Poisson-bracket approach to the dynamics of nematic liquid crystals

PHYSICAL REVIEW E 67, 061709 2003Poisson-bracket approach to the dynamics of nematic liquid crystalsH. Stark 1 and T. C. Lubensky 21 Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany2 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USAReceived 2 January 2003; published 30 June 2003We use the general Poisson-bracket formalism for obtaining stochastic dynamical equations for slow macroscopicfields to derive the equations that govern the dynamics of nematic liquid crystals in both their nematicand isotropic phases. For uniaxial molecules, we calculate the Poisson bracket between the tensorial nematicorder parameter Q and the momentum density g, as well as those between all pairs of conserved quantities. Weshow that the full nonlinear hydrodynamical equations for the nematic phase derived in this formalism areidentical to the nonlinear Ericksen-Leslie equations. We also obtain the complete dynamical equations for theslow dynamics of the tensorial nematic order parameter Q valid both in the isotropic and the nematic phase.They differ from those obtained by other techniques only in the values of kinetic coefficients and in the numberof nonlinear terms in Q, which are present.DOI: 10.1103/PhysRevE.67.061709I. INTRODUCTIONPACS numbers: 61.30.v, 47.50.dNematic liquid crystals are unique materials, since theyflow in all directions such as a homogeneous fluid and yetthey are optically anisotropic 1,2. Their low-frequency dynamicalproperties, which are characterized by orientationalrelaxation as well as by shear and compressional flow 3–5,are central to the switching behavior of liquid crystal displays6. They give rise to interesting fundamental phenomenaincluding electrohydrodynamical pattern formation 7,instabilities under shear flow 1,2, orientational fluctuationsobservable in light scattering experiments 8, and the anisotropicand nonlinear Stokes drag on a particle embedded in anematic solvent 9. Nematic liquid crystals raise fundamentalquestions about how broken symmetries affect lowfrequencyhydrodynamics, and it is not surprising that overthe years there have been a number of derivations 3–5,1 ofthe equations governing nematohydrodynamics. In this paper,we present a derivation of these equations, and of theequations governing the relaxation of the symmetrictraceless-tensornematic order parameter Q 10–12,1 in boththe nematic and the isotropic phase. We use the Poissonbracketformalism of classical mechanics 13 developed inthe framework of stochastic equations for fluctuating macroscopicfield variables 14–18 and applied to dynamical criticalphenomena 19.Using the formalism of rational mechanics, Ericksen andLeslie were the first to derive a full set of dynamic equations,commonly referred to as Ericksen-Leslie equations 3,4, forthe velocity and director field n specifying the direction ofnematic anisotropy. Later, the Harvard group argued that theinterpretation of the dynamic equation for the director theso-called Oseen equation as the balance law for the angularmomentum is not well justified 20, and they presented arigorous derivation of the linearized hydrodynamic equationsstarting from conserved and hydrodynamic Goldstone variables5. The full nonlinear theory based on this apprachwas developed by Pleiner and Brand 21. De Gennespointed out the common features of the two approaches 1.Recently, Sonnet and Virga rederived the Ericksen-Leslieequations in the framework of a variational principle of Rayleighstarting from complex fluids with a general tensorialorder parameter 22.The low-frequency hydrodynamics of the nematic phaseis determined entirely by the conserved densities, characterizingboth the isotropic and nematic phases, and the directorn. There are, however, physical situations in which the directoralone does not provide a complete description of thenematic orientational order, and a description in terms of thefull nematic order parameter Q also called the alignmenttensor 23 is required. As a result, there has been an increasinginterest in dynamical equations both in the nematicand the isotropic phases in which the nematic order is describedby Q rather than by n alone. These equations necessarilyinclude nonhydrodynamic modes, which nonethelesshave characteristic relaxation times that are slow comparedto microscopic times. In conjunction with the Landau-Ginzburg–de Gennes free energy, they provide a descriptionof dynamical processes close to the nematic-isotropic phasetransition 23–26, of processes in which the degree of orderingcan vary, e.g., by applying a shear deformation, or ofprocesses in thin liquid crystal cells subject to large externalfields 27. They also provide a detailed description of thedynamics of line and point defects 28,29 whose cores exhibita complex biaxial ordering which is beyond the simpledirector picture 30.The first complete formulation of the dynamic equationsinvolving the Q tensor was presented by Hess 23. Subsequentrefinements were carried out by Kuzuu and Doi, whostarted from a molecular kinetic equation 31, and by Olmstedand Goldbart 25, who set up the dynamic equations infull analogy to de Gennes’s derivation of the Ericksen-Leslietheory in Ref. 1. In parallel, Edwards, Beris, and Grmelapioneered a derivation based on a Hamiltonian formulationof continuum mechanics 32. Further approaches to the Qtensor dynamics are introduced in Ref. 27 by Qian andSheng, in Ref. 22 by Sonnet and Virga, and in Ref. 33 byPleiner, Liu, and Brand. The latter though purely phenomenologicalis very close in spirit to the approach we presenthere.The Poisson-bracket approach we adopt in this paper hasseveral advantages. First, it automates the derivation of1063-651X/2003/676/06170911/$20.0067 061709-1©2003 The American Physical Society

H. STARK AND T. C. LUBENSKY PHYSICAL REVIEW E 67, 061709 2003coarse-grained dynamical equations once the dynamic fieldvariables are defined. As we shall see in the following, thisapproach, however, requires more carefulness in nematic liquidcrystals than it does in other systems such as magnetsand superfluids. Second, the formalism applies to nonhydrodynamicslow variables as well as to true hydrodynamic variables.Third, it gives easy access to nonlinear terms, as theyappear, e.g., in the convective derivative of the Navier-Stokes equations. The formalism has been applied to diversesystems such as antiferromagnets 19, quasicrystals 34,or to nematic polymers 35 which demonstrate its universality.In the literature, different methods are used to calculatethe Poisson brackets. Volovik employs pure symmetry arguments36,37 that are not sufficient to provide a full descriptionof nematic liquid crystals because it fails to properlytreat the degree of nematic ordering. Edwards, Beris, andGrmela on the other hand define their Poisson brackets on acontinuum level 38.Here, in contrast, we start from a microscopic definitionof the field variables. Following the work of Forster 39, weintroduce such variables for model-liquid-crystal moleculesmade from point particles and employ the conventional definitionfrom classical mechanics to arrive at coarse-grainedPoisson brackets for the macroscopic field variables. A carefultreatment is needed to determine the central Poissonbracket between the alignment tensor Q and the momentumdensity which extends the work of Forster 39. We thenrederive the Ericksen-Leslie theory and rigorously justify theOseen equation. The Onsager and Parodi relations for theLeslie viscosities are automatically fulfilled 40. We derivegeneralized dynamic equations for the alignment tensor. Ourequations have the same form as those derived by Olmstedand Goldbart 25 and those obtained by Pleiner et al. 33.We obtain explicit forms for reactive coefficients that aremerely unspecified phenomenological coefficients in theseapproaches. Our equations have terms not present in those ofOlmsted and Goldbart, and those of Pleiner et al. containnonlinear terms that our equations do not.The paper is organized as follows. In Sec. II, we introducethe general formalism. In Sec. III, we define the molecularmodel and calculate all relevant Poisson brackets. The fullsets of dynamic equations for the director or the Q-tensordynamics are derived and discussed in Secs. IV and V, respectively.II. GENERAL FORMALISMIn this section, we summarize the general formalism onwhich our derivation of the dynamic equations in nematicliquid crystals is based. Let us consider systems whose microscopicdynamics is determined by canonically conjugatevariables x and p for each particle and a microscopicHamiltonian Ĥ(x ,p ), where x x 1 , x 2 , . . . etc.Here, we focus on a set of macroscopic field variables (x,t) for 1,2,... obtained from microscopic fieldsˆ (x,x ,p ) by coarse graining over spatial fluctuationson the microscopic level; (x,t)ˆ (x,x ,p ) c ,where the symbol ••• c specifies the coarse-grained averages.The statistical mechanics of the fields (x,t) is determinedby the coarse-grained Hamiltonian H ,which is a functional of the macroscopic variables. It is constructedin such a way that the statistical averages of observablesin the microscopic and the macroscopic representationare identical. In formulas, this means f ˆ Dx ,p fˆ )e Ĥ/k B T f „ x,t… D x,tf „ x,t…e H/k B T ,where k B is the Boltzmann constant, T is the temperature,and D(x ,p ) and D (x,t) are integration measures. Theobservable f is a function of the microscopic (ˆ ) or macroscopic( ) fields. In practice, phenomenological formsfor H are used.The macroscopic fields (x,t) describe the slow dynamicresponse of the system, i.e., they are either hydrodynamicor quasihydrodynamic variables whose characteristicdecay times in the long-wavelength limit are much largerthan microscopic decay times. Then, following the theory ofkinetic or stochastic equations, the variables evolve accordingto 17–19 x,tHVt x x ,where the reactive term V (x) is also called nondissipativeor streaming velocity and the second term introduces dissipativeeffects. These equations, which can be rigorously derivedfrom microscopic principles following the works byZwanzig 14, Kawasaki 15, and Mori, Fujisaka, andShigematsu 16, describe the low-frequency and the longwavelengthdynamics of the system. When applied to an isotropicfluid, for example, they correctly reproduce the fullynonlinear Navier-Stokes equations. Supplemented with anoise term (x,t), they provide a powerful formalism forcalculating dynamical correlation functions in equilibriumand, especially, for treating effects due to the coupling betweenmodes. As such, the stochastic equations are used extensivelyin the study of dynamical critical phenomena,where nonlinearities are important 19. In this paper, we aremainly interested in deriving the equations for the lowfrequencydynamics of the isotropic and nematic phases ofliquid crystals, and we will not give any further considerationto the noise. Disregarding noise in Eq. 1 means thatnonequilibrium averages of the macroscopic field variablesare used 41.The concrete form of the reactive term involves Poissonbrackets which are the central quantity of this formalism; itcan be expressed asHV x d 3 xP x,x x ,12061709-2

POISSON-BRACKET APPROACH TO THE DYNAMICS OF ...where Einstein’s convention on repeated indices is understoodandP x,x x, xP x,xdenotes the Poisson bracket of the coarse-grained variables.It is defined as the coarse-grained average of the microscopicPoisson bracket:where 13 x, xˆ x,ˆ x c ,ˆ x,ˆ xiˆ x ˆ xp i ˆ xx ix i34ˆ x. 5p iEquation 2 can be rigorously derived from microscopicprinciples 14–16. Roughly speaking, it can be viewed asthe Poisson bracket H, which in classical mechanicsdescribes the time evolution of an observable 13. Applyingthe chain rule to the derivatives of the Hamiltonian and notingthat is a field variable, the formal structure of Eq. 2results. However, since we only employ a restricted numberof coarse-grained macroscopic variables, all the ‘‘neglected’’microscopic degrees of freedom give rise to the dissipativeterm in the kinetic equation 1. It is proportional to thegeneralized force H/ (x) which together with (x)forms a pair of conjugate variables. The quantity iscalled the dissipative tensor that, in general, may depend onthe fields and that may also contain terms proportional to“ 2 . It is determined by three principles. First, /t canonly couple to H/ if it possesses a different sign undertime reversal, the signature of dissipation. Second, hasto reflect the local point group symmetry of the dynamicsystem, and third, it has to be a symmetric tensor at zeromagnetic field to obey the Onsager principle 42.III. POISSON BRACKETS FOR NEMATICLIQUID CRYSTALSOur goal is to use the formalism outlined above to derivethe dynamical equations for the long-lived fields in a nematicliquid crystal and the isotropic phase from which the nematicphase develops. Our most important results are the Poissonbrackets of Eqs. 4 and 5, which we will derive in thissection starting from an explicit microscopic model for theconstituent molecules.Conserved fields are always hydrodynamical variables becausetheir frequencies necessarily tend to zero with wavenumber. In systems with a continuous broken symmetry,there are additional broken-symmetry hydrodynamical variables.In addition, there can be modes that have characteristicdecay times that are much longer than any microscopic timeseven though they do not tend to infinity with wave number.The conserved variables of both the isotropic and nematicphases are the mass density (x,t), the momentum densityPHYSICAL REVIEW E 67, 061709 2003g(x,t), and the energy density (x,t), where the latter quantitywill not be considered further in this paper. In the nematicphase, the Frank director n(x,t) is the broken-symmetryhydrodynamical variable with two independent degrees offreedom. As long as there are no topological defects present,it is sufficient to describe the dynamics of the orientationalorder by n(x,t). Otherwise, one has to resort to a more generalorder parameter that is able to quantify, e.g., the biaxialordering in the core of a topological defect. We will describenematic order with the conventional Saupe–de Gennessecond-rank symmetric-traceless tensor parameter Q(x,t)10–12,1, also called alignment tensor 23. In the isotropicphase, especially near the isotropic-to-nematic transition,Q(x,t) can be a slow variable whose relaxation to equilibriumis much slower than any microscopic collision time. Inthe nematic phase, it also contains slow, nonhydrodynamiccomponents in addition to the hydrodynamic director variable.The complexity of nematic order presents problems notencountered in many other broken-symmetry systems suchas magnets and superfluids. Nematic order is associated withlocal molecular rigidity, and it only arises if constituent moleculesare anisotropic. A central quantity in our formalismare the Poisson brackets between the alignment tensorQ(x,t) or the director n(x,t) and the momentum densityg(x,t). However, in the following it will become obviousthat the calculation of these Poisson brackets is not straightforwardsince neither the nematic order parameter Q nor thedirector n can be defined as simple coarse-grained averagesof microscopic quantities.A. Model molecule and dynamic variablesOur system contains a number of identical model-liquidcrystalmolecules indexed by , which consist of a set ofmass points with masses m and position vectors x .Wealso introduce position vectors x x x relative tothe location of the center of mass of each molecule at x (m x )/m 0 , where m 0 m is the total mass of amolecule. In our derivation of the Poisson brackets, it isimportant that the molecules are not rigid, but that their constituentatoms be allowed to fluctuate around their averagepositions. For simplicity, we will assume that the moleculeshave a uniaxial shape in the sense that their moment of inertiatensor, to be defined below, is also uniaxial. Consideringthe fact that molecules perform fast rotations about theirlong axis on a time scale of 10 10 s 43, this is not a verysevere restriction. With our model we can describe a hugerange of possible molecules ranging from needlelike objectsto molecules with prolate and oblate disklike shape. Evenglobular polymers with anisotropic shape are included.We employ the conventional microscopic definition of thedensity of mass and momentum:ˆ 1ˆ x m xx ,ˆ 24ĝx p xx ,67061709-3

H. STARK AND T. C. LUBENSKY PHYSICAL REVIEW E 67, 061709 2003which, after coarse graining, result in the macroscopic variables(x)ˆ (x) c and g(x)ĝ(x) c . Note that the function has the dimension of an inverse volume. We stressthat the definition of the momentum density in Eq. 7 includesthe motion of all mass points, including, in particular,rotational motion about the centers of mass of anisotropicparticles.To describe the orientational order of the molecules, wefollow the approach of Forster 39 who extracted the anisotropicpart from the density of the moment-of-inertia tensorand used its negative part as an additional microscopic fieldvariable:Rˆ ij x where for each molecule we haveRˆ ij xx ,8The nematic order parameter Q ij (x) must reduce to n i n j 1 3 ij when all local axes ˆ are aligned along the spatiallyuniform director n, and all molecules have the same value ofR . It is straightforward to verify that the tensor Q ij (x) definedin Eq. 12 satisfies this constraint. Note that the fieldsQ ij (x) constitute the fields 59 (x) in our formalism.The parameter R can easily and usefully be related to theanisotropy in the molecular moment-of-inertia tensorI ij m x 2 ij x i x j , 14whose components parallel and perpendicular to the localanisotropy axis ˆ in a uniaxial system are, respectively,I m x 2 , 15Rˆ ij m x i x j 1 3 x k 2 ij . 9The coarse-grained variable R(x)Rˆ (x) c is a symmetricand traceless tensor of second rank, as demanded for thenematic order parameter Q(x), but it also depends on thedensity of the molecules and, more important, it includes thefast fluctuations of the molecules around their average configuration,which we are not interested in. To arrive at anexpression for Q(x), we note that Rˆ ij is a tensor that for anyparticular atomic configuration can be represented in a basisin which it is diagonal. Since this tensor is traceless, it hasonly two independent eigenvalues, one of which describesmolecular biaxiality. Here, we will consider only moleculesthat are, on an average, uniaxial. In any real molecule, therewill be fluctuations in which it becomes momentarily biaxial.We assume, however, that these fluctuations relax rapidly innonhydrodynamic times. In this case, Rˆ ij is characterized bya single parameter in addition to the unit vector ˆ itslargest-eigenvalue principal axis. ThenwhereRˆ ij R Q ij ,Q ij ˆ i ˆ j 1 3 ij1011is the local molecular alignment tensor and R will be givenbelow. When this form of Rˆ ij is valid, we can define thecoarse-grained position dependent Saupe–de Gennes orderparameter viaR ij x Rˆ ij xx RxQ ij x, 12cwhereRxRˆ xc R xx . 13cI m x 2 /2x 2 , 16where x and x are, respectively, the component ofx parallel and perpendicular to ˆ . The anisotropy I distinguishes between prolate (I 0) and oblate (I 0) molecules. For uniaxial systems, we findR I I I .17Only after calculating the Poisson brackets see Sec. III C,can we safely replace R by its averaged value I, since, inmicroscopic times, R of different molecules relax to thesame value I. In this case, we can replace R(x) withRx Ix.18m 0Then, Eq. 12 results in the conventional definition ofQ ij (x), where Q ij see Eq. 11 is averaged over many molecules.In what follows, we will also useTr I II 2I ,19where the final form is in the uniaxial case we consider whenall fast modes have relaxed.We stress that our approach is a generalization of the onedescribing a simple flexible diatomic molecule in which theprincipal moments of inertia fluctuate and therefore have anonvanishing Poisson bracket with the momentum density.Our quantity Rˆ (x) incorporates these fluctuations for a morecomplex molecule. As we will see below, the Poisson bracketsof Rˆ (x) with the momentum density will give rise to oneadditional term in the analogous Poisson brackets of Q(x).This essential contribution was not taken into account byForster 39 who considered R(x) as a constant. Instead,Rˆ (x) is a dynamic quantity since it depends on the massdensity of the molecules through the functions in Eq. 13and, more importantly, also reflects the fluctuations in theanisotropy R of the molecular moment of inertia tensor.Whereas the dependence on the mass density only produces061709-4

H. STARK AND T. C. LUBENSKY PHYSICAL REVIEW E 67, 061709 2003Q ij x,g k x 1Rx R ijx,g k x Q ijxRx Rx,g kx.Calculating the single Poisson brackets, we employx ix j ij mm 0,2526Rˆ x,ĝi x i Rˆ xxxOx 2 2 From Eq. 10, wefind3 Rˆ I ik Rˆ kj I6 Rˆ ijxx j xx.Rˆ ik Rˆ kj 1 3 I Rˆ ij 2 9 I 2 ij .3435xx x j l xx m ,m 027where, in the course of the manipulations, m /m 0 eithersums up to one or gives a vanishing term due to the definitionof the center of mass: m x 0. We also use aTaylor expansion of the function,xxx xxx l l xxOx 2 2 ,28where O¯ means ‘‘order of.’’With these comments, theevaluation of the Poisson bracket for Rˆ ij (x) is straightforwardand the coarse-grained result readsR ij x,g k x k R ij xxxOx 2 2 So after coarse graining, Eq. 34, and replacing I by I andI by I, we obtainRx,g i x i RxxxOx 2 2 12I IR ij x 1 3ij Rx j xx.36Finally, combining Eqs. 29 and 36 in Eq. 25 leads to thePoisson bracket Q ij (x),g k (x) of Eqs. 22 and 23.IV. DERIVATION OF DIRECTOR DYNAMICSIn the nematic phase, the orientational order is uniaxialand the alignment tensor assumes the formR ij x kl 2 3 R kl x ijQ ij xQˆ ij x c Sxn i xn j x ij /3,37whereR t x 1 6 R jl x ik R il x jk l xxR t x ik jl jk il 2 3 ij kl l xx,29Tr I xx . 30cwhere, on an average, the molecules point along the directorn(x). Projecting Q on n i n j in the definition Rˆ ij (x) cR(x)Q ij (x), and using Rˆ ij from Eq. 10 and R(x) fromEqs. 18 gives the conventional Maier-Saupe order parameterSx m 0 3x 2 cos 2 3 1 xx , 38cAs we discussed earlier, the components of the moment-ofinertiatensors I relax in microscopic times to theirmolecule-independent value I with components I and I .Inthis limit, we haveR t xI6I Rx.31To complete our calculation of Q ij (x),g k (x), we need toevaluate R(x),g k (x). To do so, we use the relationto calculateRˆ ij Rˆ ij 2 3 R 2R ,p k 3 2 R 1 Rˆ ij Rˆ ij ,p k .Using this result and R I , we obtain3233where is the angle of the principal axis of the molecule relative to n(x) and m 0 is the molecular mass. Note that••• c means coarse graining over distances much largerthan the molecular scale so that S is defined by averagingover many molecules. The prefactor m 0 /ϱ has to appear,since S should not depend on the number density of themolecules.A. Poisson brackets for the directorTo derive the dynamic equations in the nematic phasewith its constant S(x), we need the Poisson brackets for thedirector. However, a direct microscopic definition of the directorsuch that n(x)nˆ(x) c does not exist. The director isonly defined via the alignment tensor Q in Eq. 37. Wetherefore employ the macroscopic Poisson bracketQ ij (x),g k (x), insert the uniaxial alignment tensor of Eq.37, and apply the product rule for Poisson brackets:061709-6

POISSON-BRACKET APPROACH TO THE DYNAMICS OF ...Q ij x,g k xn i xn j x 1 3 ij Sx,g k xSxn i xn j x,g k xSxn j xn i x,g k x.39Taking the trace of this equation with respect to i, j and projectingit on n i (x) and n j (x), respectively, we obtain therelations2Sxn i xn i x,g k x0,2Sx,g k xQ ij x,g k xn i xn j x.40a40bTaking the dot product of Eq. 39 with n i (x) and using Eqs.40 give the director-momentum Poisson bracketwheren i x,g j x 1 S ikT Q kl x,g j xn l x, T ij ij n i n j4142is the projector on the space perpendicular to n(x). It has toappear since the derivative n/t has to be perpendicular ton itself due to n 2 1. After insertion of Eq. 22 into Eq.41, we arrive at the final expressionn i x,g j x j n i xxx ijk x k xx,43withwhere ijk T il n m lmjk /S 1 2 T ij n k T ik n j 1 2 T ij n k T ik n j , 1 3 1 I 1I S4445denotes a reactive coefficient. It depends on the Maier-Saupeorder parameter and, in addition, on the molecular parametersI and I. It agrees with Forster’s result so the additionaldynamic variable R(x) does not affect its value 39.In the case of needlelike molecules, I/I2 and (12/S)/3. If, in addition, all the molecules are completelyaligned (S1), we obtain 1 and ijk T ij n k . Thisagrees with Volovik’s result, who derived the Poisson bracketfor the director on pure symmetry arguments 37, but it alsodemonstrates that the symmetry arguments are not sufficientfor determining Poisson brackets in complex media. Finally,the Poisson bracket that enters the momentum balance readsg i x,n j x i n j xxx k xx jik x.46B. Nondissipative velocitiesTo calculate the nondissipative velocities from Eq. 2, weneed the HamiltonianH g2 x2x d3 xFx,nx. 47It consists of a kinetic part and a free energy F(x),n(x) f (,n,“n)d 3 x, which is Frank’s free energy plus a termpurely depending on .In Sec. III B, we have already calculated the nondissipativevelocity for the density of mass as V div g(x). Forthe director, we obtain with Eq. 43,V i n vx•n i x ijk x k v j x.48The first term on the right-hand side is the convective derivativeof n and the second term introduces a reactive couplingto the deformation rate. Note that n•V n 0, as it should besince n is a unit vector. The most complex term is the nondissipativevelocity of the momentum density. It employsEqs. 24a, 24c, and 46 to getV g i j g ixg j xFxx ix Fin j xn j x kF jik xn j x.49The divergence of the momentum flux tensor originates fromthe kinetic part of the Hamiltonian; a surface term has beendropped since we are only interested in bulk properties. Thefourth term introduces an elastic coupling to the molecularfield F/n. The second and the third term can be rewrittensuch see the Appendix that the pressureand the elastic stress tensorp f f 0 ij f j n i n k ,k5051known from the Ericksen-Leslie equations, appear. The nondissipativevelocity finally readsV g i j g ixg j x x i p j ijPHYSICAL REVIEW E 67, 061709 2003C. Dissipative terms0 kF jik xn j x.52In this section, we collect the dissipative terms for thedynamic equations following the rules outlined in Sec. II.For the mass density no such term appears since it obeys aconservation law.The time derivative n/t couples to conjugate forceswith a different sign under time reversal. A possible terminvolving H/ cannot occur since the dissipative tensor is061709-7

H. STARK AND T. C. LUBENSKY PHYSICAL REVIEW E 67, 061709 2003a vector pointing along n due to symmetry, thus violating therequirement that n/tn. A second dissipative term introducesa coupling to H/n i F/n i : 1 T Fij ,53n jwhere is a rotational viscosity.The time derivative of g can only couple to the velocityv i H/g i . Pure couplings to v i are forbidden since theywould break the Galilean invariance of the equations. Thus,we are led to j ijkl k v l j ij where we have introducedthe viscous stress tensor ij . The tensor of viscosities ijklpossesses two permutation symmetries: 1 ijkl jikl since ij should be symmetric 44 and 2 ijkl klij since it isassociated with the dissipative energy ijkl i v j k v l d 3 x.Furthermore, it has to obey the local D h symmetry of thenematic phase. It can therefore be written as a sum of termsthat only contain ij and n i : ijkl ¯1n i n j n k n l 42 ik jl il jk ¯5¯6n4 i n k jl n j n k il n i n l jk n j n l ik 1 ij kl 2 ij n k n l n i n j kl .The viscous stress tensor then takes the form 1 ¯1n i n j n k n l A kl 4 A ij ¯5¯6n2 i A jk n j A ik n k ij 1 ij A kk 2 ij n k n l A kl n i n j A kk ,5455where the symmetrized velocity gradient A ij (“ i v j“ j v i )/2 is also called strain rate tensor. The last two termsonly appear in compressible fluids renormalize the pressure.Therefore, they do not occur in the Ericksen-Leslie equations.We will comment on the bars of some of the viscositiesin the following section.D. Final equationsCollecting the reactive and dissipative terms, we arrive atthe final equations. We first obtain the conservation law forthe density of mass: ijk k v j T ij n k A jk n with 1 2 curl v58and introduce the rate of change of n relative to a vortex inthe fluid,N dndt n,59we arrive at the Oseen equation well known from theEricksen-Leslie theory: 1 N i 2 T Tij A jk n k Fij ,n j60where 1 and 2 . Note that the ratio of the twoviscosities 2 and 1 is the reactive coeffcient which onlydepends on S and the two molecular parameters I and I, asfirst derived by Forster in Ref. 39 and later by Kamien 35.This coefficient determines the angle between the flow directionand the director in a shear field; a phenomenon calledflow alignment.Finally, for the momentum balance equation we obtaing itj g ig j i p j 0 ij jF kijn k j ij .61The first two terms correspond to the familiar material derivativedv i /dt. Our dynamic equation for the momentumdensity completely agrees with the momentum balance ofEricksen and Leslie. We have already identified the elasticstress tensor 0 ij in Eq. 51. The two terms in the second linegive the complete viscous stress tensor EL ij of Ericksen andLeslie. To show this, we replace F/n k by the Oseen equation60, use Eq. 55 for ij , and, after some manipulations,arrive at EL Fij kij n ijk 1 n i n j n k n l A kl 2 N i n j 3 n i N j 4 A ij 5 n j n k A ik 6 n i n k A jk 1 ij A kk 2 ij n k n l A kl n i n j A kk ,62where the Leslie viscosities i are related to our viscositiesbyt “•g.The equation for the director readsn it v•“n i ijk k v j 1 T Fij ,n j5657 1 ¯1 2 , 2 1/2, 3 1/2, 4 4 ,63a63b63c63dwhere ijk and the reactive coefficient are given in Eqs.44 and 45. Note that dn i /dtn i /tv•“n i is the materialderivative of n i . If we write061709-8 5 ¯5¯6/21/2, 6 ¯5¯6/21/2.63e63f

H. STARK AND T. C. LUBENSKY PHYSICAL REVIEW E 67, 061709 2003It is evident that the viscous stress tensor of Eq. 71 containsseveral shear viscosities which need to be worked out.C. Final equationsIn discussing our final equations, we will compare them tothe set of equations derived by Olmstedt and Goldbart whofollowed the path outlined by de Gennes in Ref. 1 to derivethe Ericksen-Leslie equations.The density of mass obeys the mass-conservation law.The dynamic equation for the alignment tensor readsQijQ ijv•“Qtij ijkl l v k 1 FWith ijkl (x) from Eq. 23, we obtain explicitlyst. 73 ijkl l v k Q il W lj W il Q lj 1 I3 I A st ij 2 3 Q ijA kk2QA st ij 1I IQ ij Q kl A kl , 74where W ij ( i v j j v i )/2 is the antisymmetric part of thevelocity gradient. The commutator of Q and W on the righthandside of Eq. 74 describes a rotation of Q due to a fluidvortex. With this in mind, we introduce, in the spirit of theEricksen-Leslie equations, the new dynamic variable for therate of change of the alignment tensor:and reformulate Eq. 73 asK ij dQ ijQdt il W lj W il Q lj K ij 2 3 Q ijA kk 1 I3 I A ijst st2QA ij 1I IQ ij Q kl A kl 1 FQijst75. 76This result is essentially the same as that obtained by Pleineret al. 33, except that their result has additional terms quadraticin Q ij and linear in A ij . It is also very close to theresult of Olmsted and Goldbart who derived their equationby a linear expansion of the dissipative flux (K) into thestgeneralized forces A ij and F/Q st . The first term onthe right-hand side vanishes for an incompressible fluid(A kk 0) and, therefore, does not appear in the Olmsted-Goldbart approach. The second term has a reactive naturewith a known coefficient (I/3I); it appears in the Olmsted-Goldbart treatment with an arbitrary coefficient 3 . The thirdand fourth term do not appear in their treatment but, in theirspirit, could be regarded as reactive terms of higher order inQ. They necessarily appear in our approach with calculablecoefficients. The last term has a dissipative nature in bothapproaches.Finally, the momentum balance equation takes the formg itj g ig j i p j 0 ij jF klijQ kl j ij ,77where the elastic ( 0 ij ) and viscous ( ij ) stress tensors aredefined in Eqs. 67 and 71, respectively. Here, we alsohave a close resemblance with the result of Olmsted andGoldbart. The first line of Eq. 77 is identical to their expression.To investigate the second line, we write explicitly klijQklQikII Q F stijF 2 Q kl 3 ijQ kl F st2Q jk F st 1 3 12I IQ ij Q kl F st. 78QklThe first term on the right-hand side renormalizes the pressureand, therefore, does not exist in the Olmsted-Goldbartapproach. Furthermore, only the antisymmetric part of thesecond term appears in their paper. They had to introduce itin order to obtain the proper dissipation functionstst Fd3 x, 79T dSdt ijA ij 1 Q FijQijwhere 1 F/Q st is the dissipative flux introduced in Eq.69 in connection with the generalized force F/Q st .The third tensor on the right-hand side of Eq. 78 with itsreactive coefficient I/3I again corresponds to a dissipativeterm with the viscous coefficient 3 in the Olmsted-Goldbartapproach. The fourth tensor, of second order in Q, is new.Finally, Olmsted and Goldbart only introduced the isotropicpart of the viscous stress tensor ij in Eq. 71, since theywere close to the nematic-isotropic phase transition.In conclusion, we find a remarkable agreement betweenour formalism and the approach by Olmsted and Goldbart.By identifying the reactive character of two of the terms, weare able to give a concrete value for the viscous coefficent 3in the Olmsted-Goldbart paper in terms of the molecular parametersI and I. Furthermore, we arrive quite naturally atadditional terms with a reactive nature whose consequencesneed to be worked out.ACKNOWLEDGMENTSThe authors thank R. Kamien for helpful discussions andH. R. Brand and H. Pleiner for bringing Refs. 21,39,33 toour attention. H.S. acknowledges financial support from theDeutsche Forschungsgemeinschaft under Grant No. Sta 352/5-1. T.C.L. was supported by the U.S. National ScienceFoundation under Grant No. DMR00-96531.APPENDIX: PRESSURE AND ELASTIC STRESS TENSORTo introduce the pressure p of Eq. 50 and the elasticstress tensor 0 ij of Eq. 51 in Eq. 49, we rewrite the secondterm on the left-hand side as061709-10

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