Laser manipulation in liquid crystals: an approach to microfluidics and micromachines

Helen F Gleeson, Tiffany A Wood, Mark Dickinson

Abstract

Laser trapping of particles in three dimensions can occur as a result of the refraction of strongly focused light through micrometre-sized particles. The use of this effect to produce laser tweezers is extremely common in fields such as biology, but it is only relatively recently that the technique has been applied to liquid crystals (LCs). The possibilities are exciting: droplets of LCs can be trapped, moved and rotated in an isotropic fluid medium, or both particles and defects can be trapped and manipulated within a liquid crystalline medium. This paper considers both the possibilities. The mechanism of transfer of optical angular momentum from circularly polarized light to small droplets of nematic LCs is described. Further, it is shown that droplets of chiral LCs can be made to rotate when illuminated with linearly polarized light and possible mechanisms are discussed. The trapping and manipulation of micrometre-sized particles in an aligned LC medium is used to provide a measure of local shear viscosity coefficients and a unique test of theory at low Ericksen number in LCs.

Keywords:

1. Introduction

Optical trapping has become a popular tool for manipulating micrometre-sized, dielectric particles in three dimensions. This technique is particularly useful in biology and medicine since it is non-invasive and near-infrared radiation, commonly used for the traps, has low absorption in biological tissue, thus not causing damage to samples. Near-infrared optical tweezers have been used in applications, such as laser-assisted in vitro fertilization (Konig et al. 1996) and nanosurgery (Konig 2000), while the sensitivity of traps to piconewton forces has enabled the forces involved in breaking the base pairs of DNA to be measured (Bockelmann et al. 2002) and the biomechanics of molecular motor proteins to be studied (Kojima et al. 1997). Laser tweezers further provide an extremely useful tool for microrheology; for example, facilitating the study of the hydrodynamic interactions between colloids in suspension (Henderson et al. 2002). The advantages of remote control of microscopic particles and molecules and the sensitivity of the optical trap to piconewton-scale forces make optical tweezers an extremely useful tool in modern science and technology.

Optical trapping in liquid crystalline systems is still in its infancy. Liquid crystals (LCs) are fluids of wide-ranging relevance: thermotropic LCs are widely used in flat-panel display devices (Demus & Goodby 1998); many biological materials and functions rely on the liquid crystalline state (Collings 1990); and detergents and some food products are often processed via or exist in lyotropic liquid crystalline states. Since LCs are also non-Newtonian, optically anisotropic fluids, they present a particularly interesting challenge in the context of optical trapping and the technique yields some fascinating information and phenomena. There are clearly several exciting approaches to the concept of optical trapping and LCs. First, it is possible to form droplets of liquid crystalline material dispersed in an isotropic fluid, and to study the phenomena associated with such systems. Since LCs are birefringent fluids, it is possible to transfer optical angular momentum to such droplets, producing rotating droplets. This is the first topic considered in this paper. Second, it is possible to study the trapping phenomena in liquid crystalline materials themselves, using optical tweezers to study the motion and properties of colloidal particles or defects, and thereby providing a microscopic approach to the study of viscosity and elasticity in LCs. This area of work forms the second part of this paper.

2. The transfer of optical angular momentum to liquid crystalline droplets

The transfer of orbital angular momentum from light to particles has been of interest since the experiments of Bethe (1936) on quartz plates. The invention of optical trapping enabled experiments to be undertaken on micrometre-sized calcite particles that acted as wave plates (Friese et al. 1998). Nematic LC droplets in water will normally form a birefringent dipolar structure, with the LC director (defining the average direction of orientation) endeavouring to lie parallel to the LC–water interface. Juodkazis et al. (1999) demonstrated that nematic liquid crystalline droplets rotate with frequencies of the order of 103 Hz when illuminated with circularly polarized light, thus forming a fast, optically addressed, optical switch when viewed between crossed polarizers. This laser manipulation phenomenon can be considered as an ‘active’ micromachine, in the sense that the light not only traps the particle, but also exerts torque on the particle resulting in a switch. Several mechanisms have been suggested for the transfer of optical angular momentum to nematic droplets, including wave plate behaviour, absorption and anisotropic scattering. A recent study (Wood et al. 2004) of LCs with birefringence values between 0.15 and 0.26 and for droplets varying in size from 1 to 20 μm demonstrated that the rate of rotation of the droplet was highly dependent on the material birefringence and the droplet diameter. It was deduced (Juodkazis et al. 2003; Wood et al. 2004) that the retardation properties of the droplet (it acted like a wave plate) were dominant in allowing the transfer of beam angular momentum to the droplet.

The optical trapping of chiral nematic materials, systems that include chiral molecules and adopt a helicoidal superstructure, has not previously been considered, although such materials add new dimensions to the possibilities of laser-manipulated optical switches owing to their unique structure and optical properties. The pitch length, p, of a chiral nematic material describes the length over which the local director, Embedded Image, rotates by 2π. A characteristic property of chiral nematic samples aligned with their helicoidal axis parallel to the direction of incident light is that a band of wavelengths are selectively reflected from a sample owing to the Bragg reflection. The selectively reflected light is circularly polarized with the same handedness as the chiral nematic helix. The central wavelength, λ0, and the spectral width, Δλ, of the reflected light are related to the pitch, the average refractive index, Embedded Image, and the optical anisotropy, Δn, of the material by equations (2.1) and (2.2),Embedded Image(2.1)

Embedded Image(2.2)

When the wavelength of the incident light, λ, is much less than the selective reflection wavelength (i.e. λλ0), a waveguide regime exists in which the electric vector of a linearly polarized wave follows the rotation of the director about the helix (Blinov & Chigrinov 1996).

The director structure within a chiral nematic droplet has been considered (Xu & Crooker 1997), showing that the configuration within a droplet is strongly dependent on the relationship between the particle radius, r, and the helicoidal pitch, p. In all cases, the director is considered to be parallel to the droplet interface (planar alignment) at the droplet surface. In the high chirality regime (pr), the helicoidal axis is radial from the centre of the droplet, as shown in figure 1a. When p<r, the structure adopted has a single radial disclination and the droplet exhibits either a concentric ring pattern with a radial disclination (Frank–Pryce structure) or a spiral pattern with no disclination, depending on the direction from which the droplet is viewed. When p>r, a fully twisted bipolar configuration is observed, analogous to the simple bipolar configuration of a nematic droplet in which the droplet has an effective axis. However, in the chiral nematic droplet, the director twists from one side of the droplet axis to the other, as shown in figure 1b. Further, the molecular director undergoes a rotation across the axis, such that the LC director at the surface, far from the droplet axis, is normal to the droplet axis, as shown in figure 1c.

Figure 1

The director fields in chiral nematic droplets in the cases (a) pr, where a concentric ring pattern occurs, and (b, c) p>r, where a fully twisted bipolar configuration occurs (after Xu et al. 1994). In (b), the droplet axis is viewed from the side and in (c), a section is taken across the central axis. The ‘nails’ indicate the direction of twist of the chiral nematic material within the droplet by depicting the direction of the local director (a shorter ‘nail’ indicates that the director is pointing out of the plane of the diagram).

Optical trapping forces result from the reflection and the refraction of light at the surfaces of a particle. Two factors are of particular interest in the optical trapping of chiral nematic droplets: the extent to which selective reflection affects the optical trapping forces acting; and whether the optical properties of chiral nematic materials allow angular momentum to be transferred from the beam to the droplet, allowing new phenomena to be observed. In the selective reflection regime, it is expected that the scattering force acting along the direction of beam propagation is increased (owing to the Bragg reflection of light), thus decreasing the axial trapping force that allows the droplet to be trapped in three dimensions. This is readily studied experimentally through trapping experiments on chiral nematic droplets, where pr. The more complex director geometry in droplets, where the pitch and the droplet radius place the experiment in the waveguide regime, provides a novel system for the study of the transfer of angular momentum, since there is an inherent chirality within the droplets.

(a) Experiment 1. Chiral nematic droplets

Chiral nematic mixtures of different pitch lengths were produced by combining commercially available nematic and chiral nematic materials (MDA-00-1444 and MDA-00-1445, respectively). These materials were chosen because their pitch and refractive indices are not strongly temperature dependent (Roberts et al. 2004), ensuring that any possible heating in the trap had a minimal effect on the experiments performed. The reflection spectra of the mixtures that reflect in the visible region were measured directly using well-aligned planar samples. Fitting the spectra to theory allowed the pitch and the ordinary and extraordinary refractive indices of the mixtures to be accurately determined (Roberts et al. 2003). The pitch and the reflection spectra were calculated for mixtures, where the selective reflection band was above 700 nm, using equation (2.3) (Gleeson & Coles 1989) and an optical model describing the chiral nematic structure,Embedded Image(2.3)

In equation (2.3), pt is the total pitch of the mixture and pi and ci are the pitch and the concentration of individual components, respectively. For a nematic material, 1/pi=0. This approach readily identifies the mixtures that selectively reflect the trapping beam of wavelength 1064 nm in the case of normal incidence.

A suspension of chiral nematic droplets was created by dispersing a small amount of each mixture in distilled deionized water, so that spherical chiral nematic droplets are formed. A range of droplet diameters between 2 and 7 μm were studied. The optical trap was produced by focusing a linearly polarized 1064 nm beam from a Nd : YVO4 diode-pumped solid-state laser through an objective numerical aperture of 1.3. The position of the beam in the trapping plane was controlled using galvanometer-steered mirrors and the position of the optical trap focus was controlled using a stepper motor attached to the microscope objective. The sample was illuminated with white light and imaged onto a charge-coupled device (CCD) camera.

An established method for measuring optical trapping forces is to apply a known viscous drag force to the optically trapped particle and increase this applied force until it is sufficient to release the particle from the trap. At the point of release, it is assumed that the maximum optical trapping force acting on the particle is equal to the viscous drag force. In the case of low Re, the magnitude of the viscous drag force Fs, experienced by a spherical particle of radius r, moving with velocity u, relative to a fluid of viscosity μ, is governed by Stokes' law according toEmbedded Image(2.4)

The viscosity of the chiral nematic droplets is around 95 cP, two orders of magnitude greater than the viscosity of water. In the cases where the viscosity of the suspended droplet is so much greater than the surrounding medium, it may be assumed that the droplets retain their spherical shape when a viscous drag force is applied. In the case of LC droplets, viscoelasticity rather than isotropic viscosity is likely to be important, but it is nonetheless reasonable to assume that the droplets remain spherical.

The motion of the objective was computer-controlled, allowing a trapped droplet to be moved at a known speed with an amplitude of 20 μm. A sinusoidal displacement function of the microscope objective was chosen, ensuring that the maximum speed was reached at the central trap depth of 20 μm. This prevented the particle from being lost at greater trap depths, where it has been shown that spherical aberrations significantly reduce the axial trapping force. The galvanometer-controlled mirrors were simultaneously moved slowly, so that the trap described a circle of around 20 μm in diameter in the trapping plane. This process avoided direct recapture of the droplet, so it was clear when the viscous drag force had exceeded the axial trapping force. The beam power was held constant at 50 mW while the velocity of the trap in the axial direction was increased until the droplet was released from the trap.

To study droplet rotation, polystyrene particles of 1 μm in diameter were added to the suspensions, so that any rotation of the chiral nematic droplets could be observed through the orbiting motion of the polystyrene tracer particles around each droplet. This was necessary since observation of the chiral nematic droplets between crossed polarizers does not allow direct observation of the rotation process, which is possible for nematic droplets (see movies 1 and 2 in the electronic supplementary material, which illustrate the potential of these rotatable droplets as optically controlled optical microswitches and micromachines). The measurement of droplet rotation speeds was made from the observation of the orbiting motion of the polystyrene particles. The relative intensities of σ and π polarizations passing through the droplet were also measured, allowing an insight into the mechanism for angular momentum transfer to the droplets. Two quadrant photodiodes (QPDs) detecting σ and π light were centred on the position of the particle in the π trap (this was done by bringing the σ and π beams together, trapping a particle in the combined beam to centre both QPDs and, finally, blocking the σ beam before measurements). During measurements, 222 data points were collected at a rate of 20 kHz, giving information on the intensity of laser light reaching each QPD with time (around 4 min). The trapping beam was linearly polarized along the π direction, but a polarizing beam splitter allowed σ and π polarized light to be collected on separate QPDs.

(b) Trapping of chiral nematic droplets

The theoretical calculation of the reflectivity of a linearly polarized beam of wavelength 1064 nm by planar LC samples, for the various mixtures and for sample thicknesses 1, 3 and 5 μm, is presented in figure 2a. The calculations indicate that selective reflection of a 1064 nm beam occurs in mixtures, where the concentration of the chiral nematic component is between 50 and 55%. Measurements of the axial escape speed of the chiral nematic droplets are displayed in figure 2b. A greater escape speed implies that the axial force acting on the droplet is stronger. At high (60–100%) and very low (less than 9%) concentrations of the chiral nematic component, the axial speed at which the particle escapes (the axial escape speed) is high, around 40 μm s−1, corresponding to strong axial optical trapping forces of around 2 pN. At a mixture concentration of 42%, the droplets were strongly attracted to the beam focus but as they passed through the focus, they scattered rapidly with zero axial trapping force, thus making the trapping of any droplet impossible at this concentration. Figure 2b shows that the force measurements are affected by selective reflection effects across material concentrations of around 20%, corresponding to a bandwidth of approximately ±200 nm, while figure 2a shows that selective reflection of 1064 nm light occurs only for a concentration range of 10%.

Figure 2

(a) The calculated reflectivity for slabs of planar-aligned chiral nematic samples of thicknesses 3 (circles) and 5 μm (squares) for the MDA-00-1444/1445 mixtures of different concentrations (expressed as wt%) and (b) measurements of the axial escape speed required to release a chiral nematic droplet of diameter 4.5±2.5 μm from an optical trap (beam power, 50 mW) as a function of the concentration of the chiral nematic MDA-00-1445.

The measurements of the axial trapping force and the calculations of the selective reflection of 1064 nm light from the chiral nematic mixtures show reasonable qualitative agreement; there is an obvious concentration regime where it is impossible to trap a particle. The fact that the mixture compositions for the calculation and measurement did not agree exactly merits some discussion. There are several possible contributions to the mismatch. First, the droplet structure can distort the pitch. The radii of the droplets were 4.5±2.5 μm, so, for selective reflection of 1064 nm light, the droplet radius is greater than the pitch (approx. 665 nm). As already discussed, in this regime the axis of the chiral helix aligns radially, with a planar (parallel) director at the surface of the droplet. Such a structure can distort the equilibrium pitch because anchoring of the director is at the surface and there is a requirement of a whole number of half-pitches radially. This phenomenon can slightly increase or decrease the effective pitch, depending on the droplet size. However, this effect would be rather small (not more than half a pitch) and so cannot fully account for the significant effective difference in pitch from the calculated and measured data. A second possibility is that the director within the droplet undergoes an optical Freedericksz transition, as has been reported for nematic droplets (Wood et al. 2004; Murazawa et al. 2005). However, in such a case, it is most probable that the pitch would unwind somewhat, moving the concentration regime for rejecting the trap again to higher concentrations, contrary to the effect observed.

Angular effects must also be taken into account in examining the apparent shift of the selective reflection wavelength. In chiral, nematic materials, the selective reflection peak is broadened and the central reflection wavelength, λ0, is shifted to a smaller wavelength, λθ, when a sample is illuminated or viewed at an angle (θi or θr), according to equation (2.5) (Fergason 1966),Embedded Image(2.5)

Angular contributions (as a result of both the random alignment of the droplet axis and the angular dependence of the incident light) would tend to broaden the band over which light is reflected (in our case, the concentration regime) and shift the selective reflection to shorter wavelengths. Therefore, a relatively low concentration mixture that selectively reflects at normal incidence at wavelengths greater than 1064 nm will, at an angle, reflect shorter wavelengths, i.e. in this case, the laser. This is clearly the mechanism responsible for both broadening and shifting the concentration regime in which trapping is difficult or impossible, compared to the ideal case of a slab of material where only normal incidence is considered.

(c) Rotation of chiral nematic droplets in linearly polarized light

Polystyrene particles in suspension were not observed to orbit the chiral nematic droplets at any concentration. However, as shown in figure 3, there was evidence of the transfer of angular momentum to the 9.7% concentration chiral nematic droplets, though not for higher concentrations. At a concentration of 9.7%, the material has a pitch of approximately 14 μm. In figure 3, the appearance of the droplet reveals a twisted bipolar structure, similar to that illustrated in figure 1b,c (Xu et al. 1994). The droplets were observed to rotate slowly within the trap, with a rotation rate of the order of 1 Hz. This result is surprising since there is no obvious mechanism by which angular momentum can be transferred from the incident light to the droplet. The phenomenon was examined in more detail by determining the relative intensities of the σ and π polarized light transmitted by the droplets over a range of mixture concentrations for incident π polarization. These measurements demonstrated that the beam polarization was largely unaffected on transmission through the droplet. However, for droplets of low chirality (4.9–9.5%), the intensities of the σ and π signals were observed to vary with the droplet diameter, with the σ signal reaching values greater than the π signal. A power spectrum study on a random selection of droplets did not give any evidence that the polarization signals were modulating while the droplets were trapped.

Figure 3

Successive photographs of a chiral nematic droplet, 4 μm in diameter, with pλ rotating within an optical trap formed from a linearly polarized beam (approx. 1000×).

The regime where droplet rotation was observed corresponds to the situation in which the π and σ components of the light transmitted by the droplet have approximately equal intensities. In such circumstances, it is possible that the droplet has converted the incident linear light to circular polarization. Nonetheless, the observation of rotation cannot be explained purely by wave plate phenomena, even if the droplet was acting as a quarter wave plate. In that case, the droplet could rotate until one of the axes (fast or slow) aligned with the incident polarization and then, owing to viscous drag, it would stop rotating, such that plane-polarized light emerged. The droplet would continue to rotate with circularly polarized light only if the incident plane of polarization also rotates. This is in agreement with the fully twisted dipolar configuration suggested for droplets having p>r, thus the chirality of the LC is important in the process. It is also possible that an asymmetric scattering process (in addition to the anisotropic scattering) could be responsible for the slow rotation of the chiral droplet. Though this suggestion has not yet been tested in detail, it is difficult to think of another mechanism that could explain the observation. The circumstances under which such droplets rotate, and possible explanations, certainly warrant further investigation.

3. Microrheology of liquid crystals

Poulin et al. (1997) were the first to demonstrate the novel properties exhibited by composites formed from colloidal (micrometre-sized) particles dispersed in LC materials. As the LC director field is forced to distort around the particle, the global free energy of the system is minimized through particle aggregation, which condenses the defects and leaves large regions of undistorted nematic material. The LC–colloid surface interaction has a strong influence on the structures formed; chain-like aggregates form for perpendicular (homeotropic) anchoring, while parallel (planar) anchoring leads to the formation of particle clusters (Poulin & Weitz 1998).

There have been very few reports of optical trapping of colloids in LCs. In one application (Iwashita & Tanaka 2003), disclinations were created by optically dragging trapped particles through a lyotropic LC and the rate of repair of the defects subsequently measured. The anisotropic force field acting between two colloidal particles in a thermotropic nematic LC was studied by Yada et al. (2004), while Musevic et al. (2004) reported the formation of ‘ghost’ particles in a relatively high refractive index nematic material, inducing the trapping of sub-micron particles owing to optically nonlinear effects. Most recently, Smalyukh et al. (2005) use laser trapping to probe the ordered structures and elastic–capillary interactions between colloidal particles at the air–LC interface. Optical trapping is clearly finding use in the study of several fundamental aspects of LCs, and it is an ideal method for the study of local hydrodynamic effects in LCs.

Hydrodynamic properties of LCs are of particular interest, because the fluids are anisotropic and non-Newtonian. However, comparison of experimental data with theory is problematic because most experiments measure in a highly nonlinear regime, while the theory tends to concentrate on the linear regime. A convenient parameter to distinguish the regimes is the Ericksen number Er, defined by Embedded Image, where v and r are characteristic velocity and lengthscales (in this case, the particle velocity and radius); α4, a Leslie coefficient; and K, a (single) Frank elastic constant of the nematic material. When Er≪1, the elastic distortions of the director are negligible, so the viscous drag force, Fd, acting upon a spherical particle can be described by Stokes' law (Ruhwandl & Terentjev 1996; Stark & Ventzki 2001). The very low velocities at which particles can be moved using an optical trap allow measurements to be made in this regime. Further, control of the sample geometry allows the anisotropic viscosity to be probed.

(a) Experiment 2. Microrheology of nematic liquid crystals

Polystyrene spheres (6 μm in diameter) of refractive index ns=1.59 were dispersed at very low concentrations (less than 0.01% solids by weight) in the nematic LC MLC-6648, produced by Merck Ltd. Both the LC ordinary and extraordinary refractive indices are lower than the particle index, as required for trapping (n0=1.476 and ne=1.546 at 589 nm), and the low birefringence minimizes distortion of the trap. Sample cells were constructed using glass slides held 150 μm apart using Kapton spacers, with the inner surfaces treated to provide good quality planar (parallel) alignment of the director at the LC–glass interface.

The sample was viewed in the trapping apparatus (identical to that described in §2a) using polarizing microscopy to determine the LC alignment around the dispersed particles, and hence obtain an indication of the anchoring strength, related to the symmetry of the alignment observed around the particle. The particles were observed to form chain-like structures within the LC, thus suggesting that the director is probably homeotropic (perpendicular) at the colloid–LC interface (Poulin et al. 1994; Chaikin & Lubensky 1998). The symmetrical director alignment around an individual particle is shown in figure 4. This pattern suggests a ‘Saturn-ring’ defect, implying a relatively weak anchoring of the director at the particle surface (Poulin & Weitz 1998), though it is worth noting that the transition from ‘weak’ to ‘strong’ anchoring is a continuum.

Figure 4

Photograph of a 6 μm diameter polystyrene particle dispersed in MLC-6648 and viewed between crossed polarizers, showing the symmetric transmission pattern (approx. 1000×).

(b) Confirmation of the low Ericksen number regime

The particles were trapped at a trap depth of 20 μm and the sample stage was moved horizontally with a triangular displacement function of 50 μm amplitude. The optical power was held constant at values between 10 and 50 mW (as measured at the back aperture of the microscope objective) and the sample stage speed increased until the particle left the trap. It is possible to move the particles parallel or perpendicular to the alignment direction within the device, allowing measurement of either of the two effective viscosities Embedded Image or Embedded Image, i.e. parallel or perpendicular to the director. The sample stage was moved with the fastest speed of 70 μm s−1, resulting in a maximum Ericksen number, Er=0.1, i.e. Er≪1. In accordance with Stokes' law, figure 5 shows that the escape velocity of a particle is a linear function of the beam power, confirming that, where Er<0.1, colloidal dynamics within a nematic LC are linear for either translation directions. It follows that nonlinear effects that could result from the trap itself (Musevic et al. 2004) are not important in these measurements. The effect of beam polarization was minimal as rotation of the plane of polarization of the trapping beam by 90° influenced measurements of the escape velocity by less than ±1 μm s−1. From figure 5, it can also be seen that the direction of motion has a significant effect on the speed at which the particle is released from the trap. In the graph of the escape velocity as a function of the beam power, the steeper gradient suggests a weaker Stokes' drag force or a stronger optical trap. The anisotropy of the optical trapping forces is considered to be low owing to the steep angle of convergence of the incident light forming the trap.

Figure 5

The stage velocity required for a particle to escape the optical trap as a function of the beam power (measured at the back aperture of the objective) for motion along (squares) and across (circles) the nematic director in a planar sample. The effective local viscosity may be deduced from a least-squares fit to the data.

(c) Anisotropic local shear viscosity measurements

The local shear viscosity of the nematic material can be deduced by equating the force on the particle owing to the trap with the Stokes' force. The trap strength depends on the difference between the refractive indices of the particle and the surrounding medium. As already noted, the director configuration surrounding the particle (a layer of thickness around 1 μm) was homeotropic, implying an approximately isotropic distribution at the particle interface in the highly convergent trapping beam. Thus, the average refractive index (1.515) was used in calculating the strength of the trap, hence the values of local shear viscosity. Measurements were made in three different devices and table 1 presents the effective viscosities for motion parallel and perpendicular to the director. The uncertainties quoted in the table reflect the reproducibility of measurements in a single position within a device, while the slight variation in each value obtained is a result of the slightly different alignment quality and definition across the three devices.

View this table:
Table 1

The effective viscosities, Embedded Image and Embedded Image, deduced from the observation of a 6 μm diameter trapped particle moving parallel and perpendicular to the LC director.

The data show that Embedded Image, as expected. Ideally, the quantities deduced would be compared with some other measurements of local shear viscosity made using a different methodology. However, as already mentioned, this is not possible; optical trapping provides a unique approach to measuring local hydrodynamic properties of LCs in the linear regime. The measurements presented in table 1 compare well with the flow viscosity of 19 cP quoted by the manufacturer. Further, the weighted mean of the ratio Embedded Image is in excellent agreement with the theoretical prediction of Stark et al. (Stark & Ventzki 2001; Stark et al. 2003; Pasechnik et al. 2004), who suggest that the ratio for strong and weak homeotropic anchorings should be 2 and 1.3, respectively. It is worth recalling, there was some indication that the anchoring strength of the colloidal particles in this nematic LC was rather weak, as the Saturn-ring defect was observed. However, it is known that the transition from weak to strong anchoring is a continuum, and the ratio of viscosities, together with the predictions of Stark, implies that, in fact, the anchoring was relatively strong. It appears that, in addition to providing a direct method of measuring the anisotropic shear viscosity of nematic LCs, there is also some potential for deducing a quantitative measure of the anchoring strength of the LC at the interface, allowing an alternative for determining anchoring strength through director patterns which necessarily divides the anchoring conditions into simply strong or weak.

4. Conclusions

Optical trapping used in conjunction with LCs is clearly a powerful technique with many opportunities for providing novel effects and new measurement techniques. Trapping chiral nematic droplets, for the first time, has led to the confirmation that the axial trapping force is significantly reduced when the trapping wavelength is comparable to the selective reflection wavelength of the material. Indeed, when both are matched, the droplets could no longer be optically trapped axially. One of the most surprising observations was the transfer of optical angular momentum to a chiral nematic droplet in linearly polarized light. Droplet rotation occurred where the pitch length was slightly greater than the size of the droplet, such that the director configuration can be described by the fully twisted bipolar configuration. Wave plate behaviour is excluded as a mechanism leading to the rotation of the droplet, leaving an open and interesting question as to what the actual mechanism can be. This observation extends the suite of systems that can be reasonably described as light-driven micromachines, and it offers optically active materials as a fruitful avenue for further study in this context.

The power of using optical traps in studying the rheology of LCs was demonstrated. This is the one among the few techniques that allows a direct comparison between theory and experiment, as it allows accurate control of the velocity of the particle as it moves through the LC. Stark & Ventzki (2001) have reported theoretical calculations of the effective viscosities and dynamics of colloids moving through liquid crystalline materials for situations of low (Er≪1), intermediate (Er=1) and high (Er>1) Ericksen numbers. It would be particularly interesting to vary the particle size and velocity, thereby controlling the Ericksen number, and employ optical trapping techniques to carry out microrheological experiments that provide experimental verification of these theories. Theoretically, the dynamics have also been shown to depend on the alignment of the LC around the colloid, and this could also be controlled by the surface treatment of colloids, or indeed by the application of electric or magnetic fields to the system.

Discussion

H. Lekkerkerker (Debye Research Institute, The Netherlands). You described the use of optical tweezers to drag a colloidal particle through a nematic liquid and measure from the velocity the viscosity. Would the measurement of the mean square displacement of a particle fixed in a trap be viable and perhaps allow a more detailed analysis?

H. Gleeson. A measurement of the mean square displacement of the particle would be expected to show anisotropy in the viscosity. However, additional complications might arise from the optical anisotropy of the medium in determining the mean square displacement of the particle in the trap by interferometry of the scattered light. Furthermore, this method provides only a passive measurement of the dynamics within the system. Our measurement evaluates the effective viscosity under flow conditions and is comparable to the theory of Stark & Ventzki (2001) and Stark et al. (2003).

V. Götz (Department of Chemistry, University of York, UK). Is the direction of rotation induced in dural molecules dependent on the enantiomer chosen, i.e. does it rotate in the opposite sense when the enantiomer is used?

H. Gleeson. This isn't something that we actually tried since the materials used to study the pitch dependence of the trapping and rotation phenomena were of a single handedness. However, if the rotation in linearly polarized light is due to the chirality, which is very likely, I would agree that rotation would be likely to occur in the opposite direction if the droplet had the opposite chirality.

A. N. Cammidge (School of Chemical Sciences & Pharmacy, University of East Anglia, UK). Do you see Raman scattering, and if so, is the scattered beam circularly polarized (in the chiral system)?

H. Gleeson. We didn't look for Raman scattering, although we would expect some chiral component in the scattered light.

M.-H. Li (Institut Curie, France). Could you please give some details about the application of LC laser manipulation in microfluidics? Here, what is the advantage of LC relative to isotropic systems?

H. Gleeson. The advantage of a birefringent spherical particle is that, provided the applied torque to the particle is known, then the hydrodynamics around the particle are well understood. In an isotropic system, optical torque might be applied only if the shape was anisotropic or if a mode of laser beam carrying orbital angular momentum was used. Thus, the use of birefringence as demonstrated with the LC droplet provides a simple method of introducing torque to micron-scale system. A good example of how birefringent, rotating particles can enhance microfluidics applications is given in the optically driven pump described in the paper by Leach et al. (2006).

S. J. Picken (TU Delft, The Netherlands). You mentioned a variety of mechanisms to achieve rotation of the particles including dichroism. Could you comment on the efficiency of dichroism to achieve rotation of the particles? In that case, one of the waves would be absorbed inelastically which would appear to be the most efficient mechanism.

H. Gleeson. Certainly, it would seem that if a wave is absorbed inelastically, then that would provide a more efficient transfer mechanism. However, as I don't know of any work on transfer of optical angular momentum to birefringent, anisotropic particles, I can't be quantitative about this transfer efficiency. The efficiency of other mechanisms was considered by us in Wood et al. (2004).

M. P. Neal (Faculty of Science and Engineering, Manchester Metroploitan University, UK). The chiral resonator droplets rotate much more slowly. Is there a cut-off frequency/threshold effect compared to the achiral nematic droplets?

H. Gleeson. We haven't observed any threshold effects in this system, but cannot rule them out.

P. Rudquist (Department of Microtechnology and Nanoscience, Chalmers University of Technology, Sweden). Did you try circularly polarized light on chiral droplets, and if so, did you notice any difference when using left- or right-handed light?

H. Gleeson. Since chiral nematic droplets have no net birefringence when p<r, it would be difficult to observe or measure any rotation, neither would rotation be expected.

H. J. Coles (CMMPE, University of Cambridge, UK). On the length-scale of the droplets from Raman and dynamic light scattering, one might expect significant depolarization. Therefore what happens in elliptically polarized light?

H. Gleeson. We looked at the depolarization of linearly polarized light and observed significant effects, depending on the pitch of the chiral nematic mixture and the droplet size. It would seem reasonable to also expect significant depolarization of elliptically polarized light.

H. J. Coles. Secondly, it would be fun to incorporate into the nematic (chiral or achiral) a fluorosurfactant (3M-F430) this should migrate to the droplet surface and through the slippery surface mechanism, radically alter the rotation rate.

H. Gleeson. This is an interesting suggestion. If there is significantly less drag, then it may be that the rotation rate is increased, depending on the relative importance of the drag and the efficiency of the transfer of angular momentum.

S. T. Lagerwall (The Royal Swedish Academy of Sciences, Sweden). You transfer angular momentum from linearly polarized light to the particle. This means that the light scattered off the particle should be partly circularly polarized. It would be interesting to know how large or small this effect is, but I suppose you cannot measure it?

H. Gleeson. Indeed, this was not measured. We did measure the relative intensities of the π and σ polarizations after passing through droplets and we noted that droplet rotation occurred in the situation where the transmitted light was approximately equally π and σ. This could mean that the light was circularly polarized, but we can't conclude that without doing Stokes parameter measurements which are very difficult for such a small sample—and we haven't done them.

P. Palffy-Muhoray (Liquid Crystal Institute, Kent State University, USA). It seems possible that (i) the director rotates, (ii) the droplet rotates, or (iii) some combination of these. How can you distinguish between the different scenarios?

H. Gleeson. Small dust particles in the suspension have been observed to orbit droplets, thus indicating that the droplet is rotating. The viscosity of the LC is around 60 times that of the surrounding water, so it is much easier for the droplet to rotate as a whole rather than the director itself. In addition, droplets can be observed to rotate even at very low beam powers of the order of 20 mW, which would be too low to allow reorientation of the director within the droplet and measurements have shown that between 50 and 500 mW, the rotation rate is linearly proportional to the beam power, indicating that a combined nonlinear effect does not kick in within this range. Reorientation of the director within the droplet would be likely to be a nonlinear effect.

P. Palffy-Muhoray. We know the light can cause director rotation in LCs without exerting a torque or exchanging angular momentum, by causing one part of the system to expel a torque on another. Do you think that such a process could be operating here?

H. Gleeson. It must be remembered that the droplet is free to move if any force is applied to it, while in bulk samples in which light-induced director rotation has been observed require elastic deformations. For the case of the droplet, if the director is not parallel to the linear component of plane-polarized light, then the droplet can act as a wave plate and the light emitted from the droplet has a circular component. This induces a torque on the droplet which will decrease reaching zero when the droplet director is parallel to the plane of incident polarization. Thus rotation of the droplet as a whole would be energetically more favourable.

Acknowledgments

Funding for equipment and a studentship is gratefully acknowledged from the Engineering and Physical Sciences Research Council and the Jersey Education Department. Thanks are due to N. Roberts of Manchester University for carrying out the reflectivity calculations on the chiral nematic mixtures and P. Bartlett of Bristol University for the use of his apparatus for the measurements of the polarization selection of the chiral nematic droplets.

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