Soliton-induced liquid crystal enabled electrophoresis

Manipulation of particles by a uniform electric field, known as electrophoresis, is used in a wide array of applications. Of especial interest is electrophoresis driven by an alternating current (AC) as it eliminates electrode blocking and produces a steady motion. The known mechanisms of AC electrophoresis require that either the particle or the surrounding medium are asymmetric. This asymmetry is usually assured before the field is applied, as in the case of Janus spheres. We report on a new mechanism of AC electrophoresis, in which the symmetry is broken only when the field exceeds some threshold. The new mechanism is rooted in the nature of electrophoretic medium, which is an orientationally ordered nematic liquid crystal. Below the threshold, the director field of molecular orientation around a spherical particle is of a quadrupolar symmetry. Above the threshold, the director forms a polar self confined perturbation around the inclusion that oscillates with the frequency of the applied field and propels the sphere. The director perturbations are topologically trivial and represent particle like solitary waves, called"director bullets"or"directrons". The direction of electrophoretic transport can be controlled by the frequency of the field. The AC directron induced liquid crystal enabled electrophoresis can be used to transport microscopic cargo when other modes of electrophoresis such as induced charge electrophoresis are forbidden.


INTRODUCTION
Electric field acting on a nematic liquid crystal produces a number of nonlinear nonequilibrium phenomena with a rich spectrum of spatiotemporal patterns in the director field   ,t n r that specifies average local orientation of the molecules [1][2][3].
Recently, 3D particle-like dissipative solitons, called "director bullets" that represent propagating solitary waves of self-trapped oscillating director driven by an alternatingcurrent (AC) electric field, have been observed [4,5]. The director bullets are topologically unprotected self-confined configurations that lack fore-aft [4] or left-right [5] symmetry. "Topologically unprotected" means that the self-confined configuration forms by a smooth director deformation from the uniform state; there is no topological charge associated with such a soliton. Since these formations are self-confined waves of the director field that survive collisions, an appropriate term for them is "directrons" that we use as a synonym of director bullets in what follows. Directrons propagate perpendicularly to the driving electric field E and leave the background director field 0 const  n intact. The directrons exist in nematics with negative anisotropies of dielectric permittivity || 0         and electric conductivity, || 0         (the subscripts refer to the direction with respect to the director) [4,5].
In this work, we demonstrate that the directrons can develop at colloidal spheres dispersed in a nematic with 0    and 0    that feature tangential orientation of the director n at their surface. In absence of the electric field, the director field around these spheres is of a quadrupolar symmetry with two point defects-boojums at the poles [6], Fig. 1(a). Electrophoresis of these symmetric particles in an AC electric field is impossible. The so-called induced-charge electrophoresis that can transport metaldielectric Janus spheres [7][8][9][10] is unable to cause a net displacement of a homogeneous sphere. The so-called liquid crystal-enabled electrophoresis (LCEP) that relies on the dipolar asymmetry of the director configuration around particles that exists prior to the electric field application [11][12][13] is also ineffective because   ,t n r around a tangential sphere is of a higher quadrupolar symmetry. Electrically induced directron dresses around the spheres, however, bring about a necessary polar symmetry in the structure and render the tangentially anchored spheres electrophoretically active under the AC field. The structure of the directrons that form above some electric field threshold is similar to the directrons described for uniformly aligned nematics without colloids [4,5]. The directron-dressed spheres move in the plane perpendicular to E ; depending on the frequency or amplitude of the field, the spheres can move parallel, perpendicularly or at some angle to the uniform background director 0 n . The soliton-dressed particles survive head-to-head collisions with each other, restoring their mobility. The effect can be used for electrically controllable transport of microcargo when other mechanisms of electrophoresis, such as linear electrophoresis, induced charge or liquid crystal enabled electrophoresis are ineffective. Since the motility of the spheres requires a formation of the directrons, we call the effect a directron-induced liquid crystal enabled electrophoresis (DI-LCEP).

II. MATERIALS AND EXPERIMENTAL DESIGN
We used a nematic liquid crystal 4'-butyl-4-heptyl-bicyclohexyl-4-carbonitrile (CCN-47, purchased from Nematel GmbH). The material is of the     [14]. The velocities of particles were obtained by measuring the , x y coordinates of the particles as a function of time.

A. Electric field induced cargo transportation
In absence of the field, director deformations around a tangentially anchored sphere are quadrupolar, extending over distances comparable to the radius of the sphere R , Fig. 1(a) [6]. Once the AC field of a fixed frequency f exceeds some threshold th E , the sphere acquires an asymmetric director "dress" of dipolar symmetry that extends over much larger distances  

B. Director structure of directrons in high-frequency field
The director field within the field-induced dress lacks the left-right symmetry with respect to 0 n , Fig. 1(c). To decipher the director details, we used a full wave (530 nm ) optical compensator with the optic axis aligned under 45  to the polarizer ( y -axis) and analyzer ( x -axis), Fig. 1(a). In this setting, regions with a uniform background director 0 (0,1,0)  n appear red. The regions in which the actual director deviates from the y -axis in an anti-clockwise manner appear yellow, while the areas with a clockwise director tilt are blue, Fig. 1(a-c).
The director shows a dynamic behavior, oscillating with the same frequency as the frequency of the applied AC electric field, Fig. 1(c-e). The dynamics of in-plane deformations was established by observations between the polarizer and analyzer decrossed at 60  , according to the protocol described in Refs. [4,5]. The in-plane azimuthal distortions do not change their curvature when the polarity of the voltage is reversed. To determine the period and polarity of out-of-plane director oscillation, we used oblique propagation of light [4,5]. The cell is tilted so that the light beam of the polarizing optical microscope enters the cell at the angle 15    from the normal to the cell [4,5]. The dynamics of light intensity suggest that the polar tilt  oscillates in phase with the applied voltage. The overall director configuration and dynamics in the high-frequency dresses are thus similar to that of the director in 90 B h directrons reported in Ref. [4]. Because of that, we call a tangential sphere with a directron dress induced

C. Director structure of directrons in low-frequency field
At low frequency, the spheres acquire a directron dress similar to that of the previously described B l  directrons [5]; the subscript is the angle between the In the 90 B l dresses, the in-plane director tilts in segments 1, 2, 3, and 4 oscillate, changing their polarity with the frequency f , as evidenced by observations with decrossed polarizers, Fig. 3(b,c). The field-induced director deformations preserve mirror symmetry with respect to a plane perpendicular to 0 n , but lack it along 0 n , Fig.  3(b,c). As a result, a sphere dressed in a 90 B l directron moves perpendicularly to 0 n , Fig. 3(a). The dynamics of light intensity measured from the normal 0   and oblique 15    incidences according to the protocol described in Ref. [5], bring into evidence that the polar director tilt oscillates in phase with the applied voltage, Fig. 3(d). This director dynamics is thus similar to that of the director inside the particle-free 90 B l directrons at low frequency reported in [5].  15 B l dress is shown in Fig. 4. In this structure, the director oscillations are similar to those in Fig. 3, i.e., the azimuthal tilts change their polarity with the frequency f . The principal difference is that the structure shows no mirror symmetry with respect to any plane perpendicular to the cell, Fig. 4(b). The reason for the asymmetric structure is not clear, but can be tentatively associated with the increased role of surface anchoring and its plausible inhomogeneities at the surface of the spheres once the field becomes weaker; shape deviations from an ideal sphere might also be of importance.   Fig. 5(a), and 500 Hz, Fig. 5(b). In general, the directrons dressing colloids exist in a wider voltage range, especially in the case of high frequency driving, Fig.5 (b).

D. Collisions of two directron-dressed spheres
It is commonly known that solitons can survive collisions and restore their shape and propagation mode in head-to-head encounters [15]. The same is true for the standing-alone directrons [4,5]. This feature is the ultimate reason for the term "soliton", as it stresses particle-like properties of the solitary waves [16]. The directrondressed spheres in our experiments show a similar ability to survive collisions and restore their dresses, even when they collide head-to-head. Since the solid particles cannot penetrate each other, the scenarios of encounters are very peculiar, as illustrated in Fig. 6 in which two 90 B h -dressed spheres move towards each other. Their initial impact distance y  (the separation of the centers of mass along the y -axis) is small, 0.5R , Fig. 6(a, c). As the spheres approach each other, they slow down and y  decreases to zero, but once their centers arrive at the same x -coordinate, the y  distance increases to about 4R . The effective repulsion along the y -axis is caused by impermeability of the particles and by elastic repulsion between their soliton dresses.
Remarkably, after the spheres part with each other, they completely restore the soliton dresses, speed and horizontal direction of propagation, Fig. 6.

IV. DISCUSSION
It is known that colloidal particles placed in a liquid crystal electrolyte can become mobile when the director field around them is of a dipolar symmetry. The effect is called liquid crystal-enabled electrophoresis or LCEP [11][12][13][17][18]. LCEP of spheres in nematic is effective when the director is anchored perpendicularly to the surface of a sphere and the cell thickness is significantly larger than the diameter of the particle.
In this case, the director field acquires a dipolar symmetry, representing a locally radial structure in the vicinity of the sphere and a topological defect, the so-called hyperbolic hedgehog, next to it. In presence of the electric field, this dipolar structure separates electric charges that cause directional propulsion of the colloid with the velocity growing as 2 E , so that the effect can be driven by an AC field [11,12,17,18].
Nonlinear electrophoresis can also be caused by a pulsed high-frequency AC electric field that couples dielectrically to the dipolar director around a perpendicularly anchored sphere [19]. In the case of a tangentially anchored sphere, however, these mechanisms are not valid, as the director and charge separation patterns are of a quadrupolar symmetry with two planes of mirror symmetry, one parallel to the bounding plates and one normal to 0 n . The present work shows that the tangentially anchored spheres become electrophoretically active through formation of electricallytriggered directron dresses around them. These directron dresses are similar to the 3D particle-like solitary waves called directrons and described earlier for high [4] and low [5] frequencies of an electric field acting on a uniformly aligned nematic. The speed of spheres grows with the square of the field, similarly to the conventional LCEP, but with that difference that the LCEP shows no threshold behavior while the effect described in this work does show a threshold behavior. Given all these similarities and differences, we call the observed phenomenon a directron-induced LCEP, or DI-LCEP.
In the description above, we presented the data for two different geometries, Since the directron around a colloidal sphere is of a similar size as the colloid-free directrons, the ratio 2 / is expected to be in a specific range, as observed. If the colloidal sphere is too big, it over-stretches the director deformations beyond the length-scale that corresponds to a stable self-confinement. If the sphere is too small, there are two reasons why the directrons do not dress them.
First, a small sphere does not modify substantially the director field of the directron which is of a typical size 2d . Second, if the particle is smaller than the de Gennes-  [20], its surface anchoring is not strong enough to produce substantial director deformations.
As compared to the induced-charge electrophoresis in isotropic electrolytes [7][8][9][10] that requires the particles to be asymmetric (such as metal-dielectric Janus spheres), the advantage of the DI-LCEP is in the ability to move perfectly symmetric homogeneous spheres. As compared to the conventional LCEP that moves particles with dipolar director configuration [12,13,14], the advantage of the DI-LCEP is in the ability to move particles that show a higher symmetry of the director in absence of the field.
Moreover, in LCEP, the colloids move only parallel to the background director 0 n , while in DI-LCEP, the direction of motion can be tuned by the electric field.
Steering of colloidal transport is attracting a considerable interest lately. The LCEP mechanism has been demonstrated to control the direction of colloids by patterned surface director in the plane of the cell [11,21,22] or even in three-dimensional space, by combining LCEP with linear electrophoresis [23]. Hernàndez-Navarro et al. [13] reported on reconfigurable swarms of asymmetric pear-shaped colloids driven by LCEP and steered by photoactivated photo-switchable surface anchoring. Sahu, Ramaswamy, and Dhara [24] reported on an in-plane omnidirectional transport of metal-dielectric Janus spheres that is based on the asymmetries of both the particles and the surrounding director field; the direction of propulsion is controlled by varying the field frequency and amplitude [24]. In the described DI-LCEP effect, the particle is also steered in the plane of the cell by changing the frequency and voltage of the AC electric field, but the difference is that the particle is a symmetric homogeneous sphere.
The interdependency of the surface properties of the colloids, symmetry of the directron dresses, field parameters such as amplitude and frequency, material parameters of both the liquid crystal electrolyte and the colloids and the direction and speed of the particles driven by DI-LCEP suggests that the described mechanism can bring about many different dynamic scenarios worthy of further studies.