Chiral dipole induced by azimuthal anchoring on the surface of a planar elastic quadrupole

A spherical colloid with the tangential surface nematic director, aligned along the surface meridians, is known as a planar elastic quadrupole. The azimuthal anchoring, however, can induce a deviation of the planar director from the meridional lines. We show that a helical component of the planar surface director at the spherical surface of a planar quadrupole removes all the reflection symmetry planes and gives rise to a chiral elastic dipolar component. Using an ansatz approach, we consider the interplay between the quadrupole and anchoring-induced chiral dipole components. The chirality is enhanced by the bend-twist anisotropy. The interaction of the chiral components changes the attraction directions of two such colloids. In particular, a point appears at which the quadrupolar repulsion is balanced by the dipolar attraction.

Figure 1

(a) If the director lines on the spherical colloid surface lie in the vertical meridional planes passing through the poles, then this is a bipolar planar quadrupole. (b) If the surface director makes a nonzero angle $\alpha$ with the vertical meridional planes, the helical component, proportional to $\sin \alpha$, in addition gives rise to a chiral dipole component.

Figure 2

Numerical value of the dipole coefficient $c$ vs $\alpha$ for different elastic anisotropy $s=K_{11}/K_{22}$ and $b=K_{33}/K_{22}.$ 1: $s=1,b=2$; 2: $s=2,b=2$; 3: $s=2,b=2.6$; 4: $s=1,b=3$; 5: $s=2,b=3$; 6: $s=1,b=4$; 7: $s=2,b=4$; 8: $s=0.5,b=5$; 9: $s=1,b=5$; 10: $s=3.33,b=5$; 11: $s=0.5,b=7$; 12: $s=0.5,b=10.$

Figure 3

Directions of the force a colloid at the center exerts on another similar colloid; the spatial scale (indicated by integers along and across the director) is in units $r_0$. (a) Both colloids are pure PQs, the attraction direction is at $\Psi \simeq {49}^{\circ }$. (b) Two quadrupoles with PQ and CD components of like handedness, $C_1{C}_2>0$, the attraction direction is at $\Psi =0^{\circ }$. (c) Two quadrupoles with PQ and CD components of opposite handedness, $C_1{C}_2<0$, the attraction direction is at $\Psi ={90}^{\circ }$. The saddle balance point is shown by crossed arrows. The fields in panels b and c correspond to curve 3 in Figs. 2 and 4: $s=2$, $b=2.6$. Because of very different scales, the force magnitude is not indicated.

Figure 4

Numerical values of the balance distance $R_{\left|\right|}$ (in units $r_0$) vs angle $\alpha$. The curves with the same numbers in this figure and Fig. 2 correspond to the same anistropies $s$ and $b$.