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Nonionic Micelles near the Critical Point: Micellar Growth and Attractive Interaction† Otto Glatter,*,‡ Gerhard Fritz,‡ Helmut Lindner,‡ Judith Brunner-Popela,‡ Rainer Mittelbach,‡ Reinhard Strey,§ and Stefan U. Egelhaaf | Institute of Chemistry, University of Graz, Heinrichstrasse 28, A-8010 Graz, Austria, Institute of Physical Chemistry, University of Cologne, Cologne, Germany, and Department of Physics and Astronomy, University of Edinburgh, U.K. Received March 2, 2000. In Final Form: July 3, 2000 We report on a series of SANS experiments on the structure of binary water-nonionic surfactant systems accompanied by complementary ultralow shear experiments and depolarized light scattering. The analysis gives a clear picture of the temperature dependence of aqueous solutions of nonionic surfactants of the n-alkyl polyglycol ether type (CiEj) when approaching the cloud point curve. The series is based on temperature variations from 3 °C up to a temperature of about 1.5 K below the critical point Tc and concentration variations around the critical concentration cc by a factor of 3-9. Six different surfactants were studied, changing the alkyl chain length i as well as the number of ethylene oxide groups j. Excluded volume effects were taken into account in the evaluation procedure by a generalized indirect Fourier transformation procedure recently developed for the evaluation of scattering data from semidilute and dense systems. The bottom line is that all systems examined show a sphere-to-rod transition, the degree of growth and the transition temperature depending on the concentration and hydrophobicity of the surfactant. Superimposed on this transition is the onset of attractive interactions as the cloud point curve is approached, the range depending on the overall surfactant size.
Introduction Nonionic surfactants, such as oligo(oxyethylene)-n-alkyl ether (abbreviated as CiEj) show a rich phase behavior in aqueous mixtures. At very low surfactant concentrations the surfactant dissolves in the form of unimers. With an increase in the surfactant concentration, the temperaturedependent critical micelle concentration (cmc) is passed and the surfactant molecules form mostly globular micelles, at least at low temperatures. A common feature of these surfactants in mixtures with water is an upper miscibility gap with a lower critical point in the temperature-composition diagram. Above the so-called cloud curve, the solutions first become very turbid and then phase separate into two micellar solutions of extremely different surfactant contents. The position of the critical point depends on the overall chain length of the amphiphile and hydrophilic-lipophilic balance.1 The understanding of the binary phase behavior and structural properties is central for the understanding of ternary mixtures with oil (microemulsions).2 One finds conflicting interpretations of the temperature and concentration dependence of aqueous solutions of nonionic surfactants of the type CiEj in the literature, especially when approaching the cloud curve and in particular the critical point with the temperature Tc and the concentration cc. Zulauf and colleagues explained their neutron and light scattering data of C8E4 and C8E5 near * To whom correspondence should be addressed. FAX: (+43) 316 380 9850. E-mail:
[email protected]. † Part of the Special Issue “Colloid Science Matured, Four Colloid Scientists Turn 60 at the Millenium”. ‡ University of Graz. § University of Cologne. | University of Edinburgh. (1) Schubert, K.-V.; Strey, R.; Kahlweit, M. J. Colloid Interface Sci. 1991, 141, 21. (2) Kahlweit, M.; Strey, R. Angew. Chem., Int. Ed. Engl. 1985, 24, 654.
the critical point by assuming spherical micelles, constant in shape and size but with increasing attractive interaction when approaching Tc.3,4 They exclude micellar growth explicitly. However, the strong increase in light scattering intensity observed as one approaches the miscibility gap with a rise in temperature was interpreted already in early work as micellar growth.5 Degiorgio and Corti,6 on the other hand, have emphasized that the observed increase in light scattering intensity should rather be attributed to critical opalescence. Cebula and Ottewill7 have argued that both mechanisms, micellar growth and attractive interaction, are not mutually exclusive. They were able to fit their small-angle neutron scattering (SANS) data of C12E6 with a cylindrical model with growing length for increasing temperatures. The same system was studied by Strey and Pakusch.8 They were able to distinguish micellar aggregation kinetics from critical fluctuation kinetics. They found for C12E6 at the critical concentration of 2.25% (w/w) globular micelles at low temperatures transforming into large aggregates above 10 °C and critical fluctuations above 40 °C up to the critical temperature of about 48 °C. They discussed micellar growth vs the aggregation of small micelles, but were unable to resolve that question. The picture of attractive small micelles is still used for the interpretation of light scattering results,9 even though it was highly questioned in a comment, based on NMR experiments, by the Lund group some years ago.10 These authors also related the (3) Hayter, J. B.; Zulauf, M. Colloid Polym. Sci. 1982, 260, 1023. (4) Zulauf, M.; Rosenbusch, J. P. J. Phys. Chem. 1983, 87, 856. (5) Herrmann, K. W.; Brushmiller, J. G.; Courchene, W. L. J. Phys. Chem. 1966, 70, 2909. (6) Degiorgio, V. J.; Corti, M. Phys. Rev. Lett. 1980, 45, 1045; J. Phys. Chem. 1981, 85, 1442. (7) Cebula, D. J.; Ottewill, R. H. Colloid Polym. Sci. 1982, 260, 1118. (8) Strey, R.; Pakusch, A. In Surfactants in Solution; Mittal, K. L., Bothorel, P., Eds.; Plenum: New York, 1986; Vol. 4, p 465. (9) Strunk, H.; Lang, P.; Findenegg, G. H. J. Phys. Chem. 1994, 98, 11557. (10) Lindman, B.; Wennerstro¨m, H. J. Phys. Chem. 1991, 95, 6053.
10.1021/la000315s CCC: $19.00 © 2000 American Chemical Society Published on Web 09/26/2000
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absence of any growth to a misinterpretation of the scattering data. Small-angle X-ray studies recently performed on C8E411 were not able to discriminate between the different possible interpretations due to the nearly vanishing contrast of this system in water for X-rays, a common problem in structural analysis of surfactant systems.12 Finally, micellar growth with temperature has also been predicted for CiEj on the basis of a molecularthermodynamic approach.13 The most powerful methods for structural investigations on surfactant aggregates are small-angle neutron scattering and light scattering. We have performed a series of SANS experiments for different samples at various concentrations around cc at temperatures between 3 up to about 1 °C below Tc to find a clear answer to the questions if the micelles show a sphere-to-rod transition or if the micelles stay globular and become only more and more attractive when approaching Tc. We have shown recently14,15 that interpretation of scattering data not only in reciprocal space but also in real space, facilitated by the method of indirect Fourier transformation (IFT), is important for a model free evaluation. An intrinsic problem of such studies is the fact that they have to be performed at, or around, the critical concentration. So particle interactions cannot be ignored. We make use of the recently developed generalized indirect Fourier transformation (GIFT) method16-18 to separate information about the particle form and particle interaction. Combining the SANS results with complementary experiments using depolarized light scattering and ultralow shear viscosimetry, we are able to show that there is micellar growth with temperature and concentration and attrative interaction close to Tc. We would like to emphasize that this study was not planned to study the critical phenomenon itself. This is the reason we stopped our experiments about 1.5 K below Tc. Experimental Section Materials. The polyoxyethylene glycol mono-n-dodecyl ethers C12E6 and C12E5 were purchased from Fluka Chemie AG (Buchs, Switzerland), the tetraoxyethylene glycol mono-n-decyl ether C10E4 and the polyoxyethylene glycol mono-n-octyl ethers C8E5, C8E4, and C8E3 were purchased from Bachem Biochemica GmbH (Heidelberg, Germany). D2O (g99.8%) was obtained from Merck (Darmstadt, Germany). The compounds were used as purchased. The surfactants were stored in the dark at T ) -20 °C, and samples were stored in sealed glass tubes under nitrogen atmosphere in the dark at T ) 4 °C before they were transferred into the different measuring cells. Critical Temperatures. Tc and cc have been determined for a large series of different CiEj samples after application of a special purification technique.1 We used the values for cc from this study because the miscibility gaps are rather flat for all these samples. However, the substitution of D2O for H2O, needed for SANS experiments, lowers the critical temperature by about 3-5 K. We determined the actual values of Tc for our solutions by optical inspection and by ultralow shear viscosimetry. The method of determining the critical temperature optically provides (11) Lang, P.; Glatter, O. Langmuir 1996, 12, 1193. (12) Iampietro, D. J.; Brasher, L. L.; Kaler, E. W.; Stradner, A.; Glatter, O. J. Phys. Chem. B 1998, 102, 3105. (13) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 3710. (14) Glatter, O.; Strey, R.; Schubert, K.-V.; Kaler, E. W. Ber. BunsenGes. Phys. Chem. 1996, 100, 323. (15) Strey, R.; Glatter, O.; Schubert, K.-V.; Kaler, E. W. J. Chem. Phys. 1996, 105, 1175. (16) Brunner-Popela, J.; Glatter, O. J. Appl. Crystallogr. 1997, 30, 431. (17) Weyerich, B.; Brunner-Popela, J.; Glatter, O. J. Appl. Crystallogr. 1999, 32, 197. (18) Brunner-Popela, J.; Mittelbach, R.; Strey, R.; Schubert, K.-V.; Kaler, E. W.; Glatter, O. J. Chem. Phys. 1999, 21, 10623.
Langmuir, Vol. 16, No. 23, 2000 8693 Table 1. Surfactants CiEj in D2O: Critical Points, Sample Concentrations, and Measuring Temperatures concn (wt %) surfactant Tc (°C) C12E6 C12E5 C10E4 C8E5 C8E4 C8E3
44.7 29.7 16.6 55.6 37.4 6.2
exptl temp (°C) 3 3 3 3 3 5
18 15 15 36 20
cc/9
cc/3
33 43 0.251 0.75 28 0.45 0.78 45 54 0.96 2.89 36 2.14 1.57
cc
3cc
9cc
2.26 6.77 20.34 1.35 4.06 2.35 7.04 8.67 26.0 6.41 19.2 4.70 14.1
the phase transition temperatures to better than 0.01 K, if desired. For this method the sample is precisely thermostated in a water bath and the temperature is adjusted to find the critical temperature Tc. At this temperature the sample changes from transparent to turbid and finally separates into two distinct phases. The precise location of Tc, which is not always needed, may be quite time-consuming. A recently developed new instrument for ultralow shear viscosimetry19 allows the measurement of the viscosity in a sealed glass tube with a temperature control of (0.01 K. In a temperature scan the viscosity shows an abrupt decrease above Tc due to the demixing of the sample. These experiments are performed under computer control and can only be used for samples with high enough relative viscosity η/c. With this technique the critical temperature is easily located within (0.1 K. The resulting Tc values are summarized in Table 1. Samples. The sample compositions and measuring temperatures are given in Table 1 together with the values for Tc and cc. All solutions were prepared with D2O to guarantee a minimal incoherent background in the SANS experiments. All samples were measured at least at three different concentrations: besides the critical composition we prepared samples with cc/3 and 3cc, for C12E6 and C8E5 we had also prepared cc/9 and for C12E6 also 9cc in order to have an overview on the concentration dependence. We measured all samples in the temperature range starting at 3 °C up to about 1.5 K below Tc. The temperature range was widest for C8E5 (51 K) and C12E6 (40 K) and lowest for C8E3 (only measured at 5 °C). Small-Angle Neutron Scattering Measurements. The small-angle neutron scattering experiments were carried out using the D22 spectrometer at the Institute Laue Langevin (ILL) in Grenoble, France. A mean wavelength of λmean ) 0.6 nm having a triangular distribution with a full width at half-maximum (fwhm) of ∆λ/λ ) 0.1 was used. The wave vector q ) (4π/λ) sin(θ/2) ranged from 0.04 to 4.2 nm-1, where θ is the scattering angle. Collimation was chosen to maximize the neutron flux without limiting the resolution. Two sample-to-detector distances were measured (14.0 and 2.0 m) with the detector 38 cm off axis to provide a wider q-range. Samples were equilibrated (in half-filled Hellma quartz cells of 1 mm optical path length) at the temperature of interest and were then rapidly transferred to the cell holder. The cell holder, preset to and kept at the desired temperature to within 0.02 °C, allowed the samples to be tilted and thereby mixed and homogenized after mounting. More details were given previously.20 Data were collected until the noise level became insignificant. The data from the 128 × 128 two-dimensional detector (pixel size 7.5 × 7.5 mm) were corrected for dead time effects, masked, radially averaged, and normalized according to the standard procedures provided by ILL. Data sets from the different distances overlapped without scale adjustment. A few data points of the lowest and highest q values were cut from each set. Major smearing of the data is the result of the wavelength spread of 10%, and in the data analysis (see below) this effect is taken into account. The procedure of primary data handling is the same as described in recent communications15,18 and includes merging of data from different detector positions and subtraction of the scattering of the blank cell and of a background. (19) Fritz, G.; Scherf, G.; Glatter, O. J. Phys. Chem. B 2000, 104, 3463. (20) Schubert, K.-V.; Strey, R. J. Chem. Phys. 1991, 95, 8532.
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Ultralow Shear Measurements. The viscosity η of a surfactant solution does not only depend on the surfactant concentration but, most importantly above the cmc, on the size and shape of the micelles. Especially the formation of rodlike micelles has a strong influence on the viscosity. The micelles are quite fragile objects, so it is important to apply a low shear strain to avoid mechanical destruction of the micelles and shear thinning. We use a new technique to measure the viscosity of a fluid in a mechanical oscillator. The mechanical oscillator method is widely used in precision densimetry.21,22 It is based on the measurement of the natural resonant frequency of a U-shaped glass tube filled with ∼1 mL sample. We use a commercially available laboratory instrument (DMA5000, A. Paar, Graz, Austria). It is possible to measure the ratio between the damping force, introduced by the fluid, and the spring force of the instrument utilizing a phase shifted mode of operation. This damping ratio (or loss angle) can be directly related to the viscosity by calibration experiments. For further details see refs 19 and 23. In this instrument the sample is in a closed volume and is thermostated to (0.01 K. It is possible to make stepwise temperature scans; i.e., the temperature is preset to a given value. Once this temperature is reached, the experiment is performed and then the temperature can be increased or decreased with a given increment, as small as 0.01 K. The whole process is operated by a PC. This instrument allows an easy scan of the temperature dependence of the viscosity of CiEj solutions using an increment of 0.5 or 1.0 K and the determination of the critical temperature with an increment of 0.1 K or less. Depolarized Light Scattering. Particles in solution scatter light due to the difference in the refractive indices of the particles and the solvent. The scattered intensity at qD < 1, where D is the diameter of the particle, is proportional to the mass or volume of the particle at constant volume fraction of dispersed particles. This allows one to follow aggregation processes but does not give direct evidence of the shape and size of particles much smaller than the wavelength of light. Especially the integrated intensity does not allow one to discriminate between spherical micelles or aggregates of these and elongated or rodlike micelles. Here one can take advantage of the fact that spherical particles do not depolarize light in the scattering process but cylinders do. Using a vertically polarized primary beam and using a horizontally adjusted analyzer in front of the detector, one may detect the depolarized light scattering signal IVH(q) even for solutions of rodlike micelles for which the scattered light is only weakly dependent on scattering angle. The depolarized light scattering signal is very low compared to the intensity of the primary beam and even high quality polarizers have a certain leakage. This background is always observed and has to be minimized by careful alignment of the polarizers, and it must be subtracted from every experiment. We use a laboratory-built precision laser goniometer, equipped with a 5 W argon ion laser (Spectra Physics, Model 2060-5S) operated with 1 W, high-quality Glan-Thompson polarizers, extinction ratio >10-5 (Halle, Berlin, Germany).The leakage of this system is less than 100 counts/s for the laser power used, while typical experimental signals range from some hundred to some thousand counts per second. For even higher signals as from microemulsions, it is possible to determine the mean length of the micelles by depolarized dynamic light scattering.24
Small-Angle Scattering Theory The q-dependent scattering intensity I(q b) is the complex square of the scattering amplitude F(q b), which is the Forier transform of the scattering length density difference ∆F(r b), describing the scattering particle in real space. In solution scattering one measures the spatial average of (21) Stabinger, H.; Leopold, H.; Kratky, O. Monatsh. Chem. 1967, 98, 436. (22) Kratky, O.; Leopold, H.; Stabinger, H. Z. Angew. Phys. 1969, 27, 273. (23) Glatter, O. J. Phys. IV 1993, 3, 27. (24) Lehner, D.; Lindner, H.; Glatter, O. Langmuir 2000, 16, 1689.
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these functions, and we finally have
I(q) ) 4π
∫0∞ p(r) sinqrqr dr
(1)
where p(r) is the pair distance distribution function (PDDF) of the particle:
p(r) ) r2 ∆F˜ 2(r)
(2)
and ∆F˜ 2(r) is the convolution square (spatial correlation function) of ∆F(r) averaged over all orientations in space.25 The functional form of I(q) or p(r) can be used to determine the shape and the internal structure of the scattering object. In principle, data analysis can be performed in two ways. Both ways should lead to extraction of the same information contained in, and obtainable from, the scattering curves, which are limited by experimental resolution, finite q-range, structural complexity, etc. The first way to analyze the scattering data is to compare them to calculated scattering curves of model structures. If carefully done, this is a valid procedure. However, there is the danger that models with too many adjustable or unknown paramters can be used. Moreover, it is very difficult to find a way to improve incorrect models. The second procedure is the more general but theoretically more demanding route: The scattering curves are fitted (smoothing of statistical noise), the wavelength spread effect desmeared, and Fourier transformed into real space. Generally, this procedure is preferred if it is unknown what structure to expect. The Fourier transform yields the pair distance distribution function p(r) in real space which, by its very shape, allows determination of the basic geometry (spherical, cylindrical, or planar) even for inhomogeneous particles. It clearly indicates, for instance, the transition from a globular, almost spherical, symmetry to cylindrical micelles with varying and increasing lengthsa key point in this investigation. The corresponding techniques have recently been applied successfully to water-surfactant systems and microemulsions based on C12E5.14,15 These techniques, based on the indirect Fourier transformation (IFT),26-28 are, however, only applicable to very dilute systems without problems, where particle interactions can be neglected or to a certain extent suppressed by cutting the low-q part of the scattering curve. This approach is very dangerous for rodlike particles which we report to exist in this work. Thus, for semidilute systems with volume fractions above 1% it is important to take particle interactions into account during the IFT process. Otherwise the p(r) function of the particles is modified by an additional term originating from the convolution of the particle correlation function γ(r) with the total interparticle correlation function h(r).29 This would lead to strong oscillations in the p(r) function, which make its interpretation impossible. In general, the intraparticle scattering contribution are related to the particle form factor P(q) and the interparticle correlations are related to the structure factor S(q). For monodisperse spherical systems, the total scattering (25) Bracewell, R. Fourier Transform and its Applications; McGrawHill: New York, 1986. (26) Glatter, O. J. Appl. Crystallogr. 1977, 10, 415. (27) Glatter, O. J. Appl. Crystallogr. 1980, 13, 577. (28) Glatter, O. In Small Angle X-ray Scattering; Glatter, O., Kratky, O., Eds.; Academic Press: London, 1982; p 119. (29) Glatter, O.; Jamnik, A.; Bergmann, A.; Mittelbach, R.; Fritz, G.; Brunner-Popela, J. Manuscript in preparation.
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intensity I(q) can simply be written as
I(q) ) P(q) S(q)
(3)
For dilute systems, S(q) is identical to one and I(q) is simply given by P(q) and we come back to eq 1. An expression similar to eq 3 can be found which is valid also for polydisperse spherical systems and nonspherical systems by replacing S(q) by Seff(q).17 Seff(q) is then no longer only a function of the particle distribution in space but depends also on the form amplitudes of the particles. We have recently developed the so-called generalized indirect Fourier transformation (GIFT) method,16,17,30 which is based on eq 3 and on the principles of the IFT method.26-28 This method allows the simultaneous determination of the form factor and the structure factor from the scattering data with a minimum of a priori information. The method has already been successfully applied to microemulsion data.18 The great advantage of the GIFT method is the fact that influence of interparticle contributions on the p(r) function can be reduced to an insignificant level under certain circumstances without any cutoff at low q. This is essential for the present study in which scattering curves from samples have to be analyzed at given and fixed concentrations where extrapolation to zero concentration is impossible or meaningless. The method performs in one step also the smoothing and desmearing. The desmeared curve represents the scattering curve that would have been obtained from a measurement with strictly monochromatic neutrons. The desmearing process leads to only slighty more pronounced minima and maxima in the scattering curves, but the general trend remains unchanged. It has, however, to be mentioned that the GIFT method can, at least until now, only be applied to uncharged systems without attractive interactions. Thus, it cannot separate form factor and structure factor from data very close to the critical point where attractive interactions are to be expected. In this case we have to use a modified OrnsteinZernicke correlation function.31,32 It is also based on the factorization given in eq 3, the structure factor S(q) can be approximated by a Lorentzian function:
S(q) ) 1 +
npkBTχT 1 + q2ξc2
(4)
where ξc is the correlation length of the local concentration fluctuations, kB is the Boltzmann constant, T the absolute temperature, np is the number density of colloidal particles, and χT is the isothermal osmotic compressibility. ξc and χT are the critical parameters which grow with critical exponents when approaching Tc.32,33 For the form factor P(q) we use, as a first approximation, a model function determined at a lower temperature, i.e., in this case a cylinder with appropriate cross-section and length. Results The SANS experiments yield radially averaged scattering curves for different sample-to-detector positions. The scattering functions obtained by this procedure are smeared according to the triangular wavelength distribu(30) Bergmann, A.; Fritz, G.; Glatter, O. J. Appl. Crystallogr., in press. (31) Ornstein, L. S.; Zernicke, F. Proc. Ned. Akad. Sci. 1914, 17, 793. (32) Chen, S. H. Ann. Rev. Phys. Chem. 1986, 37, 351. (33) Sinn, C.; Woermann, D. Ber. Bunsen-Ges. Phys. Chem. 1992, 96, 913.
Figure 1. Scattering curves of C8E5 as a function of temperature and concentration: cc/9 (a, top), cc, i.e., critical concentration (b, middle), and 3cc (c, bottom): (O) 3, (3) 36, (0) 45, and (]) 54 °C.
tion with fwhm of ∆λ/λ ) 0.1. Smoothing of statistical noise, desmearing of the wavelength effect and transformation into real space is performed in one step by the model-independent indirect Fourier transformation (IFT) method. The resulting pair distance distribution function p(r) in real space shows best the structural information about the sample. However, for semidilute systems with volume fractions above 1% it is important to take particle interactions into account during the IFT process. This is made possible by the recently developed generalized indirect Fourier transformation (GIFT) method. A series of scattering curves of C8E5 as a function of temperature for different concentrations is shown in Figure 1a-c. Interparticle interactions can be neglected at the lowest concentration (cc/9, 0.96% (w/w)). One can clearly see the increase in the forward scattering for higher temperatures. The situation is similar for the higher concentrations (cc and 3cc), shown in Figure 1b,c. The upturn at higher
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temperatures is reduced with increasing concentration, while an interaction peak evolves. The forward scattering increases for all concentrations with temperature; the interpretation of these spectra becomes, however, increasingly difficult for higher concentrations. A very similar temperature and concentration dependence is found for all other investigated systems (results not shown). A q-1 line is added to Figure 1a,b. A growth to long, rodlike particles should lead to a q-1 behavior in the scattering curve, at least for low concentrations with negligible interaction effects. It is obvious that no such regime can be found, even not at cc/9. This is due to the fact that the cylindrical micelles formed are too short and polydisperse in length, as will be shown in the following. In the following figures we present the corresponding results for C8E5 in real space. First we consider the effect of concentration at constant temperature (T ) 3 °C). Figure 2a shows the results of IFT calculations neglecting particle interactions. The lowest concentration (cc/9) is practically free from such effects; the PDDF corresponds to the form factor and has the typical shape for a globular, almost spherical micelle. The PDDFs for higher concentrations show the influence of the structure factor, resulting in increasingly strong oscillations. The results for the same data, but evaluated with the GIFT method and thus taking interactions into account, are presented in Figure 2b. These results clearly demonstrate the possibility of separating out the influence of the structure factor nearly completely. The small tails at higher r-values are within the error bars of such calculations; i.e., all samples contain globular micelles with a diameter of about 5 nm. The corresponding structure factors are presented in Figure 2c. The S(0) value decreases dramatically with concentration, while the interaction peak becomes sharper and moves to higher q-values, reflecting the decreasing mean distance between neighboring particles. Now we want to show the effect of increasing temperature at fixed concentration. The results for the lowest concentration are depicted in Figure 3a. Again, we can use here the much simpler IFT technique. From the PDDFs one can clearly see that the globular micelles, found at 3 °C, having a diameter of about 4.8 nm, continuously transform into rather short rodlike micelles with a constant cross-section dimension of about 3.6 nm and an exponentially decaying length distribution18 with a mean length Lmean which increases with temperature. Such a decrease of the cross-section dimension in a sphere-to-rod transition has been reported in detail recently for C12E5.15,14 (See Table 2 in ref 15.) The micelles formed are rather short and have a length distribution. This is the reason no q-1 dependence can be observed in reciprocal space. This clearly demonstrates the advantages of interpretation of real space information. It should also be mentioned that the polydispersity in size is low for the spherical micelles, while the length polydispersity is large for the rodlike micelles and can be modeled with an exponential distribution. A very similar temperature dependence is found for cc/3 as shown in Figure 3b. Here we have already applied GIFT as the weight fraction is close to 3%, and we find some additional features. First of all, the mean length increases as can be seen clearly from the increase of the maximum dimension where the PDDF decays to zero. In addition, there evolves a second peak after the typical cross-section peak at the highest temperature (54 °C). This new, broad peak is the signature of the attractive interaction as will be discussed in more detail in the following section. At the critical concentration cc we see a similar temperature dependence (Figure 3c), the only difference is the fact that the second peak due to attractive
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Figure 2. (a, top) C8E5 results at 3 °C in real space (PDDFs) using IFT algorithm. The p(r) functions are divided by their actual concentration and show oscillations increasing with concentration due to the interaction effects: (s) cc/9, (--) cc/3, (-‚-) cc, and (-‚‚-) 3cc. (b, middle) C8E5 results at 3 °C in real space using GIFT algorithm (same data and line types as in a). The p(r) functions do not show oscillations and indicate that all samples contain globular micelles with a diameter of about 5 nm. (c, bottom) Structure factors determined for the C8E5 data at 3 °C: (s) cc/9, (‚‚‚) cc/3, (--) cc, and (-‚‚-) 3cc. The decreasing S(0) values are the signature for the increasing volume fraction.
interaction is already seen at 36 °C, but it is much stronger for 45 and 54 °C. In the case of 3cc the weight fraction is already 26%. Here the GIFT method does not work properly any more for nonglobular particles. This is due to the fact that the structure factor model is based on a simple averaged hard spheres interaction, and the structure factor of long rods with such a high volume fraction cannot be represented with this model. But one still finds the same trend as for the lower concentrations: a growth into elongated structures with a cross-section peak and an interaction peak at larger r-values (results not shown).
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Figure 4. PDDFs of C12E6. (a, top) The system forms globular micelles at 3 °C for all concentrations studied, i.e., for cc/3 (s), cc (--), and 3cc (-‚-). Even at 18 °C the C12E6 samples predominantly show a globular state coexisting with a few short rods, not longer than 25 nm with a cross-section diameter of about 5 nm (b, bottom).
Figure 3. Temperature induced sphere-to-rod transition of C8E5 as a function of temperature and concentration: cc/9 (a, top), cc/3 (b, middle), and cc (c, bottom): (s) 3, (--) 36, (-‚-) 45, and (-‚‚-) 54 °C. For the lowest concentrations these PDDFs clearly indicate the transition from a spherical micelle at 3 °C to rodlike micelles growing with temperature in length. In addition, there evolves a second peak after the typical crosssection peak at the highest temperature for cc/3 (b). This new, broad peak is the signature of the attractive interaction. This interaction peak evolves at even lower temperatures for cc.
In a next step we now describe the dependence on the kind of surfactant. The structure and the temperature dependence of micelles of nonionic surfactants CiEj is, of course, a function of their composition expressed by the relative chain lengths i and j. If we compare, for example, C12E6 and C12E5, we see already from Table 1 that there is a reduction of the critical temperature by 15 K when one ethylene oxide group is removed. The C12E6 system forms globular micelles at 3 °C for all concentrations studied. The resulting PDDFs for cc/3, cc, and 3cc are shown in Figure 4a. For C12E6 the samples even at 18 ˚C predominantly show a globular state coexisting with a few short rods, not longer than 25 nm with a cross-section
diameter of about 5 nm (Figure 4b). The interaction peak starts to develop at 33 ˚C and is fully developed at 43 ˚C (see Figure 6, below). In contrast, the PDDFs of C12E5 clearly indicate cylindrical structures already at 3 ˚C (Figure 5a) with a maximum length of about 50 nm. The evolution with temperature is demonstrated for this sample for the critical concentration in Figure 5b. The maximum length is at 15 °C already about 90 nm, and a shoulder indicates the onset of attraction. This interaction peak is fully developed at 28 °C. At the same time the maximum dimension increases beyond the resolution limit of 100 nm. For comparison we show also the results for cc/3 at 28 °C in this figure, where the two curves at 28 °C are scaled to the same height in the interaction peak. It is obvious from this figure that the peak position does not depend on the actual concentration of the sample. However, this interaction peak depends highly on the chemical composition of the different samples, as shown in Figure 6, for all samples measured at their highest temperatures (close to Tc) at cc/3 in order to minimize excluded volume effects. The position of the interaction peak seems to be correlated with the structure of the micelles: the more elongated the micelles, the larger the r-value of the peak, as can be seen easily from the example C12E5 and C10E4. Also, the interaction apparently reaches further out as one can see comparing C8E3, C10E4, and C12E5 which have similar relative sizes of the hydrophilic and hydrophobic groups. To see that this broad peak can be understood as the onset of the critical phenomenon, we use a model fitting procedure based on eqs 3 and 4. We take the data for C10E4 at cc/3 and start with a form factor of a cylinder with
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Figure 5. (a, top) PDDFs of C12E5 at 3 °C as a function of concentration: cc/3 (s), cc (--), and 3cc (-‚-). These data clearly indicate cylindrical structures already at this low temperature. (b, bottom) PDDFs of C12E5 as a function of temperature for cc (s) 3, (-‚-) 15, and (--) 28 °C. The results for cc/3 (‚‚‚) are shown for comparison. These functions for 28 °C are scaled to the same height in the interaction peak. It is obvious that the peak position does not depend on the actual concentration.
Figure 6. PDDFs of all surfactants measured, at their highest temperatures (close to Tc), at cc/3 in order to minimize excluded volume effects.
the right cross-section in order to fit the cross-section peak in the PDDF (Figure 7a) corresponding to the high-q data in the scattering curve (Figure 7b). The length of about 40 nm was estimated from the data at 3 °C (results not shown). This form factor alone does, of course, neither fit the scattering curve at low q-values nor the PDDF at high r-values. Optimizing S(q) by variation of the free parameters leads to the amplitude parameter npkBTχT ) 7.10 and to a correlation length ξc ) 24.3 nm. The product of the form factor and the structure factor gives an excellent fit to both functions, the scattering curve, and the PDDF, as can be seen in Figure 7. The correlation length ξc
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Figure 7. Ornstein-Zernicke fit to the C10E4 data (cc/3, 15 °C). (a, top) Real space: (s) form factor of the cylinder, determined from the data at 3 °C, (--) PDDF resulting from the product from factor (cylinder) times the optimized OrnsteinZernicke structure factor, (O) PDDF from the experimental data. (b, bottom) Reciprocal space: same symbol and line assignment.
describes the size of the density fluctuations which are driven by attractive interactions, i.e., the particles have a probability to come closer to each other than the statistical mean. This situation can also be simulated by calculations where two or three cylinders are positioned close to each other with a mean side shift of 5 ( 2 nm and a tilt angle of 0 ( 25°. The average of a hundred different configurations consisting of two cylinders (65%) or three cylinders (35%) with a length of 100 nm is shown in Figure 8 together with the PDDF of C10E4 for cc/3 at 15 °C. A typical configuration of three cylinders is shown in the inset. We can see that the results from the SANS data are in full agreement with the model of attractive cylinders. On the other hand it is impossible to describe the situation at medium temperatures by aggregating or attractive spherical micelles as shown in Figure 9. There the PDDF of a cylinder, which describes the scattering data very well, is compared with two different model PDDFs from aggregating spheres. The corresponding models are depicted in the inset. A quite globular aggregate of eight spheres has no features in common with the experimental PDDFs. Even a quite elongated aggregation of eight spheres shows a strong and sharp maximum due to the next neighbor distance of two adjacent spheres. So the model of aggregating spheres cannot describe the changes in the structure of the micellar solution when increasing the temperature from 3 °C to Tc. The transition from spherical to elongated micelles can also be followed by two independent complementary
Nonionic Micelles near the Critical Point
Figure 8. Simulation in real space. Two or three cylinders are positioned close to each other with a mean side shift of 5 ( 2 nm and a tilt angle of 0 ( 25°. The average of a hundred different configurations consisting of two cylinders (65%) or three cylinders (35%) with a length of 100 nm is shown (--) together with the PDDF of C10E4 for cc/3 at 15 °C.
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Figure 10. Viscosity ratio of surfactant solutions to water (η/ηD2O), shown as a function of temperature. For all three samples we find a nearly linear increase of the viscosity with temperature up to Tc, where the sample demixes leading to a sudden apparent reduction of the viscosity (see also inset for high-resolution data).
Figure 9. Simulation in real space: cylinder vs aggregates of spheres. The models of a globular and an elongated aggregate of eight spheres and of a cylinder are shown together with the corresponding PDDFs: (-) cylinder, (--) elongated aggregate, (‚‚‚) globular aggregate. The high-frequency oscillations in the PDDF of the cylinder are an artifact caused by the finite elements used to build up the model.
techniques: viscosimetry and depolarized light scattering. Surfactant micelles are soft and fragile objects which can easily be broken by shear forces. The viscosity measurements were therefore performed with the recently developed ultralow-shear method.19 The results are shown in Figure 10, where the viscosity ratio to water η/ηD2O is shown as a function of the temperature. C8E5 makes rather short elongated micelles, changing the trend of decreasing viscosity with temperature into a slight increase only above 36 °C. The situation is quite different for C12E6.34 Here the upturn is much stronger and starts already below 20 °C in full agreement with the SANS data which indicate growth of rodlike micelles above 18 °C. The viscosity of C12E5 is much higher already at 5 °C. For this sample we found rodlike micelles already at 3 °C. For all three samples we find a nearly linear increase of the viscosity with temperature up to Tc, where the sample demixes leading to a sudden apparent reduction of the viscosity. This effect allows an easy determination of the critical temperature with an accuracy of 0.1 °C19 (see also the inset of Figure 10). In depolarized light scattering experiments one uses a highly polarized light source with vertically polarized light (34) Strey, R. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 182.
Figure 11. Depolarized light scattering signal as a function of temperature for three different surfactants (a, top). The signal is reduced by the leakage of the polarizers I0, divided by the surfactant concentration and is the higher, the lower the ratio between the relative sizes of the hydrophilic and hydrophobic groups. Signal as a function of temperature for three different concentrations of C12E5 (b, bottom).
IV as primary beam and detects the depolarized component of the scattered light IVH. Isotropic objects such as spherical micelles do not create such a depolarized component, while rodlike micelles do. The depolarized component IVH reduced by the leakage of the polarizers I0, divided by the surfactant concentration is plotted as a function of the temperature in Figure 11a (same samples as in Figure 10). We can see the very low signal for C8E5 with a slight increase above 36 °C. The signal for C12E6 increases above
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18 °C, and the signal is strongest for C12E5 as expected. There is a perfect agreement between the results in Figure 11a, Figure 10, and the SANS analysis. In Figure 11b we show the results of the depolarized light scattering experiments for C12E5 at three different concentrations (cc/3, cc, and 3cc). The results for cc/3 and cc are in good agreement, even though there is already a slight decrease in the intensity. Please note that the measured intensities [counts per second] have been divided by the actual concentrations [wt %]. The corresponding error bars are in the size of the symbols for the 3 °C data and much smaller for the higher intensities. They can therefore not be shown in the figure. The corresponding values are much lower for 3cc. As the absorbance of the sample is negligible, this effect can only be understood as a loss of anisotropy. From our SANS data we know that the length of the rods never decreases with concentration; it even increases slightly. So this effect can only be caused by an increase in the flexibility of the wormlike micelles with concentration. It should also be mentioned in this context that 3cc is about 9c*, with c* being the overlap concentration. Additional experiments to answer this question in detail are in preparation. Discussion Our intensive SANS studies, accompanied by complementary independent ultralow shear experiments and depolarized light scattering give a clear picture of the temperature dependence of aqueous solutions of nonionic surfactants of the type CiEj when approaching the critical point. These series are based on temperature variations form 3 °C up to a temperature of about 1 K below Tc and concentration variations around cc by a factor of 3-9. Six different surfactants were studied, changing the alkyl chain length i as well as the number of ethylene oxide groups j. We find that our nonionic surfactant systems show the trend of a sphere-to-rod transition with increasing temperature. This trend is only slightly increasing with concentration, if at all. Surfactants with a relatively high ratio of i to j form rodlike micelles already at low temperatures. C12E5 and C8E4 are typical examples for this situation, while C8E5 and C12E6 form spherical micelles at low temperatures. Approaching Tc, we find a strong increase of the scattering intensity in forward direction for all samples. This can be explained as the onset of attractive interactions, independent of the actual size or shape of the micelles. This means that we find clear evidence for micellar growth and attractive interaction. A SAXS study was also performed recently to solve this question.11 However, this study suffers from two problems: first of all, micelles from nonionic surfactants of the type CiEj have a very low overall contrast in water for X-rays hindering the detection of a sphere-to-rod transition. This also can happen for mixtures of ionic surfactants.12 Second, the generalized indirect Fourier transformation method was not yet available at the time of this study, so the influence of the excluded volume effect was not taken into account. This effect can counterbalance the influence of a micellar growth.18 A combined light scattering and NMR self-diffusion study35 for C12E6 resulted in a model of micellar growth of cylindrical micelles, the observed obstruction effects in the self-diffusion were interpreted to exclude the existence of oblate micelles. A sphere-to-rod transition of a micellar system or microemulsion can be studied best by SANS.14 The (35) Brown, W.; Johnsen, R.; Stilbs, P.; Lindman, B. J. Chem. Phys. 1983, 4548.
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diameter of the rod is smaller by about 20% than the diameter of the sphere, but the scattering pattern remains nearly unchanged in the high-q region because of the different transformations involved in the theoretical description of the system. The fact that the side maximum at high q-values, defining the cross-section of the system does not change in position can therefore be misinterpreted easily by the assumption that the local structure does not change.15 Neglecting the information in the low-q part can lead to wrong models.18 A temperature driven transition from spherical micelles to rodlike micelles has also been predicted theoretically for CiEj’s by a molecular-thermodynamic approach.13 These authors predict rodlike micelles for 1% C12E5 for all temperatures above 0 °C in good agreement with our results, while 1% C12E6 should transform from spheres into rods above 15 °C, experimentally we find a growth above 18 °C in this concentration regime; i.e., theoretical predictions and experimental data agree within the typical error bars of such predictions and measurements. Corti and Degiorgio36 reported in a dynamic and static light scattering study the temperature dependence of C12E6 in the range 25 to 50 °C. They concluded from the power law dependence of these data that the observed phenomena are only due to critical phenomena and not due to a temperature dependence of the size of the micelles. However, light scattering does not allow a direct shape determination due to its long wavelength. The processes of micellar growth and of critical fluctuations can be separated by T-jump experiments as demonstrated in the case of C12E6.8 Such experiments cannot differentiate between aggregates of spheres or larger, rodlike micelles. This can be done by depolarized light scattering as shown in this contribution. Only anisotropic samples show a depolarized component in the scattered light. The results from such depolarized light scattering experiments are in perfect agreement with the results from rheological experiments and with our SANS data. Reduced signals in depolarized light scattering at high concentrations, like in the case of C12E5 at 3cc are an indication of loss of anisotropy, which can be due to bending of the micelles. Earlier papers in this field, especially the pioneering work of Zulauf and colleagues,3,4 always excluded one of the two possible effects; i.e., their discussion was always focusing on the decision: is there a growth or attractive interaction? They reported indirect evidence for both effects, but they finally came to the conclusion that it is attractive interaction at constant size of the micelles. This decision was mostly based on the problem that there was no divergence in the viscosity of the system, while the scattering intensity and the apparent radius of gyration showed such a diverging behavior. They studied C8E5 and C8E4, but SANS data were mostly recorded for C8E5 which does not exhibit growth into long cylinders. SANS data were also recorded with relatively low resolution and signal quality at that time, and sophisticated evaluation routines for semidilute solutions were not yet available. They could only conclude from the nearly unchanged high-q part of their data that the local structure did not change; i.e., the micelles remain globular. The increased forward scattering was solely attributed to the structure factor S(q). An NMR self-diffusion study37 could not find any general trend of the temperature dependence of the self-diffusion coefficient, so they rule out any growth from spheres to (36) Corti, M.; Degiorgio, V. J. Phys. Chem. 1981, 85, 1442. (37) Stubenrauch, C.; Nyde´n, M.; Findenegg, G. H.; Lindman, B. J. Chem. Phys. 1996, 17028.
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rodlike micelles or clustering. These results are however somehow in contradiction with all other studies. C8E4 has been studied by different techniques. Static and dynamic light scattering results9 were interpreted in terms of clustering of surfactant micelles being invariant in size and shape, but no direct evidence for this assumption can be found from the data presented. Such a model is in clear contradiction to our results. It has already been pointed out by Cebula and Ottewill7 that both mechanisms, micellar growth and attractive interaction, are not mutually exclusive. They found a best fit to their SANS data from C12E6 by cylindrical models. They also discuss the possibility of a linear aggregation of nonspherical micelles. Their data, however, were not of high enough quality at that time to answer such questions. Conclusions Our results based on intensive SANS studies taking into account excluded volume effects in the evaluation procedure give a much clearer picture than that discussed in the literature before. We find by the recently developed generalized indirect Fourier transformation procedure for scattering data from semidilute and dense systems that all our nonionic surfactant systems show the trend of a
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sphere-to-rod transition with increasing temperature. This trend depends on the nature of the surfactant studied and somewhat on concentration. Surfactants with a relatively high ratio of i to j form rodlike micelles already at low temperatures; C12E5 and C8E4 are typical examples for this situation while C8E5 and C12E6 form spherical micelles at low temperatures. Approaching Tc, we find a strong increase of the scattering intensity in forward direction. This can be explained as the onset of attractive interactions independent of the actual size or shape of the micelles. On the other hand the degree and range of the attractive interactions seems to correlate with the overall surfactant size and its hydrophobicity. This means that we find clear evidence for micellar growth and attractive interaction. Acknowledgment. The results report were obtained in experiments number 9-10-248 and 9-10-348 at the Institute Laue Langevin (ILL) in Grenoble, France. We are grateful for the support by the research grant P12611CHE and F0118 from the Austrian “Fonds zur Fo¨rderung der wissenschaftlichen Forschung”. We want to acknowledge Gu¨nther Scherf for the practical help in sample preparation. LA000315S
