Langmuir 1998, 14, 2297-2306
2297
A Self-Consistent Multicomponent Activity Coefficient Model for Ionic Micellar Surfactant Solutions D. A. Amos,† Scott Lynn, and C. J. Radke* Department of Chemical Engineering, University of California, Berkeley, California 94720-1462 Received June 3, 1996. In Final Form: February 4, 1998
The proposed model incorporates a distribution of micellar sizes self-consistently. The model is based on a mass-action equilibrium approach that includes micelle-micelle interactions as a function of size for a multicomponent surfactant solution consisting of micellar aggregates, monomer, counterions, and added electrolyte. The primary solution nonidealities are accounted for in the multicomponent model with excludedvolume and electrostatic interactions as a function of aggregate size. In addition, the model accounts for the Donnan equilibrium existing between an ionic surfactant solution and the electrolyte solution from which it is separated by a semipermeable membrane. Surfactant solutions of sodium dodecyl sulfate (SDS) and cetylpyridinium chloride (CPC) in 0.01 M NaCl are studied over the concentration range from the critical micelle concentration up to volume fractions of 0.19 (0.87 M) for SDS and 0.16 (0.56 M) for CPC. In comparison to the predictions of an ideal solution multiple-chemical equilibrium constant model, the activity-coefficient model predicts increased growth and polydispersity of the aggregates for both CPC and SDS at higher surfactant concentrations. Micellar interactions enhance the growth of the micelles due to excluded-volume effects that favor growth and electrostatic repulsions that oppose it. Micellar aggregates are found to be slightly globular for both SDS and CPC; however, a clear spherical to globular transition is predicted for the CPC micelles. The multicomponent model reflects experimental osmotic pressure data successfully.
Introduction Micellar aggregates of ionic surfactants frequently are assumed to be monodisperse for calculational purposes, even though they are known to exhibit varying degrees of polydispersity. The success of this simplification can be attributed to the relatively narrow width of the size distribution as observed by neutron1 and dynamic light scattering.2 In addition, although models to calculate multiple equilibrium constants exist, the experimental determination of multiple equilibrium constants is still unattainable.3 So the results of even the more complete multicomponent models ultimately must be compared to average solution properties with only a limited number of exceptions. Thus, it is reasonable to model surfactant solutions with a single equilibrium constant and average aggregation number, at least as a first approximation. Many studies show that despite these simplifying assumptions, it is still possible to capture most equilibrium thermodynamic properties of the system in comparisons between experiment and theory.3-9 However, despite the obvious attraction of assuming a single micelle size, the presence of a distribution of * To whom correspondence should be addressed. † Currently at Eastman Kodak Co., 1999 Lake Avenue, Rochester, NY 14650-2116. (1) Hayter, J. B.; Penfold, J. Colloid Polym. Sci. 1983, 261, 1022. (2) Brown, J. C.; Pusey, P. N.; Goodwin, J. W.; Ottewill, R. H. J. Phys. A: Math. Gen. 1975, 8, 664. (3) Moroi, Y. Micelles: Theoretical and Applied Aspects; Plenum Press: New York, 1992. (4) Moroi, Y. J. Colloid Interface Sci. 1988, 122, 308. (5) Vold, M. J. J. Colloid Sci. 1950, 5, 506. (6) Bendedouch, D.; Chen, S.-H.; Hoehler, W. C. J. Phys. Chem. 1983, 87, 2621. (7) Mille, M.; Vanderkooi, G. J. Coll. Inter. Sci. 1977, 59, 211. (8) Burchfield, T. E.; Woolley, E. M. J. Phys. Chem. 1984, 88, 2149. (9) Douhe´ret, G.; Viallard, A. Fluid Phase Equilib. 1982, 8, 233.
aggregate sizes is well accepted.10 Experimental evidence of the polydispersity and growth of micelles in surfactant solutions has been investigated using a wide variety of techniques over a range of different length scales, including light scattering,11 viscosity measurements,12 time-resolved fluorescence quenching,13 NMR,14 and small-angle neutron scattering.1,15 The equilibrium distribution of micelles may shift to larger or smaller average sizes, depending on the solution concentration, temperature, pressure, and other factors such as ionic strength. For example, dynamic and static light-scattering results for SDS show that the average micellar size increases with increasing surfactant concentration at high ionic strength where the electrostatic repulsions have been screened sufficiently.16,17 Other experimental evidence for growth in surfactant solutions has shown increasing micellar aggregation numbers as a function of total surfactant concentration,18 salt,12 and added organic solutes or cosurfactants.19,20 In addition, the effect of aggregate interactions on the growth of micelles has been addressed in several recent studies, (10) Mukerjee, P. J. Phys. Chem. 1972, 76, 565. (11) Mazer, N. A.; Carey, M. C.; Benedek, G. B. In Micellization, Solubilization, and Microemulsions; Mittal, K. L., Ed.; Plenum Press: New York, 1977; Vol. 1, p 359. (12) Khatory, A.; Kern, F.; Lequeux, F.; Appell, J.; Porte, G.; Morie, N.; Ott, A.; Urbach, W. Langmuir 1993, 9, 933. (13) Croonen, Y.; Gelade´, E.; Van der Zegel, M.; Van der Auweraer, M.; Vandendriessche, H.; Schryver, F. C. D.; Almgren, M. J. J. Phys. Chem. 1983, 87, 1426. (14) Leaver, M.; Furo´, I.; Olsson, U. Langmuir 1995, 11, 1524. (15) Sheu, E. Y.; Chen, S.-H. J. Phys. Chem. 1988, 92, 4466. (16) Hayashi, S.; Ikeda, S. J. Phys. Chem. 1980, 84, 744. (17) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y. J. Phys. Chem. 1980, 84, 1044. (18) Quina, F. H.; Nassar, P. M.; Bonilha, J. B. S.; Bales, B. L. J. Phys. Chem. 1995, 99, 17028. (19) Kumar, S.; Aswal, V. K.; Singh, H. N.; Goyal, P. S.; Kabir-udDin. Langmuir 1994, 10, 4069. (20) Porte, G.; Marignan, J.; Bassereau, P.; May, R. J. Phys. France 1988, 49, 511.
S0743-7463(96)00540-9 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/03/1998
2298 Langmuir, Vol. 14, No. 9, 1998
particularly for surfactants that form elongated or cylindrical micelles.15,21-24 In the present work, we investigate the importance of micellar interactions in nonideal aqueous surfactant solution. We utilize osmotic pressure data for sodium dodecyl sulfate (SDS) and cetylpyridinium chloride (CPC) coupled with our multicomponent activity coefficient model to study a wide range of surfactant concentrations above the critical micelle concentration (cmc). Unlike surfactants that form cylindrical or rodlike structures in aqueous solution, both SDS and CPC are known to form globular micelles in the ionic-strength range studied here.16,20 Thus, as a first approximation we model the aggregates as spherical micelles. Previous calculations have illustrated the importance of repulsive interactions as a function of concentration in explaining nonlinear osmotic pressure results by assuming the micelles behave as monodisperse spheres.25,26 In contrast to previous work, we examine the effect of electrostatic and excluded-volume interactions not only on the osmotic pressure of spherical aggregates but also on the distribution of micelle sizes as a function of surfactant concentration. The activity coefficient model includes surfactant monomers, counterions, added electrolyte, and a distribution of micelle sizes. The multicomponent model allows for self-consistent predictions of the micelle size distributions and the growth of the aggregates above the critical micelle concentration. Theory Method of Investigation. A multicomponent model that is consistent with a multiple-equilibrium (or massaction) description is necessary to treat a polydisperse micellar surfactant system. Thus, the equilibrium constants for the reaction between a micellar aggregate and the singly dispersed surfactant monomer must vary as a function of size or number of monomer molecules, g, present per micelle. In addition, the nonideal-solution corrections include a distribution of aggregate sizes and aggregation numbers along with counterions and added electrolyte, all in a self-consistent manner. The major assumptions of our multisize micelle solution model follow: (i) The surfactant aggregates form spherical or roughly spherical, globular micelles, which may be characterized by their effective spherical radius. (ii) At the cmc, a multiple-chemical equilibrium model, where each aggregate of size g is considered as a separate species, is a suitable reference state for the formation of an ideal, dilute solution of micellar aggregates. (iii) The equilibrium constants calculated for a surfactant solution at the cmc are sufficient for the description of the multiple chemical equilibria at surfactant concentrations above the ideal-solution concentration regime. (iv) The dominant contributions to the interactions between micelles and other solutes in solution are excluded-volume and electrostatics. For the excludedvolume interactions, the micelles, monomers, and other ions are modeled as hard spheres.27 In addition to the hydrated ionic radii used for the added electrolyte, the (21) Aswal, V. K.; Goyal, P. S.; Menon, S. V. G.; Dasannacharya, B. A. Physica B 1995, 213 & 214, 607. (22) Delsanti, M.; Moussaid, A.; Munch, J. P. J. Colloid Interface Sci. 1993, 157, 285. (23) van der Schoot, P.; Cates, M. E. Langmuir 1994, 10, 670. (24) Nagarajan, R. Langmuir 1994, 10, 2028. (25) Amos, D. A.; Markels, J. H.; Lynn, S.; Radke, C. J. J. Phys. Chem. B 1998, 102, 2739. (26) Amos, D. A. Characterization and Modeling of Aqueous Micellar Surfactant Solutions Ph.D. Dissertation, University of California, Berkeley, 1996.
Amos et al.
effective hard-sphere diameters for the micellar aggregates modeled here include an additional layer roughly equivalent to a monomolecular hydration sheath. Activity-Coefficient Model. The activity-coefficient model is applied to aqueous ionic surfactant solutions in the micellar concentration regime at or above the critical micelle concentration prior to the onset of more complicated phase behavior. Its basis is the thermodynamic equilibrium between the surfactant monomer at volume fraction, φ1, and the surfactant aggregate composed of g surfactant monomers and volume fraction, φg. This chemical equilibrium may be written as
φgγg ) (φ1γ1)g exp
[
]
-(µ°g - gµ°1) ) (φ1γ1)gKg kbT
(1)
where µ°g is the standard chemical potential of a micelle composed of g surfactant monomers in an infinitely dilute solution, γg is the activity coefficient for the micelle of aggregation number g, kb is the Boltzmann constant, and T is the absolute temperature. The total change in standard chemical potential, ∆µ°g ≡ µ°g - gµ°1, is calculated for the ideal solution according to the thermodynamic model of Nagarajan and Ruckenstein for a surfactant micelle relative to an infinitely dilute reference state.28 This thermodynamic model accounts for micelle formation by separating the free energy contributions for distinct intramolecular interactions within the micelle interior and at the micelle interface relative to its surrounding aqueous solution. Accordingly, the main energetic contributions to the micelle formation model include a transfer free energy contribution to account for the so-called hydrophobic effect where the hydrocarbon surfactant tails are removed from an aqueous environment into the oil-like interior of the micellar aggregate; a contribution to account for geometric constraints on the surfactant tails within the micelle interior; a contribution to account for the presence of the hydrocarbon/water interface present at the micellar surface where water may penetrate the charged headgroups; a contribution to account for the proximity of the charge to the hydrocarbon/water interface; a contribution to account for the steric interactions among surfactant headgroups; and finally, a contribution to account for the electrostatic interactions between the ionic headgroups at the micellar surface. Once ∆µ°g is known as a function of g, the equilibrium constant, Kg, follows by definition: Kg ) exp(-∆µ°g/kbT). With the exception of a few changes,25,26 the thermodynamic model is used in its entirety, and we refer the reader to this work for further details.28 The activity coefficients in eq 1 may be calculated in general on the basis of the total excess Helmholtz free energy such that
ln γj ) ln γj, hs + ln γj, cc
(2)
where solute j may be either a surfactant micelle of aggregation number g, surfactant monomer, counterion, or added electrolyte ion. All activity-coefficient corrections consist of an excluded-volume, hard-sphere contribution, (27) Blum, L. In Theoretical Chemistry: Advances and Perspectives; Eyring, H. Henderson, D., Ed.; Academic Press: New York, 1980; Vol. 5, p 1. (28) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934.
Coefficient Model for Surfactant Solutions
Langmuir, Vol. 14, No. 9, 1998 2299
γj,hs, and an electrostatic charge-charge contribution, γj,cc, for solute j in a multicomponent mixture of i species of volume fraction φi (or number density Fi), diameter di, and valence zi. The Boublik-Mansoori multicomponent hard-sphere equation of state accounts for the excludedvolume interactions.29 The general form of the hardsphere activity coefficient expression for the individual solute j is then
ln γj,hs )
{(
)
3ξ22dj2 2ξ23dj3 - 1 ln(1 - ξ3) + ξ32 ξ33
( )( ) ξ3 -
ξ23
3(ξ1dj2 + ξ2dj) 3ξ1ξ2dj3 dj3 + + + 1 - ξ3 (1 - ξ3) (1 - ξ )2
ξ32
}
3
3ξ22dj2
ξ23dj3
2ξ23dj3
+ ξ3(1 - ξ3)2 ξ32(1 - ξ3)2 ξ3(1 - ξ3)3
(3)
where
ξm )
π
∑Fidim
m ) 0, 1, 2, or 3
with
6 i*0
(4)
where the quantity 2Γ is analogous to the Debye-Hu¨ckel inverse screening parameter, κ, with one important distinction.30 In the MSA framework, the finite size of the solute molecules is accounted for self-consistently. In the Debye-Hu¨ckel equation, however, all ions are modeled as idealized point charges. This difference makes the MSA appropriate at concentrations far outside the range of the Debye-Hu¨ckel model. To calculate the osmotic pressure of the multicomponent surfactant solution and to make comparisons with the experimental osmotic pressure data,26 one final set of equilibrium restrictions is required. The osmotic equilibria between the surfactant and solvent solutions coupled with the electroneutrality condition for both solutions leads to the well-known Donnan contribution to the osmotic pressure due to an excess concentration of counterions on the solution side of the membrane.31 The physical origin of this effect derives from the osmotic equilibria between the surfactant and electrolyte ions, which are separated by a semipermeable membrane that excludes the surfactant micelles and monomer only. Phase equilibrium between the two solutions containing permeable microions on each side of the membrane is imposed such that
(f(F-F+)(2) - (f(F-F+)(1) ) 0
The particles are treated as hard spheres where the number density is related to the volume fraction of species i by the relation, φi ) (π/6)Fidi3. The electrostatic interactions included in the activity coefficient expression of eq 2 are accounted for using the mean spherical approximation (MSA) referenced to the hard-sphere diameters of the multiple solution components. The analytic expression for the individual activity coefficient yields30
{[
]
π P d πe2 zj zjΓ 2∆ n j + + ln γj,cc ) kbT π 1 + Γdj 1 + Γdj π 2 Pndj zj + 2∆Pndj π P d2 + ∆ 12∆ n j 2(1 + Γdj)
[
(
)
]}
(2Γ)2 )
4πe2
[
∑ Fi
kbT i*0
zi -
2∆
]
di2Pn
1 + Γdi
(5)
2
∆)1-
∑ Fkdk 6k*0
3
(6)
(7)
and
Fkdkzk
∑ 1 + Γd
Pn ) 1+
k*0
k
π
Fdk3
∑
2∆ k*0 1 + Γdk
(29) Boublı´k, T. J. Chem. Phys. 1970, 53, 471. (30) Blum, L.; Høye, J. S. J. Phys. Chem. 1977, 81, 1311.
(z-F-)(1) + (z+F+)(1) ) 0 (z-F-)(2) + (z+F+)(2) +
where
π
where f( is the mean ionic activity coefficient on a number density basis, the superscripts (1) and (2) refer to the solvent and the solution sides of the membrane, respectively, and the subscripts refer to the valence of the added salt ions. Electroneutrality on either side of the semipermeable membrane reads
solvent side
(10)
and
The MSA screening parameter, Γ, must be solved for by iteration as a function of solute number density, Fi, based on the following equation:30
π
(9)
(8)
∑g (zgFg)(2) ) 0
solution side (11)
Equations 9-11 comprise the so-called Donnan equations. Here, they are coupled with a mass balance for the added 1-1 electrolyte,
4F°( + F1 - [(F- + F+)(2) + (F- + F+)(1)] +
∑g βgFg ) 0
(12)
to specify completely the microion concentrations on both sides of the membrane, where F°( is the initial concentration of a monovalent electrolyte added prior to equilibration and β is the degree of dissociation of the counterions at the micellar interface. In eq 12, the initial salt concentration has been set equal on both sides of the membrane. The mean activity coefficient is then
ln γ( )
∑φi ln γi ∑φi
(13)
Upon rearrangement, eqs 9-12 predict an excess concentration of the surfactant counterions in the surfactant solution, which gives rise to an additional contribution to the total osmotic pressure known as the Donnan pressure. For a more detailed account of this effect on the osmotic pressure calculations, see Amos and the references therein.26 Equations 1-13 form the basis of the activity-coefficient model for the micellar surfactant solution. Activity coefficients are calculated as a function of volume fraction (31) Donnan, F. G.; Harris, A. B. J. Chem. Soc. (London) 1911, 99, 1554.
2300 Langmuir, Vol. 14, No. 9, 1998
and aggregation number, g, according to eq 2 and coupled with eq 1. There exists a vector of coupled equations, one for each aggregation number g included in the calculation. The total surfactant volume fraction is then calculated from a surfactant mass balance as
∑ φg
(14)
gmin
where the summation includes m aggregate sizes ranging from gmin to gmax. Equation 14 serves as the final relationship needed to specify completely the set of l equations and l unknowns. The model describes a distribution of m micellar aggregates of volume fraction φg as defined above, plus the surfactant monomer and the four microion concentrations representing the counterions and coions present from the added electrolyte on each side of the osmotic membrane. Thus, for each total surfactant volume fraction, φ, the total number of equations and unknowns is l ) m + 5. The equations are defined by an array of m different equilibrium equations (eq 1) coupled with m + 1 activity coefficients (eq 2) including the monomer, plus eqs 9-12 for the Donnan equilibria and the total surfactant mass balance eq 14. In practice, after a scaling change in the variables the coupled equations are solved iteratively along with eq 6 for the MSA screening parameter using the Newton-Raphson method.26 The result is a distribution of micelle volume fractions, φg, representing the equilibrium size distribution for the nonideal surfactant solution at concentration φ for m micellar aggregates ranging from gmin to gmax, as shown in eq 14. The range of micellar aggregates specified was determined sufficient when the inclusion of additional micelle sizes (i.e., increasing m) had no effect on the average properties of the size distribution. These comparisons were made based on changes in the cmc, the average aggregation number, and the standard deviation for the distribution. The modified size distributions, including the effect of interactions, were then used in the calculation of the osmotic pressure, as discussed below. The degree of counterion dissociation, β, is used in the model as an adjustable parameter. In addition, the effective diameter of the micelles includes a hydration thickness, h, which is maintained at a constant value for all micellar aggregates in the distribution, such that deff,g ) 2(rg + h). The micellar radius, rg, is calculated according to the geometrical constraints for a spherical or for a globular, ellipsoidal micelle with an effective spherical radius. The diameter, deff, reported in the figures and text that follows is calculated based on the average micelle aggregation number and size unless otherwise specified. For further calculation details, see Appendices 2A, 3A, and 3B of Amos, and the references therein.26 Osmotic Pressure. The osmotic pressure of the surfactant solution includes the ideal contributions from the dissolved solutes as well as the nonideal contributions due to excluded-volume and electrostatic interactions. Since added salt is present on both sides of the membrane,25,32 the osmotic pressure is calculated by difference as
Π ) Π(2) - Π(1)
(15)
where the osmotic pressure on each side of the membrane is defined by (32) Markels, J. H.; Lynn, S.; Radke, C. J. AIChE J. 1995, 41, 2058.
[
{( ) Fi
∑i 1 - ξ
+
3
]}
3 6 3ξ1ξ2(1 - ξ3) - (ξ2) (ξ3 + 2/ξ3 - 5)
π
gmax
φ ) φ1 +
Π(1,2) ) kbT
Amos et al.
(1 - ξ3)3
+ Pex,cc (16)
as derived previously, and the summation is extended over all solutes i.26 The third term in eq 16 is the excess pressure due to the electrostatic, charge-charge interactions, which is calculated as30
Pex,cc ) -kbT
[
( )]
Γ3 πe2 Pn + 3π 2kbT ∆
2
(17)
The Donnan contribution is calculated implicitly as a part of eq 16 based on the differences in the microion distributions between the surfactant solution and the electrolyte solvent for the coupled equations. Results and Discussion Size Distributions and Micellar Growth. Figure 1 displays the effects of the repulsive micelle interactions on the aggregate size distribution as a function of micelle concentration. In this plot, the weight-average aggregation numbers as a function of micelle volume fraction calculated according to the activity-coefficient model outlined above are compared to an ideal-solution model prediction (i.e., γg ) 1). Results are shown for aqueous solutions of sodium dodecyl sulfate (SDS) and cetylpyridinium chloride (CPC) in 0.01 M NaCl for the parameters as listed. Calculated critical micelle concentrations in terms of volume fractions are 1.5 × 10-3 (〈gw〉 ) 71) and 5.2 × 10-5 (〈gw〉 ) 87) for SDS and CPC, respectively, in agreement with experimental cmc data.25 The idealsolution calculations are based on the mass-action equilibrium model of Nagarajan and Ruckenstein for Kg28 and include a distribution of aggregation numbers from 2 to a maximum of 250 monomers per micelle.25,26 Based on the parameters determined for SDS in comparison to osmotic pressure data presented in the next section, the fractional ionization of the surfactant molecules at the micellar interface is set here as β ) 0.05. The calculated magnitude of the average micelle valence is z〈g〉 ) 4.3-5.2 over the concentration range studied where z〈g〉 ) β〈gw〉. The quantity, z〈g〉, represents the magnitude of the total valence at the micelle interface where the constitutive monomers are only partially dissociated. The effective diameter calculated based on the average size with an additional hydration layer of h ) 6.5 Å is 49-59 Å over the volume fraction range shown in Figure 1. In the multicomponent calculation shown in Figure 1 for CPC, the valence magnitude is z〈g〉 ) 8.2-11.5 for β ) 0.08 with an effective diameter of 63-70 Å and h ) 5.5 Å. Figure 1 shows an initial increase in 〈gw〉 for the idealsolution calculations involving both SDS and CPC at 30 °C. The rapid initial change in weight-average aggregation number is in the transition region of the cmc where there is a drastic shift from singly dispersed monomers to larger, self-assembled micellar aggregates. This effect is similar to that shown experimentally for the nonionic C10E8 surfactant using gel-filtration chromatography and is consistent with the use of an ideal-solution multiple
Coefficient Model for Surfactant Solutions
Figure 1. Calculated weight-average aggregation numbers for sodium dodecyl sulfate (SDS) and cetylpyridinium chloride (CPC) in 0.01 M NaCl at 30 °C. Solid lines represent the idealsolution (γg ) 1) equilibrium coefficient model, and dashed lines reflect the self-consistent multicomponent activity coefficient model. Parameters: β ) 0.05 (SDS), 0.08 (CPC); h ) 6.5 Å (SDS), 5.5 Å (CPC). Remaining parameters are available elsewhere.25,26
chemical equilibrium model.33 The period of rapid change in aggregation number is followed by a gradual increase in average size, as shown also in Figure 1. An increase in the micelle aggregation number is inherently predicted by an ideal-solution mass-action model of eq 1, regardless of the properties of the surfactant.10,34 However, even though an increase in surfactant aggregation number is expected physically with increasing surfactant concentration,10 the increase in the monomer volume fraction predicted by the ideal-solution mass-action model is in contradiction to experimental data.35-40 Experimental results show a decreasing monomer concentration as a function of increasing surfactant concentration. This apparent discrepancy in the change in monomer concentration is explained by the incorporation of solution interactions into the thermodynamic model.24,25 When micelle interactions are included via the proposed activity-coefficient model, a similar rapid growth in the micellar aggregation number is observed, as shown in Figure 1 at concentrations around the cmc. However, in contrast to the ideal-solution calculation, this initial behavior is followed by a period of continued growth in the micelles up to volume fractions of 20%. The growth predicted by the activity-coefficient model is more pronounced than in the ideal case, as illustrated in Figure 1. The largest growth is exhibited by the cationic CPC surfactant with its 16-carbon, hydrophobic tail, as compared to the smaller, anionic SDS surfactant with its 12carbon surfactant tail. The increase in average aggregation number for the CPC micelles is from 20-60% as a function of the total surfactant concentration; the calculated increase in the SDS micelle size is from 10(33) Funasaki, N.; Shim, H.-S.; Hada, S. J. Chem. Soc., Faraday Trans. 1991, 87, 957. (34) Israelachvili, J.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525. (35) Kale, K.; Cussler, E. L.; Evans, D. F. J. Phys. Chem. 1980, 84, 593. (36) Kale, K. M.; Cussler, E. L.; Evans, D. F. J. Soln. Chem. 1982, 11, 581. (37) Mysels, K. J.; Mukerjee, P.; Abu-Hamdiyyah, M. J. Phys. Chem. 1963, 67, 1943. (38) Sasaki, T.; Hattori, M.; Sasaki, J.; Nukina, K. Bull. Chem. Soc. Jpn. 1975, 48, 1397. (39) Cutler, S. G.; Meares, P.; Hall, D. G. J. Chem. Soc., Faraday Trans. 1 1978, 74, 1758. (40) Elworthy, P. H.; Mysels, K. J. J. Colloid Interface Sci. 1966, 21, 331.
Langmuir, Vol. 14, No. 9, 1998 2301
Figure 2. Ideal-solution size distribution for SDS in 0.01 M NaCl at 30 °C: (1) φ ) 0.021, 〈gw〉 ) 74; (2) φ ) 0.069, 〈gw〉 ) 75; (3) φ ) 0.117, 〈gw〉 ) 76; (4) φ ) 0.165, 〈gw〉 ) 76; (5) φ ) 0.200, 〈gw〉 ) 76.
20%. The differences in the magnitude of the growth between these two ionic surfactants is expected due to a combination of effects, including variations in the hydrophobic chain lengths, micelle size, and electrostatic headgroup repulsions. For example, it is well documented that surfactants with long hydrophobic tails have lower cmc’s and form larger minimum aggregate sizes than those with the identical headgroups and shorter chain lengths.41 However, above the cmc, differences in size and growth patterns may be attributed to changes in micelle shape, electrostatic repulsions, and the optimal surface area per surfactant headgroup based on geometric constraints.34 To interpret the behavior seen in Figure 1 better, we examine the micelle size distributions for the two ionic surfactants calculated for ideal and nonideal solution conditions. Figure 2 shows a series of size distributions for an ideal SDS solution at various total surfactant volume fractions up to 0.20. As anticipated from the discussion above, there is a slight increase in weight-average aggregation numbers for SDS from 74 to 76, and the predicted distributions are symmetric and rather narrow. These aggregation numbers are consistent with values observed from static light scattering and SANS measurements for SDS in 0.0-0.02 M NaCl in the vicinity of the cmc at 25 and 30 °C.16,18 The calculated distributions are typical of SDS and other surfactants that form relatively small, spherical or globular, micelles with a narrow, roughly Gaussian distribution of sizes.10,34 The corresponding ideal-solution size distributions are shown for CPC in Figure 3 for a similar range of surfactant concentrations. Unlike the SDS solutions, the calculations for CPC show a bimodal distribution corresponding to both small spherical micelles and larger, ellipsoidal or globular micelles. For the ideal CPC solution, the average aggregation numbers fall at roughly 87 and 120 for the spherical and globular micelles, respectively. Spherical micelles are favored at low concentrations around the cmc (φ < 0.005), but globular micelles are increasingly favored as the overall surfactant volume fraction increases and the distribution shifts toward the larger aggregates. However, a small number of spherical micelles is always predicted by the ideal solution model, even at volume fractions approaching 0.20, where the volume ratio of spherical to globular micelles is roughly one to ten. The shift in aggregation number to the larger, globular CPC aggregates can be attributed to the packing con(41) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes, 2nd ed.; John Wiley & Sons: New York, 1980.
2302 Langmuir, Vol. 14, No. 9, 1998
Figure 3. Ideal-solution size distribution for CPC in 0.01 M NaCl at 30 °C: (1) φ ) 0.005, 〈gw〉 ) 108; (2) φ ) 0.020, 〈gw〉 ) 112; (3) φ ) 0.044, 〈gw〉 ) 114; (4) φ ) 0.092, 〈gw〉 ) 116; (5) φ ) 0.140, 〈gw〉 ) 117; (6) φ ) 0.188, 〈gw〉 ) 118.
straints within the micelle. The maximum spherical aggregate is limited by the criterion that no point within the micelle can be a distance from the micelle surface greater than the length of a fully extended surfactant molecule.41 An additional constraint is that the micelle volume, Vg, and area, Ag, ratios per surfactant monomer must be consistent with the aggregation number g of the micelle, such that Vg/vo ) Ag/ao ) g, where ao is the optimum area per headgroup and vo is the molecular volume of a surfactant tail. The geometric packing constraints are critical to the consistency of the ideal-solution and activity-coefficient models. However, once these constraints are satisfied, the total change in standard chemical potential ultimately determines the average aggregation number of the micelle. Thus, under ideal-solution conditions for CPC, the larger aggregation numbers are energetically favored due to the delicate balance between the hydrophobic driving force for the micelle formation and the opposing electrostatic and steric interactions. As the average aggregation number increases as a function of concentration, as discussed above for the ideal-solution case, globular aggregates emerge in increasing concentrations. An average aggregation number of 95 was measured at 0.018 M NaCl by static light scattering at the cmc.42 This value is consistent with the present calculations for roughly spherical CPC micelles at the cmc. Although the size distribution for the ideal SDS surfactant solution shown in Figure 2 shows only a single peak, these surfactant micelles are also slightly globular.16,19 However, the globular micelles predicted for SDS in 0.01 M NaCl are small, and the predicted average aggregation number (cf. Figure 1) falls close to the maximum aggregation number of approximately 58 for a SDS micelle based on a fully extended dodecyl chain. Because of these small differences, a distinct bimodal distribution for spherical and globular micelles is not predicted for the SDS as a function of concentration at the relatively low ionic strength studied here. The effect of micelle-micelle interactions on the SDS and CPC size distributions is illustrated in Figures 4 and 5, along with the accompanying weight-average aggregation numbers at the same concentrations as those shown previously for the ideal-solution model in Figures 2 and 3. These calculations are for distributions that include the number of micelles, m, in the distribution equal to 60 (42) Anacker, E. W. J. Phys. Chem. 1958, 62, 41.
Amos et al.
Figure 4. Size distribution for SDS in 0.01 M NaCl at 30 °C calculated from the multicomponent activity-coefficient model, deff ) 49-52 Å: (1) φ ) 0.021, 〈gw〉 ) 76.0; (2) φ ) 0.069, 〈gw〉 ) 78.4; (3) φ ) 0.117, 〈gw〉 ) 80.5; (4) φ ) 0.165, 〈gw〉 ) 83.6; (5) φ ) 0.200, 〈gw〉 ) 87.0. Parameters: β ) 0.05; h ) 6.5 Å.
Figure 5. Size distribution for CPC in 0.01 M NaCl at 30 °C calculated from multicomponent activity-coefficient model, deff ) 63-70 Å: (1) φ ) 0.005, 〈gw〉 ) 121.5; (2); φ ) 0.020, 〈gw〉 ) 125.7; (3) φ ) 0.044, 〈gw〉 ) 128.7; (4) φ ) 0.092, 〈gw〉 ) 131.2; (5) φ ) 0.140, 〈gw〉 ) 138.1; (6) φ ) 0.188, 〈gw〉 ) 143.7. Parameters: β ) 0.08; h ) 5.5 Å.
and 130 aggregates for SDS and CPC, respectively, evenly spaced about the average aggregation number. In general, the increase in the micelle size observed in Figures 4 and 5 is due largely to the hard sphere, excluded-volume effects in which the surfactant aggregates shift to larger aggregation numbers, forming a smaller number of larger micelles, leading to a reduction in the number density of the aggregates. Physically, the repulsive intra- and interaggregate electrostatic interactions limit the size of the aggregates. Significant repulsive electrostatic micelle-micelle interactions strongly prevent the growth of large aggregates. This effect is illustrated by the large values of the osmotic pressures measured for SDS and CPC surfactant solutions.25 For SDS in 0.01 M NaCl, we see from Figure 4 that the average aggregate size increases from 〈gw〉 ) 76 to 87 up to volume fractions of 0.20, leading to a smaller number of larger particles. These aggregation numbers represent compact, globular micelles that have been confirmed experimentally for ionic strengths up to about 0.4 M NaCl by a number of experimental techniques,16 as well as by geometric packing arguments.34 A much larger increase in aggregation number of more than 1 order of magnitude for 〈gw〉 has been predicted due solely to excluded-volume interactions over a similar concentration range in the work
Coefficient Model for Surfactant Solutions
of Gelbart et al.43 These authors calculated the interactions based on a virial type expansion up to third order in reduced aggregate density for a model SDS surfactant solution. However, they neglect electrostatic interactions in order to account for growth of large, rod or disklike aggregates, presumably at high ionic strength where such an assumption is warranted. The shift in average aggregation numbers for SDS at 0.01 M NaCl shown in Figure 4 is modest by comparison because the growth of the micelles is limited by the large repulsive interactions between them at this relatively low ionic strength. At ionic strengths less than 0.4 M NaCl, neutronscattering results show significantly larger growth than the average aggregation numbers calculated here. For example, based on the rescaled MSA analysis of Sheu, average aggregation numbers range from 87 to 150 over a similar concentration range for SDS with no added electrolyte.44 Similar, but slightly different results were obtained by others with the same experimental technique but different data analysis.1,45 On the other hand, fluorescence quenching results for SDS with no added salt give aggregation numbers from 51 to 89 over the SDS concentration range 0.02-0.2 M (φ ) 4 × 10-3-0.04).18 Based on the uncertainties in the interpretation and data reduction of the different experimental techniques, the differences in the range of aggregation numbers between the experimental and theoretical results are not surprising. The range of aggregation numbers observed for SDS with the time-resolved fluorescence quenching results are similar to the range observed here (〈gw〉 ) 72-87). However, the net growth rate consistent with the present multicomponent interaction model is significantly smaller than the empirical growth law obtained based on both neutron-scattering and fluorescencequenching data. These results show that SDS micelles grow in proportion to the one-fourth power of the total surfactant concentration.18,45,46 However, these experiments were performed in a limited surfactant concentration range (0.02-0.2 M) where the effect of interactions is expected to be minimal. Based on our osmotic pressure data, it seems likely that growth of the micelles is severely limited at volume fractions greater than 0.05, such that the overall growth rate may be quite a bit less than the heuristic one-fourth power law extracted experimentally. In addition, there is as yet no physical explanation for the heuristic power law. For CPC, the calculated growth of the interacting micellar aggregates is more significant, as shown in Figure 5. Notably, in the presence of the repulsive interaggregate interactions, the smaller distribution of spherical micelles seen for the ideal CPC surfactant solution in Figure 3 has been drastically reduced, and larger globular micelles are predicted almost exclusively. The shift to the larger globular structures occurs based on the balance between the hard-sphere, excluded-volume interactions, which are reduced for a smaller number of larger particles and the repulsive electrostatic interactions between charged headgroups. This effect leads to an overall growth of the aggregates, and in this case, a change in the aggregate shape from spherical to globular micelles, consistent with the geometrical constraints discussed earlier. The (43) Gelbart, W. M.; Ben-Shaul, A.; McMullen, W. E.; Masters, A. J. Phys. Chem. 1984, 88, 861. (44) Sheu, E. Y.; Wu, C.-F.; Chen, S.-H.; Blum, L. Phys. Rev. A 1985, 32, 3807. (45) Bezzobotnov, V. Y.; Borbe´ly, S.; Cser, L.; Farago´, B.; Gladkih, I. A.; Ostanevich, Y. M.; Vass, S. J. Phys. Chem. 1988, 92, 5738. (46) Bales, B. L.; Almgren, M. J. Phys. Chem. 1995, 99, 15153.
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overall shift in aggregation numbers for CPC is from 121 at φ ) 0.005 to 144 at φ ) 0.18. These aggregation numbers are consistent with experimental evidence for the existence of globular micelles based on light-scattering and smallangle neutron-scattering data.20,42,47 The predicted shift to larger micelle sizes is accompanied by a broadening of the aggregate size distribution due to the growth of the micelles and the interactions between them. The combined effects of the micellar interactions lead to a competition between the aggregates of different sizes due to the opposing forces of the hard-sphere interactions, which favor growth and the electrostatic interactions which limit it. This competitive effect due to the interactions is superimposed on the increased polydispersity predicted by the ideal mass-action model with the growth of the micelles.34 Increased polydispersity as a function of increasing surfactant concentration is consistent with growth of the micelles, as confirmed by dynamic light-scattering experiments on SDS and on the cationic surfactant cetyltrimethylammonium bromide,17,22 as well as by thermodynamic theories.24,34 Experimentally, it is difficult to distinguish between the spherical and globular micellar aggregates, which is needed to confirm the bimodal distributions for CPC observed in Figures 3 and 5 due to the relatively small differences in overall size and shape of these micelles. However, aggregate shape changes may be induced experimentally by either variations in concentration or ionic strength. Shape transitions as a function of added salt tend to be more pronounced, and experimental observations of these changes based on a variety of techniques are available.6,48,49 Although direct evidence of micelle shape transformations is not available for CPC, recent measurements for SDS as a function of salt based on a fluorescence lifetime-distribution technique and analysis have been reported.49 In this study, a bimodal size distribution and a shift to larger average aggregation numbers is observed. These results show clear evidence of a bimodal distribution for SDS above 0.4 M NaCl, lending credence to the existence of bimodal distributions in surfactant solutions such as the one predicted here for CPC micelles at 0.01 M NaCl. Osmotic Pressure. The osmotic pressure calculated from the multicomponent size distributions including micelle interactions and the Donnan equilibria are shown in Figures 6 and 7 for SDS and CPC, respectively, as a function of total surfactant volume fraction. The experimental data points are those measured earlier.25,32 Overall, the average size and charge of the micellar aggregates are increased relative to a calculation considering only micellar aggregates of a single size.25 For SDS, the valence magnitude, based on a fractional ionization of β ) 0.05, is z〈g〉 ) 4.3-5.2, and the effective diameter is 49-59 Å over the volume fraction range, as shown in Figure 6. When only a single aggregate size is considered in the calculation, z〈g〉 ) 4.2 and deff ) 47 Å, where the aggregation number is held constant over the same concentration range. For CPC, the valence magnitude is z〈g〉 ) 8.2-11.5 for β ) 0.08 with an effective diameter of 63-70 Å for the multicomponent calculation shown in Figure 7 compared to q ) 7.0 and deff ) 58 Å in a monodisperse calculation. Comparison between the calculated osmotic pressure and the measured data is good for both ionic surfactants, (47) Anacker, E. W.; Ghose, H. M. J. Am. Chem. Soc. 1968, 90, 3161. (48) Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1976, 80, 1075. (49) Siemiarczuk, A.; Ware, W. R.; Liu, Y. S. J. Phys. Chem. 1993, 97, 8082.
2304 Langmuir, Vol. 14, No. 9, 1998
Figure 6. Osmotic pressure of SDS in 0.01 M NaCl at 30 °C. Closed circles represent experimental data.25 The solid line represents calculations from the multicomponent activitycoefficient model, 〈gw〉 ) 72-87. Parameters are as given in Figure 4. The inset shows the calculated osomotic pressure in the vicinity of the cmc as a function of total surfactant volume fraction.
Figure 7. Osmotic pressure of CPC in 0.01 M NaCl at 30 °C. Open circles represent experimental data.25 The line represents calculations from the multicomponent activity-coefficient model, 〈gw〉 ) 102-144. Parameters are as given in Figure 5.
as shown in Figures 6 and 7, with the exception of SDS at volume fractions below 0.06. The calculated osmotic pressure rises linearly as expected due to contributions from the monomeric surfactant up to the cmc as illustrated for SDS (see inset of Figure 6). At surfactant concentrations in the vicinity of the cmc, there is a discontinuity in the osmotic pressure, which continues to rise rapidly as a function of total volume fraction above this concentration. The discontinuity in the osmotic pressure exists also for CPC but is not as obvious due to the lower critical micelle concentration and corresponding decrease in the osmotic contribution of the monomeric surfactant. At the dilute concentrations below the cmc, it was not possible to measure the small osmotic pressure changes in this vicinity due to experimental limitations.50 The discrepancy between the calculation and the experimental data at low volume fractions in excess of the cmc up to approximately 0.06 for SDS can be explained by leakage of small amounts of surfactant monomer through the osmotic pressure membrane, as discussed in Appendix 2D of Amos.26 (50) Markels, J. H. Batch and Continuous Ultrafiltration of Micellar Solutions: Experiment, Theory, and Design. Ph.D. Dissertation, University of California, Berkeley, 1993.
Amos et al.
There are two adjustable parameters in the osmotic pressure results presented here: the degree of counterion dissociation, β, and the hydrated Stern layer thickness, h, which includes the hydrated surfactant counterions and any bound water associated with the micelle. It is useful at this point to discuss the values of the best-fit parameters obtained in the calculation. One of the basic assumptions of the calculation is to include a hydration layer for the micellar aggregate, which extends the total effective diameter of the micelle by 11-13 Å. Such a parameter is included to account for the presence of physically adsorbed counterions and water molecules in the excluded-volume calculation and therefore can only be varied within a limited range of values.26 The inclusion of this parameter also allows for the correction of limitations in the MSA theory, which lead to negative pair correlation functions at the contact diameter corresponding to the micelle interface. Evidence exists to show that the rescaled MSA (RMSA) theory is more appropriate for modeling charged surfactant solutions in the low-density limit (φ e 0.2) where significant Coulomb coupling exists.51 The values of the hydration layer thickness used for the activity coefficients calculated here are consistent throughout the model and are adjusted to fit the experimental data for the osmotic pressure within a physically realistic range. The degree of dissociation of the counterion headgroups is the second and only truly adjustable parameter in the multicomponent activity coefficient model. This parameter accounts for the dissociated counterions in the ionic surfactant solution in equilibrium with the electrolyte solution on the other side of the semipermeable membrane osmometer. We find that the values of β that represent the best fit to the osmotic pressure data in this study are 0.05 and 0.08 for SDS and CPC, respectively. These values are considerably lower than theoretical and experimental values reported in the past for SDS and other comparable ionic surfactants where experimental degrees of dissociation ranged up to 0.50 and theoretical values were typically between 0.13 and 0.20.52-54 The discrepancy in the degree of dissociation reported here versus other studies is in large part due to the nature of the calculations and the experimental data they represent. As stated previously, the parameters obtained for the model in this investigation are based on the data obtained by membrane osmometry. As a general rule, data of this kind are not readily available in the literature for surfactant solutions. In this experiment the principle determining factor in the micellar dissociation is the Donnan equilibria. The greater the degree of dissociation of the micellar aggregates, the larger the concentration of counterions in solution. This increased concentration of electrolyte ions leads to a larger Donnan contribution to the osmotic pressure due solely to the uneven distribution of microions on either side of the porous membrane.31 Depending on the degree of dissociation, the calculated Donnan contribution can be so large as to exceed that of the measured osmotic pressure. On the basis of the experimental data presented here in Figures 6 and 7, it is estimated that at most 0.10 or 10% of the total counterions present in solution are fully dissociated from the micelle interface.25,50 We note that this calculation is (51) Hansen, J.-P.; Hayter, J. B. Mol. Phys. 1982, 46, 651. (52) Frahm, J.; Diekmann, S.; Haase, A. Ber. Bunsen-Ges. Phys. Chem. 1980, 84, 566. (53) Weinheimer, R. M.; Evans, D. F.; Cussler, E. L. J. Colloid Interface Sci. 1981, 80, 357. (54) Evans, D. F.; Mukherjee, S.; Mitchell, D. J.; Ninham, B. W. J. Coll. Inter. Sci. 1983, 93, 184.
Coefficient Model for Surfactant Solutions
based solely on the equilibrium thermodynamics and thus is model independent. The degree of dissociation calculated here based on the fit to experimental data is in close agreement with the value of 0.11 obtained previously for CPC by membrane osmometry.55 However, the degree of dissociation measured with membrane osmometry may be somewhat lower than that obtained with other experimental techniques due to the additional energetic penalty for dissociation of the counterions imposed by the Donnan equilibrium for surfactant micelles in concentrated solutions. In addition, it is important to note that the values for the degree of dissociation obtained here are necessarily model dependent and as such are subject to the assumptions and limitations of the model. One possible explanation for the lower theoretical values for the degree of dissociation when compared to experiment may be rooted in the inherent nature of a model-dependent calculation. It has been shown that while results calculated from radial distribution functions using the MSA and HNC lead to excellent fits of experimental data, the effective charge obtained from such theories is consistently low, particularly for the MSA closure.56 Therefore, while such theories may represent experimental data quite well qualitatively, the quantitative values of the parameters may differ from those measured directly. One possible way around this is to use a model-independent development to describe the osmotic pressure such as that developed by Hall and more recently applied by Nagarajan for the description of ionic surfactant solutions.24,57 However, such an approach still requires the use of some model for the activity coefficients of the surfactant micelles and microions for quantitative comparisons. One advantage of the multicomponent activity coefficient model presented here is that it accounts for the polydispersity of the surfactant micelles. However, due to the polydisperse nature of these surfactant solutions, there is not a unique parameter set that describes the measured osmotic pressure data. Because the micelles are present in a distribution of sizes, a range of parameters does equally well. Here, we find that a related physically realistic set of parameters can be determined based on the full size distributions shown in Figures 4 and 5. The fact that we were also able to model the osmotic pressure data successfully as a monodisperse surfactant solution, may be attributed to the relatively limited overall growth of the aggregates, to incomplete counterion dissociation, and to the relatively sharp aggregate size distribution. These properties of the ionic surfactants allow the main features of the surfactant solution to be captured by a single aggregation number at the concentrations and ionic strength studied here while there is little change in the overall charge or net size of the aggregate. However, based on this observation alone, one can conclude neither that the micelles are monodisperse nor that there is no growth or change in aggregation number for the micelles with concentration. To examine further the effect of growth on the measured osmotic pressures, we draw attention to the solubilization of a hydrophobic organic solute in the aqueous surfactant solution. Micellar solubilization of organics is important in understanding the solution interactions because the incorporation of organic solubilizate molecules into the micelle interior is known to produce not only growth but also aggregate shape changes in many cases.19,58 Figure 8 shows experimental data for the osmotic pressure of (55) Fineman, M. N.; McBain, J. W. J. Phys. Chem. 1948, 52, 881. (56) Linse, P. J. Chem. Phys. 1991, 94, 3817. (57) Hall, D. G. J. Chem. Soc., Faraday Trans. 1 1981, 77, 1121.
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Figure 8. Comparison of experimental osmotic pressure data for CPC in 0.01 M NaCl at 30 °C with and without decane. Open circles represent CPC alone, and solid triangles represent CPC micelles with solubilized decane at the saturation limit.50 The dashed line is drawn for guidance through the data points only.
CPC in a 0.01 M NaCl solution saturated with n-decane. The data were obtained by membrane osmometry as described elsewhere.25,50 For comparison, the data for the CPC solution in the absence of organic solute are shown as well. These results show that the osmotic pressure stays the same or actually decreases slightly at concentrations up to 200 kg/m3. Similar results have also been obtained for the osmotic pressure of SDS in the presence of decane, and dynamic light-scattering results also indicate that CPC micelles grow in the presence of decane.26 Thus, comparison of the osmotic pressure data in Figure 8 with micelle growth predicted by the lightscattering results suggests that the increased electrostatic repulsion between the larger swollen micelles may be offset by the reduction in the number density of the aggregates. Conclusions To predict the osmotic pressure behavior of concentrated ionic surfactant solutions of SDS and CPC, we propose a self-consistent multicomponent activity-coefficient model that includes both excluded-volume and electrostatic interactions. We have focused our attention on the relatively low ionic strength solutions of SDS and CPC in 0.01 M NaCl. Theoretical calculations of the size distributions of these micellar aggregates reveal a shift in the size distribution and average aggregation number, which represents growth of the micelles with increasing surfactant concentration. The ideal mass-action equilibrium model used here as a reference state predicts only a relatively small increase in weight-average aggregation number (∼10%). However, the predicted growth is much larger when the electrostatic and excluded-volume interactions among all the solutes are accounted for selfconsistently as a function of aggregate size. We predict an increase in the weight-average aggregation number of between 20 and 60% for SDS and CPC, respectively, when these interactions are included in the calculation. This shift in aggregation number can be explained by the excluded-volume interactions, while the overall growth is limited due to the electrostatic repulsions between micelles. For the cationic CPC surfactant, which has a 16-carbon hydrophobic tail, two different aggregate shapes are (58) Chevalier, Y.; Zemb, T. Rep. Prog. Phys. 1990, 53, 279. (59) Porte, G.; Appell, J.; Poggi, Y. J. Phys. Chem. 1980, 84, 3105.
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predicted by the ideal equilibrium-constant model. However, when the micelle interactions are taken into account as a function of size, only the larger globular micelles are predicted as favorable at concentrations more than one or two orders of magnitude above the critical micelle concentration. These results compare favorably with dynamic light-scattering results for CPC, which show micelles with average aggregation numbers in the same range as predicted.42,47,59 In the absence of micelle-micelle interactions, the predicted average aggregation numbers are low relative to experimental data. Thus, these interactions are important not only in predicting aggregate growth but also in predicting aggregate shape as well. Such predictions are only possible when the presence of multiple aggregate sizes are accounted for and the average aggregation number is allowed to shift naturally as part of the multicomponent calculation. Finally, the multicomponent calculations predicted here compare favorably with the experimental data for the osmotic pressure of SDS and CPC surfactant solutions in 0.01 M NaCl. While the monodisperse calculation presented as part of our earlier work is able to predict the measured osmotic pressure data also, other features of the nonideal surfactant solution are lost, such as the aggregate growth and shift in shape. These features are retained by the present self-consistent model. Notably, values of the fitted degree of dissociation for the micelles are found to be less than what has been predicted in earlier studies. These differences may very well be related to approximations in the MSA theory with respect to the electrostatic interactions, which deviate from the real ionic system. Because of this, further work must be done on the electrostatic theories and model in order to make conclusive quantitative statements about the degree of dissociation and micellar charge of the aggregates. However, what the multicomponent activity coefficient model does do quite successfully is capture the essential features of the measured osmotic pressure. Our calculated results and experimental data indicate that osmotic pressure is a valuable and important measure of micelle and solution species interactions. Nomenclature Ag ao b1 b2 deff di e f( g gmin gmax 〈gw〉 h kb Kg
surface area of the hydrocarbon micellar core (Å2/ micelle) optimum area per surfactant headgroup in a micelle (Å2/molecule) semimajor axis of a prolate ellipsoid (Å) semiminor axis of a prolate ellipsoid, )lo (Å) effective spherical diameter of the micelle (Å) hard-sphere diameter of solute i (Å) electron charge ) 1.6 × 10-19 (C) activity coefficient on a solute number density basis number of surfactant monomers per micellar aggregate, Ag minimum aggregation number of aggregates included in the size distribution maximum aggregation number of aggregates included in the size distribution weight-average aggregation number thickness of hydration sheath surrounding surfactant micelle (Å) Boltzmann’s constant ) 1.38 × 10-23 (J/K) equilibrium constant of the micellar aggregate of g surfactant monomers
Amos et al. l lo m NAv Ni nc Pex,cc Pn Po Req,g T Vg vo zi z〈g〉
total number of equations/unknowns in the NewtonRaphson iteration length of a fully extended hydrocarbon surfactant tail (Å) total number of micellar aggregates included in the micelle size distribution Avogadro’s number ) 6.023 × 1023 (molecules/mol) number of molecules of solute i number of carbons in the alkane chain of the surfactant tail excess pressure due to charge-charge interactions (kPa) MSA parameter, see eq 8 (Å-2) solvent pressure (kPa) equivalent spherical radius for a micellar aggregate of size g (Å) temperature (K) volume of the hydrocarbon core for a micellar aggregate of size g (Å3/micelle) molecular volume of the surfactant tail (Å3/molecule) valence of solute species i magnitude of the effective valence of the average size surfactant micelle taking into consideration counterion binding, )〈gw〉β
Greek Symbols β ∆ δ φi φ Γ γi κ µi Π Fi ξm
degree of ionization at the micellar interface MSA parameter, see eq 7 distance from the R-carbon of the surfactant tail to the location of charge on the surfactant headgroup (Å) dielectric constant of the solvent (water) volume fraction of solute species i total volume fraction of surfactant, defined by eq 14 MSA screening parameter defined by eq 6 (Å-1) activity coefficient of solute species i inverse Debye-screening parameter (Å-1) chemical potential of solute i (J) osmotic pressure (kPa) number density of solute species i (molecules/Å3) solute volume ratio, see eq 4
Subscripts and Superscripts ° 0 1 (1) (2) cc Clex g hs i j k Na+
infinite-dilute reference state solvent surfactant monomer solvent side of the semipermeable membrane in the osmometer surfactant solution side of the semipermeable membrane in the osmometer charge-charge interaction chloride ions excess micellar aggregate composed of g surfactant monomers hard-sphere interaction multicomponent solute corresponding to a surfactant aggregate of average size, surfactant monomer, counterion, or added electrolyte ions component j in a multicomponent mixture of i solutes index for MSA screening parameter sodium ions LA9605403
