Topological transition from a hopfion to a toron via flexoelectric self-polarization in chiral liquid crystals

Abstract

The presence of topological defects in apolar chiral liquid crystals cause orientational distortions, leading to non-uniform strain. This non-uniform strain generates an electric polarization response due to the flexoelectric effect, which induces an internal electric field. Associated to this electric field is an electrostatic self-energy, which has a back-reaction on the director field. Calculation of this internal electric field and its resulting back-reaction on the director field is complicated. We propose a method to do such, adapting a method recently developed to study the magnetostatic self-interaction effect on skyrmions in chiral ferromagnets. Bloch skyrmions in chiral magnets are solenoidal and are unaffected by the magnetostatic self-interaction. However, Bloch skyrmions in liquid crystals yield non-solenoidal flexoelectric polarization and, thus, are affected by the electrostatic self-interaction. Additionally, as the flexoelectric coefficients are increased in strength, a transition from a hopfion to a toron is observed in three-dimensional confined systems.

Figures (5)

• Figure 1: Flexoelectric polarization effect on a skyrmion in a liquid crystal in the one constant approximation. A twist favored Bloch skyrmion is shown in (a), and a splay-bend favored Néel skyrmion in (b). In each subfigure is the director $\mathbf{n}(x,y)$ and the flexoelectric potential $\varphi(x,y)$ is shown alongside the electric field vector $\mathbf{E}=-\bm{\nabla}\varphi$. Also shown in is the associated self-induced electric charge density $\rho=-\bm{\nabla}\cdot\mathbf{P}_f$. It can be seen that the Bloch skyrmion has a core of negative charge, surrounded by an outer ring of positive charge. Whereas, the Néel skyrmions has a neutral core surrounded by an internal negatively charged ring and a positively charged outer ring. For material properties, we use the elastic coefficient $K=10\,\textup{pN}$ and the cholesteric pitch is chosen to be $p=7\,\mu\textup{m}$. The flexoelectric coefficients are chosen to be $e_1=2\,p\textup{Cm}^{-1}$ and $e_3=4\,p\textup{Cm}^{-1}$, with an applied electric field of strength $E_z=1\,\textup{V}/\mu\textup{m}$ and dielectric anisotropy $\Delta\epsilon=3.7$.
• Figure 2: Flexoelectric polarization effect on a hopfion in a liquid crystal in the one constant approximation $K_i=K=10\,\textup{pN}$. Shown is the relaxed director field $\mathbf{n}(x,y,z)$ solution of the flexoelectric Frank--Oseen energy \ref{['eq: Flexoelectric Frank--Oseen energy']}, starting from the hopfion ansatz \ref{['eq: Hopfion ansatz']} as an initial configuration. Plots of the director field $\mathbf{n}$ are shown for (a) a hopfion and (b) a toron. The distance between the cell plates is taken to be $d=7\,\mu\textup{m}$, with an applied voltage of $U=2\,V$. This gives an applied electric field strength of $E_z=U/d=0.29\,\textup{V}/\mu\textup{m}$ and we set the dielectric anisotropy to be $\Delta\epsilon=4.8$. The cholesteric pitch $p$ is taken to be the same length as the distance between the cell plates, $p=d$. We set the flexoelectric coefficients to be equal $e_1=e_3$. There is a phase transition from a hopfion to a toron as the flexoelectric coefficients are increased from $e_i=4\,\textup{pCm}^{-1}$ to $e_i=8\,\textup{pCm}^{-1}$.
• Figure 3: Isosurface plots of the director field: the top row is the field $n_3=0$ and the bottom row is the linking of the field $n_1=\pm0.9$. The hopfion is shown in (a) and the toron in (b). The parameter set is detailed in Figure \ref{['fig: Hopfion results - Director field']}. As the flexoelectric coefficients are increased from $e_i=4\,\textup{pCm}^{-1}$ to $e_i=8\,\textup{pCm}^{-1}$, the hopfion transitions to a toron. It is clear to see from the bottom row that the knots become unlinked.
• Figure 4: Flexoelectric polarization effect on a hopfion in a liquid crystal in the one constant approximation. The top row shows the self-induced electrostatic scalar potential $\varphi$, which is a solution of the Poisson equation $\Delta\varphi=\rho/\epsilon_0$, and the bottom row displays the electric charge density $\rho=-\bm{\nabla}\cdot\mathbf{P}_f$. It is clear to see that the electric charge density is confined within the confined geometry $\Omega$, whereas the electric scalar potential extends into $\mathbb{R}^3/\Omega$. The parameter set is detailed in Figure \ref{['fig: Hopfion results - Director field']} with flexoelectric coefficients $e_1=e_3=4\,\textup{pCm}^{-1}$.
• Figure 5: Transition from a hopfion to a toron due to the flexoelectric self-polarization effect in a liquid crystal in the one constant approximation. The top row shows the self-induced electrostatic scalar potential $\varphi$, which is a solution of the Poisson equation $\Delta\varphi=\rho/\epsilon_0$, and the bottom row displays the electric charge density $\rho=-\bm{\nabla}\cdot\mathbf{P}_f$. It is clear to see that the electric charge density is confined within the confined geometry $\Omega$, whereas the electric scalar potential extends into $\mathbb{R}^3/\Omega$. The parameter set is detailed in Figure \ref{['fig: Hopfion results - Director field']} with flexoelectric coefficients $e_1=e_3=8\,\textup{pCm}^{-1}$.