Thermodynamic and Kinetic Investigation of the Fe-Cr - DiVA Portal

Thermodynamic and Kinetic Investigation of the Fe-Cr-Ni System

Driven by Engineering Applications

Wei Xiong

Doctoral Thesis

Department of Materials Science and Engineering School of Industrial Engineering and Management

KTH Royal Institute of Technology

Stockholm, Sweden 2012

ISRN KTH/MSE--12/13--SE+THERM/AVH ISBN 978-91-7501-394-7

Materialvetenskap KTH Royal Institute of Technology

SE-100 44, Stockholm Sweden

Akademisk avhandling som med tillstånd av Kungliga Tekniska högskolan framlägges till offentlig granskning för avläggande av doktorsexamen tisdagen den 28:e augusti, 2012 kl 10:00 i sal B2, Brinellvägen 23, Materialvetenskap, Kungliga Tekniska högskolan, Stockholm.

Wei Xiong (熊伟), August 2012

Tryck: Universitetsservice US AB

To Weiwei

爱在左，同情在右，走在生命的两旁，随时撒种，随时开花，将这一径

长途，点缀得香花弥漫，使穿枝拂叶的行人，踏着荆棘，不觉得痛苦，

有泪可落，却不是悲凉。―― 冰心

黑夜给了我黑色的眼睛, 我却用它来寻找光明。 ―― 顾城

Even with these dark eyes, a gift of the dark night,

I go to seek the shining light. ―― GU Cheng

Abstract

This work is a thermodynamic and kinetic study of the Fe‐Cr‐Ni system as the core of

stainless steels. The Fe‐Cr, Fe‐Ni and Cr‐Ni systems were studied intensively using

both computational and experimental techniques, including CALPHAD (CALculation

of PHAse Diagrams), phase field simulation, ab initio modeling, calorimetry, and atom

probe tomography. The purpose of this thesis is to reveal the complexity of the phase

transformations in the Fe‐Cr‐Ni system via the integrated techniques. Due to the im‐

portance of the binary Fe‐Cr system, it was fully reassessed using the CALPHAD tech‐

nique by incorporating an updated description of the lattice stability for Fe down to

zero kelvin. The improved thermodynamic description was later adopted in a phase

field simulation for studying the spinodal decomposition in a series of Fe‐Cr binary

alloys. Using atom probe tomography and phase field simulation, a new approach to

analyze the composition amplitude of the spinodal decomposition was proposed by

constructing an amplitude density spectrum. The magnetic phase diagram of the Fe‐Ni

system was reconstructed according to the results from both ab initio calculations and

reported experiments. Based on the Inden‐Hillert‐Jarl magnetic model, the thermody‐

namic reassessment of the Fe‐Ni system demonstrated the importance of magnetism in

thermodynamic and kinetic investigations. Following this, the current magnetic model

adopted in the CALPHAD community was further improved. Case studies were per‐

formed showing the advantages of the improved magnetic model. Additionally, the

phase equilibria of the Fe‐Cr‐Ni ternary were discussed briefly showing the need of

thermodynamic and kinetic studies at low temperatures. The “low temperature CAL‐

PHAD” concept was proposed and elucidated in this work showing the importance of

low temperature thermodynamics and kinetics for designing the new generation of

stainless steels.

Keywords: phase transformation, magnetism, spinodal decomposition, stainless steel,

low temperature CALPHAD, phase field, ab initio, atom probe tomography, calorime‐

try

Contents Part I. Thesis

Preface ········································································································ i

List of appended papers and contribution statement ········································ i

Acknowledgements ················································································· iii

Nomenclature, abbreviations and denotations ················································ v

Chapter 1 Introduction ··················································································· 1

1.1. Stainless steels and materials design ························································ 1

1.2. Structure and aim of thesis ····································································· 2

Chapter 2 Methodology ·················································································· 5

2.1. Low temperature CALPHAD ·································································· 6

2.2. Phase field simulation ········································································· 10 2.2.1. Basic functional ············································································· 10 2.2.2. Numerical solver of phase field equations ··········································· 12 2.2.3. Issues related to parameters in phase field model ·································· 16

2.3. Experimental techniques ····································································· 17 2.3.1. DSC measurement ········································································· 17 2.3.2. Atom probe tomography ································································· 18

Chapter 3 Low temperature thermodynamics and kinetics of the Fe-Cr-Ni alloy ······· 21

3.1. Phase diagrams of the Fe-Cr-Ni system ··················································· 21 3.1.1. Phase diagrams of the boundary systems ············································ 21 3.1.2. Phase equilibria of the Fe-Cr-Ni ternary system ···································· 29

3.2. Improvement of the magnetic model for computational thermodynamics ····· 31 3.2.1. Improved magnetic model ······························································· 31 3.2.2. Case studies ·················································································· 37

3.3. Kinetic study of spinodal decomposition ················································ 41 3.3.1. Spinodal decomposition in the Fe-Cr system ········································ 41 3.3.2. Estimation of composition amplitude – the ADS method ························ 44 3.3.3. Kinetic issues on spinodal decomposition of the Fe-Cr system ················· 48

Chapter 4 Concluding remarks and outlook ······················································ 53

4.1. Concluding remarks ············································································ 53

4.2. Suggestions on future work ·································································· 54

Bibliography ······························································································ 57

Part II. Supplements (appended papers)

i

Preface It is during a transitional period that I am performing this work in the research field of

CALPHAD. After successful utilization of CALPHAD in practical engineering, the re‐

search interest is extending to low temperatures, at which the materials are under ser‐

vice. This is certainly a challenge not only to the CALPHAD method itself but also to

other experimental and modeling techniques. In this work, it is shown that many prob‐

lems arise when applying the state‐of‐the‐art research tools. Great efforts have been

made which is reflected by the overview of this work as the first part and the appended

papers as the second. The phase equilibria, magnetic phase diagrams and spinodal de‐

composition in the Fe‐Cr‐Ni system are the main topics. The magnetic model used in

computational thermodynamics is further improved in this thesis. A new method called

amplitude density spectrum is proposed for quantitative study of spinodal decomposi‐

tion. Additionally, the concept of the low temperature CALPHAD is introduced and

materialized.

I do not wish to increase the volume of my unpublished manuscripts in the second part

of this thesis. Instead, some not yet published results are included in the first part and

discussed in Chapter 3. The research work in this thesis was performed at the Depart‐

ment of Materials Science and Engineering, KTH Royal Institute of Technology, Sweden.

Most of the results have been published or are to be published as papers in peer‐

reviewed journals.

List of appended papers and contribution statement

I. Wei Xiong, Malin Selleby, Qing Chen, Joakim Odqvist, Yong Du, ʺPhase Equilib‐

ria and Thermodynamic Properties in the Fe‐Cr Systemʺ, Critical Reviews in Solid

State and Materials Sciences, 35 (2010) 125‐152. doi: 10.1080/104084310 03788472

Contribution statement: Wei Xiong performed the literature survey, literature

review, as well as the draft of the manuscript.

II. Wei Xiong, Peter Hedström, Malin Selleby, Joakim Odqvist, Mattias Thuvander,

Qing Chen, ʺAn improved thermodynamic modeling of the Fe‐Cr system down

to zero kelvin coupled with key experimentsʺ, CALPHAD: Computer Coupling of

ii

Phase Diagrams and Thermochemistry, 35(3) (2011) 355‐366. doi:10.1016/j.calphad.

2011.05.002 (CALPHAD Best Paper Award 2012)

Contribution statement: Wei Xiong performed the thermodynamic calculations,

result analysis, and prepared the draft of manuscript.

III. Wei Xiong, Hualei Zhang, Levente Vitos, Malin Selleby, ʺMagnetic phase dia‐

gram of the Fe‐Ni systemʺ, Acta Materialia, 59(2) (2011) 521‐530. doi:

10.1016/j.actamat.2010.09.055

Contribution statement: Wei Xiong performed the thermodynamic calculations,

designed the ab initio calculations, and performed the data analysis. The draft of

the manuscript was prepared jointly by Wei Xiong and Hualei Zhang.

IV. Wei Xiong, Qing Chen, Pavel Korzhavyi, Malin Selleby, ʺAn improved magnetic

model for thermodynamic modelingʺ, CALPHAD: Computer Coupling of Phase

Diagrams and Thermochemistry, accepted for publication.

Contribution statement: Wei Xiong proposed the possibilities to modify the

magnetic model inspired by the intensive discussion with the other authors. The

draft of the manuscript was prepared by Wei Xiong.

V. Wei Xiong, Klara Asp Grönhagen, John Ågren, Malin Selleby, Joakim Odqvist,

Qing Chen, ʺInvestigation of spinodal decomposition in Fe‐Cr alloys: CALPHAD

modeling and phase field simulationʺ, Solid State Phenomena, Vols. 172‐174 (2011)

1060‐1065. doi:10.4028/www.scientific.net/SSP.172‐174.1060

Contribution statement: Wei Xiong performed the thermodynamic modeling,

and participated in result analysis. The phase field simulation was carried out

jointly by Wei Xiong and Klara Asp Grönhagen using the open finite element

code femLego. The draft of the manuscript was prepared by Wei Xiong.

VI. Wei Xiong, John Ågren, Jing Zhou, ʺAn effecitve method to estimate composition

amplitude of spinodal decompositionʺ, available in arXiv eprints:

arXiv:1205.4195v1, http://arxiv.org/abs/1205.4195v1, submitted for publication.

Contribution statement: Wei Xiong proposed the new method by intensive dis‐

cussion with the other authors. Wei Xiong performed the phase field simulation

and experimental data analysis. The simulation was carried out using the home‐

made code by Wei Xiong. The manuscript was drafted by Wei Xiong.

Usually, it is hard to completely avoid typos and even mistakes in the thesis. I am al‐

ways grateful to suggestions and comments, which can be sent to: [email protected]

at your convenience.

iii

Acknowledgements

On the first day when I arrived at KTH, no one including myself would believe that I

will spend the next years just for the “simple and well studied” Fe‐Cr‐Ni system. Cer‐

tainly, even after this thesis, the thorough study of this system in thermodynamics and

kinetics needs to be continued. There is no doubt that more doctoral thesis will be gen‐

erated for further investigation on the Fe‐Cr‐Ni system. During my research studies, I

strongly realized that some big problems normally exist in the things that are “well‐

known”. The stay at KTH shaped my character to be more independent but also coop‐

erative. The chance to work with all of the researchers in this excellent team of the He‐

ro‐m center, KTH is certainly my great treasure, and I am grateful to all the people

working together with me in Stockholm.

Firstly, millions of thanks should be sent to my supervisors Profs. Malin Selleby and

John Ågren. They not only spent very much time on me for numerous discussions, but

did also release sufficient freedom to involve me in different research projects with my

own interests, and thus allow me to extend the expertise.

Drs. Qing Chen, Pavel Korzhavyi, and Prof. Andrei Ruban have spent much of their

time to discuss with me starting from the beginning. I really learnt quite a lot from them,

but certainly not enough! I wish they will forgive me to take vast amount of their time

in the past as well as future. Without assistance of computer genius Dr. Lars Höglund, I

probably could not run phase field simulation on the server. It was a great time to dis‐

cuss some practical problems in both modeling and experiments with Drs. Joakim

Odqvist, Peter Hedström, Mr. Jing Zhou and Moshiour Rahaman, Drs. Martin Schwind,

Mattias Thuvander, Vsevolod Razumovskiy, Drs. Xiao‐Gang Lu and Huahai Mao, Profs.

Annika Borgenstam and Bo Sundman. I want to extend my gratitude to Drs. Klara Asp

Grönhagen, Walter Villanueva, and Minh Do‐Quang, Mr. Amer Malik, and Prof. Gus‐

tav Amberg for their help on manipulating the femLego software. The committed re‐

search team with a huge sense of camaraderie will be really missed. I am grateful to the

edifying experience sharing from Prof. Emerit. Mats Hillert as well as his time for in‐

spiring discussions. And I was certainly impressed by his exemplary attitude on enjoy‐

ing research. Moreover, it is worth mentioning that, by working closely with Prof. Staf‐

fan Hertzman in the Outokumpu Stainless Research Foundation, I could always get

some extra driving force. Since the research orientations directed by him are usually full

of foresight in practical engineering, and most are enormous challenges to the state‐of‐

the‐art physical metallurgy and solid physics.

iv

Working in the Hero‐m center, there are some great opportunities to meet excellent sci‐

entists in a broad research field. It is my fortune to discuss magnetic models with Prof.

Emeritus Gerhard Inden from MPI, Düsseldorf, during the Hero‐m annual workshop.

Without the kind help from Drs. Xiaogang Lu, Qing Chen and Huahai Mao and their

families, it would be not so easy to stay in the beginning when I arrived. There are lots

of happy memories living in Stockholm with friends from different countries.

Last but not least, without the constant support from my family, this thesis would be

impossible to accomplish.

When I was studying in the primary schools in the last century, I was taught that I

should study hard in order to respond the care and kindness from my family, friends,

teachers and society. It might be “out of date” to repeat this in the 21st century. But I

still wish to reserve this chance saying that, I will keep moving towards my dream in

order to respond your care and kindness.

Wei Xiong

v

Nomenclature, abbreviations and denotations

ADS amplitude density spectrum

AFM antiferromagnetic

APT atom probe tomography

β* effective magnetic moment

b mean magnetic moment

βi magnetic moment of component i

CALPHAD CALculation of PHAse Diagrams

Cp heat capacity

DFD direct frequency diagram

DSC differential scanning calorimetry

ε Interfacial energy coefficient

FD frequency diagram

FM ferromagnetic

IHJ Inden-Hillert-Jarl

LBM Langer-Bar-on-Miller

lc critical length scale

Ms Martensite start temperature

MScale atomic mobility scale

PM paramagnetic

SGTE Scientific Group Thermodata Europe

Chapter 1 Introduction

1.1. Stainless steels and materials design

Steels are alloys composed of iron and other elements, and they are certainly one type

of the most important and versatile engineering materials acting as a symbolic mark of

the modern civilization of mankind. Steels normally serve as constructional materials,

and they exhibit numerous mysteries in their physical properties, which attract many

investigations in the field of materials, metallurgy and solid physics. In the family of

steels, the stainless steel occupies its own place for its corrosion and heat resistant prop‐

erties. Stainless steel usually contains a minimum of 10.5 wt.% chromium, which is the

essential element to gain its ability of being stainless. The major components in the

stainless steel are normally Fe, Cr, Ni, Mo, N, and C (shown in the sequence of amount

in weight).

Fundamental study on phase equilibria and phase transformations is the key to im‐

prove the performance of stainless steel, and there are tremendous amount of modeling

and experiments on topics related to stainless steels [1]. As the basis of the stainless

steels, the Fe‐Cr system has got quite much attention in both thermodynamics and ki‐

netics [2]. Therefore, even if the focus is limited to the Fe‐Cr‐Ni ternary system and its

binaries, one needs a lot of patience to review the vast amount of reported data from

various research groups [1].

With the development of theory, computation and experiments, the research on stain‐

less steels has entered a new era which is directed by the concept of materials design.

Nowadays, the computational simulation plays a vital role for designing new materials.

The simulation tools can sometimes effectively shorten the research cycle for steel in‐

dustry. The ultimate aim of materials design is to obtain the desired microstructure of

the materials, to improve chemical and physical properties, and to enhance the perfor‐

mance in service. Although there may be accumulated experimental data available, it

should be noted that new experiments are still needed, especially for the purpose to

verify the reported data when conflicts exist. Besides, the designed key experiments

often not only provide supplementary data but also verify the model of simulation.

CHAPTER 1. INTRODUCTION

2

As shown in Fig. 1.1, scientific efforts are often made in the study of theory, modeling

and experiment in order to fill the need from engineering applications. During the ma‐

terials processing, the material microstructures can be determined by phase transfor‐

mation which governs the phase distribution. As a consequence, the study of the phase

equilibria and phase transformation is the key to successful materials design in steels

research.

MaterialsDesign

MaterialsDesign

Performance

Structure

PropertyTheory

Modeling

Experiment

Research Tools and Orientation

Fig. 1.1. The concept of materials design and its basic research tools and orientation

It should be noted that, although the integration of different methods is necessary in the

modern materials research, one could not possibly learn all the techniques. However,

this will not affect the application of different methods, which is analogous to driving a

car. The detailed knowledge of the engine will facilitate maintenance of the car, but is

not a must to drive one. For instance, one may easily handle some experimental facili‐

ties with several specific functions needed, since many of them are designed quite user‐

friendly. At least, the joint work by collaboration is always possible.

1.2. Structure and aim of thesis

It is known that stainless steels are susceptible to the notorious “475 °C embrittlement”,

which is detrimental to the mechanical properties [3], e.g. impact toughness. A possible

reason for this is the segregation of the Cr rich cluster formed by phase separation.

Therefore, a fundamental study on the phase transformation and its influence on me‐

chanical properties is desired.

This thesis is dedicated to study the mechanism of phase transformations at lower tem‐

peratures through the integrated tool of modeling and experiments. Although there are

numerous reports on phase equilibria and phase transformations in the Fe‐Cr‐Ni sys‐

tem, it is found that previous studies have not paid enough attention to the phase be‐

havior at low temperatures, defined in the thesis as temperatures below 1073 K.

CHAPTER 1. INTRODUCTION

3

Since content of chromium is normally much larger than nickel in stainless steels,

enormous efforts have been made on the Fe‐Cr in the research field of steels. Moreover,

the phase separation in the Fe‐Cr binary can be regarded as the prototype of the phase

separation in the multi‐component stainless steel system. A thoroughgoing literature

review on the phase equilibria and thermodynamic properties was performed as the

initiation of the work, see paper I. It was found that the generally accepted phase dia‐

gram of the Fe‐Cr system needs to be modified especially in the low temperature region.

Besides, the reported thermodynamic modeling needs to be improved, since the mag‐

netic phase diagram cannot be correctly reproduced by the previous thermodynamic

modeling. Therefore, in paper II, the thermodynamic modeling of the Fe‐Cr system was

performed according to the comprehensive review in paper I to provide accurate ther‐

modynamic information at low temperatures. The improved thermodynamic modeling

was verified by both calorimetry and atom probe tomography, and the chemical spi‐

nodal curve was determined. Due to the problems on magnetism revealed in the Fe‐Cr

system, another systematical study of the Fe‐Ni system was performed as well. The

thermodynamic description was refined for the low temperature region of the Fe‐Ni

phase diagram assisted by ab initio calculations on the magnetic moment (see paper III).

Further investigation was performed for the magnetic model adopted in the state‐of‐

the‐start CALPHAD method. The intensive study on the magnetic model for thermo‐

dynamic modeling was finalized with an improved magnetic model in paper IV. Run‐

ning in parallel, the improved thermodynamic description of the Fe‐Cr system was tak‐

en into account during the phase field simulation of the spinodal decomposition in the

Fe‐Cr system in papers V and VI. The study proposed a new method to evaluate the

composition amplitude of the spinodal decomposition. Meanwhile, it demonstrated

that a successful application of the phase field technique in spinodal decomposition

needs more accurate estimation of the atomic mobility as well as the wavelength of the

interconnected spinodal structure.

In the first part of this thesis, an overview is given and organized as follows. Chapter 2

discusses different methods used in this study. Chapter 3 summarizes some important

results regarding thermodynamics and kinetics of the Fe‐Cr‐Ni systems at low tempera‐

tures. Outlooks for further work and conclusions are presented in Chapter 4. Main re‐

sults are presented in the appended papers or manuscripts as the second part of this

thesis.

Chapter 2 Methodology In the present thesis, both computational and experimental tools were applied to study

phase equilibria and phase transformations in the Fe‐Cr‐Ni system. However, modeling

and simulation were taken as the major research tools. In order to study phase behavior

at relatively low temperatures, the ab initio method, CALPHAD, key experiments and

phase field simulations were applied to the present study. The research concept in this

thesis is illustrated in Fig. 2.1.

WEIPHAM

FFT

Home-made

FORTRAN

Kinetics

Thermodynamics

CALPHAD &Experiment

(DSC, APT)

Ab initio Phase field

Thermodynamic propertyMagnetic propertyInterfacial energyGibbs free energy

Atomic mobilityInterfacial mobilityThermo-Calc DICTRA

EMTO

VASP

Fig. 2.1. Illustration of the relationships among different methods referred in this thesis.

The ab initio method could provide valuable information for thermodynamic and kinet‐

ic modeling to construct CALPHAD databases, which can be adopted later in the phase

field simulations as the input. For some cases, atomistic calculations can be employed to

predict some physical quantities directly for phase field simulation, e.g. interfacial en‐

ergy. However, usually, phase field simulations are performed by coupling with the

thermodynamic and kinetic information constructed using the CALPHAD approach.

Additionally, some key experiments in this work are designed for verifying the compu‐

tational results in this thesis. For example, the model‐predicted thermodynamic proper‐

ties of the alloys, e.g. heat capacity, were verified by differential scanning calorimetry

CHAPTER 2. METHODOLOGY

6

(DSC). The simulation results by phase field technique were compared with the atom

probe tomography (APT).

2.1. Low temperature CALPHAD

There are tremendous amount of exhaustive reviews on both ab initio [4, 5] and CAL‐

PHAD methods [6, 7]. Therefore, it is not necessary to take space here for a detailed in‐

troduction on how to perform ab initio and CALPHAD modeling. Instead, it is neces‐

sary to emphasize the conjunction between these two in this section, since there is a

strong tendency to perform thermodynamic and kinetic studies using a combined

method of ab initio and CALPHAD.

In this work, the concept of “low temperature CALPHAD” is introduced, driven by the

engineering application of steels at low temperatures. As mentioned in the preceding

chapter, the present work is focusing on the phase behavior of the materials at low

temperature during service.

It is interesting to note that, in the early days of CALPHAD, 1970s, many physicists per‐

forming atomic modeling would not believe that CALPHAD would prevail today.

However, as indicated by R.W. Cahn [8], CALPHAD is the typical method that can ful‐

ly demonstrate the strength of computational modeling and simulation. Besides, it may

be difficult to find any other so successful approach invented by metallurgist as CAL‐

PHAD, which attracts such wide interests from metallurgy and material scientists,

physicists, chemists, and industrial engineers.

The CALPAHD method is a semi‐empirical technique based on evaluation of experi‐

mental data, and theoretical estimates related to the thermodynamic properties of mate‐

rials. Therefore thermodynamic models are the core of this technique. A general formu‐

la to describe a phase can be expressed as:

srf config ex physG G G G G= + + + (2‐1)

in which the first term on the right hand side represents the contribution from the “sur‐

face” of reference as defined in the compound energy formalism [9]. The second term

denotes the ideal mixing which can be extended to include random arrangements on

several sublattices. The third term describes the contribution due to non‐ideal interac‐

tion between the components. In principle, the first and the third terms cover all of the

contributions from electronic, vibrational and other type of contributions. The last term

considers other physical contributions. For systems exhibiting magnetic properties, this

term could represent the magnetic contribution to the Gibbs energy of the system.


7

A classical CALPHAD‐type work would only obtain a self‐consistent thermodynamic

parameter set by optimization using, e.g. the Thermo‐Calc software package. The basic

criteria of a successful optimization would be the quality of reproducing the phase dia‐

gram and thermodynamic properties which were determined by experiments. Due to

the development of the ab initio methods, the calculated results from the ab initio mod‐

eling are considered as one type of the most important input data, which sometimes

even hold the same status as experiments. A typical example is the adoption of calcu‐

lated enthalpy of formation using ab initio methods, which is now generally considered

as a reliable input for CALPHAD modeling if the experimental value is unavailable.

The computed enthalpy of formation at 0 K can often be considered to be the same as

the one at 298 K in traditional CALPHAD modeling [10], and thus merely considering

negligible differences. Another successful application of ab initio input is the estimation

of interaction parameters and energy of the end‐members in the sublattice model used

in the CALPHAD modeling. A case study was performed in the Re‐W system by Fries

and Sundman [11], which demonstrated the important contribution from ab initio cal‐

culations to evaluation of thermodynamic properties in unstable phase regions.

As a tendency of further development, both ab initio and CALPHAD methods are mak‐

ing efforts to enter each others fields. In ab initio calculations, researchers are now in‐

terested in predicting the thermodynamic and kinetic quantities above zero kelvin. So

far, there are numerous successful examples of calculations of heat capacity for sub‐

stances using ab initio methods. For example, in the work by Grabowski et al. [12], the

heat capacity of pure Al up to the melting temperature was successfully calculated. The

ab initio predicted heat capacity of the Ni‐B compounds by Kong et al. was successfully

confirmed by calorimetry [13]. Meanwhile, researchers in the CALPAHD field start to

develop models which are validated below 298 K.

In fact, the CALPHAD community realized the importance of extending the thermody‐

namic description of the pure elements down to zero kelvin in the modeling, and orga‐

nized intensive discussion at the Schloß Ringberg workshop in 1995 entitled “Thermo‐

dynamic models and data for pure elements and other end‐members of solutions”. The

first group discussed heat capacity models for crystalline phases from 0 to 6000 K, try‐

ing to find a “universal” model to describe the thermodynamic properties over the

whole temperature range [14]. The idea is to use the Debye or Einstein model to fit the

heat capacity at low temperatures, while using a polynomial to describe the heat capaci‐

ty above 298 K.

, 2Deb Ein magnP P PC C aT bT C= + + + (2‐2)


8

where the first term on the right hand side will be expressed using the Debye or Ein‐

stein temperatures (θD or θE) :

( )

3 4

209

1

DxT

DebP

xD

T x eC R dx

e

q

q

æ ö÷ç ÷= ç ÷ç ÷çè ø -ò (2‐3)

( )

2

23

1

E

E

TEin EP

T

eC R

Te

q

q

qæ ö÷ç ÷= ç ÷ç ÷çè ø-

(2‐4)

The second term in Eq. (2‐2) is related to electronic excitations and low‐order anhar‐

monic corrections, and coefficient a could thus be evaluated using non‐thermodynamic

information, e.g. electron density of states at the Fermi level. The third term consists of

the high‐order anharmonic lattice vibrations. The last term considers the contribution

from the magnetic ordering.

In 2001, Chen and Sundman [15] performed the pioneering work on the lattice stability

of pure Fe based on the Einstein model and described Cp of the bcc phase at low tem‐

peratures by further modifying Eq. (2‐2) into:

( )

24

23

1

E

E

TmagnE

P PT

eC R aT bT C

Te

q

q

qæ ö÷ç ÷= + + +ç ÷ç ÷çè ø-

(2‐5)

Since it was found that the 4th power of the third term would make it easier to fit the

high temperature experimental Cp [15].

Although the work by Chen and Sundman [15] successfully described the lattice stabil‐

ity of Fe, it has not been used in any practical application yet. In this thesis, the im‐

proved thermodynamic description of the Fe‐Cr system can be considered as the first

attempt to adopt the updated lattice stability of pure Fe down to zero kelvin in a ther‐

modynamic description of a binary system [16] (see paper II).

It is encouraging to notice that the most recent work by Vřešt’ál et al. [17] extended the

studies on the lattice stability of 53 elements down to zero kelvin in the framework of

the Schloß Rinberg workshop in 1995. It combines the expression of Eqs. (2‐2) and (2‐5)

by adopting both temperature dependent terms: T2 and T4:


9

( )

22 4

23

1

E

E

TmagnE

P PT

eC R aT bT cT C

Te

q

q

qæ ö÷ç ÷= + + + +ç ÷ç ÷çè ø-

(2‐6)

The above expression can guarantee easy fitting at temperatures above 298 K.

Evidently, there will be an increasing amount of work on the lattice stability down to

zero kelvin in the near future, and the integration of ab initio and CALPHAD will sub‐

stantially step forward. It should also be stressed that the ab initio calculations play a

vital role in the construction of a new database of lattice stability. Because, so far, it is

probably the only alternative to obtain thermodynamic properties at low temperatures,

especially for the unstable phases, if the experimental data is unavailable.

Fig. 2.2. Relationship among the ab initio, CALPHAD and experimental methods in the low

temperature CALPHAD concept shown in the Taiji diagram.

Regarding the magnetic contribution, i.e. the last term in Eqs. (2‐2), (2‐5) and (2‐6), it is

worth noting that the estimation is based on the model proposed by Inden [18], and

later modified by Hillert and Jarl [19]. It should be pointed out that since the early focus

was normally at high temperatures, the thermodynamic modeling usually has not

enough concern with reproducing the magnetic phase diagram, which is more im‐

portant at low temperatures due to different magnetic ordering effects. This is the rea‐

son that, using the thermodynamic descriptions of some basic systems, the magnetic

properties cannot be well reproduced. In this thesis, it is found that the calculated mag‐

netic phase diagram of the Fe‐Cr‐Ni system (including its boundary binaries) according

to the reported CALPHAD work differs significantly from experimental data.

More seriously, it is found that some of the basic magnetic quantities, like Curie/Néel

temperatures of metastable or unstable phases, are still not well determined and needs


10

further investigation. This requires some efforts in both ab initio and CALPHAD com‐

munities in the near future.

In view of the above, the relationship between ab initio and CALPHAD techniques can

be considered as the Taiji diagram as shown in Fig. 2.2.

The ab initio and CALPHAD is nondetachable for further development, in the metal‐

lurgy and materials science, ab initio is making great efforts to compute the material

properties at high temperatures, while CALPHAD is trying to extend its area to the low

temperature region. The issue related to magnetism is a challenge to both methods,

since there is no perfect model yet to describe the complex magnetism in solid state

physics. In this thesis, an attempt was made to further improve the magnetic model

which is now used in many mean field approaches, proposed by the CALPHAD com‐

munity [18, 19].

2.2. Phase field simulation

2.2.1. Basic functional

Phase field simulation is a mathematical technique for studying interfacial properties of

the materials. It has been extensively developed in the last two decades. The ultimate

purpose of phase field simulation is to obtain the microstructural evolution in the mate‐

rial, in order to determine how to process a material with desired properties. The de‐

velopment of phase field simulation is due to the work by Hillert [20, 21] and Cahn and

Hilliard [22] on the spinodal decomposition. The gradient energy contribution from an

interface was extensively discussed in the above works. However, it should be noted

that the concept of the gradient energy is indeed a renaissance, since in the work by

Van der Waals on the capillarity effect of critical systems in 1894 [23], the continuum

model and the energy of a diffuse interface was proposed.

For a binary A‐B case, it is proposed that the total energy in the system can be ex‐

pressed as:

( )2

21,

2m B Bm V

G G x T x dVV

eæ ö÷ç ÷= ç + ÷ç ÷÷çè øò (2‐7)

where ( ),m BG x T can be obtained from CALPHAD databases, ab initio calculations, or

other theoretical modeling. The molar volume mV was normally considered as constant.

If the thermodynamics of the system can be described by the regular solution model,


11

the gradient coefficient e is approximately considered as a function of interatomic dis‐

tance al and regular solution parameter W [22]:

2 2ae l= W (2‐8)

Since concentration is a conserved quantity, it satisfies the solute diffusion equation:

1 B

Bm

xJ

V t

¶= - ⋅

¶ (2‐9)

The solute flux BJ can be given by the Onsager linear law of irreversible thermody‐

namics:

B AB mB

GJ L V

x

dd

æ ö÷ç ÷= - ç ÷ç ÷çè ø (2‐10)

in which ABL is the phenomenological coefficient and can be expressed as a function of

atomic mobility:

( )AB A B A B B A mL x x x M x M V= + (2‐11)

As a result, the Cahn‐Hilliard equation can be obtained by combining the above equa‐

tions:

2 21 B mAB B

m B

x GL x

V t xe

æ æ öö¶ ¶ ÷÷ç ç ÷÷= ⋅ - ç ç ÷÷ç ç ÷÷ç ç¶ ¶è è øø (2‐12)

In phase field modeling, one considers the contribution to thermodynamics and kinetics

from a diffuse interface during the phase transformation. Moreover, an order parameter

that represents a certain phase is introduced in the phase field model. A comparison

between the typical sharp interface and the diffuse interface models is illustrated in Fig.

2.3.

In the spinodal decomposition, the concentration can be considered as a kind of con‐

served order parameter, while in a grain growth or precipitation study, a non‐

conserved order parameter f is necessary to introduce. Usually, if is considered to be 0 or 1 for the individual phase i, and the diffuse interface will be determined as

0 1if< < . If there are N phases, in a certain phase region, there will be:


12

1N

ii

f =å (2‐13)

The evolution of the non‐conserved order parameters if are governed by the Allen‐Cahn equation (also known as the Ginzburg‐Landau equation):

( ), ,ii

i

F x TM

t f

d ffdf

¶= -

¶ (2‐14)

in which,Mf is the physical quantity related to the interfacial mobility. It is noteworthy

that the energy of the system can normally be defined in terms of composition and

temperature T, but the one used in the Allen‐Cahn equation, F, is a function of the non‐

conserved order parameter, composition, and temperature. Therefore, many efforts

have been made to propose a reasonable free energy function varying with both non‐

conserved order parameter and composition. Different construction of the free energy

landscape may generate different phase field models. There are some widely adopted

phase field models used for different application fields, such as the Wheeler‐Boettinger‐

McFadden model [24, 25], the Steinbach model [26], and the Kim‐Kim‐Suzuki model [27,

28].

Sharp interface Diffuse interface

Ord

er

pa

ram

ete

r

Ord

er

pa

ram

ete

r

Distance Distance

Fig. 2.3. Illustration the concept of diffuse and sharp interfaces.

More reviews on the phase field methods are available in Refs. [29‐31].

2.2.2. Numerical solver of phase field equations

There are tremendous amounts of investigations on solving the Cahn‐Hilliard and Al‐

len‐Cahn equations using different numerical methods, such as, finite difference, finite

element, finite volume and Fourier‐spectral methods. The explicit finite difference

method can be considered as a starting point for learning the phase field method. It can


13

easily handle some simple 2 dimentional cases. However, in order to be more efficient,

one would consider some other methods.

It is worth noting that nowadays several packages are available for applying the phase

field method. The first commercial software named MICRESS [32] was developed at the

Access Technology, Aachen, Germany. By collaborating with the Thermo‐Calc Software

AB in Sweden [33], thermodynamic and kinetic databases were included in the simula‐

tion. Some open codes are also available. For example, using the finite volume method,

the software called FiPy [34] written in Python was released by the National Institute of

Standards and Technology, USA. In Germany, the ICAMS institute at the Ruhr Univer‐

sity Bochum is developing another open code, OpenPhase [35], written in C++. Based

on the finite element method, a code named femLego [36] has been released at KTH

Royal Institute of Technology, Sweden. Alternatively, some commercial mathematical

tools, e.g. FlexPDE [37], for solving partial differential equations may also be suitable

for phase field simulation. Similarly to the situation in the beginning of the 80s for the

CALPHAD approach, since the phase field model is still under development, the exist‐

ing software at present could not fit all the needs of both scientific and engineering ap‐

plications. Scientifically, researchers sometimes prefer to modify the phase field

code/model with utmost freedom, and thus write their own code.

In this work, the 3D (3 dimensional) phase field simulation is mainly performed by us‐

ing the semi‐implicit Fourier‐spectral method. There are many advantages when ma‐

nipulating the phase field equations in the Fourier space. The attractive feature is that

the implicit solution of a differential equation could be simplified in the Fourier space

as a division operation instead of a matrix inversion in the real space. There are a lot of

fast Fourier transform codes available, e.g. FFTW package developed at MIT (Massa‐

chusetts Institute of Technology).

Considering the derivative of ( )f x¢ in the Fourier space, it could be manipulated as:

( ) ( )( ) ( ) ( )( )1ikx ikxf x f k e dk ikf k e dk ikf kx

¥ ¥ -

-¥ -¥

¶¢ = = =¶ò ò (2‐15)

in which, k is the wavenumber, ( )f k is the reciprocal form of ( )f x in the frequen‐

cy/reciprocal space. The above manipulation in Eq. (2‐15) is based on the following

Fourier transform between the Fourier transform ( )f k and the function ( )f x in real

space:


14

( ) ( ) ( )

( ) ( ) ( )1

1

2

ikx

ikx

f x f k f x e dx

f k f x f k e dkp

¥ -

-¥

¥-

-¥

ìïï é ùï = =ï ë ûïïíïïïï é ù = =ï ë ûïî

ò

ò

(2‐16)

In brief, the first derivative in the real space is a multiplication by ik in the Fourier space.

In this thesis, the semi‐implicit solver developed by Chen and Shen [38] has been

adopted. The Cahn‐Hilliard and Allen‐Cahn equations are solved in dimensionless

form by considering the periodic boundary conditions. Consequently, the Cahn‐

Hilliard equation shown in Eq. (2‐12) for the A‐B binary case can be written in the di‐

mensionless form as:

( ) 2 2BB

AB BB

G xxL x

xe

t

é æ öù¶¶ ÷çê ú÷ç= ⋅ - ÷çê ú÷ç ÷¶ ¶è øê úë û

(2‐17)

The Fourier transform of Eq. (2‐17) will be:

( )

( )22BBAB B

B

G xxi L i k x

xe

t¢

ì üé æ öùæ öï ï¶¶ ÷ï ï÷ççï ê ú ï÷÷ç¢ ¢ç= + ÷í ý÷çê úç ÷÷ï ç ïç ÷ ÷¶ ¶çè øê úè øï ïë ûï ïî þk r k

k k

(2‐18)

In the above equation, k and ¢k are the modes in the Fourier space, which is equal to

( )2k ¢ . Subscripts k and ¢k mean the Fourier transform, while subscript r means the

inverse Fourier transform. It should be noted that according to the analysis by Chen

and Shen [38], the above equation has a constraint of:

2 4 1Kt eD ⋅ < (2‐19)

in which, K is the number of Fourier modes in each direction. Therefore, the ideas of

splitting the variable mobility into A and M A-

can be introduced in Eq. (2‐18) to

solve the problem [38]:

( )

( ) ( )( )

2 4 1

22 4 2

1

1

nB

BnB AB B

B

A k x

G xA k x i L i k x

x

t

t t e

+

¢

+ ⋅ D Îì üé æ öùæ öï ï¶ ÷ï ï÷ççï ê ú ï÷÷ç¢ ¢ç= + ⋅ D Î + D ⋅ + ÷í ý÷çê úç ÷÷ï ç ïç ÷ ÷¶çè øê úè øï ïë ûï ïî þk r k

k k

(2‐20)

It is important to keep in mind that, in order to solve the above equation, A needs to

fulfill the requirement [38]:


15

( )max min1

2A M M³ +

(2‐21)

Where maxM

and minM

are the maximum and minimum of the atomic mobility, respec‐

tively. By doing this, the time step constrain of the form in Eq. (2‐19) will no longer exist.

As for the Allen‐Cahn equation, normally it has the form of:

( )( )2M ff ff

f et

¶= - -

¶

(2‐22)

Therefore, in the Fourier form, Eq. (2‐22) can be written as:

( )( )2 2k

dM f k

d f ff

f e ft

= - - (2‐23)

Similarly, ( )1 2 3, ,k k k=k is a Fourier vector in the reciprocal space, while the magni‐

tude of k is 2 2 21 2 3k k k+ + . Considering the explicit Euler scheme during the simula‐

tion:

( )( )1 2n n n nd M f kff f t f f+ = + ⋅ - -k

(2‐24)

Furthermore, if the semi‐implicit form is applied, we will arrive at:

( )( ) ( )1 21n n nd M f d M kf ff f t f t+ = - ⋅ + ⋅k

(2‐25)

In order to improve the accuracy in time, one would consider the higher order semi‐

implicit schemes. Thus, a second‐order backward difference (BDF) for ( )d dt f and a second‐order Adams‐Bashforth (AB) for the explicit treatment of the nonlinear term can

be applied to Eq. (2‐23) [38]:

( ) ( ) ( )2 1 1 13 2 4 2 2n n n n nd M k d M f ff ft f f f t f f+ - -é ù+ ⋅ = - - ⋅ -ê úë ûk k

(2‐26)

( ) ( )( ) ( )1 1 1 24 2 2 3 2n n n n nd M f f d M kf ff f f t f f t+ - -é ù= - - ⋅ - + ⋅ê úë ûk k

(2‐27)

Analogously, a three‐order semi‐implicit BDF/AB scheme can be derived as [38]:

( ) ( ) ( )

( )

1 2 1 2

12

18 9 2 6 3 3

11 6

n n n n n n

nd f f f

d M kf

f f f t f f ff

t

- - - -+

é ù- + - ⋅ - +ê úë û=+ ⋅

k k k

(2‐28)


16

Based on the above numerical methods, a home‐made code named ‘WEIPHAM’ has

been written in FORTRAN for simulating the 3D spinodal decomposition in this thesis.

2.2.3. Issues related to parameters in phase field model

In the phase field models, the choice of the parameters directly affects the final simula‐

tion results. Some of the parameters can be defined rigorously from mathematical deri‐

vations, but some are defined arbitrarily in view of the physical models.

In the Cahn‐Hilliard equation, it is known that the most important inputs are the Gibbs

energy, atomic mobility, and interaction parameters. In the simulation of spinodal de‐

composition, the Gibbs energy can determine phase stability regions, like the chemical

spinodal curve. The atomic mobility is one of the main factors to control the kinetic pro‐

cess. One could envisage that the atomic mobility is usually easier to obtain at higher

temperatures, where usually the diffusion is faster than that at low temperatures. Ac‐

cording to the work by Cahn and Hilliard [22], the interaction parameter of the regular

solution model can be related to the gradient coefficient as shown in Eq. (2‐8). However,

only the pairwise interactions are considered in this approximation. In fact, the gradient

coefficient can bring effects to the wavelength of the spinodal structure. According to

Hillert [20, 21], in the spinodal decomposition, assuming a spinodal wave will behave

as an ideal sinusoidal profile, the critical wavelength of the spinodal structure can be

expressed as:

22 2c mGl p e ¢¢= (2‐29)

where mG is the second derivative of the Gibbs energy with respect to composition.

In the study of grain growth or precipitation, the phase field variable (non‐conserved

order parameter) f is necessary to introduce. Therefore, the atomic mobility term used

in the Cahn‐Hilliard function also needs to be considered with respect to both composi‐

tion and phase field variables [39].

Regarding the Allen‐Cahn equation, several issues on input parameters should also be

addressed briefly. When applying the Allen‐Cahn equation to the study of grain

growth or precipitation, it is quite important to determine the way to construct the en‐

ergy curves with respect to the phase field variable f and composition variable x ,

which influence the choice of different parameters, such as interfacial energy, interfacial

thickness, and gradient coefficient. The interfacial mobility M can directly affect Mf in

the Allen‐Cahn equation as discussed in section 2.2.1. Normally, the interfacial energy

and mobility are difficult to evaluate properly. Therefore, in order to obtain reliable


17

values, one may consider both experiments [40] and atomistic simulations [41]. The dif‐

ferent choices of the related quantities may affect the final results substantially.

2.3. Experimental techniques

As mentioned in the preceding chapter, key experiments play an important role when

validating the simulation results and provide supplementary data. In this thesis, the

main experimental techniques used are calorimetry and atom probe tomography. DSC

was employed to determine the heat capacity of Fe‐Cr binary alloys, while atom probe

tomography was used to validate the thermodynamic modeling and to estimate the

intensity of the spinodal decomposition. Therefore, it is necessary to briefly discuss the‐

se two techniques. The ADS (Amplitude Density Spectrum) method invented in paper

VI can be considered as the outcome of the combined research of simulation and exper‐

iment in this thesis.

2.3.1. DSC measurement

The DSC measurement is widely used in the study of phase equilibria and thermody‐

namic properties, such as, phase transition temperature, heat capacity, enthalpy of

phase transformation, etc. Basically, there are two different kinds of DSC instruments,

one called heat‐flux DSC, and the other called power‐compensating DSC. The former

one has a single heater as the one for DTA (differential thermal analysis), while the lat‐

ter one has two separate heaters for sample and reference in order to determine the

power difference required to maintain the temperatures of the sample and reference

identically. Although the latter one is more precise than the former, in this thesis, we

used the heat‐flux DSC to determine the heat capacity, since it could measure up to

higher temperatures which was necessary for the present research.

During the measurement, it is crucial to calibrate the instrument as the first step in or‐

der to obtain a good baseline, and thus guarantee the sensitivity of the measurement.

There are some standard test methods for determining the specific heat capacity using

DSC, such as, ASTM E968, ASTM E1269, etc. Normally, the derivation of the heat ca‐

pacity of the sample will follow:

Sample BaselineSample Standard Standard

Sample Standard BaselineP P

S SmC C

m S S

-= ⋅

- (2‐30)

where m, S, and Cp denote the mass, signal and heat capacity respectively. Normally,

synthetic sapphire is used as the standard.


18

In this thesis, the DSC measurements are applied to determine Cp curves, from which

the magnetic transition temperature can also be achieved (see paper II).

It should be noted that the low temperature DSC measurement can provide valuable

experimental data for the heat capacity. Indeed, there will be great amount of require‐

ment for experimental data even though ab initio can provide some modeling inputs.

One should keep in mind that the experiments can always be applied to validate the

model‐predicted results.

2.3.2. Atom probe tomography

The atom probe tomography (APT) is an advanced technique to provide insight on so‐

lute distribution and reconstruct the structure of materials. With the development of

APT, many theories in materials science can be confirmed. For example, the quantita‐

tive composition of the small clusters as the precipitation in the Guinier‐Preston (GP)

zones can be measured by APT, while the high resolution electron microscopy can only

provide qualitative insights of the structure [42].

In the study of spinodal decomposition, APT is one of the most powerful techniques to

study phase separation especially for the early stages when phase separation is not

prominent. As mentioned before, the quantitative determination of the solute composi‐

tion is an advantage compared to electron microscopy. Different from the energy‐

dispersive X‐ray spectroscopy, since the location of atoms can be reconstructed, the

methods of determining compositions should be considered properly, especially in the

early stages of the solute clustering. In the study of nucleation and spinodal decomposi‐

tion, these methods to identify clusters and quantitatively estimate local compositions

are particularly useful when the clustering is difficult to visualize.

It should be mentioned that the results from Monte Carlo simulation are in favor of the

comparison between simulation and APT experiment in many investigations, since

both methods can provide the discrete information as configuration of individual atoms.

Therefore, only few combined research works have been reported by using both phase

field and APT.

The systematical APT experiments on Fe‐Cr alloys were performed by the Group of

G.D.W. Smith in Oxford since the early 90s [43‐48]. Meanwhile, Monte Carlo simulation

was also applied to the decomposition process. However, due to the complexity of the

Fe‐Cr system, there is still a large demand on additional APT experiments. In this thesis,

the APT technique is applied to determine the chemical spinodal curve of the Fe‐Cr

miscibility gap of the bcc phase. Comparing with the Monte Carlo simulation, the phase


19

field simulation shows its advantages in representing microstructure in real space and

time scale. Therefore, there is a demand to study spinodal decomposition by combing

the phase field and APT techniques.

In fact, it transpires that if the scale of the spinodal structure of the Fe‐Cr bcc phase is in

nano size, the analysis of the APT results is more complex that one might imagine. Alt‐

hough there are some methods available to gain composition amplitude and wave‐

length for the spinodal structure, in this work, it turns out that some modifications are

needed in order to characterize the spinodal decomposition more properly.

In the present work, the annealed Fe‐Cr alloys are shaped into needle‐like samples, fol‐

lowed by a standard two‐stage electro‐polishing. The analyses were performed under a

local electrode atom probe (LEAP 3000X HRTM, Imago Scientific Instruments, USA)

equipped with a reflectron for improved mass resolution. Since the analyses are per‐

formed on a small volume, it is natural that the average composition of the needle‐like

sample for APT experiment are somewhat different from the matrix (less than 5 at.% Cr

in this work), which is affected by intensity of phase separation. The ion detection effi‐

ciency is about 37 %, and the experiments were made in voltage pulse mode (20 %

pulse fraction, 200 kHz, evaporation rate 1.5 %) with a specimen temperature of 55 K.

Chapter 3 Low temperature thermodynamics and kinetics of the Fe-Cr-Ni alloy

3.1. Phase diagrams of the Fe‐Cr‐Ni system

Although phase diagram research regards thermodynamic properties and phase equi‐

libria as the primary issues, because of the considerable contribution to the Gibbs ener‐

gy, the magnetic properties should be taken into account meticulously as well in the

CALPHAD modeling for the systems exhibiting magnetic ordering, such as Fe‐Cr‐Ni.

As a consequence, the magnetic phase diagrams are included for discussion in this sec‐

tion. In order to specify phase diagrams representing the magnetic ordering, these

phase diagrams are entitled “magnetic phase diagrams” in this thesis.

3.1.1. Phase diagrams of the boundary systems

The research interests in the three binary systems, Fe‐Cr, Fe‐Ni, and Cr‐Ni, are spurred

enormously due to the technical importance in engineering applications. The phase

equilibria have been studied intensively by different research groups. Although the

phase relations are simple in these three binaries, the magnetic ordering introduces

most of the complexity in modeling these systems. In spite of this, in this thesis, it is

found that some parts of the phase equilibria still lack thorough investigations, which

even includes the pure elements. For example, due to easy oxidation, the melting tem‐

perature of Cr needs to be determined carefully by experiments. According to the litera‐

ture review in this work (see paper I), we suggested that the melting temperature of Cr

should be taken 2136 K rather than 2180 K as adopted in the SGTE (Scientific Group

Thermodata Europe) database [49].

In this thesis, the phase diagram of the Fe‐Cr system is considered the most important

of the three binaries, since the thermodynamic feature will be inherited in the ternary

phase region. The spinodal decomposition of the ferrite phase in stainless steels is

mainly due to the bcc miscibility gap in the Fe‐Cr binary and the brittle σ phase has its

origin from the Fe‐Cr system. Besides, chromium is the element with the second largest

CHAPTER 3. LOW TEMPERATURE THERMODYNAMICS AND KINETICS

22

amount with over 10.5 wt.% in stainless steels. Therefore, the Fe‐Cr system is the proto‐

type for studying phase equilibria and phase transformations in stainless steels.

The Fe‐Cr phase diagram and thermodynamic properties is evaluated in paper I as a

comprehensive literature review. In the review, we found that all of the reported ther‐

modynamic evaluations should be further improved. The experimental data at both

high temperatures (with the liquid phase) and low temperatures (bcc phase) should be

reassessed, except for the γ‐loop region at medium temperatures. Based on the litera‐

ture review in paper I, thermodynamic modeling is performed by using the lattice sta‐

bility down to zero kelvin from the work of Chen and Sundman [15], see paper II. As

shown in Fig. 3.1 (a), the calculated phase diagram from this work shows significant

differences with the one from Andersson and Sundman [50], except for the γ‐loop.

Considering the thermodynamic properties, one can see from Fig. 3.1 (b), where the

experimental data of the enthalpy of mixing for the liquid phase are extremely scattered.

Both previous assessments using the CALPHAD approach generate a negative value

for the enthalpy of mixing at 1960 K for the liquid phase, while the present calculation

agrees with the most recent work by Thiedemann et al. [51] as well as the earliest report

by Pavars et al. [52]. During the thermodynamic optimization, it was found that the

overestimated temperatures for liquidus at the Cr rich side will easily generate a large

negative value for the enthalpy of mixing in the liquid phase.

0

500

1000

1500

2000

2500

Tem

pera

ture

, K

0 0.2 0.4 0.6 0.8 1.0

Mole fraction Cr

γα

Liq

α α+ ’

σ

Fe Cr

Andersson and Sundman, 1987This work

-6

-4

-2

0

2

4

6

En

tha

lpy o

f m

ixin

g (

Liq

), k

J/m

ol

Experimental data:1923K: Pavars et al., 19701873K: Nobori et al., 19761973K: Shumikhin et al., 19811863K: Iguchi et al., 19821960K: Batalin et al., 19841895~2010K:Thiedemann et al., 1998

This work, 1960 K

Lee, 1993, 1960 K

1960 KAndersson and Sundman, 1987

0 0.2 0.4 0.6 0.8 1.0

Mole fraction CrFe Cr

(a) (b)

Fig. 3.1. (a) Comparison of the calculated phase diagram between this work and the one by

Andersson and Sundman [50]. (b) Comparison of the enthalpy of mixing of the liquid phase

between CALPHAD results [50, 53] and experimental data.

More importantly, the magnetic phase diagram of the Fe‐Cr alloys cannot be produced

by the work of Andersson and Sundman [50], which is considered as the most reliable


23

CALPHAD modeling taken as the basis for the thermodynamic databases of steels. In

fact, the other reported thermodynamic assessments also ignore the description of the

magnetic phase diagrams of the Fe‐Cr system. Therefore, presently no thermodynamic

assessment is validated in both thermodynamic equilibria and magnetic ordering. In

this work, we carefully evaluated the magnetic phase diagram, which is needed to be

reproduced in the CALPHAD modeling. On the basis of paper I, the reliable magnetic

transition temperatures can be well reproduced by the updated thermodynamic model‐

ing in this work (paper II), but not by the previous assessment [50] as shown in Fig. 3.2.

It should be noted that the thermodynamic assessment of the Fe‐Cr system in this work

is based on the magnetic model used in the standard CALPHAD method. More discus‐

sion on improving the standard magnetic model will be addressed in the next section.

0

200

400

600

800

1000

1200

Te

mp

era

ture

, K

Oberhoffer and Esser, 1927

Adcock, 1931

Fallot, 1936

Rajan, et al., 1960

Nevitt and Aldred, 1963

Yamamoto, 1964

Ishikawa, et al., 1965

Imai, et al., 1966

Arajs and Dunmyre, 1966

Suzuki, 1966

Arrott and Werner, 1967

Ishikawa, et al., 1967

Arajs and Anderson, 1971

Mitchell and Goff, 1972

Tsunoda et al., 1974

Loegel, 1975

Normanton et al., 1976

Suzuki, 1976

Mori, et al., 1976

Aldred and Kouvel, 1977

Burke and Rainford, 1978

Strom-Olsen, et al., 1979

Inden, 1981

Benediktsson, et al., 1982

Vilar and Cizeron, 1982

Burke and Rainford, 1983

Burke, et al., 1983

Furusaka, et al., 1983

Mirebeau, et al., 1984

Furusaka, et al., 1986

Fischer, et al., 2001

This work, DSC measurement

1043

980

1000

1020

1040

1060

1080

0 0.05 0.10 0.15

Andersson and Sundman, 1987

This work

Experimental data:

0

0.5

1.0

1.5

2.0

2.5

Me

an

ma

gn

eti

c m

om

en

t,μ

/ato

mB

Shull and Wilkinson, 1955

Aldred, 1976

Bacon, 1961

(a) (b)

0 0.2 0.4 0.6 0.8 1.0

Fe Cr0 0.2 0.4 0.6 0.8 1.0

Fe Cr

Fig. 3.2. Magnetic phase diagram of the Fe‐Cr system (a) Magnetic transition temperature;

(b) Mean magnetic moment. Experiment data are taken from Xiong et al. [2].

In the work related to the Fe‐Cr system, we should emphasize the issues on the enthal‐

py of formation at 0 K. In the ab initio calculations, it is reported [54] that the enthalpy

of formation for the bcc phase at 0 K in the ferromagnetic (FM) state shows a sign

change from negative to positive (see paper II). In addition, some experiments [55] in‐

dicate a larger solubility range of Cr in (α‐Fe) compared with the accepted phase dia‐

grams in both handbook [56] and CALPHAD databases [50] in the temperature range


24

between 600 and 800 K. This seems consistent with the prediction by ab initio calcula‐

tions.

200

300

400

500

600

700

800

900

Te

mp

era

ture

, K

0 0.05 0.10 0.15 0.20

Mole fraction CrFe

Kuwano and Hamaguchi, 1988

Bonny et al., 2008

Evaluation:CALPHAD:


This work

α α+ '

Filippova et al., 2000

α

Mirebeau et al., 1984

Kuwano, 1985

Dubiel and Inden, 1987

SRO type change in equilibria

Phase boundary in equilibria

Kuwano and Hamaguchi, 1988



Precipitation under irradiation

SRO type change under irradiation

Single Solid solution under irradiation

Experimental data:

A

Mirebeau et al., 2010

This work

-5

-4

-3

-2

-1

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1.0


Ab initio (Olsson et al., 2006):

EMTO-CPA-GGA

PAW-VASP, SQS

Ab initio (Korzhavyi et al., 2009):


CALPHAD:

This work

EMTO-CPA-GGA

EMTO-CPA-LDA

Ma

gn

eti

c o

rde

rin

g e

ne

rgy, kJ/(

mo

l·a

tom

)

-1

0

1

2

3

()

0.40 0.2-1

0

1

2

3

0.161

(a)

(b) E(DLM) E(FM)–

Fig. 3.3. (a) Comparison of the solvus at Fe‐rich side in the Fe‐Cr phase diagram among

evaluation, thermodynamic modeling and experiments. NB: a later published experimental

data by Mirebeau et al. in 2010 [57] is included. Moreover, the data published in the same

group in 1984 [55] were not interpreted correctly in the work by Xiong et al. [2] (paper I),

and are corrected in this figure. (b) Comparison of the magnetic ordering energy between

DLM (disordered local moment, similar to paramagnetic state) and FM states among differ‐

ent calculations. Cited references in the figure are available in paper I and II.


25

In order to confirm this conclusion on the enthalpy of formation at 0 K, a short litera‐

ture review of the experimental data was reported by Bonny et al. [58] with a revised

solubility curve of (α‐Fe) as shown in Fig. 3.3(a). In the work of Xiong et al. [2], another

detailed study (see paper I in the thesis) was followed up, and different conclusions

from the one by Bonny et al. [58] were made as shown in paper I. It was approved that

the enthalpy of formation for the bcc phase at ground state calculated by ab initio can‐

not be considered as evidence of non‐zero solubility of Cr in (α‐Fe) at 0 K.

Firstly, the ab initio calculations were performed for FM states, which is different from

the magnetic state of pure Cr. Secondly, the calculated energy difference between par‐

amagnetic (PM) and FM states for pure Fe using the ab initio method differs significant‐

ly from the CALPHAD description in both SGTE database [49] and new lattice stability

reported by Chen and Sundman [15]. Thirdly, as pointed out by Xiong et al. [2], the

work by Bonny et al. [58] is questionable and not judicious with some crude judgments

(see paper I). For instance, the authors [58] adopted the experimental results in the non‐

equilibrium states of commercial steels to obtain the consistency for phase equilibria in

the Fe‐Cr binary system. In fact, the analyzed solubility of Cr in (α‐Fe) by Bonny et al.

[58] seems quite artificial. Instead, Xiong et al. [2] provided a possible location of the Fe‐

rich solvus in the Fe‐Cr phase diagram as shown in Fig. 3.3(a) according to the reliable

experimental data under equilibrium [55, 59‐62].

The above debate on the solubility limit of Cr in (α‐Fe) at 0 K was partially solved in the

improved thermodynamic description of the Fe‐Cr system in this thesis (see paper II). It

has been shown that the experimental data for the solvus of the bcc phase at the Fe‐rich

side can be well described by the CALPHAD modeling without introducing any solu‐

bility limit of Cr in Fe at 0 K. Since the updated lattice stability of pure Fe down to 0 K

by Chen and Sundman [15] is adopted in the work by Xiong et al. [2] (see paper II), the

model‐prediction at 0 K will be more reliable than the one taking lattice stability from

the SGTE database [49]. It was intriguing to discover that an anomaly of the sign

change exists in the magnetic ordering energy but not in the enthalpy of formation for

the bcc phase at 0 K as shown in Fig. 3.3(b). However, further confirmations may be

needed since pure Cr is not described satisfactory at 0 K in paper II [2] due to the adop‐

tion of the lattice stability from the SGTE database, although it is expected that the Cr‐

rich side will not bring significant effects on the Fe‐rich side, since the magnetic contri‐

bution to the Gibbs energy of Cr is much smaller than in pure Fe.

It is worth mentioning that a certain solubility limit of Cr in (α‐Fe) at 0 K was also con‐

sidered as the initial attempt in the CALPHAD modeling by Xiong et al. [63], see paper

V in the thesis. However, it was found that such an assumption will generate a high


26

consolute temperature (1027 K), which is similar to the problem found in some other

atomistic modeling [64].

0

200

400

600

800

1000

1200

Te

mp

era

ture

, K

0 0.2 0.4 0.6 0.8 1.0

Mole fraction Ni

bcc

fcc

Fe Ni

Phase boundaryMagnetic transition temperature, fcc

Magnetic transition temperature, fcc

Phase boundarySSOL database, Thermo-Calc AB:

This work:

0

200

400

600

800

1000

1200

Te

mp

era

ture

, K

0 0.2 0.4 0.6 0.8 1.0

Mole fraction NiFe Ni

Swartzendruber et al., 1991 & 1992

0

200

400

600

800

1000

0 0.1 0.2 0.3 0.4 0.5

Ms

T0Tc

Evaluation for Handbooks:

Jansson, 1987; De Keyzer et al., 2009Cacciamani et al., 2010

CALPHAD:

This work

Magnetic Transition Temperature for fcc:

This workSSOL database

CALPHAD, T line:0

Experimental Curie Temperature:Symbols, Xiong, 2011 (Thesis paper)

Calculated phase diagram, CALPHAD:

(a) (b)

fcc

Fig. 3.4. (a) Magnetic phase diagram of the fcc phase in the Fe‐Ni system. Symbols are exper‐

iments taken from this work (see Xiong et al. [65] in the attached papers), (b) Comparison of

the calculated phase diagram between this work and the SSOL database from Thermo‐Calc

AB. Magnetic transition curves of the fcc phase are drawn as chained and dashed lines.

Due to the problem related to the magnetic phase diagram revealed in the Fe‐Cr system,

the Fe‐Ni magnetic phase diagram was revisited in this thesis as well. After compiling

the reported experimental data, it was found that the magnetic phase diagrams of the

Fe‐Ni system reported in both handbook and thermodynamic databases need to be re‐

vised. As shown in Fig. 3.4, the compiled experimental Curie temperatures indicate that

the kink on the magnetic transition temperature curve at 0 K should be at the 25 at.% Ni,

which can be confirmed by the variation of the mean magnetic moment manifested as

the variation in global magnetization. Ab initio calculations have been applied for cal‐

culating the mean magnetic moment of the bcc phase as the input of the current mag‐

netic model adopted in the CALPHAD approach. Afterwards, the low temperature

equilibria of the Fe‐Ni system related to the fcc and bcc phases were reassessed using

the CALPHAD technique. As shown in Fig. 3.4, the calculated phase diagram in this

work generates a different reaction temperature of the monoeutectoid equilibrium: fcc

(PM) fcc (FM) + bcc (FM), which is apparently caused by the description of the Curie


27

temperatures (see Fig. 3.4). However, it should be stressed that no experimental data is

available to confirm such an invariant temperature yet.

Furthermore, it is intriguing to point out that significant improvements have been made

due to the corrected description of the magnetic phase diagram in the Fe‐Ni system.

According to the previous thermodynamic description compiled in the SSOL database

[33], the model‐predicted T0 curve, at which fcc and bcc would have the same Gibbs

energy, will cross the Ms (Martensite start temperature) curve, while the thermodynam‐

ic model in this thesis generates a reasonable T0 curve.

Jette et al. 1934Baer, 1958

Vintaikin and Urushadze, 1969

Vintaikin and Urushadze, 1970Karmazin, 1982 (stable phase boundary)Karmazin, 1982 (unstable phase boundary)

Tomiska, 2004

Rahaman and Ruban, 2010

ab initio Monte Carlo:

Experimental data:

Te

mp

era

ture

, K

Mole fraction, NiCr Ni

1617

856

CrNi2

Liq

fccbcc

198

fcc

CrNi2

bcc+fcc CrNi2

856fcc

(a)

(b)

(c)

856

Fig. 3.5. (a) Comparison of the Cr‐Ni phase diagram according to thermodynamic modeling

in this work, experimental data [66‐71] and ab initio Monte Carlo simulation [72]. The

peritectoid reaction at 856 K is magnified in (b) and (c).

The Cr‐Ni phase diagram has also been studied by many research groups both with

experiments and modeling [73‐78]. It should be mentioned that the prevalent thermo‐

dynamic databases, e.g. TCFE database from Thermo‐Calc software, are based on the

thermodynamic descriptions performed by Lee [78]. The phase equilibria have been

reviewed extensively by Nash [73]. However, during the thermodynamic descriptions

in Ref. [78], the practical interests are at high temperatures, the phase equilibria related

to a low temperature ordered phase, CrNi2, was not included. This was the reason for

reassessments carried out recently by Turchi et al. [76] and Chan et al. [77], who consid‐

ered the thermodynamic model for the CrNi2 compound in their updated thermody‐


28

namic modeling. It should be noted that such reassessments did not intensively consid‐

er the solubility limit of the CrNi2 phase, i.e. a stoichiometric model is used in the as‐

sessment. Therefore, the thermodynamic modeling needs to be further improved.

-2

-1

0

1

2

3

4

5

6

En

thalp

yo

fm

ixin

g,

Liq

uid

,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mole fraction Ni

Experimental data:Batalin et al., 1983

Cr Ni

kJ/m

ol

This workLee, 1992

1960 KLiquid

CALPHAD:

-5000

-4500

-4000

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

0.50 0.55 0.60 0.65 0.70 0.75 0.80

Mole Fraction Ni

ab initio calculations:Rahaman and Ruban, 2010, unpublishedKorzhavyi, 2010, unpublished

Chan et al., 2006, WIEN2KTurchi et al., 2006Wang et al., 2005, VASPArya et al., 2002, WIEN97

Experimental data for CrNi :2

Hirabayashi et al., 1969, 773 K

En

tha

lpy o

f fo

rma

tio

n,

CrN

i,

J/m

ol·

ato

m2

Reference states:Cr: bcc; Ni: fcc

Data for CrNi2

(a) (b)

This work

Fig. 3.6. (a) Comparison of the enthalpy of mixing at 1960 Kin the liquid phase between ex‐

perimental data [79] and thermodynamic modeling. (b) Comparison of the enthalpy of for‐

mation of the CrNi2 phase among the CALPHAD results at 773 K in this work, experimental

data [80], and ab initio calculations at ground state [72, 76, 77, 81‐83].

In this thesis, we performed a preliminary thermodynamic modeling of the phase equi‐

libria and thermodynamic properties of the Cr‐Ni alloys. It is noteworthy that the pa‐

rameters for describing the magnetic phase diagram of the Cr‐Ni system were taken

from the previous work by Lee [78]. As will be discussed later in section 3.2, since the

magnetic Cr‐Ni phase diagram is more complex than the one assessed in all of the

available thermodynamic descriptions, a correct description of the magnetic phase dia‐

gram of the Cr‐Ni system is impossible at the moment without further improvement on

the magnetic model in the CALPHAD approach. However, there is a need at least to

point out the features and drawbacks of the available thermodynamic descriptions of

the Cr‐Ni system, which will facilitate future studies. A preliminary result of a thermo‐

dynamic assessment using the current magnetic model is presented here.

As can be seen from the reassessed phase diagram shown in Fig. 3.5, the experimental

data can be well reproduced. The degenerated reaction at 856 K is modeled as a peritec‐

toid reaction, and the composition of CrNi2 at the invariant reaction is very close to the


29

one for fcc. It should be noted that the invariant reaction related to CrNi2 is estimated as

863 K in the work by Nash [73]. A Nishizawa horn is generated at about 500 K due to

the Curie temperature in the fcc phase at the Ni‐rich side (see more discussion in sec‐

tion 3.2).

It is noticed that in the previous work by Lee [78], the experimental data on the enthal‐

py of mixing at 1960 K cannot be well described as shown in Fig. 3.6(a). As shown in

Fig. 3.6(b), the model‐predicted enthalpy of formation of the CrNi2 ordered phase agree

fairly well with the experimental data measured at 773 K [80] as well as the ab initio

calculations [72, 76, 77, 81‐83]. Moreover, the tendency for the variation in the enthalpy

of formation predicted by Rahaman and Ruban [72] is consistent with the model‐

predicted one in this work, although the slops are different as shown in Fig. 3.6(b).

Because of the complexity in the magnetic phase diagram on Cr‐Ni, it is better to intro‐

duce it in section 3.2, where the improved magnetic model in this thesis is presented.

The Cr‐Ni magnetic phase diagram will be treated as a case study in the discussion of

the magnetic model for the CALPHAD approach.

3.1.2. Phase equilibria of the Fe‐Cr‐Ni ternary system

Although many issues and challenges arise in these three binary systems, it is worth to

have some tentative discussion on the ternary Fe‐Cr‐Ni system. The thermodynamic

modeling of the Fe‐Cr‐Ni system was reported by different groups [66, 84‐86], and the

CALPHAD databases usually adopt the thermodynamic modeling by Lee [86]. Howev‐

er, it is found that, in the existing thermodynamic assessments, the phase equilibria

were well described above 1073 K, where most of the technical issues in steel processing

need to be addressed. As described in the last chapters, since the present interest is ex‐

tended to materials services at low temperatures, the phase equilibria lower than 1073

K should be described properly as well. In this thesis, the problems related to the mag‐

netic phase diagram found in the binaries have impeded further improvement to ther‐

modynamic modeling of the phase equilibria in the Fe‐Cr‐Ni system. Additionally, alt‐

hough the experimental data in the literature are available, a comprehensive compari‐

son between experiments and modeling is still lacking. Consequently, a comprehensive

comparison of the phase equilibria between available experimental data and the CAL‐

PHAD database TCFE6 (from the Thermo‐Calc software AB [33]) has been performed

as the first step. It is noteworthy that the some unpublished experimental data by

Hertzman [87] have been attached for comparison between experimental data and

thermodynamic database.


30

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

MA

SS

_FR

AC

TIO

NC

R

0 0.2 0.4 0.6 0.8 1.0

MASS_FRACTION NI

Experimental data at 1073 K

Schafmeister and Ergang, 1939

Sigma Sigma+bcc

Sigma+fcc

Sigma+(fcc/bcc)

Sigma+bcc+fcc

Rees et al., 1949

Two phases:

Sigma+fcc

et al., 2010Hertzman

tie-line

Fe Ni

1073 K

fcc

bcc

σ

bcc

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

MA

SS

_FR

AC

TIO

NC

R

0 0.2 0.4 0.6 0.8 1.0

MASS_FRACTION NI


Rees et al., 1949

fcc

bcc

fcc+sigma

bcc+fcc

bcc+sigmabcc+fcc+sigma

Hattersley and Hume-Rothery, 1966

fcc fcc+sigma

bcc+fccbcc+fcc+sigma

Hertzman et al., 2010

tie-line

Fe Ni

923 K

fccbcc

σ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

MA

SS

_FR

AC

TIO

NC

R

0 0.2 0.4 0.6 0.8 1.0

MASS_FRACTION NI


Cook and Brown, 1952

fcc bcc+sigma

fcc+sigma

fcc+bcc

fcc+bcc+sigma


tie-line

Fe Ni

823 K

bcc fcc

σ

Cr

Cr

Cr

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

MA

SS

_FR

AC

TIO

NC

R

0 0.2 0.4 0.6 0.8 1.0

MASS_FRACTION NI



tie-line

Fe Ni

973 K

fccbcc

σ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

MA

SS

_FR

AC

TIO

NC

R

0 0.2 0.4 0.6 0.8 1.0

MASS_FRACTION NI



tie-line

Fe Ni

873 K

bccfcc

σ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

MA

SS

_FR

AC

TIO

NC

R

0 0.2 0.4 0.6 0.8 1.0

MASS_FRACTION NI



tie-line

Fe Ni

773 K

bccfcc

σ

Cr

Cr

Cr

Fig. 3.7. Comparison of the isothermal sections of the Fe‐Cr‐Ni system at different tempera‐

tures between experimental data [87‐91] and thermodynamic modeling according to the

TCFE6 database [33]. The symbols in unary regions are in blue, the ones in binary phase re‐

gions are in green, and the ones in ternary regions are in red.


31

As shown in Fig. 3.7, the present thermodynamic modeling available cannot well repre‐

sent the low temperature phase equilibria, except for at 1073 K. According to the plots

at 923 and 823 K, the experimental phase boundaries of bcc and fcc phases cannot be

described by the available thermodynamic modeling. Although the data provided by

Hertzman [87] may contain some unexpected experimental results due to the inade‐

quate annealing, the model‐predicted phase diagram at low temperature is questiona‐

ble and deserve further confirmation by experiments. Regardless the possible non‐

equilibrium data by Hertzman [87], the calculated phase triangles of bcc, fcc and at different temperatures show a systematical shift from the experimental ones. The phase

boundaries of the phase at low temperatures are yet unknown, especially for the ones

close to the Cr‐rich corner. According to the comparison at 923 K, one can easily find

that the experimental data at low temperatures may not have been taken into account in

the previous assessments, since the fit of the tie‐lines between bcc and fcc on the Fe‐rich

corner are not satisfactory at all.

3.2. Improvement of the magnetic model for computational

thermodynamics

As indicated in Eq. (2‐1), some physical contributions to the Gibbs energy may need to

be included in the CALPHAD models. One of these contributions that should be con‐

sidered in the Fe‐Cr‐Ni system is due to magnetic ordering. As mentioned in the pre‐

ceding sections, when performing thermodynamic modeling in the Fe‐Cr and Fe‐Ni

systems, we found some serious problems when using the magnetic model implement‐

ed in the current CALPHAD method. Therefore, an attempt to improve the magnetic

model for computational thermodynamics was made. In addition, some case studies

were performed on the systems: Cr‐Ni and Fe‐Cr‐Al. It is worth noting that the Fe‐Cr‐

Ni system was not chosen as an example to show the validity of the modified magnetic

model. Due to its complexity, a full picture of the magnetic phase diagram of the Fe‐Cr‐

Ni system needs more ab initio inputs in addition to another comprehensive literature

review of the magnetic transition in the Fe‐Cr‐Ni system before using it as an example.

Pure Al normally will not exhibit magnetic ordering. Therefore, it is suitable to be in‐

cluded for discussing the magnetic model in this work when considering a more gen‐

eral case.

3.2.1. Improved magnetic model

Before introducing the revised magnetic model, it is necessary to briefly mention the

Inden‐Hillert‐Jarl (IHJ) model, which is used in the current CALPHAD approach. The


32

IHJ model has been available since 1978, and was originally discussed for the transition

in FM states but not for antiferromagnetic (AFM) states. The model was proposed by

Inden [18] in the annual CALPHAD conference in 1976 and later modified by Hillert

and Jarl [19], and consists of two basic ideas.

Firstly, the magnetic contribution to the heat capacity (Cp) was expressed in an analytic

form:

3 5

3 5

1 1 2 , 1

3 5

1 1 2 , 1

3 5

FM FM m m m

P

PM PM n n nP

C K R

C K R

t t t t

t t t t- - -

ì æ öï ÷ï ç ÷= ⋅ ⋅ + + <ï ç ÷çï ÷çè øïïí æ öï ÷çï ÷= ⋅ ⋅ + + >çï ÷çï ÷çè øïïî

(3‐1)

where m = 3 and n = 5, τ is T/Tc, Tc is the Curie temperature, KFM and KPM are two con‐

stants from FM and PM states, R is the gas constant.

Secondly, based on the work by Tauer and Weiss [92], Inden suggested to calculate the

maximum magnetic entropy of an element in the FM ordering transition as a function

of the mean magnetic moment (b ) [18]:

( )maxln 1magnS R b= + (3‐2)

As a consequence, the final expression of the Gibbs energy for magnetic ordering be‐

comes:

( ) ( )1magn

mG RT ln gb t= ⋅ + (3‐3)

( )

1 3 9 15

5 15 25

1 79 474 11 1 , 1

140 497 6 135 600

1 1 1 1, 1

10 315 1500

A p pg

A

t t t tt

t

t t t t

-

- - -

ì é ùæ öï æ öï ÷÷ççê ú÷ï ÷ç- + - + + <ç ÷÷ï ê úçç ÷÷ç ÷çï è øè øï ê úë û= íï æ öï ÷ç ÷ï- + + ³ç ÷ï ç ÷çï è øïî

(3‐4)

518 11692 1

11125 15975

Ap

æ ö÷ç ÷= + -ç ÷ç ÷çè ø (3‐5)

where p is called the structure factor, which is the ratio of the magnetic enthalpy in the

PM state to the total magnetic enthalpy. For the bcc structure, the accepted value is: p =

0.4, while for fcc and hcp structure, p = 0.28 [18].


33

It is noteworthy that the above functions, from (3‐3) to (3‐5), consider the disordered

state at the infinitely high temperatures as the reference state. As explained in another

work by Inden [93], an artificial miscibility gap could occur in some cases when some

region show zero for magnetic transition temperature and the FM state at low tempera‐

tures is considered as the reference state.

0 0.2 0.4 0.6 0.8 1.0

non-bcc

TN

–3TN

Tc

A B0 0.2 0.4 0.6 0.8 1.0

bcc

TN

–TN

Tc

A B

Te

mp

era

ture

, K

0 0

Te

mp

era

ture

, K

AFM factor=–3

(AFM) (FM)

non-bcc Tc

0

AFM factor=–3AFM factor=–1

(AFM) (FM)

bcc Tc

0

AFM factor=–1

0 0.2 0.4 0.6 0.8 1.0A B

(FM)

0 0.2 0.4 0.6 0.8 1.0A B

(FM)

(a) (c) (d)(b)

TN

TN

Te

mp

era

ture

, K

Te

mp

era

ture

, K

–TN –3TN

xB, at.% xB, at.% xB, at.% xB, at.%

Fig. 3.8. Illustration of describing magnetic phase diagram of a hypothetical A‐B system by

using the AFM factor, the cases shown in (a) and (c) are for the bcc structure, while the ones

shown in (b) and (d) are for non‐bcc structures.

Since the above treatment was just for the FM ordering transition, it is necessary to in‐

troduce a way to include the cases with AFM ordering transition. This task was left to

Hertzman and Sundman [94] when they investigated the Fe‐Cr system, and introduced

an empirical AFM factor proposed by Weiss and Tauer [95] and successfully applied

the IHJ model for both FM and AFM states in the Fe‐Cr system. In order to have a sin‐

gle expression for the composition dependences of both Curie and Néel temperatures at

Fe and Cr‐rich side, respectively, the parameters for AFM ordering was treated as nega‐

tive values divided by a factor of ‐3 for fcc structures and ‐1 for bcc crystals. The whole

scheme is illustrated in Fig. 3.8 (a) and (b). It should be emphasized that the same AFM

factors have also been adopted by Hertzman and Sundman [94] for describing the mean

magnetic moment in the Fe‐Cr system. Ever since then, this treatment has become an

important part of the standard magnetic model used in the CALPHAD community and

has been generalized and used for all different kinds of systems without any further

questioning.

However, in this thesis, it was found that the magnetic model used in the current CAL‐

PHAD approach could not handle all the cases in the Fe‐Cr‐Ni system properly. Firstly,

in paper III, for the Fe‐Ni system, it was found that a single Redlich‐Kister polynomial

cannot well describe the fcc magnetic phase diagram of the Fe‐Ni system by introduc‐

ing the AFM factor ‐3. More seriously, the concept of the magnetic entropy has been


34

misunderstood when applying the magnetic model to cases with AFM ordering transi‐

tion, since the mean magnetic moment was considered as the input for calculating the

magnetic entropy, which differs from the original idea of Inden when he proposed the

model. In a later work published by Inden [96], Eq. (3‐2) was carefully discussed again.

The original expression proposed by Tauer and Weiss [92] was suggested for calculat‐

ing the maximum magnetic entropy using the experimental data for the local magnetic

moment, i.e. individual magnetic moment. Therefore:

( )maxln 1magn

i ii

S R x b= +å (3‐6)

where xi is the mole fraction of component i. Eq. (3‐2) can only be considered as an ap‐

proximation to obtain the value of maximum magnetic entropy and act as a makeshift if

local magnetic moments are unavailable.

In this work, the above problem of using Eq. (3‐2) for the AFM ordering transition was

elucidated. It was demonstrated that substantial difference will occur when adopting

Eqs. (3‐2) and (3‐6) to compute magnetic entropy in the AFM ordering transition. Case

studies were performed on the Fe‐Cr‐Al and Cr‐Ni systems, and will be discussed in

the next section.

More importantly, the concept of effective magnetic entropy was proposed in this thesis.

In order to represent the local magnetic moment using a single function, the “effective

magnetic moment” was introduced and denoted as β*, which satisfies the following

relation:

( ) ( )*

maxln 1 ln 1magn

i ii

S R R xb b= + = +å (3‐7)

It is obvious that the effective magnetic moment β* can be calculated from the local

magnetic moment through:

( )* 1 1ix

ii

b b= + - (3‐8)

In this work, it is suggested that a single R‐K polynomial can still be used for describing

the effective magnetic moment in multi‐component cases.

Furthermore, it is suggested that the effective magnetic moment is considered to be a

physical quantity in order to represent the λ shape of the magnetic contribution to the

heat capacity. Therefore, for achieving a good fit to the heat capacity, sometimes a devi‐

ation from the experimental magnetic moment can be acceptable. One example can be


35

found in the case of pure Cr, it was found that due to the complexity of the magnetic

properties of pure Cr, the expected value of the effective magnetic moment cannot be in

the vicinity of the local magnetic moment determined by experiments [2]. In order to fit

the heat capacity of pure Cr, a value was assessed to be about 0.008 by Andersson [97].

Therefore, the ultimate aim of introducing the effective magnetic moment is to repro‐

duce the magnetic contribution to the heat capacity determined by experiments. In ad‐

dition, the effective magnetic moment may sometimes be different from the value gen‐

erated from the local spin magnetic moment as in Eq. (3‐8) if a good reproduction of the

heat capacity should be achieved.

Moreover, the AFM factor proposed by Weiss and Tauer [95] is not capable to handle

common cases as those shown in Fig. 3.9. Instead of adopting the AFM factor, a more

general method to manipulate the magnetic ordering transition is proposed in this the‐

sis, which is illustrated in Fig. 3.9.

0 0.2 0.4 0.6 0.8 1.0

0 0

0 0.2 0.4 0.6 0.8 1.0

0

0 0.2 0.4 0.6 0.8 1.0

TN

A

TCA

TCB

TN

B

TCB

TN

B

TN

A

TCA

TCB

TN

B

0

0 0.2 0.4 0.6 0.8 1.0

TCB

TCA

TNB

TNA

Te

mp

era

ture

, K

Te

mp

era

ture

, K

Te

mp

era

ture

, K

Te

mp

era

ture

, K

A B A BA BA B

Experimental N el temperatureé

Experimental Curie temperature

(a) (b) (c) (d)

xB, at.% xB, at.% xB, at.% xB, at.%

Fig. 3.9. Illustration of different magnetic phase diagrams described using the improved

magnetic model. The temperature curves with the negative absolute value are plotted as the

dashed lines.

Both FM and AFM states are introduced for each phase exhibiting magnetic ordering in

reality, and assuming one is unstable while the other is stable. For each magnetic state,

there exists a corresponding magnetic transition temperature, no matter if it is stable or

unstable. Furthermore, we assume that for each phase there is only one stable magnetic

state at each composition. Therefore, a negative absolute magnetic transition tempera‐

ture is assigned for each phase in its unstable magnetic state. For convenience, we take

the absolute value of this magnetic transition temperature the same as that for the stable

magnetic state. In this way, the composition dependences of the Curie and Néel tem‐

peratures will be described separately using two different R‐K polynomials, and the


36

empirical AFM factor applied in the standard magnetic model currently used in the

CALPHAD community is not necessary. As a result, the artifact caused by using one

single polynomial with help of the AFM factor shown in Fig. 3.8 (c) and (d) and Fig. 2 of

paper V will disappear. Moreover, these manipulations can handle more general cases

found in magnetic phase diagrams.

Due to the work by Chen and Sundman [15], Eq. (3‐1) was extended by using a third

term in the Taylor expansion as shown below.

3 5 7

3 5 7

1 1 1 2 , 1

3 5 7

1 1 1 2 , 1

3 5 7

FM FM m m m m

P

PM PM n n n nP

C K R

C K R

t t t t t

t t t t t- - - -

ì æ öï ÷ï ç ÷= ⋅ ⋅ + + + <ï ç ÷çï ÷çè øïïí æ öï ÷çï ÷= ⋅ ⋅ + + + >çï ÷çï ÷çè øïïî

(3‐9)

The power exponent, n, shown in the above equation was adjusted to 7, while m is kept

to 3. Therefore, FMK and PMK have been re‐optimized for Fe and yields new values for

the structural factor, p = 0.37 for the bcc crystal, and p = 0.25 for the non‐bcc structures.

It has been shown that the slight change of the p value will not bring any strong effects

to the fitting of the heat capacity for pure Co and Ni [15].

Accordingly, the basic function of the Gibbs energy of magnetic ordering has been

modified to:

( ) ( )* 1magn

mG RT ln gb t= ⋅ + (3‐10)

( )

1

3 9 15 21

7 21 35 49

0, 0

0.38438376 0.635708951

1 , 0< 11

16 135 600 1617

1 1 1 1 1, 1

21 630 2975 8232

pg

Dp

D

t

t

t tt t t t

t t t t t

-

- - - -

ìïïïï £ïïï é ùïï ê ú+ ´ï ê úïïï ê ú= - £í æ öê úæ öï ÷÷ççï ê ú÷÷ç- + + +ç ÷÷ççê ú÷÷ç ÷çè øè øê úë ûæ ö÷ç ÷- + + + >ç ÷ç ÷çè øî

ïïïïïïïïïïï

(3‐11)

1

0.33471979 0.49649686 1Dp

æ ö÷ç ÷= + -ç ÷ç ÷çè ø (3‐12)


37

It is noteworthy that since the reference state of the magnetic ordering is at the infini‐

tively high temperature, according to Ref. [15], the reference state of the magnetic dis‐

ordering can be expressed as:

( ) ( )* *ln 1 0.38438376magn

disorder

TG R T

p Db

æ ö÷ç ÷ç¥ = - + - ÷ç ÷ç ÷⋅è ø (3‐13)

where *T is the magnetic transition temperature of the pure component. Since the ex‐

pression of the Gibbs energy in the Einstein model does not contain the first order term

of T, one should keep in mind to include the term for magnetic disordering at infinitive‐

ly high temperature, because this expression is not automatically taken into account by

the software, like Thermo‐Calc. However, because the expression of the chemical con‐

tribution to the Gibbs energy used in the SGTE databases includes a first order term of

T, Eq. (3‐13) is not necessary to consider specifically.

3.2.2. Case studies

In order to demonstrate the difference of the magnetic contribution to the Gibbs energy

by using the magnetic model in the current CALPHAD method and our improved

magnetic model, case studies are performed on the Cr‐Ni and Fe‐Cr‐Al systems. Alt‐

hough phase equilibria in the Cr‐Ni system is rather simple, the magnetic phase dia‐

gram of the fcc and bcc phases in the Cr‐Ni system is found to be totally different from

the ones calculated using thermodynamic database taken from Refs. [53, 78].

As shown in Fig. 3.10 (a), according to the database TCFE6 released by the Thermo‐Calc

software AB [53, 78], the magnetic phase diagram of the bcc and fcc phases in the Cr‐Ni

system is not quite clear yet and needs further improvements. The only experimental

dataset on magnetic transition temperatures in the bcc structure is from Fukamichi and

Saito [98], which shows a drastic decrease of the Néel temperature on the Cr‐rich side.

As discussed in paper III for the Fe‐Ni system, according to a recent experimental re‐

port by Tian et al. [99], the Curie temperature of pure Ni in the SGTE database for pure

elements [49] should be corrected to be 456 K, while the magnetic moment is 0.52 μB but

not 0.85 μB as in the SGTE database [49]. The ab initio calculations performed by Ra‐

haman and Ruban [72] not only further confirmed the experimental results, but also

provided more information on the Ni‐rich side. It is shown in Fig. 3.10 (a) and (b) that

the Curie temperature of Ni‐rich alloys also exhibit a drastic decrease along with in‐

creasing Cr content. As a result, the magnetic phase diagram of the bcc phase was re‐

described by using the improved magnetic model. It was striking to find that the ener‐

gy differences between the TCFE6 database using the IHJ model in the current CAL‐


38

PHAD method and the new description using the improved magnetic model in this

work are enormous. By taking into account the improved magnetic model and the re‐

vised magnetic phase diagram, the magnetic contribution to the Gibbs energy is almost

ignorable.

Mole fraction NiCr Ni

Ma

gn

eti

c o

rde

rin

g e

ne

rgy

J/m

ol-

ato

m

200K

400K

500K

200K

400K

500K

Ma

gn

eti

c o

rde

rin

g e

ne

rgy

J/m

ol-

ato

m

Mole fraction NiCr Ni

bcc

bccTCFE6 this work

0

100

200

300

400

500

600

700

800

0 0.2 0.4 0.6 0.8 1.0

Experimental data:

Fukamichi and Saito, 1977ab initio calculations:

Rahaman , 2010, unpublished workand Ruban

bcc

Ma

gn

eti

c t

ran

sit

ion t

em

pe

ratu

re, K

312(SGTE)

575(SGTE)

456 (Exp.)

Mole fraction NiCr Ni Mole fraction NiCr Ni

Eff

ecti

ve m

ag

ne

tic m

om

en

t,μ

B

ab initio calculations:Rahaman , 2010, unpublished workand Ruban

0.52(Exp.)

0.008 (SGTE)

(a) (b)

(c) (d)

0.85(SGTE)

bcc

Fig. 3.10. Magnetic phase diagram and ordering energy of the bcc phase in the Cr‐Ni system.

The calculation based on the database of TCFE6 is in dashed line, while the one based on the

improved magnetic model of this work is in solid line. (a) Comparison of the magnetic transi‐

tion temperature between experiments and calculations. (b) Comparison of the effective mag‐

netic moments between ab initio calculations and CALPHAD modeling. (c) The model‐

predicted magnetic ordering energy from the TCFE6 database using the IHJ model imple‐

mented in the current CALPHAD approach at 200, 400 and 500 K. (d) The model‐predicted


39

magnetic ordering energy in this work by using the improved magnetic model at 200, 400

and 500 K. Experimental data for pure Ni is from the work by Tian et al. [99].

Analogous to the case of the bcc phase, serious problems are found for the fcc phase in

the accepted thermodynamic description [53, 78]. According to the experiments [100‐

104], the Curie temperatures and magnetic moments decrease rapidly on the Ni‐rich

side with increasing Cr content. It should be noted that, the ab initio Monte Carlo calcu‐

lations performed by Rahaman and Ruban [72] indicate small differences between the

mean and effective magnetic moments in the Cr‐Ni fcc alloys due to the weak magneti‐

zation of fcc Cr (see Fig. 3.11 (a) and (b)). Apparently, a Néel temperature of 369.67 K

and a magnetic moment 0.82 μB were considered as the reliable magnetic quantities in

the SGTE database [49] which does not coincide with the prediction by Rahaman and

Ruban [72]. As a matter of fact, such large values for the magnetic transition tempera‐

ture and magnetic moments were taken from the work by Chin et al. [105], who ob‐

tained some tentative values from a simple extrapolation when modeling the higher

order system Cr‐Fe‐Co‐Ni, and certainly does need to be further confirmed. Conse‐

quently, the ab initio Monte Carlo data [72] were considered to be most reliable, and

thus generate a different fit of the magnetic phase diagram as shown in Fig. 3.11 (a) and

(b). It should also be pointed out that although the experimental value of the magnetic

moments for pure fcc Ni is 0.62 μB, one may preserve the value adopted in the SGTE

database, since that value of 0.52 μB can give the best fit to the heat capacity of bcc Ni. In

this work, the fcc Cr was considered not to exhibiting magnetic ordering intrinsically,

therefore, the magnetic contribution to the Gibbs energy will be zero on the Cr‐rich side

as shown in Fig. 3.11. Moreover, one can realize that the improved magnetic model in

this thesis can handle such a case as the fcc phase correctly. However, using the stand‐

ard magnetic model adopted in the current CALPHAD method, an artificial Néel tem‐

perature curve cannot be avoided, and similar artifacts may occur as illustrated in Fig.

3.11(c).

Another case study was performed on the bcc Fe‐Cr‐Al alloys. It has been mentioned in

the beginning of this chapter that, since Al does not exhibit magnetic ordering under

ambient conditions, this system is a good example to shown the advantages by using

the improved magnetic model proposed in this work. The detailed discussion on the

magnetic phase diagram is available in paper V (see Figs. 7 and 8 in the paper). The

comparison of the calculated magnetic ordering energy using the model before and af‐

ter improvement is shown in Fig. 3.12, and significant difference of the magnetic con‐

tribution to the Gibbs energy using the magnetic models before and after the improve‐

ment can be observed.


40

Experimental data:Sadron, 1932Marian, 1937

Besnus et al., 1972Tange et al., 1981Simpson, 1982

ab initio Monte Carlo calculations:

Rahaman and Ruban, 2010, unpublished work

Mole fraction NiCr NiMole fraction NiCr

Mole fraction NiCr Ni Mole fraction NiCr Ni

(a) (b)

(c) (d)

Ma

gn

eti

c o

rde

rin

g e

ne

rgy

J/m

ol-

ato

m

Ma

gn

eti

c o

rde

rin

g e

ne

rgy

J/m

ol-

ato

mM

ag

ne

tic t

ran

sit

ion

te

mp

era

ture

, K

Eff

ecti

ve

ma

gn

eti

c m

om

en

t,μ

B0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

200K

400K500K

fcc fccTCFE6 this work

200K

400K

500K

Effective magnetic moment:


ab initio calculations:

Sadron, 1932Van Elst et al., 1962

Chiffey and Hicks, 1971

Besnus et al. 1972Tange et al., 1981

Experimental data:


ab initio calculations:

Mean magnetic moment:

633(SGTE)

0.52(SGTE)

0.62(Exp.)

369.67(SGTE)

0.82(SGTE)

fcc

fcc

Fig. 3.11. Magnetic phase diagram and ordering energy of the fcc phase in the Cr‐Ni system.

The calculation based on the databank of TCFE6 is in dashed line, while the one based on the

improved magnetic model of this work is in solid line. (a) Comparison of the magnetic transi‐

tion temperature between experiments and calculations. (b) Comparison of the effective mag‐

netic moments between ab initio calculations and CALPHAD modeling. (c) Comparison of

the model‐predicted magnetic ordering energy from the TCFE6 database using the IHJ model

implemented in the current CALPHAD approach at 200, 400 and 500 K. (d) Comparison of

the model‐predicted magnetic ordering energy in this work by using the improved magnetic

model at 200, 400 and 500 K.


41

Fig. 3.12. Case study on the bcc Fe‐Cr‐Al alloys. (a) Calculated magnetic ordering energy

according to the simplified magnetic model implemented in the state‐of‐the‐art CALPHAD

approach. (b) Calculated magnetic ordering energy according to the improved magnetic mod‐

el in this work. (c) Calculated difference of magnetic ordering energy from the models be‐

tween before and after modification. (d) The three dimensional view of the calculated differ‐

ence of magnetic ordering energy.

3.3. Kinetic study of spinodal decomposition

3.3.1. Spinodal decomposition in the Fe‐Cr system

The spinodal decomposition is a common phenomenon in materials indicating thermo‐

dynamic instability of the system. The modern theory of spinodal decomposition was

established by Hillert [20, 21] and Cahn and Hilliard [22, 106, 107]. Before that, the theo‐

ry of spontaneous segregation inside the spinodal region had been strongly advocated.

In the theory of spinodal decomposition, the concept of a macroscopically diffuse inter‐

face between the solute‐rich and solute‐poor regions is introduced with a correspond‐


42

ing interfacial energy. Therefore, the Gibbs energy of a system can be expressed as the

sum of three different energy components: the Gibbs energy per unite volume, the gra‐

dient energy, and the stress energy arising from the lattice mismatch. The last energy

term may cause significant shifts of the chemical spinodal curve (or incoherent spinodal

curve) into the coherent spinodal curve, and thus reduce the extension of the chemical

spinodal region. However, in the Fe‐Cr system, since the lattice mismatch between Cr

and Fe is very small, the location of the coherent spinodal curve is considered to be neg‐

ligible shifted from the incoherent spinodal curve.

A B

a

b c

d

1 2 34

xB

Chemicalspinodal

curve

α α’ α α’

(a)

(b)

(c)

(d)

A B A BxB xB

Te

mp

era

ture

Gib

bs e

ne

rgy

Te

mp

era

ture

Spinodalregime

Transientregimeinside

spinodal

Transientregimeoutsidespinodal

T0

T0

2

20

B

G

x

��

�

2

20

B

G

x

��

�

2

20

B

G

x

��

�

Fig. 3.13. Illustration of the spinodal decomposition (a) variation of Gibbs energy at tempera‐

ture of T0, (b) phase diagram of hypothetical A‐B system, (c) morphology in different region

near the spinodal curve, (d) practical transient region between nucleation and growth and

spinodal decomposition marked in green shadow.

As illustrated in Fig. 3.13 (a) and (b), inside the chemical spinodal region, the second

derivative of Gibbs energy is negative, which implies an uphill diffusional process. Ide‐

ally, inside the spinodal curve, the mechanism of spinodal decomposition governs the

phase transformation, while the regions outside the spinodal but inside the phase

boundary (regions between points 1 and 2, and between 3 and 4 at temperature T0 in

Fig. 3.13 (b)) are dominated by nucleation and growth. Practically, there is a transient


43

regime between these two mechanisms (spinodal and nucleation and growth), which

can be reflected by the atom map constructed from APT, see Fig. 3.13 (c). The green

shadow indicates the area of the transient regime. As shown in Fig. 3.13 (c) and (d), on

the location marked by the red circle, the morphology may show a weaker tendency to

an interconnected structure compared with the case at the point marked by a pink

square, while on the location marked with the blue diamond, isolated clustering may

dominate rather than interconnected areas, and sometimes called “’spinodal’ nuclea‐

tion” [108]. It should be noted that the location of the transient regime is difficult to

measure precisely. Therefore, the spinodal curve is still quite helpful in order to get a

general sense of the spinodal region especially for experimental data analysis. Besides,

the phase field simulation is based on thermodynamic and kinetic databases developed

by the CALPHAD method, and such a transient regime is normally disregarded.

Calculated phase boundary:

Model-predicted chemical spinodal curve:

This work; Andersson and Sundman, 1987

This work; Andersson and Sundman, 1987

Experimental data:

Marcinkowski et al., 1964

Yamamoto, 1964

Imai et al., 1966

Lagneborg, 1967

Vintaikin et al., 1969

Chandra and Schwartz, 1971

De Nys and Gielen, 1971

Miyazaki et al., 1974

Brenner et al., 1982

Katano and Lizumi, 1982

Furusaka et al., 1983 & 1986



Kuwano, 1985

Kuwano, 1985

LaSalle and Schwartz, 1986

park et al., 1986

Kuwano et al., 1987

Bley et al., 1992

Hyde et al., 1995

Okano et al., 1995

Miller et al., 1996

Cieslak et al., 2000

Ujihara and Osamura, 2000

Tem

pera

ture,

K


0 0.2 0.4 0.6 0.8 1.0

1000

800

600

400

200

0

α α’

Tc

TN

Magnetic transition temperature: This work; Andersson and Sundman, 1987

Fig. 3.14. Comparison of the spinodal curve and phase boundary of the bcc phase in the Fe‐Cr

system between experimental data and modeling results. Compilations of the experimental

data are according to the work in Refs. [2, 16]. The blue symbols are the experimental data

indicating a mechanism of spinodal decomposition, while the red symbols are for nucleation

and growth.


44

In this work, great efforts were firstly made in order to determine the spinodal region.

In the beginning, the experimental data on the spinodal region were carefully reviewed

[2] as compiled in Fig. 3.14. Since thermodynamic modeling was updated in this work

based on the correct description of the Curie temperature curve and new lattice stability

of pure Fe, it is necessary to validate the model‐predicted spinodal curve. In this thesis,

APT were performed on binary Fe‐Cr samples with three different Cr content (26.65,

31.95, 37.76 at.% Cr) annealed at 773 K for different periods, the comparison between

the microstructure evolutions in the alloys are inconsistent with the model‐predicted

spinodal region, which can be seen from paper II [16]. It was found that the alloy with

37.76 at.% Cr at 773 K is close to the edge of the transition region and mainly behave

like spinodal decomposition, while the other two samples is most likely in the composi‐

tion range at 773 K, where a mixed regime dominates showing both spinodal decompo‐

sition and nucleation and growth.

3.3.2. Estimation of composition amplitude – the ADS method

In the study of spinodal decomposition, two physical quantities should be carefully

considered for comparison between experiment and simulation: wavelength and com‐

position amplitude. So far, many efforts have been made to evaluate the wavelength of

the spinodal structure as well as the composition amplitude in order to quantify the

degree of the spinodal decomposition. In most of the cases, the mechanical failure

caused by “475 °C embrittlement” starts at an early stage of spinodal decomposition,

therefore, it requires sensitive and accurate methods to determine these two physical

quantities.

As for the estimation of dimensions, i.e. determination of wavelength, several methods

could be applied directly, such as, radius distribution function, autocorrelation function,

etc. The comparison of the different methods for determining wavelength in a case

study on the Fe‐Cr alloys was reported by Odqvist et al. [109].

However, one can envision that, in the very beginning, when traces of the interconnect‐

ed spinodal structure starts to form, it may be impossible to determine the wavelength

sensitively, whereas the composition amplitude becomes a more important factor.

Presently, the most commonly adopted method for estimating the composition ampli‐

tude in APT is the Langer‐Bar‐on‐Miller (LBM) method [110], which assumes that the

outline of the frequency distribution diagram constructed in the APT experiments is the

sum of two Gaussian distribution functions.


45

In order to evaluate the composition amplitude by the LBM method, the local concen‐

tration should be calculated first by sampling certain number of ions in a block. In this

work, the size of the block is 25 ions, and the frequency distribution diagram can be

generated as shown in Fig. 3.15 (b). Alternatively, one can generate a concentration

field as shown in Fig. 3.15 (b) using a cubic block size of 0.83x0.83x0.83 nm3 to compute

the local concentration from the atom map over the whole volume. Identically, the fre‐

quency distribution diagram can be generated by counting the number of nodes with

different concentration values within different concentration ranges from the concentra‐

tion field.

Cr atom

X

Y

Z(a)

4x20x20 nm3 (b)

25 ionsSampling block

Mole fraction, Cr

Cr

fre

qu

en

cy o

bse

rva

tio

n

33.1 at.%Cr, 100 h, 773 K

18.26x18.26x66.4 nm3

Cr at.%

Co

nce

ntr

ati

on f

ield

Fig. 3.15. The APT data for the sample with 33.1 at.% Cr isothermally heat treated at 773 K

for 100 hours. (a) Selected parts of the atom map. The distance between atoms has been en‐

larged dually in order to show the tendency of atom configuration. One sampling block for

constructing the concentration field is illustrated as a red cube on the right bottom corner. (b)

Frequency distribution histogram and the corresponding concentration field. The fitted

Gaussian distribution functions by the LBM method denoted by curves. Sampling size for

constructing the concentration field is 25 ions.

In this thesis, the LBM method was applied for the Fe‐37.76 at.% Cr alloy (local compo‐

sition in the analyzed volume is 33.1 at.% Cr) annealed at 773 K. In the case of 100 hours

annealing, the composition amplitude is to be 16.6 at.% Cr (see Fig. 3.15 (b)). The com‐

position amplitude becomes smaller to be 10.1 at.% Cr for the case with 10 hours an‐

nealing. For the as‐cast sample, the LBM method provides a value of 7.8 at.% Cr. How‐

ever, during the curve fitting using the Gaussian distribution functions, the tolerance of

the curve fitting already exceeded 3.5 at.% Cr. Therefore, the accuracy of the estimation

on the as‐cast sample cannot be guaranteed.


46

50 100 150 200 250 300

1

2

3

4

5

6

Dimensionless time0

Cr

co

nce

ntr

ati

on

, m

ole

fra

cti

on

0.8

0.6

0.4

0.2

0.0

ADS, upper limitADS, lower limitADS, amplitude

DFD, upper limitDFD, lower limitDFD, amplitude

τ = 20 τ = 60

τ = 300

(a) (b)

(c) (d)

0.298

0.498 0.654

0.209

0.856

0.180

Fig. 3.16. The frequency distribution histograms as well as concentration field simulated by

phase field method at different dimensionless time (a) 2 (b) 6 (c) 30. (d) Comparison of the

analyzed results for composition amplitude and its upper and lower limit at different simula‐

tion times. DFD stands for the direct frequency distribution method, while ADS for ampli‐

tude density spectrum.

In this thesis, a new method, called ‘the Amplitude Density Spectrum (ADS) method’

was proposed in order to estimate the composition amplitude more accurately. As in‐

troduced in Paper VI, the ADS pattern was constructed first for analysis. Empirically,

some criteria were set in order to obtain reasonable composition amplitude (See paper

VI for details). It is worth noting that the ADS method originates from the phase field

simulation on the Fe‐Cr system, since there are difficulties in analyzing the composition

amplitude at early stages based on the phase field results. As show in Fig. 3.16 (a), the

composition amplitude is impossible to evaluate by just directly observing the frequen‐

cy distribution diagram obtained in the early stages of the phase field simulation. On

the contrary, the composition amplitude can be easily read for the late stages when the

frequency distribution diagram is clearly separated into two peaks at the solute‐rich

and solute‐poor regions (see Fig. 3.16 (c)). In this work, we named this method as the

direct frequency diagram (DFD) method, which can be used to validate the ADS meth‐

od, at least for the late stages.


47

Using the ADS method proposed in this thesis (see paper VI), the accurate estimation

of the composition amplitude becomes possible. As shown in Fig. 3.16 (d), the compari‐

son of the results between the ADS and DFD methods implies the reliability of the ADS

method. More importantly, the initial noise with 5 at.% Cr set as the start condition of

the phase field simulation can be well estimated by using the ADS method (4.01 at.%

Cr).

By analyzing the experimental results of APT, it is demonstrated that the estimation of

the composition amplitude using the LBM method is systemically smaller than the es‐

timation using the ADS method. As discussed in paper VI, it is evident that the LBM

method will underestimate the composition amplitude. Using the ADS method, the

estimated composition amplitude for the analyzed volume with Fe‐33.1 at.% Cr an‐

nealed at 773 K for 100 hours is 34.9 at.% Cr, the one for 100 hours is 26.9 at.% Cr, and

the one in the as‐cast state is 23.6 at.% Cr. All of the above three values for the composi‐

tion amplitude is higher than the ones according to the LBM method as can be seen

from Table 3.1.

Table 3.1. Comparison of the analysis between the LBM and ADS methods

of the alloy with 33.1 at.% Cr.

Annealing

time

Method for

analysis

Size of

block

Composition (at.% Cr)

upper limit lower limit amplitude

100 h LBM 25 ions 41.6 24.9 16.6

ADS (8.3 nm)3 50.7 15.9 34.8

10 h LBM 25 ions 39.3 29.2 10.1

ADS (8.3 nm)3 48.0 21.1 26.9

0 h

as‐cast

LBM 25 ions 41.7 33.9 7.8

ADS (8.3 nm)3 48.9 25.3 23.6

(1.0 nm)3 45.3 28.7 16.6

One may also realize that the initial composition amplitude studied in nano size, like

Fe‐Cr alloys, could be larger than some other cases in micron dimension. Because the

present estimation of the local composition in APT only includes 25 ions, but the latter

one usually can be determined using energy dispersive X‐ray analysis containing over

hundreds or even thousands of atoms. However, one should keep in mind that the size

of the sampling block to construct the frequency distribution diagram or concentration

field can significantly affect the final results in both LBM and ADS analysis. Therefore,

the reasonable size of the sampling block for capturing the periodic feature in the 3D

interconnected structure is a formidable task, and deserves further investigation. For

example, the size of the cubic sampling block with 1.0x1.0x1.0 nm3 can obtain a compo‐


48

sition amplitude at about 16.6 at.% Cr, instead of 23.6 at.% as obtained from the

0.83x0.83x0.83 nm3, which also indicates the sensitivity of the ADS method.

All in all, it is evident that the LBM method is likely to underestimate the composition

amplitude (see paper VI). The advantages of the ADS method are summarized as be‐

low by comparing with the LBM method:

(1) The LBM method is based on the lever rule, which is only applicable in the equilib‐

rium state, while the ADS method does not require equilibrium.

(2) By comparing the results from the APT experiment, the LBM is approved to be un‐

derestimating the composition amplitude, while the ADS method does not (see paper

VI).

(3) The APT method is only applicable for the discrete information, while the ADS

method can be used for both discrete and continuum data.

(4) The APT largely depends on the outline of the frequency distribution diagram, in

order to obtain a reasonable fitting using the Gaussian distribution functions. However,

the ADS method is independent of the frequency distribution diagram. It is worth not‐

ing that the frequency distribution diagram sometimes will have non‐zero intensity at

the composition of 0 or 1, and is thus not suitable to be fitted by using the Gaussian dis‐

tribution function, since the tail of the Gaussian distribution function will extend be‐

yond the composition limit at 0 and 1.

3.3.3. Kinetic issues on spinodal decomposition of the Fe‐Cr system

In this thesis, the kinetic study on the spinodal decomposition of the Fe‐Cr system is

performed through the phase field simulation using the semi‐implicit Fourier‐spectral

method based on the thermodynamic modeling in paper II.

It is noteworthy that, as a preliminary test of the phase field model, a 2 dimensional

phase field simulation on Fe‐Cr alloys has been performed first using the femLego

software [36] (see paper V). In this work, the influence of the thermodynamic descrip‐

tion has been demonstrated, more importantly, it is found that the mobility database

considered in the simulation taken from the work by Jönsson [111] generates faster evo‐

lution of the spinodal structure than indicated by experimental data. Some further dis‐

cussions are available in the work by Grönhagen [112].

In this section, further quantitative discussion on the discrepancies between phase field

simulation and experimental data is necessary. In the theory of spinodal decomposition,


49

if the composition profile is assumed to follow the sinusoidal variation, the amplitude

A can be expressed as:

( ) ( )2 2

0exp 4 /A t A tMp l= - (3‐14)

in which, A0 is the initial composition fluctuation, M is a parameter containing the dif‐

fusivity, and λ is the wavelength of the spinodal structure.

A critical wavelength λc, below which M is positive and fluctuation will decay, can be

further defined:

( )2 2 22 2C m

d G dxl p e= (3‐15)

Only if the fluctuation with λ larger than the critical wavelength, it will grow and form

the modulated structure within the spinodal region.

Phase field simulation

Initial noise: 5 at.% Cr

Initial noise: 15 at.% Cr

Fe-35 at.% Cr

Dimensionless time (simulation)

This work (APT)

Fe-25 at.%Cr, 773 KThis work (APT)

Experimental data

Fe-45 at.%Cr, 773 K

Fe-35 at.%Cr, 773 K

Fe-45 at.%Cr, 673 K

Fe-32 at.%Cr, 746 K

Com

positio

n a

mplit

ude, at.%

Time, hours

Fe-45 at.%Cr, 773 K, phase field, 15 at.% Cr - initial noise

Fe-45 at.%Cr, 773 K, experimentsCerezo, 1992 (AP)Cerezo, 1992 (PoSAP)Hyde, 1992 (PoSAP)Hyde, 1995(a) (PoSAP)Hyde, 1995(b) (PoSAP)Hyde, 1995(b) (ECAP)Miller, 1990 (APFIM)LBMADS

(a) (b)Correction for mobilities: 1.46 magnitude lowerCorrected mobility = 0.035 x current mobilityCorrection for mobilities: 2.46 magnitude lowerCorrected mobility = 0.0035 x current mobility ‒1

Figure 3.17. (a) Comparison of the composition amplitude between experimental data and

phase field simulated results. The composition amplitude evaluated from phase field simula‐

tion is by employing the ADS method. The open symbols are the original data reported in the

previous experiments using the LBM method, the solid symbols are shifted from the original

data by 18 at.% Cr, according to the ADS method performed in this work discussed in the

last section, see Table 3.1. (b) Wavelength of the spinodal structure analyzed from experi‐

mental data and phase field simulation, The ones from simulation is shown in the dimension‐

less time. Comparison implies the wavelength can be easily fitted by simulation due to the

invariant tendency of the experimental wavelength.


50

By considering the adopted thermodynamic modeling of the Fe‐Cr system (paper II),

the critical wavelength is estimated to be 0.98 nm, and much shorter than the experi‐

mental data as given in Fig. 3.17. According to Eq. (3‐14), one can realize that the wave‐

length also contributes to the variation of the composition amplitude. If the wavelength

reaches a value which is large enough and comparable with the experimental ones, the

development of the spinodal in the simulation will also slow down. Therefore, in order

to obtain a reasonable fit between phase field simulation and experiments, the first step

is to assure that the wavelength measured in the experiments can be well reproduced

by the phase field simulation. Afterwards, the atomic mobility can be estimated in or‐

der to describe the correct tendency of the variation in composition amplitude. It is

noteworthy that the ADS method proposed in this thesis becomes an important tool in

order to estimate the magnitude of the atomic mobility.

According to the estimation shown in Eqs. (2‐8) and (3‐15), to enlarge the wavelength in

the phase field simulation, the gradient energy coefficient can be adjusted. One shall

keep in mind that the relation between the gradient energy coefficient 2e and the inter‐action parameter W is based on the regular solution model of the solution phase. There‐

fore, the magnetic effect and interactions beyond the nearest neighbors are not taken

into account. In fact, some tentative investigation was performed in the work of [113] by

considering the magnetic effects on the gradient energy coefficient. Besides, according

to the thermodynamic modeling in paper I, it is found that the second and third nearest

neighbor interactions should also be considered, since the thermodynamic parameters

of the Fe‐Cr system achieved in this thesis indicates that a sub‐sub regular solution

model is suitable to describe the bcc phase. Presently, a makeshift is adopted by directly

adjusting the gradient energy coefficient in order to fit the experimental wavelength.

Before estimating the atomic mobility needed for the simulation, it is better to consider

the dimensionless relationship among time (t and τ), length scale (lC), temperature (T)

and atomic mobility scale (MScale) as shown below, because the current phase field simu‐

lation are numerically solved in the dimensionless form.

According to the dimensionless relation:

2C

Scale

l

t M RT

t= (3‐16)

It is obvious that, at constant temperature, the time scale is only related to MScale and lC.

In other words, if one could estimate the time scale needed for fitting the composition

amplitude, a reasonable scale of the atomic mobility can be estimated. In the beginning

of the phase field simulation, the critical length scale, lC, is normally determined first to


51

capture the characteristic length of the physical picture. In the present work, since the

wavelength of spinodal decomposition in the Fe‐Cr alloys is in the nano‐scale, lC is set

to be 0.22 nm.

In the phase field simulation of the thesis, it is found that in order to get a proper fit to

the experimentally determined wavelength of the spinodal structure in the Fe‐Cr alloys

at 773 K, the gradient energy coefficient should be five times larger than the one esti‐

mated using the relation in Eq. (3‐15). The comparison of the wavelength between

phase field simulation and experimental data is shown in Fig. 3.17. Apparently, the var‐

iation of both experimental and simulated wavelength can be easily fitted, since there is

no significant increase of wavelength with time neither in simulation nor experiments.

It should be noted that the wavelength determined from the work by Brenner et al. [114]

agree well with the experimental observation performed in this work. Therefore, it has

been used in the present comparison, and the ones from Cerezo et al. [115] and Hyde et

al. [44‐47] were excluded due to the large discrepancy mutually. It is interesting to find

that all of the experimental reports summarized in Fig. 3.17 are all from the same re‐

search group, and the data reported in different years are scattered. Therefore, further

experimental data on Fe‐45 at.% Cr is urgently needed in order to confirm the previous

experimental results from the same group.

Phase Field Simulation(Cahn-Hilliard theory)

Thermodynamic database(TCFE6/2011Xiong)

Kinetic database(MOB2)

Fourier-spectral method

1. Fast development ofcomposition amplitude

2. Small wave length

Comparison between

simulation and experiment

FORTRAN code WEIPHAM

Fundamental Basis

Increasing gradientenergy coefficient

Solution in this work

Scaling atomic mobility

Fig. 3.18. Flow chart of the phase field study on the spinodal decomposition in this work.

After achieving a good agreement on the wavelength, the composition amplitude can

be studied further in order to estimate the atomic mobilities. It should be noted that,

due to the low temperature and complexity of the magnetism, the atomic mobility of Cr

and Fe in the Fe‐Cr binary is difficult to obtain from both experiments and ab initio cal‐

culations. According to different experiments compiled in Fig. 3.17, the composition

amplitude of the spinodal structure for the Fe‐45 at.% Cr shows a tendency to increase


52

from 7 to 30 at.% Cr (average value) despite that there are notable discrepancies among

different experiments.

It should be noted that the experimental composition amplitudes were all estimated

based on the LBM method. As discussed in paper VI as well as the last section in this

thesis, the LBM method may underestimate the composition amplitude. In fact, it is

almost impossible to estimate the atomic mobility without knowing the composition

amplitude. As a compromise, the difference between the LBM and ADS methods on the

analyzed composition amplitude of the Fe‐37.76 at.% Cr alloy is considered and further

applied on the present case with 45 at.% Cr. As mentioned in the previous section, for

the annealed alloy with 33.1 at.% Cr, the difference of the estimated composition ampli‐

tude from the LBM and ADS methods are 15.8, 16.8 and 18.2 at.% for 0, 10 and 100

hours, respectively. Therefore, it is assumed that the reported experimental composi‐

tion amplitude by using the LBM method may also be shifted upwards for 17 at.% Cr.

Accordingly, the magnitude of the atomic mobility adjustment can be estimated by

comparing the phase field simulation and corrected experimental composition ampli‐

tudes shown in Fig. 3.17. According to the relation showed in Eq.(3‐16), the present

atomic mobility of Cr and Fe in the bcc phase at 773 K is suggested to be 0.035 of the

current mobilities obtained from the work by Jönsson [111] (i.e. the one from the MOB2

kinetic database in the Thermo‐Calc databank). Whereas if the original data reported

from the previous experiments shown in Fig. 3.17 considered to be the accurate ones,

the mobilities at 773 K can be a factor of 0.0035 of the present ones from Jönsson [111],

which will generate the fit shown in Fig.3.17(a) with blue color.

As a summary, the strategy in this work is to obtain a reasonable value for the gradient

energy coefficient, which can be estimated phenomenologically according to the exper‐

imental observation. As shown in Fig. 3.18, in the present work, the gradient energy

coefficient was adjusted so as to increase the critical wavelength which can be estimated

according to Eq. (3‐15). Afterward, the comparison of the variation in composition am‐

plitude between simulation and experimental data will be performed in order to esti‐

mate the scale of the atomic mobility.

Chapter 4 Concluding remarks and outlook

4.1. Concluding remarks

This thesis concerns fundamental issues in the phase equilibria and phase transfor‐

mation of the Fe‐Cr‐Ni system focusing on the low temperature range. In paper I, a

comprehensive literature review was carried out for the Fe‐Cr core binary. It reveals

that the thermodynamic description of the Fe‐Cr system needs to be improved further.

In addition, the melting temperature of pure Cr is corrected to be 2136 K, which is 44 K

lower than the one adopted in the SGTE database [49].

Based on paper I, CALPHAD modeling was performed on the Fe‐Cr binary down to 0

K in paper II to obtain more accurate description of the spinodal decomposition. Model‐

predicted results were confirmed by some critical experiments, such as, APT and DSC.

Thermodynamic modeling in this thesis demonstrates the rationality of non‐solubility

of Cr in bcc Fe at the ground state, and predicts anomalous variation of the magnetic

ordering energy rather than enthalpy of formation calculated from atomic modeling.

In paper III, the low temperature equilibria of the Fe‐Ni system was reassessed by im‐

proving the description of the magnetic phase diagram of the Fe‐Ni alloys using the

current magnetic model for CALPHAD, i.e. the IHJ model with the AFM Weiss factor.

Significant improvements were made and validated by the experimental Ms tempera‐

tures. However, some problems of the current magnetic model used in the CALPHAD

approach were found, and initiate the needs of improving the magnetic model itself.

Consequently, in paper IV, the current magnetic model of the CALPHAD approach is

improved. The AFM factor adopted in the current magnetic model is abandoned. In‐

stead, dual magnetic states (FM and AFM) for each phase exhibiting magnetic ordering

were introduced to extend the Inden‐Hillert‐Jarl model to the AFM ordering transition.

The concept of effective magnetic moment is considered as the input of the improved

magnetic model for representing the magnetic entropy more accurately. The applicabil‐

ity of the improved magnetic model is demonstrated in the case study of the Al‐Cr‐Fe

CHAPTER 4. CONCLUDING REMARKS AND OUTLOOK

54

system. Another case study on the Cr‐Ni magnetic phase diagram was performed in

section 3.2 in the thesis.

The phase field modeling of the Fe‐Cr binary alloys was performed based on the updat‐

ed thermodynamic description as presented in paper V. The phase field simulation

demonstrates a much faster microstructural evolution of the spinodal structure in the

Fe‐Cr alloys, which calls for a modification of the atomic mobility at low temperatures.

In paper VI, a new method called ADS is proposed for evaluating the composition am‐

plitude from both discrete and continuum dataset by APT and phase field modeling,

respectively. The method is sensitive to determine the small composition fluctuation,

which can facilitate the study of the spinodal decomposition in the early stages.

It is worth to mention that, in section 3.3.3, a detailed discussion on how to estimate the

atomic mobility at low temperatures by phase field method is performed. The effective

ADS method plays an important role to estimate the magnitude of the atomic mobility

at low temperatures where both atomistic modeling and experimental data are unavail‐

able due to the slow kinetics and complex magnetic effects. It turns out that the combi‐

nation of phase field and APT through the ADS method is a good way to study low

temperature kinetics of the spinodal decomposition.

The improved magnetic model and the ADS method is a basis for further investigation

of the thermodynamics and kinetics in the Fe‐Cr‐Ni alloys. Although a complete ther‐

modynamic description is retarded by many fundamental aspects at the moment, the

thermodynamic description of the Fe‐Cr system in this work is a template for future

studies in computational thermodynamics.

4.2. Suggestions on future work

In order to avoid the spinodal decomposition in the stainless steels, the most effective

way is to choose the composition outside the spinodal region. Therefore, the phase

equilibria is the key to the successful design of the materials. However, according to the

review in section 3.1.2 in the thesis, the phase equilibria of the Fe‐Cr‐Ni system are still

uncertain at low temperatures. The present thermodynamic description from reported

work or commercial databases also needs to be improved, since the previous focus was

on the region above 1073 K.

One shall keep in mind that the thermodynamic modeling determines the spinodal de‐

composition region, which eventually affects the results of kinetic investigation using

phase field method.

CHAPTER 4. CONCLUDING REMARKS AND OUTLOOK

55

In order to perform a reliable thermodynamic modeling of the Fe‐Cr‐Ni system at low

temperatures, two essential aspects need to be considered. Firstly, the lattice stability of

the pure elements needs to be reassessed analogous to the work performed by Chen

and Sundman for pure Fe [15] using the Einstein model. Secondly, the magnetic phase

diagram of the ternary Fe‐Cr‐Ni alloys should be considered carefully by reviewing the

related experimental data. Atomistic modeling, like ab initio calculations, can provide

valuable data for the related study, especially for the unstable or metastable phases.

Regarding the kinetic aspect, the quantitative study of the phase behavior with long

time annealing at low temperatures is the primary concern. However, the efforts should

be devoted to the remaining issues of the Fe‐Cr alloys. As indicated in Fig. 3.17, exper‐

imental data reported from the same group for Fe‐45 at.% Cr alloys needs to be further

confirmed. This will ultimately leads to an accurate prediction of the low temperature

atomic mobility of the Fe‐Cr binary alloys. Afterwards, similar studies can be applied to

the ternary Fe‐Cr‐Ni alloys.

Regarding the phase field simulation of the spinodal decomposition, further improve‐

ment is needed for including the contribution from more interactions, like the 2nd near‐

est interaction, to estimate the gradient energy coefficient. In addition, the magnetic

effects may have considerable contributions to the gradient energy term, which needs

to be formulated and included in the phase field models.

It is believed that the remaining work will concentrate more on the application of the

model and methods proposed in this thesis, and will extend the case studies to multi‐

component systems. Although it may be time‐consuming to perform a compact re‐

search of the multicomponent system by using all of the new models or methods pre‐

sented in this thesis, it will establish a solid base for the low temperature thermody‐

namics and kinetics.

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Thermodynamic and Kinetic Investigation of the Fe-Cr - DiVA Portal