Peter Smereka

An improved level set method for incompressible two-phase flows

Abstract A level set method for capturing the interface between two fluids is combined with a variable density projection method to allow for computation of a two-phase flow where the interface can merge/break and the flow can have a high Reynolds number. A distance function formulation of the level set method enables us to compute flows with large density ratios (1000/1) and flows that are surface tension driven, with no emotional involvement. Recent work has improved the accuracy of the distance function formulation and the accuracy of the advection scheme. We compute flows involving air bubbles and water drops, among others. We validate our code against experiments and theory.

Axisymmetric free boundary problems

We present a number of three-dimensional axisymmetric free boundary problems for two immiscible fluids, such as air and water. A level set method is used where the interface is the zero level set of a continuous function while the two fluids are solutions of the incompressible Navier–Stokes equation. We examine the rise and distortion of an initially spherical bubble into cap bubbles and toroidal bubbles. Steady solutions for gas bubbles rising in a liquid are computed, with favourable comparisons to experimental data. We also study the inviscid limit and compare our results with a boundary integral method. The problems of an air bubble bursting at a free surface and a liquid drop hitting a free surface are also computed.

Semi-Implicit Level Set Methods for Curvature and Surface Diffusion Motion

In this paper we introduce semi-implicit methods for evolving interfaces by mean curvature flow and surface diffusion using level set methods.

Simulation of faceted film growth in three dimensions: microstructure, morphology and texture

We present the results of a series of simulations of the growth of polycrystalline, faceted films in three spatial dimensions. The simulations are based upon the assumptions of the well known van der Drift model in which the growth rate of each surface is fixed only by its crystallographic orientation. The simulation method is based upon the level-set formalism and the only input are the relative velocities of the different facets. We focus specifically on cubic crystals that expose only {111} and {001} facets, such as diamond. Results are presented for the temporal evolution of the surface morphology, microstructure, mean grain size, grain size distribution, and crystallographic texture. The mean grain size and surface roughness increase with film thickness h as h, in agreement with theoretical results. The grain size distribution is self-similar. The films all exhibit a columnar microstructure and a fiber texture that sharpens as the film grows. The orientation of the texture is determined by the facet growth velocity ratio. The new simulation method is equally applicable to any type of faceted film growth. 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

On the motion of bubbles in a periodic box

The motion of spherical bubbles in a box containing an incompressible and irrotational liquid with periodic boundary conditions is studied. Equations of motion are deduced using a variational principle. When the bubbles have approximately the same velocity their configuration is Lyapunov stable, provided they are arranged in such a way so as to minimize the effective conductivity of a composite material where the bubbles are treated as insulators. The minimizing configurations become asymptotically stable with the addition of gravity and liquid viscosity. This suggests that a randomly arranged configuration of bubbles, all with approximately the same velocity, cannot be stable

Mechanisms of Stranski-Krastanov growth

Stranski-Krastanov (SK) growth is reported experimentally as the growth mode that is responsible for the transition to three dimensional islands in heteroepitaxial growth. A kinetic Monte Carlo (KMC) model is proposed that can replicate many of the experimentally observed features of this growth mode. Simulations reveal that this model effectively captures the SK transition and subsequent growth. Annealing simulations demonstrate that the wetting layer formed during SK growth is stable, with entropy playing a key role in its stability. It is shown that this model also captures the apparent critical thickness that tends to occur at higher deposition rates and for alloy films (where intermixing is significant). This work shows that the wetting layer thickness increases with increasing temperature, whereas the apparent critical thickness decreases with increasing temperature. Both of which are in agreement with experiments.

Spiral crystal growth

Abstract We numerically study the spiral mode of crystal growth using a theory developed by Burton, Cabrera and Frank using a level set method. This method is novel in that it can handle not only closed curves but open curves as well. We use our method to compute interacting spirals and make estimates of growth rates. We also propose a possible coarsening mechanism for a large number of interacting spirals.

Interacting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality

The stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation is∂tU(x,t)=D∂xxU+γU(1−U)+eU(1−U)η(x,t)for 0⩽U⩽1 where η(x,t) is a Gaussian white noise process in space and time. Here D, γ and e are parameters and the equation is interpreted as the continuum limit of a spatially discretized set of Ito equations. Solutions of this stochastic partial differential equation have an exact connection to the A⇌A+A reaction–diffusion system at appropriate values of the rate coefficients and particles’ diffusion constant. This relationship is called “duality” by the probabilists; it is not via some hydrodynamic description of the interacting particle system. In this paper we present a complete derivation of the duality relationship and use it to deduce some properties of solutions to the stochastic Fisher–Kolmogorov–Petrovsky–Piscunov equation.

A Level-Set Method for the Evolution of Faceted Crystals

A level-set formulation for the motion of faceted interfaces is presented. The evolving surface of a crystal is represented as the zero-level of a phase function. The crystal is identified by its orientation and facet speeds. Accuracy is tested on a single crystal by comparison with the exact evolution. The method is extended to study the evolution of a polycrystal. Numerical examples in two and three dimensions are presented.

Coupling kinetic Monte-Carlo and continuum models with application to epitaxial growth

We present a hybrid method for simulating epitaxial growth that combines kinetic Monte-Carlo (KMC) simulations with the Burton-Cabrera-Frank model for crystal growth. This involves partitioning the computational domain into KMC regions and regions where we time-step a discretized diffusion equation. Computational speed and accuracy are discussed. We find that the method is significantly faster than KMC while accounting for stochastic fluctuations in a comparable way.