crystallography

                                     Preface

The work presented in this thesis has been carried out in Multiscale Statistical

Physics group of Aalto University School of Science and Technology. The group is

also a part of Academy of Finland’s COMP (Computational Nanoscience) Center of

Excellence. I am very grateful for my supervisor, Prof. Tapio Ala-Nissil¨a, for giving

me the opportunity to carry out doctoral studies as a part of his group. Especially

I want to thank Tapio for being supportive of my ideas, often motivating me to

push them further. I am also very grateful to all the people who I collaborated with

in the process of doing the research reported in this thesis. I want to thank Prof.

Ken Elder for introducing me the Phase field crystal model, helping me to process

my ideas further and reassuring me that the work I’ve been doing is important. I

want to thank my former office mate, Dr. Cristian Achim, for the invaluable help

he provided me in so many technical issues. Prof. Laszlo Gr´an´asy and his group

deserve many thanks for our fruitful collaboration that led to Publication II. Prof.

Seppo Louhenkilpi deserves thanks for all the inspiring discussions about practical

applications of mathematical models that we’ve had over the years.

I also want to thank all my office mates, family and friends, who are too numerous to

be listed by names here. Although most of you haven’t directly contributed in this

work, all the diverting discussions and other activities we’ve had together during

these years have helped me a lot in keeping my motivation up. I know this is the

part where I’m suppose to say that I couldn’t have finished this thesis without you.

As an objective scientist, I’m not sure I can say such thing with certainty. What I

can say with high confidence is that if I had to complete this job without having you

around, I would have become insane by now. The greatest thanks for my sanity of

course belong to my dear wife Lissu, who has always been there for me, even during

the few times that this work pushed me to the limits. So truly, with all my heart:

Thanks.

Contents.

Preface 

Contents 

List of Publications 

Author’s contribution 

List of Abbreviations 

List of Symbols

List of Figures

List of Tables 

1 Introduction 

2 Phase field crystal model 28

2.1 Mathematical formulation .

2.2 Phase diagram .

2.3 Application to diffusion controlled crystal growth .  

3 Connection between density functional theory and phase field crystal

model 

3.1 Density functional theory of classical systems . .

3.1.1 General formulation .                               3.1.2 Density expansion.  

3.1.3 Density functional theory of freezing . . . 

3.1.4 Dynamical density functional theory . . . . . . . 

3.2 Derivation of the Swift-Hohenberg phase field crystal model from the

density functional theory .

3.2.1 Approach of Elder et al. (2007) . . 

3.2.2 Phase field crystal model of Wu and Karma (2007) .  

3.3 Eighth-order phase field crystal model . 

3.3.1 Local formulation .                                                                                                                         3.3.2 Application to grain boundaries of body-centered-cubic iron 

3.3.3 Non-local formulation .  

3.3.4 Comparison to dynamical density functional theory

List of Publications

This thesis consists of an overview and of the following publications which are referred

to in the text by their Roman numerals.

I A. Jaatinen and T. Ala-Nissila, “Extended phase diagram of the threedimensional

phase field crystal model,” Journal of Physics: Condensed

Matter 22 205402 (2010)

II G. Tegze, L. Gr´an´asy, G. I. T´oth, F. Podmaniczky, A. Jaatinen, T. Ala-

Nissila and T. Pusztai, “Diffusion-controlled anisotropic growth of stable

and metastable crystal polymorphs in the phase-field crystal model,” Physical

Review Letters 103, 035702 (2009)

III A. Jaatinen, C. V. Achim, K. R. Elder, and T. Ala-Nissila, “Thermodynamics

of bcc metals in phase-field-crystal models,” Physical Review E 80,

031602 (2009)

IV A. Jaatinen, C. V. Achim, K. R. Elder and T. Ala-Nissila, “Phase field crystal

study of symmetric tilt grain boundaries of iron,” Technische Mechanik

30, 169-176 (2010)

V A. Jaatinen and T. Ala-Nissila, “Eighth-order phase-field-crystal model for

two-dimensional crystallization,” Physical Review E, accepted for publication.

Author’s contribution.

The author has had an active role in all phases of the research reported in this thesis.

He has written the first drafts of publications I, III, IV and V. Publications I and

V are based on calculations and their interpretations by the author. In Publication

II, the author’s role was defining the thermodynamical driving forces for each phase

studied. Publication III is based on calculations and their interpretations by the

author, except for the grain boundary energies. In Publication IV interpretation of

the results was done by the author.

List of Abbreviations.

BCC Body-centered cubic

BCT Body-centered tetragonal

CDFT Classical density functional theory

CSL Coincidence site lattice

DDFT Dynamical density functional theory (of classical systems)

EOF Eighth-order fit

FCC Face-centered cubic

FOF Fourth-order fit

GBE Grain boundary energy

HCP Hexagonal close-packed

MD Molecular dynamics

PF Phase field

PFC Phase field crystal

RY Ramakrishnan and Yussouff

SH Swift and Hohenberg

WK Wu and Karma

List of Symbols.

S(k) Structure factor

k Length of a wave vector

km Position of the main peak in S(k)

φ(r) Phase field variable

r Spatial coordinate

t Time

M Mobility

F Free energy

α, λ, q0, h, g Parameters of the Swift-Hohenberg PFC model

x Rescaled spatial coordinate

 Rescaled parameter of the Swift-Hohenberg PFC model

ψ Dimensionless phase field variable

˜ F Rescaled free energy

τ Rescaled time variable

ψ0 Spatial average of ψ

u Amplitude of density fluctuations

q Wave vector corresponding to density fluctuations

al Lattice spacing

x Position of a moving solid-liquid interface

d Growth velocity parameter for diffusion controlled growth

Ω Grand potential

μint(r) Intrinsic chemical potential

μ Chemical potential

u(r) External potential energy field

ρ(r) One-particle density

F Intrinsic free energy

kB Boltzmann’s constant

T Absolute temperature

λT Thermal de Broglie wavelength

c(n)(r1 . . . rn; ρ0) n-body direct correlation function

ρ0 Reference density

β Thermodynamic beta

h(r) Total pair correlation function

r Length of a spatial coordinate vector

Δρ∗ Fractional density change in freezing

G Reciprocal lattice vector

μG Amplitude of Fourier mode corresponding to G

γ Friction coefficient

F Potential force

f Gaussian random force

n Dimensionless density variable

¯ρ Spatial average of ρ(r)

C(r) Reference density times two-body direct correlation function

ES Expansion parameter in fourth- and eighth-order expansions of ˆ C(k)

a, b Parameters for local part of free energy in DFT-based PFC models

EB Expansion parameter in eighth-order expansion of ˆ C(k)

θ Misorientation angle of a grain boundary

w(r) Weighing function

ρs, ρl Densities of solid and liquid phases at coexistence

ρi Initial density

Δ Undercooling

v Velocity of a solid-liquid interface

D Effective diffusion coefficient

L Linear part of time-evolution operator

N Non-linear part of time-evolution operator

F, F−1 Forward and inverse Fourier transform operators