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Micelle size and order in lyotropic nematic phases from nuclear spin relaxation

Per-Ola Quist, Bertil Halle, and IstvAn Fur6 Physical Chemistry I, University of Lund, Chemical Center, P.O. Box 124, S-221 00 Lund, Sweden

(Received 7 August 199 1; accepted 4 November 199 1)

Nuclear magnetic resonance (NMR) relaxation of quadrupolar nuclei is introduced as a new method for determining micelle size and nematic order in lyotropic nematic mesophases from the dependence of the spin relaxation rates on molecular diffusion over the curved micelle surface. The approach is illustrated by an experimental study of two uniaxial nematic phases (the calamitic Nc and discotic ND phases of the sodium dodecyl sulphate/decanol/water system) using two nuclei: 2H in the a-deuterated surfactant and 23Na in the counterions. The two nuclei yield similar results: an apparently temperature independent axial ratio of 34 in both phases and a nematic order parameter which decreases from ca. 0.9 (0.75) at the lowest temperature to ca. 0.6 (0.5) at the highest temperature in the Nc (N, ) phase. As compared to the predictions of the Maier-Saupe theory, the nematic order parameter in the NC phase is considerably larger and decreases more strongly as the nematic-isotropic transition is approached.

I. INTRODUCTION

The rapidly growing family of lyotropic mesophases in- cludes three distinct nematic phases: the calamitic (NC) and discotic (ND ) uniaxial phases, and the biaxial ( NB ) phase. Built from nonspherical micellar aggregates, these nematic phases exhibit long-range orientational order but only short- range translational order. Since the first report’ in 1967 nematic phases have been discovered in a variety of surfactant systems, and the problem of understanding their physical properties in terms of the size, shape, orientational order, and dynamics of the micellar building blocks has developed into an active field of research.2”

Ever since the first nuclear magnetic resonance (NMR) studies7 of thermotropic nematic liquid crystals were reported nearly 40 years ago, the NMR technique has been used extensively to obtain molecular-level information about nematic liquid crystals and about molecules dissolved in such phases.‘-” The long-standing popularity of nematic liquid crystals within the NMR community derives largely from the fact that the magnetic field used to polarize the nuclear spin system also acts to align the nematic phase. NMR studies of nematic liquid crystals fall in two main categories: (i) spectroscopic studies of spectral line shapes, providing information about the symmetry of the mesophase, the sign of the magnetic susceptibility anisotropy, and the orientational or- dering of the nematogens (or solutes), and (ii) spin relaxation studies, providing information about the rate and amplitude of various molecular motions. Whereas numerous studies of both kinds have been performed on thermotropic nematics, NMR studies on lyotropic nematics2v6 have been largely confined to category (i). In fact, we know of only one” previous NMR relaxation study of a lyotropic nematic phase.

Being at the same time orientationally ordered fluids and association colloids, lyotropic nematics present a chal- lenging task of elucidating the physical basis of phase stability and relating macroscopic behavior to molecular properties. As a prerequisite for progress in this direction, it is

necessary to experimentally characterize the size (and shape) of the micellar aggregates as well as their orientational order. The bulk of the existing NMR studies of lyonematics amounts to measurements of line splittings from quadrupolar nuclei. While, in principle, the quadrupole splitting contains information about micelle size as well as micelle order, this information is not separable. In the early NMR studies2P’2P’3 it was usually postulated at the outset that the micelles are sufficiently large that the quadrupole splitting only reflects their orientational order. It is now widely recog- nized that such an assumption cannot be justified in general. More recently, quadrupole splittings have been used’4s’5 with a different objective in mind: to deduce local molecular order parameters by combining NMR data with independent information (from x-ray and neutron scattering”j or electrical conductivity17 ) about the size and order of the micelles.

The purpose of the present work is to find out if the size and order of micelles in nematic phases can be separately determined by a more complete NMR study, including not only quadrupole splittings but also quadrupolar relaxation rates. Our approach is based on the expectation that the spin relaxation behavior of nuclei residing in surfactant molecules or counterions should reflect the diffusion of the molecule or ion over the curved micelle surface and thereby con- vey information about the size and shape of the micelle. Moreover, the relaxation contribution from surface diffusion should be weighted by a factor which reflects the orientational distribution of the micelle and is thus related to the nematic order parameter. The theoretical framework re- quired to implement this approach has been presented elsewhere. la

In this work we study the recently discovered’9V20 NC and ND phases in the system sodium dodecyl sulphate (SDS)/decanol/water, using as probes the 2H nuclei in the a-deuterated SDS and the 23Na nuclei in the counterions. For both nuclei we measure the quadrupole splitting and the spin relaxation rates R rz and R , o , characterizing the return to equilibrium of the odd-rank (R ,z ) and even-rank (R , p )

J. Chem. Phys. 96 (5), 1 March 1992 0021-9606/92/053875-17$06.00 0 1992 American Institute of Physics 3875 Downloaded 30 Jan 2009 to 130.235.253.27. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

3876 Quist, Halle, and Fur& Spin relaxation in lyonematics

magnetic multipole polarizations,24 over the temperature range where the two nematic phases are stable. Unlike the more viscous hexagonal and lamellar mesophases, nematic phases can usually be investigated at only one orientation. While this limits the information derivable from the relaxation experiments, the number of independent model parameters can be reduced by invoking reference data from the recently studied 21 hexagonal phase (adjacent to the NC phase in the ternary phase diagram), and by combining the 2H and 23Na results. We thus tlnd that the micelles (modeled as spheroids) in both nematic phases are relatively small (axial ratio 3-4, or longest dimension 12-16 nm) at all temperatures, and that the nematic order parameter is high at low temperatures (0.9 in the Nc phase) and, as the Nc- isotropic transition is approached, drops more sharply than predicted by the Maier-Saupe mean-field theory.

II. EXPERIMENT

isotropic micellar (L’ ) phase was obtained (cf. Fig. 1). Next, this micellar phase and decanol were weighed into a new glass ampoule yielding a clear, viscous, birefringent nematic NC or ND phase (cf. Fig. 1) after mixing at 25 “C!.

The phases investigated by 23Na NMR were stable from ca. 15 “C to ca. 28.2 “C (N,) or 28.8 “C (N,). At temperatures below 15 “C the SDS precipitated. At the upper phase boundary the N, sample entered a two-phase region with the N, phase and the isotropic micellar phase in equilibrium, whereas the No sample entered a two-phase region with the ND and NC phases in equilibrium. These observations are in accordance with the results of Amaral et LZ~.,‘~~~~ although the lower phase boundaries are slightly shifted. These minor differences may be due to too fast temperature scanning in previous works. ‘9V20 The upper phase boundary of the Nc sample used for the 2H NMR experiments was shifted to ca. 26.2 “C!, probably as a result of the slight difference in sample composition (cf. Table I and Set II C) .

A. Materials and sample preparation

SDS (sodium dodecyl sulphate, specially pure) and de- canal (n-decanol, specially pure) from BDH Chemicals were used as supplied. SDS selectively deuterated at the a position (next to the sulphate headgroup) from Synthelec was purified by repeated recrystallization from aqueous solution. The water was either deuterium-depleted H, 0 from Sigma (samples for ‘H experiments), or millipore filtered H, I60 mixed with either 23.7 wt% 170-enriched H, 0 (Nc sample for 23Na experiments) or 5.5 wt% ‘H, 160 and 6.6 wt% ‘70-enriched H, 0 (No sample for 23Na experiments). The “O-enriched H,O, from Norsk Hydro, contained 20 mol% “0 and 80 mol% “0, while the 2H2 160, from Norsk Hydro, contained > 99.8 mol% 2H. Sample compositions in weight% and on a mole basis are listed in Table I. Checks of the reproducibility of the 23Na NMR results and phase boundaries showed no variation with the isotope composition as long as the molar ratios of the SDS, water, and de- can01 were kept constant.

B. Spectrometer characteristics The NMR experiments were performed on a Bruker

MSL- 100 spectrometer (resonance frequencies 15.37 1 and 26.487 MHz for ‘H and 23Na, respectively), equipped with a 10 mm vertical saddle-coil probe and a 2.35 T wide-bore superconducting magnet.

During the experiments the samples were contained in flame-sealed 7 mm i.d. Pyrex tubes filled to 15 mm height. The sample volume was centered in the saddle-coil yielding a spatial rf (B, ) inhomogeneity of less than f 10%. The static magnetic field (B, ) inhomogeneity was less than 4 Hz for both nuclei. The temperature was controlled by a home-built

The nematic N, and ND samples were prepared in the following manner. Either a-deuterated SDS and deuterium- depleted water (samples for 2H experiments) or regular SDS and the water mixture (samples for 23Na experiments) were weighed into a glass ampoule, tightly sealed with a teflon- coated screwcap. Upon mixing SDS and water a clear, fluid,

TABLE I. Sample compositions.

Phase NC NC ND ND Nucleus studied 2H “Na *H 13Na

SDS (wt%) ’ 24.52 24.35 24.87 24.51 Decanol (wt%) ’ 4.38 4.38 5.44 5.41 Water (wt%) *+ 71.10 71.27 69.69 70.08 n,/(n,, + nsm ) = 35.2 34.5 32.2 32.2 %cc~(Kicc + %DS) = 0.247 0.247 0.286 0.287 9d 0.268 0.266 0.283 0.280

“Less than * 0.02 wt% uncertainty. bIsotope composition as described in the text. ‘n = number of moles. d Micelle volume fraction.

Hz0 24 26 28 30 32 34 SDS b

FIG. 1. Partial phase diagram (wt%) for the system SDS/decanol/H, 0 at 25 “C showing the extension of the one-phase regions of the isotropic micellar (L, ) and the hexagonal (E), lamellar (D), and nematic (N,, N,) liquid crystalline phases. (The dashed phase boundaries have not been accurately determined.) The dots give the composition of the investigated nematic samples as well as of the previously studied (Ref. 2 1) E phase sample.

J. Chem. Phys., Vol. 96, No. 5,i March 1992

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temperature regulator or a Stelar VTC87 regulator with high airflow ( 1.5 m3/h), yielding a temperature stability of f 0.05 “C (home-built regulator, *jNa experiments) and f 0.03 ‘C (Stelar, ‘H experiments) or better. The tempera-

ture gradients within the samples were less than f 0.03 “C. Typically, the 180 a pulse lengths were 20 and 13 pus for ‘H and 23Na, respectively. Since the frequency spectrum of the resonant pulses (applied at the 23Na central peak and half- way between the two ‘H satellites) showed less than 6% attenuation at the satellite frequencies, the pulses were considered nonselective.

C. Spin relaxation experiments

The lab-frame spectral densities J f (w,, ) and J f ( 2w, ) , which refer the molecular motions to the laboratory-fixed frame defined by the static magnetic field, were determined from three relaxation experiments, all based on the variation of peak intensity with pulse delay time ( 7). For both *H and 23Na the spin relaxation can be considered as entirely quadrupolar. Typically the relaxation experiments followed the procedures described in Ref. 2 1.

Inversion recovery. The longitudinal relaxation rate, R 129 was determined by the (a)+ - r - (7r/2), - acq. pulse sequence with the phase of the first pulse cycled around the four orthogonal phases to suppress the genera- tion of coherences.** The single-exponential evolution of the satellite intensity with increasing r yields the *H and 23Na relaxation rate23*24

RI, = J:+4Jf (I= l,*H), G (I = 3/2,23Na). (2.1)

For convenience, the lab-frame spectral densities J i = J: (km,, ) are defined so as to include spin-dependent numerical factors and coupling constants.

Quadrupole polarization decay. The Jeener-Broekaert pulse sequence” (77-/Z)+ - r1 - (1~/4)+*~,~ - 7 - (r/4), - ( f > acq., with the phase cycle 4 = 0,7r/2, s-,

3n-/2,26 was used to determine the *H relaxation rate R,, = 3Jf (I= l,*H). (2.2a)

The fixed delay time r1 was set to 1/(2v,) (where vp is the quadrupole splitting in Hz) in order to maximize the conver- sion of dipole to quadrupole polarization.

In both the inversion recovery and quadrupole polarization decay experiments, an acquisition delay of 30-100 ,US was introduced to avoid distortion from probe ringing. First- order phase correction was avoided by setting the acquisition delay to a multiple of the precession time of the satellites. (A quadrupolar echo detection pulse sequence was thus not needed.) The experiments were performed with 24 different delay times in the approximate range [ 1/20- 201/R, where R is the appropriate relaxation rate. A three- parameter least-squares fit to the satellite peak intensity vs delay time, according to I(r) = A + B exp( - RT), finally yielded the desired relaxation rate. Two representative spectra from the inversion recovery experiment are shown in Fig. L.

20 spin echo. Due to the nearly complete alignment of the investigated phases, the central line of the anisotropic 23

Quist, Halle, and Fur& Spin relaxation in lyonematics 3877

I~““~“‘I’~~‘~~‘~~~~“~~‘~~‘,~~~~~~,~I1IIII.1I~I, loo00 .5cao 0 -5coo -1OOfX

Hertz

I’~““~~~,~~‘~~~~~~,~~~~~~~~~,~~~~~~III,ItI~IIIII,I1IIII,II,I1~1.,

15cHlo loo00 5ow 0 -5000 -loo00 -15oclo HlZtZ

FIG. 2. 2H (top, at 20.0 ‘C) and Z3Na (bottom, at 20.6 ‘C) spectrum from inversion recovery experiments on ND phase samples. The spectra correspond to the longest delay time rand were obtained as described in the text.

Na spectrum reports on relaxation at a well-defined director orientation 19,~. The 23Na homogeneous central line width Aem, and the corresponding relaxation rate

R ; = ITAVF~ = J f + Jt, (I = 3/2,*‘Na) (2.2b) were determined from a 2D spin echo experiment,27P28 using the pulse sequence (7r/2), - r - (q) *,, - r - acq. with 64 delay times r and a narrow (2-3 kHz) filter width. Aem was obtained from a Lorentzian fit to the (homogeneous) central line in the F 1 cross-section spectrum produced by zero-filling to 4 K and Fourier transforming the decay (with increasing 7) of the (inhomogeneous) central line in the F2 spectrum.

The signal-to-noise ratio for the shortest delay time was better than 100 (*H experiments), 250 (23Na, inversion recovery) and 700 ( 23Na, 2D spin echo). From reproducibility tests we estimate the experimental random error ( f 20) to f 1.2% in R,, (*H), f 2.0% in R,, (*H), and f 1.0% in R,, (23Na) and R; (23Na).

The reproducibility of the 23Na results was checked on the samples used for the *H experiments. For the ND phase the 23Na results from the two samples were identical within the specified experimental uncertainties. The two ND phase samples can thus be regarded as identical in their NMR

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properties. For the N, samples, however, the water-to-surfactant mole ratio differs slightly (by 2%, cf. Table I). This difference was manifested in a slightly different temperature dependence of the 23Na results from the two N, phase samples. The difference in the NMR data corresponds to a temperature shift of 1.5-2.0 “C, which also brings the No-isotropic transition temperatures for the two samples into agreement. Due to its strong temperature variation near the transition, the 23Na quadrupole splitting differed by up to 3 kHz (at 26 “C) between the two samples.

Ill. LINE SHAPE AND QUADRUPOLE SPLITTING A. Magnetic alignment and phase identification

For uniaxial nematic phases the orientation corresponding to the minimum in the potential of mean torque experi- enced by the micelles in a small volume I’ (containing many micelles) defines the (local) director. In a macroscopic nematic sample that has not been exposed to an aligning magnetic field, the director varies continuously in space. Such a sample is sometimes referred to as a powder, implying that the director is isotropically distributed within the macroscopic sample (neglecting surface alignment).

When a nematic powder sample is introduced into a uniform magnetic field B,, a small region of volume Vexpe- riences a magnetic torque that tends to orient its director so as to minimize the magnetic free energy29

A, = - vB’ -P* (COSBLD) AxD. 3PO

(3.1)

Here IL,, is the vacuum permeability, P2 is the second-rank Legendre polynomial, 8,, is the angle between the magnetic field and the director, and AxD = G - xf is the anisotropy of the uniaxial diamagnetic susceptibility tensor in the director frame. The latter may be related to the molecular susceptibility components along and perpendicular to the length of the conformationally averaged surfactant molecule, as

Ax” = SD, AxN. (3.2) The order parameter SD, = ( P2 ( cos eDN ) ) depends on the shape and order of the micelles. [We are assuming here that zN (the local normal to the micelle surface) is at least a threefold symmetry axis.] For spheroidal micelles, SD, is positive (negative) if the shortest (longest) micelle axis is along the director. ” For the saturated hydrocarbon-chain surfactants used in the present study AxN < O;30 it then follows from Eqs. (3.1) and (3.2) that if the longest (shortest) micelle axis is along the director, then A,,, is minimized when the director is parallel (perpendicular) to the magnetic field. This corresponds to the No and ND phases, respectively, studied here. (Sometimes these phases are denoted N 2 and N; , the superscript referring to the sign of AxD. )

In addition to the magnetic torque, the interaction with the surface of the sample container tends to align the director parallel (No phase) or perpendicular (ND phase) to the surface. In the absence of a magnetic field, the range of surface alignment may be of the order of mm,31 whereas in the field ofBo = 2.35 T used here the range is given by the magnetic coherence length,29 gm = (po~/AxD) “*/B. z 10 - 5 m.32

In any case, the NMR probe configuration used in the present study has the B, field along the cylindrical sample tube axis, so the magnetic and surface-induced torques coincide in (the major part of) the N, phase samples as well as in the ND phase samples. Consequently, a spatially uniform alignment is expected.

In an NMR study of a nematic phase, the external magnetic field plays a dual role: it orients the nematic director and it polarizes the nuclear spin system. For a quadrupolar nucleus such as *H or 23Na, the Zeeman levels thus established are perturbed by the nuclear electric quadrupole coupling, giving rise to a quadrupole splitting33


(3.3)

where co is a spin-dependent numerical factor (co = 3/2 for I = 1, co = l/2 for I = 3/2) and X0 is the residual quadrupole coupling constant (qcc), which is proportional to the residual electric field gradient component along the local normal to the micelle surface. The effect of motional averaging by restricted micelle reorientation and surfactant or counterion diffusion over the curved micelle surface is ac- counted for by the order parameter S,,. Equation (3.3) gives the splitting from a spatial region whose director makes an angle t9,, with the magnetic field. The powder splitting is obtained by setting e,, = 7r/2 in Eq. (3.3) .33

In order to identify the nematic phases, water 2H and counterion 23Na NMR spectra were recorded at 20 s intervals following the introduction of our powder samples in a magnetic field (B, = 2.35 T). The spectra showed a progressive evolution from the characteristic powder line shape to that expected from a uniformly oriented sample.31P33 This qualitative change in the line shapes was essentially complete in 15 min. However, the asymmetry and width of the inhomogeneous satellites continued to decrease for about an hour, reflecting a progressive director alignment. When re- moved from the magnetic field, mechanically undisturbed nematic samples retain their macroscopic alignment at least several days.

The powder line shapes observed initially confirm that our nematic samples are indeed uniaxial. For the N, phase, the splitting from the aligned sample was twice as large as the powder splitting. According to Eq. (3.3) the director is then aligned with the field (e,, = 0) and the longest aggregate axis is preferentially along the director. For the ND phase, the oriented and powder samples had the same splitting, implying that the director is oriented perpendicular to the field and that the shortest aggregate axis is preferentially along the director.

In contrast to the No sample, which is homeotropically aligned, the ND sample retains an isotropic distribution of directors in the plane perpendicular to the magnetic field. (Due to long-range surface alignment,31 the director distribution is probably radial rather than random.) The distribution in the plane was examined by orienting the ND sample in a different probe configuration with the B, field perpendicular to the sample tube axis. The sample was then gently transferred to the parallel field configuration and NMR spectra were recorded at 20 s intervals. The first spectrum showed a two-dimensional powder line shape as expected

J. Chem. Phys., Vol. 96, No. 5,1 March 1992


from an isotropic distribution of directors in a plane containing the 3, vector.3’ In subsequent spectra, the intensity of the 0,, = n/2 peaks increased at the expense of the ~9,~ = 0 peaks, as expected. Although a homeotropically aligned iVD sample could be produced by spinning the sample around the tube axis in the perpendicular field configuration,31 this was not attempted as the two-dimensional director disorder in our ND sample has no effect on the NMR observables. [The director distribution in the plane perpendicular to the B,, field has no effect on the secular part of the quadrupolar Hamiltonian (and hence does not affect the splitting) and can only contribute to the spectral density function J t (w ) . As this is probed at the frequency w = 2w, and since diffusion through the spatial inhomogeneity of the director field is expected to be much slower than the Larmor period l/w,, no contribution is expected.]

B. Quadrupole splitting and motional averaging

In analyzing our NMR data we assume that the micelles in the two nematic phases are uniaxial and, more specifical- ly, of spheroidal shape. The results of Sec. III A then imply that the iV, phase is composed of prolate spheroidal micelles with the (long) symmetry axis preferentially aligned with the B, field, while the ND phase consists of oblate spheroidal mice&es with the (short) symmetry axis preferentially oriented perpendicular to the B,, field.

The temperature dependence of the quadrupole splittings from the two nematic phases is shown in Fig. 3 for three molecular components: SDS ( 2H>, counterion ( 23Na), and water ( “0). The qualitative behavior is the same for all species: a strong reduction of the splitting with increasing temperature. This may be contrasted with the behavior in the hexagonal phase of the same system, where the SDS 2H and counterion 23Na splittings are reduced by just ,a few percent on increasing the temperature from 25 to 35 oC.2’ We may therefore conclude that the strong temperature dependence of vp in the nematic phases is associated with the order parameter S,, in F,q. (3.3)) reflecting changes in micelle size and orientational order, rather than with the residual qcc &, which should have essentially the same temperature dependence in the hexagonal and nematic phases.

Provided that the aggregates are uniaxial and the surface distribution of spin-bearing molecules is independent of aggregate orientation, the order parameter S,, in IQ. (3.3) may be decomposed as’*

SLN = SLuSAri.

Quist, Halle. and Fur& Spin relaxation in lyonematics

18

4 16

g 14

R* 2 12

P 10

15 20 25 30 T/V

8 15 20 25 30

T/Y2

FIG. 3. Quadrupole splittings vs temperature in the NC (top) and ND (bottom) phases obtained from a-d, SDS 2H spectra, counterion Z3Na spectra, and water “0 spectra.

motion relative to the local surface normal). To separate these factors, we must invoke additional experimental data.

(3.4) S,, = ( P2 (cos eDA ) ) is the nematic order parameter associated with the distribution of the angle 19,” between the director and the symmetry axis of the aggregate. S,,, = (P2 (cos BAN) ) is a shape-dependent order parameter accounting for the partial averaging of the quadrupole coupling brought about by molecular diffusion over the curved micelle surface (Q,, is the angle between the micelle symmetry axis and the local surface normal).

The residual qccfc is primarily determined by the local structure at the micelle surface and, in the case of 23Na, also by the radial counterion distribution. We have recently2’ obtained Xp ( 2H> for cr-deuterated SDS and ,&, (23Na) for counterions in the hexagonal (E) phase of the present system in the temperature range 25-35 “C and at the composition indicated in Fig. 1. As a first approximation, we shall assume that these?* data are appropriate also for the nematic phases studied here, which, as seen from Fig. 1, are very close in composition to the E phase sample. We thus have2’

,&(‘H)/kHz= 34.3-0.11 TPC, (3.5a) 2a ( 23Na)/kHz = 87.1- 0.33 TPC. (3.5b)

The quadrupole splitting yp is thus seen to be a product of three unknown factors: the order parameters S,, and SAN, and the residual qcc?o (which incorporates an “internal” order parameter describing the averaging effect of local

Using Eqs. (3.3) and (3.5) we can now obtain the composite order parameter S,, from the ‘H and 23Na quadrupole splittings in Fig. 3. The result is shown in Fig. 4. (Since the infinite-cylinder limit of S,, is - l/2, while the infinite- disc limit of S,, is 1, one should compare - 2S,, from the iVc phase with S,, from the ND phase.) The agreement between the 2H and 23Na order parameters is seen to be ex- cellent for the N, phase (if the temperature shift of the NC-L, transition is taken into account), whereas, for the ND phase, S,, ( 23Na) exceeds SD, (‘H) by lo%-15 %. We

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3880

0.9 , 1 I I, 1 I, I ( I

0.8 - -0 NC

0.7 - ,’ q 0.

-2sDN - 0 cl* 0.6 - 0 d

. 0.5 - 0

0.4 - 0

0.3 8 a * * "0 8 n " 15 20 25

T/V

T I I I I I t

; Ll 8 I 1 I I I I 0 / 1 I 0 : I I 9: I I

i

30

0.9 ,,I #,, ,,I,# III,

0.8 -

SDN .

. l

.

ND/NC IV. SPIN RELAXATION RATES A. Spectral densities

The lab-frame spectral densities J f ( w. ) and J$ (20, ), derived from the measured spin relaxation rates according to Eqs. (2.1) and (2.2)) are shown in Figs. 5 (‘H) and 6 ( 23Na) as functions of temperature for the NC and ND phases. The quantity J f (kw, ) is the cosine transform of the time auto- correlation function of the k th spherical component in the

15 20 25 30 T/V

FIG. 4. Composite order parameters - 2.SDN (Nc phase, top) and S,, (ND phase, bottom) vs temperature as deduced from *H (open symbols) and r3Na (filled symbols) quadrupole splittings. The circular symbols refer to results obtained with the residual qcc To taken from the E phase, while the square symbols refer to results obtained with ,&(*H,N,) = 1.10 &(2H,E) andfp(23Na,N,) = 1.25zo(r3Na,E). Thedashedverticallines indicate the Nc - L, (slightly shifted for *H, due to differences in the sample compositions) and ND - Nc transition temperatures. The open squares in the top figure represent the ‘H data after a 2 OC temperature shift to correct for the difference in T,.,, between the two samples.

attribute this latter discrepancy to differences in the residual qcc’s between the E and No phase, where the m icelles have significantly smaller interfacial curvature than the aggregates in the E and NC phases. The resulting smaller SDS headgroup area should lead to a higher local C-D bond order parameter, and hence a larger residual qcc j& (‘H), in the ND phase.‘4*34*3’ Similarily, we expect that je ( 23Na) is larger in the No phase than in the E and NC phases. Taking xa(‘H,No) = 1.10 Fo(2H,E) and &(23Na,N,) = 1.25 ,?Q( 23Na,E) we obtain perfect agreement between the order parameters S,, derived from the 2H and 23Na data (square symbols in Fig. 4). The particular values of these correction factors are motivated in Sec. V.

The deviation of - 2S,, (NC phase) or S,, (No phase) from unity is a measure of the combined effect of finite aggregate size (via SAN) and orientational disorder of the aggregates (via S,, ) according to Eq. (3.4). While the results in Fig. 4 tells us that this composite quantity varies more strongly with temperature in the NC phase (particular- ly near the isotropic phase), a more detailed interpretation

Quist, Halle, and Fur& Spin relaxation in lyonematics

of these results requires a separation of the factors S,, and S,, . Such a separation has not hitherto been accomplished solely on the basis of NMR data. In the early NMR studies of lyonematics2”2*13 large aggregates were postulated at the outset, in which case the data in Fig. 4 would be interpreted as giving the nematic order parameter S,, directly. More recently, however, high-resolution x-ray and neutron scattering studies’6*36*37 as well as freeze-fracture transmission electron m icroscopy studies3* of several lyonematic phases have suggested rather small m icelles (axial ratios of 2-4, typically). It is thus clear that we cannot in general set S,, to its infinite-rod or infinite-disc lim iting value, but must face the problem of separating S,, into its two factors S,, and S,, . The central idea of the present work is that this separation can be accomplished by invoking spin relaxation data, to which we now turn.

12 J ‘2

0 0 0

f . u1 r’ . +8 2

‘I B t -I 4

2

4

I

------ JJf -- 2H ! i

-------+--- -

01 ’ 8 * ’ ’ ’ ’ * ’ ’ IfI ’ * 1 15 20 25 30

T/“C

12 J ‘I

0 . 2 c i 4

r” ‘2

----_ --

0 0 :

JJf ----- ---we-

15 20 25 30 T/Y!

FIG. 5. Lab-frame spectral densities Jf(o,,) andJf(ti,) vs temperature obtained from a-d2 SDS ‘H relaxation rates in the NC (top) and ND (bottom) phases. The experimental uncertainty is smaller than the size of the symbols. The fast-motion contribution Jf, taken from the E phase, is also shown.

J. Chem. Phys., Vol. 96, No. $1 March 1992



80 ,,?,I,, “,“,,’

0 0

NC: _ i Ll- 60 JJ2

00

%I

3 ~~

l J ‘I 0 0 0 - *a 0 0.

q 2 40 l :

----__ ---_ 2 ---_ f

--_ I --- 20 Jf

15 20 25 30 T/T

80 , I I , , I I , I I / I I , ,

N,, ;Nc

60,

2

* o& ‘2

‘k Q . O 0 2 ‘1 00: - 40, 3 ---_ --__ ---_ 2 --__

f ---L- 20- Jf

_ =Na 0 a ” ” c ’ * 3 ” ’ )!I

15 20 25 30 TIT

FIG. 6. Lab-frame spectral densities Jf(w, ) and Jf( 2w, ) vs temperature obtained from counterion “Na relaxation rates in the NC (top) and ND (bottom) phases. The experimental uncertainty is smaller than the size of the symbols. The fast-motion contribution J,, taken from the E phase, is also shown.

lab-fixed frame (defined by the external magnetic field) of the irreducible electric field gradient tensor at the nuclear position. 33 In order to relate these spectral densities to the structure of the nematic phases, it is convenient to transform the field gradient components from the lab frame to the frame defined by the nematic director. Since the director is parallel with the magnetic field in the NC phase, but perpendicular to the field in the ND phase, we obtain’8~2’ for the N, phase

J:(w,) =J,+ Jfb,), (4.la)

J;Wo) =J,+Jt’(hoL (4.lb) and for the ND phase

J:bo) =J/+tJfbo) +tJi’(wo), (4.2a)

J:Ck,) =J/+~:G4,) +$JfWo) +&‘(2wo).

(4.2b) Here we have explicitly displayed the contribution Jf from fast (relative to the Larmor frequency, w0 ), local (relative to the micelle dimensions) motions. As usual,” we assume (i) that these motions are statistically independent from the slower motions contributing to the director-frame spectral densities J E (kw, ), (ii) that they are sufficiently fast that we can set J,(w,) = Jf(200) [ = J,(O)], and (iii) that they are sufficiently weakly anisotropic that we can set

J,(t?,, = 0) = Jf(& = 7r/2) and assume that J, is independent of the lab-frame projection index k.

By the same reasoning as applied to the residual qcc Fe (cf. Sec. III B), we assume that J, in our nematic samples is the same function of temperature as the Jf recently deduced from a study’l of a sample in the neighboring hexagonal phase (cf. Fig. 1). [As will be seen in Sec. V C, the present data actually allows us to determine Jf( 23Na) independently for the NC phase.] These J,( T> functions are included in Figs. 5 and 6.

As seen from Figs. 5 and 6, the lab-frame spectral densities J f (w. ) and J 2” ( 2w, ) contain substantial contributions from other (slower) motions, above the ubiquitous contribution Jf from fast local motions. These slower motions, associated with the director frame spectral densities J”, (km, ), are (i) molecular (surfactant and counterion) diffusion over the curved surface of the spheroidal micelle, and (ii) restricted reorientation of the entire micelle in the potential of mean torque established by the surrounding micelles. (We show elsewhere5’ that director fluctuationsz9 do not contribute significantly to these high-frequency spectral densities.) Before analyzing these motions in detail, we note from Figs. 5 and 6 that in the iV, phase Jf’( 20, ) > J f( w. ) for both nuclei as expected for prolate micelles aligned with the director. The ‘H NC data in Fig. 5 yield a ratio J~(2wo)/J~(wo)of1.O-1.2,ascomparedto2.4-3.0forthe E phase. ” This large difference in relaxation behavior suggests that the micelles are much smaller and/or much more orientationally disordered in the N, phase than in the E phase. [For perfectly aligned, infinitely long and straight cylinders, Jf(w, ) = 0.” ] For the ND phase, the situation is more complicated and the two nuclei give qualitatively different results.

B. Surface diffusion and micelle reorientation The theoretical problem of calculating the director-

frame spectral densities J “, (km, ) for surface diffusion on a prolate or oblate spheroid undergoing restricted rotational diffusion in a uniaxial potential of mean torque has recently been considered in some detail. *’ Here we shall merely quote some key results needed for the analysis of our data (Sec. V) .

We begin by considering the order parameters S,, and S,, , which affect the quadrupole splitting (cf. Sec. III B) as well as the spectral densities (cf. below). If the spin-bearing molecule (SDS or Na + ) is uniformly distributed over the spheroidal micelle surface, then S,, is a purely geometrical quantity and is related to the ratio p = a/b of the major (a) and minor (b) axes of the spheroid as18*39*40

s _ _ 2 +P’ + (p2 - 4)Fpr AN -

ap2- 1)(1 +I;;,,,

(prolate),

s AN

= 2p2 + 1 + (1 - 4p’x-h xp2- 1)(1 +F&)

(oblate),

F = p2 arccos( l/p) Pr (/g- I)‘/2 ’

F ob

= arccosh(p) p(p” - 1)“2 *

(4.3a)

(4.3b)

(4.4a)

(4.4b)


In Fig. 7 we show the functions - 2S’,, (p) for prolates and S,,(p) for oblates, which both increase monotonocally from zero at p = 1 (sphere) to unity at p = 00 (infinite rod or disk). For a given axial ratio the oblate curve deviates more than the prolate curve from the infinite-aggregate limit, as expected since the “end effects” are two-dimensional for oblates but only one-dimensional for prolates. For both types of aggregates, however, motional averaging by surface diffusion is expected to strongly influence the composite order parameter in Fig. 4 unless the micelles are highly aniso- metric.

A comment is in order concerning the assumption of a uniform molecular surface density. On the basis of a neutron scattering study4’ of the ND phase in the system potassium laurate/decanol/water, a nonuniform surface distribution of amphiphiles was implicated, but the degree of nonuni- formity was not quantified. However, we have reasons to believe that the effect is small. First, explicit calculations42 demonstrate that the electrostatic effect, which tends to give a higher surface charge density in regions of high curvature, is rather small. Second, this electrostatic effect is counteract- ed by chain-packing restrictions, which favor a larger headgroup area in regions of high curvature.34.35 Third, the close agreement between the reduced 2H and 23Na quadrupole splittings (Fig. 4) suggests that the surfactant and counterion surface distributions are very similar. In fact, even the reduced water “0 splittings (not shown in Fig. 4) are very close to the 2H and 23Na results.

The nematic order parameter S,, is a measure of the orientational order of the micelles with respect to the mean director and plays a key role in all theories of nematic liquid crystals.29,43 The director-frame spectral densities J”, (km, ) involve also the fourth-rank nematic order parameter,” QDA = ( P4 (cos 8, ) >, where P4 is the fourth-rank Le- gendre polynomial. To reduce the number of independent parameters, we introduce a relation between QDA and S’,, by assuming that the (reduced) potential of mean torque is of the form

w (cos&) = -ilc02e,,, (4.5)

where il is a dimensionless coupling parameter. This functional form, which may be regarded as the leading term in a multipole expansion, defines a one-to-one correspondence between S,, and QDA as illustrated in Fig. 8.

The director-frame spectral densities J fl, (ko, ) may be expressed as18*33

J:(kw,) =c,bjM2 s

- dt cos(kw,t)gD, (t), (4.6) 0

where cR is a spin-dependent numerical factor ( cR = 3/2 for I = 1, cR = 1 for I = 3/2) andXR is a residual qcc which, in principle, may differ slightly in value from the residual qcc ,& introduced in Sec. II A in connection with the quadrupole splitting. For the reasons given in Ref. 21, we hence- forth neglect any difference between XR and X0 and denote the residual qcc simply by 2.

The three director-frame time correlation functions s”, (t), describing the combined effects of diffusion (of the SDS molecule or the Na + counterion) over the spheroidal micelle surface and (restricted) reorientation of the micelle in the uniaxial nematic phase, can be decomposed as18

&tf) =%,‘f~(t) + i (2--s,) ?I=0 x [&z (Q + f4J%os2, ]g;;‘N(t), (4.7)

involving three distinct surface-diffusion correlation functions d”(t) and nine distinct reorientational correlation functions 8; ( t). All correlation functions appearing in Eq. (4.7) are defined” so as to vanish in the limit l+ CO.

Micelle reorientation will be modeled as rotational diffusion of a symmetric top in an even uniaxial potential of mean torque of the form (4.5). The rotational diffusion tensor is diagonal in the aggregate frame with distinct components D,, and Dl, referring, respectively, to the spinning motion of the spheroid around its symmetry axis and to the tumbling motion of this axis relative to the director. The

““L-----Y t I 1

0.8 1.0 I I I I t

s 2 0.8 0.6

3 QDA : 0.6 0.4 3 s 0.4 2

\ 0.2 0.2

LA 0.0 0.0

1 2 3 4 5 Axial ratio, p

0.0 0.2 0.4 0.6 0.8 1.0

sDA

3882 Quist, Halle, and Fur6: Spin relaxation in lyonematics

FIG. 7. Order parameters describing the effect of orientational averaging of P2 (cos e,,) by diffusion over the surface of a prolate or oblate spheroid of axial ratio p.

FIG. 8. Relation between the fourth-rank ( QD,, ) and second-rank (S,, ) nematic order parameters for a potential of mean torque of the form (4.5).

J. Chem. Phys., Vol. 96, No. 5,l March 1992


Quist, Halle, and Fur6: Spin relaxation in lyonematics 3883

correlation functions pm$ (t) describe pure tumbling modes (involving only D, ), whereas, for large S,, , &‘(t) and &t(i) are essentially pure spinning modes. For the high degrees of nematic order in the phases studied here (cf. below), we may, without significant loss of accuracy, use a single- exponential approximation44V45 for the correlation functions e”(t).Theex 1’ ‘t p ICI expressions for the e (t), reproduced in Ref. 18, reveal that the actual shape of the potential of mean torque enters only through the implicit relation between the order parameters S,, and QDA. Given this relation, the nine orientational correlation functions g”< (t) are fully determined by (i) the nematic order parameter S,, and (ii) the micellar axial ratio p, which, together with the viscosity of the aqueous medium and the minor semiaxis b of the micelle, determines the rotational diffusion coefficients D,, and DL according to Perrin’s equations.40*46

The three surface-diffusion correlation functions g;llN( t) are fully determined by the micellar axial ratio p [which yields S,, according to Rqs. (4.3) and (4.4) ] and the time constant b 2/D1, where D, is the surface diffusion coefficient of the spin-bearing species (SDS or Na + ) . Although neither the SDS headgroups nor the counterions are strictly confined to the micellar surface, the “radial” distribution function (dominated by hydrophobic and electrostatic interac- tions, respectively) for both species exhibits a deep minimum at or very near the surface, thereby justifying the surface diffusion approximation. Radial diffusion is poten- tially more important for counterions than for surfactants, but, as shown elsewhere,2’s47A9 it does not give rise to significant deviations from the surface diffusion approximation. The time constant b ‘/D, for our nematic samples is assumed to be the same function of temperature as deduced from the ‘H and 23Na relaxation in the Ephase.2’ For given values of p and b 2/Ds, the three surface-diffusion correlation functions g;;‘“( t) are then computed numerically to desired accuracy using the methods developed in Ref. 18. [Although an analytical single-exponential approximation for s;;‘“(t) can be formulated,‘8 it is not sufficiently accurate to be useful here. ]

In the limit of slow micelle reorientation, i.e., when d”(t) decays much faster than g$ (t) for all m and n, we may replace the latter by its initial value in the cross terms of Eq. (4.7). If the tumbling modes are also slow compared to l/o,, the first term in Eq. (4.7) does not contribute significantly to the spectral densities at w0 and 2w,, whence

Ji(bo) =c,(7q)” i (2-&) “XC

where

x [g-z (0) + LAOS:, l.rwb 1, (4.8)

- j;?N(ho) = s

dt co~(kw,t)g;;‘~(r) (4.9) 0

are the (in general, non-Lorentzian) reduced surface-diffusion spectral densities. The limiting form (4.8) is accurate for sufficiently large micelles and is exact in the limit p+ CO, where also”

x”( ko, ) = S,&“( kw, ) (prolate), (4.10a) xN(kmo) = 0 (oblate). (4.1Ob)

The following analysis is based on the more general form (4.7). While not entirely consistent, this description of micelle reorientation should be sufficiently accurate for our purposes. As discussed in Appendix B, this is because the spectral densities J f ( w. > and J,“( 2w, > reflect the orientational disorder of the micelles mainly via the equilibrium orientational averaging of the surface-diffusion correlation functions rather than via the actual reorientational dynamics. Our treatment becomes strictly consistent in the slow rotation limit (4.8), which turns out to be a good approximation under the conditions of the present study.

V. MICELLE SIZE AND NEMATIC ORDER A. Computational procedure

We now turn to the quantitative analysis of the relaxation and splitting data to deduce the micellar axial ratio p and the nematic order parameter S,, . As illustrated by the flow chart in Fig. 9, these two parameters determine the three observables yp, J f ( w. ) , and J 2” ( 2~0, ) . The axial ratio p enters mainly via the order parameter S,, and correlation functions s;;‘“(t) for surface diffusion. [The reorientational correlation functions gmt (t) also depend on p via the rotational diffusion coefficients D,, and DL, but this dependence is weak as we are nearly in the slow-rotation limit (4.8).]

A FIG. 9. Flow chart showing how the primary parameters p and S,, affect the measured quadrupole splitting ho and lab-frame spectral densities Jf(o,,) and Jk(20, ). The spectral densities also depend on the secondary parametersJ,, D, (or b’/D, ), andf, whichare taken from theEphase. The connection between p and p: (t) , via the rotational diffusion coefficients D,, and 4, is weak and disappears entirely in the slow-rotation limit (4.8).

J. Chem. Phys., Vol. 96, No. $1 March 1992 Downloaded 30 Jan 2009 to 130.235.253.27. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp


The nematic order parameter S,, enters the spectral densities mainly via the orientational weighting of the surface- diffusion correlation functions [by the factors &$ (0) in the slow-rotation limit (4.8) ] and, to a lesser extent, directly via the reorientational dynamics. The spectral densities J f and J$ depend on three additional parameters: Jr, D, (or b 2/ D, ) and 2. These parameters are considered known and taken from our previous study of the E phase (cf. above) .2’ The absolute dimensions of the micelle (as opposed to its axial ratio,p = a/b) enters via the minor semiaxis b of the spheroid, which should be close to the length of the extended SDS molecule and virtually the same as the radius of the cylindrical aggregates in the hexagonal phase. The value of b determines the rate of micelle reorientation (D,, and D, ) and of orientational modulation by surface diffusion (D,/b 2). For the calculation of D,, and D, we set b = 2.0 nm, however, this value is not critical as we are almost in the slow-rotation limit (cf. above). For the surface diffusion, we take the rate parameter D,/b 2 (rather than the diffusion coefficient D, ) from the hexagonal phase, assuming that it has the same value in the nematic phases.

The computation proceeds as follows. For a given value ofp the quadrupole splitting yp and the residual qccx yields s . The pair of parameter values (p, S,, ) is then used (zth J/, b ‘/D,, and 2) to calculate J f ( w. ) according to the scheme in Fig. 9. The process is repeated with different p values until the experimental J f (w. ) is reproduced. [Note that the measured quadrupole splitting imposes a constraint on the calculations; when p (S,, ) is increased S,, must decrease so as to maintain the experimental ho value.] By carrying out the same analysis on J 4 ( 2w, ) we obtain a second, independent, estimate of p (and S,, ) . The whole procedure is then repeated for each experimental temperature and for both the nematic phases. Finally, we compare the resulting p( T) and S,, ( 7’) curves obtained from the two nuclei ( 2H and 23Na); since p and S,, are properties of the micelles, the two nuclei should give the same results (if the investigated samples are identical). The four independent determinations ofp and S,, (from J f and Jf for both nuclei) provide a thorough check of the consistency of the analysis.

In principle, the correlation functions & (t) in Eq. (4.7) should decay more rapidly for 23Na than for 2H as a result of intermicellar counterion exchange. As discussed in Appendix C, however, this difference between the two nuclei is negligible under the conditions of the present study.

6. Analysis of *H data

Figure 10 shows the results of the analysis of the 2H data from the Nc phase. The solid curves essentially define the range of allowed J: (km, ) values, given the measured yp and the secondary parameters Jr, D,/b 2 and 2 (from the E phase). The upper curve corresponds to perfectly aligned aggregates (S,, = 1) of minimal axial ratio p (determined by vo and 2) and the lower curve corresponds to infinitely long (p-+ CO ) aggregates with minimal order parameter S,, (determined by vo and 2). The dotted curves refer to various flxed p values. [Since a plot of Jt (km, ) vs p usually

20 ( , , 3 , I , , I , ,

=: SDA= 1

3” lo- . J . . . . . .._____. . IY . . . . . . . . . . . . . . . . . ..___.._... h” . . . . . . . . .._.._...__.___....-

. . . . . ..__...._............ s- f p=m

---~----------

n 8 lJf, 1 n s t s ’ ’ ’ -16 18 20 22 24 26 28

T/T

20 ( , ) , , , , , , , ,

1.5 - S,,A=l

5 I Jf I --------__ -_--

or ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ 16 18 20 22 24 26 28

TIT

2H NC :

FIG. 10. Analysis of *H (SDS) data from the NC phase. The dots are experimental data (error bars within symbols) and the curves are calculated using the experimental vp data, the secondary parameters J/ (shown explicitly), b ‘/D,, and2 from the Ephase, and the following primary parameters: solid curves as indicated; dashed curves from top to bottom, p = 3,4, 5, and 10.

exhibits a weak maximum at small p values,“’ the dotted curve corresponding to the smallest permissiblep value may lie (partly) above the upper solid curve.]

While the model qualitatively accounts for the ‘H No data, there is a quantitative discrepancy in the case of J$ (20, ). Furthermore, since the permissible range is so small, we cannot hope to determine p (and hence S,, ) . In particular, J$ is seen to be nearly independent of the axial ratio under the present conditions. The J f data suggest axial ratios in the range 2.5-3.0, but unless the J,” discrepancy can be explained this result must be treated with caution. It does not seem possible to remove the Ji discrepancy (while keep- ing Jf within its allowed range) by altering the values of the secondary parameters.

The ‘H data for the ND phase, shown in Fig. 11, look more promising. For this phase, the lower limit of the allowed range coincides with the J, curve since J”, (kw, ) = 0 for an infinite disk. (In contrast, for an infinite rod surface diffusion around the axis remains as a major contribution to Jk and, when S,, < 1, also to J f. ) As a consequence, both the spectral densities Jf ( w. ) and J$( 2w, ) depend more strongly on the axial ratio than they do for the Nc phase.

Taking the secondary parameters J/, D,/b 2, and? from the E phase, we find that the J f data yield small axial ratios,

J. Chem. Phys., Vol. 96, No. 5,l March 1992 Downloaded 30 Jan 2009 to 130.235.253.27. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp


20 , , , ( , , , ( , , ,

SD*=1 2H ND 1

15 J s

- i

L........................... 2.5

2 10 . . . . . . . . . . . . . . . . . . . . . . . . . .._ . 0 . 3

c i

cl I.

1

J

h” . . . . . . . . . . . . . . . . . . . . . . . .._.. 4 . . . . . . . . . . . . . . . . . ..__._.._..

5 5

j . . . . . . . . . . . . . . . .._....._._._ 10 - ---------_ ----

0, I I ,Jf< I I , I , , , - 18 u) 22 24 26 28 30

T/Y

20 , , , , , I, , , , ,

2H ND

15

Ta 1 s~A=l

\ . . . . ..“..“....~......... . 2.5

. . ..-................._._... q i

18 20 22 24 26 28 30 T / “C

FIG. 11. As in Fig. 10 but for the ND phase. The residual qcc used in the calculation is i(N,) = 1.10 X(E). Note that for the ND phase the lower bound (p = 03 ) coincides with the JI curve. The p values for the dashed curves are given in the figure.

p ~2.5-3, while the Ji data fall slightly above the calculated upper limit. We believe that the (small) discrepancy in the J t data is related to the discrepancy between the reduced *H and 23Na splittings (S,, > from the ND phase (Fig. 4). It is well-known’4*34~35 that the a-C-D bond order parameter, and hence the residual qcc F( *H), increases with decreasing surfactant headgroup area, as in going from the cylindrical aggregates in the E phase to the less curved oblate-spheroi- da1 micelles in the ND phase. On the basis of existing experimental and theoretical data on C-D bond order parameters in hexagonal and lamellar phases of binary systems’4P34*35 and the calculated SDS headgroup areas in the present Band No phases, we estimate that F(‘H,N, ) exceeds t( ‘H,E) by roughly 10%. The calculated curves in Fig. 11 were obtained with this correction, which removes the Jf discrepancy. It is reassuring to note that with the 10% correction applied, the Jf and J$ data yield closely similar axial ratios: pz 3.O.k 0.5 over the full temperature range of the ND phase. Although thep value would be slightly altered by a different jj correction, the conclusion appears inescapable that the micelles are relatively small and do not change their size significantly with temperature.

In the preceding we have tacitly assumed that the quadrupole coupling is motionally averaged over the full equilibrium distribution for the orientation 8, of a micelle with

respect to the mean director. A detailed study of line shapes and transverse relaxation rates’* reveals that this is indeed the case for 23Na but not quite so for *H. Since intermicellar diffusion is much slower for SDS than for Na + (cf. Appen- dix C), the longest wavelength director fluctuation modes are not effective in averaging the *H quadrupole coupling.52 As a consequence the *H nucleus sees a narrow, static local director distribution, giving rise to an asymmetric inhomogeneous broadening and a slight shift of the ‘H satellites towards the center of the spectrum. A numerical line shape fit yields a true quadrupole splitting yfie’ that is a few percent larger than the nominal splitting yp shown in Fig. 3. The true splitting #J is proportional to an incompletely averaged order parameter S & which is a few percent larger than the fully averaged order parameter S,, , which determines the orientational averaging of the surface-diffusion correlation functions (cf. Appendix B). Since the two corrections Y* -Pi and S& -+S,, partly cancel, the net effect of in- complete motional averaging is negligible in the present analysis.

C. Analysis of *3Na data

The results of the analysis of the 23Na data from the N, phase are shown in Fig. 12. As in the case of the 2H data in

23Na NC 1

251 I”“‘* 1 ,““‘I 15 20 25 30

T/‘C

75 , , I,, I,, I I,, , I

$ 50:TeNc

c --__

f ---_ Jf --+- ---_

1 1

1 251 1”“““““’

15 20 25 TIT

-I 50

FIG. 12. Analysis of Z3Na (counterion) data from the Nc phase. The dots are experimental data (error bars for .I: within symbols) and the curves are calculated using the experimental vp data, the secondary parameters J, (slightly corrected according to Fig. 13 ), b */D,, and Xp from the E phase, and the following primary parameters: solid curves as indicated; dashed curves from top to bottom, p = 3,4,5, and 10.


Fig. 10, J$ is seen to be virtually independent ofp. In fact, the allowed range is so narrow that the Jf data can be used, for example, to determine Jf with D,/b ’ and 2 taken from the E phase. The calculated curves in Fig. 12 are actually based on a J,( T) function obtained in this way. As is evident from Fig. 13, these J, values differ very little from those obtained (by extrapolation at the lower temperatures) from the E phase. The J f data in Fig. 12 suggest axial ratios in the rangepz3-4, with the lower values at the higher temperatures.

For the N,, phase, the 23Na Jf data yield small axial ratios (p z 3)) while the J 4 data fall well above the calculated upper limit. As in the case of the *H ND data, this discrepancy is probably due to a variation in the residual qcc j? between the ND and E phases and is related to the splitting discrepancy in Fig. 4. Writing” F = P&, we may ascribe such a variation to a difference in the fraction P, of counterions in the interfacial region and/or to a difference in the local residual x.5 in that region. Poisson-Boltzmann mean- field calculations of the ion distribution outside prolate and oblate spheroidal micelles, using the computational approach of Ref. 42, suggest that P, should be nearly the same in the ND and E phases. Any significant difference in jj between the two phases should thus mainly reflect a difference in Fs. In calculating the curves shown in Fig. 14, we have taken T( 23Na,ND ) = 1.25 jj( 23Na,E). This reasonable correction factor was chosen so as to make the ‘H and 23Na reduced splittings (SD, ) coincide (cf. Fig. 4). It is also reassuring that this correction not only removes the Jf discrepancy but also leads to results (p =: 3-4) that (i) are internally consistent (J f and J $ yield similar p values) and (ii) agree reasonably well with the *H results (Fig. 11).

D. Nematic order parameter

The preceding analysis of *H and 23Na relaxation and splitting data from the Nc and ND phases show that the micelles are rather small and that their size does not vary significantly within the investigated temperature range. In

40 III ,““,““,““,““I””

23Na

35 1

72 > 30: ‘1

“,i,;,y;;i;e ,,,,,,,,,, ‘\\i 10 15 20 25 30 35 40

T/V


75

5 . 2 50

x

” -.... . . . . . .._. . .- 2.5 . . . . . . . ‘.. . .._ .

+ . . . . . 3 - . . . . .

*.....-.. * ““V.. ..-..... i”‘... . . . .‘.... *.*4 -

. . . . . . ‘..._. . ____ . . . . . . . . . . 5 -

Jf f ---_ .-...... ---_ . . .._.

---- ... lo -- 251 “““““1 #‘I 1

15 20 25 30 T/V

75 I I,, I ,,, I (I,,,

SDA= 1 23Na ND

15 20 25 30 TIT

FIG. 14. As in Fig. 12 but for the N,, phase. The residual qcc used in the calculations is z(N,) = 1.25 f(E). Note that for the ND phase the lower bound (p = CO ) coincides with the J, curve. The p values for the dashed curves are given in the figure.

order to estimate the nematic order parameter S,, and its temperature dependence, we set p = 3.5 it 0.5 for both phases. According to Eqs. (4.3 ) and (4.4)) the surface diffusion order parameter then becomes S,, = - (0.41 + 0.02) for the NC phase sample and S,, = 0.67 + 0.06 for the N,, phase sample. The nematic order parameter S, can then be obtained from the quadrupole splittings in Fig. 3 using Eqs. ( 3.3 ), ( 3.4), and the 2 values employed in the preceeding analysis.

The temperature variation of the nematic order parameter S,, obtained in this way from the 23Na splittings is shown in Fig. 15 for the two phases. The orientational order is high at the lower temperatures, especially in the Nc phase where S,, is about 0.9, as obtained in the E phase.21 As expected, the nematic order decreases with increasing temperature. This decrease appears to be linear in the ND phase but is stronger in the NC phase as the nematic-isotropic transition is approached.

FIG. 13. The “Na (counterion) fast-motion contribution J, vs temperature. The data were obtained from the E phase (squares) and from the Nc phase (circles); in the latter case by requiring that J,“(20, ) falls within the allowed range in Fig. 12. The line is a linear least-squares fit to the filled data points and is identical to the J, curves included in Figs. 12 and 14.

For the Nc phase, we have included in Fig. 15 the predictions of the Maier-Saupe mean-field theory,” based on a potential of mean torque of the form (4.5) with a temperature-independent interaction parameter E = 2k, TA / (3S,, ) . Whereas this theory is generally in good agreement with experiment for thermotropic nematics,‘i we find in the present lyotropic phase a considerably larger S,, at low

J. Chem. Phys., Vol. 96, No. 5,l March 1992


0.6

0.4

0.2

nn -.-

T/Y

I~ ) , , , , , , , , , ) , ( L

15 20 25

FIG. 15. Nematic order parameter SD, vs temperature in the N, (top) and ND (bottom) phases as obtained from the 23Na quadruple splitting with p = 3.5 f 0.5 at all temperatures in both phases (as deduced from the relaxation data). For the NC phase, we also show the prediction of the Maier- Saupc theory (with TN, = 28.2 "C).

30

temperatures and a stronger temperature dependence in the vicinity of the nematic-isotropic transition. An even stronger temperature dependence of S,, , as deduced from electrical conductivity data, has been reported by Boden et al.” for the N,+ phase of the system cesium perfluorooctan- oate/water (4 = 0.35). In that case, the strong temperature dependence of S,, was attributed to a m ice&r growth (increasing p) on cooling. In the present case, however, the axial ratio appears to be essentially independent of temperature in both nematic phases.

T/V

As discussed in Sec. III B, the reduction of the quadrupole splitting yp from the value expected for perfectly oriented spheroidal m icelles of infinite axial ratio (or infinite cylinders and bilayers, respectively) is due to motional averaging by m icelle reorientation (SD, < 1) and by surface diffusion over the finite m icelle ( - 2S”, < 1 for the NC phase, S,,., < 1 for the iVo phase). Since we estimate (cf. above) - Zs,, = 0.82 for the NC phase and S,, = 0.67 for the ND

phase, it is clear from Fig. 15 that the two kinds of motional averaging are of comparable importance.

In some previous NMR studies of lyotropic nematics 2~12~13 “end effects” were neglected (i.e., S,, was set to its i -+ CO limiting value) and the reduced splitting (cf. SD,


in Fig. 4) was identified with the nematic order parameter S,, . (As we have seen, such an approximation may be gross- ly inaccurate.) The axial ratio of the m icelles was then estimated from S,, by invoking a highly simplified geometrical interpretation of the nematic order parameter in terms of noncooperative overlap effects of m icelles positioned on a lattice. We demonstrate in Appendix A that geometrical models of this kind yield values of S,, that are much higher than observed (and, of course, cannot account for the temperature dependence of S,, ). Rather, it appears that S,, reflects cooperative orientational fluctuations, possibly cou- pled to local density fluctuations. In fact, on the basis of *H and 23Na transverse relaxation data from the present nematic phases, we have recently ” shown that the nematic order parameter S,, has important contributions from director fluctuationsz9 (related to the macroscopic elastic constant of the phase) as well as from more localized (but presumably cooperative) reorientation modes. For the ND phase the localized modes appear to be dominant.52

VI. CONCLUDING DISCUSSION

The principal experimental techniques that have been used to determine the size (axial ratio, p) and orientational order (nematic order parameter, S,, ) of m icelles in uniaxial nematic phases are NMR quadrupole splittings, x-ray and neutron scattering, and electric conductivity. Through the present work we have added a fourth method to the list: NMR relaxation. Before summarizing the main features of the NMR relaxation method, we shall briefly discuss the existing methods and some of the results obtained from them. In all of these methods the relation between the experimental observables and the parameters of interest (p and SDA ) is nontrivial. In particular, the separation of the effects of m icelle size (p) and orientational order (S,, ) is a major problem. As noted above (Sets. III B and V D), this separation cannot be accomplished using only NMR quadrupole splittings.

X-ray and neutron scattering studies have been performed on several lyonematic phases.‘6P36*37P53 For the system sodium decyl sulphate/decanol/water, which is closely related to the present system, Hendrikx et aZ. I6 deduced axial ratios of 2.8 (independent of temperature and composition) in the ND phase and > 2.3 (with indications of considerable size polydispersity) in the NC phase. (The m icelle volume fraction, 4 = 0.43, was higher in this system than in ours, 4 = 0.27.) These results were obtained (without cor- recting for the intramicellar form factor) from scattering intensity maxima, interpreted directly as Bragg peaks giving effective intermicellar separations parallel and perpendicular to the nematic director. By adopting a (highly simplified) model for the spatial distribution of m icelles, the m i- celle volume was deduced and thereby, for a given value of the m icellar short axis, the axial ratio.

The angular variation of the scattering intensity also contains information about the nematic order parameter.16*17*37P54 The S,, values deduced in this way are high, usually in the range 0.8-0.9. For the ND phase in the system decylammonium chloride/ammonium chloride/water,



Holmes et al. 37 obtained S,, = 0.83 a few degrees from the nematic-isotropic transition temperature. It has been suggested,” however, that order parameters derived from scattering data do not reflect the full distribution function for the orientation of a micelle with respect to the macroscopic phase director, and that only the long-wavelength, collective, orientational fluctuations are manifested in the scattering data.

Electric conductivity measurements have also been used to deduce p and S,, in various lyonematic phases. 17V55-58 The analysis involves a number of simplifying assumptions. In particular, the effect of the local electric field around a micelle on the local conductivity tensor is not explicitly taken into account. Consequently, the results so far obtained with this method might not be quantitatively reliable.

A seemingly more direct approach to the determination of micelle size is the use of freeze-fracture electron micros- copy. This method has been applied by Sammon et aL3* to the ND phase in the system decylammonium chloride/ammonium chloride/water. It appears difficult, however, to obtain quantitatively accurate information about p and S,, with this method. Furthermore, considering that the mean surfactant residence time in an SDS micelle is only a few microseconds,59 there is reason to be concerned about possible artifacts due to structural rearrangements during the so- lidification (on a time scale of several milliseconds).

In this work we have explored a new method for determining p and S, in lyonematics from the effect on spin relaxation rates of surfactant and counterion diffusion over the curved micelle surface. Our results demonstrate that, for the present discotic (ND ) phase, p and S,, can indeed be accurately determined from the motional spectral densities J f (o,, ) and J 2” ( 2w, ) and the quadrupole splitting ho. For the present calamitic (N 2 ) phase, a separate determination of p and S,, is more difficult, requiring highly accurate J f ( w. ) data (and accurate secondary parameters-cf. below). For the N $’ phase, J f (2w, ) is insensitive to variations in p and S,, (under the constraint of fixed yp ) and hence, provides little information above that contained in the quadrupole splitting. This is a result of (i) the dominant contribution to J 4 (20, ) from the azimuthal surface diffusion spectral density J $( 2w, >, which varies little with p for p)3 and does not, like the other J E’s, vanish in the limit p-‘cQ, ‘* and (ii) a nearly compensating effect of increasing p and decreasing S,, [the former increases xN( 2w, ) , the latter decreases g$; (0) in Eq. (4.8) 1.

parameters p and S,, requires knowledge of the “secondary” parameters D,/b ‘, Jf and 2. In principle, all five parameters can be determined from the quadrupole splitting yp and the spectral densities J f ( w0 ) and Jt (2~3, ) measured at two different magnetic fields (i.e., two different Lar- mor frequencies, w0 ). Even with spectral density data from a single magnetic field, one of the three secondary parameters can be eliminated: Jf by forming the difference J f (2w, ) - Jf(to), or X by forming the ratio [J4(2w,) - Jf] / [ J f ( w0 ) - J,] . In the present work, however, we

have chosen to analyze the J f ( w0 ) and J 2” ( 2w, ) data separately to display the internal consistency (or lack thereof) in the data. For this analysis, the secondary parameters were obtained from the nearby hexagonal phase. Some variation in these parameters can be expected between the hexagonal phase and the nematic phases. Indeed, we found it necessary to slightly adjust the 2 values deduced from the hexagonal phase. Although these corrections were guided partly by a comparison of our 2H and 23Na data and partly by independent sources of information, it would be preferable to avoid corrections by actually determining X in the nematic phases (by varying the magnetic field). The other two secondary parameters, D,/b 2 and Jf, are less critical for the analysis. For 2H the relative contribution of Jf to the spectral densities J f and J f is modest (cf. Fig. 5 ) so that a considerable uncertainty in J, can be tolerated. For both nuclei D,/b 2 z=wo, making the director-frame spectral densities J “, (w, ) and J fl, (20, ) insensitive to modest variations in D,/b *. Fur- thermore, J, and D, are not expected to be as strongly dependent on the local structure of the interface (curvature, surface charge density, etc. ) as the residual qcc X, which results from the near cancellation of large terms (i.e., 2 4 (x2) “2). In the case of 23Na, this expectation is supported by a recent molecular dynamics simulation study.48

It should be emphasized that the previous remarks ap- ply only to the nematic phases investigated here. In nematic phases with smaller micelles (p-2, say), the approach is expected to be even more powerful, as the surface-diffusion spectral densities then are more sensitive to changes in p. l8 Moreover, in nematic phases of reversed diamagnetic susceptibility anisotropy, a62 the orientation dependence of the spectral densities J i (kw, ) is different. Thus, for example, J,” (2w, ) from an NC phase, as given by Eq. (4.2b), is dominated by J~(200) and Jf(2o1,) rather than by Jf( 20, ) (as for an N $ phase).

The present approach for determining the “primary”

In conclusion, it appears that the NMR relaxation method can become a valuable complement to the existing methods for determination of micelle size and orientational order in uniaxial nematic mesophases. To exploit the full potential of the NMR method, one should perform relaxation measurements at several magnetic fields, thus obviating the need for reference data from related mesophases. (We are currently installing a field-variable resistive magnet that will permit such studies.) Biaxial nematic phases63 could, in principle, be investigated by the same approach. Although there are now four primary parameters to be determined (two axial ratios and two nematic order parameters), this is partly compensated for by the additional experimental information available.64 In the case of uniaxial nematic phases composed of orientationally disordered biaxial aggregates,36 however, the increased complexity of the theoretical analysis is not compensated by the availability of new observables.

In the only previous NMR relaxation study of a lyonematic phase, Wong and Jeffrey” presented *H J f ( w. ) and Jk (2w, ) data from the a-deuterated surfactant in the No phase of the system potassium laurate/decanol/potassium chloride/water. On the a priori assumption of very large micelles, these authors interpreted their J f ( w0 ) and J f ( 2~0, ) data in terms of collective director fluctuations.29*6s*66 It ap-

J. Chem. Phys., Vol. 96, No. 5,i March 1992


pears, however, that director fluctuations do not contribute significantly to J f ( w. ) and J f (2w, ) in lyonematics (at conventional magnetic fields).52 We thus believe that the data of Wong and Jeffrey can be reinterpreted along the lines presented here, with the probable conclusion that the mice&s are rather small. On the other hand, director fluctuations make a very large contribution to the zero-frequency spectral density Jk (0) (and, hence, to the transverse relaxation rates), which can be exploited to gain information about the viscoelastic properties of the mesophase.52

ACKNOWLEDGMENTS

We are grateful to Professor Tuck C. Wong for intro- ducing us to the investigated phases and for his kind helpful- ness and to Dr. Mikael Landgren for performing the Pois- son-Boltzmann calculations. This work was supported by grants from the Swedish Natural Science Research Council.

APPENDIX A: LOCAL CONTRIBUTION TO THE NEMATIC ORDER PARAMETER

In this Appendix we show that the nematic order parameter S,, cannot be given a simple-minded geometrical interpretation in terms of orientational fluctuations of a spheroidal micelle within the “cage” defined by the neighboring micelles in an average configuration. Neglecting any intermicellar interaction apart from the hard-core repulsion and assuming that the micelles are positioned on a regular lattice and, except for the reference micelle, are perfectly aligned with the director, we may write6’

SE” =~cosB,(l +cos8,), (AlI where 0, is the maximum angle (where the reference micelle touches one of its neighbors) between the director and the symmetry axis of the spheroidal micelle.

Next we relate the cone angle 19, to the micelle volume fraction r$ and the micelle axial ratiop = a/b, where a and b are the major and minor semiaxes of the spheroid. The cage is taken to be a spheroid of the same axial ratio p = A /B as the micelle. Assuming that the micelles are positioned on a uniformly expanded regular lattice, we obtain for the expansion factor

K = A/a = B/b = 2(4,/5~%)“~ - 1, (A21 where rPcp = rr/ ( 31/2 ) z 0.74 is the volume fraction of regu- larly close-packed (hcp or ccp) micelles. By some straight- forward geometrical considerations we can express the angle 6, in terms ofp and K as

19, = arcsin[p(? - l)/~(p~ - l)]. (A3) (This problem has previously been considered6’ in connection with dielectric relaxation, but no analytical solution was obtained. >

By combining Eqs. (Al )-( A3 ) we can now obtain the local order parameter SE” as a function of the axial ratio p and the volume fraction 4. For example, for the case p = 3 and 4 = 0.5 17, we obtain S h; = 0.978. For a hard-spheroid fluid with these parameter values (at the nematic-isotropic transition), density functional theories69 as well as Monte

Carlo simulations70 yield S,, ~0.55. It is clear therefore that, even in a hard-spheroid fluid, the nematic order parameter cannot be interpreted in the simple geometrical terms described above.

For the parameter values, p = 3.5 and 4 = 0.27, appropriate for the lyotropic nematic phases studied here, we find S E = 0.89. If electrostatic repulsive forces were included an even larger order parameter would be obtained.

APPENDIX B: LOCAL VERSUS COLLECTIVE MICELLE REORIENTATION

In the theoretical development in Sec. IV B we disre- garded a complication concerning the reorientational correlation functions g$ (t), which were associated with restricted rotational diffusion of individual micelles in the time-independent potential of mean torque (4.5) determined by the nematic order parameter S,, . Actually micelle reorientation in a lyonematic phase is a more complex process, involving local as well as highly collective modes. We describe micelle reorientation as a process with a rate gov- erned by the rotational diffusion coefficients (D,, and D, ) for individual micelles but with an amplitude determined by a nematic order parameter (S,, ) that partly reflects highly cooperative orientational fluctuations. Although this description is inconsistent, we believe that it is sufficiently accurate for our purposes. This can be seen as follows.

The contributions to the spectral densities J f (o. ) and J$ (2w, ) from the first, tumbling-mode, term in Eq. (4.7) is overestimated in our description. If micelle tumbling is treated as a single-micelle process, then the fluctuation amplitude g”$ (0) should be calculated with a local order parameter (cf. Appendix A) which is considerably higher than the nematic order parameter S,, appearing in the quadrupole splitting, thus reducing g{ (0). If, on the other hand, micelle reorientation is treated as a collective process, then the effective correlation times r Ei would be much longer than we calculate from the single-micelle tumbling diffusion coefficient DL. As war ii ) 1 already in our treatment, a further increase in r $ would reduce the tumbling-mode contributions to the spectral densities. However, since even our overestimated tumbling-mode terms contribute merely a few percent to the spectral densities J 4 (w. ) and J k (2~0, ) , the error made cannot be significant. The situation is radical- ly different for the zero-frequency spectral density J;(O), which is strongly influenced by collective tumbling modes (director fluctuations) .52

Under the conditions of the present study, the slow rotation limit is an accurate approximation so that pm< (t) in the cross terms of I$. (4.7) can be replaced by fm{ (0) as in Eq. (4.8). Our use of the nematic order parameter S,, for the fluctuation amplitudes e, (0) then amounts to averaging the orientation-dependent correlation functions g” ( t ) , due to surface diffusion, over the full micelle distribution. This is indeed the correct procedure if, as is the case here, the rate of exchange of nuclei among micelles of different orientations (cf. Appendix C) is fast compared to the corresponding difference in the spin relaxation rates.

Quist, Halle, and Furb: Spin relaxation in lyonematics 3869



APPENDIX C: EFFECTS OF INTERMICELLAR COUNTER-ION EXCHANGE

In the present work we have analyzed NMR data from the SDS molecules (‘H) as well as from the sodium counterions ( 23Na). As shown, the two sets of data yield essentially the same results for the micelle size and orientational order, thus confirming the validity of our approach (at least for the iVD phase). This agreement between the ‘H and 23Na results may appear surprising in view of the very different long- range mobility of surfactants and counterions in a nematic phase: the mean residence time of an SDS molecule in a micelle is of order 10 - 5 s (Ref. 59) while the mean time taken for a Na + counterion to diffuse from the surface of one micelle to the surface of a neighboring micelle can be estimated (using the Smoluchowski diffusion equation with a Poisson- Boltzmann mean potential, along the lines of Ref. 7 1) to ca. 10 ns in the NC phase and ca. 20 ns in the No phase. Whereas the SDS residence time is far too long to influence the spectral densities J f (w, ) and Ji( 2w, ), the counterion residence time is in a range where it might have an effect. There are two effects to consider. First, the effective correlation times for the tumbling modes [first term in Eq. (4.7) ] will be reduced. Second, the effective correlation times for the surface diffusion modes [second term in Eq. (4.7) ] are reduced.

s;;‘“(t) have decay times in the range 2-l 5 ns. [The slowest decay appears in the azimuthal correlation function g;““(t) for oblates.] Since some of the decay times are shorter than l/w, while others are longer, the effect of intermicellar counterion exchange on the spectral densities J f ( w0 ) and J 4 ( 2~0, ) is partly cancelled. An (over) estimate of the effect of exchange on the surface diffusion modes can be obtained by multiplying the second term in Eq. (4.7) by exp( - t /reX ) . Explicit calculations show that the effect on J f ( w0 ) and J i ( 20, ) is of the same order of magnitude as the experimental error. The effect of intermicellar counterion exchange can thus be neglected, at least under the conditions of the present study.

Since the tumbling modes are slow compared to l/w, [e.g., 7 fz 20-40 ns as compared to l/w, ( 23Na) = 6.0 ns] , intermicellar counterion exchange enhances the tumbling- mode contributions to the spectral densities. If one were to correct for counterion exchange simply by multiplying the first term in Eq. (4.7) by exp( - t/7,, ), then the lower bounds on the calculated J f ( o0 ) and J k (2w, ) in Figs. 12 and 14 would be significantly increased, thus leading to larg- erp values. Such a correction, however, would not be appropriate, since it corresponds to a complete randomization of the micelle orientation [within the distribution f( 8, ) determined by the potential of mean torque (4.5) ] and, hence, to a total neglect of orientational pair correlations. Adjacent micelles are likely to be strongly correlated in orientation, whence the tumbling “amplitudes” g$A (0) in Eq. (4.7) modulated by nearest-neighbor counterion exchange should be calculated with the order parameter for the relative orientation of two micelles rather than with the nematic order parameter S,, . As the former order parameter should be close to unity (strong orientational correlation), the appropriate amplitudes 2: (0) are very small, whence the effect of counterion exchange on the tumbling modes is negligible.

For small micelles (p<2, say), the surface diffusion correlation functionsdN( t) decay on a time scale given roughly by b ‘/D, z 2-3 ns, and are thus virtually unaffected by intermicellar counterion exchange with T,, 220 ns (cf. above). For large micelles, the dominant terms in the nonexponen- tial surface diffusion modes (except the n = 2 mode for prolates) are no longer fast compared to r,, , but then the fluctuation amplitudes g;;‘“( 0) are so small (except the n = 2 mode for prolates) that surface diffusion ceases to be a significant relaxation mechanism. For an intermediate axial ratio p z 3.5, the dominant (and most slowly decaying) terms in

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