Nematic order condensation and topological defects in inertial active nematics

Living materials at different length scales manifest active nematic features such as orientational order, nematic topological defects, and active nematic turbulence. Using numerical simulations we investigate the impact of fluid inertia on the collective pattern formation in active nematics. We show that an incremental increase in inertial effects due to reduced viscosity results in gradual melting of nematic order with an increase in topological defect density before a discontinuous transition to a vortex-condensate state. The emergent vortex-condensate state at low enough viscosities coincides with nematic order condensation within the giant vortices and the drop in the density of topological defects. We further show flow field around topological defects is substantially affected by inertial effects. Moreover, we demonstrate the strong dependence of the kinetic energy spectrum on the inertial effects, recover the Kolmogorov scaling within the vortex-condensate phase, but find no evidence of universal scaling at higher viscosities. The findings reveal complexities in active nematic turbulence and emphasize the important cross-talk between active and inertial effects in setting flow and orientational organization of active particles.

Figure 1

Active turbulence and vortex condensates. Snapshots of the flow vortices for incrementally decreasing viscosities: vorticity contours for (a) $\nu /K\mathrm{\Gamma }=125$ with $\mathrm{Re}\sim 0.03$ (equivalent to ${\text{Re}}_{\text{T}}=0.0002$ using the Taylor microscale and ${\text{Re}}_{\text{I}}=0.005$ using the integral scale to define the Reynolds number), (b) $\nu /K\mathrm{\Gamma }=12.5$ with $\mathrm{Re}\sim 0.7$ (equivalent to ${\text{Re}}_{\text{T}}=0.0023$ using the Taylor microscale and ${\text{Re}}_{\text{I}}=0.03$ using the integral scale), and (c) $\nu /K\mathrm{\Gamma }=1.25$ with $\mathrm{Re}\sim 250$ (equivalent to ${\text{Re}}_{\text{T}}=8.2227$ using the Taylor microscale and ${\text{Re}}_{\text{I}}=191$ using the integral scale). (d) Effect of viscosity on vorticity-vorticity correlations $C_{\omega -\omega }\left(r\right)$. The distance $r$ is normalized by the active length scale $l_a=\sqrt{K/\zeta }$. (e) Characteristic vorticity length scale as a function of viscosity. Upon decreasing viscosity, after an initial decrease in the size of vortices, condensates spanning the entire system are formed. The length scale ($\ell$) equals to the length $r$ at which $C_{\omega -\omega }\left(r\right)=0$. Green data points represent an incremental increase in viscosity and show the presence of a hysteresis loop, indicating a discontinuous transition to the vortex-condensate state.

Figure 2

Quantification of viscosity impact on flow and orientation properties of active nematics. Effect of viscosity on (a) rms velocity $V_{\text{rms}}$, (b) magnitude of the nematic order, and (c) defect number, before and after the transition to the vortex-condensate state. Inset in (a) shows the semilog plots of the rms velocity as a function of the logarithm of viscosity, highlighting its logarithmic decay with viscosity.

Figure 3

Nematic order condensation. (a) Snapshots of director field and topological defects for vortex condensate case. Color map indicates the magnitude of the nematic order $q$, and $+1/2$ and $-1/2$ topological defects are marked by yellow comets and green triangles, respectively. (b) Averaged values of the magnitude of order ${\langle q\rangle }_{\mathbf{x},t}$ and defect density calculated separately inside giant vortices and the bulk of the system excluding the giant vortices.

Figure 4

Viscosity impact on the flow field of topological defects. Velocity field and contours of velocity magnitude for the average defect flow at (a) $\nu /K\mathrm{\Gamma }=125$ and (b) $\nu /K\mathrm{\Gamma }=12.5$. Panels (c) and (d) show the average defect flow for the vortex-condensate state at $\nu /K\mathrm{\Gamma }=1.25$, calculated separately for (c) defects inside giant vortices and (d) the bulk of the system excluding the giant vortices. Blue dashed line schematically shows the alignment of the $+1/2$ defect with respect to the averaged flows. (e) Velocity profile around defects showing the magnitude of velocity along a vertical axis passing through the center of defects in (a)–(d).

Figure 5

Stability diagram for the vortex-condensate formation in active nematics. Effect of viscosity and activity on the vortex condensation formation are shown. Filled points represent the vortex-condensate state.

Figure 6

Active turbulence and vortex condensates for contractile activity ($\zeta /A=-0.03$) at $\nu /K\mathrm{\Gamma }=1.25$. Snapshots of the flow vortices for positive, zero, and negative tumbling parameter: vorticity contours for (a) $\lambda =+0.7$, (b) $\lambda =0$, and (c) $\lambda =-0.7$. (d) Effect of tumbling parameter on vorticity-vorticity correlations.

Figure 7

Kinetic-energy spectra. The wave number is nondimensionalized by the active length scale $l_a=\sqrt{K/\zeta }$. Solid and dashed lines, respectively, represent $1024\times 1024$ and $2048\times 2048$ grid resolutions. While within the vortex-condensate state a power-law decay is observed, lower viscosities manifest exponential decay with the wave number (semilog plots in the inset).