Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.

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The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

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

Despite the decade-long study of the effect of nanoconfinement on the glass-transition temperature (Tg) of amorphous materials, the quest to probe the distribution of Tgs in nanoconfined glass formers has remained unfulfilled. Here the distribution of Tgs across polystyrene films has been obtained by a fluorescence/multilayer method, revealing that the enhancement of dynamics at a surface affects Tg several tens of nanometres into the film. The extent to which dynamics smoothly transition from enhanced to bulk states depends strongly on nanoconfinement. When polymer films are sufficiently thin that a reduction in thickness leads to a reduction in overall Tg, the surface-layer Tg actually increases with a reduction in overall thickness, whereas the substrate-layer Tg decreases. These results indicate that the gradient in Tg dynamics is not abrupt, and that the size of a cooperatively rearranging region is much smaller than the distance over which interfacial effects propagate.

The distribution of glass-transition temperatures in nanoscopically confined glass formers

Controlling interparticle interactions, aggregation and cluster formation is of central importance in a number of areas, ranging from cluster formation in various disease processes to protein crystallography and the production of photonic crystals. Recent developments in the description of the interaction of colloidal particles with short-range attractive potentials have led to interesting findings including metastable liquid-liquid phase separation and the formation of dynamically arrested states (such as the existence of attractive and repulsive glasses, and transient gels). The emerging glass paradigm has been successfully applied to complex soft-matter systems, such as colloid-polymer systems and concentrated protein solutions. However, intriguing problems like the frequent occurrence of cluster phases remain. Here we report small-angle scattering and confocal microscopy investigations of two model systems: protein solutions and colloid-polymer mixtures. We demonstrate that in both systems, a combination of short-range attraction and long-range repulsion results in the formation of small equilibrium clusters. We discuss the relevance of this finding for nucleation processes during protein crystallization, protein or DNA self-assembly and the previously observed formation of cluster and gel phases in colloidal suspensions.

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Equilibrium cluster formation in concentrated protein solutions and colloids

Crystallization of concentrated colloidal suspensions was studied in real space with laser scanning confocal microscopy. Direct imaging in three dimensions allowed identification and observation of both nucleation and growth of crystalline regions, providing an experimental measure of properties of the nucleating crystallites. By following their evolution, we identified critical nuclei, determined nucleation rates, and measured the average surface tension of the crystal-liquid interface. The structure of the nuclei was the same as the bulk solid phase, random hexagonal close-packed, and their average shape was rather nonspherical, with rough rather than faceted surfaces.

https://science.sciencemag.org/content/sci/292/5515/258.full.pdf

Real-space imaging of nucleation and growth in colloidal crystallization.

Confocal microscopy was used to directly observe three-dimensional dynamics of particles in colloidal supercooled fluids and colloidal glasses. The fastest particles moved cooperatively; connected clusters of these mobile particles could be identified; and the cluster size distribution, structure, and dynamics were investigated. The characteristic cluster size grew markedly in the supercooled fluid as the glass transition was approached, in agreement with computer simulations; at the glass transition, however, there was a sudden drop in their size. The clusters of fast-moving particles were largest near the α-relaxation time scale for supercooled colloidal fluids, but were also present, albeit with a markedly different nature, at shorter β-relaxation time scales, in both supercooled fluid and glass colloidal phases.

http://science.sciencemag.org/content/sci/287/5453/627.full.pdf

Three-Dimensional Direct Imaging of Structural Relaxation Near the Colloidal Glass Transition

Confocal microscopy is used in the study of colloidal gels, glasses, and binary fluids. We measure the three-dimensional positions of colloidal particles with a precision of approximately 50 nm ~a small fraction of each particle’s radius! and with a time resolution sufficient for tracking the thermal motions of several thousand particles at once. This information allows us to characterize the structure and the dynamics of these materials in qualitatively new ways, for example, by quantifying the topology of chains and clusters of particles as well as by measuring the spatial correlations between particles with high mobilities. We describe our experimental technique and describe measurements that complement the results of light scattering. © 2001 Optical Society of America OCIS codes: 110.2960, 110.0180, 180.1790.

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Three-dimensional confocal microscopy of colloids.

We use confocal microscopy to study the motion of a magnetic bead in a dense colloidal suspension, near the colloidal glass transition volume fraction g. For dense liquid-like samples near g, below a threshold force the magnetic bead exhibits only localized caged motion. Above this force, the bead is pulled with a fluctuating velocity. The relationship between force and velocity becomes increasingly nonlinear as g is approached. The threshold force and nonlinear drag force vary strongly with the volume fraction, while the velocity fluctuations do not change near the transition.

Andrew C. Levitt

Papers

Three-Dimensional Direct Imaging of Structural Relaxation Near the Colloidal Glass Transition

Three-dimensional confocal microscopy of colloids.

Forced motion of a probe particle near the colloidal glass transition