2474 Langmuir 2006, 22, 2474-2481

Zero Spontaneous Curvature and Its Effects on Lamellar Phase Morphology and Vesicle Size Distributions

Bret A. Coldren, Heidi Warriner, Ryan van Zanten, and Joseph A. Zasadzinski*

Department of Chemical Engineering, UniVersity of California, Santa Barbara, California 93106-5080 Eric B. Sirota

Exxon Research and Engineering Company, Corporate Strategic Research, Route 22 East, Annandale, New Jersey 08801

ReceiVed September 7, 2005. In Final Form: January 2, 2006

Equimolar mixtures of dodecyltrimethylammonium chloride (DTAC) and sodium octyl sulfonate (SOSo) show a vesicle phase at >99 wt % water and a single, fluid lamellar phase for water fractions below 80 wt %. This combination is consistent with the bilayer bending elasticity κ ≈ kBT and zero bilayer spontaneous curvature. Caille´ line shape analysis of the small-angle X-ray scattering from the lamellar phase shows that the effective κ depends on the lamellar d spacing consistent with a logarithmic renormalization of κ, with κo ) (0.8 ( 0.1)kBT. The vesicle size distribution determined by cryogenic transmission electron microscopy is well fit by models with zero spontaneous curvature to give (κ + (κj/2)) ) (1.7 ( 0.1)kBT, resulting in κj)(1.8 ( 0.2)kBT. The positive value of κj and the lack of spontaneous curvature act to eliminate the spherulite defects found in the lamellar gel phases found in other catanionic mixtures. Current theories of spontaneous bilayer curvature require an excess of one or more components on opposite sides of the bilayer; the absence of such an excess at equimolar surfactant ratios explains the zero spontaneous curvature.

Introduction bilayer. To stabilize a bilayer, κ > 0, but κj>0 for surfaces that It is no surprise that oppositely charged surfactants mix in a prefer hyperbolic shapes (saddle-shaped surfaces in which the < highly nonideal way and in fact are able to form structures such centers of curvature are on opposite sides of the surface, c1c2 j< as unilamellar vesicles that are not found in other surfactant 0) and κ 0 for surfaces that prefer elliptical shapes (spheres, mixtures.1 The range of self-assembled microstructures in ellipsoids, etc. in which the centers of curvature are on the same > mixtures of cationic and anionic surfactants also includes small side of the surface, c1c2 0). For a chemically and physically 3-5 spherical micelles, cylindrical or wormlike micelles, and other symmetric bilayer, the spontaneous curvature c0 equals 0. 2 Hence, a single-component bilayer cannot have a nonzero lamellar and L3 phases. The vesicle phases discussed here form spontaneously with no external energy input, and some vesicle spontaneous curvature. Theory suggests that it is necessary for phases have been observed for well over a decade, suggesting the bilayer or its local environment to be chemically asymmetric 5 at least metastable equilibrium. for spontaneous curvature to exist. The fundamental questions remain as to why vesicles form in Purely entropically stabilized vesicles have a low bending ≈ mixtures of oppositely charged surfactants and, once formed, constant (κ kBT, where kB is the Boltzmann constant), so the whether they are at thermodynamic equilibrium. The formation enthalpic bending contribution to the free energy is small and stability of any surfactant aggregate depends on whether compared to the mixing entropy. Unilamellar vesicles are that aggregate represents the global minimum in free energy for stabilized against aggregation and formation of multilamellar a given composition, and there are different approaches to liposomes by both the entropy of mixing and the steric repulsion describing the energetics of these mixtures. Translational entropy between bilayers caused by Helfrich undulations. The Helfrich favors many unilamellar vesicles over a smaller number of undulation repulsion is a consequence of bilayer thermal multilamellar liposomes. To examine the enthalpic part of the fluctuations, which are damped when bilayers come into contact; free energy, it is often useful to start with a mechanical description this causes the bilayer entropy to be reduced and leads to a 6 of the properties of the bilayer.3-5 In this case, the elastic energy repulsive interaction. For bilayers in a lamellar phase of repeat spacing d and membrane thickness δ, the Helfrich undulation of a bilayer is described by the two principle curvatures c1 and repulsion is c2. For spherical vesicles, c1 ) c2 ) 1/R, in which R is the vesicle radius. To terms second order in curvature, the free energy of 3,4 2 2 a bilayer, per unit area, is 3π (kBT) E ) (2) und 128 2 E 1 κ(d - δ) ) κ(c + c - 2c )2 + κjc c (1) A 2 1 2 0 1 2 (For free bilayer or vesicles, (d - δ) is replaced by the interbilayer in which κ is the bending modulus, κj is the saddle splay or (2) Kaler, E. W.; Herrington, K. L.; Iampietro, D. J.; Coldren, B.; Jung, H. Gaussian modulus, and c0 is the spontaneous curvature of the T.; Zasadzinski, J. A. In Mixed Surfactant Systems, 2nd ed.; Abe, M., Scamehorn, J., Eds.; Marcel Dekker: New York, 2005; pp 298-338. * To whom correspondence should be addressed. E-mail:gorilla@ (3) Frank, F. C. Discuss. Faraday Soc. 1958, 25,19-28. engineering.ucsb.edu. Phone: 805-893-4768. Fax: 805-893-4731. (4) Helfrich, W. Z. Naturforsch. 1973, 28C, 693-703. (1) Kaler, E. W.; Murthy, A. K.; Rodriguez, B. E.; Zasadzinski, J. A. N.Science (5) Safran, S. A.; Pincus, P. A.; Andelman, D. Science 1990, 248, 354-356. 1989, 245, 1371-1374. (6) Helfrich, W. Z. Naturforsch. 1978, 33A, 305-315.

10.1021/la052448p CCC: $33.50 © 2006 American Chemical Society Published on Web 02/09/2006 Lamellar Phase Morphology and Vesicle Size Langmuir, Vol. 22, No. 6, 2006 2475 distance.) The repulsive undulation interaction can overwhelm the van der Waals attraction between bilayers (which is also proportional to (d - δ)-2) when κ is small, leading to a net repulsive interaction between bilayers and hence stable unila- mellar vesicles, especially when combined with electrostatic repulsion in charged systems.7-9 Enthalpically stabilized vesicles, however, require nonzero spontaneous curvature and a larger value of the bending constant (κ > kBT). The curvature energy of forming a multilamellar liposome with many layers far from the optimal curvature is prohibitively high. In this case, the vesicles are narrowly distributed around a preferred size set by the spontaneous curvature.8,9 Spontaneous, apparently equilibrium vesicles with both narrow8-10 and broad size distributions1,2,11-15 in aqueous mixtures of a wide range of mixtures of cationic and anionic (catanionic) surfactants have been characterized by a variety of experimental techniques including small-angle neutron, X-ray, and light scattering and cryo and freeze-fracture transmission electron microscopy. The most common cationic surfactant used has been an alkytrimethylammonium bromide or tosylate, (e.g., cetyltrimethylammonium bromide (CTAB) or tosylate (CTAT)), whereas the common anionic surfactants are the sodium alkyl sulfates (e.g., sodium octyl (SOS), decyl sulfate (SDS), or dodecylbenzene sulfonate (SDBS)), which may have either a branched or comb structure. The phase diagrams of mixtures of these surfactants typically show two vesicle lobes, which are roughly symmetric on either side of the equimolar line at high water fractions (Figure 1A). One vesicle phase has a net excess of anionic surfactant, whereas the other has a net excess of cationic Figure 1. (a) Partial phase diagram of CTAT/SDBS/water at 25 °C. surfactant. For most catanionic mixtures, an insoluble precipitate Typical of most catanionic mixtures, the phase diagram is roughly forms at equimolar anion-to-cation ratios. At lower water symmetric about the equimolar line. Two vesicle lobes (V) are found fractions, two lamellar gel phases (MLV in Figure 1A) are found at high water fractions, one with an excess of CTAT and the other that are turbid and viscoelastic because of a highly defected, with an excess of SDBS. An insoluble precipitate forms along the spherulite texture; the lamellar gel phases are also located roughly equimolar line. The symmetric MLV phases are onionlike multi- 16 lamellar spherulites with no excess water that form a lamellar gel symmetrically about the equimolar line. These gels are similar similar to DMPC/pentanol/polymer lipid gels.17-19 Not indicated to lamellar gels of dimyristoylphosphatidylcholine, pentanol, and are micellar phases at either extreme of the surfactant mixing ratio. - poly(ethylene glycol) lipids17 19 and show a transition at lower (b) Partial phase diagram of DTAC/SOSo/water at 25 °C. M water fractions to a fluid, flat bilayer phase.16,20 A simple defect- represents spherical micelle phases, R represents a viscous rodlike based theory of this structural progression requires that κ ≈ kBT micelle phase, and V represents unilamellar vesicles. Instead of two and that c * 0.17-19 Cryo-TEM analysis of the vesicle size symmetric lobes, both the vesicle and lamellar phase occur along 0 the equimolar line. The lamellar phase consists of flat stacks at all distributions and Caille´ line shape analysis of the small-angle compositionssno MLV spherulite phase is observed.

(7) Herve´, P.; Roux, D.; Bellocq, A.-M.; Nallet, F.; Gulik-Krzywicki, T. J. 16,19,21-23 ≈ - X-ray scattering confirm that κ kBT for all of the Phys. II 1993, 8, 1255 1270. 16,20 (8) Jung, H. T.; Coldren, B.; Zasadzinski, J. A.; Iampietro, D. J.; Kaler, E. W. catanionic systems studied thus far. This shows that surfactant Proc. Natl. Acad. Sci. U.S.A. 2001, 98, 1353-1357. mixing can lead to low bending constants24 and is consistent (9) Jung, H. T.; Lee, Y. S.; Kaler, E. W.; Coldren, B.; Zasadzinski, J. A. Proc. with the entropic stabilization of spontaneous vesicles and the Natl. Acad. Sci. U.S.A. 2002, 99, 15318-15322. 16,20 (10) Iampietro, D.; Kaler, E. W. Langmuir 1999, 15, 8590-8601. formation of defects in the lamellar gel phases. (11) Herrington, K. L.; Kaler, E. W.; Miller, D. D.; Zasadzinski, J. A.; Chiruvolu, However, mixtures of DTAC and SOSo do not follow this S. J. Phys. Chem. 1993, 97, 13792-13802. (12) Brasher, L. L.; Herrington, K. L.; Kaler, E. W. Langmuir 1995, 11, 4267- typical phase progression. The shorter chain lengths apparently 4277. inhibit the formation of a precipitate at equimolar concentration (13) Brasher, L. L. Phase Behavior and Microstructure of Surfactant Mixtures. and instead lead to a single vesicle phase and a single, fluid Ph.D. Dissertation, University of Delaware, Newark, DE, 1996. (14) Coldren, B. A.; van Zanten, R.; Mackel, M. J.; Zasadzinski, J. A.; Jung, lamellar phase. Caille´ line shape analysis of the small-angle X-ray - 16,19,21-23 H. T. Langmuir 2003, 19, 5632 5639. scattering from the lamellar phase shows that κ ≈ kBT (15) Kaler, E. W.; Herrington, K. L.; Murthy, A. K.; Zasadzinski, J. A. N. J. Phys. Chem. 1992, 96, 6698-6707. as in other entropically stabilized catanionic mixtures. The lack (16) Coldren, B. Phase Behavior, Microstructure and Measured Elasticity of of spherulite texture in the lamellar phase suggests that the Catanionic Surfactant Bilayers. Ph.D. Dissertation, University of California, Santa spontaneous radius of curvature is large (spontaneous curvature Barbara, CA, 2002. (17) Warriner, H.; Idziak, S.; Slack, N.; Davidson, P.; Safinya, C. Science 1996, 271, 969-73. (21) Safinya, C.; Roux, D.; Smith, G. S.; Sinha, S. K.; Dimon, P.; Clark, N.; (18) Keller, S. L., Warriner, H., Safinya, C., Zasadzinski, J. A. Phys. ReV. Lett. Bellocq, A. M.Phys. ReV. Lett. 1986, 57, 2718-2721. 1997, 78, 4781-4784. (22) Safinya, C. R.; Sirota, E. B.; Roux, D.; Smith, G. S. Phys. ReV. Lett. 1989, (19) Warriner, H. E.; Keller, S. L.; Idziak, S. H. J.; Slack, N. L.; Davidson, 62, 1134-1137. P.; Zasadzinski, J. A.; Safinya, C. R. Biophys. J. 1998, 75, 272-293. (23) Roux, D.; Safinya, C. R. J. Phys. (Orsay, Fr.) 1988, 49, 307-318. (20) Coldren, B. A.; Warriner, H. E.; van Zanten, R.; Zasadzinski, J. A.; Sirota, (24) Szleifer, I., Kramer, D., Benshaul, A., Gelbart, W. M., J. Chem. Phys. E. B. Proc. Natl. Acad. Sci. U.S.A. 2005, in press. 1990, 92, 6800-6817. 2476 Langmuir, Vol. 22, No. 6, 2006 Coldren et al. is near zero) compared to the d spacing of the lamellar phase. thickness, which are proportional to B; the second term describes The DTAC/SOSo vesicle size distribution determined by cryo- variations in the membrane splay, which are proportional to the TEM can be fit equally well by a one-parameter model with c0 splay elastic modulus, K (this term is equivalent to the first term ) 0 or by a two-parameter model with a finite c0. Hence, at in the Helfrich bending energy expression in eq 1 with K ) κ/d; equimolar ratios, the necessary composition asymmetry for terms involving κj do not alter the free energy of the bulk lamellar spontaneous curvature is small to nonexistent, but surfactant phase). Thermally induced mean square layer displacements mixing can still make κ ≈ kBT, as required for the entropic diverge logarithmically with domain size L, destroying long- stabilization of vesicles and a highly swollen lamellar phase. range order, which causes the conventional delta-function Bragg peaks in a crystal to be replaced by power law divergences.26 For Materials a powder sample, profiles of the bilayer (00l) reflections have Cetyltrimethylammonium p-toluenesulfonate (98% pure, CTAT, the asymptotic form21 Sigma), dodecyltrimethylammonium chloride (98% pure, Fluka), - ≈| - |ηl 1 sodium 1-octylsulfonate (SOSo, 99% pure, Lancaster Chemical), S(q) q q00l and dodecylbenzene sulfonate (SDBS, 98% pure, TCI America) were used as received. In all experiments, water was purified by the l2q2 k T ≡ 00l B < Milli-Q process, which results in a resistance of 18.2 MΩ. Brine ηl if ηl 1 (4) solutions were prepared from mixtures of Milli-Q water and NaCl 8πxBK salt of >99.9% purity (Aldrich) to minimize the effect of electrostatics on the X-ray line shape. At these salt concentrations, the Debye Thus, the shape of the (00l) scattering peaks is determined by screening length is <0.6 nm,25 so electrostatic interactions are the elastic properties of the bilayer as η2 ≈ 1/KB. To distinguish insignificant at lattice spacings greater than ∼3 nm.21 In addition clearly between changes in intermembrane interactions (B) and to the added salts, the cationic mixtures have significant concentra- membrane rigidity (K), a more formal analysis is used to fit the tions of free counterions in solution, so the actual Debye length is X-ray line shapes. The real-space Caille´ correlation function is less than 0.6 nm. Samples were prepared by gentle mixing and given by27 heating cycles and allowed to equilibrate for at least 2 months undisturbed before X-ray, cryo-TEM. or freeze-fracture analysis. 1 2ηl F2 Small-Angle X-ray Diffraction. Preliminary phase behavior and ∝ - - G(r) (F) exp( 2γηl ηlEl( )) (5) scattering experiments at UCSB were performed on two different 4λz custom-built instruments with 18 kW Rigaku rotating anode sources ≡ < >< > < >< > (Cu KR, λ ) 1.54 Å) and 2-D area detectors. One instrument is set in which λ scp rl K/B rlx mx , γ is Euler’s constant, up for intermediate length scales (10-100 Å), has a bent graphite and El is the exponential integral function. monochromator, 18 cm Mar image plate detector, and four sets of In a single crystal of domain size L, the structure factor is the Huber slits. The 0.8 × 0.8 mm2 beam has a flux of ∼107 photons/s. Fourier transform of G(r) multiplied by a finite-size factor The fwhm was ∼0.01 Å-1. The second instrument is optimized for - 2 2 - - ‚ very small angle work (50-600 Å), has an Osmic Confocal Maxflux ∝ πr /L i(q qz) r 3 S(0, 0, qz) ∫∫∫ G(r)e e d r (6) double-focusing multilayer mirror, an 11 cm Bruker HI-STAR multiwire area detector, and a 1.5 m sample-to-detector distance. For a powder sample, S(0, 0, q) is averaged over all solid angles ∼ -1 The fwhm was less than 0.005 Å . In all experiments, samples in reciprocal space: were loaded into 1.5 mm borosilicate glass capillaries (Charles Supper, Natick, MA) and flame sealed. Temperature control was via - ≡ ∫ ∝ circulating heated or cooled fluid through an aluminum sample holder S(q q00l) S(0, 0, qz)dΩq block, monitored by a thermistor located adjacent to the capillary. ∞ ∞ - 2 2 sin(qr) - For Caille´ line shape analysis of lamellar phases, the ExxonMobil ∫ dz ∫ G(z, F)e πr /L e q00lz dF (7) -∞ 0 X10A beamline at the Brookhaven National Laboratory’s National qr Synchrotron Light Source (NSLS) was used. A double-crystal (Ge- For each harmonic peak order of l, there are five fitted 〈111〉) monochromator, narrow slit geometry, and silicon crystal analyzer result in a typical in-plane resolution having a fwhm of just parameters: ηl, λ, L, q00l, and the peak intensity Il. For a given 0.0004 Å, corresponding to a resolution of ∼3 µm. Scattering was sample, all harmonics should give the same values of λ and L, 2 detected using a Bicron point detector (Bicron, OH). but ηl should scale as l . From these fits, κ and B are readily extracted: X-ray Line Shape Analysis 2 κ qo λ The Caille´ structure factor, originally developed to describe ≡ k T diffraction from smectic A liquid crystals, can also successfully d 8πη B 23 describe scattering from both electrostatic- and undulation- 2 19,21,22 k T q stabilized lamellar systems. The Caille´ theory relates the B ) B ( o ) (8) X-ray line shape to the membrane spacing, d, the bulk compression 8π ηλ modulus, B, and the smectic splay elastic modulus, K. The theory starts with the Landau-De Gennes expression for the free-energy After background subtraction, the SAXS peaks were fit to eq 7 density (F/V) of a smectic A liquid crystal:19,21,22 using a numerical routine that minimizes ø2 as in Warriner et al.19 F 1 ∂u 2 ∂2u ∂2u 2 ) B + K + (3) Cryo-TEM V 2{ (∂z) ( 2 2) } ∂x ∂y To prepare samples for cryogenic transmission electron microscopy, a thin (<0.5 µm) layer of the surfactant-water u(r) is the layer displacement in the z direction normal to the mixture was spread on a lacey carbon grid (Ted Pella, Redding, layers. The first term describes variations in the membrane (26) Caille, A. C. R. Hebd. Seances Acad. Sci., Ser. B 1972, 274, 891-893. (25) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic (27) Als-Nielsen, J.; Litster, J. D.; Birgeneau, R. J.; Kaplan, M.; Safinya, C. Press: London, 1992. Phys. ReV.B1980, 22, 312-320. Lamellar Phase Morphology and Vesicle Size Langmuir, Vol. 22, No. 6, 2006 2477

CA) in a temperature-controlled chamber saturated with the solution of interest.28 The grid was plunged into a mixture of liquid ethane and liquid propane cooled by liquid nitrogen. The frozen samples were transferred to a Gatan (Pleasanton, CA) cold stage and imaged directly at 100 kV using a JEOL 100CXII transmission electron microscope (TEM). Bright-field phase- contrast TEM micrographs were recorded using standard low- dose procedures on either film or with a Gatan CCD camera. Vesicle radii were measured from the outer edge of the dark rim in the image to obtain the best representation of the true radius of the vesicle. Hundreds of individual vesicles were measured from the digitized images using a commercial image analysis package (Image-Pro Plus version 4.1, Media Cybernetics, MD) to determine the size distributions. Samples for cryo-TEM were prepared and examined over the course of several weeks to ensure that neither the structures nor the size distributions were changing with time.

Results and Discussion Dodecyltrimethylammonium chloride (DTAC) and sodium octyl sulfonate (SOSo) have a phase diagram that is quite different from those of other catanionic mixtures such as CTAT/SDBS (Figure 1). We observe broad cationic and anionic micelle phases at either extreme of the DTAC/SOSo ratio (Figure 1B) as well as unilamellar vesicles at <1% total surfactant. A more viscous rodlike micelle phase appears at slightly anionic ratios, which is expected on the basis of molecular packing arguments.25 Instead of two vesicle and lamellar gel phases distributed symmetrically about the equimolar line as in CTAT/SDBS (Figure 1A), DTAC/ SOSo has a single lamellar and a single vesicle phase located along the equimolar line. Unlike the lamellar gel phases of the typical catanionic systems,14,16,20 this lamellar phase is not turbid and is fluid rather than viscoelastic. Freeze-fracture TEM images (not shown) reveal a typical lamellar texture with flat bilayers. Along the equimolar line the bilayers should have zero net charge, so the large water dilution must be stabilized solely by thermal undulations, which are characteristic of a low bilayer bending rigidity. The position and extent of the lamellar phase is essentially unaffected in 0.25 M NaCl solution. DTAC and SOSo have 12 and 8 linear carbon chains, respectively, so it is not surprising that this mixture is below the Kraft temperature of the catanionic surfactant and that the precipitate does not form. Figure 2. (a) Longitudinal SAXS profiles of the first harmonic of a series of four DTAC/SOSo, 1:1 mole ratio, lamellar phase samples Small-Angle X-ray Scattering at the 0.25 M NaCl brine wt % value indicated above each profile. The solid lines are fits of the full Caille´ power law line shape, eq Figure 2A shows SAXS profiles of the first harmonic of four 7. Peak intensities are normalized to unity. (b) Plot of the lamellar different equimolar DTAC/SOSo mixtures along a brine dilution repeat distance (measured via the peak position of the structure factor) as a function of membrane volume fraction. The slope of the path. The weight fraction of brine varies between 60 and 75%, ∼ corresponding to interlayer separations of 30-70 Å. The profiles line yields a membrane thickness of 22.5 Å. are fit by the Caille´ line shape, as shown by the solid line. Higher of stacked membranes and indicates the integrity of the harmonics were not observed in these samples, indicative of a experimental system. From the slope of this line, the DTAC/ high degree of thermal fluctuations (and a corresponding small SOSo membrane thickness δ ) 22.5 ( 0.5 Å. Logarithmic value of the bending constant). Likewise, the tail scattering deviations from this simple dilution law, due to short wavelength increases with dilution (d spacing). The full width at half- fluctuations of the membranes, are expected at high water maximum for the lower water fraction profiles are resolution- fractions. Theoretical descriptions of this deviation present a limited, consistent with well-correlated, flat lamellar bilayers straightforward way to estimate the bilayer bending modulus, without significant defects. This is very different than that κ.29,30 However, these deviations are rather small and must be observed for other catanionic lamellar gel phases, which show observed over a much larger range of dilution (1/Φmem g 50) - 16,20 broader peaks and correlations of only 5 10 bilayers. than is possible in this phase. Figure 2B shows a plot of the lamellar repeat distance Figure 3 presents a plot of the same DTAC/SOSo scattering (determined from scattering peak positions) as a function of the profiles on a logarithmic intensity scale, normalized to unity by inverse membrane volume fraction, Φmem (calculated by sample the peak intensity, I(G), where the horizontal axis is normalized composition). The linear variation is characteristic of 1-D swelling (29) Strey, R.; Schomocker, R.; Roux, D.; Nallet, F.; Olsson, U. J. Chem. Soc., (28) Bellare, J. R.; Davis, H. T.; Scriven, L. E.; Talmon, Y. J. Electron Microsc. Faraday Trans. 1990, 86, 2253-2261. Tech. 1988, 10,87-111. (30) Freyssingeas, E.; Nallet, F.; Roux, D. Langmuir 1996, 12, 6028-6035. 2478 Langmuir, Vol. 22, No. 6, 2006 Coldren et al.

Figure 3. Fits of the full Caille´ power law line shape, eq 7, to the first harmonics of SAXS scans of DTAC/SOSo in a 1:1 mole ratio at the indicated 0.25 M NaCl brine wt % values. The horizontal axis is normalized by subtracting peak position G from scattering vector q. The vertical intensity axis is normalized by the peak intensity, I(G). For 60 wt % brine, the fitting parameters yield κ ) (0.61 ( 0.19)kBT. For 65 wt % brine, κ ) (0.30 ( 0.05)kBT. For 70 wt % brine, κ ) (0.22 ( 0.02)kBT. For 75 wt % brine, κ ) (0.12 ( 0.01)kBT. In all cases, the domain size, L, is large and resolution- limited, consistent with a flat lamellar phase. by subtracting the peak position, G, from the scattering vector, q. The solid lines are fits of the Caille´ line shape in order to measure the bending constant, κ, via eq 7. For 60 wt % brine, the fitting parameters yield κ ) (0.61 ( 0.19)kBT. For 65 wt % brine, κ ) (0.30 ( 0.05)kBT. For 70 wt % brine, κ ) (0.22 ( Figure 4. Fitted parameters and calculated elasticity constants for 0.02)kBT. For 75 wt % brine, κ ) (0.12 ( 0.01)kBT. In all cases, a brine dilution series of four equimolar DTAC/SOSo lamellar phase the average lamellar domain size, L, is very large and resolution- samples. The Caille´ exponent increases with dilution, reflecting the κ limited, consistent with well-correlated, defect-free bilayers. expected inverse d dependence on B . The parameter λ is constant to within the precision given. κ shows the decrease in d as is Figure 4 shows a summary of fitted parameters and calculated theoretically expected for a fluctuating bilayer. As expected, B is elasticity constants for the series of four equimolar DTAC/SOSo a smoothly decreasing function of brine dilution (or d). lamellar phase samples. The Caille´ exponent, η, increases with dilution, reflecting the expected inverse d dependence on the the apparent rigidity at small distances approaching the membrane quantity Bκ.17,19,21,22 The parameter λ is essentially constant to thickness while decreasing the rigidity at large distances. within the precision of the fits. Relative to η, the error bars on However, it should be noted that this renormalization remains λ are much larger, demonstrating the greater difficulty in under scrutiny in the literature, and Helfrich has more recently determining peak asymmetry than in measuring power law decay. proposed that thermal fluctuations stiffen the membrane.33 Both κ and B decrease with brine dilution. As expected in the Renormalization effects become most important when the Helfrich model, B is a smoothly decreasing function of brine intermembrane spacings significantly exceed membrane thick- dilution (or d). ness, which occurs in DTAC/SOSo lamellae above 60% brine. 31 32 Helfrich and others have postulated that thermal fluctuations Limited experiments were also performed on DTAC/SOSo at renormalize the bending constant, κ, based on the length scale a 45:55 molar ratio (not shown), which is found within the same of observation, here the lamellar d spacing (for vesicles, it is R, lamellar phase envelope (Figure 1B) as the equimolar composi- the vesicle radius), relative to δ, the bilayer thickness tions. To within the accuracy of our measurements, none of the elastic parameters are different from those of the equimolar kBT d - δ κ ) κ -R ln samples, suggesting a weak compositional dependence of the o 4π ( δ ) elastic parameters within the single DTAC/SOSo lamellar phase. k T - This is also consistent with the observation that a significant j)j +Rj B d δ κ κo ln (9) fraction of the component in excess forms a micellar phase, and 4π ( δ ) the actual composition of the bilayers does not reflect small changes in the overall surfactant composition in solution.11-13,15,34 in which the prefactors R and Rj are predicted to be 3 and 10/3, respectively, although this value is model-dependent.32 Fitting Equimolar DTAC/SOSo mixtures are electrostatically neutral yet swell extensively with added water or brine, suggesting that the limited data for κ to eq 9 with δ ) 2.25 nm gives κo ) 0.8kBT and R)7.9. The effect of thermal fluctuations, then, is to increase the source of long-range repulsion interaction is Helfrich undulations (eq 2) induced by a low bilayer bending rigidity. (31) Helfrich, W. J. Phys. (Orsay, Fr.) 1985, 46, 1263-1268. (32) Peliti, L.; Leibler, S. Phys. ReV. Lett. 1985, 54, 1690-1693. (33) Pinnow, H. A.; Helfrich, W. Eur. Phys. J. E 2000, 3, 149-157. Lamellar Phase Morphology and Vesicle Size Langmuir, Vol. 22, No. 6, 2006 2479

The SAXS Caille´ line shape measurements quantitatively confirm Origins of the Spontaneous Curvature this assertion because the data are consistent with a high degree Safran and co-workers have shown that the likely origin of of thermal fluctuations leading to an undulation repulsion that a bilayer spontaneous curvature is nonideal mixing (i.e., a dominates the van der Waals attraction. Putting in some relevant concentration difference exists between the two monolayers that numbers, for a κ between 0.1kBT and 0.8kBT, the Helfrich repulsion make up the bilayer36,37). Equation 1 is rewritten (for spherical 2 scales as Efluct ) (0.2-2)kBT/dw , where dw ) d - δ is the vesicles with c1 ) c2 ) c ) 1/R, the vesicle radius) in terms of spacing between adjacent membranes. With a Hamaker constant the interior and exterior spontaneous curvatures, c and c 35 I E of about (1-2)kBT, the unretarded van der Waals attraction ≈ - 2 scales as EvdW 0.05kBT/dw . Hence, the Helfrich repulsion E ) (2κ + κj)[(c + c )2 + (c - c )2] (13) dominates the van der Waals attraction at all membrane A E I separations, leading to a purely repulsive intermembrane potential. This leads not only to the large dilutions possible in the lamellar with the convention that the curvature of the outer monolayer phase but also to the thermal stabilization of the observed is positive. For symmetric bilayers, cI ) cE and the free energy spontaneous unilamellar vesicle phase. is minimized for c ) 0 (flat bilayers). However, for surfactant However, these observations do not explain why the lamellar mixtures, nonideal mixing in the bilayer could allow the interior phase in DTAC/SOSo is fluid and the bilayers are well correlated and exterior monolayers to have equal and opposite curvatures: whereas those of other catanionic mixtures are viscoelastic, turbid cI )-cE ) c. One way for this to happen is for the average lamellar gels with a high percentage of defects.16,20 The gel- headgroup size of the mixed surfactants on the inside monolayer fluid transition has been explained via a simple defect-based to differ from that of molecules in the outside monolayer. This model that balances the curvature energy of bilayer defects with can be achieved either with a mixture of amphiphiles with widely the defect entropy; the theory predicts that gelation requires a different areas per headgroup or in a mixture in which surfactant spontaneous bilayer curvature and κ ≈ kBT. The relationship complexes form such that the complex has a small area per among the d spacing at the gel-fluid transition, dgel, the headgroup (from the ion pairing of an anionic and cationic spontaneous curvature radius, R0, and κ is given by a balance surfactant). By placing more of the smaller headgroup component between the curvature energy, E, and entropy, TS, of forming (or complex) in the inside monolayer (and more of the larger 17,19 an edge dislocation pair of opposite Burgers vector 2dgel. headgroup component in the exterior monolayer) of the vesicle, The curvature energy per unit length is (from eq 1, with R1 ) the spontaneous curvatures could be adjusted to suit a particular d/2 and R2 ) ∞) composition of the vesicle.5,37 However, for a 1:1 DTAC/SOSo mixture, there is likely not a significant excess of uncomplexed E ) κ 2 - 2 2 components with larger or smaller headgroups to create the ( ) πdgel (10) L 2 dgel R0 concentration difference, as is the case for the typical catanionic vesicle and lamellar gel phases that form with an excess of either positive or negative charged surfactant (Figure 1).16,20 Hence, The entropy per unit length, TS/L, of such a line defect is kBT/êp, the 1:1 DTAC/SOSo mixture is actually more similar to a single- in which êp is the persistence length along the main axis of the defect: component bilayer that cannot have spontaneous curvature.

πd κ Cryo-TEM Evaluation of Bilayer Elastic Constants ê ) gel p k T Simple models in which the spontaneous curvature and the B sum of the bending constants are free parameters can be used 2 to describe most catanionic vesicle size distributions.8,9,14,20 For TS (kBT) ) (11) spherical vesicles with c1 ) c2 ) 1/R (the vesicle radius) and L πd κ 5,37 gel spontaneous curvature c0 ) 1/r0, eq 13 can be simplified to be

These contributions balance at the gel point, which gives the E ) + j 1 - 1 2 17,19 (2κ κ)( ) following relation: A R Ro + j x ) 2κ κ R 2k T Ro ro (14) d ≈ 0(2 - B ) (12) 2κ gel 2 πκ Ro is the radius of the minimum-energy vesicle. The distribution Whereas the numerical values of dgel depend on the particular of surfactant between vesicles is dictated by a balance between defect model chosen, defect formation that leads to the gel- the entropy of vesicle mixing and the curvature energy:8,38 16,17,19,20 fluid transition requires a finite value of R0 and κ ≈ kBT. 2 2 κ - κ + κj R 2 R /Ro For 1:1 DTAC/SOSo, ranges from 0.12kBT to 0.61kBT with ) 4π(2 ) - o decreasing dilution as the d spacing decreases from 9.1 to 5.6 CN {CM exp[ (1 ) ]} (15) kBT R nm; however, the lamellar phase remains fluid and defect-free, hence this suggests that R is large compared to d or that there 0 CM () XM/M) and CN are the molar and number fractions of is no bilayer spontaneous curvature. For the CTAT/SDBS system, vesicles of size R and R, respectively. A consequence of eq 15 ) ≈ o with R0 55 nm and a similar range of κ kBT, a lamellar gel is that vesicles stabilized by thermal fluctuations ((2κ + κj) ≈ phase was formed that transformed to a fluid phase for water kBT) have a much broader size distribution than vesicles stabilized fractions less than 40 wt % at a d spacing of 4.4 nm.20 (36) Safran, S. A.; Pincus, P.; Andelman, D. Science 1990, 248, 354-355. (34) Yatcilla, M. T.; Herrington, K. L.; Brasher, L. L.; Kaler, E. W.; Chruvolu, (37) Safran, S. A.; Pincus, P. A.; Andelman, D.; MacKintosh, F. C. Phys. ReV. S.; Zasadzinski, J. A. J. Phys. Chem. 1996, 100, 5874-5879. A 1991, 43, 1071-1078. (35) Chiruvolu, S.; Israelachvili, J.; Naranjo, E.; Xu, Z.; Kaler, E. W.; (38) Denkov, N. D.; Yoshimura, H.; Kouyama, T.; Walz, J.; Nagayama, K. Zasadzinski, J. A. Langmuir 1995, 11, 4256-4266. Biophys. J. 1998, 74, 1409-1420. 2480 Langmuir, Vol. 22, No. 6, 2006 Coldren et al.

Figure 6. Plots of the vesicle size distributions of eq 17 using the renormalized values of the bending constants and c0 ) 0. Over the entire range of experimentally observed vesicles (20 nm < R < 200 nm), (κ + (κj/2)) varies only from 1.5kBT to 2.0kBT. The larger the mean size of the distribution, the broader the distribution. For (κ + j > Figure 5. Plots of the predicted size distributions from eq 15 (with (κ/2)) 3, the distribution is essentially flat. Compare to the distributions in Figure 5. spontaneous curvature c0 ) 1/R0) demonstrating the effects of bilayer elasticity (κ + (κj/2)) and R0. For larger values of (κ + (κj/2)), (κ + 2 (κj/2)) = kBT/(16π(R0/σ) ), in which σ is the standard deviation of the size distribution. For the smallest values of (κ + (κj/2)), the size distribution becomes more asymmetric and skewed to larger radii and is less dependent on R0. by spontaneous curvature ((2κ + κj) . kBT). Figure 5 shows size distributions predicted by eq 15 for a variety of different parameter values; for the smallest values of (2κ + κj), the distribution is almost independent of R0. In the absence of spontaneous curvature, Herve´ et al. predict that monodisperse vesicles can be achieved only in the limit of 7,25 (κ + (κj/2)) ≈ kBT, the opposite of the prediction of eq 15. With increasing (κ + (κj/2)), the polydispersity and average size grow exponentially. From eq 14, for 1/R0 ≈ 0, the curvature energy per vesicle is a constant, Eves ) 8π(κ + (κj/2)), that is independent of vesicle size. Using the same assumptions in the mass action model used to derive eq 15, the mole fraction of such Figure 7. Histogram of vesicle sizes determined by cryo-TEM. 7,25 vesicles having an aggregation number of n is The fits to eq 15 (with R0 ) 48 ( 10 nm and (κ + (κj/2)) ) (0.12 ( 0.04)kBT) and eq 17 (with (κ + (κj/2)) ) 1.7kBT) give equally good -E /k T n representations of the data. e ves B - X ) 1 - e Eves/kBT (16) n ( x ) Camp R and Rj are taken as 3 and 10/3, respectively.7 Figure 6 shows plots of eq 17 at a total surfactant concentration of 2 vol %; the However, eq 16 yields a most probable vesicle size that is very mean vesicle size and polydispersity are very sensitive to the + j ≈ 39 small, similar to that of a micelle for (κ (κ/2)) kBT. As value of (κ + (κj/2)), and the distribution is skewed toward vesicles + j (κ (κ/2)) increases, the distribution shifts to larger sizes; larger than the mean. Because essentially all catanionic vesicles + j however, regardless of the value of (κ (κ/2)), the distribution observed thus far are between 20 and 150 nm in radius, Figure is extremely polydisperse, and the most probable size is less than 6 suggests that 1.5kBT < (κ + (κj/2)) < 2kBT. Choosing a 10 nm, which is physically unreasonable for bilayer vesicles in renormalization constant of R)1orRj)0 does not significantly - which the membrane thickness is 2 3 nm. alter these findings.39 The results are similar for 0.5% surfactant, 7 Herve´ et al. showed that the renormalization of the elastic with the average vesicle size and polydispersity decreasing constants (eq 9), which makes the elastic energy of a vesicle slightly. decrease logarithmically with increasing size, provides a more Figure 7 shows the vesicle size distribution histogram realistic size distribution in the absence of spontaneous curvature. determined by cryo-TEM8 for an equimolar mixture of DTAC: Equation 16 now becomes SOSo at 0.5% total weight in water. The vesicles are broadly - -E /k T 1/ν+1 n distributed with the most probable size being about 25 30 nm. ves B - ) ν - ν!e Eves/kBT A reasonable fit to the cryo-TEM data is achieved with eq 15 Xn n (1 ( ) ) e (17) Camp with R0 ) 48 ( 10 nm and (κ + (κj/2)) ) (0.12 ( 0.04)kBT. However, there is a significant mismatch between experiment and the renormalization constant is V)1 +R-(Rj/2), where and theory at small vesicle radii. Equation 15 is relatively insensitive to R0 for small values of (κ + (κj/2)), as can be seen (39) Herrington, K. Phase Behavior and Microstructure in Aqueous Mixtures of Oppositely Charged Surfactants. Ph.D. Dissertation, University of Delaware, in Figure 5, so the spontaneous curvature has a large uncertainty. Newark, DE, 1994. Using the renormalized elastic parameters in eq 15 makes the Lamellar Phase Morphology and Vesicle Size Langmuir, Vol. 22, No. 6, 2006 2481 fit worse (data not shown). The size distribution with zero values of κ ≈ kBT, as in all other catanionic surfactant mixtures spontaneous curvature, eq 17, also fits the vesicle histogram data made from hydrogenated surfactants.24,40 The absence of the with (κ + (κj/2)) ) 1.7kBT. On the basis of the fits to the cryo- lamellar gel phase with its spherulite texture and viscoelastic TEM data alone, there is no basis for choosing one model over rheology, as found in other catanionic mixtures,20 requires zero the other. or near-zero spontaneous curvature, which is consistent with the vesicle size distribution. It is also consistent with a positive value Conclusions of κj. A lack of spontaneous curvature is related to the lack of However, the absence of the lamellar gel phase combined an excess of one or the other charged surfactant, in good agreement with the X-ray determination that κ ≈ kBT is consistent only with with Safran et al.’s theory that requires such an excess of one zero spontaneous curvature (large spontaneous radius of curvature, or more components on opposite sides of the bilayer.5,37 A second R0). In contrast, for the CTAT/SDBS system with R0 ) 55 nm conclusion is that the vesicle size distributions found are consistent and a similar range of κ ≈ kBT, a lamellar gel phase was formed with the equilibrium size distribution stabilized by the entropy that did not transform to a fluid phase until the water fractions of mixing and the Helfrich undulation repulsion. There appears was 40 wt % and the d spacing was 4.4 nm.20 The fit of the to be no good thermodynamic argument as to why unilamellar DTAC/SOSo cryo-TEM size distribution to the model that vesicles cannot be equilibrium structures in dilute solution. includes spontaneous curvature (eq 15) gives R0 ) 48 ( 10 nm, which would imply that the DTAC/SOSo system should also Acknowledgment. We thank Hee-Tae Jung, Tomas Zemb, form a lamellar gel phase containing defects at even lower water and Sam Safran for consultations and Eric Kaler and his group fractions than would the CTAT/SDBS system (eq 12). at the University of Delaware for a 20 year collaboration on Because the lamellar phase of DTAC/SOSo is fluid at all d catanionic vesicles. This work was supported by National Science spacings, R0 must be significantly greater than 48 nm; hence, we Foundation grant no. CTS-0436124 and Petroleum Research choose parameters consistent with the zero spontaneous curvature Foundation grant no. 41016-AC7. Use of the National Synchrotron fit to give (κ + (κj/2)) ) 1.7kBT. From the X-ray measurements, Light Source at Brookhaven National Laboratory was supported by the U.S. Department of Energy, Office of Science, Office of κo ) 0.8kBT,soκj)1.8kBT. (Using the values of κ obtained at higher dilution makes κj more positive.) A positive value of κj Basic Energy Sciences, under contract no. DE-AC02-98CH10886. also acts to eliminate the spherulite defects in the lamellar gel LA052448P phase,20 which have positive curvature. The vesicle size distribution and X-ray line shapes of equimolar DTAC/SOSo (40) Szleifer, I.; Benshaul, A.; Gelbart, W. M. J. Phys. Chem. 1990, 94, 5081- mixtures are consistent with surfactant mixing leading to low 5089.