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Cementite coarsening during the tempering of Fe-C-Mn martensite

Y.X. Wu1, W.W. Sun

1, M.J. Styles

2, A. Arlazarov

3, C.R. Hutchinson

1*

1Department of Materials Science and Engineering, Monash University, Clayton, 3800, VIC,

Australia

2CSIRO Manufacturing, Clayton, 3168, VIC, Australia

3ArcelorMittal Global Research and Development, Voie Romaine-BP30320, 57283, Maizières-

les-Metz, France

*Corresponding Author: christopher.hutchinson@monash.edu

Keywords: Advanced high strength steels (AHSS), cementite precipitation, martensite tempering, Mn

partitioning, coarsening.

Abstract

Cementite precipitation during martensite tempering has become an important processing step in

some advanced high strength steels (AHSS). This tempering can lead to the formation of Mn-

partitioned cementite which has a strong influence on subsequent austenite reversion during

intercritical annealing. In this contribution, both the cementite Mn content and particle size evolution

have been quantitatively monitored during the tempering of Fe-C-Mn martensite at temperatures

between 400°C and 600°C. Mn progressively partitions to cementite during the coarsening reaction

from the earliest tempering times. This coarsening of the cementite occurs in a matrix that contains

strong Mn concentration gradients that drive partitioning of the Mn to the cementite. A model is

developed to simultaneously describe the time, temperature and bulk alloy dependence of the mean

cementite size and Mn composition evolution. In the model, both the Mn flux, driven by the

composition gradient, and C flux, driven by capillarity, have been taken into account to determine the

operative tie-lines at the cementite and martensite/ferrite interface. The effect of the cementite Mn

content on the dissolution of smaller cementite particles that supply C to the larger particle growth is

considered by coupling the growth and dissolution of cementite under conditions of multi-component

diffusion.

*Text onlyClick here to download Text only: Yuxiang_BatchAnnealing_FINAL_12July.docx Click here to view linked References

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1. Introduction

Tempering is an essential step in the processing of martensitic steels. The process sacrifices strength

but provides sufficient ductility and damage tolerance to the material for engineering applications.

Because of this practical significance, the softening kinetics during tempering have been widely

studied for decades. It has been shown that alloying additions have important effects on the

tempering process [1] and these effects are often related to cementite precipitation [2,3]. Despite the

long history of studying softening during tempering, it is only recently that the question of the

cementite composition during tempering has become important.

During the tempering of steels alloyed with carbide-forming elements (e.g. Mn, Cr and Mo), cementite

can form with partitioning of these alloying elements and is often described as (Fe, M)3C where M is

the partitioned substitutional elements in cementite. Although the equilibrium partitioning coefficient

between cementite and ferrite as a function of temperature and alloy compositions [4–6] is now

reasonably well described in the thermodynamic databases, the actual level of element partitioning

during tempering is kinetically controlled [5].

The substitutional concentration in cementite has practical importance. During the sheet production of

advanced high strength steels (AHSS), annealing between hot-rolling and cold-rolling stages is a

tempering treatment and leads to Mn-partitioned cementite in a ferrite matrix. This is the starting

microstructure for austenite reversion during intercritical annealing [7,8] and the Mn-partitioned

cementite is known to have a strong influence on the austenite reversion kinetics [9–11]. In Cr-Mo

power plant steels, cementite composition changes (i.e. Mn, Cr and Mo enrichment) have been

studied as a function of time and temperature during bainite tempering [12–14]. This knowledge is

used to back calculate the thermal history of steel components during elevated temperature service

and to estimate the remaining life. From the perspectives of both steel manufacturing and life-cycle

maintenance, understanding cementite formation kinetics and especially the concomitant elemental

partitioning kinetics are important during tempering treatments.

There have been some studies on the evolution of cementite composition during tempering. It is often

observed that cementite inherits the bulk substitutional composition of the parent phase in Fe-C-Mn

and higher-order multicomponent systems when tempered at low temperatures or for short tempering

times. These measurements have been made both by chemical analysis or transmission electron

microscopy using extraction methods [5,15–21], and by atom probe field ion microscopy [22–27]. At

higher temperatures or prolonged tempering times, substitutionals can diffuse and the cementite

composition can evolve: Mn, Cr, Mo partitioning to cementite [5,19,20,28] and Si, Al being rejected

from cementite [2,22,29,30].

The early theoretical treatment of substitutional solute composition of cementite during tempering was

made by Kunitake [20]. Kunitake assumed that the substitutional (Mn, Cr and Mo) enrichment in

cementite was diffusion controlled, and the change in cementite composition was governed simply by

diffusion in the ferrite. An analytical expression was derived to describe the time evolution of

enrichment kinetics in cementite after soft impingement (i.e. overlap of diffusion fields [31]).

Bhadeshia [32] extended this by numerically solving the diffusion equations for substitutional

composition gradients in ferrite and cementite. This treatment assumed a stationary ferrite/cementite

interface and the cementite composition was determined by applying a mass balance at the interface.

These works effectively decouple the cementite growth and coarsening processes (which requires

carbon diffusion) from substitutional diffusion that leads to cementite compositional changes. During

the tempering of martensite in alloyed steels such as Fe-C-Mn, both the average cementite particle

size and its substitutional composition increase in time. Cementite evolution during tempering requires

both carbon and substitutional diffusion, and the diffusion fields in ferrite are continually changing as

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the tie-line governing the interfacial conditions changes. A more complete description of the cementite

enrichment kinetics needs to simultaneously consider the diffusion of both carbon and the

substitutional solutes as well as the evolution of the cementite size and number density. The

development of such a treatment is presented in this work.

The experimentally observed evolution of average cementite size during tempering in multicomponent

steels is often described by classic coarsening theory [19,33–36], and alloying additions have strong

effects on the coarsening kinetics of cementite. An analytical expression was proposed by Björklund

et al. [37] to describe the diffusion-limited coarsening of cementite in Fe-C-Mn ternary systems under

local equilibrium conditions. This treatment respects the fact that the chemistry of the cementite may

evolve during time, but does not simultaneously provide the average size and chemistry of the

cementite as a solution. Rather, if the size is known, an estimate of the chemistry can be made. As

pointed out by Miyamoto et al. [19], the partitioning of substitutional elements to cementite evolves

during tempering and this chemical evolution must be part of the solution of the cementite size

evolution. There is no model currently available to self-consistently describe this coupled evolution of

cementite chemistry and particle size.

In this contribution, we experimentally examine the cementite evolution during martensite tempering of

a range of ternary Fe-C-Mn steels tempered at temperatures from 400C to 600C. Both the

cementite chemistry and particle size evolution are monitored. These alloy compositions were

selected as simple ternary systems with Mn partitioning to cementite, but also comparable to the alloy

chemistries of AHSS [38]. A model is proposed to simultaneously describe the Mn partitioning kinetics

and cementite size evolution.

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2. Experimental procedure

2.1. Materials and heat treatments

The alloys listed in Table 1 were prepared by ArcelorMittal Global Research and Development

(Maizières, France). The alloys were prepared by vacuum melting and casting. The cast ingots were

hot-rolled to 3.5 mm thickness and then homogenised in flowing Ar at 1250°C for 18 hrs. Electron

probe microanalysis (EPMA) across the thickness shows that the level of Mn microsegregation after

homogenisation for each alloy is less than 10% relative variation. Samples with sizes of 10 mm x 10

mm x 3.5 mm were sectioned from the hot-rolled and homogenised plates for microstructure

characterisation. To obtain the fully martensitic starting microstructure, samples were austenitised at

900°C for 15 min followed by water quench. The compositions listed in Table 1 were measured after

austenitisation by inductively coupled plasma-atomic emission spectroscopy (ICP-AES). These

compositions were designed to separate the effects of bulk C and Mn contents on the evolution of

cementite precipitates. Samples were subsequently tempered in a salt bath at 400C (673 K), 500C

(773 K) and 600C (873 K) for times up to 2000 min followed by water quenching. Further tempering

up to 10000 min, was performed for samples encapsulated in Ar-filled silica tubes in a muffle furnace.

The examined regions in all samples were the mid-thickness where decarburization is avoided.

Table 1: Compositions (measured by ICP-AES) of the Fe-C-Mn alloys used in this study

Alloy designation C (wt.%) Mn (wt.%) Si (wt.%) Fe (wt.%)

Fe-0.1C-2Mn 0.116 1.93 0.059 Bal.

Fe-0.4C-2Mn 0.400 1.90 0.070 Bal.

Fe-0.2C-5Mn 0.173 4.91 0.048 Bal.

2.2. Transmission electron microscopy (TEM) with energy-dispersive X-ray

spectroscopy (EDS)

Transmission electron microscopy, equipped with energy-dispersive X-ray spectroscopy, was used to

monitor the substitutional composition of the cementite as a function of tempering time and

temperature. Two types of EDS data were collected: point analysis and line profile analysis. Point

analysis was performed on direct extraction carbon replica samples [39] on an FEI Tecnai G2 T20

TEM operated at 200 kV with a Bruker QUANTAX 30 mm2 silicon drift detector. The low-background

TEM holder was tilted 20 towards EDS detector to maximise the X-ray collection efficiency and the

electron probe was converged to cover the cementite particles (with probe size typically of 50-150 nm).

Each point acquisition time was 180 s to ensure at least 104 counts obtained in K peaks of Mn and

Fe. More than 20 particles from several fields of view were analysed to determine the cementite

composition distribution for each condition. EDS line profile analyses were performed on selected

samples prepared by the focused ion beam (FIB) lift-out technique to characterise the local

distributions of alloying elements across the cementite precipitates. These FIB foils were also used to

validate the EDS results from replica samples and ensure the point analysis was not modified by

back-deposition of solute during preparation of replicas. Line profile analyses were performed using

an FEI Tecnai G2 F20 scanning transmission electron microscopy (STEM) operated at 200 kV with a

Bruker QUANTAX 30 mm2 silicon drift detector.

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EDS spectra quantification was performed using the Cliff-Lorimer ratio approach [40] as embodied in

Bruker ESPRIT 2 microanalysis software. The measured cementite composition is described by the

U-fraction that measures a species substitutional site occupancy. In the Fe-C-Mn system, Mn U-

fraction is expressed as:

(1)

where XMn, XC and XFe represent atomic fractions, and the ratio between XMn and XFe was obtained

from EDS quantification. The relative error associated with the counting statistics was typically below

4% for the measured Mn concentration. As will be shown in Section 3.1, a relative error of ~5% is

much smaller than the width of the Mn concentration distribution measured from particles for a given

sample state. The reported average composition of cementite was determined by the arithmetic mean

of all the point analyses for a given sample state and the confidence interval for the average

composition was estimated by the statistics of the Student’s t-distribution.

2.3. Scanning electron microscopy (SEM)

The particle size evolutions were characterised using scanning electron microscopy (SEM) on

polished samples etched by 4% picral (4 g picric acid and 2 mL hydrochloric acid in 100 mL ethanol).

This etchant was chosen to resolve the fine cementite precipitates. A JEOL JSM-7001F field-emission

gun SEM was used and secondary electron images from several fields of view were taken with

magnifications from 15,000 to 100,000. The particle size is approximated by the radius of the circle

with area equivalent to the projected area of each particle in SEM images. The particle size measured

is the apparent size appearing in the two-dimensional polished and etched surface. The average

particle size is determined by the arithmetic mean of all the measured equivalent spherical radii, and

the confidence interval for the average particle radius is estimated by the statistics of the log-normal

distribution ([41]).

2.4. X-ray powder diffraction (XRD)

X-ray powder diffraction (XRD) was used to monitor any retained austenite phase in the as-quenched

state after austenitisation. The measured amount from whole-pattern Rietveld refinement [42] is within

the uncertainty (less than ~0.5%) and any retained austenite decomposes during tempering even for

the shortest time of 2 min. One exception is, at 600°C, the Fe-0.2C-5Mn alloy composition lies in the

ferrite+austenite+cementite three-phase field. Continuous growth of austenite was observed

simultaneously with cementite precipitation. The austenite phase fraction evolution during 600°C

tempering in Fe-0.2C-5Mn alloy was determined as a function of tempering time. The data were

collected on a Bruker D8 Advance diffractometer with Co Kradiation and a Lynxeye position

sensitive detector. The samples surfaces were finished with colloidal silica suspension polishing. The

XRD data were analysed using whole-pattern Rietveld method as embodied in the software package

TOPAS (version 5, Bruker AXS).

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3. Results

3.1. Mn partitioning during cementite growth

SEM micrographs of the Fe-0.1C-2Mn alloy after tempering for 30 min and 2000 min at 500C (Fig.

1a-b) and 600C (Fig. 1c-d) are shown below. In Fig. 1, white and light grey regions represent the

cementite particles, and the dark grey regions are the tempered martensite matrix.

Fig. 1. SEM images of the tempered martensite of Fe-0.1C-2Mn alloy (picral etched): (a) 500 C for 30 min; (b)

500 C for 2000 min; (c) 600 °C for 30 min; (d) 600 °C for 2000 min.

Within 30 min at 500C and 600C, the cementite particles in the martensite matrix already begin to

spheroidise (Fig. 1). A typical shape factor from image analysis is 0.7~0.8 (stable from 2 min to

10,000 min) and this is close to the circularity value of 1 from a perfect circle. Inter-lath precipitates

are always more elongated than intra-lath precipitates, but intra-lath precipitates dominate the

statistical measurement due to their larger number density. It is well accepted that the initial

nucleation and growth of cementite in lath martensite leads to a plate/rod-shaped morphology and

plate cementite will gradually spheroidise and coarsen with time and temperature [43,44]. This

morphological transition occurs very quickly during tempering, and XRD and TEM both show

cementite formation from the earliest times examined (i.e. 2 min).

As shown in Fig. 1, the cementite generally retains a much smaller particle size at 500C, and the size

evolution at 500C from 30 to 2000 min is obviously slower than at 600C. The time evolution of

average particle size and areal number density at 500C and 600°C (Fe-0.1C-2Mn alloy) is

summarised in Fig. 2(a-b). At both temperatures, the average particle size evolution can be

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reasonably well described by a power-law time dependence with exponent of 1/3, as expected from

Lifshitz-Slyozov-Wagner (LSW) coarsening theory [45,46]. A number of groups have also described

the cementite size evolution using LSW-type coarsening theory [19,34,36,47,48], and they have linked

the time dependence of time

1/3 to volume-diffusion controlled coarsening.

A clearer indication of cementite coarsening is the areal number density decrease (shown in Fig. 2b).

This process is driven by the total interfacial energy reduction. 600°C witnessed a faster number

density drop than 500°C. The initial areal number density during tempering is ~1014

m-2

(Fig. 2b),

which is very close to the TEM observations during ultra-fast tempering of an Fe-0.15C-1.97Mn alloy

by Massardier et al. [49].

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Fig. 2. (a) Average particle size time evolution and (b) areal number density at 500 C and 600 °C tempering in the Fe-0.1C-2Mn alloy. (c-d) Histograms of cementite Mn U-fraction distribution (from over 20 TEM EDS

measurements on extraction replicas) in Fe-0.1C-2Mn alloy tempered at (c) 600C for 30, 2000 and 10,000 min and (d) 500°C for 30 and 10,000 min. (e-f) Correlation between cementite particle sizes (nm) and Mn U-fraction in

cementite at the intermediate stage for Fe-0.1C-2Mn (e) at 600C for 2000 min and (f) at 500°C for 10,000 min (50 measurements are made for each condition and correlation coefficients are labelled). In (a), error bars of average particle sizes represent 95% confidence interval estimated from over 300 particles in the SEM image analyses. In (b), error bars of average number density represent one standard deviation from several fields of view in the SEM image analyses. The areal number densities describe the qualitative trends but should not be directly compared with the LSW equation that describes volume number density. In (c,d), the bulk U-fraction determined from bulk alloy composition (Table 1) and the calculated cementite equilibrium U-fractions (TCFE8 database, Thermo-Calc) are labelled. (For some data points, the error bars are shorter than the height of the symbol.)

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The cementite composition distributions of the Fe-0.1C-2Mn alloy during tempering at 600°C and

500°C are shown in Fig. 2(c-d), respectively. The composition distribution for each tempering

condition was measured from point analysis on replica samples by TEM EDS and all the distributions

(Fig. 2(c-d)) can be reasonably well described by Gaussian distributions. At all temperatures, the

average Mn U-fraction increases from the bulk alloy U-fraction at very short times towards equilibrium

(calculated from TCFE8 database, Thermo-Calc). Tempering for 10,000 min at 600°C (Fig. 2c) leads

to an average U-fraction close to equilibrium. In contrast, 10,000 min at 500°C (Fig. 2d), results in

cementite compositions that remain far from equilibrium.

As the average cementite U-fraction increases during tempering, the width of the composition

distribution also changes. At 600°C (Fig. 2c), the Mn U-fraction distribution becomes wider from 30

min to 2000 min, but the distribution then becomes narrower when the average Mn U-fraction

converges towards equilibrium at 10,000 min. In contrast, at 500°C (Fig. 2d) the width of the

distribution continually increases till the longest time of measurement at 10,000 min, where the

deviation of average Mn U-fraction from equilibrium is considerably larger than that of 600°C. Of

course, the distribution of Mn U-fraction in cementite will converge to equilibrium at infinitely long time.

At the intermediate times when wide distributions of cementite Mn contents are observed (e.g. 600°C

for 2000 min in Fig. 2c and 500°C for 10,000 min in Fig. 2d), the cementite Mn content is not strongly

dependent on the particle size (correlation coefficients of ~0.5), as shown in Fig. 2(e-f).

3.2. Time and temperature dependence of Mn partitioning kinetics to cementite

The time evolution of the average cementite Mn U-fraction at 600C, 500C and 400C is shown in

Fig. 3(a-c).

Fig. 3. Time evolutions of average cementite Mn U-fraction in Fe-0.2C-5Mn, Fe-0.1C-2Mn and Fe-0.4C-2Mn alloys at: (a) 600 °C, (b) 500 °C and (c) 400 °C. Error bars represent 95% confidence interval estimated from over 20 TEM EDS measurements on extraction replicas. Cementite Mn U-fraction data from Miyamoto et al. [19] in an Fe-0.6C-2Mn(wt.%) alloy tempered at 650 °C is included in (a). Equilibrium U-fractions in cementite (calculated from TCFE8 database, Thermo-Calc) are highlighted as horizontal lines and tables. (For some data points, the error bars are shorter than the height of the symbol.)

At 600C (Fig. 3a), the Mn partitioning kinetics are fast and in all the investigated alloys they tend to

approach the calculated equilibrium Mn U-fractions. The cementite Mn U-fraction data measured by

Miyamoto et al. [19] in an Fe-0.6C-2Mn (wt.%) alloy tempered at 650°C is also included in Fig. 3a as a

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comparison to the Fe-0.1C-2Mn and Fe-0.4C-2Mn alloys with similar bulk Mn contents considered in

the present study. Fig. 3a covers a range of alloy compositions that show the effect of bulk C and Mn

contents. For the 2Mn alloys (red symbols in Fig. 3a) with 0.1C, 0.4C and 0.6C, there are effectively

no differences in partitioning kinetics up till 1000 min. They start to diverge only when thermodynamic

equilibrium begins to constrain the maximum Mn U-fractions in cementite (lower bulk C leads to

higher equilibrium Mn U-fractions, Fig. 3a). The Mn partitioning shows a much stronger dependence

on bulk Mn content. The 5Mn alloy (blue symbols in Fig. 3a) has much faster partitioning kinetics than

the 2Mn alloys. The higher bulk Mn content not only introduces a higher final equilibrium U-fraction

than C, but also leads to faster partitioning kinetics from the earliest tempering times.

The errors reported in Fig. 3 represent 95% confidence interval estimated by Student t-distribution

from over 20 TEM EDS analyses on replica samples. Errors (evaluated from standard deviations and

similar numbers of analysed particles) also represent the width of the cementite Mn U-fraction

distribution (e.g. Fig. 2(c-d)). Similar trends are observed in all alloys (especially in 5Mn alloy), where

the Mn U-fraction distribution becomes broader and then narrower when approaching equilibrium.

The time evolution of the cementite Mn U-fraction in the 5Mn alloy tempered at 600°C shows an

interesting behaviour (Fig. 3a). The Mn content increases continually until ~ 2000 min and then

decreases between 2000 and 10000min. This behaviour is because tempering at 600°C occurs within

the cementite+ferrite+austenite three-phase field of the phase diagram. In Fig. 3a, two equilibrium Mn

U-fractions in cementite are highlighted for Fe-0.2C-5Mn alloy: one is the equilibrium between

cementite and ferrite (with austenite suspended) and the other is the three-phase equilibrium between

cementite, ferrite and austenite (FCC). Depending on the austenite formation kinetics during

tempering, the cementite Mn U-fraction initially can evolve towards the constrained equilibrium

between only cementite and ferrite (up to 2000 min) but once austenite forms in a significant quantity,

this affects the choice of interfacial tie-line in cementite and the Mn U-fraction evolves towards the

global equilibrium.

At 500°C and 400°C (Fig. 3(b-c)), the Mn partitioning kinetics to cementite are much slower, although

the equilibrium Mn U-fractions in cementite (Fig. 3(b-c)) are much higher at lower temperatures. The

dramatically different kinetic behaviours at different temperatures is due to the exponentially slower

Mn diffusion in martensite at lower temperatures. Even after 10,000 min, there are still substantial

deviations from equilibrium at 500°C and 400°C. For the 2Mn and 5Mn alloys, they start from different

initial bulk Mn U-fractions, but after 10,000 min they eventually end up having similar Mn enrichments.

3.3. Local distribution of Mn in cementite

The local distributions of Mn within the cementite precipitates and across the cementite/martensite

interfaces were characterised by STEM-EDS on the thin foils prepared by FIB and extraction replicas.

Example Mn U-fraction line profiles across cementite precipitates are shown in Fig. 4(a-c). Despite the

large change in cementite Mn composition which can occur as a function of time (e.g. at 600°C in Fig.

3a), Fig. 4 shows comparatively uniform Mn distributions in the cementite precipitates for the

tempering times and temperatures shown in Fig. 4. These relatively uniform Mn profiles are created

while Mn continuously diffuses into the growing cementite and the Mn concentration in cementite

increases with time (Fig. 3). Instead of piling up near the interface, Mn can diffuse fast enough within

the cementite to redistribute in the time frame of these experiments. As will be shown, Mn diffusion in

cementite has a very important effect on both the Mn enrichment kinetics in cementite and the

cementite particle size evolution. In Fig. 4, the shapes of the Mn profiles within cementite show some

differences. ‘Smiley face’ Mn profiles with higher Mn contents towards interfaces shown in Fig. 4(b&c)

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are easy to conceptually understand and their origin will be explained in Section 4.4. However, the Mn

composition gradient shown in Fig. 4(a), with lower Mn content near the interface, is more difficult to

rationalise and we will provide a tentative explanation to this observation in Section 4.4.

Fig. 4. Example Mn U-fraction profiles across cementite precipitates by STEM-EDS: (a) Fe-0.1C-2Mn tempered at 600 °C for 2000 min (FIB foil sample); (b) Fe-0.2C-5Mn tempered at 500 °C for 400 min (replica sample); (c) Fe-0.4C-2Mn tempered at 400 °C for 10,000 min (replica sample). Simulated Mn profiles for each tempering condition are included.

3.4. Austenite formation during 600°C tempering in Fe-0.2C-5Mn alloy

During the tempering of the Fe-0.2C-5Mn alloy at 600°C, austenite formation has an important effect

on the Mn partitioning to cementite (Fig. 3a). An example of the distributions of Mn between

cementite, austenite and martensite is shown at 2000 min tempering time in Fig. 5(a-b). Line profile 1

shows that Mn is distributed uniformly in both the cementite and austenite precipitates. Austenite

grows with a Mn U-fraction close to the calculated full equilibrium, and the adjacent cementite

precipitate also takes the Mn U-fraction that is closer to the cementite+ferrite+austenite three-phase

equilibrium (compared to line profile 2).

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Fig. 5. (a) STEM bright-field image of cementite (CEM) and austenite (FCC) in the martensite matrix tempered at 600 °C for 2000 minutes in the Fe-0.2C-5Mn alloy (FIB sample). (b) Mn U-fraction line profiles corresponding to (a). (c) Measured austenite volume fractions by quantitative XRD as a function of time at 600 °C. The experimental data fitted by the KJMA model. Rietveld errors are shown as error bars in the plot. (d) Average particle size time evolution in the Fe-0.2C-5Mn alloy tempered at 600 °C is highlighted in comparison with Fe-0.4C-2Mn and Fe-0.1C-2Mn alloys. Error bars of average particle sizes represent 95% confidence interval estimated from over 300 particles in the SEM image analysis. (For some data points, the error bars are shorter than the height of the symbol.)

The Mn partitioning kinetics shown in Fig. 3a, demonstrate that the cementite Mn U-fraction can be

close to the cementite-ferrite equilibrium at 2000 min. At this point, the austenite volume fraction from

XRD measurement is only half of the calculated equilibrium fraction (Fig. 5c). The austenite formation

kinetics is sufficiently slow that it only starts to compete with cementite precipitation after long time

annealing at 600°C. In the martensite matrix supersaturated with C and defects, cementite has a

much larger kinetic advantage over austenite. It has been demonstrated by Luo et al. [50] that

cementite formation is almost inevitable during heating (300 K/s) from fresh martensite to the ferrite-

austenite two phase-field at 650°C in an Fe-0.17C-4.72Mn (wt.%) steel. Austenite formation

essentially starts from the tempered martensite and slow kinetics is likely to be controlled by the Mn

diffusion in martensite. As shown in Fig. 5c, the austenite formation kinetics can be described by the

Kolmogorov-Johnson-Mehl-Avrami (KJMA) equation [51–55]. As the austenite volume fraction

increases (Fig. 5c), the austenite acts as an expanding reservoir of Mn and C and may maintain the

full equilibrium composition during growth. Due to the interaction with growing austenite, cementite

would evolve towards the reduced equilibrium volume fraction with reduced Mn concentration (Fig.

5b). The evolution of the operative tie-lines at the cementite/martensite interface towards cementite-

ferrite-austenite equilibrium is partially controlled by the austenite formation kinetics.

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13

During austenite formation, the average cementite particle size continues to increase with time (Fig.

5d) and it also shows a reasonably good dependence on time1/3

. However, the coarsening rate (slope

shown in Fig. 5d) is significantly lower than the coarsening rate of the Fe-0.4C-2Mn and Fe-0.1C-2Mn

alloys at the same temperature (Fig. 5d). There are many thermodynamic and kinetic factors built into

the coarsening rate in the LSW-based analytical treatments of two-phase mixtures [37,45,46,56,57].

However, in this case where austenite formation occurs concurrently, the coarsening rate is affected

by austenite formation kinetics because cementite coarsens in the environment where the Mn and C

concentrations for the cementite-ferrite mixture are decreasing.

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14

4. Discussion

4.1. Coarsening of cementite with time dependent Mn composition

In Section 3, we have quantitatively characterised the cementite evolution (in terms of both size and

Mn composition) during tempering of a series of Fe-C-Mn steels. The objective now is to develop a

model that is capable of simultaneously describing the time, temperature and bulk alloy composition

dependence of the mean cementite size and Mn composition.

The number density of cementite particles continually decreases from the very earliest tempering

times, in all the alloys studied in this work, at all temperatures between 400°C and 600°C (Fig. 2).

This is an indication of coarsening and it may be tempting to treat the evolution as a pure coarsening

process, as in the framework of LSW treatments [45,46]. The complication is that the cementite

coarsening is occurring in a matrix that contains strong Mn concentration gradients that drive

partitioning of the Mn to the cementite during the process. In this respect, it is not a pure coarsening

process driven by capillarity. Whilst the carbon supersaturation is consumed very quickly, and during

the time frames studied in this work the carbon diffusion is driven by capillarity, the Mn diffusion is

driven by composition gradients and results in an evolution of the cementite composition during

coarsening. It is a coupled growth and coarsening problem and must be treated as such.

The normalised particle size distribution measured from the Fe-0.1C-2Mn alloy during tempering at

600°C is shown in Fig. 6a. The particle size distribution is self-similar during the reaction, as is usually

assumed during coarsening reactions (even though the cementite composition is not constant in time).

During this reaction, particles smaller than the average will dissolve and provide a carbon flux for

those larger than the average to grow. The cementite composition is continually changing during this

time (Fig. 3a), and both the growing and dissolving particles are enriched in Mn and evolve in a matrix

containing Mn gradients (Fig. 6b). In the development of a model that can simultaneously describe

both the evolution of the particle size and its chemistry during tempering, two considerations must be

respected: the effect of the enriched Mn on the dissolution of the smaller particles (and hence the

carbon supply), and the Mn gradients in the matrix on the growth of the larger particles.

Fig. 6. (a) Smaller particles dissolve to supply carbon for the growth of larger particles: the particle size

distribution showing the relative probability as a function of particle radius scaled by average particle radius ( ). Experimental data from SEM analysis of Fe-0.1C-2Mn alloy tempered at 600 °C for 2, 30, 2000 and 10,000 min and corresponding fitting by log-normal distributions. Dashed line represents the steady-state invariant size distribution predicted from the diffusion-controlled LSW coarsening theory. (b) Schematics of the Mn and C diffusion field during coarsening of cementite: diffusion of Mn is driven by the chemical potential gradient due to compositional gradient while carbon chemical potential gradient due to the capillarity effect.

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15

One may anticipate that a model capable of describing this reaction should contain at least the

following ingredients:

The cementite evolution during tempering requires the mass transport of both C and Mn. A

model must be able to properly describe this multicomponent diffusion.

The diffusion of carbon is driven by the capillarity difference between the growing and

dissolving particles. The Mn diffusion is driven by the composition gradients in the matrix (Fig.

6b). A model must be capable of respecting these different driving forces for the interstitial

and substitutional species.

The Mn composition of the particles (both those growing and dissolving) evolve during the

reaction. Not only must the model be capable of describing multicomponent diffusion, it must

do so within both the matrix and the precipitates phases.

Models do exist to describe cementite coarsening kinetics in multicomponent systems [37,58].

However, they are originated from the LSW treatment. In these treatments the only driving force for

mass transfer comes from capillarity (i.e. describing pure coarsening). DICTRA [59,60] contains an

interesting coarsening module [61] that properly considers the required multicomponent diffusion for

steels considered in this study. However, the diffusion of all species is constrained to be due to

capillarity (for particles with compositions not far removed from equilibrium). Hence, it cannot deal with

problems where the composition of the coarsening particles is evolving significantly during the

reaction.

A new development is required that can address multicomponent diffusion in both the matrix and

precipitate phases under conditions where the interstitial diffusion is driven by capillarity and the

substitutional diffusion is driven by concentration gradients. A first attempt at such a model is

presented below and shown to provide an excellent platform for quantitatively and self-consistently

describing both the evolution of the mean cementite size and Mn composition during coarsening.

4.2. Model development

In this approach, instead of considering the full particle size distribution (Fig. 6a), we represent the

coarsening distribution by two interacting particles – one of size and the other of size 1.5 (Fig. 7a).

The former is supposed to represent an average dissolving particle and the latter, an average growing

particle. Such an approach is inspired by the coarsening module of DICTRA, and one may make the

analogy with the mean particle size models for diffusion controlled precipitate growth being a

simplified representation of the full class models for precipitation. The former has been shown to be

an excellent approximation for the latter [62]. The choice of and 1.5 is guided by the LSW

coarsening distribution, in which the maximum particle size is 1.5 times the average particle size in

the invariant LSW particle size distribution [45,46]. It is assumed that particles smaller than will

always be dissolving and those of size 1.5 are always growing. Experimental measured ratio (e.g.

Fig. 6a) between the dissolving particle size class (< ) and growing particle size class (> ) can be

deviated from 1.5, but the exact choice of the sizes of the growing and dissolving particles (used to

represent the growing and dissolving particle size distributions) has only a minor effect on the

calculations. The growing and dissolving particles are treated as separate moving boundary

calculations in DICTRA (using spherical geometry) to take the advantage of the rigorous treatment of

multicomponent diffusion present in DICTRA.

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16

The main module of this model is the growing cell, in which the C and Mn composition profiles are

continually tracked. The dissolving cell is a sub-module that is used to simulate the carbon flux

leading to the particle growth in the growing cell due to particle dissolution (Fig. 7). To couple the

growing and dissolving cells, the two separate moving boundary problems are coupled through their

geometry as well as through their boundary conditions (Fig. 7a). The boundary conditions for both the

growing and dissolving cells include a zero-flux condition for Mn. The carbon boundary condition for

the growing cell is a constant carbon flux condition (for a time increment Δt), which is the average

carbon flux out of the dissolving cell for the same time increment Δt. To simulate the carbon flux from

particle dissolution in the dissolving cell, a constant carbon activity, which is equal to the activity at the

boundary of the coarsening cell, is used for the boundary condition. The C and Mn composition

profiles and particle size in the dissolving cell are re-defined in every time increment Δt. Due to the

size difference between the dissolving and growing particles in the two cells, this leads to a carbon

potential gradient from the particle to its cell boundary that drives dissolution. Capillarity is included in

both dissolving and growing particles, and a value of 0.7 J/m2 is used for the cementite/martensite

interfacial energy [63].

By numerical integration of the two coupled DICTRA moving boundary problems (for the growing cell

and dissolving cell), one is able to approximate the coarsening of the cementite particles under

conditions that respect multicomponent diffusion, evolving Mn profiles within both the matrix and

particles as well as the different driving forces for carbon and Mn diffusion. Local equilibrium is

assumed between particle and matrix at the interface at all times. Such coupled simulations are not

possible in DICTRA. An algorithm was developed that calls DICTRA to make individual moving

boundary simulations for each time increment Δt, and then outputs the results which are then used to

define new boundary conditions, and cell sizes, for the next simulation sent to DICTRA for a time

increment Δt.

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17

Fig. 7. (a) The coupled two cell simulations of growing and dissolving cementite: growing cell and dissolving cell simulations have different types of boundary conditions for C and same zero-flux condition for Mn. (b) The iterative operations coupling individual growing and dissolving simulations. Each iterative loop controls one growing DICTRA simulation (main module) and one dissolving DICTRA simulation (sub-module). Boundary conditions are modified iteratively between these DICTRA simulations. (c) Simulated dissolving particle radius as

a function of time1/3

and simulation is based on Fe-0.1C-2Mn alloy tempered at 600 C with initial average particle size of ~25nm. The time increment Δt within one individual dissolution simulation is determined based on the dissolution time to the minimum size.

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18

4.3. Numerical implementation

The algorithm is realised and compiled in MATLAB, and it is used to communicate between growing

and dissolving cell simulations as the boundary conditions evolve with the reaction (Fig. 7b). DICTRA

outputs the results of individual simulations, and the algorithm in MATLAB processes the DICTRA

simulation results and generate the new macro files (including modified boundary conditions and new

concentration fields along the corresponding coordinates) to start the next DICTRA simulation.

4.3.1. Initial conditions

The full simulation is initiated by a pure growth simulation and growing cell simulation will continue

from this initial pure growth cell (Fig. 7b). The Mn content of the cementite is initially set equal to the

bulk Mn content, since the cementite has been shown to form very quickly without long range Mn

partitioning at short times. The Mn content of the matrix is also initially set equal to the bulk Mn

content and the distribution is uniform. The particle and cell sizes for the growing cell (Fig. 7a-b) are

initiated from a pure growth simulation. A pure growth simulation will always immediately lead to

nearly the equilibrium volume fraction of cementite via fast carbon diffusion, and particle growth is

effectively stopped very quickly under zero-flux boundary conditions. The initial cell size for the pure

growth simulation is chosen based on the experimental measurement of particle sizes and spacing at

the shortest time of 2 min. The growing cell simulation (Fig. 7b) continues from the initial pure growth

simulation, and cementite volume fraction is assumed to be constant at this nearly equilibrium value

during the entire coarsening simulation. For the dissolving cell (Fig. 7a), the particle size is always

determined as from 1.5 of the growing particle.

4.3.2. Growing cell

Each time increment Δt (i.e. one MATLAB iteration loop) contains one growing and one dissolving

simulation (Fig. 7b). The C and Mn composition profiles are continually tracked in the growing cell

simulation, and the time increment Δt and the boundary condition for the carbon flux for the growing

cell (both of which change in time) is determined from the dissolving simulation. The average Mn

composition in cementite is found by integrating the composition profile over the cementite particle,

and this is used to compare with experimental data. In an individual growing simulation, cementite

grows in a cell with a fixed size. A constant cementite volume fraction is maintained between

simulations. Therefore, as the particle size increases, the cell size must also increase to reflect the

concurrent decrease in particle number density and increase in interparticle spacing. The mass

balance is monitored after each iteration, although the change is very small. It is compensated by

slightly modifying (increasing or decreasing) part of the compositions profiles in the ferrite matrix far

from the particle. Once the growing cell simulation of time increment Δt finishes in one DICTRA

calculation, the new dissolving cell simulation will be started in the next MATLAB iteration.

4.3.3. Dissolving cell

The dissolving cell uses boundary condition of zero-flux Mn and a finite C activity from the boundary

of last growing cell. C and Mn composition profiles and particle size in the dissolving cell are re-

defined in every time increment Δt. To represent the C and Mn composition fields of the dissolving cell

(changing with time), these are approximated by the C and Mn concentration fields of last growing cell

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19

simulation, but the coordinates corresponding to C and Mn concentration fields in cementite are

scaled by 1.5 to take into account the size difference between dissolving and growing particles.

A number of assumptions are made to approximate the composition profiles of the dissolving cell:

During cementite precipitation with slow Mn partitioning, the carbon chemical potential has

time to equilibrate and can be approximated as spatially constant throughout the system (but

not constant in time). The major effect leading to different interfacial conditions (and hence

different Mn contents) at any point in time for the differently sized cementite particles is

capillarity. For the typical cementite particles sizes sampled, this gives rise to a relatively

small effect on the Mn content of the cementite and hence it is assumed that the growing and

dissolving particle have the same Mn content. It is observed in Fig. 2(e-f) that even at the

intermediate stage the correlation between cementite particle size and Mn composition is

relatively weak, which is consistent with the above reasoning.

It is assumed that the Mn composition gradient in the matrix has the same sign for both the

growing and dissolving cells. As the coarsening reaction progresses and the carbon chemical

potential of the system decreases, the average Mn content of both growing and dissolving

particles increases. The mass balance then requires the same sign of the macroscopic Mn

concentration gradient in the matrix around all particles, until the true equilibrium is reached

and no chemical gradients remain. This can also be understood by considering that those

particles that are part of the dissolving population at one point in time, were part of the

growing population at a previous point in time.

The scaling process by 1.5 to satisfy the size difference of vs 1.5 creates steeper Mn

profiles for the dissolving particle. The exact details of the Mn profile in the ferrite surrounding

the dissolving particle should not have a drastic effect on the dissolution process because the

interface is moving away from this profile. One might expect that the Mn profile within the

cementite will have a more significant effect on the dissolution process. However, it is

numerically tested that the steeper Mn gradient in the dissolving cementite would only lead to

slightly faster Mn enrichment in the growing cell and slightly slower kinetics but the

differences are relatively minor and within the experimental error of size and Mn content

measurements.

The time increment Δt for integration is set by the dissolution. An example of simulated dissolving

particle radius as a function of time1/3

for the Fe-0.1C-2Mn alloy during tempering at 600°C is shown

in Fig. 7c. In Fig. 7c, individual dissolving simulations show the dissolution gradually to a plateau. The

time increment Δt is objectively set to the point where minimum radius of dissolving particle is reached

(i.e. end of dissolution). The dissolving particle in this model is supposed to represent an ‘average’ of

all the dissolving particles (some of the real dissolving particles are smaller than the average, and

others are larger). Those that are smaller than our ‘average’ dissolving particle will dissolve faster for

the same boundary carbon activity, and those that are larger will dissolve slower than the average

representative dissolving particle. The carbon flux leaving the dissolving cell over the time increment

Δt (as output from DICTRA) is averaged to approximate the representative average boundary flux for

the next growing cell simulation for the same time increment Δt. It is important that the average

carbon flux coming from the ‘average’ dissolving particle is representative of the whole dissolving

distribution, by considering the average carbon flux from the dissolving particle over its duration from

maximum rate of dissolution to cessation of dissolution.

As the particle grows and Mn partitions to cementite, the dissolution kinetics changes and this leads to

a gradually increasing time increment Δt and decreasing boundary carbon flux. Therefore, in the

current context, time increment Δt used in the calculations should not be considered necessarily in the

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20

same sense as a Δt used in numerical integration. It is not a free variable but an internal characteristic

time for dissolution that is objectively set at each point in time so that a representative average carbon

flux from the dissolving particles is obtained as a boundary condition for the growing particles.

Following this iterative scheme, the dissolving and growing simulation are coupled by the inter-

dependence of boundary conditions for carbon, and the communications between dissolving and

growing simulations are iterated for thousands of times to create a simulation of full kinetics up to

10,000-min tempering time.

4.4. Simulation of Fe-C-Mn alloys at 600C and 650C

Using the model above, simulation results are shown in Fig. 8 for the Fe-0.1C-2Mn, Fe-0.4C-2Mn,

and Fe-0.2C-5Mn alloys tempered at 600C and an Fe-0.6C-2Mn alloy tempered at 650C from

Miyamoto et al. [19]. They are compared with experimental measurements of average Mn U-fraction

in cementite and average particle size as functions of time for each alloy chemistry. A proper model

should be able to self consistently describe both Mn enrichment kinetics and cementite growth

kinetics.

Fig. 8. Comparisons between simulated kinetics and experimental data at 600/650 C: Mn U-fraction in cementite

and average particle radius as functions of time for (a, e) Fe-0.1C-2Mn alloy at 600 C, (b, f) Fe-0.4C-2Mn alloy

at 600 C, (c, g) Fe-0.6C-2Mn alloy at 650 C (experimental data from [19]) and (d, h) Fe-0.2C-5Mn alloy at

600 C. Enhancement factors for Mn mobilities in cementite (MMn(CEM)) used in the simulations are labelled.

Mn diffusion data in ferrite has been critically assessed and stored in the DICTRA MOBFE3 mobility

database (used in the simulations) but there is lack of assessed Mn diffusion data in cementite. The

reported Mn diffusivity in cementite is inferred from cementite compositions during precipitation

[22,64]. There is uncertainty with the current knowledge of Mn mobility in cementite and a sensitivity

analysis of the effect of Mn mobility in cementite is shown in Fig. 8(a, e). Three different Mn mobilities

in cementite are shown using the Fe-0.1C-2Mn alloy as an example. It shows that higher Mn mobility

leads to lower Mn partitioning kinetics but faster growth kinetics. The distinct effect of Mn mobility in

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21

cementite on the Mn partitioning and growth kinetics makes it possible to identify an intermediate

value that can simultaneously describe both kinetics. The intermediate value for Fe-0.1C-2Mn alloy at

600C is an enhancement factor of 7.5E3, and this leads to a good description of both Mn enrichment

kinetics and cementite growth kinetics. This enhancement factor may seem large, but as

demonstrated in Fig. 4a, this factor also leads to reasonable agreement between the experimental

and simulated Mn profiles within the cementite particles measured by STEM-EDS. In principle, a

lower Mn mobility in cementite would slow the Mn enrichment kinetics but we can more easily

interpret the effect by considering the cementite dissolution. A lower Mn mobility decreases the

dissolution kinetics and generates less carbon flux for cementite growth. In addition, a lower Mn

mobility in cementite causes higher interfacial Mn concentration due to large Mn composition gradient

in cementite. High interfacial Mn reduces the carbon activity difference (i.e. the driving force) between

the coarsening and dissolving particles, and hence reduces the generated carbon flux for cementite

growth. These two effects lead to the slower cementite growth and Mn has more time to diffuse to

cementite and results in faster enrichment.

Although the Mn mobility in cementite determined here is fit from the experimental data for the Fe-

0.1C-2Mn at 600C, if the values are reasonable, the same values should be used at the same

temperatures for all alloy compositions. This is required to demonstrate the self-consistency of the

proposed model. For the simulations at 600C including Fe-0.1C-2Mn (Fig. 8(a, e)), Fe-0.4C-2Mn (Fig.

8(b, f)) and Fe-0.2C-5Mn (Fig. 8(d, h)) alloys, the enhancement factor of 7.5E3 is used. For

simulations at 650 C based on the Fe-0.6C-2Mn alloy, the same trial and error approach is used to

obtain an enhancement factor of 2E4. For all the 2Mn alloys (Fig. 8(a-c, e-g)), both simulated Mn

partitioning kinetics and cementite coarsening kinetics agree well with the experimental

measurements. At these temperatures, the available thermodynamic and kinetic descriptions for

ferrite (BCC_A2) seem sufficient to describe the cementite precipitation in the martensite matrix. The

volume diffusion assumption (for Mn diffusion) also works well at these high temperatures.

For the Fe-0.2C-5Mn alloy at 600C, the cementite formation competes with austenite precipitation

(Fig. 5) and cementite coarsens in an environment where the C and Mn contents for cementite-ferrite

mixture is decreasing. If the competition with austenite can be simply represented by a gradual

change in the available C and Mn contents for the cementite-ferrite mixture, a similar dissolution-

coarsening simulation for cementite-ferrite two phase can be implemented with a controlled bulk

chemistry during the simulation. The austenite growth kinetics was measured using XRD (Fig. 5) and

it is reasonable to assume austenite grows with constant composition (at equilibrium). Therefore, the

time evolution of effective composition for the cementite-ferrite mixture can be deduced from the

austenite growth kinetics (fitted by KJMA model in Fig. 5). By successively reducing the bulk C and

Mn for the cementite-ferrite mixture in the simulation, the simulated Mn partitioning and growth

kinetics are shown in Fig. 8(d, h). Using this treatment, the reduced C and Mn content for cementite-

ferrite due to austenite formation can reflect the evolution of tie-lines with increased but then

decreased Mn interfacial composition. This simulation also uses the same Mn enhancement factor at

600C.

An example of the time evolution of interfacial condition (i.e. the tie-lines between ferrite and

cementite) is shown in Fig. 9 from the simulation results of Fe-0.1C-2Mn at 600C (Fig. 9a) and Fe-

0.2C-5Mn at 600C (Fig. 9b). Throughout the entire the reaction, the tie-line at the cementite/ferrite

(shown as Mn U-fraction in cementite and ferrite) continuously changes. This is especially true in the

case of Fe-0.2C-5Mn at 600C where the choice of tie-line is not only affected by the C and Mn

diffusion in both cementite and ferrite but also influenced by the austenite formation. The studied

cementite coarsening reaction is not at steady state but in a transient state all the time in the

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22

examined time frames. The evolution of the tie-line has a direct effect on the Mn composition inherited

at the interface of the cementite (Fig. 4). If the Mn content inside the growing particle cannot

redistribute very quickly, then we may obtain the types of ‘smiley face’ Mn profiles shown in Fig. 4b&c

and this results from the historical time evolution with increasing Mn content at the interface.

Equivalently, the dissolving particles are dissolving in response to the same lower carbon activity in

the surroundings, and hence the cementite interfacial Mn content will also increase to satisfy local

equilibrium. Therefore, either growing or dissolving particles can show a ‘smiley face’ Mn gradient

within the cementite particle.

The negative Mn profile seen in some particles, e.g. Fig. 4a, is more difficult to rationalize. In one

respect, the profile of Mn within the cementite particle indicates that it is subjected to some

redistribution within the cementite to decrease the Mn gradient. This would tend to suggest that

cementite particles containing negative gradients, at some point during their growth, the tie-line

determining the interfacial composition moved to higher carbon activities with lower interfacial Mn

content in cementite rather than lower activities as a function of time. Such a seemingly unlikely

process is thermodynamically permissible (so long as the activity at the dissolving particle is higher),

but we think that it is only the growing particles that could possibly contain these negative gradients.

The circumstances around this behavior are likely due to the system having to choose a different tie-

line (corresponding to a lower Mn content in the growing cementite) to satisfy the compatibility

between the slow Mn diffusion in ferrite and a large carbon flux coming from the dissolving particles. It

is possible that this is the explanation but to prove this would require an extended version of the type

of model we have presented that considers the full particle size distribution with more than one

representative particle to describe the growing population and one to describe the dissolving

population.

Fig. 9. Evolution of simulated tie-lines at the cementite/ferrite interface as a function of time in (a) Fe-0.1C-2Mn at

600 C and (b) Fe-0.2C-5Mn at 600 C. UMn(CEM/BCC) representing Mn U-fraction in cementite at the interface (left y-axis) and UMn(BCC/CEM) representing Mn U-fraction in ferrite at the interface (right y-axis).

4.5. Simulation of Fe-C-Mn alloy at 500°C and 400°C

The simulations are extended to lower temperatures: 500°C (Fig. 10) and 400°C (Fig. 11). As was the

case at 600C, the Mn mobilities within cementite are enhanced to obtain the simultaneous

description of Mn partitioning kinetics and cementite growth kinetics. However, as shown in the Fig.

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23

10(a, d) and Fig. 10(b, e), a simple manipulation of Mn mobility in cementite may not be sufficient to

fully describe the experimental measurements.

Fig. 10. Comparisons between simulated kinetics and experimental data at 500 C: Mn U-fraction in cementite

and average particle radius as functions of time for (a, d) Fe-0.1C-2Mn alloy at 500 C, (b, e) Fe-0.4C-2Mn alloy

at 500 C, (c, f) Fe-0.2C-5Mn alloy at 500 C. Enhancement factors for Mn mobilities in cementite (MMn(CEM)) and ferrite (MMn(BCC)) used in the simulations are labelled.

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24

Fig. 11. Comparisons between simulated kinetics and experimental data at 400 C: Mn U-fraction in cementite

and average particle radius as functions of time for (a, d) Fe-0.1C-2Mn alloy at 400 C, (b, e) Fe-0.4C-2Mn alloy

at 400 C, (c, f) Fe-0.2C-5Mn alloy at 400 C. Enhancement factors for Mn mobilities in cementite (MMn(CEM)) and ferrite (MMn(BCC)) used in the simulations are labelled.

At these lower temperatures, it is suggested that Mn diffusion in martensite may not be adequately

approximated by volume diffusion in ferrite. This is because the measured Mn enrichment kinetics at

lower temperatures (especially at 400°C) cannot be reconciled with the limited bulk diffusion occurring

in ferrite according to the MOBFE3 Mn diffusion data. If using the diffusion data summarised by

Fridberg et al. [64], the Mn grain boundary diffusivity in ferrite at 500°C is about 6 orders of magnitude

higher than the volume diffusivity (10-15

m2/s vs. 10

-21 m

2/s). In the highly dislocated martensite,

cementite preferentially nucleates at dislocations and lath boundaries, and grows at the boundaries

during coarsening. These high-diffusivity paths are practically more important at lower temperature as

the difference between volume diffusion and dislocation/boundary diffusion becomes larger.

A simple expression to estimate the ‘effective’ diffusivity from volume and dislocation pipe diffusion

has been suggested by Dutta et al. [65]. The calculated effective diffusivity defines the relative

contribution of volume diffusivity and dislocation pipe diffusion based on the dislocation density.

Examples of simulating carbide precipitation utilising similar methods can be found in Radis and

Kozeschnik [66] and Zamberger et al. [67]. In the context of martensite tempering, the reasonable

range of dislocation density can be 1013

-1016

m-2

and it is expected that the dislocation and

substructure recover during tempering. Following the method of Dutta et al., a reasonable

enhancement factor for Mn diffusivity in ferrite can be 1-100 at 500°C and 1-300 at 400°C. A detailed

examination of the diffusion in martensite requires and deserves a separate study and here constant

Mn mobilities are fitted to the experimental data over the simulation time.

For the simulations based on Fe-0.1C-2Mn and Fe-0.4C-2Mn alloys, the enhancement factors for Mn

mobility in ferrite are 5 at 500°C (Fig. 10(a, b& d, e)) and 100 at 400°C (Fig. 11 (a, b& d, e)). All the

enhancement factors used are within the range of the calculation considering the effect from fast-

diffusion path. As shown from the 500°C simulation (Fig. 10(a, b& d, e)), enhanced Mn mobility in

ferrite significantly accelerates the Mn enrichment kinetics and at the same time the growth kinetics

are slightly slowed. In comparison, the Mn mobility in cementite remains the more dominant effect

determining the growth kinetics and fitting the Mn mobility in cementite still can be made with

confidence.

For the simulations based on the Fe-0.2C-5Mn alloy, the enhancement factors for Mn mobility in

ferrite are 1 at 500°C (Fig. 10(c, f)) and 15 at 400°C (Fig. 11 (c, f)). The enhancement factors are

consistently lower than those for the 2Mn alloys at both temperatures. In martensite, the dislocation

density is expected to correlate with carbon content [68] and if considering the dislocation density

determines the effective diffusivity we would see a stronger dependence of enhancement factor on C

instead of Mn. The chemical interaction during Mn diffusion may be altered when fast diffusion path is

involved.

There are uncertainties in the kinetic data used as inputs in the simulations. Good agreement

between experimental and simulated line profiles are obtained when the Mn mobility in cementite is

enhanced by 2.5E4 at 500°C and 1E6 at 400°C (Fig. 4b-c). This is an important indication that self-

consistency can be obtained by the proposed model and the mobility data used in the simulation is

reasonable.

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4.6. Diffusivity of substitutional elements in cementite

From the investigation of cementite precipitation at 400°C, 500°C and 600°C, different Mn mobilities in

cementite are obtained at each temperature by comparing with the experimental data. The extraction

of Mn diffusion data in cementite is possible due to its important kinetic effects. Mn tracer diffusivities

in cementite (calculated from mobilities in Fig. 8, Fig. 10 and Fig. 11) are plotted at each temperature

in Fig. 12 and the data (red circles) exhibit a linear relationship in the Arrhenius plot.

Fig. 12. Arrhenius plot showing natural logarithm of Mn tracer diffusivity in cementite as a function of 1/temperature. Red circles are the values in the simulations best describing the experimental results. Mn diffusivity in cementite (CEM), ferrite-Fe (BCC) and austenite-Fe (FCC) from MOBFE3 database and Mn diffusivity in cementite from Ågren et al. [69] and Barnard et al. [22] are included.

Compared with the current MOBFE3 database (Mn diffusion in cementite from Ågren et al. [69] (taken

from Fridberg et al. [64]) and Barnard et al. [22]), the diffusivities used in our simulation are

considerably higher. The diffusivities of Mn in cementite (that can describe the Mn partitioning

kinetics, cementite growth kinetics and Mn profile of cementite from the experimental data) is

comparable to the Mn diffusivity in ferrite. For the most important intermetallic phase, the

substitutional diffusivities in cementite are unknown. The reported Mn diffusivity are inferred from Mn

concentration in cementite from the phase transformation in steel, which resembles our method. The

lack of Mn diffusion data in cementite is mainly due to the experimental difficulty. Either conventional

radiotracer method or measuring spatial chemistry change are difficult to achieve experimentally

because synthesising cementite and maintaining its metastability at elevated temperatures is difficult.

Carbon diffusion in cementite has been studied experimentally from the formation kinetics of a

cementite surface layer by carburising fine iron particles [70,71] and it is theoretically assessed by

Hillert et al. [72]. This is not applicable for substitutional elements, like Mn. An in-depth study of

diffusion within cementite is required and may be pursued by atomistic approaches, such as density

functional theory calculation [73,74].

Although the extracted Mn diffusivities in cementite may seem high, it is certain that Mn can diffuse

much faster than expected based on the available expressions. In Fig. 12, the Arrhenius law for the

Mn diffusivities in cementite (red circles) shows a smaller slope than other phases and the slope

represents Q/R (Q is the activation energy of diffusion and R is the gas constant). If atom–vacancy

exchange mechanism is assumed for the substitutional diffusion in cementite, Q involves two

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26

energetic terms Em + Ef: Em represents the energy barrier of jumping to the neighbouring vacancy and

Ef represents formation energy of the vacancy. For cementite phase with lower Em and Ef, it is thus

speculated that the cementite is highly defected. This may be reasonable for such a fast, initial growth

of cementite from the highly distorted martensite matrix and would be a worthwhile topic for a

dedicated in-depth investigation.

5. Summary and conclusions

A detailed study of the kinetics of cementite coarsening during martensite tempering in a range of low

and medium carbon ternary Fe-C-Mn steels was conducted in the temperature range from 400C to

600C. Both Mn enrichment kinetics in cementite and cementite growth kinetics are measured

quantitatively as a function of alloy chemistry, tempering time and temperature. From the experimental

data, it is shown that the bulk C has a rather limited effect on the Mn partitioning kinetics and a strong

influence of bulk Mn was highlighted.

In the time frames examined, cementite precipitation is an interesting coarsening reaction occurring in

a matrix containing strong Mn concentration gradients that drive partitioning of the Mn to the

cementite. During the cementite coarsening, the different driving forces of C and Mn diffusion need to

be respected: C diffusion is driven by capillarity effect while Mn diffusion is driven by composition

gradients. It is important to couple the C flux and Mn flux to determine the operative tie-lines at the

particle/matrix interface, and hence describe the Mn enrichment kinetics in the cementite and

cementite growth kinetics.

A model coupling the growth and dissolution of cementite was developed to quantitatively describe

the Mn enrichment kinetics and cementite growth kinetics. The model describes well the Mn

partitioning kinetics and growth kinetics over a range of compositions and temperatures. This includes

a special condition in Fe-0.2C-5Mn alloy tempered at 600°C, where austenite growth is in competition

with cementite growth.

Finally, the uncertainty of Mn diffusivity in cementite is discussed. Due to its strong effect on the

cementite dissolution kinetics and Mn partitioning kinetics, the Mn diffusivity within cementite is

obtained at each temperature by comparing with both the experimental Mn concentration profiles

within cementite and the cementite size evolution. An Arrhenius expression of Mn diffusivity in

cementite is presented and the need for further investigation of Mn mobility in cementite is

emphasised.

Acknowledgments

The authors gratefully acknowledge the support of ArcelorMittal and the Australian Research Council

through the Linkage Grant Scheme (LP150100756). YXW gratefully acknowledges the award of the

Australian Government Research Training Program. Dr. Henrik Larsson (Thermo-Calc, Sweden) is

gratefully acknowledged for technical support regarding DICTRA. YXW gratefully acknowledges

A/Prof. Nicole Stanford and A/Prof. Laure Bourgeois for the stimulating discussions during the PhD

mid-candidature review, and Mr. Patrick Barges and Dr. Marion Bellavoine (ArcelorMittal R&D, France)

for the discussions on the replica sample preparation. The authors would also like to express thanks

for the use of equipment within the Monash Centre for Electron Microscopy (Mr. Renji Pan is

acknowledged for carbon coating) and the Monash X-Ray Platform.

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References

[1] R.A. Grange, C.R. Hribal, L.F. Porter, Hardness of tempered martensite in carbon and low-alloy steels, Metall. Trans. A. 8 (1977) 1775–1785. doi:10.1007/BF02646882.

[2] W.S. Owen, The effect of silicon on the kinetics of tempering, Trans. Am. Soc. Met. 46 (1954) 812–829.

[3] E. Biro, J.R. McDermid, J.D. Embury, Y. Zhou, Softening Kinetics in the Subcritical Heat-Affected Zone of Dual-Phase Steel Welds, Metall. Mater. Trans. A. 41 (2010) 2348–2356. doi:10.1007/s11661-010-0323-2.

[4] K. Kuo, A. Hultgren, Isothermal transformation of austenite III. Distribution of alloy steels, Jernkontorets Ann. 135 (1951) 449.

[5] R.W. Gurry, J. Christakos, L.S. Darken, Size, manganese content, and curie point of carbides extracted from manganese steel, Trans. ASM. 53 (1961) 187.

[6] M. Hillert, M. Waldenström, A thermodynamic analysis of the Fe-Mn-C system, Metall. Trans. A. 8 (1977) 5–13. doi:10.1007/BF02677257.

[7] A. Arlazarov, Evolution of microstructure and mechanical properties of medium Mn steels and their relationship (Ph.D Thesis), Université de Lorraine, 2015.

[8] J. Lis, J. Morgiel, A. Lis, The effect of Mn partitioning in Fe–Mn–Si alloy investigated with STEM-EDS techniques, Mater. Chem. Phys. 81 (2003) 466–468. doi:10.1016/S0254-0584(03)00053-1.

[9] M. Hillert, K. Nilsson, L.-E.Torndahl, Effect of Alloying Elements on the Formation of Austenite and Dissolution of Cementite, J. Iron Steel Inst. 209 (1971) 49–66.

[10] C. Atkinson, T. Akbay, R.C. Reed, Theory for reaustenitisation from ferrite/cementite mixtures in Fe-C-X steels, Acta Metall. Mater. 43 (1995) 2013–2031. doi:10.1016/0956-7151(94)00366-P.

[11] M. Gouné, P. Maugis, J. Drillet, A Criterion for the Change from Fast to Slow Regime of Cementite Dissolution in Fe–C–Mn Steels, J. Mater. Sci. Technol. 28 (2012) 728–736. doi:10.1016/S1005-0302(12)60122-4.

[12] R.B. Carruthers, M.J. Collins, Use of scanning transmission electron microscopy in estimation of remanent life of pressure parts, in: G.W. Lorimer, M.H. Jacobs, P. Doig (Eds.), Quant. Microanal. with High Spat. Resolut., The Metal Society London, London, 1981: pp. 108–111.

[13] A. Afrouz, M.J. Collins, R. Pilkington, Microstructural examination of 1Cr-0.5Mo steel during creep, Met. Technol. 10 (1983) 461–463. doi:10.1179/030716983803291569.

[14] R.C. Thomson, H.K.D.H. Bhadeshia, Changes in chemical composition of carbides in 2·25Cr–1Mo power plant steel: Part 2 Mixed microstructure, Mater. Sci. Technol. 10 (1994) 205–208. doi:10.1179/mst.1994.10.3.205.

[15] A. Hultgren, Isothermal transformation of austenite, Trans. Am. Soc. Met. 39 (1947) 915–1005.

[16] T. Sato, T. Nishizawa, Study on Carbides in Iron and Steel by Electrolytic Isolation (Report 2). Partition of Alloying Elements between Ferrite and Carbides, J. Japan Inst. Met. 19 (1955) 385–389. doi:10.2320/jinstmet1952.19.6_385.

[17] T. Sato, T. Nishizawa, Y. Honda, Study on carbides in practical special steels by electrolutica isolation (I), Tetsu-to-Hagane. 41 (1955) 1188–1192. doi:10.2355/tetsutohagane1955.41.11_1188.

[18] G. Ghosh, G.B. Olson, Precipitation of paraequilibrium cementite: Experiments, and thermodynamic and kinetic modeling, Acta Mater. 50 (2002) 2099–2119. doi:10.1016/S1359-6454(02)00054-X.

[19] G. Miyamoto, J.C. Oh, K. Hono, T. Furuhara, T. Maki, Effect of partitioning of Mn and Si on the growth kinetics of cementite in tempered Fe–0.6 mass% C martensite, Acta Mater. 55 (2007) 5027–5038. doi:10.1016/j.actamat.2007.05.023.

[20] T. Kunitake, On the Concentrating Process of the Alloying Element into Cementite During the Tempering of Steel, J. Japan Inst. Met. 30 (1966) 481–487. doi:10.2320/jinstmet1952.30.5_481.

[21] T. Sato, T. Nishizawa, M. Ohashi, STUDY ON CARBIDES IN PRACTICAL SPECICAL STEELS BY ELECTROLYTIC ISOLATION (III), Tetsu-to-Hagane. 43 (1957) 485–489. doi:10.2355/tetsutohagane1955.43.4_485.

[22] S.J. Barnard, G.D.W. Smith, A.J. Garratt-Ree, J. Vander Sande, Atom probe studies: 1) The role of silicon in tempering of steel and 2) low temperature chromium diffusivity in bainite, in: H.I. Aaronson, D.E. Laughlin, R.F. Sekerka, C.M. Wayman (Eds.), Proc. an Int. Conf. Solid-Solid Phase Transform., Pittsburgh, PA, 1981: pp. 881–885.

[23] S.S. Babu, K. Hono, T. Sakurai, Atom probe field ion microscopy study of the partitioning of substitutional elements during tempering of a low-alloy steel martensite, Metall. Mater. Trans. A. 25 (1994) 499–508.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

28

doi:10.1007/BF02651591.

[24] S.S. Babu, K. Hono, T. Sakurai, APFIM studies on martensite tempering of Fe-C-Si-Mn low alloy steel, Appl. Surf. Sci. 67 (1993) 321–327. doi:10.1016/0169-4332(93)90333-7.

[25] R.C. Thomson, M.K. Miller, The partitioning of substitutional solute elements during the tempering of martensite in Cr and Mo containing steels, Appl. Surf. Sci. 87–88 (1995) 185–193. doi:10.1016/0169-4332(94)00496-X.

[26] R.C. Thomson, M.K. Miller, An Atom Probe Study of Cementite Precipitation in Autotempered Martensite in an Fe-Mn-C alloy, Appl. Surf. Sci. 95 (1996) 313–319.

[27] R.C. Thomson, M.. Miller, Carbide precipitation in martensite during the early stages of tempering Cr- and Mo-containing low alloy steels, Acta Mater. 46 (1998) 2203–2213. doi:10.1016/S1359-6454(97)00420-5.

[28] R.C. Thomson, H.K.D.H. Bhadeshia, Changes in chemical composition of carbides in 2·25Cr–1Mo power plant steel: Part 1 Bainitic microstructure, Mater. Sci. Technol. 10 (1994) 193–204. doi:10.1179/mst.1994.10.3.193.

[29] L. Chang, G.D.W. Smith, The silicon effect in the tempering of martensite in steels, Le J. Phys. Colloq. 45 (1984) C9-397-C9-401. doi:10.1051/jphyscol:1984966.

[30] K. Zhu, H. Shi, H. Chen, C. Jung, Effect of Al on martensite tempering: comparison with Si, J. Mater. Sci. (2018). doi:10.1007/s10853-018-2037-6.

[31] C. Wert, C. Zener, Interference of growing spherical precipitate particles, J. Appl. Phys. 21 (1950) 5–8. doi:10.1063/1.1699422.

[32] H.K.D.H. Bhadeshia, Theoretical analysis of changes in cementite composition during tempering of bainite, Mater. Sci. Technol. 5 (1989) 131–137. doi:10.1179/mst.1989.5.2.131.

[33] O. Bannyh, H. Modin, S. Modin, An electron microscopic study of a eutectoid carbon steel after quenching and tempering, Jernkontorets Ann. 146 (1962) 774.

[34] G.P. Airey, T.A. Houghs, R.F. Mehl, The growth of cementite particles in ferrite, Trans. Metall. Soc. AIME. 242 (1968) 1853.

[35] K.M. Vedula, R.W. Heckel, Spheroidization of binary Fe-C alloys over a range of temperatures, Metall. Trans. 1 (1970) 9–18. doi:10.1007/BF02819236.

[36] T. Sakuma, N. Watanabe, T. Nishizawa, The Effect of Alloying Element on the Coarsening Behavior of Cementite Particles in Ferrite, Trans. Japan Inst. Met. 21 (1980) 159–168. doi:10.2320/matertrans1960.21.159.

[37] S. Björklund, L.F. Donaghey, M. Hillert, The effect of alloying elements on the rate of ostwald ripening of cementite in steel, Acta Metall. 20 (1972) 867–874. doi:10.1016/0001-6160(72)90079-X.

[38] E. De Moor, P.J. Gibbs, J.G. Speer, D.K. Matlock, J.G. Schroth, Strategies for third-generation advanced high-strength steel development, Iron Steel Technol. Trans. 7 (2010) 133–144.

[39] E. Smith, J. Nutting, Direct carbon replicas from metal surfaces, Br. J. Appl. Phys. 7 (1956) 214–217.

[40] G. Cliff, G.W. Lorimer, The quantitative analysis of thin specimens, J. Microsc. 103 (1975) 203–207. doi:10.1111/j.1365-2818.1975.tb03895.x.

[41] X.-H. Zhou, S. Gao, Confidence intervals for the log-normal mean, Stat. Med. 16 (1997) 783–790. doi:10.1002/(SICI)1097-0258(19970415)16:7<783::AID-SIM488>3.0.CO;2-2.

[42] H.M. Rietveld, A profile refinement method for nuclear and magnetic structures, J. Appl. Crystallogr. 2 (1969) 65–71. doi:10.1107/S0021889869006558.

[43] Y. Ohmori, A.T. Davenport, R.W.K. Honeycombe, Tempering of a high carbon martensite, Trans. Iron Steel Inst. Japan. 12 (1972) 112–117.

[44] G.R. Speich, K.A. Taylor, Tempering of Ferrous Martensite, in: G.B. Olson, W.S. Owen (Eds.), Martensite. A Tribut. to Morris Cohen, ASM International, 1992: pp. 243–275.

[45] I.M. Lifshitz, V.V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids. 19 (1961) 35–50. doi:10.1016/0022-3697(61)90054-3.

[46] C.Z. Wagner, Z. Elektrochem, Theory of precipitate change by redissolution, Angew. Phys. Chem. 65 (1961) 581–591.

[47] R.W. Heckel, R.L. Degregoria, The growth and shrinkage rates of second-phase particles of various size distributions, Trans. Metall. Soc. AIME. 233 (1965) 2001.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

29

[48] P. Deb, M.C. Chaturvedi, Coarsening behavior of cementite particles in a ferrite matrix in 10B30 steel, Metallography. 15 (1982) 341–354. doi:10.1016/0026-0800(82)90026-X.

[49] V. Massardier, M. Goune, D. Fabregue, A. Selouane, T. Douillard, O. Bouaziz, Evolution of microstructure and strength during the ultra-fast tempering of Fe–Mn–C martensitic steels, J. Mater. Sci. 49 (2014) 7782–7796. doi:10.1007/s10853-014-8489-4.

[50] H. Luo, J. Liu, H. Dong, A Novel Observation on Cementite Formed During Intercritical Annealing of Medium Mn Steel, Metall. Mater. Trans. A Phys. Metall. Mater. Sci. 47 (2016) 3119–3124. doi:10.1007/s11661-016-3448-0.

[51] olmogoroff, ur tatisti der ristallisationsvorg nge in Metallen, Zur Stat. Der Krist. Met. 1 (1937) 355–359.

[52] W.A. Johnson, R.F. Mehl, Reaction Kinetics in Processes of Nucleation and Growth, Trans. AIME. 135 (1939) 416–442. http://link.springer.com/10.1007/s11661-010-0421-1.

[53] M. Avrami, Kinetics of Phase Change. I General Theory, J. Chem. Phys. 7 (1939) 1103–1112. doi:10.1063/1.1750380.

[54] M. Avrami, Kinetics of Phase Change. II Transformation‐Time Relations for Random Distribution of

Nuclei, J. Chem. Phys. 8 (1940) 212–224. doi:10.1063/1.1750631.

[55] M. Avrami, Granulation, Phase Change, and Microstructure Kinetics of Phase Change. III, J. Chem. Phys. 9 (1941) 177–184. doi:10.1063/1.1750872.

[56] R.A. Oriani, Ostwald ripening of precipitates in solid matrices, Acta Metall. 12 (1964) 1399–1409. doi:10.1016/0001-6160(64)90128-2.

[57] C.-Y. Li, J.. Blakely, A.. Feingold, Mass transport analysis for ostwald ripening and related phenomena, Acta Metall. 14 (1966) 1397–1402. doi:10.1016/0001-6160(66)90159-3.

[58] J. Ågren, M.T. Clavaguera-Mora, J. Golcheski, G. Inden, H. Kumar, C. Sigli, Applications of computational thermodynamics: group 3: application of computational thermodynamics to phase transformation nucleation and coarsening, Calphad. 24 (2000) 41–54. doi:10.1016/S0364-5916(00)00014-6.

[59] J.O. Andersson, J. Ågren, Models for numerical treatment of multicomponent diffusion in simple phases, J. Appl. Phys. 72 (1992) 1350–1355. doi:10.1063/1.351745.

[60] J.-O. Andersson, T. Helander, L. Höglund, P. Shi, B. Sundman, Thermo-Calc & DICTRA, computational tools for materials science, Calphad. 26 (2002) 273–312. doi:10.1016/S0364-5916(02)00037-8.

[61] Å. Gustafson, L. Höglund, J. Ågren, Simulation of carbo-nitride coarsening in multicomponent Cr-steels for high temperature applications, in: R. Viswanathan, J. Nutting (Eds.), Adv. Heat Resist. Steels Power Gener. Conf. Proc. 27-29 April 1998 San Sebastian, Spain, IOM Communications Ltd., 1998: pp. 270–276.

[62] M. Perez, M. Dumont, D. Acevedo-Reyes, Implementation of classical nucleation and growth theories for precipitation, Acta Mater. 56 (2008) 2119–2132. doi:10.1016/j.actamat.2007.12.050.

[63] J.J. Kramer, G.M. Pound, R.F. Mehl, The free energy of formation and the interfacial enthalpy in pearlite, Acta Metall. 6 (1958) 763–771. doi:10.1016/0001-6160(58)90051-8.

[64] J. Fridberg, L.-E. Törndahl, M. Hillert, Diffusion In Iron, Jernkontorets Ann. 153 (1969) 263–276.

[65] B. Dutta, E.J. Palmiere, C.M. Sellars, Modelling the kinetics of strain induced precipitation in Nb microalloyed steels, Acta Mater. 49 (2001) 785–794. doi:10.1016/S1359-6454(00)00389-X.

[66] R. Radis, E. Kozeschnik, Numerical simulation of NbC precipitation in microalloyed steel, Model. Simul. Mater. Sci. Eng. 20 (2012) 055010. doi:10.1088/0965-0393/20/5/055010.

[67] S. Zamberger, T. Wojcik, J. Klarner, G. Klösch, H. Schifferl, E. Kozeschnik, Computational and Experimental Analysis of Carbo-Nitride Precipitation in Tempered Martensite, Steel Res. Int. 84 (2013) 20–30. doi:10.1002/srin.201200047.

[68] M. Kehoe, P.M. Kelly, The role of carbon in the strength of ferrous martensite, Scr. Metall. 4 (1970) 473–476. doi:10.1016/0036-9748(70)90088-8.

[69] J. Ågren, H. Abe, T. Suzuki, Y. Sakuma, The dissolution of cementite in a low carbon steel during isothermal annealing at 700°C, Metall. Trans. A. 17 (1986) 617–620. doi:10.1007/BF02643980.

[70] B. Ozturk, V.L. Fearing, J. a. Ruth, G. Simkovich, Self-Diffusion Coefficients of Carbon in Fe3C at 723 K via the Kinetics of Formation of This Compound, Metall. Trans. A. 13 (1982) 1871–1873. doi:10.1007/BF02647845.

[71] A. Schneider, G. Inden, Carbon diffusion in cementite (Fe3C) and Hägg carbide (Fe5C2), Calphad

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

30

Comput. Coupling Phase Diagrams Thermochem. 31 (2007) 141–147. doi:10.1016/j.calphad.2006.07.008.

[72] M. Hillert, L. Höglund, J. Ågren, Diffusion in interstitial compounds with thermal and stoichiometric defects, J. Appl. Phys. 98 (2005) 053511. doi:10.1063/1.1999833.

[73] M. Mantina, Y. Wang, R. Arroyave, L.Q. Chen, Z.K. Liu, C. Wolverton, First-principles calculation of self-diffusion coefficients, Phys. Rev. Lett. 100 (2008) 1–4. doi:10.1103/PhysRevLett.100.215901.

[74] M. Mantina, Y. Wang, L.Q. Chen, Z.K. Liu, C. Wolverton, First principles impurity diffusion coefficients, Acta Mater. 57 (2009) 4102–4108. doi:10.1016/j.actamat.2009.05.006.

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*Graphical Abstract