Langmuir 1998, 14, 7403-7409
7403
Characterization of Sodium Sulfopropyl Octadecyl Maleate Micelles by Small-Angle Neutron Scattering Hans von Berlepsch,*,† Uwe Keiderling,‡ and Heimo Schnablegger§ Max-Planck-Institut fu¨ r Kolloid- und Grenzfla¨ chenforschung, Rudower Chaussee 5, D-12489 Berlin, Germany, Hahn-Meitner-Institut, Glienicker Strasse 100, D-14109 Berlin, Germany, Technische Universita¨ t Darmstadt, Petersenstrasse 23, D-64287 Darmstadt, Germany, and Max-Planck-Institut fu¨ r Kolloid- und Grenzfla¨ chenforschung, Kantstrasse 55, D-14513 Teltow, Germany Received August 17, 1998. In Final Form: October 20, 1998 Aqueous solutions of the ionic surfactant sodium sulfopropyl octadecyl maleate (SSPOM) have been studied by small-angle neutron scattering (SANS) to characterize the micelles formed in the presence of 25 mmol/L NaCl at 50 °C. The external scattering contrast is varied by changing the composition of the solvent mixture from D2O to H2O and yields a reliable value for the volume of the monomer. The experimental scattering curves are best described by a system of interacting and polydisperse prolate ellipsoidal micelles with an aspect ratio of 1.15. The polydispersity is small, amounting to about one-tenth of its equivalent sphere radius of 3.27 nm. A mean aggregation number of about 200 is determined, confirming recent estimates from light scattering and time-resolved fluorescence quenching. The obtained quantitative data are in agreement with the theory of micellization and show that simple packing considerations are also useful for uncommon surfactants possessing a complicated conformational structure.
I. Introduction Surfactants play an important role in many industrial applications and processes due to their ability to lower interfacial tensions. In emulsion polymerization, surfactants are used to control the size of the resulting polymer particles and to stabilize the latices. Polymerizable emulsifiers possess the advantage of not desorbing during the subsequent application of the polymers.1,2 The surfactant sodium sulfopropyl octadecyl maleate (SSPOM) investigated in this study
was synthesized aiming at such application.3,4 In a series of recent papers,5-9 we investigated the aggregation behavior of SSPOM and its homologues in aqueous * To whom correspondence should be addressed. † Max-Planck-Institut fu ¨ r Kolloid- und Grenzfla¨chenforschung, Berlin. ‡ Hahn-Meitner-Institut and Technische Universita ¨ t Darmstadt. § Max-Planck-Institut fu ¨ r Kolloid- und Grenzfla¨chenforschung, Teltow. (1) Urquiola, M. B.; Dimonie, V. L.; Sudol, E. D.; El-Asser, M. S. J. Polym. Sci., Part A: Polym. Chem. 1992, 30, 2619. (2) Urquiola, M. B.; Dimonie, V. L.; Sudol, E. D.; El-Asser, M. S. J. Polym. Sci., Part A: Polym. Chem. 1992, 30, 2631. (3) Kanegafuchi Kogaku Kogyo Kabushiki Kaisha, G. B. Patent 1 427 789, 1976; U.S. Patent 3 980 622, 1976. (4) Tauer, K.; Goebel, K.-H.; Kosmella, S.; Sta¨hler, K.; Neelsen, J. Makromol. Chem., Macromol. Symp. 1990, 31, 107. (5) von Berlepsch, H. Langmuir 1995, 11, 3667. (6) von Berlepsch, H.; Hofmann, D.; Ganster, J. Langmuir 1995, 11, 3676. (7) von Berlepsch, H.; Dautzenberg, H.; Rother, G.; Ja¨ger, J. Langmuir 1996, 12, 3613. (8) von Berlepsch, H.; Sta¨hler, K.; Zana, R. Langmuir 1996, 12, 5033. (9) von Berlepsch, H.; Harnau, L.; Reineker, P. J. Phys. Chem. B 1998, 102, 7518.
solution. We found unusual mesoscopic properties which are connected with the complicated conformational structure of these amphiphilic compounds. The surfactant molecule is characterized by two hydrocarbon chains of different length which are linked with a cis double bond. The micellar properties in aqueous solution, that is, critical micelle concentration, effective micelle ionization degree, and aggregation number as a function of surfactant concentration, added salt concentration (sodium chloride), and temperature, were thoroughly investigated by means of electrical conductivity, light scattering, and timeresolved fluorescence quenching (TRFQ).7,8 It was demonstrated that the micelles forming at low ionic strength are small and show a low polydispersity and a slight shape asymmetry. Above a critical threshold ionic strength (NaCl concentration of about 50 mmol/L at 50 °C) they grow dramatically to form wormlike aggregates. This behavior is generally observed for ionic surfactants of different chemical structure. However, we were surprised at the unusually high stiffness (or persistence length) of the wormlike micelles of SSPOM. A persistence length of about 120 nm was obtained by static7 and dynamic9 light scattering. While micellar growth and flexibility could be well explained by the TRFQ and light-scattering studies, respectively, the diameter of these micelles was until then solely estimated from the measured mass density using the simple geometrical model of a homogeneous cylinder. To get more reliable data about the local structure, SANS experiments were performed, which are suited to give detailed structural data on the nanometer scale.10,11 First we investigated aqueous SSPOM solutions in the presence of 250 mmol/L NaCl.12 A detailed picture of the cross-sectional structure has been attained from the scattering data by application of a model-independent Fourier transformation technique with a subsequent (10) Chen, S. H. Annu. Rev. Phys. Chem. 1986, 37, 351. (11) Cabane, B.; Duplessix, R.; Zemb, T. J. Phys. (Paris) 1985, 46, 2161. (12) von Berlepsch, H.; Mittelbach, R.; Hoinkis, E.; Schnablegger, H. Langmuir 1997, 13, 6032.
10.1021/la9810408 CCC: $15.00 © 1998 American Chemical Society Published on Web 12/04/1998
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deconvolution technique that included polydispersity. The calculated pair-distribution function confirmed the suggested cylindrical structure of the micelles. The cylindrical symmetry considerably simplified the subsequent detailed data analysis. The derived structural data supported the simplified view of an oil droplike homogeneous core formed by alkyl chains with an effective chain length of 13.5 carbon atoms, and a relatively thick and smooth headgroup shell comprising the remaining atomic groups of the SSPOM molecule. A mean cylinder radius of 2.73 nm with a radial polydispersity of about 10%, reflecting molecular protrusions of individual surfactant molecules into the surrounding water, has been obtained. In the present work the SANS studies are expanded into the region of low ionic strength where the micelles are small and presumably spheroidal. As in the former study, we were particularly interested in characterizing their cross-sectional structure. At low ionic strength correlations between the particle positions due to their electrical charge will markedly interfere with the scattering curves. To reduce this effect, a salt concentration of 25 mmol/L is chosen, which reduces the stong intermicellar electrostatic repulsions but is held just below the concentration where the strong micellar growth occurs, that is, the so-called “spheroid-to-rod transition”. There, according to the former TRFQ and light-scattering studies,7,8 the aggregation number is about 200 (at 50 °C). From simple packing considerations8,13 a prolate ellipsoidal shape is expected. The geometrical data of the small micelles, which will be determined, permit a quantitative comparison with the corresponding data of the wormlike aggregates above the spheroid-to-rod transition for the same amphiphile. II. Experimental Section and Methods 1. Materials. The surfactant was synthesized following a procedure described in ref 14. For purification it 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 molecular mass of SSPOM is M0 ) 512.7 g/mol. The critical micelle concentration (cmc) of =6.0 × 10-5 mol/L (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-MeitnerInstitut Berlin (HMI), Germany.15 A (mean) wavelength of λ ) 0.6 nm having a triangular distribution with a full width at halfmaximum (fwhm) of ∆λ/λ ) 0.108 was used. The chosen sampleto-detector distances were 1.0 and 4.0 m for all samples and in addition 16.0 m for the two strongest scattering samples. The magnitude of the wave vector q ) (4π/λ) sin(θ/2) ranged from 0.13 to 3.6 nm-1 and from 0.035 to 3.6 nm-1 for the strongest scattering samples, respectively, where θ is the scattering angle. The surfactant and salt concentrations of the samples were C ) 20 mmol/L and CNaCl ) 25 mmol/L, respectively, and were kept constant 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. The external contrast was varied to allow an estimation of the dry volume of the monomer.16 The contrast variations were made by varying the volume fraction of D2O in the D2O/H2O mixture, β, from 1 to 0.784, 0.573, 0.376, and 0. The (13) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 1 1976, 72, 1525. (14) Goebel, K.-H.; Sta¨hler, K.; von Berlepsch, H. Colloids Surf., A: Physicochem. Eng. Aspects 1994, 87, 143. (15) Keiderling, U.; Wiedenmann, A. Physica B 1995, 213 & 214, 895. (16) Stuhrmann, H. B. J. Appl. Crystallogr. 1974, 7, 173.
von Berlepsch et al. samples were placed in rectangular 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 the absolute scale by the scattering intensity of a standard sample (1 mm H2O) using the “BerSANS” software package17 provided by the HMI. 3. Scattering Data Analysis. Structure Factor. For interacting micelles the dependence of the SANS intensity (differential scattering cross section per unit volume) on the magnitude of the scattering vector q can be expressed as a function of both the particle form factor P(q) and the interparticle structure factor S(q), as follows,10
dΣ(q)/dΩ ) I(q) ) I0 P(q) S(q) ) I′(q) S(q)
(1)
where I0 is the extrapolated zero-angle scattering intensity. This equation is strictly valid for monodisperse spherical particles, but it is a common practice to describe systems with small polydispersity by eq 1. In addition it has been empirically shown10,18 to be a reasonable approximation for charged quasispherical micelles, where strong electrostatic repulsion makes close micelle-micelle contact unlikely. The structure factor S(q) may be calculated, in principle, for any type of interaction potential. Assuming the micelle to be a charged sphere, the most widely used model potential is a screened Coulomb potential (DLVO potential), given by
U(r)/kT )
LBZ2eff exp(2κrHS) exp(-κr) r (1 + rHSκ)2
for
r > rHS (2)
with LB ) e2/(4π0kT) the Bjerrum length, Zeff the effective number of charges of the micelle, rHS its (effective) hard-core radius, κ ) (8πLBNAJ)1/2 the inverse Debye-Hu¨ckel length due to small ions, NA Avogadro’s number, J the ionic strength, e the electronic charge, k the Boltzmann constant, the solvent dielectric constant, and 0 the absolute permittivity. The degree of ionization of micelles, R, is defined as the ratio of the effective number of charges and the aggregation number: R ) Zeff/Nagg. Four parameters are needed to calculate S(q): the micellar charge (or degree of ionization), the radius, the number density of the micelles n, and the screening length of the solution 1/κ, respectively. The concept followed in this paper is to treat the S(q) function as a kind of perturbation that distorts the unknown particle form factor P(q) (or I′(q)). The goal is to find an optimum function Sopt(q) = S(q) which minimizes the perturbation and then to divide the measured raw scattering intensity I(q) by this Sopt(q). The scattering data, “corrected” in such a way, I′(q) ) I(q)/S(q), are thereafter assumed as effectively unperturbed, and they are subjected to the Fourier transformation techniques to obtain the particles’ characteristics. This approximation should be acceptable because the S(q) contributions are diminished by a screening electrolyte, which has been accomplished by adding salt (CNaCl ) 25 mmol/L). Several theoretical approaches for calculating S(q) have been applied: (1) the random phase approximation (RPA),19 (2) the rescaled mean spherical approximation (RMSA),20,21,22 and (3) the primitive-model mean spherical approximation (PM-MSA) described by Blum and Høye.23 The formulas and details of the calculation procedures can be found in the original papers. For nonspherical micelles the MSA becomes increasingly invalid and the fitted particle parameters have to be interpreted with caution. For low anisometry (e1.3) it is known, however, (17) Keiderling, U. Physica B 1997, 234-236, 1111. (18) Chevalier, Y.; Zemb, T. Rep. Prog. Phys. 1990, 53, 279. (19) Baba-Ahmed, L.; Benmouna, M.; Grimson, M. J. Phys. Chem. Liq. 1987, 16, 235. (20) Hayter, J. B.; Penfold, J. Mol. Phys. 1981, 42, 109. (21) Hansen, J.-P.; Hayter, J. B. Mol. Phys. 1982, 46, 651. (22) Snook, I. K.; Hayter, J. B. Langmuir 1992, 8, 2880. (23) Blum, L.; Høye, J. S. J. Phys. Chem. 1977, 81, 1311.
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that the error is small enough to be neglected,24 if the (effective) hard-core radius of an equivalent sphere is used for the calculation. Another important point to be discussed is the effect of polydispersity, because size and charge polydispersity are known to influence the structure factor in much the same way.25 We discuss the experimental details of determining Sopt(q) in the Results section. Here we need to outline the general procedure. (1) We select the low-q region (0.035 e q e 1 nm-1), where the S(q) contributions, though small in total strength, have their largest impact. (2) We choose a couple of trial form factors Pi(q,rff) of monodisperse simple geometrical objects (homogeneous spheres, ellipsoids, etc.) and multiply these with the theoretical structure factor Sth(q,rHS,Zeff) from one of the above-mentioned approximations. (3) Using a standard nonlinear least-squares algorithm (usually Levenberg-Marquardt26), the three unknown parameters (rff, rHS, Zeff) are calculated in order to bring the product PiSth to match the experimental I(q) in the selected low-q region. The best-fitting rHS and Zeff values now constitute Sopt(q), which is understood as a sufficiently good approximation for S(q) in eq 1. Having selected the theory (RMSA) and the bestfitting rHS and Zeff, Sopt(q) now can be calculated for any desired q-value. (4) By dividing the experimental I(q) by Sopt(q) over the whole q range, we obtain the “corrected” form factor I′(q) ) I0P(q). Guinier Analysis. For monodispersed micelles or micelles of low polydispersity the mean aggregation number Nagg, may be obtained from the zero-angle scattering intensity I0 by the expression27
I0 ) CNANagg(bm - FsolvVm)2
(3)
where bm is the known sum of the coherent neutron-scattering lengths of all nuclei constituting the monomer, Vm is the excluded volume of the monomer, and Fsolv is the scattering length density of the solvent. In the low-q region and for particles having a center of symmetry, the particle form factor P(q) may be expressed in the Guinier approximation28 as
I′(q) ) I0 exp(-q2Rg2/3)
∫
∞
r)0
p′(r)
sin(qr) dr qr
y(q,a) = I′D(q) )
∫
bmax
b)bmin
(5)
(24) Kotlarchyk, M.; Chen, S. H. J. Chem. Phys. 1983, 79, 2461. (25) Klein, R. In Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution; Chen, S.-H., Huang, J. S., Tartaglia, P., Eds.; Kluwer: Dordrecht, 1992; p 39. (26) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran. The Art of Scientific Computing, 2nd ed.; Cambridge University Press: Cambridge, 1992. (27) Lin, T.-L.; Chen, S.-H.; Gabriel, N. E.; Roberts, M. F. J. Am. Chem. Soc. 1986, 108, 3499. (28) Guinier, A.; Fournet, G. Small Angle Scattering of X-rays; Wiley: New York, 1955. (29) Teubner, M. J. Chem. Phys. 1991, 95, 5072. (30) Glatter, O. Acta Phys. Austriaca 1977, 47, 83. (31) Glatter, O. J. Appl. Crystallogr. 1977, 10, 415. (32) Glatter, O. J. Appl. Crystallogr. 1979, 12, 166.
D(b) W(b) Φ(q,a,b) db
(6)
where a and b are the semiaxes of an ellipsoid of revolution, oriented in such a way that a runs parallel to the y-axis, which is the axis of rotation. Thus, a < b means oblate geometry, and a > b is the prolate type. D(b) is the distribution function of the length b, that can be of any arbitrary type, and it is either the number, the volume, or the intensity-weighted distribution, depending on the respective weighting function W(b),
{
(ab2)2 ... number W(b) ) ab2 ... volume 1 ... intensity
(7)
The form factor Φ(q,a,b) of a randomly oriented ellipsoid of revolution is given by Guinier34
Φ(q,a,b) )
∫
π/2
β)0
[
3
]
sin u - u cos u u3
2
sin µ dµ
(8)
where u ) qbx(sin2 µ + (a/b)2 cos2 µ) and µ is the angle between the axis of revolution and the direction of the scattering vector. The procedure calculates a best-fitting size distribution D(b) for a given constant value of a. By repeated application for a set of different a-values and by comparing the mean-deviation values MD of the fit,
(4)
where Rg is the radius of gyration of the micelle. It is a useful approach to obtain quantitative structural data from contrast variation experiments. When the corrected scattering intensity I′(q) extrapolated to q ) 0 is plotted as a function of the contrast (bm/Vm - Fsolv), Vm can be deduced from the matching point. Nagg is obtained from limqf0 I′(q) and Rg from the q-dependence via eq 4. For polydisperse spherical particles the scattering intensity at the matching point is not zero and, in principle, some information about the size distribution may be obtained.29 The present measurements near the matching point are not accurate enough to permit such analysis, and the polydispersity has to be determined by form-factor analysis techniques. Fourier Transformation Techniques. We used the indirect Fourier transform (IFT) method,30-32 which calculates the threedimensionally averaged pair-distance distribution function (PDDF) p′(r) of the scattering system by transforming the corrected scattering function I′(q) into real space. The equation to be inverted is
I′(q) ) 4π
Hereby, all instrumental-broadening effects are put into the calculations and, besides the model-free p′(r), we obtain I′D(q), which is a desmeared and smoothed version of the experimental I′(q). In a second step this I′D(q) can be reanalyzed in terms of the polydispersity and the axial ratio, simultaneously. The general procedure of this step is explained elsewhere33 for the mathematically equivalent problem of the simultaneous determination of the polydispersity and the refractive index of LorenzMie particles. In this work we applied the equation
MD(a) )
∑ i
[
]
I′D(qi) - y(qi,a) σ(qi)
2
(9)
where σ(qi) is the experimental standard deviation of the i-th data point, an optimum value of a can be found provided that the stability parameter of the constrained least-squares algorithm is set to the appropriate value. Details of this optimization technique (ORT) can be found elsewhere.33
III. Results and Discussion The SANS scattering curves are the primary experimental results. Background from solvent, the sample cell, and the incoherent background is subtracted before the data analysis. 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. As outlined in the preceding paragraph, the data analysis consists of three parts. First, an appropriate estimate for the structure factor S(q) must be found. To remove the electrostatic-interaction contributions from the experimental I(q), the background-corrected [I(q) - K1] is divided by the resulting S(q). Second, a Guinier analysis is performed in order to estimate the effective scatteringlength density from the contrast-matching point, an estimate of Vm, the aggregation number Nagg, and the radius of gyration Rg. The third stage is the quantitative analysis of the particle form factor by inversion techniques. 1. Structure Factor. The effective structure factor was calculated by fitting Sth(q,rHS,Zeff)Pi(q,rff) to the experimental I(q) (100% D2O) in the range 0.035 e q e 1 nm-1. (33) Schnablegger, H.; Glatter, O. J. Colloid Interface Sci. 1993, 158, 228. (34) Guinier, A. Ann. Phys. 1939, 12, 161.
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von Berlepsch et al.
Table 1. Effective Micellar Radii rHS and rff and the Number of Charges Zeff, Estimated by Fitting Sth(q,rHS,Zeff) Pi(q,rff) to the Scattering Intensity I(q) (100% D2O) approximation
rHS (nm)
rff (nm)
Zeff
RPA RMSA PM-MSA
4.50 3.38 6.81
3.31 3.38 3.31
36.0 17.1 =0
rHS is the effective hard-sphere radius of the particle, which, depending on the applied approximation, can be identical to the radius rff that is used in the form factor Pi(q,rff). The particular type of form factor did not show any significant influence on the resulting best-fitting Sth(q,rHS,Zeff), because, when rff is adjusted appropriately, in the low-q region all form factors become indistinguishable from each other. All three approximations (RPA, RMSA, and PM-MSA) gave a similar good fit to the data, and as expected,25 the actual best-fitting values for rHS and Zeff differed a lot. We substantiate this by restating the following facts. First, the contribution of the interparticle interferences is largely reduced due to the addition of salt, and therefore it cannot be parametrized with high accuracy. Second, we neglected any polydispersity effects in Pi(q,rff ) and Sth(q,rHS ,Zeff ) so that the calculated bestfitting values are strongly biased, as was demonstrated by Klein et al.25,35A further discussion of these approximation-dependent deviations is beyond the scope of this work, and we only quote the resulting values in Table 1 for the sake of completeness. The constant parameters used in the calculations were the temperature T ) 50 °C, the dielectric constant of the solvent ) 70, the ionic strength of the solution J ) 26.3 mmol/L, and the surfactant volume fraction Φ ) 9.2 × 10-3. The ionic strength J was calculated by the equation J ) CNaCl + 0.5RC, from both salt and surfactant concentrations CNaCl ) 25 mmol/L and C ) 20 mmol/L, respectively. For the degree of micelle ionization we took the value R ≈ 0.13 estimated recently by light scattering.7 Φ was calculated from Φ ) 4/3πrHSn ) CVmNA, with the number density of micelles n and assuming a volume per molecule of Vm ) 0.763 nm3 (experimental value for rodlike micelles12). It should be emphasized once more that the actual values of the fitting parameters are of minor importance for the structural analysis. The main point is that we obtain a good estimate Sopt(q) for the structure factor S(q) that can be used to retrieve the particles’ form factor I′(q) ) I0P(q). The quality of the fit is demonstrated in Figure 1, where the error-bar symbols are the experimental I(q), the slightly oscillating horizontal line is the best-fitting Sopt(q) from the RMSA, and the full line through most of the symbols is the corrected I′(q). In what follows we exclusively used the modified scattering data I′(q) ) I(q)/ Sopt(q) calculated with the Sopt(q) from the RMSA. 2. Guinier Analysis. Figure 2 shows Guinier plots log(I′(q)) versus q2 for the contrast variation series together with the best-fitting straight lines. Apart from the very initial data points showing a slightly higher level of noise, the data points agree quite well with the fitting curves up to about q2 ∼ 0.6 nm-2. The deviations occurring for larger q reflect the internal micellar structure. At lowest contrast, that is, for pure H2O as solvent (lowest curve), the noise increases due to the high solvent background. Both the zero intensity I0 and the slope of the straight lines are functions of the contrast. The radii of gyration obtained from the slopes are listed in Table 2. The radii decrease (35) Krause, R.; D′Aguanno, B.; Mendez-Alcarez, J. M.; Na¨gele, G.; Klein, R.; Weber, R. J. Phys.: Condens. Matter 1991, 3, 4459.
Figure 1. Experimental scattering curve I(q) (error bars) of 20 mM SSPOM in 100 vol % D2O with the corrected scattering curve I′(q) (full line) obtained by I′(q) ) I(q)/Sopt(q), where the structure factor Sopt(q) from the RMSA is the slightly oscillating horizontal line.
Figure 2. Guinier plots log(I′(q)) versus q2 of corrected intensity from contrast variation experiments on 20 mM SSPOM solutions. The volume fraction β of D2O in the D2O/H2O mixture varies from 1 (O) to 0.784 (0), 0.573 (3), 0.376 (4), and 0 (]). The solid lines are least-squares fits. The error bars are well within the symbol widths and hence are not visible. Table 2. Radii of Gyration (Rg) Obtained from Contrast Variation Using the Guinier Approximation
a
βa
Rg (nm)
1.0 0.784 0.573 0.376 0
2.69 ( 0.05 2.64 ( 0.05 2.57 ( 0.05 2.23 ( 0.10 3.48 ( 0.10
β is the D2O volume fraction in the D2O/H2O mixture.
from 2.69 to 2.23 nm when the solvent changes from D2O to a mixture of H2O/D2O containing 37.6% D2O. In pure H2O an essentially larger value of 3.48 nm is found. The same effect has been reported for other SANS investigations on micellar solutions,27 including our own on the rodlike SSPOM micelles,12 and indicates a nonuniform radial scattering density distribution. When the scattering length density of the solvent mixture is expressed by the D2O volume fraction β, the linear relation between the square root (xI0 and β follows from eq 3:
(xI0 ) (CNANagg)1/2(bm - [FH2O(1 - β) + FD2Oβ]Vm) (10)
Sodium Sulfopropyl Octadecyl Maleate Micelles
Figure 3. Result of contrast variation. The solid line is a linear least-squares fit according to eq 10. The zero-intensity intercept (matching point) occurs at 16.3 ( 0.5 vol % D2O in the solvent. The error bars agree with the symbol widths.
which was utilized to determine Vm and Nagg. In Figure 3 the experimental data are plotted and approximated by a straight line. Minor systematic deviations from the regression line could signal a certain amount of polydispersity. In the polydisperse case a parabola would be expected,29 but the effect is too small, and more data points would be needed for a reliable estimation. Contrast matching occurs at β ) 0.163 ( 0.005. With the sum of scattering lengths 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 cm-2 and FD2O ) 6.34 × 1010 cm-2, respectively, a mean scattering length density of the micelle Fm ) 0.563 × 1010 cm-2 ( 5%, a volume per molecule Vm ) 0.735 ( 0.030 nm3, and the aggregation number Nagg ) 207 ( 10 for D2O as solvent are obtained. The monomer volume agrees nicely with the corresponding theoretical value of 0.736 nm3 calculated for 20 °C from experimentally determined group volume increments.36 3. Indirect Fourier Transformation Techniques. At present we know basically three methods to determine structural parameters from interacting colloidal systems. (1) The first j data points at low q can be left out for the inversion analysis. This is an arbitrary procedure which cannot be recommended. If done so with our experimental scattering function (100% D2O), the resulting p(r) still shows a tail at high r as depicted in Figure 4 (+). This tail could be falsely interpreted as a small fraction of larger aggregates, but in actuality it is simply the result of cutting off the scattering function at a q-value where S(q) starts an upward bend in its oscillations. (2) The two independent parameters rHS and Zeff constituting Sopt(q) can be determined simultaneously with p′(r) in a specialized inversion routine.37 At the time of compilation of this work this program was not yet available and we had to confine our investigations to the third method below. (3) As explained in the previous section, we chose the method of directly dividing the experimental I(q) by a best-fitting structure factor. For this purpose some kind of form factor P(q,r) is needed. We already mentioned that the actual type of form factor played a minor role. Here, we show in Figure 4 that the averaged pair-distance distribution functions p′(r) that result from the various (36) Benjamin, L. J. Phys. Chem. 1966, 70, 3790. (37) Brunner-Popela, J.; Glatter, O. J. Appl. Crystallogr. 1997, 30, 431.
Langmuir, Vol. 14, No. 26, 1998 7407
Figure 4. Pair-distance distribution function p(r) calculated from the scattering curve shown in Figure 1.The upper curve showing the tail at high r (+) is obtained from I(q) by cutting the first data points at low q. The other PDDFs are obtained from I′(q) with form factors for a homogeneous sphere (rff ) 3.31 nm, 4), an oblate ellipsoid of revolution (a/b ) 0.63, rff ) b ) 3.60 nm, ×), and a prolate ellipsoid of revolution (a/b ) 2.09, rff ) a ) 5.41 nm, ]).
Figure 5. Desmeared and smoothed experimental scattering function I′D(q) (error bars) of 20 mM SSPOM in 100 vol % D2O and reconstructed scattering function from ORT (full line).
scattering functions I′(q) obtained by varying the type of Pi(q,rff) indeed are identical except at r > 7 nm, where some small differences are encountered. The main features are identical for all three trial form factors, that is, (4) Pi(q,rff) of a homogeneous sphere with radius rff ) 3.31 nm, (×) Pi(q,rff) of an oblate ellipsoid of revolution with axial ratio a/b ) 0.63 and rff ) b ) 3.60 nm, and (]) Pi(q,rff) of a prolate ellipsoid of revolution with axial ratio a/b ) 2.09 and rff ) a ) 5.41. The previously mentioned tail in the PDDF can no longer be found in all three corrected PDDFs. In addition to the PDDF, the desmeared and smoothed scattering function I′D(q) (Figure 5, error-bars) is obtained by the inversion routine. The I′D(q) shown in Figure 5 (symbols) hereby contains the same structural information as does the p′(r) in Figure 4 (]). The full line in Figure 5 represents the best fitting y(q,aopt) of eq 6 as determined with the program ORT. aopt turned out to be about 3.6 ( 0.1 nm, and the associated intensity-weighted distribution Di(b) is displayed in Figure 6. The opposite scenario of optimizing b by simultaneously calculating Di(a) did not yield consistently stable values of bopt, which is presumably due to the systematic error introduced by imposing the oblate model. The same unstable (drifting) behavior was observed when the axial ratio (a/b) was optimized simul-
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von Berlepsch et al.
Figure 6. Intensity-weighted distribution of semiaxis b obtained from the polydispersity analysis of the experimental scattering function I′D (q) shown in Figure 5. The optimized value of the second semiaxis of the prolate ellipsoids turned out to be aopt ) 3.6 ( 0.1 nm. Table 3. Micelle Characteristics for 20 mM SSPOM Solutions in the Presence of 25 mM NaCl and at 50 °C ORT analysis aggregation number σ/〈Nagg〉 equivalent sphere radii (nm) a/x〈b2〉 A0 (nm2)
199a 0.306 3.27c 1.15 0.68e
Guinier analysis 207 ( 10b 3.47 ( 0.06d
b c R 1/3 d 1/2 e a 〈N agg〉. 〈Nagg〉w. eq ) (3〈V〉/4π) . Req ) (5/3) Rg. A0 ) 2πa(〈b2〉)1/2[x(1 - 2) + arcsin /]/〈Nagg〉, with 2 ) 1 - 〈b2〉/a2.
taneously with the calculation of Di(a) or Di(b). We thus believe that the system is best described by a polydisperse prolate system with relatively small polydispersity in a, that is, cylindrical particles with length 2aopt ) 7.2 nm and varying diameter in the range 4 e 2b e 9 nm. 4. Micelle Characteristics. To discuss the micelle characteristics, it is useful to calculate the moments of the distribution function Di(b), defined by
∫Dibn db 〈b 〉 ) ∫Di db n
(11)
The (intensity-) averaged aggregation number, 〈Nagg〉 ) 〈V〉/Vm follows from the second moment 〈b2〉 via 〈Nagg〉 ) (4π/3)a〈b2〉/Vm, and the respective relative standard deviation σ/〈Nagg〉 is given by (〈V2〉 - 〈V〉2)1/2/〈V〉 ) (〈b4〉 〈b2〉2)1/2/〈b2〉. Besides, by applying the Guinier approximation (eqs 3 and 4), the weight-averaged aggregation number 〈Nagg〉w is obtained from the zero-angle scattering intensity I0. In Table 3 are listed the various averaged aggregation numbers, the relative standard deviation, the respective equivalent sphere radii, and the mean interfacial area per amphiphile in the quasi-spherical micelle A 0. The good agreement between the mean aggregation numbers calculated by applying the Guinier approximation and from the form factor analysis demonstrates the consistency of the scattering data analysis. Because of the slight polydispersity of the micelles, the weightaveraged aggregation number is larger. Within the experimental accuracy, agreement is also reached with the (quencher-) averaged aggregation number from our former TRFQ study,8 amounting to 〈Nagg〉q ) 200 ( 20. Also the value of 180 ( 10 obtained from static light scattering by fitting the ionic strength dependence of the
structure factor7 (in the low-q limit) is roughly confirmed. The relative standard deviation of the aggregation number (or micelle volume) of 0.306 is a reasonable value for small quasi-spherical micelles11,13 and ranges within those values from the TRFQ study for SSPOM and its homologues.8 A corresponding polydispersity of about one-tenth of its average cross-sectional radius has also been obtained by SANS for the cylindrical SSPOM micelles at large ionic strength.12 The axial ratio of the anisometrical micelles differs only slightly from 1; that is, the particles may be considered as quasi-spherical. However, it should be noted that, in accordance with the normally observed sequence of going from prolate to rodlike micelles by changing the salt concentration, we also find a better fit of the scattering curve by assuming a prolate shape. The low anisometry subsequently supports the assumption of quasi-spherical particles in the S(q) calculation. The equivalent sphere radius given in Table 3 exceeds markedly the respective mean value of 2.73 nm of the cylindrical micelles at high ionic strength. This finding confirms the simple relations advanced by Israelachvili et al.13 for the packing of surfactant molecules in spherical and cylindrical micelles. For a molecular volume which is independent of the shape of the aggregate, the (cross-sectional) radii of spherical and cylindrical micelles, Rs and Rc, respectively, and the corresponding optimum areas per molecule are related by Rs/Rc ) 1.5 (A0c/A0s). While the experimentally determined molecular volume Vm is virtually identical for both micelle shapes, the estimated effective interfacial areas for the spheroid and for the cylinder12 are A0 ) 0.68 and 0.63 nm2 , respectively, and the predicted trend is confirmed. Finally, it should be emphasized that the presented scattering data analysis was performed under the assumption of a homogeneous scattering length density of the particles. It is clear that the used approach is only one of various approximative descriptions of the experimental scattering data. The result of contrast variation reveals a nonuniform radial scattering density distribution, that is, an internal core-shell structure. More information about this structure can in principle be derived by applying the Fourier transform techniques to the 0% D2O scattering data, where the highest contrast of the shell can be achieved. Unfortunately, the scattered intensity, of the sample with 0% D2O was too close to that of the background intensity so that the signal-to-noise ratio after the background subtraction was not good enough to allow for a quantitative description. IV. Conclusions The SANS study was aimed at a characterization of the small micelles of sodium sulfopropyl octadecyl maleate forming in aqueous solution containing 25 mmol/L NaCl. The external scattering contrast was varied by changing the composition of the solvent mixture of D2O and H2O and yields a volume per monomer which is in agreement with recent measurements. The experimental scattering curves are best described by a system of interacting and polydisperse prolate ellipsoidal micelles with an axial ratio of 1.15. The polydispersity is small, amounting to about one-tenth of its equivalent sphere radius of 3.27 nm. The equivalent-sphere radius exceeds that of the cylindrical micelles formed after changing over to high ionic strength. The mean aggregation number in the presence of 25 mmol/L NaCl is about 200 and confirms recent measurements with light scattering and time-resolved fluorescence quenching. The obtained quantitative data are in agreement with the theory of micellization and show that simple packing considerations are also useful for such surfactants
Sodium Sulfopropyl Octadecyl Maleate Micelles
as the polymerizable surfactant SSPOM possessing a complicated conformational structure. Acknowledgment. H.v.B. wishes to thank Prof. H. Mo¨hwald for supporting this work. We are grateful to K.-H. Goebel for providing the surfactant. The assistance
Langmuir, Vol. 14, No. 26, 1998 7409
of R. Wagner during data acquisition is gratefully acknowledged. The research was supported in part by the Fonds der Chemischen Industrie. LA9810408
