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.102.028301 PACS 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 , examples being
Cadmium-Selen  and Europium nanorods . 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  and the design of mate-
rials with novel rheological features. Moreover, the case of
polar rods also relates to polar active biomatter such as
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
huui and the local dipole mo-
?aþ ?dþ ?adþ?2
2ðrdÞ:ðrdÞ ? cfðdrÞ:a;
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 . 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-
approximation) . The last term corresponds to the
PRL 102, 028301 (2009)
PHYSICAL REVIEW LETTERS
16 JANUARY 2009
? 2009 The American Physical Society
lowest-order bilinear coupling between a and d involving
spatial derivatives. With cf> 0 this term is similar to a
flexoelectric coupling , 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
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 . In the present work we
somewhat simplify the resulting dynamic equations for a
(see, e.g., ) and d assuming that bilinear couplings
between order parameters and deformations of v and that
order parameter fluxes can be neglected . 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
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:
p ¼ ?2?isorvþ
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
ð@tþ v?? rÞv?¼ ???1r ? p?;
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  are set such that the bulk state is nematic and
nonferroelectric and displays a kayaking-tumbling motion
under shear . 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-
?K??? c~ W?1d?d?? "fard?;
d?d?þ ca?? d?
d?d?? "fdr ? a?;
?Kð?a?~ W~ E?1?a?
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  for the space discretization and an
adaptive fourth-order Runge-Kutta-Fehlberg algorithm for
time integration (see  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) . 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
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
and d ¼ jdj (assuming rods
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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16 JANUARY 2009
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 . 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
?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
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,
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
?h_PzðxÞðy;tÞdy (see  for a derivation) with P ¼
FIG. 2 (color online).
selected time units (a) and (b) correspond to defect creation (see
Snapshots of vðyÞ and dðyÞ (arrows). The
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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16 JANUARY 2009
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 .
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 ), 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
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 . 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 ). 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  (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 .
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 . This would be similar to the electrokinetic
effect used to investigate surface potentials and to con-
struct microchannel batteries .
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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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) .
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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16 JANUARY 2009