J. Phys. Chem. 1981, 85, 2142-2147

flux of the solute through the membrane, eq 1, mol local hydraulic permeability, eq 3, m4 N-’ s-l m-2 s-l concentration-independentpart of A(x,a),eq 21 and volume flux through the membrane, eq 2, m s-’ 33, m4 N-’s-l local partition coefficient in the membrane, eq 5 coefficient of inverse proportionality of A(x,a)with filtration coefficient or hydraulic permeability, eq the concentration, eq 21 and 33, mol m N-’ s-l 1 and 32, m3 N-’ s-l integral defined by eq 8, s m-l filtration coefficient corrected for concentration space coordinate in transport direction, m effects at small J,, eq 33, m3 N-’ reflection coefficient in eq 1 and 2 membrane thickness, m local reflection coefficient in the membrane, eq 3 pressure, N m-2 average reflection coefficient, eq 25 universal gas constant, N m mol-’ K-’ pore volume fraction of the membrane solute retention in a reverse-osmosis experiment, solute permeability of the membrane in eq 1, m s-l eq 42 partial solute permeability, eq 22, m s-l absolute temperature, K space coordinate in transport direction, m Acknowledgment. We are grateful to Professor Dr. A. dimensionless quantity determined by the structure J. Staverman and Dr. J. van Daalen for critical reading of of the membrane, defined by eq 30 dimensionless quantity, defined by eq 34 the manuscript. dimensionless quantity, defined by eq 35 difference between the concentrations on both sides Supplementary Material Available: Appendix A, pro- of the membrane, eq 1, mol m-3 viding a derivation of eq 3 and 4, and Appendix B, com- difference between the pressures on both sides of paring results at low P6clet values with those of Kedem the membrane, eq 2, N m-2 and Katchalsky for multilayer membranes (7 pages). position at which a discontinuity occurs in the Ordering information is given on any current masthead membrane parameters, eq 38, m page.

Hydration of Ionic Surfactant Micelles from Water Oxygen-I 7 Magnetic Relaxation

Bertil Halle’ and Goran Carlstrom

Division of Physical Chemistry 1. Chemical Center, 5-22007 Lund 7, Sweden (Received: January 16, 1981; In Final Form: March 17, 196 1)

Water oxygen-17 relaxation rates have been measured for aqueous solutions of micelles composed of ionic surfactants of varying alkyl chain length, head group, and counterion. The concentration of surfactant and salt was also varied. From these measurements, supplemented with relaxation data for short-chain molecules and with quadrupolar splittings from anisotropic mesophases, we derive structural and dynamic information about the molecular details of the water-micelle interaction. Water molecules at the micelle surface reorient anisotropically, typically 2-3 times slower than in pure water. The average lifetime for water molecules associated with sodium dodecyl sulfate micelles is 6-37 ns. The water-hydrocarbon contact in the micellar solutions is equivalent to less than two fully exposed methylene groups per amphiphile. Small head groups and small counterions produce the largest effects on the 170 relaxation rate, as expected from geometrical and electrostatic considerations.

Introduction micelles, proposed originally by Hartle~.~According to There are few systems in which the solute-solvent in- this picture, the micellar aggregate consists of a liquidlike teraction has more dramatic consequences than in aqueous hydrocarbon core, with the charged head groups residing surfactant solutions. A characteristic feature of these at the surface in contact with the surrounding aqueous systems is the cooperative self-association of surfactant, medium. Although there has been some controversyc8 i.e., amphiphilic molecules composed of a polar, often about the degree of water penetration into the hydrocarbon charged, part and a relatively large nonpolar part, to form core, it seems that no unequivocal evidence of substantial large aggregates, known as micelles. Aggregation typically water penetration exists. On the other hand, some occurs only above a well-defined surfactant concentration water-hydrocarbon contact inevitably exists since most (cmc). Several recent reviews14 provide extensive accounts head groups are simply too small to completely cover the of the phenomenon of surfactant association. curved aggregate surface. Another factor which leads to The overwhelming majority of experimental and theo- water-hydrocarbon contact is the so-called “dynamic retical studies have confirmed the classical picture of roughness” of the micelle surface. Thus, Anianssong es-

(1) C. Tanford, “The Hydrophobic Effect-Formation of Micelles and (5) G. S. Hartley, “Aqueous Solutions of Paraffin Chain Salts”, Her- Biological Membranes”, 2nd ed, Wiley, New York, 1980. mann, Paris, 1936. (2) G. C. Kresheck in “WateI-A Comprehensive Treatise”, Vol. 4, F. (6) F. M. Menger and B. J. Boyer, J. Am. Chem. Soc., 102,5936 (1980). Franks, Ed., Plenum Press, New York, 1975, Chapter 2. (7) H.Wennerstrom and B. Lindrnan, J. Phys. Chem., 83,2931 (1979). (3) H. Wennerstrom and B. Lindman, Phys. Rep., 52, 1 (1979). (8)B. Lindman, H. Wennerstrom, H. Gustavsson, N. Kamenka, and (4) B. Lindman and H. Wennerstrom, Top. Curr. Chem., 87, 1 (1980). B. Brun, Pure Appl. Chem., 52, 1307 (1980).

0022-3654/81/2085-2142$01.25/0 @ 1981 American Chemical Society Micelle Hydration from Oxygen-17 NMR The Journal of Physical Chemistry, Vol. 85, No. 14, 1981 2143 timates an average protrusion of about 1.5 methylene less than an order of magnitude slower than pure water, groups in a sodium dodecyl sulfate micelle. A further is probably indisputable, because of the well-characterized important characteristic of ionic micellar solutions is the dispersion in the gigahertz range. Finally, mention should inhomogeneous distribution of counterions, with 40-80 % be made of a recent ultrasonic relaxation studyz2in which of the counterions confined to a region extending a few a relaxation process below 2 MHz was detected at high tenths of a nanometer out from the micelle ~urface.~ surfactant concentrations. The authors suggest that water In theoretical models of micellar solutions, the aqueous exchange between the hydration and bulk regions is the phase is usually treated as a continuous dielectric medium. underlying process, however, no quantitative analysis was It is clear, however, that a detailed understanding of these attempted. systems will require explicit recognition of the molecular In the present contribution we report water 170 relax- nature of the medium. Thus we would like to quantify the ation rates for a variety of ionic surfactant solutions above strength and range of the water-micelle interaction as well the cmc, and also for a few nonassociated molecules. After as its dynamic consequences. Quantities related to the a brief expos6 of the relevant relaxation theory, which amount of interacting water, e.g., “hydration numbers”, involves a “two-step model” with explicit reference to the have been obtained mainly from transport properties, e.g., anisotropy in the “bound” water reorientation, we present intrinsic visc~sity,’~’~micellar sedimentation and diffusion and interpret the relaxation data in five sections. The first coefficients,” and water self-diffusion coefficients,’“16 and section deals with the extent of water-hydrocarbon contact also from molar volume^.'^ The dynamic behavior, on the in micelles and the second with the effect on the 170re- other hand, has been probed by various relaxation tech- laxation of alkyl chain length. In the third section we niques: rnagneti~,~~~’~dielectric20~21 or ultrasonic.22 exploit the well-established growth of sodium dodecyl The main conclusion from the viscosity and diffusion sulfate micelles upon salt addition to estimate the average studies of solutions of ionic micelles is that the deduced residence time for water molecules at the micelle surface. “hydration numbers” of 4-10 water molecules per sur- Next we consider the analogous case of the sphere-to-rod factant molecule, corresponding to the amount of water transition for hexadecyltrimethylammonium bromide moving with the micelle as a kinetic entity, can be fully micelles and, finally, we discuss the effect of head group accounted for by the hydration of the charged head groups and counterion on the hydration. and of the associated counter ion^.^^ This conclusion is in qualitative agreement with the water proton chemical Experimental Section shift and longitudinal relaxation studies of Clifford and Chemicals. The surfactants were obtained from the Pethicals and of Walker,lg which indicate a reduced following sources: acetic acid (Merck, Suprapur), propa- water-hydrocarbon contact upon micellization. noic acid (BDH, min 99%), octanoic, dodecanoic, tetra- As regards the dynamics of the associated water, e.g., decanoic, and hexadecanoic acids (BDH, specially pure), reorientational rate and anisotropy and average residence sodium dodecyl sulfate (BDH, specially purified for bio- time, considerable uncertainty remains. Due mainly to the chemical work), hexadecyltrimethylammonium bromide influence of hydrocarbon protons on the water proton (BDH and Merck). relaxation behavix-, it has not been possible to extract The potassium salts were prepared by neutralization dynamic information from proton relaxation The with KOH, and then freeze dried and recrystallized from most detailed studies of the microdynamics of water in absolute ethanol. The molar masses were checked by ti- micellar solutions are the dielectric relaxation measure- tration with perchloric acid in glacial acetic acid. The ments from the Gottingen group.20*2’As compared to sodium and cesium salts were prepared by neutralization magnetic relaxation, the dielectric relaxation technique has with NaOH (EKA) or CsOH (Ventron). Added NaCl was the advantage of accessibility to frequencies in the giga- from Merck (Suprapur). hertz range. However, the theoretical understanding of Hexadecyltrimethylammonium chloride was prepared dielectric relaxation in terms of molecular parameters is by transferring the bromide salt to the hydroxide form on at present rather limited. To interpret dielectric relaxation a Dowex 21 K ion exchanger (BDH). The hydroxide was data one is thus forced to use continuum models involving immediately neutralized with HC1 to pH 4-5, whereafter many adjustable parameters and based on several sim- the solution was lyophilized and the chloride salt recrys- plifying assumptions. Nevertheless, the main conclusion tallized from acetone. from these studies,”S1 that the “hydration water” reorients Aqueous surfactant solutions were made from doubly distilled (quartz apparatus) water, enriched to about 1% in 170by addition of 10 atom % H2170(Biogenzia Lema- (9)G. E. A. Aniansson, J. Phys. Chem., 82, 2805 (1978). (10) P. Mukerjee, J. Colloid Sci., 19, 722 (1964). nia). The solution pH was adjusted, with KOH, NaOH, (11)W. L. Courchene, J. Phys. Chem., 68, 1870 (1964). or HC1, sufficiently far from neutral (cf. Table I) for the (12)P. Ekwall, L. Mandell, and P. Solyom, J. Colloid Interface Sci., proton exchange br~adening~~v~lto be negligible. 35, 519 (1971). (13)D. E. Guveli, J. B. Kayes, and S. S. Davis, J. Colloid Interface Relaxation Measurements. Oxygen-17 magnetic relax- Sci., 72, 130 (1979). ation rates were measured at 13.56 MHz on a modified (14)B. Lindman and B. Brun, J. Colloid Interface Sci., 42,388(1973). Varian XL-100-15 Fourier transform spectrometer and at (15)N. Kamenka, B. Lindman, and B. Brun, Colloid Polym. Sci., 252, 144 (1974). 34.56 MHz on a home-built Fourier transform spectrom- (16) M.4. Puyal, ThBse, Academie de Montpellier, 1980. eter with a 6-T wide-bore magnet from Oxford Instrument (17)P. Ekwall and P. Holmberg, Acta Chem. Scand., 19,455(1965). co. (18)J. Clifford and B. A. Pethica, J. Chem. SOC.,Faraday Trans., 60, Longitudinal relaxation rates were measured by inver- 1483 (1964);61, 182 (1965). (19) T. Walker, J. Colloid Interface Sci., 46, 372 (1973). sion recovery (~-7-x/2 pulse sequences). Each R1 value (20)U. Kaatze, C. H. Limberg, and R. Pottel, Ber. Bunsenges. Phys. is the result of a least-squares fit to the magnetization vs. Chem., 78, 561 (1974). delay time for eight different 7 values. Transverse relax- (21)R. Pottel, U. Kaatze, and St. Muller, Ber. Bunsenges. Phys. Chem., 82, 1086 (1978). ation rates were obtained from the line width (Avl12)at (22)G. J. T. Tiddy, M. F. Walsh, and E. Wyn-Jones, J. Chem. SOC., half-amplitude of the absorption curve according to R2 = Chem. Commun., 252 (1979). rAvllZ. The contribution to the line width from magnetic (23)P. Mukerjee, J. R. Cardinal, and N. R. Deaai in “Micellization, Solubilization, and Microemulsions”, Vol. 1, K. L. Mittal, Ed., Plenum field inhomogeneity was eliminated by measuring at 3-6 Press, New York, 1977, p 241. concentrations and taking the slope aR2/am,or by sub- 2144 The Journal of Physical Chemistry, Vol. 85, No. 14, 1987 Halle and Caristrom tracting the relaxation rate for a sample of pure water (pH monomer concentration is merely 2%,which is less than 2.7) measured at the same temperature. the standard deviation in aRe,/am. Two to four independent measurements of R1 as well as Equation 3 contains two unknowns: n and RB. (RF is of R2 were performed on each sample. In those cases where taken to be equal to the relaxation rate for pure water, no significant difference between R1 and R2 (or the cor- which was determined to be 129.7 f 2 s-l at 27.4 “C.) The responding slopes) was observed, the reported values are quantity n is a measure of the number of water molecules, averages over all R1 and R2 data. The quantity n(RB - RF), per amphiphile molecule, whose reorientational motion is reported in Table I, was obtained according to eq 2 as significantly perturbed. The interpretation of the intrinsic explained in a footnote to the table. The experiments were relaxation rate RB for these n water molecules depends on carried out at a probe temperature of 27.4 OC, which was the model chosen to describe the molecular motions. The kept constant to within k0.2 “C by the passage of dry simplest model considers the “bound’ water molecules to thermostated air. reorient isotropically with a rotational correlation time 7&, in which case RB is given26 by eq 4. The water 170 Results and Discussion General Considerations. In aqueous surfactant solutions (4) the 170 nuclei are distributed over several motional states with different intrinsic relaxation rates but with negligible quadrupole coupling constant x = 6.67 and is chemical shift differences. If the exchange of water virtually independent of the molecular environment. The molecules between these states is fast compared to the asymmetry parameter for the electric field gradient is intrinsic relaxation rates (see below), then the total re- taken to be the same as in ice:29 q = 0.93. laxation rate may be dec~mposed~~.~~according to eq 1, As will be clear from the following, the isotropic reori- where the sum runs over all states and Pi is the fractional entation model, underlying eq 4, cannot satisfactorily ex- population in state i. plain all our data. This model has also been found inad- equate in connection with recent 170 studieP3 of aqueous solutions of macromolecules. In a more realistic model one For aqueous surfactant solutions above the cmc, it is has to recognize that the water-micelle interaction inev- reasonable to consider three states: water “bound” to itably induces some anisotropy in the reorientation of the micelles, water “bound” to monomeric (nonmicellized) interacting water molecules. This local anisotropic re- amphiphiles, and “free” water. Furthermore, we will adopt orientation can only partially average the quadrupolar the phase separation model,lW4according to which the intera~tion,~~~~~the remaining part being averaged out by monomer concentration remains constant above the cmc. a slower motion, e.g., exchange of water molecules with the For a fast exchange three-state model, the excess relaxation bulk or reorientation of the entire micelle. We will now rate is thus given by eq 2. In eq 2, m is the total am- assume that (1) the fast and slow motions occur inde- pendently and on different time scales, (2) there is local threefold symmetry around the normal to the micelle surface, and (3) the anisotropy is small. It can then be shownn$’ that RB can be decomposed into one contribution from the fast (0 motion and another contribution from the slow (9) motion, as in eq 5. 1279 RB = -x2[(1 + q2/3 - A2hCBf+ A27as] (5) 125 phiphile molality, n is the number of motionally perturbed water molecules per amphiphile, and the subscripts F, The residual anisotropy A, the magnitude of which is mon, and mic refer to “free” water and water “bound” to confined to the range [0,1], is determined by the asym- monomeric or micellized amphiphiles, respectively. For metry parameter 1 and by the orientational probability the variation of the excess relaxation rate with the total distribution for the “bound” water molecules with respect amphiphile molality we find to the normal to the micelle surface.27 For completely isotropic reorientation, A = 0 and eq 5 reduces to eq 4. n aR,,/am = -(RB - RF) (3) From eq 5, which is valid only for A2 << 1, it is seen that 55.5 for the slow motion to contribute significantly to the re- where n and RB refer to monomeric or micellized amphi- laxation rate, it must be slower than the fast motion by philes depending on whether m is lower or higher than a factor of the order of 1/A2. If both micelle reorientation (correlation time 7,B) and water exchange (average lifetime mcmc. It is well-known’-4 that the phase separation model is 7B) contribute to the slow motion, then this may be not strictly valid, in particular, the monomer concentration characterized by an effective correlation time% according decreases above the cmc.16v62For the present purposes, to eq 6. however, eq 2 and 3 are excellent approximations. This -=-111 is so because, with one exception, our data refer to con- +- (6) centrations that are so much larger than the cmc (Table 7cB8 7rB 718 I) that any variation in the small contribution to the 170 relaxation rate from monomeric amphiphiles must nec- (26)A. Abragam, “The Principles of Nuclear Magnetism”, Clarendon essarily be small. The exception is octanoate, which has Press, Oxford, 1961. (27)B. Halle and H. Wennerstrom, J. Chem. Phys., in press. a relatively high cmc. The data of Puyal16 for sodium (28)B. Halle, T. Andersson, S. Forsgn, and B. Lindman, J. Am. Chem. octanoate show, however, that the effect of decreasing SOC.,103, 500 (1981). (29)D. T. Edmonds and A. Zussman, Phys. Lett. A, 41, 167 (1972). (30)B. Halle and L. Piculell, J. Chem. SOC.,Faraday Trans. I, in (24)J. R. Zimmerman and W. E. Brittin, J. Phys. Chem., 61, 1328 press. (1957). (31) H.Wennerstrom, G. Lindblom, and B. Lindman, Chem. Scr., 6, (25) H.Wennerstrom, Mol. Phys., 24, 69 (1972). 97 (1974). Micelle Hydration from Oxygen-17 NMR The Journal of Physical Chemistry, Vol. 85, No. 14, 1981 2145

TABLE I: Water "0 Relaxation Data for Aqueous Surfactant Solutions at 27.4 t 0.2 "C n. cmc, (RB --RF),' surfactant" mol kg-l m, mol kg-' pH S

R2,ex CH, CO 2K 0.49 12.0 1.38 t 0.05 CH,CH,CO,K 0.56 12.2 1.84 + 0.05 S -' C,CO,K 0.35 0.10-0.32 12.1 3.90 t 0.14 C,CO,K 0.35 0.45-1.00 12.1 1.86 t 0.04 25 C,,CO,Na 0.023 0.42 11.8 2.93 t 0.20 CllCO,K 0.0 24 0.44 10.5 1.99 t 0.20 C,,CO,CS 0.025 0.45 12.3 2.18 t 0.20

C13C02K 0.006 0.10-0.50 12.0 2.19 t 0.09 Cl,CO,K 0.00 2 0.10-0.55 12.0 2.09 t 0.06 (Rl) m,mol kg -' 2.48 i 0.18 (R,) Figure 1. Water "0 transverse excess relaxation rate vs. molality of C, ,OSO, Na 0.008 0.10-0.90 10.9 1.77 t 0.05 potassium octanoate at 27.7 OC and 13.56 MHz. The lines were obtained by least-squares fits to the data points connected by solid C,,OSO,Na 0.001 0.10 + 0.38 10.9 3.88 t 0.30 lines. m NaCl C,,OSO,Na <0.001 0.10 + 0.72 10.9 4.42 + 0.50 If the slow motion is not fast compared to the inverse m NaCl angular resonance frequency l/wo,then the longitudinal (CH,),NBr 1.0-3.7 1-2 1.18d and transverse relaxation rates become unequal. In eq 4 (C, H, )PBr 0.6- 2.4 1-2 8.06d and 5, 7& and 7cB8,respectively, should then be multiplied C16N(CH3 )3" 0.001 0.18 3.6 2.99 t 0.30 by323 C16N(CH3 )3Br 0.001 0.18 3.4 1.88 t 0.30 Cl,N(CH,),Br 0.001 0.57 3.4 2.19 + 0.10

0.2 + 0.8 (74 C, denotes the normal alkyl chain CH,(CH,),., . 1 + (WO~,..~)~ 1 + (~woT,~~)~ From ref 34, except for C,CO,K, the cmc of which is taken from Figure 1. With one exception, we found no for the longitudinal relaxation rate, and by significant difference between R, and R, and no significant frequency dependence between 13.56 and 34.56 MHz. The exception is C,,CO,K, for which R, + R, at 34.56 MHz. The quantity n(RB - RF) was for the transverse relaxation rate. obtained, according to eq 2, from least-squares fits to Extent of Water-Hydrocarbon Contact. The variation 5-12 data points (each derived from 2-4 measurements) in the indicated concentration range (t 1standard of the water 170excess relaxation rate (R, = R,) with deviation), or from measurements at a single concentra- concentration of potassium octanoate is shown in Figure tion (c estimated uncertainty). The data for the dodeca- 1. There are two linear regions interrupted by a well- noates have been corrected for the monomer contribution defined change of slope at 0.350 mol kg-', which we identify (5-10 %), using n(R,,, - RF)= 5.5 X lo3(extrapolated as the cmc. (Literature cmc fall in the range from the monomer data). From ref 41. 0.354.45 mol kg-'.) The reduction in the quantity n(RB - RF), obtained from the slopes according to eq 3, from 3.90 area" is occupied by carboxylate groups. x lo3 below the cmc to 1.86 x lo3 above the cmc (Table The conclusion that the water-hydrocarbon contact I), indicates a substantial reduction of the water-hydro- corresponds to less than two fully exposed methylene carbon contact upon micellization. (As noted above, the groups holds also for micelles composed of the longer-chain effect of decreasing monomer concentration above the cmc alkanoates. A longer chain results in a slightly enhanced is negligible.) relaxation rate (Table I), which, however, is due to an In order to quantify this conclusion, we performed increasing contribution from slow motions. This point is measurements on solutions of two short-chain potassium discussed in the following section. alkanoates. From the data in Table I it is seen that oc- The water-hydrocarbon contact in hexadecyltri- tanoate micelles produce virtually the same effect on the methylammonium bromide micelles may be estimated with water 170relaxation as does propionate. We can thus infer the aid of 170relaxation rates for short-chain tetraalkyl- that the water-hydrocarbon contact in potassium octa- ammonium bromides reported by Fister and Hertz41 (in- noate micelles corresponds to at most two fully exposed cluded in Table I). A comparison with the micelle data methylene groups per amphiphile. This must be taken as shows that an upper limit of two fully exposed methylene an upper limit since, as discussed above (eq 5), there may groups is reasonable also for hexadecyltrimethylammonium be a contribution from slow motions to the relaxation rate bromide micelles. for the micellar solution. On the other hand, a much Effect of Alkyl Chain Length. The effect of alkyl chain smaller hydrocarbon exposure is unlikely on purely geo- length, i.e., of micelle size, on the water 170relaxation rate metrical grounds. Thus, in a molecular model with an was investigated for potassium alkanoates with 8, 12, 14, aggregation numbera of 11 only about half of the "surface and 16 carbon atoms. The data in Table I reveal a mon- otonic increase in relaxation rate with increasing chain (32)B. Halle and H. Wennerstrdm, J. Magn. Reson., in press. length. From geometrical considerations, it is clear that (33)A. D. McLachlan, Proc. R. SOC.London, Ser. A, 280, 271 (1964). the area per head group (at a surface which passes through (34)P. Mukerjee and K. J. Mysels, Natl. Stand. Ref. Data Ser., Natl. Bur. Stand., No. 36 (1971). the head groups) must decrease as the micelle grows ra- (35)P. White and G. C. Benson, J. Chem. SOC.,Faraday Trans, 55, 1025 (1959). (36)P. White and G. C. Benson, J. Phys. Chem., 64, 599 (1960). (40)B.-0. Persson, T.Drekenberg, and B. Lindman, J. Phys. Chem., (37)H. B. Klevens, J. Phys. Colloid Chem., 52, 130 (1948). 83, 3011 (1979). (38)S. H. Herzfeld, J. Phys. Chem., 56, 953 (1952). (41)F. Fister and H. G. Hertz, Ber. Bunsenges. Phys. Chem., 71,1032 (39)K. Shinoda, J. Phys. Chem., 58, 541 (1954). (1967). 2146 The Journal of Physical Chemistry, Vol. 85, No. 14, 7981 Halle and Caristrom dially.42 (This effect, which is quite small, is a consequence pating the result of the calculation, we assume that this of the assumption of a constant area per surfactant mol- is so much larger than the lifetime T~ that, according to ecule at the surface of the hydrocarbon core.) One would eq 6, 7,gS =. 71B. To proceed, we need an estimate of the thus expect the relaxation rate to decrease somewhat with residual anisotropy A, appearing in eq 5. increasing chain length, contrary to our observations. We The local environment for water molecules interacting therefore conclude that the effect of decreasing water- with the sulfate head groups is undoubtedly very nearly hydrocarbon contact is overcompensated by a contribution the same in the rod-shaped aggregates under discussion from slow motions according to eq 5. If micelle reorien- and in the hexagonal mesophase consisting of ordered tation contributes to ~,g’ (cf. eq 6),this slow contribution rod-shaped aggregates. Since this latter system is aniso- should increase with the chain length. This follows from tropic one obtains quadrupolar splittings in the water the fact that a longer alkyl chain leads to a larger micelle deuteron or 170spectra, from which the residual anisotropy per se and also increases the aggregation number. (Ac- can be extracted. Since the water 170residual anisotropy cording to the Debye-Stokes-Einstein relation, the rota- is simply twice that for water deuteron^,^^^^^ we have not tional correlation time T,B, appearing in eq 6, is propor- measured the 170splitting but have used literature dataM tional to the cube of the micelle radius.) for the deuteron splitting in the hexagonal phase of the For hexadecanoate micelles we observed a significant D20-NaDS system. The observed splitting is 560 Hz for difference between the longitudinal and transverse relax- 21.6 mol of D20per mol of NaDS. Using the equationnJ1 ation rates at a resonance frequency of 34.56 MHz (Table for the quadrupolar splitting in a powder sample and a I). This enables us to estimate the slow correlation time deuteron quadrupole coupling constantz7 of 220 kHz, we 7,Bs, if we assume that the fast contribution to the relax- find IAl = 0.29/n, where n is the water “coordination ation rates is the same as for acetate. (As noted above, number”. We have then multiplied by a factor of 2 to the water-hydrocarbon contact should be smaller for obtain the local anis~tropy~~p~land by another factor of hexadecanoate micelles than for octanoate micelles.) In- 2 to convert to the 170residual anisotropy, as noted above. serting data from Table I into eq 3,5, and 7 we thus find We can now substitute x = 6.67 MHz, 7 = 0.93, and IAl T,B’ .= 3 ns. This figure agrees well with the observed = 0.29ln into eq 5 for Rg, which on combination with the maximum in the longitudinal relaxation rate vs. chain value n(RB- RF) = (4.42 f 0.50) X lo3,corresponding to length (Table I), although this maximum is barely sig- the highest NaCl concentration in Table I, and the known nificant by itself. From eq 7a one can thus infer that R1 value RF = 129.7 s-l, yields an expression for the lifetime should exhibit a maximum at w~T,~~= 0.6158, which, at 71Bas a function of n and 7,Bf. It seems reasonable to use a resonance frequency of 34.56 MHz, corresponds to 7,g’ a “coordination number” n in the range 6-9, in accordance = 2.8 ns. with viscosity data.’Ov51 The fast correlation time 7,gf is With this estimate for T,B’, it can be shown that the certainly not shorter than 2.38 ps, which is the value for isotropic motion model, i.e., eq 4, is physically unreason- pure water at 27.4 “C. An upper limit is furnished by the able. Thus n(Rg - RF)= 2.09 X lo3 s-’ and T,B’ = 3 ns for relaxation rate for salt-free NaDS micelles (Table I), if we hexadecanoate micelles yield, with eq 4 and 7a, the com- neglect the contribution from the slow motions in this case. pletely absurd result: n = 0.03. This clearly demonstrates It is found that values of n in the range 6-9 correspond that it is necessary to take into account explicitly the to 7,gf values of 3.0 f 0.5 times the pure water value. consequences of the anisotropy in the “bound” water re- Finally, we go back to the expression for T~,to find that orientation, as we have done in eq 5. values of n in the range 6-9 together with 7,gf values 1-3 NaDS Micelles at High NaCl Concentration. From times longer than for pure water correspond to lifetimes several light-scattering studies,- it has been inferred that in the relatively narrow range 6-11 ns. Since this actually sodium dodecyl sulfate (NaDS) micelles, at high NaCl refers to an effective correlation time, in the sense of eq concentrations, grow laterally to form large rod-shaped 7,the upper limit should be multiplied by 110.3. The final aggregates exceeding 50 nm in length. If the slow corre- result is 6 < 7~ < 37 ns. lation time 7,gS, entering in eq 5, were determined solely The Sphere-to-Rod Transition for HTABr Micelles. In by aggregate reorientation, then one would expect that the absence of added salt, hexadecyltrimethylammonium NaCl addition to NaDS micelle solutions should increase bromide (HTABr) micelles undergo a change, at about 0.3 the transverse 170 relaxation rate by orders of magnitude mol kg-l, from an essentially spherical to a cylindrical or even lead to quadrupolar splittings. As can be seen from shape. Upon further increase in concentration, the rodlike Table I, this is not the case; the relaxation rate is enhanced aggregates grow along the cylindrical axis. This transition by merely a factor of 2.5. This observation implies that gives rise to dramatic changes in physical properties such T&’ is controlled, to a large extent, by a more rapid motion, as reduced viscosity,12light scattering,12small angle X-ray which we identify as exchange of water molecules between ~cattering,~~transverse magnetic relaxation rates of “N the “free” and “bound” states. On the basis of this in- (head groups)%and of 81Br (counterions),“ and anisotropy terpretation, we will now proceed to estimate the average of the electrical conductivity in oriented solutions.M The residence time TBfor water molecules associated with the increase in the water 170relaxation rate (Table I), however, micelle surface. is quite small, in fact barely significant. The explanation From the dimension^^^-^^ of the rod-shaped NaDS mi- is most probably the same as in the case of NaDS, i.e., the celles and with the aid of Perrin’s equations47we find an residual anisotropy is averaged out, not by aggregate re- effective48rotational correlation time T,~= 2 ps. Antici- (49)Y. Tricot and W. Niederberger, Biophys. Chem., 9, 195 (1979). (42)J. E. Leibner and J. Jacobus, J. Phys. Chern., 81, 130 (1977). (50)I. D. Leigh, M. P. McDonald, R. M. Wood, G. J. T. Tiddy, and (43)N. A. Mazer, G. B. Benedek, and M. C. Carey, J. Phys. Chem., M. A. Trevethan, in press. 80, 1075 (1976). (51)F. Tokiwa and K. Ohki, J. Phys. Chem., 71, 1343 (1967). (44)C. Y. Young, P. J. Missel, N. A. Mazer, G. B. Benedek, and M. (52)F. Reiss-Husson and V. Luzzati, J. Phys. Chern., 68,3504(1964). C. Carey, J. Phys. Chem., 82, 12 (1978). (53)U. Henriksson, L. Odberg, J. C. Eriksson, and L. Westman, J. (45)P. J. Missel, N. A. Mazer, G. B. Benedek, C. Y. Young, and M. Phys. Chem., 81, 76 (1977). C. Carey, J. Phys. Chem., 84, 1044 (1980). (54)G. Lindblom, B. Lindman, and L. Mandell, J. Colloid Interface (46)S. Hayashi and S. Ikeda, J. Phys. Chem., 84,744 (1980). Sci., 42, 400 (1973). (47) F. Perrin, J. Phys. Radium,5, 497 (1934);7, 16 (1936). (55)K. G. Gotz and K. Heckmann, Z. Phys. Chern. (Frankfurt am (48)D. E. Woessner, J. Chem. Phys., 37, 647 (1962). Main), 20, 42 (1959). Micelle Hydration from Oxygen-17 NMR The Journal of Physical Chemistry, Vol. 85, No. 14, 7981 2147 orientation, but by water exchange. (The effective rota- differ in hydration. The much larger effect of HTACl tional correlation time, as deduced in the 14N relaxation micelles, as compared to HTABr micelles (Table I), is thus exceeds 340 ns.) probably due partly to the different ionic radii and partly The approach used in the preceding section to estimate to exclusion of some hydration water upon Br- association. the water lifetime for NaDS micelles cannot, however, be To get an idea of the importance of the nature of the used for HTABr micelles. The reason is that no inde- head group in determining the water 170 relaxation rate, pendent information about the residual anisotropy is we may compare the data in Table I for NaDS (salt-free) available. Thus no resolved deuteron splittings could be and sodium dodecanoate micelles. It is seen that the observed in a studyS6 of the hexagonal phase of the DzO- carboxylate group produces a much larger effect than the HTABr system, indicating a very small anisotropy. This sulfate group. This difference can be accounted for is not surprising, considering that the methyl groups act qualitatively on the basis of the smaller size of the COO- to keep the water molecules at some distance from the group, relative to the OSOy group. Consequently, there charge. The small residual anisotropy for water associated is more space available for water molecules to penetrate with the trimethylammonium group is also consistent with between the head groups in the alkanoate micelles. the insignificant effect of the sphere-to-rod transition on Conclusions and Summary the 170relaxation rate, as compared to the transition in NaDS. The present study confirms our previousaJO conclusion It should be pointed out here that the extremely large that water 170relaxation in aqueous solutions of macro- “hydration numbers” (n = 60-70) suggested for HTABr molecules or large molecular aggregates cannot be satis- micelles by Ekwall et al.12 receive no support from our data. factorily explained by a simple two-state model in which They are probably an artifact arising from the electro- each state is characterized by a single correlation time. viscous effect. In fact, the water-HTABr interaction ap- Rather, one must introduce two correlation times to de- pears to be very weak. Thus Mukerjee’O has estimated, scribe the stepwise averaging of the quadrupolar interac- from viscosity data, a hydration number of 4.1 f 1.3 for tion in the “bound” state. The contributions from the fast tetradecyltrimethylammonium chloride micelles. And, as and slow motions to the relaxation rate are often of com- the data in Table I show, HTABr micelles produce a parable magnitude when the slow correlation time is in the considerably smaller 170 relaxation enhancement than do nanosecond range. Micellar systems are ideally suited for HTACl micelles. application of the two-step model, because of their simple Effect of Counterion and Head Group. The effect of structure and because the weighting factor for the slow the nature of the counterion on the water 170relaxation component, Le., the square of the residual anisotropy, often rate was investigated for dodecanoate micelles and for can be obtained from quadrupolar splittings in the cor- hexadecyltrimethylammoniummicelles. In the former case responding anisotropic mesophases. we found (Table I) that sodium counterions are substan- The rate of local water reorientation at the micelle tially more effective than either potassium or cesium ions surface is typically 2-3 times slower than in pure water. in enhancing the 170relaxation rate. It is known57that This is also true for the hydration water of small solutes all three counterions retain their primary hydration upon and agrees with results from dielectric relaxation stud- association to alkanoate micelles. Consequently, the origin ies.20B21@The residual anisotropy is averaged out mainly of the difference must be that the smaller size of the so- by micelle reorientation for small micelles and mainly by dium ions results in a stronger dynamical perturbation of water exchange for large micelles. The average lifetime the adjacent water molecules. (This effect on the I7O for water molecules associated with sodium dodecyl sulfate relaxation is likely to dominate over the effects caused by micelles is between 6 and 37 ns. the higher aggregation number and the more extensive No long-range water-micelle interaction is indicated by counterion association for CsDS as compared to NaDS, our data. For potassium octanoate micelles, the 170 re- recently reporteds2 by Frahm et al. Indeed, if these were laxation enhancement corresponds to hydration of the the only counterion effects the 170relaxation rate would, carboxylate head groups, the potassium counterions, and contrary to observation, be largest for CsDS.) The dif- a water-hydrocarbon contact equivalent to less than two ference in the 170relaxation enhancement between sodium fully exposed methylene groups. For longer-chain micelles and potassium dodecanoate micelles is about three times the water-hydrocarbon contact is of similar magnitude. larger than the difference between NaCl and KC1 in the Small head groups permit more water to penetrate between absence of surfactant (unpublished data). The micelles themselves, as evidenced by the larger effect on the I7O thus have the effect of enhancing the difference in relaxation rate of micelles with carboxylate head groups “hydration” between Na+ and K+. This may be due to the as compared to those with sulfate head groups. The in- fact that the proximity of a large hydrocarbon core with fluence of counterions on the 170 relaxation rate can be low relative permittivity acts to enhance all electrostatic explained in terms of the ionic radii and a possible dehy- interactions in the adjacent aqueous regiona5* dration accompanying Br- association to hexadecyltri- It has not been conclusively demonstrated whether methylammonium micelles. chloride and bromide ions retain their primary hydration Acknowledgment. We thank Krister Fontell for gen- upon association to alkyl trimethylammonium micelle^.^^^ erous supplies of recrystallized surfactants and Bjorn However, in view of the striking differences between Lindman for encouraging and helpful discussions. Grants HTACl and HTABr as regards micelle shape at high from Kungl. Fysiografiska Sdlskapet i Lund, from Stif- ~oncentration,5~it is plausible that associated C1- and Br- telsen Bengt Lundqvists Minne, and from the Swedish Natural Sciences Research Council are gratefully ac- (56) N.-0. Persson and A. Johansson, Acta Chem. Scand., 25, 2118 knowledged. (1971). (57) H. Gustavsson and B. Lindman, J. Am. Chem. SOC.,100, 4647 (197R).~-.. _,. (60) K. Hallenga, J. R. Grigera, and H. J. C. Berendsen, J. Phys. (58) J. D. Jackson, “Classical Electrodynamics”, 2nd ed, Wiley, New Chem., 84, 2381 (1980). York, 1975, Section 4.4. (61) S. Meiboom, J. Chem. Phys., 34, 375 (1961). (59) H. Fabre, N. Kamenka, A. Khan, G. Lindblom, B. Lindman, and (62) J. Frahm, S. Diekmann, and A. Haase, Be?. Bumenges. Phys. G. J. T. Tiddy, J. Phys. Chem., 84, 3428 (1980). Chem., 84, 566 (1980).