A theoretical perspective is provided on the glass transition in molecular liquids at thermal equilibrium, on the spatially heterogeneous and aging dynamics of disordered materials, and on the rheology of soft glassy materials. We start with a broad introduction to the field and emphasize its connections with other subjects and its relevance. The important role played by computer simulations in studying and understanding the dynamics of systems close to the glass transition at the molecular level is given. The recent progress on the subject of the spatially heterogeneous dynamics that characterizes structural relaxation in materials with slow dynamics is reviewed. The main theoretical approaches are presented describing the glass transition in supercooled liquids, focusing on theories that have a microscopic, statistical mechanics basis. We describe both successes and failures and critically assess the current status of each of these approaches. The physics of aging dynamics in disordered materials and the rheology of soft glassy materials are then discussed, and recent theoretical progress is described. For each section, an extensive overview is given of the most recent advances, but we also describe in some detail the important open problems that will occupy a central place in this field in the coming years.

Theoretical perspective on the glass transition and amorphous materials

Advances in physics

Understanding glass formation is a challenge, because the existence of a true glass state, distinct from liquid and solid, remains elusive: Glasses are liquids that have become too viscous to flow. An old idea, as yet unproven experimentally, is that the dynamics becomes sluggish as the glass transition approaches, because increasingly larger regions of the material have to move simultaneously to allow flow. We introduce new multipoint dynamical susceptibilities to estimate quantitatively the size of these regions and provide direct experimental evidence that the glass formation of molecular liquids and colloidal suspensions is accompanied by growing dynamic correlation length scales.

https://science.sciencemag.org/content/sci/310/5755/1797.full.pdf

Direct Experimental Evidence of a Growing Length Scale Accompanying the Glass Transition

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, ageing 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.

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Glassy dynamics of kinetically constrained models

Collective cell migration in tissues occurs throughout embryonic development, during wound healing, and in cancerous tumor invasion, yet most detailed knowledge of cell migration comes from single-cell studies. As single cells migrate, the shape of the cell body fluctuates dramatically through cyclic processes of extension, adhesion, and retraction, accompanied by erratic changes in migration direction. Within confluent cell layers, such subcellular motions must be coupled between neighbors, yet the influence of these subcellular motions on collective migration is not known. Here we study motion within a confluent epithelial cell sheet, simultaneously measuring collective migration and subcellular motions, covering a broad range of length scales, time scales, and cell densities. At large length scales and time scales collective migration slows as cell density rises, yet the fastest cells move in large, multicell groups whose scale grows with increasing cell density. This behavior has an intriguing analogy to dynamic heterogeneities found in particulate systems as they become more crowded and approach a glass transition. In addition we find a diminishing self-diffusivity of short-wavelength motions within the cell layer, and growing peaks in the vibrational density of states associated with cooperative cell-shape fluctuations. Both of these observations are also intriguingly reminiscent of a glass transition. Thus, these results provide a broad and suggestive analogy between cell motion within a confluent layer and the dynamics of supercooled colloidal and molecular fluids approaching a glass transition.

/pdf/glass-like-dynamics-of-collective-cell-migration-3cyisj1sw7.pdf

Glass-like dynamics of collective cell migration

We show by means of experiments, theory, and simulations that the slow dynamics of coarsening systems displays dynamic heterogeneity similar to that observed in glass-forming systems. We measure dynamic heterogeneity via novel multipoint functions which quantify the emergence of dynamic, as opposed to static, correlations of fluctuations. Experiments are performed on a coarsening foam using time-resolved correlation, a recently introduced light scattering method. Theoretically we study the Ising model, and present exact results in one dimension, and numerical results in two dimensions. For all systems the same dynamic scaling of fluctuations with domain size is observed.

/pdf/heterogeneous-dynamics-of-coarsening-systems-508hmgw3z4.pdf

Heterogeneous dynamics of coarsening systems.

The validity of mode-coupling theory (MCT) is restricted by an uncontrolled factorization approximation of density correlations. The factorization can be delayed and ultimately avoided, however, by explicitly including higher order correlations. We explore this approach within a microscopically motivated schematic model. Analytic tractability allows us to discuss in great detail the impact of factorization at arbitrary order, including the limit of avoided factorization. Our results indicate a coherent picture for the capabilities as well as limitations of MCT. Moreover, including higher order correlations systematically defers the transition and ultimately restores ergodicity. Power-law divergence of the relaxation time is then replaced by continuous but exponential growth.

https://arxiv.org/pdf/cond-mat/0602248.pdf

Cooperativity beyond caging: Generalized mode-coupling theory.

We investigate the relation between two-time multispin correlation and response functions in the nonequilibrium critical dynamics of Ising models in d=1 and d=2 spatial dimensions. In these nonequilibrium situations, the fluctuation-dissipation theorem (FDT) is not satisfied. We find FDT "violations" qualitatively similar to those reported in various glassy materials, but quantitatively dependent on the chosen observable, in contrast to the results obtained in infinite-range glass models. Nevertheless, all FDT violations can be understood by considering separately the contributions from large wave vectors, which are at quasiequilibrium and obey the FDT, and from small wave vectors where a generalized FDT holds with a nontrivial fluctuation-dissipation ratio X infinity. In d=1, we get X(infinity)=1/2 for spin observables, which measure the orientation of domains, while X(infinity)=0 for observables that are sensitive to the domain-wall motion. Numerical simulations in d=2 reveal a unique X infinity approximately equal 0.34 for all observables. Measurement protocols for X infinity are discussed in detail. Our results suggest that the definition of an effective temperature T(eff)=T/X(infinity) for large length scales is generically possible in nonequilibrium critical dynamics.

Fluctuation-dissipation relations in the nonequilibrium critical dynamics of Ising models

We give a brief review of violations of the fluctuation-dissipation theorem (FDT) in out-of-equilibrium systems; in mean field scenarios the corresponding fluctuation-dissipation (FD) plots can, in the limit of long times, be used to define a effective temperature Teff that shares many properties of the true thermodynamic temperature T. We discuss carefully how correlation and response functions need to be represented to obtain meaningful limiting FD plots in non-mean field systems. A minimum requirement on the resulting effective temperatures is that they should be independent of the observable whose correlator and response are being considered; we show for two simple models with glassy dynamics (Bouchaud's trap model and the Glauber-Ising chain at zero temperature) that this is generically not the case. Consequences for the wider applicability of effective temperatures derived from FD relations are discussed; one intriguing possibility is that at least the limit of the FDT violation factor for well separated times may generically be observable-independent and so could yield a meaningful Teff.

/pdf/fluctuation-dissipation-relations-and-effective-temperatures-2jooatnfbs.pdf

Fluctuation-dissipation relations and effective temperatures in simple non-mean field systems

We consider the dynamics of spin facilitated models of glasses in the non-equilibrium ageing regime following a sudden quench from high to low temperatures. We briefly review known results obtained for the broad class of kinetically constrained models, and then present new results for the behaviour of the one-spin facilitated Fredrickson–Andersen and East models in various spatial dimensions. The time evolution of one-time quantities, such as the energy density, and the detailed properties of two-time correlation and response functions are studied using a combination of theoretical approaches, including exact mappings of master operators and reductions to integrable quantum spin chains, field theory and renormalization group, and independent interval and timescale separation methods. The resulting analytical predictions are confirmed by means of detailed numerical simulations. The models we consider are characterized by trivial static properties, with no finite temperature singularities, but they nevertheless display a surprising variety of dynamic behaviour during ageing, which can be directly related to the existence and growth in time of dynamic lengthscales. Well-behaved fluctuation–dissipation ratios can be defined for these models, and we study their properties in detail. We confirm in particular the existence of negative fluctuation–dissipation ratios for a large number of observables. Our results suggest that well-defined violations of fluctuation–dissipation relations, of a purely dynamic origin and unrelated to the thermodynamic concept of effective temperatures, could in general be present in non-equilibrium glassy materials.

https://iopscience.iop.org/article/10.1088/1742-5468/2007/07/P07017/pdf

Peter Mayer

Papers

Heterogeneous dynamics of coarsening systems.

Cooperativity beyond caging: Generalized mode-coupling theory.

Fluctuation-dissipation relations in the nonequilibrium critical dynamics of Ising models

Fluctuation-dissipation relations and effective temperatures in simple non-mean field systems

Non-equilibrium dynamics of spin facilitated glass models