Electrokinetic effects in nematic suspensions: Single-particle electro-osmosis and interparticle interactions

Electrokinetic phenomena in a nematic suspension are considered when one or more dielectric particles are suspended in a liquid crystal matrix in its nematic phase. The long-range orientational order of the nematic constitutes a fluid with anisotropic properties. This anisotropy enables charge separation in the bulk under an applied electric field, and leads to streaming flows even when the applied field is oscillatory. In the cases considered, charge separation is seen to result from director field distortions in the matrix that are created by the suspended particles. We use a recently introduced electrokinetic model to study the motion of a single-particle hyperbolic hedgehog pair. We find this motion to be parallel to the defect-particle center axis, independent of field orientation. For a two-particle configuration, we find that the relative force of electrokinetic origin is attractive in the case of particles with perpendicular director anchoring, and repulsive for particles with tangential director anchoring. The study reveals large scale flow properties that are respectively derived from the topology of the configuration alone and from short scale hydrodynamics phenomena in the vicinity of the particle and defect.

Figure 1

Computational domain used for single-particle studies. Within a square $S_1$ is a circular domain $C_0$ in which the mesh is more finely resolved. $C_1$ represents the disk-shaped particle, and the mesh is further resolved within the circle $C_2$ to resolve the topological defect.

Figure 2

Possible director fields around a disk in a two-dimensional domain, with far-field director orientation ${\mathit{n}}_0=\widehat{\mathit{x}}$. For both homeotropic and tangential anchoring, a suspended disk has topological charge $\left(+1\right)$, requiring either one $\left(-1\right)$ defect or two $\left(-1/2\right)$ defects in the bulk.

Figure 3

Three-dimensional director configurations of interest around a spherical particle. With homeotropic anchoring (a), the sphere has the same topological charge as a point defect, and in most experimental systems it induces a hyperbolic hedgehog defect (shown left of the particle). A sphere with tangential anchoring (b) has zero topological charge and requires no defects in the bulk of the nematic, but two boojum defects are required on the particle surface.

Figure 4

Subset of the computational domain showing the numerical charge density at $t=2\pi$ (a), and the time averaged velocity (b) of the electrokinetic flow around a two-dimensional particle with electric field parallel to elastic dipole, plotted in dimensionless units. The flow field comprises six vortices as shown by the labels in the figure. The electrokinetic force on the particle, computed using Eq. (18), is found to point opposite the elastic dipole $\mathit{p}$.

Figure 5

Numerical charge density at $t=2\pi$ (a) and time averaged velocity (b) of the induced electrokinetic flow around a two-dimensional particle with electric field perpendicular to elastic dipole, plotted in dimensionless units. We observe four vortices in the flow field, and the electrokinetic force on the particle, computed using Eq. (18), is found to point along the elastic dipole $\mathit{p}$.

Figure 6

Numerical velocity field for a $\left(-1,+1\right)$ point defect pair, averaged over a period of the applied electric field, plotted in dimensionless units. Note the appearance of only two vortices here in contrast to the six vortices in Fig. 4.

Figure 7

First three Fourier components of numerical angular velocity for a system with a $\left(-1,+1\right)$ point defect set (a) and a particle-defect pair (b), averaged over a period of the field, plotted in dimensionless units as a function of $r$.

Figure 8

Two-dimensional average flow field induced by the director field, Eq. (28), at $z=0$, plotted in dimensionless units.

Figure 9

Results for two particles with homeotropic anchoring, separated by $s=0.426$, plotted in dimensionless units. (a) Director field at $t=2\pi$, with magnitude plotted in color. (b) Charge density at $t=2\pi$, with electric field pointing to the right. (c) Velocity field averaged over a period of the applied field.

Figure 10

Numerically computed difference in hydrodynamic and elastic force between two particles with homeotropic anchoring at $\text{Er}=2$, averaged over a period of the field and plotted as a function of particle separation in dimensionless units. For the range of parameters plotted, the hydrodynamic forces always result in particle attraction, while the elastic forces are more repulsive at closer distances. The total force difference is also plotted, and we see the equilibrium position occurs at $s\approx 0.429$.

Figure 11

Particle separation in dimensionless units as a function of Ericksen number. The decrease in separation as Er increases implies the hydrodynamic electrokinetic force is attractive.

Figure 12

Elastic energy (dimensionless units) for a system of two circular particles with tangential anchoring at various surface to surface separations $s$, and relative orientations $\phi$ (rad).

Figure 13

(a) Instantaneous charge density corresponding to two circular particles fixed at the separation and angle that minimize the elastic free energy. The oscillatory electric field point in the $y$ direction. (b) Time averaged electrokinetic velocity, which is shown magnified in (c) to highlight the appearance of boundary layer flows near particle surfaces. All values are in dimensionless units.