Recent studies have highlighted that oscillatory and time-dependent shear flows might help increase flowability of dense suspensions. While most focus has been on cross-flows we here study a simple two-dimensional suspensions where we apply simultaneously oscillatory and stationary shear along the same direction. We first show that the viscosities in this set-up significantly decrease with an increasing magnitude of the oscillations, contrary to previous claims. The decrease can be attributed to the large decrease in the number of contacts and an altered microstructure as the magnitude of the oscillation is increased. As a sub-result we find both an extension to the $μ(J)$-rheology, a constitutive relationship between the shear stresses and the shear rate, valid for pure oscillatory flows and with a higher shear-jammed packing fraction for suspensions composed of frictional particles compared to steady-shear conditions.
This is a comment on the recent letter by Wang and Brady on "Constant stress and pressure rheology of colloidal suspensions", Phys. Rev. Lett. 115, 158301 (2015).
Suspensions of cornstarch in water exhibit strong dynamic shear-thickening. We show that partly replacing water by ethanol strongly alters the suspension rheology. We perform steady and non-steady rheology measurements combined with atomic force microscopy to investigate the role of fluid chemistry on the macroscopic rheology of the suspensions and its link with the interactions between cornstarch grains. Upon increasing the ethanol content, the suspension goes through a yield-stress fluid state and ultimately becomes a shear-thinning fluid. On the cornstarch grain scale, atomic force microsco…
We probe the three dimensional flow structure and rheology of gravitational (non-density matched) suspensions for a range of driving rates in a split-bottom geometry. We establish that for sufficiently slow flows, the suspension flows as if it were a dry granular medium, and confirm recent theoretical modeling on the rheology of split-bottom flows. For faster driving, the flow behavior is shown to be consistent with the rheological behavior predicted by the recently developed "inertial number approaches for suspension flows.
Using a novel bi-axial confocal rheoscope, we investigate the structure and rheology of sheared colloidal suspensions under confinement. Consistent with previous work [X. Cheng \textit{et al., Proc. Natl. Acad. Sci. U. S. A.}, 2011, 109, 63], we observe a vorticity-aligned string phase in moderate concentrated colloidal suspensions under uniaxial shear. Using bi-axial shear protocols, we directly manipulate the orientation and morphology of the string structures. Simultaneously, we measure the suspension rheology along both the flow and vorticity directions with a bi-axial force measurement de…
The dynamics and rheology of suspensions of fluid vesicles or red blood cells is investigated by a combination of molecular dynamics and mesoscale hydrodynamics simulations in two dimensions. The vesicle suspension is confined between two no-slip walls, which are driven externally to generate a shear flow with shear rate $\dotγ$. The flow behavior is studied as a function of $\dotγ$, the volume fraction of vesicles, and the viscosity contrast between inside and outside fluids. Results are obtained for the encounter and interactions of two vesicles, the intrinsic viscosity of the suspension, an…
We present an experimental study of the rheology and structure of a confined suspension of deformable particles flowing in a quasi-two-dimensional Poiseuille flow. Thanks to a precise microfluidic viscosimetry technique combined with measurements of concentration profiles, our study provides the first experimental confirmation {with three-dimensional particles} of a strong relationship between structuring effects and rheology, previously only reported in numerical simulations of purely two-dimensional systems. In conditions where strong structuring effects take place due to confinement, the ev…
The rheology of suspensions of Brownian, or colloidal, particles (diameter $d \lesssim 1$ $μ$m) differs markedly from that of larger grains ($d \gtrsim 50$ $μ$m). Each of these two regimes has been separately studied, but the flow of suspensions with intermediate particle sizes (1 $μ\textrm{m} \lesssim d \lesssim 50$ $μ$m), which occur ubiquitously in applications, remains poorly understood. By measuring the rheology of suspensions of hard spheres with a wide range of sizes, we show experimentally that shear thickening drives the transition from colloidal to granular flow across the intermedia…
A first principles approach to the nonlinear flow of dense suspensions is presented which captures shear thinning of colloidal fluids and dynamical yielding of colloidal glasses. The advection of density fluctuations plays a central role, suppressing the caging of particles and speeding up structural relaxation. A mode coupling approach is developed to explore these effects.
We study the rheology of confined suspensions of neutrally buoyant rigid monodisperse spheres in plane-Couette flow using Direct Numerical Simulations. We find that if the width of the channel is a (small) integer multiple of the sphere's diameter, the spheres self-organize into two-dimensional layers that slide on each other and the suspension's effective viscosity is significantly reduced. Each two-dimensional layer is found to be structurally liquid-like but their dynamics is frozen in time.
We consider active suspensions in the isotropic phase subjected to a shear flow. Using a set of extended hydrodynamic equations we derive a variety of {\em analytical} expressions for rheological quantities such as shear viscosity and normal stress differences. In agreement to full-blown numerical calculations and experiments we find a shear thickening or -thinning behaviour depending on whether the particles are contractile or extensile. Moreover, our analytical approach predicts that the normal stress differences can change their sign in contrast to passive suspensions.
Results on the rheological behavior of novel CHO cell suspensions in a large range of concentrations are reported. The concentration dependent yield stress and elastic plateau modulus are formalized in the context of fractal aggregates under shear, and quite different exponents are found as compared to the case of red blood cell suspensions. This is explained in terms of intrinsic microscopic parameters such as the cell-cell adhesion energy and cell elasticity but also the cell individual dynamic properties, found to correlate well with viscoelastic data at large concentrations (phi>0.5).
We study relaxation and rheology of dense athermal suspensions of frictionless particles close below the jamming density. Our key quantity, the relaxation time---determined from the exponential decay of the energy after the shearing has suddenly been switched off---is argued to be a determining factor behind the algebraic divergence of various quantities as the jamming density is approached from below. We also define and measure the ``dissipation time'', which is obtained directly in shearing simulations and find that it behaves similarly to the relaxation time. Comparing shear viscosity with …
Dense suspensions of hard particles in a Newtonian liquid can be jammed by shear when the applied stress exceeds a certain threshold. However, this jamming transition from a fluid into a solidified state cannot be probed with conventional steady-state rheology because the stress distribution inside the material cannot be controlled with sufficient precision. Here we introduce and validate a method that overcomes this obstacle. Rapidly propagating shear fronts are generated and used to establish well-controlled local stress conditions that sweep across the material. Exploiting such transient fl…
We provide a detailed derivation of a recently developed first-principles approach to calculating averages in systems of interacting, spherical Brownian particles under time-dependent flow. Although we restrict ourselves to flows which are both homogeneous and incompressible, the time-dependence and geometry (e.g. shear, extension) are arbitrary. The approximations formulated within mode-coupling theory are particularly suited to dense colloidal suspensions and capture the slow relaxation arising from particle interactions and the resulting glass transition to an amorphous solid. The delicate …
We explore the rheology predicted by a recently proposed constitutive model for jammed suspensions of soft elastic particles derived from particle-level dynamics [Cuny et al., Phys. Rev. Lett. 127, 218003 (2021)]. Our model predicts that the orientation of the anisotropy of the microstructure, governed by an interplay between advection and contact elasticity, plays a key role at yielding and in flow. It generates normal stress differences contributing significantly to the yield criterion and Trouton ratio. It gives rise to non-trivial transients such as stress overshoots in step increases of s…
Shear flow of dense, non-Brownian suspensions is simulated using the discrete element method, taking particle contact and hydrodynamic lubrication into account. The resulting flow regimes are mapped in the parametric space of solid volume fraction, shear rate, fluid viscosity and particle stiffness. Below a critical volume fraction $ϕ_c$, the rheology is governed by the Stokes number, which distinguishes between viscous and inertial flow regimes. Above $ϕ_c$, a quasistatic regime exists for low and moderate shear rates. At very high shear rates, the $ϕ$ dependence is lost and soft particle rhe…
Oscillatory shear has been widely used to study the rheological properties of suspensions under unsteady shear. Furthermore, recent works have shown that oscillatory flows can improve the flowability of dense suspensions. While most studies have been done under constant volume we here study oscillatory shear flows of a two-dimensional suspensions using a normal pressure-controlled set-up. To characterise the rheology, we introduce both a complex macroscopic friction coefficient $μ^{*}$, following the convention of the complex viscosity $η^{*}$, and a shear-rate averaged viscous number $J'$. Th…
Dense suspensions of soft colloidal particles display a broad range of physical and rheological properties which are still far from being fully understood. To elucidate the role of deformability on colloidal flow, we employ computer simulations to measure the apparent viscosity of a system of droplets of variable surface tension subjected to a pressure-driven flow. We confirm that our suspension generically undergoes discontinuous shear thinning, and determine the dependence of the onset of the discontinuity on surface tension. We find that the effective viscosity of the suspension is mainly d…
Using a simple model of self-propelled particle, the effective shear viscosity of a dilute, spatially homogeneous suspension of active particles is studied. We use formulation of non-linear Fokker-Planck equation to drive a kinetic description, including the effect of external flow field, rotary diffusion, and particle-particle hydrodynamic interactions in two dimensions. Analytical expressions are obtained for the stress tensor at small Peclet numbers, in a simple shear flow, up to second order of volume fraction of the swimmers, which shows the explicit dependence of shear viscosity on the d…
We study granular suspensions under a variety of extensional deformations and simple shear using numerical simulations. The viscosity and Trouton's ratio (the ratio of extensional to shear viscosity) are computed as functions of solids volume fraction $ϕ$ close to the limit of zero inertia. Suspensions of frictionless particles follow a Newtonian Trouton's ratio for $ϕ$ all the way up to $ϕ_0$, a universal jamming point that is independent of deformation type. In contrast, frictional particles lead to a deformation-type-dependent jamming fraction $ϕ_m$, which is largest for shear flows. Trouto…
At the same solid volume fraction, bidisperse and polydisperse suspensions display lower viscosities, and weaker normal stress response, compared to monodisperse suspensions. The reduction of viscosity associated with size distribution can be explained by an increase of the maximum flowable, or jamming, solid fraction $ϕ_m$. In this work, dense suspensions are simulated under strong shearing, where thermal motion and repulsive forces are negligible, but we allow for particle contact with a mild frictional interaction with interparticle friction coefficient of $μ = 0.2$. Aspects of bidisperse s…
Many colloidal systems display very non-Newtonian and solid-like behaviour when concentrated, a striking feature being the apparition of a yield stress. After recalling some basics about the interactions between colloidal particles, I present a few experimental facts commonly observed in these systems: aging and non-linear rheology. A simple phenomenological model is then introduced, in which the local state of the system is described by a single scalar parameter, the fluidity. I proceed with comments on heterogeneous flows in some of these systems. These notes are not intended to be a compreh…
Vesicles under shear flow exhibit various dynamics: tank-treading ($tt$), tumbling ($tb$) and vacillating-breathing ($vb$). A consistent higher order theory reveals a direct bifurcation from $tt$ to $tb$ if $C_a\equiv τ\dotγ$ is small enough ($τ$= vesicle relaxation time towards equilibrium shape, $\dotγ$=shear rate). At larger $C_a$ the $tb$ is preceded by the $vb$ mode. For $C_a\gg 1$ we recover the leading order original calculation, where the $vb$ mode coexists with $tb$. The consistent calculation reveals several quantitative discrepancies with recent works, and points to new features. We…
We use large scale computer simulations and finite size scaling analysis to study the shear rheology of dense three-dimensional suspensions of frictionless non-Brownian particles in the vicinity of the jamming transition. We perform simulations of soft repulsive particles at constant shear rate, constant pressure, and finite system size, and study carefully the asymptotic limits of large system sizes and infinitely hard particle repulsion. Extending earlier analysis by about two orders of magnitude, we first study the asymptotic behavior of the shear viscosity in the hard particle limit. We co…
We probe non-equilibrium properties of an active bacterial bath through measurements of correlations of passive tracer particles and the response function of a driven, optically trapped tracer. These measurements demonstrate violation of the fluctuation-dissipation theorem and enable us to extract the power spectrum of the active stress fluctuations. In some cases, we observe $1/\sqrtω$ scaling in the noise spectrum which we show can be derived from a theoretical model incorporating coupled stress, orientation, and concentration fluctuations of the bacteria.
We investigate the nonlinear rheological behavior of colloidal suspensions of Laponite, a synthetic clay, driven by a steady and homogeneous shear strain. We show that the external drive leads to a drastic slowing down of the aging dynamics or even, in some cases, in the rejuvenation of the system. Under shear, after a surprisingly long time, the spontaneous aging process observed at rest is suppressed. The system then reaches a non-equilibrium stationary state, characterised by a complex viscosity depending on the applied shear rate. In addition, the glass exhibits a non-Newtonian shear-thinn…
Dense suspensions are widespread in nature, manufacturing and process engineering. Particle-based simulations have proven to be an invaluable complement to experimental rheological characterisation, serving as a virtual rheometer that enables rapid exploration of parameter space and detailed scrutiny of microscopic dynamics. To maximise the utility of such simulations, it can be advantageous to exploit pre-existing, well-optimised, well-documented codes. Here we provide a simple description of how to use LAMMPS to study the rheology of dense, granular suspensions.
We study the steady-state response to applied stress in a simple scalar model of sheared colloids. Our model is based on a schematic (F2) model of the glass transition, with a memory term that depends on both stress and shear rate. For suitable parameters, we find transitions from a fluid to a nonergodic, jammed state, showing zero flow rate in an interval of applied stress. Although the jammed state is a glass, we predict that jamming transitions have an analytical structure distinct from that of the conventional mode coupling glass transition. The static jamming transition we discuss is also…
We report a detailed experimental study of peculiar thixotropic dynamics of carbon black (CB, Vulcan XC-72) suspensions in mineral oil, specifically the observation of sequential stress increase then decrease at a fixed shear rate in a step-down test. We verify that such dynamics, though peculiar, come from a true material response rather than experimental artifacts. We also reveal how this long-time stress decay is associated with anti-thixotropy, rather than viscoelasticity, by using orthogonal superposition (OSP) rheometry to probe viscoelastic moduli during the step-down tests. The orthogo…
We study the rheological behavior of concentrated granular suspensions of simple spherical particles. Under controlled stress, the system exhibits an S-shaped flow curve (stress vs. shear rate) with a negative slope in between the low-viscosity Newtonian regime and the shear thickened regime. Under controlled shear rate, a discontinuous transition between the two states is observed. Stress visualization experiments with a novel fluorescent probe suggest that friction is at the origin of shear thickening. Stress visualization shows that the stress in the system remains homogeneous (no shear ban…
A systematic microscopic theory for the rheology of dense non-Brownian suspensions characterized by the volume fraction $φ$ is developed. The theory successfully derives the critical behavior in the vicinity of the jamming point (volume fraction $φ_{J}$), for both the pressure $P$ and the shear stress $σ_{xy}$, i.e. $P \sim σ_{xy} \sim \dotγη_0 δφ^{-2}$, where $\dotγ$ is the shear rate, $η_0$ is the shear viscosity of the solvent, and $δφ= φ_J - φ> 0$ is the distance from the jamming point. It also successfully describes the behavior of the stress ratio $μ= σ_{xy}/P$ with respect to the viscou…
We propose an improved effective-medium theory to obtain the concentration dependence of the viscosity of particle suspensions at arbitrary volume fractions. Our methodology can be applied, in principle, to any particle shape as long as the intrinsic viscosity is known in the dilute limit and the particles are not too elongated. The procedure allows to construct a continuum-medium model in which correlations between the particles are introduced through an effective volume fraction. We have tested the procedure using spheres, ellipsoids, cylinders, dumbells, and other complex shapes. In the cas…
In everyday life we encounter many complex fluids, from shear-thinning paint and toothpaste to shear-thickening starch suspensions. The study of their properties offers an opportunity for students to relate sophisticated physical concepts to their everyday experience. Modern rheology uses expensive equipment impractical for the teaching laboratory. Here we describe a rudimentary rheometer suitable for student laboratories that can demonstrate and quantify discontinuous shear thickening, the most dramatic property of complex fluids, and use it to measure the properties of starch suspensions.
Mud is a suspension of fine-grained particles (sand, silt, and clay) in water. The interaction of clay minerals in mud gives rise to complex rheological behaviors, such as yield stress, thixotropy and viscoelasticity. Here, we experimentally examine the flow behaviors of kaolinite clay suspensions, a model mud, using steady shear rheometry. The flow curves exhibit both yield stress and rheological hysteresis behaviors for various kaolinite volume fractions ($ϕ_k$). Further understanding of these behaviors requires fitting to existing constitutive models, which is challenging due to numerous fi…
The linear and nonlinear rheological behavior of two rod-like particle suspensions as a function of concentration is studied using small amplitude oscillatory shear, steady shear and capillary breakup extensional rheometry. The rod-like suspensions are composed of fd virus and its mutant fdY21M, which are perfectly monodisperse, with a length on the order of 900 nm. The particles are semi flexible yet differ in their persistence length. The effect of stiffness on the rheological behavior in both, shear and extensional flow, is investigated experimentally. The linear viscoelastic shear data is …
We show that particle roughness leads to changes in the number, shape and resulting capillary force of liquid bridges in capillary suspensions. We created fluorescently labeled, raspberry-like particles with varying roughness by electrostatically adsorbing silica nanoparticles with sizes between 40 nm and 250 nm on silica microparticles. Rougher particles require more liquid to fill the surface asperities before they form pendular bridges, resulting in smaller and weaker bridges. In a system where the effective bridge volume is adjusted, higher particle roughness leads to less clustered networ…
We investigate in detail the problem of confined pressure-driven laminar flow of neutrally buoyant non-Brownian suspensions using a frictional rheology based on the recent proposal of Boyer et al., 2011. The friction coefficient and solid volume fraction are taken as functions of the dimensionless viscous number I defined as the ratio between the fluid shear stress and the particle normal stress. We clarify the contributions of the contact and hydrodynamic interactions on the evolution of the friction coefficient between the dilute and dense regimes reducing the phenomenological constitutive d…
We show that Hertzian particle contacts are the underlying cause of the as-yet-unexplained noninteger power laws in weakly nonlinear rheology. In the medium amplitude oscillatory shear (MAOS) region, the cubic scaling of the leading order nonlinear shear stress ($σ_\mathrm{3} \sim γ_\mathrm{0}^{m_\mathrm{3}}$, $m_\mathrm{3}=3$) is the standard expectation. Expanding on the work by Natalia et al. [J. Rheol. 64 625-635 (2020)], we report an extensive data set of noncubical, noninteger power law scalings $m_\mathrm{3}$ for particle suspensions in two immiscible fluids with a capillary attractive …
Using a combination of theory, experiment and simulation we investigate the nonlinear response of dense colloidal suspensions to large amplitude oscillatory shear flow. The time-dependent stress response is calculated using a recently developed schematic mode-coupling-type theory describing colloidal suspensions under externally applied flow. For finite strain amplitudes the theory generates a nonlinear response, characterized by significant higher harmonic contributions. An important feature of the theory is the prediction of an ideal glass transition at sufficiently strong coupling, which is…
As a paradigmatic model of active fluids, bacterial suspensions show intriguing rheological responses drastically different from their counterpart colloidal suspensions. Although the flow of bulk bacterial suspensions has been extensively studied, the rheology of bacterial suspensions under confinement has not been experimentally explored. Here, using a microfluidic viscometer, we systematically measure the rheology of dilute E. coli suspensions under different degrees of confinement. Our study reveals a strong confinement effect: the viscosity of bacterial suspensions decreases substantially …
We present a phenomenological model for granular suspension rheology in which particle interactions enter as constraints to relative particle motion. By considering constraints that are formed and released by stress respectively, we derive a range of experimental flow curves in a single treatment and predict singularities in viscosity and yield stress consistent with literature data. Fundamentally, we offer a generic description of suspension flow that is independent of bespoke microphysics.
The shear-jamming of dense suspensions can be strongly affected by molecular-scale interactions between particles, e.g. by chemically controlling their propensity for hydrogen bonding. However, hydrogen bonding not only enhances interparticle friction, a critical parameter for shear jamming, but also introduces (reversible) adhesion, whose interplay with friction in shear-jamming systems has so far remained unclear. Here, we present atomic force microscopy studies to assess interparticle adhesion, its relationship to friction, and how these attributes are influenced by urea, a molecule that in…
We introduce a class of simple models for shear thickening and/ or `jamming' in colloidal suspensions. These are based on schematic mode coupling theory (MCT) of the glass transition, having a memory term that depends on a density variable, and on both the shear stress and the shear rate. (Tensorial aspects of the rheology, such as normal stresses, are ignored for simplicity.) We calculate steady-state flow curves and correlation functions. Depending on model parameters, we find a range of rheological behaviours, including `S-shaped' flow curves, indicating discontinuous shear thickening, and …
We study the jamming transition in a model of elastic particles under shear at zero temperature. The key quantity is the relaxation time $τ$ which is obtained by stopping the shearing and letting energy and pressure decay to zero. At many different densities and initial shear rates we do several such relaxations to determine the average $τ$. We establish that $τ$ diverges with the same exponent as the viscosity and determine another exponent from the relation between $τ$ and the coordination number. Though most of the simulations are done for the model with dissipation due to the motion of par…
We study the interplay of activity, order and flow through a set of coarse-grained equations governing the hydrodynamic velocity, concentration and stress fields in a suspension of active, energy-dissipating particles. We make several predictions for the rheology of such systems, which can be tested on bacterial suspensions, cell extracts with motors and filaments, or artificial machines in a fluid. The phenomena of cytoplasmic streaming, elastotaxis and active mechanosensing find natural explanations within our model.
We develop a constitutive model allowing for the description of the rheology of two-dimensional soft dense suspensions above jamming. Starting from a statistical description of the particle dynamics, we derive, using a set of approximations, a non-linear tensorial evolution equation linking the deviatoric part of the stress tensor to the strain-rate and vorticity tensors. The coefficients appearing in this equation can be expressed in terms of the packing fraction and of particle-level parameters. This constitutive equation rooted in the microscopic dynamic qualitatively reproduces a number of…
Recent advances in shear-thickening suspension rheology suggest a relation between (wet) suspension flow below jamming and (dry) granular physics. To probe this connection, we simulated the contact force networks in suspensions of non-Brownian spheres using the discrete element method (DEM), varying the particle friction coefficient and volume fraction. We find that force networks in these suspensions show quantitative similarities to those in jammed dry grains. As suspensions approach the jamming point, the extrapolated volume fraction and coordination number at jamming are similar to critica…
