Capillary-Stress Controlled Rheometer Reveals the Dual Rheology of Shear-Thickening Suspensions

The rheology of dense colloidal suspensions, which may undergo discontinuous shear thickening or shear jamming, is particularly difficult to analyze with conventional rheometers. Here, we develop a rheometer adapted to colloidal suspensions: the “capillarytron,” which uses the air-suspension capillary interface to impose particle (or osmotic) pressure during shear. The main virtues of this new device are that (i) it gives direct access to the suspension friction coefficient, (ii) it operates for very dense suspensions up to jamming, and, most importantly, (iii) it decouples the stresses developed within the suspension from the applied shear rate. We can, thus, smoothly move through the different frictional states of the system, precisely in the range of volume fractions where discontinuous shear thickening or shear jamming occur under volume-imposed conditions. Our results obtained with the capillarytron provide the first complete characterization of the dual frictional behavior of a model shear-thickening suspension, in agreement with the recently proposed frictional transition scenario. Based on a new concept in rheometry, the capillarytron unlocks the path to pressure-imposed rheology on colloidal and Brownian suspensions. Moreover, its fine control of the particle pressure via the soft capillary interface opens the possibility to explore the flow of “fragile” particles close to jamming, such as Brownian colloids, active particles, and living cells.

Popular Summary

Media composed of a mixture of fluid and microscopic solid particles are all around us, from mud to fresh concrete to liquid food. Yet, how these media flow is still poorly understood, as illustrated by the familiar and notorious suspension of cornstarch particles, which, while liquid at rest, can suddenly become solid upon impact. In this study, we develop a new tool, called the capillarytron, that allows us to access for the first time the frictional properties of these microscopic suspensions during flow.

Based on a new concept in rheometry, the capillarytron uses the air-suspension capillary interface to confine the particles in a controlled way by means of a liquid reservoir of adjustable height. We show that, depending on the confining force, the particles of the microscopic suspension behave as rubbing or nonrubbing solids, a feature that deeply modifies the flowability of the suspension.

These results confirm a mechanism recently proposed to explain the behavior of shear-thickening suspensions, such as the suspension of cornstarch particles, and open the way to characterizing complex suspensions such as those encountered in geophysics and in industry. The principle of the capillarytron also makes it possible to envisage its application to fragile systems, such as suspensions of living cells for biomedical applications.

Figure 1

Capillarytron: The concept. (a) Sketch of the capillarytron. A reservoir of adjustable height is connected to the suspension through a permeable grid to impose a negative liquid pressure $P_{\ell }=-\rho gH$: Here, the capillary interface is used to impose the particle pressure. (b) Schematic of the horizontal and vertical force balances in steady state: In the radial direction and at the air-suspension capillary interface, the particle pressure ${\sigma }_p^r(R)$ is balanced by the liquid pressure $P_{\ell }$. In the vertical direction, the mass $m$ measured with the scale results from the contribution of the mean particle stress in the vertical direction $\overline{{\sigma }_p^z}$ and the liquid pressure $P_{\ell }$. (c) Typical experiment under volume-imposed condition (valve closed). The liquid pressure becomes negative as a result of the particle stress developed by the dilatant suspension. Images of the smooth and corrugated capillary interface before and during shear, respectively. (d) Typical experiment under pressure-imposed condition (valve open). Under constant shear rate and upon a sudden increase of particle pressure obtained by lowering the reservoir, the shear stress and the volume fraction self-adjust to a new steady state.

Figure 2

Capillarytron: Proof of concept on a Newtonian suspension. (a) Picture of the PMMA particles of diameter $d\approx 120\mathrm{\mu }\mathrm{m}$. (b) Relevant variables to access the frictional rheology: shear stress $\tau$, particle pressure $P_p$, and shear rate $\dot{\gamma }$. (c) Frictional rheological laws. Top: suspension friction coefficient $\mu =\tau /P_p$; bottom: suspension volume fraction $\varphi$ versus viscous number $J={\eta }_{\ell }\dot{\gamma }/P_p$, obtained from stationary and quasisteady “dynamic” ramps of shear rate. The data are fitted by $\mu (J)={\mu }_c+A{J}^{{\alpha }_{\mu }}$ and $\varphi (J)={\varphi }_J/(1+B{J}^{{\alpha }_{\varphi }})$ with ${\mu }_c=0.30$, $A=2.99$, ${\alpha }_{\mu }=0.50$, ${\varphi }_J=0.57$, $B=1.10$, and ${\alpha }_{\varphi }=0.45$. Insets: (top) reduced macroscopic friction coefficient $\mathrm{\Delta }\mu =\mu -{\mu }_c$ and (bottom) reduced volume fraction $\mathrm{\Delta }\varphi ={\varphi }_J-\varphi$ versus $J$.

Figure 3

Frictional rheology of a model shear-thickening suspension. (a) Picture of the silica particles, $d=23.46\pm 1.06\mathrm{\mu }\mathrm{m}$. (b) Schematic illustrating the electrostatic double layer at the origin of the short-range repulsive force and the Debye length ${\lambda }_D$. (c) Macroscopic friction coefficient $\mu$ versus $J$ for measurement performed at low particle pressure ($P_p=5$ Pa, $P_p\ll P_{\mathrm{rep}}$), at large particle pressure ($P_p\approx 500\mathrm{Pa}$, $P_p\gg P_{\mathrm{rep}}$), in a solution of water and NaCl ($[\mathrm{NaCl}]=0.2\mathrm{mol}.{\mathrm{L}}^{-1}$, $P_p\gg P_{\mathrm{rep}}$), and for a decreasing ramp of particle pressure ($P_p=2600$ to 10 Pa, $P_p\gg P_{\mathrm{rep}}$ to $P_p\ll P_{\mathrm{rep}}$). (d) Macroscopic friction coefficient $\mu$ versus $P_p$ for the decreasing ramp of particle pressure shown in (c). (e) Reduced macroscopic friction coefficient $\mathrm{\Delta }\mu =\mu -{\mu }_c$ versus $J$ from data shown in (c) (the same symbols). Solid lines: fittings of the data by $\mu ={\mu }_c+A{J}^{{\alpha }_{\mu }}$.

Figure 4

Toward more complex colloidal suspensions. Suspension friction coefficient $\mu$ versus viscous number $J$ (a) for a suspension of slightly Brownian silica particles of diameter $d=4.7\mathrm{\mu }\mathrm{m}$ immersed in water and measured at $P_p=20\mathrm{Pa}$ ($P_p/P_{\mathrm{rep}}\ll 1$) and (b) for a suspension of potato-starch particles immersed in water and measured at small ($P_p=10\mathrm{Pa}$, $P_p/P_{\mathrm{rep}}\ll 1$) and large ($P_p=500\mathrm{Pa}$, $P_p/P_{\mathrm{rep}}\gg 1$) particle pressure.