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Shear-Induced Dynamic Polarization and Mesoscopic Structure in Suspensions

of Polar Nanorods

Sebastian Heidenreich,1Siegfried Hess,1and Sabine H.L. Klapp2

1Institut fu ¨r Theoretische Physik, Technische Universita ¨t Berlin, Hardenbergstrasse 36, D-10623 Berlin, Germany

2Institut fu ¨r Theoretische Physik, Fachbereich Physik, Freie Universita ¨t Berlin, Arnimallee 14, 14195 Berlin, Germany

(Received 31 July 2008; published 13 January 2009)

We investigate the spatiotemporal behavior of sheared suspensions of rodlike particles with permanent

dipole moments. Our calculations are based on a self-consistent hydrodynamic model including feedback

effects between orientational motion and velocity profile. The competition between shear-induced

tumbling motion and the boundary conditions imposed by plates leads to oscillatory alignment structures.

These give rise to a spontaneous time-dependent polarization generating, in turn, magnetic fields. This

novel shear-induced effect is robust against varying the boundary conditions. The field strengths are of a

measurable magnitude for a broad parameter range.

DOI: 10.1103/PhysRevLett.102.028301PACS numbers: 82.70.Dd, 47.20.?k, 47.50.?d, 47.54.?r

Complex fluids driven by shear flow display fascinating

nonequilibrium phenomena such as tumbling behavior

[1,2], rheochaos [3,4], and shear banding [5,6]. Examples

of such systems are liquid crystals, nematic polymers,

wormlike micelles, or suspensions of nanorods. Recently,

complex flow behavior has also been detected in active

biomaterials (such as bacterial swimmers and actomyosin

solutions) which can exhibit spontaneous flow even in the

absence of an external force [7–9].

In the present Letter, we consider the flow behavior of a

passive yet novel class of complex fluids composed of

nanorods with permanent dipole moments. While electric

molecular dipole moments are a rather generic feature of

liquid-crystalline molecules, there are nowadays a variety

of colloidal (i.e., nanosized) rods with magnetic or electric

dipole moments. Indeed, only in the last decade were the

experimental techniques to synthesize functionalized

nanorods developed and refined [10], examples being

Cadmium-Selen [11] and Europium nanorods [12]. As a

consequence, properties like shape, aspect ratio, and

strength of the dipole moment are now well controllable

[11–13]. Thus, investigating the flow behavior of such

substances is timely, especially in view of their possible

applications in microfluidics [14] and the design of mate-

rials with novel rheological features. Moreover, the case of

polar rods also relates to polar active biomatter such as

actomyosin [15].

Clearly, for sheared systems of polar particles the dy-

namics of the average dipole moment is an additional issue

on top of the director dynamics and the coupled flow. A

particularly intriguing aspect is whether flow can induce

spontaneous polarization, that is, a nonvanishing average

dipole moment, in a system which would be nonferro-

electric without flow. In the present Letter, we show, for

the first time to our knowledge, that spontaneous polariza-

tion can indeed occur in sheared nanorod suspensions with

mesoscopic spatial structure. We focus on systems where

the structure is induced by the combination of shear flow

and confining walls, such as in microchannels. Our inves-

tigations are based on a self-consistent hydrodynamic

model including feedback of the orientational motion on

the flow. This model generalizes earlier approaches for

homogeneous (bulk) systems of polar rods [16,17], where

a nonvanishing average dipole moment appears only if the

equilibrium state is ferroelectric. On the contrary, the

structured systems develop spontaneous, time-dependent

polarization for a wide range of parameters and boundary

conditions. Moreover, for electric dipoles the resulting

magnetic fields are of a measurable order of magnitude.

We employ a mesoscopic description of the system,

where its orientational state is characterized by the local

alignment tensor a ¼

ment d ¼ hei. Here, u and e are unit vectors describing the

molecular axis and the dipole moment, respectively, and

... indicates the symmetric traceless part of a tensor. The

system’s equilibrium behavior is governed by the free

energy density

?

þ?2

ffiffiffiffiffiffiffiffiffiffiffi

15=2

p

huui and the local dipole mo-

gðd;aÞ ¼kBT

m

?aþ ?dþ ?adþ?2

a

2ðraÞ:ðraÞ

?

d

2ðrdÞ:ðrdÞ ? cfðdrÞ:a;

(1)

where ?a¼ ?aðaÞ is a modified Landau–de Gennes po-

tential of a homogeneous nematic [18,19], and ?d¼

?dðdÞ is a purely dipole contribution describing a non-

ferroelectric system [16,17]. The coupling ?ad¼ c0d ? a ?

d was motivated in [17]. For rods with longitudinal dipole

moments, c0< 0 such that polarization parallel to the

nematic director is preferred. The two subsequent terms

in Eq. (1) measure the free energy increase due to inho-

mogeneities in a and d, with ?aand ?dbeing the corre-

spondingelasticitylengths

approximation) [18]. The last term corresponds to the

(inthe one-constant

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? 2009 The American Physical Society

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lowest-order bilinear coupling between a and d involving

spatial derivatives. With cf> 0 this term is similar to a

flexoelectric coupling [20], where the commonly appear-

ing external electric field is replaced by d. The equilibrium

and shear-induced behavior of the homogeneous system

defined by Eq. (1) with ra ¼ rd ¼ 0 was explored in

[17].

Tomodel the inhomogeneoussituation we introducetwo

infinitely extended plates located in the xz plane and

separated by a distance 2h along the y direction. We apply

a plane Couette flow by moving one plate with speed vw

relative to the other, yielding a velocity profile v ¼

ðvðyÞ;0;0Þt. For a full hydrodynamic description of the

fluid in between the plates we use the framework of irre-

versible thermodynamics [21]. In the present work we

somewhat simplify the resulting dynamic equations for a

(see, e.g., [22]) and d assuming that bilinear couplings

between order parameters and deformations of v and that

order parameter fluxes can be neglected [22]. The relaxa-

tion of the order parameters back to equilibrium is gov-

erned by the derivatives of the free energy with respect to a

and d, respectively, involving relaxation times ?a? 0 and

?d? 0.

In a realistic system the velocity profile is not an exter-

nally imposed quantity but is affected by the orientational

dynamics. This interplay is taken into account via the

constitutive equation for the pressure tensor:

ffiffiffi

?a

p ¼ ?2?isorvþ

2

p

pk?pa

?

?a??2

a?aþcfrdþc0

2dd

?

;

(2)

where pk¼ ?kBT=m is the kinetic pressure and ?a¼

@?a=@a. Combining (2) with the momentum conservation

and the equations of motion for a and d, we obtain

ð@tþ v?? rÞa?¼ 2??? a?þ~ E?1?a??~ W?1?a?

?

ð@tþ v?? rÞd?¼ 2??? d??~ W?1

?~ E?1

ð@tþ v?? rÞv?¼ ???1r ? p?;

p?¼ ?2?isorv??

þ~ W"fard?þ cd?d?=2Þ:

The asterisks denote scaled dynamic variables [17,22], ??

and ??are the vorticity and distortion of v, respectively,

and ?d¼ @?d=@d. The parameters appearing also in bulk

systems [23] are set such that the bulk state is nematic and

nonferroelectric and displays a kayaking-tumbling motion

under shear [17]. Here we focus on the role of ‘‘surface’’

parameters such as the inverse Eriksen number ~ E?1/

ð?a=2hÞ2controlling the gradient terms and the ‘‘flexo-

ffiffiffiffiffiffiffiffi

3=2

p

?K??? c~ W?1d?d?? "fard?;

d?d?þ ca?? d?

d?d?? "fdr ? a?;

ffiffiffiffiffiffiffiffi

3=2

p

?Kð?a?~ W~ E?1?a?

(3)

electric’’ coupling parameters "fa¼ cf=?aq (with q ¼

? ?vwpeljdj) and "fd¼ cf=?dq.

Equation (3) forms our self-consistent hydrodynamic

model. To solve the nine-dimensional coupled system of

partial differential equations for the five and three compo-

nents of a and d, respectively, and for vðyÞ, we use a finite-

difference scheme [24] for the space discretization and an

adaptive fourth-order Runge-Kutta-Fehlberg algorithm for

time integration (see [22] for details). We employ ‘‘no-

slip’’ boundary conditions for vðyÞ and strong anchoring

conditions such that aðy ¼ ?hÞ is fixed. Specifically, we

consider ‘‘planar’’ anchoring (uniaxial alignment within

the plate plane), ‘‘homeotropic’’ (alignment normal to the

plates), ‘‘isotropic’’ (modeling rough surfaces), and ‘‘de-

generate’’ (particle axes equally distributed in the plate

plane) [25]. In all cases we set the polarization at the walls

to be zero. The dynamic variables between the plates are

initialized according to the boundary conditions. The equi-

librium behavior is obtained by solving Eq. (3) with zero

plate speed. The resulting states are characterized by

aeqðyÞ ? 0 and deqðyÞ ¼ 0 throughout the confined system,

consistent with the nonferroelectric nematic bulk phase

[17,23].

After a relaxation time of 10 time units (tref¼ 2h=vw)

we switch on the flow. Typical stationary solutions for the

scalar variables a ¼

with longitudinal dipoles) as functions of space and time

forplanar anchoring conditionsare displayed inFig. 1. The

alignment aðy;tÞ shows a pulsating behavior characterized

by periodically occurring, mesoscopic structures (gra-

dients). The latter correspond to regions of planar biaxial

ordering, that is, platelike ‘‘defects,’’ as an analysis of the

local director reveals. Their creation is due to a mismatch

between the wagging motion close to the plates induced by

the strong anchoring and the tumbling motion of the direc-

tor in the middle of the plates (which still differs from the

kayaking-tumbling motion expected without plates).

Similar defectlike structures are found in systems of non-

polar nanorods [22,26]. The novel feature occurring for

polar rods is that the pulsating structures of the

alignment give rise to a nonzero, time-dependent dipole

moment dðy;tÞ. Its spatiotemporal behavior is illustrated in

Fig. 1 (right), revealing similar pulsating defects as those

observed in aðy;tÞ. Since, by construction, d ¼ 0 at the

ffiffiffiffiffiffiffi

a:a

p

and d ¼ jdj (assuming rods

FIG. 1.

alignment and (right) the average dipole moment at "fa¼ 10,

"fd¼ 10, ~ E ¼ 100, ?iso¼ 0:1,

circle indicates a platelike defect.

(Left) Shear-induced spatiotemporal behavior of the

~ Wd¼ 2, ? ¼ 1. The white

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boundaries (y ¼ ?h) and in the bulk, we conclude that the

dynamic polarization (or magnetization, respectively) ob-

served for jyj < h is indeed spontaneous.

The relation between the orientational structure and the

flow profile at some selected time units is illustrated in

Fig. 2. In vðyÞ one observes the formation of ‘‘jets,’’ i.e.,

spurtlike deviations from the ordinarylinear flow profile, at

specific time units correspondingto defects in aðy;tÞ. Their

occurrence is a consequence of the feedback effect in our

model [22]. We see from Fig. 2 that the jets serve as a

‘‘trigger’’ of the local dipole moment dðyÞ, which becomes

particularly large within the spurts.

The spontaneous, time-dependent polarization is not

restricted to the planar anchoring conditions considered

so far. This is illustrated by the plots of dðy;tÞ in Fig. 3.

For all boundary conditions considered, one observes re-

gions of large polarization. However, the details of the

pattern strongly depend on the anchoring. In the homeo-

tropic case [Fig. 3(b)] the defects occur mainly in the

middle rather than close to the walls as in the planar case

[Fig. 3(a)]. For isotropic anchoring (rough surfaces), par-

ticularly high peaks of dðy;tÞ occur. We also see from

Fig. 3 that an increase of~ E supports the creation of defects

due the weakening of gradient terms in the free energy.

According to Maxwell’s equations the shear-induced

time-dependent polarization (magnetization) generates

magnetic (electric) fields. These may be used as an experi-

mentally accessible measure of the predicted effects.

Focussing on electric dipoles, the relevant components of

the system-averaged magnetic field BðtÞ are BxðzÞðtÞ ¼

??0

?peld. Considering the time average ?Bzof BzðtÞ over a

suitable period ? we can systematically study the role of

the relevant model parameters. In the main part of Fig. 4(a)

we have plotted?Bzas function of the Eriksen number~ E.

Increasing~ E from small values the Laplace term in the first

member of Eq. (3) becomes less and less important, allow-

ing a larger number and more pronounced defects in a (and

thus, in d) to occur, and?Bzincreases. Only for very large~ E

does the field saturate because the defect structures ‘‘smear

out.’’ From the inset of Fig. 4(a) we also see that the field

increases monotonically with cf. This is expected since, as

indicated by Fig. 1, the flexoelectric coupling between the

gradients of a and the local dipole moment in the free

energy (1) is essential for development of an overall

polarization.

The dependence of the field on the plate separation at

fixed elasticity length is shown in Fig. 4(b). For small h the

fluid structure is essentially defect-free, and thus?Bzis very

small. Increasing 2h allows for the formation of defects,

yieldingastrongincreaseof dand?Bz.Thischangesateven

larger 2h where one observes large, homogeneous regions

between thewalls. The latter induce a decrease and, finally,

a disappearance of the spatially averaged polarization and

magnetic field, consistent with the absence of spontaneous

polarization in homogenous polar nanorod systems, i.e.,

the limit h ! 1 [16,17]. We also observe from Fig. 4 that

the field depends strongly on the viscosity ?iso/ ?iso. This

is since large viscosities suppress the feedback effect [i.e.,

thesecond term inEq. (2)] and thus deviations of vðyÞfrom

the ordinary linear Couette profile. The latter are essential

for the development of a polarization as illustrated in

Fig. 2.

Rh

?h_PzðxÞðy;tÞdy (see [17] for a derivation) with P ¼

FIG. 2 (color online).

selected time units (a) and (b) correspond to defect creation (see

Fig. 1).

Snapshots of vðyÞ and dðyÞ (arrows). The

FIG. 3.

choring, (b) homeotropic, and (c) disordered (the case of degen-

erate anchoring resembles the planar case). Parameters as in

Fig. 1, except for~ E ¼ 300 (left) and~ E ¼ 800 (right).

Spatiotemporal behavior of dðy;tÞ for (a) planar an-

FIG. 4 (color online).

function of (a) the Eriksen number and the flexoelectric coeffi-

cient (inset) with vw¼ 10?2m=s and ?a¼ 10?3s [remaining

parameters as in Fig. 1 (left)]. (b)?Bzvs plate separation 2h with

vwand ?aas in (a), and remaining parameters as in Fig. 1 (right).

(a) Time-averaged magnetic field?Bzas

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Finally, the reported effects are not specific for the bulk

parameters (shear rate W, coupling parameter ?k) consid-

ered so far. This is indicated by Table I, where we present

additional results corresponding to other (than kayaking-

tumbling) dynamic bulk ‘‘states.’’ We conclude that the

shear-induced polarization in inhomogeneous suspensions

is a rather generic phenomenon occurring in large parts of

the dynamic phase diagram of the (nonpolar) bulk [17].

For realistic systems the fields are small but nevertheless

detectable. For example, using the parameters cf¼

62 nC=m (which are typical for bent-core nematic liquid

crystals such as C1Pbis10BB [27]), c ¼ 30, vw¼

10?1m=s, ?a? ?d¼ 10?4s, ? ? ¼ 1027m?3, peljdj ¼

1:61?28Cm, ?iso¼ 10 mPs, 2h ¼ 10?4m ?a¼ ?d¼

10?6m, the resulting field is of the order 10?11T, and a

similar order could be reached with larger gaps and other

materials. This order is within the detection limit of a

SQUID ½Oð10?14ÞT?.

To summarize, our hydrodynamic model for structured

suspensions of polar nanorods predicts spontaneous time-

dependent polarization. This novel flow effect is robust

against details of the wall boundary conditions. The fact

that flow is an essential ingredient yields a similarity to

shear-induced isotropic-nematic transitions [28]. In the

present case, important additional ingredients are the

strong anchoring conditions and a feedback effect.

In the present study we assumed that the mesoscopic

structure of the sheared fluid is induced by walls such as in

nanochannels and nanopores (disregarding local confine-

ments effects such as layer formation [29]). However, one

may also think of spontaneous symmetry breaking such as

banding instabilities. Indeed, the present model gives a

nonmonotonic relation between shear stress and shear

rate [21] (as does the frequently used Johnson-Segalman

model [5,6]), and thus the essential ingredient for shear

bands. Their interplay with the spontaneous polarization

observed here will be explored in the future. Moreover, by

including ‘‘activity terms’’ we can extend our equations (3)

to polar active biomaterials [15].

The polarization effect discovered here may have poten-

tial for several applications. For nanorods, measuring the

resulting magnetic or electric field would be a convenient

way to detect and control the flexoelectric feature in a

fabrication process. Furthermore, our predictions offer a

way to generate, in a controlled manner, time-dependent

fields by streaming polar nanorods through microfluidic

devices [14]. This would be similar to the electrokinetic

effect used to investigate surface potentials and to con-

struct microchannel batteries [30].

We thank Stefan Grandner for helpful discussions on

homogeneous nanorod systems. Financial support from the

Deutsche Forschungsgemeinschaft within the Sfb 448

(project B6) is gratefully acknowledged.

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Eur. Phys. J. E 24, 353 (2007).

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Crystals (Clarendon Press, Oxford, 1993).

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100, ?K¼ 1,~ W ¼ 1, and # ¼ 0.

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TABLE I.

netic fields?B at~ E ¼ 300 and parameters (W, ?K) corresponding

to bulk tumbling (T), wagging (W), flow alignment (FA),

kayaking-wagging (KW), and kayaking-tumbling (KT) [17].

Maximal polarization dmaxand presence of mag-

Bulk solutionInhomogeneous system

T (?K¼ 0:9, W ¼ 1:0)

W (?K¼ 1:0, W ¼ 5:0)

FA (?K¼ 1:3, W ¼ 5:0)

KW (?K¼ 1:2, W ¼ 4:0)

KT (?K¼ 1:0, W ¼ 1:0)

?B ? 0, dmax¼ 0:023

?B ? 0, dmax¼ 0:021

?B ¼ 0, dmax¼ 0:023

?B ¼ 0, dmax¼ 0:023

?B ? 0, dmax¼ 0:073

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