integrated experimental and computational study of
TRANSCRIPT
Integrated Experimental and Computational Study of
Precipitation in Martensitic Steels
Tao Zhou
Doctoral Thesis, 2019
Department of Materials Science and Engineering
School of Industrial Engineering and Management
KTH Royal Institute of Technology
SE-100 44 Stockholm, Sweden
This doctoral thesis will, with the permission of Kungliga Tekniska
Hรถgskolan, Stockholm, be presented and defended at a public
dissertation on December 12, 2019, at 10 a.m., Lindstedtsvรคgen 26,
KTH Campus, Stockholm, Sweden.
Akademisk avhandling som med tillstรฅnd av Kungliga Tekniska Hรถgskolan
framlรคgges till offentlig granskning for avlรคggande av teknologie doktorsexamen
torsdag den 12 december 2019 klockan 10:00 i hรถrsal F3, Lindstedtsvรคgen 26,
Kungliga Tekniska Hรถgskolan, Stockholm. Avhandlingen presenteras pรฅ engelska.
Integrated Experimental and Computational Study of Precipitation in
Martensitic Steels
Tao Zhou ([email protected])
KTH Royal Institute of Technology
School of Industrial Engineering and Management
Department of Materials Science and Engineering
SE-100 44 Stockholm, Sweden
TRITA-ITM-AVL 2019:41
ISBN: 978-91-7873-381-1
Denna avhandling รคr skyddad enligt upphovsrรคttslagen. Alla rรคttigheter fรถrbehรฅlles.
Copyright ยฉ 2019 Tao Zhou
All right reserved. No part of this thesis may be reproduced by any means without
permission from the author.
This thesis is available in electronic version at kth.diva-portal.org
Printed by Universitetsservice US-AB, Stockholm, Sweden
i
Abstract
Precipitation is a phase transformation process in metallic materials that significantly
affects properties. The precipitation process that includes nucleation, growth and
coarsening of small particles can be tuned by alloying, deformation, thermal treatment.
This opens opportunities for optimizing the properties of metallic materials by
tailoring precipitation. An example of high-performance metallic materials with
contribution from precipitation is tempered martensitic steels. By means of highly
dispersed nanoscale precipitates within the hierarchic martensitic microstructure,
these steels achieve an excellent combination of ultra-high strength and high
toughness. With the objective of accelerating the development of these high-
performance steels, an integrated computational materials engineering (ICME)
approach, combining advanced characterization, physically based/semi-empirical
modelling, theory and databases, is used in this thesis to develop computational
linkages from heat treatment to precipitation to strength.
Two multicomponent steels, a Cu precipitation-hardened maraging stainless steel and
a carbide-strengthened low alloy CrโMoโV martensitic steel, are studied in this thesis
using quantitative characterization and modelling. The results suggest that the
precipitation simulations using Langer-Schwartz-Kampmann-Wagner (LSKW)
modelling have good agreements with the experiments and show promise for future
predictive modelling to be used for materials design. The semi-empirical models for
individual strengthening mechanisms and an integration of the strengthening
mechanisms used in this work may also represent the trends in the yield strength of
fresh and tempered martensite, but it is difficult to predict the early yielding of fresh
martensite and the correlation of hardness and strength. This indicates the need to
further develop the models. Overall, this thesis shows that the ICME approach can be
used to study and predict precipitation and precipitation-strengthening in
multicomponent steels. The applied approach differs from traditional trial-and-error
testing and has the potential to save time, money and resources in steel development.
Keywords: Precipitation; Martensite; Modelling; Thermodynamics; Mechanical
Property; Transmission Electron Microscopy; Materials Design
ii
Sammanfattning
Utskiljning รคr en fasomvandlingsprocess i metalliska material som signifikant
pรฅverkar egenskaperna. Utskiljningsprocessen som inkluderar kรคrnbildning, tillvรคxt
och fรถrgrovning av smรฅ partiklar kan styras genom legeringstillsats, deformation,
vรคrmebehandling, etc. Detta รถppnar mรถjligheter fรถr att optimera metallernas
egenskaper genom att kontrollera utskiljningarna. Ett exempel pรฅ en
vรคrmebehandlingsprocess som leder till utskiljningar รคr anlรถpning av martensitiska
stรฅl. Under denna anlรถpning kan stรฅlen uppnรฅ mycket god prestanda genom bildning
av nanopartiklar i den hierarkiska martensitiska mikrostrukturen. Med mรฅlet att
pรฅskynda utvecklingen av dessa hรถgpresterande stรฅl anvรคnds ICME (Integrated
Computational Materials Engineering) som kombinerar avancerad karakterisering,
fysikaliskt baserad modellering, teori och databaser fรถr att relatera vรคrmebehandling
till utskiljning och slutligen egenskaperna hos stรฅlet.
Tvรฅ hรถgprestanda stรฅl, ett Cu-utskiljningshรคrdat marรฅldrings stรฅl och ett Cr/Mo/V-
legerat kolstรฅl med karbidutskiljningar, studeras i denna avhandling med hjรคlp av
kvantitativ karakterisering och modellering. Resultaten visar att modelleringen med
LSKW (Langer-Schwartz-Kampmann-Wagner) har god รถverenstรคmmelse med
experimenten och visar stor potential fรถr prediktiv modellering. De fysikaliskt
baserade modellerna fรถr de individuella hรคrdningsmekanismerna som anvรคnds i detta
arbete kan ocksรฅ representera trenderna fรถr stรฅlens styrka, men det รคr svรฅrt att
prediktera utskiljningshรคrdningens effekt pรฅ styrkan som uppmรคtts via dragprov.
Detta visar pรฅ behovet av att vidareutveckla modellerna. Sammantaget visar denna
avhandling att ICME-metodik kan anvรคndas fรถr att studera och prediktera utskiljning
och utskiljningshรคrdning i kommersiella stรฅl. Tillvรคgagรฅngssรคttet som anvรคnts skiljer
sig frรฅn traditionell โtrial-and-errorโ metodik och kan leda till besparingar av tid,
pengar och resurser vid stรฅlutveckling.
iii
Appended papers and authorโs contribution
The present thesis is based on the following appended papers:
I. Quantitative electron microscopy and physically based modelling of Cu
precipitation in precipitation-hardening martensitic stainless steel 15-5 PH
T. Zhou, R. Prasath Babu, J. Odqvist, H. Yu, P. Hedstrรถm
Materials and Design 143 (2018) 141โ149
II. Exploring the relationship between the microstructure and strength of fresh
and tempered martensite in a maraging stainless steel Fe-15Cr-5Ni
T. Zhou, J. Faleskog, R. Prasath Babu, J. Odqvist, H. Yu, P. Hedstrรถm
Materials Science & Engineering A 745 (2019) 420โ428
III. Precipitation of multiple carbides in martensitic CrMoV steels -
experimental analysis and exploration of alloying strategy through
thermodynamic calculations
T. Zhou, R. Prasath Babu, Z. Hou, J. Odqvist, P. Hedstrรถm
Under review
IV. Mechanical behavior of fresh and tempered martensite in a CrMoV-alloyed
steel explained by microstructural evolution and model predictions
T. Zhou, Jun Lu, P. Hedstrรถm
Under review
V. Recent developments in transmission electron microscopy based
characterization of precipitation in metallic alloys
T. Zhou, Z. Hou, R. Prasath Babu, P. Hedstrรถm
In manuscript
iv
The contribution of the respondent to each paper is listed below:
I. Literature survey, sample preparation, data analysis, modelling and paper writing;
the TEM measurements were performed with the help of Dr. P. Babu.
II. Literature survey, major part of the experimental work, data analysis, modelling
and paper writing.
III. Literature survey, sample preparation, data analysis, calculations, modelling and
paper writing; the TEM measurements were performed with the help of Dr. P.
Babu.
IV. Literature survey, major part of the experimental work, data analysis and paper
writing; the mechanical testing was performed with the help of MSc. J. Lu.
V. Major part of literature survey, data analysis and paper writing.
Part of this work has been presented in the following international conferences:
1. EUROMAT, European Congress and Exhibition on Advanced Materials and
Processes, Stockholm, Sweden, 01 โ 05 September, 2019. Oral presentation: A
study on Cu precipitation-hardening martensitic stainless steel by ICME.
2. TMS 146th Annual Meeting & Exhibition, San Diego, California, USA, 26
February โ 02 March, 2017. Oral presentation: An experimental and modelling
study on precipitation during tempering of martensitic alloy.
v
Abbreviations
APT Atom probe tomography
CALPHAD Calculation of phase diagrams
CBED Convergent beam electron diffraction
Cc Chromatic aberration correction
CNT Classical nucleation theory
Cs Spherical aberration correction
EBSD Electron back scatter diffraction
ECCI Electron channeling contrast imaging
EDS Energy dispersive X-ray spectroscopy
EELS Electron energy loss spectroscopy
FFT Fast Fourier transform
HAADF High-angle annular dark-field
HRTEM High-resolution transmission electron microscopy
ICME Integrated computational materials engineering
LSKW Langer-Schwartz-Kampmann-Wagner
LSW Lifshitz-Slyozov-Wagner
MWHWA Modified Williamson-Hall and Warren-Averbach
PH Precipitation-hardening
PSD Particle size distribution
SAED Selected area electron diffraction
SANS Small angle neutron scattering
SAS Small angle scattering
SAXS Small angle X-ray scattering
SEM Scanning electron microscopy
SFFK Svoboda-Fischer-Fratzl-Kozechnik
STEM Scanning transmission electron microscopy
TEM Transmission electron microscopy
TEP Thermodynamic extremal principle
TKD Transmission Kikuchi diffraction
XRD X-ray diffraction
vii
Table of Contents
1 Introduction ........................................................................................................ 1
1.1 Precipitation in metallic materials .................................................................... 1
1.2 Scope of the present work .................................................................................. 2
2 Microstructural characterization ...................................................................... 4
2.1 Electron microscopy .......................................................................................... 4
2.2 Atom probe tomography .................................................................................... 6
2.3 Small angle scattering ....................................................................................... 7
3 Precipitation modelling .................................................................................... 10
3.1 Thermodynamic basis ...................................................................................... 10
3.2 Classical nucleation theory ............................................................................. 15
3.3 Diffusion-controlled growth and coarsening .................................................. 17
3.4 Precipitation modelling by mean-field method ............................................... 21
4 Materials design through precipitation hardening ....................................... 25
4.1 Precipitation hardening ................................................................................... 25
4.2 Design concepts ............................................................................................... 28
5 Summary of appended papers ......................................................................... 30
6 Concluding remarks ......................................................................................... 35
7 Future work ....................................................................................................... 37
Acknowledgements .............................................................................................. 38
References ............................................................................................................. 40
1
1 Introduction
1.1 Precipitation in metallic materials
Precipitation is a phase transformation process by which a child phase forms out
of a supersaturated mother solid-solution phase. The process of precipitation is
divided into three stages: nucleation, growth and coarsening, but these three stages
are concomitant rather than consecutive processes. In order to model the concurrent
three stages during precipitation, Langer-Schwartz [1] and Kampmann-Wagner [2, 3]
developed a numerical approach to model the evolution of particle size distribution.
This approach provides the framework for the mean-field modelling of precipitation
in multicomponent and multiphase systems and can make use of CALPHAD
databases. This mean-field method holds high potential for predictive modelling of
precipitation in metallic materials if it is based on experimental understanding of the
precipitation and calibrated by quantitative experimental information. This suggests
that systematic characterization of the whole precipitation process could pave the way
for predictive modelling of precipitation in a certain materials system. The
quantitative information that is needed includes size, morphology, structure and
number density of precipitates as well as grain size and dislocation density of the
matrix. In order to develop a complete picture of precipitation in a certain materials
system, multiple characterization techniques such as transmission electron
microscopy (TEM) [4], atom probe tomography (APT) [5] and small angle scattering
with X-rays (SAXS) [6] and neutrons (SANS) [7], are often required.
Precipitates with different compositions and lattice characteristics compared to the
matrix, can be viewed as one kind of volumetric defect within a perfect lattice, the
matrix. This means that the periodicity and perfection of the atom arrangement are
locally perturbed. This heterogeneity of the matrix caused by precipitates significantly
influences the properties of metallic materials, such as hardness, strength, toughness,
ductility, creep, wear, corrosion resistance and functional properties. This also means
that metallic materials properties can be controlled by tuning precipitation via
alloying, heat treatment and deformation. Traditionally, trial-and-error is the approach
in alloy development and optimization, however, this is a costly process which
requires significant time and resources. In order to shorten the time and lower the cost
of materials design, integrated computational materials engineering (ICME) [8, 9]
2
combining advanced characterization, modelling, theory and databases has been
proposed to create computational linkages between processing, structure, properties
and performance. ICME can be applied to tackle precipitation-hardening of metallic
materials and thus enable the design and optimization of alloys and heat treatments
for precipitation-hardened metallic materials.
1.2 Scope of the present work
In this work, an ICME approach to precipitation engineering is demonstrated (see
Fig. 1.1). Two high-performance commercial steels: a Cu precipitation-hardening
(PH) maraging stainless steel 15-5 PH [10, 11] from Outokumpu Stainless and a low
alloy CrโMoโV martensitic wear-resistant steel [12, 13] from SSAB were
investigated. To stimulate the precipitation, isothermal heat treatments were applied
to the as-quenched supersaturated martensite of both steels. Since the size of
precipitates in both steels is on the nanoscale (< 50 nm), and there are complex
structural transformations of the Cu precipitates in the PH stainless steel and multiple
carbides precipitation in the CrโMoโV steel, advanced characterization techniques
including TEM and APT are utilized for quantitative experimental characterization of
the precipitation. The experimental work tries to address thermodynamics, kinetics
and crystallography to generate a complete understanding of the precipitation
processes. The experimental information further contributes to the setup and
calibration of modelling of precipitation in multicomponent alloy systems by a mean-
field method utilizing CALPHAD databases. In addition, precipitation strengthening
is evaluated based on the outputs of precipitation modelling, and it is further correlated
to the yield strength of the material by combining with modelling of other
strengthening mechanisms like grain boundary strengthening, dislocation
strengthening and solid solution strengthening. All the evaluations of strengthening
are based on quantitative experimental characterization of the evolution of the
hierarchic martensite microstructure, such as effective grain size characterized by
electron back scatter diffraction (EBSD) [14, 15] and dislocation density
characterized by X-ray diffraction (XRD). Tensile testing is also carried out for all the
samples providing the reference for the strength modelling. By creating precipitation
modelling and property modelling linkages, the ICME approach exemplified in this
work shows that it can be possible to use for design and optimization of alloys and
heat treatments.
3
Fig 1.1 Schematic comparisons of approaches for precipitation-hardened materials
design: traditional trial-and-error (loop I), calculation-driven materials design
calibrated by experiments (loop II) and efficient computational materials design with
only very limited experimental support (loop III).
4
2 Microstructural characterization
2.1 Electron microscopy
The field of materials science has dramatically benefited over the past decade from
the continuous development of electron microscopy. Scanning electron microscopy
has been a routine tool serving materials researchers in studies of morphology,
chemical composition and orientation relationship of various materials [16, 17]. Great
improvements has been achieved by the introduction of field emission guns (FEGs),
which produce smaller electron beams with significantly higher brightness compared
to conventional thermionic electron emitters [18]. The improvement in resolution
through utilizing FEGs has also significantly contributed to the development and
application of SEM-based techniques, such as electron backscatter diffraction
(EBSD), transmission Kikuchi diffraction (TKD) [19, 20] and electron channeling
contrast imaging (ECCI) [21]. With respect to characterization of precipitates, SEM
imaging techniques using secondary electrons or backscatter electrons are most often
employed before detailed studies by higher resolution techniques such as TEM or
APT are applied. EBSD, with the resolution limit of about 50 nm for metallic
materials, has rarely been used for characterization of nanoscale precipitates. In
contrast, TKD, with a resolution limit of about 5-10 nm, due to the reduced interaction
volume between the electron beam and the thin-foil sample, has some applications for
studying the morphology and orientation of nano-sized precipitates. For example,
M23C6 with size around 30 nm in steels [22], Al-Cu precipitates with size around 50
nm in Al-Li alloys [23] and cementite particles in steels [24] have been investigated
by TKD.
TEM, with higher resolution than SEM, is a more powerful tool for the study of
very fine precipitates, i.e. particles with a size of only a few nanometers require TEM-
based characterization. Moreover, TEM is the most versatile among all the
instruments for characterization of precipitates since it can provide almost all the
desired information using various methods in the TEM (see Table 2.1). The
applications and methods are for example [25]:
i) the morphology and size information of precipitates are attainable by imaging
techniques using conventional TEM / scanning TEM (STEM) using bright-field
and dark-field imaging;
5
ii) identification of precipitating phases, structural information and orientation
relationship by diffraction techniques like selected area electron diffraction
(SAED), high-resolution TEM (HRTEM) together with fast Fourier transform
(FFT) analysis (see Fig. 2.1), nano-beam diffraction (NBD) and convergent beam
electron diffraction (CBED);
iii) chemical composition by analytical TEM, such as energy dispersive X-ray
spectroscopy (EDS) and electron energy loss spectroscopy (EELS) in STEM
mode;
iv) atomic resolution analyses by HRTEM and high-angle annular dark field
(HAADF) imaging in STEM mode;
v) in-situ studies of precipitation within different environments;
vi) 3D tomography of precipitates within the matrix phase by electron
tomography or EDS.
Table 2.1 Summary of widely used TEM techniques for characterization of
precipitation.
Imaging
Diffraction Analytical Diffraction
contrast
Phase
contrast Z-contrast
CTEM BF, DF HRTEM -- SAED, CBED,
NBD EDS
STEM BF-STEM,
DF-STEM --
HAADF-
STEM CBED
EDS,
EELS
Fig 2.1 HRTEM characterization of a twinned 9R Cu precipitate in a maraging
stainless steel 15-5 PH.
The speed of TEM method development has recently been accelerated due to
novel technologies, such as chromatic aberration correction (Cc) and spherical
6
aberration correction (Cs), which significantly improves the signal-to-noise ratio and
achieves a sub-ร ngstrรถm spatial resolution. This largely improves the capability of
TEM based techniques like in-situ characterization [26, 27], EDS-STEM [28, 29] and
HAADF-STEM [30]. Among these advanced techniques, atomic-resolution HAADF-
STEM imaging technique in Cs- or/and Cc-corrected TEM, or combining with EDS
and EELS mapping, has received considerable attention in recent years for
characterization of precipitates with size of several nanometers [31-34]. Combining
with careful sample preparations such as twin-jet electro-polishing, carbon extraction
replica, focused ion beam and electrolytic extraction, TEM is very powerful for
understanding of precipitation mechanisms in physical metallurgy. The importance of
TEM is foreseen to contribute in a similar way also in the future for in-depth
understanding of precipitation, especially considering the mentioned recent
developments of TEM.
2.2 Atom probe tomography
Atom probe tomography, with the advantage of three-dimensional (3D)
tomographic representation of a sample with atomic resolution (see Fig. 2.2), is
another powerful tool for quantitative characterization of nano-precipitates or even
clusters, including size, shape, composition, spacing, number density and volume
fraction of particles.
Fig 2.2 Cu precipitates in a maraging stainless steel 15-5 PH tempered for 5 h at 500
ยบC characterized by APT.
Two critical steps for the collection of nano-precipitate information by APT are
firstly the 3D reconstruction of the acquired dataset and secondly the determination
of precipitate-matrix interface so that the particles can be quantified. For the former,
geometric field factor kf, image compression factor and detection efficiency ฮท are
7
key parameters for reconstructing the probed volume of the needle-shaped specimen,
and the time-of-flight of ions projected from tip to detector is used to identify chemical
species based on the mass-to-charge ratio shown in the mass spectrum [35, 36]. For
the latter, the two most used methods are the maximum separation method [37] where
๐๐๐๐ฅ (any solute atom within the given distance from another solute atom is defined
as being part of the same particle), ๐๐๐๐ (removing the particles that contain less than
a minimum value of solute atoms) and ๐๐๐๐ (removing the undesired shell of solvent
atoms formed after the first step) have to be determined, and the iso-concentration
surface method [38] where an iso-concentration value of solute atoms have to be
determined to define the precipitate and matrix interface. Thereafter, for example, a
chemical profile with respect to distance from the defined interface (i.e. proxigram
[39]) of a single particle can be evaluated. However, it is challenging to perform
precise analysis of small precipitates due to local magnification effect, i.e. the
difference in field evaporation potential between the matrix and precipitates,
trajectory aberrations of emitted ions, etc.
2.3 Small angle scattering
Compared to local techniques like TEM and APT, SAS is a more global technique
which provide average information on a large population of precipitates, usually with
size scale between 1 and 100 nm. SAS measurements can be performed either with
X-rays (SAXS) or neutrons (SANS). X-rays are mainly scattered by the electrons and
thus the contrast depends on difference in atomic number, while the neutrons are
scattered by the nuclei and their magnetic moment, and therefore the contrast depends
on atomic species. SAXS and SANS are complementary with each other in most
cases. SAXS is suitable for a system with high contrast in atomic number, when fast
measurements are needed (e.g. in-situ measurements) or when a small beam size is
needed; while SANS is suitable in case of low Z-contrast, thick samples, two phases
with different magnetic properties [40]. One noteworthy advantage of SAXS over
SANS is that chemical selectivity in the measurements can be applied using the
technique called anomalous SAXS (ASAXS), since the scattering factor of a certain
element can be changed when the wavelength of the X-rays is close to the absorption
edge of that element [41].
One of the strong advantages of SAS over TEM and APT is the possibility to
perform in-situ studies of precipitation kinetics. The temporal evolution of precipitate
8
radius, volume fraction and number density can be measured continuously (see Fig.
2.3). With this data it is then possible to construct, for example, temperature-time-
precipitation (TTP) diagrams. One such example for Cu precipitation in the Fe-Cu
system, studied by SAXS, can be found in Refs. [40, 42, 43] and another example of
NbC precipitation, studied by SANS, can be found in Ref. [44]. In addition, the
mapping of precipitate volume fraction and size information is obtainable by SAXS,
see the example in Ref. [45]. It is noteworthy that the scattering intensity includes
factors for particle number density, contrast (the difference in scattering length
between particle and matrix), volume and shape, and thus a reliable interpretation of
the SAS data is usually obtained by coupling SAS with local techniques for a priori
understanding of precipitate features, which can then be used in the SAS data
analysis/modelling.
Fig 2.3 SAXS in-situ measurements of Cu precipitation kinetics in Fe-1.4 wt%Cu alloy
at three tempering temperatures: (a) precipitate radius, (b) precipitate number
density and (c) precipitate volume fraction [40]. (Copyright permission from
Elsevier)
9
The strengths and limitations of the three techniques for characterization of
precipitation are summarized in Table 2.2.
Table 2.2 A comparison of the techniques TEM, APT and SAS for characterization of
precipitation
TEM APT SAS
Strengths
Chemical info.
Structural info.
Atomic resolution
In-situ study
Chemical info.
Atomic resolution
3D reconstruction
Large volume
In-situ study
Time resolution (SAXS)
Fast data acquisition
Non-destructive
Limitations Small volume
Destructive
Small volume
Destructive
Structural info.
In-situ study
Chemical info.
Structural info.
Accessibility
10
3 Precipitation modelling
3.1 Thermodynamic basis
Gibbs energy, denoting the available energy of a solid-state system for doing
thermodynamic work, determines the state of a solid-state system. The Gibbs energy
(J) of a multicomponent system can be expressed as (most of the equations in this
chapter refer to the book by E. Kozeschnik in Ref. [46])
๐บ = โ ๐๐๐๐ , (3.1)
๐
where ๐๐ and ๐๐ are the chemical potential of component ๐ and corresponding number
of moles in the system, respectively. The Gibbs energy is an extensive quantity, and
can be expressed in the way of molar Gibbs energy (J/mol) considering only one mole
of atoms
๐ = โ ๐๐๐๐ ,
๐
(3.2)
where ๐๐ is the mole fraction of component ๐.
The molar Gibbs energy of the mixed system actually cannot be depicted as a
macroscopically weighted superposition of the two pure systems when two blocks of
pure systems are mixed, due to the mixing of the two kinds of atoms in atomic
dimensions by diffusion. The driving force for this kind of microscopic mixing is the
production of entropy. The molar mixing entropy of an ideal binary system can be
expressed as
๐ = โ๐๐ต๐(๐๐ด๐๐๐๐ด + ๐๐ต๐๐๐๐ต), (3.3)
where ๐๐ต is the Boltzmann constant with a value 1.3806504 10-23 J/K, ๐ is equal to
the Avogadros number ๐๐ด= 6.02214179 1023 mol-1 for one mole of atoms, ๐ =
๐๐ต๐๐ด is the universal gas constant with a value 8.314472 J/(molK). It is noteworthy
that the entropy is positive (note that 0 < ๐๐ < 1 makes ๐๐๐๐ < 0). The entropy
production keeps stabilizing the solid solution by minimizing the molar Gibbs energy
of the system. Consequently, the molar Gibbs energy of an ideal binary system (no
interactions between atoms exist) can be expressed as
๐๐ผ๐ = ๐๐ด๐๐ด0 + ๐๐ต๐๐ต
0 + ๐ ๐(๐๐ด๐๐๐๐ด + ๐๐ต๐๐๐๐ต), (3.4)
11
where ๐๐ด0 and ๐๐ต
0 are the molar Gibbs energy of pure system A and B, respectively.
The molar Gibbs energy of an ideal binary system ๐๐ผ๐ and the corresponding energy
of the macroscopic mixture ๐๐๐ evolving with molar fraction of the components are
comparatively shown in Fig. 3.1.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
RT
ln
(X
A)
RT
ln
(X
B)
g0
B
uB
gIS
gMM
Mo
lar G
ibb
s en
erg
y
Molar fraction of component B
XB
=0.45
uA
g0
A
Fig 3.1 Molar Gibbs energy of an ideal binary system.
By assigning ๐๐ต = 0 and ๐๐ต = 1 in Eq. 3.4, the chemical potential of A and B in
the ideal binary system are easily obtained
๐๐ด = ๐๐ด0 + ๐ ๐๐๐๐๐ด, (3.5)
๐๐ต = ๐๐ต0 + ๐ ๐๐๐๐๐ต . (3.6)
Similarly, the molar Gibbs energy and chemical potential of components in the ideal
multicomponent system can be expressed as
๐๐ผ๐ = โ ๐๐๐๐0 + ๐ ๐ โ ๐๐๐๐๐๐
๐๐
, (3.7)
๐๐ = ๐๐0 + ๐ ๐๐๐๐๐ . (3.8)
It should be noted in the Eq. 3.4 and Eq. 3.7 that the entropic contribution to the molar
Gibbs energy has a linear relationship with temperature, which means that the effect
of entropy is proportionally increased with the increase of temperature.
12
However, a real solid solution is rarely ideal due to various interactions between
atomic species. The effect of atomic interactions on the Gibbs energy is further
accounted for by the enthalpy of mixing
โ๐ป =1
2๐ง๐ฟ๐๐๐ด๐๐ต โ (ํ๐ด๐ด + ํ๐ต๐ต โ 2ํ๐ด๐ต), (3.9)
where ๐ง๐ฟ is the coordination number (the number of nearest atoms), ๐ is the total
number of atoms in a system, ํ๐ด๐ด, ํ๐ต๐ต and ํ๐ด๐ต are the bond energies for atomic bond
A-A, B-B and A-B. Using a substitution of
๐๐ด๐ต = ๐ง๐ฟ โ (ํ๐ด๐ด + ํ๐ต๐ต โ 2ํ๐ด๐ต), (3.10)
and considering only one mole of atoms, the molar Gibbs energy of a regular binary
system can be expressed as
๐๐ ๐ = ๐๐ด๐๐ด0 + ๐๐ต๐๐ต
0 + ๐ ๐(๐๐ด๐๐๐๐ด + ๐๐ต๐๐๐๐ต) +1
2๐๐ด๐๐ต๐๐ด๐ต. (3.11)
For multicomponent system,
๐๐ ๐ = โ ๐๐๐๐0 + ๐ ๐ โ ๐๐๐๐๐๐
๐
+1
2โ โ ๐๐๐๐๐๐๐
๐>๐๐๐
, (3.12)
where ๐๐๐ is called the regular solution parameter. When ๐ > 0, the bonds between
like atoms are energetically favorable, resulting in phase separation (like spinodal
decomposition in concentrated alloys and precipitation in dilute systems); when ๐ =
0, the enthalpy is equal to zero and the system is an ideal solution, i.e. the atoms are
randomly distributed; when ๐ < 0, the bonds between unlike atoms are energetically
favorable, resulting in short-range ordering.
It is noted in Eq. 3.11 and Eq. 3.12 that the molar Gibbs energy of a regular
solution includes three parts of energy, including the energy of macroscopic mixture,
the entropy of ideal mixing and the enthalpy of mixing. The entropy of ideal mixing
always negatively contributes to the Gibbs energy and has a linear dependence on
temperature, scaling with ๐๐๐๐ + (1 โ ๐)๐๐(1 โ ๐) in binary systems, while the
enthalpy of mixing to the Gibbs energy is positive if ๐ > 0, scaling with ๐(1 โ ๐) in
binary systems. The contributions of entropy and enthalpy are together called Gibbs
energy of mixing, which can show significantly different plots depends on the
temperature (see the upper part of Fig. 3.2). The Gibbs energy of mixing curves at
13
various temperature can generate a phase diagram (see Fig. 3.2). When the
temperature is high, like 800 K, the โG -๐ curve only shows single minimum due to
the dominant contribution of entropy. Below a critical temperature, ๐๐๐๐๐ก, the โG -๐
curve starts to be w-shaped.
Fig 3.2 The relationship between โ๐บ - ๐ curves and the phase diagram [46].
(Copyright permission from Momentum Press)
The corresponding phase diagram is subsequently divided into three regions
based on the two minimum and two inflections indicated by tangents in the โG -๐
curves. In region I, a homogenous solid solution of two kinds of atoms is
thermodynamically stable, without phase separation; in region II, phase separation is
thermodynamically favorable, but a small fluctuation in local chemical composition
14
increases the Gibbs energy of the system, which means a large enough perturbation,
of critical size, in local chemical composition is necessary to overcome an energy
barrier (nucleation and growth); in region III, phase separation is thermodynamically
favorable and any fluctuation, exceeding a critical wavelength, in local chemical
composition decreases the Gibbs energy of the system, which means that the phase
separation can take place without a nucleation event in the classical sense (spinodal
decomposition).
Real thermodynamic systems are much more complex than the aforementioned
ideal and regular systems due to complex interactions, e.g. chemical and magnetic. In
the CALPHAD approach, the Gibbs energy is defined by the ideal solution Gibbs
energy ๐๐ผ๐ and excess Gibbs energy ๐๐ธ๐
๐ = โ ๐๐๐๐0 + ๐ ๐ โ ๐๐๐๐๐๐
๐
+ ๐๐ธ๐๐ ๐พ + ๐๐ธ๐
๐๐๐๐+ ๐๐ธ๐
๐๐ ๐ + โฏ, (3.13)
๐
where ๐๐ธ๐๐ ๐พ , ๐๐ธ๐
๐๐๐๐ and ๐๐ธ๐
๐๐ ๐ are excess energy contributions from non-ideal
chemical interactions, magnetism and short-range ordering. For thermodynamic
studies of precipitation in solid solutions, the common tangent method is usually used
for the evaluation of chemical driving force based on the molar Gibbs energy
expression in Eq. 3.13. However, to fully describe the total energy of a solid-solution
precipitation system, the energy from capillarity effect and misfit effect has to be
accounted for. The former is due to a net force towards the center of the precipitate
caused by the interface curvature. This force causes a pressure on the precipitate,
called curvature-induced pressure
๐๐๐ข๐๐ฃ =2๐พ
๐, (3.14)
where ๐พ is interfacial energy (J/m2) and ๐ is the radius of a spherical precipitate. The
corresponding energy can be expressed as
๐๐๐ข๐๐ฃ =2๐พ
๐๐ฃ๐ฝ, (3.15)
where ๐ฃ๐ฝ is the molar volume of the precipitate. The latter is caused by the lattice
mismatch between coherent precipitate and matrix. The corresponding elastic energy
can be expressed as
15
โ๐บ๐ฃ๐๐๐๐ =
๐ธ๐
9(1 โ ๐ฃ๐)(๐ฃโ)2, (3.16)
where ๐ธ๐ and ๐ฃ๐ are the elastic modulus and the Poissonโs ratio of the matrix, ๐ฃโ
represents the volume elastic misfit with
๐ฃโ =๐ฃ๐ฝ โ ๐ฃ๐ผ
๐ฃ๐ผ, (3.17)
where ๐ฃ๐ผ and ๐ฃ๐ฝ are the molar volume of the matrix and precipitate, respectively.
Both factors have a pronounced effect at the early stage of precipitation, when the size
is small and the interface is coherent.
3.2 Classical nucleation theory
Nucleation happens at the early stage of decomposition of a supersaturated solid
solution. Nucleation theories, dealing with the formation rate of stable nuclei, i.e.
spatially localized solute-rich clusters, can be classified into classical nucleation
theory (CNT) and non-classical nucleation theory. The CNT treats the โdiscrete
dropletโ formalism usually used in nucleation and growth mechanism of phase
separation, while the non-classical nucleation theory depicts a โcomposition waveโ
picture usually employed in the spinodal decomposition mechanism of phase
separation [3]. Only CNT is discussed hereafter. Nucleation is a stochastic process
with a certain probability. Thus, it is reasonable that the theory of nucleation deals
with the nucleation rate in a unit volume and a unit time as a function of the system
state for a large number of potential nucleation events. Based on the theory of Volmer
and Weber [47], there exists a stationary distribution of clusters with size below the
critical size. The cluster distribution function can be expressed as
๐ = ๐0๐๐ฅ๐(โโ๐บ๐๐ข๐๐
๐๐ต๐), (3.18)
where ๐0 is the number of atoms in a unit volume, โ๐บ๐๐ข๐๐ is the nucleation energy
(also called the Gibbs energy change when forming a cluster of a certain size), ๐๐ต is
the Boltzmann constant and ๐ is absolute temperature. The distribution of clusters in
Eq. 3.18 shows that the number density of cluster under a certain temperature is
reduced with an exponential relationship with the increase of nucleation energy or
cluster size. The steady-state nucleation rate at the critical nucleation size is expressed
as
16
๐ฝ๐ = ๐0๐๐ฝโ๐๐ฅ๐(โโ๐บโ
๐๐ต๐), (3.19)
where ๐ is the Zeldovich factor. Since the probability of a critical cluster for growth
and decay are the same, i.e. 50%, ๐ accounts for the thermal activation of a critical
nuclei with energy ๐๐ต๐ governing the growth and decay of critical clusters. โ๐บโ is the
critical nucleation energy (also called nucleation barrier) and ๐ฝโ is the attachment rate
of monomers to the critical nucleus.
The stable-state cluster distribution is a function of temperature. At higher
temperature, the distribution is narrower due to the pronounced effect of entropy at
high temperature. At lower temperatures, the distribution becomes wider and wider,
and nucleation occurs when the entropic effect becomes lower than the phase
separation effect due to interatomic attraction and repulsion. The time that is needed
to establish a new equilibrium cluster distribution at a specific temperature is called
incubation time (the time needed to reach the steady state nucleation), ๐
๐ =1
2๐ฝโ๐2. (3.20)
Considering the incubation effect, the transient nucleation rate or time-dependent
nucleation rate in a real system can be consequently expressed as
๐ฝ = ๐ฝ๐ ๐๐ฅ๐(โ๐
๐ก) = ๐0๐๐ฝโ๐๐ฅ๐(โ
โ๐บโ
๐๐ต๐)๐๐ฅ๐(
โ๐
๐ก). (3.21)
Considering the formation of a spherical and coherent nucleus with radius ๐ under
the assumption of a sharp interface between the nucleus and matrix, the total energy
change can be expressed as
โ๐บ๐๐ข๐๐ =4
3๐๐3 โ โ๐บ๐ฃ๐๐ + 4๐๐2 โ โ๐บ๐ ๐ข๐๐ , (3.22)
where the former part, a volume-related energy change, is usually negative, while the
latter part, an interface-related energy change, is always positive. โ๐บ๐ ๐ข๐๐ is identical
to interfacial energy ๐พ. โ๐บ๐ฃ๐๐ includes chemical and mechanical contributions
โ๐บ๐ฃ๐๐ = โ๐๐โ๐๐
๐ฝ
๐ฃ๐ผ+ โ๐บ๐ฃ๐๐
๐๐ , (3.23)
17
where ๐๐โ๐๐๐ฝ
is the chemical driving force, ๐ฃ๐ผ is the molar volume of the mother
phase, the elastic energy โ๐บ๐ฃ๐๐๐๐ has been introduced in Eq. 3.16. By setting the
derivative of โ๐บ๐๐ข๐๐ in Eq. 3.22 to zero, the critical nucleation radius ๐โand critical
nucleation energy โ๐บโcan be derived
๐โ = โ2โ๐บ๐ ๐ข๐๐
โ๐บ๐ฃ๐๐
=2๐พ
๐๐โ๐๐๐ฝ
๐ฃ๐ผ โ โ๐บ๐ฃ๐๐๐๐
, (3.24)
โ๐บโ =16๐
3
(โ๐บ๐ ๐ข๐๐)3
โ๐บ๐ฃ๐๐2 =
16๐
3
๐พ3
(โ๐๐โ๐๐
๐ฝ
๐ฃ๐ผ + โ๐บ๐ฃ๐๐๐๐ )2
. (3.25)
However, real materials rarely have perfect lattice crystal where each single atom
site is a potential nucleation site for homogenous nucleation. Many kinds of defects
like vacancies, dislocations, grain boundaries and inclusions usually act as preferential
sites for heterogeneous nucleation due to local elastic stress fields or chemical
fluctuations caused by these defects resulting into lower nucleation barrier. For the
heterogeneous nucleation cases, the number density of potential nucleation sites can
be estimated based on the features of microstructure, e.g. the number density of
nucleation sites can be estimated based on the dislocation density under the
assumption that each atom along the dislocation core is a potential nucleation site, and
in a similar way the number density of nucleation sites for grain boundaries can be
estimated based on some geometrical model for grains. See details in Refs. [46, 48].
3.3 Diffusion-controlled growth and coarsening
Nucleation as a stochastic process can be well described by the CNT for a
sufficiently large volume and time interval. The supercritical nuclei formed in this
first stage of precipitation will face a deterministic process, i.e. growth and
coarsening, which is considered to be governed by the diffusion-controlled migration
of the phase boundary between precipitate and matrix. For each individual precipitate,
the diffusion-controlled migration of phase boundary can be described by the flux-
balance equation:
๐ฝ๐๐ผ๐ฝ
+ ๐ฝ๐๐ฝ๐ผ
= โฆ๐๐๐ผ๐ฝ
โง โ ๐ฃ = (๐๐๐ฝ๐ผ
โ ๐๐๐ผ๐ฝ
) โ ๐ฃ, (3.26)
18
where ๐ฝ๐๐ผ๐ฝ
is the fluxes of element ๐ into the phase boundary, ๐ฝ๐๐ฝ๐ผ
is the fluxes of
element ๐ out of the phase boundary, ๐ฃ is the velocity of phase boundary movement
and โฆ๐๐๐ผ๐ฝ
โง is the composition difference of element ๐ in the two sides of the phase
boundary expressed by concentrations at the precipitate side ๐๐๐ฝ๐ผ
and the matrix side
๐๐๐ผ๐ฝ
. For a system with ๐ kinds of components, the mass balance equation holds 2(๐ โ
1) unknown chemical compositions at two the sides of the interface and one common
interface velocity for all components, i.e. 2๐ โ 1 variables in total. This equation can
be solved with a unique solution under the widely used local equilibrium assumption,
where the mobility at the interface is assumed significantly faster than in the matrix
and a state of local equilibrium at the interface can be instantaneously established.
Under the local equilibrium, the chemical potential of each component at the two sides
of interface is identical
๐๐๐ฝ๐ผ
= ๐๐๐ผ๐ฝ
, (3.27)
where ๐๐๐ฝ๐ผ
and ๐๐๐ผ๐ฝ
represents the chemical potential of component ๐ at the
precipitate side and the matrix side of the interface, respectively. On basis of local
equilibrium assumption, Zener [49] introduced that for a stoichiometric precipitate
phase in a binary system, ๐ฝ๐ต๐ฝ๐ผ
= 0. Then the flux-balance equation, under one more
assumption that the composition gradient has a linear profile in the matrix side of the
interface, can be expressed as
(๐๐ต๐ฝ๐ผ
โ ๐๐ต๐ผ๐ฝ
) โ ๐ฃ = ๐ฝ๐ต๐ผ๐ฝ
= ๐ท๐ต๐ผ
๐๐๐ต
๐๐ฅ
, (3.28)
where ๐ท๐ต๐ผ and ๐๐ต are the diffusion coefficient and concentration of component B in
the matrix, respectively. The concentration gradient ๐๐๐ต๐๐ฅโ is a function of ๐๐ต
๐ผ๐ฝ, ๐๐ต
0
and precipitate size. For a binary system, this flux-balance equation (๐ฃ) can be easily
solved, for the ๐๐ต๐ฝ๐ผ
and ๐๐ต๐ผ๐ฝ
are determined for a specific state under local equilibrium.
However, for multicomponent system, the determination of the operating tie-line is
complicated, since countless tie-lines exist and the operating tie-line changes during
phase transformation. Fortunately, several analytical solutions have been developed
to treat the multicomponent growth of precipitates. Chen, Jeppsson and ร gren [50]
proposed an approximate growth rate equation by inducing an effective diffusion
distance factor
19
(๐๐๐ฝ๐ผ
โ ๐๐๐ผ๐ฝ
) โ ๐ฃ = โ๐ท๐๐
๐ผ(๐๐0 โ ๐๐
๐ผ๐ฝ)
๐๐๐
๐โ1
๐=1
, (3.29)
(๐๐๐ฝ๐ผ
โ ๐๐๐ผ๐ฝ
) โ ๐ฃ = ๐๐๐ผ๐ฝ
๐๐ (๐๐0 โ ๐๐
๐ผ๐ฝ) ๐๐๐โ , (3.30)
where ๐๐ is a factor adjusting the effective diffusional distance as a function of
supersaturation, ๐ท๐๐๐ผ is the chemical diffusion in the matrix. The features of this
solution are that cross-diffusion is considered and it is valid for both small and high
supersaturation conditions. Svoboda, Fischer, Fratzl and Kozechnik (SFFK) [51]
proposed an alternative approach to derive the evolution equation for multicomponent
precipitate growth based on the thermodynamic extremal principle (TEP) [52] which
suggests that a thermodynamic system evolves along the particular pathway with
maximum entropy production. Based on this theory, the total Gibbs energy of a
precipitation system can be expressed as
๐บ = โ ๐0๐๐0๐
๐
๐=1
+ โ4๐๐๐
3
3
๐
๐=1
(๐๐ + โ ๐๐๐๐๐๐
๐
๐=1
) + โ 4๐๐๐2๐พ๐
๐
๐=1
, (3.31)
where the three terms represent the Gibbs energy of matrix and precipitates and the
total interfacial energy from the left term in the summation to the right one. ๐
represents component and ๐ represents precipitate. ๐0๐ and ๐0๐ are the number of
moles and chemical potential of component ๐ in the matrix, respectively. ๐๐ is the
radius of precipitate ๐. ๐๐๐ and ๐๐๐ are the mean concentration and chemical potential
of component ๐ in precipitate ๐ . ๐๐ represents mechanical energy associated with
elastic stress field caused by the volume misfit between precipitate and matrix. ๐พ๐
designates the interfacial energy of precipitate ๐ with matrix. The total energy
dissipation during the precipitation process can be also divided into three parts, i.e.
the dissipative energies related to interface migration ๐1 , diffusion inside the
precipitates ๐2 and the diffusion in the matrix ๐3
๐1 = โ4๐๐๐
2
๐๐๐ผ๐น
๐
๐=1
๏ฟฝฬ๏ฟฝ๐2, (3.32)
๐2 = โ โ4๐๐ ๐๐๐
5
45๐๐๐๐ท๐๐
๏ฟฝฬ๏ฟฝ๐๐2
๐
๐=1
๐
๐=1
, (3.33)
20
๐3 โ โ โ4๐๐ ๐๐๐
3(๏ฟฝฬ๏ฟฝ๐(๐๐๐ โ ๐0๐) + ๐๐ ๏ฟฝฬ๏ฟฝ๐๐ 3โ )3
๐0๐๐ท0๐
๐
๐=1
๐
๐=1
, (3.34)
It is noteworthy that only the mean compositions of precipitate and matrix are used in
the models, without the interfacial compositions used in the local equilibrium
assumption.
Precipitate coarsening (or Ostwald ripening) is a size-dependent growth process
where large precipitates, having a positive growth rate, continuously grow at the
expense of smaller precipitates with a negative growth rate. The driving force for the
coarsening is the minimization of total interfacial area, since the smaller precipitates
have a higher surface area to volume ratio. The coarsening process can be well
explained by the Gibbs-Thomson effect where both the small and large precipitates
are assumed to have a local equilibrium with the matrix, but concentration of the small
precipitates is higher than that of the large ones due to pressure difference acted on
the precipitates, thus a concentration gradient exists from the small precipitates to the
large ones and diffusions from small precipitates to large ones take place. Lifshitz,
Slyozov and Wagner (LSW) [53, 54] studied the coarsening theory and derived that
the size distribution of precipitate, after a long time of coarsening in systems where
volume fractions of precipitates are close to zero (inter-particle diffusional
interactions can be neglected and a particle only interacts with the matrix), has a
stationary size distribution as
๐(๐, ๐ก) =4
9๐2(
3
3 + ๐)
73(
32
32
โ ๐)
113 ๐๐ฅ๐(โ
๐
32
โ ๐), (3.35)
where ๐ represents ๐ ๐๐๐๐๐โ . The LSW size distribution function has the maximum
when ๐ = 1.135. The time evolutions of mean radius and number density during the
coarsening stage obtained by LSW are expressed as
๐(๐ก) = ๐0 [1 +๐ก
๐๐ท
]
13
, (3.36)
๐(๐ก) = ๐0 [1 +๐ก
๐๐ท
]โ1
, (3.37)
21
where ๐๐ท is the time constant. It is noteworthy that the LSW theory is based on the
linearization version of the Gibbs-Thomson equation and under the assumption that
the supersaturation is close to zero [3]. Under such assumptions, the LSW yields that
the mean radius evolves with ๐ก1 3โ , the number density ๐กโ1 and the supersaturation
๐กโ1 3โ during coarsening. Ardell [55] extended the LSW theory where the volume
fraction of precipitates is negligibly small to applications with large volume fractions,
called modified LSW theory. The results of the modified LSW suggest that the
coarsening becomes stronger (due to the increase of growth and shrinking rates caused
by the higher concentration gradients as precipitates become closer with each other)
and the stationary size distribution becomes broader and more symmetric with the
increase of volume fraction of precipitates, but the exponents of the temporal power
laws given by LSW are still kept the same. It should be noted that the above-
mentioned coarsening theories are based on the evaporation and condensation of
single atoms from dissolving and growing precipitates. The cluster-diffusion-
coagulation mechanism [56-58] may also contribute to the coarsening, even though
the time exponent is evaluated as from 1/6 to 1/4, which is smaller than the 1/3
predicted by the LSW theory. In addition, the LSW-type theories are under the
assumption that precipitate coarsening is entirely driven by the release of interfacial
energy. In the presence of elastic strain, however, the coarsening can be driven by the
release of both the interfacial energy and the elastic energy. The elastic energy is a
function of the shape of the precipitate and may result in the sharp bifurcations of
precipitates, e.g. one individual precipitate can be split in two or eight smaller
cuboidal precipitates [59, 60].
3.4 Precipitation modelling by mean-field method
Combining the approximate theories and models for nucleation and diffusion-
controlled growth and coarsening, an approach with the capability of modelling the
concurrent nucleation, growth and coarsening stages of precipitation is possible. In
their treatment of the formation of a droplet in a supersaturated vapor, Langer and
Schwartz (LS) [1] developed a theoretical model to describe the concurrent nucleation
and growth of droplets in metastable, near-critical fluids. This theory was later used
for the treatment of precipitation in solid-state solutions as well. In their treatment, LS
proposed a size distribution function ๐(๐, ๐ก) of size and time, with the first assumption
of LS models that an arbitrary function is used to depict the shape of size distribution
22
๐โ(๐โ, ๐ก) = ๐๐ฟ๐
๐
๏ฟฝฬ ๏ฟฝ๐ฟ๐ โ ๐โ , (3.38)
where ๐โ is the critical radius, ๏ฟฝฬ ๏ฟฝ๐ฟ๐ is the mean radius, ๐๐ฟ๐ is the total number of
precipitates, ๐ is a constant and ๐ = 0.317 is chosen to make the coarsening rate
identical to that of the LSW model.
The second assumption of LS models is that only the precipitates with a size above
the critical size belong to the size distribution. Under both assumptions, LS gives the
evolution functions of number density and mean radius during the entire course of
precipitation
๐๐๐ฟ๐
๐๐ก= ๐ฝ โ ๐โ(๐โ, ๐ก)
๐๐โ
๐๐ก, (3.39)
๐๏ฟฝฬ ๏ฟฝ๐ฟ๐
๐๐ก= ๐ฃ(๏ฟฝฬ ๏ฟฝ๐ฟ๐) + (๏ฟฝฬ ๏ฟฝ๐ฟ๐ โ ๐โ)
๐โ(๐โ, ๐ก)
๐๐ฟ๐
๐๐โ
๐๐ก+
1
๐๐ฟ๐
๐ฝ(๐โ)(๐โ + โ๐โ โ ๏ฟฝฬ ๏ฟฝ๐ฟ๐), (3.40)
where ๐ฝ is the nucleation rate via steady state nucleation theory and the third
assumption of LS models is that steady-state nucleation rate is used here; ๐ฃ(๏ฟฝฬ ๏ฟฝ๐ฟ๐) is the
growth rate of precipitates with mean radius, which is correlated to the diffusion-
controlled evolution models; ๐โ(๐โ, ๐ก) โ ๐๐โ accounts for the number of dissolved
particles with radii between ๐โ and ๐โ + ๐๐โ; the third term on the right hand side of
the radius evolution function represents the change of mean radius caused by the
nucleation of new particles which must be slightly larger than the critical size. Finally,
the two evolution equations have to be solved together with the mass conservation
equation
4๐
3๏ฟฝฬ ๏ฟฝ๐ฟ๐
3 โ ๐๐ฟ๐ โ (๐๐ต๐ฝ
โ ๐๐ต0) = (๐๐ต
0 โ ๐๏ฟฝฬ ๏ฟฝ), (3.41)
where ๐๐ต๐ฝ
and ๐๏ฟฝฬ ๏ฟฝ are the mean composition of component B in the precipitate and
matrix, respectively. In the original LS model, the linearized Gibbs-Thomson
equation is applied to consider the interface composition at the matrix side deviated
from equilibrium composition caused by the capillarity effect. Later, Kampmann and
Wagner (KW) [2] suggest the linearized Gibbs-Thomson equation introduces
significant error particularly in systems with large interfacial energy and high
supersaturation and thus suggest using the non-linearized Gibbs-Thomson equation,
i.e. changing from
23
๐๐ต๐ผโฒ
โ ๐๐ต๐ผ (1 +
2๐พ๐๐ฝ
(๐๐ต๐ฝ
โ ๐๐ต๐ผ)๐ ๐
โ1
๐), (3.42)
to
๐๐ต๐ผโฒ
โ ๐๐ต๐ผ๐๐ฅ๐ (
2๐พ๐๐ฝ
๐ ๐โ
1
๐). (3.43)
In addition, KW used the time-dependent nucleation rate expression which involves
the incubation time instead of the steady-state nucleation rate expression used in the
LS models. The modified LS models by KW are usually called MLS models.
Based on all these theories, KW further proposed a framework for modelling of
precipitation kinetics, which is now widely used and implemented in programmes
including TC-Prisma [50, 61-63], MatCalc [51, 64], PrecipiCalc [65],
PanPrecipitation [66], etc. This framework usually refers to numerical Kampmann-
Wagner (NKW) model or Langer-Schwartz-Kampmann-Wagner (LSKW) model. In
this new framework, the number of critical size ๐โ(๐โ, ๐ก) is substituted by a discrete
distribution of size classes [๐๐ , ๐๐+1] with |๐๐ โ ๐๐+1| ๐๐โ โช 1 and the number of nuclei
๐๐ in each class is determined by the time-dependent CNT model at a time interval
โ๐ก. Thus, the PSD function ๐(๐, ๐ก) is split into a sequence of steps and these steps are
chosen so that in the same class, all the precipitates can be looked upon as having the
same composition, size and growth rate. Different compared to the LS and MLS
models, the NKW model accounts for the precipitates below critical size to the size
distribution as well. The modelling of precipitation kinetics by NKW model makes
use of the CALPHAD databases, i.e. the evaluation of chemical driving force and
interfacial energy from the thermodynamics database and the atomic mobility from
the kinetics database. As the compositions within precipitate and within matrix are
assumed to be homogenous, the NKW model is a mean-field method. There are more
assumptions in the original NKW model: i) the interface between the precipitate and
matrix is assumed to be sharp, i.e. one atomic layer thickness, but in reality the
interface is diffuse and the thickness varies with temperature. The effect of the
interface structure on particularly interfacial energy could be corrected by some
supplementary models [67]; ii) the interactions between particles like impingement is
not considered, but it can also be considered nowadays by combining with additional
models [58]; iii) the shape of precipitate is assumed to be spherical, which can be
24
corrected by shape factor models [68], and the curvature effect on interfacial energy,
evolving with precipitate size, can also be corrected [69]; iv) each atom site is assumed
to be the potential nucleation site (homogeneous nucleation), but in reality it is always
heterogeneous nucleation, which can also be corrected in the modelling by re-
evaluation of the number of nucleation sites and atomic diffusional coefficient at
boundaries, dislocations, etc [46]. In comparison to full diffusion field methods for
simulation of precipitation like DICTRA [70], phase field, Monte Carlo, the strengths
of the mean-field method represented by the LSKW approach are lowering saving
computational costs, possible to treat multicomponent systems as well as to treat a
large number of precipitates and at the same time with high accuracy. Therefore, the
mean-field methods are invaluable for modelling precipitation in engineering
materials and can guide the design of precipitation-strengthened materials.
25
4 Materials design through precipitation hardening
4.1 Precipitation hardening
An alloy that is precipitation hardenable will experience an increase of hardness
or yield strength with time at an aging temperature, subsequent to a heat treatment
with fast cooling from the higher temperature. The phenomenon of precipitation
hardening was first discovered by Wilm for Al alloys in 1911 and a first plausible
explanation of the precipitation mechanism was presented by Merica et al. in 1920,
who postulated that a new phase can form within a supersaturated solid solution at the
lower temperature if the solute solubility increases with increasing temperature. From
then on, a new research field in physical metallurgy started. The role of dislocations
in precipitation hardening was firstly discovered by Orowan, Taylor and Polanyi in
1934. And, 14 years later Orowan devised the famous equation relating the strength
of an alloy strengthened by hard particles to the ratio of shear modulus of matrix and
average planar spacing of particles [71]. An early review of precipitation hardening is
given by Kelly and Nicholson [72] in 1963, where the impact of TEM on the
understanding of precipitation hardening is illustrated. A further advancement in the
understanding of precipitate-dislocation interaction is presented in the seminal review
article by Brown and Ham [73].
Precipitation strengthening, coming from the barrier effect that precipitates
impose on the gliding of dislocations during deformation, usually includes two
mechanisms, Orowan looping and particle shearing. The Orowan looping mechanism
is usually used to depict the dislocation interaction with incoherent precipitates, while
the shearing mechanism is used for the dislocation interaction with small coherent
precipitates. If the precipitate is large and impenetrable by dislocations, the Orowan
looping mechanism will determine the strengthening, and in this case the strength
contribution by precipitation strengthening is determined by the inter-particle spacing
and has nothing to do with the strength of precipitate itself. However, if the precipitate
is small to be shearable or penetrable, dislocations will cut through the precipitate and
leave a glide step at the surface of precipitate and the strengthening is related to the
nature and correlation of the precipitate and the matrix, usually divided into lattice
misfit (coherency) strengthening, modulus difference strengthening, ordered domain
strengthening and chemical strengthening. With respect to stress-strain curves, a
26
gradual increase of stress up to the maximum stress value can be observed due to
working hardening by formation of dislocation loops around the precipitates in steels
strengthened by the Orowan looping mechanism [74], while a flat plateau is usually
observed in steels strengthened by the shearing mechanism [11].
The Orowan dislocation looping strengthening can be evaluated by the model [75]
โ๐๐๐๐๐ค๐๐ = ๐0.4๐บ๐
๐โ1 โ ๐
ln (2โ23
๐ ๐โ )
๐, (4.1)
ฮป = 2โ2
3๐ (โ
๐
4๐โ 1), (4.2)
where ๐ is the Taylor orientation factor, ๐บ is the shear modulus of the matrix, ๐ is
the magnitude of the Burgers vector, ๐ is the Poissonโs ratio, ฮป is the inter-particle
spacing, and the ๐ and ๐ are the volume fraction and mean radius of precipitates,
respectively.
Lattice misfit strengthening comes from the lattice difference between coherent
precipitate with matrix and this lattice misfit leads to a stress field around the interface,
which inhibits the gilding of dislocations when they encounter the stress field. The
lattice misfit strengthening can be evaluated by the model from Brown and Ham [76-
78]
โ๐๐๐๐ ๐๐๐ก = ๐๐๐บ(ํ)32โ2๐
๐
๐ , (4.3)
ฮต = |๐ฟ|[1 + 2๐บ(1 โ 2๐๐) ๐บ๐(1 + ๐๐)โ ], (4.4)
๐ฟ =๐๐ โ ๐
๐, (4.5)
where ฯ is a constant varying between 2 and 3, ฮต is the misfit strain parameter, ๐บ๐ and
๐๐ are shear modulus and Possionโs ratio of the precipitates respectively, ฮด is the
difference of lattice parameters of precipitate ๐๐ and matrix ๐.
As the name implies, the modulus difference strengthening originates from the
elastic modulus difference between precipitate and matrix. If the elastic modulus of
precipitate is larger than that of matrix, the tension of dislocation will be increased
27
when dislocation starts to cut precipitates, which makes the mutual repulsion of
precipitate and dislocation and more energy is needed for the dislocation to cut
through the precipitate; If the elastic modulus of the precipitate is smaller than that of
matrix, the tension of dislocation will be decreased when dislocation starts to cut
precipitates, which makes the mutual attraction of precipitate and dislocation and
more energy is needed to let the dislocation leave the precipitate after cutting through
it. Both cases result in strengthening. The modulus difference strengthening can be
evaluated by the Russel-Brown model [79-81]
โ๐๐๐๐๐ข๐๐ข๐ = ๐๐บ๐
๐ฟ[1 โ (
๐ธ๐
๐ธ๐
)2
]
34
, ๐๐๐ ๐๐โ1 (๐ธ๐
๐ธ๐
) โฅ 50ยฐ, (4.6)
โ๐๐๐๐๐ข๐๐ข๐ = 0.8๐๐บ๐
๐ฟ[1 โ (
๐ธ๐
๐ธ๐
)2
]
12
, ๐๐๐ ๐๐โ1 (๐ธ๐
๐ธ๐
) โค 50ยฐ , (4.7)
๐ธ๐
๐ธ๐
=๐ธ๐
โ
๐ธ๐โ
ln (๐ ๐0โ )
ln (๐ ๐0โ )+
ln (๐ ๐โ )
ln (๐ ๐0โ ), (4.8)
๐ฟ = 0.866/(๐ ๐)1 2โ , (4.9)
Where ๐ธ๐ and ๐ธ๐ are dislocation line energy in precipitate and matrix respectively, ๐ฟ
is the mean particle spacing in the gliding plane, ๐ธ๐โ and ๐ธ๐
โ are energy per unit
length of a dislocation in an infinite medium for precipitate and matrix respectively,
๐ and ๐0 are inner and outer cut-off radius of the dislocation stress field, ๐ and ๐ are
the mean radius and number density of precipitates respectively.
Ordered domain strengthening is caused by the formation of antiphase boundary
within ordered precipitate due to the atomic rearrangement when a gilding dislocation
cut through the precipitate. The energy needed for the formation of the new antiphase
boundary is called antiphase boundary energy. The ordered domain strengthening can
be evaluated by the model [82, 83]
โ๐๐๐๐๐๐๐๐๐ =๐๐พ๐๐๐
3 2โ
๐(
4๐๐ ๐
๐๐)
1 2โ
, (4.10)
where ๐๐๐๐ is the average value of the antiphase boundary energy of precipitates, ๐๐ =
(2 3โ )1 2โ ๐ is the average radius of the sheared precipitates in the gliding plane, ๐ is
dislocation line tension, approximated as ๐บ๐2 2โ .
28
Chemical strengthening results from the additional interface created by
dislocations cutting through the precipitates, leaving two ledges. Chemical
strengthening is not an important mechanism for strengthening since this mechanism
usually has a negligible contribution to the strength of the aged alloys [71].
4.2 Design concepts
Since properties of metallic materials can be significantly affected by the nature of
the precipitates and the correlation between precipitate and matrix, tuning
precipitation becomes an effective strategy for materials design. By means of
precipitation, steels can achieve excellent mechanical properties. Precipitates can also
suppress recrystallization and inhibit grain growth to assist in refinement of grains.
Kong and Liu [84] suggest that, in comparison to large incoherent precipitates, a high
number density (~1024 m-3) of nanoscale coherent precipitate (2~6 nm in diameter)
can bring a dramatic improvement of strength without a significant (<5%) or even no
loss of ductility, since fine coherent precipitates are easier to be cut through by
dislocation and subsequently reduce the stress concentration and prevents crack
propagation at the interface of precipitate and matrix. Therefore, high number density
of coherent precipitates (e.g. MC carbides, M2C carbides, carbides, Cu, NiAl and
Ni3Ti) with fine size is highly desirable for precipitation-strengthened materials. High
number density means high nucleation rate, which is directly affected by the number
of nucleation site and the nucleation energy barrier known from the CNT. To increase
the number of nucleation sites, the possible approaches include the introduction of
precursors for the ideal kind of precipitate [85] and facilitating co-precipitation of
various particles [86-93]. To lower the nucleation energy barrier for higher nucleation
rate, one can reduce the interfacial energy and elastic strain energy (misfit) by alloying
[80, 83, 94, 95], increase the chemical diving force of nucleation by increasing
supersaturation (e.g. increasing the content of precipitate forming element [94] and
lowering the temperature for precipitation [96, 97]). In addition, tuning austenite to
ferrite transformation kinetics by introduction of mechanical strain [98] or by alloying
[99] can facilitate high number density of interphase nanoscale precipitates during
phase transformation. One can also tune the type or composition of precipitates [100]
and even introduce hierarchical nanoscale precipitation in multi-principal-element
alloys [101] to increase the strengthening effect of precipitates. In contrast to the
almost constant strength of precipitation-strengthened steels for room temperature
29
applications, high temperature application of these steels are highly related to the
coarsening of precipitates during service at elevated temperature. The creep resistance
of these steels can also be improved by inhibiting the coarsening by tuning interfacial
energy, elastic strain energy, diffusion coefficient of solute, solubility limit of solutes,
etc. [84].
To bring these design concepts into practice, one efficient way is to firstly perform
thermodynamic calculations to study the variations of precipitation driving force,
interfacial energy, etc. under various compositions and, and find the ideal composition
range and temperature range before validation by experiments [102-104]. However,
in order to achieve the design and optimization of alloying and heat treatment for a
specific alloy system by computational calculations, predictive modelling of
precipitation kinetics (for instance, by the LSKW approach using CALPHAD
databases) and of precipitation strengthening are necessary. This type of modeling can
be accelerated by critical inputs and calibration from advanced characterization.
30
5 Summary of appended papers
Paper I. Quantitative electron microscopy and physically based modelling of Cu
precipitation in precipitation-hardening martensitic stainless steel 15-5 PH
The crystal structure transformation from bcc to 9R to fcc, mean radius, particle
size distribution (PSD), number density and volume fraction of Cu precipitation in a
15-5 PH steel are quantitatively characterized by TEM. All these experimental results
are used to calibrate the physically based modelling of Cu precipitation in this
multicomponent system by Langer-Schwartz-Kampmann-Wagner (LSKW) approach
utilizing CALPHAD-type databases. The LSKW modelling well represents the crystal
structure transformation, mean radius, number density and volume fraction of Cu
precipitation during aging. It is found that the LSKW modelling shows a bimodal PSD
when the size-controlled transformation models are applied, which disagrees the real
log-normal PSD and indicates that the models used for modelling structure
transformation needs further improvement. However, from a technical point of view,
this study suggests that the combination of quantitative characterization and the
LSKW modelling is a viable route for predictive modelling of Cu precipitation in an
experiment-calibrated alloy system.
Fig. 5.1 The graphical abstract of Paper I, showing the precipitation modelling
calibrated by quantitative characterization in order to achieve predictive modelling
of precipitation.
31
Paper II. Exploring the relationship between the microstructure and strength of
fresh and tempered martensite in a maraging stainless steel Fe-15Cr-5Ni
The dislocation densities of fresh and tempered martensite are characterized by
XRD and quantitatively estimated based on the modified Williamson-Hall and
Warren-Averbach (MWHWA) method and the results show that the dislocation
density experiences a sharp decrease at the very early stage of aging (~5 min). The
effective grain size for grain boundary strengthening is characterized by EBSD and
the results indicate that the effective grain size is fairly stable during aging. The
comprehensive study suggests that the dislocation annihilation and Cu precipitation
take the main responsibilities for the variation in yield strength from the quenched to
tempered states of the studied PH steel. In addition, semi-empirical models are used
to estimate individual strengthening contributions based on the quantitative
experimental data of microstructural evolution, by which and the tensile tests the
precipitation strengthening is evaluated. This evaluated precipitation-strengthening
results are further compared with the precipitation-strengthening modelling using our
previous LSKW modelling of Cu precipitation as inputs. This work exemplifies the
promising approach of combing physically based modelling of microstructural
evolution and semi-empirical modelling of strengthening mechanisms for
optimization and design of high-performance steels.
Fig. 5.2 The graphical abstract of Paper II, showing strength modelling calibrated
by combining quantitative characterization of microstructural evolution,
precipitation modelling and tensile testing.
32
Paper III. Precipitation of multiple carbides in martensitic CrMoV steels -
experimental analysis and exploration of alloying strategy through
thermodynamic calculations
Cr, Mo and V are strong carbides forming elements so they are widely used as
alloying elements to strengthen steels by precipitating carbides. Six kinds of potential
carbides, MC, M2C, M3C, M6C, M7C3 and M23C6, were found in low alloy CrโMoโV
steels and the precipitations of them are mainly determined by the content of alloying
elements and thermal treatment cycles. A complete understanding of carbides
precipitation in this kind of alloy system under various heat treatments is critical to
tailoring carbides precipitation to achieve ideal properties. A study of a steel with
alloying 1.36Cr-0.78Mo-0.14V under isothermal treatment at a selected temperature
of 550 ยบC is exemplified in this work, where the thermodynamic predictions of
carbides precipitation are well validated by experimental characterizations by TEM
and APT. Based on the validated thermodynamic calculations, a set of alloying
compositions is designed to facilitate the precipitation of fine MC and M2C with
higher strengthening capability than coarse M3C, M6C and M23C6.
Fig. 5.3 The graphical abstract of Paper III, showing the research roadmap by
combining quantitative characterization utilizing TEM and APT, and thermodynamic
calculation utilizing Thermo_Calc software in order to study precipitation of multiple
carbides and to explore alloying strategy.
33
Paper IV. Mechanical behavior of fresh and tempered martensite in a CrMoV-
alloyed steel explained by microstructural evolution and model predictions
This part of work studies the mechanical behavior of the low alloy CrMoV steel
in as-quenched condition and tempered conditions at 450 ยฐC and 550 ยฐC by combining
quantitative characterization of microstructural evolution and modelling. The results
suggest that the studied material shows an excellent combination of strength and
ductility (~1300 MPa and ~14%) after 550 ยฐC tempering for 2-8 h, mostly resulted
from carbides precipitation and high number density of dislocations (in the order of
1015 m-2). The LSKW precipitation modelling agrees quite well the experimental
results of precipitation and transition of carbides in this complex multi-component
multi-phase alloy system. The early yielding phenomenon of fresh microstructure is
found and the possible mechanisms are discussed. This work indicates that the semi-
empirical models may capture the trend of yield strength, but the early yielding
phenomenon could cause problem for strength modelling of fresh martensite and
tempered martensite by one same set of models.
Fig. 5.4 The graphical abstract of Paper IV, combining quantitative characterization
of microstructural evolution and model predictions to explore the mechanical
behaviors of martensite microstructure and to perform strength modelling.
34
Paper V. Recent developments in transmission electron microscopy based
characterization of precipitation in metallic alloys
This work is an overview of how the state-of-the-art of TEM instrumentation and
methodology can improve the precipitation analysis and facilitate the performance
optimization of steels in the traditional trial-and-error approach, and modelling of
precipitation and precipitation strengthening within the ICME approach. It is
noteworthy that TEM is the most versatile instrument for study of precipitation,
covering imaging and composition analysis down to atomic resolution. 3D
tomography and in-situ study are also possible. With the focus on TEM based
methodologies, the present work introduces the recent advancement of TEM
techniques and sample preparation methodologies for applications in precipitation
characterization in steels, exemplified by specific cases. Finally, an outlook of
challenges and opportunities for studies of precipitation from a TEM perspective is
given.
35
6 Concluding remarks
Precipitation studies using an ICME approach, different from the traditional trial-
and-error approach for materials design, is exemplified in this thesis. It is promising
for the design and optimization of precipitation-strengthened high-performance alloys
by integrating advanced characterization techniques, theories, models and databases.
Quantitative characterization of precipitation during nucleation, growth and
coarsening by techniques such as TEM, APT and SAS are critical for the
understanding of precipitation mechanisms, and for the setup and calibration of the
precipitation simulations by the mean-field method (LSKW type approach) calling
CALPHAD databases. Studies on two multicomponent steels suggest that the
precipitation simulations by the LSKW approach agree well with the experiments on
precipitation and transitions of both Cu precipitation and Cr/Mo/V-dominant carbides.
The experiment-validated precipitation modelling further provides input for
precipitation strengthening modelling. In addition, quantitative information of
microstructural evolution, including effective grain size and dislocation density in
quenched and tempered conditions is critical for the modelling of grain boundary
strengthening and dislocation strengthening. The study of microstructural evolution
in both martensitic steels suggest that the effective grain size of lath martensite is
constant from as-quenched to tempered conditions, thus resulting in constant grain
boundary strengthening in the modelling. The dislocation density decreases sharply
at the very beginning of tempering (at ~5 min), followed by a gradual decrease with
further tempering. This variation in dislocation density can be well reproduced by the
Nesโs law. This indicates that the whole trend of dislocations in a certain martensitic
steel can be produced by dislocation density measurements on merely the as-quenched
sample and one over-aged sample.
Precipitation and dislocation annihilation are the two main factors controlling the
strength variation of the martensitic steels from quenched to tempered conditions, and
variations in quantitative information of precipitation and dislocation density can be
provided by the LSKW type precipitation modelling and the Nesโs law. Moreover, the
study indicates that semi-empirical strength modelling can be applied to model yield
strength if quantitative microstructure information is available. Therefore, the
optimization and design of precipitation-hardened martensitic steels are promising by
36
combining: i) the experiment-validated LSKW type precipitation modelling, ii) the
Nesโs law based on two simple measurements of dislocation density, iii) one simple
measurement of effective grain size, and iv) semi-empirical models for individual
strengthening mechanisms and v) the integration of these mechanisms.
37
7 Future work
The precipitation engineering in high-performance steels based on the ICME
approach opens up opportunities for design and optimization of precipitation-
strengthened steels. However, predictive modelling of precipitation and strength
could be improved by further understanding of precipitation and precipitation-
hardening mechanisms and further developments of corresponding models.
A better understanding of Cu precipitate composition at the very early stage of
precipitation by TEM on extracted replica specimen, the bcc Cu to 9R Cu
transformation during cooling and deformation by in-situ TEM, and of the
interactions of bcc Cu precipitate and dislocation by in-situ TEM, can facilitate
the further optimization of precipitation modelling and strength modelling of
Cu precipitation-hardened steels.
A computational design of alloying and heat treatment parameters of Cr-Mo-
V alloyed steel may be performed based on the experimentally calibrated
modelling in this thesis to optimize the properties. Thereafter, the alloy can be
manufactured and heat treated following the design routes, and finally can be
performed for validation by tensile testing.
38
Acknowledgements
The work presented within this doctoral thesis would not have been possible to
produce without inspiration, support and help from many people. I would like to
express my sincere gratitude to those who have contributed to my doctoral studies. To
those who are not mentioned here I would like to thank you for your presence that has
contributed to my doctoral study in different ways.
Assoc. Prof. Peter Hedstrรถm, my principal supervisor, I am sincerely grateful to you
for providing me this opportunity to work on this topic. Your professional guidance,
invaluable suggestions, encouraging discussions have motivated me all the time. Your
immense support and continuous encouragement have made my doctoral study
smoother.
Assoc. Prof. Joakim Odqvist, my co-supervisor, I appreciate your professional
guidance on precipitation modelling and thermodynamic calculations, and great help
during the whole process of my doctoral study.
Dr. R. Prasath Babu, I am indebted to you for the great help on TEM measurements
and sharing your rich experience on TEM techniques. Your support, patience and
friendship have made my doctoral study enjoyable. It has been wonderful to
collaborate and work with you.
Dr. Ziyong Hou, thank you for behaving like a brother around me and back me up all
the time. I really enjoy the time working on the same topic with you and I am grateful
for your continuous assistance on the research.
Dr. Manon Bonvalet-Rolland, thank you for your professional guidance and
constructive discussions on the precipitation kinetics.
Prof. Jonas Faleskog at department of solid mechanics, I appreciate the instructive
discussions with you about the precipitation-hardening and strength modelling. The
collaboration with your group has been making the project proceed smoother.
Prof. Hao Yu at University of Science and Technology Beijing, I am grateful for your
encouragement and enlightenment for this journey
I would like to express my gratitude to Dr. Fredrik Lindberg at Swerim for the
guidance and help on TEM measurements and sample preparations and Erik Claesson
39
at Swerim for the help on SAXS measurements and data analysis. Prof. Ernst
Kozeschnik at Vienna University of Technology is thanked for the support on
precipitation modelling by MatCalc software. Jun Lu at University of Science and
Technology Beijing is thanked for the help on tensile testing. Wenli Long is
appreciated for her great support in the laboratory.
Special thanks to China Scholarship Council, Stiftelsen Jernkontorsfonden fรถr
Bergsvetenskaplig Forskning, Center for X-rays in Swedish materials science (CeXS),
Hero-m 2&2i and Jernkontoret for funding my doctoral study, and Grants from the
Foundation of Applied Thermodynamics for conference traveling. Thanks to Magnus
Andersson from SSAB Special Steels and Jan Y Jonsson from Outokumpu Stainless
AB for providing the research materials.
All the present and former Colleagues at the department of Materials Science and
Engineering, thank you for the friendly working environment, daily discussions and
substantial support and help with different matters.
I express my gratitude to the friends I met in Stockholm. It is you that make my life
in Sweden easier, full of fun and memorable.
My beloved wife Tingru Chang and parents, I am deeply grateful for your endless
love, support and understanding in my life.
Tao Zhou (ๅจๆถ)
Stockholm, October 2019
40
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