Modeling release of chemicals from multilayer materials into food

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MODELING RELEASE OF CHEMICALS FROM MULTILAYER

MATERIALS INTO FOOD

by

Xiu-Ling HUANG a,b*, Xiao ZHANG a, Xian-Tao ZHANG a, and Kit L. YAM c

a _{School of Mechatronics En gi neer ing and Au to ma tion, Shang hai Uni ver sity, Shang hai, China} b _{Jiangsu Key Lab o ra tory of Ad vanced Food Man u fac tur ing Equip ment & Tech nol ogy,}

Jiangnan Uni ver sity, Jiangsu, China

c _{School of En vi ron ment and Bi o log i cal Sci ences, Rutgers, State Uni ver sity of New Jer sey,}

N. J., USA

Orig i nal sci en tific pa per DOI: 10.2298/TSCI1603839H

The mi gra tion of chem i cals from ma te ri als into food is pre dict able by var i ous math e mat i cal mod els. In this ar ti cle, a gen eral math e mat i cal model is de vel oped to quan tify the re lease of chem i cals through multilayer pack ag ing films based on Fick's dif fu sion. The model is solved nu mer i cally to elu ci date the ef fects of dif fer ent diffusivity val ues of dif fer ent lay ers, dis tri bu tion of chem i cal be tween two ad ja cent lay ers and be tween ma te rial and food, mass trans fer at the in ter face of ma te rial and food on the mi gra tion pro cess.

Key words: model, controlled release, migration

In tro duc tion

Mi gra tion of chem i cals, such as plasticizers and other trim mer pro duced dur ing pack ag ing man u fac tur ing, has been a ma jor con cern for food pack ag ing in dus try and has been stud ied ex ten sively [19]. The abil ity to pre dict the mi gra tion is crit i cal to as sess the haz ards. How -ever, the cur rent pre dic tive mod els have many lim i ta tions, and the most ac cu rate pre dic tive val ues were ob tained from sim ple Crank's model [10, 11], which are valid only for uni form pack ag ing ma te ri als, and the pre dic tion be comes com pletely in valid for multi-layer cases, which has a com plex bound ary and ini tial con di tions, and each layer fol lows dif fer ent diffusivity of chem i cals [12, 13].

The ob jec tive of this pa per is to de velop a gen eral mi gra tion model for multi-layer film con sist ing of dif fer ent ma te ri als. In this model, one-di rec tion mi gra tion, dif fer ent diffusivities of mi grants in dif fer ent lay ers, par ti tion co ef fi cient be tween two lay ers, par ti tion co ef fi cient be tween ma te rial and food, and mass trans fer co ef fi cient be tween ma te rial and food are con sid -ered. A se ries of equa tions are de vel oped based on Fick's dif fu sion, and solved nu mer i cally by fi nite dif fer ence method [14-16].

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Mod el ing As sump tions

The as sump tions of this model are:

– all the layers are in perfect contact. Migrant is from the outermost layer to food, fig. 1, – at the initial stage, the initial concentration of migrant is uniform in the outmost layer, – the migration is in one direction, i. e. from the outermost layer into food, there is no

migration from the packaging film into air, and there is a finite coefficient of mass transfer, hm, from innermost contact layer to food,

– the migration in each layer follows Fick's diffusion, and diffusivities of migrant in each layer are constant,

– the partition coefficients of a migrant are constant at the interface of adjacent packaging layers, and between packaging and food, and

– there is no migration of food into the package.

Model de vel op ment

The model is de vel oped based on Fick's dif fu -sion model:

¶

¶ C

t D

C

x l x l j N

j j

j

j j

= _- < < £ £

2 , 1 , 1 (1)

where C_j is the con cen tra tion of mi grant in layer,

j, at po si tion, x, and time, t. D_j is the diffusivity in layer, j.

The ini tial con di tions are:

– t = 0

C₁ =C_in 0< <x l₁ (2)

C_j =0 l₁< <x l_N (3)

where C_in is the ini tial con cen tra tion of mi grant in the out er most layer.

The bound ary con di tions ex press the fact that there is no ex ter nal trans fer at x = 0; the flux of com pound is con stant whereas the con cen tra tion is not con stant at x = l_j:

– t > 0

¶

¶ C

x x

1 ₌₀_, ₌₀ ₍₄₎

D C

x D

C

x x l j N

j j

x l j

j

x l

j

j j

¶

=

+ +

=

= ₁ 1 = £ £

-1 1

, (5)

C_j+1 =k C_j _j, x=l_j 1£ £j N -1 (6)

- = æ

-è

çç ö

ø

÷÷ =

=

D C

x h C

C

k x l

x l

m N

M

N N N F

N

¶

¶ , (7)

C

C V M

V

m j

j N

F in

F

=

- å

=1

(3)

where k_j and k_N are the con stant par ti tion co ef fi cient of com pound at the in ter face be tween ad ja -cent lay ers, and be tween pack ag ing film and food, re spec tively. The h_m is the con stant mass trans fer co ef fi cient of mi grant at the in ter face be tween the in ner layer and food. The C_F is the con cen tra tion of the mi grant in food, and V_F is the vol ume of food.

The pack ag ing film is di vided into a uni form mesh in the di rec tion along film thick ness, fig. 2, in which l_N is di vided into M parts (h is mesh size, M_jrep re sents j layer ma te rial); re lease time,

T, is di vided into n parts (time step is t). The in -ter faces be tween ad ja cent lay ers are con sid ered to be sep a rated for the con ve nience of cal cu la -tion, wherein l, M_j + j, M_j_–1 + j, M_N + N + 1 in di -cate the in ter face of pack ag ing ma te rial and air, j

layer ma te rial right bound ary, j layer ma te rial left bound ary, and chem i cal lo ca tion in food, re spec

-tively. Ac tu ally, M_j + j and M_j + j + 1 are all rep re sented in the size x = l_j. The fol low ing signs are in tro duced [14, 15]:

C C C C C C

i k

t

- +

-

-= + = +

( / )

( ), ( ),

1 2 1

1 2

s C C C

C

h C C

i k

x ik ik ik

-

--

-=

-( / )

( / )

( ),

1 2 1

1

t

s s s s

x2Cik _h xCik 1 2 xCik 1 2

1

= ( ₊ - _- )

( / ) ( / )

Equa tions (1)-(8) can be con verted into:

1 2

1 2 1 2

1 2

1 2 2

s s s

t ji k

j x ji

C _--_{( / )}( / ) + C ₊-_{( / )}( / ) -D Ck-( / )1 2 =0

1£ £j N, Mj-₁ + + £ £j 1 i Mj + -j 1 1, £k £TN (9)

C₁0_i C i M

1

2 1

= _in, £ £ + (10)

C_ji0 j N M i M N

1

0 2 2

= , £ £ , + £ £ _N + (11)

s s

tCk x k

D

h C k T

1

1 2 1 1

1 2

1 1 2 1 1 2

2

1

+ +

- ₌ - _£ _£

( / ) ( / ) ( / ) ( / )_,

N (12)

Left bound ary of j layer:

s s

t jM

k j

x jM

k i

C D

h C

D

j- + +j j- + +j

- ₌ - _-

-1 1 2 1 1

1 2 1 2

( / )

( / ) ( / ) 1

1 1 2

1 1 2 1

h xCj M j N k T

k

j j s

-- + -- £ £ £ £

( / )

, , _N (13)

Right bound ary of j layer:

s s

t jM j

k j

x j M j

k i

C D

h C

D h

j+ - j

- +

+ + +

-=

-( / )

( / ) ( / )

1 2

1 2 1

1 1

1 2 _s

x jM

k

C j N k T

j+ -j

- _{£ £} _£ _£

1

1 2 ₂ ₁

( / )_, _,

N (14)

C k C j N k T

j M j

k

j jM j

k

j j

+ + +

-+

-= £ £ - £ £

1 1

1 2 1 2 ₁ ₁ ₁

( / ) ( / )_, _,

N (15)

s

t NM M

k m

NM M

k FM N

k

C h

h C

C

N N

N +

-+

- + +

-+

-( / )

( / ) ( / )

(

1 2

1 2 1 2 1

1 2

1 1 2 ₀ ₁ / )

( / ) _,

k

D

h C k T

N

x NM N

k

N

N æ

è ç ç

ö

ø ÷

÷+ + - = £ £

-s (16)

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C

C V h C C

FM N

k

in jM j

k jM j k N j j + + - + -+ -= - + -1 1 2 1 1 2 1 2 1 ( / )

( / ) (1 2 1 1 2 1 1 1 1 / ) ( / ) æ è çç ö ø ÷÷

å + å å

= -= + + + -= -j N i k

i M j

M j j N h C j j

V_F k N

- £1 £ (17)

The cor re spond ing fi nite dif fer ence scheme is:

( ) ( ) ( )

( )

1 2 2 1

1 1 1 1 - + + + - = = + - +

-r C r C r C

r C

j kji j jik j kji

j jik +(2-2r Cj) kji-1 +(1+r Cj) kji+-11, 1£ £j N, 1£k£TN (18)

Ini tial con di tions:

C₁0_i C i M

1

1 1

= £ £ +

in (19)

C_ji0 j N M i M_N N

1

0 2 2

= , £ £ , + £ £ + (20)

Bound ary con di tions:

(1 ₁) ₁₁ (1 ₁) ₁₂ (1 ₁) ₁₁1 (1 ) , 1

1 121

+_{r C}k + -_{r C}k = -_{r C}k- + +_{r C}k- £_k _T

N

£ (21)

Left bound ary of j layer:

- _- _- _{+ -} + _- _- _{+ -} + + ₊

- -

-r_j C r C r C

j M j

k

j k_j _M _j j _jM

j j j

1 ₁ ₁ ₂ 1 ₁ ₁ ₁ (1 ) k ₁ _j j k_jM _j

j j M j

k

j j

r C

r C r C

j j + - = = -+ -+ - -- + - -

-(1 )

1 1

1

1 ₁1 ₂ 1 k ₁_M _j j jM j k

j jM j

k

j-+ - r C j- r C j

-+ -+ -+ -+ - + +

1 1 1 1

1 1 1 1 1 1 2 ( ) ( )

£ £j N, 1£k £T_N (22)

Right bound ary of j layer:

(1 ) (1 )

1 1 1 1 1

-r C_j _{+ -} + +r C ₊ +r₊ C ₊ _{+ +} -r₊

jM j k

j jM j

k

j j M j

k

j

j j j

C

r C r C r

j M j

k

j jM j

k

j jM j

k j j j j + + + + -+ -+ = = + + - -1 2 1 1 1 1 1

( ) ( ) ₁

1 1

1

1 1 2

1

1 1 1

C r C

j N k T

j M j

k

j j M j

k

N

j j

+- + + + + +- + +

£ £ - , £ £ (23)

C_jk _M _j k C_j k_jM _j j N k T_N

j j + + + -+ -= £ £ - £ £ 1 1

1 2 1 2 ₁ ₁ ₁

( / ) ( / )_, _, ₍₂₄₎

1 1 2 1 1 2 1 -æ è ç ö ø

÷ +æ + +

è

ç ö

ø

÷

-+ - +

r C r h

h C

h

N NM N

k

N m NM N

k m

N N

t t

hK C

r C r h

h

N

FM N

k

N NMk N N m

N N + + + -=

=æ +

è

ç ö

ø

÷ +æ -

-1 1 1 1 1 2 1 1 2 t è ç ö ø ÷ + £ £ + -+ + -C h hK C k T NM N k m N

Mk N

N N N 1 1 1 1 t (25) 1 2 1 1 1 1 1

h C C h C

jM k

jM j k

ik

i M j

M j j j j j -+ + ₌ _{+ +} + -+ æ è çç ö ø

÷÷ + å å + =

å = - -+ + = = + + -+ -V C

V C h C

F FM N

k

j N

F FM N

k jM j N N j 1 1 1 1 1 1 2 1 k jM j k ik

i M j

M j

C h C C

j j j -+ - -= + + + -+ æ è çç ö ø

÷÷ - å +

-1 1 1

1 1

1

2 _inV

k T m j N j N N = = å å £ £ 1 1 1 (26)

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Con clu sion

The pa per re ports a gen eral math e mat i cal model aim ing to pre dict the mi gra tion of chem i cals from multilayer pack ag ing ma te rial into food. Many im por tant pa ram e ters are con -sid ered in the model, such as dif fer ent dif fu sion co ef fi cient, dif fer ent par ti tion co ef fi cient, and trans mit co ef fi cient at the in ter face of ma te rial and food. Nu mer i cal so lu tion is given by it er a -tive com pu ta tion from fi nite dif fer ence method.

Ac knowl edg ment

The work is sup ported by the Open ing Fund of Jiangsu Key Lab o ra tory of Ad vanced Food Man u fac tur ing Equip ment & Tech nol ogy at Jiangnan Uni ver sity (FM-201508).

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