Activity drives self-assembly of passive soft inclusions in active nematics

Abstract

Active nematics are out-of-equilibrium systems in which energy injection at the microscale drives emergent collective behaviors, from spontaneous flows to active turbulence. While the dynamics of these systems have been extensively studied, their potential for controlling the organization of embedded soft particles remains largely unexplored. Here, we investigate how passive droplets suspended in an active nematic fluid self-organize under varying activity levels and packing fractions. Through numerical simulations, we uncover a rich phase diagram featuring dynamic clustering, activity-induced gelation, and a novel activity-driven deformability-induced phase separation regime where activity stabilizes dense droplet assemblies. We find that droplet deformability plays a key role in enabling this regime, as it allows droplets to absorb the stress exerted by the surrounding active fluid. Crucially, we demonstrate that temporal modulation of activity enables precise control over structural morphological transitions. Our results suggest new routes to design adaptive smart materials with tunable microstructure and dynamics, bridging active nematics with applications in programmable colloidal assembly and bio-inspired material design.

Introduction

Active nematics are a class of non-equilibrium systems that consume energy at the microscopic scale, driving them far from equilibrium and enabling the emergence of spontaneous flows and complex macroscopic behaviors^1,2,3,4. At low activity levels, energy injection leads to local deformations of the active nematics, resulting in ordered flow patterns^4,5,6. However, as the rate of energy injection increases, deformations strengthen, accompanied by a corresponding rise in the magnitude of active flows. This drives the system into a chaotic regime known as active turbulence, characterized by the continuous creation and annihilation of topological defects^{2,7,8,9,10,11,12,13,14}.

The ability to control the chaotic active nematics dynamics is central to exploiting their potential for applications, as for instance the self-organization, assembly, and transport of suspended passive particles^{15,16,17,18,19,20,21,22,23,24,25,26,27}. In the turbulent regime, active nematics create a unique environment where fluid dynamics and defect-mediated transport can be exploited to orchestrate the motion and organization of passive inclusions^19,28,29,30. Many-body active interactions provide an effective avenue for creating new soft composite materials, such as colloidal crystals and self-quenched glasses. Similarly, the behavior of passive soft inclusions, such as lipid vesicles or polymeric droplets, in active nematics could enable novel applications in drug delivery, microfluidics, and programmable self-assembly. Unlocking this potential requires understanding the interplay between active flows, topology, and passive inclusions. In particular, it is essential to consider how spatial confinement and topological constraints introduced by suspended particles affect emergent behaviors at different activity levels and packing fractions.

The collective organization of two-dimensional colloidal inclusions in passive liquid crystals is strongly influenced by the liquid crystal anchoring^{31,32,33,34,35,36}. In the presence of anchoring (either tangential or homeotropic), each suspended particle introduces a local topological charge of +1, which the surrounding nematic absorbs by generating a total compensating charge of -1. This is typically achieved through the formation of two −1/2 topological defects, so that inclusions mutually interact through weak quadrupolar interactions^31,37. These interactions drive various self-assembled configurations, including linear chains of colloids^38,39, particle aggregates^40,41, and defect-stabilized colloidal gels⁴². Recent studies suggest that activity can overcome energy barriers and render certain excitations effectively gapless⁴³. This opens intriguing possibilities for designing new classes of topological soft materials, where active forces could be harnessed to reshape defect landscapes and guide structure formation beyond equilibrium constraints.

While active droplets suspended in passive fluids^{44,45,46,47,48,49,50,51,52,53,54} and colloidal suspensions in passive liquid crystals^{55,56,57,58,59,60,61,62,63} have been extensively studied, the case of passive inclusions suspended in active nematics has received far less attention. Recent research demonstrated that a single passive inclusion suspended in an active nematic exhibits anomalous diffusivity, deviating significantly from Brownian motion due to the interaction between bulk defects and the inclusion-induced nematic deformations^26,64. This behavior is strongly dependent on the ratio of the droplet radius to the active length scale, with smaller droplets experiencing significant drifts driven by active flows.

However, whether this behavior can be harnessed for material self-assembly remains an open question, requiring systematic investigation of the dynamical and morphological properties in dense droplet suspensions. In this study, we explore the full phase diagram of a system of multiple passive soft droplets, modeled as multiphase fields, suspended in an active nematic fluid. We systematically vary both the activity level and the droplet packing fraction, revealing a rich diversity of dynamical and morphological regimes.

At intermediate activity levels, we find a dynamic clustering behavior also observed in hard colloidal suspensions⁶⁵. At higher packing fractions, the clustered phase gives way to a gel-like phase where droplets organize into a percolating network spanning the entire system. Crucially, when activity exceeds the threshold for active turbulence, the system undergoes a striking transition: the suspended droplets completely cluster, exhibiting tunable diffusive and morphological properties. This regime resembles motility-induced phase separation (MIPS) observed in systems of self-propelled particles^{1,66,67,68,69,70,71,72,73}, but with a fundamental distinction. In classical MIPS, phase separation arises from the particles’ own activity, whereas in our system, it is driven by the motility of the active solvent and by the deformability of the passive soft inclusions. Accordingly, we name this regime Activity-Driven Deformability-Induced Phase Separation, or shortly active-DIPS.

Results

Morphological characterization

To investigate the collective behavior of passive soft droplets suspended in an active nematic fluid, we employ a multiphase field approach. Each droplet is represented by a distinct phase-separating field ϕ_i, where i uniquely labels each droplet. The orientational state of the active nematic fluid is described by the Q-tensor, defined as Q = S(nn − I/3). Here, S represents the nematic order parameter, quantifying local nematic order, and n is the head–tail symmetric unit vector indicating the local alignment direction of nematic molecules.

We numerically solve the continuum active nemato–hydrodynamic equations coupled to the conserved dynamics of the suspended passive droplets. This model has previously been validated for reproducing the dynamics of active suspensions confined on rigid and deformable shells, active liquid interfaces, and predicting patterns in three-dimensional active suspensions (see the “Methods” section for a detailed description of the numerical model).

Non-equilibrium energy injection is introduced into the Navier-Stokes equation via the active stress tensor σ^act = −ζQ, where ζ is the activity parameter. In the following, we refer to the dimensionless activity number defined as $A={R}_{{\rm{d}}}^{2}\zeta /{\kappa }_{{\rm{F}}}$, which quantifies the relative strength of active stress compared to the elastic stress of the liquid crystal introduced by the inclusions. Here, R_d is the droplet radius, and κ_F is the elastic constant of the active liquid crystal. [Refer to the “Methods” section for further details on the numerical model and the definition of the reference time-scale τ and energy-scale ϵ_def.]

At low activity levels (A ≲ 3), the system remains in a quiescent state dominated by passive effects. When the liquid crystal exhibits either homeotropic or tangential anchoring at the droplet interface, the director field wraps around each droplet in a way that imposes a topological charge of +1. To satisfy global topological constraints, the liquid crystal compensates by generating two −1/2 defects in the surrounding medium. These defects are not rigidly attached to the droplet interface but are energetically stabilized at a distance ${\ell }_{{\rm{p}}}=\sqrt{{\kappa }_{{\rm{F}}}/W}$—the so-called penetration length, where W denotes the anchoring strength at the droplet surface—and, in isolated droplets, they are typically located at antipodal positions (Fig. 1a(i)).

Fig. 1: Morphology and phase behaviors of passive emulsions in active nematics.

a Representative configuration of the system at different values of activity number and packing fraction: (i) quiescent state (the inset in the panel, where topological defects of charge −1/2 are highlighted in red, depicts the director field in proximity of a droplet); (ii) gelation state; (iii) dynamic clustering state; (iv) active-DIPS state. The color map corresponds to the scalar concentration field ϕ = ∑_iϕ_i, with the interior of the passive droplets shown in yellow and the surrounding active fluid shown in purple. Topological defects are marked in white. b Phase diagram showing the system’s morphological behavior as a function of packing fraction Φ and activity number A. The background color represents the time-averaged kinetic energy of the fluid E_kin normalized by the defect energy ϵ_def, as indicated by the color bar.

This configuration induces a quadrupolar short-range interaction between droplets, too weak to drive droplet attraction. As a result, the emergent structure depends sensitively on the droplet packing fraction Φ. Specifically, at low packing fraction and activity number, droplets remain near their initial positions in a quiescent stationary state (Supplementary Movie 1).

The elastic interactions mediated by the defects induced around the suspended droplets become significant when the system is sufficiently active to sustain spontaneous flows, or when the packing fraction is high enough to reduce the average inter-droplet distance to the penetration length of the liquid crystal. In these regimes, defect-mediated forces can influence droplet organization and dynamics, contributing to the emergence of collective behavior. Specifically, when two droplets approach each other, anchoring-induced deformations in the surrounding liquid crystal stabilize the formation of defect bonds. These bonds arise when droplets share two (or more) of their surrounding topological defects, which causes them to adhere (Fig. 2a, b). Such droplet dimers act as aggregation centers, facilitating the growth of larger clusters through the absorption of isolated droplets or the merging with other clusters. In particular, when the packing fraction becomes sufficiently high (Φ ≳ 0.21), the typical distance between droplets becomes small enough for elastic interactions to dominate. These short-range attractive forces are strong enough to drive a global rearrangement of the droplets, leading the system to evolve into a stationary percolating network (Fig. 1a(ii) and Supplementary Movie 2), even in the absence of activity. This behavior is indeed consistent with previous observations in colloids or droplets dispersed in passive nematics, which showed chain-like assemblies^38,39, particle aggregates^40,41 and defect-stabilized colloidal gels⁴².

Fig. 2: Morphological characterization of the passive emulsion.

Panels a, b show representative configurations of the director field in the clustered and gelation phases, respectively. Topological defects with charge −1/2 are highlighted in orange, displaying the characteristic three-fold pattern. Panels c–f present the corresponding radial distribution functions g(r) for the c gelation, d clustering, e active turbulence, and f active-DIPS phases. Insets display the associated two-dimensional structure factor ${{\mathcal{S}}}_{\phi }$. The packing fraction Φ and activity number A for each phase shown here are the same as those reported in Fig. 1a, with the addition of the active turbulent regime at Φ = 0.09 and A = 576.

Introducing activity into the system profoundly alters its morphology. A sharp transition occurs when the activity level surpasses the threshold for spontaneous flow (A ≳ 3). In this regime, droplets are advected by active jets and exhibit a characteristic stop-and-go motion⁶⁴. This persistent movement causes droplets to interact with nearby droplets, forming defect-mediated bonds that lead to the emergence of small droplet clusters. Interestingly enough, at intermediate activity levels (A ≲ 200), cluster growth is arrested due to the disruptive effects of active jets, which undermine the structural stability of the clusters. This behavior is prevalently observed at low packing fractions and intermediate values of activity (Fig. 1a(iii), b). Consequently, the system exhibits highly dynamic behavior, with clusters continuously assembling and disassembling over time (Supplementary Movie 3).

At higher packing fractions, activity accelerates the formation of a percolating network even at intermediate activity levels. As a result, the system undergoes a transition from dynamic clustering to a gelated state as the packing fraction increases (Fig. 1b). Specifically, when the packing fraction is sufficiently large, active jets mobilize droplets into chain-like structures that eventually intersect, forming a passive network of droplets that encapsulates regions of active nematic fluid. Notably, the droplet network confines the active liquid crystal into domains smaller than the overall system size, effectively suppressing unstable infrared active modes, leading the system to settle into a stationary state over long timescales (Supplementary Movie 2).

To characterize the system morphologically, we show in Fig. 2c–f the droplet radial distribution function g(r) (see the “Methods” section) for different morphological regimes. In the gelation phase, Fig. 2c displays a sharp, well-defined first peak corresponding to the nearest-neighbor droplet distance, followed by a series of secondary peaks, reflecting short-range positional correlations typical of an extended, percolated droplet network lacking long-range order. By comparison, in the clustering phase, Fig. 2d reveals a dominant first peak corresponding to the first-neighbor contribution, along with a smaller secondary peak, consistent with the finite size of the cluster. The two-dimensional structure factor ${{\mathcal{S}}}_{\phi }=\langle {\widehat{\phi }}_{{\boldsymbol{k}}}^{*}\,{\widehat{\phi }}_{{\boldsymbol{k}}}\rangle$ (with ϕ = ∑_iϕ_i the global concentration field) further exhibits broken rotational symmetry, reflecting the local arrangement of droplets within the clusters.

When the activity is increased beyond A ≳ 200 at low packing fractions (Φ ≲ 0.15), the active flow becomes large enough to prevent clustering, leading the system to transition into a new regime characterized by chaotic droplet motion (Supplementary Movie 4) and the absence of structures (isotropic concentration spectrum and flat g(r) in Fig. 2e). We identify this state as active turbulence, where the continuous active injection destabilizes the liquid crystal, resulting in deformed patterns with a characteristic length scale of ${\ell }_{{\rm{act}}}=\sqrt{{\kappa }_{{\rm{F}}}/| \zeta | }$. In this regime, the liquid crystal relaxes the elastic stress caused by activity-induced deformations by nucleating pairs of topological defects. This process leads to the proliferation of topological defects throughout the system, further contributing to the active turbulence dynamics.

Remarkably, at very high activity levels, we observe droplets reorganizing into clusters once again, despite the presence of strong active jets (Supplementary Movie 5). While defect bonds are primarily responsible for cluster stability at lower activity levels, at high activity, the pressure exerted by the active liquid crystal becomes sufficiently large to deform the suspended droplets and stabilize the clusters (Fig. 1a(iv), b). Accordingly, we refer to this regime as Activity-Driven Deformability-Induced Phase Separation, or shortly active-DIPS. As a result, these clusters remain fully separated, grow over time, and retain their constituent droplets without loss. This occurs because the steric constraints imposed by the surrounding active nematic matrix effectively trap the droplets in place, deforming and arranging them in a hexatic fashion as shown by the g(r) and 2D structure factor in Fig. 2f.

The existence of this fully clustered state at high activity is one of the most significant findings reported in this article. Notably, this behavior is largely independent of the packing fraction of the system, highlighting the robustness of steric trapping at large activity. Finally, as illustrated in the phase diagram in Fig. 1b, a transition from a percolating network to a fully separated phase also occurs at large packing fractions.

Kinetic characterization

Figure 3a presents the mean squared displacement (MSD) 〈Δr²(t)〉 of droplets across different phases. In the quiescent and gelation phases at low activity, droplets reach a stationary configuration at long times (Fig. 3a(i, ii)), while in the three higher-activity regimes, they exhibit long-time diffusive behavior (Fig. 3a(iii–v)). Notably, this diffusive dynamics emerges independently of packing fraction, as demonstrated by the behavior of the long-time MSD exponents versus activity (Fig. 3e), obtained by performing a statistical average over independent simulations for different packing fractions (see the “Methods” section).

Fig. 3: Kinetic characterization of the passive emulsion.

a Mean squared displacement (MSD) averaged over all droplets in the system for various phases. Dashed lines indicate best-fit slopes for short- and long-term dynamics. Times are normalized with respect to the timescale of elastic distortions of the liquid crystal τ. b–d Representative droplet trajectories in b the clustering phase, c the active-DIPS phase, and d the active turbulence regime. The packing fraction Φ and activity number A for each phase shown in panels (a–d) are identical to those reported in Figs. 1a, 2c–f. e Long-time MSD exponent as a function of activity number for three packing fractions, obtained by averaging over n = 5 independent simulations. f Standard error of the mean (SEM) for the long-time MSD exponent at varying activity numbers, for the same packing fraction as panel (e). g Average fluid kinetic energy versus activity number, for different packing fractions. h Average droplet kinetic energy versus activity number, for the same packing fractions as panel g. In panels g, h, error bars are contained within the marker size.

In general, the system undergoes a transition from subdiffusive behavior at low activity to diffusive behavior at higher activity, with the packing fraction influencing the specific activity required for this transition; higher packing fractions reduce the available space for the active nematic instability, necessitating greater activity to generate self-sustained flows and sustain droplet diffusive motion. Notably, near the transition between gelated and clustered states, we observe significant fluctuations in the MSD exponents (Fig. 3f), evidenced by peaks in the standard error of the mean. These fluctuations suggest phase coexistence during the transition to the fully clustered phase at high activity, whereas the transition to the turbulent state at lower packing fractions appears continuous.

Interestingly, the kinetic energy of the droplets, ${E}_{{\rm{d}}}=\langle {m}_{{\rm{d}}} \dot{\boldsymbol{r}}^{2} / {2} \rangle$, and of the fluid, E_kin = 〈ρv²/2〉, exhibit distinct behaviors as the activity level and packing fraction are varied. For a fixed packing fraction, the fluid’s kinetic energy increases monotonically with activity (Fig. 3g), consistent with the progressive amplification of active stresses. However, the dependence on packing fraction is more nuanced. At low packing fractions (e.g., Φ = 0.09) and moderate activity (A = 432), the system is in the active turbulent regime, with droplets dispersed throughout the domain. In this configuration, the active liquid crystal can develop instabilities over a limited length scale, roughly set by the inter-droplet distance, which restricts the formation of large-scale vortical structures.

By comparison, at the same activity but higher packing fraction (e.g., Φ = 0.35), the system enters the active-DIPS phase, where droplets cluster and leave extended regions of active fluid. This phase-separated configuration allows the LC to exploit the full system size to develop active instabilities, resulting in stronger and more coherent flows. Consequently, the fluid kinetic energy can be higher at larger packing fractions, contrary to the intuition that crowding suppresses flow (see also the “Turbulent Flows in active-DIPS” section in the Supplementary Information). At higher activity levels, even low packing fraction systems undergo phase separation into the active-DIPS regime. In this case, the lower packing fraction allows for a larger volume of active fluid, which supports stronger flows and thus higher fluid kinetic energy. In contrast, the behavior of the droplet kinetic energy E_d is non-monotonic with increasing activity (Fig. 3h). At low packing fractions, the system transitions from a turbulent regime—where droplets are advected independently by active flows—to a phase-separated state where droplets form a single cluster. In this clustered state, droplet motility decreases, as the diffusivity of the larger aggregate becomes smaller than the typical scale of uncorrelated active jets. Conversely, at high packing fractions, the system transitions directly from a gel-like state with minimal droplet motion to a phase-separated state where the cluster is still advected by the flow. However, due to its larger size, the cluster’s motility remains low, and the overall droplet kinetic energy is reduced.

Clustering dynamics

The dynamics of active-DIPS reveal a complex interplay between active flows and passive droplet organization. As illustrated in Fig. 4a(i-ii), the process begins with the formation of small, transient clusters similar to those observed at lower activity levels. However, at these elevated activity levels, the system exhibits a qualitatively different behavior as clusters progressively merge into larger and more stable structures (panels a(iii-iv)). These emergent superstructures serve as nucleation centers that incorporate smaller clusters and individual droplets through an ongoing coarsening process. Notably, these macroscopic clusters eventually reach a stationary state where the internal droplets are organized into hexatic configurations.

Fig. 4: Clustering dynamics.

a Time-lapse sequence of the system configuration for Φ = 0.22 (N_d = 50) and A = 864 (ζ = 0.06), resulting in a fully separated state. b, c Color maps of the hydrodynamic pressure P_h (panel b) and the active force (panel c) at time t = 2 × 10² τ. d, e Temporal evolution of the structure factor of the concentration field (panel d) and the kinetic energy spectrum (panel e). f Characteristic cluster size ℓ_ϕ and flow structure size ℓ_v—derived as the inverse of the first moment of their respective structure factors—plotted versus time. g PDF of the droplet velocity component v_x. The color scales in panels (f, g) correspond to those used in panels (d, e).

The stabilization mechanism of these clusters differs fundamentally from the defect-mediated bonding observed at lower activities. In this high-activity regime, where the interstitial space between droplets becomes depleted of liquid crystal, hydrodynamic effects dominate the collective behavior. Cluster stabilization arises from the pressure gradient that develops as a consequence of asymmetric forces acting on peripheral droplets. While interior droplets experience balanced interactions with their neighbors, peripheral droplets are subjected to strong active stresses from the surrounding turbulent fluid. This imbalance establishes a radially decreasing hydrodynamic pressure profile, P_h toward the cluster center, as clearly evidenced in Fig. 4b, c.

The dynamics of active-DIPS result from a density-dependent mobility reduction, analogously to conventional MIPS. However, in our case, passive droplets are transported by active flows and feature decreased mobility upon collisions and elastic interactions, which eventually lead to local accumulation. This process is complemented by a feedback mechanism in which accumulating droplets progressively exclude the active fluid from the cluster interior. The latter, in turn, exerts compressive stresses on the cluster boundaries, which are transmitted inward through the compression of the droplet network.

The temporal evolution of the system’s organization is quantitatively captured through spectral analysis. The concentration spectrum $\langle {\widehat{\phi }}_{{\boldsymbol{k}}}^{*}{\widehat{\phi }}_{{\boldsymbol{k}}}\rangle$ Fig. 4d reveals two important features: the development of Bragg peaks at large wavenumbers reflects the emergence of hexatic order within clusters, while the growing intensity at small wavenumbers indicates the progressive increase in cluster size. Furthermore, the kinetic energy spectrum ${E}_{k}=\langle {\widehat{{\boldsymbol{v}}}}_{{\boldsymbol{k}}}^{*}\cdot {\widehat{{\boldsymbol{v}}}}_{{\boldsymbol{k}}}\rangle$ in Fig. 4e maintains the characteristic scaling of active turbulence, demonstrating that the active fluid retains its chaotic nature despite the presence of the separated cluster. From these spectra, we extract the characteristic length scales of the concentration field ℓ_ϕ and velocity field ℓ_v as the first moments of their respective distributions. As shown in Fig. 4f, ℓ_v < ℓ_ϕ throughout the evolution, confirming that the turbulent vortices remain smaller than the emergent droplet structures. Such a length-scale separation highlights the hierarchical organization of the system, where microscopic active turbulence coexists with macroscopic droplet ordering. This is further exemplified by the evolution of the probability density function (PDF) of the droplet velocity shown in Fig. 4g as clustering progresses. At early times, the distribution is approximately Gaussian, aside from heavy tails that reflect intermittent behaviors typical of passive particles suspended in active turbulent nematics⁷⁴. As clustering advances, the Gaussian behavior progressively disappears, leading to a pronounced peak around the mean velocity. This shift indicates that droplet velocities become increasingly correlated–consistent with the emergence of collective motion, where clustered droplets behave as a single body rather than as independent particles advected by the turbulent flow.

We next examine the role of surface tension in the stability of active-DIPS, as illustrated in Fig. 5. In panel (a), we perform an experiment starting from a clustered state and vary the surface tension σ. As surface tension rises, the cluster undergoes melting, which is accompanied by a sharp decrease in the average hydrodynamic pressure gradient $\langle | \nabla {P}_{{\rm{h}}}| \rangle {R}_{{\rm{d}}}^{3}/{\kappa }_{{\rm{F}}}$ (Supplementary Movie 6). This behavior is further detailed in the inset, showing the temporal evolution of the pressure gradient for two representative values of surface tension: low tension maintains a high pressure gradient, which stabilizes the cluster, whereas high surface tension makes the droplets stiffer so that they are not able to absorb external pressure, eventually leading to melting.

Fig. 5: Impact of surface tension on the stability of active-DIPS.

a Average pressure gradient $\langle | \nabla {P}_{{\rm{h}}}| \rangle {R}_{{\rm{d}}}^{3}/{\kappa }_{{\rm{F}}}$ as a function of normalized surface tension σR_d/κ_F, highlighting regimes with active-DIPS (yellow) and melted states (purple). Measurements are taken at steady state after thermalization (t/τ > 25) from an initially clustered configuration. The error bars represent the standard deviation. For the melted states, the error bars are contained within the marker size. The inset shows the time evolution of the pressure gradient for two representative surface tension values. b Phase diagram showing the stability of clustered configurations as a function of surface tension and activity. For large enough values of surface tension, clusters melt, and the system settles into an active turbulent state. The boundary separating active-DIPS and melted states scales approximately as ~A^1/2.

The phase diagram in Fig. 5b further reveals a competition between activity and surface tension; high activity stabilizes active-DIPS, while high surface tension promotes melting and drives the system toward an active turbulent state, with the transition boundary scaling approximately as σ ~ A^1/2. Our data suggest that this scaling can be rationalized by introducing an active capillary number, defined as

$${{\rm{Ca}}}_{{\rm{act}}}=\frac{\zeta a{R}_{{\rm{d}}}^{2}}{{\sigma }^{2}}\,,$$

(1)

where a is the bulk compressibility modulus. The observed transition line $\sigma \sim \sqrt{A} \sim \sqrt{\zeta }$ corresponds to constant Ca_act, confirming that the phase behavior is controlled by the competition between active forcing and interfacial stresses that resist deformation and regulate droplet compressibility.

Discussion

Understanding how clustered and active-DIPS states respond to time-dependent activity changes is crucial for developing robust self-assembly strategies. To investigate this dynamic response, we conducted systematic numerical experiments that probe the system’s behavior under different activity quenching rates. Figure 6 illustrates the results of this investigation. A disordered configuration with packing fraction Φ = 0.18 at the initial time is evolved by increasing activity until A = 1000, to obtain a fully clustered state through active-DIPS (left panel). Starting from this configuration, we then proceed to quench the activity at various rates and observe three possible regimes.

Fig. 6: Activity quenching.

Left panel: Initial configuration of the system in the active-DIPS regime at Φ = 0.18 (N_d = 40) and A = 10³ (ζ = 0.07). The system then evolves under a linear quench of activity toward the passive limit. Right panel: Phase diagram as a function of activity number A and time t/τ. Distinct regimes emerge depending on the rate of quenching, including clustered, partially clustered, and melted states, with example snapshots illustrating typical configurations in each state. Dashed and dash-dotted lines indicate two representative quench rates, −τ dA/dt = 3.7 and 29, corresponding to the transitions between melted and partially clustered, and between partially clustered and clustered, respectively. The solid gray lines correspond to the quench rates tested through simulations to construct the phase diagram.

At rapid activity quenching rates (−τ dA/dt ≳ 29), the system maintains its clustered configuration through persistent defect-mediated bonds, with the interstitial nematic forming a hexatic matrix of defects between droplets (Supplementary Movie 7). More gradual activity quenching leads to different morphological behaviors. In particular, at intermediate quenching rates (3.7 ≲ −τ dA/dt ≲ 29), we observe the fully separated cluster melt, as decreasing active pressure renders peripheral droplets vulnerable to the advective action of turbulent flows. This creates an intriguing coexistence regime where the initial cluster persists alongside a dispersed droplet suspension (Supplementary Movie 8). At slower quenching rates (−τ dA/dt) ≲ 3.7, complete disorder is ultimately restored, with the system adiabatically returning to its initial configuration through complete melting (Supplementary Movie 9).

These rate-dependent pathways are summarized in the phase diagram of Fig. 6 (right panel), where the activity number A is plotted against normalized time t/τ, establishing temporal activity modulation as a powerful tool for designing reconfigurable activity-driven composites. The ability to select between clustered, partially clustered, or melted states through control of activity dynamics suggests new possibilities for designing materials with tunable structural memory.

From a broader perspective, our work demonstrates how active nematics can orchestrate the behavior of embedded soft particles through the interplay of hydrodynamic flows and topological constraints. The observed transitions–from turbulence-driven clustering to gelation and active-DIPS–significantly broaden the toolkit for non-equilibrium self-assembly by harnessing the chaotic dynamics of active turbulence. We anticipate these findings will stimulate experimental efforts using both biological and synthetic active nematics, particularly to explore how droplet softness, anchoring conditions, and confinement influence the reported phenomenology. Our model conceptually aligns with quasi-2D experimental systems such as active nematic layers confined at a water-oil interface or soap films, where soft inclusions like liquid-liquid phase separation (LLPS) droplets or hydrogel particles can be embedded and interact with active flows. These setups offer a promising route to experimentally realize and test the mechanisms uncovered in this study. The principles revealed in this study could ultimately guide the design of adaptive smart materials capable of programmable structural transitions.

Methods

The model

We consider a suspension of isotropic droplets in a quasi two-dimensional active nematic liquid crystal. Each droplet is described using a multiphase approach, modeled by phase fields ϕ_i(r, t), with i = 1,…,N_d, where N_d represents the total number of droplets in the system. The dynamics of the system are characterized by the incompressible two-dimensional velocity field v(r, t), constrained by ∇ ⋅ v = 0, and the nematic Q-tensor Q(r, t) = S(nn−I/3). Here, the principal eigenvector n–the so-called director field–defines the 3D direction of alignment of the liquid crystal, while its magnitude S quantifies the degree of local nematic order.

Evolution equations

The dynamics of the fields are governed by the following set of coupled PDEs,

$${D}_{t}{\phi }_{i}=M{\nabla }^{2}{\mu }_{i}\,,$$

(2)

$${D}_{t}{\boldsymbol{Q}}={\boldsymbol{S}}({\boldsymbol{W}},{\boldsymbol{Q}})+{\gamma }^{-1}{\boldsymbol{H}}\,,$$

(3)

$$\rho {D}_{t}{\boldsymbol{v}}=\nabla \cdot ({{\boldsymbol{\sigma }}}^{{\rm{hydro}}}+{{\boldsymbol{\sigma }}}^{\phi }+{{\boldsymbol{\sigma }}}^{{\rm{LC}}}+{{\boldsymbol{\sigma }}}^{{\rm{act}}})\,,$$

(4)

where the operator D_t = ∂_t + v⋅∇ denotes the material derivative. Eq. (2) is the Cahn-Hilliard equation for the conserved concentration fields ϕ_i, where M is the mobility parameter, and ${\mu }_{i}=\delta {\mathcal{F}}/\delta {\phi }_{i}$ are corresponding chemical potentials, and ${\mathcal{F}}$ is the free energy of the system introduced below. Eq. (3) is the Beris-Edwards equation ruling the dynamics of the Q-tensor. The operator

$${\boldsymbol{S}}({\boldsymbol{W}},{\boldsymbol{Q}})= (\xi {\boldsymbol{D}}+{\boldsymbol{\Omega }})({\boldsymbol{Q}}+{\boldsymbol{I}}/3)\\ +({\boldsymbol{Q}}+{\boldsymbol{I}}/3)(\xi {\boldsymbol{D}}-{\boldsymbol{\Omega }})\\ -2\xi ({\boldsymbol{Q}}+{\boldsymbol{I}}/3){\rm{Tr}}({\boldsymbol{Q}}{\boldsymbol{W}})\;,$$

(5)

denotes the co-rotational derivative and defines the dynamical response of the LC to straining and shearing. Here, D = (W + W^T)/2 and Ω = (W − W^T)/2 are the symmetric and anti-symmetric parts of the velocity gradient tensor W_αβ = ∂_βv_α, respectively. The flow alignment parameter ξ determines the aspect ratio of the LC molecules and the dynamical response of the LC to an imposed shear flow. Here, we choose ξ = 0.7 to consider flow-aligning rod-like molecules. The coefficient γ in Eq. (3) is the rotational viscosity measuring the relative importance of advection with respect to relaxation, and ${\boldsymbol{H}}=-\frac{\delta {\mathcal{F}}}{\delta {\boldsymbol{Q}}}+({\boldsymbol{I}}/3){\rm{Tr}}\frac{\delta {\mathcal{F}}}{\delta {\boldsymbol{Q}}}$ is the molecular field. Finally, Eq. (4) is the Navier-Stokes equation for the incompressible velocity field (∇⋅v = 0) with constant density ρ. Here, the stress tensor has been divided in: (i) a hydrodynamic contribution σ^hydro = −P_hI + η∇v where P_h is the hydrodynamic pressure ensuring incompressibility, and η = 5/3 is the shear viscosity; (ii) an interface contribution

$${\sigma }_{\alpha \beta }^{\phi }=\mathop{\sum }\limits_{i=1}^{{N}_{{\rm{d}}}}\left[\left(f-{\phi }_{i}\frac{\delta {\mathcal{F}}}{\delta {\phi }_{i}}\right){\delta }_{\alpha \beta }-\frac{\partial f}{\partial ({\partial }_{\beta }{\phi }_{i})}{\partial }_{\alpha }{\phi }_{i}\right]\,,$$

(6)

where ${\mathcal{F}}\equiv {\int } \, d^2{\boldsymbol{r}} \, f$ is the free energy; (iii) a LC contribution

$${\sigma }_{\alpha \beta }^{LC} = -\xi {H}_{\alpha \gamma }({Q}_{\gamma \beta }+\frac{1}{3}{\delta }_{\gamma \beta })-\xi ({Q}_{\alpha \gamma }+\frac{1}{3}{\delta }_{\alpha \gamma }){H}_{\gamma \beta }\\ +2\xi ({Q}_{\alpha \beta }-\frac{1}{3}{\delta }_{\alpha \beta }){Q}_{\gamma \mu }{H}_{\gamma \mu }+{Q}_{\alpha \gamma }{H}_{\gamma \beta }-{H}_{\alpha \gamma }{Q}_{\gamma \beta }\\ -{\partial }_{\alpha }{Q}_{\gamma \mu }\frac{\partial f}{\partial ({\partial }_{\beta }{Q}_{\gamma \mu })}\,,$$

(7)

accounting for the elastic backflow; and (iv) the active stress σ^act = −ζQ, where ζ is the activity parameter, positive for extensile systems.

Free energy

The equilibrium properties of the system are defined by the free energy

$${\mathcal{F}}={\int } \, d^2{\boldsymbol{r}}\,f={\int } \, d^2{\boldsymbol{r}}\,\left({f}^{\phi }+{f}^{{\rm{LC}}}+{f}^{{\rm{anch}}}\right)\,,$$

(8)

where f^ϕ is the emulsion free energy density, f^LC is the liquid crystal free energy (accounting for both the isotropic-nematic transition and elastic deformations), and f^anch is the anchoring energy at the droplet surfaces. In particular,

$${f}^{\phi }=\mathop{\sum }\limits_{i=1}^{{N}_{{\rm{d}}}}\frac{a}{4}{\phi }_{i}^{2}{({\phi }_{i}-{\phi }_{0})}^{2}+\frac{{k}_{\phi }}{2}\mathop{\sum }\limits_{i=1}^{{N}_{{\rm{d}}}}{(\nabla {\phi }_{i})}^{2}+\mathop{\sum }\limits_{i < j}\epsilon {\phi }_{i}^{2}{\phi }_{j}^{2}\,,$$

(9)

for a, k_ϕ > 0, describes phase separating fields with two minima at ϕ_i = ϕ₀ and ϕ_i = 0 that, at equilibrium, arrange into circular 2D droplets with surface tension $\sigma=\sqrt{8a{k}_{\phi }/9}$, and interface thickness ${\xi }_{\phi }=\sqrt{2{k}_{\phi }/a}$. The interaction term proportional to ϵ > 0 describes soft repulsion pushing droplets apart when overlapping.

The LC free energy density

$${f}^{{\rm{LC}}}= {A}_{0}\left[\frac{1}{2}\left(1-\frac{\chi (\phi )}{3}\right){\rm{Tr}}{{\boldsymbol{Q}}}^{2}-\frac{\chi (\phi )}{3}{\rm{Tr}}{{\boldsymbol{Q}}}^{3}\right.\\ \left.+\frac{\chi (\phi )}{4}{\left({\rm{Tr}}{{\boldsymbol{Q}}}^{2}\right)}^{2}\right]+\frac{{\kappa }_{{\rm{F}}}}{2}{(\nabla {\boldsymbol{Q}})}^{2}\,.$$

(10)

Here, the bulk constant A₀ > 0, and χ(ϕ) is a temperature-like parameter that drives the isotropic-nematic transition that occurs for χ(ϕ) > χ_cr = 2.7⁷⁵, where we have denoted with ϕ = ∑_iϕ_i the global concentration field. We choose χ = χ₀ + χ_sϕ, with χ₀ > 2.7 and χ_s < 0 to confine the LC outside of the droplet where ϕ = 0 (see Fig. 1a). The last term in Eq. (10), proportional to the elastic constant κ_F, captures the energy cost of elastic deformations in the single elastic constant approximation.

Finally, the anchoring contribution to the free energy,

$${f}^{{\rm{anch}}}=W\mathop{\sum }\limits_{i=1}^{{N}_{{\rm{d}}}}\left[{\partial }_{\alpha }{\phi }_{i}\,{Q}_{\alpha \beta }\,{\partial }_{\beta }{\phi }_{i}\right] \,,$$

(11)

dictates the preferred orientation of the liquid crystal at the droplet interfaces. For W < 0, homeotropic anchoring is enforced at the droplet surfaces.

Simulation protocol

We integrate the dynamics of the hydrodynamic fields in Eqs. (2–4) in a square grid of size L_g = 320 with a predictor-corrector hybrid lattice Boltzmann approach^44,76,77,78 consisting of solving Eqs. (2,3) with a finite-difference algorithm implementing a first-order upwind scheme and fourth-order accurate stencils for space derivatives, and the Navier-Stokes equation (Eq. (4)) through a predictor-corrector LB scheme on a D2Q9 lattice. For technical details on the lattice Boltzmann method here implemented, refer to References⁷⁷.

The simulation is initialized by randomly placing droplets within the two-dimensional domain. Each droplet concentration field, ϕ_i, is defined as a circular region of radius R_d, taking the value ϕ_i = ϕ₀ inside the droplet and ϕ_i = 0 outside. The nematic order parameter is initialized as S = 0 inside the droplets, while in the surrounding active fluid it is assigned a small magnitude and a random in-plane orientation. The velocity field is initially set to zero.

The key control parameters of the system are: (i) the droplets’ packing fraction $\Phi={N}_{{\rm{d}}}\pi {R}_{{\rm{d}}}^{2}/{L}_{{\rm{g}}}^{2}$; and (ii) dimensionless activity number

$$A=\frac{{R}_{{\rm{d}}}^{2}}{{\ell }_{{\rm{act}}}^{2}}=\frac{{R}_{{\rm{d}}}^{2}| \zeta | }{{\kappa }_{{\rm{F}}}}\,,$$

(12)

where ${\ell }_{{\rm{act}}}=\sqrt{{\kappa }_{{\rm{F}}}/| \zeta | }$ is the active length-scale.

Throughout the paper, results are presented in terms of the dimensionless numbers Φ and A. Furthermore, other physical quantities are normalized as follows: distances are normalized with respect to the droplet radius R_d; energies are normalized with respect to the nominal energy of an isolated defect of charge +1, ${\epsilon }_{{\rm{def}}}=\pi {\kappa }_{{\rm{F}}}ln({R}_{{\rm{d}}}\sqrt{{A}_{0}/{\kappa }_{{\rm{F}}}})$, and times are normalized with respect to the time-scale of elastic distortions of the liquid crystal $\tau=\gamma {R}_{{\rm{d}}}^{2}/{\kappa }_{{\rm{F}}}$.

Simulation parameters

We have varied two control parameters: (i) the packing fraction Φ was varied between 0.02 and 0.44, by considering systems comprising a different number of droplets, namely N_d = 5, up to N_d = 100; (ii) the activity number A was varied between 0 (ζ = 0) and 2 × 10³ (ζ = 9 × 10⁻²). The other model parameters are chosen as follows. We fixed the droplet radius to R_d = 12, the mobility parameter to M = 0.1, and the rotational viscosity to γ = 1. Unless otherwise stated, the free energy parameters are a = 0.07, ϕ₀ = 2.0, k_ϕ = 0.1, ϵ = 0.05, A₀ = 0.2, κ_F = 0.01, and W = −0.04. To confine LC outside of the droplets, the parameters χ₀ = 3.0 and χ_s = −0.25 were selected. With the chosen parameters, the liquid crystal coherence length is ${\ell }_{c}=\sqrt{{\kappa }_{{\rm{F}}}/{A}_{0}}\approx 0.25$. This length scale characterizes how rapidly the orientational order decays near a defect core, effectively setting the characteristic defect size to approximately one grid point in the simulation. Unless otherwise stated, simulations were performed at a fixed system size of L_g = 320, where we used periodic boundary conditions along both directions.

Measurements

Radial distribution function analysis

The radial distribution function g(r) measures the probability of finding a pair of droplets separated by a distance r relative to an ideal homogeneous distribution. It is defined as

$$g(r)=\frac{1}{{\rho }_{0}{N}_{{\rm{d}}}}\left\langle \mathop{\sum }\limits_{i\ne j}\delta (r-{r}_{ij})\right\rangle \,,$$

(13)

where N_d is the number of droplets, ${\rho }_{0}={N}_{{\rm{d}}}/{L}_{{\rm{g}}}^{2}$ is the average number density, and r_ij is the separation distance between i-th and j-th droplets computed using the minimum-image convention to account for periodic boundaries. The angular brackets 〈⋅〉 denote a time average over all time steps. Normalization by ρ₀N_d ensures that g(r) approaches 1 in a completely homogeneous system.

Mean squared displacement analysis

The mean squared displacement (MSD) at time t is defined as

$$\left\langle \Delta {r}^{2}(t)\right\rangle=\frac{1}{{N}_{{\rm{d}}}}\mathop{\sum }\limits_{i=1}^{{N}_{{\rm{d}}}}{\left|{{\bf{r}}}^{(i)}(t)-{{\bf{r}}}^{(i)}({t}_{0})\right|}^{2}\,,$$

(14)

where N_d is the number of droplets, r⁽ⁱ⁾(t) is the unwrapped position vector of the i-th droplet’s center of mass at time t, and t₀ = 0 is the reference time. The angular brackets 〈⋅〉 denote an average over all droplets at a given time. Short- and long-time exponents of the MSD are extracted by fitting power laws to the data.

Statistical analysis of long-time MSD exponent

To investigate the long-time dynamical behavior, we analyze the MSD across a range of packing fractions and activity levels. Specifically, we consider three packing fractions: Φ = 0.09, 0.15, 0.22. For each packing fraction, ten distinct activity levels are considered. At each combination of packing fraction and activity level, we perform n = 5 independent simulations.

For every simulation, the MSD is computed as a function of time. A power-law fit of the form MSD(t) ~ t^α is applied in the long-time regime to extract the MSD exponent α. The exponents obtained from the five independent simulations are then used to compute the mean and the standard error of the mean (SEM), plotted in Fig 3e, f, respectively.

Spectral analysis

The 2D structure factor of the concentration field ϕ(r, t) is defined in Fourier space as

$${{\mathcal{S}}}_{\phi }({\boldsymbol{k}},t)={\widehat{\phi }}_{{\boldsymbol{k}}}^{*}(t)\,{\widehat{\phi }}_{{\boldsymbol{k}}}(t)\,,$$

(15)

where ${\widehat{\phi }}_{{\boldsymbol{k}}}(t)$ denotes the Fourier transform of the concentration field, $\widehat{\phi }({\boldsymbol{k}},t)=\int \phi ({\boldsymbol{r}},t)\,{e}^{-i{\boldsymbol{k}}\cdot {\boldsymbol{r}}}\,{d}^{2}{\boldsymbol{r}}$. Accordingly, we define the concentration spectrum $\langle {\widehat{\phi }}_{{\boldsymbol{k}}}^{*}\,{\widehat{\phi }}_{{\boldsymbol{k}}}\rangle$ by integrating over circular shells of constant wavenumber magnitude q = ∣k∣

$$\langle {\widehat{\phi }}_{{\boldsymbol{k}}}^{*}\,{\widehat{\phi }}_{{\boldsymbol{k}}}\rangle (q,t)=\int {{\mathcal{S}}}_{\phi }({\boldsymbol{k}},t)\,\delta (| {\boldsymbol{k}}| -q)\,{d}^{2}{\boldsymbol{k}}\,.$$

(16)

This spectrum quantifies the spatial correlations among droplets across scales, identifying the dominant inter-droplet spacing and characteristic structural organization. The characteristic cluster size ℓ_ϕ is then defined as the inverse of the first moment of the spectrum

$${\ell }_{\phi }(t)=\frac{\int \, \langle {\widehat{\phi }}_{{\boldsymbol{k}}}^{*}\,{\widehat{\phi }}_{{\boldsymbol{k}}}\rangle \, dq}{\int \, q \, \langle {\widehat{\phi }}_{{\boldsymbol{k}}}^{*}\,{\widehat{\phi }}_{{\boldsymbol{k}}}\rangle \,dq} \, .$$

(17)

Similarly, the 2D structure factor of the kinetic energy is defined from the velocity field v(r, t) as

$${E}_{{\boldsymbol{k}}}(t)={\widehat{{\boldsymbol{v}}}}_{{\boldsymbol{k}}}^{*}(t)\cdot {\widehat{{\boldsymbol{v}}}}_{{\boldsymbol{k}}}(t)\,,$$

(18)

where $\widehat{{\boldsymbol{v}}}({\boldsymbol{k}},t)=\int {\boldsymbol{v}}({\boldsymbol{r}},t)\,{e}^{-i{\boldsymbol{k}}\cdot {\boldsymbol{r}}}\,{d}^{2}{\boldsymbol{r}}$. We then define the kinetic energy spectrum 〈E_k〉 by radial integration:

$$\langle {E}_{{\boldsymbol{k}}}\rangle (q,t)=\langle {\widehat{{\boldsymbol{v}}}}_{{\boldsymbol{k}}}^{*}\cdot {\widehat{{\boldsymbol{v}}}}_{{\boldsymbol{k}}}\rangle (q,t)=\int {E}_{{\boldsymbol{k}}}(t)\,\delta (| {\boldsymbol{k}}| -q)\,{d}^{2}{\boldsymbol{k}}\,,$$

(19)

and the characteristic flow structure size ℓ_v as

$${\ell }_{v}(t)=\frac{\int \,\langle {E}_{{\boldsymbol{k}}}\rangle \,dq}{\int \,q\,\langle {E}_{{\boldsymbol{k}}}\rangle \,dq}\,.$$

(20)

Intermittency analysis

The intermittent dynamics of droplets are characterized by the distributions of the x-component of their instantaneous velocities. Droplet velocities are obtained from the temporal derivatives of their center-of-mass positions, and the probability density function (PDF) of the x-component, v_x, is used to assess deviations from Gaussian behavior. PDFs are collected over all droplets across different time intervals to reveal the statistical features of velocity fluctuations.