J. Phys. Chem. 1984, 88, 1655-1662 smaller than those inferred from neutron scattering, calculated by assuming spherical aggregates with no water penetration, the conductance, NMR, and neutron scattering data are consistent with the predictions of the multiple equilibrium mass action model. Mysels and Princen2*established by light scattering techniques that SDS forms micelles with a mean aggregation number of 62 and a variance of about 10. Fitting our N M R data to a medium aggregation number of 65 yields a variance of 14. Mukerjee and Mysels20 have shown by conductance measurements that, below the cmc of SDS (ca. 8.5 X M), there is measurable formation of small oligomers. They inferred this from a slope of equivalent conductivity vs. the square root of surfactant concentration greater than that predicted by Onsager limiting law behavior. The best mass action model fit to the SDS data also shows a small fraction of surfactant inventory in oligomers below the cmc (Figure 8a). Williams et al.' and Rudenko et aLz argued from the break in slope of surface tension vs. the logarithm of surfactant concentration that Aerosol OT micellizes. Although the break is abrupt, which indicates cooperativity of association, the surface tension continues to decrease beyond the cmc, which indicates small aggregation numbers. Both cooperativity of association and small aggregation numbers are consistent with our model fit, although the cumulative surfactant inventory (Figure 9b) indicates the cooperativity of association is less than that for SDS. The conductance data of Fontel13 is also consistent with small aggregation numbers, because there is an abrupt but small change in slope of equivalent conductivity with the square root of surfactant concentration. All experimental data on SHBS are consistant with non- or anticooperative surfactant association, not micelle formation. Conductance dataI0*l2show a continuous variation of equivalent conductivity greater than that predicted by the Onsager limiting law. Moreover, there is no evidence of a break in slope of equivalent conductivity. This is in stark contrast to the SDS data
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and measurably different from the Aerosol OT data3 which show breaks consistent with micelle formation and slopes in agreement with Onsager limiting law below the breaks. The neutron scattering data of Magid et al.13give an apparent hydrodynamic radius of ca. 23 A. By assuming spherical surfactant aggregates with no water penetration, we calculate an apparent z-average aggregation number (A,) of about 65 from their data. This is larger than the A , of 20-30 (Figure 11) predicted from N M R data by the mass action model. However, the assumptions of spherical aggregates and no water penetration both tend to increase the value of A, calculated from the experimental radius of gyration. Thus, although our choices of K(n) distributions in the mass action model are not unique, they are fully consistent with all experimental data which provide information on the state of aggregation of SHBS.
Conclusions 23Na NMR chemical shift and line-width measurements indicate SHBS does not micellize in water at 47 "C but instead continuously associates with increase in surfactant concentration to form small aggregates. The N M R evidence is consistent with conductance and small-angle neutron scattering data, which also indicate the presence of small aggregates. N M R measurements also indicate Aerosol OT is molecularly dispersed below a concentration of about 2.5 X M at 47 "C and appears to micellize above that concentration, in good agreement with its reported cmc at 20 "C. The micellar state persists up to a concentration of about 0.10 M, whereupon a lamellar liquid-crystalline phase appears. Determination of the isotropic-liquid-crystal phase boundary with chemical shift, line-width, and line-shape measurements agrees well with the phase boundary as determined from visible turbidity. Acknowledgment. This work was supported in part by the Department of Energy and the National Science Foundation. Registry No. SHBS,67267-95-2; Aerosol OT, 577-1 1-7.
Micellar Dynamics and Organization. A Multifield 13C NMR Spin-Lattice Relaxation and {'H)'"C Nuclear Overhauser Effect Study Harald Walderhaug, Olle Soderman,* Physical Chemistry 1 , Lund University, S-220 07 Lund, Sweden
and Peter Stilbs Institute of Physical Chemistry, Uppsala University, S-751 21 Uppsala, Sweden (Received: June 27, 1983)
An extensive "C spin-relaxation study is presented for four micellar systems. These are dodecyltrimethylammonium chloride, hexadecyltrimethylammonium chloride, potassium palmitate, and sodium p-octylbenzenesulfonate. The spin relaxation data are analyzed with a relaxation model that takes into account the complexity of molecular motions in surfactant systems. The results show that the internal motions in the micelles are quite fast, occurring on a time scale of lo-'' s, which is similar to that found for liquid hydrocarbons of comparable chain lengths. The motional restrictions imposed by the aggregate-water interface resemble those found in other types of surfactant aggregates. A correlation time associated with the motions of the whole aggregate is extracted. The value of this correlation time implies that, apart from the rotational tumbling and monomer diffusion, the micelles undergo additional motions.
Introduction The measurements of nuclear magnetic resonance (NMRI) relaxation times are an important method in studying molecular motions in solutions. Carbon-13 relaxation ( T I ,T2,and nuclear (1) Abbreviations used: NMR, nuclear magnetic resonance; NOE, nuclear Overhauser effect; DOTAC, dodecyltrimethylammonium chloride; CTAC, hexadecyltrimethylammonium chloride; CTAB, hexadecyltrimethylammonium bromide; PPALM, potassium palmitate; SOBS, sodium p-octylbenzenesulfonate.
0022-3654/84/2088-1655$01.50/0
Overhauser enhancement (NOE')) is particularly useful in this respect, since it is almost exclusively related to intramolecular reorientational processes. One particular area where extensive I3C NMR relaxation studies have been made is that of surfactant system^.^^^ Surfactants, dispersed in water, form a whole hierarchy of aggre(2) R. E. London and J. Avitabile, J . Am. Chem. SOC.,99, 7765 (1977). (3) D. Canet, J. Brondeau, H. Nery, and J. P. Marchal, Chem. Phys. Lett., 72 (1980).
0 1984 American Chemical Society
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The Journal of Physical Chemistry, Vol. 88, No. 8, 1984
gate^.^^^ In general, spherical micelles initially form above a certain critical micelle concentration, followed by more complex aggregates such as rodlike or prolate or oblate micelles and ultimately liquid-crystalline mesophases of various kinds. An overall description of molecular motion for micellarly bound molecules, the subject of the present paper, is by no means simple. Consequently, the interpretation of spin relaxation data in surfactant-water systems is not straightforward either. Motional models for micelles have been suggested that have in common a separation of “fast” internal alkyl chain motions and “slow” motions, ascribed to tumbling of the whole aggregate and/or diffusion of surfactant monomers over the micellar surface. Several suggested spin-relaxation models are based on a time scale separation of this kind,2,3.6,7 In the present paper the motional model suggested in ref 7 is applied to micellar dynamics in solution, as derived from N M R spin-relaxation data. Parameters entering into that theory are directly comparable with structural and dynamic data for other types of surfactant systems. It is then possible to directly evaluate the influence of aggregate shape on the state of the hydrocarbon chains. With the dual purpose of testing the N M R spin-relaxation model7 and obtaining structural and dynamic information about micellar systems, we have performed a multifield ”C N M R T I relaxation and (‘H)I3CNOE study on the alkyl chains for four different micellar systems. Dodecyltrimethylammonium chloride (DOTAC)’ and hexadecyltrimethylamonium chloride (CTAC) I were chosen for study of the effect of different chain lengths on the motional/organizational parameters entering into the spinrelaxation model. In addition, DOTAC micelles were studied at three temperatures to monitor the temperature dependence of these parameters. Potassium palmitate (PPALM)’ was chosen for study since the lamellar lyotropic liquid-crystalline phase of this system has been extensively investigated by 2H N M R band-shape methods. This allows a direct comparison of the alkyl chain state. Sodium p-octylbenzenesulfonate (SOBS)’ was selected, with particular interest focused on the influence of the aromatic head group on the micellar dynamics and organization. It should be pointed out that all these amphiphiles have low cmc’s.* Therefore, at all the experimental concentrations used in this work, the contributions to relaxation from free monomers are negligible and need not be taken into a c c o ~ n t . ~ Theory With the usual assumption that the relaxation of a 13Cnucleus is given solely by the dipole-dipole interaction to directly bonded protons, the general expressions for Tl and the nuclear Overhauser enhancement 11 (NOE) are given bylo (SI units)
where T l jand vi are the spin-lattice-relaxation time and NOE for carbon i along the alkyl chain, respectively, ko is the perme(4) H. Wennerstrom and B. Lindman, Phys. Rep., 52C, 1 (1980). ( 5 ) G. J. T. Tiddy, Phys. Rep., 57C, 1 (1980). (6) Y. K. Levine, N. J. M. Birdsall, A. G. Lee, J. C. Metcalfe, P. Partington, and G. C. K. Roberts, J . Chem. Phys., 60, 2890 (1974). (7) H. Wennerstrom, B. Lindman, 0. Soderman, T. Drakenberg, and J. B. Rosenholm, J . Am. Chem. SOC.,101, 6860 (1979). (8) P. Mukerjee, and K. J. Mysels in “Critical Micelle Concentrations of Aqueous Surfactant Systems”, National Bureau of Standards, Washington, DC, 1970. (9) T. Ahlnas, H. Walderhaug, 0. Soderman, and B. Lindman in “Surfactants in Solution”, K . L. Mittal and B. Lindman, Eds., Plenum Press, New York, in press. (10) D. Doddrell, V.Glusko, and A. Allerhand, J . Chem. Phys., 56, 3683 (1972).
Walderhaug et al.
b:;.
0.8
I--\
I
I
I
LL 02
0.1 6
8
7
9
10
II
I2
logW
Figure 1. J(u)/J(O) as a function of log w for S = 0.2, 7,f = 1 X lo-” s, and T: = 1 X s.
ability of vacuum, yHand yc represent the magnetogyric ratios of proton and _carbon,and rc-H denotes the C-H bond length (see below). The J(w)’s are various reduced spectral densities, N i is the number of directly bonded protons to carbon i, h is Planck’s constant divided by 2a, and wH and wc are the angular Larmor frequencies of protons and carbons at a given magnetic field. Combining the result of Wennerstrom et aL7 with those of Halle and Wennerstrom,” we arrive at the following reduced spectral densities for molecular motions:
where 7; is the correlation time for the fast local motion in the alkyl chain. It is assumed that this motion is sufficiently fast to fall within the extreme narrowing range; Le., ( w T , ~
