Confinement effect on the interaction between colloidal particles in a nematic liquid crystal: An analytical study

Motivated by a recent experimental study on the interaction between colloidal particles in a confined nematic liquid crystal [M. Vilfan, N. Osterman, M. Čopič, M. Ravnik, S. Žumer, J. Kotar, D. Babič, and I. Poberaj, Phys. Rev. Lett. 101, 237801 (2008)], we discuss in an analytical manner how the interaction potential $U$ between spherical colloidal particles in a confined nematic cell behaves as a function of the interparticle distance $r$. We show that the short-range potential follows a power law $U\left(r\right)\sim r^{-5}$ as expected from the quadrupolar nature of the interaction, while the long-range potential is dominated by an exponential function $U\left(r\right)\sim \sqrt{d/r}\text{exp}\left(-2\pi r/d\right)$, where $d$ is the cell thickness. These two regimes are interchanged at $r/d\simeq 0.8$. This behavior of $U\left(r\right)$ is in a good semiquantitative agreement with the experimental finding.

Figure 1

Geometry of our nematic cell containing two colloidal particles. Rigid normal anchoring is assumed at the cell surfaces $z=0$ and $d$. In Sec. 3, we assume that the centers of the particles are located at the midplane $z=d/2$. Here weak planar anchoring on the particle surfaces is assumed.

Figure 2

(Color online) Log-log plots of the rescaled interaction potential $U\left(r\right)/\left({\Gamma }^2/\pi K{d}^5\right)$ as a function of the rescaled interparticle distance $r/d$ for the range (a) $0.05\le r/d\le 5$ and (b) $0.6\le r/d\le 1$. Solid red lines represent the full interaction potential calculated using Eq. (10) with replacing the infinite sum with ${\sum }_{m^{\prime }=1}^{100}$. Dotted straight blue lines are the power-law behavior given in Eq. (8). Dashed green lines correspond to Eq. (12), the long-distance result with ${\Delta }_{\perp }{K}_0\left(2\pi r/d\right)$. Black dashed-dotted lines represent the exponential behavior in Eq. (13).