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Self-Assembly of Ionic Surfactant in Cross-Linked Polyelectrolyte Gel of Opposite Charge. A Physical Model for Highly Charged Systems Per Hansson Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, Box 124, S-22100 Lund, Sweden Received October 28, 1997. In Final Form: February 10, 1998 A physical model is presented for the self-assembly of ionic surfactants in oppositely charged polyelectrolyte gels. The basic idea is that the formation of a micelle in an initially water-swelled gel is accompanied by a collapse of the surrounding polyion chains resulting in the formation of a complex. The strong electrostatic interaction between the micelle and the highly charged polyion produces a short-range perturbation of the gel structure in the vicinity of the micelle, leaving the complex-free regions of the gel essentially unchanged. With a fixed volume of the complex and a fixed stoichiometry of the surfactant and the polyion in the complex, a linear change in the gel volume is predicted as a function of the number of micelles in the gel. Experiments with dodecyltrimethylammonium bromide in cross-linked sodium polyacrylate support this for weakly cross-linked gels, but deviations are found at higher cross-linking densities.The distribution of the surfactant between gels and equilibrium aqueous phases is investigated and presented as binding isotherms. It is found that prior to micelle formation the surfactant binding can be described as an exchange of sodium ions in the gel with a preferential binding of the sodium ion (∆G°: 1-2 RT/mol). Cooperative binding is observed when the surfactant concentration in the gels exceeds the critical aggregation concentration in solutions of linear polyacrylate of the corresponding polyelectrolyte concentration. The details of the cooperative part of the binding isotherm are investigated by means of a simple law of mass action analysis. It is found that a pronounced effect of small additions of salt is well described at moderate degrees of surfactant binding to the gels. The abrupt reduction of the cooperativity at high degrees of binding is found to coincide with the gels reaching their fully collapsed state, which is characterized by an ordered packing of the complexes. At this point each micelle is surrounded by an excess of polyelectrolyte units.
Introduction The stability of solutions containing both polymer and surfactant depends on the effective interaction between the components. Phase separation is a common feature whether the interaction is attractive1,2 or repulsive.3-5 As the surfactant is always found to form micelles in at least one of the separated phases, such a behavior can be understood and related to the well-known “polymer incompatibility” (segregation) when the components repel each other and “complex coacervation” (association) in the case of an attractive interaction.6 This analogy derives from the fact that not only the polymer but also the surfactant molecules, when associated with micelles, contribute very little to the entropy of mixing in the system. When at least one of the components is charged, this has the important implication that the phase behavior becomes strongly influenced by the distribution of the counterions. For oppositely charged mixtures the interaction is particularly strong. When the dominating interaction between the polyelectrolyte and the surfactant is electrostatic, even small additions of one of the components to a solution of the other typically results in an associative (1) Goddard, E. D.; Hannan, R. B. J. Colloid Interface Sci. 1976, 55, 73. (2) Thalberg, K.; Lindman, B.; Karlstro¨m, G. J. Phys. Chem. 1990, 94, 4289. (3) Wormuth, K. R. Langmuir 1991, 7, 1622. (4) Robb, I. D.; Williams, P. A.; Warren, P.; Tanaka, R. J. Chem. Soc., Faraday Trans. 1995, 91, 3901. (5) Piculell, L.; Bergfelt, K.; Gerdes, S. J. Phys. Chem. 1996, 100, 3675. (6) Piculell, L.; Lindman, B. Adv. Colloid Interface. Sci. 1992, 41, 149.
phase separation.2,7,8 It has been known for some time that the resulting precipitates, or highly dense phases, can be ordered.9,10 Carnali11 found that the (fourcomponent) phase diagram of sodium polyacrylate (NaPA), tetradecyltrimethylammonium bromide (C14TAB), and water contains a hexagonal phase related to the hexagonal phase in the C14TAB/water system. However, in the former system this concentrated phase is made up primarily from PA-/C14TA+ complexes, with the dilute phase containing the majority of the simple ions. Likewise, Ilekti et al.12 found both hexagonal and lamellar phases in the phase diagram of NaPA, C16TAB, and water. Here, the latter phase was obtained by “washing” out the counterions from the hexagonal phase replacing the dilute phase with pure water. A comparison with phase diagrams for the binary surfactant/water systems suggests that cubic phases are also to be expected. Indeed, cubic structures (as well as hexagonal and lamellar) have been reported for the complexes formed between covalently cross-linked polyelectrolyte networks and surfactant of opposite charge.13-18 These results were obtained from small-angle X-ray (7) Thalberg, K.; Lindman, B.; Bergfeldt, K. Langmuir 1991, 7, 2893. (8) Lindman, B.; Thalberg, K. In Interactions of Surfactants with Polymers and Proteins; Goddard, E., Ananthapadmanabhan, K. P., Eds.; CRC Press: Boca Raton, FL, 1993; Chapter 5. (9) Harada, A.; Nozakura, S. Polym. Bull. 1984, 11, 175. (10) Antonietti, M.; Burger, C.; Effing, J. Adv. Mater. 1995, 7, 751. (11) Carnali, J. O. Langmuir 1993, 9, 2933. (12) Ilekti, P.; Piculell, L.; Cabane, B.; Tournilhac, F. J. Phys. Chem. B 1998, 102, 344. (13) Khandurina, Y. V.; Dembo, A. T.; Rogacheva, V. B.; Zezin, A. B.; Kabanov, V. A. Polym. Sci. 1994, 36, 189. (14) Khandurina, Y. V.; Alexeev, V. L.; Evmenenko, G. A.; Dembo, A. T.; Rogacheva, V. B.; Zezin, A. B. J. Phys. II 1995, 5, 337.
S0743-7463(97)01165-7 CCC: $15.00 © 1998 American Chemical Society Published on Web 03/26/1998
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(SAXS) and neutron scattering (SANS) measurements on gels equilibrated several weeks in aqueous surfactant solutions. Similar to what has been observed in polyelectrolyte solutions,19 Philippova and Starodoubtzev20 found that in the cross-linked networks the surfactant forms aggregates at concentrations far below the critical micelle concentration (cmc) in aqueous solutions of the surfactant. It is well documented that the micelle formation, as a counterpart to the macroscopic phase separation, is accompanied by a pronounced shrinking of the gel volume.17,21-27 The highly ordered structures appear in the fully collapsed state of the gel. In the present paper we present a model describing the volume changes induced by the formation of globular surfactant micelles in covalently cross-linked polyelectrolyte networks. The model is based on the notion of microscopic perturbations of the network in the vicinity of each micelle. To be more specific, the formation of a micelle is assumed to induce a collapse of the polyion network on the local level resulting in a polyelectrolytesurfactant “complex” of low net charge. This will leave the parts of the gel far from (and free from) complexes essentially unaffected. The description is in line with recent thermodynamic19 and structural studies28-38 as well as model simulations39-41 suggesting that flexible linear polyelectrolytes collapse around micelles to form complexes of low net electric charge. Here the density of the collapsed polyion layer surrounding a micelle is expected to depend on the matching of the charge density of the interactants, analogous to the binding of polyelectrolytes to oppositely charged planar surfaces.42 It is important to note that in the proposed “heterogeneous” description of the network, a clear distinction is made between the polyion chains close to and far from (15) Okuzaki, H.; Osada, Y. Macromolecules 1995, 28, 380. (16) Chu, B.; Yeh, F.; Sokolov, E. L.; Starodoubtsev, S. G.; Khokhlov, A. R. Macromolecules 1995, 28, 8447. (17) Sokolov, E. L.; Yeh, F.; Khokhlov, A.; Chu, B. Langmuir 1996, 12, 6229. (18) Yeh, F.; Sokolov, E. L.; Khokhlov, A. R.; Chu, B. J. Am. Chem. Soc. 1996, 118, 6615. (19) Hayakawa, K.; Kwak, J. C. T. In Cationic Surfactants: Physical Chemistry; Rubingh, D., Holland, P. M., Eds.; Surfactant Science Series 37; Marcel Dekker: New York, 1991; Chapter 5. (20) Philippova, O. E.; Starodoubtzev, S. G. J. Polym. Sci., Part B 1993, 31, 1471. (21) Khokhlov, A. R.; Karamarenko, E. Y.; Makhaeva, E. E.; Starodubtzev, S. G. Macromolecules 1992, 25, 4779. (22) Okuzaki, H.; Osada, Y. Macromolecules 1994, 27, 502. (23) Machaeva, E. E.; Starodoubtzev, S. G. Polym. Bull. 1993, 30, 327. (24) Machaeva, E. E.; Starodoubtzev, S. G. Makromol. Chem. Rapid Commun. 1993, 14, 105. (25) Sasaki, S.; Fujimoto, D.; Maeda, H. Polym. Gels Networks 1995, 3, 145. (26) Okuzaki, H.; Osada, Y. Macromolecules 1995, 28, 4554. (27) Philippova, O. E.; Hourdet, D.; Audebert, R.; Khokhlov, A. R. Macromolecules 1996, 29, 2822. (28) Thalberg, K.; van Stam, J.; Lindblad, C.; Almgren, M.; Lindman, B. J. Phys. Chem. 1991, 95, 8975. (29) Kiefer, J. J.; Somasundaran, P.; Ananthapadmanabhan, K. P. In Polymer Solutions, Blends, and Interfaces; Noda, I., Rubingh, D. N., Eds.; Studies in Polymer Science 11; Elsevier: Amsterdam, 1992. (30) Kiefer, J. J.; Somasundaran, P.; Ananthapadmanabhan, K. P. Langmuir 1993, 9, 1187. (31) Almgren, M.; Hansson, P.; Mukhtar, E.; van Stam, J. Langmuir 1992, 8, 2405. (32) Hansson, P.; Almgren, M. Langmuir 1994, 10, 2115. (33) Hansson, P.; Almgren, M. J. Phys. Chem. 1995, 99, 16694. (34) Hansson, P.; Almgren, M. J. Phys. Chem. 1995, 99, 16684. (35) Hansson, P.; Almgren, M. J. Phys. Chem. 1996, 100, 9038. (36) Fundin, J.; Brown, W. Macromolecules 1994, 27, 5024. (37) Fundin, J.; Brown, W.; Vethamuthu, M. S. Macromolecules 1996, 29, 1195. (38) Anthony, O.; Zana, R. Langmuir 1996, 12, 1967. (39) Wallin, T.; Linse, P. Langmuir 1996, 12, 305. (40) Wallin, T.; Linse, P. J. Phys. Chem. 1996, 100, 17873. (41) Wallin, T.; Linse, P. J. Phys. Chem. 1997, 101, 5506.
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micelles, or rather, the state of a chain is assumed to depend on its distance from a micelle. This is in contrast to other models of surfactant/network interactions, where all polyion chains in the network are treated equally; i.e., they are all subject to the same average stretching force. To make this point clear we shall briefly summarize the models by Khokhlov et al.43 and Gong and Osada.44 The former authors restricted their treatment to weakly charged polyions.43 This motivated a description of the polymer network using the simple theory for rubber elasticity45 neglecting the electrostatic free energy of the polyion. The swelling pressure is provided by the mobile ions in the gel, with a small contribution from the network chains. At equilibrium, the osmotic swelling force is balanced by the elastic force provided by the network. The micelle formation promotes gel shrinking by reducing the average concentration of mobile ions in the gel. Thus, a surfactant molecule in a micelle is treated as “immobilized”, in agreement with the phase separation approximation.46 No explicit interaction between the polyion and the micelle is considered. Still, the micelle formation is more favorable in the gel/solution system compared to the pure surfactant/water system since, in the former, the polyion counterions are released to the solution. A similar analysis was made by Gong and Osada44 who included the effect of a finite size of the surfactant assemblies. In their model the surfactant molecules are assumed to interact with nearest-neighboring surfactants bound to the polyion, which is described as a linear array of binding sites. Such a picture has been found to be incorrect for the surfactant assembly in polyelectrolyte solutions28,29,31-35,38 and should therefore be inappropriate also in a cross-linked system. However, from a comparison between model predictions and experiments, they concluded that the cooperativity of the surfactant aggregation process (i.e., the surfactant-surfactant interaction parameter in the model) had only a minor influence on the shape of the surfactant binding isotherms. As will be demonstrated in the present paper, this is true only at low salt concentrations. In the presence of excess salt the binding isotherms are almost identical to those reported for solutions of the linear polyion (this is particularly clear from the investigation by Maeda and co-workers25) where a relation between the sharpness of the binding isotherm and the aggregation number has been found.35,47 In the experimental part of the paper we test the model and its implications to the distribution of the surfactant between the network and an equilibrium aqueous solution. In particular, a detailed study of the surfactant binding close to the cac in the gels is carried out. As a model system we have chosen cross-linked NaPA/C12TAB, where the interaction between the components in dilute solutions of the linear polyelectrolyte has been thoroughly investigated and found to be strong and mainly of electrostatic nature.7,11,12,29,30,32,34,48 Furthermore, as will be shown in a subsequent paper,49 in this system the micelles remain (42) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman & Hall: London, 1993. (43) Khokhlov, A. R.; Kramarenko, E. Y.; Makhaeva, E. E.; Starodoubtzev, S. G. Makromol. Chem., Theory Simul. 1992, 1, 105. (44) Gong, J. P.; Osada, Y. J. Phys. Chem. 1995, 99, 10971. (45) Flory, P. J. Principles of polymer chemistry; Cornell University Press: Ithaca, NY, 1953. (46) Tanford, C. The hydrophobic effect: Formation of Micelles and Biological Membranes, 2nd ed.; John Wiley & Sons: New York, 1980. (47) Hayakawa, K.; Shinohara, S.; Sasawaki, S.; Satake, I.; Kwak, J. C. T. Bull. Chem. Soc. Jpn. 1995, 68, 2179. (48) Hayakawa, K.; Santerre, J. P.; Kwak, J. C. T. Macromolecules 1983, 16, 1642.
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monomer unit is there the same as prior to surfactant binding, v0 (see below for a correction). For every micelle formed the volume change of the gel is equal to Np,1v0-V1, where Np,1 is the number of monomer units per cell. Hence, the total volume, V, of the gel is
V ) V0 - nmic(Np,1v0 - V1)
(2)
To relate V to the degree of surfactant binding, β, we define
β ) ns,gel/np,tot
(3)
where ns,gel is the number of surfactant molecules in the gel. The number of micelles may then be expressed as
nmic ) Figure 1. A schematic description of the density of monomer units (F) outside a micelle of radius rm in the gel. The cell radius, rc, is defined in the model as the distance from the micelle center where F equals the average value in the micelle-free gel.
small as the network transforms from a disordered and highly water-swelled state to a cubically ordered collapsed state.
np,tot(β - βcac) Ns
where Ns is the surfactant aggregation number and βcac is the value of β at the critical aggregation concentration (cac) for the surfactant in the gel; np,totβcac is equal to the number of surfactant unimers in the gel. When all monomer units are distributed in cells, the gel is in its fully collapsed state with a volume Vmin equal to V1nmic. It follows immediately from eqs 1, 2, and 4 that
Model The model intends to describe the association of ionic surfactants in a polyelectrolyte network of opposite charge. In this first communication we describe the physical model and investigate the predicted volume changes for an idealized case. The polyion is assumed to have a highly flexible backbone and to be highly charged. The number of colvalent cross links is assumed to be small, so that the number of polyion charged groups (from now on referred to as monomer units) between two consecutive cross-links is larger than the surfactant aggregation number. As a reference state we choose the water-swollen state of the gel in equilibrium with pure water or brine. The volume of the gel is then V0 and the average volume per charged monomer unit is v0. Thus
V0 ) np,totv0
(1)
where np,tot is the total number of charged monomer units in the gel. When the gel, in its reference state, is immersed in a surfactant solution, nmic surfactant micelles form in the gel. There is a distribution of monomer units outside each micelle, characterized by a high unit density in the close vicinity of the micelle and gradually decrease down to a bulk (average) unit density far from the micelle; see Figure 1. Without the loss of generality the gel may be divided into electroneutral cells, each containing a micelle plus water, monomer units, and simple ions. We restrict the treatment to spherical, monodisperse micelles of fixed size. We assume that each micelle is located at the center of a spherical cell of volume V1, determined by the radial distribution of monomer units as depicted in Figure 1. The distribution function is expected to resemble the results from Monte Carlo simulations39-41 of model polyions outside oppositely charged spheres, perturbed however by the presence of cross-links. Since the region between the cells is, by definition, unaffected by the presence of micelles, the volume per (49) Hansson, P. Submitted to Langmuir.
(4)
V0V1 v0Np,1
(5)
Ns + βcac Np,1
(6)
Vmin ) βsat )
The subscript “sat” (for saturation) is used on β to indicate that, within the model, no further cooperative binding is possible when the gel reaches its fully collapsed state. The gel can still bind more surfactant, but only in a noncooperative fashion; see discussion. Up to this point we have neglected the fact that the binding of surfactant results in an accumulation of polyion counterions (or salt) in the solution. Under situations where the concentration of inorganic salt in the solution is low, this must be accounted for as the osmotic pressure in the solution will increase. To avoid a complicated description of the elasticity the polyion network, we simply introduce a correction function, f, so that in eqs 1 and 2 V0 and v0 are replaced by fV0 and fv0, respectively. f is thus an empirical relation describing V/V0 as a function of the electrolyte concentration in the solution (in the absence of surfactant). Note that the correction does not change eqs 5 and 6. Experimental data are most conveniently expressed in terms of the relative volume V/V0. By including the correction f we have for βcac e β e βsat
(
V1 fNp,1 V ) f - β* V0 Ns v0Ns
)
(7)
where β* ) β - βcac. In the general case, βcac, Np, V1, and Ns are functions of β. In this paper we shall explore the simple situation where they are constant. Experimental Section Materials. C12TAB from Serva (analytic grade), acrylic acid from Aldrich, pyrene from Janssen (99%+), sodium bromide from Bakers, and N,N,N′,N′-tetramethylethylenediamine (TEMED), ammonium persulfate, and N,N′-methylenebis(acrylamide), all
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Table 1 cross-linker C0a cac ∆G0 b v0 (mol %) (mM) (103 Å3) Vmin/V0 103βcac (mM) (RT/mol) 0.90 1.4 3.0 b
23 52 192
73 32 8.6
0.012 0.026 0.11
8(1 8(2 5(1
0.18 0.42 0.96
1.6 1.2 1.4
a Concentration of polyelectrolyte in the surfactant-free gel. Free energy change in (8).
from Sigma, were used as received. All solutions were prepared using high-quality Millipore water. Preparation of Gels. Solutions of acrylic acid (10 wt %) were polymerized in glass tubes (5-10 mm diameter) in the presence of N,N′-methylenebis(acrylamide) as cross-linking agent. Ammonium persulfate and TEMED were included as radical initiator and accelerator, respectively. The solutions contained 0.90, 1.4, and 3.0 mol % of the reactants as crosslinkers. Although no compositional analysis of the resulting gel samples was performed, the gels will be referred to as 0.90, 1.4, and 3.0% gels, respectively. After polymerization for 12 h at 70 °C, the gels were neutralized in 0.5 M NaOH solutions. The gels were then cut into short cylinders and placed in large baths of 10-4 M NaOH. The solution was replaced many times. Finally, the swelled up gels were stored in deareated solutions of 10-4 M NaOH in tightly sealed containers. To determine the concentration of acrylate groups, gel pieces of known mass were freezedried and then weighted. To calculate the molar concentration, we used a density of the water-swelled gels equal to 1.0 g/mL. The result, expressed as the volume per acrylate group (v0), is given in Table 1. In general, the same result was obtained by measuring the diameter of the gels immediately after the polymerization and in their final state and then calculating the concentration from the degree of swelling with the knowledge of the amount of acrylic acid in the reaction mixture. This indicates that most of the reactants were polymerized. Surfactant-Gel Systems. Cylindrical gel pieces of known mass (typically 0.4-5 g) were placed in 1.0 × 10-4 M NaOH solutions containing precisely known amounts of surfactant. The total volume of the system varied between 30 and 50 mL, but the total concentration of cross-linked polyacrylate was always 3.0, 3.1, and 2.1 mM for the 0.90, 1.4, and 3.0% gels, respectively. The surfactant concentration varied from zero to 3 mM, which is well below the cmc for C12TAB (cmc ) 15 mM). To prevent unpleasant effects of carbon dioxide, the water was purged with nitrogen, as were all sample containers before they were sealed. All samples were equilibrated 1 month at 25 °C. During that period the containers were shaken slowly several times. After the incubation period it was confirmed that phenolphthalein added to a fraction of the solution gave a strongly pink/red color (i.e., pH > 9). Mass and Volume of Gels. The equilibrated gels were taken out from the solutions and weighed after carefully removing water from the surfaces with a filter paper. No significant change of the mass or the general appearance of the gels placed in the surfactant-free solutions was observed. The density of the fully collapsed gels was estimated as 1.0 g/mL from mass/volume determinations. Thus, to a good approximation the relative change in volume V/V0 equals the mass ratio m/m0 irrespective of the amount of surfactant in the gel. Free Surfactant Concentration. The surfactant concentration of the aqueous solution in equilibrium with the gels was determined using a surfactant sensitive electrode. The electrode was of the type used extensively in the literature.50,51 The active part is a thin PVC plastic membrane separating a surfactant reference solution from the test solution. We used a membrane without a charge carrier complex. The potential difference between two silver/silver chloride standard electrodes, in contact with the two solutions via salt bridges and a system of salt trapping compartments, was measured with a Keithely 177 Microvolt DMM (10 MΩ). The response from standard C12TAB solutions in 50 mM NaBr was used to construct calibration curves. (50) Maeda, T.; Ikeda, M.; Shibahara, M.; Haruta, T.; Satake, I. Bull. Chem. Soc. Jpn. 1981, 54, 94. (51) Hayakawa, K.; Kwak, J. C. T. J. Phys. Chem. 1982, 86, 3866.
Figure 2. Measured response in millivolts (mV) from the surfactant sensitive electrode as a function of the concentration of C12TAB in the test solution. The equation represents the linear fit to the data at the higher concentrations. Temperature ) 25 °C. A typical example is shown in Figure 2. The response was Nernstian (58.9 mV/decay) at the higher concentrations, but the sensitivity was poorer in the very dilute range. However, this did not affect the reproducibility of the measurements. At the present concentration of supporting electrolyte the response was very fast. A fraction of the solution in equilibrium with the gels was filtered through a glass filter (Jena, 1G2) to remove small gel particles. A small volume of a concentrated NaBr solution was added to a certain volume of the filtrate so that the concentration of NaBr was 50 mM. The concentration of surfactant in this solution was determined with the electrode and then corrected for the dilution to yield the concentration in the initial solution. The number of moles of surfactant in the gels was obtained from the difference between the initial and the final concentration of surfactant in the aqueous phase. Here, it was necessary to consider the increase of the volume of the aqueous phase due to the gel shrinkage. Finally, the degree of surfactant binding, β, was calculated using eq 3. Fluorescence Measurements. Pyrene has a strong tendency to associate with surfactant micelles. To probe the presence of micelles in the gels, pyrene was added to the aqueous phase (10-7 M). After 2 days the signal from the third vibronic peak in the pyrene fluorescence spectra was measured on a SPEX Fluorolog 1680. The excitation wavelength was 320 nm. The emission from the solutions was detected at right angle in a 1 cm quartz cell. The emission from the gels was measured frontface directly from the gel surface. It is common to use the ratio between the third and the first vibronic peaks in the pyrene fluorescence spectra to probe the existence hydrophobic aggregates. This ratio is known to be sensitive to the polarity experienced by the probe.52 However, in the present case the observed enhanced intensity from the gels containing micelles (and the reduced signal from the solution) was found to be more sensitive.
Results and Discussion Gel Volume. Figure 3 shows the relative volume of the 0.90% gels placed in aqueous solutions of NaCl and C12TAB, respectively. After 1 month of equilibration the gel placed in the strongest solution of simple salt showed only a 20% reduction of the volume. In contrast, an equal amount (ca 2 mM) of surfactant brought about an almost complete collapse of the gel. The experiment shows that the contribution to the shrinkage from an increased concentration of ions in the aqueous phase is small but nonnegligible in the present range of concentrations. The (52) Kalyanasundaram, K.; Thomas, J. K. J. Am. Chem. Soc. 1977, 99, 2039.
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Figure 3. Relative volume (V/V0) of 0.9% NaPA gels placed in solutions of C12TAB or NaCl. C equals the concentration of these species plus 10-4 M NaOH. The gels were initially swelled in 10-4 M NaOH solutions. The arrow marks the appearance of C12TAB micelles. Temperature ) 25 °C.
NaCl curve in Figure 3, and similar ones for the 1.4 and 3.0% gels, was used to determine the correction function f defined above. Before proceeding it is important to mention that all shrunken gels in this study were transparent and had essentially maintained the shape of the initial gels. The slow process of surfactant diffusing into the swelled-up gels seems to prevent the formation of macroscopic inhomogenities. In contrast, dilute solutions of the linear polyion readily phase separate when mixed with the surfactant to give nontransparent and sticky precipitates.7,32 These may not be equilibrium structures but instead trapped in metastable states. Note that true phase separation in the gels is not possible due to the covalent cross-links. Figure 4 shows the shrinking of 0.90, 1.4, and 3.0% gels when placed in C12TAB solutions. The relative volume V/V0 is given as a function of β. The binding data will be presented in the next section, but we mention here that micelles were detected already at very low β as indicated by the arrows in the inserts to Figure 4. The volume of the gels decreases gradually with increasing β down to a minimum level. The observation that the relative volume of the fully collapsed gels increases with increasing degree of cross-linking is simply explained by the difference in v0 between the gels. In fact, the composition of the gels in the collapsed state is very similar for the three gel samples. This indicates that the cross-linking density is of minor importance for the interactions in that state. The values of Vmin/V0 in Table 1 show that the total volume change varies between 10 and100 times depending on the degree of cross-linking. Similar to what we observe with the 0.90% gels, Maeda and co-workers25 found linear relations between the gel volume and β for slightly cross-linked NaPA gels (ca. 0.2% cross-linker) interacting with dodecylpyridinium ions. They also found that dV/dβ was proportional to V0. These observations were made in the presence of 10-100 mM NaCl and for various degrees of ionization of the polyion in the range from 0.2 to 0.9, albeit the result was less clear in the latter case. It should be mentioned also that they found an empirical expression similar to eq 7. However, our measurements reveal an increasing deviation from the linear dependence on β as the degree of cross-linking increases; see discussion below. In conclusion, it seems reasonable to state that a linear behavior can be expected in the limit of zero cross-linking density where the local polyion-micelle interactions dominate in the entire volume range. This may have
Figure 4. Relative volume (V/V0) of NaPA gels versus the degree of C12TA+ binding, β. Degree of cross-linking (%): (a) 0.90, (b) 1.4, (c) 3.0. Temperature ) 25 °C. The insets show the same results on a logarithmic x-axis. Arrows indicate the appearance of micelles.
interesting implications to the phase behavior in mixtures with the linear polyelectrolyte. Binding Isotherms. General Observations. The distribution of C12TA+ between gel and solution was determined from measurements of the surfactant concentration in the aqueous phase. The resulting binding isotherms are shown in Figure 5, where β is given as a function of Cs,w, the equilibrium concentration of surfactant in the aqueous phase. We emphasize that binding isotherms have been reported by other authors for similar systems. The new contribution here is the detailed description of the binding close to the cac. When the binding isotherms are given a log-log presentation, three or four different regimes can be
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Figure 6. Pyrene fluorescence emission from gel surfaces normalized to the emission from solutions in equilibrium with the gels (Igel/Isolution). For reasons of clarity values of 25 and 50 were added to Igel/Isolution for the gels containing 1.4 and 3.0% cross-linkers, respectively.
Figure 7. Dependence of the cac on the polyelectrolyte concentration in NaPA-solutions and cross-linked gels. Data for the solutions were taken from ref 34.
Figure 5. Degree of binding to the gel, β, as a function of the equilibrium surfactant concentration in the aqueous phase, Cs,w. The points are the experimental data for 0.90 (a), 1.4 (b), and 3.0% (c) NaPA gels. Temperature ) 25 °C. The solid lines are the binding isotherms predicted by eq 8 with the following input data {K, Kex, N, Cp,tot, Csalt,tot, Ø}: (a) {70, 0.20, 60, 3.0 mM, 8.7 × 10-5 M, 0-0.14}, (b) {70, 0.29, 60, 3.1 mM, 9.4 × 10-5 M, 0.01-0.04}, (c) {140, 0.25, 60, 2.1 mM, 9.9 × 10-5 M, 0.01}. The dotted line is the monomer binding predicted by eq 11; K ) 0, the other parameters were unchanged. The broken line in (a) is the binding predicted by eq 11 for Csalt,tot ) 50 mM. The roman numerals in (c) depict the four characteristic binding regimes.
observed in the present range of concentrations; see Figure 5c. At sufficiently low surfactant concentrations the micelle formation can be neglected. With also the formation of dimers, trimers, etc., neglected, the binding is determined by the mutual distribution of sodium ions and surfactant monomers, consistent with a noncooperative binding of surfactant, regime I. The cac may be defined as the border between regime I and II, where the slope of the binding isotherms start to increase due to the
cooperative formation of micelles. As will be shown in the analysis below, the fate of regimes II and III depends on the amount of salt in the system. Regime IV is entered when there is no space left in the gel to host another micelle. Here, surfactant monomers are still expected to replace sodium ions in the gel in a noncooperative fashion. Cac in the Gel. From the binding isotherms in Figure 5 the cac for the surfactant in the gels was calculated from the value of β corresponding to the end of regime I. The relevant point on each binding isotherm was evaluated both by directly observing the change of the slope and, independently, from spectroscopic measurements. In the latter case, the fluorescence signal from the gels and the solutions were monitored after additions of small amounts of pyrene to the samples. The measurements, presented in Figure 6, show the enhanced signal from the gels containing micelles. The agreement between the two estimates of βcac is satisfactory. In Figure 7 the cac in the gel is compared with the cac previously found in solutions of the linear NaPA.34 The error bars were calculated from the estimated error in βcac indicated in Table 1. The points obtained in the gels fall closely to the line extrapolated from the other data points, indicating that the presence of cross-links apparently have little influence on the micellization in these gels at the cac. This indicates that the surfactant assembly in the gel is first of all a local event, not involving to a large extent parts of the gel far from the micelles. The interaction between polyion and micelle in the gel should thus, at the cac, be similar to that in solutions of the linear polyion.
Self-Assembly of Ionic Surfactants
Langmuir, Vol. 14, No. 9, 1998 2275
Law of Mass Action. When a surfactant-free gel is placed in a pure surfactant solution, the distribution of ions is highly nonuniform. Thus, the counterions in the gel start to mix with the surfactant monomers. However, if there is enough surfactant in the system the micellar state becomes highly populated due to the hydrophobic interactions. If these interactions are strong, the equilibrium situation will again be a highly nonuniform system, now with the majority of the surfactant molecules in the gel and the network counterions in the solution. It is easy to see that the addition of salt to the solution will shift the equilibrium to give a higher free concentration of the surfactant. The thought experiment points to the important entropy effects in polyelectrolyte/surfactant systems. To describe the partitioning of the surfactant between the micelles, the surrounding bulk gel, and the solution, we need expressions showing how the chemical potential in these subsystems changes with V, β, etc. However, if the proposed model for the gel structure is correct, a reasonable first description can be obtained by assuming fixed standard chemical potentials in the subsystems. To see this we first divide the cooperative binding in two steps: (i) the binding of surfactant monomers and (ii) the formation of a micelle from these inside the network. Now, since all micelles are assumed to be formed in identical environments, i.e., in equally swelled-up parts of the gel, the standard free energy change of the second process (≈NRT ln cac) is a function of the polyelectrolyte concentration in the bulk gel (see Figure 7) (but independent of V and β) and can thus be treated as a constant. Under conditions where salt is excluded from the gels (see below), process i is simply an exchange of polyelectrolyte counterions, M, for the surfactant ions, S. Using the subscripts w and g for reactants in the water solution and in the gel, respectively, this can be written
S w + Mg h S g + M w
Kex
(8)
where Kex is the equilibrium constant related to a conditional binding constant K0, and the concentration of counterions (sodium) in the aqueous phase, CM,w, through
K0 )
Kex CM,w
(9)
If we allow for only one aggregation number, the micellization process ii is consistent with the following equilibrium between N monomers and a micelle, SN
NSg h SN
KN
(10)
where K is the equilibrium constant (per monomer) for the micelle/monomer equilibrium in the gel. It was shown previously35,53 that eqs 8-10 can be combined to give
β ) K0Cs,w + (KK0Cs,w)N 1-β
(11)
where the first term to the right describes the binding of surfactant monomers and the second term the cooperative binding of micelles. (53) Linse, P.; Piculell, L.; Hansson, P. Polymer Surfactant Systems; Kwak, J. C. T., Ed.; Surfactant Science Series Marcel Dekker: New York, in press.
The mass balance for M in the total system of gel plus aqueous phase can be written
(1 - Ø)CM,w ) βCp,tot + Csalt,tot
(12)
where Ø is the volume fraction of gel in the system and Cp,tot and Csalt,tot are the total concentration of polyelectrolyte and added salt (of the type MX), respectively. By combination of (9), (10), and (11), the role of the counterions on the binding can be investigated. Before comparing eq 11 with experimental binding data, we discuss some of the assumptions in the above analysis. The ion-exchange mechanism (eq 8) follows immediately from the condition of electroneutrality if salt is prohibited from entering the gels as neutral components. By considering the high polyelectrolyte concentration in the gels, a pronounced exclusion is expected due to the pure Donnan effect. To check this more carefully, we investigated the partitioning of a 1:1 salt between the gel and the solution phase using the Poisson-Boltzmann (PB) cell model.54 We thus neglected the cross-links and described the polyion chains in the bulk gel as a system of charged cylinders, each one located at the center of a cylindrical cell. Numerical solutions of the PB equation showed that the salt was almost completely excluded from these cells, except at conditions corresponding to the highest surfactant concentrations in the systems with 0.90% cross-linker. However, it was assumed in the calculations that the volume of the cylindrical cell was fixed. A correction of this volume using the function f above will reduce the calculated amount of salt in the 0.90% gels as well. Within the model, the formation of a complex in the bulk gel involves the neutralization of a fixed number of polyion charges ()Np,1). If salt is excluded from the bulk gel, the electrostatic (free) energy per polyion charge in that region is constant.54 This motivates why the electrostatic energy change of process ii can be treated as constant, which in turn leads to a standard free energy change equal to -NRT ln K for the same process. Note that the standard free energy of binding a micelle from the solution becomes -NRT ln KK0. Binding below the cac. For the premicellar regime the second term on the right-hand side of eq 11 can be omitted. The remaining part, which under the present conditions can be considered as purely thermodynamic, permits the estimation of Kex from the experimental data. The dotted lines in Figure 5 represent the predicted behavior when Kex is equal to 0.20, 0.29, and 0.25, respectively. Note that prior to micelle formation Ø may be treated as a constant. From the definition Kex ) exp(-∆G°/RT), it follows that the standard free energy change for the reaction to the right in (8) is between 1.1 and 1.6 RT/mol; see Table 1. ∆G° would be equal to zero if sodium ions and C12TA+ were distributed in the same way between the gel and the aqueous phase. The preferential binding of sodium ions (3-5 times stronger than the binding of C12TA+ monomers) is expected to reflect a general difference in the distribution of the two components in the electric field of the polyion. The large size of the surfactant charged group certainly contributes to the effect. This can explain the insignificant binding of C12TA+ previously found in solutions of NaPA at surfactant concentrations below the cac.34 Cooperative Binding. The solid lines in Figure 5 are the binding isotherms calculated from eq 11 using N ) 60 (54) Jo¨nsson, B.; Wennerstro¨m, H. J. Colloid Interface Sci. 1981, 80, 483.
2276 Langmuir, Vol. 14, No. 9, 1998
Hansson
(from experiments49) and K ) 70, 70, and 140 in parts a, b, and c of Figure 5, respectively. β was solved for in an iterative calculation using eqs 9 and 12 with the above estimates of Kex. The other parameters, Cp,tot, Csalt,tot, and Ø, are given by the experimental conditions. In Figure 5 we see that the theory is able to capture the binding surprisingly well for low and medium values of β. We may thus conclude that the changes in mixing entropy have a major influence on the binding. Included in Figure 5a is the binding isotherm predicted by eq 11 when Csalt,tot is equal to 50 mM, all other parameters the same as before. In agreement with experimental findings25 the surfactant in the gel is predicted to be in equilibrium with a higher free surfactant concentration and the isotherm is much steeper than that in the absence of salt. The explanation for the latter effect is that, in the absence of salt, K0 decreases with increasing β; see eqs 9 and 12. Thus, according to eq 8 the accumulation of polyion counterions in the water phase suppresses the surfactant binding. On the other hand, in the presence of excess salt K0 is constant throughout the binding isotherm. The maximum slope will then be approximately equal to N, as can be verified from eq 11.35 Under these circumstances there is no distinction between regimes II and III. In the hypothetical case of no salt, the binding isotherms predicted by eq 11 are very flat; see below. The experiments in the present study belong to the intermediate case where the salt concentration is small but higher than the free surfactant concentration at the cac. The slope is then large immediately above the cac but decreases as the buffering effect of the added salt vanishes. As expected, region III starts when βCp,tot ≈ Csalt,tot. The characteristic slope in region III of the binding isotherm is slightly below 1. With Csalt,tot ) 0 the derivative of the cooperative part (neglecting the first term on the right-hand side) of eq 11 is
d log β ) d log Cs,w
N N+1+
β 1-β
(13)
Equation 13, which holds to a good approximation when βCp,tot . Csalt,tot, shows that the aggregation number, and hence, the mixing entropy of the micelles, has only a minor influence on the slope of the binding isotherms in region III (and IV). This can explain why the error introduced by assuming ideal mixing of the micelles is small also when β is rather large. In conclusion, the agreement between the theoretical and the experimental binding isotherms support the proposed “heterogeneous” picture of the gel where the macroscopic volume change results from microscopic collapses in the vicinity of the micelles. It is instructive to mention that Gong and Osada, to obtain a satisfying agreement between the experimental binding isotherms and their theoretical predictions, found it necessary to model the polymer network as completely rigid.44 This represents a serious problem with their model, as they intended to describe also the surfactant-induced gel collapse. However, in light of the discussion above, the result is understandable since the network in the micellefree parts of the gel remains swelled-up (i.e., it is in a sense “rigid”). Gel Microstructure and Size of the Complex. The solid lines in Figure 4 represent the volume changes calculated from eq 7 for the simple case where the size and the composition of the cell is constant. The function f, described above, was used to correct for the accumulation
of salt. The correction is responsible for the slight nonlinearity of the curves in the transition region. According to the model, the major volume change starts when micelle-polyelectrolyte complexes appear in the gel and ends when the complexes are close-packed. This defines start and stop coordinates (βcac, f) and (βsat, Vmin/ V0), respectively, in the plots of Figure 4. βcac and Vmin/V0 can be considered as well-defined parameters, easily determined from the binding isotherms and from the volume of the completely collapsed gels, respectively. In contrast, it is not straightforward to estimate βsat from the present set of data. There is, of course, no reason to believe that there should exist a well-defined saturation point, but for the present purpose we may simply put βsat ) 0.73 for all degrees of cross-linking. This choice is in agreement with the borders between regimes III and IV in the binding isotherms. Note that, by neglecting the volume of the complex, a fairly good prediction of the volume change, at least for the 0.90% gel, can be obtained just by using the information from the binding isotherm. This shows that the complex is small compared to the volume occupied by the monomer units before interacting with the micelle. In turn, this means that the rather arbitrary choice of rc (Figure 1) represents a minor problem with respect to the volume changes. For other purposes a better definition of rc (and V1) may be necessary. In a fluorescence quenching study of the present system we found that there is a small increase in the aggregation number as the gel shrinks, but the micelles remain small even in the fully collapsed state.49 The aggregation number was found to be 70 at βsat ) 0.73. By combination of this result with the estimates of Vmin/V0, βcac, and v0 in Table 1, the size of the polyelectrolyte/micelle complex can be calculated using eqs 5 and 6. Note that these expressions are essentially model independent. The radius of the micelle/polyelectrolyte complex is found to be 27, 27, and 28 Å for the 0.90, 1.4, and 3.0% gels, respectively. In the model, these values correspond to the cell radius. These estimates are, of course, only rough since a C12TA+ micelle with an aggregation number of 70 is expected to be slightly elongated. Still, by taking 20 Å as a reasonable measure of the micelle radius, the thickness of the polyelectrolyte layer surrounding the micelle is found to be 7-8 Å. This is in agreement with recent Monte Carlo simulations performed by Wallin and Linse41 on a model system containing a micelle of aggregation number 53, a highly charged flexible linear polyion (80 charges), together with their counterions. By considering exclusively electrostatic interactions using the primitive model, they estimated that the thickness of the adsorbed layer of monomer units was ca. 7 Å calculated from the surface of the micelle (radius ) 20 Å). This corresponded to a 93% screening of the micelle charge. It should be mentioned that the concentration of polyelectrolyte in the simulations was only about 1 mM. SAXS measurements on the present system show that an ordering takes place when the gels reach their fully collapsed state, i.e., when the degree of binding is close to βsat.49 Thus, in this state the gels contain close-packed complexes typically made up from micelles of aggregation number 70 surrounded by some 100 polyion monomer units and 30 sodium ions (plus water). The binding isotherms indicate that further micelle formation is not favorable, which means that there is a large cost of employing all monomer units in the gels as micelle counterions. The properties of the collapsed state appear to be independent of the degree of cross-linking. This is reasonable, however, as the constraints on the network
Self-Assembly of Ionic Surfactants
chains imposed by the cross-links should decrease as the gels shrink. On the other hand, it is evident from Figure 4 that the degree of cross-linking has an effect on the relative volume as a function of β. An explanation to this is expected to involve the compressibility of the network in a complicated way which is beyond the scope of the present contribution. Here we shall only point out that the volume of a cell V1 is small compared to Np,1v0, the initial volume of the monomer units in it (Table 1). Then it follows from eq 2 that the large change in volume per monomer unit gives the main contribution to the predicted linearity of the volume change of the gels in Figure 4. With the shrinking due to the accumulation of salt in the system neglected, deviations from the linearity should therefore be introduced by the approximation of a fixed number of monomer units per cell. A question for future studies is thus whether the cells (or the complexes) contain an excess of polyion monomer units at low β?
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when the surfactant concentration in the gel reaches the cac measured in a polyelectrolyte solution of the same concentration. The degree of cross-linking has only an indirect influence on the cac in the gels. A picture where the strong attraction between micelles and polyion results in the formation of well-defined complexes surrounded by the swelled-up network appears to give a good first-order description and makes it formally easy to the calculate the free energy of the gel by adding together the contributions from the complexes and the complex-free parts. A structural ordering in the gel can in the present system be visualized as a close-packing of complexes comprising globular micelles surrounded by a small excess of polyelectrolyte. Finally, the model discussed in this paper should be applicable to other systems, for instance, cationic networks interacting with charged colloidal silica particles.
Conclusions The surfactant self-assembly in the present gels can be classified as a polyelectrolyte-induced micelle formation. The binding of surfactant monomers prior to micelle formation is governed by the mutual distribution of the surfactant and the polyion counterions. Micelles form
Acknowledgment. The author is thankful to Bengt Jo¨nsson and Håkan Wennerstro¨m for valuable discussions. This study was supported by CAP (Center for Amphiphilic Polymers), Lund University. LA971165X
