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Transport of monomer surfactant molecules and hindered diffusion of micelles through porous membranes
Transport of Monomer Surfactant Molecules and Hindered Diffusion of Micelles through Porous Membranes KAMESWARA RAO KROVVIDI,* ANTHONY MUSCAT, *'l PIETER STROEVE, *'2 AND ELI RUCKENSTEIN'~'2 *Department of Chemical Engineering, University of California, Davis, Davis, California 95616, and "~Department of Chemical Engineering, State University of New York at Buffalo, Amherst, New York Received November 4, 1983; accepted January 24, 1984 Diffusion of Triton X-100 through Celgard 2500 m e m b r a n e s was examined. The pore permeability for m o n o m e r s was 5,0 × 10-6 cm2/sec and it was measured for upstream concentrations below the C M C value of 2.29 X 10-4 M at 30°C. This value is close to the m o n o m e r diffusion coefficient in bulk suggesting that the m o n o m e r s do not experience significant hindrance due to the pore walls. The permeability of the surfactant drops abruptly within a narrow range of reservoir solution concentrations in the vicinity of the CMC. At concentrations 10 × CMC, the permeability coefficient becomes constant and equal to 3.9 × 10-7 cm2/sec which is the pore permeability for the Triton X- 100 micelles. Compared to the diffusion coefficient of micelles in bulk water, the transport of micelles is hindered by the pore walls. In a 10-fold concentration range the micellar pore permeability is practically constant indicating no large change in micelle size. The chemical equilibrium model applied to surfactant diffusion in pores shows reasonable agreement over the entire range of the experimental data for reservoir concentrations from one-fifth times the C M C to 100 times the CMC. INTRODUCTION
cultural applications, polymeric devices have Transport of surfactant molecules through been applied in pest control. Recently, the or into membranes occurs in the controlled diffusion through membranes of drugs which release of surface active drugs or chemical have surface active properties and form miagents through polymeric devices (1, 2), the celles in solution has been studied (1, 2). The treatment of waste water containing detergents diffusion of Nonoxynol-9, a nonionic surfacby membrane separation processes (3, 4), and taut, was examined in both nonporous and the chemical modification of porous materials porous membranes. Nonoxynol-9 is a polysuch as fabrics and leather by dyeing and tan- oxyethylene alkyl phenol and is an active ning operations (5, 6). Of special interest is component in a variety of contraceptive forthe controlled release ofbioactive agents with mulations. In nonporous membranes, the flux polymeric delivery devices as a means of pro- of surfactant was found to be due to the longing and controlling the release of drugs, monomer molecules only. The flux was found pesticides, herbicides, and fertilizers (7-10). to be low due to the low diffusivities of the Polymeric delivery systems have been used in monomer in the membrane matrix and due clinical applications such as the treatment of to the low driving force for monomer diffusion ocular diseases, immunizations, contracep- since in aqueous solutions the monomer contion, and the release of antibiotics. In agri- centration almost remains constant once the total surfactant concentration exceeds the critical micelle concentration (CMC). In the I Present address: Department of Chemical Engineering, porous membranes, with the pores filled with Stanford University, Stanford, Calif. 2 Authors to w h o m correspondence should be addressed. water, both monomer and micelle diffusion 497
Journalof Colloidand InterfaceScience,Vol. 100, No. 2, August 1984
0021-9797/84 $3.00 Copyright© 1984by AcademicPress, Inc. All rightsof reproduction in any form reserved.
498
KROVVIDI
were observed :to take place. For membranes with pore diameters about three times larger than the miceUe diameter (Celgard 2500, Celanese Corp.), significantly lower micellar pore permeabilities were measured than with membranes with pore diameters two orders of magnitude larger (Mitex, Millipore Corp.). Assuming free diffusion to take place in the membrane with large size pores, the ratio of the micellar permeability in the Celgard 2500 membranes to the free diffusion coefficient was found to be in reasonable agreement with a theory for hindered diffusion in pores. From the limited experiments the pore permeability of the surfactant m o n o m e r was found to be equal to the free diffusion coefficient of the monomer. The monomeric and micellar pore permeabilities were obtained under surfactant driving force conditions where the contribution to the surfactant flux by the micelles or monomer, respectively, could be assumed negligible. Roy et al. (1) report data on the mass transfer of surfactant in porous membranes for reservoir concentrations larger than 15 × CMC and sink concentrations near the CMC. The purpose of this paper is to present surfactant diffusion data through porous membranes for a larger range of surfactant concentrations in the reservoir, extending from values lower than the CMC to greater than 100 × CMC. The diffusion of Triton X-100 in Celgard 2500 membranes has been studied. In addition, the free diffusion coefficient of Triton X-100 was measured with the Taylor dispersion technique (11). The free diffusion coefficient of micelles is necessary in order to determine the effect of hindrance of the pores on the diffusion of micelles. The data on porous membranes permeability can be interpreted on the basis of both the phase separation and equilibrium models of micellization. For reservoir concentrations greater than 10 × CMC, the two models lead to the same results because the flux of the m o n o m e r is negligible. For smaller values, the equilibrium model is the more appropriate one, since the contribution of the flux of the m o n o m e r becomes important. Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
ET AL. EXPERIMENTAL
Triton X-100 (iso-octylphenoxypolyethoxyethanol) was obtained from Sigma Chemical Co. and was used as received. The porous membranes were Celgard 2500 which are microporous polypropylene films. The pores in Celgard 2500 are "slit-like" with dimensions of 400 by 4000 ~. Membrane pretreatment was similar to that used by Roy et al.' (1). Circular membranes with an effective area of 19.6 cm 2 were cut out and laid on top of the surface of a 100 ml solution of Triton X-100 at 2.5 × CMC. After 30 min, the membranes were completely immersed into the solution and stored for several hours. The membranes were then copiously rinsed with distilled water and stored in distilled water before use in a diffusion experiment. The surfactant soaking process ensured that the walls of the pores became hydrophylic and hence filled with water. The choice of a surfactant concentration of 2.5 X CMC was based on the approximate amount needed to form a close packed layer on the pore walls for a final concentration of the solution equal to the CMC. The batch diffusion apparatus and experimental procedure are identical to those described by Roy et aL (1). The membrane separated the reservoir and the sink compartments. Magnetic stirrers at both sides of the membrane were found to eliminate the fluidside resistance when the stirring speed was greater than 90 rpm. A set of diffusion experiments (Type I) was carried out with the reservoir surfactant concentration about 10% below the CMC and the initial sink surfactant concentration at zero. Also a set of experiments (Type II) was carried out with the reservoir concentration at least a factor of 10 greater than the CMC, and the sink concentration greater than the CMC. These types of experiments can yield the pore permeability for m o n o m e r and micelle, respectively, in a manner described by Roy et al. (1). As will be emphasized later, the contribution to the flux by the "other" species, i.e., micelle or monomer, is negligible in the above experiments. Another set of diffusion experiments
DIFFUSION OF MICELLES (Type III) was conducted where the reservoir concentration was below 10 × CMC and the sink concentration was close to the CMC. The mass transfer rates in all experiments were obtained by measuring the sink surfactant concentration spectrophotometrically at 274 nm. After the diffusion experiment, a surfactant mass balance was performed to insure that no surfactant was lost from the sample holder during the experiment. The free diffusion coefficient of micelles was measured with the Taylor dispersion technique using the apparatus described by Weinheimer et al. (12). In this method a pulse of more concentrated surfactant solution is injected into a surfactant solution flowing in a precision capillary tube. The time dependence of the average concentration is recorded at the end of the tube concentration from which the diffusion coefficient can be obtained. To check the technique, the free diffusion coefficient of urea was measured and compared with available values. Surface tension measurements were conducted with a Wilhelmy plate technique to determine the CMC of the surfactant. In all experiments, doubly distilled water was used and the temperature was 30°C.
499
Assuming for the sake of simplicity that the micelles are monodisperse, then, at equilibrium, Ke nA O-- An, [1] where A and An are the m o n o m e r and micelle forms, respectively, K, is the equilibrium constant, and n is the aggregation number. These quantities are related via the expression
where Cm is the micellar concentration and C1 is the m o n o m e r concentration. The equilibrium constant Ke can be expressed in terms of the CMC and aggregation number n as follows. First let us introduce the dimensionless variables: C*
-
C CMC '
The micellization thermodynamics. The process of micellization has been modeled either by the phase separation model (13) or by the equilibrium model (14-16). In the first of these models, it is assumed that the entire amount ofsurfactant which exceeds the critical micelle concentration (CMC) aggregates as micelles. In the second, aggregates and singly dispersed m o n o m e r are in thermodynamic equilibrium, the concentration of the latter increasing slightly above the CMC as the total concentration of surfactant increases. As the phase separation model is an approximate version of the equilibrium model, the emphasis in what follows is on the equilibrium model.
C*-
C1 CMC '
and C* -
Cm CMC
[3a, b, and c]
where C is the total surfactant concentration. Because C* = C'~ + n C * , [4] Eq. [2] can be rewritten as C* - C* n(CMC)n_lf~n ,
[5]
= In(C* - C*) - n In C*.
[6]
Ke
THEORY
[21
Cm = K~C7
or ln[Ken(CMC) n-I ]
Following Stigter (17), one can define the CMC as the concentration at which the derivative of the m o n o m e r concentration with respect to the t o t a l surfactant concentration is equal to one-half. This yields for the dimensionless m o n o m e r concentration at the CMC C~' -
n
n+l'
[7]
which combined with Eq. [5] leads to (n + 1)n l K~ = nn+l(CMC)n_ 1 .
[8]
Journal of Colloid and Interface Science, Vot. 100, No. 2, August 1984
500
KROVVIDI ET AL.
The value n = 175 obtained from Corti and Degiorgio for Triton X-100 (18) was used in the calculations. The value of the CMC obtained by us from surface tension versus concentration measurements is 2.29 × 10 -4 M at 30°C. Pore diffusion. In the diffusion of surfactant through the membrane pores, the monomers and the micelles have to satisfy the species continuity equations. If quasi-steady state is assumed and Fick's expression is used for each species within the pore, then (12): d2Clp
nRm = 0
[9]
Drop dz----5- + Rm = 0,
[10]
Dip d z 2
and d2Cmp
where Rm is the rate of micelle formation, z is the distance to the face of the membrane in contact with the reservoir, and Dip and Dmp are the diffusivities of the m o n o m e r and micelle, respectively, which account for the hindrance effects in the pore. E l i m i n a t i n g Rm between Eqs. [9] and [10], one obtains D
d2Cl° daCmp lp ~dz + nDmp dz 2 - 0 .
[11]
Integrating Eq. [ 11 ] twice, with the boundary conditions Cmp=C°p,
Clp=C°p
at
Cmp =Cmp, L
Clp = CLp at
nPm
o
Here PI = DlpKI and Pm = DmpKm are pore permeability coefficients, C1 and Cm are the bulk monomer and micelle concentrations, and K1 and Km are the m o n o m e r and micelle partition coefficients. Because the equilibrium between micelles and monomers is achieved in a very short time (10) compared to the diffusion through the membrane pore, it is reasonable to consider that C ° and C o as well as CLm and C L are related by the thermodynamic equilibrium expression given by Eq. [2]. Defining the flux in terms of an effective pore permeability and total surfactant concentration
Perf
J = -L-- ( c ° - c L ) '
[15]
where C o and C L are the total surfactant concentrations at the two pore ends at the reservoir and sink sides. Equations [14] and [15] yield Peff =
(C O __
C L)
cL )
.
[16]
[12]
where L is the thickness of the membrane, gives for the total flux J = Dip - Z - ( C 0,o - cl p) Drop o + n T (Cmp -- CLp)-
PI
J = 7 (C° - C~) + - - F (Cm - CLm). [14]
P I ( C ° - C L) + r t P m ( C ° -
z=0 z=L,
because the ratios between the sizes of the surfactant molecules and micelles and the size of the pore affect their partition between the bulk and the pore mouth (20). Assuming thermodynamic equilibrium between the bulk and the pore mouth, Eq. [ 13] can be rewritten as
[131
Here the superscripts 0 and L refer to the two ends of the pore and the subscript p indicates that the concentration is on the pore side. The concentrations C°o, cOo, C~fo, and C~p differ from those in the reservoir and sink Journal of Colloidand InterfaceScience, Vol.100,No. 2, August1984
RESULTS AND DISCUSSION
At 30°C, two types of diffusion experiments were carried out by the diaphragm cell method. In the first kind (Type I), the initial reservoir concentration of Triton X-100 was at least 10% below the CMC value of 2.29 × l 0 -4 mole/liter, while that of the sink was zero (distilled water). The diffusing molecules will be mainly the monomeric form of the surfactant with negligible micelle transfer. In the second type of experiments (Type II) the initial reservoir concentration was one to two orders of magnitude greater than the CMC
DIFFUSION OF MICELLES while the initial concentration of the sink was again at zero concentration. As observed by Roy et al., in Type II experiments a distinct change in the quasi-steady flux occurs when the sink concentration becomes equal to the CMC. From this time onward, the m o n o m e r activities in the two compartments of the cell are almost equal. This implies that the surfactant concentration increase in the sink concentration above the CMC is due mainly to micellar transport. Assuming quasi-steady state within the pore, the monomeric Triton X-100 pore permeability was obtained from the expression: In
CR-- Cs CR(0) -- C s ( 0 )
= -P1/3t.
[17]
Similarly, the micelle permeability was obtained from In
CR-- Cs CR(~'c) -- CMC
= --Pm~O,
[18]
140
501 I
I
I
I
I
x Porous Membrane Experiment Least Square Pit By Eq • [ 18t
120 . . . .
I ,,.x ..x.~~ sxSx -
sx jx 100
~risx~d x''x
80 )r
X
X
-
x
X X
60
X X
40
X X
20 0 xx
I
i
50
100
I
I
I
150 re 200 250
I
300
350
t (rain) FIG. 1. Number of moles of Triton X-100 diffused through Celgard 2500 porous membrane with time in a Type 1I experiment: reservoir volume = 68 ml, sink vol = 40 ml; CR(0)= 1.73 × 10-3M; Cs(0) = 0.0M. Calculated micellar permeability is Pm = 4.63 × 10 7 cm2/secwith /~ = 25.35 cm-z and Zc = 183.8 min.
where 13 = ~
+
[19]
CR and Cs are surfactant concentrations in the reservoir and sink at time t, P1 = KID~p and Pm = gmDmp are the m o n o m e r and the micelle pore permeabilities,/£1 and Km are the m o n o m e r and micelle partition coefficients between the bulk and pore; CR(0) and Cs(0) are the initial reservoir and sink concentrations, respectively. 0 = t - re where ~c is the critical time at which the sink concentration is at the CMC; Ap is the total pore area and Lp is the pore length. VR and Vs are the volumes of reservoir and sink, respectively. The ratio Ap/Lp for a particular membrane can be obtained from a diffusion experiment with a solute of small size relative to the pore diameter and with a known free diffusion coefficient, D , . Duffey et al. (21, 22) reported an Ap/Lp value of 40 c m / c m 2 of total membrane area for Celgard 2500. Data of a typical run of a Type II experiment and the least square fit by Eq. [18] are shown in Fig. 1. The adsorption of Triton X-100 on the pore wall causes a decrease in the available pore
area to the diffusing m o n o m e r as well as the micelles. The adsorption occurs during the pretreatment of the membrane and it is assumed that a monolayer coverage on the walls takes place. Then the space between the pore walls is decreased by about 70 A which is twice the length of the surfactant molecule in the adsorbed state (23). It is assumed that the molecule will be in the extended form when adsorbed on the wall as opposed to the helical or coiled form within the solution (24). It follows that for an adsorbed monolayer the pore area of the Celgard membrane is reduced by a factor = [(400 A × 4000 ~)/(400 A - 70 ,~)(4000 A - 70 A)] = 1.23. The Ap/Lp ratio is therefore 32.5 cm. This value was confirmed by us to within 10% for the diffusion of the solute butyl-p-aminobenzoate through Celgard 2500 membranes which were pretreated by the surfactant. The radius of this molecule is 4.0/~, the molecular weight is 193, and the free diffusion coefficient is 8.0 × 10-6 cm2/ see (25). This molecule should not experience pore hindrance since its diameter is much smaller than the pore size. Journal of ColloM and Interface Science, Vol. 100, No. 2, August 1984
502
KROVVIDI ET AL.
Figure 2 shows the monomeric and micellar pore permeabilities obtained from the Type I and Type II experiments. The monomeric pore permeability of Triton X-100 is found to be about PL = 5.0 _ 0.7 × 10-6 cm2/sec. This value is near to the free diffusion coefficient of D1 = 5.35 × 10 -6 cm2/sec (26). Since the monomeric pore permeability is close to the free diffusion coefficient, the hindrance experienced by the m o n o m e r in diffusing through the Celgard pores appears indeed to be negligible. Although the extended length of the Triton X-100 m o n o m e r (about 45 A (27)) is significant with respect to pore size, the coiled configuration of the molecule is smaller (24). The data imply that the partition coefficient of the monomer is near one. Unfortunately the diffusion data are not highly accurate below the CMC because of the low driving force. The micellar pore permeability Pm = 3.93 × 10-7 cm:/sec at 3.0 × 10 -3 moles/liter. The micellar free diffusion coefficient is Dm = 6.71 X 10 -7 c m 2 / s e c . This value was obtained by interpolating between those measured by the Taylor dispersion technique. The latter value compares well with a value of 6.93 × 10-7 cm:/sec at 25°C for Triton X-100 by Weinheimer et al. (12). I f F p and Foo are the drag forces experienced by a rigid spherical particle 60
I
moving with the same steady velocity inside a pore and in an "infinite" medium, respectively, then (28-30) Dmp _ Fo~ Dm
For a particle moving between parallel walls of a pore, translating along its axis (31), 1 '
where w is half the distance between the walls and r is the radius of the particle. In our experiments, 2w = 400 - 70 = 330 N. For a micellar diameter of 92.8 N (18), from Eq. [21] Dmp/Dm = 0.728. From theory (20) the micellar pore distribution coefficient is about 0.72. Thus the theoretical ratio of micellar pore permeability to the free diffusion coefficient is 0.52 which is slightly lower than the experimental value of 0.59. These values are consistent with those obtained for Nonoxynol9 (1). The micellar aggregation number for Nonoxynol-9 is about 250 (32, 33) versus 175 for Triton X-100 (18), and the micelle diameter is about 100 A versus about 93 N for Triton X-100 (18). The experimental ratio of I
I --m--
50
Free Diffusion Experiments
" " ° - - P o r o u s Membrane
Diffusion Experiments
o
o
40
%
[2Ol
Fp
30
a.
20 CMC
10 --
I ~ - - I - ---B------- 1 O--.--C 0 C--O 0
10-5
I 10-4
I 10-3
o
I 10-2
10-1
CR (m01es/I)
FIG. 2. Pore permeabilities of Triton X-100 in Celgard 2500 membranes and free diffusion coefficients of micelles in water at 30°C. Here P is either Pr or pro. Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
503
DIFFUSION OF MICELLES
micellar pore permeability to the free diffusion coefficient was found to be 0.53 for Nonoxynol-9 which is somewhat lower than that of Triton X- 100, as is expected because of the larger size of the micelle. The effect of surfactant concentration on the micellar permeability is modest, decreasing by approximately 10% over a 10-fold concentration range (1.7 × 10-3 to 3.0 X 10-2 M). The decrease in the free diffusion coefficient is within this 10% variation. The micellar permeability data suggest that the micellar size does not vary significantly in this concentration range. Figure 3 shows the permeabilities in all the types of experiments (Types I, II, and HI) as well as those calculated by the equilibrium model. The values of the permeabilities are plotted versus the log mean reservoir surfactant concentration during the measurement. This concentration is very close to the average reservoir concentration during the course of the experiment. The theoretical curve of Fig. 3 represents Pe~ and it was obtained on the basis of the equilibrium model and Eq. [16] using for the calculations the monomeric and micellar pore permeabilities obtained in Fig. 2 for Type I and II experiments. Thispresumes that for Type I and II experiments the contribution to the flux of the other species (mi60
I
o
celle and monomer) is negligible. Table I gives the ratio between the flux of the monomer and the total flux below and above the CMC and demonstrates that for reservoir concentrations larger than 10 X CMC the flux of the monomer is indeed negligible. For reservoir concentrations 10% below the CMC the micellar flux is negligible. Figure 3 shows that the experimental pore permeability coefficients for reservoir concentration slightly above the CMC remain relatively constant and equal to the monomer permeability and then fall abruptly to the lower micellar permeabilities. The presence of high permeability values (5,0 X 10-6 cmZ/sec) for reservoir concentrations just above the CMC is expected since the concentration of micelles is not yet significant to influence the diffusional process. A similar trend is predicted by the equilibrium model coupled with Eq. [16]. However, the model predicts an earlier and smoother drop in the permeability in the transition region. The earlier drop is probably due to the fact that the equilibrium model is rather sensitive to the value of the equilibrium constant, thereby influencing the concentration of miceltes. Further, the equilibrium model as used here involves near the CMC micelles of a single aggregation number, which may not be re-
I
o o
50
0°~"'~ o 0
40--
o
-
--
PorousMembrane Diffusion Experiments Chemical EquilibriumModel
g
~
I
I
30 x ct.
20
I I I
-
I I Io
-
10
% 0 10-5
13
I
I
I
10-4
10-3
10-2
i0-1
OR, Im ( m 0 t e s / I )
FIG. 3. Pore permeabilities of Triton X-100 in Celgard 2500 membranes for Type I, II, and III experiment& The dashed curve represents the theoretical calculations with the chemical equilibrium model. Here P is Pee- The average downstream concentration for type III experiments is about 8 × 10-5 moles/1. Journal of Colloid and Interface Science, Vol. 100, No, 2, August 1984
504
KROVVIDI ET AL. TABLE I Percentage of Monomer Flux Contribution to the Total Surfactant Flux as Calculated from the Chemical Equilibrium Modela'b Reservoir concl CMC
M o n o m e r flux × 100 Total flux
Reservoir conc.
M o n o m e r flux
CMC
Total flux
100. 100. 99.7 95.8 90.7 67.9
11.3 20.6 34.8 42.3 84.5 103.1
0.52 0.64 0.90 1.42 1.96 2.02
X 100
2.7 2.2 0.86 0.77 0.44 0.43
a Triton X-100; Celgard 2500 porous membrane. bp~ = 5.0 × 10-7 cm:/sec; Pm= 3.9 × 10-v cm:/sec; temperature = 30°C; n = 175; CMC = 2.29 × 10-4 mole/ liter; Ke(CMC)n-I = 8.8 × 10-5 (mole/liter)n-l.
alistic. Nevertheless, the theory shows fair agreement With the experimental values considering the number of assumptions made. SUMMARY
The diffusion of the surfactant Triton X-100 through porous Celgard 2500 membranes was measured over a large range of surfactant concentrations, from conditions below the critical micelle concentration (CMC), around and well above the CMC. From the diffusion data, the monomeric and the micellar permeability of Triton X-100 in the water-filled pores of Celgard 2500 were obtained. The monomeric pore permeability was found to be close to the monomeric diffusion constant in bulk water, indicating unhindered diffusion in the pores. Micellar pore permeability was about one half of the value for the micellar diffusivity in bulk water. Hindered diffusion of micelles in the water-filled pores occurs because the characteristic pore dimension is only three times the micellar diameter. The chemical equilibrium model applied to surfactant transport in the porous membranes shows reasonable agreement with the experimental data. ACKNOWLEDGMENT This study was supported in part by the University of California. Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
REFERENCES 1. Roy, S., Ruckenstein, E., and Stroeve, P., J. Colloid Interface Sci. 92, 383 (1983). 2. Stroeve, P., Ruckenstein, E., Roy, S., and Lippes, J., AIChE Symp. Ser. 227, 11 (1983). 3. Bhattacharyya, D., Garrison, K. A., Jumawan, A. B., and Grieves, R. B., AIChE J. 21, 1057 (1975). 4. Schott, H., J. Phys. Chem. 68, 3612 (1964). 5. Morita, Z., and Iijima, T., J. Colloid Interface Sci. 82, 155 (1981). 6. Hod, T., Mizuno, M., and Shimuzu, T., Colloid Polym. Sci. 258, 1070 (1980). 7. Langer, R., Chem. Eng. Commun. 6, 1 (1980). 8. Robinson, J. R. (Ed.), "Sustained and Controlled Release of Biologically Active Agents." Plenum Press, New York, 1978. 9. Scher, H. B. (Ed.), "Controlled Release Pesticides." Am. Chem. Soc., Washington, D. C., 1977. 10. Cardarelli, N. F. (Ed.), "Controlled Release Pesticides Formulations." CRC Press, Boca Raton, FL, 1976. 11. Taylor, G. I., Proc. Roy. Soc. London A 219, 186 (1953). 12. Weinheimer, R. M., Evans, D. F., and Cussler, E. L., J. Colloid Interface Sci. 80, 357 (1981). 13. McBain, J. W., Trans. Faraday Soc. 9, 99 (1913). 14. Mukerjee, P., J. Pharm. Sci. 63, 972 (1974). 15. Tanford, C., "The Hydrophobic Effect." Wiley, New York, 1980. 16. Ruckenstein, E., and Nagarajan, R., J. Phys. Chem. 79, 2622 (1975). 17. Stigter, D., Recueil Trav. Sci. Pays Bas 73, 611 (1954). 18. Corti, M., and Degiorgio, V., Optics Comm. 14, 358 (1975). 19. Aniansson, E. A. G., Wall, S. N., Almgren, M., Hoffmann, H., Kielmann, I., Ulbricht, W., Zana, R., Lang, J., and Tondre, C., J. Phys. Chem. 80, 905 (1976).
DIFFUSION OF MICELLES 20. Glandt, E. D., A I C h E J. 27, 51 (1981). 21. Duffey, M. E., Ph.D. Thesis, Carnegie-Mellon University, Pittsburgh, PA (1977), 22. Duffey, M. E., Evans, D. F., and Cussler, E. L., J. Membr. Sci. 3, 1 (1976). 23. Kushner, L. M., and Hubbard, W. D., J. Phys. Chem. 58, 1163 (1954). 24. Cooney, R. P., Barraclough, C. G., and Healy, T. W., J. Phys. Chem. 87, 1868 (1983). 25. Majumdar, A., M.S. Thesis, University of California, Davis (1983). 26. Ramirez, W. F., Shuler, P. J., and Friedman, F., Soc. Petr. Eng. J. 20, 430 (1980). 27. Rosch, M., in "Nonionic Surfactants" (M. J. Schick, Ed.), pp. 753-773. Marcel Dekker, New York, 1967.
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28. Bean, C. P., in "Membranes" (G. Eisenman, Ed.), Vol. 1. Marcel Dekker, New York, 1972. 29. Brenner, H., and Gaydos, L. J., J. Colloid Interface Sci. 58, 312 (1977). 30. Anderson, J. L., and Quinn, J. A., Biophys. J. 14, 130 (1974). 31. Happel, J., and Brenner, H., "Low Reynolds Number Hydrodynamics," Chap. 7. Prentice-Hall, Englewood Cliffs, NJ, 1965. 32. Schick, M. J. (Ed.), "Nonionic Surfactants." Marcel Dekker, New York, 1967. 33. Thomas, J. K., and Kalyansundaran, K., in "Micellization, Solubilization, and Microemulsions" (K. L, Mittal, Ed.), Vol. 2. Plenum Press, New York, 1977.
Journal ~?fColloMand InterfaceScience. Vol. 100,No. 2, August1984
cultural applications, polymeric devices have Transport of surfactant molecules through been applied in pest control. Recently, the or into membranes occurs in the controlled diffusion through membranes of drugs which release of surface active drugs or chemical have surface active properties and form miagents through polymeric devices (1, 2), the celles in solution has been studied (1, 2). The treatment of waste water containing detergents diffusion of Nonoxynol-9, a nonionic surfacby membrane separation processes (3, 4), and taut, was examined in both nonporous and the chemical modification of porous materials porous membranes. Nonoxynol-9 is a polysuch as fabrics and leather by dyeing and tan- oxyethylene alkyl phenol and is an active ning operations (5, 6). Of special interest is component in a variety of contraceptive forthe controlled release ofbioactive agents with mulations. In nonporous membranes, the flux polymeric delivery devices as a means of pro- of surfactant was found to be due to the longing and controlling the release of drugs, monomer molecules only. The flux was found pesticides, herbicides, and fertilizers (7-10). to be low due to the low diffusivities of the Polymeric delivery systems have been used in monomer in the membrane matrix and due clinical applications such as the treatment of to the low driving force for monomer diffusion ocular diseases, immunizations, contracep- since in aqueous solutions the monomer contion, and the release of antibiotics. In agri- centration almost remains constant once the total surfactant concentration exceeds the critical micelle concentration (CMC). In the I Present address: Department of Chemical Engineering, porous membranes, with the pores filled with Stanford University, Stanford, Calif. 2 Authors to w h o m correspondence should be addressed. water, both monomer and micelle diffusion 497
Journalof Colloidand InterfaceScience,Vol. 100, No. 2, August 1984
0021-9797/84 $3.00 Copyright© 1984by AcademicPress, Inc. All rightsof reproduction in any form reserved.
498
KROVVIDI
were observed :to take place. For membranes with pore diameters about three times larger than the miceUe diameter (Celgard 2500, Celanese Corp.), significantly lower micellar pore permeabilities were measured than with membranes with pore diameters two orders of magnitude larger (Mitex, Millipore Corp.). Assuming free diffusion to take place in the membrane with large size pores, the ratio of the micellar permeability in the Celgard 2500 membranes to the free diffusion coefficient was found to be in reasonable agreement with a theory for hindered diffusion in pores. From the limited experiments the pore permeability of the surfactant m o n o m e r was found to be equal to the free diffusion coefficient of the monomer. The monomeric and micellar pore permeabilities were obtained under surfactant driving force conditions where the contribution to the surfactant flux by the micelles or monomer, respectively, could be assumed negligible. Roy et al. (1) report data on the mass transfer of surfactant in porous membranes for reservoir concentrations larger than 15 × CMC and sink concentrations near the CMC. The purpose of this paper is to present surfactant diffusion data through porous membranes for a larger range of surfactant concentrations in the reservoir, extending from values lower than the CMC to greater than 100 × CMC. The diffusion of Triton X-100 in Celgard 2500 membranes has been studied. In addition, the free diffusion coefficient of Triton X-100 was measured with the Taylor dispersion technique (11). The free diffusion coefficient of micelles is necessary in order to determine the effect of hindrance of the pores on the diffusion of micelles. The data on porous membranes permeability can be interpreted on the basis of both the phase separation and equilibrium models of micellization. For reservoir concentrations greater than 10 × CMC, the two models lead to the same results because the flux of the m o n o m e r is negligible. For smaller values, the equilibrium model is the more appropriate one, since the contribution of the flux of the m o n o m e r becomes important. Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
ET AL. EXPERIMENTAL
Triton X-100 (iso-octylphenoxypolyethoxyethanol) was obtained from Sigma Chemical Co. and was used as received. The porous membranes were Celgard 2500 which are microporous polypropylene films. The pores in Celgard 2500 are "slit-like" with dimensions of 400 by 4000 ~. Membrane pretreatment was similar to that used by Roy et al.' (1). Circular membranes with an effective area of 19.6 cm 2 were cut out and laid on top of the surface of a 100 ml solution of Triton X-100 at 2.5 × CMC. After 30 min, the membranes were completely immersed into the solution and stored for several hours. The membranes were then copiously rinsed with distilled water and stored in distilled water before use in a diffusion experiment. The surfactant soaking process ensured that the walls of the pores became hydrophylic and hence filled with water. The choice of a surfactant concentration of 2.5 X CMC was based on the approximate amount needed to form a close packed layer on the pore walls for a final concentration of the solution equal to the CMC. The batch diffusion apparatus and experimental procedure are identical to those described by Roy et aL (1). The membrane separated the reservoir and the sink compartments. Magnetic stirrers at both sides of the membrane were found to eliminate the fluidside resistance when the stirring speed was greater than 90 rpm. A set of diffusion experiments (Type I) was carried out with the reservoir surfactant concentration about 10% below the CMC and the initial sink surfactant concentration at zero. Also a set of experiments (Type II) was carried out with the reservoir concentration at least a factor of 10 greater than the CMC, and the sink concentration greater than the CMC. These types of experiments can yield the pore permeability for m o n o m e r and micelle, respectively, in a manner described by Roy et al. (1). As will be emphasized later, the contribution to the flux by the "other" species, i.e., micelle or monomer, is negligible in the above experiments. Another set of diffusion experiments
DIFFUSION OF MICELLES (Type III) was conducted where the reservoir concentration was below 10 × CMC and the sink concentration was close to the CMC. The mass transfer rates in all experiments were obtained by measuring the sink surfactant concentration spectrophotometrically at 274 nm. After the diffusion experiment, a surfactant mass balance was performed to insure that no surfactant was lost from the sample holder during the experiment. The free diffusion coefficient of micelles was measured with the Taylor dispersion technique using the apparatus described by Weinheimer et al. (12). In this method a pulse of more concentrated surfactant solution is injected into a surfactant solution flowing in a precision capillary tube. The time dependence of the average concentration is recorded at the end of the tube concentration from which the diffusion coefficient can be obtained. To check the technique, the free diffusion coefficient of urea was measured and compared with available values. Surface tension measurements were conducted with a Wilhelmy plate technique to determine the CMC of the surfactant. In all experiments, doubly distilled water was used and the temperature was 30°C.
499
Assuming for the sake of simplicity that the micelles are monodisperse, then, at equilibrium, Ke nA O-- An, [1] where A and An are the m o n o m e r and micelle forms, respectively, K, is the equilibrium constant, and n is the aggregation number. These quantities are related via the expression
where Cm is the micellar concentration and C1 is the m o n o m e r concentration. The equilibrium constant Ke can be expressed in terms of the CMC and aggregation number n as follows. First let us introduce the dimensionless variables: C*
-
C CMC '
The micellization thermodynamics. The process of micellization has been modeled either by the phase separation model (13) or by the equilibrium model (14-16). In the first of these models, it is assumed that the entire amount ofsurfactant which exceeds the critical micelle concentration (CMC) aggregates as micelles. In the second, aggregates and singly dispersed m o n o m e r are in thermodynamic equilibrium, the concentration of the latter increasing slightly above the CMC as the total concentration of surfactant increases. As the phase separation model is an approximate version of the equilibrium model, the emphasis in what follows is on the equilibrium model.
C*-
C1 CMC '
and C* -
Cm CMC
[3a, b, and c]
where C is the total surfactant concentration. Because C* = C'~ + n C * , [4] Eq. [2] can be rewritten as C* - C* n(CMC)n_lf~n ,
[5]
= In(C* - C*) - n In C*.
[6]
Ke
THEORY
[21
Cm = K~C7
or ln[Ken(CMC) n-I ]
Following Stigter (17), one can define the CMC as the concentration at which the derivative of the m o n o m e r concentration with respect to the t o t a l surfactant concentration is equal to one-half. This yields for the dimensionless m o n o m e r concentration at the CMC C~' -
n
n+l'
[7]
which combined with Eq. [5] leads to (n + 1)n l K~ = nn+l(CMC)n_ 1 .
[8]
Journal of Colloid and Interface Science, Vot. 100, No. 2, August 1984
500
KROVVIDI ET AL.
The value n = 175 obtained from Corti and Degiorgio for Triton X-100 (18) was used in the calculations. The value of the CMC obtained by us from surface tension versus concentration measurements is 2.29 × 10 -4 M at 30°C. Pore diffusion. In the diffusion of surfactant through the membrane pores, the monomers and the micelles have to satisfy the species continuity equations. If quasi-steady state is assumed and Fick's expression is used for each species within the pore, then (12): d2Clp
nRm = 0
[9]
Drop dz----5- + Rm = 0,
[10]
Dip d z 2
and d2Cmp
where Rm is the rate of micelle formation, z is the distance to the face of the membrane in contact with the reservoir, and Dip and Dmp are the diffusivities of the m o n o m e r and micelle, respectively, which account for the hindrance effects in the pore. E l i m i n a t i n g Rm between Eqs. [9] and [10], one obtains D
d2Cl° daCmp lp ~dz + nDmp dz 2 - 0 .
[11]
Integrating Eq. [ 11 ] twice, with the boundary conditions Cmp=C°p,
Clp=C°p
at
Cmp =Cmp, L
Clp = CLp at
nPm
o
Here PI = DlpKI and Pm = DmpKm are pore permeability coefficients, C1 and Cm are the bulk monomer and micelle concentrations, and K1 and Km are the m o n o m e r and micelle partition coefficients. Because the equilibrium between micelles and monomers is achieved in a very short time (10) compared to the diffusion through the membrane pore, it is reasonable to consider that C ° and C o as well as CLm and C L are related by the thermodynamic equilibrium expression given by Eq. [2]. Defining the flux in terms of an effective pore permeability and total surfactant concentration
Perf
J = -L-- ( c ° - c L ) '
[15]
where C o and C L are the total surfactant concentrations at the two pore ends at the reservoir and sink sides. Equations [14] and [15] yield Peff =
(C O __
C L)
cL )
.
[16]
[12]
where L is the thickness of the membrane, gives for the total flux J = Dip - Z - ( C 0,o - cl p) Drop o + n T (Cmp -- CLp)-
PI
J = 7 (C° - C~) + - - F (Cm - CLm). [14]
P I ( C ° - C L) + r t P m ( C ° -
z=0 z=L,
because the ratios between the sizes of the surfactant molecules and micelles and the size of the pore affect their partition between the bulk and the pore mouth (20). Assuming thermodynamic equilibrium between the bulk and the pore mouth, Eq. [ 13] can be rewritten as
[131
Here the superscripts 0 and L refer to the two ends of the pore and the subscript p indicates that the concentration is on the pore side. The concentrations C°o, cOo, C~fo, and C~p differ from those in the reservoir and sink Journal of Colloidand InterfaceScience, Vol.100,No. 2, August1984
RESULTS AND DISCUSSION
At 30°C, two types of diffusion experiments were carried out by the diaphragm cell method. In the first kind (Type I), the initial reservoir concentration of Triton X-100 was at least 10% below the CMC value of 2.29 × l 0 -4 mole/liter, while that of the sink was zero (distilled water). The diffusing molecules will be mainly the monomeric form of the surfactant with negligible micelle transfer. In the second type of experiments (Type II) the initial reservoir concentration was one to two orders of magnitude greater than the CMC
DIFFUSION OF MICELLES while the initial concentration of the sink was again at zero concentration. As observed by Roy et al., in Type II experiments a distinct change in the quasi-steady flux occurs when the sink concentration becomes equal to the CMC. From this time onward, the m o n o m e r activities in the two compartments of the cell are almost equal. This implies that the surfactant concentration increase in the sink concentration above the CMC is due mainly to micellar transport. Assuming quasi-steady state within the pore, the monomeric Triton X-100 pore permeability was obtained from the expression: In
CR-- Cs CR(0) -- C s ( 0 )
= -P1/3t.
[17]
Similarly, the micelle permeability was obtained from In
CR-- Cs CR(~'c) -- CMC
= --Pm~O,
[18]
140
501 I
I
I
I
I
x Porous Membrane Experiment Least Square Pit By Eq • [ 18t
120 . . . .
I ,,.x ..x.~~ sxSx -
sx jx 100
~risx~d x''x
80 )r
X
X
-
x
X X
60
X X
40
X X
20 0 xx
I
i
50
100
I
I
I
150 re 200 250
I
300
350
t (rain) FIG. 1. Number of moles of Triton X-100 diffused through Celgard 2500 porous membrane with time in a Type 1I experiment: reservoir volume = 68 ml, sink vol = 40 ml; CR(0)= 1.73 × 10-3M; Cs(0) = 0.0M. Calculated micellar permeability is Pm = 4.63 × 10 7 cm2/secwith /~ = 25.35 cm-z and Zc = 183.8 min.
where 13 = ~
+
[19]
CR and Cs are surfactant concentrations in the reservoir and sink at time t, P1 = KID~p and Pm = gmDmp are the m o n o m e r and the micelle pore permeabilities,/£1 and Km are the m o n o m e r and micelle partition coefficients between the bulk and pore; CR(0) and Cs(0) are the initial reservoir and sink concentrations, respectively. 0 = t - re where ~c is the critical time at which the sink concentration is at the CMC; Ap is the total pore area and Lp is the pore length. VR and Vs are the volumes of reservoir and sink, respectively. The ratio Ap/Lp for a particular membrane can be obtained from a diffusion experiment with a solute of small size relative to the pore diameter and with a known free diffusion coefficient, D , . Duffey et al. (21, 22) reported an Ap/Lp value of 40 c m / c m 2 of total membrane area for Celgard 2500. Data of a typical run of a Type II experiment and the least square fit by Eq. [18] are shown in Fig. 1. The adsorption of Triton X-100 on the pore wall causes a decrease in the available pore
area to the diffusing m o n o m e r as well as the micelles. The adsorption occurs during the pretreatment of the membrane and it is assumed that a monolayer coverage on the walls takes place. Then the space between the pore walls is decreased by about 70 A which is twice the length of the surfactant molecule in the adsorbed state (23). It is assumed that the molecule will be in the extended form when adsorbed on the wall as opposed to the helical or coiled form within the solution (24). It follows that for an adsorbed monolayer the pore area of the Celgard membrane is reduced by a factor = [(400 A × 4000 ~)/(400 A - 70 ,~)(4000 A - 70 A)] = 1.23. The Ap/Lp ratio is therefore 32.5 cm. This value was confirmed by us to within 10% for the diffusion of the solute butyl-p-aminobenzoate through Celgard 2500 membranes which were pretreated by the surfactant. The radius of this molecule is 4.0/~, the molecular weight is 193, and the free diffusion coefficient is 8.0 × 10-6 cm2/ see (25). This molecule should not experience pore hindrance since its diameter is much smaller than the pore size. Journal of ColloM and Interface Science, Vol. 100, No. 2, August 1984
502
KROVVIDI ET AL.
Figure 2 shows the monomeric and micellar pore permeabilities obtained from the Type I and Type II experiments. The monomeric pore permeability of Triton X-100 is found to be about PL = 5.0 _ 0.7 × 10-6 cm2/sec. This value is near to the free diffusion coefficient of D1 = 5.35 × 10 -6 cm2/sec (26). Since the monomeric pore permeability is close to the free diffusion coefficient, the hindrance experienced by the m o n o m e r in diffusing through the Celgard pores appears indeed to be negligible. Although the extended length of the Triton X-100 m o n o m e r (about 45 A (27)) is significant with respect to pore size, the coiled configuration of the molecule is smaller (24). The data imply that the partition coefficient of the monomer is near one. Unfortunately the diffusion data are not highly accurate below the CMC because of the low driving force. The micellar pore permeability Pm = 3.93 × 10-7 cm:/sec at 3.0 × 10 -3 moles/liter. The micellar free diffusion coefficient is Dm = 6.71 X 10 -7 c m 2 / s e c . This value was obtained by interpolating between those measured by the Taylor dispersion technique. The latter value compares well with a value of 6.93 × 10-7 cm:/sec at 25°C for Triton X-100 by Weinheimer et al. (12). I f F p and Foo are the drag forces experienced by a rigid spherical particle 60
I
moving with the same steady velocity inside a pore and in an "infinite" medium, respectively, then (28-30) Dmp _ Fo~ Dm
For a particle moving between parallel walls of a pore, translating along its axis (31), 1 '
where w is half the distance between the walls and r is the radius of the particle. In our experiments, 2w = 400 - 70 = 330 N. For a micellar diameter of 92.8 N (18), from Eq. [21] Dmp/Dm = 0.728. From theory (20) the micellar pore distribution coefficient is about 0.72. Thus the theoretical ratio of micellar pore permeability to the free diffusion coefficient is 0.52 which is slightly lower than the experimental value of 0.59. These values are consistent with those obtained for Nonoxynol9 (1). The micellar aggregation number for Nonoxynol-9 is about 250 (32, 33) versus 175 for Triton X-100 (18), and the micelle diameter is about 100 A versus about 93 N for Triton X-100 (18). The experimental ratio of I
I --m--
50
Free Diffusion Experiments
" " ° - - P o r o u s Membrane
Diffusion Experiments
o
o
40
%
[2Ol
Fp
30
a.
20 CMC
10 --
I ~ - - I - ---B------- 1 O--.--C 0 C--O 0
10-5
I 10-4
I 10-3
o
I 10-2
10-1
CR (m01es/I)
FIG. 2. Pore permeabilities of Triton X-100 in Celgard 2500 membranes and free diffusion coefficients of micelles in water at 30°C. Here P is either Pr or pro. Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
503
DIFFUSION OF MICELLES
micellar pore permeability to the free diffusion coefficient was found to be 0.53 for Nonoxynol-9 which is somewhat lower than that of Triton X- 100, as is expected because of the larger size of the micelle. The effect of surfactant concentration on the micellar permeability is modest, decreasing by approximately 10% over a 10-fold concentration range (1.7 × 10-3 to 3.0 X 10-2 M). The decrease in the free diffusion coefficient is within this 10% variation. The micellar permeability data suggest that the micellar size does not vary significantly in this concentration range. Figure 3 shows the permeabilities in all the types of experiments (Types I, II, and HI) as well as those calculated by the equilibrium model. The values of the permeabilities are plotted versus the log mean reservoir surfactant concentration during the measurement. This concentration is very close to the average reservoir concentration during the course of the experiment. The theoretical curve of Fig. 3 represents Pe~ and it was obtained on the basis of the equilibrium model and Eq. [16] using for the calculations the monomeric and micellar pore permeabilities obtained in Fig. 2 for Type I and II experiments. Thispresumes that for Type I and II experiments the contribution to the flux of the other species (mi60
I
o
celle and monomer) is negligible. Table I gives the ratio between the flux of the monomer and the total flux below and above the CMC and demonstrates that for reservoir concentrations larger than 10 X CMC the flux of the monomer is indeed negligible. For reservoir concentrations 10% below the CMC the micellar flux is negligible. Figure 3 shows that the experimental pore permeability coefficients for reservoir concentration slightly above the CMC remain relatively constant and equal to the monomer permeability and then fall abruptly to the lower micellar permeabilities. The presence of high permeability values (5,0 X 10-6 cmZ/sec) for reservoir concentrations just above the CMC is expected since the concentration of micelles is not yet significant to influence the diffusional process. A similar trend is predicted by the equilibrium model coupled with Eq. [16]. However, the model predicts an earlier and smoother drop in the permeability in the transition region. The earlier drop is probably due to the fact that the equilibrium model is rather sensitive to the value of the equilibrium constant, thereby influencing the concentration of miceltes. Further, the equilibrium model as used here involves near the CMC micelles of a single aggregation number, which may not be re-
I
o o
50
0°~"'~ o 0
40--
o
-
--
PorousMembrane Diffusion Experiments Chemical EquilibriumModel
g
~
I
I
30 x ct.
20
I I I
-
I I Io
-
10
% 0 10-5
13
I
I
I
10-4
10-3
10-2
i0-1
OR, Im ( m 0 t e s / I )
FIG. 3. Pore permeabilities of Triton X-100 in Celgard 2500 membranes for Type I, II, and III experiment& The dashed curve represents the theoretical calculations with the chemical equilibrium model. Here P is Pee- The average downstream concentration for type III experiments is about 8 × 10-5 moles/1. Journal of Colloid and Interface Science, Vol. 100, No, 2, August 1984
504
KROVVIDI ET AL. TABLE I Percentage of Monomer Flux Contribution to the Total Surfactant Flux as Calculated from the Chemical Equilibrium Modela'b Reservoir concl CMC
M o n o m e r flux × 100 Total flux
Reservoir conc.
M o n o m e r flux
CMC
Total flux
100. 100. 99.7 95.8 90.7 67.9
11.3 20.6 34.8 42.3 84.5 103.1
0.52 0.64 0.90 1.42 1.96 2.02
X 100
2.7 2.2 0.86 0.77 0.44 0.43
a Triton X-100; Celgard 2500 porous membrane. bp~ = 5.0 × 10-7 cm:/sec; Pm= 3.9 × 10-v cm:/sec; temperature = 30°C; n = 175; CMC = 2.29 × 10-4 mole/ liter; Ke(CMC)n-I = 8.8 × 10-5 (mole/liter)n-l.
alistic. Nevertheless, the theory shows fair agreement With the experimental values considering the number of assumptions made. SUMMARY
The diffusion of the surfactant Triton X-100 through porous Celgard 2500 membranes was measured over a large range of surfactant concentrations, from conditions below the critical micelle concentration (CMC), around and well above the CMC. From the diffusion data, the monomeric and the micellar permeability of Triton X-100 in the water-filled pores of Celgard 2500 were obtained. The monomeric pore permeability was found to be close to the monomeric diffusion constant in bulk water, indicating unhindered diffusion in the pores. Micellar pore permeability was about one half of the value for the micellar diffusivity in bulk water. Hindered diffusion of micelles in the water-filled pores occurs because the characteristic pore dimension is only three times the micellar diameter. The chemical equilibrium model applied to surfactant transport in the porous membranes shows reasonable agreement with the experimental data. ACKNOWLEDGMENT This study was supported in part by the University of California. Journal of Colloid and Interface Science, Vol. 100, No. 2, August 1984
REFERENCES 1. Roy, S., Ruckenstein, E., and Stroeve, P., J. Colloid Interface Sci. 92, 383 (1983). 2. Stroeve, P., Ruckenstein, E., Roy, S., and Lippes, J., AIChE Symp. Ser. 227, 11 (1983). 3. Bhattacharyya, D., Garrison, K. A., Jumawan, A. B., and Grieves, R. B., AIChE J. 21, 1057 (1975). 4. Schott, H., J. Phys. Chem. 68, 3612 (1964). 5. Morita, Z., and Iijima, T., J. Colloid Interface Sci. 82, 155 (1981). 6. Hod, T., Mizuno, M., and Shimuzu, T., Colloid Polym. Sci. 258, 1070 (1980). 7. Langer, R., Chem. Eng. Commun. 6, 1 (1980). 8. Robinson, J. R. (Ed.), "Sustained and Controlled Release of Biologically Active Agents." Plenum Press, New York, 1978. 9. Scher, H. B. (Ed.), "Controlled Release Pesticides." Am. Chem. Soc., Washington, D. C., 1977. 10. Cardarelli, N. F. (Ed.), "Controlled Release Pesticides Formulations." CRC Press, Boca Raton, FL, 1976. 11. Taylor, G. I., Proc. Roy. Soc. London A 219, 186 (1953). 12. Weinheimer, R. M., Evans, D. F., and Cussler, E. L., J. Colloid Interface Sci. 80, 357 (1981). 13. McBain, J. W., Trans. Faraday Soc. 9, 99 (1913). 14. Mukerjee, P., J. Pharm. Sci. 63, 972 (1974). 15. Tanford, C., "The Hydrophobic Effect." Wiley, New York, 1980. 16. Ruckenstein, E., and Nagarajan, R., J. Phys. Chem. 79, 2622 (1975). 17. Stigter, D., Recueil Trav. Sci. Pays Bas 73, 611 (1954). 18. Corti, M., and Degiorgio, V., Optics Comm. 14, 358 (1975). 19. Aniansson, E. A. G., Wall, S. N., Almgren, M., Hoffmann, H., Kielmann, I., Ulbricht, W., Zana, R., Lang, J., and Tondre, C., J. Phys. Chem. 80, 905 (1976).
DIFFUSION OF MICELLES 20. Glandt, E. D., A I C h E J. 27, 51 (1981). 21. Duffey, M. E., Ph.D. Thesis, Carnegie-Mellon University, Pittsburgh, PA (1977), 22. Duffey, M. E., Evans, D. F., and Cussler, E. L., J. Membr. Sci. 3, 1 (1976). 23. Kushner, L. M., and Hubbard, W. D., J. Phys. Chem. 58, 1163 (1954). 24. Cooney, R. P., Barraclough, C. G., and Healy, T. W., J. Phys. Chem. 87, 1868 (1983). 25. Majumdar, A., M.S. Thesis, University of California, Davis (1983). 26. Ramirez, W. F., Shuler, P. J., and Friedman, F., Soc. Petr. Eng. J. 20, 430 (1980). 27. Rosch, M., in "Nonionic Surfactants" (M. J. Schick, Ed.), pp. 753-773. Marcel Dekker, New York, 1967.
505
28. Bean, C. P., in "Membranes" (G. Eisenman, Ed.), Vol. 1. Marcel Dekker, New York, 1972. 29. Brenner, H., and Gaydos, L. J., J. Colloid Interface Sci. 58, 312 (1977). 30. Anderson, J. L., and Quinn, J. A., Biophys. J. 14, 130 (1974). 31. Happel, J., and Brenner, H., "Low Reynolds Number Hydrodynamics," Chap. 7. Prentice-Hall, Englewood Cliffs, NJ, 1965. 32. Schick, M. J. (Ed.), "Nonionic Surfactants." Marcel Dekker, New York, 1967. 33. Thomas, J. K., and Kalyansundaran, K., in "Micellization, Solubilization, and Microemulsions" (K. L, Mittal, Ed.), Vol. 2. Plenum Press, New York, 1977.
Journal ~?fColloMand InterfaceScience. Vol. 100,No. 2, August1984