Welcome message from author

This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript

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 170 spin 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 170/2H ratios 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 relaxation studies of water one has a choice of three stable isotopes:

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 oil- continuous 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 micro- emulsion phase constitutes an attractive model system for in- vestigating molecular behavior in microheterogeneous fluids.

In the following, we describe a study of the AOT/D20/iso- octane microemulsion phase by means of the spin relaxation of the 2H and 170 water 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.

( 3 ) Cabos, C.; Delord, P. J . Appl. Crysrallogr. 1979, 12, 502. (4) Robinson, B. H.; Toprakcioglu, C.; Dore, J. C.; Chieux, P. J . Chem.

( 5 ) Toprakcioglu, C.; Dore, J. C.; Robinson, B. H.; Howe, A. J . Chem.

(6) Kotlarchyk, M.; Chen, S. H.; Huang, J. S. J . Phys. Chem. 1982, 86,

SOC., Faraday Trans. 1 1979, 75, 132.

SOC., Faraday Trans. I 1984,80, 13.

SOC., Faraday Trans. 1 1984,80, 413.

3273. (7) Kotlarchyk, M.; Chen, S. H.; Huang, J. S.; Kim, M. W. Phys. Rev.

(8) Kotlarchyk, M.; Huang, J. S.; Chen, S. H. J . Phys. Chem. 1985,89, A 1984, 29, 2054.

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 Solu-

tions: 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

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 di- pole-dipole couplings, while the other two isotopes are relaxed mainly by intramolecular electric quadrupolar c0up1ings . l~~~~ Consequently, the 2H and I7O relaxation 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 com- plication 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 re- laxation 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,22 the 2H relaxation rates were used to probe water reorientation, aggregate structure, and freezing behavior a t subzero temperatures (down to -29 "C).

Of crucial importance for spin relaxation studies of micro- heterogeneous 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 con- tribute 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:

(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,

(20) Llor, A.; Rigny, P. J . Am. 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,

Oxford, U.K., 1961.

4730.

1033.

0 1989 American Chemical Society

3288

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 (Rz) . 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 modu- lation 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 ex- perimental 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 170 relaxation studies of colloidal s i l i ~ a ~ ~ , ~ ~ and . 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-trimethyl- pentane) 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 insig- nificant.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 I7O enrichment of less than 2.3%, where the scalar contribution is negligible. In order to reduce the

The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 Carlstrom and Halle

(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.

(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.

1981, 103, 500.

Chem. 1982, 86, 215.

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 res- onance frequency 15.37 MHz and 170 resonance 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 a t 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 inho- mogeneity 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 mi- croemulsion 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 a t tem- peratures ranging from 18.9 to 21.3 OC, measured with a cali- brated 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 ex- periment 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 decou- pling. 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 com- pared 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 170 relaxation is strictly exponential only under conditions of extreme narrowing (Le., when Rl = R2). The relaxation is, however, effectively exponential as long as R2/Rl is of order one, and approximate analytical expressions can then be used for the measured rates.29 The error in such a treatment can be evalu- atedz9 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 spec- trometer and the technique are described in ref 30.

The sample used in this experiment was prepared as described above, except that the water was D2I60 of pD 11.3.

The estimated uncertainty in R1 is 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

(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

TABLE I: Experimental Conditions

The Journal of Physical Chemistry, Vol. 93, No. 8. 1989 3209

series variable T/OC ”$/MHZ X Y $0’

A X 20 15.37/13.57 15.3-51.1 4.75 0.464.63 B X 20 13.57‘ 14.9-51.9 35.0 0.10-0. I9 C Y 20 13.57‘ 25.3 2.60-48.9 0.66-0.095 D T 15. 1-21.7 15.31/13.57 25.3. 50.8, 51.1 48.9. 35.0, 4.75 0.095, 0.18, 0.63 E yo 20 2.0-49.0‘ 28.2 4.75 0.53 F % 20 0.030-5.0‘ 53.2 4.75 0.63

‘Resonance frequencies are given as uo(’H)/uo(”O). bVolume fraction droplets (AOT + D20). ‘Only ”0 data. ‘Only ’H data

isooctane

‘ AOT D20 i o ’ do ’ i o ’

Figure 1. Schematic extension 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).

‘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 com- position 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 NMR 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.

--I

. OA 10 20 30 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 .

00 u 10 20 30 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

A R = 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

U a

3

w

C

TABLE 11: Temperature Dependence of AR for Three Samples" 210AR(2H)/

~ ~ ( 1 7 0 ) / ~ - 1 S-'

T/OC sample I samDle I1 sample 111 sample 111 15.1 15.9 24.6 117 121 16.1 24.6 24.9 126 126 19.2 35.8 29.0 115 1 I7 21.7 37.8 38.6 89.5 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 toy 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 en- countered 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 in- vestigated 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-@D samples 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 in- vestigated 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, also Rz( "O), and hence AR( 170), is frequency independent in 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.

I 5 fi 7

log(VO/Hz)

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 Rz data 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 re- laxation 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 re- sponsible for 4R 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 a t the nu- cleus, modulated by water exchange within a polydisperse droplet population; (ii) the scalar electron-mediated spin-spin coupling between the ZH and 170 nuclei 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 gen- erated 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

difference. In order to estimate the contribution to AR from this mechanism, we assume that the droplets are of two sizes: small (s) and large (I). If the exchange of water molecules between these two kinds of droplet can be modelled as an uncorrelated Markov process, we have3)

ARex = P , P I ( ~ o A 6 ) 2 ~ , , ( 2 ) where P, = 1 - PI is the fraction water in small droplets, wo is the Larmor frequency, and A6 = 61 - 6, is the difference in in- tradroplet-averaged (relative) chemical shift between the two types of droplet. The exchange time T~~ is related to the mean water resideace times T , and T~ in the droplets through

( 3 ) 1 1 1 _ - - - + -

Tex 's TI

Water exchange is known to proceed via a droplet fusion-fission p r o c e ~ s , ' ~ - ~ ~ characterized by a second-order coalescence rate constant k . Thus, for example,

71 = (kIcl)-' (4) where cI is the concentration of large droplets.

It is known, primarily from small-angle scattering s t ~ d i e s , 4 ~ ~ ~ ~ that large ( x > 20) AOT-stabilized droplets have a polydispersity of about 30% in the radius. The average radius of the droplets varies essentially linearly with x.I-l4 In addition, there is some e v i d e n ~ e ~ ' ? ~ ~ that a fraction of the AOT is present in the form of very small droplets (reversed micelles), which presumably contain a small amount of water. The water 'H chemical shift has been measured in AOT-stabilized microemulsions as a function of x.35936 The difference A6('H) between large (x = 60) and very small ( x = 1) droplets is about 1 ppm. As the chemical shift a t a given nuclear site should be virtually invariant to isotope substitution, this figure should apply also for 2H.

Consider now a sample with x = 30 at high volume fraction (series A), for which AR(2H) = 0.4 s-I (Figure 4). For this sample, the droplet concentration is cl = 8 X mol dm-3 and the coalescence rate constant13 is kl = 8 X lo6 dm3 mol-l s-I, whence T~ = 2 X lo4 s. We consider first the effect of the small reversed micelles. With an estimated 15% of the AOT present in the reversed micelles34 and with x, = 2, we have P, = 0.01 (this is 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 is a factor 2000 smaller than the observed AR(2H). To estimate the effect of the ca. 30% polydispersity of the large droplets, we take P, = PI = 0.5 and A6(2H) = 0.1 ppm,35.36 whence AR,,(2H) = 2 X S-I, which is, again, negligible compared to the observed AR(2H). In both cases, the fast-exchange condition W ~ A ~ T , << 1 is satisfied, as assumed in the derivation of eq 2.

Exchange modulation of the Zeeman coupling may thus be discarded as a possible relaxation mechanism under the conditions of the present study. Two further arguments in support of this conclusion may be mentioned. First eq 2 predicts that AR should increase quadratically with the magnetic field strength-contrary to our observations. Second, the observed ratio AR('70)/AR(2H) is about 210, whereas, according to eq 2, this ratio should be of order 1. This follows since wo(l7O) = w ~ ( ~ H ) and since A6(170) is of the same order of magnitude as A C ~ ( ~ H ) in concentrated aqueous electrolyte solutions.37

Scalar Relaxation. In general, there will be a contribution to AR from modulation of the electron-mediated spinspin coupling between the 2H (or IH) and 170 nuclei in the water m ~ l e c u l e . ~ " ~ ~ ~

(33) Carrington, A,; McLachlan, A. D. introduction to Magnetic Reso-

(34) North, A. N.; Dore, J. C.; McDonald, J. A.; Robinson, B. H.; Heenan,

(35) Maitra, A. J . Phys. Chem. 1984, 88, 5122. (36) Heatley, F. J . Chem. SOC., Faraday Trans. 1 1988, 84, 343. (37) Luz, Z.; Yagil, G. J . Phys. Chem. 1966, 70, 554. (38) Meiboom, S . J . Chem. Phys. 1961, 34, 375. (39) Halle, B.; Karlstrom, G . J . Chem. SOC., Faraday Trans. 2 1983, 79,

nance; Chapman & Hall, Bristol, 1979; p 208.

R. K.; Howe, A. M. Colloids Surf. 1986, 19, 21.

1047.

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~ Since the magnitude of the spin-spin coupling is small, scalar relaxation can compete with other mechanisms only when the modulation is slow. At typical Larmor frequencies, scalar re- laxation therefore contributes only to the transverse relaxation rate.

In a water-continuous system, ARsc is inversely proportional to the concentration of prototropic species and can thus be made negligible compared to RI by adjusting the pH (or pD) of the s o l u t i ~ n . ~ ~ * ~ ~ (Outside the pH range 4-10, AR,, is entirely neg- ligible.) In a droplet-type microemulsion, where the prototropic species (H30+ or OH-) are distributed over relatively small aqueous droplets, the situation is different. This difference arises because of the possibility of having a significant fraction of droplets that are free from prototropic species. For water molecules in such droplets, the spin-spin coupling is modulated as a result of fusion with another droplet containing one or more prototropic ions. We have recently exploited this intriguing phenomenon in developing a new technique, called nuclear spin quenching,a for determining exchange kinetics and droplet size in disperse fluids such as microemulsions. For our present purposes, however, scalar relaxation is an undesirable effect. Since the fusion-controlled modulation time may be much longer than the proton exchange time expected on the basis of the pH of the total aqueous pseu- dophase, AR,, may be much larger than expected for a water- continuous system. The situation is further complicated by the fact that trace amounts of acidic impurities in the AOT may radically change the pH dependence of a. It is clear, therefore, that the possibility of a scalar contribution to the transverse water relaxation rates in disperse systems must be carefully investigated in each case.

If the rate of modulation is large compared to the spin-spin coupling JOD (rad s-l), then one can s h o ~ ' ~ ~ ~ ~ that, in D 2 0 with I7O isotope fraction Pl7,

A & c ( ~ H ) = ( 3 5 / 1 2)P,7J0D2Teff ( 5 )

where

1 1 Teii Tsc - = - + ~ ~ ( ' ~ 0 ) (7)

For 2H, R2 is usually measured by a Carr-Purcell spin-echo sequence; the right-hand side of eq 5 should then be multiplied by a factor 1 - tanh ( c r ) / c r , where cr = Tp/(2Te#) and T~ is the time delay between the refocusing pulses.41 In H 2 0 , the factor 4 / 3 in eq 6 is replaced by 1/2 and JOD is replaced by JoH = ( y H / yD)JoD = 6.514JO0 (y is the nuclear magnetogyric ratio). The scalar modulation time T,, may, under certain assumptions,4° be expressed as

where k is the coalescence rate constant, c is the droplet con- centration, and f is the fraction droplets free of prototropic species.

Fortunately, several strategies are available for testing whether scalar relaxation contributes to the observed AR. The most straightforward approach (provided that the necessary instru- mentation is available) is to effectively eliminate the spin-spin coupling by double-resonance decoupling techniques. Such a decoupling experiment was performed on two microemulsion samples made from H 2 0 of neutral pH. The composition was

(40) Carlstrom, G.; Halle, B. Mol. Phys. 1988, 64, 659. (41) Lankhorst, D.; Schriever, J. ; Leyte, J. C. J . Magn. Reson. 1983, 51,

430.

3292 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989

220 03 1 Carlstrom and Halle

0.024.) Hence we conclude that the AR(2H) data for series A are virtually free from scalar contributions. (The difference compared to series B is (i) lower P17 values; and (ii) higher droplet volume fractions and, hence, shorter 7,,.) The low-frequency Rl(2H) data (series F) should also be free from scalar contribu- tions, since no 170 was added to this sample (Pi7 = 3.7 X at natural abundance). Further, the R, dispersion expected from a scalar relaxation mechanism is Lorentzian and occurs at [ W ~ ( ~ H ) - w0(170)]~,c = 1, which yields log [ V ~ ( ~ H ) / H Z ] = 3.6, i.e., much lower than observed (Figure 5).

Quadrupolar Relaxation. The quadrupolar mechanism is known to be responsible for the 2H and I7O high-frequency longitudinal spin relaxation in the microemulsion phase under study here, the modulation being caused by fast local water re- orientatiow2' Since water molecules near the droplet interface are orientationally ordered with respect to the interface, as evi- denced most directly by the water 2H and 170 quadrupolar line splitting observed in the reversed hexagonal phase of the same system (see below), a contribution to AR could come from the slow modulation of the residual quadrupolar interaction, partially averaged by fast water reorientation in the anisotropic interfacial environment. As we shall see in the next section, several slow diffusive processes could be responsible for this modulation.

We thus consider the quadrupolar spin relaxation of 2H and I7O nuclei in water molecules confined to an aqueous micro- emulsion droplet. The longitudinal and transverse quadrupolar relaxation rates are given by17

Rl = K[2J(w,) + 8J(2wo)]

R2 = K[3J(O) + SJ(w0) + 2 J ( 2 ~ 0 ) ]

(10)

(11)

where wo is the Larmor frequency and K depends on the nuclear spin I and the nuclear quadrupole moment eQ as

,80/ 3

'i . "4 0

201

o 0 0

3 4 5 6 pOH'

Figure 6. Variation of the scalar contribution AR, to the transverse "0 relaxation rate with pOH* (defined in the text) for an AOT/H20/iso- octane microemulsion sample with x = 38-48 (depending on NaOH additions) and y = 4.76. Note the scale shift in the ordinate for data points near the maximum.

x ly = 3814.75 and 29/35. Measuring R2(170) with and without proton decoupling, we obtained a scalar contribution of about 10 s-I for both samples. Using eq 6 and its counterpart for H 2 0 and accounting for the H / D isotope effect on 7,,,13 we arrive at an estimate of AR,(170) 5 2 s-l in D 2 0 for these compositions. Since the samples in series A-E were made from D 2 0 of pD = 4.1 (rather than neutral D20), the scalar contribution should be even smaller. As seen from the observed AR values in Figure 2, any scalar contribution to AR(I7O) can safely be neglected. This conclusion is supported by the observed decrease of AR with y (Figure 3). If the scalar contribution was dominant, AR would-contrary to observation-increase with y, since the droplet exchange rate, kc, should decrease as the droplet concentration c is reduced.

Another strategy of testing for a scalar contribution is to study the pH dependence of the relaxation. In simple aqueous solutions AR, has a strong pH dependence with a pronounced maximum around neutral pH.38*39 In disperse aqueous systems, however, the pH dependence of AR,, may be quite different. This is il- lustrated in Figure 6, showing ARSc(l7O), determined by the decoupling technique (cf. Experimental Section), as a function of pOH* = -log [(moles of added OH-)/(dm3 of H 2 0 in sample after addition)] for a microemulsion sample made with H 2 0 . These data can be quantitatively accounted for on the basis of the theory in ref 40, thus yielding values for the droplet size and the coalescence rate constant k . The appearance in Figure 6 of a sharp maximum in AR,, a t low pOH* (rather than at neutral pH) signals the presence of an acidic impurity (presumably a product of AOT hydrolysis) which is being titrated on addition of NaOH. From the position of the peak, the amount of impurity can be determined to be 400 ppm. Furthermore, Figure 6 reveals a small but significant scalar contribution which is nearly inde- pendent of pH over a wide range (to pH* = 3 in the present case). For lower levels of impurity, this contribution will be corre- spondingly larger.40 Thus, in disperse systems-in contrast to water-continuous systems-a pH-independent AR does not imply that AR, is negligible. In Figure 6, the pH independence comes about simply because the intrinsic prototropic impurity is in excess over the added prototropic ions (OH- or H 3 0 f ) .

Since the observed AR(2H) is 2 orders of magnitude smaller than AR(I7O), the 2H data are more susceptible to scalar con- tributions. According to eq 5-7,

AR,c(2H) = (35/16)P17[1 + T ~ ~ R ~ ( ~ ~ O ) ] ~ ' A R ~ ~ ( ~ ~ O ) (9)

The highest I7O enrichment used was about 10% (for the x = 14.9 sample in series B). With AR,(I7O) = 1 s-I, we could thus have AR,,(2H) = 0.2 s-', which is of the same order of magnitude as the observed AR(2H). Indeed, we have direct indications of a substantial scalar contribution to AR(2H) for series B, most no- tably, an observed variation of the ratio AR(2H)/AR(170) with P17. Consequently, we shall not make use of the AR(2H) data for series B. For series A, however, this ratio was independent of P I 7 . (For the data in Figure 4, P 1 7 varied between 0.007 and

K = -F(I) - 60 (Y), where

2 I + 3 F ( f l = P ( 2 I - 1)

(Equations 10 and 11 are strictly valid only for I = 1 nuclei such as 2H. However, under the conditions of the present study, the 170 ( I = 5/2) relaxation is effectively exponential and eq 10 and 11 are valid to an excellent appro~imation.~~) The spectral density function J ( w ) is the (real) Fourier transform of the electric field gradient (efg) time correlation function G(t)

J ( w ) = Jmdt cos (ut) G( t ) (14)

(15) G(t) = ( J'oL*(0) J'oL(t))

VoL is the m = 0 spherical component, expressed in the lab-fixed frame (L), of the irreducible efg tensor as defined in ref 18 and ( ) denotes an ensemble average.

As a result of the local orientational ordering of water molecules in the interfacial region, the correlation function G(t) does not decay completely on the time scale of the local (perturbed) water reorientation but acquires a long-time tail associated with those processes that randomize the orientation of the local orientational constraint axis with respect to the external L frame. Since, in a typical microemulsion, thse latter processes are much slower than the local reorientation, G ( t ) may be decomposed as18942

G(4 = Gf(t) + G,(t) (16)

Henceforth we shall focus on the part G,(t) associated with the slow dynamics, which accounts for the high-frequency AR (Figures 2-4) and the low-frequency R , dispersion (Figure 5). The in- terfacial perturbation of the local water reorientation in the microemulsion droplet, as described by Gr(t), is manifested in the

(42) Halle, B.; Wennerstrom, H.; Piculell, L. J . Phys. Chem. 1984, 88, 2482.

Shape Fluctuations in Microemulsions

frequency-independent part of R I , which is considered in detail in a separate report.21

If the Larmor frequency coo is small compared to the local reorientational rate but large compared to the slow dynamic processes that determine Gs(t), then it is only the transverse relaxation rate R2 that carries information about the slow process. The contribution from the fast local water reorientation Can then be eliminated by taking the difference R2 - R1, which under these conditions is given by

AR = 3KJs(0) (17)

which may be conveniently expressed as

where j , (w ) is a reduced spectral density function and x is the nuclear quadrupole coupling constant. The quantity A in eq 18 is the so-called residual anisotropy, which depends on the preferred orientation of a water molecule with respect to the interface and on the distribution around this orientation. The explicit forms of ( A 2 ) and j ,(O) are given in the Appendix. Since the (intra- molecular) efg principal frames are different for the 2H and I7O nuclei in the water molecule,18 the orientational distribution function is probed in different ways by these nuclei, i.e. A(2H) # A(I7O). The angular brackets in eq 18 denote a spatial average of the local residual anisotropy A(r) over all accessible water locations. From measurements of quadrupolar line splittings from the lamellar and reversed hexagonal phases of this system (Soderman, O., unpublished results; Quist, P-O., unpublished results), it is known that only water molecules in the primary hydration shell of the AOT headgroups and counterions experience a significant residual anisotropy. Accordingly, we may write

( A 2 ) = PBAB2 (19)

is the fraction of water molecules with A # 0 and AB is the root-mean-square residual anisotropy for this fraction.

By expressing AR on the form of eq 18, we have factorized out the quantities that depend on the nuclear species (2H or I7O). Using eq 13 and 19, we thus obtain

since j , (O) should be the same for 2H and I7O (cf. the Appendix). The quadrupolar line splittings, AVQ, of the water 2H and I7O nuclei in anisotropic phases also depend on the parameters AB and x and one has'*

(22)

If line splittings and relaxation rates are measured in different phases of the same system, which differ in long-range structure ( e g , spherical microemulsion droplets and cylindrical water rods in the reversed hexagonal phase) but which have essentially the same local interfacial structure, then it follows from eq 21 and 22 that

Actually, in eq 22 AB = (A)B while in eq 21 AB = (A2)B'/ ' . However, unless A(r) varies very strongly among different local environments in the interfacial region, this difference should be unimportant and eq 23 should be valid.

Equation 23 provides a convenient test of whether the observed AR data can be ascribed to quadrupolar relaxation. We therefore measured the quadrupolar splittings in the reversed hexagonal

The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3293

phase of the AOT/D20/isooctane system (x = 20.8, y = 1.07, T = 23 "C), finding AVQ('~O)/ALJQ(~H) = 5.97. (In most in- vestigated systems, but not in all,"3 this ratio is close to 6 as a result of the relative orientation of the 2H and I7O efg principal frames.I8) According to eq 23, we should thus have AR('70)/AR(2H) = 228 for quadrupolar relaxation. The data in Figure 4 yield an average ratio of 210, in excellent agreement with the prediction based on the splittings. We therefore conclude that the observed AR (and the low-frequency R1 dispersion) can be ascribed entirely to quadrupolar relaxation and that the time scale separation implicit in eq 18 is valid. To the best of our knowledge, this is the first time that the important relation eq 23 has been experimentally verified for any system. (In many other heterogeneous aqueous systems, either AR(2H) is dominated by exchanging non-water deuterons, or anisotropic phases are not readily prepared.)

The quadrupolar relaxation mechanism is also consistent with the magnitude of the observed low-frequency dispersion step in R,(2H) (Figure 5). According to eq 10 and 11

Rl(low v0) = R,(high vo) + (10/3)AR(high v 0 ) (24)

The upper dashed line in Figure 5 is the prediction of Rl(low vo), based on ey 24 and high-frequency relaxation data. The factor 10/3 in eq 24 is a consequence of the second-rank tensorial nature of the quadrupole coupling.

We return here briefly to the issue of droplet size polydispersity, which has the effect that the observed AR is an average over the droplet size distribution. As discussed elsewhere:l Rl is unaffected by polydispersity as long as all droplets are sufficiently large ( x 2 15). AR, on the other hand, depends on x, not only through the fraction (Pe = xB/x) of orientationally perturbed water, but also through the reduced spectral density js (0 ) (cf. below) and will therefore be more sensitive to size polydispersity. However, since the observed x dependence (Figures 2 and 4) is relatively weak, a small polydispersity in the large droplet fraction is of little consequence for the analysis; we can simply reinterpret x as an average yalue. As regards the small reversed micelles (cf. above), the small fraction (about 1%) of water in these is not likely to contribute significantly to the observed AR. This would require that the reversed micelles, which at room temperature are small and essentially have several orders of magnitude larger AR than the large droplets, which is hard to imagine. Further- more, in a pseudophase model, the fraction of water present in reversed micelles should increase as the microemulsion is diluted by addition of oil. Thus, if this water fraction contributed sig- nificantly to AR, we should see an increase of AR with y , in contrast to the experimental results in Figure 3.

The reduced spectral densityj,(O) in eq 18 is the time integral of the reduced time correlation function g,(t) in eq A3. If there is 3-fold rotation symmetry with respect to the local constraint axis (perpendicular to the interface), we have (cf. the Appendix)

5c~[r (0)1 p2[t(o)i ~ [ r ( t ) i p 2 [ t ( t ) 3 ) (25)

where P2 is the second-rank Legendre polynomial and E = cos 0, 0 being the angle between the local orientational constraint axis and the external static magnetic field. Since the system is isotropic, g,(t) decays to zero. The reduced time correlation function g,(t) reflects the nature of the slow dynamic processes that cause the orientation ( E ) of the local constraint axis and the magnitude of the local anisotropy A(r) to fluctuate in time. In order to evaluate g,(t), we must identify the relevant motional degrees of freedom and construct an appropriate equation of motion.

Motional Models Having established that AR and the low-frequency R1 dispersion

are due to the fluctuating residual quadrupole coupling, it remains to identify the molecular motion that is responsible for these fluctuations. We first attempted to interpret the relaxation data in terms of translational diffusion of water molecules within a

( A 2 ) =

(43) Hakala, M-R.; Wong, T. C. Langmuir 1986, 2, 83.

3294

spherical microemulsion droplet. As it became clear that this model, referred to in the following as the water diffusion (WD) model, could not account for the data, we were led to explore two other dynamic models, which we refer to as the droplet rotation (DR) model and the shape fluctuation (SF) model. Both of these models require that the microemulsion droplet can assume a nonspherical shape. In the DR model the residual quadrupolar coupling is modulated by reorientation of a permanently spheroidal droplet, while, in the SF model, the modulation is brought about by temporal shape fluctuations of a droplet with a spherical “equilibrium” shape. Although it is possible that all of these types of motion contribute to the observed relaxation behavior, the following quantitative analysis indicates that the S F model is the most likely alternative.

Water Diffwsion (WD) Model. As noted above, the orienta- tional ordering, as reflected in the residual anisotropy A, is of short range. It is therefore reasonable to model it as a step-function perturbation. Accordingly, we divide the aqueous droplet, assumed to be spherical with radius a , into two regions: a perturbed (anisotropic) region (denoted B), within a distance 6 of the in- terface, and an unperturbed (isotropic) core region of radius a - 6.

The Journal of Physical Chemistry, Vol. 93, No. 8. 1989

The fraction water in region B is then

Carlstrom and Halle

diffusivity with the same range (6) as for the orientational per- turbation.

The time correlation function g,(t) is affected not only by water diffusion but also by rotation of the spherical droplet dispersed in the oil medium. The rotational diffusion of a fluid droplet suspended in another fluid is a nontrivial hydrodynamic problem. However, since the interface is covered by surfactant molecules, the aqueous droplet is expected to rotate essentially as a rigid body.45 The effect of droplet rotation may then be incorporated by adding to Dlat(r) in eq 30 a term r2DrOt, where D,,, is the rotational diffusion coefficient of the (rigid) spherical micro- emulsion droplet (water and AOT)

(31)

where 9 is the viscosity of the oil medium and b is the droplet radius, taken to be

(32) with a given by eq 27 and 10 A being the length of the AOT hydrocarbon chain. In eq 31, we have neglected the hydrodynamic interaction (cf. below). If, furthermore, aZDrot << Do and 6 << a, we have

kBT Drot = - 87r9b3

b / A = a + 10

The geometrical parameters 6 and a can be related to the molar ratios x and xB in several ways. The radius of the spherical diffusion space within the droplet is taken to be given entirely by the composition variable x according to

(27)

which agrees roughly with the results from small-angle scatter- ing3-I0 and fluorescence quenchingI4 studies of AOT-stabilized microemulsions. (The recent study by Lang et al.14 shows that a increases slightly with temperature and droplet volume fraction; however, this variation is unimportant for our considerations.) The decay length 6 for the orientational perturbation can be chosen in several ways. If eq 20 and 26 are identified, one finds, for a constant xB, that 6 decreases with x . However, most of this x dependence occurs for x values close to xB, while, for larger x , 6 depends only weakly on x. For xB = 15, 6 thus decreases from 14 to 10 A as x goes from 20 to 50. In the following we regard 6 as independent of a.

For the WD model, the time correlation function in eq 25 may be expressed as44

a /A = 1 . 6 ~ + 6

1 gs(t) = 2 J-; dE0 P2(50) J’ dE P2(8 f(B,E;tlB,So) (28)

- I

wheref(B,l;tlB,.$o) d.$ is the probability that a water molecule, initially in region B with local constraint axis orientation to, after a time t resides in region B with constraint axis orientation to within dE of 5 . The “transition probability”f(B,[;tlB,~o) in eq 28 is obtained from the continuous space propagatorf(r,,$;tlro,.$o) by spatial integration. The latter is assumed to satisfy a diffusion equation

(29) at f(r,t;tIro,tJ = V.D(r).Vf(r.f;tlro,to)

Due to the assumed spherical symmetry of the droplet, the dif- fusion tensor takes the form

(30)

where I is the unit dyadic and i is the radial unit vector. The r dependence of the radial and lateral self-diffusion coefficients accounts for perturbations of water translation induced by the proximity to the interface. Far away from the interface, both Dmd(r) and Dlal(r) attain the value Do characteristic of bulk water. For simplicity, we assume a step-function perturbation of the

a

D(r) = Drad(r)ii + &(r)(I - it)

D J , r < a -- 6

D,,,jE + (Dht + a2DrOt)(I - rr) , a - 6 < r < a M (33) D(r) =

At 20 ‘C, the bulk D 2 0 d i f f ~ s i v i t y ~ ~ is Do = 1.65 X m2 s-’ and the shear viscosity of isooctane4’ is 9 = 0.50 cP.

The dynamic model is now complete. By solving the diffusion equation (29) with the diffusivity tensor in eq 33 and with the appropriate boundary conditions, we can obtain the diffusion propagator and, after spatial and angular integration, the desired correlation function in eq 28. However, since we shall be primarily concerned with the zero-frequency spectral density j , (O), Le., the time integral of g,(t), it is more convenient to use a Green function approach. This leads, with no further approximations, to the analytical result44

-( P + 1 - P - ) 2 + j(U-P”- 2 - U+P+)( 1 - v5) + x

where

p* = [ ( $ 2 - ;)% - (1 f 2s) v*s 1 (35)

5 s2 - 2s - - 4

A = i

s2 + 2s - 1 4

u* = s f 7 2

v = 1 - (6 /a ) (39) For the special case D = DI, eq 34 reduces to

277r2 a2 25000

AR = -F(Z) (ABx)*5[(1 - v5)(7 - 3u5) + 20v5 In v]

(40)

(45) Levich, V . G. Physicochemical Hydrodynamics; Prentice-Hall: En-

(46) Mills, R. J . Phys. Chem. 1973, 77, 685. (47) Geist, J. M.; Cannon, M. R. Ind. Eng. Chem., Anal. Ed. 1946, 18,

glewood Cliffs, NJ, 1962; section 69.

61 1. (44) Halle, B., to be submitted for publication.

Shape.Fluctuations in Microemulsions

i 40

I 20 30 40 50 60

X

Figure 7. Variation of AR(”O) with droplet size according to the W D model in the limit of uniform and isotropic diffusivity. The values used for D/Do and 6/A are: 0.1 and 10 (curve a); 0.1 and 5 (curve b); 1 and 10 (curve e ) . The filled circles are the experimental data for series B.

--I I

I -A I

r 40t 1 &

X

Figure 8. Variation of AR(”O) with droplet size according to the WD model, eq 34. The values used for D,d/Do, Dlat/Do, and S/A are 0.005, 0.1, and 10 (curves a and b); O.Ol,O.Ol, and 10 (curve c); 0.001, 1, and 10 (curve d). For curve b, droplet rotation has been frozen out (& = 0). The filled circles are the experimental data for series B.

We shall now confront the WD model with the experimental facts, beginning with x dependence of AR(I7O) for the low volume fraction series B. In all calculations for this model, we use ABx = 1.34 X lo5 s-I for I7O, as obtained from the quadrupolar splittings taking xB = 12.5. Figure 7 shows the predictions of the WD model in the limit of a uniform and isotropic diffusivity, as given by eq 40. To obtain AR values of the right order of magnitude for the large droplets, the water diffusivity must be (uniformly) reduced as compared to bulk water by an order of magnitude. Not only does such a large dynamic perturbation in the entire droplet seem implausible, but, furthermore, !he model fails qualitatively to predict the observed x dependen :e for x 5 30.

Figure 8 shows the predictions of the more elaborate WD model, eq 34, with a nonuniform and anisotropic diffusivity. Curve a corresponds roughly to the radial and lateral diffusivities for interfacial water that were recently deduced from an I7O relaxation study of colloidal silica dispersions.23* While this yields AR values of the right order of magnitude, the x dependence for small droplets is, again, qualitatively opposite to that observed.

In the WD model, as formulated above, AR is entirely inde- pendent of y , whereas experimentally there is a strong y depen- dence (Figures 2 and 3 ) . One possible cause of y dependence in AR is via the droplet radius. However, this effect is far too small. Consider, for example, a sample with x = 28, for which the difference in AR between series A 0, = 4.75) and B 0, = 35.0) is a factor 3.2. From the data of Lang et al.,14 we infer that a varies by less than 10% in this y range, which clearly cannot account for the experimental observation. Another possible source of y dependence is via the effect of hydrodynamic interaction on the rate of droplet rotation. This effect is roughly described by

(41 ) where &, is the droplet volume fraction (AOT and water) and K is a numerical coefficient estimated4* to be in the range 1 / 3 - 1 / 2

Drat = ( 1 - K ~ ) D ~ ~ ~ O

The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3295

However, since AR depends only weakly on Dmt, even a complete freezing out of the rotational degree of freedom would fall far short of explaining t h e y dependence in Figure 3. This is seen from the small difference between the curves a (Drat = Dm:) and b (Drat = 0) in Figure 8.

In the long residence time limit, when Drad << Do, the radial diffusion within the droplet may be characterized by a mean residence time T,,, of a water molecule in region B and eq 34 reduces to24349

(42) 9** 100

AR = -F(r) (ABX)2PB7B

with PB given by eq 26 and the effective correlation time 7 B given

1 1 1 1 - = - + - + - by

(43 ) 7B 71at TICS Trot

with

(44 )

(45)

where eq 45 is valid for 6 << a. The curves a, b, and d in Figure 8 are rather well reproduced by eq 42-46. In this limit, the time correlation function gs(t) is exponential with correlation time 7 B . In order to account for the magnitude of the dispersion step in Figure 5 , 78 must be about 5 X lo-* s. However, this would yield a Lorentzian (rather than extended) dispersion at a frequency higher than observed by an order of magnitude.

Because of the inability of the WD model to account for the observed x and y dependence in AR and for the low-frequency dispersion, we conclude that water diffusion does not contribute significantly to the relaxation in samples of high volume fraction. Further, for samples of low volume fraction (series B), it does not, at least, account for the whole observed effect. This follows from the incorrect x dependence predicted by the model (Figure 8) and from the fact that AR is observed to increase with temperature (Table 11), whereas the WD model predicts the opposite (the observed increase in droplet size with temperatureI4 is far too small to compensate for the temperature dependence of the water diffusion coefficient). However, a relatively small contribution to AR from water diffusion cannot be ruled out. For example, if Drad = Dlat i= O.lDo and 6 = 10 A, then AR(I7O) 5 5 s-l, which would hardly be significant experimentally.

Droplet Rotation ( D R ) Model. In the case of a spherical droplet, the effect of droplet rotation is directly superimposed on the effect of lateral water diffusion, as reflected in the combination (Dia, + a2DrOt) in eq 3 3 . If a2Drot is small compared to Dla,, the effect on AR of completely freezing out the rotational degree of freedom (e.g., by strong interdroplet interactions) will be insig- nificant. If, on the other hand, the droplet is nonspherical, droplet rotation can assume much greater importance. This is clearly seen by considering the limiting case where droplet rotation is frozen out. Even if optically isotropic, such a system will be anisotropic from the NMR point of view (unless water molecules can exchange among the droplets at a sufficient rate). Since water diffusion within a nonspherical droplet (strictly speaking: a droplet shape of lower than cubic symmetry) cannot completely average out the quadrupolar coupling, the degeneracy of the different Zeeman transitions is lifted and a quadrupolar splitting occurs in the spectrum (if the droplets are randomly oriented, a so-called powder pattern is observed).

If water diffusion within the droplet is much faster than the reorientation of the droplet, the time correlation function in eq 25 may be decomposed as

(48) Montgomery, J. A.; Berne, B. J. J . Chem. Phys. 1977, 67, 4589. (49) Halle, B. Mol. Phys. 1985, 56, 209.

3296 The Journal of Physical Chemistry, Vol. 93, No. 8. 1989

gs(t) = (1 - PBa2)gdif(r) + PBa2grol(t) (47) where, as before, we have divided the droplet into a perturbed region ( A = AB) and an unperturbed region ( A = 0). PB is the fraction of water molecules in the perturbed region as given by eq 20 and CY is the fraction of the locally averaged quadrupole coupling that remains after further averaging by translational diffusion within the droplet. If the droplet has cylindrical sym- metry, the reduced time correlation functions in eq 47 become

gdif(t) = C(A[r(Q)I%o*[Q(o)I A[r(t)I %dQ(t)I) - ( P B A B ~ ) ~

(48)

where &(ti) is a Wigner rotation matrix element and Q are the Euler angles specifying the transformation from the local constraint axis to the droplet symmeky axis, and

m

PB( 1 - PB(Y2)Ag

grot(t) = 5(P2[M0)l P2[l(t)l) = ~ X P -- (49) ( 7:01)

where { = cos O‘, 9’ being the angle between the droplet symmetry axis and the external static magnetic field. Combination of eq 18, 19, and 47 yields

AR = &if 4- ARrO1 (50)

with

9*’ 100 ARdif = --F(r) PB(1 - PBa2)(ABX)’Tdif (51)

(52) 9?r2

W o t = =-F(n ( P B ~ A B X ) ~ ~ ~ ~ ,

where the correlation times 7dif and 7rOt are defined as the time integrals of the reduced correlation functions in eq 48 and 49.

For a spheroidal (oblate or prolate) droplet with the constraint axis along the local interface normal, the diffusional averaging factor CY becomesSo

(2p2 + 1) - (4p’ - l )B

2(p2 - 1)(1 + B) (53)

(54)

“ob =

(4 - p2)C - (2 + p2)

2(P2 - 1)(1 + c) a p r =

where p is the ratio of the major and minor axes of the spheroid ( p = 1 corresponds to a sphere, p = m to an infinite rod or disk) and

arccosh p B = (55)

C = (56)

p(p2 - 1)1/2

p2 arccos (1 / p )

(p2 - 1)1/2

The rotational correlation time is

3c

v

Q

2 -

Carlstrom and Halle

0 c * 0

0 0

‘ D 8

0 . 8

(57)

for a spheroidal droplet of volume V suspended in a medium of viscosity 9. The factor If($,) accounts for hydrodynamic, ex- cluded-volume, and other interactions among the droplets; H = 1 for vanishing volume fraction dD of droplets. The shape factor S(p) in eq 57 is given byS’vs2

p2 + (p’ - 2)C

p 4 - 1 sob =

p2[ (2p2 - l )B - 11 s,, = (59)

p 4 - 1

(50) Halle, B. Mol. Phys. 1987, 61, 963.

6ol OBLATE A

P Figure 9. Variation of AR with the axial ratio p of a spheroidal droplet ( x = 35.1) according to the DR model.

l o

’IO m 20 30 40 50

X

Figure 10. Variation of the axial ratio p with x according to the DR model for oblate (0) and prolate (0) spheroidal droplets, and variation of the mean axialratio ( p ) with x according to the SF model (m). These axial ratios reproduce the experimental AR(I7O) data for series B.

In comparing the predictions of the DR model with experiment, we first consider samples a t low droplet volume fraction (series B) for which it should be reasonable to set H = 1 in eq 57. The spheroidal droplet volume Vis taken to be the same as that of a spherical droplet with radius b given by x through eq 27 and 32. The quantity PBABx appearing in eq 52 is obtained directly from the quadrupolar splittings in the reversed hexagonal phase (see above): xBABx = 1.67 X lo6 s-l for I7O.

Figure 9 shows how AR,,, increases with the axial ratio p of the spheroidal droplet for a fixed droplet volume (corresponding to x = 35). The increase in ARrot is due partly to an increase of the residual anisotropy factor CY’ and partly to an increase of the rotational correlation time T~~~ with increasing axial ratio. For a given axial ratio, the prolate has larger T~~~ while the oblate has larger a2. As the relative difference is largest for a2, AR,,, is always somewhat larger for the oblate shape. By using the ex- perimental AR(I7O) values for series B (the same data points that were considered in Figures 7 and 8) and by assuming that AR = AR,,, (Le., neglecting u d i f in eq 50 in accordance with the previous discussion), we can calculate the axial ratio of the mi- croemulsion droplets (AOT and water) as a function of their size (x). The result is shown in Figure 10. An axial ratio in the range 2.0-2.5 is obtained for the larger droplets, with a slight increase for the smallest droplets. As expected from Figure 9, the prolate model requires a somewhat larger axial ratio than the oblate model. The rotational correlation time is in the range 10-7-10-6

The observed y dependence of AR (Figure 3) can be understood, in the DR model, as a result of increasing interactions between the droplets as their volume fraction increases. These interactions can be manifested in AR either via an induced shape asymmetry (increased p and, hence, larger a’ and 7,,,) or via increasing rotational hindrance for a fixed @-independent) spheroidal shape (H < 1 and hence larger 7,0t).

S .

(51) Edwardes, D. J . Pure Appl. Math. 1892, 26, 70. ( 5 2 ) Perrin. F. J . Phys. Radium, Paris 1934, 5 , 491.

Shape Fluctuations in Microemulsions

The low-frequency dispersion in Figure 5 refers to a sample with large droplets (x = 53.2) at high volume fraction ($D = 0.63). Using the DR model, we can predict T ~ ~ ~ , and hence the frequency of the dispersion region, from the magnitude of the dispersion step (the dashed lines in Figure 5, obtained from the high-frequency AR). If we assume that H = 1 also at this high volume fraction, we obtain T~~~ 1 ps (e.g., T~~~ = 0.9 ps and p = 3.45 for an oblate spheroid), which falls in the observed dispersion range. A somewhat larger T~~~ results if a reasonable estimate of H less than 1 is used; e.g., for an oblate with H = 0.5 (0.2), T , , ~ = 1.4 (2.9) ps, and p = 2.58 (1.93). However, the DR model predicts an exponential correlation function, eq 49, and hence an essentially Lorentzian R I dispersion, whereas the observed dispersion is much more extended (Figure 5 ) . This could be taken to indicate a shape polydispersity in the population of microemulsion droplets. This point is discussed in more detail in connection with the S F model in the following section.

In conclusion, it seems that the DR model can account for the observed relaxation behavior, if we are willing to accept the notion of nonspherical microemulsion droplets. In most previous work, spherical droplets have been assumed. In particular, in the nu- merous small-angle neutron and X-ray scattering studies per- formed on AOT-stabilized mi~roemulsions,~-~~ the scattering curve has usually been analyzed in terms of spherical droplets (see, however, ref 3) with some (typically 30%) size polydispersity. However, it is known that the scattering curves from polydisperse spheres and monodisperse spheroids are virtually identicaLS3 Indeed, reanalyzing the SANS data of Kotlarchyk et aL6 from the AOT/DzO/decane microemulsion phase, we obtain an ex- cellent fit to the scattering data with the theoretical expression for prolate spheroids. For the sample with x = 24.5 and y = 94 at 33.4 OC, we thus obtained prolate semiaxes of 28 and 8 3 A, Le., an axial ratio of 3.0 for the aqueous core. Adding 10 A for the AOT layer (cf. eq 32), we thus obtain p = 2.4. However, electron microscopy images of freeze-fractured microemulsions in the same system, recently reported by Jahn and st re^,^^ do not seem to be compatible with axial ratios as large as p = 2-3.

Shape Fluctuation (SF) Model. In the DR model, we tacitly assumed that the droplets are monodisperse in shape and size. However, the extended R , dispersion was taken as an indication of shape polydispersity, Le., a distribution of axial ratios around a most probable (“equilibrium”) shape. The existence of shape polydispersity in the droplet population implies that any given droplet will have a shape that fluctuates in time. As a result, the intradroplet-averaged quadrupole coupling P&BxPz({) will fluctuate in magnitude (through a) as well as in orientation (through 0. The effect of shape polydispersity on the time correlation function g,(t) in eq 25 depends on the relative time scale of shape fluctuations and droplet rotation. If fluctuations are fast compared to rotation, then droplet rotation will modulate a shape-averaged quadrupole coupling. The DR model then applies, with the axial ratio p interpreted as an average, except that eq 50 must be supplemented with a term due to the faster shape fluctuations. If, on the other hand, fluctuations are slow compared to rotation, then g,(t) will be an average over the shape distribution of exponential correlation functions of the DR type. This could lead to an extended R , dispersion as noted above.

In this section, we shall examine whether the experimental findings can be rationalized by a model which invokes shape polydispersity and fluctuations around a spherical equilibrium shape. Since the shape fluctuations are expected to take place on the same time scale as droplet rotation (cf. below), it becomes a rather difficult problem to evaluate g,(t). The following shape fluctuation (SF) model will therefore be of a less rigorous nature than the preceding two models.

For simplicity, we assume that the shape distribution function f ( p ) is Gaussian, peaked at the spherical shape, and symmetrical

(53) Hayter, J. B. In Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Digiorgio, V., Corti, M., Eds.; North-Holland: Amsterdam, 1985; p 59.

(54) Jahn, W.; Strey, R. J. Phys. Chem. 1988, 92, 2294.

The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3291

with respect to oblate and prolate spheroidal deformations, i.e.

The mean axial ratio ( p ) is related to the width u of the distri- bution through

( p ) = 1 + (yo If water diffusion within the droplet is fast compared to those

shape fluctuation modes that contribute to the spin relaxation, then, in analogy with eq 47, the time correlation function in eq 25 may be decomposed as

gs(t) = (1 - PB(az))gdif(t) + PB(a2)gflu(t) (62)

where ( ) denotes averaging over the shape distribution in eq 60. The reduced correlation function associated with shape fluctuations may be expressed as

g d t ) = 5(az)-’(a[p(o)I p 2 [ l ( 0 ) l a[p(t)l p2[l(t)l)

= 5(a2 ) - ’S dpo dlOf(po,lo) PO) ~ z ( { o ) X

J- dp J- dlflP,C4Po,{o) &) (63)

where a subscript zero signifies the initial time. equilibrium distribution in eq 63 is simply

The joint

(64) 1

f ( P J = - s ( P )

since the system is isotropic. For the joint propagator, we use the ansatz

where ~ ~ , , ( p ~ ) is the mean time taken for an initial shape fluc- tuation of axial ratio po to decay to spherical shape. Successive shape fluctuations are assumed to be orientationally uncorrelated; hence p and {have the same rate of decay. Combination of eq 63-65 yields

with aob and apr as given by eq 53 and 54. Combination of eq 18, 19, 62, and 66 now yields, in analogy with eq 50

AR = ARdif + ARn, (67)

with

9 7lz ARfl,, =

- F ( I ) 100 ( P B A B X I Z c(lm dp f ( p ) [aot(p) + aP,2(p)l mU(p) (68)

In order to evaluate the fluctuation time q , , ( p ) , we assume that shape fluctuations may be described as a diffusion process in shape space under a restoring force consistent with the shape distribution in eq 60. (This force ensures that an ensemble of droplets of arbitrary initial shape evolves toward the correct equilibrium shape distribution.) The fluctuation time can then be identified as a so-called mean-first-passage timeSS given by

where Q ( p ) = Ds(p) /D , and D, = D , ( p = l ) . The diffusion coef-

(55) Gardiner, C. W. Handbook of Stochastic Methods; Springer-Verlag: Berlin, 1983.

3298

ficient in shape space should be closely related to the translational diffusion coefficient of the droplet. An intuitive ansatz based on the concept of displaced volume yieldss6

The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 Carlstrom and Halle

splittings and of R2-RI. Further, we have measured a 2H R, dispersion over an extensive frequency range, which appears to be the first low-frequency relaxation dispersion reported for any heterogeneous aqueous solution system where exchange-averaging with non-water nuclei is undisputably absent.

In order to give a molecular interpretation of our relaxation data, we proposed three dynamic models. The result of the quantitative analysis may be summarized as follows.

Water diffusion within the microemulsion droplet cannot ac- count for the experimental data. If the diffusivity of interfacial water is moderately reduced as compared to bulk water (by about 1 order of magnitude), then the contribution to the relaxation rates could escape detection. If, however, the radial diffusivity would be more strongly reduced (by a factor of 300), as indicated by the long residence times (ca. 50 ns) deduced from relaxation studies of interfacial water in solutions of colloidal silica particles and globular proteins,24 then a large relaxation effect would be seen but its variation with droplet size and volume fraction, temperature, and resonance frequency would deviate qualitatively from that observed here. A possible explanation is that, in the case of silica particles and proteins, a substantial amount of water is trapped in pores or clefts. In the microemulsion droplets such structures do not persist on the implicated time scale.

A satisfactory explanation of our relaxation data seems to require a substantial shape polydispersity where, at any instant, a large fraction of the droplets are distinctly nonspherical (axial ratios in the range 2-3). However, it is not necessary that the equilibrium shape is nonspherical. It seems worth considering whether the existing small-angle neutron scattering data, usually interpreted in terms of a size polydispersity (assuming a spherical shape), may equally well be accounted for by a shape polydispersity of the magnitude suggested by our NMR study.

Acknowledgment. We are indebted to Professor R. Kimmich for placing at our disposal the field-cycling instrument at Sektion Kernresonanzspektroskopie a t University of Ulm, and to Dr. G. Schauer for expert technical guidance during these experiments. Grants from the Swedish Natural Science Research Council are gratefully acknowledged. This work was not supported by any military agency.

Appendix The Residual Anisotropy of the 2H and I7O Quadrupole

Coupling. In this Appendix we give a more detailed account of the derivation of eq 18-25 in the main text. In particular, we discuss the validity of the important relation, eq 23, between the 'H and I7O quadrupolar relaxation rates and line splittings.

We begin by giving the explicit expressions for the mean square residual anisotropy ( A2) and the zero-frequency reduced spectral densityj,(O), which appear in eq 18. The latter quantity is defined, in accordance with eq 14, as

where Vis the droplet volume (AOT and water) and 17 is the viscosity of the continuous medium (isooctane), and

(71) 4 ( P Z -

= 27 p10/3(p2/3 + 1)2@/3 - 1)'

By numerically evaluating the integrals in eq 68 and 69, we can now estimate ARflu as a function of the shape polydispersity or, equivalently, the mean axial ratio ( p ) . Figure 10 shows the ( p ) values required to reproduce the experimental AR values for series B with the S F model, eq 68. The results are seen to be similar to those of the DR model, although the ( p ) values are somewhat smaller than the p values required in the DR model. As in the DR model, the observed y dependence of AR can be explained by droplet-droplet interactions, which at high volume fractions are expected to decrease the rate of the shape fluctua- tions. Further, the anomalous temperature dependence at low volume fraction (Table 11) may be ascribed to an increased fluctuation amplitude with increasing temperature, an effect that may be less pronounced at high volume fractions. Considering the rather crude approximations that we have built into the S F model, its quantitative predictions should be regarded with some caution. However, it does demonstrate that AR values of the order of magnitude observed can be accounted for by shape fluctuations of droplets with a spherical equilibrium shape.

In contrast to the (monodisperse) DR model, the S F model predicts a non-Lorentzian R, dispersion. According to eq 10 and 24, R,(o) - R,(high uo) is obtained from eq 68 after inserting in the integrand the factor

The only adjustable parameter in the S F model is the width parameter u in eq 60. Since this parameter is fixed by the ex- perimentally determined dispersion step, as given by AR(high uo) and eq 24, the position (on the frequency scale) and the detailed shape of the R, dispersion are fully determined by the S F model. Proceeding in this manner, we found that the calculated dispersion is centered around 2 X lo4 Hz (about a factor of 10 lower than the experimental dispersion in Figure 5) and that it is distinctly non-Lorentzian, although not as extended as in Figure 5. We have explored some other functional forms for the shape distribution functionf(p), besides the Gaussian form in eq 60, without being able to obtain a more extended dispersion centered at a higher frequency. A more quantitative agreement with the experimental dispersion could probably be achieved by avoiding the rather crude approximation in eq 65. It may also be necessary to include the dynamic coupling between the fluctuating droplets.

Conclusions In this work we have reported water 2H and I7O spin relaxation

data for the microemulsion phase in the AOT/D20/isooctane system. This collection of water relaxation data is probably the most extensive one available for any heterogeneous aqueous so- lution system (where there are no contributions from non-water nuclei). In addition, we have recently used 2H and I7O N M R to study local water reorientation,21 droplet exchange kinetics,40 and low-temperature behavior22 in the same system.

The focus in the present work has been on the contribution from slow molecular processes to the quadrupolar relaxation behavior. For the first time in any system, we have verified the theoretically predicted relation between the 170/2H ratios of quadrupolar line

( 5 6 ) Halle, B.; Landgren, M.; Jonsson, B. J . Phys. (Les Uh, Fr . ) 1988, 49, 1235.

('41)

with the reduced time correlation function

gAt) = G,(t)/G,(o) ('42)

where G,(t) is given by eq 15, except that the efg component VoL is now averaged over the fast motion.

Given the time scale separation implicit in eq 17, one can show that

gAt) = 5

- -Vbo(Q + 2Re[ho2(t) + h20(t)l + 2Re[h22(t) + h-22(t)lI (A2)

where we have introduced the partial time correlation functions

Here D&(QLD) is a second-rank Wigner rotation matrix element and the Euler angles R,, specify the transformation from the lab-fixed frame (L) to the director frame (D). The latter is, by

J. Phys. Chem. 1989, 93, 3299-3304 3299

definition, the principal frame for the locally averaged efg tensor. The zD axis, referred to in the main text as the local orientational constraint axis, is usually assumed to be perpendicular to the interface, but this need not be the case.

Taking into account the C2, symmetry of the water molecule, the residual anisotropy parameters A , in eq A4 may be expressed as

1 1 Re D : O ( n M F ) + s 7 [ D : 2 ( n M F ) + D : - 2 ( n M F ) l Sm2(r) I

where 9 is the asymmetry parameter of the (nonaveraged) efg tensor in its molecule-fixed principal frame (F). Since the D frame is a principal frame for the averaged efg, we have A , = A,. From A2-A4, it follows that the mean square residual anisotropy is related to A . and A2 through

( A 2 ) = (Ao2) + 2(A22) (A6) To obtain this result, we have also made use of the isotropy condition f (aLD) = 1/(8r2) and the orthogonality of the Wigner rotation matrix elements.

The second-rank orientational order parameters in eq AS are defined as

= (2 - 6,) Re(%n(nDM)) ('47) These parameters are determined by the orientational distribution function f(nDM) for the orientation of the water molecule (the molecule-fixed M frame is defined as in ref 18, with the zM axis along the molecular C2 axis) with respect to the D frame.

Since the 2H and 170 efg principal frames have different ori- entations in the water molecule,18 i.e., n M F ( ' H ) # QMF(I7O), it follows from eq A5 that the residual anisotropies A,(2H) and A,(I7O) are different linear combinations of the two order pa- rameters Sd and Sm2. However, as previously pointed out,'* the geometry of the water molecule and the 2H and I7O efg tensors are such that, unless IS,l >> ISm21, the first term in eq A5 is much smaller than the second one. We then have approximately

(A8) L

A,(I7O) = -2A,(2H) = - 61/2sm2

As a consequence, the reduced time correlation function g,(t), and hencej,(O), is the same for 2H and I7O, as assumed in the de- rivation of eq 21. The approximation inherent in eq A8 is readily tested, since eq A8 predicts a quadrupole splitting ratio, A U & ~ O ) / A U ~ ( ~ H ) , close to 6. This is indeed found for the system under study here (cf. the main text), although not in all other systems.43

Finally, we note that if the orientational distribution function f ( n ~ M ) possesses a t least 3-fold rotational symmetry with respect to the orientational constraint axis (zD), then it follows from eq A7 that

S m n = 6moSon (-49)

According to eq A3-A6, the reduced time correlation function then becomes

which is identical with eq 25 in the main text (with A = Ao). Registry No. AOT, 577-1 1-7; isooctane, 540-84-1.

Thermodynamics of the Formation of Benzene Series Polycyclic Aromatic Hydrocarbons in a Benzene Flame

Robert A. Alberty

Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received: July 14, 1988; In Final Form: October 19, 1988)

The equilibrium partial pressures of the successive isomer groups in the benzene series of polycyclic aromatic hydrocarbons are calculated at various levels in a near-sooting benzene flame under the ambient conditions on the basis of two mechanisms for the formation of the polycyclic aromatic hydrocarbons from benzene. The calculations utilize Legendre transforms of the Gibbs energy of formation that remove acetylene, molecular hydrogen, and hydrogen atoms from the fundamental equation of thermodynamics. Special consideration is given to the constraints on equilibrium compositions that are introduced by the mechanisms. Comparison of the calculations with experimental data shows that the observed maxima in the partial pressures of CloHs and C14H10 in the flame are in agreement with mechanism 1 and not with mechanism 2. Sensitivity analysis of the calculation indicates that uncertainties in the Gibbs energies of formation do not have a significant effect on this conclusion.

Introduction It is often said that chemical equilibrium is independent of

mechanism, but this is not true when the mechanism involves constraints in addition to element balances. In 1921 Jouguetl pointed out the advantages of calculating the equilibrium com- position using the reaction paths actually available for chemical change. Schott2 showed how the necessary constraints for the restricted equilibrium problem can be used in addition to the elemental formula coefficients in a general equilibrium computer program. B j ~ r n b o m ~ - ~ has discussed examples of restricted

( I ) Jouguet, E. J . Ec. Polyrech. (Paris) 1921, 61. (2) Schott, G. L. J . Chem. Phys. 1964, 40, 2065.

equilibria, and Smith and Missed have shown how to handle additional stoichiometric restrictions that arise explicitly and implicitly. Implicit restrictions can arise when a mechanism, rather than a maximal set of independent chemical reactions, is used in calculating the equilibrium composition. When there are con- straints in the mechanism other than element balances, these constraints have to be taken into account in calculating the composition of the equilibrium states that can be reached by that

(3) Bjornbom, P. H. Ind. Eng. Chem. Fundam. 1975, 14, 102. (4) Bjornbom, P. H. AIChE J . 1977, 23, 285. (5) Bjornbom, P. H. Ind. Eng. Chem. Fundam. 1981, 20, 161. (6) Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Anal-

ysis: Theory and Algorithms; Wiley: New York, 1982.

0022-3654/89/2093-3299$01 .SO/O 0 1989 American Chemical Society

Related Documents