Single Microgel Particle Studies Demonstrate the Influence of

Mar 26, 2005 - The binding of two cationic surfactants, dodecyltrimethylammonium bromide (DoTAB) and N-(1,1,2,2-tetrahydroperfluorodecanyl)pyridinium ...
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Langmuir 2005, 21, 3773-3781

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Single Microgel Particle Studies Demonstrate the Influence of Hydrophobic Interactions between Charged Micelles and Oppositely Charged Polyions Martin Andersson, Per Johan Ra˚smark, Christer Elvingson, and Per Hansson* Department of Pharmacy, Uppsala University, Box 580, S-75123 Uppsala, Sweden, and Department of Physical Chemistry, Uppsala University, Box 579, S-75123 Uppsala, Sweden Received November 2, 2004. In Final Form: February 24, 2005 The binding of two cationic surfactants, dodecyltrimethylammonium bromide (DoTAB) and N-(1,1,2,2tetrahydroperfluorodecanyl)pyridinium bromide (HFDePB), to covalently cross-linked sodium poly(styrenesulfonate) (PSS) microgels has been investigated by means of micromanipulator-assisted timeresolved light microscopy on single gels. It is demonstrated that repeated measurements on the same microgel under conditions of controlled liquid flow give highly reproducible results. The two surfactants are found to behave very differently with respect to degree of swelling, surfactant distribution in the gels, both during shrinking and at equilibrium, and kinetics of volume changes induced by them. The main difference is attributed to the presence of a hydrophobic interaction between PSS and the DoTAB micelles, absent in the case of HFDePB. Kinetic shrinking curves are recorded and analyzed using a model for steady-state transport of surfactant between the solution and the gels. Aggregation numbers for DoTAB in PSS solutions obtained from fluorescence quenching measurements are presented. A strong dependence on the surfactant-to-polyion concentration ratio is observed. Relations between surfactant binding isotherms, phase diagrams for linear polyelectrolyte/surfactant/water systems, and the binding to gels are discussed.

Introduction The swelling of polyelectrolyte gels in water is strongly diminished in the presence of multivalent ions1 or macromolecules2 of opposite charge to the network. In this respect, surfactants give rise to a number of interesting effects. It has been established that they are absorbed by oppositely charged gels through an ion-exchange reaction where simple counterions to the network are replaced by the surfactant.3-5 The well-documented collapse of initially swollen gels can be attributed to the formation of micelles inside the gel. Hence, when the osmotic swelling pressure exerted by the simple counterions is removed the gel contracts due to the elastic energy stored in the network.3 In some systems, however, phase separation is observed in gels that are only partly collapsed by the surfactant taken up. Typically, a collapsed micelle-rich surface phase (skin) is formed enclosing a swollen micelle-free core.6-10 In the excess of surfactant, the entire gel is converted to a dense phase,11 resembling the “complex coacervates” frequently observed when * Corresponding author. Address: Department of Pharmacy, Uppsala University, Box 580, S-75123 Uppsala, Sweden. E-mail: [email protected]. (1) Katchalsky, A.; Zwick, M. J. Polym. Sci. 1955, 16, 221. (2) Zezin, A. B.; Rogacheva, V. B.; Kabanov, V. A. Macromol. Symp. 1997, 126, 123. (3) Khokhlov, A. R.; Kramarenko, E. Y.; Makhaeva, E. E.; Starodoubtzev, S. G. Makromol. Chem., Theory Simul. 1992, 1, 105. (4) Khokhlov, A. R.; Kramarenko, E. Y.; Makhaeva, E. E.; Starodubtzev, S. G. Macromolecules 1992, 25, 4779. (5) Hansson, P. Langmuir 1998, 14, 2269. (6) Khandurina, Y. V.; Rogacheva, V. B.; Zezin, A. B.; Kabanov, V. A. Polym. Sci. 1994, 36, 184. (7) Khandurina, Y. V.; Dembo, A. T.; Rogacheva, V. B.; Zezin, A. B.; Kabanov, V. A. Polym. Sci. 1994, 36, 189. (8) Khandurina, Y. V.; Alexeev, V. L.; Evmenenko, G. A.; Dembo, A. T.; Rogacheva, V. B.; Zezin, A. B. J. Phys. II 1995, 5, 337. (9) Hansson, P.; Schneider, S.; Lindman, B. Prog. Colloid Polym. Sci. 2000, 115, 342. (10) Hansson, P.; Schneider, S.; Lindman, B. J. Phys. Chem. B 2002, 106, 9777. (11) There is evidence that a small core may be left in gels even at equilibrium.

oppositely charged polyelectrolytes are mixed.12 A large number of studies show that the microstructures of collapsed gels (and skins) are, in general, highly ordered, resembling the liquid crystalline phases in concentrated phases of surfactants in water, e.g., micellar cubic, hexagonal, and lamellar structures.7,13-16 The same types of structures have also been observed in concentrated phases of linear polyion together with oppositely charged surfactants.17-21 Svensson et al.22,23 found that a stoichiometric complex between linear polyacrylate (PA) and cetyltrimethylammonium ions is insoluble in pure water. The dry “complex salt”, free from simple ions, can absorb water, but the swelling stops when the water content is about 55 wt %. In this state the phase contains discrete micelles ordered on a cubic lattice with the polyion contained in the aqueous regions between them.22 The phase stability has been discussed in terms of an interaction between polyion-dressed aggregates,24 the latter considered as the repeating unit of the complex salt.23,24 The structural order should be strongly influenced by (12) Lindman, B.; Thalberg, K. Polymer-surfactant interactions: Recent developments. In Interactions of Surfactants with Polymers and Proteins; Goddard, E., Ananthapadmanabhan, K. P., Eds.; CRC Press: Boca Raton, FL, 1993; p 203. (13) Hansson, P. Langmuir 1998, 14, 4059. (14) Chu, B.; Yeh, F.; Sokolov, E. L.; Starodoubtsev, S. G.; Khokhlov, A. R. Macromolecules 1995, 28, 8447. (15) Okuzaki, H.; Osada, Y. Macromolecules 1995, 28, 380. (16) Sasaki, S.; Koga, S.; Sugiyama, M.; Annaka, M. Phys. Rev. E 2003, 68, 021504. (17) Ilekti, P.; Piculell, L.; Tournilhac, F.; Cabane, B. J. Phys. Chem. B 1998, 102, 344. (18) Ilekti, P.; Martin, T.; Cabane, B.; Piculell, L. J. Phys. Chem. B 1999, 103, 9831. (19) Kogej, K.; Evmenenko, G.; Theunissen, E.; Berghmans, H.; Reynaers, H. Langmuir 2001, 17, 3175. (20) Kogej, K.; Theunissen, E.; Reynaers, H. Langmuir 2002, 18, 8799. (21) Kogej, K. J. Phys. Chem. B 2003, 107, 8003. (22) Svensson, A.; Piculell, L.; Cabane, B.; Ilekti, P. J. Phys. Chem. B 2002, 106, 1013. (23) Svensson, A.; Piculell, L.; Karlsson, L.; Cabane, B.; Jo¨nsson, B. J. Phys. Chem. B 2003, 107, 8119.

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excluded volume interactions, i.e., repulsive forces. However, the net interaction between the micelles is also attractive at some distances.22-24 The attraction has been explained to arise from electrostatic correlation and/or polyion bridging interactions.25 In the present paper, we show that insights into the mechanisms of gel collapse and the nature of the interaction between polyions and oppositely charged surfactants can be obtained by studying two cationic surfactants, dodecyltrimethylammonium bromide (DoTAB) and N-(1,1,2,2-tetrahydroperfluorodecanyl)pyridinium bromide (HFDePB), binding to covalently cross-linked sodium poly(styrenesulfonate) (PSS) gels. Both systems have interesting relationships to PA/DoTAB5,9,10,13,26-29 and other well-characterized pairs of interactants.30,31 In contrast to the latter group, representing the “normal” behavior, aqueous mixtures of linear PSS and DoTAB are stable as long as the equivalent concentration of the polyion is in excess.27 The explanation to this appears to be that the micelles formed gain a substantial net charge from excess binding of the polyion. The major reason for this is a hydrophobic interaction between the polyion backbone and the micelle surface, in addition to the electrostatic interactions, as was demonstrated by Kwak and co-workers.32,33 (For other systems, purely electrostatic motifs for “overcharging”,34,35 as well as the suppression of the effect, have been discussed.24) With HFDePB, on the other hand, the hydrophobic interaction will be effectively removed due to the nonfavorable interaction between hydrocarbons and fluorocarbons. Therefore, as our results will show, the abnormal behavior observed with DoTAB is “normalized” when it is replaced by HFDePB.36 One purpose of this paper is thus to demonstrate that the presence of hydrophobic interactions between polyion and micelles can be probed just by observing the effect of surfactant on gels. Another is to explore the relationships between the collapse of gels and the phase equilibria observed with linear polyions. With the present choice of components, we have the opportunity to test how attractive and repulsive micelle-micelle interactions influence the distribution of surfactant in gels, including the phenomenon of phase coexistence, and the degree of swelling of the collapsed phases appearing. The main technique employed has been micromanipulator-assisted microscopy studies of microgels under conditions of controlled liquid flow rates, which allows for accurate kinetic studies of deswelling and growth of surface phases. To remove uncertainties arising from variations between individual gel beads, repeated experiments has been conducted on one single gel specimen. The results are interpreted with reference to previously (24) Hansson, P. Langmuir 2001, 17, 4167. (25) Granfeldt, M. K.; Jo¨nsson, B.; Woodward, C. E. J. Phys. Chem. 1991, 95, 4819. (26) Thalberg, K.; Lindman, B.; Bergfeldt, K. Langmuir 1991, 7, 2893. (27) Hansson, P.; Almgren, M. Langmuir 1994, 10, 2115. (28) Hansson, P.; Almgren, M. J. Phys. Chem. 1995, 99, 16684. (29) Go¨ransson, A.; Hansson, P. J. Phys. Chem. B 2003, 107, 9203. (30) Thalberg, K.; Lindman, B.; Karlstro¨m, G. J. Phys. Chem. 1990, 94, 4289. (31) Thalberg, K.; Lindman, B.; Karlstro¨m, G. J. Phys. Chem. 1991, 95, 6004. (32) Gao, Z.; Kwak, J. C. T.; Wasylishen, R. E. J. Colloid Interface Sci. 1988, 126, 371. (33) Hayakawa, K.; Kwak, J. C. T. J. Phys. Chem. 1982, 86, 3866. (34) Park, S. Y.; Bruinsma, R. F.; Gelbart, W. M. Europhys. Lett. 1999, 46, 454. (35) Mateescu, E. M.; Jeppesen, C.; Pincus, P. Europhys. Lett. 1999, 46, 493. (36) The idea to test a fluorinated surfactant was suggested to P.H. by Prof. Mats Almgren.

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Figure 1. Experimental setup for kinetic measurements.

reported phase diagrams for mixtures of DoTAB with linear PSS and PA.27 To facilitate the interpretations, we also present some previously unpublished data on DoTAB aggregation numbers in solutions of PSS at various degrees of complexation. Experimental Section Chemicals. Sodium styrene sulfonate (Aldrich), N,N′-methylene-bis-acrylamide (Sigma), N,N,N′,N′-tetramethylethylenediamine (TEMED) (Sigma), ammonium persulfate (Sigma), DoTAB (Aldrich), sodium bromide (FisherChemical), and paraffin oil (VWR International) were used as received. HFDePC was a kind gift from Prof. Tsuyoshi Asakawa, Kanazawa University. Linear sodium poly(styrene sulfonate) (Mw ) 1.5 × 106) was a kind gift from Prof. Hans Vink, Uppsala.37 Highly purified water (Millipore) was used throughout. Pyrene (Aldrich) and dimethylbenzophenone (DMBP) (Aldrich) were twice recrystallized from ethanol. Preparation of Microgels. Particles of cross-linked poly(styrene sulfonate) were synthesized in a water in oil (w/o) suspension. Two grams of styrene sulfonate, 0.13 g of N,N′methylene-bis-acrylamide, and 0.01 g of ammonium persulfate were dissolved in 10 g of water. As a last step, 0.05 g of TEMED was added to the solution. A 1.5 mL portion of the solution was carefully added from a syringe to a RB-flask containing 70 mL of paraffin oil, preheated to 70 °C. The w/o suspension was created by stirring with an egg-shaped magnet (1000 rpm). The paraffin oil and the reaction mixture were deoxygenized with nitrogen prior to use, and the entire polymerization process was held under a nitrogen atmosphere. After 30 min the paraffin oil was poured into a separating funnel, and suspended particles were extracted from the oil, using water. The particles were then washed with, and stored in, pure water. Submillimeter-sized particles can be obtained this way, with a wide size distribution, but larger gels can also be obtained. Macroscopic gel globules, some as large as 0.8 cm in diameter, were obtained when using a lower rotational speed (700 rpm) of the magnetic stirrer. Microscopy. Single gels were studied using a light/fluorescence microscope (Olympus BX-51) equipped with a micromanipulator (Narishige ONM-1), a digital camera (Olympus DP50), and software (Olympus DP-Soft); see Figure 1. All micropipets were pulled and polished using a Narishige PC-10 Puller and a MF-9 Forger. Gels were picked up with the micromanipulator by suction, using an IM-5A Injector. All test solutions containing DoTAB and HFDePC were thoroughly degassed (under low pressure) prior to use to avoid bubble formation. For the kinetic studies, particles with diameters of about 100 µm were singled out. The particles were equilibrated in 10 mM NaBr and centered inside a 90 × 1.54 mm glass tube 1 mm from the outlet. The glass tube, in turn, was connected to a pump providing the test solution (110B Solvent delivery module; Beckman). The fluorinated surfactant was added as a chloride salt. However, since NaBr is in excess in the test solutions, we will refer to it as HFDePB (B ) bromide). The mean flow rate in the tube was 0.7 cm/s as calculated from the volume flow and the tube diameter. For laminar flow, the actual flow rate (v) at the center of tube, where the particle is positioned, is a factor of (37) Vink, H. Makromol. Chem. 1981, 182, 279.

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2 larger38 but has not been explicitly determined (see below). The flow rate was the same for all measurements, and the test solutions were given time to stabilize (with respect to flow rate and dilution effects) before beginning a measurement. However, the connection between the pumping system and the glass capillary tube was made up by a duct (190 × 1.1 mm) always containing a 10 mM NaBr solution at the beginning of each measurement. The time required for the test solution to reach the particle was estimated to about 10 s, and 10 s has been subtracted from each curve in all the plots. Photographs of the particle interacting with the flowing surfactant solution were then taken at regular intervals, and particle diameters were determined from these photographs. Repeated measurement on a single particle was possible to perform as the surfactant (DoTAB) could readily be washed out from the gel matrix. The particles were rinsed, alternating with 100 mM NaBr solution and pure water, and thereafter equilibrated in 10 mM NaBr before repeating the experiment. For convenience, when repeating measurements, an additional (peristaltic) pump (P-1 Pharmacia Fine Chemicals) was used for rinsing the particles and emptying and refilling the observation vessel. All measurements were carried out in a room thermostated at 22 °C. Time-Resolved Fluorescence Quenching (TRFQ). The method was used to determine micellar aggregation numbers. Decays were recorded using the single-photon counting technique, as described in detail by Almgren et al.39 The fluorescent probe (pyrene) was excited at 325 nm by short (ps) laser pulses. The emission was selected using a monochromator (395 nm). The data analysis was performed in one step by fitting the InfeltaTachiya equation40,41 to the recorded decays. In the present case, where the quencher is stationary in the micelles during the experimental time-window, this takes the form

[

F(t) ) F(0) exp -

]

t + 〈n〉{exp(-kqt) - 1} τ0

Figure 2. Micrographs of a 104 µm PSS particle during shrinking in two different environments. The pictures are taken at t ) 0, 60, and 360 s respectively (columns). The three uppermost images show the particle in DoTAB solutions, and the three pictures below show the same particle in HFDePB solutions. Observe that for the particle shown, a higher concentration of DoTAB (0.4 mM) has been used than of HFDePB (0.1 mM). The qualitative difference as discussed in the text is retained regardless of the concentrations used.

(1)

where 〈n〉 is the average number of quenchers per micelle, F(0) is the fluorescence intensity at time zero, and kq is the intramicellar quenching rate constant. τ0 is the probe lifetime, determined in a separate experiment in the absence of quencher. The average aggregation number, N, is calculated from the relationship

N)

[S]m [Q]m

〈n〉

(2)

where [S]m and [Q]m are the concentrations of surfactant and quencher in micelles, respectively. In the present solutions, the free concentrations of both surfactant and quencher are negligible in comparison with the total concentrations. Pyrene and DMBP were mixed with the other components in the following way. Appropriate amounts of stock solutions of the probe and the quencher in ethanol were transferred to the sample vials. Ethanol was carefully removed by letting nitrogen flow over the sample. Finally, solutions containing PSS and surfactant at different ratios were added, and the samples were stirred with a magnetic stirrer for 3 days before measurements. All solutions contained 41 mM surfactant, 10-5 M pyrene, and 0.150.62 mM DMBP, the latter concentration adjusted to keep 〈n〉 below 1.

Results and Discussion General Observations. Microscopy pictures of a PSS microgel (initial diameter, 104 µm) captured at three stages during shrinking in 0.4 mM DoTAB (top) and 0.1 mM HFDePB solutions are shown in Figure 2. In both (38) Coulson, J. M.; Richardson, J. F.; Blackhurst, J. R.; Harker, J. H. Coulson & Richardson’s Chemical Engineering, 5th ed.; ButterworthHeinemann: Oxford, 1996; Vol. 1. (39) Almgren, M.; Hansson, P.; Mukhtar, E.; van Stam, J. Langmuir 1992, 8, 2405. (40) Infelta, P. P.; Gra¨tzel, M.; Thomas, J. K. J. Phys. Chem. 1974, 78, 190. (41) Tachiya, M. Chem. Phys. Lett. 1975, 33, 289.

Figure 3. Repeated measurements on a 132 µm PSS particle, the surfactant solution containing 0.2 mM DoTAB (two measurements) and 0.2 mM HFDePB (one measurement). The lines are guides to the eye.

environments, the binding of surfactant results in a substantial volume decrease of the gel. However, as illustrated by the pictures in the middle, taken after 60 s, with DoTAB a diffusion boundary rapidly moves toward the center of the gel. Once the boundary has reached the center, only one phase can be seen in the gel, but the shrinking continues to some extent until the final volume is reached. This behavior is not observed with HFDePB. Instead a dense but very thin surface phase (skin) forms that is difficult to see in the light microscope. During the major part of the shrinking, the presence of it is evident from other observations, as will be discussed below. However, already at an early stage the surface phase can give rise to patterns in the gel surface, and at late stages of the shrinking process a wrinkling may be observed, as seen earlier in a related system.29 These are features absent with DoTAB. In the experiments presented here, the relative errors in the estimates of V/V0 due to both wrinkling and deviations from a spherical gel shape are small; see error bars in Figures 3-5. Another important difference between DoTAB and HFDePB is the degree of swelling of the gels in the final state, which is typically much larger in the former case. With HFDePB the final

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Figure 4. Deswelling of a 104 µm PSS particle in two different environments. Triangles: 0.1 mM HFDePB. Squares: 0.4 mM DoTAB. Photographs of this particle are shown in Figure 2. The lines are guides to the eye.

Figure 5. Deswelling of a 96 µm PSS particle at four different DoTAB concentrations.

state of the gel is very compact. The behavior of HFDePB resembles much that observed for DoTAB binding to PA microgels, where the presence of a thin surface phase during shrinking has been proved by fluorescence microscopy.29 As pointed out in the Introduction, the hydrophobic interaction between PSS and DoTAB is believed to be responsible for a phase behavior that is rather different from that observed for PA/DoTAB. We can conclude already at this point that the odd behavior of PSS/DoTAB is directly observable also in kinetic microgel experiments, and furthermore that the “normal” behavior of PA/DoTAB is recovered when the hydrophobic interaction between micelle and PSS is removed by replacing DoTAB with the fluorinated surfactant. Single Gel Experiments. Figure 3 shows the deswelling of a PSS microgel (initial diameter, 132 µm) in 0.2 mM solutions of DoTAB and HFDePB, respectively. One should note that all data are obtained from repeated experiments on the same gel bead. This was possible as DoTAB could be effectively removed from the gel by rinsing with a salt solution and the gel restored to its fully swollen state (see Experimental Section). As a check, the DoTAB experiment was repeated under identical conditions. As shown in Figure 3, the data for the two runs nearly overlap. A comparison between DoTAB and HFDePB shows that the shrinking is faster with the latter. The shrinking rate is

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Figure 6. The relative volume (V/V0) of particles at the plateau in Figure 5 plotted against the DoTAB concentration in the solution. The diagram consists of data obtained from three particles. Four of the data points (0.1, 0.15, 0.2, and 0.3 mM) were obtained from a single 96 µm diameter particle, one (the slightly higher value at 0.2 mM) from a 132 µm particle, and one (0.4 mM) from a 104 µm particle.

expected to depend on the initial gel size, concentration in the solution, liquid flow rate, etc. (see below). Since all these parameters are the same, the result must reflect a difference in the interactions between the components. We will show below that this can be attributed to a difference in the densities of the surface phases. Figure 4 provides further information. Again the results are from repeated measurements on one gel (initial diameter, 104 µm), but the DoTAB concentration is larger than the HFDePB concentration. The result shows that the final degree of swelling is larger for the DoTAB system despite the 4 times larger surfactant concentration and the more rapid volume decrease. A comparison between the curves in Figures 3 and 4 indicates that the final gel volume in the case of DoTAB depends on the bulk concentration, rather than always reaching a fully collapsed state. To investigate this, repeated experiments at four DoTAB concentrations were performed on one PSS microgel (initial diameter, 96 µm). The result is presented in Figures 5 and 6. In the coming sections, we will analyze the kinetic data in further detail in relation to structure and dynamics. Stoichiometry of PSS/DoTAB Complexes. For the range of DoTAB concentrations investigated in Figure 5, the final state of the PSS microgel is a comparatively swollen structure with a homogeneous distribution of surfactant. As shown in Figure 6, the degree of swelling decreases with increasing surfactant concentration in equilibrium with the gel. The binding isotherms for DoTA+ to PSS in solution33,39 indicate that the free concentration of surfactant in equilibrium with the complexes increases substantially with the (average) stoichiometry of the complexes. This is in sharp contrast to the behavior of NaPA and most other polyelectrolytes investigated in the literature42 displaying highly cooperative binding isotherms. For the latter class of systems, the interpretation is that the chemical potential for the surfactant in the complexes changes very little with the composition for a range of concentrations above the critical aggregation concentration (cac), indicating that only small changes in the composition of the polyion-dressed micelles take place.24 With PSS, on the other hand, the binding isotherms indicate that the composition of the micelles changes gradually with the degree of binding to the polyion, β, except very close to cac. (β is defined as the (42) Hayakawa, K.; Kwak, J. C. T. Interactions Between Polymers and Cationic Surfactants. In Cationic Surfactants: Physical Chemistry; Rubingh, D., Holland, P. M., Eds.; Marcel Dekker: New York, 1991; Vol. 37.

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Figure 7. Relative volume of a 96 µm PSS particle as a function of β, the degree of binding of DoTAB.

number of bound surfactant molecules per polyion charged group.) Since the gels studied here are in contact with bulk solutions, the free concentration in equilibrium with the complexes is simply equal to the bulk surfactant concentration. To see how the final gel volume depends on the amount of surfactant taken up by the gel, we have used binding isotherms reported by Hayakawa and Kwak33 to obtain a relationship between surfactant concentration and β. By combining their data with the data in Figure 6, we obtain a graph showing how V/V0 depends on β in the PSS gels (Figure 7). Unfortunately, no isotherm was available for 10 mM salt (used by us), but for a range of other concentrations. The points in Figure 7 were therefore obtained from an extrapolation. The uncertainty introduced is expected to be small, as the isotherms had a similar shape at all salt concentrations; the curves were only shifted to higher concentrations with increasing salt concentration. Figure 7 shows that, at equilibrium, the molar ratio of surfactant to polyion in the gels is below 1 (in the studied range). This explains why the surfactant is homogeneously distributed in the gels. The argument comes from a previously reported phase diagram27 showing that mixtures of DoTAB and linear PSS in water do not phase separate as long as the polyion is in excess. Thus, the micelle-micelle interaction remains net repulsive at the studied compositions. As will be discussed later, this is in contrast to the “normal” behavior represented by PA/DoTAB, and, as it appears, PSS/HFDePB. A reduction of the volume of gels with increasing β is expected since the swelling pressure in the gel is strongly reduced as micelles replace sodium ions. Therefore, Figure 7 also explains why V/V0 for gels in their final state decreases with increasing bulk surfactant concentration (Figure 6). Aggregation Numbers. As already discussed, a gradual change in the stoichiometry of the complexes can also be inferred from the binding isotherms underlying Figure 7. To provide structural information to support this explanation, we present here redundant data on surfactant aggregation numbers (N) in 40 mM DoTAB solutions containing various amounts of PSS. In Figure 8, N is given as a function of the overall surfactant-to-PSS molar ratio. For comparison, we have included data for PSS/cetyltrimethylammonium bromide (CTAB) under the same conditions (previously unpublished). The data were obtained from time-resolved fluorescence quenching measurements at 25 °C using pyrene as the fluorescent probe and dimethylbenzophenone as the quencher dissolved in the micelles. As can be seen in the figure, the effect of PSS on N is large, and N varies substantially with the surfactant/PSS ratio. At high PSS concentrations, N is considerably smaller than in solutions of the pure surfactants in water. For comparison, at 40 mM surfactant, the same technique gave N equal to 63 and 120 for DoTAB and CTAB,

Figure 8. Aggregation numbers (N) for DoTAB and CTAB in PSS solutions. Surfactant concentration ) 0.04 M. Temperature ) 25 °C.

respectively.43 Importantly, the values obtained here are smaller than in the reference systems PA/DoTAB and PA/ CTAB, where N, both in solutions and in gels, has been found to be close to those of the pure surfactant systems at concentrations near the critical micelle concentration (cmc),24 and only marginally depending on the binding ratio. The effect of PSS resembles more the effect of polyethylenglycol on SDS micelles44 or poly(acrylic acid) on CTAB at low pH,45 where strong dependences on the relative amounts of surfactant and polymer have been observed. It is generally believed that the effect of PSS is caused by an incorporation of benzene moieties (part of the PSS backbone) in the headgroup region of the micelles,32,46 thereby reducing the surface free energy of the interface and, in turn, the attraction between the surfactant headgroups. (As aromatic groups are polarizable, they tend to distribute close to the micelle surface.47) The reduction of N can then be explained by force balance considerations.48 The effect is larger the lower the surfactant-to-PSS ratio, since there are more PSS available per surfactant molecule. This explains the large variation of N in Figure 8. The trend can also be explained using a slightly different argument, namely, that N adapts to increase the total area of the micelles (or, equivalently, the area per headgroup) available for PSS, a small N being consistent with a larger area for a given amount of surfactant. It is conceivable that the excess binding of PSS to the micelles must be significant to give rise to the small N observed here. The resulting net charge of the micelles explains the high water content of gels that has reached the plateau in Figure 5 and the swollen character of the (transient) surface phase during shrinking, as well as the stability of PSS/DoTAB mixtures in solutions. (The net charge of PSS/DoTAB micelles is also proved by the considerable protection of fluorescence probes in the (43) Hansson, P.; Jo¨nsson, B.; Stro¨m, C.; So¨derman, O. J. Phys. Chem. B 2000, 104, 3496. (44) van Stam, J.; Almgren, M.; Lindblad, C. Prog. Colloid Polym. Sci. 1991, 84, 13. (45) Fundin, J.; Hansson, P.; Brown, W.; Lidegran, I. Macromolecules 1997, 30, 1118. (46) Linse, P.; Piculell, L.; Hansson, P. Models of Polymer-Surfactant Complexation. In Polymer-surfactant systems; Kwak, J. C. T., Ed.; Marcel Dekker: New York, 1998; Vol. 77. (47) Almgren, M.; Grieser, F.; Thomas, J. K. J. Am. Chem. Soc. 1979, 101, 279. (48) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press Ltd.: London, 1992.

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micelles from quenching by anions present in the water.39,49 In comparison, the protection offered by PA/DoTAB was found to be quite poor.50) Along the same line of reasoning, it is understandable that phase separation is observed in solutions27 at around equimolar compositions, i.e., when the osmotic repulsion preventing phase separation is removed. Interestingly, as the concentration ratio in Figure 8 approaches unity, N approaches the values found in the PA systems. Probably, at this point only a minor fraction of the surface of each micelle is in hydrophobic contact with the polyion. The gradual change in the composition of the micelles, as shown by the variation of N, accounts for the gradual increase in the chemical potential of the surfactant with increasing surfactant-to-polyion ratio as shown by the binding isotherms. The behavior is very different from the PA case where N and the free concentration of surfactant in equilibrium with the micelles are constant for a range of concentrations above the cac (in very dilute PA solutions where phase separation is prevented). The observed reduction of N in the presence of PSS is in agreement with early results obtained for very dilute solutions.39 The values obtained in that study were somewhat larger than the present ones and showed no systematic dependence on the binding ratio; N(DoTAB) ≈ 39, N(CTAB) ≈ 67. For accurate determination of aggregation numbers in dilute solutions, it is important to know the partitioning of the quencher (and the surfactant) between micelles and water.43,51,52 Despite the efforts made to determine the concentration of quencher and surfactant in micelles in the earlier study, the reported values should be less accurate than the ones obtained here under conditions where all quencher and, essentially, all surfactant molecules are in micelles. The data presented here are also in good agreement with the synchrotron X-ray scattering data by Kogej and co-workers obtained for mixtures of linear PSS and alkylpyridinium surfactants.19 Shrinking Kinetics. In an earlier paper,29 the rate of transport of surfactant from a bulk solution to the core of a spherical gel, via a stagnant layer and a surface phase, was analyzed using the equation

4πD(C - cac)r1 dn ) dt 2/(2 + Sh) + (r1/r0 - 1)D/P

(3)

where n is the number of moles of surfactant taken up by the gel, r0 is the radius of the core, r1 is the gel radius, Sh is the Sherwood number relating the thickness of the stagnant layer to the gel radius (see Appendix), C is the surfactant concentration in the bulk, cac is the critical association concentration, D is the effective diffusion coefficient in the stagnant layer, and P is the permeability of the surface phase to the surfactant. Strictly, the equation is valid for steady-state transport in a fixed geometry; however it has been found to be applicable also to shrinking gels when the transport of surfactant is the rate-controlling process.29,53 The requirement is that other processes such as the relaxation of the network, the arrangement of the material in the surface phase, and the transport of water between gel and solution occur on much shorter time scales (for a full discussion see ref 29). When these requirements (49) Abuin, E. B.; Scaiano, J. C. J. Am. Chem. Soc. 1984, 106, 6274. (50) Hansson, P. Langmuir 2001, 17, 4161. (51) Hansson, P.; Almgren, M. J. Phys. Chem. B 2000, 104, 1137. (52) Almgren, M.; Hansson, P. Fluorescence studies of micelles. In Encyclopedia of Surface and Colloid Science; Marcel Dekker: New York, 2002; p 2255. (53) Nilsson, P.; Hansson, P. Manuscript in preparation.

Figure 9. Master plot of V/V0 as a function of tC based on the data for PSS/DoTAB in Figure 5.

are fulfilled, eq 3 can be integrated to give an equation that can be used to calculate the time for an initially swollen gel to reach a radius r1 in a deswelled state after n moles of surfactant has been taken up:

t)

(

( )) r

∫0n 2 +2Sh + DP r01 - 1

1 4πD(C - cac)

1 dn r1

(4)

It follows from the assumptions behind eq 4 that, when evaluating the integral, r0 and r1 (and Sh) can be considered as functions of n only. In a previous analysis of PA/DoTAB, relationships giving r0 and r1 as functions of β (rather than n) were obtained from a model taking into account the effect of the surface phase on the swelling of the core.29 Here we use eq 4 to show that the kinetics observed is dominated by stagnant-layer diffusion during the major part of the shrinking. The curves in Figure 5 were all obtained from experiments on the same microgel bead at a fixed liquid flow rate (fixed Sh), C being the only variable. Since the integral in eq 4 is independent of C, all curves should coincide if V/V0 is replotted against (C - cac) times t, or in the present case, since cac is negligibly small, against C times t. Figure 9 shows that this is indeed the case as long as V/V0 > 0.4. When the core is consumed the model is no longer appropriate, and so a divergence of the curves is expected after that point. The fact that the model predicts the correct concentration dependence is strong evidence that the transport of surfactant is the rate-controlling process, implying also that even for gels shrinking at different rates and in different environments, their volume at a given time is determined mainly by the amount of surfactant taken up. The result is somewhat surprising considering the rapid progression of the surface phase. The diffusion barrier provided by a thick surface phase could result in an accumulation of surfactant just outside the gel preventing a steady-state from developing, and furthermore that P changes both with time and C, giving rise to complicated kinetics. In fact, since the permeability to surfactant must depend on the structure and composition of the surface phase and since we know from previous sections that the final state of the gel depends directly on C, a reasonable first guess would be a different P for each curve in Figure 5. Such a variation is not at all in line with the simple behavior implied by the “master plot” in Figure 9. Our conclusion from this is that the permeability is so large at all studied concentrations that the kinetics is completely dominated by stagnant-layer diffusion. By

Binding of Surfactants to Microgel Particles

Langmuir, Vol. 21, No. 9, 2005 3779

(5)

of a plot of V/V0 versus n/V0. (Here, n was divided by V0 to get a number independent of gel size.) In the literature V/V0 is often plotted against β. Unfortunately, this is not possible, as the number of moles of polyion in the gel is not known. In the present case, a comparison between the behavior of DoTAB and HFDePB is straightforward anyway as the experiments were made on the same PSS gel bead. For both surfactants, the diffusion constant was put equal to 4 × 10-10 m2/s. This value has been determined for DoTAB in D2O29 but should be a good estimate also for HFDePB since it has a similar tail length. For both surfactants C is much larger than cac. (Cac ≈ 10-5 M for DoTAB/PSS in the presence of 10 mM NaBr.54 No value is available for HFDePB, but cmc is substantially smaller than for C14TAB (closer to CTAB).55 Therefore, since cac for C14TAB is equal to 5 × 10-6 M and 3 × 10-5 M in PSS54 and PA56 solutions (10 mM NaBr), respectively, cac for HFDePB/PSS should be e10-5 M.) Included in Figure 10 is also the same type of analysis for the data in Figure 4 (0.4 mM DoTAB, 0.1 mM HFDePB). The result is in qualitative agreement with the previous data set, but not quantitatively. The discrepancy is attributed to the fact that they originate from different gels (pointing to the relevance of repeating experiments on one gel bead). As can be seen in Figure 10, at a given amount of surfactant in the gel, the volume is considerably larger for DoTAB than for HFDePB. An important consequence of this is that, at a given bulk surfactant concentration, DoTAB binds faster to the gel, but the shrinking is slower than for HFDePB. This can be confirmed by combining the curves in Figures 3, 4, and 10. The faster binding with DoTAB is explained by a larger gel surface area; see eq 3. Of course, at long times, when the gels are reluctant to take up more surfactant, the situation may be different. The difference in gel volume at a given surfactant loading can be attributed to the difference in compactness of the surface phase formed with the two surfactants. We limit the discussion to V/V0 > 0.5, where the assumptions made in the kinetic analysis are best fulfilled. In this range the data originate from gels displaying a core/shell structure captured at intermediate stages of the shrinking. There are two different but related effects. First, the volume decrease depends directly on the difference in the swelling of the surface phase and the gel core, which is largest in the case of HFDePB, forming very dense phases with the polyion; see previous sections. Second, the rubber elasticity of the surface phase influences the swelling of the core network.10 In a thin surface phase, the lateral deformation of the network is essentially the same as the average deformation ratio in the core. This follows from symmetry. However, the network is also compressed in the radial direction, and when the surface phase is compact the deformation in this direction is substantial. This gives rise to an additional restoring elastic force, responsible for a further compression of the core, explaining why the PSS/HFDePB gels shrink further. Relation between Phase Diagrams and Structures in Gels. The phase diagrams of DoTAB together with linear PSS27 and PA,26 respectively, both contain a twophase region where associative phase separation takes phase. In this section we discuss how the precise location of it in the phase diagram is related to gel volume transitions and phase coexistence in gels. The phase

The equation can be used to find the amount of surfactant in a gel at any time t during shrinking from data of r1 as a function of t. The result from an analysis of the data in Figure 3 is given in Figure 10, in the form

(54) Kogej, K.; Skerjanc, J. Langmuir 1999, 15, 4251. (55) Kadi, M.; Hansson, P.; Almgren, M. J. Phys. Chem. B 2004, 108, 7344. (56) Kiefer, J. J.; Somasundaran, P.; Ananthapadmanabhan, K. P. Langmuir 1993, 9, 1187.

Figure 10. V/V0 as a function of the amount of bound surfactant (see text). The plot consists of data derived from the particles in Figures 3 and 4.

looking at eq 4, one can see that stagnant-layer-controlled kinetics is expected when the surface phase is thin (r0 ≈ r1) and/or when the transport of surfactant in the gel is intrinsically fast compared to stagnant-layer transport (P . D). According to the binding isotherms discussed above, the binding is cooperative close to the cac but the isotherm starts to flatten out already at rather low degrees of binding. It is reasonable to assume that, as long as there is a micelle free core left in the center of the gel, the surface phase maintains a composition corresponding to the cooperative part of the isotherm, i.e., an excess of polyion around the micelles, and thus that the surfactant activity at the interface between the surface phase and the core is close to cac. Hence, the surface phase is expected to maintain a uniform composition (degree of swelling) that changes little as long as there is a core left in the center. Note that a quick redistribution of surfactant in the surface phase is consistent with a large P in eq 4. The growth of the surface phase is driven, however, by a nonuniformity of the surfactant chemical potential, and the migration rate is controlled by the rate of transfer of surfactant to the gel, unless the surfactant flow is too small. When the core is consumed, it is still favorable for the surfactant to bind to the gel. However, the chemical potential will increase gradually as the average composition of the complexes changes. The binding rate therefore slows down and the kinetics becomes qualitatively different. Relation between Gel Volume and Amount of Bound Surfactant. In this section we take advantage of the stagnant-layer-controlled kinetics to obtain data of V/V0 as a function of n for DoTAB and HFDePB, respectively. When the transport of surfactant through the stagnant layer is rate determining, the kinetics can be analyzed with a simplified form of eq 3 obtained by putting r0 ) r1 or D/P ) 0. Now, since the gel swelling and the transport of surfactant to the core are treated as independent processes, we may consider r1 as a function of t (instead of n). Integration gives

n ) 4πD(C - cac)

r dt ∫0t(1 + Sh 2) 1

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Langmuir, Vol. 21, No. 9, 2005

diagrams referred to here consider three pseudo-components: surfactant, polyion, and water. In the case of PSS, the two-phase region extends out from the water corner of the phase triangle but is located only on the surfactant-rich side of the equimolar line,27 showing that a spontaneous formation of a dense phase enriched in both polyion and surfactant takes place only when the surfactant is in excess. As already mentioned, the phase stability observed when PSS is in excess can be explained by a net repulsion between micelles due to an excess binding of PSS segments to each micelle. The overcharging is due to the hydrophobic binding motive. By simple reasoning, the net charge is expected to decrease as the overall ratio of surfactant-to-polyion increases, and finally, when the repulsion is absent, phase separation sets in. Interestingly, the most concentrated phase bordering the two-phase region does not contain stoichiometric complexes, but instead a rather large excess of surfactant. Probably the hydrophobic interaction contributes to this effect, since the possibilities for the polyion chains to make contacts with micelles should increase with increasing surfactant concentration in the phase. However, this seems not to be the only contribution to the effect, since the complex salt formed by PA and CTA+ can also incorporate large amounts of CTAB without dissolving,22,23 despite the fact that hydrophobic interactions are weak here. The choice of counterion appears to be of importance.22,23 The most important consequence of the hydrophobic interaction is therefore the stability of the dispersion observed when PSS is in excess. When discussing cross-linked gels, transition between swollen and collapsed states taking place at a well-defined concentration in the solution is often described as the counterpart to associative phase separation. Therefore, it may seem strange that the phase diagram for PSS/DoTAB contains a two-phase region where associative phase separation takes place, and at the same time we observe that the gel volume is a continuous function of the bulk surfactant concentration (i.e., no volume transition). However, this is completely in order, since in the gels, the elastic forces in the network bring the micelles closer to each other also when they repel each other electrostatically. The gels shrink as the binding ratio increases, due to a reduction of the swelling pressure, but nothing dramatic happens when the molar ratio exceeds unity. Importantly, what the two-phase region in the phase diagram really means is that, for a sizable range of DoTAB/ PSS ratios exceeding unity, a concentrated complex phase does not dissolve even in the absence of covalent crosslinks. Of course, for gels, dissolution would correspond to swelling. To investigate how the gels behave at high binding ratios, we immersed spherical PSS macrogels (about the size of a pea) in DoTAB solutions more concentrated than used in the microgel studies. As expected, the gels were found to shrink considerably when placed in a 1 mM surfactant solution (overall charge ratio > 10), and in the final state only one phase could be observed in the gel. The two phases in equilibrium in the system, the gel and the liquid, correspond to the concentrated phase and the dilute solution in the phase diagram for linear PSS/DoTAB. Interestingly, at 3 mM DoTAB the behavior was different as evident from the pictures taken of a gel at different times shown in Figures 11 and 12. During the shrinking process, the initially transparent gel turned opaque, but some time after the gel had reached the collapsed state, a clear surface phase started to appear. Please note that in the first picture shown a considerable deswelling has already occurred. When the experiment was repeated at

Andersson et al.

Figure 11. Left image: a 0.8 cm PSS gel (extensively washed in pure water). Right image: the same gel after it has been equilibrated for a year in a 3 mM DoTAB solution (size, ca. 1 mm); the light source is underneath the globule, and shadow effects make the translucent shell visible.

Figure 12. Shrinking process of an initially swollen, macroscopic, PSS gel in a 3.0 mM DoTAB solution. Pictures are taken at 15 min, 2 h, 24 h, 3 days, 4 days, 10 days, and 30 days, respectively (from upper left). Observe the gradual change in opacity: The sphere remains completely transparent during the first 10 min and thereafter gradually becomes more opaque as it shrinks. With time, the globule again turns transparent from the surface and inward.

even higher concentrations, the clear phase was found to increase on behalf of the opaque core with increasing surfactant concentration, but the gel volume did not change much. Our interpretation is the following. With increasing surfactant concentration in the solution, more and more surfactant is incorporated in the gels. According to the phase diagram, this is possible without inducing a swelling (see above). However, at some point the packing of the complexes in the gel may be in favor of an ordered structure, appearing as an optically clear surface phase. The surface phase was observed to grow slowly, suggesting a small structural difference between the coexisting phases. A recent Raman study shows that the surfactantto-polyion ratio is essentially the same in the core and the surface phase.57 In the phase diagram for PA/DoTAB,26 the two-phase region extends along the polyion-surfactant equimolar line (in a pseudo-three-component representation). Notably, a concentrated phase, containing nearly equal amounts of polyion and surfactant, separates out from mixtures containing even a large excess of the polyion. This is the “normal” behavior where, in the absence of a hydrophobic motive for overcharging, the electrostatic free energy is in favor of a concentrated phase where the polyion and the surfactant neutralize each other without involving a large number of small ions. There are important consequences for gels that depend on the conditions. For a gel in contact with a bulk solution of the surfactant (typical for studies of single microgels), a volume transition from a swollen to a collapsed state takes place at a defined (57) Råsmark, P. J.; Andersson, M.; Lindgren, J.; Elvingson, C. Langmuir 2005, 21, 2761.

Binding of Surfactants to Microgel Particles

surfactant concentration (≈cac). Importantly, during shrinking the “new” phase appears as a dense surface phase gradually consuming the swollen core. For a gel in contact with a solution containing a limited amount of surfactant, not enough to collapse the whole gel (typical for macrogels), the two phases coexist in the final state. A matter of debate is often if the core/skin arrangement is an equilibrium structure or simply due to slow dynamics. First of all, the only requirement for phase coexistence in gels, in addition to those in a non-cross-linked system, is that the cohesive energy in the collapsed phase overcomes the work of deforming the two phases. Regarding the dynamics, a comparison between PSS/DoTAB and PA/ DoTAB is illuminating. Consider the shrinking in solutions where the final state, in both cases, is a dense collapsed gel. During shrinking a micelle-rich surface phase appears as an intermediate state in both systems. We have seen for PSS gels that the phase boundary reaches the center of the gel very rapidly. This is in contrast to PA/DoTAB and also PSS/HFDePB, where the movement of the phase boundary relative to the gel surface is very slow (or, in other words, both move essentially with the same rate to the gel center). In both cases micelles form at very low concentrations in the gels. Therefore, when the concentration in the solution is larger than the cac, there will always be micelles in the gels, at least near the surface. The local concentration of surfactant unimers should be approximately equal to and never exceed cacgel (defined as the concentration of surfactant in a gel when that in the solution is equal to cac).5 If the redistribution of surfactant within the network was not influenced by interactions between micelle/polyion complexes, the rate of the process would depend strongly on the concentration of unimers. Then the mobility would be largest in PA gels, since cac for DoTAB is larger with PA than PSS.24,42 However, since the phase boundary migration rate is in fact slower with PA, it can be concluded that it is more favorable for the surfactant, in this case, to form a surface phase than to distribute evenly in the gel. Hence, the appearance of dense surface phases during shrinking is due to a net attraction between the surfactant aggregates (cohesive forces), not an effect of slow dynamics. Consider now gels placed in solutions containing a limited amount of surfactant, to give, say, β ) 0.5 in the final state. Here, DoTA+ micelles are evenly distributed in PSS but exclusively in the surface phase of PA gels. In both cases the surfactant has entered the gel from outside, and the interaction between the micelles and polyion is at least as strong with PSS as with PA. Also in this case, the difference is difficult to explain by using arguments based on slow dynamics. Therefore, we are convinced that two phases can coexist in gels at equilibrium. There are arguments in favor of a core/shell arrangement in spherical gels.58 However, for gels with other forms other alternatives may be more favorable.59-61 Furthermore, if the surfactant, for some reason, could enter only via a small part of the gel surface, rearrangements leading to the most favorable state may be slow, due to long-range elastic interactions in the network. Also, volume transitions in gels are expected to involve hysteresis.60,62-64 (58) Sekimoto, K.; Kawasaki, K. Physica A 1989, 154, 384. (59) Hirotsu, S. J. Chem. Phys. 1987, 88, 427. (60) Sekimoto, K. Phys. Rev. Lett. 1993, 70, 4154. (61) Panyukov, S.; Rabin, Y. Macromolecules 1996, 29, 8530. (62) Matuso, E. S.; Tanaka, T. J. Chem. Phys. 1988, 89, 1695. (63) Onuki, A. Phys. Rev. A 1988, 38, 2192. (64) Tomari, T.; Doi, M. Macromolecules 1995, 28, 8334.

Langmuir, Vol. 21, No. 9, 2005 3781

Concluding Remarks The major results of the present work were obtained from repeated experiments on one gel bead. The advantage of this is that comparisons between the effects of the two surfactants investigated are not obscured by the difference between individual beads. The results obtained demonstrate that a previously documented hydrophobic interaction between PSS and DoTAB micelles has a large effect on the interaction between the surfactant and cross-linked gels of the polymer. Most importantly, the surfactant aggregates distribute evenly in the gels as long as the polyion is in excess. This supports the idea that the micelles have a net charge due to excess binding of polyion segments to them. The behavior is in sharp contrast to that of systems where the interaction between polyion and surfactant is purely electrostatic. This was demonstrated by reducing the attraction between micelle and polyion backbone by replacing DoTAB with a fluorinated surfactant. The effect was very clear. At the same surfactant loading, the gels were more contracted and collapsed surface phase was found to coexist with a surfactant-free swollen core. The latter behavior strongly resembles that observed for DoTAB absorbed by PA gels, a system where the interaction between polyion and surfactant is known to be mainly electrostatic. We have found that many of the intricate aspects characterizing the binding of surfactants to gel are in agreement with the behavior observed for mixtures of the surfactants and the corresponding linear polyions, both regarding phase stability and binding isotherms. However, there are effects due to the elasticity of cross-linked networks that must be taken into account, for instance, when relating gel volume transitions to associative phase separation. Acknowledgment. This work was supported by a grant from the Swedish Foundation for Strategic Research (SSF). C.E. acknowledges support from Uppsala University and Ingegerd Berghs stiftelse. We are grateful to Go¨ran Karlsson, Peter Nilsson, and Helena Bysell for skillful technical assistance. Appendix The stagnant-layer thickness d, a fictitious quantity constructed to simplify calculations of mass transfer, is defined as the ratio between D and the mass transfer coefficient.38 For the situation considered here, the stagnant-layer concept allows us to treat the steady-state transport of surfactant from the bulk solution to the gel surface as a pure diffusion process across a liquid film of thickness d. For single spherical gels,38

d)

2r1 Sh

(A.1)

where Sh is the Sherwood number. Under conditions of forced convection, Sh is a function of the Reynolds (Re) and Schmidt (Sc) numbers. For Re < 20,38

Sh ≈ 2.0 + 0.6(Re)1/2(Sc)1/3 Re )

(A.2)

2vr1F η

(A.3)

η FD

(A.4)

Sc )

where v is the liquid flow rate, F is the liquid density, and η is the liquid viscosity. LA047316V