J. Phys. Chem. 1991, 95,6053-6054
0) is called a cone, and its edges are the extreme vectors corresponding to the extreme currents. There exists a computer program due to Von Hohenbalken et al." for finding the extreme vectors based on the corral and edge methods. Here we use our own program employing the method of singular value decompoiti ion.'^ The null space of S is 8-dimensional and there are 11 extreme vectors wl,..., wllreach componding to a certain reaction subnetwork. The Jacobi matrix J of eqs A2 evaluated at the steady state j2 can be written J = S diag (8) K ( f ) diag (h) (A4) where the steady-state velocity vector V = a l w l + ... + aIIwII is a linear combination of the extreme vectors with nonnegative coefficients a,,the elements of the kinetic matrix K ( f ) are K , ~= (41) Von Hohenbalken, B.; Clarke, B.; Lewis, J. J . Compur. Appl. Marh. 1987, 19, 231.
(42) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes. The Art of Scientific Computing, Cambridge University Press: Cambridge, 1986.
6053
t3 In o,(f)/d In x, and the components of h are h, = l/X,.
The problem of finding a steady state at the Hopf bifurcation is solved by adjusting the coefficients a,such that J has a pair of pure imaginary eigenvalues. In general, only a few of the extreme currents can destabilize the steady state, and SNA gives methods of how to find them. After that it is not difficult to find the appropriate linear combination by a trial-and-error method. In the present case only two of the ais had to be varied (while the other ais are fixed at some small but arbitrary values) to find the Hopf bifurcation point as compared to 12 kinetic coefficients in the direct approach. Moreover, a numerical procedure for computation of the steady state that is necessary in the direct approach is avoided entirely. Having determined I,we obtain the kinetic parameters readily from (Al). The Hopf point obtained in this way can be then used as a starting point for the continuation method,'*J9 which generates the desired parametric dependence of steady states on A-. Acknowledgment. This work was supported in part by the DOE/BES Engineering Research Program.
COMMENTS Nonlonlc Mlcelleo &ow wlth Increadng Temperature Sir: The problem of the size of surfactant micelles has been a
matter of intense discussion for at least 30 years. In pioneering studies, Luzzati' could demonstrate that for certain ionic surfactants there is a growth from spherical to rod-shaped micelles with increasing concentration. While there was at an early stage a concensus about micellar growth, the rod shape was occasionally questioned. However, by inter alia quusi-elastic light-scattering studies2 and from the 'polymer-like" behavior of the micelles,' convincing support for flexible rods rather than disks has accumulated. The rod micelles have a continuous hydrophobic core, and 'string-of-beads" types of models,' based on aggregated That ionic micelles may grow spherical micelles, can be reje~ted.~ with increasing salt concentration and with decreasing temperature are features which are well documented and understood, whereas the strong counterion specificity of the growth is only qualitatively understood. Nonionic micelles have been extensively studied, especially during the past decade, but for the most studied class of nonionia, with the polar group being an oligo(ethy1ene oxide) chain, the literature has been confused, regarding the question of micelle size and shape. Recently in this journal, an effort was made to clarify certain problems on the basis of light-scattering experiments,6 but in our opinion a broader perspective should be used to analyze these problems. The purpose of this comment is 2-fold. First, we wish to identify the problems behind the difficulty in arriving at a concensus, even on the simplest and most general aspects of micellar growth of nonionic micelles, and, second, we (1) Reiss-Husson, F.; Luzzati, V. J . fhys. Chem. 1964, 68, 3504. (2) Mazer. N. A.; knedek, 0. B.; Carey, M.C. J. fhys. Chem. 1976,80, 1075. M k l , P.J.; Mazer, N . A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J . fhys. Chem. 1980, 84, 1044. (3) For a review see, e.&: Hoffmann, H.; Ebert, G. Angew. Chem., Inr. Ed. Engl. 1988, 27.902. Much important work in this field has been done by S. J. Candau and S. lkeda and their co-workers. See also ref 13. (4) Tanford, C. J. fhys. Chem. 1974, 78, 2469. Manohar, C.; Rao, U.; Valaulikar, B. S.;lyer, R. M. J . Chem. Soc., Chem. Commun. 1986, 379. ( 5 ) Anet, F. A. L. J . Am. Chem. Soc. 1986,108,7102. Olsson, U. Thesis, Lund, Sweden, 1988. The rapid NMR relaxation of rod micellar solutions is inconsistent with the "string-of-beads" model. (6) Wilcoxon, J. P. J . fhys. Chem. 1990, 94, 7588.
0022-3654/91/2095-6053$02.50/0
will describe our present understanding of these topics. Furthermore, we will briefly comment on certain aspects of ref 6. Important features of nonionic surfactant-water systems, to which many studies have been devoted, are (1) aggregate structure (e.g., micellization, micelle size, and shape), (2) phase behavior (e.g., the formation of liquid-crystalline phases, the miscibility gap, and nonmicellar isotropic phases), and (3) critical behavior (e.g., critical exponents, correlation lengths). Experimental quantities may be sensitive to one or (typically) more of these features. This has not always been critically examined. In particular, scattering data have often been interpreted in terms of either only micellar growth or only critical fluctuations, without properly considering both factors. Scattering data are typically dominated by critical effects in the small scattering vector ( k ) region. The critical behavior is of a universal character and, therefore, contains little information on aggregate structure. In particular, one expects little difference in experimental quantities, dominated by critical effects, between systems with or without aggregate growth; even systems without self-assembly display the same universal critical behavior. The only system specific information is in the bare correlation length, to. However, it is difficult to give a precise microscopic interpretation of Q,and the data in ref 6 show that it varies greatly between systems that are molecularly very similar. In much work, comparison is made with critical effects for polydisperse polymer solutions. However, in our opinion this is inappropriate. Polydisperse polymer solutions are true multicomponent systems, while a surfactant-water system is strictly a twecomponent system, irrespective of whether the micelles have a narrow of broad size distribution. Therefore, the critical behavior of surfactant solutions should be universal, and the observation of the absence of "polydispersity" effects should be taken as evidence for a narrow size distribution only after a careful analysis (see ref 7 for an example). The reason for our understanding of nonionic micelles being much slower to develop than for ionics can be directly traced to the fact that the scattering techniques, which were so important for ionic systems, require a much more careful analysis for nonionics. Prior to the identification, due to Corti and Degiorgio,B (7) Puwada, S.;Blankschtein, D. J . fhys. Chem. 1989, 93, 7753.
Q 1991 American Chemical Society
6054 The Journal of Physical Chemistry, Vol. 95, No. 15, I991 of the dominating contribution from critical fluctuations,scattering data were misinterpreted in terms of giant micelles on approach of the critical point. However, the tendency developed that many authors went too far and misinterpreted the scattering data to imply the absence of any growth. Since, for example, so-called diffusion coefficients deduced from dynamic light-scattering experiments are dependent on collective effects, we were led in our attempts to study the size of nonionic micelles to use approaches based directly on molecular factors and independent of collective effects? Two approaches were used, both based on monitoring the molecular motions of the surfactant molecules. In one, the long-range selfdiffusion is studied, whereas the other is based on the nuclear magnetic relaxation of the surfactant 'H nuclei. Both methods show unambiguously that, for example, C12E5 and Cf2E6micelles grow strongly with increasing temperature; for ClzEs micelles, growth is much less important although still significant. The self-diffusion data allow a quantification of the growth in terms of a hydrodynamic radius, and it could also be demonstrated that the micelles grow into elongated (rod-type) micelles rather than into disk-shaped ones. It was furthermore shown that the micellar growth is not directly related to the distance from the lower consolute curve, but it is a general temperature effect; the lower consolute curve can be very effectively displaced by small additions of ionic surfactant, changing the intermicellar interaction^.^-'^ It is a general expectation, that the lower consolute curve may be quite sensitive to small amounts of cosolutes or impurities, while micellar size is much less affected. The problem has later been followed up by several other groups. In particular, Brown and Kat0 have, by combining self-diffusion and dynamic light-scattering measurements on the same solutions, been able to separate the effects due to micellar growth from those due to critical fluctuations."J2 Very significant are the observations that the mutual diffusion coefficients of semidilute micellar solutions show the same pattern as entangled polymer chains in a good s o l ~ e n t . ' ~An alternative, and quite different technique of determining micellar sizes is based on fluorescence quenching. By this method, it has been amply demonstrated that the aggregation numbers of micelles of, for example, C12E5and C&!6 increase with increasing temperature." To our knowledge, there is no study, by a technique insensitive to critical fluctuations, that has failed to demonstrate micellar growth for these systems. The recent paper by Wilcoxon6 returns to these problems in a light-scattering study of aqueous solutions of Cf2E5and Cf2E6 (the latter with 10 wt 96 of NaCl). Inter alia, he points out that these surfactants, in solution, are sensitive to dissolved oxygen and claims that previous conclusions about micellar growth are unjustified. He bases these statements on the findings that "failure (8) Coni, M.; Degiorgio, V. opl. Commun. 1975,14,358. J. Phys. Chcm. 1981,85, 1442. Corti, M.;Minero, C.; Degiorgio, V. J. Phys. Chcm. 1984, 88,309. (9) Nilsson, P.-G.; Wenncrstrbm, H.; Lindman, 9. J . Phys. Chcm. 1983, 87, 3289; Chcm. Scr. 1985, 25, 67. Faucompr€, 9.; Lindman, B. J . Phys. Chem. 1987, 91, 383. Jonstrbmer. M.; Jbnmn, 9.; Lindman, 9. J . Phys. Chem. 1991, 95, 3293. (IO) Nilsson, P . 4 . ; Lindman, 9. J . Phys. Chem. 1984, 88, 5391. (I 1) Brown, W.; Johnson, R.; Stilbs, P.; Lindman, 9 . J. Phys. Chcm. 1983, 87,4548. Brown, W.; RymdCn, R. J. Phys. Chrm. 1987,91,3565. Brown, W.; Pu, 2.;Rymdln, R. J . Phys. Chrm. 1988. 92. 6086. (12) Kato, T.;Anzai, S.;Seimiya, T. J. Phys. Chcm. 1987,91,4655. Kato, T.;Seimiya, T. J. Phys. Chcm. 1986,90,3159. Kato, T.; Anzai, S.; Takano, S.;Seimi a, T. J . Chcm. Soc., Faraday Trans. 1989,85, 2499. (13) &to, T.;Anzai, S.; Seimiya, T. J. Phys. Chem. 1990,94,7255, which also contains many references in general to the application of scaling theory for entangled solution8 of flexible polymers to micellar systems. (14) Almgren, M.; Lbfroth, J. In Surfactants In Solution; Mittal, K. L., Lindman, B., Eds.; Plenum: New York, 1984; Vol. 1, p 627. Malliaris, A,; Le Moigne, J.; Sturm, J.; a n a , R. J. Phys. Chcm. 1985,89, 2709. Zana, R.;Weill. C. J . Phys. Lcrr. 1985, 46, 953.
Comments to remove dissolved oxygen and seal the samples in an inert atmosphere results in major changes in the low-angle scattering behavior with time" and, therefore, prepares his solutions under argon atmosphere. Since in our opinion phase transitions should be much more sensitive to impurities than should micellar growth and since it is known that nonpolar cosolutes (and presumably also argon) change the cloud point of nonionic surfactants, we were motivated to reexamine the problem. In articular, CI2E5 samples were prepared under Ar atmosphere' and the micelle self-diffusion coeffcients were obtained at different temperatures by NMR FT-PGSE experimentsf6for the surfactant 'Hnuclei. The results confirm our previous findings and show that even for samples prepared under Ar atmosphere there is a very important growth of the micelles with increasing temperature (10-25 OC)." We would also note that in our previous work we found no indications for any long-term changes in the micellar diffusion coefficients, which could indicate any long-term changes in micellar size due to dissolved oxygen. Furthermore, identical values were obtained after reverting from a higher temperature. In conclusion, we maintain that the evidence for a temperature-induced growth of C12E,and micelles is compelling. Furthermore, this growth is well correlated with the general phase behavior'*^'^ of these surfactant-water systems and can be qualitatively understood from general packing'* or spontaneous curvature arguments" in combination with the well-established temperature dependence of the effective water-head group interaction.*' These temperature effbcts are particularly apparent in the water-oil-nonionic surfactant microemulsion systems.2z
P
Nore Added in Proof. The temperature-induced growth of nonionic micelles has also been established in cryotransmission electron microscopy work. Thus, 8 major growth of C12ESmicelles with increasing temperature was documented by Y. Talmon (personal communication). Acknowledgmenr. We thank Dr. Arianeh Samii for preparing the solutions under Ar atmosphere and Dr. Mikael Jonstramer for performing the NMR measurements. (IS) Samples (0.90and 2.82 wt %of CI2E,in D20) were prepred by first keeping the N M R tubes under argon atmosphere and then filling the sample tubes by the solvent followed by bubbling Ar gas through the solvent for I S min. Subsequently surfactant was introduced, and the samples were kept for an additional 5 min under Ar atmosphere before scaling the sample tubes. (16) Fourier transform pulsed gradient spin-echo NMR mwurcments were performed on a Bruker MSL-100 spectrometer equipped with a homebuilt field gradient unit producing a field gradient of about 25 O/cm. Temperature was accurate within 0.5 OC. (17) For both concentrations studied, the micelle self-diffusion coefficient was found to decrease with increaein temperature. Estimating the micelle hydrodynamic radii from the Stokar-%instein equation, we obtain an increase in radius from 11.7 to 25.0 OC,which is 110-l70 A at 0.996 and 250-450 A at 2.8%. (18) Mitchell, J. D.; Tiddy, G. J. T.; Warin& L.; Bostock, T.; McDonald, M. P. J . Chrm. Soc., Faraday Trans. I 1983,79,975. (19) Kunieda, H.;Shinoda, K. J . Colloid Inrer/occ Sci. 1985, 107, 107. Kahlweit, M.; Strey, R. Angm. Chrm., Inr. Ed. Engl. 1985,24, 654. (20)Anderson, D.; Wennerstfim, H.; Olson, U. J. Phys. Chcm. 1983,93, 4243. (21) See, e.&: Lindman, 9.; Carlsson, A.; Karlstrbm, G.; Malmsten, M. Adv. Colloid Inrer/plce Scl. 1990, 32, 183 and references therein. (22) Olsson,U.;Shinoda, K.; Lindman, B. J . Phys. Chcm. 1986,90,4083. Lindman. B.; Shinoda, K.; Olsson, U.; Anderson, D.; Karlstrh, 0.; Wennerstrbm, H. Colloid Surf 1989, 38, 205. Olsson, U.;Nagai, K.; Wennerstrh, H. J . Phys. Chcm. 1998, 92,6675.
Physical Chemisrry 1 , Chemical Center B j h Lidman* Box 124 HAkan Wenaerstrijm S-221 00 Lund, Sweden Received: February 5, 1991
