Thermodynamic interpretation of soft glassy rheology models

Physical Review E, 2013

The Soft Glassy Rheology (SGR) model is a mesoscopic framework which proved to be very successful in describing flow and deformation of various amorphous materials phenomenologically (e.g. pastes, slurries, foams etc). In this paper, we cast SGR in a general, model independent framework for nonequilibrium thermodynamics called General Equation for the Nonequilibrium Reversible-Irreversible Coupling (GENERIC). This leads to a new formulation of SGR which clarifies how it can properly be coupled to hydrodynamic fields, resulting in a thermodynamically consistent, local, continuum version of SGR. Additionally, we find that compliance with thermodynamics imposes the existence of a modification to the stress tensor as predicted by SGR.

Soft Matter, 2009

The 'soft glassy rheology' (SGR) model gives an appealing account of the flow of nonergodic soft materials in terms of the local yield dynamics of mesoscopic elements. Newtonian, power-law, and yield-stress fluid regimes arise on varying a 'noise temperature', x. Here we extend the model, to capture the idea that the noise is largely caused by yield itself. The extended model can account for the viscosity-bifurcation and shear-banding effects reported recently in a wide range of soft materials. A variant model may shed light on shear banding and strain-rate hysteresis seen in glassy star polymers solutions.

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.

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.

Physical Review B, 2011

We discuss aging and localization in a simple "Eshelby" mesoscopic model of amorphous plasticity. Plastic deformation is assumed to occur through a series of local reorganizations. Using a discretization of the mechanical fields on a discrete lattice, local reorganizations are modeled as local slip events. Local yield stresses are randomly distributed in space and invariant in time. Each plastic slip event induces a long-ranged elastic stress redistribution. Mimicking the effect of aging, we focus on the behavior of the model when the initial state is characterized by a distribution of high local yield stress values. A dramatic effect on the localization behavior is obtained: the system first spontaneously self-traps to form a shear band which then only slowly widens. The higher the "age" parameter the more localized the plastic strain field. Two-time correlation computed on the stress field show a divergent correlation time with the age parameter. The amplitude of a local slip event (the prefactor of the Eshelby singularity) as compared to the yield stress distribution width acts here as an effective temperature-like parameter: the lower the slip increment, the higher the localization and the decorrelation time.

EPL (Europhysics Letters), 2010

We provide a clear evidence that a two species mesoscopic Lattice Boltzmann (LB) model with competing short-range attractive and mid-range repulsive interactions supports emergent Herschel-Bulkley (HB) rheology, i.e. a power-law dependence of the shear-stress as a function of the strain rate, beyond a given yield-stress threshold. This kinetic formulation supports a seamless transition from flowing to non-flowing behaviour, through a smooth tuning of the parameters governing the mesoscopic interactions between the two species. The present model may become a valuable computational tool for the investigation of the rheology of soft-glassy materials on scales of experimental interest.

Journal of Physics: Condensed Matter

Glassy solids may undergo a fluidization (yielding) transition upon deformation whereby the material starts to flow plastically. It has been a matter of debate whether this process is controlled by a specific time scale, from among different competing relaxation/kinetic processes. Here, two constitutive models of cage relaxation are examined within the microscopic model of nonaffine elasto-plasticity. One (widely used) constitutive model implies that the overall relaxation rate is dominated by the fastest between the structural (α) relaxation rate and the shearinduced relaxation rate. A different model is formulated here which, instead, assumes that the slowest (global) relaxation process controls the overall relaxation. We show that the first model is not compatible with the existence of finite elastic shear modulus for quasistatic (low-frequency) deformation, while the second model is able to describe all key features of deformation of 'hard' glassy solids, including the yielding transition, the nonaffine-to-affine plateau crossover, and the rate-stiffening of the modulus. The proposed framework provides an operational way to distinguish between 'soft' glasses and 'hard' glasses based on the shear-rate dependence of the structural relaxation time.

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

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.

The European Physical Journal E, 2014

We study the rheological response at low temperature of a sheared model disordered material as a function of the bond rigidity. We find that the flow curves follow a Herschel-Bulkley law, whatever is the bond rigidity, with an exponent close to 0.5. Interestingly, the apparent viscosity can be related to a single relevant time scale t rel , suggesting a strong connection between the local dynamics and the global mechanical behaviour. We propose a model based on the competition between the nucleation and the avalanche-like propagation of spatial strain heterogeneities. This model can explain the Herschel-Bulkley exponent on the basis of the size dependence of the heterogeneities on the shear rate.