## Abstract

Optical tweezers are exciting tools with which to explore liquid crystal (LC) systems; the motion of particles held in laser traps through LCs is perhaps the only approach that allows a low Ericksen number regime to be accessed. This offers a new method of studying the microrheology associated with micrometre-sized particles suspended in LC media—and such hybrid systems are of increasing importance as novel soft-matter systems. This paper describes the microrheology experiments that are possible in nematic materials and discusses the sometimes unexpected results that ensue. It also presents observations made in the inverse system; micrometre-sized droplets of LC suspended in an isotropic medium.

## 1. Introduction

Trapping and moving micrometre-sized particles in liquid crystals (LCs) have provided a unique way of studying some of the fundamental properties of colloidal LC systems. In particular, this experimental approach allows the study of forces between colloidal particles in LCs, the measurement of elasticity of defects and the anisotropic viscosity coefficients of LCs in the low Ericksen regime (in which the director configuration is considered not to be influenced by the particle motion) to be deduced. Rather few nematic materials have been studied using this approach, but we describe the viscosity coefficients that have been deduced for two different materials. There are two main methods of measuring viscosity: either through the release of a particle from a trap that moves at different speeds, or from an analysis of particle tracking via a power spectral density (PSD) measurement.

The inverse experiment, in which micrometre-sized LC droplets are held in laser traps with specific polarization properties, allows the transfer of angular momentum to the droplets. A torque is imparted to a birefringent particle held in an optical trap through the phase shift experienced by the transmitted light. Fast, optically driven switches, rotating at speeds more than 1 kHz, can readily be produced from nematic droplets held in a circularly polarized beam. This paper examines some of the unexpected observations caused by nonlinear interactions of the LC material with the trapping field. However, we have also shown that a chiral nematic droplet can, in some special circumstances, undergo continuous rotation in a *linearly* polarized trap, a phenomenon that is the result of optically induced changes in chirality of the system. We touch on this remarkable effect, which results in an optically driven transducer.

## 2. Laser manipulation

Optical trapping, as a means to manipulate and study micrometre-sized dielectric particles, has advanced significantly since the first steps towards the technology were taken more than 40 years ago by Ashkin [1]. Since the successful demonstration of the first single-beam optical trap for dielectric particles by Ashkin *et al.* in 1986 [2], *optical tweezers*—the popular name when referring to optical trapping—have become one of the most popular techniques used for manipulating such particles and determining their characteristics, resulting in the proliferation of further innovative optical trapping technologies. The range of optical tweezing applications is rapidly growing, with the current focus on biological and microanalytical systems as well as the integration of optical tweezers with already well-established investigative techniques [3].

The scope of achievements with optical tweezers continues to grow as the technique progresses alongside and in partnership with contemporary technology. The technique has seen phenomenal interest; at the time of writing, Ashkin’s 1986 paper [2] had been cited more than 2300 times! In addition, there have been several publications geared towards undergraduates understanding and experimenting with optical tweezers [4–6], illustrating the acceptance of the technique into mainstream optics.

At the start of 2011, a publication detailed the use of an Apple iPad—a thin, tablet computer controlled via a multi-touch display—to control multiple particles with laser tweezers [7], known as the iTweezers interface. At the end of 2011, researchers in Germany developed a microscopic heat engine, a Stirling engine, whose piston mechanism is powered by an optically trapped particle [8].

### (a) Basic principles

The magnitude of the force exerted on a trapped particle by a laser beam with power *P* travelling through a medium with a refractive index is given by
2.1
where *Q* is the optical trapping efficiency, a dimensionless constant of proportionality that can be thought of as the fraction of incident power used to exert an optical force. *Q* accounts for beam parameters, such as spot size, wavelength and beam profile, and optical properties of the trapped particle, including its size, shape and mass, and refractive index relative to the surrounding medium, *δn* [9,10,11]. In the case of LCs, is an average refractive index of the ordinary and extraordinary refractive indices, *n*_{o} and *n*_{e}, respectively, given by [12].

An optical trap is analogous to a Hookean spring where the particle behaves like an overdamped harmonic oscillator for small displacements, and the optical potential is harmonic about the equilibrium position. The optical restoring force in one dimension is given by **F**_{T}=−*κ***x**, where *κ* is the optical trap stiffness, and is analogous to the spring constant [4]. As such, **F**_{T} acts to counteract the particle’s displacement and restore it to an equilibrium position at the centre of the optical trap.

### (b) Optical set-up

The optical layout for all experiments, built around a commercially available Leica inverted optical microscope, is shown in figure 1, which can be divided into five components: (i) optical tweezers, (ii) microscope optics, (iii) illumination system, (iv) visual tracking system and (v) photodetection system.

The optical trapping system uses a diode-pumped, solid-state Nd:YVO_{4} laser operating at a wavelength of 1064 nm. The optical trap is created by tightly focusing the trapping beam, achieved using a microscope objective with a high numerical aperture of 1.30. A dichroic mirror, which selectively reflects radiation above 700 nm, directs the trapping beam towards the objective. The position of the beam in the trapping plane is controlled using galvanometer-controlled mirrors (G1, G2), and the position of the optical trap focus is controlled using stepper motors. The trapping beam is expanded to slightly overfill the back aperture of the microscope objective to ensure that the maximum gradient force is produced at the centre of the optical trap.

The sample is illuminated using a 100 W halogen bulb directed through a condenser lens. The sample can then be imaged using a digital camera, TV monitor and PC monitor or the human eye. The sample plane can also be imaged via a complementary metal oxide silicon (CMOS) camera, EC1280 manufactured by Prosilica (now Allied Vision Technologies), and an Apple Mac laptop screen that constitute the visual tracking system used for high-speed particle tracking. High-speed CMOS cameras can record the positions of multiple particles in real time simultaneously, offering frame rates reaching several kilohertz by controlling the field of view, and are relatively low cost. They also provide an accuracy reaching the thermal limit [13].

High-speed particle tracking and real-time analysis were performed using the virtual instrument software LabVIEW. Alternative particle-tracking algorithms were applied for lateral and axial tracking. For lateral tracking, the particle’s centre of mass (CoM) coordinates are determined using LabVIEW’s centroid algorithm, which computes the image’s centre of energy in pixels. Axial tracking was performed by calculating the standard deviation of the particle’s image, which is proportional to the size, or defocus, of the particle, as suggested by Crocker & Grier [14].

The illumination system can be replaced by the photodetection system, developed to detect and characterize the rotation of optically trapped LC droplets. The system consists of two linear polarizers (P1, P2) optimized for 1064 nm, and a photodetector. The first polarizer P1 is placed in the optical train prior to the sample to generate an optical trap with a high degree of linear polarization. An amplifier and discriminator circuit connect the system to a PC where the LabVIEW interface is used for measuring light transmitted through the droplet. A quarter-wave plate can be inserted to measure the Stokes parameters.

## 3. Anisotropic viscosity in liquid crystals

Defining LC characteristics plays a particularly significant role in improving liquid crystal display (LCD) technologies [15]. For example, determining a twisted nematic LC’s rotational viscosity coefficient or describing fluid flow, or *back-flow*, effects [16] is critical for improving LCD efficiency because they influence the device’s response times. Computer models for predicting back-flow effects often require several anisotropic viscosity coefficients, termed *back-flow coefficients*, which can be determined via their relation with shear viscosity coefficients measurable in classical flow experiments [16,17]. The effect of back-flow has itself been used to optically trap colloids [18], illustrating that progress in understanding both LC back-flow and optical trapping in LCs can be mutually beneficial.

In the past, viscosity coefficients have been determined by studying LC flow through a capillary via the application of a magnetic field and observing pressure differences [19,20]. Pasechnik *et al.* [17] used a decay flow method, inducing shear flow with optically controlled pressure differences, to determine back-flow viscosities for the LC mixture MLC-6609. For the majority of new LCs generated for LCD technology, research often focuses on determining LC refractive indices while other quantities, including viscosity coefficients, remain unknown. Ignoring such back-flow effects in LCDs allows poor prediction of their functionality and limits progress.

Established techniques of LC research conduct macroscopic investigations because they examine the bulk LC. Therefore, there is a strong call for microscopic, local investigations. Research involving LC–colloidal dispersions expands the knowledge and understanding of LCs and their anisotropic nature.

As in isotropic fluids, viscosity in LCs can be determined by observing the viscous, or Stokes, drag experienced by a particle moving through a fluid [21,22,23]. The Stokes drag is highly nonlinear for particles in a nematic solvent; the colloid (acting as a quadrupole) finds it easier to diffuse parallel to than perpendicular to it [24,25]. Further research by Verhoeff *et al.* [26] quantized this direction-dependent Stokes drag. They demonstrated that, for a colloid moving through a nematic LC, the viscous drag (VD) parallel to was larger than the drag perpendicular to by a factor of 2. Gleeson *et al.* [22] used the VD technique with optical tweezers to measure the effective viscosities of LC mixture MLC-6648 for motion parallel and perpendicular to it. The values allowed inference of the anchoring strength of colloidal particles within the LC mixture that was in good agreement with prediction.

Passive methods of determining LC viscosity, such as observing the anisotropic Brownian motion of colloids in LC media by analysing associated power spectra (PS) in the frequency domain, have remained largely underrepresented in the literature until recently [27]. Exploration is warranted into the idea that passive methods could be used to understand and characterize the behaviour of viscoelastic media, including LCs.

The relationship between and the nematic LC’s fluid velocity, termed *nematodynamics*, is described by the Ericksen–Leslie equations that include five independent Leslie coefficients *α*_{i} where *i*=1,2,…,5. Solutions to these equations are paramount for determining LCD switching times. These are typically calculated by determining effective viscosities in shear flow experiments, applying shear to an LC sample cell, in which is aligned electrically or magnetically.

The effective viscosity, *η*, of an LC medium is given by
3.1
where *η*_{1}, *η*_{2} and *η*_{3} are the three principal anisotropic viscosity coefficients, also known as the Mięsowicz viscosity coefficients, concerned with back-flow effects in an LC and, thus, switching times of an LC device. As shown in figure 2, the angles *ϑ* and *φ* define the orientation of relative to shear flow *v* and its gradient ∇*v*. *η*_{12} is the anisotropic viscosity coefficient associated with not being parallel to either *v* or ∇*v*.

In a three-dimensional scenario, the aniostropic viscosity coefficients *η*_{1} and *η*_{3} occur simultaneously and are inseparable, so are given by effective viscosity *η*^{⊥}_{eff} while *η*_{2} is given by *η*^{∥}_{eff}.

Experiments with optical tweezers allow the measurement of *η*^{⊥}_{eff} and *η*^{∥}_{eff}, for example, by observing three-dimensional position fluctuations of an optically trapped colloid immersed in the LC. For a homeotropic cell, fluctuations occurring in the *z* direction are parallel to , whereas those in the *x* and *y* directions are both perpendicular to . In a planar cell, fluctuations in the *z* direction are perpendicular to . Depending on the orientation of the planar cell, either the *x* or *y* will be parallel to .

### (a) Materials and methods

Details of the optical tweezing of polystrene and silica beads dispersed in different LC mixtures with homeotropic and planar alignment are given below. Active and passive measurement techniques were used to determine the anisotropic viscosities of the LC mixtures, the VD force method and PS analysis, where these are described below. More specifically, this allowed determination of effective anistropic shear viscosity coefficients. The experiments used the optical tweezing system shown in figure 1, specifically (i) optical tweezers, (ii) microscope optics, (iii) illumination system, and (iv) the visual tracking system.

#### (i) Ericksen number (*E*_{r})

In an LC, the contribution of elastic forces to the Stokes VD force can be quantified using the Ericksen number , where and *r* are velocity and length scales, e.g. the velocity and radius of a colloid immersed in the nematic LC, *α*_{4} is a Leslie coefficient, and *K*_{i} is a Frank elastic coefficient. At low Ericksen number *E*_{r}≪1, elastic forces exceed viscous forces, and so any elastic distortions of the LC director owing to the flow field, i.e. the particle’s motion, are considered negligible [22]. Therefore, the VD force acting on an optically trapped colloid immersed in an LC can be described by Stokes’ law. This allows the viscosity to be determined in the same way as for a purely viscous fluid.

The Ericksen number can be minimized by controlling the velocity length scale . In active VD force experiments, the translation stage was moved at a maximum speed of 70 μm s^{−1}. In passive microviscometry, is calculated as the speed of the particle’s Brownian motion. With characteristic values of *K*_{i}∼10 pN and *α*_{4}∼10^{−2} Ns m^{−2}, maximum values of *E*_{r} are given in table 1. These values illustrate that both experimental situations, VD and PS and the chosen LC–colloid combinations, facilitate experiments in the low Ericksen number regime. There is considerable difference between *E*_{r} for the two cases because is either calculated from the speed of the stage oscillation or given by the Brownian velocity of the trapped particle.

#### (ii) Liquid crystal cells

The LC cells used in this work consist of microspheres dispersed in aligned nematic LC mixture at relatively low concentrations—less than 5 per cent solids by weight. The specifications of the chosen LCs and colloids are given in table 1. These LCs were chosen such that they possessed: (i) sufficiently low refractive indices *n*_{e} and *n*_{o} to ensure successful optical trapping, i.e. less than that of the colloid to be trapped, (ii) a low *Δn* to facilitate efficient trapping, minimizing aberrations, (iii) a reasonable viscosity to allow colloidal motion, and (iv) a high clearing temperature (*T*_{c}) to avoid any possibility that the laser’s thermal heat caused any significant changes to the physical properties of the material. The nematic-to-isotropic phase transition temperature is of the order of 90^{°}C for both LCs in this case.

In addition, these LCs were chosen as their viscosity coefficients have been studied and measured previously, using an alternative measurement technique— shear flow—which was not in the low Ericksen number regime [17] or also using optical trapping and VD, and in the low Ericksen number regime, but using an alternative measurement technique [22].

The LC–colloid suspension is sandwiched between two coverslips, separated by Capton spacers and sealed with UV-sensitive glue. The LC cell is adhered to a microscope slide using nail varnish. Cell thickness *h*, measured using reflectance spectroscopy, varied between approximately 25 and 40 μm.

For homeotropic alignment, each coverslip was immersed in a silane (OTMS) mixture, and alignment is verified by heating and cooling the LC-filled cell past its phase transition temperature and observing a conoscopic cross. For planar alignment, polyimide is applied and a ‘rubbing’ technique used. Alignment is verified by observing minimum and maximum transmission when rotating the LC-filled cell between crossed polarizers.

#### (iii) Active microviscometry

The escape force is the minimum force required to cause a particle of radius *r* to escape from the optical trap, defined as
3.2
where **v**_{esc} is the velocity of the surrounding fluid, or in this case the sample stage, when the particle is released from the optical trap. At this point, **F**_{esc} is considered as the maximum trapping force proportional to the maximum displacement **F**_{max}=−*κ***x**_{max}. **F**_{esc} can be equated to equation (2.1) to measure **v**_{esc} as a function of *P*, thus allowing determination of *η*.

#### (iv) Passive microviscometry

The PS of an optically trapped particle’s Brownian motion as a function of frequency *f* is given by
3.3
where *k*_{B} is the Boltzmann constant, *T* is temperature and the drag coefficient is *γ*=6*πηr*. The corner frequency (*f*_{c}) describes a characteristic roll frequency *f*_{c}=*κ*/2*πγ*. For frequencies *f*≪*f*_{c}, the particle is confined by the optical trap, and the PS remains approximately constant. However, for frequencies *f*≫*f*_{c}, the particle experiences only Brownian motion and, as such, *S*(*f*) is proportional to 1/*f*^{2} giving a slope of −2, as expected for free diffusion. Equation (3.3) has an approximate Lorentzian profile in which the two distinct regions can be seen.

It is possible to use this analysis to determine the viscosity of the medium in which the particle is optically trapped, as given by
3.4
where 〈*x*^{2}〉 is the particle’s positional variance, or the distribution of positions visited by the trapped particle during thermal fluctuations [28]. It is possible to obtain 〈*x*^{2}〉 from either a histogram of positions visited by the particle, which has a Gaussian distribution, or the autocorrelation function at short time scales. The drag coefficient *γ*=6*πηr* has been incorporated into equation (3.4).

### (b) Results

Figure 3 presents three-dimensional PS for optically trapped 2 μm silica and 6 μm polystyrene beads dispersed in MLC-6648 and MLC-6609, respectively, for both homeotropic and planar alignment. Each spectrum and associated Lorenztian fit is produced from the time average of 100 consecutive PS recorded by the LabVIEW analysis program, performed three times for each dimension.

#### (i) MLC-6648

For the homeotropic cell, both the *x*- and *y*-directions are perpendicular to and so both experience *η*^{⊥}_{eff}, which is confirmed by the similarity in PS. The *z*-direction is parallel to and would, therefore, experience *η*^{∥}_{eff}. The difference in PS between the three different directions can clearly be seen when comparing the *f*_{c} associated with the position fluctuations in each direction, *x*, *y* and *z*. The values of *f*_{c}(*x*) and *f*_{c}(*y*) both exceed *f*_{c}(*z*) by approximately 45 per cent. It is known that, because *κ*_{xy} is always greater than *κ*_{z}, so *f*_{c}(*xy*)>*f*_{c}(*z*). However, we would expect a greater difference between them in water and thus the higher value of *f*_{c}(*z*) suggests that the particle is experiencing a reduced viscosity in the *z*-direction.

In comparison, for a cell of MLC-6648 with planar alignment, the *x*- and *z*-directions are parallel to , whereas *y* is perpendicular, producing clearly different PSD plots. This is again evident from the difference in corner frequencies, where *f*_{c}(*x*)>*f*_{c}(*y*), of the order of three times greater, showing that the *x*- and *y*-directions experience an alternative viscosity. Also, *f*_{c}(*x*)>*f*_{c}(*z*) because, again, lateral trap stiffness is stronger than axial trap stiffness.

#### (ii) MLC-6609

For the homeotropic cell of MLC-6609, the corner frequencies in all three dimensions are similar. This suggests that the optically trapped particle experiences a similar effective viscosity *η*^{⊥}_{eff} in the *x*- and *y*-directions, whereas the *z*-direction, although *κ*_{xy}>*κ*_{z}, experiences a reduced effective viscosity *η*^{∥}_{eff}.

For the planar cell of MLC-6609, *f*_{c}(*x*)>*f*_{c}(*y*)>*f*_{c}(*z*). Taking into account the usual differences between lateral and axial *κ*, this therefore suggests that the optically trapped particle experiences a different effective viscosity in the *x*-direction from that in the *y*- and *z*-directions, *η*^{∥}_{eff} and *η*^{⊥}_{eff}, respectively.

#### (iii) Effective viscosities *η*^{∥}_{eff} and *η*^{⊥}_{eff}

Values of *η*^{∥}_{eff} and *η*^{⊥}_{eff} were determined for homeotropic and planar cells of MLC-6648 and MLC-6609 using both passive and active microviscometry. These results are presented in figure 4, which illustrates the observed anisotropy using both methods. The labels *x*, *y* and *z* correspond to values obtained using the power spectra for that direction, whereas *xy* denotes the average of *x* and *y*, i.e. the lateral direction. On average, the value is found to be close to the average bulk viscosity values of each LC. The results also allow discussion on the anchoring of the LC at the particle surface, described by the ratio , where calculated values are given in table 2. There is considerable variation in the values of , all of which are higher than expected for weak anchoring. However, the results are consistent with the previous observation from active measurements [22] that strong anchoring is implied, where *η*^{⊥}_{eff}/*η*^{∥}_{eff}=2.0 for strong anchoring.

Chain-like structures were observed for 2 μm silica beads dispersed in MLC-6648 with planar alignment, suggesting that the LC molecules are homeotropically aligned at the colloid surface. The particles experience a strong attraction to one another in the direction of and the formation of particle chains lowers the free energy of the system.

It is most likely that a Saturn ring director field is present as these most commonly occur for particles of radii *r*<5 μm. For this type of director field, the homeotropic surface anchoring is weak, in comparison with topological defects that are due to strong homeotropic anchoring. This creates an isotropic distribution relative to the large convergence angle of the rays impinging upon the particle surface and hence minimizes any anisotropy of the trapping forces. This is the reason why one may therefore assume an effective refractive index [12].

Planar anchoring at the particle surface generates clustering of beads [29]. There was some evidence of this for silica beads in homeotropic cells of MLC-6648. The formation of clusters indicates the existence of long-range attractive forces, although repulsive forces are also present where the particles remain separated. It was possible to trap individual silica beads and detach them from the surrounding particles.

#### (iv) Trap stiffness

It is possible to estimate corresponding values of trap stiffness *κ* using values of *f*_{c} and *η*_{eff}. These are given in table 3. For the data shown in figure 3, values of *κ* were significantly higher than expected, of the order of two times higher, for any of the directions when compared with the same trapping power for a 6 μm polystyrene bead in water. This discrepancy is unlikely to be due to aberrations of the trapping beam, which has a greater effect in the axial (*z*) direction.

Similarly, in both isotropic and anisotropic samples, the particle was held away from the coverslip surface (approx. 10 μm) so as to minimize any boundary effects. As such, the increased trap stiffness values imply that the influence of the surfaces is much more important for LC systems, and perhaps other non-Newtonian fluids, than for isotropic materials.

## 4. Rotation of liquid crystal droplets in circularly and elliptically polarized optical tweezers

Advances and understanding of LC emulsions—microdroplets of LC, coated with a surfactant, dispersed in an isotropic liquid [30]—through optical manipulation has established the potential of LCs for optically driven micromachines. In a 1996 paper, Tamai *et al.* [31] reported second- and third-harmonic generation from optically trapped micrometre-sized nematic and ferroelectric LC droplets dispersed in water. This was closely followed in 1999 with Juodkazis *et al*. [32] reporting the rotation of optically trapped nematic LC droplets with circularly polarized light. Further work by the same group detailed the dependence of rotation frequency on droplet size [33] and the efficiency of optical torque transfer to droplets [34].

There have been several further reports demonstrating optical trapping and rotation of LC droplets in suspension, including nematic droplets with circularly polarized light [35,21,22], chiral nematic droplets with linearly polarized light [22,23], dye-doped nematic droplets using circularly and elliptically polarized light [36] and smectic LC droplets in circularly polarized light [37]. Interestingly, LC droplets have recently been used to fabricate micrometre-sized LC cylinders that were rotated in circularly polarized optical tweezers to measure the viscosity of the composite LC [38].

Spinning LC droplets have also been used to measure the viscosity of the host medium, such as heavy water (*D*_{2}O) [39,40]. The trapping laser beam transfers optical torque to the LC droplet and its electric field induces molecular reorganization within the LC droplet [41]. Gleeson *et al*. [22] discussed wave-plate behaviour as the mechanism responsible for observing LC droplet rotation but cited other possible mechanisms for the optical angular momentum transfer, including anisotropic scattering, absorption and the optical Fréedericksz transition. The reasons for the underlying physical processes continue to remain elusive and of interest.

### (a) Materials and methods

Details of the optical trapping of liquid droplets dispersed in water are given in the following. Experiments are performed to examine the influence of droplet geometry on trapping and rotation, with circularly and elliptically polarized trapping beams. LC droplets in optical tweezers with a linearly polarized trapping beam were also observed. The experiments used the optical tweezing system shown in figure 1, specifically (i) optical tweezers, (ii) microscope optics, (iii) illumination system, and (iv) photodetection system.

#### (i) Liquid crystal droplets

For droplets of nematic LC, E7 from Merck was used. Its high refractive index (*Δn*=0.225) ensured successful optical trapping, because **F**_{T} increases with the droplet’s refractive index. For the chiral nematic LC droplets, related chiral and achiral nematic systems were used: MDA-1444 and MDA-1445,^{1} also manufactured by Merck. These were chosen as they facilitated preparation of mixtures with a range of pitch lengths, which remained relatively constant as a function of temperature—a change in pitch length of the order of 1 nm per degree Celsius.

### (b) Observation of droplet rotation

An LC is intrinsically birefringent; however, the birefringence of a spherical droplet depends significantly on its director configuration. One may consider this a *structural birefingence*, where schematics of the director configurations considered in this work—bipolar, twisted bipolar, radial and Frank–Pryce—are shown in figure 5.

In circularly and elliptically polarized optical tweezers, nematic and chiral LC droplets were observed to rotate continuously as the beam polarization direction also varied continuously, as shown in figure 6.

The bipolar and radial nematic droplets exhibit two different rotation mechanisms: the bipolar droplet spins in the optical trap, whereas the centre radial droplet orbits about a circle. A similar observation is made for chiral nematic droplets. The Frank–Pryce droplet spins in the optical trap in the same way as the bipolar nematic droplet, whereas some chiral droplets adopt a variational Frank–Pryce director structure, whose centre orbits about a circle similar to the radial nematic droplet. Here, the variational structure occurs because the chiral droplet structure is strongly dependent on the ratio between the droplet size and pitch length. The structural transition between the twisted bipolar and Frank–Pryce structure is continuous. Finally, the twisted bipolar droplet behaves in the same way as the bipolar nematic droplet.

The rotational axis of droplets is displaced for certain droplets. We can define ℓ as the rotational axis, where ℓ=0 for the droplet centre. The optical torque depends on ℓ, which defines the passage of light through the droplet. Optical force can be calculated numerically assuming the geometric optics regime. Values of ℓ depend on shape birefringence—the symmetry of the director structure in the droplet. Further details of rotation for each droplet type are given in table 4. The dynamics of the rotating droplet depend on optical torque, moment of inertia and drag, as shall now be described in further detail.

As shown in figure 7, these observations allow definition of two general types of droplet rotation: (*a*) the spherical droplet’s rotational axis is defined as parallel to the original rotational axis passing through the droplet’s CoM or (*b*) the axes are displaced from one another by ℓ (ℓ≥0).

### (c) Optical torque

The resultant optical torque acting upon the droplet owing to the birefringent nature of the LC causes the rotation of the droplet in the optical trap. This optical torque can be quantified by calculating the change in angular momentum between the light incident upon and transmitted through the droplet.

This angular momentum change induces a torque per unit area *τ* [42],
4.1
where *P* is the laser power per unit area, *ω* is its angular frequency, *ϕ* the degree of ellipticity and *θ* is the angle between the fast axis of the wave plate producing the elliptically polarized light and the optical axis of the droplet. The torque is dependent on the birefringent material’s thickness *d*, as described by the retardation of the emergent light
4.2
Here, *Δn*_{eff} represents the effective birefringence, *n*_{o}−*n*_{e}. Thus, the retardation is the phase difference between the light field components experiencing the ordinary *n*_{o} and extraordinary *n*_{e} refractive indices of the birefringent material. *Γ* takes into account the droplet’s director structure.

The first term in equation (4.1) is the torque owing to the linearly polarized component of incident light, whereas the second term is owing to the change in polarization of the light emerging from the birefringent material.

The viscosity of the droplet is much greater than that of the surrounding medium, typically, and in this case, water. Therefore, the optically driven LC droplet can be regarded as the rotation of a rigid body until the optical torque no longer exists. The optical torque acting on the droplet is given by
4.3
where *α* is the angular velocity of the droplet, and *I* and *D* are the inertia and drag–damping factor of the rotating droplet, respectively. Equation (4.3) describes the droplet’s drag–damped rotational behaviour.

The moment of inertia for the droplet depends on the displacement of the parallel axis away from the centre axis, ℓ, and is given by *I*_{sphere}+*m*ℓ^{2}. In addition, the drag–damping factor for the rotating droplet in water is given by *D*=8*πη*(*r*+ℓ)^{3}, where *m* and *r* are the mass and radius of the droplet, and *η* is the viscosity of the surrounding medium–water. Furthermore, *τ*_{d} also depends on ℓ because *Γ* is affected by each droplet’s director configuration. Thus, it is possible to express equation (4.3) as a function of ℓ, *τ*_{d}(ℓ), by substituting expressions for *I* and *D*.

### (d) Optical reorientation of droplet director structure

The optical torque is highly dependent on the optical reorientation of the droplet’s director, which, as shown in figure 6, has the ability to break the droplet’s rotational symmetry about the optical trapping beam axis. Brasselet *et al.* [41] indicated that the optical realignment within the radial droplet depends on the beam polarization. They showed that displacement ℓ decreases with increasing beam ellipticity, tending towards zero as ellipticity approaches *π*/4 (circular polarization). Figure 8 illustrates the optical reorientation of a bipolar and radial droplet when held in an elliptical optical trap.

The director configuration for all droplet types along the beam axis is shown in figure 9. For the bipolar, twisted bipolar and Frank–Pryce droplets, ℓ=0 and is independent of beam polarization, meaning that the beam axis coincides with the rotational axis of the droplet. In these cases, *Δn*_{eff} is independent of the azimuthal angle *ϕ* where the vectors lie perpendicular to the beam axis. Therefore, the effective birefingence is *n*_{a}=*n*_{e}−*n*_{o} and, so, equation (4.2) can be applied as a simple approximation for these director structures.

Conversely, for a radial droplet, ℓ≠0 and its effective birefringence has a non-trivial expression, derived geometrically to account for all components of *Δn*(*z*) for all angles that occur when the beam is transmitted through a particle off-axis,
4.4
where and , which is the polar angle of the vector depending on the director position *z*.

### (e) Dynamics of droplet rotation: simulation and experiment

It is now possible to use the information about the droplet’s trapping position ℓ and optical torque *τ*_{d}(ℓ) to numerically simulate the dynamics of LC droplets in optical tweezers. To do so, *θ* is considered as the cumulative angle with respect to time, such that
4.5
where the time period *t* is divided into *N* equal, small periods. Here, *α*_{i} is the angular velocity of the droplet after the *i*th rotation. Substituting equations (4.2) and (4.5) into the expression for the optical torque acting on the droplet (equation (4.3)), a general formula that describes the dynamics of the optically trapped droplet is produced
4.6
Here, *N* is the number of samples, *θ*_{0} is the initial angle between the droplet director and the major axis of the polarization and *ϕ* is, again, the degree of ellipticity of the trapping beam. The subscript ‘bi’ denotes bipolar.

The numerical simulations are applied for a bipolar droplet with a trapping power *P*=50 mW and wavelength *λ*=1064 nm to replicate experimental conditions. The droplet radius *r*=10 μm, trapping position ℓ=0 and birefringence *Δn*=0.2. The viscosity of water at 20^{°}C was used (1.003 cP) and *θ*_{0}=*π*/3 for all simulations.

The numerical simulations lead to plots of angular velocity *α* and rotated angular distance *Ω* as a function of time, as shown in figure 10. For a circularly polarized optical trap (*ϕ*=*π*/4), the result indicates that the rotational dynamics of the bipolar droplet exhibits a linear behaviour—the droplet rotates at a constant angular velocity *α*. Therefore, *Ω* increases linearly with time.

For an elliptically polarized trap (*ϕ*=*π*/3), the rotational dynamics of the bipolar droplet shows unique, nonlinear behaviour, where the droplet rotates at a varying angular velocity. Furthermore, the variation in *α* follows a repeating and regular pattern: fast and slow. This, therefore, produces a nonlinear increase in *Ω*.

Measured optical transmissions of rotating bipolar droplets in circularly and elliptically polarized optical tweezers are shown in figure 11. The droplet diameters *d* are 9.7±0.3 and 8.9±0.3 μm, respectively. For circularly polarized optical tweezers (*ϕ*=*π*/4), the period of rotation as a function of time is collected and used to calculate the angular velocity *α* of the rotating droplet. The experimental result, shown in figure 11*b*, follows the trend of the theoretical simulation of figure 10*a* in that the angular velocity, *α*, is constant with time. The actual value of *α* is very sensitive to the effective birefringence and the diameter of the droplet, neither of which are known accurately, leading to large systematic errors. This can be seen from the term in equation (4.6), and renders comparison with theory difficult.

For the bipolar droplet in elliptically polarized optical tweezers, the experimental result shown in figure 11*c* is in good agreement with the theoretical simulation of figure 10*b*. Let us consider the elliptical beam, which consists of both a linear and circular component. The circular component provides the constant optical torque, whereas the linear component offers a varying optical torque depending on *θ*, the angle between the beam polarization and the droplet director (see equation (4.1)). This angle varies between 0 and 2*π*, thus the first term in equation (4.1) repeats with regular positive/negative values. This implies that the optical torque owing to the linear component is regularly driving and resisting the droplet’s rotation.

Further experiments were conducted to examine how droplet size affected the rotational velocity of the rotating droplet. The numerical simulations were performed and experimental measurements made for bipolar droplets in elliptically polarized optical tweezers. As shown in figure 11*d*, the droplet dynamics are characterized by marked nonlinear behaviour where experimental measurements agree well with the numerical simulation.

## 5. Rotation of liquid crystal droplets in linearly polarized optical tweezers

A numerical simulation for a bipolar droplet in linearly polarized optical tweezers (*ϕ*=0) is performed and shown in figure 12, with the same simulation parameters as for circularly and elliptically polarized beams. The simulation illustrates that the bipolar droplet rotates for *π*/3 for approximately 1 s to an equilibrium orientation with no further rotation, as would be observed for a wave plate.

Figure 13 shows the time evolution of a chiral nematic LC droplet in an optical trap with linear polarization. It illustrates that the droplet experiences a torque until the droplet’s director is aligned with respect to the beam polarization, which is also observed for a nematic LC droplet. At this point, *ϕ*=0, and, thus, the torque *τ*_{d}=0. In this situation, the extraordinary and ordinary rays no longer deviate into two directions with two alternate velocities.

A rather different rotation is observed for chiral nematic droplets of a critical size, which adopt a twisted bipolar structure, as reported in Yang *et al.* [23]. The equilibrium orientation is related to the configuration of the droplet and so depends on droplet size and the pitch length of the chiral material. The rotation occurs only for small droplets of a critical size and is accompanied by a distinct vertical motion of the droplet in the trap.

The mechanism responsible for the anomalous motion of the chiral droplets is quite different from any previously reported that produces rotation of birefringent particles in optical tweezers. It relies on a photo-induced molecular rotation that induces a critical reorganization of the LC in the droplet, followed by elastic relaxation of the photo-induced structure. A cycle of molecular reorganization and relaxation follows. This fascinating combination of continuous rotation and linear motion when the chiral droplets are held in optical tweezers had not previously been observed and generates interesting implications for optical switching microfluidics.

## 6. Conclusions

LCs and particles are fascinating media and the hybrid systems do far more than merely combine properties.

Microrheology allows measurement of local viscosity in anisotropic systems. In this case, experiments were performed in the low Ericksen number regime, where anisotropy in the LC’s viscosity is observed with passive microviscometry. Values of effective viscosity obtained were close to the bulk viscosity value. However, the results produced unexpectedly high values of trap stiffness. It is possible that surface effects in the LC cell, i.e. the coverslip, are influential. Therefore, it illustrates the importance of understanding how laser manipulation works in non-Newtonian systems.

The rotation of LC droplets leads to the prospect of all-optical switches. The rotation observed here depends on the droplet phase, geometry and polarization with respect to the trapping beam. The phenomena include rotation on- and off-axes; continuous, uniform rotation; nonlinear rotation; and continual rotation until some equilibrium position is reached. The experimental observations are all readily simulated by a simple model. The novel observation of continuous rotation for specific kinds of *N** droplets suggests the opportunity for an opto-mechanical transducer.

## Acknowledgements

Funding for equipment and a studentship is gratefully acknowledged from the Engineering and Physical Sciences Research Council. Thanks are due to Prof. Miles Padgett and Richard Bowman of the Optical Trapping Group at the University of Glasgow for their expertise on high-speed particle tracking.

## Footnotes

One contribution of 14 to a Theo Murphy Meeting Issue ‘New frontiers in anisotropic fluid–particle composites’.

↵1 Also known as BL131 and BL131a.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.