Barbara Niethammer

Non-Self-Similar Behavior in the LSW Theory of Ostwald Ripening

The classical Lifshitz–Slyozov–Wagner theory of domain coarsening predicts asymptotically self-similar behavior for the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field. Here we consider the long-time behavior of measure-valued solutions for systems in which particle size is uniformly bounded, i.e., for initial measures of compact support. We prove that the long-time behavior of the size distribution depends sensitively on the initial distribution of the largest particles in the system. Convergence to the classically predicted smooth similarity solution is impossible if the initial distribution function is comparable to any finite power of distance to the end of the support. We give a necessary criterion for convergence to other self-similar solutions, and conditional stability theorems for some such solutions. For a dense set of initial data, convergence to any self-similar solution is impossible.

On the initial-value problem in the Lifshitz-Slyozov-Wagner theory of Ostwald ripening

The Lifshitz--Slyozov--Wagner (LSW) theory of Ostwald ripening concerns the time evolution of the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field. We prove global existence, uniqueness, and continuous dependence on initial data for measure-valued solutions with compact support in particle size. These results are established with respect to a natural topology on the space of size distributions,one given by the Wasserstein metric which measures the smallest maximum volume change required to rearrange one distribution into another.

Ostwald ripening: The screening length revisited

Abstract. We are interested in the coarsening of a spatial distribution of two phases, driven by the reduction of interfacial energy and limited by diffusion, as described by the Mullins–Sekerka model. We address the regime where one phase covers only a small fraction of the total volume and consists of many disconnected components (“particles”). In this situation, the energetically more advantageous large particles grow at the expense of the small ones, a phenomenon called Ostwald ripening. Lifshitz, Slyozov and Wagner formally derived an evolution for the distribution of particle radii. We extend their derivation by taking into account that only particles within a certain distance, the screening length, communicate. Our arguments are rigorous and are based on a homogenization within a gradient flow structure.

On the rate of convergence to equilibrium in the Becker–Döring equations

Abstract We provide an explicit rate of convergence to equilibrium for solutions of the Becker–Doring equations using the energy/energy-dissipation relation. The main difficulty is the structure of equilibria of the Becker–Doring equations, which do not correspond to a Gaussian measure, such that a logarithmic Sobolev-inequality is not available. We prove a weaker inequality which still implies for fast decaying data that the solution converges to equilibrium as e−ct1/3.

On First-order Corrections to the LSW Theory I: Infinite Systems

We present a new method to efficiently identify the first-order correction to the classical model by Lifshitz, Slyozov and Wagner (LSW). The latter describes the evolution of second phase particles embedded in a matrix during the last stage of a phase transformation and is valid in the regime of vanishing volume fraction φ of particles. We consider a statistically homogeneous (and thus infinite) system, where the first-order correction is of order φ1/2. The key idea is to relate the full system of particles to systems where a finite number of particles has been removed. This method decouples screening and correlation effects and allows to efficiently evaluate conditional expected values of the particle growth rates.

Coarsening Rates in Off-Critical Mixtures

We study coarsening of a binary mixture within the Mullins--Sekerka evolution in the regime where one phase has small volume fraction

Transient coarsening behaviour in the Cahn–Hilliard model

\phi \ll1

Self-similar Solutions with Fat Tails for Smoluchowski’s Coagulation Equation with Locally Bounded Kernels

. Heuristic arguments suggest that the energy density, which represents the inverse of a typical length scale, decreases as

The LSW Model for Domain Coarsening: Asymptotic Behavior for Conserved Total Mass

\phi t^{-1/3}

Domain coarsening in thin films

as