Flexibility of Elongated Sodium Dodecyl Sulfate Micelles in Aqueous

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Langmuir 2000, 16, 10028-10036

Flexibility of Elongated Sodium Dodecyl Sulfate Micelles in Aqueous Sodium Chloride: A Small-Angle Neutron Scattering Study L. J. Magid* and Z. Li Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600

P. D. Butler NCNR, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8562 Received April 28, 2000. In Final Form: September 20, 2000 Aqueous salt solutions containing elongated micelles of sodium dodecyl sulfate (SDS) have been studied at 45 °C using small-angle neutron scattering. At NaCl concentrations in the 1-2 M range, the scattering data are consistent with semiflexible rather than rigid rodlike micelles, and this conclusion is consistent with previous studies of SDS micelles using small-angle scattering. However, until now quantitative determinations of micellar flexibility, that is, of persistence lengths, especially as a function of ionic strength, have not been available. By making measurements in the range of scattering vectors appropriate for this length scale and by incorporating the effect of intermicellar interactions into the fitting protocol, it has been possible to determine persistence lengths accurately. Partitioning of the values into intrinsic and electrostatic components is discussed, and the results are compared with data for the highly flexible anionic polyelectrolyte sodium polystyrenesulfonate. The one-dimensional bending moduli for the flexible SDS micelles are compared to the analogous two-dimensional bending moduli for SDS bilayers.

Background and Motivation. In the mid-1980s, certain similarities between polymer solutions and solutions containing elongated micelles (the micelle-polymer analogy) were first recognized in studies of cationic surfactants.1 This recognition has stimulated substantial interest in the relationship between micellar microstructure and macroscopic solution properties, particularly behavior under flow and the associated formation of shearinduced structures in bulk solution and near solid surfaces.2-8 This in turn naturally leads to an interest for elongated micelles in the extent of micellar flexibility and its dependence on surfactant and counterion structures, surfactant chain length, ionic strength, and so forth. It can also be useful to attempt to relate measured 1D bending moduli (flexible micelles have small moduli) to the 2D bending moduli determined for surfactant membranes in lamellar systems. The recent work of Pedersen and Schurtenberger9 has provided the robust scattering functions for semiflexible wormlike chains needed to fit small-angle scattering data over ranges of the scattering vector Q that are sensitive to micellar contour lengths and persistence lengths as

well as to local cross-sectional structure. They have successfully applied their methods to reverse lecithin micelles in hydrocarbons10 and to nonionic and mixed nonionic/ionic surfactants in water.11 More recently, we have applied them to analyze the effect of counterion specificity on micellar size and flexibility in a cationic micellar system containing halide as well as aromatic counterions.12 Our work is now being extended to investigate the impact of surfactant headgroup structure and ionic strength on the flexibility of wormlike micelles formed in aqueous solutions by other cationic surfactants as well as anionic surfactants.13 The work reported here focuses on the growth and increasing flexibility, that is, decreasing persistence length, of SDS micelles in water as the concentration of supporting electrolyte, NaCl, increases. There is significant disagreement in the literature concerning both the magnitude of persistence lengths, lp’s, for SDS micelles and their dependence on salt.14-17 To determine lp’s directly, it is necessary to access a Q range of order lp-1, which means small-angle neutron or smallangle X-ray scattering data are needed. Analysis of the small-angle neutron scattering (SANS) data reported here

(1) Candau, S. J.; Hirsch, E.; Zana, R. J. Phys. Fr. 1984, 45, 1263; J. Colloid Interface Sci. 1985, 105, 521. (2) Cates, M. W.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. (3) Magid, L. J. J. Phys. Chem. B 1998, 102, 4064. (4) Butler, P. Curr. Opin. Colloid Interface Sci. 1999, 4, 214. (5) Hamilton, W. A.; Butler, P. D.; Hayter, J. B.; Magid, L. J.; Kreke, P. J. Physica B 1996, 221, 309. (6) Keller, S. L.; Boltenhagen, P.; Pine, D. J.; Zasadzinski, J. A. Phys. Rev. Lett. 1998, 80, 2725. (7) Hamilton, W. A.; Butler, P. D.; Magid, L. J.; Han, Z.; Slawecki, T. M. Phys. Rev. E 1999, 60, R1146. (8) Gamez-Corrales, R.; Berret, J.-F.; Walker, L. M.; Oberdisse, J. Langmuir 1999, 15, 6755. (9) (a) Pedersen, J. S.; Schurtenberger, P. Macromolecules 1996, 29, 7602. (b) Pedersen, J. S.; Laso, M.; Schurtenberger, P. Phys. Rev. E 1996, 54, R5917.

(10) Jerke, G.; Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, S. Phys. Rev. E 1997, 56, 5772. (11) Jerke, G.; Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. Langmuir 1998, 14, 6013. (12) Magid, L. J.; Han, Z.; Li, Z.; Butler, P. D. Langmuir 2000, 16, 149; J. Phys. Chem. B, in press. (13) A portion of this work was presented at the 216th ACS National Meeting, Boston, MA, Aug 1998, and in May 1999 at the XIth International Conference on Small-Angle Scattering, Brookhaven National Laboratory, Upton, NY. (14) Mazer, N. A. In Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy; Pecora, R., Ed.; Plenus Press: New York, 1985; Chapter 8. (15) Mishic, J. R.; Fisch, M. R. J. Chem. Phys. 1990, 92, 3222. (16) Mishic, J. R.; Nash, R. J.; Fisch, M. R. Langmuir 1990, 6, 915. (17) Bergstro¨m, M.; Pedersen, J. S. Phys. Chem. Chem. Phys. 1999, 1, 4437.

Introduction

10.1021/la0006216 CCC: $19.00 © 2000 American Chemical Society Published on Web 11/21/2000

Flexibility of Elongated SDS Micelles in NaCl(aq)

for SDS micelles at 45 °C in aqueous solutions at various concentrations of NaCl (1, 1.2, 1.5, and 2 M) resolves these discrepancies and provides a coherent view of increasing micellar size and flexibility for SDS with increasing ionic strength. The dependence of measured lp’s on ionic strength that we find is rather similar to that observed for quenched polyelectrolytes such as sodium polystyrenesulfonate.18 To our knowledge, our work constitutes the first study that sucessfully incorporates the effect of intermicellar interactions into least-squares fitting of the small-angle scattering data for wormlike micelles at finite Q values. Case of SDS. The direct micelles formed in water and in aqueous salt solutions by the anionic surfactant sodium dodecyl sulfate [CH3(CH2)11OSO3-Na+] have been the subject of numerous studies designed to determine micellar size (weight- or number-aggregation numbers, 〈n〉w or 〈n〉n), transitions in micellar shape from globular to elongated, and intermicellar interactions. The major techniques used included static light scattering by Huisman and co-workers19 in the 1960s; early applications of dynamic light scatttering (DLS) to micelles by McQueen and Hermans,20 by Benedek, Mazer, and co-workers21 and by Corti and Degiorgio22 in the 1970s and early 1980s; the use of small-angle scattering (with neutrons,23,24,25 and/or X-rays, SAXS26) starting in the 1980s; and the use of fluorescence quenching techniques.27,28 SANS measurements using both internal and external contrast variation established that the interiors of globular dodecyl sulfate micelles are basically water-free.25,29 For the globular SDS micelles in water, 〈n〉w’s in the range of 70 to 90 are found using scattering techniques. For example, the value is 88 for 70 mM SDS in D2O at 40 °C;24 the corresponding micellar radius is ∼2.3 nm, somewhat larger than the extended length, 1.67 nm, of a C12 chain. As NaCl is added, micellar growth is initially modest and then more rapid once [NaCl] reaches 0.4 M at 25 °C and 0.5 M at 40 °C, as assessed by DLS.22 Using time-resolved fluorescence quenching (TRFQ), which measures number-average aggregation numbers, the onset of rapid growth is found at 0.45 M NaCl for 70 mM SDS at 40 °C: 〈n〉n ) 63 in the absence of NaCl, 104 at 0.3 M NaCl, 134 at 0.45 M NaCl, and 373 at 0.75 M NaCl.27 Thus, the slope of a plot of 〈n〉 versus [NaCl] equals 200 in the interval from 0.3 to 0.45 M NaCl; from 0.45 to 0.75 M NaCl, it increases to 800. Almgren and co-workers28 also found significant micellar growth for SDS in 0.8 M NaCl using SANS: for 30 mM SDS, an aggregation number of 810 at 40 °C can be interpolated from the measured values at 35 °C (980) and 45 °C (650). Using (18) Eisenberg, H. Acta Polym. 1998, 49, 534. (19) Huisman, H. F. Proc. Kon. Ned. Akad. Wet. Ser. B 1964, 67, 367, 376, 388, 407. (20) McQueen, D.; Hermans, J. J. Colloid Interface Sci. 1975, 39, 358. (21) (a) Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1976, 80, 1075. (b) Young, C. Y.; Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1978, 82, 1375. (c) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 84, 1044. (22) Corti, M.; Degiorgio, V. J. Phys. Chem. 1981, 85, 711. (23) See Chen, S.-H. Annu. Rev. Phys. Chem. 1986, 37, 351 and references therein. (24) Hayter, J. B.; Penfold, J. Colloid Polym. Sci. 1983, 261, 1022. (25) Cabane, B.; Duplessix, R.; Zemb, T. J. Phys. Fr. 1985, 46, 2161. (26) Zemb, T.; Charpin, P. J. Phys. Fr. 1985, 46, 249. (27) Almgren, M.; Lo¨froth, J.-E. J. Colloid Interface Sci. 1981, 81, 486. (28) Almgren, M.; Gimel, J. C.; Wang, K.; Karlsson, G.; Edwards, K.; Brown, W.; Mortensen, K. J. Colloid Interface Sci. 1998, 202, 222. (29) Bendedouch, D.; Chen, S.-H.; Koehler, W. C. J. Phys. Chem. 1983, 87, 153. These authors studied lithium dodecyl sulfate micelles, which show less tendency for micellar growth than SDS micelles.

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SANS, strong micellar growth was assessed to occur starting at 0.5 M NaBr for 35 mM SDS at 40 °C: the slope increased from 120 in the interval from 0.2 to 0.5 M NaBr to ∼5200 in the interval from 0.5 to 0.7 M NaBr.17 SDS micellar growth is primarily unidimensional. The growth can be modeled as a transition from spheres to rodlike micelles (prolate spherocylinders with a cylindrical body and hemispherical end-caps) with monomers being incorporated into the cylindrical body. Definition of a thermodynamic parameter related to the difference in chemical potentials for monomers in the two different environments, body and end-caps, allows calculation of SDS micellar size distributions as a function of surfactant and salt concentrations and temperature that agree well with the experimental results from scattering (at NaCl concentrations up to 0.8 M).21 Because the chemical potential difference is assumed to be independent of aggregation number, the model is generally called a ladder model.21c It predicts the classic dependence of average aggregation number on the square-root of surfactant concentration.30 By combining static and dynamic light scattering measurements, Mazer and co-workers found that SDS micelles at NaCl concentrations of 0.8-2.0 M over the temperature interval 20-60 °C have ratios of radii of gyration, 〈Rg〉, to hydrodynamic radii, 〈Rh〉, that are smaller than the ratios expected for polydisperse rigid rods.14 To reconcile this, micellar flexibility was considered; over the entire range of salt concentrations and temperatures, the ratios were consistent with semiflexible cylinders having a persistence length, lp, of 70 nm. They noted that for the cationic micelles of cetylpyridinium bromide (CPyBr, C16 tails) in aqueous NaBr (0.4-0.6 M), lp, determined by Appell and Porte31 using light scattering, was only 25 nm. Longer alkyl tails produce larger micellar radii, which should have resulted in a larger intrinsic persistence length for the C16 surfactant.32 The electrostatic contribution to lp depends on ionic strength, decreasing as the Debye screening length decreases with increasing salt. Thus, the total persistence length should decrease with increasing salt concentration, a feature not observed for either SDS or CPyBr. In 1990, Fisch and co-workers15,16 published a reexamination of the dependence of SDS micellar lp’s on temperature and salt concentration (ionic strength). They performed both static and dynamic light scattering measurements, and they used the Yamakawa and Fujii formulation to relate the dependence of 〈Rg〉/〈Rh〉 on 〈Rh〉/d (d is the micellar diameter) at various values of lp/d. Their results contradict those of Mazer and co-workers: they found a very strong dependence of lp on ionic strength, not consistent with any theory currently available from the polyelectrolyte literature. At 2 M NaCl, the micellar persistence length was found to be less than the micellar diameter, which they took as 6 nm. At 1 M NaCl, values on the order of hundreds of nanometers were observed. For completeness, we note that careful evaluation of SANS data at large Q suggests that large SDS micelles in fact have an elliptical rather than a circular crosssectional geometry.17 However, since this has essentially no impact on evaluation of micellar flexibility (vide supra), we ignore this feature. (30) Mukerjee, P. J. Phys. Chem. 1972, 76, 565. (31) Appell, J.; Porte, G. J. Colloid Interface Sci. 1981, 81, 85. (32) May, S.; Bohbot, Y.; Ben-Shaul, A. J. Phys. Chem. B 1997, 101, 8648.

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Materials and Methods Materials. Sodium dodecyl sulfate (SDS), >99% purity, was obtained from British Drug House (BDH)33 and was used as received. D2O, with an atom fraction of 99.9% D, was obtained from Aldrich or Isotec. AR grade NaCl was used as received. Small-Angle Neutron Scattering. SANS measurements were performed on the NG-7 SANS spectrometer34 at the Cold Neutron Research Facility, National Institute of Standards and Technology, Gaithersburg, MD. The range of scattering vector, Q, used was 0.019-1.2 nm-1. Q is defined as (4π/λ) sin θ, with 2θ denoting the scattering angle and λ the wavelength of the incident neutrons. The samples were contained in quartz spectrophotometric cells of 2 or 5 mm path length, mounted in a thermostated cell holder held at 45.0 ( 0.1 °C. The temperature was chosen to exceed the Krafft temperature for SDS at the highest NaCl concentration investigated, 2 M. Surfactant concentration is denoted by c, and salt concentration by cs. Each two-dimensional set of raw scattering data was corrected for detector background and sensitivity and for scattering from the empty cell and was then radially averaged. The resulting I(Q)’s were converted to absolute intensities (in cm-1) using precalibrated secondary standards provided by NIST. The incident neutron wavelengths used were 0.5 or 1.0 nm; ∆λ/λ was 0.1. Since the scattering curves (I(Q) versus Q) contain no pronounced maxima or minima over the Q range to be fit, these wavelength spreads are small enough so that the impact of instrumental resolution can be neglected. However, this point was checked experimentally for a few scattering curves by including in the weighted nonlinear least-squares fitting routines the option of convoluting the instrumental resolution functions into the computed I(Q,c)’s.35 Evaluation of Local Micellar Structure. The intermediate Q range in a bending rod (BR) or Holtzer plot36-38 can be analyzed to obtain the cross-sectional radius of gyration, Rg,cs, of the wormlike micelles, by employing a Guinier approximation for the scattering form factor of the cross section:

Q‚I(Q) ) KBR exp(-Q2Rg,cs2/2)

(1)

For a circular cross section, the micellar radius, rcs ) x2Rg,cs. The value of KBR is given by

KBR ) πc〈N/L〉w(bm - VmFs)2

(2)

where bm and Vm are respectively the sum of the neutron scattering lengths and the volume per surfactant monomer in the micelle; Fs is the scattering length density of the solvent. In what follows, the weight-average aggregation number (Nw or just N) per unit length, L, will simply be denoted as N/L. The value of N/L is derived from KBR, given a value for the contrast term, (bm - VmFs)2, and the area per surfactant headgroup, Ahg, equals 2πrcsL/N. Analysis of the Full Scattering Curves. In static scattering, the excess scattered intensity due to the micelles in the absence of intermicellar interactions is given by

I(Q,c) ) KcMw〈P(Q,c)〉 + BD

(3)

K is the usual collection of constants, equal in the case of SANS to [u(Fm - Fs)]2/NA. The specific micellar volume, u, is expressed in cm3/g, and the scattering length densities for micelles and solvent, Fm and Fs, are expressed in cm-2. The Q-independent background term BD accounts for residual scattering from solvent (33) Identification of certain equipment or materials does not imply recommendation by NIST. (34) Details of the instrument may be found at http://rrdjazz.nist.gov/ chrnsans.html. (35) (a) Pedersen, J. S. Adv. Colloid Interface Sci. 1997, 70, 171. (b) Barker, J. G.; Pedersen, J. S. J. Appl. Crystallogr. 1995, 28, 105. (36) Casassa, E. F. J. Chem. Phys. 1955, 23, 596. (37) Holtzer, A. J. Polym. Sci. 1955, 17, 433. (38) (a) Schmidt, M.; Paradossi, G.; Burchard, W. Makromol. Chem. Rapid Commun. 1985, 6, 767. (b) Denkinger, P.; Burchard, W. J. Polym. Sci. B: Polym. Phys. 1991, 29, 589.

and/or incoherent background and is typically e0.06 cm-1. 〈P(Q,c)〉 is the form factor averaged over the micellar molecular weight distribution. The impact of micellar flexibility on the scattering curves is apparent when they are presented as bending rod plots. Provided the average contour length exceeds the Kuhn length, equal to twice the persistence length, 2lp, a maximum appears in the plot.38 The three length scales (L, lp, and rcs) being reasonably well-separated in these micelles, it is reasonable to decouple the form factor of the micellar cross section, Pcs, from that of the wormlike chain, Pwc,39 so that the form factor needed in eq 3 becomes

〈P(Q,c)〉 ) Pcs(Q,rcs)〈Pwc(Q,L(c),lp)〉

(4)

The micelles are considered to be solid cylinders, comprised of surfactant tails, so that Pcs is simply [2J1(Qrcs)/(Qrcs)]2. The headgroup region is assumed to be “invisible”, having essentially the same scattering length density as the solvent, as found previously in SANS measurements on globular dodecyl sulfate micelles.25,29 A wormlike chain form factor Pwc for semiflexible chains with intrachain excluded-volume interactions was used: the parametrized expression for Pwc was recently derived by Pedersen and Schurtenberger9 on the basis of a series of Monte Carlo simulations. Polydispersity in lengths was included by computing the average 〈Pwc(Q,L(c),lp)〉, using a Schulz-Zimm distribution40 for N(L) with the polydispersity fixed at Mw/Mn ) 2 (hence z ) 1), as suggested by the ladder model:

〈Pwc(Q,L(c),lp)〉 )

∫N(L)L P 2

wc(Q,L(c),lp)

∫

dL/ N(L)L2 dL (5)

Micellar radii of gyration, 〈Rg2〉1/2, are computed according to

Rg2(L,lp) ) Rs2(L,lp)lp2〈(L/3lp) - 1 + (2lp/L) - (2lp2/L2)[1 exp(-L/lp)]〉 (6)

∫

∫

〈Rg2〉1/2 ) [ N(L)L2Rg2(L,lp) dL/ N(L)L2 dL]1/2 where Rs is the expansion factor accounting for the effect of intrachain excluded volume. Weighted (by the reciprocal of the square of the statistical error of the individual points) nonlinear least-squares fits of the scattering curves to eq 3 were performed. Fixing or varying the background has little effect on the overall quality of the fits, other than to increase somewhat the error bars on rcs when it is varied. Fixed backgrounds do produce fits that converge more rapidly and sensibly. Cross-sectional radii were varied (the usual case) or fixed; typically fixing rcs was required only for the elongated micelles of low aspect ratio, that is, those in solutions containing 1 M NaCl. The fitting parameters were 〈L〉n, lp, BD, and A, equivalent to KcMw. Because the Q-dependent parameters (rcs, 〈L〉n, and lp) can show significant correlation, a simple grid search method35a was used in the fitting in order to ensure that a global optimization (minimization of χ2) was reached: the parameters were repeatedly optimized one by one. As already observed by Schurtenberger, Pedersen, and coworkers,10,11 neglecting intermicellar interactions leads to an apparent dependence on surfactant concentration of the fitted values of lpsthey increase with c. In the present work, fits incorporating interactions succeeded only when lp was constrained to its extrapolated value at c ) 0. In that case, eq 3 is replaced by

I(Q,c) ) KcMwSRPA(Q,c) + BD

(7)

(39) Pedersen, J. S.; Schurtenberger, P. J. Appl. Crystallogr. 1996, 29, 646. (40) (a) Schulz, G. V. Z. Phys. Chem. Abt. B 1939, 43, 25. (b) Zimm, B. H. J. Chem. Phys. 1948, 16, 1099.

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Pedersen and Schurtenberger41 found that fits at finite Q to the results of their Monte Carlo simulations on chain conformations in many-chain systems suggest the following form for the full structure factor in the random-phase approximation, SRPA(Q,c):

SRPA(Q,c) ) S1(Q,c)/{1 + [S(0,c)-1 - 1]fD(Q2R h g2)}

(8)

fD(x) ) 2(e-x - 1 + x)/x2 S1(Q,c) is the single-chain scattering function, given by eqs 4 and 5 for the wormlike micelles. An appropriate expression for S(0,c) comes from applying the renormalization group theory (RGT):42

S(0,c)-1 ) 1 + 1/8[9X - 2 + 2 ln(1 + X)/X] exp{1/4[1/X + (1 - 1/X2) ln(1 + X)]} (9) X ) (cA2Mw)/(9/16 - ln(Mw/Mn)/8) where A2 is the second virial coefficient given by 4π3/2NA[〈Rg2〉3/2/ Mw2]ψ, with ψ as the penetration factor.43 The reduced concentration X ≈ c/c*, where c* is the concentration at which micellar overlap begins. Weighted nonlinear least-squares fits were performed with values for BD and rcs fixed to those obtained when interactions were neglected; the persistence length was fixed at lp,0, the value extrapolated to zero surfactant concentration. This leaves 〈L〉n and A as the only adjustable parameters. The grid-search method was used. As fitting proceeded, the values of Mw and 〈Rg2〉 needed for A2 were computed from the fitted 〈L〉n and the values of N/L previously obtained by evaluation of the local micellar structure.

Results and Discussion SANS curves were obtained for SDS micelles in D2O at NaCl concentrations of 1, 1.2, 1.5, and 2 M. Below 1 M NaCl at 45 °C, the micelles are too small for the scattering curves to show clear evidence of flexibility. Surfactant concentrations ranged from 1.4 to 30 mM; the range at a given NaCl concentration was chosen so that most of the solutions were dilute rather than semidilute. In Figure 1a the coincidence of scattering curves at Q > 0.1 nm-1 illustrates that for 4.8 mM SDS the cross-sectional micellar structure is basically constant as the NaCl content increases. This was also observed at the other SDS concentrations studied. The lower Q region makes obvious the significant micellar growth, via increasing contour length, that occurs at the same time. The same data are presented in Figure 1b as bending rod plots. At 1 M NaCl, there are evidently approximately two persistence lengths (one Kuhn length) per micellar contour length, and the curve shows a weak maximum at a Q of 0.085 nm-1, consistent with an apparent Rg of ∼21 nm (QmaxRg ≈ 1.78 when Mw/Mn ) 2). This agrees well with the apparent Rg of 21.7 nm obtained using eq 6 and the average contour and persistence lengths obtained from the least-squares fit. As the NaCl content increases, the maximum moves to smaller Q and gets much more pronounced, reflecting the increasing contour lengths of the micelles and the increasing number of persistence lengths per micelle. By 2 M NaCl, the micelles are so large that the maximum moves to a Q too small to be accessible. Local Micellar Structure. Figure 2 shows the scattering curves for SDS in 1 M NaCl, plotted as ln[Q‚I(Q)] versus Q2 and fitted in the region 0.15 nm-2 < Q < 1 nm-2 (41) (a) Schurtenberger, P.; Jerke, G.; Egelhaaf, S. U.; Pedersen, J. S. Presented at the 71st ACS Colloid and Surface Science Symposium, Newark, DE, 6/97. (b) Pedersen, J. S.; Schurtenberger, P. S. Europhys. Lett. 1999, 45, 666. (42) Ohta, T.; Oono, Y. Phys. Lett. 1982, 89A, 460. (43) The penetration factor, Ψ, can be obtained from: Huber, K.; Stockmayer, W. H. Macromolecules 1987, 20, 1400.

Figure 1. (a) SANS data, I(Q) versus Q, for the semiflexible micelles in 4.8 mM SDS at 45 °C and NaCl concentrations of 1 M (O); 1.2 M (0); 1.5 M (3); and 2.0 M (]). (b) SANS data of part a, presented as Holtzer (≡ bending rod, BR) plots. Symbols as in part a. The increase in micelle size with increasing cs is evident: at Mw/Mn ) 2, the Holtzer plot’s peak (ref 38) occurs at QRg ≈ 1.78.

Figure 2. Guinier-like plot of the SANS data in the intermediate-Q region for SDS in 1 M NaCl, fitted according to eq 1. Surfactant concentrations: 17 mM (O); 24 mM (0); 30 mM (3).

according to eq 1. Table 1 reports the resulting micellar cross-sectional radii, derived from the curves’ slopes. For the micelles in 1 M NaCl, the mean value is 1.66 nm; values between 1.72 and 1.89 nm are found at the three

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Table 1. Evaluation of Local Micellar Structure c (mM)

cs (M)

17.0 24.0 30.0

1.0 1.0 1.0

3.5 4.8 6.2 7.8 12.0 15.0

(bm - VmFs)2 (10-22 cm2)a,b

N/Lc (nm-1)

rcs (nm)

Ahg (nm2)

SDS in 1.0 M NaCl 5.45 23.3 5.45 23.8 5.45 23.5 Mean: 23.5

1.64 1.67 1.67 1.66

0.452 0.451 0.457

1.2 1.2 1.2 1.2 1.2 1.2

SDS in 1.2 M NaCl 5.43 26.2 5.43 24.5 5.43 26.6 5.43 26.7 5.43 26.9 5.43 27.3 Mean: 26.4

1.74 1.72 1.74 1.76 1.75 1.77 1.75

0.424 0.447 0.417 0.421 0.415 0.412

3.0 3.8 4.8 6.2 7.8 10.0

1.5 1.5 1.5 1.5 1.5 1.5

SDS in 1.5 M NaCl 5.40 24.5 5.40 24.6 5.40 25.0 5.40 26.0 5.40 26.2 5.40 25.9 Mean: 25.4

1.77 1.77 1.80 1.77 1.82 1.76 1.78

0.450 0.448 0.448 0.427 0.433 0.420

1.4 3.2 4.8 6.2 7.8

2.0 2.0 2.0 2.0 2.0

SDS in 2.0 M NaCl 5.36 21.8 5.36 22.5 5.36 19.3 5.36 21.1 5.36 23.7 Mean: 21.7

1.89 1.89 1.88 1.83 1.78 1.85

0.541 0.526 0.599 0.547 0.467

a The values of b and V , respectively, for the dodecyl chains m m are -1.37 × 10-12 cm and 3.50 × 10-22 cm3. b The contributions of NaCl to Fs are taken into account. c Error bars for N/L and rcs are (3%.

higher NaCl concentrations. However, fits of the full scattering curves, vide infra, give values for the radius close to the length, 1.67 nm, of an extended C12 hydrocarbon chain at all NaCl concentrations. Using micellar contrast terms computed only on the basis of the contrast between the hydrocarbon chains and the solvents, values of N/L were derived from the plots’ intercepts (KBR) using eq 2; the resulting N/L’s and the derived Ahg’s are tabulated in Table 1. Although there is considerable scatter in the mean values of N/L at the four NaCl concentrations studied, a systematic dependence on salt content is absent. Thus, the results suggest that the local packing of surfactant molecules,that is, the local micellar structure, is independent of NaCl concentration over the range from 1 to 2 M NaCl. If the contrast between micelles and solvent is instead chosen to include the sulfate headgroups and the bound sodium ions, then the contrast term increases to ∼6.2 × 10-22 cm2, and the mean N/L declines by ∼10%. The overall average, for all SDS and NaCl concentrations studied, becomes 21.1 monomers per nanometer, and the derived headgroup areas are in the range from 0.47 to 0.63 nm2. The micellar scattering length density, Fm, is -0.39 × 1010 and 0.38 × 1010 cm-2, respectively, without and with the headgroups included in the contrast. A value of Fm close to 0.38 × 1010 cm-2 is also recovered from the analysis of the fitting parameter A, vide infra. Bergstro¨m and Pedersen17 estimated a similar value of Fm for elongated SDS micelles in their recent work. Overall Micellar Size and Flexibility. The full scattering curves were first fit using eqs 3-5, that is, without taking intermicellar interactions into account. Table 2 tabulates the results; Figure 3 shows the fits to the scattering curves for SDS in 1 M NaCl. Figure 4, in which the observed concentration-dependent increases in

micellar contour lengths at all four NaCl concentrations are plotted, makes clear the large impact of increasing NaCl concentration on micellar size at all surfactant concentrations, c, investigated. With increasing salt content, the exponent in the scaling relationship, 〈Ln〉 ≈ cR, increases as follows: 0.15 at 1 M NaCl; 0.31 at 1.2 M NaCl; 0.55 at 1.5 M NaCl; 0.88 at 2 M NaCl. The value at 1 M NaCl is consistent with an exponent of 0.13 for SDS in 0.8 M NaCl, derived from the data of Almgren and co-workers.28 Other micellar systems, for example nonionic surfactants44 in water, cationic surfactants in aqueous salt solutions,12 and lecithin reverse micelles10 in organic solvents, have also shown growth law exponents that deviate strongly from the classical (mean-field) value of 0.5. The fitted micellar persistence lengths (Table 2 and Figure 5) increase with increasing surfactant concentration. The values obtained for SDS in 1 M NaCl agree rather well with those obtained recently by Bergstro¨m and Pedersen,17 but our fitted contour lengths are much smaller. They estimate the micelles to be hundreds of nanometers in length; our values of 40 to 45 nm are more consistent with the fitted lengths we observe at 1.2 and 1.5 M NaCl, as well as with the values of Almgren and co-workers28 for SDS micelles in 0.8 M NaCl: 31 nm at 30 mM SDS. As demonstrated by Schurtenberger and co-workers, the increases in persistence lengths with c are apparent: they result from the neglect of intermicellar interactions rather than from a real physical stiffening of the semiflexible, that is, wormlike, micelles. Neglecting interactions also produces fitted micellar contour lengths that are smaller than those obtained with interactions incorporated, since in the former case S(Q,c) is constrained to equal one at all values of Q. To take the interactions into account, we attempted to fit the scattering data using eqs 7-9. Because the scattering curves are sensitive to intermicellar interactions as well as to contour and persistence lengths at low Q (∼Q < 0.2 nm-1 for the micelles studied heressee Figure 1b), some constraints on the fitting parameters had to be incorporated.10,11 At each surfactant concentration, lp was set equal to the c ) 0 extrapolated value, lp,0. The micellar cross-sectional radii and background term were also fixed (vide supra), leaving 〈Ln〉 and A as the only adjustable parameters. Results of the fits for SDS in 1.2 and 1.5 M NaCl are found in Table 2 and in Figures 6 and 7. As fitting proceeds, an updated value for the reduced concentration, X (eq 9), is derived from Mw, which is in turn derived from the fitting parameter 〈Ln〉. Once X is ∼0.5, that is, the surfactant concentration is about half of c* and S(0) is of order 0.3-0.5, the fitted micelle size apparently undergoes a rapid increase. See for example the results in Table 2 for 12 mM SDS in 1.2 M NaCl and for 6.2 mM SDS in 1.5 M NaCl. We do not regard this as a real acceleration of micellar growth, since there are no marked changes in the energetics of micellization associated with a surfactant concentration of 0.5c*. The apparent rapid increase is probably due instead to the approximations used for the intermicellar interactions beginning to fail as c* is approached, and to the challenge presented by the occurrence of multiple Q-dependent factors at low Q, namely intermicellar interactions, contour lengths, and persistence lengths. As far as we are aware, the results presented in Figures 6 and 7 in fact represent the first instances of successful (44) Schurtenberger, P.; Cavaco, C.; Tiberg, F.; Regev, O. Langmuir 1996, 12, 2894.

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Langmuir, Vol. 16, No. 26, 2000 10033

Table 2. Fitted Parameters for SDS Micelles in Aqueous NaCl interactions neglected c (mM)

cs (M)

〈L〉n (nm)

2lp (nm)

interactions included; fixed 2lp rcs (nm)

BD

17.0 17.0 24.0 24.0 30.0 30.0

1.0 1.0 1.0 1.0 1.0 1.0

40.4 ( 0.4 41.5 ( 0.4 42.9 ( 0.4 43.7 ( 0.4 43.8 ( 0.4 44.7 ( 0.4

37.1 ( 1.0 36.7 ( 1.0 40.2 ( 1.3 39.7 ( 1.1 41.4 ( 1.1 40.7 ( 1.1

SDS in 1.0 M NaCl 1.27 ( 0.02 0.054 1.68 0.115 ( 0.002 1.24 ( 0.02 0.054 1.68 0.145 ( 0.003 1.20 ( 0.02 0.054 1.68 0.173 ( 0.003

3.5 4.8 6.2 7.8 12.0 15.0

1.2 1.2 1.2 1.2 1.2 1.2

65.4 ( 1.6 81.0 ( 1.7 82.6 ( 1.5 84.8 ( 1.4 101 ( 2 108 ( 2

34.8 ( 1.2 31.5 ( 0.8 30.6 ( 0.7 32.5 ( 0.6 34.4 ( 0.7 32.6 ( 0.7

SDS in 1.2 M NaCl 1.44 ( 0.03 0.054 1.48 ( 0.03 “ 1.49 ( 0.02 “ 1.57 ( 0.02 “ 1.56 ( 0.02 “ 1.56 ( 0.02 “

3.0 3.8 4.8 6.2 7.8 10.0

1.5 1.5 1.5 1.5 1.5 1.5

152 ( 4 145 ( 3 163 ( 4 221 ( 6 238 ( 7 219 ( 6

29.6 ( 0.7 29.4 ( 0.5 33.4 ( 0.6 31.4 ( 0.4 32.6 ( 0.4 33.5 ( 0.5

SDS in 1.5 M NaCl 1.63 ( 0.03 0.061 1.64 ( 0.02 0.061 1.60 ( 0.02 0.061 1.63 ( 0.02 0.061 1.68 ( 0.01 0.061 1.64 ( 0.01 0.061

1.4 2.0 3.2 4.8 6.2 7.8

2.0 2.0 2.0 2.0 2.0 2.0

156 ( 5 189 ( 7 337 ( 2 510 ( 3 516 ( 7 678 ( 49

21.8 ( 0.8 20.3 ( 0.7 25.8 ( 0.5 26.8 ( 0.3 29.5 ( 0.4 29.6 ( 0.3

SDS in 2.0 M NaCl 1.56 ( 0.06 0.061 1.66 ( 0.06 0.061 1.65 ( 0.03 0.061 1.68 ( 0.02 0.061 1.66 ( 0.02 0.061 1.64 ( 0.02 0.061

Figure 3. Bending rod plots of SANS data for SDS micelles in 1 M NaCl at 45 °C, and the corresponding fits using eqs 3-5: semiflexible micelles, neglecting intermicellar interactions. Surfactant concentrations: 17 mM (O); 24 mM (0); 30 mM (3).

least-squares fits for SANS data from interacting wormlike micelles. To reproduce the scattering curves at higher values of c/c*, the approach of Schurtenberger and coworkers10 could be used, calculating scattering curves using eqs 7-9 with fixed values for all the micellar parameters. For example, 〈Ln〉’s at higher c’s would be estimated from the micellar growth laws obtained from fits at c’s where c/c* is less than half. However, since pointby-point agreement between experimental and calculated scattering curves is often no better than 10%, we have not done this.10,11 The values tabulated in Table 2 for the adjustable parameter A, equivalent to KcMw, can be analyzed to obtain values for K at each surfactant concentration. Provided that the scattered intensities have been converted properly to the absolute scale and that the scattering function used (eq 8) properly accounts for the

X

S(0)

A

〈L〉n (nm)

0.127 0.231 0.331 0.490 1.465

0.871 0.784 0.713 0.620 0.325

3.97 ( 0.09 7.05 ( 0.18 11.2 ( 0.3 17.1 ( 0.4 61.3 ( 0.3

75.2 ( 1.8 103 ( 3 116 ( 3 140 ( 4 321 ( 17

0.201 0.262 0.355 0.781

0.807 0.760 0.697 0.495

8.03 ( 0.21 10.6 ( 0.2 14.5 ( 0.4 39.2 ( 1.5

179 ( 5 186 ( 4 203 ( 6 404 ( 16

Figure 4. Dependence of the fitted (eqs 3-5) number average contour lengths on SDS concentration at all four salt concentrations. Values of cs: 1 M (O); 1.2 M (0); 1.5 M (3); 2.0 M (]).

intermicellar interactions, then K should be a true constant from which the square of the contrast factor, (Fm - Fs)2, for the SDS micelles in D2O can be obtained. For the specific volume of the SDS micelles,45 we use 0.86 cm3/g. Values for Mw are derived from the fitted 〈L〉n’s; the mean value for N/L discussed earlier, 21.1 monomers‚nm-1, which includes the headgroups in the contrast; and the molar mass of SDS: Mw ) (N/L)‚2〈L〉n‚288 g/mol. Table 3 tabulates the results for SDS in 1.5 M NaCl. The mean value of K is 0.004 32 g-2 cm2 mol, which corresponds to a mean value for (Fm - Fs)2 of 3.52 × 1021 cm-4. Given Fs ) 6.25 × 1010 cm-2, this gives a mean value for the micellar scattering length density of Fm ) 0.35 × 1010 cm-2. This agrees rather well with the value, 0.38 × 1010 cm-2, calculated from atomic scattering lengths and volumes for SDS.25 (45) Kodama, M.; Miura, M. Bull. Chem. Soc. Jpn. 1972, 45, 2265.

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Figure 5. Dependence of the fitted persistence lengths on surfactant concentration at the four salt concentrations: 1 M (O); 1.2 M (0); 1.5 M (3); and 2.0 M (]). Extrapolation to c ) 0 gives the persistence lengths used for further fitting of the SANS curves, taking into account the intermicellar interactions.

Magid et al.

Figure 7. Bending rod plots of SANS data for SDS micelles in 1.5 M NaCl at 45 °C, and the corresponding fits using eqs 7-9: semiflexible micelles, including intermicellar interactions. Surfactant concentrations: 3.0 mM (O); 3.8 mM (0); 4.8 mM (3); 6.2 mM (]). Table 3. Derivation from Fitting Parameter A of Contrast Factors for SDS Micelles in 1.5 M NaCl Mw A(cMw)-1 (Fm - Fs)2 Fm c c (mM) (10-3 g/cm3) (106 g/mol) (g-2cm2mol) (1021 cm-4) (1010 cm-2) 3.0 3.8 4.8 6.2

Figure 6. Bending rod plots of SANS data for SDS micelles in 1.2 M NaCl at 45 °C, and the corresponding fits using eqs 7-9: semiflexible micelles, including intermicellar interactions. Surfactant concentrations: 3.5 mM (O); 4.8 mM (0); 6.2 mM (3); 7.8 mM (]); 12 mM (/).

To assess values of c* for SDS in 1.2 and 1.5 M NaCl, the values of Mw and of 〈Rg2〉1/2 (eq 6) derived from the fits including intermicellar interactions can be used in the relationship c* ) 3Mw/4π〈Rg2〉3/2. For 6.2 mM SDS in 1.5 M NaCl, a dilute solution (X ) 0.78), Mw ) 4.91 × 106 g/mol and 〈Rg〉1/2 ) 82.3 nm, from which a c* of 12 mM is predicted. For 7.8 mM SDS in 1.2 M NaCl, also a dilute solution with X ) 0.49, Mw ) 1.70 × 106 g/mol, and 〈Rg2〉1/2 ) 45.8 nm, a c* of 24 mM is predicted. Since the micelles continue to grow as c* is approached, both of these values are overestimates of the true c*’s. This can be seen by inspection of Figure 4: the apparent micelle size, apparent because the plotted micellar lengths were obtained by neglecting interactions, in 1.5 M NaCl passes through a maximum at ∼8 mM. A second method for assessment of c*’s involves the functional dependence of X on c; X reaches one, corresponding roughly to c ) c*, at 7.8 mM SDS in 1.5 M NaCl and at 14 mM SDS in 1.2 M NaCl. Dependence of Flexibility on Ionic Strength. The dependence of the SDS micelles’ total persistence lengths, lp,t, on NaCl concentration, extrapolated to zero surfactant concentration (Figure 6), is shown on a double logarithmic

0.864 1.09 1.38 1.79

2.17 2.26 2.47 4.91 Mean:

0.004 27 0.004 30 0.004 26 0.004 47 0.004 32

3.48 3.50 3.47 3.64

0.351 0.334 0.359 0.217

plot in Figure 8a. The values are 15.9 nm at 1 M NaCl; 15.4 nm at 1.2 M NaCl; 13.8 nm at 1.5 M NaCl; and 9.8 nm at 2 M NaCl. We are interested in understanding several aspects of these values: (1) how they can be partitioned into electrostatic and intrinsic components of the persistence lengths; (2) why the electrostatic component may still be nonzero at high ionic strength; and (3) how the 1D bending moduli derived from lp,t compare to the 2D bending moduli observed for SDS lamellae. To determine the electrostatic component, lp,e, of the total persistence length, an estimate is needed for the intrinsic persistence length, lp,i, of the (uncharged) micelles; we get this estimate using the approach of Appell and Porte.46 A cylindrical (or wormlike) micelle is considered equivalent to beads of diameter d on a string, and the ensemble of allowed configurations which exclude the centers of beads n - 1 and n + 1 from a separation less than d leads to a value for lp,i. The direction vectors connecting the center of bead n to bead n + 1 and to bead n - 1 each have a magnitude of d; the angle φ between them can have with equal probability any value from 0 to 120°. This leads to the relationship ) exp(2d/lp,i) ) 0.25. The magnitude of the resulting intrinsic persistence length reflects the steric hindrance, due to the finite cross-sectional size of the micelle. Since reasonable values of d range from 3.6 nm (twice the SANSdetermined radius) to 5.0 nm, a typical value for the hydrodynamic diameter of the micelles,15 possible values for lp,i are 5.6 and 7.8 nm, respectively, with an uncertainty of at least (1 nm. The larger value is perhaps more realistic, since the hydrated headgroups and their associated counterions are part of the bending micellar (46) Appell, J.; Porte, G.; Poggi, Y. J. Colloid Interface Sci. 1982, 87, 492. This is adapted from an earlier use of the pearl necklace model by: Stigter, D. J. Phys. Chem. 1966, 70, 1323.

Flexibility of Elongated SDS Micelles in NaCl(aq)

Langmuir, Vol. 16, No. 26, 2000 10035 Table 4. One-Dimensional Bending Moduli for SDS Wormlike Micelles and Derived Two-Dimensional Bending Moduli for SDS Bilayer Membranes cs (M)

Bt/kBT (nm)

Be/kBTa (nm)

Kt/kBTb

Ke/kBTb

1.0 1.2 1.5 2.0

15.9 15.4 13.8 9.8

8.1 7.6 6.0 2.0

4.4 4.3 3.8 2.7

2.2 2.1 1.7 0.56

a Assuming l b p,i ) 7.8 nm. Membrane thickness assumed to be 3.6 nm.

Figure 8. (a) Dependence of persistence lengths, total (O) and electrostatic (×), for SDS micelles on NaCl concentration. The intrinsic persistence length is set equal to 7.8 nm. (b) Dependence of persistence lengths, total (O) and electrostatic (×), for NaPSS, 84% quaternized with Mw ) 3.5 × 105 g/mol, on NaCl concentration. The intrinsic persistence length is set equal to 1.4 nm. Data from ref 47. (c) Another view of persistence lengths for the SDS micelles, illustrating the lack of scaling with cs-0.5.

objects. Figure 8a also shows the electrostatic persistence lengths, lp,e ) lp,t - lp,i, under this assumption. The value of lp,e does not reach zero, even at 2 M NaCl. Over the range of salt investigated, both lp,t and lp,e versus cs show pronounced downward curvature at high salt. This behavior has also been observed for highly flexible polyelectrolytes such as sodium polystryene-

sulfonate (NaPSS),18,47 whose lp,i is generally accepted to be 1.4 nm.48 Nordmeier and Dauwe47 observed lp,t ) 5.2 nm for 84% quaternized NaPSS at 2 M NaCl. They claim an intrinsic persistence length of 4.8 nm, but if the value is in fact 1.4 nm, then the majority of the polyelectrolyte coils’ rigidity is still of electrostatic origin, even when the Debye-Hu¨ckel screening length, κ-1, is only 0.2 nm (2 M NaCl). Figure 8b presents their data. Some more rigid polyelectrolytes, such as sodium hyaluronate49 (fitted lp,i ) 8.7 nm from data up to 0.4 M NaCl) or DNA in 5 M LiCl18 (lp,i ) 28.5 nm, θ conditions) do show scaling law behavior, lp,e ∼ cs - x. However, the exponent obtained experimentally does not generally agree with DebyeHu¨ckel-based models (x ) 1) for the electrostatic persistence length:50 x equals 0.5 for hyaluronate and other polysaccharides in aqueous salt, and 0.3 for DNA. NaPSS at low ionic strength (0.005-0.05 M salt) also yields x ) 0.5 (SDS micelles are still globular rather than elongated at those cs values, so a direct comparison is not possible). LeBret’s numerical approach51 to persistence lengths can give x ) 0.5. In Figure 8c, both the total and electrostatic persistence lengths for the SDS micelles are plotted against cs - 0.5. For the micelles at 1 M e cs e 1.5 M, neither lp,t nor lp,e is described by x ) 0.5. There is an additional consideration for both polyelectrolytes and wormlike micelles that makes partitioning between lp,i and lp,e more difficult than expected, namely the impact of excluded volume effects on lp,i. As the NaCl concentration changes, the solvent quality changes, so that lp,i may in fact not be a single, constant value at all cs. For example, 2 M NaCl at 45 °C is a poor solvent for SDS micelles (there is a proximate Krafft boundary), but 1-1.5 M NaCl solutions are appreciably better. The fact that lp,e for SDS micelles in 2 M NaCl is nonzero is not surprising, given the behavior of NaPSS at high salt. Furthermore, Daicic et al.52 have developed a flexible surface model for systems such as lamellae or sponge phases that produces significant changes in membrane rigidity as a function of surface charge density at cs values as high as 1-2 M salt. Although hydrated sodium ions are not considered to penetrate extensively into the micellar surface, at very high salt some modest penetration may alter the micellar surface charge density and potentials at the headgroup plane. Finally, we can estimate from the total 1D bending moduli, Bt, of SDS wormlike micelles the corresponding 2D moduli, Kt, given that B ∼ dK. Values of B obtained from the relationship lp,t ) Bt/kBT are tabulated in Table 4. Taking d = 3.6 nm as the thickness of a bilayer of SDS hydrocarbon chains, the values of Kt obtained from the (47) Nordmeier, E.; Dauwe, W. Polym. J. 1992, 24, 229. (48) Fo¨rster, S.; Schmidt, M. Adv. Polym. Sci. 1995, 120, 51. (49) Ghosh, S.; Li, X.; Reed, C. E.; Reed, W. F. Biopolymers 1990, 30, 1101. (50) (a) Odijk, T. J. Polym. Sci. Phys. Ed. 1977, 15, 477. (b) Skolnick, J.; Fixman, M. Macromolecules 1977, 10, 944. (51) Le Bret, M. J. Chem. Phys. 1982, 76, 6243. (52) Daicic, J.; Fogden, A.; Carlsson, I.; Wennerstro¨m, H.; Jo¨nsson, B. Phys. Rev. E 1996, 54, 3984.

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lp,t’s range from 4.4 kBT at 1 M NaCl down to 2.7 kBT at 2 M NaCl. We can estimate the electrostatic component of K in the same way (Table 4). The estimates for the total 2D bending moduli are in the same range as those found for SDS layers containing medium chain length alcohols as cosurfactants. For example, 2.2 kBT is found53 for water/SDS/pentanol/ dodecane and values ranging from 2.2kBT to 5.8kBT are observed54 for water/SDS/octanol as the molar ratio of alcohol decreases from 2.8 to 1.2. With respect to Ke, the agreement is less good: adding 0.35 M NaCl to the water/ SDS/octanol system produced no significant change in rigidity. Conclusions The flexibility of SDS micelles in aqueous solutions containing 1-2 M NaCl has been clearly established by SANS measurements over a range of scattering vectors sensitive to micellar persistence lengths as well as to micellar contour lengths and cross-sectional radii. The availability of numerical expressions for the scattering functions of wormlike chains and for a structure factor describing interactions between the chains at finite Q, both calculated by Pedersen and Schurtenberger from their Monte Carlo simulations, has allowed for a robust analysis of the SANS data. To our knowledge, the work described here is the first example of implementation of (53) Halle, Quist J. Phys. II Fr. 1994, 4, 1823. (54) Auguste, F.; Barois, P.; Fredon, L.; Clin, B.; Dufourc, E. J.; Bellocq, A. M. J. Phys. II Fr. 1994, 4, 2197.

Magid et al.

their protocol with adjustable parameters in a full leastsquares fitting mode. The micelles become more flexible as the ionic strength increases, with the electrostatic persistence length still contributing to the total in 2 M NaCl (Debye screening length of 0.2 nm). This is perhaps not surprising given the high surface charge density at the plane of the micellar headgroups. With their lower (linear) charge densities, the anionic polyelectrolytes NaPSS (highly flexible) and DNA (more rigid than our micelles) also show small contributions from the electrostatic component at high salt.18 The total and the electrostatic persistence lengths for the SDS micelles do not scale with salt concentration: neither an exponent of -1 nor one of -0.5 is observed. This suggests that theoreticians and experimentalists seeking an understanding of polyion flexibility may be interested in including data for micellar systems (equilibrium polyelectolytes) as well as quenched polyelectrolytes. Forthcoming papers from our laboratory will explore additional determinants of micellar flexibility, such as surfactant chain length and headgroup identity. Acknowledgment. Support of the work at the University of Tennessee by the National Science Foundation (CHE-9729433) is gratefully acknowledged. A portion of this material is based on activities supported at NIST by NSF under Agreement No. DMR-9423101. LA0006216