Chapter 9
Theoretical and Computational Aspects of Migration of Package Components to Food
Downloaded by CORNELL UNIV on August 16, 2016 | http://pubs.acs.org Publication Date: March 9, 1988 | doi: 10.1021/bk-1988-0365.ch009
Shu-Sing Chang, Charles M . Guttman, Isaac C. Sanchez1, and Leslie E. Smith
Polymers Division, Institute for Materials Science and Engineering, National Bureau of Standards, Gaithersburg, MD 20899 The numerical solutions and computational methods for the normal Fickian diffusion process applicable to packaging material is given in detail. Most experimental observations on the migration of small molecules from polymeric package materials into food or food simulating solvents show some non-Fickian behavior. In one case solvent absorption and swelling of the polymer have often been observed when the behavior is non-Fickian. A model for a solute diffusing in a swelling polymer is used to explain this phenomenon. In another case, where the migrant is sparingly soluble in the solvent, a stagnant solvent layer at the polymer surface may give rise to an initial migration behavior which is linear in time instead of linear in square root of time. In certain cases where the solvent is not stirred or is highly viscous, the quiescent migration is found to depend on the diffusion coefficient of the migrant in the solvent. Either alone or in combination, these models can be applied to describe most migration behavior in rubbery or semicrystalline packaging material. The amount of package components that may be leached by food or food simulating solvents depends on the o r i g i n a l concentration of the p a r t i c u l a r component or migrant i n the polymer, i t s s o l u b i l i t y i n the solvent and/or the p a r t i t i o n c o e f f i c i e n t between the polymer and solvent as well as temperature and time. I f the polymer i s thick or the time i s short, the amount migrated w i l l be less than that predicted by the equilibrium p a r t i t i o n . In these cases, dynamic modeling of the migration process i s required to predict the migration as a function of time. In this paper we describe four 1Current address: Aluminum Company of America, Alcoa Center, PA 15069 This chapter not subject to U.S. copyright Published 1988 American Chemical Society
Hotchkiss; Food and Packaging Interactions ACS Symposium Series; American Chemical Society: Washington, DC, 1988.
Downloaded by CORNELL UNIV on August 16, 2016 | http://pubs.acs.org Publication Date: March 9, 1988 | doi: 10.1021/bk-1988-0365.ch009
9. CHANG ET AL.
Aspects of Migration of Package Components to Food 107
models o f t h e d i f f u s i o n o f migrant from food packaging m a t e r i a l : t h e simple F i c k i a n d i f f u s i o n w i t h f i x e d boundary c o n d i t i o n s , d i f f u s i o n i n a m a t e r i a l w h i c h i s s w e l l i n g , t h e e f f e c t o f s t a g n a n t l a y e r due t o low s o l u b i l i t y o f migrant i n t o f o o d o r f o o d s i m u l a n t , and q u i e s c e n t m i g r a t i o n i n t o u n s t i r r e d o r v i s c o u s medium. S o l u t i o n f o r a normal F i c k i a n d i f f u s i o n process w i t h simple boundary c o n d i t i o n s , as t h emigrant migrates from a plane sheet i n t o a s t i r r e d l i q u i d , w a s s o l v e d l o n g a g o [1]. T h e n u m e r i c a l e v a l u a t i o n of t h e s o l u t i o n g e n e r a l l y depends on s e r i e s e x p a n s i o n s a n d converges v e r y s l o w l y f o r e a r l y t i m e s o r s m a l l amounts o f m a t e r i a l m i g r a t i n g . To s i m p l i f y t h e c o m p u t a t i o n a l p r o c e d u r e , v a r i o u s s p e c i a l c a s e s w e r e setup w i t h l i m i t e d ranges o f a p p l i c a b i l i t y . With t h e ready a v a i l a b i l i t y o f p e r s o n a l computers, i t i salmost as convenient t o compute t h e g e n e r a l s o l u t i o n n u m e r i c a l l y w i t h o u t h a v i n g t o w o r r y about t h e l i m i t a t i o n s imposed b y these s p e c i a l boundary c o n d i t i o n s and t h e i r ranges o f a p p l i c a b i l i t y . A form o f n u m e r i c a l procedure for t h e s o l u t i o n o f t h e F i c k i a n d i f f u s i o n a p p l i c a b l e t o a wide range i s shown i n t h i s p a p e r . For t h e d i f f u s i o n o f s m a l l migrant molecules from a polymer i n t o a s o l v e n t , o r v i c e v e r s a , there a r e o f t e n d e v i a t i o n s from t h e above d e s c r i b e d normal b e h a v i o r . T h i s i s due l a r g e l y t o t h e solvent/polymer i n t e r a c t i o n i nthe i n i t i a l stage, which leads t o a s w o l l e n polymer. The i n i t i a l s t a g e i s d e s c r i b e d b y a s m a l l e r d i f f u s i o n c o e f f i c i e n t w h i c h i s a r e s u l t o f l i t t l e o r no s o l v e n t content i n t h epolymer. A t l a t e r times t h e d i f f u s i o n c o e f f i c i e n t i n c r e a s e s t o a h i g h e r v a l u e when t h e p o l y m e r i s s w o l l e n b y t h e s o l v e n t i n c o n t a c t . A m o d e l i s g i v e n w h i c h shows t h a t t h e c h a n g e i n the d i f f u s i o n c o e f f i c i e n t as a f u n c t i o n o f time f o l l o w s c l o s e l y t h e movement o f t h e s o l v e n t f r o n t i n t h e p o l y m e r . In t h e case o f low s o l u b i l i t y o f t h emigrant i n t h e s o l v e n t , t h e r a t e o f m i g r a t i o n may b e c o n t r o l l e d b y a t h i n s t a g n a n t l a y e r o f s o l v e n t near t h epolymer s u r f a c e . This stagnant layer generally gives an i n i t i a l m i g r a t i o n behavior which i sapproximately l i n e a r l y p r o p o r t i o n a l t o t h e time, i n s t e a d o f t o t h e square-root o f time as i n normal F i c k i a n d i f f u s i o n behavior. I n some i n s t a n c e s , w h e r e t h e s o l v e n t i s n o t s t i r r e d o r i s h i g h l y v i s c o u s , a q u i e s c e n t m i g r a t i o n phenomenon i s f o u n d t o depend on t h e d i f f u s i o n c o e f f i c i e n t o f t h e m i g r a n t i n t h e s o l v e n t . A c o m b i n a t i o n o f t h e s e cases i s t h e n p o s s i b l e t o t r e a t most d i f f u s i o n cases encountered i nt h e a d d i t i v e m i g r a t i o n behavior. N o n - i n t e r a c t i n g Systems-Normal F i c k i a n
Behavior
For a n o n - i n t e r a c t i n g system, t h e d i f f u s i o n o f a migrant between a l a r g e p l a n e s h e e t ρ o f p o l y m e r o f t h i c k n e s s L o r 2i a n d s t i r r e d l i q u i d s o f f i n i t e v o l u m e V , t h e m o s t w i d e l y u s e d s o l u t i o n [1,2] f o r t h e F i c k i a n s e c o n d l a w , dC/dt =Od C/dx , i s i n t h e f o r m o f : s
2
1
~
2
2
n i l l+a+a q
z n
(
*
1
)
w h e r e a = M /Mp = V / K V , t h e p a r t i t i o n c o e f f i c i e n t Κ = C / C a t t-*» ( a s s u m i n g t h a t t h e p o l y m e r a n d t h e l i q u i d h a v e t h e same d e n s i t y ) a n d the reduced time Τ = D t / i . M, V a n d C d e n o t e t h e a m o u n t , v o l u m e s
s
p
p
2
Hotchkiss; Food and Packaging Interactions ACS Symposium Series; American Chemical Society: Washington, DC, 1988.
s
FOOD AND PACKAGING INTERACTIONS
108
and concentration respectively. The concentration of the solute i n the solution i s assumed to be uniform. The concentration of the solute j u s t within the sheet i s Κ times that i n the solution. The rate at which the solute leaves the sheet i s always equal to that at which i t enters the solution. The solution for the non-zero p o s i t i v e roots, q , of n
tan q
n
= -aq
(2)
n
l i e s between ηπ when a=0 and (η-1/2)π when a=«. q
n
At
a«l,
- ηπ/(1+α)
(3)
~ [η - α/2(1+ο)]π
(4)
Downloaded by CORNELL UNIV on August 16, 2016 | http://pubs.acs.org Publication Date: March 9, 1988 | doi: 10.1021/bk-1988-0365.ch009
For other values of a, q
n
Computation of Equation (1) i s quite straight forward with a computer. Equation (3) or (4) may be used to provide the s t a r t i n g value for q i n a r e i t e r a t i v e solution for Equation (2). At early times and small amounts of migration, the convergence of Equation (1) i s rather slow, e.g., at Τ - 0.001 a sum of about 50 terms would be required. Although the above mentioned c a l c u l a t i o n may not present any r e a l problem i n computation, there are however time savings from simpler solutions applicable for d i f f u s i o n at early stages. At T
