Stress-activated constraints in dense suspension rheology

Dispersing small particles in a liquid can produce surprising behaviors when the solids fraction becomes large: rapid shearing drives these systems out of equilibrium and can lead to dramatic increases in viscosity (shear thickening) or even solidification (shear jamming). These phenomena occur above a characteristic onset stress when particles are forced into frictional contact. Here we show via simulations how this can be understood within a framework that abstracts details of the forces acting at particle-particle contacts into general stress-activated constraints on relative particle movement. We find that focusing on just two constraints, affecting sliding and rolling at contact, can reproduce the experimentally observed shear thickening behavior quantitatively, despite widely different particle properties, surface chemistries, and suspending fluids. Within this framework parameters such as coefficients of sliding and rolling friction can each be viewed as proxy for one or more forces of different physical or chemical origin, while the parameter magnitudes indicate the relative importance of the associated constraint. In this way, a new link is established that connects features observable in macroscale rheological measurements to classes of constraints arising from micro- or nanoscale properties.

Figure 1

(a) Shear thickening is a phenomena that can be understood across a hierarchy of length scales, and unraveling the interrelationship between macroscopic rheology, microscopic stress-activated constraints that hinder relative particle motion, and nanoscopic particle surface properties poses a major challenge. Macroscale rheology can become a sensitive probe of nanometer-scale interactions between particle surfaces once the link with microscopic frictional constraints has been established. (b) How microscopic frictional constraints affect shear thickening. Viscosity $\eta$ plotted as function of applied shear stress $\sigma$ at constant volume fraction $\varphi$. Above the critical onset stress ${\sigma }_{\mathrm{on}}$, “lubricated” contacts are starting to become transformed into “constrained” frictional ones. At ${\sigma }_{\max }$ all contacts are constrained and a maximum plateau viscosity is reached. At this volume fraction, stress-activated sliding constraints alone lead only to continuous shear thickening (solid black line). Attractive central forces typically lead to a yield stress, without affecting shear thickening (dashed green line). Adding stress-activated rolling constraints can lead to discontinuous shear thickening (DST, slope 1) over a wider stress range and with a higher plateau viscosity (solid blue line).

Figure 2

Unit cell of the simulation, with 2000 total particles of two radii $a$ (red) and $1.4a$ (blue). Each size particle makes up half of the particle volume fraction. This cell is replicated in all three directions and shearing is imposed by Lees-Edwards boundary conditions.

Figure 3

Shear thickening of “standard” particle suspensions show nearly identical shear thickening after rescaling by the onset stress ${\sigma }_{\mathrm{on}}$. As there are small discrepancies in the experimentally-reported volume fractions, data with similar thickening are grouped by color. See Table 1 for details regarding particle size, solvent, and onset stress. Solid lines denote simulation data for $\{{\mu }_s,{\mu }_r\}=\{0.5,0.07\}$ and Debye length $\lambda /a=0.01$ at various volume fractions $\varphi$ as mentioned. (Inset) Beyond the onset of DST ${\varphi }_c<\varphi <{\varphi }_J^{\mu }$, simulations (solid black line) capture nonmonotonic flow curves while experimental measurements cannot (filled gray circles). For unscaled data, see Fig. S1 [45].

Figure 4

Tuning the jamming volume fraction map by changing stress-activated frictional constraints due to sliding (${\mu }_s$) and rolling (${\mu }_r$). The horizontal thick line represents the jamming volume fraction for “standard” particle suspensions, while “extreme roughness” refers to the experimentally measured jamming point for rough raspberry-type particles [13].

Figure 5

Linking changes in microscopic constraints to deviations from “standard” behavior. (a) Particle surface roughness modifies both sliding and rolling constraints. Experimental data from Ref. [29] at volume fraction $\varphi =0.5$ (symbols) for variable roughness with ${\sigma }_{\mathrm{on}}=$ 0.95 Pa (MR) and 5 Pa (VR), simulation data (lines) for combinations for $\{{\mu }_s,{\mu }_r\}$. (b) Addition of urea to carboxylic acid (-${\mathrm{CO}}_2\mathrm{H}$) coated particles disrupts interparticle hydrogen bonding and specifically reduces rolling constraints. Experimental data from Ref. [12]. The onset stress ${\sigma }_{\mathrm{on}}$ for 0 M and 6M urea concentrations is 5 Pa and 160 Pa, respectively. (c) Decreasing solution pH for -${\mathrm{CO}}_2\mathrm{H}$ coated particles dramatically increases microscopic constraints. Experimental data from Ref. [28]. Here, ${\sigma }_{\mathrm{on}}$ is 3500 (pH = 7.1), 1925 (6.2), 696.6 (5.7), and 96.3 (5.1) Pa.