Local stress relaxation and shear banding in a dry foam under shear

Rearrangements during slow compression of a jammed two-dimensional emulsion

As amorphous materials get jammed, both geometric and dynamic heterogeneity are observed. We investigate the correlation between the local geometric heterogeneity and local rearrangements in a slowly compressed bidisperse quasi-two-dimensional emulsion system. The compression is driven by evaporation of the continuous phase and causes the area packing fraction to increase from 0.88 to 0.99. We quantify the structural heterogeneity of the system using the radical Voronoi tessellation following the method of Rieser et al. [Phys. Rev. Lett. 116, 088001 (2016)]0031-900710.1103/PhysRevLett.116.088001. We define two structural quantities characterizing local structure, the first of which considers nearest neighbors and the second of which includes information from second-nearest neighbors. We find that droplets in heterogeneous local regions are more likely to have local rearrangements. These rearrangements are generally T1 events where two droplets converge toward a void, and two droplets move away from the void to make room for the converging droplets. Thus, the presence of the voids tends to orient the T1 events. The presence of a correlation between the structural quantities and the rearrangement dynamics remains qualitatively unchanged over the entire range of packing fractions observed.

Quasistatic deformation of yield stress materials: Homogeneous or localized?

We analyze a mesoscopic model of a shear stress material with a three-dimensional slab geometry, under an external quasistatic deformation of a simple shear type. Relaxation is introduced in the model as a mechanism by which an unperturbed system achieves progressively mechanically more stable configurations. Although in all cases deformation occurs via localized plastic events (avalanches), we find qualitatively different behavior depending on the degree of relaxation in the model. For no or low relaxation, yielding is homogeneous in the sample, and even the largest avalanches become negligible in size compared with the system size (measured as the thickness of the slab L_{z}) when this is increased. On the contrary, for high relaxation, the deformation localizes in an almost two-dimensional region where all avalanches occur. Scaling analysis of the numerical results indicates that in this case, the linear size of the largest avalanches is comparable with L_{z}, even when this becomes very large. We correlate the two scenarios with a qualitative difference in the flow curve of the system in the two cases, which is monotonous in the first case and velocity weakening in the second case.

Strain correlation functions in isotropic elastic bodies: large wavelength limit for two-dimensional systems

Strain correlation functions in two-dimensional isotropic elastic bodies are shown both theoretically (using the general structure of isotropic tensor fields) and numerically (using a glass-forming model system) to depend on the coordinates of the field variable (position vector r in real space or wavevector q in reciprocal space) and thus on the direction of the field vector and the orientation of the coordinate system. Since the fluctuations of the longitudinal and transverse components of the strain field in reciprocal space are known in the long-wavelength limit from the equipartition theorem, all components of the correlation function tensor field are imposed and no additional physical assumptions are needed. An observed dependence on the field vector direction thus cannot be used as an indication for anisotropy or for a plastic rearrangement. This dependence is different for the associated strain response field containing also information on the localized stress perturbation.

Micro and Macroscopic Stress-Strain Relations in Disordered Tessellated Networks

We demonstrate that for a rigid and incompressible network in mechanical equilibrium, the microscopic stress and strain follows a simple relation, σ=pE, where σ is the deviatoric stress, E is a mean-field strain tensor, and p is the hydrostatic pressure. This relationship arises as the natural consequence of energy minimization or equivalently, mechanical equilibration. The result suggests not only that the microscopic stress and strain are aligned in the principal directions, but also microscopic deformations are predominantly affine. The relationship holds true regardless of the different (foam or tissue) energy model considered, and directly leads to a simple prediction for the shear modulus, μ=⟨p⟩/2, where ⟨p⟩ is the mean pressure of the tessellation, for general randomized lattices.

Cross over to collective rearrangements near the dry-wet transition in two-dimensional foams

Liquid foams respond plastically to external perturbations over some critical magnitude. This rearrangement process is directly related to the mechanical properties of the foams, playing a significant role in determining foam lifetime, deformability, elasticity, and fluidity. In this paper, we experimentally investigate the rearrangement dynamics of foams near a dry-wet transition. When a foam transforms from a dry state to a wet state, it is found that considering collective events, separated T1 events propagate in dry foams, while T1 events occur simultaneously in wet foams. This cross over to collective rearrangements is closely related to the change in local bubble arrangements and mobility. Furthermore, it is also found that a probability of collective rearrangement events occurring follows a Poisson distribution, suggesting that there is little correlation between discrete collective rearrangement events. These results constitute progress in understanding the dynamical properties of soft jammed systems, relevant for biological and material sciences as well as food science.

Viscoelastic coarsening of quasi-2D foam

Foams are unstable jammed materials. They evolve over timescales comparable to their "time of use", which makes the study of their destabilisation mechanisms crucial for applications. In practice, many foams are made from viscoelastic fluids, which are observed to prolong their lifetimes. Despite their importance, we lack understanding of the coarsening mechanism in such systems. We probe the effect of continuous phase viscoelasticity on foam coarsening with foamed emulsions. We show that bubble size evolution is strongly slowed down and foam structure hugely impacted. The main mechanisms responsible are the absence of continuous phase redistribution and a non-trivial link between foam structure and mechanical properties. These combine to give spatially heterogeneous coarsening. Beyond their importance in the design of foamy materials, the results give a macroscopic vision of phase separation in a viscoelastic medium.

Computational Analysis of Shear Banding in Simple Shear Flow of Viscoelastic Fluid-Based Nanofluids Subject to Exothermic Reactions

We investigated the shear banding phenomena in the non-isothermal simple-shear flow of a viscoelastic-fluid-based nanofluid (VFBN) subject to exothermic reactions. The polymeric (viscoelastic) behavior of the VFBN was modeled via the Giesekus constitutive equation, with appropriate adjustments to incorporate both the non-isothermal and nanoparticle effects. Nahme-type laws were employed to describe the temperature dependence of the VFBN viscosities and relaxation times. The Arrhenius theory was used for the modeling and incorporation of exothermic reactions. The VFBN was modeled as a single-phase homogeneous-mixture and, hence, the effects of the nanoparticles were based on the volume fraction parameter. Efficient numerical schemes based on semi-implicit finite-difference-methods were employed in MATLAB for the computational solution of the governing systems of partial differential equations. The fundamental fluid-dynamical and thermodynamical phenomena, such as shear banding, thermal runaway, and heat transfer rate (HTR) enhancement, were explored under relevant conditions. Important novel results of industrial significance were observed and demonstrated. Firstly, under shear banding conditions of the Giesekus-type VFBN model, we observed remarkable HTR and Therm-C enhancement in the VFBN as compared to, say, NFBN. Specifically, the results demonstrate that the VFBN are less susceptible to thermal runaway than are NFBN. Additionally, the results illustrate that the reduced susceptibility of the Giesekus-type VFBN to the thermal runaway phenomena is further enhanced under shear banding conditions, in particular when the nanofluid becomes increasingly polymeric. Increased polymer viscosity is used as the most direct proxy for measuring the increase in the polymeric nature of the fluid.

Quasistatic rheology of soft cellular systems using the cellular Potts model

Soft cellular systems, such as foams or biological tissues, exhibit highly complex rheological properties, even in the quasistatic regime, that numerical modeling can help to apprehend. We present a numerical implementation of quasistatic strain within the widely used cellular Potts model (CPM). The accuracy of the method is tested by simulating the quasistatic strain of two-dimensional dry foams, both ordered and disordered. The implementation of quasistatic strain in CPM allows the investigation of sophisticated interplays between stress-strain relationship and structural changes that take place in cellular systems.

Unconventional rheological properties in systems of deformable particles

We demonstrate the existence of unconventional rheological and memory properties in systems of soft-deformable particles whose energy depends on their shape, via numerical simulations. At large strains, these systems experience an unconventional shear weakening transition characterized by an increase in the mechanical energy and a drastic drop in shear stress, which stems from the emergence of short-ranged tetratic order. In these weakened states, the contact network evolves reversibly under strain reversal, keeping memory of its initial state, while the microscopic dynamics is irreversible.

Mechanical properties of 2D aggregates of oil droplets as model mono-crystals

We investigate the elastic and yielding properties of two dimensional defect-free mono-crystals made of highly monodisperse droplets. Crystals are compressed between two parallel boundaries of which one acts as a force sensor. As the available space between boundaries is reduced, the crystal goes through successive row-reduction transitions. For small compression forces, the crystal responds elastically until a critical force is reached and the assembly fractures in a single catastrophic global event. Correspondingly there is a peak in the force measurement associated with each row-reduction. The elastic properties of ideal mono-crystal samples are fully captured by a simple analytical model consisting of an assembly of individual capillary springs. The yielding properties of the crystal are captured with a minimal bond breaking model.

Interparticle attraction controls flow heterogeneity in calcite gels

We couple rheometry and ultrasonic velocimetry to study experimentally the flow behavior of gels of colloidal calcite particles dispersed in water, while tuning the strength of the interparticle attraction through physico-chemistry. We unveil, for the first time in a colloidal gel, a direct connection between attractive interactions and the occurrence of shear bands, as well as stress fluctuations.

Microscopic origin of shear banding as a localized driven glass transition in compressed colloidal pillars

Here we report on compression experiments of colloidal pillars in which the evolution of a shear band can be followed at the particle level during deformation. Quasistatic deformation results in dilation and anisotropic changes in coordination in a localized band of material. Additionally, a transition from solid- to liquidlike mechanical response accompanies the structural change in the band, as evidenced by saturation of the packing fraction at the glass transition point, a diminishing ability to host anelastic strains, and a rapid decay in the long-range strain correlations. Overall, our results suggest that shear banding quantitatively resembles a localized, driven glass transition.

Connecting shear localization with the long-range correlated polarized stress fields in granular materials

One long-lasting puzzle in amorphous solids is shear localization, where local plastic deformation involves cooperative particle rearrangements in small regions of a few inter-particle distances, self-organizing into shear bands and eventually leading to the material failure. Understanding the connection between the structure and dynamics of amorphous solids is essential in physics, material sciences, geotechnical and civil engineering, and geophysics. Here we show a deep connection between shear localization and the intrinsic structures of internal stresses in an isotropically jammed granular material subject to shear. Specifically, we find strong (anti)correlations between the micro shear bands and two polarized stress fields along two directions of maximal shear. By exploring the tensorial characteristics and the rotational symmetry of force network, we reveal that such profound connection is a result of symmetry breaking by shear. Finally, we provide the solid experimental evidence of long-range correlated inherent shear stress in an isotropically jammed granular system.

Numerical Study of Shear Banding in Flows of Fluids Governed by the Rolie-Poly Two-Fluid Model via Stabilized Finite Volume Methods

The flow of viscoelastic fluids may, under certain conditions, exhibit shear-banding characteristics that result from their susceptibility to unusual flow instabilities. In this work, we explore both the existing shear banding mechanisms in the literature, namely; constitutive instabilities and flow-induced inhomogeneities. Shear banding due to constitutive instabilities is modelled via either the Johnson–Segalman or the Giesekus constitutive models. Shear banding due to flow-induced inhomogeneities is modelled via the Rolie–Poly constitutive model. The Rolie–Poly constitutive equation is especially chosen because it expresses, precisely, the shear rheometry of polymer solutions for a large number of strain rates. For the Rolie–Poly approach, we use the two-fluid model wherein the stress dynamics are coupled with concentration equations. We follow a computational analysis approach via an efficient and versatile numerical algorithm. The numerical algorithm is based on the Finite Volume Method (FVM) and it is implemented in the open-source software package, OpenFOAM. The efficiency of our numerical algorithms is enhanced via two possible stabilization techniques, namely; the Log-Conformation Reformulation (LCR) and the Discrete Elastic Viscous Stress Splitting (DEVSS) methodologies. We demonstrate that our stabilized numerical algorithms accurately simulate these complex (shear banded) flows of complex (viscoelastic) fluids. Verification of the shear-banding results via both the Giesekus and Johnson-Segalman models show good agreement with existing literature using the DEVSS technique. A comparison of the Rolie–Poly two-fluid model results with existing literature for the concentration and velocity profiles is also in good agreement.

Depletion attraction impairs the plasticity of emulsions flowing in a constriction

We study the elasto-plastic behavior of dense attractive emulsions under a mechanical perturbation. The attraction is introduced through non-specific depletion interactions between the droplets and is controlled by changing the concentration of surfactant micelles in the continuous phase. We find that such attractive forces are not sufficient to induce any measurable modification on the scalings between the local packing fraction and the deformation of the droplets. However, when the emulsions are flowed through 2D microfluidic constrictions, we uncover a measurable effect of attraction on their elasto-plastic response. Indeed, we measure higher levels of deformation inside the constriction for attractive droplets. In addition, we show that these measurements correlate with droplet rearrangements that are spatially delayed in the constriction for higher attraction forces.

Roughness-Induced Friction on Liquid Foams

Complex liquids flow is known to be drastically affected by the roughness condition at the interfaces. We combined stresses measurements and observations of the flow during the motion of different rough surfaces in dry liquid foams. We visually show that three distinct friction regimes exist: slippage, stick-slip motion, and anchored soap films. Our stress measurements are validated for slippage and anchored regimes based on existing models, and we propose a leverage rule to describe the stresses during the stick-slip regime. We find that the occurrence of the stick-slip or anchored regimes is controlled by the roughness factor, defined as the ratio between the size of the surface asperities and the radius of curvature of the Plateau borders.

Self-diffusion in plastic flow of amorphous solids

We report on a particle-based numerical study of sheared amorphous solids in the dense slow flow regime. In this framework, deformation and flow are accompanied by critical fluctuation patterns associated with the macroscopic plastic response. The former is commonly attributed to the collective slip patterns that relax internal stresses within the bulk material and give rise to an effective mechanical noise governing the latter particle-level process. In this paper, the avalanche-type dynamics between plastic events is shown to have a strong relevance on the self-diffusion of tracer particles in the Fickian regime. As a consequence, strong size effects emerge in the effective diffusion coefficient that is rationalized in terms of avalanche size distributions and the relevant temporal occurrence.

From plastic flow to brittle fracture: Role of microscopic friction in amorphous solids

Plasticity in soft amorphous materials typically involves collective deformation patterns that emerge on intense shearing. The microscopic basis of amorphous plasticity has been commonly established through the notion of "Eshelby"-type events, localized abrupt rearrangements that induce flow in the surrounding material via nonlocal elastic-type interactions. This universal mechanism in flowing disordered solids has been proposed despite their diversity in terms of scales, microscopic constituents, or interactions. Using a numerical particle-based study, we argue that the presence of frictional interactions in granular solids alters the dynamics of flow by nucleating micro shear cracks that continually coalesce to build up system-spanning fracturelike formations on approach to failure. The plastic-to-brittle failure transition is controlled by the degree of frictional resistance which is in essence similar to the role of heterogeneities that separate the abrupt and smooth yielding regimes in glassy structures.

Identifying structural signatures of shear banding in model polymer nanopillars

Amorphous solids are critical in the design and production of nanoscale devices, but under strong confinement these materials exhibit changes in their mechanical properties which are not well understood. Phenomenological models explain these properties by postulating an underlying defect structure in these materials but do not detail the microscopic properties of these defects. Using machine learning methods, we identify mesoscale defects that lead to shear banding in model polymer nanopillars well below the glass transition temperature as a function of pillar diameter. Our results show that the primary structural features responsible for shear banding on this scale are fluctuations in the diameter of the pillar. Surprisingly, these fluctuations are quite small compared to the diameter of the pillar, less than half of a particle diameter in size. At intermediate pillar diameters, we find that these fluctuations tend to concentrate along the minor axis of shear band planes. We also see the importance of mean "softness" as a classifier of shear banding grow as a function of pillar diameter. Softness is a new field that characterizes local structure and is highly correlated with particle-level dynamics such that softer particles are more likely to rearrange. This demonstrates that softness, a quantity that relates particle-level structure to dynamics on short time and length scales, can predict large time and length scale phenomena related to material failure.

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.

Long-range stress correlations in viscoelastic and glass-forming fluids

A simple and rigorous approach to obtain stress correlations in viscoelastic liquids (including supercooled liquid and equilibrium amorphous systems) is proposed. The long-range dynamical correlations of local shear stress are calculated and analyzed in 2-dimensional space. It is established how the long-range character of the stress correlations gradually emerges as the relevant dynamical correlation length l grows in time. The correlation range l is defined by momentum propagation due to acoustic waves and vorticity diffusion which are the basic mechanisms for transmission of shear stress perturbations. We obtain the general expression defining the time- and distance-dependent stress correlation tensor in terms of material functions (generalized relaxation moduli). The effect of liquid compressibility is quantitatively analyzed; it is shown to be important at large distances and/or short times. The revealed long-range stress correlation effect is shown to be dynamical in nature and unconnected with static structural correlations in liquids (correlation length ξ_s). Our approach is based on the assumption that ξ_s is small enough as reflected in weak wave-number dependencies of the generalized relaxation moduli. We provide a simple physical picture connecting the elucidated long-range fluctuation effect with anisotropic correlations of the (transient) inherent stress field, and discuss its implications.

Monitoring deformation mechanism of foam cells in polyethylene foams via optical microscopy: Effect of density and microstructure

The effect of cellular microstructure and density of low-density polyethylene (LDPE) foam during the compressive deformation were evaluated. Progressive foam cell structure deformation at crosshead speed of 10 mm/min was closely monitored and captured using a portable digital microscope with the assistance of a proposed coating technique to enhance contrast of the foam cells. Results revealed that higher density foam, LDPE1.0, experienced higher compressive stress than the LDPE0.5 foam. It was believed that thicker cell walls in the LDPE1.0 foam contributed to the foam stiffness and higher stress was required to impose cell bending, buckling, and collapse. It was also discovered that the existence of larger cells within a group or cluster of small cells in the LDPE1.0 foam acted as stress concentration points and initiated the formation of shear bands in the foam cells deformation mechanism. The extent of compressive strain on the foam cells were also investigated, and results showed that cells undergo lower compressive strain compared to the actual deformation level given directly by the machine crosshead. Cells located in the middle section of the foam recorded a strain of about 13% when the bulk foam was deformed at 20% strain. The success in capturing the actual cell strain using the proposed coating technique proved that with the right improvisation, the micromechanical deformation of optically challenged white polyethylene foam samples can be monitored effectively using optical microscopy.

Non-local stresses in highly non-uniformly flowing suspensions: The shear-curvature viscosity

For highly non-uniformly flowing fluids, there are contributions to the stress related to spatial variations of the shear rate, which are commonly referred to as non-local stresses. The standard expression for the shear stress, which states that the shear stress is proportional to the shear rate, is based on a formal expansion of the stress tensor with respect to spatial gradients in the flow velocity up to leading order. Such a leading order expansion is not able to describe fluids with very rapid spatial variations of the shear rate, like in micro-fluidics devices and in shear-banding suspensions. Spatial derivatives of the shear rate then significantly contribute to the stress. Such non-local stresses have so far been introduced on a phenomenological level. In particular, a formal gradient expansion of the stress tensor beyond the above mentioned leading order contribution leads to a phenomenological formulation of non-local stresses in terms of the so-called "shear-curvature viscosity". We derive an expression for the shear-curvature viscosity for dilute suspensions of spherical colloids and propose an effective-medium approach to extend this result to concentrated suspensions. The validity of the effective-medium prediction is confirmed by Brownian dynamics simulations on highly non-uniformly flowing fluids.

Stable small bubble clusters in two-dimensional foams

Key features of the mechanical response of amorphous particulate materials, such as foams, emulsions, and granular media, to applied stress are determined by the frequency and size of particle rearrangements that occur as the system transitions from one mechanically stable state to another. This work describes coordinated experimental and computational studies of bubble rafts, which are quasi-two dimensional systems of bubbles confined to the air-water interface. We focus on small mechanically stable clusters of four, five, six, and seven bubbles with two different sizes with diameter ratio σ_L/σ_S ≃ 1.4. Focusing on small bubble clusters, which can be viewed as subsystems of a larger system, allows us to investigate the full ensemble of clusters that form, measure the respective frequencies with which the clusters occur, and determine the form of the bubble-bubble interactions. We emphasize several important results. First, for clusters with N > 5 bubbles, we find using discrete element simulations that short-range attractive interactions between bubbles give rise to a larger ensemble of distinct mechanically stable clusters compared to that generated by long-range attractive interactions. The additional clusters in systems with short-range attractions possess larger gaps between pairs of neighboring bubbles on the periphery of the clusters. The ensemble of bubble clusters observed in experiments is similar to the ensemble of clusters with long-range attractive interactions. We also compare the frequency with which each cluster occurs in simulations and experiments. We find that the cluster frequencies are extremely sensitive to the protocol used to generate them and only weakly correlated to the energy of the clusters.

Shear banding, discontinuous shear thickening, and rheological phase transitions in athermally sheared frictionless disks

We report on numerical simulations of simple models of athermal, bidisperse, soft-core, massive disks in two dimensions, as a function of packing fraction ϕ, inelasticity of collisions as measured by a parameter Q, and applied uniform shear strain rate γ[over ̇]. Our particles have contact interactions consisting of normally directed elastic repulsion and viscous dissipation, as well as tangentially directed viscous dissipation, but no interparticle Coulombic friction. Mapping the phase diagram in the (ϕ,Q) plane for small γ[over ̇], we find a sharp first-order rheological phase transition from a region with Bagnoldian rheology to a region with Newtonian rheology, and show that the system is always Newtonian at jamming. We consider the rotational motion of particles and demonstrate the crucial importance that the coupling between rotational and translational degrees of freedom has on the phase structure at small Q (strongly inelastic collisions). At small Q, we show that, upon increasing γ[over ̇], the sharp Bagnoldian-to-Newtonian transition becomes a coexistence region of finite width in the (ϕ,γ[over ̇]) plane, with coexisting Bagnoldian and Newtonian shear bands. Crossing this coexistence region by increasing γ[over ̇] at fixed ϕ, we find that discontinuous shear thickening can result if γ[over ̇] is varied too rapidly for the system to relax to the shear-banded steady state corresponding to the instantaneous value of γ[over ̇].

Eshelby inclusions in granular matter: Theory and simulations

We present a numerical implementation of an active inclusion in a granular material submitted to a biaxial test. We discuss the dependence of the response to this perturbation on two parameters: the intragranular friction coefficient on one hand, and the degree of the loading on the other hand. We compare the numerical results to theoretical predictions taking into account the change of volume of the inclusion as well as the anisotropy of the elastic matrix.

Measurement of Stress Redistribution in Flowing Emulsions

We study how local rearrangements alter droplet stresses within flowing dense quasi-two-dimensional emulsions at area fractions ϕ≥0.88. Using microscopy, we measure droplet positions while simultaneously using their deformed shape to measure droplet stresses. We find that rearrangements alter nearby stresses in a quadrupolar pattern: stresses on neighboring droplets tend to either decrease or increase depending on location. The stress redistribution is more anisotropic with increasing ϕ. The spatial character of the stress redistribution influences where subsequent rearrangements occur. Our results provide direct quantitative support for rheological theories of dense amorphous materials that connect local rearrangements to changes in nearby stress.

Fluctuations of particle motion in granular avalanches - from the microscopic to the macroscopic scales

In this study, we have investigated the fluctuations of particle motion, i.e. the non-affine motion, during the avalanche process, discovering a rich dynamic from the microscopic to the macroscopic scales. We find that there is a strong correlation between the magnitude of the velocity fluctuation and the velocity magnitude in the spatial and temporal domains. The possible connection between this finding and the theory of the shear transformation zones is discussed based on the direct measurement of the T1 events. In addition, the velocity magnitude of the system and the stress fluctuations of the system are strongly temporally correlated. Our finding will pose challenges to the development of more rigorous theories to describe avalanche dynamics based on the microscopic approach. Moreover, our finding presents a plausible mechanism of particle entrainment in a simple system.

Experimental observation of local rearrangements in dense quasi-two-dimensional emulsion flow

We experimentally study rearranging regions in slow athermal flow by observing the flow of a concentrated oil-in-water emulsion in a thin chamber with a constricting hopper shape. The gap of the chamber is smaller than the droplet diameters, so that the droplets are compressed into quasi-two-dimensional pancakes. We focus on localized rearrangements known as "T1 events" where four droplets exchange neighbors. Flowing droplets are deformed due to forces from neighboring droplets, and these deformations are decreased by nearby T1 events, with a spatial dependence related to the local structure. We see a tendency of the T1 events to occur in small clusters.

The effect of plastic rearrangements on the flow of two-dimensional wet foam

The effect of the elementary plastic events on the flow behavior of the two-dimensional wet foam is investigated by quasistatic simulation on the bubble scale. The position where the plastic event occurs is traced by recording the coordinate at which two bubbles separate in the simulation. A localized shear band is found, and the width of this band increases with the increase of foam quality. From the displacement fields of these bubbles, it shows that the T1 plastic events can give rise to an increase in local bubble displacements due to the separation between these bubbles. The average relative pressure as well as normal stress difference of bubbles increases with the flow of foam in the initial elastic domain and then decreases as the elastic domain turns into the plastic domain. In the plastic domain, the plastic events rearrange the local structure of foam, which leads to decreasing both the average pressure and the normal stress difference. Additionally, the wall slip of foam is discussed in the simulation as well. The width of the localized shear band is narrower under the slip boundary condition. Meanwhile, the plastic events occurring between the first and second layers of bubbles change the pulling force of the films near the wall and cause an instantaneous increase in the slip velocity.

Numerical model for the shear rheology of two-dimensional wet foams with deformable bubbles

Shearing of two-dimensional wet foam is simulated using an introduced numerical model, and results are compared to those of experiments. This model features realistically deformable bubbles, which distinguishes it from previously used models for wet foam. The internal bubble dynamics and their contact interactions are also separated in the model, making it possible to investigate the effects of the related microscale properties of the model on the macroscale phenomena. Validity of model assumptions was proved here by agreement between the simulated and measured Herschel-Bulkley rheology, and shear-induced relaxation times. This model also suggests a relationship between the shear stress and normal stress as well as between the average degree of bubble deformation and applied shear stress. It can also be used to analyze suspensions of bubbles and solid particles, an extension not considered in this work.

Time-dependent elastic response to a local shear transformation in amorphous solids

The elastic response of a two-dimensional amorphous solid to induced local shear transformations, which mimic the elementary plastic events occurring in deformed glasses, is investigated via molecular-dynamics simulations. We show that for different spatial realizations of the transformation, despite relative fluctuations of order one, the long-time equilibrium response averages out to the prediction of the Eshelby inclusion problem for a continuum elastic medium. We characterize the effects of the underlying dynamics on the propagation of the elastic signal. A crossover from a propagative transmission in the case of weakly damped dynamics to a diffusive transmission for strong damping is evidenced. In the latter case, the full time-dependent elastic response is in agreement with the theoretical prediction, obtained by solving the diffusion equation for the displacement field in an elastic medium.

Mechanical and microscopic properties of the reversible plastic regime in a 2D jammed material

At the microscopic level, plastic flow of a jammed, disordered material consists of a series of particle rearrangements that cannot be reversed by subsequent deformation. An infinitesimal deformation of the same material has no rearrangements. Yet between these limits, there may be a self-organized plastic regime with rearrangements, but with no net change upon reversing a deformation. We measure the oscillatory response of a jammed interfacial material, and directly observe rearrangements that couple to bulk stress and dissipate energy, but do not always give rise to global irreversibility.

Nonlocal rheology of granular flows across yield conditions

The rheology of dense granular flows is studied numerically in a shear cell controlled at constant pressure and shear stress, confined between two granular shear flows. We show that a liquid state can be achieved even far below the yield stress, whose flow can be described with the same rheology as above the yield stress. A nonlocal constitutive relation is derived from dimensional analysis through a gradient expansion and calibrated using the spatial relaxation of velocity profiles observed under homogeneous stresses. Both for frictional and frictionless grains, the relaxation length is found to diverge as the inverse square root of the distance to the yield point, on both sides of that point.

Dislocation dynamics during plastic deformations of complex plasma crystals

The internal structures of most periodic crystalline solids contain defects. This affects various important mechanical and thermal properties of crystals. Since it is very difficult and expensive to track the motion of individual atoms in real solids, macroscopic model systems, such as complex plasmas, are often used. Complex plasmas consist of micrometer-sized grains immersed into an ion-electron plasma. They exist in solidlike, liquidlike, and gaseouslike states and exhibit a range of nonlinear and dynamic effects, most of which have direct analogies in solids and liquids. Slabs of a monolayer hexagonal complex plasma were subjected to a cycle of uniaxial compression and decompression of large amplitudes to achieve plastic deformations, both in experiments and simulations. During the cycle, the internal structure of the lattice exhibited significant rearrangements. Dislocations (point defects) were generated and displaced in the stressed lattice. They tended to glide parallel to their Burgers vectors under load. It was found that the deformation cycle was macroscopically reversible but irreversible at the particle scale.

The physics of the colloidal glass transition

As one increases the concentration of a colloidal suspension, the system exhibits a dramatic increase in viscosity. Beyond a certain concentration, the system is said to be a colloidal glass; structurally, the system resembles a liquid, yet motions within the suspension are slow enough that it can be considered essentially frozen. For several decades, colloids have served as a valuable model system for understanding the glass transition in molecular systems. The spatial and temporal scales involved allow these systems to be studied by a wide variety of experimental techniques. The focus of this review is the current state of understanding of the colloidal glass transition, with an emphasis on experimental observations. A brief introduction is given to important experimental techniques used to study the glass transition in colloids. We describe features of colloidal systems near and in glassy states, including increases in viscosity and relaxation times, dynamical heterogeneity and ageing, among others. We also compare and contrast the glass transition in colloids to that in molecular liquids. Other glassy systems are briefly discussed, as well as recently developed synthesis techniques that will keep these systems rich with interesting physics for years to come.

Non-linear oscillatory rheological properties of a generic continuum foam model: comparison with experiments and shear-banding predictions

The occurrence of shear bands in a complex fluid is generally understood as resulting from a structural evolution of the material under shear, which leads (from a theoretical perspective) to a non-monotonic stationary flow curve related to the coexistence of different states of the material under shear. In this paper we present a scenario for shear-banding in a particular class of complex fluids, namely foams and concentrated emulsions, which differs from other scenarios in two important ways. First, the appearance of shear bands is shown to be possible both without any intrinsic physical evolution of the material (e.g. via a parameter coupled to the flow such as concentration or entanglements) and without any finite critical shear rate below which the flow does not remain stationary and homogeneous. Secondly, the appearance of shear bands depends on the initial conditions, i.e. the preparation of the material. In other words, it is history dependent. This behaviour relies on the tensorial character of the underlying model (2D or 3D) and is triggered by an initially inhomogeneous strain distribution in the material. The shear rate displays a discontinuity at the band boundary whose amplitude is history dependent and thus depends on the sample preparation.

Hot spots in an athermal system

We study experimentally the dynamical heterogeneities occurring at slow shear, in a model amorphous glassy material, i.e., a 3D granular packing. The deformation field is resolved spatially by using a diffusive wave spectroscopy technique. The heterogeneities show up as localized regions of strong deformations spanning a mesoscopic size of about 10 grains and called the "hot spots." The spatial clustering of hot spots is linked to the subsequent emergence of shear bands. Quantitatively, their appearance is associated with the macroscopic plastic deformation, and their rate of occurrence gives a physical meaning to the concept of "fluidity," recently used to describe the local and nonlocal rheology of soft glassy materials.

How topological rearrangements and liquid fraction control liquid foam stability

The stability of foam is investigated experimentally through coalescence events. Instability (coalescence) occurs when the system is submitted to external perturbations (T1) and when the liquid amount in the film network is below a critical value. Microscopically, transient thick films are observed during film rearrangements. Film rupture, with coalescence and eventual collapse of the foam, occurs when the available local liquid amount is too small for transient films to be formed. Similar experiments and results are shown in the two-bubble case.

Plasticity and dynamical heterogeneity in driven glassy materials

Many amorphous glassy materials exhibit complex spatio-temporal mechanical response and rheology, characterized by an intermittent stress strain response and a fluctuating velocity profile. Under quasistatic and athermal deformation protocols this heterogeneous plastic flow was shown to be composed of plastic events of various sizes, ranging from local quadrupolar plastic rearrangements to system spanning shear bands. In this paper, through numerical study of a 2D Lennard-Jones amorphous solid, we generalize the study of the heterogeneous dynamics of glassy materials to the finite shear rate (gamma not equal to 0) and temperature case (T not equal to 0). In practice, we choose an effectively athermal limit (T approximately 0) and focus on the influence of shear rate on the rheology of the glass. In line with previous works we find that the model Lennard-Jones glass follows the rheological behavior of a yield stress fluid with a Herschel-Bulkley response of the form, sigma = sigmaY + c1gamma(beta). The global mechanical response obtained through the use of Molecular Dynamics is shown to converge in the limit gamma --> 0 to the quasistatic limit obtained with an energy minimization protocol. The detailed analysis of the plastic deformation at different shear rates shows that the glass follows different flow regimes. At sufficiently low shear rates the mechanical response reaches a shear-rate-independent regime that exhibits all the characteristics of the quasistatic response (finite-size effects, cascades of plastic rearrangements, yield stress, ...). At intermediate shear rates the rheological properties are determined by the externally applied shear rate and the response deviates from the quasistatic limit. Finally at higher shear the system reaches a shear-rate-independent homogeneous regime. The existence of these three regimes is also confirmed by the detailed analysis of the atomic motion. The computation of the four-point correlation function shows that the transition from the shear-rate-dominated to the quasistatic regime is accompanied by the growth of a dynamical cooperativity length scale xi that is shown to diverge with shear rate as xi is proportional to gamma(-nu), with nu approximately 0.2 -0.3. This scaling is compared with the prediction of a simple model that assumes the diffusive propagation of plastic events.

The continuum theory of shear localization in two-dimensional foam

We review some recent advances in the rheology of two-dimensional liquid foams, which should have implications for three-dimensional foams, as well as other mechanical systems that have a yield stress. We focus primarily on shear localization under steady shear, an effect first highlighted in an experiment by Debrégeas et al. A continuum theory which incorporates wall drag has reproduced the effect. Its further refinements are successful in matching results of more extensive observations and making interesting predictions regarding experiments for low strain rates and non-steady shear. Despite these successes, puzzles remain, particularly in relation to quasistatic simulations. The continuum model is semi-empirical: the meaning of its parameters may be sought in comparison with more detailed simulations and other experiments. The question of the origin of the Herschel-Bulkley relation is particularly interesting.

Foam as a complex system

What is a 'complex system'? The two-dimensional foam, as originally popularized by Cyril Stanley Smith, provides an ideal context in which to explore this question.

Soft Dynamics simulation. 2. Elastic spheres undergoing a T(1) process in a viscous fluid

Robust empirical constitutive laws for granular materials in air or in a viscous fluid have been expressed in terms of timescales based on the dynamics of a single particle. However, some behaviours such as viscosity bifurcation or shear localization, observed also in foams, emulsions, and block copolymer cubic phases, seem to involve other micro-timescales which may be related to the dynamics of local particle reorganizations. In the present work, we consider a T(1) process as an example of a rearrangement. Using the Soft Dynamics simulation method introduced in the first paper of this series, we describe theoretically and numerically the motion of four elastic spheres in a viscous fluid. Hydrodynamic interactions are described at the level of lubrication (Poiseuille squeezing and Couette shear flow) and the elastic deflection of the particle surface is modeled as Hertzian. The duration of the simulated T(1) process can vary substantially as a consequence of minute changes in the initial separations, consistently with predictions. For the first time, a collective behaviour is thus found to depend on a parameter other than the typical volume fraction of particles.

Comparison of low-amplitude oscillatory shear in experimental and computational studies of model foams

A fundamental difference between fluids and solids is their response to applied shear. Solids possess static shear moduli, while fluids do not. Complex fluids such as foams display an intermediate response to shear with nontrivial frequency-dependent shear moduli. In this paper, we conduct coordinated experiments and numerical simulations of model foams subjected to boundary-driven oscillatory planar shear. Our studies are performed on bubble rafts (experiments) and the bubble model (simulations) in two dimensions. We focus on the low-amplitude flow regime in which T1 events, i.e., bubble rearrangement events where originally touching bubbles switch nearest neighbors, do not occur, yet the system transitions from solid- to liquidlike behavior as the driving frequency is increased. In both simulations and experiments, we observe two distinct flow regimes. At low frequencies omega, the velocity profile of the bubbles increases linearly with distance from the stationary wall, and there is a nonzero total phase shift between the moving boundary and interior bubbles. In this frequency regime, the total phase shift scales as a power law Delta approximately omegan with n approximately 3. In contrast, for frequencies above a crossover frequency omega>omegap, the total phase shift Delta scales linearly with the driving frequency. At even higher frequencies above a characteristic frequency omeganl>omegap, the velocity profile changes from linear to nonlinear. We fully characterize this transition from solid- to liquidlike flow behavior in both the simulations and experiments and find qualitative and quantitative agreements for the characteristic frequencies.

Viscous shear banding in foam

Shear banding is an important feature of flow in complex fluids. Essentially, shear bands refer to the coexistence of flowing and nonflowing regions in driven material. Understanding the possible sources of shear banding has important implications for a wide range of flow applications. In this regard, quasi-two-dimensional flow offers a unique opportunity to study competing factors that result in shear bands. One proposal for interpretation and analysis is the competition between intrinsic dissipation and an external source of dissipation. In this paper, we report on the experimental observation of the transition between different classes of shear bands that have been predicted to exist in cylindrical geometry as the result of this competition [R. J. Clancy, E. Janiaud, D. Weaire, and S. Hutzlet, Eur. J. Phys. E 21, 123 (2006)].

Soft Dynamics simulation. 1. Normal approach of two deformable particles in a viscous fluid and optimal-approach strategy

Discrete simulation methods are efficient tools to investigate the behaviors of complex fluids such as dry granular materials or dilute suspensions of hard particles. By contrast, materials made of soft and/or concentrated units (emulsions, foams, vesicles, dense suspensions) can exhibit both significant elastic particle deflections (Hertz-like response) and strong viscous forces (squeezed liquid). We point out that the gap between two particles is then not determined solely by the positions of their centers, but rather exhibits its own dynamics. We provide the first ingredients of a new discrete numerical method, named Soft Dynamics, to simulate the combined dynamics of particles and contacts. As an illustration, we present the results for the approach of two particles. We recover the scaling behaviors expected in three limits: the Stokes limit for very large gaps, the Poiseuille-lubricated limit for small gaps and even smaller surface deflections, and the Hertz limit for significant surface deflections. We find that for each gap value, an optimal force achieves the fastest approach velocity. The principle of larger-scale simulations with this new method is provided. They will consitute a promising tool for investigating the collective behaviors of many complex materials.

Rheological properties of the soft-disk model of two-dimensional foams

The soft-disk model previously developed and applied by Durian [D. J. Durian, Phys. Rev. Lett. 75, 4780 (1995)] is brought to bear on problems of foam rheology of longstanding and current interest, using two-dimensional systems. The questions at issue include the origin of the Herschel-Bulkley relation, normal stress effects (dilatancy), and localization in the presence of wall drag. We show that even a model that incorporates only linear viscous effects at the local level gives rise to nonlinear (power-law) dependence of the limit stress on strain rate. With wall drag, shear localization is found. Its nonexponential form and the variation of localization length with boundary velocity are well described by a continuum model in the spirit of Janiaud etal [Phys. Rev. Lett. 97, 038302 (2006)]. Other results satisfactorily link localization to model parameters, and hence tie together continuum and local descriptions.