PHYSICSREPORTS(ReviewSection of PhysicsLetters)135, No. 4 (1986) 195—257.North-Holland,Amsterdam

LANDAU THEORY OF THE NEMATIC-ISOTROPIC PHASE TRANSITION

EgbertF. GRAMSBERGEN,Lech LONGA* andWim H. de JEUtSolidStatePhysicsLaboratory,Melkweg1, 9718EPGroningen, The Netherlands

ReceivedOctober1985

Contents:

1. Introduction 197 3.3. Field effects fort~x

m~<0 2271.1. Nematicliquid crystals 197 3.4. Field effectson thebiaxial nematic phase 2291.2. The nematic orderparameter 198 4. Landau theory of the nematic—isotropicphasetransition:

1.2.1. Definition of amacroscopicorderparameter 199 fluctuations 2311.2.2. Relationshipbetweenmacroscopicand micro- 4.1. The importanceof orientationalfluctuations 231

scopicorderparameters 200 4.2. Generalform of theLandaufree energy 2321.3. Phasetransitionsand critical phenomena 203 4.3. Pretransitionallight scattering 234

2. Landautheoryof thenematic—isotropicphasetransition 205 4.3.1. Introduction 2342.1. Landautheory: ingredients 205 4.3.2. Correlation functions in the Gaussian ap-2.2. The nematic—isotropicphasetransition 207 proximation 237

2.2.1. Landau—DeGennestheory 207 4.3.3. Comparison with experimentand discussion:2.2.2. Comparisonwith experimentanddiscussion 212 validity of theGaussianapproximation 239

2.3. The biaxial nematicphase 213 4.4. BeyondtheGaussianapproximation 2422.3.1. Theory 213 4.4.1. Perturbation calculation of the correlation2.32. Comparisonwith experimentanddiscussion 217 functions 242

3. Landautheory of the nematic—isotropicphasetransition: 4.4.2. Comparisonwith experimentanddiscussion 245the influenceof externalfields 219 5. Conclusions 2483.1. Introduction 219 Appendix A 2493.2. Field effects for ~Xma, >0 220 AppendixB 250

3.2.1. Theoryof thenematic—isotropicphasetransition 220 Appendix C 2513.2.2. Comparisonwith experiment 222 References 2553.2.3. Singularbehaviouron approachingT* 224

Abstract:A reviewis givenof thewide variety of predictionsthat resultsfrom a Landau-typeof descriptionof thenematic—isotropicphasetransition.This

includesadiscussionof thenatureof theorderparameterandof thevarioustypesof possiblephases,of the influenceof externalfields, and of theeffect of inclusionof spatialvariationsof theorder parameter.The variouspredictionsarecomparedwith the availableexperimentalresults.It isconcludedthat thereis still no clearpicture aboutthenatureof thesingularity nearthenematic—isotropicphasetransition.Thoughtheassumptionofclassical(mean-field)critical behaviourseemsto be incorrect,thereis no conclusiveproof which alternativeapplies.

* Permanentaddress:PhysicsInstitute,JagellonianUniversity,Reymonta4, Krakow, Poland

t Presentaddress:Institut für Theoretischeund AngewandtePhysik,UniversitãtStuttgart,Pfaffenwaldring57/VI, 7000Stuttgart80, Fed. Rep.Germany.Also: The OpenUniversity,P.O. Box 2960, 6401 DL Heerlen,The Netherlands

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PHYSICSREPORTS(Review Sectionof PhysicsLetters)135, No. 4 (1986) 195—257.

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LANDAU THEORY OF THENEMATIC-ISOTROPIC PHASE TRANSITION

Egbert F. GRAMSBERGEN, Lech LONGA and Wim H. de JEU

Solid StatePhysicsLaboratory, Melkweg1, 9718EPGroningen, TheNetherlands

INORTH-HOLLAND-AMSTERDAM

E.F. Gramsbergen et al., Landau theory of the nematic—isotropic phase transition 197

1. Introduction

1.1. Nematicliquid crystals

The liquid crystallinestateof matter is observedin certain compoundsbetweena crystallinesolidandan isotropic liquid. Liquid crystalspossessmanyof the mechanicalpropertiesof a liquid, e.g., highfluidity andthe inability to supportshear.At the sametime theyare similar to crystalsin their physicalproperties.For example, they are birefringent and have anisotropicmagnetic and electric suscep-tibilities. For this reasonthe contradictoryname“liquid crystals”was introducedalmost a centuryago[1].

Often liquid crystals are obtainedas a function of temperaturebetween the crystalline and theisotropic liquid phase:thesearecalled thermotropicliquid crystals.Another type of liquid crystalscanbe found in solutions, with soapsas a commonly known example;theseare called lyotropic liquidcrystals.For this class,concentrationis the importantcontrollableparameter.Unlessotherwisestated,we will havethermotropicliquid crystalsin mind in the presentdiscussion.

To understandthe nature of liquid crystals, we recall that a crystal consisting of anisotropicmoleculesexhibits two kindsof long-rangeordering:

(1) long-rangeorderingof the centersof massof the molecules;(2) long-rangeorderingof the orientationsof the molecules.

When a crystal melts into an isotropic liquid usually both kinds of ordering disappearat the sametemperature.Alternatively, it is possible that one of the two ordering types survives until a highertransition temperatureis reached.In caseit is the long-rangepositionalordering that survives, this iscalled a plastic crystalline phase;if the long-rangeorientationalordering survives,we havea liquidcrystal. If in addition the long-rangepositionalorderinghascompletelydisappeared,theliquid crystal iscalled nematic.The absenceof long-rangepositionalordering in the presenceof long-rangeorien-tational orderingperfectly explainsthecombinationof high fluidity ande.g.,opticalanisotropy.In someliquid crystalsthereis long-rangepositionalorderingin one dimension(the molecularcentersare, onaverage,orderedin layers);then we speakof smecticliquid crystals. In fact a greatvariety of smecticphasesexists which will not be of concernhere.When smecticand nematicphasesare found in onecompound,the nematic phase is nearly always found at higher temperature(exception: reentrantnematicphase[2]). Some liquid crystallinecompoundsandtheir phasesarelisted in table1. Evidentlystrongly anisotropicmoleculesareneededto build a liquid crystal. Often thesemoleculesarerod-like(fig. 1), but recently alsodisc-likemoleculeshavebeenfoundto give a nematicphase(fig. 2) [31.

In nematicliquid crystalsthe moleculesare,on average,alignedwith their long axesparallelto eachother.Macroscopically,a preferreddirection is thusdefined.In manycases(exceptions:seesections2.3and 3.4) there exists rotational symmetry around this direction: then the phase is uniaxial. It isconvenientto describethelocal direction of alignment by a unit vectortl(r), the director,giving at eachpoint the direction of the preferredaxis. In nematicsone alwaysfinds that ñ and —ñ are equivalent,even though the molecules may be polar. For example, no ferroelectricity has been observed.Microscopically,this meansthat an equalnumberof moleculespoints “up” and “down”. As we shallsee,the equivalenceof ñ and — ii is crucial for the constructionof the theory of the nematicphase.Itwill be shownthat this will leadto

(1) a first-ordernematic—isotropicphasetransition;(2) the possibilityof a biaxial phase;(3) a non-trivial Gaussianfluctuationspectrumin the isotropicphase(seechapter4).

198 E.F Gramsbergen eta!., Landau theory of the nematic—isotropic phase transition

‘Un i\V1 ______

Fig. 1. Structureof a conventionalnematicof elongatedmolecules. Fig. 2. Structureof adiscotic nematic.

Table 1Compoundswith liquid crystallinephases.The transition temperaturesarein °C.The phases

aredenotedasK (crystalline),S (smectic),N(0) ((discotic)nematic),I (isotropic)

/ N—~~~—-QCHCH3O—~~--N” (PAA): K 1I8N 135.51

“0

~ (MBBA): K22N471

1/ ~ C4 H9/ N —~--- N

C4H9—~~---’ (TBBA): K tI3S~144S~172SA200N2371

C8H17—~~.--t~--CN (8CB): K21.5SA33.8N40.8I

R 0C9H19 : KI80N~222l

Our concernin this review will be to provide someinsight into theseproblemsvia a systematicpresentationof the Landautheory of the nematicphase,and to comparethe resultswith experimentaldata.

1.2. Thenematicorderparameter

The transition between phases of different symmetry (crystal and liquid, different crystalmodifications)can be describedin termsof a so-calledorderparameter.It representsthe extentto which

E.F. Gramsbergen et a!., Landau theory of the nematic—isotropic phase transition 199

theconfigurationof the moleculesin the lesssymmetricphasediffers from that in themoresymmetricalone. In general, an order parameterQ describing a phase transition, must satisfy the followingrequirements:

(1) Q= 0 in the moresymmetric(lessordered)phase;(2) Q� 0 in the lesssymmetric(moreordered)phase.Theserequirementsdo not definethe orderparameterin auniqueway. In spiteof this arbitrariness,

in many casesthe choicefollows in a quite naturalway. For instance,in liquid—gassystemsthe orderparameteris the differencein densitybetweentheliquid andthegasphase.In this case,0 is a scalar.Insimple ferromagneticsystemsthe order parameteris the magnetization.In this case, 0 is a vector,proportionalto the thermalaverage(1’), where i~is a unit vectorgiving the direction of the molecularmagneticmoment.In morecomplicatedcasesthe problemof the definition of an orderparametercanbe formulated in the languageof group theory, as originally proposedby Landaufor second-orderphasetransitions[4, 51. Thechoiceof an orderparameterfor the nematicphaserequiressomecarefulconsiderations,but thereis no needto go to group-theoreticalarguments.

1.2.1. Definition of a macroscopicorderparameterThe nematicphasehasa lower symmetry than the high-temperatureisotropic liquid. In order to

expressthis in a quantitativeway we introducean order parameter0, such that(1) 0 = 0 in the isotropicphase;(2) 0 � 0 in the nematicphase.

The long-rangeorder in the nematicphasepreventsany molecularanisotropy from beingaveragedtozero. Consequentlyall macroscopicresponsefunctions of the bulk material, such as the dielectricpermittivity or the diamagneticsusceptibility,are anisotropicas well. In deriving a macroscopicorderparameterwe shall use the diamagneticsusceptibility.However, any other macroscopicproperty as,e.g.,the refractiveindex or the dielectricpermittivity, could be used.

Considerthe relationbetweenthemagneticmomentM (dueto themoleculardiamagnetism)andthefield H;

~ a,f3—x,y,z. (1.1)

HereXa$ is an elementof the susceptibilitytensorx. In eq. (1.1) summationover repeatedindices isimplied. In generalthe tensorx is symmetric,while in the isotropicphaseit hasthe simple form

= x&~. (1.2)

Here ~ is the Kroneckerdelta which equals one for a = /3 and is zero otherwise.For the uniaxialnematicphasex can bewritten in diagonalform without any loss of generality:

0 0\

x( 0 ,y~ 0 ). (1.3)\o 0 xi!

Now Xii andx±are the susceptibilitiesparallel and perpendicularto the symmetry axis, respectively.The symmetry axis is by definition a line parallel to the director ñ, i.e., the eigenvectorof xcorrespondingto the extreme(nondegenerate)eigenvalue.

200 E.F. Gramsbergen et a!., Landau theory of the nematic—isotropicphase transition

Comparingeqs. (1.2) and (1.3) we see that the requirementsimposed on an order parameterare

fulfilled by the anisotropicpart of the diamagneticsusceptibility:~Xap = Xa,s — ~&~sXyy. (1.4)

Thus an orderparametertensorQ is definedwith elements

Oats = ~Xa~3I~Xrnax, (1.5)

where~Xmax is themaximalanisotropythatwouldbeobservedfor aperfectlyorderednematicphase.Fordisc-likemolecules,thesign of ~Xmax needssomeexplanation,which isgiven in section1.2.2.The choiceofQ reflectsthattheorientationalorder is theonlygeneralaspectin which thenematicandtheisotropicstatediffer, while alsothe absenceof ferroelectricityis incorporated.The use in theLandautheory of Q thusdefined,makesthistheory quite independentof anyassumptionsabouttheconstitutingmolecules[6,7]. Inparticularconsiderationsaboutmolecularflexibility [8]do not comeinto play. Q asdefinedin eq. (1.5)canbeconsideredasthesimplestchoiceof anorderparameterthattakesthesymmetrypropertiesof thephasesinto account.

The choiceof the diamagneticsusceptibilityto defineQ is convenientfrom the point of view that xcan relatively easily be related to molecular properties.This is not always true for the possiblealternativesthat could be chosenfrom the variousmacroscopicresponsefunctions[9]. We shall comeback to this point in the nextsection.

The definition (1.5) of the order parameterQ coversa wider class of liquid crystals than simpleuniaxial nematics.In the most generalcaseQ is an arbitrary symmetric tracelesssecond-ranktensor.Henceit has five independentelements.BecauseQ is symmetric,it is alwayspossibleto find aframeofreferencein which it is diagonal,with the eigenvaluesof Q as diagonalelements:

0 0\

Q=( 0 —~(x—y) 0 ). (1.6)\ 0 0 xJ

In this generalparametrizationtheconditionof zerotraceis automaticallyfulfilled. In addition it allowsfor the possibility that all threeeigenvaluesare different (x � 0, y � 0). This correspondsto a biaxialnematic phase,which could occur in addition to the uniaxial nematicphase(x � 0, y= 0) and theisotropic phase(x = y = 0). Up to now a biaxial nematicphasehasnot beenobservedfor thermotropicliquid crystals,but it hasrecentlybeendiscoveredin soap-likesolutions[10]. In an arbitrary referenceframe, Q~in termsof the parametersx andy is given by

= ~x(n,,n8~ — (ñ x th)a(~x th)~], (1.7)

where fl, th and ñ x th are the orthogonaleigenvectorsof Q correspondingto the eigenvaluesx,

—~(x+ y) and —~(x— y), respectively.

1.2.2. Relationshipbetweenmacroscopicand microscopicorderparametersIn the previous sectiona nematic order parameterhasbeenintroducedwithout any referenceto

molecular properties.The predictions of the Landau theory based on such a macroscopicorderparameterarein this senseuniversal.However, orderparametersareoften alsoconstructedin relation

E.F. Gramsbergen et a!., Landau theory ofthe nematic—isosropic phase transition 201

to specific molecularmodels.Theseorderparameterswill be called microscopic.By definition thesewillcontain more information thanjust the symmetry of the phase.This can complicatea Landautheorybased on such order parametersconsiderably.Only for some simple molecular models there is astraightforwardconnectionbetweenthe two approaches.In order to add to the understandingof thevarious approachestheseconnectionswill be discussedin short.However, the theory to be presentedlater refersto Q as definedin the previoussection.

In order to constructa microscopicorderparameterwe takethesimplest typeof molecularmodel. Itis assumedthat the moleculescan be describedas rigid, axially symmetricrods. The reasonwhy themoleculesmay be thoughtof as axially symmetric, is that they rotate nearly isotropically along theirlong axes. The assumptionof molecular rigidity is more problematic and certainly not correct.Nevertheless,in the rigid rod model it is possibleto specify the molecularorientationby a single unitvector i, which is along the long molecularaxis. In that casescalarandvector orderparametersarebothimpossible.A scalarorderparameterwould involve theproduct(i~1) which is unity by definition.Avectorwould involve (i~)which is averagedto zeroowing to therequirementthattherebeno macroscopicpolarity. So the simplestpossibility is to considera second-ranktensor.A naturalorderparametertodescribemacroscopicallynematicorderingis then QM with elements[41

(1.8)

wherethe ~aare thecomponentsof i~in a laboratoryfixed frameof reference.The orderparameterQM

is a symmetrictensor.Due to the Kroneckerdelta6ats, which is includedto makethe elementszerointhe isotropicphase,it is also traceless.

Equation (1.8) can be arrived at in a more formal way by realizing that the information aboutthemolecularorientationalorder is containedin the one-particledistribution function. It is often con-venientto expandthe latter in a completeset of functions,that, of course,haveto reflect the symmetryof both the phasesand themolecules[11,12]. Suchan expansioncanbe made,for instance,in termsofthe direction cosinesof the director ii with respectto a frame fixed to a molecule (1 II 2). Theneq. (1.8)is the lowest-orderset of coefficientsof this expansionwhen the equivalenceof i~and — i is takenintoaccount(seealso [11]).

Like Q the microscopicorder parameterQM is the symmetrictracelesssecond-ranktensorthat canbebrought on a principal axis. Thenthe samethreephasescan beidentified asbefore:

(1) Isotropic (I): all threeeigenvaluesarezero.

(2) Uniaxial nematic(Ne): two of theeigenvaluesareequal.Takingthe z-axis of a laboratoryframeasthe axisof ordering,the orderparameterhasthe form

0 o\QMS( 0 —~ 0 J, with S~(v~)—~.

0

In an arbitrary frame of reference:

= S(nanp— ~8ats), (1.9)

S= ~(cos20) — = (P2(cos0)), (1.10)

202 E.F Gramsbergen et at, Landau theory of the nematic—isotropic phase transition

where the fla are the directorcomponents,0 is the anglebetweenthe long molecularaxis and ii, andP2(cos0) is thesecondLegendrepolynomial.Notethateq. (1.9)is identicaltoeq. (1.7) providedin thelattercasey = 0,x = ~S.Thereforetherigid rodmodelprovidesuswith asimpleinterpretationof theparameterxintroducedin eq. (1.7).

MoresystematicallytherelationbetweenQ andQM canbeobtainedfor therigid rodmodelby pointingout thattheanisotropicpartof thediamagneticsusceptibilityis in therigid rodmodelproportionalto Q~:

= NXaQ~, (1.11)

where N is the particle density numberand Xa is the anisotropyof the molecularmagnetic suscep-tibility. Sinceby definition AXmax = NXa, onefinds within the framework of the rigid rod model [9, 14,15]

= Q~. (1.12)

Almost all nematicsknowntodaybelongto the classof uniaxial nematics,despitethe fact that noneof the constituentmoleculesis axially symmetric. Becausealmost all nematics are macroscopicallyuniaxial, S, ratherthan QM, is often referredto as the orderparameter.S is normalizedsuch that forperfectorder S= 1 and for completedisorderS= 0. Negativevalues of S occur when the moleculestendto lie with their long axesperpendicularto the director.Rod-like moleculeswith a negativei~xcanbeforced into sucha phaseby a magneticfield (chapter3).

Disc-likemoleculeswill havea naturaltendencyto lie with their longaxesperpendicularto thedirector.In thissense,onecouldassignanegativevalueto Sfor thediscoticnematicphase.A properdescriptionoftheorientationalordershould,however,involve themolecularaxisof symmetry,which is, in thiscase,theshort axis. TheorderparameterS is thenpositivefor bothrodsanddiscs.In orderto distinguishbetweenthetwo cases,wewill couplethesignof Stothatof theopticalbirefringenceE~n= — n0. ne andn0 aretheindicesof refraction,takenat thesamewavelength,for the extraordinaryandordinary rays, respectively.iXn ispositivefor virtually all nematicsof rod-likemolecules,andnegativefor disc-likemolecules.Tomakethis sign conventionfor S compatiblewith the definition (1.5) of Oa$ we have to associate~Xrnax with

~Xrnax = ~Xisi—isgn(z~n), (1.13)

which is nearlyalwayspositivefor nematicsof bothrodsanddiscs.Referringtothesign of both S and~n,wewill talk about“positive” and“negative”uniaxialnematics.Unlessotherwisestated,thetermnematicrefersto the N~i,phase.

(3) Biaxial nematic(N8): all threeeigenvaluesaredifferent. Onemight be surprisedto find a biaxialphasein a systemof axially symmetricrigid rods,since no molecularinteractionin such a systemcanproducea macroscopicordering that is less symmetric than the moleculesthemselves.The secondmacroscopicaxis must then havean externalcause,for examplean externalfield perpendicularto thedirector [13].This caseis describedin moredetail in chapter3.

In theabsenceof fields, a biaxialnematicphasemaybeexpectedfor moleculesthat arenot (not even“effectively”) axially symmetric. As thereare two molecularaxes, the order parameterhas no longerthe simple form of eq. (1.9). For realisticmolecularmodelsthe relationbetweenQ andthe microscopicorder parametersmay appearto be quite complicated.The intuitively obvious relation (1.12) cancertainlynot be generalized.

E.F. Gramsbergeneta!., Landau theory of the nematic—isotropic phase transition 203

1.3. Phasetransitionsand critical phenomena

In this sectionwe introduceconceptsthat will be usedto place the NI phasetransitionin the widercontext of critical phenomena[16, 171. In general two different types of phasetransitionscan bedistinguishedin varioussystems.A phasetransitionis said to be first orderwhen the orderparameterchangesdiscontinuouslyat the transition.At a second-orderphasetransition,or critical point, the orderparameteris continuousbut is discontinuousin its first derivative(fig. 3). Numerousexamplesof first-and second-ordertransitionsexist. Often both kindsof transitionappearin onephasediagram.In the(p, T) diagram of a liquid—gassystem(fig. 4) thereis a line of first-ordertransitionsterminatingat anisolated,critical point wherethe transitionis secondorder.The (H, T) diagramof an antiferromagneticsystem (fig. 5) showsa line of phasetransitionsthat changesfrom first to secondorder at an isolatedpoint, calleda tricritical point. A point of this kind is alsofound in 3He—4Hemixtures. Another typeoftricritical point is a critical point wherethree phasesmeet eachother. In the first casea completedescriptionof a tricritical point requiresthe addition of a third thermodynamicvariable[181.If that isdone, the two typesof tricritical pointscan be seen to be only different projectionsof one tricriticalpoint in a three-dimensionalphasediagram. As an example, fig. 6 showsa three-dimensionalphasediagram of an antiferromagneticsystem.The variablesare temperature(T), magneticfield (H) and

__ __ /P

Fig. 3. OrderparameterQ nearfirst- (a)andsecond-order(b) phase Fig. 4. Liquid—gas phasediagram with a first-order transition line,transitions. endingin a critical point (cp).

H _____________________

/ IFig. 5. Antiferromagneticphasediagram with first-order (solid line) Fig.6. Antiferromagneticphasediagramshowinga tricritical point P.and second-order(dashedline) transitionlines, meetingin atricritical T is temperature,H is magneticfield and H’ is staggeredmagneticpoint (tcp). field. Thecross—hatchedsurfacesaresurfacesof first-order transitions;

theline L is aline of first-ordertransitions;thebrokenlines L1, L2, L3

arelinesof second-ordertransitions(from ref. [181).

204 E.F. Gramsbergen et a!., Landau theory ofthe nematic—isotropic phase transition

staggeredmagneticfield (H’). The tricritical point is seenas oneof the first kind in the planeH’ = 0

(which is the sameas fig. 5), andas oneof the secondkind in a planeparallelto the H’-axis and at asuitableangle in the (H, T) plane.

Close to a critical point one observesthat physical propertiesvary accordingto powerlaws withnonintegerexponents.This meansthat thesequantitiesarenonanalyticat T~.They can be expandedina generalizedpowerseriesaroundT~:

f(t)~’~+f0t’°(1+f1t~’+...) fort>0,

~ for t<0,

with t = (T— T~)/T~,p >0,~a’>0.Theexponentsu, w’ whichcanbenoninteger,arethecriticalexponentsof f They can bedefinedas

w = lim log(f—f0) T> T~, w’ = urn log(f—f~) T< T~.t-*o~ log t r-.0 log~tj

By convention,—a, —a’ arethe specific heatexponents,/3’ is the order-parameterexponent(oneoftenomits the prime), — y, — y’ arethe susceptibilityexponents.

It is remarkablethat such behaviour,though singular, hassomevery simple characteristics.Thecritical exponentsdo not assumejustanyvalue.Different systems,undergoingthe mostvariedkinds oftransitions,can be assignedto a small numberof classes,eachspecifiedby a certainset of values of theexponents.Which set of valuesis assumeddependsonly on the most generalpropertiesof the system,like the dimensionalitiesof space, d, and of the order parameter,n. Moreover, when the scalinghypothesisis applied [16], oneobservesthat primed and unprimedexponentsare equivalentandthatvery simple relationsexist, like a + 2/3 + y = 2, called scaling laws. In most casesthey reducethenumberof independentcritical exponentsto two. The reasonfor this universalbehaviourlies in theincreasingrangeof spatialcorrelationsbetweenfluctuationsasthecriticalpoint isapproached(seechapter4). For staticprocesses,thisisexpressedin theincreaseof thecorrelationlength ~. If ~divergesat T = T~,then nearT~the fluctuationsare completely dominantand all the thermodynamicquantitiesmay beexpectedto dependonly on ~. Hencemanyparametersareeliminatedfrom the pictureanda universalbehaviourresults.

Critical exponentscan be calculatedwith the Landautheory (see [16]).It gives the samevaluesasobtainedby mean-fieldtheory. In such an approachfluctuationsplay no role. Thereforethe rangeofvalidity of this approximation,which can be estimatedwith the Ginzburgcriterion [19],is restricted.Itthenfollows [18,20]thatthemean-fieldcriticalexponentsarecorrectif thedimensionof thespaceexceedstheso-calledmarginaldimensiond*. Thevalueof d* is 5 for critical and3 for tricritical points.Hence,forthepracticalcased = 3, mean-fieldexponentsarewrong for critical, but correct for tricriticritical points.We shall comebackto this point later.

When fluctuationsare takeninto account,thermodynamicquantitiescan be calculatedas a powerseries expansionin the coupling parameters(see chapter4). Near a second-orderphasetransition,however,theseexpansionsdo not convergeand one hasto use the renormalizationgroup theory toextract the critical exponents.This theory, originating from the 1950s, was formulatedin its presentform by Wilson and Kogut [21]. The marginal dimensionarises in a natural way and the criticalexponentsare calculatedas a power series expansionin the parametersr = d* — d or 1/n. In thelimiting casee = 0, such as for the tricritical point in

3He—4Hemixtures,d* = d = 3 and the mean-field

E.F. Gramsbergenet at, Landau theory of the nematic—isotropic phase transition 205

behaviouris modified by smallcorrectionsthat behavelogarithmically with reducedtemperatureanddonot influencethe critical exponents[18,22].

A simplified model that is often used to calculatecritical exponentsis the Ising model [23]. It consistsof a lattice of spins that can havethe orientations“up” and “down” and interactonly with nearestneighbours.It servesasa model for systemsunderthe influenceof short-rangeforces. It canbe handledwith different methodsand approximations,ranging from simple mean-field to an exact solution ford = 2, renormalizationgroup methodsand computersimulation “experiments”. Although the modelwas originally intendedto describethe ferromagneticcritical point, its critical exponentsapply to manymore systems of the same spatial and order parameterdimensionalities,due to the universalityproperty.The possiblerelevanceof the Ising model for the nematic—isotropicphasetransitionwill bediscussedlater. Theoretical valuesof critical exponentsfor somecasesand models that are of someimportancein connectionwith the nematic—isotropicphasetransitionaregiven in table2.

Table 2

Critical exponentsfor sometheoreticalmodels

Mean-field

Critical (B� 0, C� 0),Quasi-tricritical Tricritical

Critical exponent (B � 0, C = 0) (B = 0, C = 0) Ising d = 3

}0.5 0.25 0.324±0.006

y, y’ 1 1 1.241 ±0.004

2. Landau theory of the nematic—isotropic phasetransition

In this chapterwe develop systematicallythe Landautheory as applied by De Gennesto thenematic—isotropicphasetransition. First in section 2.1 the ingredientsof the Landautheory will besummarized.Then in section2.2 it will be shownhow the generalruleswork out in theparticularcaseconsidered.An effort will bedoneto showtheoutcomeof themathematicalstructureof thetheorybasedon the orderparameterQ. As far as theseresults refer to a biaxial nematicphasetheir treatmentispostponedto section2.3. Forboth the uniaxial andthe biaxial casea comparisonwill be madewith theexperimentalsituation.A discussionof theeffectof externalfieldsandthetreatmentof fluctuationsisgivenin laterchapters.

2.1. Landautheory: ingredients

The Landautheory [4, 5] is concernedwith oneof thesimplestand mostelegantspeculationsaboutthe form of thermodynamicpotential near a critical point. In order to clarify Landau’s originalargumentswe start by consideringthe simple caseof a macroscopicsystemwhoseequilibrium stateischaracterizedby aspatially invariant,dimensionless,scalarorderparameter0. Thoughthis situationisnot directly relatedto the NI phasetransition,it allows us to demonstratethe main ingredientsof theLandauapproach.

206 E.F. Gramsbergen et at, Landau theory of the nematic—isotropicphase transition

Implicit in the Landautheory is the hypothesisthat for temperaturesin thevicinity of the transitiontemperaturethe equilibrium propertiesof the system can be calculated from a single function,dependingon the order parameter,which can be regardedas a generalizationof the free energytononequilibriumsituations. Originally this hasbeen applied to second-orderphasetransitions.Heregeneral considerationswill be given, including first-order phase transitions. Let us consider thethermodynamicvariablesof a systemfor given deviationsfrom the symmetricalstate (i.e. for a given0). According to the Landauhypothesiswe can representthe free energyF as a function of somethermodynamicvariables,for instanceof p, T and 0, wherep is the pressureand T the absolutetemperature*.The Landautheory of phasetransitionsnow startswith an expansionof the free energyin termsof theorderparameter0 [4, 5]. In generalthe expansionreads

F(p, T, 0)= F0(p, T)— hO+ AQ2+ B03+ CO4~ (2.1)

The equilibrium statecan be obtainedfrom the minimumof thefree energywith respectto 0 for fixedp and T

The existenceof an equilibrium statefor the nonequilibriumfree energyis assuredwhen thehighestorder in 0 is evenandthe associatedcoefficient is assumedto be positive. For the caseof eq. (2.1) itmeansthat C = C(p, T)>0 for all p and T. From generalthermodynamicargumentssomeconditionsfor the coefficient A can also be derived,which is the lowest-orderterm in 0 in the expressionforÔF/80.From the requirementthat eq. (2.1) describesa phasetransitionbetweenstateswith differentvalues of 0, and from the existenceof a spinodal for a first-order phasetransition, it follows that Amust changesign as a function of temperature.The critical point (in caseof a second-orderphasetransition) or the spinodal (for a first-order phasetransition) are then determinedby the equationA(p, T) = 0. Thisequationdefinesa line T

0 = T0(p), often alsodenotedas T~,T~Kor T~.If we considerT0 for a given valueof p, one canwrite nearT0:

A(p, T)= a(T— T0),

wherea = (aA/9T)TTQ.On theotherhand,becausethesignof B andC is normallynotassumedtochangewith temperature,thesecoefficientscan betakenasweakly temperature-dependentandareconsideredasconstantsin the neighbourhoodof T0.

Next, restrictionsare put on the coefficientsin the expansiondue to symmetry considerations.Forinstance,if the stateswith 0 = 0 and0 � 0 areof adifferent symmetry,in eq. (2.1) the term linear in 0must be identically zero: h = 0. It can be nonzero when an external symmetry-breakingfield isintroducedor whenthe orderparameteris a scalar.The presenceof the cubic term (B) is alsostrictlyrelated to the symmetry of the order parameter.If the order parameteris a scalar (like for theliquid—gas phasetransition) such a term is allowed. If the order parameteris a vector (like themagnetizationof a magneticsystem)a cubic term cannotbe a scalaranymore,andsuch a term is notallowed. For such a caseone musthaveB = 0 for all T andp.

Onecan easily check that for B = 0 a phasetransitionbetweenthe states 0 = 0 and 0 � 0 takesplace,that is secondorder if C> 0. One mechanismto obtain a first-order phasetransition is to have

* strictly speakingthenonequilibriumfree energywould be assumedto bea functionalof theone-particledistribution function,p and TThen,

onecouldexpand the one-particledistribution function in terms of the irreducible representationsof thesymmetrygroupof interest,and identifytheorderparameterswith scalarcombinationsof thecoefficientsof theexpansion.For theNI phasetransitionsuchaformal and morecomplicatedtreatmentis equivalentto thetreatmentdiscussedhere,wherewe identify therelevantorderparameterfrom thebeginning.

E.E Gramsbergen et a!., Landau theory of the nematic—isotropic phase transition 207

B = 0 and C< 0. In that casea stabilizingsixth-orderterm with coefficient E> 0 is required.Anotherpossibility is the presenceof a third-order term B03 in (2.1), provided it cannot be removed byredefiningthe orderparameter.If 0 is a scalaronecan write 0 = O~+ 0’ andchooseQ~such that thecubic term disappears.The thermodynamicpotential thenbecomessimilar to that of an Ising magnetinan external field, anda second-ordertransitionis possible in a single point. However,in the situationthat the cubicterm cannotbe removed(for examplewhenthe orderparameteris a tensor),a first-ordertransitionresultsfor nonzeroB.

Apart from the hypothesisof the existenceof F(p, T, 0), the additional assumptionhasbeenmadethat the free energy is an analytic function of 0, p and T There is, a priori, no reason that thisassumptionis true,as in the vicinity of a critical point correlationsbetweenfluctuationsof the orderparameterareof greatimportance.In fact it is to be expectedthat at least for a second-orderphasetransitionthe expansionof the free energycannotbe continuedto arbitrary high order, andthat theexpansioncoefficientshavesingularitiesas a function of p and T. A completeelucidationof the natureof thesesingularitiespresentsgreatdifficulties. The situationseemsto be somewhatbetterfor the caseof a first-orderphasetransition.In that casesingularitiesareexpectedwhen approachingthe spinodal.On the otherhandwe know that in the vicinity of the spinodalall statesof the systemareoutsidetheequilibrium regime. Then one might expect that such singularities are less important when weextrapolatethe expansionof the free energyto the regimewherethe first-orderphasetransition takesplace.In general,for both first- andsecond-orderphasetransitionsit is assumedthat the presenceofsingularitiesdoesnot affect the termsof the expansionsthat areused.

Finally, it should be mentionedexplicitly that the Landau theory doesnot contain any informationaboutmolecularinteractions.This is anotherlimitation which preventsto relatemacroscopicbehaviourto molecularproperties.The coefficientsentering the Landaufree energyare arbitrary andcan at bestbefitted so asto give theobservedphysicalbehaviour.Theoriesthat do not sufferfrom thesedifficultieswill be called moleculartheories.An early oneof the nematicphaseis dueto Maièr and Saupe[98].A recentreviewthat also includesinteractionsbetweenthe moleculesof biaxial symmetry is found inref. [52].

Despiteall thesedifficulties Landautheory hasbeensuccessfullyadaptedto a greatvariety of phasetransitions;it providesa unified descriptionof almost all critical phenomenaand especiallyrevealstherole of symmetry in phasetransitions.In addition, the original theory can begeneralizedsuch that it isalsorelevantfor otherproblemsof currentinterest.

The variouspostulatessummarizedin this sectionwill be illustratedin the following sectionsof thischapter.In that context somegeneralizationsandalsomore specific discussionsof limitations will alsobe presented.This subject has beendiscussedextensivelyat variousplaces[4, 5, 9, 16—18, 24, 25], towhich referenceswe refer for furtherdetails.

2.2. Thenematic—isotropicphasetransition

2.2.1. Landau—DeGennestheoryIn this section the generalrules of the Landautheory will be applied to the NI phasetransition.

Some of the general statementsabout the coefficientsof the expansionof the free energywill besupportedby a detailedanalysisof the minimum of the free energy.The free energybeing a scalar,the expansionof F in powers of Q can only contain terms that are invariant combinationsof theelementsQ,~of the orderparametertensor.Therefore,in order to makea Landauexpansionfor thenematicfree energywith respectto Q, we first haveto constructall absoluterotationalinvariants(ARI).

208 E.F. Gramsbergen et a!., Landau theory ofthe nematic—isotropic phase transition

An ARI is a polynomialbuilt up of tensorcomponentsthat doesnot changewhentensorcomponentsinan arbitrary frame of referencewill be replaced by analogouscomponentsin any other frame ofreference.The following theoremaboutARIs is valid: Any ARI built up from a set of tensorscan beobtainedby meansof multiplication of the tensors,addition and multiplication by numbersand fullcontraction(with respectto all indices).Applying this theoremto our casewe find that the mostgeneralinvariantbuilt up from Q hasthe form

~ [ fl Tr(Q~l)], fl,~= 1, 2,.. ., m = 1, 2

m n~n,,, i=Im

As we seefrom this formula, the basicinvariantsof Q are

Tr(Q’), n = 1, 2, 3

Now we shall use the property that for any symmetric3 x 3 matrix Q, Tr(Q”) can be expressedas apolynomial in Tr(Q), Tr(Q2) and Tr(Q3). As this property is an important ingredientof the theory wegive the proof explicitly in appendixA. Togetherwith the tracelesscondition this leadsto a Landauexpansionin termsof only two ARIs:

Tr(Q2), Tr(Q3).

The reduction of the numberof degreesof freedom to two is not surprising since threeof the fivedegreesof freedomin Q~are usedto fix the axesof the liquid crystal in space,whereasthefree energyis invariant with respectto rotation of thisset of axes. It is also clear from the generalparametrizationgiven in eq. (1.7). The free energythus reads

F= F0 + ~ATr(Q

2) + ~BTr(Q3) +

+ ~D[Tr(Q2)][Tr(Q3)]+ ~E[Tr(Q2)]3+ E’[Tr(Q3)]2 +.“. (2.2)

wherethe numericalcoefficientsareintroducedfor future convenienceand F0 is the free energyfor a

given temperatureandpressureof the statewith Q = 0. In principle gradienttermsandtermsrelatedtopossibleexternalfields haveto be added.Thiswill be donein chapters3 and4, respectively.

Various propertiesof the expansion(2.2) can be obtainedfrom the generalconsiderationsgiven inthe previoussection.In particularwe note thefollowing points:

(1) There is no term linear in Q. This allows for the possibilityof an isotropic phase.In the caseofexternalfields, a linear term hasto be included(seechapter3), making theisotropicphaseimpossible.

(2) Odd terms of order three and higher are allowed. As we shall see, this causesthe NI phasetransitionto befirst order.This can be contrastedto the caseof a vectororderparameterdiscussedinsection2.1. Then odd termsarenot allowedandthe transitioncan be either first or secondorder.

(3) There are two independentsixth-order terms. As we shall see, the presenceof the E’ termintroducesthe possibility of a biaxial nematicphase.

(4) The NI phasetransition takesplacein the neighbourhoodof A = 0. Thereforewe assumethatthe temperaturedependenceof the free energyof the systemis containedin thecoefficient A alone.Todescribethe phasetransitionwe dealwith asmall temperaturerangeand linearizeto (seesection2.1):

A=a(T_T*). (2.3)

EF. Gramsbergen et a!., Landau theory of the nematic—isotropic phase transition 209

Herea is apositiveconstantand T* is atemperaturecloseto thetransitiontemperatureTNI. The othercoefficientsare assumedto be constants.However, in section2.3 wewill alsoconsidervariationsof B.

The minimumof the Landaufree energy.In calculating the minimumof the free energy,we use theorderparameterin diagonalizedform andparametrizedaccordingto (1.6):*

0

Q=( 0 —~(x—y)0 J. (2.4)

\ 0 0 xl

For uniaxial orderingalongthe z-axis, y= 0; alongthe x- andy-axes,y = 3x andy = —3x, respectively.The invariantsof Q are

Tr(Q2) = ~(3x2+ y2), (2.5a)

Tr(Q3) = ~x(x2— y2). (2.5b)

A useful relation betweenTr(Q2) andTr(Q3) is

(Tr Q3)2� ~(Tr Q2)3, (2.5c)

the equalitysign holding in the uniaxial caseonly. As a consequence,the minimumfree energygives auniaxial phaseas long as no termshigher than linear in Tr(Q3) arepresent.

Since for the momentwe areinterestedonly in the uniaxial phasewe omit the E’ termin eq. (2.2).Choosingthe z-axisalongthe director,y = 0 andthe free energyis (normalizedsuch that F

0 = 0):

~ (2.6)

We considertwo cases.In thesimplestmodelof the NI transitionD = E = 0.For the minimumto beat finite x= x0, C must be positive. As follows from the calculationsto be given, the sign of B isoppositeto the sign of x0. So for a conventionalnematicof rod-like molecules,B<0 andfor a discoticnematicB>0. Figure 7 showsthe free energyfor B<0, for different temperatures.Four temperatureregionscan be distinguished:

(1) T> Tt: the minimumcorrespondsto an isotropicphase;(2) TNI< T< Tt: the minimum correspondsto an isotropic phase. In addition, there is a local

minimumcorrespondingto a possiblesuperheatednematicstate;(3) T* < T< TNI: the minimum correspondsto a nematic phase.There is a local minimum

correspondingto apossiblesupercooledisotropicstate.(4) T< T*: the minimumcorrespondsto anematicphase.Below T* the isotropic phaseis completelyunstablewith respectto the nematic state; the local

minimum at Q = 0 doesnot exist anymore.Above Tt the nematicphaseis completely unstablewith

* An elegant form that is sometimesusedis Q = r cos(8+ 2i,13), Q~,= r cos(O— 2ir/3), Q,~,= r cos0, relatedto x and y by x = r cos0,

y = \/3r sin 0. Uniaxial casesaregiven by 9 = n,r/3 with n even(odd)for U~(U—). Theinvanantsof Q areTr(02)= ~r2, Tr(Q3)= ~ cos(39).In this

formulationeq. (2.5c) is immediatelyevident.

210 E.F. Gramsbergen et a!., Landau theory of the nematic—isotropic phase transition

1.5 T~TNI ~

j / ::~~ 0.0 . fU- S

S -I

-0.5

0.2

-1.0

-1.50.0 I I —

I I I -0.5 0.0 0.5 1.0-0.4 0.0 0.4 0.8 (T~T~)/(T T*)

S NF

Fig. 7. Thefree energyF asa function of theorderparameterS, for Fig. 8. Thenematicorderparametervs. temperature,accordingto eq.thespecialtemperaturesT~,TNt and T

t, for Spa= 0.4. (2.9). The dashedline representsthesupercoolednematicstate.

respect to the isotropic phase; the local minimum at Q � 0 does not exist anymore (see fig. 7).

Consequentlyboth temperaturesT* and T~belongto the spinodal. TNT and Tt aregiven by

TNT = T* + ~B2/aC, (2.7)

Tt = T* + ~B2/aC. (2.8)

In order to makeconnectionwith experimentsit is convenientto expresstheminimumvaluex0 in terms

of S (section1.2.2). In the nematicphase(fig. 8):

3a(Tt_ T) 1/2 8(T— T*) 1/2San~xo=St+( 2C ) SNI{1+[1_9(TT*)] }. (2.9)

TtandS~arefoundfrom thetwoequations8F/öx= 0 anda2F/3x2= 0, wherethesecondonefollows from

the requirementthat [(8S/öT)TTt]~ = 0 (seefigs. 7 and8). At TNT and T~:

SNI= —~B/C, (2.lOa)

ct__A D/c’_~C ‘210b

— ‘- — 4’-’NI, ~.

andthe transitionentropyis

= ~aB2/C2. (2.lOc)

There is anotherformulation of the NI phasetransitionwhich is closely related to the observation

E.F. Gramsbergen et a!., Landau theory of the netnatic—isotropic phase transition 211

that the constantsB and C are found to be very small [38c].Of these,C hasthe samesymmetry as Awhich changessign anyhow.In general,whentwo coefficientsof the samesymmetryin the Landaufreeenergyvanish simultaneously,such a point is called tricritical [14, 26]. Therefore the experimentalsituationhas led to the questionwhetherthe NI transition is close to the tricritical point A = C= 0.

This leads to an alternative formulation of the NI phasetransition by taking C = 0. In that caseapositive, stabilizingE term mustbe addedto the free energy:

F=~Ax2+~Bx3+~Ex6,E>0. (2.11)

In this case,

B4 1/3

TNI = T* + (1/4a)(—) , (2.12)

3B4 1/3

Tt = T* + (1/4a) (—) , (2.13)

SN!= (—3B/8E)~3, (2.14a)

St = (—3B/16E)”3 = (~Y’3SNI, (2.14b)

andthe equationto determinethe equilibrium valueof S is

(S- St)4+ 4St(S- St)3+ 6(St)2(S- St)2- (Tt- T)=0. (2.15)

As mentioned in section 1.3 various thermodynamicquantities around a critical point can becharacterizedby meansof critical exponents(seetable 2). The Landautheory gives universalvaluesfor theseexponents.This can be extendedto the Landaudescriptionof a first-orderphasetransitionasgiven above.Then the critical exponentscan be definedaroundthe spinodal, i.e., aroundT* and Tt.For instance,the temperaturedependenceof S (eq. (2.9))can be expressedas

S—St---(Tt—T)ts (2.16)

with /3 = ~. For B = 0, the transition is a (classical)critical point; the value /3 = ~is thereforecalled the(classical)critical value of /3. For the tricritical case(C = 0, eq. (2.15)) the tricritical exponent/3, forB = 0 (or equivalentlyS~= 0) is foundto be /3 = ~. For B� 0 the exponent/3 still hasthe value/3 =

i.e., the classicalcritical value. As in practiceB is very smallonecan expectthat for the caseC= 0 thelatter situationis only found in a very small temperaturerangeclose to Tt, that is not accessibleexperimentally.Thenquasi-tricriticalbehaviourwith /3 = can be observed.Valuesfor theothercriticalexponentscan be found by standardmethods,as presented,for example,in Stanley’sbook [16]. Inprinciple it is not clearhow muchof the theory of critical pointscan beusedto understandthe approachto a spinodal. From the Landautheory one can conclude,however,that this behaviouris exactly thesameas for a critical point. This is importantin connectionwith theargumentin section 1.3 that, going

212 E.F Gramsbergenet a!., Landau theory ofthe nematic—isotropic phase transition

beyondmean-fieldtheory, tricritical mean-fieldexponentsare still expectedto hold for three-dimen-sional systems,whereasclassicalmean-fieldexponentsareno longer correct.

2.2.2. Comparisonwith experimentand discussionIn spiteof the limitations mentioned,the Landau—DeGennestheory of the NI phasetransitioncan

be consideredas quite successful.In agreementwith the experimentalsituationit predictsthat the NIphasetransitionis alwaysdefinitely first order.This is dueto the cubicterm in theexpansion(2.2),whichleadsto a jump in the transitionentropyor a latent heat proportionalto B2 (eq. (2.lOc)). As it is aphenomenologicaltheory, it containsseveral phenomenologicalparameters.These can be fitted tovarious experimentaldata, thus providinga semi-quantitativedescriptionof the order parameter,thespecific heat,entropy,etc., in thevicinity of the phasetransition.As an examplein table 3 the valuesofthe coefficientsin the expansionaregiven for N-(p-methoxybenzylidene)p’-butylanilineor MBBA. Aswe see the parametersB and C haveboth very small values,andthe presenceof a sixth-ordertermseemsto be important.It is not necessaryto takea term —~Q5 into account,as it can be shown that inthe presenceof the cubic term it leadsto only minor corrections.

Table 3Expansionparametersof eq. (2.lOc)calculatedfrom variousexperimen-

tal dataof MBBA [38cJ,in units of RTNI (TNt= 318.2K, D = 0)

Restrictionon fit aT* B C E

E = 0 0.933 —0.27 0.225 0B= 0 0.933 0 —0.135 0.8775C= 0 0.933 —0.045 0 0.3713C<0 0.933 —0.0045 —0.1575 1.3163C>0 0.933 —0.405 0.405 0.6075

The very small value of B closely approximatesthe vanishingchangesof a second-orderphasetransition (B = 0). One might then expect that many physical propertiesof the isotropic phasewilldisplay critical-like behaviour(or pre-transitionalbehaviour)on cooling to temperaturesclose to T*.Similar effectscan be foundon heatingthe nematicphaseandapproachingTt. Theseeffectsare indeedobserved;a significant increaseof fluctuationphenomenais observedin the static responsefunctions,light scattering,specificheat,andvelocity andattenuationof ultrasound[20,27,28,38c]. But thesituationis still not very clearin the sensethat experimentalestimatesindicatethat theNI phasetransitionshowscharacteristicsof tricritical-like behaviour(B = C = 0). On theotherhand,thefirst-ordernatureof theNIphasetransitionmanifestsitself in asignificant jump of theorderparameterS (eq. (2.lOa)),which hasavalueof about0.3—0.4at the nematicside of the transition[20,24, 29,38c].

Though the singular-likebehaviourof variousquantitiesat the NI phasetransitionis still a puzzle,the solutioncould well be obtainedwithin the Landau—DeGennestheory,andthe predictionsof thistheory should be comparedwith experimentaldata. Owing to the first-order natureof the NI phasetransition, the part of the critical region, closestto the critical point is not accessibleto experiment.Thereforethe critical exponents,beingdefinedin this very region,cannotbe determinedwithout theuse of someextrapolation.This can causerelatively largeerrorsin their numericalvalues.For example,whereasPoggi et al. [30] measuredfor the order parameterexponent/3 = ~,Keyes [31] showedthatthesedata could almost equallywell be fitted with the tricritical value /3 = ~. Variousothermeasure-ments [32—36]show a tendencytowards/3 = ~,althoughthis valueis often somewhatunsureand the

E.F Grwnsbergenet a!., Landau theory of the nematic—isotropic phase transition 213

Ising value/3 = 0.312cannotbe excluded.As thesymmetry of thefree energy(2.6) for B D = E = 0 isthe sameas that of the Ising model,this predictionof /3 might berelevant.However,the classicalvalue/3 = ~ is definitely outsidethe rangeof experimentalerror. Recently,Thoen and Menu [37] reportedhigh-resolutionmeasurements(better than 0.01%) of the static relative permittivity of 8CB. With atemperaturestability betterthan 0.01K this allowedfor a detailedanalysisof the dielectricanisotropyin the nematic phase.For the exponent /3 a best value /3 = 0.247±0.01 was obtained.This valuestrongly supportsthe suggestionof tricritical characterof the NI phase transition for 8CB. Thereis obviouslyneedfor morehigh-qualitydataon /3 for variouscompoundsand usingalsootherresponsefunctionsthanthe dielectricpermittivity.

The remainingdoubtdrew the attentionto othercritical exponents,especiallythoseof the specificheat, a and a’. The susceptibilityexponenty = 1 in both (critical and tricritical) cases.Theoreticalvalues of critical exponentsarelisted in table 2. Concerningthe specific heat, Anisimov et al. [26,38]made a strong plea for the tricritical hypothesis,fitting their data of very precise specific heatmeasurementsof MBBA andothercompoundswith a crossoverfunction of the form

C~/R— [a1(T—T*)o.s+ a2(T— T*)ou]~+ b + c(T—T*) (2.17)

for T> T~andan analogousonefor T< T~.This accountsfor the tricritical behaviourdisturbedby anIsing contribution closeto TN!. Thoen et al. [39,40] usingvery precisespecific heatmeasurementsof8CB, arguedthat thesecould equally well be fitted with tricritical and Ising critical exponents.Asimilarly largeuncertaintyin the a values for 8OCB was reportedby Kastinget al. [41].

Adding up all experimentalresults,thereis still no unambiguousproof of what kind of singularitycan be expected near the NI transition. Values of /3 favour tricritical behaviour. If we assumecritical-like behaviour one does not expect the results to be valid in the neighbourhoodof TNT.

However,that regionis experimentallynot accessibleandthe /3 valuesremainto beexplained.We shallreturn to theseproblemsin chapters3 and4.

2.3. Thebiaxial nematicphase

2.3.1. TheoryThe orderparameterQ introducedvia the anisotropicpart of a responsefunction hasin principle

two degreesof freedom.This propertyis well expressedby meansof the parametrizationgiven in eqs.(1.7) and (2.4). From this form it is evident that the generalform of Q allows for a biaxial phase.Inorder to constructwithin the frameworkof the Landau—DeGennestheory the simplestmodel in whichboth uniaxial andbiaxial phasesarepossible,onemust retain the following termsin (2.2):—The A term to accountfor the temperaturedependenceA = a(T— T*).— The B term to accountfor the asymmetrywith respectto Q *-* —Q.— The C term to accountfor the possibilityof classicalcritical behaviournearthe NI transition.—The E’ term for the possibilityof a biaxial phase.

The free energynow reads

F = ~ATr Q2+ 4B Tr Q3+ ~C(TrQ2)2+ E’(Tr Q3)2. (2.18)

For stability we require C> 0 and E’ >0. Theseconditionsare both necessarydue to the inequality(2.5c). Insteadof consideringonly A (temperature)as the controllablevariable,we will now construct

214 E.F~Gramsbergen et a!., Landau theory ofthe nematic—isotropic phase transition

phasediagramsas a function of A and B. The physical meaningof B will be discussedin the nextsection.

For minimizing F, different setsof variablescan be used:(a) TrQ2 andTrQ3 as quasi-independentvariableswith (2.5c)as a constraint;(b) r and 0, as mentionedin the footnoteunder(2.4);(c) x and y, definedin (1.6).

Although for the presentcaseit is a somewhatclumsychoice,we shall usex andy. The advantageisthat in thesevariablesthe free energyis mosteasilygeneralizedwhen externalfields areadded(chapter3). Now eq. (2.2) reads

F(x, y) = a(x)+ y2/3(x)+ y4y(x), (2.19)

with

a(x) = ~Ax2+ ~Bx3+ ~Cx4+ ~E’x6, (2.20a)

/3(x)= ~A —~Bx+~Cx2—~E’x4, (2.20b)

y(x) = ~C+~E’x2. (2.20c)

Minimization with respectto y gives

y= 0 (uniaxial), (2.21a)

y2 = —/3(x)/y(x) (biaxial), (2.21b)

and uniaxial F~(x)= a(x), biaxial FB(x)= a(x)— /32(x)/2y(x). Becausealways y(x)>0, the biaxialsolution (2.21b) is only allowed when /3(x)<0. Furthermore,wheneverthe biaxial solution is allowed,FB(x)< F~(x),so the problemreducesto finding the minimum of the function

F(x) = Fu(x)= a(x) when /3(x)� 0 (uniaxial),

F(x)= FB(x) a(x)—/.32(x)/2y(x) when/3(x)<0 (biaxial).

At the point /3(x)= 0, F(x)and F’(x) arecontinuousand (seefig. 9)

F~(x)>F~(x).

Now concentratingon the phasetransition,we can formulatethe following conditions.

For a second-orderuniaxial—biaxialphasetransitionat x= x0 (fig. 10):

/3(xo)= 0, (2.23a)

a”(x0) = 0, (2.23b)

(a — /32/2)/)] � 0. (2.23c)dx

E.F. Gramsbergen et a!., Landau theory ofthe nematic—isotropic phase transition 215

F

Fig. 9. Schematicdrawing of the free energyvs. order parameterx Fig. 10. As fig. 9, at asecond-orderNUNB phasetransition.accordingto eq. (2.22),nearthepoint x = x

0 where~(x) changessign.

For a first-order uniaxial—biaxialphasetransition,the limit of metastabilityfor the uniaxial phaseatx= x0 is given by the sameequationsexceptthat the inequality sign in (2.23c) is reversed(fig. 11).

For the first-orderuttiaxial—isotropictransition,with x= x0 on the nematicside (fig. 7):

(2.24a)

(2.24b)

a”(xo)>0. (2.24c)

The aboveconsiderationsresultin the phasediagramof fig. 12a.Apart from the usualfirst-orderNItransition (solid line), thereare two N~N8transitionswhich are both of secondorder (broken lines).The biaxialphaseis sandwichedbetweenpositiveandnegativeuniaxial phases.All four phasesmeet inabicritical point,calledthe Landaupoint [5Jandlocatedat A = B = 0. In thispoint, the NI transitionissecondorder.

Thelines of phasetransitionsaregiven by:(a) Isotropic—nematic:

A = ~Cx2+ ~E’x4, (2.25a)

B = ~Cx+ 18E’x3. (2.25b)

F

xx 0

Fig. 11. As fig. 10, at thevanishingof the local minimum of theNu phaseneara first-orderN0N5 phasetransition.

216 E.F Gramsbergenet a!., Landau theory of the nematic—isotropic phase transition

2.4 2.1.

a / b

Nu /1.2 1.2 Nu

0.0 NB — ~ L I o.c - L I

B B

-1.2 . Nu~ -1.2 Nu+

-2.4 —2.4 I I-0.4 -0.2 0.0 0.2 —0.4 -0.2 0.0 0.2

A AFig. 12. Phasediagrams,following from thefree energy(2.2), with C = 2.67,D = E= 0. Solid lines representphasetransitionsof first order,dashedlines secondorder.A is ameasurefor thetemperature,B is thedegreeof flatnessof themolecules.(a)Phasediagramwith abiaxialnematicphase(E’ = 3.56). (b) Phasediagram withoutbiaxial nematicphase(E’ = 0).

(b) Uniaxial—biaxialnematic:

A=—~Cx2, (2.26a)

B=—~E’x3, (2.26b)

indicating that the width of the biaxialregion is proportionalto (T* — T)312~In fig. 12b, the analogousdtagram1S shownwithout the sixth-orderterm,which was essentiallycalculatedin section2.2. Herethe

0.4 N8

~O.~0Id0~0I8

Fig. 13. Orderparameters0,.,, 0,,,, Q~vs. B, for thephasediagramof fig. 12a,A = —0.3.

E.F. Gramsbergen et a!., Landau theory of the nematic—isotropic phase transition 217

width of the biaxialregion hasshrunkto zero,leavingafirst-orderN~JNjJtransition.Evenwhenthe freeenergyexpansionis carriedout to higher orders,thetopology of thephasediagramsin the vicinity of theLandaupoint remainsthat of fig. 12a or fig. 12b (seeappendixB). Only the introduction of morecomplicatedorderparameterscanleadto othertopologies[104].TheorderparametersO~(i = x, y, z)areshownin a crosssectionof the first phasediagramat constantA <0 in fig. 13.

Going beyondmean-fieldtheory,severalauthors[42—44]calculatedcritical exponentsfor the Landaupoint by meansof renormalizationgroupmethods[21].Unfortunatelysomeof the resultingexpressionsdo not seemto convergevery well for three-dimensionalsystems.Around the NUNB transitionsthe freeenergyexpansionhasthe symmetryof theXYmodel.Thereforethe critical exponentsareexpectedtobe thoseof the three-dimensionalXYmodel (interactingspinsthat can freely rotate in the XYplane)[45—47].

2.3.2. Comparisonwith experimentand discussionThere are in principle threepossibilities that may lead to the spontaneousformation of a one-

componentbiaxial nematicphase:(1) A molecularsymmetry that is not (effectively) uniaxial.(2) Strong correlation of molecules leading to aggregatesof molecules that have no uniaxial

symmetry.(3) Applicationof an externalfield.Herewe restrict thediscussionto the first two possibilities,leavingthe effectof externalfields to the

nextchapter.Thenwe mustconcludethat in thermotropicsystemsa biaxial nematicphasehasbeenthesubjectof several molecular-statisticaltheoreticalstudies[48, 49], but never has beenfound experi-mentally so far. A biaxial nematic phaseconnectedwith the secondpossibility hasbeenobserved,however, in lyotropic systemsof amphiphilic (soap-like)solutions in water. The moleculesin thesesolutionstend to clusterinto aggregatesso that the hydrophilic groupsoccupythe surfaceto optimizecontactwith water,while the lipophilic tails occupythe innerpart of the aggregates[50].Theseso-calledmicellescanhaveasphericalshape,andgrowwith increasingconcentrationto becomeeitherrod-like orplate-like [51,52]. The micellesbehavesomewhatlike large-scalemoleculesin that they tend to orderinto smecticor nematicstructuresunderasuitablechoiceof temperature,concentrationandsometimesadditional solvents.The nematic phasesthusformed can be positive or negative uniaxial [53—55]andeven biaxial [10, 56, 57], dependingon the shapeof the micelles. Becausethe shapechangeswithconcentration,complete phasediagramsinvolving several nematic phasesare observedwith tem-peratureand concentrationas variables. In this way, both phasediagramsof section 2.3 havebeenfound [10,53, 56, 57] (fig. 14). In both cases,the orderof the phasetransitionsandthe topology of theLandaupoint areas predictedby theLandautheory.Also, the orderparametersare in good agreementwith the Landautheory [57, 58]. In thesefits, the fifth-order term in the free energyis retainedand AandB areboth linear functionsof temperature.

Returningto one-componentthermotropicsystems,onemaywonderwhy biaxialnematicphasesarenot encounteredhere; especiallybecausetheoreticalstudies[48, 59] indicate that for systemswithmoleculeswithout axial symmetry,abiaxial nematicstateat lower temperatureis necessaryratherthanjust possible.The practical difficulty is, however, that smectic and crystallinephasesmay interferebefore this temperatureis reached.Hence, to bring the N8 phase into reach, the uniaxial nematictemperaturerangeshouldbereduced,possiblyby lowering the valueof IBI. Precursorsof theNB phasemay thus be found in the decreaseof discontinuitiesat the NI phasetransition [11,52]. Interestingin

218 E.F. Gramsbergen et a!., Landau theory of the nematic—isotropic phase transition

020 CONCENTRATION, VT I

68. 2 68.0 67.8 67.6 67. 450 -~- I I I I 4 4 4

ISOTROPIC

+ - — --—+ -‘ L

— — —

~ 30 ( NB Nu+

~

20+-.-.-----------+—

10 ~ +

ISOTROPIC

0 I 4 6 I 1 I I

25.6 25.8 26.0 26.2 26.4

XL CONCENTRATION, VT Ir AT 6.24 WI I OF 1—DECANOL 3

Fig. 14. Phasediagram of thepotassiumlaurate/1-decanol/D2Osystem(reproducedfrom ref. [10]).L is theapproximatelocationof the Landau

point.

thiscontext is the observationthat5N1 (the orderparameterat the NI transition)hasbeenobservedto

decreasewith increasingpressure[601.The same effect is found for the density discontinuity [61].Extrapolationresultsin a critical pressureof 2500atm. At the theoreticallevel the pressure(or density)dependenceof the Landaupoint has beenstressedby Alben [62]andShih [63].In practice,however,itis againpossible that extremepressureswill induceclosely packedsmecticand crystallinephases[64]ratherthan a biaxial nematicstate.

To circumventtheseproblemsone might consideranothersystemin which a biaxial phasecould beexpected:a mixture of rod-like andplate-likemoleculesof comparablesize andin comparableamounts[65,66]. This idea, launchedby Alben [66], hasthe advantagethat thereis no need to go to lowtemperaturesor highpressures:thusthereis no risk of unwantedcrystallizationorsmecticordering.Oneproblem is that the known nematogenicdisc-likemoleculesaremuch largerthanthe rod-like molecules.Consequently,only small amountsof discs can be solved in rod solvents (and vice versa). In earlyexperiments,trendsin thesemixtures [67] indicatedthe approachof the Landaupoint as predictedbyAlben. Morerecentlytheseresultshavebeenquestioned,andevidencehasbeenfound that themixtureundergoesa transition to two coexisting uniaxial phasesratherthan a single biaxial phase[68].

E.F. Gramsbergen et a!., Landau theory ofthe nematic—isotropic phase transition 219

3. Landau theory of the nematic—isotropic phasetransition: the influence of external fields

3.1. Introduction

In thischapterthephysicalconsequencesof the Landau—DeGennestheory will beinvestigatedwhenan externalfield is included. This is relevant becausemany of the applicationsof liquid crystals arerelatedto their ability to respondstrongly to such externalstimuli. Supposethat a staticmagneticfieldH is applied to the system.As nematicliquid crystalsarediamagneticthepresenceof the field leadstoan extraterm in the free energyof eq. (2.2) of the form [12,24]

Fm = ~ (3.1)

ExpressingXa$ in the orderparameterelementsvia eqs. (1.4) and (1.5) thiscan bewritten as

I;’ — 1—z..121A LI

m — — 2X~A — 2 ~ XmaxT1=1 1$ ‘..~a$

wherej = ~ The first term in eq. (3.2) will be omittedasit is independentof the molecularordering.In a first intuitive picture, the molecules are assumedto be elongated.When ~Xmaxis positive

(negative),theythentend to align parallel(perpendicular)to the field. Thus,whena field is addedto anN~phasewith positive~Xmax,thedirectorwill be parallelto the field andthe effect is a slight increaseof the uniaxial ordering(fig. 15). When~Xrnaxis negative,however,the directorwill be perpendiculartothe field. Thefield direction thenintroducesasecondaxisandthe phasebecomesbiaxial. In table4thefield effect on all possible phasesis summarized.In the following sections,thesepredictionswill bemadequantitative.

a) b) C)

o~®~~’--

Fig. 15. Probabilitydistributionsof the long molecularaxis.Upper: in thex, z plane.Lower: in thex, y plane.(a)in zerofield; (b) in nonzerofieldwith &t’m,., >0; (c) in nonzerofield with t~x

m~.<0. Thefield effect hasbeenexaggeratedfor clarity.

220 E.F. Gramsbergen et aL, Landau theory of the nematic—isotropic phase transition

Table 4The breakingof symmetryby a magneticfield

Phasein nonzerofield

Phasein zerofield &t’ma, >0 ~Xm,x <0

I N~ N~,N~ N~ NB

N~ NB N~3

The alignment effect of the director parallelor perpendicularto the field is dueto the presenceof avolume anisotropyin the susceptibility.This can be contrastedwith the situation in an isotropicliquid,whereany field-induced alignmentis due to the anisotropyof individual molecules,as no cooperativeeffect is present.This leadsat most to a weak induced birefringenceif an externalfield is applied(Cotton—Moutoneffect in the magneticcase,Kerr effect for an electric field). Comparingnowthe twosituationsonecan expectpre-transitionaleffectsdueto thefact thatthe NI transitionis only weakly firstorder.This can beobservedas a strongincreasein the field-inducedbirefringenceif TN! is approachedfrom theisotropicside.Theseeffectswill be discussedin section3.2.2,following a discussionof the fieldeffect on the NI transition itself in section 3.2.1. Emphasiswill be on materials with a positivediamagneticor dielectricanisotropy,as for that situationmostexperimentaldataareavailable.

The contributionto thefree energydueto an electric field hasthe sameform aseq. (3.2). Only k andl~Xrnaxhave to be replaced by the averagepermittivity and the maximum permittivity anisotropy,respectively[24].The theory with inclusion of an electric field hasthesamestructureasthe casewith amagneticfield. Experimentally,the use of an electric field to study field effects around the NI phasetransitionis somewhatlessattractivebecauseof the complicationsdueto spacechargeandflexoelectriceffects. It has neverthelessthe advantagethat due to possible permanentdipole momentslarger(absolute)anisotropiescan be obtained.Taking typical anisotropies:

~Xrnax~= 10~(CGS), (3.3a)

IL~EmaxI= 6, (3.3b)

one finds that a magneticfield of 15 T has the same effect as an electric field of about iO~V/cm.Experimentally,it is impossibleto reachcontinuouslymuchhigher magneticfields, while electric fieldsof 105_106V/cm arefeasible.

3.2. Field effectsfor z~Xma,.>0

3.2.1. Theoryofthe nematic—isotropicphasetransitionThe Landau—DeGennesfree energy in presenceof an external magnetic field is obtained by

combining eqs. (2.2) and(3.1). When the free energyis cut off at fourth order in Q~onearrivesat

F = 4 XrnaxH,.H,3Q,.,3 + ~A Tr Q2 + ~BTr Q3 + ~C(Tr Q2)2. (3.4)

With L~Xrnax>0 the phaseremainsuniaxial. With H and n alongthe z-axisand the orderparametersx,y definedby (2.4) onehasy = 0 and

E.F. Gramsbergen a a!., Landau theory ofthe nemasic—isotropic phase transition 221

F=—hx+~Ax3+~Bx3+~Cx4, (3.5)

where

h ~.~Xmax1Tt2. (3.6)

In termsof S(=~x),minimizationof F gives

a(T_T*)=h/S_~BS_~CS2. (3.7)

Hence,when the field is switchedon,the samevalueof S belongsto a highertemperature.BecausetheS(T)curve hasa negativeslope,this meansthat the order of theN~phaseincreases,as discussedinsection3.1. It is alsoseenin (3.7) that S = 0 is nevera solutionfor h � 0. Insteadof an isotropicphasethereis a small inducedN~ordering.To distinguishbetweenthe usualN~phasewith muchlargerorderparameter,this is often called the paranematicphase.

The completesolutionof eq. (3.7) for S(T)is shownin fig. 16 for a numberof valuesof the field. Aswesee,for small valuesof the field thereis a first-orderphasetransitionbetweenthe paranematicandthe nematicphase.The valueof S in the paranematicphaseis smallandthereforecan beobtainedfromeq. (3.7) while disregardingthe termswith B and C:

S(h)= h/[a(T— T*)]. (3.8)

Thejump of the orderparameterat the NI phasetransitionis directly relatedto thevalueof thefield. It

0.0 112 0.1. 0.6 0.8 1.0 t2I . cp

00.50 l.( 1.50 2.00

(T_T*)/(TNI_T*)

Fig. 16. OrderparameterS/SC,vs. temperatureT— T* for different valuesof thefield variable h/h,.,. The dashedline is theNI coexistencecurve,cpis thecritical point.

222 E.F. Gramsbergen eta!., Landau theory ofthe nematic—isosropic phase transition

decreaseswith increasingfield until thecritical value ~ of the field is reachedwherethereis no jumpanymore.At thispoint the phasetransitionbecomessecondorder.For fieldshigher than ~ thereis nophasetransition and the nematic and paranematicphasesare indistinguishable.The location of thecritical point is given by*

F’(x) = F”(x) = F”(x) = 0, (3.9)

or equivalently

h~0= 364C2= ~rC(S~i)3, (3.lOa)

T,.~—T~J=54

1C=~(T~I_T*), (3.lOb)

S~=~ (3.lOc)

Herethe superscript0 refersto zerofield. At the phasetransitiontherearetwo minima x1, x2 of equal

energy,so:

F(x1) = F(x2), (3.lla)

F’(x1) = F’(x2) = 0, (3.llb)

which is solvedwith (3.5) to givewith T = (27aC/B2)(T—T*):

x12= x~~[1±V’3—2r], (3.12a)

= 1 + ~ (3.12b)

Hence,the coexistencecurve of the nematic and the paranematicphase(broken line in fig. 16) isparabolic.Theshift in the phasetransitiontemperatureis proportionalto h:

TNI(h)— TNJ(0)= 2h/(aS~1). (3.12c)

With the substitutionT—*p, S—~V, H—* T, the structureof thecritical point is identicalto that of theliquid—gassystem.In both cases,thephasetransitioncanendin anisolatedpointbecausethephasesonbothsidesof thetransitionhavethe samesymmetry.The similarity betweenthe two systemsis furtherillustratedby the h, T diagram(fig. 17, h > 0).

3.2.2. Comparisonwith experimentThereare threegeneralpredictionsof the theory presentedin the previoussection that could be

checkedexperimentally:* The equationsF”(x) = F”(x) = 0 are equivalentto theequations

[8.~öTlr,.,,s..,,,.,[82S/êT2}~J-Ll-,.,.h_h,.,=0 that aresatisfiedat T,.~,h,., (seefig. 16).

E.F. Gramsbergeneta!., Landautheory of the nemasic—isotropic phase transition 223

2

NCp

0- I

-2

U -

N

4

C p

-6 -

0.5 1.5 2.5

(T-T ) / I - TI

Fig. 17. (T, h)phasediagramof a conventional(N/i in zerofield) nematic.Thicksolid linesarefirst-order transitions,dashedlines second-order;cpandtcp arecritical andtricritical points.

(i) The existenceof a paramagneticphasewith field-inducedorientationalorder.(ii) the increaseof TN! with increasingfield.(iii) Theexistenceof a magnetic(electric)critical point.Experimentallythe answerto thefirst point is veryclear. Applicationof a field to the isotropicphase

of a liquid crystallinematerialwith a positive anisotropyleadsindeedto inducedorientationalorder.Quantitatively,however,it is veryweak.Taking maximumfields andtypical anisotropiesas in (3.3),onearrivesfor TN! — T* 1K atvaluesfor theinducednematicorderatTN! of S iO~—iO~.This isverysmallcomparedwith 5 0.3 at the other side of the NI phasetransition. It indicates that the simpleapproximationof (3.8) is in fact quite good.

An increaseof TN! dueto an appliedfield was first suggestedby Helfrich [69],andobservedby himin the electriccase.Magneticfield experimentshavebeenreportedby Rosenblatt[70].Again the effectis small, and to raise TNt by 5 mK requiresthe applicationof the maximal possiblefields mentionedearlier.

The smallnessof the effects mentionedmakesthe intriguing possibility of observing the magneticcritical point in thermotropicnematicsquite remote.Taking the parametersof MBBA (table 2) andusingthe value(3.3a)for the diamagneticanisotropy,oneestimatesa critical magneticfield of theorderof iO~T (seealso [71]),which is beyondexperimentalreach.In searchof the critical point Rosenblatt[72]foundno decreaseof theNI latentheatfor fields up to 18.7T. We concludethat, contraryto KeyesandShane’sfindings [73],the critical point mustbe far away.

224 E.F. Gramsbergen et a!., Landau theory of the nemaxic—isosropic phase transition

In the electriccase,the critical point is estimatedat a field of the orderof magnitude106V/cm andpossiblylower whenpermanentdipolesarepresent[74,75]. Experimentally,fields of 105_106V/cm arefeasible.The first experimentalevidencefor the existenceof the electrically inducedcritical point hasrecentlybeengiven by Nicastro and Keyes [103].

In lyotropic systems,the Landaupoint (section2.3.1, fig. 12a,b) can be approachedto an arbitrarilysmall distanceby changingthe concentration.The critical field (eq. (3.lOa)) is then correspondinglysmall. In this way, Saupeet a]. [57]observedtheN~~Itransitionin the potassiumlaurate/1-decanol/D20systemto becomecompletelycontinuousat amagneticfield of 1 T.

3.2.3. Singularbehaviouron approachingT*The existenceof inducedorientationalorder in the isotropicphasein combinationwith asmall value

of TN! — T* leadsto experimentallyobservablesingular-likebehaviourof variousphysicalquantities.Inthis sectionwe restrict ourselvesto a discussionof the field-inducedbirefringence(which is relatedtothe first derivativeof the order parameterwith respectto H

2) andof thegap exponent(relatedto thesecondderivativeof the orderparameterwith respectto H2). Note that in this context field-inducedorientationalorder in the nematic phase is not very relevant, as it is negligible comparedto thedominantnematicbackground.

Field-inducedbirefringence.The formula for 5(h), eq. (3.8), showsthat to a good approximationtheinduced order in the paranematicphasedependsquadratically on H. The proportionality coefficientdefinestheparanematicsusceptibility:

= a(H~)IH2=O (3.13)

Comparingeqs. (3.8) and (3.13) onefinds

n = &ymax/[2a(T T*)]. (3.14)

The paranematicsusceptibilityis directly relatedto themagneticallyinducedbirefringence.The induced(optical)dielectricanisotropywill be proportionalto S. Thus we find for thedifferencein the refractiveindicesparallelandperpendicularto the field

n~— ~maxH2/[a(T T*)]. (3.15)

Defining the birefringenceas L~sn= n11— n4, the quantity

~Xrnax 316H

2 — a(n11+ nj(T— T*)’

is called the Cotton—Moutonconstant.As we seethe right-hand-sideof (3.16) divergesat T= T*, andfalls off as (T— T*)_l for T> T*. In the caseof an electric field the Kerr constantAntE

2 is definedsimilar to (3.16).

Birefringenceinduced by electric or magnetic fields has beenstudiedextensively [27, 30, 76—97].

E.F Gramsbergena aL, Landau theory ofthe nematic—isotropic phase transition 225

Some examples are shown in figs. 18, 19. In all cases the induced order shows the expectedpretransitionaldivergence,while the proportionality with H2 hasbeencheckedup to 10 T [99]. Thevaluesobtainedfor T — T* from electricand magneticmeasurementsagreewell with eachother [96].Contraryto the theoreticalpredictions,however,the extrapolateddivergencetemperatureT* hasbeenreportedto be field dependent[77,78]. This hasbeenshownby Palify—MuhorayandDunmur [78]to beconsistentwith Maier andSaupe’stheory [98].Comparingthis with the Landau—DeGennesapproach,theyarguedthat an appropriateLandauexpansionshouldhavea field dependencein all its coefficients.

Finally it shouldbe emphasizedthat thetemperaturedependenceof the inducedbirefringencecloseto TN! is not in agreementwith eq. (3.16) (seefig. 19). Though thedeviationsarenot large,theyare very

TNt 30 35 40 45 50 55

T/°C

Fig. 18. Reciprocalmagneticandelectricfield inducedbirefringencein theisotropic phaseof 6CB (after ref. [961).

0.8

0.6

0

a~0.2C

C0’o0.0

V.i 9~

I I I I I

0.1. 0.6 0.8 1.0 1.2

Ln(T-T)Fig. 19. InverseKerrconstantB/(ndJ2)versusreducedtemperaturefor 5CB at 633nm (after ref. [771).Note the logarithmic scales.

226 E.F. Gramsbergen etaL, Landau theory of the nemasic—isotropic phase transition

relevantbecausethey arefoundat the temperatureclosestto the phasetransition,wherethe Landauexpansionshouldholdbest.This is a first indication that the validity of theLandau—DcGennestheoryshouldnot be takenfor granted,and that othereffectsmight comeinto play. This will be discussedinsomedetail in thenextchapter.

The gap exponent.As has been mentionedin section 2.2.1 it hasbeensuggestedthat the NI phasetransitionis governedby thepresenceof a nearbytricritical point. Thedifferencebetweentricritical andcritical behaviouris difficult to verify, and the main reasonto suspecttricritical behaviouris in thevalue of the critical exponent$ (section 2.2.2). As this exponentis measuredas the behaviourof theorderparameterat the nematicsideof the phasetransition,one haslookedfor alternativesamongthevariouscritical exponents,that would havea different valuefor the critical andtricritical case.In thiscontextattentionhasfocussedon the gapexponentLI [73,99], definedvia

92S/8h2 I(T— T*)_~ for T> T*,G = hm -= (3.17a)

h—~O ~S/9h 1(Tt_T)4’ for T<Tt.

According to Keyes and Shane [73], LI should havethe value 2 in the critical (B = 0) and ~ in thetricritical case(B = C= 0). Within the frameworkof the Landautheory onefinds that

r a3F’3x3G—limI ‘ I. (3.17b)

h—*O ~(32F/3x2)2i

Now combining eqs. (2.6) and(3.1) we get

3 ~ .~L135r: 3

2 1

2L,X~2J_~XG—lim t3.17ch-.O (~A+ ~Bx+ ~Cx

2+ ~Ex4)2

or equivalentlyfor T> T*:

G1B/A

2 B/[a2(T— T*)2], B ~ 0, (3.17d)

tO, B=0,

andfor T< Tt:

x3 (Tt— T)314 — (Tt — T ~ B — C — 0

(~A+ ~Ex4)2 (yf — T)2 — ‘ — —

G— (~A+~Cx2)2 (Ttfl2 (T T)312, B=O, C=0, (3.17e)

B/(B2x2)-=(Tt—T)’, B�O.

The resulting(Landau)valuesfor LI andLI’ indicate the following:— For B = 0 it is impossibleto define LI via eq. (3.17a).

E.F. Gramsbergen etaL, Landau theory ofthe nemasic—isotropic phase transition 227

— In thecaseB � 0 any choiceof termshigher than cubicgives LI = 2.— The tricritical exponent~referredto earlier shouldbe associatedwith LI’ ratherthanLI.— For LI’ a critical exponentis foundwith the value ~for B = 0 and 1 for B� 0.

Wherethe experimentsso far referonly to LI, the most importantconclusionis that the presenceof acubic term in the expansion (3.5) leads to LI = 2 in the Landau theory. In the absenceof arenormalizationgroupcalculationit is not surewhetherthisis alsoso in thecriticalcase.The ideathat theresultsmight be correctis supported,however,by the observationthat Kumar et al. [100]arrivedat thesameconclusionfrom a different argument.

Recentexperimentalresults on a lyotropic liquid crystal give LI = 1.75±0.25 [100],which agreeswithin the experimentalerror with thevaluegiven. The error resultsmainly from theuncertaintyin T*.We mustconcludethat the problem of critical versustricritical behaviourstill seemsto be open.

3.3. Field effectsfor AXmax<0

The applicationof a magneticfield to a nematicliquid crystal with a negativemagneticanisotropy,induces, in general, biaxial ordering (seefig. 15). Therefore,in this casewe must allow for a biaxialsolutionof 0. Againcuttingoff atfourthorderin Qa$andkeepingthefield alongthez-axis,thefreeenergyreadsas in (2.19), with the orderparametersx, y definedby (2.4):

F(x, y) = a(x)+ y2$(x)+ y4y,

now with

a(x)= —hx+ ~Ax2+~Bx3+ ~Cx4, (3.18a)

(3.18b)

y=~C. (3.18c)

Applying the samerules as in section 2.3.1, we arrive at an NBNU transition that is first order forh > h

5~andsecondorder for h <h5~.The point h = h5,.~is thusa tricritical point. Its location is givenby (2.33)with (2.23c)holding as an equality,or [72]:

= ~B3/C2, (3.19a)

31B2— T~!= 864aC= ~TNI — T), (3.19b)

S~=

1B/C= ~ (3.19c)

Comparing(3.10)and (3.19)we seethat for the samevalueof IAXmaxI:

~ = ~ (3.20)

The line of second-orderphasetransitionsis given by

228 E.F Gramsbergen et aL, Landau theory of the nematic—isotropicphase transition

T — T* ~ [(hB)u12 — /l > ~ (3.21)

andthe orderparametersin theNB phaseby

x=j~[l+(1_n+~)h/~2], (3.22a)

= 2V2CE 2~ ~ (1- ~ + ~)1~], (3.22b)

with

= 24AC/B2, ~= 48C2h/B3.

The hT diagram of fig. 17 showsthe critical and tricritical points in one picture. Another pictureinwhich both pointsappear,is that with electricandmagneticsusceptibilitiesof oppositesign. Onepointis then recoveredby electric, the otheroneby magneticfield [74].Priest [101]calculatedthe completethree-dimensional(T, H, E) phasediagram andfound the topology of the tricritical point to be asin fig.6.

6-

1 2.4tCp

4- /— / 1.2 N B

U

N N1~

N D U

2-0.0W - -

BC I

-1.2 U

~Cp

Nu

-2 0.5 1.5 2.5 -2.4 I • I

-0.2 0.0IT-I )/(TNI-T I A

Fig.20. As fig. 17, for adiscoticnematic (N~in zerofield). Fig. 21. Phasediagram of fig. 12b, modified by a magnetic fieldh = 0.0018.

E.F. Gramsbergenet aL, Landautheory ofthe nematic—isotropic phase transition 229

N~andN~nematicsUntil now we only consideredthefield effect on conventionalNj~nematics.Thestudyof the effecton

discotic Nj~nematicsis madealmosttrivial by the observationthat the magneticterm andthe B termare the only odd terms in the free energy. Hence the above description remains valid undersimultaneoussign reversalof AXmax,B andQ~.The hTdiagram is then as shown in fig. 20. Figure21is a phasediagramfor constantH (AXmax > 0) andvariableB. Comparingwith fig. 12b, we seethat theLandaupoint is split up into the critical andtricritical points;thesearedriven furtherapartwhen thefield is increased.The first orderNj1~JN~transition is changedinto a secondorderN~NBtransition.

Experimentallytherearehardlyany resultsrelevantto the theory presentedin this section.Nematicswith a negativemagneticanisotropyhaveonly recentlybecomeavailable,andno experimentsin thesefields havebeenreportedyet. The tricritical point is certainlyout of reach,as it is theoreticallyexpectedat evenhigher fields thanthe critical pointsin the caseof a nematicwith a positivemagneticanisotropy(seeeq. (3.20)). In the electriccasethe tricritical point is expectedat a field of the order of 106 V/cm.Though thisis in principle feasible,no experimentshavebeenreported.

Recently,Saupeetal. [57]haveshownthat a critical pointcan be reachedmucheasierin the caseofsomelyotropic systems.The tricritical magneticfield is hereestimatedto be of the orderof 1 T.

3.4. Field effectson the biaxial nematicphase

In orderto studythe effect of externalfields on thebiaxial phase,we includein the freeenergy(2.18)amagneticfield term.With the field alongthe z-axis,a term —hx is thenaddedto a(x) in (2.20a).Theresultingphasediagramfor AXmax >0 is shownin fig. 22.The topology of the diagramis the sameas infig. 21. ThesecondorderN~NBtransitionis given by

A=~h/x—~Cx2, (3.23a)

B=~h/x2—~E’x3, (3.23b)

x�x5~~. (3.24)

At the tricritical point x = x~,,with

~ CE’$CP+ 12hE’x~9+ ~hCx~~— ~h2 = 0. (3.25)

Whenx > x1,~,,eq. (3.23) givesthe superheatinglimit of theNB phase.The isolatedcritical point is given

by

A = ~h/x+~Cx2+~E’x4, (3.26a)

B = —~h/x2— 6Cx— 18E’x3, (3.26b)

x= ~ (3.27)

with x~,given by

—h + 9Cx~~+81E’$0= 0. (3.28)

230 E.F. Gramsbergen et aL, Landau theory of the nemasic—isotropic phase transition

2.4 I

1.2 NB

: tcp

B 0.0

-1.2 Nu+

Cp

—2.4 I • I

0.4 0.2 0.0A

Fig. 22. Phasediagramof fig. 12a, modifiedby amagneticfield h = 0.0018.In thecross—hatchedarea, spontaneousbiaxiality is predominant.

When x<x,~,,eq. (3.26) is the supercoolinglimit of the paranematicphase;when x> x,~,it is thesuperheatinglimit of the nematicphase.

Although the phasediagram of fig. 22 looks very much like that of fig. 21, thereis quite a differencein the orderparametervaluesnearthe second-orderN~NBtransition. In fig. 21 thereis only induced

Spontaneous NB induced NB

Fig. 23. OrderparametersQ,.,., Q~,Q,, vs. B underthe influenceof amagneticfield h = 0.0018. Thedashedline representsthefield-free caseoffig. 13.

E.F. Gramsbergenet a!., Landau theory of the nemasic—isotropic phase transition 231

biaxiality which is very small. In fig. 22, however,the biaxiality is partly spontaneous(large)andpartlyinduced (small). In the dashedarea the spontaneousbiaxiality is predominant(compare fig. 12a).Although the two biaxial regionsareclearlydistinguishable,thereis no sharpboundarybetweenthem.This can be seenin fig. 23, showinga crosssectionof the phasediagramat constantA<0. The dashedlines refer to the field-free case(fig. 13). The roundingoff of the NBNI3 transitioninto a spontaneous-inducedNB transition is reminiscentof the field responseof a ferromagnetnearthe Curie point [102].

As far as we know no experimentshave beenreportedin relation to the theory discussedin thissection.The discoveryof a biaxial nematicphasein somelyotropic systemsmakessuch experimentsinprinciple feasible.

4. Landau theory of the nematic—isotropic phasetransition: fluctuations

4.1. The importanceof orientationalfluctuations

For a physicalsystemin thermalequilibrium the instantaneousvalueof the orderparameterwill beeither equal or close to its meanvalue. However, any disturbanceof the system such as a thermalfluctuation producesspatial variations of the order parameter.Therefore in the theory of phasetransitionsnot only at each temperaturethe equilibrium value of the order parametermust bedetermined,but also its fluctuation amplitude. A secondimportant ingredient that influences thethermodynamicfunctions neara phasetransition is the presenceof spatial correlationsbetweenthefluctuationsof theorderparameter[105,106].The averagesize of the rangeof correlationsbetweenthefluctuationsdefinesthe so-calledcorrelationlength ~. Far away from the critical point ~ is of the orderof theintermoleculardistance.On the otherhand,nearasecond-orderphasetransitionthecorrelationsdecreasevery slowly with distance,indicating a divergenceof ~ at the critical temperature(seesection4.3.2).

Responsefunctions can only be properly defined in terms of correlation functions, and thusfluctuationsmust quite generallybe taken into account.Therefore,eq. (2.2) describingthe NI phasetransitionhasto be generalized,in principle independentof the fact that this phasetransition is firstorder.More quantitativelythe importanceof including fluctuationscan beestimatedfrom the Ginzburgcriterium [19, 106] andfrom experiments.

The validity of the Landau theory of the NI phase transition (chapter2) can be estimated,assuggestedby Ginzburg [19], by examiningthe fluctuationsaroundthe solutions S = 0 and S � 0 (eq.2.9). For the NI phasetransition,governedby the free energyexpansion(2.6), the Ginzburgcriteriumcan be put into the form [72]

b~3~kBT, (4.1)

whereb is theheightof the barrierseparatingtheorderedanddisorderedstate(fig. 7). Closeto TN! theimportant fluctuations involve regionsof volume ~. The inequality (4.1) then statesthat thermallyactivatedfluctuations leading from orderedto disorderedregionsor vice versa do not occur with asignificant probability. Under theseconditionsthe Landaudescriptioncan be expectedto bevalid.

If we approachthe NI transitionfrom abovewith T closeto TN!, then b is found for T = TN! to begiven by

232 E.F. Gramsbergenet a!., Landau theory of the nemasic—isotropic phase transition

5.— ~ D4IP’3

V

11664L)/L~

The correlationlength~ ascalculatedunder theassumptionthatthe fluctuationsaresmall [16,105, 106](the Ornstein—Zernikeapproximation,see alsothe nextsection),is given by

~2 g2T*/(T_ T*),

where~ is the direct correlationradius,which is of moleculardimensions.Iaking.typic.aLvaluesoLT”,

B, C atT= TN! for MBBA (table3) or for p-azoxyanisole[107],andassumingthat ~ is of the orderofthe molecularlength,onefinds that both sidesof (4.1) are of the orderof 1021 J. We concludethat thedescription given in chapters2 and 3 may well be inadequate,and that critical behaviourmay beobservablecloseto TN!.

4.2. Genera/formofthe Landaufree energy

In order to include fluctuationsinto the Landautheory one usually first calculatesa free energydensityat each temperatureas a function of both the order parameterand its spatial derivatives.Forlow energy(long wavelength)fluctuationsthe spatialvariationsoccur on a scalemuch larger than themoleculardimensions.Therefore,for such fluctuationsaround the equilibrium statewe can restrictourselvesto only the lowest-orderspatial derivativesof the orderparameter:~ where c

9~ 9/I9x~.In this sectionwe shall give a generalformulation of the free energywithin theselimitations,which isrequired in the later sections of this chapter where the theory is taken beyond the Gaussianapproximation.

The free energy, F, for an arbitrary (but fixed) spatial variation of the order parametercan beobtainedfrom the free energydensity,f, by a volume integration

F(Q) = Jd3rf[Q(r), 8Q(r), T, p]. (4.2a)

Now assumingthat the amplitudesQ,.,,~(r)maybe conceivedas randomvariables,theequilibrium valueof the free energy,3~,can be obtainedvia a thermalaverageover all possiblespatial variationsof 0(r).Thus ~ can be calculatedfrom

~i= —kBTIim~, (4.2b)

where

Z J DQ(r)exp{_$J d3rf[Q(r), o.Q(r), T,p]}, (4.2c)

with fl_i = kBl while DQ(r) denotesFeynman’spathintegral [108]overall possibletensorfields Q(r).The Feynmanintegral can be understoodas a limit of the standardRiemanintegral in the following

E.F. Gramsbergen et a!., Landau theory of the nematic—isotropic phase transition 233

way. First let usdivide the systemwith volume V into N identical cubes,andtakethe centerof eachcubeas referencepoint. In the exponentin (4.2c)now the integrationcan be approximatedby a finitesum, andthe derivativesreplacedby finite differences.ThenZ can bewritten as

= lirn fl’ fl J dQ,.~(r~)exp[—/3~~f{Q(r1), [Q(r1)— Qfrj+a)], T, p1].

Now I, J run over all cubes,while a refersto all cubesthat are neighboursof J. fl~runsover all a$that enumeratethe independentcomponentsof Q.

Similar to eq. (2.2) the free energydensitycan againbe expanded,but now with respectto both Qandits derivatives.The invariantsformedfrom the Qa$ have the sameform as before,andwe havetoconsiderthe additional terms involving the derivatives. There is no way of forming a scalarquantitylinear in ~ Also thereis no scalarcombinationof Qa$andits spatial derivative.Hence,the lowestorderspatial derivativeinvariantsof Qa$ havethe form [24]:

ioc~ \2 j~f~ \2

Combining the Landaufree energydensityas given before with only theselowest-ordertermsin the

derivativesof the orderparameter(long-wavelengthlimit), one arrivesat

f[Q(r), o1Q(r), T, p] = ~ATr Q2 + ~BTr Q3+ ~C(TrQ2)2 + ~D(Tr Q2)(Tr Q3)+ ~E(Tr Q2)3

+ E’(Tr Q3)2 — ‘AX HQH — A Em~a,,.EQE+ ~L1(o~Q~~)

2+ IL2(3~Qp7)

2. (4.2d)

Only two new expansionparameters,L1 and L2, havebeen introduced.These are related to the

curvatureelasticconstantsK1, K2 and K3 of the nematicphase[14].The otherexpansionparametersmuststill bethe sameas usedbeforein eqs.(2.2) and(3.2).This follows by realizingthat the calculationof the physical propertiesof a spatially uniform system requiresonly information about f in theimmediatevicinity of its minimum.Thus aroundthe minimumof f, wherethe function (4.2c) is sharplypeaked,in the thermodynamiclimit (V-+ cc) ~ from eq. (4.2b) coincideswith eq. (2.2) plus eq. (3.2).*

Equations(4.2a—d)are the basicequationsof the generalizedLandau—DeGennestheory (GLGT) ofthe nematic—isotropic phase transition that includes long-wavelength fluctuations of the orderparameter.The GLGT hasvariousapplications,of which we shall concentrateon the pretransitionallight scattering.In that casefrom eq. (4.2b) correlationfunctionshaveto be calculatedof the form

(Qa~(r)Qy3(r’))=Z~JDQ(r)Q~(r)Q~s(r’)exp[_$Jd~r”f(r”)]. (4.3)

More preciselythe Fourier transformof thesefunctionsareneeded,becausetheseare directly relatedto theresponsefunctionsof the system.There is a one-to-onecorrespondencebetweenany 0~.~(r)andits set of Fouriercoefficients:

* More formally,

(Q onhlorm) (~= k8TlimlnZ =—kBTlimlnJ dQexp{flVf(Q,0, T,p)}=minf(Q,Q, T,p).

V-= {QI

234 E.F Gramsbergen et aL, Landau theory of the nematic—isstropic phase transition

Q,,~(q)= V~Jd3r Qa~fr) exp(—iq.r), (4.4a)

Q,~~(r)= V/(21T)3Jd3qQ,,8(q)exp(iq . r). (4.4b)

Consequently,the functional integrals(4.2c)and(4.3) areequivalentto integratingover all possiblesetsof Fouriercoefficients{Q~,8(q)}.As Q,~(r)is real thereis the constraintthat

Q~(q)= Qas(q),

where the asterisk denotes complex conjugate. The Jacobian of the linear transformation{Q~(r)} ~-* {Q,,,,~(q)} is independentof thecomponentsof Q.Thereforethetransformationaddsaconstantto ~ in eq. (4.2b)which cancelsin eq.(4.3). In thefollowing sectionsa numberof approximationswill beconsideredthat allows for the calculationof ~ from eq. (4.2) andof the correlationfunctions.

Finally we want to recall that the NI phasetransition is first orderwith adiscontinuousjumpat TN!

from Q = 0 in the isotropic phaseto some finite value in the nematicphase.Thereforewe expecttheLandauexpansionaroundQ = 0 to provide a betterdescriptionof phenomenasuch as fluctuationsinthe isotropic phase, than it does in the low-temperaturenematic ~ In the following_sectionsemphasiswill be on the fluctuationsin the high-temperaturephase.Morepreciselywe will deal in somedetailwith the elastic(Rayleigh) scatteringof light in the isotropicphasenear TNJ.

4.3. Pretransitionallight scattering

4.3.1. IntroductionAlthough the NI phasetransitionis first order, the changesof the entropyandof the volume at the

transitionareso small that it is sometimesincorrectlycalled “nearly secondorder”. Also reminiscentofsecond-orderphasetransitionsis thepretransitionalbehaviourof, for example,themagneticallyinducedbirefringence(Cotton—Moutoneffect, seechapter3) and the light scattering.The intensityof scatteredlight is shown schematically in fig. 24. There is a divergence towards the temperatureT~.This

--.---_-_-_~_~ ~ _Lt_~_~_

>-.

INI Temperature

Fig. 24. Pretransitionalbehaviourof the intensityof scatteredlight neartheweaklyfirst-order NI transition (schematically).

E.F. Gramsbergen et aL, Landau theory of the nemasic—isotropic phase transition 235

divergenceis cut off by the interferenceof the nematicphaseat T= TN!. As mentionedin section4.1the physicalorigin of divergencesof this kind lies in correlationsbetweenfluctuationsthat extendoverregionsthat becomevery large as T* is approached.In a pictural way, one could say that nematicdropletsare formedin the isotropic liquid ~‘. Thesedropletsareunstablebut the excessfree energyAFassociatedwith their formationis sosmall that theprobabilityexp(—AF/kBT)is comparableto unity. Asthe minimumof the free energyis extremelyflat in the vicinity of a second-ordertransition, it is clearthat larger and larger dropletscan exist on approachingT*. In principle T* gives the location of asecond-ordertransitionwhich is not observablebecauseit is precededby a first-order transitionat TN!(seefig. 7).

Forlight scattering,therelevantfluctuatingquantityis theopticaldielectricconstantr, whichcan besplitinto two parts:the isotropicpart,with elements

= ~ Tr(r)

proportionalto the density,andthe anisotropicpart

Ar = —

which is proportional to the order parameterQ. We will negect the density fluctuationsand onlyconsiderthe orderparameterQ, which is responsiblefor the critical behaviour.

The experimentalsituationunder considerationis depictedin fig. 25: k is the ingoing and k’ theoutgoing wavevector. The scatteringwavevectorq = k’ — k, 0 is the scatteringangle. For elasticscattering, lkl = ik’l and i~i= 21k1 sin ~0. Now let e and e’ be unit vectors giving the polarizationdirectionsof in- and outgoing waves,while Q,~(q)is given by eq. (4.4a). The integration is over theentire sample whose volume is normalizedto unity (V= 1). The scattering intensity 1(q) is thenproportionalto [9, 109, 110]:

1(q)— (Ie,ea,s(q)e,~I2)—~(~e~Qa~(q)e0J

2)= e,~,eçe8e~(Q~(q)Q7a(q)). (4.5)

Two specialcasesare:

(1) e lie’ liz : 111(q) — (IQ~(q)l2)

(2) e liz , e’ ii~: I±(q)—~-(IQ~~(q)i2)

FIg. 25. Connectionbetweenthe incoming andoutgoingwavevectorsk andk’, thescatteringwavevectorq = k’ — k andthescatteringangle 0.

In principle, thesefluctuationscanbe biaxial, eventhoughthereis no stablebiaxial nematicphase.Thesebiaxial fluctuationsareautomatically

includedin a descriptionin termsof Q,suchasin this chapter.The importanceof biaxialfluctuationsin thenematicphaseis discussedin ref. [25].

236 E.F. Gramsbergen et a!., Landau theory of the nemasic—isotropic phase transition

Our concernwill be to calculate the correlation functions(Q~(q)Q~~(q)),in particular thoseof thespecialcases(1) and(2). Beforedoing so, we will now derivesomeusefulsymmetryrelationsunder theassumptionthat thereareno externalfields.

Taking into accountthesymmetry andzero trace conditions,Q(q) can be parametrizedas follows:

c/i4 \Q(q)=21~’2( I/13 —~~1/V~+c/~2c/f5 ). (4.6)

(2/V3)c/~1/

In a frameof referencethat is rotatedover an anglea alongthe z-axis, the orderparametersread:

Q’(q) = MQ(q)M1,

where

/ cosa —sina

M = ( sin a cosa 0 ). (4.7)

~ 0 0 iJ

This can berewritten as

414(q)= 41i1(q), (4.8a)

= (cos2a sin2a~ (c/12(~)~~ (4.8b)

\41’~(q)J \—sin2a cos2aJ \41i3(q)J’

= ~cosa sina~ (~4(q)~ (4.8c)

\i,b~(q)! \—sin a cosa! \qi5(q)J

Now, choosingz q, thez-axis is theonly preferreddirection andall thermalaveragesmustbethe samein the primedandthe unprimedsystem.For example,(4.8b)

(41i2(q)) = s~41i~(q))= (41i2(q))cos2a + (41i3(q))sin 2a

for anya. It follows immediatelythat (41’2(q)) = (413(q))= 0. In the sameway, choosingfor a easyvalues

like ii~, IT, IT, oneobtainsthe following relations:= 0 for i � 1, (4.9)

(41i’~(q)ç/i~(q))=0for i�j, (4.10)

(l’1’2(q)12) = (k~(q)i2), (4.11)

(I~’4(q)i)= (ic/’s(q)i). (4.12)

E.F. Gramsbergen et aL, Landau theory ofthe nemasic—isotropic phase transition 237

As we shall see (section4.4.2), in the optical region it is a good approximationto take q = 0 in the

correlatiOn functions.Thenthereis completesphericalsymmetryandwe can addat once,(1) (iQ~5(O)i

2)= (IQxy(0)12) or (Ic/’3(°)l)= (lc/’(°)i

2) (4.13)

(2) (Q~~(O))= (Q~~(O)),giving (tfr~(O))= 0 , (4.14)

which states,togetherwith (4.9), that the isotropicphaseis indeedan isotropicphase.

(3) (jQ~z(0)i2)= (jQ~y(o)j2), giving (i41,i(O)i2) = (14112(0)1) (4.15)

reflecting the well-known fact that in the isotropic phasethe depolarizationratio [109, 110] has the

value

I~O)/I~(0)= ~(i41’1(0)l

2)/(i41’2(0)l

2)= ~, (4.16)

to be comparedwith I~= 0 for isotropic (density) fluctuations. The geometryfor measuringthedepolarizationratio is shown in fig. 26.

~~~~26. Geometryfor measuringthedepolarizationratio ‘i/’i~ ê

1 andê1 arepolarizationdirections.

4.3.2. Correlationfunctionsin the Gaussianapproximation

In this sectionwe shall deriveexpressionsfor the correlation functionsrelevantto light scatteringexperimentsusingthe so-calledGaussianapproximationof theGLGT. In theGaussianapproximationwetakeB~. . E’ = 0 in (4.2d).This is expectedto be a good approximationwhen(QtiaQyS)is not too large,andthe free energy(4.2d) in termsof the Fouriercomponents(4.4a)of Q~~(r)reads:

F = f (2)~[~(A + L1q’2)i Q.~(q’)l~+ ~L

2q~q~Q~ (q’)Q~~(q’)] (4.17)

Iq’I<qm

whereq~is the radiusof the Debyesphere:

= (6ir2p)1”3, where p is the particledensity. (4.18)

The Debyesphereis the “amorphous”analogonof the first Brillouin zonein solid statephysics[111].

238 E.F. Gramsbergen et a!., Landau theory of the nematic—isotropic phase transition

The expression(4.17)for F hasaGaussianform in Qa~(q’).Thereforethe Feynmanintegrals(4.2c)and(4.3) can becalculated.For instance,(4.3) nowhasthe form

= z1 f DQ(k)Qasi(k)Qva(J1)exp(—f3F),

whereF is now given by (4.17). Becauseof the Gaussianform of the integrals,the calculationsareequivalentto applicationof theextendedequipartition theorem[105,112] to (4.17). Beforedoing so wewill first expressthe freeenergyin termsof theparameters41’j, i = 1 . . 5 (eq. (4.6)),andthencalculatethecorrelationfunctions (41i’~(q)41’~(q))to which the scatteredintensity is proportional. It now follows that

IQ~(q)l2 k~1(q)I

2, (4.19)

and

qaq~0~y(q)Q,~(q)=ç/i~(q)Sq(q)ç1ij(q), (4.20)

with

~ ç~=(q~—q~)—~q~q~~

/ ~ (q~— q~) q~+ q~ 0 ~t’1xLlz

—~q~q~ 0 q~+q~ q~q~ q~q~ (4.21)

q~q~ —q~q~ q~q~ q~+ q~ q~q~

\ ~ qyqz q~q~ qxqz qxq~ q~y+

The completeGaussianpart of the free energythenbecomes

F0=~ f -~_9_41,~(q~)71(q~),fr3(qP), (4.22)

q’I<qm

where

T~j(q’)= (A + L1q’2)S~,+L

2S51(q’).

The extendedequipartition theorem[112]nowgives the Gaussiancorrelationfunctions

= kB T[T1(q)]~. (4.23)

E.F. Gramsbergen et a!., Landau theory of the nemasic—isotropic phase transition 239

The matrix inversionin (4.23)hasbeencarriedout,in a slightly different form, by GoversandVertogen[113].

However,to discusslight scatteringexperimentsit is not necessaryto usethis generalform. We canfix the frameof the experimentsuch that the momentumtransferq is parallel to the z-axis. Therefore,without loss of generality,we can take q~= q, q.. = q~= 0. Now the matrix T is in diagonalform, andcan be invertedvery easily.The result is

= (i41i~(q)l2)3~1, (4.24a)

with

(l41’1(q)i2) = kBTI[A + (L

1 + ~L2)q2], (4.24b)

(i41’2(q)12) = (i41’~(q)I2)= kBTI(A+ L1q

2), (4.24c)

(I’4’4(q)1

2) = (l41’5(q)1

2) = kBTI[A + (L1 + ~L2)q

2]. (4.24d)

We seethat threedifferent correlationlengthscomeinto play, a fact that could havebeenforeseenbythe symmetryrelations(4.11) and(4.12).

Typically L2/L1 is in the orderof unity. This follows from measurementsof the elasticconstantsK1,

K2, K3 in thenematicphaseas will be seenin thesection4.4.2. In theso-calledisotropicapproximation,which is often cited in the literature,one takesL2= 0. Consequently,all the correlationfunctionsareequal,andgiven by

(l41’a(q)12) = kBTI(A + L

1q2). (4.25)

The scattered intensity is proportional to (4.25) and is predicted to have a Lorentzian shape.When eq.(4.25) is transformedbackto real space,oneseesthat spatialcorrelationsof the orderparameterhavean Omstein—Zernikeform [16]anddecayproportionalto~ wherex issomespacecoordinateand~ isthecorrelationlength

= (L~/A)”2. (4.26)

4.3.3. Comparisonwith experimentand discussion:validity of the Gaussianapproximation

The important predictionsfollowing from the Gaussianapproximationof the Landaufree energyaregiven in eqs.(4.25)and (4.26). In practice,the term L

1q2 in (4.25) is small (seealsosection4.4.2). In

that casethe inverselight scatteringintensityis predictedto be proportionalto T— T*. Experimentalresults are shown in figs. 24 and 27. The predictedtemperaturedependenceand, though weak, theq-dependencehaveboth been verified by Stinson and Litster [27, 114]. It should be noted that thepredictedproportionality with T — T* of the inverse light scattering intensity, or equivalently thesquaredcorrelationlength,is independentof thevalues of the parametersof thefree energyexpansion.From measurementsof the light scatteringof MBBA [114] the correlation length (4.26) has beenestimatedto be given by (seefig. 28)

= (68± l0)(T/T* — l)h/2 nm.

240 E.F. Gramsbergen et aL, Landau theory ofthe nemasic—isotropic phase transition

6-

=1

>_. 11-I-

U-)zLiJI—z

2-

a ~ I

I T~ ‘12 TEMPERATURE/°C ‘18 51

Fig. 27. Inverselight scatteringintensity vs. temperature,for 8CB. The experimentaldata are from ref. [115].The solid line is the Gaussianapproximation;thedashedline is theextrapolationbelow TNL.

The light scattering measurements shown in fig. 27 for 8CB have been extended to the fullhomologousseriesof nCB, with n = 5—12 [115].Theseresultsareshownin figs. 29 and30,emphasizingthe region close to TN!. As we see, in this region thereare clear deviationsfrom the proportionalitybetween the inverse intensity and T— T*, the inverse intensity bendingdownwards on approaching TN!(seealso [116]).Though thesedeviationsare not large, theyareimportantbecausethey occurclosetothephasetransition.The deviationscannotbe accountedfor in theGaussianapproximationusedso far,independentof the choiceof the expansionparametersin the Landautheory.What is more, therearequantities,like T*, that arevery sensitiveto such deviations(seesection4.4.2).

2’l—

b

~I6 ‘V

V8

I I

38t 46 54 62I T/°C

Fig. 28. Reciprocalof thesquareof thecorrelationlength in the isotropicphaseof MBBA (after ref. (114]).

2.00 -

1.50-

I- 1.00-

zuJI-z

0.50

ob cd

‘~O0I II

39 ‘10 ‘11 ‘42 ‘13

TEMPERATURE/°C

Fig. 29. Enlargedportion of fig. 27. Theoreticalcurvesareboth undertheassumptionof theNI transitionasa critical point. Dashedline: theoreticalcurvefor Debyeq.,. Solidline: bestfit. Locationsof T*: (a)bestfit; (b) Gaussianapproximation;(c) Debyeq,,; (d) is the location of TNt.

w)

o •.~_ . — S. 12—GB.5

c •• • •_ ._ • • • • •. • —— 11—CB•

c —-• - - • . - io-ca:s~

C ~ - - • • • - S • 9—CB

C ~ • . 8—GB

D

~ C •‘ —. • • — • . . . 7—GB

C ~ — • - • : • • 6— GB

C ..~--•:..:.-...-. 5—CB•#_~ • .

I • I I0 1. 8 12

T—TNI/K *)TT/KFig. 30. Thenonlinear partof I~ for the nCBseries(from ref. [115]).

241

242 E.F. Gramsbergen er aL, Landau theory of the nemasic—isotropic phase transition

As has beennotedalreadyin section 3.2.3 the inducedbirefringencedue to a magneticor electricfield shows similar deviations from the theoretically predictedbehaviour on approaching TNJ asobservedfor the light scattering[77,97] (seefig. 19). The Gaussianapproximationdoesnot changeeq.(3.16) for the Cotton—Moutoncoefficient (and similarly for the Kerr constant), both of which arepredictedto vary linearly with T— T*. Summarizingwe must concludefrom the experimentsfor manyliquid crystals,that either the Gaussianapproximationis insufficient to describethe NI phasetransition,or that the GLGT hasto bemodified.

From table 3 one can concludethat the higher-ordertermsin the Landauexpansioncould be quiteimportant. Priest [117] has argued that the GLGT can be improved upon using a perturbationcalculationbeyondGaussian.The effectof the cubicand fourth-orderterm on the fluctuationshasbeeninvestigated by Senbetuand Woo [73]. In the next section we shall discuss the influence of aperturbationcalculation in the lowestorder beyondGaussianon the correlation function (4.3). Suchacalculationbrings out the possible importanceof non-Gaussianfluctuations,without introducinganynew phenomenologicalparametersinto the theory.Thereforesuchcalculationsarealso importantwithrespectto the internal consistencyof the GLGT. As will be shown, the deviation from linearity asobservedexperimentally,can, at leastin principle,be explainedby taking theseeffectsinto account.

However, the problem could be much morecomplicatedthan describedabove,as the introductionof other couplings potentially can lead to similar results. More precisely, one can question theassumption(4.2a) statingthat the Qas(r) are the relevantdegreesof freedom,which is only true if anyeffect of otherdegreesof freedomis averagedout. Thisalternativedirection hasrecentlybeendiscussedby Gohin et al. [1161and Anisimov et al. [118].The latter authors succeededin describing lightscatteringandspecific heatdataof 8CB in oneconsistentpicture, assuminga couplingbetweensmecticand nematic order parameters.In this description,smectic pretransitionplays a crucial role and theproximity of a smecticphaseis thusbelievedto be an importantparameter.As the smecticSA phaseapproachesthe NI transition on increasing the alkyl chain in the nCB series, the light scatteringbehaviouris then expectedto dependon the alkly carbonnumber n. This is in contradiction withexperimentalfindings [115] (seefig. 30) which indicateverysimilarbehaviourfrom n = 5 (nematicrange11 K, no smecticphase)up to n = 9 (TN, — TSN = 1.7 K). This behaviourhasbeenconfirmedby magneticbirefringencemeasurements[971.In anotherseriesof compounds,however,Gohin et a!. [116]founddeviationsfrom theGaussianlight scatteringbehaviourto becomemorepronouncedwhenasmecticphaseis nearby.Thereforethe suggestedtrendhas to be checked.

4.4. Beyondthe Gaussianapproximation

4.4.1. Perturbationcalculation of thecorrelationfunctionsIn this section higher-orderterms in the Landauexpansion(4.22) will be considered.The simplest

way to takethem into accountis to performacalculationof the correlationfunctions(4.3) thatgoesonestepbeyondthe Gaussianapproximation.We shall expandthe correlationfunctionsto lowest orderofthe coefficientsB, C, E and E’, and in addition take L2 = 0. The situationL2 � 0 is only discussedqualitatively. Various aspectsof such a calculation can already be found in the literature [119].Asimplified version, involving only the scalarorderparameter*S, was carriedout by SenbetuandWoo[120] in two ways: perturbationaroundS = 0 and aroundS = ~SNI. The first one gives resultsthat are

* strictly speaking,thereis no iso!ropicphasein adescriptionwith only S. Dueto theasymmetricF(S)curve(S)remainsnonzeroevenwhenS= 0 is

theglobal minimum.

E.F. Gramthergen et a!., Landau theory of the nemalic—isotropic phase transition 243

not very different from ours andfrom thoseof ref. [74]. The secondmethod is an expansionaroundS = ~ so the expansionparameteris not small anymore, and the disagreementthat was foundbetweenthe two methodsis not surprising.

Becauseliteratureexplicitly showing perturbationcalculationswith the full tensorQ is scarce,wepresentsometechnicaldetails in appendixC. In this sectionpart (a) containsthe perturbationfor theisotropicapproximation(L2 = 0) andin part (b) we give somecommentson the caseL2 � 0.(a) L2=0

The free energyis given in eqs. (4.2a)and(4.2d):

F=F0+V (4.27)

where

F0= Jd3r[1ATrQ2+~Li(9,~Qs

7)(i9aQs7)] (4.28)

is the Gaussianfree energyand

V = Jd3r [~BTr Q3+ ~C(TrQ2)2+ ~E(TrQ2)3 + E’(Tr Q3)2] (4.29)

is the perturbation.We will expressboth F0 and V in 41r1(r), i.. . 5, of which the 41’~(q) arethe Fourier

transforms.The invariantsare:

Tr Q2(r) = ~ 414(r) (compare(4.19)), (4.30)

Tr Q3(r) = ~ (~41,~— (41r~+ 41,~)+ ~(41s~+ ç/i~))+ (41~541’5+ ~ 4112(4115— ifr~))]. (4.31)

The expressionsfor F0 andVbecome:

Jd3r~(A+Li ~a)~ c/iL (4.32)

J d3r[v~i,çb1~i5+ v~~41JjçbJ41f~çL’,+ v,mflc/Jjç/ijc/~~~çfr,c/JmçL1fl], (4.33)

where~ n = 3, 4, 6, is given in table5. Usingthe standardmethodof Feynmangraphs[105,108, 112]onecan obtainthe correlationfunctions

244 E.F. Gramsbergen et a!., Landau theory of the nematic —isotropic phase transition

Table 5All nonzerovaluesof V~”~in eq. (4.33), n = 3,4,6

ijk ijklmn V~,,,,0 uk/mn V~0

111 ~B/V6 Of theform Not122/133 —B/\/6 iikkmm of theform

144/155 ~B/V6 11 11 11 ~E+~E’ iikkmm

244 —~B/V6 11 Ii 22/33 ~E—E’ 1 1 I 2 4 4 —~V3E’255 ~B/V~ 11 11 44/55 1 1 1 2 5 5 ~V3E’345 BJV2 11 22 22 ~E+~E’ 1 1 1 3 4 5 V3E’______________________ 11 22 33 E+3E’ 1 2 2 2 4 4 ~V3E’

11 22 44/55 E—~sE’ 1 2 2 2 5 5Uk! V)jkl 11 33 33 ~E—~E’ 1 2 2 3 4 5 —3V3E’

11 33 44/55 E—~E’ 1 2 3 3 4 4Of the form i 3 3 —

11 44 44 ~E+~E’ 1 2 3 3 5 5 —~V3E’‘Ill 11 44 55 E+~E’ 1 2 4 4 4 4 —~V3E’

Oftheform I 3 3 —11 55 55 ~E+~E’ 1 2 5 5 5 5 ~V3E’

ukk,,<k ~C I —_______________________ 22 22 22 1 3 3 3 4 5 —3’(/3E’

22 22 33/44/55 ~E 1 3 4 4 4 5 ~V3E’

22 33 33 1 3 4 5 5 5 ~sV3E’

22 33 44/55 E 2 3 4 4 4 5 —~ E’22 44 44 2 3 4 5 5 5 E’

22 44 55

22 55 55

33 33 33

33 33 44/55 ~E33 44 44

33 44 55 E4~E’

33 55 55

44 44 4444 44/55 55

55 55 55

G11(q) = (41i~(q)41i1(q)) (4.34)

in termsof the unperturbedcorrelationfunctions

G~~~(q)(ç/i5(q)c/i(q))o= AkBT Go(q)8

11. (4.35)

Here the subscript ( )~meansaveragingwith respectto the unperturbed(Gaussian)free energy.Thefree energy is isotropic in q-spaceas the Gaussianpart is the same as in section 4.3.1 and theperturbationhas only a term q = 0. Hence we can againusethe symmetry relations(4.8).-(4.15) withq = 0. So the completeG11(q), too, must beof the form

G11(q)= G(q)511. (4.36)

E.F. Gramsbergen et a!., Landau theory of the nematic—isotropic phase transition 245

To find G(q) explicitly one can apply the Feynmangraph techniqueto the perturbation(4.29).Though the calculationsare algebraicin nature,one can easily miss some graphsor take impropercombinationalfactors.Examplesof this can be found in the literature.For that reasonsomedetails ofthesecalculationsare given in appendixC. Summingup all relevantgraphsof lowestorder,we finallyobtain the correlationfunction

G1(q)= ~+j~fc~fB + + q2 ~ +gB]+ 0(q4), (4.37)

where

f~= (q5 — tan14.), f~= 24~~(tan_i ~ — +~2)’ fE = 44 (41’ — tan~~)2

7 ~(4,2 1)g~= ~ [tan~ q5 + (1 + ~2)2]’ ~ = (L

1IA)112, ~ = ~

The correlationlengthcan easily be extractedfrom (4.37) as (seealsoeq. 4.26)

~(T) = L1/k8T+ (B/Ll)

2~g~ 1 u12~ (4.38)1, CfCID,T \2c f E+~E’~

~ ~ r2t2 JE~ co ~ico

(b) L2�0

Becausein this caseinternalmomentahaveto be integratedover, we haveto invert thefull matrix(4.21) [113].The calculationthenproceedsasbefore,providedthe z-axisis takenparallelto the externalq, so that the external propagatorsstill contain delta functions 5~(see appendixC). The actualcalculationof G11(q) is a cumbersomeproceduredue to the integrationof the complicatedfunctions

over internalmomenta.For that reasonwe keptL2 = 0 andusedL = L1 +~L2insteadof L1.This is equivalentto replacingtheelastic constantsK1, K2, K3 by their meanvalue K = ~(K1+ K2+ K3)in the nematicphase.Onewould not expectthat this will changethe final resultsin an importantway.

4.4.2. Comparisonwith experimentand discussionIt is clear from eq. (4.37) for the correlation function that a perturbationcalculation aroundthe

Gaussianapproximationleadsto a departureof the inverselight scatteringintensity from the straightline in T—T*. The magnitudeof this departurecan be comparedwith the predictionsof chapter2. Inthis context the critical (E = E’ = 0) and the tricritical casewill be considered.As an example8CB istaken,but very similar resultsarefound for othermembersof the nCB series.

Critical caseFirst, someadditionaldata on the coefficientsin the free energyare needed.In mean-fieldtheory,

with the latent heat ~H and entropy 1~,both per volume: L~H= — TNtL~.X = — TNIA(8F/3T)=— TNJ~(~aS

2)= ~aTNlS~!.Furtherusingeqs. (2.7) and (2.lOa)we obtain

246 E.F. Gramsbergen et a!., Landau theory of the nematic—isotropicphase transition

a = 3~H/(TNJS~!), (4.39a)

B = —27~H(TN!— T*)/(TNJS~!), (4.39b)

C= 9AH(TN! — T*)/(TNJS~JI). (4.39c)

Neglectingtermsof orderS3 andhigher, the elasticcoefficients L1,2 arerelatedto the nematicelastic

constantsK1,2,3 by [14]= K3 = (2L1+ L2)5

2, (4.40a)

K2=2L1S

2. (4.40b)

Or, at TN! whereS is smallandthe approximationis best:

r_ij /c2

— ~iVNI/~.YN!.

Using for 8CB the set of data

(1) TN! 40.42°C[115];

(2) SNI=0.395 [121];

(3) latentheat612J/mol [40],normalizedwith adensityp = 0.98g/cm3[1221anda molecularweightof290.5, to L~H= 2.06x 106 Jm~3•

(4) elasticconstantsat TN! [123], K1 = 2.4x 1012Jm’, K2 = 1.2x 10—12Jm

1,K3 = 2.6x 1012Jm~

1wearrive at* (energiesin 10_21J, distancesin l0~m, temperaturesin K):

kB TN! = 4.33, (4.42a)

q~= 4.94 (Debyevalue), (4.42b)

a = 0.126, (4.42c)

B = —2.88(TN!— T*), (4.42d)

C= 2.43(TN!—T*), (4.42e)

L = 6.62. (4.42f)

In fact for the ratio L2/L1 a valueof about2 is obtained.A least squareprocedurewas adoptedto fit

the light scatteringdatato eqs.(4.37) and(4.42),with T* as adjustableparameter.An enlargedview ofthe neighbourhoodof TN! is shown in fig. 29. The theoreticalcurve (brokenline) doesnot explain the

* With thesevaluesand TN! — T* 1K wecanestimateforA 5000A: q2~2~ 0.01.This justifiestheapproximationq 0 in thesynrmetryrelations

(4.8)—(4.15).

E.F. Gramsbergen et aL, Landau theory of the nematic—isotropic phase transition 247

curvaturein the experimentaldata.The only significant differencewith the Gaussianapproximationisthe reductionof TN! — T* with a factorof 30.

The sourceof the discrepancybetweentheory and experimemtmight be in the Debyevaluetakenfor the momentumcut-off q~.In fact, using this value meansthat order parameterfluctuations areconsideredup to apurely molecularscale,wherethe continuumtheoryfails. Hence,thoughin principleq~should be well definedandfixed, thereis in a liquid of anisotropicmoleculessomeambiguity aboutthe propervalue. Becauseof short-rangecorrelations(clusteringof molecules),it could be justified tousealowerqm-value.Thecorrectvaluecannotbederivedfrom theforegoingtheory,aswouldbein thecaseof a solid crystal. X-ray measurementsin the nematic phase indicate clusters of 5—10 molecules[124]. Sheng [125] found from nematic director fluctuation measurementsin PAA (table 1) byLuckhurst[126]theorderof magnitudeq~ 0.6x iO~m1. This is in theneighbourhoodof the reciprocalmolecularlength l~ 0.5x iO~m’ andsome10 timesbelow the Debyevalue.SenbetuandWoo [120]usedq,,, = 1~as a first stepto include steric effects in the continuumtheory.

Keeping qm free to adjustwe found for 8CB a good fit with experimentfor q~= 0.25X i0~rn’ (seefig. 29), which is alsoin the neighbourhoodof 11 andsome20 timesbelow the Debyevalue.Hence theorder of magnitudeis in agreementwith the values in ref. [125]and ref. [120]. Including the nonzerovalue of L

2 in the calculationwill not alter thisconclusiondrastically.Table 6 gives the calculatedvalues of TN! — T*, which turn out to bevery sensitiveto the choiceof

q~.We concludethat the customaryprocedureof locating T* by extendingthe straight line, as in fig.27, is too simple. The agreementof TN! — T* from the Gaussianapproximation and the value forq~= 0.25x i0~rn_i is purely accidental.

Table 6Values of TN!— T* for different fits of the data for

8CB to eq. (4.37) with E= E’ = 0

q,,(109m1) TN!—T*(K)

Gaussianapproximation 1.044.94 (Debye) 0.031.00 0.16

0.50 0.390.25 (bestfit) 1.12

Tricritical caseIn thiscaseTN! — T andSNt aregiven by eqs.(2.12) and(2.14),andwe mustreplacein eqs.(4.39) and

(4.42) the expressionsfor B and C by

B = —6(TNI — T*)/SNJ= —15.2(TNI— T*), (4.43a)

E+ E’ = ~(TN! — T*)/S~~= 92.4(TNI — T*), (4.43b)

Noticethatin thenematicphaseE andE’alwaysappearin thecombinationE + E’whereasfor fluctuationsin theisotropicphaseE + ~E’ is theimportantparameter(seeeq. (ClO)). Thereforewewill useE’/E as an

extraadjustableparameter.Proceedingas before,we arriveat a good fit for q,,., = 0.03x 10~m’ almost independentof E’/E.

This is an order of magnitude lower than the reciprocalmolecular length l’ = 0.4X 109m1, andprobablyrepresentsan unphysicalsolution.

248 E.F. Gramsbergen ci aL, Landau theory ofthe nemasic—isosropic phase transition

ConclusionPre-transitionallight scatteringin the isotropicphaseof 8CB is in good agreementwith a model of

the NI transitionas a mean-fieldcritical point, providedthat the reciprocalmolecularlength is usedasmomentum cut-off q~,,.The latter choice, though somewhatarbitrary, seemsreasonable.With atricritical hypothesisfor the NI transition,not as good an agreementcan be obtained.

5. Conclusions

The Landau—DeGennestheory of thenematic.-isotropic(NI) phasetransitionhasbeenprovedto bevery rich in making qualitative predictions. As theseare independentof the actual values of thephenomenologicalexpansionparameters,they test the generalassumptionsof the theory. In the firstplaceit is predictedthat due to the presenceof a cubic invariant,Tr(Q3), in the free energyexpansion(2.2), the NI phasetransitionis alwaysfirst order. Indeed,experimentallyno exceptionto this rule hasbeenfound up to now. Eventhoughthe phasetransitionis only weakly first order (in the meaningthatthe transitionheatis small), the jumpin the orderparameter,S, is alwaysconsiderable.

Next the theory predictsin addition to the isotropicphasethreedifferent nematicstates:positiveuniaxial (Nj~),negative uniaxial (Nj,) and biaxial (NB). Of all possible phase transitions only thetransitionNUNB is of the secondorder.The appearanceof an uniaxial or a biaxial phaseis determinedby thecompetitionbetweenthetermsTr(Q3) and[Tr(Q3)]2in theexpansion(2.2). In theparameterspaceall thesephasesmeet at a single, isolatedpoint, the Landau point (fig. 12). Experimentallythesepredictionshaveonly beenconfirmed for lyotropic systems,though the Landaupoint has not beenobservedso far. Also, in thermotropicliquid crystalsthe biaxial nematicphasehasnot beenobserved.According to PokrovskiandKats [25] theabsenceof abiaxial phaseandthefirst-ordercharacterof theNI phasetransitionarecorrelated.

Severalpredictionsof the model refer to a situation when an externalelectricof magneticfield ispresent.When applying a field to materialswith a positive magnetic(dielectric) anisotropy,the orderparameterjump at TN! shoulddecreasewhile TN! is predictedto increasewith increasingfield. At acertain finite value of the field this leadsin the model to a field induced NI critical point. Thesepredictionshave been testedsuccessfully in lyotropic systems.The increaseof TN! hasalso beenobservedin thermotropicliquid crystals. The magnetically (electrically) inducedcritical point is inthermotropicmaterials found to be out of experimentalreach. For liquid crystals with a negativemagnetic(dielectric) anisotropythe model predictsa field inducedbiaxiality anda field inducedNUNB

tricritical point. Thesepredictionshaveso far not beenconfirmed experimentally:the estimatedvaluesof therequiredfield aretoo high.

The Landau—DeGennesmodelpredictsmean-fieldtypesingularitiesfor the variousthermodynamicquantitieson approachingthe spinodalor Landaupoint. As the first-ordercharacterof the NI phasetransition is only weak, thesesingularitiescan in principle be observedin the vicinity of TN!. Thisconclusionalsofollows from the Ginzburgcriterium. Combiningall experimentalinformationon criticalexponents,thereis still no clearpicture about the natureof the singularity nearthe NI transition. Itseems,however,that amodelassumingclassical(mean-field)critical exponentsis incorrect.There is noconclusiveexperimentalproofyet whetherthediscrepanciesaredueto thepresenceof anearbytricriticalpoint (assuggestedby Keyes),or thatcorrelationsbetweenfluctuationsof thetensororderparameterareimportant.The secondpossibilitywould agreewith amodelof a nonclassicalcritical point.

Spatialvariationsof the orderparametercan be introducedin the theory by addinggradient termsto

E.F. Gramsbergen et aL, Landau theory ofthe item asic—isotropic phase transition 249

the free energy expansion (eq. (4.2d)). The generalizedLandau—De Gennestheory (GLGT) thusobtainedallows an interpretationof the pretransitionalincreaseobservedin the light scatteringandinthe magneticand electric birefringenceon approachingTN! in the isotropic phaseof liquid crystals.They can be consistentlyand quantitatively describedby use of the GLGT in the Gaussianap-proximation,exceptfor a small temperatureregion close to TN!. As the GLGT containsat least fiveadjustableparametersit is difficult to answerthe questionhow thesedeviationsarise. In principle thepresentexperimentscan beconsistentlyexplainedby going beyondtheGaussianapproximation,i.e., byincluding the interactionbetweenorderparameterfluctuations.

Appendix A

With I,, Tr(Q”) for any symmetricreal3 x 3 matrix Q, we provethat I,,, n > 3 can be expressedas apolynomial of I~,I~and13. Becauseof its symmetry,Q can be written in diagonalform

0 0\

Q=( 0 b 0 J. (A.1)

0 cJ

Thenthe product

1m,Ln2 Imk (A.2)

is a sum of termsof the form

fl ~ N~

N5.N~N~a C

with

Na+Nb+NcNns~nsma, Na,N6,Nc0,1,2 (A.3)

Due to the invarianceof im with respectto interchangeof a, b, c, termswith permutationsof Na, Nb, ‘~will have the same coefficient ANa,N~NC• These terms will be called dependentand the sum ofpermutationsdenotedby the letter P. (For example, P(a

2b)= a2b+ a2c+ b2a + b2c+ c2a+ c2b). Soany linearcombinationof termslike I~~~’

5I~3•..of the sameorderN, can bewritten as a sum of termslike

A N,,.N~,N,,p(~N~.b~,cI’~’*) with Na � Nb � N~� 0. (A.4)

Thenumherof independenttermsis~equalto the numberof elementsin theset

{Na,Nb,Nc},Na�Nb�Nc�0, Na+N6+NcN

250 E.F. Grams/~ergenci aL, Landau theory of the nemasic—isotropic phase transition

or, with Na = N1+N2+N3, Nb = N2+N3, N~= N3, in theset

{N1,N2,N3}, N12 3�0, N1+2N2+3N3= N. (A.5)

This is “by accident” the samenumberas there are combinationsof the form I7!I~~12J~3of orderN = N1 + 2N2+ 3N3. So when all combinationsJ~1J~12J~13of order N are combinedinto the vector iandall combinationsP(a”~b”~’c”)of orderN into the vectorp. We haveprovedthat

iz~Mp (A.6)

where i andp are vectors of the samesize and M is a squarematrix. Now, supposethat a linearcombinationof the rows of M could beformed to give a zerorow-vector. This would imply a relationbetweenIl, 12 andI3~ thusreducingthe numberof degreesof freedomto two. However,wemusthavethreedegreesof freedombecausewe took a, b, c independent.So M must be a regularmatrix andpcan be uniquelyexpressedin i:

p=M’i. (A.7)

Fortunately,one of the elementsof p is P(a”) ‘N, so (A.7) meansthat ‘N can be expressedas apolynomial in .T~,‘2 and13. As an example,we will follow the procedurefor N= 6. Then, i andp areseven-elementvectorsand(A.6) reads:

/ i~ \ /~6 15 30 20 60 90 \ /P(a6)1112 \ / 1 4 7 12 8 16 18 \ / P(a

5b)I~I

3 \ / 1 3 3 6 2 3 0 \ f P(a4b2)

I~1~ = 1 2 3 2 4 4 6 P(a4bc)‘11213 / \ 1 1 1 0 2 1 0 / \ P(a3b3)I~ / \ 1 0 3 0 0 0 6 / \ P(a3b2c)I~ / \ 1 0 0 0 2 0 0/ \P(a2b2c2)

After inverting this relationwe obtain

Tr(Q6) = P(a6) ?~(I~— 3i~’2 + 4I~I3— 9I~I~+ 12111213+ 3J~+ 4I~).

Appendix B

Li this appendixit ism~deplausiblethat the phasediagramin the vicinity of the Landaupoint canhave only two distinct topol Thed ssion only refers to th~~soluteminimum of the freei~i~gy;fluctuationsandmetastablestatesarenot takeninto account.

With 12 = Tr(Q2) and 13 = Tr(Q3), the expansionof the free energyis, up to 10th order:

F a2I2+a4I~+a6I~+a8I~+a10I~+~+I3(/33+85I2+$7I~+f39J~+...)

+I~(y6+Y8I2+ y1~I~+• -)+ 19+.• .)+...

E.F. Gramsbergen ci a!., Landau theory of the nematic—isosropic phase transition 251

We considerthe phasediagramas a function of a2(=~A)andf33(=~B)in the vicinity of a2= /33= 0. Insection 2 we studiedthe cases:(1) a2, /33 variable, a4>0, all other terms zero. (2) a2, /33 variable,a4>0, Y6> 0, all othertermszero. Onemaywonderwhat happensif a4>0, Y6<0. Then,with (2.5) inmind, it can be seen that a positive a, I � 6 or y~,i � 8 term is neededfor stability andthereis nobiaxial phase.The phasediagramis qualitatively that of fig. 8a. Next, if a4� 0 andhigherordera termsareaddedfor stability, thetopologyof thephasediagramis determinedby thesignof y6 in thesameway asbefore.Only whenY6>Oanda4= 0 (thiswouldbequiteaccidental)thewidthof thebiaxialphasebecomesproportionalto (T — T)

3”4 (seefig. Bi). Next,weconsiderthehigher-orderoddtermsf3~,i � 5, 6~,i � 9, etc.The influenceof thosetermsis primarily that the symmetryof the phasediagramaroundthe A axis isbroken.For example,anonzerof3~term leadsto a “rotation” of the phasediagramaroundthe origin,clockwisefor $~> 0 andcounterclockwisefor $~<0 [42].However,in any casethe successionof phasesaroundthe Landaupoint and the orderof the transitionsremainsthat of fig. 8a or fig. 8b.

B

o N~ L I

_~~‘ Nu+

A

Fig. B!. Phasediagram for a4= 0, 16>0 (schematically).

Appendix C

In this appendixwe shall presentsome details of graphscounting for the perturbation(4.33). Asreferencebooksfor thesemethodswerecommend[108,112].In theFeynmangraphlanguageG(q)(seeeq.(4.36)) is given by

G~(q)= G~1(q)— 1(q),

where1(q) is the self-energypart,which is definedas the sum of all irreduciblegraphswith amputated

externallegs:

(C.1)

All Feynmangraphsfor our casearebuilt up from threetypesof vertices(“points”) andtwo typesofpropagators(“lines”). The rules for verticesandpropagatorsareas follows:

252 E.F. Gramsbergen et a!., Landau theory of the nemasic—isotropicphase transition

Vertices:

carriesa factor — V~

:~ carriesa factor — V~,

k carriesa factor — V~imn.

In calculationswe will meetverticeswith openpoints(suchas ), only carryinga factor 1.

Propagators:

1 internalpropagator,carriesa factor G~(q)

i externalpropagator,carriesa factor &~.

Furtherrulesof calculationare:(1) The vertex legsarethe onsetsto propagatorswith correspondingindices.(2) Momentumis conservedin the verticesby a deltafunction

or (2ir)3ô(~q5),

where~ q• is the sum of momentacarried into the vertex by the propagators.(3) All internal indices and momentaare summed,resp. integratedover, keepingthe momenta

within the Debyesphere.(4) The remainingfree indices,1,1 on the externalpropagatorsand the externalmomentumq are

thoseof the correlationfunction G~(q)underconsideration.(5) Only topologically different graphs are drawn separately; topologically identical graphsare

accountedfor by a so-calledstatisticalfactor.As an examplewecalculatethe contributionof the seconddiagramto G11(q):

18 —---~--- +

+ 2 T—i~—T + 2 iT~Ti (C.2)

+ 2 — .i~*c:II:I?.~i— + 2 — — 1

E.F. Gramsbergen ci a!., Landau theory of the nematic—isotropic phase transition 253

The factorsin front of thegraphsarethe statistical factors.The factor of the first graph arisesfrom thefact that an externalleg can be joint to any of the threevertex legslabelled “1”. This happenstwice.The remaining vertex legs can be joined to each other in two ways. So there are 3 x 3 x 2 = 18possibilities.The factor 2 of the secondto fifth graph arisesfrom the two ways in which the internalpropagatorscanbe joint. Eachvertexonly hasoneleg labelled“2” . . . “5”, so, thereis only onechoicefor eachexternalleg. The total of possibilitiescomesto 1 x 1 x 2 = 2. Notice that theô~,in (4.35)greatlyreducesthe numberof graphsto be added.

Taking into accountthe vertex factorsof table3, the graphs(C.2) add up to

r -4TI~——1= [18(B/3V6)2+ 2.2(BIV6)2+ 2.2(_B!2V6)2]—-‘El~-- ~B2HBB(q), (C.3)

where[1201

HBB(q) =-~E:i.->= J ~ G~(k)G~(q— k)

1k <q,,,— k

= 4ir2L1 {tan’ _~ - ~ [tan-i + ~ + O(q4)}, (C.4)

with

= (L/A)1”~, (C.5)

cl’ = ~oqm. (C.6)

In the sameway we find for the first graphin (C.1)

r 121~qL1+2~ _1~L~7CHc,

where[120]

Hc*ø_~ J ~Go(k)=1(~-tan~) (C.8)

16 I<q,,,

For the third graph in (C.2):

=0 (C.9)

becauseit containsthe part —~-‘(3contributing to (~(‘j(q))which is zero in the isotropic phasein theabsenceof externalfields.

254 E.F. Gramsbergen ci aL, Landau theory of the itemasic—isotropic phase transition

For the fourth graph,with (ii) standingfor T~7 (noticethat only V~6~of the form V~jkk

contribute):

r90(1~)+12~ (1~)+6~ (1~)+2~ (1~)=—(63E+21E’)HE, (C.10)j�1 j*’1 1<j<k

where

d3k d3k’= .J~ (~)3 I ~G

0(k)G0(k’)=H~. (C.11)

Notice that the one-loop diagrams(C.7) and (C.9) do not contribute to the q-dependenceof thecorrelationfunction.Adding all contributionsto eq. (C.1)wefinally arriveattheexpressionfor G(q)givenby eq. (4.37).

To do better thaneq. (4.37) we would haveto includethe self-energy(C.1) termslike:

O(C°):--~--

O(B2C):~~~_~ __e__ ~etc.

If thereis alsoa fifth-order term in the free energy,its lowest-ordercontributionis of order (BD)

O(BD):--Q~’-- 8

so its influencewill be small comparedwith the 0(C), 0(B2) and 0(E) graphscalculatedin this

appendix.For that reasonwedid not includesucha termfrom the very beginning.

Acknowledgements

Oneof the authors(L.L.) is indebtedto ProfessorM. Tosi,The InternationalCenterfor TheoreticalPhysics,Trieste,for kind hospitalityat the Centerduring the workshopon Solid StatePhysics,Summer1984. This work is part of the researchprogramof the “Stichting voor FundamenteelOnderzoekderMaterie” (Foundation for FundamentalResearchon Matter, FOM) and was sponsoredby the“NederlandseOrganisatievoor ZuiverWetenschappelijkOnderzoek”(NetherlandsOrganizationfor theAdvancementof Pure Research,ZWO).

E.F. Gramsbergen a a!., Landau theory of the nemasic—isotropic phase transition 255

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