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Simulation of release of additives from mono- and multilayer packaging
B. Roduit(1) , Ch. Borgeat(1), S. Cavin(2) ,
C. Fragnière(2) and V. Dudler(2)
Swiss Federal Office of Public Health, Division of Food Science
Advanced Kinetics and Technology Solutions
Training CourseThe use of diffusion modelling to predict migration
offered by theCommunity Reference Laboratory on Food Contact Materials
for National Reference Laboratories on Food Contact Materials7-8 November 2006, JRC, Ispra, Italy
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(1)
(2)
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Overview
• Actual limitation in simulation• Description of model• Importance of temperature control• Relevance of the partition coefficient • Mathematical verification • Experimental validation• Conclusions
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Kinetics of diffusion in polymer
2
2
x
CD
t
C
Fick’s 2nd law of diffusion
The description of the migration in a polymer requires an analytical solution of this partial differential equation
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Diffusion out of a plane sheet
02
22
22 4
)12(exp
)12(
18
nt L
tDn
nMM
time
Mt
M
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Constraints
Migrant M
0 L
C
X
C01. Initial conditions
t = 0 C = C0Ct
2. Boundary conditions
t > 0 X = L C = 0
3. The diffusivity D is constant
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Consequences
• Analytical solutions of Fick’s law are restricted to simple cases:
Single layer package
Simple initial and boundary conditions during migration
Homogeneous distribution of migrant
Migration under isothermal condition
• Complex, modern packaging requires numerical approximation
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Numerical approximations
• Monte-Carlo• Variational methods• Finite Element Analysis• Finite Differences…
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computationalphysical
ft
Discretization
Elements
FEA is the application of the Finite Element Method. In it, the object or system is represented by a geometrically similar model consisting of multiple, linked, simplified representations of discrete regions i.e., finite elements. The analysis is therefore done by modelling an object into thousands of small pieces (finite elements). The finite elements are used for solving partial differential equations (PDE) approximately.
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• Finite Element Analysis is written as a set of communicating elements• Organization of an object in a (virtual) mesh
uniform regular
Structured Grids:
rectilinear
• Grid generation in time and in space
?
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Considering one layer inside the packaging, it can be demonstrated that the mass of the layer which is taken for calculation of the diffusion of both migrant and simulant can be treated as an ‘infinite’ surface of thickness ‘d’ (i.e. ‘infinite’ in two directions and of wall thickness ‘d’ in the third).
2
2
2
2
z
c
x
c
z
c
x
c
2
2
2
2
y
c
x
c
y
c
x
c
and
2
2
x
CD
t
C=> Fick’s 2nd law of diffusion
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Model assumptions
• the migration follows a diffusive process (Fick’s law) and is not controlled by other kinetic steps
• D = f (T) [Piringer’s model, Arrhenius relationship or
customized equation]
• the equilibrium solubility of the migrant in the different layers of the structure and in the food is governed by the partition coefficients, K, between the layers of the multilayer structure and between the contact layer and food, respectively.
• the food is in intimate contact with all the package surfaces (no void space)
• the transfer of migrant at the interface material-food is rapid and the migrant is homogeneously distributed in the food.
• the transfer of migrant at the interface package-air is nil
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Diffusion in a multilayer structure
additive
PP
FO
ODmigration
0 100 200 300 400th ickness [µm ]
0
200
400
600
800co
nce
ntra
tion
[pp
m]
layer 1 layer 3 layer 4 layer 5layer 2
PE
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0 days
2 days
5 days
40 days
70 days
0 100 200 300 400th ickness [µm ]
0
200
400
600
800co
nce
ntra
tion
[pp
m]
layer 1 layer 3 layer 4 layer 5layer 2
food
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0 20 40 60 80 100m igration tim e [days]
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
con
cen
tra
tion
in fo
od
[mg
/kg
]
(B)
solubility in food = 4.3 mg/kg
functional barrier => time lag 5 days
0 100 200 300th ickness [µm]
0
200
400
600
800
con
cen
tra
tion
[p
pm
]
layer 1 layer 3 layer 4 layer 5layer 2
(a)
(b)
(c)
(d)
(e)
(A)
Simulated migration experiment in a five-layers laminate film. (A) Concentration profiles of the migrant in the multilayer material at different times: 0 (a), 0.5 (b), 5 (c), 20 (d) and 70 days (e). (B) Corresponding migration curve.
partition coefficientK3,4 = 0.7
partition coefficientK5,Food = 100
K1,2 = 1
K2,3 = 1 K4,5 = 1
Example with partition coefficient:Cylindrical package, height of 25 cm and diameter of 4 cm
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Importance of temperature control
HDPE filmd: 250 µmAdditive MW: 350g/molConc.:1000 ppm
1000cm3
Migration conditions
a) 10 days, temperature 20± 10°C, 24 hours modulation
a) 10 days, isothermal temperature 20°C
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Importance of temperature control
T isothermal20°C
T modulation20 ± 10°C,24 hours period
0 2 4 6 8 10m igration tim e [day]
0
2
4
6
8
mig
rant
con
cent
ratio
n [m
g/k
g] 12%
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Real climatic variations
0 2 4 6 8 10m igration tim e [day]
0
2
4
6
8
mig
rant
con
cent
ratio
n [m
g/kg
]
T isothermal20°C
T modulation20 ± 10°C,24 hours period
Barcelona climateNovember
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Programme validation
1. Mathematical verification
2. Experimental validation
to assess the accuracy and stability of the algorithm
measure of the migrant distribution inside multilayer structures
migration tests with temperature variation
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Mass conservation
Diffusion until equilibrium
concentration
C
C/6
error < 5 10-5
Iterative, repetitive calculation can bring rounding calculation error ?
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Strategy of mathematical validation
• Design a multilayer structure comparable to a single layer
• Calculate the migration by FEA approximation and with the “true“ analytical solution
• Determine the accuracy at different Mt/M of the migration
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Strategy of mathematical validation
Diffusion comparison
FEA (Numerical solution)10 Layers
‘TRUE’ (Analytical solution)1 Layer
C
• Determine the accuracy at different Mt/M of the migration
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Strategy of mathematical validation
Diffusion comparison
FEA (Numerical solution)10 Layers
‘TRUE’ (Analytical solution)1 Layer
C
• Determine the accuracy at different Mt/M of the migration
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Strategy of mathematical validation
• Vary parameters and repeat experimentThickness of multilayer structure: 1-1000 µmNumber of layers: 1-10Minimal layer thickness: 1 µmMigrant concentration: 100-1000 mg/kg
Diffusion coefficient: 10-15 – 10-7 cm2/sMigration time: 10 min – 100 years
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Distribution of relative error
-5 -4 -3 -2 -1 0 1 2 3 4 5relative error [% ]
0
100
200
300
400
500
Fre
quen
cy
Number of tests1200
Average error -0.4%
Std. Deviation ± 0.6%
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Diffusion experiment in multilayer
o
Benzophenone
experimental conditions
Multilayer: LDPE/LDPE/PPwith one PE layer saturated with additive Total thickness: 1100 µmDiffusion: both external surfaces are insulatedTemperature: 60°CAnalysis: IR-microspectrometryPE
PP
additive
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0 250 500 750 1000 1250th ickness [µm ]
0
10
20
30
40
50
60
con
cent
ratio
n [a
.u]
LD PE LD PE PP
time = 0
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0 250 500 750 1000 1250th ickness [µm ]
0
10
20
30
40
50
60
conc
ent
ratio
n [a
.u]
LD PE LD PE PP
time = 51 min
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0 250 500 750 1000 1250th ickness [µm ]
0
10
20
30
40
50
60
conc
ent
ratio
n [a
.u]
LD PE LDPE PP
time = 84 min
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0 250 500 750 1000 1250thickness [µm ]
0
10
20
30
40
50
60
conc
entr
atio
n [a
.u]
LD PE LD PE PP
time = 154 min
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Migration with temperature variation
experimental conditions
Polymer: LDPE, 800 µm thick film with 5% additive
Simulant: hexaneMigration: one sideT-variation: step or rampAnalysis: GC
HP 136® C-radical scavenger(Ciba Specialty Chemicals)
o
o
H
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Migration profile with a T-step
0 20 40 60 80tim e [m in]
0
0.1
0.2
0.3
0.4co
ncen
trat
ion
in f
ood
sim
ulan
t [m
g/kg
]
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Migration Profile with a double T-step
0 10 20 30tim e [m in]
0
0.04
0.08
0.12
0.16
0.2co
ncen
trat
ion
in f
ood
sim
ulan
t [m
g/kg
]
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Migration profile with a T-ramp
0 10 20 30 40tim e [m in]
0
0.1
0.2
0.3
0.4co
ncen
trat
ion
in fo
od s
imul
ant
[mg/
kg]
serie 2serie 1
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Conclusions
• Simulation of migration from multilayer laminate by numerical analysis is possible
• Temperature variation can be taken into account
• Possible implementation of partition coefficients in the model up to 10 multilayer films
• Trade-off between the complexity of use and the programme capability