PHYSICAL REVIEW APPLIED 14, 034004 (2020)

Measuring Electric-Field-Induced Dipole Moments of Metal-Dielectric Janus
Particles in a Nematic Liquid Crystal
Dinesh Kumar Sahu and Surajit Dhara*

School of Physics, University of Hyderabad, Hyderabad 500046, India(Received 27 May 2020; revised 15 July 2020; accepted 3 August 2020; published 1 September 2020)
Janus particles are special types of nanoparticles or microparticles possessing at least two surfaces with
distinct physical or chemical properties. The most-studied Janus particles are metal-dielectric particles, inwhich half the surface of the dielectric particles is coated with a very thin layer of a metal. The external
electric ﬁeld induces a dipole moment, and consequently the particles exhibit self-assembled dynamic
structures in concentrated aqueous suspensions. Here we study metal-dielectric Janus particles in a nematic
liquid crystal under an ac electric ﬁeld and demonstrate a method for measuring the eﬀective induced
dipole moments of the particles through competition between elastic and electrostatic (Coulomb) forcesof the two particles. The calculated polarizability of the particles based on a simple model agrees well withthe eﬀective polarizability measured in the experiments. Our ﬁndings have important bearing on functional
materials based on metal-dielectric Janus particles dispersed in an anisotropic medium.
DOI: 10.1103/PhysRevApplied.14.034004

I. INTRODUCTION
In recent years, Janus particles have emerged as a newclass of colloids that ﬁnd potential applications in interdisciplinary areas ranging from physics to biology, medicine,and chemistry. For metal-dielectric Janus particles, theasymmetric surfaces behave diﬀerently under an external
electric ﬁeld, and the resulting self-assembly and dynamics are markedly diﬀerent from those of regular colloids
with a symmetric surface [1,2]. Janus particles are usedas building blocks for reconﬁgurable colloidal structuresand superstructures on diﬀerent length scales, and canpotentially be used to design new and functional materials with a complex architecture [3]. Apart from theirapplications, these particles are also important in studying nonequilibrium phenomena in soft-matter physics. The
imposed electric ﬁeld induces a dipole moment, whose
magnitude depends on the electrical properties of the suspending medium and plays a crucial role in determining
the equilibrium structures and dynamics. Although several
experimental studies have been reported, the magnitude of
the induced dipole moments of such particles has not beenmeasured experimentally so far in any dispersive medium.
Here we demonstrate a method of measuring the eﬀective dipole moments of Janus particles in a thermotropic
nematic liquid crystal (NLC).
When microparticles with normal anchoring for the
director n̂ (the average direction of molecular orientation) are dispersed in a uniformly aligned NLC,*

sdsp@uohyd.ernet.in

2331-7019/20/14(3)/034004(8)the particles induce ring [4,5] or point [6,7] defects,
resulting, respectively, in quadrupolar or dipolar elastic distortions. Unlike conventional colloids in aqueous solutions, these particles experience structural forces,
which arise due to the elastic distortions and local
variations of orientational order of the liquid crystals, and exhibit a fascinating self-assembled colloidal
structure [8–13]. The interaction potentials between twoelastic dipoles and two elastic quadrupoles are given
by UDD (r, θ ) = C1 K(1 − 3 cos3 θ )/r3 and UQQ (r, θ) =C2 K(3 − 30 cos2 θ + 35 cos4 θ )/r5 , respectively, where C1
and C2 are constants, K is the average elastic constantof the medium, r is the separation between the two particles, and θ is the angle between the vector connecting
the centers of the particles and the far-ﬁeld orientation of
the nematic director [7]. Liquid crystals are weak electrolytes, and the external electric ﬁeld drives the ions in
the direction opposite to that of the ﬁeld and generates
electro-osmotic ﬂows surrounding the particles due to nonlinear electrophoresis [14,15]. Under an imposed electric
ﬁeld, the dipolar particles become motile along the nematic
director as the fore-aft (left-right) symmetry of the electroosmotic ﬂows is broken, whereas the quadrupolar particles
by contrast exhibit no motility due to the fore-aft symmetery [14–16]. When metal-dielectric Janus particles are
suspended in NLCs, from the mechanics point of view they
are either elastic dipoles or elastic quadrupoles but display
Janus character under the application of an external electric ﬁeld. Recently, it was shown that the fore-aft symmetryof quadrupolar Janus particles is broken and the particles
pick up a direction of motion (above a certain electric ﬁeld)

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depending on the orientation of the Janus vector ŝ (normal to the metal-dielectric interface) with respect to thedirector [17].
Here we study the eﬀect of small-amplitude ac electricﬁelds on a static pair of assembled quadrupolar Janus particles. The particles experience dipolar Coulomb forces aswell as elastic forces of the medium that are directed oppositely along the line joining their centers. We demonstrate acompeting eﬀect of the two forces, which allows us to measure the eﬀective induced dipole moments of the particles.
The applicability of the technique is limited to liquid crystals but is relevant to all microscopic particles, irrespectiveof their shape.
II. EXPERIMENTAL PROCEDURES
Metal-dielectric Janus particles are prepared by directional deposition of metal onto dry silica particles (SiO2 ) of
diameter 2a = 3.0 ± 0.2 μm (Bangs Laboratories, USA)in a vacuum [3]. Approximately 2% suspension (25 μl)of silica particles is spread on a half glass side, which ispretreated with piranha solution (sulfuric acid to hydrogen
peroxide 3:1; CAUTION: highly corrosive and dangerous to skin) and dried to form a monolayer. Next, a thintitanium (Ti) layer of thickness 35 nm is deposited vertically by electron-beam deposition. On top of this, athin layer of SiO2 ﬁlm (10–15 nm) is deposited. Thenthe slides with a silica monolayer are washed thoroughlywith deionized water and isopropyl alcohol. The particles
are detached from the slides by ultrasonication in deionized water for 20 s. Furthermore, the collected particles
are sonicated for 5 min to break up any agglomerationof the Janus particles. The surface of the Janus particles is coated with N ,N -dimethyl-N -octadecyl-3 aminopropyltrimethoxysilyl chloride (DMOAP) to induce a perpendicular (homeotropic) orientation of the liquid-crystal
director [18–20]. A schematic representation of side andcross-section views of a Janus particle with successive
coatings is shown in Fig. 1. A small quantity (0.001
wt %) of DMOAP-coated Janus particles is dispersed ina nematic liquid crystal, MLC-6608 (Merck). It exhibits
the following phase transitions: smectic A below −30 ◦ C,
nematic between −30 and 90 ◦ C, isotropic above 90 ◦ C.
The dielectric anisotropy of MLC-6608 is negative ( = − ⊥ = −3.3, where  and ⊥ are the dielectric permittivities for electric ﬁeld E parallel and perpendicular
to n̂, respectively) whereas the conductivity anisotropy is
positive (σ = σ − σ⊥  6 × 10−10 Sm−1 at 100 Hz)(see Appendix B). No electroconvection is observed in theexperimental ﬁeld and frequency range.
The experimental cells are prepared with use of two
indium tin oxide– (ITO) coated glass plates. The plates
are spin-coated with polyimide AL-1254 (JSR Corporation, Japan) and cured at 1800 C for 1 h. They are thenrubbed unidirectionally with a benchtop rubbing machine

FIG. 1. Side and cross-section views of a metal-dielectric
Janus particle prepared by our coating a SiO2 microparticle witha 35-nm Ti layer. The Ti layer is coated with 10 nm of SiO2 .
Magenta represents DMOAP molecules, which are anchored tothe surface.

(HO-IAD-BTR-01) for homogeneous or planar alignmentof the nematic director n̂. The ITO plates are separated byspherical spacers of diameter 5.0 μm, making the rubbing
directions antiparallel, and then sealed with UV-curing
optical adhesive (NOA-81, Norland). A schematic diagram of a cell is shown in Fig. 2(b). The output of afunction generator (AFG 3102, Tektronix) is connected
to a voltage ampliﬁer for application of sinusoidal voltage to the cell [Fig. 2(c)]. An inverted polarizing optical
microscope (Nikon Ti-U) with a water-immersion objective (NIR Apo 60/1.0, Nikon) is used for observing theparticles [21]. A laser tweezer is built on the microscopeusing a cw solid-state laser operating at 1064 nm (Tweez
250si, Aresis) for manipulating the particles. A chargecoupled-device video camera (iDs-UI, CMOSIS) at a rateof 50–100 frames/s is used for video recording of the particle trajectory. A particle-tracking program is used oﬀ-line
to track the particles with an accuracy of ±10 nm.
III. RESULTS AND DISCUSSION
We work in the dilute regime of the concentration (0.001
wt %) and disperse the particles in a cell whose gap islarger than but close to the diameter of the particles. In thiscase the particles mostly stabilize the quadrupolar director ﬁeld as shown in Fig. 2(a). Figure 2(d) shows thelight-microscope texture of a few Janus particles placed
between crossed polarizers (P,A). The four-lobed intensity pattern of the particles, a characteristic feature ofelastic quadrupoles, is further substantiated from the texture obtained by our inserting a λ-plate (full-wave plate)[inset in Fig. 2(d)]. The texture of the particles (without
polarizers) shows that the metal hemisphere (dark half) ofparticles is oriented in diﬀerent directions, always keeping
the Saturn-rings perpendicular to the macroscopic director[Fig. 2(e)]. Once the ac electric ﬁeld is switched on along
the z axis, the particles reorient themselves [22] such thatthe plane of the metal-dielectric interface lies parallel to the034004-2

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PHYS. REV. APPLIED 14, 034004 (2020)
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FIG. 2. (a) Quadrupolar elastic distortion of nematic directorn̂ around a spherical dielectric particle. The red circle representsa Saturn-ring defect. Janus quadrupolar particles (b) without and(c) with an electric ﬁeld. Note that the metal-dielectric interface
becomes parallel to the ﬁeld direction. (d) Cross-polarized micrograph of a few quadrupolar particles without an electric ﬁeld. P
and A indicates polarizer and analyzer, respectively. The inset
shows a micrograph of a particle with a λ-plate or full-wave
plate (530 nm) inserted in-between the polarizer and the sample.
(e) Image captured by a charge-coupled-device camera. The dark
(bright) hemisphere represents the metal (dielectric).

ﬁeld as shown in Fig. 2(c). Since the dielectric anisotropyof MLC-6608 is negative, the applied electric ﬁeld does
not inﬂuence the far-ﬁeld director except very close to theparticles [15].
We focus on a pair of Janus particles that are assembled
by elastic forces of the nematic liquid crystal and study theeﬀect of the ﬁeld. At zero ﬁeld, the line joining the centers of the particles makes an angle of 57◦ with respect
to the director n̂. [4]. At this angle the particles experience an attractive elastic force. Once the electric ﬁeld
is switched on, the particles rotate such that the metaldielectric interface becomes parallel to the ﬁeld due tothe eﬀective induced electric dipole moment (peﬀ ), keeping
the mutual separation unchanged as shown in Figs. 3(a)and 3(b) (see Video 1 in Appendix D). With increasingﬁeld amplitude, the particles are displaced in the opposite
directions along the joining line of their centers as shown in
Figs. 3(c)–3(e). Above a certain ﬁeld, the particles become
free from each other’s inﬂuence and swim independently.
The center-to-center separation r as a function of theﬁeld at diﬀerent frequencies is shown in Fig. 3(f). For agiven frequency f , it is apparent that there is a threshold ﬁeld E2 beyond which r increases rapidly. Carefulobservation reveals that in the low-ﬁeld region (E < E2 )there is another threshold E1 , beyond which the separation begins to increase. For clarity, the low-ﬁeld region in
Fig. 3(f) is expanded and shown in Fig. 3(g). For example at 20 Hz, the threshold ﬁelds are E1  0.2 V μm−1(g)

FIG. 3. (a)–(e) Eﬀect of an ac ﬁeld on a pair of assembled quadrupolar Janus particles (f = 100 Hz) (see Video 1 inAppendix D). (f) Center-to-center separation r with the ﬁeld
at diﬀerent frequencies. (g) Expanded rectangular region of (f)enclosed by the dashed line. E1 and E2 are the ﬁrst and second
threshold ﬁelds, respectively, and req is the equilibrium separation in the ﬁeld range E1 ≤ E ≤ E2 . Dotted lines are drawn as a
guide for the eye. Cell thickness 5.5 μm.

and E2  0.7 V μm−1 . With increasing ﬁeld the magnitude of peﬀ and consequently the repulsive dipolar force Fdincreases, which pushes the particles apart. At E1 , the magnitude of Fd just exceeds the elastic binding force Fel , andbeyond E2 the particle’s state is controlled by the surrounding electro-osmotic ﬂows. In the ﬁeld range E1 ≤ E ≤ E2 ,the dipolar Coulomb (repulsive) and elastic (attractive)
forces are balanced as shown schematically in Fig. 4(a),resulting in a stable system in which the equilibrium separation (req ) is measured. Within this range the eﬀect of
the ﬁeld is reversible (i.e., req reduces to zero when theﬁeld is decreased to zero). Below E2 , the separation occurs

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instability is observed. This indicates that the magnitudeof the induced dipole moment in silica particles is muchsmaller than that of the Janus particles. This is expected
as the polarizability of silica is lower than that of titanium metal. As a result, the dipolar Coulomb repulsion
between the two silica particles is not suﬃciently strong
in the applied-ﬁeld range to overcome the attractive elastic
forces.
The dipolar Coulomb force Fd between two identicalinduced electric dipoles (peﬀ ) of the Janus particles isexpressed as(a)2
Fd (r) = 3peﬀ
(E)/4π 0 m r4eq (E),(b)

FIG. 4. (a) The balance of elastic and dipolar Coulomb forces.
Induced dipole moments (p) are parallel to E (out of plane).
Green lines indicate far-ﬁeld director n̂. The inset shows acharge-coupled-device image of Janus particles under an ac ﬁeld.
(b) Equilibrium separation req between a pair of Janus (half-ﬁlled
blue circles) and silica (open pink circles) particles with a ﬁeld
(f = 100 Hz).

along the line joining their centers, which makes an angleof 57◦ with respect to the far-ﬁeld director n̂, irrespectiveof the orientation of the Janus vector ŝ (normal to themetal-dielectric interface). This indicates that the induced
electro-osmotic ﬂows surrounding the particles are weakand do not aﬀect their equilibrium separation. Above E2 ,the surrounding electro-osmotic ﬂows are stronger and theparticles start swimming in diﬀerent directions in the planeof the sample, depending on the orientation of ŝ as reported
recently by Sahu et al. [17].
The electric ﬁeld also induces a dipole moment in silica particles (non-Janus particles) and hence a similareﬀect is expected to be seen in a pair of silica quadrupolar particles. To see an such eﬀect, we apply an electricﬁeld and compare the results with those for the Janus
particles. Figure 4(b) shows that the equilibrium separation between two silica particles does not change up to
the applied ﬁeld (E ≤ E2 ). Even at much higher ﬁelds theseparation remains unchanged until electrohydrodynamic(1)

where m is the relative permittivity of the medium andreq (E) is the equilibrium separation. In the ﬁeld range
E1 ≤ E ≤ E2 , the Coulomb force Fd (r) is obtained byour balancing it against the elastic force Fel (r), which is
obtained by videomicroscopy in the absence of an electric ﬁeld. To measure Fel , initially two particles are held
at a distance with the help of the laser tweezers and thenallowed to interact freely, by our switching oﬀ the laser.
The center-to-center separation r(t) between two particles
obtained from the recorded movie is shown in Fig. 5(a).
The motion of the particles in the NLC is overdamped
due to high viscosity. Consequently, the Stokes drag force(FS ) in a uniformly aligned NLC is in equilibrium withthe elastic force Fel (r), resulting in no acceleration. The
elastic force between two particles is calculated from thenumerical diﬀerentiation of the trajectory, which is given
by Fel (r) = −FS = ζi [dr(t)/dt], where ζi is the drag coeﬃcient and the subscript i refers to the motion either parallelto n̂ (i =) or perpendicular to n̂ (i =⊥) [23,24]. From thestudy of the Brownian motion of a free particle [25,26], we
determine the drag coeﬃcients using the Stokes-Einstein
relations ζ = KB T/D and ζ⊥ = KB T/D⊥ by measuring
the anisotropic diﬀusion coeﬃcients D and D⊥ at room
temperature (see Appendix A). In estimating Fel (r), we
use the average drag coeﬃcient ζ = (ζ + ζ⊥ )/2 ≈ 1.2 ×
10−6 kg s−1 . The variation of Fel (r) between two Janus particles as a function of separation is shown in Fig. 5(b). If
we substitute Fd (r) = Fel (r) in Eq. (1), the magnitude ofthe eﬀective induced dipole moment of the Janus particles
can be expressed as
peﬀ (E) = (4/3)π 0 m r4eq (E)Fel (r),(2)
where m is the average relative permittivity of the sample and can be written as m = ( +2⊥ )/3 = 6.4 (seeAppendix B). The variation of peﬀ calculated with Eq. (2)as a function of the ﬁeld amplitude (E1 ≤ E ≤ E2 ) at different frequencies is shown in Fig. 6(a). We observe a
linear variation with the ﬁeld; that is, peﬀ = αeﬀ E, whereαeﬀ is the eﬀective electric polarizability of the Janus particles. The best-ﬁt lines nearly pass through the origin.

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FIG. 5. (a) Center-to-center distance r(t) between two Janus
quadrupolar particles approaching each other due to elastic interaction in the absence of an electric ﬁeld. The inset showssnap-shots at the initial time (t = 0 s) and the ﬁnal time (t  9s). (b) Variation of elastic force Fel between two Janus particles
as a function of r measured by videomicroscopy.

This means that the dipole moment is induced only by theelectric ﬁeld and eventually it means that the separation(req ) between the two particles (below E2 ) occurs primarily
due to the dipolar Coulomb repulsion. Figure 6(b) shows
αeﬀ obtained as a ﬁt parameter at diﬀerent frequencies. It
decreases with increasing frequency as expected.
In what follows we theoretically calculate the polarizability of the Janus particles. For a spherical particle,
polarizability can be expressed as [27]αM (ω) = 4π 0 m KM (ω)a3 ,αD (ω) = 4π 0 m KD (ω)a3 ,(3)where m is the relative permittivity of the dispersing
medium, K(ω) is the complex Clausius-Mossotti (CM)factor, a is the radius of the sphere and M and D stand

FIG. 6. (a) Eﬀective induced dipole moment peﬀ with the ﬁeld
at diﬀerent frequencies. Solid lines show the least-squares ﬁt toequation peﬀ = αeﬀ E. (b) Polarizability αeﬀ at diﬀerent frequencies obtained from experiments (ﬁlled red circles) and theory(ﬁlled black circles). Dotted and dashed lines are drawn as a
guide for the eye. Error bars represent the standard deviation of
the mean value.

for “metal” and “dielectric,” respectively. The CM factors of the metallic and dielectric spheres are calculated byuse of the analytical solutions given by Ramos et al. [28]and Shilov et al. [29], respectively. The real and imaginary components of the CM factors for metal and dielectric
spheres are presented in Appendix C. The average relative
permittivity of the sample obtained from the experiments
is given by m = 6.4 (see Appendix B). We use the superposition principle in which it is assumed that the induced
dipole moment of a Janus particle is equal to the sumof half the contribution from a metal sphere and half thecontribution from a dielectric sphere, so the eﬀective theoretical polarizability of the Janus particles can be written
as αeﬀ = 12 (αM + αD ). The polarizability calculated with
Eq. (3) at diﬀerent frequencies is shown in Fig. 6(b), and isin reasonably good agreement with the experiments.

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Two further remarks are in order. First, apart from
dipolar repulsion, the hydrodynamic repulsion due to theoverlap of electro-osmotic ﬂows may also contribute to theinterparticle separation [30]. We present an approximate
estimation of the magnitudes of the two forces. The swimming force of the particles can be written as Fswim =6π ηav, which approximately sets the upper limit of theeﬀective hydrodynamic force. Taking radius a = 1.5 μm,
ﬂow viscosity η = 20 mPa s, and swimming velocity v =1 μm s−1 (below E2 ), we ﬁnd the estimated hydrodynamic
force Fswim  0.6 pN. Using Eq. (1) and taking separation r = 4 μm, m = 6.4, and peﬀ = 2 × 108 D we ﬁnd
the estimated dipolar force between two particles is given
by Fd  7 pN, which is 1 order of magnitude higher
than the upper limit of the hydrodynamic force. Therefore, the eﬀect of hydrodynamic repulsion is much smaller
compared with that of the dipolar repulsion between thetwo particles. Further reasonably good agreement between
the experimentally measured and theoretically calculated
polarizabilities based on an approximate model aﬃrms thatthe eﬀect of hydrodynamic repulsion on the equilibrium
separation between two Janus particles below E2 is negligibly small compared with the eﬀect of the dipolar Coulomb
repulsion. Second, conceptually this method could also be
used to measure the induced dipole moments of silica particles but a much-higher electric ﬁeld is required. Hence,
an appropriate liquid crystal must be chosen such that noelectroconvection is observed at the desired ﬁelds.
IV. CONCLUSION
In conclusion, we study the eﬀect of an ac electric ﬁeld
on a pair of quadrupolar Janus particles that are assembled in a nematic liquid crystal by the elastic forces of thedirector ﬁeld. The imposed electric ﬁeld induces an eﬀective dipole moment in each particle that is parallel to theﬁeld direction. The dipolar Coulomb repulsion between
two assembled Janus particles increases with the ﬁeld, andbeyond a particular ﬁeld the repulsive force exceeds theelastic binding force of the particles, as a result of which
the mutual separation increases. The competing eﬀects ofthese two oppositely directed forces within a certain ﬁeld
range allows us to measure the eﬀective induced dipole
moments of the Janus particles, which vary linearly withthe ﬁeld. We measure the eﬀective polarizability at diﬀerent frequencies. The theoretically calculated polarizabilityof the particles based on the superposition principle agrees
well with the experimental ﬁndings. Our study focuseson spherical particles in nematic liquid crystals; however,this method is applicable to all microscopic Janus particles
irrespective of their shape in a variety of liquid crystals.
ACKNOWLEDGMENTS
S.D. thanks Steve Granick for hosting his visit to theInstitute for Basic Science, Ulsan National Institute of

Science and Technology, which resulted in very useful discussions. S.D. also thanks Manasa Kandula and Jie Zhang
for useful discussions during his stay at the Institute for
Basic Science. We thank K.V. Raman for help in preparing Janus particles and Sriram Ramaswamy for useful
discussions. This work was supported by the Departmentof Science and Technology, Government of India (Grant
No. DST/SJF/PSA-02/2014-2015). S.D. acknowledges a
Swarnajayanti Fellowship and D.K.S. acknowledges an
INSPIRE Fellowship from the Department of Science andTechnology.
APPENDIX A: MEASUREMENT OF
SELF-DIFFUSION COEFFICIENTS OF JANUSQUADRUPOLAR PARTICLES
Particles moving in NLCs experience a dissipative dragforce due to the coupling of orientational order to the ﬂow
ﬁelds induced by particle motion as well as the inherent anisotropy of shear viscosities of NLCs. In the Stokes
regime, the drag force on a particle moving with velocity
dr/dt can be expressed as Fdrag = −ζi [dr(t)/dt], where ζi
is a drag coeﬃcient and the subscript i refers to motion
either parallel to the director (i =) or perpendicular to thedirector (i =⊥). Using methods similar to those reported inRefs. [19,20], we determine ζ and ζ⊥ of a Janus particle inMLC-6608 by measuring its anisotropic diﬀusive motion.
Particle displacements (x and y) parallel and perpendicular to the director n̂ are measured by videomicroscopy
as a function of the delay time τ . The mean squared displacements along the x and y directions shown in Fig. 7
are linear in τ with diﬀerent slopes corresponding to two
diﬀusivities, D and D⊥ , as σ2 = 2D τ and σ⊥2 = 2D⊥ τ .
The diﬀusion constants obtained from from the leastsquares ﬁts are given by D = 4.1 ± 0.1 × 10−3 μm2 s−1
and D⊥ = 2.9 ± 0.2 × 10−3 μm2 s−1 . The corresponding

FIG. 7. Mean squared displacements (MSDs) of a quadrupolar Janus particle in two orthogonal directions (parallel andperpendicular to the director) in MLC-6608.

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drag coeﬃcients obtained by use of the Stokes-Einstein
relations ζ = KB T/D and ζ⊥ = KB T/D⊥ are ζ  1.0 ×
10−6 kg s−1 and ζ⊥  1.4 × 10−6 kg s−1 .
APPENDIX B: MEASUREMENTS OF RELATIVEPERMITTIVITY AND CONDUCTIVITYANISOTROPY
The relative permittivity and conductivity of MLC-6608
at room temperature (30 ◦ C) in the experimental frequency
range (20–500 Hz) are measured with an LCR meter. The
frequency dependence of the parallel and perpendicularcomponents of the relative permittivity ( and ⊥ ) andconductivity (σ and σ⊥ ) are shown in Fig. 8. Both permittivities remain constant [Fig. 8(a)], whereas both conductivities increase with frequency and σ increases at a
FIG. 9. Frequency dependence of the Clausius-Mossotti factors. Filled and empty circles represent the real and imaginaryparts of KM for a metal sphere (red circles) and KD for a dielectric
sphere (blue circles), respectively, at diﬀerent frequencies.

(a)much-faster rate than σ⊥ [Fig. 8(b)]. The relative permittivity anisotropy of the sample (MLC-6608) is negative( =  − ⊥ < 0), whereas the conductivity anisotropy
is positive (σ = σ − σ⊥ > 0).

APPENDIX C: CALCULATION OF COMPLEX
CLAUSIUS-MOSSOTTI FACTOR
For a metal sphere and a dielectric sphere, the CM
factors KM (ω) and KD (ω) are calculated with the approximate analytical solutions presented in Refs. [28] and [29],respectively. They depend on the frequency of the appliedﬁeld, charge density, relative permittivity, and conductivity of the medium. In the calculation we use the charge
density n  1020 m−3 [31] and the average values of therelative permittivity (m ) and conductivity (σm ) at diﬀerent frequencies obtained from Fig. 8. A typical variationof the real and imaginary parts of KM (ω) and KD (ω) withfrequency is shown in Fig. 9.

(b)

APPENDIX D: MOVIE

FIG. 8. Parallel and perpendicular components of (a) the relative permittivities ( and ⊥ ) and (b) the conductivities (σ andσ⊥ ) as a function of frequency at room temperature at an appliedﬁeld of 0.1 V μm−1 .

Video 1 shows the eﬀect of an electric ﬁeld on apair of quadrupolar Janus particles. The ﬁeld amplitude
is increased gradually from 0 to 1.6 V μm−1 , with thefrequency kept constant at 100 Hz. The interparticle separation increases under equilibrium condition in the ﬁeldrange 0.3 to 1.2 V/μm. Beyond 1.2 V/μm, the particles swim independently due to the surrounding electroosmotic ﬂows.

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