Inversion Ring in Chromonic Twisted Hedgehogs: Theory and Experiment

Abstract

Twisted hedgehogs are defects in spherical cavities with homeotropic anchoring for the nematic director that arise when twist distortions are sufficiently less energetic than splay (and bend) distortions. They bear a characteristic inversion ring, where the director texture changes the sense it spirals about the center of the cavity. This paper applies a quartic twist theory recently proposed to describe the elasticity of chromonics to explain a series of inversion rings observed in aqueous solutions of SSY at two different concentrations. The theory features a phenomenological length a, whose measure is extracted from the data and shown to be fairly independent of the cavity radius, as expected for a material constant.

Figures (12)

• Figure 1: Field lines of $\bm{n}_{\mathrm{T}}$ in \ref{['eq:twisted_hedgehog']} in a spherical cavity enforcing homeotropic boundary conditions. An inversion ring, depicted in blue, is present on the equatorial plane (orthogonal to the symmetry axis $\bm{e}$). Black lines are field lines lying on the equatorial plane; red lines are field lines coming out of the equatorial plane. The whole 3D picture is obtained by rotating this drawing about $\bm{e}$. This sketch is adapted from Fig. 2 of paparini:spiralling; it represent only one chiral variant of twisted hedgehog, the other is obtained by reversing the sign of $\alpha$ in \ref{['eq:twisted_hedgehog']}.
• Figure 2: The elastic free energy (positive) minimizer $\alpha_\lambda$ for reduced elastic constants $k_1 = 7.5$, $k_3 = 9.0$ (which apply to a SSY solution at concentration $c=30\,\mathrm{wt}\%$), and $\lambda=0.94$ is compared with the minimizer $\alpha_0$ according to the Oseen-Frank theory. The dotted line is drawn at $\alpha=\frac{\pi}{2}$; its intercepts with the graphs of $\alpha_\lambda$ and $\alpha_0$ designate the (scaled) radii of the inversion ring, $\rho^\ast:=r^\ast/R\doteq0.52$ and $\rho^\ast_0:=r^\ast_0/R\doteq0.03$, respectively. As shown in paparini:spiralling, $\widehat{\alpha}_0=\arccos(-1/4)$ is the value of $\alpha_0$ at $\rho=0$, whereas $\alpha_\lambda(0)=\pi$.
• Figure 3: The scaled radius $\rho^\ast:=r^\ast/R$ of the inversion ring for the elastic free energy minimizer $\alpha_\lambda$ is plotted against $\lambda:=a/R$ for the same reduced elastic constants chosen in Fig. \ref{['fig:alpha_profiles']}. The graph saturates at $\rho^*_\infty\doteq0.82$, while $\rho^*_0\doteq0.03$ is the limiting value as $\lambda\to0$.
• Figure 4:
• Figure 5: Molecular structure of PDMS.