Anisotropic 23Na Spin Relaxation in Liquid Crystals ...

JOURNAL OF MAGNETIC RESONANCE 100, 267-28 1 ( 1992)

Anisotropic 23Na Spin Relaxation in Liquid Crystals. Determination of All Nine Spectral Densities for a Hexagonal Lyotropic Phase

PER-OLAQUIST,INGEBLOM, AND BERTILHALLE

Pkvsical Chemistcv 1, University qf Lund, Chemical Center, P.O. Box 124, S-221 00 Lund, Sweden

Received December IS, 199 I ; revised February 19, I992

The orientation dependence of three independent 23Na relaxation rates is determined from measurements on the counterions in a magnetically aligned hexagonal lyotropic liquid crystal. The data allow us to determine the nine director-frame spectral densities that comprise the full information content ofthe spin-relaxation behavior ofa quadrupolar nucleus in a uniaxial phase. The model-independent director-frame spectral densities provide important information about the time scale and rotational symmetry of the anisotropic motions in the phase. More detailed information is obtained by invoking a specific dynamic model that separates dynamic and structural factors. The six high-frequency spectral densities are quantitatively accounted for in terms of fast local motions and counterion surface diffusion around the cylindrical surfactant aggregates, while the three zero-frequency spectral densities contain large contributions from diffusion-modulated director fluctuations. 0 1992 Academic Press. Inc.

Spin relaxation of quadrupolar nuclei has been widely used to investigate molecular dynamics and organization in anisotropic fluids, such as thermotropic liquid crystals ( I, 2) and phospholipid bilayers (3, 4), as well as in molecular solids (5). Within the motional narrowing regime (6)) the spin-relaxation behavior of any quadrupole-relaxed nucleus is completely determined by three multivariable functions: the laboratory- frame spectral density (LFSD) functions J,“( kwo, QLP) with k = 0, 1, 2. These functions embody all information about the molecular system that can be obtained from spin- relaxation experiments.

Each LFSD is a different function of Larmor frequency ( oO) and sample orientation ( fiLP). For macroscopically oriented samples both of these variables are under the control of the experimenter; however, there is an important distinction between them. Whereas the functional form of the frequency dependence, reflecting the particular molecular motions taking place in the investigated system, is not known a priori, the functional form of the orientation dependence is, in fact, known explicitly as it follows directly from the macroscopic rotational symmetry of the system. By exploiting this symmetry one arrives at a more fundamental, and more concise, characterization of the molecular system in terms of director-frame spectral density (DFSD) functions. The number of independent DFSD functions depends on the symmetry of the system.

The vast majority of spin-relaxation studies of anisotropic systems concerns uniaxial systems. (In the present context, a uniaxial system is defined as a system whose symmetry group includes an n-fold axis with n 2 5.) The orientation of a uniaxial system

267 0022-2364192 $5.00 Copyright 0 1992 by Academic Press. Inc. All rights of reproduction in any form resewed.

268 QUIST. BLOM. AND HALLE

can be specified by a single quantity: the angle eLP between the static magnetic field and the symmetry axis. For uniaxial systems there are three independent DFSD functions, J:(w), with y1 = 0, 1, 2. The model-independent information available from spin-relaxation measurements (at fixed magnetic field) on a uniaxial system thus com- prises the nine quantities JF( kwo) with k, y1 = 0, 1, 2. To determine these nine quantities, one needs to measure three linearly independent relaxation rates, each at three different orientations. Inclusion of more relaxation rates or orientations may improve the accuracy but does not provide new information.

The realization that the orientation dependence of spin-relaxation rates provides useful information about molecular dynamics is by no means new. Orientation-dependent ‘H (R, and R,,) and 2H (RI= and Rio) relaxation rates as well as orientation- dependent ESR linewidths have been extensively measured in thermotropic liquid crystals ( 7-16) and, more recently, in phospholipid bilayers ( 17-20) and in hexagonal lyotropic liquid crystals (21, 22). In most of these studies, however, individual spectral densities were not determined. As far as we know, the only reports of individual director-frame spectral densities are the determinations of five DFSDs from measurements at HLP = 0” and 90” in a discotic mesophase ( 15) and in two hexagonal lyotropic phases (21, 22) and the determination of six DFSDs from measurements at HLP = O”, 30”, and 90” in a smectic B phase (14).

In the following, we present orientation-dependent 23Na spin-relaxation rates from the counterions in the macroscopically oriented hexagonal phase of the system sodium dodecyl sulfate/decanol/water. The relaxation data allow us to determine all nine DFSDs JF( kwo). To our knowledge, this is the first report of a determination of the con?pl~te information content of the spin-relaxation behavior of any ordered fluid (at given Larmor frequency, temperature, etc.).

It is important to realize that the nine DFSDs J,“( kwo) constitute modeel-independent information and, as such, can be used to discriminate among alternative models. Taken together, these nine DFSDs inform us about the time scale as well as about the rotational symmetry of the anisotropic molecular motions in the hexagonal phase. The fast, weakly anisotropic, local motions (mainly within the counterion hydration shell) are expected to contribute equally to all nine DFSDs; counterion diffusion around uniformly oriented, straight cylindrical surfactant aggregates should contribute only to the three DFSDs with n = 2; and slow diffusion-modulated director fluctuations should contribute only to the three DFSDs with k = 0. A determination of the individual spectral densities, rather than the orientation dependence of a single spin-relaxation rate, is particularly revealing in cases where a nearly orientation- independent relaxation rate (18, 20-22), due to a balancing of spectral densities with opposite orientation dependences, might erroneously be interpreted as being due to nearly isotropic motions.

To proceed further and to separate dynamic properties (temporal correlations) from static properties (spatial correlations), it is necessary to invoke specific dynamic models. We find that a model of fast, weakly anisotropic, local motions and slower counterion diffusion around the cylindrical aggregates (22) accounts quantitatively for the six high-frequency DFSDs (k = 1, 2). In addition; the three zero-frequency DFSDs (k = 0), in particular Jy( 0), contain large contributions from director fluctuations. !

SODIUM RELAXATION IN LIQUID CRYSTALS 269

EXPERIMENTAL

All 13Na NMR experiments were performed on a sample in the hexagonal phase of composition 25.9/3.5/70.6 wt% sodium dodecyl sulfate/decanol/water. The sample was prepared and magnetically aligned (by heating from the nematic calamitic phase) as described in Ref. (22). On account of the high viscosity of the phase, the initial macroscopic orientation of the phase is essentially “frozen in,” thus allowing NMR measurements to be performed at any angle 6’ rP between the magnetic field and the phase director. In fact, no changes were observed in the 23Na lineshape at HLP = 0” after the oriented sample was kept at orientations (0t.r # O”, 90”) with nonzero magnetic torque for one week.

The ‘“Na NMR measurements were performed at 25.O”C on a Bruker MSL-100 spectrometer with a horizontal 10 mm solenoidal probe (22). The director orientation was varied by simply twisting the sample tube around its axis and BLP was measured by a homemade goniometer with an estimated accuracy of & lo. The magnetic field inhomogeneity, measured as the difference between the inhomogeneous and homogeneous central linewidths, was about 10 Hz at all orientations. The three LFSDs J:( koo, HLP) with k = 0, 1, 2 were obtained from the three independent relaxation rates RI,, RsE and R QE determined as follows.

The single-exponenfia; recovery of the central line intensity with increasing delay time 7 in the modified inversion-recovery pulse sequence (23) 180°--7-54.7’- acq. yields

RI,(~LP) = ~J~L(~wo, ALP). [II The single-exponential decay of the central line intensity with 7 in the spin-echo

pulse sequence (24-26) 90 : ---I 80;----7-acq., with the phase &I of the refocusing pulse varied according to the “Exorcycle” scheme (27-29) yields

R,SE(HLP) = J?(%? BLP) + -G-(-%I,~LP). [II Both these experiments were performed with a narrow filter width (2-3 kHz) and

24 delay times in the range (0.05-20)/R. The signal-to-noise ratio was at least 250 for the shortest delay time. The relaxation rates were obtained from a three-parameter least-squares fit to the central peak intensity, Z(r) = A + B exp( - RT).

The homogeneous satellite linewidth Auprn = RzE/r was determined from a 2D quadrupolar-echo experiment (21, 26) with the pulse sequence 90 T -7-90 & - T-acq., yielding

R?(~LP) = Jii(O, ALP) + Ji-(wo> HLP) + J5(2q1> HLP). [31

Typically. 300 delay times were used with a signal-to-noise ratio of at least 100 for the shortest delay time. The duration of the 2D QE experiments ranged from 4 to 40 hours, the longer times being necessary because of the large inhomogeneous satellite broadening at intermediate orientations ( BLp # O”, 90”). In all three relaxation experiments, the repetition time between successive accumulations was at least 10/R,,.

On the basis of reproducibility tests, we estimate the experimental random error (k2a) to +l% for Rlc, t1.4% for R,SE. and +4% for RFE at all orientations. The R,, results are susceptible to systematic errors from inaccurate setting of the flip angle of

270 QUIST, BLOM, AND HALLE

the detection pulse and from BI inhomogeneity; however, these effects are unimportant in the present study (cf. below).

SPECTRAL LINESHAPES

23Na spectra recorded at five different orientations of the macroscopically oriented hexagonal phase are shown in Fig. 1. One notes immediately the distinctly asymmetric satellites in the parallel and perpendicular orientations (Figs. la and le) and the

3)

Cd) l”~“‘~‘,,~~~‘,~,,,,~~,‘,-,,,l.l~,~lll,l~ll

2oooo 10000 0 -1OOCil -2o300 Hertz

FIG. 1. Orientation-dependent Z3Na spectra obtained from the quadrupolar echo with T = I / uo. (a) BLP = 0” vertically magnified twice (X2). (b) HLP = 20” (X8), (c) HLP = 65” (x8), (d) HLP = 78” (X8), and (e) BLp = 90” (Xl). The central peak is truncated in spectra (a)-(d).


dramatically enhanced satellite broadening at intermediate orientations ( Figs. 1 b, 1 c, and 1 d). These spectral features are the hallmark of a static (nonaveraged) distribution, f( &,) , of director orientations ( 22, 30).

The inhomogeneously broadened satellite lineshape L( w; I&) is the superposition of homogeneous (Lorentzian ) lineshapes Ln( w; 19~~) weighted according to the director distribution.f( 8rn; 19~~); i.e.,

T L(w; BLP) = s deLDsin eLDf(OLD; ~LP)LD(% OLD), [41

0

&(~LD) LD(W'BLD) = [R,(BLD)]" + [O - ~t~a(b'L,)]~' 151

where R,(OLD) = RPE( BLn) + 6/x, 6 being the 10 Hz magnetic field inhomogeneity broadening (cf. Experimental). The quadrupole frequency wo( OLD) is

aQ(dLD) = 4p2(cos OLD). [61 In Eqs. [ 4]- [ 61, &n is the angle between the static magnetic field and the director,

defined as the average orientation of the cylindrical aggregates in a volume (“domain”) wherein Na+ diffusion averages out all inhomogeneities in the static quadrupole coupling. The angle eLp, on the other hand, specifies the orientation (with respect to the magnetic field) of the macroscopic phase director, defined as the average of the “domain” directors in the sample.

The characteristic spectral features noted above can be qualitatively understood on the basis of Eq. [ 61 and its derivative

dw,- 3

d&D - - 5 wtsin( 2&n). [71

Equation [6] shows that -w$/2 < uQ < w&, which accounts for the “edge” features in the spectra in Figs. la, Id, and 1 e. Equation [ 71 shows that the satellite broadening caused by a small orientational spread vanishes (to first order) at BLp = 0” and 90” and is largest at BLP = 45”.

The inverse of [ 71 is also the frequency weighting factor associated with the well- known singular features in powder spectra. It shows that the 90” peaks in a 3D powder spectrum are nut, as widely believed, due to the predominance of 90” orientations in an isotropic distribution. In a 2D powder, with f( &n) = l/r, there are singularities at 8 Ln = 0” and 90”, where the frequency weighting factor d8rD/dwQ diverges. In a 3D powder, Withf(dLD) = sin(tiLD)/& the 6 Ln = 0” singularity is precisely cancelled, leaving the &n = 90” singularity to produce the powder satellite peaks. If the 3D powder distribution is not perfectly isotropic but exhibits a slight tendency for parallel alignment (0 Ln = 0”), then the &n = 0” singularity is not completely cancelled and one observes the well-known “rabbit ear” feature in the powder spectrum.

For a quantitative analysis of the inhomogeneous satellite lineshape, we need to consider the distribution function ,f( OLD; BLP). Since the highly viscous hexagonal- phase sample does not respond to the magnetic field, it is clear that the shape of the distribution ./( &n; HLP) depends on the macroscopic orientation of the phase, i.e., on the angle BLP between the magnetic field and the symmetry axis of the uniaxial phase.


However, f( HLD; f&) can be derived from the distribution f( &.,,), which directly reflects the orientational disorder, with respect to the symmetry axis of the phase, that was frozen in as the magnetically aligned nematic phase was transformed into a hexagonal phase (cf. Experimental). Since this is not necessarily an equilibrium distribution, its functional form cannot be specified a priori. If the tIPD distribution is suf- ficiently narrow, we might hope that a Gaussian form is adequate (22, 30). The BLD distribution can then be calculated as

.f(hD; &P) = A r&p 1 WDsin &Dew-&J&)

X G[cos 19~~ - cos eLpcos tlPD + sin BLpsin epDcos @PI

s

oLP+om ==A dfJPD

sin dpDexp( -8&/a&)

I flu- b I [sin20Lpsin2BpD - (cos 19~~ - cos B,,cos /&)*I ‘j2 ’

[81 where A is an unimportant normalization constant and uPD = (( &,)2) ‘I* is the root- mean-square orientational spread. The result [8], obtained by integrpting over the azimuthal angle &, is valid for flLP # 0; for eLP = 0, one finds trivially ,j”( 19~~; 0) = ew-WdD).

Using Eqs. [ 4]-[ 61 and [ 81 we can now attempt to reproduce the experimental satellite lineshapes in Fig. 1 by varying w$ and flPD in a nonlinear least-squares fitting procedure. The phase orientation 0 LP is taken from the goniometer readings, and R,( B,D) is approximated by R,( 19~~) with RFE values from Table 1. The result of such a fit is shown in Fig. 2a for dLp = 0”. Using the parameter values from this fit (w6/2~ = 18.38 kHz and @pD = 7.0”), we then calculate the satellite lineshape at HLP = 20”, shown in Fig. 2b. Systematic deviations are seen in both cases, suggesting that the actual BPD distribution deviates significantly from the Gaussian form.

The assumption of a Gaussian opD distribution can actually be tested in a simple way. Consider, for example, the tiLp = 20” spectrum (Figs. lb and 2b). Since the homogeneous linewidth (R,/T = 88 Hz) is much smaller than the inhomogeneous one ( AyFhorn = 2.96 kHz), we may approximate the Lorentzian LD(w; OLD) in Eq. [5] by the delta function 6[0 - wo(&)]. We then obtain from Eqs. [4], [6], and [71

TABLE I

Orientation-Dependent 2’Na Spin-Relaxation Rates

flLp (degrees) RI, (s-‘) RzE (s-l) RQE(s-') s

0 125.9 94.9 158 20 120.9 98.2 245 39 100.2 96.2 - 65 17.6 93.5 282 78 69.8 90.0 230 90 66.5 88.9 193


(a)

V/kHZ v/kHz

FIG. 2. Satellite lineshape (a) at BLP = 0” (solid line) and the fit (dashed line) resulting from Eqs. [4]- [6] and [8] and (b) at BLP = 20” (solid line) and the calculated lineshape (dashed line) obtained with parameters deduced from the fit in (a).

.f‘(hDi BLP) +.f(r - OLDi kp) = B COS ~LDLb$%(COS OLD); 0~~1, [91

where B is an unimportant normalization constant. The second term on the left, which arises from the fact that the orientations &, and P - OLD correspond to the same quadrupole frequency, can safely be neglected in the case BLP = 20”.

The director distribution functionf( HLD; BLP) obtained, by means of Eq. [ 91, from the satellite lineshape in Fig. 2b is shown in Fig. 3. The quantity W$ was taken from the fit to the BLP = 0” spectrum in Fig. 2a. (As it is determined by the edges in the spectrum, the fitted ~6 is insensitive to the shape of the OLD distribution.) Figure 3 also shows the distribution ,f( OLD; BLp) obtained from Eq. [ 81 with dLp = 20” and dpn = 5”. This distribution is very nearly Gaussian; it differs from exp[-(OLD - &,)*/a&,] essentially by a small (<lo) shift toward smaller OLD. A comparison of

0 10 20 30 40 OLD / deg.

FIG. 3. Director distribution function [(flro; 20”) obtained from the satellite lineshape (solid line) at BLP = 20”, using Eq. [9]. and from a Gaussian H ro distribution (dashed line) with gPD = 5”, using Eq. [ 81.

274 QUIST, BLOM. AND HALLE

the two distribution functions in Fig. 3 shows that the actual f( HLD; BLP) has more intensity in the “wings” than a Gaussian and is skewed toward smaller OLD. This explains the failure to obtain quantitatively accurate lineshape fits with a Gaussian t&u distribution.

Before leaving the subject of lineshapes, we note that the satellite lineshape in the BLP = 78” spectrum (Figs. Id and 4) reflects a bimodal distribution of quadrupole frequencies resulting from a unimodal distribution of director orientations. This may be understood from Eq. [ 91, which shows that (for a negligibly small homogeneous linewidth) the spectral amplitude at a frequency w = w$‘,( cos OLD) is proportional to Mb; HLP) +.f(r - h; BLP)]/cos fILD and hence may exhibit one peak near w = w?&(cos &) associated with the maximum inf( OLD) and another peak near w = - ~$12 associated with the cos 0,n factor (which accounts for the high density of quadrupole frequencies near BLD = 90”; cf. Eq. [7]). Using Eqs. [4]-[8] we can qualitatively reproduce the bimodal satellite lineshape as shown in Fig. 4; however, a quantitative agreement seems to require a non-Gaussian f&n distribution.

DIRECTOR-FRAME SPECTRAL DENSITIES

The three independent ‘3Na spin-relaxation rates R,,, RzE, and RzE were determined at six orientations eLP of the hexagonal phase. From the relaxation rates in Table 1 we obtain, by means of Eqs. [I] - [ 31, the three LFSDs Jk( kwo, BLP) with k = 0, 1,2. The pronounced orientation dependence of these LFSDs, presented in Table 2, demonstrates that anisotropic motions make important contributions to the 23Na relaxation. The opposite orientation dependence of Jk(wo, BLP) and Ji(2w,,, &) is a characteristic feature of hexagonal mesophases ( 21, 22) and causes R 2” to be nearly independent of orientation.

The imperfect director alignment, reflected in the inhomogeneous satellite lineshape (cf. preceding section), may also affect the orientation dependence of the spin-relaxation rates. However, since the relaxation rates are much smaller than the quadrupole frequency, the relaxation rate distribution is motionally averaged, whereas the quad-

-11 -10 -9 -8 -1 -6 -5 -4 v/kHz

FIG. 4. Satellite lineshape at 0 Lp = 78” (solid line) and the calculated lineshape (dashed line) obtained from Eqs. [ 4]- [ 61 and [ 8 1, In order to generate a bimodal lineshape an orientational spread pPD of 8” was used.


TABLE 2

Orientation-Dependent Laboratory-Frame Spectral Densities

BLp (degrees)

0 63 32.0 63.0 20 147 37.8 60.4 39 - 46.1 50.1 65 189 54.1 38.8 78 140 55.1 34.9 90 104 55.6 33.2

Uncertainty” +-8 k1.0 k0.6

a Propagated random errors in R,,, Rs", and RFE (cf. Experimental)

rupole frequency distribution is essentially static (21, 22). If motional averaging of the relaxation rates over the 0 LD distribution did not take place, we would expect multiexponential relaxation in the R,, and R,SE experiments involving the central line. This is not observed. In fact, RI, and 2RzE were equal, within the experimental uncertainty, to the relaxation rates R,, and R,,, which were determined from the satellite peaks (at eLP = 0” and 90” only) and therefore correspond essentially to OLD = BLP (21, 22).

For a uniaxial phase, each of the three LFSDs is a different linear combination of three DFSDs:

[lOI

The necessary requirement for the validity of Eq. [lo] is that the phase director is an n-fold axis with IZ > 5 (31). The angular functions ckn( &) can be expressed in terms of reduced Wigner functions as (21, 32)

or in explicit form, with x = cos OLP, as (14)

coo = +( 1 - 3X2)2,

Cl1 = $( 1 - 3x2 + 4x4),

c22= A(1 +6x2+x4),

co1 = 2c,o = 3x2( 1 - x2),

co2 = 2c20 = ;( 1 - x2)?,

Cl2 = C2l = i(l -x4 ).


These functions are shown in Fig. 5. It follows from Eq. [ 1 l] or [ 121 that

2

c Cd&P) = 1. n=O

[I31

To obtain the three DFSDs JF( koo) of given k, we need to determine the corre- sponding LFSD Jk( kwo, tiLp) at three orientations 0 rP. Since we have determined the LFSDs at five or six orientations (cf. Table 2)) we obtain the DFSDs by a generalized linear least-squares analysis. The fit is shown in Fig. 6 and the resulting DFSDs are collected in Table 3. These nine quantities represent the complete information available from spin-relaxation measurements on the present uniaxiul phase under the given external conditions.

As seen from Table 3, the standard deviations for the DFSDs JF( kw,) vary con- siderably with n. A more detailed insight is provided by the covariance matrices in Table 4. As expected for a data set including (ILP = O”, the “diagonal” DFSDs JF( kw,,) are most accurately determined. Consider now the case k = 0. From Fig. 5a it is seen that J:(O) and J?(O) can be accurately determined from data at fILP = 0” and 90”, while an accurate determination of J?(O) requires measurements in the

FIG. 5. The angular functions ck,(O,,) in Eq. [I l] or [12] vs 0~: (a) co,.(b) cl,,. and Cc) ~‘2~


t

FIG. 6. Fit ofEqs. [lo] and [I I] to the LFSDs Jk(O, BLP) (O), J:(wO, BLP) (o), and J4(2w0. BLP) (0) in Table 2. The deduced DFSDs JF( kw,) are collected in Table 3.

neighborhood of HLP = 45” (BLP = 43.7” is the optimal third angle). Due to the large inhomogeneous broadening, however, J,“( 0, err) cannot be accurately determined for orientations near ALP = 45”. The optimal choice of orientations, providing the best accuracy in a given experimental time, thus depends on the amount of orientational disorder in the sample. It should be noted, however, that what really matters is the relative accuracy. Although the standard deviation for Jp( 0) is 2.5 times larger than that for J,“( 0)) the relative error in Jy( 0) is roughly half of that in JF( 0). In the case k = 1, the largest standard deviation occurs for Jx( wO). As seen from Fig. 5b, data obtained in the neighborhood of BLP = 0” or 90” contain little information about JF( oo). Even with data at /ILP = 39” (f& = 4 lo is the optimal third angle), however, the standard deviation for JF( wo) is more than 3 times larger than that for Jy( oo). This is because all three JF( wo) make comparable contributions in this HLP range, in contrast to the case k = 0. The case k = 2 is the most unfavorable one, since there is no orientation, other than BLP = O”, at which one of the c2,, functions vanishes. To make things worse, the functions c 2. and c2, are either small or vary in a qualitatively similar way (cf. Fig. 5~). As a result, the covariance between J,“( 2wo) and Jp( 2wo) is large (cf. Table 4) and both are poorly determined as compared to Jy( 2~~). For- tunately, the standard deviations for J,“( 2wo) and Jp(2wo) are kept at a reasonable level by the high precision in the LFSDs J,“( 2wo, &).

TABLE 3

Director-Frame Spectral Densities Jr(kw,) (s-‘)~

k n=O n= 1 n=2

0 68 f7 300 + 17 120 +s I 30.8 f 2.9 32.3 +- 0.8 78.7 f 1.6

2 28.4 f 4.1 30.0 f 2.1 63.4 IL 0.5

u Uncertainties based on random errors in R,,, RsE, and RFE (cf. Experimental).


TABLE 4

Covariance Matrices for Director-Frame Spectral Densities”

k=O

1.00 -0.75 -0.09

- 6.38 -1.33 - - 1.38 1

k=l

Il.9 -0.58

[-

-3.23 1.00 -0.76

- - 3.37 I

k=2

63.6 -40.4 4.16 - 27.6 -3.13 - - I .oo I

n Each matrix has been normalized to make the smallest diagonal element (variance) equal to unity. The matrices are sym- metric.

DYNAMIC MODEL

The nine DFSDs in Table 3 report on molecular dynamics as well as on the equilibrium structure of the mesophase. Furthermore, since the hexagonal phase exhibits orientational order at several levels (or length scales), several kinds of anisotropic molecular motion may contribute to the DFSDs. For these reasons, a model is needed to extract the molecular-level information from the DFSDs.

Considering the high-frequency (o = wo, 2wo) 2H and 23Na relaxation behavior in the present system, we have previously (22) shown that the simplest model consistent with the relaxation data is one of fast (compared to wo), local, nearly isotropic, motions and slower surface diffusion around the symmetry axis of slightly orientationally disordered cylindrical surfactant aggregates. To account for the zero-frequency DFSDs, we now include also a contribution from slow (compared to wo) director fluctuations, i.e., counterion diffusion among spatial regions with different cylinder axis orientation. The 23Na DFSDs then take the form (22)

3a2 J;(kw,) = Jf + - X2,A, razi

4 l + Cko07aziJ2 + Gj&I~f(o). 1141

The first term in Eq. [ 141 represents the local motions; it is assumed independent of k (extreme narrowing) and IZ (weak anisotropy). As discussed in detail elsewhere (21, 33, 34)) these two approximations should be highly accurate in the present case. The second term accounts for surface diffusion around the cylindrical aggregates with a correlation time T,,i. XR is the root-mean-square residual quadrupole coupling constant, averaged by the local motions. The amplitude factors A, account for orientational disorder of the cylinder axes with respect to the phase director (32) :


A, =LSS-’ 35Q, [15bl AZ = : + $ + &Q. [15cl

The order parameters in Eq. [ 151 are S = (P*( cos &c)) and Q = ( P4( cos flpc)), where BPc is the angle between the phase director and the local cylinder axis. They reflect cylinder flexibility (flu,) as well as a nonuniform director orientation (or,). For straight cylinders parallel to the phase director, A,, = 6,*/2.

Considering the six high-frequency DFSDs (k = 1, 2 ), to which the third, director- fluctuation, term in Eq. [ 141 does not contribute, our model contains five parameters. Assuming a functional relation between the order parameters S and Q (35 ), we elim- inate Q as an independent parameter. (This has very little effect on the remaining parameters.) A nonlinear least-squares fit of the four parameters to the six DFSDs yields Jr = 29.3 + 2.7 s’ (30 5 1 SK’), ?? R = 81 + 6 kHz (79 + 2 kHz), S = 0.91 + 0.08 (0.92 * 0.03), and 7,i = 2.5 + 0.2 ns (2.3 * 0.5 ns). The figures within parentheses are the results of our previous combined 2H and 23Na relaxation study (22) at two orientations ( eLP = 0” and 90” ). While th e t wo sets of results are in close agreement, it should be noted that the present spin-relaxation data do not provide statistically significant support for ruling out the simpler model of uniformly aligned cylinders (S = Q = 1). The situation is different, however, for the previously reported 2H data (22), where surface diffusion makes a relatively large contribution also to the DFSDs with n # 2.

The values obtained for 2, and S are consistent with the information contained in the inhomogeneous satellite lineshapes. From the BLP = 0” lineshape we obtained VQ ’ = 18.38 kHz, whence i!Q = 421, - ’ - 73.5 kHz. As expected (22), XR > %Q. An orientational spread (orb = 7” corresponds to a (static) director order parameter S,i, = 1 ~ &z,, = 0.98 which, as expected, is larger than the cylinder order parameter S in Eq. [ 151. We note also that, for an effective diffusion radius b, = 2.1 nm (22)) the correlation time T,,i obtained here corresponds to a counterion surface diffusion coef- ficient D, = b,Z/(47,,i) = (4.4 k 0.4) X 10-l’ m2 SC’.

Using Eqs. [ 141 and [ 151 and the deduced parameter values, we can now calculate the combined contribution from local motions and surface diffusion to each of the

TABLE 5

Contributions to Director-Frame Spectral Densities JF (h-w,,) from Local Motions and Surface Diffusion”

k n=O n= 1 n=2

0 29.6 32.7 84.4 1 29.6 32.2 18.7

2 29.5 31.3 63.4

“Calculated from Eqs. [ 141 and [ 151 with parameter values from the fit to JF(kw,), k = I, 2.

280 QUIST, BLOM. AND HALLE

nine DFSDs (Table 5). A comparison of Tables 3 and 5 shows that, while the six high-frequency (k = 1,2) DFSDs are quantitatively accounted for, the zero-frequency DFSDs J:(O) include large contributions from an additional dynamic process. We ascribe this contribution, represented by the third term in Eq. [ 141, to translational diffusion of counterions through the orientationally disordered hexagonal phase. Sub- tracting the values in the first row of Table 5 from those in Table 3, we find Jt’( 0) = 38 f 7 s-l, Jyf(0) = 267 -t 17 ss’, and Jlf(0) = 36 f 8 s’.

A detailed interpretation of these zero-frequency DFSDs requires a theory relating the amplitude and spatial correlation of orientational fluctuations to the anisotropic elastic properties of the hexagonal phase (36). This will not be attempted here. We mention, however, two facts supporting the interpretation in terms of (diffusion-modulated) director fluctuations. First, J;” (0) 9 Ji’( 0)) Ji’( 0)) as expected for small- amplitude fluctuations ( cpc G 1 ), since Jyf (0) is of second order in upc while Ji’( 0) and Ji’( 0) are of fourth order (21, 37, 38). Second, the zero-frequency LFSD J,“( 0, BLp), which contains the director-fluctuation spectral densities .I:’ (O), was found to be correlated with the static director spread. As discussed previously (22)) a newly aligned hexagonal phase undergoes a slow “annealing” process (independently of the magnetic field), whereby the director spread decreases. This is seen most clearly in the inhomogeneous satellite linewidth at intermediate orientations (BLp # O”, 90”), but is also seen as a reduction of the transverse relaxation rate RFE. The RFE data and lineshapes reported here were recorded two months after the magnetic alignment of the sample.

ACKNOWLEDGMENTS

This work was supported by the Swedish Natural Science Research Council. We thank IstvLn Fur6 for useful comments on the manuscript,

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