Thermodynamics of Micellization of Aqueous Solutions of Binary

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Thermodynamics of Micellization of Aqueous Solutions of Binary Mixtures of Two Anionic Surfactants Katarzyna Szymczyk and Broniszaw Ja nczuk* Department of Interfacial Phenomena, Faculty of Chemistry, Maria Curie-Skzodowska University, Maria Curie-Skzodowska Sq. 3, 20-031 Lublin, Poland Received October 3, 2008 The thermodynamics of micellization of mixed anionic/anionic surfactant solutions, that is, sodium dodecyl sulfate (SDDS) and sodium decyl sulfate (SDS), have been studied by surface tension, density, and conductivity measurements. The obtained results indicate that the values of critical micelle concentration strongly depend on the composition of the mixture and that the mole fraction of surfactants in the mixed micelle calculated on the basis of Rosen and Villeneuve approaches are different from those in the bulk phase. The small negative deviation from the linear relationship between the critical micelle concentrations and composition of SDDS and SDS mixtures in the bulk phase, the values of the molecular interaction parameters, activity coefficients, and the excess Gibbs energy of mixed micelle formation calculated on the basis of Rosen and Villeneuve approaches and calculations based on the MT theory of Blankschtein proved that there is synergism in mixed micelle formation of aqueous solutions of SDDS and SDS. It was also found that the values of the standard Gibbs energy of micellization for the mixture of these two surfactants which confirm the synergetic effect can be predicted on the basis of the proposed equations which include the values of the mole fraction of surfactant in the mixed micelle and excess Gibbs energy of micellization of SDDS and SDS. Knowing the values of mole fractions of surfactants in the mixed micelle, it is also possible to calculate the volume change of surfactants after the micellization process.

1. Introduction The critical micelle concentration (CMC) represents a fundamental micellar quantity to study the self-aggregation of amphiphilic molecules in solution. The CMC value determines the industrial usefulness and biological activity of detergents as well as other interesting surfactant features like solute-solvent and solute-solute interactions. The ability of amphiphilic surfactant molecules to the aggregation and micellization in aqueous solutions is of significant interest due to a large number of various physicochemical and technological applications1,2 as well as due to peculiarities of the micellization mechanism.3,4 Commercial surfactants used in these applications invariably are mixtures. In this regard, it is important to gain an understanding of interactions leading to the distribution of individual components in the surface aggregates versus those in the bulk solutions and to model the adsorption at interfaces. Mixtures of different surfactant types often exhibit synergism in their effects on the properties of a system.5-8 *To whom correspondence should be addressed: phone (48-81) 5375649; fax (48-81) 533-3348; e-mail [email protected]. lublin.pl. (1) Kuni, F. M.; Shchekin, A. K.; Rusanov, A. I.; Grinin, A. P. Colloid J. 2004, 2, 174. (2) Ogino, K.; Uchiyama, H.; Ohsato, M.; Abe, M. J. Colloid Interface Sci. 1987, 116, 1. (3) Marangoni, D. G.; Rodenhiser, A. P.; Thomas, J. M. Langmuir 1993, 9, 438. (4) Sivakumar, A.; Samosundaran, P.; Tharch, S. Colloids Surf., A 1993, 70, 69. (5) Lucassen-Reynders, E. H.; Lucassen, J.; Giles, D. J. Colloid Interface Sci. 1981, 82, 150. (6) Rosen, J. M. Surfactants and Interfacial Phenomena; Wiley-Interscience: New York, 2004. (7) Hua, X. Y.; Rosen, M. J. J. Colloid Interface Sci. 1980, 90, 212. (8) Szymczyk, K.; Ja nczuk, B. Langmuir 2007, 23, 4972.

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This synergism can be attributed to nonideal mixing effects in the aggregates, and it results in critical micelle concentrations (CMC) and interfacial tensions that are substantially lower than would be expected on the basis of the properties of the unmixed surfactants alone. This situation has generated both theoretical and practical interest in developing a quantitative understanding of the behavior of a mixed surfactant system, driven in part by the potential applications of these systems in detergency,9 enchanced oil recovery,10 and mineral flotation.11 Moreover, micelles formed from a solution of mixed surfactants generally have a different surfactant composition than in a bulk phase.8,12 Because aggregation of the surface active agents into micelles is related to a change of their apparent molal and the partial molal volumes,13,14 it is interesting how the changes of mole fractions of surfactants in micelles influence on the synergetic effects in mixed micelle formation and their volumetric properties. Thus, the purpose of our studies was to determine the influence of the concentration and composition of aqueous solutions of binary mixtures of two anionic surfactants, sodium dodecyl sulfate (SDDS) and sodium decyl sulfate (SDS), on the values of the critical micelle concentration, mole fraction of surfactants in the mixed micelles, and the volume change upon micelle formation on the basis of surface tension, conductivity, and density measurements.

(9) Reif, I.; Samosundaran, P. Langmuir 1991, 7, 3411. (10) Rudin, J.; Bernard, C.; Wasan, D. T. Ind. Eng. Chem. Res. 1994, 33, 1150. (11) Von Rybinski, W.; Schwuger, M. J. Langmuir 1986, 2, 639. (12) Szymczyk, K.; Ja nczuk, B. Colloids Surf., A 2007, 293, 39. (13) Gonz alez-Martin, M. L.; Ja nczuk, B.; Mendez-Sierra, J. A.; Brugue, J. M. Colloids Surf., A 1999, 148, 213. (14) Benjamin, L. J. Phys. Chem. 1996, 70, 3790.

Published on Web 2/25/2009

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2.

Experimental Section

2.1. Materials. Sodium decyl sulfate (SDS) (Fluka) and sodium dodecyl sulfate (SDDS) (Sigma) were used for preparation of aqueous solutions. Aqueous solutions of individual surfactants and SDDS + SDS mixtures at different ratios of SDDS to SDS were prepared using doubly distilled and deionized water received from a Destamat Bi18E distiller. The surface tension of water was always controlled before solution preparation. 2.2. Liquid Surface Tension Measurements. Surface ten:: sion measurements were made at 293 K with a Kruss K9 tensiometer under atmospheric pressure by the ring method. The platinum ring was thoroughly cleaned and flame-dried before each measurement. The measurements were done in such a way that the vertically hung ring was dipped into the liquid to measure its surface tension. It was then pulled out. The maximum force needed to pull the ring through the interface was then expressed as the surface tension, γLV (mN/m). Measurements of the surface tension of pure water at 293 K were performed to calibrate the tensiometer and to check the cleanliness of the glassware. In all cases more than 10 measurements were carried out, and the standard deviation did not exceed (0.2 mN/m. The temperature was controlled within (0.1 K. 2.3. Density Measurements. We have measured the densities of water and aqueous solutions of the individual surfactants SDDS and SDS and their binary mixtures using a vibrating tube densimeter (Anton Paar, model DMA 5000). The apparatus consists of a glass U tube with a platinum resistance thermometer inside a thermostatic jacket. The sample density is a function of the oscillation frequency when the tube vibrates under the assumption that the sample volume trapped between the oscillation nodes is constant. The accuracy of the thermometer and the density measurements are (0.01 K and (0.005 kg/m3, respectively. The precision of the density and temperature measurements given by the manufacturer is (0.001 kg/m3 and (0.001 K. The densimeter is calibrated regularly with distilled and deionized water. After measuring the density of water, more than three measurements of density were carried out at constant temperature equal to 293 K. 2.4. Conductivity Measurements. Conductivity measurements of surfactant solutions were made by a conductivity meter (model Elmetron CX-731). After measuring the conductivity of the solvent three successive conductivity measurements of the surfactant solutions were carried out under controlled constant temperature. The accuracy of the measurements was (0.01 μS. The break point in the plot of either the equivalent conductivity versus the square root of the total surfactant concentration or the molar conductivity versus the total surfactant concentration was taken as CMC at the mole fraction.

3.

Results and Discussion

3.1. Critical Micelle Concentration. The values of the critical micelle concentration (CMC) for SDDS, SDS, and their binary mixtures were determined from surface tension isotherms (Figure 1), density, and conductivity measurements, and they are presented in Figure 2 (curves 1-3). This figure shows the cmc values of the individual surfactants and their mixtures determined on the basis of the surface tension (curve 1), density (curve 2), and conductivity (curve 3) measurements as a function of monomer mole fraction of SDDS in aqueous solutions, R. Determination of the value of CMC from density and conductivity measurements was carried out through a change in the slope when the density and conductivity versus the surfactant concentration for surfactant solutions 4378

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Figure 1. Dependence of the surface tension of aqueous SDS (curve 1) and SDDS (curve 6) solutions and their mixture at monomer mole fraction of SDDS equal to 0.2, 0.4, 0.6, and 0.8 on log C.

Figure 2. Dependence of the critical micelle concentration (CMC) determined from the surface tension (curve 1), density (curve 2), and conductivity (curve 3) measurements and calculated from eq 1 (curve 4) on the monomer mole fraction of SDDS, R. were plotted. The determined values of CMC for individual surfactants, SDDS and SDS, determined in different ways are close to each other and to those obtained by other researchers, especially for SDDS: 8.00  10-3 mol/dm3.15,16 In the case of SDS the values of CMC measured (1.86  10-3 mol/dm3) are a little different from those in the literature.17,18 The values of CMC of the all mixtures of SDDS and SDS at different monomer mole fraction of SDDS determined from density and conductivity measurements (Figure 2, curves 2 and 3) are somewhat higher than those obtained from the surface tension isotherms (curve 1), but it is well-known that the various methods used for detection of the CMC often lead to different numerical estimates, (15) Jones, M. N. Int. J. Pharm. 1999, 177, 137. (16) Gu, G.; Yan, H.; Chen, W.; Wang, W. J. Colloid Interface Sci. 1996, 178, 614. (17) Moroi, Y.; Nishikido, N.; Hiromoto, H.; Matuura, R. J. Colloid Interface Sci. 1975, 50, 254. (18) Kresheck, G. C.; Hargraves, W. A. J. Colloid Interface Sci. 1974, 48, 481.

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not only because of measurement errors but also because the CMC is the range rather than a single pinpointed value.19 Curve 4 in Figure 2 presents the values of CMC calculated on the basis of the molecular thermodynamic theory of mixed surfactant solutions.20-22 This theory expresses the CMC of a binary mixture of surfactants 1 and 2 as a function of the CMC’s of the constituent pure surfactants as follows: 1 R 1 -R ¼  þ  CMC12 f1 CMC1 f2 CMC2

ð1Þ

where CMC12, CMC1, and CMC2 are the critical micelle concentrations of the mixture, pure surfactant 1, and pure surfactant 2, respectively, R is the solution monomer composition, and the variables f1 and f2 are the micellar activity coefficients, which can be calculated from the relations ! β12 ð1 -R Þ2  ð2Þ f1 ¼ exp kT f2

β ðR Þ2 ¼ exp 12 kT

! ð3Þ

where β12 is the parameter that reflects specific interactions between surfactants 1 and 2, R* is the optimal micellar composition, i.e., the composition at which the free energy of mixed micellization attains its minimal value, k is the Boltzmann constant, and T is the absolute temperature. The value of R* can be obtained from the molecular thermodynamic theory from the relation      β12 R R CMC2  ¼ ln ð4Þ ð1 -2R Þ þ ln 1 -R CMC1 kT 1 -R The calculated values of CMC from eq 1 are presented in Figure 2 (curve 4). From this figure it appears that the changes of the calculated values of CMC as a function of R are somewhat similar to those obtained from the surface tension isotherms; however, the minimal value is obtained at R = 0.8. This minimum on curve 4 may suggest nonideal mixing of surfactants in mixed micelles, different micelle composition than that of the surfactant in the bulk phase, and synergetic effects in mixed micelle formation. 3.2. Evaluation of the of Mixed Micelle Composition and Molecular Interaction Parameters. Most of the theories dealing with the binary mixtures of surfactants are based on the regular solution theory, and they have been applied to the phase separation model for the micelles in order to estimate the interaction parameter β in various systems and the synergetic effects in mixed micelle formation.5 The molecular interaction parameter, β, for mixed micelle can be evaluated, among other things, using the equation derived by Rubingh and Rosen:6,23,24 βM ¼

M lnðRC12 =X1M C1M Þ

ð1 -X1M Þ2

ð5Þ

(19) Ysambertt, F.; Vejar, F.; Paredes, J.; Salager, J.-L. Colloids Surf., A 1998, 137, 189. (20) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 5567. (21) Sarmoria, C.; Puvvada, S.; Blankschtein, D. Langmuir 1992, 8, 2690. (22) Shiloach, A.; Blankschtein, D. Langmiur 1997, 13, 3968. (23) Hua, X. Y.; Rosen, M. J. J. Colloid Interface Sci. 1982, 87, 469. (24) Rubingh, D. N. In Mittal, K., Ed.; Solution Chemistry of Surfactants; Plenum Press: New York, 1979; p 337.

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Table 1. Values of the Mole Fraction of Surfactant 1 in the Mixed M Micelle Calculated from Eqs 6 and 7, XM 1 , X 1 , and Molecular Interaction Parameter in the Mixed Micelle, βM 1 , Calculated from Eq 5 R

XM 1

XM 1

βM 1

0.2 0.4 0.6 0.8

0.4302 0.5608 0.6923 0.8361

0.2392 0.4601 0.6586 0.8422

-2.0648 -1.3738 -1.0699 -0.8530

M where CM 1 and C12 are the critical micelle concentrations (CMC) of the individual surfactant 1 and a mixture of surfactant 1 and 2, respectively, and XM 1 is the mole fraction of surfactant 1 in the mixed micelle. XM 1 can be evaluated from the equation

M ðX1M Þ2 lnðRC12 =X1 C1M Þ M =ð1 -X M ÞC M  ð1 -X1M Þ2 ln½ð1 -RÞC12 1 2

¼1

ð6Þ

where CM 2 is the CMC of the individual surfactant 2. Villeneuve et al.25,26 claimed that the treatment of intermolecular interaction parameters by Rubingh and Rosen6,23,24 is not appropriate in the sense that they do not take into account the presence of the solvent,25 and the physical significance of the parameter β is not clear when the excess entropy of mixing is not zero.26 They proposed thermodynamic strategy for examining the miscibility of surfactants in adsorbed films and micelles by new concentration variables. The miscibility of two anionic surfactants in micelles may be examined by the equation25,26

X

M 1

X 1X 2 ¼ X 1 -2 C

!

DC DX 1

! ð7Þ T, p

where X 1 and X 2 for the mixture of two anionic surfactant types of 1:1 electrolyte (AB) are defined20,23X 1 = 2m1/m and X 2 = 2m2/m, respectively, and mh fulfills the condition given by the equation m ¼ 2m1 þ 2m2

ð8Þ

where m1 and m2 are the molalities of the anionic surfactants 1 and 2, respectively. C is equal mh at CMC. 25 The magnitude of X M 1 is defined by XM 1 ¼

N1M N1M þ N2M

ð9Þ

M where NM 1 and N2 are the excess numbers of molecules of the anionic surfactants 1 and 2, respectively, per micelle particle, of which the dividing surface between the bulk solution is defined so as to make the excess number of water zero.25,26 In Table 1 there are values of mole fraction of SDDS in the mixed micelle calculated from eqs 6 and 7. From this table it is results that the values of mole fraction of SDDS in the M mixed micelle, XM 1 and X 1 , are bigger than in the bulk phase, M but there is a difference between XM 1 and X 1 which indicates

(25) Villeneuve, M.; Sakamoto, H.; Minamizawa, H.; Aratono, M. J. Colloid Interface Sci. 1997, 194, 301. (26) Motomura, K.; Aratono, M. In Ogino, K., Abe, M., Eds.; Mixed Surfactant System; Marcel Dekker: New York, 1993; p 99.

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Figure 3. Dependence of the activity coefficient in the mixed micelle M M M of SDDS, f M 1 (eq 10) and f 1 (eq 12), and SDS, f 2 (eq 11) and f 2 (eq 13), on the monomer fraction of SDDS, R. that entropy plays an important role in the process of surfactant mixing in the mixed micelles. From Table 1 it also results that βM is negative. Because the values of βM for all mixtures are negative and their absolute values are higher M than |ln(CM 1 /C2 )|, we can state that synergism exists in the mixed micelle formation in the solution of all examined mixtures. However, taking into account the lowest value of βM, the best synergism in the range of monomer mole fraction of SDDS in the bulk phase from 0.2 to 0.8 exists at R = 0.2. 3.3. Activity Coefficient in the Micellar Phase. Knowing the values of mole fraction of surfactants in the micellar phase and molecular interaction parameters for the mixed micelles, it is possible to determine the activity coefficient of the surfactants in the mixtures. From the Rubingh and Rosen6,23,24 and Villeneuve et al.25,26 theories it results that the activity coefficients of the surfactants 1 and 2 in the mixed M micelle (f M 1 and f 2 ) fulfill the conditions ln f1M ¼ βM ð1 -X1M Þ2

ð10Þ

ln f2M ¼ βM ðX1M Þ2

ð11Þ

0 X1

12 C @ A 0 C1

M

ð12Þ

0 X2

12 C @ A

M

¼ ðf 1 Þ2 X 1

0

C2

¼

M M ðf 2 Þ2 X 2

ð13Þ

M In Figure 3, the values of f M 1 and f 2 (curves 1 and 3) calculated from eqs 10 and 11 as well as the values of f M 1 and f M 2 (curves 2 and 4) calculated from eqs 12 and 13 are presented, respectively. From calculation of the activity coefficients it appears that all their values are smaller than 1 which, according to the Villeneuve et al. model, indicates that interactions between SDDS and SDS molecules are stronger than between single surfactants; however, M M M differences between the values of f M 1 , f 2 and f 1 , f 2 indicate a important role of entropy in mixing of surfactants.

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Figure 4. Dependence of the excess Gibbs energy in mixed micelle, gM (eq 14) and gM,E (eq 15) on the monomer fraction of SDDS, R. This fact also confirms the values of the excess Gibbs energy of mixing presented in Figure 4 and calculated from the equation gM ¼ RTðX1M ln f1M þ X2M ln f2M Þ M

M

g M, E ¼ RTðX 1 ln f 1 þ X 2 ln f 2 Þ M

M

ð14Þ ð15Þ

As it appears from Figure 4, the values of gM and gM,E for each R are smaller than zero which, according to the Villeneuve et al. model, indicates that there is a synergism in mixed micelle formation of SDDS and SDS. However, taking into account the lowest value of gM and gM,E, the best synergism exists at R = 0.2 which confirms our ealier conclusion based on the lowest value of βM. 3.4. Standard Free Energy of Micellization. The tendency of surfactants to form micelles can be established on the basis of standard free energy of micelization (ΔG0mic). In the literature there are many different ways for determination of this energy. Maeda27,28 has proposed a approach of standard free energy determination for mixed micelles involving ionic species. In this approach ΔG0mic for mixtures of two surfactants including one nonionic and one ionic are given as a function of the ionic surfactant in the mixed micelle by ΔG0mic ¼ B0 þ B1 x1 þ B2 x1 2 RT

ð16Þ

where B0 is the independent term related to CMC of nonionic surfactant by B0 = ln C2. The other parameter, B1, is related to the standard free energy change upon replacement of a nonionic monomer in the nonionic pure micelle with an ionic monomer and B2 is equivalent to βM calculated from eq 5. Finally, the parameters B1 and B2 are related to the CMC values of pure systems by the equation   C1 ¼ B1 þ B2 ln C2

ð17Þ

(27) Ruiz, C. C.; Aguiar, J. Langmuir 2000, 21, 7946. (28) Maeda, H. J. J. Colloid Interface Sci. 1995, 172, 98.

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Article Table 2. Values of the Degree of Counterion Binding (δ) Calculated from Eq 21 δ

0 0.2734

0.2 0.4853

0.4 0.5119

0.6 0.5627

0.8 0.5753

1 0.6339

surfactants in the mixed micelles and excess Gibbs energy of mixing as ΔG0mic ¼ X1M ΔG0mic1 þ X2M ΔG0mic2 þ gM

ð19Þ

0 0 M, E ΔG0mic ¼ X M 1 ΔGmic1 þ X 2 ΔGmic2 þ g

ð20Þ

M

Figure 5. Dependence of the free energy of micellization ΔG0mic determined from eq 18 (curve 1), eq 19 (curve 2), eq 20 (curve 3), and eq 16 on the monomer mole fraction of SDDS, R.

We employed the Maeda approach to mixture of two anionic surfactant and calculated values of B1, B2, and ΔG0mic. The values of B1 are negative and equal to -2.882, -2.191, -1.887, and -1.670 for R = 0.2, 0.4, 0.6, and 0.8, respectively. According to Maeda,27 parameter B1 is related to the standard free energy change associated with the introduction of one ionic species into a nonionic micelle coupled with the release of one nonionic species from the micelle. The change of this standard free energy is associated with the transfer process consisting of two contributions: interaction between the head groups and between the hydrocarbon chains. When the hydrocarbon chains are of the same kind, the first contribution is predominant; however, when there is dissimilarity between the hydrocarbon tails, as in the case of SDDS and SDS, the interaction between these tails becomes more significant and makes the values of B1 negative. It is interesting that the smallest value of B1 exist at R equal to 0.2. On the basis of the values of B1, B2, and the molecular interaction parameter in the mixed micelle, βM, we determined the values of the standard free energy of micellization of SDDS and SDS mixtures from eq 16 which are presented in Figure 5 (curve 4). The points in curve 4 (Figure 5) corresponding to the values of the standard free energy of micellization of individual surfactants were determined from the following equation:29 ΔG0mic ¼ RT ln CMC

ð18Þ

These values of standard free energy of micellization for SDDS and SDS are equal to -9.71 and -11.70 kJ/mol, respectively. In Figure 5 there are also presented the values of standard free energy of micellization of SDDS and SDS mixtures estimated by other three different ways. Curve 1 presents the values of ΔG0mic calculated from the same equations as in the case of individual surfactants (eq 18). Curves 2 and 3 in Figure 5 represent the values of the standard free energy of micellization for mixtures of two anionic surfactants defined on the basis of a monomer mole fraction of (29) Carnero Ruiz, C.; Molina-Bolivar, J. A.; Aguiar, J. Langmuir 2001, 17, 6831.

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where the ΔG0mic1 and ΔG0mic2 are the values of the standard free energy of micellization of SDDS and SDS, respectively, calculated on the basis of eq 18. From Figure 5 it appears that all values of ΔG0mic are negative, and values of the standard free energy calculated from eq 19 are identical to those determined from eq 16 and nearly the same as those determined from eq 17 (curve 3). In the above-presented relationships in curves 2 and 4 there are a clear minimum of values of ΔG0mic at R = 0.2, confirming the synergetic effect in mixed micelle formation of two anionic surfactants at this monomer mole fraction of SDDS in a bulk phase. Different calculations of ΔG0mic indicate that using eq 19 it is possible in a simple way to obtain the values of the standard free energy of micellization for mixtures of two surfactants identical to those proposed by Maeda.28 Of course, the changes of ΔG0mic (Figure 5) and CMC (Figure 2) as a function of mole fraction of SDDS in the bulk phase are identical, but observed deviation from the linear relationship on the curves presenting these values can be connected with the degree of counterion binding (δ). The values of the degree of counterion binding were determined on the basis of pre- and post-CMC slopes in the specific conductance versus surfactant concentration from the expression δ ¼ 1-

s2 s1

ð21Þ

where s1 is the pre-CMC and s2 is the post-CMC slope.30,31 The values of the degree of counterion binding for individual surfactants and their mixture calculated from eq 21 are listed in Table 2. From Table 2 it results that the values of δ increase with increasing the mole fraction of SDDS in the bulk phase. If we compare the relationship between the values of CMC (Figure 2) and δ (Table 2), we can see that that there are the opposite relations between these two values, that is, maximum of CMC corresponds to minimum of δ. It means that the degree of counterion binding with surface active ion has an influence on the synergetic or antagonistic effects in mixed micelle formation of two anionic surfactants. The higher the degree of counterion binding, the easier for the micellization process to occur. In other words, the micellization process take place at a lower concentration of binary surfactant mixture than each single surfactant. 3.5. Apparent Molal and Partial Molar Volumes. Aggregation of the surface active agents into micelles is related to a change of the apparent molal and the partial molar volumes. (30) Al.-Wardian, A.; Glenn, K. M.; Palepu, R. M. Colloids Surf., A 2004, 247, 115. (31) Zana, R. J. Colloid Interface Sci. 1980, 78, 330.

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Figure 6. Dependence between the apparent molar volume of

surfactants, φv, and concentration of solution, in wt %, cp, for mixtures of SDDS and SDS at different values of monomer mole fraction of SDDS, R.

Figure 7. Dependence between the partial molal volume in the micellized state for mixtures of SDDS and SDS at different values of monomer mole fraction of SDDS, R.

In the literature there are many works dealing with the apparent molar volume of surfactants, φv, and the volume change, ΔV, upon micelle formation.32,33 Kale and Zana34 presented for a simple mixture a law for evaluation of φv and ΔV on the basis of density measurements. The φv can be evaluated from the following expression: φv ¼

MS 1000ðF0 -FÞ þ CFF0 F0

ð22Þ

where MS is the molecular weight of surfactant, C its concentration (in mol/cm3), and F and F0 are the density of solution and pure solvent, respectively. However ΔV ¼ V M -V m

ð23Þ

where V M and V m are the partial molal volumes in the micellized and dispersed states, respectively. Usually it is assumed that V m is equal to the apparent molal volume at infinite dilution. V M can be calculated from the density by the equation32 "

VM ¼

MS 100 -cp dF 1dcp F F

# ð24Þ

where cp is the solution concentration in wt %. The values of φv calculated from eq 22 are presented in Figure 6. In the case of the surfactant mixtures for calculations of φv and V M, average values of MS for SDDS and SDS were used [MS = MS(1)R + (1 - R)MS(2)]. From Figure 6 it appears that the biggest values of φv are for the SDDS (curve 6) and the smallest for SDS (curve 1). In the case of mixture of these two anionic surfactants the values become smaller from R = 0.8 to R = 0.2 (curves 2-5). For both pure surfactants and each mixture at the lowest surfactant concentrations the values of φv clearly become smaller. (32) Benjamin, L. J. Phys. Chem. 1966, 70, 3790. (33) Kato, S.; Harada, S.; Nakashima, H.; Nomura, H. J. Colloid Interface Sci. 1992, 150, 305. (34) Kale, K. M.; Zana, R. J. Colloid Interface Sci. 1977, 61, 312.

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Figure 8. Dependence between the volume change upon micelle formation, ΔV, calculated from eq 23 (curve 1), eq 25 (curve 2), eq 26 (curve 3), and eq 27 (curve 4) for mixtures of SDDS and SDS at different values of monomer mole fraction of SDDS, R. After a specific concentrations, lower than CMC, the values of φv are the same. These leveling of the values of φv before CMC indicates that at these concentrations dimers or trimers are begging to form. In Figure 7 there are presented the values of V M calculated from eq 24 for concentrations at CMC as a function of the mixture composition. From this figure it appears that with the increasing mole fraction of SDDS in the bulk phase the values of V M increasing almost linearly, but at R = 0.2 there is a positive deviation from this linear relationship between V M and R. On the basis of values of V M the values of ΔV were calculated from eq 23, and they are presented in Figure 8 (curve 1). From this figure it appears that there is a deviation of curve 1 from linear relationship between the values of ΔV for a single surfactant and their mixtures calculated from the equation ΔV ¼ ΔV1 R þ ΔV2 ð1 -RÞ

ð25Þ

and presented in Figure 8 as curve 2. Langmuir 2009, 25(8), 4377–4383

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Because we proved that for all studied mixtures the values M of XM 1 and X 1 are higher than appropriate R, we calculate the values of ΔV from the equations ΔV ¼ ΔV1 X1M þ ΔV2 ð1 -X1M Þ

ð26Þ

M

ð27Þ

ΔVM ¼ ΔV1 X M 1 þ ΔV2 ð1 -X 1 Þ

The values of ΔV calculated on the basis of the eqs 26 and 27 are presented in Figure 8 as curves 3 and 4, respectively. As is seen in Figure 8 (curves 1, 3, and 4), there is a positive deviation from the linear relationship between ΔV at CMC and monomer mole fraction of SDDS in the mixture.

4. Conclusions The results of the measurements of the surface tension, density, conductivity, and the calculations of the mixed micelle composition and standard free energy of micellization

Langmuir 2009, 25(8), 4377–4383

of aqueous solution of SDDS and SDS and their mixtures suggest the following: (a) The values of critical micelle concentration strongly depend on the composition of the mixture of two anionic surfactants, and there is a small minimum between values of CMC calculated theoretically and monomer mole fraction of SDDS in the bulk phase. (b) The values of mole fraction of SDDS in the mixed micelle calculated on the basis of the Rosen and Villeneuve approaches are bigger than in the bulk phase. (c) The negative values of the parameters of molecular interaction, activity coefficients, and the excess Gibbs energy of mixed micelle formation calculated on the basis of the Rosen and Villeneuve approaches and calculations based on the MT theory of Blankschtein proved that there is synergism in mixed micelle formation of aqueous solutions of SDDS and SDS. (d) Knowing the composition of the mixed micelle, it is possible to determine in a simple way the standard free energy of micellization process of SDDS and SDS surfactant mixtures as well as volume change, ΔV, upon micelle formation.

DOI: 10.1021/la804183n

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