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Nitrogen diffusion through cementitelayersMarc Nikolussi a , Andreas Leineweber a & Eric Jan Mittemeijer aa Max Planck Institute for Metals Research , Heisenbergstrasse 3,D-70569 Stuttgart, GermanyPublished online: 30 Mar 2010.

To cite this article: Marc Nikolussi , Andreas Leineweber & Eric Jan Mittemeijer (2010)Nitrogen diffusion through cementite layers, Philosophical Magazine, 90:9, 1105-1122, DOI:10.1080/14786430903292365

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Philosophical MagazineVol. 90, No. 9, 21 March 2010, 1105–1122

Nitrogen diffusion through cementite layers

Marc Nikolussi, Andreas Leineweber* and Eric Jan Mittemeijer

Max Planck Institute for Metals Research, Heisenbergstrasse 3,D-70569 Stuttgart, Germany

(Received 12 July 2009; final version received 26 August 2009)

Massive cementite layers with a time-dependent thickness were grownon ferrite substrates by nitrocarburising in a dedicated NH3/H2/CO/N2-containing gas atmosphere at 783K, 823K and 843K. Nitrogen diffusionthrough the cementite layer into the ferrite substrate took place inconjunction with growth of the cementite layer; a significant, i.e.measurable, solubility of nitrogen in cementite was not observed.The nitrogen concentration-depth profiles in the substrate, underneaththe growing cementite layer, were quantitatively determined using acalibrated microhardness measurement technique. The nitrogen concentra-tion-depth profiles were simulated on the basis of a model using an implicitfinite-difference method. The simulation yielded values for the diffusivity ofnitrogen through cementite, including its activation energy.

Keywords: cementite; nitrogen diffusion; activation energy; nitrocarburis-ing; nitriding

1. Introduction

The corrosion and wear resistances of iron-based workpieces as well as their fatigueendurance can be improved markedly by thermochemical heat treatments such asnitriding and nitrocarburising, which are of great technological importance [1,2].

Gaseous nitriding and gaseous nitrocarburising are usually performed attemperatures between 773K and 853K. Thus, the process temperature is locatedbelow the binary/ternary eutectoid temperature of the Fe–N/Fe–N–C solidsolution [3,4]. The nitriding/nitrocarburising gas atmosphere provides nitrogen ornitrogen and carbon, which are taken up by the surface-region of the iron-basedworkpieces [5].

The incorporation of nitrogen or nitrogen and carbon results in (i) the formationof a compound layer (several 10 mm thick) and/or (ii) the formation of a diffusionzone (several 100 mm thick). The compound layer is responsible for a considerableenhancement of the corrosion and wear resistance. Depending on the treatmenttemperature, the pressure and the composition of the gas atmosphere, the compoundlayer can consist of different phases. Typically � 0-Fe4N1�y and "-Fe3(N,C)1þx areconstituents of the compound layer [6] but also cementite can occur in the compoundlayer [7,8]. The diffusion zone is responsible for a considerable enhancement of the

*Corresponding author. Email: a.leineweber@mf.mpg.de

ISSN 1478–6435 print/ISSN 1478–6443 online

� 2010 Taylor & Francis

DOI: 10.1080/14786430903292365

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fatigue endurance [9]. Nitrogen or nitrogen and carbon are dissolved (at thetreatment temperature) as solid solution in the octahedral interstices of the iron bcclattice.

It was shown recently that, under certain conditions, pure, massive cementitelayers can also form on the ferrite substrate [8]. These cementite layers have beeninvestigated in detail with respect to their homogeneous cementite-phase micro-structure, with respect to conditions for occurrence of additional (carbo)nitridephases and the corresponding phase equilibria as well as with respect to selectedmechanical properties [8,10–14].

In the presence of cementite compound layers, it turns out that the nitrogen (andcarbon) diffusion into the substrate is strongly decelerated; cementite acts as adiffusion barrier for nitrogen [15] and carbon [16]. Nonetheless, although nitrogentransport through cementite is decelerated, it is shown in this paper that a certaintime-dependent amount of nitrogen is transported through the cementite into theferrite substrate, leading to an increase of the nitrogen content in (and the hardnessof ) the ferrite substrate. The solubility of nitrogen in the ferrite is much higher thanfor carbon at the temperatures of interest (783–843K), e.g. values of 0.29 at.% fornitrogen vs. �0.012 at.% for carbon at 823K were found [17]. Furthermore, most ofthe carbon incorporated in the solid is used to build up the cementite layer, whereasnitrogen passes through the cementite layer to be dissolved in the ferrite.

The present work focusses on the diffusion of nitrogen through cementite.Hardness-depth profile measurements in the ferrite substrate are used to determinethe nitrogen concentration-depth profiles. A model is proposed that enables thesimulation of such nitrogen concentration-depth profiles. On this basis, for the firsttime, information about the diffusivity of nitrogen through cementite, including thecorresponding activation energy of nitrogen diffusion in cementite, is obtained.

2. Experimental

2.1. Specimen preparation and gaseous nitriding/nitrocarburising

Two differently produced types of specimens were investigated in this study: ‘‘Thick-plate’’ specimens for nitrocarburising experiments were produced by cold-rolling aferrite cast rod (Alfa Aesar, 99.98wt% Fe) on both sides to a thickness of 1mm.From the resulting cold-rolled sheet, rectangular pieces (20mm� 25mm) were cut,ground, polished (final stage 1 mm diamond suspension) and cleaned ultrasonically inethanol. ‘‘Thin-plate’’ specimens for nitriding experiments were produced by cuttingfrom a 10 cm� 10 cm iron sheet (Alfa Aesar, 99.98wt% Fe) of 0.1mm thicknessrectangular pieces of 20mm� 25mm which were ground, polished (final stage 1 mmdiamond suspension) and cleaned ultrasonically in ethanol. In order to removethe cold deformation, both types of specimens were, prior to gaseous nitriding/nitrocarburising, recrystallised at 973K for 2 h under a reductive hydrogen flowof 200mlmin�1, polished (final stage 1 mm diamond suspension) and cleanedultrasonically in ethanol.

The nitriding/nitrocarburising facility had to ensure the desired gas environmentand the desired temperature during treatment. Moreover, it was necessary to quenchthe specimens after treatment in order to retain the microstructure as at the

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treatment temperature. Thus, the nitriding/nitrocarburising facility was composed ofa vertical quartz-tube furnace which was, at its end, equipped with a water (flushedwith N2) container for quenching the specimens to room temperature. Using a quartzfibre, which, to realise quenching, can mechanically be destroyed after the nitriding/nitrocarburising treatment, the specimens prepared as described above werepositioned in the middle of the quartz-tube furnace where the process temperature(controlled within �1K) prevailed.

Two types of treatments were conducted in the present work: (i) nitrocarburisingexperiments to study nitrogen diffusion through growing cementite layers, and(ii) nitriding experiments to prepare specimens of homogeneous nitrogen contentand without compound layer. For these experiments, ammonia (99.999 vol.%) asnitrogen supply, carbon monoxide (99.97 vol.%) as carbon supply, hydrogen (99.999vol.%), and nitrogen (99.999 vol.%), as inert gas, were employed. Each gas fluxwas controlled by a separate mass-flow controller. An overall linear flow rate of13.5mm s�1 (calculated for the gas volume at room temperature) through the quartzretort (diameter 28mm) was maintained. Such a flow rate ensures that ammoniadissociation, which would change the composition of the gas atmosphere, can beneglected.

For gaseous nitrocarburising, the gas atmosphere was composed of 13 vol.%NH3, 58 vol.% H2 (nitriding potential rN¼ 0.3 atm�1/2), 20 vol.% CO and 9 vol.%N2 (as inert gas). The chemical potential of nitrogen in the gas atmosphere can berelated with the nitriding potential [18]. In the present work, no carburising potentialcan be adjusted: the carburising potential for a gas atmosphere composed of NH3/H2/CO/N2 is (hypothetically) infinite [8] although an ‘‘effective’’ carburisingpotential acts at the specimen surface [10]. The experiments were performed atT¼ 783K, 823K and 843K, for treatment times of 0.5 h, 2 h, 6 h, 14 h, 24 h and 48 h,always leading to massive cementite layers on ferrite substrates [8,10] (see Figure 1).For these experiments, ‘‘thick-plate’’ specimens with a thickness of 1mm, producedas described above, were used. In fact, the 783K and 843K specimens are amongthose used in [10] to analyse the cementite-layer growth kinetics at the correspondingtemperatures.

Figure 1. Optical micrograph (cross-sectional view; bright field) of a massive cementite layer(stained) on iron (unstained). The massive cementite layer was generated on ferrite by gaseousnitrocarburising at 823K for 48 h. The metallographic cross-section was etched with 1vol.% Nital containing 1 vol.% HCl and stained with an alkaline sodium picrate solution(cf. Section 2).

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Gaseous nitriding experiments were performed at 823K using different NH3/H2-containing gas atmospheres yielding different nitriding potentials (see Table 1).The applied experimental conditions ensured the formation of compound-layer freespecimens in accordance with the Lehrer diagram [19]. Gaseous nitriding wasperformed for 42 h in order that the ferrite substrates were saturated with interstitialnitrogen with respect to the nitriding gas atmosphere. For these experiments‘‘thin-plate’’ specimens with a thickness of 0.1mm, produced as described above,were used.

2.2. Microstructural and hardness analysis

The specimens after both the nitriding and the nitrocarburising experiments were cutinto four pieces, used for optical microscopy, hardness measurements, carrier-gas hotextraction and X-ray diffraction (XRD), respectively.

For optical microscopy of and hardness measurements on specimen cross-sections, the nitrocarburised specimens were covered with an electrodepositedprotective nickel layer using a Watts bath [6,20] at 333K. The protective nickel layeravoids mechanical damage at the surface of the metallographic cross-sections andguarantees the required sharpness at the sample surface in the cross-sections. Allspecimens (nitriding and nitrocarburising experiments) were embedded using aStruers LaboPress 3. Embedding was performed with 15ml Polyfast (BuehlerGmbH), a load of 15 kN, an annealing time of 5min at 453K and a cooling time of3min down to room temperature. After embedding, the specimens were ground,polished (final stage 1 mm diamond suspension) and etched using 1 vol.% Nitalcontaining 0.1 vol.% HCl [21]. In order to differentiate between the massive

Table 1. The experimentally determined (average) hardness values and (average) nitrogencontents (results from carrier-gas extraction) of all nitrided specimens. Composition (NH3 andH2 contents and corresponding nitriding potential) of the nitriding gas atmosphere, yieldingcompound-layer free specimens, have been indicated. The experiments were performed for42 h at 823K leading to a homogeneous solid solution of interstitial nitrogen in the ferritesubstrate. The hardness of pure iron without any nitrogen dissolved determined in this workequals 100� 2HV. Similar results obtained for some selected (see text) nitrocarburisingexperiments at 823K are given as well.

Nitriding experimentsNH3 content (vol.%) 2 4 6 7 9 12 13H2 content (vol.%) 98 96 94 93 91 88 87Nitriding potential,rN (atm�1/2)

0.02 0.04 0.06 0.08 0.1 0.14 0.16

Hardness [HV] 116� 7 119� 4 133� 6 147� 3 152� 3 182� 9 205� 7Nitrogen concentration(at.%)

0.034 0.075 0.108 0.144 0.184 0.297 0.367

Nitrocarburising experimentsTreatment time (h) 6 14 24 48Hardness (HV) 147� 4 158� 2 171� 2 194� 3Nitrogen concentration(at.%)

0.136 0.176 0.228 0.307

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cementite layer and the ferrite substrate, the cross-sectional specimens were stainedusing either (i) an alkaline potassium permanganate solution (1 g NaOH, 1 g KOHand 4 g KMnO4 per 100ml distilled water) at 328K [8,22], the effect of which issimilar to that of a Murakami solution [6,22], or (ii) an alkaline sodium picratesolution (25 g NaOH and 2 g picric acid per 75ml distilled water) at roomtemperature [22]. In both cases, the massive cementite layer becomes, due to its highcarbon content, severely stained. The ferrite substrate remains unstained and cantherefore easily be distinguished from the cementite layer using optical microscopy(Zeiss Axiophot microscope). By this type of staining it is also possible todifferentiate between different nitride, carbonitride and carbide phases (alsosupported by electron back-scatter diffraction) in the compound layers (see, forexample, [11]), indicating that samples in the present work indeed were of purecementite.

Hardness measurements (micro-Vickers) were performed with a Leica VMHTMOT microhardness indenter using a load of 25 gf (ffi0.245N) for 10 s. Hardness-depth profile measurements in the ferrite substrates were performed for all specimensof the nitrocarburising experiments. Hardness indents were set in the ferrite substrateat selected distances from the interface cementite/ferrite until the middle of thespecimen was reached. The exact distance values of the hardness indents from theinterface cementite/ferrite were determined by optical microscopy using the software‘‘analySIS’’ (Soft Imaging System GmbH). For each specific distance value from theinterface cementite/ferrite, five hardness measurements were made; the average valuewas taken as the local hardness and the standard deviation was taken as indication ofthe experimental error. For all specimens of the nitriding experiments, which shouldhave – due to the small specimen thickness and due to the long treatment time –homogeneous nitrogen content, 10 hardness measurements were made across thecross-sectional specimen; their average value and their standard deviation wascalculated.

In order to relate the hardness of the ferrite substrate with its nitrogen content,values for both quantities were determined for a couple of specimens withhomogeneous nitrogen contents in the ferrite (see Section 3). The nitrogen contentsof all the nitrided specimens (compound-layer free) were determined by carrier-gashot extraction. The same procedure was applied to specimens, which werenitrocarburised at 823K with a treatment time of 6 h, 14 h, 24 h and 48 h, sincethese specimens do not exhibit a nitrogen concentration (hardness) gradient acrossthe ferrite substrate in the cross-section (see Figure 2b). Prior to nitrogen-contentdetermination by carrier-gas hot extraction on the nitrocarburised specimens, themassive cementite layer was removed by grinding so that only the nitrogen-enrichedsubstrate was analysed. After completion of the grinding procedure, it was checkedby X-ray diffraction whether the entire cementite layer had been removed.

To additionally verify the presence or absence of a (cementite) compound layerand of the possible presence of other nitride and carbonitride phases, phaseidentification was carried out by X-ray diffractometry. A PANalytical X’Pert Multi-Purpose Diffractometer, which was equipped with a graphite monochromator in thediffracted beam, was used. The diffractometer was operated with CoK� radiation inBragg–Brentano geometry. Each specimen was rotated around its surface normalduring the measurements to achieve better crystallite statistics.

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3. Experimental results and evaluation: hardness measurements and nitrogen-

content determination

Each nitrocarburising experiment performed at 783K, 823K and 843K led to the

formation of a massive cementite layer on the ferrite substrates (see Figure 1) [8,10].

The cementite layers exhibit a characteristic microstructure (partially rough

ferrite-cementite interface) and a specific orientation relationship with the ferrite

substrate [12].

Figure 2. Experimentally determined hardness-depth profiles in the ferrite substrate with amassive cementite layer of time-dependent thickness at the surface of the ferrite substrate(dots; cf. hardness axes). Hardness-depth profiles were transformed into nitrogen concentra-tion-depth profiles (dots; cf. nitrogen concentration axes) applying the calibrated hardness-nitrogen concentration relation (cf. Figure 3). The simulated nitrogen concentration-depthprofiles are shown by the continuous curves. The dashed curves connect the data pointspertaining to the hardness/nitrogen concentration-depth profiles for 0.5 h and 2 h, which werenot considered for the simulation. Experiments as well as simulations were performed at(a) 783K, (b) 823K and (c) 843K. (Figure continued).

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After cross-sectional metallographic preparation, hardness-depth profile mea-

surements were performed on the ferrite substrate (see Figures 2a–c).In order to transform hardness-depth profiles into nitrogen concentration-depth

profiles (see Figures 2a–c), the dependence of hardness on nitrogen content in ferrite

has to be known. Such dependence was obtained on the basis of the results for

hardness and nitrogen content in homogeneously nitrided/nitrocarburised ferrite

given in Table 1 and shown in Figure 3.

Figure 3. Calibration line: nitrogen concentration (determined by carrier-gas hot extraction;the error is smaller than the size of the dots) vs. the hardness for homogeneous ferritesubstrates (determined by micro-Vickers hardness indentations). The straight line wasdetermined by least-squares fitting of Equation (1) to the experimental data. Solid data pointsindicate results from nitrided specimens; open data points indicate results from nitrocar-burised specimens.

Figure 2. Continued.

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It follows that a linear dependence exists between nitrogen content and hardness

of the ferrite substrate:

C �-FeN ¼ mhþ b, ð1Þ

where C�-FeN denotes the nitrogen concentration in the ferrite substrate, and h

its hardness. Least-squares fitting of a straight line to the experimental data in

Figure 3 yielded the following values for m (slope of the straight line in Figure 3)

and b (part cut from the ordinate): m ¼ ð3:46� 0:1Þ � 10�3 at:%HV and b¼�0.354�

0.02 at.%. On this basis, the experimentally determined hardness-depth profiles

were transformed into nitrogen concentration-depth profiles (see Figures 2a–c).

This was done for all considered treatment temperatures (783K, 823K and 843K)

and treatment times.Such quantitative information on the nitrogen concentration in ferrite cannot be

obtained by other techniques. Whereas, for example, carrier-gas hot extraction can

only provide an average nitrogen concentration across the whole specimen (instead

of a location-resolved nitrogen concentration), the sensitivity of electron probe

microanalysis (EPMA) is too low for the presently determined nitrogen

concentrations.

4. Modelling nitrogen concentration-depth profile development

4.1. General assumptions

The experimentally determined nitrogen concentration-depth profiles will be

simulated to analyse, on the basis of comparison with the experimentally determined

nitrogen concentration-depth profiles, the diffusivity of nitrogen through cementite.Consider a laterally infinitely extended iron plate of finite thickness. Nitrogen can

enter this plate from both sides; a symmetrical (with respect to the centre plane of the

plate) nitrogen concentration-depth profile is built up.The process of gaseous nitrocarburising can be subdivided into four different

steps: (i) diffusion of ammonia from the gas phase to the surface of the specimen;

(ii) dissociation of ammonia at the surface; (iii) nitrogen diffusion through the

growing cementite layer of time-dependent thickness; (iv) nitrogen diffusion within

ferrite (after having diffused through the cementite layer) from the cementite/

ferrite interface to the core of the specimen. For step (i), in the present

experiments, the gas flow through the quartz-tube furnace is large; the gas

composition at the sample surface can be taken as constant for a constant

treatment temperature [10]. For step (ii), already after a very short treatment time,

dissociation of ammonia at the surface of the specimen is not rate-determining for

the nitrogen uptake; instead nitrogen diffusion through the growing cementite

layer is rate-determining (in contrast with the case of nitriding pure ferrite [23]).

The conditions indicated under steps (i) and (ii) imply that local equilibrium

prevails at the interface of the gas atmosphere and the specimen. Furthermore,

local equilibrium is assumed to hold at the interface cementite/ferrite, which in the

present context, in particular implies that there is no chemical-potential gradient

for nitrogen at that interface.

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Nitrogen diffusion within ferrite is governed by Fick’s second law,

@C �-FeN

@t¼ D�-Fe

N

@2C �-FeN

@x2, ð2Þ

where D�-FeN is the diffusion coefficient of nitrogen in ferrite (taken as concentration

independent), C �-FeN the nitrogen concentration at depth x (measured from the

cementite/ferrite interface) and t the time of diffusion. The cementite layer is thincompared to the ferrite substrate at all considered treatment times and the cementitelayer grows much slower than the extent of the nitrogen concentration-depth profilein the substrate. Therefore, the consumption of a part of the ferrite upon formationof the cementite is neglected in the modelling of the diffusion of nitrogen in ferrite.

The flux of nitrogen through cementite at the interface cementite/ferrite,JFe3CN jx¼0, differs from the flux of nitrogen into ferrite at this interface J �-FeN jx¼0 bya term considering the amount of N consumed by the moving interface cementite/ferrite due to the concentration difference at the interface:

JFe3CN

���x¼0¼ J�-FeN

��x¼0þ CFe3C

N,interface � C�N,interface

� �dSFe3C

dt

¼ �D�-FeN

@C�-FeN

@x

����x¼0

þ CFe3CN,interface � C�N,interface

� �dSFe3C

dt, ð3Þ

where x¼ 0 denotes the location coordinate which represents the interface cementite/ferrite. There are no reliable data about the N content in cementite (it is even unclearwhether the concentration difference in Equation (3) should be positive or negative).However, test calculations with nitrogen contents in cementite ranging from 0 at.%to 0.5 at.% indicate that � given the known growth rate of cementite � the nitrogenconsumption or release due to the growing cementite (at the cost of ferrite with anitrogen content of C �-Fe

N,interface) is negligible compared to the overall nitrogen fluxthrough the cementite layer.

Nitrogen enters the ferrite substrate at both sides. Hence, a symmetrical nitrogenconcentration-depth profile results and no net flux of nitrogen through the middle ofthe sample occurs, leading to the second boundary condition

@C �-FeN

@x

����x¼Z

¼ 0, ð4Þ

where x¼Z denotes the position of the centre plane.Before nitrogen can pass the interface cementite/ferrite and can diffuse through

ferrite, it must be transported through the cementite layer. Nitrogen diffusionthrough cementite is provoked by a gradient of the chemical potential of nitrogenacross the cementite layer (see Figure 4)

F ¼ �@�Fe3C

N

@x

!p,T

, ð5Þ

where F is the thermodynamic force, �Fe3CN the chemical potential of nitrogen in

cementite and x the location coordinate, T the treatment temperature and p thepressure.

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For the nitrogen flux through cementite, JFe3CN , it holds that [24]

JFe3CN ¼ vFe3CN CFe3CN ¼MFe3C

N FCFe3CN

¼ �MFe3CN CFe3C

N

@�Fe3CN

@x

!p,T

¼ �DFe3CN

@CFe3CN

@x

!p,T

, ð6Þ

where vFe3CN denotes the drift velocity of nitrogen in cementite, MFe3CN is the mobility

of nitrogen in cementite, DFe3CN represents the diffusion coefficient of nitrogen in

cementite and CFe3CN is the nitrogen concentration in cementite. The diffusion

coefficient of nitrogen in cementite can be expressed by

DFe3CN ¼MFe3C

N RT 1þ@ ln �Fe3CN

@ lnCFe3CN

!, ð7Þ

where ð1þ@ ln �

Fe3C

N

@ lnCFe3C

N

Þ is the thermodynamic factor, with �Fe3CN as the activity

coefficient of nitrogen in cementite. The solubility of nitrogen in cementite is

extremely small, e.g. for 1593K a solubility of about 0.064 at.% was reported [25].

Due to this extremely limited solubility, Henry’s law [26,27] is assumed for nitrogen

in cementite. Consequently, the thermodynamic factor equals one and DFe3CN

reduces to

DFe3CN ¼MFe3C

N RT ¼ D�,Fe3CN , ð8Þ

Figure 4. Schematic drawing of a cementite layer on a ferrite substrate (cross-sectional view).A gradient of the chemical potential of nitrogen in cementite provokes the flux of nitrogenthrough cementite. Consequently, a nitrogen concentration-depth profile develops in theferrite substrate, which can be calculated for different time steps and grid points using themodel described in Section 4 and the Appendix (Online Supplementary Material) applyingthe implicit finite-difference method. Local equilibrium at the gas-solid interface and at thecementite/ferrite interface is assumed.

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where D�,Fe3CN is the (tracer-)diffusion coefficient of nitrogen in cementite. Thus,

JFe3CN ¼ �D�,Fe3CN

@CFe3CN

@x

!p,T

: ð6bÞ

A linear concentration profile of nitrogen across cementite is assumed in

Equation (6b). The concentration difference of nitrogen across the cementite layer

is given by the difference of the nitrogen concentration in cementite at the interface

gas atmosphere/cementite, CFe3CN,surface, and the nitrogen concentration in cementite at

the interface cementite/ferrite, CFe3CN,interface. Then Equation (6b) can be given as

JFe3CN ¼D�,Fe3CN CFe3C

N,surface � CFe3CN,interface

� �SFe3C

, ð9Þ

where SFe3C is the cementite-layer thickness.The nitrogen concentrations in cementite at both interfaces (gas atmosphere/

cementite and at the interface cementite/ferrite) will be expressed in terms of

‘‘corresponding’’ (see what follows) nitrogen concentrations in ferrite. For both,

cementite and ferrite, Henry’s law can be adopted. Therefore, for cementite and

ferrite in (hypothetical) equilibrium with each other and, possibly additionally, with

the same outer gas atmosphere, the concentrations of nitrogen in cementite, CFe3CN ,

and in ferrite, C�-FeN , are in equilibrium related by

CFe3CN ¼ ’C �-Fe

N , ð10Þ

where ’ is the partition coefficient. For the chemical potential of nitrogen in

cementite it holds that �Fe3CN ¼ �0

N þ RT ln �Fe3CN xFe3CN and for the chemical potential

of nitrogen in ferrite it holds that ��-FeN ¼ �0

N þ RT ln � �-FeN x �-FeN . Assuming for

both, cementite and ferrite, the same reference state and Henry’s law (constant

activity coefficient of nitrogen), the mole fractions of nitrogen in cementite and

ferrite in equilibrium with each other are related by a partition coefficient, ’,

according to ’ ¼� �-FeN

�Fe3C

N

. It will be assumed that, for the minute range of nitrogen

concentration in cementite, ’ does not depend on nitrogen concentration. Hence,

the nitrogen concentration in ferrite at the cementite/ferrite interface in (real, local)

equilibrium with cementite at the same interface corresponds to

C �-FeN,interface ¼ CFe3C

N,interface=’ and thus CFe3CN,interface ¼ ’C

�-FeN,interface. Diffusion of nitrogen

through the cementite will come to a halt if the nitrogen concentration in the ferrite

corresponds to the nitrogen concentration which would prevail in ferrite in

(hypothetical) equilibrium with the gas atmosphere, C �-FeN,eq . Hence, the concentration

of nitrogen in cementite at the surface in (real, local) equilibrium with the gas

atmosphere, CFe3CN,surface, is given by CFe3C

N,surface ¼ ’C�-FeN,eq .

The above consideration implies that Equation (9) can be written as

JFe3CN ¼’D�,Fe3CN C �-Fe

N,eq � C �-FeN,interface

� �SFe3C

: ð11Þ

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The cementite-layer thickness SFe3C has been determined experimentally for thepresent samples. The layer thickness could be described for each temperature byadopting a modified parabolic growth law [8,10]

S2Fe3CðtÞ ¼ kðT Þtþ S2

0ðT Þ, ð12Þ

where k denotes the parabolic growth constant for cementite and S0 a (hypothetical)initial cementite-layer thickness at t¼ 0 (see discussion in [8]). Combining

Equations (11) and (12) leads to JFe3CN ¼’D�,Fe3C

NðC�-Fe

N,eq�C�-Fe

N,interfaceÞ

ðktþS20Þ1=2 .

4.2. Simulation of nitrogen concentration-depth profiles: numerical procedure

For the simulation of the nitrogen concentration-depth profiles, Fick’s first andsecond law have to be solved subject to the boundary conditions given byEquations (3) and (4). This can be done numerically on the basis of the method afterCrank and Nicolson [23,28] using the implicit finite-difference method. For thatpurpose, the nitrogen concentration profile within ferrite was calculated forequidistant grid points in space (distance Dx) (see Figure 4) at different time steps(see what follows in Section 5). Details of the numerical procedure are described inthe Supplementary Material published together with the online version.

5. Results of the simulation

The simulation was performed with M¼ 100 grid points (up to the middle of thespecimen) and different time steps. The increment of time was increased by 0.5%after each generation of a nitrogen concentration profile [23], recognising thedecrease of the nitrogen concentration gradients with time, in order to reduce

calculation time. Doing this, the condition r ¼D�,Fe3C

NDt

ðDxÞ2� 1=2 has to be fulfilled [28],

where Dt denotes the increment of the time and Dx denotes the grid-point distance.The simulation was started at the respective treatment temperatures on the basis ofthe nitrogen concentration-depth profile determined after 6 h, since for smallertreatment times the diffusion geometry and mechanism is incompatible with themodel (see Section 6.2).

The following parameters were known: (i) the diffusion coefficient of nitrogenin ferrite, which was calculated for the respective treatment temperature accordingto [29] (see Table 2) and (ii) the cementite-layer thickness, which was determinedexperimentally. Hence, the fitting parameters were the equilibrium nitrogenconcentration in ferrite, C �-Fe

N,eq ðT Þ, and the product of the partition coefficient andthe (tracer-)diffusion coefficient of nitrogen in cementite, ’D�,Fe3CN . Values for theseparameters were determined (for each considered treatment temperature) byminimising the sum of the squared differences of simulated and experimentalnitrogen concentrations for all treatment times simultaneously (see Figures 2a–c).The results obtained for C �-Fe

N,eq and ’D�,Fe3CN are presented in Table 2 and Figure 5.Adopting an Arrhenius-type temperature dependence for the (tracer-)diffusion

coefficient of nitrogen in cementite and taking the partition coefficient ’ astemperature independent, a plot of the logarithm of ’D�,Fe3CN vs. 1/T should yield a

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straight line. This is indeed observed; see Figure 5. The slope of the straight lineobtained equals �Q=R, with Q as the activation energy of nitrogen diffusion incementite and R as the gas constant. It follows: Q¼ 65�2 kJmol�1.

6. Concluding discussion

6.1. Hardness/concentration-depth profiles

The hardness of the ferrite substrates increases linearly with the nitrogenconcentration. Upon embedding (cf. Section 2), which has to be considered as aheat treatment, the nitrogen in the ferrite forms (semi-)coherent �00-Fe16N2

precipitates [30–34], which are responsible for the hardness increase as measuredat room temperature. In order to guarantee comparability of the hardness

Figure 5. Arrhenius plot: natural logarithm of the product of the partition coefficient and the(tracer-)diffusion coefficient of nitrogen through cementite, ’D�,Fe3CN , versus the reciprocaltemperature 1/T.

Table 2. Results of the simulation of nitrogen concentration-depth profiles. The equilibriumnitrogen concentration in ferrite, and the product of the partition coefficient and the(tracer-)diffusion coefficient of nitrogen through cementite were refined by matching thesimulated data and the experimental data by least-squares fitting. Furthermore, the diffusioncoefficient of nitrogen in ferrite was obtained according to [29].

Treatment temperature, T(K) 783 823 843Equilibrium nitrogen concentrationin ferrite, C�-FeN,eq (at.%)

0.599 0.613 0.700

Product of the partition coefficient and the(tracer-)diffusion coefficient of nitrogenthrough cementite, ’D�,Fe3CN (m2 s�1)

2.9� 10�15 4.8� 10�15 5.9� 10�15

Diffusion coefficient of nitrogen inferrite, D�-Fe

N (m2 s�1)7.4� 10�12 1.3� 10�11 1.8� 10�11

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measurements, the embedding has to be performed in a reproducible way. Therefore,a special standard embedding procedure was developed, which has been described inSection 2.

In the course of relating the ferrite hardness with the nitrogen content in theferrite, the effect of present carbon was neglected. The amount of carbon which canbe dissolved in ferrite at 823K is more than 20 times smaller than the amount ofnitrogen (see introduction). Already because of this recognition it is expected that amuch smaller hardening effect due to carbides in ferrite is expected than due tonitrides. Indeed, for carbon in ferrite, an increase of Vickers hardness of a maximumof 15HV was observed in [35], which is much less than the maximum increase of 100HV observed for nitrogen in ferrite (cf. Figure 3). Moreover, in [35], the solidsolution treatment prior to the age hardening was performed at temperatures as highas 973K, which certainly led to dissolution of more carbon in the ferrite than in thepresent experiments conducted at �843K. Thus the (maximum possible) hardeningeffect due to carbon in ferrite in the present work is expected to be even smaller than15HV. Further, the data points in Figure 3 do not differ systematically from thecommon straight line, independently of whether the points originate from anitrocarburised specimen (containing some carbon in the ferrite) or from purelynitrided specimen (containing no carbon in the ferrite), confirming that the hardnessdepends practically only on the nitrogen content in the ferrite.

Moreover, it must be noted that the calibration line shown in Figure 3 wasdetermined on the basis of experiments performed at 823K. This calibration line wasalso applied to the data obtained at 783K and 843K. The true relation betweenhardness and nitrogen content at 783K and 843K may differ slightly from thatpertaining at 823K. This may be the origin of the slightly negative nitrogen-concentration values in the ferrite substrate for some data points at very shorttreatment times (0.5 h and 2 h) at 783K (cf. Figure 2a).

The data obtained for the nitrogen concentration-depth profiles at the threedifferent considered treatment temperatures indicate pronounced nitrogen concen-tration gradients in the ferrite matrix at short treatment times as 0.5 h and 2 h; theconcentration gradients level off with increasing treatment time. The development ofthe nitrogen concentration-depth profile in the ferrite substrate is dominated by twoprocesses: (i) nitrogen transport through the cementite layer via diffusion. Thecementite-layer thickness increases as a function of the treatment time (according toa modified parabolic growth law [8,10]). Therefore, the nitrogen flux throughcementite decreases with increasing treatment time. (ii) The inward-diffusion ofnitrogen into the ferrite substrate remains relatively fast. From (i) and (ii) it isconcluded that the initially large nitrogen concentration gradient in the ferritesubstrate decreases with time, as observed.

6.2. Simulation of nitrogen concentration-depth profiles

In the present work, nitrogen concentration-depth profiles in the ferrite substratewith a growing cementite layer at the surface were simulated. In the literature [23]such an approach was already applied for the calculation of nitrogen concentration-depth profiles of compound-layer free ferrite specimens. In that case, local

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equilibrium between gas atmosphere and the surface of the specimen was notimmediately established. Therefore, nitrogen absorption and/or ammonia dissocia-tion and nitrogen transfer through the surface contributed significantly to theprocess kinetics. In fact, nitrogen transfer through the cementite layer in the presentcase and nitrogen transfer through the surface in case of nitriding ferrite withoutcompound layer at the surface [23] can be conceived as analogues. However,different dependences on ðC �-Fe

N,eq � C �-FeN,interfaceÞ occur in both cases: in [23],

JsurfN / kðC �-FeN,eq � C �-Fe

N,interfaceÞ, and here, JFe3CN /D�,Fe3C

NðC�-Fe

N,eq�C�-Fe

N,interfaceÞ

SFe3C, i.e. in the present

case, an extra time dependence, through SFe3C, is introduced.Treatment times of 0.5 h and 2 h were not considered in the simulation procedure.

In the early stage of cementite formation (t56 h), cementite-layer growth proceededfaster than as predicted by the developed model. This can be explained as follows:(i) in the early stage of cementite formation, an incompletely closed cementite layeris present [8,10]. Nitrogen diffusion from the surface of the specimen into ferriteproceeds directly, i.e. bypassing cementite. Transport of nitrogen in ferrite is muchfaster than transport of nitrogen through cementite. (ii) Further, just after theformation of a closed cementite layer, short-circuit diffusion of nitrogen through thinand defect-rich parts of the cementite layer occurs. In this stage of cementiteformation, cementite is expected to exhibit a relatively high grain-boundary density;for data demonstrating grain-boundary diffusion in cementite (see [12]). Due tosuch a ’bypass/short-circuit’ mechanism, nitrogen can be transported relatively fastthrough the cementite layer in the early stage of cementite formation. (A similarexplanation was used for the observed enhanced initial growth rate during carbon-diffusion controlled cementite-layer growth on ferrite [8,10].) Therefore, treatmenttimes56 h were not used in the data evaluation on the basis of the model developedin Section 4.1.

Literature data for the equilibrium nitrogen concentration in ferrite varyconsiderably: e.g. the values reported in [36] are distinctly lower than the valuesreported in [17]. The values of C �-Fe

N,eq obtained in the present work are generally largerthan those compatible with the data from [17] and [36]. These differences can beunderstood by recognising that the nitrogen concentration in the ferrite substrate,C �-Fe

N,eq , as determined in the present work, pertains to the (hypothetical) equilibriumof ferrite with the present nitrocarburising gas atmosphere. In reality, such anequilibrium cannot occur: for such high nitrogen concentrations, solubility limits fornitrogen in ferrite in equilibrium with iron (carbo-)nitrides are surpassed, andformation of � 0-Fe4N below 833K or of "-Fe3(N,C)1þx above 843K [11] occurs.Indeed, in the present work, treatment times 48 h at 853K occasionally led to theformation of these (carbo-)nitride phases at the cementite/ferrite interface.

According to the knowledge of the present authors, diffusion data of nitrogen incementite have not been reported before. The values obtained for the product of thepartition coefficient and the (tracer-)diffusion coefficient of nitrogen throughcementite are considerably lower than values for the diffusion coefficients ofnitrogen in ferrite [29,37] (cf. Table 2) and iron nitrides [38–41] at similar treatmenttemperatures. The role of the cementite layer as diffusion barrier for nitrogen can beillustrated by assuming that the cementite layer would instead consist of ferrite.In that case, in Equation (11), ’D�,Fe3CN should be substituted by D�-Fe

N . Hence, the

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nitrogen flux through the (cementite) surface layer, JFe3CN , would increase by a factorD�-Fe

N =’D�,Fe3C

N , i.e. a factor of about 2500–3000 (cf. Table 2). This clearlydemonstrates the diffusion-barrier behaviour of the cementite layer for nitrogen,which then could be due in particular to the very low solubility of nitrogen incementite, leading to very small values of ’; further, the pre-exponential factorscontained in D�-Fe

N and D�,Fe3CN may differ considerably.

The activation energy of nitrogen diffusion through cementite (here determinedas 652 kJmol�1) is relatively low as compared to real or ‘‘apparent’’ activationenergies for carbon diffusion through cementite, which are in the range 110–185 kJmol�1 [10,42–44], and for nitrogen diffusion in iron nitrides, which are in therange 92–144 kJmol�1 [40,41,45]. The activation energy for nitrogen diffusion incementite appears to be as low as the activation energy for nitrogen (or carbon)diffusion in ferrite, which is about 78–79 kJmol�1 [29,37]. One possibility is that ’,although very small, depends significantly on temperature and thus affects theactivation energy value determined. Another possibility is that the low activationenergy for D�,Fe3CN may hint at nitrogen transport favourably occurring via cementitegrain boundaries; preferred diffusion of carbon along grain boundaries in cementitewas reported very recently [12]. A role of defects (which later-on heal out) has indeedalready been discussed above for the initial fast stage of nitrogen transport throughcementite (see second paragraph of Section 6.2), but it is imaginable that still laternitrogen diffusion along remaining grain boundaries (or other extended defects) maycontribute to the occurring nitrogen flux through cementite. As long as the grain-boundary density remains constant, the grain boundaries’ contribution to thenitrogen transport will lead to same kinetic equations as derived in Section 4, andthe obtained diffusion coefficient becomes an ‘‘effective’’ diffusion coefficient andthe activation energy is an ‘‘effective’’ activation energy [46].

7. Conclusions

(1) Hardness-depth profile measurements provide a suitable method toquantitatively determine nitrogen-concentration depth profiles in ferrite(maximal nitrogen content: �0.4 at.%) after well-defined thermal post-treatment on the basis of a calibration. The relation between hardness andnitrogen content was found to be linear.

(2) Massive cementite layers with a time-dependent thickness can be generatedon ferrite substrates by gaseous nitrocarburising at 783K, 813K and 843K.The nitrogen concentration-depth profile in the ferrite substrate underneaththe growing cementite layer can be simulated on the basis of a developedmodel using the implicit finite-difference method, yielding values for thediffusivity of nitrogen through cementite.

(3) The effective activation energy of nitrogen diffusion through cementite isabout as large as the activation energy of nitrogen diffusion in ferrite. Thismay hint at nitrogen grain-boundary diffusion as a dominant transportmechanism of nitrogen through cementite. The cementite layer acts as adiffusion barrier for nitrogen in particular because of the very low solubilityof nitrogen in cementite.

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