APPLICATION OF THE CLUSTER VARIATION METHOD TO INTERSTITIAL

APPLICATION OF THE CLUSTER VARIATION

METHOD TO INTERSTITIAL SOLID SOLUTIONS

Marjon Indra PEKELHARING


METHOD TO INTERSTITIAL SOLID SOLUTIONS

Proefschrift

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen

op dinsdag 8 januari 2008 om 10.00 uur

door

Marjon Indra PEKELHARING

Doctorandus in de Geochemie, Universiteit Utrecht

geboren te Zevenaar

Dit proefschrift is goedgekeurd door de promotor:

Prof. dr. B.J. Thijsse

Toegevoegd promotor:

Dr. A.J. Böttger

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof. dr. B.J. Thijsse, Technische Universiteit Delft, promotor

Dr. A.J. Böttger, Technische Universiteit Delft, toegevoegd promotor

Prof. J. Foct, Université de Lille

Prof. dr. ir. M.A.J. Somers, Technical University of Denmark

Prof. dr. R. Boom, Technische Universiteit Delft

Prof. dr. ir. S. van der Zwaag, Technische Universiteit Delft

Dr. ir. B.J. Kooi, Rijksuniversiteit Groningen

This work is part of the research programme of the 'Stichting voor Fundamenteel

Onderzoek der Materie (FOM)', which is financially supported by the 'Nederlandse

Organisatie voor Wetenschappelijk Onderzoek (NWO)'.

ISBN 978-90-8559-304-1

i

CONTENTS

1 GENERAL INTRODUCTION.............................................................................1

1.1. INTRODUCTION.......................................................................................................... 2

1.2. ORDERING OF INTERSTITIALS IN IRON NITRIDES............................................. 3

1.3. THE CLUSTER VARIATION METHOD..................................................................... 6

1.4. OUTLINE OF THIS THESIS ........................................................................................ 7

2 MODELING THERMODYNAMICS OF Fe-N PHASES:

CHARACTERIZATION OF ε-Fe2N1-z ..............................................................13

2.1. INTRODUCTION........................................................................................................ 14

2.2. EXPERIMENTAL PROCEDURES............................................................................. 16

2.2.1. SPECIMEN PREPARATION............................................................................... 16

2.2.2. MÖSSBAUER SPECTROSCOPY ....................................................................... 16

2.2.3. X-RAY DIFFRACTION ....................................................................................... 17

2.3. RESULTS AND DISCUSSION................................................................................... 18

2.3.1. THERMODYNAMICS OF ε-Fe2N1-z; THE NITROGEN ABSORPTION

ISOTHERM........................................................................................................... 18

2.3.2. MÖSSBAUER SPECTROSCOPY ....................................................................... 20

2.3.3. X-RAY DIFFRACTION ....................................................................................... 23

2.4. CONCLUSIONS .......................................................................................................... 25

3 APPLICATION OF THE CLUSTER VARIATION METHOD TO

ORDERING IN AN INTERSTITIAL SOLUTION; THE γ-Fe[N] / γ'-

Fe4N1-x EQUILIBRIUM ......................................................................................27

3.1. INTRODUCTION........................................................................................................ 28

3.2. DESCRIPTION OF LRO AND SRO OF INTERSTITIALS BY THE

CLUSTER VARIATION METHOD (CVM)............................................................... 29

3.2.1. APPLICATION TO AN INTERSTITIAL SOLID SOLUTION........................... 29

3.2.2. THERMODYNAMICS OF γ-Fe[N] AND γ'-Fe4N1-x ........................................... 30

3.2.3. CALCULATION OF PHASE EQUILIBRIA ....................................................... 34

ii

3.2.4. APPLICATION TO THE γ-Fe[N] / γ'-Fe4N1-x MISCIBILITY GAP.................... 37

3.3. DISCUSSION............................................................................................................... 40

3.3.1. LATTICE PARAMETERS OF THE FCC Fe-N PHASES................................... 40

3.3.2. THE γ-Fe[N] / γ'-Fe4N1-x MISCIBILITY GAP ..................................................... 41

3.3.3. ORDERING OF NITROGEN ATOMS IN γ-Fe[N] AND γ'-Fe4N1-x ................... 42

3.4. CONCLUSIONS .......................................................................................................... 48

4 APPLICATION OF THE CLUSTER VARIATION METHOD TO

AN INTERSTITIAL SOLID SOLUTION: THE γ'-Fe4N1-x / ε-Fe2N1-z

EQUILIBRIUM ...................................................................................................55

4.1. INTRODUCTION........................................................................................................ 56

4.2. THERMODYNAMICS OF γ'-Fe4N1-x.......................................................................... 57

4.2.1. CONFIGURATIONAL ENTROPY...................................................................... 58

4.2.2. INTERNAL ENERGY.......................................................................................... 59

4.2.3. INTERSTITIAL INTERACTIONS IN γ'-Fe4N1-x................................................. 60

4.3. THERMODYNAMICS OF ε-Fe2N1-z........................................................................... 60


4.3.2. INTERNAL ENERGY.......................................................................................... 63

4.3.3. INTERSTITIAL INTERACTIONS IN ε-Fe2N1-z.................................................. 64

4.4. THE CALCULATION OF THE γ'-Fe4N1-x / ε-Fe2N1-z EQUILIBRIUM ..................... 66

4.4.1. CALCULATION OF PHASE EQUILIBRIA USING THE CVM ....................... 67

4.5. DISCUSSION............................................................................................................... 69

4.5.1. THE γ'-Fe4N1-x / ε-Fe2N1-z PHASE EQUILIBRIA............................................... 70

4.5.2. ORDERING OF NITROGEN ATOMS IN γ'-Fe4N1-x........................................... 72

4.5.3. ORDERING OF NITROGEN ATOMS IN ε-Fe2N1-z ........................................... 73

4.5.4. COMPARISON WITH MÖSSBAUER DATA .................................................... 77

4.5.5. LATTICE PARAMETERS ................................................................................... 80

4.6. CONCLUSIONS .......................................................................................................... 82

5 APPLICATION OF THE CVM CUBE APPROXIMATION TO FCC

INTERSTITIAL ALLOYS .................................................................................85

5.1. INTRODUCTION........................................................................................................ 86

5.2. CVM CUBE APPROXIMATION................................................................................ 87

iii

5.2.1. INTERNAL ENERGY.......................................................................................... 88


5.2.3. CALCULATION OF PHASE EQUILIBRIA ....................................................... 91

5.3. APPLICATION TO FCC INTERSTITIAL ALLOY PHASE EQUILIBRIA.............. 93

5.3.1. PAIR DISTRIBUTION VARIABLES.................................................................. 95

5.3.2. CUBE DISTRIBUTION VARIABLES ................................................................ 98

5.4. CONCLUSIONS ........................................................................................................ 106

6 PHASE TRANSFORMATIONS AND PHASE EQUILIBRIA IN THE

IRON-NITROGEN SYSTEM AT TEMPERATURES BELOW 573 K ......109

6.1. INTRODUCTION...................................................................................................... 110

6.2. EXPERIMENTAL PROCEDURES........................................................................... 111

6.3. RESULTS AND DISCUSSION................................................................................. 113

6.3.1. AS-PREPARED CONDITION ........................................................................... 113

6.3.2. PHASE TRANSFORMATIONS IN ε-PHASE SPECIMENS............................ 118

6.3.3. PHASE TRANSFORMATIONS OF γ' SPECIMENS (GROUP B) ................... 131

6.3.4. PHASE TRANSFORMATION OF THE γ SPECIMEN (GROUP C)................ 133

6.3.5. PHASE TRANSFORMATION OF γ + ε TWO PHASE SPECIMENS

(GROUP D)......................................................................................................... 138

6.4. GENERAL DISCUSSION ......................................................................................... 142

6.5. CONCLUSIONS ........................................................................................................ 145

SUMMARY ..............................................................................................................149

SAMENVATTING ..................................................................................................153

LIST OF PUBLICATIONS ....................................................................................157

DANKWOORD........................................................................................................159

CURRICULUM VITAE..........................................................................................160

1

1

GENERAL INTRODUCTION

Chapter 1

2

1.1. INTRODUCTION

Nitrogen, carbon, boron, and hydrogen are among the lightest elements in the periodic

table and have a small enough size to fit in the interstitial spaces formed by the close-

packed structure of metals. This property is put to use in gas separation technology as

well as in processes intended to change the properties of materials.

The capability of certain metal alloys to absorb interstitial atoms forms the

basis of thermochemical treatments like nitriding, carburizing, and boriding, which

are widely applied to improve performance of steels with respect to wear, fatigue, and

corrosion[1]. Other applications include hydrogen storage, hydrogen gas separation

technology, the production of rechargeable hydride batteries[2], and the use of iron

nitrides in magnetic recording and as permanent magnets[3].

Although interstitial solid solutions such as nitrides and carbides are often

metastable, non-equilibrium phases, for the most part the kinetic decomposition

process is so slow that the materials can be applied and retained successfully at room

temperature[3]. Unfortunately, direct observation of precipitates of such phases is

complicated because of their small size, and thermodynamic calculations may provide

helpful information that cannot obtained otherwise. Knowledge of the

thermodynamics of interstitial solid solutions combined with the ability to predict

experimental thermodynamic data accurately is therefore an important tool for process

and material property optimization in industrial applications[4].

Changes in composition of interstitials in a solid solution can lead to extensive

changes in volume. The resulting microstructural deformation is, for example,

assumed to be the main mechanism of failure of palladium membranes in hydrogen

gas separation technology[5,6]. Despite durability contraints due to embrittlement upon

hydrogenation, high cost, and susceptibility of the membranes to fouling[7], future

application of palladium-based membranes looks very promising.

Besides changes in volume, order-disorder transitions can occur both on the

sublattice formed by the host matrix and on the sublattice formed by the interstitial

sites, making accurate description of the thermodynamics of the phases quite a

challenge. Hydrogen-induced ordering in Pd-alloys[8-12] has been observed, as well as

ordering transitions induced by interstitial atoms in Fe-Cr and Al-Mn based alloys[13].

Suppression of ordering on the metal sublattice after introduction of interstitial atoms

has been observed in ordered Pd7M (M = Sm, Gd) alloys of a Pt7Cu type crystal

General Introduction

3

lattice annealed in a hydrogen atmosphere (pH2 > 20 bar)[14]. In addition, the

substitution of Sm and Gd appears to strongly reduce the ability of the Pd7M alloys to

absorb hydrogen, which may be related to the preferential occupation by hydrogen of

the octahedral interstices located between nearest-neighbor Pd atoms as opposed to

interstitial spaces surrounded by interstices surrounded by both Pd and M atoms[14].

Thus, not only may the presence of interstitial atoms result in order-disorder

transitions on the host sublattice: vice versa, there are indications that the substitution

of metal atoms on the host sublattice of an alloy could also lead to ordering transitions

on the interstitial sublattice[14,15]. Including short- and/or long-range ordering, order-

disorder transformations, and the interaction between the interstitial and metal host

sublatttice in the thermodynamic description of interstitial solutions is therefore

appropriate as well as necessary.

Fig. 1.1. Face centered cubic (fcc) close packed structure with tetrahedral interstices,

located between three atoms in one layer and an atom in the layer directly above or

below; and octahedral interstices, formed by the space between three atoms in one

layer and three atoms in the layer above or below[16].

1.2. ORDERING OF INTERSTITIALS IN IRON NITRIDES

An example of a system that has been the subject of extensive study because it lays

the foundation for modeling of more complex systems common in the steel industry,

and a very suitable candidate for the purpose of studying the ordering of interstitials in

solid solutions, is the Fe-N system.

Chapter 1

4

Fig. 1.2. The Fe-N phase diagram[17,18]

Iron nitrides are metastable binary interstitial solid solutions consisting of a

metal sublattice, assumed to be fully occupied with iron atoms in a close-packed

arrangement, and an interstitial sublattice, consisting of the octahedral sites occupied

by nitrogen atoms and vacancies. The presence of the nitrogen atoms in the octahedral

interstices causes pronounced strain-induced interactions, which influences the

distribution of the nitrogen atoms (and vacancies) over the available interstitial sites[3].

The interaction between the nitrogen atoms and the metal sublattice does not favor a

random distribution of the interstitials. Depending on the nitrogen content of the

phase, the interstitial atoms may display short-range (local) ordering (SRO), which

has been reported to occur in γ-Fe[N][19,20], long-range ordering (LRO), like in

γ'-Fe4N1-x[21], or a combination of both, as has been observed in ε-Fe2N1-z

[22]. In order

to describe equilibrium phase boundaries and absorption isotherms of Fe-N phases

accurately, the ordering of the interstitial atoms needs to be taken into account.


5

The first models used to calculate phase equilibria that incorporated ordering

were based on (sub)regular solution models, which describe Fe-N phases as a

(sub)regular solution of stoichiometric groups of FeaNc and FeaVc (V = vacancy). This

approach is still in use[12] but remains rather limited in accuracy because LRO of the

interstitial atoms is not incorporated in the thermodynamic description[23-29].

Application of the Gorski-Bragg-Williams (GBW)[30] approach to the calculation of

phase boundaries and nitrogen absorption isotherms has proven to be more

successful[31]. However, although the GBW approach introduces LRO into the

description of the phases, SRO is not explicitly accounted for. In this thesis, the

Cluster Variation Method (CVM)[32], a cluster-based approach (described in more

detail in Section 1.3.) capable of including both LRO and SRO in the thermodynamic

description, is applied to the calculation of phase equilibria between interstitial solid

solutions such as iron nitrides.

Because of the metastability of the iron nitrogen phases, obtaining

thermodynamic properties (that are essential input parameters for the calculations)

through experiment is problematic[33,34]. In a recent study of the γ-Fe[N] / γ'-Fe4N1-x

phase equilibrium, first principles calculations were combined with the CVM. A set of

effective cluster interactions (ECIs)[35] was thus obtained, subsequently replacing the

phenomenological Lennard-Jones potential[36]. In addition, the Debye-Grüneisen

model was applied to account for the vibrational contributions to the entropy and

internal energy, giving the overall description of the free energy more of a physical

basis than the traditional CVM approach.

Another interesting issue to be addressed remained: how to handle the

interaction between the substitutional metal host sublattice and the interstitial

sublattice, which so far had been included in the parameters of the so-called effective

pair potentials, which mimic the host-interstitial interaction through an effective

interaction on the interstitial sublattice. An innovative approach, coupling the host

sublattice with the interstitial sublattice and incorporating the sites of both sublattices

into the basic CVM cluster to model a hypothetical alloy, was published in 2006[37].

This model has formed the starting point for a closer study of the interactions between

the metal atoms and the interstitials of a hypothetical system in this thesis.

Application of this concept to a non-hypothetical system may provide a future helpful

tool for controlling ordering phenomena in alloys for industrial processes and design

of new materials.

Chapter 1

6

1.3. THE CLUSTER VARIATION METHOD

The cluster variation method (CVM), published in 1951 by Kikuchi[32], has been

applied, modified, and expanded for more than half a century. The main concept in

the original publication was the derivation of a configurational entropy expression, as

well as the description of the enthalpy (and thus the free energy) in terms of cluster

distribution variables[38].

A major advantage of the CVM, compared to the previously mentioned

thermodynamic models, is that both LRO and SRO can be taken into account if the

basic cluster is large enough. The choice of the basic cluster in the CVM therefore

depends on the range of interactions to be included. For the basic cluster and its

subclusters, all possible arrangements of atoms are assigned an individual cluster

distribution variable, which describes the fraction of that particular configuration per

(sub)cluster. Since basic clusters typically consist of more than two lattice points, and

therefore automatically involve multiparticle interactions, another problem that can be

avoided by using the CVM is that of lattice frustration. Lattice frustration is a

fascinating but problematic phenomenon for nearest-neighbor triangular structures, a

basic feature for both fcc and hcp structures, and refers to the impossibility of forming

three unlike atom pairs A-B simultaneously[39]. The CVM handles lattice frustration

by simply considering all possible distributions of atoms in order to minimize the

system’s free energy.

Shortly after the CVM was first introduced, an easier, systematic method to

obtain the CVM entropy expression was derived by Barker[40]. In 1967, the CVM

superposition approximation, which describes the basic cluster distribution variables

as a function of the distribution variables of its subclusters, was published[41].

Although progress was being made on the theoretical side, practical application of the

CVM remained quite limited until 1973, when the CVM was picked up by Van

Baal[42] and used to model the fcc substitutional Cu-Ag system. Since then, the CVM

has been widely applied to systems with different, more complex structures and

composition. Still, simplification of the minimization procedure was urgently needed

for larger clusters. After all, one of the main disadvantages of the CVM is the rapidly

increasing complexity of calculations with increasing basic cluster size. The number

of correlation functions necessary to describe a cluster probability distribution equals

the number of subclusters composing the basic cluster, and increases exponentially


7

with increasing basic cluster size[43] (and even faster for non-symmetrical clusters[44]).

To address this problem, Sanchez and de Fontaine[43] proposed a scheme to generate a

set of independent cluster variables, with the number of variables equal to the total

number of subclusters into which the basic cluster can be decomposed. A linear

correlation function and degeneracy factor (the number of indistinguishable

configurations resulting from symmetry operations applied to the cluster) are

associated with each independent cluster. The resulting set of equations to be solved

simplified the minimization of the free energy considerably for large clusters and

paved the way for further implementation of the CVM to highly complex structures

such as aluminosilicate minerals[44-46].

Another modification that has been made about a decade ago to the

conventional CVM formulation targets the local displacement of atoms. For modeling

purposes, rigidity of the lattice structure is usually assumed. In real alloys however,

alterations in atomic positional arrangement occur to accommodate local lattice

distortions, which may result from differences in size between the atoms on the

substitutional sublattice, size misfit of the interstitial atoms in the structure, thermal

vibration effects, or elastic effects[38]. Continuous Displacement[47] CVM introduces

vectors indicating the actual position of the atom with regard to its reference lattice

point, and it has been shown to significantly reduce the discrepancy between

experimental data and calculated equilibrium phase boundaries[38].

1.4. OUTLINE OF THIS THESIS

The main focus of this thesis is the use of the cluster variation method to describe the

ordering of interstitial atoms in a metal host matrix, as occurring in phases such as

iron nitrides, and comparison of the obtained results with available experimental data

to verify the validity and applicability of the model.

Chapter 2 describes long-range ordering of nitrogen on the sublattice of

octahedral interstices of the Fe sublattice of ε-Fe2N1-x by application of the Gorski-

Bragg-Williams approach. The model is fitted to experimental nitrogen absorption

data and the determined probabilities for Fe atom surroundings are verified

successfully with Mössbauer spectroscopy. Using a synchrotron source, the

occurrence of X-ray diffraction superstructure reflections is analyzed.

Chapter 1

8

In Chapter 3, the tetrahedron approximation of the Cluster Variation Method

(CVM) is applied to describe the ordering of nitrogen atoms on the fcc interstitial

sublattices of γ-Fe[N] and γ'-Fe4N1-x. A type 8-4 Lennard-Jones potential is used to

describe the strain-induced interactions caused by the misfit of the N atoms in the

interstitial octahedral sites. The γ-Fe[N] / γ'-Fe4N1-x miscibility gap, SRO and LRO of

nitrogen in γ-Fe[N] and γ'-Fe4N1-x, respectively, and lattice parameters of the γ and γ'

phases are calculated. For the first time, nitrogen distribution parameters, as

calculated by CVM, are compared directly to Mössbauer spectroscopy data for

specific surroundings of Fe atoms.

The application of the cluster variation method to establish effective

interaction potentials that describe both γ'-Fe4N1-x / ε-Fe2N1-z miscibility gaps in the

Fe-N phase diagram is described in Chapter 4. The calculated nitrogen distributions

show that LRO occurs in the γ'-Fe4N1-x phase and that SRO, as well as LRO, occurs in

the ε-Fe2N1-z phase. The calculated nitrogen distributions for the ε-Fe2N1-z, pertaining

to temperatures and concentrations at the γ' / ε phase boundaries, are compared with

available data obtained by Mössbauer spectrometry. Preferential occupation of

specific interstitial sites occurs from about 16 at.% nitrogen on; at the highest

concentration considered, about 25 at.% nitrogen, the occupation is that of Fe3N as

proposed in literature on the basis of diffraction data.

Chapter 5 describes the application of the CVM simple cube approximation to

calculate a hypothetical fcc interstitial alloy phase equilibrium. Instead of limiting the

description of the alloy to the species occupying the interstitial sublattice sites and

including the interaction with the metal sublattice in the effective pair potentials like

in the previous chapters, the basic cluster is composed of both metal and interstitial

sublattice sites. The metal sublattice is described as fully occupied by two types of

metal atoms, while the interstitial sublattice sites are filled with two interstitial

species, one representing an atomic species and the other a vacancy. The Lennard-

Jones parameters chosen to describe the interaction between the species lie within the

range typical for transition metals. Analysis of the calculated cube distribution

variables shows that phase transitions on the metal and interstitial sublattices are

coupled: ordering of interstitial species can be influenced by introduction of extra

metal species to the host matrix of the alloy, which enables purposeful adjustment or

change of the properties of a material.


9

Finally, in Chapter 6 the phase transformations of homogeneous Fe-N alloys

with nitrogen contents ranging from 10 to 26 at.% are investigated by means of X-ray

diffraction analysis after ageing at temperatures in the range of 373 to 473 K. It is

found that precipitation of α"-Fe16N2 below 443 K does not only occur upon ageing

of supersaturated α (ferrite) and α' (martensite), but also upon transformation of

γ'- Fe4N1-z and ε-Fe2N1-x (<20 at.% nitrogen). No α" is observed to develop upon

ageing of γ (austenite). Therefore, it is proposed that γ' is a stable phase at

temperatures down to (at least) 373K. Phase formation upon annealing at low

temperatures is apparently governed by the (difficult) nucleation and (slow) growth of

new Fe-N phases: α" forms as a precursor for α because of slow nitrogen diffusion,

and nitrogen-enriched ε develops as a precursor for γ' because of a nucleation barrier.

Chapter 1

10

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Metals Park, OH: ASM international, 1995.

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11

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13

2

MODELING THERMODYNAMICS OF Fe-N PHASES:

CHARACTERIZATION OF ε-Fe2N1-z

ABSTRACT

Long-range ordering of nitrogen on the sublattice of octahedral interstices of the Fe

sublattice of ε-Fe2N1-x was described by application of the Gorski-Bragg-Williams

approach. The model was fitted to experimental nitrogen absorption data and the

determined probabilities for Fe atom surroundings were verified successfully with

Mössbauer spectroscopy. Using a synchrotron source, the occurrence of X-ray

diffraction superstructure reflections was analyzed.

Chapter 2

14

2.1. INTRODUCTION

The degree and type of ordering of interstitial atoms play a very important role in

understanding and modelling of the experimentally observed absorption isotherms

and phase diagrams of binary interstitial iron alloys. In particular, ordering of nitrogen

atoms in ε-Fe2N1-x nitrides has not yet been determined unambiguously.

In a series of publications[1-3], a very good description was obtained of the

equilibrium nitrogen content in α-Fe[N][2], γ'-Fe4N1-x [1], and ε-Fe2N1-z

[3] as a function

of the chemical potential, imposed by an NH3/H2 mixture, as well as of the phase

boundaries in the Fe-N phase diagram[2], by considering these Fe-N phases as

constituted of two interpenetrating sublattices: one for the metal atoms and one for the

interstitial nitrogen atoms. According to this approach, the metal sublattice is assumed

to be fully occupied at temperatures below the melting temperature of iron, while the

interstitial sublattice, formed by the octahedral interstices of the Fe sublattice, is

occupied by nitrogen atoms, N, and vacancies, V. The thermodynamics of an Fe-N

alloy can thus be reduced to the thermodynamics of a binary “alloy” of N and V on

the interstitial sublattice. The occurrence of long-range ordering (LRO) of nitrogen

atoms on the interstitial sublattices of γ'-Fe4N1-x [1] and ε-Fe2N1-z

[3] was accounted for

by adopting the Gorski-Bragg-Williams (GBW) approach.

The ε-Fe2N1-z phase consists of an hcp iron sublattice and a simple hexagonal

interstitial sublattice. Thermodynamic analysis of the mixing of atoms N and

vacancies V on the interstitial sublattice of ε-Fe2N1-z[4] has indicated the occurrence of

two ground-state structures: configuration A for ε-Fe2N (50 at.% N) and configuration

B for ε-Fe3N (33.3 at.% N), which correspond with proposed arrangements of

nitrogen atoms in Refs. [5] and [6] (Fig. 2.1.). Mössbauer spectroscopy has indicated

that configuration B is predominant for compositions close to Fe3N and that

configuration A is predominant for compositions close to Fe2N. For intermediate

compositions, Mössbauer results have suggested the occurrence of a two-phase region

where both configurations coexist.

Long-range ordering (LRO) of the nitrogen atoms results in the occurrence of

superstructure reflections (with respect to the hcp metal sublattice), which can be

characterized by diffraction. With each configuration (A/B), a number of specific

superstructure reflections is associated[7]. Direct experimental verification of the

Modeling Thermodynamics of Fe-N Phases: Characterization of ε-Fe2N1-z

15

Fig. 2.1. The unit cell of the hcp sublattice of Fe atoms containing one unit cell of the

hexagonal interstitial sublattice. The ground-state structures A and B of the trigonal

prism for Fe3N and Fe2N, constituted by six kinds of sites A1..C2 forming the

interstitial sublattice, are shown as well.

ordered configurations, using X-ray diffraction analysis (XRD), has only been

realized for configuration B[6,3]. Both conventional X-ray and neutron diffraction

experiments could not distinguish all characteristic reflections for configuration A[3],

possibly because of the very low intensity of the superstructure reflections in

ε-Fe2N1-z. However, atomic displacements of the Fe atoms, caused by the misfit of the

interstitial nitrogen atoms occupying the octahedral interstices, also have a strong

influence on the intensity of the reflections[8]. A preliminary calculation of the

structure factor (including displacements of iron atoms) has indicated that the

intensities of the 001 and 301 reflections, which are specific for the A

configuration and have not yet been observed, are strongly reduced by such

displacements.

In this work, a new thermodynamic analysis of the absorption isotherm at

723 K, describing the nitrogen content in ε-Fe2N1-z as a function of an imposed

chemical potential of nitrogen, is given, applying the model presented in Refs. [3] and

[4]. Furthermore, a new series of ε-Fe2N1-z samples, prepared at 723 K, has been

characterized employing Mössbauer spectroscopy, XRD, and HRPD (High Resolution

Powder Diffraction – synchrotron radiation). A very good agreement between

thermodynamic model predictions and experimental results has been obtained.

Chapter 2

16

2.2. EXPERIMENTAL PROCEDURES

2.2.1. SPECIMEN PREPARATION

A series of seven homogeneous ε-Fe2N1-z powders was prepared by gaseous nitriding

of small amounts (0.2 to 0.4 g) of pure α-Fe powder (average particle size 5 ± 3

micron; composition: <0.002 wt% Ni; <0.002 wt% Mn; <0.01 wt% Al; <0.002 wt%

Cr; <0.002 wt% Ti; <0.01 wt% W; <0.002 wt% V; 0.04 wt% Si; 0.002 wt% N; 0.221

wt% O and balance Fe). Nitriding was performed in a vertical quartz-tube furnace for

16 hours, at a temperature of 723 K in an NH3/H2 gas mixture. The inlet gases NH3

and H2 were purified and dried before mixing and entering the furnace. The ratio

NH3:H2 was adjusted by thermal gas flow controllers and chosen on the basis of the

absorption isotherm for ε-Fe2N1-z at 723 K[3], such that the nitrogen content of the

samples covers the range from 26.1 to 31.4 at.% N. Nitriding was terminated by

pulling the samples into the lock-chamber on top of the vertical furnace to achieve

relatively fast cooling. An additional sample containing 24.9 at.% N was prepared

analogously by nitriding at 843 K for 5 hours.

2.2.2. MÖSSBAUER SPECTROSCOPY

Mössbauer spectroscopy uses the resonant absorption of γ-rays by a nucleus to probe

the hyperfine splitting of nuclear energy levels and thus provides information on the

atomic environment of the nucleus. Three types of hyperfine interactions between the

nucleus and its surroundings can be discerned: 1) the isomer shift, 2) the quadrupole

splitting, and 3) the hyperfine interaction.

(1) The isomer shift (δ) can be observed due to variations in the s-electron

density at the nucleus and leads to an overall shift of the pattern. The isomer shift

should have a more or less constant value, or, when Mössbauer spectroscopy is

applied to study the local surroundings of the iron atoms in Fe-N specimens, increase

a little with increasing nitrogen content.

2) The quadrupole splitting (QS) leads to splitting of the spectrum in two lines

and arises from the coupling between the quadrupole moment of the nucleus and a

non-spherical charge distribution in its immediate vicinity.


17

3) The most important interaction in magnetic materials is the hyperfine

interaction, which splits the spectrum into six lines and arises from coupling between

nuclear and atomic magnetic moments.

At temperatures below the Curie temperature, Tc, the hyperfine splitting

increases. The Tc of ε-Fe2N1-z strongly decreases with increasing nitrogen content[9].

Thus, to realize an (almost) complete hyperfine splitting into the subspectra

constituting the Mössbauer spectrum of the samples, Mössbauer spectra were

recorded at 4.2 K, with a constant acceleration spectrometer using a 57Co source.

The recorded spectra were fitted to a combination of sextuplets. Each sextuplet

represents an Fe atom at the centre of a trigonal prism of the interstitial sublattice

(Fig. 2.1.), while the prism sites are occupied by a specific number of interstitial

atoms. Magnetic texture in the powder samples was assumed to be absent.

Consequently, the ratio of the relative intensities of the peaks of each sextuplet

conform to 3:2:1:1:2:3[10,11]. In each sextuplet, the individual lines are assumed to be

Voigt functions, i.e. a convolution of Lorentzian and Gaussian components. The

Lorentzian component is due to the source and the Gaussian component is due to the

sample.

2.2.3. X-RAY DIFFRACTION

A suspension of nitrided powder and ethanol was deposited onto a Si <510> single

crystal slab. An adherent thin layer of ε-Fe2N1-z on this Si substrate was obtained by

allowing sedimentation of the powder from this suspension and by subsequent

evaporation of the ethanol. A Siemens D-500 goniometer equipped with a primary

beam monochromator set to select Co Kα1 radiation was used to scan the samples

within the angular range of 20 to 85 o2θ, employing a step size of 0.05 o2θ, and a

counting time of 500 seconds/step. Additionally, five samples were analysed also at

the Synchrotron Radiation Source (SRS) in Daresbury (UK), using the high-resolution

powder diffraction equipment. A quantitative analysis of the diffracted intensities was

made to establish the types and degrees of ordering as a function of nitrogen content.

Using synchrotron radiation, the intensity of the superstructure reflections with

respect to the background is higher than for conventional X-ray diffraction because

the wavelength can be chosen to minimize absorption by the Fe matrix, thereby

enhancing the scattered intensity of the very weak superstructure reflections. The

Chapter 2

18

reflections in the HRPD spectra were fitted using a pseudo-Voigt function. The peak

positions thus obtained were used to assess the lattice parameters a and c.

2.3. RESULTS AND DISCUSSION

2.3.1. THERMODYNAMICS OF ε-Fe2N1-z; THE NITROGEN ABSORPTION

ISOTHERM

Applying the Gorski-Bragg-Williams (GBW) approximation for long-range ordered

binary solutions to the interstitial sublattice occupied by atoms N and vacancies V, the

interstitial sublattice is subdivided into six sublattices, denoted as A1, B1, …, C2 (cf.

Fig. 2.1.). The following expression can then be derived for the Gibbs energy of

ε-Fe2N1-z [3,4]:

( )

( )

( ) ( )[ ]∑=

−−++

+++++−+

++−++=

2

1

222222111111

21212100

1ln1ln61

61

62

C

Ak

kkkk

CBCABACBCABA

NP

CCBBAA

NCNNFe

yyyyRT

yyyyyyyyyyyyyW

yyyyyyyWGyGGε

(1)

where 0FeG and 0

NG are the hypothetical Gibbs energies of pure Fe with an hcp lattice

and of pure N with a simple hexagonal lattice, respectively (cf. Fig. 2.1.), yN denotes

the fractional occupancy of the N sublattice, ky represents the fractional occupancy of

sublattice k by N atoms, and WP and WC are the exchange energies within the basal

plane of the hexagonal unit cell and in the direction perpendicular to the basal plane,

respectively.

Equilibrium of ε-Fe2N1-z implies that the chemical potentials of nitrogen on

each of the six sublattices A1..C2 are equal. If the ε-Fe2N1-z phase is in equilibrium

with an NH3/H2 gas mixture, the following equation1 is obtained for the nitrogen

absorption isotherm, i.e., the nitrogen content in the solid state as a function of the

chemical potential of nitrogen imposed by the gas mixture[3]:

1 Actually, six similar equations can be obtained from Eq. (1), one for each of the sublattices A1..C2.


19

( ) ( )RT

Wyy

RT

Wy

y

y

r

r PCBCA

A

A

N

N 1121

1

0 1211

lnln −−+−+

−= (2)

where rN is the nitriding potential of the NH3/H2 gas mixture and rN0 is defined as

0000

23 23

ln HNHNN GGGrRT +−= (3)

with 0

3NHG and 0

2HG as the Gibbs energies of NH3 and H2[4].

The experimentally determined absorption isotherm for 723 K from Ref. [3] is

given in Fig. 2.2., where the ordinate is chosen such that a Langmuir-type absorption

isotherm for the N sublattice, with only half of all sites available for occupation,

would give a horizontal line.

Fig. 2.2. Absorption isotherm data at 723 K [3], presented as the deviation from a

Langmuir-type isotherm, vs. the occupancy of the interstitial sublattice yN. Solid and

dashed line pertain to calculated absorption isotherms, using Eq. (2) with

WC/RT = -3.5 and WP/RT = -4.0 for configurations A and B.

In Ref. [3], the thermodynamics of ε-Fe2N1-z was assessed by fitting the set of

Eq. (2) for either configuration A or configuration B to the absorption isotherm data

Chapter 2

20

over the entire experimentally covered composition range, while optimizing the

values for exchange energies WP and WC (cf. Refs. [3] and [4]). As suggested by the

Mössbauer analysis in Ref. [3], configuration B is stable for compositions near Fe3N

and configuration A is stable for compositions near Fe2N. Thus, the dominant part of

the experimental absorption isotherm data may not be ascribed exclusively to

configuration A or to configuration B. This may make separate fitting of Eq. (2) for

either configuration A or configuration B problematic. Therefore, in the present

analysis, a different procedure was utilized. For one set of WP, WC, and rN0 values, a

pair of absorption isotherms corresponding to configuration A and configuration B

was calculated such that the experimental absorption isotherm data were enveloped

(note Fig. 2.2.). Substitution of the values for WC and WP and the nitrogen-content

dependent occupancies of the sublattices A1..C2, using

∑=

=2

1

C

Ak

N

k

N yy (4)

for configurations A and B, yields the Gibbs energy functions for configuration A and

B (cf. Eq. (1)). Then, the miscibility gap between configurations A and B can be

obtained straightforwardly using the common tangent construction to the

corresponding Gibbs energy functions. Thus, a two-phase region was determined

ranging from yN = 0.390 for configuration B to yN = 0.482 for configuration A.

2.3.2. MÖSSBAUER SPECTROSCOPY

All Mössbauer spectra could be satisfactorily described with 3 (or 4) sextuplets,

representing Fe atoms surrounded by 1, 2 or 3 N atoms, and denoted as sextuplet FeI,

FeII and FeIII, respectively. For the higher nitrogen contents, two sextuplets for FeIII

had to be adopted to obtain acceptable fits. By fitting the overall Mössbauer spectra

resulting from the combination of several sextuplets to the experimental data, the

values of the hyperfine field, the isomer shift, the widths of the Gaussian and

Lorentzian components for the specific sextuplets and the relative contribution of

each sextuplet were obtained.

The results of the analysis of the Mössbauer spectra are given in Table 2.1. As


21

Table 2.1. Hyperfine field (H), isomer shift (δ) and relative contribution (fi) of the

sextuplets designated by I,II, IIIa and IIIb to Mössbauer spectra of the ε-Fe2N1-z

powders, recorded at 4.2 K. Σfi(i/6) = yN is the fraction of interstitial sites occupied

by nitrogen atoms as calculated from the relative contributions of iron atoms

surrounded by i = 0,1,...,6 nitrogen atoms (there is one interstitial site per Fe atom

and each Fe atom is surrounded by six different sites) and as obtained from high

resolution powder diffraction data.

at.% N 26.1 26.8 27.4 28.4 29.5 30.4 31.4

H(kOe) 332.1 331.2 334.3 330.7 - - -

I δ 0.66 0.69 0.66 0.70 - - -

f (%) 4.1 1.3 3.0 1.5 - - -

H(kOe) 259.9 259.5 259.2 258.1 257.4 255.5 251.8

II δ 0.73 0.73 0.74 0.74 0.75 0.76 0.77

f (%) 69.9 65.7 59.5 55.1 43 30.9 21.9

H(kOe) 147.5 145.6 141.9 139.5 141.2 138.5 129.2

IIIa δ 0.83 0.81 0.80 0.81 0.81 0.82 0.81

f (%) 17.3 29.3 32.5 37.5 39.3 39.5 46.3

H(kOe) 128.0 62.2 81.6 86.7 95.2 87.8 56.5

IIIb δ 0.67 0.37 0.48 0.54 0.76 0.78 0.8

f (%) 8.7 3.7 5.1 5.9 17.7 29.7 31.7

yN Σfi(i/6) 0.3631 0.3838 0.3861 0.4006 0.4283 0.4486 0.4634

HRPD 0.3596 - 0.3830 0.3967 0.4213 0.4432 -

was demonstrated in Refs. [12] and [3], (at least) two sextuplets for FeIII of distinctly

different hyperfine field, H, are required to achieve a satisfactory fit. Within

experimental accuracy, the hyperfine field for a particular Fe environment is fairly

constant, indicating that magnetic saturation is attained at the measurement

temperature of 4.2 K (Table 2.1.).

The total fractional occupancy of the N sublattice, yN, can be obtained by

summation of the relative contributions of the sextuplets fj (j=1-3) given in Table 2.1.

to the overall spectrum; for f3 the contributions of FeIIIa and FeIIIb have been summed.

The relative contributions of the sextuplets fj for j=2,3, i.e. the relative contributions

of the iron atoms surrounded by 2 and 3 nitrogen atoms on the six sites of the trigonal

prism (note Fig. 2.1.), are given as a function of the total occupancy yN, in Fig. 2.3.

For comparison, similar results obtained here from previously published data

for Mössbauer spectra have been presented too in Fig. 2.3. Present and previous data

agree very well, despite differences in sample preparation and in procedures applied

Chapter 2

22

for fitting of the sextuplets to the measured Mössbauer spectra (cf., Refs [3], [9], and

[12]). The probabilities p2 and p3 for iron atoms to be surrounded by two and three

nitrogen atoms respectively, can be calculated straightforwardly from the occupancies ky of the sublattices A1..C2, pertaining to the absorption isotherms corresponding to

WC/RT = -3.5 and WP/RT = -4 for configurations A and B as shown in Fig. 2.2. (cf.,

Section 2.3.1.); they are represented by the dashed lines in Fig. 2.3.

Fig. 2.3. Relative abundances of Fe atoms surrounded by 2 (ƒ2) and 3 (ƒ3) N atoms as

a function of the occupied fraction of interstitial sites yN at 723 K (data points). The

probabilities p2 and p3 for iron atoms to be surrounded by 2 (p2) and 3 (p3) N atoms,

as calculated from the yN values using WC/RT = -3.5 and WP = -4.0 are shown as a

function of yN by the dashed lines for configuration A and B.

Recognizing the occurrence of the miscibility gap between configurations A

and B (hereafter designated as A+B region), in fact, a linear combination of the

probabilities p2 and p3 at the extremities of the A+B region, i.e. p2 and p3, for

configuration B at yN = 0.390 and p2 and p3 for configuration A at yN = 0.482 (see end

of Section 2.3.1.) should be presented for the dependence of p2 and p3 on yN in the

A+B region: see solid straight lines between the limiting compositions given in

Fig. 2.3. Clearly, the thus obtained calculated p2 and p3 values agree well with the


23

Fig. 2.4. X-ray diffractograms of ε-Fe2N1-z, containing 24.9 at.% N (bottom) and

31.4 at.% N (top). The positions of the superstructure reflections for configurations A

and B have been indicated (marked by s).

experimental f2 and f3 values. For yN < 0.390, where only configuration B is stable,

the values for f2 and f3 are in good agreement with p2 and p3 for configuration B. A

similar observation is not possible for configuration A, because the sample with the

highest N content is still within the A+B region.

2.3.3. X-RAY DIFFRACTION

The nitrogen content can be derived from the lattice parameters a and c of hexagonal

ε-Fe2N1-z [3]. The HRPD results are compared with the corresponding data derived

from the Mössbauer spectra (cf., Section 2.3.2.) in Table 2.1. The HRPD patterns of

two samples are given in Fig. 2.4.; the positions of the superstructure reflections

expected to be observed for the A and B configuration are indicated (marked by s).

The presence of the superstructure reflection 001 provides unambiguous evidence

for the occurrence of configuration A, but was not observed (Fig. 2.4.). Figure 2.5.

Chapter 2

24

shows the observed intensity of the 100, 200 and 102 reflections, specific for

configuration B, in samples with different nitrogen contents. With increasing nitrogen

content, the superstructure reflections specific for configuration B gradually

disappear. Note that none of the present specimens has a composition that would

correspond to pure configuration A (cf., Section 2.3.2.). If, for specimens in the A+B

region, phases A and B would diffract independently, the corresponding X-ray

diffraction patterns would display doublet peaks for the majority of the reflections:

one peak of the doublet due to phase B (yN=0.390; high 2θ) and one peak of the

doublet due to phase A (yN=0.482; low 2θ). The relative intensities of the two peaks

of the doublet would be proportional with the relative amounts of phases A and B. No

such doublet peaks are observed. Hence, it is concluded that in the specimen the

“domains” exhibiting configuration A and the “domains” exhibiting configuration B

diffract coherently.

Fig. 2.5. The superstructure reflections 100, 200 and 102, specific for ordering

of the nitrogen atoms according to configuration B, for different nitrogen contents.


25

2.4. CONCLUSIONS

The thermodynamics of ε-Fe2N1-z at 723 K can be described applying the Gorski-

Bragg-Williams approach to mixing of nitrogen atoms and vacancies on the

interstitial sublattice: long-range order of nitrogen atoms occurs. Both the nitrogen

absorption isotherm and the Mössbauer spectroscopic data demonstrate that ordering

of nitrogen takes place according to two ground-state structures: one for Fe3N

(configuration B) and one for Fe2N (configuration A). A two-phase region, where

domains of configuration B and domains of configuration A coexist, extends from

yN = 0.390 to yN = 0.482 at 723 K. These domains diffract coherently.

The present observations of superstructure reflections and local surroundings

of Fe atoms are consistent with ordering of the nitrogen atoms according to

configuration B for ε-Fe2N1-x with an Fe3N composition. The absence of the 001

superstructure reflection, specific for configuration A, in the diffraction pattern of

ε-Fe2N1-x with a nitrogen content close to Fe2N, and preliminary structure factor

calculations indicate that atomic displacements of the Fe atoms due to the presence of

N atoms in the structure occur, which may cause the 001 reflection to disappear.

Chapter 2

26

REFERENCES

1. Kooi, B.J., M.A.J. Somers, E.J. Mittemeijer, Metall. Mater. Trans. A, 1996, vol. 27A, pp.1055-61

2. Kooi, B.J., M.A.J. Somers, E.J. Mittemeijer, Metall. Mater. Trans. A, 1996, vol. 27A, pp.1063-71

3. Somers, M.A.J., B.J. Kooi, L. Maldzinski, E.J. Mittemeijer, A.A., van der Horst, A.M. van der

Kraan, N.M. van der Pers, Acta Mater., 1997, vol. 45, pp. 2013-25

4. Kooi, B.J., M.A.J. Somers, E.J. Mittermeijer, Metall. Mater. Trans. A, 1994, vol. 25A, pp. 2797-

2814

5. Hendricks, S.B., P.B. Kosting, Z. Kristallogr., 1930, vol. 74, pp. 511-33

6. Wriedt, H.A., N.A. Gokcen, R.H. Nafziger, Bull. of Alloy Phase Diagrams, 1987, vol. 8, pp. 355-

77

7. Jack, K.H., Acta Cryst., 1952, vol. 5, pp. 404-11

8. Genderen, M.J. van, A. Böttger, E.J. Mittemeijer, Metall. Trans., 1997, vol. 28A, pp. 63-77

9. Chen, G.M., N.K. Jaggi, J.B. Butt, E.B. Yeh, L.H. Schwartz, J. Phys. Chem., 1983, vol. 87, pp.

5326-32

10. Rancourt, D.G., Nucl. Instr. Meth. Phys. Res., 1989, vol. B44, pp. 199-210

11. McLean, A.B., J. Electr. Spectr., 1994, vol. 69, pp. 125-32

12. Schaaf, P., Chr. Illgner, M. Niederdrenk, K.P. Lieb, Hyperfine Interactions, 1995, vol. 95, pp. 199-

225

27

3

APPLICATION OF THE

CLUSTER VARIATION METHOD TO

ORDERING IN AN INTERSTITIAL SOLUTION;

THE γ-Fe[N] / γ'-Fe4N1-x EQUILIBRIUM

ABSTRACT

The tetrahedron approximation of the Cluster Variation Method (CVM) was applied

to describe the ordering of N atoms on the fcc interstitial sublattice of γ-Fe[N] and

γ'-Fe4N1-x. A Lennard-Jones potential was used to describe the dominantly strain-

induced interactions, caused by the misfitting of the N atoms in the interstitial

octahedral sites. The γ-Fe[N] / γ'-Fe4N1-x miscibility gap, short-range ordering (SRO)

and long-range ordering (LRO) of nitrogen in γ-Fe[N] and γ'-Fe4N1-x, respectively,

and lattice parameters of γ and γ' were calculated. For the first time nitrogen

distribution parameters, as calculated by CVM, were compared directly to Mössbauer

data for specific surroundings of Fe atoms.

Chapter 3

28

3.1. INTRODUCTION

Nitriding is a thermochemical treatment usually applied to workpieces of iron-based

alloys (steels) to improve the performance with respect to fatigue, wear and corrosion.

Knowledge of the thermodynamics of iron-nitrogen phases is a prerequisite for

process and property optimization.

Most descriptions of the Fe-N system are based on the (sub)regular-solution

model[1,2,3] and describe the Fe-N phases as a (sub)regular solution of postulated

stoichiometric groups FeaNc and FeaVc, where a and c are whole numbers and V

denotes a vacant interstitial site. In the regular solution (RS) model, an excess Gibbs

energy term, which can be physically interpreted as an excess enthalpy term due to

pairwise interaction of neighboring stoichiometric groups, is added to the Gibbs

energy of an ideal solution of these groups. If the Redlich-Kister polynomial[4] is

adopted for the description of the excess Gibbs energy, the corresponding series

expansion for the excess Gibbs energy has no physical meaning, apart from the first

term that corresponds with the RS model; incorporation of both the second and first

terms corresponds to the so-called subregular solution (SRS) model. In these models,

possible long-range ordering of atoms N on the interstitial sublattice is not taken into

account explicitly.

In Fe-N phases, the N atoms reside in the octahedral sites formed by the close-

packed Fe atoms. Pronounced strain-induced interactions occur due to misfitting of

the N atoms in the interstitial positions. Due to these interactions, the N atoms cannot

be distributed randomly over all available sites: ordering of N atoms over the sites of

the interstitial sublattice occurs. If the fraction of N atoms is low, short-range ordering

(SRO) is observed, i.e., order prevails locally as a consequence of the tendency of N

atoms not to be surrounded by N atoms at (nearest) neighboring sites. If the fraction

of N atoms is high, long-range ordering (LRO) occurs and a periodic arrangement of

N atoms on the interstitial sublattice becomes apparent.

The (sub)regular solution models cannot take into account LRO of N atoms

on the interstitial sublattice as present in γ'-Fe4N1-x (fcc Fe sublattice)[5] and ε-Fe2N1-z

(hcp Fe sublattice)[6]. Yet, the RS model has been applied to treat the FeN-phases with

bcc, fcc and hcp Fe sublattices, considering γ'-Fe4N1-x as a stoichiometric

compound[7,8]. Using the SRS model, the Fe-N system has been reevaluated, still

treating γ' as a stoichiometric phase[9-12]. The γ' phase was treated as a

Application of the CVM to an Interstitial Solid Solution: The γ-Fe[N] / γ'-Fe4N1-x Equilibrium

29

nonstoichiometric compound in the SRS model applied in Ref. 13 to account for the

presence of a composition range of the γ' phase. Recently, it was shown[14] that the

available nitrogen-absorption isotherms for γ'-Fe4N1-x cannot be accurately described

by any of the RS and SRS models, because they do not account for the presence of

LRO. Instead, it was shown that the Gorsky-Bragg-Williams approximation

(GBW)[15] could be used successfully to calculate both the phase boundaries and the

nitrogen-absorption isotherms (i.e. the N content as a function of N activity),

considering LRO of N atoms in γ'-Fe4N1-x and ε-Fe2N1-x[14,16,17]. Although the GBW

approximation provides a satisfactory description of the LRO in γ'-Fe4N1-x and

ε-Fe2N1-x, it cannot account for the presence of SRO. Short-range ordering has been

reported to occur for the N atoms in γ-Fe[N] (nitrogen austenite)[18,19].

In the present work, the tetrahedron approximation of the cluster variation

method (CVM)[20] (Section 3.2.2) is applied to describe the miscibility gap between

γ-Fe[N] and γ'-Fe4N1-x and the ordering (SRO or LRO) of nitrogen atoms N and

vacancies V on the interstitial sublattices of γ-Fe[N] and γ'-Fe4N1-x in particular. In the

thermodynamic description of γ'-Fe4N1-x, LRO has been incorporated explicitly

(Section 3.2.1). The results have been compared with available literature data.

3.2. DESCRIPTION OF LRO AND SRO OF INTERSTITIALS BY

THE CLUSTER VARIATION METHOD (CVM)

3.2.1. APPLICATION TO AN INTERSTITIAL SOLID SOLUTION

Binary solid solutions, consisting of metal atoms (M) and interstitial atoms (I), can be

described at temperatures well below the melting point by two interpenetrating

sublattices: one fully occupied by atoms M (M sublattice) and one partially occupied

by atoms I (I sublattice). The I sublattice is conceived as a solid solution of atoms I

and vacancies V[14]. Thus, both γ-Fe[N] and γ'-Fe4N1-x consist of an Fe fcc sublattice

wherein the nitrogen atoms occupy a fraction of the octahedral interstices.

In contrast with the models applied previously to the Fe-N system and as

indicated in Section 3.1, the CVM can take into account multi-particle interactions.

Therefore, a basic cluster, with reference to the previous models necessarily

containing more than two lattice points, is chosen such that the whole lattice can be

Chapter 3

30

constructed using only the basic cluster. Each possible distribution of atoms over the

sites of the basic cluster is accounted for by a cluster-distribution variable. The value

of each cluster-distribution variable equals the fractional occurrence of that particular

cluster in the crystalline solid. The enthalpy and entropy contributions to the energy

function are expressed in terms of the cluster-distribution variables[20,21].

Long-range ordering on the I sublattice implies that a distinction has to be

made between sites which are preferably occupied by atoms N and sites which are

preferably occupied by vacancies V. Hence, in order to describe LRO using the CVM,

a set of sublattices (i.e. specific lattice points in the chosen basic cluster) is indicated

such that the point group symmetry of the ordered phase is reflected. In the absence of

LRO, SRO is characterized by the discrepancy of the values obtained by CVM for the

cluster-distribution variables and those obtained straightforwardly for a random

distribution of the atomic species involved.

The CVM is usually applied to substitutional systems. In the present work

ordering of atoms N (and vacancies V) on the I sublattice is considered. The M-

sublattice is assumed to remain fully occupied by Fe atoms. Thereby, ordering of

interstitials N on the I sublattice can be described as ordering in a binary (N,V)

substitutional system.

The interaction of the Fe sublattice with the interstitials N is not accounted for

explicitly, but it is effectively incorporated in the interaction parameters for the N-N,

N-V and V-V pair interactions (see also Sections 3.2.3.1 and 3.2.4). In the Fe-N

system, a considerable part of the interaction of interstitials is based on the elastic

strains introduced by a misfitting N atom in an octahedral interstice[22]. Obviously, the

elastic interaction of interstitials is mediated by the Fe-sublattice. Using CVM indirect

interactions have also been adopted to describe the antiferromagnetic-paramagnetic

transition in α-Fe2O3[23] and ordering due to M-M electrostatic repulsion in the

hematite (α-Fe2O3) - ilmenite (FeTiO3) two-phase region[24]; in these cases the

interactions are mediated by the O sublattice.

3.2.2. THERMODYNAMICS OF γ-Fe[N] AND γ'-Fe4N1-x

Order-disorder phase transformations in substitutional binary and ternary systems

have been described extensively with the CVM[20,21,25-37]. To calculate equilibria

applying the CVM, usually the Helmholtz energy of the system is minimized.


31

Recognizing that the volume changes pronouncedly with composition for interstitial

solid solutions, relative to substitutional solid solutions, in the present work the Gibbs

energy, G, is minimized to calculate equilibrium. Then, in addition to the temperature,

during minimization of G the external pressure is kept constant, instead of the volume

per atom as in the usual CVM approximations based on the Helmholtz energy.

Further, use of the Gibbs energy, as compared to use of the Helmholtz energy, leads

to an extra minimization condition (Section 3.2.3)[25].

Using the CVM tetrahedron approximation, the ordering of atoms N and

vacancies V on the interstitial sublattice of γ-Fe[N] and γ'-Fe4N1-x is described. The

interstitial (I) sublattice exhibits a disordered (A1) structure in the case of γ-Fe[N] and

an ordered (L12) structure in the case of γ'-Fe4N1-x. In turn, the I sublattice is

subdivided in four interpenetrating simple cubic lattices. In the tetrahedron

approximation the basic cluster is a regular tetrahedron[38] (Fig. 3.1.). Six nearest

neighbor I-I interactions can be discerned on the basis of the four tetrahedron

sublattice sites, denoted by the superscripts α, β, γ and δ. Whether the sites are

occupied by N (nitrogen atoms) or V (vacancies) is indicated by the value of the

subscripts i, j, k, and l, which take a value of 1 or 2.

Fig. 3.1. The interstitial sublattice is subdivided in 4 interpenetrating simple cubic

sublattices α, β, γ, and δ. The basic cluster in the CVM tetrahedron approximation is

constituted by one site of each of the sublattices α, β, γ, and δ.

Chapter 3

32

In the A1 structure, the probability of finding N or V on a sublattice site of

type α, X iα , is equal to that of sublattice sites of type β, γ, and δ. Thus the symmetry

of the A1 structure can be described by δγβαlkji XXXX === (with i=j=k=l equal to

1 or 2). In the L12 structure N and V reside preferably on their own type of sublattice

site. If sublattice sites of type α are denoted N-type sites and, consequently, sublattice

sites of type β, γ and δ are denoted V-type sites, the probability of finding an atom N

at an N-type sublattice site is X1α and at a V-type sublattice site is X1

β (= X1

γ = X1

δ ).

The probability of finding a vacancy V at an N-type sublattice site is X2α and at a V-

type sublattice site is X 2β

(= X 2γ = X 2

δ ). In γ'-Fe4N1-x the ratio of N:V-type sublattice

sites is 1:3[5]. The symmetry of the L12 structure is described by X iα ≠ X j

β= X k

γ= X l

δ

(with i = j = k = l equal to 1 or 2).

3.2.2.1. INTERNAL ENERGY

The internal energy of the system is taken equal to the sum of the internal energies of

all occurring tetrahedrons. In the present case a total number of N lattice sites (at the I

sublattice) corresponds with a total number of 2N tetrahedrons (two tetrahedrons per

lattice site). Hence, the internal energy of the system is given by

∑=ijkl

ijklijkl ZU αβγδαβγδεN2 (1)

where εijklαβγδ

is the energy of a specific tetrahedron configuration with a frequency of

occurrence given by the distribution variable Zijklαβγδ

, which indicates the probability

that a tetrahedron has configuration ijkl on the tetrahedron sublattice sites α, β, γ, and

δ. The tetrahedron energy εijklαβγδ

is described as the sum of the pairwise interactions

within the tetrahedron:

( ) ( ) ( ) ( ) ( ) ( ) ( )rrrrrrr kljljkilikijijkl

γδβδβγαδαγαβαβγδ εεεεεεε +++++= (2)

where ( )rij

αβε is the pair interaction energy of an N-N pair (εNN) or an N-V pair (εNV)

or an V-V-pair (εVV) on the sublattice sites i and j, depending on the interatomic

distance r.


33

3.2.2.2. CONFIGURATIONAL ENTROPY

The configurational entropy in the tetrahedron approximation for an fcc lattice, as the

I sublattice considered here, is described as [20]

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

++++

++++

+−−=

∑∑∑∑

∑∑∑∑

∑ ∑∑

l

l

k

k

j

j

i

i

kl

kl

jl

jl

jk

jk

il

il

ijkl ik

ik

ij

ijijkl

XLXLXLXL

YLYLYLYL

YLYLZLS

δγβα

γδβδβγαδ

αγαβαβγδ

45

2BNk

(3)

where kB is Boltzmann’s constant and the function L(a) = a ln a - a. Yijαβ

indicates the

probability that a pair of nearest neighbour tetrahedron sublattice sites of type α and β

has configuration ij. X iα and Yij

αβ can be calculated from the tetrahedron distribution

variables Zijklαβγδ

:

X iα = Zijkl

αβγδ

jkl∑ , ... (4a)

Yijαβ

= Zijklαβγδ

kl∑ , Yik

αγ= Zijkl

αβγδ, ...

jl∑ (4b)

The ordering of N and V is assumed to take place on an undistorted interstitial

sublattice. Atomic displacements of Fe atoms due to the presence of N atoms in the

octahedral interstices are not explicitly accounted for.

The change in vibrational entropy for the phase transition γ – γ' is regarded

small in comparison to the change in configurational entropy, because both γ and γ'

are based on interpenetrating fcc M and I sublattices. Therefore, vibrational entropy

terms are neglected in the present work[39].

Chapter 3

34

3.2.3. CALCULATION OF PHASE EQUILIBRIA

For each phase a thermodynamic function Ω , referred to as the grand potential

function, is defined [25]

( ) ∑∑==

∗∗ −=−+−≡Ω2

1

*2

1

*21 ,,,

n

nn

n

nn xGxpTSUTp µµµµ V (5)

where G is the Gibbs free energy, U is the internal energy, S is the entropy and V is

the volume, all per tetrahedron-cluster site. The term T is the temperature, p is the

external pressure and xn denotes the mole fraction of component n (n = 1, 2) in the

phase considered:

( ) 4/δγβαnnnnn XXXXx +++= (6)

Furthermore, µn∗ is an effective chemical potential (Appendix A), defined as

∑−=n

nnnc

µµµ1* (7)

where µn is the chemical potential of component n and c is the number of

components in the system. The constraint that the tetrahedron distribution variables

Z ijklαβγδ

obey,

1=∑ijkl

ijklZ αβγδ (8)

is accounted for by introduction of the Lagrange multiplier λ. Then, minimization of

the grand potential function with respect to Z ijklαβγδ [20] yields

8/52/1ijkl

8exp

kT

-exp

2kTexp −

∗∗∗∗

+++

= ijklijkl

lkji

ijkl XYkT

Zµµµµελαβγδ


35

with Yijkl ≡ Yijαβ

Yikαγ

YilαδYjk

βγYjl

βδYkl

γδ

and δγβαlkjiijkl XXXXX ≡ (9a)

The volume per cluster site, V, corresponding to a particular Zijklαβγδ

for a phase at

constant p and T is obtained from

∑

=−

ij

ij

ijd

dY

wp

V

ε

2 (9b)

where w is the coordination number of the fcc lattice and εij denotes the pair

interaction energy between nearest neighbors (cf. text following Eq. (2)). (In this

work, atmospheric pressure is taken as the reference pressure).

Using Eqs. (9a) and (9b), the grand potential function in the tetrahedron

approximation is minimized with respect to Zijklαβγδ

applying the Natural Iteration (NI)

method[26,40]. At thermodynamic equilibrium, the Zijklαβγδ

(and thus the corresponding

composition and volume) of the phases involved, γ and γ', can be calculated for the

chosen range of temperatures from the following conditions (Appendix A):

minimum of Ωγ

= minimum of Ωγ '

(10)

µN∗,γ

= µN∗,γ '

µV∗,γ

= µV∗,γ '

The fit parameters (discussed in the next Section) are adjusted by trial and error, and

the procedure of calculating the composition of the phases involved is repeated until

the calculated and the experimental phase boundaries agree as well as possible.

Chapter 3

36

3.2.3.1. DESCRIPTION OF THE PAIR INTERACTION ENERGIES

3.2.3.1.1. LENNARD-JONES PAIR POTENTIAL

The dependence of the pair interaction energy ε12αβ

, for an atom 1 on a site of type α

and an atom 2 on a site of type β, on the interatomic distance r may be described by a

so-called 8-4 type Lennard-Jones interatomic potential[25]:

( )

−

=

4012

80120

1212 2r

r

r

rr εε αβ (11)

where ε12o is the pair interaction energy in the reference state and parameter r12

o

corresponds to the interatomic distance for which ( )rαβε12 has a minimum value, equal

to - o

12ε .

3.2.3.1.2. LENNARD-JONES PARAMETERS

In general, the parameters ε11o and r11

o can be estimated from the enthalpy of

formation o

f H1∆ per atom and the lattice constant (a1) of the pure element 1,

respectively: ε11o and r11

o can be written as

o

f

o Hw

111

2∆−=ε (12a)

and in the case of an fcc structure it holds for r11o

r11o =

a1

2 (12b)

In the present work the pair interaction energies ε ijαβ

etc. (cf. Eq. (2)) for the binary N,

V system on the I sublattice have to incorporate the interaction with the fully occupied

Fe sublattice. Hence, recognizing that the pure γ-Fe phase is associated with an

interstitial sublattice composed solely of vacancies, the effective pair interaction

energy in the reference state for an V-V pair at neighboring interstitial sites, εVVo , is


37

calculated from the standard enthalpy of formation of γ-Fe[41] using Eq.(12a). The

parameter rVVo is calculated from the lattice parameter of pure γ-Fe[N] [42] using

Eq.(12b).

Unfortunately the available thermodynamic data do not allow an estimation of

εNNo , εNV

o , rNNo and rNV

o in a similar way. Therefore, to model the equilibrium

between γ-Fe[N] and γ'-Fe4N1-x, εNNo , εNV

o , rNNo and rNV

o were adopted as fit

parameters.

3.2.4. APPLICATION TO THE γ-Fe[N] / γ'-Fe4N1-x MISCIBILITY GAP

In Sections 3.2.1 through 3.2.3, the tetrahedron approximation of the CVM for the

case of ordering of atoms N and vacancies V on an fcc interstitial sublattice, as in

γ-Fe[N] and γ'-Fe4N1-x, has been presented, as well as the corresponding procedure to

calculate phase boundaries. On this basis, the equilibrium compositions of the γ-Fe[N]

and γ'-Fe4N1-x phases at the γ / γ+ γ' and γ+ γ' / γ' phase boundaries in the Fe-N system

were calculated for temperatures in the range of 864 to 923 K.

The equilibrium between the γ-Fe[N] and the γ'-Fe4N1-x phase extends over a

temperature region of only 60 K. A limited number of experimental phase-boundary

data are available [43-45]. However, the composition at the eutectoid temperature (the

α / γ / γ' triple point) is known accurately, and was taken as point of suspension

during the fitting procedure.

The experimental phase-boundary data and the calculated γ / γ' miscibility gap

are shown in Fig. 3.2(a). Differences with previous attempts to model this miscibility

gap can be assessed using Figs. 3.2(b) and (c). Note the differences between

descriptions of the γ+ γ' / γ' phase boundaries for constant composition of γ'-Fe4N1-x

and those based on a variable composition of γ'-Fe4N1-x. The values for the Lennard-

Jones parameters providing the best fit are gathered in Table 3.1.

Chapter 3

38

Table 3.1. Lennard-Jones parameters used in the CVM phase-boundary calculations

Parameters Normalised parameters Pair

interaction εo (kJ/mol) ro (nm) εo

/ ε VVo r

o/ rVV

o

Method of

determination

V-V 68.27 0.25265 1.000 1.000 Eqs. (12)

V-N 66.25 0.26687 0.971 1.056 fitting

N-N 47.79 0.29459 0.700 1.166 fitting

Fig 3.2. (a) The experimental phase-boundary data and the calculated

γ-Fe[N] / γ'-Fe4N1-x miscibility gap. Differences with previous attempts to model this

miscibility gap can be assessed using


39

Figs. 3.2(b) and (c), showing the γ / γ + γ' and γ + γ' / γ' phase boundaries,

respectively.

Chapter 3

40

3.3. DISCUSSION

3.3.1. LATTICE PARAMETERS OF THE FCC Fe-N PHASES

The lattice parameters of γ-Fe[N] and γ'-Fe4N1-x, as calculated from the volume V of a

cluster site (Section 3.2.3), are compared with experimental values given in the

literature (Refs. 42 and 46) in Fig. 3.3. The calculated values are somewhat smaller

(up to 3% for γ'-Fe4N1-x) than the experimental ones.

Fig. 3.3. Calculated lattice parameters of γ-Fe[N] and γ'-Fe4N1-x as obtained by CVM

as a function of the percentage of nitrogen dissolved, and corresponding experimental

data.


41

The fit parameter rNN is directly related to the lattice parameter of an

(interstitial) fcc sublattice fully occupied by nitrogen (refer to the discussion of rVV in

Section 3.2.3.1). Thus, the value determined for rNN can be used to calculate (cf. Eq.

(12b)) the lattice parameter of γ-FeN (NaCl structure). The phase γ-FeN has actually

been prepared by magnetron sputtering. The lattice parameters of two γ-FeN phases

(NaCl and ZnS structure) have been reported [47,48]. As follows from Fig. 3.3., again,

the lattice parameter calculated by CVM is smaller than these corresponding

experimental values.

With respect to the underestimation of the actual lattice parameter by the

CVM calculations, as compared to the available experimental data, the following

remarks can be made. i) Lattice parameters are temperature dependent. The

experimental data presented regard room temperature values for the lattice

parameters, whereas the calculated CVM data pertain to ~ 900 K. However, the effect

due to thermal expansion (0.4 to 1%) cannot explain the difference observed (linear

expansion coefficient of γ'-Fe4N1-x in Ref. 46). ii) In the CVM calculations, the Fe

sublattice is considered to be undistorted, whereas displacements of the Fe atoms

neighbouring N atoms can be expected [49]. Such effects will influence the average

lattice parameter. iii) A small discrepancy (0 to 5%), as observed here, between

calculated lattice parameters and experimental values has been observed before [25].

3.3.2. THE γ-Fe[N] / γ'-Fe4N1-x MISCIBILITY GAP

The γ / γ+ γ' phase-boundary on the left-hand side of the miscibility gap is shown in

Fig. 3.2(b). Clearly, as compared to the previous attempts, the present CVM

calculation provides a satisfactory fit with the experimental data for the γ / γ+ γ'

phase-boundary.

The γ+ γ' / γ' phase-boundary on the right-hand side of the miscibility gap is

shown in Fig. 3.2(c). Most previous attempts to model the γ + γ' / γ' phase-boundary

were based on the (S)RS approach. If γ' is conceived as a stoichiometric phase with

zero homogeneity width [8-12], then the γ+ γ' / γ' phase boundary appears as a straight

vertical line in the Fe-N phase diagram (cf. Fig. 3.2(c)). Since in the present work a

composition range in γ' was taken into account, the current results can best be

compared with only those previous works that also incorporate compositional

variation of γ', i.e. Ref. 13 and the extrapolation derived from the α + γ' / γ' phase

Chapter 3

42

boundary calculated in Ref. 14. The experimental data for the γ + γ' / γ' phase

boundary show considerable scatter, which is probably due to experimental

difficulties associated with precise determination of the nitrogen content. Yet, it may

be concluded that the γ + γ' / γ' phase-boundary is well described by the one

calculated by CVM. Note that the composition of the γ' phase at the triple point, as

calculated by CVM, agrees well with the compositions derived from Refs. 29 and 31

(differences <0.2 at.%).

3.3.3. ORDERING OF NITROGEN ATOMS IN γ-Fe[N] AND γ'-Fe4N1-x

The main results of the present work concern the description of nitrogen ordering in γ-

Fe[N] and γ'-Fe4N1-x. The distribution of the nitrogen atoms over the sites of the

interstitial sublattice is different for γ' and γ. In the γ'-phase long-range ordering

(LRO) occurs: a periodic arrangement of atoms N and vacancies V on the interstitial

sublattice, as indicated by diffraction experiments [5]. In the γ-phase, LRO of

interstitial atoms is absent, but Mössbauer experiments [19,50] indicate that the

distribution of atoms N over the octahedral interstices is not random; an indication

for short-range ordering (SRO) in γ also follows from the non-Henrian behaviour of

nitrogen-absorption isotherms for γ-Fe[N] [18,51].

3.3.3.1. LONG RANGE ORDERING OF NITROGEN IN γ'-Fe4N1-x

By denoting I sublattice sites of type α as N-type sites and I sublattice sites of types β,

γ and δ as V-type sites (Fig. 3.1. and Section 3.2.2), the probability of finding an atom

N at a N-type site is X1α and at a V-type site is X1

β (= X1

γ = X1

δ ). Then, the degree of

LRO of N atoms in γ'-Fe4N1-x can be described using an order parameter ρ, defined as

(cf. Ref. 16)

+−

= βα

βα

ρ11

11

3XX

XX (13)

The order parameter, ρ, is a function of the fraction of occupied interstitial

sites, yN; see the results for γ'-Fe4N1-x in equilibrium with γ-Fe[N] in Fig. 3.4. In the

temperature range considered the value of ρ is very close to 1, indicating occurrence


43

of nearly perfect LRO of the N atoms. With decreasing temperature and increasing

occupation of the sites of the interstitial sublattice, the order parameter increases. A

similar trend can be derived from the data presented in Ref. 16 for γ'-Fe4N1-x in

equilibrium with NH3/H2 gas mixtures. The present value of ρ in γ'-Fe4N1-x at the

triple point α / γ / γ' at 863 K (0.9994) can be compared with the value derived by

extrapolation from the LRO data of Ref. 16 (0.998). Experimental data for the degree

of ordering have not been reported.

Fig. 3.4. The degree of order ρ for γ'-Fe4N1-x in thermodynamic equilibrium with

γ-Fe[N] as a function of the fraction of occupied interstitial sites, yN, as obtained

from the cluster-distribution variables, using Eq. 13, for the temperature range 864 K

to 923 K.

3.3.3.2. SHORT RANGE ORDERING OF NITROGEN IN γ-Fe[N]

Short-range ordering in γ-Fe[N] can be evaluated by comparing the site occupancies

of the tetrahedron clusters, as calculated from the cluster-distribution variables Zijklαβγδ

,

and as obtained by application of the CVM for a fixed composition and temperature,

with those for a random distribution of N for the same composition.

Chapter 3

44

Fig. 3.5. The fractions of tetrahedron clusters occupied by a) 0 or 1 atom N b) 2-4

atoms N for both the random distribution and the distribution as calculated by CVM

for γ-Fe[N], in thermodynamic equilibrium with γ'-Fe4N1-x, for a nitrogen content in

γ-Fe[N] of 9.46 at.% and at T = 888K.

The fraction of tetrahedrons with a certain number of interstitial atoms nN,

P

n N

CVM(yN ) , for an occupied fraction yN of the interstitial sublattice, as predicted by

the CVM calculations, is obtained straightforwardly by summation of the distribution

variables Zijklαβγδ

representing tetrahedrons containing nN nitrogen atoms. Since the

occupancies of the I sublattice sites of type α, β, γ and δ in γ-Fe[N] are equal, the

fraction P

n N

RND(yN) for a random distribution of atoms over the interstitial sites

constituting a cluster can be given as

( ) ( ) ( ) NN

N

nn

n1 −−

= S

S

N

n

NN

n

N

RND

n yyyP (14)

where nS is the number of cluster sites (here: ns = 4).

The fractions of tetrahedron clusters occupied by 0 up to 4 atoms N for both

the random distribution and the distribution as calculated by CVM for the γ / γ'

equilibrium are given in Figs. 3.5(a) and (b) for a nitrogen content in γ-Fe[N] of


45

9.46 at.% (and T=888K, cf. Fig. 3.2(b)). According to the CVM calculations, for this

composition only a very small fraction of the tetrahedron clusters contains more than

1 N atom (Fig. 3.5(b)); most tetrahedron clusters contain 0 or 1 N atom (Fig. 3.5(a)).

As compared to a random distribution of N atoms over all available sites, SRO in γ-

Fe[N] involves that the probability of finding tetrahedrons with no N atoms is ~ 10%

lower and that the probability of finding tetrahedrons with 1 N atom is ~ 10% higher.

This reflects a preference for N-V nearest neighbors over N-N or V-V nearest

neighbors on the interstitial sublattice, which is the driving force for ordering.

3.3.3.3. COMPARISON WITH MÖSSBAUER DATA

Quantitative data of SRO of N atoms in γ-Fe[N] can be deduced from Mössbauer

spectroscopy data reported in Refs. 19, 50, and 53 as follows. Interpretation of

Mössbauer spectra leads to values of the fractions of Fe atoms in γ-Fe[N] surrounded

by 0, 1,...6 atoms N (A nN, nN = 0 to 6) on the 6 nearest neighbouring interstitial sites

forming an octahedron surrounding each Fe-atom (Fig. 3.6.); each type of Fe

surroundings corresponds with a subspectrum (component) in the overall Mössbauer

spectrum that has to be unraveled. The occupation of one (or more) of the 8 next

nearest neighboring sites can be included as well by fitting additional subspectra in

the overall spectrum. In γ-Fe[N] the fractions of Fe atoms surrounded by 3 or more

atoms N are too small to be quantified.

The CVM calculations yield values for the cluster variables αβγδijklZ which can

be used to calculate the pair probabilities αβijY , αγ

ikY ,... (cf. Eq. (4b)). For the γ phase

there is no distinction between the sites of type α, β, γ, and δ and, thus,

...==≡ αγαβikijij YYY The relation between the Mössbauer data for

NnA and the CVM

data for Yij can be derived as follows.

The fractional occurrences of N-N, V-V and N-V neighbor pairs on the total

number Mtot of interstitial sites neighboring an Fe atom can be calculated

straightforwardly for cases of 1, 2, nN, ... atoms N distributed over the Mtot interstitial

sites (Appendix B). The fractional occurrence of nN atoms N on the Mtot sites next to

an Fe atom is denoted by NnA (refer to the preceding discussion). Taking into account

all possible surroundings of an Fe atom, the fractional occurrences of N-N, V-V and

N-V pairs in the specimen can be expressed in terms of NnA (nN=0,1,... Mtot). These

expressions for Yij are linear in NnA . The set of expressions for Yij can be rewritten as

Chapter 3

46

a set of expressions for NnA in terms of Yij. Thus, the calculated CVM results for Yij

can be transformed into calculated CVM results for NnA , which then can be compared

with experimental (Mössbauer) results for NnA . Note that since only three pair

probabilities Yij exist, at most three surroundings NnA can be solved from the set of

expressions for NnA . In the present case A0, A1 and A2 are significant (refer to

preceding discussion).

Fig. 3.6. Each Fe atom in γ-Fe[N] can be surrounded by 0,1,...6 atoms N (NnA ,

nN = 0 to 6) on the 6 nearest neighboring interstitial sites forming an octahedron

surrounding each Fe-atom in the fcc structure.

The NnA (nN = 0,1,2) values obtained from the Yij (CVM) values are compared

with the experimental NnA (i = 0,1,2) values obtained from the Mössbauer

experiments reported in Refs. 20, 50, and 53 (Fig. 3.7.). The experimental NnA data

pertain to homogeneous (quenched) γ-Fe[N] samples prepared at temperatures in the

range 923-973 K; the calculated NnA data pertain to γ-Fe[N] in equilibrium with

γ'-Fe4N1-x for the temperature range 864 to 923 K.

It can be concluded that i) the calculations and the experiments indicate that

significant SRO occurs in γ-Fe[N] (refer to the difference with the predictions of NnA

for a random distribution, indicated in Fig. 3.7.) and ii) although the experimental

conditions are not exactly equal to those relevant to the calculations, calculated and

experimental results agree well.


47

Fig. 3.7. The AnN(nN = 0,1,2) values (Fe atom surroundings) obtained from the Yij

values (pair probabilities) calculated by CVM and the corresponding experimental

AnN (nN = 0,1,2) values derived from Mössbauer data.

Chapter 3

48

3.4. CONCLUSIONS

Short- and long-range ordering (SRO and LRO) on the sublattice of interstitial sites

formed by a closed packed sublattice has been described successfully by application

of the Cluster Variation Method.

The CVM calculations demonstrate occurrence of SRO in γ-Fe[N] and distinct

LRO in γ'-Fe4N1-x, in agreement with Mössbauer data and X-ray diffraction.

Quantitative analysis of calculated CVM data and experimental Mössbauer

data for the fractional occurrences in γ-Fe[N] of certain Fe atom surroundings, in

terms of the number of neighboring nitrogen atoms, shows that the CVM predictions

agree well with the measured results.

The calculated miscibility gap between γ-Fe[N] (nitrogen austenite) and

γ'-Fe4N1-x (iron nitride) agrees well with the available experimental data.

As a side result, theoretical lattice parameter values were obtained for γ and

γ', which tended to be up to 4% smaller than available experimental values.


49

APPENDIX A:

CALCULATION OF PHASE EQUILIBRIA

Consider the equilibrium at constant pressure p and temperature T between two

phases, I and II, for a closed binary system with components 1 and 2. The so-called

grand potential function for phase I is defined as:

( ) ∑=

−=Ω2

121 ,,,

i

I

i

I

i

IIII xGTp µµµ (A-1a)

with GI as the Gibbs energy of phase I per cluster site and where µ iI is the chemical

potential of component i in phase I. At constant p and T and for fixed µ1I and µ2

I it

follows from Eq. (A-1a)

IIIIII dxdxGdd 2211 µµ −−=Ω (A-1b)

Since II dxdx 21 −= (closed system), at a minimum of IΩ ( 0=Ω Id )

III

I

I

dx

Gdµµµ ∆≡−= 21

1

(A-2a)

Similarly for phase II:

IIIIII

II

II

dx

Gdµµµ ∆≡−= 21

1

(A-2b)

In order for equilibrium to exist between phase I and phase II, at least

II

II

I

I

dx

Gd

dx

Gd

11

= (A-3)

which is a consequence of GI and GII, both a function of x1, sharing the same tangent

(however, this is not a conclusive condition for a common tangent; as shown

Chapter 3

50

subsequently). Imposing equilibrium it follows that III

11 µµ = and III

22 µµ = and thus III µµ ∆=∆ µ∆≡ . Now the minimum of IΩ and the minimum of IIΩ can be

calculated as a function of µ∆ . At the point of intersection, 'µµ ∆≡∆ , the minima of IΩ and IIΩ are equal, which yields (cf. Eq. (A-1a))

( )( ) '

11

µ∆=−−

III

III

xx

GG (A-4)

Hence for 'µµ ∆=∆ , GI and GII, both as a function of x1, share the same tangent.

Thus the equilibrium between phases I and II is described by the minimum of IΩ (or IIΩ ) at 'µµ ∆=∆ .

Finding the minimum of IΩ and the minimum of IIΩ as a function of µ∆

may be done by choosing a value of, say, 2µ and by varying 1µ . However, only a

specific value of 2µ , unknown at this stage, is compatible with equilibrium between

the phases I and II. Therefore a more efficient approach is to choose only a value for

µ∆ without specifying a value for 2µ . This can be done by redefining the zero level

of the energy scale for each value of µ∆ such that the modified ∗1µ and ∗

2µ are

known, given a value for ∗∗ −=−=∆ 2121 µµµµµ . This leads to:

+−=∗

221

11

µµµµ ;

+−=∗

221

22

µµµµ (A-5)

implying: ∗=∆ 12µµ and ∗∗ −= 21 µµ .


51

APPENDIX B:

CALCULATION OF THE PAIR PROBABILITIES FOR VARIOUS IRON

ATOM SURROUNDINGS ON THE INTERSTITIAL SUBLATTICE

Consider an Fe atom surrounded by a number of neighboring interstitial sites Mtot.

Any combination of two of these interstitial sites constitutes a pair of interstitial sites.

There are NP = (Mtot!)/(2!(Mtot-2)!) of such pairs. Only a fraction (f) of these pairs can

be designated as neighbor pairs. As an example refer to the six sites of the octahedron

of (nearest) neighbor interstitial sites surrounding an Fe atom in the fcc sublattices of

γ and γ'. There are 6!/(2!4!) = 15 pairs and 3 of the 15 pairs are not neighbor pairs,

implying f = 4/5.

Now, suppose there are nN atoms N and nV vacancies V on the Mtot interstitial

sites considered (nN + nV = Mtot). Any combination of two N atoms out of the nN

atoms N constitutes an N-N pair. There are NN-N = nN!/(2!(nN-2)!) N-N pairs. The

number of neighbor N-N pairs then is (cf. previous reasoning): fNN-N and the

fractional amount of N-N neighbor pairs is fNN-N/(fNP). Similarly it follows for the

fractional amount of neighbor V-V pairs: NV-V/NP, where NV-V is analogous to NN-N

given previously.

For the case considered, nN atoms N and nV vacancies V on Mtot sites, the

number of N-V pairs NN-V is given by the product of i) the number of ways to select

one N atom out of nN atoms (= nN!/(1!(nN-1)!) = nN) and ii) the number of ways to

select one V “atom” out of nV “atoms” (= nV!/(1!(nV-1)! = nV). The fractional amount

of neighbor N-V pairs then is NN-V/NP.

Chapter 3

52

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2. Harvig, H., Acta Chem. Scand., 1971, vol. 25(9), pp. 3199-3204

3. Kaufman, L., H. Bernstein, Computer Calc. of Phase Diagrams, NY, 1970

4. Redlich, O., A.T. Kister, Ind. Eng. Chem., 1948, vol. 40, pp. 345



7. Hillert, M., M. Jarl, Met. Trans. A, 1975, vol. 6A, pp. 553-59

8. Ågren, J., Met. Trans. A, 1979, vol. 10A, pp. 1847-52

9. Kunze, J., Steel Research, 1986, vol. 57 (8), pp. 361-67

10. Kunze, J., Nitrogen and Carbon in Iron and Steels; Thermodynamics, Phys. Res., vol. 16,

Akademie Verlag, Berlin, 1990



13. Du, H., J. Phase Eq., 1993, vol. 14(6), pp. 682-93

14. Kooi, B.J., M.A.J. Somers, E.J. Mittemeijer, Met. Mat. Trans. A, 1996, vol. 27A, pp. 1063-71

15. Gokcen, N.A., Statistical thermodynamics of alloys, Plenum Press NY, 1986

16. Kooi, B.J., M.A.J. Somers, E.J. Mittemeijer, Met. Mat. Trans. A, 1996, vol. 27A, pp. 1055-61

17. Somers, M.A.J., B.J. Kooi, L. Maldzinski, E.J. Mittemeijer, A.A. van der Horst, A.M. van der

Kraan, N.M. van der Pers, Acta Mater., 1997, vol. 45(5), pp. 2013-25

18. McLellan, R.B., K. Alex, Scripta Met., 1970, vol. 4, pp. 967-70

19. Fall, I., J.-M.R. Genin, Met. Mat. Trans. A, 1996, vol. 27A, pp. 2160-77

20. Kikuchi, R., D. de Fontaine, Appl. of phase diagrams in Metall. and Ceramics, Proc. of workshop

at NBS Gaithersburg, 1977, pp. 967-79

21. De Fontaine, D., R. Kikuchi, Appl. of phase diagrams in Metall. and Ceramics, Proc. of workshop

at NBS Gaithersburg, 1977, pp. 999-1027

22. Zener, C., Phys. Rev., 1948, vol. 74(6), pp. 639-47

23. Burton, B., R. Kikuchi, Phys. Chem. Minerals, 1984, vol. 11, pp. 125-31


25. Sanchez, J.M., J.R. Barefoot, R.N. Jarrett, J.K. Tien, Acta Met., 1984, vol. 32(9), pp. 1519-25

26. Kikuchi, R., J. Chem. Phys., 1974, vol. 60, pp. 1071-80

27. Kikuchi, R., Phys. Rev., 1951, vol. 81, pp. 988-1003

28. Colinet, C., Theory and Applications of the Cluster Variation Method and Path Probability

Methods, J.L. Morán-López and J.M. Sanchez, Plenum Press NY, 1996, pp. 313-40


53


30. Sigli, C., J.M Sanchez, CALPHAD, 1984, vol. 8(3), pp. 221-31

31. Enomoto, M., H. Harada, Met. Trans. A, 1989, vol. 20A, pp. 649-64

32. Kikuchi, R., J.M. Sanchez, D. De Fontaine, H. Yamauchi, Acta Met., 1980, vol. 28, pp. 651-62

33. Sanchez, J.M., D. De Fontaine, Phys. Rev. B, 1980, vol. 21, pp. 216

34. Sanchez, J.M., D. De Fontaine, Phys. Rev. B, 1982, vol. 25, pp. 1759

35. Kikuchi, R., J. Chem. Phys., 1977, vol. 66, pp. 3352

36. Burton, B., R. Kikuchi, Am. Min., 1984, vol. 69, pp. 165-75

37. Colinet, C., G. Inden, R. Kikuchi, Acta Met. Mat., 1993, vol. 41(4), pp. 1109-18

38. Kikuchi, R., D. de Fontaine, M. Murakami, T. Nakamura, Acta Met., 1977, vol. 25, pp. 207-19

39. De Fontaine, D., C. Wolverton, Ber. Bunsenges. Phys. Chem., 1992, vol. 96 (11), pp. 1503-12

40. Kikuchi, R., Acta Met., 1977, vol. 25, pp. 195-205

41. Kubaschewski, O., C.B. Alcock, P.J. Spencer, Materials thermochemistry, 6th ed. Perg. Oxford

1993

42. Cheng, L., A. Böttger, Th.H. de Keijser and E.J. Mittemeijer, Scripta Met. Mat., 1990, vol. 24, pp.

509

43. Eisenhut, O., E. Kaupp, Z. Elektrochem., 1930, vol. 36(6), pp. 392-404

44. Lehrer, E., Z. Elektrochem., 1930, vol. 37(7), pp. 460-73

45. Paranjpe, V.G., M. Cohen, M.B. Bever, C.F. Floe, Trans. AIME, 1950, vol. 188, pp. 261-67

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vol. 20A, pp. 1533-39

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55

4


METHOD TO AN INTERSTITIAL SOLID SOLUTION:

THE γ'-Fe4N1-x / ε-Fe2N1-z EQUILIBRIUM

ABSTRACT

The cluster variation method has been applied to establish effective interaction

potentials that describe both γ'-Fe4N1-x / ε-Fe2N1-z miscibility gaps in the Fe-N phase

diagram. The calculated nitrogen distributions show that long-range order occurs in

the γ'-Fe4N1-x phase and that short-range ordering, as well as long-range ordering,

occurs in the ε-Fe2N1-z phase. The calculated nitrogen distributions for the ε-Fe2N1-z,

pertaining to temperatures and concentrations at the γ' / ε phase boundaries, are

compared with available data obtained by Mössbauer spectrometry. Preferential

occupation of specific interstitial sites occurs from about 16 at.% nitrogen on; at the

highest concentration considered, about 25 at.% nitrogen, the occupation is that of

Fe3N as proposed in literature on the basis of diffraction data.

Chapter 4

56

4.1. INTRODUCTION

Iron nitrides are metastable solid solutions, as produced in practice by a

thermochemical treatment called nitriding, which is applied to steels to improve their

performance with respect to fatigue, wear, and corrosion. Therefore, the knowledge of

the thermodynamics of Fe-N phases is a prerequisite for understanding and

controlling the process of nitriding in (industrial) applications.

In interstitial solid solutions like the Fe-N phases, the misfit of the interstitial

nitrogen atoms results in pronounced strain-induced interactions, which lead to short-

range ordering (SRO) or long-range ordering (LRO) of the interstitials. To model the

phase diagrams and absorption isotherms of Fe-N alloys, this ordering of the

interstitial atoms needs to be taken into account[1]. LRO has been first introduced in

the thermodynamic description of γ'-Fe4N1-x[1] and ε-Fe2N1-z

[2] by applying the

Gorsky-Bragg-Williams (GBW) approximation. In the present work, the cluster

variation method (CVM) is used. The cluster variation formalism allows considering

both SRO and LRO, which implies that the same set of (atomic interaction)

parameters can be used over the whole composition range of a phase to calculate

thermodynamic data. In addition, the CVM deals with correlations between the

occupations of lattice sites within the cluster (tetrahedron, prism etc.) used in the

approximation[3,4], whereas the GBW approach, which is equivalent to a point cluster

approximation, does not. The CVM was successfully used for calculation of the

γ-Fe[N] / γ'-Fe4N1-x phase boundary[5].

Interstitial solid solutions, consisting of metal atoms and interstitial atoms, can

generally be described by two interpenetrating sublattices: the metal sublattice, fully

occupied by metal atoms, and the interstitial sublattice, partially occupied by

interstitial atoms. The interstitial sublattice is constituted by the octahedral interstices

of the metal sublattice, and is conceived as a solid solution of interstitial atoms and

vacancies (V), in case of a binary metal-interstitial system[2]. The ordering of

interstitial N atoms on the interstitial sublattice can then be described as ordering on

the interstitial sublattice of a binary system consisting of nitrogen atoms N and

vacancies V. The atomic interactions between the metal and the interstitial sublattice

are not accounted for explicitly but incorporated in effective interaction parameters for

the N-N, N-V, and V-V pair interactions. The pair interaction parameters are kept

constant in the calculations, which implies that possible changes in the vibrational

Application of the CVM to an Interstitial Solid Solution: The γ'-Fe4N1-x / ε-Fe2N1-z Equilibrium

57

contributions caused by changes of temperature and composition are not taken into

account explicitly (see discussion Section 4.5.1.).

This chapter starts with the thermodynamic description of γ'-Fe4N1-x and

ε-Fe2N1-z using the CVM, followed by the procedure used to calculate the phase

equilibrium. From the calculation of the γ'-Fe4N1-x / ε-Fe2N1-z equilibrium data on the

ordering of the interstitial atoms and values for the lattice parameters are obtained.

The ordering of N in the γ'-Fe4N1-x phase has been studied in detail in Ref. [5]. The

focus is on the ordering of N in ε-Fe2N1-z, in equilibrium with γ'-Fe4N1-x, as a function

of the nitrogen content. Finally, the results are compared with literature data on the N

distribution, as obtained using Mössbauer spectroscopy.

4.2. THERMODYNAMICS OF γ'-Fe4N1-x

The (long-range) ordering of N and V on the interstitial sublattice of γ'-Fe4N1-x is

described using the CVM tetrahedron approximation as developed in Ref. [5]. In

contrast to ε-Fe2N1-z, the γ'-Fe4N1-x phase shows only small variations in composition:

0 < x < 0.05. The structure consists of an iron (Fe) fcc sublattice, wherein N atoms

occupy preferably a specific fraction of the interstitial fcc sublattice (see further)

constituted by the octahedral interstices. The interstitial fcc sublattice is subdivided in

four interpenetrating simple cubic sublattices, denoted by the superscripts α, β, γ, and

δ. The four sites of the basic regular tetrahedron cluster[6-8] (Fig. 4.1.), denoted by the

subscripts i, j, k, and l, each represent one of the simple cubic sublattices. Whether the

sites are occupied by atoms N or vacancies V is indicated by the value of the

subscripts i up to l, which take a value of 1 (occupied by N) or 2 (vacancy).

In γ'-Fe4N1-x, the atoms N and vacancies V on the interstitial sublattice exhibit

an ordered (Ll2) structure[9]. Both N and V preferably reside on their own type of

sublattice site. Here, sublattice sites of type α are denoted N-type sites, and sublattice

sites of types β, γ, and δ are denoted V-type sites. The probability of finding N on a α-

type site sublattice site in γ'-Fe4N1-x is indicated by X1γ ' ,α while the probability of

finding V on a α-type sublattice site is X2γ ' ,α . The symmetry of the Ll2 structure is

described by Xi

γ ' ,α ≠ X j

γ ' ,β = Xk

γ ' ,γ = Xl

γ ' ,δ( ) (i, j, k, and l can take the values 1 or 2).

Chapter 4

58

Fig. 4.1. The interstitial sublattice of γ'-Fe4N1-x Is subdivided into four penetrating

simple cubic sublattices α, β, γ, and δ. The basic cluster in the CVM tetrahedron

approximation, as indicated in the figure by the dashed lines, is constructed by taking

one site of each of these sublattices.

4.2.1. CONFIGURATIONAL ENTROPY

The configurational entropy of the ordering of N and V on the fcc interstitial

sublattice of γ'-Fe4N1-x is described as a function of the cluster distribution variables,

which indicate the frequency of occurrence of all possible arrangements of N and V

on the tetrahedron cluster and its subclusters (points and pairs). The subcluster

distribution variables are dependent on the tetrahedron distribution variable αβγδijklZ :

Pairs: Yij

αβ = Zijkl

αβγδ

kl

∑

Points: Xi

α = Zijkl

αβγδ

jkl

∑ (1)

where αβijY indicates the probability that a pair of nearest neighbor tetrahedron

sublattice sites of types α and β has configuration ij. For a binary system, i, j, k, l = 1,

2. The tetrahedron distribution variables obey the constraint

Zijkl

αβγδ

ijkl

∑ = 1 (2)

In the tetrahedron approximation, the configurational entropy contribution per lattice


59

site of the fcc interstitial sublattice of γ'-Fe4N1-x is described by[7,5]:

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

++

++

++++

+−−=

∑ ∑∑∑

∑ ∑ ∑∑

∑ ∑ ∑−−

k l

lk

j

j

i

i

jk jl kl

kljljk

il

il

ijkl ij ik

ikijijklB

NFe

XLXLXLXL

YLYLYLYL

YLYLZLkS x

δγβα

γδβδβγαδ

αγαβαβγδγ

45

214'

(3)

where kB is Boltzmann’s constant and the function L a( ) ≡ aln a .

4.2.2. INTERNAL ENERGY

The internal energy of the γ'-Fe4N1-x phase is taken equal to the sum of the internal

energies of all occurring tetrahedrons. For the interstitial sublattice of γ'-Fe4N1-x a

total number of N lattice sites corresponds with a total number of 2N tetrahedrons

(each interstitial lattice site pertains to two tetrahedrons). The internal energy U of the

system per lattice site is then described as

Uγ ' − Fe4 N1−x = 2 ε ijkl

αβγδZijkl

αβγδ

ijkl

∑ (4)

where αβγδε ijkl is the energy of a specific tetrahedron configuration. The tetrahedron

distribution variable αβγδijklZ indicates the frequency of occurrence of a specific

tetrahedron configuration. The tetrahedron energy αβγδε ijkl is described as a sum of the

pair-wise interaction energies within the tetrahedron:

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]rrrrrrr kljljkilikijijkl

γδβδβγαδαγαβαβχδ εεεεεεε +++++=21

(5)

where ( )rij

αβε is the pair interaction energy of an N-N pair ( 'γε NN ) or an N-V pair ( 'γε NV )

or a V-V-pair ( 'γεVV ) on the sublattice sites i and j, depending on the distance r

between the neighboring lattice sites i and j. The factor ½ takes into account that each

pair is shared by two tetrahedrons.

Chapter 4

60

4.2.3. INTERSTITIAL INTERACTIONS IN γ'-Fe4N1-x

The effective pair interaction energies in γ', ε ij

γ 'r( ), are described by an 8-4 type

Lennard-Jones potential given by Ref. [25]:

( )

−

=

4',08',0',0' 2

r

r

r

rr

ijij

ijij

γγγγ εε (6)

The parameter rij0,γ ' corresponds to the distance between the species (N or V)

on sites i and j, for which ε ij

γ 'r( ) has a minimum value equal to -ε ij

0,γ ' . For the

description of γ'-Fe4N1-x, all the interaction parameters were obtained in previous

work[5]. ',0 γεVV and rVV

o,γ ' were estimated from experimental values of the enthalpy of

formation and the lattice parameter of γ-Fe, respectively. The parameters εNV

o,γ ' , ',0 γε NN , ',0 γ

NVr , and rNN

o,γ ' , were obtained by optimizing the correspondence between the

calculated and the experimentally observed γ-Fe[N] / γ'-Fe4N1-x phase equilibrium

data. An overview of the Lennard-Jones parameters describing the pair-wise

interactions of γ'-Fe4N1-x is given in Table 4.1.

Table 4.1. Lennard-Jones parameters γ'-Fe4N1-x

Pair interaction oε (kJ/mol) o

VV

o εε / 0r (nm) 0

0

VV

VV

r

r

V-V 68.270 1.000 0.25265 1.000

V-N 0.971 0.26687 1.056

N-N 0.700 0.29459 1.166

4.3. THERMODYNAMICS OF ε-Fe2N1-z

In the CVM calculations, the (long-range) ordering of N and V on the simple

hexagonal interstitial sublattice of ε-Fe2N1-z is described using the prism

approximation[10,11]. The ε-Fe2N1-z phase displays a large solubility domain:


61

0 < z < 0.6 [12]. The ε-phase structure is conceived as consisting of a fully occupied

hcp Fe sublattice and a simple hexagonal interstitial sublattice, constituted by the

octahedral interstices of the Fe sublattice. The simple hexagonal interstitial sublattice

is subdivided in six interpenetrating simple hexagonal sublattices, denoted by

superscripts α, β, γ, δ, η, and κ see Fig. 4.2.). Each site of the basic prism cluster

represents one type of the six simple hexagonal sublattices α up to κ. The occupation

of the prism sites by atoms N or vacancies V is indicated by the value of the

subscripts i, j, k, l, m and n, which take a value of 1 (occupied by N) or 2 (vacancy V).

Fig. 4.2. The interstitial sublattice of ε-Fe2N1-z is subdivided into six simple hexagonal

sublattices α, β, γ, δ, η, and κ. The basic cluster in the CVM prism approximation is

constructed by taking one site of each of these sublattices.

Both experimental data[13-17] and thermodynamic calculations[2] show that with

changes in composition, different types of long-range ordering of interstitial N atoms

and vacancies can occur. Because the types of ordering and their temperature and

composition ranges of occurrence are not well established yet, no a priori assumption

Chapter 4

62

has been made about the occupations by either atoms N or vacancies V of the prism

sublattice sites in the description of the thermodynamics of ε-Fe2N1-z. In advance

designation of N-type sites and V-type sites as for γ' (Section 4.2.) has not been

performed. The probabilities to find N or V on a sublattice site

Xi

ε ,α , Xj

ε , β , Xk

ε ,γ , Xl

ε ,δ , Xm

ε,η , Xn

ε ,κ (i, j, k, l, m, n equal to 1 or 2) are the outcome of the

calculations.


The configurational entropy of the ordering of N and V on the simple

hexagonal interstitial sublattice of ε-Fe2N1-z is described as a function of the cluster

distribution variables, which indicate the frequency of occurrence of all possible

arrangements of N and V on the prim cluster and its subclusters (points, pairs,

triangles and rectangles). The subcluster distribution variables are dependent on the

prism distribution variable Pijklmn

αβγδηκ :

Rectangles Rijlm

αβδη = Pijklmn

αβγδηκ

kn

∑ , ...

Triangles Tijk

αβγ = Pijklmn

αβγδηκ

lmn

∑ , ...

Horizontal pairs YHij = Pijklmn

αβγδηκ , ...klmn

∑

Vertical pairs YVil = Pijklmn

αβγδηκ , ...jkmn

∑

Points Xi = Pijklmn

αβγδηκ

jklmn

∑ , ...

(7)

The prism distribution variables obey the constraint

Pijklmn

αβγδηκ

ijklmn

∑ = 1 (8)

In the prism approximation the configurational entropy per lattice site for the simple

hexagonal interstitial sublattice is described by


63

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

+++++

−

+++

+++

+

+

+

+−

++−−=

∑∑∑∑∑∑

∑∑∑∑

∑∑∑∑

∑∑∑∑

∑ ∑∑−−

n

n

m

m

l

l

k

k

j

j

i

i

mn

mnj

lm

lm

jk

jk

ik

ik

ij

ij

kn

kn

jm

jm

il

il

lmn

lmn

ijk

ijk

ik

ik

ijklmn jkmn

jkmn

ijlm

ijlmijklmnB

NFe

XLXLXLXLXLXL

YHLYHLYHLYHL

YHLYHLYVLYVL

YVLTLTLRL

RLRLPLkS zN

κηδγβα

ηκδκδηβγ

αγαβγκβη

αδδηκαβγαγδκ

βγηκαβδηαβγδηκε

61

21

31

2

lnln

lnln

1

(9)


The internal energy is taken equal to the sum of the internal energies of all

occurring prisms. For the interstitial sublattice of ε-Fe2N1-z a total number of N lattice

sites corresponds with a total number of 2N prisms (each interstitial lattice site

belongs to two prisms). The internal energy U of the system per lattice site is then

described as

Uε −Fe2 N1−z = 2 εijklmn

αβγδηκPijklmn

αβγδηκ

ijklmn

∑ (10)

where ε ijklmn

αβγδηκ is the energy of a specific prism configuration. The prism distribution

variable Pijklmn

αβγδηκ indicates the frequency of occurrence of a specific prism

configuration. Each interstitial site in ε-Fe2N1-z is surrounded (on the interstitial

sublattice) by two nearest-neighbor sites in the vertical direction and six next-nearest-

neighbor sites in the horizontal direction (cf. Fig. 4.2.). Therefore, for the internal

energy of ε-Fe2N1-z, two types of interactions are distinguished: vertical (nearest-

neighbor) and horizontal (next-nearest-neighbor). The prism energy ε ijklmn

αβγδηκ is

described as the sum of the pair-wise interaction energies within the prism:

[] [ ])()()()(

)()()()()(41

)(

,,,,

,ln

,,,,

rrrr

rrrrrr

v

kn

v

jm

v

il

h

mn

hh

lm

h

jk

h

ik

h

ijijklmn

γκβηαδηκ

δκδηβγαγαβαβγδηκ

εεεε

εεεεεε

++++

++++= (11)

Chapter 4

64

where the factors 1/4 and 1/6 take into account that the next-nearest-neighbor

(horizontal) pairs and next-neighbor (vertical) pairs are shared by four and six prisms,

respectively. The superscripts v and h indicate whether vertical or horizontal

interactions are considered andε ij

αβr( ) is the pair interaction energy of an N-N pair

(εNN

ε ), an N-V pair (εNV ) or a V-V pair (εVV

ε ) on the sublattice sites i and j, depending

on the distance r between neighboring sites i and j.

4.3.3. INTERSTITIAL INTERACTIONS IN ε-Fe2N1-z

The effective pair interaction energies in ε are described by 8-4 type Lennard-

Jones potentials similar to that given in Eq. (6) for γ'. Two sets of Lennard-Jones

potentials are needed to account for the vertical nearest-neighbor and horizontal next-

nearest-neighbor interactions occurring in the prism cluster. The parameters rVV

0,ε ,v and

rNN

0,ε ,v (vertical interactions) and rVV

0,ε ,h and rNN

0,ε ,h (horizontal interactions) are estimated

from the lattice-parameter relationships[18]

a(yN ) = 0.44709 + 0.0673yN (nm)

c(yN ) = 0.42723 + 0.0318yN (nm) (12)

where yN is the fraction of occupied interstitial sites in the structure. The values of

rVV

0,ε ,v and rNN

0,ε ,h are obtained by extrapolation of Eqs. (12) to yN = 0 and yN = 1 and

taking:

rVV

0,ε ,h =a(yN = 0)

3,rNN

0,ε ,h =a(yN =1)

3

rVV

0,ε ,v =c(yN = 0)

2, rNN

0,ε ,v =c(yN = 1)

2

(13)

To restrict the number of fit parameters, rVN

0,ε ,h pertaining to the horizontal V-N

interactions in ε is chosen such that rVN

0,h : rVV

0,ε ,h =1.056, equal to the value of rVN

0,γ ' :rVV

0,γ ' as

obtained from the previous CVM calculation of the γ-Fe[N] / γ'-Fe4N1-x phase

equilibrium (cf. Table 4.1.). Next, the Lennard-Jones parameter rVN

0,ε ,v describing the

vertical V-N interaction in ε-Fe2N1-z is obtained from rVV

0,ε ,v by using the same ratio as

above (Tables 4.2. and 4.3.). Realizing that the energy difference between the close-


65

Table 4.2. Horizontal Lennard-Jones parameters ε-Fe2N1-z

Pair interaction ho ,,εε (kJ/mol) ho

VV

ho ,,,, / εε εε hor .,ε (nm) ho

VV

ho

r

r,,

,,

ε

ε

V-V 68.270 1.000 0.258128 1.000

V-N 0.954 0.272583 1.056

N-N 0.700 0.296983 1.151 Table 4.3. Vertical Lennard-Jones parameters ε-Fe2N1-z

Pair interaction vo ,,εε (kJ/mol) vo

VV

vo ,,,, / εε εε vor ,,ε (nm) vo

VV

vo

r

r,,

,,

ε

ε

V-V 204.811 1.000 0.213615 1.000

V-N 0.954 0.219596 1.028

N-N 0.700 0.229515 1.074

packed structures fcc and hcp is small, the parameters εVV

0,γ ' and εNN

0,γ ' obtained from the

calculation of the γ / γ' phase equilibrium[5] are taken as an estimate for those for the ε-

Fe2N1-z as follows. For γ and γ' phases the Fe sublattice and the interstitial sublattice

structures are the same and thus the sum of all effective pair interactions obtained for

the interstitial lattice is representative for the total internal energy of the system

without more ado, recognizing that the effective interaction parameters incorporate

atomic interactions of the Fe and interstitial sublattices. For the ε-Fe2N1-z phase, the

structures of the Fe sublattice and the interstitial sublattice are different (hcp vs.

simple hexagonal) and hence the number of pair-wise interactions within each of the

sublattices is different. In the Fe sublattice each atom has twelve neighbors of which

six next-nearest-neighbor (= horizontal interactions) and six nearest-neighbor (=

vertical interactions) whereas in the interstitial sublattice each site has eight

neighboring sites of which six next-nearest-neighbor (= horizontal interactions) and

two nearest-neighbor (= vertical interactions). Thus the ratio of neighboring

horizontal lattice sites in the Fe hcp lattice and in the simple hexagonal interstitial

sublattice is 1:1 and the ratio of neighboring vertical lattice sites in the Fe hcp lattice

and the simple hexagonal interstitial lattice is 3:1. The sum of all effective pair

Chapter 4

66

interaction energies on the basis of the interstitial sublattice should represent the total

internal energy of the system, i.e. including the interactions of the Fe atoms. This

implies that the vertical effective pair-wise interaction energies (εVV

0, ε, v , εVN

0, ε, v and

εNN

0, ε, v ) based on the interstitial sublattice should be about three times εVV

0,γ ' , εVN

0,γ '

andεNN

0,γ ' , whereas the horizontal pair-wise effective interactions (εVV

0, ε, h , εVN

0, ε, h and

εNN

0, ε, h ) should be equal to approximately εVV

0,γ ' , εVN

0,γ ' andεNN

0,γ ' . The parameters εVN

0, ε (with

εVN

0, ε, v ≈ 3εVN

0,γ ' and εVN

0, ε, h ≈ εVN

0,γ ' ; see previous discussion) and rVN

0,ε ,v are used as fit

variables and are determined by minimizing the difference between the experimental

and calculated γ'-Fe4N1-x / ε-Fe2N1-z phase boundaries. The Lennard-Jones parameters

for ε-Fe2N1-z applied in the calculation of the internal energy of ε-Fe2N1-z have been

gathered in Tables 4.2. and 4.3.

4.4. THE CALCULATION OF THE γ'-Fe4N1-x / ε-Fe2N1-z

EQUILIBRIUM

By applying the CVM, both phase boundaries of a miscibility gap can be calculated

simultaneously using only one set of fitting parameters. Since in case of the phase

equilibrium between γ'-Fe4N1-x and ε-Fe2N1-z two miscibility gaps occur (Fig. 4.3.),

application of the CVM to the γ' / ε equilibrium implies that even four phase

boundaries are calculated simultaneously with one set of interaction parameters.

Previous calculations of the γ'-Fe4N1-x / ε-Fe2N1-z phase equilibrium were firstly based

on a (sub)-regular solution model without incorporating LRO[19,20] and, later, based on

the Gorsky-Bragg-Williams approximation[2,21] incorporating a priori two types of

LRO indicated by A and B[2]. In the present CVM approach, occurrence of SRO

and/or LRO for the N atoms on the interstitial lattice is not incorporated in advance,

but is established as the outcome of the calculations. Next, the procedure followed to

calculate the phase equilibria is briefly indicated, a detailed description can be found

in Ref. [5]. Compared with substitutional solid solutions, interstitial solid solutions

such as Fe-N phases exhibit a pronounced dependence of the volume on the solute

(nitrogen) content. To take this volume-effect into account, in the present work the

energy is expressed in terms of the Gibbs (instead of the Helmholtz) energy. The

temperature and external pressure were kept constant during the energy minimization.


67

Fig. 4.3. Schematic Fe-N phase diagram indicating the two miscibility gaps

ε-Fe2N1-z / γ'-Fe4N1-x denoted by I and γ'-Fe4N1-x / ε-Fe2N1-z denoted by II.

4.4.1. CALCULATION OF PHASE EQUILIBRIA USING THE CVM

For each phase, a thermodynamic function Ω, referred to as the grand potential

function, is defined

Ω p,T ,µ1∗,µ 2

∗( )≡ U − TS + pV − µ n

*xn

n=1

2

∑ = G − µn

*xn

n=1

2

∑ (14)

where G is the Gibbs energy, U is the internal energy, S is the entropy and V is the

volume, all per cluster site. T is the temperature, p is the external pressure and xn

denotes the mole fraction of component n (n = 1,2; here N, V) in the phase

considered, that is for the tetrahedron approximation: xn = ( Xn

α + Xn

β + Xn

γ + Xn

δ ) / 4

and for the prism approximation xn = ( Xn

α + Xn

β + Xn

γ + Xn

δ + Xn

η + Xn

κ )/ 6 .

Furthermore, µn∗ is the effective chemical potential[8], defined as

µn

* = µn − 12 µn

n =1,2∑ (15)

where µn is the chemical potential of component n.

Chapter 4

68

The grand potential functions Ωγ', for the γ' phase, and Ωε, for the ε phase, can

be minimized with respect to αβγδijklZ and Pijklmn

αβγδηκ , respectively for an adopted value for

∆µ = µ1∗ - µ2

∗ (see Ref. [5], Appendix A). Lagrange multipliers λγ' and λε are introduced

to account for the normalization constraints of respectively the tetrahedron and prism

distribution variables (cf. Eqs. (2) and (8)). The grand potential functions are then

minimized by applying the Natural Iteration method[22,6]. The minimization condition

of the grand potential function Ωγ' with respect to αβγδijklZ yields

Zijkl

αβγδ = expλγ '

2kBT

exp

-ε ijkl

kBT

exp

µi

∗ + µ j

∗ + µk

∗ + µl

∗

8kBT

Yijkl

1 / 2Xijkl

−5 / 8 (16)

with Xijkl ≡ Xi

αX j

βXk

γX l

δ and Yijkl ≡ Yij

αβYik

αγYil

αδYjk

βγYjl

βδYkl

γδ (17)

Minimization of the grand potential function Ωε with respect to Pijklmn

αβγδηκ yields

12/16/14/12/12/1

******

12expexp

2exp

ijklmnijklmnijklmnijklmnijklmn

B

iiiiii

B

ijklmn

B

ijklmn

XYVYHTR

TkTkTkP

−−×

+++++

−

=

µµµµµµελεαβγδηκ

with

Rijklmn ≡ Rijlm

αβδηRjkmn

βγηκRikln

αγδκ (18)

Tijklmn ≡ Tijk

αβγTlmn

δηκ

YHijklmn ≡ YHij

αβYHik

αγYH jk

βγYHlm

δηYHln

δκYHmn

ηκ , YVijklmn ≡ YVil

αδYVjm

βηYVkn

γκ

Xijklmn ≡ X i

αXj

βXk

γXl

δXm

ηXn

κ

At the minimum of Ω for each phase the volume per cluster site, V, at constant

p (taken as atmospheric pressure), ∆µ and constant T is determined by –p = dU/dV.

At each temperature, a curve of Ω(minimum) versus µ* is thus obtained for each

phase. Thermodynamic equilibrium of the phases involved, γ' and ε, then is

determined by the points of intersection of the Ωγ ' (minimum) − µ *,γ ' and

Ωε(minimum) − µ*,ε curves for each temperature, representing the phase boundaries

of miscibility gap I and II, respectively (Fig. 4.4.).


69

Fig 4.4. Schematic representation of the determination of the points of intersection of

the curves Ωγ' (minimum) – µ*, γ' and Ω (minimum) – µ*, ε

at a fixed temperature. From

each point of intersection, cluster distribution variables and lattice parameters for the

ε-Fe2N1-z and γ'-Fe4N1-x phases in thermodynamic equilibrium are obtained. The point

of intersection of the miscibility gap ε / γ' is indicated by I and for the miscibility gap

γ' / ε by II.

4.5. DISCUSSION

The procedure outlined in Sections 4.2. to 4.4. allows calculation of phase boundaries

and analysis of the distribution of nitrogen atoms over the interstitial lattice

(corresponding to occurrence of SRO or LRO) of γ'-Fe4N1-x and ε-Fe2N1-z.

The calculation of the compositions of γ'-Fe4N1-x and ε-Fe2N1-z at the ε / ε+ γ'

and ε+ γ' / γ' phase boundaries (miscibility gap I) and at the γ' / γ'+ε and γ'+ε / ε phase

boundaries (miscibility gap II) (Fig. 4.5.) is presented for temperatures in the range of

550 to 950 K (Section 4.5.1.).

The presence of the N atoms in interstitial octahedral sites of the iron nitrides

results in strain-induced N-N interactions and, upon increasing the nitrogen content,

the N atoms start to order. The variation in nitrogen content in ε-Fe2N1-z is very large

compared with other Fe-N phases; both short- and long-range ordering of N atoms are

Chapter 4

70

expected to occur. The values for the cluster distribution variables, pertaining to the

γ'-Fe4N1-x / ε-Fe2N1-z phase equilibrium, are used to discuss ordering of N in

γ'-Fe4N1-x (Section 4.5.2.) and in ε-Fe2N1-z (Section 4.5.3.). Finally, on the basis of the

values of the cluster distribution variables, values for the fractions of Fe atoms

surrounded by zero, one or more N atoms can be calculated. The latter values are

compared with Mössbauer data (Section 4.5.4.).

4.5.1. THE γ'-Fe4N1-x / ε-Fe2N1-z PHASE EQUILIBRIA

The γ'-Fe4N1-x / ε-Fe2N1-z phase equilibrium consists of two miscibility gaps

(Fig. 4.3.). The calculated miscibility gaps, using the atomic interaction parameters

given in Table 4.2. and 4.3., as well as the corresponding available experimental data,

are shown in Fig. 4.5.

The calculated ε / ε+ γ' phase boundary of miscibility gap I well agrees with the

experimental data (Fig. 4.5.). For both the ε+ γ' / γ' phase boundary (miscibility gap I)

and the γ' / γ'+ε phase boundary (miscibility gap II), the present CVM calculation

results in nitrogen contents of the γ' phase within the range of the experimental data. It

should be noted that the experimental data for the composition of γ' at both γ' / ε phase

boundaries show considerable scatter. The minimum solubility limit of N in γ' in the

temperature range of 923K to 940 K (miscibility gap I) as obtained from the

calculations, 19.5 at.% N to 19.6 at.% N, is within the experimentally observed

nitrogen contents that vary between about 18 at.% N to 19.6 at.% N. The maximum

solubility limit of nitrogen in γ' as obtained from the calculations in the temperature

range of 573 K to 940 K (miscibility gap II) is between 19.98 at.% N and 19.47 at.%

N, which is close to the experimentally obtained nitrogen contents that vary from20.7

at.% N to 19.2 at.% N. The solubility limit of the ε phase (gap II) agrees well with the

experimental data at high temperatures, but deviations occur in the low-temperature

region (< 800 K). The calculated phase boundary results in a nitrogen content of the ε

phase between 24.7 at.% N and 24.9 at.% N between 800 K and

600 K, whereas the corresponding experimental data range between 24.5 at.% N and

26.3 at.% N.


71

Fig. 4.5. Part of the Fe-N phase diagram, showing the γ'-Fe4N1-x / ε-Fe2N1-z phase

equilibria calculated by CVM, as well as the available literature data. The miscibility

gap ε / γ' is indicated by I and the miscibility gap γ' / ε by II (cf. Fig. 3.3.)

In the present calculations, possible changes in the vibrational entropy, caused

by changes of temperature and composition, are not incorporated explicitly.

Neglecting changes in vibrational entropy is reasonable for γ'-Fe4N1-x because the

degree of ordering of N and the composition are almost constant along the phase

boundary (Section 4.5.2.). Indeed, in the temperature range considered, the changes in

vibrational entropy are reported to be very small for γ'-Fe4N1-x[23]. The ε-Fe2N1-z phase

has a broad composition range which corresponds to a large variation in the degree of

ordering of N atoms (Section 4.5.3.) that may cause a variation in vibrational entropy

contribution. Although contributions of the vibrational entropy are not explicitly

incorporated in the CVM calculations, their dependence on composition may be

Chapter 4

72

implicitly represented by the relative contributions of the V-V, N-V or N-N pair

interactions as a function of composition. Thus, the observed discrepancies between

the calculations and the experimental phase boundaries may be caused primarily by

neglecting possible temperature dependencies of the effective interaction parameters.

In this context it must be noted that the parameters used were optimized for the γ / γ'

phase boundary[5] (cf. Sections 4.2. and 4.3.); that is, they hold in particular for a

temperature range of about 800 to 900 K. Indeed, the discrepancies in Fig. 4.5. mainly

occur at temperatures below about 800 K. Furthermore, it should be remarked that for

the calculation of the γ' / ε phase equilibrium in the present work (Section 4.3.2.) only

the parameters εVN

0, ε and rVN

0,ε ,v describing the effective interactions in the ε phase were

adapted to optimize the correspondence of the calculated with the experimental phase

boundaries.

4.5.2. ORDERING OF NITROGEN ATOMS IN γ'-Fe4N1-x

The degree of long-range ordering of the interstitial nitrogen atoms in γ'-Fe4N1-x can

be described using an order parameter, ρ, defined as (cf. Refs. [1,5])

+−

= βγαγ

βγαγ

ρ ',1

',1

',1

',1

3XX

XX (19)

where the probability of finding N on a α-type sublattice site (i.e. N-type site) is

indicated by X1γ ' ,α while the probability of finding N on a β-type (i.e. V-type site)

sublattice site is X1γ ' ,β(= X1

γ ' ,γ = X1γ ' ,δ ) . The order parameter is a function of the

fraction of occupied interstitial sites yN and is shown in Fig. 4.6. for γ'-Fe4N1-x in

equilibrium with ε-Fe2N1-z. The order parameter is close to unity in the temperature

(and composition) range considered (Fig. 4.6.), indicating the nearly perfect long-

range ordering of the N atoms. As expected for lower N contents, a smaller order

parameter is observed[1,5].


73

Fig. 4.6. The degree of order ρ for γ'-Fe4N1-x in equilibrium with ε-Fe2N1-z as a

function of the fraction of occupied interstitial sites, yN, as obtained from the cluster

distribution variables, using Eq. (19).

4.5.3. ORDERING OF NITROGEN ATOMS IN ε-Fe2N1-z

The prism distribution variables, Pijklmn

αβγδηκ , of the main configurations occurring in the

composition range of ε-Fe2N1-z, as studied in the present work, are shown as a

function of the nitrogen content in Fig. 4.7. The data correspond to ε-Fe2N1-z in

equilibrium with γ'-Fe4N1-x in the temperature range of 573 K to 940 K (i.e. each data

point corresponds to a different temperature). The fraction of prisms containing three

atoms N is less than 0.01 (not shown) and the fraction of prisms containing more than

three atoms N is negligible. The number of possible distributions over the sites of the

prism is large when the nitrogen content is low.

The fraction of prisms containing one or no nitrogen deviates from that of a

random distribution (Section 4.5.4.) suggesting the occurrence of short-range ordering

of atoms N and vacancies V on the interstitial sublattice. Upon increasing nitrogen

content prism configurations of two nitrogen atoms become more prominent, and at

Chapter 4

74

25 at.% N only one prism configuration, in which sites β and κ are fully occupied, is

present.

Fig. 4.7. The value of the predominantly occurring prism distribution variables

pertaining to the calculated γ'-Fe4N1-x / ε-Fe2N1-z phase equilibria as a function of the

nitrogen content ( N, V).

The corresponding occupations of the six interstitial sites α up to κ

constituting the prism cluster as derived from the prism distribution variables, are

shown in Fig. 4.8. as a function of the nitrogen content. Note that Fig. 4.8. consists of

the data on the occupation of the prism sites representing the ordering of the nitrogen


75

atoms in ε-Fe2N1-z for both calculated miscibility gaps (I and II; Fig. 4.3.).

Apparently, no discontinuities occur for the point distribution variables as function of

the nitrogen content.

Clearly, the distribution of nitrogen over the interstitial lattice sites is not

random. The N atoms in ε-Fe2N1-z exhibit a preference not to be surrounded by N

atoms on nearest-neighboring interstitial sites, i.e. the sites β and κ, positioned

diagonally in the CVM basic prism cluster, are filled preferentially.

Fig. 4.8. The value of the point distribution variables representing the fractional

occupation of sites α, β, γ, δ, η, and κ, obtained by summation of the prism

distribution variables (Eq. (7)) pertaining to the calculated γ'-Fe4N1-x / ε-Fe2N1-z

phase equilibria.

Chapter 4

76

When the nitrogen content is low, i.e. only a small fraction of the interstitial sites in ε-

Fe2N1-z is occupied, the differences in occupation between the six sites are not

pronounced, but there is a slight preference for occupation of sites of type β and κ,

and α and δ. The X-ray diffraction pattern of a powder sample nitrided at 958 K

containing γ-Fe[N], γ'-Fe4N1-x and ε-Fe2N1-z is shown in Fig. 4.9. Indeed, for the ε-

Fe2N1-z phase containing less than 20 at.% N, only the main reflections (denoted by m

in Fig. 4.9.) are observed. The superstructure reflections (denoted by s in Fig. 4.9.),

generally present when long-range ordering of the N atoms on the interstitial

sublattice occurs, are absent. Even the most prominent superstructure reflection, of

which the position is indicated by ε 101s in Fig. 4.9., is not observed.

Fig. 4.9. X-ray diffraction pattern (Co Kα1 radiation) of an iron powder sample

nitrided at 958 K and containing the phases γ-Fe[N], γ'-Fe4N1-x, and ε-Fe2N1-z. The

ε-Fe2N1-z phase as a nitrogen content of <20 at. % N. The positions of the main

reflections (m) and the superstructure reflections (s) of ε-Fe2N1-z have been indicated.

Since no superstructure reflections due to the presence of N atoms are observed for

the ε phase, long-range ordering of N atoms is practically absent.


77

Upon increasing nitrogen content, in particular the fractional occupation of the

sites β and κ, and also the fractional occupation of the (nearest neighboring) sites α

and δ increases (Fig. 4.8.). When the nitrogen content becomes larger than about 16

at.%, the preference for the occupation of sites β and κ increases rapidly, whereas the

occupation of the sites α, γ, δ and η decreases gradually. Occupation of the sites β

and κ creates as much distance as possible between the N atoms. When the fraction of

occupied sites in ε-Fe2N1-z approaches a nitrogen content of 25 at.% (i.e. ε-Fe3N) a

distinctly ordered configuration is observed: the sites β and κ are fully occupied,

whereas the sites α, γ, δ and η are empty. Note that in the CVM prism approximation

applied in the present work, no distinction has been made between interstitial lattice

sites preferably occupied by either nitrogen atoms or vacancies, implying that no

specific ordered structure was included explicitly. The result obtained here at 25 at.%

N pertains to an occupation of the prism sites compatible with the structure of Fe3N as

proposed in literature on the basis of diffraction experiments[13,15]. The resulting

distribution of N atoms over the interstitial sublattice is also compatible with the so-

called B configuration[2] obtained for Fe3N by thermodynamic calculations using the

Gorsky-Bragg-Williams approximation.

4.5.4. COMPARISON WITH MÖSSBAUER DATA

Quantitative data of ordering of nitrogen atoms in an iron matrix can be deduced from

Mössbauer spectra. In this section, results of the CVM calculations of the local

surroundings of Fe atoms in ε-Fe2N1-z by N atoms and vacancies are presented and

compared to the available Mössbauer data.

The interpretation of Mössbauer spectra provides values for the fractional

occurrences of Fe atoms in ε-Fe2N1-z surrounded by 0, 1, ..., 6 atoms N (AnN, nN = 0-6)

on the six nearest neighboring interstitial sites (α, β, γ, δ, η, and κ) surrounding an Fe

atom in the ε-Fe2N1-z structure, which constitute the CVM basic cluster (Fig. 4.2.).

The overall Mössbauer spectrum is composed of several subspectra, each

corresponding to a certain type of surroundings of the Fe atoms.

The CVM calculations yield values for the prism distribution variables, which

describe the fractional occurrences of all distributions of zero to six N atoms over the

sites of the prism cluster, adding up to a total of 26 = 64 possibilities (i.e. clusters).

Next, the fraction of prisms with a certain number of interstitial atoms nN, pertaining

Chapter 4

78

to ε with an occupied fraction of interstitial sites yN, PnN

CVM (yN), is obtained

straightforwardly by summation of the distribution variables Pijklmn

αβγδηκ for the prisms

containing nN nitrogen atoms.

While calculating the surroundings of the Fe atoms from the prism distribution

variables, all prisms containing one atom N (or three atoms N) are considered as

equivalent; the various configurations for prisms containing two atoms N are treated

separately.

Since the nitrogen content of ε-Fe2N1-z in equilibrium with γ'-Fe4N1-x is

relatively low, the surroundings for zero and one Fe atoms and one of the

surroundings for two Fe atoms (A0, A1 and A2D respectively, Fig. 4.10.(a)) are most

prominent. The probability of all other possible types of Fe surroundings is less than

1% (Fig. 4.10.(b)). Starting at low nitrogen contents, the occurrence of A0 decreases

gradually with increasing nitrogen content, while that of A2D increases strongly. The

occurrence of A1 increases slightly initially, and then starts to decrease from a

nitrogen content of about 14.5 at.% N, i.e. in parallel with the tendency for the N

atoms to order stronger (Fig. 4.8.). As has been described in the previous section, the

ordering that occurs when the composition of ε-Fe2N1-z approaches Fe3N, corresponds

with the occupation of two of the six interstitial (prism) sites, positioned diagonally

with respect to each other (corresponding to A2D in Fig. 4.10.(a)). The occurrences of

A2V, A2H and A3D each have only a small maximum value depending on the nitrogen

content (Fig. 4.10.(b)). The contribution of the remaining Fe surroundings (not

shown) can be neglected.

The probabilities PnN

CVM (yN) of the calculated total Fe surroundings A0, A1 and

A2 (equal to the sum of A2D, A2V and A2H), the corresponding probabilities PnN

random(yN)

for a random distribution of nitrogen and the few available literature data[24] as

obtained from Mössbauer experiments, are shown in Fig. 4.11. The probability of the

surroundings for a random distribution of nitrogen, PnN

random(yN), were calculated from

( ) ( ) ( ) NsNs

N

N

nn

N

n

N

n

nN

random

n yyyP−−

= 1 (20)

where yN is the fraction of occupied sites of the interstitial sublattice of ε-Fe2N1-z (the

atomic percentage of N equals 100*yN/(1+yN)) and nS is the number of cluster sites

(here, nS = 6). The calculated CVM data on the Fe-surroundings pertain to ε-Fe2N1-z in


79

equilibrium with γ'-Fe4N1-x. The experimental Mössbauer data in Fig. 4.11. have been

obtained from samples prepared at temperatures different from the ones used in the

calculations. Considering the experimental error associated with fitting of Mössbauer

spectra, and, in particular, that no specific type of ordering was adopted in the CVM

description of the thermodynamics of ε-Fe2N1-z (Section 4.3.), the calculated data for

the occurrences of specific Fe surroundings describe the experimental data very well.

Fig. 4.10.(a) and (b) Calculated Fe-surroundings, CVM

nN

P (see text), as a function of the

nitrogen content of ε-Fe2N1-z (in equilibrium with γ'-Fe4N1-x). The values given pertain

to the total fraction of prisms with a certain number of interstitial atoms nN. This is

denoted by [ ]. Only for Fe surroundings by two N atoms, a distinction has been made

between the total of prism configurations with N atoms positioned horizontally,

vertically, or diagonally with respect to each other. For the range of nitrogen

contents of ε-Fe2N1-z, only one type of Fe surroundings by three N atoms occurs.

Chapter 4

80

Fig. 4.11. Calculated total surroundings of the Fe atoms by 0, 1, and 2 N atoms as a

function of the nitrogen content of ε-Fe2N1-z (in equilibrium with γ'-Fe4N1-x). The

corresponding values in case of a random distribution of N atoms on the interstitial

sublattice and the literature values as obtained by Mössbauer spectroscopy have been

indicated as well.

4.5.5. LATTICE PARAMETERS

The lattice parameters a (distance between next-nearest-neighboring N atoms) and c

(distance between nearest-neighboring N atoms) of ε-Fe2N1-z as obtained from the


81

CVM calculations are compared with the corresponding experimental data[13,26] in

Fig. 4.12. It should be noted that the small (less than 0.6%) thermal expansion effects

caused by the differences between the temperature pertaining to the CVM calculations

and room temperature, at which the values of the experimental data hold, have been

neglected.

The calculated lattice-parameter values are smaller than the experimental ones

(Fig. 4.12.). This is compatible with the general observation that CVM calculations

underestimate[25,5] the value of the lattice parameters. Although the reason for this is

not established, it may be suggested that the adoption in the CVM calculation of the

N-N and V-V distances as linear extrapolations of the experimental lattice-parameter

data (see Eq. (13)), thereby neglecting the displacements of the locally surrounding Fe

atoms caused by the N atoms, may explain the discrepancy.

Fig. 4.12. Calculated values for the lattice parameters a and c of ε-Fe2N1-z as a

function of nitrogen content. The corresponding experimental data for the lattice

parameter relations Eq. (12) have been indicated.

Chapter 4

82

4.6. CONCLUSIONS

1. The thermodynamics of the short-range and long-range ordering of nitrogen

atoms on the interstitial sublattices of γ'-Fe4N1-x and ε-Fe2N1-z has been

described successfully by the application of the cluster variation method

(CVM). As basic clusters a tetrahedron (for γ'-Fe4N1-x) and a prism (for ε-

Fe2N1-z) of interstitial sublattice sites were chosen. The interactions with the

iron sublattice were incorporated through effective interaction parameters.

2. The two ε+ γ' miscibility gaps (i.e. four phase boundaries) were described by

adopting a single set of pair interaction energy parameters.

3. The CVM calculations demonstrate the occurrence of distinct long-range

ordering of N in γ'-Fe4N1-x and the occurrence of short- and long-range

ordering of N in ε-Fe2N1-z. The long-range ordering (type B) occurring in ε-

Fe2N1-z at N contents close to 25 at.% N is in agreement with the type of

ordering for Fe3N evidenced by diffraction experiments.

4. The occurrences of the surroundings of the Fe-atoms as predicted by the CVM

calculations are in good agreement with experimental data obtained from

Mössbauer spectrometry.

5. Theoretical values obtained for the lattice parameters of the ε-Fe2N1-z phase

tend to be up to 1% smaller than the few available experimental data in the

composition range studied.


83

REFERENCES

1. Kooi, B.J., M.A.J. Somers, E.J. Mittemeijer, Metall. Mater. Trans. A, 1996a, vol. 27, pp. 1055

2. Kooi, B.J., M.A.J. Somers, E.J. Mittemeijer, Metall. Mater. Trans. A, 1994, vol. 25, pp. 2797

3. Kikuchi, R., H. Sato, Acta Metall., 1974, vol. 22, pp. 1099

4. DeFontaine, D., C. Wolverton, Ber. Bunsenges. Phys. Chem., 1992, vol. 96 (11), pp. 1503

5. Pekelharing, M.I., A.J. Böttger, M.A.J. Somers, E.J. Mittemeijer, Met. Mat. Trans., 1999, vol.

30A, pp. 1945

6. Kikuchi, R., D. De Fontaine, M. Murakami, T. Nakamura, Acta Met., 1977, vol. 25, pp. 207

7. Kikuchi, R., D. De Fontaine, Proceedings of a Workshop in the Application of Phase Diagrams in

Metallurgy and Ceramics (Gaithersburg, Maryland: National Bureau of Standards), 1977a, pp. 967

8. Kikuchi, R., D. De Fontaine, Proceedings of a Workshop in the Application of Phase Diagrams in

Metallurgy and Ceramics (Gaithersburg, Maryland: National Bureau of Standards), 1977b, pp. 999

9. Jack, K.H., Proc. Roy. Soc., 1948, vol. A195, pp. 34

10. Burton, B., Phys. Chem. Minerals, 1984, vol. 11, pp. 132

11. Burton, B. R. Kikuchi, Phys. Chem. Minerals, 1984, vol. 11, pp. 125

12. Wriedt, H.A., N.A. Gocken, R.H. Nafziger, Bull. Of Alloy Phase Diagrams, 1987, vol. 8 (4), pp.

355

13. Jack, K.H., Acta Crystallogr., 1952, vol. 5, pp. 404

14. Chen, G.M., N.K. Jaggi, J.B. Butt, E.B. Yeh, L.H. Swartz, J. Phys. Chem., 1983, vol. 87, pp. 5326

15. Leineweber, A., H. Jacobs, F. Hüning, H. Lueken, H. Schilder, W. Kockelmann, J. All. Comp.,

1999, vol. 288, pp. 79

16. Leineweber, A., H. Jacobs, F. Hüning, H. Lueken, W. Kockelmann, J. All. Comp., 2001, vol. 316,

pp. 21

17. Jacobs, H. A. Leineweber, W. Kockelmann, Mat.Sc. Forum, 2000, vol. 325-326, pp. 117


Kraan, N.M. van der Pers, Acta. Mat., 1997, vol. 45(5), pp. 2013

19. Frisk, K., CALPHAD, 1991, vol. 51(1), pp. 79

20. Kunze, J., Nitrogen and Carbon in Iron and Steels: Thermodynamics, Akademie Verlag, Berlin,

1990

21. Kooi, B.J., M.A.J. Somers, E.J. Mittemeijer, Metall. Mater. Trans. A, 1996b, vol. 27, pp. 1063

22. Kikuchi, R. J. Chem. Phys., 1974, vol. 60, pp. 1071

23. Guillermet, A.F., H. Du, Z. Metallk., 1994, vol. 85 (3), pp. 154

24. Foct, J., P. Rocheguide, Hyperfine Interactions, 1986, vol. 28, pp. 1075

25. Sanchez, J.M., J.R. Barefoot, R.N. Jarrett, J.K. Tien, Acta Met., 1984, vol. 32(9), pp. 1519

26. Burdese, A. Metall. Ital., 1957, vol. 49, pp. 195

27. Brunauer, S., M.E. Jefferson, P.H. Emmett, S.B. Hendricks, J. Amer. Soc., 1931, vol.53, pp. 1778

85

5

APPLICATION OF THE CVM CUBE APPROXIMATION

TO FCC INTERSTITIAL ALLOYS

ABSTRACT

The CVM simple cube approximation is applied to calculate a hypothetical fcc

interstitial alloy phase equilibrium. Instead of limiting the description of the alloy to

the species occupying the interstitial sublattice sites and including the interaction with

the metal sublattice in the effective pair potentials like in the previous chapters, the

basic cluster is composed of both metal and interstitial sublattice sites. The metal

sublattice is described as fully occupied by two types of metal atoms, while the

interstitial sublattice sites are filled with two interstitial species, one representing an

atomic species and the other a vacancy. The Lennard-Jones parameters chosen to

describe the interaction between the species lie within the range typical for transition

metals. Analysis of the calculated cube distribution variables shows that phase

transitions on the metal and interstitial sublattices are coupled: ordering of interstitial

species can be influenced by introduction of extra metal species to the host matrix of

the alloy, which enables purposeful adjustment or change of the properties of a

material.

Chapter 5

86

5.1. INTRODUCTION

The formation of interstitial solid solutions involves the introduction of light elements

such as nitrogen, carbon, boron, and hydrogen into the interstitial spaces formed by

the host matrix of a material. The ability of materials such as close-packed metals to

incorporate single elements into the interstitial spaces of the structure forms the basis

of industrial applications such as gas mixture separation, hydrogen storage, and

thermochemical treatments like nitriding, carburizing, and boriding of steels.

Interstitial-induced order-disorder transitions have been documented for Pd-H[1,2,3],

Fe-Cr-C, Fe-Cr-N, Al-C-Mn, and Al-B-Mn based alloys[4].

Like ordering transitions can occur on the metal host sublattice due to the

introduction of atoms into the interstitial spaces, the substitution of metals on the host

sublattice may cause order-disorder transitions on the sublattice formed by the

interstitials. Substitution of Sm and Gd has been reported to strongly reduce the

ability of Pd7M (M = Sm, Gd) alloys to absorb hydrogen, which appears to be related

to the preferential occupation by hydrogen of the octahedral interstices located

between nearest-neighbor Pd atoms as opposed to interstices surrounded by both Pd

and M atoms[5]. Modeling of the thermodynamics of interstitial systems is therefore

an important tool for process and material property optimization in industrial

application. Moreover, direct observation of interstitial solid solutions is complicated,

and thermodynamic calculations may provide helpful information that cannot be

obtained otherwise.

In this chapter, the cluster variation method (CVM)[6-11] simple cube

approximation is applied to phase equilibrium calculations of hypothetical face

centered cubic (fcc) interstitial alloys[8]. In contrast to the previous chapters, both

metal host sublattice and interstitial sublattice sites are included in the basic CVM

cluster. Subsequent analysis of the cluster distribution variables provides new support

for the coupling of the interaction between the host and the interstitial sublattices

observed in real systems.

Application of the CVM Cube Approximation to FCC Interstitial Alloys

87

5.2. CVM CUBE APPROXIMATION

For the purpose of modeling the thermodynamics of interstitial alloys, the structure of

the alloys is described as consisting of an fcc host matrix, formed by metal atoms,

with interstitial atomic species occupying the interstitial octahedral sites in between

the close-packed metal atoms. The metal atoms constituting the host matrix structure

form a substitutional metal sublattice, assumed to be fully occupied. The interstitial

species are assumed to occupy the interstitial octahedral sites only, and form an fcc

interstitial sublattice. The assumption is made that only atoms of type M1 and M2 are

found on the metal sublattice, while the interstitial sublattice sites are occupied by

interstitial species I1 and vacancies I2 exclusively. When both atoms and vacancies are

treated as interstitial species distributed over the sites of the interstitial sublattice, the

occupation of the interstitial sublattice can be considered substitutional in nature as

well.

Fig. 5.1. Fcc interstitial alloy with metal and interstitial sublattice sites, represented

by large circles and small circles, respectively. Grouping of the metal and interstitial

sublattice sites of the fcc unit cell results in two tetrahedrons, together forming a

simple cube. The vertices of the cube are denoted by α, β, γ, and δ (interstitial

sublattice sites), and η, κ, ν, and ω (metal sublattice sites). The subscripts i, j, k, and l

(interstitial sublattice sites) and m, n, p, and s (metal sublattice sites), which take a

value of 1 or 2, indicate whether the sites are occupied by species type 1 or 2,

respectively.

Chapter 5

88

The basic CVM cluster is defined in such a way that the types of ordering

occurring on both the metal and interstitial sublattices are included. Two

interpenetrating tetrahedrons, one representing metal sublattice sites, and the other

representing interstitial sublattice sites, together form a simple cube which can be

used as the basic CVM cluster. The basic cube cluster thus obtained includes nearest-

neighbor (nn) metal-interstitial interactions as well as next nearest neighbor (nnn)

metal-metal and interstitial-interstitial interactions, as shown in Fig. 5.1.


The internal energy of the system can be obtained by summation of the internal

energies of all 28 = 256 possible configurations of the cube cluster. Since each lattice

site pertains to one simple cubic cluster, the internal energy U is given by:

∑=ijklmnps

ijklmnpsijklmnpsCU αβγδηκνωε (1)

with ijklmnpsε representing the energy of a specific cube configuration with a

probability of occurrence equal to cube distribution variable αβγδηκνωijklmnpsC . The cube

cluster sites are denoted by superscripts α, β, γ, and δ, representing the interstitial

sublattice sites, and η, κ, ν, and ω, representing the metal sublattice sites. Whether the

interstitial sublattice sites are occupied by species of type I1 (atom) or I2 (vacancy) is

indicated by the subscripts i, j, k, and l, which take a value of 1 or 2, respectively.

Likewise, the subscripts m, n, p, and s indicate the occupation of the metal sublattice

sites and the values 1 and 2 represent metal atoms of type M1 and M2. Note: each cube

cluster site represents a sublattice and is therefore unique. In this work, the internal

energy is described as the sum of the pairwise interactions between the cube cluster

sites, thus including both nearest neighbor and next-nearest neighbor interactions as a

function of the distance between the lattice sites r:


89

( ) ( ) ( ) ( ) ( ) ( )[( ) ( ) ( ) ( ) ( ) ( ) ( )]

( ) ( ) ( ) ( ) ( ) ( )[ ]

( ) ( )[ ( ) ( ) ( ) ( )]nnn

ps

nnn

ns

nnn

np

nnn

ms

nnn

mp

nnn

mn

nnn

kl

nnn

jl

nnn

jk

nnn

il

nnn

ik

nnn

ij

nn

ls

nn

lp

nn

lm

nn

ks

nn

kp

nn

kn

nn

js

nn

jn

nn

jm

nn

ip

nn

in

nn

imijklmnps

rrrrrr

rrrrrr

rrrrrrr

rrrrrr

νωκωκνηωηνηκ

γδβδβγαδαγαβ

δωδνδηγωγνγκβω

βκβηανακαηαβγδηκνω

εεεεεε

εεεεεε

εεεεεεε

εεεεεε

+++++

++++++

+++++++

+++++=

2121

41

(2)

where the fractions ¼ and ½ indicate that the nearest neighbor (nn) and next nearest

neighbor (nnn) pairs are shared by 4 and 2 cubes, respectively, and e.g. ακε mn

represents the interaction energy of a pair of cube cluster sublattice sites of type α and

κ with occupation mn. Nearest neighbor pairs are combinations of metal atoms

residing on the host sublattice and interstitial atoms occupying adjacent cube cluster

sites, belonging to the interstitial sublattice. Next nearest neighbor pairs however, are

formed between species residing on the same sublattice.

The pairwise interaction energy ακε12 for an atom of type I1 occupying an

interstitial sublattice site of type α and an atom of type M2 residing on a metal

sublattice site of type κ is described using a type 8-4 Lennard-Jones interatomic

potential[12]:

( )

−

=

4012

80120

1212 2r

r

r

rr εε ακ (3)

where parameters 012ε and 0

12r are referred to as the so-called Lennard-Jones (L-J)

parameters. In principle, the values of the L-J parameters 012ε , referring to the pair

interaction energy in the reference state, and 012r , corresponding to the interatomic

distance for which ακε12 has a minimum value equal to 012ε [13,14], can be derived from

thermodynamic data such as the cohesive energy, heat of formation, and lattice

constants of the pure components. In the present work, the interactions between atoms

occupying lattice sites that are situated further apart than next-nearest neighboring

sites are assumed to be incorporated into the L-J potentials, which therefore describe

effective pair potentials.

Chapter 5

90


The configurational entropy of the ordering of M1 and M2 on the metal sublattice and

I1 and I2 on the interstitial sublattice is described as a function of the cube distribution

variables, which indicate the probability of occurrence of all possible arrangements of

the four species (M1, M2, I1, and I2) over the sites of the simple cube cluster and its

subclusters (i.e. squares, pairs, and points). The subcluster distribution variables can

be obtained by summation of the cube distribution variables αβγδηκνωijklmnpsC according to:

Squares: ...,∑=mnps

ijklmnpsijkl CW αβγδηκνωαβγδ

Pairs: ...,∑=klmnps

ijklmnpsij CY αβγδηκνωαβ (4)

Points: ...,∑=ijklmnps

ijklmnpsi CX αβγδηκνωα

with αiX indicating the probability that a site of type α has configuration i,

αβijY indicating the probability that a pair of sites of type α and β has configuration ij,

and αβγδijklW indicating that a square consisting of cube cluster sublattice sites of type α,

β, γ, and δ has configuration ijkl. The configurational entropy S for the cube

approximation can be described as[9]:

[ ]

+++++

+++

++++

++++++

++−

++

++++−=

+−+−=

∑∑∑∑∑∑

∑∑∑∑∑∑

∑∑∑∑∑∑

∑∑∑∑∑

∑ ∑∑∑

s

s

p

p

n

n

m

m

l

l

k

k

j

j

i

i

ls

ls

lp

lp

lm

lm

ks

ks

kp

kp

kn

kn

js

js

jn

jn

jm

jm

ip

ip

in

in

im

im

klps

klps

jlms

jlms

jkns

jkns

ijklmnps ilmp

ilmp

iknp

iknp

ijmn

ijmnijklmnpsB

B

XLXLXLXLXLXL

XXLYLYLYLYL

YLYLYLYLYLYL

YLYLWLWLWL

WLWLWLCLk

pointLpairLsquareLcubeLkS

ωνκηδγ

βαδωδνδηγω

γνγκβωβκβηαν

ακαηγδνωβδηωβγκω

αδηναγκναβηκαβγδηκνω

81

41

21

3 3

(5)


91

where Bk is Boltzmann’s constant and the function aaaL ln= . Lattice distortion

resulting from the accommodation of interstitial species by the displacement of metal

atoms in the host matrix is not explicitly accounted for. Vibrational entropy

contributions to the free energy function are not included in the description of the

hypothetical system in this work, but may be significant for a real system[15].

5.2.3. CALCULATION OF PHASE EQUILIBRIA

For each phase, a thermodynamic function Ω, referred to as the grand potential

function, is defined in terms of cluster distribution variables[12]:

( )

+−+−≡Ω ∑∑

==

2

1

*2

1

**2

*1

*2

*1 2

1,,,,,

m

mM

i

iIIIMM xxpVTSUTpmi

µµµµµµ (6)

where U is the internal energy, S is the configurational entropy, and V is the volume,

all per cube cluster site. T represents the temperature and p the external pressure. The

terms iIx and

mMx indicate the mole fraction of components iI (i = 1,2) on the

interstitial sublattice and mM (m = 1,2) on the metal sublattice, respectively. The

mole fractions can be obtained from the point cluster variables using:

[ ] 4/δγβαiiiiI XXXXx

i+++= (7)

[ ] 4/ωνκηmmmmM XXXXx

m+++=

where αiX represents the probability that an interstitial sublattice site of type α is

occupied by either I1 or I2, while ηmX indicates the probability that a metal sublattice

site of type η is occupied by M1 or M2. The chemical potentials of components iI and

mM are denoted by iIµ and

mMµ , respectively. The effective chemical potentials can

then be defined as:

( ) 2/21

*IIII ii

µµµµ +−= (8)

( ) 2/21

*MMMM mm

µµµµ +−=

Chapter 5

92

with the effective chemical potentials obeying 0**

21=+ II µµ and 0**

21=+ MM µµ .

The cube distribution variables constraint

1=∑ijklmnps

ijklmnpsCαβγδηκνω (9)

is accounted for by introduction of the Lagrange multiplier λ in the grand potential

function. Minimization of the grand potential function with respect to αβγδηκνωijklmnpsC yields:

( ) ( ) ( ) 8

1

4

1

2

1****

****

8

exp

8expexpexp

ijklmnpsijklmnpsijklmnps

B

MMM

B

III

B

ijklmnps

B

ijklmnps

XYWTk

TkTkTkC

spMnm

lkIji

−×

+++

+++

−

=

µµµµ

µµµµελ αβγδηκνωαβγδηκνω

(10)

with γδνωβδηωβγκωαδηναγκναβηκklpsjlmsjknsilmpiknpijmnijklmnps WWWWWWW =

δωδνδηγωγνγκβωβκβηανακαηlslplmkskpknjsjnjmipinimijklmnps YYYYYYYYYYYYY = , and (11)

ωνκηδγβαspnmlkjiijklmnps XXXXXXXXX =

The volume per cluster site, V, corresponding to a particular αβγδηκνωijklmnpsC for a phase at

atmospheric pressure p and constant temperature T is obtained from:

αβγδηκνωµδδ

ijklmnpsCTVp

,, *

Ω=− (12)

The grand potential function is minimized with respect to αβγδηκνωijklmnpsC using the Natural

Iteration (NI) Method. The thermodynamic phase equilibrium is determined by

finding the intersection of the *

mMµ and *

iIµ versus grand potential Ω graphs of both

phases at a specific temperature.


93

Table 5.1. Lennard-Jones parameters used to describe a hypothetical fcc host

sublattice occupied by metal atoms M1 and M2, with interstitial species I1 and

vacancies I2 residing in the octahedral interstices. The parameters are normalized

with respect to the reference state, in this case a pure host M1, with 0

11MMε = 62.80 kJ/mol (15 kcal/mol) and 0

11MMr =0.27 nm.

Normalized parameters Normalized parameters

Pairs norm,0ε normr ,0

Pairs norm,0ε normr ,0

M1-M1 1.000 1.000 I2-I2 0.000 1.000

M1-M2 0.980 1.050 M1-I1 0.010 1.280

M2-M2 0.950 1.080 M2-I1 0.008 1.330

I1-I1 0.021 1.280 M1-I2 0.000 1.000

I1-I2 0.036 1.230 M2-I2 0.000 1.000

5.3. APPLICATION TO FCC INTERSTITIAL ALLOY PHASE

EQUILIBRIA

In order to study the interaction between the close packed metal atoms forming the

host matrix and the species occupying the octahedral interstitial spaces more closely,

in the present CVM approach a basic cluster has been chosen that explicitly

incorporates both metal and interstitial sublattice sites. Since the availability of

thermodynamic data necessary to derive the input parameters for the calculation of a

phase equilibrium by CVM are rather limited for substitutional alloys with interstitial

species, the Lennard-Jones (L-J) parameters used in the present work were obtained

from Ref. [37] and are shown in Table 5.1. The values of the L-J parameters lie within

the range typically used for transition metals[12,13].

In the present work, the value of the effective chemical potential 1*Mµ on the

metal sublattice is fixed at a value of zero (which implies 02* =Mµ as well, since

**

21 MM µµ −= ) Then, the equilibrium state for a specific temperature is found by

varying the effective chemical potential *

1Iµ ( *

2Iµ−= ) associated with the interstitial

sublattice. Fig. 5.2.(a)/(b) represent the temperature-composition phase diagram thus

Chapter 5

94

Fig. 5.2.(a) and (b). Phase diagram of a hypothetical fcc system for an effective

chemical potential ( ) 0*1

*

2=−= MM µµ . (a) Mole fraction

1Mx of species M1 on the

metal host sublattices of phase A and B (b) Corresponding mole fraction 1Ix of

species I1 on the interstitial sublattices of phase A and B.


95

obtained for a hypothetical fcc equilibrium at temperatures in the range of 200 to

1400 K, with mole fraction 1Mx of species M1 on the metal host sublattices of the

equilibrium phases on the left-hand side, and the corresponding mole fraction 1Ix of

species I1 on the interstitial sublattices of the equilibrium phases on the right-hand

side of the figure. In fact, two separate phase equilibria occur, one above and one

below about 500 K. In both equilibria, an I1-poor phase, referred to as phase A, and an

I1-rich phase, referred to as phase B, are observed. The phases above 500 K are

indicated by the suffix [high T], while the phases below 500 K are labeled [low T].

Note: the phases A(B)[low T] and A(B)[high T] belong to separate phase equilibria and

therefore refer to different phases. The results lead to the conclusion that phase

transitions on the metal and interstitial sublattices are coupled, i.e. the composition of

the interstitial sublattice was found to depend on the occupation of the metal host

sublattice. A detailed discussion of Fig. 5.2.(a)/(b), as well as phase diagrams

obtained at some other effective chemical potential values, has been given in Ref. [8].

5.3.1. PAIR DISTRIBUTION VARIABLES

Fig. 5.3. shows the 11IIY ,

21IIY , and 22IIY pair distribution variables IIY , representing

the calculated probability of occurrence of the next nearest neighbor pairs on the

interstitial sublattice, and their random distribution in phase A and B as a function of

1Ix . Preferential occupation (short-range ordering) of the interstitial sublattice sites by

species I1 and I2 takes place in phase B, regardless of the temperature range

considered, as evidenced by the slight deviation of the IIY distribution variable values

from those of the random distribution, with a preference for (I1, I2) pairs with regard

to (I1, I1,) and (I2, I2) pairs. Meanwhile, the distribution of interstitials in phase A

remains equal, or close, to a random distribution. Based on the observed trends of the

pair distribution variables IIY , the occurrence of ordering of the interstitials does not

seem to depend on temperature. The ordering on the interstitial sublattice appears to

be predominantly determined by the fraction of interstitials 1Ix present, i.e. the greater

the fraction of species I1, the more pronounced the preferential ordering.

Note: 21IIY (

21MMY ) includes both 21IIY (

21MMY ) and 12IIY (

12MMY ) pair

probabilities, since these are equivalent and therefore indistinguishable.

Chapter 5

96

Fig. 5.3. Pair distribution variables IIY , showing the probability of occurrence of the

next nearest neighbor pairs (I1, I1), (I1, I2), and (I2, I2) on the interstitial sublattice and

the associated random distributions as a function of 1Ix in phase A and B.

Fig. 5.4. (a) and (b) show 11MMY ,

21MMY , and 22MMY , representing the calculated

probability of occurrence of the next nearest neighbor pairs on the metal sublattice,

and their probability in case of a random distribution in phase B as a function of 1I

x .

In phase B[high T], which contains a considerably higher fraction of M2 ( 4.03.02

−≈Mx

of the metal sublattice sites) than any of the other phases, 21MMY is significantly higher

than for the random distribution, indicating the occurrence of short range ordering

(SRO). Thus, the introduction of metal species M2 seems to induce SRO on the metal

sublattice.

However, the information about ordering of the metal atoms occupying the

host lattice and their interaction with the alloy’s interstitial species that can be

obtained from the pair distribution variables remains limited. Next, the cube

distribution variables αβγδηκνωijklmnpsC , calculated using the fcc simple cube approximation,

are described.


97

Fig

. 5.4

. (a

) and (

b)

Pair

dis

trib

uti

on v

ari

ab

les

YM

M,

repre

senti

ng t

he

pro

bab

ilit

y o

f occ

urr

ence

of

the

nex

t nea

rest

nei

ghbor

pair

s (M

1,

M1),

(M

1,

M2),

and (

M2,

M2)

on t

he

met

al

subla

ttic

e, a

nd t

he

ass

oci

ate

d r

and

om

dis

trib

uti

ons

as

a

funct

ion o

f x I

1 i

n p

hase

B.

Chapter 5

98

5.3.2. CUBE DISTRIBUTION VARIABLES

As shown in Fig. 5.1, the basic cubic cluster in the simple cube approximation is

constituted of two intertwined tetrahedrons: one encompassing the sites of the

interstitial sublattice and the other those of the metal host sublattice. The 256 possible

arrangements of species over the cube cluster sites can be reduced to 35 unique types

by grouping equivalent cube configurations with an equal composition and with the

interstitial and metal atoms arranged in a similar way, i.e. similar nearest neighbor and

next nearest neighbor surroundings.

5.3.2.1. PHASE A

Fig. 5.5.(a)/(b) shows the cube distribution variables of the most frequently occurring

cube configurations observed in phase A as a function of temperature. For phase

A[low T], the number of different cube configurations observed is very limited and all

configurations represent cubes with a pure M1 metal sublattice. Compared to phase

A[low T], in phase A[high T] an increase in the number of cube configurations occurs (see

Fig. 5.5.(a)). With increasing temperature, a small percentage of M2 ( %5.42

≤Mx of

the metal sublattice sites) occupies the metal sublattice sites, as shown in Fig. 5.2.,

resulting in cube configurations with either a pure M1 metal sublattice, or a metal

sublattice occupied with M1:M2 = 3:1 (Fig. 5.5.(a)/(b)). The cubes’ interstitial

sublattice sites are occupied by 0 to 4 I1. Thus, the redistribution of atoms over a

larger variety of cube configurations seems more related to the presence of M2 on the

metal host sublattice (and, of course, to the increasing temperature) than to the

fraction of interstitial atoms on the interstitial sublattice. The distribution of

interstitials in phase A remains equal to, or close to, a random distribution (not

shown.)

5.3.2.2. PHASE B [Low T]

Fig. 5.6.(a) shows the five main cube distribution variables of phase B[low T] and those

for a random distribution as a function of the composition of the interstitial sublattice

1Ix . These cube configurations are similar to those observed in phase A. The cube

distribution variables of all configurations deviate significantly from those for a


99

random distribution, indicating that preferential occupation (SRO) is taking place.

Note: the metal sublattice of phase B[low T ] is almost fully ( %9.991

≥Mx of the metal

sublattice sites) occupied by metal species M1, while the number of interstitials of

species 1 varies from 0 to 4. Cubes with 0 or 4 I1 (representing a pure I2 or I1

sublattice, respectively) occur less frequently than in the case of a random

distribution, while cube configurations with I1:I2 = 1:1 occur more frequently than in

the case of a random distribution, regardless of the fraction of I1 on the interstitial

Fig. 5.5.(a) and (b) Cube distribution variables of the most frequently occurring cube

configurations observed in phase A as a function of temperature. Metal and

interstitial sublattice sites are represented by large and small circles, respectively.

Dark colored circles represent species of type 1, while open circles represent species

of type 2. The lines are a guide to the eye.

Chapter 5

100

Fig. 5.6.(a) Main cube distribution variables of phase B[low T] and the associated

random distribution values as a function of 1Ix . Fig. 5.6.(b) Less frequently occurring

cube distribution variables of phase B[low T] as a function of 1Ix . Note: the random

distribution curves for and ∇ overlap and are therefore indistinguishable. Metal

and interstitial sublattice sites are represented by large and small circles,

respectively. Dark colored circles represent species of type 1, while open circles

represent species of type 2.


101

sublattice. Also, the deviation from the random distribution for cubes with I1:I2 = 1:1

is greater than for any of the other configurations, indicating that SRO is taking place

on the interstitial sublattice. The greatest deviation from a random distribution is

observed at low temperatures (200 K). As expected, with increasing temperature, the

cube distribution variable values approach their random probability as a result of

increasing entropy contribution.

Although in this temperature range the metal sublattice of phase B has an

almost pure ( %9.991

≥Mx of the metal sublattice sites) M1 composition, small

fractions of a variety of cube configurations with M1:M2 = 3:1 are observed, as

shown in Fig. 5.6.(b). The highest fraction of cube configurations with M1:M2 = 3:1

occurs in the range of 400-440 K, which coincides with the temperature range in

which a slight increase in M2 occupying the metal sublattice sites is observed in phase

B. Like in phase A, the introduction of even a very minor fraction of M2 to a pure M1

metal sublattice causes a great increase in the number of cube configurations

observed.

One of the cubes with I1:I2 = 1:1 (small circles) and M1:M2 = 3:1 (large

circles) occurs more frequently than in the case of a random distribution while the

other occurs less frequently, indicating SRO of the species occupying the interstitial

sublattice sites. Both of these cubes represent a unique type (see Section 5.3.2.), i.e.

they represent a group of equivalent cube configurations with an equal composition

and with similar nearest neighbor and next nearest neighbor surroundings. In addition,

both groups consist of an equal number of configurations (12 in this case), and can

therefore be compared directly to one another. Comparison of the occurrence of the

two cubes shows that there is a preference for M1 to be surrounded by I1, while there

is a preference for M2 to be surrounded by I2.

5.3.2.3. PHASE B [High T]

The main difference between the I1-rich phases B observed in the low and high

temperature regions is the occupation of their metal sublattice. While in the low

temperature range a pure ( %9.991

≥Mx of the metal sublattice sites) M1 metal

sublatttice is observed, in the high temperature range a relatively high fraction

(roughly 0.3-0.4) of the metal sublattice sites is occupied by M2, which leads to the

Chapter 5

102

Fig

. 5.7

.(a)

Cum

ula

tive

clu

ster

dis

trib

uti

on v

ari

able

s of

all

cube

confi

gura

tions

wit

h 0

up t

o 4

I1 a

nd t

he

ass

oci

ate

d r

andom

dis

trib

uti

on

valu

es o

ccurr

ing i

n p

hase

B[h

igh

T]

as

a f

unct

ion o

f x I

1.

Fig

. 5.7

.(b)

Cum

ula

tive

clu

ster

dis

trib

uti

on v

ari

able

s of

all

cube

confi

gura

tion

s

wit

h 0

up t

o 4

M1 a

nd t

he

ass

oci

ate

d r

andom

dis

trib

uti

on

valu

es o

ccurr

ing i

n p

ha

se B

[hig

h T

] as

a f

unct

ion o

f th

e fr

act

ion o

f M

1 o

n t

he

met

al

subla

ttic

e x I

1.

Met

al

and i

nte

rsti

tial

subla

ttic

e si

tes

are

rep

rese

nte

d b

y la

rge

and s

mall

cir

cles

, re

spec

tive

ly.

Dark

colo

red c

ircl

es

repre

sent

spec

ies

of

type

1, w

hil

e open

cir

cles

rep

rese

nt

spec

ies

of

typ

e 2

.


103

occurrence of all 35 cube types in more significant fractions than in any of the other

phases. This makes phase B[high T] the most suitable for detailed analysis of individual

cube configurations.

In order to study what kind of preferential surrounding of the metal and

interstitial species takes place, the cube distribution variables are first grouped by

number of I1 or M1, respectively, and the total fractions are compared to the random

distribution as a function of 1Ix . Fig. 5.7.(a) shows the cumulative cluster distribution

variables of all cube configurations with 0 up to 4 I1 and those for a random

distribution as a function of 1Ix . While the probability of finding a cube with 0 or 4

atoms I1 on its interstitial sublattice is lower than for a random distribution, regardless

of the fraction of interstitial sites occupied with I1, the probability of finding cubes

with I1:I2 = 1:1 is always higher than for a random distribution for the composition

range considered. The probability of cubes with 1 or 3 atoms I1 varies with 1Ix , and

equals the probability for a random distribution for 1Ix = 0.5. Therefore, the

interstitial atoms I1 and vacancies I2 appear to distribute themselves over the

interstitial sublattice sites in an orderly fashion with a tendency to avoid grouping

together of similar species.

Fig. 5.7.(b) shows the cumulative cluster distribution variables of all cube

configurations with 0 up to 4 M1 and those for a random distribution as a function of

1Mx (the fraction of species M1 on the metal sublattice). Unlike the interstitial

sublattice, which is occupied by atoms and vacancies, the metal sublattice is

substitutional, i.e. occupied by two types of metal atoms. However, both the

interstitial and metal sublattice cube distribution variables show that the probability of

finding a cube with 0 or 4 atoms of type 1 (i.e. a pure I1 or I2 interstitial sublattice and

a pure M1 or M2 metal sublattice) is considerably lower than for a random

distribution, indicating that grouping of similar types of species on a sublattice is

unfavorable.

Finally, the 35 types of cube configurations are categorized by number of species M1

and I1 found in each cube cluster, and configurations are selected with an identical

number of M1, M2, I1, and I2 per cube cluster. To compare the individual cube

configurations, differences in degeneration factor have to be taken into account. The

differences in probability observed can be related to preferences in the distribution of

the metal and interstitial atoms over the sites of the simple cube cluster. A comparison

of the cube configurations in Fig. 5.8.(a)/(b) shows that for all sets of configurations,

Chapter 5

104

metal species M1 has a preference to group with species I1, while species M2 prefers

to be surrounded by vacancies I2. Similar observations have been described for

configurations occurring in phase B[low T] in Section 5.3.2.2.

Fig. 5.8.(a) Cube distribution variables, normalized with respect to the degeneracy

factor of each unique type (see Section 5.3.2.), for four sets of individual cube

configurations in phase B[high T] with identical occupation in terms of the number of

interstitial and metal species but with different distributions of the species over the

sites of the cube cluster, and the associated random distributions, as a function of

1Ix . Metal and interstitial sublattice sites are represented by large and small circles,




105

Fig. 5.8.(b) Cube distribution variables, normalized with respect to the degeneracy

factor of each unique type (See Section 5.3.2.), for five sets of individual cube

configurations in phase B[high T] with identical occupation in terms of the number of

interstitial and metal species but with different distributions of the species over the

sites of the cube cluster, and the associated random distributions, as a function of

1Ix . Metal and interstitial sublattice sites are represented by large and small circles,



An interesting observation is that the cubes with M1:M2 = 1:3 occur less

frequently than in the case of a random distribution, while the cubes with 2 or more

Chapter 5

106

M1 occupying the four cube metal sublattice sites occur more frequently than the

random distribution, regardless of the number of I1 occupying the interstitial

sublattice. In addition, the greater the number of M2 per cube, the greater the

deviation from the random distribution appears to be. A significant deviation of the

normalized CVM cube distribution variables from the associated random probabilities

is observed for all cube configurations, indicating that SRO is taking place.

Furthermore, an increase of the fraction of M2 occupying the metal sublattice sites

seems to be coupled to an increase in the degree of SRO, as reflected by the deviation

from the random distribution in Fig. 5.8.(a)/(b).

5.4. CONCLUSIONS

The CVM cube approximation has been applied successfully to describe order-

disorder transitions on the metal and interstitial sublattices of fcc alloys with

interstitial species by including sites of both sublattices into the basic cluster. Analysis

of the calculated cube distribution variables shows that phase transitions on the metal

and interstitial sublattices are coupled: not only does the introduction of a second

metal atoms type on the metal sublattice induce SRO on both the metal sublattice and

interstitial sublattice, preferential grouping of metal atoms with specific interstitials is

observed as well. These phenomena have been documented for non-hypothetical

systems[1-4] but the explicit incorporation of the metal-interstitial interaction into the

thermodynamic description of the phases by CVM is of a recent date[8]. To make the

transition from a hypothetical system to real interstitial alloys and for application of

the model to modify the properties of materials in industrial applications, ab initio

calculations may be able to provide the necessary information needed for an accurate

description of the internal energy in terms of effective cluster interactions (ECI’s)

instead of phenomenological Lennard-Jones parameters. The CVM-ECI path has been

proven to be successful for the description of phase boundaries in the Fe-N and Fe-C-

N systems[15].


107

REFERENCES

1. Noh, H., T.B. Flanagan, B. Cerundolo, A. Craft, Scr. Metall. Mater., 1991, vol. 25, pp. 225-30

2. Flanagan, T.B., Y. Sakamoto, Platinum Met. Rev., 1993, vol. 37, pp. 26-37

3. Lee, S.M., T.B. Flanagan, G.H. Kim, Scr. Metall. Mater., 1994, vol. 32, pp. 827-32

4. Villars, P., A. Prince, H. Okamoto, Handbook of Ternary Alloy Phase Diagrams, 1995, Vols.

3,6,8, Metals Park, OH: ASM international

5. Sakamoto, Y., K. Takao, T.B. Flanagan, J. Phys Condens. Matter, 1993, vol. 5, pp. 4171-78

6. Burton, B., R. Kikuchi, Phys. Chem. Minerals, 1984, vol. 11, pp. 125-131


8. Nanu, D.E., Y. Deng, A.J. Böttger, Phys Rev. B, 2006, vol. 7401(1), pp. 216-24

9. Sanchez, J.M. D. de Fontaine, Phys. Rev. B, 1978, vol. 17(7), pp.2926-36

10. Kikuchi, R.A. , Phys. Rev., 1951, vol. 81, pp. 988-1003


12. Sanchez, J.M., J.R. Barefoot, R.N. Jarrett, J.K. Tien, Acta Met., 1984, vol. 32(9), pp. 1519-25

13. Pekelharing, M.I., A.J. Böttger, M.A.J. Somers, E.J. Mittemeijer, Met. Mat. Trans. A, 1999, vol.

30A, pp. 1945-53

14. Pekelharing, M.I., A.J. Böttger, E.J. Mittemeijer, Phil. Mag., 2003, vol. 83(15), pp. 1775-96

15. Shang, S., A.J. Böttger, Acta Mat., 2005, vol. 53, pp. 255-264

109

6

PHASE TRANSFORMATIONS AND PHASE

EQUILIBRIA IN THE IRON-NITROGEN SYSTEM AT

TEMPERATURES BELOW 573 K

ABSTRACT

The phase transformations of homogeneous Fe-N alloys with nitrogen contents

ranging from 10 to 26 at.% were investigated by means of X-ray diffraction analysis

upon aging in the temperature range 373 to 473 K. It was found that precipitation of

α"-Fe16N2 below 443 K does not only occur upon aging of supersaturated α (ferrite)

and α' (martensite), but also upon transformation of γ'-Fe4N1-z and ε-Fe2N1-x (<20

at.% N). No α" was observed to develop upon aging of γ-Fe[N] (austenite).

Therefore, it is proposed that γ' is a stable phase at temperatures down to (at least)

373K. Phase formation upon annealing at low temperatures is apparently governed by

the (difficult) nucleation and (slow) growth of new Fe-N phases: α" forms as a

precursor for α because of slow nitrogen diffusion, and nitrogen-enriched ε develops

as a precursor for γ' because of a nucleation barrier.

Chapter 6

110

6.1. INTRODUCTION

Phase transformations occurring in Fe-N alloys at low temperature are of particular

practical interest because of their industrial and technological importance. For

example, in industrial practice, the nitrided parts are usually cooled relatively slowly

due to the large size of the furnaces used. This slow induces transformation of phases

formed at the nitriding temperature.

The generally accepted Fe-N phase diagram† is largely based on data obtained

at temperatures above 573 K[1,2,3,4] because below this temperature, the dissociation of

ammonia proceeds so slowly that no equilibrium can be obtained between

ammonia/hydrogen mixtures and the surface of the solid. Consequently, published

data on phase boundaries at temperatures below 573 K are based on extrapolations,

and the prediction of the thermodynamic stability of Fe-N phases at these low

temperatures cannot be made with great accuracy. Recent thermodynamic calculations

indicate that γ'- Fe4N1-z may not be thermodynamically stable at temperatures below

583K[5], 449 K, or 294 K[6] depending on the model description used for the

thermodynamics of the Fe-N phases α, γ' and ε.

Most published research on the low-temperature stability of Fe-N phases is

devoted to phases with a relatively low nitrogen content (<10 at.%): supersaturated

ferrite (α)[7,8,9], martensite (α')[10-21] and austenite (γ)[22,23]. Only very few, partly

incompatible, data have been reported on the low temperature stability of Fe-N phases

with a relatively high nitrogen content[24-28]. Moreover, these results were generally

obtained by processing under far from equilibrium conditions, as, for example, by

mechanical alloying[26], laser nitriding[27] and ion implantation[24,28].

It has been reported that α" precipitates from both supersaturated ferrite and

Fe-N martensite as a supposedly intermediate phase in a temperature range up to ~473

K[9,10,12], ~493 K[14], and ~433 K[23], preceding the formation of γ' (Fe4N) and α. It has

been shown that, at 483 K, γ transforms directly into γ' (Fe4N1-z) and α, without the

precipitation of an intermediate α" phase[24]. Mössbauer spectra of aged ε-Fe2N1-z

samples indicated the presence of α”-Fe16N2 (suggested by Ref. [25]).

Thus, although some knowledge has been acquired regarding phase equilibria

and phase transformations at low temperatures in Fe-N alloys of a relatively low

† Note that the Fe-N phase diagram concerns data of Fe in equilibrium with ammonia/hydrogen gas mixtures at atmospheric pressure (equivalent to a N2 fugacity of the order of several gigapascals). This implies that, in general, Fe-N alloys are not stable with respect to decomposition into Fe and N2.

Phase Transformations / Phase Equilibria in the Iron-Nitrogen System at Temperatures below 573 K

111

nitrogen content, much less is known about the possible transformations in Fe-N

alloys of a relatively high nitrogen content: e.g., the possible coexistence of α"

(Fe16N2) and the phases γ' (Fe4N) and ε (Fe2N1-x) is undocumented.

Reviewing the current state of knowledge, the following questions arise:

1. Is α" (Fe16N2) a stable or an intermediate phase?

2. Do α "/ γ' and α "/ε phase equilibria occur?

3. Are γ' and ε stable phases at temperatures below 573 K?

The aim of the present research is to investigate phase transformations in Fe-N

phases, prepared at elevated temperature and with uniform nitrogen contents ranging

from 10 to 26 at.%, upon aging in the temperature range 373 to 473 K.

6.2. EXPERIMENTAL PROCEDURES

Specimens of different nitrogen content were prepared by nitriding pure iron powder

of the chemical composition: <0.002 wt% Ni, <0.002 wt% Mn, <0.01 wt% Al, <0.002

wt% Cr, <0.002 wt% Ti, <0.01 wt% W, <0.002 wt% V, 0.04 wt% Si, 0.002 wt% N,

0.221wt% O, balance Fe. The average size of the powder particles is about 10 micron,

and the particle sizes range from 5 to 20 µm. Disk–shaped specimens, 15 mm in

diameter and 0.28 to 0.38 mm thick, were prepared from the powder by sintering at

958 K in pure H2. After sintering, the specimens have a density of 4 to 5 g/cm3, which

corresponds to a degree of porosity between 35 and 50 vol.%. The large iron surface

area, combined with the short distance within the iron grains to be bridged by solid-

state diffusion, allows a relatively rapid establishment of a uniform nitrogen

concentration throughout the specimen characterizing a stationary state or imposed

equilibrium between nitrogen in the solid state and nitrogen in the gas mixture.

Nitriding experiments were performed in a vertical quartz tube furnace. In

order to obtain specimens and with nitrogen contents varying from 10 to 25 at.%

initial phase constitutions of γ, γ', ε + γ, and ε, different nitriding temperatures and gas

ratios of NH3/H2 were used. Nitriding occurred for 20 hours at temperatures between

723 and 963 K, and the volume ratios NH3/H2 were in the range from 97/3 to 10/90.

Chapter 6

112

For all NH3/H2 ratios, the flow rate of the NH3 + H2 gas mixture was 300

ml/min. After nitriding, the specimens were quenched in brine at room temperature.

In this way, the specimens were prepared. The nitrogen contents in the ε, γ' and γ

phases were determined from the lattice parameters measured by X-ray diffraction

using the latest published dependencies of the lattice parameters on nitrogen content:

Ref. [29] for ε, Ref. [30] for γ', and Ref. [31] for γ.

The as-prepared specimens were divided in several pieces, which were

subjected to various heat treatments. The aging of the nitrided specimens was

performed in oil-baths at specific temperatures (373, 413, 443, 463, and 473 K)

controlled at ± 2 K and for specific times (1, 3, 7, 14, 28, and 50 days). For aging at

413 to 443 K, all periods of time were applied, whereas at 373 K, only the long time

aging of 50 days was used, and, at the high temperatures of 463 K and 473 K, only the

short aging times (1 and 3 days) were applied.

The phase composition after nitriding (before aging) and after aging was

determined by X-ray diffraction. Before measurement, the specimens were cleaned in

ethanol and acetone and then powdered afterwards. The powder was put onto a <510>

single-crystal silicon wafer and placed in the specimen holder. The X-ray diffraction

measurements were made at room temperature using a SIEMENS‡ Type D-500 ω-

type diffractometer and applying Co Kα1 radiation using a monochromator in the

incident beam. The measurements were performed in the range 20 – 100 deg 2θ (in

some cases, 20 – 160 deg 2θ) with a step size of 0.05 deg 2θ, using total counting

times up to 2500 seconds per step. In order to be able to observe possible changes

occurring during the measurements, the long counting times were obtained by five

separate measurements, with 500 seconds per step in each case. In some cases, for

lattice- parameter determination, a step size of 0.01 deg 2θ was used. In order to

unravel partly overlapping peaks, the diffraction patterns were fitted§ using a

symmetric pseudo-Voigt function for the shape of peak profiles.

‡ SIEMENS is a trademark of Siemens Electrical Equipment, Toronto. § The program ProFit1.0b (1996 Philips Electronic Instruments Corp., Mahwah, NJ) was used to fit the diffraction profiles.


113

6.3. RESULTS AND DISCUSSION

6.3.1. AS-PREPARED CONDITION

The Fe[N] specimens, with a nitrogen concentration in the range from about 10 to 26

at.%, were annealed at the temperatures presented in the overview given in Fig. 6.1.

Fig. 6.1. The composition of the specimens and the temperatures of aging are

indicated in the Fe-N phase diagram[4]

.

Chapter 6

114

The nitrided specimens were divided into several groups, according to their nitrogen

content and the phases initially present (Fig. 6.1.):

1. Group A1: ε-Fe2N1-z with a nitrogen content <19.5 at.%

2. Group A2: ε-Fe2N1-z with a nitrogen content >20 at.%

3. Group B: γ' Fe4N1-z with about 20 at.% N

4. Group C: γ-Fe[N] specimen; and

5. Group D: γ-Fe[N] + ε

After nitriding and quenching, the specimens were analyzed using X-ray

diffraction in order to establish the phases present and their nitrogen contents (Section

6.2.). In the following text, the as-prepared condition of each group of specimens will

be discussed.

6.3.1.1. εεεε SPECIMENS WITH <19.5 at.% NITROGEN (GROUP A1)

According to the Fe-N phase diagram, the ε phase with nitrogen content <19.5 at.%

occurs above 923 K[1-4]. Therefore, these specimens were prepared at a relatively high

nitriding temperature of 963 K. In total, five specimens with nitrogen contents varying

from 15.6 to 19.1 at. % were investigated (Table 6.1.). In the diffraction patterns of

the quenched specimens, only the reflections pertaining to the hcp Fe sublattice of the

ε phase could be observed. Superstructure reflections due to long-range ordering of

nitrogen atoms[32] were not observed, which indicates that, for these low nitrogen

contents, no pronounced long range ordering of nitrogen atoms is present.

After quenching, all specimens of this group contained a certain amount of γ'

in the as-prepared condition. Even by rapid quenching at a rate of about 500 K/s, it

was impossible to avoid the formation of some γ' (a similar observation was reported

earlier[32]). The amount of γ' formed during quenching is estimated on the basis of the

X-ray diffraction patterns by applying the "direct comparison method,"[33] thereby

using the 200γ' and 111ε reflections. In quenched specimens with a relatively low

nitrogen content (less than 18 at.% nitrogen), the amount of γ' is estimated to be about

0.3 - 1.0 vol.% , in quenched specimens with a nitrogen content close to that of γ', the

volume fraction of γ' increases to about 10-11 vol.%.


115

Table 6.1. Phase transformation of ε specimens (Range of composition: 15 to 19.5

at.% N)

413 K 443 K Initial

Phase

Comp.

373 K

50

days

1 day 3

days

28

days

3

days

14

days

28

days

463 K

3

days

473

K

1 day

ε (15.6

at%.N)*

ε + γ'

↑

ε + γ'

↑

ε + γ'

+ α"

α" +

γ' +

ε

α" +

γ' +

ε +

α

- - γ' +

α" +

ε +

α

γ' +

α +

ε

ε (16.7

at%.N)*

- - - α" +

γ' +

ε

- - - - -

ε (17.9

at%.N)*

- - - α" +

γ' +

ε

- - - - -

ε (18.9

at%.N)*

+ 11.2

vol.%

γ'

ε + γ'

↑

- - γ' + α"

+

ε

γ' +

α" +

ε

- - γ' +

α" +

ε +

α

-

ε (19.1

at%.N)*

+ 10.4

vol.%

γ'

ε + γ'

↑

- - γ' + α"

+

ε

- γ' + ε*

+ α

γ' +

ε +

α

- -

* Amount of γ' in the quenched ε specimen is 0.3 to 1 vol.%

↑ Indicates an increased amount of γ' after aging

Enriched

Chapter 6

116

6.3.1.2. εεεε SPECIMENS WITH >20 at.% NITROGEN (GROUP A2)

Ten specimens were prepared in the concentration range from 20 to 27 at.% N, at

nitriding temperatures between 943 K and 963K (Table 6.2.). In the diffraction

patterns of the quenched specimens, the 101ε, 201ε, 211ε, and 103ε

superstructure reflections** could be observed, which indicates that a long-range order

of the nitrogen atoms is present. In the (quenched) ε specimens with a nitrogen

content in the range from 20 to 26 at.%, some γ' had also formed during quenching. In

a similar way as indicated above (Group A1), the volume of γ' was determined by X-

ray diffraction. In the specimens with a nitrogen content close to that of γ', the volume

fraction of γ' is about 0.3 to 1 vol. %; this amount is about 0.2 vol.% for specimens

with a nitrogen content of about 23 to 24 at.% . No γ' reflections were observed for

specimens with a nitrogen content higher than about 26 at.%.

6.3.1.3. γ' SPECIMENS (GROUP B)

Two pure γ' specimens were prepared by nitriding at 723 K and subsequent

quenching.

6.3.1.4. γ-Fe[N] SPECIMEN (GROUP C)

In order to obtain homogeneous, purely austenitic specimens, i.e., to prevent the

formation of martensite upon quenching to room temperature, the nitrogen content in

austenite should be as high as ~8 at.%[34].

The γ-Fe[N] specimen was prepared at a nitriding temperature of 963 K. After

quenching to room temperature, a pure γ phase was obtained, with a nitrogen content

of 9.75 at.%; the Ms temperature of this specimen is < 223 K[34].

** All Miller indices hkl for the ε-phase are given with respect to the hexagonal nitrogen sublattice, accounting for the occurrence of long range order[32].


117

6.3.1.5. γ -Fe[N] + εεεε SPECIMEN (GROUP D)

One specimen was prepared at 963 K, such that the average composition lays in the γ

+ ε two-phase region. The nitrogen content of the ε phase was estimated to be about

Table 6.2. Phase transformation of ε specimens with a nitrogen content > 20 at.%

413 K Initial phase

composition

373 K

50 days 1 day 7 days 28 days

443 K

3 days

ε (20.2 at.% N)* - - - ε + γ' ↑ -

79 vol.% γ' + ε

(20.3 at.% N)

ε + γ' ↑ - - - -

ε (21.1 at.% N)* - - - ε + γ' ↑ -

ε (21.4 at.% N)* - ε + γ' ↑ - - -

ε (22.6 at.% N)* - - - ε + γ' ↑ -

ε (22.8 at.% N)* - - - - ε + γ' ↑

ε (23.0 at.% N)* - - - - ε + (γ')

ε (23.7 at.% N)** - - - ε + (γ') -

ε (24.6 at.% N)** - - - ε + (γ') -

ε (26.2 at.% N) ε - ε ε -

* Amount of γ' in the quenched ε specimen is 0.3 to 1 vol.%

** Amount of γ' in the quenched specimen is 0.1 to 0.2 vol.%

↑ Indicates an increased amount of γ' after aging

Enriched

13 at.%. This estimate was made using a linear extrapolation of the lattice-parameter

data pertaining to the range 20 to 33 at.% N given in Ref. [29]. The observed lattice

parameters are a=0.45737 nm and c = 0.42981 nm. The γ phase contained about

10.15 at.% N. These compositions correspond to the phase boundaries of the γ + ε

two-phase region at 963 K (Fig. 6.1.), which is the nitriding temperature of the

specimen. None of the superstructure reflections due to long-range order of the

nitrogen atoms in ε was observed in the diffractogram of the as-quenched specimen.

Chapter 6

118

6.3.2. PHASE TRANSFORMATIONS IN εεεε-PHASE SPECIMENS

6.3.2.1. PHASE TRANSFORMATIONS OF ε WITH <19.5 at.% NITROGEN

UPON ANNEALING UP TO 473K (GROUP A1)

According to the Fe-N phase diagram[35] on cooling, ε with a nitrogen content below

19.5 at.% N should undergo an eutectoid reaction of ε (15.9 at.% N) into γ + γ' at 923

K, which is followed by eutectoid reaction of γ (8 at.% N) into α and γ' at 865 K.

Upon further lowering the temperature, the α and γ' phases are expected to remain

present below 865 K down to at least 573 K; the nitrogen contents in α and γ' change

with temperature.

In Table 6.1., an overview is given of the phases observed after annealing of ε,

with a nitrogen content <19.5 at.%, at temperatures in the range from 373 K to 473 K

(note also Fig. 6.2.). The results can be summarized as follows:

1 the α" phase develops,

2 a nitrogen-enriched ε phase is formed,

3 the amount of γ' increases appreciably, and

4 a long range order of nitrogen atoms develops in the ε phase.

The results suggest that in the temperature range from 373 K to 473 K, transformation

of the ε phase into γ' and α" (or α) occurs under simultaneous enrichment of the ε

phase (indicated by ε (enriched)).

ε ⇒ α" (α) + γ' + ε (enriched)

The nitrogen content of the N-enriched ε phase formed is about 22 to 24 at.% N. The

nitrogen content of the ε (enriched) phase is higher in the specimens with an initially

low-nitrogen ε. Note that the nitrogen content of ε (enriched) is not only much higher

than that of the original ε, but also higher than the nitrogen content (~20 at.%)

required for the γ' phase. Upon annealing, a shoulder develops at the low-angle side of

the γ' reflections. The 200γ' line profile could be conceived as the enveloping profile

of two peaks (note the profile fit in Fig. 6.2.): one sharp peak at the same Bragg angle


119

as for the as-quenched specimen and one relatively broad peak at a somewhat lower

Bragg angle, corresponding to a higher N-content.

Fig. 6.2. X-ray diffraction patterns and profile fits (dashed lines) in the range of 40 to

60 deg 2θ (Co Kα radiation) of an ε specimen with a composition of 16.7 at.% N

before (top) and after aging at 413 K for 28 days (bottom). After aging, α'' (Fe16N2)

and γ' (Fe4N) phases have precipitated and nitrogen-enriched ε is observed with

peaks shifted to lower diffraction angles, as compared to the original ε.

According to Gibbs’ phase rule for a binary-phase diagram, at a chosen

pressure, a particular three-phase equilibrium can occur at one particular temperature,

Chapter 6

120

where the compositions of the respective phases in equilibrium are fixed. Since the

coexistence of at least three phases (α" (α), γ', and ε (enriched)) was observed in the

entire temperature range investigated, it has to be concluded that the annealed samples

do not represent thermodynamic equilibrium. Therefore, the observation of at least

three coexisting phases for a range of temperatures in the annealed samples is

attributed to two simultaneously occurring transformation reactions:

ε ⇒ α" (α) + γ' (1)

ε ⇒ α" (α) + ε (enriched) (2)

The first reaction may occur in particular at the γ' phase particles present in the as-

prepared samples (cf., Section 6.3.1.), which could serve as easy nucleation sites for

γ'. The development of γ' from an ε phase containing less than 19.5 at%. N leads to

nitrogen depletion of the ε phase and formation of the α'' phase. The second reaction

may occur within the pure ε phase. This would imply that α'' can readily nucleate in

the ε phase, whereas formation of γ' is thermodynamically or kinetically hindered.

The validity of the previous hypothesis cold not be verified by performing the same

heat treatment with a γ'-free, homogenous ε phase specimen, because such a

specimen could not be prepared.

The effects of temperature, time of annealing, and nitrogen content on the

transformation behavior will be discussed next.

6.3.2.1.1. EFFECT OF ANNEALING TEMPERATURE

Annealing at temperatures in the range from 413 K to 473 K leads to the formation of

α" (and α). X-ray diffraction patterns corresponding to a series of annealing steps of

the specimen initially containing ε with 15.6 at.% N are shown in Fig. 6.3. The

reflections of the (cubic) α phase are very close to (overlap with) some of the

reflections of the (tetragonal) α" phase and unraveling of the diffraction patterns by

profile fitting (cf. Section 6.2.) is necessary. Especially, the 2θ range from 40 to 60

deg 2θ, including the 202α", 220α" doublet and 110α reflections, and the 2θ

range from 96 to 100 deg 2θ, including the 422 α" and 211 α reflections, are


121

appropriate to investigate the developments of α" and α. The reflections in the 40 to

60 deg 2θ range were used to investigate the (integrated) intensity ratio of α and α".

Reflections of α" were detected after annealing for 28 days at 413 K

(Fig. 6.3. (a), left-hand side). In this stage of transformation, the peak positions as

well as the ratio of the integrated intensities of the doublet 202α" / 220α"

(incorporating 110α, if present) are compatible with those for the α" phase. The

peak position (Fig. 6.3. (a), right-hand side) is about that of the 422 α" reflection,

and the integrated intensity ratio 202α" / 220α" is about 1.75, which corresponds

to that calculated for α" for a non-textured powder. Increasing the annealing

temperature leads to asymmetric broadening of the 422 α" peak toward higher

diffraction angles and to a shift of the peak maximum to higher diffraction angles.

Both observations strongly suggest the formation of ferrite during annealing.

Furthermore, the ratio of the (integrated) intensities of reflections of the doublet

202α" / 220α" decreases upon annealing at a temperature above 413 K. This hints

at the presence of a 110α peak overlapping with 220α". After annealing at 443 K

for 3 days, α" is still the predominantly precipitating low-nitrogen phase, but, at

463 K, α appears to be the dominant low-nitrogen phase. Prolonged annealing at

443 K leads to the formation of ferrite in addition to α". This last observation

indicates that the α" phase is less thermodynamically favorable than α above this

temperature. After annealing at 473 K, α" was not observed; only α was observed

after the shortest annealing time of 1 day. Ιt cannot be excluded that also for this

temperature, α" forms prior to the formation of α, since annealing times shorter than 1

day were not investigated for this temperature. Hence, it is concluded that α" is less

thermodynamically favorable than α, at temperatures above at least 413K.

Upon annealing at 373 K neither α nor α" was observed even after 50 days of

annealing. Only very small changes (an increase of the intensities of the γ' reflections

and a local enrichment of the initial ε) were observed. Because the nitrogen content of

the initial ε is smaller (<19.5 at.%) than that of γ' (about 20 at.%), an increase of the

amount of γ' should be accompanied by the simultaneous development of nitrogen-

depleted regions and/or development of a second phase with a nitrogen content less

than that of the initial ε (most probably α"), to ensure the mass balance. Apparently,

the amount of such a second phase (α") is too low to be observed.

Chapter 6

122

Fig. 6.3. X-ray diffraction pattern (Co Kα1 radiation) in the range of 40 to 60 deg 2θ and 96 to 100 deg 2θ of an initially ε specimen with a composition of 15.6 at.% N for

several annealing times and temperatures. (a) 413 K for 28 days (dashed line:

3 days): transformation of ε into nitrogen-enriched ε and α'' occurs; there is no clear

evidence for the presence of α. (b) 443 K for 3 days and (c) 463 K for 1 day: both the

α and α'' phases are present; he diffraction profiles (range: 96 to 100 deg 2θ) are

composed of the 422α'' and the 211α diffraction peaks. (d) 473 K for 1 day:

decomposition of ε into α and γ' has occurred; there is no evidence for the presence of

α''.


123

Fig. 6.4. X-ray diffraction pattern (Co Kα1 radiation) of an ε specimen with a

composition of 15.6 at.% N after annealing at 413 K for different times showing the

kinetics of the phase transformations. The intensities are given relative (a) through (c)

to the maximum intensity of the 111ε reflections and (d) to the maximum intensity of

the 111γ' reflection. The decomposition starts with an increase of the amount of γ' (compare the intensity ratios of the 300ε / 220γ' and 002ε / 111γ' in the as-

prepared state (a) and after 1 day at 413 K (b)). After 3 days at 413 K (c), the

precipitation of α'' and local enrichment of ε (the mark on the left-hand side of the

111ε peak) are observed. Upon further annealing up to 28 days (d), a nitrogen-

enriched ε phase is observed; also, the α'' and γ' phases are present.

Chapter 6

124

6.3.2.1.2. EFFECT OF ANNEALING TIME

The effect of annealing time (at 413 K) for a specimen composed of ε (15.6 at.% N) is

shown in Fig. 6.4. Clearly, the transformation starts with the formation of locally

enriched ε, as follows from the development of shoulders at the low-angle side of the

ε reflections after 1 and 3 days (Fig. 6.4.(b)/(c)). The peak position indicates a

nitrogen content only slightly higher than that of the original ε phase. Simultaneously,

the amount of γ' increases. The first, unambiguous, observation of the phase of low

nitrogen content (α") is possible after annealing for 3 days (note the insert of Fig. 6.4.

(c)). Upon continued annealing at 413 K, the nitrogen content of the enriched ε phase

increases, as evidenced by the shift of the 110ε reflection in Fig. 6.2. (left-hand

side), while the reflections of the original ε phase disappear, resulting in a specimen

constituted of α" + γ' + ε (enriched) (Figs. 6.4.(d). and 6.2.).

Of course, the time necessary to reach a certain stage of decomposition is

temperature dependent. After annealing at 413 K for 3 days, more of the initial ε had

transformed than after annealing at 373 K for 50 days, and annealing at 443 K for 3

days leads approximately to the same amount of decomposed ε as that obtained by

aging at 413 K for 28 days. These observations can be correlated with the diffusion

distances of nitrogen in ε (note the calculated values for √Dt in Table 6.3., where D is

an effective diffusion coefficient for N in ε[36] and t the annealing time) for the

corresponding cases. The mean value of the integrated intensity change of various

reflections (relative to the initial intensities) for specimens (ε (15.6 at.% N)) tempered

according to the times and temperatures given in Table 6.1., is shown in Fig. 6.5. as a

function of the diffusion distance of N in ε. The approximately linear relationship

suggests that diffusion of nitrogen indeed is the decomposition rate-determining step

during the transformation.

6.3.2.1.3. EFFECT OF NITROGEN CONTENT

The higher the amount of α" (i.e., the intensity ratio, α"/ γ'), the lower the nitrogen

content in the initial ε phase (Fig. 6.6.): for a sample with 15.6 at.% N, α" is a

dominant phase at a stage of aging where α" can hardly be discerned for a sample

with 19.1 at.% N. Further, the smaller the amount of nitrogen-enriched ε phase, the

smaller the amount of α".


125

Fig. 6.5. The mean integrated intensities (relative to the integrated intensities in the

as-prepared state) of the 110, 111, and 112 ε reflections measured after several

annealing temperatures and times as given in Table 6.1. of the ε specimen (15.6 at.%

N). An proximate linear relationship with the diffusion distance Dt is observed

(Section 6.3.).

Table 6.3. Calculated values of the diffusion distance of nitrogen, √Dt, for different

annealing temperatures and times; The values of the effective diffusion coefficient, D,

have been calculated according to Table 6.5. in Reference 36 for the temperatures of

annealing indicated and the composition of the specimens (15.6 at.% N)

Annealing time and

temperature D (m2 · s-1) Time (s) √Dt (m)

373 K, 50 days 4.78 · 10-24 4,320,000 4.55 · 10-9

413 K, 3 days 1.97 · 10-22 259,200 7.14 · 10-9

413 K, 28 days 1.97 · 10-22 2,419,200 2.18 · 10-8

443 K, 3 days 2.06 · 10-21 259,200 2.31 · 10-8

473 K, 1 day 1.60 · 10-20 86,400 3.71 · 10-8

Chapter 6

126

Fig. 6.6. X-ray diffraction patterns (Co Kα1 radiation) of ε specimens of different

nitrogen contents (15.6, 17.9, and 19.1 at.% N) after ageing at the same temperature

and for the same time (i.e. 413 K; 28 days). The intensities have been given relative to

the integrated intensity of the 111 γ' reflection. For the sake of clarity, the

diffraction patterns have been shifted with respect to each other. The amount of the α''

phase present appears to be correlated to the nitrogen content of the initial ε phase:

the lower the nitrogen content of the initial ε, the greater the amount of α'' present.

Mass-balance considerations for both Reactions (1), ε into α"+ γ', and (2)

ε into α" + ε (enriched), leads to the following conclusion. For ε with an initial

composition close to that of γ', Reaction (2) produces the largest amount of α". The

current observations, thus, suggest that the Reaction (2) is dominant for the specimens

of low initial N content whereas Reaction (1) is more dominant the higher the initial

nitrogen content.

6.3.2.2. PHASE TRANSFORMATIONS IN ε WITH >20 at.% NITROGEN

UPON ANNEALING UP TO 443 K (GROUP A2)

The transformation behavior of ten specimens with nitrogen contents in the range

from 20 at.% to 26.5 at.% was studied in the temperature range from 373 to 443 K.

The phases present after the annealing experiments have been indicated in Table 6.2.


127

The most striking results from these experiments are:

1. the amount of the γ' phase increases for specimens of nitrogen contents in the

range from 20 to 22.8 at.%.,

2. the (remaining) ε phase is nitrogen enriched, and

3. the long-range order of nitrogen atoms in the ε phase becomes more

pronounced.

6.3.2.2.1. PRECIPITATION OF γ' -Fe4N1-z FROM εεεε-Fe2N1-x: THE ε/ε + ε/ε + ε/ε + ε/ε + γ'

PHASE BOUNDARY

For the specimens with an initial composition of the ε phase in the concentration

range from 20.2 to 22.8 at.% N, an increase of the integrated intensity of the γ'

reflections is observed. Simultaneously, the reflections of the ε phase have shifted to

lower diffraction angles, indicating nitrogen enrichment of the remaining ε, in

accordance with the conservation of nitrogen in the specimen. The enrichment is

about 1 at.% in the specimen with the lowest nitrogen content (20.2 at.%) and about

0.1 to 0.05 at.% for the other specimens, depending on the initial nitrogen content††.

Hence, for temperatures in the range from 413 K to 443 K and for nitrogen contents in

the range from 20.2 to 22.8 at.%, the initial ε phase transformed according to:

ε → ε (enriched) + γ' (3)

Further, a shoulder appeared at the low angle side of the γ' reflections after 28

days at 413 K (note the 200 γ' reflection shown in Fig. 6.7.). Such an observation,

but much less pronounced, was also made for specimens in Group A1 discussed in

Section 6.3.2.1. Apparently, the reflection appears to consist of two peaks: one small

sharp peak with a full width at half of the maximum intensity (FWHM) of 0.01 deg

∆2θ and a broad one with a FWHM of 1.49 deg ∆2θ. The peak position of the small

reflection cannot be determined very accurately because of the weakness of the

†† Since the nitrogen content of the remaining (enriched) ε after annealing for a given time and temperature depends on the nitrogen content of the initial ε phase, these results cannot be used to construct the equilibrium phase boundary. Also, the dependencies of the lattice parameters on the nitrogen content are not known for the composition range concerned.

Chapter 6

128

reflection, but this peak position is always at a significantly higher diffraction angle

than that of the broad reflection (Fig. 6.7.).

Fig. 6.7. The 200 γ' reflection (Co Kα1 radiation) after ageing of an ε specimen (20.2

at.% N) for 28 days at 413K (dashed line: single peaks obtained by profile fitting). At

the low angle side of the γ' reflection (of γ' formed during specimen preparation), a

new and broad diffraction peak appears. The intensities have been given relative to

the maximum of the 111 ε reflection.

The interpretation of the composite 200γ' line profile is identical to the one

given in Section 6.3.2.1. The sharp peak is caused by a pre-existing, relatively low N-

containing γ' phase, while the broad peak originates from γ' precipitated during

annealing, which has an nitrogen content significantly higher than the original γ'.

Considering the α + γ'/ γ' phase boundary in the (assumed) Fe-N phase diagram (cf.,

Fig. 6.1., this observation indicates that the initially present γ', precipitated during

quenching of the specimen, has developed at higher temperatures than the annealing

temperatures applied here.


129

The results are consistent with the preservation of the two-phase equilibrium

ε + γ' down to temperatures as low as 413 to 443 K, for nitrogen contents of 20 (at

least) to 22.8 at.%.

For the specimens with a nitrogen content of 23 to 26.2 at.%, further

development of a γ' phase and enrichment of the ε phase were not observed,

indicating that the initial ε phase behaves as a stable phase in the temperature range

investigated.

Since no precipitation of γ' was observed after annealing of the specimen with

a nitrogen content of 23.0 at.%, and precipitation of γ' did occur for a specimen with ε

containing 22.8 at.%, this could indicate that the ε+ γ' /ε phase boundary in the

Fe-N phase diagram shifts to lower nitrogen contents for temperatures <573 K (Fig.

6.8.). At 573 K, the ε+ γ' / ε boundary occurs at about 26.5 at.% N[33]. Recent

thermodynamic calculations[5,6] suggested a similar trend. However, it could be also

argued that the transformation of ε at these temperatures is hindered (note also Section

6.4.). In this respect, it is worth mentioning that, upon (very slow) furnace cooling of

a fully nitrided powder containing 25 at.% N from a nitriding temperature of 783 K to

room temperature, a small amount of γ' phase could also be detected[29].

6.3.2.2.2. REDISTRIBUTION OF NITROGEN ATOMS AND LONG RANGE

ORDERING OF NITROGEN IN εεεε

After nitriding and quenching, the diffraction patterns of samples with a composition

near to that of Fe3N (i.e. containing 23 to 26 at.% N) showed only the 101ε, 201 ε,

211 ε, and 103 ε superstructure reflections. After aging, new superstructure

reflections were detected: 100 ε, 200 ε, 102 ε, and 201 ε. Simultaneously, an

increase in the integrated intensity of the initially present superstructure reflections

was observed.

Hence, it is concluded that the nitrogen atoms redistribute upon annealing,

thereby establishing a more ordered state[37]. The observed superstructure reflections

indicate that the type of ordering of the nitrogen atoms is typical for Fe3N[32],

indicated the B-type configuration in Refs. [29] and [38].

Chapter 6

130

Fig. 6.8. The ε / ε + γ' phase boundary according to Ref. [4] and the results of

annealing ε and γ + γ' specimens at temperatures below 443 K.

The results obtained on phase transformation in the ε phase (Sections 6.3.2.1.

and 6.3.2.2.), (1) the absence of α" in the aged specimens with a nitrogen content

higher than 20 at.% (Group A2) and (2) the precipitation of γ' in both specimens

containing less than 20 at.% (Group A1) and more than 20 at.% (Group A2), could

indicate that γ' is a stable phase in the investigated temperature range from 373 to

473 K. On the other hand, it may be argued that the development of γ' in specimens

containing less than 20 at.% N occurs because γ' nucleation sites are already present

and that the formation of α"/α + ε (enriched) leads to a thermodynamically stable

situation.


131

6.3.3. PHASE TRANSFORMATIONS OF γ' SPECIMENS (GROUP B)

In the as-prepared condition, both γ' specimens have a composition of 19.93 at.% N‡‡.

According to the Fe-N phase diagram[4,35,39], transformation of γ' of this composition

into γ' of a slightly higher nitrogen content (<20 at.%) and ferrite is expected to occur

at temperatures down to 573 K. This was indeed observed to occur in a γ' layer at 603

K[40].

In order to investigate the phase diagram below 537 K, the γ' specimens were

annealed at 373 K and 413 K for up to 50 days. An overview of the temperatures and

tempering times applied and the phase composition observed after the anneals is given

in Table 6.4.

Table 6.4. Phase transformation of γ'-Fe4N1-z (19.9 at.% N) specimen

Initial phase composition 373K

50 days

413 K

28 days

413 K 28 days + 373 K 50 days*

γ' - γ' + α" γ' + α"

γ' γ' γ' + α" -

* A heat treatment at 413 K (28 days) was followed by a heat treatment at 373 K (50

days)

After long-term aging for 28 days at 413 K, in addition to the γ' reflections,

new weak reflections pertaining to the α" phase were detected (note the 202α",

220α", and 213α" reflections in Fig. 6.9.). No evidence was obtained for the

development of α. Simultaneously, the γ' reflections shifted slightly to lower

diffraction angles, implying nitrogen enrichment of γ' (cf. the lattic-parameter data in

Ref. [30]). Hence, at 413 K the initial γ' decomposes according to

γ' ⇒ γ' (enriched) + α" (4)

‡‡ Due to the strong dependency of the lattice parameter of γ' on the nitrogen content[30], the N content can be determined with an accuracy of 0.02 at.% N.

Chapter 6

132

Fig. 6.9. X-ray diffraction pattern (Co Kα1 radiation) of γ' (Fe4N1-x) specimen in the

as-prepared state and after ageing for 28 days at 413 K. The intensity (logarithmic

scale) is given relative to the maximum intensity of the 111γ' reflection: the

diffraction patterns have been shifted with respect to each other along the vertical

axis. After annealing (weak), reflections of α'' are observed (dashed lines in top

figures: results of profile fitting).

The development of α" and an N-enriched γ' from γ' has not been reported

before. Apparently, aging at 373 K does not allow this transformation to occur even

after 50 days. This may be understood by assuming that the nitrogen diffusion is rate

controlling. The diffusion distance, √Dt, is very small for the treatment at 373 K and

is about 3 times smaller than the one for the treatment at 413 K for 28 days: about

16 nm for 50 days at 373 K vs. about 50 nm for 28 days at 413 K (the data of

diffusivity are taken from Ref. [36]).


133

Using the relation given in Ref. [30] between the lattice parameter of γ' and its

nitrogen content, the increase of the lattice parameter upon aging at 413 K (from

a = 0.37978 nm to a = 0.37989 nm)§§ corresponds to an increase in nitrogen content

from 19.93 at.% before annealing to 20 at.% after annealing. Hence, the phase

boundary of the γ' field in the Fe-N phase diagram shifts toward higher nitrogen

contents with decreasing of temperature (from the temperature of nitriding, 723 K, to

the temperature of annealing, 413 K). This is in agreement with an extrapolation of

previous evaluations (cf. Refs. [38] and [40]).

From a mass-balance consideration, the volume fractions of the two phases

after annealing (α" and γ') can be calculated as follows. The γ' phase contains

19.93 at.% N initially and contains 20 at.% N after annealing. Assuming that the

precipitated α" contains 11.11 at.% N, corresponding to Fe16N2, the volume fraction

of the two phases after annealing is calculated to be 0.8 vol.% α" + 99.2 vol.% γ'. The

amount of precipitated α" phase is small, which explains why only its strongest

reflections can be observed (Fig. 6.9.).

6.3.4. PHASE TRANSFORMATION OF THE γ SPECIMEN (GROUP C)

According to the published Fe-N phase diagram[35], γ (austenite) specimens should

transform into α and γ' upon annealing at temperatures below 865 K. In view of the

here-observed development of α" upon transformation of ε (<19.5 at.% N) and γ' at

temperatures below 573 K (Sections 6.3.2.1. and 6.3.3.), the transformation behavior

of γ at similar temperatures is of interest to establish the stability of α" at these

temperatures. The nitrogen content of the specimen is 9.75 at.%. The resulting phase

compositions after different anneals of the specimen are given in Table 6.5.

For all temperatures and times of aging investigated, in addition to γ

reflections, only α and γ' reflections were observed. No indications for the presence of

α" were obtained. The γ reflections did not shift upon aging (Fig. 6.10.), implying that

no composition change occurs for the remaining γ during transformation. Hence, in

the temperature range of 413 K to 463 K, γ decomposes according to

γ ⇒ α + γ' (5)

§§ The error in the value of the lattice parameter (derived from the 220γ' peak position) was ± 3 × 10-5 nm, corresponding to 1/5 of the employed step size in 2θ.

Chapter 6

134

Fig. 6.10. The lattice parameters of γ and γ' during the decomposition of γ into

α + γ'.

Table 6.5. Phase transformation of a γ specimen (9.75 at.% N)

413 K 443 K Initial phase

composition 7 days 28 days 3 days 14 days 28 days 50 days

463 K

3 days

γ γ + γ' +

α

γ + γ' +

α

γ + γ' +

α

γ + γ' +

α

γ' + α γ' + α γ + γ' +

α


135

Diffraction patterns recorded at various stages of the transformation, upon

annealing at 443 K, are shown in Fig. 6.11. In the initial stage (after 3 days

annealing), weak α and γ' reflections are already observed. Prolonged annealing for

up to 14 days leads to an increase of the intensities of α and γ' reflections and the

observation of γ' superstructure reflections as 100γ'. After 28 days of annealing,

reflections of γ are absent: γ is fully decomposed into α and γ'.

Additional, cumulative annealing experiments were carried out in order to

study whether the specimens consisting of α and γ' after full decomposition of γ might

experience further phase transformations: e.g., formation of α". The annealing time

(at 443 K) was prolonged to 50 days, but no further changes were observed in the

diffraction patterns. Also, to check if α" would develop initially as an intermediate

phase, annealing was performed at 413 K for shorter time (7 days). Also, after this

short heat treatment, very weak reflections of only α and γ' were observed and no

reflections of α" could be detected.

The rate of transformation of γ strongly depends on the temperature of

annealing. For example, while after 28 days at 443 K austenite is fully decomposed,

after the same time at 413 K, the transformation is still in an initial stage. Note that

the transformation of γ in this single-phase specimen considered here is much slower

than that of γ, for the same heat treatment, in the ε + γ specimen (Section 6.3.5.

addresses the transformation of ε + γ two-phase specimens).

The type of transformation pertaining to the transformation of γ into α and γ'

follows from the intensity ratio of the appearing phases and their compositions. Apart

from the peak positions of the reflections of the parent γ phase (addressed previously),

those of the product phases γ' and α also remain constant during the transformation

(Fig. 6.10.). Further, the ratio of the integrated intensities of α and γ' reflections

remains constant (e.g. Iγ'220/Iα200; Fig. 6.12.), whereas the ratio of the integrated

intensities of γ and γ' reflections obviously decreases (e.g. Iγ220/Iγ'200; Fig. 6.12.).

The combination of a constant value of the ratio of integrated intensities and constant

composition, indicates that the transformation is of a eutectoid type. Recognizing the

low temperature at which the transformation proceeds, the transformation of γ to α

and γ' is suggested to be bainitic.

Chapter 6

136

Fig. 6.11. X-ray diffraction patterns (Co Kα1 radiation) of an iron-nitrogen austenite

(γ) specimen after ageing at 443 K for different times. The diffraction patterns have

been shifted with repect to each other along the vertical axis. Upon annealing, the γ specimen (9.75 at.% N) decomposes into α + γ'. Full decomposition is observed after

28 days.

Fig. 6.12. Ratio of the integrated intensities of (a) the 220 γ' and 200α reflections,

and (b) the 220 γ and 220 γ' reflections.


137

Using the "direct comparison method"[33] (cf. Section 6.2.), the amounts of the

three phases during the different stages of the transformation were calculated, for

different times of aging at 443 K, from the integrated intensities of the 200α,

220γ', and 220 reflections, and the 110α, 200γ', and 200γ reflections (Table

6.6.). Starting with initially 100 % γ, the specimen consists of 54.8 vol.% α and 45.2

vol.% γ' after full decomposition (after 28 days at 443 K).

Table 6.6. Volume percentages of the α, γ, and γ' phases after different times of

annealing at 443 K of a γ specimen (9.75 at.% N)

Phase Initially present 3 days 14 days 28 days 42 days

α (vol.%) - 25.0 49.2 54.8 54.8

γ' (vol. %) - 20.1 40.7 45.2 45.2

γ (vol.%) 100 54.9 10.1 - -

Table 6.7. Phase transformation of the two-phase specimen ε (13 at.% N) +

γ (10.15 at.% N)

443 K Initial phase

composition

373 K

50 days

413 K

28

days

3 days 14 days 28 days

463 K

3 days

γ + ε γ + ε +

α"

γ + α

+ γ' +

α"

γ + α + γ' +

α" + ε

α + γ' + α"

+ ε

α+ γ' + α"

+ ε

α + γ' +

γ

Enriched

From the mass balance, the expected amount of α and γ' phases after the

transformation of austenite containing 9.75 at.% N can be calculated. Assuming that

the nitrogen content in the α and γ' phases at 443 K is 0.02 and 20 at.%, respectively,

the expected values for the volume percentages are calculated to be 51.3 vol.% α and

48.7 vol.% γ', which agree well with the experimental values derived from the

integrated intensities.

Chapter 6

138

From the previous above results on the transformation of γ during low-

temperature (413 to 463 K) annealing, it appears that γ' is a stable phase in the Fe-N

phase diagram in this temperature range.

6.3.5. PHASE TRANSFORMATION OF γ + εεεε TWO PHASE SPECIMENS

(GROUP D)

According to the published Fe-N phase diagram[35], γ + ε specimens should

decompose into α and γ' upon annealing at temperatures below 865 K (down to

573 K). In view of the development of α" (or α) from ε (<19.5 at.% N), the

development of α" from γ' and the development of γ' + α from γ specimens (Sections

6.3.2.1., 6.3.3., and 6.3.4.), the transformation behavior of a γ (10.15 at.% N) + ε

(13 at.% N) specimen is of interest for investigating the (meta)stability of α" at

temperatures below 473 K. With respect to the formation of α" from γ' (Section

6.3.3.) and not from γ (Section 6.3.4.), and the inevitable presence of some γ' in ε

(<19.5 at.% N) (Section 6.3.2.1.), it could be argued that the presence of γ' may be

decisive for the formation of α", for instance, by providing an easy a nucleation site.

No γ' is present in the as-prepared ε + γ specimens discussed in this section.

The phase constitution resulting after the anneals performed is given in Table 6.7.

(also note Fig. 6.13.). It follows that phase transformation of the two-phase γ + ε

specimens can be conceived as the superposition of the transformation of ε

(<19.5 at.% N) and γ, as observed before: ε → α" (α) + ε (enriched), ε → α" (α) + γ',

and γ → α + γ'. However, the kinetics of these reactions are different from those for

the "single"-phase specimens. From the results obtained after different times of

annealing at 443 K (Fig. 6.13.), it can be deduced that first the initial ε phase (13 at.%

N) transforms relatively rapidly. After 3 days at 443 K, it already has disappeared

fully. At the same time, a new, nitrogen-rich ε phase (with about 23 to 24 at.% N; cf.

Section 6.3.2.1. for ε (<19.5 at.% N)), with an appreciably higher nitrogen content

than the initial one, and α" have developed (cf. Fig. 6.13. (b)). The γ phase

decomposes into α and γ' (cf. Section 6.3.4.). This transformation in the current two-

phase ε + γ specimen is faster than in the single-phase γ specimen: the γ phase in the


139

Fig. 6.13. X-ray diffraction patterns (Co Kα1 radiation) of a two-phase specimen γ (10.15 at.% N) + ε (13 at.% N) upon decomposition at 443 K: (a) The as-prepared

Chapter 6

140

state (b) After ageing for 3 days at 443 K, the initially present ε is fully decomposed

and the α, α'', γ', and nitrogen-enriched ε phases have developed; at this stage, some

austenite is still present. Ageing for the same time at 463 K results in decomposition

into α and γ' (c) After 14 days at 443 K, the austenite is also fully decomposed (d) At

443 K, the α, α'', γ', and nitrogen-enriched ε remain present even after a longer time

(28 days).

Fig. 6.14. X-ray diffraction patterns (Co Kα1 radiation) in the range of (a) 47 to

58 ˚2θ and (b) 90 to 100 ˚2θ for a two-phase specimen γ (10.15 at.% N) + ε (13 at.%

N) after ageing at 443 K. Upon prolonged ageing, an increase of the intensities of the

α and γ' reflections and a decrease of the α'' reflections are observed.

two-phase specimen has fully decomposed after 14 days at 443 K, whereas, in the

single-phase specimen (discussed in Section 6.3.4.), full decomposition of the γ phase

requires 28 days at 443 K.

In the current γ + ε specimen, the α" phase is clearly present after the third day

at 443 K (Fig. 6.13. (b)). The α" phase seems stable at this temperature in this sample,

as shown in the diffraction patterns in Fig. 6.13. (c) and (d) after annealing for 14 and

28 days, respectively. Prolonged annealing for up to 42 days showed that a small but

significant decrease of the integrated intensities of α" reflections and a corresponding

increase if the integrated intensities of α and γ' reflections occur (Fig. 6.14.). This

indicates that α + γ' develop during transformation of α", suggesting that α" is an

intermediate phase. The relatively slow transformation of α" → α for the

ε + γ specimen may be due to a nitrogen content near that of α" (11.11 at.%) or the

absence of initial γ' in the quenched specimens, so that no nucleation sites for γ' are

present (Section 6.3.2.1.).


141

Table 6.8. Summary of the experimentally observed phase transformations during low

temperature aging of Fe-N alloys compared with the different possibilities of low

temperature Fe-N equilibria

Specimens α and α'* γ γ + ε ε (<19.5

at.% N)

γ' ε (>20

at.% N)

Experimental

ly observed

(this work)

trans-

formations

during low

temperature

(373 – 473K)

aging

- γ → α

+ γ'

γ → α + γ'

ε → α" +

γ' [1]

ε → α" +

ε [2]

if T ≥

443K

α" → α +

γ

ε → α" +

γ' [1]

ε → α" +

ε [2]

if T ≥

443K

α" → α +

γ'

γ' →

α" +

γ'

ε

(<23at.%)

→ γ + ε

ε

(>23at.%)

→ ε

+ordering

Reported

before [7-21]

Precipitation

of α"

(Fe16N2)

- - - - -

Stable phases

at low

temperature;

extrapolation

of Wriedt [4]

phase

boundaries

α+ γ' α + γ' α + γ' α + γ' γ' + α γ' + ε

(26.5 at.%

N)

* α' : martensite

Enriched

Chapter 6

142

6.4. GENERAL DISCUSSION

A summary of the phase transformations determined to occur during low-temperature

aging of Fe-N alloys in given in Table 6.8. The expected equilibrium phases,

corresponding to different proposals for the Fe-N phase diagram at low temperature,

have also been indicated in Table 6.8.

At constant pressure, at most, two phases can be in thermodynamic

equilibrium over a certain temperature range in the Fe-N system. Thus, the concurrent

presence of three phases in apparently fully transformed samples, i.e. after long-term

annealing, does not correspond with thermodynamic equilibrium: note, for instance,

the simultaneous presence of ε, γ' and α after the long-term annealing of the (initially)

γ + ε and ε specimens.

As an explanation for the simultaneous occurrence of more than two phases, it

is proposed that local structure and/or composition variations induce the system to

establish locally metastable equilibria. The low temperatures involved hinder long-

distance nitrogen diffusion to obtain a macroscopically homogeneous specimen. This

is, in particular, clear from the transformation of the two-phase γ + ε specimens, in

which the ε-phase regions and the γ-phase regions apparently behave as independent

entities upon tempering.

Apart from the development of α" from supersaturated ferrite (α) and from

martensite (α') at low temperatures, as known from the literature, this work

unequivocally demonstrates that α" also develops during low-temperature aging of γ'

and ε (<19.5 at.% N). The α" phase does not form upon low-temperature annealing of

γ specimens, which decompose into γ' + α. By aging above 413 K, the α" phase

initially formed in the ε phase transforms into α in the ε and the ε + γ specimens,

provided the ε phase contains less than 19.5 at.% N. No aging above 413 K has been

performed for γ' specimens. Prolonged annealing at 413 K or lower temperatures did

not lead to the dissolution of α" and the formation of α, neither in the γ' and ε (<19.5

at.% N) specimens nor in the ε + γ specimen.

Also, α" formed in supersaturated ferrite or martensite[9,10,12,14,23] is apparently

stable at temperatures below about 453 K. Considering these data and the present

observations of the apparent stability of α" below 413 K, it can be argued that α" is a

stable phase. Then, the peritectoid reaction α + γ' → α" is present in the Fe-N phase


143

Fig. 6.15. The phase diagram at low temperatures (below gray dashed line) proposed

on the basis of this work: for an overall composition of <20 at.% N, α + γ' are the

equilibrium phases, whereas for an overall composition of > 20 at.% N, γ' + ε or only

ε (for nitrogen contents above about 24 at.% N at 413 K) are the equilibrium phases.

The γ' phase appears to be a stable phase at temperatures as low as 373 K. (a) The α''

phase is an equilibrium phase below about 440 K (b) The α'' phase is an intermediate

phase and precursor for the formation of α.

diagram at a temperature in the range from 410 to 450 K (note the proposed phase

diagram in Fig. 6.15. (a)). On the other hand, α" could be a transition phase on the

way to the equilibrium (α + γ'), which leads to the proposed Fe-N phase diagram in

Fig. 6.15. (b).

The kinetic preference for α" as an intermediate phase can be understood as

follows. By localized rearrangement of Fe atoms, the parent (ε or γ') iron sublattice

can be transformed into another sublattice (α" or α). However, the equilibrium α

phase can only contain a minor amount of nitrogen (~ 0.02 at.%), whereas the

transition phase α" can contain an appreciable amount of nitrogen (11.11 at.%). As

nitrogen diffusion at the temperatures considered is slow, it is conceivable that α"

forms as a transition stage for α development (and, consequently, metastable

equilibria arise), in particular in the specimens ε and γ' both containing larger nitrogen

contents than that of α".

Chapter 6

144

The observation that the γ' nitride develops and stays as a final transformation

product during aging of samples of Groups A1, A2, C, and D at the low temperatures

investigated, indicates strongly that this phase is stable at low temperatures. The

observation reported in Ref. 23, where the γ' phase vanished during low-temperature

aging assisted by ion bombardment, has to be attributed to irrelevant experimental

conditions for testing the low-temperature stability of phases (note the discussion in

Ref. 6).

If, as proposed earlier for nitrogen contents smaller than 20 at.%, α + γ' are the

equilibrium phases, then the simultaneous presence of ε (enriched), γ', and α" in

decomposing ε specimens should be understood as follows. As discussed previously,

α" develops as a precursor for α. The surrounding ε matrix incorporates the excess

nitrogen and becomes enriched in nitrogen. Apparently, the nucleation of γ' is not

easily realized, despite the similarity between the cubic and hcp Fe sublattices of the

γ' and ε phase, respectively, which would suggest an easy transformation from ε to γ'

by the introduction of stacking faults in the Fe sublattice. It appears,

thermodynamically more favorable to develop the α" phase, presumably in regions

where no nucleation sites for the γ' phase are available. Because of the slow diffusion

of nitrogen atoms, locally, the nitrogen content of the ε matrix becomes higher than

the maximum solubility of N in the (equilibrium) γ' phase. Thus, the metastable

equilibrium α"/ε (enriched) is established. In the current ε specimens, some γ'-phase

regions are already present initially, as a consequence of the specimen preparation

(Section 6.2.). These γ'-phase particles serve as nucleation sites, and γ' can form

relatively easily upon tempering because no nucleation is required. At these locations,

the equilibrium α" (as a precursor for α) + γ' is established.

Then, reviewing the results of this work (Table 6.8.), it is concluded that, for

an overall composition lower than 20 at.% N, α + γ' and, for an overall composition

higher than 20 at.% N, γ' +ε or ε (for nitrogen contents above about 24 at.% at 413 K)

are equilibrium phases, at least down to about 440 K. Below 440 K, α" has been

observed as either an intermediate phase or an equilibium phase. This leads to the

proposal for two possible Fe-N phase diagrams at low temperatures, as given in Fig.

6.15. The γ' is, then, an equilibrium phase at temperatures as low as 373 K. The

current experiments do not allow a precise determination of the ε+ γ'/ε phase

boundary, although the results suggest an extension of the ε phase toward lower

nitrogen contents at lowering temperatures, because the nitrogen contents of the


145

nitrogen-enriched ε phases (at a certain temperature) depend strongly on the initial

nitrogen contents of the specimens.

6.5. CONCLUSIONS

The low-temperature part (373 K to 473 K) of the Fe-N phase diagram can be

investigated by analyzing annealed homogeneous specimens of about 10 to 26 at.% N

by X-ray diffraction. Homogeneous Fe-N alloys of a high nitrogen content could only

be prepared by nitriding highly porous specimens (here obtained by sintering pure

iron powder). The large surface area/volume ratio is essential, because the NH3/H2 –

Fe[N] equilibrium can only be imposed at the gas-solid interface.

From the results, the following conclusions are drawn:

1. The equilibrium phases below 860 K, for nitrogen contents <24 at.%, are α, γ'

(the γ' phase is stable down to at least 383 K), and ε.

2. Two possible interpretations for the formation of the α" phase at low

temperatures are that

o the α" phase develops as a precursor for α, not only from α

(ferrite) and α' (martensite), but also upon transformation of

ε (<20 at.% N) and γ'; or

o the α" phase develops as an equilibrium phase below about 440 K.

3. Because of the difficult nucleation of γ' and slow diffusion of nitrogen atoms,

the transformation of ε containing less than 20 at.%. N initially leads to the

development of the intermediate phases α" and ε, which is enriched in

nitrogen up to 24 at.% (i.e. more than the nitrogen content of γ').

Chapter 6

146

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149

SUMMARY

Obtaining accurate thermodynamic data for interstitial solid solutions is challenging.

Not only does it require incorporation of the changes in volume caused by the

introduction of small atoms (such as hydrogen, boron, carbon, and nitrogen) into the

interstitial spaces formed by the host lattice; the interaction between the atoms

forming the host lattice and the atoms occupying the interstitial sites, and the

occurrence of long range ordering (LRO) and short-range ordering (SRO) on both the

host and the interstitial sublattices need to be considered as well. Other complicating

factors concern the metastability of interstitial solid solutions such as nitrides and

carbides, which are often non-equilibrium phases, and the fact that direct observation

of precipitates of such phases is hampered by their small size. Therefore, the ability to

predict thermodynamic properties accurately renders valuable information for process

and material property optimization in industrial applications that cannot be obtained

otherwise.

(Sub)regular solution models, which are frequently used to describe the

thermodynamics of interstitial solid solutions, do not include LRO of the interstitial

atoms. The simplest approach which includes LRO in the description of the phases is

the Gorski-Bragg-Williams (GBW) model. However, this model does not explicitly

account for SRO. The main focus of this thesis is therefore the development of a

thermodynamic model for interstitial alloys, based on the Cluster Variation Method

(CVM), capable of incorporating SRO, LRO, and the mutual interaction between the

host and the interstitial sublattices. The obtained cluster-based model is then applied

to describe phase equilibria between iron nitrides.

Fe-N phases can be conceived as consisting of a metal sublattice, fully

occupied by iron atoms, and an interstitial sublattice, occupied by a mixture of

nitrogen atoms and vacancies. In other words, such a system can be approached as a

binary nitrogen-vacancy solution in the mean field of an iron host. This concept is

further explored in Chapter 2, which describes LRO of nitrogen atoms on the

sublattice formed by the octahedral interstices of the Fe sublattice of ε-Fe2N1-x by

application of the GBW approach. Both the fitting of the nitrogen absorption isotherm

to experimental nitrogen absorption data and the probabilities for Fe atoms

surroundings obtained with Mössbauer spectroscopy indicate that ordering of nitrogen

150

in ε-Fe2N1-x takes place according to two ground-state structures: Fe2N (configuration

A) and Fe3N (configuration B). Utilizing a synchrotron source, the superstructure

reflections are analyzed to verify the occurrence of both configurations. The

superstructure reflections and local surroundings of Fe atoms consistent with ordering

of the nitrogen atoms according to configuration B are observed. Interestingly, the

001 and 301 superstructure reflections, specific for configuration A, are not

detected, contrary to the expectations raised by the Mössbauer data. As shown by

preliminary structure factor calculations, the displacement of Fe atoms by the

(ordered) occupation of interstitial sites affects the diffracted intensity, which strongly

reduces the already weak reflections typical for configuration A.

Next, the Cluster Variation Method is implemented and the thermodynamic

model is expanded to include both LRO and SRO. Since the CVM is usually applied

to substitutional systems, the main adaptation of the model to be made for the Fe-N

phases is to allow for changes in volume as a function of the composition of the

interstitial sublattice and to approach the system in such a way that the limited

experimental data available suffices for an estimation of the input parameters

necessary for the calculations. Chapter 3 describes the calculation of the

γ-Fe[N] / γ'-Fe4N1-x phase equilibrium using the CVM tetrahedron approximation. A

Lennard-Jones potential is used to describe the dominantly strain-induced

interactions, caused by the misfit of the nitrogen atoms in the interstitial octahedral

sites. The CVM calculations clearly demonstrate the occurrence of SRO in γ-Fe[N]

and distinct LRO in γ'-Fe4N1-x, in agreement with Mössbauer and X-ray diffraction

data. For the first time, quantitative analysis of CVM calculations is directly

compared to experimental Mössbauer data, resulting in good agreement between the

CVM predictions and the fractional occurrences in γ-Fe[N] of specific Fe atom

surroundings in terms of the number of neighboring nitrogen atoms.

The γ'-Fe4N1-x / ε-Fe2N1-z phase equilibrium is calculated in Chapter 4, which

necessitates the introduction of the CVM prism approximation to facilitate the

description of ordering of nitrogen atoms on the interstitial sublattice of ε-Fe2N1-z.

The interactions with the iron sublattice are incorporated through effective interaction

parameters and two γ' + ε miscibility gaps (i.e. four phase boundaries) are described

by adopting a single set of pair interaction energy parameters. The CVM calculations

demonstrate the occurrence of distinct LRO of nitrogen in γ'-Fe4N1-x as well as the

occurrence of both SRO and LRO of nitrogen in ε-Fe2N1-z. The LRO occurring in

Summary

151

ε-Fe2N1-z at nitrogen contents close to 25 at.% is in agreement with the type of

ordering for Fe3N (configuration B) observed in diffraction experiments. Again, the

occurrences of the surroundings of the iron atoms, as predicted by the CVM

calculations, are in good agreement with experimental data obtained by Mössbauer

spectrometry. The theoretical values obtained for the lattice parameters of the

ε-Fe2N1-z phase tend to be up to a mere 1% smaller than the few available

experimental data in the composition range studied.

In Chapter 5, the CVM approximation is extended to explicitly include the

atom distributions over the sites of both the substitutional host sublattice and the

interstitial sublattice. By including sites of both sublattices into the basic cluster of the

cube approximation, the description of order-disorder transitions on the metal and

interstitial sublattices of hypothetical fcc alloys with interstitial species is realized.

Analysis of the calculated cube distribution variables demonstrates that phase

transitions on the metal and interstitial sublattices are coupled: a phenomenon that has

been experimentally observed for certain metal-hydrogen systems, but that has not

been explicitly incorporated into the thermodynamic description of phases by CVM

until recently. Not only the induction of SRO on both the metal and the interstitial

sublattices following the introduction of a second metal atom type on the metal

sublattice is shown, preferential grouping of metal atoms with specific interstitials is

observed as well.

Finally, a survey of the low-temperature region (373 K to 473 K) of the Fe-N

phase diagram is performed by analyzing annealed homogeneous specimens of about

10 to 26 at.% nitrogen by X-ray diffraction. A new method to prepare homogeneous

Fe-N alloys by nitriding highly porous specimens of sintered pure iron powder is

developed for this purpose, since a large surface area/volume ratio has proven to be an

essential requirement for the preparation of alloys with a high nitrogen content. Two

possible interpretations for the formation of the α"-Fe16N2 phase at low temperatures

are formulated: either the α" phase develops as a precursor for α, not only from α

(ferrite) and α' (martensite), but also upon transformation of ε (<20 at.% nitrogen) and

γ', or the α" phase develops as an equilibrium phase below about 440 K. The

decomposition process appears to be governed by the difficult nucleation of γ' and the

slow diffusion of the nitrogen atoms, causing the formation of intermediate phases.

The ideas and methods explored in this thesis combined with ab initio

calculations create an environment which supports design and development of

152

interstitial alloys exhibiting specific desirable properties and stabilities in a finite

range of temperatures for practical industrial applications. Currently, this approach is

used to improve the efficiency and life span of hydrogen gas separation membranes.

153

SAMENVATTING

Het nauwkeurig beschrijven van de thermodynamica van interstitiële vaste

oplossingen is om verschillende redenen een uitdaging. De introductie van kleine

atomen (zoals waterstof, boor, koolstof en stikstof) in de interstitiële holtes van een

(metaal)rooster heeft vervorming van de structuur en volumeveranderingen tot

gevolg. Bij de beschrijving moet rekening gehouden worden met de interacties tussen

de (metaal) atomen in de matrix en de atomen die de interstitiële plaatsen bezetten.

Daarbij is het optreden van korte-afstands orde (KO) en lange-afstands orde (LO) op

zowel het subrooster van de metaalatomen als op het subrooster van de interstitiële

atomen van belang. Het feit dat directe waarneming van interstitiële fasen moeilijk is

doordat de gevormde precipitaten erg klein zijn en dat nitrides en carbides vaak

metastabiel zijn, maakt duidelijk dat het voorspellen van thermodynamische

grootheden noodzakelijke informatie voor het optimaliseren van processen en

materiaaleigenschappen kan verschaffen die niet op andere wijze kan worden

verkregen.

Het eenvoudigste thermodynamische model dat in staat is om lange-afstands

orde in interstitiële vaste oplossingen te beschrijven is de Gorski-Bragg-Williams

(GBW) benadering. In die benadering wordt het voorkomen van KO echter niet

expliciet meegenomen. In dit proefschrift is een model ontwikkelt voor interstitiële

vaste oplossingen gebaseerd op de Cluster Variatie Methode (CVM), dat zowel KO

als LO beschrijft. Het model is vervolgens toegepast op evenwichten tussen ijzer-

stikstof fasen.

Fe-N fasen kunnen worden opgevat als bestaand uit een metaalsubrooster,

volledig bezet met ijzeratomen, en een interstitieel subrooster, dat gevormd wordt

door stikstofatomen en vacatures. Met andere woorden: een dergelijk systeem kan

worden benaderd als een binaire oplossing van stikstofatomen en vacatures in het

gemiddeld veld (“mean field”) van het metaalrooster, waarbij de interacties tussen de

individuele atomen vervangen zijn door een effectieve interactie. Dit concept wordt

uitgewerkt in hoofdstuk 2, dat de LO beschrijft van stikstofatomen op het interstitiële

subrooster van octaëderholtes in het ijzersubrooster. Zowel het fitten van de stikstof-

absorptie- isothermen aan de experimentele absorptiedata en de waarschijnlijkheden

van de verschillende omringingen van de ijzeratomen verkregen met

Mössbauerspectroscopie wijzen er op dat ordening van stikstof in ε-Fe2N1-x

154

plaatsvindt volgens twee configuraties: Fe2N (configuratie A) en Fe3N (configuratie

B). Om de aanwezigheid van deze configuraties te onderzoeken is gekeken naar de

aanwezigheid van karakteristieke superstructuurreflecties. De waargenomen

superstructuurreflecties zijn consistent met de ordening van de stikstofatomen volgens

configuratie B. De 001 en 301 superstructuurreflecties specifiek voor

configuratie A worden niet waargenomen, in tegenstelling tot wat op basis van de

Mössbauer data verwacht werd. Berekeningen van de structuurfactor geven aan dat

een mogelijke verklaring hiervoor is dat lokale vervorming van het ijzerrooster, door

de (geordende) bezetting van de interstitiële holten, zodanig is dat de intensiteit van

de toch al zwakke reflecties van configuratie A sterk afneemt.

In hoofdstuk 3 wordt de thermodynamische beschrijving uitgebreid met KO

door het toepassen van de Cluster Variatie Methode (CVM). Aangezien de CVM

oorspronkelijk is ontwikkeld voor substitutionele systemen, is de methode aangepast

voor interstitiële systemen door het implementeren van volumeveranderingen als

functie van de samenstelling en het invoeren van effectieve interacties opdat de

beperkte beschikbare experimentele data voldoende zijn voor een schatting van de

materiaalparameters. Hoofdstuk 3 beschrijft de berekening van het

γ-Fe[N] / γ'-Fe4N1-x fase-evenwicht met behulp van de CVM tetraëderbenadering. De

effectieve interacties, die hoofdzakelijk de vervorming van het rooster ten gevolge

van de mispassing van de stikstofatomen in de interstitiële octaëderholtes beschrijven,

worden weergegeven door een Lennard-Jones potentiaal. De berekeningen tonen

duidelijk aan dat KO optreedt in γ-Fe[N] en LO in γ'-Fe4N1-x. Voor het eerst is een

kwantitatieve analyse van berekende atoomverdelingen rechtstreeks vergeleken met

experimenteel verkregen atoomomringingen. De vergelijking laat zien dat er een

goede overeenkomst is tussen de CVM voorspellingen en de fracties ijzeratomen met

specifieke omringing door stikstof in γ-Fe[N].

Het γ'-Fe4N1-x / ε-Fe2N1-z fase-evenwicht wordt berekend in hoofdstuk 4,

waarbij de CVM prismabenadering wordt geïntroduceerd die de beschrijving van de

ordening van stikstofatomen op het interstitiële subrooster van ε-Fe2N1-z mogelijk

maakt. De interactie met het ijzersubrooster is wederom meegenomen door middel

van effectieve interactieparameters. Vervolgens worden twee γ' + ε ontmenggebieden

(dat wil zeggen: 4 fasegrenzen) beschreven met een enkele set paarinteractie-

parameters. De CVM berekeningen tonen onmiskenbaar aan dat zowel KO en LO in

ε-Fe2N1-z voorkomen. De LO in ε-Fe2N1-z bij stikstofgehaltes rond de 25 at.% komt

Samenvatting

155

overeen met het type ordening voor Fe3N (configuratie B) zoals waargenomen in

diffractie-experimenten. Opnieuw komen de omringingen van de ijzer atomen, zoals

voorspeld door de CVM berekeningen, goed overeen met experimentele data

verkregen via Mössbauerspectroscopie. De berekende waarden voor de

roosterparameters van ε-Fe2N1-z zijn slechts 1% kleiner dan de weinige beschikbare

experimentele data in het bestudeerde samenstellinggebied.

In hoofdstuk 5 wordt de CVM benadering zodanig uitgebreid dat de verdeling

van de atomen over zowel het substitutionele metaalsubrooster als het interstitiële

subrooster expliciet worden meegenomen in de beschrijving. Door beide subroosters

te integreren in het basiscluster van de CVM wordt het beschrijven van orde-wanorde

overgangen op het metaalsubrooster en de interstitiële subroosters mogelijk.

Berekening van de fase-evenwichten van hypothetische interstitiële legeringen en de

analyse van de berekende atoomverdelingen laten zien dat fase-overgangen op het

metaalsubrooster en het interstitiële subrooster van een fase zijn gekoppeld. Dit

fenomeen is experimenteel waargenomen voor bepaalde metaal-waterstof systemen,

maar werd tot voor kort nooit expliciet meegenomen in de thermodynamische

beschrijving van fasen met de CVM. Aangetoond wordt dat de aanwezigheid van een

tweede type metaalatoom op het metaalsubrooster KO teweeg kan brengen op zowel

het metaalsubrooster als het interstitiële subrooster. Daarnaast wordt voorkeur voor

groepering van metaalatomen met specifieke interstitiëlen waargenomen.

Tenslotte wordt het lage-temperatuurgebied (373-473 K) van het Fe-N

fasediagram experimenteel onderzocht, door homogene monsters met een

stikstofgehalte van 10 tot 26 at% te analyseren met röntgendiffractie. Voor het

prepareren van homogene ijzer-stikstoflegeringen is een nieuwe methode ontwikkeld,

waarbij een grote oppervlakte/volume verhouding (essentieel voor het produceren van

legeringen met een hoog stikstofgehalte) wordt bereikt door zeer poreuze monsters

van gesinterd zuiver ijzerpoeder te nitreren. Twee mogelijke interpretaties voor de

vorming van de α"-Fe16N2 fase worden geformuleerd: 1) de α" fase wordt gevormd

als voorloper voor α, niet alleen in het geval van het ontmengen van α' (martensiet),

maar ook bij transformatie van ε (<20 at.% stikstof) en γ', of 2) de α" fase vormt als

een evenwichtsfase bij temperaturen lager dan 440 K. Het ontmengproces lijkt te

worden bepaald door de langzame nucleatie van γ' en de trage diffusie van de stikstof

atomen, hetgeen resulteert in de vorming van intermediaire fasen.

156

De ideeën en methoden die in dit proefschrift zijn ontwikkeld maken het

mogelijk om interstitiële legeringen met specifieke eigenschappen en stabiliteit in een

gekozen temperatuur gebied te ontwerpen en te fabriceren. Momenteel wordt deze

benadering gebruikt voor het verbeteren van de efficiëntie en de levensduur van

membranen voor scheiding van waterstofgas.

157

LIST OF PUBLICATIONS

M.I. Pekelharing, A.J. Böttger, M.P. Steenvoorden, A.M. van der Kraan and E.J. Mittemeijer:

“Application of the cluster variation method to ordering in an interstitial solid solution: calculation of

the ε-Fe2N1-z / γ'-Fe4N1-x equilibrium”, Phil. Mag. A, 2003, vol. 83(15), pp. 1775-1796 – Chapter 4.

M.I. Pekelharing, A.J. Böttger, M.A.J. Somers and E.J. Mittemeijer: “Application of the cluster

variation method to ordering in an interstitial solid solution: the γ-Fe[N] / γ'-Fe4N1-x equilibrium”, Met.

Mat. Trans. A, 1999, vol. 30A, pp 1945-1953 – Chapter 3.

M.I. Pekelharing, A.J.Böttger, M.A.J. Somers, M.P. Steenvoorden, A.M. van der Kraan and E.J.

Mittemeijer: “Modeling thermodynamics of Fe-N phases: Characterization of ε-Fe2N1-z”, 5th

international conference on High Nitrogen Steels, 24th-28th May 1998, Material Science Forum, 1999,

vol. 318-320, p. 115-120 – Chapter 2.

M.I. Pekelharing, A.J. Böttger, S.S. Malinov, C.C. Tang, E.J. Mittemeijer, “Characterization of the

ordering of N atoms in ε-Fe2N1-z, analysis of superstructure-reflections using High Resolution Powder

Diffraction”, annual report 1998, SRS Daresbury Laboratory – Chapter 2.

M.I. Pekelharing and A.J. Böttger, “Application of the CVM Cube Approximation to FCC Interstitial

Alloys”, 2007, submitted for publication in Phys. Rev. – Chapter 5.

S. Malinov, A.J. Böttger, E.J. Mittemeijer, M.I. Pekelharing, M.A.J. Somers: “Phase Transformations

and Phase Equilibria in the Fe-N System at Temperatures below 573 K”, Met. Mat. Trans. A, 2001, vol.

32A(1), pp. 59-73 – Chapter 6.

159

DANKWOORD

Na enige omwegen is dan toch het moment gekomen om alle mensen te bedanken die

mij de mogelijkheid hebben geboden om mijn promotie onderzoek te verrichten.

Om te beginnen mijn promoter Barend Thijsse, voor zijn directe, enthousiaste

en menselijke werkwijze en het vertrouwen dat hij in mij gesteld heeft. Amarante

Böttger, mijn co-promotor, voor haar enorme flexibiliteit en inzet, haar kritische

benadering, hoge verwachtingen, en de zeer plezierige samenwerking. Eric

Mittemeijer, voor de goede onderzoeksfaciliteiten, het plaatsen van de juiste

kanttekeningen, en zijn beschikbaarheid en scherpe inzicht. Marcel Somers, voor zijn

gedrevenheid en betrokkenheid bij het onderzoek, het stellen van de juiste vragen, en

zijn bijdrage aan het verzette werk.

Daarnaast Nico Geerlofs voor de technische ondersteuning en de bereidheid

tot het verrichten van noodzakelijke aanpassingen aan de nitreerovens. Niek van der

Pers voor de assistentie bij en het verrichten van röntgendiffractie metingen. Adri van

der Kraan en Michel Steenvoorden van het Interfacultair Reactor Instituut voor de

Mössbauer metingen en de assistentie met het fitten en de interpretatie van de

verkregen spectra. C. Tang voor de assistentie bij de synchrotron metingen in

Daresbury. Diana Nanu voor het gebruik van de CVM kubus programmatuur.

Mijn kamergenoten Peter en Valentina: bedankt voor de afleiding, de steun, en

het nuttige en nodige commentaar. Daarnaast de andere promovendi, afstudeerders,

postdocs en stafleden in de vakgroep in de periode dat ik in Delft werkzaam was:

Camiel, Ton, Rinze, Regina, Jan-Dirk, Ludmila, Roland, Lars, Savko, Slobodan,

Jouk, Rob Delhez, Mijnheer van Lent, Jan Helmig, Ton de Graaf, Wim Sloof, Kees

Borsboom, Staf de Keijser, Anke Kerklaan-Koene, en een ieder die ik over het hoofd

heb gezien. Bedankt voor de aangename sfeer en de prettige samenwerking.

Johan, Jan en Sandra voor hun grenzeloze gastvrijheid bij gebrek aan

onderdak in Nederland en praktische en morele steun. Daniela Nicastro for her love

for research and refusal to give up. Scott Brown for network use and child care.

Tenslotte wil ik Govert en Adam bedanken: onderaan de lijst, maar jullie bijdrage aan

het proces is zeker niet de minste geweest.

160

CURRICULUM VITAE

7 augustus 1971 geboren te Zevenaar

1983 – 1988 VWO, Florens Radewijn College te Raalte

1988 – 1989 VWO, Geert Groote College te Deventer

1989 – 1990 Propaedeuse Scheikunde, Universiteit Utrecht

1990 – 1995 Doctoraal Geochemie, Universiteit Utrecht

1995 – 1999 Onderzoeker in Opleiding (FOM)

Sectie Fysische Chemie van de Vaste Stof, Laboratorium voor

Materiaalkunde, Technische Universiteit Delft

2007 – Group Structure & Change, Department of Materials Science &

Engineering, Delft University of Technology

1999 – 2001 Junior beleidsmedewerker NWO, Den Haag

2003 – Learning Center staff, Front Range Community College,

Longmont, Colorado (USA)

2007 – Nursing staff & MDS coordinator, Longmont United Hospital,

Longmont, Colorado (USA)

Documents

APPLICATION OF THE CLUSTER VARIATION METHOD TO INTERSTITIAL