2600
J. Phys. Chem. 1990, 94, 2600-2613
clusion compound of CDx, 1-propanol, and N), and NC-AC (exciplex) fluorescence by I- and IO3- have been carried out, and the results are tabulated in Table 111 together with fluorescence lifetimes for these species. As is usually the case, rate constants for fluorescence quenching of N C by I- and IO3- are reduced by about one-fourth and one-seventh relative to those of N, respectively, indicating the protection by CDx from the quenchers. The most plausible structures of N G N C and N G A C are such that the two CDx cavities are associated facing each other (a headto-head, head-to-tail, or tail-to-tail orientation). In this case, Imust approach and quench the excimer or exciplex inside the connected cavities from the two open ends of the cavities. The same is true for the quenching of N C ( N within the cavity), although in this case only one CDx molecule is relevant. Consequently, rate constants for fluorescence quenching of NC, N G N C (excimer), and NC-AC (exciplex) by I- are almost identical. For NPrC, on the other hand, I- can attack an N molecule inside the cavity only from the other uncapped (un-
floored) open end because in NPrC one of the two open ends of the cavity is capped (floored) by 1-propanol. As a result, a quenching rate constant for NPrC is about half those for NC, NCqNC, and NCsAC. Quenching of the excimer, exciplex, and NPrC by IO3- is quite different from that by I-: slight enhancement of the excimer, exciplex, and N R C emissions by 10,rather than quenching is observed. Similar quenching behavior by IO3- has been reported for a 2-methoxynaphthalene-1,2-dicyanobenzene complex (CDx-2-methoxynaphthalene-1,2-dicyanobenzene system).'* No quenching by IO3- is attributed in part to the bulkiness of IO3- compared with I-. However, the negative quenching (intensity enhancement) effect of IO3-cannot clearly be explained at present, although there is a possibility that IO< influences the geometrical structures of NPrC, NC-NC, and NC*AC. Acknowledgment. I thank Professor Fumio Hirayama for his valuable discussion.
Counterlon Surface Dtffusion in a Lyotropic Mesophase. A ""a Two-Dimensional Quadrupolar Echo NMR Relaxation Study Istvin Furti,+Bertil Halie,* Per-Ola Quist, and Tuck C. Wongt Physical Chemistry 1, University of Lund, Chemical Center, P.O. Box 124, S-22100 Lund, Sweden (Received: July 12, 1989)
In the first systematic application of the two-dimensional quadrupolar echo (2DQE) method, we study the 23NaNMR relaxation behavior of counterions in the reversed hexagonal mesophase of the AOT/D20/isooctane system, consisting of long cylindrical aggregates with the water and ions on the inside. Using a combination of relaxation experiments, performed on magnetically aligned samples, we determine the three spectral densities at two sample orientations. The orientational dependence of the spectral densities allows us to separate contributions from different types of molecular motion. In particular, we find a dominant contribution from surface diffusion of counterions along the periphery of the cylindrical aqueous regions (diameter ca. 50 A). From the frequency dependence of this contribution, we determine the diffusion coefficient of sodium ions in the interfacial region, a quantity which is important for a variety of phenomena in colloid and polymer science, electrochemistry,and biophysics. We thus obtain D(Na+) = (2.8 f 0.3)X 1O-Io m2 s-l, which is a factor of 3.6 lower than in an infinitely dilute aqueous (DzO) electrolyte solution at the same temperature (20.6 "C). In addition, our analysis of the 23Narelaxation rates and spectral line shapes provides information about the fast local ion and water dynamics in the interfacial region, as well as about the equilibrium structure of the liquid crystalline phase.
Introduction The technique of nuclear spin relaxation is one of the most versatile and powerful tools available for the study of molecular organization and dynamic processes in complex fluids. Among the variety of useful magnetic isotopes, the quadrupolar nuclei of atomic ions, such as 23Na, have found extensive use, mainly in the polyelectrolyte and surfactant fields' and in b i o p h y ~ i c s . ~ , ~ The more recent applications include studies of synthetic polyel e c t r o l y t e ~ , ~DNA,',* -~ and polysaccharide^^*'^ as well as more complex biological systems.'I-l3 Virtually all previous counterion spin relaxation work has been devoted to isotropic solution systems. In locally heterogeneous isotropic fluids, the fluctuations that drive the relaxation of the spin system are distributed over many, often quite complex, degrees of freedom of the molecular system. While this complexity enhances the information content of the spin relaxation rates, it can also become an obstacle on the, often long and treacherous, path to a unique interpretation. One strategy for avoiding interpretational ambiguities is to study systems that are simpler in structure, yet exhibit the essential 'On leave from the Central Research Institute for Physics, Budapest, Hungary. Present address: Department of Chemistry, University of Missouri, Columbia. MO 6521 1. f
features of interest, e.g., an electrolyte solution interface. In this work we have adopted this strategy, by choosing to study a lyotropic liquid crystalline mesophase. Being translationally and orientationally ordered over macroscopic dimensions, smectic liquid crystals offer the prospect of a relatively straightforward interpretation of relaxation data, provided that appropriate relaxation experiments can be designed and implemented. This is particularly (1) Lindman, B. In N M R of Newly Accessible Nuclei; Laszlo, P., Ed.; Academic Press: New York, 1983; Vol. I, p 193. (2) Gupta, R. K.; Gupta, P.; Moore, R. D. Annu. Rev.Biopbys. Bioeng. 1984, 13, 221. (3) Springer, C. S. Annu. Rev. Biopbys. Biophys. Cbem. 1987, 16. 375. (4) Levij, M.; de Bleijser, J.; Leyte, J. C. Cbem. Phys. Lerr. 1981,83, 183; 1982, 87, 34. (5) Halle, B.; Wennentrom, H.; Piculell, L. J. Phys. Cbem. 1984,88, 2482. (6) Halle, B.; Bratko, D.; Piculell, L. Ber. Bunsen-Ges. Pbys. Cbem. 1985, 89, 1254. (7) Nordenskiold, L.; Chang, D. K.; Anderson, C. F.; Record, M. T. Biochemistry 1984, 23. 4309. (8) van Dijk, L.;Gruwel, M.L. H.; Jesse, W.; de Bleijser, J.; Leyte, J. C. Biopolymers 1987, 26, 26 1. (9) Grasdalen, H.; Kvam, B. J. Macromolecules 1986, 19, 1913. (10) Piculell, L.; Nilsson, S. J. Pbys. Cbem. 1989, 93, 5602. (11) Monoi, H. Biopbys. J . 1985, 48, 643. (12) Pettegrew, J. W.; Glonek, T.; Minshew, N. J.; Woessner, D. E. J . Magn. Reson. 1985, 63, 439. (13) Urry, D. W. Bull. Magn. Reson. 1987, 9, 109.
0022-3654/90/2094-2600$02.50/00 1990 American Chemical Society
Counterion Surface Diffusion true for the many lyotropic mesophases, whose regular arrangement of surfactant aggregates have been accurately characterized by X-ray d i f f r a ~ t i 0 n . l ~ The particular system under study here is the reversed hexagonal (F) phase in the AOT/D20/isooctane system (AOT = sodium bis(2-ethylhexyl) sulfosuccinate), consisting of hexagonally arranged cylindrical aggregates, with water and sodium ions on the inside and a continuous hydrocarbon region on the 0 ~ t s i d e . l ~ A useful feature of this system is that the radius of the aqueous rods can be varied, without altering the local properties of the interface, simply by changing the relative amounts of water and AOT. While quadrupolar splittings from counterions in lyotropic mesophases have been studied we are not aware of any systematic studies of counterion relaxation in such systems. One can discern several reasons for the scarcity of counterion relaxation studies on anisotropic systems. One major reason is related to the fact that, while the structure and molecular dynamics are usually simpler in anisotropic than in isotropic heterogeneous systems, the reverse is true for the spin dynamics, particularly for nuclei with spin I > 1 (as for virtually all counterion nuclei). However, as the spin system can be extensively manipulated by coherent radio-frequency (rf) pulses, the complicated spin dynamics need not be a problem. In fact, it can be turned into an advantage. We have recently embarked on a program for designing multiple-pulse techniques to study quadrupolar relaxation of high-spin nuclei in anisotropic system^.'^-^^ Of particular relevance for the present study is the two-dimensional quadrupolar echo (2DQE) method,” which allows the zero-frequency spectral density (which is sensitive to motions on a time scale of s) to be determined from the spectral satellites, in spite of static inhomogeneity broadening due to a residual orientational disorder and/or a spatially nonuniform quadrupole coupling constant. In the system studied here, both kinds of sample inhomogeneity are present and contribute substantially to the satellite line width in the 23Naspectrum. Under such conditions, the 2DQE experiment is an indispensable tool for an accurate determination of the homogeneous line width. (A direct line shape fit to the conventional spectrum2I cannot distinguish a symmetric inhomogeneity broadening, as produced by a nonuniform quadrupole coupling, from the homogeneous line width.) Since this study represents the first systematic application of the 2DQE method, we shall devote some attention to methodological aspects. Another potential problem with relaxation studies on anisotropic systems is the need to induce, e.g., by a strong magnetic field, a (preferably, enduring) macroscopic alignment in the NMR sample. In principle, relaxation data from unaligned (powder) samples contain information about the orientational dependence of the spectral densities. In practice, however, this information is often difficult to extract from the powder spectra (cf. below). Fortunately, the reversed hexagonal phase under study here can be magnetically aligned, so the orientation-dependent spectral densities can be readily determined. In effect, we exploit the tensorial nature of the nuclear quadrupole coupling to selectively monitor different motional degrees of freedom on the basis of their symmetry properties. The analysis of our 23NaN M R data is divided into two parts. First we deal with static properties, such as sample morphology, orientational (dis)order, counterion distribution, and residual quadrupole coupling constant. These properties, which also have a bearing on the relaxation data, are deduced from an analysis (14) Fontell, K. In Liquid Crystals and Plastic Crystals; Gray, G . W., Winsor, P. A,, Eds.; Ellis Horwood: Chichester, 1974; Chapter 4. (15) Ekwall, P.; Mandell, L.; Fontell, K. J . Colloid Interface Sci. 1970, 33, 215. (16) Boden, N.; Jones, S . A. In Nuclear Magnetic Resonance of Liquid Crystals; Emsley, J . W., Ed.; D. Reidel: Dordrecht, 1985; p 473. (17) Furb, 1.; Halle, B.; Wong, T. C. J . Chem. Phys. 1988, 89, 5382. (18) Furb, I.; Halle, B. J . Chem. Phys. 1989, 91, 42. (19) Fur& 1.; Halle, 8.; Einarsson, L. J . Magn. Reson., in press. (20) Furb, I.; Halle, B. Unpublished results. (21) Chachaty, C.; Quagebeur, J. P. Mol. Phys. 1984, 52, 1081.
The Journal of Physical Chemistry, Vol. 94, No. 6,1990 2601 TABLE I: Composition, Structure, and Phase Transition Temperature of Investigated F-Phase Samples samule wt ’?k isooctane x (mol D,O/mol AOT) cylinder radius, R I A phase transition” temp/OC nonaligned fraction, f
I
I1
111
IV
13.0 18.0 22.6 28 0.21
13.1 20.9 25.3 30 0.25
13.0 24.4 28.5 32 0.23
13.4 27.3 31.2 33 0.21
“The F and L2 phases coexist in a 1-2 “C range around the given phase transition temperature.
of spectral line shapes and quadrupolar line splittings. In the second part, we analyze the various relaxation rates. As a first step, we convert the relaxation rates into orientation-dependent spectral densities. These are further decomposed, using a recently developed general theoretical framework22for spin relaxation by translational diffusion in ordered fluids, into reduced spectral densities, associated with motions of different symmetry and on different time scales. Apart from the ubiquitous fast local motions, there is a dominant contribution from lateral counterion diffusion along the cylindrical interface (around the cylindrical axis). The frequency dependence of this contribution yields directly the associated correlation time. By repeating the whole procedure for samples of different water/surfactant molar ratio, we can accurately determine the translational self-diffusion coefficient for sodium ions in the interfacial region. For the sake of continuity in presentation, we have relegated most derivations and technical considerations to five appendixes.
Materials and Sample Preparation AOT (sodium bis(2-ethylhexyl) sulfosuccinate) from Sigma and isooctane (2,2,4-trimethylpentane) from Aldrich (99%) were used as supplied. The water was a 4:l mixture of D 2 0 (>99.8% 2H) from Norsk Hydro and I70-enriched D 2 0 (22% I7O,62% I8O) from Ventron. (The 170enrichment was motivated by a parallel water I7O relaxation study, to be reported elsewhere, of our Fphase samples. The deviation from the natural oxygen isotope composition has no significant effect on the 23NaN M R data.) Samples in the reversed hexagonal (F) phase of the AOT/ D20/isooctane system were prepared by weighing the components into 10-mm-0.d. Pyrex tubes (cleaned with alkaline ethanol and rinsed with doubly distilled water), which were then flame-sealed. The samples were homogenized by shaking at 40 OC, where they exist in the form of an isotropic microemulsion (L2) phase (cf. phase transition temperatures in Table I), which has a much lower viscosity than the F phase. The samples were then brought into the F phase by cooling. The resulting liquid crystal powder samples (containing randomly oriented crystalline domains) were identified optically between crossed polarizers and through their characteristic 2H N M R quadrupolar powder spectra. Macroscopically aligned F-phase samples were prepared by slow cooling in the presence of a magnetic field (Bo= 8.5 T). The F phase has a positive magnetic susceptibility anisotropy and is consequently aligned with the director of the uniaxial phase parallel to the magnetic field. Once induced by the magnetic field, the macroscopic orientation persists (cf. below), thus allowing N M R experiments to be carried out at nonzero angle OLD between the magnetic field and the phase director. N o change in the 23Na relaxation data could be detected after storing the samples at 5 OC (still in the F phase) for up to 2 months. To prevent the aligned samples from flowing d u r i n g measurements in the horizontal NMR probe (see below), the tubes were opened and a Teflon plug was tightly fitted at the liquid crystal/air interface. The measurements with this probe were then completed within 24 h. An air-flow temperature controller provided f0.3 OC sample temperature stability (measured with a thermocouple) during the measurements. (22) Halle, B. Mol. Phys. 1987, 60, 319. *There is a misprint in eq 2.34, the last term of which should read Fk2(OLc)&(co).
2602
The Journal of Physical Chemistry, Vol. 94, No. 6, 1990
NMR Methodology All NMR measurements were performed on a Bruker MSL-100 spectrometer (23Naresonance frequency 26.49 MHz), equipped with a saddle-coil vertical 10-mm probe (for measurements at = O o ) and a solenoidal horizontal IO-" probe (for measurements at 1 9 =~90'). ~ The 23Na 90' pulse length was 6-7 I S , which provided sufficiently uniform excitation in the quadrupole-split spectra. This was confirmed by the finding that the central line and the satellites were fully inverted by rf pulses of virtually the same length (to within 0 . 3 ps). Inversion Recovery. The satellite longitudinal relaxation rate, R , , = 1 /Tis, was determined from a conventional inversion recovery experiment, using 23 delay times in the range (O.l-lO)Tls. A least-squares fit of the recovery function h = ho[l - a exp(--Rl,T)]to the satellite peak amplitude h versus delay time r yielded R,,. Reported R,, values represent averages over the two satellites. Since the central line inversion recovery is biexponential,18 it is unsuitable for accurate determination of relaxation rates. A further complication in our samples is the contribution to the central line intensity from the nonaligned part of the sample (cf. below). Consequently, we analyze the inversion recovery of the satellites only. 2D Hahn Echo. The homogeneous central line width AuPm, and the corresponding relaxation rate Rchom= TAU>"', was determined from a 2D Fr Hahn echo e~periment,'~2~ using the pulse sequence (n/2),-~-(a) -raq with 64 T values and a narrow (2-3 kHz) filter width. was obtained from a Lorentzian fit to the homogeneous central line in the F1 cross-section spectrum resulting from Fourier transformation with respect to 7 (with zero filling to 4 K) of the peak amplitude of the inhomogeneous central line in the F2 spectrum. We also determined the inhomogeneous central line width AuCinhomin the conventional (single a/2 pulse) 1 D spectrum. The difference - Au,hom, which is due to magnetic field inhomogeneity, was less than 15 and 45 H z for the vertical and horizontal probes, respectively. 2 0 Quadrupolar Echo. The homogeneous satellite line width Av,hom, which reports on slow molecular processes via the zerofrequency spectral density, cannot be accurately determined from the conventional 1D spectrum, the satellites of which are severely broadened by sample inhomogeneity (of the local director orientation OLD and of the motionally averaged quadrupole coupling constant). This problem can be circumvented by using the 2D quadrupolar echo (2DQE) method," wherein the quadrupolar dephasing due to sample inhomogeneity is refocused by the pulse sequence (a/2),-r-(~/2),-~-acq. Fourier transformation with respect to T of the peak amplitudes of the inhomogeneous lines in the F2 spectrum yields F1 cross-section spectra, the central line width of which directly provides the homogeneous (unaffected by sample inhomogeneity) width of the corresponding F2 lines. the relaxation rates obtained by the We denote by R,QEand RSQE 2DQE method using the central line or the satellite line, respectively, in the F2 spectrum. Since this is the first systematic application of the 2DQE method, it is appropriate to give an account of some of the technicalities involved in this experiment. In the first step, we generate a 1 D QE (or F2) spectrum (1-2 K) by Fourier transforming the free induction decay, starting at the echo a time T after t h e second QE pulse. As in the spin I = 1 case,24 t h e effect of the finite length of the rf pulses on the echo position has to be corrected for. This was done empirically by shifting the starting point for the acquisition period by about the pulse length. For the special choice of delay time T = I / u Q , where uQ is the quadrupolar splitting, the F2 spectrum is identical with the conventional (single ~ / pulse) 2 3:4:3 pure absorption-mode spectrum, as shown in Figure la. F2 spectra, with T-dependent phase and amplitude, were recorded for 256 or 512 different T values. Since we only need the central line in the F1 spectra to determine RcQE and RsQE, and since the quadrupolar splitting is halved in the FI dimension (cf. Figure l ) , a relatively small spectral width, and corre(23) Hahn, E. L. Phys. Rev. 1950, 80, 580. (24) Bloom, M.; Davis, J. H.; Valic, M. I. Can. J . Phys. 1980, 58, 1510
Fur6 et al.
a
A
li
d
Figure 1. *)Na 2DQE spectra from sample IV (T = 20.6 "C) oriented at OLD = 90". (a) Conventional (F2) spectrum produced with the QE sequence for T = l/uQ, where uQ = 14.45 kHz is the quadrupolar splitting. The relative line intensities are 3:4:3. The full widths at half-amplitude are 201 and 950 Hz for the central line and the satellites, respectively. (b) F1 cross-section spectrum L(w,,O), obtained by Fourier transformation with respect to T of the amplitude at the position of the central line in (a). The relative line intensities are 3:2:3. The splitting is reduced to half of that in (a), and the central line width is 168 Hz. (c) F1 cross-section spectrum L(wI,wQ),obtained by Fourier transformation of the amplitude at the position of a satellite in (a). The relative line intensities are -1:2:6:2:-1. The splitting is reduced to half of that in (a), and the central line width is 489 Hz.(d) Asymmetric L(w,,wQ) spectrum, obtained after two complex Fourier transforms (see text). The relative line intensities are 3:2:-1, and the line widths are the same as in (c). (e) The central line in (c) superimposed on a satellite in (a). The homogeneous line width is the same in both cases, but due to sample inhomogeneity, the F2 satellite is nearly twice as broad as the F1 central line.
Counterion Surface Diffusion spondingly large T increment, could be used. To avoid receiver dead time distortion, the smallest T value was 1/u9 The phases have to be set accurately and must be stable (to within 2-3') for the duration of the experiment (typically 10-15 h). Further, a large filter width should be used, to avoid distortion of the satellite decay. Fourier transformation with respect to T of the amplitude at w 2 = 0 yields, after zero filling to 8 K and phase correction, the 3:2:3 F1 cross-section spectrum L(w,,O), shown in Figure 1b. From the width of the central line in this spectrum, we obtain the relaxation rate R,QE. Similarly, Fourier transformation of the amplitude at o2 = wQ (defined by the satellite position in the T = I / V Q F2 spectrum) yields the -1:2:6:2:-1 FI cross-section spectrum L(w,,wQ),shown in Figure IC. From the central line width, we obtain R,QE. In practice, the L(w~,wQ) spectrum was obtained as follows. The phase of the second QE pulse was alternated between +y and -y. With proper phase setting, and with the central line exactly on resonance, the out-of-phase spectrometer channel then contains only which can be zeroed to improve the signal-to-noise ratio. After two complex Fourier transforms, the absorption channel of the quadrature detector contains the asymmetric F1 spectrum shown in Figure Id. The FI line at w1 = 0 is, in fact, slightly shifted from wl= 0, due to a second-order dynamic shift and/or to an asymmetric distribution of magnetic field or residual quadrupole coupling constant.20 Consequently, the spectrum was first shifted to w , = 0 and then symmetrized by adding a frequency-reversed spectrum, resulting in the L(wl,wQ) spectrum in Figure IC. This procedure yields a higher signal-to-noise ratio than the single-channel Fourier transforms used to illustrate the 2DQE method in ref 17. The usefulness of the 2DQE method is evident from Figure le, showing the central line in the L(wl,wQ) spectrum superimposed on one of the satellites in the F2 spectrum of Figure la. The large broadening and asymmetry of the latter, essentially due to sample inhomogeneity (see below), clearly prevent a determination of the homogeneous satellite line width. The central F1 line, however, directly yields the homogeneous satellite line width, except for a (relatively small) symmetric broadening due to magnetic field inhomogeneity. In principle, this residual broadening could be eliminated by using the modified QE sequenceI7 (K/2),-7/2(R),,-T / 2-( K / 2 ) y - ~/ 2-( R ) ~ - T /2-acq. Since the introduction of two 180' pulses, which, on account of the large splitting, are not perfectly nonselective, produced some spectral distortion, we refrained from using this sequence. However, as we will need only the difference RSQE- R,QE(see below), the field inhomogeneity broadening is essentially canceled out. This is true as long as the magnetic field inhomogeneity broadening is small compared to the homogeneous line width.17 Relaxation in Powder Samples. The relaxation experiments described above were performed on macroscopically aligned samples as well as on powder samples. Since the relaxation rates are orientation dependent, the two kinds of sample, in principle, furnish complementary information. For the large quadrupolar splittings in our samples, however, the broad spectral distribution of the satellite line intensity limits the accuracy with which the true powder relaxation rates can be determined. This is particularly true for the rates derived from the satellites, since the signal-to-noise ratio at the satellite peak position is much lower in the powder samples than in the aligned samples. The inversion recovery of the central line, even in the case of a magic-angle detection pulse,18 is complicated to analyze since it is rendered nonexponential by the orientational dependence of the nonsecular spectral densities. The zero-frequency spectral density is even more difficult to determine for powder samples. Thus, the RsQE experiment is not only plagued by a low signal-to-noise ratio but is further complicated by interference from the overlapping companion satellite, which mixes in a relaxation contribution corresponding to OLD = 35.26'. For these reasons, we shall restrict the analysis to relaxation data obtained from macroscopically aligned samples. We note, however, that, within the experimental uncertainty, the powder relaxation data are consistent with the
The Journal of Physical Chemistry, Vol. 94, No. 6, 1990 2603 isooctane
A
60
40
I'
20
AOT
Figure 2. Partial phase diagram (top) of the system AOT/D,O/isooctane at 20 "C, roughly showing the one-phase regions of the isotropic microemulsion (L2) phase and the lamellar (D), cubic (I), and reversed hexagonal (F) liquid crystalline phases. (Note that the diagram does not cover the entire composition space.) The dots correspond to the composition (weight percent) of the four investigated samples. The lower part of the figure is a cross-sectional view, drawn to scale, of the hexagonally arranged aqueous cylinders in sample IV. The shaded region contains AOT hydrocarbon chains and isooctane, while the water, sodium counterions, and sulfonate headgroups reside inside the cylinders. The counterion distribution is qualitatively as shown in the central cylinder, with about 85% of the ions within 5 8, of the interface.
orientation dependence of the relaxation rates measured on aligned samples.
Static Properties Before analyzing the 23Narelaxation rates, it is necessary to examine certain static properties of our F-phase samples, such as the phase structure and the extent of magnetic-field-induced alignment. This static information can be obtained from an analysis of quadrupolar splittings and line shapes. The residual quadrupole coupling constant, deduced from the splitting, will also be of relevance for the interpretation of the relaxation data. F-Phase Structure. In its pure state, the anionic surfactant AOT forms a mesophase with a unit cell consisting of hexagonally arranged, long parallel rodlike aggregates with the sulfonate headgroups and sodium counterions confined to the central core of the rod.1s The core can incorporate water up to x = 5-6. (Throughout this paper, we use x to denote moles of water per mole of AOT.) If a third nonpolar component is added, it forms, together with the AOT hydrocarbon chains, a continuous oil region, thus increasing the rod separation while maintaining the hexagonal symmetry. This reversed hexagonal liquid crystalline mesophase (denoted F) can incorporate more water, forming aqueous rods with the AOT headgroups at the cylindrical (or nearly so) oil/water interface. The ternary-phase diagrams of several AOT/water/"oil" systems have been partially determined.'5~25-27The approximate ( 2 5 ) Kunieda, H.; Shinoda, K. J . Colloid Interface Sei. 1970, 33, 215.
2604 The Journal of Physical Chemistry, Vol. 94, No. 6,1990
Fur6 et al.
TABLE 11: 23Na Quadrupolar Splittings from F-Phase Samples at 20.6 O C sample I I1 111 IV UQ(~LD=~')'/ kH2 33.05 31.52 30.08 28.93 U Q ( ~ L D = ~ O ~ ) ' / ~ H Z 16.52 15.78 15.06 14.45 YQ(erD'o0)/uQ(eLD'9O0) 2.000 I .997 1.997 2.002 uQ(powder)'/kHz 16.48 15.68 14.99 14.54 Estimated uncertainty f 5 0 Hz.
extensions of the F phase, and of the adjacent lamellar (D) and microemulsion (L,) phases, are indicated in Figure 2 for the investigated AOT/D,O/isooctane system at 20 'C. NMR studies were performed on four F-phase samples (labeled I-IV), the compositions of which are given in Table I and shown in Figure 2. With increasing temperature, the lower boundary of the isotropic L2 phase is displaced toward lower isooctane fractions, so that, above about 30 "C (cf. Table I), our samples are isotropic. The radius R of the aqueous rods is determined essentially by the water/AOT molar ratio x . If we assume that all AOT molecules reside at the cylindrical water/hydrocarbon interface of very long (end effects neglected) aqueous rods, then the geometry dictates that R = (2u,/a,)x R, (la)
+
where u, is the volume per water molecule and a, is the interfacial area per AOT molecule. R, is related to the volume of the surfactant headgroup (including the counterion), but its precise value depends on where one chooses to locate the interface between the polar and nonpolar regions. The three quantities ow, a,, and R, are expected to vary with x a t low water content and to attain constant values only above a certain level of hydration, x > xo. From X-ray diffraction studiesI5 of the F phase in the system AOT/H20/p-xylene at 20 'C, we know that a: = 65 A2 for x > 10. (The superscript zero signifies the x-independent limit.) Further, the water added to the central part of the aqueous core should behave essentially as bulk water for x > 15, as recently demonstrated2* for water droplets in the L2 phase. Hence, uWo = 30 A3 at 20 'C. For x > xo = 15, eq l a may thus be expressed as R = (2uWo/a:)x RrS
+
where Rfsnow accounts also for any deviations at x < xo of u, and a, from their limiting values. As shown below, our molecular interpretation of the 23Na relaxation data does not require the value of R', to be specified. However, in order to estimate the radial distribution of the counterions within the aqueous region, we need to know roughly what the cylinder radius is. For this purpose only, we shall use the value R: = 6 A, which is reasonable but subject to some uncertainty. The radii obtained in this way for our four samples are given in Table I. Figure 2 also shows a cross-sectional scale drawing of sample IV, with the rod spacing given by the composition and bulk densities. Orientational Order. For each of the four F-phase samples, three series of N M R measurements were carried out, viz., on powder samples (with the crystalline domains randomly oriented with respect to the static magnetic field, Bo) and on magnetically aligned samples in two probe configurations corresponding to OLD = Oo and OLD = 90°,where OLD is the angle between the lab-fixed Bo field and the phase director. The orientational dependence of the 23Naquadrupolar splitting uQ is governed by the second-rank Legendre polynomial,29 i.e. Hence, we expect that the splitting in the OLD = 0' configuration should be twice as large as that obtained for OLD = 90'. As seen ( 2 6 ) Tamamushi, B.; Watanabe, N. Colloid Polym. Sci. 1980, 258, 174. (27) Stilbs, P.; Lindman, B. J . Colloid Interface Sci. 1984, 99, 290. (28) Carlstrom, G.;Halle, B. Langmuir 1988, 4 , 1346. (29) Abragam, A. The Principles of Nuclear Magnetism; Clarendon Press: Oxford, U.K..1961.
600
0
-600 H Z
Figure 3. Water 2H spectrum of sample I oriented at OLD = O', showing superposition of spectra from magnetically aligned and nonaligned parts of the sample.
from Table 11, the data conform closely to eq 2. In powder samples the most probable polar angle is OLD = 90'. Provided that the powder splitting is much larger than the line width, it should thus coincide with the splitting from the aligned sample with OLD = 90'. According to Table 11, this is indeed the case. Although the samples were readily and reproducibly aligned on cooling from the L2 phase in a magnetic field of 8.5 T, it was always found that 20-25% of the sample did not respond to the field. This is most clearly seen in the water 2H spectrum recorded in the OLD = 0' configuration (Figure 3). In addition to the dominating doublet, corresponding to the aligned part of the sample, there is a smaller powderlike contribution with half of the OLD = 0' splitting. The integrated intensity of the latter contribution corresponds to a nonaligned fraction of 0.21. Similar results were obtained with the other three samples (Table 1). Two minor features of the main doublet in Figure 3 may be noted: (i) the slight difference in peak intensities and (ii) the shoulders on the outer flanks. Without having made a systematic study of these features, we tentatively ascribe them to magnetic field and sample inhomogeneity, respectively. The shoulders in the 2H spectrum disappear with time (cf. below) and, moreover, are not seen in the 23Naspectra. From the splitting ratio (2.02), we conclude that the nonaligned fraction consists either of randomly oriented crystalline domains or of domains preferentially oriented perpendicularly to the axis of the sample tube (and the magnetic field), but with some spread around the preferred orientation to account for the line shape asymmetry. (In the latter event, also the powder samples should be orientationally inhomogeneous.) These two cases could be distinguished by a detailed line shape analysis (requiring knowledge of the homogeneous 2H line width) or by establishing the presence of the outer satellites of the characteristic 2D powder line shape30 in the OLD = 90' configuration. As this distinction is unimportant for the analysis of the 23Narelaxation data (see below), the matter will not be pursued further here. It is worth noting, however, that we obtained a larger nonaligned fraction in sample containers with a larger surface-to-volume ratio, indicating that the nonaligned domains are located at the glass wall of the sample tube. For an inner tube diameter of 7 mm and a nonaligned fraction of 0.25, the range of the wall effect must then be about 0.5 mm. It should also be noted that the nonaligned fraction decreases with time. The spectrum shown in Figure 3 was recorded about 1 h after the alignment was induced. Another 2H spectrum, taken after the sample had been stored for about 2 months at 5 O C , yielded a nonaligned fraction of only 0.03 (and the shoulders in the spectrum had disappeared.) Hence, after the initial alignment of 75% of the sample as the two-phase region is traversed, there is apparently a slow process whereby small peripheral domains (30) Forrest, B .I.Reeves, ; L. W. Chem. Rec. 1981, 81, I .
The Journal of Physical Chemistry, Vol. 94, No. 6,1990 2605
Counterion Surface Diffusion TABLE 111: Sample Inhomogeneity As Deduced from Satellite Line Shapes
I re1 qcc spread, 1006 re1 x spread, lOOAxX/x OLD spread, u/deg re1 line shift, IOOAVQ/VQ
0.5 1.4 5 -0.2
sample 111 0.4 0.5 1.4 1.4 1.5 5 -0.3 -0.3
I1
IV 0.6 1.6 I -0.4
are aligned with, and incorporated into, the central aligned domain. Even in the macroscopically aligned part of the sample, there /** is a certain orientational disorder. It is this fact, clearly revealed -*'***.. .*._.__ ...-*.__...._.... *....------by the broadening and asymmetry of the inhomogeneous satellite in Figure 1 e, which prompted us to measure transverse relaxation 1 0 -1 kHz by the ZDQE method. Since the relaxation rates are orientation Figure 4. Inhomogeneous satellite line shape (solid curve) from sample dependent (see below), it is important to quantify this orientational IV oriented at OLD = 0'. The nearly coincident dashed curve was obdisorder. tained as described in Appendix A, with t = 0.006 and u = 7 O . The narrow dashed curve is the homogeneous line shape: a Lorentzian with Sample inhomogeneity is manifested in the N M R spectrum a width of 250 Hz (determined by a 2DQE experiment). in fundamentally different ways depending on the time scale T Q on which the quadrupolar frequency wQ (=2?rvQ) is averaged by coefficient (determined in the following) implies that the radius molecular motions. If T Q 10" s). Since these two classes of motions occur on disjoint time scales, the spectral densities may be decomposed into additive and independent contributions from each of them,5-22,34 i.e. The orientational dependence of the spectral densities j k is a consequence of the fact that the efg components VkL(t), appearing in the time correlation function in eq 7 , refer to a lab-fixed (L) coordinate frame (in which the nuclear spins are quantized). However, the molecular motions-in particular, the diffusion of counterions within the cylindrical aqueous regions-are more conveniently described with reference to a frame fixed in the cylinder (C). Deferring a discussion of the effect of curvature of the cylinder axis, we proceed under the assumption that the C frame coincides with the (local) director frame (D). Trans(35) Engstrom, S.;JBnsson, B.; Impey, R. W. J . Chem. Phys. 1984, 80, 5481. (36) Schnitker, J.; Geiger, A. 2. Phys. Chem. (Munich) 1987, 155, 29.
The Journal of Physical Chemistry, Vol. 94, No. 6, 1990 2607
TABLE VII: Reduced Spectral Densities" sample jf jd0) jlat(u0) jlat(2u0) jrad(O) jrad(2w0)
I
I1
238 1178 596 260 45 -83
236 1213 476 188 52 -117
111 208 1598 496 204 87 -82
"All spectral densities in units of
IV 190 2113 496 196 102 -62
uncertainty *I5 *95 *80
+=35 *35 *40
s-l.
forming the spherical efg tensor components from the L to the D frame, one obtainsz2 j f+ F~o(~L jrad(kWo) D) + F ~ Z ( ~jiat(kwo) LD)
(1 3) The orientational factors may be expressed in terms of reduced second-rank Wigner rotation matrix elements as jk
Fkm(OLD)
= (1 /2){[d2km(oLD)I2
+ [&-m(oLD)I2!
(14)
Explicit formal expressions for the "reduced" spectral densities jrad(w)andjlat(u)can be found in ref 22. At this stage, it suffices to note that j,, is associated with radial counterion diffusion, while jht mainly reflects lateral counterion diffusion around the cylinder axis. In writing eq 13, we have introduced two simplifications in the fast-motion contribution j f . First we have assumed that j f is orientation independent (as indicated by omitting the index k ) . This is an excellent approximation if the orientational restriction imposed by the interface on the fast motion is weak. That this is actually the case in the present system follows from the smallness of the residual qcc, i= 150 kHz (Table IV), as compared to the ~ we have root-mean-square qcc, ( x ~ ) '2/ ~4 M H Z . ~Second, assumed that j f is frequency independent in the range 0-2wo. This is certainly the case since the fast-motion time scale32 is much s. shorter than 1/(2w0) = 3 X Using the explicit expressions for the rotation matrix element^,^' we obtain from eqs 13 and 14, with OLD = 0'
and, with
OLD
jo(Oo) = j f + jrad(0)
(15a)
j,(OO) = j f
(15b)
1 j2(Oo) = if+ jjiad2wo)
( 15c)
= 90'
j0(90") = j f +
1
3
+ ijiat(0) 1
j W O ) = j f + ijiat(@o)
3 1 j2(90°) = .if + ijrad(2wO) + i$at(2wo)
(16a) (16b) (16~)
It is seen that the spectral densities at OLD = 90' carry the most information about counterion diffusion. In particular, the "lateral" spectral density is probed at three frequencies by the OLD = 90° relaxation data (recorded at a single magnetic field). Equations 15 and 16 constitute a system of six linear equations involving six unknown reduced spectral densities. Substituting the orientation-dependent spectral densities from Table VI, we can thus solve for all the reduced spectral densities. The results, collected in Table VII, will now be discussed in detail, starting with the lateral spectral density jlat(w). Counterion Surface Diffusion. With a slight modification of the treatment in ref 22, the lateral spectral density function may be expressed as jiat(w) = ( 3 + W i i ( r ) l 2 ) A m d rCOS (ut) & a t ( t )
(17)
(37) Brink, D. M.; Satchler, G. R. Angular Momentum, 2nd ed.; Clarendon Press: Oxford, U.K., 1968.
2608
The Journal of Physical Chemistry, Vol. 94, No. 6, 1990
Fur6 et al.
ji1m:lkHZ
194
TlatbInS
5.8
EflkJ mo1-I
26.6
177 7.4
188 8.8
7
r
TABLE VIII: Parameters Derived from Reduced Spectral Densities sample 1 11 111 IV
--
197
10.6 24.3
"Calculated from the fitted PXImsparameter, using P values from Table IV. Standard deviation from fit: 2-5 kHz. bStandard deviation from fit: 0.2-0.6 ns. CActivation energy for j , from Arrhenius fit. Standard deviation: 2.0 kJ mol-'.
where x(r) is the counterion qcc locally averaged by fast motions in the neighborhood of the p i n t r and the angular brackets signify a "coarse-grained" spatial average (over different local environments). The reduced correlation function in eq 17 takes the form glat(t) =
(X[r(O)l x[r(Ol cos [ 2 A $ ( t ) l ) ( [X(r)12)
I
0
o / I O s r a d L. *
W/IORIdd
Figure 5. Fits of the spectral density function, eq 23, for counterion surface diffusion to the experimentally determined jlat(w) data for samples I and IV.
(18)
where A$(?) is the azimuthal (around the cylinder axis) net angular displacement of the counterion during a time t. As argued above, the locally averaged qcc x(r) is expected to be significantly different from zero only in a small region near the cylindrical interface. Consequently, we write
5 0 -
h
h
( [X(r)Iz) = (Pxrms)z
(19)
where, as before, P is the fraction of counterions in this interfacial region, while jim is the square root of the spatial average of [ ~ ( r ) ] * over the interfacial region. The quantity xrmsis closely related to, but not identical with, the residual qcc X, introduced in eq 5 in connection with the quadrupole splitting. The appearance in eq 19 of P2, rather than P,is a consequence of the fact that radial spatial averaging (between the interfacial region and the core) is much faster than lateral (angular) averaging, Le., ?,ad
