prediction of worst case migration: presentation of a rigorous...

Prediction of worst case migration: presentation of arigorous methodology

A. Reynier, P. Dole and A. Feigenbaum*SecuriteÂ et QualiteÂ des Emballages Alimentaires; Institut Nationalde la Recherche Agronomique, CPCB Moulin de la Housse, BP1039, 51687 Reims Cedex 2, France

(Received 22 May 1998; revised 14 September 1998; accepted20 October 1998)

An improvement of the Piringer model, allowing theprediction of a worst case migration from packaging tofood is presented here. The authors are proposing otherconstants for the calculation of the upperbound value ofthe di usion coe cient, using experimental data deter-mined by a ® lm to ® lm method. Considering theplasticizing e ects of food simulants, a model involvingthe variation of the di usion coe cient versus spaceand time must be used. Future ® elds of investigation arediscussed: the relationship between di usion coe cientsand the volume of the migrant (instead of molar mass),and the variation of di usion coe cient activationenergy with temperature.

Keywords : worst case migration, di usion, partitioncoe cient, modelling, solubility parameters,plasticization

Introduction

Is it possible to calculate the migration of a substancefrom a plastic packaging material to food, withoutcarrying out experiments, having available only re-duced information on the chemical structure of themigrant? This problem requires that the two keyparameters describing the behaviour of a compoundin a polymer± food system, namely its di usion coe -cient and the amount migrated at steady state, can beestimated.

Migration at steady state is often considered to be theamount of substance initially introduced in the poly-mer, which is equivalent to a 100% migration as-sumption. This is equivalent to assuming an in® nitepartition coe cient between food and packaging(Vergnaud 1995). This is supported by some experi-mental work (Chang et al. 1982), but contradicted byother data (O’Brien et al. 1997).More work has been dedicated to estimations ofdi usion coe cients. Some authors propose directrelationships between the di usion coe cient D andthe molar mass of the migrant (Limm and Holli® eld1996). However, these methods are not accurate, andan empirical correlation between an upperboundvalue of the di usion coe cient D* and the molarmass has been proposed (equation (1) corresponds tothe more recent publications) (Baner et al. 1994, 1996,Piringer 1994, 1997, 1998).

D 10 000 exp Ap 0.01Mw10 450

Tcm2 /s

1

where Ap depends on the polymer type, M is themolar mass and T (K) the temperature.This enables the calculation of an upperbound valueof migration, called overestimated migration. If cal-culated overestimated migration values are consistentwith the legal limits, it is not necessary to carry outmigration experiments.

This is a very practical and attractive approach. Theobjective of this paper is to propose improvements tothe approach, and to enhance its consistency withscienti® c background information. For example, mod-els used should describe the migration at more than 10days, taking into account variations of the apparentdi usion coe cient, connected to kinetic limitationsat the surface, and food± polymer interaction: a modelwhich describes only the migration at 10 days is ofinterest only for the chemical industry; but for theprocessing industry or the food industry, the problemis not only that the speci® c migration limit (SML)

Food Additives and Contaminants, 1999, Vol. 16, No. 4, 137± 152

* To whom correspondence should be addressed.

Food Additives and Contaminants ISSN 0265± 203X print/ISSN 1464± 5122 online Ñ 1999 Taylor & Francis Ltdhttp://www.tandf.co.uk/JNLS/fac.htm

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should not be exceeded after 10 days, but also overthe whole shelf -life.

Four topics will be discussed here: (i) the deter-mination of the upperbound di usion coe cient;(ii) the in¯ uence of food± polymer interactions;(iii) the kinetic limitations at the interface; (iv) thepossible Arrhenius relationship for temperaturedependence.

Experimental

Materials

Four di erent matrices have been tested for di usionproperties: LDPE (Low Density Polyethylene),LLDPE (Linear Low Density Polyethylene), HDPE(High Density Polyethylene) and PP (Polypropylene)

(CERDATO, Lacq). Melting points are determinedby DSC (onset point of endothermic peak) (TAInstruments 2920).

LDPE ® lms: 25.5 m m thick, Tm = 100.6ë CLLDPE ® lms: 27 m m thick, Tm = 113ë CHDPE ® lms: 66 m m thick, Tm = 122.4ë CPP ® lms: 54 m m thick, Tm = 138ë C (onset) Ð peak at147.34ë C

All these polymers contain very low amounts ofadditives which do not interfere with di using sub-stances.

A panel of substances has been selected for di usiontests, in order to represent various kinds of molecularshapes and mass, as well as chemical functions: linearalkanes, non-linear hydrocarbons, acids, esters, aro-matic compounds, amines (table 1). This paper dealswith preliminary results obtained for molar massranging from 200 to 500 g/mol.

138 A. Reynier et al.

Table 1. Di usion coe cients measured by ® lm to ® lm method for the tested substances at 40 ë C.

Molecular D (cm2/s)weight

Compound (g/mol) PP LDPE LLDPE HDPE

Undecane 156 2.12E-09 1.66E-08 1.90E-08 1.03E-08Dodecane 170 1.86E-08 1.23E-08

9.34E-10 9.97E-09 7.76E-09 4.23E-09Tridecane 184 1.12E-09 1.60E-08 1.86E-08 8.47E-09

1.44E-08 8.98E-09Pentadecane 212 1.15E-08 7.66E-09

2.04E-09 1.23E-08 1.50E-08 7.78E-09Hexadecane 226 1.33E-09 * * 6.37E-09Heptadecane 240 1.33E-09 * * 5.86E-09Octadecane 254 8.67E-10 5.99E-09 4.01E-09 2.16E-09Docosane 310 2.45E-10 1.35E-09 9.60E-10 4.17E-10Tetracosane 338 5.61E-10 * * 5.27E-10Octacosane 394 1.78E-10 2.46E-10 1.41E-10 *Triphenylmethane 244 1.29E-10 3.85E-09 1.52E-09 6.98E-10Tetramethylpentadecane 268 2.78E-10 5.60E-09 3.70E-09 1.02E-09Squalane 316 5.24E-10 4.88E-09 3.85E-09 1.20E-09Heptadecylbenzene 422 9.99E-11 1.46E-09 7.32E-10 3.45E-10Stearyl alcohol 270 2.11E-10 3.04E-09 1.09E-09 8.19E-10Behenyl alcohol 326 * * 2.16E-10 1.69E-102-(2 -hydroxy-5 -methyl- 225 1.51E-10 * * 1.36E-09

phenyl)-2H-benzotriazolDi-2-ethylhexyl)-phthalate 390 3.77E-11 * * 1.36E-102,5-Bis-(5-tert.-buthyl- 430 4.15E-11 * * 9.33E-11

benzoxazol-2-yl)-thiophene

* No exploitable concentration pro® le data.

Di usion test

The method proposed by Moisan in 1980 has beenimproved as follows:

Stacks of 50 ® lms are stored at 40ë C under pressure(25 kg/cm2 ) for 3 days. Then the stack is put incontact with a source of additives. This source ismade with PE wax (AC617A, Allied: powder, density0.91, viscosity at 140ë C 180 cps) and contains twosubstances of the panel. The blend is made in a mouldabove the wax melting point. If the product is nothighly soluble in the wax (concentration of additiveused: 5% ), no tests are performed.The pressure during the di usion test is 0.5 kg/cm2 .After a given contact time at 40ë C (which has to belong if the di usion coe cient of additive is low),® lms are separated, extracted for 1 day by dichloro-methane at room temperature, and the extract isanalysed by GC/FID. The di usion coe cient iscalculated from the concentration pro® le in the stackthickness.

Analytical determinations

Ð CG 8000 Series (FISONS) with:Ð DB5MS column (15 m, 0.32 mm i.d., 0.25 m m,J&W Scienti® cs)Ð FID detectorÐ Temperature programme: isothermal for 1 min at70ë C, and a ramp at 15ë C/min to 320ë C.Ð Helium: 41 cm/sÐ Internal standard: tetradecane 200 ppm

Mathematical treatment of data and modelling

Calculation of a di usion coe cient by the Moisanmethod

The di usion coe cient is calculated from the ® t ofthe experimental concentration pro® le in the thick-ness of the stack, using equation (2) (Moisan 1980):

Qe /Q t 1p

K e1 Erf

xK

dx 2

with:

K 2 DtQe total quantity of product which has di used in

the thickness e during time tQt total quantity of product which has di used in

the stackD Di usion coe cientt di usion time

Equation (2) is valid if there is no kinetic limitationfrom the di usion in the source, which justi® es the useof a PE wax, and of the large concentration of the testsubstance in this wax.

Simulations of migration kinetics

The model used in this paper for migration kineticscan be used in a general case situation: the additivedi usion depends on the simulant concentration inthe polymer at each time and at each distance to thesample surface. At a given time t, at a given distanceto the surface, the di usion coe cient of the simulant(or solvent) is a function of its own local concentra-tion, and the additive di usion coe cient is also afunction of the local food simulant concentration(exponential dependence between concentration anddi usivity is assumed): at each time, a di usioncoe cient is calculated for the additive and for thesolvent at each distance from the surface of the sur-face of the sample. The model takes into account theplasticization e ect of the simulant quantitatively.

Di usion in bulk material is calculated only by fourclassic equations (Crank 1975) which are directly usedwith numerical approximations (no mathematicalresolution):

dCAdt

DA ¶2CA

¶X2dCSdt

DS ¶2CS

¶X2Fick

3

4

DA DA0 eBA CSCSeq

DS DS0 eBS CSCSeq

Simulant concentration 5

dependent dif fusivities 6

where:

CA local concentration of the additive in anelementary volume in the bulk material

139Prediction of worst case migration

CS = local concentration of the simulant in anelementary volume in the bulk material

CSe q = concentration of food simulant in thepolymer at equilibrium

DA = di usion coe cient of the additive in anelementary volume in the bulk material

DS = di usion coe cient of the additive in anelementary volume in the bulk material

BA, BS = constants expressing the sensitivity of dif-fusion coe cient versus the simulant con-centration

The original approach in equations (3) ± (6) is therelationship between DA (connected to the additive)and CS , the food simulant concentration.

At the interface, kinetic limitations may occur. Forthe simulant, the concentration may not reach instan-taneously the solubility at the surface of the polymer(Ce q ) (Vergnaud 1991). The concentration on thesurface (CSx = 0 , t) then changes at a rate which isproportional at each time to the di erence: CSe q -CSx = 0 , t . For the additive, the same phenomenonoccurs; but in this case, the concentration to bereached is not constant in time. Indeed, at each time,the additive on the surface of the material tends to bein equilibrium with the additive in the simulant. Thetarget equilibrium concentration CAF,t /KF /P (KF / Pbeing the partition coe cient) then increases whenthe migration proceeds. The proportionality coe -cient, i.e. the rate of dissolution (hA for the additive),or the rate of sorption (hS for the simulant) variesfrom zero to in® nity. This h coe cient is sometimescalled convection factor (Vergnaud 1991). When thevalue is high, di usion is the only limiting phenom-enon.

DS ¶CSx 0,t¶X hS CSeq CSx 0,t 7

DA ¶CAx 0,t¶X hACAF,tKF,P

CAx 0,t 8

At this stage, the model can only be applied topolyole® ns, i.e. rubbery polymers at room tempera-ture. In the case of glassy polymers, a kinetic e ectshould also be taken into account in that di usioncoe cient increases with food simulant concentration(relaxation of macromolecules can then not be con-sidered immediate, and equations (5) and (6) havethus to be time dependent).

A complete presentation of the principle of the soft-ware will be presented in another publication.

Results and discussion

Determination of the di usion coe cient; upper-bound di usion coe cient; worst case migration

In Piringer’ s approach (equation (1)), di usion coe -cients were determined by a statistical correlation,plotting di usion coe cients taken from literatureagainst the molar mass of the migrants. Surprisinglywhen this was tested for comparison of estimatedmigration values against real data, many migrationvalues in fats were overestimated by a factor up to104 , while others (roughly 40% ) were strongly under-estimated (Baner et al. 1996).

We therefore decided to optimize the determinationof di usion coe cients at 40ë C (the most importantconditions for testing) for PP, LDPE, LLDPE and forHDPE, which are among the most common materialson the market. In order to determine D withoutinterference of a swelling liquid like fat or a solvent,we used the ® lm to ® lm method elaborated byMoisan. In this method, additives di use from asource into a stack of ® lms maintained under press-ure, and considered equivalent to a single material(® gure 1). After a given time, the individual ® lm layersare separated, and the additive is determined in each


Figure 1. Cell used for the measurement of the di usioncoe cients.

layer, thus reconstructing the concentration gradient(® gure 2). D is obtained from mathematical treatmentof these data. Equation (2) implies a constant con-centration of the migrant in the source, and a non-limiting migrant di usion at the interface with the® rst ® lm. Moisan solved this problem by using asource ® lled with additives in amounts exceeding theirsolubility. But in such sources, phase separations andpartial crystallization of additives were likely to oc-cur, and it is not surprising to observe in Moisan’ sArrhenius plots a discontinuity in the range of themelting point of the migrants. Rather than using thesame polymer for the stack and for the source, weused a PE wax as a source, whatever the ® lms. In thisway, the conditions for the validity of equation (2) arebetter ful® lled. In order to avoid plasticization of the® rst ® lm layer, the lower molar mass compounds ofthe wax were extracted by dichloromethane beforeuse (5 ml dichloromethane/g wax; three extractionsfor 30 min at room temperature by immersion understirring). After this procedure, no peak linked to thewax appears in the GC chromatograms of the extractsof the ® lms directly in contact with the source.

The main advantages of this method are that D valuesare obtained in a short experimental time (without

having to wait for equilibrium), and that it avoids theuse of solvents which could induce both swelling ofthe polymer and a partition at the interface. The Dvalues of the compounds studied are shown in table 1.A representation of log D versus M is shown in ® gure3(a, b, c, and d for HDPE, PP, LDPE, and LLDPErespectively).

In order to compare our results with equation (1),we have drawn on these ® gures the D values asstraight lines calculated from equation (1). Itturns out that measured D values are higher thanthe `worst case’ values given in the literature (Piringer1997). This leads us to propose revised values of AP(table 2).


Figure 2. Two examples of pro® les obtained with 50 ® lms of L DPE (25.5 m m) with ( s ) pentadecane (15.75h ofdi usion) and (D ) 2,6,10,14-tetramethylpentadecane (145.16h of di usion).

Table 2. Values of Ap constants of equation (1) at40 ë C.

PP HDPE LDPE LLDPE

Measured A p (this work) 6.5 8 9 9Statistical determination 5.4 5.4 9 9

of A p (calculated fromPiringer 1997, 1998)

This is in fact the weak point of Piringer’ s approach.Instead of determining D values in a homogeneous setof data (as we have done in this work), the authorshave taken many di erent data found in litera-

ture. Most likely these data came from very di erenttypes of experiments (permeation, migration withsometimes a single experimental determination), andwere obtained from di erent mathematical treatments.


(a)

(b)


(c)

(d)Figure 3. Correlation cards between molar mass and di usion coe cient of linear alkanes ( s ) and other substances ( d )in HDPE (a), PP (b), L DPE (c), and L L DPE (d). Straight lines represent values calculated with equation (1) at 40 ë C;dashed lines (a and b) represent values calculated with equation (1) with modi® ed constants.

Determination of D from molecular properties;possible application to QM± SML relationship

The revised equation (1) applies when only themolar mass of the migrant is known. This can oftenbe obtained using liquid chromatography coupled tomass spectrometry. Can di usion coe cients becalculated with a better approximation when moreinformation about the chemical structure is avail-able? The molar mass of migrants is perhaps notthe best variable to take into account; the use ofVan der Waals volume may no longer be adequate. Infact, for low molecular mass species (gases), Berensand Hopfenberg (1982) showed (in the case of apolystyrene matrix) that a direct relationshipbetween the Van Der Waals volume and di usivitycould be de® ned. In the case of molecules with ahigher molecular mass (up to 1200 g/mol for anadditive) the use of the Van der Waals volume asthe relevant variable should lead to the samedispersity as observed with the molar mass in ® gure 3.For instance, isomers may have very di erent di u-sion coe cients (Scott 1988), and it is oftensaid that the shape of the molecule (linear, spherical)and its ¯ exibility can play a major role. Other authorsused the cross section of the di using molecule as

relevant parameter (see discussion by Limm andHolli® eld 1996).

In this work, it can be observed on ® gure 3(b) thatlinear alkanes are the faster di using substances in PPfor a given molar mass. We have investigated thepossibility of taking into account both the molar massand the shape by using as a variable the volumeneeded by the molecule for its displacements in thepolymeric matrix. This volume is obviously largerthan the Van der Waals volume. In the ® rst approachdescribed here, we used the volume of the smallestparallelepiped containing the molecule, in a prefer-ential conformation.

Two examples are shown in ® gure 4. Themolecular masses of bis(2-ethylhexyl)-phthalateand of triphenylmethane are corrected with a factorobtained by the ratio of the whole volume of themolecule (as de® ned on ® gure 5) and the volume ofthe linear alkane which has the same molecularmass; the other points are linear alkanes (the correct-ing factor for linear alkanes is of course 1). It can beobserved that this type of correction shouldsigni® cantly reduce the dispersity of the relationshipbetween D and M: the two additives have thesame di usivity as linear alkanes which involvethe same volume for di usion. If QM± SML relation-


Figure 4. Correlation card between molar mass and di usion coe cient of linear alkanes ( s ) in PP; triphenylmethane( j ) and di-(2-ethylhexyl)-phthalate ( r ) are represented too, using their real and corrected molar mass.

ships of known migrants can be calculated, these resultsopen the way for an approach to be used for known andwell identi® ed migrants.

It can be envisaged that linear alkanes are the fastestdi using species for a given molar mass; their di u-sion coe cients could be used as D . This is certainlyveri® ed in the case of polypropylene (cf ® gure 3(b)),

but not in the case of polyethylenes (cf ® gure 3(a, c,d)).

With PE, the expected di usion coe cient (consider-ing only a di usion mode by reptation) might bedecreased by speci® c interactions linked to ordering/disordering phenomena (linear alkanes are struct-urally perfect oligomers of polyethylene). This will


(a) (b)

(c)

Figure 5. Conformations of a linear alkane (a), triphenylmethane (b) and di-(2-ethylhexyl)-phthalate (c) used for thecalculation of the corrected molar mass.

be discussed in another paper (involving results forhigh molecular mass compounds).

Introducing food and packaging interactions

Interaction of food (or food simulant) with the poly-mer probably modi® es both the rate and the amountof migration. This phenomenon, called Class II mi-gration (Naylor 1989), or Type III migration (Figge1980, Adcock et al. 1984, Lum Wan et al. 1995), hasbeen treated by di erent mathematical approaches.

In the simpli® ed worst-case model, it has been as-sumed that molar mass plays the major role. It isconsidered that migrants whose molar mass is lowerthan that of triglycerides are not a ected by thisinteraction (Piringer 1990). For olive oil, the limithas once been set at 500 g/mol (Piringer 1990) andlater at 800 g/mol (Piringer 1997, 1998). For sub-stances with a molar mass of more than 800 g/mol,the di usion coe cient is calculated with a molarmass equal to 800. This is equivalent to consideringthat species having a di usion coe cient lower thanthat of olive oil di use in the worst case at the samerate. Following this presentation of di usion, olive oilacts as a `washer’ and only on species of molar massless than 800 g/mol. We do not agree with these twoa rmations. This is an oversimplifying assumptionwhich may be in contradiction with the requirementsof a worst case approach.

It is well known that oil acts as a plasticizer. Only theinteraction between oil and polymer has to be con-sidered, the direct interaction between oil and addi-tives in the polymer being less probable and leading tominor e ects. The consequence of this is that oil actsas a plasticizer to the same extent for all the sub-stances, even with M< 800 g/mol.

In order to describe food and packaging interaction,we have developed a computer program, based onnumerical analysis. The plasticizing e ect of triglycer-ides is taken into account by introducing di usioncoe cients of the migrant (DA) and of the simulant(DS) which depend on the local concentration of oliveoil (equations (5) and (6)). The experimental determi-nation of the swelling factors BA and BS will soon bepublished. In this work we have used worst caseassumptions and literature data for the calculation:

Ð DS = 7.4 10 ± 1 1 in PP (Riquet et al. 1998); DAvalues used are calculated from equation (1) (sub-

stances of molar masses of 200, 300, 400, 500, 600,700 and 800 g/mol, in PP).

Ð It is assumed that olive oil increases all di usioncoe cients in the same way by one order of magni-tude when sorption equilibrium is reached:BA = BS = 2.3.

Figure 6 shows the simulation of olive oil di usion inpolypropylene. A di usion coe cient equal to7.4 10 ± 1 1 has been taken from Riquet et al. (1998).According to equation (1), this DS value correspondsto a substance whose the molar mass is near450 g/mol; while the average molar mass of olive oilis 960 g/mol! This remark underpins that Ap valueshave to be validated by new experiments.

Using concentration dependant di usivities (equa-tions (5) and (6)) migration of substances of molarmasses of 200, 300, 400, 500, 600, 700 and 800 g/molhave been calculated. Comparing ® gures 6 and 7, itcan be observed that a classic Fickian modelization(with constant di usion coe cients) leads to verydi erent kinetic shapes. When the plasticization e ectis taken into account, di usion coe cient increasesuntil complete sorption of oil, i.e. during all thekinetics for substances of molar mass less than450 g/mol.

For substances heavier than 450 g/mol the beginningof kinetics can be ® tted with a constant di usioncoe cient ranged between DA0 and DA with a verygood approximation (® gure 8)! Of course, for highermigration times, the model is very bad. This shouldexplain why some authors occasionally obtain verygood ® ts of migration kinetics, without taking intoaccount plasticization e ects, but considering only theinitial stage of the kinetics.

In order to determine real maximum values of BS andBA (2.3 is an arbitrary example), the authors willpresent measurements of di usion coe cients inswollen polymers in a further publication.

Kinetics limitations at the interface

How far can a single measurement at t = 10 days beextrapolated to predict migration over the entireshelf -life, even with a worst case safety margin? Inthe early stages of di usion (which often includest = 10 days), several phenomena tend to lower theapparent di usion coe cient:



Figure 6. Simulation of sorption kinetic of food simulant (dashed line) and migration kinetics of substances of molar massranged from 200 to 800 g/mol (full lines). Constants used : BA = BS = 2.3; thickness = 184 ¹m; DA is calculated withequation (1); DS = 7.4 10 11 cm2/s.

Figure 7. Same as ® gure 6 but without plasti® cation e ect (BA = BS = 0).

Ð plasticization of the material by the food graduallyincreases during sorption, and the di usion coe -cients of all compounds increase;

Ð kinetic limitations at the interface: sorption of thesimulant and desorption of the additive.

These phenomena can be highlighted using simulatedmigration curves, using equations (7) and (8) inaddition to equations (3) ± (6). Two examples are givenon ® gures 9 and 10:

Ð Figure 9: kinetics limited by additive dissolution;only additive concentration is presented because noplasticization e ect is considered. A time dependentdecrease of surface concentration of the additive isobtained ( ® gure 9a): the consequence is that thekinetics obtained are slowed compared to a classickinetics ( ® gure 9b).

Ð Figure 10: kinetic limitation by a plasticizing oil: atime dependant increase of surface concentration ofthe additive is obtained ( ® gure 10a); as the sorption ofthe simulant and the desorption of the migrant are

linked by plasticization e ect, the desorption of theadditive is also slowed down ( ® gures 10b and 10c).

The observation which has to be underlined is that themigration at 10 days can be sometimes not character-istic of the whole migration process (if di usion is notwell established, due to the phenomena mentionedabove). In these conditions, a comparison between anestimated value and an experimental one for suchshort contact times will easily lead to a false conclu-sion: an apparent worst case prediction (as the appar-ent value of D is lower during the beginning ofmigration, and as this is not necessarily the case forhigher contact times). The consequence is that thevalidation of prediction models should have to be madewith long contact times.

If the aim of the model is to estimate migration forshort contact times (as de® ned by regulations!), per-haps equation (1) (with corrected constants) answersthis question.

For higher contact times other models would have tobe applied.


Figure 8. Same as ® gure 6 for migration kinetic of a 600 g/mol substance (dashed point curve). The other kinetic is builtwithout consideration of plasti® cation e ect; the di usion coe cient is chosen to ® t an hypothetical experimental value at10 days (DA = 4.9 1011 cm2/s).


(b)Figure 9. (a) Pro® les of additive in the sample thickness, considering kinetic limitation of additive dissolution at the inter-face. Constants used: DA = 1.77 10 11 cm2/s, thickness = 184 ¹m, hA = 10 8 cm/s, BA = BS = 0 (no plasti® catione ect). (b) Comparison of the migration described on ® gure 9a and the same one without kinetic limitation (hA = ).

(a)


(a)

(b)

Future ® eld of investigation

It is false to say that there is only one activationenergy for di usion of all existing substances, in avery large range of temperature. However the pro-posal of a general law covering the worst case situa-tions (equation (1)) is attractive. Two remarks have tobe made:

A larger panel of very di erent shapes of molecules,exhibiting di erent chemical properties should beused (not only common additives but also referencesubstances or surrogates) to validate equation (1).

Malaika et al. (1991) observed a variation of acti-vation energy of di usion coe cient in LDPE inthe range 50± 60ë C. Since the phenomenon appearswhatever the migrant studied; it can be assumedthat it is connected to a transition of the polymer(and not to additive properties) in this temperaturerange. A more realistic model should involve two

di erent temperature ranges from room tempera-ture to 50ë C, and from 50ë C to 100ë C (or more forpolypropylene ).

Conclusion

Di usion coe cients of representative migrants havebeen determined by ® lm to ® lm experiments. It ap-pears that worst case di usion coe cients cannot bealways obtained by published coe cients of equa-tion (1).

Concerning worst case prediction of migration, theauthors propose to improve the constants of thePiringer model for LDPE and LLDPE at 40ë C andto use models involving a variation of di usioncoe cients taking into account food and polymerinteractions. This implies establishing correlation


(c)Figure 10. (a) Pro® les of food simulant in the sample thickness, considering kinetic limitation of food simulant sorption atthe interface. Constants used: DA = 1.77 10 11 cm2/s, DS = 7.4 10 11 cm2/s, thickness= 184 ¹m, hS = 10 8 cm/s,BA = BS = 2.3 (in order to link migration to the sorption of food simulant). (b) Pro® les of additives in the samplethickness, in the same conditions as in ® gure 10a. (c) Comparison of the migration described on ® gures 10a and 10b andthe same one without kinetic limitation (hS = ).

charts between experimental di usion coe cients andmolar mass, both with swollen and not swollen poly-mers, as this has been done with virgin polymers.

We still need better understanding concerning activa-tion of di usion phenomenon (particularly on in-¯ exion points observed in the range 50/60ë C onArrhenius plot for several polyole® ns) and kineticlimitations at the surface.

The authors are proposing also a new approachwhich requires experimental validation: a correlationbetween di usion coe cients and additive volume(instead of using molar mass).

Acknowledgements

This work was performed in the ® eld of a researchprogramme on food and packaging safety supportedby Europol’ Agro. The Conseil GeÂ neÂ ral de la Marneand INRA are acknowledged for a PhD grant (forA.R.). Special thanks to Yves Germain from CER-DATO for supplying special polymers.

References

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