Tuning Steady Shear Rheology through Active Dopants

TL;DR

The paper investigates how small fractions of active dopants alter the glass–fluid transition and steady-state rheology of dense, disordered soft materials. Using 3D dry active Brownian particle simulations with Herschel–Bulkley fits to extract yield stress $\sigma_Y$, the authors show that the critical packing density $\phi_g(\alpha)$ depends nonlinearly on the active fraction $\alpha$, with even $\alpha \ll 1$ substantially fluidizing the system. They introduce the combined active-energy parameter $\alpha \mathrm{Pe}^2$ that collapses the rheological curves for $\langle \sigma_{xy} \rangle$ and the viscosity $\eta$ across a broad range of $\phi$, ${\rm Pe}$, and $\dot{\gamma}$, though the collapse can fail at very low $\dot{\gamma}$ due to heterogeneity. Additionally, the higher-order moments of shear-stress fluctuations, the skewness $\mathcal{S}_{\sigma}$ and excess kurtosis $\mathcal{K}_{\sigma}$, provide independent markers of the glass–fluid boundary that collapse with $\alpha \mathrm{Pe}^2$. The framework offers a practical route to tune mechanical properties with minimal active doping, while acknowledging limitations such as neglected hydrodynamics and particle-size heterogeneity.

Abstract

We numerically investigate the steady shear rheology of mixtures of active and passive Brownian particles, with varying fractions of active components. We find that even a small fraction of active dopants triggers fluidization with comparable efficiency to fully active systems. A combined parameter, active energy, given by dopant fraction multiplied by propulsion speed squared controls the steady shear rheology and glass transition of the active-passive mixtures. These results together provide a quantitative strategy for fine-tuning the mechanical properties of a soft material with small amounts of active dopants.

Figures (5)

• Figure 1: The shear velocity profile is shown on a cross-section in the x-y plane of the three-dimensional simulation box, representing a small fraction of active particles doped in passive particles.
• Figure 2: Steady Shear Glass-Fluid Rheology with varying packing density($\phi$) and active fraction($\alpha$) with active strength (i.e., Péclet number) $\mathrm{Pe}=10$. Flow curves (a, b) the shear stress, $\langle\sigma_{xy}\rangle$, and (c, d) the viscosity, $\eta$, at densities $\phi = 0.60, \ldots, 0.72$. (a) and (c) depict the behavior for two active fractions: $\alpha = 0.0$ (purely passive; solid symbols and solid lines) and $\alpha = 0.05$ (small active fraction; open symbols and dashed lines). (b) and (d) illustrate the results for two relatively high active fractions, $\alpha = 0.35$ (solid symbols and solid lines) and $\alpha = 0.50$ (open symbols and dashed lines). The lines represent fits of the Herschel-Bulkley form in Eq. \ref{['eq:Herschel_Bulkley_Yield']}; used to extract yield stress $\sigma_{Y}$. Yield stress $\sigma_{Y}$ as function of packing density $\phi$ in (e) and active fraction $\alpha$ in (f). (g) Phase diagram in $\phi-\alpha$ plane. The black solid line corresponds to shifted power law $\phi(\alpha) =\phi_{0} + a_1 \alpha^{b_1}$ where $a_1,b_1$ are the fitting constants ($a_1\approx 0.064$, $b_1\approx 0.16$) with passive glass transition density $\phi_{0}=0.61$Wiese2023PRL. The blue dashed dotted (red dotted) line represent skewness (excess kurtosis) of shear stress fluctuations for $\dot{\gamma}=10^{-6}$ (see Fig. 3 in Supplemental Material Supply2025). The black solid circle at $\phi_{J}=0.648$ represent passive athermal ($\alpha=0,~k_{B}T=0$) Jamming transition Wiese2023PRL.
• Figure 3: Yield stress with varying active strength Péclet $\mathrm{Pe}$ and fraction ($\alpha$) at density $\phi=0.66$. Yield stress $\sigma_{Y}$ as function of $\mathrm{Pe}$ in (a) and active fraction $\alpha$ in (b). Vertical dashed lines in (a,b) indicate threshold of glass transition. (c) Phase diagram in $\mathrm{Pe}-\alpha$ plane. The black solid line corresponds to $\alpha \mathrm{Pe}^2=(\alpha \mathrm{Pe}^2)_G\approx 22$ where $(\alpha \mathrm{Pe}^2)_G$ is the critical glass transition activity. The blue dashed dotted (red dotted) line represent the glass-fluid transition calculated from skewness (excess kurtosis) of shear stress fluctuations for $\dot{\gamma}=10^{-6}$ (see Fig. \ref{['fig:stress_skew_kurt_alpha_Pe']}).
• Figure 4: Impact of activity on the Glass-Fluid transition. (a) Shear stress, $\langle\sigma_{xy}\rangle$, and (b) the viscosity, $\eta$, as a function of activity ($\alpha \mathrm{Pe}^2$) at density $\phi = 0.66$ for varying shear rates $\dot{\gamma}$. The dashed lines in (a) are fits to Eq. \ref{['eq:Hill_emperical_form']}. (c) $\sigma_0$, (d) $(\alpha \mathrm{Pe}^2)_c$, and (e) $m$ as a function of $\dot{\gamma}$. Symbols are values extracted from the fits in (a,b), and dashed lines are Herschel–Bulkley fits. (c) The fitted yield stress in the passive limit($\alpha\mathrm{Pe}^2\to 0$) is $\sigma_{0p}\approx 3.98\times 10^{-4}$. (e) show that $m$ diverge as $\dot{\gamma}\to 0$. (c) stress (shear and yield) through the glass-fluid crossover. The symbols shape indicates active fraction ($\alpha$), color represents Péclet $\mathrm{Pe}$, and border color denotes shear rates ($\dot{\gamma}$).
• Figure 5: Impact of activity on shear stress fluctuation moments. (a) Skewness $\mathcal{S}_\sigma$ and (b) excess kurtosis $\mathcal{K}_\sigma$ versus activity $\alpha\mathrm{Pe}^2$, showing the glass-fluid crossover. (c,d) Heat maps of $\mathcal{S}_\sigma$ and $\mathcal{K}_\sigma$ over the $(\mathrm{Pe},\alpha)$ plane, with $\mathcal{S}_\sigma,\mathcal{K}_\sigma=0.5$ threshold contour delineating the glass transition. We fixed $\phi=0.66$, $\dot\gamma=10^{-6}$.