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Nuclear spin relaxation in a hexagonal lyotropic liquid crystal 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 8 April 1991; accepted 24 July 1991)

The hexagonal (E) phase in the sodium dodecyl sulphate (SDS)/decanol/water system is investigated by 2H and 23Na nuclear magnetic resonance (NMR) of the selectively deuterated SDS and the sodium counterion. Using macroscopically oriented E phase samples, prepared from the magnetically aligned nematic (Nc ) phase, we measure the orientation-dependent relaxation rates R, z and R 1p as well as the line shape of both nuclei. The orientation dependence of the lab-frame spectral densities, determined from the relaxation rates, allow us to separate contributions from different types of molecular motion. In particular, we find a dominant contribution from molecular diffusion around the cylindrical aggregate. From this contribution we determine the lateral diffusion coefficient of SDS to ( 1.4 & 0.2) X 10 - lo m2 s-’ at 25 “C (activation energy 26 -& 2 kJ mol - ’ ) and the counterion surface diffusion coefficient to (4.8 & 0.9) X 10 - lo m2 s - ’ at 25 “C (a factor 2.8 smaller than in an infinitely dilute aqueous electrolyte solution). Furthermore, the flexibility of the cylindrical aggregates in the investigated E phase (aggregate volume fraction 0.27) is quantified in terms of an orientational order parameter ~0.9.

I. INTRODUCTION Over the past three decades nuclear magnetic resonance

(NMR) techniques have been widely used to study thermo- tropic and lyotropic liquid crystals.‘-3 In the lyotropic field the emphasis has been on static quadrupole effects (line splittings and powder line shapes) in NMR spectra of I> 1 nuclei, which can provide qualitative information about mesophase symmetry as well as quantitative information about orientational order and spatial distribution of the molecular constituents. Whereas such static NMR experiments are now routinely performed in many laboratories, NMR relaxation experiments on lyotropic liquid crystals are more de- manding in several respects. First, more instrumental so- phistication is required,2*4 especially for high-spin nuclei.5-7 Second, it is essential to work with macroscopically aligned samples, which can be difficult to prepare in the case of smectic (as opposed to nematic) lyomesophases. Finally, the data analysis leading to the desired molecular-level information is often complicated and sometimes requires considerable theoretical effort. For these reasons, relatively few’ orientation-dependent nuclear spin relaxation studies of aligned smectic lyomesophases, other than phospholipid bilayers,’ have been reported to date. Over the past two decades a number of relaxation studies have been performed on surfactant” and water” nuclei in lyotropic mesophases, but then either using unoriented (powder) samples or investi- gating only a single orientation.

Lyotropic liquid crystals are well suited for nuclear spin relaxation studies. As with isotropic fluids, magnetic field variation can be used to separate relaxation contributions on the basis of the time scale of the underlying motions. With aligned smectic mesophases, one can also symmetry-select molecular motions by varying the orientation of the phase director with respect to the magnetic field.* Furthermore, the task of separating structural and dynamic factors in the

spin relaxation rates can be simplified by making use of the complementary structural information provided by spectral line splittings and by small-angle x-ray diffraction data. Fin- ally, the long-range order and simple geometry of the surfactant aggregates enables sufficiently realistic, yet mathemat- ically tractable, structural-dynamic models to be used for calculating the time correlation functions that govern the relaxation behavior.

In the present work we study the hexagonal mesophase in the system sodium dodecyl sulphate (SDS)/decanol/water. Besides the classical lamellar (D) and hexagonal (E) liquid-crystalline phases and isotropic solution phases (L, ,L, ), the ternary phase diagram of this system exhibits two uniaxial nematic phases (N,,N, ) .I* Macroscopically oriented E phase samples were prepared simply by increasing the temperature of a magnetically aligned N, phase sample. Two quadrupolar nuclear species were used for the NMR experiments: ‘H at the CT position of the selectively deuterated SDS surfactant and 23Na in the counterion. For each nucleus we measured the Zeeman (R ,z ) and quadrupolar (R, o ) longitudinal spin relaxation rates as well as line shapes and quadrupole splittings. (Transverse relaxation

rates R, were also measured, but will not be considered in detail in this work.) All measurements were performed at two different mesophase orientations and over a range of temperatures. To facilitate the NMR data analysis, we also performed an x-ray diffraction experiment.

The present study was undertaken with two objectives in mind: (i) to identify the major nuclear spin relaxation mechanisms in hexagonal lyomesophases, and (ii) to deduce molecular-level dynamic and structural information. As re- gards the first objective, we found that the longitudinal relaxation (R 1z and R ,o ) is mainly due to molecular surface diffusion around the (slightly orientationally disordered) cylindrical aggregate axes. As to the second objective, we

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6946 Quist, Halle, and Fur& Spin relaxation in liquid crystals

could determine the lateral (surface) diffusion coefficients of the surfactant (SDS) and the counterion (Na + ).

The outline of this paper is as follows. In Sec. II we describe the NMR experiments, with special attention de- voted to potential sources of systematic error, as well as the x-ray experiments, and present all the primary data. The orientational dependence of the NMR observables is analyzed in Sec. III, while the further analysis of the relaxation data is contained in Sets. IV (‘H) and V ( 23Na). The effect on the 23Na relaxation of (3-dimensional) translational diffusion of counterions is considered in an Appendix.

micellar phase were weighed into a new glass ampoule yielding a clear, viscous, anisotropic, birefringent, nematic (N, ) phase’* after mixing at 19 “C. Finally, the sample was brought into a clear, highly viscous, anisotropic, birefiin- gent, hexagonal (E) phase by simply increasing the temperature to 25 “C!. The composition of sample I (II) was 26.3(25.9)/3.5(3.5)/70.2(70.6) wt% SDS/decanol/water, which, taking into account the different isotopic composition of the water and SDS, yields identical mole ratios in the two samples: water/(SDS + decanol) = 34.6 and decanol/( SDS + decanol) = 0.196.

II. EXPERIMENT 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 (Lund, Sweden) was purified by repeated recrystallization from aqueous solution. The water was either deuterium-de- pleted H,O from Sigma, or millipore-filtered H,O with 10% D, 0 added (for easy mesophase identification by *H NMR).

For sample II, the Nc phase was found to be stable from 17 to 21 “C (below 17 “C the SDS precipitates). The two- phase coexistence region extended from 2 1 to 22.5 “C and the E phase (which at this stage consists of randomly oriented crystalline domains) was stable from 23 “C up to at least 40 “C. For sample I, the phase boundaries were shifted by less than 1 “C, probably due to minor differences in SDS purity and sample composition.

6. Magnetic alignment

Liquid-crystalline samples were prepared in the following manner. Either a-deuterated SDS and deuterium-deplet- ed H, 0 (sample I, for surfactant ‘H NMR) or regular SDS and the H, O/D, 0 mixture (sample II, for counterion 23Na NMR) were weighed into a glass ampoule, tightly sealed with a teflon-coated screwcap. Upon mixing SDS and water a clear, fluid, isotropic, micellar (L, ) phase was obtained (cf. the partial phase diagram in Fig. 1) . Next, decanol and

12

68 /#A

Ha0 24 26 28 30 32 34 SDS b

FIG. 1. Partial phase diagram (weight percent) of the system SDS/decanol/Hz 0 at 25 “C showing the extension of the one-phase regions of the isotropic micellar (L, ) phase and the hexagonal (E), lamellar (D), and nematic (N,, N,) liquid-crystallinephases. (Thedashedphase boundaries are only indicative.) As compared to a previously published (Ref. 12) phase diagram for this system, our Nc phase extends to higher SDS content while the E phase terminates at lower decanol content. The dot in the E phase corresponds to the composition of the investigated sample. At 20 “C this sample is in the Nc phase.

Macroscopically aligned E phase samples were prepared by first inserting the N, phase sample at 19 “C in a magnetic field (2.35 T) for at least 2 h. During this period the local nematic directories are magnetically aligned to produce a homeotropic ZVc phase. The alignment process was followed via the change in the water, SDS, and counterion NMR spectra from powder line shapes to homeotropic line shapes with two (‘H) or three (23Na) apparently Lorent- zian peaks. Since the quadrupole splittings in the spectra of the aligned Nc phase were twice as large as the powder splittings, it follows that the No phase-is uniaxial with the phase director parallel to the magnetic field, as expected from previous studies of this phase.12 When no more changes could be detected in the spectra, the temperature was raised to 25 “C. During this temperature-induced phase transition the alignment of the Nc phase was essentially preserved, yielding a macroscopically aligned E phase with the director along the magnetic field. In the NMR spectra, the phase transition was seen as an increase of the splitting (by 28% for a-SDS ‘H and by 26% for 23Na) and an asymmetric broadening of the satellites. However, the satellite linewidths in the newly aligned E phase decreased with time (at constant temperature) approaching an apparent equilibrium value after a few days, e.g., at 25 “C the 23Na satellite linewidth at half height decreased from ca. 1500 to 370 Hz after 9 days. These features imply that the transfer of alignment during the phase transition is incomplete (cf. Sec. III A). Since the line narrowing was faster at elevated temperatures, the E phase was equilibrated at 31 ‘X! until no further change in the spectrum could be detected (usually after 12-24 h ) . Interestingly, the efficiency of this annealing process was not dependent on the presence of a magnetic field. The alignment procedure was reproducable and temperature cycling did not improve the alignment transfer.

Once induced, the alignment of the E phase persists (due to the high viscosity), thus allowing NMR experiments to be performed at nonzero angle BLD between the magnetic

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field and the phase director. This angle was changed simply by twisting the sample tube around its axis inside the coil of the NMR probehead. No hysteresis effect could be seen in the line shape when the sample was returned to the 19,~ = 0” configuration after several days at 8,, = 90”.

C. Spectrometer characteristics

All NMR experiments were performed on a Bruker MSL- 100 spectrometer (resonance frequencies 15.371 and 26.487 MHz for ‘H and 23Na, respectively), equipped with a horizontal 10 mm solenoid probe and a 2.35 T wide-bore superconducting magnet.

During the NMR experiments the E phase sample was contained in a sealed glass ampoule (8 mm i.d., 18 mm length) centered in the 25 mm long solenoid coil, yielding a spatial rfinhomogeneity less than + 3%. The magnetic field inhomogeneity for *“Na was ca. 10 and 20 Hz at 19,~ = 0” and 90”, respectively. (The difference may be due to the sus- ceptibility anisotropy of the oriented sample.) A temperature regulator (Stelar VTC87) with high ( 1.5 m3/h) airflow provided + 0.04 “C temperature stability and ~0.08 “C temperature gradients within the sample.

Considerable care was taken to minimize systematic errors in the various relaxation experiments. A large filter width (500 kHz) was used to avoid line shape distortions. All phase cycles were extended to include the CYCLOPS cycle to minimize the effects of detector offset. The software- controlled rf modulator as well as quadrature detector set- tings were finely tuned to provide the same response for nominally equivalent transmitter phase-receiver phase combinations. The accuracy of the setting of the four orthogonal phases was checked and found to be better than 1”. Correc- tions were made for the nonrectangular rf pulse shape. For pulse lengths longer than 2 ,us, we found a linear relationship with nonzero intercept between the nominal pulse length and the effective pulse angle (obtained by measuring the signal intensity variation with increasing pulse length). This linear relationship was determined from at least 10 points, and the nominal pulse lengths corresponding to the required pulse angles were obtained by interpolation. This procedure enabled the pulse angles to be set with better than 0.5” accuracy.

Typically, the 180” pulse lengths were 17 and 9 pus for *H and ‘“Na, respectively. As the frequency spectrum of these resonant pulses (applied at the central line for 23Na and half- way between the two satellites for *H) shows less than 5% attenuation at the satellite positions, the pulses were considered nonselective. This is further supported by the reproducibility of the *“Na results (at 8,, = 0” and T = 25 “C) with decreased pulse power (up to 14 ,us pulse length for a 180” pulse).

D. Relaxation experiments

All relaxation experiments reported here are based on the variation of satellite peak intensity with pulse delay time (T). The central line in the 23Na spectrum was not used as the imperfect alignment (especially at the higher tempera-

tures) leads to partial averaging of the 8,, dependence of the relaxation rates.8 For both *H and 2”Na the spin relaxation can be considered as arising exclusively from the coupling of the nuclear quadrupole moment with the electric field gradient at the nucleus.

Inversion recovery. The longitudinal relaxation rate, R IZ? was determined by the ( a-)m - T - (a-/2), - acq. pulse sequence where the phase of the first pulse was cycled around the four orthogonal phases to suppress the genera- tion of coherences.‘3 The single-exponential evolution of the satellite intensity with increasing 7 yields the relaxation rate4v6

R,z = Jf + 4J[ (I= l,*H),

2.J; (I = 3/2,23Na). (2.1)

For convenience, the lab-frame spectral densities J i E J k (kw, ,eLD ) are defined so as to include spin-dependent numerical factors and coupling constants. For I= 1 nuclei R,, is usually referred to as the Zeeman relaxation rate since it reflects the evolution of rank-l (dipolar) magnetic polarization. For I = 3/2 nuclei, however, this term is inappropriate since R, z then reflects the coupled evolution of dipolar and octopolar polarization and only in the special case of a 90” detection pulse angle is the satellite inversion recovery exponential.6

Quadrupole polarization decay. The Jeener-Broekaert pulse sequencei (r/2)4 - 71 - (T/4),,,, - 7- - (r/4), - ( f )acq., with the phase cycle 4 = 0,7r/2, rr,

31r/2,6*‘5 was used to determine the relaxation rate4p6

R,, = 35: (I= l,*H),

Wf-bWf (I = 3/2,23Na). (2.2)

The fixed delay time r, was set to 1/(2~o) or l/( 4~~) (where yQ is the quadrupole splitting in Hz) for 2H and 23Na respectively, in order to maximize the conversion of dipoiar to quadrupolar polarization.6

In both relaxation experiments, an acquisition delay of 30-100 ,us was introduced to avoid distortions due to probe ringing and receiver deadtime. In order to avoid baseline offsets due to first-order phase corrections, the acquisition delay was set to a multiple of l/~o or 1/(2~o) for *H and 23Na, respectively. Since the investigated E phase was aligned, the line shape distortion introduced by this acquisition delay is negligible (Fig. 2). The experiments were performed with 24 different delay times r in the approximate range [1/20-201/R, where R is the appropriate relaxation rate. A peak signal-to-noise ratio of better than 100 was obtained for the satellite peaks with the shortest delay time (Fig. 2). The repetition time between successive accumula- tions was always at least 10/R. Finally, a 3-parameter least- squares fit to the satellite peak intensity vs delay time, according to I(T) = A + B exp( - Rr), yielded the desired relaxation rate. The experimental random error ( +- 2~) was estimated to + 1.2% in R ,z and + 2.0% in R ,o for both nuclei and at both orientations.

Quadrupolar echo. The *H transverse relaxation rate R t was determined in a quadrupolar echo experiment. We

Quist, Halle, and Fur6: Spin relaxation in liquid crystals 6947

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6948 Quist, Halle, and Fur6: Spin relaxation in liquid crystals

FIG. 2. ‘H (a) and Z3Na (b) spectrum from the quadrupole polarization decay experiment performed at El,, = 0” and 25 “C. The spectra corresponds to the shortest delay time 7 and were obtained as described in the Sec. II D.

(b)

t # I I a_ . 20000 10000

H&Z -10000 -20000

used the phase-cycled pulse sequence7v’6 (r/2), - r - (B), - r - acq., which refocuses both quadrupolar and magnetic field inhomogeneous broadening and R : was obtained directly from the decay of the echo amplitude with r. The uncertainty in R F was estimated to + 5% at both orientations.

In the absence of dipole-dipole couplings, R z equals the transverse quadrupolar relaxation rate4

R, =;J,t+$Jf+JL 2’ (2.3)

However, on account of the residual static *H-*H dipole coupling within the a-CD, segment of the SDS molecule, each quadrupole satellite is actually composed of three ab- sorption lines. “*” Furthermore, each of the deuterons is

dipole coupled to the aliphatic protons further down the surfactant chain, thus masking the fine structure due to the *H-*H dipole coupling. Eliminating the residual static *H-‘H couplings by broadband proton decoupling, we found that the larger of the two 2H-2H dipole splittings is ca. 90Hz(atQ,, = 0”)) which is comparable to the (nondecou- pled) linewidth R T/n deduced from the quadrupolar echo decay (cf. Table I). In the present work we have not at- tempted to determine the quadrupolar zero-frequency spectral density Jk in Eq. (2.3) .19 The quantity R T is used only as a measure of the linewidth corresponding to a uniform domain orientation for the purpose of estimating the orientational spread of the domains in our partially aligned samples (Sec. III A).

J. Chem. Phys., Vol. 95, No. 9,i November 1991


TABLE I. Surfactant ‘H NMR results from aligned E phase sample I.

TX B,,/deg uo/kHz Au’$/Hz R,,/s-’ R,,/s- ’ RT/s-’

25.0 0 21.22 680 65.7 18.3 416 90 10.68 305 35.3 46.3 280

27.0 0 21.04 780 65.0 17.0 396 90 10.64 295 34.2 43.6 255

29.0 0 20.99 810 63.4 16.2 393 90 10.62 290 33.0 41.0 233

31.0 0 20.92 900 61.5 15.2 381 90 10.58 305 31.6 39.5 224

35.0 0 20.68 1200 58.9 13.4 365 90 10.52 310 28.9 35.0 215

E. NMR results

The primary NMR data comprise the spectral line shape, partially characterized by the quadrupole splitting up and the inhomogeneous satellite linewidth Alj;/h the spin relaxation rates R,, and R,,, and (for ‘H) the effective transverse relaxation rate R F. Our results for these quantities are collected in Table I (surfactant ‘H NMR, sample I) and Table II (counterion 23Na NMR, sample II). (Al- though not included in Table II, 23Na R, measurements, using a 2D quadrupolar echo technique,7 were performed at two temperatures; however these results are only used in Sec. III A to deduce the static orientational disorder. ) All experiments have been performed at two mesophase orientations (19,~ = 0” and 90”) and at several temperatures in the range 23-35 “C. The estimated random errors are f 30 Hz in up and f 5% in A$;. The reproducibility of the data in Ta- bles I and II was checked by repeating all measurements at BLo = 0” and 25 “C on replicas of samples I and II and was found to be better than the estimated random errors given in the tables. This was also the case when comparing the 23Na quadrupole splitting, inhomogeneous satellite linewidth, and relaxation rates R ,= and R 1p measured on samples I and II. The two samples may thus be regarded as identical.

With the aid of Eqs. (2.1) and (2.2)) the spectral densities J f and J f can be calculated from the relaxation rates R,, and R,, at each orientation. The results in Table III

TABLE II. Counterion “Na NMR results from aligned E phase sample II.

TfC

23.0

25.0

27.0

29.0

31.0

35.0

@DDE uo/kHz Avlnh /Hz 112 R IL /s-’ R&s ’

0 18.47 230 131.2 203.7 90 9.24 140 68.0 184.3

0 18.31 370 123.2 192.3 90 9.16 150 63.5 171.6

0 18.14 610 117.1 181.0 90 9.08 160 59.7 162.1

0 18.01 730 112.4 167.2 90 9.02 170 56.2 149.2

0 17.89 620 106.4 158.0 90 8.97 140 53.8 141.2

0 17.63 940 95.3 141.2 90 8.85 150 48.2 124.7

TABLE III. 2H and 23Na lab-frame spectral densities from aligned E phase samples.

‘H *“Na

TPC e,,/deg Jf/SK’ J:/s-’ J:/s-’ J:/s-’

23.0

25.0

27.0

29.0

31.0

35.0

Error ’

0 36.3 65.6 90 58.2 34.0

0 6.10 14.90 34.6 61.6 90 15.42 4.98 54.1 31.8

0 5.67 14.83 32.0 58.6 90 14.54 4.9 1 51.2 29.9

0 5.40 14.50 27.4 56.2 90 13.67 4.84 46.5 28.1

0 5.07 14.11 25.8 53.2 90 13.17 4.61 44.1 26.9

0 4.47 13.61 23.0 47.7 90 11.67 4.31 38.3 24.1

0 * 0.13 + 0.17 f 1.9 +0.7 90 + 0.32 +0.12 + 1.6 + 0.4

Quist, Hall@ and Fur& Spin relaxation in liquid UyStalS 6949

“Propagated random errors in R,, and R,,.

reveal a strong orientational dependence in both spectral density functions: Jf(O”) <Jf(90”) andJf(O”) >Jf(90”). Although the orientation dependence in the 23Na data is somewhat weaker it is qualitatively similar to that in the ‘H data.

F. X-ray diffraction A small-angle x-ray diffraction experiment was per-

formed on sample II at 25 “C. This E phase sample was not magnetically aligned as the NMR samples, i.e., it was a powder sample consisting of randomly oriented crystalline domains. The Bragg reflections were recorded directly on film in a Kiessig camera, using monochromatic Cu K, radiation (;1 = 0.154 nm) . The sample, in a sealed glass capillary (i.d. 1 mm), was exposed for 24 h with a temperature stability of + 1 “C.

The diffraction pattern clearly revealed three reflections with Bragg spacings of 5.94, 3.44, and 2.97 nm, in close agreement with the theoretical Bragg spacing ratios 1: l/1/3: l/2 expected from a two-dimensional hexagonal lattice.20 The reflections were, however, somewhat more diffuse than commonly observed from hexagonal mesophases.2’ This may indicate some structural disorder in our rather dilute (aggregate volume fraction, 4 = 0.27) E phase sample. The lattice parameter, i.e., the separation between adjacent cylinder axes, is d = 2d,,/2/3, where d,, = 5.94 nm is the Bragg spacing of the innermost reflection. Hence, d=6.87+0.1 nm.

The radius b of the cylindrical aggregates, consisting of SDS and decanol, may be calculated from the lattice parameter d as

b = 4 (A)“‘, (2.4) 2 \+o/

where 4 is the volume fraction aggregates in the sample and $. is the volume fraction at close packing. From the known sample composition (cf. Sec. II A) and the partial specific volumes20*22 0.85, 1.18, and 0.99 ml g - ’ for the SDS, de-

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.i ;,,,;;., “+-,.2. i. 4 .,,.,li, ..,,.,. ‘.‘$..5:.!;.,.. .,..: ;., ::.,: ;...;, T,.>$ ..;,,..;, .‘~‘;.~~.~.i;. z .,.. i I. t.,/.._ . . . _... I. ..,I ._..,._.,_ r: ..i __ .,i I .,,,.y ,%6 ~..:..:‘...,,:~.~.~.:~...;.,.~;.,~; ~+~:.,.:...:. . . . . . . . . . ., ., , .‘i ‘.~~~~~:~~~;;~;;v;~,~~~~.

,. i;;‘>.<,,$.,:;.. . . . . . . ),, . . J ;‘,,;<,$+ .’

snm I I

FIG. 3. Cross-sectional view (drawn to scale) of the hexagonal phase at the composition used in this work. The radius of the cylindrical aggregates, containing SDS and decanol in molar ratio 4: I, and the lattice parameter were determined by x-ray diffraction. The two-dimensionally projected counterion distribution shown around the central aggregate is quantitatively consistent with Poisson-Boltzmann calculations (Appendix).

canal, and water in sample II, we obtain 4 = 0.272 -& 0.002. For infinitely long cylindrical aggregates do = r/(22/3), where Eq. (2.4) yields a cylinder radius b = 1.88 & 0.03 nm. This value, which includes the contribution from the sulphate head groups, may be compared to the length, 1.67 nm, of an all-frans C,, hydrocarbon chain.23 For spherocylinders of axial ratio p with the hemisphere-capped ends in contact, we have r$, = [ 1 - 1/(3p)]?r/(2-\/3). Taking p = 10 (cf. Sec. IV B), we obtain the equally reasonable radius b = 1.91 nm.

Figure 3 gives a cross-sectional view of our E phase sample drawn to scale with the lattice parameter and aggregate radius as deduced above from the x-ray data.

Ill. NMR ANISOTROPY A. Static quadrupole effects

Although our E phase samples are extensively aligned, there are several indications of a small but significant residual orientational disorder. First, the inhomogeneous satellite linewidth, Au’,;, is considerably larger than the effective linewidth, R r/rr (cf. Table I). Magnetic field inhomogeneity and partial refocusing of the static dipolar effects in the QE experiment5’ only accounts for a small part of this difference. Second, the inhomogeneous broadening is much larger when the mesophase is parallel than when it is perpendicular to the magnetic field. Third, the satellite lines are distinctly

Quist, Halle, and Fur6: Spin relaxation in liquid crystals

asymmetric (as expected for a static director distribution). Fourth, the quadrupole splitting ratio up (o”)/uo (900) is slightly (0.6-1.7 % for 2H) smaller than 2, which is the ratio expected for a uniformly aligned uniaxial mesophase.

In the presence of a static, uniaxial distributionf( 8, ) of “local” directors the satellite line shape is given by the orientational average

xfl~, & (~,QLD,eDd,~D ), (3.1) where 6,, is the angle between the magnetic field and the mean director of the mesophase (controlled by the experi- menter), while the angles 19~ and #D specify the orientation of the local director with respect to the phase director. (The azimuthal angle 4D is randomly distributed in a uniaxial phase. ) Since the orientational disorder is small it is reasonable to use a one-sided Gaussian director distribution, f( BDd ) - exp ( - 0 2Dd/o& ) , where 13~ is the angle between the phase director and the local director and oDd = (~9 id ) 1’2 is the rms orientational spread. The line shape associated with a local director orientation has the Lorentzian form

L (W~LD,~Ddl~D 1

R :CfL,) = [R:(&d]2+ [w-~,(e,,,e,,,&,)]~’ (3*2)

where R : (19,~ ) is the effective transverse relaxation rate for the satellite, including the effect of (a small) magnetic field inhomogeneity. For a uniaxial phase the quadrupole frequency in Eq. (3.2) is given by

Q(%Dd%d,4,) =@;p,(cos 6,,)

=oop[~(3cos2BLD - 1)(3cos2B, - 1)

- $ sin(28,,)sin(28, )cos(4D)

+ 3 sin2 S,, sin2 8, cos(24, ) 1, (3.3)

where w$ is the angular frequency of the satellite (relative to the center of the spectrum) for a sample uniformly aligned along the magnetic field, i.e., woe =: rruo (19~~ = 0”) for ‘H and I& =: 2rrvo (8,, = 0”) for ‘“Na.

By numerically computing the satellite line shape according to Eqs. (3.1)-( 3.3) we find that all the four experimental observations referred to above can be quantitatively accounted for. The ‘H data yield an orientational spread uDd = lo”-15” in the temperature range 25-35 “C, while the 23Na data yield a, = 49-l loin the range 23-35 “C!. The pro- gressive broadening and asymmetry of the OLD = 0” satellites with increasing temperature is thus due to an increasing orientational disorder. The much smaller width of the IY,, = 90” satellites is due mainly to a reduced inhomogeneous broadening and, to a lesser extent, to a smaller effective linewidth (cf. Table I). The reduction of the inhomogeneous broadening can be understood by picturing the 6, spread as a cone around the phase director. When this cone is perpendicular to the magnetic field (e,, = 90”) the 19, spread is smaller than the 0, spread, whereas they are identical when

J. Chem. Phys., Vol. 95, No. 9.1 November 1991


the cone is along the magnetic field (0,, = 0’). The inhomogeneous broadening results from the wo spread, which is determined by the 8,d spread, where the broadening is smaller in the 8,, = 90” configuration. (Note that for a given 6 Ld spread the inhomogeneous broadening is precisely the same for Bra = 0” and 90” although the splittings differ by a factor 2.) The orientational spread not only broadens the satellite lines but also shifts the satellite peaks towards the center of the spectrum, thereby reducing the quadrupole splitting. For the same reasons as for the broadening, this effect is smaller at Bto = 90” than at 6’,, = 0” thus account- ing for the slight reduction of the splitting ratio uo (o”)/vo (90”) from its 0, = 0 limit of 2.

Orientational disorder is manifested in the NMR spectrum in fundamentally different ways depending on the time scale r. on which the inhomogeneous quadrupole frequency wQ is averaged by molecular motions. Let Awe denote the spread in wo associated with the orientational disorder. If roAtip Q 1 then the quadrupole splitting is reduced by the additional motional averaging but the satellites remain sym- metric and are not broadened (i.e., their width is determined by R T). If, on the other hand, rQAwp ) 1 then the orientational disorder is not motionally averaged and one observes broadened, asymmetric satellite lines which are superposi- tions of Lorentzians corresponding to different wo values.

The motional averaging condition can be used to estimate a lower bound for the correlation length lo associated with the residual (nonaveraged) orientational disorder. Ac- cording to Eq. (3.3), Awe = (3/2)w$& for 13~~ = 0” and small uDd. In the ‘H case motional averaging occurs mainly by surfactant diffusion along the local director. We thus identify the motional averaging time rQ with the mean time taken for a surfactant molecule to diffuse out of a spatial region of linear dimension co, whencez4 rp = 6 L/ ( 120, ), where D, is the lateral diffusion coefficient of SDS (cf. Table V). We thus obtain co > [8D,/(~~d~)]“~~O.75 ,um at 25 “C and 0.6Opm at 35 “C. From the 23Na data we estimate in a similar way (taking into account both longitudinal and transverse counterion diffusion) la > 2.5 pm at 25 “C and 1.5 pm at 35 “C. Now if la were much larger than these lower bounds we should have obtained the same orientational spread oDd from the ‘H and 23Na spectra. The fact that crDd ( 23Na) < o’Dd ( 2H) implies that go is a few,um so that the more mobile counterions can motionally average part of the orientational disorder which appears static for the less mobile SDS molecules. A correlation length of this magnitude suggests that the static orientational disorder responsible for the inhomogeneous broadening and asymmetry of the satellites is associated with spatial domains with dimensions governed by the density of orientational defects. An alternative interpretation would be in terms of thermally excited elastic deformations.25P26 However, the equipartition theorem would then require that the motionally averaged orientational disorder (on length scales < go ) should have an amplitude that is far too large to be consistent with the observed quadrupole splittings (cf. Sec. IV B) and homogeneous linewidths.

It should be clear from the foregoing that the local director is an operational construct that defines the direction of

preferred alignment within a spatial region of sufficiently small size to allow complete motional averaging of all wo inhomogeneity within the region.

B. Quadrupole relaxation

The spectral densities J f =J f ( o0 ,BLo ) and J $ = J f (2w, ,8,, ) given in Table III refer to the lab-fixed frame (defined by the static magnetic field) in which the nuclear spins are quantized. In order to analyze these spectral densities in terms of microstructure and molecular motions, it is convenient to first transform to the director frame, defined by the sixfold axis of the aligned hexagonal phase. By means of this transformation the orientational dependence of the lab-frame spectral densities is made explicit. For a uniaxial phase one obtains2’

JfW’) = J,+JPb+,,, (3.4a)

J;(W = J, + J%ho), (3.4b)

J:(90”) =J,+:J?ko,, +:Jfb,L (3.k)

J,L(90”) = J/ + $JJ,D(2w,) + 4Jf’(2w,)

+ ,kJ?W, 1. (3.4d)

In Eq. (3.4) we have included a contribution J, from fast (relative to the Larmor frequency, w,, ) , local (relative to the aggregate radius) motions. In the case of surfactant ‘H relaxation, the fast motions comprise rotational bond isomerization and rotation around the molecular long axis.9*‘o*28 In the case of counterion 23Na relaxation, small- amplitude dynamics within the primary hydration shell as well as collective hydration dynamics and local counterion and surfactant displacements are included among the fast motions.8.2q We assume that the contribution Jf from these motions is independent of resonance frequency and mesophase orientation. Under the present experimental conditions, the assumed frequency independence implies that the fast-motion time scales are short compared to 10e9 s. If the slower motions, contributing to the director-frame spectral densities Jfl, (kw, ) in Eq. (3.4), have correlation times of the order low9 s or longer, then the fast and slow motions can be considered statistically independent so that no cross terms appear in Eq. (3.4).

Without invoking a specific structural and dynamic model, spin relaxation data can in principle yield the quantity J, as well as the 9 distinct director-frame spectral densities J”, (kw, ). Our data, however, comprise only two orientations (e,, = 0” and 90”) and a single magnetic field (defin- ing w. ) . Consequently we cannot separately determine any of the 6 spectral densities appearing in Eq. (3.4). The ulti- mate aim, however, is not to determine the director-frame spectral densities per se, but rather to extract the underlying structural and dynamic information. To do this we need to invoke certain model assumptions. If the model is sufficiently simple, the number of unknown quantities can be reduced. Indeed, some of the director-frame spectral density functions may vanish identically due to symmetries inherent in the model.

For reasons that will become apparent, we analyze the 2H and 23Na relaxation data separately in Sets. IV and V.




The main theoretical results needed for the analysis are de- veloped in Sec. IV. The modifications of these results as required for the analysis of the 23Na data are then presented in Sec. V.

TABLE IV. Parameters derived from ‘H relaxation rates and quadrupole splitting using the classical model.

IV. SURFACTANT ‘H RELAXATION A. Classical aggregates

The simplest structural model consistent with the hexagonal mesophase symmetry (as established by x-ray diffraction) is the classical picture of infinitely long, straight, circular cylinders arranged on a two-dimensional hexagonal lattice.30 Given this structure, the only relevant motion on the time scale l/w, is surface diffusion of the surfactant molecule around the cylinder axis. This motion modulates the orientation of the principal frame for the residual electric field gradient (efg) tensor, the components of which have been partially averaged by the faster local motions (truns- gauche isomerization and rotational diffusion around the surfactant long axis).

T/-C J/s ’ r,/ns j&HZ ,&JkHZ

25.0 6.1 5.4 24.8 28.29 21.0 5.7 5.0 25.3 28.05 29.0 5.4 4.6 25.2 21.99 31.0 5.1 4.6 25.2 27.89 35.0 4.5 3.8 25.8 27.51

Error +O.l” * 0.3” + 0.4” * 0.04

“Propagated random errors in R,, and R,,.

The residual efg component along the cylinder axis is not affected by surface diffusion and remains to produce a static quadrupole splitting

up = p2 (~0s e,, )x0, (4.1)

where To is the residual quadrupole coupling constant (qcc).

RIQ(O”) + l/3 R,,(90”), which follows from Eqs. (2.1), (2.2), (3.4), and (4.2a), and to compare the result with the measured value. In this way we find that the ratio of the calculated to the measured R ,= (90”) varies (monotonical- ly) from 1.25 + 0.02at25 “Cto 1.18 f 0.02at 35 “C!. Amore direct way of exhibiting the inconsistency is to note that, according to Eqs. (3.4) and (4.2a), we must have J,L(90”) > Jf(0’). The 2H data in Table III clearly do not satisfy this inequality.

The residual efg components (in a cylinder-fixed frame) orthogonal to the cylinder axis are averaged to zero by (azimuthal) surface diffusion, which thus provides an efficient mechanism of spin relaxation. For symmetry reasons this process contributes only to one of the three director-frame spectral density functions43

It might be argued that these inconsistencies result from some unknown source of systematic error in the relaxation measurements. However, qualitatively the same inconsistencies emerge from the analysis of the 23Na relaxation data (cf. Sec. V B), where the lab-frame spectral densities are obtained from a different linear combination of relaxation rates (cf. Sec. II D). Furthermore, the inconsistencies are not likely to be artifacts generated by the approximation that Jf is independent of frequency and orientation, since there is little doubt that this approximation is valid for the counterions.“*29

J: (~1 = 4n,Jf(~),

which takes the form (4.2a)

Jf(kw,) =5X; rc

1 + (kw,rc.,)2 ’ (4.2b)

with a correlation time

b2

rc =z- (4.3)

Another way of assessing the validity of the classical model is to compare the deduced parameter values with related results from other sources. The calculated classical- model parameters are collected in Table IV. (Since the data are not fully consistent with the classical model, the calculated parameter values depend somewhat on how the data are analyzed. The differences are rather small, however, and the conclusions drawn in the following are not affected.)

for azimuthal surface diffusion with diffusion coefficient D, on a cylinder of radius 6. As is evident from Eqs. (2.1)-( 2.3 ) and (3.4), all spectral density functions appearing in this work include coupling constants and spin-dependent factors. The numerical factor in Eq. (4.2b) pertains to I = 1 nuclei. The subscripts Q and R on the residual qcc’s in Eqs. (4.1) and (4.2b) are a reminder of the fact that these quantities are not necessarily identical (cf. below). The model defined by Eqs. (3.4) and (4.1)-(4.3) will henceforth be referred to as the classical model

Within the classical model, the 4 measured lab-frame spectral densities in Eq. (3.4) are fully determined by the 3 parameters Jf, TR, and rc. This overdeterminacy allows us to test whether the data are consistent with the classical model. One way to apply this test is to calculate R,, (90”) from the relation R,,(90”) = l/8 R,,(V) + 9/8

Consider first the fast-motion spectral density Jp This quantity has previously been determined3’ in a variable-field ‘H relaxation study of an isotropic solution of spherical micelles composed of a-deuterated SDS with the results Jf(20”C) =2.7s-‘andJJ(34”C) = 1.9s-‘.TheJjvalues in Table IV are a factor 2.5 larger than these micellar values at the same temperature. Since the local surfactant chain dynamics should be closely similar in spherical and cylindrical aggregates, the large difference between the J, values in the two kinds of aggregate is a further indication that the classical model is not quantitatively adequate. (The find- ing 3’ that the J, values for the a- and y-deuterons in SDS micelles differ by merely 30% suggests that tram-gauche isomerization dynamics at a rate governed by the intrachain dihedral potential barrier makes a dominant contribution to Jp,

The residual qcc To has previously been determined3’ from the ‘H quadrupole splitting of a-deuterated SDS in the

J. Chem. Phys., Vol. 95, No. 9,1 November 1991


hexagonal phase of the binary system SDS/water (mole ratio H,O/SDS = 18.4) with the results Fe (25 “C) = 31.2 kHz and Xu (34 “C) = 30.9 kHz. The zSa values in Table IV are seen to be ca. 10% smaller. While this discrepancy may be due to the different composition of our E phase sample (cf. Sec. II A), it may also reflect the inadequacy of the classical model for our sample. Furthermore, the residual qcc XR derived from our relaxation data is even smaller (ca. 10% below our ,pu ). Whereas xa and XR should be equal in aggregates where the local surface normal is a threefold (or higher) symmetry axis, they may differ in cylindrical aggregates where the surface normal is merely a twofold axis. In fact, it can be shown that43

Ye = ( 1 - 77,v SNFX7 (4.4a)

,pR = (1 + 11,/3)SNFX, (4.4b) where x is the static qcc (ca. 170 kHz), S,, is the usual C-‘H bond order parameter (relative to the surface normal), and r], is the asymmetry parameter for the residual efg tensor. [With the x axis along the cylinder and the z axis normal to its surface, we define vN = (TX, - FY,,)/TZZ. Since the axes are labeled without reference to the relative magnitudes of the residual efg components Faa, it follows that v,,, is unrestricted; in particular, T,,, may be positive or negative.] While the difference between To and XR in Table IV may reflect a nonzero asymmetry parameter, it may equally well be an artifact caused by deficiencies in the classical model. Mean field model calculations32 on cylindrical surfactant aggregates indicate that the surfactant chain is preferentially tilted along the cylinder axis (as expected) but that the asymmetry of the C-H bonds is small.

B. Disordered aggregates The conclusion of the analysis in Sec. IV A is that the

classical model does not quantifariuely account for the ‘H relaxation data, although it does explain the gross features of the data, e.g., the strong orientational dependence in the relaxation rates. If the structural assumptions in the classical model are accepted, it is difficult to conceive of any other surfactant motion that could contribute significantly to the OLD-dependent parts of the spectral densities J f ( w0 ,eLD ) and J f (20, ,8,, ). However, if we admit certain deviations from the idealized picture of infinitely long, straight, circular cylinders, the relaxation data can be satisfactorily accounted for. We can envisage three basic types of structural deviation: (i) a noncircular aggregate cross section, (ii) a non- uniform cylinder orientation, and (iii) a fragmentation of the cylinders into finite aggregates.

According to a recent study33 of the closely related system sodium decyl sulphate/decanol/water, the cylindrical aggregates in (part of) the hexagonal phase have noncircular cross sections. Although an adjacent rectangular (“rib- bon”) phase has not been found in the system under study here, it seems pertinent to ask whether a noncircular cross section could explain the anomalies in our relaxation data. The answer is no. This follows2’ simply from the observa- tion that as long as the local surface normal at every point is perpendicular to the director, the director-frame spectral

density functionsJ i (w) vanish identically for m # 2, just as for the circular cylinder.

To quantitatively account for the relaxation data we clearly need a structure where the local surface normal is not everywhere perpendicular to the phase director. This can be achieved in two ways: (I) by allowing the (local) cylinder axis to deviate from the (average) director, or (II) by allowing the radius of the cylinder to vary along its length.

Structural deviations of type I may be thought of in terms of a wormlike aggregate orientationally confined by an effective potential of mean torque. The long-wavelength configurational fluctuations of such aggregates may be described by continuum theory, relating them to the macroscopic bend elastic constant. 25.26 Another potential source of structural deviations of type I is orientational defects leading to domain formation and a consequent distribution of domain directors around the average phase director (cf. Sec. III A).

An example of structural deviations of type II is fragmentation of the cylinders into finite aggregates. For a quantitative treatment, the detailed shape of the aggregate must be specified. Relatively long aggregates (rodlike micelles) are most appropriately modeled as spherocylinders, whereas small aggregates may be modeled as prolate spheroids.34 Since the orientation of a finite aggregate may fluctuate around the director, structural deviations of type I are also involved in this case.

Clearly, there are several models that might account for our relaxation data. However, a quantitative analysis in terms of such models is problematic. Although an efficient numerical procedure for computing the spectral density functions for surface diffusion on prolate spheroids undergo- ing restricted rotational diffusion is available,27 the corresponding problem for the more realistic (for large axial ratios) spherocylindrical aggregate shape has not been solved. A further problem is that these more elaborate models in- volve more parameters than can be determined from the data. For these reasons we shall perform a simplified analysis which allows us (i) to extract a correlation time roZi for motion around the aggregate axis with a minimum of model assumptions, and (ii) to estimate in a simple way the extent of orientational fluctuations and/or aggregate fragmentation needed to quantitatively account for the data.

In the models discussed above there are basically two types of slow motion. First, and most importantly, there is the surfactant surface diffusion around the aggregate axis and the (nonseparable) spinning rotational diffusion of the (possibly finite) aggregate around its axis. Second, there are the motions that modulate the angle between the local surface normal and the director, such as surface diffusion along the length of a finite aggregate or curved cylinder and restricted tumbling of a finite aggregate. As the motions in this second class contribute less to the high-frequency (wO and 2w,) relaxation and since their treatment is more model- dependent, we focus here on the effect of the azimuthal motions. Consequently, we consider only the spectral densities J f ( w0 ) and J ,“( 2w, ) , which dominate the longitudinal relaxation rates.

In order to simplify the analysis we make two observa-


J. Chem. Phys., Vol. 95, No. 9,i November 1991 Downloaded 07 Mar 2006 to 130.235.253.49. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

tions. First, the spectral density function J f( w) at w = tic, and 2w, is completely dominated by azimuthal motions un- less the aggregates are quite small (and explicit calcula- tions2’ for a prolate spheroidal geometry definitely rule out this possibility), i.e., the contributions from motions in the second class are entirely negligible. This assertion is based on a detailed analysis of the exact spectral density function for a rotating prolate spheroid2’ and should also hold for spherocylinders and curved cylinders. Second, for all practical purposes the time correlation function for aggregate spinning is exponential for any degree of orientational order,35v36 while the time correlation function for azimuthal surface diffusion is very nearly exponential for any axial ratio.27 On the basis of these observations, we arrive at the following generaliza- tion of Eq. (4.2b)

xqd:,vLv)]2) Td l + tkwCJTazi)’ ’

(4.5)

where 8,, is the angle between the phase director and the (local) cylinder axis and 8, is the angle between that axis and the surface normal. Apart from the 2: factor, the spectral density J f’(kw, ) is seen to be a product of three factors: (i) an ordering factor ( [d g, (8,, ) ] ‘) measuring the degree of alignment of the aggregate axis with the director, (ii) a shape factor ( [d $, ( 8c,v ) ] ‘) determined by the curvature of the aggregate surface, and (iii) a dynamic factor involving the azimuthal correlation time raZ,. For an infinitely long cylinder 8, = 7~/2 and ( [d :, ( 0,, ) ] ‘) = 3/8. If the cylinder is also straight and perfectly aligned with the director, then by symmetry Boc = 0 or 7~ with equal probability so that ([d:, (e,,)12) = l/2 and Eq. (4.5) reduces to Eq. (4.2b) as required. It is important to realize that the result (4.5) has been obtained with a minimum of model assumptions; it applies equally well to curved cylinders and to finite aggregates of any reasonable (uniaxial) shape.

To obtain the correlation time roZi we define the quantity

ar Wf(9P) - Jf(O’) -J,

J,“(V) -J, * (4.6)

For a given value of the fast-motion spectral density J, we can, according to Eqs. (3.4), (4.5), and (4.6), determine the azimuthal correlation time through

1 a--l”2 Tazj = - -

( > .

w. 4-a (4.7)

Returning to Eq. (4.5) we note that the ordering factor can be expressed (exactly) in terms of the 2nd-rank and 4th- rank order parameters S,, = (P, (cos f3,c ) ) and Q,, = ( P4 ( cos e,, ) ) as27*37

([d:,(&,,)]2) =:+Fs,c+&Q,,. (4.8) For simplicity we drop the last term which always contributes less than 3%. The shape factor in Eq. (4.5) is model dependent. Since our data indicate that the aggregates are very long, we model them as spherocylinders. For a uniform surface distribution, we then have

+GaLJJ2) =$-‘, @P


(4.9)

wherep = 1 + L /(2b) is the axial ratio of a spherocylinder of radius b and total length L + 26.

The BLD = 0” quadrupole splitting is now given by

UQ (03 = :s,, I&N IjQ, (4.10) where the “order parameter” Sc, = (P2 (cos 8, ) ) depends on the axial ratio of the spherocylinder (assuming a uniform surface distribution) through

s,-, = - i( 1 - l/p). (4.11)

In the cylinder limit (p --) m),Eqs. (4.10) and (4.11) reduce to Eq. (4.1) as expected.

Next we define the quantity

ps 3[uf(900) - J:C@‘, -J,] 1 + (wo~ozi)2

[277-qan]2 . (4.12)

rori

For a given value of J,- we obtain T,,, from Eq. (4.7) and thus Pis known. We now adopt the widely used3’*32.38+39 approximation vN = 0, which according to Eq. (4.4) implies that Xp = ,,& . The residual qcc (henceforth denoted 2) then cancels out in Eq. (4.12) and we obtain

P = F(S,,)G@L (4.13)

where we have introduced the ordering function

FG,,) = (4 + ~~,,/S;c (4.14)

and the shape function

G(p) =(++)(I -;)-2. (4.15)

The quantity p provides a convenient measure of deviations from the classical model, for which p = 3/4. [Due to the omission of the Q,, termin Eq. (4.8), weactually obtain p-O.73 from Eqs. (4.13)-(4.15) in the limit S,, = 1, p + CO. ] Any structural deviation from the classical model (S,, < 1 and/or p < 00 ) is manifested as a larger fi value. Conversely, the fact that fi cannot be smaller than 3/4 may be used to establish an upper bound for the fast-motion spectral density JJ. This is illustrated in Fig. 4(a), showing that Jf < 3 s - ’ at 35 “C. Actually Jf has to be significantly smaller than 3 s - ’ since this upper bound corresponds to the classical model (p = 3/4), which cannot account for the relaxation data in a quantitatively consistent way (cf. Sec. IV A).

It appears therefore that J,, in our E phase sample cannot be much larger than in spherical micelles (J, = 1.9 s - ’ at34”C). 3’ In the following we report results of calculations for two values ofJr at 35 “C!: 1.5 and 2.5 s- ‘, which are likely to bracket the true value. In both cases the temperature dependence of Jf is assumed to follow an Arrhenius law with an activation energy of 20 kJ mol - ’ as in the micellar system.3’ As shown in Fig. 4(b), the deduced correlation time raori is not very sensitive to the choice of Jf within the allowed range.

Once P has been fixed, Eq. (4.13) can be used to construct a curve in the S,, - p plane linking the allowed pairs of values of these parameters. According to Eqs. (4.13) and (4.14), this curve is given explicitly by

J. Chem. Phys., Vol. 95, No. 9.1 November 1991


0.9

0.8

P 0.7

0.6


0.5 ' I I 1 I I 0 1 2 3 4 5

Jf / s-l

7.0 I I I ' 1 I I (b) 6.0 I - I

2 5.0 . .-

[d e 4.0

, 3.0

r i i I

2.0 I t I I I 0 1 2 3 4 5'

Jf / s-l

FIG. 4. Analysis of surfactant ‘H NMR data at 35 “C. (a) Variation offi with J,. Since the model requires p> 3/4, it follows that J, < 3 s '. (b) Variation of the azimuthal correlation time r,,, with J,. The variation of rUZ, within the allowed J, range is of the same magnitude as the experimental error. In both figures the hatched area corresponds to the propagated random errors in R,, and R,,.

S,,(p) = 49p “2 1+-

I I 5G(p) ' (4.16)

with G(p) defined by Eq. (4.15). An example of such a curve is shown in Fig. 5. It is seen that the classical model limit is approached very slowly, i.e., the effect of the hemi- spherical end caps of the model spherocylinder remains significant even for very large axial ratios. The lower bound on the aggregate size is pz 10, corresponding to perfectly

,x 2 0.96

t g B

0.92

0.88 ' ' 10 20 50 100

Axial ratio, p

FIG. 5. Allowed combinations of aggregate order parameter S,, and axial ratiop as obtained from the surfactant ‘H NMR data at 25 “C. The hatched area corresponds to the propagated random errors in R,, and R,,. The dashed line at S,, = 0.894 is thep- CO limit.

aligned rigid spherocylinders, but if orientational disorder is allowed (SD, < 1) the aggregates may be much longer.

The results of the data analysis at all temperatures (using both sets of Jr values) are collected in Table V. It is reassuring to note that with Jf values in the most probable range (close to the micellar values3’ ) , the residual qcc 2 is, as expected, close to the value 3 1 .O _+ 0.2 kHz deduced from the quadrupole splitting in the E phase without decanol.31 (Actually the calculated JJ value varies slightly along the S,, -p curve. This variation is taken into account in the error limits specified in Table V.) Within the investigated range there is no significant temperature dependence in the structural parameters p and S,,. At all temperatures the aggregates are very long (effectively infinite or finite but longer than 40 nm and probably quite polydisperse34 ) and highly ordered (SDc > 0.9) with respect to the macroscopic phase director. [If we retain the Q,, term in Eq. (4.8) we obtain SD, values that are l-2 % larger than those given in Table V. ]

The order parameter SD, which enters into Eq. (4.5) via Eq. (4.8) is a measure of the degree of orientational disorder that is motionally averaged on the relaxation time

TABLE V. Parameters derived from *H relaxation rates and quadrupole splitting assuming spherocylindrical aggregates of axial ratio p and orientational order parameter S,.

J,(35"C) = 1.5~' J,(35"C) =2.5s-'

T/-C r,Jns Pmm s min DC ,i+Hz r,,/ns Pmin s min DC z/kHz D,/lO- “‘m* s-’ b

25.0 6.0 10 0.89 31.5 6.5 24 0.95 29.6 1.41 27.0 5.5 10 0.89 31.4 5.9 21 0.95 29.5 1.55 29.0 5.1 11 0.90 31.0 5.4 28 0.96 29.1 1.68 31.0 5.0 12 0.91 30.6 5.3 38 0.97 28.7 1.70 35.0 4.3 11 0.90 30.5 4.5 28 0.96 28.6 2.02

Error ’ f 0.3 *1 f 0.01 + 0.4 f 0.3 f (5 - 15) f 0.01 f 0.4 +0.15

‘Propagated random errors in R,, and R,p. bD, = b2/(4(r,,)) with b = 1.88 nm.

J. Chem. Phys., Vol. 95, No. 9,i November 1991 Downloaded 07 Mar 2006 to 130.235.253.49. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp


scale. The condition for motional averaging of the orientation dependence in the longitudinal relaxation rates is G-&R I ( 1, where 7p is defined as in Sec. III A and AR, is the spread in R,, or R,, resulting from the orientational disorder. Since AR, 4 AwQ the relaxation rate spread associated with the local director (domain) disorder is completely motionally averaged, whereas the corresponding quadrupole frequency spread is effectively static. [For example, witha, = 5”wehaveAR,Q(90”)<1s-‘while~Q=:5ms.] If we associate an order parameter S, z 1 - (3/2)a’, with the domain spread, we must therefore have S,, < S,, . In other words, the orientational disorder within a domain is characterized by an order parameter S&S,,. According to Sec. III A, a, < 15” so that S, > 0.90. From what has been said here and in Sec. III A it follows that the order parameter in Eq. (4.10) should really be S&S,. Consequently, thea value calculated from Eq. (4.12) should really be multiplied by S &* < 1.2 before it is inserted into Eq. (4.13 ) . This would slightly increase the upper bound of J, as determined from Fig. 4(a).

Since the correlation functions for azimuthal surface diffusion and aggregate-spinning rotational diffusion are exponential (cf. above), the correlation time ~~~~ is given by

40

>

-I 7cni = 2 + 49

b2 , (4.17)

where b is the aggregate radius and O,, is the rotational diffusion coefficient for motion of the aggregate around its axis. In the most probable case of effectively infinite aggregates (p>p,,,,), we can safely set O,, = 0 in Eq. (4.17). Taking b = 1.88 nm, as obtained from the x-ray study (Sec. II F), we obtain the lateral diffusion coefficients given in Table V. Even in the less probable case of spherocylindrical aggregates of minimal size (p = lo), aggregate rotation contributes very little to Eq. (4.17). Using the hydrodynamic result”’ for O,,, including a hydrodynamic interaction factor,” and taking b = 1.91 nm, as obtained from our x-ray data assuming spherocylinders with p = 10, we thus find that the D, values in Table V are reduced by less than 2%, which is well within the experimental uncertainty. Assum- ing an Arrhenius-type temperature dependence for D,, we obtain an activation energy of 26.0 + 2.5 kJ mol. ‘. Our results for the E phase are remarkably close to the previously reported”’ D, values for spherical SDS micelles (assuming b=2.0 nm): D, = (1.0+0.2)x10-‘” m*s-’ at 20°C (1.2X10 ‘“m’s-’ in the E phase by extrapolation of our data)andD, =(1.8+0.2)X10-‘“m*s-‘at34”C.

V. COUNTERION 23Na RELAXATION

The “Na spectral densities J f ( w0 ,eLD ) and J k( 20,~,0~~ ) and quadrupole splitting vQ in Tables II and III can be analyzed in basically the same manner as the surfactant ‘H data. Some modifications are required however. First, because of the difference in nuclear spin quantum number (I = 1 for ‘H, I = 3/2 for ‘“Na), the equations appearing in Sec. IV should be changed according to: z-+x/3 in expressions for the quadrupole splitting andF* + 22*/3 in expressions for the spectral densities. Second, since the

counterions are not confined to the aggregate surface to the same extent as the surfactant headgroups, the surface diffusion approximation (implicit throughout Sec. IV) is not ob- viously valid for the 23Na data.

A. Radial diffusion

The accuracy of the surface diffusion approximation depends primarily on the radial equilibrium distribution of the nucleus and on the origin of the quadrupole coupling constant. In the present system both the surfactant headgroup distribution and the counterion distribution are strongly peaked at the cylindrical aggregate surface, but the former decays more rapidly at large distances. Since the surfactant *H qcc is intramolecular, radial motion affects the *H relaxation only to the extent that it alters the reorientational motions. The effect of small-amplitude protrusions can be in- corporated into the fast-motion spectral density J, in Eq. (3.4), while large-amplitude protrusions (and complete ex- its from the aggregate) are too infrequent to significantly affect the *H relaxation.

The counterion qcc is intermolecular and the residual qcc vanishes except in the near vicinity of the surface where the local environment is significantly anisotropic.8v29*42 Whereas in the surface diffusion approximation J ,“( w ) is the only nonzero director-frame spectral density function (cf. Sec. IV A), radial counterion diffusion makes a contribution to J:(w) by modulating the magnitude of the residual qcc. [J f (w ) still vanishes by symmetry. ] We show in the Appen- dix, however, that this contribution is of the same order of magnitude as the experimental uncertainty and hence negligible. Moreover, as seen from Eq. (3.4) J;(o) contributes only to J k (2w, ,90”), which is not used in the following analysis.

A potentially more important consequence of radial counterion diffusion is its effect on the spectral density function J?(w), which is no longer strictly Lorentzian as in Eq. (4.2b). Following a recent treatment43 of spin relaxation by counterion diffusion in cylindrical geometry, we show in the Appendix that the effects of radial diffusion on Jf(w) are (i) to spatially average the magnitude of the local residual qcc, and (ii) to alter the rate of azimuthal averaging of the orientation of the residual efg tensor. The quantitative analysis carried out in the Appendix, using parameter values appropriate for the present system, shows that radial averaging of the local residual qcc is sufficiently fast compared to azimuthal averaging of the orientation of the residual efg tensor that we can write

x = prxs (5.1)

in the expression for J,“( w) . By means of this relation, which is strictly valid for the quadrupole splitting, we can compare the local residual qcc j& in the interfacial region obtained for the present system with Xs values from other systems with a different fraction P, of counterions in the interfacial region. Taking the interfacial region (where f:s #O) to extend 0.5 nm out from the aggregate surface, we obtain P, = 0.77 for the present system.

Furthermore, the analysis in the Appendix shows that J?(o) is very nearly Lorentzian and that the rate of azi-



muthal averaging is the same as if the counterions were confined to the aggregate surface. (This somewhat surprising result is rationalized in the Appendix.) Consequently, the surface diffusion approximation is highly accurate for the counterions in the present system.

B. Surface diffusion

Like the surfactant *H relaxation data, our 23Na data cannot be quantitatively accounted for in terms of the classical surface diffusion model with infinitely long, straight cylindrical aggregates. According to Eqs. (2.1), (2.2), (3.4), and (4.2a), the measured relaxation rates must satisfy the relationR,.(90”) E~/~R,~(O”) -3/4R,z(O”).The23Na data in Table II exhibit deviations from this relation by 8- 20 %, whereas the experimental random errors admit only 6% deviation.

We are thus, once again, led to consider a more general model which allows for orientational disorder (cylinder curvature) and/or end effects (spherocylindrical aggregates). As is evident from Table III and Eq. (3.4)) the fast-motion contribution J, and the surface-diffusion contributions J F(ko, ) are of similar magnitude in the *“Na case, whereas the latter are dominant in the ‘H case. For this reason we cannot use the procedure of Sec. IV B to narrow down the permissible range of J, values. Instead we make use of the fact that the quantity fi, which according to Eq. (4.13) is fully determined by the size (p) and the orientational order (SDc ) of the surfactant aggregates, must have the same value in the *H and *“Na analyses. [This approach, of course, presupposes that the samples I and II (cf. Sec. II A) used for the ‘H and ‘“Na NMR measurements are identical. This was checked and found to be the case. For example, the ‘“Na quadrupole splitting at 25 “C was 18.31 and 18.20 kHz in samples I and II, and the ‘“Na relaxation rates R ,o and R ,z of the two samples coincided within the experimental uncer- tainties specified.] The procedure is then to vary Jf until the resulting p value, calculated from Eqs. (4.6), (4.7), and (4.12)) is equal to the p value obtained from the ‘H data. The results of this analysis are presented in Table VI. [The small uncertainty in J, and hence,9 in the *H analysis prop- agates to the 23Na parameters, mainly Jf and?*. This effect is included in the error estimates in Table VI. Further, as dis-

TABLE VI. Parameters derived from “Na relaxation rates and quadrupolar splitting with B from ‘Hdata.

T/-C J/s ’ F,/kHz ’ r”,,/ns D,/lO ‘” m2 s ’ d

23.0 33 104 2.5 25.0 30 102 2.3 27.0 28 102 2.3 29.0 27 100 2.1 31.0 25 99 2.1 35.0 22 99 1.7

Error + 1” + 3” + 0.9

4.4 * 0.7 4.8 + 0.9 4.8 + 0.9 5.3 + 1.2 5.3 * 1.2 6.4 + 2.2

‘From uncertainty in ‘H fi value. ‘Propagated random errors in R,, and R,,. ’ i, = c/P, with P, = 0.77. “D, = bj/(4r,,,,) with b, = 2.10nm.

cussed in the Appendix, the effect of incomplete radial averaging in J?(o) is that ,& = P” j& differs slightly from To = P ps.As a consequence, the *‘Na p value calculated from Eq. (4.12) should really be divided by (jq&$ = P2(“- ‘) = 1.07. This small correction has been ignored as it is partly cancelled by the opposite correction associated with the interdomain order parameter S,, (cf. Sec. IV B).]

The fast-motion spectral density may be decomposed as8*43 J, = P, Jj + ( 1 - P, ) Jy, where the superscripts refer to the interfacial region (s) and the surrounding bulklike region (0). In dilute aqueous solutions of simple sodium salt? J; = 8 s - ’ at 25 “C. With P, = 0.77 and Jf = 30 s - ’ at 25 “C (Table VI), we thus obtain JJ/Jy = 4.6. This increase in J; can be partly ascribed to the high local ion concentration in the interfacial region (4-5 mol dm - 3). In bulk electrolyte solutions of similar concentration the increase in JJ. ranges from a factor 1.4 (NaCl) to 3.5 (NaClO, ) .44 In isotropic solutions of vinylic polyelectrolytes the ratio J;/Jy is of similar magnitude as found here, e.g., 2.5 for sodium polystyrene sulfonate and 7 for sodium polyacrylate.45 Much larger ratios have been reported for certain other systems, e.g., 25 in the reversed hexagonal phase of the Aerosol OT/isooctane/water system’ and 40 in solutions of poly- phosphate. 46 In view of the intermolecular nature of the residual qcc, such large differences may reflect differences in counterion hydration. The apparent activation energy deduced from the J/ data in Table VI is 26 + 2 kJ mol - ‘, which is essentially the same value as found in the reversed hexagonal phase.*

The local residual quadrupole coupling constant j& = 100 &- 3 kHz found in the present system may be compared to the value x.7 = 79 kHz deduced4* from the 23Na splitting4’ in the hexagonal phase of the binary system SDS/water. The difference can probably be ascribed to the lower water content in the binary samples (mole ratio D,O/SDS = 10-20). The insignificant temperature dependence in x5 (cf. Table VI) is in accord with the findings from the binary system.47

The surface diffusion coefficient D, for sodium ions, calculated from the azimuthal correlation time as D, = b :/(47,,i), is also included in Table VI. Two com- ments are in order regarding this relation. First, the effect of aggregate rotation on rQZi is entirely negligible (cf. Sec. IV B). Second, the modelcalculations in the Appendix show that the effective diffusion radius 6, is virtually identical to the aggregate radius b. However, these calculations are based on a model where the counterions are treated as point charges and the surfactant aggregate as a uniformly charged, smooth cylinder. In reality, the (hydrated) counterions cannot be located precisely at the surface defined by the sulphate headgroups. We therefore use a value 6, = 2.10 nm, slightly larger than the radius b = 1.88 nm deduced from the x-ray data. The limiting (infinite dilution) Na + diffusion coefficient in H,O is 1.33~ 10e9 m* s-’ at 25 “C4* whence we obtain Do/D, = 2.8 + 0.5. This mobility reduction factor is similar to the value 3.6 + 0.4 found in the reversed hexagonal phase in the Aerosol OT/isooctane/water system.* As a reference, the Na + diffusion coefficient in bulk NaCl solu-




tions48,49 at 25 “C is reduced from its limiting value by a factor 1.7 at a concentration of 5 mol dm - 3. The Arrhenius activation energy for D, in the range 23-35 “C is 22 + 3 kJ mol - ‘, slightly higher than the value48 18 kJ mol - ’ for the limiting Na + diffusion coefficient (in the same temperature range ) .

Finally, we make some further remarks with reference to our recent 23Na NMR study’ of the counterions in the reversed hexagonal (F) phase of the Aerosol OT ( AOT)/ isooctane/water system. As in the present case, the F phase relaxation data were not quantitatively consistent with the classical model. Applying the R I= (90”) test (cf. above), one finds that the deviation is very nearly the same in the two systems. As shown here, this deviation can be accounted for by a slight orientational disorder of the cylindrical aggregates with respect to the mean director. (An approximate treatment of this effect was presented in Appendix E of Ref. 8.) In the Fphase study, the classical model was nevertheless used to extract r,,, and D,. As already noted’ the error thus introduced is very small. This is also the case for the 23Na data in the present study. A comparison of the results of the two studies clearly reveals a difference in the local interfacial environment of the counterions: in the F phase the residual qcc X., is twice as large as in the E phase (which, together with the longer r,,;, makes the azimuthal spectral densities 5 times larger), while the fast-motion spectral density is 5-6 times larger in the Fphase. We believe that these differences can be ascribed to the chemical nature of the large AOT headgroup. Comparing a variety of systems one finds42 that the AOT surfactant is exceptional in giving a large 23Na residual qcc. This may also explain why we obtained different (by 25-40 % ) /Ijs values from the quadrupole splitting and the relaxation data in the Fphase study. If there are two (or more) counterion sites (e.g., the sulphonate group and the ester linkages in the AOT headgroup) where the local qcc X(r) is of substantially different magnitude (or even of opposite sign), then one expects that Xs = (x(r))* obtained from the quadrupole splitting should be significantly smaller than Xs = ([~(r)]‘)f’” obtained from the relaxation data.

tural parameters is often complicated. In this work we have adopted an integrated approach, where relaxation data are analyzed in conjunction with quadrupole splittings and x- ray data. In addition, it proved helpful to combine relaxation data from nuclei belonging to different molecular components in the system.

In a previous 23Na relaxation study8 of a reversed hexagonal (F) phase, the sample was magnetically aligned on cooling from an isotropic (microemulsion) phase. However, a substantial fraction (20-25 % ) of the sample did not align with the field but was instead aligned by the glass surface of the sample tube. In this fraction the domain directors are randomly distributed within a plane parallel to the glass surface, giving rise to a characteristic two-dimensional powder pattern in the NMR spectrum. (This is suggested by the *H spectrum shown in Ref. 8 and we have subsequently demon- strated this more explicitly by studying the orientation dependence of the ‘H spectrum from an F phase sample sand- wiched between glass plates.) After storing the sample for several months (outside the magnet) an essentially homeotropic alignment was obtained with virtually no sign of surface alignment. Similar observations have been reported for the alignment of the hexagonal phase in the dipotassium hexadecanedioate/water system.50*5’

In contrast to the previously reported cases, the present hexagonal phase was formed by heating from a homeotropi- tally aligned nematic (Nc) phase. As a consequence, the surface alignment can only penetrate a distance comparable to the magnetic coherence length2s*52 6, = (,u~K/Ax~) “*/B. ~20 ,um into the sample. In the case of the Fphase, however, alignment occurs in the bipha- sic microemulsion/mesophase region so that kinetic effects may determine the initial (nonequilibrium) mesophase alignment.

VI. CONCLUDING REMARKS

The present study demonstrates that nuclear spin relaxation rates from macroscopically aligned lyotropic mesophases can provide quantitative information about molecular dynamics in the interfacial region as well as about the structure and ordering of the surfactant aggregates. Al- though our data were obtained at a single magnetic field, the orientation dependence of the lab-frame spectral densities (determined by a combination of two independent relaxation experiments) enabled a (partial) separation into contributions from different motional degrees of freedom. We thus found that molecular surface diffusion around the cylindrical aggregates makes a dominant contribution to the longitudinal relaxation rates, which therefore could be used to determine the surface diffusion coefficients of the spin-bearing molecular species.

Our NMR data suggest the following large-scale structure of the investigated hexagonal phase. The sample consists of homeotropically aligned domains with linear extension of a few ,um and with a narrow distribution of domain directors (a=: lo” at 25 “C). The cylindrical surfactant aggregates within each domain are very long but are not entirely rigid. The intradomain orientational disorder is characterized by an order parameter S,,/S, ~0.9, which is related to the macroscopic bend elastic constant of the domain.

Since nuclear spin relaxation rates may contain so much information, the task of extracting the dynamic and struc-

As seen from the phase diagram in Fig. 1 our hexagonal (E) phase sample is very close to the nematic No phase. In certain ionic surfactant systems, lamellar phase samples close to a nematic ND phase have been shown to consist of a layered arrangement of small disklike aggregates rather than the classical continuous bilayers.53,54 Although the E-N, phase transition in the present system is clearly first-order (as there is an intervening two-phase coexistence region), one might suspect that our E phase sample should exhibit nematiclike structural defects or fluctuations in the form of cylinder fragmentation and/or orientational disorder. Our analysis shows that fragmentation is unimportant (i.e., the aggregates are very long) but that some orientational disorder does exist. To decide whether this disorder CSLJ& Dd =: 0.9 ) is typical of hexagonal mesophases in general or is a result of the proximity to a nematic phase, one

J. Chem. Phys., Vol. 95, No. 9,l November 1991


would have to study E phase samples of different composition. It is also possible that the orientational disorder is related to the unusually high dilution (aggregate volume fraction, 4 = 0.27) of our E phase sample, rather than to the proximity to a nematic phase. The SDS/decanol/water system appears to be unusual in that a small amount of decanol allows the E phase to swell to much lower volume fractions than can be achieved in the binary system. In the closely related system sodium decyl sulphate/decanol/water this is not possible and 4 > 0.38 throughout the E phase.55 This E phase should also be readily aligned since, like the present one, it can be prepared by heating a nematic Nc phase.56*57 In other systems” an aligned E phase may be obtained by cooling an Nc phase.

Spin relaxation studies along the lines described here should also prove valuable as a means for elucidating aggregate size and orientational order in lyotropic nematic phases. The present work was, in fact, undertaken partly to obtain reference data for a study of the adjacent nematic phases (cf. Fig. 1). But that is another story.5g

ACKNOWLEDGMENTS

We are grateful to Dr. Krister Fontell for performing the x-ray diffraction experiment. This work was supported by grants from the Swedish Natural Science Research Coun- cil.

APPENDIX A: SPIN RELAXATION DUE TO COUNTERION DIFFUSION IN AN HEXAGONAL MESOPHASE

In this Appendix we present the results of numerical calculations of the director-frame spectral density functions J z (0) associated with counterion diffusion in the hexagonal phase investigated here. Since this problem has already been treated in some detail,43 only a brief account of the theory (adapted to the present notation) is given here before presenting the numerical results.

Since the residual efg is short-ranged, being determined essentially by the surface-induced asymmetry in the primary hydration shell of the counterion and by (anisotropically distributed) ionic species in the immediate vicinity of the reference counterion, the components of the residual efg tensor should vanish outside an interfacial layer of thickness 6 of a few Angstrom. Accordingly we model the local residual qcc X(r) as a step function, taking the value ?.$ in the interfacial region (b < r < b + S, where b is the cylinder radius) and vanishing elsewhere.

The director-frame spectral density functions (for a spin-3/2 nucleus) may then be expressed as43

J,“(w) =+P$(l -P,) I

=dtcos(ot)j$‘(f), (Ala) 0

J?(w) = 0, (Alb)

Jf(o) = pP, s

02 dt cos(ot)g(t), (Alc)

0

where P, is the fraction counterions in the interfacial region (where the local qcc is nonzero). The reduced time correla-

tion functions in (A 1) , defined so that gP, (0) = 1, are given by

i%(f) = l s

2a

Ps(l -St&P,) 0 4 cos(md)

I b+6

X dro rdlro 1 b I

b+6

dr r[f(r,$,t Ire 1 b

- b.fC r) /2p], (A21 where r is the radial cylindrical coordinate and 4 is the azimuthal angle measuring the displacement of the ion around the cylinder. Further, f( r) and f( r&t jr, ) are the normal- ized equilibrium counterion distribution and counterion diffusion propagator, respectively. The latter satisfies the diffusion equation

+ D, 09 -!- a2f~r,~,~ lr, 1, r* dq5’

(443)

where D, (r) and D, (r) are the radial and azimuthal components of the diffusivity tensor and w(r) is the potential of mean force experienced by the counterion (due to the ionic surfactant headgroups and the other counterions). The partial differential equation (A3) is solved subject to reflecting boundary conditions at the aggregate surface (r = b) and at the cylindrical cell boundary (r = R = b /+I”, where 4 is the volume fraction of cylindrical aggregates in the hexagonal phase). The replacement of the total system by a represen- tative cylindrical cell is an approximation4” that should be acceptable for the present purposes.

The potential of mean force (in units of k, T) is ap- proximated by the Poisson-Boltzmann mean-field poten- tialbO

w(r) =21n L- cos slnr -lsin sInI- (R [ ( R) s ( R)]]’ (A4)

where the parameter s is obtained by solving the transcen- dental equation

(l+S2)[1+~cot(sln~)]~‘=I~2~~~aT, (A5)

where o is the surface charge density of the cylindrical aggregate and E, is the relative permittivity of the surrounding aqueous medium. For the present hexagonal phase we have b= 1.88nm, R = 3.60nm,andF= 16.4, whences=2.69. The fraction P, of counterions within a distance S of the surface can be expressed in closed form as a function of 6, R, S, I?, and s.~’ Taking 6 = 0.5 nm, we obtain P, = 0.77. In the limit of very strong electrostatic coupling ( I? -+ 03 ), we can replace f(r) by S(r- b)/r and f(r,& Ire) by 6(r - b)f(qS,t)/r, in which case Eqs. (Al) and (A2) reduce to the surface diffusion approximation result (4.2) for I = 3/2.

The numerical calculation of the spectral density func- tionsJE (~),definedbyEqs. (Al)-(A3),can bereduced to a matrix eigenvalue problem by discretizing the radial coordinate. The azimuthal counterion diffusivity is taken to be D, (r) = D, in the interfacial region and D4 (r) = Do else-

Quist, Halle, and Furb: Spin relaxation in liquid crystals 6959



0 107 108 109

Angular frequency, w / rad s-1

dial averaging is much faster and v = l/2 if it is much slower than azimuthal averaging. By requiring Eq. (A6) to coin- cide with the numerically calculated Jf(0) we obtain v = 0.86, i.e., we are close to the limit of fast radial averaging. This is also evident from Fig. 6, which shows that the J:(w) dispersion occurs at nearly an order of magnitude higher frequency than the J:(w) dispersion. Since P,” thus differs by merely 3.7% from P,, we can set v = 1. In conclusion, the surface diffusion approximation, as defined by Eqs. (4.2)-(4.3), is an excellent approximation, not only for the surfactant molecules, but also for the counterions in the hexagonal phase investigated here.

FIG. 6. Spectral density functions for ‘“Na counterion diffusion in the aqueous part of an hexagonal phase composed of infinitely long, straight cylinders. The azimuthal diffusivity is Dm = D, within 0.5 nm of the surface and D,, elsewhere, whereas the radial diffusivity is D, = D,, uniformly. The surface diffusion coefficient is D, = 4.8 X 10 ” m’ s- ‘, as determined in this work at 25 “C, and the bulk diffusion coefficient is 4, = 1.33~ 10 “’ mz s ‘. The local residual qcc is X$ = 100 kHz and the other parameter values are given in the text. The vertical dashed lines are at the Larmor frequency w,, and at 2w,, where the spectral density functions are probed by the longitudinal relaxation rates. The dashed (Lorentzian) curve corresponds to the modified surface diffusion approximation (A6) with b,. = b and u = 0.86.

where, with b */(4D,) = 2.5 ns and OS/Do = 0.35 (cf. Sec. V B) . For simplicity the radial diffusivity is taken to be uniform, D,(r) = Do. (The effect of having D, = D, in the interfacial region is small.) The resulting spectral density functions are shown in Fig. 6. It is evident that the direct effect of radial diffusion through J:(o) is negligible. Further, the spectral density function J;(w) turns out to be very nearly Lorentzian, with only a small high-frequency tail from fast small-amplitude radial diffusion.

The near-Lorentzian shape of the exact J;(w) suggests a modified surface diffusion approximation of the form

J;(w) = T (P;?g* Q-C 1+ (wT,)* ’

(A6)

b2 7-c:, c

40, (A7)

with an effective diffusion radius 6,. The dashed Lorentzian dispersion in Fig. 6 is obtained from Eq. (A6) with 6, and v determined as described below. Regarding the calculated exact J:(w) as experimental data, we can use Eq. (A6) to obtain the correlation time 7, from the ratio J f ( w0 )/J f( 2w, ) as described in the Sec. IV B. Comparing r’, with the true value of b */(4D, ) (which was used as input for the calculation), we obtain 6, = 1 .OOl b. Consequently, we can set b, = b. The surprisingly small difference between 6, and b results from the cancellation of two effects: although diffusion around the cylinder now involves longer trajectories than if the ion were confined to the surface r = 6, the diffusion coefficient is larger (by a factor l/0.35) outside the interfacial region. [If, in conflict with our experimental results (cf. Sec. V B), we set D, (r) = D, uniformly then we find that b, is 10% larger than b.]

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