Langmuir 2001, 17, 4211-4215
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Temperature Dependence of the Sizes of Tetradecyltrimethylammonium Bromide Micelles in Aqueous Solutions N. Gorski†,‡ and J. Kalus*,§ Frank Laboratory of Neutron Physics, Joint Institute of Nuclear Research JINR, 141980 Dubna, Moscow Region, Russia, Institut fu¨ r Physikalische Chemie, Universita¨ t Go¨ ttingen, Tammannstr. 6, D-37077 Go¨ ttingen, and Experimentalphysik I, Universita¨ t Bayreuth, D-95440 Bayreuth, Germany Received December 22, 2000. In Final Form: April 6, 2001 By small-angle neutron scattering experiments at a concentration of c ) 105.3 mM/dm3, a decrease of the size of tetradecyltrimethylammonium bromide micelles (C14H29N(CH3)3+Br-) in aqueous solutions as a function of increasing temperature was observed. It seems that the shape of the micelles is not, as usually assumed, spherical but is slightly anisometric.
1. Introduction The self-organization of surfactants in aqueous solutions results in thermodynamically stable structures, which change their size and shape in response to changes in concentration, composition, counterion species, pressure, temperature, and other conditions. By small-angle neutron scattering (SANS), we studied the change of shape of cationic tetradecyltrimethylammonium bromide micelles (C14H29N(CH3)3+Br-, (TTA)Br) in aqueous solutions as a function of the temperature. Previous work has shown that at the concentration c chosen for our experiments (c ) 105.3 mM/dm3) and at ambient temperature (TTA)Br micelles in aqueous solution with and without added salt are of spherical shape. Aggregation numbers in aqueous solution alone were determined by classical light scattering experiments,1-6 fluorescence techniques,7-9 and viscosity measurements.10 The aggregation numbers Agg ranged from 62 to 98 around room temperature. The temperature dependence of Agg was measured, too, showing always a decrease with increasing temperature.6-9 From SANS experiments at c ) 55 mM/dm3 in aqueous solution without salt,11 a radius of 2.18 and 2.46 nm was deduced, respectively, depending on the approximations applied for the calculation of the structure factor S(Q). Q is the value of the momentum * To whom correspondence should be addressed. † Joint Institute of Nuclear Research JINR. ‡ Universita ¨ t Go¨ttingen. § Universita ¨ t Bayreuth. (1) Debye, P. Ann. N.Y. Acad. Sci. 1949, 51, 575. (2) Trap, H. J. L.; Hermans, J. J. K. Ned. Akad. Wet., Proc., Ser. B 1955, 58, 97. (3) Venable, R. L.; Nauman, R. V. J. Phys. Chem. 1964, 68, 3498. (4) Jones, M. N.; Reed, D. A. Kolloid. Z. Z. Polym. 1969, 235, 1196. (5) Kryukova, G. N.; Kasaikin, V. A.; Sineva, A. V.; Markina, Z. N. Colloid J. 1978, 40, 381. (6) Beesley, A. H. Ph.D. Thesis, University of Minnesota, Minneapolis, MN, 1990. (7) Dorrance, R. C.; Hunter, T. F. J. Chem. Soc., Faraday Trans. 1 1974, 70, 1572. (8) Malliaris, A.; LeMoigne, J.; Sturm, J.; Zana, R. H. J. Phys. Chem. 1985, 89, 2709. (9) Miller, D. D. Ph.D. Thesis, University of Minnesota, Minneapolis, MN, 1988. (10) Guveli, D. E.; Kayes, J. B.; Davis, S. S. J. Colloid Interface Sci. 1979, 72, 130. (11) Gorski, N.; Gradzielski, M.; Hoffmann, H. Langmuir 1994, 10, 2594.
transfer of the neutrons and is given by Q ) 4π sin(φ/2)/λ, where λ is the incident neutron wavelength and φ is the scattering angle. Interaction between the micelles results in a structure factor S(Q) being quite different from one at low values of Q. In fact, this behavior complicates any determination of the micellar shape. Zana et al.12 performed a detailed SANS contrast variation measurement at 20 °C, c ) 14 mM/dm3, in aqueous solution of 0.1 M KBr. Tabony13 carried out the same kind of experiments at 26 °C, c ) 120 mM/dm3, without salt. Both groups made use of partial deuteration of parts of the surfactant molecule, getting deep insight into the internal structure of the micelles. Both experiments were analyzed by assuming that the micelles are of spherical shape. The spheres were modeled as consisting of a spherical core built by the tails (C14H19), surrounded by a shell incorporating the headgroups (N(CH3)3Br) and eventually some water molecules. Tabony13 stated cautiously that there is a strong evidence that these micelles are close to being spherical. The radius of the core and the thickness of the shell were around 2 and 0.2 nm, respectively. The evaluation of the scattering length densities in the core, F1, and in the shell, F2, gave for the fully protonated (TTA)Br surfactant values of F1 ) -0.4 × 1010 cm-2 and F2 ) (0.0 ( 0.5) × 1010 cm-2 (ref 13) and F1 ) -0.36 × 1010 cm-2 and F2 ) 0.191 × 1010 cm-2 (ref 12). By use of known volumes of N(CH3)3- (0.117 nm-3),14 Br+ (0.0292 nm-3),15 the C14H29 tails (0.404 nm-3),16,17 and the well-known neutron scattering lengths of the elements, we can give estimates for the scattering length densities, giving F1 ) -0.383 × 1010 cm-2 and F2 ) -0.373 × 1010 cm-2 or F2 ) 0.165 × 1010 cm-2. The two values of F2 are calculated for ionized headgroups and un-ionized headgroups, respectively, assuming furthermore that no water molecules are incorporated into the headgroup shell. The latter was found experimentally by Tabony.13 These (12) Zana, R.; Picot, C.; Duplessix, R. J. Colloid Interface Sci. 1983, 93, 43. (13) Tabony, J. Mol. Phys. 1984, 51, 975. (14) Morini, M. A.; Minard, R. M.; Schulz, P. C.; Rodriguez, J. L. Colloid Polym. Sci. 1998, 276, 738. (15) Mukerjee, P. J. Phys. Chem. 1961, 65, 740. (16) Backlund, S.; Bergenstahl, B.; Molander, O.; Wa¨rnheim, T. J. Colloid Interface Sci. 1989, 131, 393. (17) Corkill, J. M.; Goodman, J. F.; Walker, T. Trans. Faraday Soc. 1967, 63, 768.
10.1021/la0017882 CCC: $20.00 © 2001 American Chemical Society Published on Web 06/06/2001
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estimated values for the scattering length densities coincide nicely with the results of the SANS results of both Zana12 and Tabony.13 The values of F1 and F2 have to be compared with the scattering length density of pure D2O, F(D2O) ) 6.38 × 1010 cm-2, and pure H2O, F(H2O) ) -0.558 × 1010 cm-2. For a proper evaluation of the experimental results, Tabony13 had furthermore to assume a size polydispersity of 15%. It is well-known that the critical micelle concentration (cmc) and the size distribution of micelles depend on thermodynamic values.18,19 The value of the cmc is around 3.8 mM/dm3 at ambient temperature16,20,21 but increases steadily by a factor of ∼11 between 5 and 166 °C.6,19 Evans and Wightman19 found that the Gibbs free energy of micellization ∆G increases weakly with temperature, whereas both the enthalpy ∆H and the entropy times temperature T∆S decrease strongly with temperature. Therefore, large but compensating changes in the enthalpy and entropy are characteristic of the micellation processes in (TTA)Br. Large but compensating changes in the enthalpy and entropy are characteristic of many possible processes in aqueous solutions and impede one from extracting any useful conclusion from the observed dependence of ∆H and T∆S on temperature,22 but because of experiments21 with an ethyleneglycol-water binary mixture of (TTA)Br the dependence of ∆H on temperature is probably related to the destruction of the ordered aqueous regions surrounding the hydrocarbon chain of the surfactant with increasing temperature. 2. Experimental Details (TTA)Br was purchased from Aldrich in 99% purity and used without further purification. D2O was obtained from VEB Berlin Chemie, Berlin Adlershof, in 99.9% isotopic purity. The SANS experiments were performed at the time-of-flight spectrometer MURN23 of the pulsed reactor IBR-2 in Dubna, Russia. The samples were contained in 1 mm path length quartz glass Hellma cells. The temperature of the cells was kept constant in a range of (0.5 °C by means of a thermostat. Conversion of the scattered intensities into absolutely differential cross sections was done using an internal calibration standard (vanadium). For all the samples, D2O/H2O was used as solvent in order to perform contrast variation experiments. Background scattering was subtracted by comparison with corresponding pure D2O/H2O samples. The data treatment was done according to the standard procedures.24 The calculations of the theoretical scattering intensity distributions were performed by a numerical convolution with the instrumental resolution function. This function shows, contrary to steady-state reactors, a complicated but well-known dependence of Q.
3. Theory The absolute scattering cross section dσ/dΩ used for our data evaluation was due to a shell model of ellipsoids of revolution:25 (18) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 84, 1044. (19) Evans, D. F.; Wightman, P. J. J. Colloid Interface Sci. 1982, 86, 515. (20) Candau, S.; Hirsch, E.; Zana, R. J. Colloid Interface Sci. 1982, 88, 428. (21) Ruiz, C. C. Colloid Polym. Sci. 1999, 277, 701. (22) Evans, D. F.; Miller, D. D. In Organized Solutions, Surfactants in Science and Technology; Friberg, S. E., Lindman, B., Eds.; Marcel Dekker, Inc.: New York, 1992; p 40. (23) Ostanevich, Yu. M. Makromol. Chem., Makromol. Symp. 1988, 15, 91. (24) Vagov, V. A.; Kunchenko, A. B.; Ostanevich, Yu. M.; Salamatin, I. M. JINR Report, P14-83/898; Joint Institute of Nuclear Research: Dubna, Russia, 1983. (25) Guinier, A. X-ray Diffraction; Freeman: San Francisco, CA, 1963.
Gorski and Kalus
dσ ) S(Q) N dΩ
∫0π/2F2(δ) cos δ dδ
(1)
F ) (F1 - F2)V1 F1(A1) + (F2 - F3)V2 F2(A2) Av ) Qbv/xsin2 Rv + ((bv/av)cos Rv)2 Rv ) arctan((bv/av)tan δ) Fv(Av) ) 3(sin(Av) - Av cos(Av))/(Av)3 Vv ) 4πav2bv/3 v ) 1, 2 Formally, δ is the angle between the scattering vector Q B B and the axes of revolution b1 of the ellipsoid. Basically, Q is the momentum transfer of the scattered neutrons and is oriented perpendicular to the direction of the incoming neutrons. The core has the shape of an ellipsoid of revolution with axes 2a1, 2a1, and 2b1, volume V1, and scattering length density F1. The shell, surrounding the core and having the scattering length density F2, is modeled by an ellipsoid of revolution, too, with axes 2a2, 2a2, and 2b2. (a2 ) a1 + d; b2 ) b1 + d; d is the thickness of the shell.) The scattering length density of the solvent (H2O/ D2O) is F3. (Notice that for spheres with a shell bv/av ) 1. Furthermore, for a homogeneous sphere with radius a2 and with scattering length density F2 one has to put F1 ) F2 and d ) 0.) N is the number density of the micelles; S(Q) is the structure factor and is related to the interaction between the micelles. For a sufficently high value of Q or for noninteracting micelles, S(Q) becomes 1. For the data evaluation, we used a structure factor as proposed by Hayter and Penfold.26 S(Q) requires as input the volume fraction of the micelles, the charge Z0, the Debye screening length 1/κ, and the radius R of the micelles. In this theory, the micelles are assumed to be monodisperse spheres, but as long as (bv/av) is near 1, this theory can be used as a good approximation for prolate or oblate micelles. The volume of a surfactant molecule bound in micelles is V0 ) 0.550 nm3.16,17 One has to be aware of the fact that for example Z0 depends on the chosen theory used for S(Q). An example for this behavior can be seen in ref 11. 4. Experimental Results and Discussion In Figure 1, we present two intensity distributions showing the result of our fit procedure for D2O as solvent and assuming that the micelles are homogeneous spheres. This is a good approximation because F1 and F2 are nearly equal and much less than F3. As can be seen in Table 1, the radius R of the spheres decreases with increasing temperature. The screening length (1/κ), the charge Z0, and the absolute scattering cross section, extrapolated to Q f 0, (dσ/dΩ)0 ) ((dσ/dΩ)/S(Q))Qf0, were treated as fit parameters, too, and depend on temperature. If we assume that the degree of dissociation R of the bromine ions is around 0.25, we can estimate (1/κ) ≈ 1.6 nm. This value is of the order of the fitted values. The aggregation numbers Agg are given by the spherical volume and the known volume of a surfactant molecule V0. Both classical light scattering and fluorescence techniques give smaller aggregation numbers. In the case of the classical light scattering technique, the aggregation numbers are obtained by way of a measurement of an absolute scattering (26) Hayter, J. B.; Penfold, J. Mol. Phys. 1981, 42, 109.
Temperature Dependence of Micelle Size
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Figure 1. Scattering intensity for (TTA)Br micelles in D2O. The solid line is due to a fit described in the text. 4, 278 K; 0, 354 K. Notice that most of the error bars are smaller than the size of the symbols.
cross section, whereas in our case the size of the spherical volume is obtained by a straightforward analysis of the shape of the structured scattering curve, irrespective of the absolute scattering power. Therefore, in the latter case one can expect to get more reliable results. The degree of counterion dissociation of micelles is given by R ) Z0/ Agg and increases with increasing temperature. The values of R as well as their temperature dependence are in reasonable accordance with the results of an electrical conductivity measurement.19,21 The charge density at the micellar surface depends on R and the size of the micelles and increases steadily with increasing temperature. Using eq 1, we can calculate (dσ/dΩ)0, giving with the number density of the (TTA)Br molecules, c,
(dσ/dΩ)0 ) (∆F)2(c - cmc)V0 4πR3/3
(2)
where ∆F ) F3 - F1 and we assumed that F1 ) F2. Equation 2 gives the possibility to calculate ∆F, because all other values are known. ∆F is nearly independent of temperature (see Table 1), but according to the observations of refs 12 and 13 we had expected that ∆F would be around F3 ) 6.38 × 1010 cm-2. (F1 and F2 are compared with F3 nearly zero.) Taking slightly different models, such as the introduction of a variance of the radius, or assuming that the variation of the scattering length density at the border of the micelles is not steep but smooth within a transition region of some tenth of a nanometer does not change this discrepancy at all. We tested the absolute scale of our results by a SANS experiment performed at the SANS apparatus in Ju¨lich,27 where the determination of the absolute value of the scattering cross section is treated differently, but again the result did not change. In consideration of this, we tested whether another shape of the micelles, namely, a core-shell model of prolate micelles of ellipsoidal shape, can explain the measured differential cross section, too. In Figure 2, we present the results of fits to the measurement at 293 K. We can explain the scattering intensity distribution by assuming spheres (upper curve in Figure 2), prolate ellipsoids of revolution (middle curve in Figure 2), and oblate ellipsoids of revolution (lower curve in Figure 2). Furthermore, we show in Figure 2 the structure factor S(Q) as fitted to the lower curve. For (27) Schwahn, D.; Meier, D.; Springer, T. J. Appl. Crystallogr. 1991, 24, 568.
convenience, we present in Table 2 some relevant values as obtained by the fit procedures. Within the restricted Q-region in our experiment (0 < Q < 2 nm-1) and taking into account the statistical errors, we are not able to make a distinct decision as to which one of the models is better. To improve the situation, one should increase the statistics and increase the range of Q going up to at least Q ∼ 5 nm-1 (where the intensity is very low), but even then it is not straightforward to get a reliable result, because the fit parameters (for example the charge Z0) describing the theory of the structure factor S(Q) can give results which contradict each other, as suggested by the results of Table 2. If the micelles are close to being spherical, it is extremely difficult to prove whether they are really spherical, provided they have a polydispersity of the radius, as assumed by Tabony,13 or whether they are slightly anisotropic, showing eventually no polydispersity. The thickness of the shell is fixed to a value of 0.17 nm. This value is near Tabony’s and Zana’s values of d ) 0.20 ( 0.01 and 0.18 nm, respectively,12,13 and guarantees, as tested by us, a good reproduction of Tabony’s measurements with differently deuterated surfactants as shown in his paper13 (see his Figures 1, 2, 6, and 8). In both cases, Tabony’s and ours, the number of fit parameters is the same. Instead of Tabony’s parameter describing polydispersity for spheres, we introduced the length of the prolate (or oblate) ellipsoids of revolution. The screening length (1/κ) was fixed to a value of 1.8 nm; this is the mean value of (1/κ) of preceding fits, where (1/κ) was a fit parameter and is near the precalculated value shown above. We observed (see Table 3) that the two small axes a1 of the ellipsoid of revolution change their values slightly from 1.86 to 1.74 nm between 278 and 333 K. These values are not far from the expected length of the hydrocarbon chain, which yields 1.92 nm, using Tanford’s equation.28 The large axis b1 obviously depends strongly on temperature. Compared to the model with spherical micelles, we found within the margins of the errors no change of the degree of dissociation R, but both Z0 as well as the aggregation numbers Agg are less by about 10-19%. We observed as mentioned above that for micelles of spherical or ellipsoidal shape the goodness of the fits was nearly the same; therefore, no decision as to which of the two shapes the (TTA)Br micelles adopt in reality can be obtained at this level of knowledge. We are still left with the problem of the absolute value of (dσ/dΩ)0. If we take for both F1 and F2 their mean value Fm as found by the contrast variation experiment (which will be explained below) or the individual values of F1 and F2 obtained by Zana12 and Tabony13 and use the fitted values of the axes of the ellipsoid of revolution, one can calculate the absolute cross section (see eq 1), but again the calculated intensity differs from the measured one. The difference between calculated and measured absolute cross section is nearly the same as found for the model of spherical micelles. One can argue that the shell of our shell model contributes significantly differently to the scattered intensity, but the experiments made by Zana12 and Tabony13 using likewise partial deuteration of the shell support strongly the simple model assumed by both and by us. The hope to solve the discrepancy of measured and calculated absolute intensity by using anisotropic micelles instead of spherical micelles therefore was fruitless. One possible explanation for this discrepancy might be that the number density of monomers in solution, which traditionally is assumed to depend only very weakly on concentration c and therefore is approximated by good (28) Tanford, C. J. Phys. Chem. 1974, 78, 2469.
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Table 1. The Absolute Cross Section (dσ/dΩ)0 at Q ) 0, the Radius of Spheres R, the Debye Screening Length (1/K), the Charge Z0 (Given in Units of the Elementary Charge), the Aggregation Number Agg, the Degree of Counterion Dissociation r, the cmc, the Contrast ∆G (As Determined via the Absolute Scattering Cross Section), and the Goodness of Fit Parameter χ2 of Spherical (TTA)Br Micelles at Different Temperaturesa T/°C
(dσ/dΩ)0 cm-1
R/nm
(1/κ)/nm
Z0
Agg
R
278 ( 0.5 285.5 ( 0.5 293 ( 0.5 302 ( 0.5 312 ( 0.5 335 ( 0.5 354 ( 0.5
8.59 ( 0.05 7.79 ( 0.05 7.08 ( 0.04 6.42 ( 0.04 5.87 ( 0.04 4.77 ( 0.04 4.17 ( 0.04
2.74 ( 0.01 2.64 ( 0.01 2.55 ( 0.01 2.46 ( 0.01 2.38 ( 0.01 2.23 ( 0.01 2.11 ( 0.01
2.9 ( 0.1 2.8 ( 0.1 2.7 ( 0.1 2.5 ( 0.1 2.5 ( 0.1 2.3 ( 0.1 2.0 ( 0.1
28 ( 1 29 ( 2 28 ( 1 28 ( 1 26 ( 1 26 ( 1 26 ( 1
157 ( 2 140 ( 2 126 ( 2 113 ( 2 103 ( 2 84 ( 1 72 ( 1
0.18 ( 0.01 0.21 ( 0.01 0.22 ( 0.01 0.25 ( 0.01 0.25 ( 0.01 0.31 ( 0.01 0.36 ( 0.01
a
cmcb mM/dm3
∆F/1010 cm-2
χ2
3.7 3.9 4.2 5.5 7.1
5.43 ( 0.03 5.47 ( 0.03 5.50 ( 0.04 5.53 ( 0.04 5.57 ( 0.04 5.57 ( 0.05 5.71 ( 0.05
2.7 2.3 2.0 1.8 2.1 1.6 1.6
The spheres are assumed to be homogeneous. b Data are from ref 19.
Figure 2. Scattering intensity for (TTA)Br micelles in D2O at 293 K. The solid lines are due to fits for different models. The upper curve is due to spheres, the middle curve is due to prolate ellipsoids of revolution, and the lower curve is calculated for oblate ellipsoids of revolution. For convenience, the curves are shifted for one unit. For the lower curve, the calculated structure factor S(Q) (see related scale on the right-hand side of the figure) is presented. A list of important fit parameters is given in Table 2. Table 2. The Goodness of Fits χ2 for the 293 K Scattering Curve for Different Shapes of the Micellesa a/nm b/nm Z0 sphere 1 sphere 2
2.6 2.5
2.6 2.5
χ2
28 2.0 prolate 26 1.8 oblate
a/nm b/nm Z0 2.0 3.0
3.6 1.8
χ2
23 1.8 25 1.6
a Sphere 1 is a homogeneous sphere with radius a, sphere 2 is a homogeneous sphere up to a with constant scattering length density F1, followed by a transition layer, where the scattering length density F(r) changes monotonically between F1 and F3, the scattering length density of D2O. F(r) is modeled by a Gaussian with a width of 0.14 nm. Prolate and oblate are prolate and oblate ellipsoids of revolution with axes a and b, where b is the axis of symmetry. Z0 is the charge of the micelle.
reasons by the value of the cmc itself, is much higher in the high-concentration (TTA)Br solution used in our experiments. “Much higher” means a factor in the region of 6-7! This hypothesis seems to be improbable, but it is well-known that the values of the cmc of (TTA)Br micelles depend distinctly on the content X of D2O of the solvent,20 indicating that cmc can be influenced easily by some slight changes of the near neighborhood of the micelles. Furthermore, we mention that small-sized aggregates, consisting of a few molecules only (oligomeres), were found
in tetraethylammoniumperfluorooctylsulfonate micelles,29 in gemini surfactants,30 and probably in dodecyltrimethylammonium-hydroxid.31 In reference 29, it was shown that about one-fifth of the molecules were bound to these small aggregates at an overall concentration of 100 mM/ dm3. Under such conditions, small aggregates will give in a SANS experiment vanishing intensities in the observed Q-region, because basically the scattered intensity of a micelle at Q f 0 is proportional to the square of the volume of the aggregate and the typical width of the scattering curves of such small aggregates becomes very large, giving in the restricted Q-region of our experiment a flat, unstructured background intensity. Such a mechanism can indeed be an explanation of the missing intensity, if about one-fourth of the (TTA)Br molecules are bound in small-sized aggregates. Another possibility may be related to the evaluation of S(Q). S(Q) influences the scattering intensity quite drastically at Q-values below 0.8 nm-1 (see Figure 2). We tested this by repeating fits, where all measured points below 0.9 nm-1 were omitted. This procedure gives more weight to the intensity at high Q-values, where the influence of S(Q) diminishes, but again, the discrepancy did not disappear. To clear up the ∆F problem, we performed contrast variation experiments at different temperatures to get more information about the mean values Fm of the scattering length density of the micelles. Because the size of the micelles depends strongly on temperature, eventually Fm depends strongly on this parameter, too. The relative shape of the intensity distribution of the scattered neutrons of these contrast variation experiments is within the errors independent of the chosen X-values. X is the mole fraction of D2O of the solvent. (Notice that the relative shape should change drastically as soon as the scattering length density of the water (D2O-H2O mixture) has a value near F1 or F2,32 but in our experiments the scattering intensity was lost in the background at the low relevant X-values where X < 0.2.) Of course, the absolute intensity depends strongly on X. From the contrast variation experiments, we can extract the mean scattering length density Fm ) ∫F(r) dV/∫dV of the micelles, where the integral has to be taken over the volume of the micelle. This experimentally determined value does not depend on any models of the internal structure or shape of the micelles. Introducing the shell model, we get Fm ) (F1V1 + F2V2)/(V1 + V2). If F1 ≈ F2, as found experimentally, we get Fm ≈ F1. Fm is obtained from an evaluation of the socalled matching point, defined as giving (dσ/dΩ)0 ) 0. Fm shows values between -0.12 × 1010 and -0.38 × 1010 cm-2 (29) Bossev, D. B.; Matsumoto, M.; Nakahara, M. J. Phys. Chem. B 1999, 103, 8251. (30) Song, L. D.; Rosen, M. J. Langmuir 1996, 12, 1149. (31) Morini, M. A.; Minardi, R. M.; Schulz, P. C.; Rodriguez, J. L. Colloid Polym. Sci. 1996, 274, 854. (32) Herbst, L.; Kalus, J.; Schmelzer, U. J. Phys. Chem. 1993, 97, 7774.
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Table 3. The Axes a1 and b1 of the Ellipsoids of Revolution, the Charge Z0, the Aggregation Number Agg, the Degree of Counterion Dissociation r, and the Goodness of Fit Parameter χ2 as Function of Temperature Ta T/°C
a1/nm
b1/nm
Z0
Agg
R
Fm/1010 cm-2
χ2
278 ( 0.5 283 ( 0.5 293 ( 0.5 313 ( 0.5 333 ( 0.5 354 ( 0.5
1.86 ( 0.05 1.80 ( 0.05 1.79 ( 0.05 1.77 ( 0.10 1.74 ( 0.10 1.80 ( 0.15
4.00 ( 0.20 3.95 ( 0.20 3.45 ( 0.20 2.90 ( 0.30 2.60 ( 0.20 2.3 ( 0.2
23 ( 1 24 ( 1 23 ( 1 22 ( 1 21 ( 1 19 ( 2
131 ( 10 122 ( 10 106 ( 9 88 ( 8 74 ( 7 73 ( 11
0.17 ( 0.01 0.19 ( 0.02 0.22 ( 0.02 0.25 ( 0.03 0.28 ( 0.03 0.26 ( 0.04
-0.38 ( 0.17 -0.23 ( 0.06 -0.14 ( 0.04 -0.12 ( 0.02 -0.15 ( 0.04
2.5 2.8 1.8 2.1 1.7 1.7
a The thickness of the shell and the Debye screening length (1/κ) were kept constant for all fits and temperatures and had the values of 0.17 and 1.8 nm, respectively. Fm is the value of the mean scattering length density of the micelles. More details are explained in the text. At 354 K, no contrast variation experiment was performed and the values of a1 and b1 are highly correlated.
like prolate ellipsoids of revolution, where the degree of anisotropy decreases with increasing temperature, because then one of the axes, the a1-axis, can stay nearly constant, having a value near the length of the hydrocarbon tail, 1.92 nm, whereas the other axis, the b1-axis, can increase with decreasing temperature, with the constraints of equal number density of the tails. In reality, taking into account our limited Q-region and the error bars of the measured intensities, we cannot give a clearcut argument as to whether the micelles are of prolate or oblate shape. With both assumptions, we can reproduce Tabony’s results,13 but up to now micelles of oblate shape are rare for surfactants having hydrocarbon tails33 and more common for surfactants with fluorocarbon tails.34 5. Conclusion Figure 3. The results of contrast variation experiments for fully protonated (TTA)Br micelles at 283 and 333 K. Notice that the abscissa is given by the mole fraction X of D2O of the solvent. The ordinate is the square root of the absolute scattering cross section. Error bars are smaller than the sizes of the symbols.
and seems to depend not or only weakly on temperature (see Table 3 and Figure 3). This is an indication that the number density of the surfactants in the core and in the shell is independent of temperature. An almost constant number density can be expected for prolate- or oblateshaped micelles. To the contrary, for spherical micelles with radii changing with temperature between 2.74 and 2.11 nm (see Table 1), such a behavior is improbable on reason of packing constraints. (Notice that the length of the hydrocarbon tail amounts to 1.92 nm.) The results of the contrast variation experiments therefore are more in favor of anisometric micelles, but we mention that this is an indirect conclusion, based only on packing constraints. The contrasts ∆F ≈ F3 - F1 or F3 - F2 (see eq 2) for the measurements as presented in Table 1 with X ) 1, that is, in pure D2O, therefore should be around 6.5 × 1010 cm-2, which is by far distinct from the values quoted in this table (F1 ≈ F2 ≈ Fm and F3 ) 6.38 × 1010 cm-2). Therefore, our experimental results give an indication that (TTA)Br micelles are of slightly anisotropic shape
Between 278 and 354 K, the aggregation number of the (TTA)Br micelles decreases to about one-half. Despite such a large volume effect, we observed that the scattering length density, which is intimately related to the number density of the (TTA)Br molecular tails, remains constant. Indirectly, by reasons of packing constraints, we found evidences that (TTA)Br micelles have an anisometric shape. Apart from micelles, small-sized aggregates (oligomers) were present. The presence of oligomers was again indirectly deduced by an evaluation of the absolute scattering intensities. Electric birefringence experiments give no indication of any alignment of these micelles in an electric field,35 but this is not a contradiction to our model of slightly prolate micelles because birefringence might be very weak and therefore it can become difficult to see a birefringence signal. Acknowledgment. This research was supported by the BMBF of the Bundesrepublik Deutschland under Grant 03-DUBBAY-1. We thank the Forschungszentrum Ju¨lich for their support. LA0017882 (33) Bergstro¨m, M.; Pedersen, J. S. Langmuir 1999, 15, 2250. (34) Boden, N.; Corne, S. A.; Holmes, M. C.; Jackson, P. H.; Parker, D.; Jolley, K. W. J. Phys. 1986, 47, 2135. (35) Valiente, M.; Thunig, C.; Munkert, U.; Lenz, U.; Hoffmann, H. J. Colloid Interface Sci. 1993, 160, 39.
