Theory of elastic interaction of colloidal particles in nematic liquid crystals near one wall and in the nematic cell

We apply the method developed [Chernyshuk and Lev, Phys. Rev. E 81, 041701 (2010)] for theoretical investigation of colloidal elastic interactions between axially symmetric particles in the confined nematic liquid crystal near one wall and in the nematic cell with thickness $L$. Both cases of homeotropic and planar director orientations are considered. Particularly, dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole interactions of the one particle with the wall and within the nematic cell are found as well as corresponding two particle elastic interactions. A set of results has been predicted: The effective power of repulsion between two dipole particles at height $h$ near the homeotropic wall is reduced gradually from inverse 3 to 5 with an increase of dimensionless distance $r/h$; near the planar wall, the effect of dipole-dipole isotropic attraction is predicted for large distances $r>r_{\mathit{dd}}=4.76h$; maps of attraction and repulsion zones are crucially changed for all interactions near the planar wall and in the planar cell; and one dipole particle in the homeotropic nematic cell was found to be shifted by the distance ${\delta }_{\mathrm{eq}}$ from the center of the cell. The proposed theory fits very well with experimental data for the confinement effect of elastic interaction between spheres in the homeotropic cell [Vilfan et al., Phys. Rev. Lett. 101, 237801 (2008)] in the range $1-1000$ kT. The influence of the $K_{24}$ and $K_{13}$ terms as well as connection with other theoretical approaches are discussed.

Figure 1

Topological defects. (a) Dipole configuration. Point defect: hyperbolic hedgehog. (b) Saturn ring quadrupole configuration with $c>0$ and (c) two point defects: boojums at the poles of the sphere, quadrupole configuration with $c<0$.

Figure 2

Cones of repulsion and attraction for bulk interactions from (13): The minus sign means attraction and the plus sign means repulsion. (a) Dipole-dipole interaction, ${\mathit{pp}}^{\prime }>0$. (b) Dipole-quadrupole interaction with greater particle in the center, ${\mathit{pc}}^{\prime }>{\mathit{cp}}^{\prime }$. (c) Quadrupole-quadrupole interaction, ${\mathit{cc}}^{\prime }>0$.

Figure 3

Repulsion of one particle from the wall with homeotropic director orientation. (a) $p$ configuration, (b) $h$ configuration.

Figure 4

Repulsion of one particle from the wall with planar director orientation.

Figure 5

Image interpretation of the interaction between two particles near the homeotropic wall. This barely supplements the exact Eq. (12).

Figure 6

Effective power of repulsion between two particles at the same height $h$ near the homeotropic wall in (26) vs dimensionless distance $r/h$. The red line 1 corresponds to the dipole-dipole interaction, the first term in (25); the black line 2 corresponds quadrupole-quadrupole interaction, the third term in (25).

Figure 7

Effective power of repulsion between two particles at the same height $h$ near the homeotropic wall in (26) vs dimensionless distance $r/h$. Corresponds to the sum of all interactions (25) for $\alpha =2,\hspace{0.5em}\beta =0.5$ and radius of particles $a_1=a_2=0.5h$.

Figure 8

Different configurations of particles near the planar wall. (a) Usual dipole-dipole, (b) p-p configuration, and (c) h-h configuration according to [7].

Figure 9

Map of the attraction and repulsion zones for dipole-dipole interaction between two particles at the same height $h$ near the planar wall, ${\mathit{pp}}^{\prime }>0$. At short distances $r<h$, attraction and repulsion cones coincide with the unlimited nematic case [see Fig. 2a] and make the angle $\theta =\arccos (1/\sqrt{3})$ with the $z$ axis. But, at great distances $r>h$, repulsion lateral cones dramatically collapse from the right and left sides into a dumbbell-shaped region. The minus sign means attraction and the plus sign means repulsion (radial force ${\text{f}}_r=-\frac{\partial V_{\mathit{dd},\mathrm{plan}}}{\partial r}$, ${\text{f}}_r<0$ and ${\text{f}}_r>0$, respectively). On the thick blue line, ${\text{f}}_r=0$. This effect is absent in usual electrostatics for dipoles near the grounded wall.

Figure 10

Dimensionless energy of dipole-dipole interaction vs dimensionless distance $r$/$h$, ${\mathit{pp}}^{\prime }>0$. (a) Energy for all angles $\theta \in (0,\pi )$, clear attraction for $\theta =0,\pi$. (b) Energy $U_{\perp }$ in the perpendicular direction $\theta =\pi /2$ between $r$ and rubbing direction $z$. New effect: For $y>r_{\mathit{dd}}=4.76h$, attraction appears.

Figure 11

Energy of interaction $U_{\perp }$ in kT between two dipole beads near the planar wall in the perpendicular direction to the rubbing vs distance $r$ in $\mu$m for different heights from the wall: (1) $h$ $=$ 2 $\mu \mathrm{m}$, (2) $h$ $=$ 3 $\mu \mathrm{m}$, (3) $h$ $=$ 4 $\mu \mathrm{m}$. Here, $a_1=a_2=1\hspace{0.5em}\mu \mathrm{m},\hspace{0.5em}\alpha =2.04,\hspace{0.5em}\beta =0.5$.

Figure 12

Map of the attraction and repulsion zones for dipole-quadrupole interaction (33) between two particles at the same height $h$ near the planar wall with the larger particle in the center, ${\mathit{pc}}^{\prime }>{\mathit{cp}}^{\prime }$.

Figure 13

Map of the attraction and repulsion zones for quadrupole-quadrupole interaction (36) between two particles at the same height $h$ near the planar wall, ${\mathit{cc}}^{\prime }>0$.

Figure 14

One particle in the homeotropic cell. $\delta$ is the shift from the center of the cell. Equilibrium shift $\delta \approx \frac{\beta }{\alpha }a$ almost does not depend on the thickness $L$ of the cell (48). See Fig. 15 as well.

Figure 15

One particle with radius $a_1=2\hspace{0.5em}\mu$m in the homeotropic cell with the thickness $L=10\hspace{0.5em}\mu$m. Dependence of the energy shift $U-U_{z=L/2}$ on the shift $\delta$ from the center of the cell. Blue line 1 corresponds to the dipole particle $\alpha =2,\beta =0.5$. Equilibrium shift $\delta =0.5$ is in line with (48). Thick dashed black line 2 corresponds to the quadrupole particle $\alpha =0,\beta =0.5$.

Figure 16

One particle in the planar cell. $\delta$ is the shift from the center of the cell. Equilibrium shift $\delta =0$. See Fig. 17 as well.

Figure 17

One particle with radius $a_1=2\hspace{0.5em}\mu$m in the planar cell, with thickness $L=10\hspace{0.5em}\mu$m. Dependence of the energy shift $U-U_{z=L/2}$ on the shift $\delta$ from the center of the cell. Blue line 1 corresponds to the dipole particle $\alpha =2,\hspace{0.5em}\beta =0.5$. The thick dashed black line 2 corresponds with the quadrupole particle $\alpha =0,\hspace{0.5em}\beta =0.5$.

Figure 18

Two particles in the homeotropic cell with thickness $L$.

Figure 19

Experimental data taken from [16]: energy of elastic interaction between two spherical particles with diameter $D=2a=4.4\hspace{0.5em}\mu$m in the homeotropic cell with thicknesses $h=6.5$ and $8\hspace{0.5em}\mu$m. The solid blue and green lines are taken from Eq. (59). The only unknown parameter $\beta =0.28$ fits both thicknesses very well in the energy scale $1-1000$ kT.

Figure 20

Two particles in the planar cell with thickness $L$.

Figure 21

Map of the attraction and repulsion zones for dipole-dipole interaction (61) between two particles at the center of the planar cell with thickness $L$, ${\mathit{pp}}^{\prime }>0$. Director ${\text{n}}_0\left|\right|z$. For short distances, attraction and repulsion cones coincide with the unlimited nematic case [see Fig. 2a] and make the angle $\theta =\arccos (1/\sqrt{3})$ with the $z$ axis. But, for long distances $r⩾L$, attraction cones are deformed to the parabola $z=\frac{\pi y^2}{L}$. The black thin line is the parabola $z=\frac{\pi y^2}{L}$. The minus sign means attraction and the plus sign means repulsion. On the thick blue line, radial force ${\text{f}}_r=-\frac{\partial V_{\mathit{dd},\mathrm{cell}}^{\mathrm{plan}}}{\partial r}=0$.

Figure 22

Map of the attraction and repulsion zones for dipole-quadrupole interaction (62) between two particles at the center of the planar cell with thickness $L$ with the larger particle in the center, ${\mathit{pc}}^{\prime }>{\mathit{cp}}^{\prime }$. The black thin line is the parabola $z=\frac{\pi y^2}{L}$.

Figure 23

Map of the attraction and repulsion zones for quadrupole-quadrupole interaction (63) between two particles at the center of the planar cell with thickness $L$, ${\mathit{cc}}^{\prime }>0$. The black thin lines are the parabolas $z=\frac{\pi y^2}{L}$ and $y=\frac{\pi z^2}{L}$, respectively.