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Pressure Dependence of the Chemical Potential of Tetradecyldimethylaminoxide Micelles in D2O. A SANS Study N. Gorski,†,‡ J. Kalus,*,‡ and D. Schwahn§ Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research JINR, 141980 Dubna, Moscow Region, Russia, Experimentalphysik I, Universita¨ t Bayreuth, D-95440 Bayreuth, Germany, and Institut fu¨ r Festko¨ rperforschung, Forschungszentrum Ju¨ lich, D-52425 Ju¨ lich, Germany Received April 12, 1999. In Final Form: July 23, 1999 By means of small-angle neutron scattering (SANS) experiments the dependence on pressure of the mean aggregation number 〈N〉 of rodlike tetradecyldimethylaminoxide (TDMAO) micelles in D2O was determined for pressures up to 2000 bar, at temperatures of 280.8, 294.9, and 308.1 K and at a TDMAO concentration c ) 118 mM L-1. By an analysis via the so-called ladder model, the dependence of the difference of the chemical potentials on pressure for a monomer being located in the end cap and in the cylindrical part of the spherocylindrical micelles, respectively, was extracted. This difference depends on concentration c, pressure P, and strongly on temperature T. For the two lower temperatures, pressureinduced phase transition was observed.
1. Introduction The influence of thermodynamic parameters on the mechanism of self-organization of surfactants in aqueous solutions are of current interest giving a deep insight into the “new science of complex fluids”.1 The rich phase behavior of the complex fluid systems is associated with the fact that particles are molecular aggregates rather than simple molecules. The tails of the constituent amphiphiles form a hydrophobic core, assumed to be approximately liquidlike in first-order approximation, while the polar headgroups are located at their surface, facing to the aqueous surroundings. The aggregates are thermodynamical stable structures, which change their size and shape in response to changes in concentration, composition, counterion species, temperature, pressure, and other conditions. Of all variables which have been found to influence micellar size, the one which has been studied the least is pressure. If a phase transition to a new structure is known to be induced with temperature, if for instance temperature influences the structure of a bicontinuous phase, or if the size or size distribution of micelles depend distinctly on temperature, then it is likely the properties of these substance depend on pressure, too, even if pressure is not unusually high. Therefore tetradecyldimethylaminoxide (TDMAO) micelles are good candidates, because at ambient pressure the sizes strongly depend on temperature.2 The TDMAO micelles typically show a rodlike shape, which is described as a spherocylinder. A spherocylinder consists of a cylinder of length L and radius R capped with two hemispheres of the same radius at the ends. The number of pressure experiments using neutron- or X-ray small-angle scattering techniques on surfactant/water systems is still limited.3-12 We present †
Frank Laboratory of Neutron Physics. University of Bayreuth. § Forschungszentrum Ju ¨ lich. ‡
(1) Gelbart, W. M.; Ben-Shaul, A. J. Phys. Chem. 1996, 100, 13169. (2) Gorski, N.; Kalus, J.; Meier, G.; Schwahn, D. Langmuir 1999, 15, 3476. (3) Winter, R.; Pilgrim, W.-C. Ber. Bunsen-Ges. Phys. Chem. 1989, 93, 708.
a SANS study where in addition to temperature pressure was one of the external parameters. Small-angle scattering with neutrons (SANS) or with X-rays (SAXS) is one of the most direct methods for obtaining information about structural details and their changes and also may help to clarify results obtained by other experimental techniques.13-26 The combination of pressure, temperature, and SANS or SAXS in the study of physical properties of micellar systems opens up new prospects for examining amphiphilic solutions on a microscopic level. In dilute solutions of zwitterionic tetradecyldimethylaminoxides (C14H29N+O-(CH3)2), abbreviated by TDMAO, (4) Gorski, N.; Kalus, J.; Kuklin, A. I.; Smirnov, L. S. J. Appl. Crystallogr. 1997, 30, 739. (5) Gorski, N.; Ivanov, A. N.; Kuklin, A. I.; Smirnov, L. S. High Press. Res. 1995, 14, 215. (6) Takeno, H.; Nagao, M.; Nakayama, Y.; Hasegawa, H.; Hashimoto, T.; Seto, H.; Imay, M. Polym. J. 1997, 29, 931. (7) Eastoe, J.; Steytler, D. C.; Robinson, B. H.; Heenan, R. K. J. Chem. Soc., Faraday Trans. 1994, 90, 3121. (8) Nagao, M.; Seto, H.; Komura, S.; Takeda, T.; Hikosaka, M. Prog. Colloid Polym. Sci. 1997, 106, 86. (9) Nagao, M.; Seto, H.; Okuhara, D.; Okabayashi, H.; Takeda, T.; Hikosaka, M. Phys. B 1998, 241-243, 970. (10) Nagao, M.; Seto, H. Phys. Rev. E 1999, 59, 3169. (11) Nagao, M.; Seto, H.; Okuhara, D.; Matsushita, Y. Submitted for publication. (12) Czeslik, C.; Erbes, J.; Winter, R. Europhys. Lett. 1997, 37, 577. (13) Offen, H. W. Rev. Phys. Chem. Jpn. 1980, 50, 97. (14) Nishikido, N.; Shinozaki, M.; Sugihara, G.; Tanaka, M. J. Colloid Interface Sci. 1981, 82, 352. (15) Nishikido, N.; Shinozaki, M.; Sugihara, G.; Tanaka, M.; Kaneshina, S. J. Colloid Interface Sci. 1980, 74, 474. (16) Kato, M.; Ozawa, S.; Hayashi, R. Lipids 1997, 32, 1229. (17) Wong, P. T. T.; Mantsch, H. H. J. Chem. Phys. 1983, 78, 7362. (18) Eastoe, J.; Young, W.; Robinson, B. H. J. Chem. Soc., Faraday Trans. 1990, 86, 2883. (19) Fulton, J. L.; Smith, R. B. J. Phys. Chem. 1988, 92, 2903. (20) Beckman, E. J.; Smith, R. D. J. Phys. Chem. 1991, 95, 3253. (21) Rudolph, E. S. J.; Cac¸ ao Pedroso, M. A.; de Loos, Th. W.; de Swaan Arons, J. J. Phys. Chem. B 1997, 101, 3914. (22) Braganza, L. B.; Worcester, D. L. Biochemistry 1986, 25, 7484. (23) Dawson, D. R.; Offen, H. W.; Nicoli, D. F. J. Colloid Interface Sci. 1981, 81, 396. (24) Tamara, K.; Nii, N. J. Phys. Chem. 1989, 93, 4825. (25) Okawanchi, M.; Shinozaki, M.; Ikawa, Y.; Tanaka, M. J. Phys. Chem. 1987, 91, 109. (26) Peck, D. G.; Johnston, K. P. J. Phys. Chem. 1991, 95, 9549.
10.1021/la990437o CCC: $18.00 © 1999 American Chemical Society Published on Web 09/21/1999
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micellar rodlike aggregates exist. The mean aggregation number 〈N〉, which is related to the mean lengths 〈L〉 of the spherocylindrical micelle, shows a complex behavior in dependence of thermodynamic parameters, concentration, temperature, and pressure. As found by light scattering methods27 〈N〉 first increases with temperature and then decreases above 315 K. Small-angle neutron scattering experiments (SANS) have shown2 that at ambient pressure and at 278 K < T < 370 K and for TDMAO concentrations of 2.69 mM L-1 < c < 118 mM L-1 they are of rodlike shape with a radius R ) 1.85 ( 0.02 nm. The mean lengths increase with concentration c below around 350 K and decrease above that temperature, whereas the difference of the chemical potentials for a monomer being located in the spherical end cap and in the cylindrical part of the spherocylindrical micelles always increases with temperature. Previous work has shown that the zwitterionic surfactant exhibit properties which could be described as being intermediate between a typical nonionic and a typical cationic surfactant of equal chain length:27 For example, no cloud point is observed as is commonly found for nonionics.28 In this paper we discuss on the basis of SANS experiments and their detailed analysis how pressure influences the aggregation number and the chemical potential of micelles. Furthermore, phase transitions and their time evolution will be shown. 2. Experimental Details TDMAO was obtained from Hoechst AG, Gendorf, and recrystallized twice from acetone. It was dissolved in D2O giving a concentration of 118 mM L-l. D2O of 99.8% isotopic purity was obtained from “ISOTOP” in Moscow. The SANS experiments were performed at the Dido reactor in Ju¨lich, Germany.29 Neutrons were used with a wavelength λ ) 0.7 nm and with a monochromatization of ∆λ/λ ) 0.2. The samples were contained in a pressure cell with a 0.1 cm gap and with an electrical heater to control the temperature of the samples to (0.1 K.30 The data treatment followed standard procedures.30 The raw data were corrected for the cell background and detector variation and put on an absolute scale. Background scattering was subtracted by comparison with a corresponding D2O sample. Conversion of the scattered intensities into absolute differential cross-sections was performed by a calibrated polyethylene specimen.
3. Theory 3.1 Neutron Scattering. The scattering cross section per particle dσ/dΩ is given by
dσ/dΩ ) (F - Fs) V 〈F 〉S(Q) 2
2
2
(1)
provided the particles are homogeneous and monodisperse. V is the volume of the particle. F and Fs are the scattering length densities of the micelles and solvent, respectively. For homogeneous cylinderlike micelles the mean-squared form factor is given by31 (27) Hoffmann, H.; Oetter, G.; Schwandner, B. Prog. Colloid Polym. Sci. 1987, 73, 95. (28) Nakagawa, T.; Shinoda, K. In Colloidal Surfactants; Shinoda, K., Nakagawa, T., Tamamuski, B., Isemara, T., Eds.; Academic: New York, 1963; p 129ff. (29) Schwahn, D.; Meier, G.; Springer, T. J. Appl. Crystallogr. 1991, 24, 568. (30) Frielinghaus, H.; Schwahn, D.; Mortensen, K.; Almdal, K.; Springer, T.; Macromolecules 1996, 29, 3263. (31) Guinier, A.; Fournet, G. Small-Angle Scattering of X-rays; Wiley: New York, 1955.
〈F2〉 )
∫0π/2(sin(Q(L/2) cos φ)/
(Q(L/2) cos φ)2J1(QR sin φ)/(QR sin φ))2 sin φ d φ (2) φ is the angle between the axis of symmetry of the micelle and the vector of the momentum transfer Q B . J1 is the Bessel function of the first kind and of order 1. L and R are the length and the radius of the micelle, respectively. The value of |Q B | is given by Q ) 4π sin(φ/2)/λ, where λ is the incident neutron wavelength, and φ the scattering angle. S(Q) is the structure factor and is related to the interaction between the micelles. For a sufficient high value of Q or for non interacting micelles S(Q) f 1. In our case we did not observe a so-called correlation peak in the intensity distribution of the scattered neutrons and so we could assume for our data evaluations that S(Q) is near one in the studied Q range. From static light scattering experiments one knows that correlation between the micelles exists above a concentration of about 100 mM L-1,27 which is slightly less than the concentration studied. 3.2 Theoretical Concepts for the Micellar Aggregation. It is well-known that rodlike micelles show a wide distribution of the lengths L. We use the so-called ladder model,32,33 to analyze the measured intensity distribution curves neglecting for a moment any possible influence of intermicellar interactions. By assuming the ideal conditions of chemical equilibrium, the mole fraction of a N-mer XN is given by
XN ) βN/K
(3)
where N ) N0, N0 + 1, N0 + 2, .... N0 and N are the aggregation numbers of the spherical and rodlike micelle, respectively. Notice that X1 is the mole fraction of monomers still present in the solution and being in equilibrium with the N-mers. Furthermore all N-mers with 2 e N e N0 - 1 are forbidden within the framework of this model. The rodlike micelle has spherical end caps on each side. The radius R of the cylindrical section of the micelles is assumed to be equal to the radius of the spherical micelle independent of the lengths L of the micelles. Assuming the value V0 ) 0.480 nm3 for the volume of a monomer, as quoted by Gradzielski,34 and a radius of R ) 1.85 ( 0.02 nm as quoted in ref 2, N0 turns out to be 4/3πR3/V0 ) 55 ( 2. β ) X1 exp(-δ/(kBT)) determines the width of the distribution function, the number-averaged aggregation number 〈N〉, and the number-averaged squared aggregation number 〈N2〉 according to
〈N〉 ) N0 + β/(1 - β)
(
〈N2〉 ) 〈N〉 N0 +
(
β β 1+ 1-β N0(1 - β) + β
(4a)
))
(4b)
The parameter K ) exp((∆ - N0δ)/(kBT)) represents a Boltzmann factor corresponding to a chemical potential gain of a surfactant molecule to be situated in the cylindrical section of the micelle, δ, compared to the chemical potential of a molecule in the spherical end caps, ∆/N0. Both, δ and ∆, have their origin in a substantial lowering of the chemical potential of the aggregates compared to having all the monomers dispersed in water. To get rodlike micelles ∆ and δ must in addition have (32) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 84, 1044. (33) Chen, S.-H.; Sheu, E. Y.; Kalus, J.; Hoffmann, H. J. Appl. Crystallogr. 1988, 21, 751. (34) Gradzielski, M. Diplomarbeit, Universita¨t Bayreuth, 1989.
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negative values and the condition |δ| > |∆|/N0 must be fulfilled. If |δ| gets close to |∆|/N0 the mean aggregation number is near N0 and more or less only spherical micelles are present. The width of an actual length distribution is due to a subtle interplay between the energy advantages associated with transferring monomers from the solvent into the cylindrical and spherical regions of the micelles, respectively, and an entropic contribution, depending on the number concentration of the micelles of different sizes. Within the framework of the ladder model one finds35
(
)
〈N2〉 - N0 /(X - X1)1/2 ) exp((∆ - N0δ)/ 〈N〉 (2kBT))β1-N0/2(1 + 1/(β + N0(1 - β)))/ (β + N0(1 - β))1/2 ≈ 2xK for β f 1 (5)
∞ X ) ∑N)1 N‚XN is the total concentration of TDMAO molecules given in mole fractions. For high X one can approximate (X - X1) by (X - Xcmc). (The critical micellar concentration (cmc) is 0.122 ( 0.002 mM at 298 K.36) For high concentrations β f 1 and X . Xcmc, and eq 5 can be expanded giving in first order 〈N〉 - N0 ≈ xK‚X(1 + ((1 - β)N0/2)2) ≈ xK‚X. Such a behavior was indeed observed.37 An intermicellar interaction Wint can influence the size distribution {XN}. Wint can be taken into account by means of a mean-field type form.38 It is known, however, that under certain restrictions the mathematical form of an exponential size distribution as expressed by eq 3 is not influenced by intermicellar interactions.38 Therefore, for our evaluations we assumed that the size distribution can still be described by an exponential (see eq 3) where β is chosen such that the experimental results are reproduced.
Figure 1. SANS scattering intensity I for the 118 mM L-1 TDMAO solution in D2O at 308.1 K. Notice the ln I versus Q plot. The solid lines are due to a fit described in the tex: 0, 1 bar; O, 500 bar; ∆, 1000 bar; ∇, 1500 bar; ), 2000 bar. For convenience, the curves from top to bottom are shifted by 0, 0.25, 0.5, 0.75, and 1.0 units. (One unit is log 10.)
4. Results of the SANS Experiments and Their Discussion In Figures 1-3, we present the SANS intensity distribution I at different pressures and temperatures. Notice that ln I versus Q is shown. From the expressions for the size distribution of the N-mers in eq 3 and the form factors squared 〈F2〉 (see eq 2) we obtain the length distribution of the micelles. The SANS intensity distribution I(Q) depends only (apart from the radius R of the rodlike micelles) on one unknown parameter, i.e., β, and on the value of the structure factor S(Q) (see eq 1). In our case the intensity distribution I(Q) is sensitive to details of the length distribution at values below Q ≈ 0.4 nm-1, whereas for larger Q values, I(Q) is sensitive to the size of the radius R of the rodlike micelles. We mention that for concentrations above 50 mM L-1 the mean length of the micelles is not very different from the mean distance between the micelles. Therefore, our TDMAO sample represents a semidilute solution, where interactions between micelles may become important, giving a structure factor S(Q) * 1. We tested this by use of a structure factor as proposed by Hayter and Penfold.39 S(Q) requires as input the volume fraction VF of the amphiphiles, the charge Z0, the Debye screening length 1/κ, and the radius R of the micelles, which are assumed to be monodisperse and have spherical shape. It is not (35) Sheu, E. Y.; Chen, S.-H. J. Phys. Chem. 1988, 92, 4466. (36) Po¨ssnecker, G. Dissertation, Universita¨t Bayreuth, 1991. (37) Gorski, N.; Kalus, J. J. Phys. Chem. B 1997, 101, 4390. (38) Blankschtein, D.; Thurston, G. M.; Benedek, G. B. J. Chem. Phys. 1986, 85, 7268. (39) Hayter, J. B.; Penfold, J. Mol. Phys. 1981, 42, 109.
Figure 2. SANS scattering intensity I for the 118 mM L-1 TDMAO solution in D2O at 294.9 K. The solid lines are due to a fit described in the text: 0, 1 bar; O, 500 bar; ∆, 1000 bar; ∇, 1500 bar; ), 2000 bar. For convenience, the curves from top to bottom are shifted by 0. 0.25, 0.5, 0.75, and 1.0 units. (One unit is log 10.)
possible to calculate a structure factor for interacting nonspherical particles in a simple closed form. As long as the micellar shape does not differ too much from spheres, we can assume that calculations according to the model of Hayter and Penfold can fit the experimental data quite well, because the center-to-center interaction potential for large distances between the micelles depends only weakly on the actual orientation of the nonspherical particles. It is evident that in the case of rather long particles an agreement with this theory cannot be expected. Nevertheless, we adopted this theory for the nonspherical, polydisperse micelles in our samples assuming that the radius R in the theory of Hayter and Penfold can be modeled by an effective radius Reff. For such an effective radius we adopted 4πR3eff/3 ) 〈N〉‚V0‚γ, where γ is a correction term of order unity. γ ) 1 would mean that the influence of interaction works as if the
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Figure 3. SANS scattering intensity I for the 118 mM L-1 TDMAO solution in D2O at 280.8 K. The solid lines are due to a fit described in the text: 0, 1 bar; O, 500 bar; ∆, 1000 bar. For convenience, the curves from top to bottom are shifted by 0, 0.25, and 0.5 units. (One unit is log 10.) Table 1. The Chemical Potential ∆µ ) ∆ - N0δ in Units of kBT0 (T0 ) 298 K) and the Mean Aggregation Number 〈N〉 as a Function of Pressure P for Different Temperatures T (TDMAO Concentration ) 118 mM L-1). T/K P/bar 1 500 1000 1500 1900 2000
∆µ 〈N〉 ∆µ 〈N〉 ∆µ 〈N〉 ∆µ 〈N〉 ∆µ 〈N〉 ∆µ 〈N〉
280.8
294.9
308.1
13.4 ( 0.3 174 ( 12 11.1 ( 0.3 160 ( 10 10.6 ( 0.3 160 ( 10
16.7 ( 0.1 267 ( 9 16.3 ( 0.1 231 ( 8 15.9 ( 0.1 200 ( 7 15.1 ( 0.1 156 ( 6 14.8 ( 0.2 142 ( 7 14.6 ( 0.1 133 ( 5
18.2 ( 0.2 357 ( 15 17.9 ( 0.2 327 ( 12 17.8 ( 0.1 309 ( 8 17.6 ( 0.1 287 ( 8 17.5 ( 0.1 271 ( 8
volume of one rodlike micelle can be transformed to an effective spherical micelle of the same volume. This is an oversimplification for long micelles. The mean distance between the micelles, which is needed for the evaluation of the structure factor according to the Hayter and Penfold model, was obtained from the volume of a mean micelle, 〈N〉‚V0, and the volume fraction VF of the amphiphiles in the aqueous solution. Taking the input data Z0 ) 0 and κ ) ∞ (which indeed is a hard core model, neglecting any other attractive or repulsive forces) and VF, we realized that S(Q) itself shows practically no correlation peak for c ) 118 mM L-1, but has of course a value less than 1 in the limit of Q f 0. For our evaluation we treated γ and β as fit parameters to get the best agreement between the calculated and all measured SANS intensity distributions I(Q). It turned out that γ ) 0.5 ( 0.1 gives the best agreement between experiment and theory. Having obtained β by fitting the experimental intensity distribution we can calculate 〈N〉, 〈N2〉, and K, using eqs 4a, 4b, and 5 and finally ∆µ ) (∆ - N0δ) itself. The values of 〈N〉 and ∆µ depend on temperature, concentration, and pressure. The pressure dependences of ∆µ are given for three temperatures in Table 1 and Figure 4. We observe that ∆µ ) (∆ - N0δ) and 〈N〉 decreases with increasing pressure. At large Q-values, the intensity distribution I(Q) at 280.8 K differs somewhat
Figure 4. The pressure dependence of the chemical potential ∆µ at 280.8, 294.9, and 308.1 K.
from the I(Q)-curves at 294.9 and 308.1 K. According to the results of the fits the radii of the rodlike micelles at 280.8 K are given by 2.27 ( 0.03 nm at p ) 1 bar, 2.40 ( 0.03 nm at p ) 500 bar, and 2.42 ( 0.03 nm at p ) 1000 bar. The solid lines in Figure 3 as well as the results of Table 1 at 280.8 K are due to these radii. It seems therefore that near the freezing point of water the interaction between the molecules in the micelles and between the micelles and the surrounding water molecules changes. If we approximate the pressure dependence of ∆µ by a linear function we see that δ(∆µ)/δp is given by -(10.8 ( 0.8) × 10-4 bar-1 for T ) 294.9 K and by -(3.3 ( 0.4) × 10-4 bar-1 for 308.1 K. The pressure dependence of ∆µ at 280.8 K is more complex and cannot be described by a linear function. It turns out that the absolute value of the correction term γ (see above) is quite insensitive to the quoted values of δ(∆µ)/δp. In contrast, the absolute values of ∆µ depend on γ. If γ ) 1, for example, the values of ∆µ increase for about 0.6 units. Our results show that the pressure dependence of ∆µ depends strongly on temperature between 280.8 and 308.1 K. Furthermore at 1 bar ∆µ increases from 13.4 ( 0.3 at 280.8 K to 18.2 ( 0.2 at 308.1 K. The temperature dependence of ∆µ was discussed in detail in ref 2. We conclude that with respect to the growth of TDMAO micelles an increase of pressure leads qualitatively to the same result as a reduction of temperature (see Table 1). For other micellar systems the growth behavior with pressure was found to be just as for TDMAO, but the reverse behavior was observed, too. Even a change with pressure of ∂〈N〉/∂p can occur.14,23 The sign of ∂〈N〉/∂p or ∂(∆µ)/∂p reflects mainly that hydrophobic bonding of the monomers is weakened or enforced by a compression. This behavior depends sensitively on the specific interaction between the monomers in the micelles and between micelles and surrounding water. At T ) 280.8 K a phase transition was observed between 1500 and 1600 bar, which manifests itself by a dramatic change in the SANS distribution (see Figure 5). At these pressures the clear surfactant solution undergoes an abrupt transition in which macroscopic aggregates of solid spontaneously precipitate out of solution. We are not able to give a well-based interpretation of the shape and size of these precipitates. Apart from that one observes a
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Figure 5. SANS scattering intensity I for the 118 mM L-1 TDMAO solution in D2O at 280.8 K and at pressures of 1500 bar (0) and 1600 bar (×).
Figure 6. Time dependence of the SANS scattering intensity I of the 118 mM L-1 TDMAO solution at 280.8 K in D2O at low Q values by applying a pressure step from 1400 to 1600 bar at time t ) 0. The time (t) of the measurements was as follows (in minutes) -1 (0), 1 (O), 5 (∆), 10 (∇), 20 ()), 31 min (+). The sampling time ts for all measurements was 1 min. Sampling occurred always between times t and t + ts.
maximum in the SANS intensity distribution around Qmax ≈ 1.6 nm-1. Such a maximum was previously also found for other concentrations.4 This means that in these precipitates a characteristic length of l′ ) 2π/Qmax ≈ 3.9 nm is present. It might be that l′ is related to a repeat distance of a lamellar structure, but we cannot prove this at the moment. In Figure 6 we present time-dependent SANS measurements at low Q-values after applying a step in pressure from 1400 to 1600 bar. Notice that the sampling time was always one minute. The complete transition from
Gorski et al.
Figure 7. Temperature vs pressure phase diagram. The solid line represents the transition line from the micellar phase (above) to the phase in which macroscopic aggregates of solid precipitate out of solution (below). The dashed lines should indicate that no data are available to show how the borderline develops. The crosses in this phase diagram indicate regions where the existence of the micellar phase was found by SANS measurements. At a pressure of 1 bar the high-temperature phase is stable down to at least 270 K.
the low-pressure phase to the high-pressure phase needs a time which is on the order of 30 min. The transition is reversible. This was observed by applying a step in pressure from 1600 to 1400 bar. However, the characteristic relaxation time is then even higher being on the order of several hours. We have to suspect that segregation of the precipitates in the scattering cell might occur, but on the time scales of our experiment (several hours) we have no observations supporting this assumption. At 277.2 K a similar phase transition occurs between 1000 and 1100 bar. If we compare these results with a temperature versus pressure phase diagram as published by Gorski et al. 4 for some lower TDMAO concentrations, we observe noticeable changes with respect to the temperatures where phase transitions occur. It was shown in ref 2 that δ, which represents the energy advantage associated with transferring the monomer from the solvent into the cylindrical region of the micelle, is around δ ) -13 (in units of kBT) for 283 K < T < 313 K. Furthermore, it was found40 that if a long molecule of length L penetrates a micellar surface for a distance X, an energy advantage of roughly Ep ≈ δ‚X/L is involved (0 < X < L). Taking this as a guide, we can estimate that the protrusion of the monomers is quite low, being of the order of less than 0.2 nm. This might change for higher temperatures, where |δ| probably declines. 5. Conclusions We observed a decrease of the length respectively of the mean aggregation number 〈N〉 and of the chemical potential ∆µ of the rodlike micelles with increasing pressure at 280.8, 294.9, and 308.1 K. As shown by Gorski et al. 2 one observes for the same and other concentrations an increase of 〈N〉 and ∆µ with increasing temperature at ambient pressure, meaning that increasing pressure influences the growth of the TDMAO micelles in the same direction as decreasing temperature. For the temperature of 277.2 K (280.8 K), we observed phase transitions between 1000 and 1100 bar (1500 and 1600 bar). The (40) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffmann, H.; Kielmann, I.; Ulbricht, W.; Zana, R.; Lang, I.; Tondre, C. J. Phys. Chem. 1976, 80, 905.
Tetradecyldimethylaminoxide Micelles
precipitates which are found at high pressure probably show a lamellar structure. These results allow us to construct a very rudimentary temperature versus pressure phase diagram as presented in Figure 7. The dashed lines in this figure should indicate that we do not know how the border between the two phases appears in reality for the temperature below 277 K and above 281 K. The crosses in this figure indicate regions where we found the micellar phase by means of our SANS measurements.
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Acknowledgment. We thank the Forschungszentrum Ju¨lich for getting access to the SANS diffractometer, to Professor Dr. Heinz Hoffmann for the gift of TDMAO, and to Dr. H. Frielinghaus and M. Heidrich for their assistance of the SANS experiments. This research was supported by the BMBF of the Bundesrepublik Deutschland under grant 03-DU4BAY-6. LA990437O
