A nematic liquid crystal with an immersed body: equilibrium, stress and paradox

Abstract

We examine the equilibrium configurations of a nematic liquid crystal with an immersed body in two dimensions. A complex variables formulation provides a means for finding analytical solutions in the case of strong anchoring. Local tractions, forces and torques on the body are discussed in a general setting. For weak (finite) anchoring strengths, an effective boundary technique is proposed which is used to determine asymptotic solutions. The energy-minimizing locations of topological defects on the body surface are also discussed. A number of examples are provided, including circular and triangular bodies, and a Janus particle with hybrid anchoring conditions. Analogies to classical results in fluid dynamics are identified, including d'Alembert's paradox, Stokes’ paradox and the Kutta condition for circulation selection.

Defect pairs in nematic cell alignment on doubly connected domains

“…The existing studies [14,24] also showed the defects preferred to be located around the sharp corners, which is consistent with the present simulations. To further investigate the effects of sharp corners on the defect positions, it would be suitable to use Schwartz-Christoffel maps in the same way as the existing studies [14,24]. However, it is practically difficult to compute the energy numerically due to the existence of singularities on the integration path coming from the conformal map.…”

“…According to our simulations, the defects tended to be located around rounded corners after the energy-minimizing procedure. The existing studies [14,24] also showed the defects preferred to be located around the sharp corners, which is consistent with the present simulations. To further investigate the effects of sharp corners on the defect positions, it would be suitable to use Schwartz-Christoffel maps in the same way as the existing studies [14,24].…”

“…Thus, another criterion to determine Q n is required from the physical point of view since n is the free parameter in the formula (equation 3.3). It is noted that this non-uniqueness problem was also mentioned in Chandler & Spagnolie's study [24], and the free parameter n is related to the 'period' in their study.…”

“…Miyazako & Nara [13] derived an analytic formula for the orientation angle of nematic liquid crystals in a unit circle, thereby constructing successfully a general formula for simply connected domains with conformal mapping. Chandler & Spagnolie also used complex functions to derive analytic solutions for calculating nematic order with an immersed body and considered the existence of the topological defects on the surface of the body [14]. The next simplest shape of domains with non-trivial geometric features is concentric annuli.…”

“…In this paper, we take the same approach as Miyazako & Nara [13] to the case of doubly connected domains using a transcendental function called the Schottky-Klein-Prime (SKP) function [22,23]. It is noted that the SKP function was also used in Chandler & Spagnolie's study, which considered concentric annuli to derive an analytic solution of nematic order with multiple immersed bodies [24]; however, they considered the existence of the defects on the surface of the bodies. Compared with this study [24], our analytic formulae allow defects to exist in any place inside doubly connected domains, and do not have limitations on the number of defects.…”