Efficient phase field simulations of multiple crystal orientations [Elektronische Ressource] / vorgelegt von Jürgen Hubert

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• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2009
• Nombre de lectures: 15
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

“Efficient Phase-field Simulations of Multiple
Crystal Orientations”
Von der Fakult¨at fu¨r Georessourcen und Materialtechnik der
Rheinisch-Westf¨alischen Technischen Hochschule Aachen
zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften
genehmigte Dissertation
vorgelegt von Dipl. Phys. Ju¨rgen Hubert
aus Erlangen
Berichter: Univ.-Prof. Dr.-Ing. Heike Emmerich
Prof. Dr. rer.nat. Britta Nestler
Tag der mu¨ndlichen Pru¨fung: 27. Juli 2009
DieseDissertationistaufdenInternetseitenderHochschulbibliothekonlineverfu¨gbarContents
Introduction 1
1 A Phase-field Model for Crystal Growth with Multiple Ori-
entations 7
1.1 The Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Eliminating Lattice Effects . . . . . . . . . . . . . . . . . . . . 11
1.3 Phase-field Modelling of Vein Formation . . . . . . . . . . . . 15
1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 A Model for Nanoparticle Growth - Analysis 25
2.1 The Energy Functional . . . . . . . . . . . . . . . . . . . . . . 28
2.2 The Model Equations . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Interface Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 The Outer Expansion . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 The Inner Expansion . . . . . . . . . . . . . . . . . . . . . . . 39
2.6.1 Leading Order . . . . . . . . . . . . . . . . . . . . . . . 40
2.6.2 First Order . . . . . . . . . . . . . . . . . . . . . . . . 41
iii
2.6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 A Model for Nanoparticle Growth - Numerical Simulations 45
3.1 Particle Growth with Infinite Reservoir . . . . . . . . . . . . . 46
3.2 Particle Growth with Periodic Boundary Conditions . . . . . . 50
3.3 Particle Growth with Multiple Orientations . . . . . . . . . . 51
3.3.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 MultipleCrystalOrientationsUsingthePhase-FieldCrystal
Method 57
4.1 Underlying Simulation Approach . . . . . . . . . . . . . . . . 59
4.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 60
4.2.1 Simulation Set-up and Numerical Results . . . . . . . . 60
4.2.2 Analysis and Interpretation of our Results . . . . . . . 63
4.3 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . 67
Summary and Outlook 71
Appendix 75
Acknowledgements 77
Bibliography 78
Kurzfassung 91
Abstract 93
Curriculum Vitae 95Introduction
The desire to describe phase transitions analytically and numerically has
long been a strong motivating force in the Materials Science community,
as trying to understand materials systems merely by experimental studies
requires a great deal of effort, time, and resources. Earlier attempts of de-
scribing solidification processes used so-called sharp interface formulations,
which are derived from phenomenological observations [24] and treat the
boundary between phases as a line of infinite thinness. A typical sharp in-
terface formulation uses a diffusion equation for the bulk of the material, the
Gibbs-Thomson relation for the local equilibrium near the interface, and the
Stefan condition for the conservation of mass and energy at the interface [2].
However, while such approaches have been successfully used for numerical
simulations, they suffer from several drawbacks [24] - as they need to track
a moving interface line over a discretized grid describing the temperature
and/orconcentrationfield, theexactlocal conditionsattheinterfaceneedto
be interpolated, which is a likely source for numerical errors. Furthermore,
as the number of discretization points describing the path of the interface
varies from time step to time step, it is difficult to introduce parallelization
techniques for simulating complex problems. Finally, this approach is diffi-
12 Introduction
culttoextendtothreedimensions,asthatwouldrequireaconstantlymoving
surface instead of just a one-dimensional line.
Thelogicalnextstepinmodelingphasetransitionswasthereforeadiffuse
interface formulation - the so-called phase-field method. First developed by
Langer [44], it has become a powerful tool for numerical simulations of phase
transitions in the past two decades [11, 25, 41, 44, 72] and no longer tracks
the interface directly. Instead, it treats the interface as a diffuse boundary
of finite, but not infinitely small thickness, and derives the model equations
from an energy functional describing the total energy of the examined sys-
tem. Thus, it is no longer necessary to work out the precise parameters
for the sharp interface formulation for numerical simulation. However, it is
possible to derive the sharp interface formulation from the phase-field model
equations via an asymptotic analysis by expanding and matching the model
equations both in the bulk and near the interface (in the latter case via a
coordinate transformation in which the width of the interface is approaching
zero) - which is a useful tool for making quantitative statements about the
kinetics of solidification phenomena [24, 74]. Still, as the dynamics of phase
transitions are usually highly nonlinear, numerical simulations are usually
required for deeper understanding. In general, the phase-field method has
proved to be a powerful tool for studying front evolution problems under
multiphysical influences [23, 28], oftentimes accompanied by the formation
ofseveraldistinctphases[28,62]orgrainsofmultipleorientations[3,35]and
interactions of different governing mechanisms over several time and length
scales [17, 26].
When the phase-field method initially became established for modelingIntroduction 3
andsimulatingsolidificationprocesses,itwasprimarilyusedforpurematerial
systems [40] and binary systems [74]. Generic phase-field model descriptions
for a variable number of phases only arrived years later [66]. The delay is no
coincidence,forwhileitmayhavebeenpossibletocreateaccuratephase-field
model descriptions for complex material systems earlier, the computational
poweravailableatthetimesimplywasnotsufficienttosimulatesuchsystems,
as there is little point in using a model that can describe multiple phases if
simulations running for plausible time periods are forced to use simulation
grids which are so small that they cannot display solids of all the phases
possibleaccordingtothemodel. Thiswasexacerbatedbythefactthatmulti-
phase phase-field models (such as the one described by Tiaden et al. in [66])
are significantly more computationally demanding than pure material and
binarysystems,astheyusuallyinvolveagreaternumberofpartialdifferential
equations which have to be computed for every time step.
However, these restrictions are falling away rapidly. As Moore so fa-
mously discovered in 1965 [47] (which was later popularized in the so-called
“Moore’s Law”), the number of transistors on state-of-the-art computer pro-
cessors doubles within less than two years, with a corresponding increase in
computational power. With these technological advances, complex models
are becoming increasingly easy to implement, and even three-dimensional
simulations are becoming feasible thanks to parallelization techniques.
Phase-field models describing not truly different phases, but different ge-
ometrical orientations of the same crystal type have appeared in the same
time period as the models for large numbers of phases, starting with Tikare
et. al. in 1999 [67]. This is not surprising, as they require comparable com-4 Introduction
putational effort. However, as material systems consisting of multiple grains
of the same basic type represent one of the most common forms of materi-
als, these types of models have a vast range of applications and need to be
explored and expanded further to make them suitable for a large range of
different applications.
This work strives to build on the body of knowledge in this field by
examing three different models using such approaches and their different
applicationsviaextensionoftheirbasicmodels,analysisoftheirkinetics,and
numerical simulations (see the Figure below). It is hoped that these studies
will provide the foundation for future examinations of growth phenomena
involving multiple orientations across different scales and for a large number
of different applications.
In the first chapter, we examine a combined phase-field/Monte CarloIntroduction 5
approach first developed by Assadi [3], discuss its limitations, and propose a
waytosolvethem. Theexpandedmodelisthenmodifiedinordertosimulate
the growth of vein microstructures which form over geological time scales,
but the same model approach can be used to describe polycrystalline growth
in general and thus has a vast range of applications.
The second chapter describes an extension of a model for solutal growth
first proposed by Wheeler et al. [74]. This variant adds the effects of a
short-range electrical field generated by the growth of metal nanoparticles
within an ionic liquid and examines the effects of this field on the kinetics
of an isotropic particle analytically. The third chapter uses this model for
numerical simulations and compares the analytical values with the simula-
tion results. Finally, the model is combined with the Monte Carlo algorithm
discu