Analytical description of the Saturn-ring defect in nematic colloids

We derive an analytical formula for the Saturn-ring configuration around a small colloidal particle suspended in nematic liquid crystal. In particular we obtain an explicit expression for the ring radius and its dependence on the anchoring energy. We work within Landau–de Gennes theory: Nematic alignment is described by a tensorial order parameter. For nematic colloids this model had previously been used exclusively to perform numerical computations. Our method demonstrates that the tensorial theory can also be used to obtain analytical results, suggesting a different approach to the understanding of nematic colloidal interactions.

Figure 1

Direction field ${\mathit{n}}^0$ (eigenvector associated with the largest eigenvalue of ${\mathit{Q}}^0$) as explicitly computed below (7) for different values of the effective anchoring coefficient: (a) $w=\infty$, (b) $w=3$, (c) $w=1.732\approx \sqrt{3}$, and (d) $w=1$.

Figure 2

Ring radius dependence on the effective anchoring coefficient (6). Strong anchoring yields ${\rho }_{\infty }\approx 1.47$. The surface ring ${\rho }_w=1$ corresponds to $w=\sqrt{3}\approx 1.73$.