APPLICATION OF THE CLUSTER VARIATION
METHOD TO INTERSTITIAL SOLID SOLUTIONS
APPLICATION OF THE CLUSTER VARIATION
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
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)'.
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
3.2.4. 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
4.3.1. CONFIGURATIONAL ENTROPY... 62
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
5.2.1. INTERNAL ENERGY... 88
5.2.2. CONFIGURATIONAL ENTROPY... 90
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
1
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
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].
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.
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'-Fe
4N1-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].
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
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.
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.
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 473K. 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.
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2
MODELING THERMODYNAMICS OF Fe-N PHASES:
CHARACTERIZATION OF ε-Fe
2N
1-zABSTRACT
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
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
2.2.
EXPERIMENTAL PROCEDURES
2.2.1. SPECIMEN PREPARATIONA 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.
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
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 2 2 2 2 2 2 1 1 1 1 1 1 2 1 2 1 2 1 0 0 1 ln 1 ln 6 1 6 1 6 2 C A k k k k k C B C A B A C B C A B A N P C C B B A A N C N N Fe y y y y RT y y y y y y y y y y y y y W y y y y y y y W G y G Gε (1) where 0 FeG and G are the hypothetical Gibbs energies of pure Fe with an hcp lattice N0 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]:
(
)
(
)
RT W y y RT W y y y r r A C B C P A A N N 2 1 1 1 1 0 ln 1 1 2 1 ln + − + − − − = (2)where rN is the nitriding potential of the NH3/H2 gas mixture and rN0 is defined as 0 0 0 0 2 3 2 3 lnrN GN GNH GH RT = − + (3) with 0 3 NH
G and GH02 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
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 A k N k N y y (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.
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.
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
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
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.
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.
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
3
APPLICATION OF THE
CLUSTER VARIATION METHOD TO
ORDERING IN AN INTERSTITIAL SOLUTION;
THE γ-Fe[N] / γ'-Fe
4N
1-xEQUILIBRIUM
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-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
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 SOLUTIONBinary 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
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-xOrder-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.
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.
In the A1 structure, the probability of finding N or V on a sublattice site of type α, Xiα, is equal to that of sublattice sites of type β, γ, and δ. Thus the symmetry of the A1 structure can be described by α β γ δ
l k j
i X X X
X = = = (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 X1β (= X1γ = X1δ). The probability of finding a vacancy V at an N-type sublattice site is X2α and at a V-type sublattice site is Xβ2 (= X2γ = X2δ). 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 Xiα ≠ Xβj = Xγk = Xδl
(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 ijkl ijkl Z U 2N εαβγδ αβγδ (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:
( )
r ij( )
r ik( )
r il( )
r jk( )
r jl( )
r kl( )
r ijkl γδ βδ βγ αδ αγ αβ αβγδ ε ε ε ε ε ε ε = + + + + + (2) where ij( )
r 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
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 ij ijkl X L X L X L X L Y L Y L Y L Y L Y L Y L Z L S δ γ β α γδ βδ βγ αδ αγ αβ αβγδ 4 5 2 B Nk (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. Xiα and Yijαβ can be calculated from the tetrahedron distribution variables Zijklαβγδ: Xiα = Zijklαβγδ jkl
∑
, ... (4a) Yijαβ = Zαβγδijkl kl∑
, Yikαγ = Zijklαβγδ, ... 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'
3.2.3. CALCULATION OF PHASE EQUILIBRIA
For each phase a thermodynamic function Ω, referred to as the grand potential function, is defined [25]
(
)
∑
∑
= = ∗ ∗ ≡ − + − = − Ω 2 1 * 2 1 * 2 1, , , n n n n n nx G x p TS U T p µ µ V µ µ (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:
(
α β γ δ)
/4 n n n n n X X X X x = + + + (6)Furthermore, µ∗n is an effective chemical potential (Appendix A), defined as
∑
− = n n n n c µ µ µ* 1 (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 Zijklαβγδ obey, 1 =
∑
ijkl ijkl Zαβγδ (8)is accounted for by introduction of the Lagrange multiplier λ. Then, minimization of the grand potential function with respect to Zijklαβγδ[20] yields
8 / 5 2 / 1 ijkl 8 exp kT -exp 2kT exp − ∗ ∗ ∗ ∗ + + + = ijkl ijkl l k j i ijkl Y X kT Zαβγδ λ ε µ µ µ µ
with Yijkl ≡YijαβYikαγYilαδYjkβγYjlβδYklγδ and α β γ δ l k j i ijkl X X X X X ≡ (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 ij d d Y w p 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 Ω
γ
= minimum of Ωγ ' (10)
µ∗,γN = µN∗,γ '
µ∗,γV = 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.
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]:
( )
− = 4 0 12 8 0 12 0 12 12 2 r r r r r ε εαβ (11)where ε12o is the pair interaction energy in the reference state and parameter r12o corresponds to the interatomic distance for which
ε
12αβ( )
r has a minimum value, equalto - o
12
ε .
3.2.3.1.2. LENNARD-JONES PARAMETERS
In general, the parameters ε11o and r11o can be estimated from the enthalpy of formation o
fH1
∆ per atom and the lattice constant (a1) of the pure element 1,
respectively: ε11o and r11o can be written as
o f o H w 1 11 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 εijαβ 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
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 , εNVo , rNNo and rNVo in a similar way. Therefore, to model the equilibrium between γ-Fe[N] and γ'-Fe4N1-x, εNNo , εNVo , rNNo and rNVo 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
Table 3.1. Lennard-Jones parameters used in the CVM phase-boundary calculations
Parameters Normalised parameters Pair interaction εo(kJ/mol) ro(nm) ε o /εVVo ro /rVVo 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
3.3.
DISCUSSION
3.3.1.
LATTICE PARAMETERS OF THE FCC Fe-N PHASESThe 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.
