Onset of flow in a confined colloidal glass under an imposed shear stress

Slow Fatigue and Highly Delayed Yielding via Shear Banding in Oscillatory Shear

We study theoretically the dynamical process of yielding in cyclically sheared amorphous materials, within a thermal elastoplastic model and the soft glassy rheology model. Within both models we find an initially slow accumulation, over many cycles after the inception of shear, of low levels of damage in the form strain heterogeneity across the sample. This slow fatigue then suddenly gives way to catastrophic yielding and material failure. Strong strain localization in the form of shear banding is key to the failure mechanism. We characterize in detail the dependence of the number of cycles N^{*} before failure on the amplitude of imposed strain, the working temperature, and the degree to which the sample is annealed prior to shear. We discuss our finding with reference to existing experiments and particle simulations, and suggest new ones to test our predictions.

Scaling Description of Creep Flow in Amorphous Solids

Amorphous solids such as coffee foam, toothpaste, or mayonnaise display a transient creep flow when a stress Σ is suddenly imposed. The associated strain rate is commonly found to decay in time as γ[over ˙]∼t^{-ν}, followed either by arrest or by a sudden fluidization. Various empirical laws have been suggested for the creep exponent ν and fluidization time τ_{f} in experimental and numerical studies. Here, we postulate that plastic flow is governed by the difference between Σ and the transient yield stress Σ_{t}(γ) that characterizes the stability of configurations visited by the system at strain γ. Assuming the analyticity of Σ_{t}(γ) allows us to predict ν and asymptotic behaviors of τ_{f} in terms of properties of stationary flows. We test successfully our predictions using elastoplastic models and published experimental results.

Creep failure of amorphous solids under tensile stress

Applying constant tensile stress to a piece of amorphous solid results in a slow extension, followed by an eventual rapid mechanical collapse. This "creep" process is of paramount engineering concern, and as such was the subject of study in a variety of materials, for more than a century. Predictive theories for τ_{w}, the expected time of collapse, are incomplete, mainly due to its dependence on a bewildering variety of parameters, including temperature, system size, tensile force, but also the detailed microscopic interactions between constituents. The complex dependence of the collapse time on all the parameters is discussed below, using simulations of strip of amorphous material. Different scenarios are observed for ductile and brittle materials, resulting in serious difficulties in creating an all-encompassing theory that could offer safety measures for given conditions. A central aim of this paper is to employ scaling concepts, to achieve data collapse for the probability distribution function (pdf) of lnτ_{w}. The scaling ideas result in a universal function which provides a prediction of the pdf of lnτ_{w} for out-of-sample systems, from measurements at other values of these parameters. The predictive power of the scaling theory is demonstrated for both ductile and brittle systems. Finally, we present a derivation of universal scaling function for brittle materials. The ductile case appears to be due to a plastic necking instability and is left for future research.

Yielding in an Integer Automaton Model for Amorphous Solids under Cyclic Shear

We present results on an automaton model of an amorphous solid under cyclic shear. After a transient, the steady state falls into one of three cases in order of increasing strain amplitude: (i) pure elastic behavior with no plastic activity, (ii) limit cycles where the state recurs after an integer period of strain cycles, and (iii) irreversible plasticity with longtime diffusion. The number of cycles N required for the system to reach a periodic orbit diverges as the amplitude approaches the yielding transition between regimes (ii) and (iii) from below, while the effective diffusivity D of the plastic strain field vanishes on approach from above. Both of these divergences can be described by a power law. We further show that the average period T of the limit cycles increases on approach to yielding.

Elastoplastic Approach Based on Microscopic Insights for the Steady State and Transient Dynamics of Sheared Disordered Solids

We develop a framework to study the mechanical response of athermal amorphous solids via a coupling of mesoscale and microscopic models. Using measurements of coarse-grained quantities from simulations of dense disordered particulate systems, we present a coherent elastoplastic model approach for deformation and flow of yield stress materials. For a given set of parameters, this model allows us to match consistently transient and steady state features of driven disordered systems with diverse preparation histories under both applied shear-rate and creep protocols.

Flow heterogeneities in supercooled liquids and glasses under shear

Using extensive nonequilibrium molecular dynamics simulations, we investigate a glass-forming binary Lennard-Jones mixture under shear. Both supercooled liquids and glasses are considered. Our focus is on the characterization of inhomogeneous flow patterns such as shear bands that appear as a transient response to the external shear. For the supercooled liquids, we analyze the crossover from Newtonian to non-Newtonian behavior with increasing shear rate γ[over ̇]. Above a critical shear rate γ[over ̇]_{c} where a non-Newtonian response sets in, the transient dynamics are associated with the occurrence of short-lived vertical shear bands, i.e., bands of high mobility that form perpendicular to the flow direction. In the glass states, long-lived horizontal shear bands, i.e., bands of high mobility parallel to the flow direction, are observed in addition to vertical ones. The systems with shear bands are characterized in terms of mobility maps, stress-strain relations, mean-squared displacements, and (local) potential energies. The initial formation of a horizontal shear band provides an efficient stress release, corresponds to a local minimum of the potential energy, and is followed by a slow broadening of the band towards the homogeneously flowing fluid in the steady state. Whether a horizontal or a vertical shear band forms cannot be predicted from the initial undeformed sample. Furthermore, we show that with increasing system size, the probability for the occurrence of horizontal shear bands increases.

Rheological Characterization of Carbopol® Dispersions in Water and in Water/Glycerol Solutions

The influence of the solvent type on the rheological properties of Carbopol ® NF 980 dispersions in water and in water/glycerol solutions is investigated. The material formulation, preparation procedure, common experimental challenges and artifact sources are all addressed. Transient and steady-state experiments were performed. For both solvent types, a clearly thixotropic behavior occurs slightly above the yield stress, where the avalanche effect is observed. For larger stresses, thixotropy is always negligible. Among other findings, it is observed that, for a given Carbopol concentration, the dispersion in the more viscous solvent possesses a lower yield stress and moduli, a larger power-law index, and a longer time to reach steady state.

Creep dynamics of athermal amorphous materials: a mesoscopic approach

Yield stress fluids display complex dynamics, in particular when driven into the transient regime between the solid and the flowing state. Inspired by creep experiments on dense amorphous materials, we implement mesoscale elasto-plastic descriptions to analyze such transient dynamics in athermal systems. Both our mean-field and space-dependent approaches consistently reproduce the typical experimental strain rate responses to different applied steps in stress. Moreover, they allow us to understand basic processes involved in the strain rate slowing down (creep) and the strain rate acceleration (fluidization) phases. The fluidization time increases in a power-law fashion as the applied external stress approaches a static yield stress. This stress value is related to the stress over-shoot in shear start-up experiments, and it is known to depend on sample preparation and age. By calculating correlations of the accumulated plasticity in the spatially resolved model, we reveal different modes of cooperative motion during the creep dynamics.

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.

On the existence of thermodynamically stable rigid solids

Customarily, crystalline solids are defined to be rigid since they resist changes of shape determined by their boundaries. However, rigid solids cannot exist in the thermodynamic limit where boundaries become irrelevant. Particles in the solid may rearrange to adjust to shape changes eliminating stress without destroying crystalline order. Rigidity is therefore valid only in the metastable state that emerges because these particle rearrangements in response to a deformation, or strain, are associated with slow collective processes. Here, we show that a thermodynamic collective variable may be used to quantify particle rearrangements that occur as a solid is deformed at zero strain rate. Advanced Monte Carlo simulation techniques are then used to obtain the equilibrium free energy as a function of this variable. Our results lead to a unique view on rigidity: While at zero strain a rigid crystal coexists with one that responds to infinitesimal strain by rearranging particles and expelling stress, at finite strain the rigid crystal is metastable, associated with a free energy barrier that decreases with increasing strain. The rigid phase becomes thermodynamically stable when an external field, which penalizes particle rearrangements, is switched on. This produces a line of first-order phase transitions in the field-strain plane that intersects the origin. Failure of a solid once strained beyond its elastic limit is associated with kinetic decay processes of the metastable rigid crystal deformed with a finite strain rate. These processes can be understood in quantitative detail using our computed phase diagram as reference.

Mean-Field Scenario for the Athermal Creep Dynamics of Yield-Stress Fluids

We develop a theoretical description based on an existent mean-field model for the transient dynamics prior to the steady flow of yielding materials. The mean-field model not only reproduces the experimentally observed nonlinear time dependence of the shear-rate response to an external stress, but also allows for the determination of the different physical processes involved in the onset of the reacceleration phase after the initial slowing down and a distinct fluidization phase. The fluidization time displays a power-law dependence on the distance of the applied stress to an age-dependent yield stress, which is not universal but strongly dependent on initial conditions.

Transient inhomogeneous flow patterns in supercooled liquids under shear

Supercooled liquids and other soft glassy systems show characteristic spatial inhomogeneities in their local dynamical properties. Using detailed molecular dynamics simulations, we find that for sufficiently low temperatures and sufficiently high shear rates supercooled liquids also show transient inhomogeneous flow patterns (shear banding) in the start-up of steady shear flow, similar to what has already been observed for many other soft glassy systems. We verify that the onset of transient shear banding coincides quite well with the appearance of a stress overshoot for temperatures in the supercooled regime. We find that the slower bands adapt less well to the imposed deformation and therefore accumulate higher shear stresses compared to the fast bands at comparable local shear rates. Our results also indicate that the shear rates of the fast and slow bands are adjusted such that the local dissipation rate is approximately the same in both bands.

Depinning and heterogeneous dynamics of colloidal crystal layers under shear flow

Using Brownian dynamics (BD) simulations and an analytical approach we investigate the shear-induced, nonequilibrium dynamics of dense colloidal suspensions confined to a narrow slit-pore. Focusing on situations where the colloids arrange in well-defined layers with solidlike in-plane structure, the confined films display complex, nonlinear behavior such as collective depinning and local transport via density excitations. These phenomena are reminiscent of colloidal monolayers driven over a periodic substrate potential. In order to deepen this connection, we present an effective model that maps the dynamics of the shear-driven colloidal layers to the motion of a single particle driven over an effective substrate potential. This model allows us to estimate the critical shear rate of the depinning transition based on the equilibrium configuration, revealing the impact of important parameters, such as the slit-pore width and the interaction strength. We then turn to heterogeneous systems where a layer of small colloids is sheared with respect to bottom layers of large particles. For these incommensurate systems we find that the particle transport is dominated by density excitations resembling the so-called "kink" solutions of the Frenkel-Kontorova (FK) model. In contrast to the FK model, however, the corresponding "antikinks" do not move.

Yielding of glass under shear: A directed percolation transition precedes shear-band formation

Under external mechanical loading, glassy materials, ranging from soft matter systems to metallic alloys, often respond via formation of inhomogeneous flow patterns, during yielding. These inhomogeneities can be precursors to catastrophic failure, implying that a better understanding of their underlying mechanisms could lead to the design of smarter materials. Here, extensive molecular dynamics simulations are used to reveal the emergence of heterogeneous dynamics in a binary Lennard-Jones glass, subjected to a constant strain rate. At a critical strain, this system exhibits for all considered strain rates a transition towards the formation of a percolating cluster of mobile regions. We give evidence that this transition belongs to the universality class of directed percolation. Only at low shear rates, the percolating cluster subsequently evolves into a transient (but long-lived) shear band with a diffusive growth of its width. Finally, the steady state with a homogeneous flow pattern is reached. In the steady state, percolation transitions also do occur constantly, albeit over smaller strain intervals, to maintain the stationary plastic flow in the system.

Two-dimensional magnetic colloids under shear

Complex rheological properties of soft disordered solids, such as colloidal gels or glasses, inspire a range of novel applications. However, the microscopic mechanisms of their response to mechanical loading are not well understood. Here, we elucidate some aspects of these mechanisms by studying a versatile model system, i.e. two-dimensional superparamagnetic colloids in a precessing magnetic field, whose structure can be tuned from a hexagonal crystal to a disordered gel network by varying the external field opening angle θ. We perform Langevin dynamics simulations subjecting these structures to a constant shear rate and observe three qualitatively different types of material response. In hexagonal crystals (θ = 0°), at a sufficiently low shear rate, plastic flow occurs via successive stress drops at which the stress releases due to the formation of dislocation defects. The gel network at θ = 48°, on the contrary, via bond rearrangement and transient shear banding evolves into a homogeneously stretched network at large strains. The latter structure remains metastable after switching off of the shear. At θ = 50°, the external shear makes the system unstable against phase separation and causes a failure of the network structure leading to the formation of hexagonal close packed clusters interconnected by particle chains. At a microcopic level, our simulations provide insight into some of the mechanisms by which strain localization as well as material failure occur in a simple gel-like network. Furthermore, we demonstrate that new stretched network structures can be generated by the application of shear.

Creep and aging of hard-sphere glasses under constant stress

We investigate the aging behavior of glassy suspensions of nearly hard-sphere colloids submitted to a constant shear stress. For low stresses, below the yield stress, the system is subject to creep motion. As the sample ages, the shear rate exhibits a power-law decrease with time with exponents that depend on the sample age. We use a combination of rheological experiments with time-resolved photon correlation spectroscopy to investigate the time evolution of the sample dynamics under shear on various time and length scales. Long-time light-scattering experiments reveal the occurrence of microscopic rearrangement events that are linked with the macroscopic strain deformation of the sample. Dynamic time sweep experiments indicate that while the internal microscopic dynamics slow down continuously with waiting time, the storage and loss moduli are almost constant after a fast, weak decrease, resembling the behavior of quenched systems with partially frozen-in stresses.

Length scales in glass-forming liquids and related systems: a review

The central problem in the study of glass-forming liquids and other glassy systems is the understanding of the complex structural relaxation and rapid growth of relaxation times seen on approaching the glass transition. A central conceptual question is whether one can identify one or more growing length scale(s) associated with this behavior. Given the diversity of molecular glass-formers and a vast body of experimental, computational and theoretical work addressing glassy behavior, a number of ideas and observations pertaining to growing length scales have been presented over the past few decades, but there is as yet no consensus view on this question. In this review, we will summarize the salient results and the state of our understanding of length scales associated with dynamical slow down. After a review of slow dynamics and the glass transition, pertinent theories of the glass transition will be summarized and a survey of ideas relating to length scales in glassy systems will be presented. A number of studies have focused on the emergence of preferred packing arrangements and discussed their role in glassy dynamics. More recently, a central object of attention has been the study of spatially correlated, heterogeneous dynamics and the associated length scale, studied in computer simulations and theoretical analysis such as inhomogeneous mode coupling theory. A number of static length scales have been proposed and studied recently, such as the mosaic length scale discussed in the random first-order transition theory and the related point-to-set correlation length. We will discuss these, elaborating on key results, along with a critical appraisal of the state of the art. Finally we will discuss length scales in driven soft matter, granular fluids and amorphous solids, and give a brief description of length scales in aging systems. Possible relations of these length scales with those in glass-forming liquids will be discussed.

Mechanical fluctuations suppress the threshold of soft-glassy solids: The secular drift scenario

We propose a dynamical mechanism leading to the fluidization by external mechanical fluctuations of soft-glassy amorphous material driven below the yield stress. The model is based on the combination of memory effect and nonlinearity, leading to an accumulation of tiny effects over a long term. We test this scenario on a granular packing driven mechanically below the Coulomb threshold. We provide evidence for an effective viscous response directly related to small stress modulations in agreement with the theoretical prediction of a generic secular drift. We propose to extend this result more generally to a large class of glassy systems.

Creep and flow of glasses: strain response linked to the spatial distribution of dynamical heterogeneities

Mechanical properties are of central importance to materials sciences, in particular if they depend on external stimuli. Here we investigate the rheological response of amorphous solids, namely colloidal glasses, to external forces. Using confocal microscopy and computer simulations, we establish a quantitative link between the macroscopic creep response and the microscopic single-particle dynamics. We observe dynamical heterogeneities, namely regions of enhanced mobility, which remain localized in the creep regime, but grow for applied stresses leading to steady flow. These different behaviors are also reflected in the average particle dynamics, quantified by the mean squared displacement of the individual particles, and the fraction of active regions. Both microscopic quantities are found to be proportional to the macroscopic strain, despite the non-equilibrium and non-linear conditions during creep and the transient regime prior to steady flow.

Manipulating shear-induced non-equilibrium transitions in colloidal films by feedback control

Using Brownian Dynamics (BD) simulations we investigate non-equilibrium transitions of sheared colloidal films under controlled shear stress σxz. In our approach the shear rate [small gamma, Greek, dot above] is a dynamical variable, which relaxes on a time scale τc such that the instantaneous, configuration-dependent stress σxz(t) approaches a pre-imposed value. Investigating the dynamics under this "feedback-control" scheme we find unique behavior in regions where the flow curve σxz([small gamma, Greek, dot above]) of the uncontrolled system is monotonic. However, in non-monotonic regions our method allows to select between dynamical states characterized by different in-plane structure and viscosities. Indeed, the final state strongly depends on τc relative to an intrinsic relaxation time of the uncontrolled system. The critical values of τc are estimated on the basis of a simple model.

Poiseuille flow of soft glasses in narrow channels: from quiescence to steady state

Using numerical simulations, the onset of Poiseuille flow in a confined soft glass is investigated. Starting from the quiescent state, steady flow sets in at a time scale which increases with a decrease in applied forcing. At this onset time scale, a rapid transition occurs via the simultaneous fluidization of regions having different local stresses. In the absence of steady flow at long times, creep is observed even in regions where the local stress is larger than the bulk yielding threshold. Finally, we show that the time scale to attain steady flow depends strongly on the history of the initial state.

Universal and non-universal features in coarse-grained models of flow in disordered solids

We study the two-dimensional (2D) shear flow of amorphous solids within variants of an elastoplastic model, paying particular attention to spatial correlations and time fluctuations of, e.g., local stresses. The model is based on the local alternation between an elastic regime and plastic events during which the local stress is redistributed. The importance of a fully tensorial description of the stress and of the inclusion of (coarse-grained) convection in the model is investigated; scalar and tensorial models yield similar results, while convection enhances fluctuations and breaks the spurious symmetry between the flow and velocity gradient directions, for instance when shear localisation is observed. Besides, correlation lengths measured with diverse protocols are discussed. One class of such correlation lengths simply scale with the spacing between homogeneously distributed, simultaneous plastic events. This leads to a scaling of the correlation length with the shear rate as γ̇(-1/2) in 2D in the athermal regime, regardless of the details of the model. The radius of the cooperative disk, defined as the near-field region in which plastic events induce a stress redistribution that is not amenable to a mean-field treatment, notably follows this scaling. On the other hand, the cooperative volume measured from the four-point stress susceptibility and its dependence on the system size and the shear rate are model-dependent.