408
Langmuir 2008, 24, 408-417
Ambiguity in Determining the Shape of Alkali Alkyl Sulfate Micelles from Small-Angle Scattering Data Szabolcs Vass* Research Institute of Applied Chemistry, UniVersity of Miskolc, P.O. Box 2, H-3515 Miskolc-EgyetemVa´ ros, Hungary
Jan Skov Pedersen Department of Chemistry and iNano Interdisciplinary Nanoscience Center, UniVersity of Aarhus, 140 Langelandsgade, DK-8000 Aarhus C, Denmark
Josef Plesˇtil Institute of Macromolecular Chemistry, 2 HeyroVsky Square, 16202 Prague, Czech Republic
Peter Laggner Institute of Biophysics and X-ray Structure Research, 6 Schmiedlstrasse, A-8042 Graz, Austria
Eszter Re´tfalvi Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary
Imre Varga and Tibor Gila´nyi Department of Colloid Chemistry, Eo¨tVo¨s Lora´ nd UniVersity, P.O. Box 32, H-1518 Budapest, Hungary ReceiVed July 17, 2007. In Final Form: September 25, 2007 Model fitting to small-angle scattering patterns from a series of dilute sodium- and cesium alkyl sulfate micellar solutions results in two significantly different sets of best-fit parameters for each solution. One of the sets defines nearly monodisperse prolate ellipsoids; the other defines slightly, but significantly, polydisperse oblate ellipsoids. In the prolate and oblate minimum locations, the mean form and structure factors as well as the mean core volumes are equal within the experimental error such that the axial ratios are approximately the reciprocals of each other. The experimental finding is numerically generalized: it is demonstrated that, in a Q range, the upper limit of which depends on the axial ratio, the squared mean and the mean square of the scattering amplitude from homogeneous ellipsoids with equatorial radii and axial ratios, respectively (r,η) and (rη2/3,1/η), are indistinguishable in practice. In dilute solutions without added salt, neither the best-fit values of the model parameters nor the available thermodynamic models provide direct evidence for the conformation, although the prolate ellipsoidal shape is indirectly supported by experiment. The elongated conformation of ionic micelles in dense and/or salinated systems seems realistic.
I. Introduction Reliable determination of the micellar shape from scattering experiments is a difficult task both for conceptual and technical reasons. Numerous calculations have been presented for the form factor B2(Q) of single scatterers of different shape. If the characteristic dimensions of the scatterers are not widely different, their form factors differ from one another only slightly in the Q range reachable in practice;1-4 for colloids, the problem is discussed in detail in ref 3. In micellar solutions, more than 1017-1018 micelles are present, causing the scattering cross section (intensity) dΣ/dΩ from the sample to be strongly affected by the intermicellar interference * Corresponding author. (1) Guinier, A.; Fournet, G. Small-Angle Scattering of X-Rays; John Wiley: New York, 1955. (2) Glatter O., Kratky O, Eds. Small-Angle X-Ray Scattering; Academic Press: New York, 1982. (3) Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys. 1983, 79, 2461-2469. (4) Svergun, D. I.; Feigin, L. A. Small Angle X-ray and Neutron Scattering; Nauka: Moscow, 1986 (in Russian).
expressed by the structure factor S(Q) ) 1 + (nmic - 1/V)∫gmic(r) exp[-iQr]dr, where nmic is the number of micelles per unit volume, V is the volume of the solution, gmic(r) is the paircorrelation function of the micelles, Q ) (4π/λ) sin(θ/2) is the magnitude of the scattering vector Q, λ is the wavelength, and θ is the scattering angle. For monodisperse micelles (which have the same aggregation number), dΣ/dΩ is given by
dΣ ) nmic[〈∆B2(Q)〉 + 〈B(Q)〉2 × S(Q)] dΩ
(1)
〈∆B2(Q)〉 ) 〈B2(Q)〉 - 〈B(Q)〉2, and 〈B(Q)〉 and 〈B2(Q)〉 are the mean and mean square amplitudes, respectively, calculated over the micellar manifold. The scattering cross section depends on quantities related to the mean micellar shape and to its mean square fluctuation. Realistic micellar systems, however, may consist of micelles with different aggregation numbers (i.e., they may be polydisperse). Let wi be the Gaussian probability of finding a micelle with aggregation number i:5
10.1021/la702139n CCC: $40.75 © 2008 American Chemical Society Published on Web 12/13/2007
Determining the Shape of Alkyl Sulfate Micelles
wi )
1
x2πp
[
exp -
(i - nj)2 2p2
]
Langmuir, Vol. 24, No. 2, 2008 409
(2)
where nj is the mean aggregation number, and the squared polydispersity parameter p is its variance. The mean and mean square amplitudes in polydisperse systems equal B(Q) ) ∑i wi〈Bi(Q)〉 and B2(Q) ) ∑i wi〈Bi2(Q)〉, respectively. If p/nj < 0.1, which is a frequent case, it is believed that the interference from the micelles can be described by an average structure factor S(Q), and, if one introduces the notation ∆B2(Q) ) B2(Q) B(Q)2, the intensity from the polydisperse system takes the same form as that from the monodisperse system:
dΣ ) nmic[∆B2(Q) + B(Q)2 × S(Q)] dΩ
(3)
The functions 〈B(Q)〉, 〈B2(Q)〉, and S(Q) are subject to modeling; after smearing the right-hand side (rhs) of eq 3 with the collimation and wavelength spread correction functions, the smeared dΣ/dΩ is fitted to the experimental intensity I(Q). The reliability of the shape that can be deduced from smallangle scattering (SAS) patterns is thought to be limited for various reasons. First, the mean scattering amplitude is the thermodynamic average of the Fourier transform of F(r), the spatial distribution of the scattering contrast: B(Q) ) ∫〈F(r) exp[-iQr]〉dr. Since the contrast is determined by hydration, its connection with the micellar structure is not exactly known, especially if ionic components are present. Provided that the scatterers are homogeneous, the mean amplitude is defined by orientational averaging:
〈B(Q)〉 )
∫0π/2 B(Q;ϑ) sin ϑdϑ
(4)
Because of the lack of information on the intermolecular correlations inside the micelles, the accepted way of calculating the mean square of the amplitude is to evaluate the integral
〈B2(Q)〉 )
∫0π/2 B2(Q;ϑ) sin ϑdϑ
(5)
ϑ is the angle between Q and some director of the scatterer. In polydisperse systems, both B(Q) and B2(Q) are double averages, thus smearing the original Q dependence of B. Second, B(Q)2 is multiplied by another double average, the mean structure factor S(Q). Because of the crude simplifications needed when modeling the intermicellar interactions, S(Q) can usually be only poorly modeled. An exception is the case of ionic surfactant micelles: the analytically solvable model6 of intermicellar interactions is based on the Deryaguin-LandauVerwey-Overbeek (DLVO) theory.7 The model was successfully tested by small-angle neutron scattering (SANS) in a series of sodium alkyl sulfate micelles.8 Third, the scattering patterns are recorded in a finite Q range, which limits the resolution, the number of the effective model parameters, and the reliability of their best-fit values. (5) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525-1568. (6) Hayter, J. B.; Penfold, J. SQHP: A FORTRAN package to calculate S(Q) for macroion solutions. ILL Report 80HA07S, Grenoble, France, 1980. Hayter, J. B.; Penfold, J. The structure factor of charged colloidal dispersions at any density. ILL Report 82HA14T, Grenoble, France, 1982. (7) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: New York, 1948. (8) Vass, Sz.; Gila´nyi, T.; Borbe´ly, S. J. Phys. Chem. B 2000, 104, 20732081.
Under the above conditions, a priori thermodynamic knowledge would help. It is a commonly shared opinion that, because of the strong hydrophobicity of the core-forming alkyl chains, the micellar cores are compact, consisting entirely (of portions) of the alkyl chains.9,10 Provided that no long-range intermicellar interactions are present, the average shape of the aggregated surfactants is a sphere; the shape of the individual micelles fluctuates around this average. The aggregation number of spherical micelles is determined solely by the alkyl chain length. However, in solutions of ionic micelles (even without added salt present), the experimental aggregation numbers turned out to be significantly greater than the hypothetical spherical values. The reason for the discrepancy is that the local electrostatic field distorts the spherical symmetry and the alkyl chains can fit the constraint of compactness only if the average micellar shape is no longer a sphere, thereby promoting the formation of prolate or oblate ellipsoids.10,11 The conformation of ionic micelles can be studied by conventional SANS or/and small-angle X-ray scattering (SAXS). However, these methods are ambiguous in this respect in that the results they provide may be compatible with each of the possible spheroid shapes. A high-resolution SANS study12 of a 0.07 M sodium dodecyl sulfate (SDS) solution concludes that SDS micelles are spheres or ellipsoids whose axial ratio η is near unity. In a more recent work8 for sodium nonyl sulfate (SNS) (η ∼ 1) and sodium decyl sulfate (SDeS) micelles (η ∼ 1.1) were reported in 0.073 M solutions. At higher concentrations13 and with increasing alkyl chain length, η increased up to 1.8.8,13 A similar trend versus alkyl chain length was also observed in micellar solutions of cationic alkyltrimethylammonium bromide micelles.14 According to a SANS/SAXS study,15,16 cesium dodecyl sulfate (CsDS) micelles are also prolate ellipsoids, with a somewhat smaller axial ratio. The above picture, which seems to unequivocally support the prolate ellipsoidal symmetry in the structure of ionic micelles, is disturbed by the classical thermodynamic guess10,11 that the formation of oblate ellipsoids is more likely than prolate ones. What is more, the goodness of the fit turns out even better if the scattering patterns are evaluated under the condition that the micelles should be oblate.17-19 The present work was motivated by the conflicting conclusions from different laboratories. Our aim is to seek a deeper insight into the factors behind the scene, and, to this end, we have re-evaluated (previously published) SANS patterns from sodium8 and SANS/SAXS patterns from cesium15,16 alkyl sulfate micellar solutions. II. Experimental A. Materials and Methods. Although the scattering patterns and the related experimental details have been published previously in another context,8,15,16 for the sake of completeness, they are shortly (9) Tartar, H. V. J. Phys. Chem. 1955, 59, 1195-1199. (10) Tanford, C. J. Phys. Chem. 1972, 76, 3020-3024. (11) Tanford, C. Proc. Natl. Acad. Sci. U.S.A. 1974, 71, 1811-1815. (12) Cabane, B.; Duplessix, R.; Zemb, T. J. Phys. (Paris) 1985, 46, 21612178. (13) Vass, Sz. Struct. Chem. 1991, 2, 375-397 (167-189). (14) Berr, S. S. J. Phys. Chem. 1987, 91, 4760-4765. (15) Vass, Sz.; Plesˇtil, J.; Laggner, P.; Gila´nyi, T.; Borbe´ly, S.; Kriechbaum, M.; Ja´kli, Gy.; De´csy, Z.; Abuja, P. M. J. Phys. Chem. B 2003, 107, 1275212761. (16) Vass, Sz.; Plesˇtil, J.; Laggner, P.; Borbe´ly, S.; Pospı´sˇil, H.; Gila´nyi, T. Physica B 2000, 276-278, 406-407. (17) Berr, S. S.; Jones, R. R. M. Langmuir 1988, 4, 1247-1251. (18) Bergstro¨m, M.; Pedersen, J. S. Phys. Chem. Chem. Phys. 1999, 1, 44374446. (19) Bergstro¨m, M.; Pedersen, J. S. J. Phys. Chem. B 1999, 103, 8502-8513. Bergstro¨m, M.; Pedersen, J. S.; Schurtenberger, P.; Egelhaaf, S. U. J. Phys. Chem. B 1999, 103, 9888-9897.
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repeated here. The surfactants were prepared from high-purity alcohols (Aldrich) by sulfation with chlorosulfonic acid (Merck) at 0-5 °C; the salts were twice recrystallized from hot 1:1 benzene/ ethanol mixtures.20 Gas chromatographic analysis of the products showed an ∼3 × 10-5 mole fraction of alcohol; the sum of all other impurities was of a similar amount. The solutions were prepared from dried (∼72 h at 55 °C) salt and from bidistilled 99.6% D2O. In SANS and SAXS experiments the same solutions were studied. SANS experiments were performed in the Budapest Neutron Center at 313 K; the mean neutron wavelength λh was 3.46 Å, defined by a helical slot selector producing a nearly Gaussian wavelength distribution of relative spread ∆λ/λh ∼ 13%. Scattered neutrons were registered in pinhole geometry by a 3He gas detector, built up from a 64 × 64 array of 1 × 1 cm2 pixels. The applied Q range was 0.015-0.45 Å-1. Raw SANS patterns, recorded at 1.3 and 5.5 m sample-to-detector distances, were converted into absolute units using a 1 mm thick H2O standard. In the fitting procedure, corrections for collimation distortions and wavelength smearing were included.21 SAXS measurements were carried out on the same solution at the same temperature in the Institute of Biophysics and X-ray Structure Research, Graz, using a compact Kratky camera (MBG, Austria) attached to a rotating-anode X-ray generator (Rigaku-Denki, Japan) that produced Ni-filtered CuKR rays; the camera was equipped with a one-dimensional position-sensitive detector (MBRAUN, Germany). The values of Q varied in the range 0.01-0.6 Å-1. Experimental scattering curves were converted to absolute experimental (smeared) intensities by means of a LUPOLEN standard;15,16,22 the long-slit collimation correction4 was included in the model function. B. Empirical Model of Contrast Distribution in Micelles. In order to model the contrast distribution of the micellar structure, the monomer surfactants are divided into characteristic molecular groups, and, for each type of group, a function φj(r) is defined, which stands for the probability density of finding a group of type j in position r. The alkali alkyl sulfates are divided into methyl (CH3-, j ) 1), methylene (-CH2-, j ) 2), and sulfate headgroups (-SO4 -, j ) 3) and counterions (M+, j ) 4). In spherical micelles,23 both hydrophobic groups are assumed to be distributed according to the function
[ ( )]
r - rc nj φj(r) ) 1 - erf , for j ) 1, 2 2fj x2σc
φ3(r) )
x2πσc f3
[
exp -
]
(7)
∆rh is the distance between the terminal methylene group and the headgroup. The spatial density of the counterions is approximated by an exponential convoluted by the density of the headgroup ions: φ4(r) )
n4Λ
x2πσc f4
∫
r
0
[
exp -
(t - rc - ∆rh)2 2σc2
G)4
B(Q) )
∑ n ∆b ∫ 〈φ (r) exp[-iQr]〉dr j
j
j
(9)
(6)
(r - rc - ∆rh)2 2σc2
Λ is the damping constant of the counterion density. The mean scattering amplitude B(Q) from the micelles equals the linear combination of the Fourier transforms of the functions φj:
j)1
nj is the total number of the groups type j in the micelle, and the normalization factor fj ) ∫ φj(r)dr equals the volume occupied by them; rc and σc respectively stand for the mean core radius and the thickness of the core profile. Depending on rc and σc, both φ1(r) and φ2(r) model constant distribution inside the core and a smooth, Gaussian, hydrocarbon/water interface. The position of the headgroup ions is determined by the Gaussian density function of the terminal methylene groups: n3
Figure 1. Prolate (O)8 and oblate (4) equatorial core radii (a) and axial ratios (b) of sodium alkyl sulfate micelles from SANS as a function of nC, the number of carbon atoms in the aliphatic alkyl chain. Alkyl chain lengths from reference data25 are plotted with a solid line, and the equatorial core radii and axial ratios from the oblate-to-prolate transformation are plotted with ] (see the text); the latter are presented without error bars.
]
- Λ(r - t) dt (8)
(20) Dreger, E. E.; Keim, G. I.; Miles, G. T.; Shedlowsky, L.; Ross, J. J. Ind. Eng. Chem. 1946, 36, 610. (21) Pedersen, J. S.; Posselt, D.; Mortensen, K. J. Appl. Cryst. 1990, 23, 321-333. (22) Kratky, O.; Pilz, J.; Schmitz, P. J. Colloid Interface Sci. 1966, 21, 24-34. (23) Vass, Sz. J. Phys. Chem. B 2001, 105, 455-461.
and the coefficients ∆bj equal the excess scattering length of the corresponding groups. The considerations leading to the functions φj(r) and the (analytical) Fourier transforms of the latter were published in ref 23. The numerical generalization of the model for ellipsoids was made by dividing the mean micelle into congruent ellipsoidal shells with constant scattering length densities, which were calculated from the values of the group densities in the equatorial plane of the ellipsoid, presented in detail in ref 15.
III. Results and Discussion A. Results from Model Fitting. The fitting of the scattering patterns was made over a series of fixed values of the equatorial core radius: the changes in the conformation, if any, could be safely spotted in this way. First the SANS patterns published in ref 8 from a series of sodium alkyl sulfate micelles have been reinterpreted in terms of the core/shell model applied there. In each SANS pattern, two local minima were found in the sum of the squared differences, which respectively defined the prolate and oblate shapes. The equatorial core radii as well as the axial ratios defined in the prolate and oblate minimum locations are plotted in Figure 1 as a function of nC, the number of carbon atoms in the aliphatic alkyl chains. Although the fit confirmed the existence of two local minima, the significance of the results (compare the error bars in Figure 1) required further improvement. This was achieved by (i) changing the core/shell model for the multicomponent model outlined in eqs 6-9, (ii) using alkyl sulfates with cesium counterions, and (iii) simultaneously fitting the model to SANS
Determining the Shape of Alkyl Sulfate Micelles
Langmuir, Vol. 24, No. 2, 2008 411 Table 1. Equatorial Core Radius rc, Axial Ratio η, Core Volume Vc, Mean Aggregation Number n j , Aggregation Number n0 of the Spherical Micelle of Radius rc, and the Polydispersity Parameter p of CsDeS and CsDS Micelles in the Prolate and Oblate Minimum Locationsa
CsDeS
CsDS
rc [nm]
η
Vc [nm3]
nj
n0
1.389 (0.021 1.723 (0.029 1.617 (0.022 1.916 (0.059
1.28 (0.05 0.73 (0.02 1.27 (0.04 0.78 (0.05
14.40 (0.89 15.58 (0.88 22.46 (1.05 22.90 (2.24
48.9 (1.7 52.7 (1.8 64.1 (1.5 64.4 (1.2
38.2
p
0.037 (0.021 72.2 3.85 (2.20 50.5 0.006 (0.002 82.6 6.78 (3.25
χ2
N
Z
292 148
8.36
241 148
5.41
470 149 14.88 406 149 16.21
a The sum χ2 of squared differences with the degree of freedom N of the corresponding probability density function fχ2,N and the values of the variable Z ) (χ2/N - 1)/x2/N are also given.
Figure 2. Sum of the squared differences χ2 (a), axial ratio η (b) and polydispersity parameter p (c) as a function of the equatorial core radius rc. Results were obtained from simultaneously fitting the multicomponent model to SANS and SAXS patterns from CsDeS (open symbols) and CsDS (filled symbols) micellar solutions. For the sake of compactness, the CsDeS χ2 curve is shifted upward by 150.
and SAXS patterns recorded under the same conditions from the micellar solutions. Because in cesium decyl sulfate (CsDeS) and CsDS solutions the scattering information from SANS is mainly from the core and that from SAXS is from the shell structure, the combined use of the two types of scattering pattern constitutes more reliable, equilibrated information. Practical aspects of model fitting, molecular volume, and scattering length data are found in ref 15. The sum χ2 of the squared differences, the axial ratio η, and the polydispersity parameter p are plotted in Figure 2 as a function of the equatorial core radius rc. Following expectations, χ2 has two local minima in each micellar solution. The corresponding values of the best-fit parameters are listed in Table 1: in the lower (and relatively narrow) minimum location, they indicate the presence of prolate ellipsoids, whereas, in the higher (and wider) minimum location, they indicate oblate ellipsoids. The method of determining the best-fit values and their confidence limits is demonstrated in Figure 3 and will be discussed later in more detail. The prolate and oblate structure factors Spr(Q) and Sob(Q) and form factors Bpr(Q)2 and Bob(Q)2 respectively show very good agreement (see Figure 4). It follows from the agreement of Spr(Q) and Sob(Q) that the corresponding aggregation numbers njpr and njob should also be equal; consequently, so should the prolate and oblate micellar volumes Vpr and Vob. If one calculates the volumes from the best-fit values of rc and η as
Vpr )
4π 3 4π 3 r η ≈ r η ) Vob 3 pr pr 3 ob ob
(10)
Figure 3. Determination of the best-fit parameters from parabolic interpolation of χ2 around the minimum locations. Confidence intervals are defined by the EP parameter () 13.8); for its choice, see the text and refs 26 and 28.
they are found to be equal within a 1σ-2σ limit. As a very important feature of the prolate and oblate axial ratios, they turned out to be the reciprocals of each other, again within a 1σ-2σ limit (cf. Table 1 and Figures 1 and 3):
ηpr ≈ 1/ηob
(11)
The validity of the empirical eqs 10 and 11 have been confirmed in ref 18 where, among other things, the structural parameters of SDS micelles are discussed. Although the results (obtained from 0.5 wt % concentration solutions at temperature 40 °C, in the presence of 0.1 M NaBr) stem from fitting a structural model different from that used in the present work, both conformations have also been found. The prolate and oblate ellipsoids are both monodisperse, their aggregation numbers and axial ratios respectively equal 83 and 81, as well as 1.671 and 0.577, such that the latter figures are reciprocals of each other: 1.671 × 0.577 ) 0.97. The difference between the aggregation numbers published in the present work and that in ref 18 is due to the presence of brine in the latter.
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form:
{
〈Bn(x,η)〉 ) 3 x(η - 1)1/2 if η > 1 3 x(1 - η2 )1/2 if η < 1 2
∫1η
sin(xt) - xt cos(xt)
∫η
sin(xt) - xt cos(xt)
1
(xt)
dt (t - 1)1/2
2
(xt)2
2
dt (1 - t2)1/2
} }
(13a)
Similar equations are obtained for the squared mean of the normalized scattering amplitude:
{
〈Bn2(x,η)〉 ) 3 2 x(η - 1)1/2 if η > 1 3 x(1 - η2 )1/2 if η < 1
Figure 4. Structure factors (a), form factors of CsDeS (b) and CsDS micelles (c), obtained from the fit in the prolate and oblate minimum locations.
B. Numerical Generalization of the Empirical Results for Homogeneous Spheroids. On the basis of the relationships in eqs 10 and 11, one may expect that a spheroidal micelle of equatorial core radius rc and axial ratio η produces the same scattering functions as another spheroidal micelle of equatorial core radius r′c ) rcη2/3 and axial ratio η′ ) 1/η, and vice versa. The results of transforming the oblate equatorial core radii and axial ratios of the sodium alkyl sulfate micelles into the prolate quantities confirm this expectation: the transformed radii and the prolate best-fit values show excellent agreement, while the axial ratios show an acceptable one (see Figure 1). The empirical agreement of the best-fit values and the transformed quantities suggests that the observed phenomenon is the special case of a more general property of the scattering functions. We are seeking this property (i) by replacing the complex micelle by a homogeneous spheroid of equatorial radius r and axial ratio η and (ii) by evaluating the integrals in eqs 4 and 5 with the parameter sets (r,η) and (rη2/3,1/η). The normalized scattering amplitude Bn(Qrc,ϑ) from a homogeneous spheroid of scattering contrast ∆F equals
Bn(Q,ϑ) ) )3
3B(Q,ϑ) 4r3πη∆F
sin(Qru(ϑ)) - (Qru(ϑ)) cos(Qru(ϑ)) (Qru(ϑ))3
(12a)
where
u(ϑ) ) [sin2ϑ + η2 cos2ϑ]1/2
(12b)
Upon now setting 〈Bn(Q,ϑ)〉 in eq 4, introducing the notation x ) Qr and the new variable t ) u(ϑ), we see that the spatial average of the normalized scattering amplitude has the following
∫1η
[sin(xt) - xt cos(xt)]2
∫η
[sin(xt) - xt cos(xt)]2
1
(xt)
5
(xt)5
dt (t - 1)1/2 2
dt (1 - t2)1/2
(13b)
Because the integration in eq 13 is known to be unsolvable in analytical form, we determined the values of 〈Bn(x,η)〉, 〈Bn2(x,η)〉 and 〈Bn(xη2/3,1/η)〉, 〈Bn2(xη2/3,1/η)〉 numerically for η ) 3/2, 2, 3, and 5 in the range 0 e x e 10. From these quantities, the prolate and oblate form factors 〈Bn〉2 and fluctuation terms 〈∆Bn2〉 ) 〈Bn2〉 - 〈Bn〉2 were calculated and plotted in Figure 5a-d. The differences ∆〈Bn〉2 ) 〈Bn(x,η)〉2 - 〈Bn(xη2/3,1/η)〉2 of the prolate and oblate form factors, and ∆〈∆Bn2〉 ) 〈∆Bn2(x,η)〉 - 〈∆Bn2(xη2/3,1/η)〉, those of the prolate and oblate fluctuation terms, were also plotted in Figure 5e-h. As seen from the curves in Figure 5e-h, their difference increases with increasing deviation from the spherical shape characterized by |η - 1| (or |1 - 1/η|). The maximum deviation of the prolate form factors from the oblates at η ) 3/2, 2, 3, and 5, respectively, equals ∼ -0.003, -0.015, -0.05, and -0.13; those of the prolate fluctuation terms equal ∼0.003, 0.015, 0.04, and 0.07. The graphs in Figure 5a-d satisfactorily explain in monodisperse systems that, after the parameter transformation (r,η) f (rη2/3,1/η) the initial and the transformed form factors and fluctuation terms are equal. In the present case, however, the transformation formula (r,η) f (rη2/3,1/η) has been derived from comparing the best-fit values of fitting parameters that stem from a monodisperse and a polydisperse data set. This makes it unavoidable to check the relevance of the transformation formula in polydisperse systems, which cannot be done without a knowledge of the given micellar model. In the ladder model24 of micellar growth we used for interpreting SAS patterns in a somewhat modified form,8,15 the growth of micelles may take place in the axial direction only, and the equatorial radius r is expected to equal or, at least, to be very close to lc. The prolate equatorial radii and the lc data plotted in Figure 1 as a function of the number nC of carbon atoms in the aliphatic alkyl chains correspond to this expectation. The slopes of the straight lines fitted to the two data sets show (24) Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1976, 80, 1075-1085. Young, C. Y.; Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1978, 82, 1375-1378. Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 84, 1044-1057. Mazer, N. A.; Olofsson, G. J. Phys. Chem. 1982, 86, 4584-4593. Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1983, 87, 1264-1277.
Determining the Shape of Alkyl Sulfate Micelles
Langmuir, Vol. 24, No. 2, 2008 413
Figure 5. Prolate and oblate scattering functions 〈Bn〉2 and 〈∆Bn2〉 (a-d) and their differences ∆〈Bn〉2 and ∆〈∆Bn2〉 (e-f) as a function of x ) Qr. The prolate functions are calculated with axial ratios η ) 3/2 (a,e), 2 (b,f), 3 (c,g), and 5 (d,h); the oblate functions are obtained by replacing (r,η) by (rη2/3,1/η) (see the text). Table 2. Parameters of the Straight Lines A + B × nC Fitted in Figure 1 to the Prolate and Oblate Equatorial Core Radii versus nC, the Number of Carbon Atoms in the Alkyl Chains prolatea oblate ref datab () length of a -CH2- group) a
A [nm]
B [nm]
-0.073 (0.028 -0.621 (0.163
0.137 (0.002 0.222 (0.012 0.127
Taken from ref 8. b Taken from ref 25.
a surprisingly good experimental agreement (cf. Table 2), indicating that the growth in the equatorial radius per carbon atom almost equals the length of a -CH2- group. These findings (in agreement with the initial assumptions of the model of micellar growth) support the elongated form,8 especially if considering that the lc data stem from an acknowledged source.25 (25) Molecular Structure and Dimensions; Kennard, O., Watson, D. G., Allen, F. H., Isaacs, N. W., Motherwell, W. D. S., Peterson, D. G., Town, W. G., Eds.; International Union of Crystallography; A. Ousthoek: Utrecht, The Netherlands, 1972; Vol. A1.
The best-fit values of oblate radii obtained after re-evaluating the SANS patterns seem to contradict the earlier accepted elongated form. The transformation (r,η) f (rη2/3,1/η) applied to the oblate radii and axial ratios resulted in the corresponding prolate quantities such that the axial extension of the micelles 2robη < 2lc. Either set of the best-fit values may be the real value, as both meet the requirement stemming from a more general principle. Because the ionic (polar) head-groups energetically prefer the aqueous surrounding, the micellar dimensions are limited by the length lc of the core-forming alkyl chains, thereby promoting either an elongated structure of maximum radius lc or a disk-shaped form of maximum thickness 2lc; the micelles are allowed to grow without limitation in both conformations. Provided that the equatorial radius remains limited, any change in the aggregation number goes hand in hand with a change in the axial ratio η. In spheroidal micelles, the aggregation number, equatorial radius, and axial ratio are interdependent: denoting the volume of an aggregated alkyl chain by V0, they should meet the constraint (4π/3)r3η ) nV0 of the micellar volume. The volume (4π/3)r3 equals that of the spherical micelle of radius r, it defines the aggregation number n0 ) (4π/3)r3/V0, and it leads to the
414 Langmuir, Vol. 24, No. 2, 2008
Vass et al.
relationship η ) n/n0. In order to simplify the calculation of B h, the weights wi in B ) ∑i wi〈Bi〉 are replaced by a continuous normal distribution and the summation by integration, 〈B〉 is expressd by the integral in eq 13a and the axial ratio as n/n0. Because polydispersity was found in the oblate minimum location, the calculations are made for oblate systems:
3n × r3∆F ∫-∞n 4π 3 Qr(n 2 - n2)1/2
B h)
0
[
0
∫n/n 1
0
sin(Qrt) - Qrt cos(Qrt) (Qrt)2(1 - t2)1/2
dt
1
x2πp
exp -
(n - nj)2 2p2
]
dn
Let us replace n by the new variable nj + y, denote the inner integrand by φ(x, t) and its primitive function by Φ(x, t); with some algebra, we arrive at the relationships
(
2
)
n nj y n0 ≈ 2 1+ ) 2 1/2 2 1/2 2 n j n 2 - nj2 (n0 - n ) (n0 - nj ) 0
(
)
y 1 η j 1+ 2 1/2 n j1-η (1 - η j) j2
and developing the inner integral into a first-order Taylor series with respect to y,
∫n/n1
(
0
φ(x, t)dt ) Φ(x, 1) - Φ(x, (nj + y)/n0) ) Φ(x, 1) -
Φ(x, nj/n0) + φ(x, nj/n0)
)
y ) n0
∫nj1/n
0
φ(x, t)dt - φ(x, nj/n0)
y n0
The conditions for the above manipulations are that n0 > nj . p and n0 - nj > 3p; the parameters obtained from both micellar solutions meet them. Setting these results in the equation for B h n0 (y/x2πp) exp[-y2/2p2]dy ≈ 0 and that and considering that ∫-∞ n0 ∫-∞ (y2/x2πp) exp[-y2/2p2]dy ≈ p2, the mean scattering amplitude from a slightly polydisperse oblate micellar systems has the following form:
It follows from this result that, in the case of slightly polydisperse micelles, the scattering pattern can be well fitted by two form factors: (i) by the form factor from micelles of radius r and mean axial ratio η j and (ii) by a “ghost” form factor j . Considering now the flux, the of radius rη j 2/3 and axial ratio 1/η resolution, and the upper limit of the applicable Q range (j4-5 nm-1) of the recent small-angle diffractometers, the prolate and oblate ionic surfactant micelles with hydrophobic chain length j2-3 nm are indistinguishable if their mean axial ratio falls within the range 1/2 j η j 2. These conditions hold for the majority of ionic surfactant micellar systems without added salt. C. Are Ionic Micelles Prolate or Oblate? 1. Information From Parameter Fitting. The numerical generalization of the experimental results can be interpreted such that a spheroidal scatterer has, in the Q space, a real and a conjugate “ghost” image. If the spheroids are not too elongated (or not too suppressed), the respective scattering functions are very similar. Consequently, the question arises as to which conclusions can be drawn from the fitting on the micellar shape. In order to discuss the conclusions drawn from parameter fitting, the main features of least-squares fitting are summarized following a textbook devoted to the application of statistical methods in experimental physics.26 Model fitting means finding N ξi2 the minimum location χ02 ) χ2(P0) of the sum χ2(P) ) ∑i)1 of the squared and weighted differences ξi ) (yi - m(Qi,P))/xyi between the experimental observations yi and the model m(Q,P) at the points Qi of observation, i ) 1, ... N, with respect to the model parameters P ) {P1, P2, ... PK}. Provided that ξi’s are independent and normal variables with mean value 0 and variance 1, as will be assumed in the following treatment, χ2(P) has a chi-square distribution of N degrees of freedom with probability density function fχ2,N(u). The second-order Taylor approximation around the minimum location results in a pure quadratic expression of the quantities ∆Pk ) Pk - Pk0 (cf. Figures 2 and 3), because the first-order partial derivatives vanish in the minimum location: ∂χ2/∂Pk|P0 ) 0, k ) 1, ... K, and we have
χ (Q,P) ≈ χ (Q,P0) + 2
4π 3 3 j ob × r ∆Fη 3 ob Qrob(1 - η j ob2)1/2 1 sin(Qrobt) - Qrobt cos(Qrobt) p2 dt × η j ob (Qrobt)2(1 - t2)1/2 njn0(1 - η j ob2) j ob) - Qrobη j ob cos(Qrobη j ob) sin(Qrobη (14a) 2 2 1/2 (Qrobη j ob) (1 - η j ob )
B(Q;rob,η j ob) )
[∫
]
Provided that the polydispersity index p/nj j 0.1, the mean scattering amplitude from shape-polydisperse oblate micelles is equal to the monodisperse scattering amplitude (eq 13a) taken at the mean axial ratio η j ob of the polydisperse micelles plus a small correction term proportional to (p/nj)2. For prolate micelles, similar results are obtained; in this case, the conditions for the approximations are that nj > n0 . p and nj - n0 > 3p:
3 4π 3 r ∆Fη j pr × 3 pr Qrpr(η j pr2 - 1)1/2 η j pr sin(Qrprt) - Qrprt cos(Qrprt) p2 × dt + 1 (Qrprt)2(t2 - 1)1/2 njn0(η j pr2 - 1) j pr) - Qrη j pr cos(Qrprη j pr) sin(Qrprη (14b) (Qrprη j pr)2(η j pr2 - 1)1/2
B(Q;rpr,η j pr) )
[∫
]
2
∑ k,l
∂2χ2(Q,P) ∆Pk |P ∆Pl ) ∂Pk ∂Pl 0 χ2(Q,P0) +
∆PkVkl-1∆Pl ∑ k,l
(15)
where the second-order partial derivatives are equal to the elements of the inverse correlation matrix ||V-1||. The variables ∆Pk are, at least asymptotically, normally distributed random variables; their quadratic form ∑k,l∆PkVkl-1∆Pl has a chi-square distribution with K degrees of freedom. This property enables us to assign a probability content to the variables ∆Pk as follows: Fixing the value of the quadratic form at EP, one defines a K-dimensional hyperellipsoid within which all ∆Pk’s vary, and the cumulative chi-square distribution defines the probability of 2 this event as β ) ∫EP 0 fχ ,K(u)du. Conversely, fixing β at 0.67, one can calculate the corresponding EP, defining thus the 1σ confidence limits of the fitting parameters. When evaluating the CsDeS and CsDS patterns, the number of the variable fitting parameters was K ) 12, defining EP ) 13.8 (see Figure 3). An important question related to model fitting is how to use the experimental data to verify or disprove the fitting model or, as in the present case, to choose between two sets of best-fit values of the model parameters. If the deviations ξi are normally (26) Eadie, W. T.; Dryard, D.; James, F. E.; Roos, M.; Sadoulet, B. Statistical Methods in Experimental Physics; North-Holland, Amsterdam, 1971.
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Langmuir, Vol. 24, No. 2, 2008 415
distributed independent random variables with mean value 0 and variance 1, the least-squares fitting enables one to examine the plausibility of the χ2 value. For N > 30, the variable Z ) (χ2/N - 1)/x2/N is of normal distribution exp(-Z2/2)/x2π, the quantity χr2 ) χ2/N is called reduced chi-square; the reduced chi-squares and the values of the Z variable from the present interpretation are listed in Table 1. The Z variable values show that the deviation of the experimental χ2 values from their expectation values is 5-16 times greater than the square root of the variance, indicating that both the prolate and oblate models, in the statistical meaning of the word, are wrong, and the information arising from comparing two wrong models may be deceiving. This property of the experimental χ2 values is generally characteristic of the interpretation of SAS patterns: in the works referred to previously in the present paper,8,13-19 the χr2 data, where published, characteristically fall in the range of 2-10 with n > 50, resulting in Z > 5. Under these conditions, the comparison of χ2 values serves only for deciding which model better approaches the experimental data, but no really measurable probabilistic conclusion can be drawn on the validity of the hypotheses represented by the models. Paradoxically, the best-fit values stemming from “wrong” models in the cited papers are physically relevant and sometimes supported by independent methods. This contradiction between the statistical inefficiency and the physical applicability can be understood by bearing in mind that all micellar models published up till now are incomplete, since the molecular fluctuations of the micelle-forming components are disregarded. The fluctuations F(Q,P) are presumably represented by a slowly varying function of the structural parameters P and, as concluded from contrast variation studies,27 appear in the model as an additive term. Because the slow variation causes ∂F(Q,P)/∂Pk ≈ 0 and ∂2F(Q,P)/ ∂Pk∂Pl ≈ 0 for k, l ) 1, ... K, the condition for finding a local minimum of χ2 with the “proper” model N
∂ ∂Pk
∑i
∂ ∂Pk
(yi - m(Qi,P) - F(Qi,P))2
N
∑i
)
yi (yi - m(Qi,P))2 yi
N
+2
∑i
F(Qi,P) ∂m(Qi,P) yi
) 0 (16)
∂Pk
Keeping in mind that the fluctuations contribute to the high-Q region of the scattering patterns where the micellar model tends to vanish as ∼Q-4, the second term can be disregarded on the rhs of eq 16, and the minimum locations determined with the “proper” and the “wrong” models, at least in practice, are sufficiently close together to accept the “wrong” best-fit values as physically significant. By similar arguments the second derivatives of χ2 do not consist of F, and thus neither the best-fit values of the fitting parameters nor their confidence intervals are noticeably affected by the omission of the fluctuations from the fitting model. Let us now summarize the knowledge we can deduce from fitting the SAS patterns of the ionic micellar systems studied. On the basis of two sets of best-fit parameter values of an incomplete micellar model, a transformation rule was conceived between these values. Although the χr2 values fall out of even the 5σ limit, the numerical analysis has justified the rule. On this (27) Stuhrmann, H. B.; Miller, A. J. Appl. Cryst. 1978, 11, 325-345. (28) James, F.; Roos, M. Comput. Phys. Commun. 1975, 10, 373. James, F. MINUIT Function Minimization and Error Analysis, reference manual, version 94.1; CERN: Geneva, Switzerland, 1994.
ground, the best-fit values themselves are considered physically relevant, supported by the fact that the prolate radii result in an acceptable value for the length of the -CH2- group, which is a strong argument for the prolate conformation. From the best-fit parameter values obtained in the oblate minimum location, such a direct indication for the oblate conformation cannot be derived, although their transformed values approximate very well the corresponding prolate radii, making the origin of these latter values questionable. Under the given circumstances, only the χr2’s from a complete model could decide, which is not available; for the time being, the ambiguity in the conformations cannot be surmounted by model fitting. Even so, the presence of the observed minimum locations offers an experimental support (i) for the spheroidal shape of the scatterers and (ii) for the idea that the length of the alkyl chain plays a dominant role in those constraints that determine the possible conformations. 2. Shape of Sodium Alkyl Sulfate Micelles from the Literature. Experimental groups using various techniques conclude in an elongated spheroidal (cylindrical) form of SDS micelles. Coherent arguments have been published for the existence of elongated micellar aggregates in the dynamic light scattering (DLS) study of sodium alkyl sulfate micelles as a function of the alkyl chain length, solute concentration, and solution temperature. Since ionic micelles without added salt can be treated as point-like light scatterers, NaCl was added in various concentrations. Interpretation of the experiments is based on the assumption that the ground state of the micellar shape at cM is a sphere, the radius of which equals ∼lC, the length of the alkyl chain of the surfactant molecule. By increasing the surfactant concentration, micelles grow such that the “ground-state” sphere splits into two hemispheres, and monomolecular, circular layers of radius lC are inserted between them.24 The micellar model referred to in refs 8, 13, and 15 follows the philosophy of this model. A deeper (empirical) insight into the structure of elongated aggregates was obtained from more recent SANS studies. Partly deuterated SDS and lithium dodecyl sulfate (LiDS) micelles formed from 0.05 M surfactant in pure aqueous solvent and in dilute NaCl solutions turned out to be oblate ellipsoids. By increasing the salt concentration and keeping the cross section constant circular, both types of micelles exert oblate ellipsoidto-sphere-to-prolate ellipsoid transition.17 SDS and LiDS micelles exhibit very different responses to added salt: the oblate ellipsoids become spheres at j0.1 and 0.3 M NaCl, respectively; the sphereto-prolate ellipsoid transition takes place correspondingly at 0.10.2 and 0.6 M NaCl. From 0.6 M NaCl onward, the growth of SDS micelles is extremely rapid; substantial growth of LiDS has been reported only at higher (0.15 M) LiDS concentrations in the presence of 1.5 M LiCl.29 The aggregation of SDS and dodecyltrimethylammonium bromide (DTAB) surfactants has been investigated in the absence of added salt and in the presence of NaBr as a function of surfactant mole fraction and salt concentration.18 Up to 0.1 M NaBr, the SDS micelles were oblate ellipsoids of revolution, in the range 0.2-0.5 M NaBr, they were triaxial ellipsoids, and beyond 0.7 M NaBr they had strongly elongated structures having an elliptical cross section with semi-minor axes of 1.22-1.24 nm and semimajor axes of 1.92-2.02 nm, irrespective of surfactant and salt concentration. The DTAB micelles were oblate ellipsoids of revolution up to 0.5 and triaxial ellipsoids up to 1.0 M NaBr. In mixtures of anionic SDS and cationic DTAB surfactants (without and with added salt), different types of aggregates are formed, depending mainly on the surfactant volume fraction and mixture composition.19 If one of the components is dominant, (29) Bendedouch, D.; Chen, S.-H. J. Phys. Chem. 1984, 88, 648-652.
416 Langmuir, Vol. 24, No. 2, 2008
the shape of the aggregate is oblate ellipsoid of revolution. Upon increasing the ratio of the minor component, at small surfactant volume fractions (0.125-0.25 wt %), the formation of vesicles and lamellar structures is observed. At higher (0.5-1.0 wt %) surfactant volume fractions, one of the equatorial radii starts to increase (triaxial ellipsoids), and the aggregate undergoes a transition into an elongated structure such that the other two semi-axes suffer only a minor change, thereby ensuring a more or less unchanged elliptical cross section. The structure of SDS micelles was studied also in the presence of NaCl30 and NaBr.31 At high (>1 M) salinity, the aggregates were found to be flexible, “worm-like” structures in both studies: in the former having a circular cross section of mean radius 1.65-1.85 nm, depending on the salinity; in the latter having an elliptical cross section of semi-minor axis of 1.2 nm and axial ratio η ) 1.43, irrespective of the salt concentration. SANS experiments in 0.3 M SDS solutions without added salt resulted in monodisperse, prolate ellipsoidal micelles of equatorial core radius 1.67 nm and of axial ratio η ) 1.98; upon adding different ammonium salts (NH4X, X ) Cl, I, Br, NO3, and SCN) of 0.1 M concentration, the axial ratio, depending on X, increased to 2.52 e η e 2.70. The increase in the concentration of the added salt strongly increases the length of the aggregates, without affecting the radius of their cross section.32 In the presence of 0.1 M alkali halide (NaBr, KBr, CsBr, KCl, and KI) salts, the equatorial core radius of the micelles in 0.3 M SDS solution was found to be 1.7 nm, such that the axial ratio varied in the range 2.11 e η e 3.28.33 3. Approximate Thermodynamic Model Calculations. The incompleteness in the information deduced from model fitting on the micellar conformation could be remedied by thermodynamic models of micellization. However, these models, including the most sophisticated ones, consist of necessary simplifications, which may strongly limit their applicability. One of the limitations is summarized in ref 35 as “A complete theory of micellar structure should have the power to predict the joint shape and size distribution function f(Γ,N) and its dependence on the thermodynamic state (chemical composition, temperature, pressure). Even for the simplest case of a dilute (ideal) micellar solution, such a theory has not been developed”. Moreover, in the case of ionic micelles, the electrostatic field from the other micelles is disregarded in all models published: just that particular factor that is believed to be responsible for the deviation from the spherical shape is considered. Although the electrostatic field from the other micelles is disregarded in the present approach as well, the opening move has been done toward the prediction of the joint shape and size distribution required in ref 35 by making use of an iterative method applied to micellization in a somewhat different context for analyzing surface fluctuations.36 In the present form, the model can result in the joint shape and size distribution of micelles around the critical concentration of micelle formation. The chemical potential µmic,n of a micelle with aggregation 0 0 + kBT ln xmic,n, where Gmic,n number n is given by µmic,n ) Gmic,n is the standard free energy of the micelle with a fixed center of (30) Magid, L. J.; Li, Z.; Butler, P. D. Langmuir 2000, 16, 10028-10036. (31) Arleth, L.; Bergstro¨m, M.; Pedersen, J. S. Langmuir 2002, 18, 53435353. (32) Kumar, S.; David, S. L.; Aswal, V. K.; Goyal, P. S.; ud-Din, K. Langmuir 1997, 13, 6461-6464. (33) Aswal, V. K.; Goyal, P. S. Phys. ReV. E 2000, 61, 2947-2953. (34) Weisstein, E. W. Ellipsoid. From MathWorld: A Wolfram Web Resource. http://mathworld.wolfram.com/Ellipsoid. (35) Halle, B.; Landgren, M.; Jo¨nsson, B. J. Phys. (Paris) 1988, 49, 12351259. (36) Gila´nyi, T. Colloids Surf. 1995, 104, 111-118; Colloids Surf. 1995, 104, 119-126.
Vass et al.
mass, and xmic,n is the mole fraction of the micelles. The standard free energy of the micelles is approximated as 0 Gmic,n ≈ n∆µ0chain + σ0(A - nA1) -
[
nkBT ln
]
Γ0 - 1 + Gel (17) Γ
∆µ0chain is the standard state free energy of the alkyl chains in the micelle (the hydrophobic “driving force”), σ0(A - nA1) is the free energy of the micelle from the contact of the alkyl core with water, σ0 is the standard alkane/water interfacial free energy, A is the surface area of the micelle, and A1 is the area per monomer shielded by the polar head-groups from the micellar surface. The third term is the configurational entropy: Γ stands for the number of head-groups per surface area, and Γ0 stands for that at close packing: Γ0/Γ ) A/nA1. The surface Apr of prolate ellipsoids is given in terms of req and rax, the equatorial and axial radii (req < rax), respectively, as
[
]
arcsin(e) e
Apr ) 2πreq2 1 + η
(18a)
with the notations η ) rax/req > 1 and e ) (1 - 1/η2)1/2. The surface Aob of oblate ellipsoids (req > rax) takes the form
[
Aob ) 2πreq2 1 +
)]
η2 1 + e′ ln 2e′ 1 - e′
(
(18b)
with the notations η ) rax/req < 1 and e′ ) (1 - η2)1/2.34 Finally, Gel is the electrostatic contribution of the ions to the free energy of micelle formation. It was obtained from solving the Poisson-Boltzmann equation for a sphere having a surface area equivalent with that of the ellipsoid in question. The solution to the Poisson-Boltzmann equation needs a knowledge of the ionic strength: it was calculated from the monomer mole fraction xs. Consequently, the chemical potential of the micelles explicitly depends on the monomer concentration as well: µmic,n ) µmic,n(xs). The chemical potential µs(xs) of the monomers in solution equals µs(xs) ) µ0s + kBT ln xs, where µ0s is the standard-state chemical potential of the monomers. The chemical potentials in the monomeric and aggregated state of the surfactants should be equal in equilibrium, expressed by the relationship µmic,n(xs) ) nµs(xs). The joint shape and size distribution is obtained from three key variables: xs, η, and n; after fixing η, the equilibrium condition is determined at varying values of xs. At the equilibrium value of xs, we are in possession of µmic,n and, consequently, of xmic,n, for n ) 1, 2, ...∞; the latter series defines the distribution ∞ xmic,k of the aggregation number. The equilibwn ) xmic,n/∑k)2 rium (critical) monomer concentration is that value of xs at which ∞ xmic,k of the aggregated surfactants the mole fraction ∑k)2 abruptly starts to increase. It should be mentioned that all methods used to experimentally determine the critical micelle concentration (cmc) are similarly arbitrary. By making use of the parameters of SDS, the calculations result in 14 mM (xs ) 2.5 × 10-4) for the equilibrium monomer concentration (∼cmc), instead of the experimental 8 mM; this deviation, however, is acceptable for a self-consistent model. The mean aggregation number is realistic: it is equal to nj ∼ 66; the aggregation number distribution turned out to be rather wide, its width falling in the range p ∼ 10. The shape analysis was made at a fixed equilibrium surfactant concentration of 14 mM, selecting aggregation numbers n ) 30, 66, 100, 125, 130, and 135 from the distribution defined by the series µmic,n. Under this choice of the fixed key parameters, the
Determining the Shape of Alkyl Sulfate Micelles
Langmuir, Vol. 24, No. 2, 2008 417
not capable of predicting the actual values of micellar parameters; however, they agree in that (i) the electrostatic free energy of deformation promotes the prolate conformation for (ii) increasing aggregation numbers.
IV. Conclusions
Figure 6. Probability density of finding micelles with axial ratio η at different aggregation numbers: 66 (solid line), 125 (dotted line), and 130 (dashed line).
condition of equilibrium determines the mole fraction xη(∆η,n of micelles with axial ratio η ( ∆η. The probability density Pn(η) of finding a micelle with axial ratio η is given by
Pn(η) )
xη+∆η,n - xη-∆η,n 2xmic,n∆η
(19)
Selected results for n ) 66, 125, and 135 are plotted in Figure 6. All Pn(η) curves are asymmetric such that ηn ) ∫Pn(η)η∆η; the conditional mean value of the axial ratio at aggregation number n indicates that the average ellipsoid is slightly prolate: for the above-mentioned aggregation numbers, ηn ) 1.02, 1.02, 1.03, 1.07, 1.19, and 1.31, respectively. The mean axial ratio taken over the manifold of the micelles equals η j ) ∑n xmic,nηn/∑k xmic,k ∼ 1.02-1.03. An interesting comparison can be made with another thermodynamic model based on entirely different basic assumptions,35 with the final aim to characterize the shape fluctuations. The authors disregard the intermicellar electrostatic field, but solve the Poisson-Boltzmann equation with meticulous care in elliptic coordinates and calculate the free energy necessary for deforming a spherical micelle to obtain a particular (prolate or oblate) spheroid. This spherical micelle is assumed to have the maximum aggregation number N0 in the system, and the analysis of the free energy of deformation in three systems with different maximum aggregation numbers results in shape distributions (Figure 12 in ref 35), which resemble in their trends the curves plotted in Figure 6. In the system with the smallest N0 () 74), a symmetric shape distribution is obtained (cf. the curve with n ) 66 in Figure 6); in the intermediate case (N0 ) 104) the distribution becomes wide and flat (cf. the curve with n ) 125 in Figure 6); finally, at N0 ) 113, there is a sharp peak in the prolate region (cf. the curve with n ) 135 in Figure 6). The models presented here are
Analysis of SANS patterns from a series of sodium alkyl sulfate micellar solutions, and the combined analysis of SANS and SAXS patterns from cesium decyl and dodecyl sulfate micellar solutions led to the conclusion that the scattering patterns from all systems could be fitted by micellar models both with prolate and oblate spheroidal symmetry. The main result from the fitting was hypothesizing the relationship between the equatorial radii and axial ratios in the prolate and oblate conformations of the micelles. The relationship has been numerically proven and its applicability was analytically discussed for the case of slightly polydisperse micellar solutions such as the systems studied. It was shown that, in most ionic micellar systems without added salt, the prolate and the conjugate oblate conformations (or vice versa) are indistinguishable in relation to the present small-angle diffractometers. In order to surmount the problem caused by the ambiguity in determining the conformation of ionic micelles, we analyzed all available data. Although the results and the conclusions are diversified and sometimes contradictory, in one regard we found a significant agreement. At sufficiently high ionic surfactant and/or salt concentration (the “sufficient” depending on the type of solutes), the aggregates become elongated structures; the dimensions of their circular or ellipsoidal cross sections are limited by the length of the hydrocarbon segment of the surfactant. The thermodynamic model calculations harmonize with these experimental findings as long as the prolate conformation was found to correlate with increasing aggregation numbers. Finally, a practical aspect of the problem should be pointed out. Colloidal systems are frequently used for producing nanoparticles via emulsion polymerization37 or other38,39 methods; the growth of the particles is sometimes controlled by the scattering technique. The shape determination of these particles by the scattering technique is also limited by the ambiguities discussed above. Acknowledgment. Support from the National Foundation for Scientific Research, Hungary (OTKA, Contract Nos. T38050 and K63046), and from the NATO Science Fellowship Programme, Hungary, ID No. 2007/NATO/2 is gratefully acknowledged. LA702139N (37) Gilbert, R.G. Emulsuion Polymerization. A Mechanistic Approach; Academic Press: London, 1995. (38) Fendler, J. H. Chem. ReV. 1987, 87, 877-899. (39) Bo´ta, A.; Klumpp, E. Colloids Surf., A 2005, 265, 124-130
