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extrapolated from values obtained for similar but different molecules (e.g. from a different
antioxidant), a significant uncertainty in the likely D value is expected. Besides the variations in
polymer density and in the morphology of the crystalline phase are other possible sources of
variation in D (Vitrac et al., 2006).
3.2 Contamination from monolayer materials
3.2.1 Thermodynamic equilibrium

In the polymer, molecular diffusion acts macroscopically to smooth concentration differences
between internal regions and the external surface, which looses migrant quantities due to a
pervious contact with food. At equilibrium, the residual concentration is assumed homogeneous
in the packaging material, noted P, and in the food product, noted F. Microscopically, the random
walk of molecules is reversible and the molecules can migrate back from F into P. However,
since the volume of F is conventionally higher than the volume of the P, the probability of
crossing back the F/P interface, when the substance diffuses in the food product, is low. As a
result, at equilibrium, the total amount of substance is generally higher in the food product than in
the packaging material. This effect can be modulated by the chemical affinity of the substance for
both compartments. If both the desorption isotherm of the substance in P and the sorption
isotherm of the substance in F are reversible (no hysteresis) and obeys to the Henry law, the ratio
of the concentrations between both compartments at equilibrium is a constant independent on the
initial concentration in P. This ratio is called partition coefficient, noted K, and is conventionally
defined by the ratio of concentrations noted
{ }
=
,
j eq
j P F
C
and expressed in mass of substance by
mass of phase j, with j=P or F :
=
=
F eq
P
P
P
P
F
F
F
eq
C
V
K
C
V
(1)
where
{ }
=
,
j j PF



,
{ }
,
j
j P F
=
and
{ }
,
j
j P F
V
=
are respectively the activity coefficients in j, the density
of j and the molar volume of j.

By assuming that no reaction and no mass loss between P and F occur, the mass balance between
the initial and equilibrium states is given by:
=
+
=
0
P
P
P
F
F
F
P
P
P
eq
eq
t
C
l
C
l
C
l
(2)
where l
P
is the thickness of the packaging material and l
F
is the characteristic dimension of the
food product defined by the ratio between the volume of food, V
F
, and the surface area of food, A,
in contact with the packaging material:
=
F
F
V
l
A
(3)
By noting the dilution factor

=
P
P
F
F
l
L
l
,
{ }
=
,
j eq
j P F
C
are expressed from Equations (1) and (2)
in function of the initial concentration in P,
=
0
P t
C
, as: