Dynamic Coupling between a Multistable Defect Pattern and Flow in NematicLiquid Crystals Confined in a Porous Medium

Takeaki Araki

Department of Physics, Kyoto University, Kyoto 606-8502, Japan(Received 18 June 2012; revised manuscript received 27 October 2012; published 20 December 2012)

When a nematic liquid crystal is confined in a porous medium with strong anchoring conditions,

topological defects, called disclinations, are stably formed with numerous possible configurations. Since

the energy barriers between them are large enough, the system shows multistability. Our lattice Boltzmann

simulations demonstrate dynamic couplings between the multistable defect pattern and the flow in a

regular porous matrix. At sufficiently low flow speed, the topological defects are pinned at the quiescent

positions. As the flow speed is increased, the defects show cyclic motions and nonlinear rheological

properties, which depend on whether or not they are topologically constrained in the porous networks. In

addition, we discover that the defect pattern can be controlled by controlling the flow. Thus, the flow path

is recorded in the porous channels owing to the multistability of the defect patterns.

DOI: 10.1103/PhysRevLett.109.257801 PACS numbers: 61.30.Pq, 61.30.Jf, 83.60.Df, 83.80.Xz

Topological defects are observed in various types ofphases associated with spontaneous symmetry breakingsuch as crystals and ferromagnetic materials [1]. In theearly stage of the phase transitions, many defects are cre-ated, and their total amount then decreases with time toreduce the free energy [2]. If frustrations are internallyincluded or imposed by external fields, the defects aresometimes stabilized. On the other hand, reorganizationof topological defects is often important under nonequilib-rium conditions. For instance, pinning and depinning ofquantized vortices in a type-II–superconductor induce elec-trical resistance [3]. Motions of quantized vortices also leadto mutual frictional interactions in a flowing superfluid [2].

A nematic liquid crystal (NLC) is an ideal system forstudying behaviors of topological defects because of itsslow dynamics and large length scale [4–6]. When NLCsare confined in cavities [7,8], microfluidic devices [9,10],and porous media [11–17], the anchoring interactionbetween the director and the solid surface imposes frus-trations on the elastic field. In a complex geometry, manytopological defects in NLCs are thus sustained for a longtime [6,8,18–21]. A NLC in a porous material exhibitsslow glassy behavior and the resulting memory effects[15,16,19]. In a previous paper, we studied the memoryeffect by Monte Carlo simulations, focusing on the role ofthe bicontinuity of the matrix [21]. After cooling from anisotropic state, many disclination lines running through thechannels of the porous network remain, with numerouspossible trajectories. Because each trajectory is at one ofthe local energy minimum states and the energy barriersamong them are often larger than the thermal energy, thesystem shows nonequilibrium behaviors similar to those inspin glasses. We found that the memorization is attributedto the reconfigurations of the defect structure. The appli-cations of an electric or magnetic field can bring the systeminto a new different state, and the new configuration

remains even after the field is removed. Here, irreducibledisclinations, which are topologically concatenated to thesolid matrix, tend to enhance the memory effect.In this Letter, we study a NLC flowing in a regular

porous medium. In particular, we focus on dynamic cou-plings between the flow and the nonergodic disclinationpattern. We hope our findings will facilitate research onmicrofluidics using liquid crystals [9,10].First, we prepare a pattern for a porous material by

introducing a scalar variable � in a cubic lattice accordingto Ref. [21].� varies smoothly in space. Then, we partitionthe cubic lattice into three portions by using �: P is theensemble of lattice sites representing the solid material ofthe porous matrix, N denotes the nematic fraction, and Sis the ensemble of lattice sites at the interface. For theportions N and S, we introduce a nematic tensorial orderparameter Q�� [22]. We employ the Landau–de Gennes

(LdG) free energy density [23],

fLdGðQ��Þ ¼ �F

2Q��Q�� � �FQ��Q��Q��

þ �FðQ��Q��Þ2 þ L

2ðr�Q��Þ2

� �"E�E�Q��; (1)

where �F, �F, �F, L, and �" are phenomenologicalmaterial constants. E is an external field. The repeatedsubscripts indicate summations over Cartesian indices.The free energy functional of the system is given by

F fQ��g¼ZNþS

drfLdGðQ��Þ�wZSdrQ��d�d�; (2)

where d is the normal vector of the porous matrix, d ¼r�=jr�j. The second term represents the molecular an-choring interaction between the surface of the porousmatrix and the nematic orientational order, and w is its

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material constant. We assumew> 0, i.e., that homeotropicanchoring is preferred.

The hydrodynamic flow u is determined by a momentumequation [23],

�

�@

@tþ u�r�

�u� ¼ r�ð�p��� þ �d

�� þ �0��Þ; (3)

where � is the material density and p is the pressure. Here,�d

�� is the distortion stress tensor, which is given by

�d�� ¼ � @fLdG

@ðr�Q�Þ ðr�Q�Þ: (4)

Further, �0�� is the viscous stress tensor, which is given by

�0�� ¼ �1Q��Q�A� þ �4A�� þ �5Q��A��

þ �6Q��A�� þ 12�2N�� ��1Q��N��

þ�1Q��N��; (5)

whereA�� ¼ ðr�u� þr�u�Þ=2 is the symmetric velocity

gradient tensor, and N�� is the time rate of change of Q��

with respect to the background flow. The latter is defined as

N�� ¼ @

@tQ�� þ u�r�Q�� þ!��Q�� �Q��!��;

(6)

where!�� ¼ ðr�u� �r�u�Þ=2 is the asymmetric veloc-

ity gradient. In Eq. (5), �i (i ¼ 1; 4; 5; 6) and �i (i ¼ 1; 2)are phenomenological parameters having the dimension ofviscosity. The time development of the nematic order pa-rameter is given by

N�� ¼ � 1

�1

�F�Q��

� �2

2�1

A��: (7)

The above nematohydrodynamic equations are solved withnonslip boundary conditions at the surfaces of the porousmatrix.

Lattice Boltzmann simulation is one of the most efficientnumerical methods of investigating flow behaviors in acomplex geometry [24]. The bounceback method realizesnonslip boundary conditions on the velocity fields at thesurfaces of the porous matrix [24]. To our knowledge, twolattice Boltzmann algorithms for simulating nematohydro-dynamics have been proposed [25–28]. In this study, weadopt a numerical scheme developed by Care and co-workers. Here, we omit the details and follow their nota-tions (see Refs. [26,27]). We performed three-dimensionalsimulations with four speeds and coordination number 27(D3Q27). The weight factors of the four speeds ti are givenby ti ¼ 8=27 for ci ¼ ð0; 0; 0Þ, ti ¼ 2=27 for ci ¼ð�1; 0; 0Þ, ti ¼ 1=54 for ci ¼ ð�1;�1; 0Þ, and ti ¼1=216 for ci ¼ ð�1;�1;�1Þ with appropriate permuta-tions. Here, ci is the nondimensional velocity vector.

In our simulations, we impose an external force den-sity f to flow the NLC. Then, we modify Eq. (31) inRef. [26] to

Xj

Mi��j�gðneqÞj�

¼ �gðneqÞi��

iþ �mi�� þ �pi��

þXi

�tiff�ðci� � u�Þ þ f�uci�cig���; (8)

whereP

j represents summation over 27 velocity vectors

cj. We employ �F ¼ �0:133, �F ¼ 2:133, �F ¼ 1:6,

w ¼ 0:1, L ¼ 0:1, �" ¼ 1:0,�1 ¼ 3:0,�2 ¼ �1:0, 0 ¼1:1, �0 ¼ 0:3, and �2 ¼ 0:5. This set of parameters gives acorrelation length � ¼ 0:87�, an anchoring penetrationlength da ¼ �, and a bulk nematic orderQb ¼ 0:37, where� is the unit cell length. The resulting viscosity ratio of thisanisotropic liquid is �?=�k ffi 1:1. Here �k and �? are the

effective viscosities when the director is parallel to the flowand parallel to the velocity gradient, respectively. We adopta bicontinuous pattern with cubic symmetry, which we havestudied and is denoted as BC in Ref. [21]. The size of theunit cell of the bicontinuous cubic is L ¼ 32�. We use asimulation box of 643 with periodic boundary conditions.In NLCs, the flow speed is characterized well by the

Ericksen number Er, which is the ratio of the viscous andthe elastic forces. It is given by Er ¼ jf j‘3=K, where K isthe Frank elastic modulus expressed as K ¼ 9LQ2

b=2 in a

one-constant approximation, and ‘ is the characteristiclength of the distortion. Because it is difficult to determine‘ quantitatively in a complex geometry, we assume that ‘ isof the order of the radius of the narrowest channels,namely, ‘ ¼ L=4.We quenched the system from an isotropic state to a

nematic state without any external field E or bulk force f .Because of the anchoring effect, the director was stronglydeformed; therefore, disclination lines remained even afterlong duration annealing. Figure 1(a) shows one of thestable defect configurations, where the defect positionsare represented by the isosurfaces of the elastic energydensity fel ¼ Lðr�Q��Þ2=2. Here, the remaining

b

FIG. 1 (color). (a) Snapshot of disclination lines of nematicliquid crystals in a porous medium after zero field cooling froman isotropic state. Solid green objects represent the porousmatrix, and red curves are the remaining disclinations.(b) Defect pattern remaining after an external field pulse isapplied along the z axis. Red and blue curves represent reducible(type-I) and irreducible (type-II) disclinations, respectively.

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disclinations run through the channels rather randomly.They are closed without ends, and some are concatenatedto the solid matrix. Then, we applied a strong external fieldpulse along the z axis (Ez ¼ 0:1). After the field wasremoved, two types of disclination loops formed in thex-y planes, as shown in Fig. 1(b). Half of the remainingdisclinations [red loops in Fig. 1(b)] are localized at thenarrowest sections of the channels. The line defects locallyhave the topological charge s ¼ �1=2, so the loops arereducible continuously into hyperbolic point defects withs ¼ �1. On the other hand, the other half of the loops [blueloops in Fig. 1(b)] consists of disclinations, which locallyhave the topological charge s ¼ 1=2. However, they areirreducible to point defects, because the solid necks of theBC penetrate the loops. Hereafter, the former and latterdefect groups are referred to as type-I and type-II, respec-tively. This structure is the same as that obtained in theMonte Carlo simulations [21]. It also appears that thisconfiguration is probably one of the global energy mini-mum states, which are also degenerated along the x andy-axes. We employ this configuration as an initial state.

First, we impose force fz to flow the NLC toward the zdirection. In NLCs, defect motion is caused by not onlyhydrodynamic convection, but also rotations of the directorfield. If the imposed force is weak enough, both types ofdefects show only small displacements, which are deter-mined by the balance between hydrodynamic convectionand director rotation (not shown here). As the flow speed isincreased, the director rotation can no longer compensatefor the hydrodynamic convection, so the defect patternbecomes unstable against the flow. Figure 2 shows thetemporal changes in (a) the defect pattern and (b) the crosssection of Qzx at an x-z plane that cuts the centers of thenecks of the BC. The applied force strength is fz ¼ 0:006,which corresponds to Er ffi 50. Under this force, the type-Idefects start to exhibit cyclic behavior. The correspondingmovies are available in the Supplemental Material [29]. Asshown in Fig. 2(b), the director field is gradually distortedwith time. When the elastic distortion energy becomescomparable to the anchoring energy, the anchoring at thedownstream side of the porous surface is transiently broken[see t ¼ 270 in Fig. 2(b)]. When the anchoring breaks, newdefects are born from the surface. Then, the new defectsand the old type-I loops fuse into new type-I disclinationloops. During this cycle, the type-II loops keep to be at thequiescent positions. With a further increase in the flowspeed, the type-II loops also become unstable. Figure 2(c)shows snapshots of the defect pattern at fz ¼ 0:013.Because they are concatenated to the channels, they cannotmove without a topological transformation. The type-IIdefects repeat another cycle, in which they are deformed,scissored, and reconnected sequentially [29].

In Fig. 3(a), we plot the temporal changes in the aver-aged orientational order along the z axis hQzzi. When theforce is weak (fz & 0:004), the orientational order is almost

fixed at the same value as that at rest. Subject to an inter-mediate force (0:006 & fz & 0:008), the orientational ordershows an oscillation mode, which corresponds tothe repeated cycle of the type-I defects shown in Figs. 2(a)and 2(b). For strong force (fz * 0:010), two oscillationmodes are observed in hQzzi. The mode of the shorter periodis the same as that for fz ¼ 0:006; on the other hand, that ofthe longer period corresponds to the cycle for the type-IIdefects shown in Fig. 2(c). The frequencies of the twooscillation modes are shown in Fig. 3(b). Both frequenciesdecreased with decreasing Er. It is likely that they showcritical behaviors as ! / jEr� Ercjk with k ¼ 1:0. Thecritical values of Er are estimated to be ErIc ffi 26 for thetype-I defects and ErIIc ffi 64 for type-II defects by extrap-olating straight lines. However, the true critical points arenot accessible because of our simulation’s constraints.Therefore, we cannot yet conclude whether these bifurca-tions are continuous or not, and determine the exponent k.Figure 3(c) shows the temporal changes in the flow

speeds huzi averaged inside the porous matrix. Above thecritical strength for the cyclic defect behavior, the flowspeed exhibits oscillation, the period of which coincideswith that of hQzzi. In Fig. 3(d), we plot the apparentviscosity �z, which is obtained by averaging it over1000 � t � 2000. Here, the viscosity is normalized bythat of an isotropic liquid �iso, which is determined bythe same simulation method. Below the critical force, the

FIG. 2 (color). Time evolutions of (a) the defect pattern and(b) Qzx at an x-z plane under fz ¼ 0:006. In (b), the black areasare the solid matrix, and white crosses indicate the defectpositions. At t ¼ 270, the anchoring condition is broken.(c) Time evolution of the defect pattern under fz ¼ 0:013. Thetype-II (blue) defects also exhibit cyclic behavior.

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apparent viscosity decreases marginally with Er. That is,the NLC exhibits a shear-thinning rheology. At the twocritical values of Er for the onset of the oscillations, smallkinks in the apparent viscosity are observed, indicating thatthe defect pattern influences the flow properties.

Next, we impose an external force toward the x direc-tion. Under a weak force, all the defects are pinned at theiroriginal positions, showing marginal elongations along theflow (not shown here). Above a certain force, we found thatthe defect pattern switched to an orientation along the flowdirection. The switching process of the defect pattern underfx ¼ 0:005, which corresponds to Er ffi 42, is shown inFig. 4. Once the orientation is switched, the new defectpattern is sustained owing to its multistability even after theflow is stopped. The apparent viscosity for the x-orientedflow, �x, is also plotted in Fig. 3(d). Below the criticalEricksen number for switching, the apparent viscosity alongthe x axis is larger than �z. At the critical Ericksen number(Erc? ffi 25), �x decreases abruptly to �z. After switching,

the system exhibits the same oscillations as those for theparallel flow in Fig. 2.Note that the oscillation frequencies and the correspond-

ing Ericksen number depend on the anchoring strengthbecause the anchoring breaking plays a crucial role in thecyclic behaviors of the topological defects. Thus, the val-ues displayed are not universal for any anchoring strengthand pore size. Moreover, qualitatively distinct behaviorsmay be observed in extreme cases. For instance, defectsmay be transported unboundedly without the cyclic motionsif the anchoring strength is infinitely large, although oursimulations cannot access this regime. Because the Ericksennumber does not factor in the anchoring effect, it cannotdescribe all the flow properties. If the morphology of theporous matrix is the same, the director pattern is character-ized by a nondimensional parameter W ¼ wL=K in theabsence of flow. Two dimensional mapping of the nonequi-librium behaviors onto a W-Er plane will help to under-stand the flow properties of NLCs in confined systems. Weneed to study them more quantitatively.We studied the flow behaviors of a NLC in a porous

medium by lattice Boltzmann simulations. In the case ofstrong anchoring, the pattern of the director field can adoptmany (meta-)stable configurations. We found that thismultistability leads to peculiar flow properties of the con-fined NLCs. The dynamic behaviors of topological defectsunder flow depend on whether they are topologically con-catenated to the matrix.We also found that the apparent viscosity depends on the

mutual direction of the flow; i.e., the NLC flows easily alongthe orientation of the averaged director field. Imposing anintermediate flow along an incommensurate directionswitches the global director field to the new direction.Upon this switching, the apparent viscosity decreasesabruptly to that for the commensurate flow. Because thisdefect pattern is maintained even after the flow is stopped,we can say the flow path can be recorded in the porousnetwork. As reported in a previous paper, the emergence oftopological defects enhance the efficiency of memorization[21]. We also expect that the flow path in a porous matrixcan be dynamically selected by changing the multistabledefect patterns. In our simulations, the difference betweenthe apparent viscosities of the parallel and perpendicularflows is not large compared to that in Ref. [26]. In otherwords, our system is more isotropic in viscosity. Thissmall viscosity difference is attributed to the employedparameters, which are restricted in our present scheme.When a NLC having a large viscosity anisotropy is used,the difference in the apparent viscosity will be moreremarkable.We employed a bicontinuousmatrix with cubic symmetry

as an example. Because the configuration of topologicaldefects depends on the morphology of the network, ourresults cannot be directly applied to any matrices.However, we consider that dynamic coupling between the

0

t

(a)

fz=0.002ff

z

z

=0.004=0.006

fz=0.008fz=0.010

0 1000 2000 0 50 1000

0.01

0.02

Er

(b)

type-II( 01 )

1.04(d)

0 50 1001

1.01

1.02

1.03

osi/

Er0 1000 2000

0

0.005

0.010

0.015

t

(c)

fz=0.004

=0.006fz

fz=0.002

fz=0.008

fz=0.010

zu

0.1

0.2

0.3b

/Q

Qz z

type-I

I IcEr

IcEr

parallel flowperpendicular flow

switching

type-II

type-I

z

x

FIG. 3. (a) Temporal changes in the averaged remnant orderhQzzi. (b) Frequencies of the cyclic motions of the type-I andtype-II defects as a function of the Ericksen number. Brokenlines are provided for better visualization. (c) Temporal changesin the flow speed huzi of the liquid crystal in the porous matrix.The force density is increased from fz ¼ 0:002 to 0.01.(d) Effective viscosities for the parallel and perpendicular flowsplotted with respect to Er. The viscosity is normalized by that ofan isotropic liquid �iso. At the arrows, the defect patterns arereorganized.

FIG. 4 (color). Switching behaviors of topological defectsunder a perpendicular flow. A force fx ¼ 0:005 is imposed forthe defect pattern aligned along the z axis.

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multiple stabilities and the flow will be found ubiquitouslyfor NLCs confined in any porous network. We also hope ourresults will improve the understandings of the flow proper-ties of NLCs in porous media and microfluidic channels.

We acknowledge valuable discussions with H. Tanaka, T.Bellini, F. Serra, M. Buscaglia, and A. Sengupta. This workwas supported by the JSPS Core-to-Core Program‘‘International Research Network for NonequilibriumDynamics of Soft Matter’’ and KAKENHI (Nos. 24540433and 23244088). The computational work was performedusing the facilities at the Supercomputer Center, Institutefor Solid State Physics, University of Tokyo.

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[29] See Supplemental Material http://link.aps.org/supplemental/10.1103/PhysRevLett.109.257801 for mov-ies corresponding to Fig. 2.

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