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 $c\left(l,\varphi \right)\approx 1/e$.