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 a 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 nano-scale properties.

Shear thickening and jamming of dense suspensions: the roll of friction

Particle-based simulations of discontinuous shear thickening (DST) and shear jamming (SJ) suspensions are used to study the role of stress-activated constraints, with an emphasis on resistance to gear-like rolling. Rolling friction decreases the volume fraction required for DST and SJ, in quantitative agreement with real-life suspensions with adhesive surface chemistries and rough particle shapes. It sets a distinct structure of the frictional force network compared to only sliding friction, and from a dynamical perspective leads to an increase in the velocity correlation length, in part responsible for the increased viscosity. The physics of rolling friction is thus a key element in achieving a comprehensive understanding of strongly shear-thickening materials.

Structure and dynamics of hydrodynamically interacting finite-size Brownian particles in a spherical cavity: spheres and cylinders

The structure and dynamics of confined suspensions of particles of arbitrary shape is of interest in multiple disciplines, from biology to engineering. Theoretical studies are often limited by the complexity of long-range particle-particle and particle-wall forces, including many-body fluctuating hydrodynamic interactions. Here, we report a computational study on the diffusion of spherical and cylindrical particles confined in a spherical cavity. We rely on an Immersed-Boundary General geometry Ewald-like method to capture lubrication and long-range hydrodynamics, and include appropriate non-slip conditions at the confining walls. A Chebyshev polynomial approximation is used to satisfy the fluctuation-dissipation theorem for the Brownian suspension. We explore how lubrication, long-range hydrodynamics, particle volume fraction and shape affect the equilibrium structure and the diffusion of the particles. It is found that once the particle volume fraction is greater than $10%$, the particles start to form layered aggregates that greatly influence particle dynamics. Hydrodynamic interactions strongly influence the particle diffusion by inducing spatially dependent short-time diffusion coefficients, stronger wall effects on the particle diffusion towards the walls, and a sub-diffusive regime --caused by crowding-- in the long-time particle mobility. The level of asymmetry of the cylindrical particles considered here is enough to induce an orientational order in the layered structure, decreasing the diffusion rate and facilitating a transition to the crowded mobility regime at low particle concentrations. Our results offer fundamental insights into the diffusion and distribution of globular and fibrillar proteins inside cells.

Shape induced segregation and anomalous particle transport under spherical confinement

Colloid or nanoparticle mobility under confinement is of central importance to a wide range of physical and biological processes. Here, we introduce a minimal model of particles in a hydrodynamic continuum to examine how particle shape and concentration affect the transport of particles in spherical confinement. Specifically, an immersed boundary-General geometry Ewald-like approach is adopted to simulate the dynamics of spheres and cylinders under the influence of short-and long-range fluctuating hydrodynamic interactions with appropriate non-slip conditions at the confining walls. An efficient $it{O(N)}$ parallel finite element algorithm is used, thereby allowing simulations at high concentrations, while a Chebyshev polynomial approximation is implemented in order to satisfy the fluctuation-dissipation theorem. A concentration-dependent anomalous diffusion is observed for suspended particles. It is found that introducing cylinders in a background of spheres, i.e. particles with a simple degree of anisotropy, has a pronounced influence on the structure and dynamics of the particles. First, increasing the fraction of cylinders induces a particle segregation effect, where spheres are pushed towards the wall and cylinders remain near the center of the cavity. This segregation leads to lower mobility for the spheres relative to that encountered in a system of pure spheres at the same volume fraction. Second, the diffusive-to-anomalous transition and the degree of anomaly--quantified by the power-law exponent in the mean square displacement vs. time relation-both increase as the fraction of cylinders becomes larger. These findings are of relevance for studies of diffusion in the cytoplasm, where proteins exhibit a distribution of size and shapes that could lead to some of the effects identified in the simulations reported here.

Stress fluctuations and shear thickening in dense granular suspensions

We experimentally investigate the rheology and stress fluctuations of granules densely suspended in silicone oil. We find that both thickening strength and stress fluctuations significantly weaken with oil viscosity $eta_0$. Comparison of our rheological results to the Wyart-Cates model for describing different dynamic jamming states suggests a transition from frictional contacts to lubrication interactions as $eta_0$ increases. To clarify the contribution from viscous interactions to the rheology, we systematically measure stress fluctuations in various flow states. Reduction of stress fluctuations with $eta_0$ indicates that a strong lubrication layer greatly inhibits force correlations among particles. Measuring stress fluctuations in the strong shear thickening regime, we observe a crossover from asymmetric Gamma to symmetric Gaussian distributions and associated with it a decrease of lateral (radial) correlation length $xi$ with increasing shear rate.

Interaction network analysis in shear thickening suspensions

Dense, stabilized, frictional particulate suspensions in a viscous liquid undergo increasingly strong continuous shear thickening (CST) as the solid packing fraction, $phi$, increases above a critical volume fraction, and discontinuous shear thickening (DST) is observed for even higher packing fractions. Recent studies have related shear thickening to a transition from mostly lubricated to predominantly frictional contacts with the increase in stress. The rheology and networks of frictional forces from two and three-dimensional simulations of shear-thickening suspensions are studied. These are analyzed using measures of the topology of the network, including tools of persistent homology. We observe that at low stress the frictional interaction networks are predominantly quasi-linear along the compression axis. With an increase in stress, the force networks become more isotropic, forming loops in addition to chain-like structures. The topological measures of Betti numbers and total persistence provide a compact means of describing the mean properties of the frictional force networks and provide a key link between macroscopic rheology and the microscopic interactions. A total persistence measure describing the significance of loops in the force network structure, as a function of stress and packing fraction, shows behavior similar to that of relative viscosity and displays a scaling law near the jamming fraction for both dimensionalities simulated.

Shear jamming and fragility in dense suspensions

The phenomenon of shear-induced jamming is a factor in the complex rheological behavior of dense suspensions. Such shear-jammed states are fragile, i.e., they are not stable against applied stresses that are incompatible with the stress imposed to create them. This peculiar flow-history dependence of the stress response is due to flow-induced microstructures. To examine jammed states realized under constant shear stress, we perform dynamic simulations of non-Brownian particles with frictional contact forces and hydrodynamic lubrication forces. We find clear signatures that distinguish these fragile states from the more conventional isotropic jammed states.

From yielding to shear jamming in a cohesive frictional suspension

Simulations are used to study the steady shear rheology of dense suspensions of frictional particles exhibiting discontinuous shear thickening and shear jamming, in which finite-range cohesive interactions result in a yield stress. We develop a constitutive model that combines yielding behavior and shear thinning at low stress with the frictional shear thickening at high stresses, in good agreement with the simulation results. This work shows that there is a distinct difference between solids below the yield stress and in the shear-jammed state, as the two occur at widely separated stress levels, separated by a region of stress in which the material is flowable.

Microscopic origin of frictional rheology in dense suspensions: correlations in force space

We develop a statistical framework for the rheology of dense, non-Brownian suspensions, based on correlations in a space representing forces, which is dual to position space. Working with the ensemble of steady state configurations obtained from simulations of suspensions in two dimensions, we find that the anisotropy of the pair correlation function in force space changes with confining shear stress ($sigma_{xy}$) and packing fraction ($phi$). Using these microscopic correlations, we build a statistical theory for the macroscopic friction coefficient: the anisotropy of the stress tensor, $mu = sigma_{xy}/P$. We find that $mu$ decreases (i) as $phi$ is increased and (ii) as $sigma_{xy}$ is increased. Using a new constitutive relation between $mu$ and viscosity for dense suspensions that generalizes the rate-independent one, we show that our theory predicts a Discontinuous Shear Thickening (DST) flow diagram that is in good agreement with numerical simulations, and the qualitative features of $mu$ that lead to the generic flow diagram of a DST fluid observed in experiments.

A constitutive model for simple shear of dense frictional suspensions

Discrete particle simulations are used to study the shear rheology of dense, stabilized, frictional particulate suspensions in a viscous liquid, toward development of a constitutive model for steady shear flows at arbitrary stress. These suspensions undergo increasingly strong continuous shear thickening (CST) as solid volume fraction $phi$ increases above a critical volume fraction, and discontinuous shear thickening (DST) is observed for a range of $phi$. When studied at controlled stress, the DST behavior is associated with non-monotonic flow curves of the steady-state stress as a function of shear rate. Recent studies have related shear thickening to a transition between mostly lubricated to predominantly frictional contacts with the increase in stress. In this study, the behavior is simulated over a wide range of the dimensionless parameters $(phi,tilde{sigma}$, and $mu)$, with $tilde{sigma} = sigma/sigma_0$ the dimensionless shear stress and $mu$ the coefficient of interparticle friction: the dimensional stress is $sigma$, and $sigma_0 propto F_0/ a^2$, where $F_0$ is the magnitude of repulsive force at contact and $a$ is the particle radius. The data have been used to populate the model of the lubricated-to-frictional rheology of Wyart and Cates [Phys. Rev. Lett.{bf 112}, 098302 (2014)], which is based on the concept of two viscosity divergences or textquotedblleft jammingtextquotedblright points at volume fraction $phi_{rm J}^0 = phi_{rm rcp}$ (random close packing) for the low-stress lubricated state, and at $phi_{rm J} (mu) < phi_{rm J}^0$ for any nonzero $mu$ in the frictional state; a generalization provides the normal stress response as well as the shear stress. A flow state map of this material is developed based on the simulation results.