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Leticia Cugliandolo
Publication date
2014
fields
Physics
and research's language is
English
Authors
Leticia F. Cugliandolo
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This article gives a short description of pattern formation and coarsening phenomena and focuses on recent experimental and theoretical advances in these fields. It serves as an introduction to phase ordering kinetics and it will appear in the special issue `Coarsening dynamics, Comptes Rendus de Physique, edited by F. Corberi and P. Politi.
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We review a few representative examples of granular experiments or models where phase separation, accompanied by domain coarsening, is a relevant phenomenon. We first elucidate the intrinsic non-equilibrium, or athermal, nature of granular media. Thereafter, dilute systems, the so-called granular gases are discussed: idealized kinetic models, such as the gas of inelastic hard spheres in the cooling regime, are the optimal playground to study the slow growth of correlated structures, e.g. shear patterns, vortices and clusters. In fluidized experiments, liquid-gas or solid-gas separations have been observed. In the case of monolayers of particles, phase coexistence and coarsening appear in several different setups, with mechanical or electrostatic energy input. Phenomenological models describe, even quantitatively, several experimental measures, both for the coarsening dynamics and for the dynamic transition between different granular phases. The origin of the underlying bistability is in general related to negative compressibility from granular hydrodynamics computations, even if the understanding of the mechanism is far from complete. A relevant problem, with important industrial applications, is related to the demixing or segregation of mixtures, for instance in rotating tumblers or on horizontally vibrated plates. Finally, the problem of compaction of highly dense granular materials, which has many important applications, is usually described in terms of coarsening dynamics: there, bubbles of mis-aligned grains evaporate, allowing the coalescence of optimally arranged islands and a progressive reduction of total occupied volume.
We argue that a strict relation exists between two in principle unrelated quantities: The size of the growing domains in a coarsening system, and the kinetic roughening of an interface. This relation is confirmed by extensive simulations of the Ising model with different forms of quenched disorder, such as random bonds, random fields and stochastic dilution.
Coarsening kinetics is usually described using a linear gradient approximation for the underlying interface migration (IM) rates, wherein the migration fluxes at the interfaces vary linearly with the driving force. Recent experimental studies have shown that coarsening of nanocrystalline interface microstructures is unexpectedly stable compared to conventional parabolic coarsening kinetics. Here, we show that during early stage coarsening of these microstructures, IM rates can develop a non-linear dependence on the driving force, the mean interface curvature. We derive the modified mean field law for coarsening kinetics. Molecular dynamics simulations of individual grain boundaries reveal a sub-linear curvature dependence of IM rates, suggesting an intrinsic origin for the slow coarsening kinetics observed in polycrystalline metals.
Equilibrium and non-equilibrium relaxation behaviors of two-dimensional superconducting arrays are investigated via numerical simulations at low temperatures in the presence of incommensurate transverse magnetic fields, with frustration parameter f= (3-sqrt{5})/2. We find that the non-equilibrium relaxation, beginning with random initial states quenched to low temperatures, exhibits a three-stage relaxation of chirality autocorrelations. At the early stage, the relaxation is found to be described by the von Schweidler form. Then it exhibits power-law behavior in the intermediate time scale and faster decay in the long-time limit, which together can be fitted to the Ogielski form; for longer waiting times, this crosses over to a stretched exponential form. We argue that the power-law behavior in the intermediate time scale may be understood as a consequence of the coarsening behavior, leading to the local vortex order corresponding to f=2/5 ground-state configurations. High mobility of the vortices in the domain boundaries, generating slow wandering motion of the domain walls, may provide mechanism of dynamic heterogeneity and account for the long-time stretched exponential relaxation behavior. It is expected that such meandering fluctuations of the low-temperature structure give rise to finite resistivity at those low temperatures; this appears consistent with the zero-temperature resistive transition in the limit of irrational frustration.
The collective behaviour of statistical systems close to critical points is characterized by an extremely slow dynamics which, in the thermodynamic limit, eventually prevents them from relaxing to an equilibrium state after a change in the thermodynamic control parameters. The non-equilibrium evolution following this change displays some of the features typically observed in glassy materials, such as ageing, and it can be monitored via dynamic susceptibilities and correlation functions of the order parameter, the scaling behaviour of which is characterized by universal exponents, scaling functions, and amplitude ratios. This universality allows one to calculate these quantities in suitable simplified models and field-theoretical methods are a natural and viable approach for this analysis. In addition, if a statistical system is spatially confined, universal Casimir-like forces acting on the confining surfaces emerge and they build up in time when the temperature of the system is tuned to its critical value. We review here some of the theoretical results that have been obtained in recent years for universal quantities, such as the fluctuation-dissipation ratio, associated with the non-equilibrium critical dynamics, with particular focus on the Ising model with Glauber dynamics in the bulk. The non-equilibrium dynamics of the Casimir force acting in a film is discussed within the Gaussian model.
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