DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2011.15.457DYNAMICAL SYSTEMS SERIES BVolume 15, Number 2, March 2011 pp. 457–473

SHEARING THE I-N PHASE TRANSITION OF LIQUID

CRYSTALLINE POLYMERS: LONG-TIME MEMORY OF

DEFECT INITIAL DATA

Ke Xu

Department of Biomedical Engineering

University of North Carolina at Chapel HillChapel Hill, North Carolina 27599-7575, USA

M. Gregory Forest

Departments of Mathematics and Biomedical EngineeringInstitute for Advanced Materials

University of North Carolina at Chapel HillChapel Hill, North Carolina 27599-3250, USA

Xiaofeng Yang

Department of MathematicsUniversity of South Carolina

Columbia, South Carolina 29208, USA

Abstract. Liquid crystalline polymers have been extensively studied in shear

starting from an equilibrium nematic phase. In this study, we explore the

transient and long-time behavior as a steady shear cell experiment commencesduring an isotropic-nematic (I-N) phase transition. We initialize a localized

Gaussian nematic droplet within an unstable isotropic phase with nematic,

vorticity-aligned equilibrium at the walls. In the absence of flow, the sim-ulation converges to a homogeneous nematic phase, but not before passing

through quite intricate defect arrays and patterns due to physical anchoring,

the dimensions of the shear cell, and transient backflow generated around thedefect arrays during the I-N transition. Snapshots of this numerical exper-

iment are then used as initial data for shear cell experiments at controlledshear rates. For homogeneous stable nematic equilibrium initial data, the Leal

group [4, 5, 6] and the authors [12] confirm the Larson-Mead experimental

observations [7, 8]: stationary 2-D roll cells and defect-free 2-D orientationalstructure at low shear rates, followed at higher shear rates by an unstable tran-sition to an unsteady 2-D cellular flow and defect-laden attractor. We show at

low shear rates that the memory of defect-laden data lasts forever; 2-D steadyattractors of [4, 5, 12] emerge for defect free initial data, whereas 1-D unsteady

attractors arise for defect-laden initial data.

1. Introduction. The Larson-Mead experiments on sheared nematic polymers[7, 8] identified roll cell formation at low shear rates, followed by roll cell insta-bility. The corresponding orientational morphology showed stationary defect-free

2000 Mathematics Subject Classification. Primary: 76A15, 82D30.Key words and phrases. Crystal defects, isotropic-nematic phase transition, hydrodynamics,

nematic liquid crystal, shear deformation, topology.The authors are supported by NSF grants DMS-0908423 and DMS-0502266, DOE grant DE-

SC0001914 and ARO grant W911NF-09-1-0389.

457

458 KE XU, M. GREGORY FOREST AND XIAOFENG YANG

structure, followed by transient, defect-laden structure. These features were mod-eled successfully by the Leal group [4, 5, 6] and later by our group [12], each withmodels that represent second-moment closure approximations of the Doi-Marrucci-Greco theory for nematic polymers coupled to the Navier-Stokes equations in twospace dimensions.

The experiments and model simulations assert specific initial data and bound-ary anchoring conditions: vorticity-aligned (so-called logrolling) orientation at theplates, and a uniform nematic equilibrium in the interior of the shear cell, matchingthe logrolling wall alignment. Here we are interested in how sensitive the experi-ments are to initial data; this study is motivated by material processing conditionswhich would naturally present heterogeneous initial data. Rather than fabricatedifferent initial data, we choose to simulate the isotropic-nematic phase transitionbeginning from a small seed of the nematic phase (a localized Gaussian droplet) inthe middle of the cell, at a rod concentration where the nematic phase is the uniquestable equilibrium, whereas the rest of the interior of the cell is in the unstableisotropic phase, and the logrolling nematic phase is imposed at the walls with asmooth boundary layer to match the isotropic interior. This data, inspired by [9],is simulated without imposed flow, following the orientation tensor model of Doi-Marrucci-Greco. Snapshots of the texture are then stored to initialize a sequenceof data to perform the shear cell experiments. The dynamics of this simulation areinteresting in their own right, with a remarkable array of 2-D defects and backflowcreated in the transient passage to a spatially uniform equilibrium nematic phase.

The initial data is constructed as follows. The orientation tensor Q is the tracelesspart of the second-moment tensor M of the orientational probability distributionf(m,x, t). M is positive semi-definite with principal values 0 ≤ d3 ≤ d2 ≤ d1 ≤1. All nematic equilibria are uniaxial, which means d2 = d3, and d1 is a simpleeigenvalue of M with value dictated by the rod geometry and volume fraction. Forour model, the Maier-Saupe potential yields the Flory order parameter s = d1−d2 =0.809, and the biaxiality parameter β = d2−d3 = 0. The isotropic phase is specifiedby di = 1/3 for all i.

Following Yang et al.[12], we monitor disordered phases by level sets of the localdefect metrics d1 − d2 and d1 − d3. If s > 0, the local phase is ordered, with auniquely defined principal axis of orientation given by the norm 1 eigenvector ofd1, called the nematic director. If s = 0 and β > 0, this corresponds to an oblatedefect, for which the principal axis of M is the circle spanned by the eigenvectorsof the multiplicity two, largest eigenvalue d1. If s = β = 0, d1 has multiplicity 3,all directions of rod orientation are equally preferred, which is the isotropic phase.We emphasize that these order parameter defect metrics are locally defined, andtherefore easily monitored by level sets, which in our graphs are color coded so thatthe zero level sets of d1 − d2 (oblate defect metric) and d1 − d3 (isotropic defectmetric) are visible. (The authors note that a gray scale instead of a rainbow scaleis used in black and white prints, where the darkest domains are defects and thelightest domains are strongly ordered. )

We now turn to the traditional focus on the topology of defects. We cautionthat while most attention to defects has centered on topological winding numbers,isotropic and oblate disordered phases can occur in homogeneous and 1D struc-tures, where the physical dimension does not support closed curves and topology!Furthermore, once topological defects annihilate or when they are spawned, order

SHEARING I-N PHASE TRANSITION OF LIQUID CRYSTALLINE POLYMERS 459

parameter metrics evolve smoothly and continue to reveal domains of local disorder.We refer to [12, 13] for more detailed illustrations and examples.

It is well established that all degree ± 12 topological defects, based on the winding

number of the principal axis defined by d1, have cores within which d1 has multi-plicity 2 (oblate cores) or 3 (isotropic cores). Our initial data therefore begins froma large isotropic domain with a small nematic uniaxial equilibrium droplet whoseprincipal axis is vorticity-aligned, in agreement with the wall anchoring condition.Note this initial data has trivial topological degree, but the structure is nontrivialfrom the point of view of ordered and disordered phases. As the nematic dropletexpands into the isotropic phase, it does not do so as an expanding 2-D circulardomain. Rather, Figures 2 - 4 show a sequence of snapshots of the structure.

2. Governing equations. The fundamental kinetic theory of rod-likemacromolecules in a viscous solvent is developed by Doi and Hess, in which thedynamics of these molecules is described by the orientational probability distri-bution function (PDF). Instead of working with the full PDF, Doi, Marrucci andGreco derived closure models for the second moment tensor M coupled to the flowequations, which we adopt for this paper to make contact with our recent study forhomogeneous initial data [12]. The orientation tensor Q is defined as

Q = M− I

3, M =<mm >, (1)

where <> indicates averaging with respect to the PDF. We refer to [1, 12] for detailof the model and an historical literature review.

Figure 1. Sketch of the computational cell and the flow direction.Horizontal planes: shearing plates; vertical plane: plotting area;black arrows: initial flow profile.

2.1. Flow-orientation equations and non-dimensionalization. As in [12], weconsider a shear cell between two shearing plates at y = −1 and y = 1, movingat the same speed but in opposite directions: Vplates = [±v0, 0, 0]. Thus, thevelocity field is assumed to be a simple linear shear initially, as shown in Figure 1.We assume periodicity in the vorticity(z) direction and homogeneity in the flow(x)

460 KE XU, M. GREGORY FOREST AND XIAOFENG YANG

Table 1. Parameters

Re α Er µ1 µ2 µ3 a10 2 25 9× 10−3 9× 10−3 1 0.8

direction. Because of these simplifications, all velocity components, pressure, andtensor components are computed as functions of y, z, t, as we will discuss in the nextsection. Thus, all simulation snapshots are presented in the y-z plane (the verticalplane in Figure 1).

The DMG model is nondimensionalized using the gap depth 2h (h=1 in Figure 1),

the nematic rotational diffusion time scale tn, and the characteristic stress τ0 = ρh2

t2n,

where ρ is the density of the nematic polymer liquid. The dimensionless velocity,position, time, stress, and pressure variables become:

U =tnhU, x =

1

hx, t =

t

tn, τ =

τ

τ0, p =

p

τ0. (2)

The speed v0 at which the plates are moving and the depth of the gap defines a bulkflow time scale t0 = v0

h ; the average rotary diffusivity Dr of the rods defines another

time scale tn = 16Dr , whose ratio defines the Deborah number: De = tn

t0= v0

6hDr .The following seven dimensionless parameters arise:

Re =τ0tnη, α =

3ckT

τ0, Er =

8h2

αNl2, µi =

3ckTζitnτ0

, i = 1, 2, 3, (3)

where Re is the solvent Reynolds number; the solvent viscosity is η; α measuresthe strength of entropy relative to kinetic energy; c is the number density of rodmolecules; k is the Boltzmann constant; T is absolute temperature; Er is theEricksen number which measures the ratio of viscous stresses to stresses arisingfrom Frank distortional elasticity; l is the persistence length and N is the dimen-sionless rod volume fraction and dimensionless concentration of nematic polymers;1µi, i = 1, 2, 3 are the three nematic Reynolds numbers, themselves dependent on

the three shape-dependent viscosity parameters due to the polymer-solvent inter-action, 3ckTζi, i = 1, 2, 3 . In the next section, we give the model equations basedon these non-dimensionalized variables and drop the . Note that the values of theseparameters are provided in Table 1.

2.2. The Doi-Marrucci-Greco (DMG) model. The dimensionless flow (Navier-Stokes) equation and the extra stress constitutive equation are given by:

DV

Dt= ∇ · (−pI + τ), (4)

τ = (2

Re+ µ3)D + aαF (Q)

+a

3Er(∆Q : Q(Q +

I

3)− 1

2(∆QQ + Q∆Q)− 1

3∆Q)

+1

3Er(1

2(Q∆Q−∆QQ)− 1

4(∇Q : ∇Q−∇∇Q : Q))

+ µ1((Q +I

3)D + D(Q +

I

3)) + µ2D : Q(Q +

I

3),

(5)

SHEARING I-N PHASE TRANSITION OF LIQUID CRYSTALLINE POLYMERS 461

where DVDt = ∂V

∂t + (V ·∇)V is the material derivative and D = 12 (∇V+ (∇V)T ) is

the symmetric part of the velocity gradient tensor, also known as the rate of straintensor. a is a dimensionless parameter depending on the molecular aspect ratio rof spheroidal molecules:

a =r2 − 1

r2 + 1. (6)

The short-range excluded volume effects are captured by

F (Q) = (1− N

3)Q−NQ2 +NQ : Q(Q +

I

3), (7)

where N is a dimensionless concentration of nematic polymers, which controls thestrength of the mesoscopic approximation, F (Q), of the gradient of the Maier-Saupepotential.

The orientation tensor equation is

DQ

Dt= ΩQ−QΩ + a(DQ + QD) +

2a

3D

− 2aD : Q(Q +I

3)− (F (Q) +

1

3αEr(∆Q : Q(Q +

I

3)

− 1

2(∆QQ + Q∆Q)− 1

3∆Q)),

(8)

where DQDt = ∂Q

∂t + (V · ∇)Q and Ω = 12 (∇V − (∇V)T ). The physical boundary

conditions for the no-slip velocity at the plates are scaled to

V|y=±hy= (±De, 0, 0), (9)

whereas periodic boundary conditions are imposed in the vorticity (z) direction.The above equations are solved by an effective spectral-Galerkin method [10] andwe refer the reader to [12] for more details.

3. The isotropic-nematic transition from a small Gaussian droplet in thecenter of a square domain.

3.1. Spatio-temporal simulation of the isotropic-nematic transition (De=0). We first show data arising from a simulated isotropic-nematic phase transi-tion. The initial condition at t=0 is shown in Figure 2 as discussed before. At thecenter of the drop and the two plates, the orientation tensor Q has the followingrepresentation:

Q = s0(n1(x, t)n1(x, t)− I

3) (10)

where s0 = 0.809 corresponds to the equilibrium order parameter value at N = 6and n1 is along the vorticity direction (z-axis). From the center of the drop, themajor director is fixed while the order parameter s decays to 0 exponentially:

Q = s0e−15[y2+(z−1)2](n1(x, t)n1(x, t)− I

3). (11)

To avoid anomalous behavior due to a discontinuity at the walls, we likewise intro-duce a smooth but sharp boundary layer at each plate, where the order parameters in equation 1 decays linearly from .809 at the walls (y=+1 or -1) to s=0 over adistance of .05 in normalized units.

Figures 3 and 4 provide snapshots at t=2.5, 4 and 15 of the dynamics of theI-N transition for this data, illustrating a highly heterogeneous depletion of theunstable isotropic domain. The morphology is described with a combination of

462 KE XU, M. GREGORY FOREST AND XIAOFENG YANG

Figure 2. Color-coded isotropic defect metric d1 − d3 and oblatedefect metric d1 − d2 (note that in black and white prints, thedarkest domains are defects, while the lightest domains are stronglyordered) of a nematic drop in an unstable isotropic environmentat t=0 with a vorticity-aligned nematic equilibrium at the walls(y = ±1), and a smooth transition to the isotropic phase over alayer of depth .05 at each wall. Note that isotropic domains yieldpositive tests for both metrics. Subsequent figures show oblatedefect domains, which fail the isotropic metric test.

local metrics that detect and resolve the defect cores, then given a positive test,we print snapshots and determine topology from nonlocal winding number of theprincipal axis of the second moment tensor M. In Figure 3, column 1 gives snapshotsof major director topology superimposed with the color coded isotropic defect metricd1 − d3, while column 2 shows the orientational ellipsoids superimposed with theoblate defect metric d1 − d2 across the 2-D domain, which gives full orientationspace information in each snapshot. Comparing the two representations, we gleanall morphology features: 1. defect topology, where the ovals surround +1/2 degreedefects while the rectangles surround -1/2 degree defects; 2. the nature of the defectcore, where dark blue (dark gray in black and white) domains in the left columndetect isotropic cores while lighter blue (light gray in black and white) on the leftwith dark blue on the right detect oblate cores; as well as 3. domains of relativedisorder which would fail any topological metric. Again, the authors note that agray scale is used in black and white prints.

Because of the non-uniformity of the initial and boundary data and the geom-etry of the computational domain, the I-N transition is far from boring. Isotropicdomains separated from the nematic walls and Gaussian droplet simply follow thelinearized growth rate and by t=2.5 most of the cell has saturated into the nematicequilibrium. However, several small domains of disorder persist at t=2.5, whichstill pass the test for isotropic cores and for which the nonlocal topological degree is+1/2 and -1/2 in equal proportion. The evolution from t=2.5 to t=4 is intriguing:one finds nearly stationary director morphology and therefore stationary topology

SHEARING I-N PHASE TRANSITION OF LIQUID CRYSTALLINE POLYMERS 463

across the cell, whereas the defect cores transition from isotropic (complete disor-der) to oblate (partial disorder). We also observe coalescence of local domains ofdisorder. After t=4, the isotropic domains are completely gone, whereas the t=15snapshot reveals that topological defects and oblate cores are still prevalent. Wecaution that the resolution of ellipsoids is rather sparse (higher density preventsvisibility of individual ellipsoids). As a result, it is likely that the ellipsoid figuremisses the actual defect core since the zero level set of either metric may not havebeen sampled. By t = 80, the nematic transition is complete, giving a uniformnematic equilibrium in the cell that matches the vorticity-alignment of the walls.

Backflow generated by the I-N transition. Since the initial data is heterogeneous,and the morphology during the I-N transition is laden with defects, these spatialgradients are seen from formula (5) to create distortional stresses which in turngenerate flow as seen from the momentum equations (4). In Figure 4 we show thesecondary flow field (vy, vz)(y, z) at times t=2.5, 4, 15. This transient backflowfor De=0 has an order of magnitude 10−2, which is a factor of 10 stronger thanstationary roll cells at De=1.5 for homogeneous initial data!

In the next section, we show results of simulations of the full 2-D flow-nematicinital-boundary value problem, for initial data shown in Figures 2 - 4. We only showthe results for De=1.5 since they provide the long-lived dependence on initial data.Suffice to say that for De=3.5 the simulations show that by t=400 normalizedtime units (set by the rotational relaxation rate of the rod-like macromolecules)the 2-D space-time attractor reported in detail in [12] has been reached for eachinitial condition. This attractor is transient with irregular dynamics and complexfluctuating morphology, so what we mean is that the basic space-time features aresimilar for each simulation at De=3.5. There is nothing special about De=3.5, sincea wide range of Deborah numbers sufficiently greater than De=1.5 yield similarattractors. Thus a sufficiently strong shear destroys the dependence on initial data.Another way to state this result is that there appears to be a unique space-timeattractor for a range of Deborah numbers including De=3.5; quite disparate dataconverge to the same attractor, indicating loss of memory of defect-laden initial dataon computational timescales. This result will be shown to not prevail for De=1.5.

3.2. Commencement of shear from I-N transition snapshots. In [12], foruniform nematic initial data with logrolling anchoring at the shear plates, theDe=1.5 attractor consists of stationary 2-D roll cells (in the secondary flow (vy, vz)transverse to the primary flow component vx) with a modulated 2-D director mor-phology in the y-z computational domain. If we allow the I-N evolution of Section2 to run for long enough (t > 100) until the secondary flow is negligible and thedefects have disappeared, then upon commencement of the motion of the plates, werecover this 2-D attractor, which is omitted to save space. This result is insensitiveto noisy 2-D perturbations in the initial data. As motivated in the Introduction,we now consider the same shear experiment at De=1.5 but with initial data takenfrom the t=0, 2.5, 4, 15 snapshots of Section 2. We find that all of these initialdata, which possess significant defect morphology, converge to the same attractor.Thus, we report the evolution for only one initial data, that of the t=0 snapshot,and omit the rest to save space.

3.2.1. Shearing the t=0 snapshot of the I-N transition simulation. In running thesenumerical experiments, we discovered a remarkable consequence of the symmetriesof the t=0 data of Figure 2. The data consists of a perfectly circular nematic

464 KE XU, M. GREGORY FOREST AND XIAOFENG YANG

(a) (b)

(c) (d)

(e) (f)

Figure 3. Transient defect morphology snapshots during the I-Ntransition. Column 1: major director topology with color-coded(or gray scale) isotropic defect metric d1 − d3. Column 2: orienta-tion ellipsoid texture with color-coded (or gray scale) oblate defectmetric d1 − d2. Topological defects are labelled by degree: − 1

2

(rectangles) and + 12 (ovals).

SHEARING I-N PHASE TRANSITION OF LIQUID CRYSTALLINE POLYMERS 465

(a) t = 2.5. (b) t = 4. (c) t = 15.

Figure 4. Backflow generated by the I-N transition of Figure 3.Secondary flow field (vy, vz)(y, z) superimposed with color codingby the level sets of the oblate defect metric d1 − d2.

Gaussian droplet in the center of the square domain, with reflection-symmetricboundary layers at each plate. When we simulate this data, our code preservesreflection symmetry in all directions. This turns out to constrain the evolutionin a dramatic fashion, as shown in Figure 5. Namely, the simulation convergesto an unsteady attractor with four stationary roll cells symmetrically organized inthe square domain, and topological defects trapped between the plates and in theupdrafts and downdrafts between neighboring roll cells. Remarkably, the topologyof these defects fluctuates between ± 1

2 while their cores fluctuate between oblateand isotropic with a period of about 12 time units. Any perturbation that breaksthe symmetry leads to unpinning of these roll cells and defects, and convergence tothe attractor we show next. This result underscores the necessity to add asymmetricnoise to any exquisitely symmetric initial and boundary data, which we do in theremaining simulations.

3.3. Shearing the t=0 snapshot of the I-N transition simulation plusasymmetric noise. With De=1.5, we start the steady shear experiment with thet=0 data of Figure 2 and random spatial noise added. We first present transientbehavior before showing the eventual attractor because it has novel defects anddynamics that are markedly different from the evolution in Figure 5 and results forhomogeneous nematic initial data reported in [12]. Figure 6 shows snapshots of theearly evolution at t=2 and 7 of the degree of order across the shear cell, employingthe level sets of the isotropic (d1 − d3, left column) and oblate (d1 − d2, right col-umn) defect metrics. The nematic order in the plate boundary layers and dropletare spreading, and the unstable transition of the isotropic domain is evident. Att = 2, the isotropic metric shows there are no isotropic domains remaining, whilethe oblate metric shows an oblate strip near each plate and an oblate ring formsaround the growing nematic drop. At t=7, the oblate defect strips and ring havepinched off to create oblate droplets as the nematic phase invades these disordereddomains. The number of oblate drop domains that form is a consequence of theaspect ratio of the cell and the initial location of the mother droplet.

Next we move forward about 70 time units, shown in Figure 7, where we alsofocus on director morphology metrics and secondary flow. The novel features are:

466 KE XU, M. GREGORY FOREST AND XIAOFENG YANG

(a) Vx (b) Vx − V 0x (c) Vy

(d) Vz (e) p

Figure 5. De=1.5. Symmetric 2-D unsteady attractor arisingfrom shearing the t=0 symmetric Gaussian droplet of Figure 2.The defects are fluctuate between degree + 1

2 and − 12 and the de-

fect cores periodically fluctuate between isotropic and oblate do-mains. Snapshots at t = 506.5 of: (a)full 2-D velocity field Vx,(b)the departure from the pure linear shear flow imposed at t=0,Vx−Vx(t = 0), (c)the secondary “vertical” velocity along the platenormals, Vy, (d)the secondary vorticity flow, Vz and (e)the pressurefield.

1. Two pairs of degree 12 topological defects form in the updrafts and downdrafts

between the plates and 4 primary roll cells. In the t=70.9 and 71.25 snapshots,the middle pair of defects has charge − 1

2 , the edge defects have charge 12 .

2. The charge − 12 defects have isotropic cores at t = 70.9 and oblate cores at

t = 71.25, whereas the + 12 defects have oblate cores at t = 70.9 and isotropic

cores at t = 71.25! We remark that in our earlier studies and those of Leal’sgroup, both from uniform nematic initial data, isotropic cores were neverdetected.

3. There are two small satellite roll cells between the isotropic cores and theplates in both snapshots. These two small cells together with the larger inte-rior cells create a quadrupolar flow structure that apparently causes a strongerdegree of disorder in the core.

4. Between snapshots, the two satellite roll cells translate along the plates, lead-ing to the oscillation of the cores without modifying the basic defect topology.

SHEARING I-N PHASE TRANSITION OF LIQUID CRYSTALLINE POLYMERS 467

(a)

(b)

Figure 6. De=1.5. Order and disorder metrics of the weakshear evolution at Deborah number 1.5 from the Gaussian nematicdroplet plus random noise, presented in terms of color-coded levelsets of the isotropic defect metric (d1− d3, left column) and oblatedefect metric (d1−d2, right column) at t=2 and t=7. These metricsreveal the original unstable isotropic domain is completely gone,mostly replaced by the stable nematic equilibrium. The remnantsof the initial data at t=2 consist of oblate defect walls and an oblatedefect ring, which then break up and pinch off oblate droplets byt=7.

5. Unlike Figure 5, where this defect oscillation phenomenon persists for hun-dreds of time units, the broken symmetry of Figures 6 and 7 leads to a tran-sition out of these metastable states around t=300.

468 KE XU, M. GREGORY FOREST AND XIAOFENG YANG

(a) (b)

(c) (d)

Figure 7. Transient defect morphology at De = 1.5. Column 1:the major director topology superimposed with color coding bythe level sets of the isotropic defect metric d1 − d3. Column 2:secondary flow field (vy, vz) superimposed with color coding by thelevel sets of the oblate defect metric d1 − d2.

Destabilization of the 2-D flow-orientation structure and convergence to a 1-D un-steady attractor. We now show the remarkable coarsening transition from 2-D flowand orientation to a 1-D unsteady attractor. The breakup process is illustratedthrough a series of snapshots in Figure 8 starting at t=360. We superimpose thesecondary flow structure (vy, vz) and the oblate defect metric, which shows the dis-integration of the four roll cells along with the breakup, migration and dilution ofthe oblate defect domains. We emphasize that the level sets of the oblate defectmetric reveal the dynamics and structure of the degree of order in the shear cellirrespective of whether the defect topology is trivial, complex, or changing.

SHEARING I-N PHASE TRANSITION OF LIQUID CRYSTALLINE POLYMERS 469

Figure 8 shows the defect merger and coarsening process, which is facilitated bythe breakup of the roll cell structure from Figures 6 and 7. The flow sweeps theoblate defect domains toward one another, leading to strong deformation and mergerinto one oblate domain and eventually, by t=388, disappearance of all disordereddomains. Meanwhile, the flow has transformed into a large but weak central celland two cells in the top right and lower left corners. Note that this breakup processclearly breaks reflection symmetry with respect to z, and also is weakly asymmetricwith respect to y. The simulation of Figure 5 does not seed these asymmetries, andthus this breakup process is prevented.

Figures 9 and 10 show the longtime behavior after the 2-D flow and orientationbehavior has disappeared. The z-gradients of all computed quantities are essentiallyzero, leaving only y-dependent flow and orientation. Likewise, the “vertical” flowcomponent vy is reduced to O(10−8), whereas the vorticity component vz is smallbut non-negligible with amplitude O(10−3), the same strength as roll cells from[12]. Figure 9 shows a series of snapshots of the orientation ellipsoids across theplate gap. One observes a kayaking orbit at each gap height, where the principalaxis rotates around the vorticity (z) axis. One also observes a longwave spatialmodulation that oscillates with the period of the central kayaking orbit. When themid-gap returns to vorticity alignment, the entire cell is nearly uniformly aligned.

Figure 10 shows the remarkable unsteady nature of the flow field. Note that thenonlinear measure of the primary shear flow, vx − vx(t = 0), switches in time frompositive to negative in each half of the shear gap. This means the concavity of theflow profile vx(y, t) oscillates in time. Likewise, the vorticity component vz changessigns, implying a sloshing motion that is correlated with the longwave kayakingmotion in the orientational configuration field.

To our knowledge, this is the first observation of bi-stable attractors in 2-Dspace-time studies of shear cell experiments. Both attractors have been confirmedto be stable to random 2-D perturbations. These are not the first evidence of bi-stability in sheared nematic polymers. Indeed, these results are reminiscent of bi-stable monodomain attractors in imposed simple shear [14, 15]. We also note thatthe in-plane 1D attractors in [16] arise for in-plane anchoring, whereas kayaking1D attractors arise for logrolling or vorticity-aligned anchoring considered here.These 1D results are consistent with the monodomain phase diagrams in [14] and[15], which show tumbling and wagging orbits that are stable for confined in-planeorientation yet unstable to kayaking orbits in the full orientational space. It wouldbe interesting to confirm that the 1-D attractors of [16] are unstable if the anchoringconditions are shifted even marginally out of the shear plane, breaking the in-planesymmetry.

4. Conclusion. In this paper, we revisit the liquid crystalline polymer shear cellsimulations of Leal et al. [2, 3] and Yang et al. [12], which both emulate theLarson-Mead experiments on roll cell formation [7, 8]. The sole purpose of thepresent paper is to explore what happens in these shear cell experiments if theinitial data is non-homogeneous and defect-laden, with identical boundary condi-tions as previously reported. Our expectation was to determine a finite memorytimescale of the initial data, measured by a delayed convergence to these previouslyreported attractors at low and intermediate Deborah numbers (normalized shearrates). We choose data arising from an isotropic-nematic phase transition, whichcontains a complex mixture of isotropic and nematic domains, and for early times

470 KE XU, M. GREGORY FOREST AND XIAOFENG YANG

(a) t = 360. (b) t = 363.5.

(c) t = 364.75. (d) t = 365.25.

(e) t = 386.75. (f) t = 388.

Figure 8. De=1.5. Dynamics of breakup of the 2-D flow andorientation behavior. A series of snapshots of the secondary flowfield (vy, vz) superimposed with the color coded level sets of theoblate defect metric d1 − d2.

SHEARING I-N PHASE TRANSITION OF LIQUID CRYSTALLINE POLYMERS 471

Figure 9. The “1-D kayaking attractor” for De=1.5. The space-time plot of orientation ellipsoids projected onto the y-z plane afterconvergence to the 1-D atractor at t = 514. The time sequenceshows logrolling alignment at the plates, and a gradually increasingoscillation amplitude around the vorticity axis with distance fromthe plates.

replete with disordered defect domains. As shown in the Figures and descriptions ofthem, however, the memory of the initial data is effectively infinite at low Deborahnumbers; at higher Deborah numbers where the 2-D attractors are unsteady andirregular, all data converge to the previously reported space-time attractors. Wehave shown that shearing of defect-laden initial data seeded with random spatialnoise to break arbitrary symmetries of the initial data, converges eventually to a1-D unsteady attractor. This 1-D attractor is highlighted by a kayaking orienta-tional dynamics at each gap height, where the principal axis rotates around thevorticity alignment of the plates, with increasing amplitude of rotation from plateto mid-gap. The flow is likewise 1-D, with an oscillating nonlinear primary shearcomponent vx and a weak but non-negligible vorticity component vz that likewiseoscillates in sign. Together, these flow components create a slow sloshing motionin the shear cell synchronized to the kayaking orientational structure. Thus, eitherthere are bi-stable attractors in the shear cell experiment at low Deborah numbers,or we must go to full 3-D spatial resolution and physical boundary conditions inthe shear cell where we may find that 3-D perturbations alter these predictions. To

472 KE XU, M. GREGORY FOREST AND XIAOFENG YANG

(a) u− u0 at t = 514. (b) vz at t = 514. (c) Secondary flow field(vy , vz) superimposed by

the color-coded oblate defect

metric d1 − d2 at t = 514.

(d) u− u0 at t = 518. (e) v at t = 518. (f) Secondary flow field

(vy , vz) superimposed bythe color-coded oblate defect

metric d1 − d2 at t = 518.

(g) u− u0 at t = 520. (h) v at t = 520. (i) Secondary flow field (vy , vz)

superimposed by the color-

coded oblate defect metric d1−d2 at t = 520.

Figure 10. The flow structure of the 1-D attractor at De=1.5.

settle this question will require a 3-D flow-orientation solver (Leal’s group has theonly such code to date [6]) and careful physical experiments.

SHEARING I-N PHASE TRANSITION OF LIQUID CRYSTALLINE POLYMERS 473

Acknowledgments. Partial research support is gratefully acknowledged from theNational Science Foundation through grants DMS-0908423 and DMS-0502266, De-partment of Energy through grant DE-SC0001914, Army Research Office throughgrant W911NF-09-1-0389 and National Institutes of Health.

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Received December 2009; revised March 2010.

E-mail address: xuke@email.unc.edu

E-mail address: forest@amath.unc.edu

E-mail address: xfyang@math.sc.edu