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Langmuir 1997, 13, 6032-6039
Cross-Section Structure of Rodlike Sodium Sulfopropyl Octadecyl Maleate Micelles from Small-Angle Neutron Scattering Hans von Berlepsch,*,† Rainer Mittelbach,‡ Ernst Hoinkis,§ and Heimo Schnablegger| Max-Planck-Institut fu¨ r Kolloid- und Grenzfla¨ chenforschung, Rudower Chaussee 5, D-12489 Berlin, Germany, and Kantstrasse 55, D-14513 Teltow, Germany, Institute of Physical Chemistry, Karl-Franzens University Graz, Heinrichstrasse 28, A-8010 Graz, Austria, and Hahn-Meitner-Institut, Glienicker Strasse 100, D-14109 Berlin, Germany
Aqueous solutions of the ionic surfactant sodium sulfopropyl octadecyl maleate (SSPOM) have been studied by small-angle neutron scattering (SANS) to characterize the rodlike micelles formed in the presence of 250 mmol/L NaCl at 50 °C. The external scattering contrast is varied by changing the composition of the solvent mixture from D2O to H2O. Reliable values for the volume of the monomer, the mean aggregation number per unit micellar length and the mean cross-sectional radius of gyration have been obtained by the conventional cross-section Guinier analysis of the data for varied contrast. A more detailed picture of the cross-sectional structure is attaind by application of a model-independent Fourier transformation technique with a subsequent novel deconvolution technique that includes polydispersity. It is shown for the first time that the cross-sectional polydispersity of rodlike micelles can be appreciable. A polydispersity of about 10% of the mean radius of 2.73 nm is obtained for SSPOM, reflecting molecular protrusions of individual surfactant molecules into the surrounding water. The model-independent data analysis further supports the simplified general view of an oil drop-like homogeneous core of the cylindrical micelles formed by alkyl chains with an effective chain length of 13.5 carbon atoms, and a relatively thick and smooth head group shell comprising the remaining atomic groups of the SSPOM molecule.
I. Introduction Surfactants are used in emulsion polymerization to control the size of the growing polymer particles and to stabilize the latices. Polymerizable emulsifiers that covalently bind to the particle surface possess the advantage not to desorb during the subsequent application of the polymers.1-3 Sodium sulfopropyl octadecyl maleate (SSPOM) investigated in this study HCCOO(CH2)3SO3Na HCCOOC18H37
was synthesized in view of such a technological application.4-6 The type and structure of surfactant aggregates formed in aqueous solution depend generally on their geometrical packing properties.7 SSPOM possesses a rather complicated conformational structure, character* To whom the correspondence should be addressed. † Max-Planck-Institut fu ¨ r Kolloid- und Grenzfla¨chenforschung, Berlin. ‡ Institute of Physical Chemistry, Karl-Franzens University Graz. § Hahn-Meitner-Institut. | Max-Planck-Institut fu ¨ r Kolloid- und Grenzfla¨chenforschung, Teltow. X Abstract published in Advance ACS Abstracts, October 1, 1997. (1) Greene, B. W.; Sheetz, D. P.; Fisher, T. D. J. Colloid Interface Sci. 1970, 32, 90. (2) Urquiola, M. B.; Dimonie, V. L.; Sudol, E. D.; El-Asser, M. S. J. Polym. Sci., Part A: Polym. Chem. 1992, 30, 2619. (3) Urquiola, M. B.; Dimonie, V. L.; Sudol, E. D.; El-Asser, M. S. J. Polym. Sci., Part A: Polym. Chem. 1992, 30, 2631. (4) Kanegafuchi Kogaku Kogyo Kabushiki Kaisha, G. B. Patent, 1 427 789, 1976; U. S. Patent 3 980 622, 1976. (5) Tauer, K.; Goebel, K.-H.; Kosmella, S.; Sta¨hler, K.; Neelsen, J. Makromol. Chem., Macromol. Symp. 1990, 31, 107. (6) Goebel, K.-H.; Sta¨hler, K.; von Berlepsch, H. Colloids Surf., A: Physicochem. Eng. Aspects 1994, 87, 143. (7) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 1 1976, 72, 1525.
S0743-7463(97)00475-7 CCC: $14.00
ized by two chains of different lengths that are connected by a cis double bond. This conformation leads to several unusual mesoscopic properties which became visible in recent studies8-10 focusing on the characterization of the compound and its homologues. Thus, it could be demonstrated that SSPOM forms a highly ordered lamellar gel state below its Krafft boundary at TK ) 37 °C. The gel state is a colloidally structured dispersion of crystal hydrates in the diluted liquid phase with interdigitated molecules forming a bilayer, where the terminal hydrophilic SO3- group is buried in the interior of the bilayer.8,9 On the other hand, strikingly stiff wormlike micelles are formed in the micellar phase above TK at high ionic strength. These wormlike micelles were well characterized by light scattering techniques.10 Application of the Koyama theory of wormlike micelles gave a mean persistence length of about 120 nm. The reduced Rayleigh ratio extrapolated to zero scattering angle follows a power law dependence versus surfactant concentration with characteristic limiting exponents of around 0.5 and -1.0 for low and high concentrations, respectively, in excellent agreement with the result of a mean-field treatment of polydisperse rodlike micelles interacting via excluded volume interactions.11 In addition, the hydrodynamic correlation length obtained from dynamic light scattering decreases with increasing concentration according to a power law with a limiting exponent of -0.5, as suggested from scaling arguments for rodlike particles. While micellar growth and flexibility could be well explained by the light scattering studies, reliable data about the local structure of the cylindrical particles were not available until now. The present small-angle neutron scattering (SANS) investigations should help to close this gap and (8) von Berlepsch, H. Langmuir 1995, 11, 3667. (9) von Berlepsch, H.; Hofmann, D.; Ganster, J. Langmuir 1995, 11, 3676. (10) von Berlepsch, H.; Dautzenberg, H.; Rother, G.; Ja¨ger, J. Langmuir 1996, 12, 3613. (11) van der Schoot, P.; Cates, M. E. Langmuir 1994, 10, 670.
© 1997 American Chemical Society
Structure of Na Sulfopropyl Octadecyl Maleate Micelles
provide us with more detailed information on the internal micellar structure. Several groups have attempted to prove the existence of cylindrical micelles and to characterize their local structure using SANS experiments.12-19 Thereby, the measured scattering patterns have been analyzed in different ways. The simplest one uses asymptotic expressions for the scattering intensity I(q) in different regimes of the wave vector q. Thus, a cross-section Guinier plot enables one to extract, from a contrast variation experiment,20 the radius of gyration of the cross-sectional area of the micelle, the dry volume of the monomer, and the mass per unit micellar length. We will apply this method below. In the second step the experimental data are mostly compared with calculated scattering curves of model structures. This analysis requires a reasonable model for the micellar structure. For many surfactants, simple core-shell models are easily constructed with a minimum of adjustable parameters and the reached data fits often are perfect.16-18,21 Yet it is unknown which chemical groups of the present surfactant molecule belong to the “head group” of the amphiphile and which form the hydrophobic “core”. In such a situation we can make use of an alternative procedure which is model-independent and is based on the indirect (Fourier) transformation method (IFT).22,23 The IFT method yields the pair distance distribution function and, after applying a deconvolution technique, the scattering length density profile. The latter enables estimates of cross-section area, degree of order, penetration of solvent, etc. This method has been applied recently to SANS investigations on micoemulsions24,25 and will be utilized below to characterize the present more simple binary SSPOM/water system. While it is generally accepted that rodlike micelles possess a broad distribution of individual lengths, their cross-section polydispersity is commonly assumed to be low and is often neglected. However, because of the dynamic nature of the surfaces of amphiphilic aggregates, fluctuation forces, so-called molecular protrusion effects,26-28 are acting, which make the surfaces molecularly rough and should lead to smoothed radial density profiles. In this work we compare and demonstrate the structure analysis available to the scientific community according to the present state of the art. We start with simple Guinier analysis considerations and then apply modern (12) Lin, T.-L.; Chen, S.-H.; Gabriel, N. E.; Roberts, M. F. J. Phys. Chem. 1987, 91, 406. (13) Marignan, J.; Appell, J.; Bassereau, P.; Porte, G.; May, R. P. J. Phys. (Paris) 1989, 50, 3553. (14) Schurtenberger, P.; Scartazzini, R.; Magid, L. J.; Leser, M. E.; Luisi, P. L. J. Phys. Chem. 1990, 94, 3695. (15) Schurtenberger, P.; Magid, L. J.; King, S. M.; Lindner, P. J. Phys. Chem. 1991, 95, 4173. (16) Hjelm, R. P.; Thiyagarajan, P.; Alkan-Onyuksel, H. J. Phys. Chem. 1992, 96, 8653. (17) Herbst, L.; Kalus, J.; Schmelzer, U. J. Phys. Chem. 1993, 97, 7774. (18) Long, M. A.; Kaler, E. W.; Lee, S. P.; Wignall, G. D. J. Phys. Chem. 1994, 98, 4402. (19) Schmitt, V.; Schosseler, F.; Lequeux, F. Europhys. Lett. 1995, 30, 31. (20) Stuhrmann, H. B. J. Appl. Crystallogr. 1974, 7, 173. (21) Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. J. Phys. Chem. 1995, 99, 1299. (22) Glatter, O. J. Appl. Crystallogr. 1977, 10, 415. (23) Glatter, O. J. Appl. Crystallogr. 1980, 13, 577. (24) Glatter, O.; Strey, R.; Schubert, K.-V.; Kaler, E. W. Ber. BunsenGes. Phys. Chem. 1996, 100, 323. (25) Schurtenberger, P.; Jerke, G.; Cavaco, C.; Pedersen, J. S. Langmuir 1996, 12, 2433. (26) Aniansson, G. A. E.; Wall, S. N.; Almgren, N.; Hoffmann, H.; Kielmann, I.; Ulbricht, W.; Zana, R.; Lang, J.; Tondre, C. J. Phys. Chem. 1976, 80, 905. (27) Karaborni, S.; O’Connell, J. P. Langmuir 1990, 6, 905. (28) Israelachvili, J. N.; Wennerstro¨m, H. J. Phys. Chem. 1992, 96, 520.
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inversion techniques to obtain information about the radial scattering length density profile of polydisperse micelles. Two methods are currently available or will be available in the near future. One is the model-free determination of the size distribution29 assuming a radial profile consisting of two steps with different scattering length densities, and the other one is the model-free determination of the radial profile30 assuming a predefined size distribution with adjustable width. II. Experimental Section and Methods 1. Materials. SSPOM was synthesized following a procedure described in ref 6. For purification the surfactant was recrystallized at least three times from a water-acetone mixture. 1H NMR and thin-layer chromatography showed no impurities, indicating at least 99% purity. Sodium chloride (p.a.) was purchased from Merck (Darmstadt, Germany). The solvents were mixtures of Milli-Q quality water (conductivity < 1 µS/cm at 20 °C) and D2O (isotopic purity > 99.95%) from Merck. The critical micelle concentration (cmc) of =3.0 × 10-5 g/mL (50 °C) was estimated from conductivity measurements. 2. Small-Angle Neutron Scattering Measurements. The small-angle experiments were carried out with the V4 SANS instrument at the research reactor BER II at the Hahn-Meitner-Institut Berlin (HMI), Germany. A mean wavelength of λmean ) 0.6 nm having a triangular distribution with a full width at halfmaximum (FWHM) of ∆λ/λ ) 0.108 was used. Two sample-to-detector distances were chosen (1.0 and 4.0 m). The magnitude of the wave vector q ) (4π/λ) sin(θ/2) ranged from 0.13 to 3.6 nm-1, where θ is the scattering angle. Surfactant and salt concentrations of samples were c ) 0.01 g/mL and cNaCl ) 250 mmol/L, respectively, and were fixed during the whole study. The cmc has been neglected due to its low value compared with c. The measuring temperature was T ) 50.0 ( 0.2 °C. Under these conditions the micelles are entangled and form a network with a “mesh” size (hydrodynamic correlation length) of about 35 nm.10 The external contrast variations were made by varying the volume fraction of D2O in the D2O/H2O mixture from 1 to 0.785, 0.590, 0.376, and 0. The samples were placed in disk-shaped quartz cells (Hellma) of 1 mm optical path length and equilibrated at the measuring temperature over 24 h before they were rapidly transferred to the cell holders for measurement. The scattering data were collected by a two-dimensional position sensitive detector of 64 × 64 elements with 10 × 10 mm2 area, masked, corrected for background, azimuthally averaged, and converted to absolute scale by the scattering intensity of a standard sample (1 mm H2O) according to the standard procedures provided by the HMI. 3. Scattering Data Analysis. Guinier Analysis. For the SANS intensity (differential scattering crosssection per unit volume) of rodlike micelles of length L and cross-sectional radius of gyration Rg, randomly oriented in space, an asymptotic expression in the low-q region (cross-section Guinier plot) may be derived12,17,31
( )
2 2 π NA Nagg (bm/Vm - Fsolv)2Vm2e-q Rg /2 q Mo L (1)
I(q) ) (c - cmc)
where bm is the known sum of the coherent neutron scattering length of all nuclei constituting the monomer, (29) Schnablegger, H.; Glatter, O. J. Colloid Interface Sci. 1993, 158, 228. (30) Mittelbach, R.; Glatter, O. Manuscript in preparation for J. Appl. Crystallogr. (31) Porod, G. Acta Phys. Austriaca 1948, 2, 133.
6034 Langmuir, Vol. 13, No. 23, 1997
von Berlepsch et al. ∞ pc(r) J0(qr) dr ∫r)0
Vm is the excluded volume of the monomer, Fsolv is the scattering length density of the solvent, NA is Avogadro’s number, Mo is the molecular mass of SSPOM, and the ratio Nagg/L is the aggregation number per unit micellar length. This formula holds under the made assumptions for both monodisperse and polydisperse systems as long as the micelles have a similar radial structure. It is a useful approach to obtain quantitative structural data from contrast variation experiments. Here the product qI(q) extrapolated to q ) 0 is plotted as a function of the contrast (bm/Vm - Fsolv) and from the matching point where the product vanishes Vm can be deduced. Nagg/L is obtained from limqf0(qI(q)) and Rg from the q dependence via eq 1. For polydisperse cross-sectional radii, the scattering intensity at the matching point is not zero and information about the size distribution may be obtained in principle from the experiment.32 For a system of wormlike surfactant micelles in water, like those under investigation, the effect should be small. The present measurements are not accurate enough near the matching point to permitt such analysis, and the polydispersity has to be determined from the mathematical analysis of measured scattering curves I(q). Indirect Transformation Techniques (Monodisperse Systems). The indirect Fourier transform method (IFT), developed by Glatter22,33,34 in the seventies, uses the long known equation
where J0(x) is the zero-order Bessel function and the subscript letter c indicates two-dimensionally averaged quantities. Equation 4 is strictly valid for cylinders of infinite length but can be safely applied to cylindrical objects with an axial ratio of 1:10 or higher.23 With the calculated pc(r) function the radial scattering length profile ∆Fc(r) of the cylinder can be computed using eq 5. Indirect Transformation Techniques (Polydisperse Systems). We have to point out that any polydispersity contributions in pc(r) will have obscuring effects in the density profile ∆Fc(r). But these effects can be identified and distinguished from pure geometrical effects, as will be demonstrated below. The PDDF of a polydisperse system of cylinders p j c(r) can be written as an average over the size distribution DS(y,σ) so that eq 5 is reformulated
sin(qr) I(q) ) 4π r)0 p(r) dr qr
where y ) R/Rmean, R is the particle’s radius and Rmean is the radial distance serving as reference and it is usually set equal to the peak position of the size distribution DS(R,σ). The normalization constant N is calculated according to
∫
∞
(2)
relating the coherently scattered intensity I(q) to the threedimensionally averaged pair-distribution function (PDDF), p(r), of the scattering system. Naturally, the PDDF of an arbitrary scattering system contains both, the FourierBessel transformed contributions of intraparticular (form factor) and interparticular scattering (structure factor). In those special cases where the interparticular contributions are negligible, the experimental scattering function I(q) is identical with the form factor and the PDDF can be expressed as
p(r) ) r2
∞ ∆F(x) ∆F(x-r) dx ∫x)-∞
(3)
where ∆F(x) is the scattering length density difference between the particle and the surrounding medium. The IFT method calculates the PDDF from the experimental I(q) by inverting eq 2. It takes into full account instrumental and spectral broadening and minimizes termination effects usually accompaning inverse-transform methods. The same inversion concept, though in a nonlinear way, can be applied to calculate ∆F(r) from the previously calculated p(r) via square-root deconvolution35,36 of eq 3. The reader is referred to the original literature for details of the inversion method used. The PDDF is calculated in a model-free way and thus it can indicate special geometrical facts of the scattering system like, e.g., spherical, cylindrical, or lamellar symmetry with homogeneous or inhomogenous structure. Once the geometry of the scattering system is identified as, e.g., cylindrical with an axial ratio of about 1:10 (see Figure 3 below), alternative equations can be applied to investigate the radial density profile of these cylinders. Instead of eqs 2 and 3 we can use23 (32) Teubner, M. J. Chem. Phys. 1991, 95, 5072. (33) Glatter, O. Acta Phys. Austriaca 1977, 47, 83. (34) Glatter, O. J. Appl. Crystallogr. 1979, 12, 166. (35) Glatter, O. J. Appl. Crystallogr. 1981, 14, 101. (36) Glatter, O.; Hainisch, B. J. Appl. Crystallogr. 1984, 17, 435.
qI(q) ) 2π pc(r) ) r
(4)
∞ ∆Fc(x) ∆Fc(x-r) dx ∫x)-∞
(5)
p j c(r) ) (r/N)
∞ ∞ DS(y,σ) ∫x)-∞ ∆Fc(xy) ∆Fc((x-r)y) dx dy ∫y)0
N)
∞ DS(y,σ) dy ∫y)0
(6)
(7)
and the distribution function, DS(y,σ), will be assumed to be monomodal and of Schulz type
DS(y,σ) )
{
xt exp[t(1 - y + ln y) - y] exp[t(1 - y + ln y) - y]
t>1 te1
(8)
where t ) σ-2 - 1 and σ is proportional to the half width at half-maximum: HWHM ) σ x2 ln 2. The inversion of eq 6 subject to a simultaneous determination of the width parameter, σ, was recently accomplished.30 A second way to attack the structure analysis problem of polydisperse samples is to assume a distribution of layered cylinders and to determine this size distribution under subject of finding the best-fitting core-shell parameters by inverting the following equation
Idesm(q) )
R ∫R)R
max min
D(R) W(R) Φ(q,R) dR
(9)
where the distribution function D(R) can be of any arbitrary type and it is either the number, the volume, or the intensity distribution of cylinder radii, respectively, depending on the corresponding weighting function W(R),
{
W(R) ) [R2∆F(s) - Rc2(∆F(s) - ∆F(c))]2L2 2∆F(s)
[R 1
- Rc
2(∆F(s)
- ∆F(c))]L
number volume (10) intensity
where Rc and R are the core and the outer radius, L is the length of the cylinder, and ∆F(c) and ∆F(s) are the core and shell excess contrast parameters, respectively. The
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Langmuir, Vol. 13, No. 23, 1997 6035
form factor Φ(q,R) of randomly oriented layered cylinders is given by37
Φ(q,R) )
π/2 [4 sin2[H cos β] [x∆F(s) J1(x sin β) ∫β)0
xc(∆F(s) - ∆F(c)) J1(xc sin β)]2]/[H2 cos2 β[x2∆F(s) xc2(∆F(s) - ∆F(c))]2 sin2 β] sin β dβ (11) where xc ) qRc, x ) qR, H ) qL/2, J1(x) is the Bessel function of first order, and β is the angle between the cylinder axis and the direction of the scattering vector. The desmeared scattering function Idesm(q) in eq 9 can be obtained as a byproduct by inverting eq 2. It is subsequently submitted to a polydispersity analysis with an optimized regularization technique (ORT)29 to determine the ratios of inner and outer radius (Rc/R) and inner to outer contrast parameters (∆F(c)/∆F(s)). Although the values of the contrast and radial ratios can be constrained to each other, we chose to independently adjust both parameters in order to obtain effective ratios which, in the case of polydispersity, can deviate from the ideal monodisperse values. The ORT procedure was initially introduced to simultaneously determine particle size and refractive index of spherical scattering objects from light scattering data and, for the present work, it had to be extended with the form factor of layered cylinders (eqs 9-11). Because the analysis techniques used in this work are very detailed, we have to refer the reader to the original literature.
Figure 1. Cross-section Guinier plots log(qI(q)) vs q2 of intensity from external contrast variation experiments on 1.0 wt % SSPOM solutions. The volume fraction, R, of D2O in H2O varies from 1 (9) to 0.785 (b), 0.590 (3), 0.376 (0), and 0 (O). The solid lines are least-squares fits. The error bars are well within the symbols width and hence not visible. Table 1. Cross-Sectional Radii of Gyration (Rg) and Aggregation Numbers per Unit Micellar Length (Nagg/L), Obtained from Contrast Variation Using the Cross-Sectional Guinier Plot
III. Results and Discussion The SANS scattering curves are the primary experimental results. As discussed in the Introduction, their quantitative theoretical analysis has to be performed in a model-independent way because of the unknown distribution of the scattering length densities. Before doing so in the second part of this section, we start with the simpler Guinier analysis. It permits the estimation of an effective scattering length density from the contrastmatching point, the estimation of Vm, and the determination of the micellar mass per unit length. We can do this Guinier analysis because interparticular interferences are negligible, as it can be seen later (Figure 4a) from an almost vanishingly small negative portion (correlation hole, excluded volume, etc.) in the high-r region of the PDDF. Also, we note that in Figure 1 the intensity at low q values is only slightly bending down, which yields this above mentioned negative contribution in real space. Coulomb interactions can be neglected as well due to the high salt content of the solutions. 1. Guinier Analysis. Figure 1 shows Guinier plots of qI(q) for the contrast variation series together with the best fitting straight lines. The experimental scattering data are the raw data because smearing effects are negligible in that q range. The data points agree well with the fitting curves in the range 0.1 < q2 < 0.6 nm-2. Outside that range deviations occur. Those on the low-q side may result from finite size effects as well as the above mentioned interaction effects. The deviations on the high-q side reflect the internal micellar structure. The deviations increase on lowering the contrast, especially when pure H2O is used as solvent (lowest curve), due to the high solvent background. We did not perform additional background corrections in these cases. This can be thought of as the main error source in the values for the cross-sectional radii of gyration. The resulting radii are listed in Table 1. Nevertheless, the values show a (37) Mittelbach, P.; Porod, G. Acta Phys. Austriaca 1961, 14, 405.
a
Ra
Rg (nm)
Nagg/L (nm-1)
1.0 0.785 0.590 0.376 0
1.83 ( 0.04 1.77 ( 0.04 1.62 ( 0.08 1.72 ( 0.08 1.52 ( 0.08
26.6 26.0 27.8 32.3 24.1
R is the D2O volume fraction in the H2O/D2O mixture.
tendency to decrease with decreasing amount (R) of D2O in the solvent mixture, which was found also in other SANS studies on micellar solutions.38 These authors considered this effect as an indication of a nonuniform radial scattering density distribution, and it should be supported by the IFT/deconvolution calculations. With the intercept values from Figure 1 the quantity (xqI(q)lim qf0 can be determined, which according to eq 1, should depend linearly on the D2O volume fraction, designated R,
( xqI(q)lim qf0 ) NA Nagg π(c - cmc) Mo L
(
)
1/2
(bm - [FH2O(1 - R) + FD2OR]Vm (12)
This relation was utilized to determine Vm and Nagg/L. In Figure 2 the experimental points are plotted and a linear regression is possible with high quality, yielding contrast matching at R ) 0.16 ( 0.008. With the sum of scattering length bm ) 41.37 × 10-13 cm of all nuclei (without sodium) of the monomer, the scattering length densities of H2O and D2O, FH2O ) -0.562 × 1010 and FD2O ) 6.34 × 1010 g cm-2, respectively, a mean scattering length density of the micelle Fm ) 0.542 × 1010 g cm-2 ( 5%, a volume per molecule Vm ) 0.763 ( 0.038 nm3 , and the aggregation numbers per unit micellar length (Nagg/L) listed in Table 1 are obtained. The monomer volume is slightly higher than the corresponding theoretical value of 0.736 nm3 calculated for 20 °C from experimentally determined group volume increments.39 Quite a good agreement is reached, bearing in mind a certain contribution from thermal expansion. The mean mass per unit length, ML ) Mo〈Nagg/L〉 ) 1.40 × 104 g/(mol‚nm) ( 5%, where Mo )
6036 Langmuir, Vol. 13, No. 23, 1997
von Berlepsch et al.
a
Figure 2. Results of external contrast variation. The solid line is a linear least-squares fit according to eq 12. The zerointensity intercept (matching point) occurs at 16.0 ( 0.8 vol % D2O in the solvent. The error bars agree with the symbols width.
512.7 g/mol is the molecular mass of the monomer and 〈Nagg/L〉 ) 27.3 nm-1, confirms the respective value of 1.38 × 104 g/(mol‚nm) of the light scattering study. 2. Inverse Transformation Analysis. In the following IFT analysis we concentrate on the sample with the highest external contrast (100% D2O) because of the distinctly lower range of experimental errors for larger q-values as compared with the other samples. The respective scattering function is shown in Figure 3a, where the data points are the symbols (error bars) and the full line is the best fitting curve obtained from the inversion of eq 2. Background from solvent, sample cell and incoherent background are subtracted before the inversion procedure. For a proper determination of the incoherent background level (K1 ) we applied Porod’s law I(q) = K1 + K2/q4 , where K1 and K2 are least-squares parameters. The corresponding model-free PDDF (p(r) of eq 2 is depicted in Figure 3b, showing the typical characteristics of cylindrical particles, i.e., a narrow peak at low r from correlations within the cross-section and a long, almost linear, decreasing tail evolving from correlations along the cylinder axis. Figure 3b indicates that the scattering objects are cylindrical with the largest correlation length of 35 nm (Figure3b, arrow 1) and with an approximate diameter of 5-6 nm (Figure 3b, point of inflection, arrow 2). Although the diameters of the cylinders are polydisperse (see Figure 5), an identification of the cylindrical geometry is feasible without problems. It should be noticed that the length of this rodlike structure is of the same order of magnitude as the so-called “mesh” size, which was obtained for the present system under identical experimental conditions from dynamic light scattering (hydrodynamic correlation length).10 Both lengths reflect distances beyond which the interactions become important. The indicated cylindrical symmetry leads directly to the subsequent evaluation of the scattering data with eq 4. The result is the cross-sectional correlation function pc(r) as plotted with symbols (error bars) in Figure 4a. The pc(r) function does not intercept the y-axis at the origin (pc(r)0) * 0), which indicates resolution problems below r ≈ 0.5 nm. Indeed, the sampling theorem of the Fourier transformation (∆r g π/qmax) tells us that the smallest resolvable details are approximately ∆r g 0.87 nm. The interpretation of the pc(r) function in terms of monodisperse cylinder cross-sections, i.e., by inverting eq 5, yields a smoothly decaying apparent radial profile ∆Fc(r), as shown in Figure 4b. The corresponding best fitting pc(r) is shown in Figure 4a as a full line. This curve does
b
Figure 3. (a) Experimental scattering curve of SSPOM in 100 vol % D2O with the fit function (full line) obtained from the inversion of eq 2. The experimental data points are the symbols (error bars). (b) Corresponding pair distance distribution function p(r). Arrow 1 marks the spatial extent of the cylindrical structure. The inflection point (arrow 2) provides an approximate cross-sectional diameter between 5 and 6 nm.
not follow the offset in the data at low r, but otherwise has no difficulties in reproducing the data by assuming a monodisperse but accordingly inhomogeneous radial profile. The small oscillation of the data at high r comes from the approximation of infinitly long cylinders and it is also neglected by the inversion procedure; i.e., the fitting curve in Figure 4a does not follow this oscillation. Also, we note that no (or negligibly small) excluded volume contributions are present in the scattering signal, which otherwise would also lead to negative pc(r) values at high r. In Figure 5 we show the corresponding theoretical scattering function (crosses) of this assumed monodisperse cylinder with the calculated apparently smoothly decaying radial profile. In the same figure we have included the desmeared experimental scattering function Idesm(q) (full line) to show the discrepancy between the best-fitting monodisperse model and the actual experiment. It is evident from the figure that radial polydispersity has to be taken into account for a correct interpretation of the experimental data and neglecting its influence25 may lead to an erroneous interpretation of the scattering length density profile. Before representing the results of this analysis, we discuss the effect of external contrast variation on the PDDF pc(r). Figure 6 shows these profiles for D2O volume fractions, R, in the solvent mixture of 1 (∆), 0.785 (+), and 0.590 (×). It is seen that for decreasing contrast, R, pc(r) is lowered with increasing distance, r, and becomes negative. There are several arguments for such a behavior. One of those is a small penetration of solvent
Structure of Na Sulfopropyl Octadecyl Maleate Micelles
Langmuir, Vol. 13, No. 23, 1997 6037
a
b
Figure 6. Cross-sectional correlation function pc(r) for different D2O volume fractions, R, in the solvent mixture: 1 (∆), 0.785 (+), and 0.590 (×). Table 2. Cross-Sectional Radii of Gyration Rg, Obtained by IFT R
Rg (nm)
1.0 0.785 0.590
1.781 ( 0.020 1.733 ( 0.020 1.652 ( 0.020
same trend of radii of gyration. The respective radii of gyration computed from the moment equation 2
Rg Figure 4. (a) Cross-sectional correlation function pc(r) for the same sample as in Figure 3 calculated by using eq 4. The result of the calculation is plotted as symbols (error bars). The full curve is the best fit to the data obtained by the deconvolution procedure (eq 5). (b) Normalized cross-section excess scatteringlength density profile for monodisperse cylinder cross-sections calculated by inverting eq 5.
Figure 5. Desmeared experimental scattering curve of the sample shown in Figure 3a and reconstructed scattering function (symbols) under the assumption of monodisperse cylinder cross-sections.
into the head group region of the micelle, which should yield a smaller apparent radius for lower contrast.38 Another argument could be the increasing importance of excluded volume interactions for lower contrast, which leads to the same pattern as demonstrated in ref 34. Also a simple step-profile in F(r) as in ref 17 could lead to the (38) Lin, T.-L.; Chen, S.-H.; Gabriel, N. E.; Roberts, M. F. J. Am. Chem. Soc. 1986, 108, 3499. (39) Benjamin, L. J. Phys. Chem. 1966, 70, 3790.
∞ r2pc(r) dr ∫r)0 ) ∞ 2 ∫r)0 pc(r) dr
(13)
are given in Table 2. Indeed, these radii of gyration gradually decrease with decreasing R, confirming the trend and the magnitude of the corresponding values estimated from the simple Guinier approximation, given in Table 1. We used eq 13 as a cross-check. Differences could arise because eq 13 constitutes the physically exact definition of Rg, whereas Guinier’s equation is a Taylor series valid only in a limited q range. In view of the huge differences between the desmeared experimental curve and the reconstructed scattering function (crosses in Figure 5), we tried a polydispersity analysis and started with ORT. In general, this program allows us to calculate the size distribution D(R) and one (or two) additional parameter of an assumed form factor, i.e., in this case with the form factor of a layered cylinder (eq 11) with the, yet to be determined, core-shell ratios of inner to outer radius (Rc/R) and of inner to outer contrast factor (∆F(c)/∆F(s)). The parameters of the two-step model of the scattering length profile have been estimated in the following way with reference to the oil drop model of spherical micelles.40 Here it is assumed that the hydrophobic core of the micelle is a cylinder of radius equal to the length, lc, of a fully stretched alkyl chain and of volume Vc ) Nminν, where ν is the volume of the alkyl chain. lc and ν (in nm and nm3), respectively, are approximated by Tanford’s formulas:41 lc ) 0.15 + 0.1265m , ν ) 0.0274 + 0.0269m, where m is the carbon number of the alkyl chain and Nmin is given by Nmin ) πlc2/(Nagg/L)ν. With the parameters 〈Nagg/L〉 and Vm estimated by the Guinier analysis and the known atomic scattering length, the effective carbon number meff ) 13.5, the radii Rc ) 1.86 nm and R ) 2.58 nm, and the mean scattering length density of the shell Fs ) 1.526 × 1010 g cm-2 can be obtained. (40) Tartar, H. V. J. Colloid Interface Sci. 1959, 14, 115. (41) Tanford, C. J. Phys. Chem. 1972, 76, 3020.
6038 Langmuir, Vol. 13, No. 23, 1997
a
b
c
Figure 7. Results of the polydispersity analysis of the experimental scattering curve shown in Figure 3a using ORT and deconvolution techniques, eqs 9-11 and 6-8, respectively. (a) Calculated size distributions of cross-sectional radii from ORT (full line) and deconvolution technique (full line with symbols). (b) Reconstructed scattering functions from ORT (triangles) and the deconvolution technique (crosses) compared with the experimental desmeared scattering function (full line). (c) Calculated radial scattering length density profiles from ORT (triangles) and the deconvolution technique (crosses) compared with the starting profile, constructed from the data obtained above by the Guinier analysis (full line).
They are used as starting values for the nonlinear calculation procedure. Thus, the starting values are Rc/R ) 0.721 and ∆F(c)/∆F(s) ) 1.39, where ∆F(c) is normalized to unity. The starting profile is shown in Figure 7c as the full thick line. After 10 iterations of alternatively varying one of the parameters, the calculations indeed did converge, producing the size distribution shown in Figure 7a (full line) with the peak position at R ) 2.726 nm, the HWHM of 0.28 nm and with the best-fitting model
von Berlepsch et al.
parameters of Rc/R ) 0.753 ( 0.010 and ∆F(c)/∆F(s) ) 1.83 ( 0.02. In Figure 7b the corresponding scattering function (triangles) can be compared with the experimental desmeared scattering function Idesm(q) (full line). We note a considerably improved agreement compared to the corresponding curves in Figure 5. As an alternative way of structure analysis we also used the model-free deconvolution procedure (eq 6), which is subjected to the constraint that the polydispersity of the cross-sections follows a Schulz-type distribution. The width of this distribution is adjustable and can be determined simultaneously with the radial profile. This method of deconvoluting the pc(r) function was reported36,42 to be insensitive to systematic errors like, e.g., residual incoherent background, etc. The calculated radial profile is shown in Figure 7c (crosses) together with the estimated theoretical profile (full line) and the best-fitting two-step profile (triangles) from the ORT calculation above, and the corresponding scattering function is added to Figure 7b (crosses). We note that both procedures yield reasonable profiles, whereas the smooth profile in Figure 7c (crosses) seems to be better suited for describing the real scenario. The nearly homogeneous scattering length density profile in the inner part of the micelle is a result that should be pointed out in particular. It supports the simplified view of a homogeneous oil droplike core of the rodlike micelle. The calculated half-width of the assumed Schulz distribution is HWHM ) 0.30 nm. It can be compared with the previously (via ORT) determined size distribution in Figure 7a, where the line with the symbols represents the Schulz distribution after scaling it back from DS(y,σ) to DS(R,σ) with R ) 2.726y . The scaling factor is thus chosen such that the peak positions of both distribution functions in Figure 7a coincide, revealing nearly full agreement between both distributions. For the mean interfacial area per molecule in the rodlike micelle, Ao ) 2πR/〈Nagg/L〉 , we then find =0.63 nm2 . This effective area is, as expected, much higher than the corresponding one for the bilayer structure (gel state) of 0.40 nm2.9 For the packing parameter of the geometrical micelle model of Israelachvili et al.,7 P ) Vm/(AoR), a value of =0.43 is obtained, which is quite close to the limiting value for cylinders of 0.5 and supports the simple geometrical arguments. A standard deviation of the cylinder radius of about one-tenth of its average radius is a reasonable value, which compares with experimentally determined polydispersities of small spheroidal micelles,43 in particular with those estimated for SSPOM at low ionic strength.44 Theoretical estimates of molecular protrusion decay length for micellar aggregates give values of the same order of magnitude.28 VI. Conclusions The aim of these SANS investigations was to further characterize the ionic surfactant sodium sulfopropyl octadecyl maleate, which forms rodlike micelles in aqueous solutions after addition of high amounts of sodium chloride. The external scattering contrast was varied by changing the composition of the solvent mixture of D2O and H2O. The reported results demonstrate in a nearly perfect way the cylindrical micellar structure. In the first step of interpreting the scattering data, reliable values for the volume of the monomer, the mean aggregation number per unit micellar length, and the mean cross-sectional radius of gyration have been obtained from a simple (42) Glatter, O. J. Appl. Crystallogr. 1988, 21, 886. (43) Cabane, B.; Duplessix, R.; Zemb, T. J. Phys. (Paris) 1985, 46, 2161. (44) von Berlepsch, H.; Sta¨hler, K.; Zana, R. Langmuir 1996, 12, 5033.
Structure of Na Sulfopropyl Octadecyl Maleate Micelles
Guinier analysis. These quantitative data support recent estimates provided from other experimental methods, in particular light scattering. A much more detailed picture of the cross-sectional structure is attaind by application of modern inversion techniques. It is shown for the first time for rodlike micelles that their cross-sectional polydispersity can be appreciable. A polydispersity of about 10% of the mean radius of 2.73 nm is obtained for SSPOM, reflecting molecular protrusions of individual surfactant molecules into the surrounding water. The data analysis further supports the simplified general view of an oil droplike homogeneous core of the cylindrical micelles formed by alkyl chains, in the present case with an effective chain
Langmuir, Vol. 13, No. 23, 1997 6039
length of 13.5 carbon atoms, and a relatively thick and smooth head group shell comprising the remaining atomic groups of the complicated SSPOM molecule. Acknowledgment. H.v.B. wishes to thank Prof. H. Mo¨hwald for supporting the work and Prof. R. Strey for useful discussions. We are grateful to K.-H. Goebel for providing the surfactant. The assistance of R. Wagner in the course of data taking is further gratefully acknowledged. Our thanks go further to U. Keiderling (HMI) for the use of his data reduction software. LA970475C
