Spectral Densities of Motion in Liquid

Crystals by Deuterium NMR Relaxation StudyRonald Y. Dong

Department of Physics and AstrononyBrandon University

Brandon, Manitoba, R7A 6A9Canada

ContentsI. Introduction 29

II. Theory 30A. Anisotropic molecular reorientation in an ordering potential 32B. Restricted rotation in a cone model . . 33C. Director fluctuations • • 33

III. Experimental Methods 33

IV. Results And Discussion 35A. Core dynamics 35

B. Chain Dynamics . . . 40

V. Conclusions > 41

References 42

I. INTRODUCTIONIn this paper we discuss some recent advances inrnolecular dynamics of liquid crystals by meansof NMR relaxation studies [1,2]. In particular,pulsed NMR techniques using a deuteron probeenable separation of spectral densities of motion[3-8] in liquid crystals. These spectral densitiescontain information on the frequency spectra offluctuations from motions such as director fluctu-ations, molecular tumbling and, in case of flexiblemolecules, internal motions. It is customary in

formulating theoretical models to assume that themolecular tumbling and internal motions are de-coupled [9]. The ability to separate spectral den-sities and the site specificity of the deuteron probehave rendered deuterium NMR as one of the morepowerful techniques in unravelling various modesof motions. Liquid crystals serve well for study-ing dynamical processes in an anisotropic medium.Moreover, their richness in the mesomorphic typesand in the molecular structures has caused increas-ing interests in these materials. Besides the ther-motropic liquid crystals discussed here, there arelyotropic [10] and polymeric [ 11] liquid crystals.

Vol. 9, No.1/2 29

Thermo tropic liquid crystals give mesophasesupon heating and are formed by one type of or-ganic molecules or by a mixture of two or morecomponents. It is now well-known that both mol-ecules of rod (or lath) shape [12] and disk shape[13] can form liquid crystals. We will examine asexamples a rodlike liquid crystal and a discoticliquid crystal. In Figure 1, some of the commonmesophases formed by rodlike and discotic liquidcrystals are illustrated. It would be interestingto see whether or not motional behavior of mole-cules can reflect the packing and structure of themesophases.

Standard NMR relaxation theory [14-16] by nu-clear quadrupole interaction (assume axial sym-metry) for a spin / = 1 gives the following expres-sion for Tiz'j the Zeeman spin-lattice relaxationrate, T[Q, the quadrupole spin-lattice relaxationrate and T^1, the spin-spin relaxation rate:

T[J{9) = (1)

(2)

= f[3Jo(O,0)(3)

where A = {3ir2/2)(e2qQ/h)2,e2qQ/h is the quad-rupole coupling constant, 9 is the angle betweenthe director fi(optic axis of the mesophase) andthe external magnetic field, and

Jp{poj) = / coslput) x Gp(t)dt (4)Jo

Gp(t) is the autocorrelation function of secondrank tensor components which appears in the re-laxation Hamiltonian. It should be noted thatT^1 given by eqn. (3) is for a deuteron whosequadrupolar splitting is large. In the isotropicphase (or for small quadrupolar splitting), thecoefficient of Ji(w0) in eqn. (3) should become(5/2) A [15]. When one measures these relaxationrates at one Larmor frequency (u>0/2n) and at thesame temperature, one can evaluate JP(JXJJ0,9) forp - 0,1 and 2. We define Jp(pw0,0) = Jp{puo),the spectral densities for h being aligned along themagnetic field. From an angular dependence study

of these relaxation rates and through the followingexpressions [17,18] (for a uniaxial phase):

JO(O,0) = -{1 - 3 cos2 0)2 J0(0)4

+3cos20sin20J1(O) + -(1 - cos20)2J2(O) (5)

J^w, 0) = - cos2 9 sin2 9 J0(w)L

(6)

-2

J2(2u;,0) = -(1 - cos2 0)2 J0{2UJ)8

8(7)

one can derive nine spectral parameters in total,i.e. J0(w), Ji(w) and J2(w) at w = 0,w0 and 2w0,when one uses a spectrometer operating at oneradiofrequency. The frequency and temperaturedependences of Jp(w) may provide a critical testof motional models for liquid crystals.

II. Theory

In this section, we will describe several motionalmodels for processes that are known to relaxdeuteron spins in liquid crystals. Unfortunatelya comprehensive treatment of segmental isomer-ization and internal motions for alkyl chains ofliquid crystals is still lacking despite recent re-ports on deuterium spectral densities of the alkylchains [3,5,8]. However, the site dependence of thespectral densities strongly suggests that the usualassumption of rigid rod or disk in standard spinrelaxation theory [17-19] of liquid crystals is inap-propriate. Calculations of molecular reorientatiorto include internal phenyl ring rotation in a rodlikemolecule will be presented. Besides the moleculaireorientation, a slow and collective mode of mations, known as director fluctuations [20] and characteristic to liquid crystals, may also contribute t(deuterium spin relaxation in these materials.

30 Bulletin of Magnetic Resonano

CLASSICAL MESOPHASES(AX > 0)

NEMATIC (N)

I)M

DISCOTIC MESOPHASES(Ax < 0)

NEMATIC (ND)

SMECTIC (S) PHASESSA' SC

//Ill/Ill/

V SH' SG' SE

OQO©OQ-0

COLUMNAR (D) PHASESDhO

unioxial phcses

Oho °hd

Drd DOb.d

biaxial proses

Drd 0,'ob.d

Dhd

Figure 1. Sketch of some common mesophases formed by rodlike and discotic liquid crystals.

Vol. 9, No. 1/2 31

A. Anisotropic molecular reorientation inan ordering potential

Calculations [18,21] which involve solving equa-tions for small step rotation diffusion of a liquidcrystal molecule in an anisotropic restoring poten-tial have been done. Here we extend this modelwith an ordering potential U((30) = — \P2(cos Po)to include internal ring rotation. This willbe used later to understand the spectral densi-ties of motion in the mesophases of 50.7-d4(4-n-pentyloxybenzylidene-4'-n-heptylaniline-2,3,5,6-d4). The following assumptions are used in thiscalculation: 1) electric field gradient of a CD bondis axially symmetric, 2) internal ring rotation isuncorrelated to molecular tumbling, 3) rotationaldiffusion tensor for reorientation has axial symme-try (i.e. treat the molecule as a symmetric top),4) rotation of phenyl ring(s) is free with D' beingthe diffusion constant for rotation about its paraaxis, and 5) a strong collision model is assumed forrotation diffusion of the molecule (i.e. moleculesperform uncorrelated large angle jumps). It canbe shown that the autocorrelation function Gp(t)for the ring deuterons is given for the director par-allel to the magnetic field by [22,23].

G,{t) =

where(8)

- \(Dlq(no))\Hp06q0 (9)are the mean square averages of the Wigner rota-tion matrices and rqi, is a correlation time writtenin terms of the correlation time rq for molecular re-orientation (with D|| and Dj_, being the diffusionconstants for rotational diffusion about the par-allel and perpendicualar molecular axis, respec-tively) and the correlation time T& for internal ringrotation:

(10)

is the angle between the CD bond and thepara axis and jSpi is the angle between the paraaxis and the long molecular axis. /c(p, q) are givenby Freed [18] in terms of the order parameters

S = {DUilo)) and (DKUo)) where fio is the Eu-ler angles which transform between the directorframe and the molecular frame. These factors ac-count for effects of static order in an anisotropicmedium and arise from the fact that the autocorre-lation functions decay to non-zero averages whichare determined by equilibrium order parameters.Since (3m is usually small (~ 10°) for rodlike meso-gen, [d2

6(/?)]2 is nonzero when q = b. From eqn.(4), Jp{pu>) is given by

Jp{pu>) = K( T-2'00

In the limit that D' and (3Di, go to zero,one recovers the spectral densities reported ear-lier [1,18,24]:

4=0,1,2(12)

where

When the ring rotation is not free but is re-stricted to oscillate between ±7o> then Gp(t) canbe obtained in a manner given in refs. [22,23]:

GP(t)=

where

V 1 - r;1 + r-

(13)

- DL))

and f bb<n{lo) is given by eqn. (3.12) of ref. [22]Since rn -* 0 as n -* oo, the sum over n in Gp{t

32 Bulletin of Magnetic Resonano

can be truncated after a few terms. The spec-tral densities for the ring deuteron can again beevaluated using eqn. (4). In particular, if the ringundergoes -K flips as in the smectic E and G phases,then Tbtin(n) ̂ 0 when n is odd only and

(15)

q,b,b'

X(62 - 1/4) (6'2 - 1/4) 1 + p2^2

1 U s(62 - 9/4) (6'2 - 9/4) 1 + p2w2r2

3

B. Restricted rotation in a cone modelThe reorientation of a liquid crystal molecule as arigid rod in a restoring potential of the form

V{0)=O for 6<0O

= oo for 0 > 0o (16)has also been considered [6,25,26]. 0o is the half-angle of the cone in which the molecule rotation-ally diffuses. The autocorrelation functions ofspherical harmonics of rank two have been eval-uated by Wang and Pecora [25]. Thus one has thefollowing correlation functions

(17)

- i / JM + l)DRt) (18)

l)DRt) (19)

n=2

j oo

G±i(0 = z En = l

G±t{t) = ~1 n = l

where DR is the rotation diffusion constant andVn(p — 0,1,2), the nth eigenvalue of the diffusionoperator for each value of p are functions of no =cos'00• For 0o < 80 , only one term (or possiblytwo) is dominant among the sum of exponentialsin eqns. [17-19]. In consequence, one may write

(20)

where C£ were plotted versus 80 in ref. [25] andT'P = [t>l{i>l + with n = 1 or 2. Again thespectral densities can be obtained using eqns. [4]and [20] as

J0(0) ^ C°2r0 (21)

— r*1 (22)

(23)

C. Director fluctuationsContributions to the spectral densities of motionfrom thermal fluctuations of director in liquidcrystals were first calculated by Pincus [20] andlater refined by others to include high frequencycutoff [27] in the mode spectrum, to eliminatethe one-constant approximations [28] and to in-clude a cross-term between director fluctuationsand molecular reorientation [17,18,24]. For dis-cotic liquid crystals, this mode may be viewed asslow fluctuations in the ordering of the columnswhich are formed by stacking of disks. We willassume that the theory developed for the classicalnematics can be carried over to the discotics. Inthe one-qonstant approximations and small ampli-tude in the director fluctuations, one has

(24)

where A = SkT^/Ay/^K^2, 17 and K are aver-age viscosity and elastic constants. U(pu/wc) isa function depending on the so-called cutoff fre-quency wc and its form is given in ref. [17]. Inparticular, when 0 = 0, d2

pq{0)2 can be replaced by6pi. Thus with n||jff, only Ji{u>) is nonzero, whileJo(w) and Ji{2u) are zero in the first approxima-tion.

III. Experimental Methods

The separation of spectral densities of motion wasdemonstrated earlier by NMR studies of small

Vol. 9, No.1/2 33

a. C3J X

-30

a

H H

C5HII°H H

D D

50.7= 0C6D|3

THE6Figure 2. Deuterium NMR spectra of chain perdeuterated discotic liquid crystal THE6. The pealassignment gives the numbering of the methylene along the chain with the a-methylene which correspondto 1; Figure A is of a planar sample in which all domains lie in a plane with the director perpendiculato the magnetic field; Figure B is of a single domain with the director alsong the field. The temperatureof both samples was kept at 80° C. Figure C shows the molecular structure of THE6-ald and 50.7-d4.

34 Bulletin of Magnetic Resonanc

deuterated solute dissolved in liquid crystals [29].The experimental techniques have been extendedand directly applied to liquid crystal molecules [3-8]. Figure 2 shows a well-resolved DMR spectrumof a chain-deuterated discotic liquid crystal [THE-ald]. Since discotics have negative diamagneticsusceptibility anisotropy (Ax < 0), the director isaligned perpendicular to the external field. Thisis illustrated by the spectrum in Figure 2a. Figure2b shows a spectrum recorded at the same temper-ature as that of Figure 2a for the sample orientedwith its director parallel to the field. As shown inthe figure, all quadrupolar splittings (AI/Q) havedoubled their values as expected. Similar spec-tra can be obtained from deuterated rodlike liq-uid crystals [3,30]. These allow us to study themotional behavior of different atomic sites of a liq-uid crystal molecule. The various relaxation timesT\Zi T\Q and T2 can be determined by a variety ofmethods according to the experimental conditions(the magnitude of the quadrupolar splitting, theavailable r.f. power etc.). To measure T\z andT\Q simultaneously, one can use either the Jeener-Broekaert (JB) method, as suggested by Void etal. [31] or by a modification of the selective inver-sion [SI] method as proposed by Beckmann et al.[3]- I

In the JB method [(TT/2) - T - (n/4) - tx -(TT/4)], a separate experiment is needed for each \th

deuteron since the separation r = (2n + l)/(2AvQ)must be chosen to maximize the quadrupole or-der. The appropriate phase cycle [31] was usedto suppress double and single quantum coherencesformed after the second TT/4 pulse. T\z and T\Qwere derived from the sum and difference of thedoublet components' intensities (Figure 3) accord-ing to:

V2[l -

D{t1)oc--exp{-t1/T1Q) (25)

In the SI method [3] [n - tx - (w/4) -T- (TT/4)],a selective TT pulse was applied to one of the dou-blet components, following after a time £1 by amonitoring TT/4 pulse irradiated at exact reso-nance. A second monitoring TT/4 pulse was then

applied at resonance after a fixed time intervalT » TiZ,T1Q. The FID signal obtained by sub-tracting the signals of these monitoring pulses wasFourier transformed to give the difference spec-trum. Figure 4 shows a series of SI spectra for thearomatic deuterons of a ring-deuterated discoticliquid crystal (THE-ard). Again the sum and dif-ference of the doublet components' intensities fol-low the equations [3]

oc --=V2

D{h) ex - (26)

This method is particularly useful when the r.f.power is not sufficient to irradiate the doubletcomponents non-selectively because of its largequadrupolar splitting.

The spin-spin relaxation time can be measuredfrom the t dependence of the peak intensities inthe Fourier transform spectra of the quadrupoleecho, using the {ft/2)x — t — {TT/2)V sequence. Theintensity of these peaks is given by

I±(t) ocexp-2t/T2 (27)

The results are very sensitive to imperfections inthe 7r/2 pulses. The deuterium relaxation mea-surements at 46.07 MHz and 15.35 MHz weremade on a Bruker CXP-300 spectrometer [5] andon a home-made superheterodyne pulsed spec-trometer [6,7], respectively. The accuracy of thesemeasurements was estimated to be better than10%.

IV. Results And DiscussionHere we discuss our recent data for a rodlike liq-uid crystal 50.7 and a discotic liquid crystal hexa-hexyloxytriphenylene (THE6) in terms of currentunderstanding of motional models.

A. Core dynamics

The molecular structure of 50.7-d4 is given in Fig-ure 2c. Plots of spectral densities Ji(w) and J2(w)

Vol. 9, No.1/2 35

7T/2 IT/A

rIT/A

11 ocquisifion

2KHj

Figure 3. A series of Jeener-Broekaert spectra of the methyl deuterons in THE6-ald at 80° C. The phase-cycled JB sequence is depicted in the insert. The coordinates give the phases of the corresponding pulsesand the + and — signs under acquisition mean addition and subtraction of the FID signal respectively.T = 180MS.

of the aniline deuterons versus the the recipro-cal temperature in the nematic (N), smectic A(SA), SC and SB phases of 50.7 are shown in Fig-ure 5. Ji(wo) and J2(2o;o) were determined fromeqns. [1,2] in all the mesophases (SG data are notshown). In addition, an angular dependence studyof the relaxation rates [7] was possible and allowedthe determination of Jx(2wo) and ^(^o) m t n e $Bphase. These are also shown in the diagram. Itis interesting to see that the temperature behav-iors of these spectral densities are quite similar inthe N and SB phases, i.e. Ji(wo) increases whileJ2(2a>o) decreases with decreasing temperature. Itis well known that the director fluctuations arenot effective in relaxing the ring deuterons [32,33]. The two dominant relaxation mechanisms forthe aniline deuterons would be reorientation of themolecule around its short axis and internal ring ro-tation about its para axis. In the nematic phase,the internal ring-rotation is very fast to be effec-tive in spin relaxation and has the effect of aver-aging the nuclear quadrupolar interactions for the

aniline deuterons to lie along the para axis. In con-sequence, only q = 0 term needs to be consideredin eqn. (12) where we assumed that the para axislies close to the long molecular axis [6] and with0 = 60°, one has

(28)

These expressions are very similar to those givenby the cone model (eqns. (21-23) with the ex-ception that only one correlation time r0 = r^ =[6DJ.]"1 is used here. The fc(p,O) coefficients playthe same role as the C parameters in the conemodel, i.e. to limit the amplitudes of the spectraldensities because of static order in the medium.Both models have been used [6] to explain theobserved temperature dependences of Ji(wo) and«M2wo) m the nematic phase. One can see fromeqn. (28) that apart from the temperature de-pendence of Z?±, Ji(wo) increases while J2(2OJO)decreases with decreasing temperature according

36 Bulletin of Magnetic Resonance

t,-lO2(sec)

Figure 4. A series of modified selective inversion spectra for the aromatic deuterons in a single domainsample of THE6-ard at 80° C. The first •n pulse shown in the insert consisted of 20 short pulses of durationtp = 0.9/zs, separated by tr = 3^s. tp and tr were chosen to satisfy (tp + tr) AUQ = 1/2.

Vol. 9, No.1/2 37

xlO1

5r

11oa>a. N

29 3.0 3.1103/ T {K"1)

32

1 Figure 5. Plots of calculated spectral densitiesJi(w) and Ji{w) of the aniline deuterons versusthe reciprocal temperature in the mesophases of50.7-d4. x and • denote u> = w0 and 2wo, respec-tively. WO/2TT = 15.35 MHz.

to

K{2, 0) - (7 -respectively. This is due to the temperature de-pendence of the orientation order parameter (-D^Q)which increases with decreasing temperature. Onecan also explain qualitatively the temperature de-pendences of Ji(wo) and J2(2wo) if the model pa-rameter in the cone model is chosen to lie between60° and 80°, since C\, increases while G\ decreaseswith 0o varying from 80° to 60° [25]. The conemodel predicts the ratio T[/T'2 to be essentiallyindependent of do and equal to ~ 2.8 to 2.9 for5° < <?o < 85° and n = 1. The calculated value ofT[ and r'2 using eqns. (22,23) shows that rotationdiffusion of a 50.7 molecule inside a conical vol-ume defined by 0Q is in the intermediate motionalregime (wor/ « 1). Furthermore the ratio T[[T2 isin agreement with the prediction. It would ap-pear that Ji(u>) and ^(w) should be dependenton frequency in the nematic phase though thesehave not been experimentally confirmed. As seenin Figure 5, both Ji(w) and JZ{<JJ) are frequencydependent in the SB phase. A recent dielectric re-laxation study [34] has directly measured the cor-relation time T<(|| for molecular rotation about itslong axis in the SB phase of a liquid crystal. Thevalue of rd|| is of the order of 10~9 — 10~10s in thismesophase. Thus DMR relaxation through thisrotation mechanism is still in the fast motion limit.The reorientation of a molecule around its shortaxis has also been extensively studied by dielec-tric relaxation measurements [35]. The associatedcorrelation time, Tdx has a value of ~ 5 x 10~8sin the N phase, but it is much longer (~ 10~5sin the SB phase. While this end-to-end rotationmechanism is effective for spin relaxation in theN phase (as seen above), it is too slow in com-parison with the internal ring rotation to effectspin relaxation in the S^ phase. It is known thatthe ring rotation is hindered and perhaps perform-ing n flips [36] in the S<s phase of 50.7 because ofshort range herringbone structure. One would ex-pect that the aniline ring rotation may be sloweddown enough in the SB phase for it to becomedominant in spin relaxation. We cannot use the

38 Bulletin of Magnetic Resonance

restricted rotation in a cone model here. To applyeqn.(ll) to the S^ phase, we note that the firstterm involves the correlation time Too = T± and isnegligible in comparison with the two remainingterms due to the condition OJT± >> 1. In terms ofR = D||/JDJ_, the anisotropic rotational diffusionfactor, and r|j = (QD^)'1, then

T

6 "Let us suppose that R is a large number (at least100) in the SB phase, then rn = 4r22 = 6r||e//where

r,|e//= [6D,, + P T 1

is an effective correlation time for internal ring ro-tation. If we, for simplicity, also suppose that UTUand UJT22 are less than one in eqn. (11) despite theobserved frequency dependence in Jp{w) and use0Dl = 10° and P1F = 60°, then

= 27Kprl]ef 7 / 280 (29)

where

Kt = 7.76 - 1A1(D2OO) +.

K2 = 7.76-2.81{D200)-

These K factors qualitatively account for the ob-served increase in JI(UJ) and decrease in «/2(w) withdecreasing temperature as well as the fact that• J^w) > J2(w). Using {Dlo) a (D*o) ̂ 0.9, oneobtains

in the fast motion limit. The experimental ratioin the S# phase ranges from 2.7 to 7.0. In viewof the ring rotation being in the intermediate mo-tion regime, the agreement between the model andexperiment appears to be satisfactory.

Figure 6 shows plots of spectral densities ver-sus the reciprocal temperature for the aromaticdeuterons of THE6 (see Figure 2c for molecularstructure) in its columnar (Dho) phase. Here onecan measure Ji(u,Q0°) and J2(o;,90o) easily since

xltf

lOr

to 5

to

-a

Din

J,(UJO,90°)

J2(2GOO,90P)

2.8 2.9

IO 3 /T . ( K"1

3.0

Figure 6. Plots of calculated spectral densitiesJI(OJO,0) and J2(2wo,0) of the aromatic deuteronsversus the reciprocal temperature in the Dfc0 phaseof THE6-ard. The smooth and dash curves repre-sent data at 0 = 90° and 0°, respectively. WO/2TT =46.07 MHz.

Vol. 9, No.1/2 39

Ax < 0. The value of these decreases with de-creasing temperature. The director fluctuationsshould play a role in spin relaxation because theangle /? between the aromatic CD bond and themolecular symmetry axis is 90°. The experimen-tal J1(w0,900)/J2(2wo,90°) ratio is about six incomparison with the theoretical prediction of y/2from the director fluctuations alone [eqn. (24)].Thus one has to include relaxation due to rota-tion diffusion of the entire disklike molecule. Wehad also measured Ji(w0) and J2(2w0) at severaltemperatures [5]. Again they decrease with de-creasing temperature and Ji(u>o) < ^ftwo)- The«Mwo)/«M2wo) ratio lies between 0.6 and 0.8 overa narrow temperature range. To account for this,one needs the contribution from reorientation ofthe entire molecule to the spectral densities suchthat

xicr5 -

since there is no contribution from the directorfluctuations (DF) to J2{w) in the first approxima-tion. We now examine the applicability of the conemodel and the strong collision model.

In the Bh0 phase of THE6, (Dg0) for the aro-matic deuterons [37] is rather high (£>Q0 = 0.9) andgives a value of 21° for 60 in the cone model. Thecorrelation times r[ and T'2 are related by T[ ~ 3T'2(25) and C\ » G\ at this 90. Therefore the conemodel fails to give J2/?(2wo) > JIR(WO) accordingto egns. [22, 23]. In using egn. [12], we again makeuse of the fact that rotation of a molecule aboutthe symmetry axis perpendicular to the aromaticcore is fast. This motion effectively averages thenuclear quadrupole interactions of the aromaticdeuterons to lie along this axis and hence only theq = 0 term needs to be considered. Thus,

(30)

For (Dl0) = 0.9, One may use 0.9 < (D*o)< 0.96. The ratio of eqn. [30] changes from~ 0.1 to ~ 0.4 for the fast motion to slow mo-tion regimes if the upper bound of (DQ0) was used.Thus one can easily rationalize the observed ratio°f JIR{WO)/ J2{2UJO) by allowing some contributionfrom the director fluctuations.

cCD

O<DOLto .05

.01 ar I 2 3 4segment no.

6

Figure 7. Variation of the spectral densitiesJi{uo), -M^o) a n d «M2u>o) w i t n t n e deuteron po-sition in the THE6 molecule at 80° G.

B. Chain Dynamics

The spectral densities of motion for deuteron siteson the side chains of THE6 were determined [5]from measurements of relaxation rates as a func-tion of temperature at 0 = 0° and 90°. The tem-perature dependence of Ji(a;o5^) an<A ^2(2wo> )̂(both at 9 - 0° and 90°) of the aliphatic sites wasobserved to be opposite to that of the aromaticdeuterons (Figure 6). The monotonous decreasein the Jp's on going from the aromatic deuteronsalong the flexible aliphatic chain to the methyldeuterons (Figure 7) must reflect an increase inthe segmental motion as one approaches the endmethyl group. This observation is similar to thatobserved for the end chain of rodlike molecules5CB [3] and 40.8 [8]. There is yet no detailedtheory of spin relaxation in liquid crystals that

40 Bulletin of Magnetic Resonance

treats segmental isomerisation and internal rota-tions vigorously. From measurements of relax-ation rates at two sample orientations [0 = 0° and90°), we obtained [5] in total five spectral parame-ters Jo(O),Ji(wo),^2(0)5^2(^0), and J2(2w0)- It isinteresting to note

J2(0) >> J2(w0) > ^2(2^0)

for all deuteron sites on the chain (see Figure 7,J2(0) not shown). If the molecular reorientationand segmental isomerisation can be treated usinga strong collision model in the fast motion limit,then J2 should be frequency independent. Theobserved frequency dependence in J2 cannot beexplained by director fluctuations under the as-sumption of small amplitude fluctuations in thedirector. It is conceivable that this assumptionmay not be valid as the director in the discoticsample is oriented perpendicular to the externalfield. As a consequence, large amplitude fluctu-ations in the director orientation cannot be pre-cluded in the plane normal to the externnl field.We will not consider chain dynamics further ex-cept to point out that more theoretical work onspin relaxation in liquid crystals are needed to ac-count for segmental isomerisation [22, 23] and toallow for large angle fluctuations of the director[39].

IV. Conclusions

We have presented some essential features of thetheory of spin relaxation for liquid crystals. Inparticular we give new results by explicitly con-sidering internal ring rotation in the treatment ofmolecular reorientations of a rodlike molecule inan anisotropic ordering potential. Furthermorewe discuss experimental methods for separatingthe spectral densities of motion and illustrate byDMR relaxation measurements on a smectic liquidcrystal 50.7 and a discotic liquid crystal THE6.Whereas director fluctuations are insignificant inrelaxing the aniline deuterons of 50.7, these fluctu-ations are not negligible in relaxing the aromaticdeuterons of THE6. In the discotic mesophases,the director fluctuations may involve large ampli-

tude fluctuations which require further theoreticalattention. The simple "rotational diffusion in acone" model may probably be used in the nematicphase of 50.7 to account for molecular reorienta-tions, but it is inappropriate in the columnar phaseof THE6 and in the SB phase of 50.7. Its failuremust be a result of the simple ordering potentialand higher degree of orientational order in thesephases. Generally speaking, the strong collisionmodel which uses a more realistic restoring poten-tial (Maier-Saupe) can better describe the highlyrestricted rotational diffusion (or librational mo-tion) in mesophases with nearly saturated orien-tational order. In the disordered smectic phases(S^ and Sc) of 50.7, both molecular reorienta-tions and internal ring rotation contribute to spinrelaxation of the aniline deuterons. A frequencydependence study may be required to allow quan-titative analysis of the relative contributions. Ourpreliminary data [7] in the S© phase indicate thatmore theoretical work is needed to treat spin re-laxation in biaxial phases. Further studies are alsorequired to fully understand the spectral densitiesof deuterons on flexible alkyl chains in rodlike anddiscotic liquid crystals. In this regard, theory de-veloped for 13C in macromolecules [22, 23] maybe useful. In conclusion, we feel that the temper-ature and frequency dependences of the spectraldensities of motion can provide a critical test ofvarious motional models for liquid crystals and en-hance the understanding of molecular dynamics inanisotropic media.

Acknowledgments

We wish to thank our collaborators over thepast few years, K. R. Sridharan, D. Goldfarb, Z.Luz and H. Zimmerman and the Natural Sciencesand Engineering Research Council of Canada forfinancial support.

Note added in proof. The discussions on 50.7-d4were based on earlier data which suffered from alack of phase-cycling in the JB method and poorsignal-to-noise ratio, and require some changes in

Vol. 9, No.1/2 41

view of more recent data [40]. Also the effect of in-ternal motions of the alkyl chains in liquid crystalshas been discussed [41] using the superimposed ro-tations model.

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