Massimo Conti

Phase ordering with a global conservation law: Ostwald ripening and coalescence

Globally conserved phase ordering dynamics is investigated in systems with short range correlations at t=0. A Ginzburg-Landau equation with a global conservation law is employed as the phase field model. The conditions are found under which the sharp-interface limit of this equation is reducible to the area-preserving motion by curvature. Numerical simulations show that, for both critical and off-critical quench, the equal-time pair correlation function exhibits dynamic scaling, and the characteristic coarsening length obeys l(t) approximately t(1/2). For the critical quench, our results are in excellent agreement with earlier results. For off-critical quench (Ostwald ripening) we investigate the dynamics of the size distribution function of the minority phase domains. The simulations show that, at large times, this distribution function has a self-similar form with growth exponent 1/2. The scaled distribution, however, strongly differs from the classical Wagner distribution. We attribute this difference to coalescence of domains. A theory of Ostwald ripening is developed that takes into account binary coalescence events. The theoretical scaled distribution function agrees well with that obtained in the simulations.

Phase field theory of crystal nucleation in hard sphere liquid

The phase field theory of crystal nucleation described in L. Granasy, T. Borzsonyi, and T. Pusztai, Phys. Rev. Lett. 88, 206105 (2002) is applied for nucleation in hard-sphere liquids. The exact thermodynamics from molecular dynamics is used. The interface thickness for phase field is evaluated from the cross-interfacial variation of the height of the singlet density peaks. The model parameters are fixed in equilibrium so that the free energy and thickness of the (111), (110), and (100) interfaces from molecular dynamics are recovered. The density profiles predicted without adjustable parameters are in a good agreement with the filtered densities from the simulations. Assuming spherical symmetry, we evaluate the height of the nucleation barrier and the Tolman length without adjustable parameters. The barrier heights calculated with the properties of the (111) and (110) interfaces envelope the Monte Carlo results, while those obtained with the average interface properties fall very close to the exact values. In contrast, the classical sharp interface model considerably underestimates the height of the nucleation barrier. We find that the Tolman length is positive for small clusters and decreases with increasing size, a trend consistent with computer simulations.

Breakdown of Scale Invariance in the Phase Ordering of Fractal Clusters

Our numerical simulations with the Cahn-Hilliard equation show that coarsening of fractal clusters (FCs) is not a scale-invariant process. On the other hand, a typical coarsening length scale and interfacial area of the FC exhibit power laws in time, while the mass fractal dimension remains invariant. The initial value of the lower cutoff is a relevant length scale. A sharp-interface model is formulated that can follow the whole dynamics of a diffusion controlled growth, coarsening, fragmentation and approach to equilibrium in a system with conserved order parameter.

Scaling anomalies in the coarsening dynamics of fractal viscous fingering patterns.

We analyze a recent experiment of Sharon et al. (2003) on the coarsening, due to surface tension, of fractal viscous fingering patterns (FVFPs) grown in a radial Hele-Shaw cell. We argue that an unforced Hele-Shaw model, a natural model for that experiment, belongs to the same universality class as model B of phase ordering. Two series of numerical simulations with model B are performed, with the FVFPs grown in the experiment and with diffusion limited aggregates as the initial conditions. We observed Lifshitz-Slyozov scaling t(1/3) at intermediate distances and slow convergence to this scaling at small distances. Dynamic scale invariance breaks down at large distances.

Normal scaling in globally conserved interface-controlled coarsening of fractal clusters.

We find that globally conserved interface-controlled coarsening of diffusion-limited aggregates exhibits dynamic scale invariance (DSI) and normal scaling. This is demonstrated by a numerical solution of the Ginzburg-Landau equation with a global conservation law. The general sharp-interface limit of this equation is introduced and reduced to volume preserving motion by mean curvature. A simple example of globally conserved interface-controlled coarsening system: the sublimation/deposition dynamics of a solid and its vapor in a small closed vessel, is presented in detail. The results of the numerical simulations show that the scaled form of the correlation function has a power-law tail accommodating the fractal initial condition. The coarsening length exhibits normal dynamic scaling. A decrease of the cluster radius with time, predicted by DSI, is observed. The difference between global and local conservation is discussed.

Interfacial dynamics in rapid solidification processes

The broad interest in rapid solidification processes originates from the variety of non-equilibrium microstructures found in the solidified materials; most familiar examples are dendrites, lamellar eutectics, cellular and banded structures. In spite of the deep efforts devoted to this subject by the scientific community, the interfacial dynamics is far from being well understood. The aim of this article is to review some recent theoretical developments, which give a diffuse picture of the interfacial region. A new microscopic approach, based on the stochastic dynamics of a lattice of Potts spins, is also discussed.

Fingering in slow combustion

We report numerical simulations of the structure of advancing burning fronts under conditions where the dominating transport mechanisms is the diffusion of oxygen and heat. The model we study describes the interplay between two diffusive fields: one which accounts for the destabilizing mechanism leading to the production of a large surface area and the other for a stabilizing mechanism. The typical length scale associated with the observed fingering instability turns out to be a combination of the diffusion lengths associated with the two competing processes.

A microscopic model for solidification

We present a novel picture of a non-isothermal solidification process starting from a molecular level, where the microscopic origin of the basic mechanisms and of the instabilities characterizing the approach to equilibrium is rendered more apparent than in existing approaches based on coarse-grained free-energy functionals a la Landau. The system is composed of a lattice of Potts spins, which change their state according to the stochastic dynamics proposed some time ago by Creutz. Such a method is extended to include the presence of latent heat and thermal conduction. Not only the model agrees with previous continuum treatments, but it allows to introduce in a consistent fashion the microscopic stochastic fluctuations. These play an important role in nucleating the growing solid phase in the melt. The approach is also very satisfactory from the quantitative point of view since the relevant growth regimes are fully characterized in terms of scaling exponents.

Novel Monte-Carlo lattice approach to rapid directional solidification of binary alloys

A statistical approach is developed to study the phenomena related to the rapid solidification of binary alloys. We consider a microscopic lattice Hamiltonian containing an Ising-like variable to account for the alloy character of the model and a Potts-like variable to describe the presence of a solid or liquid spot. By means of a suitable set of Monte-Carlo evolution rules for these variables we are able to describe faithfully the advancing of a solid/liquid interface.

Density Effects and Fluid Flow in Phase-field Models

In this paper we discuss a method to incorporate hydrodynamic effects into phase-field models for the solidification of both pure substances and binary alloys. We start from a generalised thermodynamic potential with squared gradient terms for the associated fields; the condition of local positive entropy production is then utilised to derive a set of equations which drive the system towards equilibrium. The models are numerically solved in one dimension; the effects of the flow field on the growth dynamics are presented and discussed.