Elastic interaction between colloidal particles in confined nematic liquid crystals

The theory of elastic interaction of micrometer-sized axially symmetric colloidal particles immersed into confined nematic liquid crystal has been proposed. General formulas are obtained for the self-energy of one colloidal particle and interaction energy between two particles in arbitrary confined nematic liquid crystals with strong anchoring condition on the bounding surfaces. Particular cases of dipole-dipole interaction in the homeotropic and planar nematic cell with thickness $L$ are considered and found to be exponentially screened on far distances with decay length ${\lambda }_{\text{dd}}=\frac{L}{\pi }$. It is predicted that bounding surfaces in the planar cell crucially change the attraction and repulsion zones of usual dipole-dipole interaction. As well it is predicted that the decay length in quadrupolar interaction is two times smaller than for the dipolar case in the homeotropic cell.

Figure 1

(Color online) Log-log plots of the interaction potential in $kT$ units as a function of the rescaled interparticle distance $\rho /L$. Here particle’s radius $a=2.2\mu \text{m}$, cell thickness $L=7\mu \text{m}$, $K=7\text{pN}$, $p=p^{\prime }=2.04a^2$, $c=c^{\prime }=0.2a^3$. Blue thick line 3 is quadrupole potential in homeotropic cell, dashed thick line 4 is its power-law asymptotics $\propto 1/{\rho }^5$. All thin lines are dipole potentials in planar cell. Coming anticlockwise, purple line 1 is attraction along the direction $\phi =40°$, brown line 2 is along $\phi =20°$, red line 5 is along $z$ axis $\phi =0$. Black thick line 6 is dipole repulsion in homeotropic cell from Eq. (7) and the last green thin line 7 is repulsion along $\phi =\frac{\pi }{2}$. Thin dashed line 8 is the power-law asymptotics $U=\frac{4\pi K{p}^2}{{\rho }^3}$.

Figure 2

(Color online) Blue thick line is the border of the attraction (inside) and repulsion (outside) zone for parallel dipoles in the planar cell from Eq. (10). Director ${\mathbf{n}}_0\parallel z$. Black thin line is the parabola $z=\frac{\pi y^2}{L}$. Dashed lines make angle $\phi =\text{arccos}\left(\frac{1}{\sqrt{3}}\right)$ with $z$ and are borders of repulsion and attraction zone for unlimited nematic.