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TRANSCRIPT
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: [email protected]
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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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Figure 12