Spiralling defect cores in chromonic hedgehogs

ABSTRACT

An elastic quartic twist theory has recently been proposed for chromonic liquid crystals, intended to overcome the paradoxical conclusions encountered by the classical Oseen-Frank theory when applied to droplets submerged in an isotropic fluid environment. However, available experimental data for chromonics confined to cylindrical cavities with degenerate planar anchoring on their lateral boundary can be explained equally well by both competing theories. This paper identifies a means to differentiate these theories both qualitatively and quantitatively. They are shown to predict quite different core defects for the twisted hedgehogs that chromonics generate when confined to a fixed spherical cavity with homeotropic anchoring. In the quartic twist theory, the defect core is estimated to be nearly one order of magnitude larger (few microns) than in the other and, correspondingly, the director field lines describe Archimedean spirals instead of logarithmic ones.

GRAPHICAL ABSTRACT

KEYWORDS:

• Chromonic liquid crystals
• hedgehog defects
• core structure
• elastic theories
• curvature elasticity

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. The classification of the most general uniform distortions, which can fill the whole three-dimensional space, is given in [36] and recalled in Sect. 2.1.

2. Tactoids are elongated, cylindrically symmetric shapes with pointed ends as poles.

3. It should also be noted that other theories are known as “quartic’’ (see, for, example, the classical paper [53] and the more recent contribution [54]), but they owe this name to an elastic term globally quartic in de Gennes’ order tensor and its derivatives, added to the commonly considered version of the Landau de Gennes theory to resolve the splay-bend elastic constant degeneracy in the reduction to the Oseen-Frank theory. These theories serve a different purpose.

4. Also, a paper by Zocher [55], mainly concerned with the effect of a magnetic field on director distortions, is often mentioned among the founding contributions. Some authors go to the extent of also naming the theory after him. Others, in contrast, name the theory only after Frank, as they only deem his contribution to be fully aware of the nature of $n$ as a mesoscopic descriptor of molecular order.

5. This requirement amounts to assume that $W(Qn,Q(\nabla n)Q^{\text{T}})=W(n,\nabla n)$, for all rotations $\mathbf{Q}$ in three-dimensional space.

6. It is argued in [56] that $q$ should be given the name tetrahedral splay, to which we would actually prefer octupolar splay for the role played by a cubic (octupolar) potential on the unit sphere [57] in representing all scalar measures of distortion, but $T$.

7. With opposite chiralities, one for each sign in (10).

8. In opposite senses, according to the sign of chirality.

9. Here, we adopt the terminology proposed by Selinger [56] (see also [58]) and distinguish between single and double twists, the former being uniform and the latter not.

10. In the elastic model proposed in [36] for twist-bend nematics, a quartic free energy was posited that admits as the ground state either of two families of uniform heliconical fields with opposite chirality. There too, a length scale appears in the equilibrium pitch. The distortion state characterised by this length is the same everywhere.

11. The persistence length of a flexible aggregate is the shortest length over which unit vectors tangent to the aggregate’s contour lose correlation. For CLCs, it is estimated on the order of tens to hundreds of $\mathrm{n}\mathrm{m}$ [59].

12. Via the naive geometric argument that represents chiral molecules as cylindrical screws and derives the pitch of their assemblies by close packing them so as to fit grooves with grooves.

13. For lyotropic cholesterics, the mismatch between microscopic and macroscopic pitches, which has recently received new experimental evidence in systems of basic living constituents [60,61], is still debated. Interesting theories based on either molecular shape fluctuations [62,63] or surface charge patterns [64] have met with some experimental disagreement [65].

14. This is a tensor whose representative matrix is the cofactor matrix of the matrix representing the original tensor, see [66, p. 22] for a formal definition.

15. This equation follows from Appendix A2 and the general property of (18) stating that $N(-n)=-N(n)$, which stems from being ${(\nabla \text{s}n)}^{*}$ even in $n$.

16. Actually, in [40], $n_{\text{H}}$ was defined to be precisely $\overline{n_{\text{R}}}$, so that, being opposite to the field in (16), would form with it a defect-anti-defect pair, as would also be clear from Figure 1 once the field lines orientation in panel (b) are reversed.

17. In a temperature regime where the twist constant $K_{22}$ is sufficiently small.

18. A result which was independently rediscovered in [67].

19. We shall continue to adopt the same old symbol for the function $\alpha$, even if it is expressed in the new variable.

20. Which requires that $\rho \alpha { }^{\mathrm{\prime }}(\rho )$ be bounded as $\rho \to 0^{+}$.

21. It is equivalent to the equation of motion (B6) for the effective dynamical system described in Appendix B.

22. The absolute measured values are $K_{11}\approx 4.3\mathrm{p}\mathrm{N}$, $K_{22}\approx 0.7\mathrm{p}\mathrm{N}$, and $K_{33}\approx 6.1\mathrm{p}\mathrm{N}$.

23. It is explicitly dependent on time.

24. Asymptotically autonomous dynamical systems have an interesting literature, recalled for example in Chapter. 17 of [68].

25. See, for example, Sect. 2.2.2 of [49].

26. The $\omega$-limit set of a forward solution to a dynamical system is the collection of all limiting points attained by the solution on any diverging time sequence (see, p. 242 of [68] for a formal definition).