Protocol to evaluate the viscoelastic response of a polymer suspension to an active agent via oscillatory shear rheometry

TL;DR

The paper addresses the challenge of quantifying the local viscoelastic response generated by microswimmers in polymeric media, where self-induced flows cause polymer deformation and nonlinear rheology. It introduces a protocol that maps swimmer-induced shear to an equivalent oscillatory shear rheometry problem by defining a characteristic length ξ at which the swimmer’s flow vanishes, yielding an effective frequency ω = \frac{\pi U_0}{2 ξ} and an effective strain γ_0 linked to the active stress via the mean-shear-rate mapping; this mapping is validated with lattice Boltzmann simulations of a squirmer in polymer solutions. A key analytical contribution is the derivation of the swimmer-induced shear rate ⟨γ̇_s^2⟩ and the corresponding relation for γ_0, providing a practical bridge between active-matter dynamics and rheological characterization. The framework enables extraction of dynamic moduli, G′ and G″, from swimmer activity and is demonstrated to produce predominantly viscous responses in the tested regimes, with potential extension to active microrheology. Overall, the work offers a generic, quantitative route to quantify swimmer-induced viscoelasticity and connect active propulsion to local rheological properties, with implications for understanding motility in complex fluids.

Abstract

Microorganisms inhabit viscoelastic environments, where their locomotion can deform polymers and trigger local complex viscoelastic responses. However, a systematic approach to quantify such responses remains lacking. Here, we propose a protocol that maps the shear effect induced by an active agent to oscillatory shear rheometry. The central idea is to establish a correspondence between the mean shear rate generated by swimming and that produced by an oscillating plate. In this mapping, the swimming velocity and active stress are translated into an effective oscillation frequency and strain amplitude. The resulting viscoelastic response can then be evaluated by standard oscillatory rheometry. The protocol is validated using lattice Boltzmann simulations of a squirmer embedded in polymer solutions. Our framework is generic and can be naturally extended to active microrheology, providing a pathway to quantify swimmer-induced viscoelasticity.

Figures (4)

• Figure 1: Schematic of polymers subjected to the shear flow generated by a microswimmer. The swimmer approaches and departs from the central polymers at a characteristic length scale $\xi$. At this distance, the perpendicular flow field has effectively vanished, introducing the equivalent of a static, imaginary wall at that location. The inset demonstrates the squirmer model. (b) Schematic of polymers embedded in an oscillatory shear flow. Two parallel walls are separated by a distance $H$. The bottom wall remains fixed, while the top wall oscillates with frequency $\omega$ and displacement $d(t)$.
• Figure 2: Time series of the imposed oscillatory shear strain $\gamma_w$ and the corresponding measured (red line) and fitted (blue line) polymeric stress components $\sigma^{p}_{xz}$, respectively.
• Figure 3: (a) Influence of the squirmer active stress $\beta$ on the dynamic moduli of the polymer suspension, shown as an effective oscillatory-shear response at fixed frequency $\omega = 1.38\times10^{-4}\,\tau^{-1}$, corresponding to a swimming speed $U_{0}=3\times10^{-3}\,a\tau^{-1}$. Swimmers with $|\beta|\le 5$ generate effective strain amplitudes $\gamma_{0}=0.8\text{--}3.7$ (green region). (b) Influence of the squirmer swimming speed $U_{0}$ on the dynamic moduli at fixed effective strain amplitudes $\gamma_{0}=0.8$ (neutral swimmer) and $\gamma_{0}=2.3$ (pusher/puller with $|\beta|=3$). The range $U_{0}=3\times10^{-4}\text{--}3\times10^{-2}\,a\tau^{-1}$ corresponds to effective oscillatory frequencies $\omega = 8.3\times10^{-6}\text{--}8.3\times10^{-4}\,\tau^{-1}$ (green region).
• Figure 4: (a) Influence of polymer packing fraction $\phi$ on the dynamic moduli of polymer suspensions under shear induced by squirmer activity. The swimming velocity is fixed at $U_{0} = 0.003\,a\tau^{-1}$, corresponding to an effective oscillation frequency $\omega = 1.38 \times 10^{-4}\,\tau^{-1}$. For clarity of presentation, the dynamic moduli at $\phi = 0.06\,a^{-3}$ and $\phi = 0.19\,a^{-3}$ are rescaled by factors of $1/2$ and $1/6$, respectively. (b) Influence of polymer length $L_{p}$ on the dynamic moduli under the same conditions. Here, the polymer packing fraction $\phi= 0.06\,a^{-3}$ is fixed. Swimmers with $|\beta|\leq 5$ correspond to effective strain amplitude $\gamma_{0}=0.8 \text{--} 3.7$ (green region).