The Journal of Physical Chemistry, Vol. 83, No. 19, 1979 2471
Composition of Mixed Micelles
urface ~ e ~ ofs A ~ueous o ~Solutions of Surfactant Mixtures. The
omposition of
Noriaki Funasaki" and Sakae Hada Kyoto College of Pharmacy, Yamashina-ku, Kyoto 607, Japan (Received September 29, 1978; Revised Manuscript Received March 20, 1979) Publication costs assisted by the Kyoto College of Pharmacy
A method for determining the composition of mixed micelles by surface tension measurements is described, and the ideality of mixing in micelles is discussed. The systems investigated were (i) decyl methyl sulfoxide (DeMS)-heptaoxyethylene dodecyl ether (DE7) in water at 30 O C , (ii) DeMS-cetyltrimethylamonium bromide (CTAB) in 1 mM sodium bromide solution at 25 "C, and (iii) dodecyl methyl sulfoxide (DMS)-CTAB in 20 mM potassium bromide solution at 30 "C. Using the observed relationshipsof critical micellization concentration (cmc) and monomeric composition x b to the surface tension at cmc, one can determine the micellar composition x, from the surface tension of solutions at a higher concentration than cmc. Comparison of the theory of ideal mixing in micelles with observed values of cmc, x b , and I, revealed that the mixing of system i is ideal whereas those of systems ii and iii are not. From values of activity coefficients of each component and the excess free energy of mixing in micelles, the nonideality for the latter two systems may be ascribed to a decrease in electrostatic energy of micelles with the comicellization of the ionic and nonionic surfactants.
Introduction Extensive studies have been done on the comicellization of two or more surfactant^.^-^^ The objective of most of these s t ~ d i e s l was - ~ to determine the critical micellization concentration (cmc) of mixed surfactants and to provide a thermodynamic explanation of the value. Mysels and Otterlodetermined, by the conductivity method, not only the cmc of mixtures of sodium decyl sulfate (SDeS) and sodium dodecyl sulfate (SDS) in water, but also the composition of the mixed micelle. These authors also discussed the ideality of mixing in micelles.ll Shedlovsky et al. determined, by the pNa method, the cmc of mixtures of SDS and sodium tetradecyl sulfate (STS) in water as a function of the compositions of SDS in water and in micelles,12and the results were explained in terms of the ideal mixing of the two surfactants in mi~e1les.l~ This report deals with the determination of the composition of mixed micelles by the surface tension method, and the results obtained herein are discussed in terms of the ideality of mixing in micelles. The systems investigated are n-decyl methyl sulfoxide (DeMS)-heptaoxyethylene n-dodecyl ether (DE7) in water a t 30 "C, DeMS-cetyltrimethylammonium bromide (CTAB) in 1 mM sodium bromide solution at 25 "C, and n-dodecyl methyl sulfoxide (DMS)-CTAB in 20 mM potassium bromide solution a t 30 "C. The0KV
Composition of Mixed Micelles. The composition of mixed micelles as well as that of oily mixtures15 can be determined by the surface tension method. Above the cmc of surfactants, one assumes that the surface tension of aqueous solutions containing a single surfactant remains constant and that the compositions of bulk solutions and micelles containing two surfactants are a function of the surface tension alone, regardless of the amount of micelles. T h e symbols used hereafter are listed in the Appendix. In aqueous solutions above the cmc of mixtures, C12,of two surfactants 1 and 2, one can obtain from the material balance of surfactant 2 CtxZ C1212b (ct - C12)xZm (1) From this equation, one can obtain an expression for the composition of surfactant 2 in micelles OO22-3654/79/2083-2471$01 .OO/O
X2m
= (Ctxz - Cl*xZb)/(Ct - C12)
(2)
and for total concentration of surfactants 1 and 2 (3) ct = C12(X2b - xZm)/(xZ - XZm) According to eq 1, we found that the overall mole fraction x2 equals the monomeric one X2b a t the cmc, ViZ., C, = C12. Moreover, according to eq 2, we found that the micellar mole fraction xZm approaches the overall value x 2 at infinite total concentration. That is to say, as the total concentration of surfactants with the overall mole fraction x2 increases, a t the cmc the mole fraction of the bulk solution equals the overall value. With further increases in the total concentration, the mole fractions of bulk solution and micelles change continuously until that of micelles equals the overall value x2 As a result, the surface tension of aqueous solutions of mixed surfactants changes with an increase in total concentration above the cmc, e.g., as shown in Figure 1. Conversely, a t a concentration above the cmc, the mole fractions of bulk solution and micelles and total monomer concentration Cl2 are a function of the surface tension alone and are independent of the amount of micelles. Thus from the break point of surface tension vs. total concentration curves, one can determine c12 and x2b (equal to n2) as a function of the surface tension. Furthermore, measuring the surface tension of aqueous solutions a t a given concentration Ct above the cmc, one can calculate the mole fraction of surfactant 2 in micelles from eq 2; into and X2b corthis equation one substitutes values of c12 responding to a measured value of surface tension. Since the surface tension of pure ionic surfactants in water d e c r e a ~ e s labove ~ , ~ ~ the cmc, the above method is inapplicable to salt-free systems containing ionic surfactants. Let us consider the effect of an added salt MX on the surface tension of a pure ionic surfactant DX. In this system the Gibbs adsorption equation can be written as -dy/RT = I'D d In U D + rXd In ax -tM r d In U M (4) where I'i and aidenote the surface excess and activity of species i, respectively. Sasaki et al.17 and Cutler et al.ls found in aqueous SDS solution above the cmc that aDax@ = constant (5)
0 1979 American
Chemical Society
N. Funasaki and S.Hada
Tho Journal of Physical Chemistry, Vol. 83,No. 79, 1979
2472
0,2 0.4 0.6 Q8 mole fraction of DE7
0
2
- 4.0
-3.5 -3.0 log Ct (Ct ,mole/ I )
-2.5
Flgure 2. Typical pSots of surface tension against log total concentration for the DeMS-DE7 system in water at 30 O C . The overall mole fraction
of DE7 is (a) 1, (b) 0.500,(c) 0.350, (d) 0.149, (e) 0.049, (f) 0. The dashed lines are calculated from eq 3 by using data shown in Figures 2 and 3.
where p is the degree of counterion binding to the micelle. Moreover, the activity, aD, of dodecyl ions in sodium chloride solutions remains almost constantlg with increasing SDS Concentration above the cmc, as the salt concentration increases. Assuming that eq 5 holds also for salt solutions, one expects the activity, ax, of counterions to be constant above the cmc. In addition, since the surface excess20and the variation of the activity of co-ions above the cmc are expected to be small, the surface tension of the solution can remain unchanged with increasing surfactant concentration above the cmc. Thus, the above method is expected to be applicable to most mixture systems containing swamping salts. Ideality of Mixing in Micelles. Homologous surfactants are known to mix almost ideally in m i ~ e l l e s l - ~whereas J~l~ nonhomologues do not always ideally do As is well when two nonionic surfactants mix ideally in micelles in the absence of salts, the following applies: Xim
= Xib(@12/COi)
(6)
Substitution of this equation into xlm + x~~ = 1 and x l b X2b = 1 yields
+
l/clZ
=
nlb/COI
-k
(7)
x2b/c02
On the other hand, when two ionic homologues mix ideally in micelles in the presence of an added salt,14one finds Xim
= Xib(C12/COi)[(CIZ
+ cB)/(COi
-k Cs)IK
(9)
where K is a constant independent of salt concentration C,. The Kth power term in this equation takes into account changes in the surface potential of mixed micelles with the molar ratio of two surfactants. This equation is reduced to eq 6 when salt concentration is much higher than the cmc.14 On the basis of the above consideration, it is reasonable to define the activity coefficient of the nonionic component in mixed micelles as fim = (Xib/Xim)(c12/COi) (10) and that of the ionic component in mixed micelles of ionic and nonionic surfactants as
C~,)/(coi-k C,)IK (11) Were it should be noted that the activity coefficient of surfactants in bulk solution is assumed to be 1 and that
fim
= (Xib/Xim)(C12/601)[(C1ZXib
1.0
Figure 2. Plot of surface tension against the mole fractions of DE7 in monomers (solid circles) and in micelles (open circles) for the DeMS-DE7 system in water at 30 O C . The dashed line Is calculated from eq 6.
eq 11 approaches eq 10 with an increase in salt concentration. An alternative measure of nonideality in mixed micelles is the excess free energy of mixing defined as =
RT(x1m
In f l m
+ X2m In f2m)
(12)
Experimental Section Materials. CTAB from Nakarai Chemicals Co. was recrystallized three times from a mixt,ure of ethanol and acetone (1:l)after extraction with diethyl ether. DeMS and DMS were synthesized by the hydrogen peroxide oxidation of the corresponding sulfides, using tungstic acid as the catalyst.21*22The sulfides were synthesized by alkylation of longer chain alkyl thiols with methyl iodide and purified by distillation under reduced p r e s ~ u r e The .~~~~~ sulfoxides were purified by column chromatography and recrystallization from petroleum ether'3*22and were shown to be at least 99.8% pure by gas chromatography. DE7 (99% pure or more) was purchased from Nikko Chemicals Co. Impurities in sodium and potassium bromides from Merck were removed by adsorption with active charcoal after extraction with diethyl ether. The ion-exchanged water was twice distilled before use. Method. Surface tension was measured by the Wilhehy method as already reported.15 The value was read 20-40 min after each aqueous solution had been transferred to a water-jacketed vessel thermostated within 0.2 "C. esults DeMS-DE7 System. In Figure 1,the surface tension, y,of aqueous solutions of DeMS-DE7 mixtures at 30 "C is plotted against the logarithm of total surfactant concentration, C,,while the molar ratio of the surfactants is kept constant in each curve. The break point in each curve is the crnc. The surface tension of pure DeMS and DE7 remained constant regardless of an increase in concentration at least ten times as high as the cmc, whereas that of their mixtures decreased wit,h increase in total concentration. The solid circles in Figure 2 show the relationship between the surface tension and overall mole fraction, ~2 (yiz. %&), at the break point in Figure 1. From the break point shown in Figure 1,one can also obtain the relat,ionship of log cmc to the mole fraction of DE7 in monomers, XZb, as shown by the solid circles in Figure 3. Now, measuring the surface tension of a DeMS-DE7 mixture at a higher concentration Ct than the cmc and with the overall mole fraction x 2 , one can calculate the micellar composition, xZm,from eq 2, as described in the theory; here one reads a x 2 b value from the upper solid line shown in Figure 2 and also the cmc from the lower dashed line shown in Figure 3. Values of the micellar mole fraction
The Journal of Physical Chemistry, V d . 83, No. 19, 1979 2473
Composition of Mixed Micelles
mole fraction of DE7 Figure 3. Plot of log cmc against the mole fractions of DE7 in monomers (solid circles) and in micelles (open circies) for the DeMS-DE7 system in water at 30 OC. The upper and lower dashed lines are calculated from eq 8 and 7, respectively.
r
- 3.0
0
a2
0.4 0.6 08 mole fraction of CTAB
10
Flgure 5. Plot of surface tension against the mole fractions of CTAB in monomers (solid circles) and in micelles (open circles) for the DeMS-CTAB system in 1 mM sodium bromide solution (upper curves) and for the DMS-CTAB system in 20 mh4 potassium bromide solution (lower curves). The dashed line is calculated from eq 6.
I
-2.5 -2.0 log Ct ( Ct ,mole/ I 1
Figure 4. Typical plots of surface tension against log total concentration for the DeMS-CTAB system in 1 mM sodium bromide solution at 25 OC. The overall mole fraction of CTAB is (a) 1 in water, (b) 1, (c) 0.750, (d) 0.500, (e) 0.198, (f) 0.052, (9) 0. The dashed lines are calculated from eq 3 by using data shown in Figures 5 and 6a.
thus obtained are shown by the open circles in Figures 2 and 3. The open circles shown in these figures lie on a line regardless of the values of C, and x 2 employed, showing the validity of the present procedure for determining the micellar composition. In addition, the micellar composition which was calculated by using the surface tension of solutions a t relatively low concentration just above the cmc scattered around the lower solid line in Figure 2. The reason for this inaccuracy is not considered to be the invalidity of the present procedure but rather the experimental precision of measured values of the surface tension; when the micellar composition is calculated at a lower Concentration, a more accurate value of surface tension is required. This explanation is supported, as Figure 1 shows, by a good agreement of observed values with the dashed line. In this line, total concentration at a given surface tension is calculated from eq 3, employing values of not only jC2b and xZm,at the same surface tension, read from the solid lines in Figure 2 but also CI2,at the same %&,read from Figure 3. In general, an accurate value of the micellar composition can be evaluated for solutions at concentrations five times or more higher than the cmc. In Figure 2, intersections of two solid lines with a horizontal line represent the compositions of coexisting monomers and micelles at a given surface tension; the two compositions differ considerably. Thus, even if the overall mole fraction is kept constant, the compositions of monomers and micelles greatly change with increasing total concentration. The dashed lines shown in Figure 3 are calculated from eq 7 and 8. A good agreement of experiment and theory indicates that DeMS and DE7 form the ideal solution in
I \'\
DMS-C TA B
I
mole fraction of CTAB Figure 6. Plot of log cmc against the mole fractions of CTAB in monomers (solid circles) and in micelles (open circles) (a) for the DeMS-CTAB system in 1 mM sodium bromide solution and (b) for the DMS-CTAB system in 20 mM potassium bromide solution. The dashed line is calculated from eq 6.
micelles. Other workers have reported that homologous nonionic surfactants form ideally mixed m i c e l l e ~ . ~ ~ ~ J ~ DeMS-CTAB System. As curve a in Figure 4 shows, the surface tension of CTAB solutions without salts slightly decreases when the concentration is increased above the cmc. Addition of 1 mM sodium bromide brought the surface tension to an almost constant value within a tenfold concentration over the cmc (curve b). In the same way as was done for the DeMS-BE7 system, one can obtain Figures 5 and 6a; these figures show, respectively, plots of the surface tension and log cmc against the mole fractions of CTAB in monomers (solid circles) and in micelles (open circles). Here, the micellar composition is calculated from eq 2, employing data on solutions at 10-15-fold concentration over the cmc. The dashed line shown in Figure 4 is calculated from eq 3, in the same way as was done for the DeMS-DE7 system. DMS-CTAB System. Figure 7 shows plots of the surface tension against log total concentration for the DMS-CTAJ3 system in 20 mM potassium bromide solution at 30 "C. The dashed line is calculated from eq 3. Figures 5 and 6b, respectively, show plots of the surface tension
2474
The Journal of Physical Chemistry, Vol. 83, No. 19, f979
N. Funasaki and S. Hada
h
a,
.o” 30
v
)c
25 -4.0
-3.5
- 3.0
-2.5
log Ct ( C t , d e / I ) Figure 7. Typical plots of surface tension against log total concentration for the DMS-CTAB system in 20 mM potassium bromide solution at 30 O C . The overall mole fraction of CTAB is (a) 1, (b) 0.792, (c) 0.667, (d) 0.358, (e) 0. The dashed lines are calculated from eq 3 by using data shown in Figures 5 and 6b.
and log cmc against the mole fractions of CTAB in monomers (solid circles) and in micelles (open circles) for this system. In the DMS-CTAB system, the mole fractions of monomers and micelles are closer than those in the DeMSDE7 and DeMS-CTAB systems. In Figure 6, the cmc has a minimum a t a molar ratio. Similar observations have been reported for mixtures of ionic and nonionic surfactants with close cmc value^.^^^
Discussion Composition of Mixed Micelles. Mysels and Otterlo and Shedlovsky et a1.12determined the micellar composition of mixed sodium alkyl sulfates by conductivity and pNa methods, respectively. These methods may be limited to application in cases of homologous ionic surfactants, probably with very similar chains and in the absence of salts. The present method, although somewhat widely applicable, cannot be adopted or is inaccurate in the following cases: (i) Systems containing one or more ionic surfactants in the absence of or in the presence of a small amount of salt, since the surface tension of the pure surfactants decreases with increasing concentration above the cmc. A certain amount of scattering of the micellar composition for the DeMS-CTAB system may reveal this tendency (Figures 4-6a). (ii) Systems in which a variation of the surface tension at the cmc with the molar ratio of surfactants is small compared with experimental errors. In a CTAl3-poor region of Figure 5 (DMS-CTAB), the surface tension changed little with the monomeric composition, and an accurate composition of micelles could not be determined. (iii) Systems in which the solubility is close to the cmc, since the present method requires values of the surface tension a t a fairly higher concentration than the cmc, or a very accurate value of the surface tension just above the cmc is required. (iv) Moreover, the present method is inapplicable to surfactants which form dimers below the cmc. It was reported that CTAB does not dimerize in aqueous solutions.24 Ideality of Mixing i n Micelles. Using relations of the cmc to the compositions of micelles as well as monomers, one can discuss the ideality of mixing in micelles in detail. Studies along this line have been reported for SDeS-SDSll and SDS-STS14 systems. Homologous nonionic surfactants are shown to mix ideally in micelles on the basis of the cmc vs. X2b relat i ~ n . ’ , ~ ,From ~ ’ the relations of the cmc vs. X2b and vs. ~2~ shown in Figure 3, it can be concluded that DeMS and
o 0.2 a4 0.6 0.8 1.0 mde fraction of CTAB in micelles Flgure 8. (a) Excess free energy of mixing plotted against the mole fraction of CTAB in micelles for the DeMS-CTAB (half-closed circles) and DMS-CTAB (open circles) systems. (b) Activity coefficients plotted against the mole fraction of CTAB in micelles: DeMS (solid circles) and CTAB (open circles) for the DeMS-CTAB system; DMS (half-closed circles) and CTAB (halved circles) for the DMS-CTAB system.
DE7, nonhomologues, mix ideally in micelles. Moroi et al.7,9proposed a theory which explains the relation of the cmc to the monomeric composition for mixtures of ionic and nonionic surfactants, e.g., as shown in Figure 6. According to this theory, eq 6 is expected to hold for the nonionic surfactant in mixed micelles. Substituting the monomeric composition into eq 6, one can calculate the micellar composition as shown by the dashed lines in Figures 5 and 6. A significant deviation from the observed values means that the activity coefficient of the nonionic surfactant in the mixed micelles is not equal to 1. Substituting the observed compositions of monomers and micelles into eq 10 and 11, one can determine the activity coefficient of each component in micelles. Here, a value of K = 0.60 was used, as determined in solutions of sodium and potassium bromides at 25 “C. As Figure 8 shows, all activity coefficients of ionic and nonionic components in the DeMS-CTAB and DMS-CTAB systems are smaller than 1;the attraction occurs between the two components in the micelles. Moreover, substituting these activity coefficients into eq 12, one can obtain the excess free energy, AG,,. In Figure 8a, AG,,/RT is plotted against the mole fraction of CTAB in micelles. Since Figure 8a,b provides similar information on the mixing in micelles, let us consider only the excess free energy data. Negative values of AG,, indicate the attraction between the two components in micelles, most of which may result from a decrease in the electrostatic energy of the micelles. This energy is expected to depend much on the surface charge density of micelles and the ionic strength and little on the size and shape of micelles. Here, it should be noted that DeMS and DMS occupy a nearly equal area per molecule at the micellar surface. In Figure 8, therefore, the surface charge density of micelles at the same xZmis almost equal in both DeMS-CTAB and DMS-CTAB systems. When compared at the same xZm, the extent of electrostatic stabilization accompanying the comicellization is greater in the DeMS-CTAB system than
Composition of Mixed Micelles
that in DMS-CTAB. This is explicable in light of the fact that the ionic strength of the former is lower than that, of the latter. Rigorous treatments of the electrostatic energy for micelles of pure surfactants have been reported by Overbeek and Stigter.25 Extension of their treatments for mixed micelles of ionic and nonionic surfactants, such as CTAB-DMS and CTAB-DeMS systems, is very hard. In the present treatment, therefore, the change in the electrostatic energy with comicellization is implicitly included in the excess free energy together with others. 'The K value may represent the degree of counterion binding to micelles1' or the effective coefficient of electrostatic energy as given by the Gouy-Chapman theory for a very large flat s u r f a ~ e In . ~mixed ~ ~ ~ micelles of ionic and nonionic surfactants, the K value may depend on the micellar c o m p o ~ i t i o n . ~The , ~ activity coefficients of ionic components and the excess free energy also change with a change of the K value. Such changes do not affect the above discussion because of the presence of an excess salt. The employment of a K value of zero, as the extreme case, instead of 0.60 produced little change in the AG,, value for the CTAB-DMS system and a decrease of about 20% in the absolute AG,, value for the CTAB-DeMS system. In summary, the method for determining the composition of mixed micelles by surface tension measurements was described together with the scope and limitation of its application. Using data on compositions of micelles as well as monomers, one can elucidate to some extent phenomena of the comicellization in detail. Although the DeMS-DE7 system formed ideally mixed micelles, the DeMS-CTAB and DMS-CTAB systems did not because of electrostatic stabilization upon mixing.
Acknowledgment. Thanks are due to M. Ohara for assistance with the manuscript.
Appendix I. List of Symbols ai activity of species i
xi gib
xim fi,
overall mole fraction of surfactant i in solution mole fraction of surfactant i in monomers of surfactants mole fraction of surfactant i in micelles activity coefficient of surfactant i in micelles
The Journal of Physical Chemistry, Vol. 83,No. 19, 1979 2475
C, Ct
Coi C12 AG,,
K R
T 7
concentration of uni-univalent salts added total concentration of surfactants 1 and 2 cmc of pure surfactant i in a particular solution cmc of mixed surfactants in a particular solution excess free energy of mixing in micelles slope of log Coi vs. log (Coi 9 C,) plots for ionic surfactants gas constant absolute temperature degree of counterion binding to micelles surface tension of aqueous solutions surface excess of species i at the air-water interface
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