Rheology of hard glassy materials

Physical Review Letters, 1997

We attribute similarities in the rheology of many soft materials (foams, emulsions, slurries, etc.) to the shared features of structural disorder and metastability. A generic model for the mesoscopic dynamics of "soft glassy matter" is introduced, with interactions represented by a mean-field noise temperature x. We find power law fluid behavior either with (x < 1) or without (1 < x < 2) a yield stress. For 1 < x < 2, both storage and loss modulus vary with frequency as ω x−1 , becoming flat near a glass transition (x = 1). Values of x ≈ 1 may result from marginal dynamics as seen in some spin glass models.

We solve exactly and describe in detail a simplified scalar model for the low frequency shear rheology of foams, emulsions, slurries, etc. [P. Sollich, F. Lequeux, P. Hébraud, M.E. Cates, Phys. Rev. Lett. 78, 2020]. The model attributes similarities in the rheology of such "soft glassy materials" to the shared features of structural disorder and metastability. By focusing on the dynamics of mesoscopic elements, it retains a generic character. Interactions are represented by a mean-field noise temperature x, with a glass transition occurring at x = 1 (in appropriate units). The exact solution of the model takes the form of a constitutive equation relating stress to strain history, from which all rheological properties can be derived. For the linear response, we find that both the storage modulus G ′ and the loss modulus G ′′ vary with frequency as ω x−1 for 1 < x < 2, becoming flat near the glass transition. In the glass phase, aging of the moduli is predicted. The steady shear flow curves show power law fluid behavior for x < 2, with a nonzero yield stress in the glass phase; the Cox-Merz rule does not hold in this non-Newtonian regime. Single and double step strains further probe the nonlinear behavior of the model, which is not well represented by the BKZ relation. Finally, we consider measurements of G ′ and G ′′ at finite strain amplitude γ. Near the glass transition, G ′′ exhibits a maximum as γ is increased in a strain sweep. Its value can be strongly overestimated due to nonlinear effects, which can be present even when the stress response is very nearly harmonic. The largest strain γc at which measurements still probe the linear response is predicted to be roughly frequency-independent.

Physical Review E, 1998

We solve exactly and describe in detail a simplified scalar model for the low frequency shear rheology of foams, emulsions, slurries, etc. [P. Sollich, F. Lequeux, P. Hébraud, and M. E. Cates, Phys. Rev. Lett. 78, 2020 (1997)]. The model attributes similarities in the rheology of such ``soft glassy materials&#39;&#39; to the shared features of structural disorder and metastability. By

The Journal of Chemical Physics, 2013

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

2010

As the glass transition is approached from the high temperature side, viewed as a liquid, the properties of the ever more viscous supercooled liquid are continuous functions of temperature and pressure. The point at which we decide to classify the fluid as a solid is therefore subjective. This subjective decision does, however, have discontinuous consequences for how we determine the rheological properties of the glass. We apply the recently discovered relaxation theorem to the time independent, nondissipative, nonergodic glassy state to derive an expression for the phase space distribution of an ensemble of glass samples. This distribution is then used to construct a time dependent linear response theory for aged glassy solids. The theory is verified using molecular dynamics simulations of oscillatory shear for a realistic model glass former with excellent agreement being obtained between the response theory calculations and direct nonequilibrium molecular dynamics calculations. Our numerical results confirm that unlike all the fluid states, including supercooled liquids, a solid glass ͑in common with crystalline states͒ has a nonzero value for the zero frequency shear modulus. Of all the states of matter, a supercooled fluid approaching the glass transition has the highest value for the limiting zero frequency shear viscosity. Finally, solid glasses like dilute gases and crystals have a positive temperature coefficient for the shear viscosity whereas supercooled and normal liquids have a negative temperature coefficient.

2006

We review models for the rheology of soft glasses, a class of materials including e.g. emulsions, foams, colloidal glasses and possibly-but with substantial caveats-gels. The main focus is on the soft glassy rheology (SGR) model, and in particular on the occurrence of rheological aging effects. We first review appropriate definitions of rheological response functions suited to aging samples (in which time translation invariance is lost). These are then used to study aging effects within the SGR model. Its constitutive equations relate shear stress to shear strain among a set of elastic elements, with distributed yield thresholds, undergoing activated dynamics governed by a "noise temperature", x. For 1 < x < 2 there is a power-law fluid regime in which transients occur, but no aging. For x < 1, the model has a macroscopic yield stress. So long as this yield stress is not exceeded, aging occurs, with a sample's apparent relaxation time being of the order of its own age. The (age-dependent) linear viscoelastic loss modulus G (ω, t) rises as frequency is lowered, but falls with age t, so as to always remain less than G (ω, t) (which is nearly constant). Significant aging is also predicted for the stress overshoot in nonlinear shear startup and for the creep compliance. We discuss an extension of the model to include a proper tensorial description of stress and strain, and survey some related rheological models that have recently been developed.

Physical Review Letters, 2005

We investigate by rheology and light scattering the influence of the elastic modulus, G0, on the slow dynamics and the aging of a soft glass. We show that the slow dynamics and the aging can be entirely described by the evolution of an effective viscosity, η ef f , defined as the characteristic time measured in a stress relaxation experiment times G0. At all time, η ef f is found to be independent of G0, of elastic perturbations, and of the rate at which the sample is quenched in the glassy phase. We propose a simple model that links η ef f to the internal stress built up at the fluid-to-solid transition.

Based on numerical simulations of a lattice kinetic model for soft-glassy materials, we characterize the global rheology of a dense emulsion-like system, under three representative load conditions: Couette flow, time-oscillating Strain and Kolmogorov flow. It is found that in all cases the rheology is described by a Herschel-Bulkley (HB) relation, $\sigma = {\sigma}_{Y} + A S^{\beta}$, with the yield stress ${\sigma}_{Y}$ largely independent of the loading scenario. A proper rescaling of the HB parameters permits to describe heterogeneous flows with space-dependent stresses, based on the notion of cooperativity, as recently proposed to characterize the degree of non-locality of stress relaxation phenomena in soft-glassy materials.

2012

The nonlinear rheology of a soft glassy material is captured by its constitutive relation, shear stress vs shear rate, which is most generally obtained by sweeping up or down the shear rate over a finite temporal window. For a huge amount of complex fluids, the up and down sweeps do not superimpose and define a rheological hysteresis loop. By means of extensive rheometry coupled to time-resolved velocimetry, we unravel the local scenario involved in rheological hysteresis for various types of well-studied soft materials. We introduce two observables that quantify the hysteresis in macroscopic rheology and local velocimetry respectively, as a function of the sweep rate \delta t^{-1}. Strikingly, both observables present a robust maximum with \delta t, which defines a single material-dependent timescale that grows continuously from vanishingly small values in simple yield stress fluids to large values for strongly time-dependent materials. In line with recent theoretical arguments, these experimental results hint at a universal timescale-based framework for soft glassy materials, where inhomogeneous flows characterized by shear bands and/or pluglike flow play a central role.

Soft Matter, 2011