Volume 4, Number 3-4, Pages 460–477 · Volume 4, Number 3-4, ... We use the mesoscopic Doi-Marrucci-Greco model, which incor- ... 2) the rotary diﬀusivity is explicitly

INTERNATIONAL JOURNAL OF c© 2007 Institute for ScientificNUMERICAL ANALYSIS AND MODELING Computing and InformationVolume 4, Number 3-4, Pages 460–477

NEMATIC LIQUIDS IN WEAK CAPILLARY POISEUILLE FLOW:

STRUCTURE SCALING LAWS AND EFFECTIVE

CONDUCTIVITY IMPLICATIONS

HONG ZHOU AND M. GREGORY FOREST

Abstract. We study the scaling properties of heterogeneities in nematic (liq-

uid crystal) polymers that are generated by pressure-driven, capillary Poiseuille

flow. These studies complement our earlier drag-driven structure simulations

and analyses. We use the mesoscopic Doi-Marrucci-Greco model, which incor-

porates excluded-volume interactions of the rod-like particle ensemble, distor-

tional elasticity of the dispersion, and hydrodynamic feedback through orien-

tation dependent viscoelastic stresses. The geometry likewise introduces an-

choring conditions on the nano-rods which touch the solid boundaries. We

first derive flow-orientation steady-state structures for three different anchor-

ing conditions, by asymptotic analysis in the limit of weak pressure gradient.

These closed-form expressions yield scaling laws, which predict how lengthscales

of distortions in the flow and orientational distribution vary with strength of

the excluded volume potential, molecule geometry, and distortional elasticity

constants. Next, the asymptotic structures are verified by direct numerical

simulations, which provide a high level benchmark on the numerical code and

algorithm. Finally, we calculate the effective (thermal or electrical) conductiv-

ity tensor of the composite films, and determine scaling behavior of the effective

property enhancements generated by capillary Poiseuille flow.

Key Words. Liquid crystal (nematic) polymers, asymptotic expansions, par-

tial differential equations, capillary Poiseuille flow, conductivity

1. Introduction

The rheological behavior of Poiseuille flow of liquid crystal polymers is of interestfor technology applications in processing of high performance fibers and films. Reyand co-workers have studied capillary Poiseuille flows using the Leslie-Ericksentheory for discotic nematic liquid crystals ([4, 5, 6]). Denniston et al. have usedlattice Boltzmann simulations of Landau-deGennes orientation tensor models toexplore the behavior of liquid crystal polymers subject to Poiseuille flow ([9]). Inthis work we study capillary Poiseuille flow using the Doi-Marrucci-Greco model,which also employs an orientation (second-moment) tensor description of the roddistribution.

We extend our early work ([3], [17]) in the following ways: 1) asymptotic resultsfor tilted anchoring condition are derived; 2) the rotary diffusivity is explicitlycoupled in the asymptotic results; 3) direct numerical simulations are carried outto validate asymptotic results; and 4) effective conductivity properties of Poiseuilleflow-generated composite films are calculated.

Received by the editors February 27, 2006, and, in revised form, March 2, 2006.

460

NEMATIC LIQUIDS IN WEAK CAPILLARY POISEUILLE FLOW 461

The paper is organized as follows. First, we describe the model of liquid crystalpolymer hydrodynamics proposed by Doi, Marrucci and Greco ([10, 25, 26, 27]).Second, we present the structure scaling properties of the orientation tensor fordifferent anchoring conditions; this is achieved by asymptotic analysis that yieldsexact orientation modes with spatial variations controlled by material and boundaryconditions in the limit of weak pressure gradient. We perform direct numericalsimulations to verify the asymptotic results, and then observe new properties asthe asymptotic conditions are violated. Finally, we calculate the effective (thermalor electrical) conductivity tensor of the composite films, and determine scalingbehavior of the effective property enhancements generated by capillary Poiseuilleflow.

2. Model Formulation

We consider capillary Poiseuille flow of nematic liquid crystal polymers (LCPs).Capillary Poiseuille flow is the flow between two non-slip boundaries at y = −hand y = h driven by a pressure gradient in this context. The flow is describedby a velocity field v = (vx(y, t), 0, 0) with the centerline of the pipe located aty = 0. Figure 1 depicts the cross-section of the Poiseuille flow on the (x, y) plane.For simplicity here we suppress variations in the direction of flow (x) and primaryvorticity direction (z), and transport in the vertical (y) direction.

V=0 , Q=Q0

V=0 , Q=Q0

Directors

y

x

Figure 1. Plane Poiseuille flow geometry. Non-slip boundary con-ditions for the velocity and boundary anchoring for the orientationtensor given by the stable nematic rest state are prescribed, withmajor director angle ψ0 = 0 shown here.

The dimensionless governing equations consist of the balance of linear momen-tum, stress constitutive equation, continuity equation, and the equation for the

462 H. ZHOU AND M.G. FOREST

orientation tensor ([17]):

dvdt = ∇ · (−pI + τ), p = ǫ x,

τ =(

2Re + µ3

)

D + aαF (Q)

+ aα3Er

∆Q : Q(Q + I

3 ) − 12 (∆QQ + Q∆Q) − 1

3∆Q

+ α3Er

12 (Q∆Q − ∆QQ) − 1

4 (∇Q : ∇Q −∇∇Q : Q)

+µ1

(Q +I

3)D + D(Q +

I

3)

+ µ2D : Q

(

Q +I

3

)

,

∇ · v = 0,

ddtQ = ΩQ − QΩ + a [DQ + QD] + 2a

3 D − 2aD : Q(Q + I

3 )

−1

Λ

F (Q) +1

3Er[∆Q : Q(Q +

I

3) −

1

2(∆QQ + Q∆Q) −

1

3∆Q]

,

(1)

where v is the velocity field for the flowing LCP, p is the pressure, I is the unit tensor,τ is the total stress, a is a dimensionless parameter depending on the molecularaspect ratio r of spheroidal molecules

a =r2 − 1

r2 + 1,

Re is the solvent Reynolds number, Er is the Ericksen number which is a reciprocalelasticity constant, µi (i = 1, 2, 3) are three nematic Reynolds numbers, α is anormalized entropic parameter, D is the rate-of-strain tensor and Ω is the vorticitytensor,

(2) D =1

2(∇v + ∇vT ), Ω =

1

2(∇v −∇vT ).

Here the superscript T denotes the transpose of a second order tensor.

(3) F (Q) = (1 −N/3)Q −NQ2 +NQ : Q(Q + I/3),

(4) Λ =

1, with constant rotary diffusivity,

(1 − 32Q : Q)2, with orientation-dependent rotary diffusivity.

The no slip condition at the bounding surfaces is used for the scaled axial velocityvx:

vx|y=±1 = 0.(5)

Homogeneous mesophase anchoring is assumed at the boundaries, which assumesthat the rod ensemble is in an equilibrium bulk phase. We work at a concentrationwhere the rest state is a stable ordered equilibrium, called the nematic phase:

Q|y=±1 = s0(nn − I

3 ),

s0 = 14 (1 + 3

√

1 − 83N ),

(6)

where s0 is the stable uniaxial order parameter specified by the nematic concen-tration N > 8

3 , and n is the equilibrium uniaxial director. We will fix N > 3 so


that the nematic phase is the unique stable equilibrium, limiting the chances ofpotential hysteresis and bi-stability in flow-induced structures. We assume n liesin the flow-flow gradient plane at some experimentally dictated anchoring angle ψ0

with respect to the flow direction,

n = (cosψ0, sinψ0, 0).(7)

ψ0 = 0, ψ0 = π2 , and 0 < ψ0 <

π2 correspond to tangential, homeotropic and tilted

anchoring, respectively.We consider the in-plane mesophase orientation of LCPs with two directors of Q

confined to the plane (x, y), but still admitting biaxiality. This restriction is not eas-ily lifted for the analysis to follow; numerically, there is not limitation. The point,however, is to glean some prediction of the types of spatial structures that arise,out of equilibrium phases, due to the combined influences of a pressure gradientinduced flow, and anchoring of the rod ensemble at the walls. We will find, consis-tent with our earlier analyses, two types of structure modes: extended structures inthe orientational distribution which accompany the Poiseuille flow geometry; and,boundary layers which arise due to wall confinement. Drilling somewhat deeperin these analytical, though asymptotic, insights into structures, a more importantquestion is a prediction of the scaling laws of each type of structure mode. Namely,it is impossible to explore this multi-parameter space even with large-scale com-putations; guidance as to how the lengthscales of spatial heterogeneity vary withparameters is of paramount value in guiding our subsequent numerical simulations.Thus, we make no apologies either for the reduction in physical space dimensionsfrom three to one, or for the reduction in orientational degrees of freedom, namelyfrom a full probability distribution function with infinite degrees of freedom to asecond-moment tensor with generally five degrees of freedom, and now to a second-moment whose peak axis of orientation must lie in the plane of the primary flowand flow-gradient directions. These assumptions shall lead to explicit structuremodes and scaling laws, in flow and particle orientation, as well as an indication ofwhen these asymptotic approximations will break down. Those insights will thenindicate grid-scale resolution of algorithms across parameter space.

The in-plane orientational constraint implies that two components in the Carte-sian representation of Q must vanish, Qxz = Qyz = 0. Alternatively, the orientationtensor can be written in terms of directors and order parameters:

Q = s(y, t)(nn −I

3) + β(y, t)(n⊥n⊥ −

I

3),(8)

with the directors n,n⊥ confined to the (x, y) plane and parametrized by the in-plane Leslie angle ψ(y, t),

n = (cosψ, sinψ, 0),n⊥ = (− sinψ, cosψ, 0).(9)

The third director is rigidly constrained along the vorticity axis.The explicit coordinate change between these Q representations is:

Qxx = s(cos2 ψ −1

3) + β(sin2 ψ −

1

3) ,

Qxy = (s− β) sinψ cosψ ,

Qyy = s(sin2 ψ −1

3) + β(cos2 ψ −

1

3) .

(10)


With the biaxial representation (8) of Q, the tensor equation in (5) can bewritten explicitly in terms of the two order parameters (s, β) and the Leslie angleψ:

∂s

∂t= −

1

Λ

U(s) +2N

3sβ(1 + β − s) +

2

9Er(s− β) g(s, β) (

∂ψ

∂y)2

+1

9Er

[

−(1 − s)(1 − β + 2s)∂2s

∂y2+ s(1 − s+ 2β)

∂2β

∂y2

]

+a

3

∂vx

∂yg(s, β) sin 2ψ,

∂β

∂t= −

1

Λ

U(β) +2N

3sβ(1 + s− β) +

2

9Er(β − s) g(β, s) (

∂ψ

∂y)2

+1

9Er

[

β(1 − β + 2s)∂2s

∂y2− (1 − β)(1 − s+ 2β)

∂2β

∂y2

]

−a

3

∂vx

∂yg(β, s) sin 2ψ,

∂ψ

∂t=

1

6(s− β)

∂vx

∂y(3β − 3s+ a(2 + s+ β) cos 2ψ)

+1

3ErΛ(2 + s+ β)

[

(s− β)∂2ψ

∂y2+ 2

∂ψ

∂y(∂s

∂y−∂β

∂y)

]

,

(11)

where

U(s) = s(1 − N3 (1 − s)(2s+ 1)) ,

g(s, β) = 1 + 3sβ − β + 2s− 3s2 .(12)

The momentum equation is reduced to a single equation for the axial velocitycomponent vx,

∂vx

∂t=∂τxy

∂y− ǫ,

τxy =aα

2

[

U(s) − U(β) +4Nsβ

3(β − s)

]

sin 2ψ

+aα

18Er

[

−h(s, β)∂2s

∂y2+ h(β, s)

∂2β

∂y2+ 2(s− β)γ(s, β)(

∂ψ

∂y)2

]

sin 2ψ

+aα

18Er(s+ β + 2)[(β − s)

∂2ψ

∂y2− 2(

∂s

∂y−∂β

∂y)∂ψ

∂y] cos 2ψ

+α

6Er(s− β)[(s− β)

∂2ψ

∂y2+ 2(

∂s

∂y−∂β

∂y)∂ψ

∂y]

+

[

µ1

6(s+ β + 2) +

µ2

8(s− β)2(1 − cos 4ψ) +

1

Re+µ3

2

]

∂vx

∂y,

(13)


where

γ(s, β) = g(s, β) + g(β, s),

h(s, β) = (1 − β + 2s)(1 + β − s).(14)

We seek steady solutions of the dimensionless governing equations (11-14) subjectto the boundary conditions

vx|y=±1 = 0, s|y=±1 = s0, β|y=±1 = 0, ψ|y=±1 = ψ0.(15)

We posit a formal asymptotic expansion in the small ǫ limit, consistent with theabove boundary conditions:

vx =

∞∑

k=1

v(k)x (y)ǫk, ψ =

∞∑

k=0

ψ(k)(y)ǫk,

s =∞∑

k=0

s(k)(y)ǫk, β =∞∑

k=1

β(k)(y)ǫk.

(16)

Alternatively, the Cartesian representation of Q is expanded in the form:

Q =∞∑

k=0

Q(k)(y) ǫk ,(17)

with Q(0) a quiescent, homogeneous equilibrium given by (6). ψ(1), s(1), β(1) andthe components of Q(1) are explicitly related by :

ψ(1) =2Q

(1)xy cos 2ψ0 − (Q

(1)xx −Q

(1)yy ) sin 2ψ0

2 s,

s(1) =3

2(Q(1)

xx +Q(1)yy ) +

1

2(Q(1)

xx −Q(1)yy ) cos 2ψ0 +Q(1)

xy sin 2ψ0,(18)

β(1) =3

2(Q(1)

xx +Q(1)yy ) −

1

2(Q(1)

xx −Q(1)yy ) cos 2ψ0 −Q(1)

xy sin 2ψ0.

We remark that anchoring conditions at the two boundaries are assumed iden-tical. Different conditions at each boundary require a non-homogeneous steadystructure at leading order ([34]), which we will not pursue for this study.

3. Steady Asymptotic Structures

From either representation of Q, the above expansions are inserted into the fullsystem of flow-nematic equations, (1)-(4) or (11)-(12), equations for the O(ǫ) andO(ǫ2) variables derived, and then systematically solved. We now state the results,following the notation in expansion (16).

3.1. Velocity Structure. The axial velocity at O(ǫ), v(1)x , reproduces the simple

Poiseuille flow structure:

v(1)x = Cv (1 − y2),(19)


where

Cv =3

6Re + µ1(2 + s0) + 3µ3 + 3

2µ2s20 sin2(2ψ0) + (M1M2 +M3)α,

M1 =3Λ

2 + s0(1 − λL cos 2ψ0),

M2 = 3s20 (1 − λL cos 2ψ0),

M3 = a2 sin2(2ψ0) Λ (2 + s0 − 3s20),

λL =a(2 + s0)

3s0.

(20)

3.2. Director Structure. The first consequence of these exact mesoscopic so-lutions is quite striking: the director is seen to only participate in the long-rangeextended structure and to not vary in the boundary layers. From equation (18), thedirector angle ψ is explicitly given by

ψ = ψ0 + ǫCp y (1 − y2) +O(ǫ2), Cp = Cv M1Er.(21)

By inference, the confinement-induced boundary layer structures are therefore re-sponsible only for focusing of the orientational distribution.

It is easy to show that M1M2 + M3 > 0. Then, in the denominator of Cv, allterms are positive definite, so that Cv is positive definite. M1, however, is not:it depends on whether the Leslie parameter λL is greater or less than one. As inCouette (plate-driven) flow ([17]), this controls the direction of twist of the majordirector from plate to plate, from formula (21).

3.3. Order Parameter Structure. We now extract degree and scales of molec-ular elasticity induced by slow Poiseuille flow. The striking feature is extremesensitivity to anchoring angle. From the s, β formulas, (18), it follows that s(1) andβ(1) vanish if and only if ψ0 = 0, π

2 (mod π). These were the sole boundary condi-tions considered in ([3]), so the tilted structures presented here will have differentfeatures, again matching the similar phenomenon discovered in Couette flow ([17]).

• Tangential anchoring (ψ0 = 0)Order parameter distortions enter at O(ǫ2) through two boundary layers and the

long-range permeation structure:

s = s0 + ǫ2 s(2) +O(ǫ3),

β = ǫ2 β(2) +O(ǫ3),

β(2) = C(2)1 cosh(Er1/2Gy) +A

(2)2 y4 + C

(2)2 y2 + E

(2)2 ,

s(2) = 2C(2)3 cosh(Er1/2H y) + 2C

(2)5 cosh(Er1/2Gy)

+C(2)7 y4 + C

(2)9 y2 + C

(2)11 ,

(22)


where

G =

√

9 − 3N + 6Ns20 + 6Ns01 − s0

, H =

√

9 − 3N + 18Ns20 − 6Ns01 + s0 − 2s20

,

A(2)2 =

2Cp(1 − s0)(3s0Cp − 2aCvErΛcos 2ψ0)

Er(3 −N + 2Ns20 + 2Ns0),

C(2)2 =

12(A(2)2 − C2

ps0 + aCvCpErΛcos 2ψ0)

ErG2,

E(2)2 =

2(C(2)2 + C2

ps0)

ErG2, C

(2)1 = −

A(2)2 + C

(2)2 + E

(2)2

cosh(Er1/2G),

C(2)5 =

s0(G2 + 6N)C

(2)1

(1 + 2s0)(G2 −H2),

C(2)7 = −

6[ErNs0A(2)2 + Cp (3s0 + 1) (3s0Cp − 2aCvErΛcos 2ψ0)]

ErH2(1 + 2s0),

C(2)9 =

6

ErH2(1 + 2s0)[2(1 + 2s0)C

(2)7 − s0(2A

(2)2 + ErNC

(2)2 )

+2Cp (1 + 3s0) (Cps0 − aCvErΛcos 2ψ0)]

C(2)11 =

2(1 + 2s0)C(2)9 − 2s0(2C

(2)2 + 3ErN E

(2)2 ) − 2C2

ps0(1 + 3s0)

ErH2(1 + 2s0),

C(2)3 = −

2C(2)5 cosh(Er1/2G) + C

(2)7 + C

(2)9 + C

(2)11

2 cosh(Er1/2H).

Combined with the director results, (21), we conclude that tangential anchoringleads to a director-dominated, non-uniform, long-range structure with Er−1 meanscaling, the same behavior as in Couette flow. This implies that pressure-drivenand drag-driven flows, which are respectively parabolic and linear shear layers, donot alter the molecular elasticity-dominated orientational gradients.

• Normal (homeotropic) anchoring (ψ0 = π/2)

The structure model (11)-(15) for in-plane Q tensors admits a symmetry: (a, ψ)→ (−a, π

2 +ψ). This property implies the asymptotic solution with ψ0 = π/2 can beobtained directly from (22). Thus normal and tangential anchoring structures arequantitatively different, with changes in numerical pre-factors, yet have qualitativelysimilar scaling laws. The figures will illustrate these features.

• Tilted anchoring (0 < ψ0 < π/2 or π/2 < ψ0 < π)Any pre-tilt induces an order parameter response of the same amplitude, O(ǫ),

as the director distortion, yet localized in the two boundary layers:

s = s0 + ǫ s(1) +O(ǫ2),

β = ǫ β(1) +O(ǫ2),

(23)


where

β(1) = 2A1 sinh(Er1/2Dy) +A2 y,

s(1) = 2C1 sinh(Er1/2Fy) − C2 y + 2C3 sinh(Er1/2Dy),

B = 9 + 6Ns0 − 3N + 6Ns20, D =

√

B

1 − s0,

A2 =6aCvΛ sin(2ψ0) (1 − s0)

B, A1 = −

A2

2 sinh(Er1/2D),

B1 = 18Ns20 − 3N − 6Ns0 + 9, B2 = 6aCvΛ sin(2ψ0) (1 + 2s0 − 3s20),

B3 =6Ns0(1 − s0)A2 +B2

B1,

F =

√

B1

1 + s0 − 2s20, C2 =

6Ns0(1 − s0)A2 +B2

B1,

C3 =(s0 − s20)A1 (ErD2 + 6ErN)

(1 + s0 − 2s20 −B1Er)ErD2,

C1 =B3 − 2C3 sinh(Er1/2D)

2 sinh(Er1/2F ).

This result for Poiseuille flow is qualitatively similar to Couette flow in the slowflow limit ([3], [17]). It suggests a flow-nematic structure mechanism: molecularelasticity is amplified in local boundary layers when the major director is pinnedat an angle tilted with respect to the flow-flow gradient axes. This result forebodesa phenomenon that only occurs at much higher driving conditions or with much”softer” elasticity constants, namely the appearance of defects in the orientationaldistribution. These structures correspond to local regions of isotropy (randomness)of the orientational distribution. In such regions, the local principal axes of orienta-tion are not able to align with the primary flow axes, which causes sharp boundarylayers with even smaller lengthscales, on the order of Er−

1

2 according to the predic-tions above. Since Er for nematic polymers are on the order of 106, this suggests avery stringent grid-scale will be required to resolve these structures, and, an adap-tive mesh refinement technique will be required since there is no a priori knowledgeof where these defects will arise. Finally, in nematic polymers as opposed to liquidcrystals, defects are transient as opposed to preserved, and attractors are dynamicrather than predominantly steady; this means under-resolution in time integratorsor simply a focus on steady structures are essentially useless.

4. Effective Conductivity Tensors

We now take the results from above on pressure-driven processing flows, andpredict what kinds of conductivity features are associated with these orientationaldistributions. This is part of an effort within our larger research group to linkprocessing of nematic polymer nano-composites (NPNCs) to effective properties.In ([32]), we developed a connection between the probability distribution functionof nematic polymers and the effective conductivity tensor of low-volume-fraction


nanocomposite materials based on homogenization theory. One simply needs toposit the properties of the nano-rods and of the matrix, where each phase is assumedto be an isotropic material. Extensions to anisotropic nano and matrix phases arestraightforward, but a detail that only clouds the essential predictions we wish toconvey here.

This approach considers spheroidal molecules (rods in this instance) with molec-ular orientation m, aspect ratio r, isotropic conductivity σ2 and volume fractionθ2 (assumed quite small, indeed in practical nano-composites on the order of 1%)populated in a matrix of conductivity σ1 according to the orientation distributionfunction f(m) of the ensemble. Conductivity obeys elliptic equations which are vir-tually identical for thermal, electric, and dielectric properties. The volume fraction,θ2, of the nematic polymer inclusions is proportional to the dimensionless strength,N , of the Maier-Saupe excluded volume potential:

(24) θ2 =Nπ

8r,

where r = l/d is the aspect ratio of the molecular spheroids with length l anddiameter d. Then, the effective conductivity tensor in closed form is derived in thelow-volume-fraction limit (θ2 << 1) as:

(25)

σe = σ1I + σ1θ2(σ2 − σ1)

2

σ2 + σ1 − (σ2 − σ1)LaI

+(σ2 − σ1)(1 − 3La)

[(σ2 + σ1) − (σ2 − σ1)La][σ1 + ((σ2 − σ1)La]M(f)

+O(θ22),

where I is the 3 by 3 identity matrix, La is the spheroidal depolarization factordepending on the nematogen aspect ratio r through the relation

(26) La =1 − ǫ2

ǫ2

[

1

2ǫln

(

1 + ǫ

1 − ǫ

)

− 1

]

, ǫ =√

1 − r−2.

Here the orientation probability distribution function of the LCP inclusions, f , isgoverned by the Smoluchowski equation of the Doi-Hess kinetic theory for quiescentor flowing nematic polymers ([18]); M(f) is the second-moment of the orientationprobability density f . Recall that the orientation tensor Q of the previous devel-opment is simply Q = M − 1

3I.From equation (25) it is obvious that the effective conductivity shares the same

three principal axes with the second moment tensor M(f). We denote the threeprinciple values of conductivity (i.e. eigenvalues of σe) as σe

1 ≥ σe2 ≥ σe

3, wherethe largest principal value, σe

1, is the maximum effective conductivity. The relativeprincipal value enhancements are given by

(27) εi =σe

i − σ1

σ1, i = 1, 2, 3,

where ε1 ≥ ε2 ≥ ε3. The value ε1 − ε2 depicts property anisotropy with respect tothe two principal conductivity directions, whereas ε1 captures the maximum scalarenhancement due to the nematic polymer inclusions. For example, ε1 = 1 impliesσe

1 = 2σ1 or 100% gain of the bulk composite conductivity.Heterogeneous effective properties of sheared and extension-enhanced nematic

polymer nanocomposites have been studied in ([18, 33]). In the next section wewill study the anisotropy and effective properties enhanced by Poiseuille flows.


5. Numerical Results

5.1. Numerical Methods. In order to numerically find the steady state solutionsof the boundary value problem (11-15), we use Newton’s method to iteratively ob-tain the steady state solutions. Our codes cannot resolve parameter regimes whereEr is above a critical threshold value; this may be either due to singular structuresor due to structure transitions where steady structures give rise to transient at-tractors. In the high Ericksen number limit, where the lengthscales of distortionalelasticity are predicted to scale like Er−1 and Er−1/2, adaptive gridding will berequired since one does not know a priori where these internal boundary layers willform. Some of these numerical tools have been developed in our group, but nonehave yet been applied to pressure-driven flows.

5.2. Numerical validation of the asymptotic results and vice versa. Fig-ures 2-4 compare the exact asymptotic formulas for the director angle ψ, orderparameters s, β, and flow velocity vx, with direct numerical solutions of the steady,boundary value problem. Figure 4 also gives shear stress and light scattering in-tensity predictions, which provide a basis for experimental validation. For thecomputations shown here, we fix a concentration in the nematic regime N = 6 andthe aspect ratio r = 3 (i.e., a = 0.8), consistent with our earlier studies ([14, 17]).

We begin with confirmation of the formulas for 0 < ǫ ≪ 1 and 0 < ǫ · Er ≪1, Figure 2, where the asymptotic formulas are nearly identical to the numericalsolutions. For tangential and normal anchoring, we set ǫ = 0.1, Er = 1; for tiltedanchoring, ψ0 = π

6 , we have to lower ǫ = 0.05 with the same Er = 1 to maintainagreement between the asymptotic formulas and the numerical solutions.

Next we illustrate breakdown of the asymptotic formulas with direct numericalsolutions when either of the two asymptotic conditions, ǫ ≪ 1 and ǫ · Er ≪ 1, aregradually pushed out of the asymptotic regime. Our purpose here is to monitoremergence of new, non-asymptotic structures, and to identify sources of additionalscaling behavior.

Figure 3 imposes tangential (Figure 3c and d) and normal anchoring (Figure 3eand f) with ǫ = 0.3, Er = 10. For tilted anchoring, Figures 3a, 3b, a more stringentǫ · Er condition is needed for asymptotic agreement, with ǫ = 0.5, Er = 1. Onefinds quantitative asymptotic accuracy.

In Figure 4a,b, we raise the pressure gradient to (ǫ = 0.5), with Er = 20 (forwhich ǫ ·Er = 10), and present results only with normal anchoring. Note that thevelocity profile departs from being parabolic near the boundaries. The concavityof the velocity profile changes from concave-up to concave-down and then concave-up again. We further give predictions of shear-gap stresses (Figure 4c) and lightintensity patterns (Figure 4d) associated with these steady structures. Neither N1

nor N2 changes sign across the rectangular pipe. The light scattering intensityacross the pipe associated with the numerical solutions is given in Figure 4d. Thisintensity pattern can be viewed as a combination of pure nematic patterns of theDoi-Marrucci-Greco theory, ([15, 16]), where the flow coupling has led to selectionof order-parameter structures near the plates and director structures in the interior.

5.3. Effective conductivity properties. Our main focus in this subsection isthe determination of effective properties associated with capillary Poiseuille flow.We consider typical model NPNCs with the nanorod aspect ratio r = 100 (andone with r = 3 for comparison) and the conductivity contrast between the matrixsolvent and nanophase as σ1/σ2 = 10−5. This ratio is, in fact, quite modest;some composites have contrasts of ten or greater orders of magnitude. Figure 5


ψ-ψ0s-s0

y y

β

vx

(a) (b)

(c) (d)

(e) (f)

-1 -0.5 0 0.5 1

0

5

10

-2

-1

0

1

2x 10-3

-5

-10-1 -0.5 0 0.5 1

-4

-2

0

2

4

x 10-4

0

1

2

3

x 10-3 x 10-2

ψ-ψ0

y

s-s0

-1 -0.5 0 0.5 1

-4

-2

0

2

4

-4

-3

-2

-1

0x 10-5x 10-3

-1 -0.5 0 0.5 10

3

6

9

x 10-6

0

0.5

1

1.5x 10-1

βvx

y

-1 -0.5 0 0.5 1-2

-1

0

1

2

-3

-2

-1

0

1

x 10-5x 10-2

ψ-ψ0

y

s-s0

-1 -0.5 0 0.5 1

-5

0

5

10x 10-6

0

1

2

x 10-2

β

vx

y

Figure 2. Comparison of the numerical solutions (circles or trian-gles) of the steady boundary value problem and the exact asymp-totic formulas (solid or dashed lines ). Top row: tilted anchoring(ψ0 = π

6 ) with ǫ = 0.05 and Er = 1. Middle row: parallel an-choring (ψ0 = 0) with ǫ = 0.1 and Er = 1. Bottom row: normalanchoring (ψ0 = π

2 ) with ǫ = 0.1 and Er = 1. In all figures, circlesand solid lines use the left (vertcal) axis; diamonds and dashedlines use the right axis.

shows the maximum scalar conductivity enhancement (εmax) across the gap forthree different anchoring conditions. The top row corresponds to the aspect ratior = 100 whereas for the bottom row r = 3. It is apparent that one can achievedifferent effective conductivity enhancements across the gap by tuning anchoringconditions. The nematogen aspect ratio effect is also significant from this figure.The larger the aspect ratio, the more enhancement in effective conductivity.

It is also interesting to note that the shape of the solution for tangential anchoringis sensitive to the value of aspect ratio. In Figure 6 we plot the maximum effectiveconductivity enhancement and anisotropy versus Poiseuille flow rate (Figure 6a and


ψ-ψ0

s-s0

y y

β

vx

(a) (b)

(c) (d)

(e) (f)

-1 -0.5 0 0.5 1-10

-5

0

5

10

-2

-1

0

1

2x 10-2x 10-2

-1 -0.5 0 0.5 1

-5

0

5

10x 10-3

0

0.1

0.2

0.3

-1 -0.5 0 0.5 1

-1.5

-0.5

0.5

1.5

-3

-2

-1

0x 10-3x 10-1

ψ-ψ0

s-s0

y -1 -0.5 0 0.5 10

3

6

9

x 10-4

0

1

2

3

x 10-1

y

β

vx

-1 -0.5 0 0.5 1

-4

-2

0

2

4

-8

-5

-2

1

4

x 10-3x 10-1

ψ-ψ0

s-s0

y -1 -0.5 0 0.5 1

-0.5

0.5

1.5

x 10-3

0

2

4

6x 10-2

2

y

β

vx

Figure 3. Comparison of the numerical solutions (circles or trian-gles) of the steady boundary value problem and the exact asymp-totic formulas (solid or dashed lines). Top row: tilted anchoring(ψ0 = π

6 ) with ǫ = 0.5 and Er = 1. Middle row: parallel anchor-ing (ψ0 = 0) with ǫ = 0.3 and Er = 10. Bottom row: normalanchoring (ψ0 = π

2 ) with ǫ = 0.3 and Er = 10.

c) and volume fraction (Figure 6b and d) for three different anchoring conditions.The parameters used here are r = 100 and Er = 1. In Figure 6a and c, the volumefraction is fixed to be θ2 = 0.0236 and we vary the Poiseuille flow rate. In thiscase, tilted anchoring gives the largest conductivity enhancement and anisotropy.In Figure 6b and d, the flow rate ǫ is fixed to be 0.1 and we vary the volumefraction θ2. It turns out that the conductivity enhancement and anisotropy areboth increasing functions of θ2 and insensitive to anchoring conditions.

6. Conclusions

We have studied the structure scaling properties of nematic liquid crystal poly-mers under pressure-driven Poiseuille flow both analytically and numerically. A


(a) (b)

(c) (d)

y

x

ψ-ψ0

s-s0

y y

β

vx

y

τ11−τ22

τ22−τ33

τ12

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

-6

-3

0

3

6x 10-2

-1 -0.5 0 0.5 1-0.5

0.5

1.5

2.5

0

1

2

3x 10-1x 10-2

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

Figure 4. Top row: Comparison of the numerical solutions (cir-cles or diamonds) of the steady boundary value problem and theexact asymptotic formulas (solid or dashed lines) for normal an-choring (ψ0 = π/2) with ǫ = 0.5 and Er = 20. Bottom row:Rheological structure profiles associated with the numerical re-sults of the top row. Here we provide the first (N1 = τ11 − τ22)and second (N2 = τ22 − τ33) normal stress differences, the shearstress (τ12), and the light scattering intensity function ([28]),I = I0 sin2(2πa1s), across the plate gap, where a1 is the ratioof the sample thickness to the wavelength of incident light. Thenormalized birefringence s measures the difference between orien-tation along the flow direction and along the vorticity axis.

rich phenomenology is apparent because of the coupling between the rod ensemble(through excluded volume and distortional elasticity) and the flow field. When thenormalized pressure gradient ǫ and the scaled Ericksen number ǫ·Er are both small,the mesoscopic Doi-Marrucci-Greco model predicts: an Er−1/2 scaling associatedwith boundary layer modes residing only in the order parameters; and, an Er−1


46.7 46.75 46.8 46.85−1

−0.5

0

0.5

1

y

εmax

46.7 46.75 46.8−1

−0.5

0

0.5

1

y

εmax

45 46 47 48−1

−0.5

0

0.5

1

y

εmax

6.525 6.526 6.527 6.528 6.529−1

−0.5

0

0.5

1

y

εmax

6.52 6.525 6.53 6.535−1

−0.5

0

0.5

1

y

εmax

6.4 6.45 6.5 6.55 6.6−1

−0.5

0

0.5

1

y

εmax

Figure 5. Effective conductivity enhancements for 3 different an-choring conditions. Top row: r = 100; Bottom row: r = 3. Firstcolumn: parallel anchoring; Second columm: normal anchoring;Third column: titled anchoring (ψ0 = π

6 ). Here ǫ = 0.5, Er = 1.

mean scaling law across the rectangular pipe residing in the principal axes of ori-entation. The self-consistent, leading order, flow velocity is a parabolic Poiseuilleflow, attenuated by an explicit prefactor which accounts for the stored energy inthe rod dispersion. Anchoring conditions play an important role in the amplituderesponse of order parameters, which describe the relative focusing or defocusing ofthe rod ensemble distribution function.

This paper presents a combined approach to understanding flow-induced behav-ior of rod-like macromolecular dispersions, namely blending modeling, asymptoticanalysis, homogenization theory, numerical analysis, and scientific computation.This is, however, just the beginning in an overall strategy to design nano-compositematerials from the bottom up. The ideal strategy consists of a control wrapper code,where one is specifying performance features such as current distributions across


0 0.5 1 1.5 246.5

47

47.5

48

48.5

ε

ε ma

x(a)

tangential

normal

tilted

0 0.01 0.02 0.03 0.040

20

40

60

80

100

θ2

ε ma

x

(b)

tangential

normal

tilted

0 0.5 1 1.5 243

43.5

44

44.5

45

45.5

46

ε

ε 1−

ε 2

(c)

normal

tangential

tilted

0 0.01 0.02 0.03 0.040

20

40

60

80

θ2

ε 1−

ε 2(d)

tangential

normal

tilted

Figure 6. Effective conductivity enhancement and anisotropyversus Poiseuille flow rate ǫ (a and c) or volume fraction θ2 (band d) for 3 different anchoring conditions.

a material during operating conditions. Using composition information, process-ing controls, and effective property characterizations, the challenge is to be ableto modify composition and processing controls to achieve a specified performancetarget. It is this challenge that we have only begun to put into a framework, andwhere the guidance of modern applied mathematics can play a critical role. Thispaper is dedicated to Max Gunzburger, whose work and consultation over the yearshas inspired our group to envision such a control strategy in the first place, andwhose contributions will be a key to our anticipated success.

Acknowledgments

Effort sponsored by the Air Force Office of Scientific Research, Air Force Materi-als Command, USAF, under grant number FA9550-06-1-0063, the National ScienceFoundation through grant DMS-0604891, 0604912, 0605029, 0626180, the Army


Research Office, under grant number W911 NF-04-D-0004, and the NASA Uni-versity Research, Engineering and Technology Institute on Bio inspired Materials(BIMat) under award No. NCC-1-02037, are gratefully acknowledged. HZ’s work issupported in part by the Naval Postgraduate School Research Initiation Program.

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Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943-5216,USA

E-mail : [email protected]

Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC

27599-3250, USAE-mail : [email protected]

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Volume 4, Number 3-4, Pages 460–477 · Volume 4, Number 3-4, ... We use the mesoscopic Doi-Marrucci-Greco model, which incor- ... 2) the rotary diﬀusivity is explicitly