The Role of Elastic Anisotropy in Active Nematics

TL;DR

This work shows that elastic anisotropy in active nematics, captured by two dimensionless parameters, can generate a net active torque on monopole distortions and markedly alter dipole propulsion in three and two dimensions. By extending the active nematic multipole framework to unequal elastic constants $K_1$, $K_2$, $K_3$, the authors derive explicit corrections to far-field distortions, flows, and forces, including a torque on monopoles and potential propulsion reversal for certain dipoles. In two dimensions, dipole propulsion speeds can change by up to 50% while defect speeds remain robust (changes <5%), but anisotropy introduces a separation-independent torque on defect pairs, biasing their orientation. Collectively, these results quantify how elastic geometry shapes active flows and defect interactions, with implications for controlling ordering and transport in synthetic and biological active materials.

Abstract

We analyse the effect of anisotropy in elastic constants on the hydrodynamics of active nematics. Building on the multipole framework for a single elastic constant, we determine the leading effect of elastic anisotropy on the active response of generic distortions. The key findings are a new active torque, proportional to the anisotropy, in response to monopole distortions, and modifications to the propulsion of dipoles in both the direction of motion and changes in speed of up to 50\%. For point defects in two dimensions we find that, despite the large morphological changes in the director field, elastic anisotropy has only a minor impact on their hydrodynamics, with the self-propulsion speed of $+1/2$ defects lowered by less than 5\%. Finally, we determine the elastic torques exerted on defect pairs due to elastic anisotropy.

Figures (10)

• Figure 1: The anisotropic distribution of elastic energy around a monopole. For a monopole distortion in the $x$-direction, shown top left, the distributions of splay, twist and bend is given by dipoles along $\mathbf{e}_x$, $-\mathbf{e}_y$ and $\mathbf{e}_z$ respectively. Unequal elastic constants therefore result in a quadrupole-like contribution to the elastic energy distribution, with the plane of the quadrupole depending on the type of anisotropy. This quadrupolar symmetry is inherited by the leading director perturbation, and hence is the root of many of the effects of elastic anisotropy in active nematics.
• Figure 2: The fundamental distortions and their associated active responses. The director field (above) and corresponding active flow (below) for, from left to right, the harmonic monopole, the perturbation due to parallel, or bend, anisotropy and the perturbation due to perpendicular, or splay-twist, anisotropy. The director field is shown as blue rods in a planar cross-section, with black arrows indicating its behaviour out of this plane where appropriate. The red arrows indicate the net active torque.
• Figure 3: The active flow response to the UPenn dipole. From left to right: the contribution due to the harmonic distortion, the perturbation due to parallel (bend) anisotropy and that due to perpendicular (splay-twist) anisotropy. In each case the streamlines of the director are shown in orange, the induced active flow is shown with blue arrows and the green arrow indicates the net active force.
• Figure 4: Leading corrections due to elastic anisotropy for distortions up to quadrupole order in two-dimensional active nematics. The integral curves of the director (black lines) and the active flow (white arrows) are superposed on the pressure field. The panels are labelled according to the appropriate derivative of the fundamental distortion. The red and green arrows indicate the self-rotation or -propulsion that would result in an extensile system. For both the director and the associated active response we show only the leading contribution due to elastic anisotropy, neglecting the harmonic part. Therefore the director stems from the coefficient of $\kappa$ in \ref{['eq:DisortionLeadingOrder2D']}, while the active response is derived from \ref{['eq:AnisotropicMonopolePressure2D']} and \ref{['eq:AnisotropicMonopoleFlow2D']}.
• Figure 5: Active propulsion velocity as a function of dipole distortion orientation for different values of elastic anisotropy $\kappa$. a) The propulsive speed, $v$, given by the magnitude of $\mathbf{v}$ in \ref{['eq:ActiveVelocityDipole2D']}, normalised by the prefactor $\zeta\alpha c/(6\mu_{\parallel}\log(a/R))$. The orientation of the dipole distortion, $\theta_d$, is measured anti-clockwise from the far-field direction. b) The misalignment, in radians, between the dipole orientation and that of the active velocity, $\theta_v$.