MESOSCALE STRUCTURES IN FLOWS OF ... - Semantic Scholar

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS–SERIES B Volume 6, Number 2, March 2006

Website: http://AIMsciences.org pp. 291–310

MESOSCALE STRUCTURES IN FLOWS OF WEAKLY SHEARED CHOLESTERIC LIQUID CRYSTAL POLYMERS

Zhenlu Cui Department of Mathematics University of North Carolina at Chapel Hill Chapel Hill, NC 27599-3250

M. Carme Calderer School of Mathematics University of Minnesota,Minneapolis, MN 55455

Qi Wang Department of Mathematics Florida State University,Tallahassee, FL 32306-4510; School of Mathematics, Nankai Univeristy, Tianjin, PR China

Abstract. We revisit the permeation flow issue in weakly sheared cholesteric liquid crystal polymers in plane Couette and Poiseuille flow geometries using a mesoscopic theory obtained from the kinetic theory for flows of cholesteric liquid crystal polymers [2]. We first present two classes of equilibrium solutions due to the order parameter variation and the director variation, respectively; then, study the permeation mode in weakly sheared flows of cholesteric liquid crystal polymers employing a coarse-grain approximation. We show that in order to solve the permeation flow problem correctly using the coarse-grain approximation, secondary flows must be considered, resolving a long standing inconsistency in the study of cholesteric liquid crystal flows [7]. Asymptotic solutions are sought in Deborah number expansions. The primary and secondary flow as well as the director dynamics are shown to dominate at leading order while the local nematic order fluctuations are higher order effects. The leading order solutions are obtained explicitly and analyzed with respect to the cholesteric pitch and other material parameters. The role of the anisotropic elasticity in equilibrium phase transition and permeation flows is investigated as well.

1. Introduction. Cholesteric nematic liquid crystals also known as chiral nematics are mesophases, where the average direction of molecular orientation exhibits a chiral (twisted) pattern along its normal direction [7, 10]. Examples include cholesterol esters, DNAs, colloidal suspensions of bacteriophages [16, 30] and solutions of chiral nematic mixtures. Cholesteric phases are also manifested in smectic liquid crystals, where the positional order of molecular centers of mass exhibits a periodic structure. It is noted that the formation of the mesoscopic cholesteric phase is in many cases determined by the chiral structure inherent in the molecules, although chiral phases of achiral molecules may also occur, for instance, in the case 2000 Mathematics Subject Classification. 76A15, 76M45. Key words and phrases. Asymptotic expansions, cholesterics, liquid crystals, polymers, Couette flows, Poiseuille flows, phase-transition.

291

292

ZHENLU CUI, M. CARME CALDERER, QI WANG

of bent-core molecule liquid crystals (B2-phases). In this paper, we first examine the phase behavior and transition between isotropic and cholesteric phases in equilibrium. In particular, we investigate how the phase behavior changes with respect to the concentration, molecular conformation, and the coupling between the local nematic order and the distortional elasticity in equilibrium. We then address issues on rheology of sheared cholesteric liquid crystal polymers. Phase transition from the isotropic to the cholesteric phase can take place as concentration or temperature changes [7, 10, 42]. The isotropic-nematic phase transition was first explored theoretically by Onsager considering the steric (excluded volume) effect of rigid rod molecules [28]. The pathway to cholesteric phase in isotropic- chiral nematic phase transition is often accompanied by the existence of the intermediate blue phases however [42]. It is well-known that the cholesteric pitch is sensitive to the change of temperature and concentration in cholesteric liquid crystals [10]; in fact, it can also be affected by the long-range distortional elasticity and other material parameters. One of the important issues in the study of polymeric nematic liquid crystals is the coupling of the distortional elasticity with the local nematic order including the averaged orientational direction and the degree of orientation. Such effects are not fully taken into account in the theory for flows of nematic liquid crystals by Ericksen and Leslie [23, 22]. An extension of the Leslie-Ericksen theory to include a scalar degree of orientation variable was later developed by Ericksen [13]. The Ericksen model has been successful in the study of static singularities [24] as well as line defects and patterns in flow phenomena [3, 4, 29]. However, Ericksen’s approach does not account for biaxiality normally present in flows of liquid crystal polymers. Recently, a continuum model in terms of Landau-DeGennes second order tensor has been developed to study the coupling between the distortional elasticity and local nematic order in flows of chiral nematics [9]. This model naturally takes care of the biaxiality in inhomogeneous states and the one induced by external fields and/or flow fields. It is shown that the chiral structure effectively couples the distortional elasticity to the local nematic order to noticeably impact on the phase behavior of cholesteric liquid crystal polymers (CLCPs). As all the continuum mechanics theories, the coefficients for the distortional elasticity in the second order tensor model are prescribed (linear) functions of the second order orientation tensor, which need to be calibrated by a molecular theory. This paper revisits the modeling issues from another perspective. We derive an approximate tensor model from the kinetic theory by employing a closure approximation to reveal the nonlinearity in the distortional elasticity and relate the macroscopic material parameters to molecular parameters. The cholesteric phase is delicate in that it can be sustained only under balanced conditions; otherwise, defects and disclinations can form easily. As a result, the rheology of CLCPs has not been widely studied. Most available studies are based on Leslie-Ericken(LE) continuum theory, and use perturbation techniques with the base helical axis in the flow direction [17, 18, 25, 19, 10, 7, 37], in the velocity gradient direction [23, 20], and in the vorticity direction in simple shear [11, 31, 32, 33]. A very brief review on cholesteric rheology can be found in the books [10, 21]. A particular striking example in flows of cholesterics is permeation where a cholesteric liquid crystal is subject to an imposed flow in the direction of the helix but the helical structure remains nearly intact. The main feature of the permeation is its ultrahigh viscosity (by a factor of ∼ 105 of pure nematic liquid crystals) in

MESOSCALE STRUCTURES

293

small shear. But, in strong shear, the helical structure is completely destroyed so that cholesterics behave essentially like pure nematic liquid crystals. An explanation of this was first given by Helfrich [17]. He argued that the usual parabolic profile is replaced by plug-like flow, with a constant velocity across the capillary (Poiseuille flow). De Gennes and Prost[10] added that this occurs over a length scale ∼ P where P is the pitch of the helix. Recently Rey revisited the issue and carried out a series of studies on cholesteric liquid crystals to clarify and extend the previous studies [31, 32, 33, 34, 35, 36, 37]. He also studied the linear viscoelasticity of CLCs [34, 35] and extended the previous works to linear and oscillatory shear flows [35, 36, 37]. All the studies mentioned above are limited to “in-plane orientation” with only primary flows considered and based on the uniaxial director description of the LE theory. When only the primary flow is considered, we notice that the asymptotic studies in the literature all lead to an inconsistent and over-determined system of equations and then a solution is sought by ignoring a seemingly, arbitrarily selected set of equations [7]. This is mathematically unjustifiable. Recently Marenduzzo et al. [26] studied the sheared cholesterics using a continuum theory based on Beris and Edwards’ Poisson bracket formalism. They conducted 3-D numerical calculations of the governing system of equations of the continuum hydrodynamic equations and showed that the boundary anchoring condition as well as the gap width affect the flow and orientation structure in the shear cell. They also pointed out the importance of the secondary flow in their studies. In this paper, we will address the consistency issue in the coarse-grain approximation using a mesoscopic closure model obtained from a kinetic theory [2] and derive consistently the asymptotic solutions up to the leading order. The goal is to (i) explore how the coupling among the chirality, distortional elasticity and local nematic order affects the orientation as well as the flow structure; (ii) show the indispensable role of the secondary flow in the approximation clarifying the inconsistency issue in the previous studies in the literature; (iii) derive the coarse-grain approximate solution for the permeation flow in the presence of secondary flows and analyze its scaling behavior with respect to important material parameters. In section 2, we present dimensionless mesoscopic tensor model obtained from the kinetic theory for flows of cholesteric liquid crystal polymers. In section 3, we explore two special classes of uniaxial equilibrium solutions. Section 4 is devoted to investigations of the permeation mode in weak plane Couette flows and section 5 to studies of the permeation mode in weak plane Poiseuille flows. 2. Mesoscopic model formation. We begin with a brief review of the kinetic theory for cholesteric liquid crystal polymers (CLCPs) developed in [2]. We present the dimensionless governing system of equations along with the dimensionless parameter group only and refer readers to [2] for the detailed theory, the dimensional parameters, and the derivation. We nondimensionalize the equations in the model using the characteristic length scale h and the LCP relaxation time scale t0 = tn = D10 , where Dr0 is the rotary r diffusivity, and denote the position vector by x, the velocity vector by v, the extra stress tensor by τ , and the pressure by p, respectively. The dimensionless flow and stress variables are defined by: ˜= v

t0 1 t h2 h2 ˜ = x, t˜ = , τ˜ = v, x τ, p˜ = p, h h t0 f0 f0

(1)

294

ZHENLU CUI, M. CARME CALDERER, QI WANG

where f0 = ρh4 /t20 is an inertial force and ρ is the CLCP density. Let c be the CLCP number density, k the Boltzmann constant, T absolute temperature, N a dimensionless concentration, η the solvent viscosity, ζi , i = 1, 2, 3 three friction coefficients related to CLCP-solvent interaction, and L, L, Lq three length scales for the isotropic long-range interaction, anisotropic long-range interaction, and cholesteric length scale, respectively. The following 9 dimensionless parameters in the model arise: Re = µi =

ρh2 α t0 η , 3ckT ζi t0 , h2 ρ

=

3ckT t20 h2 ρ ,

8h2 N Dr0 t0 L2 , 8Lq h L2 L2 , θ q = N L2 .

Er =

i = 1, 2, 3, θ =

(2)

α measures the strength of the elastic energy relative to that of the kinetic energy; Re is the solvent Reynolds number; Er is the Ericksen number which measures the strength of the short-range nematic potential relative to that of the isotropic distortional elastic energy; θ measures the strength of anisotropic distortional elasticity relative to the isotropic one whose range is limited to [−1, ∞); 1/µi , i = 1, 2, 3 are three nematic Reynolds numbers, and θq parameterizes the chiral free energy relative to the isotropic distortional elastic one. We drop the tilde˜on all variables from now on so that all equations and figures in the following correspond to normalized (dimensionless) variables. The dimensionless governing equations consisting of the continuity, momentum balance, orientation tensor equation and the constitutive equation for the extra stress tensor are given below. Continuity equation ∇ · v = 0.

(3)

d v = ∇ · (−pI + τ ). dt

(4)

Momentum balance equation

Orientation tensor equation  d  dt M − Ω · M + M · Ω − a[D · M + M · D] = −2aD : M4 −   6[Q 1  − N (M · M − M : M4 )] + Er [∆M · M + M · ∆M − 2∆M : M4 ]+    . .   θ [(∇∇M)..M + ((∇∇M)..M )T + M ...∇∇M + (M ...∇∇M)T + 4 4 4 4 2Er .. ..  T  M∇∇.M4 + (M∇∇.M4 ) − 4M6 :: ∇∇M − 2M4 ∇∇ :: M4 ]−    θq     Er [(M4,ikβj + M4,jkβi )Mkγ,µ ²µγβ + (M4,ikαj + M4,jkαi )Mαγ,µ ²µγk − (Mβi Mjγ,µ + Mβj Miγ,µ )²µγβ − Mαi Mαγ,µ ²µγj − Mαj Mαγ,µ ²µγi ],

(5)

where M is the second moment of the orientational probability distribution function 2 in the kinetic theory (called the structure tensor), and a = rr2 −1 +1 parameterizes the aspect ratio r of the spheroidal molecules (0 < a ≤ 1 corresponds to a rod-like molecule and −1 ≤ a < 0 for platelets [41],) the deviatoric part of M Q = M − I/3.

(6)

. is called the orientation tensor. The symbols ·, :, .., :: denote the contraction operations on one pair, two pairs, three pairs, and four pairs of tensorial indices, respectively. More delicate index contractions are given explicitly for clarity in the equations whenever necessary.

MESOSCALE STRUCTURES

295

Constitutive equation for the extra stress tensor τ = (2η + µ3 (a))D + aα[M − 3I − N2 ((I + 3N1Er ∆)M · M +M · (I + 3N1Er ∆)M − 2(I + 3N1Er ∆)M : M4 )] α α − 6Er (∆M · M − M · ∆M) − 12Er [∇M : ∇M − (∇∇M) : M] . . aαθ + 12Er [4M6 :: ∇∇M + 2M4 ∇∇ :: M4 − ∇∇M..M4 − (∇∇M..M4 )T . . . . −M4 ..∇∇M − (M4 ..∇∇M)T − (M∇∇..M4 )T − M∇∇..M4 ] . . . . αθ − 12Er [∇∇M..M4 − (∇∇M..M4 )T − M4 ..∇∇M + (M4 ..∇∇M)T . . −M∇∇..M4 + (M∇∇..M4 )T ] + [µ1 (a)(DM + MD) + µ2 (a)D : M4 ] αθ − 3Erq [ a+1 2 (Mjβ Miγ,µ ²µγβ + Mαj Mαγ,µ ²µγi + Mjγ,µ Miβ ²µγβ +Mjα,µ Mαβ ²µiβ ) + a−1 2 (Miβ Mjγ,µ ²µγβ + Mαi Mαγ,µ ²µγj +Mjγ,µ Mαβ ²µiβ + Mjα,µ Mαβ ²µiβ ) − 2aM4,ijαβ Mαγ,µ ²µγβ )].

(7)

Since the system is not closed, we would have to employ closure approximations on the higher moments. We use two simplest ones, quadratic and cubic closures, here. Namely, we approximate the fourth order tensor M4 using a tensor product of two second order tensors and the sixth order tensor M6 using a tensor product of three second order tensors: M4 = MM, M6 = MMM.

(8)

These primitive closures respect the trace condition imposed on the orientation tensor equation. Also, coupled with the geometry parameter a, the quadratic closure gives a good approximation to M4 provided a normalized concentration is adopted [15]. In this approximation, the Frank elastic constants in the linearized limit is given by C 2 C 2 s (1 + θ(1 − s)/3), K3 = s (1 + θ(4s + 1)/6), (9) Er Er where C is a parameter proportional to the concentration and s is the equilibrium uniaxial order parameter [41]. For rodlike CLCPs, K1 = K2 ≤ K3 ; whereas the order is reversed for discotics. K1 = K2 =

3. Uniaxial nematic equilibria. We study solutions of the orientation tensor equation in quiescent state by exploring uniaxial symmetry within the governing system of equations. 3.1. Nematic equilibrium due to order parameter variations. We seek uniaxial solutions of the orientation tensor equation, whose variation is through the uniaxial order parameter s in a fixed direction m (kmk = 1): I Q = s(ζ, t)(nn − ), 3

(10)

where ζ = m · x and n is a constant uniaxial director. When m||n, through a tedious calculation, the orientation tensor equation collapses to a scalar equation for the uniaxial order parameter s: ds 1 ∂2s ∂s = [−U (s) + (1 − s)(2s + 1)[(3 + 2(1 + 4s)θ) 2 + 4θ( )2 ]], dt 27Er ∂ζ ∂ζ

(11)

296

ZHENLU CUI, M. CARME CALDERER, QI WANG

where U (s) = s[1 −

N (1 − s)(2s + 1)]. 3

(12)

The steady state is governed by −

27U (s) 1 d2 s ds + [(3 + 2(1 + 4s)θ) 2 + 4θ( )2 ] = 0, (1 − s)(2s + 1) Er dζ dζ

which can be integrated to yield ds 1 (3 + 2θ + 8θs)( )2 ] = const H = G(s) − [ 2Er dζ with Z 27U (s) G(s) = . (1 − s)(2s + 1)

(13)

(14)

(15)

We denote ds 1 (3 + 2θ + 8θs)( )2 ]. (16) 2Er dζ The time dependent problem for the order parameter dynamics can be rewritten as 27 ∂s δH =− . (17) (1 − s)(2s + 1) ∂t δs H = G(s) + [

The constant equilibria are given by [40]  N < 8/3,  0, s0 = p  1 4 (1 ± 3 1 − 8/(3N )), N ≥ 8/3.

(18)

When N < 8/3, the isotropic state s0 = 0 is the only attractor. When 8/3 ≤ N ≤ 3, there are three constant equilibria: one isotropic and two prolate uniaxial ones [39, 14], in which the isotropic and the larger prolate equilibrium are stable, and a family of periodic steady states centered around the smaller and unstable prolate constant equilibrium. When N > 3, there are one isotropic (unstable), one oblate (unstable) and one prolate (stable) constant equilibria as well as a family of periodic steady states centered around the unstable isotropic constant equilibrium. The representative phase portraits of the steady state system in four distinct concentration regimes are depicted in Figure 1. The phase diagrams are qualitatively the same as the ones obtained using the DMG model for nematics [14]. The system also admits a family of traveling wave solutions: s = s(Z), where Z = ζ − ct is the traveling wave coordinate, and c is a constant speed to be determined. The governing equation for the traveling wave solution is given by, 0

−cs = [−U (s) +

2 27Er (1

0

− s)(2s + 1)[(3 + 2(1 + 4s)θ)s + 4θ(s )2 ],

(19)

where the prime denotes d/dZ. The special case c = 0 corresponds to the previous steady state. For c 6= 0, we have 0

dH 27c(s )2 = . dZ (1 − s)(2s + 1)

(20)

which is strictly increasing for c > 0 and decreasing for c < 0. This means that no periodic solutions survive for c 6= 0, but infinite period traveling wave emerge for N > 83 , which connects one equilibrium phase at Z = −∞ to another at Z = ∞. Specifically, the highly aligned nematic phase and the isotropic one connect to the

MESOSCALE STRUCTURES

297

less aligned prolate one as c > 0 for 8/3 < N < 3. The nematic phases connect to the isotropic one at N > 3. These connections reverse as c < 0. This can be shown using a phase plane analysis. 3.2. Nematic equilibrium patterns due to director variations. We next seek solutions of the orientation tensor given by director variations. Without loss of generality, we seek orientation tensor solutions of the general form I I Q = s(t)(n(z, t)n(z, t) − ) + β(n⊥ (z, t)n⊥ (z, t) − ), (21) 3 3 where n = (cos qz, sin qz, 0), n⊥ = (− sin qz, cos qz, 0),

(22)

with q the wave number of the solution. This ansatz represents an orientation tensor moving in the direction of ez in a helical fashion. Substituting this into the orientation tensor equation, we have  2 q (−6 β 2 + 25 θ s2 β − 25 θ sβ 2 − 18 θ s3 β     +18 θ β 3 s + 6 s2 + 12 s − 12 β + 18 β 3 − 18 s3     −54 β 2 s + 54 s2 β − 9 θ β 4 + 9 θ s4 + 2 θ β 2     −2 θ s2 − 11 θ s3 + 11 θ β 3 + 4 θ s − 4 θ β)    +θq q(−26 s2 β + 26 sβ 2 − 8 s + 8 β + 2 β 2 − 2 s2 + 10 s3 − 10 β 3 ) (23) +Er(27U (s) − 27 U (β) − 36 s2 N β + 36 s N β 2 ) = 0,     q 2 (2 β 2 − 2 θ s2 β 2 − 4 sβ + θ s2 β + θ sβ 2 + 2 s2 − 2 β 3      −2 s3 + 2 β 2 s + 2 s2 β + θ β 4 + θ s4 − θ s3 − θ β 3 )     +θq q(4 sβ − 2 s2 β − 2 sβ 2 − 2 β 2 − 2 s2 + 2 s3 + 2 β 3 )   +Er(3U (s) + 3 U (β) + 4 N s β) = 0. This system of algebraic equations admits the following sets of solutions: β = 0, q = 0,

4θq (1 − s) , 6 + θ(s + 2)

(24)

and s satisfies (2s + 1)[2 + (2 − s)θ] 2 2 s q = 0. (25) 6Er The first set given by q = 0, U (s) = 0 gives the equilibrium nematic order parameter solution with an arbitrary constant director; whereas the second set given by U (s) −

q=

8θq2 (2s + 1)[2 + (2 − s)θ](1 − s)2 2 4θq (1 − s) , U (s) − s =0 6 + θ(s + 2) 3Er[6 + θ(s + 2)]2

(26)

yields the solution for cholesterics with pitch q. Using symmetry argument, we notice that the above system of equations is invariant under the following transformations: (s, β) → (β, s) and (s, β) → (−(s + β)/2, (s + β)/2). Therefore, there are two other families of solutions that can be obtained following the linear transformations. The behavior of the equilibria for pure nematics given by U (s) = 0 have been well-documented [39]. We focus on the order parameter behavior in equilibrium cholesterics next. θq2 , measuring the strength We introduce a new dimensionless parameter Kq = Er of chirality relative to the distortional elasticity. If Kq >> 1, the equilibria of s are s ≈ −1/2, 0, 1; whereas if 0 ≤ Kq 3. The stability of the equilibria is similar to the pure nematic one after all [12, 39, 40]. In contrast to the sensitivity of the turning point bifurcation point to Kq , the bifurcation diagram is less sensitive to variations of the anisotropic elasticity θ. However, there is still a noticeable effect of θ on the prolate nematic order parameter at large Kq . For example, the nematic order at the turning point bifurcation point is higher at negative values of θ shown in Figure 2. The above observations indicate that phase transition from isotropic to cholesteric may take place by changing the anisotropic properties of the CLCP and/or its chirality, both of which can depend on temperature and weakly on concentration. We remark that the governing system of equations in the closure model also may admit unphysical solutions at higher concentration which is clearly an artifact of the closure approximation we use [39].

4. Permeation structures in weakly sheared cholesterics. We next study the flow and orientation structure in weakly sheared CLCPs in a plane Couette cell, characterized by a small dimensionless shearing speed. In particular, we investigate the structure using an ad hoc “coarse-grain” approximation [7, 37]. We consider the steady permeation flow of cholesterics with the helix along the flow direction z, between two parallel plates at x = ±1 in Cartesian coordinates (x, y, z) and moving with corresponding velocity v = (0, 0, ±²), where ² = De is the Deborah number defined by De = Dv00h with the choice of h the half gap width in r the shear cell. The equilibrium state is given by (26), i.e., I Q0 = s0 (nn − ), n = (cos qz, sin qz, 0), 3

(27)

4θ (1−s )

q 0 . P0 = π/q defines the equilibrium pitch of the helix. Figure where q = (6+θ(s 0 +2)) 3 depicts the permeation flow in the plane Couette flow geometry. When the weak shear is imposed, Q is expected to exhibit biaxiality. We thus parameterize the directors with three angles ζ, ξ and χ

n1 = (cosζcosξ, sinζcosξ, sinξ), (28) n⊥ = (sinχsinζ − cosχsinξcosζ, −sinχcosζ − cosχsinξsinζ, cosχcosξ), ζ is an in-plane tilt angle, ξ and χ are out of plane tilt angles [39, 9].

MESOSCALE STRUCTURES

299

At the equilibrium, ζ0 = qz ,ξ0 = 0 and χ = χ0 (undefined). When Q is uniaxial, χ is irrelevant. We seek asymptotic solution in small Deborah number expansions: ∞ ∞ X X s(x, z) = s0 + sn (x)²n , β(x, z) = βn (x)²n , (ζ, ξ, χ)(x, z)

=

n=1 ∞ X

(qz +

n=1

φn (x)²n ,

n

v(x, z)

=

(0,

∞ X

vy(n) (x)²n ,

n=1

∞ X

ωn (x)²n , χ +

n=1 ∞ X

∞ X

ψn (x)²n ),

(29)

n=1

vz(n) (x)²n ),

n=1

where the secondary flow in y direction is sought as well. The boundary conditions on the perturbation variables are (n)

φn (±1) = ψn (±1) = ωn (±1) = sn (±1) = βn (±1) = vy (±1) = 0, (1) (n) vz (±1) = ±1, vz (±1) = 0.

(30)

First, we expand the governing system of equations in powers of ² to obtain the asymptotic equations at each order of O(²n ), n = 0, 1, 2, · · · , in which we focus on the equations at order O(²), the linearized equations. Secondly, we employ an ad hoc “coarse-grain” approximation, in which we average the linearized equations over a pitch of the cholesteric structure in z. We remark that the coarse-grain approximation is equivalent to the Galerkin projection onto the zeroth order Fourier space in z-variable. Here we assume an out-of-plane flow along with an out-of-plane orientation structure. The coarse-grain linearized equations at order O(²) are given in the appendix. Eq. (45), (47), (48) and (49) along with boundary conditions give s1 (x) = β1 (x) = ω(x) = 0 and χ is undetermined at this order. (50) automatically holds. Thus, the system reduces to 0 A1 − ϕ + 2qvz − vy = 0, Er 0 ϕ − B1 Ervy = 0, (31) Ervz − B2 qϕ = 0, where 0 denotes

d dx

and A1 = B1 = B2 =

[6+θ(s0 +2)](s0 +2) , 9 6(4 µ1 s0 +3 µ2 s0 2 +24η+12µ3 )) , (12+a θ+2 a θ s0 2 )αs0 2 12−(a(4−s0 )+3s0 )θ 2 6(12η+6µ3 −µ1 s0 ) αs0 .

(32)

Since −1 < θ < +∞ [41], A1 > 0 and B1 > 0 for both rods and discs. If θ is not too big, i.e., θ < a(4−s12 , B2 > 0. Eqs (31)1 , (31)2 and (31)3 imply 0 )+3s0 0

0

vz − Kq 2 vz = 0, where K =

2B1 B2 1+A1 B1

(33)

> 0 is a function of µi , a, α, θ and s0 . Hence

√ sinh √Kqx , sinh Kq√ sinh Kqx ( sinh√Kq − x), φ(x) = BEr 2q √ √ √ √Kqx vy (x) = (− A1B2K + √2K )coth Kq( cosh cosh Kq

vz (x) =

(34) − 1).

These solutions exhibit boundary layers near the shearing plates whose thickness is of order O(q −1 ), which shrinks with respect to the polymer concentration N since

300

ZHENLU CUI, M. CARME CALDERER, QI WANG

s is an increasing function of N . Outside the boundary layer region, the velocity is essentially a constant and the angle profile is practically linear. Figure 4 depicts the typical variation of vz , vy and φ as functions of the gap width x. We note that all variations are insensitive to the shape of the molecule. The secondary flow is nonzero and nearly constant away from the boundaries. From (31), we notice that the three unknowns (vy , vz , φ) are coupled. Therefore, it is wrong to assign vy = 0 arbitrarily, which had been done unfortunately in previous studies. The rheological property of interest here is the apparent viscosity ηapp , defined by ηapp = R1

τzx |x=1 2F

(35)

where F = 0 vz (x)dx is the flow rate per unit length and τzx is the shear stress, which is a constant at this order across the Couette cell. τzx =

0 12η+6µ3 −µ1 s0 (Ervz 6Er

0

− B2 qφ ) =

12η+6µ3 −µ1 s0 . 6

(36)

The resulting apparent viscosity is given by q √ √ sinh 12η+6µ3 −µ1 s0 2B1 B2 √ Kq (37) ηapp = 12η+6µ63 −µ1 s0 cosh Kq ≈ 6 1+A1 B1 q, as Kq >> 1 . Kq−1 This result shows the same scaling behavior with respect to q as obtained in [37] using the LE theory. However, (37) reveals more details about the impact of the local nematic order and the molecular configuration. The first and second normal stress differences can be calculated directly by the stress formulas and are shown to be zero in this order. If θ > a(4−s12 , this is the large anisotropic elastic regime, K maybe negative 0 )+3s0 (K < 0). The leading order solutions are spatially oscillatory. This shows the impact of a dominant anisotropic elasticity on the permeation structure. In this case, the solutions are given by √ sin √−Kqx , sin −Kq √ Er sin √−Kqx φ(x) = B2 q ( sin −Kq − x), √ √ √ 2 √−Kqx vy (x) = (− A1 B2−K − √−K )cot −Kq( cos cos −Kq

vz (x) =

(38) − 1),

assuming Kq 6= 2mπ for some integer m. If Kq = 2mπ, however, the solution formula has to be modified, which is omitted here. The shear stress remains the same as that for K > 0. Two solutions, one for rodlike and the other for discotic CLCPs are depicted in Figure 5. We notice that this periodic structure occurs at the highly anisotropic regime, which perhaps can only happen at the concentration and temperature regime close to smectic transition. Therefore, this could be a precursor of the transitional behavior. 5. Permeation structures in weak plane Poiseuille flows. Next, we study the steady flow and orientational structure of cholesterics in the direction of the velocity gradient under an imposed, small pressure gradient ∂p ∂z in plane Poiseuille flows. Figure 6 depicts the geometry of the plane Poiseuille flow on the (z, x) plane. The orientational boundary condition for the orientation tensor is identical to that used for plane Couette flows while the velocity boundary condition is the no-slip one: vz (±1) = 0. Like in weak plane Couette flows, we use h, the half width of the channel, and the relaxation time t0 as the characteristic length and time scale, respectively. We adopt the same dimensionless symbols used in the weak plane

MESOSCALE STRUCTURES

301

2 Couette flow and assume ∂p ∂z = −² = −De in the dimensionless form, where the Deborah number is defined by s ∂p t20 De = − , ² = De2 . (39) ∂z ρh

We seek asymptotic solutions in powers of ². The other governing equations are identical to those derived for plane Couette flows except that the third component from the momentum balance equation now becomes 12η+6µ3 −µ1 s0 (Ervz 6Er

− B2 qϕ) − 1 = 0.

(40)

The boundary conditions on the perturbation variables are the same as the ones in the plane Couette flows except that (1)

(41)

vz (±1) = 0. When K > 0, the solutions are given by

√ cosh √Kqx ), cosh √ Kq cosh √Kqx ErC1 3Er φ(x) = B2 q (1 − cosh Kq ) + 12η+6µ (x2 − 1), 3 −µ1 s0 √ √ (42) Kqsinh Kqx 6Er √ vy (x) = − BA21q (− C1 cosh + x) 12η+6µ −µ s 3 1 0 Kq √ Kqx √ +2C1 q(x − √sinh ). Kqcosh Kq √ 6A1 K 1 √ √ ≈ (12η+6µ33A −µ1 s0 )B2 q 2 , as (12η+6µ3 −µ1 s0 )[2B2 Kq+(A1 K−2B2 )tanh Kq]q

vz (x) = C1 (1 −

where C1 = q À 1. Like in the plane Couette flows, these solutions exhibit boundary layer behavior at the walls with the thickness in the order of q −1 . Outside the boundary layer region, the primary velocity component is essentially a constant similar to the result obtained using the LE theory [7] while the angle profile is practically quadratic and the secondary velocity component vy is linear. Figure 7 depicts the typical variation of vz , vy and φ as functions of the gap width x. We note that the velocity variations are insensitive to the shape of a molecule while the angle variation for discotics are much more sensitive than that for rodlike ones. The shear stress is calculated as τzx = −x. The apparent viscosity in this case is given by ηapp =

−2τzx |x=1 3F

=

2 sinhKq 3C1 (1− KqcoshKq )

≈

2(12η+6µ3 −µ1 s0 )B2 2 q , 9A1

(43)

which is always positive, correcting the result in [7] using the LE theory. As in Couette flows, both the first and second normal stress differences are zero at order O(²). When K < 0, the steady state solutions are spatially periodic and given by √ cos √−Kqx ), cos √ −Kq cos √−Kqx ErC1 3Er φ(x) = B2 q (1 − cos −Kq ) + 12η+6µ (x2 − 1), 3 −µ1 s0 √ √ −Kqx √ vy (x) = − BA21q ( C1 −Kqsin + 12η+6µ63 −µ1 s0 x) cos −Kq √ sin −Kqx √ ). +2C1 q(x − √−Kqcos −Kq √ 6A1 −K √ √ . Figure 8 (12η+6µ3 −µ1 s0 )[2B2 −Kq+(A1 K−2B2 )tan −Kq]q

vz (x) = C1 (1 −

(44)

where C1 = depicts the typical solutions for some chosen parameters, in which the primary velocity flows backwards. When the flow direction is reversed due to the extreme anisotropic elasticity,

302

ZHENLU CUI, M. CARME CALDERER, QI WANG

N=2.8 0.2

0.5

0.1

sζ

sζ

N=2 1

0

−0.5

−1

0

−0.1

−0.2

0

0.2 s

0.4

0.6

−0.2 −0.1

0

0.1 s

0.3

N=10

1

1

0.5

0.5

sζ

sζ

N=6

0.2

0

−0.5

0

−0.5

−1 −0.4 −0.2

0

0.2 s

0.4

0.6

0.8

−1 −0.5

0

0.5 s

Figure 1. Four phase portraits of steady states in phase space (s, sζ ) corresponding to Er = 1.0, θ = 1. For N < 38 , there is a stable isotropic equilibrium (attractor) but no periodic solutions. For 83 < N < 3, there exist three constant equilibria and a family of periodic solutions p oscillating about the only unstable equilibrium s− = 14 (1 − 3 1 − 8/(3N )). For N > 3, there exist three constant equilibria along with a family of periodic solutions oscillating about the unstable p isotropic phase s = 0. The only stable equilibrium is s+ = 14 (1 + 3 1 − 8/(3N )). the winding direction of the director also reverses as shown in the figure. Again, if this were to happen, it would be a precursor to transition to smectics. 6. Conclusion. We have constructed two classes of equilibrium solutions with uniaxial symmetries using a mesoscopic closure model obtained from the kinetic theory for flows of cholesteric liquid crystal polymers and studied the permeation mode in plane Couette and Poiseuille flows using a coarse-grain approximation. We show that chirality in cholesteric liquid crystal polymers couples the local nematic order with the distortional elasticity leading to equilibrium isotropic-nematic phase transition at reduced concentration, enhanced distortional elasticity as well as chirality, among which the influence of the chirality on phase behavior is quite pronounced. By solving the flow orientation coupled equations with a nonzero secondary flow, we show that it is necessary to include the secondary flow in the coarse-grain approximation required by the chiral structure, correcting a long standing inconsistency

MESOSCALE STRUCTURES

303

1

θ=−0.5

0.5

θ=1.0

s

Solid line: Stable θ=0.0

Dotted line: Unstable

0

−0.5

0

1

2

3 N

4

5

6

Figure 2. The uniaxial order parameter equilibria as functions of polymer concentration N with selected values of θ = 1.0, 0.0, −0.5, respectively. Red curves corresponds to Kq = 1 which show that θ has little effect. Blue ones correspond to Kq = 10. A saddle node bifurcation of equilibria occurs at N1 = U (s) − 8Kq (2s+1)(2+(2−s)theta)(1−s)2 s2 , and at N2 = 3 a double-saddle node 3(6+θ(s+2))2 ) bifurcation takes place. Between N1 and N2 , first order phase transition may occur.

Q=Q 0

v0

x=1

x h z P0 Q=Q 0

x=−1 −v 0

Figure 3. The permeation flow geometry in a plane Couette cell. The gap width in the cell is 2h. The CLC polymers in the cell is sheared by moving the upper plate with a constant speed v0 and the lower one with −v0 . The helix of CLCPs is oriented along z−direction. At the bounding surfaces, the orientation tensor is assumed to equal to its equilibrium value.

304

ZHENLU CUI, M. CARME CALDERER, QI WANG

1 a=0.8, θ=0.001

0

−0.5

−1 −1

−0.5

0 x

0.5

−20 −1

1

a=−0.8, θ=−0.1

0 x

0.5

1

a=−0.8, θ=−0.1

20

φ

velocities

−0.5

40 vz vy

0

−0.5

−1 −1

0

−10

1

0.5

a=0.8, θ=0.001

10

φ

velocities

0.5

20 vz vy

0

−20

−0.5

0 x

0.5

1

−40 −1

−0.5

0 x

0.5

1

Figure 4. The primary and secondary velocity field vz , vy and director angle φ as functions of x. The upper two plots are for rodlike CLCPs, where the parameter values are a = 0.8, θ = 0.001, µ1 = 0.007, µ2 = 0.095, µ3 = 0.052. The lower two plots are for discotics, where the parameter values are a = −0.8, θ = −0.1, µ1 = −0.058, µ2 = 0.062, µ3 = 0.1. The parameter values common to all are N = 6.0, Er = 1000.0, θq = 100.0, µ0 = 1.0, α = 1.0, η = 0.001. Both the primary and the secondary flows are essentially constant away from the walls. The director angle is nearly linear in the middle. on the solution method for this problem in the literature. For the asymptotic solutions at the linear order of the small Deborah number expansion, we show that the flow and director dynamics dominate while the dynamics of the orientation order parameters and out-plane angles are suppressed to the next order. The scaling behavior of the solution and the apparent viscosity with respect to the pitch and other parameters is explored. The presence of an internal length scale (pitch of the helix) leads to the boundary layers in both flows, whose thickness is proportional to the size of the pitch when K > 0. The apparent viscosity scales like O(q) in plane Couette flows while the apparent viscosity varies like O(q 2 ) in weak plane Poiseuille flows. Moreover, our analysis shows that spatially periodic structures

MESOSCALE STRUCTURES

1.5 1

0

100 a=0.8, θ=4.0

a=0.8 θ=4.0

50

vz vy

0

φ

velocities

0.5

305

−0.5 −1

−50

−1.5 −2 −1

−0.5

0 x

0.5

1

−100 −1

1

0 x

0.5

1

0.5

1

30 vz vy

a=−0.8, θ=−1.0

20

a=−0.8, θ=−1.0

10 0

φ

velocities

0.5

−0.5

0 −10

−0.5 −20 −1 −1

−0.5

0 x

0.5

1

−30 −1

−0.5

0 x

Figure 5. The primary and secondary velocity field vz , vy and director angle φ as functions of x. The upper two plots are for rodlike CLCPs, where the parameter values are a = 0.8, θ = 4.0, µ1 = 0.007, µ2 = 0.095, µ3 = 0.052. The lower two plots are for discotics, where the parameter values are a = −0.8, θ = −1.0, µ1 = −0.058, µ2 = 0.062, µ3 = 0.1. Other parameters common to all are N = 6.0, Er = 1000.0, θq = 100.0, µ0 = 1.0, α = 1.0, η = 0.001. All profiles for rodlike CLCPs show oscillatory behavior due to K < 0 while those for discotics do not since K > 0. may emerge at extreme values of the anisotropic elasticity corresponding to K < 0. In this regime, the primary flow can reverse its direction locally across the shear cell leading to opposite director winding patterns compared with the case of K > 0. Further studies on this anomalous behavior will be continued in the future. Acknowledgment and Disclaimer Effort sponsored by the AFOSR through grants F49620-02-1-0086 and FA9550-05-1-0025, and National Science Foundation through grant DMS-0128832 and DMS-0204243 are gratefully acknowledged.

306

ZHENLU CUI, M. CARME CALDERER, QI WANG

Q=Q 0

x=1 x

h z

dp/dz

P0

x=−1 Q=Q 0

Figure 6. The permeation flow geometry in a plane Poiseuille cell. The gap width in the shear cell is 2h. A pressure gradient ∂p ∂z = −² is imposed in the channel. The helix is oriented along z direction. At the bounding surfaces, the orientation tensor is assumed to equal to its equilibrium value. 0.08

0 a=0.8, θ=0.001

a=0.8 θ=0.001

−50

0.04 −100 0.02

φ

velocities

0.06

vz vy

−150 0 −200

−0.02 −0.04 −1

−0.5

0 x

0.5

−250 −1

1

0.06

0 x

0.5

1

0.5

1

0

0.04

−50

a=−0.8, θ=−0.1

a=−0.8, θ=−0.1

0.02

−100

φ

velocities

−0.5

0

−150

v z vy

−0.02 −0.04 −1

−0.5

0 x

0.5

−200

1

−250 −1

−0.5

0 x

Figure 7. The primary and secondary velocity field vz , vy and director angle φ as functions of x. The upper two plots are for rodlike CLCPs, where the parameter values are a = 0.8, θ = 0.001, µ1 = 0.007, µ2 = 0.095, µ3 = 0.052. The lower ones are for discotics, where the parameter values are a = −0.8, θ = −0.1, µ1 = −0.058, µ2 = 0.062, µ3 = 0.1. Other parameters common to all are N = 6.0, Er = 1000.0, θq = 100.0, µ0 = 1.0, α = 1.0, η = 0.001. The primary flow is essentially constant away from the walls while the secondary one is nearly linear. The director angle is quadratic in the middle.

MESOSCALE STRUCTURES

2

1400 a=0.8, θ=4.0

1

1200 1000

−1

800

vz vy

−2

φ

velocities

0

400

−4

200 −0.5

0 x

0.5

0 −1

1

0.01

−0.5

0.5

1

0.5

1

−20

a=−0.8, θ=−1.0

a=−0.8 θ=−1.0

−40 0 vz vy

−0.005

−0.01 −1

0 x

0

φ

velocities

0.005

a=0.8 θ=4.0

600

−3

−5 −1

307

−0.5

0 x

0.5

−60 −80 −100

1

−120 −1

−0.5

0 x

Figure 8. The primary and secondary velocity field vz , vy and director angle φ as functions of x. The upper two plots are for rodlike CLCPs, where the parameter values are a = 0.8, θ = 4.0, µ1 = 0.007, µ2 = 0.095, µ3 = 0.052. The lower ones are for discotics, where the parameter values are a = −0.8, θ = −1.0, µ1 = −0.058, µ2 = 0.062, µ3 = 0.1. Other parameters common to all are N = 6.0, Er = 1000.0, θq = 100.0, µ0 = 1.0, α = 1.0, η = 0.001. The profiles for rodlike CLCPs show oscillatory behavior due to K < 0 while the ones for discs do not since K > 0. Appendix: The governing equations at order O(²) [24(1 − s0 )(1 − s0 − 3 cos2 χ) + θ(8 − 53 s0 − 11 s20 + 47 s0 3 d2 +(−3s20 + 33 s0 − 6s0 3 − 24) cos2 χ)] dx 2 β1 d2 +(24s0 − 48 s20 + 24 + 8 θ + 73 θs0 + 25 θ s20 − 88 θs0 3 ) dx 2 s1 1 d 2 3 3 2 + π (96 θq s0 − 384 θq s0 − 96 θ s0 q − 192 θ s0 q) dx ω +72(4 s0 q 2 + 9 Er cos2 χ − 4 s0 Er N − 2 s20 cos2 χq 2 −4 cos2 χs0 q 2 + 2 θq s20 q cos χ2 −θ s20 q 2 cos2 χ − 4θq s0 q + 4 θq q s0 cos2 χ +ErN − 2 s20 q 2 − 3 Er + 6 Er N s0 cos2 χ +6 Er N s20 cos2 χ − 3Er N cos2 χ − θ s20 q 2 + 4 θ s0 3 cos2 χq 2 + 2 θq s20 q)β1 +72(Er N − 4 s0 q 2 + 6 s20 q 2 − 3 Er − 6 s0 2 Er N + 2 s0 Er N +3 θ s20 q 2 − 4 θ s0 3 q 2 + 4 θq s0 − 6 θq s20 q)s1 = 0,

(45)

308

ZHENLU CUI, M. CARME CALDERER, QI WANG

4 2 π (−27 s0 θ q

d + 32θq s20 + 11 θq s0 + 20θq − 48 θ qs0 + 12 θ s0 3 q) sin 2χ dx β1 d2 2 +15s0 (−6 s0 − 4 θ s0 − 4 θ − θ s0 − 12) dx2 ϕ d (46) −135 s0 Er dx vy + 270 s0 qErvz 2 +60s0 (12q + 6 s0 q 2 − 18 s20 q 2 + 27 Er + 18 s20 Er N + 10 θq s20 q + 9 θ s0 3 q 2 −8 θq q − 9 s0 Er N − 2 θq s0 q − 11 θ s20 q 2 − 9 Er N + 4 θ q 2 − 2 θ s0 q 2 )ϕ = 0, 16 d 2 2 2 2 2 π (2 θq s0 + 2 θq s0 − 8 θq s0 cos χ − 4 θq + 4 θq cos χ − 5 θq s0 cos χ) dx2β1 d d 2 2 + 16 π θq (4 + 4s0 + s0 ) dx s1 + 3s0 (−14 θ s0 − 5 θ s0 − 8 θ − 24 − 12 s0 ) dx2 ω 3 2 2 +12s0 (18 θ s0 q + 36 s0 Er N −18 N Er − 18 s0 Er N + 20 θq s20 q − 16 θ s20 q 2 − 4θq q +5 θ s0 q 2 − 7 θq s0 q + 2 θ q 2 − 36 s20 q 2 + 21 s0 q 2 + 6 q 2 + 54 s0 Er)ω = 0,

(47)

(24 s20 − 48 s0 + 24 − 24 θ cos2 χ + 72 s0 cos2 χ − 72 cos2 χ + 8θ + 25θs0 d2 +15 θ cos χ2 s20 − 21 θ cos2 χs0 + 19 θ s20 − 43 θ s0 3 + 30 θ s0 3 cos2 χ) dx 2 β1 2 d +(24 s0 − 48 s20 + 8 θ + 24 − 29 θ s0 − 53 θs20 + 56 θ s0 3 ) dx 2 s1 d 3 2 + 96 π s0 (3 θq s0 + 4 θq − 4 θq s0 + θ s0 q + 2 θ s0 q) dx ω 2 2 2 2 +72(4 s0 q + 9 Er cos χ − 4 s0 Er N − 2 s0 cos χq 2 − 4 cos2 χs0 q 2 +2 θq s0 2 q cos2 χ − θ s20 q 2 cos2 χ − 4 θq s0 q + 4 θq s0 q cos2 χ + 2 θq s20 q +Er N + 4 θ s0 3 cos2 χq 2 + 9 Er N s0 cos2 χ + 9 Er N s20 cos χ2 −3Er N cos2 χ − 3 Er − θ s20 q 2 − 2 s20 q 2 )β1 +72( Er N − 4 s0 q 2 + 6 s20 q 2 − 3 Er − 6 s0 2 Er N + 2 s0 Er N +3 θ s02 q 2 − 4 θ s0 3 q 2 + 4 θq s0 q − 6 θq s20 q)s1 = 0,

(48)

d β1 πθq (−2s0 2 − 2 s0 + 3 s0 cos2 χ + 4 − 12 cos2 χ) dx d2 d 2 2 +πθq (s0 + 4s0 + 4) dx s1 + 4(−6s0 − 3 s0 − 2 θ s0 − θ s20 ) dx 2ω +4s0 (6 q 2 + 54 Er + 21 s0 q 2 + 20 θq s20 q + 2 θ q 2 + 5 θ s0 q 2 + 36 s20 Er N −18 s0 Er N − 7 θq s0 q − 16 θ s20 q 2 + 18 θ s0 3 q 2 −36 s20 q 2 − 18 Er N − 4 θq q)ω = 0,

(49)

πa(−48 s0 − 24 θ cos2 χ − 3 θ cos2 χs20 π − 11 θ s20 − 6 θ cos2 χs0 3 − 53 θ s0 +72 cos2 χs0 + 47 θ s0 3 + 33 θ cos2 χs0 + 24 d3 +36 s0 + 24 s20 + 8 aθ − 72 cos2 χ) dx 3 β1 +72πa(−2 θq s20 q − qθ s20 cos2 χ + 4 θq s0 − 4 θq s0 cos2 χ − 2 θq s0 2 cos2 χ −q 2 θ s20 + 9 cos2 χEr − 4s0 N Er + 4 q 2 s0 − 4 q 2 s0 cos2 χ − 3Er −2 q 2 s20 − 216 cos2 χN Er + 6 cos2 χs0 N Er + 6 cos χ2 s20 N Er d −2 q 2 s20 cos2 χ + N Er + 4 q 2 θ s0 3 cos2 χ) dx β1 +π(24as0 + 24a − 72 s0 + 73 aθ s1 s0 + 25 aθ s20 − 88 aθ s0 3 (50) d3 −48 as20 + 8 aθ ) dx 3 s1 +72a(2s0 N Er − 4 q 2 s0 + 3 q 2 θ s20 − 4 θq q s0 + 6 θq q s20 d −4 q 2 θ s0 3 + 6 q 2 s20 − 6s20 N Er + N Er − 3 Er) dx s1 +48s0 (3 aθq s0 + 9 s0 q − 21 θq s0 − 3 as0 q − 2 aθ q d2 −4 aθ s20 − 8 aθ s0 − 6 a)q dx 2ω +48s0 (−36 q 2 as20 + 6 q 2 a − 9 q 2 θ s0 2 + 21 q 2 as0 + 21 qθq as0 +9 q 2 θq s0 − 36 qθq as20 + 6 qθq a + 2 q 2 aθ + 18 q 2 aθ s0 3 + 36 as0 2 N Er −18 as0 N Er − 18 aN Er + 54 aEr − 16 q 2 aθ s20 + 5aθ s0 q 2 + 9 q 2 s0 )qω = 0,

MESOSCALE STRUCTURES

309

4α[3θq ( as0 + 2a − s0 ) + q(4 aθ s0 3 − 6 a + 6 as0 d2 −9 θ s20 + 3 a θ s20 − 2 a θ − 2 a θs0 ) sin 2χ dx 2 β1 3 d d2 2 2 +3παs0 (− a θ − 2 aθ s0 − 12) dx3 ϕπ + 18π(12 η + 4 µ1 s0 + 3 µ2 s20 )Er dx 2 vy (51) +α(−72 q 3 s0 − 4 aθ q 3 s20 − 40 aθ q 3 s0 − 96 aq 3 s0 3 3 3 +24 aq − 72 aqEr N + 72 aθ q s0 + 144 aqs0 Er N −36 θ s20 q 3 + 216 aqEr − 144 aq 3 s20 + 24 θq q 2 a − 36 θq q 2 s0 +8 aθ q 3 − 144 θq q 2 as0 2 + 144 aqs20 Er N − 60 θq q 2 as0 ) sin 2χβ1 = 0, 2

2

1 d d − 108πEr [3παs20 (−12 + 4 aθ − aθ s0 + 3θ s0 )q dx 2 ϕ + 18πEr(6η − µ1 s0 ) dx2 vz 3 2 2 2 2 2 +α(44θ as0 q + 12aq − 12a θq q − 42a θq qs0 − 24a θ s0 q − 72 as0 q +18 θ s20 q 2 + 4 aθ q 2 − 72 θq as20 q + 108 aEr + 72 as20 Er N (52) −36 aEr N − 6 aθ s0 2 q 2 d +36 s0 q 2 + 90 θq qs0 − 48 as0 q 2 + 72 as0 Er N ) sin 2χ dx β1 d3 +4aα(3s0 + θ s0 − θ − 3) sin 2χ dx3 β1 ] = 0,

REFERENCES [1] A. N. Beris and B. J. Edwards Thermodynamics of Flowing systems, Clarendon, Oxford(1994). [2] M. C. Calderer, M. G. Forest, and Q. Wang, Kinetic Theories and Mesoscopic Models for Solutions of Nonhomogeneous Liquid Crystal Polymers, J. Non-Newtonian Fluid Mechanics, 120 (2004), 69-78. [3] M.C. Calderer and B. Mukherjee, Chevron Patterns in Liquid Crystal Flows, Physica D, 98 (1996), 201-224. [4] M.C. Calderer , B. Mukherjee and S. Muzumder, Poiseuille flow of liquid crystals: highly oscillatory regimes, J. Non-Newtonian Fluid Mech., 99 (2001), 37-55 (2001). [5] T. Carlsson, Theoretical investigation of the shear flow of nematic liquid crystals with the Leslie viscosity α > 0: hydrodynamic analogue of first order phase transitions, Mol. Cryst. Liq. Cryst., 104 (1984), 307-334. [6] T. Carlsson, Unit-sphere description of nematic flows, Physical Review A, 34 (1986), 33933404 (1986). [7] S. Chandrasekhar, Liquid Crystals, 2nd ed. Cambridge University Press, Cambridge, 1992. [8] Z. Cui, M. G. Forest, and Q. Wang, On weak plane Couette and Poiseuille flows of rigid rod and platelet ensembles, to appear Siam. J. Appl. Math., 2005. [9] Z. Cui and Q. Wang, A continuum model for flows of cholesteric liquid crystal polymers and permeation flows, Submitted to J. Non-Newtonian Fluid Mechanics 2005. [10] P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, Oxford University Press (1993). [11] G. Derfel, Shear flow induced cholesteric-nematic transition, Mol. Cryst. Liq. Cryst., 92 (1983), 41-47. [12] M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Oxford U. Press (Clarendon), London-New York (1986). [13] J.L., Ericksen, Liquid Crystals with Variable Degree of Orientation, Arch. Rational Mech. Anal., 113 (1991), 97-120. [14] M.G. Forest, Q. Wang and H. Zhou, Exact banded patterns from a Doi-Marrucci-Greco model of nematic liquid crystal polymers, Physical Review E, 61 (2000), no. 6, 6655-6662. [15] M. G. Forest and Q. Wang, Monodomain response of finite-aspect-ratio macromolecules in shear and related linear flows, Rheological Acta 42 (2003), 20-46. [16] E. Grelet, What Is the Origin of Chirality in the Cholesteric Phase of Virus Suspensions?, Phys. Rev. Lett., 90 (2003), 198302. [17] W. Helfrich, Capillary flow of cholesteric and smectic liquid crystals, Phys. Rev. Lett.,23 (1969), 372. [18] W. Helfrich, Capillary viscometry of cholesteric liquid crystals, Proceedings Am. Chem. Soc., 2nd ed., 1969, 405-418 (1970). [19] U. D. Kini, G.S. Ranganath, and S. Chandrasekhar, Flow of cholesteric liquid crystals-I: Flow along the helical axis, Pramana, 5 (1975), 101-106.

310

ZHENLU CUI, M. CARME CALDERER, QI WANG

[20] U. D. Kini, Shear flow of cholesterics normal to the helical axis, J. Phys. (France), 40 (1979),62-65. [21] R.G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press (1999). [22] F. M. Leslie, The theory of flow phenomena in liquid crystals, Adv. Liq. Cryst., 4 (1979), 1-81. [23] F. M. Leslie, Continuum theory of cholesteric liquid crystals, Mol. Cryst. Liq. Cryst. 7(1969), 407-420. [24] F.H. Lin, On Nematic Liquid Crystals with Variable Degree of Orientation, Comm.Pure Appl.Math., 44 (1991), 453–468. [25] T. Lubenski, Hydrodynamics of cholesteric liquid crystals, Phys. Rev. A, 6 452-461(1969). [26] D. Marenduzzo, E. Orlandini, and J. M. Yeomans, Permeative flows in cholesteric liquid crystals, Physical Review Letters, V. 92 (18) (2004), 188301-1-188301-4. [27] E. H. MacMillan, Ph. D thesis, A Theory of Anisotropic Fluid, Department of Mechanics, University of Minnesota, 1987. [28] L. Onsager, The effects of shape on the interaction of colloidal particles, Ann. N. Y.. Acad. Sci., 51 (1949), 627-659. [29] A. Panchenko and M.C. Calderer, Young Measures and Order-Disorder Transitions in Nematic Liquid Crystals, SIMA Jour. Math. Anal., submitted. [30] K. R. Purdy, and S. Fraden, Isotropic-cholesteric phase transition of filamentous virus suspensions as a function of rod length and charge, Phys. Rev. E, 70 (2004), 061703. [31] A. D. Rey, Flow-alignment in the helix uncoiling of cholesteric liquid crystals, Phys. Rev. E, 53 (1996), 4198-4201. [32] A. D. Rey, Structural transformations and viscoelastic response of shear fingerprint cholesteric textures, J. Non-Newt. Fluid Mech., 64 (1996), 207-227. [33] A. D. Rey, Helix uncoiling modes of sheared cholesteric liquid crystals , J. Chem. Phys., 104 (1996), 789-792. [34] A. D. Rey, Theory of linear viscoelasticity for chiral liquid crystals, Rheol. Acta, 35 (2000), 400-409. [35] A. D. Rey, Theory of linear viscoelasticity in cholesteric liquid crystals, J. Rheol., 44 (2000), 855-869. [36] A. D. Rey, Generalized cholesteric permeation flows, Phys Rev E,65(2002) , 022701. [37] A. D. Rey, Simple shear and small amplitude oscillatory rectilinear shear permeation flows of cholesteric liquid crystals, J. Rheol., 46 (2002), no. 1, 225-240. [38] T. Shimada, M. Doi, and K. Okano, Concentration fluctuation of stiff polymers. III. Spinodal decomposition, J Chem. Phys., 88(1988), 7181-7186. [39] Q. Wang, Biaxial steady states and their stability in shear flows of liquid crystal polymers, J. Rheol., 41 (1997), no. 5, 943-970. [40] Q. Wang, Comparative Studies on Closure Approximations in Flows of Liquid Crystal Polymers. I. Elongational Flows, Journal of Non-Newtonian Fluid Mechanics, 72 (1997), 141162. [41] Q. Wang, A hydrodynamic theory of nematic liquid crystalline polymers of different configurations, J Chem. Phys., 116 (2002), no. 20, 9120-9136. [42] D. C. Wright, and N.D. Mermin, Crystalline liquids: the blue phases, Rev. Mod. Phys., 61 (1989), 385-432.

Received March 2005; revised September 2005. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]