Topological defects lead to energy transfer in active nematics

Abstract

Active nematics are fluids in which the components have nematic symmetry and are driven out of equilibrium due to the microscopic generation of an active stress. When the active stress is high, it drives flows in the nematic and can lead to the proliferation of topological defects, a state we refer to as defect chaos. Using numerical simulations of active nematics, we observe energy transfer between length scales during defect chaos. We demonstrate that this energy transfer is driven by the exchange between variations in the orientation and degree of order in the nematic that predominantly occur during defect creation and annihilation. We then show that the primary features of energy transfer during defect chaos scale with the active length scale. Finally, we identify a second regime that features few defects, but rather bend walls. Similar to topological defects, these bend walls co-localize variations in the orientation with variations in the scalar order parameter leading to energy transfer.

Figures (4)

• Figure 1: Defect chaos. (a&b) Snapshot of a simulated active nematic showing the director field (lines) and scalar order parameter (color) (a) and vorticity (b). (c) Time-averaged spectra for the Landau De-Gennes free energy showing the distortion (blue) and scalar order parameter (red) contributions separately. The dashed line shows an exponential distribution with $k_0 \sim 17.5$; this scale is predominantly governed by the defect core radius, $\epsilon$, rather than the activity, see section III ofSI. (d) Time-averaged dissipation spectra for a simulated active nematic with total energy transfer given by the black dashed line. The wave number associated with the peak energy injection ($i$) is identified as $k^{*}\approx 7.5$, which also corresponds to the peak in stored elastic energy; see Fig. S2 in SI. The length scale associated with this wave number is indicated by the black circle in (a)&(b). (d inset) Magnified image of (d) showing better the features of $d^Q$ and $t$.
• Figure 2: (a) Time-averaged spectra of the components of the dissipation of elastic energy. The vertical line indicates the previously identified peak in energy injection, $k^*$. Flow alignment, $e^f(k)$, is not plotted here as it is zero. (b) Total Landau De-Gennes free energy and its components during defect nucleation via bend instability. The difference between the two lines shows the energy stored as deviations from $S=1$. (c&d) Director field (lines) and scalar order parameter (color) in the time following defect creation via bend instability.
• Figure 3: (a) Peak wavenumber, $k^*$, of the energy transfer spectra as a function of activity, $\alpha$, and defect core radius, $\epsilon$. (b) Same as (a) with $k^*$ scaled by the active length scale. (c) Fraction of energy dissipated through viscosity as a function $\alpha$ and $\epsilon$. The yellow diamonds indicate the location of simulations shown in Figs. \ref{['fig:f1']},\ref{['fig:f2']}. The green stars indicate the location of simulations shown in Fig. \ref{['fig:f4']}. The pink lines indicate the ratio between defect density and defect size observed in microtubule based experiments, see section VI ofSI for details.
• Figure 4: Bend wall chaos. (a) Director field and scalar order parameter and (b) vorticity. The black circle indicates the length scale of energy injection. (c) Time averaged total energy dissipation spectra showing energy transfer across scales. (d) Time averaged spectra for the Landau De-Gennes free energy showing the distortion (blue) and order (red) separately. The black dashed lines show the emergence of two exponential distributions. (e) Time averaged elastic energy dissipation spectra. (f) Energy of the different components of the Landau De-Gennes free energy during bend instability. (g) Bend instability leads to the generation of bend walls which extend over large scales. All results in this figure are simulated with $\alpha = -0.325\times512^2$, $\epsilon = 0.35/512$ indicated by the green star in Fig. \ref{['fig:f3']}b.