N NQR and Orientational Ordering in Smectic Liquid CrystalsR. Blinc, M. Vilfan and J. Seliger

J. Stefan InstituteE. Kardelj University of Ljubljana

Ljubljana, Yugoslavia

PageI. Introduction 51

II. Smectic Liquid Crystals 52A. Structure of Smectic Phases 52B. Microscopic Theories 5*»C. Ferroelectric Liquid Crystals 58

III. Proton - Nitrogen Level Crossing Double Resonance 59

IV. The Relation Between the NQR Data and the Orientational Order Parameters 61A. Averaging Due to Rotation Around the Long Axis. 62B. Additional Averaging due to Fluctuations in the Direction

of the Long Axis. 63

V. Experimental Results and Discussion 65A. Achiral Smectic TBBA 65B. Chiral Smectic TBACA 69C. The Smectic B Phase of IBPBAC 71D. (CioH2iNH3) zCdCU: a Lipid Bi layer Embedded in a Crystalline Matrix 72

References 75

I. INTRODUCTION

Liquid crystals are systems whichflow like liquids while retaining themechanical and optical anisotropy ofcrystals. On a microscopic level theyare characterized by molecular orienta-tional order accompanied by a partialor complete translational disorder ofthe molecular positions (1~3) •

Thermotropic liquid crystals areformed upon heating many anisotropicorganic compounds, while lyotropic liq-uid crystals are obtained by mixingamphiphilic compounds with water.

Lyotropic liquid crystals are espe-cially important in biology in view oftheir role in the structure of cellmembranes.

Both thermotropic and lyotropicmaterials may exhibit more than oneliquid crystalline phase. Smecticphases, whose specific property is thearrangement of the molecules into par-allel layers, have been most inten-sively investigated during the last fewyears. Studies by means of miscibilityrelations (4-6), X-ray diffraction(7~15). nuclear magnetic resonance(16-21), nuclear quadrupole resonance,

Vol. 5, No. 1/2 51

and neutron scattering (22-27) havebeen very successful in the determina-tion of the smectic structures andmolecular dynamics.

The aim of this article is to reviewthe contribution of l*N nuclear quadru-pole resonance (NQR) to the determina-tion of orientational ordering in smec-tic liquid crystals. We shall discussthe orientational ordering in the smec-tic phases of achiral terephtal-bis-bu-tyl-aniline (TBBA) and chiral tereph-ta1-b i s-am i no-methy1-buty1-c i nnamate(TBACA) . The results will be comparedwith those obtained in the until tedsmectic B phase of isobutyl-4-(V-phenylbenzylidene-ami no) cinnamate(IBPBAC). A perovskite-type layerstructure compound (Ci0H21NH3)2CdCl4,the hydrocarbon part of which repre-sents a smectic bilayer embedded in acrystalline matrix, has also beeninvestigated to explain the occurrenceof two structural phase transitionssimilar to those in biological mem-branes.

In this review we describe first thestructure of different smectic phases.To emphasize the importance of orienta-tional molecular ordering for theunderstanding of liquid crystals, wealso sketch the microscopic theoriesand describe the phenomenon of ferroe-lectricity in liquid crystals. SectionIII is devoted to the level crossingNQR double resonance method, whichenables one to measure the 14N spectrain liquid crystals. Section IVdescribes how to extract information onmolecular ordering from the experimen-tal NQR data. Finally, the 14N NQRresults, their analysis and discussionare presented in Section V.

I I SMECTIC CRYSTALS

Structure of Smectic Phases

The term "smectic" includes a widevariety of liquid crystalline phaseswith the common feature that the cen-ters of mass of the molecules arearranged in parallel equidistantplanes. It is usually used for thermo-tropic liquid crystals. However, thelyotropic lamellar phase can also be

described as "smectic" in view of itslayered structure.

The most important aspects in theconsideration of the structures of dif-ferent smectic phases are:

i) the molecular translational orderor disorder within the smecticlayers,

i i) the possible occurrence of a tiltof the long molecular axes withrespect to the normal of thesmectic planes,

iii) the orientational ordering of the"short" molecular axes, and

iv) the stacking of the smectic lay-ers one upon another.

We shall use the nomenclature of Helf-rich (28) for the different smecticphases. According to the translationalordering of molecular positions withinthe layer, the smectic phases can bedivided into:

1. disordered phases (Sm A and Sm C)and

2. ordered phases (Sm B, Sm H, Sm E,Sm G) .

Beside these, there are several lessinvestigated smectic structures whichwe shall not include here.

1. Disordered Smectic Phases

The smecti c A phase. There is no longrange translational order of the molec-ular positions within each layer. Sm Asystems thus resemble a two-dimensionalfluid. The preferred direction of thelong axes of molecules, N, is perpendi-cular to the smectic layers (Figure 1).The layer thickness is close to thefull length of the constituent mol-ecules. The Sm A phase is opticallyuniaxial with the optical axis beingnormal to the plane of the layers.

The smectic C_ phase, which resemblesthe Sm A phase by the two-dimensionalfluid character of the smectic layers,is optically biaxial. This is a

Bulletin of Magnetic Resonance

l /I/rSmA

SmC SmC*

Figure 1. Schematic presentation ofthe Sm A, spatially homogeneous Sm Cand spatially inhomogeneous, twisted SmC* phase. The unit vector N points inthe local preferred direction of thelong molecular axes. The pitch of thehelix in the Sm C* phase is much largerthan the layer thickness (103 is a typ-ical factor for their ratio).

consequence of thetive tilt of theaway from thesmectic Cconstantin chi ralthe moleculesatomic group,

spontaneous collec-long molecular axes

layer normal. In achiralphases the tilt direction isn space. On the other hand,Sm C systems (Sm C>>) , where

have a noncentrosymmetricthe direction of the tilt

precesses around the layer normal ongoing from one layer to another and anincommensurate helicoidal Sm C* config-uration is obtained (Figure 1). Themolecular chirality induces thus a chi-rality in the structure of the Sm C*phase. The chiral Sm C* phase will befurther discussed in connection withthe phenomenon of ferroelectricity in1iquid crystals.

2. Ordered Smectic Phases

In the translationally ordered smec-tic phases the molecular positionswithin a layer form a two-dimensional

SmB StnE

Figure 2. Schematic presentation ofthe molecular arrangement within asmectic plane in the hexagonallyordered Sm B (and Sm H) phases and inthe "herringbone packed" Sm E (and SmG) phases. Ellipses represent themolecular cross-section perpendicularto the long molecular axis.

lattice. In the orthogonal smectic Bphase and the tilted smectic H phase,the two dimensional lattice is hexago-nal or approximately hexagonal. Leve-lut et al. (12) have suggested on thebasis of X-ray scattering that the mol-ecules probably rotate by TT/3 orienta-tional jumps about their long axes andthat the phenyl rings arrange them-selves locally in a herring-bone typepacking. The lattice parameter (5-18A) in TBBA is appreciably smaller thanthe size of the phenyl ring (~ G.k A ) .They argue that the local orientationalorder is of higher symmetry than theaverage one (Figure 2 ) .

By analogy with the twisted disor-dered Sm C* phase, an ordered Sm H*phase [called Sm Ml by the authors(29)] has been discovered in which thehelicoidal arrangement of the tilttakes place. The chiral Sm H* phasehas a similar hexagonal structurewithin the layer as the achiral Sm Hphase except that the plane determinedby the layer normal and the directionof the molecular tilt lies parallel toone edge of the hexagonal two-dimen-sional lattice in the chiral Sm H*

Vol. 5, No. 1/2 53

phase and perpendicular to one edge inthe achiral phase.

The proposed two-dimensional latticefor the orthogonal stnectic £ phase andfor the tilted smecti c G phase is aherringbone arrangement of long range.The molecules should be able to rotatefor an angle n/2 about their long axes(Figure 2) .

The sequence of the various smecticphases with increasing temperature issuch that the least ordered phasesoccur at the highest temperatures(Table 1) .

B. Microscopic Theories of_Smectic Phases

Concurrent with experimental inves-tigations of smectic structures, theo-ries have been developed to describethe properties of different smecticphases and the transitions among them.The main contribution of NQR to thephysics of liquid crystals has been thedetermination of the local orienta-tional ordering of the short molecularaxes, which is of basic importance forthe microscopic theories of the smecticphases.

McMillan and Meyer (30-33) havedeveloped the microscopic theory of theSm A phase, the Sm C phase, of theordered Sm B and Sm H phases and of theSm E phase. Other authors have studiedthe Sm A •> Sm C transition and the cor-responding structure of the smectic Cphase. The models of the Sm C phasehave been developed by McMillan (31).Wulf (3^4-35), Priest (36), Cabib andBenguigui (37). and Van der Meer andVertogen (38). In those theories dif-ferent forces among molecules arestated to be responsible for the onsetof the Sm C phase. According to someof them the freezing out of molecularrotation around the long axis isrequired for the onset of the tiltedphases, while others leave the rotationfree or nearly free throughout the Sm Cphase as shown in Table 2. It isexactly this problem that has been suc-cesfully solved by using 14N NQR.

In the following a short descriptionwill be given of two microscopic theo-ries of smectic phases: that of McMil-lan and Meyer (30,32), which includes

most of the known smectic phases, andthat of Van der Meer and Vertogen (38),which is the newest and seems to solvemany long-standing problems.

1. McMillan-Meyer Theory

McMillan first developed the micro-scopic theory of the Sm A phase (30) byassuming a special interaction betweenanisotropic molecules which is propor-tional to:

exp-[(ri2/ro) 2][ (3/2)cos20i2 - 1/2].

Here ri2 is the distance between cen-ters of mass of two molecules, #12 theangle between the directions of theirlong axes and ro is roughly a molecularlength. With this potential he showedthat it is energetically favorable forthe oriented molecules to arrange them-selves below a definite temperatureinto smectic layers instead of beingrandomly distributed in space as in thenematic phase. The orientational orderparameter S (often called the "nematicorder parameter") is defined as

S = (!/2)<3cos20 - (1)

with 8 denoting the angle between thelong axis of one molecule and the localpreferred direction. S increasesnearly continuously from the nematic tothe smectic A phase. On the otherhand, the "Sm A order parameter" cr,representing the amplitude of a densitywave in the direction perpendicular tothe long molecular axes, assumes valuesdifferent from zero only in the smecticphases.

With the aim of developing further acomprehensive theory, which wouldinclude both disordered smectic phases(Sm A and Sm C) and the two orderedphases with hexagonal structure, McMil-lan and Meyer (32) studied the systemin which both arrangement of the mol-ecules into smectic planes and perfectorientational order are already wellestablished. The intermolecular poten-tial is then taken to be due to elec-tric dipole-dipole interactions due topermanent dipole moments and soft-corerepulsion between the molecules. Amolecule is supposed to be a long rod

Bulletin of Magnetic Resonance

Table 1. The Scheme of Established Smectic Liquid CrystallinePhases and their Structures (28)

Smectic liquid crystals

Ordered

Herringbone Hexagonal(or nearlyhexagonal)

Disordered

orthogonal

tilted

tilted andtwisted

G

E B

H

H*(l II)

A

C

c*

rising temperature

Table 2. Models of the Sm C Phase

Authors Interactions between molecules Orientational ordering

McMi11 an

Wulf

Priest

Cabib, Benguigui

Van der Meer,Vertogen

Dipole-dipole forces (betweenpermanent electric dipoles onneighboring molecules)

Steric forces

General coupling

Dipole-dipole forces (onlybetween components parallel tothe long molecular axes)

Forces between permanent andinduced dipoles

Rotation around the longmolecular axis is frozenout

Rotation is frozen out

Free rotation or nearlyfree

Free rotation

Free rotation

Vol. 5, No. 1/2 55

of diameter D with oppositely directedpermanent dipoles on opposite sites ofthe molecular centre (Figure 3 ) .According to this model the interactionU12 between two molecules in the planecan be written as:

U12[* Mi-M2 3 (Av? i 2) (/U2-r 1 2) j

[_ ri23 M 2 5 J X

S(n2) + T(n2) . (2)

—I

1 '1{i

SmA SmC

Figure 3. McMillan's model of theorthogonal Sm A and of the tilted Sm Cphase. The orientational ordering ofpermanent electric dipoles is responsi-ble for the onset of the tilt.

The first term in the above potentialrepresents dipole-dipole interactionbetween two dipoles at a distance ri2apart. S(ri2) is the two-particle cor-relation function which is zero forri2 < D and different from zero forri2 > D. The second term, T(ri2), inequation 2 is a soft-core repulsivepotential whose magnitude is Vo for ri2< D and zero outside this region. Theinterlayer interactions as well as theinteraction between dipoles on differ-ent levels have been neglected as thespacing between the two dipoles on onemolecule is larger than the

intermolecular spacing.The authors have solved the above

model within the mean field approxima-tion. The self-consistency requirementgives the following equations for theorder parameters:

a = (1/3)<COSKI-X + cosK2-x +

cosK3-x> (3a)

3 = <cos\p>

7 = (l/3)<cos<HcosKvx + cosK2-x

cosK3-x)> .

(3b)

(3c)

Here x is the coordinate in the smecticplane on the level of moleculardipoles, Ki, K2, and K3 are the recip-rocal vectors of the hexagonal latticeand ^ is the azimuthal angle determin-ing the instantaneous molecular orien-tation during its rotation about thelong axis.

The orientational order parameter is3- When 3 is finite, the rotationaround the long molecular axis isbiased and the dipoles are aligned par-allel to an axis lying in the smecticplane (Figure 3)- Though there is nonet dipole moment, the liquid crystalis biaxial and tilted. With 3 = 0, theliquid crystal is uniaxial, untilted,and there is no orientational order.

The translational order parameter isa. When a is finite, the molecules lieon a two-dimensional hexagonal latticein each smectic plane. When a = 0, themolecules move as a two-dimensional1iqui d.

The order parameter 7 describes thecoupling between the translational andorientational order and is finite onlywhen both a and 3 are finite.

Depending on the values of theparameters involved in the calculation,four characteristic sets of solutionsfor the order parameters can beobtai ned:

i) a - (3 = 7 = 0, no translationaland orientational order in thesmectic plane (Sm A)

ii) a # 0, 3 = 7 = 0 , translationalorder in smectic Sm plane, no

Bulletin of Magnetic Resonance

orientational order (Sm B)

iii) a = y = 0 , P r 0, no transla-tional order, orientational orderof dipoles in plane (Sm C)

iv) a * 0, 3 * 0, y # 0, transla-tional and orientational order inplane (Sm H ) .

The temperature dependence of orderparameters and the phase diagrams havebeen determined for some definite val-ues of the soft-core repulsive poten-tial Vo and parameter B which dependson the two-particle positional correla-tion function. In Figure U a typical

Figure U . Temperature dependence oforder parameters a, (3 and y predictedby the Meyei—McMillan theory and aphase diagram showing the onset of dif-ferent smectic phases with increasingvalue of the intermolecular soft-corerepulsive potential Vo (32).

temperature dependence of the orderparameters is presented for the phasesequence Sm A *-* Sm C «•* Sm H. Clearly

Vol. 5, No. 1/2

the calculated Sm A *•* Sm C transitionis second order while the Sm C «•» Sm Htransition is first order. It is alsofound that the Sm B «•* Sm H transitionis of second order while others are offirst order. The predicted orienta-tional order parameter @ in the Sm Hphase is between 0.3 and 0.5- In thelower part of Figure k a phase diagramis shown for B = 23-1 - The Sm C phaseappears at low values of Vo. Accordingto this theory the Sm C phase may beexpected in compounds where the molecu-lar structure is dominated by two out-board oppositely directed dipoles. TheSm B phase occurs at high Vo, that isin the cases where the molecular struc-ture is dominated by the repulsivesoft-core potential.

Meyer extended the above theory toobtain the onset of the Sm E phase aswell (33)• He included in addition todipole-dipole and soft-core intermole-cular forces a phenyl-phenyl interac-tion of the Van der Waals type.

2. Van der Meer-Vertogen Theory

Van der Meer and Vertogen (38)developed a microscopic theory which isbased on induced dipole forces betweenmolecules and accounts for the nematic,Sm A, and Sm C phase. Their theoryconsiders a model in which the moleculeis a rod-like object with an isotropi-cally polarizable unit at its center.Any atomic group within a given mol-ecule with a permanent electric dipolewill induce a dipole in the center of aneighboring molecule. The inductionforces of this type are responsible forthe onset of the tilt in the Sm Cphase. Even free rotation of the mol-ecule about its long axis does notdestroy this concept because inductionforces, in contrast to McMillan'sdipole-dipole forces, are not averagedout by this rotation.

The above model has been solvedwithin the mean field approximation toyield the values of the nematic orderparameter S, "Sm A order parameter" crand the tilt angle do. Four differentsets of solutions refer to the follow-ing phases:

S = 0, a = 0, 80 = 0 - isotropic phase

57

S * 0, a = 0, #o = 0 - nematic phase

S s* 0, <7 * 0, 0o = 0 - smectic A phase

S * 0, a * 0, So * 0 - smectic C phase.

C. Ferroelectric Liquid Crystals

Ferroelectricity in liquid crystalscan occur in two different situations:

In uniaxial liquid crystals, such asthe nematic and smectic A phase, thespontaneous polarization can be non-zero only if "head to tail" orderingtakes place. In such a case the spon-taneous polarization would be parallelto the average direction of the longmolecular axes. No direct experimentalproof of ferroelectricity in uniaxialliquid crystals has been obtained untilnow.

In biaxial liquid crystals, such as thechiral smectic C* and smectic H*phases, there may be a non zero compo-nent of the molecular dipole momentnormal to the direction of the longmolecular axis and parallel to thesmectic layer. This follows from sym-metry considerations (39)-

In such systems ferroelectricity andin-plane spontaneous polarization haveindeed been observed by Meyer et al.(39)• The system under study was p-de-cyloxybenzylidene-p'-ami no-2-methyl-bu-tylcinnamate (DOBAMBC)

-CH=N-

OII- CH=CH-C-O-CH2-CH-CH2-CH3

The methyl-butyl group is responsiblefor the chirality of the molecule. Inits vicinity is the polar ester group.The sequence of the phase transitionsin DOBAMBC is the following (T in °C):

76° 95° 115°Cr > Sm C* > Sm A > Isotropic

Sm H*^ 63

Ferroelectric properties have beenobserved in the Sm C* and the Sm H*phase. The direction of the spontane-ous polarization, which is perpendicu-lar to the direction of the long molec-ular axis, precesses from ona layer tothe next around the normal to the lay-ers in the same way as the moleculartilt. Each layer is ferroelectricallyordered but the whole sample representsa helicoidal antiferroelectric withzero net polarization. Since the dis-tance between the smectic layers is anirrational fraction of the pitch of thehelix, ferroelectric liquid crystalsrepresent structurally incommensuratesystems. The effective dipole momentper molecule is equal to ~ 0.05 Debyein the Sm C* phase. This value is10-100 times smaller than in solid fer-roelectr ics and represents only a fewpercent of the calculated value for thepermanent electric dipole per molecule(~ 1 Debye). This discrepancy can beexplained by the partial averaging outof the dipoles due to the partiallybiased rotation of molecules abouttheir long axes.

The interactions among effectivemolecular dipoles in liquid crystalsare too weak to produce the ferroelec-tric ordering themselves. The tran-sition from the Sm A to the Sm C* phaseis driven by intermolecular forces pro-ducing the onset of the tilt. Thespontaneous polarization which appearsin the Sm C* phase is only a secondaryeffect induced by the Sm C* liquidcrystalline ordering. Ferroelectricliquid crystals are thus improper fer-roelectr ics.

It is easily shown (hO) that thepolarization is proportional to thetilt angle 0o, and decreases conti-nously to zero at the transition temp-erature from the Sm C* to the Sm Aphase, T c. Above T c the freeenergy density g of a non-equilibriumstate can be expanded in powers of thein-plane polarization P± and tiltangle 6 = v x N, which is considered asthe Sm. C order parameter. Here v is

58 Bulletin of Magnetic Resonance

the normal to the smectic planes and Nthe unit vector specifying the direc-tion of the long molecular axis.

In the spatially homogeneous case wefind

g = gA + (1/2) A (T) \6\* +

(}/k)B\d\* + (1/2) K(T) |P\| 2 "

The coefficient A (T) = a(T - To)accounts for the short range drivingforce of the pure Sm A •» Sm C* tran-sition whereas K (T) = a'(T - T O 1 )accounts for the driving force of thepure ferroelectric transition. Allother coefficients are assumed to betemperature independent. The linearcoupling between the tilt and thepolarization is allowed by symmetry.

The problem can be simplified bychoosing a special coordinate system,where the two-fold rotation axis, andhence ?±, is perpendicular to theplane determined by the layer normal vand N.

The equilibrium values do and Po ofthe tilt and polarization are obtainedfrom the requirements:

(5)•a>g/dPj0O f P o = 0

It can be seen immediately that thespontaneous polarization is propor-tional to the tilt angle:

Po = (|A)0o (6)

To take into account the spatialinhomogeneity of the Sm C* phase theo-ries have been developed (1»1) whichinclude the inhomogeneous Lifshitz termin addition to the terms in equation h.Interesting effects such as the exis-tence of the Lifshitz tricritical pointin the E-T phase diagram, which havebeen predicted ( 2) , are beyond thescope of this paper.

III. PROTON-NITROGEN LEVEL CROSSINGDOUBLE RESONANCE

The signal of the nitrogen nuclei in

Vol. 5, No. 1/2

liquid crystals is too weak to bedetected by standard NQR techniques.Therefore a nuclear double resonancetechnique, introduced by Hartmann andHahn (1»3) and later Lurie and Slichter(kk), is employed which uses the"strong" NMR signal of the spins I todetect the "weak" NMR or NQR signal ofthe spins S. The sensitivity of thismethod may be several orders of magni-tude greater than the sensitivity ofclassical magnetic or quadrupole reso-nance techniques. The spectra of thespins S can be measured with practi-cally the same sensitivity as the spec-tra of the I spins.

The double resonance techniques arebased on the existence of nucleardipole-dipole coupling between the twospin systems, which takes place whenthe energy levels of the I and S spinsare equally spaced. The "resonant"coupling can be established either inthe rotating or dipolar frame or in thelaboratory frame. In the latter caseat least one of the spin species mustexhibit a non-zero quadrupole splittingin addition to the Zeeman splitting forresonant energy transfer to occur.

Measurements of 14N NQR spectra insmectic liquid crystals have been per-formed with the help of a simple pro-ton-nitrogen double resonance techniqueinvolving level crossing in the labora-tory frame. Since an excellent reviewof the double resonance method has beenpublished recently in this journal(1»5) , we shall only briefly sketch thebasic principles of our experiment.

14N is a nucleus with spin S = 1.The interaction of its electric quadru-pole moment with the electric fieldgradient V; j = 0 2V) / (3x;axj) atthe nuclear site gives rise to threepure quadrupole energy levels in theabsence of external magnetic field.Their energies are (k(>) :

E. = (lA)e2qQ(l + v)

E- - (lA)e2qQ(l - ij)

Eo = -(l/2)e*qQ.

(7)

Here eQ denotes the scalar quadrupolemoment of the 1 4N nucleus and eq thelargest eigenvalue VZ£ of the local

59

electric field gradient (EFG) tensor.Their product e2qQ/h is the 1 4N quadru-pole coupling constant. The asymmetryparameter »j reflects the deviation ofthe EFG tensor at the nuclear site fromcylindrical symmetry. It is defined as

|(vxx - vYY)|/vzz« where VXXand Vyy are the smaller two eigenva-lues of the EFG tensor. All tran-sitions between the three quadrupoleenergy levels are allowed. The corre-sponding resonant frequencies are:

v* = (3A) (e2qQ/h) (1 + r?/3)

v- = (3A) (e2qQ/h) (1 - T?/3) (8)

vo = v* - v- = (l/2)?7e2qQ/h .

If the sample is put in a magneticfield of strength B, the Zeeman inter-action of the 14N magnetic dipolemoments with the magnetic field has tobe taken into account as well. Themagnetic coupling can be treated as asmall perturbation up to fields of ~100Gauss. The magnetic interaction causesa slight increase of the E* energylevel and a slight decrease of the E-and Eo energy levels, as presented inFigure 5> It can be shown (47) thatfor a powder sample the transition fre-quencies v*1, v-1, and vo1 between theZeeman perturbed quadrupole energy lev-els of nitrogen are:

v + i - „• + (l/3)*L>N2(l/>

v.i = v. + (l/3)"LfN2(-l/vo+2/K-+l/*O

vo1 = vo + (1/3) v, Kl2(2/i

Here N 's

(9)

pure Zeeman tran-sition frequency given by v^ ^ =7^B/27r. The values of v^ ^ arerelatively small in view of the lownitrogen gyromagnetic ratio -y , thusjustifying the perturbation treatmentused in the evaluation of equations 9>

In contrast to the 14N case the pro-tonic quadrupole moment is zero. Asthe proton gyromagnetic ratio is 13-4times larger than the nitrogen one, theprotonic energy levels cross the levelsof the 1 4N nuclei at a sufficientlyhigh value of the magnetic field. At

1 un

its ]

J2

I"-

n

/ \B,,. / V | . Bre,('" y/ l '" ("

—¥- I—X, , , I , . 1 ,

Figure 5- Energy levels and the cor-responding transition frequencies of14N nuclei and protons in low magneticfield. B r e s denotes the values ofthe magnetic field at which thecoupling between nitrogen nuclei andprotons takes place.

three definite values of the magneticfield B the nitrogen and proton("L hP transition frequencies becomeequal (Figure 5 ) . and cross relaxationbetween the proton and nitrogen systemtakes place, thus enabling one to meas-ure the quadrupole resonance frequen-cies of 14N via the proton signal.

The experiment is performed as fol-lows. The sample is placed first in ahigh magnetic field Bo where the pro-tons are polarized, creating a netmagnetization Mjjo« Then the sampleis moved with a pneumatic post to a lowmagnetic field B, where the protons areallowed to relax for a fixed timet = T. The sample is subsequentlymoved back to Bo and the remaining pro-ton magnetization M is measured with a90° rf pulse. The magnetization is

Bulletin of Magnetic Resonance

given by

M(t) = MBo(l - B/Bo)exp(-t/T1B) +

MBo(B/Bo) , (10)

where T j B is the proton spin-latticerelaxation time in the field B. Thesame procedure is repeated for differ-ent values of the low magnetic field Buntil a complete dependence

•yvrv—

0.02 0.04 0.06 0.08 BIT]

BBo

i

1

f. f.

i \

K*

time

"TlM(t.B.AB)

1 time

Figure 6. Magnetic field and r.f.pulse sequences used in the proton-ni-trogen level crossing double resonance.

M(t) = M(t,B) is obtained (Figure 6 ) .The Zeeman perturbed 14N NQR frequen-cies v1 are determined by observingthe dips in the magnetic field depen-dence of the proton magnetization. Anexample is shown in Figure ~].

Because of the short proton-nitrogencross relaxation times (usually smallerthan 1 msec), the depth of the dip isdetermined by the actual change in theproton spin-lattice relaxation timeTj B at the cross-over. The changestrongly depends on the differencebetween the proton T ° 1 B (the spin-lattice relaxation time due to allother processes but cross relaxation tothe 14N nuclei) and the 14N quadrupolespin-lattice relaxation time 1Q*

thThe depth of the dip and thus the

Figure 7- An example of proton-14Ncross-over relaxation spectra.

sensitivity of the method is highestfor T ° 1 B » Tin.. The width of. thedip is determined by the broadening ofthe proton Larmor frequency v\_ ^ dueto the dipole interaction with neigh-boring nuclei and by Zeeman broadeningof the i4N NQR powder spectrum.

The advantages of this direct levelcrossing proton-nitrogen detection com-pared to other double resonance techni-ques are its simplicity, the fact thatpowdered samples can be used instead ofwell aligned single crystalline sam-ples, and the fact that the spinquenching effect is avoided. It shouldbe noted, however, that the method isapplicable only for samples in whichthe proton T^ is long enough (>, 200msec) to allow for transfer of the sam-ple from the high to the low .magneticfield in a time short compared to T^.

IV. THE RELATION BETWEEN THE NQR DATAAND THE ORIENTATIONAL ORDER

PARAMETERS

If the correlation frequency of themolecular motion is much larger thanthe 1 4N NQR frequencies, which is usu-ally the case in liquid crystals, theobserved value of the EFG tensor repre-sents a time average over the molecularmotion. This is smaller than the rigid

Vol. 5. No. 1/2 61

lattice value observed in solids. Thecentral problem concerns the nature ofthe molecular motion producing a giventime average.

Some qualitative facts on the molec-ular motion can be immediately obtainedfrom the measured values of e2qQ/h and7j. A comparison between the observedand rigid lattice values of e2qQ/hcharacterizes the degree of motionalaveraging and reflects the amplitudesof the molecular motion. The asymmetryparameter 17 is zero for unhindered.rotation around the long molecularaxis, while non-zero value indicates adeviation from cylindrical symmetry atthe nucleus and thus an anisotropy inthe molecular motion.

To quantitatively describe themolecular motion, the relations linkingthe values of e2qQ/h and r\ to the orderparameters should be properly deduced.These relations have been discussed

In the rigid crystal thetensor can be written (kS)molecular frame xo, yo, zo as

14Nin

xoxo xozo

v. . =1 V V VIJ I vxoyo yoyo vyozo

EFGthe

(11)

yozo

The molecular frame is normally chosenwith the zo axis parallel to the longaxis of the molecule. The choice ofthe yo direction depends on the partic-ular molecular group under study. Thexo axis is perpendicular to yo and zo,so that xo, yo, zo represent an orthog-onal right handed system.

If the observed 14N nucleus belongsto the nearly planar C-N=CH-C linkagegroup, as in the present case, it isconvenient to choose yo normal to thisplane. It is known from work on otherSchiff's bases that the largest princi-pal axis of the 1 4N EFG tensor liesthen in the xozo plane and makes anangle ofwhi lesmalI.tensor canplete setdata.

If the

62

aboutVxoyoThe exactbe

of

and vyozo

60° with the zo axis,should be

values of the Vdetermined from asingle-crystal rotation

• Jcorn-

five independent EFG tensor

components in the molecular frame areknown and a model for molecular motionis assumed, the components of the timeaveraged EFG tensor can be evaluated(J»9) . The principal values of the timeaveraged tensor, obtained by diagonali-zation, provide the relations betweenthe observed values of e2qQ/h and r\ onone side and the order parameters onthe other.

In the smectic phases the molecularrotation about the long axis isexpected to govern the time averagingof the EFG tensor at l*U sites. Insome phases, especially in the disor-dered ones, the fluctuations in thedirection of the long axis can be sig-nificant as wel1.

A. Averaqi ng due to Rotationaround the Long Axi s

The molecule is assumed to perform ahindered rotation around the longmolecular axis, i.e. it jumps betweenseveral equilibrium orientations. Totake into account this motion the EFGtensor (equation 11) is transformedfrom the molecular xo, yo, zo frameinto a frame x, y, zjzo, where z|zo isthe axis of rotation and \p = \p(t) isthe angle between x and xo. The EFGtensor is expressed in the x, y, zframe as:

V x x = -0/2)Vxx 20Z0

(1/2)(VXOXO

V y o y o)cos2^ -

V x y - 0/2)(V X O X O- V y o y o)sin2*

xz

V y y = -(1/2)V

(12)

ZOZO

0/2)(VX0X0

yz

"zozo

When equation 12 is averaged over themolecular motion, we have to replace

Bulletin of Magnetic Resonance

by <cos^> etc. The mean values^ <cos2^>, etc. play the role of

order parameters which are characteris-tic for the given type of orientationalorder ing.

1. Uniaxial Rotation without Biasing

In the case when all equilibriumorientations by molecular rotationaround the long axis are equivalent,the above order parameters equal zero:

<cos\k> = <cos2\l/> - <sint/'> =

0.

The .largest eigenvalue oftensor isyielding a

zztemperature

= <V2Z>averaged

= VEFG

2Z 2o2o,independent qua-drupole coupling constant e2qQ/h and a

zero value= 0.

of the asymmetry parameter

2. Uniaxial Rotation with Polar Biasing

In the case of a polar partial rota-tional freeze-out the effective poten-tial felt by each molecule has the formV = - V (cos^). The resulting orienta-tional ordering is polar with charac-teristic order parameters <cos^> * 0,<cos2^> * 0, and <sini/<> = 0, <sin2^> =0. The values of <cos^> and <cos2^>can be given in terms of modified Bes-sel functions as

li(a)/lo(a), a = V'/(kT)(13)

<cos2^> = I2(a)/lo(a), a = V'/(kT) .

If the orientational ordering is weak(i.e. if V'«kT) the Bessel functionsmay be expanded to obtain:

= (l/2)<cosi/'>2

The only relevant order parameter forweak polar ordering is thus <cos^>,while <cos2i^> can be neglected. Theresulting values for e2qQ/h and t) are:

e'qQ/h = (eQVZ020/h) [1 +

(2/3)(V X 0 2 0> + V y o 2 o*)/V 2 o 2 o> X

v - (2/3)C(vX020

v y o 2 02 ) /

We thus see that for biased rotation rjis different from zero. The tempera-ture dependence of r\ reflects the temp-erature change of the square of thepolar order parameter <cos^>.

3. Uniaxial Rotation withBipolar Biasing

Bipolar orientational order takesplace in a potential of the form V = -V (cos2$ . In this case <cos2iA> * 0,<cos^> = 0, and <sin^> = 0, <sin2^> =

0.If the bipolar orientational order-

ing is weak ( <cos2^> « 1) the largestprincipal axis of the 14N EFG tensorstill points along the axis of rotation(20). The quadrupole coupling constantand the asymmetry parameter are:

eQV22/h = eQVZ020

/h

(16)

where

zozo

We see that v depends linearly on<cos2\i'> and hence is temperature depen-dent, whereas e2qQ/h is constant.

For large values of <cos2^> thelargest principal axis of the 14N EFGtensor no longer points along the axisof rotation. For the nitrogen in theC-CH=N-C group it lies in the xozoplane. For (r<cos2^>) >1 we find:

Vol. 5, No. 1/2

eQVzz/h = -(l/2h)eQVzo2o(l +

and (17)

v = I r<cos2iA> - 3 Ij r<cos2vJ» + 1 I ,

so that both e2qQ/h and v depend on

63

<cos2i/'> and are temperature dependent.

B. Addi tional Averaging due to Fluctua-tions jjn the Direction of the Long

Axis

The fluctuations in the direction ofthe long molecular axis, in addition torotations around this axis, are takeninto account by performing stillanother transformation of the EFG ten-sor given by equation 12 to a fixed x1,y1, z' frame, where z1 is normal to thesmectic planes, x' is the direction ofthe average projection of the longmolecular axis on the smectic plane,and y1 is perpendicular to x1 and z 1.The angles G = G(t) and <i> = 4>{t) arethe polar and azimuthal angles, respec-tively, determining the direction ofthe long molecular axis in the x1, y',z1 frame. G(t) can be approximatelywritten as 0(t) = Go + 56 (t) , where Godenotes the well known tilt angle (Fig-ure 8) .

After this second transformation hasbeen performed the time averaged valuesof the 14N EFG tensor are (49):

< V X I X I > = <(VZ2 - Vxx)sin2Gcos2<*»

+ <Vxxcos2<£> + <Vyysin

2<£>

+ <VX2sin26cos2<£> ,

<Vx,y.> = <Vy2sinGcos2<*»

+ <Vxycos6cos2<£> ,

<Vx.7i> = <(V77 - Vxx)sin6cos0cos4»'x'z' zz xx'

+ <Vxzcos26cos4» ,

<V y, y l> = <(V Z 2 - Vxx)sin2Gsin24»

<VXYsin24» + <Vvvcos

2<£>'xx y y v

+ <Vxzsin2Gsin2<£>,

<VV 1 -, 1 > = - <V Y Vs i n6cos<£>

+ <Vyzcos6cos<£> ,

<V 2, z l> = - <(V 2 2 - Vxx)sin2G>

+ <V Z Z> - <VX2sin26> . (18)

Figure 8. Schematic presentation of atilted molecule within a smectic layer.Go is the average tilt angle, <56(t)denotes the fluctuations in the orien-tation of the long molecular axis inthe polar direction, and <£(t) the fluc-tuations in the azimuthal direction.<Ht) denotes the orientation of theshort molecular axes during the molecu-lar rotation around its long axis.

In this set of equations a large numberof different order parameters appear.Those that are functions of \f/ describethe order around the long molecularaxis, while those that are functions ofG and (j> describe the ordering of thelong molecular axis itself. The alge-braic relations between e2qQ/h and rjand the values of the EFG tensor in thex1, y1, z1 frame are rather compli-cated. If, however, the fluctuationsin G, <t> and ^ are assumed to be inde-pendent and are small enough to allowthe expansion of the sine and cosine ofG and 4> in a power series, relativelysimple expressions for e2qQ/h and r\ canbe obtained. For the case of weakbipolar ordering we find:

61* Bulletin of Magnetic Resonance

e*qQ a eQVZ0Z0[l -

- (3/2)Go2 (1 - <cos0>2)]

and

rj ~ (3/2)Go2(<cos4»2 -

- <cos2<£>) -

Equation 19 shows the combined effectsof both bipolar biasing and anisotropicfluctuations in the direction of thelong axis on the values of e2qQ/h and•t). In the first approximation e2qQ/hdoes not depend on the orientationalordering <cos2v£> but is mainly deter-mined by the mean square fluctuationsin the polar angle <<5G2>. In contrastto e2qQ/h, 17 does depend on the or ien-tational order parameter <cos2^>through the last term in equation 19.This term is however rather small inthe Sm C phase for which the abovemodel is appropriate. The prevailingterm which produces the characteristicmaximum in the temperature dependenceof v is (3/2)<6G2><cos2tf». In the lowtemperature part of the smectic Cphase, <cos2<£> = 1 and 77 increases withincreasing temperature due to theincrease in <<562>. At higher tempera-tures, t] decreases as <cos2<£> •* 0, sothat there has to be a maximum inbetween.

Anisotropic fluctuations in thedirection of the long molecular axis bythemselves result in a nonzero value of77 though there i s no biasing of therotation around the long axis. In thiscase we find

e2qQ = eQV2020[l +

(3/2) (<sin©cosGcostf»2 -

<cos2G><sin20cos24» -

<sin2Gsin2<£>)] ,

and (20)

v = (3/2) (<sinGcosGcos<*»2 -

<cos2G><s?n2Gcos2<£> +

<sin2Gsin2<?i> ) .

The two different mechanisms leading toa finite value of 77, or ientationalordering and anisotropic fluctuationsin the long molecular axis, can be eas-ily discriminated in view of the dif-ferent temperature dependences theypredict for 17 and e2qQ/h.

A different approach to the problemof relations among e2qQ/h and 77 andorder parameters has been developed byAllender and Ooane (51). They obtainedthe expression for 77 without assuming aspecial model for molecular motion andevaluated 77 for the general case andfor the limiting cases of uniform rota-tion or biased rotation without fluctu-ations.

It should be stressed that it is notpossible to determine by NQR the valuesof all possible different order parame-ters as there are at each temperatureonly two experimental parameters avail-able (e2qQ/h and 77). The problem isoften simplified by additional informa-tion on the symmetry of the molecularmotion deduced from X-ray or otherexper iments.

V. EXPERIMENTAL RESULTS AND DISCUSSION

A. Achiral Smectic TBBA

Terephta1-b i s-buty1 an i1i ne (TBBA)has been the most extensively studiedsmectic liquid crystal in the past fewyears. The sequence of the smecticphases in this system is as follows (Tin ° C):

C4H9- -N=CH- •CH=N- C4H9

113r

Cr<— Cr<— VI l<—>Sm G<—>SmvH-33 52 68 Bk

>Sm C< >Sm A<172 199 235

V o l . 5 , No. 1/2

TBBA possesses two equivalent nitrogennuclei in the central part of the mol-ecule, so that the conclusions drawnfrom 14H NQR data will concern the ori-entational ordering of the centralmolecular part or more precisely theordering of C-CH=N-C linkage group. Asit is exactly the ordering of the C=Nelectric dipoles in the linkage groupwhich is of prime importance in themicroscopic theory of the smecticphases, 14N is an excellent probe for.this study. The ordering of the molec-ular tails will not be discussed at

5

4

g 3E

=s

" 1

1 1!" • " JJ

Cr

-

20

0.6

0.4F"

0.2

n

" Cr

-

-

--

1 4N NQ.R in IBBA

VIItSmGl SmH SmCj I

I

i i |I ^—»+»- .i i i

11 , i60 100 140

VIljsmG| \ SmH1 1 \

SmC

!i i \i i \i i \

! i, 1 , 1 , , , ^ , Tl

SmA

1

180 T[»C]

SmA

20 60 HO 180 T[*C]

Figure 9- Temperature dependence ofthe 14N quadrupole coupling constante2qQ/h and asymmetry parameter r\ in thesmectic and crystal 1ine phases of TBBA.

present.The measurements of e2qQ/h and i\ in

TBBA (if8~5O) are presented in Figure 9.It can be seen that 17 is zero in theuniaxial smectic A phase. It is small,but non zero, in the Sm C phase andexhibits a maximum in the middle ofthis phase. On going to the Sm H

increases byof this hugeat the end ofis still sig-in the sol id

phase, v discontinuousiy increases andreaches a very high value of about 0.7at the low temperature end of thisphase. Proceeding to the smectic Gphase 77 abruptly decreases to about0.18, and then slowly increases withdecreasing temperature. In the solidphase the value of 7? i s 0.28 and ispractically temperature independent.

The quadrupole coupling constantshows a completely different tempera-ture behavior. It increases slowly andcontinuously with decreasing tempera-ture throughout the Sm A, Sm C and Sm Hphases. On going to the Sm G phase,e2qQ/h discontinuousiynearly 300%. In spitejump the value of e2qQ/hthe Sm G phase (3.28 MHz)nificantly smaller than(4.22 MHz), demonstrating that motionalaveraging is still present.solid state the quadrupoleconstant and n are temperaturedent over the range +50° toshowing that the orientationalcomplete.

A comprehensive analysisexperimental data in the Sm C,Sm G phases shows that thevalues of e2qQ/h and n and in particu-lar the large and temperature dependentvalue of 7? i n the Sm H phase, are notcompatible with either free rotation ofthe molecule around its long axis orpolar ordering where one position wouldbe preferred. The data are consistentwith a bipolar ordering with <cos2\p> *0, <cos\p> - 0, <s i rvp> = 0, <s i n2i/'> = 0.The physical model for such motion,which is in agreement with NQR (48-50)and neutron scattering data (52), is asfollows. The rigid "body" of the TBBAmolecule is reorienting in a six-wellpotential around the long molecularaxis. Four equilibrium sites areequivalent and have an occupation prob-ability p2 while the other two, sepa-rated by l80°, have a lower energy andan occupation probability pi > p2. Thebipolar order parameter is now given by

Incoupl

t heing

indepen--20°order

o fSm H

c,i s

t heand

measured

<cos2yt> = 2 (pi - P2) ,

2pi + kp2 = 1(21)

In addition to reorientations around

Bulletin of Magnetic Resonance

the long axis, fluctuations in thedirection of the long axis also occur.

The degree of orientational order-ing, which is given by the values of<cos2^> for ordering of the shortmolecular axis and by <cos<£> and S forordering of the long molecular axis,increases from the high temperaturesmectic phases towards the solid crys-talline state.

1. Smectic A Phase

The zero value of i? in the Sm Aphase shows that the molecules arefreely rotating in this phase, or reo-rient between six equivalent potentialwells around their long axes. The uni-axiality of the Sm A phase is thus notan "average" phenomenon but holds foreach individual molecule. The fluctua-tions in the direction of the long axisare isotropic. Their magnitude<302>1/2 = <sin*S9>l/a can be straight-forwardly determined from the quadru-pole coupling constant through therelation

e2qQ/h = (eqV2020/h)X

[1 " (3/2)«59*>] (22)

which follows from equations 19 if itis taken into account that in the Sm Aphase y = 0, <cos<£> = <cos2<£> = <cos2^>= <cos\t> = 0, Go = 0, and <592> # 0.In TBBA the value of v

2 o 2 o equals1.17 MHz in frequency units. Thisyields <S92>1/2 a 23° at 175° C corre-sponding to the nematic order parameterS = 0.75 for C-N=CH-C groups. Thisvalue is in excellent agreement withthe results obtained from neutron scat-tering (52) and nuclear magnetic reso-nance experiments (18,52).

2. Smectic C Phase

In the biaxial Sm C phase the non-zero value of r} demonstrates the occur-rence of anisotropy in the molecularmotion. A detailed analysis of e2qQ/hand T) shows that both partial rota-tional biasing of reorientations aroundthe long axis and ariisotropic fluctua-tions of this axis contribute to thenon-zero value of i\ .

The degree of bipolar biasing isvery small, the upper limit of <cos2^>being 0.005. This small value of theorder parameter, though indicating anon-equivalency among the six-equilib-rium positions, shows that the mol-ecules still rotate nearly uniformlyaround their long axes (kS).

The characteristic maximum in thetemperature dependence of J? shows thatthe orientational ordering of the longmolecular axis in the Sm C phase isstrongly perturbed by thermal agita-tion. The order parameters for theorientational ordering of the longmolecular axis, <cos<£> and S1 =( 1 / 2 ) < 3 C O S 2 ( 9 - eM) - 1>, whichdescribes the polar fluctuations of thelong molecular axis similarly as the"nematic order parameter" S (equation1), have been calculated by assumingthat (52):

(i) the 9 and 4> motions are indepen-dent,

(ii) the <i> distribution is of the formg (<£) a exp(Y'cos<£) (y1 describesthe width of azimuthal fluctua-tions, i.e. the motion of longaxes inside the smectic plane),

(i i i) the 9 distr ibut ion is of the formf(9) a exp [5'cos (9 - 9M)](9^ is the angle where the 9distribution is peaked, 8'describes the width of the 9 dis-tribution around 0^, i.e. themotion of the long axes outsidethe smectic plane),

(iv) the root mean square fluctuationsin the polar angle <<592>1/2 sa 22°are constant through the whole SmC phase.

The results of this analysis arepresented in Figure 10 (52). The temp-erature dependence of S1, whichincreases from 0.75 at the Sm A * Sm Ctransition to 0.82 at the Sm C + Sm Htransition, is mainly due to the temp-erature dependence of 0^. 9^increases in the Sm C phase withdecreasing temperature similarly to theaverage tilt angle 9o. Fluctuations inthe polar angle <502>1/2 = 22° are

Vol. 5, No. 1/2 67

——

SmH

SmH

SmH

SmC \

I i i

— • o ~ _ _ _ ^

SmC

SmA

— . _

SmA

SmA

110 150 160 170 180 190

biasing without any fluctuations in thedirection of the long molecular axis (S= 1) , are already a rather good approx-imation to the experimental data (50).The calculated values of the bipolarorder parameter <cos2vt> vary from 0.03at T = UO 0 C to 0.16 at the low temp-erature end of the Sm H phase (Figure11). For <cos2^> = 0.1, the occupation

w o •

0.95 -

0.90-

0 » 5 -

0.15

0.10-

0.05-

• K

•

•

\

*«

SmG

•

•

SmH (Zl

p

J o o S , |

(

• —

/ " '

\

> >

70 80 90 HO 110 120 130 U>T["C1

Figure 10. Temperature dependence ofthe azimuthal (<cos<£>) and polar (S1)fluctuations in the direction of thelong molecular axis in the smecticphases of TBBA (52). The values of theaverage tilt angle Go and the tiltangle 0^, at which the 8 distributionis peaked, are plotted as well.

smaller than those in the azimuthalangle <£, where the root mean squarefluctuation is of the order of ~6O° atT = 165° C (<cos<*>> = 0.5) and ~30° at T= 145° C (<cos<*» = 0.85) •

3. Smectic H Phase

The large temperature variation in 17and the small temperature variation ine2qQ/h in this phase demonstrate a sig-nificant bipolar biasing of the rota-tion around the long molecular axis.The six potential wells for this reo-rientation, which are compatible withthe X-ray structure (12), are not allequivalent. Equations 16 with r =J».*t9» which have been deduced for thecase of uniaxial rotation with bipolar

Figure 11. Temperature dependence ofthe bipolar orientational order parame-ter (<cos 2ij/>) i n the Sm G and Sm Hphases of TBBA. The values (1) havebeen calculated under the assumption ofrigid long molecular axes, and (2) ifthe fluctuations in the direction ofthe long molecular axis are taken intoaccount as wel1.

probability of each of the two lowenergy sites is pi = 0.2 whereas theoccupation probability of each of thefour high energy sites is P2 = 0.15-Occupancy of the two preferred poten-tial wells is thus about 30% more prob-able than that of the other four. Itshould be pointed out that neutron qua-sielastic scattering and X-ray datacannot discriminate between such smallbipolar biasing and a uniform rotationmodel. Thus NQR, where the rotationalbias is expressed in terms of a split-ting frequency, provided the firstdirect evidence for biasing of the

Bulletin of Magnetic Resonance

uniaxial rotation of the central partof the TBBA molecules in the biaxialsmectic phases.

The slight change (-20%) of e2qQ/hin the Sm H phase indicates that fluc-tuations in the direction of the longaxis are still present. If they aretaken into account in the analysis (50)of the experimental data, the calcu-lated values of <cos2^> are somewhat,though not significantly, higher at thehigh temperature part of the Sm H phase(Figure 11). The corresponding fluctu-ations in the direction of the longaxis, which have been assumed to beroughly isotropic, range from -15°close to the Sm C phase down to only~2° at the boundary of the Sm G phase,where they freeze out completely.

4. Smectic G Phase

In the smectic G phase the orderingof the long molecular axis is perfectand the averaging of the EFG tensor isentirely due to the molecular reorien-tations around its long axis. The hugediscontinuity inthe Sm H •» Sm Gthe direction ofaxis of the EFG

both e2qQ/h and attransition shows thatthe largest principal

tensor has changed dueto the large increase of the bipolarbiasing. It is not anymore parallel tothe axis of reorientation. The valueof <cos2\i'> calculated from equations 17with r = 4.49 is already as high asO.85 at the high temperature end of theSm G phase and increases to 1 at thelow temperature end (50). The fourhigh energy states in the sixfoldpotential well are now nearly empty aspj = 0.02 at 82.5° C and p2 = 7 x 10"

3

at 72° C, whereas pi approaches 0.5-The molecular motion in the Sm G phasecan be thus indeed described as 180°orientational jumps of the central partof the TBBA molecule.

5. Solid Crystalline Phase

On going from the Sm G to Sm VII andfurther to the crystalline phase, thei80° jumps freeze out so that <cos^> =<cos2^> = 1 and <sin^> = <sin2^> = 0.The freeze out of the l80° jumps proba-bly happens in the Sm VII phase whereno measurements could be made because

the proton Ti is too short.In the crystalline solid phase the

C-N=CH-C groups are already completelyrigid, as demonstrated by the fact thatneither the 14N quadrupole couplingconstant nor the 14N asymmetry parame-ter vary with temperature. The EFGtensor in the molecular frame is(49,50):

(eQV ij/h)

\±

B.

(MHz) =

3-15

0.15

2.41

Chiral

± 0,

1

± 0

.15

.98

.40

Smectic

± 2

± 0

1

TBACA

.41

.40

.17

The central part of the chemicalstructure of terephtal-bis-amino-me-thyl- butyl cinnamate (TBACA) is thesame as that of TBBA. The tails con-tain the asymmetric carbon which isresponsible for the chirality and opti-cal activity of this substance in itsright- and left-handed optically activeforms. In the vicinity of the chiralgroup is the strongly polar -COO group.The sequence of phase transitions inTBACA, first reported by Urbach andBillard (53), is the following (T in °0:

C2H5-CH-CHj-COO-CH=CH-

CH3

-CH=N-CH3

-O^CH-COO-CHJ-CH-CJHJ

130 149 180C n > Sm H* (Cr2) > Sm C* >

242 288Sm A > N > I .

The structure of the various mesophasesof TBACA has not yet been determined byX-ray crystallography as this has beendone for TBBA. Electron-diffractionstudies and NQR data (39.54) show thatthe phase Cr2 is a liquid crystallinephase, most probably the Sm H* phase.Thus this name will be used for the

Vol. 5, No. 1/2

smectic phase obtained by cooling fromthe Sm C* phase.

The temperature dependence of e2qQ/hand v of both equivalent nitrogens inthe central part of TBACA (u9) is pre-sented in Figure 12 for the Sm A, Sm C*

180 T[°C)

Figure 12. Temperature dependence ofthe 14N quadrupole coupling constante2qQ/h and asymmetry parameter TJ in thesmectic phases of TBACA.

and Sm H* phases. The 14N quadrupolecoupling constant slowly increases withdecreasing temperature without showingany discontinuities at the Sm A Sm C*or Sm C* •* Sm H* transitions. The val-ues of rj in the Sm H* phase of TBACAare less than one tenth of these inTBBA.

The proton spin-lattice relaxationtime in the solid phase of TBACA is tooshort to allow a determination of thedouble-resonance spectra. This makesthe analysis of the 14N NQR data inTBACA more difficult than in TBBA asthe rigid lattice VJ: tensor in themolecular frame cannot be determined.However, it should not be appreciably

different from that determined for TBBAas the central parts of both moleculesare the same. In the following weshall assume that Vj: is the same inboth TBACA and TBBA. The same holdsfor the magnitude of the tilt angle 6o.

In view of these approximations, theanalysis of the 14N data in TBACA ismore tentative than in TBBA. We fur-ther assume that the 0 and 4> motionsare independent and that the magnitudeof fluctuations in the long molecularaxis is small enough to allow theexpansion of sine and cosine of 66 and<t> in a power series with two or threeterms. Thus one is able to determinethe values of the order parameters<cos^> or <cos2^>, as well as the mag-nitude of the mean square fluctuationsin the long molecular axis <692> and<cf>2> C»l) .

1. Smectic A Phase

The value of •>) i s zero in the Sm Aphase of TBACA as in TBBA. The uniax-ial rotation around the long molecularaxis and the "nematic" fluctuations inthe direction of this axis are iso-tropic.

2. Smectic C* Phase

The value of r\ is smaller than inTBBA but non-zero. The occurrence of amaximum in the temperature dependenceof 7j indicates that the anisotropy inmolecular motion is produced mainly byanisotropic fluctuations in the direc-tion of long molecular axes. If theorientational ordering of the shortmolecular axes is neglected, <<592>1/2

and «t>2>x'2 can be calculated at eachtemperature. In Figure 13 they areplotted for V 2 o 2 o = 1.15 MHz. Thefluctuations in the azimuthal angle arelarger than in the polar angle through-out the smectic C* phase. The rootmean square value of the polar fluctua-

increases slowly fromboundary between theC* phase to ~15° at

A transition. Thisagreement with that

*qQ/h at the boundary

tions, <59 2> 1 / 2,~1O° close to theSm H* and the Smthe Sm C* * Smvalue is in goodcalculated from e1

between the Smwhere <cos<£> =

C* and Sm A phase (<cos2\J> = 0) ,

70 Bulletin of Magnetic Resonance

0.15

v 0.10 -

0.05 •

0

Cr

rder porometer <cos v> for |

TBACA _ ' «

P3 y / Pj S~

\ / -C. V P -*-1 A Pi -A

P/ Pj

SmH*

-

30

20

10

/< , /

SmC* [smA

!150 170 190

SmC*

<cos v ><0.05

130 U0 150T[*C]

160

Figure 13^ The temperature dependenceof the polar order parameter <cosi/'> andof the fluctuations in the direction ofthe long molecular axis (<692>1/2 and<^2>1'2) in the Sm C* and Sm H* phasesof TBACA (full line represents the cal-culated values of <cos^> for V 2 o Z o =1.08 MHz and the broken line for

'zozo = 1 . 1 5 MHz)

which turns out to be 15-5°• The fluc-tuations in the azimuthal angle,«t>2>1'1, increase rapidly in the Sm C*phase and diverge as the Sm A phaseboundary is approached.

On the basis of the NQR data in theSm C* phase it is not possible to saywhether or not the rotation around thelong axis in the Sm C* phase of TBACAis biased. Nevertheless the biasingshould be polar and very small, inaccordance with the conclusions drawnfor the Sm H* phase. Its upper limitis estimated as <cos<^> < 0.02.

3. Smectic H* Phase

The NQR data in the Sm H* phase

Vol. 5, No. 1/2

can be explained by polar biasing ofthe uniaxial rotation around the longmolecular axis with <cos^> * 0 and<cos2^> = 0 . If, on the other hand,bipolar ordering with only <cos2^> * 0or a uniform rotation around the longmolecular axis is assumed, no physi-cally reasonable solution which wouldagree with the experimental data can beobtained. The physical model for polarordering is that the molecule reorientsaround the long axis in a sixfoldpotential which is biased due to thecoupling of the rotatable electricdipole with the local electric field.Due to coupling of the dipole to thelocal electric field (-/*E) • the six ori-entations are nonequivaient, with

(23)P i oc

For such a set of probabilities* 0, while <cos2^> « <cosvJ> i f thecoupling term is smaller than kT.

The temperature dependence of thepolar order parameter <cos^> in the SmH* phase is shown in Figure 13- The

values have been calculated fortwo different values of V Z o 2 o

: 1•15MHz and 1.08 MHz. The higher value isclose to that determined for TBBA (seeabove), and the lower one is the lowestwhich still gives reasonable solutions.In the whole Sm H'V phase the values of<cosi/'> are small, ranging from ~0.1 atthe low temperature end to ~0.02 at thehigh temperature end of this phase.For <cos^> = 0.07 the difference in theprobabilities between the most favora-ble position (pi = 0.19) and the leastfavorable position (pa = O.\h) is thusonly ~30%. In the chiral Sm H* phaseof TBACA the polar biasing of themolecular rotation around its long axisis thus rather weak. This is compati-ble with the small value of the sponta-neous polarization observed in ferroe-lectric liquid crystals (39).

The accompanying fluctuations in thedirection of the long molecular axisare smaller than 10°. They can beignored and the experimental dataexplained by polar ordering only.

C. The Smectic B Phase of IBPBAC

The above results have shown that in

71

the two dimensional translationailyordered biaxial smectic phases of TBBAand TBACA the tilting of the moleculesis associated with a biasing of themolecular rotation around the longaxis. To establish whether this rela-tion is general we investigated theorientational ordering of the shortmolecular axes in the until ted twodimensional translationaily ordered SmB phase of isobutyl-4- (41 -phenyl benzyl -ideneamino)-cinnamate (IBPBAC). Thesequence of the smectic phases in thissystem is as follows (T in ° C) (55):

- CH=N -

- CH=CH - COO- CH2CH (CH^j

86 1 U 162Cr < > Sm E < > Sm B < >

206 214Sm A < > N < > I

The X4N quadrupole coupling constant inthe Sm B phase of IBPBAC (1.11 MHz at137° C and 1.09 MHz at 146° C) (56) iscomparable to the values found in TBBA,whereas the value of the asymmetryparameter is much smaller ( <, 0.015 at137° C and ~ 0 at 146° C) . This demon-strates that the rotation around thelong molecular axis is practically"free" in the Sm B phase, i.e. the mol-ecules seem to reorient between 6equivalent sites in a hexagonallypacked structure. The fluctuations inthe direction of the long molecularaxis ( <5 9 2 > ) 1 / 2 are estimated to ~12°at 140° C.

The above results show that there isindeed a close relation between themolecular tilt and the biasing ofmolecular rotation around the long axesin two dimensional ordered smectic sys-tems. Biasing is polar in tilted chi-ral systems, bipolar in tilted achiralsystems, and absent in untilted sys-tems .

D. (C10H21NH3)?CdCl4: A Lipid BilayerEmbedded j_n a Crystal 1 i ne Matrix

The occurrence of phase transitions

in cell membranes (57t58) has recentlyattracted a great deal of attention.It has been convincingly demonstrated(61,62) that the transitions in livingcell membranes are due exclusively tothe lipid component of the bilayer mem-brane. Most phase transition studieshave therefore been performed on modellipid membranes, which are similar intheir structural properties to biomem-branes but can be better characterizedphysically and exhibit sharper phasetransitions. Lipid bilayers of dipal-mitoyl phosphatidyl cholihe (DPPC)exhibit two first-order phase tran-sitions (63), the main transition atT C 2 = 42° C connected with a meltingof the chains, and the pretransition atT C 1 = 33° C. Most model lipid mem-branes are difficult to obtain in asingle crystalline form. This is how-ever not the case with(Ci0H21NH3)2CdCl4 (abbreviated asClOCd), the hydrocarbon part of whichrepresents a smectic lipid bilayerexhibiting two structural phase tran-31tions at 'Ci 35° C and T C 2 =39 C in analogy to biomembranes.

The projection of the structure ofClOCd on the b-c plane at room tempera-ture is shown in Figure 14. The struc-ture consists of CdCU 2" layers sand-wiched between well orderedalky 1 ammonium chains which are tiltedby 40° with respect to the normal tothe layer. The ammonium end of eachchain is linked to the layer byN-H'"'C1 hydrogen bonds, and eachchain is coordinated by six others.The entropy change at the lower phasetransition T C 1, (0.9 ± 0.3)R per molechains, can be explained by an order-disorder transition of rigid chainsbetween two equivalent sites. However,the entropy change at the higher phasetransi tion 'C2 corresponds to 0.8 Rper R-C-C-R bond and can be explainedonly by a "melting" of the chains, orwhat is equivalent, by rapid chainisomerization via kink diffusion.

14N quadrupole resonance data showthat on going from the low temperatureto the intermediate phase the polar-NH3groups and the alkyl chains start flip-ping by 90° around their chain axes, sothat neighboring chains move in oppo-site directions as in a two-dimensional

72 Bulletin of Magnetic Resonance

600

500

too

300

10 20 30

\

14 N NQRC10Cd

40 50

\>

Figure ]k. Projection of the structure Figure 15.of ClOCd on the b-c plane at room temp- the 14N NQRerature.

Temperature dependence offrequencies in ClOCd.

array of connected gears (64,65)•As the motion of the ammonium polar

heads between the two equilibriumsites, characterized with 14N EFG ten-sors T(l) and T (2) , is fast on the NQRtime scale, the observed 14N NQR spec-trum is determined by the time averagedEFG tensor:

from about 1 atzero as T,

<T(t)> = (1/2) [1 + p]T(l) +

(1/2) [1 - p]T(2) .(2*0

Here p is the orientational orderparameter for the 90° flipping of thealkyl-ammonium chains. As T(l) andT(2) are known from single crystalexperiments at low temperature in thecompletely ordered phase, the tempera-ture dependence of p = p(T) can bedetermined from the observed tempera-ture dependence of the NQR frequencies(Figure 15). The results are presentedin Figure 16. In the low temperature

phase p decreases10° C towards zero as T C 1 isapproached. In the intermediate andthe high temperature phases p = 0,indicating that in these two phases thepolar heads are disordered between twoequi1ibrium sites.

The average orientation of thehydrocarbon chains and their degree ofmelting has been determined by 1 3C NMR.In the intermediate phase the chainsare still tilted and relatively rigidwhereas in the high temperature phasethey are on average normal to the smec-tic layers. The transition to the hightemperature phase is also connectedwith a significant decrease in thenematic order parameter S of the alkylchains though S is non-zero even in thehigh temperature phase.

The hydrocarbon part of ClOCd thusrepresents a smectic liquid crystalwith a structure similar to theinterior of the bilayer lipid mem-branes. We shall therefore try to

Vol. 5, No. 1/2 73

describe {(>h,GS) the two phase tran-sitions in ClOCd by a Landau typeexpansion of the non-equilibrium freeenergy F in terms of order parametersGo, S - Sc

of smecticand p, used in1iquid crystals:

the theory

(25)

F = (1/2) AGo2 + (lA)BGo*

+ e(T - TC2)(S - S c )

- (1/2) c(S - S c )2 + (lA)d(S - Sc)«

+ (1/2) ap*

+ (lA)bp* - (l/2)A'6o2(S - Sc)

- (l/2)a'p*(S - Sc) .

Here S stands for the average nematicorder parameter and A, B, e, c, d, a,b, A1 and a1 are all positive con-stants. The Sm C order parameter Gogives the average tilt of the moleculeswith respect to the layer normal,whereas p is the orientational orderparameter for the 90° flipping of thechains and the terminal NH3 groupsbetween the two equilibrium orienta-tions corresponding to p = ± 1. Sc

is the critical value of the averagenematic order parameter S, i.e. thearithmetic average of Sand rigid phases coexifirst-order transition

C2(Figure 16) In

in the meltedisting at the

temperaturederiving equa-

tion 25 we assumed that the melting ofthe chains is the important drivingmechanism which induces the transitionsin S as well as in Go and p. It shouldbe noted that here S is, in contrast tothe isotropic-nematic transition, not asymmetry breaking order parameter andis different from zero both above andbelow T C 2. Therefore the linear termin the expansion of F i n powers of(S - Sc) is not equal to zero, as inthe case of nematic liquid crystals,but has the same structure as the freeenergy at liquid-gas phase transitions.

Minimizing the total free energywith respect to p, Go and S we obtainthe following stable solutions:

b10

0.8

0.6

0.4

0.2

0

P1.0

0.8

0.6

0.4

0.2

n

__a o _ o 1

-

ClOCd

h

10 20 30

- X\10 20 30

H-ci

T

~~\J

t 1

40 50 ,

a

40 50

-40°

-30"

•20°

•10°

0°

Figure 16. Comparison between the cal-culated and the observed temperaturedependences of the order parameters S,p and Go for ClOCd.

T > TC2

C1 < T < TC2

T < TC 1

S < S C , G o = 0 , p = 0 ,

S > S c , Go * 0 , p = 0

S > S C , Go # 0 , p # 0 .

(26)The high temperature transition at

T C 2 corresponds to a partial meltingof the chains which simultaneouslydestroys the tilting of the molecules,whereas the low temperature transitionat T C 1 corresponds to an orienta-tional transition and a disordering ofthe polar "heads". The temperaturedependence of the order parameters S,Go and p is shown in Figure 16 togetherwith the experimental values determinedin this study.

N0TE:This review covers the fielduntil April 1980 when it was submittedfor publication. Revised May, 1983-

Bulletin of Magnetic Resonance

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76 Bulletin of Magnetic Resonance