Lubricated-to-frictional shear thickening scenario in dense suspensions

Shear thickening is the increase of viscosity as shear rate or stress increases. For concentrated or dense suspensions of solid particles, this phenomenon may take the extreme form known as discontinuous shear thickening (DST). In DST, the relative viscosity ${\eta }_r$ exhibits a discontinuous variation when presented as a function of the shear rate $\dot{\gamma }$, typically with a large jump in viscosity at the discontinuity; ${\eta }_r\left(\varphi \right)=\eta \left(\varphi \right)/{\eta }_0$, with ${\eta }_0$ the suspending fluid viscosity and $\varphi$ the solid volume fraction. Rate dependence implies that ${\eta }_r$ is a function not only of $\varphi$ but also of $\dot{\gamma }$ or the shear stress $\sigma$. A scenario in which the close interactions between particles undergo a stress-driven lubricated-to-frictional (LF) transition provides a coherent mechanistic basis for the shear thickening seen in dense suspensions. Prior study of shear thickening leading to the proposal of the LF transition is briefly reviewed. The LF scenario and its predictions are presented, along with a perspective on unresolved issues on widely different scales, from contact interactions to system-spanning force networks.

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Figure 1

Shear thickening regimes. Data from simulation by the method described by Mari et al. [5] illustrate weak or hydrodynamic shear thickening at volume fraction $\varphi =0.45$, continuous shear thickening at $\varphi =0.5$, strong CST at $\varphi =0.54$, and discontinuous shear thickening at $\varphi =0.56$, for a mixture of equal volume fraction non-Brownian particles of radii $a$ and $1.4a$. The CST condition differs from the weak or hydrodynamic thickening by having dominant frictional contact stress above the thickening shear rate. The scaling of the shear rate is ${\dot{\gamma }}_0=F_R/6\pi {\eta }_0{a}^2$, where $F_R$ is the maximum magnitude of the repulsive interparticle force (at contact) and ${\eta }_0$ is the fluid viscosity.

Figure 2

Lubrication breakdown. The average minimum gap between particle surfaces normalized by the particle radius $h_{\text{min}}/a$ is plotted as a function of the imposed strain in a simulated simple shear flow of a monodisperse hard-sphere suspension of solid volume fraction $\varphi =0.51$ in Stokes flow. These representative curves, adapted from Ref. [13], have the same time step $\dot{\gamma }\delta t=5\times {10}^{-4}$ for predictor-corrector (red circles) and Runge-Kutta (blue squares) integration schemes, which are $O\left(\delta t^2\right)$ and $O\left(\delta t^4\right)$, respectively.

Figure 3

Illustration of the two-state concept underlying the lubricated-to-frictional scenario. The lubricated ($\mu =0$) and frictional ($\mu =1$) jamming points are indicated by the labeled vertical dashed lines. At very low shear rate or stress, all close interactions are completely lubricated so that $\mu =0$, while at very large shear rate or stress, all possible interactions which can do so have become frictional. A transition driven by shear rate of stress as indicated by the arrow takes the material to a state with a viscosity curve of lower jamming fraction and thus increases the viscosity.

Figure 4

Viscosity from simulation at $\varphi =0.54$ for two protocols, one using a constant spring stiffness, at the value determined to maintain a maximum linear deformation of $0.03a$, with $a$ the particle radius, at the highest stress, and the other using a variable stiffness sufficient to maintain a maximum deformation of $0.03a$ at each condition (hence a lower stiffness at lower stress). The stress scaling is ${\sigma }_0=F_R/6\pi a^2$, where $F_R$ is the maximum repulsive force magnitude at contact.

Figure 5

Results from shear rate- (blue dashed lines) and shear stress-controlled (symbols and red solid lines) simulations [52], showing equivalence of results up to onset of DST, with shear rate control allowing exploration of the $\mathsf{\text{S}}$-shaped curves associated with DST, as well as shear jamming.

Figure 6

Flow states. (a) Constitutive model [9] compared with simulation for $\mu =1$ with equal parts of particles of relative size 1 and 1.4. (b) Flow state diagram for conditions of (a). The dashed vertical lines indicate the frictional jamming fraction of ${\varphi }_{\mathrm{J}}^{\mu }\approx 0.585$ and the frictionless jamming ${\varphi }_{\mathrm{J}}^0\approx 0.65$. ${\text{DST}}_1$ (inverted triangles) indicates a transition between two flowing states and ${\text{DST}}_2$ (diamonds) implies thickening to a shear-jammed state (squares); regions of shear thinning (+) and rate-independent viscosity (open and closed circles at low and high stress, respectively) are indicated. (c) Combined flow state diagram boundaries for different friction coefficients, with jamming fractions illustrated by the dashed vertical lines: The rightmost is ${\varphi }_{\mathrm{J}}^0$ while the others are ${\varphi }_{\mathrm{J}}^{\mu }$ decreasing through the sequence of $\mu =0.1$, 0.5, 1, and 10.

Figure 7

Particle configurations and frictional contact force network for $\varphi =0.54$ and $\mu =1$ (strong CST) in the low-stress low-viscosity (high-stress high-viscosity) state on the left (right); the low-viscosity state is in the initial weak increase of the viscosity curve $[\sigma /{\sigma }_0\approx 0.5$ in Fig. 6]. Shear flow velocity increases from the bottom to the top of the image, indicating the orientation of force chains near the compressional axis in the low-viscosity state.