thermodynamic and kinetic investigation of systems related to...
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Thermodynamic and kinetic investigation
of systems related to lightweight steels
Weisen Zheng
Doctoral Thesis in
Materials Science and Engineering
Stockholm, Sweden, 2018
Weisen Zheng
Thermodynamic and kinetic investigation of systems related to lightweight
steels
KTH Royal Institute of Technology
School of Industrial Engineering and Management
Department of Materials Science and Engineering
SE-100 44 Stockholm, Sweden
ISBN 978-91-7729-840-3
Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i
Stockholm framlägges till offentlig granskning för avläggande av Teknologie
Doktorsexamen den 21 september 2018 kl 10.00 i rum 4301, Brinellvägen 8,
Kungliga Tekniska Högskolan, Stockholm
© Weisen Zheng (郑伟森), 2018
This thesis is available in electronic version at kth.diva-portal.org
Printed by Universitetsservice US-AB, Stockholm, Sweden
i
Abstract Lightweight steels have attracted considerable interest for automobile
applications due to the weight reduction without loss of high strength and with
retained excellent plasticity. In austenitic Fe-Mn-Al-C steels, the nano-
precipitation of the κ-carbide within the austenitic matrix significantly
contributes to the increase in yield strength. In the present work, the precipitation
strengthening simulation has been carried out within the framework of the ICME
approach. Thermodynamic assessments of the quaternary Fe-Mn-Al-C system as
well as its sub-ternary systems were performed with the CALPHAD method. All
available information on phase equilibria and thermochemical properties were
critically evaluated and used to optimize the thermodynamic model parameters.
By means of the partitioning model, the κ-carbide was described using a five-
sublattice model (four substitutional and one interstitial sublattice), which can
reflect the ordering between metallic elements and reproduce the wide
homogeneity range of the κ-carbide. Based on the present thermodynamic
description, a thermodynamic database for lightweight steels was created. Using
the database, the phase equilibria evolution in lightweight steels can be
satisfactorily predicted, as well as the partition of alloying elements. In order to
accelerate the development of a kinetic database for multicomponent systems, a
high-throughput optimization method was adopted to optimize the diffusion
mobilities. This method may largely reduce the necessary diffusion-couple
experiments in multicomponent systems. Based on the developed
thermodynamic and kinetic databases for lightweight steels, the precipitation of
the κ-carbide was simulated using TC-PRISMA. The volume fraction and
particle size were reasonably reproduced. Finally, the precipitation strengthening
contribution to the yield strength was predicted. The calculation results show that
ii
the anti-phase boundary effect is predominant in the precipitation strengthening.
Overall, the relationship between the composition, processing parameters,
microstructure and mechanical properties are established in the thesis.
Keywords: Fe-Mn-Al-C, CALPHAD, MGI, ICME, Precipitation strengthening,
Lightweight steels.
iii
Sammanfattning
Lättviktsstål har rönt stor uppmärksamhet som material för användning i bilar på
grund av den viktminskning som erhålls utan förlust av den höga hållfastheten
och med bibehållen utmärkt plasticitet. I austenitiska Fe-Mn-Al-C stål, bidrar
nano-utskiljningen av κ-karbid i den austenitiska grundmassan väsentligt till
ökningen av sträckgränsen. I det nuvarande arbetet har simuleringen av
härdningsbidraget från utskiljningarna utförts inom ramen för ICME metoden.
Termodynamiska utvärderingar av det kvartenära Fe-Mn-Al-C systemet och
dess undersystem utfördes med CALPHAD-metoden. All tillgänglig information
om fasjämvikter och termokemiska egenskaper utvärderades kritiskt och
användes för att optimera de termodynamiska modellparametrarna. Med hjälp av
den s.k. partitioneringsmodellen beskrevs κ-karbiden med användning av en
fem-subgittermodell (fyra substitutionella och ett interstitiellt subgitter) som kan
återspegla ordningen mellan de metalliska elementen och även reproducera det
breda homogenitetsintervallet för κ-karbiden. Baserat på den nuvarande
termodynamiska beskrivningen har en termodynamisk databas för lättviktiga stål
skapats. Med hjälp av databasen kan fasjämviktssutvecklingen i lättviktsstål
förutses på ett tillfredsställande sätt, såväl som partitioneringen av
legeringselementen. För att påskynda utvecklingen av en kinetisk databas för
multikomponentsystem, har en ”high-throughput”-optimeringsmetod använts
för att optimera diffusionsmobiliteterna. Denna metod kan i stor utsträckning
minska antalet nödvändiga diffusionsparsprover i multi-komponentsystem.
Baserat på de utvecklade termodynamiska och kinetiska databaserna för
lättviktsstål, simulerades utskiljningen av κ-karbid med hjälp av TC-PRISMA.
Volymfraktionen och partikelstorleken reproducerades rimligt. Slutligen kunde
även härdningsbidraget till sträckgränsen uppskattas. Beräkningsresultaten visar
iv
att antifasgränseffekten dominerar vid utskiljninghärdningen. Slutligen kan
konstateras att förhållandet mellan sammansättning, processparametrar,
mikrostruktur och mekaniska egenskaper fastställts i avhandlingen.
v
Appended papers
The present thesis is based on the following appended papers:
I. Thermodynamic assessment of the Al-C-Fe system
Weisen Zheng, Shuang He, Malin Selleby, Yanlin He, Lin Li, Xiao-
Gang Lu, John Ågren
CALPHAD, 2017, vol. 58, pp. 34-49.
II. Thermodynamic modeling of the Al-C-Mn system supported by
ab initio calculations
Weisen Zheng, Xiao-Gang Lu, Huahai Mao, Yanlin He, Malin
Selleby, Lin Li, John Ågren
CALPHAD, 2018, vol. 60, pp. 222-230.
III. Thermodynamic investigation of the Al-Fe-Mn system over the
whole composition and wide temperature ranges
Weisen Zheng, Huahai Mao, Xiao-Gang Lu, Yanlin He, Lin Li,
Malin Selleby, John Ågren
Journal of Alloys and Compounds, 2018, vol. 742, pp. 1046-1057.
vi
IV. Computational materials design for lightweight steels with ICME
approach: thermodynamics and precipitation strengthening
simulation
Weisen Zheng, Huahai Mao, Malin Selleby, John Ågren
In manuscript.
V. Experimental investigation and computer simulation of diffusion
in Fe-Mo and Fe-Mn-Mo alloys with different optimization
methods
Weisen Zheng, John Ågren, Xiao-Gang Lu, Yanlin He, Lin Li
Metallurgical and Materials Transactions A, 2017, vol. 48, pp. 536-
550.
Author contributions:
I. Literature survey, thermodynamic modelling and optimization, the
draft manuscript
II. Literature survey, thermodynamic modelling and optimization, the
draft manuscript
III. Literature survey, thermodynamic modelling and optimization, the
draft manuscript
IV. Literature survey, thermodynamic modelling and optimization,
precipitation strengthening simulation, the draft manuscript
V. Literature survey, mobility modelling and optimization, all the
experimental work, the draft manuscript
vii
Contents Appended papers ...................................................................... v
Contents ................................................................................... vii
Chapter 1 ................................................................................... 1
Introduction ............................................................................... 1
1.1 Lightweight steels ............................................................................ 1
1.2 CALPHAD ....................................................................................... 6
1.3 Scope of this thesis ........................................................................... 8
Chapter 2 ................................................................................. 11
CALPHAD modelling ............................................................. 11
2.1 Thermodynamic modelling ............................................................ 11
2.1.1 Compound energy formalism (CEF) ................................................. 12
2.1.2 Substitutional solution model ............................................................ 13
2.1.3 Partitioning model (order-disorder model) ....................................... 14
2.1.4 κ-carbide ............................................................................................ 17
2.1.5 Intermetallic compounds ................................................................... 19
2.2 Mobility modelling ........................................................................ 20
Chapter 3 ................................................................................. 23
viii
Thermodynamic investigation of the Fe-Mn-Al-C system .. 23
3.1 Binary systems ............................................................................... 25
3.1.1 Fe-Mn ................................................................................................ 25
3.1.2 Fe-Al ................................................................................................. 27
3.1.3 Fe-C ................................................................................................... 28
3.1.4 Mn-Al ................................................................................................ 29
3.1.5 Mn-C ................................................................................................. 31
3.1.6 Al-C ................................................................................................... 32
3.2 Ternary systems ............................................................................. 33
3.2.1 Fe-Mn-Al ........................................................................................... 33
3.2.2 Fe-Mn-C ............................................................................................ 35
3.2.3 Fe-Al-C ............................................................................................. 36
3.2.4 Mn-Al-C ............................................................................................ 38
3.3 Quaternary system ......................................................................... 39
Chapter 4 ................................................................................. 41
Kinetic investigation of the Fe-Mn-Mo system .................... 41
4.1 Diffusion couple experiments ........................................................ 42
4.2 Mobility optimization method ....................................................... 43
4.2.1 Traditional method ............................................................................ 43
4.2.2 Direct method .................................................................................... 46
Chapter 5 ................................................................................. 49
Application of Thermo-Kinetic database .............................. 49
5.1 Partition of alloying elements ........................................................ 49
ix
5.2 Precipitation of the κ-carbide ......................................................... 51
5.3 Precipitation strengthening simulation .......................................... 52
Chapter 6 ................................................................................. 57
Concluding remarks and future work .................................. 57
6.1 Concluding remarks ....................................................................... 57
6.2 Future work .................................................................................... 58
Acknowledgements ................................................................. 59
Bibliography ............................................................................ 61
x
1
Chapter 1
Introduction
1.1 Lightweight steels
Traditionally, automakers have regarded steels as the main material for autobody
panels, having given ideal mechanical properties and safety at a low cost.
However, the increasingly stringent regulations of energy consumption and CO2
emission have compelled automotive manufacturers to look for effective
methods to reduce vehicle weight in order to increase fuel efficiency. Therefore,
steels are facing a big challenge, partly from light alloys such as aluminium or
magnesium alloys and carbon fibre composites, and partly from the requirement
of excellent mechanical properties. In general, lightweight design of steels can
be achieved by enhancing their strength and reducing their density. The high-
strengthening of steels has become the prevailing trend of lightweight design
research by steel suppliers within the past decades. For instance, the 1st, 2nd and
3rd generation of advanced high strength steels (AHSS) have been developed to
obtain an excellent combination of strength and ductility. Recently, an
alternative method of using low density alloying elements in steels such as
aluminium and silicon, is attracting increasing attention in the automotive steel
field. Such steels are called “lightweight steel” or “low-density steel”. In the
2000s, the group of Frommeyer has worked on lightweight steel, providing a
good understanding of deformation mechanisms and mechanical properties [1,2],
which makes material researchers see the enormous potential of lightweight
steels for automotive applications [3–7]. Frommeyer and Brüx [2] developed a
CHAPTER 1
2
kind of multiphase lightweight steel consisting of a matrix austenite phase, ferrite
and nano-size κ-carbide. The composition range of the steels investigated was
18-28 Mn, 9-12 Al, 0.7-1.2 C (wt%). In their study, the Fe-28Mn-12Al-1C steel
exhibits a yield strength of 730 MPa, an ultimate tensile strength of 1000 MPa
and total elongation of around 55 % at room temperature. Moreover, the addition
of 12 wt% Al reduces the density of the steel to about 6.5 g/cm3. Since then, the
research on Fe-Mn-Al-C lightweight steels has become a hotspot in automotive
industry due to the high specific strength and excellent ductility.
Important factors in the design of lightweight steels are the low density derived
from the lower atomic masses of light elements and the effect of light elements
on the lattice parameter of the steel. Fig. 1.1 shows the effect of common alloying
elements on the density of α-Fe (bcc) and γ-Fe (fcc) at room temperature using
the volume data from Thermo-Calc Software TCFE9 Steels/Fe-alloys database
[8]. Among the metallic elements, both the density reduction of α-Fe and γ-Fe
caused by Al is largest, followed by Si, which is the reason why Si is sometimes
added in lightweight steel design. Mn and Cr slightly lower the density, while Ni
and Cu slightly increase the density. For the micro-alloying elements, Ti and V
have a positive effect on the density reduction, but the density with the addition
of Nb and Mo exhibits an increasing trend. Although some alloying elements
play a harmful role in reducing density, they are used in lightweight steels. For
instance, Kim et al. [9] added 5 wt% Ni into the Fe-Mn-Al-C alloy and obtained
a high specific strength which is comparable to titanium alloys. They indicated
that the addition of Ni improves the stability of finely dispersed nano-size B2
phase and thereby the non-shearable nature of the ordered B2 phase enhances the
work hardening effect, resulting in a high specific strength.
CHAPTER 1
3
Fig. 1.1. Effect of alloying elements on the density of (a) α-Fe (bcc) and (b) γ-Fe (fcc) at room temperature calculated using Thermo-Calc Software TCFE9 Steels/Fe-alloys
database [8].
Lightweight steels can form a variety of microstructures due to many processing
variants such as hot and cold rolling, solution and annealing treatment, and
coiling. The matrix phase of lightweight steels can be single ferrite or austenite,
or duplex ferrite + austenite, which strongly depends on the contents of the
alloying elements such as Mn, Al and C, and the heat treatment temperature. In
general, austenitic steels contain higher contents of alloying elements, for
example, Mn, Al and C up to 30, 12 and 2 (wt%) [2,10–19], respectively.
Because of this, the κ-carbide, β-Mn, ordered B2 or even D03 phases based on
the bcc lattice can precipitate in the austenite based alloys at low temperatures
[20–31], which makes the physical metallurgy complex. Fortunately, many
researchers have already performed experimental observation of the phase
transformation by means of optical microscopy (OM), scanning electron
microscopy (SEM), X-ray diffraction (XRD), transmission electron microscopy
(TEM) and atom probe tomography (APT).
In general, the κ-carbide is precipitated in the austenitic Fe-Mn-Al-C alloys aged
at temperatures between 773 and 1173 K. According to the precipitation sites,
the κ-carbide is classified into two types: intragranular and intergranular. The
intragranular κ-carbide homogeneously forms within the austenite matrix
[20,21,26]. Cheng et al. [26] studied the sequence of the phase transformations
CHAPTER 1
4
of the Fe-17.9Mn-7.1Al-0.85C (wt%) alloy through annealing at temperatures
ranging from 873 to 1173 K. In their work the transformation from the L12 phase
to the κ-carbide was observed for the first time. The formation sequence of the
κ-carbide is as follows:
γ→ γʹ+ γʹʹ→ γʹ+L12→ γʹ+κ.
The transformation starts with the spinodal decomposition from the high
temperature austenite to two low temperature austenitic phases, i.e. solute-lean
and solute-rich. Upon cooling, the solute-rich austenite transforms into the L12
phase through an ordering reaction between the metallic elements. Finally the
ordered L12 phase transforms into the κ-carbide by the ordering of the carbon
atoms. This type of the κ-carbide is coherent with the austenite matrix. The lattice
misfit between the κ-carbide and the matrix is very small [32]. Moreover, the
coherent κ particle has a significant impact on the movement of dislocations
during the deformation. The strong dislocation-κ interactions can result in an
effective improvement of the strength of lightweight steels [16,17,33].
On the other hand, the intergranular κ-carbide generally precipitates along the
austenite grain boundaries after prolonged annealing at higher temperatures.
There are three types of reactions that can form the intergranular κ-carbide:
precipitation reaction, eutectic reaction and cellular transformation, depending
on the contents of alloying elements and the annealing temperature [34,35]. This
kind of the κ-carbide has a detrimental influence on the strength and ductility.
Therefore, in order to obtain the excellent mechanical properties, the
intergranular κ-carbide should be avoided through controlling the steel
compositions, annealing temperature and holding time.
In the case of β-Mn, its formation needs a much longer aging time through the
reaction γ→κ+β-Mn [24,36]. Chao et al. [24] investigated the precipitate along
grain boundaries in the Fe-31.7Mn-7.8Al-0.54C (wt%) alloy by means of TEM.
The cold-rolled samples were solution heat-treated at 1323 K for 2 hours and
aged at 823 K for various times. The grain boundary precipitates are formed in
the samples aged for longer than 16 hours. Moreover, only the κ-carbide was
CHAPTER 1
5
precipitated after aging for less than 72 hours. For longer times than 72 hours,
the β-Mn phase was observed surrounding the κ-carbide along the grain
boundaries. By comparing with their previous study of the Fe-31.5Mn-8.0Al-
1.05C (wt%) alloy [23], Chao et al. [24] revealed that the precipitation of the κ-
carbide induces manganese enrichment of the surrounding austenite phase and
thereby the β-Mn phase forms from the austenite phase enriched manganese. The
brittle β-Mn phase at the grain boundaries is detrimental to the ductility of
lightweight steels, it is therefore necessary to avoid its formation.
With the addition of alloying elements such as Ni, Cu and Si into the Fe-Mn-Al-
C alloys, the ordered B2 and D03 phases based on the bcc lattice can be formed
[9,28,30,31]. Ni and Cu can promote the stability of the B2 phase, while Si can
catalyze the formation of the D03 phase. Kim et al. [9] found that the B2 phase
has three different morphologies: stringer bands in the austenitic matrix (type 1),
fine particles with the size of 200-1000 nm (type 2) and finer particles with the
size of 50-300 nm (type 3). The type 2 particles were formed along the grain
boundaries of the recrystallized austenite, while the type 3 particles precipitated
at the shear bands in the non-recrystallized austenite. For the D03 phase, its
formation mechanism is not yet as clear as for the B2 phase.
Recently, since lightweight steels exhibit huge potential to achieve high specific
strength and ductility, many researchers have attempted to further improve the
specific mechanical properties by adding some other alloying elements than Mn,
Al and C. Consequently, a few studies have been performed to explore the effect
of the alloying elements such as Ni, Cr and Mo on the microstructural evolution
and mechanical properties [9,27,28,37–41]. For example, as mentioned before,
the addition of Ni and Cu enhances the stability of the B2 phase and then by
controlling the nano-precipitation of the B2 phase, an excellent property
combination of specific strength and ductility can be obtained [9,27,28,37].
Bartlett et al. [38] indicated that Si accelerates the formation kinetics of the κ-
carbide, but does not increase the amount of the κ-carbide. Sutou et al. [39]
revealed that Cr inhibits the precipitation of the κ-carbide and reduces the
CHAPTER 1
6
strength. Moon et al. [40] investigated the influence of Mo on the precipitation
of the κ-carbide by means of TEM, APT and first principle calculations. They
found that the addition of Mo delays the precipitation of the κ-carbide, leading
to a change of aging hardening behavior. Overall, the utilization of the
appropriate alloying elements can optimize the microstructures and then enhance
the mechanical properties.
1.2 CALPHAD
The Materials Genome Initiative (MGI) has drawn extensive attention of
materials engineers since the US former Present Barack Obama proposed MGI
in the announcement of the Advanced Manufacturing Partnership in June 2011.
He distinctly pointed out that the goal of MGI is to discover, develop,
manufacture and deploy advanced materials twice as fast, which implies a great
shift from the trial-and-error route to the rational materials design approach. The
specific objectives of MGI are the following: 1) to find the intrinsic basic factor
that can determine the external phenomenon and 2) to find the relationships
between material composition/processing-microstructure-property, which also is
the ultimate goal of the Integrated Computational Materials Engineering (ICME)
approach. Unfortunately, the distinct definition of the materials genome was not
given. Subsequently, Kaufman and Ågren [42] suggested that the materials
genome should be a set of information in a form of CALPHAD (CALculation of
PHAse Diagrams) thermodynamic and kinetic databases that can be used to
predict a material’s structure and its responses to processing and usage
conditions. In their paper, the authors reviewed the development of CALPHAD
and indicated that CALPHAD is consistent with all the features of the materials
genome. For instance, a CALPHAD thermodynamic database can calculate the
equilibrium state of a material and a CALPHAD kinetic database can predict the
material’s response to processing and usage condition. Therefore, the
CALPHAD databases are major parts of MGI.
CHAPTER 1
7
CALPHAD databases are established with the CALPHAD method in which each
phase in a materials system is modelled and then the model parameters are
adjusted to reproduce experimental or ab initio data. As a matter of fact, in its
early years the CALPHAD method only handled thermodynamic problems. The
theoretical basis for CALPHAD is thermodynamics of materials. At the moment,
only the phase diagram and thermochemical properties such as enthalpy of
mixing, chemical potential and activity could be calculated. Later, Ågren [43]
introduced the simulation of the non-equilibrium phenomenon such as diffusion-
controlled phase transformation, which is a critically important improvement of
the CALPHAD scheme. Since then, the diffusion behavior of materials systems
such as various diffusion coefficients and diffusion profiles can be simulated
combining thermodynamic and kinetic databases. Nowadays, the CALPHAD
method has expanded to describe thermophysical properties such as elastic
moduli, viscosity and thermal expansion. There is no doubt that the physical
property database is necessary for the calculations, but which is beyond the scope
of this thesis work.
To develop the CALPHAD thermodynamic and kinetic databases, the following
procedure is performed in general, as shown in Fig. 1.2. It starts with the
collection of all experimental and theoretical information, and estimated data on
the system studied. In the case of thermodynamic assessment, for instance, the
data used for the optimization include phase equilibrium data, solubilities,
thermochemical quantities (enthalpy, activity, etc.) and crystallographic
information from experimental measurements, first principle calculations or
empirical estimations. Then a critical evaluation of all available data should be
performed to eliminate the unreliable and contradictory data, and to identify the
phases present in the system. After that, a reasonable model for each phase is
built according to the crystal structure, site occupation, order-disorder transition
or magnetic properties, which is extremely important for the final achievement
of the self-consistent model parameters. By using the least-squares method, the
model parameters are adjusted to fit the reliable experimental data. By
comprehensively comparing the calculation results with all reliable experimental
CHAPTER 1
8
data, it can be found that whether a set of self-consistent model parameters is
obtained or not. If not, the assessment has to be performed again since the
weights of some experimental data may be needed to be adjusted or even a more
advanced model for a phase may be used. Finally, when a satisfactory agreement
with reliable experimental data is achieved, the model parameters are compiled
to a database. This procedure is known as the CALPHAD assessment or
optimization.
Fig. 1.2. CALPHAD assessment procedure.
1.3 Scope of this thesis
Due to the weight reduction and excellent specific strength and ductility, the
lightweight steels have arouse great interests in the automotive industry. In order
to optimize their microstructure, such as catalysing the nano-precipitation of the
coherent κ-carbide and ordered B2 phase and avoiding the formation of the brittle
β-Mn phase, and then the mechanical properties, a detailed understanding of
thermodynamic properties and phase equilibria of the lightweight steel system is
fundamental. Moreover, the phase transformation in lightweight steels is
generally a diffusion-controlled process, which significantly depends on
diffusivity. Thus, a fundamental knowledge of diffusivity is indispensable to
CHAPTER 1
9
understand phase evolution, optimize processing parameters and design a novel
lightweight steel. However, the assessment of diffusion mobilities in a
multicomponent system requires a large number of diffusion couple experiments
with the traditional method, which largely consumes much time and cost.
Therefore, it is extremely important to develop a ‘materials genome’ database
with a high-throughput method. In this thesis, the specific objectives are as
follows: Ⅰ) to develop and refine the thermodynamic database of the Fe-Mn-Al-
C system, in particular the descriptions of the Al-containing systems; Ⅱ) to
improve the mobility optimization method with a case study of the Fe-Mn-Mo
system and Ⅲ) to predict the precipitation strengthening due to the nano-
precipitation of the κ-carbide based on the thermo-kinetic database.
Firstly, the assessment work of the Fe-Al-C system was performed in the
beginning of the thesis. This ternary is the most basic constitute subsystem of the
lightweight steel system due to the addition of Al into Fe-based alloys. The study
of this ternary system mainly focuses on the phase equilibria between the fcc,
bcc and κ-carbide, because these three phases strongly impact the microstructure
and then the mechanical properties of lightweight steels.
Then, the Mn-Al-C system was assessed within the framework of CALPHAD
owing to the new experimental data. The wide homogeneity ranges of the κ-
carbide, fcc and hcp phases are satisfactorily reproduced, which provides a
foundation for the construction of the thermodynamic database for Fe-Mn-Al-C
lightweight steels.
Thirdly, the Fe-Mn-Al metallic system was re-optimized considering the
experimental information over the entire composition range from room
temperature to well above the melting temperature. Both the new ternary phases
and the extension of the binary intermetallic compounds towards the Fe-Mn-Al
ternary system are well described. Meanwhile, the phase equilibria between the
fcc, bcc and β-Mn phases are reproduced, ensuring a reliable prediction of the β-
Mn formation in high-Mn lightweight steels.
CHAPTER 1
10
Based on the reliable descriptions of the ternary systems, the thermodynamic
investigation of the quaternary Fe-Mn-Al-C system was carried out using all
available experimental information. This part of the work aims to refine the
thermodynamic database for lightweight steels and improve the prediction
accuracy of phase equilibria in lightweight steels, in particular the precipitation
of the κ-carbide.
In order to speed up the construction of the kinetic database (i.e. mobility
database), a high-throughput method, i.e. the direct optimization method, was
improved and employed to optimize the diffusion mobility in fcc and bcc Fe-
Mn-Mo alloys. Compared with the traditional optimization method, the direct
method fits mobility parameters to the diffusion profiles without the extraction
of the interdiffusion coefficients. It may be helpful for the investigation of
diffusion mobilities in multicomponent alloys.
Finally, based on the thermo-kinetic database for lightweight steels, the phase
evolution, the partition of alloying elements and the precipitation of the κ-carbide
were simulated using the Thermo-Calc software including the precipitation
module (TC-PRISMA). According to the simulated volume fraction and particle
size of the κ-carbide with TC-PRISMA, the precipitation strengthening
contribution to the yield strength was predicted. Thus, the reasonable
quantitative relationship between composition/processing-microstructure-
property was preliminarily established for lightweight steels, which significantly
contributes to the development of lightweight steels.
11
Chapter 2
CALPHAD modelling
As mentioned in the preceding chapter, during the CALPHAD assessment, a
reasonable and suitable model for each phase is a necessary precondition for a
self-consistent description of the entire system, which is more important in a
multicomponent system since the model should be compatible within all the
lower order systems. In this chapter, the thermodynamic and kinetic modelling
are described in detail, respectively.
2.1 Thermodynamic modelling
The thermodynamic modelling is built based on the Gibbs free energy with
temperature, pressure and composition as the natural variables. It should be noted
that all the calculations presented in this thesis are performed at atmosphere
pressure. Therefore, the contribution to the Gibbs energy due to the pressure is
ignored in this thesis. The total Gibbs energy of a phase is divided into four part,
i.e.
𝐺" = 𝐺"$%& − 𝑇 ∙ 𝑆"
+,& + 𝐺". + 𝐺"/01$ (2.1)
The first part 𝐺"$%& is the contribution to the Gibbs energy from the mechanical
mixture of the constituents of the phase, where the pre-superscript “srf” denotes
“surface of reference”. The second part is the contribution to the Gibbs energy
from the configuration entropy 𝑆"+,& of the phase, depending on the number of
CHAPTER 2
12
possible arrangements of the constituents in the phase. The third term 𝐺". is the
excess Gibbs energy which accounts for the deviation from the ideal mixture of
the constituents. The last part 𝐺"/01$ represents the contribution to the Gibbs
energy due to physical effects such as magnetic transition.
2.1.1 Compound energy formalism (CEF)
The compound energy formalism (CEF) is based on the two-sublattice model
proposed by Hillert and Staffanson [44]. Later, Sundman and Ågren [45]
extended the model to an arbitrary number of sublattices and constituents on each
sublattice by introducing the concept of a “constituent array”. A constituent array
I describes one or more constituents on each sublattice, and a constituent 𝑖
represents an individual constituent on a certain sublattice (denoted as s). The
constituent arrays can be of different orders. The zeroth order 𝐼: represents just
one constituent on each sublattice, i.e. the so-called end-member. The Gibbs
energy of a phase by CEF is expressed by
𝐺" = 𝑃<=(𝑌) 𝐺<=?
<=
+ 𝑅𝑇 𝑎$
,
$BC
𝑦E($)𝑙𝑛𝑦E
($)
E
+ 𝐺". + 𝐺"/01$ (2.2)
where 𝑃<= denotes the product of the constituent fractions specified by 𝐼:. 𝐺<=?
is the Gibbs energy of the end-member 𝐼:. 𝑎$ and 𝑦E($) are the number of sites
and the site fraction of 𝑖 on sublattice s, respectively.
The excess Gibbs energy 𝐺". is expressed by
𝐺". = 𝑃<H(𝑌)𝐿<H<H
+ 𝑃<J(𝑌)𝐿<J<J
+∙∙∙ (2.3)
where 𝐼C and 𝐼L are a constituent array of first order and second order,
respectively. A constituent array of first order denotes that there are two
constituents in one sublattice, i.e. a binary parameter. A constituent array of
CHAPTER 2
13
second order includes two possible cases: three interacting constituents on one
sublattice or two interacting constituents on two different sublattices, i.e. a
ternary parameter or a reciprocal parameter. Binary interaction parameters are
described by the use of Redlich-Kister polynomial [46], given by
𝐿E,N = 𝑦E − 𝑦N, ∙ 𝐿E,N,
,
(2.4)
where 𝐿E,N, = 𝑎 + 𝑏𝑇.
According to Hillert [47], ternary interaction parameters may be composition
dependent, given by
𝐿E,N,Q = 𝑣E ∙ 𝐿E,N,QE + 𝑣N ∙ 𝐿E,N,QN + 𝑣Q ∙ 𝐿E,N,QQ (2.5)
where
𝑣E = 𝑦E + (1 − 𝑦E − 𝑦N − 𝑦Q)/3(2.6a)
𝑣N = 𝑦N + (1 − 𝑦E − 𝑦N − 𝑦Q)/3(2.6b)
𝑣Q = 𝑦Q + (1 − 𝑦E − 𝑦N − 𝑦Q)/3(2.6c)
2.1.2 Substitutional solution model
In this thesis, the liquid phase is described using the substitutional solution model
which is the simplest form of the CEF. The Gibbs-energy expression for the
liquid phase is
𝐺"Y = 𝑥E 𝐺EY?
E
+ 𝑅𝑇 𝑥E𝑙𝑛𝑥EE
+ 𝑥E𝑥N𝐿E,NYN[EE
+ 𝑥E𝑥N𝑥Q𝐿E,N,QY
Q[NN[EE
(2.7)
where 𝑥E is the mole fraction of element 𝑖. 𝐺EY? is the Gibbs energy of element 𝑖
in the liquid state. 𝑅 is the gas constant and 𝑇 the absolute temperature. 𝐿E,NY and
𝐿E,N,QY are the binary and ternary interaction parameters, respectively.
CHAPTER 2
14
2.1.3 Partitioning model (order-disorder model)
In the Fe-Mn-Al-C system, the ordered compounds B2 and D03, both based on
the bcc lattice, are stable, while the L12 or L10 ordered phases based on the fcc
lattice are not stable. However, the fcc ordered phases may form with the
addition of nickel or titanium. In order to easily extend the present database to
multicomponent systems in the future, this thesis took both the fcc and bcc
ordered phases into account. Considering the lack of the systematic investigation
of the effect of interstitial element, i.e. carbon, on the ordering of both bcc and
fcc phases, it is usually assumed that the ordering contribution to the Gibbs
energy is zero when the carbon atoms fully occupy the interstitial sublattice. In
other words, only the ordering parameters in the metallic system are necessary
to optimize.
In order to describe the Gibbs energies in both the ordered and disordered states
with a single expression, the partitioning model is employed with the aid of the
four-sublattice model (4SL). This model divides the Gibbs energy into two parts:
one is the disordered part depending upon the composition of the phase and the
other is the contribution due to the chemical ordering calculated as a function of
site fraction.
𝐺" = 𝐺"]E$ 𝑥E + ∆𝐺"?%](2.8a)
∆𝐺"?%] = 𝐺"`$a 𝑦E − 𝐺"`$a 𝑦E = 𝑥E (2.8b)
where ∆𝐺"?%] becomes zero if the phase is disordered.
Here, the fcc phase and its ordered forms in the A-B system are taken as an
example. The fcc lattice and its possible ordered configurations are shown in Fig.
2.1. It can be seen that one cubic corner site and three face centered sites form a
regular tetrahedron in an fcc unit cell. Since the four sites are equivalent in the
disordered state, the number of each sublattice must be identical. Thus, the model
for the fcc ordering is formulated as (A,B)0.25(A,B)0.25(A,B)0.25(A,B)0.25. If the
CHAPTER 2
15
site fractions on each sublattice are the same, i.e. 𝑦E(C) = 𝑦E
(L) = 𝑦E(b) = 𝑦E
(`), the
phase is disordered. If the site fractions on three sublattices are the same but
different from that on the remaining one, i.e. 𝑦E(C) ≠ 𝑦E
(L) = 𝑦E(b) = 𝑦E
(`) , the
phase is ordered and the structure is L12. If the site fractions on two sublattices
are the same and the other two the same, but different from the first two, i.e.
𝑦E(C) = 𝑦E
(L) ≠ 𝑦E(b) = 𝑦E
(`), the phase is also ordered and the structure is L10.
Fig. 2.1. The structures of the disordered fcc_A1, ordered L12 and L10 phases.
The Gibbs energy of the disordered part 𝐺"]E$ 𝑥E can be calculated as the
substitutional solution model, i.e.
𝐺"]E$ 𝑥E = 𝑥E 𝐺E]E$?
E
+ 𝑅𝑇 𝑥E𝑙𝑛𝑥EE
+ 𝑥E𝑥N𝐿E,N]E$N[EE
+ 𝑥E𝑥N𝑥Q𝐿E,N,Q]E$
Q[NN[EE
(2.9)
The term 𝐺"`$a 𝑦E in Eq. (2.8b) is expressed as
𝐺"`$a 𝑦E = 𝑦EC 𝑦N
L 𝑦Qb 𝑦a
` 𝐺E:N:Q:a?
aQNE
+14𝑅𝑇 𝑦E
$ 𝑙𝑛𝑦E$
E
`
$BC
+ 𝑦f$ 𝑦g
$ 𝑦fh 𝑦g
h 𝑦Ei 𝑦N
j 𝐿f,g:f,g:E:NNBf,gEBf,gh$
(2.10)
where 𝑦E$ is the site fraction of 𝑖 on sublattice s and 𝐺E:N:Q:a? is the Gibbs energy
of the end-member compound. 𝐿f,g:f,g:E:N is the reciprocal parameter describing
CHAPTER 2
16
the short-range order (SRO) [48]. The term 𝐺"`$a 𝑦E = 𝑥E in Eq. (2.8b) is also
calculated by Eq. (2.10) using the mole fraction instead of the site fraction.
When the phase is in the disordered state, the sublattices are identical due to the
crystallographic symmetry. Therefore, the compound energies should fulfil the
following relations:
𝐺f:f:f:g? = 𝐺f:f:g:f? = 𝐺f:g:f:f? = 𝐺g:f:f:f? = 3𝑢fg + 𝛼fng(2.11𝑎)
𝐺f:f:g:g? = 𝐺g:g:f:f? = 𝐺f:g:f:g? = 𝐺f:g:g:f? = 𝐺g:f:f:g? = 𝐺g:f:g:f?
= 4𝑢fg(2.11𝑏)
𝐺f:g:g:g? = 𝐺g:f:g:g? = 𝐺g:g:f:g? = 𝐺g:g:g:f? = 3𝑢fg + 𝛼fgn(2.11𝑐)
where 𝑢fg denotes the nearest neighbour pair interaction energy between the
atom A and B at the equi-atomic composition. 𝛼fng and 𝛼fgn are the adjustable
parameters due to different environment.
As an approximation, all reciprocal parameters are identical regardless of the
occupation on the other two sublattices, i.e.
𝐿f,g:f,g:f:f: = 𝐿f,g:f,g:f:g: = 𝐿f,g:f,g:g:g: = 𝐿f,g:f:f,g:f: = ⋯
= 𝐿fgfg(2.11𝑑)
where 𝐿fgfg is assumed to be equal to the nearest neighbour bond energy in most
cases.
Additionally, the model for the disordered phase is (A,B)1. The interaction
parameters for the disordered phase are generally assessed based on the
experimental data on the disordered phase. However, when the disordered phase
is not stable without any experimental data, the following relations can be
utilized to obtain the interaction parameters from the ordered parameters [49],
i.e.
𝐿f,g]E$: = 𝐿0 + 𝐺f:f:f:g? + 1.5 𝐺f:f:g:g? + 𝐺f:g:g:g? + 0.375 𝐺f,g:f,g:f:f?
+ 0.75 𝐿f,g:f,g:f:g: + 0.375 𝐿f,g:f,g:g:g: (2.12a)
CHAPTER 2
17
𝐿f,g]E$C = 𝐿1 + 2 𝐺f:f:f:g? − 2 𝐺f:g:g:g? + 0.75 𝐿f,g:f,g:f:f:
− 0.75 𝐿f,g:f,g:g:g: (2.12b)
𝐿f,g]E$L = 𝐺f:f:f:g? − 1.5 𝐺f:f:g:g? + 𝐺f:g:g:g? − 1.5 𝐿f,g:f,g:f:g: (2.12c)
𝐿f,g]E$b = −0.75 𝐿f,g:f,g:f:f: + 0.75 𝐿f,g:f,g:g:g: (2.12d)
𝐿f,g]E$` = −0.375 𝐿f,g:f,g:f:f: + 0.75 𝐿f,g:f,g:f:g: − 0.375 𝐿f,g:f,g:g:g: (2.12e)
where 𝐿0 and 𝐿1 are adjustable parameters independent of the ordering.
2.1.4 κ-carbide
The κ-carbide has a cubic structure with transition elements M (M=Fe or Mn in
the Fe-Mn-Al-C system) at the face centres, Al at the cubic corners and C in the
central octahedral site. The metallic atoms form the L12-type structure based on
the fcc lattice, as shown in Fig. 2.2. In the Fe-Mn-Al-C system, the description
of the κ-carbide is extremely important partly because it coexists in equilibrium
with most solid phases such as fcc and bcc, and partly because its nano-
precipitation strengthening effect on the mechanical properties of the quaternary
alloys. Therefore, a reasonable model for the κ-carbide should be carefully built
based on its crystallographic structure. There are three models proposed in
literature. The simplest model is M3Al1(C,Va)1 [50,51], which describes the κ-
carbide only along the stoichiometric composition of M3Al (green line in Fig.
2.3a). Kang et al. [52,53] introduced the transition element onto the Al sublattice,
i.e. M3(Al,M)1(C,Va)1, which can describe a wider homogeneity range of the κ-
carbide but only on the left hand side of the green line (see Fig. 2.3b). A five-
sublattice model (5SL), i.e. (Al,M)0.25(Al,M)0.25(Al,M)0.25(Al,M)0.25(C,Va)0.25,
that can reflect the ordering between the metallic elements was proposed by
Connetable et al. [54]. The model can describe the solubility in κ-carbide both
on the left and right hand sides of the green line (see Fig. 2.3c). Therefore, this
model was employed for the κ-carbide in this thesis work.
CHAPTER 2
18
Fig. 2.2. The stoichiometric E21-M3AlC and L12-M3Al.
Fig. 2.3. The isothermal section of the Fe-Al-C system at 1273 K calculated using the model-parameters in Ref. (a) [51], (b) [52] and (c) [54]. The green line shows the
ratio 𝑥tu/𝑥fa = 3 .
Although the 5SL model has 162 end-members in the Fe-Mn-Al-C system, there
are two strategies to reduce the number of the end-member compound energies
that need to be assigned. On the one hand, when the interstitial sublattice is fully
CHAPTER 2
19
occupied by vacancies, the model becomes identical to the ordered fcc phases in
the Fe-Mn-Al system. Thus, these parameters can be taken directly from the
description of the ternary system. On the other hand, due to the crystallographic
symmetry, some of the end-members are equivalent, which further reduces the
number of end-member parameters that need to be assessed.
Additionally, the disordered part of the κ-carbide is modelled as
(Al,M)1(C,Va)0.25. The energies for the end-member compounds with carbon on
the interstitial sublattice such as 𝐺fa:vfC_Qx//x are optimized based on the formation
energies calculated by ab initio methods. The energies for the end-member
compounds with vacancies on the interstitial sublattice such as 𝐺fa:yxfC_Qx//x are
identical to the Gibbs energies of the pure elements in the fcc structure but with
an addition of 100 J/mol to prevent them from becoming more stable than the
fcc phase. The interaction parameters 𝐿fa,z:yxfC_Qx//x are identical to those for the fcc
phase, while the interaction parameters 𝐿fa,z:vfC_Qx//x are optimized based on the
experimental phase equilibria involving the κ-carbide.
2.1.5 Intermetallic compounds
There are many intermetallic compounds in the Fe-Mn-Al-C system, especially
in the Al-rich part, such as Al13Fe4, Al6Mn, φ and D3. In this thesis, the
intermetallic phases were modelled based on their crystal structure and solubility
range. For example, the D3 phase was observed in Fe-Mn-Al alloys at the
temperature from 1148 to 1299 K by Balanetskyy et al. [55,56] and Priputen et
al. [57,58]. In the binary Mn-Al system, the D3 phase is metastable but can form
at the composition of Al0.78Mn0.22 by solidification [59]. However, the site
occupancy of atoms in the D3 phase has not been clearly identified. In
consideration of the atomic radius, the Fe atoms probably occupy the Mn sites.
Thus, the D3 phase can be simply modelled as (Al)0.78(Fe,Mn)0.22. Additionally,
the D3 phase was found in the Fe-Cr-Al system in Ref. [56]. Combining with the
measured composition range of the D3 phase in the Fe-Mn-Al system [55–58],
CHAPTER 2
20
it can be found that Fe or Cr extends the D3 phase to lower Al content but above
70 at%. Therefore, the model for the D3 phase is changed to
(Al)0.70(Al,Fe)0.08(Fe,Mn)0.22 in order to ensure that it can describe a large enough
composition range. Furthermore, the present model is flexible and compatible as
to describe the phase in other ternaries such as Fe-Cr-Al.
Accordingly, the Gibbs energy of the D3 phase is expressed as
𝐺"{b = 𝑦fa(L)𝑦tu
(b) 𝐺fa:fa:tu{b? + 𝑦fa(L)𝑦z,
(b) 𝐺fa:fa:z,{b? + 𝑦tu(L)𝑦tu
(b) 𝐺fa:tu:tu{b?
+ 𝑦tu(L)𝑦z,
(b) 𝐺fa:tu:z,{b? + 0.08𝑅𝑇 𝑦faL 𝑙𝑛𝑦fa
L + 𝑦tuL 𝑙𝑛𝑦tu
L
+ 0.22𝑅𝑇 𝑦tub 𝑙𝑛𝑦tu
b + 𝑦z,b 𝑙𝑛𝑦z,
b + 𝑦fa(L)𝑦tu
(L)𝑦z,(b)𝐿fa:fa,tu:z,{b
+ 𝑦tu(L)𝑦tu
(b)𝑦z,(b)𝐿fa:tu:tu,z,{b + 𝑦fa
L 𝑦tuL 𝑦tu
b 𝑦z,b 𝐿fa:fa,tu:tu,z,{b (2.13)
where 𝑦E($) is the site fraction of element 𝑖 on sublattice s. 𝐿 represents the
interaction parameters, which were fitted to the experimental information.
𝐺E:N:Q{b? is the compound energy of the end-member 𝑖:.|:𝑗:.:~𝑘:.LL. The four end-
member parameters obey the following relation according to the reciprocal
reaction:
𝐺fa:fa:tu{b? + 𝐺fa:tu:z,{b? = 𝐺fa:tu:tu{b? + 𝐺fa:fa:z,{b? (2.14)
2.2 Mobility modelling
According to the absolute reaction rate theory, Andersson and Ågren [60]
suggested a formalism for the diffusion mobility, expressed as
𝑀E = 𝑀E: exp
−𝑄E𝑅𝑇
1𝑅𝑇(2.15)
where 𝑀E: represents the frequency factor and 𝑄E the activation energy. R and T
are the gas constant and the absolute temperature, respectively. Both 𝑀E: and 𝑄E
depend on the composition and temperature. Jönsson [61] indicated that the
logarithm of the frequency factor (𝑙𝑛𝑀E:) rather than 𝑀E
: varies linearly with the
composition according to the empirical relation between the frequency factor and
CHAPTER 2
21
the composition found by Kučera and Million [62]. Thus, the mobility can be
written as
𝑀E = exp𝑅𝑇𝑙𝑛𝑀E
:
𝑅𝑇 exp−𝑄E𝑅𝑇
1𝑅𝑇(2.16)
As suggested by Andersson and Ågren [60], 𝑅𝑇𝑙𝑛𝑀E: and −𝑄E can be
represented with a linear combination of the end-point values and a Redlich-
Kister expansion, i.e.
𝛷E = 𝑥/𝛷E/
/
+ 𝑥/𝑥�[ 𝛷E/,�%
%B:,C,L,…
(𝑥/ − 𝑥�)%]�[//
+ 𝑥/𝑥�𝑥h[ 𝑣/�h$ 𝛷E/,�,h$
$
]h[�/[�/
, (𝑠 = 𝑝, 𝑞, 𝑡)(2.17)
where 𝛷E stands for 𝑅𝑇𝑙𝑛𝑀E: or −𝑄E . 𝛷E
/ is the value of 𝛷E for 𝑖 in species p.
𝛷E/,�% and 𝛷E
/,�,h$ are the binary and ternary interaction parameters,
respectively. 𝑣/�h$ is introduced for the ternary system, given by 𝑣/�h$ = 𝑥$ +
(1 − 𝑥/ − 𝑥� − 𝑥h)/3.
In order to take into account the effect of the ferromagnetic transition in bcc
alloys, Jönsson proposed a model [63,64], i.e.
𝑀E = exp𝑅𝑇𝑙𝑛𝑀E
:
𝑅𝑇 exp−𝑄E𝑅𝑇
1𝑅𝑇 𝛤"� (2.18)
where 𝛤"� is a factor considering the ferromagnetic ordering, expressed as
𝛤"� = exp 𝛼𝜉 6 −𝑄E𝑅𝑇 (2.19)
where 𝛼 is 0.3 for the bcc phase and 𝜉 is the state of the magnetic order, given
by
𝜉 =∆𝐻 𝑇"�
∆𝐻 0"� (2.20)
CHAPTER 2
22
where ∆𝐻 𝑇"� is the magnetic enthalpy calculated using the corresponding
thermodynamic description.
There are three types of diffusion coefficients based on the experimental
conditions and the frame of reference:
1) tracer diffusion coefficient 𝐷E∗ determined from diffusion studies using
radioactive isotopes in the lattice-fixed frame of reference;
2) intrinsic diffusion coefficient 𝐷E measured by the use of inert markers in the
lattice-fixed frame of reference;
3) interdiffusion coefficient or chemical diffusion coefficient 𝐷 extracted from
concentration profile of diffusion couple in the number-fixed frame of reference.
The tracer diffusion coefficient is directly correlated with the mobility by the
Einstein relation, i.e.
𝐷E∗ = 𝑅𝑇𝑀E(2.21)
The interdiffusion coefficient is calculated by
𝐷/�, = (𝛿E/ − 𝑥/)𝑥E𝑀E(𝜕𝜇E𝜕𝑥�
−𝜕𝜇E𝜕𝑥,
)E
(2.22)
where 𝛿E/ is the Kronecker delta (𝛿E/ = 1 when 𝑖 = 𝑝, otherwise 𝛿E/ = 0) and 𝑛
is the dependent species. 𝜇E is the chemical potential derived from the
corresponding thermodynamic description.
23
Chapter 3
Thermodynamic investigation of the
Fe-Mn-Al-C system
The establishment of a self-consistent thermodynamic description of a higher
order system first and foremost depends on that of the lower order systems. In
particular, a reliable unary description is of the essence. Fortunately, the
descriptions of the pure elements in both stable and metastable states were
compiled by Dinsdale [65], known as SGTE (Scientific Group Thermodata
Europe) database, which is widely accepted and applied to thermodynamic
databases for various kinds of materials. At present a new generation model with
more physical meaning and more reliable description at temperatures below RT
is being developed for pure elements [66–68]. However, if the new descriptions
of the elements were used, all higher order systems including the binaries must
be re-assessed, which is beyond the scope of this work. Instead, this work
employed the Gibbs energies of the pure elements compiled by Dinsdale [65].
All the phases in the Fe-Mn-Al-C system are summarized in Table 3.1 with their
thermodynamic models.
CHAPTER 3
24
Table 3.1: Stable phases in the Fe-Mn-Al-C system.
Phase Strukturbericht
Pearson symbol
Space group
Prototype
Model
liquid - - - - (Al,C,Fe,Mn)1 bcc A2 cI2 𝐼𝑚3𝑚 W (Al,Fe,Mn)1(C,Va)3 fcc A1 cF4 𝐹𝑚3𝑚 Cu (Al,Fe,Mn)1(C,Va)1 hcp A3 hP2 P63/mm
c Mg (Al,Fe,Mn)1(C,Va)0.5
α-Mn A13 cP20 P4132 α-Mn (Al,Fe,Mn)1(C,Va)1 β-Mn A12 cI58 𝐼43𝑚 β-Mn (Al,Fe,Mn)1(C,Va)1 graphite A9 hP4 mmc C (C) Al2Fe - aP18 P1 Al2Fe (Al)2(Fe,Mn)1 Al4C3 D71 hR7 𝑅3𝑚 Al4C3 (Al)4(C)3 Al4Mn - hP574 P63/mm
c Al4Mn (Al)4(Fe,Mn)1
Al5Fe2 - oC? Cmcm - (Al)5(Fe,Mn)2 Al6Mn D2h oC28 Cmcm Al6Mn (Al)6(Fe,Mn)1 Al12Mn - cI26 Im3 Al12W (Al)12(Mn)1 Al13Fe4 - mC102 C2/m - (Al)0.6275(Fe,Mn)0.235(Al,Va
)0.1375 bcc_B2 B2 cP8 𝑃𝑚3𝑚 CsCl (Al,Fe,Mn)0.5(Al,Fe,Mn)0.5(
C,Va)3 bcc_D03 D03 cF16 𝐹𝑚3𝑚 BiF3 (Al,Fe,Mn)0.25(Al,Fe,Mn)0.2
5
(Al,Fe,Mn)0.25(Al,Fe,Mn)0.2
5(C,Va)3 cementite D011 oP18 Pnma Fe3C (Fe,Mn)3(C)1 D3 - - P10/m
mc - (Al)0.70(Al,Fe)0.08(Fe,Mn)0.2
2 HTAl11Mn4 - oP156 Pn21a Al3Mn (Al,Mn)29(Fe,Mn)10 LTAl11Mn4 - aP15 𝑃1 Al11Mn4 (Al)11(Mn)4 Mn5C2 - mC28 C12/c1 Pd5B2 (Fe,Mn)5(C)2 Mn7C3 D101 oP40 Pnma Cr7C3 (Fe,Mn)7(C)3 Mn23C6 D84 cF116 𝐹𝑚3𝑚 Cr23C6 (Fe,Mn)20(Fe,Mn)3(C)6 RAl4Mn - hP586 P63/m Al4Mn (Al)461(Mn)107 γ1 D82 cI52 𝐼43𝑚 Cu5Zn8 (Al,Fe,Mn)8(Al,Fe,Mn)5
CHAPTER 3
25
γ2 D810 hR26 R3m Al8Cr5 (Al)12(Fe,Mn)5(Al,Fe,Mn)9 ζ - hP227 P63/m - (Al)4(Fe,Mn)1 φ - - P63/mm
c - (Al)0.70(Al,Mn)0.15(Fe,Mn)0.
15 κ E21 cP5 𝑃𝑚3𝑚 CaTiO3 (Al,Fe,Mn)0.25(Al,Fe,Mn)0.2
5
(Al,Fe,Mn)0.25(Al,Fe,Mn)0.2
5(C,Va)0.25
3.1 Binary systems
There are six sub-binaries in the Fe-Mn-Al-C system: Fe-Mn, Fe-Al, Fe-C, Mn-
Al, Mn-C and Al-C. All systems have been assessed more than once with the
CALPHAD technique. In order to obtain reliable thermodynamic descriptions of
higher order systems, it is very important to select a reasonable assessment of
each binary. Sometimes the binary description has to be modified due to the
extrapolation of ternary experiment information, for example, the Mn-Al binary
in this thesis. The descriptions of the binary systems used in this work are
summarized in Table 3.2.
Table 3.2: Thermodynamic descriptions of binary systems used in this work.
Binary system Ref. Fe-Mn Huang [69] and Djurovic et al. (hcp) [70] Fe-Al Paper I Fe-C Gustafsson [71] and Hallstedt et al. (Fe3C) [72] Mn-Al Du et al. [73] and Paper III (γ1) Mn-C Djurovic et al. [74] Al-C Gröbner et al. [75] and Connetable et al. (fcc and bcc) [54]
3.1.1 Fe-Mn
Since Mn is one of the most common elements in steels, the Fe-Mn system has
been assessed many times in the literature [69,76–79]. The early assessments
[69,77,78] were performed due to the updated description of pure Mn. Among
them, the evaluation by Huang in 1989 [69] is widely accepted and used in most
CHAPTER 3
26
databases. Witusiewicz et al. [79] re-assessed the binary based on their own
experimental data on the enthalpy of mixing of liquid Fe-Mn alloys, the enthalpy
of formation of fcc Fe-Mn alloys, and the heat capacity of fcc and bcc Fe-Mn
alloys [80]. Special focus was given to the martensitic transformation fcc→hcp
on the Fe-rich side. However, the Gibbs energy of the hcp phase on the Mn-rich
side seems much lower than that of the fcc phase in the assessment of
Witusiewicz et al. [79]. Nakano and Jacques [81] studied the influence of the
thermodynamic parameters of the hcp phase on the stacking fault energy and
then revised the hcp phase. Their description of the hcp phase leads to a
miscibility gap at high Mn compositions. Djurovic et al. [70] slightly modified
the parameters for the hcp phase by Huang [69] to agree better with the
martensitic transformation temperatures measured by Cotes et al. [82]. The
Gibbs energy of the hcp phase is more reasonable on the Mn-rich side than that
by Witusiewicz et al. [79] and Nakano and Jacques [81]. Therefore, the revision
of the hcp phase by Djurovic et al. [70] was used in this work as well as the
description of the stable phases by Huang [69]. Fig. 3.1 shows the stable and
metastable phase diagrams of the Fe-Mn system. In Fig. 3.1b, the calculation
only considers the fcc and hcp phase, which is useful for the prediction of the
martensitic transformation fcc→hcp.
Fig. 3.1. (a) Stable and (b) metastable Fe-Mn binary phase diagrams [69,70].
CHAPTER 3
27
3.1.2 Fe-Al
The Fe-Al binary is a very important constituent system for both steels and
aluminium alloys. The first complete thermodynamic assessment of this system
was made by Seiersten [83] in the COST 507 project [84], describing the B2
ordered phase using the two-sublattice model. Du et al. [85] revised the binary
to reproduce the congruent melting behaviour of the Al13Fe4 phase measured by
Griger et al. [86], which is inconsistent with the more recent experiment data
[87,88]. Connetable et al. [54] modified the parameters for the fcc phase
according to their analysis of the ternary phase equilibria in the Al-C-Fe system,
i.e. the Gibbs energy of the metastable fcc phase at higher Al contents is too
positive. Jacobs and Schmid-Fetzer [89] re-assessed the description of the
fcc_A1, bcc_A2 and bcc_B2 phases based on the evaluation of Seiersten [83].
The vacancy defect was considered in the model for the bcc phase. Sundman et
al. [90] performed the most complete assessment of the binary system including
the ordered D03 phase, and that description is now the most widely accepted. In
their assessment, the four-sublattice model was employed for the bcc and its
ordered forms. However, they neglected the experimental information on the
enthalpy of mixing of the liquid phase from Refs. [91,92], which was indicated
by Phan et al. [52]. For this reason, Phan et al. [52] re-assessed the Fe-Al system
describing the liquid phase with the modified quasi-chemical model, which is
different from the substitutional model used in this work. Therefore, the
description of the liquid phase by Sundman et al. was slightly revised using the
substitutional model based on the experimental information in Refs. [91,92] in
this work. Accordingly, the parameters for the solid phases were also slightly
modified. In the assessment of this work, the analysis result on the fcc phase by
Connetable et al. [54] was considered. Details on the revision of the Fe-Al system
can be found in Paper I. Fig. 3.2 shows the stable phase diagram of the Fe-Al
system.
CHAPTER 3
28
Fig. 3.2. Stable Fe-Al binary phase diagram.
3.1.3 Fe-C
As the most important system, the Fe-C binary has been assessed many times
[71,93–98]. The most widely used evaluation was made by Gustafson [71],
regardless of the excessively strong stability of the bcc phase and the inverse
miscibility gap in the liquid at high temperatures. Due to this, the description was
revised by Shubhank and Kang [98], but the liquid phase was described with the
modified quasi-chemical model. Hallstedt et al. [72] re-assessed the Gibbs
energy of the cementite according to the enthalpy of formation and the heat
capacity data from both ab initio calculations and experiments. Naraghi et al. [97]
re-evaluated the entire Fe-C system based on the new generation descriptions of
pure iron and carbon and could therefore not be used in this work. More recently,
a new thermodynamic model for the cementite was proposed by Göhring et al.
[99] to describe the non-stoichiometry and the vacancy content of the cementite.
In order to keep the compatibility of the model for the cementite (M3C-type
carbide) in the lower order systems, the new description by Göhring et al. was
not considered in this work. To sum up, the description by Gustafson [71]
together with the revision of the cementite by Hallstedt et al. [72] were utilized
CHAPTER 3
29
in this work. Fig. 3.3 shows the stable and metastable phase diagrams of the Fe-
C system.
Fig. 3.3. (a) Stable and (b) metastable Fe-C binary phase diagrams [71,72].
3.1.4 Mn-Al
The Mn-Al binary is a very complex system with many intermetallic phases.
Numerous thermodynamic assessments have been carried out for this system.
Kaufman and Nesor [100] made a first assessment of the Mn-Al binary, which
largely deviates from the experimental one. McAlister and Murray assessed the
liquid, fcc and intermetallic phases in the Al-rich part [101,102]. The first
complete assessment of the binary was performed by Jansson [103]. However,
the three phases (bcc, γ1 and γ2) in the middle part of the diagram were described
as one phase, Al8Mn5. Later, the bcc phase in the middle part was taken into
account in the assessment of Liu et al. [104] based on the experimental phase
equilibria in the Mn-rich portion by themselves [105]. The A2/B2 ordering
reaction was also considered using the two-sublattice model in their work.
However, they neglected the high and low temperature forms of the Al11Mn4.
Furthermore, there exists an invariant reaction between the Al11Mn4, Al8Mn5 and
Al4Mn phases and the B2 ordered phase at low temperatures in their description,
which were never found experimentally. Consequently, Ohno and Schmid-
Fetzer [106] revised the parameters for the Al11Mn4 and Al8Mn5 phases to
CHAPTER 3
30
remove the invariant reaction and the B2 phase at low temperatures. In order to
avoid reassessing the entire Mn-Al system, much more model parameters were
introduced for the two phases. Du et al. [73] re-assessed the entire system based
on the reliable literature data and their own experimental data on the Al-rich side.
The problem in the work of Liu et al. [104] was avoided and the high temperature
form of the Al11Mn4 was considered, but they neglected the B2 ordering, which
was later added by Djurovic and Hallstedt [107].
To sum up, no assessments describe the high temperature form of the Al8Mn5,
i.e. γ1, which may be due to the experimental difficulty in obtaining the γ1 phase
by even very rapid cooling. It also seems impossible to perform the in suit
experiments because of both the low melting point of Al and the high volatility
of Mn. Fortunately, an experimental study of the γ1 phase was carried out by
Göedecke and Köster [108] using metallographic examination and thermal
analysis but no X-ray diffraction. The transition temperatures of
γ1↔γ2+HTAl11Mn4 (1230 K) and γ1+bcc↔γ2 (1261 K) were obtained, which
was confirmed by Murray et al. [102]. Later, the crystal structure of the γ1 phase
was identified by Grushko and Stafford [109] with XRD. It is a cubic γ-brass
structure, the same as that of the Al8Fe5 phase in the Fe-Al system. Furthermore,
Balanetskyy et al. [55] found the γ1 phase in ternary Fe-Mn-Al alloys. The
extrapolation of the ternary phase equilibria involving the γ1 phase shows that
the γ1 should be stable in the Mn-Al binary, which necessitates the assessment
of the γ1 phase. Therefore, this work introduced the γ1 phase with the formula
(Al,Mn)8(Al,Mn)5 into the description of the Al-Mn binary by Du et al. [73].
Details on the modification of the Mn-Al system can be found in Paper III. In
addition, the ordering parameters by Djurovic and Hallstedt [107] were used in
this work. Fig. 3.4 shows the stable phase diagram of the Mn-Al binary and the
phase diagram with different ordered forms based on the fcc structure. Fig. 3.5
shows the comparison between the model-predicted and experimental diagram
around the γ1 phase.
CHAPTER 3
31
Fig. 3.4. (a) Stable Mn-Al binary phase diagram and (b) metastable phase diagram with only fcc based phases.
Fig. 3.5. Assessed phase diagram of the Mn-Al system around the γ1 phase.
3.1.5 Mn-C
The Mn-C binary system has been assessed three times [74,78,110]. Lee and Lee
[78] assessed the binary based on their own description of pure Mn without
considering any magnetic properties. Huang [110] made an assessment of the
binary based on the SGTE description of pure Mn and C. Recently Djurovic et
al. [74] re-assessed the binary system combining the available experimental
information and ab initio data on the enthalpies of formation of the manganese
carbides. The liquidus and the stabilities of the carbides were well reproduced.
CHAPTER 3
32
The description of the Mn-C system by Djurovic et al. [74] was used in this work.
Fig. 3.6 shows the stable phase diagram of the Mn-C binary system.
Fig. 3.6. Stable Mn-C binary phase diagram [74].
3.1.6 Al-C
The Al-C binary is the simplest sub-system only including four stable phases, i.e.
liquid, fcc, graphite and Al4C3. The first assessment of the system was performed
by Gröbner et al. [75]. Later, Ohtani et al. [111] re-assessed this binary since the
description by Gröbner et al. slightly deviates from experimental solubility of C
in the liquid phase. However, Connetable et al. [54] indicated that the heat
capacity of the Al4C3 phase assessed by Ohtani et al. [111] differs considerably
from that by Gröbner et al. [75] above 1500 K, and the experimental data in the
ternary Fe-Al-C system did not support the modification by Ohtani et al. [111].
Moreover, in order to describe the ternary experimental information in the Fe-
Al-C system, a slight modification of the fcc phase and a description of the
metastable bcc phase were introduced into the description by Gröbner et al. [75].
Therefore, this work used the revision of the Al-C system by Connetable et al.
[54]. Fig. 3.7 shows the stable phase diagram of the Al-C binary system.
CHAPTER 3
33
Fig. 3.7. Stable Al-C binary phase diagram [54,75].
3.2 Ternary systems
There are four sub-ternaries in the Fe-Mn-Al-C system: Fe-Mn-Al, Fe-Mn-C,
Fe-Al-C and Mn-Al-C. New stable phases appear in the ternary systems except
in Fe-Mn-C. It is therefore necessary to describe the new phases reasonably. The
descriptions of the ternary systems assessed and used in this work are
summarized in Table 3.3.
Table 3.3: Thermodynamic descriptions of ternary systems used in this work.
Ternary System Ref. Fe-Mn-Al Paper III Fe-Mn-C Djurovic et al. [70] Fe-Al-C Paper I Mn-Al-C Paper II
3.2.1 Fe-Mn-Al
The Fe-Mn-Al system has been experimentally studied to determine the phase
equilibria over the whole composition range between room temperature and the
melting temperature. Similarly, the ternary system has been assessed many times
CHAPTER 3
34
based on updated experimental information and descriptions of the binary sub-
systems. Jansson [112] made the first assessment of this ternary using the Fe-Mn
system by Huang [69], the Fe-Al system by Seiersten [83] and the Mn-Al system
by Jansson [103,112]. The thermodynamic optimization mainly focuses on the
Al-rich part of the Fe-Mn-Al system. On the contrary, Liu et al. [113] assessed
the Fe-rich part of the ternary system to reproduce the phase equilibria between
the bcc, fcc, β-Mn and liquid phases. The binary descriptions used were from
Huang (Fe-Mn) [69], Jansson (Mn-Al) [103] and Kaufman (Fe-Al) [100] with
the revision of the bcc and β-Mn phases in their work. Later, the Fe-rich part of
the ternary system was re-assessed by Umino et al. [114] based on the new
experimental phase equilibria between the bcc, fcc and β-Mn phases over a wide
temperature range. The A2/B2 and B2/D03 ordering reactions were also studied
with the diffusion couple method and differential scanning calorimetry (DSC).
The two-sublattice model was employed to describe the A2/B2 ordering. The
assessment was based upon the binary Fe-Mn by Huang [69], Fe-Al by Ohnuma
et al. [115] and Mn-Al by Liu et al. [104]. Recently Lindahl and Selleby [116]
assessed the Fe-Mn-Al system based on the new descriptions of the binary
systems, i.e. Fe-Mn by Huang [69] with the revision of the hcp phase by Djurovic
et al. [70], Fe-Al by Sundman et al. [90] and Mn-Al by Du et al. [73] with
addition of ordering parameters by Djurovic and Hallstedt [107], considering all
available experimental data in the whole system. Moreover, the four-sublattice
model was used to describe A2/B2 and B2/D03 ordering reactions [117]. More
recently, the phase equilibria in the Al-rich part of the ternary system over a wide
temperature range from 923 to 1343 K were measured using SEM, TEM, XRD
and differential thermal analysis (DTA) [55,56,58]. The solubilities of the third
element in the binary intermetallic compounds such as Al6Mn and Al13Fe4 were
found to be very wide and four new ternary phases, i.e. φ, ζ (designated as κ in
their paper), D3 and Z, were observed in the ternary alloys. Given these new
experimental data, the Fe-Mn-Al system was re-assessed over the entire
composition range from RT to the melting temperature in this work. The
descriptions of the binary systems used are shown in Table 3.2. Details on the
CHAPTER 3
35
assessment of the Fe-Mn-Al system can be found in Paper III. Fig. 3.8 shows
the isothermal section at 1073 K calculated using the present thermodynamic
parameters, including the A2/B2 ordering transformation.
Fig. 3.8. Isothermal section of the Fe-Mn-Al system at 1073 K.
3.2.2 Fe-Mn-C
The Fe-Mn-C ternary system has been completely assessed twice times in the
literature [70,118]. Huang [118] performed the first assessment based on the
binary descriptions of the Fe-Mn by Huang [69], Fe-C by Gustafson [71] and
Mn-C by Huang [110]. However, the liquidus surface projection was not
satisfactorily described in her assessment. Djurovic et al. [70] re-assessed the
ternary systems with the aid of ab initio calculations. The enthalpies of formation
of the metastable iron carbides were calculated using the Vienna Ab Initio
Simulation Package (VASP). The description reproduced the experimental
liquidus data well and was thus used in this work. Fig. 3.9 shows the calculated
isoplethal section at 50 wt% Mn using the thermodynamic description by
Djurovic et al. [70].
CHAPTER 3
36
Fig. 3.9. Isoplethal section of the Fe-Mn-C system at 50 wt% Mn [70].
3.2.3 Fe-Al-C
The Fe-Al-C ternary system has been assessed several times [52,54,119–121].
The earlier assessments [119,120] are not reliable because of the lack of
experimental information available at that time. After an experimental
investigation was carried out by Palm and Inden [122], Ohtani et al. [121]
performed a complete assessment of the ternary system combined with ab initio
data on the formation enthalpies of the end-members of the κ-carbide. The
thermodynamic model for the κ-carbide is (Al,Fe)3(Al,Fe)1(C,Va)1, but the
homogeneity range is not well described. Li et al. [123] slightly modified the bcc
and fcc phases based on the evaluation of Kumar and Raghavan [120] for the
development of Al-containing TRIP steels. In the same year, Connetable et al.
[54] made a re-assessment supported by ab initio calculations. They indicated
that the ab initio data by Ohtani et al. [121] were not reliable in comparison with
other similar data in the literature. In their assessment, the κ-carbide was
described using a more complex model, i.e.
(Al,Fe)0.25(Al,Fe)0.25(Al,Fe)0.25(Al,Fe)0.25(C,Va)0.25. This model can reflect the
ordering between metallic elements and describe the wide homogeneity range of
CHAPTER 3
37
the κ-carbide. However, their description cannot reproduce the new experimental
phase equilibria, particularly at higher temperatures [52]. Phan et al. [52]
recently performed an experimental investigation of phase equilibria involving
the κ-carbide and re-assessed the system. The κ-carbide was described using a
simpler model (Fe)3(Al,Fe)1(C,Va)1, which can only describe the limited
solubility range of the κ-carbide as discussed in Section 2.1.4. In order to
reproduce the wide solubility range of Al and C, the five-sublattice model was
used in this work. Based on all available experimental information and the ab
initio data in this work, the Fe-Al-C ternary system was re-assessed, see Paper
I for details. Fig. 3.10 shows the comparison between the calculated and
experimental isothermal sections at 1273 K. Particularly, the phase regions of
the κ-carbide calculated using the present description and the previous
assessments [52,54] are overlapped in the figure.
Fig. 3.10. Isothermal section of the Fe-Al-C system at 1273 K. The dash-dotted and dashed lines represent 𝑥fa = 0.2 and 𝑥tu/𝑥fa = 3, respectively.
CHAPTER 3
38
3.2.4 Mn-Al-C
The Mn-Al-C system has only been assessed twice [50,53] due to the lack of
experimental information on this ternary. Moreover, simple models were used
for the κ-carbide in the assessments, i.e. (Mn)3(Al)1(C,Va)1 by Chin et al. [50]
and (Mn)3(Al,Mn)1(C,Va)1 by Kim and Kang [53]. Recently, Bajenova et al.
[124,125] measured the phase equilibria at 1373 and 1473 K as well as the
liquidus and solidus projections using SEM, EPMA, DTA and XRD. Wide
solubility ranges of Mn and C in κ-carbide were found, which cannot be
described using the two simple models [50,53]. Therefore, the five-sublattice
model was employed for the κ-carbide in this work. Combined with the ab initio
calculations for the κ-carbide, the Mn-Al-C ternary system was assessed in
Paper III. Fig. 3.11 shows the comparison between the calculated and
experimental isothermal sections of the Mn-Al-C system at 1373 K. The wide
homogeneity ranges of the κ-carbide, fcc and hcp phases are reproduced
satisfactorily.
Fig. 3.11. Isothermal section of the Mn-Al-C system at 1373 K.
CHAPTER 3
39
3.3 Quaternary system
The Fe-Mn-Al-C quaternary system is the extremely important basis of
lightweight steels. Thus, a satisfactory description of this quaternary system is
of the essence for the design of lightweight steels. In this quaternary system, the
κ-carbide is of critical importance partly because it participates in equilibrium
with most phases, particularly the bcc and fcc phases (ferrite and austenite in
steels), and partly because its precipitation behaviour such as precipitation site
and particle size significantly effects the mechanical properties of steels.
Therefore, the thermodynamic assessment of the quaternary system mainly
focuses on the phase equilibria involving the κ-carbide [50,53]. Chin et al. [50]
performed the first assessment of the quaternary based upon the Fe-Al-C ternary
by Connetable et al. [54] with the modification of the κ-carbide, their own
description of the Mn-Al-C system, and the existing database for Fe-based alloy
systems (TCFE2000 [126] or TCFE5 [127]). Recently, Kim and Kang [53]
experimentally investigated the phase equilibria of the quaternary alloys using
EPMA and then assessed the quaternary system. The modified quasi-chemical
model was applied to the liquid phase in their assessment. More recently,
Hallstedt et al. [51] introduced the ternary descriptions of the Fe-Mn-C by
Djurovic et al. [70] and the Fe-Mn-Al by Lindahl and Selleby [116] into the
database by Chin et al. [50]. However, all the above mentioned assessments
described the κ-carbide using simple models, i.e. (Fe,Mn)3(Al)1(C,Va)1 or
(Fe,Mn)3(Al,Fe,Mn)1(C,Va)1, which cannot describe the wide homogeneity
range of κ-carbide. For this reason, the 5SL model, i.e.
(Al,Fe,Mn)0.25(Al,Fe,Mn)0.25(Al,Fe,Mn)0.25(Al,Fe,Mn)0.25(C,Va)0.25, was
employed for the κ-carbide when re-assessing the quaternary system based on
the newly presented descriptions of the sub-systems. Details on the assessment
of the Fe-Mn-Al-C system can be found in Paper IV. Fig. 3.12 shows the
isothermal section of the Fe-20Mn-Al-C (wt%) system at 1373 K calculated by
the present and previous descriptions [51,53] as well as experimental data.
CHAPTER 3
40
Fig. 3.12. Isothermal section of the Fe-20Mn-Al-C (wt%) system at 1373 K. The solid, dashed and dash-dotted lines represent the calculation results using the
thermodynamic parameters by this work, Kim and Kang [53], and Hallstedt et al. [51], respectively.
41
Chapter 4
Kinetic investigation of the Fe-Mn-
Mo system
As stated in Chapter 1, due to the tremendous application potential of lightweight
steels into transportation equipment, the alloying elements have been introduced
into the Fe-Mn-Al-C alloys to further improve the mechanical properties
[9,27,28,37–41]. For instance, Moon et al. [40] explored the effect of Mo on the
precipitation of the κ-carbide and then the hardness. The existence of Mo in κ-
carbide is unfavourable from the perspective of formation energies calculated by
ab initio calculations, but the partitioning of Mo is not clearly found between the
κ-carbide and austenite, which is caused by two factors: the small driving force
for Mo ejection from the κ-carbide and the low diffusivity of Mo. In
consideration of this finding, Mo may be used in lightweight steels to prevent
the precipitation of the κ-carbide along the grain boundaries. Therefore, the
diffusion behaviour of Mo is essential for the understanding of phase
transformation and the element distribution. In this chapter, the experimental and
computational study of diffusion mobilities in the Fe-Mn-Mo system are
presented. Detailed information discussed in this chapter could be found in
Paper V.
CHAPTER 4
42
4.1 Diffusion couple experiments
In general, the interdiffusion coefficients, i.e. chemical diffusion coefficients, are
extracted from the concentration profiles of diffusion couples. The diffusion
couple experiment starts with the preparation of the two alloys which are later
bound together to allow the diffusion of atoms. In the case of the solid diffusion
couples, the alloys are usually prepared using high purity raw materials using arc
melting, induction melting or levitation melting. The last two melting approaches
are more suitable for volatile elements such as Mn. After melting, the
homogenization annealing is performed to remove the micro-segregation during
the solidification and to avoid the influence of the grain boundary on diffusion,
followed by quenching in water to retain the high temperature microstructure.
The ingots are cut to obtain suitably sized blocks using wire-electrode cutting,
which are then mechanically ground to remove the surface contamination. In
particular, the mating surface of each alloy block is polished to a mirror-like
surface using standard metallographic techniques. After that, the two blocks of
the terminal alloys are bound to assemble the diffusion couple with a Mo clamp.
Subsequently, the assembled diffusion couple is annealed at a certain
temperature for a period of time, followed by quenching in water. It should be
noted that all melting and heat treatment are performed under the protection of
Ar atmosphere. After annealing, the diffusion couple is ground and polished
along the direction of diffusion and the concentration profile is measured by
EPMA. Fig. 4.1 shows the experimental concentration profile on the diffusion
couple of Fe/Fe-16.3Mn-2.81Mo (at%) annealed at 1373 K for 72 hours as well
as the microstructure.
CHAPTER 4
43
Fig. 4.1. Microstructure and concentration profile of the diffusion couple Fe/Fe-16.3Mn-2.81Mo (at%) annealed at 1373 K for 72 hours. The symbols are the EPMA
data.
4.2 Mobility optimization method
Andersson and Ågren [60] indicated that the most efficient way to establish a
kinetic database for diffusion is to use diffusion mobilities instead of diffusion
coefficients since the number of the diffusion coefficients in a system is much
larger than that of the mobilities. Therefore, the diffusion mobilities are stored
in the kinetic database and optimized to fit experimental diffusion data.
4.2.1 Traditional method
In the traditional method, the optimization of the diffusion mobility is based on
the experimental data on the tracer diffusion coefficients and the interdiffusion
coefficients using the DICTRA software [128]. Particularly, the interdiffusion
coefficients have to be extracted from the concentration profiles of diffusion
couples before the optimization.
CHAPTER 4
44
The interdiffusion coefficients in the binary and ternary systems are determined
with the Sauer-Freise method [129] and the Whittle-Green method [130],
respectively. The advantage of these two methods is that the evaluation of the
position of the Matano plane is not required. Before that, the measured
concentration profiles are smoothed using the error function expansion equation,
i.e.
𝑋 𝑧 = 𝑎E erf 𝑏E𝑧 − 𝑐E + 𝑑EE
(4.1)
where 𝑋 is the concentration and 𝑧 the diffusion distance. a, b, c and d are the
adjustable parameters.
In both the Sauer-Freise method and the Whittle-Green method, a normalized
concentration variable is introduced by
𝑌E =𝑥E − 𝑥E�
𝑥E� − 𝑥E�(4.2)
where 𝑥E� and 𝑥E� are the mole fraction of 𝑖 at the far left and far right end of the
diffusion couple.
According to Fick’s second law of diffusion, an equation for the binary
interdiffusion coefficient 𝐷EE, using the Sauer-Freise method can be derived as
following
𝐷EE, =12𝑡𝑑𝑧𝑑𝑌E
1 − 𝑌E 𝑌E𝑑𝑧�
��+ 𝑌E 1 − 𝑌E 𝑑𝑧
��
�(4.3)
where 𝑡 is the diffusion time.
Similarly, with the Whittle-Green method, the Fick’s second law in the ternary
case can be expressed as
12𝑡𝑑𝑧𝑑𝑌E
1 − 𝑌E 𝑌E𝑑𝑧�
��+ 𝑌E 1 − 𝑌E 𝑑𝑧
��
�
= 𝐷EE, + 𝐷EN,𝑥N� − 𝑥N�
𝑥E� − 𝑥E�𝑑𝑌N𝑑𝑌E
(4.4𝑎)
CHAPTER 4
45
12𝑡𝑑𝑧𝑑𝑌N
1 − 𝑌N 𝑌N𝑑𝑧�
��+ 𝑌N 1 − 𝑌N 𝑑𝑧
��
�
= 𝐷NN, + 𝐷NE,𝑥E� − 𝑥E�
𝑥N� − 𝑥N�𝑑𝑌E𝑑𝑌N
(4.4𝑏)
where 𝐷EE, and 𝐷NN, are the main interdiffusion coefficients, 𝐷EN, and 𝐷NE, the cross
interdiffusion coefficients. 𝑛 is the dependent element. Note that the
determination of the four interdiffusion coefficients in the ternary system needs
four equations to get a unique solution. Thus, one has to analyse two diffusion
couples whose diffusion paths intersect at one composition point and then only
obtain the interdiffusion coefficients at that single composition. Therefore, a
large number of diffusion couples are needed in order to derive the relationship
between the interdiffusion coefficients and composition in the ternary system.
Fig. 4.2 shows the comparison between the calculated main interdiffusion
coefficients at 1373 K in fcc Fe-Mn-Mo alloys using the mobility parameters
obtained with the traditional method and the extracted ones with the Whittle-
Green method.
Fig. 4.2. Comparison between the calculated and extracted main interdiffusion coefficients at 1373 K in fcc Fe-Mn-Mo alloys. Dashed lines refer to the main
interdiffusion coefficients with a factor of 2 or 0.5.
CHAPTER 4
46
4.2.2 Direct method
A MATLAB optimization program was developed by Höglund [131] to directly
fit the diffusion mobilities to the concentration profiles of diffusion couples
instead of the interdiffusion coefficients. The optimization is performed through
the application programming interface (TC-Toolbox for MATLAB) for the
DICTRA software [128]. Using the programming interface, the MATLAB
program can retrieve the concentration profiles simulated by the DICTRA
software. With the least-squares method, the mobility parameters are optimized
until the deviation between the calculated and measured profiles is minimum.
The optimization script was successfully applied by Lindwall and Frisk [132] in
the optimization of diffusion mobilities in bcc Cr-Mo-Fe alloys.
In order to consider more experimental diffusion data such as tracer diffusion
coefficients, the MATLAB program was improved in this work. The present
program can fit mobilities to experimental diffusion profiles and diffusion
coefficients simultaneously. Since the diffusion mobilities are directly fitted to
the original data from diffusion couple experiments, this kind of optimization is
referred to as the direct method.
Specifically, the direct method is performed in the following steps:
1) The workspace files which store the calculation conditions for the diffusion
coefficients and concentration profiles are first created using the Thermo-Calc
and DICTRA softwares, i.e. a POLY-3 file in Thermo-Calc and a DIC file in
DICTRA.
2) The experimental diffusion data and the corresponding calculation workspace
files are read in the beginning of the MATLAB program. In the meantime, the
initial mobility parameters are entered into the workspace files.
CHAPTER 4
47
3) The diffusion coefficients and concentration profiles are calculated using the
mobility parameters entered in the last step with the Thermo-Calc and DICTRA
software and then returned to the MATLAB program.
4) In the MATLAB program, the mobility parameters are optimized with the
least-squares method by fitting the calculated diffusion data to the experimental
ones.
A typical fitting to the measured concentration profile of the diffusion couple Fe-
10.94Mn-3.35Mo/Fe-16.1Mn (at%) at 1373 K is shown in Fig. 4.3, where a great
agreement is achieved.
Fig. 4.3. Comparison between the simulated and measured concentration profile of the diffusion couple Fe-10.94Mn-3.35Mo/Fe-16.1Mn (at%) at 1373 K.
Since the direct method optimizes the diffusion mobilities based on the original
experimental data, i.e. the concentration profiles of diffusion couples, the errors
during the derivation of the interdiffusion coefficients are avoided, which
improves the accuracy of the diffusion mobilities to a certain extent. In particular,
this method considerably enhances the assessment of mobilities in
multicomponent systems where the determination of the interdiffusion
coefficients requires a large number of diffusion couples. Moreover, the
CHAPTER 4
48
concentration profiles of the multi-phase diffusion couples can be used to
optimize mobilities using the direct method.
49
Chapter 5
Application of Thermo-Kinetic
database
A reliable thermo-kinetic database can provide various kinds of thermodynamic
information, phase diagram data and diffusivity, which is very useful for the
design of composition and processing parameters for new materials. In this
chapter, several examples will be given for the application of the thermo-kinetic
database to lightweight steels.
5.1 Partition of alloying elements
As the partition of alloying elements has a strong effect on the stability of the
phases and then the properties of the alloys such as the solid solution
strengthening, it is important to be able to predict the alloying element content
in each phase during the phase evolution. Chen et al. [133] investigated the phase
transformation from the high-temperature ferrite to austenite in several Fe-Mn-
Al-C alloys by solution treatment at 1373, 1473 and 1573 K for 1 hour. The Mn
and Al concentration distributions in the bcc and fcc phases of the Fe-26Mn-
7.5Al-0.34C (wt%) alloy were determined with EDS. Mn and Al are rich in the
fcc and bcc phase, respectively. Fig. 5.1 shows the contents of Mn and Al in the
bcc and fcc phases against the temperature calculated using the present
thermodynamic database. Although the holding time is not long enough to reach
CHAPTER 5
50
the equilibrium state, the calculation results are in good agreement with the
experimental data.
Fig. 5.1. Calculated element contents in the bcc and fcc phases of Fe-26Mn-7.5Al-0.34C (wt%) together with the experimental data [133].
Cheng et al. [34] studied the phase transformation in the Fe-13.5Mn-6.3Al-0.78C
(wt%) alloy by isothermal holding at temperatures ranging from 773 to 1123 K
for 100 hours. The M23C6 carbide was observed in the lamellae of ferrite and κ-
carbide for the first time. The eutectic reaction of fcc→bcc+κ+M23C6 was
determined in their study. They measured the metallic compositions in each
phase by means of TEM-EDS. Although the carbon content was not determined,
the distribution of other elements in the phases was utilized for comparison. Fig.
5.2 shows the calculated element contents in each phase of the Fe-13.5Mn-6.3Al-
0.78C alloy. The overall agreement is good considering the large uncertainty of
composition analysis by EDS and the neglect of carbon in the phases.
CHAPTER 5
51
Fig. 5.2. Calculated element contents in each phase of Fe-13.5Mn-6.3Al-0.78C (wt%) together with the experimental data [34].
5.2 Precipitation of the κ-carbide
As stated in Chapter 1, the precipitation of the κ-carbide significantly influences
the mechanical properties of lightweight steels. Huang et al. [22] studied the
effect of Al and C on the precipitation of the κ phase in Fe-Mn-Al-C alloys using
XRD, SEM and TEM. The specimens were subjected to a solution treatment at
1423 K for 2 hours, followed by oil quenching. A single austenite phase was
obtained. After that, the specimens were aged at 923 K for up to 360 hours. The
κ-carbide was observed within the austenite matrix or along the austenite grain
boundaries, i.e. intergranular and intragranular. Table 5.1 summarizes the
precipitation of the κ-carbide in different Fe-Mn-Al-C alloys at 923 K, as well
as the calculation results using the previous and present thermodynamic
descriptions. Apparently, the κ-carbide is not formed in the Fe-30.70Mn-2.28Al-
0.90C alloy according to all thermodynamic descriptions, which may be
attributed to the low content of aluminium. Although the κ-carbide precipitates
in the Fe-32.07Mn-5.43Al-0.57C alloy according to the calculation using the
CHAPTER 5
52
present database, the amount of the κ-carbide is very low (1.4%). Therefore, the
present calculation is acceptable.
Table 5.1: Calculated precipitation of the κ-carbide in Fe-Mn-Al-C alloys at 923 K with experimental data [22].
Alloys (wt%) κ-carbide* Experimental [22]
Calculated Chin et al. [50]
Kim et al. [53]
Hallstedt et al. [51]
This work
Fe-30.70Mn-2.28Al-0.90C √ × × × ×
Fe-30.40Mn-5.32Al-0.96C √ × √ × √ Fe-27.11Mn-5.33Al-0.95C √ × √ × √ Fe-29.21Mn-6.95Al-0.94C √ √ √ √ √ Fe-23.46Mn-7.38Al-0.94C √ √ √ √ √ Fe-30.71Mn-5.08Al-0.40C × × × × × Fe-32.07Mn-5.43Al-0.57C × × × × √ Fe-32.84Mn-5.56Al-0.77C √ × √ × √ Fe-32.43Mn-5.32Al-1.04C √ × √ × √
* The symbols √ and × stand for the precipitation and non-precipitation of the κ-carbide, respectively.
5.3 Precipitation strengthening simulation
The κ-carbide can precipitate within the austenitic matrix or along the grain
boundaries in austenitic lightweight steels during aging. In general, the κ-carbide
precipitates coherently in the austenite after aging for a short time. With further
aging the κ-carbide forms also along the grain boundaries. The coherent
precipitation of the κ-carbide in the austenite enhances the strength largely, while
the precipitate at the grain boundaries destroys the strength as well as the
ductility. Therefore, for a specific steel, the aging temperature and time are
important for the precipitation of the κ-carbide including the volume fraction and
the particle size, which have a strong influence on the precipitation strengthening.
From the point of view of materials design, if the relationships between steel
CHAPTER 5
53
composition, heat treatment parameter, microstructure and the precipitation
strengthening contribution are established, the optimal composition and heat
treatment for a steel can be easily obtained, which actually is the ultimate goal
of MGI and ICME. Therefore, the precipitation of the κ-carbide is simulated
using the TC-PRISMA software, and then the yield strength contribution due to
the precipitation strengthening is predicted in this thesis.
Song et al. [32] studied the formation of the κ-carbide and the mechanical
property of the austenitic Fe-29.8Mn-7.65Al-1.11C (wt%) lightweight steel. The
samples were heat treated at 1473 K for 5 hours, followed by cooling in air. After
that, the aged samples were heat treated at 873 K for various times. The lattice
misfit between the κ-carbide in the matrix and the austenite was found to be very
small. The yield strength increased with aging time and decreased after the κ-
carbide formed at the grain boundaries. Based on the present thermodynamic
database and the commercial kinetic database MOBFE4 [134], the precipitation
of the κ-carbide within the austenitic matrix and at the grain boundaries at 873
K was simulated using TC-PRISMA. The grain size of the austenitic matrix was
set to 110 µm in the simulation. The precipitate was assumed to be spherical.
The interfacial energy was set to 0.025 and 0.024 J/m2 for the intragranular and
intergranular κ-carbide, respectively. Detailed input parameters in the simulation
can be found in Paper Ⅳ.
The volume fraction and mean particle radius of the intergranular and
intergranular κ-carbide with the variation of the aging time are plotted in Fig. 5.3.
It can be seen from the built-in figure in Fig. 5.3a that the κ-carbide firstly
precipitates within the austenitic matrix. In Fig. 5.3b the calculated particle
radius is in reasonable agreement with the estimated radius from the
experimental information in Ref. [16].
CHAPTER 5
54
Fig. 5.3. Simulated volume fraction and mean particle radius of the κ-carbide in Fe-29.8Mn-7.65Al-1.11C (wt%) steel during aging at 873 K using TC-PRISMA.
In general, the strength increment in a precipitation-hardened alloy depends on
the morphological features of the precipitate such as particle size, volume faction
and inter-particle spacing. When the particle size exceeds a critical value, the
particles become impenetrable and are only by-passed by dislocations, which is
the so-called Orowan bypassing mechanism. As the particle size increases, the
strength increment due to the precipitation strengthening decreases. On the
contrary, when the particle is small or coherent with the matrix, the shearing
mechanism takes place. The increase in yield strength primarily comes from
three aspects: order strengthening, coherency strengthening and modulus
strengthening. When the dislocations shear the ordered precipitate, an anti-phase
boundary is formed. Thus, the order strengthening is closely related to the anti-
phase boundary energy. The coherency strengthening arises from the interaction
between the moving dislocation and the strain field which is produced due to the
lattice misfit between the precipitate and matrix. The modulus strengthening
results from the difference between the shear moduli of the precipitate and matrix.
Yao et al. [135] reported a relatively high APB energy (0.35-0.70 J/m2) of the κ-
carbide in an austenitic steel and then indicated that the order strengthening is
the predominant strengthening mechanism. Therefore, only the order
strengthening contribution to the yield strength was calculated in this thesis.
CHAPTER 5
55
The dislocations usually travel in pairs in an alloy with ordered precipitates. The
anti-phase boundary forms in the ordered particle during the shearing of the
leading dislocation and is removed by the trailing dislocation. When the particle
size is small and hardly comparable with the spacing of the two paired
dislocations, i.e. weakly coupled dislocations, the strength contribution can be
expressed as [136]:
𝜎¡uxQ = 𝑀𝛾f£g2𝑏
12𝛾f£g𝑓𝑟𝜋𝐺𝑏L − 𝑓 (5.1)
where 𝑀 is the Taylor factor to derive the tensile stress from the shear stress for
an fcc polycrystalline matrix. 𝛾f£g represents the anti-phase boundary energy. 𝐺
is the shear modulus of the austenitic matrix. 𝑏 is the magnitude of the Burgers
vector of the matrix, given by 𝑏 = 𝑎&++ 2, where 𝑎&++ is the lattice parameter
of the matrix. 𝑓 and 𝑟 are the volume fraction and mean particle radius of the
precipitate, respectively.
When the particle size becomes comparable with the spacing of the two paired
dislocations, i.e. strongly coupled dislocations, the expression for the order
strengthening contribution is
𝜎$h%?,� = 𝑀32𝐺𝑏𝑟 𝑓
𝑤𝜋b
2𝜋𝛾f£g𝑟𝑤𝐺𝑏L − 1(5.2)
where the dimensionless constant 𝑤 is of the order of unity.
Yao et al. [135] indicated that the dislocation pile-up stresses at the κ-carbide
interfaces would have an effect on the strength and thus 4~8 dislocations are
piled up in the grain interior, which introduced a correction factor (1/N) into the
equation for the strength calculation, where N=4~8. In this thesis, 1/N=0.4 was
used. Based on the present simulated volume fraction and particle radius of the
κ-carbide, the strength increment due to the order strengthening is calculated as
shown in Fig. 5.4 together with the experimental data [32]. A satisfactory
agreement is achieved before 15 hours. The predicted strengthening contribution
CHAPTER 5
56
after 15 hours is slightly higher than the experimental data, which is attributed
to the formation of the coarse κ-carbide along the grain boundaries.
Fig. 5.4. Prediction of the precipitation strengthening contribution with a correction factor in Fe-29.8Mn-7.65Al-1.11C (wt%) steel aged at 873 K.
57
Chapter 6
Concluding remarks and future
work
6.1 Concluding remarks
To accelerate the advancement of lightweight steels, the core system Fe-Mn-Al-
C has been thermodynamically assessed with the CALPHAD method. The sub-
ternary systems (Fe-Al-C, Mn-Al-C and Fe-Mn-Al) have been re-evaluated
based on the critical review of all experimental information in each system as
well as ab initio data. The κ-carbide is described using a five-sublattice model
with four substitutional sublattices and one interstitial sublattice. This model can
describe the ordering between the metallic elements and the wide homogeneity
range of the κ-carbide. In order to describe the entire composition range of the
system, the intermetallic compounds in the Al-rich part have been modelled.
Their stability upon temperature and composition have been satisfactorily
reproduced, which makes it possible to predict the phase equilibria in aluminium
alloys. Based on the present thermodynamic description, a reliable ‘materials
genome’ database for lightweight steels has been established.
Within this work a high-throughput optimization method for diffusion mobility
has been employed to optimize the mobility parameters for the fcc and bcc phase
in the Fe-Mn-Mo systems. This method directly uses the diffusion profiles
instead of the interdiffusion coefficients, which can avoid the errors introduced
during the extraction of interdiffusion coefficients. More importantly, the direct
CHAPTER 6
58
optimization method can be utilized in multicomponent systems to largely
reduce the number of diffusion-couple experiments.
Combining the present thermodynamic database and the commercial mobility
database, the precipitation of the κ-carbide has been successfully simulated using
TC-PRISMA. The calculated volume fraction and particle size of the κ-carbide
within the matrix have been used to predict the precipitation strengthening
contribution to yield strength. The calculated results show that the order
strengthening is predominant for the enhancement of the yield strength.
Overall, this work preliminarily established a reasonable quantitative
relationship between the composition, processing parameter, microstructure and
property, which is the ultimate goal of MGI and ICME.
6.2 Future work
In order to prefect the thermodynamic description of the quaternary Fe-Mn-Al-
C system, more experimental work should be carried out for alloys containing
higher C and Al content. It would be interesting to measure the upper limit of
carbon solubility in the κ-carbide and explore the effect of the carbon on the
ordering in the ordered bcc phase, which may significantly enlarge the alloy
composition range of lightweight steels used in automotive industry.
The present study of the strength prediction mainly focuses on the order
strengthening effect. As a matter of fact, the final yield strength of the aged alloys
is constituted by solid solution effect, coherency effect, modulus effect,
interfacial effect and grain boundary effect. For instance, the concentration of
solutes in the matrix will change during the precipitation of the κ-carbide, leading
to the change of solid solution strengthening contribution with aging time.
Therefore, further studies should be performed on the models for these parts to
obtain a more reliable model for the strength.
59
Acknowledgements
During the three years, there are many people who have helped me in various
ways. I would like to give all my thanks to them.
First of all, I would like to express my sincere gratitude to my main supervisor
Prof. Malin Selleby. She gives me constant support and sufficient freedom
throughout this work. Many thanks are given to my co-supervisor Prof. John
Ågren for his qualified guidance and inspiring lectures. I would like to give my
deepest gratitude to my other co-supervisor Dr. Huahai Mao for always being at
hand to answer questions, discussions and careful guidance on my work.
I would also like to thank Prof. Xiao-Gang Lu at Shanghai University for his
help with ab initio calculations. Dr. Lars Höglund is gratefully acknowledged for
his invaluable help during the study of the mobility optimization method. Special
thanks are given to Dr. Bonnie Lindahl for valuable discussions and for
providing data during the thermodynamic assessment work and Dr. Bartek
Kaplan for his thorough review of the first paper.
All my colleagues and friends within the Department of Materials Science and
Engineering are greatly acknowledged for providing a nice working atmosphere.
Special thanks are given to my fellow PhD students in the Computational
Thermodynamics group.
I would like to thank all my friends in Sweden for their kind help and making
time fly.
60
I gratefully acknowledge the financial support from China Scholarship Council
(CSC), Stiftelsen för Tillämpad Termodynamik and Stiftelsen
Jernkontorsfonden för Bergsvetenskaplig Forskning.
Last but not least, I would like to give my greatest gratitude to my family for
their endless support and encouragement. Special thanks to my beloved wife,
Jingya. This thesis is dedicated with my deepest love to you.
61
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