Theory of phase-ordering kinetics

The theory of phase-ordering dynamics that is the growth of order through domain coarsening when a system is quenched from the homogeneous phase into a broken-symmetry phase, is reviewed, with the emphasis on recent developments. Interest will focus on the scaling regime that develops at long times after the quench. How can one determine the growth laws that describe the time dependence of characteristic length scales, and what can be said about the form of the associated scaling functions? Particular attention will be paid to systems described by more complicated order parameters than the simple scalars usually considered, for example vector and tensor fields. The latter are needed, for example, to describe phase ordering in nematic liquid crystals, on which there have been a number of recent experiments. The study of topological defects (domain walls, vortices, strings and monopoles) provides a unifying framework for discussing coarsening in these different systems.     

Dynamic scaling behaviour of late-stage phase separation in Ni75AlxV25-x alloys

The dynamic scaling behaviour of late-stage phase separation and coarsening mechanisms of L12 and D022 in Ni75AlxV25-x （3 ≤ x ≤ 10, at.%） alloys are studied using the microscopic phase-field dynamic model. The microelaso ticity field is incorporated into the diffusion dynamic model. The results show the morphology and coarsening dynamics being greatly changed by the elastic interactions among different precipitates, the particles aligning along the dominant directions, the average domain size （ADS） of L12 and D022 deviating from the exponent of temporal power-law, and the growth slowing down due to the increasing of elastic interactions. The dynamic scaling regime of late-stage coarsening of the precipitates is attained. Thus the scaling behaviour of structure function is also applicable for elastic interaction systems. It is also found that the variations of ADS and scaling function depend on the volume fraction of precipitates.     

Comparison of molecular dynamics and Monte Carlo computer simulations of spinodal decomposition

The main results of recent computer simulations of spinodal decomposition in various systems are summarized and compared. Both Monte Carlo simulations of the kinetic Ising system and molecular dynamics simulations of phase separation in Lennard-Jones systems yield power law growth for the coarsening of the decomposition pattern and scaling of the spinodal peak of the structure factor. These similarities and also distinct differences of the dynamics of one-and two-component systems and of the different simulation techniques are discussed.     

Stabilization and scalable block preconditioning for the Navier–Stokes equations

This study compares several block-oriented preconditioners for the stabilized finite element discretization of the incompressible Navier–Stokes equations. This includes standard additive Schwarz domain decomposition methods, aggressive coarsening multigrid, and three preconditioners based on an approximate block LU factorization, specifically SIMPLEC, LSC, and PCD. Robustness is considered with a particular focus on the impact that different stabilization methods have on preconditioner performance. Additionally, parallel scaling studies are undertaken. The numerical results indicate that aggressive coarsening multigrid, LSC and PCD all have good algorithmic scalability. Coupling this with the fact that block methods can be applied to systems arising from stable mixed discretizations implies that these techniques are a promising direction for developing scalable methods for Navier–Stokes.     

Coarsening in inhomogeneous systems

This article is a brief review of coarsening phenomena occurring in systems where quenched features—such as random field, varying coupling constants or lattice vacancies—spoil homogeneity. We discuss the current understanding of the problem in ferromagnetic systems with a non-conserved scalar order parameter by focusing primarily on the form of the growth law of the ordered domains and on the scaling properties.     

Heterogeneous dynamics of coarsening systems

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.     

Motility-induced phase separation and coarsening in active matter

Active systems, or active matter, are self-driven systems that live, or function, far from equilibrium – a paradigmatic example that we focus on here is provided by a suspension of self-motile particles. Active systems are far from equilibrium because their microscopic constituents constantly consume energy from the environment in order to do work, for instance to propel themselves. The non-equilibrium nature of active matter leads to a variety of non-trivial intriguing phenomena. An important one, which has recently been the subject of intense interest among biological and soft matter physicists, is that of the so-called “motility-induced phase separation”, whereby self-propelled particles accumulate into clusters in the absence of any explicit attractive interactions between them. Here we review the physics of motility-induced phase separation, and discuss this phenomenon within the framework of the classic physics of phase separation and coarsening. We also discuss theories for bacterial colonies where coarsening may be arrested. Most of this work will focus on the case of run-and-tumble and active Brownian particles in the absence of solvent-mediated hydrodynamic interactions – we will briefly discuss at the end their role, which is not currently fully understood in this context.     

Coarsening of cracks in a uniaxially strained solid

We discuss the stress relaxation in a uniaxially strained solid due to the coarsening of a system of parallel cracks. We emphasize similarities and differences of this process to Ostwald ripening in a first order phase transition. A conventional mean-field approximation breaks down and several independent length scales have to be taken into account. Strong elastic interactions between the cracks determine the growth behavior. We derive scaling laws for the coarsening of the different length scales involved and the time evolution of stress relaxation, finally leading to the equilibrium state of a fractured body. The characteristic size of the cracks grows linearly in time which is much faster than in usual Ostwald ripening.     

Computer simulation study of phase separation in a binary mixture with a glass-forming component

We present a computer simulation study of spinodal decomposition with one of the two phases freezing in a glassy state during phase separation. As a model we used the Cahn-Hilliard equation with a concentration-dependent mobility coefficient which decreases rapidly with increasing concentration of the glass-forming component. We solved the Cahn-Hilliard equation numerically for two dimensions. The domain growth depends crucially on the volume fraction of the glassy phase. For high volume fractions, when the glassy phase forms a percolating matrix, a novel coarsening mechanism is discovered, which arises from the migration and coalescence of liquid droplets within the glassy matrix. Various quantities characterizing the time-dependent domain pattern, like droplet size distribution, one- and two-point distribution function and structure factor of the concentration field, are computed. We checked the validity of the dynamic scaling hypothesis.     

Influence of the glass transition on the liquid-gas spinodal decomposition

We use large-scale molecular dynamics simulations to study the kinetics of the liquid-gas phase separation if the temperature is lowered across the glass transition of the dense phase. We observe a gradual change from phase separated systems at high temperatures to nonequilibrium, gel-like structures that evolve very slowly at low temperatures. The microscopic mechanisms responsible for the coarsening strongly depend on temperature, and change from diffusive motion at high temperature to a strongly intermittent, heterogeneous, and thermally activated dynamics at low temperature, leading to logarithmically slow growth of the typical domain size.     

Domain growth and finite-size-scaling in the kinetic Ising model

This paper describes the application of finite-size scaling concepts to domain growth in systems with a non-conserved order parameter. A finite-size-scaling ansatz for the time-dependent order parameter distribution function is proposed, and tested with extensive Monte-Carlo simulations of domain growth in the 2-D spin-flip kinetic Ising model. The scaling properties of the distribution functions serve to elucidate the configurational self-similarity that underlies the dynamic scaling picture. Moreover, it is demonstrated that the application of finite-size-scaling techniques facilitates the accurate determination of the bulk growth exponent even in the presence of strong finite-size effects, the scale and character of which are graphically exposed by the order parameter distribution function. In addition it is found that one commonly used measure of domain size-the scaled second moment of the magnetisation distribution-belies the full extent of these finite-size effects.     

Thermodynamics of small systems embedded in a reservoir: a detailed analysis of finite size effects

We present a detailed study on the finite size scaling behaviour of thermodynamic properties for small systems of particles embedded in a reservoir. Previously, we derived that the leading finite size effects of thermodynamic properties for small systems scale with the inverse of the linear length of the small system, and we showed how this can be used to describe systems in the thermodynamic limit [Chem. Phys. Lett. 504, 199 (2011)]. This approach takes into account an effective surface energy, as a result of the non-periodic boundaries of the small embedded system. Deviations from the linear behaviour occur when the small system becomes very small, i.e. smaller than three times the particle diameter in each direction. At this scale, so-called nook- and corner effects will become important. In this work, we present a detailed analysis to explain this behaviour. In addition, we present a model for the finite size scaling when the size of the small system is of the same order of magnitude as the reservoir. The developed theory is validated using molecular simulations of systems containing Lennard-Jones and WCA particles, and leads to significant improvements over our previous approach. Our approach eventually leads to an efficient method to compute the thermodynamic factor of macroscopic systems from finite size scaling, which is for example required for converting Fick and Maxwell–Stefan transport diffusivities.     

Fluid-fluid phase separation in colloid-polymer mixtures studied with small angle light scattering and light microscopy

Mixtures of colloidal silica spheres and polydimethylsiloxane in cyclohexane with a colloid-polymer size ratio of about one were found to phase separate into two fluid phases, one which is colloid-rich and one which is colloid-poor. In this work the phase separation kinetics of this fluid-fluid phase separation is studied for different compositions of the colloid-polymer mixtures, and at several degrees of supersaturation, with small angle light scattering and with light microscopy. The small angle light scattering curve exhibits a peak that grows in intensity and that shifts to smaller wave vector with time. The characteristic length scale that is obtained from the scattering peak is of the order of a few μm, in agreement with observations by light microscopy. The domain size increases with time as , which might be an indication of coarsening by diffusion and coalescence, like in the case of binary liquid mixtures and polymer blends. For sufficiently low degrees of supersaturation the angular scattering intensity curves satisfy dynamical scaling behavior.     

New structures in complex systems

Networks represent a major modelling tool in complex systems and the natural sciences. When considering systems of interacting units, networks can only model pair interactions as represented by edges between nodes. This is a severe limitation when one tries to model higher order interactions, like triple interactions etc. Some higher order interactions may be reduced to systems of pair interactions, but as we will illustrate there are, for example, triple interactions which are not reducible to pair interactions for quite deep mathematical reasons (Borromean structures). Therefore there is a need for a new kind of structure extending and encompassing networks in such a way that we can describe and model truly higher order structures. We suggest that this can be done by the concept of a Hyperstructure as introduced in [1]. Hyperstructures encompass networks and hierarchies and incorporate the phenomenon of levelwise emergence. They represent a design principle for higher order structures. It is natural to ask how hyperstructures occur in the natural sciences and complex systems and how they may be synthesized. We will discuss this, and relate it to recent work in synthetic chemistry, nuclear physics, quantum mechanical many body systems and ultracold gases. Furthermore, we will introduce the notion of hyperstructured (higher order) molecular architectures and hyperstructured (higher order) materials. We will present suggestions and conjectures on these matters.     

Coarsening in granular systems

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 is relevant in many practical situations, is usually described in terms of coarsening dynamics: there, bubbles of misaligned grains evaporate, allowing the coalescence of optimally arranged islands and a progressive reduction of the total occupied volume.     

Coarsening kinetics with elastic effects

We discuss the coarsening process of melt inclusions inside a solid phase. Elastic effects lead to an oblate shape of the particles, resulting in a system with strong diffusional and elastic interactions between inclusions. The usual mean-field approximation breaks down and several independent length scales have to be taken into account. In a system of parallel oriented particles we find scaling laws for the coarsening of the different length scales involved. In particular, the lateral size of the particles obeys a nontrivial growth law, R approximately t(5/12).     

Some recent advances in the understanding of phase-ordering dynamics

We review some of our recent studies of phase ordering dynamics. Specifically, we describe results from numerical simulations of domain growth in systems with quenched disorder. We also present representative results from numerical studies of phase ordering dynamics in anisotropic systems.     

Forces that Drive Nanoscale Self-assembly on Solid Surfaces

Experimental evidence has accumulated in the recent decade that nanoscale patterns can self-assemble on solid surfaces. A two-component monolayer grown on a solid surface may separate into distinct phases. Sometimes the phases select sizes about 10 nm, and order into an array of stripes or disks. This paper reviews a model that accounts for these behaviors. Attention is focused on thermodynamic forces that drive the self-assembly. A double-welled, composition-dependent free energy drives phase separation. The phase boundary energy drives phase coarsening. The concentration-dependent surface stress drives phase refining. It is the competition between the coarsening and the refining that leads to size selection and spatial ordering. These thermodynamic forces are embodied in a nonlinear diffusion equation. Numerical simulations reveal rich dynamics of the pattern formation process. It is relatively fast for the phases to separate and select a uniform size, but exceedingly slow to order over a long distance, unless the symmetry is suitably broken.