Interaction between two spherical particles in a nematic liquid crystal

We numerically investigate the interaction between two spherical particles in a nematic liquid crystal mediated by elastic distortions in the orientational order. We pay attention to the cases where two particles with equal radii $R_0$ impose rigid normal anchoring on their surfaces and carry a pointlike topological defect referred to as a hyperbolic hedgehog. To describe the geometry of our system, we use bispherical coordinates, which prove useful in the implementation of boundary conditions at the particle surfaces and at infinity. We adopt the Landau–de Gennes continuum theory in terms of a second-rank tensor order parameter $Q_{ij}$ for the description of the orientational order of a nematic liquid crystal. We also utilize an adaptive mesh refinement scheme that has proven to be an efficient way of dealing with topological defects whose core size is much smaller than the particle size. When the two “dipoles,” composed of a particle and a hyperbolic hedgehog, are in parallel directions, the two-particle interaction potential is attractive for large interparticle distances $D$ and proportional to $D^{-3}$ as expected from the form of the dipole-dipole interaction, until the well-defined potential minimum at $D\simeq 2.46R_0$ is reached. For the antiparallel configuration with no hedgehogs between the two particles, the interaction potential is repulsive and behaves as $D^{-2}$ for $D\lesssim 10R_0$, which is stronger than the dipole-dipole repulsion $\left(\sim D^{-3}\right)$ expected theoretically as an asymptotic behavior for large $D$.

Figure 1

Illustration of a typical mesh in real space generated by an equally spaced mesh in the $(\zeta ,\mu )$ space.

Figure 2

The numerical grids in the real space in the case of Fig. 3b below.

Figure 3

The orientation profiles of a nematic liquid crystal shown by gray-scale plots of $Q_{zz}^2$ for (a) $D=5.0R_0$, (b) $D=3.0R_0$, and (c) $D=2.3R_0$ in the “parallel-dipole” configurations. The $z$ axis is along the horizontal direction and in the black regions, the nematic liquid crystal aligns along the $z$ axis, while $Q_{zz}=0$ in the white region.

Figure 4

The dimensionless free energy of a nematic liquid crystal $\overline{F}$ as a function of the distance $D$ between the centers of the particles in the case of the “parallel-dipole” configuration. The inset is a magnified plot around $D∕R_0=2.45$.

Figure 5

Log-log plot of the dimensionless force $\overline{f}$ as a function of $D$ in the case of the “parallel-dipole” configuration. The dashed line corresponds to $0.0150{(D∕R_0)}^{-4}$. Only the data for the attractive force $\left(-\overline{f}>0\right)$ are shown.

Figure 6

The orientation profiles of a nematic liquid crystal shown by gray-scale plots of $Q_{zz}^2$ for (a) $D=5.0R_0$, (b) $D=3.0R_0$, and (c) $D=2.3R_0$ in the “antiparallel-dipole” configurations.

Figure 7

The dimensionless free energy of a nematic liquid crystal $\overline{F}$ as a function of the distance $D$ between the centers of the particles in the case of the “antiparallel-dipole” configuration.

Figure 8

Log-log plot of the dimensionless force $\overline{f}$ as a function of $D$ in the case of the “antiparallel-dipole” configuration. The dashed line corresponds to $0.00345{(D∕R_0)}^{-3}$.