Odd elasticity in disordered chiral active materials

Abstract

Chiral active materials are abundant in nature, including the cytoskeleton with attached motor proteins, rotary clusters of bacteria flagella, and self-spinning starfish embryos. These materials break both time reversal and mirror-image (parity) symmetries due to injection of torques at the microscale. Recently, it was found that chiral active materials may show a new type of elastic response termed `odd' elasticity. Currently, odd elasticity is understood microscopically only in ordered structures, e.g., lattice designs of metamaterials. It still remains to explore how odd elasticity can emerge in natural or biological systems, which are usually disordered. To address this, we propose a minimal generic model for disordered `odd solids', using micropolar (Cosserat) elasticity in the presence of local active torques. We find that odd elasticity naturally emerges as a nonlinear effect of internal particle rotations. Exploring the viscoelasticity of this solid, when immersed in active self-spinning solvent (`odd fluid'), we discover both dynamically unstable regions and regions in which bulk waves can propagate even in an overdamped solid.

Figures (6)

• Figure 1: (a) Illustration of an elastic material composed of rigid rod-like particles. Importantly, our model applies for any other complex rigid particles (granules, colloids, fiber composites, etc.). (b) Coarse-graining at position $\bm{r}$ in the undeformed/Lagrangian space. In this work we consider a locally-disordered, isotropic elastic material, in the presence of local active torque $\tau^\alpha$ ($\alpha$ being the particle index). The various fields ${\bm X}({\bm r})$ are the average of the particle's ${\bm X}^\alpha$ within the coarse-graining volume. (c) Particle displacement $\bm{u}^\alpha$ and internal rotation $\theta^\alpha$ away from the rest position and rest orientation (blue line) of the undeformed state. The internal rotation is measured with respect to the individual rest orientation (not a universal direction).
• Figure 2: Regions of dynamic instabilities and propagating displacement waves in an odd viscoleastic solid. (a) Illustration of the odd solid immersed in an odd solvent. (b) The underdamped case. Arrows indicate increasing $\tilde{\eta}$ approaching the overdamped limit, where only $\tilde{\eta}^o$- and $\tilde{K}^o$-induced instabilities remain. (c) The overdamped case with focus on the nature of mechanical waves. Inset: Absolute value of the relative phase $|\Phi|$ between longitudinal and transverse modes as function of the coupling $\tilde{\eta}^o\tilde{K}^o$. Color coding matches the arrows in the main figure, which indicate increasing $\tilde{\eta}^o$ for various $\tilde{K}^o$. $|\Phi|$ shows non-monotonous behavior and vanishes at the tripoint (purple circle).
• Figure 3: Illustration of the micropolar elasticity model eringen1999. (a) A single particle before deformation is described by the CM position $\bm{r}$ (gray) plus a rigid director $\bm\nu$ (light blue). Upon deformation, the endpoint position of the particle changes from $\bm{r}+\bm\nu$ (light red) to $\hat{\bm{\mathcal{R}}}(\bm{r},\bm\nu)$ (a function of $\bm{r}$ and $\bm\nu$, red). $\bm{R}(\bm{r})$ is the new CM position and $\bm{O}(\bm{r})$ is a rigid rotation operation on $\bm\nu$, both of which depend on $\bm{r}$. (b) Strains are accordingly defined via the distance change between the endpoints of two neighboring particles from $d(\bm{r}+\bm\nu)$ to $d\hat{\bm{\mathcal{R}}}(\bm{r},\bm\nu)$ (red dashed) upon deformation. Here $d$ denotes the small differential change.
• Figure 4: Odd solid under uniaxial stress $\sigma_{yy}$ from the top. The sides are stress-free $\sigma_{xx}=\sigma_{yx}=0$. Due to the presence of the odd elastic modulus $K^o$, the solid is tilted (blue shaded). This tilt is captured by the odd ratio $\nu^o$. This tilt is in addition to the extension measured by the Poisson ratio $\nu$ (red dashed). The undeformed shape is shown in gray line.
• Figure 5: Numerical confirmation of the instability boundaries in the $\tilde{K}^o$ -- $(\tilde{\eta}^B/\tilde{\eta})$ space with increasing viscosity $\tilde{\eta}$ in each row for various combination of $\tilde{\eta}^B$ and $\tilde{\eta}^o$, (a) to (c), using the predictions: $1+\tilde{B}-\tilde{K}^{o^2}=0$ (blue), Eq. \ref{['appeq:BD line express']} (purple and red) and their asymptotic expressions (black dashed, Eq. \ref{['appeq:ko express at small viscosity']} for the small-$\tilde{\eta}$ limit and Eqs. \ref{['appeq:overdamped bd']} and \ref{['appeq:overdamped bd disappear']} for the large-$\tilde{\eta}$ (approximately overdamped) limit. Row (a) is the same case as Fig. \ref{['fig:instability']}(b) in the main text. Gray dots indicate unstable regions, deduced from numerical solution of Eq. \ref{['appeq:secular uniform torque dimensionless']}. All the plots use the same range in axes.