Diffusional release of a solute from a polymer matrix

Journal ofMembmne Science, l(l976) 33-48 o Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

DIFFUSIONAL

RELEASE OF A SOLUTE FROM A POLYMER

MATRIX

D.R. PAUL and S.K. McSPADDEN Department (U.S.A.)

of Chemical Engineering,

The University of Texas at Austin, Austin, Tex. 78712

(Received July 3, 1975; in revised form September 23, 1975)

Summary The theory of diffusional release of a solute from a polymeric matrix where the initial loading of solute is less than or greater than the solubility limit has been reviewed and extended. The conceptual model for the saturated case proposed by Higuchi has been refined to remove the inaccuracies caused by the “pseudosteady-state” assumption. A finite external mass transfer resistance has also been incorporated into the present analysis. For all solute loadings, the asymptotic release profile is seen to be a linear plot of total amount of solute released versus the square-root of time which has a finite intercept on the square-root of time axis because of the external resistance. The dependence of the slope and intercept on solute loading and other system parameters can be predicted for all cases with the models presented. Experimental release rates of an organic dye from a silicone polymer into acetone were measured for a range of solute loadings in order to test the applicability of these equations. All system parameters except the external mass transfer coefficient were measured by independent experiments. The experimental release data were described very well by the computed results.

Introduction

The kinetics of extraction or evaporation of a solute dispersed in a polymeric matrix is a problem of interest in areas ranging from the undesired loss of additives like antioxidants, stabilizers, etc. [1,2] to the controlled release of fertilizers, pesticides [ 31, or drugs. Usually the solute has a limited equilibrium solubility, C, , in the matrix. The initial uniform loading of solute per unit volume, A, may be less than C,, in which case all of the solute is molecularly dissolved in the matrix at equilibrium, or A could exceed C, and thus at equilibrium the excess solute, A - C, , is present as discrete particles. We wish to examine theoretically and experimentally the kinetics of solute release when A traverses the entire range, but special emphasis is placed on A > C,. The latter has been of particular interest in the past for the controlled release of drugs (see, for example, refs. [4--151) and has been reviewed recently by Baker and Lonsdale [ 161 and Flynn [ 171. The mode of extraction in this case is considerably different than when A < C,, and T. Higuchi [4] appears to have offered the first conceptual model for this process. He devel-

34

oped an approximate mathematical analysis of this model which has been used extensively by others with considerable success. Higuchi employed a “pseudosteady-state” analysis which makes the final result an approximation. In the limit of A + C, this result does not precisely match the prediction of classical theory for A < C!, applied to the limit A + C,. It is our first objective to review these two results, and then to reformulate the Higuchi conceptual model to remove the “pseudosteady-state” approximation which gives a result that merges correctly with the unsaturated case in the limit A + C,. The present problem represents a classical case of mass transfer between two phases, and, in general, rate processes in both phases could be significant, although most analyses assume that kinetics in the matrix is limiting since it is a solid whereas the external phase is a fluid. Nevertheless, the latter can have a significant effect on the release kinetics. It is our second purpose to include in the theoretical analysis the effect of an external fluid boundary layer. We show how this modifies the release kinetics for both A < C, and A >C,.

Our third objective is to present experimental release rate data for a model system in which the solute is an organic dye and the matrix is a solventswollen silicone polymer to demonstrate the utility of the theoretical models including the boundary layer resistance. Appropriate ancillary experiments have been used to determine the system parameters so that in the comparison of release data with theory no adjustable parameters are employed, except for the boundary layer mass transfer coefficient, which could not be measured directly. Theoretical models for release kinetics In the following discussion we will be concerned with the diffusion of a solute from a polymer matrix into a fluid which is assumed to be a perfect sink, i.e., the solute concentration never builds up enough to alter the diffusion driving force. The matrix will have a planar, membrane geometry which loses solute through both faces; however, the results will be expressed in terms of the amount of solute, Mt, which has escaped a unit area in time, t. We will consider only the early stages of loss before the diffusion fronts from the opposite faces begin to overlap; hence, the experimental geometry will be equivalent to the semi-infinite slab analysis used throughout. Results will be developed first for the limiting case of no resistance to solute transport by the external liquid (infinite mass transfer coefficient) which may be approximated experimentally by very vigorous agitation. Next, these results will be modified to include an external mass transfer coefficient which is finite, but not so small as to control totally the release kinetics [ 151. In all cases, time t=O corresponds to the immersion of the matrix which has a uniform solute loading per unit volume, A, into the liquid extraction bath.

35

Loading less than saturation (A < C,) Solutions to Fick’s law when all of the solute is dissolved are well known, and the particular expression for semi-infinite geometry with no external resistance is [ 181 M,=ZA

v-

4.

(1)

The assumption of an infinite external mass transfer coefficient, cyO,is equivalent to the following surface boundary condition for the solute concentration in the matrix C, = 0, fort>O.

(2)

Mathematical solutions for finite membranes of thickness, 1, which apply to the entire time scale are also well known, and they approach the following limit at t = = M,

= %Al.

(3)

Medley and Andrews [19], in a consideration of fiber dyeing, adapted results from Crank [ 181 for a finite mass transfer coefficient which replaces eqn. (2) with

where C,” is the liquid phase solute concentration (assumed zero here). Solute concentrations on either side of the membrane interface are related by the solute distribution coefficient, K, so that eqn. (4) can be written (5) Their solution to Fick’s law with eqn. (5) as the surface boundary condition rather than eqn. (2) is Mt = 2A

AD - . 41

(‘3

For large values of t, the second term on the right becomes small so the amount of solute released approaches the following asymptotic relation (7) =2A

36

The latter says that Mt us. *becomes linear and extrapolates back to a finite intercept on the fi axis given by

+A= 0

(9)

2a

rather than passing through the origin as occurs when cy= = (see eqn. (1)). Equation (8) is a useful result for analyzing data which suggests, for times sufficiently greater than to, that dMt /dfi is the same as in the case when (Y= =. Equation (9) allows one to calculate the induction period, provided the external hydrodynamics permits a priori knowledge of a. It is shown next that the idea of an induction time also applies to the more complex case with A > C,. Loading greater than saturation (A > C,) 1. Pseudosteady-state analysis This case differs from the earlier one in that initially the matrix is saturated with dissolved solute, with some excess solute dispersed uniformly as small particles. In order to be extracted, the dispersed solute must become dissolved in the matrix. Higuchi [4] pictured the physical situation after a certain time of extraction to be that illustrated in Fig. 1, where there is a surface zone 0 < 3t t , the solute loading per unit volume is still A. As time progresses, the boundary between the two zones moves inward and reduces the size of the core containing undissolved solute. Higuchi [4] effected an approximate mathematical analysis of this model by using a “pseudosteadystate” method which ignores diffusion dynamics in the outer zone - this assumes that the concentration profile here is linear. This approach uses the following

for Fick’s first law, with eqn. (2) as the surface boundary condition (01= -) and the following mass balance d.K

dt

= (A-%CS)$

(11)

and does not directly employ Fick’s second law at all in the zone 0
- XC,) t .

(12)

37

:-------I

I, Undissolved,

I

,

1

Solute

CS

0

Fig, 1. Schematic profile of dissolved and undissolved solute concentrations A >C,.

for case when

This equation predicts Mt to be linear in fi just as in eqn. (1) for A < C,,but the dependence on A is not the same. This dependency is not quite correct because of the approximate “pseudosteady-state” analysis, as can be seen in the limit of A + C, where eqn. (12) gives (13)

Mt = C, 0, whereas eqn. (1) gives

(14) The discrepancy is not great, only 11.3% too small, but this difference can be eliminated by use of an approach presented later. The pseudosteady-state analysis can be extended to include a finite external mass transfer resistance by replacing eqn. (10) with

-=

dMt

dt

D-

G t

co

(15)

38

and using the following surface boundary condition (analogous to eqn. (5) ) dMt -

=

dt

crc,

(16)

Equation (11) may be retained in its present form provided the external resistance is not too large. Equations (15) and (16) may be combined to give CO in terms of the boundary position, E, as follows (17)

When eqns. (ll), give

(15) and (17) are combined, the result can be integrated to

From this, dMt/d t can be computed, and Mt evaluated as follows:

M, =

+

(- (2)

dt

BDC,t

= (A-W,)

+ (A-%42,)

Mcs i,/*cs)

-- I + D

a

(19)

In the limit of large enough t, eqn. (19) approaches the following asymptotic form

Mt = d2DC,

(A -%Cs)t-A;

(20)

or Mt = d2DC,

(A - ?hC,)

(fi-

&i&

(21)

where

(22) These results are the same in their time functionality as eqns. (8) and (9) but both the slope and intercept differ in their dependence on A. In the limit of A + C,, eqn. (22) predicts

39

(23) while eqn. (9) gives

Thus, this approximate treatment fails to match up with the rigorous result for the unsaturated case at the common point of A = C, by predicting a slope too small by a factor of G/2 and an intercept which is too large by the factor 2/G. While this discrepancy is not large, it is undesirable for accurate calculations and will be removed in the following section. 2. Exact analysis To remove the approximations of the “pseudosteady-state” analysis embodied in eqns. (lo), (11) and (15), we must apply Fick’s second law

ac -=

D-

at

a2c (25)

ax2

to the region O
C=O

(26)

c=c,

(27)

and (A-C.):

= D&z

.

(28)

Equation (28) is a local condition for x = t that is analogous to eqn. (11) which was applied globally in the previous approach. The solution to eqns. (25-28) is

c=c,---

erf(vl) erfh *I

,

(29)

where X

r) =-

20

and q*

=---

E

24m

(30)

and fin*

exp(17*2)erf(fl*)

f? = Vs A - C,

(31)

40

The latter transcendental expression quantitatively defines q* in terms of A/C,. Equation (29) gives a nonlinear concentration profile in contrast to the linear form inherent in the Higuchi treatment. The expression for Mt can be shown to be (32) This shows Mt to be linear in fi just as eqns. (1) and (12) but the dependence on A is different. It will be of interest to examine some limiting cases to illustrate the properties of eqn. (32). When A + C,, eqn. (31) shows n* becomes very large and thus erf (‘17 *) approaches unity so

which is exactly the result from eqn. (1) in this limit. For A >> C,, eqn. (31) shows Q* is small and a series expansion permits eqn. (31) to give erf(n*) explicitly as follows erf(77*) = E

/s

,

(34)

which can be combined with eqn. (32) to give Mt =

vm

.

(35)

This limiting form is identical to eqn. (11) except for the l/2 coefficient to C!, in the latter; however, this is krconsequential when A >> C,. Thus, we may conclude that the “pseudosteady-state” approach that resulted in eqn. (12) is an adequate approximation in this limit. The effect of a finite external mass transfer resistance can be incorporated into this treatment through a rather laborious calculation or by writing the following asymptotic form in analogy to eqns. (7) and (20) (36) or (37) with

A=--

A

flii

c, 201

erf(s*) .

(33)

41

The expression for &reduces

A+Cs,

as follows for special cases

G=S,

(39)

which is the same as eqn. (9), and

which is equivalent to eqn. (22) for A/C, >> l/2. Thus we see that eqns. (36)-(38) are equivalent to the modified Higuchi results, eqns. (20)-(22), in the limit of large A/C,, and for A = C, they reduce to the correct limiting slope and intercept given by eqns. (7)--(g). Appropriate use of eqns. (7)-(g) and (36)-(38) should describe the asymptotic release kinetics for all values of A/C, when the external boundary layer is a significant but not totally dominating factor. Experimental It is the purpose of this section to describe the materials and techniques used to generate solute release data from a polymeric matrix for a range of A/C, values to compare with the theoretical models developed in the previous section. In addition, experiments to define the pertinent parameters of these models, C, and D, are discussed. All experiments were done at 25” C. Materials The polymer selected for this work was a room temperature vulcanizing silicone rubber, RTV-602, which was crosslinked by addition of 0.83% of a commercial curing agent, SRC-05. Membranes were formed by casting on a mercury surface. The cured membranes contained approximately 5% extractable material, which was always removed prior to use. The solvent selected for solute extraction was acetone, which swells the cured polymer to the extent that at equilibrium the membrane contains 31.3% acetone by volume. Prior to any diffusion experiments, the membrane was allowed to come to swelling equilibrium with the acetone, and all membrane dimensions used in calculations were the fully swoUen ones. The solute was a red organic dye known as Sudan III with the following molecular structure

42

This dye was obtained in a very pure form, and the dry powder was sieved to yield a uniform and small particle size. Its solubility in acetone was greater than 1.3 g/liter. Analysis of dye concentrations in acetone was accomplished by a photometer using an experimentally established calibration curve. In the subsequently described diffusion experiments, the dye concentration in the liquid phase was generally less than 0.005 g/l, which is more than sufficiently small to guarantee that the “sink” conditions required are never violated. A dye was selected for the solute because of the ease and accuracy of photometric analysis.

Solute distribution coefficient The distribution coefficient for the dye between the swollen membrane and that in the external acetone phase at equilibrium, X, was determined by changes produced by transferring a membrane from a sorption to an extraction bath as described previously [20]. Nine repetitions gave a value of 0.148 with a standard deviation of 0.009.

Time lag-permeation

measurement

The parameters C, and D for this system were determined by a technique consisting of a single transient permeation experiment. A swollen membrane was installed in the permeation cell shown schematically in Fig. 2 (see ref. [20] for details). A saturated solution containing a large excess of undissolved solute was introduced to the compartment on the left while pure solvent was added to the compartment on the right. Solute then diffused through the membrane from left to right. The amount of dye that had accumulated in the right-hand compartment per unit area of membrane after time t, Qt, was determined by analysis of the dye concentration in that compartment and is plotted US.time in Fig. 2 for a typical experiment. In order to keep the concentration low, or to maintain a sink condition, this compartment was emptied and refilled with pure solvent several times during the course of an experiment. While, of course, dye was removed from the left-hand compartment, the concentration of dissolved dye there never changed during the course of an experiment because of the excess of undissolved dye that was present. This creates a classical time lag situation which after enough time yields a steady rate of dye diffusion from left to right given by

=-

(41)

with the time lag resulting from the build up of the solute concentration within the membrane from zero to the steady state gradient. The time lag, 8, is related to the diffusion coefficient, D, by [18]

l2 0 =gp

(42)

43

I

I

I

I

I

6

1/

Sat.

SoIn;) .’

.

iSink -_--

I.

-.. I,‘.’

----

___

l!iB .,

.

1.

.

-

I

:

*:-..

-

.::.:

--_.

*. -o,-: .

.

.

**

.

‘.I

.*

.

..’

-.

. . . ..* . . . . ..* :. ‘* -. . . .’ ‘. . _

. . . . . . . *.. . . . I

.

--

.

-

_

___

----

.

I

I

8

IO

I:2

t (hours)

Fig. 2. Transient permeation experiment and data for determination

of B and C,.

where I is the swollen membrane thickness. Thus, from the measurement of the steady state slope and the time lag from such plots as Fig. 2, one can obtain both D and C,. Duplicate experiments gave average values of D = 2.68 X lob6 cm2/s and C, = 0.274 g/l, the difference between the duplicate values being less than 4%. So Eute-release kinetics For measurement of the rate of solute release from the matrix by extraction, a known quantity of solute was pre-mixed into the prepolymer prior to casting and curing of the membrane. In order to allow this loaded membrane to reach swelling equilibrium prior to the extraction experiment, it was “activated” by immersion in an acetone solution containing excess undissolved dye for a sufficient time to reach equilibrium. This eliminated the simultaneous swelling that would have otherwise accompanied extraction, which would undoubtedly have altered the kinetics of extraction. The use of solute-saturated solvent for this purpose eliminated unwanted extraction during the activation step. For the extraction experiment, the loaded membrane (diameter - 8 cm,

44

thickness - 0.15 cm) was suspended vertically in a round-bottom flask (1,2 or 5 liter) containing pure acetone initially. The liquid contents of the flask were agitated by a magnetic stirring bar which kept the extracted dye adequately mixed with the solvent but did not create enough motion to totally eliminate the external fluid boundary layer or cause large movements of the membrane. These hydrodynamic conditions were reasonably consistent for each experiment. The amount of dye extracted was determined by analyzing the concentration of dye in the solvent as a function of time. The results were then expressed as total dye extracted up to time t per unit area of exposed membrane surface (two sides), Mt. Typical results are shown plotted us. fi in Fig. 3. These data reach a plateau corresponding to complete removal of dye from the membrane. This value was used to calculate the original solute loading A since this was more accurate than using the amount of dye originally mixed into the polymer because of small losses, trimmings, and inaccuracies in weighing the small quantities of solute used.

Fig. 3. Typical solute-release-rate

data for various values of solute loadings.

45

Results and discussion There are several pertinent features of the sample results shown in Fig. 3. All of the release profiles approach an asymptote where Mt becomes linear in the square-root of time, just as all of the models considered predict. Second, these linear asymptotes extrapolate back to a positive and finite intercept on the square-root of time axis, i.e. &, just as predicted by the models which incorporate a finite external mass transfer resistance. It is interesting to note that the linearity of Mt us. fi does not persist up to total exhaustion of solute from the matrix but rather Mt trails off and approaches its limit asymptotically. This trailing off begins when the core thickness goes to zero and all undissolved solute has disappeared. After the two fronts in Fig. 1 meet, marked by the time when ,$ = $1, the models for A > C, no longer apply and the rate of release falls off owing to the decline in driving force caused by removal of dissolved solute. It is now of interest to see if the slopes and intercepts, shown by the data such as that in Fig. 3, i.e. -cw

and 6,

depend on the initial solute loading,

dfi

A, as the models described earlier predict. For the slopes, it will be useful to

examine them in the form of L d”, us. A/C!,. For A/C, < 1, this normalized A dfi slope should be independent of A/C, according to eqns. (1) and (8), and its theoretical value can be calculated from knowledge of just D. For A/C, > 1, this normalized slope is a declining function of A/C, , which can be calculated theoretically via eqns. (32) and (37) with knowledge of both C, and D. The solid lines in Fig. 4 have been calculated from the experimental values of C, and D given above using the applicable equation for each region. The points represent the experimental release data obtained from plots such as those in Fig. 3 plotted at the appropriate value of A/C,. The agreement between the points and the computed lines is excellent and has not involved the use of any adjustable parameters. This is convincing proof that these models accurately describe the phenomena that are occurring. Next, we will see how well these models predict the A dependence of the experimental intercepts, 6. There is one unknown parameter, viz. a, so this test cannot be an absolute one. For A/C, < 1, eqn. (9) predicts fi should be independent of A, and this value is determined by both D and (Y. For A/C, > 1, eqn. (38) predicts a0 should be an increasing function of A/C,, whose value depends on D and cy. The points in Fig. 5 are the experimentally deduced 6, while the solid lines were computed using the appropriate equation for each region from the known D and a value of cythat gave the best fit. Note that (Y is simply a multiplicative value in those equations and therefore cannot alter the dependence on A/C, but just the magnitude of 6. The value of LYselected was 3.4 X 10T4 cm/s. The agreement between the form of the experimental

46

_ IO lu 1 $3 x cu o M

6 4

-la 2 0

0

I I

I

I

1

I

I

I

2

4

6

8

IO

12

14

WCs Fig. 4. Comparison of theoretical and experimental solute release rates. Solid lines computed using C, = 0.274 g/l and D = 2.68 X lo+ cm2/s. Equation (8) used for A/C, and eqn. (37) used for A/C, > 1.

< 1

data and the theory is considered good. The scatter is undoubtedly due to the failure to achieve precisely the same hydrodynamic situation, i.e. Q, in each experiment using this relatively crude apparatus. Conclusions We conclude that the refinements of the Higuchi model offered here (via the relaxation of the “pseudosteady-state” assumption) have some advantages for describing release kinetics for loadings where A/C,is slightly greater than one, but become virtually identical with Higuchi’s equation for large values of A/C,. The incorporation of a finite external mass transfer resistance into the models for release kinetics makes these results more valuable for describing situations normally encountered in most applications. The refined model in conjunction with the classical solution to Fick’s law for A/C,< 1 offer an accurate set of equations for describing release rates over the entire range of A/C,,provided of course the condition of rapid dissolution of undissolved solute is applicable.

47

20

! 0

1 2

I 4

I 6

I 8

I IO

I \ I2

A/Cs Fig. 5. Comparison of theoretical and experimental induction times for solute release. Solid lines computed using C, and D as in Fig. 4, with LX= 3.4 x 10e4 cm/s. Equation (9) used for A/C, < 1 and eqn. (38) used for A/C, > 1.

Further, we conclude that the transient permeation experiment described here offers a novel and simple way to determine the important system parameters, C, and D. There is an interesting, limiting relationship between this experiment and the release rate that might be very useful for rapid evaluation of matrix release rates. For large values of A/C,,the release rate is given by

(43) where 1 is the thickness of the membrane used in the permeation experiment. Thus in this limit, one need not measure C, or D independently, since to estimate the release rate it is only necessary to know the loading and I [(dQ,)/(dt)], from a permeability experiment. References 1 H.E. Bair, Polym. Eng. Sci., 13 (1973) 435. 2 R.J. Roe, H.E. Bair, and C. Gieniewski, Am. Chem. Sot. Polym. Preprints, 14 (1973) 530.

48

3 G.G. Allan, C.S. Chopra, J.F. Friedhoff, R.I. Gara, M.W. Maggi, A.N. Neogi, S.C. Roberts, and R.M. Wilkens, Chem. Tech., (Mar., 1973) 171. 4 T. Higuchi, J. Pharm. Sci., 50 (1961) 874. 5 T. Higuchi, J. Pharm. Sci., 52 (1963) 1145. 6 J. Lazarus, M. Pagliery, and L. Lachman, J. Pharm. Sci., 53 (1964) 798. 7 S.J. Desai, A.P. Simonelli, and W.I. Higuchi, J. Pharm. Sci., 54 (1965) 1459. 8 S.J. De&, P. Singh, A.P. Simonelli, and W.I. Higuchi, J. Pharm. Sci., 55 (1966) 1224, 1230, and 1235. 9 P. Singh, S.J. Desai, A.P. Simonelli, and W.I. Higuchi, J. Pharm. Sci., 56 (1967) 1542 and 1548. 10 J.B. Schwartz, A.P. Simonelli, and WI. Higuchi, J. Pharm. Sci., 57 (1968) 274 and 278. 11 T.J. Roseman and W.I. Higuchi, J. Pharm. Sci., 59 (1970) 353. 12 J. Haleblian, R. Runkel, N. Mueller, J. Christopherson, and K. Ng, J. Pharm. Sci., 60 (1971) 541. 13 T.J. Roseman, J. Pharm. Sci., 61 (1972) 46. 14 F. Sjuib, A.P. Simonelli, and W.I. Higuchi, J. Pharm. Sci., 61 (1972) 1374 and 1381. 15 Y.W. Chien, H.J. Lambert, and D.E. Grant, J. Pharm. Sci., 63 (1974) 365 and 515. 16 R.W. Baker and H.K. Lonsdale, in A.C. Tanquary and R.E. Lacey (Eds.), Controlled Release of Biologically Active Agents, Vol. 47 of Advances in Experimental Medicine and Biology, Plenum Press, New York, 1974, pp. 15-72. 17 G.L. Flynn, S.H. Yalkowsky, and T.J. Roseman, J. Pharm. Sci., 63 (1974) 479. 18 J. Crank, The Mathematics of Diffusion, Oxford Univ. Press, London, 1956. 19 J.A. Medley and M.W. Andrews, Text. Res. J., 29 (1959) 398. 20 D.R. Paul, M. Garcin, and W.E. Garmon, J. Appl. Polym. Sci., in press.

Nomenclature A

c

co G

c,L

(2 D K

1 Mt

M, Qt

initial solute loading in matrix, g/l concentration of solute dissolved in matrix, g/l solute concentration in matrix at x = 0, g/l solubility of solute in matrix, g/l solute concentration in liquid phase at matrix boundary, g/l solute concentration in liquid phase far from matrix, g/l solute diffusion coefficient in matrix, cm2/s solute distribution coefficient, Co/C? swollen membrane thickness, cm amount of solute removed in time, t, per unit area, g/cm2 amount of solute removed in infinite time per unit area, g/cm2 amount of solute which has diffused through unit area of membrane in time, t, g/cm2 time, s intercept on Mt = 0 axis in Mt us. 4 plot, s position measured from edge of matrix, cm mass transfer coefficient, cm/s so/K dimensionless group, x/29 dimensionless group, .5/2m time lag, s x location of core containing undissolved solute, cm