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.Submitted to Physical Review 83.70.Hq, 05.40+j Many soft materials, such as foams, emulsions, pastes and slurries, have intriguing rheological properties. Experimentally, there is a well-developed phenomenology for such systems: their nonlinear flow behavior is often fit to the form σ = A + Bγ n where σ is shear stress andγ strain rate. This is the Herschel-Bulkeley equation [1,2]; or (for A = 0) the "power-law fluid" [1][2][3]. For the same materials, linear or quasi-linear viscoelastic mesurements often reveal storage and loss moduli′ is usually about 0.1) with a frequency dependence that is either a weak power law (clay slurries, paints, microgels) or negligible (tomato paste, dense emulsions, dense multilayer vesicles, colloidal glasses) [4][5][6][7][8][9][10]. This behavior persists down to the lowest accessible frequencies (about 10 −3 -1 Hz depending on the system), in apparent contradiction to linear response theory [11], which requires that G ′′ (ω) should be an odd function of ω.That similar anomalous rheology should be seen in such a wide range of soft materials suggests a common cause. Indeed, the frequency dependence indicated above points strongly to the generic presence of slow "glassy" dynamics persisting to arbitrarily small frequencies. This feature is found in several other contexts [12][13][14], such as elastic manifold dynamics in random media [15,16]. The latter is suggestive of rheology: charge density waves, vortices, contact lines, etc. can "flow" in response to an imposed "stress". In this Letter we argue that glassy dynamics is a natural consequence of two properties shared by all the soft materials mentioned above: structural disorder and metastability. In such materials, thermal motion alone is not enough to achieve complete structural relaxation. The system has to cross energy barriers (for example those associated with rearrangement of droplets in an emulsion) that are very large compared to typical thermal energies. Therefore the system adopts a disordered, metastable configuration even when (as in a monodisperse emulsion or foam) the state of least free energy would be ordered [17]. While the importance of disorder has been noted before for specific systems [7,11,18-21], we feel that its unifying role in rheological modelling has not been appreciated.To test these ideas, we construct a minimal "generic model" for soft glassy matter. For simplicity, we ignore tensorial asp...

We review the use of kinetically constrained models (KCMs) for the study of dynamics in glassy systems. The characteristic feature of KCMs is that they have trivial, often non-interacting, equilibrium behaviour but interesting slow dynamics due to restrictions on the allowed transitions between configurations. The basic question which KCMs ask is therefore how much glassy physics can be understood without an underlying "equilibrium glass transition". After a brief review of glassy phenomenology, we describe the main model classes, which include spin-facilitated (Ising) models, constrained lattice gases, models inspired by cellular structures such as soap froths, models obtained via mappings from interacting systems without constraints, and finally related models such as urn, oscillator, tiling and needle models. We then describe the broad range of techniques that have been applied to KCMs, including exact solutions, adiabatic approximations, projection and mode-coupling techniques, diagrammatic approaches and mappings to quantum systems or effective models. Finally, we give a survey of the known results for the dynamics of KCMs both in and out of equilibrium, including topics such as relaxation time divergences and dynamical transitions, nonlinear relaxation, aging and effective temperatures, cooperativity and dynamical heterogeneities, and finally non-equilibrium stationary states generated by external driving. We conclude with a discussion of open questions and possibilities for future work.

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, 2020Lett. 78, (1997]. 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.PACS numbers: 83.70.Hq, 05.40+j.

We study theoretically the role of ageing in the rheology of soft materials. We define several generalized rheological response functions suited to ageing samples (in which time translation invariance is lost). These are then used to study ageing effects within a simple scalar model (the "soft glassy rheology" or SGR model) whose 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. (Between yields, each element follows affinely the applied shear.) For 1 < x < 2 there is a power-law fluid regime in which transients occur, but no ageing. For x < 1, the model has a macroscopic yield stress. So long as this yield stress is not exceeded, ageing occurs, with a sample's apparent relaxation time being of order 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 ageing is also predicted for the stress overshoot in nonlinear shear startup and for the creep compliance. Though obviously oversimplified, the SGR model may provide a valuable paradigm for the experimental and theoretical study of rheological ageing phenomena in soft solids.

We consider ensembles of trajectories associated with large deviations of time-integrated quantities in stochastic models. Motivated by proposals that these ensembles are relevant for physical processes such as shearing and glassy relaxation, we show how they can be generated directly using auxiliary stochastic processes. We illustrate our results using the Glauber-Ising chain, for which biased ensembles of trajectories can exhibit ferromagnetic ordering. We discuss the relation between such biased ensembles and quantum phase transitions.