Langmuir 1998, 14, 4081-4087
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Enthalpy Measurements in Aqueous SDS/DTAB Solutions Using Isothermal Titration Microcalorimetry Robert J. Meagher and T. Alan Hatton Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Arijit Bose* Department of Chemical Engineering, University of Rhode Island, Kingston, Rhode Island 02881 Received August 21, 1997. In Final Form: April 27, 1998 Isothermal titration microcalorimetry has been used to measure enthalpies of aggregate formation and binding, at 25 °C, for aqueous solutions of the single surfactants sodium dodecyl sulfate (SDS) and dodecyltrimethylammoniumbromide (DTAB) and mixtures of these at 1 wt % total surfactant concentration, over accessible composition ranges. The enthalpy of formation for SDS micelles is 0.6 kJ g-1 mol-1, and that for DTAB micelles is -2.1 kJ g-1 mol-1. This calorimetric experiment also produces a direct measurement of the critical micelle concentration (cmc) for each surfactant. The enthalpy of formation for 5:95 SDS/DTAB mixed micelles is -5.5 kJ g-1 mol-1 total surfactant and -6.7 kJ g-1 mol-1 total surfactant for 85:15 SDS/DTAB mixed micelles. Vesicles at an SDS/DTAB ratio of 62:38 have an enthalpy of formation of -13.4 kJ g-1 mol-1 total surfactant, indicating a strong enthalpic contribution to their free energy of formation. The enthalpies of aggregate formation are in good agreement with predictions from a molecular-thermodynamic model. Within the context of regular solution theory (RST), the interaction energy parameter is -51.7 kJ g-1 mol-1, again in excellent agreement with the prediction from the molecularthermodynamic model. Monomer solutions of each surfactant show a measurable enthalpy of dilution with a positive deviation from ideality. Using RST, the activity coefficients for SDS and DTAB monomers in water are 1.035 and 1.025, respectively. The enthalpy of dilution of concentrated micellar solutions (∼5 cmc) of each surfactant up to its cmc is zero, implying that these solutions behave “ideally” over this concentration range. The enthalpy of binding of DTAB monomers to SDS micelles is -28.4 kJ g-1 mol-1, to 20:80 SDS/DTAB mixed micelles is -36.4 kJ g-1 mol-1, and to 62:38 SDS/DTAB vesicles is -28.6 kJ g-1 mol-1. The enthalpy of binding of SDS monomers to DTAB micelles is -24.7 kJ g-1 mol-1.
Introduction The amphiphilic character of surfactant molecules causes them to self-assemble into a variety of aggregate structures when added to water. These range from spherical and rodlike micelles, through a range of lamellar phases to vesicles.1 The relationships between surfactant architecture, chemical/physical characteristics, and equilibrium aggregate structures as well as properties of the aggregate suspensions have been investigated thoroughly for single surfactant systems.2 It is now possible to predict a multitude of thermodynamic properties from a knowledge of the molecular structure of the constituent molecules and their concentration, as well as from knowledge of the temperature and salt concentration in the solution.3 The addition of a second amphiphilic component creates mixed surfactant systemssthese offer the possibility of a much wider range of aggregate microstructures than those exhibited by single surfactant solutions. These variations are the result of alterations in intermolecular and interaggregate forces in ways that are not possible in single surfactant solutions. By a suitable choice of surfactants, interactions between the constituent molecules can be deliberately tuned to impact the geometry of the resulting * Corresponding author. Telephone: (401)874-2804. Fax: (401)874-4689. E-mail:
[email protected]. (1) Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1982. (2) Rosen, M. J. Surfactants and Interfacial Phenomena; J. Wiley and Sons: New York, 1989. (3) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1991, 92, 3710.
complexessthis has a direct effect on the resulting aggregate structure. Furthermore, in mixed systems, the additional compositional degree of freedom plays a key role in determining phase behavior, opening up a rich range of microstructures that are difficult to find in single surfactant systems. These include the spontaneous formation of vesicles in mixtures of single-tailed cationic and anionic surfactants4-6 and a vesicle phase in equilibrium with the bicontinuous L3 phase7 in these catanionic suspensions. The covalent zwitterionic surfactant that is nearly identical to the cat-anionic complex does not display a vesicle phase,8 showing that composition variation between the inner and outer leaflets is crucial for the formation of the stable vesicle structure.9 Intralayer composition variations also have consequences for the size distribution and mechanical properties of mixed micellar systems, with implications for their stability and their ability to solubilize solutes.10 If the chain lengths (4) Kaler, E. W.; Murthy, A. K.; Rodriguez, B.; Zasadzinski, J. Science 1989, 245, 1371. (5) Kaler, E. W.; Herrington, K.; Murthy, A. K.; Zasadzinski, J. J. Phys. Chem. 1992, 96, 6698. (6) Herrington, K.; Kaler, E. W.; Miller, D. D.; Chirovulu, S.; Zasadzinski, J. J. Phys. Chem. 1993, 97, 13792. (7) Cates, M. E.; Roux, D.; Andelman, D.; Milner, S. T.; Safran, S. A. Europhys. Lett. 1988, 5, 733. (8) Fukuda, H.; Kawata, K.; Okuda, H.; Regan, S. J. Am. Chem. Soc. 1990, 112, 1635. (9) Safran, S. A.; Pincus, P.; Andelman, D. Science 1990, 248, 354. (10) Nagarajan, R.; Ruckenstein, E. In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; 1982; 923. Leodidis, E. B.; Bommarius, A. S.; Hatton, T. A. J. Phys. Chem. 1991, 95, 5943.
S0743-7463(97)00941-4 CCC: $15.00 © 1998 American Chemical Society Published on Web 07/01/1998
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for the two surfactants differ significantly, protrusions of the heads or tails roughen the aggregate surface and introduce strong short-range stabilizing forces.11 Mixed surfactant vesicles and mixed micelles can therefore be remarkably robust, retaining their stability even in the presence of reasonably high concentrations of divalent salts.12 Although all of the observed phases form spontaneously, their stability from a thermodynamic perspective has not yet been quantified experimentally. An important contribution to the free energy of formation of aggregates is their formation enthalpies. This is linked directly to intermolecular interactions and is most directly accessible to tuning by changing the nature of the surfactants. In this paper, we report measurements of the enthalpies of formation, at 25 °C, of single component and mixed surfactant aggregates formed in aqueous solutions of sodium dodecyl sulfate (SDS) and dodecyl trimethylammonium bromide (DTAB). These measurements are compared to recent predictions from a molecularthermodynamic model which combines a molecular model of micelle formation with a thermodynamic free energy description of micellar solution behavior.13 The model is focused on developing free energies of aggregate formation. Therefore, comparisons to experimentally determined enthalpies over a wide composition range represent an extremely stringent test of the robustness of the theory since all of the contributions to the free energy of micelle formation over a wide composition range as well as their temperature sensitivity must then be described accurately. We use regular solution theory to estimate the interaction energy between the anionic and cationic surfactants and the activity coefficients for monomers of each surfactant in aqueous solution. This interaction energy is compared to the interaction parameter arising from fitting the predicted13 free energies of mixed micelle formation. We measure enthalpies of binding of DTAB monomers to SDS micelles, SDS/DTAB mixed micelles, and SDS/DTAB vesicles, as well as those of SDS monomers to DTAB micelles. Experimental Methods Materials. SDS was obtained from International Biotechnologies (molecular biology grade, 99.9%) and was used without further purification. DTAB was purchased from Tokyo Kasei at 99% purity. Recrystallization from acetone14 showed no significant increase in purity (determined from measurement of the critical micelle concentration (cmc) and demicellization enthalpy using titration microcalorimetry as well as the absence of a surface tension minimum around the cmc), and the DTAB was subsequently used as purchased. Distilled water, passed through a Milli-Q+ water system, was used to make all of the solutions. The SDS/DTAB/water ternary phase diagram6 in the dilute surfactant region (up to 1 wt % total surfactant) is shown in Figure 1. Calorimetry. Experiments were performed in a ThermoMetric isothermal titration microcalorimeter operating at 25.00 ( 0.01 °C. The calorimeter was controlled through an interface to a personal computer using ThermoMetric Digitam software. Experiments began with 3.00 g of water or aqueous surfactant solution in the titration cell, which was equipped with a specially (11) Israelachvili, J. N.; Wennestro¨m, H. Langmuir 1990, 6, 873. (12) Yaacob, I.; Nunes, A. C.; Bose, A. J. Colloid Interface Sci. 1995, 171, 73. (13) Shiloach, A.; Blankschtein, D. Langmuir 1998, 14, 1618. The general techniques used to calculate free energies of aggregate formation in mixed surfactant solutions are detailed in Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 5567 and 5579; additional references are Zoeller, N.; Blankschtein, D. Ind. Eng. Chem. Res. 1995, 34, 4150 and Zoeller, N.; Shiloach, A.; Blankschtein, D. CHEMTECH 1996, 24. (14) Deardon, L. V.; Wooley, E. M. J. Phys. Chem. 1987, 91, 2404.
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Figure 1. Ternary phase diagram for SDS/DTAB/water at 25 °C (adapted from ref 6). The unshaded regions contain isotropic single phases (micelles, mixed micelles, and vesicles), the shaded regions consist of two or more phases. Enthalpy of binding and enthalpy of formation of mixed surfactant aggregates are measured by choosing surfactant concentrations in the syringe and titration cell along the lower edge (1 wt % total surfactant) of this phase diagram. designed Teflon stirrer rotating at 30 rpm. The stirrer maximizes mixing while minimizing heat generation by viscous dissipation. Null experiments, in which the solution contained in the titration cell was also injected through the syringe, were conducted at different stirring speeds to confirm that viscous heating effects were negligible in all of our experiments. A calibration was performed before each experiment to determine the dynamic thermal characteristics of the filled cell.15 A total of 200 µL of surfactant solution was added in increments of 10 or 20 µL using a 250-µL Lund syringe pump, with 20 min between injections. The calorimeter measured the difference in heat flux into or out of the titration cell with respect to a reference cell containing 3.00 g of water, and the instantaneous power (in microwatts) was recorded by the computer. The instrument accurately measures heat flows within (0.1 µW. To speed the recovery time between injections, the data were corrected dynamically using time constants calculated by the computer during the calibration.15,16 The heat flow data were integrated with respect to time to give the total heat evolved for each injection in kJ g-1 mol-1 of total surfactant added. Three sets of experiments were performed. To determine the enthalpy of micelle formation for the pure surfactants, we titrated concentrated (8 wt %) micellar solutions (cmcDTAB ) 0.46 wt %, cmcSDS ) 0.23 wt %) of the single surfactants into water. In the case of DTAB, a series of two experiments were required to completely surpass the cmc-titration of 8 wt % DTAB solution into water, followed by titration of 8 wt % DTAB solution into 0.25 wt % DTAB solution. These experiments follow well-known techniques for determining enthalpies of micelle formation.17 We determined the enthalpy of binding of DTAB monomers to aggregates in the SDS-rich portion of the phase diagram at a total surfactant concentration of 1 wt %. These experiments involved titrating a 1 wt % micellar solution of DTAB into a range of aggregates, including a SDS micellar solution, mixed micelles with a weight ratio of 75:25 SDS/DTAB, and a vesicle suspension with a ratio of 62:38 SDS/DTAB. A 1 wt % SDS micellar solution was titrated into a 1 wt % DTAB solution to determine the enthalpy of binding of SDS monomers to DTAB micelles. (15) Randzio, S. L.; Suukursk, J. In Biological Microcalorimetry; Beezer, A. E., Ed.; Academic Press: New York, 1980; p 311. (16) Backman, P.; Bastos, M.; Hallen, D.; Lonnbro, P.; Wadso, I. J. Biochem. Biophys. Methods 1994, 28, 85. (17) van Os, N. M.; Daane, G. J.; Haandrikman, G. J. Colloid Interface Sci. 1991, 141, 199.
Enthalpy Measurements in Aqueous SDS/DTAB Solutions The third set of experiments involved titration of mixed aggregates at 1 wt % total surfactant into 1 wt % micellar SDS solution. The titrated solutions contained surfactants in weight ratios of 5:95 SDS/DTAB (DTAB-rich mixed micelles), 85:15 SDS/ DTAB (SDS-rich mixed micelles), and 62:38 SDS/DTAB (vesicles). The data from these experiments were used with the enthalpy of binding determined earlier to find the enthalpy of formation of the mixed aggregates. To arrive at the enthalpy of formation of the vesicles by an alternate path (enthalpy changes must be path-insensitive), this vesicle solution was also titrated into 1 wt % DTAB micelles. Note that all mixture compositions used in the experiments lay along the lower horizontal edge (1 wt % total surfactant) of the phase diagram shown in Figure 1. Molecular-Thermodynamic Model.13 The molecularthermodynamic model,13 to which our experimental results have been compared, combines a molecular model of micelle formation with a thermodynamic free energy description of micellar solution behavior. The molecular model of micellization accounts explicitly for the effects of surfactant molecular structure, mixture composition, and solution conditions on the physical driving forces which control micelle formation and growth. The thermodynamic free energy description incorporates the salient features of the micellar solution, including a distribution of micellar aggregates in chemical equilibrium with each other and with the monomers. To describe the micelle/monomer chemical equilibrium, the free energy of micellization (gmic) is evaluated. Specifically, gmic represents the free energy change per molecule associated with transferring surfactants from the aqueous solvent to a micelle and is a function of the micelle shape, size, and composition. The magnitude of gmic can be evaluated using a thought process that describes micelle formation from individual surfactant monomers as a series of reversible steps, each of which is associated with a physicochemical contribution to the micellization process. These contributions include (i) hydrophobic interactions between surfactant hydrocarbon chains and water, (ii) interfacial effects associated with the creation of the micellar core-water interface, (iii) conformational effects associated with hydrocarbon chain packing in the micellar core, (iv) repulsive steric interactions between the surfactant heads, (v) electrostatic interactions between charged surfactant heads, and (vi) entropic effects associated with mixing the two surfactants in the mixed micelle. From the variation of the free energy of micellization with temperature, the enthalpy of micellization can be derived. In addition, the molecular interaction energy between the oppositely charged surfactants can be obtained by fitting the predicted gmic over the entire composition range to the regular solution model. We compare the predicted enthalpies of mixed micelle formation over a wide composition range and the predicted interaction energy to our experimentally obtained values.
Results and Discussion The most direct method for measuring the enthalpy of aggregate formation (∆Haggregate formation) is to titrate the concentrated aggregate into a predetermined amount of solution such that the surfactant composition in the titration cell initially remains below the critical aggregate concentration (cac) and is above the cac by the end of the experiment (for single surfactant systems, we replace the term cac with the cmc). The enthalpy change for each injection is then plotted versus surfactant concentration in the cell. Below the cac, the final solution in the sample cell contains monomers. The observed enthalpy change in this region is due to dilution of the aggregates to the cac, their breakup to monomers at the cac, and dilution of the resultant monomers. Aggregate dilution is the key contributor to observed enthalpy changes when the surfactant concentration in the titration cell goes beyond the cac. If there is a difference between the monomer concentration in the aggregate solution being titrated and that in the aggregate solution in the titration cell, an additional contribution from monomer dilution arisess this dilution enthalpy goes to zero at the cac. In addition, we ignore monomer concentration changes beyond the
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Figure 2. Power versus time enthalpogram for demicellization of DTAB. Each peak corresponds to a dynamically corrected signal of the difference in power flowing between the titration and reference cells upon injection of 10 µL of 8 wt % DTAB into a titration cell initially containing 3 g of water.
Figure 3. Enthalpy change versus DTAB concentration. The difference in enthalpy between the two sections of the curve at the cmc represents the enthalpy of demicellization. The nonzero slope of the curve below the cmc is indicative of nonidealities in the monomer solution. Note that a single such experiment produces both the enthalpy of micelle formation and the cmc.
cmc,18 since this effect is small and because the enthalpy change/injection when the surfactant concentration in the ampule is beyond the cmc is negligible. This results in separate curves above and below the cac, with a sharp transition between the two at the cac (for ideal solutions, these two regions are flat since the enthalpy of dilution is zero). ∆Haggregate formation is then determined by the enthalpy difference between the two curves at the cac. This method was used to determine ∆Hmicelle formation for SDS and DTAB. We find ∆Hmicelle formation at 25 °C to be 0.6 kJ g-1 mol-1 (0.242 kT/molecule) for SDS and -2.1 kJ g-1 mol-1 for DTAB (-0.848 kT/molecule). These are in agreement with the results of other researchers.19,20 A representative experimental power versus time enthalpogram obtained for DTAB is shown in Figure 2, while Figure 3 shows the measured enthalpy change for each injection versus the concentration of surfactant in the titration cell. This calorimetric technique simultaneously reveals the cmc for DTAB ) 15.1 mM (0.47 wt %), in close (18) Johnson, I.; Olofsson, G.; Jonsson, B. J. Chem. Soc., Faraday Trans. 1 1987, 83, 3331. (19) Bashford, M. T.; Wooley, E. M. J. Phys. Chem. 1985, 89, 3173. (20) Clint, J. H.; Walker, T. J. Chem. Soc., Faraday Trans. 1975, 71, 946.
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agreement with the number obtained from surface tension measurements.21 Activity Coefficients of SDS and DTAB Monomers. The plot of ∆H/injection versus concentration for both surfactants shows a positive slope below the cmc. This indicates that there is measurable heat of monomer dilution below the cmc, and hence, the solution is nonideal. As a first step toward quantifying this nonideality, we invoke regular solution theory22 and then estimate activity coefficients for the dilute surfactants. For a nonideal process, the excess Gibbs free energy, GE, is given by
GE ) HE - TSE
(1)
Regular solution theory holds that the excess entropy of mixing is equal to 0; therefore, the excess Gibbs energy equals the excess enthalpy of mixing. The reference state excess enthalpy is that at infinite monomer dilution. For a species i in solution, the activity coefficient γi is defined in terms of its excess Gibbs free energy as
ln γi )
[
]
∂(nGE/RT) P, T, nj ∂ni
(2)
For our dilute surfactant solutions, we found that nHE/ RT and, thus, nGE/RT increased almost linearly with surfactant concentration or moles of surfactant. The slope of this line can be measured to give the value of ln γi. For SDS and DTAB monomers in water, we report γSDS ) 1.035 and γDTAB ) 1.025. Within the context of the regular solution approximation, these dilute monomers show a very small positive deviation from ideality. Titration microcalorimetry, therefore, represents a very sensitive tool for probing nonidealities of mixing.18,23-25 In contrast to the monomer solution, the micellar solutions for each surfactant show no excess enthalpy of mixing at surfactant concentrations up to ∼5 cmc, indicating that these solutions behave “ideally”. Micelle-micelle interactions are significantly lower than monomer-monomer interactions because of the lower concentration of micelles in solution. In addition, exposure of hydrophobic tails to water in monomer solutions contributes to nonidealities that would not be evident in micellar systems. The contribution from the enthalpy of monomer dilution in micellar solutions is negligible because the difference in monomer concentration between different micellar solutions is small. Measurement of ∆Hbinding of DTAB Monomers to Aggregates. The enthalpies of binding monomers of DTAB to SDS micelles and to SDS-rich mixed micelles and vesicles were measured by titrating a 1 wt % DTAB micellar solution into a range of aggregates at a total surfactant concentration of 1 wt %. The resulting changes can be broken down conceptually into four sequential stepssdilution of the titrated micelles to the cmc, breakup of these micelles to monomers at the cmc, dilution of the monomers from cmc to infinite dilution, followed by binding of the infinitely dilute monomers to the existing aggregates in the titration cell. An additional contribution to this measured enthalpy change arises because of the (21) Lucassen-Reynders, E. H.; Lucassen, J.; Giles, D. J. Colloid Interface Sci. 1981, 81, 150. (22) Rubingh, D. N. In Solution Chemistry of Surfactants; Mittal, K. L., Ed.; Plenum Press: New York, 1979; Vol. 1, p 337. (23) Birch, B. J.; Hall, D. G. J. Chem. Soc., Faraday Trans. 1 1972, 68, 2350. (24) Archer, D. G.; Albert, H. J.; White, D. E.; Wood, R. H. J. Colloid Interface Sci. 1984, 100, 68. (25) Kay, R. L.; Lee, K.-S. J. Phys. Chem. 1986, 90, 5266.
Figure 4. Illustration of the measurement of the enthalpy of binding of DTAB monomers to aggregates in solution. (×) represents the sum of the enthalpy of micelle dilution, demicellization, and monomer dilution. (O) represents the sum of the enthalpy of micelle dilution, demicellization, monomer dilution, monomer binding to the aggregates in the titration cell, and aggregation of monomers in the titration cell because of the difference in cac between single component and mixed micelles. The measured enthalpy change shows no significant variation between the first and second injections when titrating into an SDS micellar solution, indicating that electrostatic contributions from binding of surfactants to oppositely charged micelles dominate over aggregation effects produced by a change in the cac. The abscissa is the surfactant concentration in the titration cell. The difference between the two curves, therefore, represents the enthalpy of binding of DTAB monomers to aggregates in the solution.
variation of the critical aggregation concentration with mixture composition. This effect is most significant when a micellar solution of a surfactant is first titrated into an ampule containing micelles of oppositely charged surfactant because of the potentially large difference between the cac of a mixed surfactant solution and the cmc for a single surfactant. Our measured enthalpy of binding therefore includes this contribution. The overall (measured) enthalpy change for each injection can then be described as follows
∆H ) ∆Hmicelle dilution(fcmc) + ∆Hdemicellization(@cmc) + ∆Hmonomer dilution(cmcf0) + ∆Hbinding (3) The first three terms on the right-hand side, ∆Hmicelle dilution + ∆Hdemicellization + ∆Hmonomer dilution, are available directly from the experiments in which micellar solutions of DTAB are titrated into water. The fourth term, ∆Hbinding, is thus equal to the difference between the total measured enthalpy change and the enthalpy change for DTAB dilution at the same concentration. This is illustrated in Figure 4. In our experiments, we do not notice a measurable difference in the heat evolved between the first and second injections when titrating a surfactant into a solution containing oppositely charged micelles, indicating that the electrostatic contribution from binding of ionic surfactants to oppositely charged aggregates dominates over enthalpy changes produced by the variation in the cac. Therefore, we retain the nomenclature, ∆Hbinding, for reporting this measurement, recognizing the possibility that this number could contain additional, potentially significant contributions in other systems. The enthalpy of binding DTAB monomers to SDS-rich aggregates is large and exothermic. It is most exothermic for mixed micelles over a ratio of SDS/DTAB ranging from
Enthalpy Measurements in Aqueous SDS/DTAB Solutions
85:15 to 80:20, equal to -36.4 kJ g-1 mol-1 DTAB (-14.7 kT/molecule). The enthalpy of binding to pure SDS micelles is -28.4 kJ g-1 mol-1 DTAB (-11.5 kT/molecule). This is approximately the same as the enthalpy of binding of DTAB monomers to vesicles (SDS/DTAB 62:38) at -28.6 kJ g-1 mol-1 DTAB (-11.6 kT/molecule). Binding enthalpies are affected by the potential at the surface of these aggregates, which is related in a complex way to the ratio of the two surfactants, the counterion concentration, and the morphology. An analogous experiment was performed to determine the enthalpy of binding of SDS monomers to DTAB micelles. We obtain ∆Hbinding ) -24.7 kJ g-1 mol-1 SDS (-9.98 kT/molecule), somewhat smaller than that of binding DTAB to any of the SDS-rich aggregates. Enthalpy of Formation of Mixed Aggregates. For SDS/DTAB mixed micelles and vesicles, it is not possible to measure ∆Haggregate formation directly by titrating the concentrated aggregates into water. At very low concentrations, the DTA+ and DS- ions form a low solubility complex that precipitates. Upon addition of mixed SDS/ DTAB aggregates to water, the total enthalpy change then contains an unknown contribution from the enthalpy of precipitation of the complex. In addition, the instantaneous power versus time curves display large fluctuations, making their integration to obtain enthalpies very irreproducible. We thus look for a different way to obtain the enthalpy of formation of aggregates. We require isotropic single phases during each stage of the titration, and we thus choose a path over which there is no precipitate formed, and where all of the relevant enthalpies are measurable. To obtain the enthalpy of formation for the mixed surfactant aggregates, we titrate these at 1 wt % into 1 wt % SDS-containing micelles, to form mixed micelles, and measure the total enthalpy change associated with this process. Conceptually, the path followed by the aggregates in these experiments can be broken down into four stages. First, the aggregates are diluted from 1 wt % to a cac. At the cac, the aggregates break up, resulting in SDS and DTAB monomers. The monomers are then diluted from the cac to an infinitely dilute state. Finally, the infinitely dilute monomers bind to SDS micelles in solution, resulting in mixed micelles. The total observed enthalpy change is thus composed of four parts
∆H ) ∆Haggregate dilution(1 wt %fcac) + ∆Haggregate breakup(@cac) + ∆Hmonomer dilution(cacf0) + ∆Hbinding (4) In the SDS/DTAB system, the presence of the intervening solid phase between the dilute monomer phase and the mixed aggregate phase makes calorimetric determination of the cac impossible. Thus, it is not possible to measure ∆Haggregate dilution all the way to the cac, and therefore, it is not feasible to calculate ∆Haggregate breakup at the cac. It is, however, possible to calculate ∆Haggregate breakup at 1 wt % total surfactant. In this case, we conceptualize a slightly different path for the aggregates. First, the aggregates are assumed to break up at 1 wt %, leaving SDS and DTAB monomers at 1 wt % total, in the same ratio as was present in the aggregate. Next, these monomers are diluted from 1 wt % to an infinitely dilute state. Finally, the monomers bind to the existing SDS micelles in solution. The total enthalpy change can be represented as
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∆H ) ∆Haggregate breakup(@1 wt %) + ∆Hmonomer dilution(1 wt %f0) + ∆Hbinding (5) As discussed in the description of the enthalpy of binding measurements, the last term in eq 5 contains contributions, if any, from the conversion of monomers in the ampule to mixed micelle aggregates because of the difference between the cac of the mixed system and that of single surfactants. Note that
∆Haggregate breakup(@1 wt %) ) ∆Haggregate dilution(1 wt %fcac) + ∆Haggregate breakup(@cac) + ∆Hmonomer dilution(cacf1 wt %) (6) ∆Hbinding of DTAB monomers to SDS micelles was determined to be -28.4 kJ g-1 mol-1 DTAB. This must be multiplied by the mole fraction of DTAB (moles of DTAB per mole of surfactant) in the aggregate to give one of the contributions to the ∆Hbinding in eq 5. The remaining contribution comes from binding SDS monomers to SDS micelles and is the product of the mole fraction of SDS in the aggregate and the enthalpy of micellization of SDS. In the absence of more refined knowledge of the exact surfactant composition in the aggregates, we assume, for the purpose of this calculation, that it is equal to the overall surfactant composition. Because DTAB and SDS have equal tail lengths, their solubilities can be expected to be similar, and thus, their composition within the aggregate should not differ significantly from the overall composition. ∆Hmonomer dilution was taken to be equal to the enthalpy of dilution of each monomer separately from its concentration in the aggregate to an infinitely dilute state. The enthalpies of dilution of the individual surfactant monomers come from experiments where the single surfactants were titrated into water to concentrations below the cmc. The calculated activity coefficients could be used to determine ∆Hmonomer dilution; alternatively, this number can be determined directly from the data. We note again that for both surfactants, the partial excess enthalpy below the cmc is a nearly linear function of concentration. We thus interpolate (or extrapolate) to determine the partial excess enthalpy of each surfactant at the concentration present in the aggregate. These are used to calculate ∆Hmonomer dilution
h EDTAB + xSDSH h ESDS (7) ∆Hmonomer dilution ) xDTABH where the partial excess enthalpy of each surfactant is evaluated at its concentration in the aggregate, and the mole fractions xSDS and xDTAB are the moles of SDS or DTAB per total moles of surfactant. Note that ∆Haggregate formation ) -∆Hbreakup. These enthalpies of formation have monomers at 1 wt % total surfactant as their reference state. We titrated DTAB-rich mixed micelles (5:95 SDS/ DTAB), vesicles (62:38 SDS/DTAB), and SDS-rich mixed micelles (85:15 SDS/DTAB) into the 1 wt % SDS solution. The enthalpy of formation for the DTAB-rich mixed micelles was found to be -5.49 kJ g-1 mol-1 surfactant (-2.22 kT/molecule), more exothermic than the enthalpy of formation of pure DTAB micelles. The enthalpy of formation of the SDS-rich mixed micelles was -6.23 kJ g-1 mol-1 surfactant (-2.52 kT/molecule), also more exothermic than that of formation of pure SDS micelles. The vesicles showed a much more exothermic enthalpy of formation of -12.98 kJ g-1 mol-1 (-5.24 kT/molecule). Enthalpies of aggregate formation versus composition are plotted in Figure 5.
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Figure 5. Enthalpy of formation (0) of a variety of aggregates formed in single surfactant solutions and mixtures of SDS/ DTAB at 1 wt % total surfactant. More than one phase forms over the concentrations spanned by the shaded regions, and therefore, aggregate formation enthalpies cannot be measured in those ranges. The errors associated with the enthalpy measurements are within the symbol heights. The solid curve is the prediction from regular solution theory, with an interaction energy W ) -54.6 kJ g-1 mol-1 (-22.1 kT/molecule).
The enthalpy of aggregate formation can be interpreted as the excess enthalpy of mixing when monomers at a specified ratio and overall surfactant concentration are mixed to form the aggregates. Invoking the regular solution approximation, the enthalpy of formation is then given by
∆Haggregate formation ) xSDSxDTABW + xSDS∆HSDS micelle formation + xDTAB∆HDTAB micelle formation (8) where xSDS and xDTAB are the mole fractions of SDS and DTAB (per total moles surfactant), respectively, and W is an energy interaction parameter. Fitting the data in Figure 5 to eq 8, after conversion of mass fractions to mole fraction units, we get W ) -54.6 kJ g-1 mol-1 (-22.1 kT/ molecule). The prediction from the RST model over the entire composition range is shown in Figure 5. Such large interaction energies are symptomatic of strong electrostatic interactions between oppositely charged surfactants in the mixed aggregates. Fitting the predicted free energies of micelle formation from the molecularthermodynamic model13 to eq 8 (under the regular solution approximation, excess free energies can be replaced by excess enthalpies), the interaction energy W ) -57.1 kJ g-1 mol-1 or -23.1 kT/molecule, in remarkably good agreement with the experiments. ∆Haggregate formation should be independent of the path over which it is measured. Thus, an analogous experiment in which the mixed aggregate is titrated into a 1 wt % DTAB solution containing micelles should produce the same value. The experimentally measured enthalpy change is still given by eq 5, but ∆Hbinding now refers to binding of SDS monomers to DTAB micelles and DTAB monomers to DTAB micelles. This experiment was performed by titrating 62:38 SDS/DTAB vesicles into 1 wt % DTAB, and the corresponding enthalpy of vesicle formation was determined to be -14.3 kJ g-1 mol-1 surfactant (-5.78 kT/molecule). This represents a difference of about 9% from the enthalpy of formation determined by titrating vesicles into 1 wt % SDS, and confirms the viability of
Meagher et al.
Figure 6. Comparison of experimentally determined and calculated13 enthalpies of formation of aggregates formed in single surfactant solutions and mixtures of SDS/DTAB at 1 wt % total surfactant. The calculation for the 62:38 SDS/DTAB composition was done, assuming mixed micelles in solution while the experimental data are for vesicles.
each titration path for producing formation enthalpies of these aggregates. The free energies of micelle formation, gmic, of mixed surfactant aggregates over a range of temperatures that span 25 °C are now available.13 Using the GibbsHelmholtz relationship
∆Hmic ) {∂(gmic/T)/∂(1/T)}P
(9)
Enthalpies of formation, calculated from the predictions of gmic from the molecular-thermodynamic model and eq 9, are compared to our experimental values in Figure 6. To replicate the experiments, this comparison is made for a specified overall surfactant composition and composition ratio. The agreement between the experiments and model predictions are good over a wide range of surfactant compositions. The calculations at the SDS/DTAB ratio of 62:38 are for mixed micelles, while the experimental mixture at that composition contains vesicles. Because the two surfactants have similar chain lengths, there should only be a small difference in surfactant composition between the inner and outer vesicle leaflets. Therefore, the predicted enthalpy of formation of mixed micelles should not be very different from that of vesicles. Comparisons of experimentally measured enthalpies of formation to calculations from models that are focused on developing Gibbs free energies represent an extremely stringent test for model predictions because precise computations of the free energy of aggregate formation, as well as its variation with temperature, are required in order for the derivative in eq 9 to be obtained accurately. The quality of the agreement points directly to the robustness of the theory, while simultaneously validating the experimental method. Conclusions Isothermal titration microcalorimetry represents a powerful technique for obtaining enthalpies of aggregate formation and binding in surfactant colloids. By a suitable choice of titration paths such that isotropic single phases
Enthalpy Measurements in Aqueous SDS/DTAB Solutions
are maintained throughout an experiment, enthalpies of aggregate formation are measured, even in mixed surfactant systems that form a precipitate at low amphiphile concentrations. In the SDS/DTAB/water system, electrostatic interaction between oppositely charged surfactants represents a dominant contribution to measured enthalpies and observed nonidealities. The experimental measurements of the interaction energy parameter contributing to nonidealities in aqueous mixtures of the oppositely charged surfactants and enthalpies of aggregate formation over a wide composition range agree remarkably well with predictions from a molecular-thermodynamic model.13
Langmuir, Vol. 14, No. 15, 1998 4087
Acknowledgment. We thank the MIT UROP Office for financial support for R.J.M. and the NSF for partial support of this research. We are grateful to N. Zoeller, A. Shiloach, and D. Blankschtein for providing us with a succinct description of their molecular-thermodynamic model and predictions from the model prior to their publication, as well as providing important insights. A.B. thanks T.A.H. for his hospitality and financial support during a sabbatical leave at MIT.
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