Langmuir 1986,2,354-361
354
poise and are very small relative to the dynamic ekticities. Since the dynamic modulus e* is defined36 as e* = e + iWK (15) for a monolayer, for example, with a a of 2.5 dyn/cm and f , equal to 9596 Hz, the estimated surface dynamic viscosity is 7.5 X lo4 surface poise. This corresponds to a contribution of viscosity to dynamic modulus of about 0.45 dyn/cm (wK),as opposed to 28.5 dyn/cm from the dynamic elasticity.
Conclusions 1. A surface light scattering method for studying spread monolayers, developed in our laboratory,%has been used to give precise measurements with pentadecanoic acid. The demonstration of the first-srder LE/LC phase transition, with measurements of frequency shifts and damping coefficients, reinforces the importance of extensive purification of the sample in the study of monolayers and confirms that the equipment, as designed, should be useful in assessing the behavior of other monolayer systems.36 2. A dynamic analogue of the static surface preeaure has been proposed and used for testing affirmatively the hypothesis that the monolayer-coveredwater surface can be (36) Kawaguchi, M.; Sano, M.; Chen, Y.-L.; Zografi, G.; Yu, H. submitted for publication in Macromolecules.
approximated as another homogeneous liquid surface having an effective density higher than the subphase liquid. 3. By carrying out studies from 500 A2per molecule to just beyond the plateau region it was possible to observe dynamic properties not previously reported by using light scattering studies. Of particular interest are the changes in fr uency shift and damping coefficient from about 125 to 50 2, believed to be the end of the region of coexistence between the "gaseous" and liquid-expanded phases.24 4. In general, from an application of the dispersion equation, it has been shown that surface elasticity plays a much greater role than surface viscosity in determining the dynamic properties of this monolayer system.
1
Acknowledgment. This study was supported in part by the University Exploratory Research Program of Procter and Gamble Co. and by the Research Committee of the University of Wisconsin-Madison. We are most grateful to Dr.John C. Eamshaw of Belfast and Dr. Ronald D. Neuman of Auburn University for fruitful discussions. We also thank our colleague, Bryan B. Sauer, for his constructive critique of the data analysis and interpretation. .One of the referees for this paper is also acknowledged for a number of critical comments which were used to revise the original manuscript. Registry No. Pentadecanoic acid, 1002-84-2.
Electrostatic Model To Describe Mixed Ionic/Nonionic Micellar Nonidealities James F. Rathman and John F. Scamehorn* School of Chemical Engineering and Materials Science and Imtitute for Applied Surfactant Research, University of Oklahoma, Norman, Oklahoma 73019 Received October 23, 1985. In Final Form: February 5, 1985 Critical micelle concentrations were measured as a function of compmition for three binary ionic/nonionic surfactant mixtures. These systems exhibit large negative deviations from ideality. Two models based on electrostatic considerations alone were developed to describe mixed micellar nonidealities. One model considers the micelle pseudophase to consist of surfactant components only, while the other also includes bound counterions. Both models give a priori predictions of mixture behavior and give excellent agreement with experimental data. These results indicate that the factors giving rise to the thermodynamic nonidealities are primarily electrostatic in nature.
Introduction The understanding of physical mechanisms involved in the formation of micelles composed of mixtures of surfadants and the modeling of this process are areas of great theoretical and practical interest. One model commonly used to describe micelle formation is the pseudophase separation modell which considers the micelles as a thermodynamic phase in equilibrium with the monomer. By treatment of the monomer and micelle as pseudophases,the cmc of a mixture of similarly structured ionic surfactants'* or nonionic s~rfactantalJ-'~ can be predicted reasonably well by assuming that ideal solution theory is obeyed in the micellar phase. However, the cmc of ionic/nonionic surfactant mixtures can be much less than that predicted by ideal solution This *Towhom correspondence should be addressed.
indicates that mixed-micelle formation between these dissimilar surfactants is enhanced, relative to that between (1) Shinoda, K. In Colloidal Surfactants; Shinoda, K., Tamamushi, B., Nakagawa, T., Isemura, T., Eds.; Academic Press: New York, 1963; Chapter 1. (2) Scamehom, J. F.; Schechter, R. S.; Wade, W. H. J.Colloid Interface Sci. 1982,85, 479. (3) Shinoda, K. J. Phys. CheM. 1954,58,641. (4) Mysels, K. J.; Otter,R. J. J. Colloid Sci. 1961, 16, 474. (6) Barry. B. W.; Morrison,. J. c.;. Russell, G. F. J. J. Colloid Interface Scii iwo, 33,554. (61 Shedlovekv. _ .L.:. Jakob. C. W.:. Eostein. . . M. B. J.Phvs. Chem. 1963. 67, '2075. (7) Nishikido, N.;Moroi, Y.; Matuura, R. Bull. Chem. SOC. Jpn. 1975, 48,1387. (8) Lange, H.; Beck, K. H. Kolloid 2.2.Polym. 1973,251, 424. (9) F u n d , N.; Hada, 5.J. Phya. Chem. 1979,83, 2471. (10) Holland, P. M.; Rubingh, D. N. J. Phys. Chem. 1983,87, 1984. (11) Nishikido, N. J. Colloid Interface Sci. 1977, 60, 242. (12) Moroi, Y.; Akiaada, H.; Saito, M.; Matuura, R. J.Colloid Interface Sci. 1977, 61, 233.
0743-7463/S6/2402-0354$01.50/00 1986 American Chemical Society
Electrostatic Model To Describe Micellar Nonidealities
Langmuir, Vol. 2, No. 3, 1986 355
maximize accuracy.34 With the ring held near detachment, up surfactants of similar structure. to 4 h of equilibration were sometimes required to obtain an Regular solution theory has been very widely used to equilibrium surface tension reading. model the thermodynamic nonidealities of ionic/nonionic The fractional counterion bindings were calculated from EMF mixed micelles; it has been shown to accurately model cmc data by using specific ion electrodes with an Orion Research Inc. ~ a l u e s ~ O and J ~ ~monomer-micelle ’ equilibpium composidigital pH/mV meter, Model 701A. Since chloride and sodium t i o n ~ ” in , ~surfactant ~ systems exhibiting negative deviawere the counterions in this work, the electrodes used were tions from ideality. However, there is substantial and Corning solid-state chloride and Fisher sodium glass working varied evidence that regular solution theory is not therelectrodes and a Corning silver/silver chloride glass doublemodynamically valid in describing nonideal mixed mijunction reference electrode with an internal 4 M KC1 saturated AgCl solution and an external 1 M KN03 solution. Details on celles.% Therefore, the regular solution theory equations the procedure are given elsewhere.33 are useful empirical relationships but have no theoretical Materials. Highly pure hexadecylpyridinium chloride mofoundation. Alternative models to describe nonideal minohydrate (CPC), obtained from HEXCEL Specialty Chemicals, celle formation have been proposed, based on statistical was used as received. Hexadecyltrimethylammonium chloride m e c h a n i ~ s , ~ the l ” - ~mass-action model of micelle forma(CTAC)from Eastman Kodak Co. was purified by recrystallization t i ~ n , ~and O a group contribution method.31 A from ethyl alcohol and dried at low heat. Sodium dodecyl sulfate based on electrical effects has been proposed, but it is not (CI2SO4)from Fisher Scientific Co. was recrystallized from ethyl a predictive model. alcohol and water and dried at low heat. CTAC and CI2SO4were The most important mechanistic reason for the nonistored in vacuo. The nonionic surfactant used, NP(EO)lo,trade name IGEPAL CO-660, GAF Corp., is a polydisperse nonylphenol deality of the ionic/nonionic mixed micelles is believed to polyethoxylate with an average of 1 0 mol of ethylene oxide per be the reduction of the repulsion between the ionic head mol of nonylphenol. It was used as received. The purity of all groups due to the insertion of the nonionic hydrophilic groups between the charged g r o ~ p s In. this ~ ~ surfactants ~ ~ ~was~confirmed ~ ~ by~HPLC ~ analysis ~ ~ and ~ the~absence ~ ~of ~ minima in the surface tension curves. The sodium chloride was work, two models which include electrostatic effects are Fisher reagent grade and the water was distilled and deionized. developed to describe the thermodynamics of formation of mixed ionic/nonionic micelles. The resulting models Theory are fundamentally based and can aid in understanding the One approach that may be used to quantify the thermechanisms of interaction between surfactants in mixed modynamic nonidealities of mixed micelle formation is to micelles. One model assumes the micelle pseudophase to introduce an activity coefficient for each surfactant combe composed only of surfactant while the other model ponent in the micelle. On the basis of pure surfactant includes bound counterions as an additional component component micelles as standard states, these activity in the micelle. Unlike regular solution theory,these models coefficients are unity for ideal systems and generally less require no mixture parameters. The models are then apthan unity for ionic/nonionic systems because of negative plied to experimental data from one anionic/nonionic and deviations from ideality in mixed-micelle formation.le two cationic/nonionic surfactant systems. Only systems with swamping electrolyte will be considered here. Each surfactant in monomeric form is assumed Experimental Section to be so dilute as to obey Henry’s Law; i.e., based on the infinite dilution standard state, surfactant monomer acMethods. The cmc was determined from the sharp break in tivity coefficients are unity. Since the monomer is conthe surface tension vs. logarithm of the total surfactant concensidered to be in equilibrium with micelles, the fugacity of tration plots. The surface tension was measured using a CENCO each surfactant component in the monomer phase is equal Du-Nouy ring tensiometer, taking the necessary precautions to to that component’s fugacity in the micelle. For a binary mixture of ionic and nonionic surfactants, this results i d 6 (13) Kurzendorfer, C. P.; Schwuger, M. J.; Lange, H. Ber. Bumenges. Phys. Chem. 1978,82,962. 1966.43.133. (14) Schick, M. J.: Manning. D. J. J.Am. Oil Chem. SOC. (15) Schick, M. J. J. Am. ail Chem. SOC. 1966, 43, 681. (16) Scamehorn,J. F.;Schechter, R. S.;Wade, W. H. J.Dispersion Sci. Technol. 1982, 3, 261. (17) Rubingh, D.N.In Solution Chemistry of Surfactants; Mittal, K. L., Ed.; Plenum Press: New York, 1979; Vol. I, p 337. (18) Rosen, M. J.; Hua, X. Y. J. Am. Oil Chem. SOC.1982, 59, 582. (19) Holland, P. M. In Relation Between Structure and Performance of Surfactants; Rosen, M. J., Ed.; ACS Symposium Series; American Chemical Society: Washington, DC, 1984; Vol. 253, p 141. (20) Yoesting, 0.E.; Scamehorn, J. F. Colloid Polym. Sci., in press. (21) Osbome-Lee, I. W.; Schechter, R. S. J.Colloid Interface Sci. 1985, 108, 60. (22) Hua, X.Y.;Rosen, M. J. J. Colloid Interface Sci. 1982,90, 212. (23) Rosen, M. J.; Zhu, B. Y. J. Colloid Interface Sci. 1984, 99, 427. (24) Zhu, B. Y.; Rosen, M. J. J. Colloid Interface Sci. 1984,99,435. (25) Holland, P. M. Adu. Colloid Interface Sci., in press. (26) Nguyen, C. M.; Rathman, J. F.; Scamehorn, J. F. J. Colloid Interface Sci., in press. (27) Kamrath, R. F.;Franses, E. I. Ind. Eng. Chem. Fundam. 1983, 22, 230. (28) Scamehom, J. F.In Phenomena in Mized Surfactant Systems; Scamehorn, J. F., Ed.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986; Vol. 311, Chapter l. (29) Nagarajan, R. Langmuir, 1985, 1, 331. (30) Kamrath, R. F.; Franses, E. I. J. Phys. Chem. 1984, 88, 1642. (31) Asakawa, T.;Johten, K.; Miyagishi, S.; Nishida, M. Langmuir, 1985, 1, 347. (32) Moroi, Y.; Nishikido, N.; Saito, M.; Matuura, R. J . Colloid Interface Sci. 1975, 52, 356. (33) Rathman, J. F.; Scamehorn, J. F. J . Phys. Chem. 1984,88,5807.
xISYIcmcI = YIcmcM
(1)
~ N s Y N C ~ C= N YNC~CM (2) where subscripts I and N refer to ionic and nonionic surfactant, respectively, y is the activity coefficient of the component in the micelle, y is the mole fraction of the component in the monomer, and X I S and xNS are the mole fractions of ionic and nonionic surfactant in the micelle. The micellar mole fractions are surfactant-only based, as denoted by the letter S in the subscript. The mole fractions in the monomer phase, yI and YN, are also surfactant-only based. CmcMis the mixture cmc, and cmcI and cmcN are pure component cmc values measured at the same counterion concentration used for the mixtures. Since a component mole fraction is the ratio of the number of moles of that component to the total number of moles of surfactant in the phase of interest, then, by definition, XIS
+ xNS = 1
(3)
(4) + YN = 1 It is important to note that eq 1 and 2 apply only to mixtures at the cmc; thus, xIS and xNS are the mole fracYI
(34) Lunkenheimer, K.; Wantke, K. D. Colloid Polym. Sci. 1981,259, 354.
~
356 Langmuir, Vol. 2, No. 3, 1986
tions of ionic and nonionic surfactants in the first micelle formed. If the activity coefficients are known, eq 1-4 can be solved for the mixture cmc and the micellar composition as a function of monomer composition. A valid thermodynamic model for the mixed ionic/nonionic micelles must result in values of yI and YN that accurately predict the mixture cmc (an experimentally measurable quantity). Three models to obtain activity coefficients will be discussed here: regular solution theory, an electrostatic model, and a modified electrostatic model. Regular Solution Theory. In applying regular solution theory to mixed micelles, the micellar activity coefficients are given by35
Rathman and Scamehorn d U = T dS - P dV + CziniNAed\ko + Cpi dni (8) where pi is the chemical potential of species i in the micelle and d\ko is the difference in the electrical Stern layer potential betwsen the two phases of interest. It is straightforward to show that the proper form of the Gibbs-Duhem equation derived from eq 8 is C n i dpi = -S d T + V dP
(35) Balzhiser, R. E.; Samuels, M. R.; Eliassen, J. D. Chemical Engineering Thermodynamics; Prentice-Hall, Inc.; Englewood Cliffs, NJ, 1972; Chapter 9. (36) Newman, J. S. Electrochemical Systems; Prentice-Hall, Inc.; Englewood Cliffs, NJ, 1973; p 32
(9)
A t constant T and P and dividing by the total number of moles of surfactant in the micelle
cxis dpi = xzixi,NAed\ko
(10) For a binary mixed micelle containing ionic and nonionic surfactants, eq 10 reduces to
+ XNS dpN
XIS
where ym and ym are the regular solution theory activity coefficients for the ionic and nonionic surfactants in the micelle, respectively, WR is the regular solution theory interaction parameter, R is the gas constant, and T is absolute temperature. Theoretically, WR is independent of both temperature and the composition of the micelle. Electrostatic Model. First consider the ionic surfactant. Although the hydrophobic effects will be very similar for ideal (anionic/anionic or cationic/cationic) and nonideal (ionic/nonionic) mixed micelles, the chemical potential of the ionic surfactant will be very different in each case because of the difference in the electrical potential. For example, a nonionic surfactant generally has a much lower cmc than an ionic surfactant having the same hydrocarbon chain length1 because repulsion of the charged head groups inhibits micelle formation in the latter case. Since a micelle composed only of nonionic surfactant has zero electrical potential, the work required to insert an ionic surfactant molecule into the micelle, and thus the chemical potential of the ionic surfactant in the micelle, would be much less than in the case of inserting the ionic molecule into a micelle composed of another similarly charged ionic surfactant in the same proportion. As a result of this effect, the activity coefficient of the ionic surfactant in the micelle will generally be less than unity for ionic/nonionic mixtures and will vary with composition. One consequence of treating the mixed micelle as a separate thermodynamic phase is that, since the fractional counterion binding varies with composition, the electrical potential of the micellar phase is an additional intensive variable. The Gibbs-Duhem equation must be derived with this in mind. The differential amount of electrical work, 6 WEL,required to transfer ni moles of a charged species i between two phases having different electrical potentials iss6 6 WE, = -zin,iNAed\k (7) where NA is Avogadro's number, zi is the valence of species i,e is the charge per electron, and d\k represents the difference in electrical potential between the two phases. For a multicomponent surfactant system, the expression for the change in the internal energy for transferring molecules from one micelle phase to another micelle phase of different composition and different electrical potential must include the appropriate terms to account for electrical work. Using eq 7 for this purpose gives
+ CziniNAed\ko
(zIxIsNAe)d*o
(11)
where subscripts I and N denote ionic and nonionic surfactants, respectively, and !Po is the electrical potential in the Stern layer of the micelle. From the definition of activity coefficient based on pure component standard states for all micellar species, the chemical potential of species i in the micelle is pi = pio + R T In (-pi,) (12) where pio is the standard-state chemical potential. Thus dpi = R T d In yi + R T d In xis (13)
Since Cxisd In xL = 0, then substitution of eq 13 into eq 11 yields XIS
d In
71 .f xNS
d In
YN
= (zIxIsNAe/RT)d\ko
(14)
Equation 14 is the Gibbs-Duhem equation for a binary ionic/nonionic surfactant micelle. Since \ko varies with composition, thermodynamic consistency tests should be based on this expression and not on the more conventional Gibbs-Duhem equation in which the right-hand side of eq 14 is equal to zero. If we now assume that the nonionic surfactant micellar activity coefficient is constant, then its value must be unity at all compositions, using the pure-component standard state. YN
(15)
=
Equation 14 then simplifies to d In yI = (zINAe/RT) d\ko
(16)
Integration yields In
)'I'
= (Z,NAe\ko/RT)
+ KI
(17)
The assumption that YN = 1 is made independently of the derivation presented here. I t is not a necessary assumption; however, it allows us to obtain a predictive model, as will be seen. I t is important to note that this assumption does not violate the Gibbs-Duhem equation; from eq 14 it is apparent that it is possible for one activity coefficient to be constant while the other varies. One of the consequences of the assumption that the nonionic activity coefficient is constant is that we are implicitly assuming ideal entropy of mixing. If the mixing of surfactants in the micelle is highly nonrandom, then the nonionic surfactant would exhibit significantly entropic nonideality. The assumption that the mixing is approximately random does not conflict with the proposed mechanism which gives rise to the nonidealities (the insertion of nonionic groups decreasing the repulsive forces between ionic groups), since even in a randomly mixed micelle, the charge density and thus the ionic-ionic repulsions are less than in a pure ionic micelle.
Electrostatic Model To Describe Micellar Nonidealities The electrical potential at the micellar surface &e., in the Stern layer) is closely related to the fractional counterion binding on the micelle. A recently proposed model3 has been shown to predict bindings on mixed ionic/nonionic micelles very well by treating binding as the localized adsorption of counterions onto the charged hydrophilic surfactant groups. This model requires no mixture data but only the fractional counterion binding on the pure ionic micelle to obtain the value of the one adjustable parameter. The model predicts values of the Stern layer electrical potential, q0,as a function of micellar composition. This can be combined with eq 17 to predict the value of the activity coefficient of the ionic surfactant in the micellar phase at any composition. The constant KI can be determined from the pure-component standard-state boundary condition at =1
y1 =
1 (18) where \koa is the Stern layer electrical potential of the pure ionic micelle. Equations 1-4, 15, 17, and 18 and the electrical Stern layer potential of the micelle (from the counterion binding model) result in predicted values of the mixture cmc and micellar compositions from the electrostatic model. Since neither the binding model nor the equations given here require mixture data, this theory gives a priori predictions from pure-component data alone, unlike regular solution theory. Modified Electrostatic Model. Nearly all models used to describe mixed micelles regard the micelle as being composed only of the various surfactants in the system; counterions, co-ions, and water are neglected. Several s t ~ d i e s on ~ ~mixed-micelle, ~ ~ ~ - ~ ~ mixed-coacervate, and mixed-microemulsion systems have suggested that experimentally determined activity coefficients calculated on a surfactant-only basis may not be thermodynamically consistent; i.e., they violate the Gibbs-Duhem equation. In this modified electrostatic model, the bound counterions are considered as an additional component of the micelle pseudophase. Therefore xIM + xNM X, = 1 (19) XIS
9 0
=
*oo
+
where x I M , xNM, and x , are the mole fractions of ionic surfactant, nonionic surfactant, and counterion in the micelle. Since the counterion is included, xm and xm are, by definition, different from the composition terms used in regular solution theory and the electrostatic model, xIS and x N S , which are surfactant-only based. As before, we choose the standard state for the ionic surfactant to be the pure ionic micelle; however, since we are now including the counterion, the standard-state mole fraction of ionic surfactant is less than unity. Fractional counterion binding, /3, is defined as the number of counterions in the micelle per ionic surfactant molecule in the micelle P = Xc/XIM (20) The standard state for the counterion is chosen to be counterion bound to a pure ionic micelle. Thus, at the standard state x,o + XIMO = 1 (21) where xCo and x I M O are the micellar mole fractions of counterion and ionic surfactant, respectively, in the pure (37) Funasaki, N.;Hada, S. J. Phys. Chem. 1980,84,736-744. Scamehorn, J. F. J. Dispersion Sci. Technol., in press. (38) Hague, 0.; (39) Scamehorn, J. F. In Phenomena in Mixed Surfactant Systems; Scamehorn, J. F., Ed.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986; Vol. 311, Chapter 24.
Langmuir, Vol. 2, No. 3, 1986 357 ionic micelle. We can experimentally measure the fractional counterion binding on the pure ionic micelle, Po, and from eq 20
Po
= x,O/XIMO
(22)
Combining eq 21 and 22, we obtain an expression for the standard-state composition xIMO
= 1/(1
+ 0")
(23)
From the definition of activity coefficients, we can now calculate the standard-state activity coefficients for ionic surfactant and counterion ?Io = l / x I M o = 1 + Po (24) yco
=
= (1
l/xco
+ p")/p"
(25)
Note that in the standard-state pure ionic surfactant micelle, the activity coefficients of both the surfactant and counterion are not equal to unity, even though the respective activities are equal to unity. It is common to make cmc measurements on mixedsurfactant systems at a constant counterion concentration by using a swamping electrolyte; e.g., in this work, all systems were measured at 0.03 M NaCl. In this case, the chemical potential of the counterion in the monomer phase is constant; since the counterion bound to the micelle is assumed to be in equilibrium with counterion in the monomer phase, the chemical potential of counterion in the micelle, pC, must also be constant Pc = K2 (26) where K2 is a constant. We now write an equation for the counterion analogous to eq 12
+ R T In (Ycxc)
(27) where pco is the chemical potential of bound counterion on a mice!le containing only ionic surfactant. Equating eq 26 and 27 and then simplifying yield Puc = Pc O
Ycxc
= K3
(28)
where K3 is a constant. Applying the boundary condition given by eq 25, it is readily seen that K 3 = 1, so Yc
= l/xc
(29)
Equation 29 gives the relationship between the electrical Stern layer potential and the counterion activity coefficient in the micelle since the counterion mole fraction, x,, is a function of the Stern layer potential. The derivation of this expression for the counterion is similar to the derivation of eq 1 and 2 for the surfactant components in the micelle. To calculate the activity coefficient of the counterion and ionic surfactant in the micelle, we write eq 10 including terms for the ionic and nonionic surfactants and the counterion. Applying eq 13 and simplifying yield X I M d In 71 + xNM d In YN + X, d In yc = (ZIXIMNAe/RT + z , x a ~ e / R T d*, ) (30) Note that the surfactant mole fraction terms, xm and xm, are used in place of xIS and xNS in eq 10 and 13 in order to derive eq 30. Equation 30 is the Gibbs-Duhem equation for a mixed micelle in which the bound counterions are considered as a component of the micellar phase. Substituting eq 15 and 29 into eq 30, and then simplifying, yields d In yI = (21 + @z,)(NAe/RT)d\Eo + @ d In x , (31) As stated earlier, the electrostatic model for counterion binding gives both the fractional binding and the electrical
358 Langmuir, Vol. 2, No. 3, 1986
Rathman and Scamehorn
Table I. Summary of Counterion Binding Parameters for Localized Adsorption Model unbound counterion KB, system T,'C concn, M m3/kmol CPC/NP(EO)lo 30 0.030 1.51 CTAC/NP(EO)lo 30 0.030 0.81 CnSO,/NP(EO)io 30 0.030 1.16
UNBOUND [NACL] = 0.03 M IDEAL SOLUTION THEORY REGULAR SOLUTION THEORY ELECTROSTATIC YODEL MODIFtED ELECTROSTATIC MODEL
*----
/
/'
I
i IO0
80
60
..----CTAC/NP(
10
.-.--C
12
EO),
SO /NP(EO)lo 4
-CPC/NP( EO)
20
q 0.0
0.1
0.2
1 0.0
0.1
0.2
0.3
0.Y
0.5
0.6
0.7
0.8
0.9
1.0
MOLE FRACTION IONIC SURFACTANT IN MICELLE
Figure 1. ElectricalStem layer potential for mixed ionic/nonionic micelles calculated from localized adsorption model of counterion binding. Stern layer potential as a function of composition; thus, we know how both x , and @o vary with composition xIM. Equation 29 is used to calculate yc and eq 31 is integrated to give the ionic surfactant activity coefficient, yI, using the boundary condition at XIM
= XIMo
\kO
= \kOo
71
= TIo
(32)
where yIo is defined by eq 24. Equations 1,2,4, 15,19, 29, 31, and 32, when combined with the electrical Stern layer potential from the counterion binding model, can be solved to give mixture cmc values and micellar compositions as a function of monomer composition. Note that xIhl and xNhl must be used in place of XIS and xNS in eq 1and 2. In this model, the counterion has been included as part of the micelle phase without introducing any new parameters; as is the case for the first model presented, and only quantities needed are the pure ionic and nonionic cmc values and the fractional counterion binding on the pure ionic micelle. In the following discussion, the surfactant-only based model will be referred to as the electrostatic model and the model including counterion will be called the modified electrostatic model. Results and Discussion Following the procedure outlined fractional counterion binding measurements on the three ionic surfactants studied here were used to obtain the parameters given in Table I for the localized adsorption model. With these parameters, the binding model predicts the Stern layer electrical potential as a function of the surfactantonly based micellar composition; results are shown in Figure 1. Critical Micelle Concentration of Mixture. The experimentally measured cmc values for mixtures of CPC and NP(EO)lo as well as the values predicted by ideal solution theory, regular solution theory, the electrostatic model, and the modified electrostatic model are shown in
0.3
0.4
0.5
0.7
0.6
0.8
0.9
u
MOLE FRACTION CPC IN MONOMER
Figure 2. Comparison of experimentally measured cmc values with values calculated from ideal solution theory, regular solution theory, electrostatic model, and modified electrostaticmodel for a mixture of CPC and NP(EO),@
UNBOUND [NACL]
---
n
I
1 -----
*-.
= 0.03
M
IDEAL SOLUTION THEORY REGULAR SOLUTION THEORY ELECTROSTATIC MODEL MODIFIED ELECTROSTATIC MODEL
( /
/'I ,,/"
//
/'
/' /'
l l 10
0.0
,/'-
0.1
0.2
0.3
0.5
0.5
0.6
0.7
0.0
0.9
1.0
MOLE FRACTION CTAC IN MONOMER
Figure 3. Comparison of experimentally measured cmc values with values calculated from ideal solution theory, regular solution theory, electrostatic model, and modified electrostaticmodel for a mixture of CTAC and NP(EO)l,,. Figure 2. Similar results for the CTAC/NP(EO)lo and C12S04/NP(EO)losystems are given in Figures 3 and 4, respectively. The regular solution theory interaction parameter, WR, was determined by least-squares regression on the mixture cmc data. WR and the percent deviation of each model from experimental data are summarized in Table 11. As expected, all three systems show significant negative deviations from ideality. Regular solution theory does a very good job of fitting the mixture data; deviations between regular solution theory and actual data are generally
Langmuir, Vol. 2, No. 3, 1986 359
Electrostatic Model To Describe Micellar Nonidealities Table 11. Summary of Parameters for Cmc Models
AAD,"% modified
electrostatic
electrostatic
system
model
model
CPC/NP(EO),o CTAC/NP(Edjlo Ci804/NP(EO)io
4.30 6.81 10.33
4.46 6.27 9.67
regular solution theory WR/RT AAD, % -1.28 -1.66 -2.72
0.99 6.61 5.49
ideal solution theory AAD, % 23.9 35.0 26.2
AAD = average absolute deviation from experimental data.
1 = 3ooc
UNBOUND [NACL]
= 0.03
M
-------IDEAL SOLUTION THEORY
1
200
--SOLUTION THEORY - REGULAR ELECTROSTATIC YODEL
-------- MODIFIED ELECTROSTATIC MODEL
I ELECTROSTATIC YODEL
MOLE FRACTION C,2S0,
IN MONOMER
SURFACTANT-ONLY BASED MOLE FRACTION CPC IN MICELLE
Figure 4. Comparison of experimentally measured cmc values with values calculated from ideal solution theory,regular solution theory, electrostatic model, and modified electrostatic model for a mixture of C12S04and NP(EO),O.
Figure 5. Comparison of ionic surfactant activity coefficient calculated from regular solution theory and electrostatic model for a mixture of CPC and NP(EO)lo.
on the order of the experimental error. It is important to remember, however, that regular solution theory does not actually predict mixture cmc data; WRis calculated from the mixture data, so regular solution theory simply provides an excellent method of empirically fitting binary mixture data. Holland and Rubingh'O have successfully used regular solution theory as a predictive model when extrapolating from binary systems to ternary systems. The electrostatic and modified electrostatic models developed in this work provide a true method of making a priori predictions of mixed ionic/nonionic micelle behavior based only on pure-component data. Both models predict mixture cmc data extremely well. It is remarkable that the deviations given in Table I1 for both models are very close to the level of experimental error, especially when compared to the large deviations found when using ideal solution theory, which also uses only pure-component data. Interestingly, the predicted cmc curves for the electrostatic and modified electrostatic models very nearly overlap, particularly for the CTAC/NP(EO)lo and C12S04/NP(E0)10systems. Micellar Activity Coefficients. The ionic surfactant micellar activity coefficient calculated from regular solution theory and the electrostatic model for the CPC/NP(EO)lo system are plotted as a function of mole fraction of ionic surfactant in the micelle in Figure 5. Both models predict that yI increases monotonically with increasing ionic surfactant mole fraction. The activity coefficient of CPC predicted by the electrostatic model is significantly lower
than that calculated by regular solution theory at any given composition;this is expected since the electrostatic model attributes all nonidealities to the ionic component while regular solution theory assumes both ionic and nonionic components have activity coefficients less than unity in the mixture. Since the activity coefficients in the modified electrostatic model are based on a different standard state, they can not be compared directly with results from regular solution theory or the electrostatic model. The modified electrostatic model activity coefficients for counterion and ionic surfactant in the CPC/NP(EO)lo system are plotted in Figures 6. While the ionic surfactant shows negative deviation from ideality, the counterion shows a large positive deviation. This is expected since the work required to remove a counterion from the micelle is highest for the pure ionic (most highly charged) micelle and decreases as the mole fraction ionic surfactant decreases. The activity coefficients for the other two systems show behavior similar to the results given in Figures 5 and 6. Micelle Composition. Although the electrostatic and modified electrostatic models and regular solution theory describe mixture cmc data very well, the various techniques predict different micellar composition values. Because the micellar composition is defied differently for the modified electrostatic model (counterion included) than for regular solution theory and the electr&atic model (surfactant-only based), values from the modified electrostatic model must first be converted to a surfactant-only basis in order to compare with the other models. Given values of xIM, x m ,
360 Langmuir, Vol. 2, No. 3, 1986
Rathman and Scamehorn
100.0~
7
1.0:
I
I
IDEAL SOLUTION THEORY . --REGULAR SOLUTION THEORY ELECTROSTATIC MODEL
1.00-
/&-q II I
I I I
PURE CPC MICELLE --+: I I
0.01
-..._ 0.0
0. I
0.2
0.5
I
.. 0,s
0.Y
0.0
0 5
MOLE FRACTION CPC IN MICELLE
Figure 6. Micellar activity coefficienta of counterion and ionic surfactant calculated from modified electrostatic model for a mixture of CPC and NP(EO),,,.
0.2
0.Y
0.6
0.8
1.0
MOLE FRACTION CTAC IN MONOUER
Figure 8. Comparison of micellar composition as a function of monomer composition calculated from ideal solution theory, regular solution theory, electrostatic model, and modified electrostatic model for a mixture of CTAC and NP(EO)lo.
IDEAL SOLUTION THEORY REGULAR SOLUTION THEORY MODIFIED ELECTROSTA
0.0 0.D
0.2
0.Y
0.6
0.8
0.2
0.Y
MOLE FRACTION C,2S0, MOLE FRACTION CPC IN MONOMER
Figure 7. Comparison of micellar composition as a function of monomer composition calculated from ideal solution theory, regular solution theory, electrostatic model, and modified electrostatic model for a mixture of CPC and NP(EO)lo.
and x , from the modified electrostatic model, the surfactant-only based mole fraction ionic surfactant, xis, is XIS
= XIM/(XIM
+ XNM)
= X I M / ( ~ - XIMP)
0.6
0.8
1.c
1.0
(33)
In Figure 7, the surfactant-only based mole fraction CPC in the micelle is shown as a function of monomer composition for ideal solution theory, regular solution theory, and the electrostatic and modified electrostatic models. Similar results for the CTAC and CI2SO4systems are shown in Figures 8 and 9, respectively. All three nonideal models predict azeotropic behavior. There is limited evidence that the monomer-micelle equilibrium compositions from regular solution theory do agree with experimental data
IN MONOMER
Figure 9. Comparison of micellar composition as a function of monomer composition calculated from ideal solution theory, regular solution theory, electrostatic model, and modified electrostatic model for a mixture of CI2S4and NP(EO),@
for these types of system^.'^^^^ The monomer-micelle equilibrium compositions from regular solution theory and the modified electrostatic model are very similar in Figures 7-9; therefore, the modified electrostatic model may be more accurate than the electrostatic model, but additional data are needed to confirm this. Regular solution theory empirically describes mixed cmc values and micellar compositions. The modified electrostatic model, based on fundamental mechanisms, predicts essentially the same results for mixture cmc data and monomer-micelle equilibrium as regular solution theory. Since regular solution theory is based on incorrect assumptions for these systems, it is satisfying to demonstrate that the quantification of the forces believed to cause the
Electrostatic Model To Describe Micellar Nonidealities nonidealities of mixing can be used to develop a model that accurately describes experimental results. Such a fundamentally based model should, for example, be able to predict temperature effects much better than regular solution theory, which has been shown to be inaccurate for such prediction^.'^ Future studies will test this hypothesis.
Conclusions It is generally agreed that the large negative deviations from ideality exhibited by ionic/nonionic surfactant micelles is in large part due to the reduced repulsion between ionic head groups in the micelle upon addition of nonionic molecules. The electrostatic and modified electrostatic models presented in this work have a sound theoretical basis and can be used to accurately predict mixture cmc values from pure-component data alone. The fact that these proposed models do not require mixture parameters supports the initial hypothesis that the nonidealities of mixed systems can in large part be explained by electrostatic considerations alone.
Acknowledgment. Financial support for this work was provided by the Oklahoma Mining and Minerals Resources Research Institute and the OU Energy Resources Institute. Cuong M. Nguyen and Kevin L. Stellner helped obtain the cmc data. We also acknowledge George J. Hirasaki for his encouragement to develop a mixed-micelle model based on electrostatics.
Langmuir, Vol. 2, No. 3, 1986 361 xc
xco
Xis xIM
xIMo XIS ~ N M
xNS
YI YN
zi ZI
? Po Yc
YCO YI YIO
Glossary critical micelle concentration, kmol/m3 cmc of pure ionic surfactant, kmol/m3 cmc of pure nonionic surfactant, kmol/m3 cmc of ionic/nonionic surfactant mixture, kmoi/m3 charge per electron, 1.6 X C constant from localized adsorption model of counterion binding, m3/kmol dimensionless constants
PC
Avogadro's number, 6.023 X molecules/ kmol moles of species i gas constant, 8314 J/(kmol K) or 1.987 kcal/(mol
PI0
K)
temperature, K differential work required to move a charged particle through a differential electrical potential gradient, J regular solution theory interaction parameter, kcal/ kmol
YIR YN
YNR PC0
Pl PI
\k *O \EO0
mole fraction counterion in micelle (modified electrostatic model) mole fraction counterion in a micelle composed only of ionic surfactant (modified electrostatic model) mole fraction of species i in micelle mole fraction ionic surfactant in micelle (modified electrostatic model) mole fraction ionic surfactant in a micelle composed only of ionic surfactant (modified electrostatic model). surfactant-onlybased mole fraction ionic surfactant in micelle mole fraction nonionic surfactant in micelle (modified electrostatic model) surfactant-only based mole fraction nonionic surfactant in micelle surfactant-onlybased mole fraction ionic surfactant in monomer surfactant-only based mole fraction nonionic surfactant in monomer valence of species i valence of ionic surfactant valence of counterion fractional counterion binding on mixed ionic/nonionic micelle fractional counterion binding on pure ionic micelle activity coefficient of counterion in micelle (modified electrostatic model) standard-state activity coefficientof counterion in micelle (modified electrostatic model) activity coefficient of ionic surfactant in micelle standard-state activity coefficient of ionic surfactant in micelle (modified electrostatic model) 71 calculated from regular solution theory activity coefficient of nonionic surfactant in micelle yN calculated from regular solution theory chemical potential of counterion in micelle (modified electrostatic model), J/kmol chemical potential of counterion in pure ionic micelle (modified electrostatic model), J/kmol chemical potential of species i, J/kmol chemical potential of ionic surfactant in micelle, J/kmol chemical potential of ionic surfactant in pure ionic micelle, J/kmol electrical potential, V electrical Stern layer potential of mixed micelle, V electrical Stern layer potential of pure ionic micelle,
v
Registry No. CPC, 123-03-5;CTAC, 112-02-7; SDS, 151-21-3; NP(EO),, 9016-45-9.
