J . Phys. Chem. 1989, 93, 3287-3299
3287
Shape Fluctuations and Water Diffusion in Microemulsion Droplets. A Nuclear Spin Relaxation Study Goran Carlstrom and Bertil Halle* Physical Chemistry I , University of Lund. Chemical Center, P.O. Box 124, S-221 00 Lund. Sweden (Received: September 27, 1988)
Water 2H and 170spin relaxation data for the microemulsion phase in the AOT/D20/isooctane system are reported. The difference between the transverse (R2)and the longitudinal (R1) relaxation rates has been measured as function of droplet size, droplet volume fraction, temperature, and resonance frequency. The 2H longitudinal relaxation rate dispersion has been measured over an extensive frequency range, using the field-cycling technique. The focus in the study is on the contribution from slow molecular processes to the quadrupolar relaxation behavior. For the first time in any system, the theoretically predicted relation between the 170/2Hratios of quadrupolar line splittings and of R2- Rl is verified. The extensive experimental data are used to discriminate among three different dynamic models. It is found that water diffusion within the microemulsion droplet cannot account for the experimental data. Instead, a substantial shape polydispersity seems to be required, where at any instant a large fraction of the droplets are nonspherical. However, it is not necessary that the equilibrium shape is nonspherical.
Introduction Water-in-oil microemulsions are thermodynamically stable systems which, under certain conditions, consist of discrete aqueous droplets dispersed in a continuous oil medium. One of the most well-characterized droplet-type microemulsions is the one formed in the ternary system A O T / ~ a t e r / o i l ' - ' ~(AOT denotes the surfactant sodium bis(2-ethylhexyl) sulfosuccinate). This oilcontinuous microemulsion phase is stable over relatively wide concentration and temperature ranges and has been shown to consist of closed, essentially spherical, aggregates with an aqueous core of water and counterions surrounded by an interface of surfactant headgroups. By changing the overall composition, the size and volume fraction of the aqueous droplets can be varied independently. For these reasons, the AOT/water/oil microemulsion phase constitutes an attractive model system for investigating molecular behavior in microheterogeneous fluids. In the following, we describe a study of the AOT/D20/isooctane microemulsion phase by means of the spin relaxation of the 2H and 170water nuclei. Nuclear spin relaxation has become one of the most powerful techniques available for investigations of structure and dynamics in microheterogeneous fluids in general, and in surfactant systems in p a r t i ~ u l a r . ' ~ , 'For ~ nuclear spin (1) Zulauf, M.; Eicke, H. F. J . Phys. Chem. 1979, 83, 480. (2) Day, R. A.; Robinson, B. H.; Clarke, J. H. R.; Doherty, J. V. J . Chem. SOC.,Faraday Trans. 1 1979, 75, 132. ( 3 ) Cabos, C.; Delord, P. J . Appl. Crysrallogr. 1979, 12, 502. (4) Robinson, B. H.; Toprakcioglu, C.; Dore, J. C.; Chieux, P. J . Chem. SOC.,Faraday Trans. I 1984,80, 13. ( 5 ) Toprakcioglu, C.; Dore, J. C.; Robinson, B. H.; Howe, A. J . Chem. SOC.,Faraday Trans. 1 1984,80, 413. (6) Kotlarchyk, M.; Chen, S. H.; Huang, J. S.J . Phys. Chem. 1982, 86, 3273. (7) Kotlarchyk, M.; Chen, S. H.; Huang, J. S.; Kim, M. W. Phys. Rev. A 1984, 29, 2054. (8) Kotlarchyk, M.; Huang, J. S.; Chen, S. H. J . Phys. Chem. 1985,89, 4382: (9) Assih, T.; Larch$, F.; Delord, P. J . Colloid Interface Sci. 1982,89, 35. (10) Pileni, M. P.: Zemb, T.; Petit, C. Chem. Phys. Lett. 1985, 118, 414. ( 1 1 ) Bridge, N. J.; Fletcher, P. D. I. J . Chem. Soc., Faraday Trans. 1 1983, 79, 2161. (12) Eicke, H. F.; Shepherd, J. C. W.; Stelnemann, A. J . Colloid Interface Sci. 1976, 56, 168. (13) Fletcher, P. D. I.; Howe, A. M.; Robinson, B. H. J . Chem. SOC., Faraday Trans. I 1987, 83, 985. (14) Lang, J.; Jada, A.; Malliaris, A. J . Phys. Chem. 1988, 92, 1946. (15) Lindman, B.; Sijderman, 0.;Wennerstrom, H. In Surfactant Solutions: New Methods of Investigation; Zana, R., Ed.; M. Dekker: New York, 1987; p 295. (16) Chachaty, C. Prog. Nucl. Magn. Reson. Spectrosc. 1987, 19, 183.
0022-3654/89/2093-3287$0 1.50/0
relaxation studies of water one has a choice of three stable isotopes: the proton (IH), the deuteron (2H), and the oxygen-17 isotope (I7O). The principal relaxation mechanism for the IH isotope is due to fluctuations in intra- and intermolecular magnetic dipole-dipole couplings, while the other two isotopes are relaxed mainly by intramolecular electric quadrupolar c 0 u p 1 i n g s . l ~ ~ ~ ~ Consequently, the 2H and I7Orelaxation rates reflect the motion of single water molecules, whereas the 'H rate is also affected by relative water-water and water-surfactant motions. In the early IH studies19 of AOT-stabilized microemulsions, this complication was not recognized. More recently:0 isotopic substitution experiments were used to separate out the water-surfactant contribution to the 'H relaxation rate. On account of the fast proton exchange in water, this approach cannot be used to separate the intra- and intermolecular contributions to the water 'H relaxation rate. It is therefore not possible to extract the water reorientational correlation time, unless simplifying assumptions are made about the water pair distribution and relative motion in the interfacial region. In two recent spin relaxation studies21*22 of the AOT/D20/isooctane microemulsion phase this complication was circumvented by using the quadrupolar nuclei 2H and 170. In one of these studies,21 we used the 2H and I7O longitudinal relaxation rates ( R , ) to determine the magnitude and range of the surface-induced perturbation of water reorientational dynamics in large (10-50 mol of D20/mol of AOT) microemulsion droplets. In the other study,22the 2H relaxation rates were used to probe water reorientation, aggregate structure, and freezing behavior at subzero temperatures (down to -29 "C). Of crucial importance for spin relaxation studies of microheterogeneous fluids is the fact that such systems are locally a n i s o t r ~ p i c ' (although ~ ~ ' ~ ~ ~ ~macroscopically isotropic). Although the 2H and I7O relaxation rates are governed by single-molecule reorientational time correlation functions, the fact that molecular reorientation is measured in a lab-fixed frameI7 (in which the nuclear spin is quantized) means that, in a locally anisotropic system, other motions besides local water reorientation can contribute to the spin relaxation rates. In a microemulsion system, the *H and I7O relaxation rates may thus depend on the rate of (1 7) Abragam, A. The Principles of Nuclear Magnetism; Clarendon Press: Oxford, U.K., 1961. (18) Halle, B.; Wennerstrom, H. J . Chem. Phys. 1981, 75, 1928. (19) Wong, M.; Thomas, J. K.; Nowak, T. J . Am. Chem. SOC.1977,99, 4730. (20) Llor, A.; Rigny, P. J . A m . Chem. SOC.1986, 108, 7533. (21) Carlstrom, G.; Halle, B. Langmuir 1988, 4, 1346. (22) Quist, P-0.; Halle, B. J . Chem. SOC.,Faraday Trans. 1 1988, 84, 1033.
0 1989 American Chemical Society
3288 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 water diffusion within the droplet or on the rate of droplet rotation and shape fluctuations. Since these motions typically are slow compared to the Larmor frequencies of fixed-field spectrometers, they are manifested only in the transverse relaxation rate ( R z ) . However, with the aid of field-cycling techniques, these slow motions can also be studied through the low-frequency dispersion of the longitudinal relaxation rate. The aim of the present work is to identify and to characterize the slow molecular processes (on a time scale longer than lo-* s) that are manifested in the water 2H and I7O spin relaxation behavior in the AOT/D20/isooctane microemulsion phase. The analysis is based on a rather extensive experimental material involving mesaurements of AR = R2 - R , for both water nuclei as a function of droplet size, droplet volume fraction, temperature, and resonance frequency (2-50 MHz). Further, we report a low-frequency dispersion curve (30 kHz-5 MHz) for the 2H longitudinal relaxation rate. In addition, we make use of 2H and I7O quadrupolar line splittings from the reversed hexagonal phase in the investigated system. The analysis of these relaxation data is divided into two parts. First, we establish the mechanism of spin relaxation, considering, besides the quadrupolar mechanism, scalar relaxation by modulation of the 2H-170 spin-spin coupling as well as a mechanism based on modulation of the Zeeman coupling. Having shown that the quadrupolar mechanism in fact prevails under the conditions of the present study, we turn our attention to the time correlation function governing the spin relaxation behavior. This correlation function is evaluated for three motional models, involving (i) water diffusion within a spherical microemulsion droplet; (ii) rotation of a spheroidal droplet; and (iii) spheroidal shape fluctuations of a droplet with a spherical equilibrium shape. The extensive experimental data allow us to distinguish among these models. In particular, we find that water diffusion-even if much slower than in bulk water-cannot account for the experimental observations. This is an important finding, since previous water 170relaxation studies of colloidal s i l i ~ aand ~ ~. p, r~~~t e i n ~solutions ',~~ have been interpreted in terms of rather long (10-8-10-7 s) residence times of interfacial water. The other two models can explain all the present data provided that either the microemulsion droplets are distinctly nonspherical (axial ratio 2-3) or the droplets undergo substantial shape fluctuations. Experimental Section Materials. AOT (sodium bis(2-ethylhexyl) sulfosuccinate) of 98% purity was obtained from Fluka. It was further purified as described by Kunieda and S h i n ~ d a ?except ~ that active charcoal, rather than ether, was used to extract impurities from a solution of AOT in methanol. After evaporation the residual AOT was vacuum-dried at 40 OC for 12 h. Isooctane (2,2,4-trimethylpentane) from Ega-Chimie (99%) was used as supplied. The water used was a mixture of D 2 0 (>99% 2H) from Norsk Hydro and 170-enriched D 2 0 (22% I7O, 62% I8O) from Ventron. Microemulsion samples were made by weighing the components into N M R tubes, which were then shaken until an isotropic, transparent solution was obtained. The droplet size was varied by successive additions of D i 6 0 to the samples. As a consequence, the oxygen isotope composition of the water in the samples varied somewhat (1-10% I7O). The effect of this variation on the I7O relaxation rates and the longitudinal *H relaxation rate is insignificant.2E However, the transverse 2H relaxation rate is sensitive to contributions from scalar relaxation at high enrichments of I7O in the water. Consequently, we report 2H relaxation data only from samples having an I7Oenrichment of less than 2.3%, where the scalar contribution is negligible. In order to reduce the (23) Piculell, L. J. Chem. SOC.,Faraday Trans. 1 1986,82, 387. (24) Halle, B.; Piculell, L. J . Chem. Soc., Faraday Trans. 1 198682, 415. ( 2 5 ) Halle, B.; Andersson, T.; Forstn, S.; Lindman. B. J. Am. Chem. SOC.
Carlstrom and Halle contribution from scalar relaxation to the 2H and I7O transverse relaxation rates, the pD of the added D 2 0 was adjusted to 4.1. The volume fraction of droplets (AOT water) was calculated using the weighed-in sample composition, the molar masses (20.0, 114.2, 444.6 g mol-'), and bulk densities (1.105, 0.692, 1.138 g for DzO, isooctane and AOT, respectively. As a reference for the relaxation measurements, we used a sample of acidified D 2 0 (1.5% I7O) with pD 2.4. High-Field Relaxation Measurements. Water 2H and I7O relaxation rates were measured on a Bruker MSL 100 (2H resonance frequency 15.37 MHz and 170resonance frequency 13.57 MHz). The longitudinal relaxation rates (R,) were determined by the conventional inversion recovery method. Each R, value is the result of a least-squares fit of the magnetization plotted against delay time ( 7 ) for at least 20 different T values. The transverse relaxation rates (R,) for I7O were obtained from the line widths at half-height (Au) of the absorption spectra, according to the relation R2 = aAv, while for 2H they were determined with the Carr-Purcell-Meiboom-Gill experiment, taking only even echoes into account. At least 16 different echoes were accumulated and Rz was obtained from a least-squares fit of the peak intensity of the Fourier-transformed spectrum plotted as a function of delay time. The contribution to R2(I7O)from magnetic field inhomogeneity was always less than 1.4 s-I, as determined from the difference R2('70)-Rl('70) or the deuterium line width of the reference water sample. From the reproducibility of the measured relaxation rates for the reference water sample and for the microemulsion samples, using two different samples of the same composition, we estimate the uncertainty in the relaxation rates to be less than 3%. The experiments were carried out at temperatures ranging from 18.9 to 21.3 OC, measured with a calibrated thermocouple. All relaxation rates were subsequently corrected to 20.0 "C, using the measured temperature dependence. These corrections were always less than 4%. During each experiment the probe temperature was kept constant to within f0.2 OC by the passage of thermostated air. The experiments were done within 2 days from sample preparation. The scalar contribution, A&, to the transverse relaxation rate of microemulsion samples made from H 2 0 was obtained from the difference in R,(170)measured without and with proton decoupling. By adding (by weight) small amounts of NaOH solutions (made from Merck titrisol ampules) to the microemulsion sample, the pH dependence of the scalar contribution was studied. For the proton decoupling, we used a broadband decoupler of 0.7 W power. The decoupling power raised the sample temperature by less than 0.4 "C. A sample of H 2 0 of neutral pH was used to test the decoupling efficiency. The measured RI(l7O)was compared to R2(I7O),obtained from a decoupled spectrum. After subtracting the contribution from field inhomogeneity, obtained from the difference between R,(I7O)and Rl(l7O)for the acidic reference sample, the contribution to RZ(l7O)from incomplete decoupling was found to be 3 s-'. The 170relaxation is strictly exponential only under conditions of extreme narrowing (Le., when Rl = R2). The relaxation is, however, effectively exponential as long as R 2 / R lis of order one, and approximate analytical expressions can then be used for the measured rates.29 The error in such a treatment can be evaluatedz9 and is negligible in the present study. Field-Cycling Experiments. The frequency dependence of the water ,H longitudinal relaxation rate was measured at Sektion Kernresonanzspektroskopie, University of Ulm, West Germany, using a superconducting field-cycling spectrometer. The spectrometer and the technique are described in ref 30. The sample used in this experiment was prepared as described above, except that the water was D2I60of pD 11.3. The estimated uncertainty in R1is less than lo%, based on the reproducibility at different frequencies. During the experiments the probe temperature was kept constant at 19.9 OC to within f0.2
+
1981, 103, 500.
(26) Piculell, L.; Halle, B. J. Chem. SOC., Faraday Trans. 1 1986,82,401. (27) Kunieda, H.; Shinoda, K. J. Colloid Inlerface Sci. 1979, 70, 577. (28) Lankhorst, D.; Schriever, J.; Leyte, J. C . Ber. Bunsen-Ges. Phys. Chem. 1982, 86, 215.
(29) Halle, B.; Wennerstrom, H. J. Magn. Reson. 1981, 44, 89. (30) Schauer, G.; Nusser, W.; Blanz, M.; Kimmich, R. J . Phys. E Sci. Instrum. 1987, 20, 43.
Shape Fluctuations in Microemulsions
The Journal of Physical Chemistry, Vol. 93, No. 8. 1989 3209
TABLE I: Experimental Conditions series
variable
T/OC
A
X X
20 20 20 15.1-21.7 20 20
B C
Y
D
T
E
yo
F
%
”$/MHZ 15.37/13.57 13.57‘ 13.57‘ 15.31/13.57 2.0-49.0‘ 0.030-5.0‘
X
Y
15.3-51.1 14.9-51.9 25.3 25.3. 50.8,51.1 28.2 53.2
4.75 35.0 2.60-48.9 48.9.35.0,4.75 4.75 4.75
$0’
0.464.63 0.10-0. I9 0.66-0.095 0.095,0.18, 0.63 0.53 0.63
‘Resonance frequenciesare given as uo(’H)/uo(”O). bVolume fraction droplets (AOT + D20). ‘Only ”0 data. ‘Only ’H data
isooctane
--I
. OA
D20
io
’
do
’
io
’
‘
AOT
Figure 1. Schematicextension of the isotropic microemulsion phase (b) in the system AOT/D,O/isooctane at 20 “ C Concentrations are ex-
pressed as percentage by weight. The composition of the investigated samples (series A. B. C and E, F) are indicated ( 0 ) .
10
30
20
40
50
,
Figure 2. Variation of AR with droplet size (x = moles of D,O/mole AOT) for the high volume fraction series A (0, 0) and for the low volume fraction series B (n, 0). The open and filled symbols refer to separate experimental runs with different samples.
.
150
‘C by heating against the cold surroundings. The temperature
was measured with a thermistor. The relaxation measurements were made within 18 days from sample preparation. No time dependence was observed. The results from measurements on a reference sample of D,O having pD 11.3 indicated no significant frequency dependence in the range 30 kHz to 5.0 MHz.
Results The ternary phase diagram of the system AOT/H,O/isooctane has been partially determined” at 25 OC. In terms of the composition variables x = moles of H,O/mole of AOT and y = moles of isooctane/mole of AOT, the isotropic microemulsion phase (usually denoted L,) extends from the i s m t a n e corner down to y = 1-2 and from the isooctane-AOT axis up to x = 40-140 (depending on y). The composition of our samples is given in the schematic phase diagram of Figure 1. (The indicated boundary of the stability region of the L, phase is not precisely known; it depends sensitively on temperature and trace amount of impurities. However, all our samples were in the single-phase L, region as evidenced by the optical isotropy and the absence of quadrupolar splittings in the N M R spectra.) Relaxation data from six experimental series are reported (Table I). In series A and B we vary the size of the aqueous droplets, while in series C the droplet volume fraction is varied a t constant droplet size. (The droplet size is determined essentially by x,’-“) Apart from series D, which is a temperature dependence, all data pertain to 20.0 “C. In the four series A-D, we have measured longitudinal ( R , ) as well as transverse ( R , ) relaxation rates for both the water nuclei ‘H and I’O. (As noted in the Experimental Section, we do not consider 2H data from series B and C since these data are significantly affected by scalar relaxation.) Series E and F refer to measurements a t variable resonance frequency; the former of R,(”O) and R,(”O) over the hieh-fremencv ranee 2-49 MHz and ;he latter to R,(,H) over the liw-frequency range 30 kHz to 5 MHz. With the exception of the low-frequency ,H dispersion (series F), the R , data from these experiments have been discussed in a separate publication.2’ There we showed that R , is due to fast local water reorientation, slightly perturbed by the nearby in-
. , -
~~~
(31) Tamamushi, B.;Walanabe, N. Colloid Polym. Sei. 1980,258,174.
u
00
10
30
20
40
50
Y
Figure 3. Variation of AR(”O) with droplet volume fraction (v = moles of isooctane/moIe of AOT) far series C (x = 25.3).
terface. Further, we showed that this perturbation is of short range, being limited to the primary hydration of the AOT headgroups and counterions (about 15 water molecules per AOT). The point of departure for the present work is the observation that, under all investigated experimental conditions, the transverse relaxation rate is significantly larger than the longitudinal one. This crucial observation indicates that R2 is due not only to local water reorientation, but also to one or more dynamic processes taking place on a much longer time scale. The primary aim of this work is to identify and to characterize this slow molecular process. In the high-frequency regime (series A-E), the slow dynamics is manifested in the difference AR= R,-R, (1) which will be the primary observable for series A-D. At lower frequencies, however, the slow dynamics contributes also to R , , which then becomes frequency dependent (series F). The variation of AR(I7O)with the size of the aqueous droplets is shown in Figure 2. (The droplet radius varies linearly with x; see below.) The x-dependence is seen to be rather complex; in particular, AR is not proportional to I/x as is the excess longitudinal rate.21 Two features are immediately apparent: (i) AR does not ippear to vanish as the aqueous droplet approaches a water-free reversed micelle (x = 0); and (ii) AR depends strongly on the droplet volume fraction. The latter fact is more directly illustrated in Figure 3, where AR(l’O) is shown as a function of volume fraction (variable y at constant x = 25.3). For a droplet
3290 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989
Carlstrom and Halle
3
w
C
U
a
~
5
fi
7
I
log(VO/Hz)
TABLE 11: Temperature Dependence of AR for Three Samples" 210AR(2H)/ ~~(170)/~-1
S-'
T/OC
sample I
samDle I1
sample 111
sample 111
15.1 16.1 19.2 21.7
15.9 24.6 35.8 37.8
24.6 24.9 29.0 38.6
117 126 115 89.5
121 126 1 I7 93
'Sample I: x = 25.3, y = 48.9, &, = 0.10. Sample 11: x = 50.8, y = 35.0, &, = 0.18. Sample 111: x = 5 1 . 1 , ~= 4.75, &D = 0.63.
volume fraction @D below about 0.25 (corresponding t o y 2 15) 4R is seen to be insensitive to @D, while, at higher @D, 4 R increases sharply. The qualitative interpretation of the observations (i) and (ii) is that the molecular process responsible for 4 R cannot be exclusively confined to the aqueous core of the droplet, since then (i) 4R would tend smoothly to zero as x 0; and (ii) AR would be virtually independent of the droplet volume fraction. This interpretation is quantitatively elaborated in the following. Figure 4 is a comparison of 4R(ZH) and 4R(I7O) for series A. Within experimental accuracy, the data from the two nuclei coincide over the investigated x range when scaled by a constant factor. As shown in the following section, the value of this scaling factor provides strong evidence for a quadrupolar relaxation mechanism. The temperature dependence of AR (series D) is not of the Arrhenius type (a linear increase of In R with 1/T) usually encountered in water spin relaxation studies of microheterogeneous aqueous solution systems. In fact, for samples on the low-@, plateau of Figure 3, AR increases with temperature in the investigated range 15-22 "C (Table 11). Roughly speaking, AR may be regarded as a product of two temperature-dependent factors. One of these is related to the amplitude of the fluctuating spin-lattice coupling; this quantity thus reflects the equilibrium structure in the system (local orientational order, droplet size and shape). The other factor is a purely dynamic quantity; for a diffusive process it is expected to be roughly proportional to the viscosity of the fluid medium in which the process takes place. Since the viscosity decreases with temperature, the anomalous temperature dependence of hR for the low-@Dsamples suggests that the fluctuation amplitude increases strongly with temperature. The picture is further complicated by the observation (Table 11) that, at high volume fraction, AR decreases with temperature. Further, it is seen from Table I1 that the scaling behavior of AR(ZH) and AR(I7O), as seen in Figure 4, is obeyed at all investigated temperatures. Within experimental accuracy, RI(l7O) is independent of resonance frequency in the range 2-49 MHz. The experimental data for series E are given in Table I11 of ref 21. (As excepted, is frequency independent in also Rz( "O),and hence AR( 170), this range.) The observation of such a high-frequency plateau implies that the slow motion responsible for AR (in this sample) takes place on a time scale that is long compared to the inverse of the lowest investigated resonance frequency, i.e. (2rv0)-l = 8
-
x 10-8 s.
Figure 5. Frequency dependence of R1(2H)for a sample with x = 53.2 at high volume fraction (@D = 0.63). The dashed lines indicate the limits of the dispersion step as expected on the basis of the high-frequency R, and Rzdata and a second-rank tensorial spin-lattice coupling. The three curves are Lorentzian dispersions based on an exponential correlation function with correlation times as indicated. The open and filled symbols refer to two separate experimental runs.
The frequency dependence of R1(2H)in the vo range 30 kHz to 5 MHz for a sample of high volume fraction is shown in Figure 5. This low-frequency dispersion was recorded on a field-cycling instrument (see Experimental Section) and the accuracy of the data is somewhat inferior to that of the fixed-field measurements of series A-E. Nevertheless, several important conclusions can be drawn. First, the dispersion data are consistent with a relaxation mechanism involving a second-rank tensorial spin-lattice coupling (such as the quadrupole coupling). This follows from the fact that the dispersion data fall within (except for some scatter) the range (indicated by the dashed lines in Figure 5) expected on the basis of the high-frequency R I and AR data. The second conclusion that may be drawn from Figure 5 is that the slow motion does not give rise to an exponential time correlation function (and a Lorentzian spectral density function). The three curves in Figure 5 are the dispersions expected from an exponential correlation function with the indicated correlation time. The more extended observed dispersion shows that the slow dynamics responsible for 4 R has a range of correlation times; for this sample, it is approximately in the range 0.1-1.5 p s . On the basis of this rather extensive experimental material, we shall now try to elucidate the molecular origin of AR and, hence, extract information about the structure and dynamics in the investigated microemulsion phase. In order to establish the mechanism of spin relaxation responsible for the observed 4 R , we must first identify the relevant spin-lattice coupling and, second, formulate a dynamic model which describes how this coupling fluctuates in time.
Mechanism of Spin Relaxation In this section we consider three different mechanisms of spin relaxation that could conceivably give rise to the observed AR. These involve (i) the Zeeman coupling of the nuclear magnetic moment with the externally generated magnetic field at the nucleus, modulated by water exchange within a polydisperse droplet population; (ii) the scalar electron-mediated spin-spin coupling between the ZHand 170nuclei in the water molecule, modulated by acid- or base-catalyzed proton exchange; and (iii) the coupling of the nuclear quadrupole with the electric field gradients generated by the surrounding (mainly i n t r a m o l e c ~ l a r ~charge ~) distribution, modulated by a variety of molecular processes that affect the orientation of the water molecule with respect to a lab-fixed frame. Zeeman Coupling Modulated by Droplet Exchange. This relaxation mechanism presupposes (i) that the droplet population is polydisperse in size, (ii) that the chemical shift averaged within the droplet is size dependent, and (iii) that the rate of water exchange between the droplets is large compared to the shift (32) Cummins, P. L.; Bacskay, G. B.; Hush, N. S . ; Halle, B.; Engstrom, S. J . Chem. Phys. 1985.82, 2002.
Shape Fluctuations in Microemulsions
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3291
This modulation is due mainly to intermolecular proton exchange catalyzed by the ubiquitous ions H 3 0 + and OH- or by other prototropic species. In the case of AR,(2H), the fast longitudinal I7O relaxation may also contribute to the modulation of the spin-spin coupling. These two mechanisms are usually referred to as scalar relaxation of the first and second kind, re~pective1y.l~ ARex = P , P I ( ~ o A 6 ) 2 ~ , , (2) Since the magnitude of the spin-spin coupling is small, scalar where P, = 1 - PIis the fraction water in small droplets, wo is relaxation can compete with other mechanisms only when the the Larmor frequency, and A6 = 61 - 6, is the difference in inmodulation is slow. At typical Larmor frequencies, scalar retradroplet-averaged (relative) chemical shift between the two types laxation therefore contributes only to the transverse relaxation of droplet. The exchange time T~~ is related to the mean water rate. resideace times T , and T~ in the droplets through In a water-continuous system, ARsc is inversely proportional to the concentration of prototropic species and can thus be made 1 1 1 _ -- + negligible compared to RI by adjusting the pH (or pD) of the (3) Tex 's TI s o l u t i ~ n . (Outside ~ ~ * ~ ~ the pH range 4-10, AR,, is entirely negWater exchange is known to proceed via a droplet fusion-fission ligible.) In a droplet-type microemulsion, where the prototropic p r o c e ~ s , ' ~characterized -~~ by a second-order coalescence rate species ( H 3 0 + or OH-) are distributed over relatively small constant k . Thus, for example, aqueous droplets, the situation is different. This difference arises because of the possibility of having a significant fraction of droplets 71 = (kIcl)-' (4) that are free from prototropic species. For water molecules in where cI is the concentration of large droplets. such droplets, the spin-spin coupling is modulated as a result of It is known, primarily from small-angle scattering s t ~ d i e s , 4 ~ ~ ~fusion ~ with another droplet containing one or more prototropic that large ( x > 20) AOT-stabilized droplets have a polydispersity ions. We have recently exploited this intriguing phenomenon in of about 30% in the radius. The average radius of the droplets developing a new technique, called nuclear spin quenching,a for varies essentially linearly with x.I-l4 In addition, there is some determining exchange kinetics and droplet size in disperse fluids e v i d e n ~ e ~that ' ? ~a~fraction of the AOT is present in the form of such as microemulsions. For our present purposes, however, scalar very small droplets (reversed micelles), which presumably contain relaxation is an undesirable effect. Since the fusion-controlled a small amount of water. The water 'H chemical shift has been modulation time may be much longer than the proton exchange measured in AOT-stabilized microemulsions as a function of x.35936 time expected on the basis of the pH of the total aqueous pseuThe difference A6('H) between large (x = 60) and very small ( x dophase, AR,, may be much larger than expected for a water= 1) droplets is about 1 ppm. As the chemical shift at a given continuous system. The situation is further complicated by the nuclear site should be virtually invariant to isotope substitution, fact that trace amounts of acidic impurities in the AOT may this figure should apply also for 2H. It is clear, therefore, radically change the pH dependence of Consider now a sample with x = 30 at high volume fraction that the possibility of a scalar contribution to the transverse water (series A), for which AR(2H) = 0.4 s-I (Figure 4). For this sample, relaxation rates in disperse systems must be carefully investigated the droplet concentration is cl = 8 X mol dm-3 and the in each case. coalescence rate constant13 is k l = 8 X lo6 dm3 mol-l s-I, whence If the rate of modulation is large compared to the spin-spin T~ = 2 X lo4 s. We consider first the effect of the small reversed ' ~ in~D~2 0~with coupling JOD (rad s-l), then one can s h o ~that, micelles. With an estimated 15% of the AOT present in the I7O isotope fraction Pl7, reversed micelles34 and with x , = 2, we have P, = 0.01 (this is A & c ( ~ H= ) ( 3 5 / 12)P,7J0D2Teff (5) probably a high estimate). Using eq 2-3 and the principle of detailed balance (Plr, = P , T ~and ) inserting the above values of T~ and P,,as well as wo = 1 X lo8 rad s-I (cf. Table I) and A6(2H) = 1 ppm (cf. above), we obtain AR,,(2H) = 2 X SKI,which where is a factor 2000 smaller than the observed AR(2H). To estimate 1 1 the effect of the ca. 30% polydispersity of the large droplets, we -=- ~ ~ ( ' ~ 0 ) (7) take P, = PI = 0.5 and A6(2H) = 0.1 ppm,35.36whence AR,,(2H) Teii Tsc =2 X S-I, which is, again, negligible compared to the observed For 2H, R2 is usually measured by a Carr-Purcell spin-echo AR(2H). In both cases, the fast-exchange condition W ~ A ~
