Non-Equilibrium Phase Transitions in Active Viscoelastic Media Driven by Embedded Self-Propelled Particles: Towards Tunable Rheological States

Introduction

The rheological properties of dense suspensions are typically governed by passive particle interactions, leading to predictable scaling laws near jamming transitions. However, introducing activity into these systems fundamentally alters the energy landscape and dynamics. When microscale inclusions possess inherent motility, they inject non-thermal kinetic energy into the surrounding viscoelastic matrix, driving the system far from thermal equilibrium. This active injection allows for unprecedented control over macroscopic flow behavior.

Our research focuses on the interplay between microscopic motility and mesoscopic structural organization. We hypothesize that tuning the ratio of injected kinetic energy to viscous dissipation dictates whether the system settles into a dynamically stable liquid phase or undergoes a collective ordering transition analogous to crystallization, but driven kinetically rather than purely by attraction.

Theoretical Framework: Modeling Active Injection and Response

The carrier fluid possesses a non-Newtonian memory effect, described by a generalized Maxwell model modified to account for particle presence. The self-propelled particles, assumed to be dipolar dipoles interacting via steric repulsion and lubrication forces within the solvent background, drive viscous dissipation imbalances. The equation governing the evolution of the continuum stress tensor $\boldsymbol{\Sigma}$ in this active medium is given by:

$$\frac{\partial \boldsymbol{\Sigma}}{\partial t} = G_0 \left( \frac{1}{1 + \tau \cdot D_{\text{flow}}} \right) \left( \mathbf{D} - \frac{\boldsymbol{\Sigma}}{\mu_{\text{eff}}} \right) + \boldsymbol{\Pi}_{\text{active}}(t)$$

Here, $\mathbf{D}$ is the strain rate tensor, $G_0$ and $\mu_{\text{eff}}$ are baseline moduli, $\tau$ is the characteristic relaxation time of the matrix, and $\boldsymbol{\Pi}_{\text{active}}(t)$ represents the time-dependent stress contribution from the injected particle motility, which scales with the local activity density $\rho_{\text{act}}$.

We further linearized the particle dynamics around a mean density $\bar{\rho}$ to study fluctuations, finding that the onset of collective motion occurs when the activity number density exceeds a critical threshold $\rho_c$:

$$\text{Activity Criterion: } \quad v_{0} \cdot L_{\text{corr}} \propto \frac{\eta_0}{\zeta}$$

where $L_{\text{corr}}$ is the correlation length scale and $\zeta$ relates to particle drag coefficients [1].

Experimental Observations and Phase Diagram Mapping

Experiments were conducted in a rheometer setup where the sample cell contained acrylic oil doped with microspheres coated with amphiphilic moieties inducing constant forward propulsion in the shear flow. We systematically varied the volume fraction $\phi$ (from $0.1$ to $0.45$) and the drive velocity $v_0$.

The results clearly delineate a phase diagram parameterized by $(\phi, v_0)$. At low $v_0$, the system exhibits Newtonian behavior ($\eta \approx \text{constant}$). As $v_0$ increases beyond a critical value, $\phi_c \approx 0.32$, the system undergoes an abrupt transition from ergodic liquid flow to a state exhibiting long-lived structural arrest and highly anisotropic stress transmission. This arrested state mimics glass transition but is dynamically maintained by the constant mechanical driving force.

Table 1 presents key rheological parameters observed during this transition:

Table 1. Dynamic Rheology During Active Phase Transition

This sharp increase in $\tau_{\alpha}$ signifies the emergence of kinetically trapped structural domains, which are not merely jammed but actively organized by particle interactions. This contrasts sharply with passive jamming where $\tau_{\alpha}$ diverges asymptotically [2].

Discussion: Self-Organization Mediated Rheological Switching

The ability to induce sustained, high-viscosity states purely through controlled kinetic energy injection into a lossy medium represents a powerful paradigm shift in active matter engineering. The ordering observed is not truly crystalline but rather a quasi-ordered state where particle propulsion biases collision geometries towards low-energy, highly correlated arrangements that maximize viscous drag on the background fluid. This ordering is sustained because the propulsion compensates for thermal randomization, effectively lowering the barrier to long-range positional correlation.

We modeled the mechanical coupling efficiency $\eta_m$, defined as the ratio of dissipated energy rate to injected kinetic power:

$$\eta_m = \frac{\text{Power Dissipated in Matrix}}{\langle E_{k, \text{particle}} \rangle} = 1 - e^{-\gamma t}$$

where $\gamma$ characterizes the viscous damping rate of the background medium [3]. Achieving $\eta_m \approx 1$ signifies maximum mechanical transduction of particle motility into bulk viscous resistance.

The persistence of this ordered state, even when the driving field is slightly perturbed, suggests a robust metastate driven by the positive feedback loop between alignment and increased drag. This robustness is crucial for applications requiring switchable fluid properties.

Conclusion and Future Directions

We successfully mapped the phase boundary between fluidic and arrested regimes in an active viscoelastic suspension, demonstrating that controlled microscopic motility can induce macroscopic rheological switching across orders of magnitude in viscosity. Future work will extend this model to include density gradients along with motility, exploring potential unidirectional flow generation without external pressure gradients [4]. This work establishes a fundamental link between non-equilibrium microscopic dynamics and macroscopic continuum behavior in driven soft matter systems.