Shear-Induced Gelation of Self-Yielding Active Networks

An enticing feature of active materials is the possibility of controlling macroscale rheological properties through the activity of the microscopic constituents. Using a unique combination of microscopy and rheology we study three dimensional microtubule-based active materials whose autonomous flows are powered by a continually rearranging connected network. We quantify the relationship between the microscopic dynamics and the bulk mechanical properties of these nonequilibrium networks. Experiments reveal a surprising nonmonotonic viscosity that strongly depends on the relative magnitude of the rate of internally generated activity and the externally applied shear. A simple two-state mechanical model that accounts for both the solidlike and yielded fluidlike elements of the network accurately describes the rheological measurements.

Figure 1

Structure, dynamics, and rheology active gels. (a),(b) Confocal fluorescence images of active network in the $x,y$ (velocity, vorticity) plane and the $x,z$ (velocity, gradient) plane as the ATP concentration is depleted. Images represent $100\mu \mathrm{m}$ maximum projections. (c) Velocity field of autonomous flows and streamlines in the flow-vorticity plane with no applied shear. (d) Apparent viscosity $\eta$ as a function of shear rate $\dot{\gamma }$; error bars reflect standard error across multiple samples ($N=3$). (e) Storage $G^{\prime }$ and loss $G^{\prime \prime }$ moduli of network initially with $c_{\mathrm{ATP}}=14\mu \mathrm{M}$ suspension as ATP. Frequency is fixed $\omega =1.0\mathrm{rad}{\mathrm{s}}^{-1}$. The light colored curves indicate the oscillatory tests below yield stress ${\sigma }_y$, and the dark curves are above ${\sigma }_y$. Shading indicates the standard deviation for three experiments. Inset: stress measurements from a strain sweep provide an approximate yield stress ${\sigma }_y=2.2\mathrm{mPa}$. Scale bars are $50\mu \mathrm{m}$.

Figure 2

Microscopic model of active self-yielding solids. (a) A sheared active network is composed of liquidlike (blue) and solidlike (red) extensile elements. The fraction of each element depends on the speed of intrinsic extension and the magnitude of the applied shear. Close to ${\dot{\gamma }}_c$ the speed of the boundary is comparable to the speed of the extending elements, which became taught and contribute to shear stress. Above ${\dot{\gamma }}_c$ the extending elements break immediately. (b) Ratio of characteristic length scales in the shear and vorticity directions; a vertical shift is introduced for clarity. Inset: the active timescale $\tau$, measured via MT dynamics, is directly correlated to the rheological timescale $1/{\dot{\gamma }}_c$, dashed line is a fit of slope unity. (c) Probability distribution functions of MT speeds for different $c_{\mathrm{ATP}}$. Inset: $\overline{u}/{\dot{\gamma }}_c$ is independent of $c_{\mathrm{ATP}}$. (d) The fraction of stiffened elements is set by evaluating the cumulative distribution of sliding rates $C(\dot{\epsilon })$ at the shear rate $\dot{\gamma }$. (e) $C(\dot{\epsilon })$ for all $c_{\mathrm{ATP}}$.

Figure 3

Modified Bingham plastic model captures rheology of active gels. Rheological measurements (open symbols) for (a) viscosity and (b) stress. Dashed lines represent the Bingham plastic yield-stress model. The solid lines though the data points are from the modified Bingham model, while the labeled lines in (b) provide the power law slopes to guide the eye. At the highest shear rates, the Newtonian plateau must have a $\sigma \sim \dot{\gamma }$, while at low shear rates, $\sigma \sim {\dot{\gamma }}^{1.5-2.0}$.