Elastic octopoles and colloidal structures in nematic liquid crystals

We propose a simple theoretical model which explains the formation of dipolar two- (2D) and three-dimensional (3D) colloidal structures in nematic liquid crystals. The colloidal particles are treated as effective hard spheres interacting via their elastic dipole, quadrupole, and octopole moments. It is shown that the octopole moment plays an important role in the formation of 2D and 3D nematic colloidal crystals. We generalize this assumption to the case of an external electric field and theoretically explain a giant electrostriction effect in 3D crystals observed recently.

Figure 1

1D colloidal structure. Particles with parallel dipole moments aggregate in linear chains along ${\mathbf{n}}_0$, $b\approx 2.4r_0$ [21, 22]. Each particle is surrounded by a coat containing strong director deformations which cannot be described by the multipole expansion.

Figure 2

The attractive part of the elastic force between two parallel dipoles. The points depict the experimental results from [21]. The solid red line corresponds to the case ${\alpha }_{\mathrm{expt}}=2.05$, ${\beta }_{\mathrm{expt}}=0.2$, and $\gamma =0$. The dashed blue line corresponds to the coefficients ${\alpha }_{\mathrm{theor}}=2.04$, ${\beta }_{\mathrm{theor}}=0.72$, and $\gamma =0.157$. The inset shows the selected region on a larger scale.

Figure 3

(a) A zigzag vertical cross section of a quasi-2D checkerboard colloidal crystal formed by particles with antiparallel hedgehog ordering in a homeotropic cell, with $a\approx 2.3r_0$ and ${\theta }_{\min }\approx {60}^{\circ }$ [10]. (b) 2D structure formed by antiparallel linear chains in the planar cell. The lattice constants are $a=2.54r_0$, $b=2.46r_0$, and ${\theta }_{\min }={61}^{\circ }$ [7]. Note that in such a structure the hedgehogs transform into small rings.

Figure 4

Schematic representation of a quasi-body-centered Bravais lattice of a 3D colloidal crystal. The lattice constants are $A=(3.2\pm 0.1)r_0$, $B=(2.3\pm 0.2)r_0$, and $\Psi =(1.3\pm 0.1)r_0$ [10]. Due to such a dense packing the central particle interacts only with its nearest neighbors located at the vertices.

Figure 5

Lateral shrinking of a 3D colloidal crystal under an electric field applied along ${\mathbf{n}}_0$. The solid lines correspond to the theoretical calculations with the parameters of E7: dielectric anisotropy $\Delta \epsilon =13.8$, elastic constant $K=13.7\mathrm{pN}$, the cell thickness $L=25\mu$m. The particles' radii $r_0=2.16\mu$m. The points depict experimental results from [10].