Theory of elastic interaction between colloidal particles in a nematic cell in the presence of an external electric or magnetic field

The Green's function method developed previously [S. B. Chernyshuk and B. I. Lev, Phys. Rev. E 81, 041701 (2010)] is used to describe elastic interactions between axially symmetric colloidal particles in a nematic cell in the presence of an external electric or magnetic field. Formulas for dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole interactions in the homeotropic and planar nematic cells with parallel and perpendicular field orientations are obtained. A set of predictions has been made: (1) The deconfinement effect for dipole particles in the homeotropic nematic cell when an electric field is approaching its Freedericksz threshold value $E\Rightarrow E_t$. This means cancellation of the confinement effect found in [M. Vilfan et al., Phys. Rev. Lett. 101, 237801 (2008)] near the Freedericksz transition. In the planar nematic cell this deconfinement effect exists for both dipole and quadrupole particles and depends on the field orientation as well as on the sign of dielectric anisotropy $\Delta \varepsilon$. (2) The effect of tunable stabilization of the particles is predicted. The equilibrium distance between two particles, which are attracted along the electric field parallel to the planes of a homeotropic nematic cell with $\Delta \varepsilon <0$, depends on the strength of the field. (3) Attraction and repulsion zones for all elastic interactions are changed dramatically under the action of the external field.

Figure 1

(a) Homeotropic nematic cell of thickness $L$. (b) Planar nematic cell of thickness $L$.

Figure 2

Map of the attraction and repulsion zones for dipole-dipole interaction (14) between two particles in the infinite NLC in the presence of the electric field $\mathbf{E}\left|\right|z,\Delta \varepsilon >0$ as the function of the dimensionless distance $kr$. Dashed lines are zone borders in the unlimited NLC, corresponding to (17). $-$ means attraction zone, $+$ means repulsion zone. Here $p{p}^{\prime }>0$.

Figure 3

Map of the attraction and repulsion zones for dipole-quadrupole interaction (15) between two particles in the infinite NLC in the presence of the electric field $\mathbf{E}\left|\right|z,\Delta \varepsilon >0$ as the function of the dimensionless distance $kr$. Dashed lines are the zone borders in the unlimited NLC, corresponding to (18). $-$ means attraction zone, $+$ means repulsion zone. The largest particle is in the center, $p{c}^{\prime }>c{p}^{\prime }$.

Figure 4

Map of the attraction and repulsion zones for quadrupole-quadrupole interaction (16) between two particles in the infinite NLC in the presence of the electric field $\mathbf{E}\left|\right|z,\Delta \varepsilon >0$ as the function of the dimensionless distance $kr$. Dashed lines are zone borders in the unlimited NLC, corresponding to (19). $-$ means attraction zone, $+$ means repulsion zone. Here $c{c}^{\prime }>0$.

Figure 5

Log-log plots of the interaction repulsion energy in $kT$ units as a function of the rescaled interparticle distance $\rho /L$ in the middle of the homeotropic cell in the external field perpendicular to the cell walls $\mathbf{E}\left|\right|z$ [see Fig. 1a]. Here the particle radius $a=2.2\mu \text{m}$, cell thickness $L=7\mu \text{m}$, $K=7\text{pN}$, $p=p^{\prime }=2.04a^2$, $c=c^{\prime }=0.2a^3$. Black dashed line (3) is the repulsion in the infinite NLC. Blue line 1 corresponds to the case $\Delta \varepsilon >0\text{and}E=0.99E_t$. Violet dot-dashed line (2) corresponds to the zero-field case $E=0$. Red line (4) corresponds to $\Delta \varepsilon <0\text{and}E=0.99E_t$. (a) Dipole-dipole interaction. Red line (4) $=$ deconfinement effect. (b) Quadrupole-quadrupole interaction. Red line (4) $=$ no deconfinement effect.

Figure 6

Qualitative explanation of the existence of deconfinement effect for dipole particle and absence of it for quadrupole particles. (a) Possible deformation modes in the cell. (b) Symmetry of the dipole deformations fits the symmetry of the first and main sine mode in the cell. (c) Symmetry of the quadrupole deformations fits the symmetry of the second cosine mode and does not fit the symmetry of the main sine mode in the nematic cell.

Figure 7

Log-log plots of the repulsion energy in $kT$ units as a function of the rescaled interparticle distance $\rho /L$ for dipole-dipole repulsion in the homeotropic cell in the presence of the parallel electric field $\mathbf{E}\left|\right|x$ along the $x$ direction [see Fig. 1a] for positive $\Delta \varepsilon >0$ [see formula (28) for $\varphi =0$]. Here the particle's radius $a=2.2\mu \text{m}$, cell thickness $L=7\mu \text{m}$, $K=7\text{pN}$, $p=p^{\prime }=2.04a^2$. Violet dot-dashed line (1) corresponds to the zero-field case $E=0$. Red line (2) corresponds to the $E=0.99E_t$ $=$ deconfinement effect.

Figure 8

Rescaled equilibrium distance between dipoles (blue lines 1) and quadrupoles (red lines 2) as a function of the rescaled field strength in the homeotropic cell in the presence of the parallel to the planes electric field $\mathbf{E}\left|\right|x$ [see Fig. 1a].

Figure 9

Map of the attraction and repulsion zones of interaction between two particles in the middle of the homeotropic cell in the external field parallel to the cell walls $\mathbf{E}\left|\right|x$ [see Fig. 1a], $E=0.99E_t$. Blue lines (1) correspond to the positive $\Delta \varepsilon$; red lines (2) correspond to the negative $\Delta \varepsilon$. Dashed lines depict zones for $E=2E_t$ and $\Delta \varepsilon <0$. $-$ means attraction, $+$ means repulsion. (a) Dipole-dipole interaction, $p{p}^{\prime }>0$. (b) Quadrupole-quadrupole interaction, $c{c}^{\prime }>0$. (c) Dimensionless energy of dipole-dipole interaction for $E=2E_t$ and $\Delta \varepsilon <0$. (d) Dimensionless energy of quadrupole-quadrupole interaction for $E=2E_t$ and $\Delta \varepsilon <0$.

Figure 10

Map of the attraction and repulsion zones of interaction between two particles in the middle of the planar cell with rubbing along $z$ and thickness $L$ placed in the perpendicular field $\mathbf{E}\left|\right|x$ [see Fig. 1b], $E=0.99E_t$. Dashed lines are the zone borders in the unlimited NLC. Thick red lines $a$ correspond to the positive $\Delta \varepsilon$; thick blue lines $b$ correspond to the negative $\Delta \varepsilon$. Thin black lines are the zone borders in the zero field. $-$ means attraction, $+$ means repulsion. (a) Dipole-dipole interaction, $p{p}^{\prime }>0$. (b) Dipole-quadrupole interaction. The largest particle is in the center, $p{c}^{\prime }>c{p}^{\prime }$. (c) Quadrupole-quadrupole interaction, $c{c}^{\prime }>0$.

Figure 11

Map of the attraction and repulsion zones of interaction between two particles in the middle of the planar cell with rubbing along $z$ and thickness $L$ placed in the electric field applied along the $y\text{axis},\mathbf{E}\left|\right|y$ [see Fig. 1b], $E=0.99E_t$. Dashed lines are zone borders in the unlimited NLC. Thick red lines $a$ correspond to the positive $\Delta \varepsilon$; thick blue lines $b$ correspond to the negative $\Delta \varepsilon$. Thin black lines are the zone borders in the zero field. $-$ means attraction, $+$ means repulsion. (a) Dipole-dipole interaction, $p{p}^{\prime }>0$. (b) Dipole-quadrupole interaction. The largest particle is in the center, $p{c}^{\prime }>c{p}^{\prime }$. (c) Quadrupole-quadrupole interaction, $c{c}^{\prime }>0$.

Figure 12

Log-log plots of repulsion between two particles along the $y$ direction ($\theta =\pi /2$) in the middle of the planar cell, placed in the electric field applied along the $y\text{axis},\mathbf{E}\left|\right|y$ (see Fig. 11) (a) Log-log plots of the dipole-dipole repulsion ($F_1$ function) as a function of the rescaled distance $\rho /L$. Black dashed line (3) is the power-law asymptotics $\frac{1}{4}{\left(\frac{L}{\rho }\right)}^3$. (b) Log-log plots of the of the quadrupole-quadrupole repulsion ($D_1$ function) as a function of the rescaled distance $\rho /L$. Black dashed line (3) is the power-law asymptotics $\frac{9}{4}{\left(\frac{L}{\rho }\right)}^5$. Blue lines (1) correspond to $E=0.99E_t$ and $\Delta \varepsilon <0$. Violet dot-dashed line (2) correspond to the zero-field case $E=0$. Red lines (4) correspond to $E=0.99E_t$ and $\Delta \varepsilon >0$ and reveals the deconfinement effect for both dipole-dipole and quadrupole-quadrupole interactions in the planar cell.

Figure 13

Change of the attraction and repulsion zones of dipole-dipole interaction between two particles in the middle of the planar cell with $\Delta \varepsilon <0$ in the electric field applied along the $y\text{axis},\mathbf{E}\left|\right|y$ [see Fig. 1b] with increasing of the field strength $E$. The upper right quarter of the map is depicted. Dashed line is the zone border in the unlimited NLC. Thin black line (1) corresponds to the $E=0$, thick blue line (2) to $E=0.99E_t$, thick green line (3) to $E=1.6E_t$, thick black line (4) to $E=1.73E_t$, thick red line (5) to $E=1.74E_t$, thick brown line (6) to $E=2E_t$. Attraction zone is above each line, repulsion zone is below each line, $p{p}^{\prime }>0$ (see Figs. 14 and 15 as well). Collapse of the lateral tails into dumbbell-shaped region occurs at $E=\sqrt{3}{E}_t$.

Figure 14

Map of the attraction and repulsion zones of dipole-dipole interaction between two particles in the middle of the planar cell with $\Delta \varepsilon <0$ placed in the electric field applied along the $y\text{axis},\mathbf{E}\left|\right|y$, $E=1.73E_t$. Dashed lines are zone borders in the unlimited NLC. $-$ means attraction, $+$ means repulsion, $p{p}^{\prime }>0$.

Figure 15

Map of the attraction and repulsion zones of dipole-dipole interaction between two particles in the middle of the planar cell with $\Delta \varepsilon <0$ placed in the electric field applied along the $y\text{axis},\mathbf{E}\left|\right|y$, $E=1.74E_t$. Dashed lines are zone borders in the unlimited NLC. Collapse of the lateral tails into dumbbell-shaped region occurs at $E_{\mathrm{col}}=\sqrt{3}{E}_t$. $-$ means attraction, $+$ means repulsion, $p{p}^{\prime }>0$.

Figure 16

Map of the attraction and repulsion zones of quadrupole-quadrupole interaction between two particles in the middle of the planar cell with $\Delta \varepsilon <0$ placed in the electric field applied along the $y\text{axis},\mathbf{E}\left|\right|y$, $E=1.73E_t$. Dashed lines are the zone borders in the unlimited NLC. $-$ means attraction, $+$ means repulsion, $c{c}^{\prime }>0$.

Figure 17

Map of the attraction and repulsion zones of quadrupole-quadrupole interaction between two particles in the middle of the planar cell with $\Delta \varepsilon <0$ placed in the electric field applied along the $y\text{axis},\mathbf{E}\left|\right|y$, $E=1.74E_t$. Dashed lines are the zone borders in the unlimited NLC. Collapse of the lateral tails into dumbbell-shaped region occurs at $E=\sqrt{3}{E}_t$. $-$ means attraction, $+$ means repulsion, $c{c}^{\prime }>0$.

Figure 18

Map of the attraction and repulsion zones of quadrupole-quadrupole interaction between two particles in the middle of the planar cell with $\Delta \varepsilon <0$ placed in the electric field applied along the $y\text{axis},\mathbf{E}\left|\right|y$, $E=2E_t$. Dashed lines are zone borders in the unlimited NLC. Compression of the dumbbell-shaped regions occurs with increasing of the field strength for $E>\sqrt{3}{E}_t$. $-$ means attraction, $+$ means repulsion, $c{c}^{\prime }>0$.

Figure 19

Map of the attraction and repulsion zones of interaction between two particles in the middle of the planar cell with rubbing along $z$ and thickness $L$ placed in the electric field parallel to the rubbing $\mathbf{E}\left|\right|z$ [see Fig. 1b], $E=0.99E_t$. Dashed lines are the zone borders in the unlimited NLC. Thick red lines $a$ correspond to the positive $\Delta \varepsilon >0$, thick blue lines $b$ correspond to the negative $\Delta \varepsilon <0$. Thin black lines are the zone borders in the zero-field case. $-$ means attraction, $+$ means repulsion. (a) Dipole-dipole interaction, $p{p}^{\prime }>0$. (b) Dipole-quadrupole interaction. The largest particle is in the center, $p{c}^{\prime }>c{p}^{\prime }$. (c) Quadrupole-quadrupole interaction, $c{c}^{\prime }>0$.