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Anomalous transport of penetrants in glassy polymers: Amended equations
Marhf Compur. Modelling, Vol. 12, No. 8, pp. 947-957, Printed in Great Britain. All rights reserved
1989 Copyright
0
1989 Maxwell
0895-7 177/89 $3.00 + 0.00 Pergamon Macmillan plc
ANOMALOUS TRANSPORT OF PENETRANTS IN GLASSY POLYMERS: AMENDED EQUATIONS S. Department
of Chemical
(Received
K. SINGH and L. T. FAN?
Engineering,
Durland Hall, Kansas KS 66506. U.S.A.
State University,
Manhattan,
November 1986; accepted for publication November 1988) Communicated
by X. J. R. Avula
Abstract-The sorption of penetrant in a glassy polymer and the subsequent polymer relaxation give rise to a moving boundary problem. The governing equations for these phenomena, proposed by Peppas and Sinclair have been critically examined. It has been determined that these equations violate the mass conservation criterion. Thus, they have been amended, and analytical solutions obtained for some of the resultant equations. The cases considered are Fickian and non-Fickian transport with/without density change between gel and glassy polymer.
INTRODUCTION
Transport of a thermodynamically compatible penetrant in a glassy polymer results in the lowering of the glass transition temperature of the polymer. The polymer may then undergo a glass-gel transition due to the increased mobility of the macromolecular chains. While glassy polymers have been extensively studied to understand the mechanism of such a transition [l, 21, considerable attention has also been focused on these polymers because of their widespread application in biomedical, pharmaceutical, environmental, and agricultural fields [3]. Fickian or non-Fickian transport of a penetrant may be observed depending on the rate of macromolecular chain relaxation and characteristic diffusion time [4]. Due to the state transition from glass to gel, a moving front separating the glassy region from the gel region is observed [5, 61; thus, modeling of these phenomena is effected by moving front analysis, thereby yielding numerous models [7-l I]. Among the models, the one proposed by Peppas and Sinclair [12] has considerable utility because of the ease with which the governing equations and their solutions can be adopted for modeling transport of both the penetrant and an active agent through a controlled release device. These equations and solutions for a semi-infinite glassy polymer, therefore, have been critically examined in this work. The necessary corrections have been proposed and exact solutions obtained for some of the resultant equations. This work is a continuation of our earlier works on mathematical modeling and simulation of controlled release devices [13-l 51.
MASS
CONSERVATION
CRITERION
The system under consideration consists of a semi-infinite glassy polymer in contact with a uniform external medium, M, containing a liquid penetrant at concentration C,. Due to dynamic relaxation and swelling of glassy polymer, a moving front is formed at x = X(r); this front separates the gel region (layer 1) from the glassy polymer region (layer 2), as illlustrated in Fig. 1. It is assumed that a critical concentration of the penetrant is required for the glass-gel transition to occur. Thus, the front concentrations are at this critical level; these concentrations (on the glassy and gel sides) are related through a partition coefficient, k,, i.e. C,, = k;Czt t To whom
all correspondence
should
be addressed. 941
(la)
948
S. K. SINGH and L.T.
FAN
M
2
C,
0 Fig. 1. Penetrant transport in semi-infinite glassy polymer. Region M-external medium with penetrant concentration C,; region l-gel; region 2-glassy polymer; X(t)-interface between regions 1 and 2 at penetrant concentration C,.
where
c,,= C,(X,
l)
(lb)
C,, = C,(X, 0.
(lc)
and
C,, and C,, are characteristic of the polymer/penetrant system. The polymer is initially free of the penetrant. The instantaneous penetrant mass balance over the semi-infinite polymer ranging from the gel-medium front at x = 0 to boundary at x = cc may be written as (when the penetrant is not consumed/produced by any chemical reaction) penetrant flux into the polymer at the gel-medium front The rate of penetrant accumulation
I[ =
Rate of penetrant accumulation in the polymer
in the polymer is m d ’ C, (x, t) dx + C,(x, r) dx zi; [S 0 sX and denoting the penetrant flux into the polymer by J, we obtain x z C,(x, t)dx + Cz(x, t) dx . JIY4=; [s 0 sX Applying Liebniz’s rule to the right-hand side of this expression, we obtain
1’
1
1
-
HS
C, (x, r) dx - C,(X, t) d$
+
(2)
=a
- Cz(x, t) dx x at
or
s xa
1
xa
dX -C,(x,~)dx+(C,,-Cz,) . (3) o at s x at [ dt The model equations for the system under consideration as presented by Peppas and Sinclair [12], may now be examined in the light of the mass conservation criterion represented by equation (3).
JI.x=o=
- C,(x, t) dx +
Fickian difSusion without density change Fickian diffusion of a penetrant through the two regions is described by the governing differential equations [ 121,
ac, -=
at
D
a2c, o
l-p3
t>O
Anomalous transport of penetrants in glassy polymers
949
and
ac2_
x-
a2c2 x+
D
t>o.
23p3
(5)
[ 1e
Substituting equations (l), (4) and (5) into equation (3), we obtain
s 0~
x
a2c2
-dx
D2a2
(6)
+ (C,, - C,,) !$
Integrating term by term and rearranging, the mass balance equation reduces to _D
3
1 ax
+D,5 x=x-
=(C,rax .Y=x+
(W
c2d
which represents the flux continuity requirement at the gel-glassy polymer front x = X(t) (see Fig. 1). However, Peppas and Sinclair [12] have also assumed that C I, =
c2, = c,,
which requires that the right-hand side of the above equation be zero. The corresponding equation given by Peppas and Sinclair [12], however, has a non-zero right-hand side as shown below:
-D,!S! ax
_x
.I -
_+D22~x=x_=c’~]
(7b)
Fickian dz$&sion with density change
For the case when the glass-gel transition is accompanied by a change in density, the governing differential equations, as presented by Peppas and Sinclair [12], are
acl-D w
dt -
(P,-P~) dx acl
1 a.9+
p2
[
-z-z 1
o
cx_;
t
,.
(8)
”
and
ac2_
at-
a2c2 x+
D
-
2 ax23
t>O
where p, and p2 are the densities of the gel and glassy polymers, respectively. Substituting equations (I), (8) and (9) into equation (3) gives
+
s
a2c2
mD 2&x
X
+ (G, -
c2A
g [
1 WV .
Since the front position, X(t), is a function of time only, the mass balance equation above reduces to an expression for the flux continuity at x =X(t), i.e. -D
ac,
I ax
x=x-
+
(114 D22 lrcx+ =(~c1~-c2~)[t$]*
Under the assumption that [12] C I,=
c2, = c,,
the above equation reduces to
(1lb) The corresponding
equation by Peppas and Sinclair [12], however, does not contain the factor, side, i.e.
(p, - p2)/p2, on the right-hand _D
i%
1 ax
X-X-
+L&jx+=Cf3
(1lc)
950
SINGH
S. K.
and L. T.
FAN
Non -Fickian d@ision The governing differential equations for the non-Fickian diffusion of a penetrant based on Wang et al. [ll] are [12]
ac,
ax-, ac,
ar=D,ax’-vax,
O
t>o
(12)
t>O
(13)
and
ac2
a2c2
D
at=
ac2
2p-v-$
x+dx
where v represents the so-called pseudoconvective contribution to transport, arising from the normal component of stress tensor of the penetrant. Substituting equations (l), (12) and (13) into equation (3) we obtain
a2c, -*,g+,c, = ’ 1 S[ --a2 x
.V=0
D
1mDac +S[ 2 1
ac,
v-
0
dx
ax
x
--vz ax2
dx+(C,,-C,,)
[1 g
.
(14)
Integrating by parts and rearranging, this mass balance equation reduces to
-D,dci ax
_
(154
_+D2c? ax
I-X
x=x+
For c,, = c,, - c,, as assumed by Peppas and Sinclair [12], the right-hand side of the above equation is zero. The corresponding equation by Peppas and Sinclair [12], however, has a non-zero right-hand side as shown below:
-D,ac, ax
_
x-x
AMENDED
_+o,~~~_x+=C,~~]-C,v. Wb)
SYSTEM
OF
GOVERNING
EQUATIONS
The amended governing equations for Fickian and non-Fickian transport of the penetrant in the system under consideration are given below along with the exact analytic solutions for the Fickian diffusion cases. Fickian difSuon
without density change
Fickian transport, without change in density of the glassy polymer when transformed into gel, is described by the following governing differential equations (kinetics of the chain relaxation phenomena being neglected).
ac,
D
-jy=
ml
qp?
o
t>O
(16)
t>o.
(17)
and
ac2_
at-
D
a2c2 x+~xxco;
237’
The polymer is assumed to be initially free of the penetrant: therefore, we have C&x, 0) = 0
(18)
X(0) = 0.
(19)
and
Anomalous transport of penetrants in glassy polymers
951
Assuming that the gel and medium are in equilibrium at x = 0, we have c, (0, t) = c, = k&s
(20)
where k0 is the gel-medium interface partition coefficient, and C, the bulk medium concentration. For the semi-infinite system, C,(co, t) = 0
(214
and
ac*
-D2dxx=~ The concentration continuity equation (l), reiterated here
= 0.
@lb)
at the gel-glassy polymer front [x =X(t) C,, = k,C,,,
in Fig. I] is given by
x =X(t)
(22)
where C,, and Cz,, defined by equations (1 b) and (lc) are constants, characteristic of the polymer-penetrant system. They represent the critical penetrant concentration at which the glass-gel transformation occurs. The continuity of penetrant flux at this front gives rise to
-D,!S! aX
_ _+D2dc, ax x-x
x=x+
= cc,,
-
C2,)
g [
1 .
(23)
This is identical to equation (7a) derived from the criterion of overall penetrant mass conservation at any instant. By assuming that the position of the front, X(t), is given by X(t) = at “2
(24)
where tl is a constant to be determined, and by performing a similarity transformation
the concentration
profiles are obtained as [I& 171,
(25)
Czb,
l)
=
(26)
c2,
The value of a is recovered from the following transcendental equations (25) and (26) into equation (23).
equation, derived by substituting
= -
f (C,, - C,,).
(27)
Fickian d@usion with density change
Assuming that the relaxation and swelling phenomena are instantaneous, and taking into account the change of densities between the two phases, the governing differential equations for the system are given as
ac, -= at
D
a2c, o
I=’
t>o
(28)
S. K. SINGHand L. T. FAN
952
and dC2 _ D -_a=c2
dt-
=
@=-PI)dX -
ax2
P2
[
ac=
x+
dt 1 ax’
tbO
”
(29)
The second term on the right-hand side of the second equation arises from the fact that when observed from the origin (see Fig. l), each point in the glassy region appears to move in the (+x) direction as a consequence of the volume expansion at X [18]. Since the polymer is initially free of the penetrant, the initial conditions given by equations (18) and (19) are applicable. Equations (20) and (22) describing the equilibria at the gel-medium and gel-glassy polymer fronts are also valid for this case. The requirement of penetrant flux continuity at x -+ co (equal to zero flux) gives rise to
c2 =o 11
ac,+(~z-PI) dX = ax P2 [ dt
_D
while
(30)
.x= x
C,(co, t) = 0.
The continuity of penetrant flux at the front, x =X(t),
(21) given by
[flux inId= *- - [flux out], = X+= accumulation yields _D
over [dX/dt](,,
- 11
ac,+b2-P,)
dX
. = cc,, - G,) (31) s=x+ By substituting equations (22), (28) and (29) into equation (3) and integrating by parts, an expression identical to equation (31) is obtained from the criterion of overall penetrant mass conservation. The above system of equations can be solved by assuming that = ax
~2
[ dt
c2
X(t) = CltI,= as in equation
(24). Solving the above system of equations by the procedure described previously,
the concentration
profiles
are obtained
as
(32)
and X erfc ___
(
(P2-PII
-
2JD,t
Cz(x,
t)
=
c2,
a
P2
h/z
1.
(33)
c(
erfc E!! ( Pz 2JD,
)
The transcendental equation for c1is obtained by substituting equations (32) and (33) into equation (31).
I--* D
I
7r
(--$1 ( >
(Cl, - C&v
erf 2fi
a
’ +
Gexp($$-)
$ T’
erfc(fi P2
a
)
=-;(~G~-G).
(34)
2Jrds
Non -Fickian d@usion without density change
Anomalous transport effects due to macromolecular relaxations were accounted for by a pseudoconvective term, v, in the generalized transport equation by Wang et al. [l I]. It represents
Anomalous transport of penetrants in glassy polymers
953
the convective velocity created as a result of stress-gradients in the system. We will take “v” to be a characteristic of the polymer-penetrant combination. For the sake of generality, the value of this stress-induced drift will be denoted by v, in the gel region and by v2in the glassy polymer region. In a constant density system, the transport of penetrant in the gel and glassy regions is then governed by, respectively,
ac,_ D azc,
at-
ac, o
lp-“‘-g
t
>o
(35)
and
ac2
D
j-y=
a2c2
ac,
23742p
x+
t>o.
(36)
Equations (18)-(22) also hold, along with
-D2Z+v2c2 [
= 0. 1
(37)
X=30
This expression implies that no transport occurs across the boundary at x+00. penetrant flux at the front, x = X(t) (see Fig. 1) gives rise to
-D,z __+D,s X-X
ax
The continuity of
(38) x=X+
When v, = v2z v, this is identical to equation (15), derived from the overall penetrant mass conservation criterion. The technique of similarity transformation is not applicable to equation (35), since no two initial and boundary conditions for this equation may be combined into one. However, an approximate analytic solution can be obtained under the pseudo steady-state assumption. We have earlier stated that the glass-gel transition occurs at a critical concentration C,, (in equilibrium with Clt); thus, C,, and C,, are constant at the gel-glassy polymer front. For the penetrant concentration at any interior point to attain this finite critical concentration, the diffusion of penetrant and the development of the concentration profile must precede the movement of the gel-glassy polymer front. The concentration profile in the gel region may, therefore, be taken to be at the steady state with respect to the position of the gel-glassy polymer front. Under this assumption, the governing equation in the gel region, equation (35), is rewritten as D
ax ,,!$o, --
1 a2
o
t z-0.
(39)
This equation can be readily integrated subject to the boundary conditions, equations (20) and (22), to yield the concentration profile as exp F - 1 exp (g) - exp(g) ( > C,(x, t) = C,, v; + C, VJ exp + - 1 exp - 1 ( J ( D, > The concentration assuming that
(40)
profile in the glassy polymer region can be obtained from equation (36) by X(t) = at”’ + v,t
(41)
and by resorting to a similarity transformation (42)
954
S. K. SINGH and L. T.
FAN
Following a procedure similar to that described earlier, the solution is obtained as
(43)
The parameter tx is recovered by substituting equations (40), (41) and (43) into equation (38), thereby yielding VI (Cl, -
1 - exp Non-Fickian
(
-
ce>
av,t”2 + v, v,t D, d@ision
$> ( =J1 ( ) D
+ s.
Cztev -
2
CL
ci
(Cl,
-
C2,)
+
C,,(v,
-
VA.
(44)
2J;
erfc 2Jz
with density change
The governing differential equations for this type of polymer/penetrant behavior may be written by combining those of the two preceding cases, to account for both the density change and anomalous transport effects. These are
ac,
at=
D
a*c,
ac,
o
rg-kg
and LX,
-=
at
D
a*c2
-2 2x2
ac2 \?2dx
(Pz-LQ) -
p2
-1 dX
[
dt
ac2
t>O
x+
dx’
(45)
’
f >&
,
Equations (18) and (19) are the initial conditions while equations (20), (21) and (22) are the boundary conditions for the above set of differential equations. The requirement that no transport occurs across x -+ co leads to -D2~+v2C2+(p2-p1’[~]C2] P2
Continuity
=O. I=z
(47)
of penetrant flux at x = X(t) (see Fig. 1) gives rise to
-D,%! ax
_ _+.;~~~=~+=(~,~-~~~~)[~]-V~~II+~*~~I .‘i-x
(48)
which is identical to the equation obtained by substituting equations (45) (46) and (47) into equation (3) and integrating. The governing equation for the gel region, equation (45) may be solved by resorting to the pseudo steady-state assumption, as in the previous case. The solution is again given by equation (40). However, equation (46) for the glassy polymer region cannot be solved by resorting to similarity transforms since an explicit expression for X(t), analogous to equation (41), cannot be written for this case. A numerical approach may be required for the exact solution of this case. RESULTS
AND DISCUSSION
Comparing the governing equations of the models and their solutions presented in this work with those by Peppas and Sinclair [12] (under the assumption that C,, = C,, z C,), we observe that the difference in the first case (Fickian diffusion without density change) lies only in the forms of the tlux continuity expression at x = X(r) [equations (7a) and (7b)]. It has been demonstrated that the expression given by Peppas and Sinclair [12], equation (7b), does not satisfy the mass conservation criterion. The resultant expressions for concentration profiles in both works are similar but not identical in that the values of the parameter a in the expressions are different; the transcendental equations giving rise to the values of c1in both works are not in agreement [16]. This parameter, determines the extent of the gel region and that of the glassy polymer region, and the shape of the concentration profile.
Anomalous
transport
of penetrants
in glassy polymers
955
For the case of Fickian diffusion with density change, the governing differential equations presented in this work differ substantially from those given by Peppas and Sinclair [12]. Their flux continuity expression for this case, equation (1 lb), again has been shown not to satisfy the mass conservation criterion. The amended governing equations presented in this work lead to the expressions for the concentration profiles and the transcendental equations for evaluating the parameter c( which are at variance with those presented by Peppas and Sinclair [12]. In the case of non-Fickian diffusion, we observe that the solution for the concentration profile in the gel region, C, (x, t), presented by Peppas and Sinclair [12] (for v, = v2= v) x - vt 1 + erf ___ ( 24% ) c, (x, t> = c, + IC, - c,>
1 + erf -L ( 2JD, > does not satisfy the boundary condition given by equation (20). Their expression for the flux continuity at X(t), equation (15b), has been shown not to satisfy the mass conservation criterion. While the expressions for the concentration profile in the glassy polymer region are similar in both works, they are not identical (even for v, = v2= v) due to the difference in the value of the parameter c(. This is due to the fact that the pseudo steady-state assumption is imposed on the gel region in this work, and that the flux continuity expressions at X(t) in the two works, under the condition that v, = v25 v, are at variance. It must be mentioned that the assumption
c,, = c,, = c,, made by Peppas and Sinclair [ 121 is an oversimplification. In general, the concentrations in equilibrium across an interface separating two distinct phases cannot be expected to be equal. Values of the various parameters need to be determined to obtain the concentration profiles from the expressions presented in this work. Experiments may have to be performed with the specific polymer-penetrant system to determine the nature of the phenomenon (Fickiamnon-Fickian) and the values of D, , D,, p,, pz and “v” since the same system can behave differently under different ambient conditions. Solution of the transcendental equations [especially equation (231 is greatly facilitated by the graphs presented in [ 161.The value of the parameter CLfor the case of non-Fickian diffusion is dependent on t [see equation (44)]. An iterative approach will thus be needed for 1 .o
0.9
CYW.
F
0,
1
10-6
2
146
D2
1.0
0
7x10-’ 109
7.9583
Id
9.3238
x10-4
0.9 0.6
cur”*
D,
1
106
2
lo*
4 7x10-'
0
1.0137110-3
16
1.3299~10-'
...-7
4.3278x10-'
I 0
0.01
0.02
Distance
0.03
0.04
0.05
0.06
into polymer
Fig. 2. Concentration profiles at r = 100 s for Fickian diffusion without density change for different D, and D, (diffusivities are in cm2/s; concentration is given as fraction of the bulk concentration; distance into the polymer is in cm).
Distance
into polymer
Fig. 3. Concentration profiles at I = 100 s for Fickian diffusion with density change (p,/p, = 1.5) for different D, and D, (diffusivities are in cm2/s; concentration is given as fraction of the bulk concentration; distance into the polymer is in cm).
S. K.
956 D~-W6,
1.0
0.9
0
0.02
D, .lCP,
1
10 100
8.46‘4
7.9309
0.04
Distance
I
W4
1.0
8.3222 x lO-4
,000
0.06
into
‘J,*TxlD-‘,
C”W*
a
t
2 3
L. T. FAN
~‘7x10-‘,~,*~~10-~,“~.5~10~~
C”,“.
c
and
SINGH
0.9
I 10-4
0.06
0.10
0
0.02
1
,D
8.6973
x10-4
100
9.0738
x lo-’
1.043,
I 10-3
,000
0.04
Distance
Fig. 4. Concentration profiles at different times for diffusion without density change non-Fickian (diffusivities are in cm2/s; velocities are in cm/s; concentration is given as fraction of the bulk concentration; distanceinto the polymer is in cm).
u,.O.D
a
2 3
polymer
u,.5~10-~,
t
0.06
into
0.08
0.10
polymer
Fig. 5. Concentration profiles at different times for non-Fickian diffusion without density change (diffusivities are in cm?/s; velocities are in cm/s; concentration is given as fraction of the bulk concentration; distance into the polymer is in cm).
evaluating the concentration profiles. It appears that a numerical solution is required to obtain the exact solution for this case, although the system is singular at t = 0 [19,20]. Some representative simulations have been performed; the results are shown in Figs 2-6.These results are based on the order of magnitude values of D, , D,, p,,pz, v, and v2, assumed. The concentration profiles for three different sets of diffusivities for Fickian diffusion (at time t = 100 s)
t
CW”.
1.0 r
100 ‘O
i
0.9 t
3
,000
3.3363
x lo-”
4
1ODOO
4.0386
I 10-a
E j_
9 5
s2
0.4
-
0.3
-
0.2
-
0.1
-
0
0.02
0.04
Distance
Fig. 6.
0.06
into
0.06
J 0.10
polymer
Concentration profiles at different times for non-Fickian (diffusivities are in cm*/s; velocities are in cm/s; concentration concentration; distance into the polymer
diffusion without density change is given as fraction of the bulk is in cm).
Anomalous transport of penetrants in glassy polymers
951
are presented in Fig. 2; the corresponding profiles for Fickian diffusion with density change are shown in Fig. 3. The effect of density change becomes increasingly significant with an increase in D,;this is due to the fact that the increased gel-region diffusivity accelerates the rate of movement of the front. Concentration profiles for non-Fickian transport (without density change) at various times are illustrated in Figs 4-6 for different sets of parameters. A simple model of penetrant transport in glassy polymers is of considerable importance due to its applicability to polymers used in containers, storage tanks, etc., and also to polymers employed for controlled rate delivery of active agents such as pharmaceuticals, fertilizers, and pesticides 131. The solutions presented here can be employed to predict the concentration profiles of the penetrant and to determine the position of the gel-glassy polymer front. Thus, by fixing the extent of the gel region and that of the glassy region, the behavior of the active agent may be analyzed. Acknowledgemenrs-This is contribution No. 86-141-J, Department Experiment Station, Kansas State University, Manhattan, KS 66506.
of Chemical Engineering,
Kansas Agricultural
REFERENCES I. H. L. Frisch, Sorption and transport in glassy polymers-a review. Polym. Engng Sci. 20, 2-13 (1980). 2. W. R. Vieth, J. M. Howell and 3. H. Hsieh, Dual sorption theory. J. Membrane Sci. 1, 177-220 (1976). 3. R. W. Korsmeyer and N. A. Peppas, Macromolecular and modeling aspects of swelling-controlled systems. In Controlled Release Delivery Systems (Edited by T. J. Roseman and S. Z. Mansdorf), pp. 77-90. Marcel Dekker, New York (1983). 4. J. S. Vrentas, C. M. Jarzebski and J. L. Duda, A Deborah number for diffusion in polymer solvent systems. A.Z.Ch.E. JI 21, 894-901 (1975). 5. N. Thomas and A. H. Windle, Case II swelling of PMMA sheet in methanol. J. Membrane Sci. 3, 337-342 (1978). 6. N. Thomas and A. H. Windle, Transport of methanol in poly(methy1 methacrylate). Polymer 19, 255-265 (1978). 7. G. Astarita and G. C. Sarti, A class of mathematical models for sorption of swelling solvents in glassy polymers. Poiym. Engng Sci. 18, 388-395 (1978). 8. G. Astarita and L. Nicolais, Physics and mathematics of heat and mass transfer in polymers. Pure app/. Chem. 55, 727-736 (1983). 9. A. Peterlin, Diffusion with discontinuous swelling-V. Type II diffusion into sheets and spheres. J. Polym. Sci., Polym. Phvs. Edn 17. 1741-1756 (19791. IO. Y.10. Tu, A multi-phase Stefan’problem describing the swelling and the dissolution of glassy polymer. Q. appl. Math. XXX!‘, 269-285 (1977). II. T. T. Wang, T. K. Kwei and H. L. Frisch, Diffusion in glassy polymersIII. J. Polym. Sci. A27, 2019-2028 (1969). 12. N. A. Peppas and J. L. Sinclair, Anomalous transport of penetrants in glassy polymers. CoNoid Polymer Sci. 261,
404408 (1983). 13. L. S. Fan, K. Tojo and L. T. Fan, Cyclic dynamics of a nonisothermal controlled release. Math. Biosci. 45,69-94 (1979). 14. K. Tojo and L. T. Fan, Membrane moderated controlled release. Math. Biosci. 57, 279-300 (1981). 15. K. Tojo, K. Miyanami and L. T. Fan, Mathematical simulation of membrane-moderated controlled release. Powder Technol. 35, 89-96 (1983). 16. J. Crank, The Mathematics of Diffusion, pp. 29&293. Clarendon Press, Oxford (1975). 17. L. I. Rubinstein, The Stefan problem. In Translations of Marhematical Monographs, Vol. 27, pp. 52-55. American
Mathematical Societv. Providence. RI. (I 971). 18. L. N. Tao, On solidification problems including the density jump at the moving boundary. Q. J. Mech. appl. Math. XXX& 175-185 (1979). 19. J. Crank, Two methods for the numerical solution of moving-boundary problems in diffusion and heat flow. Q. J. Mech. appl. Math. X, 22@231 (1957). 20. S. K. Singh and L. T. Fan, A generalized model for swelling-controlled release systems. Biofechnol. Prog. 2, 145-156 (1986).
1989 Copyright
0
1989 Maxwell
0895-7 177/89 $3.00 + 0.00 Pergamon Macmillan plc
ANOMALOUS TRANSPORT OF PENETRANTS IN GLASSY POLYMERS: AMENDED EQUATIONS S. Department
of Chemical
(Received
K. SINGH and L. T. FAN?
Engineering,
Durland Hall, Kansas KS 66506. U.S.A.
State University,
Manhattan,
November 1986; accepted for publication November 1988) Communicated
by X. J. R. Avula
Abstract-The sorption of penetrant in a glassy polymer and the subsequent polymer relaxation give rise to a moving boundary problem. The governing equations for these phenomena, proposed by Peppas and Sinclair have been critically examined. It has been determined that these equations violate the mass conservation criterion. Thus, they have been amended, and analytical solutions obtained for some of the resultant equations. The cases considered are Fickian and non-Fickian transport with/without density change between gel and glassy polymer.
INTRODUCTION
Transport of a thermodynamically compatible penetrant in a glassy polymer results in the lowering of the glass transition temperature of the polymer. The polymer may then undergo a glass-gel transition due to the increased mobility of the macromolecular chains. While glassy polymers have been extensively studied to understand the mechanism of such a transition [l, 21, considerable attention has also been focused on these polymers because of their widespread application in biomedical, pharmaceutical, environmental, and agricultural fields [3]. Fickian or non-Fickian transport of a penetrant may be observed depending on the rate of macromolecular chain relaxation and characteristic diffusion time [4]. Due to the state transition from glass to gel, a moving front separating the glassy region from the gel region is observed [5, 61; thus, modeling of these phenomena is effected by moving front analysis, thereby yielding numerous models [7-l I]. Among the models, the one proposed by Peppas and Sinclair [12] has considerable utility because of the ease with which the governing equations and their solutions can be adopted for modeling transport of both the penetrant and an active agent through a controlled release device. These equations and solutions for a semi-infinite glassy polymer, therefore, have been critically examined in this work. The necessary corrections have been proposed and exact solutions obtained for some of the resultant equations. This work is a continuation of our earlier works on mathematical modeling and simulation of controlled release devices [13-l 51.
MASS
CONSERVATION
CRITERION
The system under consideration consists of a semi-infinite glassy polymer in contact with a uniform external medium, M, containing a liquid penetrant at concentration C,. Due to dynamic relaxation and swelling of glassy polymer, a moving front is formed at x = X(r); this front separates the gel region (layer 1) from the glassy polymer region (layer 2), as illlustrated in Fig. 1. It is assumed that a critical concentration of the penetrant is required for the glass-gel transition to occur. Thus, the front concentrations are at this critical level; these concentrations (on the glassy and gel sides) are related through a partition coefficient, k,, i.e. C,, = k;Czt t To whom
all correspondence
should
be addressed. 941
(la)
948
S. K. SINGH and L.T.
FAN
M
2
C,
0 Fig. 1. Penetrant transport in semi-infinite glassy polymer. Region M-external medium with penetrant concentration C,; region l-gel; region 2-glassy polymer; X(t)-interface between regions 1 and 2 at penetrant concentration C,.
where
c,,= C,(X,
l)
(lb)
C,, = C,(X, 0.
(lc)
and
C,, and C,, are characteristic of the polymer/penetrant system. The polymer is initially free of the penetrant. The instantaneous penetrant mass balance over the semi-infinite polymer ranging from the gel-medium front at x = 0 to boundary at x = cc may be written as (when the penetrant is not consumed/produced by any chemical reaction) penetrant flux into the polymer at the gel-medium front The rate of penetrant accumulation
I[ =
Rate of penetrant accumulation in the polymer
in the polymer is m d ’ C, (x, t) dx + C,(x, r) dx zi; [S 0 sX and denoting the penetrant flux into the polymer by J, we obtain x z C,(x, t)dx + Cz(x, t) dx . JIY4=; [s 0 sX Applying Liebniz’s rule to the right-hand side of this expression, we obtain
1’
1
1
-
HS
C, (x, r) dx - C,(X, t) d$
+
(2)
=a
- Cz(x, t) dx x at
or
s xa
1
xa
dX -C,(x,~)dx+(C,,-Cz,) . (3) o at s x at [ dt The model equations for the system under consideration as presented by Peppas and Sinclair [12], may now be examined in the light of the mass conservation criterion represented by equation (3).
JI.x=o=
- C,(x, t) dx +
Fickian difSusion without density change Fickian diffusion of a penetrant through the two regions is described by the governing differential equations [ 121,
ac, -=
at
D
a2c, o
l-p3
t>O
Anomalous transport of penetrants in glassy polymers
949
and
ac2_
x-
a2c2 x+
D
t>o.
23p3
(5)
[ 1e
Substituting equations (l), (4) and (5) into equation (3), we obtain
s 0~
x
a2c2
-dx
D2a2
(6)
+ (C,, - C,,) !$
Integrating term by term and rearranging, the mass balance equation reduces to _D
3
1 ax
+D,5 x=x-
=(C,rax .Y=x+
(W
c2d
which represents the flux continuity requirement at the gel-glassy polymer front x = X(t) (see Fig. 1). However, Peppas and Sinclair [12] have also assumed that C I, =
c2, = c,,
which requires that the right-hand side of the above equation be zero. The corresponding equation given by Peppas and Sinclair [12], however, has a non-zero right-hand side as shown below:
-D,!S! ax
_x
.I -
_+D22~x=x_=c’~]
(7b)
Fickian dz$&sion with density change
For the case when the glass-gel transition is accompanied by a change in density, the governing differential equations, as presented by Peppas and Sinclair [12], are
acl-D w
dt -
(P,-P~) dx acl
1 a.9+
p2
[
-z-z 1
o
cx_;
t
,.
(8)
”
and
ac2_
at-
a2c2 x+
D
-
2 ax23
t>O
where p, and p2 are the densities of the gel and glassy polymers, respectively. Substituting equations (I), (8) and (9) into equation (3) gives
+
s
a2c2
mD 2&x
X
+ (G, -
c2A
g [
1 WV .
Since the front position, X(t), is a function of time only, the mass balance equation above reduces to an expression for the flux continuity at x =X(t), i.e. -D
ac,
I ax
x=x-
+
(114 D22 lrcx+ =(~c1~-c2~)[t$]*
Under the assumption that [12] C I,=
c2, = c,,
the above equation reduces to
(1lb) The corresponding
equation by Peppas and Sinclair [12], however, does not contain the factor, side, i.e.
(p, - p2)/p2, on the right-hand _D
i%
1 ax
X-X-
+L&jx+=Cf3
(1lc)
950
SINGH
S. K.
and L. T.
FAN
Non -Fickian d@ision The governing differential equations for the non-Fickian diffusion of a penetrant based on Wang et al. [ll] are [12]
ac,
ax-, ac,
ar=D,ax’-vax,
O
t>o
(12)
t>O
(13)
and
ac2
a2c2
D
at=
ac2
2p-v-$
x+dx
where v represents the so-called pseudoconvective contribution to transport, arising from the normal component of stress tensor of the penetrant. Substituting equations (l), (12) and (13) into equation (3) we obtain
a2c, -*,g+,c, = ’ 1 S[ --a2 x
.V=0
D
1mDac +S[ 2 1
ac,
v-
0
dx
ax
x
--vz ax2
dx+(C,,-C,,)
[1 g
.
(14)
Integrating by parts and rearranging, this mass balance equation reduces to
-D,dci ax
_
(154
_+D2c? ax
I-X
x=x+
For c,, = c,, - c,, as assumed by Peppas and Sinclair [12], the right-hand side of the above equation is zero. The corresponding equation by Peppas and Sinclair [12], however, has a non-zero right-hand side as shown below:
-D,ac, ax
_
x-x
AMENDED
_+o,~~~_x+=C,~~]-C,v. Wb)
SYSTEM
OF
GOVERNING
EQUATIONS
The amended governing equations for Fickian and non-Fickian transport of the penetrant in the system under consideration are given below along with the exact analytic solutions for the Fickian diffusion cases. Fickian difSuon
without density change
Fickian transport, without change in density of the glassy polymer when transformed into gel, is described by the following governing differential equations (kinetics of the chain relaxation phenomena being neglected).
ac,
D
-jy=
ml
qp?
o
t>O
(16)
t>o.
(17)
and
ac2_
at-
D
a2c2 x+~xxco;
237’
The polymer is assumed to be initially free of the penetrant: therefore, we have C&x, 0) = 0
(18)
X(0) = 0.
(19)
and
Anomalous transport of penetrants in glassy polymers
951
Assuming that the gel and medium are in equilibrium at x = 0, we have c, (0, t) = c, = k&s
(20)
where k0 is the gel-medium interface partition coefficient, and C, the bulk medium concentration. For the semi-infinite system, C,(co, t) = 0
(214
and
ac*
-D2dxx=~ The concentration continuity equation (l), reiterated here
= 0.
@lb)
at the gel-glassy polymer front [x =X(t) C,, = k,C,,,
in Fig. I] is given by
x =X(t)
(22)
where C,, and Cz,, defined by equations (1 b) and (lc) are constants, characteristic of the polymer-penetrant system. They represent the critical penetrant concentration at which the glass-gel transformation occurs. The continuity of penetrant flux at this front gives rise to
-D,!S! aX
_ _+D2dc, ax x-x
x=x+
= cc,,
-
C2,)
g [
1 .
(23)
This is identical to equation (7a) derived from the criterion of overall penetrant mass conservation at any instant. By assuming that the position of the front, X(t), is given by X(t) = at “2
(24)
where tl is a constant to be determined, and by performing a similarity transformation
the concentration
profiles are obtained as [I& 171,
(25)
Czb,
l)
=
(26)
c2,
The value of a is recovered from the following transcendental equations (25) and (26) into equation (23).
equation, derived by substituting
= -
f (C,, - C,,).
(27)
Fickian d@usion with density change
Assuming that the relaxation and swelling phenomena are instantaneous, and taking into account the change of densities between the two phases, the governing differential equations for the system are given as
ac, -= at
D
a2c, o
I=’
t>o
(28)
S. K. SINGHand L. T. FAN
952
and dC2 _ D -_a=c2
dt-
=
@=-PI)dX -
ax2
P2
[
ac=
x+
dt 1 ax’
tbO
”
(29)
The second term on the right-hand side of the second equation arises from the fact that when observed from the origin (see Fig. l), each point in the glassy region appears to move in the (+x) direction as a consequence of the volume expansion at X [18]. Since the polymer is initially free of the penetrant, the initial conditions given by equations (18) and (19) are applicable. Equations (20) and (22) describing the equilibria at the gel-medium and gel-glassy polymer fronts are also valid for this case. The requirement of penetrant flux continuity at x -+ co (equal to zero flux) gives rise to
c2 =o 11
ac,+(~z-PI) dX = ax P2 [ dt
_D
while
(30)
.x= x
C,(co, t) = 0.
The continuity of penetrant flux at the front, x =X(t),
(21) given by
[flux inId= *- - [flux out], = X+= accumulation yields _D
over [dX/dt](,,
- 11
ac,+b2-P,)
dX
. = cc,, - G,) (31) s=x+ By substituting equations (22), (28) and (29) into equation (3) and integrating by parts, an expression identical to equation (31) is obtained from the criterion of overall penetrant mass conservation. The above system of equations can be solved by assuming that = ax
~2
[ dt
c2
X(t) = CltI,= as in equation
(24). Solving the above system of equations by the procedure described previously,
the concentration
profiles
are obtained
as
(32)
and X erfc ___
(
(P2-PII
-
2JD,t
Cz(x,
t)
=
c2,
a
P2
h/z
1.
(33)
c(
erfc E!! ( Pz 2JD,
)
The transcendental equation for c1is obtained by substituting equations (32) and (33) into equation (31).
I--* D
I
7r
(--$1 ( >
(Cl, - C&v
erf 2fi
a
’ +
Gexp($$-)
$ T’
erfc(fi P2
a
)
=-;(~G~-G).
(34)
2Jrds
Non -Fickian d@usion without density change
Anomalous transport effects due to macromolecular relaxations were accounted for by a pseudoconvective term, v, in the generalized transport equation by Wang et al. [l I]. It represents
Anomalous transport of penetrants in glassy polymers
953
the convective velocity created as a result of stress-gradients in the system. We will take “v” to be a characteristic of the polymer-penetrant combination. For the sake of generality, the value of this stress-induced drift will be denoted by v, in the gel region and by v2in the glassy polymer region. In a constant density system, the transport of penetrant in the gel and glassy regions is then governed by, respectively,
ac,_ D azc,
at-
ac, o
lp-“‘-g
t
>o
(35)
and
ac2
D
j-y=
a2c2
ac,
23742p
x+
t>o.
(36)
Equations (18)-(22) also hold, along with
-D2Z+v2c2 [
= 0. 1
(37)
X=30
This expression implies that no transport occurs across the boundary at x+00. penetrant flux at the front, x = X(t) (see Fig. 1) gives rise to
-D,z __+D,s X-X
ax
The continuity of
(38) x=X+
When v, = v2z v, this is identical to equation (15), derived from the overall penetrant mass conservation criterion. The technique of similarity transformation is not applicable to equation (35), since no two initial and boundary conditions for this equation may be combined into one. However, an approximate analytic solution can be obtained under the pseudo steady-state assumption. We have earlier stated that the glass-gel transition occurs at a critical concentration C,, (in equilibrium with Clt); thus, C,, and C,, are constant at the gel-glassy polymer front. For the penetrant concentration at any interior point to attain this finite critical concentration, the diffusion of penetrant and the development of the concentration profile must precede the movement of the gel-glassy polymer front. The concentration profile in the gel region may, therefore, be taken to be at the steady state with respect to the position of the gel-glassy polymer front. Under this assumption, the governing equation in the gel region, equation (35), is rewritten as D
ax ,,!$o, --
1 a2
o
t z-0.
(39)
This equation can be readily integrated subject to the boundary conditions, equations (20) and (22), to yield the concentration profile as exp F - 1 exp (g) - exp(g) ( > C,(x, t) = C,, v; + C, VJ exp + - 1 exp - 1 ( J ( D, > The concentration assuming that
(40)
profile in the glassy polymer region can be obtained from equation (36) by X(t) = at”’ + v,t
(41)
and by resorting to a similarity transformation (42)
954
S. K. SINGH and L. T.
FAN
Following a procedure similar to that described earlier, the solution is obtained as
(43)
The parameter tx is recovered by substituting equations (40), (41) and (43) into equation (38), thereby yielding VI (Cl, -
1 - exp Non-Fickian
(
-
ce>
av,t”2 + v, v,t D, d@ision
$> ( =J1 ( ) D
+ s.
Cztev -
2
CL
ci
(Cl,
-
C2,)
+
C,,(v,
-
VA.
(44)
2J;
erfc 2Jz
with density change
The governing differential equations for this type of polymer/penetrant behavior may be written by combining those of the two preceding cases, to account for both the density change and anomalous transport effects. These are
ac,
at=
D
a*c,
ac,
o
rg-kg
and LX,
-=
at
D
a*c2
-2 2x2
ac2 \?2dx
(Pz-LQ) -
p2
-1 dX
[
dt
ac2
t>O
x+
dx’
(45)
’
f >&
,
Equations (18) and (19) are the initial conditions while equations (20), (21) and (22) are the boundary conditions for the above set of differential equations. The requirement that no transport occurs across x -+ co leads to -D2~+v2C2+(p2-p1’[~]C2] P2
Continuity
=O. I=z
(47)
of penetrant flux at x = X(t) (see Fig. 1) gives rise to
-D,%! ax
_ _+.;~~~=~+=(~,~-~~~~)[~]-V~~II+~*~~I .‘i-x
(48)
which is identical to the equation obtained by substituting equations (45) (46) and (47) into equation (3) and integrating. The governing equation for the gel region, equation (45) may be solved by resorting to the pseudo steady-state assumption, as in the previous case. The solution is again given by equation (40). However, equation (46) for the glassy polymer region cannot be solved by resorting to similarity transforms since an explicit expression for X(t), analogous to equation (41), cannot be written for this case. A numerical approach may be required for the exact solution of this case. RESULTS
AND DISCUSSION
Comparing the governing equations of the models and their solutions presented in this work with those by Peppas and Sinclair [12] (under the assumption that C,, = C,, z C,), we observe that the difference in the first case (Fickian diffusion without density change) lies only in the forms of the tlux continuity expression at x = X(r) [equations (7a) and (7b)]. It has been demonstrated that the expression given by Peppas and Sinclair [12], equation (7b), does not satisfy the mass conservation criterion. The resultant expressions for concentration profiles in both works are similar but not identical in that the values of the parameter a in the expressions are different; the transcendental equations giving rise to the values of c1in both works are not in agreement [16]. This parameter, determines the extent of the gel region and that of the glassy polymer region, and the shape of the concentration profile.
Anomalous
transport
of penetrants
in glassy polymers
955
For the case of Fickian diffusion with density change, the governing differential equations presented in this work differ substantially from those given by Peppas and Sinclair [12]. Their flux continuity expression for this case, equation (1 lb), again has been shown not to satisfy the mass conservation criterion. The amended governing equations presented in this work lead to the expressions for the concentration profiles and the transcendental equations for evaluating the parameter c( which are at variance with those presented by Peppas and Sinclair [12]. In the case of non-Fickian diffusion, we observe that the solution for the concentration profile in the gel region, C, (x, t), presented by Peppas and Sinclair [12] (for v, = v2= v) x - vt 1 + erf ___ ( 24% ) c, (x, t> = c, + IC, - c,>
1 + erf -L ( 2JD, > does not satisfy the boundary condition given by equation (20). Their expression for the flux continuity at X(t), equation (15b), has been shown not to satisfy the mass conservation criterion. While the expressions for the concentration profile in the glassy polymer region are similar in both works, they are not identical (even for v, = v2= v) due to the difference in the value of the parameter c(. This is due to the fact that the pseudo steady-state assumption is imposed on the gel region in this work, and that the flux continuity expressions at X(t) in the two works, under the condition that v, = v25 v, are at variance. It must be mentioned that the assumption
c,, = c,, = c,, made by Peppas and Sinclair [ 121 is an oversimplification. In general, the concentrations in equilibrium across an interface separating two distinct phases cannot be expected to be equal. Values of the various parameters need to be determined to obtain the concentration profiles from the expressions presented in this work. Experiments may have to be performed with the specific polymer-penetrant system to determine the nature of the phenomenon (Fickiamnon-Fickian) and the values of D, , D,, p,, pz and “v” since the same system can behave differently under different ambient conditions. Solution of the transcendental equations [especially equation (231 is greatly facilitated by the graphs presented in [ 161.The value of the parameter CLfor the case of non-Fickian diffusion is dependent on t [see equation (44)]. An iterative approach will thus be needed for 1 .o
0.9
CYW.
F
0,
1
10-6
2
146
D2
1.0
0
7x10-’ 109
7.9583
Id
9.3238
x10-4
0.9 0.6
cur”*
D,
1
106
2
lo*
4 7x10-'
0
1.0137110-3
16
1.3299~10-'
...-7
4.3278x10-'
I 0
0.01
0.02
Distance
0.03
0.04
0.05
0.06
into polymer
Fig. 2. Concentration profiles at r = 100 s for Fickian diffusion without density change for different D, and D, (diffusivities are in cm2/s; concentration is given as fraction of the bulk concentration; distance into the polymer is in cm).
Distance
into polymer
Fig. 3. Concentration profiles at I = 100 s for Fickian diffusion with density change (p,/p, = 1.5) for different D, and D, (diffusivities are in cm2/s; concentration is given as fraction of the bulk concentration; distance into the polymer is in cm).
S. K.
956 D~-W6,
1.0
0.9
0
0.02
D, .lCP,
1
10 100
8.46‘4
7.9309
0.04
Distance
I
W4
1.0
8.3222 x lO-4
,000
0.06
into
‘J,*TxlD-‘,
C”W*
a
t
2 3
L. T. FAN
~‘7x10-‘,~,*~~10-~,“~.5~10~~
C”,“.
c
and
SINGH
0.9
I 10-4
0.06
0.10
0
0.02
1
,D
8.6973
x10-4
100
9.0738
x lo-’
1.043,
I 10-3
,000
0.04
Distance
Fig. 4. Concentration profiles at different times for diffusion without density change non-Fickian (diffusivities are in cm2/s; velocities are in cm/s; concentration is given as fraction of the bulk concentration; distanceinto the polymer is in cm).
u,.O.D
a
2 3
polymer
u,.5~10-~,
t
0.06
into
0.08
0.10
polymer
Fig. 5. Concentration profiles at different times for non-Fickian diffusion without density change (diffusivities are in cm?/s; velocities are in cm/s; concentration is given as fraction of the bulk concentration; distance into the polymer is in cm).
evaluating the concentration profiles. It appears that a numerical solution is required to obtain the exact solution for this case, although the system is singular at t = 0 [19,20]. Some representative simulations have been performed; the results are shown in Figs 2-6.These results are based on the order of magnitude values of D, , D,, p,,pz, v, and v2, assumed. The concentration profiles for three different sets of diffusivities for Fickian diffusion (at time t = 100 s)
t
CW”.
1.0 r
100 ‘O
i
0.9 t
3
,000
3.3363
x lo-”
4
1ODOO
4.0386
I 10-a
E j_
9 5
s2
0.4
-
0.3
-
0.2
-
0.1
-
0
0.02
0.04
Distance
Fig. 6.
0.06
into
0.06
J 0.10
polymer
Concentration profiles at different times for non-Fickian (diffusivities are in cm*/s; velocities are in cm/s; concentration concentration; distance into the polymer
diffusion without density change is given as fraction of the bulk is in cm).
Anomalous transport of penetrants in glassy polymers
951
are presented in Fig. 2; the corresponding profiles for Fickian diffusion with density change are shown in Fig. 3. The effect of density change becomes increasingly significant with an increase in D,;this is due to the fact that the increased gel-region diffusivity accelerates the rate of movement of the front. Concentration profiles for non-Fickian transport (without density change) at various times are illustrated in Figs 4-6 for different sets of parameters. A simple model of penetrant transport in glassy polymers is of considerable importance due to its applicability to polymers used in containers, storage tanks, etc., and also to polymers employed for controlled rate delivery of active agents such as pharmaceuticals, fertilizers, and pesticides 131. The solutions presented here can be employed to predict the concentration profiles of the penetrant and to determine the position of the gel-glassy polymer front. Thus, by fixing the extent of the gel region and that of the glassy region, the behavior of the active agent may be analyzed. Acknowledgemenrs-This is contribution No. 86-141-J, Department Experiment Station, Kansas State University, Manhattan, KS 66506.
of Chemical Engineering,
Kansas Agricultural
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