Non-fickian diffusion with chemical reaction in glassy polymers with swelling induced by the penetrant: a mathematical model

Chemical En&ecrine Sckrtce, Vol. 48, No. 16, pp. 2957-2971, Printed in Great Britain.

1993.

OlO’-2509/93 $6.00 + 0.00 0 1993 Pergamon Press Ltd

NON-FICKIAN DIFFUSION WITH CHEMICAL REACTION IN GLASSY POLYMERS WITH SWELLING INDUCED BY THE PENETRANT: A MATHEMATICAL MODEL Department

N. J. M. KUIPERS and A. A. C. M. BEENACKERS’ of Chemical Engineering, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands

(First received

10 February

1992; accepted

in revised form for publication

1 March

1993)

Abstract-A mathematical model is presented which describes non-Fickian diffusion with chemical reaction of a penetrant A with some reactive group B of a granular glassy polymer. The diffusion process cannot be described by Fick’s law due to the swelling of the glassy polymer grain caused by the penetration of the diffusing species. Therefore, the kinetics of swelling must be taken into account. A new model is presented, allowing for mass transfer with chemical reaction whilst assuming power-law kinetics for the velocity of the swelling front:

dro -= dr

- K(C,,,=,o - c.+)”

where c.’ is the threshold concentration for swelling, ea.,=, the concentration of penetrant A at the position re of the swelling front and K a constant independent of the penetrant concentration. Two cases of reaction kinetics are analysed: first order in A only and first order in both A and B (overall second order). Criteria for the occurrence of homogeneous addition and for B shrinking core type of reaction are given. For a low conversion of B, the reaction can be pseudo-first-order in A. Both Fickian diffusion (high K, n r 0 and c.’ = 0) and the so-called case II diffusion (low K, n r 0) are shown to be asymptotic solutions of the model. It is also shown that in some cases an excellent agreement can be obtained between the numerical unsteady-state solution and the so-called pseudo-steady-state solution in which the accumulation term of the mass balance of A is neglected. This PSS-solution could be obtained analytically. Both the case without reaction and with first-order kinetics in A are analysed. The approximation is rather good for high first-order reaction rates, short diffusion times and if case II diffusion is approached.

INTRODUCTION

It is known that the diffusion in many polymers cannot be described adequately with Fick’s law, especially not when diffusing species cause an extensive swelling of the polymer (Crank, 1975). Particularly this is the case with glassy polymers. The physical reason for the deviation of Fick’s law is the time dependency of the properties of a glassy polymer, due to the finite rate of adjustment of the polymer chains to the presence of the penetrant, often a low-molecular-weight solvent. Alfrey et al. (1966) distinguished three classes of diffusion based on the relative rates of diffusion and polymer relaxation:

(1) Case I or Fickian diffusion in which the

ary between a swollen shell and a glassy core. In anomalous diffusion both the diffusion coefficient and the velocity of the swelling front play a role. Extensive reviews of the experimentally observed diffusion anomalies in various polymer systems are available (Alfrey et al., 1966, Hopfenberg and Stannet, 1973; Astarita and Sarti, 1978; Park, 1983). The most characteristic features for diffusion in a planar sheet have been obtained at temperatures below the glass transition temperature; there exists a threshold solvent concentration above which the following features are reported (Astarita and Sarti, 1978):

(4 There is a sharp discontinuity in the polymer,

rate of

diffusion is much less than that of relaxation.

(2) Case II or relaxation-balanced diffusion in

which ‘the diffusion is very fast compared to the relaxation processes. (3) Non-Fickian or anomalous diffusion which occurs when the diffusion and relaxation rates are comparable.

Case I systems are controlled by the diffusion coefficient. In case II diffusion the parameter is the constant velocity of an advancing front which forms the bound‘Author

to whom correspondenceshould be addressed. 2957

03 (c) (4

which separates a glassy region, where the solvent concentration is negligible, from a swollen rubbery region with a high solvent content. Thomas and Windle (1982) observed that a small amount of solvent is present ahead of the front for case II diffusion; its concentration profile looks Fickian. The discontinuity initially moves through the polymer with a constant velocity [see also e.g. Weisenberger and Koenig (1991)]. Initially, the amount of solvent absorbed also increases linearly with time. The activation energy for the initial front velocity is not comparable with that for a diffusion process but close to that for craze formation.

2958

N. J. M. KUIPBRS and A. A. C. M. BEENACKERS

(e) The front position vs time curve can be presented by a power-law relation with an exponent ranging between 0.5 and 1. However, according to Jaques et al. (1974) some polymers exhibit swelling behaviour corresponding to a rate of the swelling front that increases with time. They called this as “super cast II”. Klech and Simonelli (1989) also attributed the apparent acceleration of the water swelling front movement in spherical gelatin beads to super case II transport. According to Lee and Kim (1992), however, this is simply the result of spherical geometry and not a super case II transport behaviour. (f) Deviations from feature (c) occur already before deviations of feature (b) are observed. According to Lasky et al. (1988b) another feature of case II diffusion is: (g) A critical solvent concentration must be reached before case II diffusion occurs. An induction time for case II diffusion also exists: time is needed to create sites in the polymer for the penetrant. Most of the existing mathematical models cannot describe these phenomena. Often they try to take into account the discontinuous morphological change at the position of the front by introducing a stressdependent diffusion coefficient (Crank, 1953), a concentration-dependent (Crank, 1975) or a time-dependent diffusivity (Petropoulos and Roussis, 1978) but nonetheless feature (c) cannot be predicted. Astarita and Sarti (1978) could explain most of the phenomena described above by taking the kinetics of the phase transition into account. They used a power law to describe the swelling behaviour of the polymer which takes place at the moving discontinuity and assumed that the rate of swelling depends on how much the local penetrant concentration exceeds some threshold value. Others tried to find an expression of the velocity of the swelling front, related to physical parameters. Sarti (1979) based his model on the premise that the ratecontrolling step at the advancing penetrant front is the formation of crazes in response to the solvent osmotic stress. The rate of the swelling front is proportional to the difference of the stress build up at the front and some critical tensile stress, above which crazes are formed. The value of the stress at the front is a non-linear function of the solvent concentration at this position. Obviously, this model is only appropriate, in case solvent penetration is accompanied by crazing. Thomas and Windle (1977, 1978, 1980, 1981, 1982) have proposed a rate-controlling step at the penetrant front which is the time-dependent viscoelastic deformation of the polymer (relaxation of the polymer chains) in response to the thermodynamic swelling (osmotic) stress. Based on their model Lasky et al. (1988b) obtained for case II diffusion an expression of

the velocity of the swelling front which is proportional to the square root of the ratio of penetrant diffusivity and viscosity of the glassy polymer. Another model of case II diffusion is by Rudolph and Peschel(l990). However, they have not taken the kinetics of swelling into account, but assume an equilibrium at the interface between swollen and glassy polymer: this causes the well-known moving discontinuity at the position of the swelling front. To our knowledge, van Warners et al. (1990) were the first in publishing on non-Fickian diffusion with chemical reaction. However, their analysis is restricted to a constant velocity of the swelling front and to reaction kinetics which are first order in the difiusing reactant and zero order in the reactive group of the polymer. In this paper their model is extended by introducing a chemical reaction not only of first-order kinetics in the diffusing species but also in the reactive group of the polymer. Further, we no longer assume a constant velocity of the swelling front but adopt the swelling kinetics of Astarita and Sarti (1978). In addition, a spherical geometry of the polymer is assumed instead of one-dimensional diffusion in a semi-infinite body as described by Astarita and Sarti (1978). It will be shown that both Fickian behaviour and case II diffusion are asymptotic solutions of our model of diffusion and chemical reaction of a penetrant in a glassy polymer, respectively. The gas-solid hydroxyethylation of starch, below its glass transition temperature, is an example of such a non-Fickian process (van Warners et al., 1990).

THE MODEL

A glassy polymer grain with radius R, in which absorption of a penetrant A causes swelling of the polymer, can be schematically presented as shown in Fig. 1. The grain consists of a swollen shell (due to the plasticizing effect of A) and a glassy core. The timedependent position of the moving front between rubbery and glassy polymer is ro. It is assumed that the increase in diameter of the grain, caused by swelling, is negligible. As a first approximation, this seems to be reasonable because the maximum increase in diameter of a starch granule due to water sorption is 12.7% with respect to the vacuum-dry diameter (Hellman and Melving, 1950). Assuming a chemical reaction between the diffusing species (penetrant) A and some reactive group B of the polymer we can write A+B-*P. In this paper two cases are considered: (1) penetrant A reacts according to a first-order reaction with reaction rate R, = Rb = - klc,; and (2) the diffusing species react according to a (1,l) reaction with reaction rate R,, = Rb = -

k, I CA.

Non-Fickian diffusionwith chemicalreactionin glassypolymers

‘O -

R

Fig, 1. Schematicdiagram of the glassy core and rubbery shell for a swelling promoting penetrant diffusing into a grain of a glassypolymer.

If the convective flux of pen&rant A in the polymer can be neglected, the unsteady, isothermal mass balance of A in the swollen part of the polymer grain is +R.,

r,GrfR

(1)

where c, is the concentration of A in the swollen polymer at radius r at time t, and IID. the diffusion coefficient of A in the rubbery part of the polymer. Only transport in the swollen region is considered, whereas the lower diffusion rate in the unpenetrated glassy core is neglected. A similar approach was successfully applied by Gostoli and Sarti (1983) in predicting characteristic features of the anomalous transport behaviour. In practice, the diffusion coefficient in the rubbery part of the polymer often increases with increasing penetrant concentration (Park, 1983). At low penetrant concentration an exponential increase may be expected, according to the free volume theory (Meares, 1986). However, in this paper, in agreement with Astarita and Sarti (1978), we assume a concentration-independent diffusion coefficient to describe this problem. If derived, the extension to a model with a concentration-dependent D, is relatively simple. In case of a (1,l) reaction, the partial differential equation for reactive group B must be solved simultaneously to find the concentration profile of A:

ac,

dt

=

R, = - h,

L~oct,r

f-,
R

(2)

where cI is the concentration of reactive group B of the polymer. The boundary conditions of this problem are: r = R, t 2 0:

c, = CT,,,,

r = r,(t):

~ac”=_cd’o aar

(3)

adt

(4)

where c,,, is the equilibrium solubility of the solvent in the swollen polymer while r. is the time-dependent position of the interface between rubbery and glassy polymer. Equation (3) is valid because it can be as-

2959

sumed that the induction period [see Thomas and Windle (1982) and Lasky et al. (1988a)] is relatively short, i.e. equilibrium occurs almost instantaneously (Cole and Lee, 1989). Equation (4) follows from a mass balance across the discontinuous interface with the assumption that no penetrant enters the glassy polymer: this means that the diffusion coefficient in the glassy core is zero (Sarti, 1979) which often is an acceptable assumption (Waywood and Durning, 1987; Tong et al., 1989; Weisenberger and Koenig, 1990). According to Astarita and Sarti (1978) we may generally use a power law for the kinetics of the swelling front:

dro dt=

-WC, ,op=, -c.*)”

-

So, a driving force for swelling is assumed which equals the difference between the penetrant concentration at the front and some threshold concentration c: [see also Lasky et al. (1988a)]. A high penetrant concentration results in more swelling and a higher velocity of the front (Harmon et al., 1988). K is a positive constant which is a function of temperature (Lasky et al., 1988b), the content of other plasticizers present, e.g. water in case of biopolymers (Klech and Simonelli, 1989), and of the molecular diameter of the solvent (Aithal and Aminabhavi, 1991; Gall et al., 1990; Papanu et al., 1990). n cannot be negative. According to Berens (1989), Fickian diffusion is observed when the temperature or the concentration of the penetrant are so low, i.e. c, < c.‘. that the polymer remains glassy and almost no plasticization (swelling) occurs (Berens and Hopfenberg, 1980; Wessling et al., 1991). Fickian behaviour may also be expected when the temperature exceeds the unplasticized glass transition temperature of the polymer as is reported by Storey et al. (1991). Astarita and Joshi (1978) extended the model of Astarita and Sarti (1978) by allowing for some diffusion in the glassy polymer core too. This results in a slightly adapted boundary condition at the front [eq. (4)]. The initial conditions for this problem are t=0:

l&=0,

O
cb = c,,,,,

0 4 r < R

(6)

r. = R

where cbO is the concentration of the reactive group B before reaction starts. For the glassy part of the polymer the conditions are c, = 0,

O
cb = cbO.

0 d r ( r0.

(7)

Once the front has reached the centre of the grain, all the glassy polymer has disappeared and Fick’s law is valid everywhere in the grain. Since then the equation of the moving boundary (5) is useless, and boundary

N. J.

2960

M.

KUIPERS and A,

r=0:

dc.= 0. ar

A. C. M.

BEENACKERS

Substitution of equations (12)-(16) into equations (i)-(5) gives

condition (4) has to be replaced with

(8)

The conversion lb of the reactive polymer group B is found from

rb < r’ < 1 for a first-order reaction

ac, x _=__

r-

ae

8 dr’

( aca1

dr’ - 41. IXGG

r,l

rh z$ r’< 1 for a (1.1) reaction

(17b)

COO

aCb

The amount of penetrant A taken up by the polymer grain at time t is equal to

(17a)

x=

-

&‘l.,X-&C.Cb

rb C r’ < 1 for a (1,l) reaction

(18) (19)

The maximum amount of A which can be taken up by the polymer consists of a physical absorption term and a reaction term M,

= f nR’cao +

;nR3cbo =

with dimensionless boundary conditions r‘=l:

C,=l

and initial conditions

(11) where cL,-,= 0 in case of no reaction. In practice, M, is the parameter which has to be measured to validate the model. The differential equations can be rewritten by introduction of the following dimensionless groups:

(12) (13) (141 D, ’ = K(c,,,, - c:)“R

(1% (16)

where x is a reciprocal P&let number equal to the ratio of the diffusion velocity and the maximum velocity of the swelling front; r& is related to the so-called Thiele number Thl for a first-order reaction in a sphere by & = 9*Z’Jz: [for a (1,l) reaction, this relation is: bl, 1 = 9*7%:, J. A dimensionless time 8 is obtained from dividing by the minimum time necessary for the front to reach the centre of the grain: with no diffusion limitation. So, all the polymer is in the rubbery state at 8 = 1. Sometimes the Fourier number, Fo, is chosen as the dimensionless time, particularly for easy comparison with the asymptotic case of Fickian diffusion. However, in most cases, 8 is used because of a favourable scaling compared to Fo. Note that values of 0 can be converted into Fo because FO = x*13.

e=o:

c,=o c*= 1

(22)

i-b = 1.

Note that C: participates in the parameters 8, x, the equation of the moving swelling front [eq. (19)] and in boundary condition (21). When the influence of the value of C: is investigated, e.g. on the concentration profiles, the value of x is kept constant by changing the value of D, or K, i.e. only the effect of Cj’ on the moving front equation (19) and the boundary condition (21) is considered. The conversion of B is I’= 1 (r’)‘Cb dr’. cb = 1 - (rb)’ - 3 (23) s “=‘; The amount of A which is absorbed by the polymer grain at time B or Fo is equal to MO = =

8=8 s

de = MFO

B=O Fd=Fo

dFo.

s Fo=O

(24)

Only a numerical solution of the set of equations [eqs (17a)-(22) J can be obtained. However, it is possible to get an approximate solution by using a pseudosteady-state approximation (PSS). IMPLEMENTATION

OF THE MODEL

moving boundary problem as described by the set equations (1) to (6) can be converted in a fixed boundary problem by introducing the parameter The

I* =

s +

1.

(25)

2961

Non-Fickian diffusion with chemical reaction in glassy polymers

In eq. (25) the number 1 is added to r* to overcome problems with the NAG-routine DO3PGF (NAG, 1982) which the set of transformed equations with fixed boundaries (r* = 1 and r* = 2) can be solved. In this routine the non-linear parabolic partial differential equations are approximated by a system of ordinary differential equations in the time variable. The equations are obtained by approximating the space derivatives by finite differences. The solutions of the simulations will be presented by using the dimensionless parameters as given in eqs (12)-(16) for the cases of diffusion without reaction, diffusion and first-order reaction and diffusion and (1,l) reaction for r. z 0. NON-FICKIAN

DIFFUSION 41 = +1.1=

WiTHOUT

REACTION:

0

The concentration profile of penetrant A in the polymer follows from the solution of the eqs (17a)-(22) with r#~t= $t,t = 0. Figure 2 shows such profiles and its dependency of the parameters x and C: for 8= 1 andn= 1. Increasing x implies a relative increase of the diffusivity compared to the velocity of the swelling front and results in a higher concentration of penetrant A because of less diffusion limitation, and therefore a flattening of the’concentration profile. If C, at the swelling front is high then also the front velocity is relatively high and this means that, for a given time 6, the front has penetrated relatively far into the polymer grain; the value 6 = 1 theoretically corresponds with the minimum time necessary to reach the centre of the sphere, i.e. for no diffusion limitation of A and n > 0. Note that Fig. 2 also can be seen as a graph which presents the concentration profile of A for different Fo times because Fo = X*&J= x. According to eq. (19) a lower threshold cancentration C: results in a higher driving force for swelling (keeping x constant) and as a consequence in a faster penetration into the polymer. Obviously the actual concentration of A at the position of the front is always higher than Cz because of a positive flux of penetrant into the glassy polymer core [see eq. (21)7. Figure 2 further shows that, for low x, the shape of the

concentration profiles looks rather similar to those obtained with Fickian diffusion. To faciiitate comparison with Fickian behaviour, Fig. 3 presents the concentration profiles for various values of x together with the profile for diffusion according to Fick’s law in case of Fo = x*8 = 0.01, C: = 0 and n = 1. The graph shows that for low values of x (this means K + 0~) Fickian behaviour is approached: therefore, the case of Fickian diffusion is an asymptotic solution of the model for x + 0 and C: = 0 for n z=0. The other boundary of this model, I+ co, results in case II diffusion behaviour, where the velocity of the swelling front is constant. Between these two asymptotic solutions, the model describes anomalous diffusion. For Fickian diffusion in a sphere (Crank, 1975) an almost linear relation is found between the position of an imaginary front re (at which is assumed C, = 0.01) and the Fourier number Fo, for Fo b 0.001 (rb = 0.865 for Fo = 0.001): rbqFicL= 0.85 - 20_112Fo and = - 20.112 for 0 s rb & 0.865. Fickian diffusion

Substitution

(26a)

occurs when

of eqs (S) and (26a) into (27) gives

or dimensionless

D. ’ = ZC(c,,, -

-

1

c,*)“R 4 20.112

z 0.05.

(29a)

Thus, Fickian diffusion occurs when x ( 0.05, about x Q 0.001; case II diffusion may be expected to take place for x & 0.05, about x 2 10 (see below).

,a c,

t

0.8

0.6

0.0 0.0

0.2

0.4

Il.6

OA

I.0

1' -----c

Fig. 2. Concentration profiles of pcnctrant A for varying _y and C: for 8=1 with n=l and Q1=&.I=O. (a) Cl = 0, (b) Cz = 0.1, (c) C: = 0.5.

Fig. 3. Concentration profiles of penetrant A compared to Fick’s solution for various values of x with Fo = 0.01, 0, = @1.1 = 0, n = 1 and C: = 0.

2962

N.J.

M. KUIPERS and A.A.C.

M. BEENACKERS

Equation (26a) is valid only for Fo z- 0.001. This is confirmed by Lee and Kim (1992) who noted theoretically and observed experimentally that only the initial stage of penetration in a sphere exhibits the same transport characteristics as in sheet samples. For small Fo (Fo c O.OOl), i.e. for a small penetration of A into the polymer, the penetration velocity of the “Fickian front” can be derived, using the theory of one-dimensional diffusion into a semi-infinite body of polymer from a plane interface. This gives for small Fo:

In case of Fickian diffusion equation (27) must be valid. For a very small penetration the velocity of the swelling front in case of non-Fickian diffusion is maximal because the concentration of A at the swelling front equals the equilibrium concentration at the sur. . face of the grain, i.e. C a,rs=,b= 1. Substltutlon of eq. (26b) into eq. (27) gives, therefore, that Fickian diffusion occurs for small Fo, when 2

J

?

a K(c,, - c,+)” or dimensionless xfi

+ 2&

= 3.5.

(29b)

Both eqs (29a) and (29b) show that the diffusion behaviour is more Fickian like for lower values of x. Equation (29b) predicts that initially (when Fo 4 0) case II diffusion occurs. During the sorption process of A a transition from case II diffusion to Fickian diffusion may occur for low values of x [x < 0.001, see eq. (43a)]. So, from eq. (29b) it follows that initially always case II diffusion occurs, for any value of x, but the lower the value of x, the lower the Fo time above which case II diffusion no longer holds. This is in accordance with feature (c) of non-Fickian diffusion of a penetrant into a glassy polymer as presented above and is also confirmed by Astarita and Sarti (1978). As noted above, Fickian behaviour is expected if the temperature exceeds the glass transition temperature of the polymer. Combined with the fact that the dXusion behaviour becomes more Fickian for decreasing x. it follows that the apparent activation energy for the propagation of the swelling front should be greater than the activation energy of diffusion. This is confirmed by data of Nicolais et al. (1978). The time dependency of the concentration profiles for different values of x is given in Fig. 4. Notably is the fact that the concentration of A at the position of the swelling front initially decreases and subsequently increases: there exists a minimum in penetrant concentration at the swelling front. This also implies that the velocity of the swelling front has a minimum. The decrease of the front concentration in time is caused by diffusion limitation: the increase is a consequence of the spherical geometry. These results are in accord-

Fig. 4. Concentration profiles of pen&rant A for different x and time fI with ~1 = 4 1,1 = 0, n = 1 and C: = 0.1. High front concentration: x = 1. (1) 8 = 0.3, (2) 0 = 0.7 (minimum), (3) B = 1.2. Intermediate front concentration: x = 0.1. (4) 0 = 0.5, (5) 8 = 1.5 (tiinimum), (6) 0 = 2.4. Low front concentration: x = 0.01. (7) 0 = 2, (8) 0 = 6.6 (minimum), (9) ti = 8.

ante with that of Lee a6d Kim (1992), who investigated the effect of geometry on solvent penetration in glassy polymers. They found that for a spherical geometry only the initial state of the penetration has the same transport characteristics as in sheet samples. This is followed by an intermediate transition region with an apparent linear front movement prior to an accelerated front penetration towards the core, which is a natural outcome for spherical geometry. In Fig. 5 the influence of the swelling parameters n and Ct is reported. For n = 0 the front velocity is constant and independent of the penetrant concentration: the front reaches the centre of the grain at 8 = 1. van Warners (1992) has described this case. For n = 1 and n = 2 the concentration profiles are about the same but different from n = 0: the penetration into the polymer is less deep. As shown above, the lower the threshold concentration C.* the higher the front velocity and the lower the penetrant concentration at the swelling front (keeping x constant). The relationship between the relative amount of A which is absorbed by the polymer, MJM, and Fo is given in Fig. 6 for various n and x. As expected a decreasing x results in an increasing amount of A absorbed by the polymer; for x -+ 0, the Fickian solution is approached. For n = 0, MFJMm is higher than for n = 1; this is the consequence of a relatively high flux of A into the grain. Figure 6 also presents the influence of Ct on the relation between M&M, and Fo. It is shown that the value of C.S influences the profiles; the higher Cz the lower is the absorption rate of A under otherwise the same conditions (so x is kept constant). No influence of the value of C.* is found for x > 10. Again, the penetration of the front is accelerated towards the centre of the grain, due to spherical geometry. However, since the total penetrant uptake is related to the increase in the cube of the swollen polymer, this acceleration of the front does not translate to any corres-

Non-Fickian diffusion with chemical reaction in glassy polymers

2963

0.6

0.4

CL2

0.0

I’

-

Fig. 5. Concentration profiles of penetrant A for different swelling behaviour for x = 0.1 and 10 with 0 = 1 and &=#,.,=O. (1) x=0.1, n=l; c:=os. (2) x=0.1, n = 1; C: = 0.1. (3) x = 0.1, n = 1 and 2; C: = 0. (4) x = 0.1, andn=0.(5)~=10,n=1and2;C~=O.S.(6)~=10,n=1 and 2; Cz = 0 and 0.1. (7) x = 10 and n = 0.

Fig. 7. Concentration profiles of penetrant A for various 41 and C: with x = 1.0 = 1 and n = 1. (1) #I = 100:C: = 0.5. (2) #Q = loo; c: = 0. (3) 4, = lo; c: = 0.5. (4) 4, = lo; c: = 0. (5) 4, = 1; C,, = 0.5. (6) 4, = 1; C: =O. (7) & = 0.1; C: = 0.5. (8) 4, - 0.1; c: = 0. (9) $1 = 0.01; c: = 0.5. (10) & = 0.01; c: = 0.

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 0.0

0.1

0.3

0.2

PO

Fig. 6. M,/M, as function of Fo for various x, C: with 01 = 41, i = 0. (0) Fick. (1) x = 0.01; n = C~=O.(2)~=0.1;n=landC~=0.(3)~=l;n=1and C: = 0 and 0.5. (4) x = 10;n = 0 and 1; Cz = 0 and x = 0.01; n = 1 and C: = 0.5. (6) x = 0.1; n = C: = 0.5. (7) x = 0.1; n = 0. (8) x = 1; n = 0.

0.2

0.4

0.6

0.8

1.0 f-

n and 1 and 0.5. (5) 1 and

ponding acceleration of mass uptake. This is also confirmed by Lee and Kim (1992).

NON-FICKIAN

0.0

-

DIFFUSION WITH FIRST-ORDER +1 > 0

REACTION:

The concentration profile of penetrant A in the polymer follows from the solution of eqs (17at_(22) with &I > 0. Figure 7 shows such profiles and its dependency of C: and the new parameter 4, which is equal to 9* Th:. As expected an increase of @I results in less penetration into the polymer. The reaction of the penetrant causes a decrease in the concentration also at the position of the front which results in a reduction of the velocity of the expanding swelling zone. In case of a fast reaction (+1 = 100) it is even possible for the concentration at the position of the front to approach the threshold concentration C.+. Figure 7 also shows the decreasing influence of #I for @I Q 1. This is accompanied by a change of shape of the concentration profiles. An increase in C.+ results in less deep penetration as is the case with no reaction.

Concentration profiles of penetrantA for different 0 and for #1 = 100 and 10 with x = 1, n - 1 and Cf = 0.1. (1) & = loo, 0 = 0.1. (2) 41 = loo; e - 0.3. (3) 41 = loo; 0 = 0.5. (4) &, = loo; B = 2. (5) 41 = loo; t’= 7.5. (6) & = lo: 8 = 0.5. (7) #1 = lo; 0 = 1. (8) & = lo; B = 1.5.(9) #Jl = 10; e = 2. (10) 41 = lo; 8 = 2.5. Fig. 8.

Figure 8 shows the variation of the concentration profiles with time for & = 10 and $I = 100. In the latter case the front does not reach the ccntre of the grain: because of the fast reaction the front conccntration approaches the threshold concentration C: resulting in a zero-front velocity. Then a pseudo-steadystate is obtained. Another remarkable effect is the increase of the concentration gradient at the surface of the grain with time. Note also that an acceleration of mass transfer occurs due to the chemical reaction as is clear from the substantial increase in the concentration gradients at the surface of the grain for & = 100 relative to 41 = 10. NON-FICKIAN

DlFFUSlON

WITH (1.1) REACTION:

4 1.1 z-0

Addition and substitution reactions between a solvent A and some reactive polymer group B can often be described by (1,l) kinetics. This is e.g. the case with the gas/solid hydroxyethylation of potato starch (van Warners et al., 1990). In such a case the assumption of a first-order reaction in A is only valid if the

2964

N. J. M.

KUIPERS and A.

concentration of B during the reaction remains approximately constant, i.e. for a low conversion of B. In case of a (1,l) reaction the concentration profiles of penetrant A and reactive group B of the polymer are found from the solution of the eqs (17b)-(22) with +1,1 > 0, and $1.1 = 9*Th:,, with 7%,,, the Thiele number for a (1,l) reaction. Compared to the preceding cases the mass balance of B [eq. (18)] has to be solved simultaneously with that of A to find the concentration profile of the latter. Two new dimensionless groups appear in the solution: 4i.r and ~,&~e. The influence of these groups on the concentration profiles and the conversions are discussed below. Also criteria for homogeneous addition, surface coating and pseudo-first-order kinetics are presented. With eq. (17b) the concentration of B at the surface of the polymer can be found, when it can be assumed that the surface concentration of A is not affected by the reaction, i.e. C,,.,.,= i remains 1 during the reaction:

=

exp

[ -

41.

(30)

I(&&bO)~~~~

Figure 9 presents typical concentration profiles of penetrant A and reactive group B of the polymer as a function of time 0. Still there is a minimum in the concentration of penetrant at the position of the front. Always the reaction time is zero at the front; therefore the dimensionless concentration of B equals 1 for r’ < r-b. The influence of the ratio c@&&, on the concentration profiles is shown in Fig. 10. Equation (18) predicts a more rapid depletion of the concentration of B with time for higher values of c,,e/cb,,.In contrast, eq. (17b) does not include this parameter directly so that less influence on the concentration profile of A is to be expected. Figure 10 shows indeed a more significant depletion of B with increasing c,&~~ For very high values of this parameter, the reaction may take place at the position of the swelling front only even for a kinetically rather slow reaction with 4i.i = 1. If so, the asymptotic solution of a shrinking core reaction is approached. No reaction occurs for r’ > rb because of lack of reactant B which has been converted completely while for r’ < rb no reaction takes place be-

,a C

B

A. C. M.

BEENACKERS

cause no penetrant is present. Indeed Fig. 10 also shows that, for &,i = 1, the ratio c&&be has little influence on the concentration profile of penetrant A, and therefore on the position of the swelling front in time (high c,e/c,, results in a relatively low C, so that the front penetrates slightly faster into the grain). In Fig. 11 concentration profiles are shown for various values of 4i.i. It follows that also the value of strongly affects the concentration profiles of 4 Br&d has less influence on the concentration profile of the penetrant (and therefore on the position of the swelling front vs time). For 4i.r = 100 the reaction almost takes place at the swelling front only, and again we have a shrinking core reaction. A difference with the asymptotic case of (pseudo-) first-order kinetics is that for high 4i.i no pseudosteady-state exists because of the ever decreasing concentration of B until all B has reacted. It is of great industrial interest to have a criterion for which a surface reaction or surface coating can be realized. Suppose we want to coat the particle with a layer of thickness A = 1 - r; of reaction product. Assuming no external mass transfer limitation, so C,+=i = 1, eq. (30) can be used as a starting point. Taking a diffusion (reaction) time 0,,,,, such that the position of the swelling front is r& and c,,,,=,; = co,.,=1 = 1, surface coating will take place,

1.0

C

P

t 0.8

LO

t

0.8

ca

0.0

0.4

0.2

0.0

Fig. 10. Concentration profiles of A and B for varying c.,&,~ with 0 = 1, x = 1, @1,1= 1, n = 1 and C: = 0.1.

1.0

t 0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

t

cb

“.O 0.0

0.2

0.4

0.6

0.8

1.0 i-

Fig.

9. Concentrationprofiles of A and B for varying times 8 with x = 1, +,,1 = t, n = 1, C: = 0.1 and c,&,, = 1.

Fig. 11. Concentrationprofiles of A and B for varying &,. , with 2 = 1, B = 1, n = 1, C: = 0.1 and c.,,/ccO = 1.

Non-Fickian diffusionwith chemical reaction in glassy polymers

2965

if c *.r,- 1 p= C il.,‘=,&

=

C

1 - t-w= 1 C b,r’=r&

B

expC-41.1 (c.o/cco)x~~~.~-= 11 exp [- ‘$1.1(c.O/cbO)xe.u~.~'=r~l

with &,,,. ,. , the dimensionless reaction time at r’ = 1 and 0 rc1.z.r’=r; the dimensionless time of reaction at r’ = rb which is zero: therefore CbSr1=,6= 1. In case of surface coating CbVVzzlapproaches zero, and therefore
t 1.0

1.0

% t

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0 0.0

0.0 0.2

0.4

0.6

0.8 r’ -

1.0

Fig. 12. Concentration profiles of A and B for various x with 0 = 1 (8 = 0.95 for x = lOO), 4,. , = 1, n = 1, C: = 0.1 and c.O/cbO

=

1.

1.0

t

0.8

0.6

0.4

From this equation the maximal value of +1,1 xcao/ cbo will be obtained if 8,,.,,,., 1 is as low as possible, i.e. diffusion limitation. for no Then 8IO..% I’= 1 = 1 - rl, = A as follows from eq. (19). Substitution into eq. (32) gives that the surface will be coated with a layer of reaction product of relative thickness A, if 4l,1zx

+ _

lnfl -~b.r'=i)_

(33)

For example, for A = 0.01 surface coating occurs if for CbSr.=1 = 0.9 (34)

h--x

c.0

> 460

for lb./= 1 = 0.99.

cbO

The coating thickness A and the coating “intensity” usually will be dictated by product specifications. Often the manufacturer then has several degrees of freedom to meet criterion (33) by selecting a sufficiently high partial pressure of A and a suitable reaction temperature that affects both #l,i and x. Equation (33) can also be used as a first estimate whether a shrinking core reaction occurs: the same derivation is valid as used above, only with the difference that the concentration C, of penetrant A in the polymer will be less than 1. This is confirmed by Fig. 11 in which ~l,i~c&~o = 100 for I$~,~ = 100, the same order of magnitude as predicted by eq. (34). From this equation it is clear that case II diffusion has a positive influence on a core reaction: the higher the value of x, the more heterogeneous the reaction. This is also presented in Fig. 12, where the influence of x on the concentration profiles of A and B is
0.2

0.0

Fig. 13. Me/M, as function of time B for varying 4,. 1 and with x = 1, Ct = 0.1 and n = 1. (1) ‘q%1e, = 1; C.&o = 10. (2) Q ,.I = 100; c,&*o = 1. (3) #,.* = lo; c.o/c*o= 1. (4) #I.1 = 1: c,rJ/c*o= 1. (5) 4J1.1= 1; c&& = 0.1.

c,Jcbo

investigated. This figure shows that in case x = 100 almost a shrinking core reaction occurs as is predicted by eq. (34). Figure 13 shows the influence of #Q,, and c,,O/cbOon the relation of the relative absorbed amount of penetrant MB/M, and time 8. It is clear that.an increase of $l.l results in a considerable increase of M*JM, because of a higher reaction rate. The same is true for the influence of c,,~/c~~: the higher this parameter, the higher the reaction rate [see eq. (17b)], and therefore the absorption rate MB/M,. From the above it becomes clear that usually the conversion of B not only depends on time, but also on the radial position in the grain. Homogeneous conversion of B occurs when the concentration of B is independent of the position in the grain. The homogeneity of the conversion can be quantified by determining the concentration of B at the surface of the polymer grain, i.e. Cb,,P=l, at the moment that the swelling front has reached the centre of the grain for which Cb,,,=O = 1. The closer the value of Cb,,, = 1 approaches 1 the more homogeneous is the conversion of B. Figure 14 shows the conversion of B, [b.r’= I, at the surface of the polymer grain as function of x for

N. J. M. KUIPERSand A. A. C. M. BEENACKERS

where ti is an arbitrary constant (@ < 1): the lower the value of 9, the more homogeneous is the conversion of B. Substitution of eq. (37) into (36) gives the criterion for homogeneous conversion: 0.6

0.4

for n = 1 and C.’ c 0.5

0.2

k&b0 0.0

.i

Fig. 14. Conversion cb,II= i at the outside of the polymer grain as function of x for various 4,. , and c.~/c~~with n = 1 and C,t = 0.1. (1) $,VI = 10; c,o/cbO= 1. (2) &1,, = 1; C.+/C~ = 1. (3) &,,I = 0.1; c.&bo = 1. (4) &I., = 1; C.&W =O.f. (5) 41.1 = 0.1; c.&bo = 0.1. (6) r& = 1; C.&o = 0.01. (7) 41.1 = 0.1; C.&o = 0.01.

p&d

= 0.17 + x

for n = 1 and C.* -c 0.5

0.001 < 2

< 1 and 4r.r < 1 _ (35)

> Equation (35) gives also and x for n = 1 and C: = simulations (see e.g. Fig. Substitution of eq.

the relation between Fo.,~ 0.5 as is shown by computer 2). (35) into (30) gives for

Fo = Foend: 6b..’= I = 1 = 1-

C*,,‘=, exp{-

&,r(c.&&CO.l7

+

xl1

for n = 1 and C.+ d 0.5 (0.001 -z c.~/c~~ c 1 and 4 1,1 x 1). If [b,.wlUird is the total required conversion homogeneous conversion occurs if
<

*

(36) of B,

< 1).

In case of Fickian diffusion x = 0 and, therefore, the critical ~I.Ic,,O/cbO is relatively high compared to anomalous and case II diffusion: this means that it is easier to get homogeneous conversion of B in case of Fickian diffusion than in case of non-Fickian diffusion. Secondly, it can be concluded that 4r.r not only determines the homogeneity of addition but also h/cbO

different values of 4r.r at the moment that the swelling front just arrives in the centre of the grain. Also the influence of c.,,/cco on this local conversion, which is equal to 1 - Cb,,,= 1, is presented in this figure. This graph predicts that Fickian diffusion (x + 0) always leads to a more homogeneous addition than diffusion with slow swelling kinetics. But homogeneous addition can also be realized by decreasing 4r.r and/or c,,&,~ as is reported in Fig. 14. A mathematical criterion for homogeneous conversion can be found by determining the time Fo_, which is necessary for the front to reach the centre of the grain. We may exclude conditions for 4r.i > 1 because the homogeneous conversion is impossible afterwards (Westerterp et al., 1987). From computer simulations it is found that the influence of c,,&,~ and on F0.n.r are small if 0.001 < c,,~/c~~ x 1 and %::: < 1: this is also shown in Figs 10 and 11. In this case F0e.d is merely determined by x (for n = 1 and C: < 0.5). The relation between Fo.,,,, and x can be fitted by

< 1 and &

and

x,

Equation (38) is only valid for $r,r < 1. Higher values of #r, f result in a higher “end” time Fo,,,~ in which case eq. (35) underestimates the value of Fo..~. However, as discussed above, for #t.t % 1 homogeneous conversion is not possible. Finally, the question remains to be answered under what conditions this overall second-order diffusion with reaction problem reduces to the ease of pseudofirst-order kinetics described in the previous section. This will be so if the amount of B does not change much during the reaction. This is the case as long as 4 l.Fromeq. the concentration Cb.,, = 1-b lor[b.p,=i (30) it is clear that a pseudo-first-order reaction in A results in case I$ I,IFoc,&bo is lower than some arbitrary value whose value is determined by the total conversion, with Fo based on the total reaction time. COMPARISON

BETWEEN

PSEUDO-STEADY43TATE

NUMERICAL SOLUTION

WITH A FIRST-ORDER

SOLUTION WITHOUT

AND AND

REACTION

In some cases an excellent agreement can be obtained between the numerical unsteady-state solutions and the so-called pseudo-steady-state solution in which the accumulation term of the mass balance of A is neglected. This PSS-solution can be obtained analytically. Below, both the situation of no reaction and first-order reaction with kinetics R, = Rb = - klc, are discussed. For first-order kinetics in both A and B, no analytical solutions can be obtained with the PSS-approximation. The PSS-model neglects the accumulation term in eq. (1). The physical significance of this approximation is that the interface between glassy and rubbery polymer moves so slowly that the penetration rate of A into virgin polymer is low relative to the total rate of mass transfer into the particle. The technique was first introduced by Wen (1968) to describe Fickian diffusion with non-catalytic gas-solid reactions. Assuming a pseudo-steady-state, eq. (17a) changes to

O=‘d

eati

(pac.>

rb -27 - dJlC.7.

< r’ < 1. (39)

Non-Fickian diHiuion with chemical reaction in glassy polymers The eqs (19~(22) are still valid in the PSS-approximatied. For “n = 0 and qjl = O”, the total set of equations can he solved analytically: /.

11

with A = eJ9,(f -“‘,

B = (A#,

C = _ AA

TO

_ 4 4

DsJK+q+~-~ r& Brb’

For “II = 1 and 41 = O”, both C. as function of r’and r& and the relation between 8 and rl, can be determined analytically:

2967

E=z(l-CC:)

(49)

After choosing values of &, rb, x and C$, first the parameters A and E can be computed from eq. (49) and subsequently the values of the other parameters that all depend on the value of A. From eq. (45) the flux of penetrant A follows:

(42) where =+c-l-J;j;;+2FJ&e*,

1-C..+F )

A=

and

B = FU - C3

(1 - C.l)w

x

* (43)

The constant F can be found by substituting eq. (41) into eq. (21) and solving the quadratic expression: ;(l-$-F’+ry(l-;) (1 - C3 -r

c, =

_ Fe26)

/&‘*

+ F&r’

.

r’

4nR2.

(51)

(45)

With the help of eqs (51), (12), (13), (15) and (50) for the flux of A through the surface of the polymer we may write in dimensionless terms:

(47)

(r6)2

Substituting eq. (45) into eq. (21) gives

eJK(l-6)

where F is a function of r& and, therefore, of time 0 [see eq. (49)]. As shown below, the PSS-solutions give rather good results for high values of x and &. This can be explained mathematically e.g. for the case of “lt = 1 and 41 > 0”. A pseudo-steady-state occurs when relatively much A diffuses through the polymer surface into the polymer grain with respect to the amount of A, used for the propagation of the swelling front, so

(44) 1 -c:)=o. F + (1

For “n = 0 and & B 0”. the total set of equations can be solved anaiytically: (&

(50)

(-+Gg

and r;, = 1 - 8.

For “n = 1 and dI > O”, only C, as function of r’ and r& can be determined analytically. Again the general solution is represented by eq. (45) and the constant F is found implicitly by substituting eq. (45) into eq. (21): the resulting quadratic equation has the following solution (for the relevant solution of F the negative term in front of the square-root term must be used). Hence. 1 He6

F=--

-

J

+c”++ 2AHe6

- 4AHED

x

(

C,,.*=, - C.* C = 1 _ C: 0.r’ rb > G(-1-,&+2F&eG).‘(52)

Because both r& and C,.,. =rb are less than 1, criterion (24) can be simplified, resulting in i@(-1-&+2F,,&eG).

ED 2AH2eG

+ 2HC.* ED + EfD2 + (HC.+)’ + 4H2EC 2AH’eG

(48)

(53)

N. J. M. KUIPERS and A. A. C. M. BEENACKERS

2968

A further simplification can be made if the minimum value of F as function of r& is substituted in eq. (53). It can be seen that the value of F for which rh = 0, is a good approximation for the minimum value of F (Kuipers, Ph.D. thesis, to be submitted). This value follows from eq. (48) after substituting rb = 0. This results in F rainhal=

1 + xe-“ml(l

- C.‘)

(eG-f?“-4q

(54)

*

Substitution of eq. (54) into eq. (53) means that the PSS-approximation is rather good if

$3{ -

Cea

1-J&+2&iG

Equation (55) can be further C: < 1. This results in

l-J1+

+ x(1 - C.‘)]

@G-e-G)

I.

simplified

2&

l?JK

(eG-e-6).

(55) because

1

(56)

This equation reduces for fast reactions to

According to eq. (56) the PSS-solution the numerical solution when: l l

l

(b) r& approaches 0, i.e. for long times 8: this is clear because for high 0, the polymer particle is saturated with A. From then on a real steady state occurs.

In Fig. 15 a comparison is made between the numerical solution and the PSS-solution for the same value of r6 in case of no reaction (the time 0 necessary to get at the values of r& shown in Fig. 15 is according to the numerical solution, 0 = 0.2, 0.6 and 1; according to the PSS-model these are 0 = 0.181, 0.533 and 0.849, respectively [see eq. (42)]. This figure shows that the PSS-model is a good approximation particularly, for B < 0.2. With increasing t7the differences between the two models grow somewhat with the PSS-solution giving higher concentrations of reactant A. This is also confirmed by the different values of 8 from the numerical and the PSS-solution; these differences grow with decreasing rb. as predicted by eq. (60). Note that both the PSS-solution and the numerical solution predict a minimum in the concentration of penetrant A at the swelling front vs time. Figure 16 shows the differences between the two solutions with varying x at B = 1 (according to the numerical solution) and no reaction. The solutions

approximates

the value of x increases, the value of $I increases.

0.6

For absence of reaction, i.e. for “+1 the PSS-approximation is better for This can be explained as follows. @1 = 0 relation (23) can be rewritten, (41):

= 0 and n = l”, low values of 0. For n = 1 and with help of eq.

P -

where F is given by eq. (44). F decreases with increasing C.*: a minimum value of F occurs when C.* = 1 (see Kuipers, Ph.D. thesis, to be submitted). For this case the minimum value of F follows from eq. (44):

Fig. 15. Comparison between numerical solution and PSSsolution for B = 0.2, 0.6 and 1 with x = 1, g& = #Q. I = 0, Ct = 0.1 and n = 1. (a) Numerical solution, (b) PSS-solulion.

I.0

Ca Substitution of eq. (59) into eq. (58) and using the fact that C,,,Vz.b.< 1 means that the PSS-solution and the numerical solution approach each other if r&

(1 - r-h) x

l

06

0.4

* 1.

This equation predicts that the PSS-solution proaches the numerical solution when: l

t 0.8

0.2

ap-

the value of x increases, which is in accordance with eq. (56). (a) r& approaches 1, i.e. for short times 8: this is shown in Fig. 16, see below.

Fig. 16. Comparison between numerical solution and PSSsolution at 0 = 1 for x = 0.01, 0.1, 1 and 10 with & = Q1,, = 0, C: = 0.1 and n = 1. (a) Numerical solution, (b) PSS-solution.

Non-Fickian

t C

diffusion with chemical reaction in glassy polymers

1.0

a 0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Fig. 17. Comparison between numerical solution and PSSsolution at 0 = 1 for #I~ = 0, 1, 10 and 100 with x = 1, C: = 0.1 and n = 1. (a) Numerical solution, (b) PSS-sotution.

agree very well for high values of x. Based on the same value of r6 the PSS-solution predicts: 8 = 0.277 for x = 0.01, 0 = 0.532 for x = 0.1, 0 = 0.849 for x = 1 and 0 = 0.978 for x = 10. This also confirms that the agreement between the two solutions improves with increasing x, which is in accordance with eqs (56) and (60). The effect of a different reaction rate is presented in Fig. 17, based on the same value of rb. As already shown before a steady state exists for +1 & 1: then the differences between the numerical and analytical solutions can be neglected. With decreasing 4, the differences grow, but even for +1 = 1 the PSS-solution is reasonably accurate. Note that this behaviour is predicted by eq. (56). CONCLUSIONS

When a plasticizing penetrant A diffuses into a glassy polymer and subsequently reacts with some reactive group B of this polymer a transition of glassy into rubbery polymer is induced, which is accompanied by a swelling of the polymer particles. This has a large influence on the diffusion process of the penetrant into the glassy granular polymer particles. In the present contribution a first analysis is given by the following: l

l

l

diffusion with power-law swelling kinetics for spherical particles, diffusion with first-order chemical kinetics and power-law swelling kinetics and diffusion with power-law kinetics and reaction kinetics of order (1,l).

Practical criteria have been derived which must be satisfied to obtain a homogeneous conversion throughout the polymer particles and, alternatively, to realize a surface coating. For non-Fickian diffusion without reaction, computer simulations have shown that for high values of x (x 2 10) case II diffusion occurs, resulting in a constant velocity of the advancing swelling front. Fickian diffusion is another asymptotic solution of the model forX+O,C.+=Oandn>O.

2949

Non-Fickian diffusion with a first-order reaction results in the existence of a steady state for values of & 2 10. Non-Fickian diffusion with a (1,l) reaction has two asymptotic limits: one asymptotic is a shrinking core reaction which is interesting if a surface treatment is aimed at. This regime occurs for @l,l x c,,,,/c~,-, 2 100:this implies that the occurrence of case II diffusion has a positive influence on a core reaction. The other asymptotic, homogeneous reaction occurs if &1,1 (0.17 + x) c,&.,, < F in which the value of F is determined by the relative inhomogeneity that is still acceptable and by the total conversion required. From this relation it is clear that homogeneous conversion throughout the particles is more difficult to obtain if case II diffusion occurs. Non-Fickian diffusion without or with first-order chemical reaction in glassy polymers with swelling induced by the penetrant according to power-law kinetics can be reasonably described by a pseudosteady-state mode1 for which analytical solutions have been derived. The physical significance of this approximation is that the interface between glassy and rubbery polymer moves such slowly that the penetration of A into virgin polymer is slow relative to its total mass transfer into the particle. For non-Fickian diffusion without reaction the approximation between the numerical results of the nonsteady-state model and the pseudo-steady-state solution is particularly good for short contact times (0 < 1) and for high x (x > 10). The higher x is the closer case II diffusion is approached. In case of non-Fickian diffusion with first-order reaction kinetics, the approximation is particularly good for a high reaction rate 41 (41 > 1) and high 0 (8 > 1). In the latter case, a real steady state may occur. Criteria have been derived in order to predict when the pseudo-steady-state solution approximates the numerical solution. Acknowledgements-The authors wish to thank H. van den Bogaard for writing the computer program and for the fruitful discussions about the results of the simulations. They also thank Auebe and the Dutch Innovation Program--Carbohydrates for their financial support.

NOTATION

A A, B B C, Go

C.+ =b

plasticizing penetrant which diffuses into a (glassy) polymer constants reactive group of a (glassy) polymer concentration of penetrant A in a polymer grain, mol mp3 concentration of penetrant A at the external surface of a polymer grain, mol mm3 threshold concentration of penetrant A for case II diffusion, mol mm3 concentration of reactive group B in a polymer, mol m ’

2970

N.

J. M.

KUIPERS

and A. A. C. M. BEENACKERS

initial concentration of reactive group B of a polymer, mol m- 3 dimensionless concentration of penetrant A in a polymer grain ( = c&.~) dimensionless threshold concentration of penetrant A ( = cj$/c,,,) dimensionless concentration of reactive group B of a polymer ( = Q./C& constants diffusion coefficient of penetrant A for diffusion into the polymer, m2 s-’ integration constant Fourier number [ = (IID,,t)/R’] constant flux of penetrant A into the polymer grain, mol me2 s-l first-order reaction rate constant, s- ’ reaction rate constant for a (1,l) reaction, m3 mol-’ s-l swelling constant in the power-law equation of a swelling front, m3”+ 1 mol” s-l amount absorbed A by the polymer grain at dimensionless time Fo, mol amount absorbed A by the polymer grain at time t, mol amount absorbed A by the polymer grain at dimensionless time 0, mol maximum amount of A which can be taken up by the polymer grain I: = 4xR3&, + c,,)/31, mol exponent in the power-law equation of an advancing swelling front ( b 0)

CbO

c*

c:

G

G&E m

F Fo H J kl k 1.1 K Ml% M, Ml9 M,

n I

distance m position

r0

coordinate

in a polymer

of the front

between

grain,

glassy

and

rubbery polymer, m dimensionless distance coordinate in a polymer grain ( = r/R) dimensionless position of the front between glassy and rubbery polymer ( 7 r&) f;rnrsyess radius [ = 1 + (R - r)/

r’

I rcl

r0

radius of a polymer grain, m reaction rate of penetrant

A,

mol

m-3s-l

reaction rate of reactive group B of the polymer, mol m-3 s-l time, s Thiele number for a first-order reaction C = (R/3) ,/‘ml Thiele number C = (R/3) Greek

A ct.

for

a

(1,l)

reaction

d-1

letters

dimensionless thickness of reaction layer in case of polymer coating conversion of the reactive group B in case of a (1,l) reaction dimensionless time [ = K (c,,~ - ~2)” t/R] proportional with the square of the

@I, 1

X

$

Thiele modulus for a first-order reaction [ = (k, R2/D,) = 9Th:] proportional with the square of the Thiele modulus for a first-order reaction in A and B [ = (kIvl R’/D,,) = 9Thf.l] dimensionless parameter which equals the quotient of diffusion rate and the maximum velocity of the moving boundary { = WCWC,O WRI) arbitrary constant which determines the homogeneity of the addition (reaction) REFERENCES

Aithal, U. S. and Aminabhavi,

T. M., 1991, Molecular

trans-

port of some industrialsolventsthrough a polyurethane membrane.J. appl. Polym. Sci. 42. 2837-2844. Alfrey,T., Gurnee, E. F. and Lloyd, W. G., 1966, Diffusion in glassy polymers. J. Polym. Sci. C12, 249-261. Astarita, 0. and Joshi, S., 1978, Sample-dimension effects in the sorption of solvents in polymers-a mathematical model. .I. Membrane Sci. 4, 165-182. Astarita, G. and Sarti, G. C., 1978, A class of mathematical models for sorption of swelling solvents in glassy polymers.Polym. Engng Sci. 18, 388-395. Berens, A. R., 1989, Sorption of organic liquids and vapors by rigid PVC. J. appl. Polym. Sci. 37, 90-913. Berens, A. R. and Hopfenberg, H. B., 1980, The induction and measurement of glassy state relaxation by vapor sorption techniques. Stud. Phys. Theor. Chem. IO, 17-94.

Cole, J. V. and Lee., H. H., 1989, A criterion for Case II diffusion in resist development by dissolution. J. Electrochem. Sot. 136,3872-3873. Crank, J., 1953, A theoretical investigation of the influence of molecular relaxation and internal stress on diffusion in

polymers.J. Polym. Sci. 11, 151-168.

Crank, J., 1975, The Mathematics of D$iision, 2nd Edition. Clarendon, Oxford. Gall, T. P., Lasky, R. C. and Kramer, E. J., 1990, Case II diffusion: effect of solvent molecule size. Polymer 31, 1491-1499. Gostoli, C. and Sarti, G. C., 1983, Influence of rheological properties in mass-transfer phenomena, super case 11sorption in glassy polymers. Chem. Et~gng Commun. 21, 67-79. Harmon, J. P., Lee. S. and Li, J. C. M., 1988, Anisotropic methanol transport in PMMA after mechanical defonnation. Polymer 29, 1221-1226. Hellman. N. N. and Melving, E. H., 1950, Surface area of starch and its role in water sorption. J. Am. Chem. Sot. 72,

51865188. Hopfenberg, H. B. and Stannett,

V., 1973, The Physics of Glassv Polvmers (Edited bv R. N. Hawardb I Aoolied .L Science Pu6lishers: Barking. ’ Jaaues. C. H. M.. Hopfenbera, H. B. and Stannett, V., 1974, &&ability ojP&ric F&s and Coatings. Plenum, New

York. Klech, C. M. and Simonelli, A. P., 1989, Examination of the moving boundaries associated with non-Fickian water swelling of glassy gelatin beads: &act of solution pH. J. Membrane Sci. 43, 87-101. Kuipers, N. J. M., Ph.D. thesis, University of Groningen. The Netherlands (to be submitted). Lasky, R. C., Kramer, E. J. and Hui, C. Y., 1988a, The initial stages of Case II diffusion at low penetrant activities. Polymer 29, 673-679. Lasky, R. C., Kramer, E. J. and Hui, C. Y., 1988b, Temperature dependence of Case II diffusion. Polymer 29, 1131-1136. Lee, P. 1. and Kim, C. J., 1992, Effect of geometry on solvent front penetration in glassypolymers.J. Membrane Sci. 65,

77-92. M eares, P., 1986, Fundamental mechanisms of transport of small molecules in solid polymers, in Proceedings 01 the

Non-Fickian

diffusion with chetmiical reaction in glassy polymers

Fourth BOC Priestley Conference on Membranes in Gas Separation and Enrichment. R. Sot. Chem. Special Publication No. 62, 1-25. NAG. 1982 FORTRAN Librarv Routine Document. NAGFLIP:1831/OzMk9:26th Feb;uaty. Nicolais, L. Drioli, E. Hopfenberg, H. B. and Caricati, G., 1978, Diffusion-controged penetration of polymethylmethacrylate sheets by monohydric normal alcohols. J. Membrane Sci. 3, 231-245. Papanu, J. S., Hess, D. W., Soane, D. S. and Bell, A. T., 1990, Swelling of poly(methyl methacrylate) thin films in low molecular weight alcohols. J. appl. Polym. Sci. 39,803-823. Park, G. S., 1983, Advanced Study Znstitute on Synthetic Membranes. North Atlantic Treaty Organization. Petropoulos, I. H. and Roussis, P. P., 1978, Influence of transverse differential swelling stresses on the kinetics of sorption of penetrants by polymer membranes. J. Membrane Sci. 3. 343356. Rudolph, F. and Peschel, G., 1990, A theoretical study of the time dependency of solvent diffusion in a Polymer. Ber. Bunsenges. Phys. Chem. 94,45&461. Sarti, G. C., 1979, Solvent osmotic stresses and the prediction of Case II transport kinetics. Polymer 20, 827-832. Storey, R. F., Mauritx, K. A. and Cole, B. B., 1991, Diffusion of bis(2ethylhexyl)phthalate above and below the glass transition temperature of poly(viny1 choride). Macromolecules 24, 450-454. Thomas, N. L. and Windle, A. H., 1977, Discontinuous shape changes associated with Case II transport of methanol in thin sheets of PMMA. Polymer 18, 1195. Thomas, N. L. and Windle, A. H.. 1978, Case II swelling of PMMA sheets in methanol. J. Membrane Sci. 3, 337-342. Thomas, N. L. and Windle. A. H., 1980, A deformation

CES 48:16-J

2971

model for Case II diffusion. Polymer 21, 613619. Thomas, N. L. and Windle, A. H., 1981, Diffusion mechanics of the system PMMA-methanol. Polymer 22, 627-639. Thomas. N. L. and Windle, A. H., 1982, A theory of Case II diifitsion. Polymer 23, 529542. Tong, H. M., Saenger, K. L. and Duming, C. J., 1989, A study of solvent diffusion in thin polyimide films using laser interferometry. J. Poiym. Sci. Polym. Phys. Ed. 27, 689-708. van Warners, A., Lammers, G., Stamhuis, E. J. and Beenackers. A. A. C. M., 1990, Kinetics of the diffusion and chemical reaction of ethylene oxide in starch granules 42,427-431. in a gas solid system_ Starch/St&ke van Warners, A., 1992, Modification of starch by reaction with ethylene oxide in liquid-sobd and gas-solid reactors. Ph.D. thesis, University of Groningen, The Netherlands. Waywood, W. J. and Duming, C. J., 1987, Solvent-induced crystallization of a compatible polymer blend. Polym. Engng Sci. 27, 126>1274. Weisenberger, L. A. and Koenig, J. L., 1990, NMR imaging investiaations of Case Ii diffusion in nolvmers. Macro. _ moJeca&s 23, 2445-2453. Weisenberger, L. A. and Koenig, J. L., 1991. in Solid State NMR ofPolymers (Edited by Math& L. J.), pp. 377-386. Plenum Press, New York. Wen, C. Y., 1968, Non+ttalvtic heterogeneous solid-fluid reaction models. Iad. Engr& Chem. 60: 3&54. Wessling, M., Schoeman, S., Boomgaard, Th. van der and Smolders, C. A., 1991, Plasticixation of gas separation membranes. Gas Sea. Purif: 5. 222-228. Westerterp, K. R., van Swaaij,.W. P. M. and Beenackem, A. A. C. M.. 1987, Chemical Reactor Design and Operation, Student Edition. Wiley, New York,