~j9f Langmuir 1996, 12, 5i90-5i95

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Changes in l\tIicelle Compositions and MonomerConcentrations in Mixed Surfactant Solutions

Lei Huang and P. Somasundaran.~

lAngmuir C~nter for Colloids and Interfaces, Columbia Universil.v,Neu' York, .~.ew York lOO2i

Received .\lay 10, 1996. In Final Form: August 20. 199~

Monomer concentration changes in mL~res of a cationic surfactant. tetradecyltrimethylammoniumchloride\TTAC), and a nonionic surfactant. pentadecylethoxylated nonylphenol (NP-1S), have been studiedwith the help of the ultrafiltration technique. While the pseudophase separation model is valid for describingthe micellization behavior of single surfactanta, it was found to fail for mixed systems. In the case ofn ACand NP-15 mixed systems, both the monomer concentrations and mixed micellar composition changedcontinuously above the critical micelle concentration (cmc) of the mi.ttures. The measured changes ofmonomer concentrations above the mixed l"mC were different from those predicted by the regular solutiontheory. Two different mixed micelles, nAC-rich and NP-l5-rich micelles, are proposed to form above aconcentration range which in turn depends on the ratio of the two surfactants in the mLxed system.

Introduction

Surfactants are used widely for different applicationslJuch a.i detergents, paints, cosmetics, flotation, and~nh.mced oil recovery- Such commercialsurfactants areinv3riably mL'ttures of different surfactants due to eco-nomical as well as beneficial effects of mixtures overindividual surfactanta. To maximize such beneficial~tTt?Ct$- it is therefore helpful to understand the interac-tit'n$ .,mt>ng surfactants in the mixtures. Mixed 5urfac-~Int ~\'stem8 are also of theoretical interest. since colloidalIk'h.t\inN such as mi~llization in mL'ted surfactant~yst~ms can be markedly different from those in single~\Irf;tct.tnt systems.

However. while the interfacial and colloidal behavior of~ingt~ $urfactant systems has been investigated widelyl)Vl'r d~c.tdes. that of surfactant mL'ttures has beeninvt!"tig.tted only to a limited extent.l-" Mixtures ofdjtT..'rl'nt :turfactant types often exhibit s)-nergism in their,'ll""I..t~ ,'n various properties of a system.3.5 Such syner-~""m i" u:tually attributed to nonideal mixing effects int h..,j r ;t...~rrl'gates and can be modeled well by using regular",'luri,m theory \..ith a negative interaction parameter.",!,During the last two decades, fluorocarbon-hydrocarbont4\lrf.t..-t.mt matures have been suggested to exhibitd..'mi.xing. i.e., theformanon of more than one type of stablepl)pul.\til\n or micelles in mixed systems.6-10 Treatment,If rhl' critical micelle concentration (cmc) data for fluo-n, tm'n -hydrocarbon mL'ttures using regular solution

theory has usually }ielded a positive interaction param-eter.l1.l2 Since mL'ting in micelles is entropically favored,the possibili ty of demL'ring warrants a significan t physicalbasis such as immiscibility of surfactant hydrophobicgroups in the core of the micelle or steric effects whichrestrict mixing on the basis of the molecular geometriesof the surfactants. By considering these effects, ~ agarajanet a}.l3 successfully predicted the formation of two typesof coexisting micelles in fluorocarbon-hydrocarbon mL~edsystems.

Even though the phenomenon of two types of coexistingmicelles has been :suggested also for hydrocarbon surf-actant mixtures. 1"' no direct evidence for such a phenom-enon has been reported in the past. Since it is themonomer-micelle equilibrium that governs and therebyreflects the surfactant aggregation behavior of mixedsurfactant systems, direct monitoring of monomer con-centrations should help to identify the formation ofdifferent types of micelles. To our knowledge directmeasurements of micelle composition and monomerconcentrations in mL~ed surfactant systems are :scarce inthe literature. In this work, the micellization beha\ior ofcationic-nonionic hydrocarbon surfactant mi."<tures hasbeen studied using the ultrafiltra tion technique .9.15-1.. andon the basis of the data obtained, a model involvingcoexisting mL~ed micelles is proposed.

Materials and ~Iethods

Surfactant-. The cationic surfactant, n-tetradeccyl trimeth).I-ammoniwn chloride .nACI. [CH~CH~'I3"~(CK1>3JCI. ~m .-\meri-can Tokyo Kaseci. Inc.. and the nonionic surfactant. pentadecylethoxylated nonylphenoltNP-151. CiHI9CsH.&OCCH.:CH2OII~H.from Nikko Chemicals. Japan, ~.ere used as received.

Reagents. The oiOdium chloride used was ACS reagent gradefrom Fisher Scientific.

- --. T., ,~h,'m corre~pondence should be addressed. La von

I'\I.I.!!"".'" Krumb Professor. School of Engineering and Applied,,'i,'","', .'\111 W 1:20th Street, )Iudd Building 918, Columbial'ni,'..r,cit~. ~ew York. NY l002i.

,. .\boIlr:~t published in AdL-a~ ACS Abstract,- No\"ember 1.I ~~1t'.

,t. H,'II'Ind, P. M.; Rubingh, D. N. In Mixed surfactant systems;H,'II.I"'I. PM., Rubingh. D. N., Ed.; ACS Symposium Series 501;An...n.':\" l'h.'mical Society; Wuhington, DC, 1992. Chapter 1.

:!' X~'r-..jan, R. .-\dt.. Colloid interface Sci. 1986.26, 205.;; ft._n.~. J.: Zhu. B. Y. J. Colloid int~rfacr Sci. 1984, 99. 42,

,~ H.,II'Ind. P. ~.. .-\dr.'. Colloid intelfa« Sci. 1986, 26, Ill.. ~ R._n, ~1. J. In PMnom~1\Q in mixed sut(actant syst~ms: Scame-

I\.'n1. J. F. Ed.; ACS Symp08ium Series 31 t: .-\merican Chemical~"lt'I\'; ,"."nington. DC, 1986; p 144.

..~. ~lukt'rjH. P.; Y8ni, A. Y. S. J. Phy,. Ch~m. 1978,BO, 1388,

. :" ~Iy""". K. J. J. Colloid inut(acr Sci. 1978. 66. 331.

.." FUn.i.~ki, N.: Hada, 5, J. Plays. Clttm. 1980,84,736.\~.. .~wa. T.; Johten. K.; Miyagiahi, 5.; Nishida. M. LoIIBmuir

1-' -I. I-111)' C.rlfors. J,: Subs, P. J. Phys. CMm. 1984,88.4410.

50743-7463(96)00459-3 CCC: $12.00

(11) Funaski. N.: Hada. S. J. Phvs. Chern. 1M3. 8';. 3-42.(12) Yoda. K; Tamori. K; Meauro. K. J. Colloid Inter;Q("e S"i. 1986.

104. 279.(13) Nagaraj.n. R. [n Mi.ud surfa"tant systems; Holland. P. :\I.;

Rubingh. D. ~.. Eds.: ACS Symposium Series 50 1: Ameri"3n ChemicalSoeietv: WuhinitQn. DC. 1992; Chapter 4.

114) Abe. :\I.; Tsubaki. N.; Ogino. K J. Colloid lnter;c,,'e Sci. 1986.107.503.

(15) Warr. G. G.; Grieser, F.; Healy. T. W. J. Phys. Chern. 1983.87,1220.

(16) Osborne-Lee. I. W.; Seheehter. R. S.: Wade. W. H. J. ColloidlnIerfa« Sci. 1M3. 94. 179.

(17)Makay8i. A.; Leawrdant, D.; Treiner. C. La.",uir 181. 9.2808.

C> 1996 American Chemical Society

Mi.md Surfactant Solutions Langmuir, Vol. 12, .\"0. 24. 1996 5791

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Surfactant Concentration ADalyses. Tetradecyltrimethyl-ammonium chloride (TTAC) concentrations were detenninedusing a two-phase titration technique.i' The pentadecyl ethoxyl-ated nonylphenol (NP-15) concentration was analyzed by de-tennining UV absorbance at 223 or 275 nm using a Shimadzu1201 UV-vis spectrophotometer.

UltratUtration. All ultrafiltration tests were done at roomtemperature (22 :!: 2 °C) using two Amicon membranes, ~-3and'Y)I-1 membranes. specified to exclude molecules withmolecular weights greater than 3000 and 1000, respectively. Thefiltration was carried out using an Amicon model 8050 filter atan 860 rnmHg nitrogen pressure. The ThI.1 and YM-3 memobranes were used to separate tetradecyltrimethylammomumchloride <TTAC) and pentadecyl ethoxylated nonylphenol (NP-15) monomers from single and mixed surfactant micellarsolutions. ,.,0"

Resulu and DiscussionUltrafiltration was first tested for the separation of

monomers from micellestd for both tetradecyltrimethyl-ammonium chloride ('M'AC> and pentadecyl ethoxylatednonylphenol (NP-15) single surfactant systems, and theresults obtained are shown in Figures 1 and 2, respectively.It can been seen that the monomer concentrations (orfiltrate concentrations) are the same as the mother liquorconcentrations when the latter are lower than the cmc ofthe surfactants. At mother liquor concentrations higherthan the cmc. as the filtrate is removed, the mother liquorconcentration increase markedly. but the monomer con-centration remains at the cmc value. It can be concludedthat the ultrafiltration method is suitable for monitoringmonomer concentrations in the micellar systems studiedhere. The experimental results also obey the phaseseparation model except around the cmc. Just below thecmc, the monomer concentrations obtained from thefiltration are somewhat lower than those dictated by thephase separation model. This suggests formation ofmicellar embryos below the cmc; the phase change in thesesurfactant systems is not as sharp as that expectednormally. The effect of salt addition was also detennined.For the cationic surfactant 'M'AC, the cmc is higher atlower ionic strength, and the plateau valueoCthe monomerconcentration is correspondingly higher. However for the

nonionic surfactant NP-15. there is no effect of salt on thecmc. These results are in agreement "ith prior results,further suggesting the suitability of the ultrafiltrationmethod for separating monomer from micelles in thepresent systems and the lack of need to maintain highionic strength conditions during ultrafiltration process-ing.19

Monomer concentrations of both tetradecyltrimethyl-ammonium chloride (Tr AC) and pentadecyl ethoxylatednonylphenol (~'P-15) in their mixtures at an ionic strengthof 0.2 '1¥I NaCI are shown in Figures 3 and 4. It can beseen that the monomer concentrations of both com-ponents are lower in mixtures than those for singlecomponent systems, suggesting synergism in mi.xed mi-celle formation. With an increase in the second componentin the mixtures, the monomer concentration of the firstcomponent in binary mixtures is further lowered. It isinteresting to note that the monomer concentrations ofboth components in the mixtures keep increasing even

:iII

- --(19) Rohde. A.; Sackmann. E. J. Colloid~rf~Sci. 1979.10.494(18) Li, z.; Rosen, M. J. Anol.CII.m. IN1, 53. 1516.

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above the cmc of the mixtures. This is in contrast to thebehavior of the single surfactant system, where thesurfactant monomer concentration is relatively constantabove the cmc. In order to test the validity of the regularsolution theory for the present system, the cationic TTACand nonionic NP-15 monomer concentrations were cal.culated using such a theory.20 The results obtained havebeen published in a previous paper ,21 and the calculationshave been described in detail there. It is clear that whilethe cationic TTAC monomer concentrations of the mixedsurfactant system measured by ultrafiltration and cal-culated using regular solution theory are similar, nonionicNP.15 monomer concentrations measured by ultrafiltra-tion are very different from those calculated using thetheory. The ultrafiltration results show the monomerconcentrations to be constant at concentrations im-mediately above the mixed cmc (depending on the mixtureratio) and then to increase at higher concentrations.Regular solution theory, however, predicts a decrease inNP-15 monomer concentration above the mixture cmc.Thus regular solution theory can be considered unsuitablefor the TTAC/NP-15 mixed system.

Ionic strength can be expected to have a significant effecton the aggregation of the cationic surfactant, which inturn will affect coaggregation with the nonionic surfactantspecies. To detennine such effects, monomer concentra-tions of'M'AC and NP-15 were measured at an ionicstrength of 0.03 M NaCI at different mixing ratios, andthe results obtained are shown in Figures 5 and 6.Comparing the results given in Figures 3 and 4, it can beseen that the monomer concentrations for both TT AC andNP-15 at the lower ionic strength (0.03 MNaCI)are higherthan those at the higher ionic strength (0.2 M NaCI). Thisis anticipated for TT AC, since the cmc of ionic surfactantsdoes decrease upon the addition of salt, which will producea decrease in the monomer concentrations. For NP-15,however, the monomer concentrations are identical at boththe ionic strengths for the single surfactant system (seeFigure 2). In the case ofmL'Ctures with TTAC, the changesin the monomer concentrations ofNP-15 can be attributedto the fonnation of mixed micelles, since any change inthe activity of the cationic surfaCtant can be expected to

(20) Nithikido. N. In MWd "urfactant ","Ulns; Olino. K., Abe, M.,Edt.; Surfactant Science Series 46; Marcel Dekker, Inc.: New York,

1992; p 23.

(21) Huang, L.; Maltesh, C.; Somasundaran, P. J. Colloid InterfaceSd. 1",177,222.

8NP.1S ONLY-I!. 4:1 TrAC:NP-IS !,A 1:1 TTAC:NP.\' II. I~ TTAC:NP-\' :

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alter the aggregation between the cationic TrAC and thenonionic NP-15.

Since the mi.~ed micelles are in equilibrium ~ith themonomers, like the monomer concentration, the composi-tion of the mL~ed micelles can be expected to var)- as thetotal surfactant concentration is increased above the cmcof the mixtures. The results obtained for the monomerconcentrations of pentadecyl ethoxylated nonylphenol(NP-15) show that the monomer concentrations remainconstant over a short concentration range above the mixedcmc (depending on the ratio of mixing) and increase athigher concentrations (see Figures 4 and 6). The nonionicNP-15 concentrations at the mixed cmc are 8.8 x 10-5,8.5x 10-5, and 5.4 x 10-5 mol/L, respectively, for TTACINP-15 ratios of 1:4, 1:1, and 4:1 at an ionic strength of0.03 M NaCI.21 The nonionic NP-15 concentration at themixed cmc decreased with an increase in the TrAC contentof the mixture, suggesting enhanced synergism withincreasing TI'AC concentration.

Since NP-15 is more surface active than TI'AC, it isreasonable to believe that NP-15 will fonn micelles

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tration with the ~P-15 concentration fixed at 2.1 x 10-~M. This concentration is higher than the cmc of thenonionic surfactant, NP-15. by itself. In other words, ro.""P-15 micelles e.~st in this case even without the addition ofthe cationic TT_-\C. From Figure 7 the NP.15 monomerconcentration can be seen to decrease with an increase inthe cationic TrAC concentration. This is reasonable whenpossible dissolution of added cationic TrAC in NP-l5-rich micelles is considered. With such dissolution of thecationic surfactant, the concentration of the nonionicsurfactant in the NP-15-rich micelle will decrease. Thechemical potential of the nonionic surfactant in micelleswill hence decrease, and this in turn \\;11 make theconcentration of the nonionic ~""P-15 monomers which arein equilibrium \\c;th micelles decrease. With a furtherincrease in the cationic surfactant concentration to about10-3 ~ (this is about the cmc of single TrAC under thesame conditions), the nonionic surfactant monomer con-centration becomes constant. This is proposed to be dueto the formation om AC-rich micelles in the system. Uponthe formation ofTT .-\C-rich micelles. no additional cationicTTAC species \\ill go into the nonionic ~"P-15-rich micelles,and hence the composition of~-P-15 in micelles does notchange further. This phenomenon can be seen moreclearly in Figure 8, where the ratio ofTT_-\C in micellesis plotted as a function of total cationic Tr.-\C.

It can be seen that, in the low cationic TI'AC concen-tration region, about 80% of the TTAC is dissolved in theNP-15 micelles. With an increase in total TTAC concen-tration. the percent of the cationic surfactant in thenonionic surfactant micelles decreases. This is due to thelimited solubility of the cationic TTAC in the nonionicNP-15 micelles. With a further increase in the cationicTTAC concentration to about 10-3 ~I (about the cmc ofsingle TTAC at the same ionic strength). the percent ofcationic TTAC in micelles begins to increase. This isattributed to the formation of the cationic TT AC-richmicelles; i.e., more and more cationic surfactant speciesform micelles.

In Figures 9 and 10, the monomer concentrations ofcationic TTAC and nonionic NP-15 are given with thecationic TTAC concentration fixed at 4.29 x 10-3 M. Thisconcentration is higher than the cmc of the cationicsurfactant TTAC itself. It can be seen from Figure 9 thatthe addition of'the nonionic NP -15 has no significant effecton the change of the cationic TI'AC monomer concentrationat low NP-15 concentrations. At such low NP-15 con-centrations, some of the nonionic surfactant molecules

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immediately above the mi.~ed cmc and then TTACmolecules will dissolve in the NP-15 micelles. At a fixedionic strength, the TTAC that can be dissolved in NP-15micelles will depend upon the packing constraints of thetWo sunactanta. With a further increase in cationic TTACconcentration, the TTAC itself may form aggregates,leading to the formation ofTT AC- rich micelles. This mayeven cause some of the TTAC species dissolved in the~P-I5-rich micelles to rearrange into TTAC-rich micelles.This is proposed to be the reason for the increase in NP -15monomer concentrations at high concentrations.

In the 1:4 TTAC/NP.15 mixed system, NP-15-richmicelles will form at a NP-15 concentration of about 8 x10-5 mol/L, into which added cationic TTAC species candissolve. This is indicated by the almost constantmonomer concentration of TTAC over a certain concen-tration range. Above a TTAC concentration of 5 x 10-"mo1l1., TTAC-rich micelles may fonn. This can be seenmore clearly in the case of 1:-1 TTAC/NP-15 mLuures atan ionic strength of 0.03 M NaCI. At this low ionicstrength, the electrical repulsion between the cationicTTAC head groups will be significant, with the result ofa decrease in the free ener~' of micellization of ~-P-15 inthe mixed systems. Under conditions when the cationicTT AC-rich micelles also form, some of the TTAC moleculesthat would have been dissolved in NP-15-rich micelles atlower concentrations will stay out ofNP-15-rich micellesto enhance the micellization of the nonionic NP-15. Thisis proposed to be the reason for the TTAC monomerconcentration increase to a new plateau upon the forma-tion of the TTAC-rich micelle.

This phenomenon was not observed with mixturescontaining a higher proportion of the cationic TTAC. Insolutions containing Ii larger ratio of the cationic surfac-tant, due to the limited solubility of cationic TTAC intothe nonionic NP-15-rich micelles, the TTAC monomerconcentrations in solutions are higher. Even though somecationic TTAC species can get released from the NP-l5-rich micelles upon the fonnation of cationic TTAC-richmicelles, its effect on the micellization of TTAC-richmicelles will be minimal due to the large quantity ofcationic TTAC monomers already existing in the mi.~edsurfactant solutions. Ho\vever the fonnation of cationicTTAC-rich micelles can be clearly seen from the changein monomer concentration of the nonionic NP-15 (seeFigures 4 and 6). At a higher cationic TTAC to nonionicNP.15 ratio, the increase of the nonionic NP-15 monomerconcentration upon the formation of cationic TTAC-richmicelles is more significant than that at lower cationicTTAC ratios. This is the result of more TTAC moleculesstaying out from the NP-15-rich micelles upon thefonnation of cationic TT AC-rich micelles and of the changein nonionic NP-15 composition in NP-15-rich micellesbeing more significant.

In the 4:1 TTAC/NP-15 mixed system, mi.~ed micelleswill be fonned first at a NP-15 concentration of 5 x 10-5mol/L. Due to the synergism between NP-15 and TTACin ~-P-15-rich micelles, the monomer concentration o~-P-15 will decrease above the mi.~ed cmc. Upon the fonnationof mixed micelles in the system, nonionic NP-15 will besolubilized in micelles and hence ita monomer concentra.tion will decrease.

To obtain more infonnation on the fonnation of the twotypes of coexisting mixed micelles, the monomer concen.trations of the cationic surfactant tetradecyltrimethyl-ammonium chloride <TrAC) and the nonionic pentadecylethoxylated nonylphenol (NP-15) in their mixtures weremeasured with one component concentration fixed. InFigure 7, the monomer concentration of the nonionic NP.15 is plotted as a function of the cationic TTAC concen-~

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Huang and Somasundaran-6794 f:.angmuir, Vol. 12, No. 24. 1996

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may dissolve in the cationic TTAC-rich micelles, but thesolubility of the nonionic surfactant in the cationicTTAC-rich micelles is small. Hence the possible change in TT ACcomposition in TT.I\C-rich micelles is also small and theconcentration of TT.I\C monomer does not change mea-surablv. With an increase in the nonionic surfactantconcen"tration to about 8 x 10-5 M (this is about the cmcof the nonionic NP.15 under the same conditions), themonomer concentration of the cationic TTAC begins todecrease significantly. This is proposed to be due to theformation of non ionic NP-15-rich micelles with more andmore cationic TT ACdissolving in the NP-15-rich micelles.Since this will alter the equilibriu~ between the cationicTT AC monomer and micelles, the monomer concentrationofTT AC decreases. The formation ofNP-15-rich micellesis also suggested in Figure 10. As mentioned before,cationic TrAC micelles exi5t in this system already evenwithout any added nonionic NP-15. In the presence ofadded nonionic NP-15, the nonionic surfactant NP-15monomer concentration is lower than that ofNP-15 alone.This suggests that some of the nonionic NP-15 is dissolved

Figure 11. Schematic model for a binary surfactant system:region I. no micelles in the mixed 30lution: region II. one typeof mixed micelles formed: region III. two type.s of coexistingmixed micelles formed.

in the cationic TTAC-rich micelles. When the nonionicNP-15 concentration increases to about 8 x 10-5 M. NP-15-rich micelles form and then the increase of the nonionicNP-15 monomer concentration beconies less.

On the basis of the results presented above. a schematicmodel for this type ofbinary surfactant system is presentedin Figure 11 with the corresponding phase diagram inFigure 12. The micellization behavior is divided into threeregions. In region I, the surfactant concentration is lowand only monomers exist. In region II, surfactants beginto form aggr~ates. Since the nonionic surfactant NP-15is more surface active than the cationic surfactant TT AC,it is reasonable to consider that nonionic NP-15-richmicelles form first for most mixing ratios. Some cationic

.Vi.tl!d Surfactant Solutions Langmuir, Vol. 12, No. 24, 1996 5795

concentration of the cationic TT AC in the mixed solutionsexceeds the cmc of the single cationic TTAC,and TTAC-rich micelles form. Above curve b, the concentration ofthe nonionic NP-15 in the mixed solutions exceeds thecmcofthe single nonionicNP-15, andNP-15-rich micellesform. This diagram illustrates the situation where morethan one type of micelle can form, leading to differentvariations in the monomer concentrations of the indi..idualcomponents.

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Tr.-\C dissolves in the NP-15-rich micelles, and the micellecomposition changes. The boundary separating regionst and II is determined by the mixture cmc.21 In region III,two t)-pes of coensting mi.,,<ed micelles are proposed to.;oe:rist in the system. The boundary separating regionsII and III consists of curves a and b. Neglecting possibleinteractions between these surfa&..ants, curves a and bcan be described as follows:

ConclusionsMonomer concentration changes in mixtures of a

cationic surfactant, tetradecyltrimethylammonium chlo-ride (TTAC), and a nonionic surfactant, pentadecylethoxylated nonylphenol (~'P-15), have been studied usingthe ultrafiltration technique. While the pseudophaseseparation model was found to be valid to describe themicellization behavior of single surfactants, it failed formixed systems. In the TTACftoi"P-15 mixed systems boththe monomer concentration and the mi."'Ced micellarcomposition changed continuously above the cmc of themixed surfactant solution, but the change of the nonionicNP-15 monomer concentration above the mixed cmc wasdifferent from that predicted by regular solution theory.While the regular solution theory predicts that nonionicNP.15 monomer concentrations will keep decreasing abovethe mixture cmc, ultrafiltration experiments showed themonomer concentration of nonionic NP-15 first to remainconstant just above the mi.'rture cmc and then to increaseat higher concentrations.

The micellization behavior of mixed surf act ants isdifferent in different concentration regions: In region I,there are no micelles and only surfactant monomers existin the mi."'Ced solution. In region II, one type of mi."'Cedmicelles forms; for most mi.'rture ratios, nonionic ~"P-15-rich micelles first form and then some cationic TTACspecies dissolve in the NP-15-rich micelles. In region III,two different mixed micelles, TrAC-rich and NP-15-richmicelles, are proposed to be formed. On the basis of theexperimental results, a schematic model is proposed todescribe the behavior of binary surfactant systems.

Acknowledgment. The Department of Energy I GrantDE-AC22-92BC14884), the National Science Foundation(Grant CTS-9212759), and ARCO are acknowledged forsupport of this work.

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where v is the molar fraction of the nonionic surfactantNP-15 'in the mi.."ted solution. The constants 1.2 x 10-3and 9.8 x 10-5 are the cmc's of the cationic TTAC and thenonionic ~-15, respectively.:!l The physical meaning ofcur\'es a and b is as follows: Above curve a, the