der Waals volume in a thermoplastic poly-

mer (groups A and B refer to substances deﬁned in Fig. 18).

32 Risk Assessment of Migration From Packaging Materials Into Food

Figure 18 Scaling of diffusion coefﬁcients of 49 substances (n-alkanes, two groups of molecules A and B with similar D

values) in LDPE at 23



with molar mass, M, and the van der Waals volume. 1: methane [1]; 2: ethane [1]; 3: propane [1]; 4: n-pentane [1,2]; 5: n-hexane [1,2]; 6: n-heptane

[1,2]; 7: n-octane [1,2]; 8: n-decane [1,2,A]; 9: n-octadecane [1]; 10: n-dodecane [2,A]; 11: 2-trans-3,7-dimethyl-2,6-octadien-8-ole (geraniol) [A];

12: 3,7-dimethyl-6-octen-1-ol (citronellol) [A]; 13: n-decylaldehyde or n-decanal (aldehyd c10) [A]; 14: 3,7-dimethyl-1-octanol [A]; 15: decylalcohol

or 1-decanol [A]; 16: cis-undecen-8-al (aldehyd c11 inter) [A]; 17: n-undecen-2-al (aldehyd c11) (2-undecenal) [A]; 18: n-undecylaldehyde (aldehyde

c11) [A]; 19: ethyloctanoate [A]; 20: 2-methoxy-4-propenylanisol (methylisoeugenol) [A]; 21: citronellyl formate or 6-octen-1-ol, 3,7-dimethyl-,

formate [A]; 22: 2-methyl-3-(4-isopropyl)phenylpropanal (cyclamen aldehyde) [A]; 23: 2,6-octadien-1-ol, 3,7-dimethyl-, acetate, (2e)- (geranyl

acetate) [A]; 24: 3,7-dimethyl-1,6-octadien-3-ylacetate (linalylacetate) [A]; 25: allyl-3-cyclohexylpropionate [A]; 26: amylcinnamicaldehyde or

2-phenylmethylene-heptanal [A]; 27: 3-methyl-3-phenylglycidate (aldehyde c16) [A]; 28: iso-amylsalicilate [A]; 29: benzylbenzoate [A]; 30: diethylph-

thalate (dep) [A]; 31: 2-hydroxy-4-methoxybenzophenone (chimassorb 90) [A]; 32: 2-methyl-undecanal (aldehyde c12 mna) [B]; 33: 3,7-dimethyl-

6-octen-1-ylacetate [B]; 34: 3-[4-tert,-buthylphenyl]-2-methylpropanale (lilial) [B]; 35: 2,4-di-t-butylphenol [B]; 36: 2,6-di-t-butylphenol [B]; 37:

4-(2,6,6-trimethyl-2-cyclohexen-1-yle)-3-methyl-3-buten-2-one (methylionone-gamma) [B]; 38: 5-(2,6,6-trimethyl-2-cyclohexen-1-yle)-3-methyl-3-

buten-2-one (methylionone-alpha) [B]; 39: 4-[4-methyl-4-hydroxyamyl]-3-cyclohexen-carboxaldehyde (lyral) [B]; 40: 2-hexyl-3-phenyl-2-propenal [B];

41: 2,5-tertbutyl-4-hydroxy-toluene [B]; 42: 2,6-di-tert-butyl-4-methylphenol [B]; 43: 2,6-di-tert-butyl-4-methylphenol (ionol or BHT) [B]; 44: phenyl-

ethylphenylacetate [B]; 45: nonane-1,3-dioldiacetate (jasmelia) [B]; 46: 2-hydroxy-4-ethanediolbenzophenone [B]; 47: 2,6-dinitro-1-methyl-3-methoxy-

4-tert,-butylbenzole (moschus ambrette) [B]; 48: 2-hydroxy-4-butoxybenzophenone [B]; 49: 2,4,6-trinitro-1,3-dimethyl-5-tert,-butylbenze ne (moschus

xylol) [B]. Data from [1] Flynn (1982), [2] NIST and Hinrichs and Piringer (2002b).

Risk Assessment of Migration From Packaging Materials Into Food 33

Vitrac et al. (2006) showed that the group of substances denoted A and B presented differences in diffusivities which could not be

explained by the differences between molecules. Three parameters were considered: the van der Waals volume (V

vdW



), the

gyration radius and the shape factor. Linear alkanes scaled as a power law with the molecular mass M, D

a

with a of 1.58

(range: 1.09–1.68) for [1] NIST data and 1.84 (range: 0.75–2.01) for [2] EC data. For bulky solutes, the large uncertainty in the

experimental D

values suggests that diffusivities can be considered signiﬁcantly different only if the V

vdW

differences between mole-

cules are larger than 35



A(Kortenkamp et al., 2009). The correlation with volume is consistent with the free-volume theory of diffu-

sion (Fang and Vitrac, 2017), and as V

vdW

is also correlated with the number of heavy atoms (see Fig. 18E, the correlation depends

on the type of substances), a reasonable upper envelope of D

values has been proposed to overestimate diffusivities (Fang et al.,

2013). The equation is coined the Equation of Piringer:

ln D ¼A’

 0:1351M

2=3

þ 0:003M 

s þ 10454

(43)

with the key parameters A’

¼ 11.5 and s ¼ 0 for LDPE. The values for other polymers are reported in the EU report (Hoekstra et al.,

2015b). The model of Piringer provides only a variable overestimation ranging from a factor 0.63 (underestimation) to 100 (see the

inset Fig. 18A).

Effect of the Polymer

The effects of the polymer on the diffusion of migrants are twofold: i) the relaxation of polymer segments affects the renewal of free-

volumes around the solute and ii) speciﬁc interactions between rigid blocks and the polymer may increase the release time of rigid

blocks. In the original free-volume theory of Vrentras and Duda (Vrentas and Vrentas, 1998; Vrentas et al., 1996), the translation of

rigid blocks is assumed to be associated with the local reorganization of special free-volumes, so-called hole free-volume (hFV),

whose redistribution is not activated by temperature. As an approximation of the thermal expansion of polymers, the amount

of hFV available for the diffusion of migrants is proportional to T T

þ K

, where T

is the glass transition temperature and K

a constant possibly dependent on the polymer and the shape of the translating block. This interpretation is illustrated in the diffu-

sion of various migrants in rubber (T  T

) and glassy (T < T

) polymers in Fig. 19. The apparent scaling exponent aðT; T

Þdecreases

rapidly with T  T

; for linear solutes, Fang et al. (2013) demonstrated that its general expression was:



T  T



¼1 þ

T  T

þ K

with T > T

þ K

(44)

with K

essentially a polymer-dependent constant and K

a positive constant for rigid blocks smaller than the cross-section of

polymer segments and otherwise negative.

Eq. (44) and Fig. 19A demonstrate that mass dependence increases rapidly when T approaches or goes below T

. In elastomers

and rubbers (with very low T

) as well as in polymer melts, the mass dependence for linear migrants approaches unity. This theo-

retical value corresponds to the independent displacements of jumping units. Due to the dependence of K

with the type of rigid

blocks, Eq. (44) describes different mass dependence (see the three series of linear molecules in Fig. 17) for different molecules.

Although approximate, the technique can be used to extrapolate the results of lnD

from one polymer to a new polymer (denoted

R) via the correction

aðT;T

ln M. The principles are illustrated in Fig. 19E by showing that almost any of the presented polymers

(except rubbers) can be used to predict the diffusion coefﬁcients of the same substances in polypropylene only by setting the T

of the source and destination polymer. The same correction can be used to predict diffusivities in plasticized polymers from their

values in non-plasticized ones.

Activation of Diffusion by Temperature

Increasing temperature affects ﬁrstly the structure of the polymer, which, in return, facilitates the translation of the migrants.

Temperature activation is consequently higher in polymers with high thermal expansion coefﬁcients (higher in rubber state than

in glassy state, higher in plasticized than in non-plasticized polymer, higher in thermoplastics than in thermosets). Below or

near T

, apparent activation energies Ea ¼ RT

vln D

are dominated by free-volume effects and are higher than at higher tempera-

tures. The effects of temperature for linear solutes near T

are shown in Fig. 20. If the behavior was strictly Arrhenian, the dependence

of lnD

would be linear with 1/T, which is not obviously the case for large diffusants near T

. Apparent activation energies increase

with the size of diffusants as E

flnM. The interested reader will ﬁnd a consistent review of free-volume and speciﬁc-migrant acti-

vation in Fang and Vitrac (2017). In any case, it is not recommended to extrapolate D

values from low temperatures to high temper-

atures for polymer recycling, as it would overestimate diffusivities and consequently the rate of decontamination of the polymer.

Sorption Properties and Partition Coefﬁcients

Highlights on the sorption and partition properties

•

Along diffusivities, sorption and partitions coefﬁ cients are essential properties to assess the kinetics of migration in food.

34 Risk Assessment of Migration From Packaging Materials Into Food

•

Along with fugacities, partial pressures offer the best representation of the force enabling a substance to move from one phase to

the other. Practical approximations are given for all practical cases of mass transfer (from a material to a liquid, from a material

to a gas phase, etc.) and possible substances (volatile or not).

•

Partition coefﬁcients are deﬁned respectively to activity coefﬁcients.

•

Binary and ternary Flory isotherms offer estimations of activity coefﬁcients for all applications of migration modeling: mixture of

food simulants, wet or plasticized polymers, homo, and copolymers

•

Temperature effects are less important between two condensed phases, but dominant between a condensed and a gas phase.

•

Activity coefﬁcients and their activation by temperature can be calculated at atomistic scale for arbitrary solutes (non-charged)

As shown in Underlying Microscopic Assumptions section and Mutual Diffusion Coefﬁ cients section, sorption properties and

partitioning between materials, food and polymer affect both the kinetics of migration and the distribution of migrants at thermo-

dynamic equilibrium. Previous descriptions were essentially counting the number of substances in a representative elementary

volume or a compartment (material, food). The present description will show explicitly the role of pair interactions between the

migrant indexed i and all other constituents (P for the polymer, F for the contact phase, e for ethanol, w for water). Generic phases

in equilibrium (in contact or not) are denoted a, b and d regardless of whether they are a solid (polymer or solid food), a liquid

contact phase or a gas (headspace, storage environment).

Figure 19 Diffusivities of various substances at 25



C in glassy and rubber polymers: (A) raw values, (B) normalized data to remove polymer

effects (standardized to T

value of 0



C corresponding to atactic polypropylene PP). Filled symbols correspond to n-alkanes (scaling laws D

f M

a

as dashed lines) and empty symbols to various solutes including gases and plastic additives (scaling laws D

a

as continuous lines). Data from

Schwope et al. (1990).

Risk Assessment of Migration From Packaging Materials Into Food 35

Some Deﬁnitions

Chemical Potentials, Fugacities, and Activities

Two phases, a and b, are said to be at thermodynamic equilibrium if they are at the same temperature T, pressure P and the same

chemical potential for any migrant i: m

i;a

ðT;PÞ¼m

i;b

ðT;PÞ. The concept of thermodynamic potential was proposed by G. N. Lewis,

with initially in mind pure ideal gases, verifying that

i;a



¼ V

, where V

is the molar volume of the gas i. After integration at

a constant temperature, the change in chemical potential from pressure P

ref

to P is RT ln

ref

for a pure ideal gas. Lewis generalized to

phase mixtures (ideal or not) by replacing pressure by a function f

, called fugacity and assessing the capacity of a substance literally

to ﬂee:

i;a

m

ref

i;a

¼ RT ln

i;a

ref

i;a

(45)

Reference values at the temperature T m

ref

i;a

and f

ref

i;a

are physically related, but the choice of the reference chemical potential or the

reference fugacity is arbitrary. The ratio a

i;a

ref

i;a

deﬁnes the activity of the solute i in the phase a; it provides a measure of the differ-

ence of the substance(s) chemical potential in a with its reference state. To describe mass transfer between the phases a and b (i.e.,

with f

i;a

¼ f

b;a

) at equilibrium, it is convenient to choose the same reference state m

ref

i;a

¼ m

ref

i;b

in both phases. Following the intuition

of Lewis, equivalent partial pressures in an ideal gas offer an almost universal potential to estimate the potential of transfer of any

substance (e.g., a volatile organic compound, a liquid plasticizer, a crystalline pigment or a poorly soluble additive or mineral oil

residues). The reference fugacity needs to be adapted accordingly as shown in Table 7.

Migrants with a reference solid state in the conditions of migration are very common in plastics and thermosets. They encompass

almost all antioxidants, colorants and pigments. They are liquid in conditions of processing of the polymer (e.g., above 160



C), but

solid in the conditions of service of the ﬁnished material (at T and P). The partial pressures are lower in this case but non-zero. The

variation of fugacities between a pure crystal and its pure amorphous state depends on the temperature difference between T and its

melting temperature, T

i,m

as shown in Fig. 21.

It is worth noting that thermodynamic models and simulation calculations may use different reference states (e.g., amorphous

reference states even if the reference one is solid). The choices presented here are consistent with the deﬁnition of the molar solva-

tion energy:

solvatation

i;solvated in a

¼ m

i;a

 m

d¼ideal gas

i;d

¼ RT ln

i;sat

ðTÞV

i;ai

i;a

i;d

¼ RT ln

i;sat

ðTÞV

i;a=d

(46)

Figure 20 Arrhenius plot of diffusivities of n-alkanes in polyethylene terephthalate. Data from Ewender and Welle (2014), (2016), and (2018).

36 Risk Assessment of Migration From Packaging Materials Into Food

with K

i;a=d

being the effective partition coefﬁcient between the condensed phase a and the theoretical ideal gas phase d. With respect

to mass transfer, the use of partition coefﬁcients should be preferred to solvation energies. Free energies need to be preferred to

analyze and interpret quantitatively the nature of the interactions (van der Waals, electrostatic, hydrogen bonding) in the phase a.

Thermodynamic integration at molecular scale offers direct access to

solvatation

i;solvated in a

without requiring a theoretical separation of

enthalpic and entropic effects.

Figure 21 Ratio of fugacities between pure solid and amorphous states for 11 model migrants and its continuous approximation proposed in

Figure S1 of the Supporting Information of Nguyen et al. (2017a). Data from Fornasiero et al. (2002).

Table 7 Expressions of practical partial pressures and saturation concentrations in relationship with the reference state of the substance in the

conditions where its migration its studied

Application Choice for the fugacity: f

Choice for the reference

fugacity: f

ref

Relationship with the concentration of

substance i in the phase a: C

i;a

(SI Unit

is mol m

3

)

Volatile substance in a gas phase

(a ¼ gas phase)

partial pressure: p

total pressure: P p

¼ RTC

i;a

Dissolved substance in

a condensed phase (polymer,

liquid) with a liquid reference

state (a ¼ PorF)

partial pressure (equivalent partial

pressure in the theoretical gas

phase d in equilibrium with a): p

saturation pressure of the pure

substance at the same

temperature: p

i,sat

(T)

i;sat

¼ g

i;a

¼ g

i;a

with g

i;a

the activity coefﬁcient of the

substance i in the phase a relatively

to its volume fraction f

i;a

At saturation, one gets: C

sat

i;a

As above but with a solid reference

state

partial pressure (equivalent partial

pressure in the theoretical gas

phase d in equilibrium with a): p

Partial pressure at the surface of

the crystal expressed as:

i;pure

ðT Þ

i;sat

ðT Þ with

i;pure

ðT Þ

being the ratios of

fugacities of the pure solute

between solid (pure crystalline)

and molten (pure amorphous)

state.



i;pure



ðT Þ

i;sat

¼ g

i;a

¼ g

i;a

Risk Assessment of Migration From Packaging Materials Into Food 37

Effective Partition Coefﬁcients Between P and F

From relationships presented in Table 7, effective partition coefﬁcients between a material P and the contacting phase is given by the

ratio of activity coefﬁcients. For semi-crystalline polymers, it is well accepted that crystallites and crystalline phases are impenetrable

to migrants. By assuming that no migrating substance has been trapped during processing in the crystalline phase, the effective parti-

tion coefﬁcient reads:

i;F=P

ð1 cÞð1  εÞ

i;F

i;P

ð1 cÞð1  εÞ

i;P

i;F

(47)

where f

i;P

refers to the volume fraction in the amorphous phase of the material (possibly porous and semi-crystalline); cand ε are

the volume crystallinity and porosity of the material. Thermoplastics are not porous materials and ε ¼ 1.

For very porous materi als such as papers, i t is prefera ble not to homogenize concentratio n between all phases ( solid crys-

talline, sol id amorphous and gas) and to choose the ﬁbers as P and to adapt the exchange surface area to the shape of the

ﬁbers. The characteristic dimension of the material should be chosen accordingly and be commensurable to the half-diameter

of ﬁbers.

Sorption Isotherms

Sorption isotherms are experimental or theoretical curves, which relate the amounts absorbed by a condensed phase in rela-

tionship to an applied activity at constant temperature and pressure. Their use is more general than partition coefﬁcients,

when the values of fg

i;k

k¼P;F

depend on concentrations. In this section, we c all isotherm a curve relating the activity coef-

ﬁcient or the excess chemical potential (i.e., in excess respectively to the ideal contribution RT ln f

) with the compo sition in

the mixture.

Linear Isotherms

For most applications at inﬁnite dilution, activity coefﬁcients can be assumed independent of composition. The relationship

between mass uptake and partial pressure is linear and governed by a Henry isotherm (see Eq. 17). The relationship between

the Henry constant

i;k

k¼P;F

and

i;k

k¼P;F

is given by:

i;k

¼p

i;sat

ðTÞV

i;k

for k ¼ P; F (48)

It is worth noting that if the condition of inﬁnite dilution is well veriﬁed in foods regardless of the considered substance (migra-

tions are expected to be low), it may not be veriﬁed in the material for substances used close to saturation (e.g., pigments) and for

plasticizing substances (e.g. used at weight fractions up to 50%). The proposed description assumes that the substances are well

mixed and exclude, by deﬁ nition, surfactants and substances causing blooming.

Binary Flory Isotherms

General Formulation

The Flory-Huggins theory offers a robust framework to account for concentration effects. The theory extends the regular solution

theory for liquid mixtures to mixtures with molecules with dissimilar sizes such as solutes mixed with segments of polymers or large

additives dispersed in a food simulant. Enthalpic and entropic interactions are calculated on a lattice assuming that the mixture is

incompressible and that molecules ﬁll space commensurably to their molar volumes. For binary mixtures, i þ P or i þ F, the activity

coefﬁcient is given by:

ln g

i;k



i;k

; T





1 



i;k

 n

compressible

k around i





1  f

i;k



þ c

ðTÞ

i;k



1  f

i;k



(49)

where r

i,k

is the size of the host molecule with respect to the size of the solute i. In a polymer, the chain is considered inﬁnitely long

i;P

/N. In foods and a liquid simulating food, r

1

i;F

represents the number of molecules of F displaced by the insertion of the

substance i in F. For most of the migrants, r

1

i;F

is expected to be larger than unity in water and ethanol and lower than unity in oil.

i,F

can be approximated as

with n

compressible

k around i

accounting for the partial compressibility of the molecules of i and F. Rigorously,

partial molar volumes should be used instead of molar volumes.

Approximation at Inﬁnite Dilution

At inﬁnite dilution and pending estimations of the Flory-Huggins coefﬁcients fc

i;k

k¼P;F

, Eq. (49) provides an estimator of activity

and partition coefﬁcients in amorphous regions:

ð1  cÞð1  εÞK

ðTÞ

i;F=P

i;P

i;F

exp



1 þ c

ðTÞ

i;P



exp



1  r

i;F

þ c

ðTÞ

i;F

þ compressible correction



¼ exp



ðTÞ

i;P

 c

ðTÞ

i;F

þ r

i;F

 compressible correction



(50)

38 Risk Assessment of Migration From Packaging Materials Into Food

Effect of Temperature on Partitioning

According to Eq. (50), the variation of K

i,F/P

with temperature depends on the variation of the difference c

ðTÞ

i;P

 c

ðTÞ

i;F

with T. When

ðTÞ

i;P

and c

ðTÞ

i;F

have the same sign, their variations with T are similar and compensate each other. When c

ðTÞ

i;P

ðTÞ

i;F

< 0, there is no

compensation and the effect of temperature is maximal (case of polar solute distributed between a polar and an apolar phase).

In both cases, the absolute values of

i;k

k¼P;F

tend to decrease with temperature, so each mixture becomes progressively ideal

(i.e., with no enthalpy of mixing). The temperature dependence is predicted via Eqs. (54)–(56).

Ternary Flory Isotherms

Eq. (49) can be generalized to ternary mixtures with two practical applications: i) the estimation of activity coefﬁcients in polar

polymers which contain some amount of water, and ii) the estimation of activity coefﬁcients in water-ethanol mixtures.

The activity coefﬁcient in a wet polymer associated with a volume fraction of water f

, depends on the three pairs of Flory-

Huggins coefﬁ cients, c

i;P

, c

w;P

(water in dry P), c

i;w

(solute in water), as (see Eq. 9a in Fornasiero et al., 2002):

ln g

i;P

wet

¼ð1  f

Þf

ð1  f

 f

i;P



w;i

þ c

i;P

ð1  f

 f



ð1  f

c

w;P

ð1  f

 f

(51)

The activity coefﬁcient in water-ethanol mixtures or in any mixture of two miscible liquids F

and F

can be estimated similarly

(Gillet et al., 2010):

ln g

i;F

þF

¼ð1  f

Þr

1

i;F

 r

1

i;F



i;F

þ c

i;F



þ f



c

vdw

 f



i;F



 f



i;F



f



i;F



 f



i;F





vdw







vdw





 f



i;F



 f



i;F



c

i;F

ð1 2f

(52)

with c

i;F

being a ternary Flory-Huggins coefﬁcient whose contribution can be neglected in the absence of a speciﬁc ternary

complex in the solution. The binary Flory-Huggins coefﬁcient between water (F1) and ethanol (F2) is obtained from the molar heat

of mixing of the mixture,

molar

þF

,asc

¼ y

ð1  f

molar

þF

, using the polynomial approximation proposed for water-ethanol

mixtures in Boyne and Williamson (1967). Fig. 22 reports the value of c

water;ethanol

with the volume fraction in ethanol as well as the

values for common food simulants expressed in alcohol-by-volume (abv) instead of volume fraction. The volume fractions ethanol

and water in the mixture are tabulated from their partial molar volumes between 10



C and 60



CinTable 8.

Figure 22 Variation of the binary Flory-Huggins coefﬁcient in water-ethanol mixtures. “abv” values represent the equivalent alcohol strength

(alcohol-by-volume at 20



C and atmospheric pressure). Simulant C: 10% ethanol for alcoholic foodstuffs; simulant D1: 50% ethanol for high alco-

holic and milk. Data from Gillet et al. (2010).

Risk Assessment of Migration From Packaging Materials Into Food 39

Binary Flory-Huggins Coefﬁcients in a Copolymer AB

Based on calculations at the molecular scale, Nguyen et al. (Nguyen et al., 2017a) demonstrated that Eq. (12) in Fornasiero et al.

(2002) was acceptable for copolymers consisting of repeated blocks shorter than the persistence length of polymer segments.

Beyond its persistence length, the polymer losses memory of its conﬁguration and a block polymer can be treated as a polymer

blend. The corresponding Flory-Huggins interaction coefﬁcient in a copolymer AB, c

i;ðABÞ

, reads:

i;ðABÞ

¼c

i;A

þ c

i;B

 c

(53)

Eq. (53) is based on an averaging of all possible contacts between iA and iB, where the term c

represents the addi-

tional cohesion energy brought about by the interactions between A and B. It can be generalized to more complex copolymers while

the contacts can be assumed perfectly random (i.e., no macro- or microphase separation).

High Throughput Calculations of Flory-Huggins Coefﬁcients at Atomistic Scale

Justiﬁcation and Limitations

Molecular modeling offers a good alternative to time-consuming and complex experiments to estimate Flory-Huggins coefﬁcients

for various substances: monomers, oligomers, solvent, additives, residues, breakdown products, and non-intentionally added

substances. The results can be tabulated in advance and used directly with Eqs. 49 –53 for a broad range of applications. In this

respect, they are more intrinsic than partition coefﬁcients. In detail, molecular modeling can be seen as an alternative to earlier

group contribution methods relying on estimating Flory-Huggins coefﬁcients from solubility parameters (see van Krevelen and

te Nijenhuis, 2009, Hansen, 2007). The limits of the approach have been discussed in Gillet et al. (2009b) and compared with

calculations at the atomistic scale. In short, group contribution methods provide only an average picture of the interactions. The

real conformation of molecules and the distance between interacting chemical groups are, in particular, not preserved. The princi-

ples of calculation of c

ðTÞ

i;P

and c

ðTÞ

i;F

have been reviewed in Gillet et al. (2009b), (2010), Vitrac and Gillet (2010), Nguyen et al.

(2017a,b). They apply to any homo- and copolymer as well as almost any solute while no net charge is present on the polymer,

on the food simulant or the solute. The main limitations are intrinsic to the Flory approximation itself: no energetic barrier should

exist in the host system (P and F) so that all states are accessible. It is not true in glassy polymers where hysteresis is frequent. The

contribution of the subcooling and accumulated elastic energy can be introduced a posteriori by combining Flory and free-volume

theories (Kadam et al., 2014; Krüger and Sadowski, 2005). These extensions are beyond the scope of this article.

Principles

The Flory-Higgins coefﬁcient

i;k

k¼P;F

is deﬁned as the dimensionless mixing energy (enthalpy) in excess relative to pure

compounds:

ðn

;TÞ

i;k



iþk





kþi







kþk





iþi

2RT

for k ¼ P ; F (54)

where hXi

represents an ensemble-average of X, n

is the length used in the approximation of the polymer and n

¼ 1. In agreement

with the original Flory approximation, enthalpies hh

AþB

are estimated by summing energies of contact ε

when B is contacting

Table 8 Density (kg m

3

) of water-ethanol mixture and corresponding volume fraction of ethanol (f

ethanol

) between 10 and 60



C. Volume

fractions are calculated from partial molar volumes

abv



C20



C30



C40



C60



mixture

(kg,m

3

) B

ethanol

mixture

(kg,m

3

) f

ethanol

mixture

(kg,m

3

) f

ethanol

mixture

(kg,m

3

) f

ethanol

mixture

(kg,m

3

) f

ethanol

10%

12%

15%

20%

30%

40%

50%

60%

70%

80%

90%

95%

99%

992.6087

990.0815

986.5772

984.3969

981.3355

976.6829

967.1218

954.5007

937.5325

917.0055

893.7478

867.6778

837.7669

819.9502

802.7596

0.0463

0.0640

0.0903

0.1080

0.1339

0.1773

0.2695

0.3722

0.4773

0.5812

0.6847

0.7889

0.8940

0.9478

0.9900

991.0594

988.4460

984.7554

982.3832

978.9701

973.5916

962.2368

948.0405

930.1519

909.1314

885.5598

859.2628

829.2000

811.3845

794.2436

0.0465

0.0643

0.0912

0.1092

0.1356

0.1800

0.2739

0.3755

0.4795

0.5828

0.6858

0.7896

0.8946

0.9479

0.9900

988.4199

985.7365

981.8636

979.3060

975.5866

969.5825

956.6964

941.2003

922.4957

901.0173

877.1417

850.6228

820.4500

802.6589

785.6217

0.0466

0.0646

0.0920

0.1103

0.1372

0.1826

0.2775

0.3789

0.4816

0.5844

0.6869

0.7906

0.8950

0.9481

0.9901

984.8805

982.1110

978.0521

975.3416

971.3450

964.7720

950.5944

933.9794

914.5695

892.6443

868.4696

841.7478

811.4700

793.7375

776.8538

0.0467

0.0650

0.0928

0.1113

0.1386

0.1847

0.2807

0.3816

0.4837

0.5859

0.6883

0.7915

0.8954

0.9482

0.9901

975.3125

972.3164

967.9100

965.0044

960.6287

953.2020

937.1834

918.9400

898.2414

875.3261

850.5234

823.5828

792.9257

774.9002

758.1182

0.0473

0.0660

0.0938

0.1125

0.1407

0.1883

0.2857

0.3863

0.4877

0.5894

0.6908

0.7929

0.8971

0.9491

0.9900

40 Risk Assessment of Migration From Packaging Materials Into Food

the seed molecule A. hh

BþA

represents the same energy when B is used as a seed molecule. In practice, ε

is calculated by choosing

an orientation randomly for the contact molecule and by translating it along a random line until at least one point of contact is

established between the van der Waals surfaces of the contact and seed molecules. The process is repeated for all conformers and

stereoisomers considered. Finally, h

AþB

is estimated as the product of contact energies and the number of neighbors z

(number of

B molecules surrounding A):

AþB



cooperative



cooperative

(55)

Eq. (55) assumes that ε

and z

are statically independent (zero covariance). For polymers, the property of independence is

achieved from a sufﬁciently large n

value so that the surface of contact of the polymer is independent of the length of the considere d

polymer. The main advantage of the whole approach is that there is no need to represent entanglements in the polymer and free-

volume. The shape of the backbone of an inﬁnitely long chain with shorter oligomers prevents head and tail atoms from coming in

contact with any van der Waal surface. Cooperative hydrogen bonding is accounted for by using a value of n

cooperative

greater than

one. The latent heat of vaporization of water can be correctly approximated by hh

waterþwater

using a value of n

cooperative

different to

unity. This value depends on the type of forceﬁeld used to simulate water. As an example, the rigid water model governed by the

TIP4P forceﬁeld gives an acceptable value with n

cooperative

¼ 1 whereas n

cooperative

¼ 4 is required with the same forceﬁeld but using

three-point charges (forceﬁeld TIP3P). The number 4 reinforces in this case that any water molecule is on average involved in 4

hydrogen bonds of similar strength.

Contact energies are calculated irrespective of any temperature consideration. The effect of temperature is recovered by weighting

the distribution of contact energies with the Boltzmann factor BðεÞ :

þN

N

prðεÞBðεÞεdε

þN

N

prðεÞBðεÞdε

with BðεÞ¼expðε

ðRTÞÞ (56)

At the price of calculati ng two integrals, Eq. (56) canbeusedtoestimatec

ðn

;TÞ

i;k

at several temperatures. Conformers need to be

generated in a way that they are representative of their conformations (radial distribution, constraints of torsion) in the corre-

sponding condensed phase. In practice, they are sampled from molecular dynamics simulations of the equivalent condensed

phase.

Probabilistic Modeling of the Migration

Highlights on probabilistic modeling of the migration.

•

Conventional modeling calculates point estimates, associated with the most likely combination of inputs. Probabilistic modeling

calculates the statistical distribution of the amount transferred for all combination of the parameters (likely or not).

•

Theoretical results make the calculations of sensitivities and statistical distributions almost as fast as point estimates.

•

The combined effects of uncertainty and variability (e.g. residence times at speciﬁc temperatures) can be analyzed together and

used to provide conservative estimates with a controlled risk of underestimation.

•

The same approach can identify the most inﬂuential parameters acting on the value of the migration or the ﬁnal decision value.

Beyond Intuition

Any risk assessment procedure needs to account for the possible variabilities in the considered scenario (e.g. variable tempera-

ture, contact time) and the numerous sources of uncertainties inherent in the limitations of our knowledge and oversimpliﬁca-

tions. Va riability and u ncertainty can be easily recognized and separated by noticing that only uncertainty can be reduced by

bringing additional knowledge or reﬁnement. By contrast, variability r epresents multiple values of several instances (lots, compo-

sitions, ﬁnal use), storage c onditions, etc. For compliance testing, conservative assumptions are mandatory, but the relationship

between the maximization of parameters (or their minimiza tion depending on the case) and the maximiz ation of the amount

transferred is straightforward only in simple con ﬁgurations: one material or one single l ayer, one step, and no variable

conditions.

The intuitive approach is illustrated in Fig. 8 for a single component and monolayer packaging in contact with food. When the

whole food-packaging system is perfectly impervious (no loss to the outside), the cumulative amount leaving the packaging-food

interface is a monotonic function of the time, the initial concentration, the diffusion coefﬁcient in the polymer, the chemical afﬁ nity

for food, the temperature, etc. As a result, choosing a conservative or upper bound for all inputs guarantees an overestimation of the

food contamination. In the presence of multiple materials or steps, the property of monotonicity between parameters and inputs is

not mathematically veriﬁed anymore. In particular, food contamination can be maximal before reaching equilibrium. For example,

overestimating all diffusion coefﬁcients or partition coefﬁcients in laminates will spread migrants everywhere instead of bringing

them faster to the contacting phase. For laminates, methods described in Typologies of Migration Behaviors section and theorized

in (Vitrac et al., 2007b) can be used to recover conservative estimates. The calculation procedure consists of splitting the

Risk Assessment of Migration From Packaging Materials Into Food 41

contribution of n components/materials into m  n independent simulations and in accumulating the concentration in food:

i;F

k¼1::m

. The algorithm is sketched for a substance i present simultaneously in the walls (bilayer AB, B is contact) and the

cap of the bottle (C). The problem comprises n ¼ 3 materials and requires n ¼ 2 simulations:

•

One-dimension mass transfer simulation from B to the food (without A and C) to calculate C

i;F

•

One-dimension mass transfer simulation from C to food (without A and B, i.e. no walls).

It is worthwhile noting that the scenario described is also conservative if it is assumed that the substance has been distributed

between A and B before being put in contact with the food. It also covers the case when i is initially located in A and not in B

(see Fig. 10), but with a higher safety margin.

Epistemic Uncertainty

In systems engineering, reliability and safety are quantiﬁed with respect to some safety margins, deﬁned as the differences between

reference values accepted by the regulating body and calculated values. A system is considered safe when the differences calculated

for a set of postulated scenarios verify a minimum distance or when the probability of the distance being zero or negative is lower

than some prescribed value. Introducing conservatism randomly by mixing worst-case bounds may propagate uncertainty and lead

to uncontrolled overestimation of the amount transferred to the food. At the beginning of the supply chain, the chemical industry

and compounders face mainly variability with the different applications of their chemicals and raw materials. On the opposite side

of the supply chain, the packaging ﬁller and the retailers face a more different situation with strong uncertainties on the nature of the

materials, their thicknesses, and their composition. In 2009 and despite the possibilities offered by EU directive 2002/72/EC, migra-

tion modeling was evaluated to see if it could be helpful to demonstrate compliance in ﬁnished products; it was only in less than 5%

of cases (Gillet et al., 2009b). The chief reason was the loss of compositional information along the supply chain. Calculations

could be done on part supplies and compounds to produce certiﬁcates of food contact compliance, but not on the full system

assembled in the intended conditions of use of the packaging. Compositional information is currently better shared in the EU

and new deformulation techniques provide grounds for spreading calculation practices from the chemical industry to the food

industry (Nguyen et al., 2015).

The best practice for industry relies on sharing minimal information so that a signiﬁcant safety margin remains for the end-user,

and keeping the utility of migration modeling. As an illustration, Fig. 23 compares the safety margins when over-conservative esti-

mates and realistic overestimates are used. It is worth noting that the location of the “real” value is not usually known so that the

overestimation factor cannot be guessed a priori. Only the safety margin is directly accessible to calculations. The deﬁnition of the

safety margin and its use in various technical guides and supporting risk assessment documents can be inconsistent and confusing.

In particular, the concept of safety margin is frequently confused with the concept of overestimation. The deﬁnition of safety margin

used here is applied in medicine to evaluate drugs, in structural engineering, and in nuclear engineering, The concept of overesti-

mation applied to some factors including diffusion (D

) and partition coefﬁcients (K

F/P

) can be misleading. Indeed, their overes-

timation by a factor Q

causes an overestimation of the concentration in the medium in contact which is not proportional. For the

Figure 23 Illustration of safety margins (SM), overestimation factors (Q) and uncertainty according to the method of calculation: real, likely and

very conservative.

42 Risk Assessment of Migration From Packaging Materials Into Food

layer in contact, it varies from

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

(short contact times) down to 0 (equilibrium) for diffusion (X ¼ D

). For X ¼ K

F/P

, it increases

from 0 to a value which depends on the volume of the food.

By evaluating the uncertainty associated with realistic estimates, probabilistic modeling offers a robust methodology to assess

the effects of the combined sources of uncertainties and ﬁnally to have no safety margin at all. The example depicted shows that

the upper limit of likely overestimates including uncertainty (95th percentile) offers a higher safety margin than the very conserva-

tive overestimate. The distinction between overestimations and realistic conservative estimates can be exempliﬁed by considering

a long contact at a variable temperature (e.g. due to transportation). A conservative estimate will calculate migration at the highest

temperature, whereas a realistic conservative value will be provided by replacing time with its integral dimensionless version, Four-

ier number, Fo ¼

DðTðtÞÞdt

. The proposed approximation does not introduce any signiﬁcant approximation and can be carried

out at the same cost as standard simulations (see Strategies and Equations to Simulate Multiple Steps and Conditions section and

Eq. 25).

Sensitivity Analysis of Migration Models

Local Sensitivity Analysis

Deterministic modeling and simulation yield the same output (concentration in food, concentration proﬁle in the packaging mate-

rial) for the same set of inputs. The analysis of the sensitivity to input parameters entails evaluating the effect of a modiﬁcation of

each parameter

€

k¼1::N

(initial concentration, diffusion coefﬁcient, partition coefﬁcient, etc.) on the safety margin, SM. Since the

parameters have different units, it is convenient to calculate the derivatives of SM with respect to the logarithm of each parameter:

JðpÞ¼J



½p





vSM



::p

vSM



::p

N11

vSM

vln p



::p

vSM

vln p



::p

N11

(57)

J(p) is the Jacobian matrix of the migration model and it can be used to evaluate a linear approximation of the safety margin when

inputs are changed from p

to p:

SMðpÞz SM ðp

ÞþJðp

Þðlnp  lnp

Þ (58)

Eq. (58) provides analytical solutions only in simple conﬁgurations, but its application is very general. Additionally, implement-

ing the difference lnp

 lnp

k,0

as ln

k;0

prevents the problem of step size known as subtractive cancellation. Its usage is shown for

a variant of the realistic but conservative model for monolayer materials presented in Governing Equations for Monolayer Mate-

rials (see Eq. 15):

Fo;K



ðFoÞ



ﬃﬃﬃ

min



ﬃﬃﬃ

;

ﬃﬃﬃﬃﬃ



(59)

When the input values are changed to

Fo ¼

; K

F=P

; C

from an initial set

Fo ¼

; K

F=P

; C

, the safety margin

ðFo;K

F=P

zSML 

ðFo;K

F=P

becomes:

Fo;K

F=P

 SM



Fo;K

F=P



z1 

¼

F=P

þ ln

min



ﬃﬃﬃ

;

ﬃﬃﬃﬃﬃﬃ



min



ﬃﬃﬃ

;

ﬃﬃﬃﬃﬃﬃ



(60)

with Fo dC

e¼

ﬃﬃﬃ

min



ﬃﬃ

;

ﬃﬃﬃﬃ





F=P

P=F



Besides showing the interactions and additivity of the different sources of uncertainty, it offers a rapid methodology to identify

the main inﬂuencing parameters without requiring any simulation or software. Eq. (57) also applies to numerical simulations, but

it requires N þ 1 simulations (the reference one and N variations). When the number of inputs increases, it is preferable to reduce

the computational effort by using similitude principles and dimensionless numbers. In the detailed example, the use of a dimen-

sionless time Fo ¼

enables simultaneous testing of the effects of t, D

and l

. Eq. (57) is a ﬁrst order approximation and, despite

the use of a logarithm scale, it looses accuracy as soon as parameters are tested beyond several factors.

Risk Assessment of Migration From Packaging Materials Into Food 43

Global Sensitivity Analysis via Stochastic Simulation

When the number of variables becomes large as well as the intervals to be explored, the statistical sampling of inputs is preferable.

Statistical analysis of the outputs can be used to extract the inﬂuence of each variable and the probability to have the prescribed

threshold exceeded. Each component of the vector p needs to be sampled randomly and uniformly over its interval of interest.

The technique is the so-called Monte-Carlo trials and its numerical implementation stochastic simulation. By denoting

k;i

i¼1::M

M samples chosen around the likely vector p so that only the parameter

k¼1::N

is modiﬁed at a time, and by denoting

SMð

k;i

Þg

ï ¼ 1. M

the corresponding safety margins, the sample covariance is given by:

M  1

i¼1



k;i







k;i

E



k;i







k;i

E

(61)

where hSMð

k;i

Þi ¼

i¼1

SMð

k;i

Þ is the average safety margin, which does not coincide with SMðpÞ in the general case. SMðpÞ

represents the prediction associated with the 50th percentile.

Eq. (61) generalizes the local sensitivity analysis performed in Eq. (57) , based on small variations and partial derivatives. The

concept of covariance enables screening of the whole input spectrum to identify the interaction and dependency structure on all

parameters including the analysis. If the geometry, temperature and contact time are introduced, the design and conditions of

use can also be explored.

A probabilistic interpretation is achievable but, as means and covariances provide only the ﬁrst and second moments, a likely

distribution of the safety margin or the concentration in food is required. If the concentration in food is normally distributed, the

problem is fully determined with the ﬁrst and second moments. This assumption is valid only in the presence of a low range of

variabilities and uncertainties. Indeed, the Gaussian distribution is unbounded, and it implies, even with very low probabilities,

that the concentration in food could also be negative and the amount transferred could be higher than the amount in the material.

The next section removes these limitations for risk assessment and the evaluation of consumer exposure.

Principles of the Probabilistic Interpretation of Mass Transfer

Global sensitivity analysis presented in Global Sensitivity Analysis via Stochastic Simulation introduces the ﬁrst interpretation of

mass transfer with a marginal distribution on each input variable, which is assumed to be uniform. The combination of these vari-

ables and its interpretation is known as a copula in probability theory and statistics. Copulas describe well the dependence between

inputs on the output(s) of a model, but they fail to describe the joint distribution of contamination in realistic situations. The diffu-

sion and partition coefﬁcients, as well as the initial concentration in food, are not distributed uniformly. The industry is not

applying randomly any concentration value or the molecules do not have random properties. It is because of limited knowledge

and the variability of practices that spread the inputs around a likely value. In short, the sensitivity analysis is a perfect tool to opti-

mize the geometry, formulation, etc. but it is not appropriate to get a reliable estimate of the probability to have a concentration

threshold exceeded.

The principles of probabilistic modeling of migration have been described in (Vitrac and Hayert, 2005) and applied to various

cases (Vitrac et al., 2007a; Vitrac and Leblanc, 2007). The central idea is to combine a dimensionless formulation (with a reduced

number of input variables) along with random numbers. For the same reason as invoked for the local decomposition in Local

Sensitivity Analysis section, each quantity X is written as the product of a scaling value (with units), denoted

X (usually the likeliest

value), and a random dimensionless number X*distributed around unity. As an illustration, the dimensionless time reads:

Fo ¼

















2

¼ FoFo



(62)

Input Distributions

The distributions ofX*can be chosen either from experimental measurements or from prior guesses or beliefs. In this second alter-

native, distributions which have a shape factor and are non-negative, denoted f

(1, s

), should be preferred (beta, Erlang, exponen-

tially modiﬁed Gaussian, exponential, gamma, inverse-gamma, inverse-Gaussian, lognormal, Weibull, etc.). For some quantities,

such as concentrations or thicknesses, symmetric distributions are more realistic; truncated normal distributions can be used for

this purpose. A non-exhaustive list of practical distributions is given in Table 9.

Estimation of Probabilities via Monte-Carlo Sampling

Probabilistic modeling aims at determining the cumulative density function (cdf), which can be written for monolayer materials as

the probability to get a value of C

lower than an arbitrary number x:

prðC

 xÞ¼F



Fo; Bi; K

F=P

; C

; a

; b

; s



(63)

For the sake of efﬁciency, Fo (see Eq. 62) was used instead of ðt;D

Þ. Indeed using Eq. (10), it can be shown that the

distribution a posteriori of

ﬃﬃﬃﬃﬃﬃﬃ



converges in law to a Gamma distribution with parameters ða

Þ. Nguyen et al. (2015) tabulates

their values with s

, s

and s

44 Risk Assessment of Migration From Packaging Materials Into Food

The cumulated probability pr(C

 x) (or its complement pr(C

> x)) can be estimated by repeating the simulations for different

values of input parameters and by counting the number of occurrences for which the inequality C

 x (or C

> x) is veriﬁed. If the

intent is to demonstrate that pr(C

> x) is low for a sufﬁciently large x, it might be thought that it sufﬁces to apply some worst-case

scenarios (bounds of intervals) and to demonstrate that the value x is never exceeded. This approach is correct only if pr(C

> x ) ¼ 0,

that is for a value of x larger than the one corresponding to a total extraction. In the general case, the intervals of all parameters need

to be sampled with each value chosen according to its theoretical prescribed distribution. This technique of randomly picking input

values and launching the corresponding simulation is known as Monte-Carlo simulation.

In practice, for each input quantity X, a random number gischosen uniformly between 0 and 1 (function rand() in many

programming languages). The speciﬁc value to be included in the considered simulation vector will be F

1

ðgÞ, with F

(x) ¼

pr(X  x) being the cdf of the variable X. Depending on the size of the intervals, 10

to 10

simulations are required. For multilayer

materials, the sampling effort can be even higher. The total cost of simulations can be reduced dramatically by tabulating the results

in advance for a signiﬁcant range of dimensionless numbers and by subsequently interpolating the values of interest. These concepts

are now justiﬁed mathematically. They make the cost of probabilistic modeling close to the cost of deterministic modeling.

Estimation of Joint Probabilities via the Composition Theorem

The calculation of probability density functions (pdf) associated with a combination of variables X

, /, X

(e.g., Eq. 62) or with the

resolution of partial differential equations is particularly expensive computationally and requires a speciﬁc treatment of the joint

density f

;/;X

ðx

; /; x

Þ¼prðX

¼ x

; /; X

¼ x

Þ. The composition theorem offers a very efﬁcient computational approach

for invertible and differentiable transformations:

¼ h

ðX

; .; X

Þ; i ¼ 1; .; n

¼ h

1

ðY

; .; Y

Þ; i ¼ 1; .; n

(64)

Mathematical functions

i¼1::n

represent either a variable transformation (e.g., Eq. 62) or the full mass transfer model. The

joint density of Y

, /, Y

is given by the determinant of the Jacobian matrix vh

=vx

jsi

, denoted J

;/;Y

ðy

; /; y

Þ¼

;/;X



1

ðy

; /; y

Þ; /; h

1

ðy

; /; y







1

ðy

; /; y

Þ; /; h

1

ðy

; /; y





1

(65)

Eq. (65) can be generalized to non-monotonic functions by splitting the transformation into intervals which are locally mono-

tone. For example, in the special case where the dimensionless concentration in food



is not a continuous function of Fo (some

multilayer or multicomponent conﬁgurations), the pdf of



¼ hðFoÞ is obtained by the accumulation of the p transformations over

p contiguous intervals:



ðvÞ¼

k¼1



1



Fo˛Y

ðvÞ





1



Fo ˛ Y

ðvÞ



1

(66)

with

k¼1::p

the partitions of Fo where v



¼ hðFoÞ is piecewise monotonic.

Some Illustrations

Probabilistic modeling must be envisioned as the generalization of deterministic modeling, but its clear deﬁnition depends on how

the normalization of X* is performed, how deterministic inputs are set (likeliest values or averaged ones) and how the transforma-

tion stretches or contracts the probability space. Deterministic modeling provides the unique solution of the initial-value problem.

Table 9 Recommended distributions for probabilistic modeling of the migration from monolayer materials. The

distributions of



and

ﬃﬃﬃﬃﬃﬃﬃ



are posterior distributions

Random contribution

Distribution

Recommendations

Rubber polymers Glassy polymers

Diffusion coefﬁcient log



Norm(O, s

) s

¼ 0.1 s

¼ 0.5

Contact time t* Weib(O, s

)

to be determined

Initial concentration C

0

Normð1; s

truncated

1  s

 5

Thickness l* Norm(1, s

) s

to be determined

Mass Biot number log

Bi* Norm(O, s

) s

/ 0 s

/ 0

Partition coefﬁcient log



F =P

Norm(O, s

) < 0.2 < 0.2

Fourier number Fo



Gamma(a

, b

) a

, b

: to be calculated

Concentration in food v



Beta(a

, b

) a

, b

: to be calculated

To be normalized to get a unitary expectation.

Risk Assessment of Migration From Packaging Materials Into Food 45

Probabilistic modeling generates the distributions of outputs (e.g., C

) for any combination of input parameters and initial condi-

tions. The use of dimensionless numbers such as a cumulated Fourier number including temperature variations

ðTðtÞÞ

dt or

the ones associated with homologous solutes

i;p

i˛homologuous

solutes

, offers an even broader interpretation by including random

variations of temperature or uncertainty of diffusion coefﬁcients (see Vitrac et al., 2006 for discussion). In the context of risk assess-

ment, the scaling quantities

Fo , Bi and K

F=P

as acceptable conservative estimates and the effects of uncontrolled variations and

uncertainties can be captured with the distributions of Fo*, Bi* and K



F=P

or by choosing them equal to unity.

Typical Probabilistic Migration Kinetics

Without loss of generality for estimating multivariate distributions of concentrations, the principles of composition are illustrated in

Fig. 24 for the dimensionless migration kinetics from monolayer materials (see section Governing Equations for Monolayer Mate-

rials). In this example, Fo* is the only random variable and all other parameters remain ﬁxed (not distributed). The distribution of

the dimensionless concentration



is inferred from Eq. (66) with p ¼ 1 (strictly monotonic curve). It can be viewed as the projection

of the support ofFo ¼

FoFo



onto the curve v



¼ f ðFo



Þ. When the transformation is repeated for different values for Fo, the upper

and lower percentiles of the migration kinetics can be interpolated continuously with Fo. Such curves are not accessible to the direct

simulation and are not parallel to the deterministic or 50

percentile. The vertical distance between the extreme percentiles repre-

sent the resulting uncertainty in the amount transferred according to the original dispersion in Fo values. It is worthwhile noting that

dispersion increases with time but that its effect on



is decreasing after Fo >

ﬃﬃﬃ

. Close to equilibrium, Fo ceases to have a signiﬁcant

effect.

Effect of Bi and s

The overall mass transfer resistance, <, of any compartment (either the food or the packaging layer) is evaluated as

, whether

the chemical afﬁnity of the substance is considered or not. For monolayer materials, the packaging-to-food mass transfer resistance

Figure 24 Probabilistic modeling of the contamination from a monolayer material via Eqs. (10) and (62): (A) point distribution for Fo ¼ 0:5; (B) cor-

responding 10th and 90th percentile curves. The likeliest migration curve corresponding to the maximum probability (mode) of the Fo distribution

appears in bold.

46 Risk Assessment of Migration From Packaging Materials Into Food

ratio is given by K

F/P

1

and Bi

1

, respectively. Bi values are expected to be large above 10

with well-mixed and low viscous liquids

in contact with thick or barrier polymers. Low values ranging between 5 and 10

were observed only in polyoleﬁns in contact with

liquids (Vitrac et al., 2007c). In semi-solids, solids and dry foods in contact with polyoleﬁns and plasticized polymers, migration

kinetics linearize with time (Till et al., 1987) and Bi approaches unity.

The effects of Bi and s

are illustrated in Fig. 25. Increasing the mass transfer resistance on the food side (i.e. decreasing Bi) affects

non-linearly the dispersion. As justiﬁed by Eq. (66) and because the contamination is strictly increasing, the dispersion of the

contamination is weighted by the term



1

ðvÞ



1



dFo

ðFoÞ. As low Bi values and large Fo ones lead to low slopes of v



Fo, the dispersion of concentration values are, as expected, maximal for intermediate concentration values far from the initial

and equilibrium states. Increasing s

modiﬁes strongly the shape of the median kinetics, the upper limits and the overall distribu-

tion in food. The depicted example demonstrates that the migration can be overestimated reliably by considering a likely value for

and by taking for example the 95th percentile of v



along with a proper value of s

. Doing the reverse, calculating v



from the

95th percentile of D

values does not guarantee that the value of the contamination is overestimated in 95% of cases. It is not veri-

ﬁed in the case when the slope of



changes rapidly with Fo. This is true only when Bi / 1, as shown in Fig. 25a.

What Will Be the Future?

Highlights on new trends

•

Because migration modeling is cost effective, its application spreads between countries and beyond its original application

domain: food packaging in plastics.

•

Migration modeling turns the fate of contamination into forecastable phenomena with causes and responsibilities.

•

Migration modeling paves the way for food safety to be managed in the cloud.

Extending the Legal Scope of Migration Modeling

A robust validation of the macro scopic equations of mass transfer (transport equati ons and boundary conditions) has been

central to the development of the US legal system authorizing migrat ion model ing in t he nine ties. The European system

focused during the two past decades on diffusion coefﬁcients with much smal ler attention on partition coefﬁcients. In both

Figure 25 (A) Effect of Bi on the dimensionless migration kinetics. (B) Effect for Bi/N (the percentiles are represented as equivalent kinetics;

distributions of



for Fo ¼ 0.5, 1, 2, 3 and 4.

Risk Assessment of Migration From Packaging Materials Into Food 47

cases, migration modeling is recognized and well accepted but only for thermoplast ics. Its ap plication to thermosets, elasto-

mers, p aper and board is comparatively support ed by a much small number of scientiﬁc and technical publications. Recently,

Chinese regulations adopted migrati on modeling and equiv alent calculations without material restriction. This move is a l ogical

step after the substitution of negative lists by positive ones and the adoption of speciﬁc migration limits for a broad range of

applications.

At t he expense of proper signiﬁcant safety margins, migration modeling has unlimited scope and can cover multiple mate-

rials and complex conditions of use of food contact materials. The chief difﬁculty remains in practice the lack of information on

the substances i nitially present in the materials and their amounts. Parado xically, migration modeling offers an efﬁcient and

secure solution to the delicate problem of the declaration of conformity according to the p ositio n of the operator in the supply

chain. At each step, the essential information to demonstrate compliance and safety assessment respectively to intentionally

added substances can be encoded into almost anonymous numbers (worst-case values of concentrati ons, properties, and

thresholds), without revealing the exact details of formulations, chemical structures, etc. The amount of data could grow al ong

the supply chain by adding new records (association of substances, materials, logistics data, date of production) and reﬁned

scenarios which correspond to the ﬁnal application of the packaging or the targeted market (e.g., country, food/pharmaceu-

tical). In a very near future, the whole process could be included in a blockchain resistant to falsiﬁcation and not requiring

any third-party authorit y. Migration modeling softwa re c ould process any node and verify rapidly hundreds of co nstraints cor-

responding to the inte grity of all decisions. Date stamps and default expiration dates could alert operators in advance of the

need to revi se either risk assessment or risk management decisions. All the effort related to typing inputs, looking into databases

and estimating properties coul d be performed by s canning a QR code or equivalent. The result could be a new QR code veriﬁ -

able b y local authorities.

After several decades of pioneering developments and validation, the chemical industry, producers, converters, recyclers, food

industry, retailers, authorities and consumer associations could turn to interoperable systems promoting both safety and economic

efﬁciency. The deﬁnition of common standards (CEN, ISO) and training could be the ﬁrst step. With respect to modeling, the legal

systems initiated for food contact materials and articles could also be used as templates for future evolutions of the regulation of

materials used in cosmetic, pharmaceutical, medical, biotechnological, and clothing products.

The possibility of managing cross-contamination via modeling and simulation is expected to impact rapidly on good

manufacturing and handling practices. Migration modeling can be used, indeed, to establish the causality between practices and

migration regardless of their position in the supply chain and the distance between the incriminated material and the food. Demon-

strating causality will bring some form of responsibility irrespective of whether the material or the practice are intended to be in

direct contact with food or not. As an example, the contribution of secondary packaging materials acting as large reservoirs of

contaminants can be evaluated, as the foreseen effects of corrective actions, including air renewal in storage places, separation of

printed and non-printed materials, and separation of laminates from monolayer materials ..

Extending the Capacity of Migration Modeling

The validations of functional barriers and recycled materials have been the main successful applications of migration modeling. The

last advance in migration modeling including multiscale modeling offers the only viable solution to evaluate complex problems

met by the food industry and the food packaging supply chain:

a.NIAS: non-intentionally added substances (no need for standards or analytical methods; hypothetical molecules can be accepted);

b.cross-contamination between materials at any stage of the supply chain (all conﬁgurations can be included);

c.post-consumer contaminations, including misuse

d.optimization of decontamination steps in mechanical recycling processes

e.materials and articles with repeated use (no need for long experiments)

f.materials and devices used with ﬂows (no need for any setup)

g.materials subjected to aging and long-term storage.

NIAS include hypothetical and unknown substances (e.g., breakdown products), but also known impurities and substances inten-

tionally present or added to third-party materials (printing inks, adhesives, lacquers, overpackaging, secondary packaging). All the

cases can be evaluated by combining molecular and migration modeling. The same approaches can be used to optimize decontam-

ination conditions (solvent choice, temperature, duration) in recycling processes.

Test conditions for articles associated with repeated use and ﬂows have not been detailed, and total migration scenarios prevail

as a general recommendation in most of the regulations. Migration modeling can be used to evaluate articles and devices (gaskets,

hoses, tubings, reservoirs, tanks, conveyors .) in more realistic conditions and all through their lifetime. As an illustration, plas-

ticized materials are prone to release substances only for short-time contacts. Once the surface is depleted of plasticizer(s), the glass

transition temperature increases sufﬁciently to transform the contact surface into a temporary functional barrier. Any period without

liquid in contact (e.g., stopping or cleaning period) redistributes the plasticizer(s) uniformly and causes a new risk of leaching.

Long-term storage of materials and material aging redistribute contaminants and may bring new breakdown products. Aging

before use is not considered in current regulations. Modeling can offer a very cost-efﬁcient method to evaluate the risk associated

with the redistribution of contaminants from tie layers, printing inks, adhesives, and lacquers before they are processed into food

contact materials.

48 Risk Assessment of Migration From Packaging Materials Into Food

Bridging Migration Modeling, Safe-by-Design, and Eco-Design Approaches

This article encourages addressing safety issues at early stages in the design of food packaging and food contact materials before they

even become integrated into a ﬁnished product. Instead of checking the compliance of the ﬁnished product, additives, designs

(shapes, surface-to-volume) and conditions of use (shelf life, storage) can be used to minimize the risk of contamination and

cross-contamination. Safe-by-design approaches (Nguyen et al., 2013) are particularly relevant for foods devoted to babies and

infants as well as for all applications maximizing the amount of recycled materials. The new generation of simulation tools

(Zhu et al., 2019) will bridge safe-by-design approaches with eco-design, integrate 3D simulation of mass transfer and explicit

food representation.

Online Resources Ease Risk Assessment

Safety decisions supported by calculations, modeling and simulation are only acceptable if they demonstrate that the migration

amount, estimated by

, is below the threshold, C

thresh

, prescribed by regulation, good manufacturing practices or other consid-

erations. Otherwise, migration testing (experiments) must be applied. Without any threshold to compare, (i.e., non-evaluated

substances),

or its conversion into chronic exposure migration modeling loses its interest. The utility of migration increases

when

is not greatly overestimated (i.e., too conservative) and when the value of C

thresh

is not the default value for a very toxic

compound. The development of online resources (computational tools, databases, training, guidance, etc.) offers to make migra-

tion modeling and risk-assessment readily accessible, transparent, and cost-effective.

Lower Bounds of Toxicological Thresholds for Non-evaluated Substances

For non-evaluated substances or in the absence of speciﬁc chemical data, the lowest (safest) value of C

thresh

is determined by the

concept of toxicological threshold of concern (TTC) (Kroes et al., 2004). It has been recently endorsed jointly by EFSA and

WHO (EFSA and WHO, 2016). It is pragmatic ersatz to identify situations, where migration can be presumed to present no appre-

ciable human health risk (risk of cancer lower than 1 in one million). By excluding high potency carcinogens, i.e., aﬂatoxin-like,

azoxy- or N-nitroso-compounds and benzidines, the TTC value for potentially genotoxic compounds is:

TTC;genotoxic

thresh

¼0:0025 mg$kg

1

$day

1



body weight

daily intake

(67)

Eq. (67) relies on a chronic exposure estimate: the same food is in contact with the same packaging, and 1 kg (EU rule) or 3 kg

(US rule) of food is daily consumed by an adult. For an adult of 60 kg (EU rule), the default value is 0.15 mgkg

1

of food. The

default value for infants (EU rule: infant up to 11 months old, weighing 5 kg and eating 0.75 kg) is 0.0167 mgkg

1

of food. For

infants under three months of age, additional considerations may be considered when

values are close to C

TTC;genotoxic

thresh

, including

metabolism, frequency, and duration of exposure.

Similar principles apply to substances which are not part of the exclusionary categories and without structural alert of chemical-

speciﬁc genotoxicity data suggesting a potential DNA reactivity. The values of TTC and C

TTC

thresh

are tabulated in Table 10 for adults.

The classiﬁcation of substances relies on the Cramer decision tree (Cramer et al., 1976), which has been implemented in the

program Toxtree, commissioned by the JRC Computational Toxicology and Modelling, and in the OECD QSAR Toolbox. Both tools

require chemical structures to be validated independently. Practical tools and databases are listed in Table 11.

Migration Modeling Tools

Several generations of tools have been developed during the last decades using either analytical or numerical schemes. All current

tools are capable of describing transfer across multilayer structures and offer the possibility to simulate variable storage conditions

(liquid contact, setoff, etc.). Similar results can be obtained with generic solvers of partial differential equations (PDE), such as

ANSYS, Cast3M, Comsol Multiphysics



, Fluent, FreeFem, Modelica, OpenFoam, etc. The difﬁculty for multilayer structures is

the implementation of the internal conditions deﬁned by Eq. (20), which require a weak formulation of the PDE problem.

The most notable migration modeling software are listed in Table 12 with their corresponding license (open-source, freeware

and commercial). It is worth noting that they implement Fickian diffusion and that they assume uniform and isotropic distribution

within each layer. These tools need to be combined with internal or external databases and mathematical relationships in order to

Table 10 Threshold concentration values based on TTC approach for an adult of 60 kg and a daily intake of 1 kg

Type of TTC value TTC in mg/kg body weight per day C

TTC

thresh

in mg/kg food (assuming 1 kg daily intake)

With structural alert for genotoxicity 0.0025 0.15

organophosphates and carbamates 0.3 18

Cramer class III 1.5 90

Cramer class II 9.0 540

Cramer class I 30 1800

Risk Assessment of Migration From Packaging Materials Into Food 49

Table 11 Main tools and databases to derive acceptable toxicological threshold

Type of tools Online resources

TTC tools - OECD toolbox: http://www.oecd.org/chemicalsafety/risk-assessment/theoecdqsartoolbox.

htm

- Toxtree (TTC and related data) https://eurl-ecvam.jrc.ec.europa.eu/laboratories-research/

predictive_toxicology/qsar_tools/toxtree

- EFSA scientiﬁc opinion on TTC https://www.efsa.europa.eu/en/consultations/call/181112

Toxicological

databases

- CEFIC LRI Toolbox (http://www.ceﬁc-lri.org/lri-toolbox) including RepDose, FeDTex and

CEMAS

- ChemIDplus (Toxnet, USA) (http://chem.sis.nlm.nih.gov/chemidplus/),

- Cosmetic Ingredient Review (CIR) database (http://www.cir-safety.org/ingredients), SCCS

opinions

- Council of Europe Database (not publicly available) https://fcm.wiv-isp.be/

- ECHA Final decisions on compliance checks and testing proposals in REACH registration

dossiers: https://echa.europa.eu/addressing-chemicals-of-concern

- ESIS http://esis.jrc.ec.europa.eu/

- GESTIS substance database for Occupational Exposure Limits (OELs) http://www.dguv.de/

dguv/ifa/Gefahrstoffdatenbanken/GESTIS-Stoffdatenbank/index-2.jsp

- HPV-Program http://webnet.oecd.org/hpv/ui/SponsoredSubstances.aspx

- IARC: list of carcinogenic substances (https://monographs.iarc.fr/agents-classiﬁed-by-the-

iarc/)

- NICNAS (Australia) http://www.nicnas.gov.au/chemical-information

- NTP: US/NIH list of carcinogenic substances (https://ntp.niehs.nih.gov/pubhealth/roc/

index-1.html)

- OSHA: EU list of carcinogenic substances (https://osha.europa.eu/en/legislation/directives/

directive-2004-37-ec-carcinogens-or-mutagens-at-work)

- Toxline (http://toxnet.nlm.nih.gov/cgi-bin/sis/htmlgen?TOXLINE)

- TSCA http://yosemite.epa.gov/oppts/epatscat8.nsf/reportsearch?openform

General

databases

- Chemspider http://www.chemspider.com

- PubChem https://pubchem.ncbi.nlm.nih.gov/

Table 12 Notable physics-based software and tools to evaluate migration

License

Stand-alone compliance

testing programs

Migratest

EXP commercial (demo available)

https://www.fabes-online.de/en/software-en/migratest-exp/

AKTS-SML version 6

https://www.akts.com/sml-diffusion-migration-multilayer-packaging/download-diffusion-

prediction-software.html

Compliance testing

client/server

Client-server SFPP3

application (SafeFoodPackaging portal version 3) to be used by one

to 25 simultaneous users.

freeware, partly opensource,

online access or standalone

installation.

http://sfpp3.agroparistech.fr:443/cgi-bin/login.cgi

SFPP3 includes all public data of the European task force TF-MATHMOD.

Interactive training on SFPP3 tools with case studies (French):

http://rmt-propackfood.actia-asso.eu/

Expandable preventive

and safe-by-design

tools

FMECAengine

and key2key() language enabling simulation from one to thousands food

packaging systems, an entire supply chain, etc.

open-source (written in

Matlab

, Octave language)

https://github.com/ovitrac/FMECAengine

FMECAengine includes and expands all features implemented in SFPP3.

Online Thermophysical properties of polymers free access

databases http://polymerdatabase.com/

Diffusion and partition coefﬁcients of the European task force TF-MATHMOD

http://modmol.agroparistech.fr/Database/

Guidance EU rules: http://publications.jrc.ec.europa.eu/repository/handle/JRC98028

US rules: https://nepis.epa.gov/Exe/ZyPURL.cgi?Dockey¼P100BCMB.TXT freely accessible

Generic: http://modmol.agroparistech.fr/home/

Generic:https://www.foodpackagingforum.org/food-packaging-health/migration-modeling

50 Risk Assessment of Migration From Packaging Materials Into Food

take into account t he nature of the migrant and the physical pr operties of the polymer. Open-source software and devoted

programming language, as FMECAengine and key2key() language, are part of a global effort to f oster interoperability, collabo-

rative open-source projects and free templates for common problems. The European Committee for Standardiz ation has already

identiﬁed that “a standardized t erminology will improve future exchanges among experts in the entire area of materials

modeling, facilitate the exchange with industrial end-users and experimentalists and reduce the barrier utilizing materials

modeling” (CEN, 2018).

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Table 12 Notable physics-based software and tools to evaluate migrationdcont'd

License

MOOC: Massive Open

Online

Course (USA) – 7 hours

MOOC lectures from the School of Packaging (Michigan State University) no registration

Overview

https://www.canr.msu.edu/cpis/research/downloadable_presentations

Diffusion coefﬁcients

https://mediaspace.msu.edu/media/Dr.

þOlivierþVitracþpresentsAþDiffusionþcoefﬁcientsþofþorganicþsolutesþinþ

polymersA/1_zz20dgt9

Partition coefﬁcients

https://mediaspace.msu.edu/media/Dr.þOlivierþVitracþpresentsAþAnþatomisticþ

Flory-Hugginsþformulationþforþtheþtailoredþpredictionþofþactivityþandþ

partitionþcoefﬁcients/1_uzi6h91k

Migration modeling and decision-making workshop

https://mediaspace.msu.edu/media/WorkshopAþPredictionþofþtheþmigrationAþ

beyondþconventionalþestimates%2A/1_won1m7aw

MOOC (EU) > 50 hours The European project ERASMUS þ Fitness “Food packaging open courseware for higher

education and staff of companies”

is preparing a MOOC on all aspects of packaging design

including compliance testing, food safety, and risk assessment. Main page (building):

http://www2.agroparistech.fr/FITNess-Project.html

to be released by late 2019

Risk Assessment of Migration From Packaging Materials Into Food 51

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