FIRST-PRINCIPLES INVESTIGATIONS OFPLANAR DEFECTS

SONG LU

Licentiate ThesisSchool of Industrial Engineering and Management, Department of

Materials Science and Engineering, KTH, Sweden, 2012

MaterialvetenskapKTH

ISRN KTH/MSE–12/09–SE+AMFY/AVH SE-100 44 StockholmISBN 978-91-7501-313-8 Sweden

Akademisk avhandling som med tillstand av Kungliga Tekniska Hogskolan framlaggestill offentlig granskning for avlaggande av licentiatexamen torsdagen den 8 May2012 kl 10:00 i konferensrummet, Materialvetenskap, Kungliga Tekniska Hogskolan,Brinellvagen 23, Stockholm.

c⃝ Song Lu, May, 2012

Tryck: Universitetsservice US AB

iii

Abstract

Two types of planar defects, phase interface and stacking fault, are addressed in thisthesis. The first-principles exact-muffin orbitals method in combination with thecoherent-potential approximation is the main density functional theory (DFT) toolfor our studies. The investigation is mainly carried out for stainless steels which arefundamental materials in modern society. Ferritic and austenitic stainless steels arethe two largest subcategories of stainless steels.

In ferritic stainless steels, the interface between Fe-rich α and Cr-rich α′ phasesformed during spinodal phase decomposition is studied. This decomposition isknow to increase the hardness of ferrites, making them brittle (also called the ”475◦Cembrittlement”). We calculate the interfacial energies between the Cr-rich α′-FexCr1−x

and Fe-rich α-Fe1−yCry phases (0 < x, y < 0.35) and show that the formation energyis between ∼0.02 and ∼0.33 J m−2 for the ferromagnetic state and between ∼0.02and ∼0.27 J m−2 for the paramagnetic state. Although for both magnetic states,the interfacial energy follows a general decreasing trend with increasing x and y,the fine structures of the γ(x, y) maps exhibit a marked magnetic state dependence.The subtleties are shown to be ascribed to the magnetic interaction between the Feand Cr atoms near the interface. The theoretical results are applied to estimate thecritical grain size for nucleation and growth in Fe-Cr stainless steel alloys.

In close-packed alloys possessing the face centered cubic crystallographic lattice ,stacking faults are very common planar defects. The formation energy of a stackingfault, named stacking fault energy, is related to a series of mechanical properties.

Intrinsic stacking fault energy for binary Pd-Ag, Pd-Cu, Pt-Cu and Ni-Cu solid so-lutions are calculated using the axial interaction model and the supercell model. Bycomparing with experimental data, we show that the two models yield consistentformation energies. For Pd-Ag, Pd-Cu and Ni-Cu, the theoretical SFEs agree wellwith those from the experimental measurements. For Pt-Cu no experimental resultsare available, and thus our calculated SFEs represent the first reasonable predictions.We also discuss the correlation of the SFE and the minimum dmin in severe plasticdeformation experiments and show that the dmin values can be evaluated from firstprinciples calculations.

After gaining confidence with the axial interaction model, the alloying effects of Mn,Co, and Nb on the stacking fault energy of austenitic stainless alloys, Fe-Cr-Ni withvarious Ni content, are investigated. In the composition range (cCr = 20%, 8 ≤cNi ≤ 20%, 0 ≤ cMn, cCo, cNb ≤ 8%, balance Fe) studied here, it is found that Mndecreases the SFE at 0 K, but at room temperature it increases the SFE in high-Ni (cNi & 16%) alloys. The SFE always decreases with increasing Co. Niobiumincreases the SFE significantly in low-Ni alloys, however this effect is strongly di-minished in high-Ni alloys. The SFE-enhancing effect of Ni usually observed inFe-Cr-Ni alloys is inverted to SFE-decreasing effect in the hypothetical alloys con-taining more than 3% Nb in solid solution. The revealed nonlinear composition

iv

dependencies are explained in terms of the peculiar magnetic contributions to thetotal SFE.

v

Preface

List of included publications:

I First-principles determination of the α-α′ interfacial energy in Fe-Cr alloysSong Lu, Qing-Miao Hu, Rui Yang, Borje Johansson, and Levente Vitos, Phys. Rev.B 82, 195103 (2010).

II Composition and orientation dependence of the interfacial energy in Fe-Cr stain-less steel alloysSong Lu, Qing-Miao Hu, Borje Johansson, and Levente Vitos, Phys. Status Solidi B248, 2087 (2011).

III Determining the minimum grain size in severe plastic deformation process viafirst-principles calculationsSong Lu, Qing-Miao Hu, Erna Krisztina Delczeg-Czirjak, Borje Johansson, andLevente Vitos. Acta Mater. Accepted.

IV Stacking fault energies of Mn, Co and Nb alloyed austenitic stainless steelsSong Lu, Qing-Miao Hu, Borje Johansson, and Levente Vitos, Acta Mater. 59, 5728(2011).

Comment on my own contributionPaper I: half of the calculations, data analysis, literature survey; the manuscript waswritten jointly.Paper II: all calculations, data analysis; the manuscript was written jointly.Paper III: all calculations, data analysis, literature survey; writing the manuscript (90%).Paper IV: all calculations, data analysis, literature survey; writing the manuscript (90%).

Publications not included in the thesis:

V Static equation of state of bcc ironH. L. Zhang, S. Lu, M. P. J. Punkkinen, Q.-M. Hu, B. Johansson, and L. Vitos, Phys.Rev. B 82, 132409 (2011).

VI Mechanical Properties and Magnetism: Stainless Steel Alloys from First-principlesTheoryL. Vitos, H. L. Zhang, N. Al-Zoubi, S. Lu, J.-O. Nilsson, S. Hertzman and B. Johans-son. MRS Proceedings, 1296, mrsf10-1296-o02-01 doi:10.1557/opl.2011.1448,(2011)

VII First-principles Quantum Mechanical Approach to Stainless Steel AlloysL. Vitos, H. L. Zhang, S. Lu, N. Al-Zoubi, B. Johansson, E. Nurmi, M. Ropo, M.

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P. J. Punkkinen, and K. Kokko, in Alloy Steel: Properties and Use, Book edited byEduardo Valencia Morales, ISBN 978-953-307-888-5, Publisher: InTech, pp. 3-28,Dec. (2011).

VIII Stainless Steel Alloys from First-principles TheoryL. Vitos, H. Zhang, N. Al-Zoubi, S. Lu, J.-O. Nilsson, S. Hertzman, G. Nilson, andB. Johansson, 7th European Stainless Steel Conference: Science and Market, ComoItaly 21-23 Sep. (2011).

Contents

Preface v

Contents vii

1 Introduction 1

2 Spinodal Decomposition in BCC Iron-Chromium alloys and Stacking Faultin FCC Metals 42.1 Spinodal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 What is spinodal decomposition? . . . . . . . . . . . . . . . . . . . . 42.1.2 Critical nucleus size . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Alloying effects on spinodal decomposition . . . . . . . . . . . . . . 6

2.2 Stacking Fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 What is stacking fault? . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Methods for measuring the SFE . . . . . . . . . . . . . . . . . . . . . 62.2.3 SFE and mechanical properties . . . . . . . . . . . . . . . . . . . . . 82.2.4 Two models for calculating SFE . . . . . . . . . . . . . . . . . . . . . 9

3 Theoretical Methodology 113.1 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Exchange-Correlation Functionals: LDA and GGA . . . . . . . . . . . . . . 143.3 Computational methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.1 Exact Muffin-tin Orbital method . . . . . . . . . . . . . . . . . . . . 163.3.2 Exact muffin-tin orbitals wave function . . . . . . . . . . . . . . . . 163.3.3 The full charge density (FCD) total energy . . . . . . . . . . . . . . . 173.3.4 Coherent potential approximation (CPA) . . . . . . . . . . . . . . . 17

4 α/α′ Interfacial Energy in Ferrite Fe-Cr Alloys 194.1 Interface Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Interfacial Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Anisotropy of The Interfacial Energy . . . . . . . . . . . . . . . . . . . . . . 214.4 Magnetic Structure at The Interface . . . . . . . . . . . . . . . . . . . . . . . 224.5 Predicted Critical Nucleus Size . . . . . . . . . . . . . . . . . . . . . . . . . 24

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CONTENTS viii

5 Stacking Fault Energy of Binary Alloys: Pd-Cu, Pt-Cu, Pd-Ag and Ni-Cu 255.1 Stacking Fault Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 The Non-linear Composition Dependence of Stacking Fault Energy . . . . 275.3 Minimum Grain Size vs. Stacking Fault Energy . . . . . . . . . . . . . . . . 28

6 Stacking Fault Energy of Austenitic Stainless Steels 316.1 Comparing with Previous Results . . . . . . . . . . . . . . . . . . . . . . . . 316.2 Magnetic Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.3 The SFE of Fe-Cr-Ni-X Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 Concluding Remarks and Future Work 37

Acknowledgements 39

Bibliography 40

Chapter 1

Introduction

Stainless steel belongs to the most important engineering materials since its inventionat the beginning of 20th century. When corrosion or oxidation are curial, stainless steelis the primary choice. Chromium is the essential element which imparts the ability ofcorrosion and oxidation resistance to steels. Generally, a minimum of ∼12% Cr is nec-essary to form a passive, Cr-rich oxidation film on the surface of steel. The protectivefilm is extremely thin, ∼1-5 nanometers, and invisible, however it adheres firmly, and ischemically stable under conditions which provide sufficient oxygen to the surface. Fe-Cr alloys form the basis of stainless steels. By alloying metallurgists have developed alarge and steadily growing family of stainless steels that can offer unique combinationsof corrosion resistance and properties such as high strength, low temperature tough-ness, creep strength and formability. Some of the most common alloying elements areNi, Al, Mn, Co, Mo, Cu, N, etc. Several important sub-categories of stainless steelshave been developed. They can be classified as austenitic, martensitic, ferritic, duplex,precipitation hardening and super alloys.

Ferritic grades have been developed to provide a group of stainless steel to resist corro-sion and oxidation, while being highly resistant to stress corrosion cracking. The fun-damental player of ferritic steels are Fe-Cr alloys which crystallize into body-centeredcubic (BCC) structure. There are several alloying elements which act as ferrite stabi-lizer, eg., Cr, Mo, Al, Ti and Nb, etc. These steels are magnetic but cannot be hardenedor strengthened by heat treatment. They can be cold worked and softened by anneal-ing. They are used for decorative trim, sinks, and automotive applications, particularlyexhaust systems. Common grades of ferritic stainless steels are 430, 405, 409, 434, etc.

Spinodal decomposition is a big headache when ferritic stainless steels serve at interme-diate to high temperature. Above ∼12% Cr, these alloys are known to undergo harden-ing and embrittlement after thermal aging due to phase separation under the spinodaldecomposition mechanism. The phenomenon is well known as ”475◦C embrittlement”.By spinodal decomposition, the uniform Fe-Cr solid solution decomposes into the Fe-rich α phase and the Cr-rich α′ phase. The toughness of the material decreases with thegradually developed interconnected vein-like microstructure with aging time. Brenner

1

CHAPTER 1. INTRODUCTION 2

et al. [1] studied the spinodal decomposition in Fe-32 at.% Cr alloy. They found that theirregular micro-structure coarsens very sluggishly which was presumably attributed tothe low interfacial energy between the α and α′ phases. Though the interfacial energybetween α and α′ phases is crucial on determining the kinetic process of spinodal de-composition, experimentally it is very difficult to measure.

Although tremendous efforts have been made to investigate the phase decompositionof Fe-Cr alloys [2–9], due to the complexity of the interface the accurate determinationof the composition-dependent interfacial energy, either experimentally or theoretically,has been very limited. The first part of this thesis is to evaluate the interfacial energybetween the α and α′ phases of the Fe-Cr alloys and investigate the effects of chem-istry and magnetism. The method I use is the first-principles exact muffin-tin orbitals(EMTO) method. Random chemical and magnetic disorders are dealt with the coher-ent potential approximation (CPA). Our study provides an insight into the fundamentalphysics behind the phase decomposition which is not accessible by the phenomeno-logical theories. Results and discussions are presented in Chapter 4 and supplementedPaper 1 and 2.

Austenitic stainless steels are Fe-Cr-Ni alloys which are hardenable only by cold work-ing. Nickel is the main substitutional alloying element in this class of steels and car-bon is kept at low levels. The Ni content may be varied from about 4% to 22% - higheramount of nickel leads to higher ductility of the metal. It shows that 8 wt% Ni is enoughto ensure that the alloy is fully austenitic at high temperature and can then be quenchedto ambient temperature while retaining austenite. When Ni content is reduced, intersti-tial elements such as C and N are used to stabilize the face-centered-cubic (FCC) struc-ture. When Cr content is increased to raise the corrosion resistance of the metal (Cr is aferrite stabilizer), Ni must also be increased to maintain the austenitic structure. Thesealloys are slightly magnetic in the cold-worked condition, but usually at ambient con-ditions they are paramagnetic metals with randomly oriented individual magnetic mo-ment on Fe atoms. Tiny magnetic moment develops on Ni and Cr atoms at room tem-perature [10]. The austenitic types feature adaptability to cold forming, ease of welding,high-temperature service, and, in general, the highest corrosion resistance. Austeniticstainless steels are widely known as the 300 series (eg., 304 (18wt%Cr-8wt%Ni), 304L,316 (16wt%Cr-11wt%Ni-2wt%Mo), 317 etc.).

In the present thesis, one problem addressed in austenitic stainless steels is the deter-mination of stacking fault energies. Stacking fault is one of the most common planardefect in FCC metals. The formation energy of a stacking fault (stacking fault energy(SFE or γSFE)) relates to the behavior of dislocations (formation, cross-slip, recombina-tion, etc.). Thus the stacking fault energy is a parameter of significant importance onthe mechanical properties of FCC alloys, such as strength, toughness and fracture. Inaustenitic stainless steels, it has been demonstrated that γSFE correlates closely with theplastic deformation mechanisms [11, 12]. It has been recognized that plastic deforma-tion is mainly realized by martensitic transformations (γ (fcc) → α′ (bcc/bct) or γ → ϵ

CHAPTER 1. INTRODUCTION 3

(hcp)) at low γSFE values and by twining at intermediate γSFE ( 18 . γSFE . 45 mJ m−2).At even higher γSFE, plasticity and strain hardening are controlled solely by the slideof dislocations [13–15]. Experimentally, there is no precise method for measuring theSFE. In Chapter 6, using the EMTO-CPA method, we investigate the alloying effects ofMn, Co and Nb on the stacking fault energy in austenitic stainless steels. The reader isreferred to Chapter 6 and supplemented Paper 4 for details of that study.

Binary solid solutions, Pd-Ag, Pd-Cu, Pt-Cu and Ni-Cu, are also investigated in thepresent thesis. The stacking fault energy as a function of composition has been deter-mined using first-principles methods. These systems are relative simple, comparing tothe stainless steels, therefore they are good candidates to test the two models we use forcalculating the SFE. Details of the two models, the supercell model and the axial inter-action model, are given in Chapter 2. Experimentally, efforts have been made to achievenanocrystalline or ultra-fine grain bulk materials which may posses high strength andhigh ductility using severe plastic deformation in these alloys. [16, 17] The critical roleof stacking faults in the process of microstructure evolution in severe plastic deforma-tion has been recognized. However, the stacking fault energy data for these alloys hasnot been well established. A detailed literature survey shows that, for Pd-Ag alloys,merely Harris et al. [18] and Rao et al. [19] have measured the SFEs by different tech-niques. Although they observed a similar compositional dependence of the SFE, thereported values are rather different. Namely, for the alloys with Ag content less than60 wt.%, the SFE values measured by Rao et al. are about half of those values obtainedby Harris et al. The only available data on the SFEs of Pd-Cu alloys are also due toHarris et al. [18]. For the Pt-Cu system, despite of the numerous investigations on theorder-hardening [20–22], deformation twinning [23], etc., no SFE has been measured upto now. The SFE has been extensively measured for Ni-Cu alloys. However, besides thesimilar composition dependence, the experimental values from different measurementsmay differ by one order of magnitude [24]. Composition dependence of the SFE andelectronic structures are discussed in Chapter 5 and supplemented Paper 3.

Chapter 2

Spinodal Decomposition in BCCIron-Chromium alloys and StackingFault in FCC Metals

In this chapter, I introduce the spinodal decomposition and its effects on the mechanicalproperties of ferrite stainless steels in the first part. The significance of the interface en-ergy between the decomposed phases is also discussed. In the second part, I introducethe concepts of stacking faults and their effects on mechanical properties of FCC metals.In FCC metals, stacking faults are very common planar defects, especially after severeplastic deformation.

2.1 Spinodal Decomposition

2.1.1 What is spinodal decomposition?

Fe-Cr is a typical system exhibiting miscibility gap at low temperature. The low-temperaturepart of the Fe-Cr phase diagram is shown in Figure 2.1. The solid and dashed curvesinclose the miscibility region and the spinodal decomposition region, respectively. Forthe composition range between the two lines, phase separation is due to the nucleus andgrowth mechanism. Inside the spinodal line, it is the spinodal decomposition mecha-nism that works. Fe and Cr fully solute with each other with body-centered-cubic crys-tallography structure at high temperature (T& 475◦C). Aging at lower temperature, theFe-Cr alloy is thermodynamically unstable and decomposes into α Fe-rich and α′ Cr-rich phases. The decomposition mechanism depends on the concentration of Cr, whichis nucleus and growth in Fe-dilute/Cr-dilute alloys (two ends of the composition range)and spinodal decomposition in intermediate Cr-rich alloys (∼15-75% Cr). The decompo-sition degrades seriously the mechanical properties of ferritic stainless steels, namelyhardening the materials and making the material brittle, which is usually denoted ”475◦C embrittlement”. For example, the hardness of an Fe-28Cr steel can increase by more

4

CHAPTER 2. SPINODAL DECOMPOSITION IN BCC IRON-CHROMIUM ALLOYSAND STACKING FAULT IN FCC METALS 5

Figure 2.1. Low temperature part of the Fe-Cr phase diagram by Calphad calcula-tion. [25]

than 300 Hv over an exposure of 10,000 h at 450 ◦C [26]. This results in a severe dropof impact toughness and ductility. An ultrafine-scale interconnected network structureof α and α′ phases develops from the uniform matrix of solid solution upon aging. Thecomposition separation can be detected by atom probe experiments, from which we canobtain the concentration profile for each component element, showing the characteris-tic wave-like concentration variation of spinodal decomposition [27]. Duplex stainlesssteels which approximately consist of 50% ferrite and 50% austenite also are also sub-jected to the ”475 ◦C embrittlement” because of the presence of ferrite.

2.1.2 Critical nucleus size

For both phase separation mechanisms, the interfacial energy γ between the decom-posed phases α and α′ plays an important role. The interfacial energy is crucial ineither determining the critic nucleus radius in the nucleation and growth theory [28]or the critical wave-length of the fluctuations in the spinodal decomposition mecha-nism [29–35]. According to the classical nucleation theory [28], the work W required toform a heterogeneous spherical grain of radius R in an otherwise homogenous phase is

W = 4πR2γ − 4

3πR3∆P, (2.1)

where ∆P is the volumetric driving force for the decomposition. The extremum ofW (R)determines the critical nucleus size

Rcrit = 2γ/∆P. (2.2)

A nucleus will grow continuously with initial size R > Rcrit and disappear with R <Rcrit. When the solution is brought into the spinodal region by changing its temperatureand composition, the nucleus critical size becomes unrealistically small, which makes

CHAPTER 2. SPINODAL DECOMPOSITION IN BCC IRON-CHROMIUM ALLOYSAND STACKING FAULT IN FCC METALS 6

the nucleation process meaningless. Small microdomains or clusters rapidly appear andgrow by diffusion and coalescence induced by thermofluctuations. The critical size inthe spinodal region is so small that almost all the concentration fluctuations could serveas a nucleus and grow spontaneously. Therefore, the spinodal decomposition does notinvolve a nucleation barrier mechanism.

2.1.3 Alloying effects on spinodal decompositionAlloying has significant effects on the shape of spinodal line and the kinetic process ofphase separation. Cobalt increases the critical decomposition temperature and enlargesthe two phases field [27], while Al decreases the critic temperature and shrinks the twophases field [36]. Addition of Ni appears to increase the rate of hardening and alsoaccelerates the kinetics of spinodal decomposition [37]. Ni is a very common contentin the metal filler for welding ferritic stainless steels. Therefore, when post-weld heat-treatment is not possible, there is higher possibility of weld embrittlement.

2.2 Stacking Fault

2.2.1 What is stacking fault?

Along the<111> direction of FCC crystal structure, the close-packed {111} planes stackwith a sequence which may be denoted as ...ABCABC.... If the stacking sequence, bymistake (eg., produced by plastic deformation), becomes ...ABCABABC..., an intrinsicstacking fault forms which is bordered by an edge partial dislocation with a Burgersvector b = a

6<112> (Shockly dislocation), where a is FCC lattice constant. The stacking

fault area can be seen as thin HCP phase with stacking sequence ...ABAB.... If the stack-ing sequence is ...ABCABACABC..., an extrinsic stacking fault is included, which is alsobordered by a partial dislocation with a different Burgers vector b = 1

3<111> (Frank

dislocation). A twinning is formed when the stacking sequence is symmetrical with re-spect to plane C, e.g., ...ABCABCBACBA.... Considering only interactions between thenearest neighboring layers leads to the approximate relation [38]

2∆Ehcp−fcc ≃ 2γtwin ≃ γintrinsic ≃ γextrinsic, (2.3)

where ∆Ehcp−fcc is the structural energy difference between the HCP and the FCC struc-tures. In the present work, only the formation energy of an ideal intrinsic stackingfault, which is assumed laterally infinitely expanded, is calculated. Strain for realisticdislocation-terminated stacking fault will be discussed in Chapter 5.

2.2.2 Methods for measuring the SFE

Partial dislocation nodes with enclosed stacking faults in austenitic Fe-Cr-Ni stainlesssteels are shown in Figure 2.2. The width of these stacking faults is a consequence of the

CHAPTER 2. SPINODAL DECOMPOSITION IN BCC IRON-CHROMIUM ALLOYSAND STACKING FAULT IN FCC METALS 7

Figure 2.2. Dislocation nodes in austenitic stainless steels. [39]

balance between the repulsive force between two partial dislocations on one hand andthe attractive force due to the surface tension of the stacking fault on the other hand.The equilibrium width is determined by the stacking fault energy. Experimentally, afteryou measure the width of dissociated dislocations, the SFE can be evaluated using thefollowing equation [40]

γ =Gb2p8π∆

· 2− ν

1− ν(1− 2ν

2− νcos 2θ), (2.4)

where G is shear modulus, bp is the Burgers vector of the partial dislocations, ∆ is thewidth of partials, ν is the Poisson ratio and θ is the angle between the dislocation lineand the Burgers vector of the perfect dislocation. Using a transmission electron micro-scope (TEM), a direct observation of stacking fault configurations (in particular triplenodes and stacking fault ribbons) in thin foils can be realized. The width of partial canthen be measured. In practice, however, the dimensions of stacking fault nodes andribbons observed in a given sample at room temperature generally exhibit considerablescatter and standard deviations in the experimental measurements of the order of 10 -20% of the mean value are quite common [41].

Another frequently used method for measuring the SFE is by means of X-ray diffrac-tion [42]. Stacking faults in a FCC lattice produce a slight shift in the angular positionof most of the diffraction peaks. And then, the stacking fault probability, α, can bemeasured by comparing the separation between (111) and (200) peak positions for bothannealed and cold-worked specimens. Thus

∆2θ = (2θ200 − 2θ111)cold−worked − (2θ200 − 2θ111)annealed

=−45

√3

π2[tan θ200 +

1

2tan θ111]α, (2.5)

where θ is the peak position in degrees. Then the SFE can be calculated from [43]

γ =K111ω0Gb111aA

−0.37

√3π

⟨ϵ250⟩111α

, (2.6)

CHAPTER 2. SPINODAL DECOMPOSITION IN BCC IRON-CHROMIUM ALLOYSAND STACKING FAULT IN FCC METALS 8

where K111ω0 = 6.6 is a proportionality constant, G111 = 13(c44 + c11 − c12) is the shear

modulus in the (111) fault plane, a is the unit cell edge dimension andA = 2c44/(c11−c12)is the Zener elastic anisotropy. With X-ray diffraction method, one can measure veryhigh SFE which is beyond the ability of TEM method (. 70 mJ m−2).

Although various technologies have been developed to measure the SFE, none of themis precise. The scattering of the SFE values has been studied and discussed in manypapers, please refer to Refs. [42, 44–49].

2.2.3 SFE and mechanical propertiesThe stacking fault energy is known to be closely related to mechanical properties (suchas plasticity, strength, ductility, twinning ability, and creep resistance, etc.) of metalsand alloys [50, 51]. When the SFE is high, the dissociation of a perfect dislocation intotwo partials is unlikely and the material deforms only by dislocation glide. LowerSFE materials display wider stacking faults and have more difficulties for cross-slipand climb because before crossing a defect partials need to reunite. The SFE modifiesthe ability of a dislocation in a crystal to glide onto an intersecting slip plane. Whenthe SFE is low, the mobility of dislocations in a material decreases. One example isshown by the role of the SFE played in severe plastic deformation (SPD) experiments.Nanocrystalline/ultrafine-grained materials may possess both high strength and highductility [16, 17]. Significant grain refinement is achieved by processing the materialsunder severe plastic deformation. Under SPD, the hardness and the grain size satu-rate to steady state levels at high strains. In various systems, e.g., Cu-Al [52–55], Cu-Zn [56, 57], Pd-Ag [58–60], Al-Mg [61] and stainless steels [62], the minimum grain size(dmin) is found to be related to the stacking fault energy. Namely, dmin decreases withdecreasing the SFE. A relationship between dmin and the SFE was proposed by Mo-hamed [63],

dmin

b= A

( γ

Gb

)q

, (2.7)

where b denotes the Burgers vector and G represents the shear modulus. A is a dimen-sionless constant. A and q are parameters which depend on external loading conditions.For different SPD processes, e.g., High Pressure Torsion (HPT), equal-channel angularpressing (ECAP), dynamical plastic deformation (DPD), etc., they are different [52, 64].This equation indicates that alloying effects present in the stacking fault energy andshear modulus should also show up in the minimum grain size.

In order to optimize the mechanical properties as desired, γSFE has to be adjusted to anappropriate value. Several variables, such as composition [13, 42, 48], temperature [39,41,47,65,66], grain size [67,68] and deformation ratio [12,14], have been shown to affectγSFE. Alloying effects on γSFE are most important and are quite complicated. Experi-mental observations on the alloying effect commonly contradict to each other [42,48,69].

Based on the existing databases, several empirical relationships between γSFE and chem-ical compositions have been proposed [42, 44, 48]. However, the application of these

CHAPTER 2. SPINODAL DECOMPOSITION IN BCC IRON-CHROMIUM ALLOYSAND STACKING FAULT IN FCC METALS 9

empirical relationships is limited and in most cases they are unable to reproduce thecomplex nonlinear dependence of γSFE on the composition [45, 70, 71]. Moreover, theseempirical relationships can hardly address appropriately the interactions between dif-ferent alloying elements.

2.2.4 Two models for calculating SFE

Based on first-principles calculations, there are two models commonly used for calcu-lating the SFE. Within the supercell model (SM), the SFE is determined by the energydifference between structures with and without stacking fault. Then the intrinsic stack-ing fault energy (γ) is computed as

γ = (Fisf − F )/A, (2.8)

where Fisf is the free energy of the supercell consisting of FCC (111) layers with oneintrinsic stacking fault and A is the area of stacking fault. Large supercells have to beadopted to calculate the SFE so as to reduce the interaction between the faults due to theperiodic boundary conditions, which, however, demands heavy computational efforts.In the present calculations, the reference bulk energy (F ) is calculated from the energydifference between two supercells each containing a stacking fault: one with 8 layersand the other with 11 layers. This treatment eliminates the errors induced by k-pointsampling and structure relaxation.

Alternatively, γ can be calculated by the use of a parameterized method, i.e., the axialinteraction model (AIM) [45, 72, 73]. Within the axial interaction model, along the [111]direction, any stacking sequence of the close-packed (111) planes can be mapped by aset of variable Si, where i is the layer index. Si may take two possible values: +1 if thelayer at site (i+1) follows the ideal FCC stacking sequence, else a value of −1 is assigned.Using pair-like interaction between layers, the energy of a structure with any particularstacking sequence can be expanded as

F = J0 − J1∑i

SiSi+1 − J2∑i

SiSi+2 − J3∑i

SiSi+3 − o(J4), (2.9)

where the sums run over the atomic layers. J0 is the energy per unit cell in one layer ifthe interactions between layers are disregarded. J1, J2,...are the nearest-neighbor, nextnearest-neighbor, etc., layers interaction parameters. o(J4) stands for the contributionfrom high order terms. Using this expression, the energies of an intrinsic stacking fault,FCC, HCP and double-HCP structures can be expressed as

CHAPTER 2. SPINODAL DECOMPOSITION IN BCC IRON-CHROMIUM ALLOYSAND STACKING FAULT IN FCC METALS 10

Fisf − Ffcc = −4J1 − 4J2 − 4J3 − o(J4)

Ffcc = J0 + J1 + J2 + J3 + o(J4)

Fhcp = J0 − J1 + J2 − J3 + o(J4) (2.10)Fdhcp = J0 − J2 − o(J4).

Taking into account the interactions between the (111) layers up to the third nearestneighbors, the SFE can then be extracted from the energies of FCC, HCP and DHCPstructures

γ = (Fhcp + 2Fdhcp − 3Ffcc)/A. (2.11)

The AIM adopted by Vitos et al. [45, 70, 71, 74] in the case of paramagnetic steel alloys,led to SFE in good agreement with experimental results.

Normally, austenite stainless steels are paramagnetic with local magnetic moments [75].In the present work, the paramagnetism is represented by the disordered local mag-netic moments (DLMs) approximation [76–78]. Within this approximation, at 0 K nolocal magnetic moments develop on Cr and Ni sites, except on Fe sites. Thermal spin-fluctuations, however, can induce non-zero local magnetic moments on the Cr and Nisites as well.

The disordered local magnetic moments contribute to the stacking fault energy in termsof the magnetic entropy which is calculated by the following expression

γmag = −T (Smaghcp (µ) + 2Smag

dhcp(µ)− 3Smagfcc (µ))/A2D, (2.12)

where Smaghcp (µ), Smag

dhcp(µ) and Smagfcc (µ) denote the magnetic entropies in HCP, DHCP and

FCC phases, respectively. In a completely disordered paramagnetic alloy the mag-netic entropy may be estimated using the mean-field expression Smag = ΣikBcilog(µi+1)(where µi denotes the local magnetic moments of atom i, kB is the Boltzmann constantand ci is the concentration of atom i [79]. The contribution of electronic entropy andlattice vibrational entropy at room temperature were verified to be relatively insignifi-cant [45, 70, 71] and thus are neglected in the present work. γmag is thus the dominanttemperature dependent part of the total stacking fault energy.

The definition of γmag implies that a large γmag is expected if the difference in the localmagnetic moments from the hcp (µhcp

i ) or dhcp (µdhcpi ) and fcc (µfcc

i ) phases are signifi-cant. In the present study we neglect the thermal induced spin fluctuations and assumethat µi is constant at low temperature. This approximation means that γSFE dependslinearly on the temperature.

Chapter 3

Theoretical Methodology

In this chatter, we outline the basic aspects of the first-principles or ab initio theory whichis used to calculate the properties of materials here. This chapter is based on Refer-ences [80, 81].

3.1 Density Functional TheoryThe starting point of the first-principles theory is the hamiltonian for the system of elec-trons and nuclei

H = −~2

2

∑I

∇2I

MI

− ~2

2me

∑i

∇2i −

∑i,I

e2ZI

|ri − RI |+

+1

2

∑i =j

e2

|ri − rj|+

1

2

∑I =J

e2ZIZJ

|RI − RJ |, (3.1)

where electrons are denoted by lower case subscripts and nuclei, with charge ZI andmass MI , denoted by upper case subscripts. It is essential to include all the effects ofmany-body interactions, namely the electron-electron, electron-nuclei and nuclei-nucleiinteractions. The central issue of electronic structure theory is to develop methods thatcan treat electronic correlations with sufficient accuracy. The first step to simplify theproblem is given by the Born-Oppenheimer approximation. Because the nuclei aremuch heavier than the electrons, then it is assumed that the electronic subsystem is al-ways in its stationary state on the time scale of nuclear motion. Then the motion of thenuclei is solved separately, and the fundamental hamiltonian for the theory of electronicstructure can be written as

H = T + Vext + Vint + EII . (3.2)

11

CHAPTER 3. THEORETICAL METHODOLOGY 12

Adopting Hartree atomic units ~ = me = e = 4π/ϵ0 = 1, the kinetic energy operator forthe electrons T is

T =∑i

−1

2∇2

i . (3.3)

Vext is the potential acting on the electrons due to nuclei,

Vext =∑i,I

VI(|ri − RI |). (3.4)

Vint is the electron-electron interaction

Vint =1

2

∑i=j

1

|ri − rj|, (3.5)

and the final term EII is the classic interaction of nuclei and any other terms that con-tribute to the total energy of the system but are not germane to the problem of describ-ing electrons. With the above hamiltonian, the Schrodinger equation for the interactingelectrons gas becomes,

HΨ = EΨ, (3.6)

which is still far too big to be solved for any realistic solid. The second step to solvethis problem is density functional theory (DFT). The original density functional theory ofquantum systems is developed by Thomas [82] and Fermi [83]. In the original work,the kinetic energy of the electron system is approximately expressed as an explicit func-tional of electron density, idealized as non-interacting electrons in a homogeneous gaswith density equal to the local density at any point. However, the Thomas-Fermi ap-proximation is not accurate enough for electronic structure calculations. In 1964, Ho-henberg and Kohn [84] showed particle density in the ground state of a many-bodysystem plays a special role that the density is a basic variable for any properties. TheHohenberg-Kohn theorems are summarized as,

Theorem I: For any system of interacting particles in an external potential Vext(r), thepotential is determined uniquely, except for a constant, by the ground state particledensity n0(r).

Theorem II: A universal functional for the energy E[n] in terms of density n(r) can be de-fined valid for any external potential Vext(r). For any particular Vext(r), the exact groundstate energy of the system is the global minimum value of this functional, and the den-sity n(r) that minimizes the functional is the exact ground state density n0(r).

In 1965, Kohn and Sham [85] came up with an approach to replace the original many-body problem by an auxiliary independent particle problem. Within the Kohn-Shamapproach, it is assumed that the ground state density of the original interacting system

CHAPTER 3. THEORETICAL METHODOLOGY 13

is equal to that of some chosen non-interacting system. It results in the independent-particle Schrodinger equations for non-interacting system that can be considered ex-actly solvable and all the complexity of many-body interactions is incorporated intoan exchange-correlation functional of the density. The famous Kohn-Sham single-particleSchrodinger equations are

(HσKS − εσi )ψ

σi (r) = 0, (3.7)

where εσi are the eigenvalues for spin σ, andHKS is the effective hamiltonian (in Hartreeatomic units)

HσKS(r) = −1

2∇2 + V σ

KS(r), (3.8)

withV σKS(r) = Vext + VHartree(r) + V σ

xc(r). (3.9)

Then the ground state energy functional can be expressed as,

EKS = Ts[n] +

∫drVext(r)n(r) + EHartree[n] + EII + Exc[n], (3.10)

where Ts is the kinetic energy of a non-interacting reference system, given explicitly asa functional of the non-interacting electron orbitals,

Ts = −1

2

∑σ

Nσ∑i=1

< ψσi |∇2|ψσ

i >= −1

2

∑σ

Nσ∑i=1

∫d3r|∇2ψσ

i (r)|2. (3.11)

EHartree is the classic Coulomb interaction energy of an electron density interacting withitself

EHartree[n] =1

2

∫d3rd3r′

n(r)n(r′)|r − r′|

. (3.12)

The density of the auxiliary system is given by sums of squares of the orbitals for eachspin

n(r) =∑σ

n(r, σ) =∑σ

Nσ∑i=1

|ψσi (r)|2. (3.13)

All many-body effects of exchange and correlation are grouped into the exchange-correlationenergy Exc

Exc =< T > −Ts[n]+ < Vint > −EHartree[n]. (3.14)

The above equation explicitly shows that Exc consists of the first part of the kineticenergy difference between the true interacting and the auxiliary non-interacting systemsand the second part of the difference between true internal energies of the interactingelectrons and the classic Coulomb Hartree energy.

CHAPTER 3. THEORETICAL METHODOLOGY 14

If the universal functional Exc were known, then the exact ground state energy and thedensity of the many-body electron problem could be found by solving the Kohn-Shamequations for independent-particles self-consistently.

3.2 Exchange-Correlation Functionals: LDA and GGAThe exact formula for exchange-correlation functional,Exc[n], is unfortunately unknown.However, the exchange energy can be calculated exactly within the Hartree-Fock ap-proximation. The exchange interaction acts only between electrons with same spin,which stems from Pauli exclusion principle, while the correlation is much important forantiparallel spins. Considering the fact that in a solid the electron density can often beconsidered as close to the limit of the homogeneous electron gas, the local density approx-imation (LDA) (or generally the local spin density approximation (LSDA)) assumes that ina real system the exchange-correlation energy is simply an integral over all space withthe exchange-correlation energy density at each point exactly the same as in a homoge-neous electron gas with that density [85]

ELSDAxc [n↑, n↓] =

∫d3rn(r)ϵhomxc (n↑(r), n↓(r))

=

∫d3rn(r)[ϵhomx (n↑(r), n↓(r)) + ϵhomc (n↑(r), n↓(r))]. (3.15)

ϵhomx is the exchange energy density for the homogeneous electron gas,

ϵσx = − 3

4πkσF = −3

4(6nσ

π)1/3. (3.16)

In a polarized system, it has the form

ϵx(n, ζ) = ϵx(n, 0) + [ϵ(n, 1)− ϵ(n, 0)]fx(ζ), (3.17)

where ζ = n↑−n↓

nis the fractional polarization, and

fx(ζ) =1

2

(1 + ζ)4/3 + (1− ζ)4/3 − 2

21/3 − 1. (3.18)

The explicit analytic form of correlation energy (ϵc[n]) of homogeneous electron gas asa functional of density was fitted to the numerical results from quantum Monte Carlocalculations [86]. There are several fitted forms for ϵc. One widely used functional dueto Perdew and Zunger (PZ) [87] is

ϵPZc (rs) = −0.0480 + 0.031ln(rs)− 0.0116rs + 0.0020rsln(rs), rs < 1

= −0.1432/(1 + 1.9529√rs + 0.3334rs), rs > 1. (3.19)

CHAPTER 3. THEORETICAL METHODOLOGY 15

where rs = ( 34πn

)3 describes the average distance between electrons. The LDA is ex-pected to be best for solids close to a homogeneous gas and worst for very inhomo-geneous cases like atoms. Generally, the LDA overestimates the bonding and conse-quently overestimates the bulk modulus.

Inspired by the success of LSDA, various generalized-gradient approximations (GGAs)with marked improvement over LSDA for many cases have been developed. WithinGGA, the exchange-correlation energy depends not only on the local electron density,but also on its local density gradient

EGGAxc [n↑, n↓] =

∫d3rn(r)ϵxc(n↑, n↓, |∇n↑|, |∇n↓|, ...)

≡∫d3rn(r)ϵhomx (n)Fxc(n

↑, n↓, |∇n↑|, |∇n↓|, ...). (3.20)

Fxc is a dimensionless exchange enhancement factor which is a functional of the localdensity and its gradient. And ϵhomx [n] is the exchange energy of the unpolarized homo-geneous gas. For most materials, the enhancement factor has a value ∼1.3-1.6. All thedifferent types of GGA lead to lower exchange energies than the LDA, which resultsin a reduced binding energy, a correction of the LDA overbinding and an improvedagreement with experiment.

3.3 Computational methodsMany DFT methods have been developed based on solving the single-electron Kohn-Sham equations. The full potential methods can give exact local densities and lead tovery accurate descriptions for many physical properties of solid materials. However,these techniques are computationally expensive. Pseudopotential methods are devel-oped to give fast computational speed. Within pseudopotentials, a full-potential de-scription is kept only for interstitial regions, where the bonds are located, whereas thetrue Coulomb-like potential is replaced with a weak pseudopotential in the region nearthe nuclei. Another important group of methods is established based on the muffin-tinapproximation. Based on the observation that in solids the potential and the electrondensity vary strongly and are spherically symmetrized in the vicinity of nuclei, whilethey are smoother between the atoms, theoretically, space may be divided into two re-gions, the spheres around atomic sites and the interstitial region between spheres, wheredifferent kinds of potential representations and basis functions are used when solvingthe Kohn-Sham equation. Due to the adopted approximations, the muffin-tin methodsmostly restrict themselves to densely packed systems.

A ”new” muffin-tin formalism developed in the 1990s is called exact muffin-tin orbitals(EMTO) theory, which distinguishes itself by applying the optimized overlapping muffin-tin (OOMT) potential [81]. The EMTO method, together with the coherent potential

CHAPTER 3. THEORETICAL METHODOLOGY 16

approximation (CPA), using the Green function formalism other than the Hamiltonianformalism, is suitable for studying disordered systems. In the following part of thischapter, the general aspects of the EMTO method are introduced.

3.3.1 Exact Muffin-tin Orbital method

Within the overlapping muffin-tin approximation, the effective single-electron potentialis approximated by spherical potential wells υR(rR) centered on lattice sites R plus aconstant potential υ0, i.e.,

υKS(r) ≈ υmt(r) ≡ υ0 +∑R

[υR(rR)− υ0]. (3.21)

The spherical and the constant potentials (υR(rR) and υ0, respectively) are determinedby optimizing the mean of the squared deviation between the muffin-tin potential (υmt(r))and υ(r), which leads to the so-called optimized overlapping muffin-tin (OOMT) poten-tial.

3.3.2 Exact muffin-tin orbitals wave function

The single-electron Kohn-Sham equations, with the effective potential vKS(r) approx-imated by the muffin-tin potential (vmt(r)), are solved by expanding the Kohn-Shamorbital in terms of the exact muffin-tin orbitals (ψ

a

RL), viz.

Ψj(r) =∑RL

ψa

RL(ϵj, rR)υaRL,j. (3.22)

The expansion coefficients, υaRL,j , are determined in a way that Ψj(r) is a solution forEq. (3.7) in the entire space.

The exact muffin-tin orbitals, ψa

RL(ϵj, rR), are constructed using different basis functionsinside the potential sphere (rR ≤ sR) and in the interstitial region (rR > sR). sR is theradius of the potential sphere centered at site R.

Inside the potential sphere (rR < sR) the basis functions, so-called partial waves (ϕaRL),

are constructed from solutions of the scalar-relativistic, radial Dirac equation for thespherical potential (ϕRl) and the real harmonics (YL(rR))

ϕaRL(ϵ, rR) = Na

Rl(ϵ)ϕRl(ϵ, rR)YL(rR). (3.23)

The normalization factor NaRl(ϵ) is determined by the matching conditions at the poten-

tial sphere boundary with the basis function outside of the potential sphere.

In the interstitial region the basis functions are solutions of the free electron Schrodingerequation, called screened spherical waves (ψa

RL(ϵ − υ0, rR)). The boundary conditions forthe free electron Schrodinger equation are given in conjunction with non-overlapping

CHAPTER 3. THEORETICAL METHODOLOGY 17

spheres, called hard spheres, centered at lattice siteRwith radius aR. The screened spher-ical waves are just defined as being free electron solutions which behave as real harmon-ics on their own a-spheres centered at site R and vanish on all the other sites.

The partial waves and the screened spherical waves must join continuously and dif-ferentiable at aR. This is implemented using additional free electron wave functions(φa

Rl(ϵ, rR)), by which the connection between the screened spherical waves and the par-tial waves is obtained. It joins continuously and differentiable to the partial wave at sRand continuously to the screened spherical wave at aR. Since aR < sR the additionalfree-electron wave function should be removed, which is realized by the so-called kink-cancelation equation.

3.3.3 The full charge density (FCD) total energy

Within the present EMTO method, to obtain the total charge density, the total numberof states and the total energy, the Green’s function formalism is employed, instead ofcalculating all possible wave functions and single electron energies. This is because thatboth self-consistent single electron energies and the electron density can be expressedwithin Green’s function formalism [81].

The total charge density is obtained by summations of one-center densities, which maybe expanded in terms of real harmonics around each lattice site

n(r) =∑R

nR(rR) =∑RL

nRL(rR)YL(rR). (3.24)

The total energy of the system is obtained using the total charge density via the fullcharge density (FCD) technique, where the space integrals over the Wigner-Seitz cellsin Eq. (3.10) is solved via the shape function technique, the interaction energy betweenremote Wigner-Seitz cells is taken into account through the Madelung term, a particu-larly delicate contribution arising from Wigner-Seitz cells with overlapping boundingspheres is calculated by the so-called displaced cell technique [81].

The FCD total energy is decomposed in following terms

Etot = Ts[n] +∑R

(FintraR[nR] + ExcR[nR]) + Finter[n] (3.25)

where Ts[n] is the kinetic energy, FintraR the electrostatic energy due to the charges insidethe Wigner-Seitz cell, Finter the electrostatic interaction between the cells (Madelungenergy), and ExcR the exchange-correlation energy.

3.3.4 Coherent potential approximation (CPA)

The coherent potential approximation (CPA) is implemented using two main approxi-mations. First, it is assumed that the local potentials (Pi) around a certain type of atom

CHAPTER 3. THEORETICAL METHODOLOGY 18

from the alloy are the same, i.e., the effect of local environments is neglected. Second, thesystem is replaced by a monoatomic set-up described by the site-independent coherentpotential (P ). In terms of the Green function, the real Green function is approximated bya coherent Green function (g = [S − P ]−1, S denotes the structure constant matrix.). Foreach alloy component, a single-site Green function (gi) is introduced, which is expressedmathematically via the real-space Dyson equation

gi = g + g(Pi − P )gi, i = A,B,C · · · . (3.26)

The average of the individual Green functions should reproduce the single-site part ofthe coherent Green function, viz.

g = agA + bgB + cgc + · · · . (3.27)

After iterative solution, the output of g and gi’s equations are used to determine theelectronic structure, charge density and total energy of the random alloy.

Chapter 4

α/α′ Interfacial Energy in Ferrite Fe-CrAlloys

In this chapter, we present the interfacial energies between the Cr-rich α′-FexCr1−x andFe-rich α-Fe1−yCry phases (0 < x, y < 0.35) evaluated by using the first-principles exactmuffin-tin orbital method in combination with the coherent potential approximation.Two orientations of the interface, (001) and (110) have been taken into account. Themagnetism of the interface has been studied and its effect on the interfacial energy hasbeen discussed. The theoretical results are applied to estimate the critical grain size fornucleation and growth in Fe-Cr stainless steel alloys. The composition and orientationdependence of the interfacial energy is discussed in relation with the morphology ofprecipitations.

4.1 Interface ModelThe interfaces are modeled by a supercell which contains equal number of atomic layers(N ) for the α and α′ phases. The two phases are assumed to be fully commensurableand the lattice constant of the supercell (a) is fixed to the average of those obtained forthe single α and α′ phases. This approximation is plausible since alloying has smalleffect on the lattice constant of Fe-Cr [88]. The interfacial energy is defined as

γ(x, y) = (∆E(x, y)−N [∆Eα(y) + ∆Eα′(x)]) /2S, (4.1)

where ∆E(x, y) is the formation energy of the supercell, ∆Eα(y) and ∆Eα′(x) are theformation energies per site of the single phases, and S is the area of the interface. Theaccuracy of the interfacial energy was tested with respect to the number of atomic layersN and the k-point mesh, and finally we chose N = 5 and 13 × 13 × 3 k-points in theirreducible part of the Brillouin zone. With this setup the numerical accuracy of thecalculated interfacial energy is ∼ 1 mJm−2.

Former density-functional calculations performed on the Fe/Cr interfaces found no ten-dency for interface alloy formation between Fe and Cr [89]. We also find that at lower

19

CHAPTER 4. α/α′ INTERFACIAL ENERGY IN FERRITE FE-CR ALLOYS 20

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

y (

-Fe 1-

yCr y)

x ( '-FexCr1-x)

0.00

0.07

0.14

0.21

0.28

0.350.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

y (

-Fe 1-

yCr y)

x ( '-FexCr1-x)

0.03

0.08

0.14

0.19

0.24

0.29

(a) (b)

Figure 4.1. Interfacial energy (in J m−2) between the Cr-rich α′ phase (FexCr1−x)and the Fe-rich α-(Fe1−yCry) phase for (a) (001) interface and (b) (110) in-terface.

temperature, the interface favors sharp composition transition. Considering the mixingentropy, the diffuse interface (mixing interface) will be stabilized at higher temperature.

4.2 Interfacial EnergyFigures 4.1 (a) and (b) show the maps of the interfacial energies for the low temperatureferromagnetic (ferromagnetic α and ferromagnetic or antiferromagnetic α′, dependingon the composition) state versus composition for the (001) and (110) α-Fe1−yCry/α′-FexCr1−x (0≤x, y≤0.35) interfaces, respectively. Generally, the two maps have very sim-ilar features. The interfacial energy decreases with increasing x and y beyond the regiony.0.1 and almost vanishes for x≈y≈0.35. The general trend is due to the fact that, withincreasing x and y, the compositional difference (composition gap) between the α andα′ phases decreases, and consequently the interfacial energy decreases. The anomalousincrease of the interfacial energies when going from y=0 to y≈0.1 at fixed x stems fromthe magnetic frustration at the interface.

For paramagnetic state, the interfacial energy, Figure 4.2, changes nearly linearly with xand y. Notice that in general interfacial energies for both magnetic states decrease fasterwith x than with y. This means, for instance, that the surface pressure is smaller for anFe grain in Fe0.20Cr0.80 than for a Cr grain in Fe0.80Cr0.20, indicating that it is easier tocreate Fe nuclei in Cr-rich alloys than to create Cr nuclei in Fe-rich alloys.

CHAPTER 4. α/α′ INTERFACIAL ENERGY IN FERRITE FE-CR ALLOYS 21

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

y (

-Fe 1-

yCr y)

x ( '-FexCr1-x)

0.00

0.07

0.14

0.21

0.28

0.35

Figure 4.2. Interfacial energy (in Jm−2) between the Cr-rich α′ phase (FexCr1−x)and the Fe-rich α (Fe1−yCry) phase for paramagnetic state.

4.3 Anisotropy of The Interfacial EnergyIn Figure 4.3, we plot the interfacial energy anisotropy defined by the ratio γ001/γ110 asa function of the chemical compositions of the decomposed phases. In general, fromsimple bond-cutting arguments, one would expect that the formation energy for the

1.13 1.14

1.151.17 1.181.19

1.211.22

1.23 1.251.26 1.27

1.291.30

1.311.33

1.34

1.35

1.37

1.38

1.25

1.39

1.38

1.27

1.41

1.35

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

x ( FexCr1-x)

y (

Fe1-

yCr y)

1.10

1.17

1.23

1.30

1.37

1.43

1.50

Figure 4.3. The interfacial energy anisotropy ratio γ001/γ110 for Fe-Cr alloys plottedwith respect to the chemical composition of the α and α′ phases.

CHAPTER 4. α/α′ INTERFACIAL ENERGY IN FERRITE FE-CR ALLOYS 22

(110) surface/interface is smaller than that for the (001) one. One exception is bcc Crfor which the (001) facet has lower surface energy (the bulk-vacuum interfacial energy)than the close packed (110) one, which has been ascribed to the enhanced surface mag-netism [90–92]. For the present system, the (110) interfacial energy is smaller than thatfor the (001) interface for the whole composition range considered in this study and thuswe conclude that in Fe-Cr the interfacial energy shows the normal anisotropy. On theother hand, from Figure 4.3 we observe a strong nonlinear composition dependence forγ001/γ110. At the bottom-left corner (x, y.0.2) the interfacial energy anisotropy is small,approaching 1. With increasing x and y, the anisotropy becomes larger, especially forthe up-right corner (x, y&0.2), where γ001 is larger than γ110 by more than 30%.

According to Figure 4.3, for x, y.0.2, the negligible anisotropy implies nearly spheri-cal equilibrium shape for the nanometer-sized crystals precipitated in the matrix. Thatis, the pure Cr grains in Fe-rich matrix and the pure Fe-grains in Cr-rich matrix arepredicted be nearly spherically symmetric at low temperature. On the other hand, forx, y&0.2, the anisotropy becomes significant (&1.2) and a polyhedral equilibrium shapeis expected.

4.4 Magnetic Structure at The InterfaceFigure 4.4 shows the magnetic moments of the atoms in the supercells of α′-FexCr1−x/α-Fe1−yCry (001) interface with x = y = 0.35 and x = y = 0.20 in FM and DLM states. Forboth interfaces in FM state (Figure 4.1 a and b), Fe atoms are less magnetized at theCr-rich side than at the Fe-rich side. At the Cr-rich side, the magnetic moments of thesurface Fe atoms (layer 1 and 5) are larger (in absolute value) than those of the bulkFe atoms (layer 2, 3, and 4) and vice verse for the Fe-rich side. All the Fe atoms areferromagnetically aligned. The bulk Cr atoms (layer 7, 8, and 9) are ferromagneticallycoupled to each other but antiferromagnetically coupled to Fe at the Fe-rich sides ofboth interfaces. However, the magnetic orientations of the bulk Cr atoms (layer 2, 3,and 4) at the Cr-rich side are composition dependent: for the x = y = 0.35 interface(Figure 4.4a), they are ferromagnetically aligned whereas for the x = y = 0.20 interface(Figure 4.4b), they are antiferromagnetically aligned.

The main features of the magnetic moments of Fe in DLM are similar to those in FMstate. However, there are notable differences: (i) the absolute values are generallysmaller; (ii) Although the absolute magnetic moments of the bulk Fe atoms are larger inFe-rich side than in Cr-rich side, the DLM magnetic moments of the interface atoms aresmaller in Fe-rich side than in Cr-rich side.

The maximum of the interfacial energy for ferromagnetic state around 0.05 < y <0.15 for a fixed x (Figure 4.1(a)) can also be ascribed to the peculiar magnetic inter-actions. Within the Fe rich interface layer, the Cr atoms experience a locally high Cr-concentration environment due to the α′-FexCr1−x side of the interface. To minimize theferromagnetic Cr-Cr interactions, the Cr atoms within the interface layers decrease their

CHAPTER 4. α/α′ INTERFACIAL ENERGY IN FERRITE FE-CR ALLOYS 23

-2.10

-2.00

-1.90

-1.80

1 2 3 4 5 6 7 8 9 10

-0.05

0.00

0.05

0.10

-Fe0.65Cr0.35

Fe '-Fe0.35Cr0.65

Layer

Cr

-2.20

-2.00

-1.80

-1.60

1 2 3 4 5 6 7 8 9 10

-0.20

0.00

0.20

0.40

-Fe0.80Cr0.20

Fe '-Fe0.20Cr0.80

Layer

Cr

(a) (b)

1 2 3 4 5 6 7 8 9 10

-1.80

-1.65

-1.50

-1.35

-Fe0.65Cr0.35

Layer

Fe

'-Fe0.35Cr0.65

1 2 3 4 5 6 7 8 9 10

-2.00

-1.75

-1.50

-1.25

-Fe0.80Cr0.20

Layer

Fe'-Fe0.20Cr0.80

(c) (d)

Figure 4.4. Magnetic moments of the atoms in the supercells of α′-FexCr1−x/α-Fe1−yCry (100) interfaces with compositions x = y = 0.35 (a and c) andx = y = 0.20 (b and d) in ferromagnetic (a and b) and disordered localmagnetic states (c and d), where 1 and 5 are the Cr-rich interface layersand 6 and 10 are Fe-rich ones.

0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

8 FexCr1-x

FM

Form

atio

n En

ergy

(mR

y/at

om)

cFe (at.%)

DLM

Fe1-yCry

Figure 4.5. Formation energy of FecCr1−c in ferromagnetic and disordered localmoment states.

CHAPTER 4. α/α′ INTERFACIAL ENERGY IN FERRITE FE-CR ALLOYS 24

0 20 40 60 80 1000

60

120

180

240

R Crit

(A)

CFe

of Homogenous Phase (%)

FM DLM

Solubility Limit

Spinodal Line

Figure 4.6. Critical nucleus size (Rcrit in A as a function of the composition of thehomogenous phase FecCr1−c for ferromagnetic and paramagnetic states.The vertical lines show the experimental spinodal and solubility limits.

magnetic moments. For ferromagnetic bulk α-Fe1−yCry the formation energy shows ananomalous negative region for y.0.07, Figure 4.5. However, for the Fe-rich interfacelayer, such a minimum in the formation energy disappears because of the Cr-rich co-ordination. Hence, according to Eq. 4.1, the negative bulk formation energy at low yyields a large positive contribution to the interfacial energy. For the paramagnetic mag-netic state, the bulk formation energy shows no anomalous region (Figure 4.5), whichexplains the monotonous structure of the interfacial energy map in Figure 4.2.

4.5 Predicted Critical Nucleus SizeFigure 4.6 shows the critical nucleus size, Rcrit which is calculated according to Eq. 2.2,with respect to the composition of the alloy with homogeneously distributed Fe and Cr.It can be seen from the figure, for both ferromagnetic and DLM paramagnetic states,within the Spinodal line, Rcrit is very small. This means that any size of compositionfluctuation may result in the phase decomposition, corresponding to the Spinodal de-composition for which no nucleation is needed. When the composition of the homoge-neous phase exceed the Spinodal line, Rcrit increases drastically, which indicates that anucleation process is necessary.

Chapter 5

Stacking Fault Energy of Binary Alloys:Pd-Cu, Pt-Cu, Pd-Ag and Ni-Cu

In this chapter, we will calculate the intrinsic stacking fault energy (infinitely expandedintrinsic stacking fault) of binary alloys, Pd-Ag, Pd-Cu, Pt-Cu and Ni-Cu. These systemsare investigated for reasons summarized as following,

1. In order to get optimized combination of mechanical properties, eg., high strengthenand high ductility, these alloys are frequently processed under severe plastic deforma-tion to form nano-crystalline or ultra-fine-grain microstructure. During the process, thestacking fault energy is the most important variable which determines the saturate min-imum grain size obtainable. However, the SFEs of these alloys have not been measuredaccurately.

2. Two models, the supercell model and the axial interaction model, which are com-monly applied for calculating the SFE have not been systematically compared with eachother. Therefore, these two models are applied in those binary alloys to check if the twomodels can give consistent SFE results.

3. The four systems are isoelectronic. By comparing the composition dependence of theSFE with each other, the effects of the electronic topological transitions with compositiondiscovered previously on the SFE is interesting.

This chapter is arranged as following: first, we present the calculated stacking fault en-ergies, then the origin of the non-linear composition dependence of SFE is discussedwith Fermi surface topological transitions. As an immediate exhibition of the impor-tance of SFE, a quantitative relation between the SFE and the minimum grain size isdiscussed.

25

CHAPTER 5. STACKING FAULT ENERGY OF BINARY ALLOYS: PD-CU, PT-CU,PD-AG AND NI-CU 26

0 10 20 30 40 50 60 70 80 90 100020406080100120140160180200 Present. Supercell Model

Present. Axial Interaction Model KKR-CPA, Crampin1 et al. (1993) Exp. Rao et al. (1968) Exp. Harris et al. (1966)

SFE

(mJ m

-2)

Ag at.%

Pd-Ag

0 10 20 30 40 50 60 70 80 90 100

60

80

100

120

140

160

180

200

220 Present. Supercell Model Present. Axial Interaction Model Exp. Harris et al. (1966) Fitting Line to Exp. Data

SFE

(mJ m

-2)

Cu at.%

Pd-Cu

(a) (b)

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

300

350

400

Present. Supercell Model Present. Axial Interaction Model Exp. Harris et al. (1966) Exp. Vishnyakov et al. (1970) Exp. Mitra et al. (1975) Exp. Noskova et al. (1965) Exp. Henderson (1963) Exp. Nakajima (1965)

SFE

(mJ m

-2)

Cu at.%

Ni-Cu

0 10 20 30 40 50 60 70 80 90 100

50

100

150

200

250

300

350

400

SFE

(mJ m

-2)

Cu at.%

Present. Supcell Model Present. Axiel Interaction Model

Pt-Cu

(c) (d)

Figure 5.1. Variation of the stacking fault energy with composition in (a) Pd-Ag,(b) Pd-Cu, (c) Ni-Cu and (d) Pt-Cu alloys. The quoted experimental andtheoretical data are from Refs. [18, 19, 93–98].

5.1 Stacking Fault EnergyThe calculated SFE together with the available experimental and theoretical data forPd-Ag, Pd-Cu and Ni-Cu are plotted in Figure 6.5 (a), (b) and (c), respectively. Wefind that the AIM and SM give consistent results. The theoretical SFEs are in perfectagreement with those measured by Harris et al. [18] using a texture method for bothPd-Ag and Pd-Cu. For the Pd-Ag alloys, the SFE values at high Pd contents (>60 at.%Pd) are strongly underestimated in the work by Rao et al. using an X-ray method [19].Previous theoretical prediction of the SFE by Crampin et al. [93] using a Korringa-Kohn-Rostoker CPA method shows good agreement with our result except for the alloys withintermediate composition.

For the Pd-Cu alloys, the experimental SFEs at high Cu content are dispersive as mea-sured by Harris et al. [18]. They ascribed the SFE scattering from 58% to 100% Cu to the

CHAPTER 5. STACKING FAULT ENERGY OF BINARY ALLOYS: PD-CU, PT-CU,PD-AG AND NI-CU 27

ordering effect. The measured SFE values may be altered by the presence of long/short-range ordering.

For the Ni-Cu alloys, the experimental measurements largely differ from each other(Figure 6.5 (c)). Our theoretical results for Ni-Cu represent some intermediate values.With Cu alloying into Ni, the SFE decreases monotonously. We note that, for Cu con-centration less than 20 at.% the measured SFE experiences a rapid decrease. For highercontent of Cu the measured SFE saturates to the value of pure Cu. The fast decreasein the SFE for the dilute Cu content is not captured by our calculated composition de-pendence of the SFE. This may be ascribed to the fact that we treat the alloy as randomsolid solution, whereas, in experiments, Cu atoms segregate to the stacking fault area(Suzuki effect) or the partial dislocation core (Cottrell pinning). The segregation of Cuatoms reduces the SFE [99]. In Ni-Cu alloys, the segregation of Cu atoms to surfaceand grain boundary is a well-documented phenomenon [100–102]. Atomistic simula-tion in Ni90Cu10 alloy performed by Smith et al. [103] showed that Cu atoms segregatesstrongly to the partial dislocation core and weakly to the stacking fault between the par-tial dislocations. For low Cu concentration alloys with high stacking fault energy, thedistance between the partials is limited and it is difficult to distinguish between the twotypes of segregation.

In a 11-layers supercell of Ni (Cu) which contains one stacking fault, we substitute onelayer of Ni (Cu) with Cu (Ni) and calculate the total energy of the supercell for Cu (Ni)layer locating at different distance away from the stacking fault. We find that energet-ically Cu atoms favor the position near a stacking fault and Ni atoms repel a stackingfault. This may explain the observation in experiments that the alloying of Ni in Cuchanges barely the SFE.

For Pt-Cu alloys, the calculated SFEs are shown in Figure 6.5 (d). There is no experi-mental data for the alloys to compare. Our results represent the first evaluation.

5.2 The Non-linear Composition Dependence ofStacking Fault Energy

We plot the deviation of our calculated SFE from the linear behavior, ∆γ, with respectto the concentration of Cu/Ag in Figure 5.2. For Pd-Cu, Pd-Ag and Pt-Cu systems,we observe a minimum at similar Cu/Ag concentration (around 53 at.%). For Ni-Cualloys, the minima locates at ∼25 at.% Cu. The reason is that the Ni-Cu alloy is param-agnetic for Ni concentration less than ∼44 at.% and for higher Ni content it experiencesa magnetic transition to ferromagnetic state [104]. Spin-polarized calculation of the SFEresults in a smaller value than the non-spin-polarized one [105], therefore it shifts theminima of the SFE curve for Ni-Cu to lower Cu content.

The minima can be explained considering the electronic structure evolution upon al-loying. Advanced electronic structure calculations have shown that the Fermi surface

CHAPTER 5. STACKING FAULT ENERGY OF BINARY ALLOYS: PD-CU, PT-CU,PD-AG AND NI-CU 28

0 10 20 30 40 50 60 70 80 90 100-70

-60

-50

-40

-30

-20

-10

0

Ni-Cu Pt-Cu Pd-Ag Pd-Cu

(mJ m

-2)

Ag/Cu at.%

Figure 5.2. Deviation of the calculated SFE (∆γ) using the AIM from the linearbehavior.

topological changes are responsible for the anomalies in equilibrium properties such asequilibrium volume, specific heat, elastic constants, etc. [106–109]. Bruno et al. [106–108]calculated the Fermi surface topologies of both Pd-Ag and Pd-Cu systems with varyingcompositions. By alloying nobel atoms into the transition metals, the Fermi surfacesmove from the d states towards the sp states. The most important and prominent Fermisurface topological change occurs when the pocket of d holes at the X point disappearsat ∼53 at.% Ag in Pd-Ag and slightly below 50 at.% Cu in Pd-Cu. Further increasingthe concentration of the nobel element, only very weak singularities may occur in the sand p states components. After d band holes get filled, the alloy has a simple-metal-likeFermi surface topology.

5.3 Minimum Grain Size vs. Stacking Fault EnergyIn this section, we discuss the relation between the SFE and the minimum grain sizein severe plastic deformation processes. We reanalyze the experimental data accordingto equation (2.7) without mixing data from different types of SPD experiments (e.g.,Ref. [61]) or including any data for pure metals from other SPD experiments as per-formed previously in Refs. [52, 61, 64], considering that the solute-dislocation effect isunclear now in this case as suggested by Edalati et al. [110]. We linearly fit three setsof experimental data (e.g., SFE, G, dmin) of Cu-Al [52], Cu-Zn [57] and Pd-Ag [58] alloysprocessed through HPT to equation (2.7), from which we evaluate the average HPTexperimental conditions in terms of parameters A and q. Note that all the SFE, shearmodulus and minimum grain size dmin used for fitting are experimental values. Thefitting result is shown in Figure 5.3 (a). A nice linear correlation is achieved betweenthe normalized minimum grain size (log(dmin/b)) and the normalized stacking fault en-ergy (log(γ/Gb)) with a slop of q≈0.653 and A≈1.31×104. Figure 5.3 (b) is similarlyplotted with data of Cu-Al [54, 55, 64] and Al-Mg [111–116] from ECAP experiments as

CHAPTER 5. STACKING FAULT ENERGY OF BINARY ALLOYS: PD-CU, PT-CU,PD-AG AND NI-CU 29

performed in Figure 5.3 (a). The linear fitting parameters we get are A≈3.03×104 andq≈0.696.

-3.2 -2.8 -2.4 -2.0 -1.62.0

2.2

2.4

2.6

2.8

3.0

Cu-10Zn

Cu

Cu-16Al

Cu-8Al

Cu-5Al

Cu-2.3AlCu

Pd-40AgPd-20Ag

Pd-10Ag

Cu-Al Cu-Zn Pd-Ag

log(

d min/b

)

log( /Gb)

0.653

1

Pd

A=1.31 104

-3.2 -2.8 -2.4 -2.0

2.2

2.4

2.6

2.8

3.0

3.2Al-0.12MgAl-0.9Mg

Al-3.3MgAl-2.86Mg

Al-5.3MgCu

Cu-2.3Al

Cu-5Al

Cu-8Al

Cu-11.6Al

Cu-Al Al-Mg

log(

d min/b

)

log( /Gb)

0.696

1Cu-16Al

A=3.03 104

(a) (b)

Figure 5.3. Normalized minimum grain size (dmin/b) as a function of the normal-ized stacking fault energy (γ/Gb) after processing by HPT [52, 57, 58] (a)and ECAP [54, 55, 64, 111–116](b), respectively (logarithmic scale). The in-dividual compositions are shown on both figures.

As long as the parameters of A and q determined, the composition dependence of dmin

for alloys subjected to the same experimental conditions can be calculated using thefirst-principles stacking fault energy and shear modulus. As an example, in Figure 5.4,we plot the predicted dmin values for HPT and ECAP experiments as a function of com-position. The dmin values are calculated according to equation (2.7) using our calculated

0 10 20 30 40 50 60 70 80 90 100

100

150

200

250

300

350

400

HPT Cal. ECAP Cal. HPT, Kurmanaeva et al. (2010) HPT, Choi et al. (2009) HPT, Yang et al. (2010)

d min (n

m)

Ag at.%

Figure 5.4. Calculated minimum grain size (dmin) as a function of Ag concentrationfor the Pd-Ag alloys after processing by HPT and ECAP. The displayedHPT experimental values are from Refs. [58–60]. The error bars of the HPTexperimental values are estimated from Ref. [58].

CHAPTER 5. STACKING FAULT ENERGY OF BINARY ALLOYS: PD-CU, PT-CU,PD-AG AND NI-CU 30

SFE and shear modulus obtained with the same first-principles approach [109] for Pd-Ag and using our fitting parameters A and q. The available dmin values from HPT ex-periments are also included. We observe that dmin gradually decreases with increasingAg. It also shows that after processing by HPT the obtained dmin is roughly half of thevalue obtained after processing by ECAP, which is because of the less severity of ECAPcompared to the HPT process. There is no ECAP data for Pd-Ag alloys to compare withour prediction, therefore we encourage experimentalists to perform these experiments.Similar procedure may be applied for other solid solutions.

It should be emphasized that the model behind equation (2.7) is built on the conceptthat the minimum grain size obtained in SPD is a result of a balance between dislo-cation formation and its recovery [63]. The grain refinement mechanism may changewith the intrinsic parameter of materials, such as, the stacking fault energy and also theexternal loading conditions (lnZ). It is reported that there is a transition of the grainrefinement mechanism from dislocation subdivision to twin fragmentation in pure Cuwith increasing lnZ [117] and in Cu-Al alloys with decreasing the SFE (increasing Alcontent) [64]. When dynamic recrystallization, grain boundary migration or twinningcontrol the grain refinement mechanism, equation (2.7) may become invalid.

Chapter 6

Stacking Fault Energy of AusteniticStainless Steels

Manganese, cobalt and niobium are commonly used alloying elements to improve thespecific mechanical and/or corrosion-resistive properties of stainless steels. The effectsof Mn on the properties of stailess steels have been comprehensively studied, however,its effect of the SFE is controversial [42,48,69]. For Co and Nb, there is no adequate datato judge their effects on γSFE. Though it is noticed that significant interactions amongalloying elements occur and extra care should be taken when extending the empiricalequations beyond the composition ranges used to derive the equations [118], there isa clear need to set up the interaction profiles between alloying elements and to under-stand the fundamental mechanisms controlling the alloying effects on γSFE.

Encouraged by results in previous chapter, we are confident to apply the axial inter-action model in the more complex systems, austenitic stainless steels. In this chapter,the alloying effects of Mn, Co, and Nb on the stacking fault energy (SFE) of austeniticstainless steels, Fe-Cr-Ni with various Ni content, are investigated.

6.1 Comparing with Previous ResultsIn order to verify our assumptions about the temperature effect on the SFE in Chapter 2,in figure 6.1 we compare the theoretical temperature dependence of γSFE calculated forFe70Cr18Ni12 with the available experimental and theoretical data . The almost perfectcorrespondence between the three sets of data confirms the validity of our approxima-tion.

We also present the calculated γSFE for Fe80−nCr20Nin at 300 K in figure 6.2, compar-ing with previous theoretical and experimental results. Taking into account that theexperimental data were measured for samples containing small amount of other impu-rities, such as N, C interstitials and Si, Mn substitutes, the agreement between theoryand experiment can be considered reasonable. Nickel was found to increase γSFE quite

31

CHAPTER 6. STACKING FAULT ENERGY OF AUSTENITIC STAINLESS STEELS 32

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

30

35

Present Vitos et al. (2006 PRL) Olson et al. (1976)

SFE (m

J m-2

)

T (K)

Fe70

Cr18

Ni12

Figure 6.1. Theoretic temperature dependence of the stacking fault energy forFe70Cr18Ni12 (solid line) compared to the available experimental [13](black square) and previous theoretical data [71] (red dashed line).

6 8 10 12 14 16 18 20 22 24 26 28-5

0

5

10

15

20

25

30

35

40Fe80-nCr20Nin

SFE (m

J m-2

)

Ni at. %

Present Fawley et al. (1968) Kaneko (1996) Vitos et al. (2006 PRL) Breedis (1964) Latanision et al. (1969) Murr (1969) Dulieu et al. (1964)

Figure 6.2. Comparison between the theoretical and experimental stacking faultenergies for Fe80−nCr20Nin. References in the legend can be found inRef. [39, 40, 42, 46, 71, 119, 120].

significantly in alloys with less than ∼ 20 at.% Cr.

For alloys with 20 at.% Cr, the linear relationship between γSFE and Ni concentration isreproduced for n . 16 with a slop of ∂γSFE/∂n ≈ 3 mJ m−2 per at.% Ni. At higher Ni,γSFE slightly decreases, which is in line with some experimental or theoretical results [46,48, 66], but not supported by others [45, 69, 71]. We note that in the iso-SFE contourplots in the Fe-rich corner of the Fe-Cr-Ni ternary diagram, at around 20 wt% Cr, theSFE exhibits a saturation/declining trend with increasing Ni exceeding ∼15 wt% inRefs. [48, 66, 121].

CHAPTER 6. STACKING FAULT ENERGY OF AUSTENITIC STAINLESS STEELS 33

1.851.861.871.881.89

(Fe72Cr

20Ni

8)

0.00

0.52

1.04

1.57

2.09

2.61

3.13

1.851.861.871.881.89

a hcp (B

ohr)

(Fe64Cr

20Ni

16)

1.58 1.59 1.60 1.61 1.62 1.631.851.861.871.881.89

(Fe60Cr

20Ni

20)

c/ahcp (ahcp=a(111)fcc )

1.851.861.871.881.89

(Fe72Cr

20Ni

8)

0.00

0.36

0.72

1.08

1.44

1.80

1.851.861.871.881.89

(Fe64Cr

20Ni

16)a hc

p (Boh

r)

1.58 1.59 1.60 1.61 1.62 1.631.851.861.871.881.89

(Fe60Cr

20Ni

20)

c/ahcp (ahcp=a(111)fcc )

(a) (b)

Figure 6.3. The total energy (panel a) (in units of mRy) and the Fe magnetic momentµFe (panel b) (in units of µB) of hcp Fe-Cr-Ni as a function of ahcp and c/ahcpfor 8%, 16% and 20% Ni at 0 K. Upon changing the c/ahcp ratio the ahcp isfixed to a(111)fcc .

6.2 Magnetic TransitionIn figure 6.3 (a) we show the total energies of hcp Fe80−nCr20Nin as functions of latticeparameters a and c/a for various Ni contents at 0 K. The corresponding local magneticmoments of Fe atoms are shown in (b). The red solid and dash lines mark two differentways of relaxing the hcp structures: the prior corresponds to a path when ahcp is keptfixed to a(111)fcc and c is relaxed (route I), and the latter relaxes the volume while keepingc/a ideal (route II). These two routes represent two typical ways when applying the axialinteraction model to calculate the SFE in literature. The dominant feature of the energycontours is that the global minimum state gradually moves from the low volume region(small c/a and small a) to the high volume region (large c/a and large a) with increasingNi content. These two regions correspond to the low spin (LS) and high spin (HS) statesof Fe atoms, respectively, as seen in figure 6.3 (b). The high spin area spreads fromthe high a, high c/a corner to the low a, low c/a corner with increasing Ni. It meansthat the local magnetic moment of Fe in hcp phase experiences a transition (LS → HS)with Ni at around cNi = 16 at.% [45]. For alloys with intermediate content of Ni, e.g.,Fe64Cr20Ni16, following either of the relaxation routes we observe two metastable states,which are also relevant to the LS and HS states, respectively. The difference is that thesetwo routes will not show the magnetic transition at exactly the same composition.

CHAPTER 6. STACKING FAULT ENERGY OF AUSTENITIC STAINLESS STEELS 34

0.00.20.40.60.8

0.00.20.40.60.8

1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.640.00.20.40.60.8

Fe70

Cr20

Ni8Mn

2

Fe64

Cr20

Ni8Mn

8

(a)LS

HS

(b)

Fe70

Cr20

Ni8Co

2

Fe64

Cr20

Ni8Co

8

Fe70

Cr20

Ni8Nb

2

Fe68

Cr20

Ni8Nb

4

Rel

ativ

e En

ergy

(mR

y)

(c)

c/ahcp (ahcp=a(111)fcc )

Figure 6.4. Total energies of the hcp Fe-Cr-Ni-X alloys as a function of hexagonallattice parameter c/ahcp. The energies are plotted relative to the respectiveequilibrium values. X represents Mn (a), Co (b), Nb (c), respectively, andthe actual alloy compositions are shown in the legends. For notations, seelegend for figure 6.3.

Normally, the local magnetic moments of Fe in fcc and dhcp phases change smoothlywith the concentration of Ni or Cr [45]. Then the transition of the magnetic state of hcpphase implies a transition in γmag (large γmag → small γmag). Therefore, one should bevery careful when choosing the state of the hcp phase used to calculate the stacking faultenergy, especially for the compositions close to the magnetic transition. The differencein the energies for these two states may be very small, giving insignificant difference inthe SFE at 0 K (γ0SFE ), but the difference in the local magnetic moment is relatively largeand leads to a big difference in γmag.

The details of the magnetic transition in the hcp phase are expected to depend stronglyon the additional alloying elements. The total energies of hcp Fe-Cr-Ni-X (X=Mn, Nband Co) alloys as a function of c/a for various compositions are plotted in figure 6.4.Generally, we can observe two local minima as a function of c/a in line with figure 6.3 (a)and their relative stability changes with the amount of the additional alloying elementX. The local minima at small c/a are associated with nearly vanishing local magneticmoments of Fe (LS), consequently contribute large γmag to the total SFE, whereas thelocal minima at higher c/a have larger local magnetic moments (HS) and relatively smallγmag. No local magnetic moment develops on the sites of Mn, Co and Nb.

From figure 6.4, Co and Nb are found to favor the HS state as they lower the energy ofthe HS state relative to that of the LS state. Manganese tends to stabilize the LS state

CHAPTER 6. STACKING FAULT ENERGY OF AUSTENITIC STAINLESS STEELS 35

as Cr [45]. Niobium is very efficient in shifting hcp phase from LS to HS state. As seenin figure 6.4 (c), with m = 2, the stable state of hcp Fe72−mCr20Ni8Nbm is LS, whereasit becomes HS with m = 4. The reason is that Nb strongly increases the equilibriumvolume. Manganese and cobalt have negligible effects on the equilibrium volume andconsequently change the spin state of the hcp phase very weakly. By manipulating thelocal magnetic state, alloying addition to Fe-Cr-Ni shows a greater capability in alteringthe formation energy of the stacking fault, besides the intrinsic chemical effect at 0 K.This will be discussed in detail in the following part of this chapter.

6.3 The SFE of Fe-Cr-Ni-X AlloysFigures 6.5 (a), (c) and (e) display the calculated γSFE maps as a function of the chemicalcomposition for alloying elements Mn, Co and Nb at 300 K and figures 6.5 (b), (d) and(f) show the corresponding magnetic contribution to the stacking fault energies (γmag),respectively.

For a fixed concentration of Ni, from figure 6.5 (a), we observes that γSFE decreases withMn concentration in alloys with n . 16 and slightly increases with Mn at higher Ni.The magnetic entropy contribution (γmag) to the SFE always increases with increasingMn content as shown in figure 6.5 (b). At 0 K, Mn is found to decrease the SFE, but thisdecreasing effect is gradually weakened with increasing Ni content, which is the reasonwhy Mn increases the SFE at high Ni content at 300 K.

The dependence of γSFE on Ni is also altered upon adding Mn as shown in figure 6.5 (a).The magnetic transition from LS to HS in hcp phase with increasing Ni is found to bepostponed at high Mn. In particular, the critical point cNi where γSFE begins to decreaseis ∼18 at.% at 8 at.% Mn, compared to cNi ≈ 16 at.% obtained for the case without Mn.

Cobalt is a hcp-stabilizing alloying element and always tends to decrease γSFE in thewhole concentration range studied as shown in figure 6.5 (c). The slop ∂γSFE/∂m isestimated to be about -0.5 (mJ m−2 per at.% Co) in low-Ni (n . 16) alloys and about-2 (mJ m−2 per at.% Co) in high-Ni (n & 16) alloys. The Ni-enhanced decreasing effectof Co on the SFE is also present at 0 K, meaning that this effect has a chemical origin.In figure 6.5 (d), we can see that γmag is only slightly decreased by Co in the wholeconcentration range of Ni (8 ≤ n ≤ 20). The behavior pattern of Ni on γSFE is notacutely altered by Co.

In low Nb alloys (m . 3), at 300 K γSFE increases with Ni for n . 16, while in highNb alloys γSFE decreases with Ni for the whole concentration range of 8 ≤ n ≤ 20(in figure 6.5 (e)). This phenomenon is present at 0 K and thus stems from the intrinsicchemical effect of Nb. Niobium strongly increases γSFE in the low-Ni alloys, but in high-Ni alloys this increasing effect is weakened. On the other hand, Nb always decreasesthe magnetic contribution γmag to the SFE as shown in figure 6.5 (f). Because Ni andNb are both HS stabilizing elements in the hcp phase, γmag has large values only in the

CHAPTER 6. STACKING FAULT ENERGY OF AUSTENITIC STAINLESS STEELS 36

810

1214

1618

20

-10

0

10

20

30

0 1 2 3 4 5 6 7 8

SFE (m

J m-2

)

Mn at. %

Ni at. %

8 10 1214

1618

20

0

10

20

30

012345 6

78

mag

(mJ m

-2)

Mn at. %

Ni at. %

(a) (b)

8101214161820

0

10

20

30

0 1 2 3 4 5 6 7 8

SFE (m

J m-2

)

Co at. %

Ni at. %810

1214

1618

20

10

20

30

01

23 4 5 6 7 8

mag

(mJ m

-2)

Co at. %Ni at. %

(c) (d)

8 10 12 1416

1820

10

20

30

40

50

60

0123456

78

SFE (m

J m-2

)

Nb at. %

Ni at. %

8101214161820

0

10

20

30

0 1 2 3 4 5 6 7 8

mag

(mJ m

-2)

Nb at. %

Ni at. %

(e) (f)

Figure 6.5. The calculated stacking fault energy (γSFE) and the magnetic contribu-tion to the stacking fault energy (γmag) maps of Fe-Cr-Ni-X alloys plottedas a function of composition for T=300 K. X represents Mn in (a) and (b),Co in (c) and (d), Nb in (e) and (f), respectively.

low-Ni, low-Nb region (n . 16, m . 3), beyond which γmag is rather small.

Chapter 7

Concluding Remarks and Future Work

In this thesis, using the exact muffin-tin orbitals method, I investigated the interfacialenergies for the (001) and (110) interfaces between the Cr-rich FexCr1−x alloys and theFe-rich Fe1−yCry alloys. For ferromagnetic alloys, the (110) interface energy is, on theaverage, slightly smaller than the (001) interface energy. However, the anisotropy ratioγ001/γ110 is found to dependent significantly on the composition of the α and α′ phases.For the (001) interfaces, the formation energy decreases by ∼30% when going from theferromagnetic to the paramagnetic interface. The ferromagnetic interfacial energy ex-hibits a strong nonlinear concentration dependence, whereas the paramagnetic interfa-cial energy follows a smooth composition dependence. These trends can be explainedon the basis of the peculiar magnetic interactions in Fe-Cr alloys. Using a continuummodel, we estimated the critical grain size for phase separation and found that criticalradii depend very strongly on the composition of the initial homogeneous alloy. Therapidly increasing Rcrit between the spinodal and solubility lines is in good agreementwith experiments.

I calculated the stacking fault energy for binary Pd-Ag, Pd-Cu, Pt-Cu and Ni-Cu alloysusing the axial interaction model and the supercell model. Our results agree well withexperiments. The two models were shown to be able give consistent results.

Alloying effects of Mn, Co and Nb on the stacking fault energy in austenitic stainlesssteels are also studied with the axial interaction model. From the SFE maps with respectto composition we have shown the interesting effect of the interactions between differ-ent alloy components on the SFE. Manganese is found to decrease the SFE in alloys withless than 16 at.% Ni, beyond which the SFE slightly rises with Mn. Cobalt always tendsto decrease the SFE and the decreasing effect is enhanced in high-Ni alloys. Niobiumstrongly increase the SFE in low-Ni alloys, but the increasing effect is weakened by Ni.Niobium is found to overwrite the effect of Ni on the SFE of Fe-Cr-Ni ternary alloys.

In the future, another type of planar defect, FCC/BCC interface, will be studied usingfirst-principles methods. The FCC/BCC phase interface is present in many systems, eg.,duplex stainless steels, and is of fundamental importance to many mechanical proper-

37

CHAPTER 7. CONCLUDING REMARKS AND FUTURE WORK 38

ties.

Acknowledgements

I am most grateful to Prof. Levente Vitos for giving me the opportunity to join the AMPgroup in KTH, for always showing a great interest to my work and for professionalguidance.

Grateful acknowledgement also goes to my co-supervisor Prof. Qing-Miao Hu andProf. Borje Johansson, the leader of the AMP group, for great support and professionalguidance.

Many thanks to group members: Krisztina, Marko, Noura, Hualei, Chunmei, Stephan,Andreas, Wei, Xiaoqing, Guisheng, Fuyang, etc., for helpful discussions and co-working.I would like to thank all other people at the Department of Material Sciences and Engi-neering for creating a nice working atmosphere.

I would also like to thank my friends, Zhiyun, Shaoteng, Bo Wei, Zi Yang, Jing Sun etc.,for the time spent together. Special thanks to my families.

The Swedish Research Council, the Swedish Steel Producers’ Association, the SwedishEnergy Agency and the Chinese Scholarship Council are acknowledged for financialsupports. The National Supercomputer Center in Sweden (NSC) and UPPMAX areacknowledged for providing high performance computing resources.

39

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