Mixed Micellar Aggregates of Anionic and Nonionic Surfactants with

7130

Langmuir 1998, 14, 7130-7139

Mixed Micellar Aggregates of Anionic and Nonionic Surfactants with Short Hydrophobic Tails. A PGSE-NMR Study Donato Ciccarelli, Lucia Costantino, Gerardino D’Errico, Luigi Paduano, and Vincenzo Vitagliano* Dipartimento di Chimica, Universita` di Napoli, Federico II, via Mezzocannone 4, 80134 Napoli, Italy Received April 21, 1998. In Final Form: September 25, 1998 Intradiffusion coefficients for the ternary system heavy water-sodium hexanesulfonate (C6SNa)pentaethylene glycol monohexyl ether (C6E5), in both the premicellar and micellar composition ranges, were measured by the PGSE-NMR technique at 25 °C. Experimental data show the formation of mixed micelles and allow the determination of the cmc. In the premicellar composition range the monomermonomer interactions are analyzed. The compositions of the aqueous and micellar pseudophase have been computed in the whole micellar composition range; they indicate that C6E5 has a larger tendency to form aggregates than C6SNa. The aggregation number and the free energy of micellization are calculated and interpreted in terms of interactions among the tensides in the micellar aggregates. The experimental results have been interpreted in terms of a regular solution model, and the interaction parameter β is computed. β increases by increasing the number of micellar aggregates in the system; this effect can be interpreted in terms of intermicellar interactions.

I. Introduction Mixtures of surfactants are important because of their wide use in industry; in fact in practical fields mixed surfactants work better than a single surfactant. Furthermore, the composition and concentration can be optimized for each particular application. Since different types of surfactants exist, various kinds of combinations are possible, with different properties and application fields. The physicochemical properties and functions of surfactant mixtures in solution are essentially guided by the headgroups’ nature. Even though various studies have been conducted on these systems, with the aim of relating their properties to those of the corresponding binary mixtures water-surfactant, theories and models, summarized in a recent publication,1 are not yet adequate. This is due to the difficulty of finding experimental methods suitable for these kinds of studies and at the same time to elaborate theoretical treatments representative of the process. Nonionic surfactants of the poly(ethylene oxide) class are known to form mixed micelles with ionic surfactants in aqueous solution.2 In the present study, the ternary system pentaethylene glycol monohexyl ether-sodium hexyl sulfonate-water is considered. These two surfactants have a short hydrophobic chain so that the critical micelle concentration (cmc) values are high enough to allow an experimental study in the premicellar composition range. The binary solutions of these two surfactants were widely studied in our laboratory.3-7 As will be shown in this paper, the knowledge of the physicochemical proper* Corresponding author. Fax: +39081 5527771. E-mail: VITA@ chemna.dichi.unina.it. (1) Ogino, K., Abe, M., Eds. Mixed Surfactant Systems; Marcel Dekker: New York, 1993. (2) Tokiwa, F.; Moriyama, N. J. Colloid Interface Sci. 1969, 30, 338. (3) Ambrosone, L.; Costantino; L.; D’Errico, G.; Vitagliano, V. J. Solution Chem. 1996, 25, 755. (4) Ambrosone, L.; Costantino, L.; D’Errico, G.; Vitagliano, V. J. Solution Chem. 1996, 26, 737.

ties of the micellar systems formed by each surfactant is of considerable help in studying the mixed system. Surfactants with short hydrophobic tails form aggregates with loose structure (hydrotropes) and less extensive association behavior;8,9 they provide an option for fundamentally important analysis of the mechanism of surfactancy. In our previous papers5,6 we showed that, for these systems, the experimental data can be interpreted in terms of the models used for the micellar systems. In this paper we will show that these aggregates are also able to solubilize hydrophobic molecules, just as common micellar systems are. In the present work evidences of the formation of mixed micelles are reported, and the compositions of the aqueous and micellar pseudophases are determined using the intradiffusion coefficients measured by the pulsed gradient spin-echo (PGSE)-NMR method. The mixed micellization process can be described either as a phase separation or as a chemical equilibrium. Both models are good enough to allow reasonable insight into the behavior of surfactant solutions through the micellization process,10 although more sophisticated models are described in the literature.11,12 The choice of the model is mainly related to the discussion of experimental results. In this work, using the phase separation model, the micelle intradiffusion coefficient has been related to the hydrodynamic size of the aggregates, and the aggregation (5) Ambrosone, L.; Costantino, L.; D’Errico, G.; Vitagliano, V. J. Colloid Interface Sci. 1997, 189, 286. (6) Paduano, L.; Sartorio, R.; Vitagliano, V.; Costantino, L. J. Colloid Interface Sci. 1997, 189, 189. (7) Ortona, O.; Costantino, L.; Paduano, L.; Vitagliano, V. J. Colloid Interface Sci. 1998, 203, 477. (8) Jo¨nsson, B.; Edholm, O.; Teleman, O. J. Chem. Phys. 1986, 85, 2259. (9) Laaksonen, L.; Rosenholm, J. B. Chem. Phys. Lett. 1993, 216, 429. (10) Zana, R. Surfactants Solutions, New Methods of Investigation; Marcel Dekker: New York, 1987. (11) Mukerjee, P. J. Phys. Chem. 1972, 76, 565. (12) Desnoyers, J. E.; De Lisi, R.; Roberts, D.; Roux, A.; Perron, G. J. Phys. Chem. 1983, 87, 1397.

10.1021/la9804583 CCC: $15.00 © 1998 American Chemical Society Published on Web 11/07/1998

Mixed Micellar Aggregates

Langmuir, Vol. 14, No. 25, 1998 7131

number of the mixed micelles has been evaluated. All these data are compared with those of the binary micellar systems and analyzed in terms of the regular solution theory. The analysis of the mutual diffusion coefficients of the same surfactant mixtures has also evidenced the formation of mixed micelles.13 Nilsson et al.,14,15 with different aims, studied a similar system with the same technique, interpreting the results in terms of intermicellar interactions. They considered surfactants with long hydrophobic chains, in the composition range in which monomers can be completely neglected. In the last part of this paper, we analyze how the pseudophase composition is affected by the number of micellar aggregates. II. Experimental Section A. Material. Surfactants were reagent grade and were used without further purification: Pentaethylene glycol monohexyl ether (C6E5) was a Bachem product with a declared purity >99%. Sodium hexanesulfonate (C6SNa) was a Sigma product with a declared purity ∼98%. The solvent was D2O obtained from Sigma (>99.96% isotopic purity). All solutions were prepared by weight. B. Apparatus and Methods. The intradiffusion coefficients were obtained by using the FT-PGSE-NMR technique.16,17 A spectrometer operating in the 1H mode at 80 MHz, and equipped with a pulsed magnetic field gradient unit, made by Stelar (Mede, Italy), was employed. The temperature was controlled within 0.1 °C with a Stelar variable-temperature controller, model VTC87. The individual spin-echo peak amplitude A for a given line is given by the equation

[

(

A ) A0 exp -γ2g2Daξ2 ∆ -

ξ 3

)]

Figure 1. C6E5 molality dependence of the intradiffusion coefficient of C6SNa (b), C6E5 (O), and TMS (0) in the mixed -1 micellar system D2O-C6SNa-C6E5. (mC6SNa ) 0.08 mol kgD ). 2O

(1)

where A0 is a constant for a given set of experimental conditions, γ is the gyromagnetic ratio of the proton, Da is the intradiffusion coefficient of the species responsible for the NMR signal, g is the strength of the applied gradient, and ∆ and ξ are time parameters in the pulse sequence. The time between the 90° and 180° pulses ∆ was kept constant. The duration of the two gradient pulses ξ was varied over a suitable range to observe the decay of the spin-echo signal A. The above equation was fitted by a nonlinear least-squares routine to the decay of A as a function of ξ. To evaluate the values of the intradiffusion coefficients, g must be known. Measurements to establish its value were performed on a reference sample with a known intradiffusion coefficient; we used heavy water with trace amounts of light water (DHDO ) 1.872 × 10-9 m2 s-1).18 The experimental errors for the intradiffusion coefficients were generally less than 2%. Both the premicellar and micellar ranges were considered. Different sets of measurements were made at constant C6SNa molality, varying the C6E5 molality. Eleven C6SNa molalities were considered; five of them (0.04; 0.08; 0.15; 0.3; 0.4) include also samples where micelles are absent. As will be discussed later, in the micellar composition range solubilized tetramethylsilane (TMS, Sigma product, purity 99.9%) was used to measure the micelle intradiffusion coefficient. In the elaboration of the experimental data, the solution density and viscosity were required. In these cases we approximated them with those of the binary solutions C6SNa-D2O, considering that the C6SNa concentration is generally larger than that of C6E5 in our measurements. Density and viscosity data for the C6SNa-H2O system were given in a previous work.6 To consider the isotopic effect, these experimental data were multiplied by / / / / ratios. /FH or ηD /ηH the FD 2O 2O 2O 2O (13) Castaldi, M.; Costantino, L.; Ortona, O.; Paduano, L.; Vitagliano, V. Langmuir 1998, 14, 5994. (14) Nilsson, P. G.; Lindman, B. J. Phys. Chem. 1984, 88, 5391. (15) Gue´ring, P.; Nilsson, P. G.; Lindman, B. J. Colloid Interface Sci. 1985, 105, 41. (16) Stilbs, P. Prog. NMR Spectrosc.1987, 19, 1. (17) Tanner, J. E.; Stejskal, E. O. J. Chem. Phys. 1964, 42, 288. (18) Mills, R. J. Phys. Chem. 1973, 77, 685.

Figure 2. C6E5 molality dependence of the intradiffusion coefficient of C6SNa (b), C6E5 (O), and TMS (0) in the mixed -1 micellar system D2O-C6SNa-C6E5. (mC6SNa ) 0.7 mol kgD ). 2O The isotopic substitution of the solvent might result in an alteration of the structural properties of the micellar aggregates. In fact D2O is thought to be slightly more structured than H2O.19 Berr20 showed that these differences are very small and become appreciable only for surfactants with long hydrophobic chains. For this reason we neglected this effect.

III. Results and Discussion The PGSE-NMR method allows us to measure the intradiffusion coefficients of both the nonionic and anionic surfactants. For the nonionic surfactant the signal intensities for the protons of the ethoxilic group were followed (δ ) 3.6), while for the anionic surfactant that of the CH2 near to the sulfonic group (δ ) 2.9) was used. All the experimental results are summarized in Table 1; some of them are reported in Figures 1 and 2. The analysis of the intradiffusion coefficients trend offers a good instrument for an understanding of the association mechanism. In the five sets of measurements that start at compositions below the cmc the intradiffusion coefficients of both surfactants show a slope change where the molecules start to aggregate. This is evidence of their comicellization. As -1 , shown in Figure 1 for the set at mC6SNa ) 0.08 mol kgD 2O this effect allows us to determine the cmc of the mixture, defined as mC6E5,cmc + mC6SNa,cmc. The problems and errors (19) Nemethy, G.; Scheraga, H. A. J. Chem. Phys. 1964, 41, 680. (20) Berr, S. S. J. Phys. Chem. 1987, 91, 4760.

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Table 1. Intradiffusion Coefficients and Computed Molalities in D2O-C6SNa-C6E5 Mixtures at 298.15 K mC6E5 -1 (mol kgD ) 2O

109DC6SNa (m2 s-1)

0.0000 0.0219 0.0419 0.0591 0.0757 0.0842 0.1033 0.1211 0.1458 0.2023 0.2895 0.4166 0.5129 0.6098

0.524 0.526 0.525 0.523 0.525 0.520 0.510 0.501 0.460 0.398 0.300 0.179 0.133 0.110

0.0000 0.0226 0.0420 0.0641 0.0737 0.0834 0.1016 0.1207 0.1432 0.1923 0.2966 0.4072 0.4796 0.6282

0.515 0.511 0.508 0.503 0.503 0.499 0.486 0.483 0.424 0.385 0.283 0.159 0.119 0.0935

0.0000 0.0177 0.0293 0.0594 0.0763 0.0998 0.1029 0.1184 0.1405 0.1773 0.2632 0.4221 0.4721 0.5818

0.500 0.498 0.496 0.496 0.485 0.457 0.446 0.430 0.412 0.384 0.194 0.116 0.0884 0.0814

0.0000 0.0146 0.0199 0.0275 0.0382 0.0500 0.0658 0.0950 0.0998 0.1216 0.1280 0.1591 0.1986 0.2843 0.3525 0.4682 0.5870

0.5000 0.4988 0.4970 0.4954 0.4906 0.4714 0.4108 0.3437 0.3292 0.2955 0.2832 0.2491 0.2160 0.1415 0.1131 0.1090 0.1005

0.0000 0.0096 0.0118 0.0146 0.0199 0.0274 0.0382 0.0500 0.0658 0.0950 0.0998 0.1216 0.1280 0.1590 0.1986 0.2843 0.3525 0.4682 0.5870

0.500 0.501 0.498 0.496 0.488 0.470 0.396 0.380 0.358 0.330 0.329 0.307 0.300 0.276 0.240 0.183 0.136 0.114 0.108

109DC6E5 (m2 s-1)

109DM (m2 s-1)

-1 mC6SNa ) 0.04 mol kgD 2O

0.365 0.364 0.365 0.362 0.358 0.343 0.319 0.276 0.223 0.166 0.132 0.110 0.0974

0.0928 0.0832 0.0679 0.0513 0.0449 0.0359

mCM6SNa -1 (mol kgD ) 2O

mCM6E5 -1 (mol kgD ) 2O

0.00481 0.0103 0.0193 0.0297 0.0315 0.0337

0.0438 0.0991 0.192 0.308 0.408 0.496

0.00357 0.00801 0.0152 0.0222 0.0445 0.0616 0.0644 0.0701

0.0144 0.0345 0.0644 0.101 0.214 0.312 0.374 0.528

0.00171 0.00586 0.0164 0.0202 0.0266 0.0330 0.0425 0.106 0.128 0.135 0.135

0.00334 0.0128 0.0371 0.0402 0.0584 0.0734 0.111 0.221 0.396 0.436 0.502

0.0711 0.121 0.130 0.159 0.168 0.188 0.209 0.254 0.267 0.264 0.267

0.0360 0.0678 0.0717 0.0918 0.101 0.132 0.171 0.259 0.323 0.423 0.541

0.119 0.130 0.156 0.182 0.181 0.198 0.206 0.224 0.256 0.294 0.333 0.349 0.347

0.0315 0.0405 0.0592 0.0856 0.0933 0.115 0.114 0.140 0.179 0.262 0.329 0.438 0.551


0.354 0.351 0.347 0.346 0.341 0.311 0.276 0.234 0.212 0.148 0.123 0.111 0.0885

0.103 0.0962 0.0927 0.0852 0.0692 0.0523 0.0422 0.0370


0.344 0.345 0.325 0.301 0.253 0.248 0.224 0.216 0.185 0.112 0.0690 0.0667 0.0789

0.115 0.112 0.105 0.103 0.102 0.0990 0.0894 0.0680 0.0501 0.0428 0.0353


0.315 0.309 0.304 0.295 0.278 0.221 0.182 0.179 0.166 0.161 0.144 0.127 0.100 0.0885 0.0838 0.0743

0.128 0.120 0.118 0.111 0.113 0.104 0.0923 0.0757 0.0649 0.0558 0.0512


0.295 0.292 0.290 0.264 0.211 0.180 0.179 0.155 0.151 0.138 0.128 0.138 0.131 0.117 0.0945 0.0826 0.0715 0.0678

0.145 0.141 0.140 0.129 0.123 0.116 0.111 0.101 0.0910 0.0718 0.0627 0.0515 0.0480


Langmuir, Vol. 14, No. 25, 1998 7133 Table 1. Continued

mC6E5 -1 (mol kgD ) 2O

109D

C6SNa

(m2 s-1)

0.0000 0.0287 0.0395 0.0596 0.0833 0.0976 0.1155 0.1186 0.1500 0.1959 0.2855 0.3791 0.4921 0.5637

0.470 0.400 0.375 0.353 0.333 0.320 0.305 0.300 0.276 0.235 0.184 0.137 0.117 0.102

0.0000 0.0173 0.0415 0.0551 0.0775 0.0958 0.1047 0.1195 0.1439 0.2057 0.2575 0.3710 0.4706 0.6057

0.370 0.357 0.331 0.324 0.306 0.290 0.282 0.270 0.244 0.202 0.163 0.124 0.111 0.110

0.0000 0.0204 0.0394 0.0580 0.0808 0.1063 0.1041 0.1106 0.1360 0.1970 0.3301 0.4160 0.5149 0.6081

0.249 0.242 0.231 0.218 0.211 0.206 0.206 0.204 0.196 0.172 0.120 0.106 0.0909 0.0754

0.0000 0.0184 0.0373 0.0595 0.0746 0.1001 0.1057 0.1185 0.1361 0.2040 0.2786 0.3818 0.4995 0.5967

0.185 0.180 0.175 0.170 0.165 0.158 0.156 0.154 0.151 0.132 0.112 0.100 0.0715 0.0673

0.0000 0.0257 0.0473 0.1050 0.1542 0.2005 0.2976 0.3824 0.4811 0.5933

0.128 0.113 0.108 0.102 0.0908 0.0830 0.0710 0.0607 0.0580 0.0535

0.0000 0.0199 0.0475 0.1003 0.1359 0.1936 0.2840 0.3948 0.5131 0.5590

0.0702 0.0696 0.0680 0.0649 0.0625 0.0590 0.0555 0.0512 0.0497 0.0497

109D

C6E5

(m2 s-1)

109DM (m2 s-1)


0.228 0.202 0.185 0.167 0.158 0.144 0.142 0.136 0.118 0.102 0.0918 0.0782 0.0681

0.191 0.181 0.162 0.139 0.125 0.106 0.108 0.0936 0.0757 0.0602 0.0551 0.0355 0.0321

mCM6SNa -1 (mol kgD ) 2O

mCM6E5 -1 (mol kgD ) 2O

0.158 0.201 0.220 0.237 0.245 0.260 0.251 0.277 0.309 0.359 0.407 0.412 0.425

0.0207 0.0336 0.0512 0.0713 0.0824 0.0994 0.0969 0.123 0.164 0.243 0.331 0.424 0.499

0.272 0.320 0.315 0.353 0.366 0.376 0.402 0.442 0.507 0.563 0.618 0.634 0.633

0.0133 0.0352 0.0476 0.0699 0.0865 0.0950 0.111 0.133 0.199 0.254 0.368 0.461 0.583

0.632 0.654 0.681 0.694 0.699 0.699 0.702 0.716 0.755 0.842 0.857 0.878 0.908

0.0189 0.0366 0.0541 0.0753 0.0986 0.0967 0.103 0.126 0.182 0.312 0.397 0.495 0.588

1.04 1.03 1.02 1.02 1.02 1.03 1.03 1.03 1.05 1.08 1.10 1.17 1.18

0.0186 0.0379 0.0589 0.0727 0.0962 0.101 0.113 0.129 0.192 0.263 0.366 0.482 0.581

1.50 1.53 1.53 1.57 1.59 1.63 1.65 1.64 1.68

0.0250 0.0463 0.102 0.150 0.194 0.288 0.370 0.465 0.573

2.11 2.12 2.13 2.14 2.15 2.16 2.16 2.16

0.0462 0.0981 0.133 0.190 0.278 0.385 0.497 0.540


0.173 0.155 0.149 0.136 0.131 0.129 0.120 0.116 0.0946 0.0859 0.0737 0.0708 0.0691

0.131 0.125 0.121 0.115 0.111 0.108 0.104 0.0981 0.0865 0.0825 0.0721 0.0653 0.0584


0.108 0.106 0.102 0.0994 0.0971 0.0972 0.0962 0.0940 0.0852 0.0639 0.0537 0.0462 0.0422

0.0921 0.0893 0.0865 0.0831 0.0794 0.0797 0.0787 0.0749 0.0658 0.0486 0.0402 0.0343 0.0323


0.0977 0.0871 0.0830 0.0806 0.0757 0.0751 0.0731 0.0705 0.0638 0.0517 0.0405 0.0359 0.0321

0.101 0.0908 0.0804 0.0743 0.0657 0.0642 0.0609 0.0573 0.0484 0.0351 0.0284 0.0254 0.0241


0.0524 0.0487 0.0456 0.0397 0.0374 0.0336 0.0313 0.0302 0.0295

0.0446 0.0424 0.0365 0.0314 0.0276 0.0234 0.0210 0.0196 0.0188


0.0365 0.0343 0.0326 0.0314 0.0307 0.0301 0.0300 0.0300

0.0289 0.0279 0.0272 0.0261 0.0245 0.0224 0.0202 0.0194

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Table 2. Critical Micellar Composition and Aggregation Number in D2O-C6SNa-C6E5 Mixtures at 298.15 K cmca -1 (mol kgD ) 2O

mC6SNa,cmc -1 (mol kgD ) 2O

mC6E5,cmc -1 (mol kgD ) 2O

XC6E5b

YC6E5c

s

0.56d 0.41 0.33 0.21 0.16 0.13 0.10e

0.56 0.40 0.30 0.15 0.08 0.04 0.00

0.000 0.010 0.027 0.059 0.081 0.089 0.100

0.000 0.024 0.082 0.28 0.49 0.69 1.00

0.000 0.15 0.33 0.68 0.80 0.90 1.00

10 13 16 22 25 25 25

a Critical micelle composition defined as m 25,28 b Mole fraction of C E in the solution bulk, as defined by eq 7. c Mole C6E5,cmc + mC6SNa,cmc. 6 5 fraction of C6E5 in the micellar pseudophase, as defined by eq 8. d Reference 34. This datum is in good agreement with those reported in standard literature (0.46/0.56 mol dm3, in light water35). e Reference 5. This datum is in good agreement with those reported in standard literature (0.0925/0.10 mol dm3, in light water35).

In the presence of micelles, the experimental data are mean values of the free and micellized molecules:

mCF 6SNa mCM6SNa F DC6SNa ) D + DM mC6SNa C6SNa mC6SNa DC6E5 )

Figure 3. C6E5 intradiffusion coefficient in D2O-C6SNa-C6E5 mixtures versus C6E5 molality at constant C6SNa molality. mC6SNa ) 0.00 (O); mC6SNa ) 0.04 (b); mC6SNa ) 0.08 (+); mC6SNa ) 0.15 (×); mC6SNa ) 0.3 (4); mC6SNa ) 0.4 ([). The inset shows the limiting DC∞6E5 values (9) versus the C6SNa molality.

connected with the cmc determination from intradiffusion measurements were analyzed in a previous work.5 In this work we use the C6E5 curves to measure the mC6E5 values at the cmc. This is because the C6E5 intradiffusion coefficient trend shows a much more marked slope change (Figure 1), as a consequence of its a higher tendency to micellize. The cmc values and the corresponding compositions of the ternary solutions are collected in Table 2. In all runs taken at compositions over the cmc, the sharp decreasing of the intradiffusion coefficients can be observed immediately upon adding C6E5 to the solution (Figure 2). In the premicellar composition range the intradiffusion coefficient of the anionic surfactant DC6SNa is quite constant within each set of measurements and does not change appreciably with the C6SNa concentration. These evidences indicate that the anionic surfactant is not influenced by the small amount of C6E5 present and that the C6SNa-C6SNa interactions are negligible. On the contrary the intradiffusion coefficient of the nonionic surfactant DC6E5 is a decreasing function of the C6E5 concentration and it is largely affected by the presence of C6SNa (Figure 3); the slope increase is much higher in -1 . As the systems with mC6SNa ) 0.15, 0.3, and 0.4 mol kgD 2O ∞ shown in the inset of Figure 3 the limiting DC6E5 values are linearly dependent on the C6SNa concentration. A little discontinuity can be noted in the extrapolated values -1 . These (Figure 3, inset) between 0.08 and 0.15 mol kgD 2O evidences could suggest the formation of premicellar aggregates, as has been often noted for surfactants with short hydrophobic tails.6 However, considering the experimental errors, we prefer not to argue about this point.

mCF 6E5 mCM6E5 DCF 6E5 + DM mC6E5 mC6E5

(2)

(3)

DCF 6SNa and DCF 6E5 are the intradiffusion coefficients of the molecules in the solution bulk; DM is the micelle intradiffusion coefficient, mF and mM are the molalities of free and micellized surfactant, and m is the total surfactant molality in the system. DM can be determined experimentally by the addition of TMS to the system. TMS is a strongly hydrophobic molecule and is solubilized in the micellar core. Following its NMR signal, the micelle intradiffusion coefficient can be measured directly.21 To be sure that the TMS insertion does not change the shape and dimension of micelles, two measurements were performed for each solution, before and after the TMS addition, checking that the surfactants’ intradiffusion coefficients were the same. For all samples, a single-exponential decay of the echo amplitude was observed for the TMS signal. The fact that the micelle intradiffusion can be described by a single intradiffusion coefficient is strong evidence for mixed micelle formation.14 Both DC6SNa and DC6E5 approach DM as mC6E5 increases; the still lower value of DM at the highest considered C6E5 concentration must be ascribed to the presence of a nonnegligible fraction of free surfactant molecules. All measured DM values are collected in Table 1, and some of them are shown in Figures 1 and 2. In the phase transition model, assuming that the mobility of free surfactant molecules is unaffected by the presence of micelle aggregates, the DF terms can be computed from the measurements taken at compositions where micelles are absent. Accordingly the DF values are assumed to be those measured at the cmc composition Dcmc depending on the mC6SNa/mC6E5 ratio. Provided that DF and DM are known, eqs 2 and 3 allow the determination of mF and mM for both surfactants. However, because of the obstruction effect due to the micelles, DF could be lower than Dcmc. It is possible to account for this effect by an iterative calculation. In the literature, expressions have been given22 for the obstruction effect in systems with spherical and spheroidal obstructing particles. Assuming a spherical shape for the (21) Lindman, B.; Puyal, M. C.; Kamenka, N.; Brun, B.; Gunnarson, G. J. Phys. Chem. 1982, 86, 1702. (22) Bell, G. M. Trans. Faraday Soc. 1965, 60, 1753.


Langmuir, Vol. 14, No. 25, 1998 7135

Figure 4. Free and micellized surfactant molalities in D2OC6SNa-C6E5 mixtures versus C6E5 molalities: free (2) and micellized (4) C6SNa; free (b) and micellized (O) C6E5. mC6SNa -1 ) 0.08 mol kgD . 2O

Figure 5. Free and micellized surfactant molalities in D2OC6SNa-C6E5 mixtures versus C6E5 molalities: free (2) and micellized (4) C6SNa; free (b) and micellized (O) C6E5. mC6SNa -1 ) 0.7 mol kgD . 2O

micelles under consideration, the following equation can be used:22

(

DF ) Dcmc 1 +

)

φM 2

-1

(4)

φM is the micellar volume fraction; it is computed from the molalities obtained neglecting the obstruction effect. / h CM6SNa + hC6SNaVD )+ φM ) cCM6SNa(V 2O

/ cCM6E5(V h CM6E5 + hC6E5VD ) (5) 2O

/ h CM6SNa + hC6SVD ) + mCM6E5(V h CM6E5 + φM ≈ [mCM6SNa(V 2O

F

/ hC6E5VD )]‚ 2O 1000 + mC6SNaMWC6SNa + mC6E5MWC6E5

(6) Here V h M is the volume of the micellized surfactant, assumed to be the same as that in its binary solutions; / VD is the solvent volume; h is the hydration number of 2O the hydrophilic heads;5,23 and F is the deuterated ternary mixture density. In eqs 5 and 6 the only nonideality was assumed to be the hydration of the polar heads of the micelles, considered to be independent of the comicellization and of the aggregation number change. From eq 4 a value of DF corrected for the obstruction factor was obtained and used to calculate mF and mM through eqs 2 and 3, in an iterative procedure. The final results are weakly affected by the obstruction correction (

Mixed Micellar Aggregates of Anionic and Nonionic Surfactants with

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