DOI: 10.1021/la9018426 12101Langmuir 2009, 25(20), 12101–12113 Published on Web 08/11/2009

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© 2009 American Chemical Society

Thermodynamic Modeling of the Duality of Linear 1-Alcohols

as Cosurfactants and Cosolvents in Self-Assembly of

Surfactant Molecules

Livia A. Moreira and Abbas Firoozabadi*

Department of Chemical Engineering, Mason Laboratory, Yale University, New Haven,Connecticut 06520-8286

Received May 22, 2009. Revised Manuscript Received July 11, 2009

The effect of adding an alcohol to surfactant systems dependsmuch on the alcohol chain length. Investigations on theeffect of alcohols in micellar systems point out that medium-chain alcohols are appreciably incorporated in the micellarphase whereas short-chain alcohols are localized mainly in the aqueous phase. Nonetheless, penetration of thehydrocarbon chain of alcohols in the micellar shell has been experimentally observed for the entire homologous series oflinear 1-alcohols. We present a thermodynamic model in which the alcohol molecules play two roles: cosurfactant andcosolvent. The cosurfactant effect of the alcohols is included by assuming that the alcohol molecules are nonionicsurfactants. The cosolvent effect is modeled by accounting for the changes in the free energy to relocate the surfactanttail from the solvent to the aggregate core. The effects of short-chain alcohols in the macroscopic interfacial tension anddielectric constant of the solvent medium are also taken into account. For short-chain alcohols the partition coefficientof the alcohols between water and liquid hydrocarbons provides knowledge of the fraction of the molecules thatparticipate in each function. Our proposed thermodynamic model improves the modeling of the effect of short- andmedium-chain alcohols in self-assembly ofmolecules that are of increasing importance inmodern scientific research andtechnological processes.

Introduction

Alcohol molecules are well-established additives on surfactantself-assemblies. Linear alcohols are found to be an effectivepromoter of microemulsions. Medium-chain alcohols such as 1-butanol and 1-pentanol are regularly used as efficient promo-ters. Even though medium-chain alcohols are very effective innumerous jobs, these cosurfactants show acute toxicity and,therefore, their applications are limited. For processes thatdemand low-toxic cosurfactants and cosolvents, short-chainalcohols hold a privileged place. For example, short-chain alco-hols are often added to cleaning products, and ethanol is added toedible microemulsions to improve stability and to lower viscosityof formulations.

The effect of alcohols on the properties of micellar solutionshas been the subject of extensive experimental and theoreticalinvestigations.1-6 Rao and Ruckenstein4 are pioneers in applyingthe thermodynamic model for mixed micelles developed byNagarajan7,8 to medium-chain alcohols (nC4-nC7). The modelbased on the assumption that the alcohol molecules behave as acosurfactant molecule correctly predicts the primary effect ofthese alcohols to decrease the critical micelle concentration (cmc)

due to incorporation in the micellar shell. The quantitativeagreement, however, is poor for systems containing 1-butanol.

Studies on short-chain alcohols (C1-nC3) have shown thatthese alcohols mainly appear in the aqueous phase, thus influen-cing the micellar structure by modifying the solvent properties.However, an experimental investigation by Caponetti and co-workers3 on the distribution of linear 1-alcohols between theaqueous phase and the micellar shell has found that short-chainalcohols are also incorporated in the micellar shell, although inlesser extent than medium-chain alcohols. Li and co-workers6

have recently applied the thermodynamic model for mixedsolvents developed by Nagarajan and Wang9 to hexadecyltri-methylammoniumbromide (C16TAB) inwater-ethanolmixtureswhere ethanol is a short-chain alcohol. The model assumes thatethanol behaves exclusively as a cosolvent. The model developedby Li and co-workers describes their own experimental results;10

however, it does not predict the initial depression in the cmcdue toincorporation of the alcohol in the micellar shell that wasexperimentally observed and reported years earlier by Cipicianiand co-workers.11

In this work, we present a theoretical model for the effect oflinear 1-alcohols inmicellar solutions. The thermodynamicmodelis predictive since there are no free parameters. In our model thealcohol molecules are allowed to play two roles: cosurfactant andcosolvent. The cosurfactant effect of the alcohols is accounted forby assuming that the alcohol molecules are nonionic surfactants.The cosolvent effect of the alcohols is modeled by accountingfor the changes in the free energy to relocate the surfactant tailfrom the solvent to the aggregate core. The cosolvent effect ofshort-chain alcohols in the macroscopic interfacial tension and

*Correspondence concerning this article should be addressed to A.F.E-mail: abbas.firoozabadi@yale.edu.(1) Zana, R. Adv. Colloid Interface Sci. 1995, 57, 1–64.(2) Stuart, M. C. A; Van de Pas, J. C.; Engberts, J. B. F. N. J. Surfactants

Detergents 2006, 9, 153–160.(3) Caponetti, E.; Martino, D. C.; Floriano, M. A.; Triolo, R. Langmuir 1997,

13, 3277–3283.(4) Rao, I. V.; Ruckenstein, E. J. Colloid Interface Sci. 1986, 113, 375–387.(5) Enders, S.; Kahl, H. Fluid Phase Equilib. 2007, 261, 221–229.(6) Li, W.; Han, Y. C.; Zhang, J. L.; Wang, L. X.; Song, J. Colloid J. 2006, 68,

304–310.(7) Nagarajan, R. Langmuir 1985, 1, 331–341.(8) Nagarajan, R. In Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N.,

Eds.; ACS Symposium Series 501; American Chemical Society:Washington, DC, 1992;pp 54-95.

(9) Nagarajan, R.; Wang, C. C. Langmuir 2000, 16, 5242–5251.(10) Li, W.; Han, Y. C.; Zhang, J. L.; Wang, B. G. Colloid J. 2005, 67, 159–163.(11) Cipiciani, A.; Onori, G.; Savelli, G. Chem. Phys. Lett. 1988, 143, 505–509.

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12102 DOI: 10.1021/la9018426 Langmuir 2009, 25(20), 12101–12113

Article Moreira and Firoozabadi

dielectric constant of the solvent medium are also taken intoaccount. For short-chain alcohols we calculate the partitioncoefficient of the alcohol in auxiliary water/hydrocarbon systemusing themodifiedUNIFAC (Dortmund)model12 for the activitycoefficients. The partition coefficient provides the fraction of thealcohol that acts as cosolvent and cosurfactant. By applying thedual model, we predict micellar properties such as cmc andmicellar size in agreement with experimental data that can notbe accurately predicted by other thermodynamicmodels from theliterature.

At constant temperature, pressure, and global composition, theequilibrium state is the one that minimizes the Gibbs free energy.To the best of our knowledge, the method of minimization ofGibbs free energy has only been used in a thermodynamicmicellization model for asphaltene aggregation,13 while thermo-dynamic models developed for self-assembly of surfactant mole-cules14-19 use the equality of chemical potentials at equilibrium inan isothermal system, i.e., the chemical potential of a surfactant infree solution and the chemical potential of a surfactant in amicelleof arbitrary size are the same. The equality of chemical potentialsapproach uses the size distribution equation in the calculations.Although the equality of chemical potentials, evaluated forsystems with inert constituents, is properly regarded as a neces-sary requirement for thermodynamic equilibrium, it is not asufficient condition. In our calculations of self-assembly ofsurfactant molecules we use the minimization of total Gibbs freeenergy to guarantee equilibrium.

In this work, “alcohol” denotes the linear 1-alcohol. And werefer to alcohols from methanol to propanol as short-chainalcohols; to alcohols from butanol to heptanol as medium-chainalcohols; and to alcohols larger than heptanol as long-chainalcohols.

The rest of this paper is structured as follows: in the nextsection, we introduce the minimum of the Gibbs free energyapproach to thermodynamic calculations on self-assembly ofsurfactants, report the geometries presupposed, and describehow the duality of alcohols is accounted for. In the Results andDiscussion section, we compare model predictions and experi-mental data of cmc, aggregation number and micellar size.Subsequently, the conclusions are drawn. For completeness, wepresent an Appendix with expressions for the computation of thestandard free energy of micellization.

Thermodynamics of Micellization

In this description, the minimum of the Gibbs free energy ispursued for a systemwith fixed temperature, pressure, and globalcomposition. The system is composed of water, surfactant andalcohol. In this work,we present amodel for systems consisting ofone type of surfactantmolecule and one type of alcohol molecule.However, the same methodology can be readily extended tosystems consisting of two or more types of surfactant molecules

and any other molecule that behaves as both cosurfactant andcosolvent simultaneously.

Consider a surfactant solution as a multicomponent systemconsisting of NW water molecules, NsA molecules of surfactantA, andNsB molecules of alcohol B, at temperatureT and pressurep. When the concentration of surfactant is lower than the criticalmicelle concentration (cmc), all surfactantmolecules in the systemwill be singly dispersed. However, when the concentration ofsurfactant exceeds the cmc, then due to surfactant self-associa-tion, a distribution of micellar sizes {Ng} is realized. Ng denotesthe number of micelles composed of g surfactant monomerscontaining gA molecules of surfactant A and gB molecules ofsurfactant B (or cosurfactant). Moreover, N1A and N1B, respec-tively, are the number of singly dispersed surfactants A and B inthe aqueous phase. (The subscript W refers to water, 1 to thesingly dispersed surfactant, and g to the aggregate containingg surfactant molecules.)

The totalGibbs free energy of the solution,G, ismodeled as thesum of two contributions: the free energy of formation, Gf, andthe free energy of mixing, Gm.

G ¼ Gf þ Gm ð1Þ

In this work, we assume dilute surfactant solution; therefore,the free energy of interaction between the various species isneglected. The free energy of formation includes the manycomplex physical factors that contribute to the formation ofmicelles. These contributions are evaluated for the dilute referencesolution with negligible intermicellar interactions. Gf is expressedas

Gf ¼ NWμoW þN1Aμo1A þN1Bμ

o1B þ

X¥g¼2

Ngμog ð2Þ

where μWo (T, p) is the standard state chemical potential of water

molecules taken as the chemical potential of pure water, μ1Ao (T, p)

and μ1Bo (T, p) are the standard state chemical potentials of singly

dispersed surfactants A and B, respectively, and μgo(T, p) is the

standard state chemical potential of themicelles with the aggrega-tion number g (=gA + gB). The standard states of all speciesother than water are taken as those corresponding to infinitelydilute solution conditions.

Since the total number of surfactantmolecules is fixed, we maywrite material balance equations:

NsA ¼ N1A þX¥g¼2

gANg ð3Þ

NsB ¼ N1B þX¥g¼2

gBNg ð4Þ

Using the abovematerial balance equations, eq 2 can be rewrittenas:

Gf ¼ NWμoW þNsAμo1A þNsBμ

o1B þ

X¥g¼2

NggΔμog ð5Þ

where gΔμgo = μg

o - gAμ1Ao - gBμ1B

o . Δμgo is the difference in the

standard chemical potentials between a surfactantmolecule in theaggregate that contains gA and gB surfactant molecules of typesA and B, respectively, and a singly dispersed surfactant in water.

(12) (a) Gmehling, J.; Li, J.; Schiller, M. Ind. Eng. Chem. Res. 1993, 32, 178–193.(b) Gmehling, J.; Lohmann, J.; Jakob, A.; Li, J.; Joh, R. Ind. Eng. Chem. Res. 1998, 37,4876–4882. (c) Gmehling, J.; Wittig, R.; Lohmann, J.; Joh, R. Ind. Eng. Chem. Res.2002, 41, 1678–1688. (d) Jakob, A.; Grensemann, H.; Lohmann, J.; Gmehling, J. Ind.Eng. Chem. Res. 2006, 45, 7924–7933.(13) Pan, H.; Firoozabadi, A. SPE Prod. Facilities 1998, 13, 118–127.(14) Tanford, C. J. Phys. Chem. 1974, 78, 2469–2479.(15) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday

Trans. II 1976, 72, 1525–1568.(16) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934–2969.(17) Nagarajan, R. In Structure-performance relationships in surfactants; Esumi,

K., Ueno, M. Eds.; Marcel Dekker: New York, 2003; pp 1-81.(18) Blankschtein, D.; Thurston, G.M.; Benedek, G. B. J. Chem. Phys. 1985, 54,

955–958.(19) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 3710–3724.

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Moreira and Firoozabadi Article

Δμgo is here referred to as the standard free energy ofmicellization.

Besides the dependenceof standard free energyofmicellization onthe nature of the surfactant, it also depends on the aggregationnumber, micelle composition, aggregate shape, mole fractionof singly dispersed ionic surfactants, and bulk and interfacialproperties such as dielectric constant and interfacial tension,respectively. The formulation for the standard free energy ofmicellization used in this work is based on the expressionsdeveloped by Nagarajan and co-workers,8,9,16,17 which is pre-sented in the Appendix for completeness.

The free energy of ideal mixing has the following form:

Gm ¼ kT½NW ln XW þN1A ln X1A

þN1B ln X1B þX¥g¼2

Ng ln Xg� ð6Þ

Here k is the Boltzmann constant, XW is the mole fraction ofwater, X1A and X1B are the mole fractions of the singly dispersedsurfactants A and B, and Xg is the mole fraction of g-mers.

The total Gibbs free energy is then written as follows:

G ¼ NWμoW þNsAμo1A þNsBμo1B þX¥g¼2

NggΔμog

þ kT½NW ln XW þN1A ln X1A þN1B ln X1B þX¥g¼2

Ng ln Xg�

ð7ÞThe terms of eq 7 that only depend on the fixed variables, T, p,

NsA, NsB, and NW, can be moved to the left side. G0 is defined asfollows:

G0 ¼ G - NWμoW - NsAμo1A - NsBμo1B ¼X¥g¼2

NggΔμog

þkT ½Nw ln Xw þN1A ln X1A þN1B ln X1B þX¥g¼2

Ng ln Xg�

ð8ÞDividing the expression by kT:

G0

kT¼X¥g¼2

gNg

ΔμogkT

!þNW ln XW þN1A ln X1A

þN1B ln X1B þX¥g¼2

Ng ln Xg ð9Þ

TheMaximum-TermMethod. The maximum-term methodhas been widely employed in calculations of micellar solutions.The approach is built on the recognition that for spherical andglobular micelles, the size distribution is usually narrow. Theconcentrations of aggregates other than that corresponding tothe maximum in the size distribution are relatively small. Becausethe average properties of the solution are strongly influenced bythe species present in the largest amount, the average aggregationnumber gn can be taken as the value of g for which the total Gibbsfree energy is minimum.

If the concentration of surfactant is lower than the cmc,all surfactant molecules in the system are singly dispersed and

eq 9 reduces to

G0

kT¼ NW ln XW þNsA ln X1A þNsB ln X1B ð10Þ

where

XW ¼ NW

NW þNsA þNsB,

X1A ¼ NsA

NW þNsA þNsB, and X1B ¼ NsB

NW þNsA þNsB

However, if the concentration of surfactant exceeds the cmc,part of the surfactant molecules associate to form aggregates andthe rest remains singly dispersed. Then, using the maximum-termapproximation, G0 is given by

G0

kT¼ gnNgn

ΔμognkT

!þNW ln XW þN1A ln X1A

þN1B ln X1B þNgn ln Xgn ð11Þwhere the aggregation number gn can, theoretically, vary from 2to infinity and where

Xw ¼ Nw

Nw þN1A þN1B þNgn

,

X1A ¼ N1A

Nw þN1A þN1B þNgn

,

X1B ¼ N1B

Nw þN1A þN1B þNgn

, and

Xgn ¼ Ngn

Nw þN1A þN1B þNgn

ð12Þ

Geometrical Relations for Aggregates. The aggregatesstudied here are small enough so they can bemodeled as sphericalor globular. We use the geometrical relations developed byNagarajan.8 For spherical and globular aggregates containingg surfactant molecules, the geometrical relations are listed inTable 1, including the volume of the hydrophobic domain of the

Table 1. Geometrical Properties of Aggregates

spherical micelles (radius Rs e ls)

Vg ¼ 4πRS3

3¼ gvS

Ag = 4πRs2 = ga

Ag = 4π(Rs + δ)2 = gaδ

P ¼ Vg

AgRS¼ vS

aRS¼ 1

3

globular micelles (semiminor axis Rs = ls, semimajor axis b e 3ls,eccentricity E)

Vg ¼ 4πbRS2

3¼ gvS

Ag ¼ 2πrS2 1 þ sin - 1 E

Eð1 - E2Þ1=2

� �¼ ga, E ¼ 1 - Rs

b

� �2� �1=2

Agδ ¼ 2πðRS þ δÞ2 1 þ sin - 1 Eδ

Eδð1 - Eδ2Þ1=2

� �¼ gaδ

Eδ ¼ 1 - Rs þδbþδ

� �2� �1=2

P ¼ Vg

AgRS¼ vS

aRS, Req ¼ 3Vg

4π

� �1=3

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Article Moreira and Firoozabadi

aggregate, Vg, the surface area of contact between the aggregateand water (or the solvent mixture), Ag, the surface area at adistanceδ fromthe aggregate-water interface,Agδ, and a packingfactor, P. For an ionic surfactant, the molecular constantδ depends on the size of the headgroup, the size of the hydratedcounterion, and the proximity of the counterion to the charge onthe surfactant ion.8 The δ values used in this work are from ref 17and are given inTable 2 togetherwith othermolecular parametersto be described in the Appendix.

In the geometrical relations, the tail volume vS is given by vS=RgAvSA + RgBvSB, where vSA and vSB denote the volumes of thehydrophobic tails of surfactants A and B, and RgA and RgB standfor the compositions of the mixed micelle:

RgA ¼ gA

gA þ gB¼ gA

g, RgB ¼ gB

gA þ gB¼ gB

gð13Þ

The tail molecular volumes vSA and vSB are calculated from themethylene and methyl group contributions. These group molec-ular volumes are estimated from the density versus temperaturedata available for aliphatic hydrocarbons16 and are given by thefollowing expressions for T in K:

νðCH3Þ ¼ 0:0546 þ 1:24� 10 - 4ðT - 298Þ nm3 ð14Þ

νðCH2Þ ¼ 0:0269 þ 1:46� 10 - 5ðT - 298Þ nm3 ð15ÞThe extended length of the surfactant tail lS at 298 K is

calculated using a group contribution of 0.1265 nm for themethylene group and 0.2765 nm for the methyl group.20 Tosatisfy packing condition, the upper limit of the radius of themicelles RS is taken to be the composition averaged tail length,8

RS e (ηAlSA + ηBlSB), where ηA and ηB are the volume fractionsof surfactant and cosurfactant tails in the micellar core.

ηA ¼ RgAνSARgAνSA þ RgBνSB

, ηB ¼ RgBνSBRgAνSA þ RgBνSB

ð16Þ

Alcohols as Cosurfactant and Cosolvent. The central themeof our work relates to the dual nature of alcohol molecules thatleads these molecules to play two roles: cosurfactant and cosol-vent. The cosurfactant effect is included by assuming that thealcohol molecules are nonionic surfactants. In the dualmodel, forboth short- and medium-chain alcohols, the cosolvent effectassociated to the transfer free energy is accounted for all alcoholmolecules singly dispersed in solution, N1B.

For short-chain alcohols, besides the changes in the transferfree energy of the surfactant tail from the solvent to the aggregatecore, two other cosolvent effects are accounted for: the change inthe aggregate core-solvent macroscopic interfacial tension, andthe dielectric constant of the solventmedium. In these systems,wecalculate the partition coefficient of the alcohol in a hypotheticalwater/liquid hydrocarbon system as a strategy to estimate the

fraction of the short-chain alcohols which acts as a cosurfactantand the fraction that acts as cosolvent changing the macroscopicinterfacial tension and the dielectric constant of the solution.

Short-chain alcohols are miscible in water at all proportions,they also partition between an aqueous and nonaqueous phasedue to their miscibility in both media. Long-chain alcoholsare known to solubilize within the micelles as micelles start toform,21,22 but short- and medium-chain alcohols are not expectedto solubilize in the core of the micelle, they make part of themicellar shell. We assume that the fraction of the short-chainalcohol that partitions to the nonaqueous phase in this hypothe-tical system is the fraction of alcohol that is available to partici-pate on the micellar shell; therefore, it is the fraction that is actingas a cosurfactant in the dual model.

In order to calculate the partition coefficient, we proceed withan auxiliary liquid-liquid equilibrium calculation between anaqueous and nonaqueous phase. The auxiliary system is com-posed of water, hydrocarbon, and alcohol molecules. This phaseequilibrium calculation is a separate computation from the Gibbsfree energy minimization of the micellization process. The aux-iliary system is assumed to be at the same temperature andpressure of the surfactant system of interest. The hydrocarbonmolecule is assumed to have the same number of carbon atoms asthat of the tail of the surfactant A. The auxiliary system has thesame ratio of NW to NsB as that of the surfactant system ofinterest. To have the nonaqueous phase comparable in size withthe aqueous phase, the number of hydrocarbon molecules in theauxiliary system is taken to be a fraction of the sum of water andalcohol molecules (NH = 0.7(NW + NsB)).

For the purpose of alcohol-hydrocarbon-water phase split-ting, we proceed with a liquid-liquid equilibrium calculationbetween the aqueous phase (R) and the nonaqueous phase (β).The three species i present are water (W), hydrocarbon (H), andalcohol (B). We calculate the compositions of the two coexistingphases by solving the three equations of phase equilibrium (eq 17)using the Rachford-Rice approach as described by O’Connelland Haile.23 The condition of equilibrium is given by,

ZiRi γRi ¼ Z

βi γ

βi for i ¼ W,H,B ð17Þ

whereZik is themole fraction of component i in phase k (k=R,β).

The activity coefficients γi are calculated using the modifiedUNIFAC (Dortmund) method.12

The thermodynamic partition coefficient for a chemical com-ponent i between aqueous and nonaqueous phases is defined as24

Ki ¼ Zβi

ZRi

¼ γRiγβi

ð18Þ

We relate the thermodynamic partition coefficient to an opera-tional partition coefficient, pc:

pc ¼ 1

KB þ 1ð19Þ

HereKB is the thermodynamic partition coefficient of the alcoholin the auxiliary system. The operational partition coefficient

Table 2. Molecular Constants for Surfactant Headgroups4,17

surfactant headgroupap

(nm2)ao

(nm2)δ

(nm)vh

(nm3)

sodium sulfate 0.17 0.17 0.545 0.2trimethylammonium bromide (TAB) 0.54 0.21 0.345 0.19alcohol 0.08 0.08

(20) Tanford, C. The hydrophobic effect: formation of micelles and biologicalmembranes, 2nd ed.; John Wiley & Sons, Inc: New York, 1980.

(21) Holmberg, K.; J€onsson, B.; Kronberg, B.; Lindman, B. Surfactants andpolymers in aqueous solution, 2nd ed.; John Wiley & Sons, Inc: England, 2003.

(22) Salager, J.-L.; Ant�on, R.; Forgiarini, A.; M�arquez, L. In Microemulsions:Background, new concepts, applications, perspectives, 1st ed.; Stubenrauch, C. Ed.;Wiley: New York, 2009; pp 84-121.

(23) O’Connell, J. P.; Haile, J. M. Thermodynamics: Fundamentals for Applica-tions, 1st ed.; Cambridge University Press: New York, 2005; pp 488-491.

(24) Leo, A.; Hansch, C.; Elkins, D. Chem. Rev. 1971, 71, 525–616.

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Moreira and Firoozabadi Article

varies from 0 to 1. When pc = 0, all alcohol molecules contributeas cosurfactant, andwhen pc=1, all alcoholmolecules contributeas cosolvent. At the extreme when pc = 0, the dual model mergesto the mixed micelle model developed byNagarajan.8 We refer tothe extreme when pc = 0 as cosurfactant model, and to theextreme when pc = 1 as cosolvent model. At the other extremewhen pc = 1, the dual model does not merge to the mixed solventmodel developed by Nagarajan and Wang9 due to differentexpressions in the free energy associated with the formation ofthe hydrophobic core-solvent interface. The mixed solventmodel estimates the interfacial tension between mixed solventandhydrocarbon liquid on the basis of the Prigogine theorywhichrecognizes that the surface composition that determines theinterfacial tension is different from the bulk composition.9 Thismodel works well for systems containing a mixture of solvents asshown by Nagarajan and Wang. In the case of linear alcohols,these surface active alcohol molecules are not only concentratingon the interface region, but also making part of the surfactantshell. In the dual model, we use the bulk composition to infer theinterfacial tension for we assume that the alcohol that is in theinterfacial region is acting as a cosurfactant.

Medium-chain alcohols have limited solubility in water, thusonly small concentrations of these alcohols are usually used in themicellar systems.We assume the effect of medium-chain alcoholson the dielectric constant and the macroscopic surface tension tobe negligible. Thus, formedium-chain alcohols, there is noneedofcalculating pc as the cosolvent effect accounted for is only thechange in the transfer free energy of the solvent medium. Eventhough the effect of medium-chain alcohol on the macroscopicinterfacial tension is being neglected, the effect of these alcoholson the interfacial energy is taken into account by the cosurfactanteffect, i.e., by the penetration of these alcohols in the micellarshell.Gibbs Free Energy Minimization. In order to find the

minimum of the total Gibbs free energy of a system consistingof Nw water molecules, NsA molecules of surfactant A, and NsB

molecules of alcohol B, at temperature T and pressure p weproceed with the following algorithm. First, eq 11 is minimizedwith respect to the independent variables subject to the materialbalance constraint under the maximum term approximation.

NsA ¼ N1A þ gnANgn ð20Þ

NsB ¼ N1B þ gnBNgn ð21ÞFor short-chain alcohols, an inequality constrain related to the

cosurfactant fraction of the alcohol is added. For these alcohols,there is amaximum fraction ofmolecule that can be assimilated inthe micellar shell, as follows:

N1B ¼ NsBð1 - pcÞ - gnBNgng0 ð22ÞFor systems without alcohols (NsB = 0) or with alcohols

acting only as cosolvent, theGibbs free energy is minimizedwithrespect to two independent variables, g andNg. For systemswithalcohol acting as cosurfactant and cosolvent concomitantly,or with alcohols acting only as cosurfactant, the Gibbs freeenergy is minimized with respect to three independent variables,g, RgA, and Ng. The minimization is executed using the FSQPalgorithm.25

The result of the minimization of eq 11 is compared with theresult of eq 10 and the minimum of the total Gibbs free energy isdetermined as well as the aggregation number gn and the numberof aggregates in solution Ngn of composition RgA.Calculating the Cmc. The cmc is a key thermodynamic

quantity of surfactant systems. Knowledge of this quantity iscrucial for both scientific and practical understanding of howsurfactants behave. The importance of the cmc has led to itsmeasurement by many authors, for a wide range of surfactantsand under different solvent conditions. Experimentally, the cmc isdetermined as the concentration atwhich a sharp changeoccurs ina wide variety of physicochemical properties. In order to comparethe model predictions to experimental data, we calculate the cmcby constructing a plot of X1A against the total concentrationXtot (= X1A + gXg) (see Figure 1). The Gibbs free energyminimization is performed for different values of NsA for fixedvalues of NW and NsB. The plot is marked by a sharp change inslope as the concentration reaches the cmc. The extrapolationprocedure is used to define the cmc.

Results and Discussion

Wehave successfully applied theGibbsminimization approachto the thermodynamic models for single micelle,16,26 mixedmicelles,8 and mixed solvents9 of spherical and globular geome-tries. However the maximum-term approximation may not be agood assumption for the mixed solvent model when short-chainalcohols are present in surfactant solution for the averageaggregation number decreases to only a few surfactant moleculesper micelles. A stable micelle cannot be formed by a very smallnumber of surfactant molecules.20 Nagarajan and Wang9 haveshown that a low hydrocarbon-mixed solvent interfacial tensionpromotes micelles with small average aggregation numbers, andthat the decrease in the average aggregation number is accom-panied by a simultaneous increase in the polydispersity.

The maximum-term approximation can be used in the dualmodel, because even though the number of molecules of surfac-tant A is decreasing in the micellar shell, the penetration ofalcohol molecules maintains the average aggregation numberhigh enough to use the maximum-term approximation properly.

Because we use the maximum-term approximation in thiswork, we do not compare the results of the dual model withthe mixed solvent model developed by Nagarajan and Wang.9

We use the term ‘cosolvent model’ to refer to the case when allalcohol molecules in solution contribute to the cosolvent effect as

Figure 1. X1A versus Xtot. Each circle corresponds to one Gibbsfree energy minimization calculation. The cmc corresponds tothe X1A for which the extrapolated slopes (solid and dotted lines)cross.

(25) Zhou, J. L.; Tits, A. L.; Lawrence, C. T. Electrical Engineering Departmentand Institute for Systems Research; University of Maryland: College Park, MD, 1997. (26) Nagarajan, R.; Wang, C. C. J. Colloid Interface Sci. 1996, 178, 471–482.

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Article Moreira and Firoozabadi

described in the section on the duality of alcohols, and in theAppendix.Cmc. We compare experimental data and model predictions

for the cmc of sodiumdodecyl sulfate (SDS) in aqueous-alcoholicsolutions from methanol to heptanol and for the cmc of tetra-decyltrimethylammonium bromide (C14TAB) in aqueous-alco-holic solutions from ethanol to hexanol. We also compareexperimental data and model predictions for the cmc of thehomologous family of CnTAB in ethanol-water solutions (n =10, 12, 14, 16). Some of the results are not presented for the sakeof brevity.

The experimental data for cmc in SDS-methanol-watersolutions show that methanol often increases the cmc withmethanol concentration increase.27,28 However some authorsreport a nonmonotonic behavior for the cmc.29,30 For SDS-ethanol-water mixtures, the nonmonotonic behavior is moreaccentuated as shown by several authors.31-34 For SDS-propa-nol-water mixtures, the nonmonotonic behavior has also beenexperimentally observed34 even though most of the experimentalstudies are focused in the concentration range that the cmcdecreases.28,35-38

In Figures 2, 3, and 4, the assumption that the short-chainalcohols act exclusively as cosurfactant results in a monotonicdecrease of the cmc while the assumption that the short-chainalcohols act exclusively as cosolvent results in a monotonicincrease of the cmc. However, the dual model captures theminimum in the cmc with increasing concentration of short-chainalcohols. The inability of any of the two exclusive models topredict the minimum in the cmc indicates that the alcoholmolecules have a dual role.

The prediction of the cmc of the homologous family of Cn

TAB in ethanol-water solutions (n = 10, 12, 14, 16) is com-pared to experimental data10,11,39 in Figure 5. The results for

Figure 2. Cmc of SDS in methanol-water mixtures. Data: Leeand Huang27 at 25 �C (square), L�opez-Grio et al.28 (temperaturenot reported in the paper) (upward-pointing triangle), Parfitt andWood29 at 25 �C (sphere), and Emerson and Holtzer30 at 25 �C(diamond). Predictions at 25 �C:methanol exclusively as cosolvent(dashed line), methanol exclusively as cosurfactant (dotted line),and methanol with dual role (solid line).

Figure 3. Cmc of SDS in ethanol-watermixtures. Data: Suzuki32

at 30 �C (upward-pointing triangle), Javadian et al.33 at 25 �C(square and diamond), Rafati et al.34 at 29.85 �C (sphere). Predic-tions at 25 �C: ethanol exclusively as cosolvent (dashed line),ethanol exclusively as cosurfactant (dotted line), and ethanol withdual role (solid line).

Figure 4. Cmc of SDS in propanol-water mixtures. Data: Rafatiet al.34 at 29.85 �C (square), Romani et al.38 at 24.85 �C (upward-pointing triangle), Jain and Singh37 at 25 �C (downward-pointingtriangle), Shirahama and Kashiwabara35 at 25 �C (sphere). Pre-dictions at 25 �C: propanol exclusively as cosolvent (dashed line),propanol exclusively as cosurfactant (dotted line), and propanolwith dual role (solid line).

Figure 5. Cmc of CnTAB in ethanol-water mixtures. Data:Huang et al.39 at 30 �C (downward-pointing triangle forC10TAB, plus sign for C12TAB), Cipiciani et al.11 at 25 �C(upward-pointing triangle for C14TAB, square for C16TAB), Liet al.10 at 25 �C (sphere forC16TAB). Predictions:C10TABat 30 �C(dashed line), C12TAB at 30 �C (dash-dot line), C14TAB at 25 �C(dotted line) and C16TAB at 25 �C (solid line).

(27) Lee, D. J.; Huang, W. H. Colloid Polym. Sci. 1996, 274, 160–165.(28) L�opez-Grı́o, S.; Baeza-Baeza, J. J.; Garcı́a-Alvarez-Coque, M. C. Chro-

matographia 1998, 48, 655–663.(29) Parfitt, G. D.; Wood, J. A. Koll. Z. Z. Polym. 1968, 229, 55–60.(30) Emerson, M. F.; Holtzer, A. J. Phys. Chem. 1967, 71, 3320–3330.(31) Ward, A. F. H. Proc. R. Soc. London, Series A: Math. Phys. Sci. 1940, 176,

412–427.(32) Suzuki, H. Bull. Chem. Soc. Jpn. 1976, 49, 1470–1474.(33) Javadian, S.; Gharibi, H.; Sohrabi, B.; Bijanzadeh, H.; Safarpour, M. A.;

Behjatmanesh-Ardakani, R. J. Mol. Liquids 2008, 137, 74–79.(34) Rafati, A. A.; Gharibi, H.; Rezaie-Sameti, M. J. Mol. Liquids 2004, 111,

109–116.(35) Shirahama, K.; Kashiwabara, T. J. Colloid Interface Sci. 1971, 36, 65–70.(36) Manabe, M.; Koda, M. Bull. Chem. Soc. Jpn. 1978, 51, 1599–1601.(37) Jain, A. K.; Singh, R. P. B. J. Colloid Interface Sci. 1981, 81, 536–539.(38) Romani, A. P.; Gehlen, M. H.; Lima, G. A. R.; Quina, F. H. J. Colloid

Interface Sci. 2001, 240, 335–339.(39) Huang, J. B.;Mao,M.; Zhu, B. Y.Colloids Surf. A: Physicochem. Eng. Asp.

1999, 155, 339–348.

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DOI: 10.1021/la9018426 12107Langmuir 2009, 25(20), 12101–12113

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C16TAB-ethanol-water mixtures show that the dual modelpredicts the initial cmc drop that was experimentally observedby Cipiciani and co-workers11 (squares in Figure 5), but was notpredicted by the mixed solvent model by Li and co-workers6

(circles in Figure 5).The mixed micelle model correctly predicts the decreasing

trend of the cmc in SDS-butanol-water mixtures28,35-37,40,41

(see Figure 6) and in C14TAB-butanol-water mixtures42

(see Figure 7), but it does not predict the cmc for the entire rangeof concentration for which the cmc has been experimentallymeasured forC14TAB-butanol-watermixtures. The dualmodelpredicts the cmc for the entire range of concentration for bothsystems with good agreement.

The predictions of cmc of SDS in pentanol-water, hexanol-water, and heptanol-water mixtures are compared with experi-mental data36,37,40,41,43 in Figures 8, 9, and 10.With the increaseof the alcohol tail length, the cosolvent effect reduces progres-sively as the concentration of the alcohols in the systemdecreases.

The results in Figures 2-10 reveal that both cosolvent andcosurfactant effects are important in short-chain alcohols. Theynot only have an imperative function altering the properties of thesolvent, but also penetrating the micellar shell, whereas alcoholswith long hydrocarbon chains work primarily as cosurfactants,

influencing the properties of the aggregates. However, the transi-tion between both extremes is gradual.Aggregation Number and Micellar Size. Other important

micellar properties which can be computed and compared withexperimental data are micellar molecular weight, number ofsurfactant molecules in the micelles, micellar size and composi-tion. In Tables 3, 4, and 5, model predictions are compared withexperimental data for micellar molecular weight and number ofsurfactant molecules in the micelles in alcohol-water mixtures atcmc for short-chain alcohols.

Figure 6. Cmc of SDS in butanol-water mixtures at 25 �C. Data:JainandSingh37 (square), ShirahamaandKashiwabara35 (sphere),Hayase and Hayano40 (upward-pointing triangle). Predictions:butanol exclusively as cosurfactant (dotted line), butanol with dualrole (solid line).

Figure 7. Cmc of C14TAB in butanol-water mixtures at 25 �C.Data: Zana et al.42 (spheres). Predictions: butanol exclusively ascosurfactant (dotted line), butanol with dual role (solid line).

Figure 8. CmcofSDS inpentanol-watermixtures at 25 �C.Data:Jain and Singh37 (upward-pointing triangle, square), Hayase andHayano40 (sphere). Predictions: pentanol exclusively as cosurfac-tant (dotted line), pentanol with dual role (solid line).

Figure 9. Cmc of SDS in hexanol-water mixtures. Data: Jain andSingh37 at 25 �C (square), Hayase andHayano40 at 25 �C (sphere),Abuin and Lissi43 at 20 �C (upward-pointing triangle). Predictionsat 25 �C: hexanol exclusively as cosurfactant (dotted line), hexanolwith dual role (solid line).

Figure 10. Cmcof SDS in heptanol-watermixtures.Data:Hayaseand Hayano40 at 25 �C (sphere), Abuin and Lissi43 at 20 �C(square), Manabe and Koda36 at 25 �C (dash-dot line), Manabeet al.41 at 25 �C (dashed line). Predictions at 25 �C: heptanolexclusively as cosurfactant (dotted line), heptanol with dual role(solid line).

(40) Hayase, K.; Hayano, S. Bull. Chem. Soc. Jpn. 1977, 50, 83–85.(41) Manabe, M.; Kawamura, H.; Yamashita, A.; Tokunaga, S. J. Colloid Int.

Sci. 1987, 115, 147–154.(42) Zana, R.; Yiv, S.; Strazielle, C.; Llanos, P. J. Colloid Interface Sci. 1981, 80,

208–223.(43) Abuin, E. B.; Lissi, E. A. J. Colloid Interface Sci. 1983, 95, 198–203.

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The predicted micellar molecular weight for SDS in water is insatisfactory agreement with experimental data as shown pre-viously by Nagarajan and Ruckenstein.16 In Table 3, the “appar-ent” micellar molecular weight reported by Parfitt andWood29 iscompared with model predictions. The results from the cosurfac-tant model show that micelles would stop being formed atconcentrations for which micelle formation has been experimen-tally verified (see Table 3). The results from the cosolvent modeldisagree with the experimental trend showing an increase inmicellar molecular weight with increasing methanol concentra-tion. By comparing the micellar molecular weight in Table 3 withthe micellar radius in Figure 11, we verify that for SDS inmethanol-water the increase in the micellar molecular weightfor alcohol acting exclusively as cosolvent is coupled withan increase in the micellar radius. Additionally, in Figure 11,

we observe the dual model and the cosurfactantmodel predict themicellar radius decrease with different rates.

The same trend of micellar molecular weight for SDS inmethanol-water is also seen in Table 4 for the number ofsurfactant molecules in C16TAB micelles in ethanol-water mix-tures. For SDS in propanol-water mixtures (see Table 5), boththe cosurfactant model and the dual model predict the correcttrend in the range of concentration of the data, but the dualmodelgives an improved prediction. This range of concentration in theexperimental data for SDS in propanol-water mixtures is in theregion of decreasing cmc.

Caponetti et al.3 have done size and compositional measure-ments of SDSmicelles at constant SDS concentration and differentalcohol concentrations. We selected the same conditions used byCaponetti et al. to compare the dual model predictions with theirexperimental data. In Figures 12 to 15, the SDS concentration iskept constant at 0.2 M in aqueous-alcoholic solutions, frommethanol to heptanol, with alcohol concentration up to 2% v/vat 25 �C. Caponetti et al. show that the alcohols affect the micellarsize in the following sequence: butanol< pentanol< propanol<ethanol < methanol < methanol < hexanol < heptanol. Thealcohol that promotes the smallest micelle is butanol, and in thepresence of hexanol and heptanol, the mixed micelles are largerthan the SDS single micelles. The micellar size sequence predictedby the dualmodel is in agreementwith the experimental data exceptfor pentanol and propanol (see Figure 12).

In order to investigate the distribution of alcohols between theaqueous and the micellar phases we plot the micellar composition

Table 3. Micellar Molecular Weight of SDS Micelles at Cmc in

Methanol-Water Mixtures at 25 �C

micellar molecular weight

mole fraction ofmethanol

exptldata29

dualmodel

cosurfactantmodel

cosolventmodel

0 14300 21052 21052 21052

0.06 12200 10604 5898 22205

0.12 3930 7655 23070

Table 4.Number of SurfactantMolecules inC16TABMicelles at Cmc

in Ethanol-Water Mixtures at 25 �C

gA

concentration of ethanol(% v/v)

exptldata10

dualmodel

cosurfactantmodel

cosolventmodel

0 60 43 43 4310 41 20 8 4850 19 19 55

Table 5. Number of Surfactant Molecules in SDSMicelles at Cmc in

Propanol-Water Mixtures at 25 �C

gA

concentration ofpropanol (M)

exptldata38

dualmodel

cosurfactantmodel

cosolventmodel

0.2 65 ( 3 47 44 740.4 56 ( 3 35 29 750.6 47 ( 4 29 19 760.8 44 ( 7 25 12 77

Figure 11. Predictions of the micellar radius of SDS micelles atcmc inmethanol-watermixtures at 25 �C.Methanol exclusively ascosolvent (dashed line), methanol exclusively as cosurfactant(dotted line), and methanol with dual role (solid line).

Figure 12. Dual model predictions of the micellar radius forSDS-alcohol-water systems as functions of the alcohol concen-trations (% v/v): methanol (diamond), ethanol (plus sign), propa-nol (downward-pointing triangle), butanol (square), pentanol(sphere), hexanol (upward-pointing triangle), heptanol (cross).

Figure 13. Dual model predictions of RgA versus RsA at the rangeof concentration up to 2% v/v of alcohol: methanol (diamond),ethanol (plus sign), propanol (downward-pointing triangle), buta-nol (square), pentanol (sphere), hexanol (upward-pointing tri-angle), heptanol (cross), and equidistributed (solid line).

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RgA (eq 13) versus RsA in Figure 13. Here RsA is the fraction ofsurfactant A in the system on a water-free basis:

RsA ¼ NsA

NsA þNsBð23Þ

In Figure 13, for a given alcohol, RgA decreases with decreasingtotal surfactant fraction in the system RsA. For methanol(diamond) the fraction of SDS in the micelles is above 0.9 eventhough the fraction of SDS in the system RsA goes down to levelsbelow 0.5. Pentanol is the alcohol closest to the equidistribution(solid line). For alcohols shorter than pentanol, RgA is greaterthanRsA ; while for alcohols longer than pentanol,RgA is less thanRsA. As expected, short-chain alcohols accumulate in the aqueousphase and long-chain alcohols penetrate the micellar phase.

Figures 14 and 15 show that number of SDSmolecules formingthe micelles decreases with alcohol concentration. In agreementwith experimental data,3 pentanol is the alcohol that promotes thelargest decrease in the number of SDS molecules in the micelles(Figure 14). The predictions of the numberof alcoholmolecules inthe micelles (Figure 15) are also in satisfactory agreement withexperimental data of Caponetti and co-workers.3

Influence of Alcohol on Aggregation. The addition ofalcohol may affect three solvent dependent free energy contribu-tions. The effect of added alcohol on these solvent dependent freeenergy contributions may help to explain the observed behavior

of the cmc and aggregation number. The first solvent dependentfree energy contribution is the surfactant tail transfer free energy.The absolute value of the transfer free energy is lower in alcoholsthan in water. Furthermore, the absolute value of the surfactanttail transfer free energy decreases with the length of the aliphatictail of the alcohol. The decrease in the absolute value of thetransfer free energy alone increases the cmc.

The second solvent dependent free energy contribution is thatassociated with the formation of the aggregate core-solventinterface. The addition of alcohol to the aqueous solution reducesthe interfacial tension between the aqueous solution and thehydrocarbon core, therefore promoting smaller aggregates. How-ever, even though the lower hydrocarbon-solvent interfacialtension causes the cmc and the size to decrease, the larger areapermolecule associatedwith small aggregation numbersmay giverise to an opposite tendency of increasing cmc.9

The third solvent dependent contribution is associated with theelectrostatic headgroup interactions. The lower dielectric con-stants of alcohols than that of water leads to an increase in therepulsion between ionic charges. An increase in the repulsionbetween charges causes an increase in both the double layerenergy and cmc. However, the calculations show that the ionicinteractions decrease with increasing concentration of alcohol.The lower value of the ionic interactions is due to the participationof the alcohols on themicelle shell which lowers the surface chargedensity, decreasing the electrostatic repulsion force between ionicheadgroups, thereby decreasing the cmc.

The addition of short-chain alcohols to ionic surfactant sys-tems causes twoopposite effects: at lowalcohol concentration, thealcohol molecules will predominantly be incorporated in themicelle shell (cosurfactant effect), decreasing the electrostaticrepulsion energy and causing the cmc to decrease. However, asthe alcohol concentration increases, the transfer free energyincreases and the dielectric constant of the solution decreases-(cosolvent effects). At some critical concentration, the cosolventeffect overcomes the cosurfactant effect and the cmc starts toincrease.

For medium-chain alcohols, the change in the transfer freeenergy due to the singly dispersed alcohol in the bulk phase causesthe cmc to increase compared to the cosurfactant model that doesnot include solvent change due to the presence of the alcohol.

Concerning the micelle size and aggregation number of med-ium-chain alcohols, there is a strong dependency on the length ofthe surfactant and alcohol tails, and also on the surfactantheadgroup. This is because contributions to the standard freeenergy of micellization depend on these surfactant and alcoholcharacteristics, and have an important influence on the size,aggregation number, and micelle composition. For the systemscontaining short-chain alcohols investigated in this work, the sizeand the number of surfactant molecules decrease. The decrease inmicelle size may be due to the decrease of the interfacial tension(cosolvent effect) and the incorporation of alcohols in the micelleshell (cosurfactant effect) with increasing concentration. Botheffects promote smaller aggregates, and overcome the decrease ofthe dielectric constant (cosolvent effect) that would promotelarger aggregates. The incorporation of alcohols in the micelleshell would also cause a decrease in the number of surfactants inthe micelles.

Conclusions

The Gibbs free energy minimization approach used in thiswork is a powerful alternative to the size distribution approachin thermodynamic modeling for surfactant self-assembly. Theapproach is applied with remarkable success to thermodynamic

Figure 14. Dual model predictions of the number of SDS mole-cules in the micelles of different alcohols vs concentration (% v/v):methanol (diamond), ethanol (plus sign), propanol (downward-pointing triangle), butanol (square), pentanol (sphere), hexanol(upward-pointing triangle), and heptanol (cross).

Figure 15. Dual model predictions of the number of alcoholmolecules in the micelles of different alcohols vs concentration(% v/v): methanol (diamond), ethanol (plus sign), propanol(downward-pointing triangle), butanol (square), pentanol(sphere), hexanol (upward-pointing triangle), and heptanol(cross).

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modeling for spherical and globular single micelles, mixedmicelles, and mixed solvents developed by Nagarajan and co-workers.

We propose a thermodynamic dual model for alcohols that actas cosurfactant and cosolvent concomitantly. The dual model isapplied to linear 1-alcohols from methanol to heptanol in ionicsurfactant systems. We show a quantitative improvement by thedual model to predictions of micellar properties such as cmc andmicellar size of surfactants in alcoholic-aqueous solutions. Thedual model has the advantage to provide structural informationand distribution of the alcohols between the aqueous and themicellar phases. It grants access to information on the extensionof penetration of the alcohol in the micellar shell.

Appendix: Standard Free Energy of Micellization

The expressions for the different contributions to the standardfree energy of micellization, Δμg

o, are based on the work byNagarajan and co-workers.16,7

Δμog ¼ ðΔμogÞtr þ ðΔμogÞdef þ ðΔμogÞint þ ðΔμogÞstericþ ðΔμogÞionic þ ðΔμogÞmix ðA1Þ

The first contribution (Δμgo)tr represents the transfer of the

surfactant hydrophobic tail from the solvent to the aggregate coreassumed to be like a hydrocarbon liquid. In this work we use themixed solvent approach of Nagarajan andWang9 for the transferfree energy. We provide the methyl and methylene group con-tribution to transfer free energy expressions for pure alcoholswithexplicit temperature dependence.

The second contribution is the packing and deformation of thesurfactant tail, (Δμg

o)def. This free energy accounts for the fact thatsurfactant tail inside the aggregate core deforms locally in order tosatisfy both the packing and the uniform density constraints. Thethird contribution is the free energy of formation of aggregatecore-solvent interface, (Δμg

o)int. This term incorporates theinfluence of short-chain alcohols on the macroscopic interfacialtension of the solvent.

The fourth contribution is the steric free energy, (Δμgo)steric,

which accounts for the steric repulsion between the surfactantheadgroups on the aggregate surface. The fifth contribution is theelectrostatic repulsions between the ionic headgroups at theaggregate surface, (Δμg

o)ionic. For short-chain alcohols, the influ-enceof the alcohol on the dielectric constant of the solvent is takeninto account. Finally, the free energy of mixing term, (Δμg

o)mix,accounts for the enthalpy and entropy ofmixing of surfactant tailsin the hydrophobic core of the micelle whenever there are twodifferent tail lengths.

Explicit analytical expressions for (Δμgo)tr, (Δμg

o)int, and(Δμg

o)ionic are presented in this Appendix in terms of themolecularcharacteristics of the surfactants and alcohol. Explicit analyticalexpressions for (Δμg

o)def, (Δμgo)steric, and (Δμg

o)mix are used fromrefs 16 and 17.

1. Transfer of the Surfactant Tail. The transfer free energyof the surfactant tail from the solvent to the hydrophobic core ofthe aggregate is accounted for here. The transfer free energy fromthe mixed solvent depends on the transfer free energies from purewater and pure cosolvent and also on the interactions betweenthe two solvents:9

ΔμogkT

!tr

¼ jW

ΔμogkT

!tr,W

þ j1B

ΔμogkT

!tr, solB

- jW lnVW

V- j1B ln

VB

Vþ χWBjWj1B ðA2Þ

where subscriptsW and B refer to water and cosolvent B,jW andj1B are the respective volume fractions in the mixture, V is themolar volume of the mixed solvent calculated from the molarvolumes VW and VB and mole fractions YW and Y1B of the twocomponents, and χWB is the Flory interaction parameter betweenwater and cosolvent B.

VW ¼ MW

FW, VB ¼ MB

FBðA3Þ

where MW and MB are the molecular weight of water and thealcohol, respectively, and FW and FB are the density of water andthe alcohol, respectively. The atomic weights of the elements usedare from ref 44 and the densities as function of temperature arefrom ref 45. Themole fractions and volume fractions are given by:

YW ¼ NW

NW þ N1B, Y1B ¼ N1B

NW þ N1BðA4Þ

jW ¼ YWVW

V, j1B ¼ Y1BVB

VðA5Þ

V ¼ YWVW þ Y1BVB ðA6ÞThe Flory interaction parameter χWB is obtained by fitting

the activity data for water-alcohol mixtures to the Flory-Huggins equation to be presented shortly. In the absenceof adequate experimental data, we calculate the activity ofwater (aW) in the solvent mixture as a function of compositionusing the modified UNIFAC (Dortmund) method12 and fitthese data to the expression given by the Flory-Hugginsmodel.9

ln aW ¼ lnð1 - jsolBÞ þ 1 -VW

VB

� �jsolB þ χwBjsolB

2 ðA7Þ

The Flory interaction parameters used in the calculationspresented in this work are given in Table 6.

For amixedmicelle of composition (RgA,RgB), the transfer freeenergy per surfactant molecule is given by:8

ΔμogkT

!tr,W

¼ RgA

ΔμogkT

!tr, tA,W

þ RgB

ΔμogkT

!tr, tB,W

ðA8Þ

Table 6. Water-Alcohol Flory Interaction Parameter

cosolvent B T (�C) χWB

methanol 25 0.5604ethanol 25 1.0484ethanol 30 1.0568propanol 25 1.4372butanol 25 1.7476pentanol 25 2.0033hexanol 25 2.2194heptanol 25 2.4056

(44) Lide, D. R., Ed. CRC Handbook of Chemistry and Physics, 84th ed; CRCPress: Boca Raton, FL, 2004.

(45) Perry, R. H., Green, D. W., Maloney J. O’H., Eds. Perry’s ChemicalEngineers’ Handbook, 7th ed; McGraw-Hill, 1997.

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ΔμogkT

!tr, solB

¼ RgA

ΔμogkT

!tr, tA, solB

þ RgB

ΔμogkT

!tr, tB, solB

ðA9Þ

where (Δμgo/kT)tr,tA,W and (Δμg

o/kT)tr,tB,W are the transfer freeenergy of the tail of surfactants A and B, respectively, from purewater, and (Δμg

o/kT)tr,tA,solB and (Δμgo/kT)tr,tB,solB are the free

energy of transferring the tail of surfactantsA andB, respectively,from pure alcohol.

The expressions for the methylene and methyl group contribu-tion to the transfer free energy of an aliphatic tail as a function oftemperature T (in kelvin) from pure water is estimated byNagarajan and Ruckenstein:16

ΔμogkT

!tr,CH2,W

¼ 5:85 ln T þ 896

T- 36:15 - 0:0056T

ðA10Þ

ΔμogkT

!tr, CH3,W

¼ 3:38 ln T þ 4064

T- 44:13 - 0:02595T

ðA11ÞThe transfer free energy of the surfactant tail from pure

alcohols is estimated from independent experimental data onthe solubility of alcohols in hydrocarbonsusing experimental dataof infinite dilution activity coefficient from Dechema.46 In addi-tion to that, the infinite dilution activity coefficients predicted bythe modified UNIFAC (Dortmund)12 method are used to fill upthe temperature range for which there is not experimental dataavailable. The expressions for the methylene and methyl groupcontribution to the free energy of transfer of an aliphatic tail as afunction of temperature T (in kelvin) ranging from 283.15 to363.15 K from pure alcohols are estimated and given here.

For methanol:

ΔμogkT

!tr, CH3,methanol

¼ 1:0355 ln T þ 174:3589

T- 7:5749

ðA12Þ

ΔμogkT

!tr, CH2,methanol

¼ 3:0669 ln T þ 852:8705

T- 20:6123

ðA13ÞFor ethanol:

ΔμogkT

!tr, CH3, ethanol

¼ 5:8958 ln T þ 1623:9480

T- 39:9229

ðA14Þ

ΔμogkT

!tr, CH2, ethanol

¼ - 1:9634 ln T -627:32613

Tþ 13:1197

ðA15Þ

For propanol:

ΔμogkT

!tr, CH3, propanol

¼ 1:3828 ln T þ 361:0902

T- 9:7412

ðA16Þ

ΔμogkT

!tr, CH2, propanol

¼ 1:3335 ln T þ 372:7471

T- 8:9903

ðA17Þfor butanol:

ΔμogkT

!tr, CH3, butanol

¼ - 0:8138 ln T -233:8105

Tþ 4:8891

ðA18Þ

ΔμogkT

!tr, CH2, butanol

¼ 1:5672 ln T þ 443:3190

T- 10:5381

ðA19ÞFor pentanol:

ΔμogkT

!tr, CH3, pentanol

¼ 5:3438 ln T þ 1658:9049

T- 36:5469

ðA20Þ

ΔμogkT

!tr, CH2, pentanol

¼ 0:07967 ln T -8:4656

T- 0:5166

ðA21Þfor hexanol:

ΔμogkT

!tr, CH3, hexanol

¼ - 1:6493 ln T -575:6458

Tþ 10:9191

ðA22Þ

ΔμogkT

!tr, CH2, hexanol

¼ 1:3309 ln T þ 395:3028

T- 9:0001

ðA23Þ

For heptanol:

ΔμogkT

!tr, CH3, heptanol

¼ 2:5102 ln T þ 737:1626

T- 17:1426

ðA24Þ

ΔμogkT

!tr, CH2, heptanol

¼ 0:3052 ln T þ 74:4594

T- 2:0704 ðA25Þ

2. Formation of Aggregate Core-Solvent Interface. Thefree energy associated with the formation of the hydrophobic

(46) Tiegs, D.; Gmehling, J.; Medina, A.; Soares, M.; Bastos, J.; Alessi, P.Kikic,I. DECHEMA Chemisry Data Series, Activity Coefficients at Infinite Dilution;Parts 1-2; DECHEMA: Frankfurt/Main, Germany, 1986; Vol. IX.Gmehling, J.; Menke,J.; Schiller, M. DECHEMA Chemisry Data Series, Activity Coefficients at InfiniteDilution; Parts 3-4; DECHEMA: Frankfurt/Main, Germany, 1994; Vol. IX.

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12112 DOI: 10.1021/la9018426 Langmuir 2009, 25(20), 12101–12113

Article Moreira and Firoozabadi

core-water interface is given for binary mixtures by the expres-sion:8

ðΔμogÞintkT

¼ σagg

kTða - RgAaoA - RgBaoBÞ ðA26Þ

Here, σagg is themacroscopic interfacial tension, a is the surfacearea of the hydrophobic core per surfactant molecule, and aoAand aoB are the areas per molecule of the core surface shieldedfrom contact with water by the polar headgroups of surfactants Aand B. aoB is estimated from the chemical structure of thesurfactant molecule by Nagarajan17 and is listed in Table 2. Alsoin Table 2 are listed the cross-sectional areas of the headgroups ofsurfactants A and B respectively, apA and apB.

For medium- and long-chain alcohols the aggregate core-water interfacial tension σagg is taken to be equal to the interfacialtension between water and the aliphatic hydrocarbon of the samemolecular weight as the surfactant tail (σtail,W). The water-normal alkane interfacial tension data47,48 are fitted to

σtail,W ¼ 0:7562ðσW þ σtailÞ - 0:4906ðσWσtailÞ0:5 ðA27Þwhere the interfacial tension and surface tension are in mN/m.The surface tension of pure water is given by49

σW ¼ 235:8 1 -T

647:15

� �1:256

1 - 0:625 1 -T

647:15

� �" #ðA28Þ

where σW is expressed in mN/m and the temperature in kelvin.The surface tension of pure normal alkanes is fitted from experi-mental data:50

σtail ¼ 29:7003½1 - expð - 0:1532ncÞ� - 0:0896ðT - 298:15ÞðA29Þ

where nc correspond to the number of carbonatoms of the normalalkane tail. For short-chain alcohols σagg is taken to be equal tothe interfacial tension between the solvent and the aliphatichydrocarbon of the same molecular weight as the surfactant tail,given by

σagg ¼ ð1 - YsolBÞðηAσtailA,W þ ηBσtailB,WÞþ YsolBðηAσtailA, solB þ ηBσtailB, solBÞ ðA30Þ

where YsolB is the mole fraction of short-chain alcohol that actsexclusively as cosolvent.

YsolB ¼ pcNsB

NW þ pcNsBðA31Þ

The liquid-liquid interfacial tension between the short-chainalcohols and normal alkanes is modeled by the following expres-sion:51

σtail, solB ¼ σsolB þ σtail - 2ΦsolBðσsolBσtailÞ0:5 ðA32Þ

where the surface tension and the interfacial tension are in mN/mand ΦsolB is a the interaction parameter at the alcohol-alkaneinterfaces. The interaction parameters are estimated51 to beΦmethanol = 0.82, Φethanol = 0.90, and Φpropanol = 0.93 foralcohol-n-dodecane interfaces. In this work, the dependence ofthese parameters on the surfactant tail length is neglected.

The surface tension of pure methanol is given by52

σmethanol ¼ 66:03 1 -T

512:6

� �1:062

1 - 0:364 1 -T

512:6

� �" #

for 279:11 K e T e 333:81 K ðA33ÞThe surface tension of pure ethanol is given by53

σethanol ¼ 45:80 - 0:0804T for 288:15 K e T e 308:15 K

ðA34ÞThe surface tension of pure propanol is given by:50

σpropanol ¼ 46:48 - 0:0777T for 283:15 K e T e 363:15 K

ðA35Þ3. Headgroup Ionic Interactions. The ionic free energy

expression is proposed by Nagarajan for mixed micelles.8 Thefree energy expression has the form:

ðΔμogÞionickT

¼ 2 lnS

2þ 1 þ S2

4

!1=28<:

9=;

-4

S1 þ S2

4

!1=2

- 1

8<:

9=; -

4C

KSln

1

2þ 1

21 þ S2

4

!1=28<:

9=;

ðA36Þwhere

S ¼ e2

εoεmixKaδ, ionkTðA37Þ

and κ is the reciprocal Debye length. The area per molecule aδ,ionis evaluated at a distance δ from the hydrophobic core surface (seeTable 1), where the center of the counterion is located. The valuesfor δ are given in Table 2.

For the casewhenA is an ionic surfactantwhileB is a non-ionicsurfactant

Rg, ion ¼ RgA, Rδ, ion ¼ aδag, ion

, δ ¼ δA ðA38Þ

The first two terms on the right side of eqA36 constitute the exactsolution to the Poisson-Boltzmann equation for a planar geo-metry, and the last term provides the curvature correction. Thecurvature-dependent factor C is given by

C ¼ 2

RS þ δðA39Þ

(47) Aveyard, R.; Haydon, D. A. Trans. Faraday Soc. 1965, 61, 2255–2261.(48) Zeppieri, S.; Rodrı́guez, J.; Ramos, A. L. L. J. Chem. Eng. Data 2001, 46,

1086–1088.(49) Vargaftik, N. B.; Volkov, B. N.; Voljak, L. D. J. Phys. Chem. Ref. Data

1983, 12, 817–820.(50) Jasper, J. J. J. Phys. Chem. Ref. Data 1972, 1, 841–1009.(51) Janczuk, B.; Bialopiotrowicz, T.; Wojcik, W. Colloids Surf. 1989, 36, 391–

403.

(52) Souckov�a, M.; Klomfar, J.; P�atek, J. J. Chem. Eng. Data 2008, 53, 2233–2236.

(53) Calvo, E.; Pintos, M.; Amigo, A.; Bravo, R. J. Colloid Interface Sci. 2002,253, 203–210.

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DOI: 10.1021/la9018426 12113Langmuir 2009, 25(20), 12101–12113

Moreira and Firoozabadi Article

The reciprocalDebye length κ is related to the ionic strength of thesolution via

K ¼ 2nοe2

εoεmixkT

!1=2

, no ¼ 103C1NAv ðA40Þ

In the above equation, no is the number of counterions insolution per m3, C1, is the molar concentration of the singlydispersed surfactant molecules in moles per liter, and NAv isAvogadro’s number.

The molar concentration of the singly dispersed surfactantmolecules, C1, is calculated from

C1 ¼ 1000 νA þ 1 - X1A

X1A

� �V

" # - 1

ðA41Þ

where vA (= vsA þ vhA) is the total volume of the surfactant A,X1A is the mole fraction of singly dispersed surfactant A (eq 12),and V is the total molar volume (eq A6). The headgroup volume,vhA, is given in Table 2.

The dielectric constant εmix is available from experimentalmeasurements54 for several compositions of water-alcohol mix-tures at temperatures ranging from 283.15 to 313.15 K. We havecorrelated the dielectric constant of the mixture to the dielectricconstant of the pure components.

ln εmix ¼ ð1 - YsolBÞ ln εW þ YsolB ln εsolB þ ψsolBð1 - YsolBÞYsolB

ðA42Þ

where ψsolB is the interaction parameter given by ψmethanol =0.694, ψethanol = 0.783, and ψpropanol = 0.694 for alcohol-watermixtures. The dielectric constant of pure water and pure short-chain alcohols as function of temperature (in kelvin) are givenby44

εW ¼ - 1:0677 þ 306:4670 exp½ - 4:52

� 10 - 3T � for 273:15 K e T e 373:15 K ðA43Þ

εmethanol ¼ 0:1934� 103 - 0:92211T þ 0:12839

� 10 - 2T2 for 177 K e T e 293 K ðA44Þ

εethanol ¼ 0:15145� 103 - 0:87020T þ 0:19570

� 10 - 2T2 - 0:15512

� 10 - 5T3 for 163 K e T e 523 K ðA45Þ

εpropanol ¼ 0:98045� 102 - 0:3686T þ 0:36422

� 10 - 3T2 for 193 K e T e 493 K ðA46Þ

Acknowledgment.We thank Dr. Ramanathan Nagarajan forhelpful discussions; the Electrical Engineering Dept. and theInstitute Systems Research (ISR) at University of Maryland forkindly providing the code of the FSQP algorithm; and themember companies of the Reservoir Engineering Research In-stitute (RERI) in Palo Alto, CA, for financial support.

(54) Wohlfarth, C. Static Dielectric Constants of Pure Liquids and Binary LiquidMixtures; Landolt-B€ornstein; Springer-Verlag: Berlin, 1991.

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