Spatiotemporal Chaos and Defect Proliferation in Polar-Apolar Active Mixture

TL;DR

Problem: understand how interspecies coupling in active nematic mixtures drives complex spatiotemporal states. Approach: a dry, coarse-grained 2D model of a polar-apolar mixture with coupled density and order-parameter fields $(\rho_p,\mathbf{P})$ and $(\rho_n,\mathbf{Q})$, analyzed by large-scale simulations across a phase diagram. Findings: a reentrant inhomogeneous phase with high-density nematic bands, spontaneous $\pm \frac{1}{2}$ defects, and spatiotemporal chaos evidenced by the spectral properties of density fluctuations and positive maximal Lyapunov exponents ($\Lambda > 0$). Significance: extends understanding of non-equilibrium transitions in active matter and suggests experimental tests in bacterial suspensions or synthetic microswimmer assemblies; implications: demonstrates a route to externally tune active states via minority polar components in dry active systems.

Abstract

Chaotic transitions in inertial fluids typically proceed through a direct energy cascade from large to small scales. In contrast, active systems, composed of self propelled units, inject energy at microscopic scales and therefore exhibit an inverse cascade, giving rise to distinctly unconventional flow patterns. Here, we investigate an active mixture consisting of both apolar and polar self driven components, a setting expected to display richer behaviours than those found in living liquid crystal (LLC) systems, where the apolar constituent is passive. Using numerical solutions of the corresponding hydrodynamic equations, we uncover a variety of complex dynamical states. Our results reveal a non-monotonic response of the apolar species to changes in the density and activity of the polar component. In an intermediate regime, reminiscent of LLC-induced disorder, the system develops a dynamically disordered phase characterised by high-density, chaotically evolving band-like structures and by the continual creation and annihilation of half integer topological defects. We show that this regime exhibits spatiotemporal chaos, which we quantify through two complementary measures: the spectral properties of density fluctuations and the maximal Lyapunov exponent. Together, these findings broaden the understanding of complex transitions in active matter and suggest potential experimental realisations in bacterial suspensions or synthetic microswimmer assemblies.

Figures (20)

• Figure 1: Global characteristics of active nematics containing an active polar component. Panel (a) shows the phase diagram of the system based on the steady-state characteristics of the active nematic. In the homogeneous regime, both $\rho_n$ and $\boldsymbol{Q}$ are homogeneous, whereas in the IN-regime $\rho_n$ and $\boldsymbol{Q}$ exhibit significant spatial fluctuations. The insets show the probability density function of $Q = |\boldsymbol{Q}|$, $P(Q)$. In the homogeneous regime, $P(Q)$ shows a single peak, whereas in the IN-regime, $P(Q)$ shows two peaks showing the coexistence of high and low ordered regions. The panel (b) shows the variation of nematic scalar order parameter, $S$, with polar density, $\rho_{p0}$, for different values of $v_p$. The entire range of $\rho_{p0}$ is partitioned into distinct states according to the magnitude of $S$, as illustrated in the inset. The panel (c) showcases the plot of $S_{min}$$vs.$$v_p$ on a log–log scale, with the dashed line representing a power-law fit. The panels (d-f) show the snapshots of the density and orientation field of apolar species in different phases : (d) Phase-I, (e) Phase-II, (f) Phase-III. The heat map represents the density field, while the lines indicate the local nematic director $\hat{n}_n=(\cos(\theta_n),\sin(\theta_n))$, with their length proportional to the local nematic order $\vert \boldsymbol{Q}\vert$. The PDF $P(Q)$ and $P(\delta \rho_n)$ corresponding to the snapshots (d-f) are shown in figure \ref{['fig:histdennop']}. Parameters: System size, $L = 512$.
• Figure 2: Snapshots of the magnitude of $\boldsymbol{Q}$-field (i.e. $Q$) for the apolar species for $\rho_{p0} = 0.09$ in Phase-II for different values of polar activity : (a) $v_p = 0.001$, (b) $v_p = 0.005$, (c) $v_p = 0.05$, (d) $v_p = 0.10$, (e) $v_p = 0.20$, and (f) $v_p = 0.50$. The heatmap depicts the magnitude of the nematic order parameter field of active nematics $Q = |\boldsymbol{Q}|$. Parameters: System size, $L = 512$. The rest of the parameters are the same as in figure \ref{['fig:pd_nop']}.
• Figure 3: Variation of structural properties of the bands with changing control parameters in the IN regime. Panel (a) shows the variation of averaged correlation length, $<l_{\rho_n}>$, with polar density, $\rho_{p0}$, in Phase-II for two different system sizes $L =512$, $\&$$1024$ for $v_p = 0.25$. The inset shows the time series of $l_{\rho_n}(t)$ for $v_p = 0.25$ and $\rho_p = 0.07$; Panel (b) presents $\overline{l}_{\rho_n}$$vs.$$\rho_{p0}$ plot for different values of $v_p$. The inset of subplot (b) depicts the plot of $\langle l_{\rho_n} \rangle_{min}$$vs.$$v_p$. The dashed line represents the power law behaviour; Panel (c) depicts the $\overline{l}_{\rho_n}$$vs.$$v_p$ plot for two different values of $\rho_{p0}$ in Phase-II. Parameters: System size, $L = 512$ for subplots (b) and (c). The rest of the parameters are the same as in figure \ref{['fig:pd_nop']}.
• Figure 4: Plot of the autocorrelation function of the fluctuations in $l_{\rho_n}(t)$, $C_{acf}(t)$,vs.time, $t$, for a different set of control parameters. Panel (a) shows $C_{acf}(t)$vs.$t$ for different values of $\rho_{p0}$ for $v_p = 0.20$. The inset shows the plot of the correlation time, $\tau_c$, vs.$\rho_{p0}$. Panel (b) shows $C_{acf}(t)$vs.$t$ for different values of $v_p$ for $\rho_{p0} = 0.07$. The inset shows the plot of the correlation time, $\tau_c$, vs.$v_p$. The error bars in the inset of (a) and (b) represent the standard deviation of $\tau_c$ calculated over independent realisations. System Size, $L=512$. The rest of the parameters are the same as in figure \ref{['fig:pd_nop']}.
• Figure 5: Mechanism of the formation of chaotic bands following an instantaneous change in the $v_p$. The system is prepared to have an initial condition that corresponds to the steady-state for $v_p=0.001$, as shown in panel (a), which shows the configuration of the nematic order parameter field $\vert \boldsymbol{Q}(\boldsymbol{r}) \vert$. The system is then instantaneously changed to $v_p =0.07$. The subsequent snapshots in the top row (b-e) show the temporal evolution of the magnitude of the nematic order parameter field $\vert Q(\boldsymbol{r}) \vert$, and those in the bottom row (f-j) displays the local stress magnitude corresponding to the snapshot shown in the top row, $\sigma_{loc}(\boldsymbol{r})$. Parameters : $L = 400$. The rest of the parameters are the same as in figure \ref{['fig:pd_nop']}.