PHYSICAL REVIEW RESEARCH 2, 032009(R) (2020)
Rapid Communications

Omnidirectional transport and navigation of Janus particles through a nematic liquid crystal film
Dinesh Kumar Sahu,1 Swapnil Kole,2 Sriram Ramaswamy ,2 and Surajit Dhara

1,*

1

2

School of Physics, University of Hyderabad, Hyderabad 500 046, India
Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560 012, India(Received 29 November 2019; revised 20 February 2020; accepted 17 June 2020; published 8 July 2020)
We create controllable active particles in the form of metal-dielectric Janus colloids which acquire motility
through a nematic liquid crystal film by transducing the energy of an imposed perpendicular AC electric field. We
achieve complete command over trajectories by varying the field amplitude and frequency, piloting the colloids
at will in the plane spanned by the axes of the particle and the nematic. The underlying mechanism exploits
the sensitivity of electro-osmotic flow to the asymmetries of the particle surface and the liquid crystal defectstructure. We present a calculation of the dipolar force density produced by the interplay of the electric field withdirector anchoring and the contrasting electrostatic boundary conditions on the two hemispheres, which accounts
for the dielectric-forward (metal-forward) motion of the colloids due to induced puller (pusher) force dipoles.
These findings open unexplored directions for the use of colloids and liquid crystals in controlled transport,
assembly, and collective dynamics.
DOI: 10.1103/PhysRevResearch.2.032009

Electrophoresis, the use of electric fields to transport tiny
particles through fluids, is an important technology for macromolecular sorting, colloidal assembly, and display devices
and a challenging area of soft-matter research [1–8]. Classic
electrophoresis is linear: ions in the electrical double layerdrag the fluid, and hence the particle itself, with a velocity
proportional to and parallel to the applied field. Inducedcharge electro-osmosis of particles is nonlinear: the appliedfield itself creates the double layer. Polarity in the shape orsurface properties of the particle results in a flow pattern thatpicks out a direction of motion, with a velocity quadratic inand normal to the field [9–13]. Neither effect offers the optionof continuously tuning the direction of transport and hence thedesired motility of the microscopic particles [14,15].
When the ambient fluid is a nematic liquid crystal (NLC),
the anchoring of the mean molecular orientation or director n̂
normal to the surface of a suspended homogeneous sphericalparticle mandates Saturn-ring [16,17] or asymmetric [18,19]
defect structures, resulting, respectively, in quadrupolar or
dipolar elastic distortions in the NLC [19]. The nonlinear
electro-osmotic flow resulting from an imposed electric field
yields bidirectional transport of dipolar particles parallel tothe local director, thanks to their broken fore-aft symmetry, aneffect termed liquid-crystal-enabled electrophoresis [20–24].
The Saturn-ring particles, by contrast, maintain the quadrupolar symmetry of the flow and hence display no motility[23,24].

*sdsp@uohyd.ernet.in

Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Further
distribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.
2643-1564/2020/2(3)/032009(6)032009-1

Our study focuses on spherical particles with two hemispherical faces: one metal, the other dielectric (see Supplemental Material Fig. S1 [25]). Their “Janus” character issensed only by the electrostatics of the medium; as far as themechanics of the ambient NLC is concerned they are elastic
quadrupoles. Our central result is that purely by tuning theamplitude and frequency of an imposed electric field, andnot its direction, we can achieve guided transport of Janus
colloids in the direction of our choosing perpendicular to thefield, amounting to the realization of controllable phoretic
active particles [26]. Our findings suggest novel possibilities
at the interface of colloids and liquid crystals for controlled
transport, assembly, nonequilibrium phenomena, and collective dynamics.
We work with a room-temperature NLC mixture, MLC6608. Macroscopic alignment in the x direction is imposedby the treated surfaces of the bounding electrodes parallelto the xy plane [Fig. 1(b)]. Their separation is larger than,but close to, the diameter 2a of the suspended particles [27],which produce a quadrupolar elastic field in the nematic[Fig. 1(a)]. We work in the dilute regime and do not consider
cases of higher concentration where aggregation and network
formation are important [28,29]. Due to the elastic distortionof the director, the particles resist sedimentation and levitatein the bulk [30]. This feature, and therefore liquid-crystalenabled electrophoresis as well, is absent in the isotropic
phase. The dielectric anisotropy ( = || − ⊥ , where || and⊥ are the dielectric permittivities for the electric field paralleland perpendicular to n̂) of the sample is negative so that theelectric field E, applied in the z direction, does not influence the macroscopic director except near the particles [23].
Figure 1(c) shows the optical microscope texture of a Janus
quadrupolar particle with cross polarizers. The four-lobed
intensity pattern of the particle, a characteristic feature of theelastic quadrupole [27], is further substantiated by the texture
obtained by inserting a λ plate (inset). The texture without
Published by the American Physical Society

SAHU, KOLE, RAMASWAMY, AND DHARA

PHYSICAL REVIEW RESEARCH 2, 032009(R) (2020)

FIG. 2. Quadrupolar director field surrounding a Janus particle.
The applied electric field E is transverse to the director. The red circle
denotes the Saturn-ring defect. The black hemisphere represents
metal.

FIG. 1. (a) Quadrupolar distortion of the nematic director arounda spherical particle. The red circle represents the Saturn-ring defect.
(b) Janus quadrupolar particles in a planar cell with an AC field. The
dark hemisphere represents metal. (c) Optical microscope texture of a
Janus quadrupolar particle in the NLC between crossed polarizer (P)and analyzer (A) in the xy plane. Inset: Texture with a λ plate (530nm). (d) Time-coded trajectories (Tmin = 0 s to Tmax = 5 s), labeledfrom I to VIII, under an AC electric field (1.54 V μm−1 , 30 Hz).
Trajectories of selected particles are grouped. Pink arrows denotethe direction of motion. (See Supplemental Material Movie S1 [25]).
−→
The direction of the electric field ( E ) and the director (n̂) for all
trajectories are shown in the center. Cell thickness: 5.2 μm.

polarizers shows that the metal hemisphere of particles in
the absence of an AC field is oriented in different directions,always keeping the Saturn rings perpendicular to the macroscopic director (see Supplemental Material Fig. S2A [25]).
Depending on the anchoring, Janus particles can also induce
other types of defects [31,32], which are not considered in thisstudy.
Once the AC electric field is switched on, the particles
reorient [14] so that the plane of the metal-dielectric interface
lies parallel to the field [Fig. 1(b) and Fig. 1(d)]. With increasing field, they start moving in specific directions in the planeof the sample, depending on the orientation of the Janus vectorŝ (normal to the metal-dielectric interface) (Supplemental
Material Movie S1 [25]). Real-time trajectories of selectedparticles are grouped in Fig. 1(d). The dielectric hemispheres[Fig. 1(d), III and VII] lead when movement is parallel to, andthe metal hemisphere [Fig. 1(d), I and V] when it is perpendicular to, the macroscopic director. For particles moving atother angles the Janus vector ŝ interpolates smoothly between

these two extremes [Fig. 1(d), II, IV, VI, and VIII], and theparticles can thus move in any direction in the plane of thesample as shown in Supplemental Material Fig. S2B [25].
To understand the motility of a Janus particle in an ACelectric field we calculate the electrostatic force density induced by the field in the region around the particle. This force
density drives fluid flow over the surface of the particle in sucha way as to turn the particle into a swimmer, as sketched inFig. 3.
We consider a single spherical Janus particle of radius ain an NLC. The particle’s surface consists of a conducting
and a dielectric hemisphere, both of which are taken to
impose identical uniform homeotropic surface anchoring on
the ambient nematic, so as to produce an elastic quadrupolar
distortion in the director field, that is, a radial hedgehog
compensated by a Saturn-ring defect. An electric field E0
along z is externally imposed on the system as shown in
Fig. 2. The mean macroscopic director field lies parallel tothe x axis and local deviations from this mean direction
are described by an angle θ , positive for counterclockwise
rotation. We consider a system with conductivities σ|| , σ⊥ anddielectric constants || , ⊥ , for electric fields parallel (||) andperpendicular (⊥) to the director n̂ and define the anisotropies
σ = σ|| − σ⊥ and  = || − ⊥ . Our strategy, generalizing[24], is to use charge conservation and Gauss’s law to obtainthe total electric field E and the charge density ρ and thus
the electrostatic force density, F (r) = ρ(r)E(r), separately
for the case of a dielectric and a conducting sphere and tocombine these results to infer the character of the induced flow
around the Janus sphere.
For our particles of radius a  1.5 μm moving at speedv  10 μm/s through a liquid crystal of mass density ρ 
103 kg/m3 , shear viscosity η  0.02 Pa · s, Frank elastic
constants K  20 pN (values quoted by the supplier Merck
KGaA for our sample of MLC-6608), Reynolds number Re =
ρva/η  10−7 , and Ericksen number Er = ηva/K  10−2 .
We can therefore ignore the effect of fluid inertia and we
can take the director configuration around the particles tobe negligibly influenced by fluid flow. As in [24], we workat zero Peclet number, i.e., take the charge currents to be
purely ohmic and not advected by fluid flow. For simplicity

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we work at low frequencies so that time dependence can beneglected in the induction equation and thus the electric field
E = −∇ , whereis a potential. We work to first orderin the anisotropies σ and . We begin by evaluating thepotential for a conducting or dielectric particle in the absenceof anisotropy, for which the electrostatic boundary conditions
imply

is dielectric or conducting, while F λ depends, through λ, onthe electrical nature of the sphere,F0  −0 − + σ G−1
 Gσ (n̂0 ∇ · δ n̂ + n̂00 ≡ −E0 z + λE0

E0 a3 z≡ −E0 z + λ 3 ,r⎧⎨1,
λ = r − 1< 1,⎩
r + 2

conductor,
dielectric,(σ⊥ I + σ n̂n̂) : ∇(∇ ) = −σ (n̂∇ · n̂ + n̂ · ∇ n̂) · ∇ ,(2)where the colon (:) denotes contraction with both indices of∇(∇ ). For small deviations δ n̂ about a mean alignmentn̂0 and corresponding deviations δE from a field E = −∇ 0
imposed from the boundaries, writing δE = −∇δ in termsof a potential δ , (2) becomes= −σ [(n̂0 ∇ · δ n̂ + n̂0 · ∇δ n̂) · ∇+ (n̂0 δ n̂ + δ n̂n̂0 ) : ∇(∇

0 )],

0

(3)

where || and ⊥ denote components along and transverse ton̂0 and = δ + 0 . Next, Gauss’s law ρ = ∇ · D = 0 ∇ ·(E) reads, in the same linearized approximation,
ρ = −0 (⊥ ∇⊥2 + || ∇||2 )· ∇δ n̂) · ∇− 0 [(n̂0 ∇ · δ n̂ + n̂0

0 + (n̂0 δ n̂ + δ n̂n̂0 ) : ∇(∇

0 )].

(4)

Solving (3) for

allows us to write the force densityF = ρE  0 − + σ G−1
 Gσ [(n̂0 ∇ · δ n̂n̂0 · ∇δ n̂)·∇

0 + (n̂0 δ n̂ + δ n̂n̂0 ) : ∇(∇0 )]E0 ,(5)where we have defined the Green’s functions
Gσ = (σ⊥ ∇⊥2 + σ|| ∇||2 )−1(6)G = (⊥ ∇⊥2 + || ∇||2 )−1 .

(7)

and

At the lowest order in δ n̂ the charge density is driven by theexternally imposed electric field E0 . We show below that thesecond term (which we call FII ) in brackets on the right-handside of (5) contributes only a higher multipole to the force
density. The force density thus takes the formF  0 − + σ G−1
 Gσ [(n̂0 ∇ · δ n̂ + n̂0·∇δ n̂) · ∇0 ]E0 .

and(1)

where r is the dielectric constant.
To calculate the induced charge density in the nematicdue to anchoring, we impose steady-state charge conservation∇ · J = 0 for a current J = σ · E = (σ⊥ I + σ n̂n̂) · (−∇ ),where I is the unit tensor. This implies(σ⊥ ∇⊥2 + σ|| ∇||2 )· ∇δ n̂) · ẑE02 ẑ

(8)

It is useful to decompose the force density as F = F 0 + F λ ,where F 0 is a contribution independent of whether the sphere

(9)



Fλ  0 λ − + σ G−1
 Gσ (n̂0 ∇ · δ n̂ + n̂0· ∇δ n̂) · ∇ E02 ẑ,

(10)

where
is defined in (1) above, and F 0 is the same forboth dielectric and conducting surfaces, while the symmetrybreaking piece is F λ . From (9) and (10), we see that theeffect is proportional to the square of the electric field andthus survives time averaging over a period. Also, from (10),the force dipole is larger for a conducting sphere than for adielectric sphere by a factor of 1/λ. Therefore, for a Janus
particle, the center of the force dipole shifts from the geometric center towards the conducting side, breaking the symmetry
and hence rendering the particle motile in an applied AC
electric field.
For Fig. 2, we consider a local Cartesian coordinate
system and small-angle approximation, n̂0  x̂ and δ n̂ ∼(0, θ (x, z)). The second term in (5) reduces to FII 
20 (− + σ G−1
 Gσ )θ ∂x ∂z 0 E0 , with signs (− + −+) for
positive z and (+ − +−) for negative z, as one moves frompositive x towards negative x, and hence has only a higher
multipole contribution. The force density of (8), on the other
hand, reads ∂θ ∂ 0∂θ ∂ 0+E0 ẑ.
GF  0 − + σ G−1
σ

∂z ∂x
∂x ∂z
(11)
In Fig. 2 we see that the director curvatures in (11) arecomposed of bend concentrated just outside the Saturn ring(∂x θ ) coupled to (∂z 0 ) and splay on the particle surface (∂z θ )coupled to (∂x 0 ). We see that the signs in the four quadrants
are ∂x θ (− − ++), ∂z 0 (− − −−), ∂z θ (+ − −+), and∂x 0 (− + −+). From (6) and (7) we see that although3
G−1
 Gσ has a nonlocal piece decaying as 1/r , it is formally apositive operator if examined in Fourier space. For our system < 0 and σ > 0. Therefore, the splay contribution produces a force dipole of the contractile or puller type, while thebend produces a force dipole of the extensile or pusher typewith respect to the electric-field axis.
Given that the electric field, via (11), results in a force
density along z, we can understand trajectories III and VII [seeFig. 1(d)] by asking how the Janus character breaks symmetry
in the xz plane. In this plane the splay contribution is greaterthan the bend, as it is present over a larger part of the particle
surface. Shifting the force dipole towards the metallic side
yields a puller-type force dipole resulting in motility with thedielectric face forward as shown in Fig. 3(a). To test whether
this idea makes sense let us apply it to the case of trajectories
I and V in Fig. 1(d). Here the breaking of symmetry in the yz
plane is of relevance. Bend all along the Saturn ring is in play,which gives a pusher force dipole. Shifting this towards themetal face results in motility with the conductor face forward
as shown in Fig. 3(b). The consistency of our explanation
for the cases of the Janus axis parallel and perpendicular to

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FIG. 3. Liquid-crystal-enabled electro-osmotic flows around aquadrupolar Janus particle in two orthogonal planes parallel tothe electric field. Red circles represent Saturn-ring defects. The
black hemisphere represents metal. The large curved arrows aroundthe metal hemisphere indicate stronger flows. The off-center force
dipoles F (shown in green) are (a) pullers and (b) pushers. The
propulsion direction of the particles is indicated by a pink arrow for(a) trajectory III/VII and (b) trajectory I/V.

the macroscopic nematic alignment is reassuring. For othertrajectories motility is due to a combination of both the effects
stated above and therefore interpolates in direction.

When the Janus vector ŝ is oriented neither parallel nor
perpendicular to n̂, its direction of motion can be controlled
by changing the amplitude and frequency of the field as shownin Figs. 4(a) and 4(d), respectively. The moving direction ischanged recursively at different points [Fig. 4(a)] by altering
the field amplitude between 1.5 and 2.0 Vμm−1 at a fixed frequency (Supplemental Material Movie S2 [25]). Figure 4(b)shows the trajectories of a Janus particle at different fields
for a fixed orientation of the metal-dielectric interface. The
angle θ their velocity makes with the director n̂ decreases withincreasing field [Fig. 4(c)], with a linear dependence over therange explored. Figure 4(d) shows the variation in directionof motion along a trajectory as the frequency is changed from30 to 50 Hz (Supplemental Material Movie S3 [25]) at a fixed
field amplitude. Figure 4(e) shows the trajectories of a particle
at various frequencies in which θ increases linearly [Fig. 4(f)].
Particles with an appropriately chosen orientation of theJanus vector can be transported to any predetermined placein the plane of the sample by varying the amplitude andfrequency of the AC field. Figure 5(a) shows the trajectory of aparticle whose Janus vector ŝ is oriented at an angle ϕ = 135◦with respect to n̂ (Supplemental Material Fig. S2C [25]).
The particle is piloted to a predetermined location on the left
with respect to the starting point by increasing the amplitude

FIG. 4. (a) Change in the direction of motion of a quadrupolar Janus particle upon altering the field amplitude recursively between
1.5 V μm−1 (purple) to 2.0 V μm−1 (yellow), keeping the frequency constant at 30 Hz. (See Supplemental Material Movie S2 [25].) Arrows
indicate the direction of motion. (b) Time-coded trajectories of a particle at different field amplitudes and a fixed frequency (30 Hz). (c)The angle between the trajectory and the director (θ ) decreases linearly with the field at a slope of −43.2◦ ± 2.3◦ μm V−1 . (d) Change in the
direction of motion of a particle upon altering the frequency recursively between 30 Hz (purple) and 50 Hz (yellow), keeping the field amplitude
constant at 1.5 V μm−1 . (See Supplemental Material Movie S3 [25].) (e) Time-coded trajectories of a particle at different frequencies and afixed amplitude (1.5 V μm−1 ). (f) θ increases linearly with f at a slope of 1◦ ± 0.05◦ Hz−1 . By “relative coordinate” we mean relative to thestarting point of each trajectory. Error bars represent the standard deviation of the mean value. Cell thickness: 5.2 μm.

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depend strongly on the frequency ( f ) of the field [seeFig. 6(b)]. Both the components show the frequency dependence given by [11,23]ω2 τe2,Vi (ω) = Vio 
1 + ω2 τ p2 1 + ω2 τe2FIG. 5. (a) A particle with ϕ = 135◦ is piloted to a predeterminedplace by changing the amplitude of the field (see Supplemental Material Movie S4 [25]). The field amplitude is increased in a continuous
manner from 1.2 to 2.0 V μm−1 , keeping the frequency fixed at
30 Hz. (b) A particle with ϕ = 315◦ is transported to a predeterminedplace by increasing the frequency in a continuous manner from 20 to
90 Hz, keeping the amplitude fixed at 1.6 V μm−1 (see Supplemental
Material Movie S5 [25]). Arrows near the particles show the initial
direction of motion. Cell thickness: 5.2 μm.

of the field at a fixed frequency. Similarly, a particle with
ϕ = 315◦ is guided to a specified destination by increasing thefrequency while keeping the amplitude fixed [see Fig. 5(b)].
In both cases the direction of motion changes continuously
while the orientation of the Janus vector ŝ remains unchanged(see Supplemental Material Movies S4 and S5 [25]).
To understand the navigation of the Janus quadrupolar
particles, we measured the field dependence of the velocity(see Supplemental Material Fig. S3 [25]). The motion of
the particles shown in Figs. 4(a) and 4(d) has two velocity
components, namely, Vx and Vy , along the x and y directions,
respectively, both of which are proportional to E 2 , but theslope of Vx is larger than that of Vy [see Fig. 6(a)].
When the field is increased, the relative enhancements areunequal, i.e., Vx is larger than Vy , and consequently, θ =tan−1 (Vy /Vx ) decreases. The velocity components also

FIG. 6. (a) Dependence of the velocity components Vx and Vy of a
Janus particle [trajectory II; Fig. 1(d)] on the electric-field amplitude
E at f = 30 Hz. Solid lines show least-squares fits to V ∝ E 2 . The
slopes of Vx and Vy are 3.2 and 2.2 μm3 V−2 s−1 , respectively. (b)Frequency-dependent (E = 1.54 V μm−1 ) velocity components Vx
and Vy . Red and blue lines show the theoretical fit to Eq. (2). Fitparameters are τe = 0.25 s, τ p = 0.032 s for Vx and τe = 0.08 s,
τ p = 0.04 s for Vy . Error bars represent the standard deviation of themean value. Cell thickness: 5.2 μm.

(12)where i = x, y, ω = 2π f is the angular frequency of theapplied field, and τe and τ p are the characteristic electrodeand particle charging time, respectively. Both the velocity
components increase as f 2 in the low-frequency regime but
decrease as f −2 at higher frequencies because the ions cannot
follow the rapidly changing field. For frequencies above 15 Hz
both Vx and Vy are proportional to 1/ f 2 but the coefficient of
the decrease in Vx is larger than that for Vy , resulting in an
increase in the angle θ when f is increased.
Two further remarks are in order. First, for induced-charge
electro-osmosis in an isotropic medium the flows are alwaysof the puller type with respect to the electric-field axis [33],ruling out directional control of the type we discuss. Second, we expect that hydrodynamic torques arising from thecoupling between the squirmer flow field and the anisotropic
viscosity of the NLC [34] cannot operate here, as the orientation of the force dipole driving our swimming particles isdetermined by the direction of the imposed electric field. This
rules out spontaneous change in the direction of motion ofparticles due to such torques at a fixed field and frequency.
We have shown that metal-dielectric Janus particles in an
NLC film subjected to a perpendicular AC electric field behave like steerable active particles whose direction of motion
can be dictated purely by varying the field amplitude and frequency. The underlying mechanism involves the contrasting
electrostatic boundary conditions on the two Janus faces of theparticles, the dielectric anisotropy of the nematic, and anchoring on the particle surfaces. We show that the time-averaged
electrostatic force density produced around the Janus particle
by the AC field is that of a force dipole whose center isshifted towards the conducting face, causing the particle toswim in the plane transverse to the field, with the dielectric(metal) face forward for a particle axis parallel (perpendicular) to the nematic director and interpolating smoothly
for intermediate orientations. Studies on motility at higher
concentrations, as well as collective dynamics as for diffusiophoretic active colloids [35,36], are natural experimental
and theoretical challenges. Our study has focused on sphericalparticles in nematics with perpendicular surface anchoring
and linear macroscopic alignment, but we have preliminary
results for particles with planar surface anchoring, as well as
for a racetrack director configuration. The abundance of newparticles with controlled shapes, surface anchoring [37], andgenus [38] now available and their extraordinary topological[39] and dynamical properties [40,41] promise a wide rangeof as yet unexplored physical effects and their applications.
S.D. thanks Steve Granick for hosting his visit to IBS,UNIST, which resulted in very useful discussions. S.D.
also acknowledges Myeonggon Park and Joonwoo Jeong of
UNIST for various discussions. We thank O. D. Lavrentovich,
I. Muševič, S. Bhattacharya, and P. Anantha Lakshmi foruseful discussions. We also thank K. V. Raman for help in

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preparing Janus particles. This work was supported by the
DST, Government of India (DST/SJF/PSA-02/2014-2015).
S.D. acknowledges a Swarnajayanti Fellowship and D.K.S. an

INSPIRE Fellowship from the DST. S.R. was supported by aJ. C. Bose Fellowship from the SERB, India, and by the Tata
Education and Development Trust.

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