Self-organized flows break morphological symmetry in active/passive systems

TL;DR

The paper addresses symmetry breaking in active/passive phase-separating fluids by coupling a generalized Navier–Stokes description for the active region with a Cahn–Hilliard framework for phase separation in a two-dimensional, symmetric setup. The authors demonstrate that self-organized flows in the active region arrest coarsening to form dynamic bicontinuous emulsions, with energy injected at intermediate scales near $K=(k_0+k_1)/2$ and exhibiting a partial inverse cascade toward larger structures. A key finding is that larger space in the active region yields stronger vortices and interface fluctuations, increasing pinch-off events for passive protrusions and producing more passive droplets while preserving active-region connectivity. They introduce a geometric measure, the maximum inscribed circle radius $r_{ m MIC}$, to link local geometry to flow scales, showing the dominant flow structure aligns with available space and thereby tightly couples morphology and flow. Importantly, the mechanism does not require nematic order and may extend to heterogeneous active systems, offering a route to tunable soft-material microstructures through controlled activity and geometry.

Abstract

We consider a phase-separating mixture of active and passive fluids and explore morphological asymmetries of the emerging dominantly bicontinous dynamic emulsion. Two-dimensional numerical simulations reveal that the geometric and topological asymmetries can solely be explained by self-organized flows in the active region. As in inertial turbulence an inverse energy cascade in the active region leads to the formation of condensates. The size of these mesocales vortices is determined by the locally available space in the emulsion. As these condensates accumulate energy they impact the fluctuation of the surrounding interface and thus form a tight coupling between the flow field and the dynamic morphology. While explored for active/passive systems the symmetry-breaking mechanism can be generalized to heterogeneous active systems and proposes a way to control the morphology of various functional soft materials.

Figures (6)

• Figure 1: A.0-A.3: LIC (line intensity convolution) visualization of a time instance of a two-phase system in dynamic equilibrium state at activity $\alpha_{\rm act}\xspace = 2.5, 3, 3.5,$ and $4$, respectively. The interface is shown in orange. The active region appears in light gray and can also be identified by the micro-vortices. A Movie is available in the Supplementary Information for $\alpha_{\rm act}\xspace = 3.5$, corresponding to (A.2). B: Starting from initial random noise, after spinodal decomposition the interface length decreases over time, indicating coarsening of the structure. Depending on activity $\alpha_{\rm act}$, this coarsening is arrested, with the gray boxes marking the corresponding dynamic equilibrium states. The averaged mean interface length in the arrested state increases approximately linearly with $\alpha_{\rm act}$ (inlet). C: Structure factor of the phase field $c$, normalized by the activity scale $K$ and its maximum value at that scale. At small $k$, the distribution flattens with increasing $\alpha_{\rm act}$. D: Velocity increment distributions in the active region show only a weak dependence on $\alpha_{\rm act}$. The deviation from Gaussian statistics (gray dashed lines) is quantified by the kurtosis ${\cal K}$ (inlet). E: Mean kinetic energy density in the active phase during coarsening. After the initial build-up of the flow ($t<10$), the kinetic energy density increases until the system reaches a dynamic equilibrium state, indicated by gray boxes. The averaged mean energy density in the dynamic equilibrium state increases approximately linearly with $\alpha_{\rm act}$ (inlet). F: The energy spectrum peaks at values of $k$ smaller than the forcing scale $K$, indicating the presence of flow structures at larger scales. At small $k$, the energy spectrum decreases linearly. G: Probability distribution (pdf) for number of active (solid lines) and passive (dashed lines) domains in the dynamic equilibrium state. At high activity $\alpha_{\rm act}$, the bicontinuous emulsion breaks up, resulting in the formation of active and passive droplets, thus increasing the number of domains. The number of passive domains is significantly higher than that of active domains (inlet).
• Figure 2: A: Local averaged mean kinetic energy $\frac{1}{2}\langle \mathbf{u}^2 \rangle_{\rm C}$ in the active region. The $\mathbf{u}^2$ field is averaged inside the active region within a circle of radius $r = 1.23$ (yellow circle). Highest energies are mostly found deep inside the active region. Narrow protrusions and bridges tend to exhibit lower kinetic energy. In general, $\frac{1}{2}\langle \mathbf{u}^2 \rangle_{\rm C}$ decreases toward the interface. The passive region is shown in black and the interface in orange. B: Isolines of the distance to the interface $d$. The passive region is shown in black and the interface in orange. C: Mean kinetic energy density $\frac{1}{2}\langle \mathbf{u}^2 \rangle$ (orange) and phase field $c$ (blue) as functions of the distance to the interface $d$. In the active region, $\frac{1}{2}\langle \mathbf{u}^2 \rangle$ increases sharply for $d < 1$, forming a boundary layer. In the bulk region ($d > 1$), $\frac{1}{2}\langle \mathbf{u}^2 \rangle$ further increases with increasing $d$, but much more moderately. In the passive region, $\frac{1}{2}\langle \mathbf{u}^2 \rangle$ is significantly smaller than in the active region and decreases with increasing $d$. The interface width of the phase field is much smaller than the thickness of the fluid boundary layer. The fluctuations and increasing error bars for large $d$ for the active region result from limited available data.
• Figure 3: Interface height fluctuation of a phase separated situation in a thin rectangular domain within dynamic equilibrium states. A: Geometric setting. B: Interface height fluctuation $\langle [ h - \langle h \rangle ]^2 \rangle$ for two different settings with varying height $w_{\rm act}$ and $w_{\rm pas}$, respectively. Closed symbols: $1 \leq w_{\rm act} \leq 3.5$ and $w_{\rm pas} \gg 3.5$. Open symbols: $w_{\rm act} = 2$ and $1.5 < w_{\rm pas}< 3.5$). No fluctuation is measured for $w_{\rm pas} = 1$, as in this case the interface becomes unstable and a passive droplet forms. For both cases $l_\Omega=\pi$. C: LIC visualization of the flow field and the interfaces (orange lines) at a time instance within the dynamic equilibrium states for various $w_{\rm act}$ (corresponding to closed symbols in B, demonstrating the fluctuation increase with increasing $w_{\rm act}$. The corresponding flow fields to the open symbols in B are not shown as there is no affect on the interface fluctuation for increasing $w_{\rm pas}$.
• Figure 4: A: Definition of the available space for fluid organization using maximum inscribed circles (MIC). Three inscribed circles are shown in the closeup. The available space at a point is measured by the radius of the MIC, $r_{\rm MIC}$, which is color coded. At $\mathbf{x}$, the available space is defined by the largest circle, while at $\mathbf{y}$, it is defined by the smallest circle shown. B: Mean kinetic energy as a function of $r_{\rm MIC}$. The kinetic energy is averaged over the entire region ($d > 0$, orange), as well as excluding the boundary layer ($d > 1$, green). Crosses show mean kinetic energies for the simplified setting from Figure \ref{['fig:slabfluc']} (closed symbols), with $r_{\rm MIC}\xspace \approx 0.5 w_{\rm act}$. The inlet considers a different scaling to highlight the increase of the mean kinetic energy for larger $r_{\rm MIC}$, independent of the boundary layer.
• Figure 5: Comparison of fluid structure, $\langle E(k) \rangle_T$$_{\rm max}$, with the available space $\langle r_{\rm MIC}\xspace \rangle$. The mean energy spectrum is shown for the emulsion (thick line) and for single slabs with width $w$ (thin lines, see Fig. \ref{['fig:slabfluc']}). Horizontal lines indicate the length scale defined by the available space, $k = 2 \pi / r_{\rm MIC}\xspace$, for each setup. As slab width $r_{\rm MIC}\xspace$ increases, the maximum of $E(k)$ shifts to larger length scales and grows in magnitude, closely following the available space. For the emulsion, the mean available space $\langle r_{\rm MIC}\xspace \rangle$ is shown as a thick line, with light orange shading representing the standard deviation.