18802
J. Phys. Chem. 1996, 100, 18802-18807
Micellar Surface Charge Modification for Regular Solution Theory: Application to Mixed System of Cationic Fluorocarbon-Nonionic Hydrocarbon Surfactants Kenji Takasugi and Kunio Esumi* Department of Applied Chemistry and Institute of Colloid and Interface Science, Science UniVersity of Tokyo, Kagurazaka, Shinjuku-ku, Tokyo 162, Japan ReceiVed: May 16, 1996; In Final Form: September 18, 1996X
A micellar surface charge modification for the regular solution theory was performed by considering the counterion bind effect. For mixed micelles of cationic fluorocarbon-nonionic hydrocarbon surfactants, this modified equation can be tested by measuring the mixed cmc, surface potential, aggregation number, and molar volumes. The mixed cmc curves are well fitted with the calculated curves by the modified equation as a function of cationic surfactant molar fraction. The interaction between the surfactants is not so repulsive as expected from the regular solution theory. In addition, the mixed micelles with various mole fractions exist in a wide mixing region and the size of mixed micelles having charge is fairly small. ∆ψ ) 4πF/
Introduction well-known1-3
that fluorocarbon surfactants form miIt is celles in more dilute solution than other hydrocarbon surfactants, since the fluoromethylene chain has a larger exclusion volume than the methylene chain. In addition, the attractive force of fluorocarbon chains is not so large compared to that of hydrocarbon chains so that the fluorocarbon surfactants show very high surface activity. The mixtures of fluorocarbon and hydrocarbon surfactants in aqueous solutions have exhibited various unique physicochemical properties. To discuss the interaction of surfactant mixtures, regular solution theory has often been employed. Theories of nonideality in mixed surfactant systems have been reported by many investigators.4-12 In particular, the regular solution theory developed by Rubingh et al. has been applied to many homogeneous and nonhomogeneous surfactants mixtures.5,11 However, since the interaction parameter β in this theory is not included in the electrical repulsive energy of the ionic surfactant, the regular solution theory is not suitable for surfactant mixtures containing ionic surfactants. On the other hand, Shinoda et al.13 considered the counterion effect in micellization and they successfully applied their theory for ionic-nonionic mixtures. In addition, to estimate the micellar surface potential Drummond et al. have employed the Nernstian equation by using acid-base indicator probing.14 Hobson et al. applied these techniques of acid-base indicator probing for ionic-nonionic mixtures and demonstrated the importance of electrical work in aggregate formation.15 Given the electrostatic effect in surfactant mixtures, the regular solution theory including electrostatic effect can be suitable for various mixed systems. In this study, a modified regular solution theory including electrostatic effect has been developed and applied for a mixed surfactant system of cationic fluorocarbon and nonionic hydrocarbon. Theory Relation between the Potential and Charge Density of Micellar Surface. The electrostatic potential of a particle is represented by the Poisson equation16 X
Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(96)01413-X CCC: $12.00
(1)
where ψ is the electrostatic potential, F is the charge density, and is the relative permittivity. If the electrostatic potential is isotropic, eq 1 is solved as
-
{ [
]}
4 cosh(y0/2) - 1 dy | ) 2 sinh(y0/2) 1 + dx κr sinh2(y /2) 0
(2)
where y, and κ are dimensionless quantities replaced with y ) eψ/kT, κ2 ) 8πn0e2/kT. At the micellar surface, the absolute value of the gradation of electrical potential may be equal to surface charge density, and the surface potential will be zero at infinite point. Moroi and Katsuura16 theoretically derived the relation between micellar radius and surface potential. Applying their procedure for cationic-nonionic mixed micelles can lead to
-
{ [
]}
4πe2(x1 - Kg) 4 cosh(y0/2) - 1 ) 2 sinh(y0/2) 1 + eκakT κr sinh2(y /2) 0
(3)
where a is the mean micellar surface area per ionic surfactant, r is the micellar radius, and Kg is the degree of anion binding to the mixed micelle. Modified Regular Solution Theory with Micellar Charge. In the regular solution theory, the chemical potential of the surfactant i is given by17,18 0 + RT ln fixi µmici ) µmic i
(4)
where µmic is the chemical potential of surfactant in the mixed 0 is the standard chemical potential of surfactant micelles, µmic i in micellar state, and fi and xi are the activity coefficient and mole fraction in the mixed micelles of ionic surfactant, respectively. In the modification of Shinoda’s approach,13 the chemical potential of ionic surfactant is the sum of the chemical potential induced from hydrophobic effect and micellar charged effect. Considering their modification for the regular solution theory, the chemical potential of ionic surfactant in the micelles can be represented as 0 + RT ln fixi + (xi - Kg)Fψ0 µmici ) µmic i
© 1996 American Chemical Society
(5)
Micellar Surface Charge Modification for Regular Solution where ψ0 is the potential of micellar surface and F is Faraday constant. In a single surfactant system, it can be derived that 0 0 µmic + (1 - Kgi)eψ0i/kT ) µmono + RT ln cmci i i
(6)
0 , Kgi, ψ0i, and cmci are the standard chemical where µmono i potential in the micelles, counterion binding degree, surface potential of the micelles, and cmc in the single surfactant solution. As a result, the activity is expressed as
Ri )
fixicmci exp[(xi - Kg)eψ0/kT] exp[(1 - Kgi)eψ0i/kT]
(7)
where e is the unit charge. Thus, for mixed surfactant systems, the mixed cmc is related to the activity as
Ri exp[(1 - Kgi)eψ0i/kT] Rn 1 + ) cmc ficmci exp[(xi - Kg)eψ0/kT] fncmcn
(8)
where Ri is the total mole fraction of ionic surfactant. Approximating the activity by the lattice model, the cmc can be evaluated by eqs 9 and 10,where β is the interaction parameter
fi ) exp[β(1 - xi)2]
(9)
fn ) exp[β(1 - xn)2]
(10)
of surfactants in the mixed micelles. Experimental Section Materials. DEFUMAC (bis(2-hydroxyethyl)(2-hydroxy-3perfluorooctylpropyl)methylammonium chloride) was prepared by an anion-exchange method from the corresponding iodine salt kindly supplied by Daikin Industries, Ltd. and was purified by recrystallizing more than five times from mixtures of hexane and 2-proponal. C10E8 (octaethylene glycol mono-n-decyl ether) and C10E6 (hexaethylene glycol mono-n-decyl ether) were supplied by Nikko Chemical Co., Ltd. and were used without purification. Both surfactant’s purity was certified by static surface tension measurement. Acid-base indicator ET30 (Reichard’s Dye) was obtained from Aldrich Chemical Co. Inc. and was used as received. Methods. Surface tension was measured with a KRU ¨ SS K12 surface tensiomer. ET30 spectra in aqueous solutions with various pH adjusted by using hydrochloric acid and sodium hydroxide were measured with a Hewlett Packard 8452A diode array spectrophotometer. The microscopic potential of micellar surface was estimated from dissociation degree of ET30 and Nernstian equation. The scattering intensity of aqueous surfactant solutions was measured with Otsuka Electronics Co., Ltd. DLS-700 (Ar-Laser, 75 mW), and mean aggregation number was determined from the Debye plot. Density measurements of surfactant solutions were performed with an Anton Paar DMA60/602 vibrational densimeter. All measurements were carried out at 25 °C. Results and Discussion Molar Volume of Surfactants. In order to study the interaction C10E8, C10E6, and DEFUMAC in the mixed micelles, at first the partial molar volume was measured. Partial molar volume is related to density with the following equation.19,20
J. Phys. Chem., Vol. 100, No. 48, 1996 18803
φav )
1000(d - d0) Mixi + Mnxn mdd0 d
(11)
where d is the density of surfactant solution, φav is the apparent average partial molar volume, m is the total surfactant molality, Mi and Mn are the molecular weight of ionic and nonionic surfactants, and subscripts 0, i, and n represent the solvent, ionic surfactant, and nonionic surfactant, respectively. The molar volumes of C10E8- or C10E6-DEFUMAC mixed system calculated from the density measurements are shown in Figure 1. One may easily note that the linearity of partial molar volume with the molar fraction is held since the interaction between the surfactants is weak. If the interaction between surfactants is very much, the partial molar volume would be deviated downward. Accordingly, the linearity of the molar volumes of C10E8 or C10E6-DEFUMAC system indicates that the interaction of these surfactants is fairly small in these mixed systems. Further, the partial molar volumes thus obtained are used to estimate the micellar radius of mixed micelles. Cmc Evaluation with the Regular Solution Theory. Generally, the surface tension of a surfactant solution decreases with increasing surfactant concentration. That is, the surfactant adsorbs at the air/liquid interface and decreases the free energy of surface, and the surface tension stays constant above the cmc due to saturation of surfactant adsorption. However, in mixed systems, because of the exchange between the surfactant adsorbed at the air/liquid interface and the other surfactant in bulk, the surface tension would sometimes change even above the cmc. From Figures 2 and 3, it is suggested that the C10E8 or C10E6 adsorbed at the air/liquid interface is replaced to DEFUMAC even in the solution whose concentration is greater than the cmc, so the surface tension is decreased even above the cmc. However, since the dγ/d ln c changes abruptly at the cmc, the mixed cmc can be determined from the surface tension curves for this mixed system. Mixed cmcs thus obtained with the molar fraction of DEFUMAC are shown in Figures 4 and 5. There are many theories reported for mixed surfactant systems.17,21-22 At first, we apply the regular solution theory for the present mixed systems because of its simplicity. As previously explained in the Theory section, the regular solution theory does not include the effect of micellization against electrical repulsion, and the fitted curve is fairly deviated from the experimental results. It can be easily predicted that the surfactant interaction may be more weaker in the DEFUMACrich region than in the C10E8- or C10E6-rich region. Since the difference from the calculated value is more positive in the higher mole fraction of DEFUMAC,17 the following Guo’s theory is appreciable for this system. Guo17 approximates the activity coefficient by using second empirical parameter with the regular solution theory.
fi ) exp[xn2(β + δxi)]
(12)
fn ) exp[xi2(β - δ/2 + δxi)]
(13)
where β and δ are the empirical parameters. The ionic strengths can be expressed with following equations.
Ii ) [cmci/(cmcxi)]Kg
(14)
In ) [cmcn/(cmcxn)]Kg
(15)
We can obtain the parameters of β and δ from least-squares fitting with eqs 12-15.17 The obtained parameters are 0.0 and
18804 J. Phys. Chem., Vol. 100, No. 48, 1996
Figure 1. Averaged apparent molar volume of the C10E8- (O) or C10E6-DEFUMAC (b) mixed system.
Figure 2. Surface tension of the C10E8-DEFUMAC mixed system. R is the total mole fraction of DEFUMAC.
Figure 3. Surface tension of the C10E6-DEFUMAC mixed system. R is the total mole fraction of DEFUMAC.
1.1 for β and δ for the DEFUMAC-C10E8 system, while they are 0.0 and 1.0 for the C10E6-DEFUMAC system. This theory is successfully fitted with the experimental mixed cmcs as shown in Figures 4 and 5. It is understood from Figures 4 and 5 that the mole fraction of DEFUMAC in the micellar state is much lower than that in the bulk solution; that is, DEFUMAC is much bulky than C10E8 and C10E6. In other words, it is inferred that C10E8 and C10E6 easily form micelles and DEFUMAC is
Takasugi and Esumi
Figure 4. Mixed cmc of the C10E8-DEFUMAC system. The normal lines are calculated from Guo’s theory and dashed lines are from eq 8. R is the total mole fraction and X is the micellar mole fraction of DEFUMAC.
Figure 5. Mixed cmc of the C10E6-DEFUMAC system. The normal lines are calculated from Guo’s theory and dashed lines are from eq 8. R is the total mole fraction and X is the micellar mole fraction of DEFUMAC.
solubilized in C10E8 or C10E6 micelles, resulting in a formation of mixed micelles of DEFUMAC and C10E8 or C10E6. Further, since the value of δ is much smaller than that of the other mixed systems,16 it can be said that the electrostatic effect between DEFUMAC and C10E8 or C10E6 is fairly weak compared with the other mixed systems. The reason why the parameter δ has a positive value cannot be explained because this parameter loses its theoretical meaning like the regular solution theory. In the latter section, it is discussed again with the modified equation taking account of electrical repulsion. Micellar Radius of Mixed Micelles. Static light scattering measurements were performed to obtain the size of micelles, and the interaction between the surfactants was evaluated by eq 8. The mean micellar weight was determined by a static light scattering method. Scattering intensity can be related to mean micellar weight with the following equation23
(
)
〈Rg〉 2 K(C - cmc) 1 ) 1+ q + B2(C - cmc) + ... R - Rcmc Mw 3 (16) 2
where K is the optical constant as K ) 4π2n02(dn/dc)2/NAλ04, R is the Rayleigh ratio, Mw is the weight average micellar weight, 〈Rg〉2 is the gyration radius of mixed micelle, q is the scattering vector, and B2 is the secondary virial coefficient. The Debye
Micellar Surface Charge Modification for Regular Solution
J. Phys. Chem., Vol. 100, No. 48, 1996 18805
Figure 6. Debye plot of C10E8, C10E6, and DEFUMAC obtained from static light scattering measurement.
Figure 9. Radius of mixed micelle calculated from molar volume and aggregation number: (open symbol) C10E8-DEFUMAC system; (closed symbol) C10E6-DEFUMAC system.
Figure 7. Mean micellar aggregation of the C10E8- or C10E6DEFUMAC system.
is shown in Figure 9. It is seen that the micellar radius is controlled by the fluorocarbon chain of DEFUMAC in the C10E8-DEFUMAC mixed micelles. Furthermore, the radius of DEFUMAC micelles corresponds to about 0.95 times of the fully extended length of DEFUMAC calculated as 1.9 nm for transformed undecane, and the packing ratio of DEFUMAC is obtained to be about 0.86. This suggests that DEFUMAC micelles are excluded from solvent water. In addition, the radius of mixed micelles is much smaller for the C10E6-DEFUMAC system than that for the C10E8-DEFUMAC system because the interactions between the hydrophilic groups for the former system are smaller than that for the latter system. Accordingly, it is concluded that the mixed micelles of C10E8-DEFUMAC are easily formed compared with those of C10E6-DEFUMAC. Evaluation of Surfactant Interaction with the Modified Regular Solution Theory. Considering the results of static light scattering and the density measurements, it can be said that the radius of mixed micelles is not significantly affected by interference of the nonionic surfactant because of the stiffness of fluorocarbon chain, so that the mean surface area of the surfactant can be written as
a ) 4πrf3/3nw
Figure 8. Secondary virial coefficient of the C10E8- or C10E6DEFUMAC system.
plot of single species solution is shown in Figure 6. Since these values near the cmc include monomer weight, the mean micellar weight can be obtained by extrapolating this value to the cmc. The intercept is associated to the micellar weight by the following equation.
K(C - cmc) 1 ) + B2(C - cmc) R - Rcmc Mw
(17)
The mean micellar aggregation number and secondary virial coefficient vs molar fraction are plotted in Figures 7 and 8. These figures suggest that the interaction between the surfactants is so weak that the micelles may not grow appreciably in both the mixed systems. Combining the mean micellar aggregation number with the partial molar volume obtained from the density measurements, we can calculate the micellar radius whose result
(18)
where rf is the micellar radius of fluorocarbon surfactant, nw is the mean aggregation number of the mixed micelles. The electrostatic potential of mixed micellar surface is calculated from eq 3. Then, the estimated surface potential is inserted to eq 8 to obtain β. By our fitting, β is determined to be -0.62 for the C10E8-DEFUMAC system, indicating that the mixing interaction is not so interactive and the β value is nearly same order with nonionic-nonionic surfactant mixture in a range of 0 to -0.8. It is understood that the calculated parameter β from the regular solution theory indicates the interaction of surfactants with subtracting the charge interaction. The calculated β is obtained to be -0.10; i.e., the interaction energy between a pair of ethylene oxides is about 0.26kT. From this result, it is assumed that the mixed micelles are stable by the interaction between the hydrophilic groups and the repulsive force works between the hydrophobic chains. In addition, the surface potential of mixed micelles is theoretically calculated by the fitting and is shown in Figure 10. It is seen that for both the mixed systems, the work of aggregation against the electric repulsive force is much large in higher regions of DEFUMAC mole fraction. We applied the regular solution theory and modified the regular solution theory with electrostatic potential to our mixed
18806 J. Phys. Chem., Vol. 100, No. 48, 1996
Takasugi and Esumi TABLE 1: Micellar Surface Potentials for the C10E8- or C10E6-DEFUMAC Systems Obtained from ET30 Probing system DEFUMAC +C10E8
DEFUMAC +C10E6
Figure 10. Micellar surface potential calculated with eq 3 at the cmc: (open symbol) C10E8-DEFUMAC system; (closed symbol) C10E6DEFUMAC system.
Figure 11. Maximum absorption wavelength of ET30 and effective dielectric constant: (open symbol) C10E8-DEFUMAC system; (closed symbol) C10E6-DEFUMAC system.
systems. The results of fitting indicate that the fitted curve with the regular solution theory is positively deviated in the nonion-rich region and even repulsive in the cation-rich region because of asymmetrical electrostatic effect. On the other hand, the fitting by the modified regular solution theory is better suited even in the cation-rich region. The interaction parameter β is not unity with the regular solution theory fitting, but with our modified theory, β remains constant successfully in the whole region. The above results suggest that the surfactant interaction observed must be almost electrostatic effect, and the net interaction of the surfactants is more interactive. Surface Charge of Mixed Micelles. Maximum absorption wavelengths of ET30 in the surfactant solution are shown in Figure 11 where the total surfactant concentration is 50 mmol dm-3. In this figure, the effective dielectric constant predicted from the ethylene glycol-water system reported by Drummond et al.24 is calculated. This result shows that the microscopic dielectric constant around the probe changes continuously, suggesting that the probe is not solubilized at a specified position but informs us of its macroscopic property. Then we calculate the potential around the probe by the Nernstian equation. i pKobs a ) pKa -
Fψprobe 2303RT
(19)
are the intrinsic and observed pKa of where pKia and pKobs a ET30. The results obtained from probing ET30 are summarized in Table 1 and are shown in Figure 12. It can be asserted that the measured potential is not suited for the nonionic system, but a relative change in the surface potential of the micelle is estimated. Though the potentials obtained from ET30 spectra give positively deflected values in high mole fraction of
a
RDEFUMAC λmax/nm effa pKai a pKaobs ψprobe/mV 0.0 0.2 0.4 0.6 0.8 10 0.0 0.2 0.4 0.6 0.8 1.0
534 510 499 494 486 478 522 512 504 496 490 478
53 56 57 58 59 60 39 46 49 53 57 60
9.36 9.30 9.26 9.23 9.22 9.16 9.57 9.48 9.42 9.36 9.23 9.16
8.68 7.60 7.21 6.88 6.60 6.34 9.56 8.62 8.12 8.01 7.72 6.34
40 100 121 139 155 167 0 51 77 80 91 167
Calculated from the intrinsic pKa and eff from ref 23.
Figure 12. Micellar surface potential obtained from ET30 probing: (open symbol) C10E8-DEFUMAC system; (closed symbol) C10E6DEFUMAC system.
DEFUMAC, the results of potential calculation indicate the existence of ionic micelle whose surface is similarly charged as DEFUMAC micelle for both systems. Moreover, the mixed micelles formed in the C10E8-DEFUMAC system contain more DEFUMAC than for the C10E6-DEFUMAC system. We also calculated the surface charge from the virial coefficients determined by light scattering measurements with a following equation. If the hydrophobic effect of micelles is considerably small and neglected compared with the electrical charge effect, the surface charge per one micelle (P) can be expressed as eq 2025 The results of calculation from eq 20 are
P ) Mwx(2B2 × cmc)
(20)
shown in Figure 13. It is apparent that the surface charge of the mixed micelles is not so large as expected from Figure 12. That is, the results of the surface potential and surface charge are different and the existence of mixed micelles with various mole fractions is predicted in the mixed region. The surface potential has a much larger value because ET30 is considerably solubilized in the positive site of charged micelles and, furthermore, if the micelles with various molar fractions are formed, ET30 tends to be solubilized for the micelles whose fraction of ionic surfactant is higher. Therefore, the surface potential obtained from ET30 has a fairly higher value. On the other hand, the light scattering is caused from the difference in the refractive index between the solute and solvent, so the intensity of scattering from the micelles abounding in hydrocarbon surfactant contributes appreciably to the total intensity of light scattering. Accordingly, the virial coefficient has a comparably lower value than expected from the results of ET30 probing; namely, these results show that mixed micelles with
Micellar Surface Charge Modification for Regular Solution
J. Phys. Chem., Vol. 100, No. 48, 1996 18807 present in a wide mixing region and are charged, having a fairly small size. These micelles are stable due to the interaction between the hydrophilic groups. References and Notes
Figure 13. Effective surface repulsion P of the C10E8- or C10E6DEFUMAC system: (open symbol) C10E8-DEFUMAC system; (closed symbol) C10E6-DEFUMAC system.
various compositions are formed in the C10E8- or C10E6DEFUMAC systems. Conclusion Although the calculated cmcs of ionic and nonionic surfactant mixtures from the regular solution theory are considerably different from the experimental cmcs, the calculated cmcs obtained by the modified regular solution theory including electrostatic effect are in fair agreement with the experimental cmcs in the whole mole fraction region. The calculation of the interaction parameter β from the modified regular solution theory is adjustable for the system containing ionic and nonionic surfactants. Applying this modified regular solution theory, it is found that the interaction between DEFUMAC and C10E8 or C10E6 is nearly the same as nonionic-nonionic mixtures. Further, the mixed micelles with various mole fractions are
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