# Electric field driven controllable motility of metal-dielectric Janus
particles with boojum defects in a nematic liquid crystal

Dinesh Kumar Sahu, and Surajit Dhara∗ School of Physics, University of
Hyderabad, Hyderabad 500 046, India

###### Abstract

In a sharp contrast to the response of silica particles we show that the
metal-dielectric Janus particles with boojum defects in a nematic liquid
crystal are self-propelled under the action of an electric field applied
perpendicular to the director. The particles can be transported along any
direction in the plane of the sample by selecting the appropriate orientation
of the Janus vector with respect to the director. The direction of motion of
the particles is controllable by varying the field amplitude and frequency.
The command demonstrated on the motility of the particles is promising for
tunable transport and microrobotic applications.

Transport of tiny particles in fluids by electric field, the so-called
electrophoresis ramos , is an active area of research in soft matter science
and widely used in segregating macromolecules drop , colloidal-assembly and
transport stv1 ; div2 , biomedical morgan ; drop , microfluidic alex  and
electrophoretic display devices com ; rc . Active control and piloting of the
particles in the suspension is of crucial importance for developing new
technologies. In linear electrophoresis, the ions in the electrical double
layer drag the fluid (electro-osmosis) as a result the freely suspended
charged particles move along the applied field direction with a velocity
proportional to the field (${\bf E}$). In nonlinear electrophoresis, known as
induced-charge electroosmosis (ICEO), the applied field creates a double layer
and drags the fluid, setting a flow pattern of quadrupolar symmetry
surrounding the particles tm1 ; baz1 ; baz2 ; baz3 ; tm . The quadrupolar
symmetry of the ICEO flows is broken for polarisable particles with nonuniform
surface properties or polarity in the shape, consequently the particles move
with a velocity proportional to $E^{2}$baz3 . However, the transport of
particles by the electric field in either type is unidirectional, without
having any navigational capabilities sum .

Figure 1: (a,b) Schematic diagram of a metal-dielectric Janus particle. MAP
coating is indicated by magenta strands. (c) Optical micrograph of the Janus
particles. Darker (brighter) region represents the metal (dielectric). (d)
Director field surrounding a Janus particle with Boojum defects. (e) Diagram
of a cell made of parallel electrodes. (f) Polarising optical micrograph of a
Janus Boojum-particle. (g) Micrograph with a full-wave plate (530 nm). Double
headed arrows represent director $\bf{\hat{n}}$. Scale bar: 3 $\upmu$m. Figure
2:  (a) Janus vector ${\bf\hat{s}}$ is normal to the metal-dielectric
interface. $\varphi$ is the angle between ${\bf\hat{s}}$ and the far field
director ${\bf\hat{n}}$. (b) Particles with selected $\varphi$ chosen for the
experiments. (c) Time coded trajectories in the $xy$-plane, labelled from I to
VIII, under ac electric field (2.38 V$\upmu$m-1, 30 Hz). Trajectories of
selected particles are grouped. Pink arrows denote the direction of motions.
(Movie S1 sup ). The direction of electric field E (along $z$-direction) and
the director ${\bf\hat{n}}$ for all the trajectories are shown at the centre.
(Tmin=0 s to Tmax=5 s)

The origin of electrophoretic motion of colloidal particles in nematic liquid
crystals (NLCs) is quite different from that of the isotropic counter part od
; od1 ; oleg ; oleg1 ; oleg2 ; id . In NLCs, the particles with perpendicular
anchoring of director $\hat{\bf n}$ (the average direction of molecular
orientation) create elastic deformation in the medium, inducing point lub ;
stark  or equatorial disclination ring defects (so-called Saturn ring) ram ;
abot . Such particles are known as elastic dipoles and quadrupoles,
respectively stark . When the director anchoring is planar, two antipodal
surface defects (boojums) are induced along the far-field director zuhail .
The particles with dipolar director configuration exhibit bidirectional
transport in which the fore-aft symmetry of the electro-osmotic flows is
broken due to the point defect od ; od1 . Electrophoresis of such defect-
decorated particles is nonlinear and known as LC-enabled electrophoresis
(LCEEP)oleg . The particles with quadrupolar director configuration inducing
symmetric defects such as Saturn ring or Boojums are immobile as the fore-aft
symmetry of the electroosmotic flow is retained oleg2 . Very recently Sahu et
al. have shown that Janus particles decorated with Saturn-ring defect can be
transported and navigated at will by external ac electric fields sd . Here, we
study the effect of ac electric field on Janus particles with planar anchoring
of the LC director that induces two antipodal surface defects. We show that
the Janus Boojum-particles become motile and the direction of motion depends
on the relative orientation of the metal-dielectric interface with respect to
the far-field director $\hat{\bf n}$. Further, the direction of motion of the
particles is controllable by changing the frequency and the amplitude of the
ac field.

The metal-dielectric Janus particles are synthesized by adapting the following
the procedure. To begin with, an aqueous suspension ( 2$\%$) of silica
particles (SiO2) of diameter $2a=3.0\pm 0.2$ $\upmu$m (Bangs Laboratories,
USA) are spread over a glass slide treated with Piranha solution to form a
colloidal monolayer. A layer of Titanium (Ti), thickness approximately 35 nm
is deposited on the top of the colloidal monolayer using an electron-beam
deposition technique so that only the exposed upper-hemispheres of the
particles get coated (Fig.1(a)). Further, a thinner layer (10 nm) of SiO2 is
deposited to make the overall particle’s surface chemically isotropic
(Fig.1(b)). Finally the particles are extracted from the slides through
ultrasonication in distilled water. An optical micrograph of a few Janus
particles is shown in Fig.1(c). The metal hemisphere appears dark under
transmitted light.

Next the Janus particles are coated with N-Methyl-3 aminopropyl
trimethoxysilane (MAP) to induce planar or homogeneous anchoring of the LC
molecules (Fig.1(a,b)). A small quantity of Janus particles (0.01 wt%) is
dispersed in the nematic liquid crystal (MLC-6608, Merck) that exhibits the
following phase transitions: smectic-A -30oC nematic 90oC Isotropic. At room
temperature the dielectric anisotropy is negative
($\Delta\epsilon=\epsilon_{\parallel}-\epsilon_{\perp}$=-3.3) and and the
conductivity anisotropy is positive
($\Delta\sigma=\sigma_{||}-\sigma_{\perp}\simeq 6.1\times 10^{-10}$ S/m) pra .
The subscripts $\parallel$ and $\perp$ denote the quantities measured parallel
and perpendicular to the director $\bf\hat{n}$. The colloidal mixtures are
studied in cells made of two indium-tin-oxide (ITO) coated glass plates
(Fig.1(e)). The ITO coated glass plates are pretreated with a polyimide
AL-1254 (JSR Corporation, Japan) and cured at 180∘C for 1 hour and rubbed
unidirectionally using a bench top rubbing machine (HO-IAD-BTR-01) for planar
alignment of the director. Typical cell thickness used in the experiment is in
the range of 5-7 $\upmu$m.
We used an inverted polarising optical microscope (Nikon Ti-U) with water
immersion objective (Nikon, NIR Apo 60/1.0) for the experiments. For particle
manipulation, a dynamic laser tweezer with a cw solid-state laser operating at
1064 nm (Aresis, Tweez 250si) was used. A camera (iDs-UI) attached to the
microscope is used for recording the particle trajectories at the frame rate
of 50-60 per second. The position of the particle is tracked off-line using a
computer program with an accuracy of $\pm$10 nm.
The MAP coated Janus particles in nematic LC induce two antipodal point
defects (Boojums) as shown in Fig.1(d). Figure 1(f) shows the polarising
optical micrograph of a Janus Boojum-particle in the nematic LC (MLC-6608).
The surface anchoring of the LC molecules and the resulting elastic
deformation surrounding the Janus particle are confirmed from the texture
obtained by inserting a full-wave plate or $\lambda$-plate (530 nm) as shown
in Fig.1(g). The magenta colour corresponds to the parallel orientation of the
long axis of the molecules in the rubbing direction (e.g., the far field
director ${\bf\hat{n}}$), whereas the bluish and yellowish colours surrounding
the particle correspond to the anticlock-wise and clockwise rotation of the
director with respect to the rubbing direction zuhail .

When no electric field is applied the Janus vectors ${\bf\hat{s}}$ of the
particles (normal to the plane of the metal-dielectric interface, see
Fig.2(a)) are oriented in arbitrary directions. When the electric field is
switched on the particles reorient such that the metal-dielectric interface
becomes parallel to E to minimise the torque due to the induced dipole moment
sd ; pra . The applied field, however does not influence the macroscopic
director except near the particles as the dielectric anisotropy of the sample
is negative oleg1 . In Fig.2(b) we present orientation of the Janus vectors of
a few selected particles, which we investigated. As the field is increased
beyond a particular value the particles start moving along some specific
directions in the plane of the sample, depending on the orientation of the
Janus vector ${\bf\hat{s}}$ with respect to the director. Real-time
trajectories of the particles at a fixed field are grouped (I-VIII) and shown
in Fig.2(c). It is noted that for all trajectories the particles are moving,
facing the the metal hemisphere [Movie S1 sup ]. This is in contrast to the
motion of the quadrupolar particles in which the motion is not restricted to
metal facing. With the help of the laser tweezer, the angle $\varphi$ between
the Janus vector ${\bf\hat{s}}$ and the director $\bf\hat{n}$ can be changed
by photothermal quenching of the surrounding director field and the particle
can thus be transported in any predetermined direction as desired.

Figure 3:  Director field and proposed electroosmotic flows surrounding a
Janus Boojum-particle in two orthogonal planes parallel to the electric field
E. The large curved arrows on the metal hemisphere indicate stronger flows.
The off-centered force dipoles $\bm{\mathcal{F}}$ (shown in green colour) are
pushers in both the planes. The propulsion directions of the particles are
indicated by pink arrow.

The motility of the Janus particles with Saturn-ring defect was accounted for
theoretically from the electrostatic force density induced by the field which
drags the fluid in the vicinity of the particles sd . The force depends on the
field strength, dielectric anisotropy, conductivity anisotropy and the
curvature of the director around the particles. Within a linearised theory,
for particles with homeotropic boundary condition and assuming the deviation
of director from the mean alignment is small, it was shown that the resulting
force dipole was either a puller or a pusher depending on the orientation of
the Janus vector. However, for particles with planar anchoring the director
deviation near the defects is not small and the force density equation can not
be simplified analytically. A numerical solution of the complete nonlinear
theory will be required for getting quantitative values of the force dipoles,
which is beyond the scope of the present work. It is observed in Fig.2(c) that
the particles move with the metal hemisphere forward irrespective of the
orientation of the Janus vector in relation to the nematic axis. We conjecture
that the effective force dipole is pusher type with respect to the direction
of the electric field. Based on this assumption, we propose possible flow
directions in two orthogonal planes as shown in Fig 3. Since the metal
hemisphere is highly polarised the flow is stronger on that side relative to
the dielectric hemisphere resulting in an off-centred force dipole as shown by
the green arrows in Fig 3.

Figure 4: (a) Time-coded trajectories of Janus particles at different field
amplitudes and at a fixed frequency (30 Hz) for a fixed orientation
($\varphi=45^{\circ}$) of the Janus vector. Tmax$=14.5$ s and Tmin$=0$ s.
$\theta$ is the angle of the trajectories with respect to the director. (b)
Velocity components $V_{x}$ and $V_{y}$ at different fields obtained from (a).
Slopes of $V_{x}$ and $V_{y}$ are 1.85 and 1.01 $\upmu$m3 s-1 V-2
respectively.  Figure 5: (a) Time coded trajectories of a particle at
different frequencies and at a fixed amplitude (2.38 V$\upmu$m-1) for a fixed
orientation ($\varphi=250^{\circ}$) of the Janus vector. Tmax$=10$ s and
Tmin$=0$ s. $\theta$ is the angle between the far field director and the
trajectory. (b) Velocity components $V_{x}$ and $V_{y}$ at different
frequencies at $E=2.38$ V$\upmu$m-1. Red and black lines show theoretical fit
to equation (1). Fit parameters are: $\tau_{e}=0.05$ s, $\tau_{p}=0.014$ s for
$V_{x}$ and $\tau_{e}=0.045$ s, $\tau_{p}=0.008$ s for $V_{y}$. Error bars
represent the standard deviation of the mean value.

For a fixed orientation of the Janus vector the trajectory also depends on the
amplitude and frequency of the applied electric field. Figure 4(a) shows that
for a fixed orientation of the Janus vector, the trajectory depends on the
amplitude of the field. In particular, for the Janus vector
$\varphi=45^{\circ}$, the angle ($\theta$) between the trajectory and
$\bf{\hat{n}}$ at a fixed frequency decreases with increasing field amplitude
from 2.29 V$\upmu$m-1 to 2.47 V$\upmu$m-1. This effect can be explained from
the relative variation of the velocity components $V_{x}$ and $V_{y}$ as shown
in Fig.4(b). As expected both $V_{x}$ and $V_{y}$ increases linearly with the
squared of the field but $V_{x}$ increases at a much faster rate than $V_{y}$.
When the field amplitude is increased, the relative enhancements of the
velocity components are unequal, namely $\Delta V_{x}$ is larger than $\Delta
V_{y}$, and as a result the angle between the director $\bf{\hat{n}}$ and the
trajectory, $\theta=\tan^{-1}(\Delta V_{y}/\Delta V_{x})$ decreases.

For a particle with fixed orientation of the Janus vector the trajectory also
depends on the frequency as shown in Fig.5(a). For example, when the frequency
of the field is increased from 30 to 90 Hz, the angle between the trajectory
of the particle and the director ($\theta$) is increased. This effect can be
explained from the relative variation of the velocity components $V_{x}$ and
$V_{y}$ with frequency ($f$) as shown in Fig.5(b). The motion of the particles
is limited within a narrow range of frequency (30 - 170 Hz). The frequency
dependence of both the components are fitted to the following equation baz1 ;
oleg1

$V_{i}(\omega)={V^{o}_{i}}\frac{\omega^{2}\tau^{2}_{e}}{(1+\omega^{2}\tau^{2}_{p})(1+\omega^{2}\tau^{2}_{e})}$
(1)

where $i=x,y$; $\omega=2\pi f$ is the angular frequency of the applied field,
$\tau_{e}$ and $\tau_{p}$ are the characteristic electrode and particle
charging time, respectively. In the low frequency region both the components
increase as $f^{2}$ but decrease as $1/f^{2}$ in the high frequency region.
Beyond the peak frequency (47Hz), the velocity decreases as $1/f^{2}$ but the
coefficient of decrease of $V_{x}$ is larger than that of $V_{y}$ as a result
the angle $\theta$ increases with the increasing frequency. Similar effect was
observed in case of quadrupolar particles at slightly different field
amplitude and frequency sd .

Figure 6: (a) Change of direction of motion of a Janus particle with a fixed
orientation ($\varphi=45^{\circ}$) by altering the field amplitude recursively
between 2.29 V$\upmu$m-1 (black) to 2.38 V$\upmu$m-1 (red), keeping the
frequency constant at 30 Hz (see Movie S2 sup ). (b) Change of direction of
motion of a janus particle ($\varphi=315^{\circ}$) by altering the frequency
recursively between 30 Hz (Black) to 60 Hz (Red) at a fixed field 2.38
V$\upmu$m-1. (see Movie S3 sup ).

The effect of the changing field amplitude and frequency is exploited for
navigating the direction of motion of the particles. Figure 6(a) demonstrates
the command over the trajectory by varying the field amplitude recursively
between two slightly different amplitudes, namely 2.29 and 2.38 V$\upmu$m-1.
For example, when the field amplitude is increased from 2.29 to 2.38
V$\upmu$m-1, the particle changes it’s direction of motion with respect to the
initial direction by nearly 37∘ (Fig.6(a)). Figure 6(b) demonstrates the
navigation capability by changing the frequency recursively between 30 and 60
Hz. For example, the change of angle in it’s direction of motion due to the
change of frequency from 30 to 60 Hz is about 50∘. Thus, by changing either
the amplitude or the frequency of the field or both, we can navigate the
particles as desired. It may be mentioned that the effect of inertia is
negligibly small and the director configuration around the particle remains
unaffected as the Reynolds and the Ericksen numbers both are much less than 1
sd .

In conclusion we have studied the effect of ac electric field on metal-
dielectric Janus particles inducing boojum defects. Although the director
deformation induced by the particles is symmetric, the surface asymmetry of
the particles break the fore-aft symmetry of the surrounding electroosmotic
flows facilitating their motility. In all the trajectories the particles move
facing the metal hemisphere and their direction of motion is dictated by the
relative orientation of the Janus vector. The direction of motion of the
particles is manoeuvrable by changing the amplitude and frequency of the
field. Such a spectacular control originates from the relative sensitivity of
the electroosmotic velocities to the asymmetric surface and local director
structure. Our work deals with the spherical particles however there are
several new shape asymmetric particles whose Janus character under applied
field is unexplored. Further our study is limited to dilute concentration of
particles. It is expected that studies on the motility at higher concentration
and their collective dynamics will be interesting saha1 ; saha2 .

ACKNOWLEDGEMENTS: S.D. thanks Steve Granick for hosting his visit to IBS,
UNIST which resulted in very useful discussions. We thank K.V. Raman for help
in preparing Janus particles This work is supported by the DST, Govt. of India
(DST/SJF/PSA-02/2014-2015). S.D. acknowledges a Swarnajayanti Fellowship and
D.K.S. an INSPIRE Fellowship from the DST. We thank Swapnil Kole and Sriram
Ramaswamy from Indian Institute of Science, Bangalore for very useful
discussion.

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