Atomistic Simulations and Experimental Studies of Transition Metal Systems

Involving Carbon and Nitrogen

Jiaying Xie

Doctoral Dissertation

Department of Materials Science and Engineering Division of Materials Process Science

Royal Institute of Technology SE-100 44 Stockholm

Sweden Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm, framlägges för offentlig granskning för avläggande av teknologie

doktorsexamen, fredagen den 15 december 2006, kl. 10.00 i Salongen, KTHB, Osquars Backe 31, KTH, Stockholm.

ISRN KTH/MSE--06/66--SE+THMETU/AVH

ISBN: 91-7178-487-X

ABSTRACT

The present work was initiated to investigate the stability, structural and thermodynamic properties of transition metal carbides, nitrides and carbo-nitrides by atomistic simulations and experimentations. The interatomic pair potentials of Cr-Cr, Mn-Mn, Fe-Fe, C-C, Cr-C, Mn-C, Fe-C, Cr-Fe, Cr-N and Mn-N were inverted by the lattice inversion method and ab initio cohesive energies, and then employed to investigate the properties of Cr-, Mn- and Fe-carbides by atomistic simulations in this work. For the binary M7C3 carbide, the structural properties of M7C3 (M = Cr, Mn, Fe) were investigated by atomistic simulations. The results show that the stable structure for these compounds is hexagonal structure with P63mc space group. The cohesive energy of M7C3 calculated in this work indicates that the stability of carbides decreases with the increasing in metal atomic number. Further, the vibrational entropy of Cr7C3 was calculated at different temperatures and compared with the entropy obtained by experimentations. The comparison demonstrates that the main contribution to the entropy is made by the vibrational entropy. For the binary τ-carbides, the structural properties of Cr23C6 and Mn23C6, as well as the vibrational entropy of Cr23C6 were computed. Further, the site preference of ternary element Fe among 4a, 8c, 32f and 48h symmetry sites in Cr23-xFexC6 was studied. It has been seen that Fe atoms would firstly occupy 4a sites and then 8c sites. The lattice constant and stability of Cr23-xFexC6 were also computed with different Fe content. In order to understand the relative stability of the transition metal carbides and nitrides, the standard formation Gibbs energies of carbides and nitrides for Cr, Mn and Fe were compared. The order of carbon and nitrogen affinities for Cr, Mn and Fe was further clarified by the comparison of the interatomic pair potentials among Cr-C, Mn-C, Fe-C, Cr-N and Mn-N. It was found that Cr-N interaction was very strong in comparison with other binary interactions above and consequently, nitrogen addition would lead to a strong decrease in the thermodynamic activity of chromium in Cr-containing alloys. This was confirmed by the investigations of thermodynamic activities of Cr in the Fe-Cr-N and Fe-Cr-C-N alloys. The activities were measured in the temperature range 973-1173 K by solid-state galvanic cell method involving CaF2 solid electrolyte under the purified N2 gas. In addition, the analysis of nitrogen content and phase relationships in the Fe-Cr-N and Fe-Cr-C-N alloys equilibrated at 1173 K were carried out by inert-gas fusion thermal conductivity method, X-ray diffraction and scanning electron microscopy technique. The experimental results show that the solubility of nitrogen in the alloys decreases with the decreasing chromium content, as well as the increasing temperature. The addition of nitrogen to the alloys was found to have a strong negative impact on the Cr activity in Fe-Cr-N and Fe-Cr-C-N systems. Keywords: Transition metal carbides; Transition metal nitrides; Transition metal carbo-nitrides; Lattice inversion method; Interatomic potential; Atomistic simulation; Thermodynamic activity; Galvanic cell; Solid electrolyte; Phase equilibrium

i

ACKNOWLEDGEMENTS

A man can only attain knowledge with the help of those who possess it. This must be understood from the very beginning. One must learn from him who knows. George Gordjieft In search of the Miraculous

The work presented here is the result of years of dedicated research. Even accounting for this effort, it would not have been possible, or taken the form it now has, if not with the help and support of many people and organizations. This is time and place where I give them their due acknowledgements.

First and foremost, my heartfelt thanks go to my supervisor, Prof. Seshadri Seetharaman and Prof. Nanxian Chen. Prof. Seetharaman, thank you for employing me as a Ph.D. student at KTH, guiding my quest for scientific knowledge and encouraging me to appreciate life in itself. Prof. Chen, thank you for introducing me into an interesting and promising field and continuous supporting during these years.

My thanks also go to my co-supervisor, Dr. Lidong Teng. Thank you for sharing your knowledge of materials science, and creating a friendly atmosphere to work in. Special thanks to Prof. Jiang Shen who helps me to grasp calculation method and encourages me constantly.

At the same time, many thanks to Prof. Wenchao Li, Prof. Benfu Hu, Prof. Mats Hillert, Dr. Anna Delin and Dr. Ganaphy Vaitheeswaran, for their valuable discussions; Dr. Mselly Nzotta, UDDEHOLM TOOLING AB, for the kind help with the carbon and nitrogen analysis of the alloys; Mr. Peter Kling for the help with the experimental equipments as well as all nice coffee breaks; Mrs. Tong He for her suggestion and encouragement.

Financial supports for this project from the Royal Institute of Technology and the Foundation for Applied Thermodynamics, and Conference grants from Walfrid Peterssons Minnesfond are gratefully acknowledged.

On a more personal lever, it goes without saying that I must acknowledge my husband Tiejun Zhao, thanks for all your support, encouragement and taking care of everything at home.

Finally, I would like to thank all my friends both in China and Sweden for their help and encouragements.

Jiaying Xie (解家英)

Stockholm, October 2006

ii

SUPPLEMENTS

This thesis is based on the following papers:

Supplement 1: “Atomistic study on the structure and thermodynamic properties of Cr7C3, Mn7C3, Fe7C3 ”

Jiaying Xie, N.X. Chen, J. Sheng, L.D. Teng and S. Seetharaman

Acta Materialia, 53 (2005) 2727-2732.

Supplement 2: “Atomistic simulation on the structural properties and phase stability for Cr23C6 and Mn23C6 ”

Jiaying Xie, L.D. Teng, N.X. Chen and S. Seetharaman

Journal of Alloys and Compounds, 420 (2006) 269-272.

Supplement 3: “Atomistic study on the site preference and thermodynamic properties for Cr23-xFexC6 ”

Jiaying Xie, N.X. Chen, L.D. Teng and S. Seetharaman

Acta Materialia, 53 (2005) 5305-5312.

Supplement 4: “Thermodynamic investigations of the Cr-Fe-N system with solid-state galvanic cell method”

Jiaying Xie, L.D. Teng, N.X. Chen and S. Seetharaman

Sent to Metallurgical and Materials Transaction A for publication.

Supplement 5: “Thermodynamic studies of the Fe-Cr-C-N system by EMF measurements”

Jiaying Xie, L.D. Teng, N.X. Chen and S. Seetharaman

Sent to Metallurgical and Materials Transaction B for publication.

iii

iv

The author’s contribution to the different supplements of this thesis:

Supplement 1: Literature survey, calculation and major part of the writing.

Supplement 2: Literature survey, calculation and major part of the writing.

Supplement 3: Literature survey, calculation and major part of the writing.

Supplement 4: Literature survey, experimental work, analysis and evaluation of the experimental results, and major part of the writing.

Supplement 5: Literature survey, thermodynamic measurements, analysis and evaluation of experimental results, and part of the writing.

Parts of the work have also been presented at the following conferences:

I. “Stability and thermodynamic investigations for the Cr23-xTxC6 (T=Mo, W) system”

Jiaying Xie, J. Shen, L.D. Teng, N.X. Chen and S. Seetharaman

TOFA 2006, Discussion Meeting on Thermodynamics of Alloys, Beijing, China, June 18-23, 2006.

II. “Thermodynamic investigations of Cr-Fe system containing carbon and nitrogen”

L.D. Teng, Jiaying Xie and S. Seetharaman

TOFA 2006, Discussion Meeting on Thermodynamics of Alloys, Beijing, China, June 18-23, 2006.

v

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Other contributions:

Appendix A: “Solubility of Co in Cr7C3--ab initio studies and experimental verification”

T. Sterneland, Jiaying Xie, N.X. Chen and S. Seetharaman

Scripta Materialia, 54 (2006) 1491-1495.

Appendix B: “Site preference and mechanical properties of Cr23-xTxC6 and Fe21T2C6 (T=Mo,W)”

Jiaying Xie, J. Shen, N.X. Chen and S. Seetharaman

Acta Materialia, 54 (2006) 4653-4658.

vii

CONTENTS

ABSTRACT…………………………………….............................................. i

ACKNOWLEDGEMENTS……………………………................................ ii SUPPLEMENTS………………………………………………………......... iii 1. INTRODUCTION…………………………………………………......... 1 2. CALCULATION METHODS……………………………………......... 3

2.1. Extraction of interatomic potential…………………...…………….. 3

2.2. Calculation method for thermodynamic properties…………............ 6 3. EXPERIMENTAL METHODS……………………………………...... 9

3.1. ElectroMotive Force (EMF) measurement……………………........ 9 3.2. Cell arrangement and reaction …………………………………….. 11

3.3. Preparation of alloys and electrodes………………………...……... 12 3.4. Gas cleaning system.…………………………………….…………. 14

3.5. Experimental apparatus and procedure…………………….…......... 16

4. RESULTS AND DISCUSSIONS ……………………………….......... 19

4.1. Studies of Cr7C3, Mn7C3 and Fe7C3 by atomistic simulations……... 19

4.2. Investigations of binary τ-carbides by atomistic simulations……… 21

4.3. Investigations of ternary τ-carbides Cr23-xFexC6…………………… 22

4.4. The stability of carbides and nitrides ……………………………… 24

4.5. Phase analysis of the Fe-Cr-N alloys.……………………………… 25

4.6. The activities of chromium in the Fe-Cr-N system………………… 28

4.7. Phase analysis of the Fe-Cr-C-N alloys……………………………. 29

4.8. The activities of chromium in the Fe-Cr-C-N system……………… 30

5. CONCLUSIONS…………..……………………………………………... 33

6. FUTURE WORK........................................................................................ 35

REFERENCES……………………………………………………... 37

1. INTRODUCTION

The transition metal carbides, nitrides and carbo-nitrides play an important role in the stainless steels and other alloy steels [1-3]. The knowledge of the properties, viz. structural, mechanical and thermodynamic properties, etc, of these compounds is essential for the understanding of the properties of steels and the prediction of their chemical and physical behaviors in various environments. Many theoretical and experimental investigations have been made to get some information of these kinds of compounds [4-9].

In the recent years, with the remarkable improvement in numerical algorithms and computer hardware, theoretical materials modeling has become the fastest growing part and a very important research tool within modern materials science. Theoretical materials modeling is most often used in collaboration with experimental research to support and explain experimental observations [10-13]. Presently, quantum mechanical (QM) methods based on the predictions of electronic structure have become prevailing and the most accurate modeling methods [14-16]. However, QM methods can be exceedingly computationally expensive, rendering the study of some complex systems impractical. As an alternative method, atomistic simulation offers a more approximation, but often more practical approach [17-19]. It takes the atom as its fundamental unit, discarding the electronic detail, and assumes that these atoms can be described by relatively simple equations.

The obtaining for accurate and realistic potentials constitutes a challenging problem for atomistic simulation. In the mid-1990s, Chen et al. [20] developed an inversion method to obtain the interatomic pair potential, with which the interatomic pair potential could be extracted directly from the ab initio calculations. Therefore, it seems compelling to construct a bridge between two research lines, making use of the large amount of information that can be obtained by ab initio methods to construct potentials for computations on a much larger scale.

On the other hand, experimental study is a significant and basic method for materials investigation at high temperature all the time. The thermodynamic properties of transition metal systems including carbon and nitrogen have been extensively studied by experimentations. It is generally known that solid-state galvanic cell measurement with CaF2 single crystal as the solid electrolyte is a reliable method for thermodynamic studies [21-25]. CaF2 is an ionic conductor (tion

1

>0.99) over a wide range of fluorine partial pressure and temperature [26]. The experimental errors of this method are mainly attributed to the control of experimental atmosphere. A gas purification train has been developed and successfully used in the thermodynamic investigations for some transition metal systems containing carbon and nitrogen in the present laboratory [23-25].

The present work is part of a systematic program at the present laboratory which aims at investigating the properties of carbides, nitrides, carbo-nitrides by the collaboration of atomistic simulation, ab initio calculation, CALPHAD and experimentation.

In this work, due to the structural complexity of some transition metal carbides, atomistic simulations involving inverted interatomic pair potentials were employed to investigate the stability, structural and thermodynamic properties for Cr-, Mn- and Fe-carbides.

In view of the lack of experimental data on the chromium activities in the Fe-Cr-N and Fe-Cr-C-N systems, the solid-state galvanic cell measurements with CaF2 single crystals as solid electrolytes were performed to investigate the activities of Cr in the Fe-Cr-N and Fe-Cr-C-N alloys.

The structure of thesis is as follows: in chapter 2, the calculation methods used to obtain the interatomic potentials and the thermodynamic properties are introduced. In chapter 3, some information of the experimentations (EMF measurement, preparation of alloys and electrodes, gas cleaning system and cell reaction and arrangement) will be described. In chapter 4, the results will be presented and discussed. Conclusions will be shown in chapter 5. Finally in chapter 6, suggestions for future work will be given.

2

2. CALCULATION METHODS

2.1 Extraction of interatomic pair potential

While the ab initio methods have been vigorously developed for a long time [27-30], interatomic potentials continue to constitute the advantageous way to perform computations on the systems with a complex structure, or very large size [31-36]. With the advent of massively parallel machines and the proper computer code, simulations on the mesoscopic scale appear feasible, allowing one to address a whole new range of problems in the physics of defects, surfaces, clusters, liquids and glasses.

For atomistic simulation, the key issue is the determination of interatomic potential. It is now well recognized [37] that fairly elaborate analytic expressions are necessary for a realistic description of most materials under different conditions (geometries, structures, thermodynamic phases). A typical potential is thus constituted by a number of functions combined in a complex way, and often nested one into another. Unfortunately, such powerful forms can make the task of fitting a potential to a given material quite formidable and cumbersome.

In the mid-1990s, Chen et al. developed a concise inversion method based on the modified Möbius inversion transform in number theory to obtain the interactomic potentials. This method has been applied successfully in many complex systems [38-40]. For our purpose, this method is selected to study the properties of transition metal carbides in the present work.

2.1.1 Lattice inversion method

Here, we take a single element crystal as an example to explain how to use Chen's lattice inversion technique to obtain the interatomic pair potential. In this method, the crystal cohesive energy, , is approximately expressed as sum of pair potentials.

)(xE

))(()(21)( 0

10 xnbnrxE

n

φ∑∞

=

= (1)

Where x is the nearest–neighbor interatomic distance, b is a monotonically

increasing series that represents the distance between the origin on which the reference atom is located and the n-th set of lattice points, is the

)n(0

)(0 nr

3

)1(0coordination number of the n-th neighbor. For example, 1=b corresponds to

the nearest-neighbor distance. The inverse lattice problem is to determine )(xφ from fitting curve which can be obtained from the ab initio calculations. In general, is not a multiplicative semi-group. The trick for solving this

problem is to work with the multiplicative closure { of the set of shell radii , in other words, { has become a multiplicative semi-group containing

not only all the elements of { but also all products of them. Simultaneously

the set of occupation numbers { is then extended to { , which is

assigned zero to all the elements {

)(xE)}n

nb)}nb

0 nb

0 nr )}(nr

0 nbnb

({ 0b

)}(

)}({ 0 nb (

)}(

)}(

)}({)}( ∉ , and called virtual lattice point

shell.

r (2) )}({ )( if 0)}({ (n) if ) )]([(

)(0

01

00

⎩⎨⎧

∉∈

=−

nbnbnbbnbbr

n

Here refers to the inverse of the arithmetic function. With these extensions,

the problem can be expressed as:

10−b

∑∑∞

=

∞

=

==1

01

0 ))(()(21))(()(

21)(

nn

xnbnrxnbnrxE (3) φφ

With the multiplicative relation that for any m and n, there must be a k such that:

b )()()( nbmbk = (4)

Thus the pair potential, )(xφ , can be written as:

(5) ∑∞

=

=1

))(()(2)(n

xnbEnIxφ

where satisfies the recursive relation: )(nI

1

)()(

1

)()()( k

kbnb nbkbbrnI δ=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡∑ − (6)

I(n) is related only to crystal geometrical structure, not to concrete element category. Thus the interatomic pair potential can be obtained from the known cohesive energy function.

4

2.1.2 Acquisition of effective potentials in carbides

AφFor a simple binary compound AB, three cohesive energy curves are needed to extract the pair potentials, , and . Due to the structural complexity of carbides,

)(xAA−φ )(xBB−φ )(xB−

the ab initio calculations of the cohesive energies for these compounds are difficult. Hence, a practical method is designed to perform the calculation of cohesive energy curve, in which some simple and virtual structures are involved. The structures, B1, B2, B3 and B10, have been used to obtain the pair potentials in our work. Extensive applications of pair potentials extracted from simple B2 structure have indicated that these potentials are valid to investigate the properties of transition metal carbides with M7C3 or M23C6 structures. The method employed to obtain the potentials from B2 structure is introduced as the following.

In this work, the B2 structure is constructed to obtain the interatomic pair potential between the distinct atoms A and B. The cohesive energy function of one A atom with all the B atoms can be obtained based on the ab initio calculations and following equation:

)])

21()

21()

21[(

34(

)()()()(

222

0,,

2

xkji

xExExExE

kjiBA

SCB

SCA

BABBA

−+−+−=

−−=

∑∞

≠−

−

φ (7)

where x is the nearest-neighbor distance in the B2 structure, represents the

total energy curve for AB with a B2 structure, or is the total

energy function for A or B with a simple cubic structure. Now, automatically becomes the cohesive energy function of one A atom with all the B atoms. Then the

)x(2E BAB

E (E SCB

)(xE BA−

)x

)(xSCA )x

(BA−φ can be extracted from by the use of lattice inversion technique.

)(xB−EA

A−

In order to obtain pair potential of identical atoms, the BCC structure is constructed and considered as B2 structure. Then the )(xAφ can be obtained similarly as the acquirement of potential of distinct atoms above with following equation:

)])

21()

21()

21[(

34(

)()()()(

222

0,,

xkji

xExExExE

kjiAA

SCA

SCA

BCCAAA

−+−+−=

−−=

∑∞

≠−

−

φ (8)

5

In this work, the total energy ab initio calculations (ESOCS 4.0 program provided by Materials Simulation Incorporation) are performed at 0 K on the basis of the augmented spherical-wave (ASW) method [41, 42] within the local density functional theory. A series of total energies E(x) are calculated with various lattice constants at equal intervals of 0.1Å. The obtained pair potentials can be generally described by the Morse function:

( )[ ] [ ]{ })1(2exp21exp)( 000 −−−−−= RxRxDx ααφ (9)

where x is the distance between two atoms, D0, α and R0 are potential parameters. The calculated potential parameters are given in Table 1.

Table 1. Potential parameters acquired by the lattice inversion method

Potential type

D0 (eV) α R0 (Å) Potential

type D0

(eV) α R0 (Å)

C-C 0.278 6.243 3.185 C-Cr 0.779 7.785 2.564 Cr-Cr 0.690 6.942 3.102 C-Mn 0.767 8.069 2.496

Mn-Mn 0.668 7.083 2.979 C-Fe 0.721 7.938 2.490

Fe-Fe 0.624 7.245 2.903 Cr-Fe 0.809 8.474 2.820

With the lattice inversion method, the interatomic pair potentials are directly extracted from ab initio calculations without any experimental data and the prior potential functional forms, which reduce some uncertainties in the derivation of pair potentials. Furthermore, the pair potentials inverted from simple structures can be transferred to some calculations of complex compounds to some extent. On the other hand, due to limitations of pair potentials, it should be pointed out that some properties dependent on many-body potentials may not be well described. And for the defects, vacancies and surfaces, the pair potentials could not give the satisfactory results because the structural local anisotropy deformation cannot be correctly described within the context of pair potential. Consequently, the many-body potentials are planed to be introduced in the future work.

2.2 Calculation method for thermodynamic properties [43]

In the harmonic approximation of the lattice dynamics, the secular equation of the lattice vibration can be written as:

6

( ) 0det 2, =− μαβμαβ δδω kk qD v (10)

ω is the angular frequency and qv is the wave vector. where ( )qD kv

μαβ , is a

dynamical matrix and its elements are written as:

( ) ( )[ ]jiRR

jikkk RRqiMMqDji

v vv v −⋅−Φ= ∑−

− exp)( ,,2

1, μαβμμαβ

μαβ jik ,,

(11)

where Φ is the force constant, Ri

v, jRv are positions in the ith and jth cell, and

, k μ represent different atoms in the same cell. According to the interatomic pair potential, the force constant can be expressed as:

⎪⎪⎩

⎪⎪⎨

⎧

===Φ−Φ−

≠⎥⎦

⎤⎢⎣

⎡−−

=Φ (12)

∑ ∑∑≠ ≠μ

μμαβμμαβ

αβμμβα

μβα

μαβ

δφφφ

k jikji

kjikji

kkk

jik

rRR

RR

RR

RRR

RRR

)r ,( 0R if

0R if )('1)('')('21

.,,,

,,2,3

,, vvvv

where μrrRRRR kjivvvvv

−+−== , krr vv ,μ are the relative positions of the different

atoms in the same cell, )(Rkμφ is the pair potential between the kth and μth atoms,

and 'φ and ''φ are the first and second derivative of φ respectively.

The phonon dispersion )(qvω can be given by Eq. (10). Then the density of states

)(ωg can be acquired from )(qvω . The specific heat and vibrational entropy can be written as:

ωωωω

ω

dge

eTKNkTC (13) TK

TKB

BV B

B

)()1(

)(3)(0

2

2

∫∞

−=

h

hh

TdT

TCTS (14) T

V ′′′

= ∫0

)()(

)(where ωg is the normalized density of states (DOS).

7

8

3. EXPERIMENTAL METHODS

3.1 ElectroMotive Force (EMF) measurement

3.1.1 The principle of EMF measurement

The solid-state galvanic cell consists of two electrodes (working electrode and reference electrode) connected by a solid electrolyte between the individual half-cells. The electric potential difference between the working electrode and reference electrode will be introduced because of the difference of the chemical potential between two electrodes. If the chemical potential at one electrode is known, the chemical potential of another electrode will be acquired by the measured EMF value.

E , is given by Wagner [26] as: The electro-chemical potential of a galvanic cell,

i

a

a ion adtnFRTE ln2

1

⋅= ∫ (15)

Where is the transference number of the conducting ion species of the

electrolyte, the number of electrons that are participating in the electrode reactions, and the activities of the species at different electrodes and F, the Faraday’s constant.

iontn

a 2a1

The Gibbs energy change of the galvanic cell reaction can be calculated by the equation:

nEFG −=Δ (16)

The establishment of cell equilibrium ideally requires that the conduction is purely ionic and by a single species, which is dependent on the temperature and activities of species involved in the electrolyte. The presence of even a small electronic contribution to the conductivity will give rise to internal short circuiting and thereby disturb the thermodynamic equilibrium within the cell. It has been suggested that the ionic transference number t must be larger than 0.99.

Furthermore, Schmalzried has pointed out that the electrolyte should have a conductivity of not less than approximately 10

ion

-6 ohm-1cm-1 to enable a stable potential to be established in a galvanic cell.

Solid electrolytes have been extensively used to measure the thermodynamic or kinetic properties of the materials [21, 22, 44, 45]. Many kinds of solid electrolytes are

9

listed in Table 2. Among these electrolytes, ZrO2 (stabilized), ThO2 and CaF2 are the three most frequently employed solid electrolytes. These electrolytes exhibit the same cubic fluorite crystal structure, and their useful ionic conduction is attributable to the ready creation of lattice defects within this structure. In the present work, the galvanic cell method with CaF2 single crystal as solid electrolyte has been employed to investigate the chromium activities in the Fe-Cr-N and Fe-Cr-C-N systems.

Table 2. The most commonly used solid electrolytes

Type of electrolyte Example

Fluorite-type oxides ThO2, CeO2, ZrO2, HfO2

β-alumina-type oxides Na2O·11Al2O3, Na2O·MgO·5Al2O3

Sliver –Iodide-type materials AgCl, AgBr, AgI, Ag3SI

Sulphides CaS, MgS, CuS, SrS

Fluorides NaF, CaF2, SrF2, BaF2, MgF2

3.1.2 CaF2 solid electrolyte

CaF2, a fluoride-ion conducting electrolyte, is one of the most successfully used solid electrolytes. The simple fluoride-ion conduction mechanism of this material was first established by Ure in 1957 [46], which indicated that the electrical conduction in CaF2 solid electrolyte in the temperature range 963-1193 K was ionic with −F ion carrying almost all of the current. These results were confirmed and the temperature range was further extended to 823-1373 K [22, 47, 48]. Furthermore, CaF2 is applicable over a wide fluorine pressure, , range. The

lower limit of pure ionic conduction with respect to fluoride partial pressure for CaF

2FP

42

×−=

2 can be determined by the following equation [22]:

log (17) TPF /)1034.6(0.9

It has been reported that the use of polycrystalline CaF2 electrolyte could result in unsteady EMF values due to the penetration of the metal fluoride into the electrolyte, and better result has been obtained by using CaF2 single crystal [49]. On the other hand, the vaporization of the fluoride and the softening of the electrolyte usually will produce unreliable results at higher temperature, and the low temperature limit is fixed by the conductivity of CaF2 [50-51]. Hence, the

10

investigations with this electrolyte are usually carried out in the temperature range 873 to 1273 K. It has been established that CaF2 is sensitive to moisture [52] which seems to affect the EMF measurement to some extent.

The galvanic cell method with CaF2 solid electrolyte has been used to determine the thermodynamic properties of oxides, alloys, borides, phosphides, sulphides, carbides and nitrides, as for example, Ca-O, Mg-O, Zn-O and Al-O systems [53], Th-Co system [54], Fe-Mn-C alloy [55], Cr-C [51], Mn-C [56] and Mn-Ni-C systems [24].

3.2 Cell arrangement and reaction

The galvanic cell used in the present work can be represented as follow:

(-) Pt │ N2, Cr-alloy+CrF2 +CaF2 ║CaF2║ CrN+CrF2+CaF2, N2 │ Pt (+) (18)

The individual electrode reactions can be written as:

Left-hand side electrode: (19) eCrFFalloyCr 22 2 +↔+− −

Right-hand side electrode: −+↔++ FCrNNCrFe 2212 22

(20)

Overall: CrNNalloyCr ↔+− 221 (21)

The change of molar Gibbs energy in reaction (21) is given by equation (22):

nEFGr −=Δ )21( (22)

where n denotes the number of electrons that are participating in the electrode reactions, E the electromotive force of the cell and F, the Faraday´s constant.

In order to calculate the Cr activity in the alloy, the standard Gibbs energy change in reaction (23) given below needs to be considered.

CrNNbccCr ↔+ 221)( (23)

According the thermodynamic data [57], the Gibbs energy of formation corresponding to equation (23) can be expressed as:

(TGof ⋅+−=Δ 351.72111892)23( ) 1500 800 KTK ≤≤ (24)

11

(23)-(21) gives: alloyCrbccCr −↔)( (25)

Thus (26) )21()23()25( GGG ro

fr Δ−Δ=Δ

The activity of Cr in the alloy can be obtained by equation (27):

nEFTaRTG alloyCrr +⋅+−==Δ − 351.72111892ln)25( (27)

where R is the gas constant and T is the temperature. The standard state for the activity of Cr, a , in Eq. (27) is pure Cr with bcc structure. alloy−Cr

3.3 Preparation of alloys and electrodes

Table 3. The raw materials used for the preparation of alloys and electrodes

Name Purity (wt. %) Particle size origin Cr powder 99.95 -200 mesh Alfa Aesar Fe powder 99.9 <10 μm Alfa Aesar Graphite powder 99.999 -325 mesh Aldrich CrF2 powder 97 Fine Aldrich CaF2 powder 99.95 Fine E. Merck, W. Germany Nitrogen gas 99.9999 AGA, Stockhlom Cr2N Analytical Reagent -325 mesh Sigma-Aldrich

Table 4. The sample numbers of the Fe-Cr-N and Fe-Cr-C-N alloys and the compositions (in weight %) of starting Fe-Cr powder mixtures for the Fe-Cr-N

alloys, as well as the starting Fe-Cr-C powder mixtures for the Fe-Cr-C-N alloys

Sample No. Cr Fe Sample No. C Cr Fe CFN-1 80 20 CCFN-11 2.8 77.2 20 CFN-2 60 40 CCFN-12 2.1 57.9 40 CFN-3 40 60 CCFN-13 1.4 38.6 60 CFN-4 20 80 CCFN-14 0.7 19.3 80 CFN-5 92 8 CCFN-22 4.5 55.5 40 CFN-6 94 6 CCFN-24 1.5 18.5 80 CCFN-05 32 39.5 28.5

CCFN-06 3 38 59

12

The raw materials used in this work for the preparation of alloys and electrodes are presented in Table 3. The compositions of starting Fe-Cr powder mixtures for the Fe-Cr-N alloys, as well as Fe-Cr-C powder mixtures for the Fe-Cr-C-N alloys, with corresponding sample numbers are listed in Table 4 (CFN donates Cr-Fe-N, CCFN donates C-Cr-Fe-N). The starting powder mixtures in appropriate weight ratios were prepared in 20 gram batches. The samples were mixed thoroughly, compacted into shallow alumina boats, and nitrided in an atmosphere of purified N2 gas (101 325 Pa) for two weeks at 1473 K for Fe-Cr-N alloys (1573 K for Fe-Cr-C-N alloys). The thickness of the powder mixture bed was less than 5 mm in order to get sufficient in-depth nitridation. The synthesized alloys were cooled in the furnace, crushed and ground to powder form before being examined by XRD to confirm the existence of the expected phases. The microstructures of these alloys were also measured by SEM technique.

Further, some of synthesized Fe-Cr-N alloys, as well as Fe-Cr-C-N alloys in the form of plates with the maximum thickness of 4mm were kept at 1173 K under purified N2 gas (101 325 Pa) for two different time intervals, viz. 240 or 720 hours, and then quenched in liquid nitrogen. The choice of the equilibration time was based on the estimation of the diffusion rate of nitrogen. The average diffusion distance of N in bcc-Cr is given by tD N2 where D is the diffusivity

of N in bcc Cr (3.16×10N

t-12

m2/s-1 at 1173 K) at the given nitriding temperature [59] and is the nitriding time under consideration. The time needed to diffuse a distance of 2 mm at 1173 K was estimated to be approximately 180 hours. Thus, the choice of the heat-treatment time would be more than adequate for the attainment of equilibrium. The nitrogen content and equilibrium phases in these alloys were determined by inert-gas fusion thermal conductivity (IGFTC) method and XRD analysis.

In the case of EMF measurements, the working electrodes were prepared by mixing the synthesized alloys, CrF and CaF2 2 powders in the weight ratio of 75:15:10, compacting the mixtures into cylindrical pellets with a diameter of 16mm. Earlier experiments carried out in the present laboratory have shown that CrF and CaF2 2 do not dissolve in each other in the solid-state at the experimental temperatures [58].

CrN used for the reference electrodes in this work was prepared by heating the Cr2N powder at 1273 K in 101 325 Pa purified N2 gas for two weeks. A consideration of the thermodynamic data for chromium nitrides [57] reveals that Cr2N phase would transfer into CrN phase below 1350 K in 101 325 Pa N gas. 2

13

The XRD spectra corresponding to the original Cr2N sample and final sample obtained after the heat-treatment indicated that the Cr2N phase was totally transferred into CrN phase after heat-treatment. The CrN, CrF and CaF2 2 powders used as the reference electrodes were mixed in a weight ratio of 75:15:10, and then compacted into pellets with a diameter of 16 mm.

All pellets of working- and reference electrodes were placed in boron nitride crucibles, dried under vacuum at 373 K for 12 hours, and then sintered at 1123 K for 4 hours in 101 325 Pa purified nitrogen gas before being mounted in the galvanic cell. Cylindrical single crystal discs of CaF2 with 22mm in diameter and 4mm in thickness (supplied by Alfa Aesar) were employed as solid electrolytes.

3.4 Gas cleaning system

2O

0

From section 3.2, it is understood that the galvanic cell must be kept in the nitrogen gas at 101 325 Pa pressure in order to keep the nitrogen potential in the system constant. The phase stability diagram of ternary system Cr-O-N at 1173 K was calculated from thermodynamic data [57] and presented in Fig. 1 ( P , and

represent the partial pressure of oxygen and nitrogen respectively, and 2NP

P is defined as 101 325 Pa). Commercial nitrogen gas ordered from AGA has an oxygen impurity level of less than 0.1 Pa. The cell components would be oxidized significantly under this oxygen pressure. Hence, the nitrogen gas must be purified extensively in order to meet the experimental requirement.

-30 -25 -20 -15 -10 -5-30

-20

-10

0

10

20

Phase stability Diagram at 1173K

Cr2O3

CrN

Cr2N

Cr

log (PO2/P0)

log (PN2/P0)

Figure 1. Cr-O-N phase stability diagram at 1173 K calculated based on the thermodynamic data [57]

14

A gas purification train has been developed, standardized and successfully used in the thermodynamic investigations for some transition metal systems containing carbon and nitrogen in the present laboratory. As shown in Fig. 2, this system consists of columns of silica gel, copper turnings, ascarite, magnesium perchlorate and magnesium chips. Trace of H2O in the gas was removed by passing the gas through silica gel and magnesium perchlorate, CO2 by ascarite and O2, by copper turnings at 873 K and magnesium chips at 773 K. Finally, the oxygen partial pressure in the gas stream was monitored by passing the outgoing gas from the cell furnace through an oxygen sensor involving partially stabilized zirconia electrolyte with a slow stream of dry air as the reference electrode.

1 2 3 4 5 6 7

Gas inlet Gas outlet

1. Silica gel; 2. Copper turnings; 3. Silica gel; 4. Ascarite; 5. Magnesium pechlorate; 6, 7. Magnesium chips

Figure 2. The schematic diagram of the nitrogen gas cleaning system

By carefully renewing the copper turnings and magnesium chips in the columns, the oxygen partial pressure in the gas stream was maintained to be less than 10-17 Pa. It was necessary to point out that the value of oxygen partial pressure measured by the oxygen sensor was probably higher than the exact value of the oxygen partial pressure in the exhaust, since the measured value was likely to be below the ionic conduction region of the ZrO2-7.5mol % CaO electrolytic tube used in the oxygen sensor. The electrodes of the galvanic cell were examined by optical microscope after the EMF measurements and were found to be free from oxidation.

15

3.5 Experimental apparatus and procedure

A schematic diagram of the galvanic cell arrangement used in this work is presented in Fig. 3. As can be seen from this figure, the single crystal electrolyte is kept sandwiched between the two electrodes in an open cell arrangement. The cell was placed in a vertically-mounted alumina reaction tube inside a Kanthal A1 resistance furnace, and positioned in the uniform temperature zone of the furnace (± 0.5 K over a length of 6 cm). The temperature of the cell was measured by a K-type thermocouple positioned under the cell which had earlier been calibrated against the melting point of pure gold. The reaction tube was closed at both ends with water-cooled copper lids provided with O-ring seals. The cell leads, the gas inlet and outlet tubes, as well as the thermocouple sheath were taken out of the reaction tube through O-ring seals.

1. Pt wire 2. N2 gas inlet 3. Gas outlet 4. Copper cap 5. O-rings 6. Spring 7. Reaction tube 8. Mo wires 9. Al2O3 tube 10. Al2O3 crucible 11. Al2O3 disk 12. Pt foil 13. Working electrode 14. CaF2 single crystal 15. Reference electrode

678

9

1314 15

1

2

3

45

12 11

10

Figure 3. Schematic diagram of the apparatus used for EMF measurement

In order to remove the moisture absorbed in the ceramic components around the cell, the reaction tube of the furnace as well as Al2O3 components in the cell arrangement were dried at 423 K under a pressure of 1.0 Pa for 10 hours, and then flushed with purified nitrogen gas for 2 hours. This vacuum-drying and cleaning

16

process was repeated three times before the cell was posited into the reaction tube in the furnace. The reaction tube with cell was also dried at 373 K under a pressure of 1.0 Pa for 12 hours, and then flushed with purified nitrogen gas for 4 hours in order to improve the accuracy of EMF measurements. When the partial pressure of oxygen in the outgoing gas monitored by oxygen sensor was less than 10-17 Pa, the furnace was turned on. The cell temperature was raised to 973 K at a rate of 6 K/min. The time needed for the cells to obtain equilibrium during the first heating was more than 3 days, and the equilibrium time thereafter was dependent on the temperature.

The EMF measurements were carried out in the purified nitrogen gas at 101 325 Pa pressure and the values were measured by a high impedance digital Newport millivoltemeter within an accuracy of ± 0.01 mV. In general, the EMF values were considered to be stable if the values were constant within ± 0.3 mV over a period of 4 hours. The reproducibility of the EMF value was verified by measuring the EMF of the cell during both heating and cooling cycles, and by polarizing the cell in both directions and observing that the cell EMF returned to its original value. The galvanic cell measurements were repeated in a number of cases in order to confirm the reliability of the results.

17

18

4. RESULTS AND DISCUSSIONS

C , Mn C and Fe C

4.1 Studies of Cr7 3 7 3 7 3 by atomistic simulations

4.1.1 The structure of Cr C , Mn7 3 7C and Fe3 7C3

According to the literatures, there are three structures proposed to describe Cr7C3 carbide: trigonal by Westgren (1935), orthorhombic by Bouchaud and Fruchart (1964) and hexagonal by Herbstein and Snyman (1964) [60-63]. All the structures are very similar and can be considered as built up from the same structural element called ‘triads’ and differ by the arrangement of these triads in the elementary unit cell. In order to investigate the structural properties of M C7 3, three corresponding structures were constructed respectively. The energy minimizations were then performed using the conjugate gradient method under the control of the inverted pair potentials. The relaxed structures of Cr C7 3, Mn7C3 and Fe C7 3 with lowest energy are hexagonal structure with P63mc space group which is consistent with the structure proposed by Herbstein and Snyman [61].

The structural stability of M7C3 (M = Cr, Mn, Fe) with hexagonal structure was tested. The initial lattice constants were chosen arbitrarily in a certain range, and then the energy minimizations were performed. The structures finally stabilized to hexagonal structure with P63mc space group. A certain range randomness of the initial structures and the stability of the final structures illustrate the validity of inverted potentials for the structure study.

Table 5. The lattice parameters and cohesive energies calculated in this work, as well as the experimental lattice parameters taken from the literatures

a(Å) c(Å) Cohesive energy

(eV/atom) Cal. Exp. Err. (%)

Err. (%) Cal. Exp.

Cr7C3 7.371 6.910[61] 6.671 4.337 4.495[61] 3.515 -8.626 Mn7C3 7.263 6.944[62] 4.805 4.262 4.542[62] 6.123 -8.022 Fe7C3 7.187 6.882[61] 4.432 4.235 4.540[61] 6.718 -7.356

Based on the stable structure of M C7 3 obtained from the calculation above, the supercell (M7C3)64 was constructed to evaluate the lattice parameters and cohesive energy of M C7 3. In the calculation, the configuration and energy average were taken from 30 samples. The calculated results are presented in Table 5. The

19

average deviation of calculated results from the experimental data is 5.303 % for a, as well as 5.452 % for c. In addition, the calculated cohesive energies listed in Table 5 indicate that the stability of carbides decreases with the increasing atomic number of the metallic component involved in the carbides, which has been found by some experimental works [64-65].

4.1.2 The thermodynamic properties of Cr7C3

Phonon density of states (DOS) reflects the lattice dynamic properties, from which some important thermodynamic parameters can be derived, for example, specific heat, vibrational entropy and Debye temperature. In this work, based on the method described in section 2.2, the total phonon DOS and partial phonon DOS of different component were calculated for Cr7C3 by the use of inverted pair potentials. The results listed in Fig. 4 of Supplement 1 indicate that the metal atoms contribute a major part to acoustic modes, while the carbon atoms contribute a major part to optic modes. The optic modes are separated by a very large gap from the acoustic ones and have very high frequencies, which are partly due to the large mass difference between metallic component and carbon, and partly due to strong force constants between nearest neighbor metal atoms and carbon [66]atoms .

400 600 800 1000 1200 1400

6

8

10

12

14

16

18

20

22

Vib

ratio

nal E

ntro

py (c

al/k

.gm

-ato

m M

e)

Temperature(K)

experimental entropy taken from the literature vibrational entropy calculated in this work

Figure 4. Comparison between the calculated vibrational entropy in this work and experimental entropy taken from the literature [67]

Based on the calculated phonon DOS, the dependence of the vibrational entropy on the temperature of Cr7C3 was derived and compared with experimental entropy

20

taken from the literature [67]. The comparison is presented in Fig. 4. The comparison shows that the calculated vibrational entropy of Cr7C3 is slightly lower than the experimental entropy, which means that dominating contribution to entropy is made by vibrational entropy.

4.2 Investigations of binary τ-carbides by atomistic simulations

4.2.1 Structural properties and cohesive energies of Cr

23C6 and Mn23C6

Basic features of the crystal structure of all τ-carbides were first determined by Westgren [68]. The carbides crystallize in the cubic system with a face-centered lattice in which 92 metal atoms are located at the 4a, 8c, 32f and 48h symmetry sites of space group Fm-3m. Carbon atoms are most reasonably placed on 24e sites.

Table 6. Atomic coordinates of Cr23C6 and Mn23C6

Atomic coordinates (Exp.) (Cal.) Cr23C Cr6 23C (Cal.) Mn6 23C6

x y z x y z x y z 4a 0, 0, 0 0, 0, 0 0, 0, 0 8c 0.25, 0.25, 0.25 0.250, 0.250, 0.250 0.250, 0.250, 0.250 32f 0.385, 0.385, 0.385 0.380, 0.380, 0.380 0.383, 0.383, 0.383 48h 0, 0.165, 0.165 0, 0.171, 0.171 0, 0.172, 0.172 24e 0.275, 0, 0 0.272, 0, 0 0.271, 0, 0

With the inverted pair potentials, the atomic coordinates and lattice parameters of Cr23C and Mn6 23C6 were calculated. The comparison of atomic coordinates of Cr23C6 between the calculated results and experimental data is listed in the Table 6. It can be seen from the comparison that calculated results are close to the experimental date. The calculated crystal constants are 10.903 Å for Cr23C6 and 10.729 Å for Mn23C , which are larger than the experimental data [68, 69]

6 and the deviations are lower than 3 %. In addition, the cohesive energies of these compounds were also evaluated. The values are Ecoh= -8.404 eV/atom for Cr23C6 and Ecoh= -7.744 eV/atom for Mn23C6, which means that Cr C23 6 is more stable than Mn23C6.

21

4.2.2 The thermodynamic properties of Cr23C6

The dependency of vibrational entropy of Cr C)(TS 23 6 on temperature was derived from the phonon DOS calculated by the use of inverted pair potentials. The vibrational entropy calculated was also compared with the experimental entropy taken from the literature [67]. The comparison presented in Fig. 5 indicates that the entropy is constituted largely by vibrational entropy. This result is in agreement with the reference [70].

400 600 800 1000 1200 1400

6

8

10

12

14

16

18

20

experimental entropy taken from the literatrue vibrational entropy calculated in this work

Ent

ropy

(Cal

K-1 g

m-a

tom

Me-1

)

Temperature (K)

Figure 5. Comparison between the vibrational entropy obtained in

this work and experimental entropy taken from the literature [67]

4.3 Investigations of ternary τ-carbides Cr23-xFexC6

enerally contains more The τ-carbide precipitated from multicomponent alloys gthan one metallic element, although an element that forms a stable binary τ-carbide (e.g. Cr or Mn) often predominates. Since the four symmetry sites available for metal atoms differ in their coordination numbers, coordination geometries and proximity to carbon, one may anticipate non-random elemental distributions on them due to variations in atomic sizes and chemical affinities. Hence, the site preference of Fe atoms in Cr23-xFe Cx 6 was evaluated by the use of inverted pair potentials in this work.

22

-8.28

0.0 0.2 0.4 0.6 0.8 1.0

-8.40

-8.38

-8.36

-8.34

-8.32

-8.30

(a)

Coh

esiv

e en

ergy

(eV/

atom

)Content of Fe atom in Cr23-xFexC6

4a x<=1 8c x<=1 32f x<=1 48h x<=1

Cr23-xFexC6

1.0 1.5 2.0 2.5 3.0

-8.30

-8.28

-8.26

-8.24

-8.22

-8.20

-8.18

-8.16

-8.14

-8.12

-8.10

(b)

C

ohes

ive

ener

gy (e

V/a

tom

)

Content of Fe atom in Cr23-xFexC6

8c 1<x<=3 32f 1<x<=3 48h 1<x<=3

Cr23-xFexC6

Figure 6. Cohesive energy variation with x when Fe atoms occupy different sites in Cr23-xFexC6

In the calculation, the supercell (Cr C23 6)8 was chosen as the crystal cell with periodic boundary in order to reduce statistical fluctuation. Within the composition range x<1, the Fe atoms randomly occupied 4a, 8c, 32f or 48h sites respectively, and then energy minimizations were performed under the interaction of pair potentials. The average cohesive energies with 30 samples are shown in Fig. 6(a). The cohesive energies of Cr Fe C23-x x 6 within x<1 are similar when Fe atoms occupy different sites, but the tolerances presented in Fig. 6 of Supplement 3 are lowest when the Fe atoms occupy the 4a sites. The tolerance characterizes the deviation of the atomic position distribution of the ternary compound Cr23-xFexC6 from the space group Fm-3m, which is determined by the intrinsic binary structure Cr23C6. The calculated results indicate that Fe atoms preferentially occupy the 4a sites when x<1 based on the cohesive energy and tolerance considerations.

23

In the composition range 1<x<3, four Fe atoms have already occupied the 4a sites based on the above analysis, so the remaining Fe atoms can only occupy the 8c, 32f or 48h sites. The cohesive energies and tolerances are lowest when Fe atoms are at the 8c sites. So when 1<x<3, Fe atoms preferentially occupy the 8c sites besides four Fe atoms occupying the 4a sites. The calculated results show that iron atoms do not randomly replace chromium atoms on the four independent metal atom sites. The iron atoms to enter structure fill a high percentage of 4a sites, after which occupation of the 8c sites begins, which are consistent with some experiment results [71-73]. The calculated results also show that the cohesive energy of Cr

23-xFe Cx 6 increases with the increasing Fe content, which indicates that the addition of Fe decreases the stability of Cr23C [74]. 6

According to the result of site preference, the lattice constants of Cr23-xFe Cx 6 were calculated with different Fe content. The results indicate that the lattice constants decrease with increasing Fe content.

4.4 The stability of carbides and nitrides

In the industrial applications, the Fe-Cr alloys usually contain carbon and nitrogen. It would be interesting to examine if the relative stability of separate carbide phases is consistent with the stability of nitrides in transition metal alloys. In order to understand the relative formation of the transition metals carbides and nitrides, the standard formation Gibbs energies of the carbides and nitrides of Cr, Mn and Fe were compared, in which only the carbides in equilibrium with graphite, as well as the nitrides in equilibrium with pure nitrogen gas, have been considered. Such comparisons using the ΔfGo from literature [57] are presented in Fig. 6 and 9 of Supplement 4. It is found that Cr3C2 and CrN have the largest negative Δ Go

f values among these carbides and nitrides respectively, and the Δ Go

f of carbides and nitrides for Cr, Mn and Fe increase with the increasing in the atomic number.

It is interesting to examine if the order of carbon affinities is consistent with nitrogen affinities for Cr, Mn and Fe. This is clarified by the interatomic pair potentials of Cr-C, Mn-C, Fe-C, Cr-N and Mn-N. These pair potentials inverted by the lattice inversion method and ab initio cohesive energies are presented in Fig. 7. It can be seen from this figure that the order of interaction intensity is

(I donates the intensity of interatomic

interaction), which indicates the decreasing carbon and nitrogen affinities for Cr, Mn and Fe atom with the increasing atomic number. Hence, in regard to thermodynamics aspect, Cr carbides and Cr nitrides are likely to precipitate out

CFeCMnCCrNMnNCr IIIII −−−−− >>>>

24

preferentially among Cr, Mn and Fe components. Further, due to the stronger interaction of Cr-N than Cr-C, it can be foreseen that the formation of Cr nitrides will be more preferred than the formation of Cr carbides in Cr alloys containing carbon and nitrogen.

the strong interaction o egative effect of nitrogen on hromium activity in Cr alloys can also be inferred, while there is no

nalysis of the Fe-Cr-N alloys

, CFN-2, CFN-3 and CFN-4 alloys synthesized at 1473 K were examined by XRD at room temperature. According to

e presented in Fig. 8. Combining with the

and Mn-N inverted by lattice inversion method and ab initio

2 3

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

Fe-C Mn-C Cr-C

Mn-N

Figure 7. Interatomic pair potentials of Cr-C, Mn-C, Fe-C, Cr-N

f Cr-N, the strong n

cohesive energies

Based onthe cexperimental validation of such estimations. Further, as the calculations are performed at 0 K, the adoption of nitrides in industrial applications requires the verification at higher temperatures. In view of this, it was decided to carry out experimental measurements of the thermodynamic activities of Cr in the Fe-Cr-N and Fe-Cr-C-N systems.

4.5 Phase a

The main phases coexisting in the CFN-1

the XRD spectra, the main phases in these alloys are the same and identified as (Cr, Fe)2N nitride and α solid solution phase. Further, the intensity of (Cr, Fe)2N peaks decreases, while intensity of α solid solution phase peaks increases with the increasing Fe content in the alloys.

The microstructures of synthesized alloys were also examined by SEM measurements. The SEM micrographs ar

Cr-N

Pair

pote

ntia

l (eV

)

Interatomic distance (Å)

25

XRD results, the black phase in this figure is identified as (Cr, Fe)2N phase, the light grey one as α solid solution phase. Since the nitrogen content in the various phases could not be determined accurately in view of the limitations of the EDS analysis, the chemical compositions were analyzed for Fe and Cr only. The results listed in Table III of supplement 4 indicate that the main metallic component in (Cr, Fe)2

this work, the equilibrium nitrogen content in Fe-Cr-N alloys quenched from

(c) CFN-3

N nitride is chromium, while the iron is the main component in α solid solution phase.

(d) CFN-4

(a) CFN-1 (b) CFN-2

Figure 8. SEM micrographs of the CFN-1, CFN-2, CFN-3 and CFN-4 alloys synthesized at 1473 K in 101 325 Pa purified N2 gas for two weeks

In1173 K was analyzed by IGFTC method, and presented in Table 7. It can be seen from this table that there is very little difference of the nitrogen content in the alloys heat-treated at 240 and 720 hours. This indicates that the alloys with 4mm thickness kept at 1173 K for 720 hours would have reached equilibrium with nitrogen gas. It can also be seen that the nitrogen content increases with the increasing chromium content in the alloys, which indicates the presence of nitrogen as compounded with Cr.

26

and 720 hours) were investigated by XRD

phases of the Fe-Cr-N alloys at 1173 K.

The phases coexisting in the alloys after equilibrium treatment at 1173 K at different time interval (240 hoursanalysis. The results in the case of CFN-1 are shown in Fig. 9. The equilibrium phases are the same in these two samples and they are (Cr, Fe)2N, (Cr, Fe)N and α solid solution. Thus, the XRD analysis results and nitrogen analysis indicate that 720 hours is long enough for the alloys to reach equilibrium state at 1173 K. The equilibrium phases of the alloys at 1173 K are listed in Table 7.

Table 7. Nitrogen content (in weight %) in the Fe-Cr-N alloys kept at 1173 K for 240 hours (alloy-240) and 720 hours (alloy-720) and equilibrium

Nitrogen content Sample

loy-720Main phases by XRD

Alloy-240 AlCFN-1 9.5 (Cr, Fe)2N, (Cr, Fe)N, α solid solution 10.3 CFN-2 7.1 (Cr, Fe)2 lution N, (Cr, Fe)N, α solid soCFN-3 2.89 2.62 (Cr, Fe)2N, α solid solution CFN-4 0.93 (Cr, Fe) N, α solid solution 2

CFN-5 (Cr, Fe) N, (Cr, Fe)N, α solid solution 2

CFN-6 (Cr, Fe) N, (Cr, Fe)N, α solid solution 2

.

Figure 9. XRD patterns of the CFN-1 alloy held at 1173 K for different period, viz240 hours (CFN-1-240) and 720 hours (CFN-1-720)

40 50 60 70 800

30

60

90

120

α

α

α

M

MMM

M

M

M2

M

M2

M2M

2

M2

2 Theta/degree

CFN-1-2400

30

60

90

120 M2: (Cr, Fe)2NM: (Cr, Fe)NM

2

α: Solid solution

αα

α

M M MM

M

M2

M2

M2

M2M2

M2

Inte

nsity

CFN-1-720

27

.6 The activities of chromium in the Fe-Cr-N system

based on the EMF measurements are presented in Table 8. As shown in this table, Cr activities in the

e (CFN) alloys obtained by EMF measurements in the present work, as well as that in the Cr-Fe (CF) alloys

calcu

4

The activities of Cr in the Fe-Cr-N alloys calculated

Fe-Cr-N alloys are very low and the values in different alloys are close in the measured temperature range, which can be contributed to very strong interaction between Cr and N. Further, the Cr activities are found to increase with the increasing temperature. This can partly be attributed to the decreasing nitrogen solubility in the alloys [75] and partly to the increasing Gibbs formation energy of nitrides with the increasing temperature [57].

Table 8. Activities of Cr in the Cr-F -N

lated using Thermo-Calc software. The standard state for Cr activity is pure bcc-Cr

a a aCr Cr CrT(K) CFN-1 CF-1

T(K) CFN-2 CF-2

T(K) CFN-3 CF-3

983 0.016 0.851 0.010 0.777 0.009 0.632 988 983 1023 1031 1023 0.017 0.848 0.015 0.745 0.012 0.617 1073 0.022 0.844 1083 0.020 0.734 1073 0.024 0.615 1123 0.031 0.837 1120 0.032 0.707 1118 0.032 0.599

aCr Cr Cra a

T(K) CFN-4 CF-4

T(K) CFN-5 CF-5

T(K) CFN-6 CF-6

993 0.007 0.534 0.010 0.932 0.012 0.945 983 993 1024 1023 1023 0.009 0.502 0.017 0.932 0.024 0.944 1072 0.022 0.453 1073 0.028 0.931 1073 0.030 0.944 1123 0.035 0.421 1123 0.039 0.930 1123 0.036 0.944

The effect of nitrogen on the Cr activities in the Fe-Cr-N alloys was studies by the comparison of Cr activities between the Fe-Cr and Fe-Cr-N alloys in which the

28

weight ratios of Fe and Cr in the starting powders for the Fe-Cr-N alloys and Fe-Cr alloys are the same. The data of Cr activities in the Fe-Cr alloys are calculated by using Thermo-Calc software [76]. The comparison presented in Table 8 indicates that the addition of nitrogen into the Fe-Cr-N alloys obviously decreases the chromium activities, and the difference of Cr activities between the Fe-Cr and Fe-Cr-N alloys decreases with the decreasing Cr content in the alloys.

s in which

The main phases of some Fe-Cr-C-N alloys synthesized at 1573 K were identified resented in Fig. 10 (M represents

The fac te similar en content in 720 hours) and the occurrence of the same coexisting phases examined by XRD, in the

Figure 10. XRD patterns of CCFN-11, CCFN-12 and CCFN-22 alloys synthesized at 1573 K in 101 325 Pa purified N2 gas for two weeks

The effect of nitrogen on the chromium activity was further investigated by comparing the Cr activities between the Fe-Cr-C and the Fe-Cr-N alloythe weight ratio of Fe in the starting mixtures for the Fe-Cr-C and Fe-Cr-N alloys is the same in each comparison group, while weight ratio of Cr is slightly different. The data of Cr activities in the Fe-Cr-C alloys were taken from the earlier work in the present group [25], in which the standard state for the Cr activity is pure bcc-Cr. The comparison presented in the Fig. 11 of Supplement 4 shows much lower Cr activities in the Fe-Cr-N alloys than those in the Fe-Cr-C alloys.

4.7 Phase analysis of the Fe-Cr-C-N alloys

by XRD at room temperature. The results are pFe and Cr). According to the XRD spectra of these alloys, it can be found that (1) M2(C, N) nitride are formed during the nitridation procedure; (2) The intensity of α phase peaks increases, while intensity of M2(C, N) nitride peaks decreases with increasing Fe content in the alloys; (3) The intensity of M2(C, N) nitride phase peaks decreases with increasing C content; (4) The M7(C, N)3 carbide phase is detected in CCFN-22 alloy.

t that the bulk nitrogen contents obtained by IGFTC method are quibetween the heated-treated times of 240 and 720 hours (The nitrog

CCFN-11 alloy is 9.3 wt. % for 240 hours and 10.2 wt. % for

30 40 50 60 70 800

100200300400500600

M2

α

M2

M2

M2

M2M

2

M2

2 Theta/degree

CCFN-11

0100200300400500600

αα M2

M2

M2M2 M2

M2

Cou

nts/

s

0100200300400500600

CCFN-12

M2: M2(C, N)M7: M7(C, N)3

α: Solid solution

α

α

M2

M2

M2

M2

M2

M7

M7

M7

M7

M7

M7

M2

CCFN-22

29

alloys during the two heat-treatment intervals suggest that the alloys have reached equilibrium after heat-treatment for 720 hours at 1173 K. The equilibrium phases are M(C, N), M

of Cr in Fe-Cr-C-N alloys were calculated and are presented in Table 9. As shown in this table, Cr activities in the

ange due to strong interaction of Cr-N. It can also be found that the Cr activities increase with

CCFN-12 CCFN-13 CCFN-14

2(C, N), M7(C, N)3 and α solid solution for CCFN-11 and CCFN-12, M (C, N), M2 23(C, N) and α solid solution for CCFN-13, and M6 2(C, N) and α solid solution for CCFN-14. Further, the nitrogen content analysis results indicate that the nitrogen content increases with the increasing in chromium content in the alloys as well as the decreasing in temperature.

4.8 The activities of chromium in the Fe-Cr-C-N system

Based on the EMF measurements, the activities

Fe-Cr-C-N alloys are very low in the measured temperature r

increasing temperature.

Table 9. Activities of Cr in the Fe-Cr-C-N alloys obtained by the EMF measurements with pure bcc-Cr as standard state in the present work

CCFN-11 T(K) aCr T(K) aCr T(K) aCr T(K) aCr

9 91 0.009 973 0.006 974 0.004 973 0.0081 032 0.011 1023 0.015 1033 0.011 1023 0.02410 0 10 6 10 9 1078 0.02 53 0.02 57 0.01 53 0.043 1111 0 0 0 0 .035 1073 .038 1071 .028 1073 .0481141 0.067 1123 0.066 1103 0.068 1123 0.085 1169 0.082 1133 0.081 1173 0.102

1159 0.091

CCFN N N FN--22 CCF -24 CCF -05 CC 06 T(K) a T(K) a T(K) a T(K) aCr Cr Cr Cr

10 3 0.0 2 10 3 0.0 9 0.0 5 2 1 2 10 983 0.003 83 01053 0.019 1071 0.026 23 0.005 1023 0.008 1010 8 11 9 10 8 1083 0.02 19 0.05 53 0.00 53 0.011 1113 0 0 0 0 .035 1158 .079 1073 .012 1073 .0151143 0.047 1123 0.043 1123 0.024

1153 0.061 1153 0.031

30

The iu vities in the quatern -C allo asu he present work are also compared w the on r ac s in Cr-

Figure 11. The comparison of Cr activities between the CFN-22 and CCFN-22 alloys (the weight ratios of C, Cr, Fe in starting

mixtures for CCF-22 and CCFN-22 alloys are the same)

chrom m acti ary Fe r-C-N ys me red in tith corresp ding C tivitie the Fe-

C ternaries. The data of Cr activities in the Fe-Cr-C system were taken from earlier work in the present group [31]. It should be noted that the weight ratios of C, Cr and Fe in the starting mixtures for the Fe-Cr-C [31] and Fe-Cr-C-N alloys are the same in the each comparison group. As an example, the comparison of Cr activities between CCF-22 and CCFN 2 alloys is presented in Fig. 11. The comparison shows that the Cr activities in the CCFN-22 alloy are much lower than those in the CFN-22 alloy, which indicates the strong negative effect of nitrogen on the Cr activity.

-2

950 1000 1050 1100 1150 1200

0.0

0.2

0.4

0.6

0.8

1.0

Temperature (K)

Activ

ity o

f Cr

CCF-22 CCFN-22

31

32

. CONCLUSIONS

tomistic simulations with inverted pair potentials and the EMF measurements with CaF2 single crystals as solid electrolytes were employed

alculated in this work is hexagonal structure with P63mc space group, which is consistent with the

(2) s of M7C3 (M = Cr, Mn, Fe), it is established that the stability of carbides decreases with the increasing

(3) culated in this work with the entropies obtained by experimentations, it can be seen that

(4) d lattice constants of Cr23C6 and Mn23C6 calculated in this work are in good

(5) e, the distribution of Fe atoms is not random. Fe atoms will firstly substitute for Cr atoms at 4a

(6) a tials of Cr-C, Mn-C, Fe-C, Cr-N and Mn-N suggest that both carbon and nitrogen affinities for Cr, Mn and Fe

(7) thesized Fe-Cr-N and Fe-Cr-C-2 formed in the Fe-Cr-N and the

33

5

In the present work, the a

to investigate the properties of transition metal systems containing carbon and nitrogen. The conclusions can be drawn as following.

(1) The structure of M C (M = Cr, Mn, Fe) c7 3

structure proposed by Herbstein and Snyman. The calculated lattice constants are close to the experimental data.

According to the calculated cohesive energie

atomic number of metallic component within the carbides.

By comparing vibrational entropies of Cr7C3 and Cr C23 6 cal

the vibrational entropy has a large contribution to total entropy.

For the binary τ-carbides, the atomic coordinates of Cr23C an6

agreement with the experimental data. The calculated cohesive energies indicate that Cr23C6 is more stable than Mn23C6.

When Fe atoms dissolve into the Cr C23 6 carbid

sites and then 8c sites. The lattice constant decreases with the increasing Fe content in Cr23-xFexC6.

The calculated p ir poten

decrease with the increasing atomic number, the formation of Cr nitrides will be more preferable than the formation of Cr carbides in Cr alloys containing carbon and nitrogen, and strong negative effect of nitrogen on the chromium activity can be inferred.

It is seen from the XRD spectra of the synN alloys that the Cr N-based nitride wasFe-Cr-C-N alloys during the nitridation procedure. The main metallic

(8)

(9) ts indicate that the nitrogen content

(10) e Fe-Cr-N and Fe-Cr-C-N alloys in

(11) ctivities between the Fe-Cr and Fe-Cr-N alloys,

34

component in carbides and nitrides is chromium, while iron is main metallic component in α solid solution.

The equilibrium phase relationships in the Fe-Cr-N and Fe-Cr-C-N alloys at 1173 K have been established.

The nitrogen content analysis resulincreases with the increasing chromium content in the alloys, as well as the decreasing temperature.

The activities of chromium in thtemperature range 973-1173 K are very low and increase with the increasing temperature.

The comparisons of Cr athe Fe-Cr-C and Fe-Cr-N alloys, and the Fe-Cr-C and Fe-Cr-C-N alloys in temperature range 973-1173 K indicate that the addition of nitrogen to the alloys obviously decreases the activity of chromium.

. FUTURE WORK

rt of an ongoing program at the present laboratory on the

onsidered in the form of interatomic ab initio

(2) ations will be performed to get the phase equilibrium in the Fe-Cr-N and the Fe-Cr-C-N systems at different

(3) ulation results acquired by atomistic simulations and ab initio calculations at low temperature and

35

6

The present work is pasystemization of thermodynamic dat f different alloys systems. Further calculations, experimentations, as well as careful thermodynamic assessments need to be carried out in order to produce consistent thermodynamic descriptions of systems related to the Cr-C, Fe-C, Fe-Cr-C, Cr-N, Fe-Cr-N and Fe-Cr-C-N. In view of this, further work should involve:

(1) Many-body potentials will be c

a o

potentials. Atomistic simulations, calculations and experimentations will be further carried out in order to investigate the properties of carbides and nitrides.

More experimentations and calcul

compositions and temperatures.

The bridge between the calc

experimental results at high temperature will be explored. The thermodynamic properties of the Cr-C, Fe-C, Fe-Cr-C, Cr-N, Fe-Cr-N and Fe-Cr-C-N systems at high temperature will be studied in the future by the collaboration of these methods.

36

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