Effect of Salt and Surfactant Concentration on the Structure of

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J. Phys. Chem. B 2007, 111, 10959-10964

10959

Effect of Salt and Surfactant Concentration on the Structure of Polyacrylate Gel/Surfactant Complexes Peter Nilsson,*,† Johan Unga,‡ and Per Hansson† Department of Pharmacy, Uppsala UniVersity, Box 580, S-75123 Uppsala, Sweden, and Department of Chemical and Biological Engineering, Chalmers UniVersity of Technology, S-41296 Go¨teborg, Sweden ReceiVed: May 28, 2007; In Final Form: July 9, 2007

Small-angle X-ray scattering was used to elucidate the structure of crosslinked polyacrylate gel/ dodecyltrimethylammonium bromide complexes equilibrated in solutions of varying concentrations of surfactant and sodium bromide (NaBr). Samples were swollen with no ordering (micelle free), or they were collapsed with either several distinct peaks (cubic Pm3n) or one broad correlation peak (disordered micellar). The main factor determining the structure of the collapsed complexes was found to be the NaBr concentration, with the cubic structure existing up to ∼150 mM NaBr and above which only the disordered micellar structure was found. Increasing the salt concentration decreases the polyion mediated attractive forces holding the micelles together causing swelling of the gel. At sufficiently high salt concentration the micelle-micelle distance in the gel becomes too large for the cubic structure to be retained, and it melts into a disordered micellar structure. As most samples were above the critical micelle concentration, the bulk of the surfactant was in the form of micelles in the solution and the surfactant concentration thereby had only a minor influence on the structure. However, in the region around 150 mM NaBr, increasing the surfactant concentration, at constant NaBr concentration, was found to change the structure from disordered micellar to ordered cubic and back to disordered again.

1. Introduction Complexes between charged polymers and oppositely charged macroions have for the last few years drawn attention as potential vehicles for targeted drug delivery, a thrilling prospect for facilitating the administration of labile or highly toxic drugs.1-7 One important aspect in understanding the mechanisms behind these polymer/surfactant systems is to investigate the orderedstructures,suchasmicellarcubic,8-15 hexagonal,8-10,12,15-17 and lamellar18-21 phases, that are often formed by them. To expand our earlier work with polyacrylate (PA) microgels,22,23 we have chosen to continue to investigate the system of lightly crosslinked PA gels and dodecyltrimethylammonium bromide (C12TAB), this time using macroscopic gel/surfactant complexes and small-angle X-ray scattering (SAXS). We have limited the scope by choosing to vary only two parameters: the surfactant concentration (Csurf) and the electrolyte concentration. Sasaki and Koga have earlier investigated the relationship between electrolyte concentration and structure for PA, but using only one concentration of dodecylpyridinium chloride (DPC) and only a few electrolyte concentrations.12 There have also been studies of PA and C12TAB but only at a fixed electrolyte concentration.8,11,13,15 A recent review on polyelectrolyte gels/ oppositely charged surfactants also considers the aspects of electrolyte concentration and structure.24 To complement these earlier findings, we now conduct a more thorough exploration of the structure of PA/C12TAB complexes mainly at two fixed surfactant concentrations while varying the electrolyte concentration and at one fixed electrolyte concentration while varying * To whom correspondence should be addressed. E-mail: Peter.Nilsson@ farmaci.uu.se. † Uppsala University. ‡ Chalmers University of Technology.

the surfactant concentration. The aim is to find out what ordered structures exist under different conditions and try to elucidate the parameters governing their formation. 2. Experimental Methods 2.1. Materials. Acrylic acid (99%) from Aldrich, dodecyltrimethylammonium bromide (99%), ammonium persulfate (AP) (g98%), N,N′-methylenebisacrylamide (NMBA), and N,N,N′,N′tetramethylethylenediamine (TEMED) (99%) from Sigma, sodium hydroxide (NaOH) from Eka Chemicals, and sodium bromide (NaBr) from Kebo were all used as received. Solutions were prepared using high-quality Millipore water. 2.2. Gel Synthesis. A solution of 1.6 M acrylic acid, 14 mM NMBA (crosslinker), 6 mM TEMED (accelerator), and 6 mM AP (initiator) was prepared and degassed in vacuum. The solution was transferred into glass capillaries (ca. 5 cm length, one end capped) and heated to 65 °C for 3 h. The gels were removed from the capillaries and transferred to a 0.5 M NaOH solution over night. After being washed (3 × 12 h) in large excess of water, the gels were dried to prevent degradation. 2.3. Sample Preparation. Dried cylindrical gels (dry weight ∼1 mg) were put into solutions of varying C12TAB (0-200 mM) and NaBr concentrations (0-400 mM), and the pH was kept >10 by addition of small amounts of NaOH. The C12TAB concentration and solution volume were chosen such that fully collapsing the gel would change the C12TAB concentration by less than 1%. The gels were allowed to equilibrate for at least one month, after which the fully collapsed PA/C12TAB complexes were weighed prior to SAXS measurements. Three main sets of samples were analyzed, two with fixed C12TAB concentration and one with fixed electrolyte concentration, as described in Table 1.

10.1021/jp074112p CCC: $37.00 © 2007 American Chemical Society Published on Web 08/23/2007

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TABLE 1: Summary of the Three Main Setsa of Samples Analyzed set

Csurf (mM)

CNaBr (mM)

I II III

5 3 1-200

5-395 7-397 199-0

a Sets I and II have fixed surfactant concentration, and set III has fixed total electrolyte concentration (C12TAB + NaBr).

2.4. X-ray Measurements. The structure of the samples was determined by SAXS using two different diffractometers. The gel samples were cut to suitable size and mounted in a vacuumsealed sample holder. All samples were examined on a Kratky camera with line collimation. Cu KR radiation of λ ) 1.542 Å was generated using a Seifert Iso-Debyeflex 3003 generator working at 50 kV and 40 mA. Either a tungsten (W) beamstop filter or a nickel (Ni) Kβ filter was used. The diffraction pattern was recorded by a MBraun linear position sensitive detector PSD 50M and stored digitally. The camera uses a multiplex to record both small- (SAXS) and wide-angle scattering (WAXS) if needed. Selected samples were also examined using SAXS with a synchrotron X-ray source at beamline I711 at Maxlab (Lund University).25 The radiation wavelength was 1.125 Å, and diffractograms were recorded under vacuum at room temperature (20 °C) using a MAR-CCD 165 2D detector. The exposure time was 300 s for each sample. The resulting CCD images were integrated using Fit2D software provided by Dr. A. Hammersley.26 As expected, no peaks were found in WAXS measurements in any sample, and therefore only SAXS data will be presented below. According to Bragg’s law, the angle between the scattered and the incident beams, 2θ, is related to the scattering vector q and the lattice spacing d as

dhkl-1 )

2 θ q ) sin 2π λ 2

Figure 1. Scattering data for 3 mM of C12TAB and 7 mM of NaBr, collected with SAXS with a synchrotron X-ray source. Nine peaks are easily discernible at 21/2, 41/2, 51/2, 61/2, 81/2, 101/2, 121/2, 131/2, and 141/2, consistent with a cubic Pm3n arrangement. The inset shows the magnification of a part of the q range.

(1)

where h, k, and l are the Miller indices of the reflecting planes and λ is the wavelength of the radiation. The unit cell length a for a cubic structure can then be calculated as

(2)

Figure 2. Scattering curves for set I: PA gels in 5 mM of C12TAB at different electrolyte concentrations (10-400 mM), as indicated to the right of each curve. For clarity, the spectra are displaced by an appropriate integer. Note the shift to lower q values as the electrolyte concentration is increased and also the shift from cubic (black) to disordered micellar (dark gray) to micelle free (light gray).

3.1. General Structure. At the lowest surfactant concentration (3 mM) and the lowest electrolyte concentration (10 mM), see Figure 1, nine distinct peaks are found at 21/2, 41/2, 51/2, 61/2, 81/2, 101/2, 121/2, 131/2, and 141/2 indicating a primitive cubic arrangement consistent with the Pm3n structure found earlier for the PA/C12TAB system.8,14 The unit cell length a is found to be 85.2 Å, which is similar to earlier found data.11 3.2. Electrolyte Concentration. The electrolyte concentration (C12TAB + NaBr) was increased in a stepwise manner from 0 to 400 mM at two different C12TAB concentrations (set I, 5 mM and set II, 3 mM). From set I, depicted in Figure 2, we see that when the electrolyte concentration is increased the peaks shift slightly to the left (lower q), showing an increased micellemicelle distance. In a certain electrolyte concentration interval (150-170 mM) the distinct peaks disappear as the cubic phase is replaced by a disordered micellar one that only shows an indistinct correlation peak. The same effect has been seen in the PA/DPC system (5 mM surfactant), where a Pm3n cubic structure was found below 200 mM NaCl concentration but was

replaced by a single broad peak at concentrations of 300 mM or more.12 Also, for poly(diallyldimethylammonium) chloride/ sodium dodecyl sulfate, a similar effect was noted where a lamellar structure gradually disappeared as the sodium chloride concentration was increased.18 An excellent parallelism can also be found in the findings of Ashbaugh and Lindman,11 where a stepwise lowering of micelle surface charge by addition of nonionic octaethylene glycol monododecyl ether (C12E8) to PA/ C12TAB resulted in a shift of the Pm3n peaks toward lower q. When enough C12E8 was added, the cubic structure disappeared completely to be replaced by a single indistinct correlation peak. A similar effect has been found for linear PA/cetylpyridinium chloride when the PA was partially neutralized. For highly charged PA, a cubic Pm3n structure was found, which shifted to higher q as more polymer was neutralized and finally melted into a broad maximum at complete neutralization.13 A transition from hexagonal to cubic to disordered micellar structure has also been noted by Norrman et al. when lowering the charge fraction of linear PA interacting with cetyltrimethylammonium

a ) dhkl xh2 + k2 + l2 3. Results and Discussion

Polyacrylate Gel/Surfactant Complexes

Figure 3. Theoretical binding isotherm for PA/C12TAB, calculated with Cp ) 2100 mol/m3, φ ) 0.34,8,41 a0 ) 67 Å,2 γ ) 0.02 N/m, and ∆µS0 + ) 4.5 RT/mol (in mM concentration units).41 Added NaBr concentration is indicated next to each line. Increasing the surfactant concentration increases β with a plateau around β ) 1. The sigmoid character of the curve decreases with increased salt concentration.

bromide, either by partially neutralizing the PA or by copolymerizing it with N-isopropylacrylamide or dimethylacrylamide.27 As more simple ions are added to the solution, the penalty for overcharging the gel/surfactant complex decreases; by overcharge we mean a complex with a surfactant/polymer charge ratio (β) significantly higher or lower than 1, giving either a positive or negative overcharge. This is most easily visualized by looking at the theoretical binding isotherms in Figure 3. The isotherms, which show the effect of changing salt concentration, are calculated as outlined in the Appendix. Note that the isotherms in themselves do not cover gel swelling, as the gel volume is considered constant in the calculations, and that each curve deals with a different concentration of NaBr and not total electrolyte concentration. Had the isotherms instead been calculated for constant electrolyte concentration, they would have been symmetrical around β ) 1. We will use the isotherms to draw conclusions regarding overcharging and then use these to discuss the effects on gel swelling. In Figure 3, we see that increasing the salt concentration means shifting from one curve to another less sigmoid one. At β < 1, this shift between the curves will decrease β, giving the gel a negative overcharge resulting in more simple ions in the structure. This will promote an osmotic swelling of the gel resulting in slightly larger micelle-micelle distances. A similar shift of the binding isotherm has been shown for the neighboring system of PA/ DPC.28 Also, for samples with Csurf > critical micelle concentration (cmc), adding salt will decrease the cmc and therefore lower the free surfactant concentration. This will move our position on the binding isotherm to the left, lowering β and increasing the overcharge as discussed above. Another factor promoting swelling is that the salt screens polyion mediated attractions between micelles. It has been demonstrated recently that such attractions, due to electrostatic correlation and polyion bridging forces, stabilize the dense structures formed by complex salts of linear polyacrylate and C12TA+.29 Together, all these effects will increase the micelle-micelle distance and promote swelling of the gel. A similar increase of the intermicellar distance has been seen when the micelle11 or the polymer charge13 has been reduced. As a result of the micelle-micelle distance becoming too large the ordered cubic structure will finally collapse and melt into a disordered micellar phase. To illustrate this swelling of the gel as the electrolyte concentration increases, Figure 4 shows both the macroscopic swelling of the

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Figure 4. Small, filled symbols show the characteristic micellemicelle distance (a/2 for cubic; ξ ) 2π/qmax for disordered micellar) as a function of the NaBr concentration. Also, included as large, open symbols is the macroscopic swelling in one dimension (cube root of the gel volume) scaled into arbitrary units so as to start at the same value as the microscopic distance. Circles represent set I, 5 mM of C12TAB and triangles set II, 3 mM of C12TAB. The filled and dotted lines are just guides for the eye. Note that the curves coincide until the transition from cubic to disordered micellar phase takes place (indicated by the dotted line) at ∼150 mM of NaBr.

gel in one dimension (large, open symbols) and, as a parallel, the characteristic correlation length (small, filled symbols). According to Fontell et al. the Pm3n unit cell consists of one micelle in the middle, one in each corner, and two on each side, for a total of eight micelles per unit cell, all slightly elongated 30 with an axial ratio 1.35.31 With this geometry the mean volume per micelle will be a3/8 or (a/2)3, and a/2 will therefore be used as the characteristic micelle-micelle distance. a/2 is also approximately the distance between the two micelles on each side of the unit cell and should therefore be the most common micelle-micelle distance found. To get a corresponding value for the disordered micellar phase, the maximum of the correlation peak (ξ ) 2π/qmax) is used instead. Comparing a/2 for the cubic structure with ξ ) 2π/qmax for the disordered micellar structure was earlier suggested by Sasaki and Koga.12 At low NaBr concentration, both the macroscopic and the microscopic swelling increase with increased NaBr concentration, but after the transition from cubic to disordered micellar, around 150 mM, there is a drop in the SAXS correlation length that is not matched by the macroscopic swelling. This change in micellemicelle distance is due to the change in geometry of the complex, as a cubic lattice of rotating, slightly rodlike micelles30,31 is replaced with a random distribution of micelles. This resembles the order-disorder transition in a hard sphere system, where the average distance between micelles is larger in the ordered state.32 The same drop in volume has been noted by Sasaki and Koga for PA/DPC.12 In parallel Ashbaugh and Lindman found that when they lowered the micelle surface charge through addition of nonionic surfactant, they got steeper macroscopic than microscopic swelling at low micelle charge, although it should be noted that they used a slightly different approach in calculating the microscopic swelling. They attributed their discrepancy between macro- and microscopic level to increased heterogeneity in the gel, with the formation of micellerich and micelle-lean domains.11 When the electrolyte concentration is further increased (above 250 mM), the driving force for micelle formation in the gel finally weakens to the point where forming micelles in the gel is no longer favorable, resulting in an absence of polyion stabilized micelles and thereby

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Figure 5. Scattering curves for set II: PA gels in 3 mM of C12TAB at different electrolyte concentrations (10-400 mM), as indicated to the right of each curve. For clarity, the spectra are displaced by an appropriate integer. Note the shift to lower q values as the electrolyte concentration is increased and also the shift from cubic (black) to disordered micellar (dark gray) to micelle free (light gray).

Figure 6. Scattering curves for set III: PA gels in constant total electrolyte concentration (C12TAB + NaBr ) 200 mM) at different C12TAB concentrations (1-200 mM), as indicated to the right of each curve. For clarity, the spectra are displaced by an appropriate integer. Note the shift to lower q as well as from cubic (black) to disordered micellar (dark gray) to micelle free (light gray) as the surfactant concentration is decreased, similar to the effect seen when the electrolyte concentration is increased.

no ordering. At 3 mM C12TAB (set II, see Figure 5), the same trend as for 5 mM is visible with a Pm3n cubic phase at low electrolyte concentration, an increase in micelle-micelle distance, transformation to disordered micellar phase, and finally disappearance of micelle-micelle correlations, in that order, as the electrolyte concentration is increased. The only difference is that the transformation from cubic to disordered phase takes place between 130 and 150 mM, and the micelles disappear above 230 mM electrolyte. 3.3. Surfactant Concentration. The surfactant concentration was raised in a stepwise fashion in the range 1-200 mM, while the total electrolyte concentration (C12TAB + NaBr) was kept constant at 200 mM (set III, see Figure 6). At low concentrations, Csurf < critical aggregation concentration (cac), no micelles are present and no ordering can be found. When the surfactant

Nilsson et al.

Figure 7. Summary of all analyzed samples. The graph shows the correspondence between microstructure, C12TAB concentration, and NaBr concentration. Cubic samples are indicated by open squares, disordered micellar ones by filled diamonds, and micelle free swollen ones by open circles. The solid line represents calculated cmc values for pure C12TAB, and the dotted line shows extrapolated cac values for PA/C12TAB. The straight dotted lines and the numbers indicate the four different regions discussed.

concentration is raised above cac, micelles appear and, due to the high electrolyte concentration, form a disordered micellar phase in the gel. As the C12TAB concentration is increased further, the NaBr concentration is simultaneously decreased by an equal amount. As we early on exceeded the cmc, see Figure 7, adding more surfactant will only increase the amount of micelles in the solution, while the amount of salt available to screen their charges will decrease. This decreased screening will increase the free surfactant concentration and thereby move our position on the binding isotherm, see Figure 3, toward higher β (β closer to 1). The decreased salt concentration as such will also give rise to a higher β (β closer to 1) as the screening between the polymer and the surfactant decreases, which is easily visualized in Figure 3 on moving from a higher salt curve to a lower salt curve at constant (or rather increasing) surfactant concentration. As β approaches closer to 1, the cubic structure can be formed instead of the disordered micellar one. Also promoting this effect is the fact that the polyion mediated attractive forces in the complex will become stronger as the salt concentration decreases. 3.4. Summary of Analyzed Samples. To summarize all the samples in sets I-III as well as some additional ones at other surfactant/electrolyte concentrations, we have plotted C12TAB concentration versus NaBr concentration in Figure 7, and indicated whether the PA/C12TAB complexes were found to be cubic, disordered micellar, or micelle free swollen ones. For reference, cmc values for pure C12TAB according to the equation proposed by Garcia-Mateos et al. are shown as the solid line33 and cac values extrapolated from data by Hansson and Almgren are shown as the dotted line.34 The cac values should only be considered as a rough guidance, since they are extrapolated from data up to 10 mM NaBr. But according to data by Sasaki et al., the relationship between cac and salt for PA/DPC is linear on a log-log scale between 10 and 400 mM NaCl,28,35 and it is reasonable to assume that the same is valid for PA/C12TAB. In the graph four different regions can be distinguished. To the left, at low NaBr concentration, is region [1] consisting of pure cubic Pm3n structure, at all surfactant and electrolyte concentrations studied. As the NaBr concentration is increased, this turns into a critical region at around 150 mM, region [2], with both cubic and disordered micellar phase. At the lower surfactant concentrations, the high NaBr concentration allows the complex

Polyacrylate Gel/Surfactant Complexes to become overcharged, with more polymer than surfactant (i.e., β < 1), and the increased swelling due to this is enough to get a disordered micellar phase. Such nonstoichiometric complexes have been reported earlier for the poly(diallyldimethylammonium chloride) (PDADMACL)/sodium dodecylbenzenesulfonate (SDBS) system.19 Claims have also been made that this effect cannot be seen for PA,36 but unlike the situation here those experiments were not performed with excess simple salt. As the surfactant concentration is increased, at constant NaBr concentration, the free C12TA+ concentration in the solution will be approximately constant as the added surfactant is used up for micelle formation while the Br- concentration will increase. Since this will raise the activity of the neutral surfactant species, C12TAB, which is the relevant parameter affecting the binding isotherm,24 the result will be a β closer to 1, no overcharging, and thereby a cubic structure. This is also supported by the simple binding model predicting, at fixed C12TA+ concentration, a monotonic increase of β with increasing C12TAB concentration (not shown). As the surfactant concentration is increased even more, moving to the right in the binding isotherm in Figure 3, β will be larger than 1 and once again the complex will be overcharged, although this time with a positive charge (more surfactant than polymer). The overcharging will increase the micelle-micelle distance here as well and transform the complex into a disordered micellar structure. This type of overcharged complexes has been reported for the PDADMACL/ SDBS system in excess salt as well.19 At even higher NaBr concentrations, region [3], the binding isotherm is steeper, as shown in Figure 3, making the surfactant concentration interval with β close to 1 smaller and thereby decreasing the possibility of finding a cubic structure. Even more important though is that the change in β and subsequent slight swelling is drowned out by the swelling due to decreased polyion mediated attractive forces in the complex, which is caused by the high salt concentration. All samples are thereby swollen enough to prohibit the formation of a cubic structure, and the disordered micellar phase is found at all surfactant concentrations (and all values of β). Finally we find region [4] at the highest NaBr concentrations, above ∼250 mM, where the salt concentration in the solution is so high that the benefit of releasing counterions from the gel is not enough to enable micelle formation in the gel and no micelles exist, resulting in swollen gels with no ordering, which is consistent with earlier found data.34 Interesting to note is that this NaBr concentration is quite similar to the critical electrolyte concentration for phase separation in the linear PA/C12TAB system.37 A complication that might be significant, which has not been addressed in the discussion above, is the twin roles played by the bromide ion. While increasing the salt concentration generally weakens the polymer/surfactant interaction, bromide ions have the specific effect of lowering the curvature of the micelles through stronger screening of the headgroups and thereby increase the polyion bridging forces in the complex.29,38,39 This means that changing the bromide concentration could promote either swelling through the general salt effect or deswelling through the specific bromide effect depending on the concentration interval studied. We have, however, found no direct evidence suggesting that this should be a determining factor here. To verify the dependence of structure on microdistance and electrolyte concentration, we plot the scattering distance d as a function of the NaBr concentration in Figure 8. For the cubic samples, the three main peaks (41/2, 51/2, and 61/2) are plotted,

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Figure 8. Repetitive distance for the three main peaks of the cubic phase and the characteristic distance of the disordered micellar phase as a function of added NaBr concentration. Disordered micellar samples are indicated by filled symbols and the cubic ones by open. Set I is indicated by circles, set II by triangles, set III by squares, and additional samples with varying C12TAB and NaBr concentrations by diamonds. Note that all samples coincide on the same line, more or less independent of C12TAB concentration, and also that the shift from ordered cubic phase to disordered phase takes place at 41 Å for the main peak (corresponding to a ) 92 Å).

whereas for the disordered micellar samples, the highest point of the curve, interpreted as an indistinct correlation length, is used. Here, we see that all samples, almost independent of surfactant concentration in the range 3-200 mM, fall on the same curve and more importantly that at a certain repetitive distance (41 Å), corresponding to a critical NaBr concentration (150 mM), a border between the two structures exists. As a side note, one might notice that this is curiously close to the physiological salt concentration, 154 mM.40 At all intermicellar distances greater than this, the cubic structure melts into a disordered micellar phase. Also, worth noting is the fact that the strongest peak of the cubic structure, 51/2 with the Miller index 210, seems to correspond closely to the correlation peak of the disordered micellar structure. 4. Conclusions At low electrolyte concentrations nine distinct peaks, 21/2, 41/2, 51/2, 61/2, 81/2, 101/2, 121/2, 131/2, and 141/2, consistent with a cubic Pm3n structure are found. Increasing the NaBr concentration is shown to increase the distance between scattering planes d until at a certain d value (41 Å for the strongest peak, 51/2, corresponding to unit cell length a ) 92 Å), coinciding with a critical NaBr concentration (∼150 mM), the cubic structure melts into a disordered micellar one. This disordered correlation peak will in turn broaden and shift to lower q, indicating further swelling, as the NaBr concentration increases until all ordering finally disappears, when the salt concentration is high enough to make micelle formation in the gel unfavorable. The surfactant concentration, as long as Csurf > cac, has only a minor influence on the structure, because of most of the surfactant being present in the form of micelles in the solution, as Csurf > cmc for most samples. However, in a certain critical interval, the increase in surfactant concentration increases the activity of the neutral C12TAB species so that the subsequent change in β is enough to affect the structure. In this region, the structure goes from a disordered micellar structure to a cubic Pm3n structure and back to a disordered structure as the surfactant concentration is increased at fixed NaBr concentration. With support from a simple binding model, we suggest that this sequence reflects the swelling as β increases from below

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to above 1. At higher NaBr concentrations, the salt decreases the attractive forces in the gel at all surfactant concentrations (as long a Csurf > cac) resulting in a gel too swollen to retain the cubic structure and only a disordered micellar structure is found. Appendix To derive a binding isotherm, we consider a gel with a volume fraction of micelles φ and a concentration of polyion charged groups Cp. The partitioning of the surfactant (S+Br-) and the simple salt (Na+Br-) between the water subdomains in the gel (w) and the solution (sln) is given by w sln sln CSw+CBr - ) CS+ CBr-

(A1)

w w sln sln CNa +CBr- ) CNa+CBr-

(A2)

where C is the concentration (or activity). The condition for electroneutrality in the gel can be written as w w CBr - + Cp/(1 - φ) ) βCp/(1 - φ) + CNa+

(A3)

The factor (1 - φ) is introduced to correct for the ions being excluded from the hydrocarbon core (calculating the concentrations in the aqueous domain only). The chemical potential of S+ in the aqueous regions in the gel is w µSw+ ) µS0,w + + RT ln CS+

(A4)

where µS0,w + is the standard contribution. The chemical potential in the micelles formed in the gel depends on several factors. For qualitative purposes, we use a very simple expression derived earlier:41 0,mic + 2a0γ µSmic + ) µS+

(A5)

where a0 is the optimal area per surfactant in the head group region of the micelle and γ is the “interfacial tension” in this region. Equation A5 is derived from considerations of the balance of repulsive and attractive forces at the micelle surface.42 The equilibrium requirement that µSw+ ) µSmic + gives 0 )/RT) CSw+ ) exp((2a0γ - ∆µS+

(A6)

0,mic where ∆µS0 + ) µS0,w + - µS+ . Equations A1-A3 and A6 allow β to be calculated as a function of CSsln+ for a given composition of the solution and specified values of φ, Cp, a0, γ, and ∆µS0 +. In simple descriptions of micellar equilibrium, the last two parameters are usually treated as constants.41,43 To highlight the purely entropic effect of salt on the distribution of surfactant and salt between gel and solution, one can put in values of the other three determined from experimental data. Also, a more thorough theoretical exploration is being prepared and will be published shortly.

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