p. huang ken&, carlstrbm; i. fur&* and b. halle* · 2009. 12. 25. · 9524 j. phys. chem....

9524 J. Phys. Chem. 1992, 96, 9524-9531

Counterion Spln Relaxatlon in Mlcroemulslon Droplets

P. Huang Ken&, C. Carlstrbm; I. Fur&* and B. Halle* Physical Chemistry 1, University of Lund, Chemical Center, P.O. Box 124, S-221 00 Lund, Sweden (Received: May 6, 1992; In Final Form: July I O , 1992)

Counterion 23Na spin relaxation data are reported from the microemulsion phase in the AOT/water/isooctane system as a function of the water/AOT ratio, which determines the size of the aqueous droplets. The W a NMR measurements comprise three independent relaxation rates, allowing the individual spectral density values to be determined, as well as the seumd-ordcr quadrupolar dynamic shift. The effect on the relaxation obwables of counterion diffusion within the aqueous droplet core is calculated by solving the diffusion equation in the presence of the electrostatic mean field. This allows the 23Na NMR data to be interpreted in terms of the Na' lateral diffusion coefficient D, in the surface region, the residual quadrupole coupling constant XI, and a spectral density contribution Jf due to fast local motions. These quantities have also been determined in a previous 23Na NMR study of the reversed hexagonal phase of the same system, with similar results. Although the prescnt data are basically consistent with the expected microstructure of closed droplets with highly mobile counterions, a model of monodisperse spherical droplets cannot account quantitatively for all the data. This discrepancy is tentatively ascribed to the existence of small reversed micelles coexisting with the classical microemulsion droplets.

Introduction In a typical water-in-oil microemulsion, the water is dispersed

in the form of surfactant-coated droplets whose size is controlled primarily by the water/surfactant ratio.'+ Among the many systems that form microemulsions, the ternary system AOT/ water/alkane has long been a favorite candidate for experimental study. (AOT refers to the surfactant sodium bis(2-ethylhexy1)- sulfoeuccinate.) Information about droplet size and polydispersity has come mainly from studies of tight scattering and small-angle neutron and X-ray scattering.5-'2 Droplet size, clustering, and coalescence dynamics in AOT-based microemulsions have also

kinetics,'* fluorescence quenching,1w1 dielectric spectrmpy12u3 and several other techniques.

While many aspects of this microemulsion phase are by now well characterized, relatively less is known about the internal structure and dynamics of the aqueous droplet core. Such information can be obtained by studying the spin relaxation of nuclei residing in the water molecules or the counterions contained in the droplet. In several recent studies24-26 the 2H and 170 relaxation of water in microemulsion droplets has been measured and ana- lyzed in terms of water Wbsion and droplet structure. The Z3Na spin relaxation of the counterions in AOT-stabilized microemulsion droplets was studied in an early but due to instrumental limitations and the incomplete understanding of the relaxation proc*ls at that time, the results of that study are of limited value.

In the present work we report the results of an extensive 23Na spin relaxation study of the sodium counterions in the microemulsion phase of the AOT/H20/isooctane system. By meesUring thne independent spin relaxation rates, we can determine the value of the motional spectral density function at three frequencies. Additional independent information is obtained from the second-order quadrupolar dynamic shift. These data, determined for a range of droplet sizes (core radius 2-1 1 nm) at two temperatures, allow us to decisively test various structural and dynamic models.

The relaxation data clearly reveal the existence of a dynamic process on the time scale 10-* s, which we identify as counterion diffusion within the droplet. Since the counterions are strongly accumulated near the oppositely charged AOT headgroup, it is natural to model this motion as diffusion on a spherical surface. However, this simple model is not adequate for a quantitative analysis. Numerical calculations reported here show that counterion diffusion into the core causes significant deviations from

been studied by c ~ n d ~ ~ t i v i t y , ~ ~ * ~ ~ ~elfdiff~~i~n,~~~'~~' StOppad-flOW

To whom correspondence should be addreassd. ' Rcaent address: Department of Molecular Biology, Scripps Research

$On leave from the Central Rtsearch Institute for Physics, B u d a p t , Institute, 10666 North Torrey Pines Road, La Jolla, CA 92037.

Hungary.

sample xw* 4b b/nmc Pd 1 9.4 0.087 2.11 0.90 2 17.7 0.108 3.42 0.86 3 28.3 0.133 5.12 0.84 4 34.0 0.147 6.04 0.83 5 43.6 0.168 7.58 0.82 6 48.2 0.177 8.31 0.82 7 61.7 0.204 10.5 0.81 8 65.6 0.212 11.1 0.8 1

'Molar ratio H,O/AOT. bDroplet volume fraction at 20 O C (H20 + AOT). 'Aqueous core radius for monodisperse spherical droplets, calculated according t01-*2~19~20 b/nm = 0 . 1 6 ~ ~ + 0.6. dFraction counterions (at 20 "C) within 6 = 0.5 nm of the spherical interface, calculated from eq 14.

the predictions of the surface diffusion model. Even when this complication is taken into account, however, the data are not fully consistent with a microstructure of monodisperse spherical dpplcts. Neither a unimodal droplet size distribution nor droplet shape fluctuations can remove the inconsistency. Rather, the data suggest a bimodal size distribution, where microemulsion droplets coexist with small reversed micelles. Such a distribution has previously been invoked to explain small-angle X-ray scattering and kinetics data.43 Finally, we compare the results of this study with those of a recent 23Na spin relaxation study of the reversed hexagonal phase of the same system.29

Materials and sample Repntioa AOT (sodium bis(2-ethylhexyl) sulfosuccinate) from Sigma

and isooctane (2,2,4trimethylpentane) from Aldrich (99%) were used as supplied. The water was Millipore-filtered H20.

Microemulsion samples were made by weighing the components into 'I-mm-i.d.-Pyrex tubes, which were then flamesealed. All samples were made from a stock solution of molar ratio isooctane/AOT no = 35.0, to which water was added to obtain the

for the investigated samples are given in Table I. Figure 1 shows the relevant part of the x r T phase diagram

(at fmed xo = 35.0), with the position of the inveatigated samples indicated. The boundarim of the onaphase microemulsion region were established by visual inspection and agree closely with previous results for this system.'* The two-phase region at low temperature contains microemulsion in equilibrium with e x a s water (or normal micellar phase), whereas the two-phase region at high xw, at least at 10 and 20 O C , contains microemulsion in equilibrium with lamellar liquid-crystalline phase. The samples marked by open circles were first thought to be in the onaphase microemulsion region but, on closer inspection, turned out to be

desired mdar ratio HzO/AOT, denoted XW. The cOmpaPition data

0022-3654/92/2096-9524$03.00/0 63 1992 American Chemical Society

23Na NMR in Microemulsions The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 9525

100

80

60

X W

40

4 1 i 20[ : , : I 1 0 0 5 10 15 20 25 30

T/"C Figure 1. Apparent stability region of the microemulsion phase (L2) in the AOT/H20/isooctane system with xo = 35.0. In the shaded two- phase regions the " u l s i o n phase is in equilibrium with excess water (L2 + aq) or with a lamellar phase (b + D). In the latter case, the (dashed) phase boundary is only apparent; careful examination re- vealed that the samples marked by open circles are biphasic. Samples marked by solid circles were found to be homogeneous.

biphasic. This was established either by extended centrifugation (several days) leading to phase separation. or by a significant deviation from the biexponential transverse 23Na relaxation expected for a homogeneous microemulsion sample. In the former case, the bottom phase had a texture (observed in a polarizing microscope) characteristic of a lamellar phase, but the sample reverted spontaneously to a clear solution on a time scale of days. Whereas the phase-separated sample exhibited a uNa quadrupole splitting of ca. 20 Wz, no splitting could be detected from the clear sample, indicating that the dissolved lamellar microcytallites are small (lateral dimensions less than ca. 0.2 pm).

Diffusion Measurements To ensure that the microemulsion samples investigated by uNa

NMR consist of discrete water droplets rather than having a bicontinuous topology, we performed water diffusion measurements on several samples using the 'H pulsed-field-gradient spin-echo technique.30 Since this experiment measures the ma- croscopic water diffusion coefficient, corresponding to root- mean-square displacements of the order 1od m, it readily dis- tinguishes between closed droplets and a bicontinuous microstructure.

The samples, contained in 4-mm4.d. glass ampules, were located in a horizontal solenoid probe centered in a quadrupole gradient coil providing an essentially homogeneous magnetic field gradient of ca. 4 mT an-'. The conventional spin-echo pulse sequence with two gradient pulsesm was used to record the intensity of the water 'H signal with increasing gradient pulse length. The 180' pulse length was ca. 12 ps at a 'H Larmor frequency of 100.13 MHz. The separation between the gradient pulses and the 180' pulse or the echo was at least 10 ms; no phase distortions were seen in the spectra. The water self-diffusion coefficient D in the microemulsion sample was determined from a three-parameter nonlinear least-squares fit. (The other parameters were the initial intensity and the base line.) No significant changes in D were found by varying the delay between the gradient pulses from 20 to 200 ms, indicating unrestricted diffusion on the micrometer length scale.

The results of diffusion measurements on samples 4,5, and 7 are shown in Figure 2. A transition from closed droplets to bicontinuous topology is evident for sample 4 above 40 OC as the boundary of the one-phase region is approached. At 10 and 20 OC, however, all the samples investigated by 23Na NMR showed water diffusion coefficients characteristic of closed droplets. In fact, the measured water diffusion ooefficients deviate by less than 20% from the theoretical value D = (1 - 2+)kBT/(6q&) for

I I 4 " I - j, 1 % 4 t

2 t @ i

0 1 I- 0 -I p . b . @ o @ @

O i Po i o i o i o sb i o T PC

Figure 2. Temperature dependence of water self-diffusion coefficient D in microemulsion samples 4 (O), 5 (0), and 7 (A).

hard spheres of radius R = 6 + 1.0 nm at volume fraction + (cf. Table I) in a medium with the viscosity vo of isooctane.

fjNa Spin Relaxation Measurements The 23Na NMR measurements were performed on a Bruker

MSGl00 spectrometer equipped with a 10-mm vertical s a d d l d probe and a 2.35-T (26.485-MHz resonance frequency for 23Na) wide-bore superconducting magnet.

The sample volume (ca. 15-mm height in the Pyrex tube) was centered in the coil yielding a spatial rf (B1) inhomogeneity of ca. &lo%. The static magnetic field inhomogeneity was minimized by shimming on the narrow 'H spectrum; the resulting inhomogeneous broadening in the 23Na spectrum was typically ca. 2 Hz. The temperature was controlled by a Stelar VTC 87 regulator with high airflow (1.5 m3/h), yielding a temperature stability of f0.03 O C . The temperature gradients within the sample are estimated to be less than h0.03 OC. The 180' pulse length (typically around 12 ps) was determined from the zero crossing of the signal with varying pulse length. The 90' and 63.4' pulse lengths were determined by extrapolating from 180' and 360' pulse lengths. Free Induction Decay. When rccarding the free induction decay

(FID) signal, the following steps were taken to minimize errors. First, spurious (ghost) signals due to receiver imbalance were suppressed (i) by fine-tuning the amplitude and phase balance of the receiver and (ii) by using the CYCLOPS phase cycle.31 The unsuppressed spurious signals were measured in an off-re- sonant experiment and were found negligible (S0.296). Second, the aquisition delay was set to 230 ps. A good suppression of any rf-coherent ringing was accomplished by varying the filter width, pulse length, and acquisition delay. The 230-ps delay also ensures that the 23Na signal from the Pyrex tube has decayed completely before signal aquisition starts. The filter width (100 W z ) was chosen so as to cause negligible distortions in the pass band.

To achieve a satisfactory precision, 48 000-96 OOO scans were collected with 120-ms repetition time. (To avoid sample heating, no shorter repetition times were used.) The full spectral width was 71.4 or 83.3 Wz, and 1600-2400 points were digitized in the decay. (The signal was recorded in sequential digitization mode; i.e., data points were alternately digitized in the "real" and "imaginary" channels. This has been taken into account when fitting the data.) A typical FID is shown in Figure 3. The obtained signal-tu-noise ratio (calculated as the maximum intensity of the signal in the "real" channel divided by one standard deviation characterizing the assumedly Gaussian noise) was 300-600.

The FID signal S(t) following the single-pulse excitation of an I = 3/2 spin system consists of two components of fixed relative a m p l i t ~ d e s ~ ~

The transverse quadrupolar relaxation rates of the slowly and rapidly decaying components are denoted by R2- and R2+, respectively. The two decaying signal components are modulated

9526 The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 Huang KenQ et al.

Longitudinal Relaxation. These experiments were carried out with a modified inversion recovery sequenceKx with a detection pulse angle of 63.4". This particular pulse angle cancels the coherence transfer from octupole polarization, which develops under nonextreme narrowing conditions, to the detected single- quantum coherence. Since this experiment exclusively monitors the evolution of the dipole polarization, the line shape does not change with delay time and the two longitudinal relaxation rates R1- and R1+ can be obtained from a double-exponential fit with fixed amplitudes according to

I I 3 1

I I 0 5 10

t /ms Figure 3. A typical FID signal recorded on sample 6 at 10 OC.

TABLE U "Na Relamtion Rates rad Dynamic Shift"

9.4 1264 f 20 17.7 778 f 12 28.3 564 i 9 34.0 495 f 7 43.6 437 f 6 48.2 419f 5 61.7 379 f 5 65.6 373 f 5

9.4 9 1 4 i 14 17.7 5 9 0 f 8 28.3 432 f 6 34.0 378 f 5 43.6 329 f 4 48.2 314f 4

1372 f 25 1960 f 35 260 h 40 852 f 15 1268 f 20 240 f 30 616 f 10 1106 f 12 266 f 20 5 4 0 * 9 1 1 1 7 f 12 2 5 9 f 2 0 471 f 9 1197 f 13 249 f 20 453 f 9 1256 f 13 237 f 20 402 f 9 1509 f 15 226 f 25 401 i 8 1614f 15 232f 25

T = 2 0 ° C 980 f 20 1340 * 30 130 f 25 645 i 10 914 f 12 154 f 12 4 7 7 i 8 837f 1 1 1 8 3 f 7 4 2 4 * 8 916f 12 2 0 2 i 8 368 f 8 1105 * 15 220 i 15 354 f 6 1280 f 20 230 f 20

OAll quantities in units of s-l.

by the second-order quadrupolar dynamic shifts and a+. These four quantities are related to the one-sided Fourier transform of the time correlation function of the fluctuating electric field gradient at the nuclear site, as described in the following.

The data in the "real" and "imaginary" channels were fitted simultaneously using the Levenberg-Marquardt nonlinear least-squares alg~rithm.'~ The fit provided the relaxation rates R2+ and R< as well as the relatiue dynamic shift

(2)

The results for the single-phase microemulsion samples are collected in Table 11. The relaxation rates have been corrected for the small magnetic field inhomogeneity by subtracting off the inhomogeneous 'H broadening scaled down to the =Na frequency. (For the small inhomogeneous broadening encountered here a spin-echo experiment, which refocuses this broadening, was not considered advantageous, since the longer experimental time in- troduces systematic errors due to spectrometer instabdity.) The experimental uncertainties given in Table I1 include the random error (k2a) from the fit as well as the uncertainty (4-6 s-') introduced by the approximate inhomogeneity correction. The random error was established by Monte Carlo simulation. The FID fit involved five additional parameters of no physical interest: the absolute intensity S(O), the frequency offset, the receiver phase, and the base line offsets in the two channels. The FID's were recorded with a receiver phase that directed most of the signal into the 'real" quadrature channel. Although the "imaginary" channel contained a relatively small signal, a simultaneous fit of both channels substantially increased the accuracy of the dynamic shift, which otherwise has large covariances with the frequency offset and the receiver phase. The effect of relaxation during the rf pulse was checked and found to be negligible. Accordingly, we held the relative amplitudes in eq 1 fixed (0.4 and 0.6) in the fit. (Monte Carlo simulations demonstrate increased systematic errors in the relaxation rates when the relative amplitudes are treated as adjustable parameters.)

ndyn = n+ - n-

Since R i and Rl+ do not differ much, we actually performed a single-exponential fit according to

with an effective longitudinal relaxation rate S(T) A(1 - 2 exp(-Rl*~)] + B (3b)

(The intensity A and the base line B were parameters of no physical interest.)

Monte Carlo simulations showed that the (random and systematic) error in Rl*, obtained from eq 3b, was smaller than the random errors in the individual rates obtained from a double- exponential fit to eq 3a. Furthermore, the simulation results could be used to correct for the small (<1%) systematic error in R1*. The resulting Rt* values are given in Table 11.

The spectrometer setup for the longitudinal relaxation experiments was the same as for the FID measurements. A total of 4000-8000 scans were recorded at each of 35-40 delay times T.

To optimize precision, the T values were distributed more densely around the expected l/R1*."

Quadrupole Relaxation 23Na is a spin Z = 3/2 nucleus with a large quadrupole moment,

and under the conditions of the present study, the relaxation of the 23Na spin system is due entirely to the fluctuating quadrupole coupling. In an isotropic system, like the microemulsion phase investigated here, the spin dynamics is then governed by the complex-valued spectral density function K(o), which is the one-sided Fourier transform of the time autocorrelation function G(T) of the fluctuating quadrupole coupling

K(w) = JmdT exp(iwr)G(r)

The real part J(o) gives rise to spin relaxation, while the imaginary part Q(o) is associated with the second-order dynamic ~hift.~*J*

The maximum amount of information that can be derived from relaxation experiments at a fixed magnetic field consists of the five quantities J(O), J(oo), J(2w0), Q(wo), and Q(2wo), where wo is the Larmor frequency. The three single-quantum relaxation rates RI*, R2+, and R2- and the relative dynamic shift SI,,, are related to these quantities as32,39

2 8 R'* = -J(wo) 5 + 3J(2w0) ( 6 4

R2- = J(u~) + J ( 2 ~ 3 ) (6b) R2+ = J(0) + J(oo) (6c)

nap = 2Q(ao) - Q ( b o ) (7) To separate Q(wo) and Q(2wo), we would also need to determine the double-quantum or triple-quantum dynamic ~ h i f t . ~ ? ~ '

The real spectral densities derived from the relaxation rates are shown as functions of the composition variable xw in Figure 4. As previously observed in the water ZH and 170 relaxation?6 J(0) exhibits a minimum as a function of droplet size. In contrast, J(oo) and J(2wo) decrease monotonically with increasing droplet size and appear to level off at large xw. Remarkably, the difference J(wo) - J(2wo) is nearly constant at 20 'C, although J(0)

= J(w) + iQ(w) ( 5 )


7 1.0 E . 2 - 24

0.5

TABLE IE Parameters in the Spherical Droplet Model Derived from Real Spectral Densities4 10 OC 20 OC

xw JfJS-' X / W Z %t/ns JfJP' X / M Z %t/nS 9.4 495 f 50 220 f 28 7.2 f 1.4 373 f 37 172 f 27 7.2 f 1.8

17.7 295 f 28 182 f 19 7.4 f 1.2 222 f 22 157 f 17 6.6 f 1.1 28.3 228 f 15 157 f 12 10.9 f 1.4 168 f 12 145 f 10 10.3 f 1.2

11.8 f 1.2 34.0 201 f 12 154 t 10 13.0 f 1.5 142 f 10 150 f 9 43.6 186 f 10 147 f 11 17.6 f 2.4 128 f 7 152 f 8 16.6 f 1.7 48.2 177 f 9 150f 11 18.6 f 2.5 120 f 7 161 f 7 18.5 f 1.5 61.7 169 f 8 144 f 14 27.4 f 5.3 65.6 160 f 8 156 f 11 25.7 f 3.5

"The given errors refer to propagated random errors in the spectral densities.

-

-

varies substantially. This intriguing observation, which will turn out to be important in the quantitative analysis of the data, is supported by relaxation measurements at a higher magnetic field (*"a Larmor frequency 95.70 MHz). Although those data were deemed of insufficient accuracy to be included in the quantitative analysis, they clearly reveal a frequency dependence in J(uo) and J(2oO) suggesting a correlation time of the order s for large droplets (xw = 30-60).

Surface Diffusion in Spherical Droplets To extract molecular-level information from the real spectral

densities and the dynamic shift, we need to formulate a model for the field-gradient time correlation function G(r). We begin by assuming that the microemulsion droplets are spherical and monodisperse in size. Under certain conditions (cf. below), G(T) then takes the simple form

(8) where the numerical constant r2/5 pertains to a spin I = 3 /2 nucleus.

The first term in eq 8 accounts for fluctuations of the magnitude and orientation of the electric field gradient (efg) at the Na+ nucleus due to fast local motions, such as small-amplitude dynamics within the Na+ hydration shell and local diffusive motions of ionic species within the interface r e g i ~ n . ~ ~ . ~ ~ We make no assumptions about the functional form of GLT) but require only that it decays on a time scale that is short compared to the Larmor period 1 / W O = 6.0 ns. (This condition is referred to as extreme narrowing.)

In the second term of eq 8, x denotes the residual quadrupole coupling constant (qcc) averaged by the fast motions. The orientation of the residual efg tensor fluctuates as a result of lateral diffusion of the counterion at the interface of the spherical droplet. This process gives rise to an exponentially decaying contribution to G(T) with a correlation time

where 6 is the radius of the aqueous droplet core (cf. Table I) and D8 is the lateral diffusion coefficient of Na' in the interface region. The modeling of counterion diffusion within the droplet as diffusion on a spherical surface is a simplification. However, a more realistic treatment, presented in the next section, shows that the exponential form in eq 8 is extremely accurate and that only the interpretation of x and r8 needs to be revised.

Rotational diffusion of the entire droplet, with correlation time T ~ , also contributes to the decay of the second term in eq 8, but as rm is an order of magnitude longer than rkt even for the smallest investigated droplets, we may write

G(T) = G,-(T) + ( r Z / 5 ) x 2 exp(-r/r8)

7bt = b2/6Ds (9 )

7 1.0 1 .

Tlat Trot

Tlat + Trot = rlat 7, = -

-

The real spectral density function corresponding to the time correlation function in eq 8 is

The three parameters Jf, x, and skt in this model can be calculated from the real spectral densities J(O), J(q), and J(2w0) shown in Figure 4. The results are c o U d in Table 111. Before discussing

1.5 1 I

i 0

10°C

0.0 ' I 0 10 20 30 40 50 60 70

XW

1.5 I I , I I I 1

t 0

0

a n a

J(h$

2O0C

0

J(%) a 0

0.0 ' 8 I I 1 0 10 20 30 40 50 60 70

XW

FIgm 4. Variation of the 23Na spactral densities J(O), J(q,) , and J(&) with the H20/AOT mole ratio xw at 10 O C (top) and 20 OC (bottom).

these results, we shall examine the validity of the model underlying eq 11.

Counterion Musion in Microemulsion Droplets Although the Na+ counterions are strongly accumulated near

the spherical droplet interface, their diffusive motion is in fact threedimensional rather than two-dimensional, as assumed in the previous section. We now investigate the consequences of relaxing this constraint. We model the microemuhian droplet as a dielectric medium of relative permittivity p bounded by a spherical surface carrying a uniform surface charge density -e/u, where u is the surface area available to an AOT surfactant molecule. The counterions are treated as point charges. If the spatial correlation among the counterions is neglected, the potential of mean force w(r) can be obtained by solving the Poisson-Boltzmann equa- t i ~ n . ~ ~ , ~ As this equation must be solved numerically for the present geometry, we shall use a highly accurate analytical a p proximation to the Poisson-Boltzmann In units of kBT

I" is a dimensionless coupling parameter $b r = -

UWrkBT The fraction, P, of counterions residing within a distance 6 of the charged interface is obtained from eq 12 and Gauss' law as

9528 The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 Huang Ken& et al.

4(1 -a3 r + 4 - r ( i - I ) ~

P - 1 -

where Q = 6/b. The fraction P (with 6 = 0.5 nm) for the investigated droplet sizes is given in Table I.

The contribution from counterion diffusion to the time correlation function G(T) arises because (i) the magnitude of the residal qcc x fluctuates as the ion moves radially and (ii) the orientation of the principal frame for the residual efg tensor fluctuates as the ion moves laterally. Although the first contribution turns out to be negligible (cf. below), radial motion has a significant indirect effect on the second contribution. This effect is twofold: first, the orientational randomization is speeded up by allowing the ion to diffuse through the droplet core (the diffusion path is shorter and the diffusion coefficient is higher), and second, the effective residual qcc & that is modulated by lateral motion is reduced by radial averaging. As the residual efg is of short range," we model the residual qcc as a step function, taking the value % in a surface shell of thickness 6 and vanishing elsewhere.

The contribution to G(T) from counterion diffusion, corte- sponding to the second term in eq 8, can be expressed asM

5 g(r,rlro) = i l l l d t o P 2 ( t 0 ) S 1 d t - I P 2 ( t ) A r , t , ~ l r ~ , t o ) (15b)

where P2(O = (3t2 - 1)/2, t = cos 8, and B is the angle between the current radius vector of the ion and the external magnetic field. (As noted above, droplet rotation can be neglected since it is too slow to compete with ion diffusion.) Further&) andf(r,{,Tlr&) are the normalized equilibrium counterion distribution and mter ion diffusion propagator, respectively. The former is related to the potential in eq 12 asfir) - exp[-w(r)], while the latter is taken to satisfy the mean-field diffusion equationM

where D d and ht are the radial and lateral components of the counterion diffusivity tensor.

The numerical evaluation of the time correlation function G,(7), or its one-sided Fourier transform, according to eqs 15 and 16 can be reduced to a matrix eigenvalue problem by discretizing the radial coordinate.& For simplicity we set Dnd = DO (the limiting Na' diffusion coefficient) uniformly, while ht = D, in the surface shell (within a distance 6 of the spherical interface) and ht = Do elsewhere. (The effect of having Dnd = D, in the surface shell is very small.)

Figure 5 shows the calculated real spectral density contribution J,(w), i.e., the cosine transform of G,(T), for droplet sizes corresponding to samples 2 and 6. For these calculati~n~ we have used 6 = 0.5 nm, u = 0.60 nmZ, and D,/Do = 0.3. To a remarkably high degree of accuracy, the spectd density function is Lorentzian. There is a very small deviation from Lorentzian shape, hardly visible in Figure 5, in the form of a high-frequency tail in the w range 109-10L0 rad 6' associated with fluctuations in the magnitude of the residual qcc due to radial counterion diffusion. The initial value of the time correlation function in eq 15 is G,(O) = (?rz/S)~,~p, which follows directly from the initial condition on the propgator: Ar.t,Okottd = r-26(mJ Nt- 6). m p o n d i n g to the high-frequency tail in the spectral density function there is a fast initial decay of GS(7) in the range 10-10-10-9 s, which reduces its value to ( d / 5 ) ~ ~ . Since the contribution from this initial decay to the spectral density function is negligible, we may write to an excellent approximation

The effective residual qcc falls in the range GAT) = ( r 2 / S ) x : exp(-7/7J (17)

(18) Pxs I xc I P'/2x,

20

15

z 3 B

. 10 - 5

n

w I rad s-l

I .

I 1

I I I I I I I I I I

Sample 6

107 to8 w, 2w0 109 w /rad s-I

Figure 5. Reduced spectral density function Jd(w) = J , (w) / (*2b2 /5 ) for counterion diffusion within spherical droplets of size corresponding to samples 2 (top) and 6 (bottom). The curves refer to numerical calculations based on cqs 15 and 16, while the dots corrcppond to the exponential correlation function in q 17.

Numerical calculations show that, in the present system, we are close to the lower limit of &, which corresponds to the case where radial averaging of the qcc magnitude is much faster than orientational averaging by lateral diffusion.

Since G,(r) in eq 17 is of the same form as the corresponding term in eq 8, we can simply reinterpret the quantities x and rht in Table I11 as the effective quantities xc and T ~ . The latter quantities can then, by means of numerical calculations based on eqs 15 and 16, be related to the two model parameters x, and D,, provided that the third unknown model parameter 6 is specified. Figure 6 shows the result for a reasonable choice of 6 = 0.5 nm. For this particular 6 value, the given T~ value for the smallest droplet (sample 1) is too large to be accounted for by the model. (The upper limit of re corresponds to D, = 0, when orientational averaging takes place exclusively by lateral diffusion in the core with diffusion coefficient D,,.) A change in 6 by a factor 2 produces a ca. 10% variation in x,, whie D, is virtually unaffected for the largest droplets but changes by as much as a factor 2 for the smallest ones. Although the absolute values of and D, are thus somewhat uncertain, an important conclusion can be drawn from Figure 6. Irrespective of the choice of 6 value, there is a strong increase of D, with droplet size with no sign of a leveling off even for the largest droplets with a diameter of ca. 20 nm. This behavior is difficult to understand and leads us to question the model of monodisperse spherical droplets. This conclusion is supported by the analysis of the dynamic shifts in the next section.

Dynamic shifts The time correlation function defined by eqs 8 and 17 gives

rise to a relative dynamic shift &,, which, rtccording to eqs 5 and 7, takes the form

with no contribution from the fast local motions which are in the extreme narrowing limit. According to q 19, n d y n increases

23Na NMR in Microemulsions The Journol of Physical Chemistry, Vol. 96, No. 23, 1992 9529 Mo 1 I ! I I ,

400- 10°C

\

8 . 2 0 0 -

1 .o

0.8

0.6

0.4

0.2

0.0

250 r I I I 1

50 "1 XW

F¶gure 6. Variation with droplet size of the Na+ diffusion coefficient D, in the surface layer (6 = 0.5 nm) relative to the bulk Na+ diffusion coefficient Do (top) and the residual qcc in the swface layer (bottom). Filled symbols refer to 10 O C and open symbols to 20 O C . These results were obtained from the measured real spectral densities by numerically calculating the spectral density for countcrion diffusion within a spherical droplet according to eqs 15 and 16.

monotonically with the correlation time T, toward the static limit (for o07, >> 1)

This is simply the isotropic orientational average of the relative static second-order quadrupolar ~ h i f t . 4 ~ 9 ~ Unlike the real spectral density function J(o), the dynamic shift thus provides information (about x,) even in the limit WOT, >> 1.

The quadrupolar dynamic shift has long been regarded as a theoretical Curiasity, and it is only in the p t few years that actual measurements have been perf0rmed."'8'~*~ Yet, if accurately determined, the dynamic shift can be a valuable source of information. In our case the dynamic shift thus provides a check on the spherical droplet model and, indeed, of any model giving rise to a time correlation function consisting of one term in extreme narrowing and another exponentially decaying term. Figure 7 shows the measured dynamic shifts from Table I1 along with f& calculated from eq 19 with xs and T, (listed as x and 7kt in Tabc 111) as obtained from the real spectral densities. The spherical droplet model clearly fails to account quantitatively for the ob served dynamic shifts.

Beyond tbe spherical Droplet Model The model considered so far involves three adjustable param-

eters: the fast-motion spectral density Jf , the effective residual qcc xe, and the effective correlation time T,. There are two problems with this model. Fit, it is not quantitatively consistent with the four measured quantities J(O), J(oo), J(2uo), and Qd, for any sample (xw) at any of the two investigated temperatures (cf. Figure 7). Second, the Na' diffusion coefficient 0, in the surface region, deduced from T, by. numerically solving the diffusion problem for a counterion subject to a realistic electrostatic potential of mean force, increasca linearly with droplet size without

" 0 10 20 30 40 50 60 70 X W

O b io io 30 40 sb i o ,b X W

Ftprpe 7. Variation with droplet size of the measured (0) and calculated (0) dynamic shift &, at 10 O C (top) and 20 O C (bottom). The calculated Qdya was obtained from the real spsctral densities by way of cq 19 and the parameter values in Table I11 deduced within the spherical droplet model.

any sign of approaching a limiting value for large droplets (cf. Figure 6). This second discrepancy can be traced back to the unexpectedly small variation with droplet size of the difference J(oo) - J(2w0) (cf. Figure 4), which causes the calculated 7, to increase linearly rather than quadratically with the droplet core radius 6.

In this section we discuss several modifications of the thrce- parameter model for J(o) and Qya, based on plausible deviations of the microstructure from the idealized picture of monodisperse spherical droplets. As these more sophisticated models involve several additional parameters, the present 23Na NMR data do not justify a detailed quantitative analysis. Rather than attempting to fit additional parameters, our aim will be to examine whether certain microstructural features can explain our data without invoking unphysical variations in the microscopic model parameters

Shupe Flucruathns. In most previwS studies of AOT/water/oil " u l s i o n s , experimental data have been interpretad in terms of spherical droplets, although direct information about the shape is rarely available. However, shape fluctuations have been im- plicated in the m a t recent scattering studies,11J2 while more direct evidence has come from recent e le~t roopt ica l ,~~*~~ neutron spin echo,s3.s4 and water ?H and "IO spin relaxationM studies.

The effect of droplet shape fluctuations on the counterion spin relaxation depends crucially on the relative time scala of shape fluctuations (~t f ) , droplet rotation (SA, and counterion diffusion (T,). To gain a qualitative understanding of the effect of shape fluctuations, we consider three limiting cases: (i) the static deformation case with T~ << T~ << ~ d , (ii) the slow fluctuation case with T, << 7,f << T ~ ~ ~ , and (iii) the fast fluctuation case with r,f I 7, << Tm.

The most pronounced effect on the spin relaxation occurs in the static deformation c8se. Since a typical droplet is nonspherical on the time scale of counterion diffusion, the time correlation function G(T) acquires a longtime tail whose decay reflects the reorientation of the entire (nonspherical) droplet. Since T~ >> T,, even a small deviation from spherical shape can give a significant contribution to J(0) . If the droplets are modeled as

(4, XI, and 0,).

9530 The Journal of Physical Chemistry, Vol. 96, No. 23, 1992 Huang Kenbz et al.

monodisperse prolate or oblate spheroids, only one additional parameter is introduced: the axial ratio p of the aqueous core. It is thus possible to determine the four model parameters Jf, xe, De, and p from the four quantities measured for each sample. Using a recently developed numerical approach,55 we have camed out this analysis. & and De were then rescaled to XI and D, values on the basis of numerical calculations (for spherical geometry) of counterion diffusion as described above. The obtained Jf and XI are within ca. 10% of their values for the spherical droplet model (cf. Table I11 and Figure 6), and the axial ratio is ca. 1.3 for all droplet sizes irrespective of whether a prolate or oblate spheroidal shape is assumed. While the additional parameter (p) obviously removes the inconsistency between spectral densities and dynamic shift, the problem with the droplet size dependence of D, remains. Since rw increases essentially as xw3, the additional contribution to J(0) from droplet rotation causes D, to increase even more strongly than in Figure 6 until, absurdly, it exceeds the bulk value Do for the largest droplets. The slow fluctuation case26*56,57 is qualitatively similar to the static deformation case, except that shape fluctuations take the place of droplet rotation as the isotropic motion. We conclude, therefore, that the 23Na NMR data cannot be satisfactorily accounted for in terms of static deformation or slow shape fluctuation.

The failure of the static deformation and slow fluctuation limits to account for our data is due to the condition re << r,f. With this constraint, shape fluctuations (and droplet rotation) affect only J(O), and the virtually unaffected difference J(oo) - J(2u0) then forces a strong droplet size dependence on D,. On the other hand, the effect of fast shape fluctuations on J(wo) - J(2w0) is again negligible. Since shape fluctuations are of small amplitude?'-% they can only cause a minute variation of the orientation of the local residual efg tensor. An effect on the spin relaxation [J(O)] in the static deformation and slow fluctuation limits is then due to the long correlation time, while a small fluctuating coupling with a short (r,f I re) correlation time, as in the fast fluctuation limit, has no significant effect on the spectral densities. In conclusion, it appears that droplet shape fluctuations (irrespective of their time scale) cannot resolve the problem of the droplet size dependence of 0,.

Size Polydispersity. Recent studies indicate a polydispersity in droplet radius of 1 C k m in AOT/water/oil microemulsions."~'* The exchange of counterions between different droplets, mediated by a droplet fusion-fission takes place on a time scale of s in the present system. Since this is fast compared to the difference between the 23Na spin relaxation rates of droplets of different size, the effect of size polydispersity is to average the time correlation function G(r) (or the spectral density) over the droplet size distribution. With a polydispersity of 10-2096, the effect is not expected to be significant.

An extreme form of size polydispersity is a bimodal distribution where traditional droplets coexist with essentially anhydrous small reversed micelles. The existence of such small micelles has been proposed to explain scattering and kinetics data from AOT/ water/oil microemulsions.43 In the system AOT/D20/n-heptane at 20 "C, with xw = 20 and xo = 135 it was inferred that ca. 15% of the AOT was present in the form of small There is also some theoretical support for the existence of small micelles, stabilized by the entmpy of mixing.% A simple equilibrium model yields for the fractionfof counterions in small reversed micella

where N , is the number of AOT molecules in a reversed micelle and p, is the electrostatic contribution to the chemical potential (in units of kBT) of an AOT molecule in a microemulsion droplet. An accurate Pad5 approximant representation of p, as a function of the electrostatic coupling parameter r (which de nds on xw via the droplet radius b) in eq 13 has been given.r Since the electrostatic interaction favors large droplets (-p, increases with xw), the fractionfdecreases with increasing droplet size. On the basis of eq 21 and the citedfvalue,28 we estimate for our samples f = 0.04 at xw = 20 decreasing to 0.03 (or less if N , > 1) at xw = 70.

A model of Na' ions in fast exchange between droplets and small reversed micelles contains at least six parameters: Jf , xo, ref , xm, and r,, where the last two quantities refer to the micelles. While a quantitatively meaningful analysis cannot be performed with so many adjustable parameters, it is interesting to examine certain qualitative features of the model. First, assuming that the structure of the small micelles is indepndent of xw, the nearly constant difference J(wo) - J(%) can be accounted for provided that 7, is of the order l/wo c3 6.0 ns, which is not unreasonable. If we assume, at least for the largest droplets (xw = 65.6), that counterion diffusion is too slow to contribute to J(wo) - J(2w0) (i.e., wore >> l), then for f = 0.02 and 0.5 < worm < 5 we obtain the reasonable values 800 kHz < xm < 1200 kHz. (A subtantially enhanced residual coupling in the micelles is expected if the counterions are incompletely hydrated.) With the J(wo) - J(2uo) difference accounted for by the small micelles, we can estimate the effective correlation time for counterion diffusion by plausibly assuming that the contributions from the small micelles to J(0) - J(2w0) and to ad,,,, are negligible for large xw

(22)

Setting re = b2/(6De) and relating the effective Na+ diffusion coefficient De to the actual diffusion coefficient D, in the surface layer as described above, we find D, values roughly a factor 2 smaller than those in Figure 6 for large xw, and with a much smaller dependence on xw. For samples 5 and 6, we obtain 4 / 0 0 ;z: 0.35 at 10 OC and 0.18 at 20 OC. Comprlsan with the Reversed Hexagonal Phase

It is of interest to compare the parameters estimated from this study of the microemulsion phase with the parameter values deduced in a recent 23Na NMR study of the reversed hexagonal (F) phase of the same system (but with D20) with xw in the range 18-27 (aqueous cylinder radius 2.3-3.1 nm).

The fast-motion contribution, Jf, found here using the spherical droplet model (cf. Table 111) is nearly the same as in the F phase at the same temperature and xw value. (The water isotope effect on Jf is complex, but probably small.) If 7, k l/wo, similar Jf values as in Table I11 are obtained for the bimodal droplet model.

The residual qcc in the F phase was xs = 190 f 10 kHz with no clear-cut xw dependence in the range 18-27. In the microemulsion phase we obtain, for the spherical droplet model, x, = 180 f 20 kHz with no significant xw dependence (cf. Figure 6). For the bimodal droplet model, however, smaller xa values are obtained.

In the F phase we found D,/Do = 0.28 f 0.03 at 20.6 OC with no apparent xw dependence in the range 18-27. (Numerical calculations show that the effect of radial counterion diffusion on the spectral density is negligible for the cylindrical geometry and under the conditions of the F phase study.) This is larger than the D,/Do value obtained here at the same xw and temperature but smaller than the values obtained with the spherical droplet model for the largest droplets (cf. Figure 6).

COaChlSiOaS

The present study shows that accurate 23Na spin relaxation data from the counterions in a microemulsion can give information about dynamic proceases and about microstructure that is difficult to obtain by other methods. Despite some problems with the detailed quantitative interpretation, the data are basically consistent with a microstructure of closed spherical droplets with highly mobile COUIltCriOhP. The close agreement of the fast-motion spectral density Jr and the residual qcc x, between the present microemulsion phase and the reversed hexagonal phase. in the same system indicates that the local structure and dynamics at the interface is closely similar in the two systems.

The intqwetation of spin relaxation data from isotropic complex fluids is always difficult since, in contrast to anisotropic fluids, only a single spectral density function is probed. In the present work this difficulty was partly alleviated by measuring also the dynamic shift related to the imaginary part of the complex-valued

3 J(0) - J(2wo) = 2 won*y"


spectral density function. The accuracy of the data allows us to identify deviations from the simple spherical droplet model, which we tentatively ascribe to a dynamic equilibrium between classical microemulsion droplets and small mersed micelles. More definite, and quantitative, conclusions about this matter would require more extensive data, in particular a study of the magnetic field dependence of the spin relaxation rates and dynamic shift.

Acknowledgment. This work was supported by grants from the Swedish Natural Science Research Council.

Registry No. AOT, 577-11-7; Na, 7440-23-5; isooctane, 540-84-1.

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p. huang ken&, carlstrbm; i. fur&* and b. halle* · 2009. 12. 25. · 9524 j. phys. chem....

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