Coarsening Systems Research Articles

Nonlinear energy dissipation and transfers in coarsening systems.

During coarsening, small structures disappear, leaving behind only large ones. Here we study the spectral energy transfers in Model A, where the order parameter ϕ evolves via nonconserved dynamics. We show that the nonlinear interactions dissipate fluctuations and facilitate energy transfers among the Fourier modes so that only ϕ(k=0), where k is the wave number, survives at the end and approaches the asymptotic value +1 or -1. We contrast the coarsening evolution for the initial conditions with 〈ϕ(x,t=0)〉=0 and with uniformly positive or negative ϕ(x,t=0).

Phase Field Modelling of Precipitate Coarsening in Binary Alloys with Respect to Atomic Mobility of Solute in the Precipitate Phase

In the current work, a phase field model is used to study the effect of atomic mobility inside the precipitate phase on coarsening behaviour in two-dimensional (2D) systems. In all the coarsening theories until now, the diffusivity inside the precipitate phase has not been explicitly considered which would imply that there is no chemical potential gradient inside the precipitate. The aim of this study is to assess the potential effect of diffusivity inside the precipitate on coarsening behaviour by examining systems with moderate volume fractions, i.e. $$f_{\mathrm{v}}=0.2$$ , $$f_{\mathrm{v}}=0.3$$ and variable atomic mobilities inside the precipitate phase. The atomic mobility inside the matrix is considered to be constant. The Cahn–Hilliard (C–H)-type nonlinear diffusion equation with variable mobility is numerically discretized and solved for a two-phase coarsening system, using a semi-implicit Fourier spectral method. Coarsening behaviour is studied and analysed by keeping in track, the variation of the number density of precipitates, their average size and size distribution with respect to time. The key observation is that, the atomic mobility inside the precipitate phase has a substantial effect on the coarsening kinetics of precipitates. The effect is more clearly observed when the atomic mobility inside the precipitate phase is greater than that of matrix.

Perturbing cyclic predator-prey systems: How a six-species coarsening system with nontrivial in-domain dynamics responds to sudden changes

Cyclic predator-prey systems have been shown to give rise to rich, and novel, space-time patterns, as for example coarsening domains with non-trivial in-domain dynamics. In this work we study numerically the responses of a cyclic six-species model, characterized by the formation of spirals inside coarsening domains, to two different types of perturbations: changing the values of the predation and reproduction rates as well as changing the interaction scheme. For both protocols we monitor the time evolution of the system after the onset of the perturbation through the measurement of dynamical correlation functions and time-dependent densities of empty sites. In this way we gain insights into the complex responses to different perturbations in a system where spirals, which are due to the formation of cyclic alliances, dominate the dynamics inside the coarsening domains.

Coarsening and mechanics in the bubble model for wet foams.

Aqueous foams are an important model system that displays coarsening dynamics. Coarsening in dispersions and foams is well understood in the dilute and dry limits, where the gas fraction tends to zero and one, respectively. However, foams are known to undergo a jamming transition from a fluidlike to a solidlike state at an intermediate gas fraction ϕ_{c}. Much less is known about coarsening dynamics in wet foams near jamming, and the link to mechanical response, if any, remains poorly understood. Here we probe coarsening and mechanical response using numerical simulations of a variant of the Durian bubble model for wet foams. As in other coarsening systems we find a steady state scaling regime with an associated particle size distribution. We relate the time rate of evolution of the coarsening process to the wetness of the foam and identify a characteristic coarsening time that diverges approaching jamming. We further probe mechanical response of the system to strain while undergoing coarsening. There are two competing timescales, namely the coarsening time and the mechanical relaxation time. We relate these to the evolution of the elastic response and the mechanical structure.

Critical percolation in bidimensional coarsening

I discuss a recently unveiled feature in the dynamics of two dimensional coarsening systems on the lattice with Ising symmetry: they first approach a critical percolating state via the growth of a new length scale, and only later enter the usual dynamic scaling regime. The time needed to reach the critical percolating state diverges with the system size. These observations are common to Glauber, Kawasaki, and voter dynamics in pure and weakly disordered systems. An extended version of this account appeared in 2016 C. R. Phys. . I refer to the relevant publications for details.

On a relation between roughening and coarsening

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 andstochastic dilution.

Molecular dynamics study of phase separation in fluids with chemical reactions.

We present results from the first d=3 molecular dynamics (MD) study of phase-separating fluid mixtures (AB) with simple chemical reactions (A⇌B). We focus on the case where the rates of forward and backward reactions are equal. The chemical reactions compete with segregation, and the coarsening system settles into a steady-state mesoscale morphology. However, hydrodynamic effects destroy the lamellar morphology which characterizes the diffusive case. This has important consequences for the phase-separating structure, which we study in detail. In particular, the equilibrium length scale (ℓ(eq)) in the steady state suggests a power-law dependence on the reaction rate ε:ℓ(eq)∼ε(-θ) with θ≃1.0.

The envelope of size distributions in Ostwald ripening and grain growth

In the present paper the scaling properties of microstructure coarsening by Ostwald ripening or grain growth are studied by means of the envelope of the family of evolving size distribution functions, which complements the usual scaling analysis by some new aspects and results. For a self-similar family of size distribution functions the envelope is uniquely determined by the growth exponent and an envelope parameter, where the latter is a new characteristic quantity of the coarsening system associated with the scaled size distribution function. If the family of size distributions obeys the continuity equation their envelope is the location of the maximum particle flux density in size space, while for the corresponding cumulative size distribution the envelope is the location of maximum size that a particle or grain can take during the passage through its growth path. The construction of the envelope curve therefore allows the independent determination of some coarsening parameters without recourse to the full time development of the individual growth paths. Numerical studies by means of the Monte Carlo Potts model of grain growth confirm and complement the analytical results. Especially they reveal that at the early stage of growth the envelope has a much reduced and non-integer exponent.

Structural evolution of nanoporous gold during thermal coarsening

The three-dimensional evolution of nanoligaments of nanoporous gold created by Ag–Au dealloying was studied during isothermal coarsening by X-ray nanotomography and microbeam Laue diffraction. The surface normal orientation, curvature and size of the gold nanoligaments were measured as a function of coarsening time (from 2 to 320min). The following observations were made at 550, 600 and 650°C. First, the distribution of orientations for the surfaces of the nanoligaments becomes more anisotropic with coarsening time, with an increasing area of the surfaces having a low surface energy, consistent with the growth of facets. Second, the curvature distribution of the nanoligaments (scaled by their size) also evolves during coarsening. The evolution of both surface orientation and scaled surface curvature indicates that coarsening does not occur in a self-similar manner, i.e. the interfacial shape distribution of the gold nanoligaments is not self-similar over time as they coarsen. This is consistent with the ligament size not being described by a classical temporal power law for coarsening systems. All three effects, and in particular the increased prevalence of surfaces with a low surface energy at long coarsening times, may affect the surface functionalities and properties of nanoporous gold in various applications, e.g. as catalysts, sensors and actuators.

Heat exchanges in coarsening systems

This paper is a contribution to the understanding of the thermal properties of ageingsystems where statistically independent degrees of freedom with greatly separated timescales are expected to coexist. Focusing on the prototypical case of quenched ferromagnets,where fast and slow modes can be respectively associated with fluctuations inthe bulk of the coarsening domains and in their interfaces, we perform a set ofnumerical experiments specifically designed to compute the heat exchanges betweendifferent degrees of freedom. Our studies promote a scenario with fast modesacting as an equilibrium reservoir to which interfaces may release heat through amechanism that allows fast and slow degrees to maintain their statistical propertiesindependently.

Modified fluctuation-dissipation theorem for general non-stationary states and applicationto the Glauber–Ising chain

In this paper, we present a general derivation of a modified fluctuation-dissipation theorem(MFDT) valid near an arbitrary non-stationary state for a system obeying Markoviandynamics. We show that the method for deriving modified fluctuation-dissipation theoremsnear non-equilibrium stationary states used by Prost et al (2009 Phys. Rev. Lett. 103090601) is generalizable to non-stationary states. This result follows from both standardlinear response theory and from a transient fluctuation theorem, analogous to theHatano–Sasa relation. We show that this modified fluctuation-dissipation theorem can beinterpreted at the trajectory level using the notion of stochastic trajectory entropy, in away which is similar to what has been done recently in the case of the MFDT nearnon-equilibrium steady states (NESS). We illustrate this framework with two solvableexamples: the first example corresponds to a Brownian particle in a harmonic trapsubjected to a quench of temperature and to a time-dependent stiffness; the secondexample is a classic model of coarsening systems, namely the 1D Ising model with Glauberdynamics.

Fluctuations and effective temperatures in coarsening

We study dynamic fluctuations in non-disordered finite dimensionalferromagnetic systems quenched to the critical point and the low temperaturephase. We investigate the fluctuations of two two-time quantities, calledχ andC, theaverages of which yield the self-linear response and correlation functions. We introduce a restricted averageof the χs, summing over all configurations with a given value ofC. We find that therestricted average ⟨χ⟩C obeys a scaling form, and that the slope of the scaling function approaches the universalvalue of the limiting effective temperature in the long time limit and for . Our results tend to confirm the expectation that time-reparameterization invariance isnot realized in coarsening systems at criticality. Finally, we discuss possible experimentaltests of our proposal.

Nonlinear response and fluctuation-dissipation relations

A unified derivation of the off-equilibrium fluctuation dissipation relations (FDRs) is given for Ising and continuous spins to arbitrary order, within the framework of Markovian stochastic dynamics. Knowledge of the FDRs allows one to develop zero field algorithms for the efficient numerical computation of the response functions. Two applications are presented. In the first one, the problem of probing for the existence of a growing cooperative length scale is considered in those cases, like in glassy systems, where the linear FDR is of no use. The effectiveness of an appropriate second order FDR is illustrated in the test case of the Edwards-Anderson spin glass in one and two dimensions. In the second application, the important problem of the definition of an off-equilibrium effective temperature through the nonlinear FDR is considered. It is shown that, in the case of coarsening systems, the effective temperature derived from the second order FDR is consistent with the one obtained from the linear FDR.

Towards a Statistical Theory of Texture Evolution in Polycrystals

Most technologically useful materials possess polycrystalline microstructures composed of a large number of small monocrystalline grains separated by grain boundaries. The energetics and connectivity of the grain boundary network play a crucial role in determining the properties of a material across a wide range of scales. A central problem in materials science is to develop technologies capable of producing an arrangement of grains—a texture—that provides for a desired set of material properties. One of the most challenging aspects of this problem is to understand the role of topological reconfigurations during coarsening. Here we propose an upscaling procedure suitable for large complex systems. The procedure is based on numerical experimentation combined with stochastic tools and consists of large-scale numerical simulations of a system at a microscopic level, statistical analysis of the microscopic data, and formulation of the model based on stochastic characteristics predicted by the statistical analysis. This approach promises to be valuable in establishing the effective model of microstructural evolution in realistic two- and three-dimensional systems. To test ideas we use our upscaling procedure to study the mesoscopic behavior of a reduced one-dimensional network of grain boundaries. Despite the simplicity of its formulation, this model exhibits highly nontrivial behavior characterized by growth and disappearance of grain boundaries and develops probability distributions similar to those observed in higher-dimensional simulations. Here we focus on the grain deletion events which are common to all coarsening systems.

Kinetics for coarsening co-controlled by diffusion and a reversible interface reaction

The kinetics for coarsening co-controlled by diffusion and a reversible interface reaction were investigated theoretically. Equations for the particle size distributions and particle growth were deduced, through which the effects of the competition, in terms of the rate-controlling strength, between diffusion and the interface reaction on the kinetics behaviors were revealed. When the coarsening was mainly controlled by the interface reaction the particle size distribution was broad and squat but, along with the enhancement of the rate-controlling strength of diffusion, it changed continuously to be narrower in shape and finally became the form suggested in previous research. The particle growth formula was the sum of the third and second power laws. It can be approximately expressed as R ¯ n = Kt in a relatively short period of time, with the exponent varying between 2 and 3, depending on the coarsening stage and/or coarsening system.

Scaling and universality in the aging kinetics of the two-dimensional clock model

We study numerically the aging dynamics of the two-dimensional p -state clock model after a quench from an infinite temperature to the ferromagnetic phase or to the Kosterlitz-Thouless phase. The system exhibits the general scaling behavior characteristic of nondisordered coarsening systems. For quenches to the ferromagnetic phase, the value of the dynamical exponents suggests that the model belongs to the Ising-type universality class. Specifically, for the integrated response function chi(t,s) approximately or = s(-a)chif(t/s), we find a(chi) consistent with the value a(chi)=0.28 found in the two-dimensional Ising model.

Effect of shear on persistence in coarsening systems

We analytically study the effect of a uniform shear flow on the persistence properties of coarsening systems. The study is carried out within the anisotropic Ohta-Jasnow-Kawasaki (OJK) approximation for a system with nonconserved scalar order parameter. We find that the persistence exponent theta has a nontrivial value: theta = 0.5034 in space dimension d = 3, and theta = 0.2406 for d = 2, the latter being exactly twice the value found for the unsheared system in d = 1. We also find that the autocorrelation exponent lambda is affected by shear in d = 3 but not in d = 2.

Fluctuations in the coarsening dynamics of the O(N) model with N → ∞: are they similar to those in glassy systems?

We study spatio-temporal fluctuations in the non-equilibrium dynamics of thed-dimensionalO(N) model inthe large N limit. We analyse the invariance of the dynamic equations for the global correlation andresponse in the slow ageing regime under transformations of time. We find that theseequations are invariant under scale transformations. We extend this study to the action inthe dynamic generating functional, finding similar results. This model therefore fallsinto a different category to glassy problems in which full time reparametrizationinvariance, a larger symmetry that encompasses timescale invariance, is expected to berealized asymptotically. Consequently, the spatio-temporal fluctuations of the largeN O(N) model should follow a pattern different from that for glassy systems. We compute thefluctuations of local, as well as spatially separated, two-field composite operators andresponses, and we confront our results with the ones found numerically for the 3DEdwards–Anderson model and kinetically constrained lattice gases. We analyse thedependence of the fluctuations of the composite operators on the growing domain lengthand we compare to what has been found for supercooled liquids and glasses. Finally, weshow that the development of time reparametrization invariance in glassy systems isintimately related to a well-defined and finite effective temperature, specified from themodification of the fluctuation-dissipation theorem out of equilibrium. We then conjecturethat the global asymptotic time reparametrization invariance is broken down to timescaleinvariance in all coarsening systems.

Maximally fast coarsening algorithms

We present maximally fast numerical algorithms for conserved coarsening systems that are stable and accurate with a growing natural time step Deltat=At2/3s. We compare the scaling structure obtained from our maximally fast conserved systems directly against the standard fixed time-step Euler algorithm, and find that the error scales as square root of A--so arbitrary accuracy can be achieved. For nonconserved systems, only effectively finite time steps are accessible for similar unconditionally stable algorithms.

Screening in Interacting Particle Systems

We consider the Green's function of the Laplace operator in domains with spherical holes (particles). Under natural assumptions on the distribution of particles we show that the Green's function decays exponentially over distances larger than the screening length. This result is fundamental for example when deriving effective equations for coarsening systems in unbounded domains.