Coulomb-like interaction in nematic emulsions induced by external torques exerted on the colloids

An external mechanical torque on colloids immersed in a nematic liquid crystal can induce a Coulomb-like $1∕r$ interaction between them [Lev and Tomchuk, Phys. Rev. E 59, 591 (1999); Lev et al., ibid. 65, 021709 (2002)]. In this paper we show that the director-mediated Coulomb-like interaction of two colloids is determined by the vectors ${\mathbf{\Gamma }}_{\perp }^{\left(1\right)}$ and ${\mathbf{\Gamma }}_{\perp }^{\left(2\right)}$ of the transverse external torques exerted upon these colloids. We derive the $1∕r$ potential in which the scalar product $-({\mathbf{\Gamma }}_{\perp }^{\left(1\right)}\bullet {\mathbf{\Gamma }}_{\perp }^{\left(2\right)})$ of the two torques plays the role of the product of two electrostatic charges. The $1∕r$ interaction is attractive for $({\mathbf{\Gamma }}_{\perp }^{\left(1\right)}\bullet {\mathbf{\Gamma }}_{\perp }^{\left(2\right)})>0$ and repulsive for $({\mathbf{\Gamma }}_{\perp }^{\left(1\right)}\bullet {\mathbf{\Gamma }}_{\perp }^{\left(2\right)})<0$ (“parallel torques” attract whereas “antiparallel torques” repel each other). The vector of transverse torque determines the two-component “elastic charge” (dyad), which is illustrated by the $1∕r^2$ and $1∕r^3$ terms in the elastic energy (the elastic analogs of the monopole-dipole and dipole-dipole interactions). The general status of the pairwise approach to nematic emulsions is considered in terms of the elastic charge density.

Figure 1

Particlelike distortion domain with nonlinear core (darkened). At the enclosing spherical surface $S$ the director component normal to the unperturbed uniform director ${\mathbf{n}}_{\infty }$ is small, whereas at the nonlinear core it can be large. $\mathbf{i}$ is the image of the point $\mathbf{r}$ in $S$.

Figure 2

Two spheres enclosing particles 1 and 2.