DYNAMICAL PHENOMENA IN LIQUID-CRYSTALLINE MATERIALS

29 Oct 2001 17:38 AR AR151-10.tex AR151-10.SGM ARv2(2001/05/10)P1: GJC

Annu. Rev. Fluid Mech. 2002. 34:233–66Copyright c© 2002 by Annual Reviews. All rights reserved

DYNAMICAL PHENOMENA IN

LIQUID-CRYSTALLINE MATERIALS

Alejandro D. ReyDepartment of Chemical Engineering, McGill University, Montreal, Quebec H3A 2A7,Canada; e-mail: [email protected]

Morton M. DennThe Benjamin Levich Institute for Physico-Chemical Hydrodynamics, City College of theCity University of New York, New York, New York 10031; e-mail: [email protected]

Key Words liquid crystal, nematic, Leslie-Ericksen theory, liquid-crystallinepolymer, interface

■ Abstract Recent progress in modeling and simulation of the flow of nematicliquid crystals is presented. The Leslie-Ericksen (LE) theory has been successful inelucidating the flow of low molar-mass nematics. The theoretical framework for theflow of polymeric nematic liquid crystals is still evolving; extensions of the Doi theorycapture qualitative features of the flow of polymeric nematics in simple geometries, butthese theories have not been shown to predict texture development in flow. Mesoscopictheories for textured materials based on spatial averaging capture only some quali-tative features of nonrectilinear liquid-crystalline polymer flow. Interfacial effects inliquid-crystalline systems have begun to receive attention in the context of interfacialviscoelasticity and the dynamics of dispersed liquid-crystalline polymers in immiscibleblends.

1. INTRODUCTION

The wide range of applications of liquid-crystalline materials has created new ar-eas of academic and industrial research. Some of the more important applicationsof low molar-mass liquid crystals include displays, light valves, and temperatureand pressure sensors, whereas some of the potential uses include chromatographyand smart fluids for brakes and clutches. One of the most important new develop-ments in display technology is the emergence of polymer-dispersed liquid crystalsfor applications in flat-panel television technology and switchable windows. Thesynthesis of polymer liquid crystals has enlarged the range of applications of thesematerials to areas where unique properties are important, including high-strengthfibers and injection-molded parts for electronic interconnects and extreme chem-ical and thermal environments. Some naturally existing materials, like coal andpetroleum pitches, are spun from the liquid crystalline state into high-strength

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carbon fibers. A large number of reviews, textbooks, and monographs on the theory,applications, and rheology of liquid-crystalline materials are available (de Gennes& Prost 1993, Kleman 1983, Ciferri 1991, Noel & Navard 1991, Beris & Edwards1994, Srinivasarao 1995, Marrucci & Greco 1993, Burghardt 1998, Larson 1999,Demus et al. 1999, Sonin 1995, Rey 1993a, Tsuji & Rey 1998, Sawyer et al. 1998).

The modeling of these structured materials must take the partial positional andorientational order into account, which necessitates adding new balance equationsto those that govern structureless fluids. For a uniaxial nematic liquid crystal,for example, an internal momentum balance equation is required to describe theaverage macroscopic orientation of the liquid. As is the case in other continua,constitutive equations reflecting the symmetry properties of the phases are requiredto specify relations between forces and fluxes. These requirements give rise to avariety of complex macroscopic theories applicable to different phases.

The nematic phase, which is characterized by orientational order but positionaldisorder, is perhaps the most-studied liquid-crystalline phase and is the focus ofthis chapter. Nematics are usually anisotropic molecules with rigid segments inthe structure, often because of the presence of parasubstituted phenyl rings. Thespatial and temporal distribution of the average orientation in liquids made up ofrod-like or disk-like molecules can often be described by a unit vector fieldn, thedirector. The presence of order admits the possibility of defects, from which thename nematic is derived (Kleman 1983). The spatial distribution of defects definesthe texture of a liquid crystal; texture is a “fingerprint” of a liquid-crystalline phaseand plays a significant role in the rheology. The imperfect molecular alignmentalong the average orientation is described by one or more scalar order parame-ters that, in polymeric nematic liquid crystals, are likely to be affected by strongfields. The Leslie-Ericksen (LE) vector theory, which is applicable to rigid-rod anddiscotic nematics in the absence of spatio-temporal variations of the scalar-orderparameter, is successful in describing the flow of low molar-mass nematics (deGennes & Prost 1993). The theory may also apply to some slow flows of nematicpolymers. Sufficiently strong polymer flows affect the order parameters, and mod-els based on the dynamics of the nonequilibrium orientation distribution function,such as the Hess theory (Hess 1975) and Doi theory (Larson 1999) as well asits extensions (Feng et al. 2000, Kupferman et al. 2000, Lhuillier 2000), are nec-essary. The mesoscopic Landau–de Gennes tensor model, which is based on thesecond moment of the orientation distribution function, generalizes the LE theoryto variable order parameters and is successful in describing flow-induced texturaltransformations (Tsuji & Rey 1998) up to the defect-core length scale (Schopohl& Sluckin 1987, 1998; Hudson & Larson 1993).

2. CLASSIFICATION OF LIQUID-CRYSTALLINE PHASES

2.1. Low Molar-Mass and Polymeric Liquid Crystals

Phase transitions to the ordered fluid state can be effected through changes intemperature (thermotropic liquid crystals) or concentration (lyotropic liquid


LIQUID-CRYSTALLINE MATERIALS 235

crystals). The molecular orientational order can best be achieved with molecu-lar shapes that are disc-, lath-, or rod-like. Typical thermotropic low molar-massliquid crystals are N-(p-methoxybenzylidene)-p′-n-butylaniline (MBBA), 4,4′-Di-methoxyazoxybenzene (p-azoxyanisole), 4-pentyl-4′-cyano-biphenyl (5CB),and 4-octyl-4′-cyano-biphenyl (8CB) (de Gennes & Prost 1993). Macromole-cules can also form similar liquid crystals (Donald & Windle 1992); a typi-cal example is the random copolymer of 73%p-hydroxybenzoic acid and 27%p-hydroxynaphthoic acid, sold commercially as Vectra A. The mesogenic groupscan be attached to the macromolecules as side chains or part of the main chain. So-lutions of anisotropic molecules in an isotropic solvent form liquid-crystal phasesfor sufficiently high solute concentration; a typical example of a lyotropic liquid-crystalline polymer is poly (1,4-phenyleneterephthalamide), sold commercially asKevlar. The most thoroughly characterized lyotropics are liquid-crystalline poly-mers formed by concentrated solutions of synthetic polypeptides. Polybenzyl-L-glutamate (PBLG) assumes a rod-like alpha-helical conformation in manysolvents. Other naturally occurring phases are obtained from certain linear viruses,such as tobacco mosaic virus (TMV) (Lee & Meyer 1991). The natural parameterthat effects the transition is the concentration because the principal interactionproducing long-range order is the solute-solvent interaction (Donald & Windle1992).

Symmetry considerations led Friedel (de Gennes & Prost 1993) to distin-guish the three main classes of liquid crystals: nematic, cholesteric, and smectic.Schematics are shown in Figure 1. (a) Uniaxial nematic order: The two main fea-tures of the nematic phase are the long-range orientational order and the fluidity.This phase has cylindrical symmetry and is therefore uniaxial. The direction of theaxis of cylindrical symmetry is arbitrary in space and is described by the directorn.Equilibrium biaxial nematic phases are discussed in de Gennes & Prost (1993). Inthis review, we abbreviate liquid crystals by LCs, nematic liquid crystals by NLCs,low molar-mass nematic liquid crystals by LMMNLCs, and liquid-crystalline ne-matic polymers by LCNPs. (b) Cholesteric order: Chiral nematic molecules giverise to the cholesteric phase. The cholesteric phase lacks positional order, as doesthe nematic phase, but the director follows a helical path. The strong modulationof the refractive index due to the helical deformation causes Bragg scattering of

Figure 1 (A) The uniaxial rod-like nematic phase, (B) the helical structure of thecholesteric phase, and (C) the smectic A phase.


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various colors of light, a property that is exploited in the use of liquid crystals astemperature sensors. Polymers possessing sufficient rigidity and chiral centers onthe chain backbone can exhibit cholesteric behavior. The sense of the twist is sol-vent dependent, and the nematic phase may be formed from the cholesteric by thecompensatory effect of a binary mixture; an example is poly (benzyl-glutamate)(PBG), which admits a nematic phase from the racemic mixture of cholesteric Dand L compounds. (c) Smectic order: The remaining liquid-crystalline phases areall smectics, and as many as 11 are known to exist. The smectics are distinguishedby having an intermediate degree of positional order in addition to orientationalorder. The best-known smectic phases are the A and C, which have one degree oftranslational ordering and hence a layered structure. A schematic of the smectic Ais shown in Figure 1C.

2.2. Micellar Liquid Crystals

Lyotropic liquid-crystalline phases can be formed by amphiphilic molecules insolution. Amphiphilic molecules consist of a hydrophilic (polar) head that attractswater and a lipophilic tail that avoids water. The lipophilic tail is usually a longhydrocarbon chain. Potassium stearate, phospholipids, and sodium dodecyl sulfateare examples of amphiphilic molecules. Amphiphilic molecules can be cationic,ionic, and nonionic surfactants, and some have double lipophilic tails. At a water-hydrocarbon interface, the amphiphilic molecules form a smectic layer, whereasin the two bulk phases, if the concentration is above the critical micellar concentra-tion, microdomains known as micelles form. In the water phase, normal micelleswith the head on the outside form, whereas in the hydrocarbon phase, inversemicelles with the head at the inner side form. The micelles may be spherical,rod like, or disk like. If the surfactant concentration is further increased, orderedmesophases such as nematic, smectic, hexagonal, or cubic can arise dependingon the particular surfactant(s)-solvent(s) system (Boden 1994, Larson 1999). Forin-depth discussion of the conditions for formation of the various micellar LCphases, see Boden (1994) and Larson (1999).

3. DEFECTS AND TEXTURES IN NEMATICLIQUID CRYSTALS

A nematic phase exhibits broken symmetry when compared to the isotropic phasebecause it is defined by the average molecular orientation,n. Because any orien-tation of n is permissible in principle, the degeneracy leads to the possibility ofdefects, or spatial discontinuities ofn. Defects are classified in terms of strength(S) and dimensionality (D). The strength captures the degree of rotational discon-tinuity when encircling the defect, whereas the dimension refers to points, lines,and walls (Kleman 1983, 1989; Bouligand 1998, Noel 1998). The spatial arrange-ment of defects is called texture, and each class of LC displays a distinguishing



number of textures; textures in many liquid-crystalline materials have beenthoroughly characterized (Slaney et al. 1998, Demus & Richter 1978, Bouligand1998, Kleman 1991, Zimmer & White 1982). Defects play a determining role inthe suitability of a liquid crystal as a material of choice for product manufacturing,and the defect texture is undoubtedly a factor in difficulties encountered in control-ling spatial inhomogeneity in the industrial production of large molded parts fromliquid-crystalline polymers (Sawyer & Jaffe 1986). Moreover, measured phys-ical properties are intrinsic material properties only when the material is a liquidmonocrystal, and the physical properties of textured liquid crystals are functions oftextural parameters, such as defect type and density. A large body of research hasbeen directed to understanding the physics of defect nucleation and coalescence forlow molar-mass liquid crystals (de Gennes & Prost 1993, Bouligand 1998, Kleman1983) and for liquid crystalline polymers (Noel 1998, Kleman 1991, O’Rourke &Thomas 1995, Larson 1999).

The nematic liquid monocrystal (monodomain) state appears only with com-patible surface effects or in the presence of highly orienting external fields, suchas strong electromagnetic fields or extensional flow. Nematic textures are ubiqui-tous and are generated, in the absence of external fields, by (a) the nematic-phaseformation process, as in the thermally induced phase transition of the isotropicphase into the nematic phase (Bouligand 1998) and (b) by incompatible orient-ing surface couples, arising from untreated surfaces, geometric singularities, orsuspended second phases (i.e., filled nematics, emulsions, etc.). In the absenceof external flows, texture coarsening is the result of defect-defect and defect-surface interactions and defect reactions; the reactions are governed by the con-servation of topological charge, such as one S=+1 disclination line decayinginto two S=+1/2 lines, or a disclination loop decaying into a point defect (Rey1990a). Imposed external flows also create (de Gennes & Prost 1993; Rey 1993b;Mather et al. 1996, 1997; Alderman & Mackley 1985, Rey & Tsuji 1998), trans-form, and coarsen textures (Larson 1999, Kleman 1991, Noel 1998, Rey & Tsuji1998).

3.1. Classification and Stability of Defects

The strength S of a defect is equal to the number of rotations the director experienceson a path encircling the defect. Points are D= 0 defects, disclinations are D= 1line defects, and walls are D= 2 defects. Walls and disclination ends cannot befound in the bulk of the sample. Wedge and twist disclinations are possible ac-cording to the rotation axis: Wedge (twist) lines are parallel (perpendicular) to therotation axis (Noel 1998). The cores of disclinations can be singular or nonsin-gular; S= ±1/2 disclinations have singular cores, whereas S= ±1 disclinationscan have either. Figure 2 shows schematics of some representative disclinations.In singular cores, the orientational order decreases, whereas in nonsingular cores,the director escapes into the third dimension. Because defects have associatedcore and distortion energies, short- and long-range energy calculations are used to


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Figure 2 Schematics of director distributions (full lines) around some wedge dis-clinations.

establish their stability (Kleman 1991, Noel 1998); these calculations involve theOseen-Frank long-range elastic energy density fg:

2fg = K11(∇.n)2+ K22(n.∇ × n)2+ K33|n×∇ × n|2, (1)

where K11, K22, and K33 are the splay, twist, and bend Frank elastic constants,respectively, whose corresponding distortions are shown in Figure 3. Defect energyis proportional to S2 and to the appropriate{Kii}. Anisimov & Dzyaloshinskii(1972) showed that elastic anisotropy controls the stability of disclinations and therelative abundance of certain types of defects.

3.2. Texture Creation During theIsotropic→Nematic Phase Transition

During the isotropic-nematic mesophase phase tranformation, when nematic sphe-rules grow in an isotropic matrix and coalesce into larger mesophase regions, a largenumber of disclinations nucleate (Bouligand 1998, Noel 1998) because the lack oforientation registry between the uncoalesced mesophase regions is resolved by thenucleation of disclinations. Defect nucleation during phase change is a universalfeature of phase transitions with broken symmetry; Trebin (1998) has shown thecorrespondence between defects in nematics and defects in cosmology, wherethe defect nucleation process is known as the Kibble mechanism (Kibble 1976).The defect density at the end of the phase-transition stage was calculated by Kibble

Figure 3 Schematic of the three elastic modes: (A) splay (K11), (B) twist (K22), (C) bend(K33). The segments represent the directorn, and the arrows the rotation ofn.



within the cosmological model and by Rey & Tsuji (1998) using the Landau–deGennes model for LCNPs. In the latter, the single parameter that controls defectdensity at the phase transition is the ratio R of short-range (homogeneous) to long-range (Frank) nematic elasticity. For relatively large R, long-range elasticity isunable to transmit orientation information effectively, and thus, lack of orientationregistry between coalescing mesophase spherules is more prevalent than when Ris relatively smaller.

3.3. Texture Coarsening

Unstable textures undergo coarsening processes driven by, among several possi-bilities, defect-defect reactions and annihilations and defect recombinations thatemit loops that shrink (Trebin 1998). Thus, texture coarsening is driven by a de-fect density reduction. Texture coarsening is a self-similar process whose scalinglaw gives the mean defect separation distanceξd in a space of dimension d as apower law in time:ξd≈ ρ(t)−1/d ≈ tn, whereρ is the defect density. For nematicpolymers, Rojstaczer et al. (1988) and Shiwaku et al. (1990) showed experimen-tally that for d= 2, n≈ 0.35–0.37. Chuang et al. (1991) performed studies of theisotropic rod-like nematic transition of 5CB by pressure quenches. Texture coars-ening followed the self-similar process, with the power-law exponent n= 0.5, asexpected in dispersive systems (Trebin 1998). String recombination and decay of atexture into monopole-antimonopole pairs was also observed. Texture coarseningwas simulated using the Landau–de Gennes equations by Rey & Tsuji (1998), whofound that the coarsening-law exponent is a function of the ratio R of short-rangeto long-range elasticity, such that the coarsening rate increases with increasingR. As R→∞, the results of the LE theory for a dispersive system are recovered.Texture coarsening has been simulated at the vector level by Rieger (1990), using areaction-diffusion model, and by Greco (1989), using a phenomenological dipolemodel; initial conditions for the number, type, and strength of the defects haveto be arbitrary for these models, which cannot capture texture formation. Energyminimizations of long-range elasticity for nematic textures have been performedby Bedford et al. (1991), Bedford & Windle (1993), and Kimura & Gray (1993);these static simulations are unable to capture the self-similarity of the coarseningprocess.

Abundant experimental data on textural transformation and texture coarseningunder flow clearly indicate that defects may be nucleated under flow and thatdefect-defect annihilation processes are modified by the presence of flow (Noel1998). Defect nucleation under flow has been characterized for rod-like nematics(Graziano & Mackley 1984; Alderman & Mackley 1985; Mather et al. 1996, 1997;Noel 1998; Srinivasarao 1995; Burghardt 1998; and references therein) and is alsorelated to the texture refinements reported for liquid-crystalline polymers (Larson& Mead 1993, Larson 1999, Kleman 1991, Noel 1998, Srinivasarao 1995, Vermantet al. 1994, Walker & Wagner 1994, Ugaz et al. 1997, Hongladarom & Burghardt1998, Burghardt 1998). Simulations of texture coarsening under flow are discussedsubsequently.


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4. THEORY AND SIMULATIONS BASEDON LESLIE-ERICKSEN THEORY

4.1. Leslie-Ericksen Equations

In the Leslie-Ericksen (LE) theory for NLCs (Leslie 1983, de Gennes & Prost 1993,Larson 1999), the microstructure of the material is explicitly taken into accountthrough a director of unit magnitude. Assuming that the fluid is incompressible,the mass, linear momentum, director, and internal energy balance equations ofmicrocontinuum mechanics are as follows:

∇.v = 0; ρv = f +∇.T; ρ1n = G+ g+∇.π ; U = T : A + π : M − g.N,

(2)

wheref is the body force per unit volume,T the total stress tensor,v the linearvelocity,ρ1 the moment of inertia per unit volume,G the external director bodyforce (torque per unit volume),π the director stress tensor, U the internal energyper unit volume,N the angular velocity of the director relative to that of the fluid,Mthe gradient ofN, andg the intrinsic director body force. The kinematic measuresare

N = n+W.n; M = ∇n+W.∇n; 2A = ∇v+∇vT ; 2W = ∇v−∇vT,

(3)

whereA andW are the rate of deformation tensor and the spin tensor, respectively.The LE constitutive equations for the stress tensor and the director body force aregiven in terms of objective linear functions ofN andA. Expanding in these vari-ables and using transversely isotropic tensor coefficients that reflect the materialsymmetry, the following equations are found:

T = −pI − ∂ fg

∂∇n· ∇nT + α1A : nnnn + α2nN

+α3Nn+ α4A + α5nn.A + α6A.nn, (4)

g= an− b.∇n− ∂fg

∂n− γ1N− γ2n.A; π = bn− ∂fg

∂∇n. (5)

p is the pressure;I the unit tensor;{αi }, i = 1 . . .6, the six Leslie viscosity co-efficients; a the director tension;b a Lagrange multiplier vector;γ1=α3−α2 therotational viscosity; andγ2=α6−α5=α2+α3 the irrotational torque coefficient,where the last equality follows from the Onsager reciprocal relations.

The boundary conditions for the velocity are usually no slip at the boundingsurfaces. Boundary conditions for the director are given (Rey 2000) by the balanceof the surface elastic and the surface viscous torque:

n× hse− n× hs

v = 0. (6)



The surface elastic molecular fieldhse is given by

hse = −

∂fs

∂n− k.

∂fg

∂∇n, (7)

wherek is the unit normal, and the surface free energy density fs is given by

fs(n.k,T) = τo(T)+ τan(n.k,T); τan(n.k,T) = τ2(T)[n.k]2+ τ4(T)[n.k]4,

(8)

where T is the temperature,τ0 is the isotropic interfacial tension, andτan isthe anisotropic contribution to the surface free energy, known as the anchor-ing energy. For LMMNLCs, the isotropic surface interfacial tensionτo is of theorder of 10 ergs/cm2, whereasτan varies from 10−4−1 erg/cm2 (Sonin 1995).The director orientation that absolutely minimizes the surface free energy den-sity is known as the easy axis of the interface. For nondeforming interfaces, theviscous surface molecular field takes dissipation due to orientational slip intoaccount:

hsv = γ s

1//I s.dndt+ γ s

1⊥kk .dndt, (9)

whereI s is the surface unit tensor andγ s1// andγ s

1⊥ are surface viscosities associatedwith tangential and normal rotations of the director. According to the generalboundary condition (Equation 6), two particular static-director surface conditionsare possible: (a) no-torque condition,k.∂fg/∂∇n= 0, corresponding to the case ofinsignificant surface anchoring energy, and (b) strong-director surface anchoring,n= nfix, corresponding to the case in which bulk gradient elasticity is insignificantwith respect to surface anchoring energy. All the terms in Equation 6 must beretained when both anchoring and gradient energies are equally significant andorientational slip occurs.

4.2. Scaling and Dimensionless Numbers

The characteristic static and dynamic behavior of NLCs is found by using the timescales, length scales, and dimensionless numbers of the LE equations. The set ofmaterial parameters in the generalized LE theory are{αi, i = 1, . . . ,6; Kii , ii =11, 22, 33;γ s

1//; γs1⊥; τ0; τ2; τ4}. If the characteristic geometric length scale is L, the

extrinsic timescale of the LE theory is the orientation diffusion timeζor = γ1L2/K,where K is a characteristic Frank constant. In the generalized LE theory, thereis an intrinsic length scale, given by the ratio of gradient to anchoring energy:`ga = K/τ2. In the absence of flow, orientation gradients can exist only if L> `ga.In the presence of a flow whose time constant isζflow, flow-induced orientation ispossible only ifζflow < ζor. This last condition is expressed in terms of the Erick-sen number, which for a shear flow of rate ˙γ is E = γ1L2γ /K. Thus it is foundthat, if E À 1, flow-induced orientation will be homogeneous only if L< `ga.Comparing the transient apparent shear viscosityη of a NLC in simple shear


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between two plates with separation Lα, Lβ, shear rates ˙γα and γβ , and shearstrainsγ α andγ β, it follows that η(γαL2

α, γα) = η(γβL2β, γβ) if γαL2

α = γβL2β

andγ α= γ β.

4.3. Flow-Induced Orientation

According to the LE theory, in the absence of elastic effects due to director gra-dients, a flow orients the director such that the total viscous torque densityΓv

vanishes:

Γv = γ1n× [−n+ (λA −W).n] = 0, (10)

where the first term represents the effect of transient rotation and the secondthe effect of the deformation. The temperature-dependent reactive parameterλ

establishes the relative magnitude of the rotational (W) to the aligning (A) floweffects.

Both A and W are nonzero for simple shear. For rod-like NLCs,λ>0 andtwo modes are possible: (a) 0<λ<1, called the tumbling or nonaligning mode,and (b) λ>1, called the aligning mode. Consider a simple-shear flow along the zdirection, withv= (0, 0,γ x). We assume the director orients in the shear plane, sowe can writen= (sinθ , 0, cosθ ), in which case Axz= γ /2, Nx=−γ sinθ/2,Nz=−γ cosθ/2. The viscous torque along the y axis computed from Equa-tion 10 gives the following director angle equation:

∂θ/∂t+ γ1γ [1− λ cos 2θ ]/2= 0, (11)

with solutions

(a) nonaligning mode: 0< λ < 1, θ = arctan

{√1+ λ1− λ tan

[γ t

2

√1− λ2

]}(12)

(b) aligning mode:λ > 1, θL = 1

2arc cos

1

λ; (13)

θL is known as the Leslie angle. The possible stable alignment angles are restrictedto 0<θ <π/4. For nonaligning materials the shear flow attempts to rotate thedirector with a period equal to [2π (1− λ2)−1/2/γ ].

The orienting behavior of uniaxial nematics in shear-free flows (Rey & Denn1988) is simpler becauseN= 0, andA has a simple diagonal form at steady state:

A =

e1 0 0

0 e2 0

0 0 e3

; e1+ e2+ e3 = 0. (14)



According to Equation 10, the steady-state orientation occurs whenn is colin-ear with the eigenvectors ofA, and the orientation is stable whenn is normal(parallel) to the compression (extension) axis or plane. For uniaxial extensionalflows, e1= e, e2= e3=−e/2, rod-like nematics align with the director alongthe stretching direction (1-axis). For equibiaxial stretching flows,e1=−2e,e2= e3= e, andn aligns normal to the compression direction (1-axis).

The generalization of the expression for viscous torques in a binary miscibleNLC mixture was derived by Rey (1996). The concentration-dependent reactiveparameter of the mixture at low deformation rateλmix

0 is given by

λmix0 =

8C

8(C − 1)+ 1λn + 1−8

8(C − 1)+ 1λm, (15)

where8 is the relative volume fraction, C(8) is the concentration-dependentratio of the rotational viscosities of the pure components, and{λi ; i= n,m} arethe reactive parameters of the pure components. If C is independent of8, thenEquation 15 is in agreement with low deformation-rate experimental results (Liu& Jamieson 2000, Ternet et al. 1999).

4.4. Shear Flows of Nonaligning LE Fluids

The presence of long-range elasticity and possible strong anchoring of the directorat the bounding surfaces modifies the planar (two-dimensional orientation) directordynamics of nonaligning NLCs (0<λ<1) under simple-shear flow. The resultingorientation must satisfy the torque balance equation

Γe+ Γv = 0; Γe = −n×(∂fg

∂n−∇. ∂fg

∂∇n

), (16)

whereΓe is the elastic torque andΓv is the viscous torque defined in Equation 10.The elastic torques arising from in-plane orientation gradients stabilize the flowand result in steady-state planar solutions, with a series of limit-point bifurcationsknown as tumbling instabilities, but at high shear rates, stationary solutions existonly if the director escapes from the shear plane. Studies using the full nonlinearLE equations with fixed orientation at the walls (n = nw = constant) (Luskin &Pan 1992, Zuniga & Leslie 1989, Han & Rey 1993) found that out-of-plane (±OP)solutions bifurcate supercritically from in-plane (IP) solutions at a critical Ericksennumber; the± denotes the fact that the director can escape the shear plane intwo equally possible directions. This out-of-plane transition generally precedesthe two-dimensional tumbling instability. Both OP and partially IP modes ofsheared nonaligning NLCs have been experimentally observed in 8CB. Pieranski& Guyon (1974) reported that OP orientation exists throughout the sample thick-ness, whereas Cladis & Torza (1975) detected OP orientation at the interfacebetween the center and wall regions, with IP orientation at the centerline region.These two sets of experimental observations were predicted by Han & Rey (1994),using 8CB material data at 35◦C and homeotropic director anchoring, as shown


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Figure 4 Computed visualizations of director orientation of sheared 8CB with homeo-tropic anchoring. (a) E= 133, (b) E= 130.

in Figure 4. The results of Cladis & Torza (1975) correspond to a chiral OPsolution branch that displays IP orientation at the center gap region but has ori-entation along the vorticity close to the two bounding surfaces, when E= 133,as shown in Figure 4a. Pieranski & Guyon’s (1974) results were replicated whenE= 130, when the stable solution branch is an achiral OP mode, as shown inFigure 4b.

The thermal dependence of the transient and steady rheological material func-tions of 8CBP has been studied experimentally by Gu & Jamieson (1994) andsimulated by Han & Rey (1995a) using the full nonlinear LE equations. This NLCcompound becomes isotropic at T=TNI= 40.8◦C, exhibits a lower temperaturesmectic-A phase at T=TNSm= 33.2◦C, and undergoes the aligning↔nonaligningrheological transition at T=Tan= 38.45◦C. Differences between the experimentalcone-and-plate geometry and the parallel-plate geometry of the model lie in the Er-icksen number, which is position dependent in the former and constant in the latter.For T>Tan (T<Tan) the behavior is aligning (nonaligning). For shear steps, thesimulations reproduce the emergence and oscillatory dissipation of characteristicdouble peaks in the transient shear viscosity, driven by the IP→OP transition.The simulations also reproduce smectic-A pretransitional phenomena. Quantita-tive agreement between simulation and experiments was observed for the highlynonlinear temperature dependence of the apparent steady-shear viscosityη∞, in-cluding the viscosity jump at Tan and the cross-overη∞=α4/2 at T= 36.2◦C, asshown in Figure 5a. The computed steady-state nematodynamic phase diagram,in terms of the Ericksen number and temperature, and characteristic orientationstructure visualizations for sheared 8CB with homeotropic director anchoring, areshown in Figure 5b. Close to the isotropic phase, the NLC displays the alignedboundary-layer IP mode for all E, whereas at T< Tan, the splay-bend distorted,IP mode exists only if E< 100. At higher E, only the twisted OP mode with fewsplay or bend distortions is stable.



Figure 5 (a) Temperature dependence of the apparent steady-shear viscosityη∞ (largedots) as a function of1T= TNI-T. At 1Tan=TNI-Tan, the viscosity exhibits a jump, andclose to1TMV =TNI-TMV, η∞= ηa= ηb. (b) Nematodynamics phase diagram of sheared8CB with homeotropic anchoring. The phase fields correspond to the basin of attractionsfor initial conditions ny = 1. The vertical line at T= 38.5◦C corresponds to the aligning-nonaligning transition (a first-order transition), whereas the slanted dashed line gives theE (T) at which the IP-OP transition occurs (a second-order transition). From Han & Rey(1995a), with permission of theJournal of Rheology.

4.5. Nonviscometric Flows

Simulations of microstructure evolution in nonviscometric flows of aligning ne-matics using the LE equations have been performed for Jeffrey-Hamel (Rey &Denn 1988) and radial outflow between disks (Rey 1990b). Viscous torques in-clude shear and elongation, and the stable orientation is achieved by balancingthe three torques. The Jeffrey-Hamel flow geometry is best described in (r,ψ , z)cylindrical coordinates. The shear plane is spanned by (r,ψ), and z is along thevorticity axis. The IP director angle isθ , and the director and velocity fields aren(ψ)= (cosθ , sinθ , 0); v(ψ)= (u, 0, 0). For converging (diverging) flow u< 0(u> 0). The stable centerline director is along the extension direction, which forconverging (diverging) flow is along radial (azimuthal) direction. The viscoustorque balance yields, at steady state, the following director equation:

[λ cos 2θ − 1]u′

2+ [λ sin 2θ ]u = 0. (17)


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The stable solutions, consistent with the centerline orientations for diverging andconverging flows are, respectively,

9 > 0, θ+a = tan−1

[2λu−

√4λ2u2+ (λ2− 1)u′2

u′(1+ λ)

]; 9 < 0, θ−a = −π − θ+a

(18)

9 > 0, θ+a = tan−1

[2λu+

√4λ2u2+ (λ2− 1)u′2

u′(1+ λ)

]; 9 < 0, θ−a = −θ+a .

(19)

In the centerline (wall) region, elongation (shear) dominates. Both shear and ex-tension promote aligning close to the radial direction for converging flow, whereasfor diverging flow, shear promotes radial alignment and elongation azimuthal align-ment. Thus, at high E, converging (diverging) flows have weak (strong) orienta-tion gradients. These results are a consequence of the fact that extension andcompression are co-planar with shear, and their magnitudes have the same radialdependence.

Two-dimensional converging flow of an aligning LE fluid with rectilinear up-stream and downstream regions has been simulated by Chono et al. (1994). Directoralignment in the converging region is essentially along the streamlines, as expected.The recirculating region near the corner is larger than that in a Newtonian fluidbecause of the entropic elasticity terms in the constitutive equation.

Nonviscometric radial outflows of aligning NLCs between two parallel diskshave been characterized experimentally by Hiltrop & Fisher (1976) and simulatedwith the LE equations by Rey (1990), assuming homeotropic (orthogonal) directoranchoring. For flow-aligning NLCs, the net effect of shear is to try to keep thedirector aligned close to the flow direction in the shear plane, whereas the effect ofelongation is to align the director along the extension direction. Inhomogeneousshear-elongational flows will exhibit transitions from IP modes to OP modes as aresult of these competing orienting effects. The increase in cross-sectional area inthe pressure-driven radial outflow between parallel disks introduces a stretchingdeformation normal to the shear plane and drives the emergence of three possibleorientation modes, shown as side-view schematics in Figure 6. For homeotropicdirector anchoring there is a critical pressure drop for each radial distance atwhich the original IP bow mode becomes unstable and any of two OP screwmodes appear within a cylinder centered at the axis of the disks. The radius of thecylinder containing the OP modes is pressure dependent; increasing the pressuredrop increases the radius. Further increase in pressure drop above another criticalvalue produces the emergence of a radially aligned IP peak mode.

There have been few simulations of nonviscometric flows of tumbling nematicsusing the LE equations. Rey & Denn (1989) studied planar converging flow of atumbling nematic and found a cascade of transitions. Burghardt & Fuller (1990)



Figure 6 Side view of the sequence of orientational modes of a flow-aligning NLCin radial outflow, with homeotropic anchoring, as the pressure drop increases: (A) in-plane bow mode, (B) left-screw mode, (C) right-screw mode, (D) in-plane peak mode.The arrows represent changes due to increasing pressure drop.

studied transients in plane shear. Chono et al. (1998) computed velocity and texturedevelopment in pressure-driven planar channel flow. A typical result is shown inFigure 7 for parameters characteristic of 8CB at 34◦C and for Ericksen numbersranging from 10 to 70. The velocity profile is essentially independent of the directororientation. The number of tumbling transitions in the fully developed regionincreases with Ericksen number. Regions of different orientation are separated bysplay-bend walls; singularities are not possible at steady state in a LE fluid becauseof the diffusive nature of the director equations. It is likely that some of these statesare unstable to out-of-plane perturbations.

Figure 8 shows the coating thicknessedeposited on a fiber of radiusr draggedthrough a bath of a LMMNLC mixture (E7) at 25◦C. The coating flow consists ofmixed extension and shear. The thickness follows a scalinge/r ∝ Ca0.94, where Cais the capillary number, rather than the classical Landau-Levich-Derjaguin scalinge/r ∝ Ca2/3 for Newtonian fluids. The 2/3 power scaling is observed for coatingabove the nematic-isotropic transition temperature. This scaling is predicted bythe LE equations in the absence of elasticity (Park et al. 2001) and originates inthe coupling between orientation and stress.

4.6. Banded Textures

Light transmission patterns of sheared LCPs under crossed polars show a bandedtexture that arises during and/or after cessation of flow (Elliott & Ambrose 1950,Toth & Tobolsky 1970, Kiss & Porter 1980, Donald et al. 1983, Navard 1986,Srinivasarao & Berry 1991, Larson & Mead 1992, Vermant et al. 1994, Larson1999, Harrison & Navard 1999). The formation of banded textures under a mag-netic field has been simulated with the full LE theory under fixed anchoring con-ditions (Rey & Denn 1990) and under orientational slip (Rey 1991), with goodagreement with experiments. Simulations of banded textures after cessation offlow using extensions of the LE and Landau–de Gennes equations (Rey & Tsuji


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Figure 7 Developing flow between parallel plates for a LE fluid with parameters char-acteristic of 8CB at 34◦C. The wall is at y= 0 and the midplane at y= 0.5. E= 10 (top),50 (middle), and 70 (bottom). From Chono et al. (1998), with permission from ElsevierScience.

1998) indicate that molecular and defect elastic storage drive the pattern-formationprocess.

Yan & Labes (1992, 1994) observed a clear banded-texture image during weak-shear start-up flow of PBG, using strong homeotropic director anchoring and amonodomain initial condition; the relations between the banded-texture charac-teristics, the molecular weight, and the shear rates were experimentally determined.These studies have established that the banded texture is a result of a periodic spa-tial modulation of the orientation, leading to a spatial variation of the effectivebirefringence. Han & Rey (1995b) simulated the experiments of Yan & Labes(1994) by computing the transient nonplanar orientation pattern formation andcorresponding light transmission during shear flow of PBG, using the full LEequations and experimentally measured viscoelastic material constants. Figure 9ashows a visualization of the nz director component, where x is horizontal (velocitydirection), y vertical (gradient direction), and z (vorticity direction) into the page.The figure shows the presence of an array of tubular orientation inversion wallsimmersed in a matrix of planar (nz≈ 0) orientation, whose axes are orthogonal to



Figure 8 Reduced-film thickness for wetting of a polymer fiber by a nematic liquid-crystal mixture (E7). The data follow a power law of Ca0.94, whereas Newtonian fluidsfollow the Landau-Levich-Derjaguin 2/3 scaling. From Park et al. (2001).

the x direction. Figure 9b shows the corresponding light transmission intensity asa function of dimensionless distance along the x axis under crossed polars, clearlyshowing periodic behavior along the flow direction. Yan & Labes (1994) found thatas the shear rate increases the wavelength decreases from infinity to a saturationvalue equal to the half-gap thickness. The infinite wavelength occurs at a criticalshear rate, or equivalently at a critical Ericksen number, corresponding to the IP-OPorientation transition. The transition for periodic pattern formation is found to bevery close to the critical Ericksen number for the nonperiodic planar↔ nonplanarorientation transition (Han & Rey 1995b). Figure 9b also shows that there are ap-proximately eight major peaks in the light transmission distribution, which meansthat the wavelength of the banded texture for the Ericksen number shown is closeto the gap thickness. The simulations show that the transmission intensity is peri-odic and that the maxima occur between two adjacent tubular orientation inversionwalls.

4.7. Variable Order Parameter

The LE theory assumes that the nematic structure is defined only by the director,n,and that the degree of molecular alignment is uniform. Ericksen (1991) relaxed thesecond assumption by introducing a scalar order parameter into the free energy in


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Figure 9 (a) Visualization of the nz director component for E= 2918 and an appliedshear strain of 240. The long and short axes represent the flow and thickness direc-tions, respectively. Dark and bright stripes representnz≈−1 andnz≈+1 (orientationinversion walls), respectively, and a midgray area represents a planar orientation ma-trix (nz≈ 0). (b) Computed light transmission intensity as a function of dimensionlesslength along the flow direction. The intensity is periodic, and the wavelength is of theorder of the shear-cell thickness. From Han & Rey (1995b), with permission of theAmerican Chemical Society.

an attempt to replicate the behavior of liquid-crystalline polymers. The extendedEricksen theory has been explored in a variety of rectilinear flows (e.g., Calderer &Mukherjee 1998, Calderer & Liu 2000), but it is unlikely to offer as much insightinto the flow of LCNPs as extensions of the Doi theory, discussed in the followingsections.

5. DOI THEORY AND EXTENSIONS

Liquid-crystalline polymers exhibit behavior that cannot be described by the LEtheory. There are qualitative differences between lyotropic LCNPs, which un-dergo an isotropic-nematic transition in solution at a critical concentration, andthermotropic LCNPs, which undergo an isotropic-nematic transition in the meltat a critical temperature (but which, in fact, often decompose before reaching thetransition temperature). The backbone chemistry that enables thermotropic LCNPsto be processed in the melt typically results in molecules that are less stiff thanthose that form lyotropic LCNPs, and it is likely that entanglements occur in ther-motropic LCNPs. Lyotropes often exhibit a “three-region” viscosity curve likethat shown in Figure 10 (Walker & Wagner 1994): The viscosity in Region I, atlow rates, is shear thinning, with a power-law dependence that is close to 0.5; the



Figure 10 Three-region viscosity curve in two samples of a liquid-crystalline solution of60% hydroxypropylcellulose in water. Data from Walker & Wagner (1994), reprinted withpermission from theJournal of Rheology.

viscosity is insensitive to shear rate in Region II, at intermediate rate, whereasthe viscosity is again shear thinning in Region III. Lyotropes often exhibit an in-termediate interval, near the transition between Regions II and III, in which thefirst normal-stress difference is negative. Three-region viscosity curves are lessobvious in thermotropes, and negative normal stresses are not observed.

Polymeric liquid crystals are highly textured, with a correlation length for ne-matic order that is typically a few microns in size. The submicron defect structuresare not well understood, but they are believed to play a major role in the rheology atlow and intermediate stresses. Texture cannot be removed in thermotropic LCNPs,either by shearing or by the application of strong fields. The texture is sensitiveto the intensity and magnitude of shear, as illustrated in Figure 11 for Vectra B,a co-poly(ester amide) consisting of 60 mole % 6-hydroxy-2-naphthoic acid, 20mole % hydroxybenzoic acid, and 20 mole % aminophenol, but the initial textureis recovered upon cessation of shear. Although the free energy minimum shouldbe a monodomain, LCNPs, especially thermotropes, appear to have a glass-likeenergy landscape with deep minima that retain texture. Texture development in theflow of LCNPs has been probed by a variety of methodologies in the work of Ritiand coworkers (Riti et al. 1997, Riti & Navard 1998) and references cited therein.


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Figure 11 Development of texture in Vectra B at 5000 d/cm2 as viewed under crossedpolarizers at (a) 0, (b) 5, (c) 115, and (d) 500 strain units (Kim 1996).

5.1. Constitutive Equation

The Doi theory is based on an evolution equation for the probability distribution ofthe orientation of a suspension of monodisperse rods.ψ(u, R, t) is the probabilityof finding a rod at positionR in an orientationu. The Smoluchowski equation forψ(u) includes the rotational diffusivityD and a nematic mean-field potential V(u),as follows:

Dψ

Dt= D

∂

∂u•[∂ψ

∂u+(ψ

kBT

)∂V(u)

∂u

]− ∂

∂u•[

DuDt

]. (20)

The nematic potential is usually taken as the Maier-Saupe form,

V(u) = −3/2UkBT〈uu〉 : uu, (21)

where U is a nondimensional potential, kB the Boltzmann constant, T the temper-ature, and〈. . .〉 denotes an average over the distribution functionψ :

〈y . . . z〉 =∫ψ(u, R, t) y(u) . . . z(u) du. (22)



The evolution equation foru for infinite rods is

DuDt= u.∇v−∇v : uuu. (23)

The theory is usually written in terms of the order parameter tensorS, defined

S=⟨uu− 1

dI⟩, (24)

where d is the dimension of the space (two or three). The evolution equation forSand the equation for the stress involve the fourth moment〈uuuu〉, and all variantsof the theory in use explicitly or implicitly require a closure.

This set of equations has been solved without closure by Faraoni and coworkers(1999) for unbounded simple shear by employing an expansion in spherical har-monics, using a Galerkin approximation with truncation at 10 terms. The solutionshows rich dynamics above a critical value of U, where the isotropic solution forS becomes unstable. A typical result is shown in Figure 12 for U= 5.33, whereb2,2 is a coefficient in the expansion forψ and G is the shear rate. Zero values ofthe real part of b2,2correspond to isotropic solutions. The nomenclature describingthe various regimes is intended to be descriptive of the motion of the rods. Logrolling is an orientation orthogonal to the plane of shear, and it has been observedfor some thermotropic LCNPs (Romo-Uribe & Windle 1996), but the transitionseems to vanish with increasing molecular weight. Grosso et al. (2001) have shownthat chaotic behavior is possible.

The continuum equation forS is obtained by multiplying Equation 20 byuuand by averaging over the distribution. Using the closure approximation〈uuuu〉 =〈uu〉〈uu〉 results in the equation

DS/Dt = F(S)+G(S), (25)

where

F(S) = −2DS+ 6DU

[S · S+ 1

dS− (S : S)S− 1

d(S : S)I

](26)

and

G(S) = 2

dA +∇vT.S+ S.∇v− 2∇v : S

(S+ 1

dI). (27)

The corresponding stress tensor has the form

T = −pI − (ckBT/2D)F(S)+ 2ηsA, (28)

where c is the concentration of rods andηs the viscosity of the suspending fluid.The rotational diffusion coefficientD is sometimes taken to depend onS. Closureapproximations have been studied by Chaubal et al. (1995), Chaubal & Leal (1998),and Feng et al. (1998), who recommend the Bingham closure in preference toquadratic closures.


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Figure 12 Solution diagram for the Doi theory with U= 5.33. Solid lines represent stablestationary solutions, dashed lines unstable stationary solutions, filled circles stable periodicsolutions, open circles unstable periodic solutions, filled squares stable periodic solutionswith double period, and open squares unstable periodic solutions with double period. FromFaraoni et al. (1999), with permission by theJournal of Rheology.



The Doi theory, which implicitly assumes a monodomain, cannot describe tex-tures seen in liquid-crystalline polymers. This weakness has motivated a numberof authors to incorporate longer-range interactions through enhanced potentials.Marrucci & Greco (1991), for example, suggested a potential of the form

V = −2UkBT

{S+ 1

24R2∇2S+ L2(uu : ∇∇S)

}: uu, (29)

where R is a characteristic interaction distance between molecules and L is themolecular length. This potential neglects inhomogeneous terms associated withthe shape of the molecule, which is equivalent to taking the three Frank elas-tic coefficients to be equal. R is expected to be much larger than L. Kupfermanand coworkers (2000), Feng and coworkers (2000), and Lhuillier (2000) have de-veloped similar continuum formulations based on incorporation of the Marrucci-Greco potential into Equation 20. Their results, in turn, are essentially equivalentto an equation obtained by Rey & Tsuji (1998) using the Landau–de Gennes ap-proach and to an equation obtained by Beris & Edwards (1994) using a Poissonbracket formalism. [The Poisson bracket formalism for a tumbling parameter dif-ferent from±1 does not obey the Jacobi identity (Kats & Lebedev 1994), andthe equations cannot be rewritten in terms of canonically conjugated variables.]This formalism introduces terms in∇2S into both the evolution equation for S andthe equation for the stress. The continuum theories reduce to the LE theory in thelimits of weak flows and small distortions (Feng et al. 2000), and all predictthe orientation dynamics of the LE equations and of the Doi equation, includingthe steady tumbling mode of the LE model (Carlsson 1984) and the tumbling-wagging-aligning cascade predicted by the Doi equation (Rey & Tsuji 1998,Kawaguchi 1998).

5.2. Solved Flows

Solutions to the continuum Doi equations have been obtained for viscometricflows in many of the studies cited above and in extensional flows (Forest et al.1999). Negative first normal stresses are predicted in steady shear, and the flowtransitions shown in Figure 11 are retained in the averaged equations, althoughtransition parameters are dependent on the specifics of the closure approxima-tion used (e.g., Feng et al. 1998). Ramalingam & Armstrong (1993) and Forestet al. (1997) have solved the thin-filament equations for the isothermal spinningof liquid-crystalline polymer fibers, whereas Mori et al. (1997b) have solved thetwo-dimensional axisymmetric equations. Mori and coworkers found an unusualconcavity in the computed velocity profile that had been observed experimentally(Mori et al. 1997a). Feng and coworkers have solved the Doi equations with a vari-ety of closure approximations for time-dependent flow between rotating eccentriccylinders, and they report the development of director singularities of order±1/2in some cases. There are significant quantitative differences between the variousclosures, although all give qualitatively similar results.


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Figure 13 Simulation of transient shear flow for a fluid with a Marrucci-Greco potentialat De= 1, E= 400. (a) The structure-tensor component Sxx(y, t), where y is the coordinatebetween the plates; (b) the velocity component vx(y, t); (c) visualization of the structure tensorduring one cycle; and (d) the velocity profile over one cycle. From Kupferman et al. (2000),reproduced with permission by Elsevier Science.

Kupferman and coworkers (2000) have reported the development of structurein two-dimensional transient shear flow using the continuum formulation withthe Marrucci-Greco potential. Figure 13 shows some results for the velocity andorientation fields at a Deborah number of unity and an Ericksen number of 400.Under these conditions there is no steady state. The director tumbles in the middlesection of the channel, generating smooth orientational waves. The outermostdefect line is periodically annihilated as it is squeezed between the propagating



wave and the anchoring wall. Velocity gradients are very large. Their results showthat it is essential to incorporate coupling between the velocity and structure fields.One disappointing feature of this study was that structure with a length scaleindependent of the macroscopic dimension did not emerge, and the transverseorientational patterns are reminiscent of those obtained from the LE theory (Chonoet al. 1998).

5.3. Textured Systems

Larson & Doi (1991) attempted to deal with the presence of texture in LCPs byformulating a mesoscopic theory in which the textured fluid is assumed to becomposed of “domains” in which the director orientation is uniform. Field equa-tions for the stress and orientation are obtained by spatial averaging of the LEequations over regions large relative to the domain size, with phenomenologicalevolution equations for mesoscopic parameters. Kawaguchi & Denn (1999) de-veloped a mesoscopic theory for textured fluids that reduces to the Larson-Doitheory at lowest order, but with a different meaning for the model parameters.Ugaz et al. (2001) developed a “polydomain Ericksen model” that is similar tothe Larson-Doi model when contributions from director elasticity are neglected.Rey (1996) developed a coarse-grained model using the tensor order parameterthat predicts entropy-driven stable textures and showed that under certain condi-tions it converges to the Larson-Doi model. Burghardt and coworkers (Bedford &Burghardt 1996; Burghardt 1998; Cinader & Burghardt 2000a,b) have carried outa series of experimental studies of LCP structure development in channel flowswith expansions and contractions, together with simulations using the Larson-Doitheory. The general conclusion is that the theory captures some of the qualitativefeatures of the orientation development but is quantitatively inadequate.

Several other models aim at explaining the rheological implications of collectivedefect behavior in nematic polymers (Wissbrun 1985, Marrucci 1991, Yamakaziet al. 1991). Detailed descriptions of defect annihilations, reactions, defect coretransitions, and other local phenomena are not captured because the averaging isover distances orders of magnitude greater than defect cores. Rey (1993b) incor-porated the loop emission model of de Gennes (1976) into population balanceequations to describe the birth, death, and deformation of the loop population inthe presence of arbitrary flows. The predictions for steady and oscillatory shearwere shown to be consistent with the experiments of Graziano & Mackley (1984)and Alderman & Mackley (1985). Rey & Tsuji (1998) used the Landau–de Gennesmodel to study defect nucleation and coarsening in the presence of shear flow. Thesequence of phase transition→ texture creation→ texture coarsening was simu-lated when shear flows of various strengths were activated at different stages ofthe process cascade. They found that flow speeds up coarsening by optimizing thedefect annihilation process.

The only three-dimensional flow visualization study of a LCP in a complexgeometry seems to be that of Kawaguchi & Denn (1997), who used tracers to


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follow the velocity profile of thermotropic Vectra A through a conical contractionbetween two cylindrical channels. They observed a breakdown of simple streamlineflow, with an asymmetric flow in the contraction and regions that appear to bestagnant. It is likely that substantial changes in microstructure and material occurover a small spatial region, possibly because of changes in texture. Gentzler et al.(1998) used solid-state nuclear magnetic resonance (NMR) spectroscopy to imagedirector orientation in frozen samples from a contraction flow of this polymer. Thetechnique showed orientational differences in different parts of the samples, but thevolumes were too large to resolve meaningful structural information. A recent studyby Gentzler et al. (2000) used NMR to study velocity and orientation developmentin situ for tube flow of a lyotropic LCNP, and it is likely that investigators will finduses for NMR in geometries that cannot be probed by optical techniques becauseof the opacity of the liquids.

6. INTERFACIAL EFFECTS

Interfaces involving NLCs are characterized by nematic ordering as well as by thegeometry, with the result that even the interfacial tension of a planar interface maychange owing to changes in the nematic ordering in textured materials. The studyof nematic interfaces is just beginning.

6.1. Interfaces in LC Blends

Liquid-crystalline materials are employed as the dispersed phase in immiscibleblends in a variety of technologies. Low molar-mass liquid crystals are used indisplay technology (Drzaic 1995), whereas polymeric liquid crystals are used asflow modifiers to reduce the pressure drop in extrusion (Cogswell 1983) and asthe fibrous phase in “self-reinforced” composites (Weiss et al. 1987, Handlos &Baird 1995, Qin 1996). Liquid-crystalline polymers are also of interest as barrierlayers in multilayer films (Sawyer et al. 1998). The interface between LCs andisotropic phases has not been extensively studied, but limited observations suggesta strong effect of the LC anisotropy. Droplets smaller than the correlation lengthfor nematic order do not seem to contribute to the linear viscoelasticity in poly-mer blends where the LCNP is a dispersed phase, for example, whereas largerdroplets contribute as expected (Lee & Denn 1999, Riise et al. 1999). The relax-ation dynamics of LCNP droplets following a step strain scale differently fromthose of a flexible polymer in the same suspending fluid (Lee & Denn 2000) andshow an unexpected dependence on the initial strain. Kernick & Wagner (1999)have suggested that the interfacial tension might be affected by morphologicalchanges induced during droplet deformation, and they employ a theory of vanOene (1972) regarding the effect of normal stresses on interfacial tension to arguethat the parallel orientation induced by droplet stretching may reduce the interfacialtension.



Li & Denn (2001) have carried out a Monte Carlo simulation of the inter-face between a liquid-crystalline polymer and a flexible polymer using the three-dimensional Bond Fluctuation Method (Carmesin & Kremer 1988, Deutsch &Binder 1990), from which the equilibrium interfacial tension, the interfacial bend-ing elasticity, and the chain configurations can be extracted. Two far-field orienta-tions were assumed in the calculations, parallel to the interface and orthogonal tothe interface (homeotropic). The interface was more diffuse for the homeotropicorientation because of the easier penetration of chain ends into the isotropic phase.Some chains in the LCNP phase moved out of plane and adopted a parallel ori-entation near the interfacial plane, and a parallel orientation was induced in theisotropic polymer over a length of the order of one radius of gyration. The computedequilibrium interfacial tension for the parallel far-field orientation was consider-ably larger than that for the homeotropic orientation. This computation suggestsan effect opposite to that proposed by Kernick & Wagner (1999).

6.2. Interfacial Viscoelasticity

Interfacial viscoelastic theories of nematic liquid crystals using the LE and Landau–de Gennes formalisms have recently been formulated (Rey 2000, 2001) and usedto analyze static and dynamical phenomena. Static interfacial phenomena includedisjoining pressures and contact angles in nematic films. Nonequilibrium phenom-ena include wetting and spreading, Marangoni flows, dynamic interfacial tension,interfacial Miesowiczs viscosities, and interfacial dilational viscosities. The ne-matic Rayleigh fiber instability has also been analyzed (Rey 1997). We limit thediscussion here to the disjoining pressure in nematic films and Marangoni surfaceflows predicted by the interfacial LE equations.

The interfacial LE equations consist of the interfacial linear momemtum balanceequation and the interfacial torque balance equation, the latter given by Equations 6and 7. The interfacial linear momentum balance equation for a nematic (N)-viscousfluid (A) interface is

− k.(tA − tN) = ∇s.tse+∇s.ts

v, (30)

wheretN is the total stress tensor in the nematic phase at the N/A interface,tA

is the total stress tensor in the isotropic fluid phase,tse is the elastic surface stress

tensor,tsv is the viscous extra surface stress tensor,∇s is the surface gradient, and

k is the unit normal. The elastic stress tensortse is a 2× 3 tensor whose gradients

represent tangential and normal elastic forces. The surface elastic stress tensor isgiven by the usual 2× 2 symmetric interfacial tension contributionts

en (normalstresses) and the 2× 3 anisotropic contributionts

eb (bending stresses):

tse = ts

en+ tseb; ts

en= FsI s; tseb= −I s.

[∂τan

∂kk]= −I s.

[(dτan

d(n.k)

)nk]. (31)


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The physical significance of the normal and bending components of the surfacestress tensor is clearer when one considers the net surface forces engendered bytheir surface gradients:

f = ∇s.t ={[

dτan

d(n.k)

]k.(∇sn)T

}.I s + {2Hfs}k︸︷︷︸

surface gradients in normal stresses

+{−2H

(∂τan

∂k.k)−∇s.

(∂τan

∂k

)}k︸︷︷︸

surface gradients in bending stresses

, (32)

where H = −∇s.k/2 is the average curvature. The first term on the right isthe tangential nematic Marangoni force, which arises from orientation gradientsand drives interfacial flows. The remaining terms comprise the surface normalforce, which contributes to the disjoining pressure and plays a fundamental rolein thin-film stability. For a planar (H= 0) interface, the normal force may tendto stabilize or destabilize a nematic thin film, depending on the sign of surfacegradients and the nature of the anisotropic anchoring energies. Stabilization of thinfilms by anisotropic surface energy has practical utility in the use of liquid-crystalsurfactants (Larson 1999). The LE and Landau–de Gennes interfacial theories andthe equilibrium molecular simulations have not yet been integrated.

7. CONCLUDING REMARKS

The LE theory has been successful in elucidating the flow of low molar-massnematics in simple geometries. The theory has not been tested against relevantexperiments in complex flow fields, but complex flow fields are not likely to beencountered in applications of LMMLCs. The theoretical framework for the flowof nematic liquid crystals is still evolving. Various extensions of the Doi theory thatdiffer in final form only in detail capture many of the qualitative features of the flowof LCNPs in simple geometries. These theories have not been shown to predicttexture development in flow; hence, they cannot be expected to predict behaviorin processing flows. Mesoscopic theories based on spatial averaging capture onlysome qualitative features of nonrectilinear LCNP flow. Interfacial effects in liquid-crystalline systems have just started to receive attention. The outstanding fluid-dynamical challenge is the development of a more complete theoretical basis for thedescription of texture evolution in flow and its consequences in complex flow fields.The development of a rational theory of blends for immiscible systems containing aliquid-crystalline phase, incorporating new developments in describing interfacialeffects, is another important challenge, but real progress for polymeric systemswill require an adequate description of the dynamic structure distribution in theLC phase.



ACKNOWLEDGMENTS

A.D.R. acknowledges support from the Natural Sciences and Engineering Re-search Council (Canada), Air Force Office of Scientific Research-MathematicalDirectorate, the Donors of The Petroleum Research Fund (American Chemical So-ciety), and the NSF Center for Advanced Fibers and Films at Clemson University.

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