Omnidirectional transport and navigation of Janus particles through a nematic liquid crystal film

In a striking departure from conventional electrophoresis, we show that metal-dielectric Janus particles can be piloted at will through a nematic liquid crystal film, in the plane spanned by the axes of the particle and the nematic, and perpendicular to an imposed AC electric field. We achieve complete command over particle trajectories by varying field amplitude and frequency, exploiting the sensitivity of electro-osmotic flow to the asymmetries of particle and defect structure. Our results include a new method for measuring the induced electrostatic dipole moment of the Janus particles, through competition between elastic and electrostatic interactions. These findings open unexplored directions for the use of colloids and liquid crystals in controlled transport and assembly.

electrostatic dipole moments.We work with a room-temperature nematic liquid crystal mixture (SM).Macroscopic alignment in the x direction is imposed by the treated surfaces of the bounding electrodes parallel to the xy plane (Fig. 1C).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. 1B).
Due to the elastic distortion of the director, the particles resist sedimentation and levitate in the bulk (28).This feature, and therefore electrophoresis as well, are absent in the isotropic phase.
The dielectric anisotropy ∆ of the sample is negative so that the electric field E applied in the z direction does not influence the macroscopic director except near the particles (24).Figure 1E shows the light-microscope texture of the particles placed between crossed polarisers.The fourlobed intensity pattern of the particles, a characteristic feature of elastic quadrupoles, is further substantiated from the texture obtained by inserting a λ-plate (inset to Fig. 1E).A texture of the particles (without polarisers) shows that the metal hemisphere (dark-half) of particles is oriented in different directions, always keeping the Saturn rings perpendicular to the macroscopic director (Fig. 1F).
Once the AC electric field is switched on, the particles reorient so that the plane of the metaldielectric interface lies parallel to the field (Fig. 1, D and G).With increasing field, they start moving in specific directions in the plane of the sample, depending on the orientation of the normal to the metal-dielectric interface (Movie S1).Real-time trajectories of selected particles are grouped in Fig. 1G.The dielectric hemispheres (Fig. 1G, III and VII) lead when movement is parallel to, and the metal hemisphere (Fig. 1G, I and V) when it is perpendicular to, the macroscopic director.For particles moving at other angles the orientation of the metal-dielectric interface interpolates smoothly between these two extremes (Fig. 1G, II, IV, VI and VIII), and the particles can thus move in any direction in the plane of the sample as shown in figs.S1 and S2 (SM).As we show in SM, the electric field, the director-dependent anisotropies in the dielectric and conductivity tensors, and the anchoring of the director together give rise to an effective force dipole of contractile or puller type with respect to the electric field axis, consisting of a pair of point forces of strength ∼ ∆ E 2 a 2 and separation ∼ a, pointing towards each other.Proportionality to the square of the AC field means the effect survives time-averaging.
For Janus quadrupolar particles, the fore-aft symmetry of the resulting electro-osmotic flow is broken by the stronger polarisation in the vicinity of the metallic hemisphere, and in general the particle will now "swim" due to the squirming flow close to it surface.When the Janus particles are oriented neither parallel nor perpendicular to n, their direction of motion can be controlled by changing the amplitude and frequency of the field as shown in Fig. 2, A and D, respectively (Movies S2 and S3).The moving direction is changed recursively at different points by altering the field amplitude between 1.5 Vµm −1 to 2.0 Vµm −1 .Figure 2B shows the trajectories of a Janus particle at different fields for the fixed orientation of the metal-dielectric interface.The angle θ their velocity makes with the director n decreases with increasing field (Fig. 2C), with a linear dependence over the range explored.Figure 3D shows the variation in direction of motion along a trajectory as frequency is changed from 30 Hz to 50 Hz (Movie S3) at fixed field amplitude.Figure 2E shows the trajectories of a particle at various frequencies in which θ increases linearly (Fig. 2F).The motion of the particles shown in Fig. 2, A and D have two velocity components, namely V x and V y along the x and y directions respectively, both of which are proportional to E 2 but the slope of V x is larger than V y (fig.S3D).When the field is increased, the relative enhancements are unequal, i.e., ∆V x is larger than ∆V y , and consequently θ = tan −1 (∆V y /∆V x ) decreases.The velocity components also depend strongly on the frequency (f ) of the field (fig.S4).For frequencies above 15 Hz both V x and V y are proportional to 1/f but the coefficient of the decrease of V x is larger than that for V y , resulting in an increase in the angle θ when f is increased.
In what follows, we show the effect of field on a pair of particles which are assembled by elastic forces of the nematic medium.At zero field, the joining line of the pair makes an angle of 57 • (15) with respect to the director.Once the field is switched on, the particles rotate such that the metal-dielectric interface is parallel to the field due to the induced electric dipole moment, keeping the mutual separation unchanged as shown in Fig. 3, A and B (Movie S4).With increasing field amplitudes, the particles are displaced in opposite directions with increasing separation as shown in Fig. 3, C to E. At higher fields, they are completely separated and move apart with a velocity (Movie S4).The centre-to-centre equilibrium separation r as a function of field at different frequencies is shown in Fig. 3F.For a given frequency f , it is apparent that there is a threshold field E 2 , beyond which r increases rapidly.Careful observation reveals that in the low field region, there is another threshold (E 1 ) beyond which the separation begins to increase (Fig. 3G).For example at 20 Hz the threshold fields are and E 2 0.7 Vµm −1 .With increasing field the induced dipole moments and consequently the mutual repulsive electrostatic force increase, which pushes the particles apart.At E 1 , the magnitude of the electrostatic repulsive force (F d ) just exceeds the elastic binding force (F el ) and beyond E 2 , the separation between the particles is dominated by the electro-osmotic flow.
For comparison we measured the separation between two non-Janus silica particles and found it to be independent of field (Fig. 3H), substantiating our claim for the mechanism at work in the case of Janus particles.
The repulsive force between two adjacent identical electric dipoles p separated by a distance r is given by , where is the permittivity of the medium.In the field range obtained by balancing it against the elastic force F el which is measured as a function of separation using the videomicroscopy technique (inset to Fig. 3H) (SM).This allows us to estimate the magnitude p = (4/3)π r 4 F el of the effective induced dipole moments of the Janus particles, at different frequencies (Fig. 3I).We find a linear variation with field, p = αE, where α is the effective electric polarisability of the Janus particles.The polarisability obtained at different frequencies is shown in the inset of Fig. 3I.It decreases with frequency, as expected.
The calculated polarisability using a physical model (29,30).at a representative frequency of 20 Hz is obtained as α 1.5 × 10 −27 Cm 2 V -1 , which is in close agreement with the experiment (SM).
We have thus demonstrated an exceptionally rich dynamics in the electrically induced motility of Janus particles in a nematic liquid crystal.Our most remarkable findings are autonomous motion in any direction in the plane perpendicular to an imposed AC electric field, through a subtle interplay of Janus character, electrostatics and the physics of liquid crystals, and the ability to dictate the direction of motion purely through field amplitude and frequency.A valuable bonus of our studies is a new method of measuring the induced dipole moment, applicable to all microscopic particles irrespective of shape.Our results offer prospects for practical applications where accurate control on particle positioning, delivery, sorting and mixing is indispensable.A detailed theory of the single-particle motility of our particles, as well as their pair and collective dynamics as for active colloids (34,35), are natural theoretical challenges.Our study has focused on spherical particles; the plethora of new nematic colloids (31)(32)(33) now available promises a wide range of as yet unexplored physical effects and their applications.

Figs. S1 to S7
Movies S1 to S5 References (36)(37)(38)(39)(40)(41)(42) Liquid crystal cell preparation: The cells are prepared using two glass plates (15 mm×10 mm), coated with indium-tinoxide (ITO).The thickness of the ITO film is 150 nm and the resistivity is about 15-20 Ω/sq.The plates are spin coated with polyimide AL-1254 (JSR Corporation, Japan) and cured at 180 0 C for 1 hour.They are then rubbed unidirectionally using a bench top rubbing machine (HO-IAD-BTR-01) for homogeneous or planar alignment of the nematic director, n.The plates are separated by spherical spacers of diameter 5.0 µm, making the rubbing directions antiparallel, and then sealed with UV-curing optical adhesive (Norland, NOA-81).Two thin electrical wires are soldered onto the ITO plates using an ultrasonic soldering machine (Sunbonder, USM-IV).The output of a function generator (Tektronix, AFG 3102) is connected to a voltage amplifier for applying sinusoidal voltage to the cells.

Particle synthesis and sample preparation:
Metallo-dielectric Janus particles are prepared using directional deposition of metal onto dry silica particles (SiO 2 ) of diameter 2a = 3.0 ± 0.2 µm (Bangs Laboratories, USA) in vacuum (4).Approximately 2% suspension (25 µl) of silica particles is spread on a half glass side (40 mm×25 mm), which is pretreated with Piranha solution and dried to form monolayer.Next, a thin Titanium layer of thickness 35 nm is deposited vertically using electron-beam deposition, at a pressure of 3 × 10 −6 torr, and deposition rate of 0.5 Ås -1 .On top of this, a thin layer of SiO 2 film (10-15 nm) is deposited.Then the slides with silica monolayer are washed thoroughly with deionized water (DI) and isopropyl alcohol.The particles are detached from the slides by ultrasonication in deionized water of volume 20 ml for 100 s.Further, the collected particles are sonicated for 30 minutes to breakup any agglomeration of the Janus particles.After the sedimentation of Janus particles at the bottom of the centrifuge tube, the concentrated suspension is used for the next step.The surface of the Janus particles is coated with N, N-dimetyl-N-octadecyl-3 aminopropyl-trimethoxysilyl chloride (DMOAP) in order to induce perpendicular (homeotropic) orientation of the liquid crystal director (36)(37)(38)(39)(40).A small quantity (1 wt%) of DMOAP coated Janus particles is dispersed in nematic liquid crystal, MLC-6608 using a vortex mixture and sonicator.The liquid crystal obtained from Merck is used directly without any further purification.It exhibits the following phase transitions: SmA −30 0 C N 90 0 C Iso.The dielectric anisotropy of MLC-6608 is negative (∆ = − ⊥ = −4.2,where || and ⊥ are the dielectric permittivities for electric field E, parallel and perpendicular to n) whereas the conductivity anisotropy is positive (∆σ = σ − σ ⊥ = 6 × 10 −10 Sm -1 at 100 Hz).There is no electroconvection observed in the experimental field and frequency range.

Experimental Setup:
An inverted polarizing optical microscope (Nikon Ti-U) with water immersion objective (Nikon, NIR Apo 60/1.0) is used for observing the particles.A laser tweezer is built on the microscope using a cw solid-state laser operating at 1064 nm (Aresis, Tweez 250si).An acousto-optic deflector (AOD) interfaced with a computer is used for trap movement.The cell is mounted on a motorized stage (PRIOR Optiscan) of the microscope.A charge-coupled device (CCD) video camera (iDs-UI) at a rate of 50-100 frames per second is used for video recording of the particle trajectory.A particle tracking program is used off-line to track the centres of the particles, with an accuracy of ±10 nm.A colour camera (Nikon DS-Ri2) is used for capturing the texture of the sample.The elastic force between two Janus particles is measured using videomicroscopy technique.Initially two particles are held at a distance with the help of the optical tweezers and then allowed to interact freely, by switching off the laser light.The motion of the particles in the nematic liquid crystal (NLC) is overdamped due to high viscosity.Consequently, the Stokes drag force (F s ) in a uniformly aligned NLC is in equilibrium with the elastic force (F el ), resulting in no acceleration.The elastic force between two particles is calculated from the numerical differentiation of the trajectory, which is given by dt , where ζ i is the drag coefficient and the subscript, i refers to the motion either parallel to n (i = ) or perpendicular to n (i =⊥).The interparticle separation r(t) is obtained from the recorded movies.Using methods similar to those discussed in ref. (27,36,37), we determine the drag coefficients ζ 1.0 × 10 −6 kg/s and ζ ⊥ 1.4 × 10 −6 kg/s, by measuring the anisotropic diffusion coefficients D and D ⊥ at room temperature (T = 298 K).It may be mentioned that during the Brownian motion no rotational diffusion of the particles is observed.In estimating the elastic force, the average drag coefficient ζ = (ζ + ζ ⊥ )/2 ≈ 1.2 × 10 −6 kg/s is used.Neglecting the coupling of the nematic director to the flow field induced by the motion of the particle, one obtains ζ = 6πηa ≈ 0.6 × 10 −6 kg/s, which is in reasonable agreement with the experiment.

Onmidirectional transport of quadrupolar Janus particles:
The direction of motion of the particles in the sample plane depends on the orientation of the metal-dielectric interface with respect to the nematic director, n. Figure S1 shows the trajectories of several particles with different orientations of the metal-dielectric interface, which is summarized in Fig. S2.Thus, transport of particles in all directions in 2D can be achieved by continuously varying ϕ about the field direction (Fig. S2).Electric field dependent velocity of quadrupolar Janus particles: Figure S3, A, B and C show the electric field dependent velocity of three Janus particles in trajectories-III, I and II, respectively (see Fig. 1G).In all these trajectories V ∝ E 2 , with their respective slopes given by 3.5 µm 3 V −2 s −1 , 2.1 µm 3 V −2 s −1 and 3.6 µm 3 V −2 s −1 .The velocity components of the particle in trajectory-II is shown in Fig. S3D.It is noted that at a given field, V x is always larger than V y , owing to the fact that the particles can move easily along the director rather than in the perpendicular direction.Moreover, V x increases at a much faster rate (slope: 3.2 µm 3 V −2 s −1 ) with field than V y (slope: 2.2 µm 3 V −2 s −1 ).Frequency dependent velocity of the quadrupolar Janus particles: The frequency dependent velocity components V x and V y , of the Janus particle in trajectory-II are measured at a fixed field amplitude of 1.54 Vµm −1 , which are shown in Fig. S4.Both the components show the following frequency dependence, given by (10,24,41) where i = x, y; ω = 2πf is the angular frequency of the applied field, τ e is the characteristic electrode charging time and τ p is the characteristic charging time of the particles.Both the velocity components increase as f 2 in the low frequency regime but decrease as f −1 at higher frequencies because the ions cannot follow the rapidly changing field.It is observed that the frequency beyond 15 Hz, V x < V y and V x decreases at a faster rate with frequency than V y .
Fig. S4.Frequency dependence of V x and V y of the particle moving in trajectory-II (Fig. 1G).Red and blue lines show theoretical fit to Eq. (S1).Fit parameters: τ e = 0.25 s, τ p = 0.032 s for V x and τ e = 0.08 s, τ p = 0.04 s for V y .

III. Calculation of polarizability (α) of the Janus particles
The polarizability of the Janus particles is calculated using the superposition principle in which it is assumed that the induced dipole moment of one hemisphere is equal to the half of that induced on a sphere.The polarizability α for a spherical particle can be written as (29,30): where, m is the dielectric permittivity of the dispersing medium, K(ω) is the complex Clausius-Mossotti factor, a is the radius of the sphere and M , D stand for metal and dielectric, respectively.The Clausius-Mossotti factor of the metallic and dielectric sphere is calculated by using the analytical solutions reported in ref. (29).It depends on the frequency (f ) of the AC field, permittivity ( m ) and the conductivity (σ m ) of the medium.We measured the parallel and perpendicular components ( , ⊥ ) of the dielectric permittivity and the conductivity (σ , σ ⊥ ) of MLC-6608 at a frequency of 100 Hz.The medium conductivity and permittivity obtained from the experiments are given by σ m ≈ σ=(σ +2σ ⊥ )/3 2.0 × 10 −10 Sm −1 and m ≈ = ( +2 ⊥ )/3 6.4, respectively.Considering a typical number density of ions in NLCs, n ≈ 10 20 m −3 (42), the estimated Debye screening length to be λ D 0.2 µm.Theoretically calculated polarizability using Eq.S2 at a representative frequency 20 Hz is given by α 1.5 × 10 −27 C m 2 V −1 , which is in good agreement with the experimentally measured value (inset to Fig. 3I).

IV. Transport of quadrupolar Janus particles in a twisted cell
Chiral ordering of the liquid crystals often alters the defect structure induced by the particles, with a possible impact on the liquid crystal-enabled electrophoresis.To examine the effect of chirality on particle transport, we studied quadrupolar Janus particles in a twisted cell, which is prepared by arranging the rubbing directions orthogonal to each other as shown in Fig. S5A.This provides a π/2-twisted director configuration between the two plates.The handedness of the twisted structure is determined by inserting a λ−plate (530 nm) in the polarising optical microscope.The plane of the Saturn ring is found to be rotated clockwise (Fig. S5B) in comparison to the orientation in untwisted cell (see Fig. 1E).Once the field is switched on, the particle gets attracted to the bottom plate and the Saturn ring is further rotated nearly by 45 0 , while going in the downward direction (Fig. S5C).Then, it starts moving along a trajectory which depends on the orientation of the metal-dielectric interface.Trajectories of a few selected particles are grouped and shown in Fig. S5D.For comparison, the trajectories of some particles in the untwisted cell (see Fig. 1G) are also shown in Fig. S5E.The trajectories in the twisted cell are found to be rotated clock-wise nearly by 45 0 in the viewing plane in comparison to the untwisted cell.Thus, chirality provides an additional control on the field-induced motility of Janus particles in a liquid compares advection with orientational relaxation.Here ρ 10 3 kg/m 3 is the mass density, η 0.02 Pa.s the shear viscosity and K 20 pN a Frank elastic constant, the values quoted being those provided by Merck KGaA for our sample of MLC-6608.For a 1.5 µm and v 10 µm/s we then see Re 10 −7 and Er 10 −2 .Hence, the effect of inertia can be neglected, and the director configuration around the particles is negligibly influenced by fluid flow.

VI. Force density around a particle in nematic with transverse electric field
We consider a uniform quadrupolar particle in a nematic in an electric field in the geometry of Fig. S7.The mean macroscopic director field lies parallel to the x axis and local deviations from this mean direction are described by an angle φ, positive for counterclockwise rotation.The imposed electric field E points along z.
We consider a system with conductivities and dielectric constants σ || , || and σ ⊥ , ⊥ for electric fields E parallel and perpendicular to the director n, and define the anisotropies ∆σ = σ || − σ ⊥ and ∆ = || − ⊥ .We will calculate the charge density ρ and hence the electrostatic force density F(r) = ρ(r)E(r).First, steady-state charge conservation ∇ • J = 0 for a current J = σ • E = (σ ⊥ I + ∆σnn), where I is the unit tensor, implies where : denotes contraction with both indices of ∇E.For small deviations δn about a mean alignment n0 , and δE away from a field E 0 = E 0 ẑ imposed from the boundaries, writing where From Fig. S7 we see that the director curvatures in Eq. (S6) are composed of bend concentrated just outside the Saturn ring and splay on the particle surface.In our notation both give contributions of order 1/a, positive above and negative below the particle.n0 x for the former and n0 ẑ for the latter.From Eq. (S8) and (S7) we see that G −1 G σ in Eq. (S6), although it has a nonlocal piece decaying as 1/r 3 , is formally a positive operator.For our system ∆ < 0 and ∆σ > 0, which qualitatively implies an overall minus sign in Eq. (S6).Multiplying by the volume of the particles, we see that the particles behave like force dipoles, of contractile or puller type with respect to the electric field axis, consisting of a pair of point forces of strength ∼ (1 + |∆σ/σ|/|∆ / |)∆ E 2 0 a 2 and separation ∼ a, pointing towards each other.The effect is proportional to the square of the electric field and thus survives time-averaging over a period.The resulting puller flow is of course symmetrically disposed about a uniform sphere and will not lead to propulsion of the particle.If the sphere is replaced by a Janus quadrupolar particle, this symmetry is broken by the stronger polarization in the vicinity of the metallic hemisphere, and in general the particle will now "swim" due to the squirming flow close to its surface.

Fig. 1 .
Fig. 1.Omnidirectional transport.(A) Dipolar and (B) quadrupolar elastic distortions of nematic director around spherical particles.Janus quadrupolar particles (C) without and (D) with field (metal-dielectric interface becomes parallel to the field direction).Red circles represent Saturn-ring defects.(E) Optical microscope texture of particles in the nematic liquid crystal between crossed polariser (P) and analyser (A) in the xy plane.(Inset) Texture with an additional λ-plate (530 nm) oriented at 45 • .(F) Texture without cross polarisers.Metal hemisphere appears darker under transmitted light.(G) Trajectories of selected particles (labelled from I to VIII) under AC electric field (1.54 Vµm −1 , 30 Hz).Pink arrows denote the direction of motion (Movie S1).Textures of a few particles are grouped.Time variation is coded in the trajectories from T min = 0 s to T max = 5 s.

Fig. 2 .
Fig. 2. Navigation by electric field amplitude and frequency.(A) Change of direction of a Janus particle by altering the field amplitude from 1.5 Vµm −1 (purple) to 2.0 Vµm −1 (yellow) at a fixed frequency of 30 Hz (Movie S2).Arrows indicate the direction of motion.(B) Timecoded trajectories at different fields at the same frequency.(C) Angle between the trajectory and the director (θ) decreases linearly with field at a slope of −43.2 ± 2.3 • µmV −1 .(D) Change of direction by altering the frequency from 30 Hz (purple) to 50 Hz (yellow) at an amplitude of 1.5 Vµm −1 (Movie S3).(E) Time coded trajectories at different frequencies.(F) θ increases linearly with the frequency at a slope of 1 ± 0.05 • Hz −1 .By "relative coordinate" we mean relative to the starting point of each trajectory.

Fig. 3 .
Fig. 3. Measurement of induced dipole moment.(A-E) Effect of increasing AC field on a pair of quadrupolar Janus particles (f = 100Hz) (Movie S4).(F) Variation of r with field at different frequencies.Dotted region of Fig. 3F is expanded and shown in Fig. 3G for clarity.(H) Field dependent variation of r of a pair of Janus (half-filled circles) and non-Janus silica (open circles) particles.(Inset) Elastic force, F el between two particles.(I) Effective induced dipole moment p with field at different frequencies.Solid lines show the least squares fit to the equation p = αE.(Inset) Variation of fit parameter α with frequency.Dotted line is a guide to the eye.
V.S.S. Sastry for useful discussions.Funding: This work is supported by the DST, Govt. of India (DST/SJF/PSA-02/2014-2015).SD acknowledges a Swarnajayanti Fellowship and DKS an INSPIRE Fellowship from the DST.SR was supported by a J. C. Bose Fellowship of the SERB (India), and by the Tata Education and Development Trust.Author contributions: SD designed and supervised the project.DKS and SD performed the experiments.SR provided theoretical support including supplementary calculations.All three authors analyzed the results and prepared the manuscript.Competing interests: The authors declare that they have no competing interests.Data and materials availability: Data from the main text and supplementary materials are available from Surajit Dhara.

Fig. S1 .
Fig. S1.Time coded trajectories of particles in many directions.Directions of n and E (f = 30 Hz, E = 1.54 Vµm −1 ) are shown in the central region.Magenta arrows indicate the direction of motion of the particles.By "relative coordinate" we mean relative to the starting point of each trajectory.

Fig. S2 .
Fig. S2.ϕ is the angle between the director n and Janus axis ŝ (normal to the metal-dielectric interface).Orientation of a particle in the xy plane at different ϕ.

Fig. S3 .
Fig. S3.Electric field dependent velocity of particles in (A) trajectory-III, (B) trajectory-I and (C) trajectory-II (see Fig. 1G).Arrows near the spheres indicate the direction of motion.(D) Velocity components, V x and V y for the particle in trajectory-II.Solid lines show least squares fit to V ∝ E 2 .Frequency = 30 Hz.

Fig. S7 .
Fig. S7.Quadrupolar director field of a spherical particle in an NLC.Applied electric field E is transverse to the director.Red circle denotes Saturn ring defect.