Interfacial mass transport properties which control

the migration of packaging constituents into foodstuffs

O. Vitrac*, A. Mougharbel, A. Feigenbaum

INRA, UMR Fractionnement des Agro-Ressources et Emballage, Moulin de la Housse,

51687 Reims cedex 2, France

ABSTRACT

Interfacial mass transport properties, such as partition coefficients (K ), mass transfer coefficients (h )

and diffusion coefficients (D ) were estimated from controlled desorption kinetics at 40°C, which

would be realistic of extreme conditions of use of the packaging materials in contact with a liquid food.

The experiments were carried out on formulated pieces of low density polyethylene (LDPE) with

variable thicknesses (50, 100 and 150 µm) dispersed in ethanol, in controlled conditions of stirring,

with Biot mass number between 5 and 103. The 3 transport properties for 8 homologous molecules (n-

alkanes and n-alcohols), 2 antioxidants and 2 fluorescent tracers were estimated from a set of 78

desorption kinetics. Concentrations in ethanol samples were measured by GC-FID. The uncertainty

related to analytical errors was quantified by Monte-Carlo sampling consisting in adding an arbitrary

white noise to results in the range of experimental errors. The physical interpretation of the external

transport resistance is finally discussed.

Keywords: mass transport properties, packaging, diffusion, parameter identification

*: Corresponding author. Tel.: +33.3.26.91.85.72; fax: +33.3.26.91.39.16

E-mail address: olivier.vitrac@reims.inra.fr

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Notations

A surface area of liquid in contact with the polymer phase (m2)

Bi Biot mass number (-)

Pc concentration in the polymer phase (kg.kg-1)

Lc concentration in the liquid phase (kg.kg-1)

D diffusion coefficient (m2.s-1)

Fo dimensionless time or Fourier number (-)

h mass transfer coefficient (m.s-1)

i sampling index

j mass flux density at the interface (kg.m-2.s-1)

*j dimensionless mass flux (-)

K partition coefficient ( eq eqL PC C ) (-)

L dimensionless dilution factor ( P P

L L

ll

rr× ) (-)

Ll characteristic length scale of the liquid phase (m)

Pl characteristic length scale of the polymer phase (m)

M number of samples used in the identification procedure

P vector of parameters to be identified

Lq liquid mass flow rate (m3.s-1 )

DR global resistance to diffusion (s.m-1)

Re Reynolds number (-)

HR mass transport resistance at the interface (s.m-1)

t time (s)

it sampling time (s)

u ² dimensionless concentration in the solid phase (-)

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v dimensionless concentration in the liquid phase (-)

LV volume of the liquid phase at time t (m3)

x spatial coordinate (m)

*x dimensionless coordinate (-)

Greek letters and symbols

α Critical exponent

( )xd Dirac function

l2 positive constant used in the numerical identification

$x estimated value of x

expx experimentally assessed value of x

c2 distance criterion

Lρ density of the liquid phase (kg.m-3)

Pρ density of the polymer phase (kg.m-3)

1. INTRODUCTION

Constituents of packaging materials (additives, monomers etc.) can diffuse into food products.

Significant concentrations of packaging substances with potential health concern were mainly identified

in foods packed in plasticized PVC cling-films (Harrison 1998, Sharman et al., 1994, Petersen and

Breindahl 2000) and in canned foods (Cottier et al. 1998, Biederman et al. 1996, Simoneau et al. 1999,

Fontani and Simoneau, 2001; EC-DG SANCO-D3, 2002a). The European regulation defined positive

lists of substances authorized for the formulation of materials intended to be put in contact with food,

and Specific Migration Limits (SML) for the substances. Based on 400 SML data, Baner et al. (1996)

used migration modeling to derive maximum amount that could be acceptable in plastics materials

intended to be in contact with food. A recent European directive (EC 2002) makes it possible the use

of mathematical modeling based on diffusion and mass balance equations to check the compliance of

single layer plastic packaging materials against SML. Franz (2005) proposed also migration modeling

as new tool for consumer exposure estimation. Besides, probabilistic modeling based on realistic

physical assumptions of the contamination of food has been recently proposed to perform sanitary

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surveys (Vitrac and Hayert, 2005a). In both situations, realistic transport properties must be

encouraged rather than a combination of rough overestimations that would draw invaluable

conclusions (Chatwin and Katan, 1989; Vitrac 2003). The effect of interfacial mass transport

properties in the assessment of the risk of contamination of packaged foodstuffs and in the assessment

of consumer exposure are discussed in details in Vitrac et al. (2005d) and Vitrac and Leblanc (2005).

Previous research mainly focused on diffusion coefficients, noted D , in plastic materials, aimed at

identifying molecular descriptors and mathematical relationships to predict migration (EC-DG

SANCO-D3 2002b, Helmroth et al. 2002, Reynier et al. 2001a and 2001b, Vitrac et al. 2005c). In

absence of general model to predict diffusion coefficient in polymers, some authors privileged semi-

empirical relationships that overestimate true diffusion coefficients (Brandsch et al., 2002, Begley et

al., 2005). Although, they are not predictive of real situations and although the safety margin is highly

variable according to the considered diffusant and polymer, they are of practical use to check the

compliance of food contact materials (Begley et al., 2005; a general discussion and numerical tools are

available at the safe food packaging portal - http://h29.univ-reims.fr).

By contrast, little work has been performed on determination or prediction of apparent partition

coefficients between packaging materials and food simulants, noted K. Previous experimental

determinations of K were reviewed by Tehrany and Desobry (2004). A rigorous definition of partition

coefficients from fugacities is detailed in Baner (1995). The analogy between partitioning between

packaging materials and food and between food packaging and hydrophobic food simulants is discussed

in Baner et al. (1992). Its effect is generally not taken into account via an appropriate boundary

condition assuming a local thermodynamical equilibrium but implicitly by using the analytical solution

proposed by Crank (1975) for closed systems. A demonstration updated to food packaging materials

can be found in Chatwin and Katan (1989), Vergnaud (1991) and more recently in Han (2004). The

analytical solution proposed for closed systems cannot be extended to open systems since it assumes

the mass conservation in diffusing species between both compartments. In addition, this solution

neglects a possible mass transfer resistance at the food packaging interface, which may contribute

significantly to the overall transfer resistance. The effect of mass transfer coefficient at the food

packaging interface, noted h , has been discussed in Gandek et al. (1989a and 1989b) and Vergnaud

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(1995), in Reynier (2002) and in Vitrac and Hayert (2005b) but very few values are available in the

literature for food packaging application. It must be notice that the definition of h varies according to

the authors. Vergnaud (1995) includes thermodynamical effects whereas Gandek et al. (1989a) and

Vitrac and Hayert (2005b) separate the thermodynamic contribution from the interfacial mass

transport property. Both last works describe the boundary layer diffusion with or without bulk

convection as described in Bird et al. (2002).

The objective of this work is to provide a method for simultaneous determination of the three

transport properties: D , h and K . Two sets of homologous molecules and some typical additives are

studied in reference conditions: a same polymer material (low density polyethylene) in contact with a

liquid polar food simulant (ethanol) at 40°C. The contributions of the molecular structure and

polarity is particularly discussed on the two properties that control the mass transfer at the interface:

K and h . The effect of stirring (with and without) is examined to bring bounds to h values. Besides,

since the simultaneous estimation of D and h from desorption kinetics is a feasible but a stiff

identification problem (Vitrac and Hayert 2005), this work discusses also the effect of different

experimental strategies on the confidence on h values. The possible mass transfer in vapor phase and

the connected h value as described in Bellobono et al. 1984, will be not considered. Only the mass

transfer between the polymer matrix and the liquid food simulant is described in this work. In

addition, it is underlined that although h is conventionally connected to an equivalent mass transfer

resistance on the liquid side (as considered in this work), the scaling relationship between the diffusant

size and determined h values suggested that this description should be revised.

The paper is organized as follows. Section 2 describes the physical problem, which controls the

desorption of packaging food substances into liquid food. Due to the design of our experimental setup,

a dimensionless formulation is proposed, taking into account the variation in the liquid volume in

contact with the packaging material during the experiment time. Section 3 details experimental

conditions and numerical methodology used to identify the 3 transport properties from one or several

desorption kinetics. The results obtained for plastic layers with different thicknesses and in the

presence of stirring are discussed in section 4 in particular according to the molecular structure of

considered diffusants. Finally, a general conclusion and different possible descriptions of the mass

transfer at the interface between low density polyethylene and ethanol are given in section 5.

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2. THEORITICAL BACKGROUND

2.1.Physical description of the desorption of packaging substances into liquid food or

food simulants

2.1.1 Transport equations

The transport of additives and monomers in the plastic matrix is governed by the process of molecular

diffusion. If the liquid in contact is non-interacting with the polymer (no swelling, no plasticization),

the 1D transport in a monolayer material can be macroscopically written as second Fick’s Law with an

assumed uniform D in the packaging material:

( ) ( ), ,

2

² with 0∂ ∂= ≤ ≤∂ ∂

x t x tP P

Pc cD x lt x

(1)

where Pl is the characteristic dimension of the plastic layer. It is equal to the half of the thickness for

double sided contact and equal to the thickness for single sided contact. The local concentration in

surrogate contaminant in the polymer, Pc , is expressed in practical units: kg.kg-1.

At the position = Px l , The previous boundary contact assumed a condition of local thermodynamical

equilibrium (LTE), which is represented in figure 1. The sorption and desorption in each phase are

assumed reversible and controlled by a relation an isothermal relationship similar to Henry’s Law so

that the condition of LTE can be written:

( )

( )

,

,

+

−

→

→=

P

P

x l tLx l tP

cKc

(2)

where Lc is the local concentration in surrogate contaminant in the liquid phase. Equation (2) is

consistent for low concentrations in diffusing species (Vitrac and Hayert, 2005b) and assumes that the

density of the liquid phase does not change with Lc .

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According to the boundary layer approximation, Lc is assumed to vary sharply close to the interface

and to be almost homogeneous far from the interface. By assuming a convective mass flux at the

interface controlled by the parameter h :

( )( ) ( ) ( )

,, ,ρ ρ

+

−

= → ∞

=

∂ = − ⋅ ⋅ = ⋅ ⋅ − ∂P

P

x tx l t x tP

P L L L

x l

cj t D h c cx

(3)

At the position 0=x (figure 1a), a symmetry plane or boundary condition is applied:

( ),

0

0=

∂ =∂

x tP

x

cx

(4)

2.1.2 Macroscopic mass balances in the diffusing species

By noting the bulk concentration in the liquid phase ( ) ( ),→ ∞=t x tL LC c , the mass balance in the liquid

phase is written for a continuous sampling of the liquid phase (the liquid is removed continuously):

( ) ( )( ) ( )

( )t t t tr r== + × × - × × ×ò ò14444244443 14444444244444443

00

0cumulative

cumulativediffusingsampledamountamount

1 1 1 1( ) ( )t

ttt tLL L Lt t

L LL LC C j d C q d

l V(5)

where ( ) ( )t tL Ll V A= is the characteristic dimension of the liquid phase and ( )tLq is the liquid mass flow

rate due to sampling. ( )tLV is the volume of liquid phase at time t . A is the surface area of liquid in

contact with the polymer, it assumed to be constant (the plastic is always submerged in the liquid).

In practice, ( )tLC is measured by sampling and the interpretation is based on the residual

concentration in the solid phase ( ) ( ),

0

= ⋅∫Pl

t x tP P PC c dx l . The mass balance in the solid phase is given

by:

( ) ( )

( )( ) ( )0

0

1 1 1 ( )t

tt t tLP P L L

P PC C C C q d

L t Vt t

r== - × + × × ×ò (6)

where ( ) P P

L L

lL tl

rr

= × is the dilution factor.

Discontinuous sampling was preferred in this study. It leads to discontinuous variations of ( )tLq

between sampling times it such that:

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( ) ( )1

111for

i j jt L L

L j i iLj jj

V Vq t t t t t

t tr d

-

+-=

-= × - × £ <

-å (7)

where ( )xd is the Dirac function verifying ( ) 1x dxd+¥

-¥

× =ò .

2.2. Dimensionless formulation for a discontinuous sampling in time

A dimensionless formulation was obtained for 1i it t t +£ < by introducing a dimensionless position

*P

xx

l= , a dimensionless time

( )2

ii

P

D t tFol× -= also known as Fourier number, a dimensionless

residual concentration in the plastic layer ( )

( )

,

, i

x tP

i x tP

cu

C= , a dimensionless bulk concentration in the

liquid phase ( )

( )

,

,

i

i

x tL L

i x tP P

Cv

Crr= × , a Biot mass number D P

H

R h lBiR D

×= = that assesses the ratio between

the global resistance to diffusion and the mass transport resistance at the liquid interface (see figure

1), and a dimensionless dilution factor: ( )iP P

i tL L

VLV

rr

= .

The system of dimensionless transport equations is expressed as:

2

2

, * 1* 1 * 1* 1 0

*

* **

i

i i

iFo

ii x i ix x

x

i

u uFo x

uj Bi K u v L j dx

t-

-== =

=

ì ¶ ¶ïï =ïï¶ ¶ïïï æ öï ÷¶ çïï ÷ç= - = × × - - × × ÷í ç ÷çï ¶ ÷çï è øïïï¶ïïï ¶ïïî

ò

**x =0

u = 0x

(8)

3. MATERIALS AND METHODS

3.1 Surrogate contaminants

Three sets of surrogate contaminants including a total of 12 different molecules were tested (table

1): (i) 4 homologous linear alkanes, (ii) 4 homologous linear alcohols and (iii) 2 hindered phenolic

antioxidants (2,6-di-tert-butyl-4-hydroxytoluene and octadecyl 3-(3,5-di-tert-butyl-4-hydroxy phenyl)

propionate) and 2 rigid molecules (laurophenone and triphenylene) conventionally used as tracers in

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diffusion experiments. To analyze the effect of set, octadecane was used as common surrogate

contaminant in the first and third set as noted in table 1.

3.2 Formulation and processing of plastic materials

Low density polyethylene (LDPE), supplied by Atofina (France), was formulated with the three sets of

surrogates (see table 1). Virgin LDPE granules were ground in fine powder within an experimental

grinder (model Retsch ZM100, USA) cooled with liquid nitrogen to prevent the thermal degradation of

the polymer. Formulated resins were prepared by soaking the powder into a formulated

dichloromethane solution which contains the set of surrogates with a ratio of 1:1 (1 g of powder for 1 g

of dichloromethane) and vaporizing subsequently the solvent under vacuum.

Plastic films with thickness, Pl , ranged between 50 and 150 µm were processed by extrusion in a 4

temperature zones mono-screw extruder (model Scamia RHED 20.11.D, France; set zone temperatures:

120, 125, 130 and 135°C) combined with a cylindrical die and subsequently calendered as a 40 mm

width ribbon. To prevent any plasticization effect by the surrogates, each formulation was setup to

achieve a final concentration lower than 0.3 % (w/w) in LDPE. Due to variable evaporation rates of

surrogates during the extrusion process, final concentrations were measured on final samples; they

ranged between 100 and 1000 mg.kg-1 (see table 1).

3.3 Characterization of the processed films

3.3.1 Density

The density of the processed films was calculated by the displacement of liquid water and was

estimated at 924 ± 3 kg.m3. It did not significantly vary with the ribbon thickness.

3.3.2 Crystallinity

The degree of crystallinity of the polymer was derived from the melting enthalpy assuming an

enthalpy of the wholly crystalline sample of 295,8 J.g-1 (Mandelken and Alamo, 1996). The melting

enthalpy was measured by differential scanning calorimetry (TA instruments DSC 2920) under a

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nitrogen atmosphere. The rate of heating was fixed at 10°C.min-1 and the weight of analyzed samples

was around 10 mg.

3.3.3 Thermomechanical Analysis

A possible orientation of polymer chains due to calendering was analyzed by TMA (tension mode).

The thermal expansion and shrinkage in the longitudinal direction was assessed during an extension

test carried out in a Dynamic Mechanical Analyzer (model DMA 2980, TA instruments, USA) under

minimal load (12 mN). This load was assumed not to induce any significant creep in the sample. The

sample with size 10 mm (length) × 5 mm × 2 Pl⋅ was thermally equilibrated at 30°C during 10 min

prior heating up to 120°C at a constant rate of 1°C.min-1. The longitudinal strain was monitored

versus time. Each measured was repeated 4 to 5 times.

3.4 Desorption experiments

Desorption experiments were performed on suspensions of plastic strips (5 mm × 5 mm × 2. Pl ) were

dispersed under gentle stirring in ethanol at 40°C (figure 2). An initial ratio of 1:10 (1 g of plastics for

10 g of ethanol) was used for the experiments. The stirring was obtained by rotating a 15 ml flask as

depicted in figure 2a with a rotation rate of 40 ± 2 rpm. The stirring effect was mainly related to the

vertical displacement of the gaseous volume in the flask during rotation. Assuming that the

displacement of each strip is controlled by the size of the gas bubble in the flask, the particle Reynolds

number was estimated between 10 and 50. The concentration in the liquid phase was monitored by

sampling aliquots of 0.1 ml. 15 samplings were performed with a sampling period increasing as the

square root of time to mime the variation of the mass flux with time. At the end of the experiment,

the variation of mass of ethanol varied between 30 and 40 %. All desorption experiments were

duplicated.

3.5 Concentration measurements

Initial concentration in LDPE and residual concentration after the last sampling were determined by

extraction with dichloromethane at 40°C with a ratio of 1:50 (w:w). Concentrations in

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dichloromethane extracts and in ethanol migrates were determined by gas chromatography with flame

ionization detection (model GC 8000 series, Fisons Instruments, USA, equipped with an on-column

injector and an autosampler). All samples were filtered before chromatographic analysis. The

separation was carried out on a 30 m length column (model DB-5HT, J&W Scientific, USA) with an

internal diameter of 0.32 µm and a film thickness of 0.1 µm and connected to a 1 m x 0.53 mm

retention gap (J&W Scientific, USA). Helium at a flow rate of 1.8 ml.min-1.was used as carrier gas.

Samples of 1 µl were injected on-column into the retention gap by an autosampler. The oven

temperature was set to hold a temperature close the boiling point of the solvent (40°C and 70°C for

dichloromethane and ethanol respectively) after injection. The temperature was subsequently increased

up to 350°C (at rate of 20°C.min-1) and up to 320°C (at 15°C.min-1) and held for 2 minutes for

samples containing dichloromethane and ethanol respectively. The on-column injector and FID

detector temperatures were kept at 150°C and 360°C respectively .

3.6 Numerical identification strategy of [ ], ,p D K h ′= and sensitivity analysis

For each desorption kinetic including in M samples, the identification of properties [ ], ,p D K h ′= was

obtained by minimizing the following least-square criterion:

( ) ( ) ( )2 22 2exp

1

ˆ ˆ,M

P P iii

p C C t p K K pχ λ=

= − + ⋅ − ∑ (9)

where P iC is the residual concentration in LDPE as determined by equation (6) from experimental

data. ( )ˆ ,P iC t p is the corresponding concentration in LDPE as calculated by solving the equation

system defined in (8). Since the partition coefficient was also experimentally assessed from the ratio of

residual concentration in the liquid and LDPE phases at the end of the experiment, the term

( ) 2exp

ˆK K p − entails the closeness between the experimental value derived at equilibrium, expK ,

and the value identified from whole kinetic data. This formulation increase the well-poseness of the

simultaneous identification of three properties without promoting arbitrary the use of all kinetic data

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instead of punctual extractions and vice-versa to estimate K . 2λ acts as a positive Lagrange

multiplier, which controls the trade off between both contributions.

Due to the high non-linearity of the distance function and efficiency, equation (9) was minimized using

a downhill simplex method that does not use gradient information of ( )2 pχ . After an initial raw

exploration, optimization proceeded by successive contractions towards a minimum p that may be a

local minimum and possibly different of the true one if ( )2 pχ is biased. A confidence interval on each

parameter was analyzed by Monte-Carlo sampling. The identification of properties was reiterated on

kinetics including a significant non biased noise. The noise was setup to fit within the common

experimental error in concentration measurement. For each parameter, the final result is a distribution

of values, the 5th and 95th percentiles were chosen as confidence interval. Between 50 and 150 Monte-

Carlo sampling per experimental curve were used in this study. This method gave more reliable results

than conventional treatment of the information matrix JJF ′= at the minimum p as a qualitative

interpretation of the variance of p, where J are the Jacobian of the model.

4. RESULTS AND DISCUSSION

4.1 Typical desorption kinetics

4.1.1 Repeatability of desorption kinetics

Typical desorption curves in ethanol obtained for a set of alkanes (C14, C16, C18) are presented in figure

3 for a suspension of 1 g of LDPE strips ( Pl =75 μm) dispersed in an initial volume of 14 cm3. The

surrogate contaminants were included in the same formulation of LDPE with initial concentrations

ranged between 100 and 1000 mg.kg-1 (table 1). The kinetics were repeated in two independent flasks.

Concentration of surrogates in ethanol was determined in samples of about 0.25 ml. Successive

sampling (up to 15) was responsible for a decrease of about 30 % of the total amount of liquid. It was

verified that no additional mass losses occurred during the experiment due to possible leakages or

evaporation during sampling. This variation in volume was assumed no to modify significantly the

hydrodynamic conditions in the flasks.

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The stirring effect was generated by a gyration motion (figure 2) and was mainly related to the

displacement of the air bubble created by the head space along the flask height. As a result the

relative velocity of the fluid in contact with particles was almost the velocity of the bubble. These

considerations based on velocity 75 mm.s-1 and an hydrodynamic radius of 7 mm leads to a typical

particle Reynolds number of about 350 in ethanol. This value much higher than 1 confirmed that the

inertia forces were assumed to dominate the viscous forces in the bulk flow. These hydrodynamic

conditions were obviously more severe than in conventional packaging applications but they ensured i)

a correct homogenization of the concentration in the liquid phase in particular for low or sparingly

soluble substances and ii) conditions, which make possible to derive overestimates of the interfacial

mass transport resistance. The effect of our stirring protocol is analyzed later (see 4.1.2).

Sampling times were chosen to generate approximately similar desorption rates between successive

samples as those observed theoretically for a pure diffusive process. The typical error was assessed by

the fluctuation of the concentration in the liquid phase at equilibrium. It was estimated at maximum

at 10 %. All kinetics were satisfactorily repeated since the variation between repetitions was within the

same range as the systematic error. The residual concentration in the solid phase (figure 3d) was

inferred from the mass balance in ethanol (figure 3a) and in surrogate (figure 3b). The change in liquid

mass was considered via equations (6) and (7). A quantitative interpretation of desorption kinetics was

made by fitting the residual concentration in the solid phase by equation (8). The fitted curves

demonstrate that the proposed model was able to describe accurately the residual concentration in the

solid phase and in the liquid phase. At the beginning of the kinetic, it is underlined that concentrations

in both phases varied not linearly with the square root of time. The sigmoid shape depicted in figures

3c and 3e confirmed that an external mass transfer resistance contributed to the overall mass transfer

rate even when the experiments were performed under significant stirring.

4.1.2 Effect of stirring

The contribution of the external mass transfer resistance was identified by comparing the desorption

kinetic obtained with and without stirring for the set of linear alkanes. In absence of rotation of the

flask, a slight manual stirring was only performed before sampling to homogenize the bulk

concentration. It is worth to notice that plastics strips were sedimented in absence of stirring. Figure 4

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compares the mass flux at the interface ( j ) versus the residual concentration in the solid phase for

both stirring conditions. j was calculated as follows ( , )

( )x t

PP P

dcj t l

dt= − ⋅ ⋅ρ

where ( , )x tPdc

dt and its confidence interval were numerically calculated from the methodology described

in Vitrac and Hayert (2005b).

For all tested molecules, it is showed that stirring improved significantly the desorption rate in

particular at the beginning of the kinetic, when the residual concentration in the polymer was the

highest. The effect of stirring was also discernible on the smallest curvature of j with Pc . Indeed, it is

straightforward to demonstrate that j varies linearly with Pc when external mass transfer dominates.

However it was not possible to derive a h value in absence of stirring since the effective surface

contact area between sedimented polymer strips and ethanol might differ from the value achieved

when the flask was rotating. Besides, the figure 4 demonstrates also that, for a same initial

concentration, the desorption rate was higher for smaller molecules than for larger ones.

4.1.3 Effect of Pl on desorption kinetics

The contribution of the internal resistance DR (figure 1) controlled by diffusion to the overall mass

transfer rate is theoretically expected to be lower for strips with a lower characteristic length

(thickness), whereas the external mass transfer resistance, HR , remains unchanged. This description is

in particular valid if both the polymer remains unchanged and the hydrodynamic conditions are not

modified. The figure 5 tests these assumptions by plotting the dimensionless residual concentration

versus the scaled time 2Pt l for typical surrogates belonging to different sets (LDPE) and Pl ranged

between 25 and 75 μm. If transport phenomena are controlled only by diffusion, the scaled desorption

rate is expected to be similar for all thicknesses.

The experimental curves demonstrated that only the scaled thicknesses obtained for the highest

thicknesses ( Pl = 50 and 75 μm) were similar. For the 3 sets, the scaled kinetics obtained with Pl =

25 μm were significantly above the ones obtained with larger thicknesses. The lower slope of scaled

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kinetics accompanied by a higher sigmoid shape for Pl = 25 μm hinted that an additional mass

transfer resistance contributed also to the desorption rate. Its contribution would less significant for

higher Pl values.

For almost similar desorption and sampling conditions, the residual concentrations at equilibrium

showed qualitatively the affinity of tested molecules for ethanol. Thus, C18OH (figure 5c) and BHT

(figure 5e) exhibited a higher affinity for ethanol than C18 (figure 5a).

4.2 Sensitivity analysis on identified transport properties

The transport properties identified from desorption kinetics described in 4.1 are discussed on the basis

of Monte Carlo sensitivity analysis, which consisted in adding arbitrary white noise to experimental

kinetics. The amount of white noise (5 %) was adjusted to fit within the common experimental error.

This approach was preferred to other techniques such as bootstrap, because it could be applied to our

experimental results without changing the mass balance in ethanol. As a result, the sensitivity analysis

performed on a given desorption kinetic assessed only the effect of experimental errors related to our

analytical protocol (concentration measurements by GC-FID ). The effect of sampling and the effect of

the formulation were analyzed separately. The former one was derived by comparing the results

obtained for different repetitions. Possible bias due to of the formulation of LDPE with different

surrogates (i.e. plasticization of LDPE or interactions between surrogates) could be analyzed by

comparing the results obtained with a surrogate (C18) belonging to different formulations.

The sensitivity analysis performed on each desorption kinetic was based on more than 130 Monte

Carlo trials. All fits had a regression coefficient higher than 0.97. The final result was a distribution in

values of each estimated parameter. This methodology is very powerful since it does not require any

assumption on the distribution of experimental errors and on the independence between identified

transport properties.

4.2.1 Uncertainty due to analytical errors and repetitions

The distributions of transport properties (D , h and K ) for the 3 alkanes identified from desorption

kinetics depicted in figure 3 are plotted in figure 6. The distributions of Bi defined by /D HR R are

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also represented. For consistency, the distributions of K were estimated from a non constrained

criterion ( 2 0λ = in equation (9)) whereas an optimal 2λ was used to calculate more accurately the

distributions of D and h . It is emphasized that the results were not dramatically changed by setting

2 0λ = since the thermodynamic equilibrium was reached in all kinetics. All distributions were

approximatively centered around the values identified without adding noise. The effect of repetition

was fitted almost within the uncertainty related to the analytical errors.

The diffusion coefficients of 3 tested surrogates were spread between 3.10-13 and 2.10-12 m2.s-1. The

relative high sensitivity of this property to the experimental error did not make it possible to detect a

significant effect of the number of carbon on the value of the diffusion coefficient in a same polymer.

By contrast, the dispersion of h values was lower with values ranged around 10-7 m.s-1. The

corresponding calculated values of Bi , typically ranged between 3 and 40 with likely values close to

10, confirmed the contribution of both internal and external mass transfer resistances to the overall

desorption rates.

The differences due to the surrogates were more significant on K values. The uncertainty on partition

coefficient was higher when its expected value was higher. This trend is related to the low sensitivity

of the desorption model to partition coefficients close to 0.7 for the considered range of dilution

factors.

4.2.2 Variations of transport properties with Pl

Following the idea detailed in Vitrac and Hayert (2005), the well-posedness of the identification

problem is expected to be improved by combining the information derived from kinetics obtained for

different thicknesses or equivalently for different Bi values. The necessary condition is that similar

transport properties can be experimentally achieved. This condition is tested on results depicted in

figure 5a (surrogate C18) for Pl = 25, 50 and 75 μm.

Surprisingly, D decreased significantly when the thickness (2 pl⋅ in this work) decreased (figure 7a).

This outstanding effect but reproducible for the 3 tested sets and almost all surrogates was related to

a possible drawing of the polymer during calendering of ribbons. Drawing is known to decrease the

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transport properties such as diffusion coefficients (Peterlin, 1975) and permeabilities (Wang and Porter

1984) in the direction perpendicular to the plane of orientation of polymer chains. This orientation

may be accompanied by an increase in crystallinity. In our work, a possible modification of the

crystallinity fraction was not detectable between the thinner and the thicker samples with 95%

confidence intervals between 19 and 21%.

A possible orientation of polymer chains in thinner samples was tested during a thermal analysis. The

method consisted in measuring the thermal expansion and subsequently the recrystallisation of the

sample during a heating between 30 and 120°C. Figure 8 plots the relative strain versus temperature

in the longitudinal direction for samples of thickness 2 pl⋅ = 50, 100, 150 µm. To check the reversibility

of a possible drawing, the deformation profile of the thinner material was compared to the profile

obtained after annealing at 120°C during 12h. The first slope of the strain increase was related to the

expansion property of the material, while the final shrinkage was connected to recrystallisation process

prior melting. Similar expansion properties were observed for all materials. Differences were noticed

only during the recrystallisation period. The temperature of recrystallisation was lower for the thinner

material. By contrast, the same material presented after annealing a recrystallisation temperature close

to the one measured for thicker ones. It was therefore hinted that the main differences between 50 µm

thick materials and thicker ones were mainly related to changes in crystallinity morphology in the

calendering direction.

The unexpected consequence was that the lowest Bi value was obtained for the largest Pl value, that

is 75 μm. As a result, h values, which were not related to the structure of the polymer, were better

estimated for Pl =75 μm (figure 7). For Pl =25 and 50 μm, slightly higher h values were obtained.

This tends to demonstrate that h did not limit significantly the desorption rate. Consequently, in this

work, only h values based on Bi values lower than 50 were be considered.

4.3 Distribution of transport properties and effect of the molecular structure

This sub-section summarizes the transport properties obtained for the 3 sets and the 3 geometries. The

previous distributions including both repetition, thickness and uncertainty effects are in particular

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analyzed as boxes and whiskers plots in figures 9 and 10. Boxes have lines at the lower, median and

upper quartile values. The notches represent the uncertainty in median estimates. Lines extended from

each box end have a length equal to 1.5 the interquartile distance. Outliers are plotted beyond these

lines. The uncertainty is assessed as previously on the basis of at least 50 Monte-Carlo trials per

repetition. All kinetics were duplicated.

4.3.1 Diffusion coefficients

As already mentioned in 4.2.1, diffusion coefficients estimated for the set of alkanes were not very

different (figure 9a). Displaying all results obtained for different thicknesses within a same distribution

made the effect the number of carbons more discernible. A decrease in both median and lower quartile

values were noticed. Besides, the good repeatability of results was supported by the very similar span

of D values obtained for C18 in two independent sets: “alkanes” (figure 9a) and “others” (figure 9c).

D values of tested linear alcohols and alkanes are not significantly different for a same number of

carbon atoms. The difference between molecules was however more evident, because the upper limit

decreased also with the number of carbons.

Except C18, surrogates of the third set “others” showed D values with similar magnitudes and high

relative confidences (figure 9c). In particular, the variation in D values with Pl was significantly

lower than the one observed for aliphatic surrogates. The effect related to calendering and observed

mainly for aliphatic surrogates could be related to a process of co-crystallization during rapid cooling.

The correlation of D values with molecular mass on a log-log scale is presented in figure 9d. The

reference line with a slope of -2 is also plotted. The critical exponent α in a model D M α−∝

characterizes the molecular transport mechanism as discussed in Lodge, 1999 for polymer in solutions

and in Vitrac et al. (2005) for polyolefines. Since the molecular diffusion coefficient is related to the

fluctuation (second moment) of the position of center-of-mass due to the thermal-agitation, the central

limit theorem implies that the diffusion coefficient of a molecule, whose atoms positions fluctuate

independently and homogeneously, decreases as the reciprocal of the number of atoms in the molecule

or equivalently as the reciprocal of the molecular mass. This mechanism related to α =1 is known as

Rouse regime and is particular alike for liquid mixtures of aliphatic molecules. Confinement as assessed

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in amorphous polymers whose chain length is larger than the intermingling length leads to a diffusion

mechanism by reptation and a theoretical α value of 2. Values higher than 2 are related to very

strong entropic effects due to a combination of confinement, frictions and restrictions.

On the basis of the values averaged for overall Pl values, the model α = 2 was very likely for all

tested molecules. By contrast, the scaling relationship related to the lowest Pl value suggested α

values possibly higher than 2, which could corroborate an assumption of stronger interactions at

molecular level within thin plastic films as formulated and processed in this study.

4.3.2 Mass transfer coefficients h at the LDPE-ethanol interface

Calculated h values were significantly different for almost all molecules. The likely value was ranged

between 6.10-8 and 2.10-7 m.s-1 for alkanes (figure 10a). Slightly lower values, ranged between 2.10-8 and

2.10-7, were obtained with alcohols (figure 10b). Other molecules yielded intermediate h values (figure

10c). They were higher up to 2.10-7 m.s-1 for TRI and lower down to 3.10-8 m.s-1 for IRGA. LAU and

BHT gave h values spread over one decade between 2.10-8 – 2.10-7 m.s-1. The values obtained for BHT

are one magnitude order below the ones derived by Gandek et al. (1989b) in a desorption stirred cell

filled with water. Since h values are dependent on the experimental desorption device and or the

liquid in contact with plastic material, it was not possible to compare both results further.

As for diffusion coefficients in the polymer, the variation of the averaged h values with the molecular

mass is analyzed on a log-log scale as h M αα −. Three correlations with the critical exponent α are

also plotted: 1α = , 2α = and 3α = . It is emphasized that α values higher than 1 were never

reported for diffusion coefficient of medium sized molecules in liquid phase. In liquids, α values ranged

between 0.5 and 1 are expected. A value of 0.5 is connected in particular to a dragging force

proportional to the gyration of the molecule (Bird et al., 2002). From our experimental values, the

likely critical exponent would be close to 2 or slightly above 2. It is worth to notice that such an

unexpected α value for h could not be explained by possible correlations between the determined h

and D values. Indeed figure 7 shows that both h and D estimates were poorly correlated together.

Two scenarios can be proposed depending on the considered side of the interface where the strong

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interaction with the polymer is expected to occur. Since the likely α value was very similar to the

critical exponent related to the diffusion coefficients in the polymer itself, it should be envisioned that

the interface transport resistance is generated by a specific interactions between the diffusant and the

polymer at or close to the interface. At the solid-liquid interface, the migrant could diffuse through a

layer where polymer chains could swell in presence of ethanol. Thermal desorption experiments

demonstrated that the sorption of ethanol was not detectable in the polymer for all samples

thicknesses. As a result, only an adsorption at the interface could occur. An alternative scenario would

be related only to interactions on the polymer side due to a gradient of diffusion coefficients close to

the interface. In the thickest films, this effect would be only related to an external resistance whereas

it would contribute to generate lower bulk diffusion coefficient in the thinnest films. By assuming a

diffusion coefficient for the superficial layer close to the value obtained in the bulk for Pl =25 μm, it is

found that this layer is about 1% of the whole thickness for Bi≈10.

4.3.3 Bi values

The distributions of mass transfer coefficients calculated for Bi values lower than 50 are plotted in

figure 10. Since h was estimated by assuming the boundary condition detailed in equation (3), h was

assumed to be mainly related to the capacity of surrogates to leave the interface on the liquid side as a

result of the thermal agitation (i.e. molecular diffusion) and/or as a result of external dragging forces

(e.g. stirring). In statical physics, h is also related to the group velocity of surrogates when they leave

the interface. In the boundary layer approximation (figure 1), h is defined as the ratio between the

molecular diffusion coefficient in the liquid phase divided by the theoretical thickness of the boundary

layer. The thickness of the boundary layer is expected to be controlled by the hydrodynamic

conditions and/or the local composition of the liquid interface. h is in particular expected to be

disconnected with partitioning effects since the diffusant is assumed to be on the liquid side. This

simplified description should however be modulated if the interface was expected to be irregular due to

the presence of disentangled chains.

The importance of interfacial transport properties on the overall desorption rate was estimated by the

magnitude of Bi values. It is conventionally admitted that diffusion in the polymer dominates for Bi

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values higher than 102. Figure 11 recapitulates the variation of Bi values estimated for all molecules

and all tested geometries. The absence of apparent correlation between Bi and the molecular mass

strengthened the assumption of similar transport mechanisms in the polymer and at the interface. The

plot highlights again the outstanding increase in Bi values when the sample thickness was reduced.

Since this phenomenon is observed for all molecules and formulations, it seems to be only related to

the polymer structure. In practice, h values were best estimated on thicker samples.

4.3.4 Partition coefficients between the LDPE and ethanol

For a given molecule, partition coefficient values provide a quantitative interpretation of its relative

chemical affinity for LDPE or ethanol. Its value was either numerically identified from the transport

model detailed in equation (8), or experimentally assessed according to equation (2) from the residual

concentration in the solid phase and the concentration accumulated in the liquid phase at equilibrium.

Both results were very well correlated while the expected K values were lower than 0.7. For larger K

values, as already mentioned in part 4.2.1, the effect of K on both the kinetic and mass balance is not

sufficient to be accurately estimated. As a result, estimated K based on concentrations at equilibrium

were preferred. Figure 12 displays all individual values of K for all molecules versus two possible

explicative quantities: the molecular mass and the hydrophobicity expressed by the octanol-water

partition coefficient, logP , as calculated with the ACD/Labs PhysChem Software (Version 7,

Advanced Chemistry Development, USA). Since our sets of surrogates included homologous series of

molecules, both properties were partially correlated together. Since our partition concentrations were

expressed as a ratio of concentration in mass per mass, a surrogate with a greater affinity for ethanol

was detected by K values higher than the ratio of densities 1.17P Lρ ρ ≈ .

Except TRI and alcohols inclding less than 18 carbons, all surrogates had a higher affinity for LDPE.

The higher affinity of LDPE was obtained for C18 with a K value as low as 0.2. Only K values of

alcohols varied as a power law of the molecular mass. It was noticeable that the correlation between

( )10log K and logP (figure 12) was poor. The discrepancy was mainly related to the different

behaviors of C18 and IRGA.

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5. CONCLUSIONS

This work investigated the interfacial transport properties of 12 surrogates in formulated LDPE during

desorption experiments in a food simulant with intermediate polarity (ethanol). Double side contact

experiments based on suspension of plastic strips were carried out in rotating flasks that somewhat

exacerbated the hydrodynamic conditions (Re<50) comparatively to real food-packaging contacts. By

varying the typical characteristic length scale related to mass transfer between 25 and 75 μm, Bi

values ranged between 5 and 103 were achieved. Contrary to the expected variation of Bi with the

thickness, the lowest values of Bi were obtained for thick materials. This effect was related to the

calendering process which was able to modify significantly the diffusion coefficients in LDPE.

Experiments related to Bi values lower than 50 provided reliable estimates of the interfacial mass

transport coefficient, h , for all 12 molecules. The values were ranged between 2.10-8 and 5.10-7 m.s-1

with an uncertainty always lower than a factor 5. Since the critical exponent related to the mass

dependence of h had a likely value close to 2, which was very similar to value observed for the

diffusion coefficient in LDPE, the estimated h values were mainly related to an interfacial mass

transport resistance within a dense and entangled phase (i.e. on the LDPE side) rather than in the

liquid phase (i.e. on the ethanol side). This could be related to a very localized sorption or adsorption

of ethanol at the solid liquid interface and to a subsequent local reordering of the polymer. A

significant sorption in the bulk region of ethanol is by contrast very unlikely since no detectable

sorption of ethanol was measured in the bulk polymer. Besides, a swelling or plasticization effect in the

core region would contradict the higher diffusion coefficients assessed in thinner samples. An

alternative description would consist in envisioning that calendering might yield a drawing mainly at

the immediate surface of the polymer. This effect would be consistent with a possible change in crystal

morphology in thin materials as detected in thermal analysis after annealing as well as with higher

diffusion coefficients in thin materials.

From both scenarios, the interface should be envisioned as an intermediate region where diffusion

coefficients should differ significantly either from the bulk liquid phase or from the bulk polymer. In

the current version of the boundary condition between the polymer and simulant (see equations 3 and

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equation 8), the dimensionless Bi number represents the ratio between the internal mass resistance and

a superficial one, whose driven potential is assumed located in the liquid. If the superficial resistance

was located in the polymer, the driven potential should be expressed in the solid phase and the ratio

between the bulk and superficial resistance would be best estimated by the product ⋅Bi K with

experimental K values varying between 0.2 and 1.6. From these considerations and by assuming a

rough diffusion coefficient related to superficial resistance close to the diffusion coefficient in the

polymer bulk phase (median approximation), the thickness of the resisting region would be ranged

between 0.5 μm and 2 μm or between 1 and 4 µm according to it was located on the liquid side or on

the solid side.

Even in the corresponding mass transport description is not completely established – non uniform

distribution of diffusion coefficients in the polymer close to the interface or specific interaction between

LDPE and ethanol – it is emphasized that the detected superficial mass transport resistance is not an

artifact as confirmed by the sigmoid shape of the desorption kinetics plotted versus the square root of

time. With a similar physical description as used in Gandek et al. (1989b), estimated h values in our

tested experimental conditions are one decade lower than those obtained by the authors during the

desorption of a phenolic antioxidant in water.

Current work aims at identifying in confocal microscopy and with fluorescent surrogates such as TRI

and LAU the existence of this layer (location, thickness and related transport properties). Besides,

further theoretical work seems desirable to relate previously h values to the real molecular mechanism

of transport in the ternary mixture (polymer, surrogate and ethanol) which would contribute to the

“boundary” layer.

Acknowledgement

This work was supported by the "Grand Bassin Parisien" program, and funded by the Région

Champagne Ardenne including a PhD grant (for AM) and by the Mission interministérielle et

interrégionale d'aménagement du territoire.

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Table 1. Composition in surrogates of the 3 tested sets.

Figure 1. Physical interpretation of mass transport between a packaging material and a liquid food simulant: a) local interpretation of the concentration profile close to the solid-liquid interface, b) macroscopic interpretation as a serial association of mass transport resistances.The surface area in light gray represents the residual content in the packaging material. The surface area in dark gray represents the amount, which diffused in the liquid.

Figure 2. Experimental device for desorption experiments: a) top view, b) side view, O: rotation axis.

Figure 3. Typical desorption kinetics ( Pl = 75 μm) in a variable volume of ethanol: a) volume of ethanol, b and c) accumulated concentration in the liquid phase, d and e) residual concentration in the solid phase. b and d) kinetics are plotted against a linear time scale; c and e) kinetics are plotted versus a time scale varying as the square root of time.

Figure 4. Effect of stirring on the variation of the normalized interfacial mass flux ( 0tP

jc = ) with the

residual concentration in the polymer ( 0P

tP

cc = ) for surrogates a) C14, b) C16 and c) C18. Continuous

lines plot likely values and dotted lines plot 80 % confidence intervals. Values obtained with and without stirring are depicted as filled and open symbols.

Figure 5. Effect of Pl on scaled desorption curves for typical surrogates a) and b) C18 (alkanes), c) and d) C18OH (alcohols), e) and f) BHT (others).

Figure 6. Distribution of parameters of a) D , b) h , c) Bi and d) K identified after adding 5 % white noise to experimental kinetics plotted in figure 3.The results are plotted for surrogates C14, C16, and C18 ( Pl = 75 μm) and two trials. The distributions are based on more than 130 Monte-Carlo trials. The values identified from non modified kinetics (i.e. without adding noise) are depicted as vertical lines.

Figure 7. Distribution of parameters of a) D , b) h , c) Bi after adding 5 % white noise to experimental kinetics plotted in figure 4a (surrogate C18, set “alkanes”).The results are plotted for 3 different thicknesses corresponding to Pl = 25, 50 and 75 μm and two trials. The distributions are based on more than 130 Monte-Carlo trials. The values identified from non modified kinetics (i.e. without adding noise) are depicted as vertical lines.

Figure 8. Variation of relative strain vs temperature for 3 different thicknesses corresponding to Pl = 25, 50 and 75 m. Analysis was equally realized to an annealed thin film (25 m).μ μ

Figure 9. Distribution of diffusion coefficients of a) alkanes, b) alcohols and c) others. d) Variation of diffusion coefficients vs molecular mass. Values averaged over all tested conditions are depicted as filled symbols. Averaged values obtained for the smallest and highest thickness are plotted as open symbols, whose size reflects the Pl value.

Figure 10. Distribution of mass transfer coefficients of a) alkanes, b) alcohols and c) others determined for 50Bi ≤ . d) Variation of the averaged mass transfer coefficients vs molecular mass.

Figure 11. Variation of averaged Bi values vs molecular mass for the 3 tested sets of surrogates. The size of symbols reflects the thickness values.

Figure 12. K values based on residual concentrations in the solid phase for two trials measured at 40°C.

Set surrogate Code( )0tPC

=

(mg⋅kg-1)

Properties

M

(g⋅mol-1)logP(*)

supplier

Alkanes Dodecane C12 109 ± 10 170.3 7.1 A

Alkanes Tetradecane C14 280 ± 16 198.4 8.2 A

Alkanes Hexadecane C16 385 ± 13 226.4 9.3 A

Alkanes Octadecane C18 417 ± 14 254.5 10.3 A

Alcohols Dodecanol C12OH 338 ± 15 186.3 5.1 F

Alcohols Tetradecanol C14OH 424 ± 15 214.4 6.2 F

Alcohols Hexadecanol C16OH 415 ± 16 242.5 7.3 F

Alcohols Octadecanol C18OH 401 ± 15 270.5 8.3 F

Others2.6-di-tert-butyl-4-

hydroxytoluene

(BHT)BHT 136 ± 11 220.4 5.3 F

Others Octadecane C18 417 ± 13 348 10.3 A

Others Laurophenone LAU 352 ± 15 260.4 7 O

Others Triphenylene TRI 438 ± 12 228.3 5.9 F

Others

Octadecyl 3-(3,5-di-tert-butyl-4-hydroxy phenyl)

propionate

(Irganox 1076)

IRGA 995 ± 24 530.9 13.9 C

* calculated with the ACDD/Labs PhysChem Software(Version 7, Advanced Chemistry Development, USA).

Supplier: Aldrich (A), Acros (O), Ciba (C), Fluka (F).

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