A combined rheometry and imaging study of viscosity reduction in bacterial suspensions

Abstract

Suspending self-propelled "pushers" in a liquid lowers its viscosity. We study how this phenomenon depends on system size in bacterial suspensions using bulk rheometry and particle-tracking rheoimaging. Above the critical bacterial volume fraction needed to decrease the viscosity to zero, [Formula: see text], large-scale collective motion emerges in the quiescent state, and the flow becomes nonlinear. We confirm a theoretical prediction that such instability should be suppressed by confinement. Our results also show that a recent application of active liquid-crystal theory to such systems is untenable.

Keywords: Escherichia coli; active matter; particle image velocimetry; particle tracking; rheology and imaging.

Fig. 1.

Confinement effect on flow stability and collective motion. Lines indicate calculated confined critical volume fraction ${\varphi }_c\left(H\right)$, according to CKT using Eq. 4 and the experimentally measured ${\varphi }_c\left(H=400\hspace{0.17em}\mathrm{\mu }\text{m}\right)\approx 0.75\%$ (Fig. 4 I and J) as ${\varphi }_c^{\infty }$. ${\varphi }_c\left(H\right)$ defines the boundary between stable (below) and unstable (above) pusher suspensions for different persistence time $\tau$ of the swimmers, as indicated in the key, with speed $v=15\hspace{0.17em}\mathrm{\mu }\text{m}\hspace{0.17em}{\text{s}}^{-1}$. Symbols: experimental observation of flow with (filled squares) and without (open squares) banding; and with (filled circles) or without (open circles) large correlation length-scale $l$, based on rheo-imaging (squares) and phase-contrast imaging (no flow; circles), respectively. See Fig. 4 and related main text for more details. Red: Data from ref. at $H=60\hspace{0.17em}\mathrm{\mu }\text{m}$ (an $x$ offset is applied to squares for better visualization).

Fig. 2.

Schematic of the three experimental setups used (not to scale). (A) Rheo-imaging setup using cone-plate geometry for visualization during shear. (B) Couette cell for bulk rheometry, after ref. . (C) Phase contrast imaging without applied shear.

Fig. 3.

(A–C) The viscosity of E. coli suspensions as a function of shear rate, $\eta \left(\stackrel{\cdot }{\gamma }\right)$, at gap sizes $H=240$, 500, and $730\hspace{0.17em}\mathrm{\mu }\text{m}$ for $\varphi =0.2\%$ (A), $0.4\%$ (B), and $0.75\%$ (C). (D) The measured viscosity at $\stackrel{\cdot }{\gamma }\approx 0.04\hspace{0.17em}{\text{s}}^{-1}$ for three bacterial concentrations (symbols) compared to the predictions (color-matched) of ALCT (lines) using parameters from ref. and a cell volume ${\mathcal{V}}_{\mathcal{B}}=1.4\hspace{0.17em}\mathrm{\mu }{\text{m}}^3$ (34) and buffer viscosity ${\eta }_0=0.90\hspace{0.17em}\mathrm{c}\mathrm{P}$.

Fig. 4.

(A) The viscosity of E. coli suspensions measured with a gap $H=500\hspace{0.17em}\mathrm{\mu }\text{m}$ at a shear rate $\stackrel{\cdot }{\gamma }\approx 0.04\hspace{0.17em}{s}^{-1}$, as a function of volume fraction, normalized to the viscosity of the buffer ${\eta }_0\left(c_{\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{e}}\right)=\left(0.87+2.7\times 10^{-4}{c}_{\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{e}}\right)$ cP, with $c_{\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{e}}$ the concentration of serine in mM used to prepare the solutions. The gray area defines the presence of large-scale collective motion, observed above ${\varphi }_c\approx 0.75\%$ (vertical dashed line), as characterized in I–L. (B–G) Examples of velocity profiles measured by using rheoimaging in cone-plate geometry of bacterial suspensions at progressively higher volume fraction $\varphi$, as indicated, and for $\stackrel{\cdot }{\gamma }\approx 0.04\hspace{0.17em}{s}^{-1}$ and $H=170\hspace{0.17em}\mathrm{\mu }\text{m}$. (H) SD $\sigma$ of $\mathrm{\Delta }{V}_x\left(z\right)=V_x\left(z\right)-{\stackrel{\cdot }{\gamma }}_{\mathrm{a}\mathrm{p}\mathrm{p}}z$ over the entire $z$ range, with the applied shear rate ${\stackrel{\cdot }{\gamma }}_{\mathrm{a}\mathrm{p}\mathrm{p}}=V_x\left(z_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}}\right)/z_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{e}}$, as a function of volume fraction. Open and filled symbols indicate linear and nonlinear flow profile, respectively, for $H=100\hspace{0.17em}\mathrm{\mu }\text{m}$ (red), $170\hspace{0.17em}\mathrm{\mu }\text{m}$ (black; each symbol corresponds to an independent experimental campaign), and $200\hspace{0.17em}\mathrm{\mu }\text{m}$ (blue). Gray area defines the linear flow range based on an arbitrary threshold of $\sigma \lesssim 0.43$. (I and J) Examples of velocity vector fields from PIV at two $\varphi$ below (I) and above (J) ${\varphi }_c\approx 0.75\%$. Image width is $\approx 700\hspace{0.17em}\mathrm{\mu }\text{m}$. (K) Velocity correlation functions $c\left(r\right)$ calculated from Eq. 5 and averaged over $5\lesssim t\lesssim 15$ min at various $\varphi$ measured via PIV analysis of phase-contrast microscopy movies of cell suspensions in sealed capillaries with $H=400\hspace{0.17em}\mathrm{\mu }\text{m}$. Error bars are $\pm$1 SD representative of the time dependency. (L) Characteristic length $l\left(\varphi \right)$ for which $c\left(l,\varphi \right)\approx 1/e$.