Instability of toroidal nematics

ABSTRACT

Toroidal nematics are nematic liquid crystals confined within a circular torus and subject to planar degenerate anchoring on the boundary of the torus. They may be droplets floating in an isotropic environment or cavities carved out of a solid substrate. A universal solution of Frank’s elastic free energy is an equilibrium configuration for the nematic director field, irrespective of the values of the elastic constants, whose vector lines are the coaxial parallels of the torus. We explore the local stability of this configuration and identify a range of parameters where the main drive towards instability does not come from the surface-like elastic constant $K_{24}$ being large, but from the ratio $K_2/K_3$ of the twist to bend elastic constants being small, which also makes our study relevant to chromonic liquid crystals.

Graphical Abstract

KEYWORDS:

• Frank’s elastic energy
• universal solutions in liquid crystals
• chromonic liquid crystals
• linear stability

Acknowledgements

The work of A. P. has been supported by the University of Pavia under the FRG initiative, meant to foster research among young postdoctoral fellows. E. G. V. acknowledges the kind hospitality of the Oxford Centre for Nonlinear PDE, where part of this work was done while he was visiting the Mathematical Institute at the University of Oxford. We are both grateful to A. Zarnescu for several enlightening discussions on the subject of this paper while our project was in its early stages.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. We write the curvature tensor as

${\mathrm{\nabla }}_{\mathrm{s}}v={\kappa }_1{e}_1\otimes e_1+{\kappa }_2{e}_2\otimes e_2$

where $e_1$ and $e_2$ are unit vectors along the principal directions of curvature of $\mathrm{\partial }\mathfrak{B}$.

2. We learn from Truesdell ([23], p.332) that this definition of $\mathrm{\Omega }$ was introduced by Zhukowsky and named abnormality by Levi-Civita. We shall use the modern name of helicity.

3. Here and in the following, $f_{,x}$ denotes the partial derivative of a function $f$ with respect to one of its variables, $x$.

4. We were not able to prove that a divergence-free field n represents the optimal way to probe the stability of the axial configuration. At this stage, the example of chromonics is only meant to be suggestive. They indicate that nematic liquid crystals with high elastic anisotropies are not just a theoretical curiosity but real-life materials.

5. Often, the integral of $|a^{\ast }{|}^2$ is prescribed as a normalisation condition. Here, we chose to prescribe instead the derivative of $a^{\ast }$ at a point; the special value in Equation (33) was taken to represent the ratio of the only two lengths present in the problem.

Additional information

Funding

The work of A. P. has been supported by the University of Pavia under the FRG initiative.