Theory of elastic interaction between arbitrary colloidal particles in confined nematic liquid crystals

We develop the method proposed by Chernyshuk and Lev [Phys. Rev. E 81, 041701 (2010)] for theoretical investigation of elastic interactions between colloidal particles of arbitrary shape and chirality (polar as well as azimuthal anchoring) in the confined nematic liquid crystal (NLC). General expressions for six different types of multipole elastic interactions are obtained in the confined NLC: monopole-monopole (Coulomb type), monopole-dipole, monopole-quadrupole, dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole interactions. The obtained formulas remain valid in the presence of the external electric or magnetic fields. The exact equations are found for all multipole coefficients for the weak anchoring case. For the strong anchoring coupling, the connection between the symmetry of the shape or director and multipole coefficients is obtained, which enables us to predict which multipole coefficients vanish and which remain nonzero. The particles with azimuthal helicoid anchoring are considered as an example. Dipole-dipole interactions between helicoid cylinders and cones are found in the confined NLC. In addition, the banana-shaped particles in homeotropic and planar nematic cells are considered. It is found that the dipole-dipole interaction between banana-shaped particles differs greatly from the dipole-dipole interaction between the axially symmetrical particles in the nematic cell. There is a crossover from attraction to repulsion between banana particles along some directions in nematic cells. It is shown that monopoles do not “feel” the type of nematic cell: monopole-monopole interaction turns out to be the same in homeotropic and planar nematic cells and converges to the Coulomb law as thickness increases, $L\to \infty$.

Figure 1

Helicoid cylinders with the same handedness $p_{\text{hel}}{p}_{\text{hel}}^{\prime }>0$ attract along the $\mathit{\text{z}}$ axis and repel in a perpendicular direction and vice versa for helicoids with different handedness $p_{\text{hel}}{p}_{\text{hel}}^{\prime }<0$ [see (34)].

Figure 2

Helicoid cones with different dipole moments $p$ and $p_{\text{hel}}$ produced by shape and azimuthal helical anchoring, respectively [see (38)].

Figure 3

(a) General banana-shaped particles, (b) front view (see Figs. 5 and 8), (c) side view (see Fig. 4), (d) top view (see Fig. 9).

Figure 4

Banana-shaped particles in the nematic cell, side view (see Fig. 3 as well).

Figure 5

Map of the attraction and repulsion zones for two identical banana-shaped particles, $p_x=p_x^{\prime }$ and $p_y=p_y^{\prime }=\alpha p_x$, $\alpha >\sqrt{2}$, according to (42). Black line (a) corresponds to the case $p_y=1.5p_x$, blue line (b) corresponds to the $p_y=2p_x$, and red line (c) corresponds to the $p_y=3p_x$. The particles are located in the center of the homeotropic cell $z=z^{\prime }=\frac{L}{2}$. Their orientations are shown in Figs. 3b and 4a. The sign “–” means attraction (inside of the dumbbell-shaped regions), and “+” means repulsion. If $p_y<\frac{p_x}{\sqrt{2}}$, then the attraction will appear along the $y$ axis.

Figure 6

The crossover from the attraction to the repulsion between two banana-shaped particles located in the middle of the homeotropic cell $z=z^{\prime }=\frac{L}{2}$ along the $x$ axis [see Figs. 4a and 5], $p_x=p_x^{\prime }$, $p_y=p_y^{\prime }=\alpha p_x$ for $\alpha >\sqrt{2}$. $\tilde{U}=U_{\text{dd},\perp }^{\text{hom}}{L}^3/8\pi K(p_x{p}_x^{\prime }+p_y{p}_y^{\prime })$, where $U_{\text{dd},\perp }^{\text{hom}}$ is given by (42) and $\varphi =0$. Solid black line 1 corresponds to $p_y=p_y^{\prime }=1.5p_x$, orange line 2 corresponds to $p_y=p_y^{\prime }=1.7p_x$.

Figure 7

Log-log plot of the dimensionless energy of the dipole-dipole interaction as a function of the distance between two banana-shaped particles located in the middle of the homeotropic cell $z=z^{\prime }=\frac{L}{2}$. Solid blue line 2 corresponds to the particle repulsion for the orientation along the $y$ axis depicted in Fig. 4a, $p_x=p_x^{\prime }$, $p_y=p_y^{\prime }=2p_x$, $2\varphi =\pi$, and $\tilde{U}=U_{\text{dd}}{L}^3/8\pi K(p_x{p}_x^{\prime }+p_y{p}_y^{\prime })$, where $U_{\text{dd}}>0$ is given by (42). Dashed line 4 is an appropriate power-law asymptotic $U_{\text{dd}}^{\text{unc}}{L}^3/4\pi K(p_x{p}_x^{\prime }+p_y{p}_y^{\prime })\propto {\left(\frac{L}{\rho }\right)}^3$. Solid red line 1 corresponds to the particle attraction along the $y$ axis in Fig. 4b, $p_y^z=p\ne 0$ and $\tilde{U}=U_{\text{dd}}{L}^3/16\pi Kp{p}^{\prime }$, where $U_{\text{dd}}<0$ is given by (47). Dashed line 3 is the appropriate power-law asymptotic $\frac{1}{4}{\left(\frac{L}{\rho }\right)}^3$.

Figure 8

Map of the attraction and repulsion zones for two identical banana-shaped particles, $p=p^{\prime }$, according to (51). The particles are located in the center of the planar cell $x=x^{\prime }=\frac{L}{2}$. Their orientation are shown in Figs. 3b and 4c. The sign “–” means attraction, and “+” means repulsion.

Figure 9

Map of the attraction and repulsion zones for two identical banana-shaped particles $p_x=p_x^{\prime }$ and $p_y=p_y^{\prime }$, according to (53). The particles are located in the center of the planar cell $x=x^{\prime }=\frac{L}{2}$. Their orientations are shown in Figs. 3d and 4d. Black line (a) $p_x=p_y$. Red line (b) $p_x=2p_y$. Blue line (c) $p_x=5p_y$. Green line (d) $p_x=10p_y$. The sign “–” means attraction, and “+” means repulsion.

Figure 10

Ellipsoidal particles in the homeotropic (a) and planar (b) nematic cell.

Figure 11

Monopole-monopole interaction in a nematic cell. Elastic monopoles do not “feel” the type of cell. Blue line 1 corresponds to $\tilde{U}=-U_{\text{qq}}^{\text{plan}}L/16\pi Kq{q}^{\prime }$. Here $U_{\text{qq}}^{\text{plan}}$ is given by (58). The dashed line 2 is the Coulomb-like $\tilde{U}=\frac{1}{4\rho /L}$ asymptotics for $\rho \ll L$.