Langmuir 1997, 13, 3909-3912
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A New Approach to Micellization Parameters: Its Application to Sodium Dodecyl Sulfate Micelle Yoshikiyo Moroi* and Nobuyuki Yoshida Department of Chemistry, Faculty of Science, Kyushu University 33, Higashi-ku, Fukuoka 812-81, Japan Received March 6, 1997. In Final Form: June 2, 1997X A new approach to determine the three micellization parameters for ionic surfactants, the micellization constant Kn, the micelle aggregation number n, and the number of counterions per micelle m, was developed by combination of electric conductivity and either surfactant ion or counterion concentration. The approach employed two slopes of the plots of specific conductivity against surfactant concentration below and above the critical micelle concentration and the mass action model of micelle formation. This approach can be used to determine the other two parameters when the n value is available. This method was applied to the aqueous solution of sodium dodecyl sulfate whose micellization parameters have been precisely determined and was found to be quite successful to obtain the three parameters.
Introduction The mass action model is quite reasonable to discuss micelle formation and thermodynamically to treat micelles whose aggregation number is relatively small, say less than a few hundreds, since the model is consistent with the phase rule.1-5 Unfortunately, however, the three micellization parameters, Kn, n, and m, are quite difficult to determine precisely. This is the reason why micelles have been mostly dealt with as a separate phase.6-8 Therefore, the phase separation model has been adopted to obtain thermodynamical parameters of micellization, where the temperature dependence of the critical micelle concentration (cmc) was the most crucial factor for the parameters, although it is quite often that the temperature dependence of n is much larger than that of cmc.9 The degree of counterion binding to micelle (m/n) is rather easy to obtain from the slope of log(cmc) against log(counterion concentration), the Corrin-Harkins plots,10 when ionic micelles formed are relatively large in the aggregation number and do not change much with total surfactant concentration. The aggregation number can be determined by the static light scattering and by the photochemical methods.11,12 Then, the micellization constant Kn can be enumerated from cmc or from either surfactant ion or counterion concentration.13 However, it is not always the case for any ionic surfactant that these parameters can be determined without much difficulty. Recently, many kinds of newly synthesized amphiphiles
are in wide-spread use,14-16 and therefore, an easy but reliable method to determine the parameters is quite helpful to develop and characterize the properties of aggregates of these amphiphiles. The present approach was devised to answer the above request and was found to be quite successful to obtain the reliable parameter values. This was substantiated by applying the approach to micelle formation of sodium dodecyl sulfate solution. Theoretical Section The following equilibrium can be given for micelle formation of monodisperse ionic micelle17 Kn
nS- + mG+ y\z Mn-(n-m)
where S-, G+, and Mn are surfactant ion, counterion, and micelle of aggregation number n, respectively. From eq 1 the micellization constant Kn is written as
Kn ) [Mn]/([S-]n[G+]m)
(1) Mukerjee, P. J. Phys. Chem. 1972, 76, 565. (2) Moroi, Y. J. Phys. Chem. 1980, 84, 2186. (3) Moroi, Y.; Sugii, R.; Matuura, R. J. Colloid Interface Sci. 1984, 98, 184. (4) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1987; Vol. 1, Chapter 10. (5) Moroi, Y. J. Colloid Interface Sci. 1988, 112, 308. (6) Shinoda, K.; Hutchinson, E. J. Phys. Chem. 1962, 66, 577. (7) Moroi, Y.; Nishikido, N.; Uehara, H.; Matuura, R. J. Colloid Interface Sci. 1975, 50, 254. (8) Rosen, M. J. Surfactants and Interfacial Phenomena; Wiley-Interscience: New York, 1989. (9) Binana-Limbele, W.; van Os, N. M.; Rupert, L. A. M.; Zana, R. J. Colloid Interface Sci. 1991, 141, 157. (10) Corrin, M. L.; Harkins, W. D. J. Am. Chem. Soc. 1947, 69, 683. (11) Infelta, P. P. Chem. Phys. Lett. 1979, 61, 157. (12) Moroi, Y.; Humphry-Baker, R.; Gratzel, M. J. Colloid Interface Sci. 1987, 119, 588. (13) Moroi, Y.; Sakamoto, Y. J. Phys. Chem. 1988, 92, 5189.
S0743-7463(97)00255-2 CCC: $14.00
(2)
where [S-], [G+], and [Mn] are the corresponding concentrations, respectively, and the electroneutrality of solution holds:
[G+] - [S-] - (n - m)[Mn] ) 0 * E-mail:
[email protected]. Fax: 81-92-6422607. X Abstract published in Advance ACS Abstracts, July 1, 1997.
(1)
(3)
The mass balance for surfactant ion and counterion is respectively expressed as
Ct ) [S-] + n[Mn]
(4)
Ct ) [G+] + m[Mn]
(5)
where Ct is the total concentration of 1-1 ionic surfactant (14) Hoffmann, H.; Platz, G.; Ulbricht, W. J. Phys. Chem. 1981, 85, 1418. (15) Zana, R. Langmuir 1996, 12, 1208. (16) Furuya, H.; Moroi, Y.; Kaibara, K. J. Phys. Chem. 1996, 100, 17249. (17) Moroi, Y. Micelles: Theoretical and Applied Aspects; Plenum Press: New York, 1992; Chapter 4.
© 1997 American Chemical Society
3910 Langmuir, Vol. 13, No. 15, 1997
Letters
(G+S-). From eqs 3, 4, and 5, [S-] and [G+] are written as
[S-] ) (1 - n/m)Ct + (n/m)[G+]
(6)
[G+] ) (1 - m/n)Ct + (m/n)[S-]
(7)
From the logarithm of eq 2 and with eqs 4 and 7 or with eqs 5 and 6, one obtains
ln Kn ) ln{(Ct - [S-])/n} - n ln[S-] m ln{(1 - m/n)Ct + (m/n)[S-]} (8)
Hence, the slope (S2) of molar conductivity above the cmc becomes
S2 ) 1000(κ - κ0)/(Ct - C0) ) λS-{1 - (Ct - [S-])/(Ct - C0)} + λG+{1 - (m/n)(Ct - [S-])/(Ct - C0)} + (λM/n)(Ct - [S-])/(Ct - C0) (16) On the other hand, the molar conductivity of micelle with charge n - m can be given from the equivalent conductivity20 as
λM ) 0.820(n - m)2/(rη) ) R(n - m)2
or
ln Kn ) ln{(Ct - [G+])/m} - m ln[G+] n ln{(1 - n/m)Ct + (n/m)[G+]} (9) On the other hand, the following relation as to Kn results from the Phillips definition of cmc:18
ln Kn ) -ln{n(n + m)(2n + 2m - 1)/(n + m - 2)} (n + m - 1) ln[cmc] (10) Now, if the value of [S-] is known at a fixed total surfactant concentration Ct, the values of m and ln Kn can be determined as the solution of the simultaneous equations 8 and 10 for a certain value of n. The m/n value then becomes a function of n for a combination of [S-] and Ct, and vice versa. This is also the case for the combination of [G+] and Ct by eqs 9 and 10. Electrical conductivity below cmc is written as
1000κ ) λS-[S-] + λG+[G+];
[S-] ) [G+] ) Ct
(11)
where κ is the specific conductance and λS- and λG+ are the corresponding equivalent conductivities, respectively, where perfect dissociation of ionic surfactant is assumed below the cmc.19 Then, the slope (S1) of molar conductivity becomes
S1 ) 1000κ/Ct ) λS- + λG+
(12)
For concentrations above the cmc, a micellar term comes in:
1000κ ) λS-[S-] + λG+[G+] + λM[M]
(13)
Combination of κ0 at the cmc (C0) and eq 13 leads to
1000(κ - κ0) ) λS-([S-] - C0) + λG+([G+] - C0) + λM[M] (14) where λM is the molar conductivity of micelle. Introduction of eqs 4 and 7 into eq 14 results in
1000(κ - κ0) ) λS-{Ct - C0 - (Ct - [S-])} + λG+{Ct C0 - (m/n)(Ct - [S-])} + λM(Ct - [S-])/n (15) (18) Phillips, J. N. Trans. Faraday Soc. 1955, 51, 561. (19) Sasaki, T.; Hattori, M.; Sasaki, J.; Nukina, K. Bull. Chem. Soc. Jpn. 1975, 48, 1397.
(17)
where r is a micellar radius and η is a solvent viscosity. The sum of λS- + λG+ is the slope S1 below the cmc as given above, and eq 16 can be rewritten as
S1 - S2 ) {λS- + (m/n)λG+ - Rn(1 - m/n)2} × (Ct - [S-])/(Ct - C0) (18) In this electric conductivity case too, the relationship between n and m/n values can be obtained by the following procedure. The values of S1, S2, λS-, λG+, and C0 have been already determined as for SDS (see Table 1). The R value can be estimated from the micellar diameter 2.53 nm21 and the solvent viscosity at 298.2 K. If the value of [S-] is known at a fixed surfactant concentration (Ct), the value of m/n can be calculated as a function of n by eq 18. This is also the case for the combination of [G+] and Ct, as is clear from eq 7. In the above processes, two kinds of relationships between n and m/n have been obtained: one from the mass action model and the other from the electric conductivity. When the value of n is correct, the above two methods should lead to an identical value of m/n. In other words, the three unknown micellization parameters, n, m, and Kn, can be evaluated by solving eqs 8, 10, and 18 simultaneously. The practical procedure to solve the three equations will be given in the following section. Evaluation of the Three Parameters Concentrations of Na+ and DS- were separately determined up to 10 times cmc at 298.2 K by cation and anion selective electrode for the aqueous solution of sodium dodecyl sulfate (NaDS or SDS).19 The concentration change of the two ions with total surfactant concentration has been already analyzed by the mass action model. The most reliable values for the parameters were found to be n ) 64, m ) 46.7, and log Kn ) 230.5 These parameter values could perfectly reproduce the concentration changes of the two ions with SDS concentration up to 10 times cmc. These changes are summarized in Table 1. On the other hand, electric conductivity change with SDS concentration has been precisely determined at 298.2 K22 and analyzed to obtain the equivalent conductivity of Na+ and DS- up to the cmc using the λNa+ value in our previous study.22 These values are also summarized in Table 1 together with the slopes of conductance below and above the cmc (8.1 mmol dm-3). Now all the values for evaluation of the three parameters are available. First of all, the procedure to determine the parameters from [G+] at a (20) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworths: London, 1970; Chapter 2. (21) Corti, M.; Degiorgio, V. J. Phys. Chem. 1981, 85, 711. (22) Moroi, Y.; Matsuoka K. Bull. Chem. Soc. Jpn. 1994, 67, 2057.
Letters
Langmuir, Vol. 13, No. 15, 1997 3911
Table 1. Electric Conductivity Changes of Sodium Dodecyl Sulfate with Total Surfactant Concentration (Ct) at 298.2 K (K, specific conductivity; λ, equivalent conductivity)a Ct, mmol dm-3
κ, S cm-1
λNa+, S cm2 equiv-1
λDS-, S cm2 equiv-1
0.529 1.051 1.566 2.075 2.576 3.072 3.561 4.044 4.521 4.992 5.457 5.916 6.370 6.818
0.3723 0.7333 1.085 1.429 1.766 2.103 2.423 2.743 3.063 3.373 3.676 3.973 4.273 4.563
49.27 48.93 48.67 48.46 48.27 48.10 47.95 47.81 47.67 47.55 47.43 47.32 47.22 47.12
21.11 20.84 20.63 20.42 20.30 20.37 20.10 20.03 20.09 20.02 19.88 19.84 19.86 19.81
13.49 19.84 25.95 31.82 37.48 42.93 48.19 53.26 58.16 62.89 67.46 71.88 76.17 80.03 a
6.701 8.221 9.711 11.17 12.59 14.00 15.36 16.67 17.95 19.20 20.48 21.68 22.88 23.98
S1, S cm2 mol-1
S2 , S cm2 mol-1
66.56 below cmc
25.98 above cmc
[Na+], mmol dm-3
[DS-], mmol dm-3
0.529 1.051 1.566 2.075 2.576 3.072 3.561 4.044 4.521 4.992 5.457 5.916 6.370 6.818
0.529 1.051 1.566 2.075 2.576 3.072 3.561 4.044 4.521 4.992 5.457 5.916 6.370 6.818
8.859 10.12 11.37 12.61 13.85 15.08 16.29 17.49 18.68 19.86 21.01 22.16 23.29 24.32
7.174 6.547 6.001 5.529 5.123 4.778 4.488 4.248 4.052 3.898 3.781 3.698 3.645 3.622
λNa+ at the cmc ) 46.8 S cm2 equiv-1. λDS- at the cmc ) 19.74 S cm2 equiv-1. R ) 3.642 from r ) 2.53 nm and η ) 0.008 902 poise.
Figure 1. Procedure to determine the aggregation number (n) and the degree of counterion binding (m/n) for [G+] ) 15.08 and Ct ) 42.93 mmol dm-3: a, from eq 18; b, from eqs 9 and 10.
Figure 2. Change of the aggregation number (n) and the degree of counterion binding (m/n) values with surfactant concentrations calculated for the combination of [G+] and Ct; cmc ) 8.1 mmol dm-3.
fixed Ct will be presented; [G+] ) 15.08 at Ct ) 42.93 mmol dm-3. As for the combination of [G+] and Ct, eqs 9 and 10 give rise to one relationship between n and m/n (curve b in Figure 1), while eq 18 brings about the other relation between them (curve a in Figure 1). In this procedure the equivalent conductivities at the cmc were employed (see the footnotes in Table 1). The cross point is the solution of the simultaneous eqs 9, 10, and 18, and therefore, the three parameters can be automatically determined from the cross point. This process can be continued for different combinations above the cmc. The changes of n and m/n values with surfactant concentration are shown in Figure 2. The values are very near the reported values given above and remain almost constant, although they gradually decrease with surfactant con-
centration, which is, strictly speaking, contrary to the mass action model of micelle formation. However, the small variation from 0.73 to 0.74 for m/n and from 60 to 65 for n can be regarded to be within experimental error. The parameter values can be made constant at the reported values by using the cmc value of 7.4 instead of 8.1 mmol dm-3. This is not a trick but a fault resulting from the mass action model. In other words, it is impossible to do with only one set of the parameter values over the wide range of surfactant concentration, as far as the model is employed.5 Secondly, another procedure is for the case where the n value is known. It is quite often that the micellar
3912 Langmuir, Vol. 13, No. 15, 1997
Figure 3. Change of the bulk concentrations [Na+] and [DS-] and the degree of counterion binding (m/n) with surfactant concentration. For surfactant concentrations above the cmc, n ) 64. Experimental values are designated as open circles (ion selective electrode19).
aggregation number n is known from the static light scattering or photochemical process.12,23 In this case too, the values of [Na+], [DS-], and m/n are determined likewise by the n value. The values thus determined are shown in Figure 3, where good agreements between the reported values5 and the present ones can be confirmed. For ionic surfactants, electric conductivity is most commonly used to determine the cmc, and the slopes below (23) Atik, S. S.; Nam, M.; Singer, L. A. Chem. Phys. Lett. 1979, 67, 75.
Letters
and above the cmc are quite easy to obtain. At the same time, the equivalent conductivity of counterion and surfactant ion are not difficult to evaluate, because the values of many ionic species have already been determined and summarized in the table.20 On the other hand, micellar aggregation numbers have often been determined too for deep insight into micelle formation. Therefore, the combination of n and electric conductivity data of aqueous solution of ionic surfactants can make all the parameters available by the present procedure. Ion selective electrode is another choice of instrument for further investigation on micelle formation of ionic surfactant as is evident from the present report. At any rate, the degree of counterion binding to micelle cannot be easily determined only from the electric conductivity measurements.24,25 Of course, the present method should be applicable to a spherical micelle whose maximum aggregtion number is mainly controlled by the chain length and the molecular volume of hydrocarbon tail of surfactant ion.26 In other words, when an aggregation number obtained is far above the expected value from them, this method is not usable. In practice, when this method was applied to determine the micellization parameter values of lithium perfluoroundecanoate whose micelle aggregation number is quite hard to determine by the static light scattering method, the present method turned out to be quite successful.27 LA970255Z (24) Evans, H. V. J. Chem. Soc. 1956, 579. (25) Sepulveda, L.; Cortes, J. J. Phys. Chem. 1985, 89, 5322. (26) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; John Wiley & Sons: New York, 1973; Chapter 9. (27) Take’uchi, M.; Yoshida, N.; Yamauchi, A.; Moroi, Y. Submitted for publication in J. Colloid Interface Sci.
