Integrated Experimental and Computational Study of ...

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

Högskolan, Stockholm, be presented and defended at a public

dissertation on December 12, 2019, at 10 a.m., Lindstedtsvägen 26,

KTH Campus, Stockholm, Sweden.

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan

framlägges till offentlig granskning for avläggande av teknologie doktorsexamen

torsdag den 12 december 2019 klockan 10:00 i hörsal F3, Lindstedtsvägen 26,

Kungliga Tekniska Högskolan, Stockholm. Avhandlingen presenteras på 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 är skyddad enligt upphovsrättslagen. Alla rättigheter förbehålles.

Copyright © 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

mailto:[email protected]

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 Cr–Mo–V 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 är en fasomvandlingsprocess i metalliska material som signifikant

påverkar egenskaperna. Utskiljningsprocessen som inkluderar kärnbildning, tillväxt

och förgrovning av små partiklar kan styras genom legeringstillsats, deformation,

värmebehandling, etc. Detta öppnar möjligheter för att optimera metallernas

egenskaper genom att kontrollera utskiljningarna. Ett exempel på en

värmebehandlingsprocess som leder till utskiljningar är anlöpning av martensitiska

stål. Under denna anlöpning kan stålen uppnå mycket god prestanda genom bildning

av nanopartiklar i den hierarkiska martensitiska mikrostrukturen. Med målet att

påskynda utvecklingen av dessa högpresterande stål används ICME (Integrated

Computational Materials Engineering) som kombinerar avancerad karakterisering,

fysikaliskt baserad modellering, teori och databaser för att relatera värmebehandling

till utskiljning och slutligen egenskaperna hos stålet.

Två högprestanda stål, ett Cu-utskiljningshärdat maråldrings stål och ett Cr/Mo/V-

legerat kolstål med karbidutskiljningar, studeras i denna avhandling med hjälp av

kvantitativ karakterisering och modellering. Resultaten visar att modelleringen med

LSKW (Langer-Schwartz-Kampmann-Wagner) har god överenstämmelse med

experimenten och visar stor potential för prediktiv modellering. De fysikaliskt

baserade modellerna för de individuella härdningsmekanismerna som används i detta

arbete kan också representera trenderna för stålens styrka, men det är svårt att

prediktera utskiljningshärdningens effekt på styrkan som uppmätts via dragprov.

Detta visar på behovet av att vidareutveckla modellerna. Sammantaget visar denna

avhandling att ICME-metodik kan användas för att studera och prediktera utskiljning

och utskiljningshärdning i kommersiella stål. Tillvägagångssättet som använts skiljer

sig från traditionell “trial-and-error” metodik och kan leda till besparingar av tid,

pengar och resurser vid stålutveckling.

iii

Appended papers and author’s 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. Hedström

Materials and Design 143 (2018) 141–149

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. Hedström

Materials Science & Engineering A 745 (2019) 420–428

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. Hedström

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. Hedström

Under review

V. Recent developments in transmission electron microscopy based

characterization of precipitation in metallic alloys

T. Zhou, Z. Hou, R. Prasath Babu, P. Hedström

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 – 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 – 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 Cr–Mo–V 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 Cr–Mo–V 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-Ångström 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

ºC 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 η 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

𝑑𝑚𝑎𝑥 (any solute atom within the given distance from another solute atom is defined

as being part of the same particle), 𝑁𝑚𝑖𝑛 (removing the particles that contain less than

a minimum value of solute atoms) and 𝑑𝑒𝑟𝑟 (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])

𝐺 = ∑ 𝜇𝑖𝑁𝑖 , (3.1)

𝑖

where 𝜇𝑖 and 𝑁𝑖 are the chemical potential of component 𝑖 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

𝑔 = ∑ 𝜇𝑖𝑋𝑖 ,

𝑖

(3.2)

where 𝑋𝑖 is the mole fraction of component 𝑖.

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

𝑆 = −𝑘𝐵𝑁(𝑋𝐴𝑙𝑛𝑋𝐴 + 𝑋𝐵𝑙𝑛𝑋𝐵), (3.3)

where 𝑘𝐵 is the Boltzmann constant with a value 1.3806504 10-23 J/K, 𝑁 is equal to

the Avogadros number 𝑁𝐴= 6.02214179 1023 mol-1 for one mole of atoms, 𝑅 =

𝑘𝐵𝑁𝐴 is the universal gas constant with a value 8.314472 J/(molK). It is noteworthy

that the entropy is positive (note that 0 < 𝑋𝑖 < 1 makes 𝑙𝑛𝑋𝑖 < 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

𝑔𝐼𝑆 = 𝑋𝐴𝑔𝐴0 + 𝑋𝐵𝑔𝐵

0 + 𝑅𝑇(𝑋𝐴𝑙𝑛𝑋𝐴 + 𝑋𝐵𝑙𝑛𝑋𝐵), (3.4)

11

where 𝑔𝐴0 and 𝑔𝐵

0 are the molar Gibbs energy of pure system A and B, respectively.

The molar Gibbs energy of an ideal binary system 𝑔𝐼𝑆 and the corresponding energy

of the macroscopic mixture 𝑔𝑀𝑀 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 𝑋𝐵 = 0 and 𝑋𝐵 = 1 in Eq. 3.4, the chemical potential of A and B in

the ideal binary system are easily obtained

𝜇𝐴 = 𝑔𝐴0 + 𝑅𝑇𝑙𝑛𝑋𝐴, (3.5)

𝜇𝐵 = 𝑔𝐵0 + 𝑅𝑇𝑙𝑛𝑋𝐵 . (3.6)

Similarly, the molar Gibbs energy and chemical potential of components in the ideal

multicomponent system can be expressed as

𝑔𝐼𝑆 = ∑ 𝑋𝑖𝑔𝑖0 + 𝑅𝑇 ∑ 𝑋𝑖𝑙𝑛𝑋𝑖

𝑖𝑖

, (3.7)

𝜇𝑖 = 𝑔𝑖0 + 𝑅𝑇𝑙𝑛𝑋𝑖 . (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

∆𝐻 =1

2𝑧𝐿𝑁𝑋𝐴𝑋𝐵 ∙ (휀𝐴𝐴 + 휀𝐵𝐵 − 2휀𝐴𝐵), (3.9)

where 𝑧𝐿 is the coordination number (the number of nearest atoms), 𝑁 is the total

number of atoms in a system, 휀𝐴𝐴, 휀𝐵𝐵 and 휀𝐴𝐵 are the bond energies for atomic bond

A-A, B-B and A-B. Using a substitution of

𝜔𝐴𝐵 = 𝑧𝐿 ∙ (휀𝐴𝐴 + 휀𝐵𝐵 − 2휀𝐴𝐵), (3.10)

and considering only one mole of atoms, the molar Gibbs energy of a regular binary

system can be expressed as

𝑔𝑅𝑆 = 𝑋𝐴𝑔𝐴0 + 𝑋𝐵𝑔𝐵

0 + 𝑅𝑇(𝑋𝐴𝑙𝑛𝑋𝐴 + 𝑋𝐵𝑙𝑛𝑋𝐵) +1

2𝑋𝐴𝑋𝐵𝜔𝐴𝐵. (3.11)

For multicomponent system,

𝑔𝑅𝑆 = ∑ 𝑋𝑖𝑔𝑖0 + 𝑅𝑇 ∑ 𝑋𝑖𝑙𝑛𝑋𝑖

𝑖

+1

2∑ ∑ 𝑋𝑖𝑋𝑗𝜔𝑖𝑗

𝑗>𝑖𝑖𝑖

, (3.12)

where 𝜔𝑖𝑗 is called the regular solution parameter. When 𝜔 > 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 𝜔 =

0, the enthalpy is equal to zero and the system is an ideal solution, i.e. the atoms are

randomly distributed; when 𝜔 < 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 𝑋𝑙𝑛𝑋 + (1 − 𝑋)𝑙𝑛(1 − 𝑋) in binary systems, while the

enthalpy of mixing to the Gibbs energy is positive if 𝜔 > 0, scaling with 𝑋(1 − 𝑋) 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 ∆G -𝑋 curve only shows single minimum due to

the dominant contribution of entropy. Below a critical temperature, 𝑇𝑐𝑟𝑖𝑡, the ∆G -𝑋

curve starts to be w-shaped.

Fig 3.2 The relationship between ∆𝐺 - 𝑋 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 ∆G -𝑋

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 𝑔𝐼𝑆 and excess Gibbs energy 𝑔𝐸𝑋

𝑔 = ∑ 𝑋𝑖𝑔𝑖0 + 𝑅𝑇 ∑ 𝑋𝑖𝑙𝑛𝑋𝑖

𝑖

+ 𝑔𝐸𝑋𝑅𝐾 + 𝑔𝐸𝑋

𝑚𝑎𝑔𝑛+ 𝑔𝐸𝑋

𝑆𝑅𝑂 + ⋯, (3.13)

𝑖

where 𝑔𝐸𝑋𝑅𝐾 , 𝑔𝐸𝑋

𝑚𝑎𝑔𝑛 and 𝑔𝐸𝑋

𝑆𝑅𝑂 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

𝑃𝑐𝑢𝑟𝑣 =2𝛾

𝜌, (3.14)

where 𝛾 is interfacial energy (J/m2) and 𝜌 is the radius of a spherical precipitate. The

corresponding energy can be expressed as

𝑔𝑐𝑢𝑟𝑣 =2𝛾

𝜌𝑣𝛽, (3.15)

where 𝑣𝛽 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

∆𝐺𝑣𝑜𝑙𝑒𝑙 =

𝐸𝑎

9(1 − 𝑣𝑎)(𝑣∗)2, (3.16)

where 𝐸𝑎 and 𝑣𝑎 are the elastic modulus and the Poisson’s ratio of the matrix, 𝑣∗

represents the volume elastic misfit with

𝑣∗ =𝑣𝛽 − 𝑣𝛼

𝑣𝛼, (3.17)

where 𝑣𝛼 and 𝑣𝛽 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 ‘discrete

droplet’ formalism usually used in nucleation and growth mechanism of phase

separation, while the non-classical nucleation theory depicts a ‘composition wave’

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

𝑁 = 𝑁0𝑒𝑥𝑝(−∆𝐺𝑛𝑢𝑐𝑙

𝑘𝐵𝑇), (3.18)

where 𝑁0 is the number of atoms in a unit volume, ∆𝐺𝑛𝑢𝑐𝑙 is the nucleation energy

(also called the Gibbs energy change when forming a cluster of a certain size), 𝑘𝐵 is

the Boltzmann constant and 𝑇 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

𝐽𝑠 = 𝑁0𝑍𝛽∗𝑒𝑥𝑝(−∆𝐺∗

𝑘𝐵𝑇), (3.19)

where 𝑍 is the Zeldovich factor. Since the probability of a critical cluster for growth

and decay are the same, i.e. 50%, 𝑍 accounts for the thermal activation of a critical

nuclei with energy 𝑘𝐵𝑇 governing the growth and decay of critical clusters. ∆𝐺∗ is the

critical nucleation energy (also called nucleation barrier) and 𝛽∗ 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), 𝜏

𝜏 =1

2𝛽∗𝑍2. (3.20)

Considering the incubation effect, the transient nucleation rate or time-dependent

nucleation rate in a real system can be consequently expressed as

𝐽 = 𝐽𝑠𝑒𝑥𝑝(−𝜏

𝑡) = 𝑁0𝑍𝛽∗𝑒𝑥𝑝(−

∆𝐺∗

𝑘𝐵𝑇)𝑒𝑥𝑝(

−𝜏

𝑡). (3.21)

Considering the formation of a spherical and coherent nucleus with radius 𝜌 under

the assumption of a sharp interface between the nucleus and matrix, the total energy

change can be expressed as

∆𝐺𝑛𝑢𝑐𝑙 =4

3𝜋𝜌3 ∙ ∆𝐺𝑣𝑜𝑙 + 4𝜋𝜌2 ∙ ∆𝐺𝑠𝑢𝑟𝑓 , (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. ∆𝐺𝑠𝑢𝑟𝑓 is identical

to interfacial energy 𝛾. ∆𝐺𝑣𝑜𝑙 includes chemical and mechanical contributions

∆𝐺𝑣𝑜𝑙 = −𝑑𝑐ℎ𝑒𝑚

𝛽

𝑣𝛼+ ∆𝐺𝑣𝑜𝑙

𝑒𝑙 , (3.23)

17

where 𝑑𝑐ℎ𝑒𝑚𝛽

is the chemical driving force, 𝑣𝛼 is the molar volume of the mother

phase, the elastic energy ∆𝐺𝑣𝑜𝑙𝑒𝑙 has been introduced in Eq. 3.16. By setting the

derivative of ∆𝐺𝑛𝑢𝑐𝑙 in Eq. 3.22 to zero, the critical nucleation radius 𝑟∗and critical

nucleation energy ∆𝐺∗can be derived

𝑟∗ = −2∆𝐺𝑠𝑢𝑟𝑓

∆𝐺𝑣𝑜𝑙

=2𝛾

𝑑𝑐ℎ𝑒𝑚𝛽

𝑣𝛼 − ∆𝐺𝑣𝑜𝑙𝑒𝑙

, (3.24)

∆𝐺∗ =16𝜋

3

(∆𝐺𝑠𝑢𝑟𝑓)3

∆𝐺𝑣𝑜𝑙2 =

16𝜋

3

𝛾3

(−𝑑𝑐ℎ𝑒𝑚

𝛽

𝑣𝛼 + ∆𝐺𝑣𝑜𝑙𝑒𝑙 )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:

𝐽𝑖𝛼𝛽

+ 𝐽𝑖𝛽𝛼

= ⟦𝑐𝑖𝛼𝛽

⟧ ∙ 𝑣 = (𝑐𝑖𝛽𝛼

− 𝑐𝑖𝛼𝛽

) ∙ 𝑣, (3.26)

18

where 𝐽𝑖𝛼𝛽

is the fluxes of element 𝑖 into the phase boundary, 𝐽𝑖𝛽𝛼

is the fluxes of

element 𝑖 out of the phase boundary, 𝑣 is the velocity of phase boundary movement

and ⟦𝑐𝑖𝛼𝛽

⟧ is the composition difference of element 𝑖 in the two sides of the phase

boundary expressed by concentrations at the precipitate side 𝑐𝑖𝛽𝛼

and the matrix side

𝑐𝑖𝛼𝛽

. For a system with 𝑛 kinds of components, the mass balance equation holds 2(𝑛 −

1) unknown chemical compositions at two the sides of the interface and one common

interface velocity for all components, i.e. 2𝑛 − 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

𝜇𝑖𝛽𝛼

= 𝜇𝑖𝛼𝛽

, (3.27)

where 𝜇𝑖𝛽𝛼

and 𝜇𝑖𝛼𝛽

represents the chemical potential of component 𝑖 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, 𝐽𝐵𝛽𝛼

= 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

(𝑐𝐵𝛽𝛼

− 𝑐𝐵𝛼𝛽

) ∙ 𝑣 = 𝐽𝐵𝛼𝛽

= 𝐷𝐵𝛼

𝜕𝑐𝐵

𝜕𝑥

, (3.28)

where 𝐷𝐵𝛼 and 𝑐𝐵 are the diffusion coefficient and concentration of component B in

the matrix, respectively. The concentration gradient 𝜕𝑐𝐵𝜕𝑥⁄ is a function of 𝑐𝐵

𝛼𝛽, 𝑐𝐵

0

and precipitate size. For a binary system, this flux-balance equation (𝑣) can be easily

solved, for the 𝑐𝐵𝛽𝛼

and 𝑐𝐵𝛼𝛽

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 Ågren [50]

proposed an approximate growth rate equation by inducing an effective diffusion

distance factor

19

(𝑐𝑖𝛽𝛼


) ∙ 𝑣 = ∑𝐷𝑖𝑗

𝛼(𝑐𝑗0 − 𝑐𝑗

𝛼𝛽)

𝜉𝑗𝜌

𝑛−1

𝑗=1

, (3.29)

(𝑐𝑖𝛽𝛼


) ∙ 𝑣 = 𝑐𝑖𝛼𝛽

𝑀𝑖 (𝜇𝑖0 − 𝜇𝑖

𝛼𝛽) 𝜉𝑗𝜌⁄ , (3.30)

where 𝜉𝑖 is a factor adjusting the effective diffusional distance as a function of

supersaturation, 𝐷𝑖𝑗𝛼 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

𝐺 = ∑ 𝑁0𝑖𝜇0𝑖

𝑛

𝑖=1

+ ∑4𝜋𝜌𝑘

3

3

𝑚

𝑘=1

(𝜆𝑘 + ∑ 𝑐𝑘𝑖𝜇𝑘𝑖

𝑛

𝑖=1

) + ∑ 4𝜋𝜌𝑘2𝛾𝑘

𝑚

𝑘=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. 𝑖

represents component and 𝑘 represents precipitate. 𝑁0𝑖 and 𝜇0𝑖 are the number of

moles and chemical potential of component 𝑖 in the matrix, respectively. 𝜌𝑘 is the

radius of precipitate 𝑘. 𝑐𝑘𝑖 and 𝜇𝑘𝑖 are the mean concentration and chemical potential

of component 𝑖 in precipitate 𝑘 . 𝜆𝑘 represents mechanical energy associated with

elastic stress field caused by the volume misfit between precipitate and matrix. 𝛾𝑘

designates the interfacial energy of precipitate 𝑘 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 𝑄1 , diffusion inside the

precipitates 𝑄2 and the diffusion in the matrix 𝑄3

𝑄1 = ∑4𝜋𝜌𝑘

2

𝑀𝑘𝐼𝐹

𝑚

𝑘=1

�̇�𝑘2, (3.32)

𝑄2 = ∑ ∑4𝜋𝑅𝑇𝜌𝑘

5

45𝑐𝑘𝑖𝐷𝑘𝑖

�̇�𝑘𝑖2

𝑛

𝑖=1

𝑚

𝑘=1

, (3.33)

20

𝑄3 ≈ ∑ ∑4𝜋𝑅𝑇𝜌𝑘

3(�̇�𝑘(𝑐𝑘𝑖 − 𝑐0𝑖) + 𝜌𝑘 �̇�𝑘𝑖 3⁄ )3

𝑐0𝑖𝐷0𝑖

𝑛

𝑖=1

𝑚

𝑘=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

𝑓(𝑟, 𝑡) =4

9𝜉2(

3

3 + 𝜉)

73(

32

32

− 𝜉)

113 𝑒𝑥𝑝(−

𝜉

32

− 𝜉), (3.35)

where 𝜉 represents 𝑟 𝑟𝑚𝑒𝑎𝑛⁄ . The LSW size distribution function has the maximum

when 𝜉 = 1.135. The time evolutions of mean radius and number density during the

coarsening stage obtained by LSW are expressed as

𝑟(𝑡) = 𝑟0 [1 +𝑡

𝜏𝐷

]

13

, (3.36)

𝑁(𝑡) = 𝑁0 [1 +𝑡

𝜏𝐷

]−1

, (3.37)

21

where 𝜏𝐷 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 𝑡1 3⁄ , the number density 𝑡−1 and the supersaturation

𝑡−1 3⁄ 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 𝑓(𝑟, 𝑡) 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

𝑓∗(𝑟∗, 𝑡) = 𝑁𝐿𝑆

𝑏

�̅�𝐿𝑆 − 𝑟∗ , (3.38)

where 𝑟∗ is the critical radius, �̅�𝐿𝑆 is the mean radius, 𝑁𝐿𝑆 is the total number of

precipitates, 𝑏 is a constant and 𝑏 = 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

𝑑𝑁𝐿𝑆

𝑑𝑡= 𝐽 − 𝑓∗(𝑟∗, 𝑡)

𝑑𝑟∗

𝑑𝑡, (3.39)

𝑑�̅�𝐿𝑆

𝑑𝑡= 𝑣(�̅�𝐿𝑆) + (�̅�𝐿𝑆 − 𝑟∗)

𝑓∗(𝑟∗, 𝑡)

𝑁𝐿𝑆

𝑑𝑟∗

𝑑𝑡+

1

𝑁𝐿𝑆

𝐽(𝑟∗)(𝑟∗ + ∆𝑟∗ − �̅�𝐿𝑆), (3.40)

where 𝐽 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; 𝑣(�̅�𝐿𝑆) is the

growth rate of precipitates with mean radius, which is correlated to the diffusion-

controlled evolution models; 𝑓∗(𝑟∗, 𝑡) ∙ 𝑑𝑟∗ accounts for the number of dissolved

particles with radii between 𝑟∗ and 𝑟∗ + 𝑑𝑟∗; 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

4𝜋

3�̅�𝐿𝑆

3 ∙ 𝑁𝐿𝑆 ∙ (𝑐𝐵𝛽

− 𝑐𝐵0) = (𝑐𝐵

0 − 𝑐�̅�), (3.41)

where 𝑐𝐵𝛽

and 𝑐�̅� 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

𝑋𝐵𝛼′

≈ 𝑋𝐵𝛼 (1 +

2𝛾𝑉𝛽

(𝑋𝐵𝛽

− 𝑋𝐵𝛼)𝑅𝑇

∙1

𝜌), (3.42)

to

𝑋𝐵𝛼′

≈ 𝑋𝐵𝛼𝑒𝑥𝑝 (

2𝛾𝑉𝛽

𝑅𝑇∙

1

𝜌). (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 𝑓∗(𝑟∗, 𝑡) is substituted by a discrete

distribution of size classes [𝑟𝑗 , 𝑟𝑗+1] with |𝑟𝑗 − 𝑟𝑗+1| 𝑟𝑗⁄ ≪ 1 and the number of nuclei

𝑛𝑗 in each class is determined by the time-dependent CNT model at a time interval

∆𝑡. Thus, the PSD function 𝑓(𝑟, 𝑡) 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]

∆𝜎𝑜𝑟𝑜𝑤𝑎𝑛 = 𝑀0.4𝐺𝑏

𝜋√1 − 𝜐

ln (2√23

𝑟 𝑏⁄ )

𝜆, (4.1)

λ = 2√2

3𝑟 (√

𝜋

4𝑓− 1), (4.2)

where 𝑀 is the Taylor orientation factor, 𝐺 is the shear modulus of the matrix, 𝑏 is

the magnitude of the Burgers vector, 𝜐 is the Poisson’s ratio, λ is the inter-particle

spacing, and the 𝑓 and 𝑟 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]

∆𝜎𝑚𝑖𝑠𝑓𝑖𝑡 = 𝑀𝜒𝐺(휀)32√2𝑓

𝑟

𝑏 , (4.3)

ε = |𝛿|[1 + 2𝐺(1 − 2𝜐𝑃) 𝐺𝑃(1 + 𝜐𝑃)⁄ ], (4.4)

𝛿 =𝑎𝑃 − 𝑎

𝑎, (4.5)

where χ is a constant varying between 2 and 3, ε is the misfit strain parameter, 𝐺𝑃 and

𝜐𝑃 are shear modulus and Possion’s ratio of the precipitates respectively, δ is the

difference of lattice parameters of precipitate 𝑎𝑃 and matrix 𝑎.

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]

∆𝜎𝑚𝑜𝑑𝑢𝑙𝑢𝑠 = 𝑀𝐺𝑏

𝐿[1 − (

𝐸𝑝

𝐸𝑚

)2

]

34

, 𝑖𝑓𝑠𝑖𝑛−1 (𝐸𝑝

𝐸𝑚

) ≥ 50°, (4.6)

∆𝜎𝑚𝑜𝑑𝑢𝑙𝑢𝑠 = 0.8𝑀𝐺𝑏

𝐿[1 − (

𝐸𝑝

𝐸𝑚

)2

]

12

, 𝑖𝑓𝑠𝑖𝑛−1 (𝐸𝑝

𝐸𝑚

) ≤ 50° , (4.7)

𝐸𝑝

𝐸𝑚

=𝐸𝑝

∞

𝐸𝑚∞

ln (𝑟 𝑟0⁄ )

ln (𝑅 𝑟0⁄ )+

ln (𝑅 𝑟⁄ )

ln (𝑅 𝑟0⁄ ), (4.8)

𝐿 = 0.866/(𝑅𝑁)1 2⁄ , (4.9)

Where 𝐸𝑝 and 𝐸𝑚 are dislocation line energy in precipitate and matrix respectively, 𝐿

is the mean particle spacing in the gliding plane, 𝐸𝑝∞ and 𝐸𝑚

∞ are energy per unit

length of a dislocation in an infinite medium for precipitate and matrix respectively,

𝑟 and 𝑟0 are inner and outer cut-off radius of the dislocation stress field, 𝑅 and 𝑁 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]

∆𝜎𝑜𝑟𝑑𝑒𝑟𝑖𝑛𝑔 =𝑀𝛾𝑎𝑝𝑏

3 2⁄

𝑏(

4𝑟𝑠𝑓

𝜋𝑇)

1 2⁄

, (4.10)

where 𝑟𝑎𝑝𝑏 is the average value of the antiphase boundary energy of precipitates, 𝑟𝑠 =

(2 3⁄ )1 2⁄ 𝑟 is the average radius of the sheared precipitates in the gliding plane, 𝑇 is

dislocation line tension, approximated as 𝐺𝑏2 2⁄ .

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 Cr–Mo–V

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 ºC 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 °C and 550 °C 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 °C 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

Nes’s 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 Nes’s 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

Nes’s 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 Hedström, 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 för

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 (周涛)

Stockholm, October 2019

40

References

[1] J. Langer, K. Schwartz. Kinetics of nucleation in near-critical fluids, Physical

Review A 21 (1980) 948.

[2] R. Kampmann, R. Wagner. Kinetics of precipitation in metastable binary

alloys: theory and application to copper-1. 9 at.% titanium and nickel-14at.%

aluminium (Alloys). in: Decomposition of alloys: the early stages edited by

P. Haasen et al. (pp. 91–103). Oxford: Pergamon Press, 1984.

[3] R. Wagner, R. Kampmann, P.W. Voorhees. Homogeneous second‐phase

precipitation. in: Phase ransformations in materials, Wiley-VCH, Weinheim,

2001.

[4] D.B. Williams, C. Carter Barry. Transmission electron microscopy: a

textbook for materials science. New York: Springer, 2009.

[5] M.K. Miller, R.G. Forbes. Atom-probe tomography: the local electrode atom

probe. Springer, 2014.

[6] L. Feigin, D.I. Svergun. Structure analysis by small-angle X-ray and neutron

scattering, Springer, 1987.

[7] B. Hammouda. Probing nanoscale structures-the SANX toolbox, National

Institute of Standards and Technology, 2008.

[8] J. Allison, D. Backman, L. Christodoulou. Integrated computational

materials engineering: a new paradigm for the global materials profession,

JOM 58 (2006) 25-27.

[9] G.B. Olson. Computational design of hierarchically structured materials,

Science 277 (1997) 1237-1242.

[10] T. Zhou, R.P. Babu, J. Odqvist, H. Yu, P. Hedström. Quantitative electron

microscopy and physically based modelling of Cu precipitation in

precipitation-hardening martensitic stainless steel 15-5 PH, Materials &

Design 143 (2018) 141-149.

[11] T. Zhou, J. Faleskog, R.P. Babu, J. Odqvist, H. Yu, P. Hedström. Exploring

the relationship between the microstructure and strength of fresh and

tempered martensite in a maraging stainless steel Fe–15Cr–5Ni, Materials

Science and Engineering: A 745 (2019) 420-428.

[12] T. Zhou, Z. Hou, R. Prasath Babu, J. Odqvist, P. Hedström. Precipitation of

multiple carbides in martensitic CrMoV steels - experimental analysis and

exploration of alloying strategy through thermodynamic calculations

(unpublished).

[13] T. Zhou, J. Lu, P. Hedström. Mechanical behavior of fresh and tempered

martensite in a CrMoV-alloyed steel explained by microstructural evolution

and model predictions (unpublished).

[14] A.J. Schwartz, M. Kumar, B.L. Adams, D.P. Field. Electron backscatter

diffraction in materials science, 2nd edition, Springer, 2009.

41

[15] O. Engler, V. Randle. Introduction to texture analysis: macrotexture,

microtexture, and orientation mapping, CRC press, 2009.

[16] T. Zhou, H. Yu, J. Hu, S. Wang. Study of microstructural evolution and

strength–toughness mechanism of heavy-wall induction bend pipe, Materials

Science and Engineering: A 615 (2014) 436-446.

[17] T. Zhou, H. Yu, S. Wang. Effect of microstructural types on toughness and

microstructural optimization of ultra-heavy steel plate: EBSD analysis and

microscopic fracture mechanism, Materials Science and Engineering: A 658

(2016) 150-158.

[18] R. Borrajo-Pelaez, P. Hedström. Recent developments of crystallographic

analysis methods in the scanning electron microscope for applications in

metallurgy, Critical Reviews in Solid State and Materials Sciences 43 (2018)

455-474.

[19] R.R. Keller, R.H. Geiss. Transmission EBSD from 10 nm domains in a

scanning electron microscope, Journal of Microscopy 245 (2012) 245-251.

[20] P.W. Trimby. Orientation mapping of nanostructured materials using

transmission Kikuchi diffraction in the scanning electron microscope,

Ultramicroscopy 120 (2012) 16-24.

[21] S. Zaefferer, N.N. Elhami. Theory and application of electron channelling

contrast imaging under controlled diffraction conditions, Acta Materialia 75

(2014) 20-50.

[22] S. Suzuki. Features of transmission EBSD and its application, JOM 65

(2013) 1254-1263.

[23] N. Brodusch, H. Demers, R. Gauvin. Nanometres-resolution Kikuchi

patterns from materials science specimens with transmission electron

forward scatter diffraction in the scanning electron microscope, Journal of

Microscopy 250 (2013) 1-14.

[24] A.A. Saleh, G. Casillas, E.V. Pereloma, K.R. Carpenter, C.R. Killmore, A.A.

Gazder. A transmission Kikuchi diffraction study of cementite in a quenched

and tempered steel, Materials Characterization 114 (2016) 146-150.

[25] T. Zhou, Z. Hou, R.P. Babu, P. Hedström. Recent developments in

transmission electron microscopy based characterization of precipitation in

metallic alloys (unpublished).

[26] M.L. Taheri, E.A. Stach, I. Arslan, P.A. Crozier, B.C. Kabius, T. LaGrange,

A.M. Minor, S. Takeda, M. Tanase, J.B. Wagner. Current status and future

directions for in situ transmission electron microscopy, Ultramicroscopy 170

(2016) 86-95.

[27] H. Zheng, Y. Zhu. Perspectives on in situ electron microscopy,


[28] Z. Chen, D. Taplin, M. Weyland, L.J. Allen, S. Findlay. Composition

measurement in substitutionally disordered materials by atomic resolution

42

energy dispersive X-ray spectroscopy in scanning transmission electron

microscopy, Ultramicroscopy 176 (2017) 52-62.

[29] F. Krause, A. Rosenauer, J. Barthel, J. Mayer, K. Urban, R. Dunin-

Borkowski, H. Brown, B. Forbes, L. Allen. Atomic resolution elemental

mapping using energy-filtered imaging scanning transmission electron

microscopy with chromatic aberration correction, Ultramicroscopy 181

(2017) 173-177.

[30] X. Xu, P. Liu, Z. Tang, A. Hirata, S. Song, T. Nieh, P. Liaw, C. Liu, M. Chen.

Transmission electron microscopy characterization of dislocation structure

in a face-centered cubic high-entropy alloy Al0.1CoCrFeNi, Acta Materialia

144 (2018) 107-115.

[31] C. Liu, H. Chen, H. Liu, X. Zhao, J. Nie. Metastable precipitate phases in

Mg–9.8 wt% Sn alloy, Acta Materialia 144 (2018) 590-600.

[32] E.L. Solomon, A.R. Natarajan, A.M. Roy, V. Sundararaghavan, A. Van der

Ven, E.A. Marquis. Stability and strain-driven evolution of β' precipitate in

Mg-Y alloys, Acta Materialia 166 (2019) 148-157.

[33] Y. Zhao, H. Chen, Z. Lu, T. Nieh. Thermal stability and coarsening of

coherent particles in a precipitation-hardened (NiCoFeCr) 94Ti2Al4 high-

entropy alloy, Acta Materialia 147 (2018) 184-194.

[34] Y.R. Wen, A. Hirata, Z.W. Zhang, T. Fujita, C.T. Liu, J.H. Jiang, M.W.

Chen. Microstructure characterization of Cu-rich nanoprecipitates in a Fe–

2.5 Cu–1.5 Mn–4.0 Ni–1.0 Al multicomponent ferritic alloy, Acta Materialia

61 (2013) 2133-2147.

[35] B. Gault, M.P. Moody, F. de Geuser, G. Tsafnat, A. La Fontaine, L.T.

Stephenson, D. Haley, S.P. Ringer. Advances in the calibration of atom probe

tomographic reconstruction, Journal of Applied Physics 105 (2009) 034913.

[36] F. Vurpillot, M. Gruber, G. Da Costa, I. Martin, L. Renaud, A. Bostel.

Pragmatic reconstruction methods in atom probe tomography,


[37] D. Vaumousse, A. Cerezo, P. Warren. A procedure for quantification of

precipitate microstructures from three-dimensional atom probe data,


[38] M.K. Miller, R. Forbes. Atom probe tomography, Materials Characterization

60 (2009) 461-469.

[39] O.C. Hellman, J.A. Vandenbroucke, J. Rüsing, D. Isheim, D.N. Seidman.

Analysis of three-dimensional atom-probe data by the proximity histogram,

Microscopy and Microanalysis 6 (2000) 437-444.

[40] F. De Geuser, A. Deschamps. Precipitate characterisation in metallic systems

by small-angle X-ray or neutron scattering, Comptes Rendus Physique 13

(2012) 246-256.

43

[41] T. Marlaud, A. Deschamps, F. Bley, W. Lefebvre, B. Baroux. Influence of

alloy composition and heat treatment on precipitate composition in Al–Zn–

Mg–Cu alloys, Acta Materialia 58 (2010) 248-260.

[42] A. Deschamps, C. Genevois, M. Nicolas, F. Perrard, F. Bley. Study of

precipitation kinetics: towards non-isothermal and coupled phenomena,

Philosophical Magazine 85 (2005) 3091-3112

[43] M. Perez, F. Perrard, V. Massardier, X. Kleber, A. Deschamps, H. De

Monestrol, P. Pareige, G. Covarel. Low-temperature solubility of copper in

iron: experimental study using thermoelectric power, small angle X-ray

scattering and tomographic atom probe, Philosophical Magazine 85 (2005)

2197-2210.

[44] F. Perrard, A. Deschamps, F. Bley, P. Donnadieu, P. Maugis. A small-angle

neutron scattering study of fine-scale NbC precipitation kinetics in the α-Fe–

Nb–C system, Journal of Applied Crystallography 39 (2006) 473-482.

[45] M. Dumont, A. Steuwer, A. Deschamps, M. Peel, P. Withers. Microstructure

mapping in friction stir welds of 7449 aluminium alloy using SAXS, Acta

Materialia 54 (2006) 4793-4801.

[46] E. Kozeschnik. Modeling solid-state precipitation, Momentum press, 2013.

[47] M. Volmer, Α. Weber. Keimbildung in übersättigten Gebilden (Nucleation

in supersaturated formations), Zeitschrift für physikalische Chemie 119

(1926) 277-301.

[48] Precipitation Module (TC-PRISMA) User Guide, 2016.

[49] C. Zener. Theory of growth of spherical precipitates from solid solution,

Journal of Applied Physics 20 (1949) 950-953.

[50] Q. Chen, J. Jeppsson, J. Ågren. Analytical treatment of diffusion during

precipitate growth in multicomponent systems, Acta Materialia 56 (2008)

1890-1896.

[51] J. Svoboda, F. Fischer, P. Fratzl, E. Kozeschnik. Modelling of kinetics in

multi-component multi-phase systems with spherical precipitates: I: Theory,

Materials Science and Engineering: A 385 (2004) 166-174.

[52] J. Svoboda, I. Turek. On diffusion-controlled evolution of closed solid-state

thermodynamic systems at constant temperature and pressure, Philosophical

Magazine B 64 (1991) 749-759.

[53] I.M. Lifshitz, V.V. Slyozov. The kinetics of precipitation from

supersaturated solid solutions, Journal of Physics and Chemistry of Solids 19

(1961) 35-50.

[54] C. Wagner. Theory of precipitate change by redissolution, Z. Elektrochem

65 (1961) 581-591.

[55] A. Ardell. Effect of volume fraction on particle coarsening: theoretical

considerations, Acta Metallurgica 20 (1972) 61-71.

44

[56] K. Binder. Theory for the dynamics of "clusters" II. Critical diffusion in

binary systems and the kinetics of phase separation, Physical Review B 15

(1977) 4425-4447.

[57] K. Binder, D. Stauffer. Theory for the slowing down of the relaxation and

spinodal decomposition of binary mixtures, Physical Review Letters 33

(1974) 1006-1009.

[58] P. Warczok, J. Ženíšek, E. Kozeschnik. Atomistic and continuums modeling

of cluster migration and coagulation in precipitation reactions,

Computational Materials Science 60 (2012) 59-65.

[59] Y. Yoo, D. Yoon, Henry, MF. The effect of elastic misfit strain on the

morphological evolution of γ’-precipitates in a model Ni-base superalloy,

Metals and Materials 1 (1995) 47-61.

[60] K. Chen, P.A. Korzhavyi, G. Demange, H. Zapolsky, R. Patte, J. Boisse, Z.

Wang. Morphological instability of iron-rich precipitates in CuFeCo alloys,

Acta Materialia 163 (2019) 55-67.

[61] Z. Hou, R.P. Babu, P. Hedström, J. Odqvist. Early stages of cementite

precipitation during tempering of 1C–1Cr martensitic steel, Journal of

Materials Science (2018) 1-13.

[62] Z. Hou, P. Hedström, Q. Chen, Y. Xu, D. Wu, J. Odqvist. Quantitative

modeling and experimental verification of carbide precipitation in a

martensitic Fe–0.16 wt% C–4.0 wt% Cr alloy, Calphad 53 (2016) 39-48.

[63] Z. Hou, P. Hedström, Y. Xu, W. Di, J. Odqvist. Microstructure of Martensite

in Fe–C–Cr and its Implications for Modelling of Carbide Precipitation

during Tempering, ISIJ International 54 (2014) 2649-2656.

[64] E. Kozeschnik, J. Svoboda, P. Fratzl, F. Fischer. Modelling of kinetics in

multi-component multi-phase systems with spherical precipitates: II:

numerical solution and application, Materials Science and Engineering: A

385 (2004) 157-165.

[65] G.B. Olson, H.J. Jou, J. Jung, J. Sebastian, A. Misra, I. Locci, D. Hull.

Precipitation model validation in 3rd generation aeroturbine disc alloys,

Superalloys 2018 in TMS conference, 2008.

[66] W. Cao, F. Zhang, S.-L. Chen, C. Zhang, Y. Chang. An integrated

computational tool for precipitation simulation, JOM 63 (2011) 29-34.

[67] B. Sonderegger, E. Kozeschnik. Interfacial energy of diffuse phase

boundaries in the generalized broken-bond approach, Metallurgical and

Materials Transactions A 41 (2010) 3262-3269.

[68] E. Kozeschnik, J. Svoboda, F. Fischer. Shape factors in modeling of

precipitation, Materials Science and Engineering: A 441 (2006) 68-72.

[69] B. Sonderegger, E. Kozeschnik. Size dependence of the interfacial energy in

the generalized nearest-neighbor broken-bond approach, Scripta Materialia

60 (2009) 635-638.

45

[70] A. Borgenstam, L. Höglund, J. Ågren, A. Engström. DICTRA, a tool for

simulation of diffusional transformations in alloys, Journal of Phase

Equilibria 21 (2000) 269-280.

[71] A. Ardell. Precipitation hardening, Metallurgical Transactions A 16 (1985)

2131-2165.

[72] A. Kelly, R.B. Nicholson. Precipitation hardening, Progress in Materials

Science 10 (1963) 151-391.

[73] L. Brown, R. Ham. Dislocaton-particle interactions in Kelly, A and

Nicholson, RB. Strengthening methods in crystals, J. Wiley and Sons, New

York, 1971.

[74] N. Kamikawa, Y. Abe, G. Miyamoto, Y. Funakawa, T. Furuhara. Tensile

behavior of Ti, Mo-added low carbon steels with interphase precipitation,

ISIJ International 54 (2014) 212-221.

[75] H. Wen, T.D. Topping, D. Isheim, D.N. Seidman, E.J. Lavernia.

Strengthening mechanisms in a high-strength bulk nanostructured Cu–Zn–

Al alloy processed via cryomilling and spark plasma sintering, Acta

Materialia 61 (2013) 2769-2782.

[76] V. Gerold, H. Haberkorn. On the critical resolved shear stress of solid

solutions containing coherent precipitates, Physica Status Solidi (b) 16

(1966) 675-684.

[77] L.M. Brown, R.K. Ham. Dislocation-Particle interactions. In: A. Kelly, R.B.

Nicholson, editors. Strengthening methods in Crystals, Applied Science

Publisher LTD, London (1971) 9-135.

[78] A.J. Ardell. Precipitation hardening, Metallurgical Transactions A 16 (1985)

2131-2165.

[79] K.C. Russell, L. Brown. A dispersion strengthening model based on differing

elastic moduli applied to the iron-copper system, Acta Metallurgica 20

(1972) 969-974.

[80] Z. Jiao, J. Luan, Z. Zhang, M.K. Miller, W. Ma, C. Liu. Synergistic effects

of Cu and Ni on nanoscale precipitation and mechanical properties of high-

strength steels, Acta Materialia 61 (2013) 5996-6005.

[81] J. Takahashi, K. Kawakami, Y. Kobayashi. Consideration of particle-

strengthening mechanism of copper-precipitation-strengthened steels by

atom probe tomography analysis, Materials Science and Engineering: A 535

(2012) 144-152.

[82] T. Gladman. Precipitation hardening in metals, Materials Science and

Technology 15 (1999) 30-36.

[83] S. Jiang, H. Wang, Y. Wu, X. Liu, H. Chen, M. Yao, B. Gault, D. Ponge, D.

Raabe, A. Hirata, M. Chen, Y. Wang, Z. Lu. Ultrastrong steel via minimal

lattice misfit and high-density nanoprecipitation, Nature, 544 (2017) 460-

464.

46

[84] H. Kong, C. Liu. A review on nano-scale precipitation in steels,

Technologies 6 (2018) 36.

[85] F. Qian, S. Jin, G. Sha, Y. Li. Enhanced dispersoid precipitation and

dispersion strengthening in an Al alloy by microalloying with Cd, Acta

Materialia 157 (2018) 114-125.

[86] Z. Jiao, J. Luan, M. Miller, Y.W. Chung, C. Liu. Co-precipitation of

nanoscale particles in steels with ultra-high strength for a new era, Materials

Today 20 (2017) 142-154.

[87] J.S. Wang, M. Mulholland, G. Olson, D. Seidman. Prediction of the yield

strength of a secondary-hardening steel, Acta Materialia 61 (2013) 4939-

4952.

[88] M.D. Mulholland, D.N. Seidman. Multiple dispersed phases in a high-

strength low-carbon steel: An atom-probe tomographic and synchrotron X-

ray diffraction study, Scripta Materialia 60 (2009) 992-995.

[89] R.P. Kolli, D.N. Seidman. Co-precipitated and collocated carbides and Cu-

rich precipitates in a Fe–Cu steel characterized by atom-probe tomography,

Microscopy and Microanalysis 20 (2014) 1727-1739.

[90] D. Jain, D. Isheim, A.H. Hunter, D.N. Seidman. Multicomponent high-

strength low-alloy steel precipitation-strengthened by sub-nanometric Cu

precipitates and M2C carbides, Metallurgical and Materials Transactions A

47 (2016) 3860-3872.

[91] M. Kapoor, D. Isheim, G. Ghosh, S. Vaynman, M.E. Fine, Y.W. Chung.

Aging characteristics and mechanical properties of 1600 MPa body-centered

cubic Cu and B2-NiAl precipitation-strengthened ferritic steel, Acta

Materialia 73 (2014) 56-74.

[92] Z. Jiao, J. Luan, M. Miller, C. Liu. Precipitation mechanism and mechanical

properties of an ultra-high strength steel hardened by nanoscale NiAl and Cu

particles, Acta Materialia 97 (2015) 58-67.

[93] Z. Zhang, C.T. Liu, M.K. Miller, X.L. Wang, Y. Wen, T. Fujita, A. Hirata,

M. Chen, G. Chen, B.A. Chin. A nanoscale co-precipitation approach for

property enhancement of Fe-base alloys, Scientific Reports 3 (2013) 1327.

[94] Z. Jiao, J. Luan, M.K. Miller, C. Yu, C. Liu. Effects of Mn partitioning on

nanoscale precipitation and mechanical properties of ferritic steels

strengthened by NiAl nanoparticles, Acta Materialia 84 (2015) 283-291.

[95] Z. Wang, H. Zhang, C. Guo, W. Liu, Z. Yang, X. Sun, Z. Zhang, F. Jiang.

Effect of molybdenum addition on the precipitation of carbides in the

austenite matrix of titanium micro-alloyed steels, Journal of Materials

Science 51 (2016) 4996-5007.

[96] C. Chen, H. Yen, F. Kao, W. Li, C. Huang, J. Yang, S. Wang. Precipitation

hardening of high-strength low-alloy steels by nanometer-sized carbides,

Materials Science and Engineering: A 499 (2009) 162-166.

47

[97] Y.J. Zhang, G. Miyamoto, K. Shinbo, T. Furuhara, T. Ohmura, T. Suzuki, K.

Tsuzaki. Effects of transformation temperature on VC interphase

precipitation and resultant hardness in low-carbon steels, Acta Materialia 84

(2015) 375-384.

[98] I. Bikmukhametov, H. Beladi, J. Wang, P.D. Hodgson, I. Timokhina. The

effect of strain on interphase precipitation characteristics in a Ti-Mo steel,

Acta Materialia 170 (2019) 75-86.

[99] P. Gong, X. Liu, A. Rijkenberg, W. Rainforth. The effect of molybdenum on

interphase precipitation and microstructures in microalloyed steels

containing titanium and vanadium, Acta Materialia 161 (2018) 374-387.

[100] T. Yang, Y. Zhao, Y. Tong, Z. Jiao, J. Wei, J. Cai, X. Han, D. Chen, A. Hu,

J. Kai. Multicomponent intermetallic nanoparticles and superb mechanical

behaviors of complex alloys, Science 362 (2018) 933-937.

[101] Z. Fu, L. Jiang, J.L. Wardini, B.E. MacDonald, H. Wen, W. Xiong, D. Zhang,

Y. Zhou, T.J. Rupert, W. Chen. A high-entropy alloy with hierarchical

nanoprecipitates and ultrahigh strength, Science Advances 4 (2018) 8712.

[102] H.M. Lee, S.M. Allen, M. Grujicic. Coarsening resistance of M2C carbides

in secondary hardening steels: Part I. Theoretical model for multicomponent

coarsening kinetics, Metallurgical Transactions A 22 (1991) 2863-2868.

[103] H.M. Lee, S.M. Allen, M. Grujicic. Coarsening resistance of M2C carbides

in secondary hardening steels: Part II. Alloy design aided by a

thermochemical database, Metallurgical Transactions A 22 (1991) 2869-

2876.

[104] H.M. Lee, S.M. Allen. Coarsening resistance of M2C carbides in secondary

hardening steels: Part III. Comparison of theory and experiment,

Metallurgical Transactions A 22 (1991) 2877-2888.

Integrated Experimental and Computational Study of ...

Documents

Transcript of Integrated Experimental and Computational Study of ...