Diffusion Coefficients of Additives in Polymers.I. Corrélation with Géométrie Parameters

iALAIN REYNIER, PATRICE DOLE, STEPHANE HUMBEL, ALEXANDRE FEIGENBAUM

INRA SQuAIE CPCB Moulin de la Housse, BP 1039, F 51687 Reims Cedex 2, France

Received 12 September 2000; accepted 29 December 2000Published online 19 September 2001; DOI 10.1002/app.2093

ABSTRACT: Diffusion coefficients of a broad range of molécules with molecular weightranging from 100 to 800 g/mol hâve been measured in polypropylene, by solid/solidcontact methods (without liquid contact), at 40°C. The behaviors of thé différentmolécules are compared to those of linear alkanes. The diffusion coefficients are corre-lated to parameters describing size, shape, and flexibility of thé molécules. The conceptof weighted fractionated volume is introduced using molecular modeling. It enables théclassification of thé molécules according to modes of molécule displacement (crawling,jumps, or dual mode). © 2001 John Wiley & Sons, Inc. J Appl Polym Sci 82: 2422-2433, 2001

Key words:ated volume

diffusion coefficient; volume; additive; shape factor; weighted fraction-

INTRODUCTION

If prédiction of thé diffusion coefficient D of anorganic molécule in a polymer could be achieved,it would be a very useful tool in areas like food,drug, and cosmetics packaging, and textiles. Sev-eral corrélations hâve been proposed, mainly em-pirical, between thé Log of D and thé molecularsize. They emphasize thé importance of molecularsize, but do not allow a précise understanding ofthé influence of changes in molecular structure.From corrélation in homologous séries, it hasbeen recognized a long time ago that thé largerthé size of thé organic molécule, thé lower itsdiffusion coefficient. In thèse homologous séries,thé molecular size can be expressed by thé molec-ular weight,1"3 thé van der Waals volume, thé"diameter of thé molécule" (calculated from thémolar volume, assimilating thé molécule to asphère),4 or thé length of thé molécule.5 However,thé size parameters do not describe phenomenathat hâve also been shown to influence thé dis-

Correspondence to: Alexandre Feigenbaum.Journal of Applied Polymer Science, Vol. 82, 2422-2433 (2001)© 2001 John Wiley & Sons, Inc.

2422

placement of a molécule in a polymer: its mini-mum cross section,6 its shape,7"9 its interactionwith thé polymeric matrix,9 its flexibility.10 Suchchanges in thé backbone cause déviations fromthé corrélations by several orders of magnitude. Itis only estimated that for a given molecularweight, linear molécules would diffuse faster thanothers, spherical molécules being thé slowest.

Recently a compilation of hundreds of pub-lished values of D including thé molecular weightwas made.11 It shows a large scatter of values.The conclusion was that it was only possible todefine an upper value of thé diffusion coefficient Dfor a given molecular weight of thé diffusing mol-écule. Such an upper value of D could be useful toestimate thé worst case diffusion of additivesfrom packaging to food. However, thé authorsconcluded that thé scatter could at least in part beattributed to thé large number of différent expér-imental procédures used. Many data are based ondiffusion into a solvent, but this solvent may in-teract with thé polymer, and influence thé resuit.Furthermore, it should be noted that there areonly few, if any, data available with high molec-ular weight compounds. Obviously high molecu-lar weight compounds could hâve a strong influ-

DIFFUSION COEFFICIENTS OF ADDITIVES IN POLYMERS. I 2423

ence on thèse Log D = f[M) corrélations. In apreliminary study we undertook,12 we hâve con-cluded that thèse data were thé most difficult toobtain, as they require very long experiments andspécifie expérimental procédures.

Since we are involved in a program on foodsafety related to migration of additives from pack-aging, we decided to undertake a study that couldfill thèse gaps. We therefore decided to measurediffusion coefficients of additives and of moléculesselected on thé basis of their molecular structureand functional groups, including compounds up tomolecular weight 800 g/mole. Our studies are con-ducted without any suivent in order to improvethé consistency of thé results.

A rationalization of thé results was then un-dertaken, taking into account possible mecha-nisms of displacement of molécules in polymericnetworks.

MATERIALS AND METHODS

Diffusing Substances

Molécules of molecular weight ranging from 500to 807 are added (cf. Table I) to thé previouslystudied set12; thèse molécules were chosen to dis-play a broad variety of chemical structures:

• différent molecular weights,• linear/nonlinear alkanes, !• flexible/rigid molécules, and• polar/nonpolar molécules. j

Common polymer additives are also added tothé set.

Polymer

Polypropylene (PP) (supplied by CERDATOFrance/ELTEX PHV001PF) was tested for diffu-sion properties. The melting point was deter-mined by differential scanning calorimetry (DSC)(TA Instruments 2920) using onset point of endo-thermic peak. |

PP films: 54 pim thick, Tm = 138°C (onset)

This polymer contains a very low amount of ad-ditives which does not interfère with diffusingsubstances.

Measuring Diffusion Coefficients

An original test (trilayer test) was defined to mea-sure very low diffusion coefficients.

A virgin film is pressed at 40°C between twofilms of thé same thickness containing thé testedadditive. If thé concentration in thé two latéralfilms is lower than thé solubility at testing tem-pérature, thé equilibrium quantity in inner layerequals 1/3 of thé total quantity (in thé threefilms). So it is not necessary to run thé kineticsuntil equilibrium. The diffusion coefficient is ob-tained by fitting thé expérimental curve (relativequantity of thé inner film versus contact time)with a Fick's law resolution program: we assumethere an instantaneous mass trarisfer at inter-faces, constant diffusion coefficient, and homoge-neous additive repartition in latéral films.

To détermine thé relative concentration in théinner layer each films of thé trilayer stack isdichloromethane extracted and quantified by gaschromatography/flame ionization detector (GC/FID).

In order to obtain suitable concentration films(homogeneous and lower than solubility concen-tration), thé following procédure is chosen: filmsare immersed for 5 days at 40°C in a saturatedsolution of thé diffusing molécule in hexane. Hex-ane is desorbed in a ventilated oven for 1 day at80°C. The previous experiments lead to concen-trations in thé films that were too high: bloomingoccurs after annealing at 80°C. Films are thenwashed with ethanol (a nonswelling solvent), andplaced under argon at 100°C for 5 days betweentwo virgin films. Thèse last two films are used inthé diffusion test.

The trilayer test was designed to déterminevery low diffusion coefficients, without having towait for equilibrium, as thé plateau value can bepredicted [Fig. l(a)]. However, for some samplesreaching thé plateau, thé quantity at equilibriumin thé inner film was not equal to thé theoretical1/3 of thé total quantity [Fig. l(b)]. Although pre-cautionary measures were taken, thé procédure ofadditive filling in external films (cf. above) leadsto a concentration higher than thé solubility at40°C (certainly because thé solubility at 100°C isin some cases much higher than thé solubility at40°C). If thé plateau is known, thé diffusion coef-ficient is calculated considering its expérimentalvalue.

But when thé experiment is not conducted to théplateau [Fig. l(c)], it may not be correct to extrapo-late to 33%. An error bar on thé diffusion coefficient

2424 REYNIER ET AL.

0 500 1000 1500 2000 2500 3000 3500

Square root of time (s"!)

(a)

? 0.33oo1

500 1000 1500 2000 2500 3000

Square root of time (s"2)

(b)o

5000 10000 15000

Square root of time {s1'2)

(c)

Figure 1 Example of calculation of diffusion coeffi-cients with trilayer test, (a) Irganox 1076: (•) expéri-mental values; (—) calculated curves with D = 7X 10~12 cm2/s. (b) Tetracontane: (•) expérimental val-ues; (—) calculated curves withD = 1 X 10'11 cm2/s. (c)Irganox 1330: (•) expérimental values; (—) calculatedcurves with D = 1 X 10 "^ cm2/s; (-) calculated curveswith D = 4 x HT13 cm2/s.

is obtained considering that thé real plateau is in-cluded between thé lowest extrapolable plateau(leading to thé highest possible value of D), and thémaximum concentration possible (33%, leading tothé lowest possible value of D).

The data used in this work are partially issuedfrom previous publication of authors.12 In thispaper, diffusion coefficients were obtained usingthé "stack method": D is calculated using thé fit ofdiffusion profile in thé thickness of a stack offilms. The method is précise, expérimental varia-tions on D are below a factor two. An example offitted concentration profile is given Figure 2.

Analytical Détermination

Additive concentrations in polymer films intrilayer tests are gas chromatography (FID) de-termined: thé film is extracted by immersion indichloromethane (1 day at 40°C), and this extractis analyzed by GC.

Injecter: On-column (2 /iL) T = 40°CDétecter: FID T = 400°CColumn: DB5-HT (J&W Scientifics) 15 m—

0.32 mm i.d.-O.l /nmVector Gas: He at 2 mL/minTempérature program:

Injection at 40°CIsothermal température: 2 minRamp at 15°C/minUntil 400°C

Calculation of Whole Molécule Volume

Optimization of thé Molécule Conformation

The conformation searches were performed usingMacromodel (Macromodel 6—Colombia Univer-sity) in a stochastic investigation. Most of théstructures were minimized with thé MM3 forcefield, except for Tinuvin P: AMBER was used tohandle thé N—N—N containing cycle. The moststable conformation of each molécule studied waschosen as a référence for volume calculations,even if in thé polymer matrix conformation is at ahigher energy state. Moreover, thé energy of amolécule during displacement must be higher.

A program was created for thé déterminationof thé smallest parallelepiped volume, which con-tains ail thé atoms of a molécule or of a moléculefragment.

The following steps are taken: \

• There is a 90° rotation by steps of 1° aroundx axis: to détermine thé orientation leadingto thé lowest rectangle surface by projectionon yz plane. This orientation is kept for théfollowing steps.

• Same procédure as before for y axis (projec-tion on xz).

• Same procédure as before for z axis (projec-tion on ry).

• Calculation of thé parallelepiped volume.• Thèse four steps are repeated until thé con-

stant volume of thé parallelepiped is ob-tained (< 0.1%).

y mm

DIFFUSION COEFFICIENTS OF ADDITIVES IN POLYMERS. I 2425

40 45 50

Film nurnber

Figure 2 Diffusion profile of squalane after 26 days of contact at 40°C with apolypropylene stack (50 films of 54 /xm). (•) Expérimental values; (—) theoretical profilecalculated with D = 9.9 X lO^11 cm2/s.

In ail steps of thé calculation, thé real dimen-sions of thé atoms are considered (atoms are notrepresented as points, but as sphères; thé dimen-sion of thé van der Waals radius is used).

Examples

Four examples of whole volume déterminationare given Figure 3.

RESULTS AND DISCUSSION

Previous data12 together with thé results of théprésent study are given in Table I.

Behaviour Croups

When considering thé expérimental errors, thédiffusion coefficients of linear alkanes can be

Figure 3 Examples of whole volume calculations. (A) Undecane, (b) tripalmitin, (c)tinuvin P, and (d) Chimasorb 81.

2426 REYNIER ET AL.

Table I Diffusion Coefficients of Studied Molécules at 40°C in Polypropylene

Linear Alkans Other Molécules Commercial Additives

UndecaneTridecane

2.1 X 10"9a

2.1 X 10"9aTriphenyl méthaneTetramethyl

1.3 x 10~10a

4.9 x 10~10aTinuvin PChimasorb 81

1.5 x I0-10a

1.5 x nr10

pentadecanePentadecaneHexadecaneHeptadecaneOctadecaneDocosane

Tetracosane

OctacosaneHexatriacontaneTetracontane

2.0 x 10"9a

1.3 X 10~9a

1.3 x 10~~9a

8.7 x 10~10a

2.5 X HT10a

5.6 x HT10a

1.8 x 10~10a

2.0 x 10"11

1.0 X HT11

OctadecanolHeptadecyl benzèneDocosanolSqualaneTrilaurin

Tripalmitin

2.1 x 10~10a

5.2 x 10"10a

9.9 x 10~lla

7.0 X 10~12

from 3 X 10" 13

to 2 X 10" 12

DEHPUvitex OBIrganox PS800Irganox 1076Irgafos 168

Irganox 1330

3.8 x 10"lla

4.1 X 10~lla

2.0 X 10"11

7.0 x 10~"12

from I X 10" x

to 4 X 10"13

from 1 X 10"]

to 4 X 10"1

' Studied in previous paper.

bracketed between thé two straight lines of Fig-ure 4. This area can be taken as a référence forthé classification of thé différent molécules, ac-cording to their Log(Z?) and thé différences ofshape and mobility.

Three différent families are thus defmed: (1)diffusion behavior is thé same as that of linearalkanes with thé same molecular weight, (2) dif-fusion behavior is lower than that of linear al-kanes with thé same molecular weight, and (3)intermediate behavior s.

1. The behavior of molécules 1, 3—5, 7, and12 (Fig. 4) are close to linear alkanes: Ailthèse molécules hâve long alkyl chains andno bulky groups. Only heptadecyl benzènehas an aromatic group, but thé backbone isnot substituted. Ail thèse molécules hâve ahigh mobility and flexibility, and theirstructure suggests a crawling diffusionmode, as it is supposed to be thé case forlinear alkanes. It is also logical to obtaindiffusion coefficients close to that of thélinear alkanes which hâve thé same molec-ular weight.

2. Molécules 8, 9, 11, and 13-15 (Fig. 4) hâvetotally différent géométrie characteristics:Their shape tends to be spherical, becauseof

• several aromatic groups (molécules 11,13, 15) or chain branchings on smallchains (molécule 9); i

• rigid heterocycles (molécules 8, 14).

Thèse molécules are supposed to diffuse byjumps (from a free volume to another). Thisdiffusion mode implies an instantaneousdisplacement of thé molécule in its totality(this is not thé case for thé crawling modeof molécules having many degrees of free-dom). Thèse diffusion coefficients of thèsecompounds are lower than those of linearalkanes. The diffusion rate of UVITEX14

should be predicted to be specially low.However, this molécule is close to alkanes,and that is doubtless related to its lowcross section (planar molécule).

Molécules 2, 6, and 10 (Fig. 4) lead to inter-mediate behavior: Two différent cases can beobserved: j

• Some of thèse molécules (2 and 10) hâveboth flexible groups and rigid parts. It is notpossible to décide whether only one or twomechanisms of displacement occur at thésame time in thé same molécule. Thèse mol-écules diffuse by jumps but a part of thémolécule can relax during their displace-ment, as it is thé case for crawling. Thisprésentation of thé diffusion suggests adual mode

• The diffusion coefficient of a primary linearalcohol is lower than that of thé linear al-kane of thé same molecular weight (case ofmolécule 6). Although thé polypropylene isnot a polar matrix, thé polarity seems tohâve an effect on diffusivity. A possible in-terprétation is thé formation of dimers in

DIFFUSION COEFFICIENTS OF ADDITIVES IN POLYMERS. I 2427

C18H37-OH(6)\

-8 T

o

u>o-9 - .

- 1 0 - -

-11 - •

- 1 2 - -

-13 ••

-14100

Po

300 500 7QO 900

M (g/mol)

CH

j C4H, O o C4H,

C12H,5-0

(10) \

(12)

(11)

(13)

Figure 4 Corrélation Log D = /(M) in polyproylene at 40°C. (•) Linear alkanes; (O)other molécules (with formula).

(15)

2428 REYNIER ET AL.

Table II Whole Volumes of Molécules (in A3)

LinearAlkans

CilC13C15C16C17C18C22C24C28C36C40

120140159169179189229249289368408

Other Molécules

Triphenyl méthaneTetramethyl pentadecaneHeptadecylbenzeneSqualaneOctadecanolTrilaurinTripalmitinTinuvin PChimasorb 81DEHPUvitex OBIrganox PS 800Irganox 1076Irgafos 168Irganox 1330

444646404

1440198

30405380

193802880551554

163819621983

thé material by hydrogen bonding betweentwo alcohol molécules.

The corrélation between diffusion coefficientsand molecular weight of thé migrant allows toverify a linear dependence of Log(Z)) = f(M) in ahomologous séries of molécules. Except for thécase of linear alcohols, thé gap between any mol-écule and a linear molécule of thé same molecularweight is only connected to its geometry.

Whole Volume of Molécules

The volume of thé molécule has to be consideredfirst. As discussed previously, thé van der Waalsvolume should give thé same results as thé mo-lecular weight since they are proportional. Butvan der Waals volume is sometimes much lowerthan thé real volume occupied by thé molécule.Our choice was to détermine thé whole volume ofthé molécules using thé following systematic pro-cédure:

• The molécule conformation used is thé sta-blest conformation of thé molécule calculatedusing MACROMODEL.

• The whole volume is defined by thé lowestvolume parallelepiped containing thé molé-cule in thé above conformation.

Whole volumes (V) are given in Table II andthé corrélation with Log(Z)) is shown (Fig. 5). Thedispersion is very high. Compared to thé volumeof linear alkanes, thé whole volumes of other com-pounds are largely overestimated. The followingconclusions are drawn:

• The choice of studying thé molécules in theirmost stable conformation is not représenta-tive of reality. In this conformation, thé mol-écule tends to occupy thé maximum volumein order to minimize repulsion between at-

oO) -9

5

-10

-11 --

-12 --

-13 -

-14

1 O o

O o• o

2000 4000 6000WHOLE VOLUME (À3)

Figure 5 Corrélation Log D = /(Whole volume) in polypropylene at 40°C. (•) Linearalkanes; (O) other molécules.

DIFFUSION COEFFICIENTS OF AUDITIVES IN POLYMERS. I 2429

Table III Fractionated Volume Calculations

Molécule

Linear alkansTriphenyl méthane

Tetramethyl pentadecane

Heptadecylbenzene

Squalane

OctadecanolTrilaurinTripalmitinTinuvin P

Fragmentation

Dimensions of théFragment Box (À)

C TT ann-2n + 2

u

C TT a15rl32

4 X CH4Ci7H36

a

©

2.60 x 2.99 x L^ 4.96 X 9.04 x 9.91

2.60 X 2.99 x 20.512.04 x 2.04 X 2.04

: 2.60 x 2.99 X 23.07: 1.93 X 5.06 x 5.72

C24H50a 2.60 X 2.99 x 32.02

6 x CH4 2.04 X 2.04 X 2.04C18H37OHa 2.62 x 2.99 X 25.343 x CnHag— COO— CH3

a 2.62 x 3.90 x 18.683 x C15H31— COO— CH3

a 2.64 x 3.94 x 23.78nTT b 2.41 x 6.80 X 11.83

FractionatedVolume

(À3)

7.774 x L444

193

235

300

199573742194

Chimasorb 81

DEHP

Uvitex OB

Irganox PS 800°

Irganox 1076

2 x C2H5—COO—C12H25a

SC-oHir: COO Cl oHo-T

2.60 X 2.99 x 11.57 3363.70 x 6.13 X 10.84

3.59 x 6.53 x 10.92 568

1.93 X 5.06 X 5.72

4.95 x 5.12 X 21.75 551

2.61 x 4.11 x 21.20 4732.62 x 2.62 X 2.622.71 x 4.14 X 28.87 6244.96 X 5.88 x 10.28

2430 REYNIER ET AL.

Table III Continued

Molécule

Fragmentation

Fragment

FractionatedDimensions of thé Volume

Box (À) (À3)

Irgafos 168

Irganox 1330

POc 3.05 X 3.95 X 4.274.59 x 7.29 X 8.32

2.55 X 7.25 x 7.79

4.96 x 5.88 x 10.28

887

1043

a Linear conformation.b Stable conformation.c Fractionation linked to thé impossibility to hâve a linear conformation with thé S atom.

-8 T

-10 -

-11 --

-12

-13 --

-14

O O

800 1200FRACTIONATED VOLUME (À3)

Figure 6 Corrélation Log D = /[Fractionated volume) in polypropylene at 40°C. (•)Linear alkanes; (O) other molécules.

DIFFUSION COEFFICIENTS OF ADDITIVES IN POLYMERS. I 2431

oms. In a polymer matrix, diffusion mecha-nisms imply conformations characterized bya higher energy. But it is difficult to givesystematic criteria for thèse molécule défor-mations.

• Considering thé whole molécule volume byinsertion of thé molécule into a box amountsto overestimating thé real volume taken bythé molécule for thé diffusion. The most re-alistic picture is certainly intermediate be-tween thé van der Waals volume and théwhole volume (cf. overestimated box of tri-palmitin in Fig. 3).

• The sole volume criterion leads to consider-ing that thé diffusion of ail thé molécules(except linear alkanes) occurs by jumps,while "alkanes like" and intermediate molé-cules are supposed to diffuse rather by crawl-ing (cf. for example thé case of tripalmitin at5380 À3 in Fig. 5).

Thus thé proposed fragmentation of thé molé-cule volume by considering separately thé mobil-ity of each part of thé molécule.

Fractionated Volume

In each molécule, groups of atoms can belong totwo différent classes:

• Groups with high degrees of freedom: Longand flexible chains that can take several con-formations making thé displacement easy.Thèse volume parts are also calculated tak-ing into account linear conformations (lead-ing to thé lowest possible volume). The boxshape of thé volume is kept.

• Rigid parts: They are generally constitutedby cycles or short chains (for which thé con-formation is locked). Thèse volume parts arecalculated considering steric effects, so usingthé most stable conformation.

The fractionated volume of thé molécule corre-sponds to thé sum of thé différent partial volumesobtained.

The molécules were fragmented as showri inTable III. The corrélation with D is better thanthat obtained using thé whole volume (cf. Fig. 6).

The dispersion increases with thé volume ofthé molécule.

Linear alkanes hâve thé lowest diffusivities fora given fractionated volume. The shape charac-

teristics, and by extension thé diffusion mode areleft out when using fractionated volume.

Shape Factor

The shape of a molécule has an incidence on thédiffusion mode: thé probability for a long moléculeto hâve many degrees of freedom is high, and thisfacilitâtes molécule displacements by crawling.

There are two ways of approaching thé shape ofthé molécules looking at thé following:

• The shape of thé whole molécule. This possi-bility is not in good agreement with thé re-marks made on thé whole volume: thé realshape of thé whole molécule in a polymermatrix during thé diffusion does not corre-spond to thé most stable conformation. Thisreal case cannot be predicted. Moreover thereare many différent conformations corre-sponding to thé real cases.

• The shape of each parts of thé molécule asdefined in thé previous paragraph. This is animprovement of thé fractionated volume ap-proach. This was thé approach chosen.

The cube was chosen as thé référence volumeto describe spherical molécule parts. A shape fac-tor is defined by équation 2, for a part of thémolécule:

9 =Surface of thé parallelepipedic boxSurface of thé cube of same volume

ab + bc + ça(2)

where a, b, and c are thé dimensions of thé lowestvolume box.

The shape factors of each part of thé moléculeare calculated. The shape factor of thé whole mol-écule is thé average of thé shape factors of eachpart weighted according to thé number of atoms(except hydrogen atoms). The results of this cal-culation are given in table 4.

The shape factor increases from 1 (sphericalmolécule incorporated into a cube) to 1.8 (thélongest molécule of thé studied panel: linear C40).Figure 7 shows thé values of <p (represented by thédiameter of thé dots) in a Log(Z)) = /(fractionatedvolume) graph. For a given diffusivity, thé high-est shape factor (linear molécule) corresponds tothé lowest fractionated volume.

2432 REYNIER ET AL.

Table IV Shape Factors (<p) of StudiedMolécules

LinearAlkanes

CilC13C15C16C17C18C22C24C28C36C40

1.291.341.391.411.431.451.541.571.641.771.82

Other Molécules

Irgafos 168Triphenyl méthaneIrganox 1330DEHPChimasorb 81Uvitex OBTinuvin PIrganox 1076Tetramethyl pentadecaneTrilaurinHeptadecylbenzeneIrganox PS 800TripalmitinSqualaneOctadecanol

1.031.051.071.111.131.211.251.311.311.331.361.361.411.461.47

The product of thèse two parameters (y X frac-tionated volume) gives a "weighted fractionatedvolume" that leads to an acceptable corrélationwith Log(Z>) (cf. Fig. 8): most of thé dots thusobtained are located in thé area defined by linearalkanes.

The molécules having a différent behavior areas follows:

• Octadecanol: We hâve seen that its particu-lar behavior may be connected to thé forma-tion of dimers associated by hydrogen bonds.

Tinuvin P: The corrélation proposed consid-ers indirectly thé effect of thé rigidity—bothby thé shape factor and by thé calculation ofthé volume from molécule groups that canmove "independently" from thé other parts ofthé molécule. Such rigid structures (rigid butnot spherical) are imperfectly described bythis model.

CONCLUSION

Diffusion coefficients of a broad set of molécules inPP at 40°C hâve been correlated to various mo-lecular parameters: thé molecular weight and anefficient diffusion volume. In order to evaluate théinfluence of thé flexibility and of thé mode ofdisplacement, thé empirical concept of "weightedfractionated volume" was introduced. Even if thécorrélations are satisfactory, thé approach is notdesigned to provide tools nor équations to predictD from molecular parameters. It rather aims at abetter understanding of thé mechanism of dis-placement in polymeric matrixes: two main mech-anisms seem to be effective: crawling, based onthé large number of degrees of freedom of longalkyl chains, and jumps for rigid molécules. In-termediate behaviors are obtained. They can bedescribed by jump displacements facilitated bythé more or less easy relaxation of other parts ofthé molécule.

Température and swelling of thé polymer bysolvents and foodstuffs are likely to change thé

jr -s"e

O)o

-10

-11 --

-12 --

-13

-14

Ç>0 o

s° O O

400 800 1200

FRACTIONATED VOLUME (A3)

Figure 7 Eepresentation of thé shape factor ip (size of point proportional to tpz) onLoglD) = /(Fractionated volume) corrélation. (•) Linear alkanes; (O) other molécules.

DIFFUSION COEFFICIENTS OF ADDITIVES IN POLYMERS. I 2433

-8 T

D)O

-10 --

-11 --

-12

-13 --

-14

TINUVIN P

400 800 1200PONDERED VOLUME (A)

Figure 8 Corrélation Log D = /(Pondered volume) in polypropylene at 40°C. (•)Linear alkanes; (O) other molécules.

corrélations proposed hère. Thèse two factors arestudied in Part II of this paper.

This work is included in a program about migrationmodeling supported by Europol'Agro. AR acknowledgesINRA and Europol'Agro for a Ph.D. grant.

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