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J. Non-Newtonian Fluid Mech. 159 (2009) 115–129

Contents lists available at ScienceDirect

Journal of Non-Newtonian Fluid Mechanics

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

uench sensitivity to defects and shear banding in nematic polymer film flows

iaofeng Yanga, M. Gregory Foresta,b,∗, William Mullinsa, Qi Wangc

Department of Mathematics, University of NorthCarolina at Chapel Hill, Chapel Hill, NC 27599-3250, United StatesInstitute for Advanced Materials, Nanoscience and Technology, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, United StatesDepartment of Mathematics, University of South Carolina, Columbia, SC 29208, United States

r t i c l e i n f o

rticle history:eceived 17 October 2008eceived in revised form 28 January 2009ccepted 3 February 2009

eywords:ematic polymershear flowshermal quench

a b s t r a c t

A flow-orientation algorithm [S. Heidenreich, S. Hess, M.G. Forest, X. Yang, R. Zhou, Robustness ofpulsating jet-like layers in sheared nano-rod dispersions, J. Non-Newtonian Fluid Mech., in press,doi:10.1016/j.physletb.2003.10.071; X. Yang, Z. Cui, M.G. Forest, J. Shen, Q. Wang, Dimensional robust-ness and instability of sheared, semi-dilute, nano-rod dispersions, SIAM Multiscale Model. Simul. 7(2008) 622–654] for nematic (liquid crystalline) polymer films is coupled with the homogenized effec-tive conductivity tensor of the rod-matrix composite [H. Zhou, X. Zheng, M.G. Forest, Q. Wang, R. Lipton,Extension-enhanced conductivity of liquid crystalline polymer nano-composites, Macromol. Symp. 28(2005) 81–85; X. Zheng, M.G. Forest, R. Lipton, R. Zhou, Q. Wang, Exact scaling laws for electrical con-ductivity properties of nematic polymer nano-composite monodomains, Adv. Funct. Mater. 15 (4) (2005)627–638; X. Zheng, M.G. Forest, R. Zhou, Q. Wang, R. Lipton, Anisotropy and dynamic ranges in effectiveproperties of sheared nematic polymer nano-composites, Adv. Funct. Mater. 15 (2005) 2029–2035; X.Zheng, M.G. Forest, R. Lipton, R. Zhou, Nematic polymer mechanics: flow-induced anisotropy, ContinuumMech. Thermodyn. 18 (2007) 377–394; X. Zheng, M.G. Forest, R. Vaia, M. Arlen, A strategy for dimensionalpercolation in sheared nano-rod dispersions, Adv. Mater. 19 (22) (2007) 4038–4043] to simulate a simul-taneous thermal quench and arrest of the film flow. The final orientational morphology yields nematicpolymer film features that determine film performance: anisotropy and heterogeneity in the thermalconductivity tensor, and residual shear and normal stresses. With this numerical toolkit, we amplify aninherent source of nematic polymer film non-uniformity for identical processing conditions. It is well-known that steady parallel plate driving conditions yield unsteady nematic polymer behavior such asdirector tumbling and wagging, defects, and shear bands. We show that the timing of initial cooling and

arrest of plate motion produces different solid films, even with identical isothermal and quench protocols!Simulations show flow reversal and gradient elasticity bands are created when the quench and platearrest are applied in phase with isothermal shear bands and defect layers, reminiscent of effects reportedfor small molar mass liquid crystals [R.S. Akopyan, R.B. Alaverdian, E.A. Santrosian, Y.S. Chilingarian, Ther-momechanical effects in the nematic liquid crystals, J. Appl. Phys. 90 (2001) 3371–3376; R.S. Hakobyan,G.L. Yesayan, B.Y. Zeldovich, Thermomechanical oscillations in hybrid nematic liquid crystals, Phys. Rev. E 73 (2006) 061707].

. Introduction

The remarkable properties that have been observed in nematicolymers and related nano-rod composite materials are contin-

ent upon the orientational distribution of the rod ensemble, whichictates how the unique properties of individual particles scalep to effective properties. Strong, uniform, uniaxial alignment ofano-rod dispersions has only been achieved in fibers through

∗ Corresponding author at: Institute for Advanced Materials, Nanoscience andechnology, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250,nited States. Tel.: +1 919 962 9606.

E-mail address: forest@amath.unc.edu (M.G. Forest).

377-0257/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2009.02.005

© 2009 Elsevier B.V. All rights reserved.

elongational flow processing. For films, however, non-uniformityin property tensors appears inescapable, with very poor quantita-tive or predictive understanding, and the processing of nano-rodcomposites and nematic systems in general remains empirical.

There are many and diverse reasons for the limitations on pro-cessing control, but the heart of the problem lies in the complexdynamics and spatial morphology generated when nano-rod disper-sions are exposed to shear-dominated flows in confined geometries.The typical shear experiment (with steady plate motion) yields

three anomalous features of the rod orientational distribution:it is transient, anisotropic, and heterogeneous with orientationaldefects. We emphasize that these features arise even in the absenceof curved or sharp geometries, which will only exacerbate the com-plexity of the nano-particle orientational distribution.

1 an Flu

samtpiaacdttfdctp

dsn[[(dplwmfltmGnRmteec

pttumtsssnpt

f

•

•

•

t

16 X. Yang et al. / J. Non-Newtoni

Since the rod distribution function dictates all film property ten-ors, it is apparent that nano-rod and nano-platelet films will inheritnisotropy and heterogeneity. The transient nature of the isother-al processing phase introduces a new element of non-uniformity:

he timing of a quench to a solid film is prone to yield different bulkroperties from the same processing conditions. The goal in this papers to assess this variability with high-fidelity numerical modelingnd simulations; we produce extremely different nano-rod filmsnd their effective conductivity tensors from identical processingonditions. To do so, we implement the homogenized thermal con-uctivity tensor of nano-rod dispersions [41–44], which provideshe full coupling of flow, orientation, and temperature in space andime. The local conductivity tensor is updated in space and timerom the orientation-flow solver, while also serving as the tensorialiffusion coefficient in the energy equation. This modeling featureaptures thermal feedback mechanisms due to local (in space andime) variations in the direction and degree of order of the nano-article phase.

We now briefly review related literature and results. Shear-ominated isothermal flow simulations of nano-rod disper-ions have been performed by several research groups, mostotably Rey and collaborators [29,24–27], Leal and collaborators19,20,31,32,36], Armstrong and collaborators [2], and the authors3,4,9–12,18,37,39,40,46,45]. Post-processing of the isothermal dataassuming an instantaneous quench) has been implemented toetermine electrical, thermal, dielectric [41,42] and mechanicalroperties [43]. In this paper, we build the modeling and simu-

ation capability and explore the implications of thermal quenchith respect to thermal conductivity and residual shear and nor-al stresses. This involves the coupling to our nematic polymer

ow solvers [45,46,18] of: a temperature field, an energy equation,emperature-dependent material parameters, and dynamic ther-

al and flow controls at solid boundaries. The classical paper byuyon et al. [5] explores convective instabilities, whereas the phe-omena we address are far below the Rayleigh–Benard threshold.elated work by Han and Rey with a small molar mass liquid crystalodel [16], and by Grecov and Rey [15] with a Landau–deGennes

ensor orientation model, solve the coupled orientation model andnergy equation, with an imposed linear shear. Our work alsoxtends thermomechanical phenomena of flow reversal in liquidrystals [1,17,21].

Temperature control is an essential processing design com-onent. Materials are processed in a liquid phase at elevatedemperatures and then quenched (cooled at some controlled rate)o the solid phase. In fiber flows, strong extensional flow achievesniform downstream alignment of the rods along the axis of sym-etry of the fiber (cf. [2,6,26,29]). The thermal quench then serves

o lock in the alignment achieved by the flow [8,24,25]. By contrast,ince isothermal film flows of nematic polymers lead to tran-ient, anisotropic, and heterogeneous property tensors and residualtresses [13,14,28,30,33–35,46], the role of quench or annealing isot simply to lock in what the flow has rendered. Rather, thermalrotocols have a potentially dramatic impact on final film proper-ies.

We turn to the focus of this paper, beginning with the salienteatures of isothermal LCP film flows [18]:

complex morphology: anisotropy (directional variability) at anyspatial site coupled with strong gradients in the nano-rod orien-tational distribution on a spectrum of lengthscales;transient morphology fluctuations under steady processing con-

ditions; anddefects created in shear-dominated flows.

We start from a representative isothermal film flow that reflectshese three features. We then impose a rapid quench phase whereby

id Mech. 159 (2009) 115–129

the liquid composite is cooled to a solid film at room temperaturewhile bringing the moving plates to rest. The final film is evaluatedon the basis of anisotropy and heterogeneity of its effective conduc-tivity tensor and residual stresses. Next, the quench evolution andfinal film properties are re-simulated, changing only the timing ofthe onset of quench. We sample six times of onset of quench withinthe period of the dynamic isothermal attractor. The extreme fluctu-ations of the isothermal solution are shown to be inherited by thefinal film property tensor and residual stress distributions. Finally,we explore longer quench phases to see if this might make final filmproperties more robust; this turns out to enhance a thermal flowreversal effect that again leads to sensitivity in final film properties.

2. The model and numerical method

2.1. The orientation-flow-temperature model

We define m(x) as the axis of symmetry of a rod macromoleculeof aspect ratio r � 1, T(x) as the absolute temperature, and V(x) =(v1(x), v2(x), v3(x)) as the local fluid velocity, where x = (x1, x2, x3)are spatial coordinates. The rate of strain and vorticity tensors aredefined by

D = 12

(∇V + (∇V)T) and � = 12

(∇V − (∇V)T), (2.1)

respectively. The gradient is defined by ∇ = (∂x1 , ∂x2 , ∂x3 ), and(∇V)ij = ∂xjvi = vij .

We introduce the necessary descriptive variables of the orienta-tional distributions. In Landau–deGennes models, the probabilitydistribution function f (m,x, t) of Doi–Hess kinetic theory is pro-jected onto the second moment tensor by

M = 〈mm〉 =∫

‖m‖=1

mmf (m,x, t) dm. (2.2)

M is a symmetric, trace 1, positive semi-definite tensor of rank 2.The fourth moment tensor M4 is defined by

M4 = 〈mmmm〉 =∫

‖m‖=1

mmmmf (m,x, t) dm. (2.3)

The orientation tensor Q is the trace zero form of M, Q = M − (I/3),where I is the identity matrix. By the spectral theorem, M admitsthe representation

M =3∑i=1

dinTi ni, (2.4)

where {di,ni}3i=1 are the principal values and corresponding

orthonormal principal axes of M,∑3

i=1di = 1,0 ≤ d3 ≤ d2 ≤ d1 ≤ 1,ni ⊥ nj if i /= j. Non-degenerate ordered phases arise whenever d1 isa simple eigenvalue; the tensor M for all non-degenerate phasestherefore defines a prolate, triaxial ellipsoid with principal axislengths di along axes ni. For example, the uniaxial nematic equi-librium phase has d1 > d2 = d3, defining a prolate spheroid, whilesheared nematic phases typically are biaxial, with d1 > d2 > d3,defining a triaxial prolate ellipsoid. Local defect phases are detectedby degeneracies in the principal value d1: the isotropic phase ariseswhen d1 has multiplicity 3, so that d1 = d2 = d3 = (1/3), where Mdefines a sphere; the oblate defect phase arises when d1 has mul-tiplicity 2, with d1 = d2 > d3, where M defines an oblate spheroid(platelet).

These distinct local orientational phases (non-degenerate pro-late, degenerate oblate or isotropic) are easily detected numericallyby monitoring the level sets of d1 − d2 and d1 − d3 [18,45,46]; like-wise, they are graphically easily detected by either color-coding thelevel sets of the metrics d1 − d2, d1 − d3, or presenting the spatial

X. Yang et al. / J. Non-Newtonian Flu

Table 1Parameter values.

Parameter

Er(T0) Re(T0) �1(T0) �2(T0) �3(T0) N(T0) ˛(T0) H�/K

10 50 0.0002 0.003 0.001 6 2 8500

Parameter

a

0

piseta

s

wQ∂

t∑

Fbp

N1 De T0 �1 �2 � t0/(�Cph2)

.8 18,585 0.5 590 K 1 1000 0.2 0.1

attern of M ellipsoids. Both graphical tools are used in Fig. 1 forllustrative purposes, and then in the flow-orientation-temperatureimulations. Note that the relative orientation of the viewer and thellipsoid can lead to mistakes in interpretation; by superimposinghe scalar metric color code on the ellipsoids, we avoid mistakingn oblate platelet on edge for a prolate thin ellipsoid.

The Doi–Marrucci–Greco (DMG) model for the dynamics andtructure of M is [22,23,45]

DMDt

= � · M − M · � + a[D · M + M · D − 2D : M4]

− 6Dr[Q − N(M · M − M : M4)] + DrNL2

8[�M · M + M ·�M

− 2�M : M4], (2.5)

here (D(·)/Dt) = (∂(·)/∂t) + (V · ∇)(·) is the material derivative,= M − I/3 is the traceless orientation tensor, and � = ∇ · ∇ =

2x1

+ ∂2x2

+ ∂2x3

is the Laplacian operator. We also explicitly definehe multiplication operations (A · B)ij =∑kAikBkj and A : B =i,jAijBji is a scalar, and (A : M4)ij =

∑k,lAklM4lkij .a = (1 − r2)/(1 +

ig. 1. Sketch of the geometry and defect metric color-coding of three distinct phases dearcode is based on level set values of the defect metric d1 − d2, which will be adopted in shase: d1 = d2 � d3. (For interpretation of the references to color in this figure legend, th

id Mech. 159 (2009) 115–129 117

r2) is the rod aspect ratio parameter, where r � 1 is the rod aspectratio of the macromolecules;Dr is the nematic average rotary diffu-sivity; N is a dimensionless rod volume fraction which governs thestrength of short-range excluded volume interactions; L representsthe persistence length of distortional elasticity in the one elasticconstant approximation.

We normalize time by a time scale t0 (chosen below) whichdefines the dimensionless time t = t/t0, and by a characteristiclengthscale h which defines the dimensionless space variable x =x/h, which lead to a dimensionless form of the DMG model:

DQ

Dt= � · Q − Q · � + a

[D · Q + Q · D + 2

3D − 2D : M4

]− 6Drt0

[Q − N

(Q · Q + 2

3Q + I

9−(

Q + I3

): M4

)]

+ Drt0NL2

8h2[�Q · Q + Q · �Q + 2

3�Q − 2�Q : M4], (2.6)

where the “hatted” parameters are dimensionless: � =∑i∂

2xi

= h2�, ∇ = (∂x1, ∂x2

, ∂x3) = h∇, V = t0/hV, D = (1/2)(∇V +

(∇V)T) = t0(1/2)(∇V + (∇V)T) = t0D.

We normalize the stress tensor = /0 by a characteristic stress0 = �h2/t20 where � is the density of the solvent. Then we have thenormalized flow equation

�h2

0t20

DV

Dt= ∇ · , (2.7)

where the stress tensor has isotropic, viscous, and anisotropicviscoelastic components arising from non-equilibrium and gradient

termined from the spectral properties of the second-moment tensor M. The colorubsequent figures. (a) Prolate phase: d1 � d2; Isotropic phase: d1 = d2 = d3; Oblatee reader is referred to the web version of the article.)

118 X. Yang et al. / J. Non-Newtonian Fluid Mech. 159 (2009) 115–129

Fig. 2. Transient variability of isothermal flow-orientation simulations [18]. Six snapshots of the 1D heterogeneous orientation tensor ellipsoids across the plate gap arepresented, with the shear profile across the gap superimposed. The ellipsoids are color-coded in terms of the oblate defect metric, d1 − d2, as indicated by the bar code. Darkred is the most highly ordered local phase, while dark blue is the partially disordered oblate phase. The lower frame (g) is a blow-up of one of the snapshots which showst highlyt 5 for ol

o

wmdpAs

∗

he dark blue oblate defect domain and the small lengthscale of transition between= 36.45, (e) t = 36.75, (f) t = 39.75, and (g) A blow-up of the snapshot of t = 36.7

egend, the reader is referred to the web version of the article.)

rientation of the rod phase:

ˆ = pI + 2�s0t0

D + 3KT0t0

[�1(D · Q + Q · D + 2

3D) + �2D : M4 + �3D

]+ 3akT

0

[(1 − 2N

3

)Q − NQ · Q − N

(I9

− (Q + I3

) : M4

)]

− aKTNL2

160h2

[�Q · Q + Q · �Q + 2

3�Q − 2�Q : M4

]

− KTNL2

160h2

[Q · �Q − �Q · Q + 1

2(∇Q : ∇Q − ∇∇Q : Q)

]. (2.8)

here p is the pressure, T is the absolute temperature, K is the Boltz-ann constant,�s is the viscosity of the solvent, and is the number

ensity of rod macromolecules. The fourth moment tensor of the

article distribution, M4, represents a difficulty in the calculation.common approach is to posit M4 as a functional of M to close the

ystem on M. The Doi closure is

: M4 = ∗ : MM, (2.9)

ordered and oblate disordered phases. (a) t = 9.75, (b) t = 19.75, (c) t = 29.75, (d)ne-fourth of the domain. (For interpretation of the references to color in this figure

which we adopt in this paper to be consistent with ourearlier isothermal studies [18]. The resulting is called theDoi–Marrucci–Greco (DMG) model. Finally, we have the normalizedtensor equation for Q:

DQDt

= � · Q − Q · � + a[

D · Q + Q · D + 23

D − 2(D : Q)(

Q + I3

)]− 6Drt0

[(1 − N

3

)Q − NQ · Q + N(Q : Q)

(Q + I

3

)]

+ Drt0NL2

8h2

[�Q · Q+Q · �Q + 2

3�Q − 2(�Q : Q)

(Q + I

3

)]Drt0NL2

= G1 − 6Drt0F1 +8h2

F2, (2.10)

which defines the expressions G1, F1 and F2 which will be usedsubsequently. The normalized flow equation likewise is specified

X. Yang et al. / J. Non-Newtonian Fluid Mech. 159 (2009) 115–129 119

F hear sd

n

w

T

wph�

ig. 3. Viscous and elastic stress features for each snapshot in Fig. 2. (a) Viscous sifference.

ow:

�h2

0t20

DV

Dt= ∇ · , (2.11)

ith

ˆ = pI+ 2�s0t0

D + 3KT0t0

[�1(D · Q + Q · D + 2

3D) + �2D : Q

(Q + I

3

)+ �3D

]+ 3akT

0

[(1 − N

3

)Q − NQ · Q + N(Q : Q)

(Q + I

3

)]

− aKTNL2

160h2

[�Q · Q + Q · �Q + 2

3�Q − 2�Q : Q

(Q + I

3

)]

−KTNL2

160h2

[Q · �Q − �Q · Q + 1

2(∇Q : ∇Q − ∇∇Q : Q)

]

= pI + 2�s0t0

D + 3KT0t0

G2 + 3aKT0

F1 − KTNL2

160h2(aF2 + F3). (2.12)

he heat transfer equation is then given by

DTˆ

= t0h2

∇ · (A · ∇ · T), (2.13)

Dt

here A is the thermal diffusivity tensor for the rod-solvent com-osite which we describe next. Zheng et al. [38,41,43,44] usedomogenization theory to derive the effective conductivity tensorfor highly conductive rods of aspect ratio r, scalar conductivity �2

tress = localDe, (b) orientation-dependent shear stress, and (c) first normal stress

and volume fraction � in an isotropic conducting matrix of isotropicscalar conductivity �1:

� = �1

(I + 2�ˇ

1 + ˇ(1 − Lr)(

I + ˇ 1 − 3Lr1 + 2ˇLr

M))

, (2.14)

where

Lr = 1r2 − 1

(r√r2 − 1

log(r +√r2 − 1) − 1

),

ˇ = �2 − �1

2�1.

(2.15)

The conductivity tensor gives the thermal diffusivity tensor as

A = �

�Cp(2.16)

using the bulk composite density and the bulk composite specificheat. We call attention to the consequences of the homogenizationformula (2.14) for the thermal conductivity tensor. Namely, since� is a linear combination of the identity and second-moment ten-sor M, it follows that � shares the same principal axes with M.Thus, heat flows anisotropically according to the nano-rod orien-

tation distribution. In the simulations to follow, we find importantconsequences of these thermal properties when a heterogeneousfilm flow is cooled toward the solid phase. In particular, note thathighly aligned rod phases impart highly anisotropic thermal trans-port, while oblate defects (disordered orientation phases that arise

120 X. Yang et al. / J. Non-Newtonian Fluid Mech. 159 (2009) 115–129

F from at oughta stressl

id

D

w

�

at

�

T

D

Wlbbw

t

ig. 4. The locked-in orientational morphology (a) and residual stored stresses (b)emperature is lowered from 590 K to 300 K while simultaneously the plates are brnd (b) orientation-dependent (and therefore stored) shear stress and first normalegend, the reader is referred to the web version of the article.)

n film shear flows) lead to isotropy of heat transport in a plane ofirections and weaker heat flow in the transverse direction.

The rotational diffusivity Dr is assumed to obey

r ∝ T

�s, (2.17)

here �s, the viscosity of the solvent, is thermally activated

s(T) = �0 e(H�/KT), (2.18)

nd H� is the activation enthalpy for viscosity. Given a referenceemperature T0 then

s(T) = �s(T0) e(H�/K)((1/T)−(1/T0)). (2.19)

he rotational diffusivity is then

r = Dr(T0)T

T0e−(H�/K)((1/T)−(1/T0)). (2.20)

e now are in position to specify the characteristic timescale t0 andengthscale h from which previous dimensionless groups will thene explicit. To link to our isothermal results [18], we normalize time

y the rotational diffusivity at the reference temperature T0, whichill be the isothermal flow temperature:

0 = 1Dr(T0)

, (2.21)

rapid quench applied to each of the six snapshots of Fig. 2. In each simulation, theto rest over 0.1 time units. (a) Steady state orientational morphology after quenchdifference after quench. (For interpretation of the references to color in this figure

then

Drt0 = T

T0e−(H�/K)((1/T)−(1/T0)). (2.22)

We choose h, which is the half the channel thickness so that thephysical gap domain is −h ≤ x2 ≤ h, or in dimensionless units,−1 ≤ x2 ≤ 1. We define the D eborah number, De = (v0/h/6Dr) asthe normalized velocity of the shearing plates, where v0 is the veloc-ity of the plates, so that the velocity boundary condition during theisothermal part of the process is

V1(x2 = ±1) = ±De. (2.23)

We posit the nematic “strength” of the excluded volume interactionpotential, N, to be a function of temperature,

N(T) = N0 + N1 = N(T0) + N1

(1 − 1

). (2.24)

T T T0

These choices lead to the following dimensionless parameters: aReynolds number Re, an entropic energy parameter ˛, nematicReynolds numbers�i and an Ericksen number Er (which is a dimen-

X. Yang et al. / J. Non-Newtonian Fluid Mech. 159 (2009) 115–129 121

Fig. 5. The locked-in anisotropic and heterogeneous thermal conductivity tensor morphology, represented by the geometry and color-coding of the final film conductivitytensor corresponding to Fig. 4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

Fig. 6. Correlations between initial and final quenched film defect metrics, and final film conductivity anisotropy metric, for the six initial onset times of quench in Figs. 2–5.(a) Oblate metric d1 − d2 for the isothermal snapshots of Fig. 2, which are the initial data for the final quenched film depicted in (b) and (c). (b) Oblate defect metric d1 − d2

for the quenched states in Fig. 4. (c) Conductivity anisotropy metric (�1 − �2)/�1 for the final quenched films in (b).

122 X. Yang et al. / J. Non-Newtonian Fluid Mech. 159 (2009) 115–129

F evolvis

s

T

wr

2

hl

c

ig. 7. Dynamics of quench, starting from the t = 9.75 snapshot of Figs. 2 and 3 andtress, and (d) first normal stress difference.

ionless elasticity constant in the DMG model):

Re(T) = 0t0�s

= Re(T0) e−(H�/K)((1/T)−(1/T0)),

˛(T) = 3KT0

= 3KT

�h2Dr(T0)2= ˛(T0)

T

T0,

�i(T) = 3KT�i0t0

= �i(T0)T

T0,

Er(T) = 8h2

L= Er(T0)

N(T0)N(T)

.

(2.25)

hen the stress tensor can be rewritten as

ˆ = pI + 2Re(T)

D + 3KT0t0

G2 + a˛(T)F1 − ˛(T)6Er(T)

(aF2 + F3), (2.26)

here F1, F2 and F3 are defined above in Eq. (2.12), and G2 can beewritten as3KT0t0

G2 = �1(T)(

D · Q + Q · D + 23

D)

+�2(T)(D : Q)(

Q + 13

I)

+�3(T)D. (2.27)

.2. Numerical algorithm

In (2.10), �Q is multiplied by the function Q and N(T(t,x)). We

andle these products by simply subtracting cQ (t)�Q from both

eft and right hand sides, where

Q (t) = maxx

[T

T0

N(T)Er(T)

e−(H�/K)((1/T)−(1/T0))]. (2.28)

ng to the final film features in Figs. 4–6. (a) Primary flow, (b) temperature, (c) shear

Similarly, for (2.11), we subtract cV (t)�V from both sides, where

cV (t) = maxx

[1

Re(T)+ �3(T)

2

]. (2.29)

For (2.13), we subtract cT (t)�T from both sides, where

cT (t) = maxxtr(t0h2

A), (2.30)

and tr is the trace.We then rewrite the system (2.10)–(2.13) in the following linear

form and drop all ,

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

∂Q(t,x)∂t

− cQ (t)�Q(t,x) = Qn(V(t,x),Q(t,x), T(t,x)),

∂V(t,x)∂t

− cV (t)�V(t,x) + ∇p = Vn(V(t,x),Q(t,x), T(t,x)),∇·

V = 0,∂T(t,x)∂t

− cT (t)�T(t,x) = Tn(V(t,x),Q(t,x), T(t,x)),

(2.31)

where Qn, Vn and Tn contain all corresponding nonlinear terms.

Assuming �k is the numerical approximation of a function �

at time t = kıt, where ıt is the time step size, (Qk,Vk, pk, Tk) and(Qk−1,Vk−1, pk−1, Tk−1) are known, the second-order time discretescheme of (2.31) with no-slip boundary conditions can be describedas follows:

X. Yang et al. / J. Non-Newtonian Fluid Mech. 159 (2009) 115–129 123

Fig. 8. Dynamics of quench: flow, temperature, and elastic stresses, starting from the t = 29.75 snapshot of Figs. 2 and 3 and evolving to the final film features for this initiald mal s

•

•

•

ata in Figs. 4–6. (a) primary flow, (b) temperature, (c) shear stress, and (d) first nor

Find Qk+1 such that

⎧⎪⎨⎪⎩

3Qk+1 − 4Qk + Qk−1

2ıt− cQ (tk)�Qk+1 = 2Qn(Vk,Qk, Tk)

−Qn(Vk−1,Qk−1, Tk−1),Qk+1|x2=±1 = Q(t0)|x2=±1.

(2.32)

Find the intermediate velocity Vk+1 = (vk+1

1 , vk+12 , vk+1

3 ) such that

⎧⎪⎨⎪⎩

3Vk+1 − 4Vk + Vk−1

2ıt− cV (tk)�V

k+1 + ∇pk = 2Vn(Vk,Qk, Tk)

−Vn(Vk−1,Qk−1, Tk−1),vk+1

1 |x2=±1 = ±De, vk+12 |x2=±1 = 0, vk+1

3 |x2=±1 = 0.

(2.33)

Find the auxiliary function n+1 to satisfy

⎧⎨⎩

−� k+1 = 32ıt

∇ · Vk+1,

∂ k+1

∂x2|x2=±1 = 0.

(2.34)

tress difference.

• Update (pk+1,Vk+1) by{pk+1 = pk + k+1 − cV (tk)∇ · V

k+1,

Vk+1 = Vk+1 − 2ıt

3∇ k+1.

(2.35)

• Find Tk+1 such that⎧⎪⎨⎪⎩

3Tk+1 − 4Tk + Tk−1

2ıt− cT (tk)�Tk+1 = 2Tn(Vk,Qk, Tk)

−2Tn(Vk−1,Qk−1, Tk−1),Tk+1|x2=±1 = T0.

(2.36)

Several remarks are in order:

1 1

• One can initialize (Q ,V , p1, T1) by using the first-order versionof (2.32)–(2.36).

• We impose no-slip boundary conditions atx2 = ±1 for all functioncomponents. We note that this time discretization scheme is alsosuitable for 2D and 3D simulations [18,45].

124 X. Yang et al. / J. Non-Newtonian Fluid Mech. 159 (2009) 115–129

F nd evs

•

•

3

3

b

ig. 9. Dynamics of quench, starting from the t = 36.45 snapshot of Figs. 2 and 3 ahear stress, and (d) first normal stress difference.

At each time step, one only needs to solve a sequence of 1DPoisson-type equations of the form:{u− �u = f,u|x2=±1 = 0.

(2.37)

For the quench simulations presented in next section, the bound-ary condition of the velocity of vk+1

1 is set to be⎧⎪⎨⎪⎩

vk+11 |x2=±1 = ±De, t ≤ tb,

vk+11 |x2=±1 = ±De(t − ts)

tb − ts , tb ≤ t ≤ ts,vk+1

1 |x2=±1 = 0, t ≥ ts,(2.38)

where tb, ts are the time nodes when the quench starts and endsrespectively. Similarly, the boundary condition of the tempera-ture Tk+1 is⎧⎪⎨⎪⎩Tk+1|x2=±1 = T0 t ≤ tb,Tk+1|x2=±1 = T1 − T0

ts − tbt + T0ts − T1tb

ts − tbtb ≤ t ≤ ts,

Tk+1|x2=±1 = T1 t ≥ ts,(2.39)

where T0, T1 are the temperatures before and after quenchrespectively.

. Numerical simulations

.1. Isothermal baseline simulation

For the purposes of this paper, we select all parameters of theaseline isothermal simulation consistent with [18]. All dimension-

olving to the final film features in Figs. 4–6. (a) primary flow, (b) temperature, (c)

less parameters are given in Table 1, where we also set the thermalparameters using data derived from our fiber spinning studies onthe Hoechst-Celanese nematic polymer Vectra [7]. We note that theDeborah number chosen here is of order unity, which means thatthe plate conditions and gap height give a bulk shear rate that iscommensurate with the rotational diffusion rate of the rod macro-molecules. This is a realistic regime for nematic polymer film flowsand typical parallel plate experiments. With these parameters, ourrecent isothermal simulations reveal orientational, flow and storedstress features in explicit detail [18,45].

We start with a representative stable long-time solution of theisothermal flow-orientation system. Figs. 2 and 3 illustrate the threeshear-processing features under steady plate-driven conditionsthat we emphasized earlier: the attractor is transient, anisotropic,and heterogeneous with local defect phases appearing and disap-pearing periodically. In Fig. 2, the rod orientational distribution M isgraphically depicted across the shear gap for a series of snapshots.As explained in Fig. 1, the shape and color-coding of the ellipsoiddepicts the degree of nematic order: higher aspect ratio prolate (andthus dark red) ellipsoids are the most focused distributions aboutthe principal axis; platelet (and thus dark blue) shapes correspondto an oblate defect phase with a circle of principal axes; and spher-ical shapes (also dark blue) correspond to an isotropic phase, butthere are no isotropic phases present in these simulations. The cor-

responding primary flow profile across the gap is superimposed oneach orientational structure snapshot. Significant shear banding isstrongly correlated with the formation of localized oblate defectlayers that transiently appear and disappear. In Fig. 3, for each cor-responding snapshot of Fig. 2, we show the viscous shear stress

X. Yang et al. / J. Non-Newtonian Fluid Mech. 159 (2009) 115–129 125

F d evoa ate des

(rosmn

3

atitwu4a

3

srfiiFtv

ig. 10. Dynamics of quench, starting from the t = 36.75 snapshot of Figs. 2 and 3 annd residual stresses arise when onset of quench is tuned with the isothermal obltress, and (d) first normal stress difference.

which by our non-dimensionalization is simply the local Debo-ah number) and the residual orientational stress components (therientational contribution to the shear stress, and the first normaltress difference). There is no mistaking the abject lack of unifor-ity (in orientation, in space and in time) of the isothermal flow of

ematic polymers!

.2. Rapid quench protocol

We now apply “downstream” boundary conditions on temper-ture and plate motion after the isothermal film flow (at constantemperature 590 K) has converged to the dynamic attractor shownn Figs. 2 and 3. We impose the following rapid quench protocol:he temperature at each plate is lowered from 590 K to 300 Khile the plates are brought to rest over 0.1 dimensionless timenits. Since the isothermal flow is unsteady with a period of about0 time units, we first explore sensitivity to when the quench ispplied.

.2.1. Final quench film featuresFirst, in Fig. 4 we report the o utcome of six different quench

imulations, showing the fully quenched orientational distribution,esidual shear and normal stresses. The only difference among these

nal films is the starting time of the quench during the transient

sothermal flow: t = 9.75, 19.75, 29.75, 36.45, 36.75 and 39.75 ofigs. 2 and 3. Clearly, the locked-in anisotropy and heterogeneity ofhe orientation and normal stresses in each film inherit the dynamicariability from the isothermal flow. The shear stress is uniform in

lving to the final film features in Figs. 4–6. The most enhanced gradient morphologyfect domains and strong shear bands. (a) Primary flow, (b) temperature, (c) shear

space but varies in amplitude for each film. We observe that the ori-entational morphology is essentially preserved by the rapid quench,although the defect-laden structures are somewhat modified.

To assess the final quenched film properties, we now considerthe locked-in heterogeneous thermal conductivity tensor �, whichfrom (2.14) follows directly from the orientation tensor ellipsoidsdepicted in Fig. 4. �, like M, is non-negative definite and symmetric,with the same principal axes as M, and so � can also be depictedwith triaxial ellipsoids based on the spectral representation

� =3∑i=1

�intini, (3.1)

where the �i are the principal conductivities and ni are identical tothe second-moment principal axes.

In Fig. 5, we present the heterogeneous film conductivity tensorof each quenched film. Note the strong correlations, as expected,between the M and �morphologies. The color scheme for � in Fig. 5is based on the metric (�1 − �2)/�1, analogous to the defect metricused for M.

We now make the orientation and conductivity anisotropy morequantitative in Fig. 6: graph (a) shows the oblate defect metric d1 −d2 across the film for the onset of quench snapshots from Fig. 2, (b)

shows the corresponding post-quench snapshots of d1 − d2 fromFig. 4, and (c) shows the post-quench snapshots of the conductivityanisotropy metric (�1 − �2)/�1.

The orientation and conductivity correlations are stark, with theconsequence that the final film conductivity features (heterogeneity

1 an Fluid Mech. 159 (2009) 115–129

av

3

tFtsfladuclpcsbnuas

tin

Fig. 11. Analogs of figures for an order of magnitude slower arrest of the plate

Ft

26 X. Yang et al. / J. Non-Newtoni

nd anisotropy of principal axes, principal values, and residual stresses)ary dramatically with the timing of the onset of quench.

.2.2. Dynamics of quenchWe now explore the dynamics of the quench process between

he initial data in Figs. 2 and 3 and the quenched features inigs. 4–6. In Fig. 7, we show the evolution from the t = 9.75 snapshothrough the initiation of cooling and arrested plate motion and finaltationary, room temperature film. From Fig. 7(a), we see that theow throughout the gap essentially is arrested by t ≈ 11 time unitsfter the plates have come to rest. From Fig. 7(b), the temperatureistribution uniformizes to the wall temperature in about 20 timenits. During the first 11 time units in Fig. 7(a) (recall the plates haveome to rest), the initial simple shear profile develops a Poiseuille-ike nonlinear shear in each half of the gap, with no motion at thelates and at the mid-gap. The flow gradually weakens through vis-ous dissipation, while the solidification layer at the plates progresseslower than the arrest of the flow. The orientational gradient contri-ution to the shear stress, Fig. 7(c), and the orientation-inducedormal stress difference, Fig. 7(d), show the final features of resid-al stress in the film. Note the residual normal stress difference islways negative, and progressively growing during the quench to a

olid film.

We now follow the quench dynamics which starts from the= 29.75 snapshot of Figs. 2 and 3. Note the flow profile at this

nstant has a strong shear band near each plate and a very weak,early linear shear across the rest of the gap. The local Deborah

motion. The quenched orientation and conductivity tensor morphologies (a) start-ing from the t = 36.45 snapshot of Figs. 2 and 3 and (b) starting from the t = 36.75snapshot of Figs. 2 and 3. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of the article.)

ig. 12. Dynamics of slower quench, starting from the t = 36.45 snapshot of Figs. 2 and 3 where the plates are brought to rest in one time units. (a) Primary flow, (b)emperature, (c) shear stress, and (d) first normal stress difference.

X. Yang et al. / J. Non-Newtonian Fluid Mech. 159 (2009) 115–129 127

F ropy mF d1 −r the ar

nsfibiantttfbiF

mfiWpa

tfiflecsptt

ioditptfrpq

ig. 13. Final quenched film defect metrics (left), and final film conductivityanisotigs. 2 and 3: 0.1 time unit (red) and one time unit (blue). (a) Oblate defect metriceferences to color in this figure legend, the reader is referred to the web version of

umber is therefore extremely high near the plates, and the elastictresses are likewise non-uniform. These conditions set the stageor a relatively fast arrest of the flow, which is already weak in thenterior and the quick stoppage of the plates arrests the local shearand. Fig. 8(c) shows the elastic shear stress is constant except

n the plate boundary layer and this localized gradient relaxes inbout one time unit, well before the film has solidified. The firstormal stress difference, however, exhibits a non-trivial evolutionhat eventually leads to a stratified residual normal stress distribu-ion in the solid film, with negative values of N1 = 11 − 22 nearhe plates and large positive values in the interior. This is quite dif-erent from the t = 9.75 quenched film of Fig. 7, and appears toe correlated with the apparent stronger gradients in the locked-

n orientational distribution shown in the third quench profile ofig. 4.

This sign change in the residual normal stress across the filmight have significant implications for mechanical loading on a

lm, an example of the issues surrounding active nano-materials.e re-emphasize that these residual normal stresses arise inde-

endent of curved boundaries or sharp features in molds, whichre the traditional sources of strong residual stress concentrations.

Next, we explore the dynamics of quench starting from the= 36.45snapshot. Note the flow profile in Fig. 2, repeated as therst snapshot in Fig. 9(a), is almost identical to the previous initialow for the t = 29.75 quench simulation. However, the initial ori-ntational distributions are significantly different. First, from theolor-coding by the oblate defect metric, d1 − d2, the t = 36.45napshot shows the formative stages of a defect layer near eachlate, corresponding to a defocusing of the principal axes of orien-ation. The t = 29.75 snapshot, by contrast, has gradients only inhe principal axis of orientation, not the principal values.

Notice from Fig. 9(a), that during the quench a backflow is createdn each half of the shear gap, which did not occur in the previ-us simulations. This is related to the dynamics surrounding theefect domains as the plates are brought to rest. Since the solid-

fication front propagates much slower than the duration time ofhe defect domain in isothermal flows, the defect layer near eachlate intensifies, which is evident in the final dark blue value of

he defect metric in Fig. 4. Notice that the first normal stress dif-erence has shifted now to all positive values, with the strongestesidual normal stresses again in the final quenched film. Com-aring this simulation and the previous two starting times of theuench, the final morphology ofN1 has varied from all negative val-

etrics (right) for two plate arrest timescales applied to the t = 36.45 snapshot ofd2 and (b) Conductivity anisotropy metric (�1 − �2)/�1. (For interpretation of theticle.)

ues, to sign changes across the gap, to all positive values. We alsonote that the flow is fully arrested after nine time units in Fig. 9, com-pared to about one time unit in Fig. 8. This underscores the strong flowfeedback due to defect domains. Note that the temperature distribu-tion uniformizes on the same timescale in each simulation so far. InFig. 10, we start the quench in phase with the peak shear band andthe strongest defect domain, the t = 36.75 snapshot of Figs. 2 and 3.Notice already from the quenched orientational and stress distribu-tions of Fig. 3, the strongest gradient features are associated with thissimulation. Fig. 10(a) shows that a strong backflow is generated ineach half of the shear gap, which then subsides over about nine timeunits. The temperature takes, once again, about nine time units touniformize. The first normal stress difference has features similarto Fig. 9, but with higher values.

These Figs. 2–10 show the remarkable disparity in the dynamicsof quench and in the final film morphology and residual stressesthat arise simply by shifting the timing of the onset of thermalquench and arrest of the plate motion. The simulations further impli-cate the presence of oblate defect domains and shear bands at the timeof onset of quench with the most severe gradient morphology of thefinal film.

3.2.3. A slower quench protocolWe slow the arrest of the plate speeds by an order of magnitude

from the previous protocol, to one time unit, while retaining thesame temperature protocol. Fig. 11 gives the orientation tensor andthermal conductivity tensor of the final, quenched film, starting thequench from the snapshots t = 36.45 and 36.75 of Figs. 2 and 3.These film properties are to be compared with the 4th and 5thmorphologies in Figs. 4 and 5. It is clear that the morphology is dif-ferent, but still highly non-uniform in anisotropy and heterogeneity,and with more extreme gradients. From the color-coding based onthe oblate defect metric, the oblate defect layers near each plateappear to have been mollified. However, gradients in directions ofanisotropy, or distortions in the principal axes of orientation andconductivity, appear to have been enhanced.

Fig. 12 gives further insight into the dynamics of the slowerquench process for the t = 36.45 snapshot. The strong and almost

immediate backflow in each half of the shear gap is amplified by slow-ing the arrest of the plates. The slower plate deceleration generatesmuch stronger stress amplification in the interior, as evidenced bythe factor of 3 higher extreme negative values of N1, and muchstronger gradients and smaller lengthscales of morphology. Thus

1 an Flu

to

tecerfidiHt

atpep

4

ifassritbamwopdsci

A

o104

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

28 X. Yang et al. / J. Non-Newtoni

he residual stresses in the final film are amplified by the slower arrestf the plate motion.

To complete the comparison of the two different protocols onhe plate motion, in Fig. 13 we compare the final film order param-ter morphologies (using the defect metric), and the correspondingonductivity measure of anisotropy (using the normalized differ-nce between the two leading principal conductivity values). Theseesults are to be compared with Fig. 6. Consistent with Fig. 11, wend that the oblate defect layer and corresponding conductivityefect layer are mollified by a slower plate arrest timescale; there

s a clear peak axis of orientation and conductivity across the gap.owever, the gradients in the principal axis are amplified across

he film.These simulations imply that varying the timescale of plate

rrest relative to the timescales of the isothermal dynamic attrac-or can shift gradient morphology between principal values andrincipal axes, but not remove the spatial non-uniformity. The pref-rence of one protocol over another would have to be decided byerformance of the different films under realistic applications.

. Concluding remarks

We have coupled quench protocols, consisting of thermal cool-ng and arrested plate motion, to the flow-orientation equationsor nematic polymers in a parallel plate shear cell. A numericallgorithm is developed to solve the full system of equations, andimulations of an isothermal process followed by quench are pre-ented. The upshot of the numerical results is that the transientesponse of nematic polymers in a steady shear cell leads to annherent source of non-uniformity in the quenched film—namely,he timing of onset of quench and the duration of the quench areoth sensitive control parameters. The final film property metricsre the thermal conductivity tensor and the residual shear and nor-al stresses. These metrics reveal anisotropy and heterogeneityithin the film, which vary significantly with the timing of the

nset of the quench conditions and the relative timescale of arrestedlate motion. The key culprits in film variability are the fluctuatingefect layers and strong shear bands inherent in nematic polymerhear flows. It becomes evident that nano-rod composite film pro-esses will require sophisticated control strategies to deal with thisnherent non-uniformity.

cknowledgements

This research has been supported in part by the Air Force Officef Scientific Research contracts F49550-05-1-0025, FA9550-06--0063, the National Science Foundation Grants DMS-0605029,626180,0548511 and 0604891, the Army Research Office contract7089-MS-SR, and NASA URETI BIMat award no. NCC-1-02037.

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