Heterogeneous colloidal particles immersed in a liquid crystal

In this paper, we explore anisotropic interactions between particles with heterogeneous boundary conditions inside both nematic and cholesteric liquid crystals. The results show that when particles are put at different distances and angles with respect to each other, new types of defect structures are produced, depending on the relative distances and directions. In a cholesteric liquid crystal, the value of the pitch affects the defect structures and induced forces. Moreover, it was observed that it is energetically favorable for the particles to remain in a plane parallel to the far-field director in a nematic liquid crystal, while for particles immersed in a cholesteric there are multiple energy minima not all located in the same plane.

Figure 1

Schematic of our system where the periodic boundary conditions are imposed in the $x$ and $y$ directions, the fixed walls are located in the $z$ direction, and the liquid crystal molecules are set to be parallel on the walls (along $y$ axis for the nematic case and along the $x$ axis for the cholesteric).

Figure 2

The defects and their images for isolated particles in a nematic liquid crystal. In (a) and (c) the sphere has planar anchoring, and in (b) and (d) the sphere has normal anchoring. The director is shown on the surface of the sphere in (a) and (b) and the sphere is not shown in (c) and (d), so that the image defects are visible.

Figure 3

Plot of the interaction energy surface for particles immersed in a nematic liquid crystal and confined to the $xy$ plane. Defect structures are shown for particles at a separation of $0.125\mu \text{m}$. The energy is measured at discrete separations on arcs with radii: $1/16, 1/8, 3/16, 1/4, 3/8, 1/2, 5/8, 15/16, 5/4, 3/2, 7/4$, and $31/16\mu \text{m}$. The energies from the discrete points are then interpolated to produce the free energy surface. The blue (dark) and red (light) points show the final relative position of freely moving particles when the initial positions have a separation of $0.25\mu \text{m}$ at the angles $0^{\circ }$ and ${35}^{\circ }$, respectively.

Figure 4

Plot of the director field around the particles inside a nematic LC and located on the $xy$ plane with separation of (a) $0.25\mu \text{m}$ at $\theta ={35}^{\circ }$, and (b) $0.125\mu \text{m}$ at $\theta ={90}^{\circ }$.

Figure 5

(a) 2D plot of interaction energy for the particles immersed in the cholesteric liquid crystal in the $xy$ plane with pitch value of $1.5\mu \text{m}$ along with some corresponding defect structures and (b) a 3D plot of the interaction energy. The energy is measured when the particles are located at several points on arcs with radii: $0.0625\mu \text{m}, 0.125\mu \text{m}, 0.1875\mu \text{m}, 0.25\mu \text{m}, 0.375\mu \text{m}, 0.5\mu \text{m}, 0.625\mu \text{m}, 0.9375\mu \text{m}, 1.25\mu \text{m}$. The energies from the discrete points are then interpolated to produce the free energy surface. The red (light), blue (dark), black (darkest), and green (lightest) points are the final relative position of freely moving particles when the initial positions are: separation $0.125\mu \text{m}$ at ${90}^{\circ }, 0.25\mu \text{m}$ at ${55}^{\circ }, 0.375\mu \text{m}$ at ${25}^{\circ }$, and $0.375\mu \text{m}$ at ${90}^{\circ }$.

Figure 6

(a) 2D plot of interaction energy for the particles immersed in the cholesteric liquid crystal in the $xy$ plane with pitch value of $1.125\mu \text{m}$ with their defect structures and (b) a 3D plot of the interaction energy. The energy is measured when the particles are located at several points on arcs with radii: $0.0625\mu \text{m}, 0.125\mu \text{m}, 0.1875\mu \text{m}, 0.25\mu \text{m}, 0.375\mu \text{m}, 0.5\mu \text{m}, 0.9375\mu \text{m}, 1.25\mu \text{m}$. The energies from the discrete points are then interpolated to produce the free energy surface. The blue (dark), red (light), and black (darkest) points correspond to the final relative position of freely moving particles when the initial positions are: separation $0.375\mu \text{m}$ at ${90}^{\circ }, 0.25\mu \text{m}$ at ${55}^{\circ }$, and $0.375\mu \text{m}$ at $0^{\circ }$.

Figure 7

(a) 2D plot of elastic energy for the particles immersed in the cholesteric liquid crystal in the $xy$ plane with pitch value of $1.125\mu \text{m}$ at (a) particle separation of $0.25\mu \text{m}$ and the angle ${55}^{\circ }$, and (b) separation of $0.0625\mu \text{m}$ at the angle $0^{\circ }$. The scale has been multiplied by ${10}^3$.

Figure 8

3D plot of interaction energy when the cholesteric pitch is $3\mu \text{m}$. The energy is measured when the particles are located at several points on arcs with radii: $0.0625\mu \text{m}, 0.125\mu \text{m}, 0.1875\mu \text{m}, 0.25\mu \text{m}, 0.375\mu \text{m}, 0.5\mu \text{m}, 0.9375\mu \text{m}, 1.25\mu \text{m}$.

Figure 9

(a) 2D plot of elastic energy for the particles immersed in the cholesteric liquid crystal in the $xy$ plane with pitch value of $3\mu \text{m}$ at (a) particle separation of $0.375\mu \text{m}$ and the angle ${10}^{\circ }$, and (b) separation of $0.0625\mu \text{m}$ at the angle $0^{\circ }$. The scale has been multiplied by ${10}^3$.

Figure 10

2D contour plot of interaction energy of two particles in a nematic in the $xz$ plane along with some corresponding defect structures. The energy is measured when the particles are located at several points on arcs with radii: $0.0625\mu \text{m}, 0.125\mu \text{m}, 0.25\mu \text{m}, 0.5\mu \text{m}, 0.9375\mu \text{m}$.

Figure 11

Plot of director field of particles in the $xz$ plane for the particle separations of (a) $0.125\mu \text{m}$ and (b) $0.5\mu \text{m}$ when $\theta =0^{\circ }$. The colloids/defects are viewed along the $z$ axis.

Figure 12

2D contour plot of interaction energy of 2 particles in the $xz$ plane with the defect structures in a cholesteric LC with a pitch of $1.5\mu \text{m}$. The energy is measured when the particles are located at several points on arcs with radii: $0.0625\mu \text{m}, 0.125\mu \text{m}, 0.25\mu \text{m}, 0.5\mu \text{m}, 0.9375\mu \text{m}$.