Theory of Surfactant Aggregation in Water/Ethylene Glycol Mixed Solvents. R. Nagarajan * and Chien-Chung Wang

Transcription

1 5242 Lanmuir 2000, 16, Theory of urfactant Areation in Water/Ethylene Glycol Mixed olvents R. Naarajan * and Chien-Chun Wan Department of Chemical Enineerin, 161 Fenske Laboratory, The Pennsylvania tate University, University Park, Pennsylvania Received Auust 9, In Final Form: March 1, 2000 The areation behavior of surfactants in mixed solvents composed of water and ethylene lycol is predicted usin an extension to our theory of aqueous surfactant solutions. The extension accounts for the dependence of (i) the surfactant tail transfer free enery, (ii) the areate core-solvent interfacial free enery, and (iii) the headroup interaction free enery on the composition of the mixed solvent. As the proportion of ethylene lycol in the mixed solvent increases, the model predicts an increase in the critical micelle concentration (cmc), a decrease in the averae areate size, an increase in the areate polydispersity, and a stroner dependence of the averae areation number on the total surfactant concentration. The model reveals that (a) lare values for cmc oriinate primarily from the smaller manitude of the surfactant tail transfer free enery in the mixed solvent, (b) the areation numbers are small mainly because of the smaller manitude of the hydrocarbon-mixed solvent interfacial tension, and (c) neither the cmc nor the areate size are reatly affected by the lower dielectric constant of the mixed solvent. The predicted areation characteristics of cetyl pyridinium bromide, cetyl trimethylammonium bromide, and sodium dodecyl sulfate at various compositions of the mixed solvent are presented for illustrative purposes and are compared with available measurements. Introduction The self-assembly of surfactants in polar oranic solvents such as formamide, N,N-dimethyl formamide, dimethyl sulfoxide, dimethyl acetamide, N-methyl acetamide, lycerol, and ethylene lycol, and in some aqueous-oranic mixed solvents has been investiated experimentally to explore how the solvent characteristics influence areation. Amon these solvents, one of the most widely studied 1-12 is ethylene lycol. Both inter- and intramolecular hydroen bonds are formed in ethylene lycol, althouh not as stron as those formed in water. With use of mixtures of water and ethylene lycol as solvents, the areation behavior of some surfactants includin Aerosol OT, 7 cetyl pyridinium bromide, 8,10 cetyl trimethylammonium bromide, 9 tetradecyl trimethylammonium bromide, 11 sodium dodecyl sulfate, 9 and octyl and nonyl phenyl polyoxyethylene ethers 12 have been investiated experimentally. However, no commensurate effort has been made to develop and apply theoretical models to predict the areation behavior of surfactants in polar oranic solvents and in aqueous-oranic mixed solvents. * To whom correspondence should be addressed. (1) Ray, A. J Am Chem oc. 1969, 91, (2) Ionescu, L. G..; Fun, D.. J. Chem. oc. Faraday Trans , 77, (3) Binana-Limbele, W.; Zana, R. Colloid Polym. ci. 1989, 267, 440. (4) Gharibi, H.; Palepu, R.; Bloor, D. M.; Hall, D. G.; Wyn-Jones, E. Lanmuir 1992, 8, 782. (5) Gharibi, H.; Palepu, R.; Bloor, D. M.;, Hall, D. G.; Wyn-Jones, E. Lanmuir 1992, 8, (6) Warnheim, T.; Jonsson, A. J. Colloid Interface ci. 1988, 125, 627. (7) Mukerjee, K.; Mukerjee, D. C.; Moulik,. P. J. Phys. Chem. 1994, 98, (8) Callahan, A.; Doyle, R.; Alexander, E.; Palepu, R. Lanmuir 1993, 9, (9) Bakshi, M.. J. Chem. oc., Faraday Trans , 89, (10) Palepu, R.; Gharibi, H.; Bloor, D. M.; Wyn-Jones, E. Lanmuir 1993, 9, 110. (11) Backlund,.; Berenstahl, B.; Molander, O.; Warnheim, T. J. Colloid Interface ci. 1989, 131, 393. (12) Ray, A.; Nemethy, G. J. Phys. Chem. 1971, 75, 809. In a recent article, 13 we showed that quantitative predictions of the areation behavior of surfactants in ethylene lycol can be obtained by adaptin our theory of micellization in water 14 to polar oranic solvents. In this article, we extend the theory to binary mixtures of water and ethylene lycol. Illustrative predictions of the critical micelle concentration (cmc), the averae areation numbers, and the areate size polydispersity are obtained for cetyl pyridinium bromide (CPBr), cetyl trimethylammonium bromide (CTAB), and sodium dodecyl sulfate (D). The predicted cmc values are compared with available experimental data. Thermodynamics of Micellization The thermodynamic relations describin the areation behavior of surfactants in solutions are by now well established. 15 The equilibrium size distribution of areates can be calculated from X ) X 1 exp - ( µ o o - µ 1 ) ) X 1 exp - ( µ o ) (1) where X 1 and X are the mole fractions of the sinly dispersed surfactant molecules and areates of size, and µ 1 o and µ o are their standard chemical potentials, defined as those correspondin to infinitely dilute solution conditions. The factor µ o represents the difference in the standard chemical potential between a surfactant molecule (13) Naarajan, R.; Chien-Chun Wan J. Colloid Interface ci. 1996, 178, 471. (14) Naarajan, R.; Ruckenstein, E. Lanmuir 1991, 7, 2934; Naarajan, R. In tructure-performance Relationships in urfactants; Esumi, K., Ueno, M., Eds.; Marcel Dekker: New York, 1997; pp 1-81; Naarajan, R.; Ruckenstein, E. elf-assembled tructures. In Equations of tate for Fluids and Fluid Mixtures; eners, J. V., Ed.; Elsevier: New York, in press. (15) Tanford, C. The Hydrophobic Effect; Wiley: New York, /la CCC: $ American Chemical ociety Published on Web 04/29/2000

2 urfactant Areation in Mixed olvents Lanmuir, Vol. 16, No. 12, Table 1. Geometrical Properties of Areates spherical micelles (radius R e l ) V ) 4πR 3 ) v 3 A ) 4πR 2 ) a A δ ) 4π(R + δ) 2 ) a δ P ) V A R ) v ar ) 1 3 lobular micelles (semiminor axis R ) l, semimajor axis b e 3l, eccentricity E) 2 b V ) 4πR ) v 3 A ) 2πR 2[ 1 + sin-1 E A δ ) 2π(R + δ) 2[ 1 + E δ ) [ 1 - ( R + δ b + δ ) 2 ] 1/2 P ) V A R ) v ar, E(1 - E 2 ) 1/2] ) a, E ) [ 1 - ( R 1/2] sin-1 E δ E δ (1 - E 2 δ ) ) a δ 1 3 e P e 0.406, R eq ) ( 3V 4π ) 1/3 b ) 2] 1/2 in an areate of size and a sinly dispersed surfactant molecule in the solvent. To calculate X, an explicit expression for µ o as a function of the size and shape of micelles is needed. mall micelles are spherical. When the areation number exceeds the value correspondin to the larest spherical micelle (i.e., the radius of the sphere equals the extended lenth of the surfactant tail), nonspherical lobular or rodlike micelles form. Our calculations show that only small micelles form for the systems and conditions considered, and hence, we limit our discussion to spherical and lobular micelles. The averae eometrical characteristics of lobular micelles are represented usin the prolate ellipsoidal shape. 14 Table 1 provides a summary of equations to calculate the volume of the hydrophobic core of the areate (V ), the total surface area of the core (A ), the total surface area at a distance δ from the core surface (A δ ), and a packin parameter P, as functions of the areation number. The areas per surfactant molecule a ()A /) and a δ ()A δ / ), decrease with increasin areation number. The areation number alon with the surfactant tail volume v and the extended tail lenth l, completely describes all the eometrical properties of areates. For hydrocarbon surfactants, v and l can be calculated (at T K) from the roup contributions of the methylene and methyl roups: v(ch 3 ) ) (T - 298) Å 3, l(ch 3 ) ) Å v(ch 2 ) ) (T - 298) Å 3, l(ch 2 ) ) Å (2) Free Enery of Micellization in Mixed olvents Various contributions to µ o have been identified previously 14 by considerin the chanes experienced by a surfactant molecule when it is transferred from the solvent to an areate of size. These contributions account for the followin factors: (i) the surfactant tail is removed from contact with the solvent and transferred to the hydrophobic core of the micelle ( µ o ) tr, (ii) the surfactant tail inside the micelle has a conformation different from that in a pure hydrocarbon liquid because of the molecular packin requirements inside the micelle ( µ o ) def, (iii) the formation of the micelle creates an interface allowin for contact between the hydrophobic core and the solvent ( µ o ) int, (iv) the headroups of surfactants at the micelle surface exhibit steric repulsions ( µ o ) ste, and (v) the headroups, if they are ionic or zwitterionic, also exhibit electrostatic repulsions, ( µ o ) ionic or ( µ o ) dipole. Expressions for each of these free-enery contributions, applicable to ethylene lycol-water mixtures, are iven below. More detailed discussion concernin the oriin of each of the expressions can be found in ref 14. Transfer of the urfactant Tail. The transfer free enery from the mixed solvent depends on the transfer free eneries from pure water and pure ethylene lycol and also on the nature of interactions between the two solvents. Applyin the Flory-Huins equation to the mixed solvent, one ets: 16 ( µ o W( )tr,mix ) φ µ o EG( )tr,w + φ µ o )tr,eg - φ W ln V W V - φ EG ln V EG V + χ WEG φ W φ EG (3) where subscripts W and EG refer to water and ethylene lycol, φ W and φ EG are their volume fractions in the mixture, V is the molar volume of the mixed solvent (V ) X W V W + X EG V EG ) calculated from the molar volumes V W and V EG and mole fractions X W and X EG of the two components, and χ WEG is the Flory interaction parameter between water and ethylene lycol. Expressions for the transfer free enery from water have been developed previously usin experimental solubility data for hydrocarbons in water. The roup contributions of methyl and methylene roups to the transfer free enery are iven 14 by the expressions: ( µ o 896 ) 5.85 ln T + )tr,w T T for CH 2 ( µ o 4064 ) 3.38 ln T + )tr,w T T for CH 3 (4) Expressions for the transfer free enery from ethylene lycol have been developed 13 usin the infinite dilution activity coefficient data for paraffinic hydrocarbons in ethylene lycol 17 available at four temperatures ranin between 298 K and 363 K. The roup contributions to the transfer free enery from ethylene lycol, of methyl and (16) Nitta, T.; Tatasuishi, A.; Katayama, T. J. Chem. En. Jpn. 1973, 6, 475. (17) Pierotti, G. J.; Deal, C. H.; Derr, E. L. Ind. En. Chem. 1959, 51, 95.

3 5244 Lanmuir, Vol. 16, No. 12, 2000 Naarajan and Wan Here, N EG is the ratio of the molar volume of ethylene lycol to that of water and has the value of 3.1 at 298 K, for which all the computations have been performed. An estimate of χ WEG )-2.3 for the Flory interaction parameter fits the activity data well as indicated by Fiure 1. The neative value for the interaction parameter sinifies stroner water-ethylene lycol attractive interactions compared to the averae of water-water and ethylene lycol-ethylene lycol interactions. The transfer free enery stronly influences the cmc, but does not play a direct role in determinin the equilibrium size of the micelles formed, because it does not depend on the areate size. However, in the ionic surfactants, the cmc determines the ionic strenth, and thereby influences the free-enery contribution from ionic headroup interactions (discussed below). This latter contribution depends on the areate size, and consequently, the transfer free enery plays an indirect role in influencin the equilibrium size of ionic surfactant areates. Deformation of the urfactant Tail. The surfactant tail has to assume a conformation inside the micelle consistent with the maintenance of liquidlike density in the core while keepin that end of the tail connected to the headroup at the micelle-solvent interface. This molecular packin requirement can be satisfied only by allowin a variation of chain deformation alon its lenth. The followin expression for the free-enery contribution due to this factor was developed in our earlier work 14 by adaptin the approach used for polymers with a lare number of sements: ( µ o ) def ) ( 9Pπ2 80 )( R 2 N L 2) (7) Fiure 1. Activity of water vs weiht fraction of ethylene lycol in the solvent mixture at 25 C. The points are activities calculated usin the UNIFAC roup contribution approach. The line shows the fittin of calculated activity data by the Flory-Huins model (eq 6) for a water-ethylene lycol interaction parameter of χ WEG )-2.3. methylene roups, are iven 13 by: ( µ o ) tr,eg ( µ o ) tr,eg ) ln T T T for CH 2 ) ln T T T for CH 3 (5) The Flory interaction parameter χ WEG is obtained by fittin the activity data for water-ethylene lycol mixtures to the Flory-Huins equation. In the absence of adequate experimental data, we calculate the activity of water (a W ) in the solvent mixture as a function of composition usin the UNIFAC roup contribution method 18 and fit these data to the expression iven by the Flory-Huins model: ln a W ) ln(1 - φ EG ) + ( 1-1 N EG ) φ EG + χ WEG φ 2 EG (18) Gmehlin, J.; Rasmussen, P.; Fredunsland, A. Ind. En. Chem. Process. Des. Dev. 1982, 21, 118; Larsen, B. L.; Rasmussen, P.; Fredunsland, A. Ind. En. Chem. Res. 1987, 26, (6) where P is the packin parameter dependent on the areate shape as defined in Table 1, L is the size of a sement taken equal to 4.6 Å (correspondin to 3.6 methylene roups), N ()l /L) is the number of sements in the surfactant tail. Formation of Areate Core-olvent Interface. On areation, an interface is formed between the micelle and the surroundin solvent medium. Because the headroup-solvent interaction at the interface is not different from the headroup-solvent interaction of a sinly dispersed surfactant molecule, it makes no new contribution. However, at the interface, there is residual contact between a part of the hydrophobic core and the solvent. The free enery due to this hydrocarbon core-solvent contact can be calculated 14 from: ( µ o ) int ) σ a (a - a o ) (8) where σ a is the interfacial tension between a hydrocarbon liquid (representin the micelle core) and the solvent mixture, a o is the area per molecule shielded from contact with the solvent because of the presence of the headroup, and (a - a o ) is the residual hydrocarbon-solvent contact area per molecule. If the headroup has a cross-sectional area a p larer than the cross-sectional area v /l, (also equal to L 2 ) of the tail, then a o is equal to L 2.Ifa p is smaller than L 2, then the headroup does not shield the entire tail cross-section, and the shielded area a o is equal to a p. The interfacial tension σ a between the mixed solvent and a hydrocarbon liquid is estimated on the basis of the Prioine theory, 19 which reconizes that the surface composition should determine the interfacial tension and that it must be different from the bulk composition, in solutions. The surface composition is calculated via the relation (φ ln[ EG /φ EG ) 1/N )] EG (1 - φ EG )/(1 - φ ) (σ W - σ EG ) v 2/3 W + EG 3 4 χ WEG [(1 - φ EG ) - φ EG ] χ WEG [(1 - φ EG) - φ EG ] (9) where the superscript refers to the surface, σ W is the surfactant tail-water interfacial tension, and σ EG is the (19) Defay, R.; Prioine, I.; Bellmans, A.; Everett, D. H. urface Tension and Adsorption; Wiley: New York, 1966; iow, K..; Patterson, D. J. Phys. Chem. 1973, 77, 356.

4 urfactant Areation in Mixed olvents Lanmuir, Vol. 16, No. 12, Fiure 2. The composition of the ethylene lycol-water system in contact with the surface of the hydrophobic micelle core as a function of the composition of the bulk solvent mixture. The surface composition is calculated usin the Prioine theory as discussed in the text. surfactant tail-ethylene lycol interfacial tension. Once the surface composition φ EG is known, the interfacial tension σ a is calculated from ( σ a - σ W ) v 2/3 W ) ln( 1 - φ 1 - φ EG) + ( N EG - 1 N EG ) (φ EG - φ EG ) + χ WEG[ )2] 1 2 (φ EG) (φ EG (10) The interfacial tensions of the two pure solvents aainst a hydrocarbon liquid, σ EG and σ W, are related to pure component surface tensions via the relations: σ EG ) σ + σ EG - 2ψ EG (σ σ EG ) 1/2, ψ EG ) 0.78 σ W ) σ + σ W - 2ψ W (σ σ W ) 1/2, ψ W ) 0.55 (11) where ψ EG and ψ W are constants whose values have been estimated from knowlede 13,14,20 of all the tensions appearin in eq 11. The surface tensions σ of the surfactant tail (of molecular weiht, M), σ EG of ethylene lycol, and σ W of water can be calculated (in units of dyn/cm or mn/ m) from the relations: 13,14,21 σ ) M -2/ (T - 298) σ EG ) (T - 298) (12) σ W ) (T - 298) (20) Janczuk, B.; Wojcik, W.; Zdziennicka, A. J. Colloid Interface ci. 1993, 157, 384. (21) Panzer, J. J. Colloid Interface ci. 1973, 44, 142. Fiure 3. The interfacial tension between the solvent mixture and the hydrophobic micelle core as a function of the composition of the bulk solvent mixture. The interfacial tension is calculated usin the Prioine theory as discussed in the text. The competition between the surface and the bulk for ethylene lycol is determined by the relative manitudes of the ethylene lycol-hydrocarbon and water-hydrocarbon interfacial tensions and by the water-ethylene lycol interactions. The fact that the ethylene lycolhydrocarbon interfacial tension is lower than the waterhydrocarbon interfacial tension should favor the surface bein enriched in ethylene lycol. The fact that stron attractive interactions exist between water and ethylene lycol in the bulk (as represented by the neative value for χ WEG ) should counteract such surface enrichment. The surface composition predicted by the Prioine theory as a function of the bulk composition of the mixed solvent is shown in Fiure 2 and the resultin interfacial tension is plotted in Fiure 3. The calculated results from Prioine theory show that the two effects discussed above mutually compensate one another, which results in the surface composition bein not dramatically different from the bulk composition and σ a bein practically equal to the composition averae of the pure component interfacial tensions. Headroup teric Interactions. The crowdin of surfactant headroups at the micelle surface leads to steric repulsions. The correspondin free-enery contribution is calculated from a simple model based on excluded hard core areas of the headroups, in analoy with the excluded volume effect appearin in the van der Waals equation of state: 14 Headroup Ionic Interactions. For ionic surfactants, one has to consider the electrostatic interactions between the chared headroups located at the areate surface. These interactions are calculated 14 usin approximate analytical solutions to the Poisson-Boltzmann equation: ( µ o ) ionic where ( µ o ) ste )-ln 1 - a p a (13) ) 2 [ ln ( 2 + [ 1 + ( 2) 2 ] 1/2)] - ([ ( 2) 2 ] 1/2-1 ) 4 - κ(r + δ) ln ( [ ( 2) 2 ] ) 1/2) (14) 4πe2 ɛ mix κa δ (, κ ) 8πn o e2 1/2, n ɛ mix ) o ) (C 1 + C add )N Av 1000 (15)

5 5246 Lanmuir, Vol. 16, No. 12, 2000 Naarajan and Wan of the capacitor is equal to the dipole lenth d in the zwitterionic headroup. Consequently, one ets for spherical micelles 14 the expression ( µ o ) dipole ) 2πe2 R d (18) ɛ mix a δ [ R + δ + d] Fiure 4. Dielectric constant of water-ethylene lycol mixed solvent as a function of the mole fraction of ethylene lycol, at 25 C. The points are experimental data from ref 22, whereas the line is calculated usin pure solvent properties and the correlation iven by eq 16. Here, e is the electronic chare ( esu), ɛ mix is the dielectric constant of the mixed solvent, δ is the distance from the hydrophobic core surface to the surface where the center of the counterion is located, κ is the reciprocal Debye lenth, n o is the number of counterions per cm 3 of the solution, C 1 is the molar concentration of the sinly dispersed surfactant molecules, C add is the molar concentration of the uni-univalent salt added to the solution, and N Av is the Avoadro s number. For prolate ellipsoids, the radius R s is replaced with the equivalent radius R eq (defined in Table 1). The dielectric constant ɛ mix is available from experimental measurements for various compositions of the water-ethylene lycol mixture and at various temperatures from 273 K to 353 K. 22 We have correlated the dielectric constant of the mixture to that of the pure components by the composition-dependent relation ln ɛ mix ) X W ln ɛ W + X EG ln ɛ EG X W X EG (16) where no explicit dependence of temperature appears, althouh the relation is accurate throuhout the entire temperature rane of 273 K to 353 K. The quality of the correlation above can be seen from Fiure 4 where the experimental and correlated dielectric constants for the solvent mixture are shown at 298 K. The dielectric constants of water and ethylene lycol appearin in the equation above are iven by 13,14,23 ɛ W ) exp[ (t - 273)] ɛ EG ) 46.6 exp[ (t - 273)] (17) Headroup Dipole Interactions. For zwitterionic surfactants, one has to consider the electrostatic interactions amon the dipoles of the headroups located at the areate surface. This interaction free enery is estimated by considerin that the poles of the dipoles enerate an electrical capacitor and the distance between the planes (22) Corradini, F.; Marcheselli, L.; Tassi, L.; Tosi, G. J. Chem. oc., Faraday Trans. 1993, 89, 123. (23) Weast, R. C., Ed. Handbook of Chemistry and Physics; CRC Press: Boca Raton, FL, where δ is the distance from the core surface to the place where the dipole is located. For prolate ellipsoids, the same equation is used but with the radius of the spherical core R replaced with the equivalent radius R eq of the ellipsoidal core. Molecular Constants for urfactants. To perform size-distribution calculations only a sinle molecular constant (a p ) is needed for nonionic surfactants, two molecular constants (a p and δ) are needed for ionic surfactants, and three constants (a p, d, and δ) are needed for zwitterionic surfactants. The area a p is a hard-core area of the headroup. It is estimated from knowlede of the bond lenths, bond anles, and atomic volumes usin a molecular model of the headroup. The dipole lenth d of the dipolar headroup and the distance δ from the core surface at which it is located are also estimated from knowlede of the bond lenths and bond anles. The distance δ from the core surface at which the center of the counterion center is located an ionic headroup is also determined usin knowlede of bond lenths and bond anles. Once these constants are estimated usin molecular structural information on the headroup, they are used to predict the behavior of surfactants of various tail lenths in a variety of calculations such as those pertainin to cmc, micelle size, temperature effects, electrolyte effects, solvent effects, sphere-to-rod transition, mixed micelles, solubilization, etc. 14 The molecular constants are thus not fittin parameters, because a very wide rane of solution properties for a homoloous family of surfactants have to be predicted usin the same molecular constant. The surfactants selected for the predictive calculations are the three most recently studied in water-ethylene lycol mixtures: cetyl pyridinium bromide (CPBr), cetyl trimethylammonium bromide (CTAB), and sodium dodecyl sulfate (D). For the pyridinium bromide headroup, we estimate a p ) 34 Å 2 and δ ) 2.2 Å, for the trimethylammonium bromide headroup, a p ) 54 Å 2 and δ ) 3.45 Å and for the sodium sulfate headroup, a p ) 17 Å 2 and δ ) 5.45 Å. 13,14 As mentioned above, these molecular constants previously have been used to perform a wide rane of calculations involvin these surfactants. For example, the two molecular constants a p ) 17 Å 2 and δ ) 5.45 Å for the sodium sulfate headroup have been used to calculate the cmc and micelle size of sodium alkyl sulfates (from C 8 to C 16 ), how they chane with ionic strenth, the dependence of the sphere-to-rod transition parameter as a function of the alkyl chain lenth, temperature and ionic strenth, the cmc of mixtures of alkyl sulfates, composition of these mixed micelles, cmcs of mixtures of D with a nonionic surfactant, the extent of solubilization of hydrocarbons in D micelles, the cmc and micelle size of D in the presence of solubilized hydrocarbons, etc. 14 Computational Approach. The free-enery expressions are introduced in eq 1 where X 1 and are the independent variables and X is the dependent variable. We calculate the size distribution X as a function of for a specified value of the monomer concentration X 1. The total surfactant concentration X tot is then calculated to be (X 1 + X ) in mole fraction units, where the summation extends over all values of. The surfactant concentrations

6 urfactant Areation in Mixed olvents Lanmuir, Vol. 16, No. 12, expressed as molar concentrations C and mole fractions X in this article are interrelated via the conversion relation C ) 1000 V surf + (1 - X)/X[V EG X EG + V W X W ] where V surf is the molar volume of the surfactant. The molecular volume of the surfactant is obtained by addin the headroup volume (approximately 200 Å 3 for pyridinium bromide, 190 Å 3 for trimethylammonium bromide, and 60 Å 3 for sodium sulfate) and the tail volume (calculated usin eq 2). The molar volume of ethylene lycol is taken to be 56 cm 3 /mol whereas the molar volume of water is 18 cm 3 /mol. From the calculated size distribution (X vs ), the true weiht-averae and the number-averae areation numbers ( w and n ), the apparent weiht-averae and the apparent number-averae areation numbers ( w,app and n,app ) can be calculated based on the definitions 2 X X w ), n ) X X (19) w,app ) X X X 1 + X, n,app ) X 1 + X X 1 + X (20) where the summations extend to all values of > 1. The polydispersity in the areate size is iven by the ratio ( w / n ), known as the polydispersity index. The cmc usually is defined as the surfactant concentration at which an experimentally measured solution property displays a sharp transition in behavior. Because of this operational definition, the cmc usually depends on the type of experimental measurement bein made. For example, surface tension depends larely on the concentration of the monomeric surfactant as the total surfactant concentration is altered. Consequently, a sharp transition in the plot of X 1 (or C 1 ) aainst the total concentration X tot ) X 1 + X (or C tot ) can be used to determine the cmc. Properties such as osmotic pressure, vapor pressure, and freezin point depression depend on the total number of distinct species present in the solution. Therefore, the cmc measured usin such techniques can be determined from a plot of the apparent number-averae areation number n,app versus the total surfactant concentration. The solubilization of dyes which is a common technique for determinin the cmc depends on the concentration of surfactant present in the areated form. The cmc measured by the dye solubilization technique can thus be determined by plottin X aainst the total surfactant concentration. In usin liht scatterin to determine the cmc, the scatterin intensity is dependent on the averae mass of the species in solution. Thus, in this case, the cmc can be determined from a plot of the apparent weihtaverae areation number w,app versus the total surfactant concentration. In addition to the approaches discussed above, other methods can also be listed for the experimental measurement and/or calculation of the cmc. In this article, the cmc is estimated by plottin X 1, X, w,app,or n,app aainst the total surfactant concentration and identifyin the concentration at which a sharp transition in the plotted variable is observed. The computations have been performed for many mixed solvent compositions. As limitin compositions of the mixed solvent system, the areation behavior in water and in ethylene lycol are also computed for the three surfactants. Comments on Model Parameters, Their Variability and Consequences for Predicted Results. In developin the quantitative free-enery model, we have either directly estimated model parameters or used models to estimate some parameters. First are the molecular constants such as the volume and lenth of surfactant tail and volume of headroup, the constants a p, δ, and d associated with the headroup (dependin on the headroup type), and molar volumes of water and ethylene lycol. We have already commented on the three constants a p, δ, and d associated with the headroup and all the other constants are accurately known from experimental density data. Concernin the second kind of molecular parameters, for calculatin the transfer free enery in the mixed solvent, we have used the Flory theory. Errors in the estimated transfer free enery will affect the manitude of the calculated cmc but not the areation number. For calculatin the interfacial tension between the micelle core and the mixed solvent, we have used the Prioine theory. Errors in the estimated interfacial tension will affect both the predicted cmc and the micelle areation number. In both the Flory model and the Prioine model, we have used the water-ethylene lycol interaction parameter estimated from the UNIFAC roup contribution method. A chane in value of this parameter will thus affect both the transfer free enery and the interfacial tension, and thereby affect the calculated results. Finally, for the dielectric constant of the solvent mixture, we have used available experimental data correlated by an empirical equation. In our view, whenever direct experimental information on any of the model parameters is available, it is best to use it for the calculations. However, in the absence of such data, the best estimates are made usin available models such as those of Flory and Prioine and the UNIFAC method. The advantae of the present theory is that it is predictive. Therefore, no attempt will be made to view any of the model parameters as adjustable constants. As discussed below, the experimental determination of the cmc in systems containin a lare amount of ethylene lycol in the solvent mixture is not as accurate as in water, especially when the surfactant tail lenth decreases. This factor should be kept in mind; it would be necessary to compare experimental results from many independent sources to validate and develop the present theory. Results and Discussion Critical Micelle Concentration of CPBr. The four different ways of determinin the cmc discussed above are used to estimate the theoretical cmc of CPBr in mixed solvents. The calculated values of X 1, X, n,app, and w,app are plotted as functions of the total surfactant concentration X tot ) X 1 + X, in Fiures 5 to 8 for ethylene lycol composition in the mixed solvent of 0 wt % (pure water), 40 wt %, 80 wt %, and 100 wt % (pure ethylene lycol), respectively. The plots in Fiure 5 for water show sharp transitions in behavior indicatin a clearly identifiable cmc of the surfactant. imilar sharp transitions in the shapes of the plotted properties can be observed for both the 40 and 80 wt % ethylene lycol solutions, althouh the sharpness of the transition for the 80 wt % solution is not as pronounced as for the 0 and 40 wt % ethylene lycol systems. Fiure 8 for pure ethylene lycol shows markedly differin behavior in that it is very difficult to observe a sharp transition in most of the plotted properties as the total surfactant concentration is increased. Althouh, all the curves in Fiure 8 reflect the presence of areates, the physical properties (represented by the computed variables) chane only radually

7 5248 Lanmuir, Vol. 16, No. 12, 2000 Naarajan and Wan Fiure 5. Calculated size-dependent variables for cetyl pyridinium bromide solutions in water as a function of the total surfactant concentration. All concentrations are expressed in mole fraction units. ee text for the definitions of the apparent areation numbers. Fiure 7. Calculated size-dependent variables for cetyl pyridinium bromide solutions in the water-ethylene lycol mixed solvent containin 80 wt % ethylene lycol as a function of the total surfactant concentration. The lines refer to the same size-dependent variables as in Fiure 5. Fiure 6. Calculated size-dependent variables for cetyl pyridinium bromide solutions in the water-ethylene lycol mixed solvent containin 40 wt % ethylene lycol as a function of the total surfactant concentration. The lines refer to the same size-dependent variables as in Fiure 5. over a wide rane of surfactant concentrations. To identify a sinle concentration in this rane as the cmc becomes difficult and there is considerable ambiuity in the determinations made. The cmc values determined by the different approaches for aqueous solutions from Fiure 5 are quite close to one another, whereas there is appreciable variation amon the estimates of cmc in ethylene lycol from Fiure The cmc values estimated from a plot of X 1 vs X tot, are presented in Fiure 9 as a function of the amount of ethylene lycol in the mixed solvent. Also shown Fiure 8. Calculated size-dependent variables for cetyl pyridinium bromide solutions in ethylene lycol as a function of the total surfactant concentration. The lines refer to the same size-dependent variables as in Fiure 5. for comparison are available experimental data based on electromotive force (emf), 10 conductivity, 8 and surface tension 8 measurements. Micelle ize and Polydispersity of CPBr. The calculated weiht averae areation numbers and the index of polydispersity are plotted in Fiure 10 as a function of the ethylene lycol content in the mixed solvent. The calculated results correspond to a total surfactant concentration of 0.2 M. The averae areation number of the micelle decreases sinificantly as the amount of ethylene lycol is increased. This is accompanied by a

8 urfactant Areation in Mixed olvents Lanmuir, Vol. 16, No. 12, Fiure 9. Critical micelle concentration of cetyl pyridinium bromide as a function of the composition of the mixed solvent system. The experimental data indicated by squares are from emf measurements, 10 circles from conductivity measurements, 8 and trianles from surface tension measurements. 8 Fiure 11. Dependence of the weiht-averae areation number of cetyl pyridinium bromide micelles in the waterethylene lycol mixed solvent as a function of the total surfactant concentration. Results for the solvent systems containin 0, 40, 80, and 100 wt % ethylene lycol are presented. Fiure 10. True weiht-averae areation number and the polydispersity index of cetyl pyridinium bromide areates as a function of the composition of the mixed solvent, at a total surfactant concentration of 0.2 M. simultaneous increase in the areate polydispersity index. In pure ethylene lycol, the solution contains mainly small oliomers resultin in an averae areation number of about 6. The calculated averae areation numbers are shown as functions of the total surfactant concentration in Fiure 11 for the four solvent compositions. Even at very lare surfactant concentrations, the areation numbers are small in the mixed solvents compared with the typical behavior in water (where areation numbers of about 60 can be observed). Further, as the ethylene lycol content in the mixed solvent is increased, the dependence of the averae areation number on the total surfactant concentration becomes stroner. This is anticipated for polydispersed areates, as shown by eneral thermodynamic considerations. 14 Fiure 12. Critical micelle concentration of cetyl trimethylammonium bromide as a function of the composition of the mixed solvent. The experimental data 9 are from conductance measurements. Areation Behavior of CTAB. The cmc of CTAB calculated from the plot of X 1 versus X tot, are shown in Fiure 12 alon with experimental data based on conductance measurements. 9 For CTAB in water, experimental cmc values in the rane 0.7 to 1 mm have been reported based on different measurement techniques, 24 and one may therefore anticipate even larer variations in the mixed solvents. The difference between the calculated and measured cmcs in Fiure 12 for all solvent compositions lies within this rane of experimental uncertainty. The predicted weiht-averae areation number of CTAB is shown in Fiure 13, as a function of (24) Van Os, N. M.; Haak, J. R.; Rupert, L. A. M. Physicochemical Properties of elected Anionic, Cationic and Nonionic urfactants; Elsevier: Amsterdam, 1993.

9 5250 Lanmuir, Vol. 16, No. 12, 2000 Naarajan and Wan Fiure 13. Dependence of the weiht-averae areation number of cetyl trimethylammonium bromide micelles in water-ethylene lycol mixed solvent as a function of the total surfactant concentration. Results for the solvent systems containin 0, 25 50, and 75 vol% ethylene lycol are presented. Fiure 14. Critical micelle concentration of sodium dodecyl sulfate as a function of the composition of the mixed solvent. The experimental data based on conductance measurements are from ref 9 (squares), ref 25 (trianles), and ref 26 (inverted trianles). the total surfactant concentration, for four solvent compositions (0, 25, 50 and 75 vol% ethylene lycol). The results are qualitatively similar to those presented above for CPBr. Areation Behavior of D. The cmc of D calculated from the plot of X 1 versus X tot, are shown in Fiure 14 alon with experimental data based on conductance measurements. 9,25,26 The predicted areation numbers of the D areates are smaller than those of CPBr and CTAB, at comparable solution conditions, because of the shorter tail lenth of D. Indeed, in pure (25) Lee, D. J.; Huan, W. H. Colloid Polym. ci. 1996, 274, 160. (26) Gracie, K.; Turner, D.; Palepu, R. Can. J. Chem. 1996, 74, Fiure 15. Contributions to the standard free-enery chane associated with areation of cetyl pyridinium bromide in water as a function of the areation number. ee text for a discussion of the various contributions. ethylene lycol, only dimers, trimers and tetramers of D are indicated even at very hih surfactant concentrations. In Fiure 14, we observe relatively lare deviations between calculated cmc and that measured by conductance, at hiher ethylene lycol concentrations in the mixed solvent. An obvious explanation is that because of the small size of the D areates in mixed solvents (compared with the CPBr and CTAB), the identification of a sharp transition in the measured property (which is the basis of the experimental cmc) becomes ambiuous as the amount of ethylene lycol in the mixed solvent increases. Even in water, where the areates are larer and a sharper transition in the measured properties is possible, a rane of cmc values between 7 and 8.4 mm has been reported. 24 The chane in slope of the conductance curve which is used for the experimental identification of the cmc becomes increasinly difficult to observe clearly when the ethylene lycol content in the solvent mixture is lare. We speculate that the cmc determined by conductance is systematically lower than the calculated cmc because of the different way by which the presence of small areates influences conductivity from the way they influence the X 1 versus X tot relation (which is the basis for theoretical predictions). Influence of olvent on Areation Characteristics. The contrastin areation behavior of surfactants in water and in water-ethylene lycol mixed solvents can be understood quantitatively on the basis of the differences in the properties of the two solvent systems. These differences are responsible for important variations in the free-enery contributions to micelle formation in the two solvent systems. The free-enery curves calculated for CPBr are shown in Fiure 15 for water and Fiure 16 for ethylene lycol. The three solvent-dependent free-enery contributions show major differences between their values in water and those in ethylene lycol. The first is the surfactant tail transfer free enery which accounts for the solvophobic effect. The manitude of the transfer free enery is considerably smaller in ethylene lycol than in water. This

10 urfactant Areation in Mixed olvents Lanmuir, Vol. 16, No. 12, Fiure 16. Contributions to the standard free-enery chane associated with areation of cetyl pyridinium bromide in ethylene lycol as a function of the areation number. ee text for a discussion of the various contributions. results in the larer manitude of the cmc in ethylene lycol compared with that in water and an increase in the cmc as the fraction of ethylene lycol in the solvent mixture is increased. The second solvent-dependent free-enery contribution is that associated with the formation of the areate core-solvent interface. The interfacial free enery is smaller in the ethylene lycol solutions than in water because of the considerably smaller ethylene lycolhydrocarbon interfacial tension compared with the waterhydrocarbon interfacial tension (see Fiure 3). This is primarily responsible for the smaller areation numbers of the equilibrium areates formed in ethylene lycol solutions. Further, the lower hydrocarbon-ethylene lycol interfacial tension causes the cmc to decrease, whereas the larer area per molecule (associated with small areation numbers) causes the cmc to increase. The competition between these two factors can cause an increase or a decrease in the cmc in ethylene lycol solutions compared with that in water. However, the dependence of cmc on this free-enery contribution is always much weaker when compared with the dependence of the cmc on the tail transfer free enery. Therefore, one always observes an increase in cmc with increasin ethylene lycol content in the solvent mixture. The third solvent-dependent contribution is associated with the electrostatic headroup interactions. The dielectric constant of ethylene lycol is lower than that of water. This should lead to an increase in the manitude of ionic interaction enery in ethylene lycol solutions compared with aqueous solutions. But the calculated freeenery values show that this is not the case, and the ionic interactions in the low dielectric constant medium are indeed smaller and not larer than those in water. This is because the concentration of the sinly dispersed surfactant (X 1 ) at which the areates become possible is much larer in ethylene lycol and the mixed solvents than in water; consequently, the ionic strenth is substantially hiher in mixed solvents than in water. The decrease in ionic headroup repulsions due to hiher ionic strenth more than compensates for the increase in ionic headroup repulsions due to the lower dielectric constant of ethylene lycol. The net effect is a decrease in the ionic headroup interaction enery in mixed solvents compared with in water. Thus, these interactions contribute to a reduction in the cmc in the mixed solvents. However, the influence of these interactions on the cmc is neliible when compared with the influence of the tail transfer free enery. Hence, the observed cmc always increases with increasin amounts of ethylene lycol in the mixed solvent. Concernin the micelle size, one may expect that the decrease in the ionic repulsions in ethylene lycol and in mixed solvents compared with in water will result in larer areation numbers. But this effect is more than compensated by the decrease in the interfacial enery in ethylene lycol and mixed solvents compared with water, which causes the areation numbers to become much smaller. Conclusions The areation behavior of surfactants in waterethylene lycol mixed solvents is investiated theoretically on the basis of a simple extension to our model of micelle formation in aqueous solutions. The free enery of micellization includes three solvent-dependent contributions. Expressions for these free-enery contributions have been developed as a function of the composition of the mixed solvent. With use of these free-enery expressions, the cmc, the size distribution of areates, their averae areation numbers, and polydispersity have been calculated for cetyl pyridinium bromide, cetyl trimethylammonium bromide, and sodium dodecyl sulfate for various compositions of the mixed solvent. The calculated results show that the cmc increases, the averae areation number of the micelle decreases, and the polydispersity increases as the amount of ethylene lycol in the solvent mixture is increased. The lare manitude of the cmc oriinates primarily from the small manitude of the tail transfer free enery from ethylene lycol compared with that from water. The small equilibrium areation numbers are a result of the low ethylene lycol-hydrocarbon interfacial tension compared with the water-hydrocarbon interfacial tension. Althouh the dielectric constant of ethylene lycol is smaller than that of water, the ionic interactions at the micelle surface decrease rather than increase in the mixed solvents, because of the hiher monomer concentrations (typified by the larer cmcs) which ive rise to hiher ionic strenths. However, despite the larer ionic strenths and weaker ionic headroup repulsions, the equilibrium areation numbers are small because of the dominatin influence of the interfacial enery contribution. LA