Cooperativity flows and shear-bandings: a statistical field theory approach

Interplay between wall slip and shear banding in a thixotropic yield stress fluid

We study the local dynamics of a thixotropic yield stress fluid that shows a pronounced non-monotonic flow curve. This mechanically unstable behavior is generally not observable from standard rheometry tests, resulting in a stress plateau that stems from the coexistence of a flowing band with an unyielded region below a critical shear rate c. Combining ultrasound velocimetry with standard rheometry, we discover an original shear-banding scenario in the decreasing branch of the flow curve of model paraffin gels, in which the velocity profile of the flowing band is set by the applied shear rate  instead of c. As a consequence, the material slips at the walls with a velocity that shows a non-trivial dependence on the applied shear rate. To capture our observations, we propose a differential version of the so-called lever rule, describing the extent of the flowing band and the evolution of wall slip with shear rate. This phenomenological model holds down to very low shear rates, at which the dimension of the flowing band becomes comparable to the size of the individual wax particles that constitute the gel microstructure, leading to cooperative effects. Our approach provides a framework where constraints imposed in the classical shear-banding scenario can be relaxed, with wall slip acting as an additional degree of freedom.

Continuum modeling of shear startup in soft glassy materials

Yield stress fluids (YSFs) display a dual nature highlighted by the existence of a critical stress σ_{y} such that YSFs are solid for stresses σ imposed below σ_{y}, whereas they flow like liquids for σ>σ_{y}. Under an applied shear rate γ[over ̇], the solid-to-liquid transition is associated with a complex spatiotemporal scenario that depends on the microscopic details of the system, on the boundary conditions, and on the system size. Still, the general phenomenology reported in the literature boils down to a simple sequence that can be divided into a short-time response characterized by the so-called "stress overshoot," followed by stress relaxation towards a steady state. Such relaxation can be either (1) long-lasting, which usually involves the growth of a shear band that can be only transient or that may persist at steady state or (2) abrupt, in which case the solid-to-liquid transition resembles the failure of a brittle material, involving avalanches. In the present paper, we use a continuum model based on a spatially resolved fluidity approach to rationalize the complete scenario associated with the shear-induced yielding of YSFs. A key feature of our model is to provide a scaling for the coordinates of the stress overshoot, i.e., stress σ_{M} and strain γ_{M} as a function of γ[over ̇], which shows good agreement with experimental and numerical data extracted from the literature. Moreover, our approach shows that the power-law scaling σ_{M}(γ[over ̇]) is intimately linked to the growth dynamics of a fluidized boundary layer in the vicinity of the moving boundary. Yet such scaling is independent of the fate of that layer, and of the long-term behavior of the YSF, i.e., whether the steady-state flow profile is homogeneous or shear-banded. Finally, when including the presence of "long-range" correlations, we show that our model displays a ductile to brittle transition, i.e., the stress overshoot reduces into a sharp stress drop associated with avalanches, which impacts the scaling σ_{M}(γ[over ̇]). This generalized model nicely captures subtle avalanche-like features of the transient shear banding dynamics reported in experiments. Our work offers a unified picture of shear-induced yielding in YSFs, whose complex spatiotemporal dynamics are deeply connected to nonlocal effects.

Stress Overshoots in Simple Yield Stress Fluids

Soft glassy materials such as mayonnaise, wet clays, or dense microgels display a solid-to-liquid transition under external shear. Such a shear-induced transition is often associated with a nonmonotonic stress response in the form of a stress maximum referred to as "stress overshoot." This ubiquitous phenomenon is characterized by the coordinates of the maximum in terms of stress σ_{M} and strain γ_{M} that both increase as weak power laws of the applied shear rate. Here we rationalize such power-law scalings using a continuum model that predicts two different regimes in the limit of low and high applied shear rates. The corresponding exponents are directly linked to the steady-state rheology and are both associated with the nucleation and growth dynamics of a fluidized region. Our work offers a consistent framework for predicting the transient response of soft glassy materials upon startup of shear from the local flow behavior to the global rheological observables.

Plasticity in Amorphous Solids Is Mediated by Topological Defects in the Displacement Field

The microscopic mechanism by which amorphous solids yield plastically under an externally applied stress or deformation has remained elusive in spite of enormous research activity in recent years. Most approaches have attempted to identify atomic-scale structural "defects" or spatiotemporal correlations in the undeformed glass that may trigger plastic instability. In contrast, in this Letter we show that the topological defects that correlate with plastic instability can be identified, not in the static structure of the glass, but rather in the nonaffine displacement field under deformation. These dislocation-like topological defects (DTDs) can be quantitatively characterized in terms of Burgers circuits (and the resulting Burgers vectors) that are constructed on the microscopic nonaffine displacement field. We demonstrate that (i) DTDs are the manifestation of incompatibility of deformation in glasses as a result of violation of Cauchy-Born rules (nonaffinity); (ii) the resulting average Burgers vector displays peaks in correspondence of major plastic events, including a spectacular nonlocal peak at the yielding transition, which results from self-organization into shear bands due to the attractive interaction between antiparallel DTDs; and (iii) application of Schmid's law to the DTDs leads to prediction of shear bands at 45° for uniaxial deformations, as widely observed in experiments and simulations.

Rayleigh-Bénard convection of a model emulsion: anomalous heat-flux fluctuations and finite-size droplet effects

We present mesoscale numerical simulations of Rayleigh-Bénard (RB) convection in a two-dimensional model emulsion. The systems under study are constituted of finite-size droplets, whose concentration Φ₀ is systematically varied from small (Newtonian emulsions) to large values (non-Newtonian emulsions). We focus on the characterisation of the heat transfer properties close to the transition from conductive to convective states, where it is well known that a homogeneous Newtonian system exhibits a steady flow and a time-independent heat flux. In marked contrast, emulsions exhibit non-steady dynamics with fluctuations in the heat flux. In this paper, we aim at the characterisation of such non-steady dynamics via detailed studies on the time-averaged heat flux and its fluctuations. To quantitatively understand the time-averaged heat flux, we propose a side-by-side comparison between the emulsion system and a single-phase (SP) system, whose viscosity is suitably constructed from the shear rheology of the emulsion. We show that such local closure works well only when a suitable degree of coarse-graining (at the droplet scale) is introduced in the local viscosity. To delve deeper into the fluctuations in the heat flux, we furthermore propose a side-by-side comparison between a Newtonian emulsion (i.e., with a small droplet concentration) and a non-Newtonian emulsion (i.e., with a large droplet concentration), at fixed time-averaged heat flux. This comparison elucidates that finite-size droplets and the non-Newtonian rheology cooperate to trigger enhanced heat-flux fluctuations at the droplet scales. These enhanced fluctuations are rooted in the emergence of space correlations among distant droplets, which we highlight via direct measurements of the droplets displacement and the characterisation of the associated correlation function. The observed findings offer insights on heat transfer properties for confined systems possessing finite-size constituents.

A multi-component lattice Boltzmann approach to study the causality of plastic events

Using a multi-component lattice Boltzmann (LB) model, we perform fluid kinetic simulations of confined and concentrated emulsions. The system presents the phenomenology of soft-glassy materials, including a Herschel-Bulkley rheology, yield stress, ageing and long relaxation time scales. Shearing the emulsion in a Couette cell below the yield stress results in plastic topological re-arrangement events which follow established empirical seismic statistical scaling laws, making this system a good candidate to study the physics of earthquakes. One characteristic of this model is the tendency for events to occur in avalanche clusters, with larger events, triggering subsequent re-arrangements. While seismologists have developed statistical tools to study correlations between events, a process to confirm causality remains elusive. We present here, a modification to our LB model, involving small, fast vibrations applied to individual droplets, effectively a macroscopic forcing, which results in the arrest of the topological plastic re-arrangements. This technique provides an excellent tool for identifying causality in plastic event clusters by examining the evolution of the dynamics after 'stopping' an event, and then checking which subsequent events disappear. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Unified Theoretical and Experimental View on Transient Shear Banding

Dense emulsions, colloidal gels, microgels, and foams all display a solidlike behavior at rest characterized by a yield stress, above which the material flows like a liquid. Such a fluidization transition often consists of long-lasting transient flows that involve shear-banded velocity profiles. The characteristic time for full fluidization τ_{f} has been reported to decay as a power law of the shear rate γ[over ˙] and of the shear stress σ with respective exponents α and β. Strikingly, the ratio of these exponents was empirically observed to coincide with the exponent of the Herschel-Bulkley law that describes the steady-state flow behavior of these complex fluids. Here we introduce a continuum model, based on the minimization of a "free energy," that captures quantitatively all the salient features associated with such transient shear banding. More generally, our results provide a unified theoretical framework for describing the yielding transition and the steady-state flow properties of yield stress fluids.

Plastic flow and localization in an amorphous material: Experimental interpretation of the fluidity

We present a thorough study of the plastic response of a granular material progressively loaded. We study experimentally the evolution of the plastic field from a homogeneous one to a heterogeneous one and its fluctuations in terms of incremental strain. We show that the plastic field can be decomposed in two components evolving on two decoupled strain increment scales. We argue that the slowly varying part of the field can be identified with the so-called fluidity field introduced recently to interpret the rheological behavior of amorphous materials. This fluidity field progressively concentrates along a macroscopic direction corresponding to the Mohr-Coulomb angle.

Wall fluidization in two acts: from stiff to soft roughness

Fluidization of soft glassy materials (SGMs) in microfluidic channels is affected by the wall roughness in the form of microtexturing. When SGMs flow across microgrooves, their constituents are likely trapped within the grooves' gap, and the way they are released locally modifies the fluidization close to the walls. By leveraging a suitable combination of experiments and numerical simulations on concentrated emulsions (a model SGM), we quantitatively report the existence of two physically different scenarios. When the gap is large compared to the droplets in the emulsion, the droplets hit the solid obstacles and easily escape scrambling with their neighbors. Conversely, as the gap spacing is reduced, droplets get trapped inside, creating a "soft roughness" layer, i.e. a complementary series of deformable posts from which overlying droplets are in turn released. In both cases, the induced fluidization scales with the grooves' density, although with a reduced prefactor for narrow gaps, accounting for the softness of the roughness. Both scenarios are also well distinguished via the statistics of the droplets displacement field close to the walls, with large deviations induced by the surface roughness, depending on its stiffness.

Fluidization and wall slip of soft glassy materials by controlled surface roughness

We present a comprehensive study of concentrated emulsions flowing in microfluidic channels, one wall of which is patterned with micron-size equally spaced grooves oriented perpendicularly to the flow direction. We find a scaling law describing the roughness-induced fluidization as a function of the density of the grooves, thus fluidization can be predicted and quantitatively regulated. This suggests common scenarios for droplet trapping and release, potentially applicable for other jammed systems as well. Numerical simulations confirm these views and provide a direct link between fluidization and the spatial distribution of plastic rearrangements.