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J. Non-Newtonian Fluid Mech. 165 (2010) 687–697

Contents lists available at ScienceDirect

Journal of Non-Newtonian Fluid Mechanics

journa l homepage: www.e lsev ier .com/ locate / jnnfm

ynamic texture scaling of sheared nematic polymers in the large Ericksenumber limit

.Gregory Foresta, Sebastian Heidenreichb, Siegfried Hessc, Xiaofeng Yangd, Ruhai Zhoue,∗

Departments of Mathematics and Biomedical Engineering, Institute for Advanced Materials, University of North Carolina at Chapel Hill, Chapel Hill,C 27599-3250, United StatesRudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United KingdomInstitute for Theoretical Physics, Technische Universität Berlin, Hardenbergstrasse 36, D-10623, GermanyDepartment of Mathematics, University of South Carolina, Columbia, SC 29083, United StatesDepartment of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, United States

r t i c l e i n f o

rticle history:eceived 16 November 2009eceived in revised form 11 March 2010ccepted 16 March 2010

ACS:1.30.v1.30Vx7.40.Fd7.55.Fa

eywords:ematic polymersod dispersionsigh Ericksen numberength scaleeterogeneous propertiesefect core

a b s t r a c t

In steady shear experiments, nematic polymers and rigid Brownian rod dispersions develop unsteadyfine structure (Larson and Mead, 1992 [33]; Larson and Mead, 1993 [34]; Tan and Berry, 2003 [42]).These features are representative of the dynamic morphologies generated during film and mold pro-cesses, which translate to the two banes of materials engineering: heterogeneous properties on unknownlength scales, and dynamic fluctuations in processing outcomes. Our goal here is to quantify the lengthscales of these dynamic textures in the realistic parameter regime of extremely high Ericksen number,Er, the ratio of viscous to elastic stress. To avoid model-specific or numerical anomalies, we simulatethree distinct models of flowing nematic polymers employing three different algorithms run on differ-ent computer systems, following our earlier studies at moderate Ericksen numbers (Forest et al., 2008[11,12]). Our strategy recognizes the rich history on this problem as well as practical prohibitive con-straints. First, the relevant experimental and processing conditions dictate Er ≈ O(105) and order unityDeborah number (De, the bulk shear rate normalized by the rotational relaxation rate of the nematicphase). Second, theoretical estimates of the smallest length scales of morphology and their sources existonly for steady-state structures, originally from liquid crystal models (cf. Carlsson, 1984 [2]; Carlsson,1976 [3]; Cladis and Torza, 1976 [4]; de Gennes, 1974 [21]; Manneville, 1981 [35]) and subsequentlyfrom nematic polymer orientation tensor and full kinetic models (cf. Cui, 2006 [5]; Forest and Wang,2003 [13]; Forest et al., 2004 [15]; Forest et al., 2007 [19]; Marrucci, 1985 [36]; Marrucci, 1990 [37]; Mar-rucci, 1991 [38]; Marrucci and Greco, 1993 [39]; Zhou and Forest, 2006 [50]; Zhou et al., 2007 [52]). Third,numerical experiments for Er ≈ O(102,3) and De ≈ O(1) confirm experimental observations of unsteadylong-time attractors (Denn and Rey, 2002 [6]; Feng et al., 2001 [10]; Forest et al., 2008 [11,12]; Forestet al., 2005 [17]; Han and Rey, 1995 [23]; Klein et al., 2005 [30]; Kupferman et al., 2000 [32]; Sgalariet al., 2002 [40]; Tsuji and Rey, 1997 [43]; Yang et al., 2008 [46]; Yang et al., in press [47]). Fourth, theunsteady regime persists to Er ≈ O(105) (Sgalari et al., 2002 [40]), where single runs, much less two orthree decades of Er, are prohibitively expensive in two or three space dimensions due to severe spatialand temporal resolution constraints. Here we propose and implement a new strategy to finesse theselimitations. The key observation arises from comprehensive 2D simulations (Klein et al., 2005 [30]; Yanget al., 2008 [46]; Yang et al., in press [47]) which implicate transient defect cores (local isotropic or oblatedisordered phases) as the source of the finest texture, or equivalently, the strongest gradients. In lightof these results, we simulate 1D flow-orientational heterogeneity spanning the plate gap, which pre-

serves the fine-scale transient defect cores and associated shear flow and stress features, while screeninghigher-dimensional, nonlocal defect topology and cellular flow patterns. We simulate spatio-temporalattractors for Er out to 105 for all three models. Each model reveals a transient staircase of tumbling-wagging layers spanning the shear gap, where the smallest length scales of morphology are associatedwith intermittent oblate defects between adjacent tumbling and wagging layers, which form preciselywhile neighboring layers become out of phase. With increasing Er, the number of tumbling, waggingand oblate defect layers grows, meanwhile layers propagate, collide, merge and reform ad infinitum.

∗ Corresponding author.E-mail address: rzhou@odu.edu (R. Zhou).

377-0257/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2010.03.003

688 M.Gregory Forest et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 687–697

To quantify these dynamic textures, we calculate a time-averaged morphology length scale distributionfunction, defined by the reciprocal of the gradient of the oblate defect metric. The peak of this PDF char-acterizes a dominant morphology length scale which, for all three models, remarkably adheres to theMarrucci scaling law Er−(1/2), consistent with earlier numerical studies (Sgalari et al., 2002 [40]) on yetanother nematic flow-tensor model. This result yields an a priori estimate for the finest length scalesin shear-dominated nematic polymer films and molds based on processing conditions and Frank elas-ticity constants, which confirms a prediction of Marrucci, 25 years ago (Marrucci, 1985 [36]), based on

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dimensional analysis in th

. Introduction

One of the challenging problems in film and mold process-ng of nematic polymers and nano-rod dispersions is to predict,

priori, the dominant length scales of shear-induced morphol-gy [9,13,20,21,35]. We refer to the review by Tan and Berry [42]nd the subsequent review by Rey and Denn [6]. These textureeatures determine the length scales of variability in material per-ormance, for optical and electromagnetic transmission propertiess well as conductive and mechanical behavior. By contrast, fiberextension dominated) processing achieves highly uniform orien-ational alignment along the fiber axis. Another challenge is tonderstand the sources of small scale heterogeneity arising fromhear-dominated hydrodynamics and physical confinement asso-iated with film and mold processing; such information revealshether or not the heterogeneity is inevitable in shear process-

ng. There is evidence that disclination line and loop defects areesponsible for the fine scales of structure [10,23,28,31-34,37-9]. If topological defects are indeed associated with the smallesttructure length scales, then the finest texture should reside inheir cores, where local disordered phases form to obviate anctual singularity in the principal axis of nematic orientation. Two-imensional simulations of the Leal group [30] and the authors19,46–49] confirm that local defect cores are always present inalf-integer degree topological defects, and that the largest gra-ients of the orientational distribution arise in the cores. Thesetudies for Er ≈ 102, 103 are prohibitively expensive and generateuite large data sets in the realistic regime of Ericksen numbers inwo space dimensions due precisely to the proliferation of smallcales, requiring much higher spatial resolution, and the unsteadyttractors which require extremely small time steps at such highr. Data analysis for the purpose of extracting scaling behavior ofhe morphology therefore seems impractical.

The above evidence, however, suggests a numerical strategyhich we implement in this paper to estimate the dominant length

cales of texture in the dynamic regime of very high Ericksenumbers. Namely, we restrict to one-dimensional heterogeneity

n the flow-gradient direction spanning the gap in a parallel-late shear cell. This dimensional reduction suppresses topologicalefects, but allows defect cores in the form of locally disorderedblate or isotropic phases. These phases arise naturally in shearedematic polymers as the minimum energy orientational configu-ation that can smoothly accommodate nearby strongly orderedematic phases whose principal axes are out of phase. Instead ofreating a discontinuity in the director (principal axis of orienta-ion), the oblate phase corresponds to a multiplicity two eigenvalueegeneracy of the second moment of the orientational distribution,

n which the director is no longer identifiable and spreads to a circle.he isotropic phase is a fully degenerate phase, where the directories anywhere on the sphere, and the eigenvalue degeneracy is mul-iplicity 3. Whereas oblate and isotropic phases are always present

n the cores of two and three dimensional topological defects, its important to recognize that disordered phases exist in lowerimensional (zero and one) structures where topological defectsre not possible. Both of these disordered phases play a prominent

amic tumbling regime.© 2010 Elsevier B.V. All rights reserved.

role in the Onsager hysteresis diagram for the isotropic-nematicphase transition, and in 1D shear flow studies of the authors atmoderate Ericksen numbers [5,11,12,18,50,51].

Information on the length and time scales and sources offine scale structure is critical for all applications of nano-rodand nano-platelet materials. If heterogeneity is unavoidable inshear-dominated flows, which is supported by experimental andnumerical evidence, then quantitative estimates of the structureare needed. We present results from numerical simulations of threedifferent flow-orientation models, together with post-processingdiagnostics, which reveal both morphology scaling behavior andthe sources of the strongest gradients, or equivalently, the smallestlength scales. Our results are for one-dimensional heterogeneousstructures arising from parallel plate driving conditions in the real-istic regime of moderate Deborah numbers and very high Ericksennumbers, where the responses are not only unsteady but stronglyintermittent.

The theoretical formulation of the morphology length scaleproblem begins with a choice of flow-nematic model and bound-ary conditions [14,16,22,37–39]. Key non-dimensional parametersare identified, notably the Deborah number (ratio of bulk flow rateto microstructure relaxation rate) and Ericksen number (the ratioof viscous to elastic stresses). From a given model formulation,one can attempt to ascertain how the length scales of morphologyscale with the fundamental dimensionless parameters. Rarely canone infer this information from the equations and boundary condi-tions, so the primary approach has been to study special solutionsto these equations. The main results of this nature have focused on1D heterogeneous steady state solutions in parallel plate shear cells,with steady plate driving conditions and strong anchoring on themicrostructure (nematic director, or rod orientational distribution).We refer to [15,17] where the literature is reviewed, and to thereviews of Tan and Berry and Rey and Denn cited above. The upshotregarding steady-state structure scaling behavior is that all evidencepoints to a modified Marrucci scaling of the smallest length scale,Er−˛, where ˛ is between 1/4 and 1. Since the Ericksen numberis typically quite large, nailing down the exponent and confirminga power law behavior would be valuable. In [15], explicit steadysolutions are derived in the dual limit of low Deborah number(equivalent to slow plates) and low Ericksen number (equivalentto very strong distortional elasticity which arrests tumbling). Theseasymptotic solutions reveal two distinct scaling properties of ori-entational morphology: Er−1 average scaling of director dominatedstructure that spans the shear gap, and an Er−1/2 scaling of orderparameter dominated boundary layers. In [17], these scaling prop-erties are upheld with full kinetic-flow simulations for Er ≈ O(103).Sgalari et al. [40] used a different nematic tensor model and carried1D flow-nematic simulations out to Er ≈ 106. They used Fouriertransforms of flow and morphology snapshots to identify dominantlength scales, supporting the Marrucci Er−1/2 scaling. To our knowl-edge, these results have not been improved upon nor extended.

The limitations of the available analytical results with respectto materials applications are fairly severe. First and foremost, theybreak down when either the Deborah or Ericksen number exceedsorder unity, whereas realistic processing or rheological conditions

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M.Gregory Forest et al. / J. Non-New

ave order unity Deborah number and very large Ericksen num-er, on the order of 105. Second, the dual limit of small De andr yields steady solutions, whereas typical experimental condi-ions yield dynamic or unsteady flow-orientation responses. Allvidence, both experimental and numerical, points to transientnset and development of small scale structure, followed by fluc-uations with coarsening periods and then reappearance of finetructure. We will not emphasize the point here, but the non-tationary structures for all three models at high Ericksen numberxhibit chaotic dynamics. Thus, our metrics for length scale selec-ion are necessarily time averages. There are no analytical resultsn such transient space–time attractors, and experimental probesave not even been attempted to our knowledge. The only presentecourse is to approach the problem with numerical experiments.ven for numerical studies, the absence of an a priori bound on themallest length scales places significant limitations. To be sure toapture the features of interest, one has to set the spatial resolu-ion extremely low. As well, one has to set the time step extremelymall to avoid both numerical instability and error, and to avoidissing rapid dynamic transitions with strong gradients in the ori-

ntational distribution—precisely the features of interest. All ofhese challenges motivate the study reported in this paper. Weevelop and implement a strategy to determine the scaling behav-

or of the length scales of one-dimensional morphology in a parallellate shear cell experiment on nano-rod dispersions, in the rangef realistic Ericksen numbers. The strategy overcomes the inherentifficulties noted above by identifying the sources of the dominant

ength scales, defining and computing a length scale distributionunction of the morphology, and averaging over the chaotic fluctu-tions in the morphology.

We present results of numerical experiments with three dif-erent models for the hydrodynamics of nematic polymers andigid nano-rod dispersions, simulating a parallel plate shear cellith steady translating plates. The same three models were bench-arked for consistency in previous studies of order unity Deborah

umber and Ericksen numbers of order 102–103, where the attrac-ors are unsteady but the length scales of morphology are relativelyoarse. Here we extend those simulations to two more decadesf Ericksen number and focus on scaling behavior of the tran-ient morphology. We show consistent qualitative behavior fromll three flow-orientation models, which is important in thate gain insight into the source of small length scale structuretransient and non-stationary oblate defect phases, removing the

oncern that the results are model-specific. We surmise that theseame fluctuating oblate defect structures are likewise responsi-le for the Marrucci scaling behavior reported by Sgalari et al.40].

. The model equations for a spatially inhomogeneousnisotropic fluid

In this section, we recall the 3 models from [11,12] and defineonsistent dimensionless parameters for each.

.1. The Doi–Hess–Smoluchowski (DHS) kinetic model,arrucci–Greco gradient elasticity, and flow coupling

We consider plane shear flow between two plates located at= ±h, in Cartesian coordinates x = (x, y, z), and moving with cor-

esponding velocity v = (±v0, 0, 0), respectively.

There are two apparent length scales in this problem: the gap

idth 2h, an external length scale, and the finite range l of molecularnteraction, an internal length scale, set by the distortional elastic-ty in the Marrucci–Greco potential given below. When the plates

ove relative to each other at a constant speed, it sets a bulk flow

n Fluid Mech. 165 (2010) 687–697 689

time scale (t0 = h/v0); the nematic average rotary diffusivity (Dr)sets another (internal) time scale (tn = 1/Dr), and the ratio tn/t0defines the Deborah number De:

De = tn

t0= v0

h Dr. (1)

There are other time scales associated with solvent viscosity and thethree nematic viscosities, and with elastic distortion, which due tothe flow-nematic-plate interaction are not a priori known.

We nondimensionalize the DHS–Navier–Stokes system usingthe length scale h, the time scale tn and the characteristic stress�0 = �h2/t2

n . The dimensionless flow and stress variables become:

v = tn

hv, x = 1

hx, t = t

tn, � = �

�0, p = p

�0. (2)

The following 7 dimensionless parameters arise:

Re = �v0h

�, ˛ = 3ckT

�0, Er = 8h2

˛Nl2, �i = 3ckT�iDe

�v,

i = 1, 2, 3, � = L2

L2, (3)

where N is the normalized volume fraction of the rod phase, whichdictates the strength of the Maier–Saupe intermolecular potentialdefined below; Re is the solvent Reynolds number; ˛ measures thestrength of entropic relative to kinetic energy; �v = �h2/t2

0 is theviscous stress; Er is the Ericksen number which measures the ratioof viscous stresses to stresses arising from Frank distortional elas-ticity; �i, i = 1, 2, 3 are three nematic Reynolds numbers; and � isa distortional elasticity parameter that ranges between 0 and 1 tomodel equal (� = 0) or distinct (� /= 0) Frank elasticity constants[45]. For other parameters, we refer to [45].

We drop the tilde ˜ on all variables; all figures correspond tonormalized variables and length, time scales. We will first presentthe kinetic theory model for the full orientational distribution func-tion, followed by two separate second-moment orientation tensormodels.

We neglect the effect of translational diffusion and employan approximate rotary diffusivity [8]. The dimensionlessDoi–Hess–Smoluchowski equation for the probability distributionfunction (PDF) f (m, x, t) becomes

Df

Dt= R · [(Rf + fRV)] − R · [m × mf ],

m = ˝ · m + a[D · m − D : mmm],(4)

where D/Dt = ∂/∂t + v · ∇ , R = m × ∂/∂m is the rotational gradi-ent operator, D and ˝ are the dimensionless rate-of-strain andvorticity tensors, respectively, and a = (r2 − 1)/(r2 + 1), where ris the rod aspect ratio. The coupled Maier–Saupe (first term) andMarrucci–Greco potential (distortional elasticity) is

V = −3N

2M : mm − 1

2˛Er[M + � ∇∇ · M] : mm, (5)

where N, the Ericksen number Er and anisotropic elasticity parame-ter � are defined earlier. The rank-2 tensor M is the second momentof f,

M = M(f ) =∫

||m||=1

mmf (m, x, t) dm. (6)

Next we recall the oblate and isotropic defect diagnostics [53].By virtue of being the second moment of a PDF, M is symmet-ric, trace 1, and positive semi-definite, with ordered eigenvalues

0 ≤ d3 ≤ d2 ≤ d1 ≤ 1. When the leading eigenvalue d1 is simple,then the distribution is non-degenerate with a unique principalaxis of orientation n1. Degenerate, or locally disordered phases, arisewhen d1 is degree 2 (the oblate defect phase) or degree 3 (the fullydisordered, isotropic phase). Thus M generically defines a triaxial

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2.3. Irreversible thermodynamical (IT) alignment tensor model

We next introduce a model for the alignment tensor, denoted by

90 M.Gregory Forest et al. / J. Non-New

llipsoid (a uniquely defined orthonormal frame of principal axeshose lengths are given by the principal values). So long as d1 is

imple, the major director (principal axis) is uniquely identified,nd the ellipsoid is “prolate”. When d1 is not simple, the ellipsoidefined by M which degenerates to an oblate spheroid for the oblateefect phase and to a sphere for the isotropic phase. It follows thathe local defect diagnostics are the differences of the eigenvaluesf M, which we label s = d1 − d2 and ˇ = d1 − d3. The oblate phase

s characterized by s = 0 and ˇ /= 0, leading to our reference to “s”s the oblate defect metric; whereas the isotropic phase arises when= 0, which automatically implies s = 0 because the di are ordered.

hus ˇ is the isotropic defect metric. The stable nematic equilib-ium phases are always uniaxial and non-degenerate, with s > 0nd d2 = d3. In all simulations reported below, we will show theblate defect metric s but skip the isotropic metric ˇ because weever find isotropic phases in any of these simulations.

The dimensionless forms of the balance of linear momentum,he continuity equation, and the stress constitutive equation are

dvdt

= ∇ · (−pI + �), ∇ · v = 0,

� =(

2De

Re+ �3

)D

+�1(D · M + M · D) + �2D : M4

+a˛(M − 13

I − N M · M + N M : M4)

−a1

6Er(M · M + M · M − 2 M : M4)

− 112Er

[2(M · M − M · M) + Mc]

−a�

12Er[M · Md + Md · M − 4(∇∇ · M) : M4]

− �

12Er[Md · M − M · Md − Me],

(7)

here

c = (∇M : ∇M − (∇∇M) : M), (8)

d = ∇∇ · M + (∇∇ · M)T , (9)

e = (∇∇ · M) · M − Mˇj,˛Mij,i, (10)

4 =∫

||m||=1

mmmm f (m, x, t) dm. (11)

n the simulations presented below, we fix the following parameteralues: N = 6, � = 0.5, ˛ = 2, �1 = 0.0004, �2 = 0.15, �3 = 0.01.e also impose tangential anchoring conditions on the equilibrium

ematic phase at the plates (cf. [53]). This value of N determines thequilibrium value of s at each plate for each model, and a color codeor level sets of s is then used for the range 0 ≤ s ≤ smax < 1. In this

anner, oblate defects are easily detected with red used for theblate defect value s = 0.

We consider one-dimensional physical space (the gap betweenhe two parallel plates). The boundary conditions on the velocity= (vx, 0, 0) are given by the Deborah number

x(y = ±1, t) = ±De. (12)

e assume homogeneous tangential anchoring at the plates, giveny the quiescent nematic equilibrium,

(m, y = ±1, t) = fe(m), (13)

here fe(m) is an equilibrium solution of the Smoluchowski equa-ion corresponding to tangential anchoring at the plates when= 0. For the tensor models, the second moment equilibrium is

gain selected with tangential anchoring.

n Fluid Mech. 165 (2010) 687–697

2.2. Doi–Marrucci–Greco (DMG) second-moment orientationmodel

In Landau–de Gennes models, the probability distribution func-tion f (m, x, t) of Doi–Hess kinetic theory is projected onto thesecond moment tensor, M. The orientation tensor Q is the tracezero form of M, Q = M − I/3. The distinctions among models canbe framed in terms of closure rules necessary to reduce the abovecoupled flow-kinetic orientation model and stress formula to a sys-tem that closes on the second-moment orientation tensor Q and theflow variables v and p. The dimensionless forms of the stress con-stitutive equation and orientation tensor equation that are oftencalled the Doi–Marrucci–Greco (DMG) model are given as follows[52]:

� =(

2De

Re+ �3

)D + a˛F(Q)

+ a

3Er

(Q : Q

(Q + I

3

)− 1

2(QQ + QQ) − 1

3Q

)

+ 13Er

(12

(QQ − QQ) − 14

(∇Q : ∇Q − ∇∇Q : Q))

+�1

((Q + I

3

)D + D

(Q + I

3

))+ �2D : Q

(Q + I

3

), (14)

where Er is the Ericksen number, and �i are the three nematicReynolds numbers (normalized viscosities) all defined earlier, andthe short-range excluded volume effects are captured by

F(Q) =(

1 − N

3

)Q − NQ2 + NQ : Q

(Q + I

3

), (15)

where N once again is a dimensionless concentration of nematicpolymers, which controls the strength of the mesoscopic approx-imation, F(Q), of the gradient of the Maier–Saupe potential. Theclosure approximation effectively modifies N, but we abuse nota-tion and keep the same symbol.

The orientation tensor equation for the DMG closure rule is:

DQDt

= ˝Q − Q˝ + a(DQ + QD) + 2a

3D − 2aD : Q

(Q + I

3

)

−(

F(Q) + 13˛Er

(Q : Q

(Q + I

3

)

− 12

(QQ + QQ) − 13

Q))

. (16)

The boundary condition for the scaled velocity is the same as inthe kinetic simulations (12). We assume homogeneous tangentialanchoring at the plates, given by the quiescent nematic equilibrium,

Qeq = s(

exex − 13

I)

, (17)

where ex = (1, 0, 0) and s = 0.86 since we have chosen N = 6.

Q in a slight abuse of notation clarified just below, that was derivedin the framework of irreversible thermodynamics. We present themodel equations in the notation of the DMG model for a more trans-parent comparison. This notation therefore differs from the originalliterature cited below. We refer to [24] for a detailed discussion ofall parameters and terms in the equations to follow.

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M.Gregory Forest et al. / J. Non-New

The equation for the alignment tensor Q [25,1] in theoi–Marrucci–Greco model notation reads

D

DtQ = ˝Q − Q˝ +

(DQ + QD − 2

3(D : Q)I

)

+Da

(F(Q) − �

Er2Q

)− 1

�F(Q) + 1

ErQ −

√32

�KD.(18)

he parameters � , Er = ˛ De N Er/48 and �K are related to the DMGodel parameters (see [24]). The function F denotes the deriva-

ive of the spatially homogeneous part of the amended Landau–deennes potential (analogous to the Maier–Saupe potential)

∂

∂Q AM(Q) = F(Q) = ϑQ − 3

√6(QQ − 1

3(Q : Q)I)

+ 2(Q : Q)Q

1 − ((Q : Q)2)/a4max

,

hich is a modification of the Landau–de Gennes potential

LDG = 12

ϑQ : Q −√

6(QQ) : Q + 12

(Q : Q)2. (19)

n previous studies it was demonstrated that the Landau–de Gennesotential does not restrict the order parameter to physically admis-ible values. From [24], the value 2.5 is implemented for thecutting parameter” amax.

The parameter ϑ in (19) has been used with ϑ(N) = A0(1 −/N∗)/(1 − NK /N∗) [27,31]. The pseudocritical concentration N∗

nd the concentration at phase coexistence NK are also modelarameters. The value of A0 depends on the proportionality coef-cient chosen between Q and M − (1/3)I. The choice made above

mplies A0 = 1, cf. [25]. The coefficients, on the one hand, are linkedith measurable quantities and, on the other hand, can be related toolecular quantities within the framework of a mesoscopic theory

26,27,31,7].Note the alignment tensor Q in (18) differs from the DMG

ensorial order parameter by the constant factor aK

√15/2, i.e.

IT = aK

√15/2 QDMG . The value aK = 2B/(3C) is related to the

oefficients of the Landau–de Gennes potential and the amendedotential, respectively and is material specific. For MBBA (see [41])= 64 × 104 J/m3 and C = 35 × 104 J/m3 yields aK ≈ 1.22.

In (18) the spatial derivative∇Q of the alignment tensor is whereontact is made with Frank elasticity. For simplicity, the ‘isotropic’ase is considered which implies that, in the nematic phase, allhree Frank elasticity coefficients are equal. This is consistent withhe choice made in the DMG tensor model, whereas the kinetic

odel allows for two distinct elasticity constants; the phenomenae report are insensitive to one versus two elasticity constants.

The orientation-dependent stress constitutive equation is giveny

= �isoD −√

32

�K

ˇF(Q) +

√32

�K

ˇErQ

+

ˇ

(QF(Q) + F(Q)Q − 2

3(F(Q) : Q)I

)

− 1

Er

ˇ

(Q(Q) + (Q)Q − 2

3((Q) : Q)I

). (20)

he parameter ˇ is inversely proportional to ˛:

= 3(v0)2

L2DraK AK ˛, (21)

here Dr is the rotational diffusion constant, as in the other twoodels.

n Fluid Mech. 165 (2010) 687–697 691

Again, we assume homogeneous tangential anchoring at theplates, given by the quiescent nematic equilibrium,

Q(−h, t) = Q(h, t) =√

32

qeq

(exex − 1

3I)

, (22)

where qeq = √5aK s. For the velocity, no-slip boundary conditions

are used as in the other models,

u(−h, t) = −v0, u(h, t) = v0. (23)

3. Numerical results on the large Ericksen number limit

We present flow-nematic perspectives on the dynamic mor-phology evolution of the Ericksen number cascade from previousstudies at Er ≈ 102–103 to Er ≈ O(105). Since the models have dif-ferent approximate potentials for the same physics, there is noprecise link in respective numerical values of parameters, and sen-sible comparisons are thereby made only on scaling behavior withrespect to Er, the primary focus of this paper. We present resultsfirst from the full kinetic model coupled to Navier–Stokes, andthen confirm the results qualitatively and quantitatively with thetwo different second-moment tensor models coupled to the 1DNavier–Stokes equations.

• Previous results on the source of minimum length scale structure andintermittency at moderate Er

For fixed De, there is a critical Er below which the attractorsare steady. As Er increases, the distortional elasticity potential ofthe nano-rod dispersion is too weak to store all elastic distortions,leading to director oscillations (wagging) or continuous rotations(tumbling). One finds unsteady flow-orientation 1D attractors withlimit cycle behavior at fixed locations in the interior of the sheargap, a phenomenon that has been widely recognized since experi-ments of Kiss and Porter [29]. Numerically, the boundaries betweensteady and unsteady one-dimensional (1D) heterogeneous attrac-tors in the (Ericksen number, Deborah number) parameter spacehave been mapped out by Tsuji and Rey [43,44] and the authors [17]using a tensor nematic theory and the Doi–Hess–Marrucci–Grecokinetic theory, respectively. More detailed investigations intounsteady 1D heterogeneous attractors have been explored for mod-erate Er in [32,11,12], with the following phenomena that are ofrelevance to the present paper:

• The flow is nearly linear most of the time, during which the struc-ture of the orientational distribution is dominated by a smoothrotation of the major director from one plate to the mid-gap andan unwinding from the mid-gap to the other plate.

• Short periods of departure from this structure arise, during whichthe middle layer of the gap experiences director tumbling (con-tinuous rotation) while layers near both walls oscillate (wag).Gap locations where the director transitions from wagging totumbling exhibit two remarkable features.

• A defect forms in the orientational distribution. This struc-ture is not a topological defect, since the heterogeneity is onlyone-dimensional. Rather, an oblate defect phase arises betweenneighboring tumbling and wagging layers. This defect formsto obviate a singularity in the orientational distribution whichwould otherwise occur as the neighboring tumbling and waggingorbits have principal axes that are rapidly separating across an

extremely small distance. When the tumbling layer rotates andresets in-phase with the wagging layer, the oblate defect layerdisappears.

• As the defect disappears, the strong distortional elastic stressstored by the defect is released and transferred to the flow, cre-

692 M.Gregory Forest et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 687–697

Fd

eoitsaaaatmsnesa

ig. 1. Doi–Hess–Smoluchowski kinetic-flow model. The oblate defect metric s =1 − d2 in space and time. De = 5. From top to bottom: Er = 5000, 10,000, 25,000.

ating a hydrodynamic burst that lasts for a short while untilviscously dissipated.

Thus, the attractors for moderate Er consist of a banded or lay-red flow-nematic structure, where each layer either tumbles orscillates, and localized oblate defects form when the orientationn neighboring layers becomes out of phase. It is worthy of notehat in the moderate Er regime, the tumbling, wagging bands aretationary, so that the locations of the fluctuating oblate defectsre likewise stationary. These observations reveal a clear mech-nism and source of the smallest length scale features of the flownd orientation in the unsteady, moderate Ericksen number regime,nd they explain why these smallest length scales of structure areransient. We now establish through numerical simulations that thisechanism and source of small length scale, transient structure per-

ists in the high Ericksen number limit. In this paper, we devise aew dynamic strategy for detection and assessment of the small-st length scales based on two elements: (1) an efficient measure ofhort length scales is the reciprocal of large gradients of a structure;nd, (2) based on the results for intermediate Er, a strong candi-

Fig. 2. Irreversible thermodynamics tensor-flow model. Spatio-temporal behaviorof the order parameter s = d1 − d2 for the parameters De = 1, �iso = 0.1 and theEricksen number (from top to bottom) Er = 6000, 10,000, 20,000.

date for the mechanism and source of the minimum length scalesis the transient oblate defect layers that arise between layers withcontinuously rotating and oscillating orientational distributions.

• The high Ericksen number limit: a cascade of transient oblate defect-mediated tumbling and wagging layers

The strategy begins with a focus on oblate defect detec-tion and tracking in the Ericksen number cascade. In Fig. 1, wepresent level sets of the oblate defect metric, s = d1 − d2, versusgap height and time, for Er = 0.5 × 104, 1 × 104, 2.5 × 104 for theSmoluchowski–Navier–Stokes system. Fig. 2 presents results of theIT–Navier–Stokes system for Er = 2.4 × 104, 4 × 104 and 8 × 104,and Fig. 3 presents results of the DMG–Navier–Stokes system forEr = 0.5 × 104, 1 × 104, 2.5 × 104.

We only show half of the shear gap since we impose sym-metry about the mid-gap. The darkest blue domains correspondto strongly ordered phases where s ≈ s , whereas the dark red

eq

domains correspond to oblate defects. From this metric, we observea persistence of the qualitative transient banding or layering struc-ture that was reported previously at Er = 102. The new effects athigher Er are: the number of bands (and therefore the number of

M.Gregory Forest et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 687–697 693

Fd2

dtdtTdeaflogTtbT

•

Fig. 4. Doi–Hess–Smoluchowski kinetic-flow model. The time-averaged lengthscale distribution based on the reciprocal gradient of the oblate defect metric,(ds/dy)−1. Red line: Er = 5000; blue line: Er = 10,000; green line: Er = 25,000. (Forinterpretation of the references to color in this figure legend, the reader is referredto the web version of the article.)

starting with bounds on the largest gradient. We are interested inhow the distribution of values of this reciprocal gradient functionbehaves as we pass to the large Ericksen number limit. We bin thevalues of the length scale metric for each snapshot in time, andnormalize by the bin number to construct a distribution function.

ig. 3. Doi–Marrucci–Greco tensor-flow model. Space–time behavior of the oblateefect metric s = d1 − d2 when De = 2.5, Er = 5000 (top); Er = 10,000 (middle); Er =5,000 (bottom).

efects) that appear and disappear in an almost periodic fashion inhe shear gap grows proportional to Er; and, the width of the oblateefect transition layers is non-monotone for two decades of Er, buthen the defect width sharply decreases once Er reaches 50,000.he lack of a uniform trend in the defect widths and line density ofefects across the gap is partially explained by the fact that at mod-rate Ericksen numbers, the tumbling, wagging and defect layersre stationary, and therefore the transient oblate defect domainsorm and disappear at the same gap heights. However, as Er getsarge, the tumbling and wagging layers and interlacing defects notnly grow in number, but they propagate up and down in theap. An apparent propagation speed can be inferred from Fig. 1.his observation suggests that once there are sufficiently manyumbling–wagging transitions across the gap, the individual tum-ling or wagging orbits acquire translational degrees of freedom.

he next graphics will amplify these new features.

Morphology length scale distributions versus Ericksen number: thepeak obeys the Marrucci law Er−1/2

Fig. 5. Irreversible thermodynamics tensor-flow model. The texture length scaledistribution based on the time-averaged metric (ds/dy)−1.

Figs. 4 (Doi–Hess–Smoluchowski-flow model) and 5 (irre-versible thermodynamics-flow model) and 6 (Doi–Marrucci–Grecotensor-flow model) probe statistics of the structure in the largeEricksen number limit. We introduce a natural metric for the lengthscale distribution of the orientational morphology: the reciprocal

gradient of the oblate defect metric,(

∂s/∂y)−1

. It is standard inanalysis to estimate length scales of a function in this manner,

Fig. 6. Doi–Marrucci–Greco tensor-flow model. The texture length scale distribu-tion based on the time-averaged metric (ds/dy)−1.

694 M.Gregory Forest et al. / J. Non-Newtonian Fluid Mech. 165 (2010) 687–697

Fig. 7. Doi–Hess–Smoluchowski kinetic-flow model. “Lagrangian” time series offlow and orientation features at 4 nearby gap heights, for Er = 25,000 and De = 5.The local primary velocity (top) and shear rate (second from the top) show flowfluctuations in this thin layer. The major director (Leslie) angle in panel 3 showstumbling at all four locations for 2 time units, then at t ≈ 52, the director at twolocations reverses direction (wagging) while the other two directors tumble, andthen a resetting of all four heights in phase until t ≈ 57, when only the y = 0.376height continues to tumble while the other heights each reverse direction (wag-gsp

SsTt

nffg

L

Fig. 8. Irreversible thermodynamics tensor-flow model. The Lagrangian time seriesof just the Leslie angle of the major director at 3 nearby gap heights, for Er = 10,000and De = 1. As in Fig. 7, the directors tumble in phase for some time, then the waggingtransition passes from the blue to red to green and blue labeled heights, then eachheight regains phase coherence. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of the article.)

Fig. 9. Doi–Hess–Smoluchowski kinetic-flow model. Snapshots of the spatial struc-ture at t = 51.99 for the Er = 25,000 and De = 5 simulation. The top panel shows the

ing). Panel 4 shows the oblate defect metric s at each height over this time period,howing sharp drops toward 0 when the nearby directors become strongly out ofhase.

ince the structure is transient with frequent bursts of small scaletructure, we time average the length scale distribution function.he results for the nematic flow system are shown in Figs. 4–6 forhe attractors of Figs. 1–3, respectively.

There is a clear shift toward smaller length scales as Er grows. Weow take the peak of the length dimensionless scale distribution

unction for each model, and perform a power law fit to the peak as a

unction of the Ericksen number. The kinetic-flow model simulationives the exponential fit:

= 0.74Er−0.4338;

nonlinear shear profile; panel 2 shows the local Deborah number (normalized shearrate) variations across the gap; panel 3 shows director distortions, or winding andunwinding, from the mid-gap to the top plate; panel 4 shows oblate defect domainssituated precisely at the locations of sharp director gradients, and mildly disordereddomains near other significant director gradients.

M.Gregory Forest et al. / J. Non-Newtonia

Fig. 10. Irreversible thermodynamics tensor-flow model. Snapshots at t = 81.9 oftDwd

t

L

a

L

Tmsuw

•

vorFatdsroebt

details of each model snapshots cannot be expected to match,the fundamental flow and orientational features are remarkablyconsistent. One finds order-of-magnitude fluctuations in the localDeborah number, which is reflected in the gap flow profile bymany small shear bands. The principal axis of orientation n1, or

he flow, oblate defect metric and first normal stress difference for Er = 10,000 ande = 1. Consistent with Fig. 9, oblate defect domains are evident at the gap heightsith strong director gradients. Panel 3 shows the correlations between oblate defectomains and strong fluctuations and sign changes in N1.

he IT-flow model gives the fit

= 0.4Er−0.49;

nd the DMG-flow model gives the fit

= 1.98Er−0.54.

his constitutes rather remarkable evidence that the dominant nor-alized length scale L of the orientational morphology obeys the

o-called Marrucci scaling law, L ≈ Er−(1/2). Furthermore, we nowse defect diagnostics to show that this structure scale is associatedith transient oblate defects.

Transient oblate defects during phase incoherence of tumbling andwagging layers

Figs. 7 and 8 show the evolution (time series) of the primaryelocity vx, the local Deborah number dvx/dy, the Leslie angle �f the major director, and the order parameter (oblate defect met-ic) s = d1 − d2 at several different nearby locations. For example,ig. 7 shows these quantities at 4 locations y = 0.356, 0.364, 0.372,nd 0.376, for the particular attractor at Er = 25,000. Every instancehe oblate defect metric s collapses to 0, we clearly observe theefect occurs at the gap height of a local tumbling/wagging tran-ition. For example: around t = 51.99, the director continues to

otate at y = 0.356, 0.364 (local tumbling), but it reverses directionf rotation at y = 0.372, 0.376 (local wagging). The order param-ter collapses toward 0 at y = 0.364 at that instant, which is theoundary layer between the tumbling and wagging layers. Anotherumbling/wagging transition happens around t = 57.33, indicating

n Fluid Mech. 165 (2010) 687–697 695

an almost periodic oblate defect and recovery cycle of around 5.3dimensionless time units in the kinetic model. This approximateperiod between transient oblate defects remains almost constantfor the range of Ericksen numbers we have explored. Comparisonof the two snapshots also shows the location of the oblate defecthas moved, so that the tumbling-wagging transition translates orpropagates.

• Snapshots of flow and orientation morphology at Er = 25,000 duringan active oblate defect phase

Figs. 9 and 10 show snapshots of the spatial profiles of the pri-mary velocity vx, local Deborah number dvx/dy, primary alignmentor Leslie angle �, and the oblate defect metric s = d1 − d2. While

Fig. 11. Doi–Hess–Smoluchowski kinetic-flow model. Spatio-temporal behavior ofthe first (N1, Panel 1) and second (N2, Panel 2) normal stress differences and theshear stress (Panel 3) for the simulation with Er = 25,000 and De = 5.

696 M.Gregory Forest et al. / J. Non-Newtonia

FoD

mttbltdiadoptaitb

•

e�fisait

[

[

[

[

ig. 12. Irreversible thermodynamics tensor-flow model. Spatio-temporal behaviorf first (upper) and second (lower) normal stress differences for the simulation withe = 1, �iso = 0.1 and Er = 20000.

ajor director, has in-plane Leslie angle (�) which reveals direc-or distortions across the plate gap. Viewing from the mid-gap inhe kinetic-flow snapshot, the Leslie angle makes abrupt rotationsoth clockwise and counter-clockwise in relatively thin boundary

ayers, corresponding to winding and unwinding of director dis-ortions. These sharp distortion layers separate much smootherirector modulations in space. Thus, the dynamical feature shown

n Fig. 6 where nearby layers are tumbling and wagging, cre-tes spatial tumbling and wagging structures across the gap. Theirector distortions are amplified with the oblate defect metric, orrder parameter, s = d1 − d2, which shows oblate defects formingrecisely in the sharp layers where the principal axis of orienta-ion has large gradients. Thus, one finds multiple oblate defectscross the shear gap which arise precisely between neighbor-ng tumbling and wagging layers. Closer inspection further showshat the tumbling and wagging layers correlate with the shearands.

Space–time rheological stress features for Er = 25,000 over a 10 timeunit period bracketing an active oblate defect phase

Figs. 11 and 12 show the first and second normal stress differ-nces N1 = �xx − �yy, N2 = �yy − �zz , and the apparent shear stress= �xy across the shear gap and as time evolves. Here we find therst and second normal stress differences N and N experience

1 2ign changes across the gap during the periods when the tumblingnd wagging layers are most out of phase, or equivalently, dur-ng periods when there are multiple oblate defects in the layeredexture. The shear stress is nearly uniform most of the time, but

[

[

n Fluid Mech. 165 (2010) 687–697

the magnitude fluctuates during the appearance of strong directordistortions and oblate defects across the gap.

4. Concluding remarks

Simulations of the large Ericksen number regime for three dif-ferent flow-orientation models reveal transient texture length scaledistributions of the long-time attractors with consistent statisticalscaling behavior. There is a shift in the texture distributions towardsmaller length scales as Er increases to O(105), where the dominantlength scale remarkably obeys an Er−1/2 scaling, predicted by Mar-rucci in 1985 [36], for all three models. These results agree withthe study by Sgalari et al. [40] using a different model and Fouriertransforms of spatial snapshots to infer scaling behavior. Detailedspace–time graphics of flow, stress and orientation features reveala dynamic layered or banded structure, with either director tum-bling or wagging within each layer, and intermittent, thin oblatedefect domains between layers when the tumbling and wagginglayers are strongly out of phase. The oblate defect boundary layersare the source of the largest texture gradients, and therefore thesmallest length scales. While restricted to one space dimension,these attractors preserve local defect domains which are present inthe cores of two and three dimensional structures [30,48,47].

Acknowledgements

This work was supported by the Deutsche Forschungsge-sellschaft, Grant HE5995/1-1. Effort also sponsored by AFOSR grantFA9550-06-1-0063, NSF grants DMS-0908409 and DMS-0908423,Army Research Office grant W911NF-09-1-0389, and the Depart-ment of Energy Multiscale Mathematics grant DE-SC0001914. M.G.Forest acknowledges an influential conversation with Peter Con-stantin regarding estimation strategies for texture length scales.

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