Multiscale models of colloidal dispersion of particles in nematic liquid crystals

We use homogenization theory to develop a multiscale model of colloidal dispersion of particles in nematic liquid crystals under weak-anchoring conditions. We validate the model by comparing it with simulations by using the Landau–de Gennes free energy and show that the agreement is excellent. We then use the multiscale model to study the effect that particle anisotropy has on the liquid crystal: spherically symmetric particles always reduce the effective elastic constant. Asymmetric particles introduce an effective alignment field that can increase the Fredericks threshold and decrease the switch-off time.

Figure 1

Domains used in defining scale separation. The macroscopic domain consists of an open region $\mathcal{D}$, with external boundary $\partial \mathcal{D}$, constructed from unit cells $\Omega$ with outer boundaries $\mathcal{B}$ and metallic inclusions of volume ${\Omega }_{np}$ and boundary ${\Gamma }_i$.

Figure 2

Comparison of numerical simulation of the full and homogenized equations for spherical and ellipsoidal particles inside a twisted geometry, zero pretilt at boundaries, and homeotropic anchoring on dopants. (a) Spherical particles of radius $r=0.3\mu \mathrm{m}$ at applied voltages of $1.5$ and 3 V (lower and upper curves, respectively). (b) Ellipsoidal dopants $V=0$ with anchoring energy $W_1=50$ (bottom) and $W_1=100$ (top) orientation given by $({\theta }_p,{\varphi }_p)=({45}^{\circ },{30}^{\circ })$. The semimajor axis of the ellipsoids is $0.3\mu \mathrm{m}$ and the two minor axes are $0.1\mu \mathrm{m}$. [(a), (b)] Red points are from homogenization, broken black line is undoped, and blue line is from comsol numerical simulations. (c) Absolute error $\mathbb{E}=|{\theta }_H-{\theta }_N|$ where ${\theta }_H$ is the tilt angle from the homogenized equations and ${\theta }_N$ is extracted from numerical simulations of the full system. (d) Schematic diagram of the system studied: planar cell of size $L_x$ with spherical or ellipsoidal dopants with spacing $L_y$.

Figure 3

Bifurcation diagrams for spherical and ellipsoidal inclusions. Top spheres of radius from $0.05\mu \mathrm{m}$ in steps of $0.05$ to $0.45\mu \mathrm{m}$. Bottom: ellipsoids with major axis $r_1=0.3\mu \mathrm{m}$ and minor axis' $r_2=r_3=0.1\mu \mathrm{m}$, oriented in the plane of the director at an angle of $0^{\circ },{30}^{\circ },{60}^{\circ }$, and ${90}^{\circ }$ to the initial director alignment. Circles indicate the position of the Fredericks transition computed using Eq. (24).

Figure 4

Diagonal component of the elasticity tensor for spherical inclusions and components of the source term $\mathbf{q}$ for ellipsoidal particles confined to the $x_1\text{−}{x}_3$ plane. The angle ${\theta }_p$ is measured from the ellipsoids major axis to the $x_3$ axis, $q_3=q_5=0$ (not shown). The approximate linear relation between $D_{33}$ and $r^3$ indicates that the excluded volume effect is dominant.

Figure 5

Switch-off (left) and -on (right) time at $V=2$ V for colloidal nematic with ellipsoidal particles as a fraction of undoped on or off times in the splay geometry. Ellipsoids oriented orthogonal (top) or parallel (bottom) to initial director with homeotropic anchoring on inclusions and anchoring strength $W_1=10$. Vertical dopants reinforce stability of the ${\theta }_H=0$ state while horizontal dopants reduce it.

Figure 6

Color-scaled image of the solution ${\chi }_3$ of Eqs. (A22) in the domain around a spherical particle of radius $0.4L_y$. The arbitrary constant has been chosen so that the average modulation of ${\chi }_3$ is zero.