phase behavior and microstructure in aqueous mixtures of ...ceweb/ce/people/faculty/...form...

[31.8.2004–9:22pm] [289–338] [Page No. 289]{Books}4380-Abe/4380-Abe-009.3d 4380-Abe

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Phase Behavior and Microstructure inAqueous Mixtures of Cationic andAnionic Surfactants

ERIC W. KALER

Department of Chemical Engineering,

University of Delaware, Newark, DE, USA

KATHI L. HERRINGTON

W. L. Gore and Associates, Newark, DE, USA

DANIEL J. IAMPIETRO

Merck and Co., PO Box 2000,

RY814-200, Rahway, NJ, USA

BRET A. COLDREN

Advanced Encapsulation, Santa Barbara,

CA, USA

HEE-TAE JUNG

Department of Chemical and Biomolecular

Engineering, Korea Advanced Institute of

Science and Technology, Daejon 305-701, Korea

JOSEPH A. ZASADZINSKI

Department of Chemical Engineering and

Materials, University of California,

Santa Barbara, CA, USA

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1. SYNOPSIS

Mixtures of anionic and cationic surfactants in water displayinteresting phase behavior and a range of microstructures,including small micelles, rod-like micelles, lamellar phases andvesicles. This chapter reviews the properties of these mixtureswith a focus on the experimental and theoretical aspects ofvesicle formation and stability.

I. INTRODUCTION

Electrostatic forces are typically the dominant interaction incolloidal systems, and application of surfactants in solution orat surfaces calls for early consideration of the desired sign ofthe surface charge. It is no surprise then that oppositelycharged surfactants in solution mix in a highly non-ideal way,and in fact are able to form spontaneously structures (inparticular, unilamellar vesicles) that are uncommon in othersurfactant mixtures. The range of self-assembled microstruc-tures in mixtures of cationic and anionic surfactants alsoincludes small spherical micelles, cylindrical or worm-likemicelles, and other bilayer lamellar or L3 phases. The state ofequilibrium of all of these microstructures has not yet beenfully established, but some vesicle phases have been observedto be stable for well over a decade. Although the folklore insurfactant formulations suggests that oppositely chargedsurfactants should never be mixed because of the potentialfor precipitation of the insoluble surfactant ion pair, precipita-tion is generally found only near equimolar compositions or insamples below their Krafft temperature, and even thisprecipitation can sometimes be blocked.

Of particular interest in this review are mixtures ofoppositely charged surfactants from which the companionco-ion simple salt has not been removed. This is in contrastto true ‘catanionic’ surfactants, as first described by Jokela andWennerstrom [1], in which solutions contain only water andthe two oppositely-charged surfactant ions. These catanionic

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surfactants are also called ‘‘ion-pair amphiphiles’’ (IPAs), andthey are most easily formed by combining an organic acid andan organic hydroxide, so the common ancillary product is waterrather than a salt. IPA properties have been reviewed recentlyby Tondre and Caillet [2]. A spectacular example of theassembly of IPAs is the observation by Zemb et al. [3] of self-assembled structures about 1 mm in dimension having a regularhollow icosahedral form. The icosahedral particles are stabi-lized by pores located at their vertices, and they dissolve whensalt is added.

Historically the addition of oppositely charged moietieshas been used most frequently to control the growth of rod-likeor worm-like micelles. This is typically done by adding ananionic hydrotrope (such as salicylate or tosylate salt) to asolution of cationic surfactant [4]. The hydrotrope partitionsinto the surfactant headgroup region and results in dramaticmicellar growth. Similarly, a cationic hydrotrope can drive thegrowth of anionic micelles [5]. As the molecular weight (orlength) of the hydrophobic portion of the hydrotrope isincreased, the hydrotrope begins to behave more like a secondsurfactant and can alter the self-assembly in solution, leadingto the formation of vesicles. For example, when alkyl sulfates(CnSO4

�–Naþ) with n greater than six are added to alkylammonium halide surfactants, vesicles can form on both thecationic and anionic-rich side of the phase diagram [6]. Suchvesicles form spontaneously and without the input of shear,and their structural and thermodynamic properties arediscussed below. Both the static and dynamic properties ofthese mixtures have recently been reviewed [7].

Vesicles are, of course, not newly discovered. Nearly fortyyears ago, Bangham et al. showed that phospholipids dispersedin water formed closed, multibilayer aggregates called lipo-somes, capable of separating an internal compartment from thebulk solution [8]. As bilayers are relatively impermeable tomany ions and nonelectrolytes, it became possible to createsmall domains of different composition and to explore someof the properties that nature has designed into cells andorganelle membranes. Unilamellar vesicles are distinguishedfrom multilayer liposomes by their single-bilayer closed shell

Phase Behavior and Microstructure in Aqueous Mixtures 291

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structure that encapsulates a single aqueous compartment.While vesicles often form spontaneously in vivo, they have onlyrarely been observed to form in vitro without the input ofconsiderable mechanical energy or elaborate chemical treat-ments. Hence, a variety of methods have been developed tocreate vesicles with sizes ranging from about 20 nm to morethan 20mm [9].

The stability of such vesicles is limited because the bulklamellar phase, which at high water fractions can exist as acolloidal dispersion of multilamellar liposomes, is the equilib-rium form of aggregation under typical conditions. Hence,unilamellar vesicles formed from lamellar phases are meta-stable and eventually will revert to multilamellar liposomes.This reversion is invariably accompanied by a release of thevesicle contents and failure of the vesicle carriers. For multi-lamellar liposomes to be stable at high water fractions, thebilayers must have a net attractive interaction. The stabilityof mechanically formed vesicles against aggregation can beenhanced in many of the same ways that other colloidalsystems are stabilized. These include the incorporation ofbulky, polymer-like head groups on some fraction of the lipids[10–12], or the addition of charged lipids or surfactants to thebilayers to enhance electrostatic repulsion [13–15]. Bothadditives decrease the net bilayer attraction and help vesiclesform spontaneously or with little energy input [16]. Onceformed, unilamellar vesicles can also be made mechanicallystable by polymerization of the head or tail groups [17,18].These methods limit the kinetics of reversion, but leave thequestion of equilibrium stability unanswered.

There have been surprisingly few demonstrationsof equilibrium vesicle phases in the literature, in comparisonto the immense body of literature on vesicles and liposomesin general. By equilibrium, we refer to the following threecriteria:

1. Unilamellar vesicles are formed spontaneously upondispersing dry surfactant into water without mechan-ical or chemical perturbation;

2. Vesicles do not aggregate with time; and

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3. Any physical or chemical process to which the vesiclesare exposed will result in spontaneous reformation ofunilamellar vesicles on reversing the process.

Surprisingly, nearly all of the several reports of sponta-neous vesicle formation in the literature have involvedsurfactant mixtures. Single component bilayers usually formmultilamellar phases, although there are reported exceptions[19]. Nonetheless, the question of equilibrium is a difficult oneto address in these mixtures because the state of aggregationchanges in some cases only on the time scale of years, andmay thus reflect chemical degradation of the surfactant (e.g.,hydrolysis) rather than a physical progression to a morethermodynamically stable structure.

The fundamental questions are why vesicles form inmixtures of oppositely charged surfactants, and why, onceformed, do they remain stable? The formation and stability ofany surfactant aggregate depends on whether that aggregaterepresents the global minimum in free energy for a givencomposition, and there are different approaches to describingthe thermodynamics of these mixtures. As a way to begin, it isuseful to start not with a molecular-level description, butinstead with the more mechanical description of the propertiesof the bilayer first given by Helfrich [20]. In this case the elasticfree energy of a bilayer is written in terms of its local state ofcurvature, described by the two principle curvatures c1 and c2.For spherical vesicles, c1¼ c2¼ 1/R, in which R is the vesicleradius. To terms second order in curvature, the free energy of abilayer, per unit area, is

E=A ¼ 12 �ðc1 þ c2 � 2c0Þ

2þ ��c1c2 ð1Þ

in which k is the bending modulus, �� is the saddle splay orGaussian modulus, and the spontaneous curvature of thebilayer is c0. By definition, k> 0 but ��> 0 for surfaces thatprefer hyperbolic shapes (saddle shaped surfaces in which thecenters of curvature are on opposite sides of the surface,c1c2 < 0) and ��< 0 for surfaces that prefer elliptical shapes(spheres, ellipsoids, etc., in which the centers of curvature are


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on the same side of the surface, c1c2 > 0) [21]. For a chemicallyand physically symmetric bilayer, the spontaneous curvaturec0¼ 0. Hence, a single component bilayer cannot have anonzero spontaneous curvature. It is necessary for the bilayeror its local environment to be chemically asymmetric forspontaneous curvature to exist [22].

Thermal fluctuations of the bilayers lead to a repulsiveinteraction as the bilayers come into contact and the aredamped [23]. For bilayers separated by a distance, d [23], theso-called undulation interaction energy is

Efluct ¼3�2

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ðkBT Þ2

kd2ð2Þ

The repulsive undulation interaction can overwhelm the vander Waals attraction between bilayers (which is also propor-tional to d�2) when k is small, leading to a net repulsive inter-action between bilayers and hence, stable unilamellar vesicles,especially when combined with electrostatic repulsion incharged systems [24].

Within this Helfrich framework, vesicles may be madestable by either an entropic or an enthalpic mechanism.Entropically stabilized vesicles have a low bending constant(k� kBT, where kB is Boltzmann’s constant). The bendingenergy is therefore low and the population of vesicles isstabilized both by the entropy of mixing and the undulationinteraction. The resulting size distribution is broad.Enthalpically stabilized vesicles, on the other hand, require anon-zero spontaneous curvature and a larger value of thebending constant (k>kBT ). In this case the vesicles arenarrowly distributed around a preferred size set by thespontaneous curvature. These theories are discussed in moredetail below.

Work in our laboratories has demonstrated the existenceof spontaneous, apparently equilibrium vesicles with bothnarrow [25–27] and broad size distributions [6,25,28–32] inaqueous mixtures of a wide range of surfactants. The mostcommon cationic surfactant used has been an alkytrimethy-lammonium bromide or tosylate, e.g., cetyltrimethylammonium

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bromide (CTAB) or tosylate (CTAT) as well as cetylpryidiniumchloride (CPCl), while the common anionic surfactants arethe sodium alkylsulfates, e.g., sodium octyl (SOS) or decylsulfate (SDS), or dodecylbenzene sulfonate (SDBS), which mayhave either a branched or comb structure [6,33]. In addition,vesicles form with CTAB when the anionic surfactant is a saltof perfluorinated carboxylic acid such as sodium salts ofperfluorohexanote (FC5) or perfluorooctanoate (FC7) [25,34].Unilamellar monodisperse equilibrium vesicles also formspontaneously when cholinergics (for example, choline chlo-ride) are added to aqueous solutions of sodium bis[2-ethylhexyl]sulphosuccinate (AOT) [35]. In this case, the cationic ‘‘surfac-tant’’ is actually a hydrotrope and is water-soluble whilethe anionic surfactant alone forms bilayers in water. Thus,vesicle formation in this case is due to a change in curvature ofthe AOT bilayer induced by addition of the cholinergiccompound.

A substantial number of other researchers have alsoexplored vesicle formation in other mixtures of oppositelycharged surfactants, and the recent review of Gradzielski [7] isespecially comprehensive. Vesicles also form when a single-tailed anionic surfactant (SDS) is added to a double-tailedcationic surfactant (DDAB) [36], and these vesicles also forminteresting phases and complexes in the presence of DNA andother polyelectrolytes [37]. Certain bacterial surfactants knownas siderophores can undergo micelle to vesicle transitions oncomplexation of multivalent ions [38], which may be importantfor sequestering sufficient iron in marine environments.A range of zwitterionic/ionic pairs have been studied, ashave surfactant pairs when one of the ionic species is producedby a chemical reaction [39]. The kinetics of formation ofvesicles in these mixtures has been probed by light and smallangle scattering as well as by cryo-TEM [40–42]. A variety ofunstable rod and disk-like structures are found as theaggregate morphology progresses from simple micelles tovesicles.

This chapter is organized first around experimentalobservations of the surface chemistry of dilute mixtures,phase diagrams, and microstructure characterization. This


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section is followed by a review of the relevant theories and adiscussion of the origin of stability of vesicles in cationic/anionicmixtures.

II. EXPERIMENTAL OBSERVATIONS

A. Surface Tensions and Nonideal Mixing

The first striking observation about cationic–anionic mixturesis their high degree of nonideality. This is clearly shown by thestrong variation in critical aggregation concentration given bysurface tension measurements [33]. For example, mixtures ofCTAT and SDBS display critical aggregation concentrations(cacs) orders of magnitude lower than the individual CriticalMicellization Concentration (CMC) values (Fig. 1). Mixtures ofCTAT and SDBS also produce a lower surface tension than isobserved for either surfactant alone. Visual and light scatteringobservation of samples with intermediate concentration a fewtimes the cac are revealing. The samples appear somewhat bluein color, suggesting the presence of colloidal structures largerthan micelles. This is borne out by light scattering measure-ments of dimensions of ca. 100 nm, and, as described below,unilamellar vesicles are the first aggregates to form in thesehighly dilute solutions.

Similar results hold for mixtures of SOS and CTAB. TheCMC of pure SOS is 3 wt% (120 mM) and that of pure CTAB is0.03 wt% (0.88 mM). The critical aggregation concentrationsfor mixtures of CTAB and SOS are significantly lower andrange from 0.001 wt% to 0.002 wt% (3–7� 10�5 M) [31], andmixtures of CTAB and SOS also produce a lower surfacetension than either pure surfactant. Precipitate appears insamples in the vicinity of the cac after equilibration for severaldays, thus the cac for mixtures of CTAB and SOS correspondsto the solubility of the equimolar precipitate. The value of thesolubility product [¼ (aCTAþ)(aSOS�)] for this salt, calculatedfrom the dependence of the cac on bulk mixing ratio, is9� 10�10 mol2/l2, and similarly low values are expected forother surfactant ion pairs.

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B. Phase Behavior

Aqueous mixtures of cationic (RþX�) and anionic surfactants(R�Xþ) are actually five-component systems according to theGibbs phase rule: RþX�, R�Xþ, RþR�, XþX�, and water, andare subjected to an electroneutrality constraint. Therefore, the

Figure 1 Surface tensions of SDBS, CTAT, and their mixtures at25�C (top). The mixtures are highly nonideal, as shown by thevariation of the critical aggregation concentration with CTATfraction (bottom).


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pseudoternary phase diagram for RþX�, R�Xþ, and water atconstant temperature represents only a portion of the phaseprism. The full prism is needed to represent compositions inmultiphase regions when the surfactant ion and associatedcounterion separate into different phases, as is the case whenprecipitate forms. Nonetheless, when precipitate is not presentthe pseudo-ternary map is a useful guide, although the phasealternation rule [43] need not apply.

A canonical phase diagram for anionic and cationicmixtures is that of CTAT and SDBS (Fig. 2), where in thiscase the SDBS hydrophobe is of the ‘‘soft’’ type and has rake-like branching. (Hard-type SDBS has approximately a dodecanetail bonded to the aromatic ring at a single carbon.) The phaseboundaries in Fig. 2 and all other phase diagrams of catanionicmixtures can only be established after visual observationsremain unchanged over an extended period of time. Mostcompositions equilibrate within one to two weeks, but samplesthat contain vesicles or a viscous phase require longerequilibration times. Vesicle phases are identified first by theircharacteristic isotropic blue appearance and then by determin-ing the mean diameter using quasielastic light scattering (QLS)to verify that aggregate sizes are in the range typical of vesicles.Cryo-transmission and/or freeze-fracture electron microscopy(cryo-TEM) can be used to confirm the unilamellar vesiclestructure [26,32], as can small-angle neutron scatteringmethods [26,30]. The long equilibration time is necessary forthe samples containing vesicles in order to distinguish betweensingle-phase vesicle regions and two-phase vesicle/lamellarregions, since small amounts of lamellar structure developslowly compared to other phases. Visual observations and QLSmeasurements over a time period of nine months or longer (inone case over 10 years) confirm the stability of the vesicle phaserelative to a lamellar phase in one-phase regions.

At higher surfactant concentrations, Fig. 2 shows that thevesicle phases are in equilibrium with one of two lamellarphases: (1) CTAT-rich vesicle solutions (Vþ) are in equilibriumwith a CTAT-rich lamellar phase (Laþ) of lower density; and(2) SDBS-rich vesicle solutions (V�) coexist with an SDBS-richlamellar phase (La�) which undergoes an inversion in density

F2

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at � 35/65 CTAT/SDBS. Isotropic one-phase regions bordereach binary surfactant–water boundary. The rod-like micelleregion on the CTAT side extends into the ternary diagram andis separated from the vesicle lobe by several multiphase regions.On the other side of the diagram, SDBS forms sphericalmicelles in water at the water concentrations shown in thephase diagram. Within the micellar region, the viscosity is low

Figure 2 Ternary phase map for CTAT/SDBS/water at 25�C. Onephase regions are marked by M, R, or V. The V region on the CTAT sideof the diagram contains CTAT-rich vesicles that are positively charged,while the V region on the SDBS side of the diagram contains SDBS-richnegatively charged vesicles. Two-phase regions show vesicle andlamellar phase coexistence, and the ‘‘multi’’ region contains severalviscous phases that are slow to separate. R corresponds to CTAT-richrodlike micelles, and M to small globular SDBS-rich micelles. Aprecipitate forms on the equimolar line. Compositions are on a weightpercent basis. The ‘‘packing limit’’ indicates the concentration atwhich vesicles of this size become close-packed (see Section IV.D).Modified from [33].


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and constant and there is no increase in scattered lightintensity as CTAT is added. Thus it is unlikely that addingCTAT causes significant micellar growth in SDBS-rich micellarsamples. A two-phase region, in which La

� is in equilibriumwith an isotropic solution presumably containing vesicles,separates the one-phase micellar region from the vesicle lobe.The general qualitative features of two vesicle lobes, which maydiffer significantly in size, an equimolar precipitation line,lamellar phases at higher concentration and micellar phases onthe edges of the diagram are found in most cationic–anionicsurfactant phase diagrams.

The phase diagrams evolve in a systematic way as thechain lengths of the two surfactant molecules change. Figure 3shows the diagram for CTAT/SDBS compared to that forCTAB/SOS, which have highly asymmetric tails [31] and toDTAB/SDS, which have symmetric tails [6]. The CTAB/SOSdiagram displays a very large vesicle lobe on the SOS-rich(shorter hydrophobic chain) side of the diagram, while there isonly a small composition range where in CTAB-rich vesiclesform. The DTAB/SDS phase diagram is dominated by a largecomposition range wherein precipitate forms. As the ionic headgroups are the same in the two surfactant pairs, this stronglysuggests an important thermodynamic role in vesicle stabilityfor the chain packing configurations within the hydrophobiccore of the bilayer.

F3

Figure 3 Ternary phase maps of three cationic-anionic mixtures inwater at 25�C. CTAB/SOS (left) shows a large vesicle lobe on the SOS-rich side of the diagram along with other phases identified in Fig. 2(see Fig. 4). The DTAB/SDS mixture (right) shows a wide range ofprecipitate formation.

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The implications of the CTAB/SOS diagram are seen moreclearly when plotted in rectangular coordinates (Fig. 4) [29].When samples are initially prepared, vesicles appear to formover a wider range of compositions. However, as the samplesage, the range of compositions that yields stable vesiclesshrinks considerably, particularly at the lamellar phase

F4

Figure 4 Phase behavior of CTAB/SOS/H2O with no added saltplotted on a rectangular diagram. Dotted lines represent equimolarCTAB/SOS composition, at 61.4% CTAB. One-phase vesicle lobes (V)exist at dilute CTAB-rich and SOS-rich compositions. Samples hereappear bluish and are isotropic. One-phase rod-like (R) and spherical(M) micelles form near the CTAB and SOS axes, respectively. Rod-like and spherical micellar phases are both clear, yet scatter morelight than pure water. Rod-like micellar samples are viscous andviscoelastic. At intermediate mixing ratios, much of the phasebehavior is dominated by vesicles in equilibrium with a lamellarphase (L), which appears as birefringent clouds above the vesicles.The CTAB-rich R and V phases are separated by a narrow two-phaseregion of rods and vesicles in equilibrium. SOS-rich micelles trans-form abruptly to vesicles at most concentrations, though around3.0 wt%, an intervening region of rod-like growth occurs. Unresolvedmultiphase regions are at concentrations above those of the vesiclelobes. Modified from [29].


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boundary. Small amounts of the lamellar phase form in two-phase samples close to the vesicle phase, and become visibleonly after aging for days or weeks. Boundaries are assignedonly when the sample appearance does not change with time.As expected, micellar phases exist on the binary surfactant–water axes of the diagram. Vesicles form in the water-richcorner of the phase diagram, and are found in both CTAB-richand SOS-rich samples as confirmed by QLS and cryo-TEM. TheCTAB-rich vesicle lobe is small and narrow in extent, while theSOS-rich vesicle lobe is considerably larger. Note that the lobeextends to nearly the CMC value of SOS, so that compositionswhere vesicles form all have compositions such that the SOSconcentration is below its pure component CMC. As SOS isadded to CTAB-rich micellar solutions, there is strong rod-likemicellar growth as indicated by increased viscosity andviscoelasticity. Samples become increasingly more viscoelasticat high surfactant concentrations. SOS-rich micelles arespherical at low amounts of added CTAB, while at higherratios of CTAB to SOS and higher concentrations, rod-likemicelles form.

The micelle-to-vesicle phase transition is of considerableinterest. For CTAB-rich samples and SOS-rich samples athigher surfactant concentration, there is an intervening two-phase region of rod-like micelles and vesicles. Samples separateinto two phases: one phase scatters more light than a micellarsolution and is viscous, while the other phase is not viscous.Depending on the composition, the appearance of the secondphase ranges from clear and colorless to bluish and somewhatturbid. At higher surfactant concentrations, samples containmore than two phases and may contain vesicles, rod-likemicelles, and liquid crystalline microstructures. SOS-richsamples exhibit different behavior for low surfactant concen-trations (see Fig. 4). In these samples there is limited micellargrowth with added CTAB or increased dilution, and thecolorless micellar solutions progressively scatter more lightthan micellar solutions of pure SOS as the micellar phaseboundary is approached. Samples become noticeably turbidover a very narrow increment of concentration, and the phaseboundary between micelles and vesicles is set at the point at

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which samples appeared turbid. In no case is an intermediatetwo-phase region observed, so this is not a first-order phasetransition.

A crystalline precipitate, presumably the equimolar saltCTAþ : OS�, forms in equimolar mixtures as well as in dilute(<0.1–0.2 wt%) solutions at all mixing ratios. Dilute samples inthe vesicle phase close to the precipitate phase boundary oftencontain turbid wisps that are so easily dispersed by mixing thatattempts to characterize them with optical microscopy areunsuccessful. Because of the proximity of the precipitate phaseboundary, the wisps may be a fine precipitate, or a dilutedispersion of multilamellar vesicles (MLVs).

Electrostatic attractions between the cationic and anionicsurfactants are responsible for this interesting phase behavior,so it is no surprise that addition of excess electrolyte markedlyalters the phase behavior of catanionic solutions [29]. This hasbeen examined in some detail for mixtures of CTAB/SOSwith added NaBr (Fig. 5). The most conspicuous effect is thedestabilization of the large one-phase vesicle lobe with increas-ing amounts of added NaBr. The vesicle region shrinks in alldimensions. Vesicles still form in the water-rich corner of thelobe, but small turbid clouds form over the vesicle phase afterthe addition of salt. These clouds are either lamellar ormultilamellar vesicle (MLV) phases, or a fine dispersion ofprecipitate. Adding NaBr also drives the two-phase lamellar/vesicle region on the CTAB-rich side of the vesicle lobe to moreSOS-rich compositions. The lamellar phase becomes increas-ingly favorable as salt is added, as observed visually by theincreasing predominance of a turbid white birefringent phaseover the vesicle phase. This is consistent with a decreasedelectrostatic interaction between the vesicle bilayers leading toa more attractive interaction [44]. The vesicle/micelle phaseboundary, however, shifts to compositions richer in CTAB withincreasing NaBr. Thus, overall, the most SOS-rich section ofthe larger vesicle lobe becomes unstable with respect to theneighboring micelle phase, while the most CTAB-rich section ofthe lobe becomes unstable with respect to the lamellar phase. Itis striking that the addition of salt in some cases drives thevesicles to form small micelles rather than larger MLVs or

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lamellar phases, as would be the prediction based on eithercolloidal stability arguments (aggregation) or by appeal toanalogy to charged phospholipids, in which salt addition drivesfusion.

The differences in behavior of mixtures of homologoussurfactants with identical headgroups but different hydrocar-bon tails further points to the important role of the packing ofsurfactant tails in the bilayer. This can be explored moredirectly by examining mixtures of a hydrogenated cationicsurfactant with a fluorinated anionic surfactant [25,26]. Theantipathy of the tails, and even their potential demixing, isbalanced by the attractions between the head groups. Figure 6shows the phase map for the mixture of CTAB/FC5/water inthe water-rich corner (<8 wt%) at 25�C. The CMC of each

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Figure 5 Phase behavior of CTAB/SOS/H2O with 4.0 wt% NaBrplotted on a rectangular diagram. The entire CTAB-rich side of thephase diagram is dominated by a clear isotropic liquid in equilibriumwith precipitate. Samples here are no longer viscous or viscoelastic.The one-phase micellar region M extends farther toward more CTAB-rich mixing ratios. The multiphase region is at some compositions aclear streaming birefringent phase over an isotropic bluish phase andat other compositions is a clear isotropic phase that scatters morelight than water over a clear phase. Modified from [29].

304 Kaler et al.

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Figure 6 Phase maps for mixtures of CTAB/FC5 (a) and CTAB/FC7

(b) measured at 25�C. The gray lines in each graph correspond toconstant weight mixing ratios (d) as indicated. The CMC for eachsurfactant is shown by the arrows on the binary axis. At intermediatemixing ratios in each system, the phase behavior is dominated by atwo phase region consisting of vesicles in equilibrium with a denserlamellar phase (VþLa). Rodlike micelles (R) exist on the CTAB–water binary axis for both FC5 and FC7 mixtures. The mostimportant feature is the existence of a single vesicle phase (V) onthe fluorinated surfactant rich side of the phase map.


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surfactant is indicated by an arrow on the appropriate binaryaxis and the gray lines represent constant values of thesurfactant mixing ratio, d(¼wt. FC5/(wt. FC5þwt. CTAB)).The majority of the phase space at intermediate values of d,including equimolar mixtures, is composed of a two-phaseregion denoted by (VþLa) containing a vesicle phase inequilibrium with a denser, birefringent lamellar phase. WhenCTAB is present in large excess, a viscoelastic solution of longrodlike micelles is formed. The phase behavior on the FC5-richside on the phase map is more interesting. Along the FC5–waterbinary axis (d¼ 1), at concentrations below the CMC, thesurfactant presumably exists as monomers. Over the majorityof this range, the addition of small amounts of CTAB leads tothe formation of larger aggregate including vesicles between 2and 4 wt%. The vesicle region extends to about d¼ 0.8. Vesiclesdo not form in FC5–water binary mixtures, but quasielasticlight scattering measurements and cryo-TEM images indicatethat large structures form with the addition of only asmall amount of CTAB (d� 0.995). Thus on this scale, thevesicle lobe appears to almost touch the binary axis. For d belowabout 0.80, the samples also contain a more dense birefringentphase, with the phase boundary between the vesicle lobe (V)and the two-phase region (VþLa) set by the formation of thisdense phase. Mixtures of CTAB and FC7 show a similar phasediagram, and neither of these mixtures shows a substantialchange in phase behavior with temperature, suggesting thatthere is no demixing of the tails.

C. Microstructure Characterization

Once phase behavior has been established the microstructurecan be explored with a variety of techniques. Especially usefulmethods are quasielastic light scattering, which provides arapid size estimate and can be used to track the evolution ofstructure with time; small angle neutron scattering, which canprovide information about aggregate shape, structure and,under ideal conditions, size distribution; and cryo-TEM, whichcan provide unambiguous size, shape, and size distributioninformation.

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As an example of the evolution of apparent vesicle radiusdetermined by QLS as a function of time, consider theproperties of CTAB/FC5 and FC7 mixtures. Figure 7(a) showsthe measured hydrodynamic radius plotted as a function of thesample age for CTAB/FC5 concentrations 3 wt% and d¼ 0.80,0.90, and 0.99. At the two higher d values, vesicle growth isobserved over the first 50 days with a leveling off to anequilibrium size seen at longer times. For the lowest d, there isa more dramatic change in size initially and the measured sizecontinues to increase even after 110 days. In these samples alamellar phase is eventually observed visually. The situation inthe FC7 case is different, with vesicles at 2 wt% concentrationreaching a constant size more quickly at d values of 0.8 and0.85.

Cryo-TEM provides the simplest and most model-freemeasure of microstructure, although artifacts due to samplepreparation are possible [45]. Figure 8 is a cryo-micrograph of a2 wt% CTAB/SOS mixture (7 : 3 wt : wt). This sample showsunilamellar vesicles with radii varying from about 20 to>100 nm. Note that the bilayers appear flexible and thevesicles are not all spherical. Such shapes may be the equi-librium conformations of vesicles either because of the lowbending rigidity of mixed bilayers or may reflect deformationsdue to shear during sample preparation. The radii of thevesicles imaged are in good agreement with those measuredwith QLS, although the cryo-TEM average radius is typicallyless than the QLS average. It is extremely important to notethat QLS sizes measure a higher moment of the sizedistribution than the number average measured by cryo-TEM, so the QLS size is very sensitive to the presence ofnegligible (below 1 part in 1000) numbers of larger aggregates[32], and this may explain some size growth reported by others[46]. Such large aggregates can result from even a slight degreeof chemical or physical degradation of the surfactants.

Small-angle neutron scattering (SANS) is also a usefulmethod for determining average vesicle dimensions and, tosome extent, size distributions [47] and is very complementaryto cryo-TEM. SANS scattering spectra (intensity I as a functionof the scattering vector magnitude q) can be analyzed either

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Figure 7 Quasielastic light scattering results for CTAB/FCm

vesicles. (a) CTAB/FC5 vesicles at a surfactant concentration of3 wt% and d values of 0.99, 0.90, and 0.80. (b) CTAB/FC7 vesiclesmeasured at a surfactant concentration of 2 wt% and d values of 0.80and 0.85. The lines are shown to guide the eye. All measurementswere performed at 25�C.

308 Kaler et al.

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by fitting a model function describing the presumed structureto the data, or by using an indirect fourier transform (IFT)method to extract model-free estimates of microstructuredimensions. Both methods have been used to analyze thespectra from samples known (by cryo-TEM) to contain vesicles.Generally spectra are fitted using a polydisperse core-shellmodel [48] where the vesicles are assumed to have a poly-disperse core with constant shell thickness (t). For a poly-disperse system of unilamellar non-interacting vesicles, thescattered intensity as a function of the scattering vector isgiven by

IðqÞ ¼dP

dðqÞ¼n

Z 1

0

GðrcÞP2ðqrcÞ drc ð3Þ

where n is the number density of vesicles, P(qrc ) is the formfactor of a single particle (e.g., vesicles) consisting of a core andan outer shell, and G(rc ) is the normalized probability of findinga particle with a core radius between rc and rcþ drc. G(rc ) ismodeled as a Schulz distribution, so

GðrcÞ ¼r Z

c

�ðZþ 1Þ

Zþ 1

�rrc

� �Zþ1

exp�rc

�rrcðZþ 1Þ

� �

ð4Þ

Figure 8 Cryo-TEM image of 7 : 3 (wt : wt) CTAB : SOS vesiclephase at 2 wt% total surfactant in water. The broad size distributionand formation of floppy and odd shaped vesicles is consistent witha bending constant, K¼ 0.2kBT, and a spontaneous curvature,R0¼ 30 nm [27].


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where �rrc is the mean core radius and Z is related to thepolydispersity of the particles ( p) and the variance (�2) of thecore radius by

p2¼1

Zþ 1¼�2

�rr2c

ð5Þ

The form factor is

PðxÞ¼4pq3ðrb � �cÞ J1 xþ

t

rcx

� �

� J1ðxÞ

� �

ð6Þ

where x is the dimensionless variable qrc, J1ðxÞ ¼ sinðxÞ�ðxÞcosðxÞ, and rb and rc are the scattering length densities(SLDs) of the bilayer and the core (taken as the solvent),respectively. The SLDs were calculated by adding the scatter-ing amplitudes of each group or atom in a molecule and dividingthe total by the corresponding molecular volume. The SLDof the bilayers is calculated assuming the bilayer is made ofan equimolar composition of the oppositely charged compo-nents. This assumption has little influence on the results.Interestingly, use of one deuterated surfactant allows the SLDof the bilayer to be changed, and combining this informationwith mass balances allows the composition of the vesicle to bedetermined [49]. It is not, of course, generally equal to thebulk composition.

Typical scattering spectra for polydisperse vesicles arerelatively featureless and show only a broad q�2 decay ofintensity, which is consistent with the presence of a two-dimensional bilayer (Fig. 9a). More monodisperse populations(Fig. 9b) yield spectra with undulations that reflect theoscillations of the form factor. In this case, the vesicles areabout 20–22 nm in number average radius with ca. 20%polydispersity. Further analysis shows the bilayers to berather thin (2–3 nm), and IFT methods allow the cross-sectional SLD profile to be determined [34].

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Figure 9 SANS results for CTAB/FCm vesicles plotted as thescattering intensity in absolute units versus the scattering vector q.(a) CTAB/FC5 vesicles measured at a surfactant concentration of3 wt% and d values ranging from 0.85 to 0.99. The top three curveshave been offset by the indicated scale factors for clarity. Eachscattering curve shows the characteristic q�2 decays as indicated bythe line in the graph. (b) CTAB/FC7 vesicles measured at a surfactantconcentration of 2 wt% and d values of 0.80 and 0.85. The top curve isoffset by a factor of 10 for clarity. The q�2 region is observed at higherq values and minima from the form factor are evident aroundq� 0.01–0.02.


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III. THEORETICAL APPROACHES

Much work has focused on development of theoreticalmodels that can account for the behavior of surfactant solu-tions. These models range from the simple but useful geometricpacking model of Tanford [50] and Israelachvili [44] tomore quantitative molecular thermodynamic models. Thelevel of detail required from a model varies widely with thecircumstance of its application. A simple conceptual model isuseful when attempting to organize results or guide experi-mentation, and both the geometric packing and the curvatureelasticity models fulfill this need. These models requirethe input of parameters that characterize the surfactantproperties, and given these parameters, both models can beused to rationalize observations and to suggest an experimentalplan.

Often, solution properties such as micellar composition ormonomer concentrations for mixtures of surfactants arerequired for a given surfactant application. In this case, amodel based on classical thermodynamics in which aggregatesare treated as a single phase in equilibrium with an aqueousmonomer phase may be suitable. In this approach, deviationsfrom ideal mixing in the aggregate phase are accounted for byactivity coefficients for each surfactant in the mixed aggregate[51,52]. Activity coefficients can be calculated using a suitabletheory for nonideal mixing (such as regular solution theory)with interaction parameters determined from the experi-mental values of the CMC of each surfactant and the cacmeasured for at least one mixing ratio. Finally, there arepowerful predictive models based on a detailed statisticalthermodynamic analysis of aggregate formation [53–58].These molecular thermodynamic models are capable of pre-dicting the equilibrium state of aggregation, aggregate sizesand composition, and equilibria between multiple phases,given data about the pure surfactants that is readily availablein the literature.

The surfactant inventory in monomeric and aggregated(micellar or vesicular) form is needed in many applications ofsurfactant mixtures. The inventory may be calculated in

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several ways, with each method having a different degree ofcomplexity and accuracy. In a first approach, micellar sizedistributions can be computed by considering each aggregate tobe a distinct chemical species. From thermodynamics, thechemical potential of a surfactant molecule has the same valueregardless of the size of the aggregate it is in. Combining thecondition of chemical equilibrium with a mass balance onsurfactant yields the aggregate size distribution. This approachis the basis of the mass action model, which has been widelyapplied in the analysis of surfactant properties to predict themicellar size distribution, monomer concentration, electricalconductivity of micellar solutions, and degree of counterionbinding to micellar aggregates (see, e.g., Kamrath and Franses[59]).

In a second model, the effect of finite aggregate size onsolution properties is neglected. Instead, the collection ofaggregates is viewed as a separate phase with uniform proper-ties. This approach, known as the pseudo-phase separationmodel, can account for many properties, such as the micellarcomposition and monomer concentration in mixtures of sur-factants [60]. Since this model ignores the finite size ofaggregates, the monomer concentration above the CMC ispredicted to be constant, which is at odds with experimentalresults and the predictions of more realistic theories, both ofwhich show a moderate decrease in monomer concentrationabove the CMC. This decrease is a consequence of the massaction effects for a collection of micelles. Also, the pseudo-phaseseparation approach yields no information about the optimalaggregate size or distribution of sizes.

A third class of models is based on a molecular-levelthermodynamic analysis, and provides monomer and micellarcompositions as well as detailed information on aggregatestructure and stability. This last approach requires signifi-cantly more computational effort than the other two, but yieldsmore information, ideally without requiring data for mixturesof the two surfactants.

Some theoretical work has been directed at modeling theimportant properties of mixtures of anionic and cationicsurfactants [1,51,61,62]. Regular solution theory, combined


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with the pseudo-phase separation model, has been used topredict the monomer concentrations, micellar compositions,and critical aggregation concentrations (cac) for anionic andcationic surfactant mixtures [51]. In the pseudo-phase separa-tion approach, the contribution to the solution entropy byindividual aggregates is assumed small and the aggregate phaseis treated as a bulk phase. Although realistic predictions aremade for the dependence of the cac on the solution composition,the theory is strictly applicable only to molecules of similar sizeand functionality [63], which is clearly not the case for mixturesof oppositely-charged surfactants (also see Hoffmann andPossnecker [52]). Further, some of the reported interactionparameters actually characterize precipitate/monomer equi-libria rather than micelle/monomer equilibria [61].

The phase equilibria between monomer, micelles, andprecipitate in mixtures of SDS and dodecylpyridinium chloridehas been reported and interpreted in terms of a model thatcombines regular solution theory to describe the micellarphase and a solubility product to characterize the precipitatephase [61]. The thermodynamic machinery needed to combinean expression for Gibbs free energy for micellar aggregateswith the mass action law has been developed for the calculationof micellar properties, size distributions, and phase separation[58]. This model was used to make qualitative predictions oftrends for micellar solutions of anionic and cationic surfac-tants. Application of Puvvada and Blankschtein’s model [62] tomixtures requires the input of interaction parameters for eachof the binary components. Finally, a statistical thermodynamiccell model has been used to predict the phase equilib-rium between various phases for mixtures of amphiphilicmolecules [64]. Of these approaches, the cell model structureprovides the most careful way to account for the maincontributions to the aggregate free energy. In the cell model,the nonidealities due to electrostatic interactions are calcu-lated using the nonlinearized Poisson–Boltzmann equationand the effect of excluded volume is accounted for by thedependence of cell volume on surfactant concentration andaggregate size.

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A. Thermodynamics of Aggregation

The surfactant solution consists of monomeric surfactant, sur-factant ions contained in aggregates of a given geometry withaggregation number N, water, and counterions. Each aggregateis treated as a distinct species with chemical potential mN, andat equilibrium, the Gibbs free energy of the system is mini-mized. For a mixture of two surfactants, A and B, dispersed inwater, the Gibbs free energy is

G¼nw�0wþnA,mon�

0A,monþnB,mon�

0B,monþ

X

N>1, XA

nN, XA�0

N

þ kBT

�

nw ln xwþnA,mon ln xA,monþnB,mon ln xB,mon

þX

N>1, XA

nN, XAln xN, XA

�

þNXA�AþNXB�B

ð7Þ

where XA is the fraction of component A in the mixedaggregate, ni is the number of molecules of species i in theaqueous solution: ni,mon is the number of surfactant ions i (i¼Aor B) in monomeric form, nw is the number of water molecules,and nN, XA

is the number of surfactant aggregates withaggregation number N (an ‘‘N-mer’’) with composition XA, m0

i

is the standard chemical potential of species i (i refers to water,monomeric surfactant A or B, or N-mer of composition XA), kB

is the Boltzmann constant, T is the temperature, and the molefraction of species i, xi, is defined as

xi¼ni

nw þ nA, mon þ nB, mon þP

N>1, XA

nN, XA

ð8Þ

The summation (N > 1, XA) includes all possible combinationsof aggregates of aggregation number N and composition XA,excluding monomeric surfactant since this is already accounted


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for by the terms nA,mon and nB,mon. The Lagrangian multipliers,�A and �B, are chosen to satisfy the mass balance constraintsfor components A and B

X

N, XA

NXAnN, XA�A � nA, total¼ 0

X

N, XA

N 1� XAð ÞnN, XA�B � nB, total ¼ 0 ð9Þ

In this equation, ni,total is the total number of molecules ofspecies i in the sample. Minimization of Eq. (7) relative to nN, XA

yields the size distribution for mixed aggregates [65]

xN,XA¼ xNA

A,monxNB

B,mon

� exp � �0N,XA�NA�

0A,mon�NB�

0B,mon

� �.kBT

h ið10Þ

where NA and NB are the number of surfactant ions of type Aor B contained within the mixed aggregate having aggrega-tion number N and composition XA. Note that for monomers,N¼ 1.

The aggregate size distribution is calculated by simulta-neously solving Eqs (7)–(9) for the monomer mole fractions,xA,mon and xB,mon. Typically, the mole fraction of monomericsurfactant is very small and the exponent of the difference inchemical potentials is large, so that solution of these equationsmay present numerical problems [66]. A good initial guess forsolution of the equations is the mole fraction of the puresurfactant at the CMC, which is often available in theliterature.

With the help of the size distribution defined in Eq. (10),various useful quantities can be calculated, including theaverage aggregation number (neglecting the contributionfrom monomeric surfactant)

Nh i¼X

N>1, XA

NxN, XA

� X

N>1, XA

xN, XAð11Þ

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the mixing ratio, Y¼ nA, total= nA, totalþnB, total

� , in the bulk

solution

Y¼

xA, monþP

N>1, XA

NXAxN, XA

xA, monþxB, monþP

N>1, XA

NxN, XA

ð12Þ

the average mole fraction of component A in aggregate form

XAh i¼X

N, XA

NAxN, XA=X

N, XA

NxN, XAð13Þ

and the monomer mixing ratio

Ymon ¼ xA, mon= xA, mon þ xB, mon

� ð14Þ

The equations developed in this section are very generaland may be used to predict size distributions of aggregates ofvarious geometries, given a suitable function for the variationof the chemical potential of the aggregate as a function of sizeand composition. Thus, size distributions of micelles or vesiclesmay be calculated. In addition, this approach can be used topredict the distribution of surfactant between coexistingmicrostructures such as small micelles and large vesicles.

B. Curvature Elastic Energy

1. Overview

Models based on the concept of the curvature elastic energyof a surfactant film are well suited to describe the propertiesof aggregates composed of monolayers (microemulsions) orbilayers (lamellar bilayers, vesicles, sponge phases) [20,23,67–70]. In this approach, the bilayer is modeled as a two-dimensional elastic film. Trends in morphology are thenpredicted as a function of the elastic constants and thespontaneous curvature of the film. This model is of limiteduse in the treatment of very small aggregates, such as micelles,because the quadratic curvature expansion breaks down athigh curvature. To apply this model quantitatively, the


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variation of the elastic constants and spontaneous curvaturewith composition or surfactant properties is required.

Helfrich wrote the curvature elastic energy of a bilayer perunit area, E/A, in a quadratic approximation in curvaturesas given in Eq. (1). For spherical deformations, c1¼ c2¼ 1/R,where R is the radius at the surface of inextension or ‘‘neutral’’surface. For a symmetric bilayer, the area per molecule isinvariant with curvature at the midplane, and so the neutralsurface is the midplane. The energy to bend a bilayer away fromthe spontaneous curvature is proportional to k and the cost ofmaking a saddle splay deformation is proportional to �kk.

Application of the Gauss–Bonnet theorem shows that theintegral of the Gaussian curvature, c1c2, over the bilayersurface is independent of the shape and size of a closed surfaceand depends only on the number of handles, nh , or compo-nents, nc, of the surface

Z

c1c2d2S¼ 4�ðnc � nhÞ ð15Þ

Handles are defined as pores or passages from one side of thesurface to the other and each closed surface is one component.A vesicle has no handles and one component per vesicle. Hence,shape variations of simple closed vesicles produce no change inthe Gaussian curvature energy. However, for spherical vesicles,the saddle splay (‘‘Gaussian’’) contribution to the bendingenergy is 4� �� per vesicle, while for infinite lamellar bilayersit is zero [71]. Hence, the Gaussian curvature can play asignificant role in determining vesicle size distributions [26,71].

At a molecular level, the tendency of a surfactant moleculeto bend towards or away from the aqueous region is determinedby the chemical make-up of each surfactant [22,71]. Forexample, a monolayer composed of single-tailed ionic surfac-tants with large head groups and small tail volumes will tend tobend away from the water, while a monolayer of a double-tailedzwitterionic surfactant such as lecithin will tend to curve moretowards the aqueous region. By convention, positive curvatureis away from the aqueous region. If a bilayer is made up of twochemically and physically identical monolayers, the bilayer has

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no net spontaneous curvature, c0¼ 0 in Eq. (1). As the bilayer isbent, the molecules in one monolayer may be approaching theirpreferred curvature, but the molecules in the other monolayerare far from theirs. The bending energy is given by Eq. (1) withc0¼ 0. It is important to note that, in addition to allowing fora nonzero spontaneous curvature, a multicomponent bilayerconsisting of monolayers with different compositions may havebending constants, k and �kk, considerably lower than for auniform bilayer [71–73].

2. Implications of Curvature Elastic Energyfor Vesicle Formation

Vesicles form spontaneously typically only in mixtures of twoor more surfactants in water. Considering curvature elasticityalone, energy is required to bend a symmetric bilayer awayfrom the planar configuration [Eq. (1)]. Thus, for singlecomponent (i.e., symmetric) bilayers with large bendingrigidity, vesicles are formed only with the input of chemicalor mechanical energy. However, this need not be the case forbilayers containing two or more components. Two factors are atwork here. First, the elastic constants are strongly affected bybilayer composition, as observed experimentally [24,74] and assupported by theoretical calculations [72,73]. In mixtures, thebending rigidity can be reduced an order of magnitude, thusreducing the bending penalty that opposes the formation of anensemble of vesicles. This mechanism gives rise to vesiclesstabilized by the entropy of mixing. Alternately, a bilayercomposed of two surfactants could have an asymmetriccomposition, thus producing a bilayer with a spontaneouscurvature. This could occur if the two components in thebilayer have different spontaneous curvatures, so that whenmixed the surfactants assemble into monolayers of equal andopposite curvature, resulting in an effective bilayer sponta-neous curvature. This situation corresponds can lead to anenthalpic stabilization.

Consider first the case of a vesicle with a small bendingconstant ð�, �� kBTÞ in the absence of a spontaneous curva-ture. Unilamellar vesicles can be stabilized if the bending


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energy per vesicle relative to the infinite lamellar phase½8��þ 4� ��, Eq: ð1Þ� is offset by the increased translationalentropy of the larger number of independent vesicles. If thereis a nonzero spontaneous curvature, the effective bendingenergy is zero for a vesicle population with a mean radius of�1/c0 and entropy dictates that vesicles are the stable phase.A small value of k also promotes unilamellar vesicles as therepulsive undulation force (Eq. 2) reduces attraction betweenbilayers, leading to stable unilamellar vesicles, especially whencombined with electrostatic repulsion in charged systems [24].Theory and experiment have shown that surfactant mixing canlead to sufficiently low values of k for entropic stabilization tobe effective [24,72,73].

Undulations of the bilayer may also decrease the effectivebending constant at long length scales. Helfrich and othershave developed an expression relating the effective bendingrigidity of a membrane to the length scale of observation, Z(¼Rfor vesicles) and a molecular distance, d (�bilayer thickness)[68,69,75].

k¼ k0 1� �kBT

4��0lnð�=0Þ

�

ð16Þ

The numerical value of a is probably between 1 and 3 [68,75].Similarly, the effective value of the Gaussian modulus is

�� ¼ ��0 1þ �kBT

4� ��0lnð�=0Þ

�

ð17Þ

where the prefactor � is estimated to be zero [69] or 10/3 [75].The form of the bending constants, renormalized for thermalfluctuations, represents a logarithmic decay in bilayer rigiditywith distance. The effect of these thermal fluctuations is toincrease the apparent membrane rigidity at short distance(high curvature or small vesicles) and to decrease the effectiverigidity at long distances (large vesicles or free bilayers). Whenboth entropy and undulations are accounted for, there arevarious predictions of how the vesicle size distribution should

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vary with surfactant concentration [24,76–78], but these havenot been verified experimentally except for a single system [24].

Safran and coworkers have studied in detail enthalpicstabilization that results if the bilayer has a spontaneouscurvature and if the bending rigidity is of the appropriatemagnitude [22,79]. They recast Eq. (1) in terms of the interiorand exterior spontaneous curvatures, cI and cE

E=A ¼ 12 K ðcþ cEÞ

2þ ðc� cIÞ

2�

ð18Þ

again with the convention that the curvature of the outermonolayer is positive. For symmetric bilayers, cI¼ cE and thefree energy is minimized for c¼ 0 (flat bilayers). However, forsurfactant mixtures, nonideal mixing of surfactant molecules inthe bilayer could allow the interior and exterior monolayers tohave equal and opposite curvatures: cI¼�cE¼ c. For this tohappen, the effective head group size of the mixed surfactantson the inside monolayer must differ from that of molecules inthe outside monolayer. This can be achieved either with amixture of amphiphiles with widely different areas per headgroup, or in a mixture in which surfactant complexes form suchthat the complex has a small area per head group. By placingmore of the smaller head group component (or complex) in theinside monolayer (and more of the larger head group oruncomplexed component in the exterior monolayer) of thevesicle, the spontaneous curvatures could be adjusted to suit aparticular composition of the vesicle [22,79].

Vesicles with such a curvature would then be stable withrespect to flat, symmetric bilayers, especially in the limit thatthe bending modulus of the bilayer is large compared to kBT[22,79]. The net result is a composition-dependent spontaneouscurvature for the bilayer that determines the radius and sizedistribution of the vesicle population. In the event that vesicleformation is promoted by a spontaneous bilayer curvature, thesize distribution is predicted to be a Gaussian peaked at a sizenear 1/co with a relative standard deviation that is inverselyproportional to the sum of the Helfrich bending constants,K ¼ �þ ð ��=2Þ. The model does not treat bilayer interactionsthat undoubtedly become important at higher surfactant


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concentrations, but it does account for several features of theexperimental phase diagram [79].

In the spontaneous curvature model, interactions betweensurfactants are critical in stabilization of a phase of vesicles. Inparticular, attractive interactions are necessary to alter thebalance between the area required for head groups relativeto that needed for the tail groups. Applied to mixtures ofoppositely charged surfactants, the interaction between theoppositely charged head groups can produce a surfactant pairthat occupies a smaller area per molecule at the interface thanthe two individual surfactants when separated. Within thismodel, spontaneous vesicle formation with a well-definedsize distribution is predicted for mixtures of surfactants withequal length tails as well as in mixtures with asymmetric tailgroups.

IV. EXPERIMENTAL MEASUREMENTSOF BENDING CONSTANTS

A. Size Distributions

To distinguish between these two mechanisms of vesiclestability, the bending elasticity and bilayer spontaneouscurvature has been measured for a number of different systemsof spontaneous, unilamellar vesicles [26,27,32] by analysis ofthe vesicle size distribution using cryo-microscopy and freeze-fracture replication. The size distribution is calculated asfollows.

The equilibrium curvature, calculated from the minimiza-tion of the curvature energy given by Eq. (1), is

ceq¼c02�

2�þ ��ð19Þ

The stability criterion for the bilayer bending energy is2kþ ��> 0. When k>� ��/2, this theory predicts that the stablestate of the bilayer, even for the case of c0¼ 0, is a sphericaldeformation, with higher order terms in the free energyexpansion required to limit the vesicle size from becoming

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vanishingly small. For ��> 0, infinite films are unstable towardsformation of a surface with many handles and in this case cubicor sponge-like phases are predicted to be the stable bilayerconfiguration. For sufficiently dilute systems with finite bilayerfragments, vesicles are still possible [27] as systems with onlynegative Gaussian curvature cannot close on themselves, whichresults in bilayers with energetically unfavorable edges incontact with water.

The equilibrium size distribution of a population ofvesicles is determined by a subtle competition between theentropy of mixing and the curvature elasticity of the bilayers.For the case of spherical vesicles, R1¼R2¼R, and Eq. (1) canbe simplified [22,79] to

E=A ¼ 2K1

R�

1

R0

� �2

2K ¼ 2�þ ��R0 ¼2�þ ��

2�r0 ð20Þ

R0 is the radius of the minimum energy vesicle (¼1/ceq), and Kis an effective bending constant [22,79].

The distribution of surfactant between vesicles of aggrega-tion number M, corresponding to the minimum energy radius,R0 (M¼ 8�R0

2/A0, in which A0 is the mean molecular area),relative to vesicles of aggregation number N and radius R, isdictated by a balance between the entropy of vesicle mixing andthe curvature energy [80], and can be written in terms of thelaw of mass action [cf. Eq. (10)]

XN

N¼

XM

Mexp

M m0M � m0

N

�

kBT

�� N=M

ð21Þ

XM, m0M and XN, m0

N are the mole fraction of surfactant andthe standard chemical potential per molecule in vesicles ofaggregation numbers M and N, respectively. Equation (21)assumes ideal mixing of the vesicles (not the molecules withinthe bilayers, which will be assumed to be nonideal allowing fora spontaneous curvature) and is valid for dilute vesicledispersions in which the Debye length is small in comparisonto the inter-vesicle distance. In practice, this is always the casefor catanionic vesicles as the surfactant co-ions result in at least


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10–100 millimolar electrolyte concentrations. The chemicalpotential difference is due to the change in curvature energyper molecule for surfactant distributed between a vesicle ofradius R and aggregation number N and the minimum energyvesicle of radius R0 and aggregation number M

m0N � m0

M

� ¼

4pR2ðE=AÞ

N¼

8pK½1� ðR=R0�2

Nð22Þ

Inserting Eq. (22) into Eq. (21) and substituting M¼ 8�R02/A0

and N¼ 8�R2/A0, in which A0 is the area per surfactantmolecule, gives a two parameter vesicle size distribution as afunction of R0 and K [80,81]

CN ¼ CM exp�8�K

kBT1�

R0

R

� �2" #( )R2=R2

0

ð23Þ

CM (¼XM/M), and CN are the molar or number fractions ofvesicles of size M and N, respectively. A consequence of Eq. (23)is that vesicles stabilized by thermal fluctuations (K� kBT)have a much broader size distribution than vesicles stabilizedby the spontaneous curvature (K� kBT). This is the opposite ofvesicle size distribution models that do not include a sponta-neous curvature [24,44]; larger bending constants predict morepolydisperse vesicles of larger size.

B. Experimental Results

Cryo-TEM images were used to determine the radius of vesicles[32]. Histograms of the vesicle size distributions were built upby measuring the size of �3000 spherical vesicles per con-centration ratio taken from many different samples overseveral weeks. The measured distribution was fit to Eq. (23)(solid line) to determine R0 and K. Figure 10(a) shows atypical image of cetyltrimethylammonium bromide (CTAB)/sodium octyl sulfate(SOS)/water (0.3/0.7/99% by wt) vesicles,while Fig. 10(b) shows the measured size distribution of vesiclesand excellent agreement with the equilibrium distribution. Thebest fit to Eq. (23) gives K¼ 0.7� 0.2 kBT and R0¼ 37 nm,

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indicating these vesicles have a low bending constant and thusare entropically stabilized. Even though the K� kBT, singleparameter size distributions with no spontaneous curvaturecould not fit the experimental size distribution [24,82].

Figure 11(a) shows that the vesicles in a 2 wt% 2/8 CTAB/perflurooctanoate (FC7) dispersion are smaller, and muchmore monodisperse (Fig. 11b) than the CTAB/SOS dispersion(Fig. 10b). Fitting this distribution to Eq. (23) gives R0¼ 23 nm

F11

Figure 11 (a) Cryo-TEM image of fluorinated CTAB/FC7 vesiclesshow a much more narrow size distribution and smaller mean size(b), but that is also fit by Eq. (23). The bending constant, K¼ 6kBT, isabout an order of magnitude greater than the CTAB/SOS vesicles.The spontaneous radius of curvature is 23 nm.

Figure 10 (A) Cryo-TEM image of 0.3/0.7/99% by wt CTAB/SOSvesicles in water and the histogram (B) of their size distributions. Thesolid line is a fit to the size distribution predicted by a mass-actionmodel [i.e., Eq. (23)] using a spontaneous curvature, determined to be37 nm, and an elastic constant K¼ 0.7kBT as fitting parameters.


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and K¼ 6� 2kBT, indicating that the hydrocarbon–fluorocar-bon (CTAB/FC7) bilayers are much stiffer than the hydro-carbon–hydrocarbon bilayers (CTAB/SOS). Replacing the FC7

fluorocarbon surfactant with the shorter chain sodium per-fluorohexanoate (FC5) lowers K to 0.5kBT while increasing R0

to 56 nm [26,27]. For the CTAB/SOS and CTAB/perflurohex-anoate (FC5) systems, thermal undulations due to the smallvalue of K stabilize the hydrogenated vesicles against formationof multilamellar liposomes, even in the absence of electro-statics. CTAB/SOS vesicles are stable even with 1.4 wt% addedsalt, at which point a phase transition to micelles occurs. Thelarge bending constant and narrow size distribution suggestthat CTAB/FC7 vesicles are stabilized by the energy costs ofdeviations from the spontaneous curvature.

C. Theoretical Estimation of Elastic Constants

Factors such as the chemical identity of the surfactants, theaddition of cosurfactants, and the value of the ionic strengthstrongly influence the elastic properties of surfactant films.Theoretical treatments of both electrostatic [83–87] and chaincontributions [73,88] to the overall bending moduli areavailable. To summarize the results of theoretical predictions,the contribution to the bending moduli from the chain region isfound to be the dominant factor, with bending moduli rangingfrom >10kBT for pure surfactant bilayers to several kBT inmixed bilayers. The electrostatic contribution to the bendingmodulus is found to be of order kBT, and is predicted to exceedkBT only for highly charged membranes with low saltconcentrations [83–86,89,90]. However, in mixtures of short-and long-chained surfactants, and in microemulsions, the chaincontribution to the bending moduli is reduced considerably,and in these cases, the electrostatic contribution can be ofsimilar magnitude and should not be neglected.

Szleifer and coworkers have calculated the elastic modulias a function of surfactant chain length, area per molecule, andcomposition for mixed surfactant bilayers [73,88]. The model isbased on a mean field treatment of the organization of thesurfactant chains within an aggregate of a given geometry.

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The model contains no adjustable parameters and the onlyassumption made is that the chains pack at uniform densitywithin the aggregate, as is seen experimentally [91,92].Calculations based on chain conformation free energy showthat the bending constants increase strongly with increasingchain length or decreasing area per molecule. The bendingmodulus k scales with chain length as n3:2

c and with area permolecule as a�7:5 [73,88]. The bending constants are predictedto decrease sharply in mixtures of short- and long-chainsurfactants. For example, in mixtures of chains with 16 (C16)and 8 (C8) carbons, the bending constant is reduced from�40kBT for pure C16 bilayers to as low as �kBT in mixtures[72,88].

D. Close-packed Vesicle Dispersions andVesicle ‘‘Nesting’’

The hollow bilayer shells of vesicles exclude a substantialvolume, and so vesicles can overpack at relatively lowsurfactant concentrations. Overpacking occurs roughly whenthe volume fraction of solution enclosed by vesicles, �

� ¼4�

3Vtotal

X

N, XA

R3NnN ð24Þ

is of order 60%. Clearly, when the distribution shifts towardslarge vesicles, the surfactant mole fraction at which vesiclesbecome close packed, x*, will drop considerably. This concen-tration is relevant experimentally, and will occur at surfactantconcentrations of a few percent for larger vesicles. It probablymarks the high concentration end of the vesicle ‘‘lobes’’ in thephase diagram, as shown in Fig. 2.

At surfactant concentrations above x*, the added surfac-tant can be accommodated either by shifting the vesicle sizedistribution to lower radii, by forming multilamellar vesicles,or by forming a lamellar phase. Simons and Cates give athermodynamic analysis of the stability of the various phases inthe region of the overlap concentration c* [76]. Their modelcombines curvature elasticity with entropy of mixing and


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focuses on symmetric bilayers with zero spontaneous curva-ture. They find that as the overlap concentration is approached,the large vesicle sizes are eliminated from the size distribution.This is accompanied by a decrease in polydispersity and anaccompanying decrease in the entropic contribution. At thesame time, there is an energetic penalty for forcing the sur-factant to assemble into more highly curved aggregates.Consequently, the entropic term that favors formation ofunilamellar vesicles decreases, and the lamellar phase, whichhas a lower bending energy than an ensemble of vesicles,becomes more favorable. This corresponds to a first order phasetransition from vesicles to a lamellar liquid crystalline phase[76]. Another possibility is that above the overlap concentra-tion, the vesicles can ‘‘nest’’ within each other, forming multi-lamellar vesicles. In this case, surfactant can more efficientlypack into the available volume, and the favorable entropiccontribution arising from a polydisperse distribution of vesiclescontinues to stabilize the vesicle phase relative to the lamellarphase. Simons and Cates found that MLVs can be stable atconcentrations intermediate to the unilamellar vesicle phaseand the lamellar phase [76].

There is an additional interesting possibility for vesiclenesting that arises for dispersions stabilized by spontaneouscurvature. In this case it is possible for vesicles to form with adiscrete number of bilayers, depending on the magnitude andsign of the bilayer interactions. This was observed experimen-tally when 1 wt% NaBr was added to screen any residual short-range electrostatic interactions between the bilayers, the resultwas the spontaneous formation of a population of primarilytwo-layered vesicles (Fig. 12). In Fig. 12, two layer vesicles aredistinguished from one layer vesicles by the darker rim on theinside edge of the apparent vesicle membrane (arrows). Thisdark rim is due to the greater projection of the electron beamthrough both the interior and exterior vesicle bilayers. Fromexamining many images, about 90% of the vesicles with addedsalt have two bilayers, while the rest have one bilayer. Therewere essentially no vesicles with three layers or more. Thevesicles in 1% NaBr sample also had a greater tendency toadhere to each other and the carbon coated electron microscope

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grid and flatten, consistent with the enhanced attractionbetween the vesicle bilayers [93].

The distribution between one layer and two-layer vesiclescan be derived using the mass action model for vesicles with aspontaneous bilayer curvature. The analysis shows thatvesicles stabilized by spontaneous curvature can have anarrow distribution of the number of bilayers when the attrac-tive interactions just balance the curvature energy. In theabsence of a spontaneous bilayer curvature, each additionallayer added to a vesicle has a decreasing curvature energy, butthe attractive interaction energy grows with the net bilayerarea in contact, and a polydisperse population of multilamellarliposomes result. Hence, typical phospholipid vesicles, with1/R0¼ 0, are unstable relative to multilamellar liposomes. Thecombination of a narrow size distribution, a large bending

Figure 12 Cryo-TEM image of CTAB : FC7 (2 wt% total surfactant,CTAB : FC7 ratio of 2/8 by weight) in 1 wt% NaBr. Two layer vesiclesare distinguished from one layer vesicles by the darker rim on theinside edge of the vesicle membrane. This dark inside rim is due to theincreased projection of the electron beam through both the interiorand exterior vesicle bilayers; the single bilayer vesicles havemembranes with a uniform intensity and do not show the interiordark rim. From examining many images, about 90% of the vesicleswith added salt have two bilayers, with the rest appearing to have onebilayer. There were essentially no vesicles with three layers or more.


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elastic constant, and the formation of two-bilayer vesiclesshows that the CTAB/FC7 vesicles are stabilized by sponta-neous curvature. Interesting other shapes (rods and disks) canalso be analyzed quantitatively [27].

V. EQUILIBRIUM?

Unilamellar vesicles form in a wide variety of mixtures ofcationic and anionic surfactants, and do so spontaneously; thatis, without substantial input of energy. Whether or not thevesicles are the equilibrium state of aggregation has beenexplored by several authors [46,94,95], all of whom conclude forvarious reasons that the vesicles are unstable with respect to alamellar phase. A corollary argument is that the vesicles thatare observed are the consequence of shear forces, which mightbe vanishingly small, that disrupt the stacked bilayers of alamellar phase. Experimental conformation is hampered by theextremely slow evolution of these structures with time, whichin turn must reflect a combination of small thermodynamicdriving forces and slow mass transfer processes.

The question is clearer from a theoretical point of view.Consider first a patch of bilayer with zero spontaneouscurvature in water in the limit of low concentrations. Thepatch of bilayer has hydrophobic edges exposed to water. This‘‘edge energy’’ is recovered by closing the bilayer into a vesicle,with the energy penalty given by Eq. (1). For large enoughvalues of edge energy and small enough values of the bendingconstant, the vesicle is stable. The situation is more interestingwhen the bilayer has a preferred curvature, as can happen for amixed bilayer. Several theoretical approaches all yield theresult that the vesicle is the thermodynamically preferredstructure.

Thus, at least at low concentrations, bilayers with either alow bending constant or with a non-zero spontaneous curva-ture form vesicles are absolutely the preferred thermodynamicstate compared to a stacked bilayer for the reasons laid outabove. The experimental data show that vesicles are stabilizedby one of two distinct mechanisms depending on the value of

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the bending constant. Helfrich undulations ensure that theinterbilayer potential is always repulsive when the bendingconstant, K, is of order kBT. When K� kBT, unilamellarvesicles are stabilized by the spontaneous curvature that picksout a particular vesicle radius; other radii are energeticallydisfavored. Measurements of the bilayer elastic constant andthe spontaneous curvature, R0, for three different systems ofvesicles by an analysis of the vesicle size distributiondetermined by cryo-transmission electron microscopy showcases for both entropically and enthalpically stabilized vesicles.

An absolute resolution of this question is probablyimpossible, but there are several relevant observations athand. In particular, Hoffmann and co-workers have producedboth lamellar phases and vesicles by chemical synthesis toproduce one of the surfactant species in situ, thereby avoidingshear. They see the formation of lamellar phases that are stablein time but that form vesicles upon a single inversion of thesample tube. These vesicles are also stable with time, so theexperiment simply does not allow resolution of the question ofwhich is the stable phase. Further, the lamellar phases areobserved by freeze-fracture electron microscopy, which requiresexposing a thin layer of solution to a solid surface, and solidsurfaces are well known to nucleate lamellar sheets fromvesicles [96]. Thus the question of the metastability of thelamellar phase is open. For the reasons given above, creating alamellar phase by specific chemical or physical treatments thatis unstable to small perturbations does not show that thelamellar phase is the preferred state of organization.

Almgren [46] recently summarized many observationsaround swollen lamellar phases, and notes that they can beeasily dispersed to form vesicles. This observation was alsooffered by Laughlin as the explanation for the observed vesicles[94]. Almgren also reports QLS measurements of mixtures ofCTAB and SOS at a given mixing ratio that show an increase inmean radius with time, which agrees with our results at thatmixing ratio [31]. Other ratios yield vesicles whose average sizedoes not change with time. Careful, long term studies ofcatanionic mixtures at certain well-defined compositions haveshown uniform phases of unilamellar vesicles for well over a


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decade [28]. The size distributions of these vesicles remainconstant over time and are well described by equilibriumtheories of self-assembly [26,27,32]. Of course, not everymixture of anionic and cationic surfactants will form equilib-rium vesicles and care must be taken to understand the phasediagrams and carefully delimit the concentration ranges thatdo form such vesicles. That various experimental observationsare often at odds highlight the experimental challenges inresolving the issue of equilibrium vesicles.

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