6683Langmuir 1997, 13, 6683-6688

Theoretical Model and Phase Behavior for BinarySurfactant Mixtures I~

Lei Huang and P. Somasundaran*

Langmuir Center for Colloids and Interfaces, Columbia University,lVew York, New York 10027

Received May 5, 1997. In Final Form: September 3, 199~

A new model for micellization beha..-ior of binary surfactant mixtures in solution is presented. A packingparameter, P* , has been introduced to describe the nonrandom mixing in mixed micelles. The significantfeature of this model is that it considers asymmetric behavior ofmicellization in binary surfactant mixturesand predict5 the changes in the structures of mixed micelles with concentration that are dependent onthe overall ratio of the surfactant components and their chemical characteristics. On the basis of theresults obtained, a schematic model for the structural changes of mixed micelles has been proposed here.This model can be considered as a generic model for binary surfactant mixtures, \\ith regular solutiontheory being a special case for a packing parameter approaching infinity.

I. IntroductionMixed surfactants are of considerable interest because

of their wide commercial use. The properties of surfactantmixtures have received some attention in the past;1-3however. their behavior cannot yet be predicted fromavailable theory. As in the case of individual surfactants,the formation ~f mixed micelles in aqueous solution leadsto marked changes in the physicochemical properties ofthe system, but the micellization behavior of mixedsurfactant systems is indeed much more complex thanthat I>f single surfactants. In this work, a model whichcan predict the behavior of mixed surfactants is developedby taking the major interactions between the componentsinto consideration.

The thermodynamic models of mixed micelle formationcan be divided into two broad categories. At one level,which may be considered the more fundamental, thepr'.1cess is treated as a reversible chemical reaction andt1:-. mass action approach is used.4-s The reactants arethe surfactants. the bound,. counterions, and, when ap-propriate, solubilizates. The standard free energy as-sociated with this reaction is in reference to a standardstate in which the reactants are dispersed at infmitedilution. This type of model is appropriate because micelleformation is reversible. Micelles have a finite lifetime. asdo the individual components that make up a micelle.7-9The aggregation number is a dynamic variable and, atarlY instant, various individual miceUes will be composedof iiffering numbers of surfactant molecules. Further-

more, in the case of mixed micelles, the number ofsurfactant molecules of a given structure may also varyfrom micelle to micelle. In principle, this kind of treatmentcan be extended to multicomponent systems, but thecalculations required for the model then become complex.Many parameters and assumptions have to be made toget the data for the changes in standard free energyassociated in the micellization process.9,10

The second approach is to consider the micelle as aseparate bulk phase having an infinite lifetime.l.11-13 Thechemical potentials of the components in the micellarphase, sometimes called a pseudophase to emphasize thefact that it is not quite a true thennodynamically differentphase, can then be equated to those of the molecules thatare dispersed in the aqueous phase. Various models forchemical potential can be used to account for the nonidealbehavior in both the micellar and solution phases.

The phase separation model defines the proportions ofvarious components that exist within the micellar phase,and it also provides equations for the cmc of mi.'ttures; butit does not address factors that relate to the size of themicelles or their aggregation number. This type of modelis usually simple to use. Calculation using a monodispersemass action model suggests that for micelles of 50molecules or more, the pseudo phase separation modelshould provide a good approximation. One of the mostpopular models for this type of treatment is the regularsolution theory introduced first by Rubingh.13 The basicassumption in regular solution theory is that the excessentropy and mi.'tingvolume are zero and hence the excessenthalpy of mi.'Cing is equal to the excess free energy, whichcan be expressed using the results applied to "regularsolution" mL'ttures.

Even though the regular solution treatment is simple,it is to be noted that the basic assumption in this treatment,i.e. mixing entropy 1S zero, is not thermodynamically validfor describing nonideal mixed micelles. Therefore, thismodel should be viewed only as a useful empirical model.Furthennore, regular solution treatment is successfulmairily with respect to the correlation of mi.",ed criticalmicelle concentrations with one interaction parameter {3,13

l10) Nagarajan. R. In Mixed surfactant systems; Holland. P. M..Rubingh. D. N., Eds.;ACS Symposium Series SOl; American ChemicalSociety: Washington, DC, 1992; Chapter 4, pp 54-95.

(11) Holland, P. M.; Rubingh, D. N. J. Phys. Chern. 1983,87,1984.(12) Motomura, K,; Ando, N.; Matsuki. H.; Aratona, M. J. CoUoid

Interface Sci. 1990, 139, 188.(13) Rubingh, D. N.InSolution ChemistryofSurfactants; Mitall, K

L.. Ed.; Plenum Press: New York., 1979; Vol. 1, P 337.

te) 1997 American Chemical Society

Corresponding Author: Professor P. Somasundaran, La vonDuddleson Krumb Professor, School of Engineering and AppliedScience, 500 W. 12Oth Street, Mudd Building 918, ColumbiaUniversity, New York, NY 10027.

8 Abstract published in Advance ACS Abstracts, November 15,

1997.\1) Holland, P. M. Adv. Colloid lnterface Sci. 1986,26, Ill.(2) Scamehom. J. F. In Phenomena in Mixed Surfactant Systems;

Scamehom. J. F.. Ed.; ACS Symposium Series 311; American ChemicalSociety: Washington, DC, 1986; P 1-27.

;', Nishikido, N. J. Colloid Interface Sci. 1977,60, 242., Nishikido, N. In Mixed Surfactant Systems; Ogino, K., Abe, M..

Ed.: Surfactant Science Series; Marcel Dekker, Inc.: New York,l99'2;Vol. 46. p 23.

(5) Mukeljee. P. In Micellization, Soiubilizationand ,VicroemuLsions;MiUal. K. L., Ed.; Plunum Press: New York, 1977.

(6) T anford, C. In ,Vicellization, Solubilization and M icroemulsions;Nittal, K. L.,Ed.; Plenum Press, New York, 1977.

(7) Aniansson. E. A. G.; Wall. S. N. J. Phys. Chern. 1974,78, 1024.(8) Wall. S. N.; Elvingston, C. J. Phys. Chern. 1985,89, 2695.(9) Israelachvili, J. N. In Surfactants in solution; Mitall, K. L.,

Bothorel. P.. Eds.; Plenum: New York. Vol. 4. p 3.

~n'7A'1_'1A~~a'1)(\(\4~1.7 rrr. t1d. 00

6684 Langmuir, Vol. 13, No. 25, 1997 Huang and Somasunda>an

cm)Yl~; Y2C2 (9}

It is not suitable for describing many other micellizationproperties in surfactant mixtures.13-18

The aim of this modeling work was to develop a modelfor describing properties such as monomer concentrationsand micellar composition in binary surfactant systems.As a first step, this model will be limited to ionic-nonionicsurfactant systems, where the nonionic surfactant has alower cmc than the ionic surfactant.

(d = Y2C2 - i'tCJ

II. Development of the Model2.1. Pseudophase Separation Approach for Treat-

ing Nonideal Mixed Micellization. A general modelcan be developed for nonideal mixed micellization pro-cesses by considering the chemical potentials of theindividual surfactant components. For mixed surfactantcomponent i in solution:

where Xl is the mole fraction of surfactant 1 in mixedmicelles, y. (i = 1,2) is the acti..ity coefficient ofi in mixedmicelles, C* is the cmc for the mixed surfactant systems,C is the total concentration of the surfactant in the mixture,and a. is the overall mixing ratio of surfactant 1 in binarymixture.

The mathematical calculations using all the aboveequations are relatively simple. The key element for thistype of calculation is the actual definition of the activitycoefficients Yi in the system. This will be discussed next.

2.2. Excess Properties for !\fixtures.. The behaviorof mixtures can be conveniently characterized by intro-ducing the thermOdynamic function of mi.'ting, 6m.Denoting a generic molar mixture property by the symbolm,6m can be defined as the difference between the valueof any function in a mixture m and the sum of its valuesof unmixed components at the same pressure and tem-perature:

,ui = ,uio + RT in c;m (1)

where ,tlio is the standard chemical potential and Cjm isthe monomer concentration of component i.

For a micelle of pure ith component:

11.oM = 11.° + RT In C. (2)r-& r-& & f1Jn = m - Lximi (constant T and P)where Cj is the cmc of pure component i.

For a mixed micelle:

At = II.oM + RTln r;Xii r. (3)/1.,

where Xi is the mole fraction of surfactant i in mixedmicelles and Yi is the activity coefficient of i in mixedmicelles. From phase equilibrium:

..(4).Ill =.ui

Expressions for 6m are particularly simple for idealmixing systems. The basic results are summarized inTable 1. It can be seen that both the volume change ofmixing, 6.vid, and the enthalpy change of mixing, MIld.are zero, reflecting the molecular assumptions inherentin the ideal mixing model. However, neither the entropychange of mixing, 6.Sid, nor the Gibbs energy change ofmixing, 6.Gid, is zero: ~ is always positive, and 6.Gid isalways negative. The composition dependence of thescaled quantities ~d/R and ~Gd/RT for binary systemsis shown in Figure 1.

In general, real mixtures are not ideal. Nevertheless.one can take the ideal mixture as a basis for the definitionof a class of functions mE called excess properties:

Equations 1-3 can be combined to give the general result:20

c.m = '.I y.c. I "'~L . (5)

id (constant T, P, and X)mE=m -mBy simply rean-anging this basic equation and mass

balance, monomer concentrations and mixed micellarcomposition in binary surfactant systems can be easilycalculated using the following equations:

or equivalently

mE=6m-.1.mid (constantT,P,andX> (12)

A common feature of all excess properties is that mEapproaches zero as any component approaches purity:

n OJ= I.-

i-1YiCi(6)

1

C'

X1= (1

~

C ID_'

1 -, (8)2.(Y2Ct/Y1C1 - 1)

The excess properties pro\;de global measures ofdeviations from the ideal solution behavior, and they areinterrelated. In fact, choosing T, P, and composition asindependent variables, one can view the excess Gibbsenergy GE as a generating function for all the other excessproperties, as well as for the acti..ity coefficients. All theserelations can be easily derived using classical thermo-dynamics. .

The activity coefficient is related to GE in a special way.The logarithm of activity coefficient is a partial molarproperty with respect to the scaled excess Gibbs energy,and the quantity In Yi therefore has those features commonto all partial molar properties. It obeys an additiverelationship:.~~

t~

(14) Huang, L.; Somasundaran, P. Langmuir 1996, 12,5790.(15) Hey, M. J.; MacTaggart, J. M.; Rochester, C. H. J. Chem. Soc.,

Faraday Trans. 1 1985,81,207.(16) Meyer, M.; Sepulveda, L.J. ColloidlnterfaceSci. 1984,99,536.(17) Rathman, J. F.; Scamehom, J. F. Langmuir 1987,3,372.(18) Treiner, C.; Fromon, M.; Mannebach, M. H. Langmuir 1989, 5,

283.(19) Shinohara, K. In Handbook of Powder Science and Technology;

Fayed, M. E., Otten, L. Ed&.; Van Nostrand Reeinhold Company Inc.:New York, 1983; p 138.

(20) Clint, J. H.1n Surfactant aggregation; Chapman and Hall Inc.:New York, 1992; p 130. GE = RTLXi In Yi

6685LangmuIr, VOi. I.", l'IIo. Z:J, 1:1:11

Table 1. Change in Properties for Mixing of the IdealMixture Systems

~...=o6JlW=oMid: = -RIX. lnX.~Gid = RTIX. In x.

For binary systems, the activity coefficient can be readilyfound from an expression for GE by the application of thefollowing equations, which come from eq 14:

1 IGE+X ~]R1\ 2 dXt

lnY2=~GE -Xt~

in Yl =- 15)

16'

It becomes clear that if one can find a way to model theexcess free energy GE, all the other properties of mixedsystem can be easily predicted using the above equations.This is the advantage of using GE to model nonidealmixture systems.

2.3. Evaluation of Excess Free EnerlY GE inBinary Surfactant Mixtures. There exists no generaltheory yet that can adequately describe the compositiondependence of the excess thermodynamic properties ofliquid mixtures. As mentioned earlier in the Introduction.regular solution theory is one of the most popular modelswhich has been used so far to examine the behavior ofsurfactant mi.xtures. It will be useful to analyze theproblems existing in the application of this model, so thatguidelines for developing a new model could be obtained.Holland 1.15 has measured the heat of mixing in surfactantmixtures using calorimetric methods. Comparison of hisexperimental results with the expected behavior of theexcess heat of mixing from the regular solution ap-proximation shows poor agreement. These results dem-onstrate that, in spite of the success of this model forpredicting critical micellar concentrations (cmc's) innonideal mixtures, it does not properly account for theheats of mixing in these systems. This implies that theremust be a large TSE contribution to the excess free energyof mixing. Synergism between different surfactant speciesis usually observed in the formation of mixed micelles.

This apparently indicates that the interaction betweendifferent surfactant species is stronger than that betweenthe same surfactant species. Due to this specific interac-tion, the mixing process in real systems is not completelyrandom. On the other hand, the basic assumption of theregular solution theory is that the excess entropy of mixingis zero. This means that regular solution theory treatsthe mixed systems as ideal mixing from the vie\\-point ofrandomness in mixing. Since the micellar phase is not areal separate phase, the aggregation number of micellesis limited. This fact will also make totally random mixingnot valid for such cases. So when mL~ed surfactantsystems are modeled, some constraints must be appliedto account for the nonrandom mixing in the formation ofmixed micelles.

Instead of using a very complex statistical thermody-namical model to treat this nonrandom mi.'ting phenom-enon, the concept of the relative solubility of eachsurfactant component in mixed micelles is considered. Apacking constraint parameter can be introduced todescribe such nonrandom mixing. Fore:'Cample, for thecase of a mixture of an ionic surfactant A and a nonionicsurfactant B, where the latter is more surface active thanthe former, it is reasonable to assume that the nonionicsurfactant B will form micelles first when the totalconcentration exceeds the mixture cmc, and then speciesA can dissolve in B micelles to fonn mix:ed micelles. Whena small quantity of A is dissolved in the B-rich mi.xedmicelles, the fonnation of mixed micelles will be enhanceddue to the synergism between the two surfactants. Buta further increase in the A component in mix:ed micellesmay then decrease the stability of mixed micelles, sinceexcessive A in mixed micelles may cause packing problemsand introduce strong electrostatic repulsion amor.g headgroups in the micelles. It is clear that there will be apacking limit for surfactant A in mixed micelles. Whenthe A composition in the micelles is lower than this limi t,the micellization process will be enhanced due to syner-gism between the t\\.o surfactants. On the other hand.when the A composition exceeds the packing limit in themicelles, the micellization process will become energeti-cally unfavorable. Here, the packing limit of ionic A inmi.xed micelles is considered. Instead of surfactant B, ifA can fonn micelles by itself, the solubility of non ionic Bin A micelles can be treated as infinity. In other words,there is no packing limit for nomonic B species in Amicelles. This hypothesis indicates that the micellizationprocess is not a symmetrical process in terms of themicellar composition. In other words, the mixed micellewith 20% of component A and the mix:ed micelle with 20%of component B will have different excess free energiesGE. This kind of asymmetrical effect has been reportedin the literature for surfactant mLxtures.1 In regularsolution theory. however. the micellization process ofmixed surfactant is always a symmetrical process. It isclear that the asymmetrical behavior in the mixedmicellization process is an important aspect which shouldbe taken into account in any attempt to model surfactantmixtures.

On the basis of the above analysis of mixed surfactantsystems, several fundamental assumptions for modelingbinary surfactant mixtures A and B are proposed:

(1) Interactions between two types of surfactant speciescan be represented using an interaction parameter, W.Here, the interaction ~ill arise mainly from van derwaal'sattractive forces, and W for binary surfactant mixtures isalways ~O. When W = 0, there is no special interactionbetween two different surfactant species, and the sur-factant mixture can be treated as an ideal system.

6686 Langmuir, VoL 13, No. 25, 1997"

Huang and Somasundaran

(2) The fonnation of mixed micelles is affected by thepacking properties of the surfactants. For binary sur-factant mixtures, a packing parameter P* can be employedto indicate the packing constraint of surfaCtant A insurfactant B-rich micelles. This packing constraint isdirectly related to the nonrandom mixing in the system.On the other hand. the packing limit of surfactant B insurfactant A-rich micelles is taken as infinity, where thesurfactant B is defined to be more surface active than thesurfactant A

(3) By adapting the equation in the lattice model for theenthalpy of mixing in liquid mixtures, the excess freeenergy G£ for the fonnation of mixed micelles can berepresented by the following equation:

K~p;)GE = RTWX(l - X>( 1 - (17)

where a is the overall surfactant ratio of component A inthe mixture,f<a) is a function of a, Xis the molar fractionof surfactant A in mixed micelles, W is the interactionparameter, and P*is the packing parameter. The functionf<a) is used here to modify the possible change of thepacking parameter as a function of the overall mixingratio a.. Similar to the relationship of packing density oftwo different particles as a function of mixing ratio, 19 f<a)can be described by the following equation:

0 0.\ 0.2 0.3 0.4 O.S 0.6 0.7 0.8 0.9 \Molac fraction of component A in mixed micelles

Figure 2. Effect of interaction parameter on the excess freeenergy.

0 Rei*. ~--3. p. =0.5. p. :1.2. P.=2

. p.=s

. . ....

oj .i;;

",-~

... .. .. .. ..

-O.j

/(a.) = { a., a. > 0.5 (18)1 - a., a :s 0.5

This equation indicates that the packing constraint willkeep changing with any change in the overall surfactantmixing ratio. When the overall mixing ratio approaches0 or 1, the packing ability in mixed micelles will be betterthan that when the ratio is about 0.5 (1:1 miXture). Forthe same total concentration of the binary mixture, thepossibility for A and B surfactant to contact with eachother will be maximum when the overall mixing ratio is0.5, and the packing constraint will then become moresignificant.

1lI. Characteristics of the Proposed Model. Tobetter understand the micellization behavior predictedby eq 17, the excess free energy GE is plotted at differentlevels of relevant parameters as a function of the molefraction of component A in mixed micelles. For the purposeof comparison, the excess free energy calculated fromregular solution theory with the interaction parameter {J= -31 is also plotted. The effect of the interactionparameter on the excess free energy GE is presented inFigure 2. In this figure, the packing parameter P* is flXedat 1.2, and the overall mixing ratio of surfactant A is takenas 0.5 (1:1 mi'Cture). It can be seen that the excess freeenergy GE predicted by eq 17 is not symmetric with respectto the molar fraction of components in mixed micelles.Regular solution theory predicts that excess free energyis symmetric and will reach a minimum when the molefraction of surfactant A is 0.5. But this is not the casewith the model proposed in the form of eq 17. Accordingto the new model proposed here, the excess free energyGE will reach a minimum when the mole fraction ofcomponent A in mixed micelles is about 0.25. In otherwords, the mole fraction corresponding to the minimumexcess free energy is shifted to a lower value in this case.This suggests that by taking the packing effect into accountthe optimum mole fraction of component A in mixedmicelles is lower than that predicted by the regularsolution theory. In other words, the mixing of componentA in mixed micelles is not as easily achieved as predictedby the regular solution theory. This may be one of the

-I 1. " , i"" ,I0 0.1 0.2 0.3 0.4 0..5 0.6 OJ 0.8 0.9 I

Molar fraction of c:ompooent A in mixed micelles

F~ 3. Effect of packing parameter on the excess free energy.

reasons that regular solution theory cannot properlypredict the micellization behavior above the cmc. It canbe seen from Figure 2 that the asymmetry of the excessfree energy becomes more marked with a decrease in theinteraction parameter W. This will have a very significanteffect on the activity coefficients of the mi'ted micellesystem, and this ~ill be discussed in detail later.

The effect of packing parameter on the excess free energyis illustrated in Figure 3. In this plot, the interactionparameter W is fixed at -3, and the overall mixirig ratioa at 0.5. It can be seen that the excess free energy GEcalculated using equation 17 decreases with an increasein packing parameter and the GE curve approaches thecurve predicted by the regular solution theory (/J = -3).It is clear that the asymmetry phenomenon in mixedmicelles is controlled mainly by the packing parameterdefined by eq 17. When the packing parameter approachesinfinity I the model represented by eq 17 will be same asthat in the regular solution theory. Hence the modelpropo.sed in the form of eq 17 can be regarded as a genericequation for binary surfactant mixtures, and the regularsolution theory, as only a special case of eq 17. It is noted

Binary 8urfactant Mixtures Langmuir, Vol. 13, No. 25, 1997 6687

~"'.

.0 0.1 0.2 0.3 0.4 0..5 0.6 0.7 0.8 0.9 1

Mol. fractioo of com~ot A in mixed micelles

Figure 4. Effect of overall mixing ratio on the excess freeenergy.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 08 0.9 1Molar Frxlion of Componeat A in Mixed Micelle

Figure 5. Excess free energy in the surfactant A and B mixtureas a function of mole fraction of component A in mixed micelles.

)

1.,

zc..

',.

IS

8"IS?;-

:~u<

fA

~~ ~ ~7 /

~

1a

~~ ~0..5

that the minimum excess free energy shifts to the lowermole fraction region of component A in mixed micelleswhen the packing parameter for the mixture becomessmaller, and the negative value of the excess free energybecomes smaller also. This suggests that the formationof mixed micelles will be more difficult with a decrease inthe packing parameter. This may be the case for thefluorocarbon-hydrocarbon surfactant mixtures. The ef-fect of overall mixing ratio on the excess free energy GEis shown in Figure 4. In this figure, the interactionparameter W is fiXed at -3, and the packing parameterP* , at 1.2. As mentioned above, the possibility for A andB surfactants to contact with each other will be maximumwhen the overall mixing ratio is 0.5, and the packingconstraint will also become significant. This is what canbe seen from Figure 4.

For binary surfactant mixtures of A and B, the activitycoefficients of both the components in mixed micelles canbe easily calculated using the following equations derivedfrom eqs 15 and 16 and based on the excess free energydescribed by eq 17:

00 0.1 0.2 0.3 0.4 O.S 0.6 0.7 0.8 0.9 I

Mo\ac Fraction of CompCMlent A in Mixed Micelles

Figure 6. Activity coefficients of components Aand B in mixedmicelles.In y. = WX(l -X> ( l - -L

) +... fto.~

packing parameter of 1.2, is plotted in Figure 5. Theactivity coefficients of components A and B can then becalculated using eqs 19 and 20, and the results obtainedare plotted in Figure 6. As mentioned before, surfactantB is more surface active than surfactant A in the systemdiscussed here. When mL~ed micelles first form in thesystem, it is reasonable to assume that component B willbe dominant in the initial micelles and then somesurfactant A can dissolve in B-rich micelles. Due to thesynergism between surfactants A and B, the excess freeenergy GE for the system will decrease with the increaseof component A in the mixed micelles, and GE will reacha minimum point as the mole fraction of component Aapproaches 0.25. This minimum point is determinedmainly by the packing parameter for the system. Abovethe A mole fraction of 0.25, the packing constraint in themixed micelles becomes significant, and the excess freeenergy begins to increase, and this means that the stabilityof mixed micelles begins to decrease. Upon examiningthe change in activity coefficients in Figure 6, it can beseen that the activity coefficient of component A keepsincreasing with an increase of component A in mixed

~) (19)~f{o.)P*

+

In /8 = WX<t-xI(t"-

~) (20)2X

f<a}.p*,..

The total surfactant concentration in the mixtures can berelated to the mole fraction of component A, X, and theactivity coefficients using the following equation:

(YBCB - YACA)x(l-X)C;otal = (21)X-a

where CA and CB are the cmc's of pure components A andB under the same conditions.

The excess free energy GE for a binary surfactantmixture of A and B with an overall mixing ratio of 0.5,calculated using an interaction parameter of -1.5 and a

Huang and Somasundaran" t'l'

layer is B-rich micelles, and the composition of inner layer Umicelles will revert to the point at which the excess free ~:energy is minimum. Since surfactant B is m.ore surface -active and the outer layer is loosely packed. th~ solubilityof component B in the outer layer micelles is unlimited,and the existence of the outer layer micelles Will have noeffect on the chemical potential of compon~t B in themicellar phase. In other words, surfactaPt B can gothrough the outer layer freely and the chemical potentialof surfactant B in the micellar phase will be dependentupon the properties of the inner layer only. The formationof the inner layer corresponds to the decrease of A in theinner layer and the increase of the activity coefficient ofB in the micellar phase. Thus, the chemical potential ofsurfactant B in the micellar phase will increase, and this,in turn, MIl make the monomer concentration ofB in thesystem increase. This phenomenon was e,,"ident in theexperimental results discussed in our previous paper.14For surfactant A, once the outer layer fonns, its behaviorin it ,viII be similar to that of pure A. The activitycoefficient of surfaCtant A in the micellar phase will bedependent mainly on the outer layer and will stay constantat 1.

6688 Langmuir, Vol. 13, No. 25, 1997

.

I

~, L ;,.:oJ T ""

. ,

'~~

~ I

Fieure 7. Schematic model for the micelle structural changeswhen the activity coefficient of component A in mixed micellesreaches 1.

micelles. When the mole fraction of component A is 0.31,the activity coefficient of A tends to 1. This must be thecritical point for the formation of mixed micelles. Theactivity coefficient has the important limiting feature

lim.~-lY;

(22)

In other words. the activity coefficient y, approaches 1when component i approaches purity. It can be seen thatthe activity ~fficient of component A in the mi~edmicelles will have the same value as pure A when themole fraction of A in mixed micelles is 0.31. Therefore,it is reasonable to conclude that component A in mixedmicelles under this condition will begin to acquire someproperties of pure component.o\. On the basis of theexperimental results discussed in our previous paper ,1" itis proposed that the point at which fA = 1 is the point atwhich mixed micelles begin to change their structure.Possible structural change is schematically shown inFigure 7. With the increase in component A in mixedmicelles, the excess free energy will go through theminimum point when the mole fraction of component Ain mixed micelles is 0.25; a further increase in the molefraction of component A in micelles will make the stabilityof the mixed micelles decrease. \Vhen the activitycoefficient of component A reaches 1. the structure of themi."ed micelles is proposed to split into an outer layer andan inner layer. Surfactant A will form a relatively looselypacked outer layer. This outer layer is similar to thevesicles and consists mainly of surfactant A The inner

IV. Conclusions and SummaryIn this work, a new model for binary surfactant mixtures

of A and B and their micelles in solution has been proposedin the {onn of eq 17. Considering nonrandom mwng inthe formation of mixed micelles, the concept of packingconstraint is introduced, and this is represented by thepacking parameter, P*. The significant characteristic ofthis model is that it examines the asymmetric behaviorof micellization in binary surfactant mixtures. ThisasY1:lmetric behavior is controlled mainly by the value ofthe packing parameter. With an increase in the packingparameter, the formation of mixed micelles becomes morerandom.

This model also predicts that the structures of mixedmicelles can change above the mixture cmc depending onthe properties of the surfaCtant components and the overallmixing ratio; a schematic model for the mixed micelleshas been presented. Once the activity coefficient ofsurfactant A approaches I, the stnlcture of the mi~edmicelles is proposed to consist of an outer layer and aninner layer. The outer layer is surfactant A-rich micellesand is loosely packed with a structure similar to that ofvesicles, while the inner layer consists of surfactant B-richmicelles. This new model can be considered as a genericmodel of binary surfactant mixtures with the regularsolution theory being a special case of it, when the packingparameter approaches infinity.

Acknowledgment. We acknowledge the support ofthe National Science Foundation (Grant CTS-9212759)and The Department of Energy (Grant DE-AC22.92BCl4884).

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