Deswelling Kinetics of Polyacrylate Gels in Solutions of

27 Jul 2007 - Peter Nilsson* andPer Hansson. Department of Pharmacy, Uppsala University, Box 580, S-75123 Uppsala, Sweden. J. Phys. Chem. B , 2007 ...
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J. Phys. Chem. B 2007, 111, 9770-9778

Deswelling Kinetics of Polyacrylate Gels in Solutions of Cetyltrimethylammonium Bromide Peter Nilsson* and Per Hansson Department of Pharmacy, Uppsala UniVersity, Box 580, S-75123 Uppsala, Sweden ReceiVed: April 11, 2007; In Final Form: June 1, 2007

The deswelling kinetics of single sodium polyacrylate gel beads (radius 40-160 µm) in aqueous solutions of cetyltrimethylammonium bromide under conditions of forced convection are investigated using micromanipulator assisted light microscopy. The purpose of the study is to further evaluate a previously published model (J. Phys. Chem. B 2003, 107, 9203) using a higher homolog surfactant. For gels with expected fast deswelling (small gel size/low surfactant concentration) and/or in low electrolyte concentration, the model is found to correctly predict the deswelling characteristics of the gel beads. However, for some gels with expected slow deswelling, especially in high electrolyte concentration (10 mM NaBr), the model widely underestimates the required deswelling time. The reason for this is argued to be the longer time frame and high bromide concentration allowing the formation of a denser, more ordered structure in the surface phase, which resists the deformation and reorganization of material necessary for deswelling. Unexpectedly long lag times before the start of deswelling are also found for gels in low surfactant concentration, indicating that a relatively high surfactant concentration in the gel, greatly exceeding the critical aggregation concentration, is needed to start formation of a collapsed surface phase. This critical surfactant concentration is found to be dependent on initial gel radius, as small gels require a relatively higher concentration to initiate collapse.

1. Introduction The use of responsive polymer microgels for targeted delivery of toxic and/or labile drugs has during the past few years been shown to be a promising concept.1-5 Oral delivery to a selected part of the gastrointestinal tract has been demonstrated in vitro,6 and the concept of nanogels as drug carriers,7 especially for protein drugs, opens up thrilling new avenues with the possibility of targeting to and drug release into specific cell types, of which perhaps cytotoxic drugs aimed at cancer cells is the most interesting.8-10 The close connection to secretory granules and the possibility of mimicking their action makes the concept even more tantalizing.11 Understanding the gel swelling behavior and the gel-drug interaction is of crucial importance in order to allow the future rational design of such microgel based drug vehicles. To further this process we have chosen to study the interaction between cross-linked anionic sodium polyacrylate (PA) gels and oppositely charged surfactants. Previous studies in the area of charged polymer gel/oppositely charged surfactants have focused both on macroscopic12-17 and microscopic gels.18-20 For a recent review, see ref 21. Similarities can also be found with the interaction between PA gels and positively charged peptides22 and proteins.23,24 In this paper, we will focus on the deswelling kinetics of PA microgels in the presence of cetyltrimethylammonium bromide (C16TAB). A charged swollen anionic gel, such as a PA gel, that is exposed to a cationic surfactant, such as C16TAB, of a concentration exceeding the critical association concentration (cac) needed for micelle formation in the gel, will deswell due to the change in swelling pressure as the simple counterions in the gel are replaced by surfactant micelles.13,14 Interestingly, the outer part of the gel will collapse first forming a dense, micelle-rich outer surface phase around a still swollen core, * Corresponding author. Phone: +46(0)18 471 4368. Fax: +46(0)18 471 4223. E-mail: [email protected].

thereby giving rise to a phase separation within the still intact network. As more surfactant is absorbed by the gel the surface phase grows at the expense of the core.13,15 Since the surface phase is much denser than the remaining core, it will for the major part of the deswelling process be very thin compared to the total gel radius. Even so it influences the gel significantly both in that it may act as a diffusion barrier for the surfactant, and in that it compresses the core below its normal equilibrium volume.13,18 As has been shown previously, single gel beads can be handled using a micropipet technique,25 allowing for the deswelling kinetics to be evaluated under controlled conditions.19,20 Previous experiments with dodecyltrimethylammonium bromide (C12TAB)19 have shown that the kinetics are dominated by ion exchange from the bulk solution to the gel surface. We will now perform similar experiments with a longer chain surfactant, C16TAB, which requires lower critical concentrations for micelle formation, both in the gel (cac) and in solution (cmc). The lower cac, which will be measured in a separate experiment, is synonymous with a lower monomer concentration in the surface phase, which will possibly make the transport through the surface phase the rate-determining step. We will attempt to use the same semiempirical ion exchange model that has earlier been shown to correctly describe the stagnant layer controlled deswelling kinetics of PA/C12TAB,19 and test its applicability to C16TAB by using different surfactant concentrations, flow velocities, and gel sizes, as well as two different electrolyte concentrations (thereby affecting cmc and cac). 2. Experimental Section 2.1. Materials. N-Cetyl-N,N,N-trimethylammonium bromide (C16TAB) (p.a.) from Merck; acrylic acid (99%) from Aldrich; ammonium persulfate (AP) (g98%), N,N′-methylenebisacrylamide (NMBA), and N,N,N′,N′-tetramethylethylenediamine

10.1021/jp0728151 CCC: $37.00 © 2007 American Chemical Society Published on Web 07/27/2007

Deswelling Kinetics of Polyacrylate Gels (TEMED) (99%) from Sigma; sodium hydroxide (NaOH) from Eka Chemicals; sodium chloride (NaCl) (Baker analyzed) from Tamro; sodium bromide (NaBr) from Kebo; methanol (HPLC grade) from Fisher Scientific; cyclohexane (p.a.) from Riedelde Hae¨n; poly(acrylic acid) 170000 sodium salt from Fluka; and Span 60 from Carl Roth OHG were all used as received. Pyrene 98+ % from Janssen Chimica was twice recrystallized from ethanol. Solutions were prepared using high quality Millipore water. 2.2. Microgel Preparation. Submillimeter gels were prepared using an adaptation of the inverse suspension polymerization method used by Wang et al.26 A 10 mL portion of the reaction mixture containing acrylic acid, 1% cross-linker (NMBA), radical initiator (AP), accelerator (TEMED), NaCl(s), and 2 M NaOH was injected into 30 mL cyclohexane (with 0.3 wt % SPAN 60), in a round-bottom flask under heavy stirring. The polymerization was allowed to continue at 60 °C for 15 min under a nitrogen atmosphere, after which the suspension was poured into excess amounts of methanol. The precipitated gels were redissolved in water and bubbled with nitrogen to remove organic solvent, after which they were repeatedly washed with water on a 0.80 µm filter. 2.3. Micromanipulation and Light Microscopy. Single gels were studied using a light microscope (Olympus BX-51) equipped with a micromanipulator (Narishige ONM-1). All micropipets were pulled and polished using a Narishige PC-10 Puller and a MF-9 Microforge. All gels were in contact with a bulk solution containing either 10 or 0.5 mM NaBr and small amounts of NaOH (to ensure pH > 10 and thereby a fully charged network). All experiments were performed at room temperature. A small sample of gels was allowed to equilibrate in a petri dish with surfactant-free solution until no change in gel size could be observed. After equilibration, a gel of suitable size was chosen and picked up with the micromanipulator by suction, using an IM-5A injector. A larger pipet (the flow pipet), connected to a HPLC pump (Beckman 110B Solvent Delivery Module), was placed under the microscope. The gel was positioned centered inside the flow pipet, and a solution containing C16TAB was pumped through the flow pipet. As the gel deswelled, digital images were captured and stored every 10-20 s using an Olympus DP-50 digital camera. The images were analyzed with respect to gel size using the Olympus DPSoft software. For each image, the projected gel diameter in, as well as perpendicular to, the flow direction was measured, and the gel volume (V) was calculated assuming ellipsoid gels. In one set of experiments, the gels were first equilibrated and measured in 1 mM NaBr pH 10, before they were introduced to the bulk solution (10 mM NaBr), to establish the ratio of gel volume in 10 and 1 mM NaBr (V10/V1), a measure of the swelling characteristics of the network. 2.4. Steady-State Fluorescence Measurements. Samples containing 10-8-10-4 M C16TAB, 0.5 mM (monomer concentration) linear poly(acrylic acid) molar mass 170000 g/mol (noncross-linked), 10 or 0.5 mM NaBr, 0.1 µM pyrene, and small amounts of NaOH (to ensure pH > 8) were prepared by mixing stock solutions, one containing C16TAB and one surfactant-free. Pyrene emission spectra between 360 and 400 nm were recorded on a SPEX Fluorolog 1680 at an excitation wavelength of 325 nm and at a temperature of 20 °C. The ratio between the third and first vibronic peak was measured and used to determine the onset of micelle formation.27

J. Phys. Chem. B, Vol. 111, No. 33, 2007 9771 3. Theory 3.1. Swelling Equilibrium. According to a previously published model,18 there are three contributions to the osmotic pressure of the gel core: (1) the osmotic pressure exerted by the difference in ion activity between the gel core and the bulk solution (∆πion), which can be calculated by solving the Poisson-Boltzmann equation for a range of polymer concentrations; (2) the pressure induced by the deformation of the gel core network (πnet), which can be extracted from gel swelling experiments in different electrolyte concentrations as described by Ricka and Tanaka28 (data from an earlier publication has been used19); and (3) the pressure exerted on the core by the deformed surface phase (πskin), which can be calculated using the theory of rubber elasticity. When the gel is in swelling equilibrium, these three should cancel each other out, so

∆πion + πnet + πskin ) 0

(1)

For a gel in surfactant-free solution at equilibrium, the first two terms will always balance each other out, resulting in a certain degree of swelling which is dependent on the electrolyte concentration. Adding surfactant and thereby forming a surface phase will add the third term to the equation, further constricting the gel. A detailed description of how to calculate each of these terms can be found in an earlier paper.19 3.2. Deswelling Kinetics. The rate of surfactant binding to the spherical gel can, as verified in an earlier publication,19 be described by

dβ 3V0r1r0DIC ln(C1/cac) ≈ dt R 3(r - r ) 0

1

(2)

0

where β is the number of surfactant molecules per polyion charge in the gel, V0 is the volume per mole of polyion charges in the gel prior to deswelling, r0 and r1 are the core and gel radii at any given time, DI is the diffusion coefficient of the surfactant in the surface phase, C is the concentration of surfactant in the surface phase, R0 is the initial gel radius, and C1 is the surfactant concentration in contact with the gel surface. Equation 2 is valid for steady-state flow of surfactant. The theory behind eq 2 assumes that the free monomer concentration in the core is equal to the critical aggregation concentration (cacg), where the index g is used to denote the concentration in the gel, as opposed to cac, which is the corresponding concentration in the bulk solution. The total amount of surfactant in the core at the start of deswelling can be calculated from the lag time (see section 3.3). To reach the gel core and form new micelles, the surfactant has to diffuse first through the water layer around the gel, which can be modeled by stagnant layer diffusion, and then through the existing surface phase, which grows progressively thicker. It is assumed that the gel volume at any time during deswelling is equal to the volume of a gel containing the same amount of surfactant at equilibrium, and that the transport of surfactant to the gel core controls the gel deswelling rate. It is further assumed that since we have a surface phase containing closely packed micelles, we can approximate the DI and C as constant. Under these conditions, eq 2 can be integrated to give the time (t) needed for a gel to reach a shrunken state (β, r1) as

t)

R03 3V0DIC

∫0β

(

r 1 - r0

r1r0 ln(C1/cac)

)

dβ

(3)

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The integral in eq 3 will be calculated from the relationship between gel dimensions (r0, r1) and absorbed amount of surfactant (β), provided by the swelling equilibrium model described in section 3.1 and the measured lag time (section 3.3). C1 can be obtained by numerical solutions to the equation19

C 1 ) C2 -

r0CDI ln(C1/cac) (r1 - r0)(1 + Sh/2)DII

(4)

where C2 is the surfactant concentration in the bulk, DII is the effective diffusion coefficient in the stagnant layer, and Sh is the dimensionless Sherwood number describing the thickness of the stagnant layer, which can be calculated from the liquid flow velocity.29 A detailed description of how to perform these calculations can be found in an earlier paper.19 3.3. Lag Time. The time before micelle induced gel deswelling starts, during which the gel volume is constant, is denoted the lag time. In the simplest case, the formation of the surface phase coincides with the formation of micelles, and the lag time (tL) is simply the time needed to establish a steady-state diffusion profile in the stagnant layer plus the time needed for enough surfactant to diffuse into the gel to reach cacg (tcac). Thus

tL ≈ tcac +

(r2 - r0)2 2DII

(5)

where tcac can be calculated from18,29

tcac )

r02(1 - r0/r2)Cg [(R - 1)U - ln(1 - U)] 3DIIRCsalt

and

U(t) )

(

cacg Csalt +1-R RCg C2

)

(6)

(7)

where r2 is the radius of the gel including the stagnant layer, Cg is the total concentration of positive ions in the gel, Csalt is the total electrolyte concentration in the bulk, and R is the selectivity coefficient defined as

R)

CNa,0 Cs,g × CNa,g Cs,0

(8)

where Cs and CNa are the concentrations of surfactant and sodium in the gel (g) and in the liquid in contact with the gel (0). Should surface phase formation not coincide with cac, the additional uptake of surfactant during the lag phase can be calculated using eq 5 from ref 19 which under the simplification that gel volume is constant (R0 ) r0 ) r1, valid during the lag phase) becomes

t)

2R02β 3V0DII(C2 - cac)(2 + Sh)

(9)

This allows the β that fits the measured lag time to be found and used as the starting point for the calculation of the deswelling part of the curve (section 3.1, eqs 3 and 4). 4. Results and Discussion 4.1. Cac Determination. The pyrene 3/1 ratio has been measured by steady-state fluorescence on solutions containing

Figure 1. I3/I1 ratio of the fluorescence of pyrene in solutions of 0.5 mM PA 170000, 10-8-10-4 M C16TAB, pH > 8, and 10 or 0.5 mM NaBr. Cac is determined as the C16TAB concentration at the onset of I3/I1 increase (2.0 and 3.6 µM at 0.5 and 10 mM NaBr, respectively).

linear PA and different concentrations of C16TAB (10-8-10-4 M) at two different NaBr concentrations (10 and 0.5 mM). The ratios are plotted against the logarithm of the surfactant concentration in Figure 1. Since the ratio is sensitive to the polarity of the environment around the probe, such as the difference between the polar water and the apolar micelle core,27 the increase of I3/I1 can be used to detect cac.30 For PA/C12TAB, Hansson et al. have earlier shown that the onset of I3/I1 increase31 closely corresponds to the cac measured by surfactant-sensitive electrode.32 Since this should be valid also for the higher homolog C16TAB, cac has been determined as the concentration showing the first sign of I3/I1 increase. From the graph, cac has been determined as 3.6 µM in 10 mM NaBr and 2.0 µM in 0.5 mM NaBr. The latter value is somewhat higher than expected, but then again the relation between salt concentration and cac has not been thoroughly investigated in this salt concentration range. Data for DNA/C12TAB even suggest that the linear relationship normally expected may be invalid at low salt concentration,33 at least for some polymers. Finally, it has been shown that the cac of linear PA closely matches the cac of crosslinked PA at the corresponding polymer concentration,34 allowing the use of these values in the kinetic calculations later on. 4.2. Kinetic Analysis. An example of a deswelling gel (R0 ) 89 µm) held by the micromanipulator in a constant flow of 0.015 m/s of 0.5 mM C16TAB, 10 mM NaBr, can be found in Figure 2. The pipet of the micromanipulator can be seen to the left in each picture, and the liquid flow comes from the upper right corner. The pictures were captured in 15 s intervals, and the corresponding deswelling curve can be found as curve 3 in Figure 3. In summary, the kinetic experiments performed can be divided into the following three main sets: (I) high electrolyte concentration (10 mM), normal flow velocity (0.015 m/s) (data shown in Figures 3-6); (II) low electrolyte concentration (0.5 mM), normal flow velocity (0.015 m/s) (data shown in Figures 7-9); (III) high electrolyte concentration (10 mM), high flow velocity (0.028 m/s) (data shown in Figures 10-12). For each of these, several surfactant concentrations have been tested, and for each such concentration, gels of several different sizes have been used. The lines in Figures 3-13 are the predictions from the kinetic model. For all model calculations, the following values have been used unless otherwise stated: DI ) 10-14 m2/s,35 DII ) 4.16 × 10-10 m2/s,36 flow velocity ) 0.015 m/s (calculated from the volumetric flow rate assuming laminar flow; see ref 19 for details), and cac ) 3.6 µM. The value for Vskin was set to 0.0007 m3/mol, which is strictly

Deswelling Kinetics of Polyacrylate Gels

Figure 2. Example of gel deswelling. A gel with R0 ) 89 µm (corresponding to the 0 s picture) deswelling in 0.5 mM C16TAB, 10 mM NaBr, normal flow velocity during 165 s. The micromanipulator pipet holding the gel can be seen in the left part of each picture. Indicated in each picture is the time and relative gel volume, V/V0. The corresponding deswelling curve is 3 in Figure 3.

Figure 3. Gels deswelling in 0.5 mM C16TAB, 10 mM NaBr, pH 10 at 0.015 m/s flow velocity. Note that the lag time is about 9.5 s, a mere 1 s more than the time needed for the solution to travel through the tubing to the gel (8.5 s).

Figure 4. Gels deswelling in 0.3 mM C16TAB, 10 mM NaBr, pH 10 at 0.015 m/s flow velocity. Note the lag time of about 11 s.

speaking only valid at high β, as measured by Hansson et al.13 However, since the surface phase is very thin at low β, the introduced error will be quite negligible. Figure 14 shows an equilibrium swelling isotherm determined for PA/C16TAB earlier,13 with the line being the theoretical swelling calculated

J. Phys. Chem. B, Vol. 111, No. 33, 2007 9773

Figure 5. Gels deswelling in 0.2 mM C16TAB, 10 mM NaBr, pH 10 at 0.015 m/s flow velocity. Note the slightly increased lag time of 13 s, and that the fit between model and experiment is markedly worse for the larger gels.

Figure 6. Gels deswelling in 0.1 mM C16TAB, 10 mM NaBr, pH 10 at 0.015 m/s flow velocity. Note the longer lag time of on average 37 s, and that the fit of the larger gels is worse here as well.

Figure 7. Gels deswelling in 0.5 mM C16TAB, no additional NaBr, pH 10 at 0.015 m/s flow velocity. Note the lag time of 11 s, slightly higher than the corresponding time for 10 mM NaBr.

from eq 1 with Vskin ) 0.0007 m3/mol. Since the fit to the experimental data is quite good, using a fixed Vskin seems like a justified simplification. V0 has been used as a global fitting

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Figure 8. Gels deswelling in 0.3 mM C16TAB, 0.2 mM NaBr, pH 10 at 0.015 m/s flow velocity. The lag time has increased by ca. 1 s compared to 0.5 mM, to about 12 s.

Figure 11. Gels deswelling in 0.3 mM C16TAB, 10 mM NaBr, pH 10 at 0.028 m/s flow velocity. The lag time has increased by 0.5 s, to a total of 6 s.

Figure 9. Gels deswelling in 0.1 mM C16TAB, 0.4 mM NaBr, pH 10 at 0.015 m/s flow velocity. The lag time is noticeably higher, around 15.5 s.

Figure 12. Gels deswelling in 0.2 mM C16TAB, 10 mM NaBr, pH 10 at 0.028 m/s flow velocity. The lag time has further increased by 2 s, to a total of 8 s. Note that the larger gels do not fit the model especially well.

Figure 10. Gels deswelling in 0.5 mM C16TAB, 10 mM NaBr, pH 10 at 0.028 m/s flow velocity. The lag time is 5.5 s, a mere 1 s more than the transport time through the tubing, which is 4.5 s at this higher flow velocity.

parameter for each of the two electrolyte concentrations, since it is difficult to measure experimentally for microgels. The values obtained, 0.020 m3/mol at 10 mM NaBr and 0.055 m3/ mol at 0.5 mM NaBr, are in the same range as V0 values earlier

Figure 13. Gels deswelling in 0.3 mM C16TAB, 10 mM NaBr, pH 10 at 0.015 m/s flow velocity. Swelling ratio in 10 mM/1 mM NaBr surfactant free solution (V10/V1) for each gel bead as indicated in the legend. Note the largest two gels, 4 and 5, which have similar V10/V1 ratio and thereby cross-linking density, but fit the model to a radically different degree.

determined for macrogels.13 The same parameters have been used for the calculations of all deswelling curves (no individual fitting parameters), with only the experimental input parameters

Deswelling Kinetics of Polyacrylate Gels

Figure 14. Swelling isotherm for PA/C16TAB, with experimental points and the corresponding theoretical line, calculated from eq 1 with Vskin ) 0.0007 m3/mol. The satisfactory fit between theory and experiment shows that approximating Vskin as constant does not significantly influence the deswelling isotherm. Adapted from Hansson et al.13

(gel size, surfactant concentration, flow velocity, electrolyte concentration) changing between gels according to the experimental conditions. Case I will be treated as the “normal” case, as the conditions there are the most similar to our earlier work on C12TAB. In case II we have used a much lower electrolyte concentration which, apart from increasing the swelling of the gel core, mainly has an effect by raising the cmc and lowering the cac. Case III is similar to case I except that the flow velocity is increased, resulting in a faster transport of surfactant from bulk to gel surface or, to use in-model terminology, thinner stagnant layers. In case I, the general trend is that the theoretically calculated deswelling curves fit the experimental points reasonably well, as would be expected if the deswelling is controlled by stagnant layer mediated ion exchange kinetics. Increasing the surfactant concentration or decreasing the gel size increases the deswelling rate, as can be seen in Figures 3-6. However, it should be noted that the highest surfactant concentration, 0.5 mM (Figure 3), is above the cmc of C16TAB at 10 mM electrolyte concentration. Using 0.5 mM as input value in the theoretical calculations will widely overestimate the deswelling rate, and would not correctly describe the experimental conditions, as the micelles diffuse slower than the free surfactant molecules. Therefore, the concentration used in the theoretical calculations was set equal to cmc (0.3 mM),37 which is approximately the concentration of free monomers, giving a rather good fit as can be seen in Figure 3. In addition to this, two trends worth noticing are clearly apparent in case I. The first one is that the lag time increases as the surfactant concentration is lowered, which of course is what one would intuitively expect. However, if the deswelling starts when cac is reached in the gel, as has been earlier proposed for C12TAB,18 the lag times should all be in the magnitude of 1 s (eqs 5-8), which is clearly not the case. This will be discussed further in a separate section, section 4.3. The second trend is that although most gels fit the theoretical curves, the slower deswelling ones usually have a markedly worse fit, and the theoretical curve always overestimates the deswelling rate for these gels. It can further be noted that this trend is primarily visible for gels in high electrolyte concentration (10 mM). The reasons for these deviations, which we consider a major finding of this study, will be discussed further in section 4.4.

J. Phys. Chem. B, Vol. 111, No. 33, 2007 9775 In case II, where the electrolyte concentration is lower, we see shorter lag times and faster deswelling (for equal size gels), as illustrated in Figures 7-9. The faster deswelling is mainly due to the gels being more swollen, i.e., containing less polymer per volume unit (V0 ) 0.055 m3/mol) and thereby requiring less surfactant to be transported into the gel for it to collapse. The lowered cac (2.0 µM) will also have a small effect, mainly on the latter part of the deswelling curve, although this will be quite negligible in comparison to the change in V0. Naturally, we also see the expected trends that the deswelling rate increases with increasing surfactant concentration and decreasing gel size. In case III, the flow velocity of the surfactant solution has been increased, from 0.015 to 0.028 m/s. The increased convection makes transport of surfactant from the bulk to the gel surface faster, or to use in-model terminology, the stagnant layer around the gel, through which the surfactant needs to diffuse, becomes thinner. As expected, this leads to faster deswelling rates, as can been seen in Figures 10-12. The lag times also become shorter, although it is difficult to draw conclusions regarding the different lag times in case III as they all are very short (0.5-2 s). It is safe to say, though, that they are noticeably shorter than the corresponding times in case I (for the corresponding gel sizes and electrolyte and surfactant concentrations). 4.3. Lag Time. The lag time observed in the experiments consists of two parts: the time needed for the surfactant solution to travel through the tubing to the gel after switching to the surfactant solution, and the time from the surfactant solution reaching the gel to the start of deswelling (the latter being what we normally mean by lag time). To find the time needed for transport through the tubing, which only depends on the flow velocity as long as the same experimental setup is used, an additional experiment was performed where the solution was changed from low to high electrolyte concentration (no surfactant present), a change to which the gel responds with a very fast deswelling. The time from switching the solution to the onset of volume decrease was approximately 8.5 s at flow velocity 0.015 m/s and 4.5 s at flow velocity 0.028 m/s, and this time should be subtracted from the experimental data to find the true lag time. Note that the theoretical line in all figures has been set at the apparent start of deswelling for each individual gel. In earlier studies with PA microgels and C12TAB,18,19 it was found that the lag time could be accounted for by adding the time needed to establish a steady-state flow of surfactant through the stagnant layer to the time required to transport enough surfactant into the gel to reach an average concentration equal to cacg (eqs 5-7). According to this theory, the present experiments with C16TAB should all have lag times of around 1 s or less, which is clearly not in accordance with the experimental results (which show much longer lag times). To account for this, we propose that even while micelles are formed in the gel at concentrations above cacg, possibly accompanied by a slight reduction in gel size, a certain minimum amount of micelles, corresponding to a higher surfactant concentration in the gel, is needed to start formation of the collapsed surface phase. To compensate for this, we calculate the amount of surfactant absorbed into the gel during the lag time, using eqs 5-9, and then use this β as the starting point for the deswelling calculation. Note that even when the critical phase separation concentration is larger than cacg, cac is still the relevant quantity to be used in the calculation of theoretical deswelling curves, since it corresponds to the activity of the surfactant in the gel core.

9776 J. Phys. Chem. B, Vol. 111, No. 33, 2007

Figure 15. βcore, calculated from the measured lag time, as a function of R0 for all studied gels. Note the steep increase in βcore with decreasing gel radius for the smaller gels.

In the earlier study with C12TAB,19 no discrepancy was found between the start of micelle formation and formation of the surface phase. The reason for this is probably due to different experimental conditions; such a discrepancy would be very hard to detect. In the former study, the lowest surfactant concentration was 0.56 mM, which should be put into relation to the cac of C12TAB, 0.36 mM, giving a concentration difference (C2-cac) of 0.2 mM. The lowest concentration studied for C16TAB, where the longer lag times were found, has been 0.1 mM, giving a C2-cac of 1 in the surface phase), making surfactant diffusion into the gel somewhat more difficult, but more importantly making Na+ diffusion out through the surface phase much more difficult and thereby making gel deswelling equally slower. According to the model (eqs 3-4), both higher surfactant concentration and larger gel size will increase C1, and thereby increase the risk of inducing an extra diffusion barrier. However, if this was important here, then most gels at 0.5 mM C16TAB should be deviant, and practically none at 0.1 mM, as the C1 values at 0.5 mM will be much higher. The actual situation on the other hand is that we find both deviant and regular gels at each concentration, disproving this explanation. After these possibilities were refuted, two factors common to the deviant gels still remain: high electrolyte concentration and slow deswelling. Keeping these two in mind, we will look at some other clues before formulating an explanation. PA/C16TAB complexes at equilibrium have been shown to exhibit two types of ordering, either a cubic Pm3n or a hexagonal packing of rodlike micelles. It has been shown earlier in the literature that macroscopic gels that at equilibrium show a hexagonal microstructure also often exhibit anomalous deswelling behavior.13 Hansson et al. have shown that for PA macrogels in 0.5 mM C16TAB (limited amount of surfactant) partially collapsed gels (β < 0.8) exhibit cubic structures, while most fully collapsed ones (β > 0.8) show a hexagonal structure.12 SAXS data from another study with macroscopic gels in 0.5 mM C16TAB corroborate a cubic structure at β < 0.8 and hexagonal or hexagonal/cubic mixture at β > 0.8. It also shows that the transition from cubic to hexagonal for PA/ C16TAC (cetyltrimethylammonium chloride) takes place as the surfactant (micelle) content of the surface phase increases.13 In the partially collapsed gels, the presence of the core creates a local excess of polyion and a high lateral deformation of the surface phase, which both should favor a cubic or disordered packing,21 while in the fully collapsed gels, we instead find the expected hexagonal structure. Earlier studies by Svensson et al. with linear PA/C16TAB have shown that increasing the Brcontent will push a transition from cubic to hexagonal phase via a two-phase region, by increasing the polyion-mediated attractive forces.38 Fluorescence measurements show that this is valid also for cross-linked gels.13 To summarize, an excess of polyion and/or high lateral extension of the surface phase favor cubic packing, while excess of Br- and higher micelle density favor hexagonal packing.

Deswelling Kinetics of Polyacrylate Gels

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Figure 16. Relative error, calculated as the difference in time needed to reach V/V0 ) 0.4 for the experimental and the theoretical deswelling curves, normalized by the theoretical time, plotted as a function of k, the deswelling rate constant for completely stagnant layer controlled kinetics. Zero relative error, indicated by the dotted line, corresponds to a perfect fit between experiment and theory. The three different sets of experimental conditions used are indicated by I-III. Note the (positive) increase in relative error for slower deswelling (low k) gels.

As a collapsing gel has an excess of polyion, due to the availability of surfactant being kinetically limited, and a large deformation of the surface phase due to the swollen core, deswelling gels should have a swollen, disordered (or at equilibrium cubic) structure. However, it is plausible that a high bromide concentration, such as in some of the earlier mentioned experiments, could push a transition to a denser, more ordered hexagonal phase (or if there is not enough time to induce hexagonal ordering, then at least elongated wormlike micelles). Finally, we have the slow deswelling to consider. To get a value for the expected deswelling rate that can be compared between different experimental conditions (cases I-III) we will calculate a deswelling rate constant k as

k)

V0

DII(Sh/2 + 1)(C2 - cac) R02

(10)

The expected initial deswelling rate can then be described using k as

V ≈ (1 - kt)6 V0

(11)

where V/V0 is the gel volume (V) at any given time relative to the gel volume in a surfactant-free solution (V0). In the limit of an infinitely thin surface phase, eqs 10 and 11 describe the same deswelling behavior as eq 3. A detailed derivation of these equations can be found in an earlier publication by Go¨ransson et al.18 We will also calculate the relative error as the difference between the time required to reach V/V0 ) 0.4 for the experimental deswelling and the theoretical curve, respectively, divided by the theoretical time, i.e.

texp - ttheory ttheory

(12)

By plotting these two against one another in Figure 16, we see that the gels with low k, that are expected to exhibit a slow deswelling, show a larger relative error. These gels are also mainly present in cases I and III, where the electrolyte concentration is high (10 mM).

The hypothesis is then simply that if the gel volume transition, or rather the reorganization of the polymer network, is fast enough then formation of a dense, ordered phase is prohibited, and we instead get a cubic or just disordered micellar structure. Earlier findings by Mironov et al. have shown that formation of a hexagonal phase might be a relatively slow process.39 This would explain why the gels for which slow deswelling is expected (large R0, low surfactant concentration, high electrolyte concentration) show a deviant deswelling behavior, since a denser, more ordered hexagonal (or partially hexagonal) surface phase would impose a greater resistance to the deformation and reorganization of material necessary for gel deswelling, thereby limiting the deswelling process. The connection between anomalous deswelling behavior and the formation of a hexagonal surface phase has been recognized earlier for PA/C16TAB.12 Further favoring this explanation is the fact that during the first stages of deswelling, ∼0-30 s, when the surface phase is very thin and should not impose much of a resistance even with a hexagonal structure, even the deviant gels are often in agreement with the model. 5. Conclusions The deswelling kinetics of PA microgels in the presence of C16TAB can, for gels with expected fast deswelling (small gel size, high surfactant concentration, high flow velocity) and in not too high electrolyte concentration, be explained and correctly predicted using ion exchange kinetics and stagnant layer diffusion. The limiting step controlling deswelling, as has been shown to be the case with PA/C12TAB microgels,19 is the transport of surfactant from the bulk to the gel surface. However, for the gels with expected slow deswelling, the experimental deswelling deviates markedly from the model prediction. Crosslinking density, gel size, and high C16TAB concentration in contact with the gel surface are shown to be individually unable to explain this deviant deswelling. The anomalous deswelling behavior seems to be connected to high bromide concentration and slow deswelling (large gels and/or low C16TAB concentration), which both should favor the formation of a denser and more ordered surface phase. This difference in the structure of the surface phase makes the reorganization and deformation of said phase more difficult and limits the deswelling rate of these gels. Remarkably long lag times have been noted for gels in low surfactant concentration, pointing toward a certain minimum amount of micelles being needed in the surface of the gel in order to start formation of the surface phase. Finally, we would like to point at a few questions that still remain to be answered, which will provide the basis of future investigations: What is the distribution of micelles in the gel during the lag time and the formation of the surface phase? What is the microstructure of the surface phase during surfactant binding, and how does this connect to the deswelling kinetics of the gel? How general is the model used, or more precisely, at what gel size does gel deswelling stop being governed mainly by surfactant transport and instead become dominated by the rate of network relaxation? Acknowledgment. This work has been financially supported by the Swedish Research Council and by the Swedish Foundation for Strategic Research. References and Notes (1) Bromberg, L.; Temchenko, M.; Hatton, T. A. Langmuir 2003, 19, 8675.

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