Hydrodynamic Modeling of Odd Nematic Elasticity in Liquid Crystals

Abstract

There is a recent interest in studying odd elasticity in soft solids. Current focus has been on simple solids. However, many soft solids are structured and can exhibit nematic elasticity or viscoelasticity. Here we generalize the concept of odd elasticity to nematic elasticity. By rewriting the governing equation for two-dimensional nematic liquid crystals (LCs) in terms of complex Ginzburg--Landau equation, we propose an odd nematic elastic term and its stress term in the hydrodynamic model of nematic LCs. The odd nematic elasticity can be physically interpreted as non-reciprocal interactions between neighboring directors. In odd nematics we find that domain walls become self-propelled and are accompanied by a bidirectional flow, and point defects can self-spin, develop a spiral pattern, and induce a vortical flow. Interactions of a pair of defects show rich dynamics that are distinct from those in active nematics. As such, we have developed an odd general elasticity, which can be further generalized to other viscoelastic materials, and proposed a novel way to manipulate topological defects in nematic LCs.

Figures (4)

• Figure 1: Locomotion of a Néel wall with odd nematic elasticity. (a) Illustration of nematic elasticity (NE) and odd nematic elasticity (ONE) as reciprocal and non-reciprocal interactions between directors, respectively. (b) Illustration of a domain wall self-propelled by ONE-induced collective rotation of directors. (c) Schematic of clockwise (CW) and counter-clockwise (CCW) Néel. (d) Self-propulsion velocities $u_x^{\mathrm{wall}}$ of CW and CCW Néel walls as functions of $\lambda$. Markers are simulation data with and without hydrodynamic effects, and solid lines are from Eq. \ref{['wall_speed']}. (e) Director and the corresponding flow field of a CCW Néel wall with hydrodynamic effect. Director orientations are indicated by black ticks, with the background color representing $|n_x|$. Arrows indicate flow direction, with color-coded intensity representing reduced velocity $v_y/v_{\mathrm{ref}}$.
• Figure 2: Director angle $\theta$ of a Néel wall with and without hydrodynamic effects.
• Figure 3: Dynamics of a pair of $\pm1/2$ defects without hydrodynamic effects. (a) Defect trajectories for initial separation $D_0=40$ for $0\le \lambda \le 0.25$. (b) Director field colored by $|n_x|$ (top) and by the total elastic energy density (bottom). Red and blue curves indicate the trajectories of $+1/2$ and $-1/2$ defects, respectively; the cyan skeleton marks the highly deformed domain wall, and light green arrows denote the its normal direction. (c) Phase diagram in the $(D_0,\lambda)$ plane, showing annihilation, oscillatory, and gliding regimes; colors indicate the averaged gliding velocity $u_y^\mathrm{def}/u_\mathrm{ref}$ evaluated at the equilibrium separation $D(t)=D(t_e)$, with $u_\mathrm{ref}=2\times10^{-3}$. (d) Dependence of the transverse velocity $v_y$ and the equilibrium separation $D(t_e)$ on $\lambda$.
• Figure 4: Hydrodynamic effects on defect dynamics. (a) "Phase" diagram of a defect pair for $\theta_{d0}=\pi$ in the hydrodynamic simulation. The dashed black line denotes the phase boundary separating the annihilation (Ann.) phase from the other phases shown in \ref{['Defect_Pair']}, while the solid blue line indicates the new phase boundary. (b) Snapshots of the director field, colored by $|n_x|$, and the flow field, colored by the vorticity, at $D_0 = 80$ and $\lambda = 0.075$ for initial defect orientations $\theta_{d0} = \pi$ (top) and $\theta_{d0} = 0$ (bottom). Magenta arrows indicate the odd force $\mathbf{f}^{o}$ integrated around the defect cores, which are bounded by cyan circles. The right panel shows time evolution of the defect separation $D$, the angle $\theta_d$ between the $+1/2$ defect orientation and the line connecting the $+1/2$ and $-1/2$ defects, and the angle $\phi_c$ of the connection line with initial defect orientations $\theta_{d0} = \pi$ and $0$, respectively. (c) Time evolution of the director and flow fields at $D_0 = 80$ and $\lambda = 0.1$. (d) Snapshots of the director and velocity fields illustrating the mutual orbital motion of the defects about their midpoint at $D_0 = 80$ and $\lambda = 0.125$.