Dissipation and microstructure in sheared active suspensions of squirmers

Abstract: We study the energy expenditure and structural correlations in semi-dilute to concentrated suspensions of squirmers using active fast Stokesian dynamics simulations. Specifically, we simulate apolar active suspensions of squirmers, or 'shakers,' and show that shear enhances the total dissipation but reduces the relative viscosity for both puller- and pusher-type shakers. At low shear rates where activity dominates, pushers dissipate more energy than pullers, and more so at higher volume fractions, in contrast to bacterial suspensions displaying a 'superfluid' transition. At high shear rates where shear dominates, pullers and pushers behave effectively as passive spheres, generating negative normal stress differences due to shear-induced collision. Remarkably, the rate-dependent rheological responses are accompanied by unusual microstructural signatures of an enhanced nematic order and anisotropic pair correlation, both of which contribute to a higher viscosity under shear. Further simulations of self-propelled, neutral squirmers exhibit similar but weaker shear-thinning, highlighting the importance of activity over motility, underpinned by hydrodynamic interactions. Overall, our results elucidate the interplay of internal activity and external flow on the dissipation and microstructure in sheared active suspensions of squirmers.

• The paper uses large-scale active fast Stokesian dynamics simulations at volume fractions of 0.05–0.45 to show that shear increases total dissipation while both puller and pusher squirmers shear-thin without becoming superfluid.
• Dissipation is activity-dominated for Pe < 1, transitions through a microstructure-driven regime near Pe ≈ 1–100, and becomes passive-like with Φ/Φ₀ ∼ Pe² for Pe ≳ 100, as nematic order and tilted doublets emerge at intermediate shear.
• The simulations challenge the swim stress–diffusivity relation for active Brownian particles, finding a second normal-stress difference near 0.02 rather than the predicted 0.96 and identifying particle shape and hydrodynamic interactions as key rheological controls.

Overview

This paper presents large-scale hydrodynamic simulations of active spherical particles under simple shear, using an active fast Stokesian dynamics (FSD) method, to characterize energy dissipation and microstructural correlations in the semi-dilute to concentrated regime ($\phi \in [0.05, 0.45]$). The authors study two particle classes: apolar squirmers ("shakers"), which are individually immotile but hydrodynamically interacting through their imposed strain fields ($B_1 = 0$, $B_2 \neq 0$), and neutral squirmers equivalent to active Brownian particles (ABPs) with self-propulsion and rotational diffusion but negligible leading-order hydrodynamic interactions ($B_2 = 0$). The central findings are that shear enhances total dissipation while reducing relative viscosity for both puller- and pusher-type shakers, and that these rheological responses are accompanied by nonmonotonic nematic order and anisotropic pair correlations. Notably, pushers dissipate more than pullers at low shear rates—opposite to their relative diffusivities—and no "superfluid" transition is observed for spherical particles.

Dissipation: activity- and shear-dominated regimes

The total dissipation per unit volume is decomposed into external and internal contributions, $$\Phi_\text{ext} = \eta\dot\gamma^2 + \langle\Sigma_{p,xy}\rangle\dot\gamma$$ and $$\Phi_\text{int} = \langle\bm\Sigma_p : \bm E_s\rangle$$, with a relative dissipation $\varepsilon_r = \Phi/\Phi_f$. Without shear, interactions enhance dissipation above the dilute value $\Phi_\text{iso} = 6\eta\phi/(5\tau_a^2)$ in all cases; this enhancement is entirely attributable to hydrodynamic interactions, since short-range repulsion contributes negatively by keeping aligned neighbors apart. A notable asymmetry emerges: although pullers generate more hydrodynamic dissipation individually, pushers dissipate more in total at steady state, reversing the ordering seen in activity-induced diffusivity.

Under shear, two regimes appear. For Pe $< 1$, all data collapse and $\Phi \approx \Phi_0$: activity dominates both dynamically and mechanically. For Pe $B_1 = 0$0, $B_1 = 0$1 and $B_1 = 0$2 becomes constant, corresponding to a shear-dominated regime where particles behave effectively as passive spheres. The crossover Pe increases with volume fraction because $B_1 = 0$3 grows with $B_1 = 0$4. This mechanical transition mirrors the dynamical transition previously identified in the same system's diffusivity, suggesting a common underlying mechanism tied to the competition between the activity time scale $B_1 = 0$5 and the shear time scale $B_1 = 0$6.

Rheology: shear-thinning without a superfluid transition

Both puller and pusher shakers exhibit shear-thinning over $B_1 = 0$7, more strongly for pushers and at higher $B_1 = 0$8. This result contradicts the common impression that rate-dependence is dictated primarily by swimming mechanism (pullers thin, pushers thicken); here both types thin, indicating that particle shape, not just the sign of the stresslet, governs the rheology of spherical active suspensions. Fitting the Krieger–Dougherty relation with $B_1 = 0$9 fixed from passive simulations yields intrinsic viscosities $B_2 \neq 0$0 (pullers) and $B_2 \neq 0$1 (pushers) at Pe $B_2 \neq 0$2, qualitatively consistent with algal (C. reinhardtii) experiments reporting $B_2 \neq 0$3 for live cells versus $B_2 \neq 0$4 for dead ones [Rafaï et al.]. Unlike bacterial suspensions, however, $B_2 \neq 0$5 always increases with $B_2 \neq 0$6—no viscosity reduction or superfluid transition occurs for these spherical, apolar particles at any accessible shear rate.

Normal stress differences are negligible for Pe $B_2 \neq 0$7 but become negative ($B_2 \neq 0$8, $B_2 \neq 0$9) at higher Pe, consistent with shear-induced collision structures accumulating pairs in the compression quadrant, as in passive suspensions.

The ABP simulations separate motility from activity. ABPs across persistence lengths $B_2 = 0$0 collapse onto a single weakly shear-thinning curve at $B_2 = 0$1, much weaker than the shaker response. This supports the interpretation that the enhanced low-Pe viscosity observed in algal experiments stems from activity rather than motility or bottom-heaviness.

Tension with swim-stress theory

The results directly challenge the swim stress–diffusivity relation $B_2 = 0$2 proposed by Takatori and Brady. That theory predicts shear thickening at small Pe' for ABPs—and even negative viscosities at long persistence lengths—neither of which appears in the simulations. By computing the cross-diffusivity $B_2 = 0$3 from mean-square displacements, the authors show that applying the relation via measured dynamics reproduces the theory's own prediction, implying the particle kinematics are captured correctly but the stress–diffusion constitutive link itself may be untenable. A sharper test comes from normal stresses: the theory implies $B_2 = 0$4 for persistent ABPs at Pe $B_2 = 0$5, whereas the simulated value is approximately $B_2 = 0$6—a discrepancy of nearly two orders of magnitude. One caveat is that the $B_2 = 0$7 estimates are noisy and show no clear trend, so this assessment rests partly on the NSD comparison rather than on a precise stress measurement.

Microstructural origins

The rheology traces to two microstructural signatures, both peaking in the intermediate regime $B_2 = 0$8:

• Nematic orientational order: the scalar order parameter $B_2 = 0$9 of the tensor $$\Phi_\text{ext} = \eta\dot\gamma^2 + \langle\Sigma_{p,xy}\rangle\dot\gamma$$0 rises nonmonotonically, peaking near Pe $$\Phi_\text{ext} = \eta\dot\gamma^2 + \langle\Sigma_{p,xy}\rangle\dot\gamma$$1 and increasing with $$\Phi_\text{ext} = \eta\dot\gamma^2 + \langle\Sigma_{p,xy}\rangle\dot\gamma$$2. At intermediate Pe, pullers preferentially occupy the extension quadrant ($$\Phi_\text{ext} = \eta\dot\gamma^2 + \langle\Sigma_{p,xy}\rangle\dot\gamma$$3) and pushers the compression quadrant ($$\Phi_\text{ext} = \eta\dot\gamma^2 + \langle\Sigma_{p,xy}\rangle\dot\gamma$$4); substituting these correlations into the dilute stresslet formula reproduces the observed shear-thinning qualitatively. At larger Pe, pullers align with the flow and pushers perpendicular to it, driving $$\Phi_\text{ext} = \eta\dot\gamma^2 + \langle\Sigma_{p,xy}\rangle\dot\gamma$$5. Polar order in ABPs remains negligible ($$\Phi_\text{ext} = \eta\dot\gamma^2 + \langle\Sigma_{p,xy}\rangle\dot\gamma$$6) at all Pe.
• Anisotropic pair correlation: at Pe $$\Phi_\text{ext} = \eta\dot\gamma^2 + \langle\Sigma_{p,xy}\rangle\dot\gamma$$7, pair distributions show excess probability in the tension quadrant roughly along the flow direction, extending into the second particle layer. Combined with the contact alignment preferences (head-to-head for pullers, side-by-side for pushers), this indicates formation of tilted doublets whose activity-induced flow opposes the imposed shear—an interaction-level mechanism raising the viscosity. At high Pe, the structure reverts to the passive signature of contact accumulation in the compression quadrant, generating the negative NSDs.

These pair-correlation effects are weaker in ABPs and require $$\Phi_\text{ext} = \eta\dot\gamma^2 + \langle\Sigma_{p,xy}\rangle\dot\gamma$$8; colliding ABPs additionally generate an effective pusher-like dipole through excluded-volume forces, though the causal link between this alignment and the mild ABP shear-thinning remains difficult to establish quantitatively.

Limitations and open questions

Several caveats bound the conclusions. The FSD formulation truncates hydrodynamic interactions at the level considered, and short-range repulsion models excluded volume rather than resolving lubrication fully; the ABP idealization sets $$\Phi_\text{ext} = \eta\dot\gamma^2 + \langle\Sigma_{p,xy}\rangle\dot\gamma$$9, removing internal dissipation entirely. Results at very small Pe suffer from large finite-size fluctuations, mitigated but not eliminated by scaling to $$\Phi_\text{int} = \langle\bm\Sigma_p : \bm E_s\rangle$$0. The absence of a superfluid transition here applies specifically to spherical, apolar particles under homogeneous simple shear; whether elongated particles in non-dilute suspensions reproduce the bacterial viscosity collapse—and how hydrodynamic versus steric interactions control it—remains unresolved. Finally, the mechanism connecting ABP collision-induced dipoles to their weak shear-thinning is inferred rather than demonstrated.

Conclusion

This work establishes, through systematic simulation, that shear monotonically enhances total dissipation while producing shear-thinning in suspensions of spherical active particles of either stresslet sign, governed by an activity-dominated regime (Pe $$\Phi_\text{int} = \langle\bm\Sigma_p : \bm E_s\rangle$$1), an intermediate regime of enhanced nematic order and tilted doublet formation ($$\Phi_\text{int} = \langle\bm\Sigma_p : \bm E_s\rangle$$2), and a passive-like regime (Pe $$\Phi_\text{int} = \langle\bm\Sigma_p : \bm E_s\rangle$$3). The contrast between shaker and ABP responses isolates hydrodynamic activity, rather than motility, as the dominant driver of the strong rheological signatures, and the NSD measurements provide concrete evidence against the swim stress–diffusivity relation. The persistence of finite viscosity at all concentrations leaves the superfluid transition as a phenomenon requiring particle elongation or collective effects beyond this model.