Slow Coarsening in Jammed Athermal Soft Particle Suspensions.

We simulate a densely jammed, athermal assembly of repulsive soft particles immersed in a solvent. Starting from an initial condition corresponding to a quench from a high temperature, we find nontrivial slow dynamics driven by a gradual release of stored elastic energy, with the root mean squared particle speed decaying as a power law in time with a fractional exponent. This decay is accompanied by the presence within the assembly of spatially localized and temporally intermittent "hot spots" of nonaffine deformation, connected by long-ranged swirls in the velocity field, reminiscent of the local plastic events and long-ranged elastic propagation that have been intensively studied in sheared amorphous materials. The pattern of hot spots progressively coarsens, with the hot-spot size and separation slowly growing over time, and the associated correlation length in particle speed increasing as a sublinear power law. Each individual spot, however, exists only transiently within an overall picture of strongly intermittent dynamics.

Dynamic Vorticity Banding in Discontinuously Shear Thickening Suspensions.

It has recently been argued that steady-state vorticity bands cannot arise in shear thickening suspensions because the normal stress imbalance across the interface between the bands will set up particle migrations. In this Letter, we develop a simple continuum model that couples shear thickening to particle migration. We show by linear stability analysis that homogeneous flow is unstable towards vorticity banding, as expected, in the regime of negative constitutive slope. In full nonlinear computations, we show, however, that the resulting vorticity bands are unsteady, with spatiotemporal patterns governed by stress-concentration coupling. We furthermore show that these dynamical bands also arise in direct particle simulations, in good agreement with the continuum model.

Viscoelastic and elastomeric active matter: Linear instability and nonlinear dynamics.

We consider a continuum model of active viscoelastic matter, whereby an active nematic liquid crystal is coupled to a minimal model of polymer dynamics with a viscoelastic relaxation time τ(C). To explore the resulting interplay between active and polymeric dynamics, we first generalize a linear stability analysis (from earlier studies without polymer) to derive criteria for the onset of spontaneous heterogeneous flows (strain rate) and/or deformations (strain). We find two modes of instability. The first is a viscous mode, associated with strain rate perturbations. It dominates for relatively small values of τ(C) and is a simple generalization of the instability known previously without polymer. The second is an elastomeric mode, associated with strain perturbations, which dominates at large τ(C) and persists even as τ(C)→∞. We explore the dynamical states to which these instabilities lead by means of direct numerical simulations. These reveal oscillatory shear-banded states in one dimension and activity-driven turbulence in two dimensions even in the elastomeric limit τ(C)→∞. Adding polymer can also have calming effects, increasing the net throughput of spontaneous flow along a channel in a type of drag reduction. The effect of including strong antagonistic coupling between the nematic and polymer is examined numerically, revealing a rich array of spontaneously flowing states.

Active viscoelastic matter: from bacterial drag reduction to turbulent solids.

A paradigm for internally driven matter is the active nematic liquid crystal, whereby the equations of a conventional nematic are supplemented by a minimal active stress that violates time-reversal symmetry. In practice, active fluids may have not only liquid-crystalline but also viscoelastic polymer degrees of freedom. Here we explore the resulting interplay by coupling an active nematic to a minimal model of polymer rheology. We find that adding a polymer can greatly increase the complexity of spontaneous flow, but can also have calming effects, thereby increasing the net throughput of spontaneous flow along a pipe (a "drag-reduction" effect). Remarkably, active turbulence can also arise after switching on activity in a sufficiently soft elastomeric solid.

Shear banding in soft glassy materials.

Many soft materials, including microgels, dense colloidal emulsions, star polymers, dense packings of multilamellar vesicles, and textured morphologies of liquid crystals, share the basic 'glassy' features of structural disorder and metastability. These in turn give rise to several notable features in the low frequency shear rheology (deformation and flow properties) of these materials: in particular, the existence of a yield stress below which the material behaves like a solid, and above which it flows like a liquid. In the last decade, intense experimental activity has also revealed that these materials often display a phenomenon known as shear banding, in which the flow profile across the shear cell exhibits macroscopic bands of different viscosity. Two distinct classes of yield stress fluid have been identified: those in which the shear bands apparently persist permanently (for as long as the flow remains applied), and those in which banding arises only transiently during a process in which a steady flowing state is established out of an initial rest state (for example, in a shear startup or step stress experiment). Despite being technically transient, such bands may in practice persist for a very long time and so be mistaken for the true steady state response of the material in experimental practice. After surveying the motivating experimental data, we describe recent progress in addressing it theoretically, using the soft glassy rheology model and a simple fluidity model. We also briefly place these theoretical approaches in the context of others in the literature, including elasto-plastic models, shear transformation zone theories, and molecular dynamics simulations. We discuss finally some challenges that remain open to theory and experiment alike.

Modeling the relaxation of polymer glasses under shear and elongational loads.

Glassy polymers show "strain hardening": at constant extensional load, their flow first accelerates, then arrests. Recent experiments under such loading have found this to be accompanied by a striking dip in the segmental relaxation time. This can be explained by a minimal nonfactorable model combining flow-induced melting of a glass with the buildup of stress carried by strained polymers. Within this model, liquefaction of segmental motion permits strong flow that creates polymer-borne stress, slowing the deformation enough for the segmental (or solvent) modes then to re-vitrify. Here, we present new results for the corresponding behavior under step-stress shear loading, to which very similar physics applies. To explain the unloading behavior in the extensional case requires introduction of a "crinkle factor" describing a rapid loss of segmental ordering. We discuss in more detail here the physics of this, which we argue involves non-entropic contributions to the polymer stress, and which might lead to some important differences between shear and elongation. We also discuss some fundamental and possibly testable issues concerning the physical meaning of entropic elasticity in vitrified polymers. Finally, we present new results for the startup of steady shear flow, addressing the possible role of transient shear banding.

Simple model for the deformation-induced relaxation of glassy polymers.

Glassy polymers show "strain hardening": at constant extensional load, their flow first accelerates, then arrests. Recent experiments have found this to be accompanied by a striking and unexplained dip in the segmental relaxation time. Here we explain such behavior by combining a minimal model of flow-induced liquefaction of a glass with a description of the stress carried by strained polymers, creating a nonfactorable interplay between aging and strain-induced rejuvenation. Under constant load, liquefaction of segmental motion permits strong flow that creates polymer-borne stress. This slows the deformation enough for the segmental modes to revitrify, causing strain hardening.

Nonlinear dynamics and rheology of active fluids: simulations in two dimensions.

We report simulations of a continuum model for (apolar, flow aligning) active fluids in two dimensions. Both free and anchored boundary conditions are considered, at parallel confining walls that are either static or moving at fixed relative velocity. We focus on extensile materials and find that steady shear bands, previously shown to arise ubiquitously in one dimension for the active nematic phase at small (or indeed zero) shear rate, are generally replaced in two dimensions by more complex flow patterns that can be stationary, oscillatory, or apparently chaotic. The consequences of these flow patterns for time-averaged steady-state rheology are examined.

Interfacially driven instability in the microchannel flow of a shear-banding fluid.

Using microparticle image velocimetry, we resolve the spatial structure of the shear-banding flow of a wormlike micellar surfactant solution in a straight microchannel. We reveal an instability of the interface between the shear bands, associated with velocity modulations along the vorticity direction. We compare our results with a detailed theoretical study of the diffusive Johnson-Segalman model. The quantitative agreement obtained favors an instability scenario previously predicted theoretically but hitherto unobserved experimentally, driven by a normal stress jump across the interface between the bands.

Shearing active gels close to the isotropic-nematic transition.

We study numerically the rheological properties of a slab of active gel close to the isotropic-nematic transition. The flow behavior shows a strong dependence on the sample size, boundary conditions, and on the bulk constitutive curve, which, on entering the nematic phase, acquires an activity-induced discontinuity at the origin. The precursor of this within the metastable isotropic phase for contractile systems (e.g., actomyosin gels) gives a viscosity divergence; its counterpart for extensile suspensions admits instead a shear-banded flow with zero apparent viscosity.

Nonlinear dynamics of an interface between shear bands.

We study numerically the nonlinear dynamics of a shear banding interface in two-dimensional planar shear flow, within the nonlocal Johnson-Segalman model. Consistent with a recent linear stability analysis, we find that an initially flat interface is unstable with respect to small undulations for a sufficiently small ratio of the interfacial width l to cell length L(x). The instability saturates in finite amplitude interfacial fluctuations. For decreasing l/L(x) these undergo a nonequilibrium transition from simple traveling interfacial waves with constant average wall stress, to periodically rippling waves with a periodic stress response. When multiple shear bands are present we find erratic interfacial dynamics and a stress response suggesting low dimensional chaos.

Linear instability of planar shear banded flow.

We study the linear stability of planar shear banded flow with respect to perturbations with wave vector in the plane of the banding interface, within the nonlocal Johnson-Segalman model. We find that perturbations grow in time, over a range of wave vectors, rendering the interface linearly unstable. Results for the unstable eigenfunction are used to discuss the nature of the instability. We also comment on the stability of phase separated domains to shear flow in model H.

Spatiotemporal oscillations and rheochaos in a simple model of shear banding.

We study a simple model of shear banding in which the flow-induced phase is destabilized by coupling between flow and microstructure (wormlike micellar length). By varying the strength of instability and the applied shear rate, we find a rich variety of oscillatory and chaotic shear banded flows. At low shear and weak instability, the induced phase pulsates next to one wall of the flow cell. For stronger instability, high shear pulses ricochet across the cell. At high shear we see oscillating bands on either side of central defects. We discuss our results in the context of recent experiments.

Kinetics of the shear banding instability in startup flows.

Motivated by recent light scattering experiments on semidilute wormlike micelles, we study the early stages of the shear banding instability using the nonlocal Johnson-Segalman model with a "two-fluid" coupling of flow to micellar concentration. We perform a linear stability analysis for coupled fluctuations in shear rate gamma;, micellar strain W, and concentration phi about an initially homogeneous state. This resembles the Cahn-Hilliard (CH) analysis of fluid-fluid demixing (although we discuss important differences). First, assuming the initial state to lie on the intrinsic constitutive curve, we calculate the "spinodal" onset of instability in sweeps along this curve. We then consider start-up "quenches" into the unstable region. Here the instability in general occurs before the intrinsic constitutive curve can be attained, so we analyze the fluctuations with respect to the time-dependent start-up flow. We calculate the selected length and time scales at which inhomogeneity first emerges. When the coupling between flow and concentration is switched off, fluctuations in the "mechanical variables" gamma; and W are independent of those in phi, and are unstable when the intrinsic constitutive curve has negative slope; but no length scale is selected. Coupling to the concentration enhances this instability at short length scales, thereby selecting a length scale, consistent with the recent light scattering experiments. The spinodal region is then broadened by an extent that increases with proximity to an underlying (zero-shear) CH fluid-fluid (phi) demixing instability. Far from demixing, the broadening is slight and the instability is still mechanically dominated (by deltagamma; and deltaW) with only small deltaphi. Close to demixing, instability sets in at a very low shear rate, where it is dominated instead by deltaphi. In this way, the model captures a smooth crossover from shear banding instabilities that are perturbed by concentration coupling to demixing instabilities that are induced by shear.

Early stage kinetics in a unified model of shear-induced demixing and mechanical shear banding instabilities.

We present a unified model of shear-induced demixing and "mechanical" shear banding instabilities in polymeric and surfactant solutions, by combining a simple flow instability with a two-fluid approach to concentration fluctuations. Within this model, we calculate the "spinodal" limit of stability of initially homogeneous shear states to demixing/banding, and predict the selected length and time scales at which inhomogeneity first emerges after a shear start-up "quench" into the unstable region, finding qualitative agreement with experiment. Our analysis is the counterpart, for this driven phase transition, of the Cahn-Hilliard calculation for unsheared fluid-fluid demixing.

Equivalence of driven and aging fluctuation-dissipation relations in the trap model.

We study the nonequilibrium version of the fluctuation-dissipation (FD) relation in the glass phase of a trap model that is driven into a nonequilibrium steady state by external "shear." This extends our recent study of aging FD relations in the same model, where we found limiting, observable independent FD relations for "neutral" observables that are uncorrelated with the system's average energy. In this work, for such neutral observables, we find the FD relation for a stationary weakly driven system to be the same, to within small corrections, as for an infinitely aged system. We analyze the robustness of this correspondence with respect to non-neutrality of the observable, and with respect to changes in the driving mechanism.

Flow phase diagrams for concentration-coupled shear banding.

After surveying the experimental evidence for concentration coupling in the shear banding of wormlike micellar surfactant systems, we present flow phase diagrams spanned by shear stress Sigma (or strain rate gamma) and concentration, calculated within the two-fluid, non-local Johnson-Segalman (d-JS-phi) model. We also give results for the macroscopic flow curves Sigma(gamma,phi) for a range of (average) concentrations phi. For any concentration that is high enough to give shear banding, the flow curve shows the usual non-analytic kink at the onset of banding, followed by a coexistence "plateau" that slopes upwards, dSigma/dgamma>0. As the concentration is reduced, the width of the coexistence regime diminishes and eventually terminates at a non-equilibrium critical point [Sigmac,phic,gammac]. We outline the way in which the flow phase diagram can be reconstructed from a family of such flow curves, Sigma(gamma,phi), measured for several different values of phi. This reconstruction could be used to check new measurements of concentration differences between the coexisting bands. Our d-JS-phi model contains two different spatial gradient terms that describe the interface between the shear bands. The first is in the viscoelastic constitutive equation, with a characteristic (mesh) length l. The second is in the (generalised) Cahn-Hilliard equation, with the characteristic length xi for equilibrium concentration-fluctuations. We show that the phase diagrams (and so also the flow curves) depend on the ratio r congruent with l/xi, with loss of unique state selection at r=0. We also give results for the full shear-banded profiles, and study the divergence of the interfacial width (relative to l and xi) at the critical point.