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Formation of Mixed Micelles in Salt-Free Aqueous Solutions of Sodium Dodecyl Sulfate and C12E6 Vasil M. Garamus† GKSS Research Centre, Max-Planck Strasse, 21502 Geesthacht, Germany Received March 20, 2003. In Final Form: June 4, 2003 Mixtures of the ionic surfactant sodium dodecyl sulfate (SDS) and nonionic surfactant dodecyl hexa(ethylene oxide) (C12E6) in heavy water were studied by small-angle neutron scattering. The ratio of SDS to C12E6 was varied under the constant total concentration of 25 mM and temperature of 15 °C. The formation of mixed micelles was observed. The structural and electrostatic properties of the mixed micelles change monotonically with varying surfactant ratio. Micelles become smaller and the degree of dissociation of SDS molecules in the micelles decreases when SDS is added. The observed behavior was compared with the results from light and neutron scattering studies of SDS/C12E6 in an aqueous saline solution (0.1 M NaCl, 25 °C) and with the predictions of molecular-thermodynamic theory.
Introduction Mixtures of different surfactants in water or oil are the objects of intensive experimental and theoretical studies.1 More than 60 recent significant contributions, using new experimental techniques and theoretical models, are cited.1 For industrial application, all commercial products are mixtures of surfactants whose properties depend strongly on the delicate balance between the surfactants’ ratio and concentration and the ionic strength, pH, and temperature of the solutions.2 Modern theories try to extend the classical regular solution treatment (RST)3 where only one adjustable parameter is needed to predict the properties of mixtures. Recent investigations, however, have shown serious limitations of RST4,5 in the case of strong interactive surfactants. The errors of prediction of the critical micelle concentration (cmc) and micelle compositions in these studies could be about 100%. In this work, the molecular-thermodynamic (MT) theory developed by Puvvada and Blankschtein is used.6 The MT approach allows various energetic contributions of a micellar solution as a function of the micellar size and shape to be calculated. Also, important characteristics of micellar solutions, such as the cmc, aggregation numbers, and phase separation, can be predicted. The main advantage of the MT approach is the absence of adjustable parameters. Attempts to compare the experimental data, the cmc by a conductivity technique7 and an enthalpy of micellization,8 with the RST and MT approaches were performed, displaying a similar order of agreement for both models. † Phone: +49 4152 871290. Fax: +49 4152 871356. E-mail:
[email protected].
(1) Hines, J. D. Curr. Opinion Colloid Interface Sci. 2001, 6, 350356. (2) Rosen, M. J. In Phenomena in Mixed Surfactant Systems; Scamehorn, J. F., Ed.; ACS Symposium Series 311; American Chemical Society: Washington, DC, 1986; p 144-160. (3) Rubingh, D. N. In Solution Chemistry of Surfactants, Mittal, K., Ed.; Plenum: New York, 1979; p 337-362. (4) Eads, C. D.; Robosky, L. C. Langmuir 1999, 15, 2661-2668. (5) Huang, L.; Somasundaran, P. Langmuir 1997, 13, 6683-6688. (6) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 55795592. (7) Lopez-Fontan, J. L.; Suarez, M. J.; Mosquera, V.; Sarmiento, F. Phys. Chem. Chem. Phys. 1999, 15, 3583-3587.
For further development of the MT approach, it is necessary to “visualize” the different energetic contributions of mixed micelle formation. The contributions of electrostatic and excluded volume interactions among polar groups in mixed micelles of ionic/nonionic surfactants were studied and compared with the MT predictions.9 In the study, good agreement between the behavior of two mixtures, sodium dodecyl sulfate/dodecyl hexa(ethylene oxide) (SDS/C12E6) and sodium dodecyl hexa(ethylene oxide)/dodecyl hexa(ethylene oxide) (SDE6S/C12E6) in 0.1 M NaCl solutions with the predictions of the MT theory is reported. The delicate balance between the changes in the electrostatic and excluded volume interactions gives an increase in the aggregation number of the mixed SDS/ C12E6 micelle at low contents of SDS. This phenomenon is not observed in SDE6S/C12E6 mixtures; there, the excluded volume interaction with varying surfactant ratio is constant. Qualitatively, the effect is explained by the different cross-sectional area of the polar groups of SDS (∼25 Å2) and C12E6 (∼40 Å2) molecules. It is necessary to clarify the role of the electrolyte (screening of the electrostatic interaction) in the abovementioned investigations. The use of 0.1 M NaCl is thought to decrease the intermicellar interaction9 influencing the static light scattering (SLS) data. The aim of the present work is to determine the behavior of a salt-free SDS/C12E6 mixture. We use small-angle neutron scattering (SANS). This technique has shown a significant contribution in studies of strong interactive micelles in solutions.10 Mixed micelles, containing C12E6 or SDS as one of the components, have been widely studied. The aggregation numbers of mixed SDS/C12E8 micelles vary monotonically with the composition from the value of the aggregation number of pure C12E8 to that of pure SDS.11 Addition of 0.1 M NaCl to a system of SDS/dodecylmalono-bis-Nmethylglucamide has no effect on the cmc, micelle size, and shape.12 The formation of rodlike micelles in mixed (8) Meagher, R. J.; Hatton, T. A.; Bose, A. Langmuir 1998, 14, 40814087. (9) Shiloach, A.; Blankschtein, D. Langmuir 1998, 14, 7166-7182. (10) Chen, S. H. Annu. Rev. Phys. Chem. 1986, 37, 351-370. (11) Alargova, R. G.; Kochijashky, I. I.; Sierra, M. L.; Kwetkat, K.; Zana, R. J. Colloid Interface Sci. 2001, 235, 119-129. (12) Griffiths, P. C.; Whatton, M. L.; Abbott, R. J.; Kwan, W.; Pitt, A. R.; Howe, A. M.; King, S. M.; Heenan, R. K. J. Colloid Interface Sci. 1999, 215, 114-123.
10.1021/la034481m CCC: $25.00 © 2003 American Chemical Society Published on Web 08/09/2003
Mixed Micelles in Salt-Free Solutions
solutions of hexadecyltrimethylammonium bromide and C12E6 after the addition of salt is observed.13 Tablet-shaped and ribbonlike structures have been discovered in SDS/ dodecyltrimethylammonium bromide.14 These few examples show a variation of the experiments and results that are at times contradictory. The SDS/C12E6 mixtures were previously studied by several experimental methods and analyzed by various theoretical models. The surface tension15,16 was used to measure the mixture cmc and micelle composition of mixed micelles in aqueous solutions. The RST describes the experimental surface-tension measurements well.3 But titration calorimetry, which was also used to determine the enthalpy of forming mixed micelles, indicates that the RST cannot follow the mixed micelle formation.15 SDS/ C12E6 mixtures were compared with dodecyltrimethylammonium chloride (DTAC)/C12E6 using electron spinecho modulation measurements.17 DTAC headgroups are located deeper inside the mixed micelle than SDS headgroups, perhaps reflecting specific interactions.17 The very small amounts of SDS/C12E6 solutions caused significant increases in the cloud-point temperatures as a result of electrostatic intermicellar interactions.18 Structural studies (aggregation number, size, and composition) of mixed micelles in SDS/C12E6 in 0.1 M NaCl solutions by SANS and neutron reflectivity19 and by SLS20 were performed. The experiments were limited to a few compositions with high contents of SDS (>50%). The characteristic radius of formed aggregates was obtained as 24-25 Å, which is equal to the length of the C12E6 molecule in the extended conformation. The results were discussed in the frame of RST. In ref 9, the same value of the minor radius of prolate micelle aggregates SDS/ C12E6 is reported. After publication of the last MT developments,9 the systematic SANS studies of the structure of mixed micelles in SDS/C12E6 in 0.1 M NaCl heavy water solutions were performed by Penfold and co-workers.21,22 For solution compositions with low fractions of SDS,22 the variation of the micelle aggregation number with the composition exhibited a pronounced maximum, in qualitative agreement with the SLS data and MT predictions.9 The reasons of the significant difference between the absolute values of the aggregation numbers and the position of the maximum obtained by SLS9 and SANS22 are not clear and require the examination of the experimental conditions and the data-analysis procedures. As one can see, the all-structural studies of SDS/C12E6 solutions were performed in the presence of supporting electrolyte. Electrolyte decreases the intermicellar interaction and the “nonideality” of the SDS/C12E6 mixtures, which allows the analysis of the experimental data in (13) McDermott, D. C.; Lu, J. R.; Lee, E. M.; Thomas, R. K.; Rennie, A. R. Langmuir 1992, 8, 1204-1210. (14) Bergstro¨m, M.; Pedersen, J. S. Phys. Chem. Chem. Phys. 1999, 1, 4437-4446. (15) Fo¨rster, Th.; von Rybinski, W.; Schwuger, M. J. Tenside, Surfactants, Deterg. 1990, 27, 254-260. (16) Pegiadou, S.; Eleftheridias, I. Tenside, Surfactants, Deterg. 2001, 38, 234-237. (17) Baglioni, P.; Del, L. G.; Rivaraminten, E.; Kevan, L. J. Am. Chem. Soc. 1993, 115, 4286-4290. (18) De Salvo Souza, L.; Corti, M.; Cantu, L.; Degiorgio, V. Chem. Phys. Lett. 1986, 131, 160-166. (19) Penfold, J.; Staples, E.; Thompson, L.; Tucker, I.; Hines, J.; Thomas, R. K.; Lu, J. R. Langmuir 1995, 11, 2496-2503. (20) Nishikido, N. J. Colloid Interface Sci. 1987, 120, 495-501. (21) Penfold, J.; Staples, E.; Thompson, L.; Tucker, I.; Hines, J. D.; Thomas, R. K.; Lu, J. R.; Warren, N. J. Phys. Chem. B 1999, 103, 52045211. (22) Penfold, J.; Staples, E.; Tucker, I. J. Phys. Chem. B 2002, 106, 8891-8897.
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simpler way. From another point of view, the addition of electrolyte makes the system more complicated (from a three-component system to a four-component one) and probably shifts the delicate balance of electrostatic and steric interactions between the polar groups of SDS and C12E6. It is known that in solutions of SDS and β-dodecyl maltoside, the shift of the position of the maximum of the aggregation number versus the surfactants’ ratio was observed when supporting electrolyte had been added.23 In the present paper, we will show the import role of the electrostatic screening in the formation of mixed micelles in SDS/C12E6 mixtures at low fractions of SDS. It will be clear that the maximum of the aggregationnumber dependence disappears and the MT predictions can follow these observations.9 Experimental Section The nonionic surfactant C12E6 (lot 0040) was obtained from Nikko Chemicals, Japan. The ionic surfactant SDS (lot 79H0114) was obtained from Sigma, Germany. Heavy water (99.8%) was obtained from Merck, Germany. The surfactants have dodecyl hydrocarbon tails so that interactions within the mixed micelles are only due to the differences in the structures of the surfactant headgroups. SANS experiments were performed at the SANS1 instrument at the FRG1 research reactor at GKSS Research Centre, Geesthacht, Germany.24 The neutron wavelength was 8.5 Å. The range of scattering vectors (0.01 < q < 0.25 Å-1, q ) 4π sin(θ)/λ, where 2θ is the scattering angle and λ is the neutron wavelength) was obtained using three sample-to-detector distances (0.7-5 m). The wavelength resolution was 10% (full width at halfmaximum). The samples were kept at 15.0 ( 0.5 °C in quartz cuvettes with path lengths of 1 and 2 mm. According to a phase diagram of SDS, the Kraft point should not be higher than 10 °C.25 The raw spectra were corrected for backgrounds from the solvent, sample cell, and other sources by conventional procedures.26 The two-dimensional isotropic scattering spectra were azimuthally averaged, converted to an absolute scale, and corrected for detector efficiency by dividing by the incoherent scattering spectra of pure water,26 which was measured with a 1-mm-path-length quartz cell. The smearing induced by the different instrumental setups is included in the data analysis. For each instrumental setting, the ideal model scattering curves were smeared by the appropriate resolution function when the model scattering intensity was compared to the measured one by means of least-squares methods.27 The parameters in the models were optimized by conventional least-squares analysis, and the errors of the parameters were calculated by conventional methods.27,28
Results and Discussion The total concentration of the surfactants was kept constant at 25 mM (∼1 wt %). This concentration is much higher than the cmc value of C12E6 (0.07 mM) and wellabove the cmc value of SDS (8 mM).9 Measurements at higher concentrations are not reasonable because micelles of pure nonionic surfactant grow as the concentration increases.29 Micelle growth leads to the formation of (23) Bucci, S.; Fagotti, C.; Degiorgio, V.; Piazza, R. Langmuir 1991, 7, 824-826. (24) Shuhrmann, H. B.; Burkhardt, N.; Dietrich, G.; Ju¨nemann, R.; Meerwinck, W.; Schmitt, M.; Wadzack, J.; Willumeit, R.; Zhao, J.; Nierhaus, K. H. Nucl. Instrum. Methods 1995, A356, 133-137. (25) Laughlin, R. G. The Aqueous Phase Behaviour of Surfactants; Academic Press: London, 1994; p 111. (26) Wignall, G. D.; Bates, F. S. J. Appl. Crystallogr. 1986, 20, 2838. (27) Pedersen, J. S.; Posselt, D.; Mortensen, K. J. Appl. Crystallogr. 1990, 23, 321-328. (28) Pedersen, J. S. Adv. Colloid Interface Sci. 1997, 70, 171-193. (29) Kato, T.; Kanada, M.; Semiya, T. Langmuir 1995, 11, 18671872.
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interaction among particles S(q):
dΣ(q)/dΩ ) n[〈|P(q)|〉2 S(q) + 〈|P(q)|2〉 - 〈|P(q)|〉2]
Figure 1. SANS data and model fits of a 25 mM mixture of SDS and C12E6 in heavy water at 15 °C varying the fraction of ionic surfactant.
nonspherical (rodlike) aggregates, which makes the analysis more complicated. The composition of mixture R is described in the terms of the molar ratio of SDS to total surfactant (SDS + C12E6) concentration. It was observed in ref 9 that the cmc of mixtures at R < 0.6 is quite low with respect to the cmc value of pure C12E6. Hence, at the studied compositions of mixtures (R between 0 and 0.5), the total concentration of the surfactants is more than 100 times higher than the cmc values, and the composition of micelles is the same as the average solution composition. The solution of pure SDS was also measured. The temperature of the solution was 15 °C, which is 10 °C lower than those in refs 9 and 22. This was done to be well within the region of spherical C12E6 micelle formation. C12Ej micelles show a sphere-to-rod transition with increasing temperature.30,31 The sphere-to-rod transition temperature increases in accordance with the number of ethylene oxide units per surfactant molecule. For C12E6 it is 15 °C with about 10 °C of variation.32 An example of the SANS patterns obtained in this work is shown in Figure 1. The addition of ionic surfactant depresses the scattering intensities at low q, and the broad interference maximum at intermediate q appears. For the C12E6/heavy water solution, scattering from the population of noninteractive particles is observed. The scattering intensities change monotonically; for example, the increasing fraction of ionic surfactant gives a more pronounced maximum and the decreasing of scattering intensity values. We did not observe any anomalous increase in the scattering at low fractions of SDS (R ) 0-0.2), as it was observed for the solution of SDS/C12E6 in 0.1 M NaCl.9,22 The experimental data were analyzed via the fitting of scattering intensities by the model of two-shell ellipsoids of rotation interacting by excluded volume interaction (R ) 0) and by screened Coulomb potential (R > 0). This approach was successfully applied for many micellar solutions.10,33 In the case of slightly polydisperse or nonspherical particles, scattering intensities dΣ(q)/dΩ can be written as a function of scattering from a single particle P(q) and (30) Jo¨nsson, B.; Lindman, B.; Holmberg, K.; Kronberg, B. Surfactants and Polymers in Aqueous Solution; John Wiley & Sons: New York, 1998; p 95-100. (31) Maccarini, M.; Briganti, G. J. Phys. Chem. A 2000, 104, 1145111458. (32) Corti, M.; Degiorgio, V. J. Phys. Chem. 1981, 85, 1442-1449. (33) Chevalier, Y.; Zemb, Th. Rep. Prog. Phys. 1990, 53, 279-371.
(1)
The decoupling approximation,34,35 that there is no correlation between interparticle separation and particle size and there is no correlation in the separation between particles and their orientation, could be used to calculate the second term of eq 1. For the calculations of scattering from a single micelle [P(q), formfactor], we assume that the micelles are monodisperse, core-and-shell ellipsoids of rotation of volume V2 with the semiaxes a, b, b (a/b ) γ). The volume of the core, which consists of hydrocarbon chains, is V1, and its scattering length density is F1. The volume of the shell, which contains the polar headgroups, is V2 - V1, and its scattering length density is F2. Then, the single particle scattering function is given by28
P(q) )
∫01[V1(F1 - Fs)F(q, R1) +
V2(F2 - Fs)F(q, R2)] sin β dβ (2)
where Fs is the scattering length density of the solvent and
F(q, R) ) 3(sin X - X cos X)/X3
(3)
X ) qR(sin2 β + γ2 cos2 β)1/2
(4)
with
The mean volume of the core, V1, can be calculated from molecular group volumes according to
V1 ) Na[ν(CH3) + (n - 1)ν(CH2)]
(5)
where Na is the mean aggregation number of the micelle (sum of SDS and C12E6). The volume of the shell, V2 - V1, is given by
V2 - V1 ) Na{Rν(SO4-) + (1 - R)ν(E6) + (1 - κ)ν(Na+) + ν(D2O)[R$HG,SDS + (1 - R)$HG,C12E6 + R(1 - κ)$Na]} (6) where n ) 12 is the number of carbon atoms in the hydrocarbon chains of the surfactant molecule; ν(CH2), ν(CH3), ν(SO4-), and ν(E6) are the volumes of methylene groups, methyl groups, SDS headgroups, and C12E6 headgroups, respectively; ν(Na+) and ν(D2O) are the volumes of sodium ions and solvent molecules bound to the surfactant; $HG,SD, $HG,C12E6, and $Na are the hydration numbers of the headgroups of SDS, C12E6, and the sodium ion, respectively; and κ is the degree of dissociation of SDS molecules in the micelle. The numerical values for the volumes and hydration numbers were taken from refs 9 and 36. The interaction among micelles S(q), the structure factor of the solution, is described in the form of the hard sphere interaction for the pure C12E6 solution. The S(q) values have been calculated with the Percus-Yevick approximation for closure relation between direct and total correla(34) Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys. 1983, 79, 24612467. (35) Hayter, J. B.; Penfold, J. Colloid Polym. Sci. 1983, 261, 1022. (36) Berr, S. S.; Coleman, M. J.; Jones, R. R. M.; Johnson, J. S. J. Phys. Chem. 1986, 90, 6492-6497.
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Table 1. Values of Fit Parametersa R
Na
1 64 ( 1 0.5 75 ( 1 0.4 82 ( 1 0.3 84 ( 1 0.25 87 ( 1 0.2 92 ( 1 0.15 97 ( 1 0.1 107 ( 1 0.05 114 ( 1 0 122 ( 1
κ
γ
Q, e
a, Å
b, Å
1/λD, Å-1
0.28 ( 0.01 0.40 ( 0.02 0.46 ( 0.05 0.59 ( 0.06 0.68 ( 0.07 0.72 ( 0.07 0.73 ( 0.07 0.82 ( 0.08 0.78 ( 0.08 0
1.0 ( 0.3 1.4 ( 0.1 1.45 ( 0.1 1.5 ( 0.1 1.5 ( 0.1 1.6 ( 0.1 1.6 ( 0.1 1.6 ( 0.1 1.7 ( 0.1 1.9 ( 0.1
18 15 15 15 15 13 11 9 4 0
25.2 33.6 35.3 36.4 36.8 39.1 40.2 41.8 42.8 46.6
25.2 24.0 24.4 24.3 24.5 24.5 24.8 25.6 25.2 24.5
0.034 0.016 0.0166 0.0161 0.0156 0.0145 0.0132 0.0111 0.0079 ∼0.0001
a N , aggregation number; γ, axis ratio; and κ, degree of a dissociation of SDS molecules in micelles. Calculated parameters of micelles and solution: Q, electrical charge; a, larger semiaxis; b, smaller semiaxis; and inverse Debye-Hu¨ckel screening length 1/λD versus the mole fraction of SDS in 25 mM SDS/C12E6/heavy water at 15° C. Other parameters of the micelle aggregates: the thickness of the shell is in the interval of 7-8 Å; in D2O the scattering contrast of the core is equal to -6.7 × 1010 cm-2 and that of the shell is equal to -1.2 × 1010 cm-2.
tion functions.37,38 SHS(q) is the function of the volume fraction of micelles η and the hard sphere radius RHS. RHS was calculated as an average for a micelle of volume V2. In the case of mixtures containing SDS, the micelles are charged aggregates, and the structure factor was derived in a rescaled mean spherical approximation,39,40 using the Debye-Hu¨ckel theory to calculate the repulsive potential between two macroions surrounded by a diffuse double layer of counterions. S(q) is the function of the volume fraction of micelles η and degree of dissociation of surfactant molecules in micelle κ. In model fits, four (pure C12E6 solution) or five parameters (SDS containing solutions) were used (aggregation number Na, axes ratio γ, degree of dissociation of the SDS molecule, the correction of absolute unit normalization, and residual incoherent background). The variations of the last two parameters were small: less the 10% for absolute unit normalization and less than 0.02 cm-1 for residual incoherent background. The models describe scattering data satisfactorily (Figure 1), and the obtained fitting parameters are presented in Table 1. These values support our observations of monotonic changes of the micelle structure in saltfree solutions of SDS and C12E6. The aggregation number decreases with the addition of SDS from ∼120 to ∼60. No maximum is observed in mixtures with a low content of SDS that points to a principal difference between the saline solution (0.1 M NaCl) of SDS/C12E69,22 and the salt-free solution studied here. The obtained aggregation number (∼120) for pure C12E6 is significantly lower than the one (∼250) reported in ref 9, probably owing to the temperature effect (10° difference). It should be mentioned that the high value of the aggregation number was unable to be calculated9 by the MT theory without changing some input parameters. When the cross-sectional area of C12E6 was changed from 42.3 to 41.7 Å2, the theoretical aggregation number increased from 150 to 250. At a high content of SDS, however, the difference in the values of the aggregation numbers between salt-free and saline solutions (0.1 M NaCl) is minor. The minor axis of the micelle aggregate is practically constant (24-25 Å), which agrees with previous studies9,19,22 and corresponds to the sum of the length of a fully (37) Kinning, D. J.; Thomas, E. L. Macromolecules 1984, 17, 17121718. (38) Ashcroft, N. W.; Lekner, J. Phys. Rev. 1966, 145, 83. (39) Hayter, J. B.; Penfold, J. Mol. Phys. 1981, 42, 409-414. (40) Hansen, J. P.; Hayter, J. B. Mol. Phys. 1982, 46, 651-656.
extended C12 hydrocarbon tail and globular head with six ethylene oxide groups.9 The anisotropy of the aggregates decreases with the increasing of the SDS content from slightly prolate objects (∼2) to spheres (or even oblate objects) for pure SDS micelles, which sounds quite reasonable. Obtained parameters of micelles in binary solutions of SDS/water should be compared with the conclusions of ref 14, where intensive SANS studies were performed to get the shape of the SDS micelles. In the experimental conditions, there is just one difference, that is, T ) 40 °C14 whereas T ) 15 °C is the present studies. The decreasing of the temperature (25°) gives the slightly increasing aggregation number of micelles from 60 to 64 that is reasonable for charged micelles formed by ionic surfactants. The degree of ionization of SDS molecules is the same within experimental errors, 0.26-0.28. The main difference in the obtained parameters is that the oblate form (axis ratio ∼0.6) of the SDS micelles is reported in ref 14. In the present studies, the data in the smaller interval of scattering vectors (qmax ) 0.25 Å-1 in the present studies and qmax ) 0.5 Å-1 in ref 14) are obtained, which is why the errors of estimation in parameters such as the axis ratio are quite high for data on the SDS solution presented here. Nevertheless, one can conclude that present studies are in qualitative and quantitative agreements with the investigation of pure SDS solutions.14 Electrical properties of micelles (κ, the degree of dissociation of SDS molecules, and Q, the total electrical charge) show opposite tendencies. κ decreases with increasing SDS content, but Q increases with SDS addition. Its complex behavior is subject to the electrolyte strength of the solution (Debye-Hu¨ckel screening length) and indirectly to the size of the micelles. The next consideration is the comparison of experimentally obtained data (aggregation numbers) for saltfree mixture of SDS/C12E6 with the predictions of the MT theory for mixtures of ionic/nonionic surfactant.9,41 Detailed quantitative comparison demands significant computation effort. Here, we just show the qualitative comparison between the experimentally obtained dependence of aggregation numbers and the theoretical prediction of the shape of the free-energy function of micelle formation. It was shown in refs 9 and 41 that in the case of the SDS/C12E6 mixture the main contributions, which change the free-energy function of micelle formation, are the steric interaction among polar groups (gst) of surfactant molecules and electrostatic free energy (gel). From numerical calculations performed at T ) 25 °C,9,41 one can see that gst is the linear function of the micelle composition gst ≈ 1.2 - 0.6R (in units of kT, and R is fraction of SDS). In the gel case, one can use the calculations performed for 1/λD ) 0.01 Å-1 (salt-free solution) and 1/λD ) 0.05 Å-1 (saline solution of 0.1 M NaCl, ref 9), and a good approximation is a quadratic function gel ≈ 1.2R + 4.1R2 in units of kT for the salt-free solution and gel ≈ 0.2R + 2.8R2 for the saline solution of 0.1 M NaCl. Behavior of the SDS/C12E6 mixture should depend on the shape of the total free-energy function of micelle formation gmic (sum of gel and gst). And it is easy to see (Figure 2) that for the saline solution, gmic has a minimum at R ≈ 0.10, and for a salt-free solution, the minimum of gmic is shifted to R ≈ -0.02 (nonphysical solution). A physically reasonable minimum at gmic should correspond to the maximum of (41) Shiloach, A.; Blankschtein, D. Langmuir 1998, 14, 1618-1638.
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a ) a0[1 - H(T - 298)]
(7)
where a0 is the average cross-sectional area at 298 K and H is the dehydration temperature coefficient reflecting the decrease in hydration (per ethylene oxide group) with increasing temperature, and T is the absolute temperature in K. The value of the dehydration coefficient of ∼0.002 K-1 is calculated theoretically42 and is supported by experimental results.43 Taking into account this value, one can get the increasing cross section of a polar group of the C12E6 molecule from 41.7 to 42.6 Å2 when the temperature varies from 25 to 15 °C. This change in the size of the polar group of C12E6 gives a slightly different dependence of the steric energy of mixed micelle SDS/C12E6 formation versus the composition R Figure 2. Schematic representation of the free energy of micelle formation gmic (sum of gel and gst) for the salt-free solution (filled symbols) and 0.1 M NaCl (empty symbols) versus the R fraction of SDS at 25 °C (triangles) and at 15 °C (squares). Only a part of the low-R values is presented to better view the minimum.
the aggregation number, which was observed in ref 9 at R ≈ 0.05-0.10 and in ref 22 at R ≈ 0.20. Absence of a minimum should correspond to the monotonic changes of the aggregation number versus the mixture’s composition, which is reported in the present paper. The comparison between the present results and the results of other group’s experimental and theoretical calculations9,22 should be corrected for the difference in temperature (10 °C) in the experimental conditions. We have decreased the temperature and have changed the values of the contributions into the free energy of the micelle formation. The 10 °C difference should have a minor effect on electrostatic contribution gel, which is supported by the comparison with the results of the pure SDS solution at 40 °C (there the difference is 25 °C). We have found only a small decreasing of the aggregation number at higher T, and the change of the degree of dissociation of the SDS molecule is within experimental errors. It is necessary to discuss and to estimate the changes of the steric interaction gst between polar groups of C12E6 molecules. The change in the size of the polar group versus the temperature is a well-known effect of the dehydration of the nonionic surfactants. According to Blankschtein,42 the good approximation of the cross-sectional area a of the polar group of nonionic surfactant is
gst ≈ 1.2 - 0.6R
at 25 °C
gst ≈ 1.3 - 0.65R
at 15 °C
It is clear that steric interaction between the polar group of C12E6 increases when the temperature of the solution is lowered to 10 °C, but even taking into account these corrections the total free energy of micelle formation gmic does change the main feature of behavior; that is, there is minimum for the saline solution and there is no minimum for the salt-free solution (Figure 2). It demonstrates that the comparison of the present studies (15 °C) with the results of light and neutron scattering data9,22 measured at 25 °C is reasonable and can be done. In summary, it should be concluded that the MT theory can qualitatively describe (predict) the behavior of SDS/C12E6/ D2O mixtures. Conclusion The main results of the present work are a comparison of the behavior of mixed micelles formed by ionic and nonionic surfactants in salt-free and saline (0.1 M NaCl, refs 9 and 22) solutions. It shows one more time that the formation of mixed micelles is subject to the delicate balance of different contributions. The final sign of interaction among surfactants (same alkyl chains) strongly depends not only on the size and charge of polar groups but also on other solution characteristics (temperature, pH, added salt, etc.). In the present case, this is the salinity of the solution. In the absence of salt, the electrostatic interaction dominates over the steric interaction, giving a decrease in the aggregation number over the whole range of the surfactant ratio. The MT theory describes the overall behavior of mixed micelle formation in salt-free solutions. LA034481M
(42) Sarmoria, C.; Puvvada, S.; Blankschtein, D. Langmuir 1992, 8, 2690-2712.
(8)
(43) Garamus, V. M. Chem. Phys. Lett. 1998, 290, 251-254.
