Evaporation of droplets in the two-dimensional Ginzburg–Landau equation

We consider the problem of coarsening in two dimensions for the real (scalar) Ginzburg–Landau equation. This equation has exactly two stable stationary solutions, the constant functions +1 and −1. We assume most of the initial condition is in the “−1” phase with islands of “+1” phase. We use invariant manifold techniques to prove that the boundary of a circular island moves according to Allen–Cahn curvature motion law. We give a criterion for non-interaction of two arbitrary interfaces and a criterion for merging of two nearby interfaces.     

Dissociaton and energy transfer in molecular impact on surfaces: Experimental and theoretical studies of I2/Mg(100) and I2/sapphire

Solute segregation to antiphase boundaries (APBs) in long-range ordered alloys and its effects on antiphase domain coarsening kinetics have been investigated theoretically, and calculations have been carried out to model the structure and properties of APBs in B2-ordered FeAl alloys. Equilibrium segregation was studied by using the continuum diffuse-interface model of Cahn and Hilliard to calculate profiles of order parameter and composition, as well as interfacial free energy. The migration kinetics of APBs with segregation have been investigated theoretically for the low-velocity regime. A differential equation describing concentration deviations from the equiibrium profile is derived, and approximate solutions to the equation are determined to predict segregation profiles for migrating APBs in FeAl alloys. Measurements of domain coarsening kinetics in FeAl alloys are presented for a temperature range in which segregation was predicted theoretically. A marked slowing of domain coarsening kinetics in this range was observed.     

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.     

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.     

Domain wall motion in nanowires

We investigated the motion of domain walls in ferromagnetic cylindrical nanowires by solving the Landau–Lifshitz–Gilbert equation numerically for a classical spin model in which energy contributions from exchange, crystalline anisotropy, dipole–dipole interactions, and a driving magnetic field are considered. Depending on the diameter, either transverse domain walls or vortex walls are found. A transverse domain wall is observed for diameters smaller than the exchange length of the given system. In this case, the system effectively behaves one dimensionally and the domain wall velocity agrees with the result of Slonczewski for one-dimensional walls. For larger diameters, a crossover to a vortex wall sets in which enhances the domain wall velocity drastically. For a vortex wall the domain wall velocity is described by the Walker formula.     

Local discontinuous Galerkin methods for the Cahn–Hilliard type equations

In this paper, we develop local discontinuous Galerkin (LDG) methods for the fourth order nonlinear Cahn–Hilliard equation and system. The energy stability of the LDG methods is proved for the general nonlinear case. Numerical examples for the Cahn–Hilliard equation and the Cahn–Hilliard system in one and two dimensions are presented and the numerical results illustrate the accuracy and capability of the methods.     

Formal determination of the nucleation probabilities in binary alloys: A hyperfinite approach and thermodynamic justification of the lattice models

We obtained an equation of the Burgers type modeling the glass transition process in binary alloys with inhomogeneous inclusions. The proposed equation is thermodynamically justified; conditions are indicated under which this equation converts into the classical Cahn–Hillard equation.     

A Simple Kinetic Model for the Phase Transition of the van der Waals Fluid

A simple kinetic model, which is presumably minimum, for the phase transition of the van der Waals fluid is presented. In the model, intermolecular collisions for a dense gas has not been treated faithfully. Instead, the expected interactions as the non-ideal gas effect are confined in a self-consistent force term. Collision term plays just a role of thermal bath. Accordingly, it conserves neither momentum nor energy, even globally. It is demonstrated that (i) by a natural separation of the mean-field self-consistent potential, the potential for the non-ideal gas effect is determined from the equation of state for the van der Waals fluid, with the aid of the balance equation of momentum, (ii) a functional which monotonically decreases in time is identified by the H theorem and is found to have a close relation to the Helmholtz free energy in thermodynamics, and (iii) the Cahn–Hilliard type equation is obtained in the continuum limit of the present kinetic model. Numerical simulations based on the Cahn–Hilliard type equation are also performed.     

Glassy dynamics and coarsening

In this paper I present a short review of the present theoretical understanding of the dynamics, global and fluctuating, in the glassy out of equilibrium regime. The first part of the talks dealt with a summary of the results of mean-field models and served to introduce the conjecture that time-reparametrization invariance controls the fluctuating dynamics. Finally, I briefly discussed very recent results on the domain size fluctuations in coarsening problems and how these and similar studies of the dynamics of elastic lines could help us in understanding the evolution of more complex glassy systems.     

From bell shapes to pyramids: A reduced continuum model for self-assembled quantum dot growth

A continuum model for the growth of self-assembled quantum dots that incorporates surface diffusion, an elastically deformable substrate, wetting interactions and anisotropic surface energy is presented. Using a small slope approximation a thin-film equation for the surface profile that describes faceted growth is derived. A linear stability analysis shows that anisotropy acts to destabilize the surface. It lowers the critical height of flat films and there exists an anisotropy strength above which all thicknesses are unstable. A numerical algorithm based on spectral differentiation is presented and simulations are carried out. These clearly show faceting of the growing islands and a power law coarsening behavior.     

Surface magnetization,grain and domain structure in laser-scribed soft magnetic sheets

The local magnetic softening and simultaneous grain coarsening are observed in low carbon sheets due to the laser-scribing process as the results of the net mechanical stress relief. A very fine surface domain structure with domain walls of higher mobility is created within the scribed zone, in spite of the fact that local increase of microhardness of the laser-treated zone was detected.     

Mechanism for coarsening of P-mediated Ge quantum dots during in-situ annealing

The coarsening of phosphorus-mediated Ge quantum dots (QDs) on Si(0 0 1) during in-situ annealing at 550 °C is studied. In-situ annealing makes the as-grown sample morphology be remarkably changed: the larger dots are formed and the dot density is greatly reduced. The results of chemical etching and Raman spectra reveal that the incorporation of Ge atoms which originate from the diminishing dots, rather than substrate Si atom incorporation is responsible for the dot coarsening at the incipient stage of in-situ annealing. Besides, Raman spectra suggest that the larger dots formed during in-situ annealing are dislocated, which was confirmed by cross-sectional high-resolution electron microscopy observation. Through the generation of dislocations, the strain in the dots is relaxed by about 50%.     

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.     

A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios

We present an efficient time-stepping scheme for simulations of the coupled Navier–Stokes Cahn–Hilliard equations for the phase field approach. The scheme has several attractive characteristics: (i) it is suitable for large density ratios, and numerical experiments with density ratios up to 1000 have been presented; (ii) it involves only constant (time-independent) coefficient matrices for all flow variables, which can be pre-computed during pre-processing, so it effectively overcomes the performance bottleneck induced by variable coefficient matrices associated with the variable density and variable viscosity; (iii) it completely de-couples the computations of the velocity, pressure, and the phase field function. Strategy for spectral-element type spatial discretizations to overcome the difficulty associated with the large spatial order of the Cahn–Hilliard equation is also discussed. Ample numerical simulations demonstrate that the current algorithm, together with the Navier–Stokes Cahn–Hilliard phase field approach, is an efficient and effective method for studying two-phase flows involving large density ratios, moving contact lines, and interfacial topology changes.     

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.     

A multi-scale model of domain wall velocities based on ab initio parameters

A new approach is presented to evaluate the velocity of field-driven domain walls by means of ab initio parameters. This approach makes intensive use of multi-scaling by means of (a) mapping of domain wall formation energies obtained in terms of a fully relativistic method onto a Landau–Ginzburg-type expression, and (b) applying the Landau–Lifshitz–Gilbert equation to evaluate the time needed to move domain walls. In comparison with the “classical” expression for the domain wall velocity originally proposed by Landau and Lifshitz, according to which the velocity increases with increasing width of the domain wall, three different types of magnetic alloys, namely permalloy (Ni₈₅Fe₁₅), Co _x Ni_1?x and Co _x Pd_1?x , are analyzed. It is shown that the Landau–Lifshitz expression for the velocity seems to be valid whenever the slopes of the exchange and the anisotropy energy with respect to the concentration are either both increasing or both decreasing.     

An Integrodifferential Model for Phase Transitions: Stationary Solutions in Higher Space Dimensions

We study the existence and stability of stationary solutions of an integrodifferential model for phase transitions, which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity. As such, this model is a nonlocal extension of the Allen–Cahn equation, which incorporates long-range interactions. We find that the set of stationary solutions for this model is much larger than that of the Allen–Cahn equation.     

Topics in coarsening phenomena

These lecture notes give a very short introduction to coarsening phenomena and summarize some recent results in the field. They focus on three aspects: the super-universality hypothesis, the geometry of growing structures, and coarsening in the spiral kinetically constrained model.     

Comparison between the generalized tanh–coth and the (<Emphasis Type="Italic">G</Emphasis>′/<Emphasis Type="Italic">G</Emphasis>)-expansion methods for solving NPDEs and NODEs

In this paper, we find exact solutions of some nonlinear evolution equations by using generalized tanh–coth method. Three nonlinear models of physical significance, i.e. the Cahn–Hilliard equation, the Allen–Cahn equation and the steady-state equation with a cubic nonlinearity are considered and their exact solutions are obtained. From the general solutions, other well-known results are also derived. Also in this paper, we shall compare the generalized tanh–coth method and generalized (G ^′/G )-expansion method to solve partial differential equations (PDEs) and ordinary differential equations (ODEs). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important roles in engineering fields. The generalized tanh–coth method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the generalized tanh–coth method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems.