First-principles description of planarfaults in metals and alloys

Wei Li

Licentiate ThesisSchool of Industrial Engineering and Management, Department of

Materials Science and Engineering, KTH, Sweden, 2014

MaterialvetenskapKTH

SE-100 44 StockholmISBN 978-91-7595-282-6 Sweden

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolanframlägges till offentlig granskning för avläggande av licentiatexamen torsdagenden 6 Nov. 2014 kl 14:00 i konferensrummet, Materialvetenskap, KungligaTekniska Högskolan, Brinellvägen 23, Stockholm.

© Wei Li, 2014

Tryck: Universitetsservice US AB

Abstract

Phase interface and stacking fault are two common planar defects in metallicmaterials. In the present thesis, the interfacial energy and the generalized stack-ing fault energy of random alloys are investigated using density functional theoryformulated within the exact muffin-tin orbitals (EMTO) method in combinationwith the coherent-potential approximation (CPA).The interfacial energy is one of the key physical parameters controlling the

formation of the Cr-rich α’ phases during the phase decomposition in Fe-Crferrite stainless steels. This decomposition is believed to cause the so-called“475℃ embrittlement”. Aluminum addition to ferritic stainless steels was foundto effectively suppress the deleterious 475oC embrittlement. The effect of Alon the interfacial energy and the formation energy of Fe-Cr solid solutionsare studied in this thesis. The interface between the decomposed Fe-rich αand Cr-rich α′ phases carries a positive excess energy, which represents a bar-rier for the process of phase separation. Our results show that for the α-Fe70Cr20Al10/α′-Fe100−x−yCryAlx (0≤x≤10, 55≤y≤80) interface, the Al con-tent (x) barely changes the interfacial energy. However, when Al is partitionedonly in the alpha phase, i.e. for the α-Fe100−x−yCryAlx/α′-Fe10Cr90 (0≤x≤10,0≤y≤25) interface, the interfacial energy increases with Al concentration dueto the variation of the formation energies of the Fe-Cr alloys upon Al alloying.The intrinsic energy barriers (IEBs) of the γ surface (also called generalized

stacking fault energy, GSFE) provide fundamental physics for understanding theplastic deformation mechanisms in face-centred cubic metals and alloys. In thisthesis, the GSFEs of the disordered Cu-X (X=Al, Zn, Ga, Ni) and Pd-X (X=Ag,Au) alloys are calculated. Studying the effect of segregation of the solutes tothe stacking fault planes shows that only the local chemical composition affectsthe GSFEs. Based on the calculated GSFEs values, the previously revealed“universal scaling law” between these IEBs is demonstrated to be well obeyedin random solid solutions. This greatly simplifies the calculations of the twinningparameters or the critical twinning stress. Adopting two twinnability measureparameters derived from the IEBs, we find that in binary Cu alloys, Al, Zn andGa increase the twinnability, while Ni decreases it. Aluminum and gallium yieldsimilar effects on the twinnability. Our theoretical predictions are in line withthe available experimental data. These achievements open new possibilities inunderstanding and describing the plasticity of complex alloys.

iv

PrefaceList of included publications:

I The effect of Al on the 475℃ embrittlement of Fe-Cr alloysWei Li, Song Lu, Qing-Miao Hu, Huahai Mao, Börje Johansson andLevente Vitos, Comput. Mater. Sci. 74, 101 (2013).

II Generalized stacking fault energies of alloysWei Li, Song Lu, Qing-Miao Hu, Se Kyun Kwon, Börje Johansson andLevente Vitos, J. Phys.: Condens. Matter. 26, 265005 (2014).

Comment on my own contribution:Paper I: 90% calculation, part of the data analysis, literature survey; the manuscriptwas written jointly.Paper II: all calculations, full data analysis, literature survey; the manuscriptwas written jointly.

List of papers not included in the thesis:

III Elastic properties of vanadium-based alloys from first-principles theoryXiaoqing Li, Hualei Zhang, Song Lu, Wei Li, Jijun Zhao, Börje Johans-son, and Levente Vitos, Phys. Rev. B 86, 014105 (2012).

IV Influence of alloying elements Nb, Zr, Sn, and oxygen on structural sta-bility and elastic properties of the Ti2448 alloyJ. H. Dai, Y. Song, W. Li, R. Yang, and L. Vitos, Phys. Rev. B 89,014103 (2014).

Contents

List of Figures vii

List of Tables ix

1 Introduction 1

2 Phase Interface and Stacking Fault 5

2.1 Phase Interface in Fe-Cr Alloys . . . . . . . . . . . . . . . . . . . 5

2.1.1 Phase Separation Mechanisms in Fe-Cr Alloys . . . . . . . 5

2.1.2 Critical Nucleus Size . . . . . . . . . . . . . . . . . . . . . 5

2.2 Stacking Fault Energy . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Plastic Deformation Mechanism in fcc Metals . . . . . . . 6

2.2.2 Generalized Stacking Fault . . . . . . . . . . . . . . . . . 7

2.2.3 The “Universal Scaling Law” . . . . . . . . . . . . . . . . 7

2.2.4 Twinnability . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Theoretical Methodology 11

3.1 First-principles Calculation of the Electronic Structure . . . . . . 11

3.2 Exact Muffin-Tin Orbitals Method . . . . . . . . . . . . . . . . . 12

3.3 Coherent Potential Approximation . . . . . . . . . . . . . . . . . 12

3.4 Some computational details . . . . . . . . . . . . . . . . . . . . . 13

4 Al Effect on γα/α′ Interphase Energy in Fe-Cr Alloys 15

4.1 Effects of Al on the Interfacial Energy at the Initial Stage of thePhase Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2 Effects of Al on the Interfacial Energy at the Later Stage of thePhase Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2.1 Predicting the Rcrit from First-principles Calculation . . . 19

vi CONTENTS

5 Generalized Stacking Fault Energy of fcc Metals and Alloys 215.1 Elemental fcc Metals . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Random Solid Solutions . . . . . . . . . . . . . . . . . . . . . . . 21

5.2.1 Cu-Al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2.2 Cu-Ga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2.3 Cu-Zn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2.4 Cu-Ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2.5 Pd-Ag and Pd-Au . . . . . . . . . . . . . . . . . . . . . . 28

5.3 The “Universal Scaling Law” in Alloys . . . . . . . . . . . . . . . 295.4 Alloying Effects on Twinnablity . . . . . . . . . . . . . . . . . . . 30

6 Concluding Remarks and Future Work 33

Bibliography 37

List of figures

2.1 Generalized stacking fault in fcc structure. . . . . . . . . . . . . . . 8

4.1 Interfacial energy map as function of Al and Cr contents in theα′ phase. The composition of the α phase is fixed to Fe70Cr20Al10. 16

4.2 Interfacial energy map with respect to the composition of α-Fe100−x−yCryAlx. The composition of the α′ phase is fixed toFe10Cr90. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 Interfacial energy map with respect to the compositions of theα-Fe94−yCryAl6 and α′-FexCr100−x phases. . . . . . . . . . . . . 17

4.4 Formation energies (∆E) of Fe-Cr-Al alloys with respect to Cr contentfor different Al concentrations. . . . . . . . . . . . . . . . . . . . . 18

4.5 The calculated critical nucleus radius as function of temperaturefor Fe80Cr20 and Fe70Cr30 alloys. . . . . . . . . . . . . . . . . . . 20

5.1 Calculated γisf of Cu-Al alloys as a function of Al concentration. 235.2 The γ surface for non-homogenous Cu0.9943Al0.0057 alloy. . . . . . 265.3 Scaling plot of stable and unstable fault energies for different

metals and alloys. . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.4 Alloying effects on twinnability measures (rd and T ) in Cu-X

(X=Zn,Al,Ga,Ni) alloys. . . . . . . . . . . . . . . . . . . . . . . . 31

List of tables

5.1 The present (EMTO) stable and unstable stacking fault energiescompared with the available theoretical and experimental data. . 22

5.2 Theoretical lattice parameters, and stable and unstable stackingfault energies of Cu-Al alloys. . . . . . . . . . . . . . . . . . . . . 23

5.3 The effect of segregation of 4 at.% Al on the stacking fault ener-gies in Cu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.4 Theoretical lattice parameters, and stable and unstable stackingfault energies of Cu-Ga alloys. . . . . . . . . . . . . . . . . . . . . 27

5.5 Theoretical lattice parameters, and stable and unstable stackingfault energies of Cu-Zn alloys. . . . . . . . . . . . . . . . . . . . . 27

5.6 Theoretical lattice parameters, and stable and unstable stackingfault energies of Cu-Ni alloys. . . . . . . . . . . . . . . . . . . . . 28

5.7 Theoretical lattice parameters, and stable and unstable stackingfault energies of Pd-Ag/Au alloys. . . . . . . . . . . . . . . . . . 29

Chapter 1

Introduction

Interface and stacking fault are two common types of planar faults in materials.Phase interfacial energy and the intrinsic energy barriers (IEBs) of the gener-alized stack fault energy (GSFE) are very important parameters in the theoryof phase decomposition and plastic deformation, respectively. Studying theseplanar fault energies by first-principles calculations opens new opportunities foroptimization and design of alloys.Due to the excellent mechanical and corrosion-resistant properties, the Fe-

Cr alloys which form the basis of stainless steels are extensively used in do-mestic and industrial areas. However, ferritic stainless steels suffer from thephenomenon so called “475oC embrittlement” which degrades the mechanicalproperties seriously. The 475oC embrittlement is caused by the phase decompo-sition when the Fe-Cr alloys are aged at the temperature range of 300-500oC [1].The decomposition produce a Fe-rich α and a Cr-rich α′ phase. By adding alloyelements, the metallurgists could tune the performance of the Fe-Cr stainlesssteels. Al is one of the candidates which could not only enhance the oxidationand corrosion resistance [2], but also improve the mechanical service at hightemperature of the ferritic stainless steels by suppressing the formation of theσ phase [3] and the 475oC embrittlement with an appropriate Al concentra-tion. Spear and Polonis [4] reported that adding 3-5 wt.% Al to Fe-18 wt.%Cr alloys retards the onset of α′ precipitation very significantly, especially attemperatures above 495oC. Sana and Cortie [3] observed that additions of Alup to 3 wt.% in Fe-29 wt.% Cr-4 wt.% Mo-1.5 wt.% Ni alloy promotes the475oC embrittlement, however, for higher Al content, the 475oC embrittlementis suppressed. In a US patent published in 1970 [5], the author claimed thatreplacing 0.5 wt.%-4.0 wt.% Al and/or Co for the same amount of iron con-tent in the ferritic, martensitic and ferrite-austenitic steels (13-18 wt.% Cr)compensates or prevents effectively the detrimental 475oC embrittlement. Thetest specimens of steels with additions of Al showed a ductile break with aconsiderable contraction even for 1000h aging at 475oC. However, some otherexperiments demonstrated that the addition of Co was more likely to acceleratethe phase separation kinetics rather than slow it down [6–9]. It was reportedthat the addition of Co to the Fe-Cr system could produce an asymmetric misci-bility gap, widen the α+α′ miscibility gap and raise the critical decompositiontemperature [6–9]. The effects of Al on the phase diagram of Fe-Cr alloys is

2 CHAPTER 1. INTRODUCTION

differently. Capdevila et al. [10, 11] investigated the effect of Al on the spinodalphase decomposition of the Fe-Cr-Al alloy by thermodynamic calculation, andfound that Al decreases the critical temperature of spinodal decomposition andshifts the miscibility gap and spinodal region to lower Cr contents. Kobayashiand Takasugi [12] recently measured the 475oC embrittlement in ferritic alloyswith a wide composition range of Fe-(10-30) Cr-(0-20) Al (at.%) using a dif-fusion multiple technique, and they found that a large amount of addition ofAl (&2 at.%) suppresses the 475oC embrittlement, while a small solid solutionof Al promotes embrittlement. However, it is still unclear how Al (and/or Co)addition imposes the ferrites resistance to the 475oC embrittlement. The inter-facial energy between α and α′ phases represents the major resistance againstthe formation of new α′ particles from the matrix and against the coarseningprocess. Within the classical nucleation and growth theory [13] for describingthe phase separation process, Lu et al. [14] calculated the critical nucleus size(Rcrit) of α′ precipitation as a function of the alloy composition using the calcu-lated interfacial energies and showed that Rcrit experiences a rapid increase foralloys with compositions from inside the spinodal line to between spinodal andsolubility limit lines. In lack of available interfacial energy data, an estimatedinterfacial energy was incorporated for evaluating the thermal contribution tothe increment of yield strength due to phase separation in Fe-Cr-Al alloys [15].In the present thesis, the effect of Al on the interfacial energy and formation

energy of Fe-Cr alloys are evaluated by using the first-principles exact muffin-tinorbitals (EMTO) method. The random disordering is dealt with the coherentpotential approximation (CPA). The calculated composition dependence of in-terfacial energy may shed light on understanding the role of Al in the phaseseparation. The results and discussions are presented in Chapter 4 and supple-mented Paper 1.It was fund that the generalized stacking fault energy surface (GSFE or γ

surface) which represent the energy dependency of the rigidly sheared crystalwithin the fcc {111} plane along the <112> direction, can provide sufficientfundamental informations to properly classify the deformation modes in atom-istic simulations [16, 17]. On the γ surface, there are four extrema, intrinsicstacking fault energy (γisf), unstable stacking fault energy (γusf), unstable twinfault energy (γutf), and extrinsic stacking fault energy (γesf), which itself or theircombinations represent the intrinsic energy barriers (IEBs) that have to be over-come for nucleating or moving a partial dislocation. Often, γusf , γutf − γisf , andγusf − γisf are used as the intrinsic energy barriers for activating stacking fault,twinning and full slip, respectively [17, 18]. Some anomalous experimental ob-servations can be well understood in terms of the γ surface. For instance, innanocrystalline Ni, experiments showed that decreasing grain size first promotestwinning and then suppresses twinning due to the so called inverse grain-sizeeffect. However, no inverse grain-size effect exists in the formation of stackingfaults [19]. Li et al. [20] also showed that the information provided by the γsurface is crucial for understanding the nucleation mechanism of deformationtwins in pure Al.The γ surface becomes more subtle in alloys. Ebrahimi et al. [21] found that

solely based on the alloying effects on γisf , the strain-hardening rate in Cu orFe alloyed Ni can not be well understood. Namely, Cu and Fe reduce γisf to asimilar extent at dilute concentrations, however, the strain-hardening rates they

3

induced are different. On the other hand, ab initio γ surface calculations [22]demonstrated that adding Cu or Fe to Ni results in quite different γusf whichmight account for the different strain-hardening rates observed in the experi-ments. An et al. [23] also pointed out that the plastic deformation mechanismsin nanocrystalline Cu-Al alloys processed by high-pressure torsion could be bet-ter understood based on the γ surface than merely on γisf .So far, however, most first-principle calculations of the γ surfaces have been

performed for elemental metals. There are only a limited number of studies ofthe alloying effects on the γ surface in Cu [24], Ni [22], Al [25] alloys, austeniticsteels [26, 27] and Mg alloys [28]. Most of these studies adopted the super-cell method dealing with alloying and their results are not for fully randomsolid solutions. Ab initio calculations have shown that in the Cu-Al and Al-Agalloys, the segregation of alloying element, Al in Cu-Al and Ag in Al-Ag, tothe stacking fault area would significantly affect the γ surface (the Suzuki ef-fect) [29, 30]. Therefore, one must pay special attention when dealing with thelocal concentration and configuration of alloying atoms near the slip plane usingthe supercell method. Gallagher [31] reviewed the experimental studies on theeffect of the solute segregation on the stacking fault energy, and suggested thatone should be very careful when interpreting the observations as a Suzuki ef-fect. Specifically, in the Cu-Al or Cu-Zn systems, there are controversial resultsregarding the occurrence of the Suzuki effect [31]. Therefore, we consider it isimportant to have the γ surface data for fully solid solutions which may serveas the basis for further discussions, for example, regarding the Suzuki effect.

Ab initio alloy theory formulated within the EMTO method [32] has previ-ously been applied to calculate the planar fault energies in various alloy systems.For instance, using this tool, researchers successfully described the intrinsicstacking fault energies in fully random fcc Pt-Cu, Pd-Cu, Pd-Ag, Ni-Cu binaryalloys [33], austenitic stainless steels [34–36] or Ti alloy [37] with hexagonalcrystal structure. It was shown that the EMTO method yields fault formationenergies in good agreement with experiments and other theoretical calculations.The fact that a muffin-tin approach performs well for the metastable faultedalloys is not surprising since the lattices involved in such calculations are allclose-packed [33, 34] and thus the errors associated with the shape approxi-mation can be kept at a minimal level. However, in order to compute thegeneralized stacking fault energy, one needs to displace one part of the supercellagainst the other within the fcc {111} plane along the <112> direction. Dur-ing this process, the crystal symmetry is lowered and the lattice becomes moreopen relative to the ideal fcc host. Whether the EMTO method can handle suchproblem properly is of great interest for the scientific community working withdensity functional methods and alloy modeling.In Chapter 5, first, the GSFEs of several elemental fcc metals are presented

and compared with the data from the literature. And then, the GSFEs ofrandom solid solutions are studied by the EMTO-CPA method. For the selectedCu and Pd alloys, Cu-X(X=Al, Zn, Ga, Ni), Pd-X (X=Ag, Au), the EMTO-CPA results were compared with the limited available data obtained in previouscalculations employing the plane-wave pseudopotential method (e.g., Vienna Abinitio Simulation Package [38–40], VASP). In the latter case, alloying is treatedwithin the supercell model suffering from certain restrictions regarding the soluteconcentration, configuration or number of alloy components. These limitations

4 CHAPTER 1. INTRODUCTION

are not present in the EMTO-CPA method.

Chapter 2

Phase Interface andStacking Fault

2.1 Phase Interface in Fe-Cr Alloys

In a neutron irradiated Fe-Cr alloy (Fe-14Cr-0.3Mo-1.0Ti-0.25Y2O3 (MA975)),Ribis and Lozano-Perez [41] observed a semi-coherent interface between α and α′phases with the orientation relationship, 〈100〉α′//〈100〉α and {100}α′//{100}α.They also reported other types of orientation relationship, [100]α′//[111]α and[100]α′//[112]α. It is somehow curious that they did not report the {110}α//{110}α′interface, which is most close-packed and expected to be energetic favorable [14].In the present study, we only focus on the {001} type of interface with orienta-tion 〈100〉α′//〈100〉α.

2.1.1 Phase Separation Mechanisms in Fe-Cr Alloys

In the phase diagram of the Fe-Cr system, there is a wide miscibility gap at lowtemperature, with the spinodal decomposition region inside it. Fe and Cr arefully soluble to each other in the body-centered cubic structure at high tempera-ture. While aging at lower temperatures (.500℃), the single phase decomposesinto two bcc phases, Fe-rich α and Cr-rich α′ phases. For Cr concentrationbetween ∼15 at.% and ∼75 at.%, the phase separation is caused by spinodaldecomposition. In alloys containing less Cr (or Fe), the phase separation occursvia the so called “nucleation and growth” (NG) mechanism.

2.1.2 Critical Nucleus Size

According to the classical nucleation theory [13], the work W required to forma heterogeneous spherical grain of radius R in a homogeneous phase is

W = 4πR2γ − 4π3 R3∆P, (2.1)

6 CHAPTER 2. PHASE INTERFACE AND STACKING FAULT

where ∆P is the volumetric driving force, and γ is interfacial energy. Theextremum of W determines the critical nucleus size,

Rcrit = 2γ∆P . (2.2)

If the initial nucleus size R > Rcrit, it will grow continuously. While thenucleus size R < Rcrit, it will disappear.Spinodal decomposition is totally different with the NG, but we could consider

that when the concentration of Fe-Cr alloys in the spinodal region, if the rcrit isso small, any thermo fluctuations can make it disappear, also any concentrationfluctuations could serve the nucleus and grown.

2.2 Stacking Fault EnergyThe stacking fault (SF) energy, γ, is defined as the free energy difference perarea (A2D) between the structure containing the “wrong” stacking sequence(Fsf ) and the perfect one (Ffcc). In the present thesis, we only focus on the SFin the fcc structure, and calculate the γ by,

γ = (Fsf − Ffcc)/A2D (2.3)

Along the 〈111〉 direction, the stacking sequence of fcc structure should be". . . ABCABCABC. . . ". Missing an atomic plane, for example, “B”, a so-called“intrinsic stacking fault” (ISF) is formed, ". . . ABCA|CABCA. . . ". While if weinsert an atomic plane, e.g. type “C”, we get an “extrinsic stacking fault”(ESF),". . . ABCACBCABCA. . . ". However, we can not really “insert” or “subtract”one atomic plane in a crystal. These ISF and ESF usually formed during theplastic deformation in fcc materials by the movement of dislocations. The ISFand ESF energies are close to each other, and we only focus on the ISF in thepresent work.

2.2.1 Plastic Deformation Mechanism in fcc Metals

In the fcc structure, the most common slip system is 〈110〉 {111}[42]. Thesmallest dislocation introduced by this shear deformation has a Burgers vectorb = 1/2 〈110〉. The energy of a dislocation is proportional to |b|2. So this fulldislocation energetically prefers to dissociate into two partial dislocations (PD),e.g.

12[110]

= 16[211]

+ 16[121]. (2.4)

When the leading PD (LPD), one of the above two PDs, sweeps over one of the{111} planes, the planar fault left behind is the ISF. If a second PD follows theLPD and slides on the same plane, the ISF area is recovered by this trailing PD(TPD). If the second LPD nucleates, and slides on the adjacent slip plane next tothe first LPD, an extrinsic stacking fault (ESF) is formed. Since we can considerthe ESF as a two-layer twin nucleus, the second partial is usually call twinningpartial (TWP). As several subsequent twining partials are continuously formed,the twining is growing. The situations described above are the three common

2.2 Stacking Fault Energy 7

deformation mechanisms of the fcc metals via the movements of dislocations:stacking fault, full slip (SL), and twining (TW).It is commonly accepted that the stacking fault energy (SFE) plays an im-

portant role in determining the plastic deformation mechanisms of metals. De-formation twinning is favored at low SFE and dislocation slip at high SFE.However, this kind of correlation is not strong, especially at the nanoscale [43].For example, twinning is usually believed to be difficult in Al due to the highintrinsic stacking fault energy (104-142 mJ/m2) [16]. Nevertheless, twinning inAl has been observed in both atomistic simulations and experiments [16, 44,45], indicating that the SFE alone cannot properly account for the deformationmechanism in fcc metals.

2.2.2 Generalized Stacking Fault

The Generalized stacking fault (GSF) energy (GSFE) is defined as: the energyvariation referred to the energy of the fcc structure with respect to a displace-ment vector, b, when shifting one half of the crystal against the other on theatomically close-packed {111} planes. People sometimes also call it γ-surface.Due to the high symmetry of {111} slip planes, researchers only need to calcu-late the GSFE along 〈112〉 and 〈110〉 directions and estimate the whole γ-surfacewith fitting to a two-dimensional Fourier series [26, 46]. Because the 〈112〉 di-rection usually is the lowest energy path in most systems, we therefore onlyconsider the GSFE curve along this direction. When changing b from 0 to16[211](Fig. 2.1), the GSFE encounters a maximum value, which is usually

called unstable stacking fault (USF) energy(γusf ). γusf is considered as theenergy barrier related to the emission of the leading partials. In order to in-vestigate the TW nucleation, people usually also calculate the GSFE curve forb from 1

6[211]to 1

3[211]on an adjacent atomic plane. We can find another

maximum in between, which is called unstable twin fault (UTF) energy, γutf .It is normally related to the emission of twining partials.

2.2.3 The “Universal Scaling Law”

Based on the calculated γ surfaces for elemental fcc metals, Jin et al. [47] sug-gested that there may exist an “universal scaling law” between the fault energies,

γutf/γusf ' γisf/(2γusf ) + β, (2.5)

where β is close to 1 when the minimum interaction principle is satisfied, whichmeans the planar fault configuration is characterized by localized bonding ef-fects. They defined the ration Λ = γisf/γusf as a characteristic parameter andshowed that for most elemental fcc metals (except Pt) and Cu-Al alloys [48],Eq. (2.5) holds very well. Based on the interlayer interaction analysis, Jo etal. [18] showed that the relationship described by Eq. 2.5 is established up tothe second-nearest-neighbor layers by interlayer interaction analysis. WhetherEq. 2.5 holds for random solid solutions is an important question of great prac-tical implications, because many analytical equations based on the GSFE couldbe greatly simplified, for instance, the activation energies for trailing or twinningpartials [49, 50] or the twinnability measures [47, 51].

8 CHAPTER 2. PHASE INTERFACE AND STACKING FAULT

Figure 2.1: Generalized stacking fault in fcc structure.

2.2.4 Twinnability

Twinning and full dislocation slip are two competing plastic deformation mech-anisms in fcc materials. A dislocation slip emerges when a leading partial dis-location is followed by a trailing partial dislocation on the same slip plane,which creates a full dislocation and erases the stacking fault. Twinning occurswhen a leading partial dislocation is followed by a twinning partial dislocationof the same Burgers vector on the adjacent slip plane, creating a two-layer twinnucleate [49].

Several parameters have been proposed to measure the twinnability of fccmaterials based on the information of the γ surface. Kibey et al. [52] devel-oped an analytic expression to calculate the critical twinning stress based solelyon the calculated γ surface in coarse-grained fcc metals or alloys, where twinnucleation in the bulk occurs without grain-boundaries as twin sources whilethe pre-existing defects, such as stacking faults and fault pairs exists in the in-terior of the grain. Their expression yields quantitatively good predictions onthe critical twinning stress for fcc metals and Cu-Al alloys [29]. The formulawas also applied to γ-TiAl alloys [53]. In nanocrystalline structure, however,the partials mainly initiate from the grain boundaries [16, 54]. Assuming thatthe dislocation nucleation at grain boundaries in nanostructrued fcc materialsis similar to that from a crack-tip [43, 55], Asaro and Suresh [51] proposed atwinnability measure T ,

2.2 Stacking Fault Energy 9

T =√

(3γusf − 2γisf)/γutf . (2.6)

Based on their model, twin is favored over dislocation slip if T > 1, and viceversa if T < 1. If the relation given in Eq. 2.5 holds, then the twinnabilitymeasurement will only rely on one single parameter, Λ = γisf/γusf , which leadsto a very simple equation for the calculation of the twinnability, namely,

T '√

(3− 2Λ)/(1 + Λ/2). (2.7)

Recently, Jo et al. [18] proposed an improved dimensionless parameter rd toclassify the deformation mechanisms,

rd = γisf/(γusf − γisf), (2.8)

which may also be expressed in terms of Λ. Within this model, systems withlow rd favor twinning.

Chapter 3

Theoretical Methodology

3.1 First-principles Calculation of the ElectronicStructure

Many physical and chemical properties of materials are ultimately determinedby the interactions of electrons and nuclei. The “first principle” usually meansonly employing the most fundamental physics, the quantum mechanics (QM),rules, and only the nature constants to solve the problems, but in practice, theincredible complexity of the “many-body” problem introduced by solving theQM equations of the real materials makes us to have to use some reasonableapproximations and simplifications. The Born-Oppenheimer approximation [56]allows people to decouple the motion of nuclei from the electrons’. In the ma-terials with periodic crystal structures, by applying the Bloch theorem we cansolve the QM equations just in the first Brillouin zone and unit cell, but not thewhole lattice. 1964, Hohenberg and Kohn [57] proved the electrons density ofthe system uniquely determines the potential up to a constant which meaningthe density can be used as the basic variable, and start the modern densityfunctional theory (DFT) which reduced the number of degrees of freedom to aminimum. Kohn and Sham also provide a computational scheme [58], by pro-duce a non-interacting electrons system, to decouple the problem into solvingsingle electron (Schrödinger or Dirac) equations. Kohn-Sham scheme have tointroduce the exchange-correlation energy (Exc) which is unknown and must beapproximated to computing the ground-state of the system in practice. Startingfrom a uniform electron gas, a “not good” approximation, a local density ap-proximation (LDA), suggested by Kohn and Sham [58], gives “not bad” resultseven for some strongly inhomogeneous systems. There is also a gradient approx-imation, e.g. generalized gradient approximation (GGA), suggested to improvethe case of non-homogeneous system. LDA and GGA perform well on differ-ent materials. There also are several parameterizations of LDA and GGA havebeen used. Perdew and Wang [59] for LDA and Perdew, Burke and Ernzerhof(PBE) [60] for GGA are two very popular exchange-correlation functionals.

12 CHAPTER 3. THEORETICAL METHODOLOGY

3.2 Exact Muffin-Tin Orbitals Method

More detail and useful information about the EMTO method can be found inprofessor Levente Vitos’s book [61]. Here I only give a very simple, brief andincomplete introduction.EMTO belongs to the third generation Muffin-tin methods [62–64]. By em-

ploying an optimized overlapping Muffin-Tin (OOMT) potential, EMTO canprovide the results having almost the same accuracy as full potential methods,but more computational efficiently. The space is divided into different regionsby overlapping spherical potential wells, VR(rR)−V0 which are centered on lat-tice sites R with the notation rR = r−R, rR = |rR|. When the radii rR = sR,VR(rR) = V0. In the interstitial region, the potential has a constants value V0.So the effective potential can be written as:

V (r) ≈ VMT = V0 +∑R

[VR(rR)− V0]. (3.1)

VR(rR) and V0 are obtained by optimizing the mean of the squared deviationof VMT and the full potential under the Spherical Cell Approximation [65].To solve the KS equation with above OOMT potential, we could construct

the KS orbital by EMTOs.

ψi(r) =∑RL

ψa

RL(εi, rR)υaRL,i, (3.2)

where ψi(r) is the KS orbital, and ψa

RL(εi, rR) is the EMTOs, L = (l,m) isthe combined quantum numbers and υaRL,i are the expansion coefficients. TheEMTOs are constructed using different basis functions in the different regions:

ψa

RL(εi, rR) = φaRL(εi, rR) + ψaRL(εi − υ0, rR)− ϕaRL(εi, rR)YL(rR). (3.3)

The φaRL(εi, rR) is the partial wave inside the OOMT potential sphere (rR ≤ sR).The ψaRL(εi − υ0, rR) is the screened spherical wave in the interstitial regionwhich defined outside of the non-overlapping sphere aR (aR < sR), or called hardsphere. The ϕaRL(εi, rR)YL(rR) is the backward extrapolated fee-electron wavefunction which matched continuously and differentiable to the partial waves atsR and continuously to the screened spherical waves at aR. By solving theso-called kink-cancellation equation related to the boundary condition in theregion aR ≤ rR ≤ sR, we can find the solution of the KS equation.

3.3 Coherent Potential Approximation

In order to model completely random substitution alloys by the first-principlecalculation, using supercells is not a very feasible way to cover all the configura-tions and concentrations. The coherent potential approximation (CPA) [66–69]is based on the assumption that the alloy may be replaced by an ordered effec-tive medium, and the parameters are determined self-consistently. It is assumedto be the most powerful technique to model the random disordered system. Inthe CPA, the impurity problem is treated within the single-site approximation.

3.4 Some computational details 13

This means that one single impurity is placed in an effective medium and no in-formation is provided about the individual potential and charge density beyondthe the sphere or polyhedra around this impurity. In contrast to the tradi-tional KKR-CPA and LMTO-CPA, EMTO-CPA is more suitable to reproducethe structural energy differences and energy changes related to small distortionsand more open structure in random alloys systems with high accuracy [63, 68].This is very important for the GSFEs calculation.

3.4 Some computational detailsIn all of the calculations performed in Chapter 4 and Chapter 5, the totalenergies were calculated using density functional theory (DFT) [70] in combi-nation with the generalized gradient approximation (GGA) [71]. To solve theKohn-Sham equations, we adopted the EMTO theory [32, 72–75]. The effectivepotential was treated by using the optimized overlapping muffin-tin approxima-tion and the total energy was computed by using the full charge-density tech-nique [76]. The EMTO basis included s, p, d, and f orbitals. The one-electronequations were solved within the scalar-relativistic approximation and soft-corescheme. The EMTO Green’s function was calculated self-consistently for 16complex energy points distributed exponentially on a semi-circular contour. Inthe one-center expansion of the full charge density, we adopted an l-cutoff of8. The k-mesh was carefully tested. The electrostatic correction to the single-site CPA was described using the screened impurity model [77] with screeningparameter 0.6 and 0.9 for Fe-Cr-Al system and all stacking fault calculation,respectively.

Chapter 4

Al Effect on γα/α′Interphase Energy in Fe-CrAlloys

In this chapter, two composition profiles for Al were adopted to investigate theeffects of Al on the interface formation for the initial and final (equilibrium)phase separation stages. For the first case, the composition of the α phase wasfixed to Fe70Cr20Al10, close to the composition of PM 2000TM [15], and the in-terfacial energy was calculated with respect to the concentrations of Cr and Alin the α′ phases Fe100−x−yCryAlx (0≤x≤10, 55≤y≤80). This model, to someextents, represents the experimental observation that before the α′ precipitatesheavily coarsen or the phase separation reaches equilibrium, the composition ofthe α phase barely changes. With increasing aging time, the concentration ofCr/Al in α′ gradually increases/decreases [15]. For the second case, the inter-facial energy was calculated for the interface α-Fe100−x−yCryAlx/α′-Fe10Cr90(0≤x≤10, 0≤y≤25), where the composition of the α′ phase is fixed and Al isonly doped in the α phase. α′-Fe10Cr90 is selected because it is close to thecomposition of the α′ phase obtained after long time aging PM 2000TM [15].This model is applied to study the interface at the thermodynamic equilibriumstage when Al atoms fully segregate to the α phase.The interfacial energy (γ) is calculated as

γ = (∆Einterface −N∆Eα −N∆Eα′) /2S, (4.1)

where ∆Einterface is the formation energy of the supercell, ∆Eα and ∆Eα′ are theformation energies per site of the single phases, respectively. N is the numberof layers of each phase (5) and S is the area of the interface (2 stands for thetwo interfaces per unit cell). The ferromagnetic bcc Fe, anti-ferromagnetic B2Cr [78], and nonmagnetic fcc Al were taken as reference states for calculatingthe formation energies. The formation energy of a single-phase Fe1−x−yCrxAlyalloy is defined as

∆E = EFe1−x−yCrxAly − (1− x− y)EbccFe − xEB2

Cr − yEfccAl , (4.2)

16CHAPTER 4. AL EFFECT ON γα/α′ INTERPHASE ENERGY IN FE-CR

ALLOYS

0 1 2 3 4 5 6 7 8 9 1055

60

65

70

75

80

Cr a

t.%

Al at.%

0.01

0.05

0.09

0.13

0.17

0.21

Figure 4.1: Interfacial energy map (in Jm−2) as function of Al and Cr contents in theα′ phase (Fe100−x−yCryAlx, 0≤x≤10, 55≤y≤80). The composition of the αphase is fixed to Fe70Cr20Al10.

where EFe1−x−yCrxAly, Ebcc

Fe , EB2Cr and Efcc

Al are the corresponding total energiesper atom for Fe1−x−yCrxAly, bcc Fe, B2 Cr and fcc Al, respectively. Theformation energy of the supercell α/α′ is calculated in the same way.

4.1 Effects of Al on the Interfacial Energy at theInitial Stage of the Phase Separation

The calculated interfacial energy for the interface α-Fe70Cr20Al10/α′-Fe100−x−yCryAlx

(0≤x≤10, 55≤y≤80), with respect to x and y is shown in Figure 4.1. This in-terface is set up to mimic the preliminary stage of the phase separation at whichthe composition of the matrix α phase does not change notably, while the pre-cipitates are getting rich in Cr and depleting Al. We observe that Cr increasesthe interfacial energy, which is in line with the effects of Cr in binary Fe-Crsystem [79]. On the other hand, when the Cr amount if fixed, Al is found tohave a nearly vanishing effect on the interfacial energy. However, with pro-longed aging time, Cr atoms segregate to the α′ phase and simultaneously theAl atoms deplete from the α′ phase [15]. As a result, when the 10 at.% Al inα′-Fe35Cr55Al10 is replaced by Cr (α′-Fe35Cr65) the interfacial energy increasesfrom 0.06 Jm−2 to 0.09 Jm−2. Hence, the overall effect of Al and its preferentialpartition is that the interfacial energy increases with aging.

4.2 Effects of Al on the Interfacial Energy at theLater Stage of the Phase Separation

In this section, we study the effect of Al on the interfacial energy at the laterstage of the phase separation, when Al has fully segregated to the α-phase.To model this situation, we fix the concentration of the Cr-rich α′ phase to

4.2 Effects of Al on the Interfacial Energy at the Later Stage of the PhaseSeparation 17

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

Cr a

t. %

Al at. %

0.13

0.17

0.20

0.23

0.26

0.29

Figure 4.2: Interfacial energy map (in Jm−2) with respect to the composition of α-Fe100−x−yCryAlx. The composition of the α′ phase is fixed to Fe10Cr90.

0 5 10 15 20 25 30 350

4

8

12

16

20

24

28

y (F

e 1-yC

r yAl 6)

x (FexCr1-x)

0.00

0.14

0.28

0.35

Figure 4.3: Interfacial energy map (in J/m2) with respect to the compositions of theα-Fe94−yCryAl6 and α′-FexCr100−x phases. Aluminum content in the α phaseis fixed to 6 at.%.

Fe10Cr90 (i.e. considering Al-free α′ particles), and add different amounts of Al(0-10 at.% Al) to the α phase. The calculated interfacial energy with respectto the composition of the α phase is shown in Figure 4.2. It is observed thatgenerally Al increases the interfacial energy, especially for Cr concentrationless than ∼15%. For higher Cr content, the interfacial energy changes onlyweakly with increasing Al amount. The interfacial energy experiences a localmaximum with increasing Cr. This maximum is located around y = 10% forAl-free interface. Adding Al gradually shifts the maximum to lower Cr contents.Namely, for x = 10% the formation energy maximum is around y = 5%. Wenote that starting from another composition for the α′ phase keeps this generalpicture the same.

18CHAPTER 4. AL EFFECT ON γα/α′ INTERPHASE ENERGY IN FE-CR

ALLOYS

0 10 20 30 40 50 60 70 80 90 100

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

10Al

8Al6Al

4Al2Al

E (R

y)

Cr at. %

0Al

Figure 4.4: Formation energies (∆E) of Fe-Cr-Al alloys with respect to Cr content fordifferent Al concentrations.

In the case when all Al atoms have segregated to the Fe-rich phase, whichis modeled by replacing 6 at.% Fe in Fe-rich phase by Al, we studied how theinterfacial energy changes with respect to the compositions of α and α′ phases.These results are shown in Figure 4.3. Comparing the unalloyed [79] and Al-alloyed (Figure 4.3) interfacial energy maps, we find that apart from a smallincrease in magnitude, Al does not change the general trend of the interfacialenergy as a function of Cr and Fe. This finding is consistent with the results inFigure 4.1.In binary Fe-Cr systems, the local maximum in the interfacial energy map (lo-

cated around ∼10 at.% Cr) was related to the local minimum of the formationenergy curve [80] and was ascribed to the magnetic frustrations at the inter-face [14]. In the following, we show that both the increment of the maximumof the interfacial energy and its shift toward low Cr contents with increasing Alstem from the effect of Al on the formation energy.Figure 4.4 shows the calculated formation energy of Fe-Cr-Al alloys for dif-

ferent Al contents according to Eq. 4.2. For low Cr contents, increasing Aldecreases the formation energy to negative values. This means that the Fe-rich Fe-Al alloys have negative formation energies, which is consistent with theexperimental phase diagram. In the binary Fe-Cr system, there is a negativeformation energy within the range of 0-10% Cr [80]. This is related to the tran-sition from the Cr-miscible to the Cr-immiscible region in the Fe-Cr bulk phasediagram [81] and originates from the complex magnetic interactions in the Fe-Crsystem. It is interesting to notice that adding Al shifts the local minimum inthe formation energy toward lower Cr-containing alloys, which suggests that Aldecreases the solubility of Cr in Fe within this composition range.Magnetic frustrations at the interface play an important role on the interfacial

energy map in Figure 4.2. We may assume that the layer-resolved formationenergies in Eq. (4.1) cancel each other for layers which are far away from theinterface and only those close to the interface give contribution to the formationenergy. The Cr atoms in the interface layer on the Fe-rich side feel the Cr

4.2 Effects of Al on the Interfacial Energy at the Later Stage of the PhaseSeparation 19

neighbors in the interface layer on the Cr-rich side. The local Cr concentrationof the interface layer on the Fe-rich side may therefore be considered as Cr-enriched. Then according to the large positive slope of the formation energycurves of Fe-rich alloys (Figure 4.4), the formation energy of this interface layershould correspond to higher Cr content, i.e. be more positive than the oneof the corresponding bulk alloy. Hence, according to Eq. (4.1), the negativebulk formation energy at low Cr contents yields a large positive contributionto the interfacial energy. This contribution increases with Al concentration,which explains the γ versus x trend from Figure 4.2. Similarly, the shift of localmaximum in the interfacial energy (Figure 4.2) to low Cr contents is due to thefact that Al addition shifts the local minimum on the formation energy curvesto low Cr contents.

4.2.1 Predicting the Rcrit from First-principles Calculation

We can calculat γ of the α-α′ Fe-Cr interface, and the temperature dependent∆P by the first-principles calculation. The calculated critical nucleus size withrespect to temperature for Fe80Cr20 and Fe70Cr30 alloys is presented in Fig. 4.5.The details of calculation are described in Chapter 3 and supplemented Paper1.Rcrit for Fe70Cr30 is smaller than that for Fe80Cr20, which is in line with pre-

vious results [14] and also with the observations. Namely, Fe70Cr30 decomposesspinodally, and therefore from the point of view of the critical nucleus size,Rcrit is expected to be small because there is no nucleation barrier. It is foundthat Rcrit decreases monotonously with lowering the temperature (Figure 4.5).However, Rcrit decreases much faster at temperatures above ∼700 K than atlower temperatures. This is consistent with the experimental observation thataging Fe-Cr at around 475℃ (748 K) induces the ductile-brittle transition muchfaster than at other temperatures. Namely, at higher temperatures the largeRcrit leads to a large barrier for the nucleation of the α′ phase, while at lowertemperatures the decomposition is suppressed by the reduced atomic diffusionrate. Indeed, the diffusion coefficient of Cr in Fe-Cr alloys increases exponen-tially with temperature (e.g., for Fe-17Cr, D = 0.46 exp(−52500/(RT )) cm2/s,R expressed in cal/mole/K) and decreases with increasing Cr concentration [82].Hence, it seems plausible to conclude that it is the interplay between the smallcritical nucleus size (for T . 700 K) and the rapidly increasing diffusion ratewith temperature that determines the fast phase separation rate observed at∼475℃.

20CHAPTER 4. AL EFFECT ON γα/α′ INTERPHASE ENERGY IN FE-CR

ALLOYS

200 300 400 500 600 700 800 900 10000.0

0.5

1.0

1.5

2.0

2.5

Fe80Cr20

Fe70Cr30

Rcr

it (nm

)

T (K)

Figure 4.5: The calculated critical nucleus radius (Rcrit) as function of temperature(T) for Fe80Cr20 (Triangle) and Fe70Cr30 (Square) alloys.

Chapter 5

Generalized Stacking FaultEnergy of fcc Metals andAlloys

5.1 Elemental fcc MetalsThe calculated stable/unstable stacking fault energies for elemental fcc metalsare listed in Table 5.1. All values are obtained using the theoretical equilibriumlattice parameter obtained for the fcc lattice. The present results (EMTO) arecompared with the available experimental and theoretical data. To illustratethe effect of lattice relaxation on the generalized stacking fault energies, theunrelaxed values are also listed. Comparing with the previous theoretical re-sults, we can estimate the typical error bars associated with such calculations.In particular, we point out the relatively large differences between the resultsobtained in Ref.[47] and those in Ref.[52]. Since these two sets of calculationsemployed the same ab initio method, the large deviations seen, e.g., for Au γusfand γutf , illustrate the sensitivity of the computed fault energy on the details ofthe calculations. Within the above error bars, the present method reproducesvery well the results obtained by previous calculations or experiments. We alsonotice that the present EMTO results are somewhat closer to those reported byKibey et al.[52] than those by Jin et al. [47].It is found that relaxation strongly decreases γusf and γutf (by ∼ 10%), but

barely effects γisf or γesf , except Au (unrelaxed γesf not listed). We would liketo point out that previous studies of austenitic stainless steel alloys showed thatin magnetic systems relaxation may have a significant effect on γisf as well [34].In the rest of thesis, all γ surfaces are calculated taking into account the layerrelaxation near the stacking fault.

5.2 Random Solid SolutionsThis section presents the calculated stable and unstable stacking fault energiesof Cu-X (X=Al, Ga, Zn, Ni) and Pd-M (M=Ag, Au) noble-metal alloys.

22CHAPTER 5. GENERALIZED STACKING FAULT ENERGY OF FCC

METALS AND ALLOYS

Table5.1:

The

present(E

MTO)stable

andunstable

stackingfault

energiescom

paredwith

theavailable

theoreticaland

experimental

data.A

compilation

oftheexperim

entalandtheoreticaldata

canbe

foundin

Ref.[83].

The

presentunrelaxed

resultsare

includedin

parenthesisfor

reference.Allfault

energiesare

expressedin

mJ/m

2.The

secondcolum

nshow

sthe

theoreticalequilibriumlattice

constantsa(units

ofÅ)

.

γisf

γusf

γesf

γutf

aEMTO

Theory

Exp.

EMTO

Theory

EMTO

Theory

EMTO

Theory

Cu

3.63747(47)

36a,41

c45

f,43.5±2

i184(200)

158a,181

c53

40a

217(225)179

a,200c

Au

4.17631(37)

25a,33

j30

g123(152)

68a,134

j32

27a

150(179)79

a,148j

Ag

4.16417(18)

16a,18

j21

g120(135)

91a,133

j18

12a

137(152)100

a,143j

Al

4.046107(108)

112a,122

e,130j

120-142b,120

d161(166)

140a,162

j113

112a

221(225)196

a,215j

Pt

3.988310(317)

286a,324

j322

h362(391)

286a,339

j312

284a

508(563)305

a,486j

Pd

3.957143(145)

134a,168

j175

h260(289)

202a,287

j-

129a

-261

a,361j

aRef.[47]

bRef.[44]

cRef.[24]

dRef.[84]

eRef.[85]

fRef.[86]

gRef.[87]

hRef.[88]

iRef.[89]

jRef.[52]

5.2 Random Solid Solutions 23

0 2 4 6 8 10 12 14 16 18

-10

0

10

20

30

40

50

60

70

80

! isf (

J m

-2)

Al at.%

Present

Calc.

Exp.1

Exp.2

Exp.3

Figure 5.1: Calculated γisf of Cu-Al alloys with respect to Al concentration. Thepresent results (EMTO) are compared with the available theoretical [24] andexperimental data (Exp.1 from Reference [88], Exp.2 and Exp.3 from Refer-ence [31] and references therein).

5.2.1 Cu-Al

Table 5.2: Theoretical lattice parameters (a), and stable and unstable stacking faultenergies of Cu-Al alloys. Available theoretical and experimental results arelisted for comparison. The present hcp-fcc structural energy differences arealso listed for Cu-Al. Unit of lattice parameter is Å and of energies is mJ/m2.

Cu-Al a γisf 2(Ehcp − Efcc)/A γusf γutf γesf

Cu 3.637 47 50 184 217 53Cu-4 at.% Al 3.646 24 29 171 192 29Cu-6 at.% Al 3.650 15 19 165 180 17Cu-8 at.% Al 3.655 8 9 165 173 7Cu-12 at.% Al 3.664 -9 -5 153 153 -11Cu-5.0 at.% Ala 3.650 20 - 170 179 -Cu-8.3 at.% Ala 3.650 7 - 169 176 -Cu-12.5 at.%Alb 3.664 - -11 - - -a Reference[24].b Present VASP-SQS result (see text).

The present theoretical intrinsic stacking fault energies of Cu-Al alloys withrespect to Al concentration are compared with previous data in Fig. 5.1. It isfound that our results are in good agreement with the available theoretical val-

24CHAPTER 5. GENERALIZED STACKING FAULT ENERGY OF FCC

METALS AND ALLOYS

ues obtained for 5 at.% and 8.3 at.% Al [24] and with the experimental data forAl content less than ∼ 10 at.%. At higher Al concentration, the present theoret-ical γisf becomes negative indicating that the fcc lattice becomes unstable withrespect to ISF formation or hcp phase at 0K. Experimental γisf first decreasesrapidly with increasing Al concentration and then remains nearly constant, butpositive, at Al content above ∼10 at.%. The slope of SFE versus compositionchanges at ∼10 at.% Al. This pronounced slope change has not been explainedin experiments. To cross-check the present EMTO-CPA prediction for Al con-centration larger than 10 at.%, here we adopt the Vienna Ab initio SimulationPackage (VASP) [38–40] and study the phase stability of fcc against hcp for12.5 at.% Al. The numerical parameters of these additional calculations arepresented in the note. [90]. Special quasirandom structures (SQSs) [91] with 32atoms (28 Cu and 4 Al) for fcc [92] and 48 atoms (42 Cu and 6 Al) for hcp [93],respectively, are built to model the random fcc and hcp Cu-Al solid solutions.All VASP-SQS calculations take into account the local lattice relaxation effects,i.e. both fcc and hcp SQSs are fully relaxed. From the VASP-SQS results wefind that the hcp phase is more stable than the fcc phase by ∼2 meV/atom,which, in virtue of Eq. (5.1) (see our discussion below), corresponds approxi-mately to -11 mJ/m2. This result is fully consistent with the negative γisf ofCu-12 at.% Al obtained with the present EMTO-CPA approach. Therefore, abinitio alloy theory suggests that the fcc phase is not the true stable state of theCu-Al alloys with Al content above ∼ 12 at.%. This suggestion apparently goesagainst the existing phase diagrams of the Cu-Al system, and call for futurein-depth investigations. We notice that the fcc phase could still survive at lowtemperatures as a result of the atomic short range order effects, which were notaccounted here.

We speculate that the discrepancy between our SFE results and the exper-imental values at high Al content is partly due to temperature effect on theSFE. We recall that the present results correspond to static conditions whereasthe experimental measurements refer to room temperature. The temperaturedependence of the SFE ( dγisf/dT ) for Cu is negative [88]. However, dγisf/dT ofCu-15 at.% Al was measured to be positive within the temperature interval of293K to 593K (dγisf/dT = 0.02 mJ/m2) [94]. If a similar level of temperaturedependence is also valid for Cu-12 at.% Al between 0K and room temperature,then the theoretically predicted SFE should become almost zero at 300K. Unfor-tunately, a complete account for the temperature dependence of SFE for randomalloys within first-principles theory is very complicated [95]. We mention thattaking into account merely the thermal lattice expansion would increase the SFEof Cu-12 at.% Al from -9 to -7.6 mJ/m2 as the temperature increases from 0Kto 300K. Hence there must be factors other than simple lattice expansion whichare responsible for the deviation between theory and experiment for Cu-12 at.%Al alloy. One possible missing term is the explicit phonon contribution, whichassumes the knowledge of the phonon spectra for fcc lattice and for the supercell.Considering that the SFE is roughly proportional to the hcp-fcc energy differ-ence (see Eq. (5.1)), and that these two close packed structures are expectedto posses very similar Debye temperatures [96], it seems less likely that explicitphonon terms could indeed resolve the above discrepancy. Systematic phononcalculations are needed to give a definite answer to this question. Another sourceof discrepancy between theory and experiment may come from the homogenous

5.2 Random Solid Solutions 25

chemistry as briefly discussed above. In practice, the behaviour of the partialdislocations/stacking fault depends on the deformation and heat treatments (de-formation temperature, annealing temperature and time, etc,) which may resultin non-homogenous chemical distributions. In particular, short range order maydevelop in concentrated Cu-Al alloys, which may weakly increase the SFE [31].Before making a final verdict on this matter, one should perform further the-oretical studies for concentrated alloys accounting for inhomogenous chemistryas well as further experimental investigations which could eventually reveal theactual mechanism behind the pronounced change in the slope of SFE versus Alcontent (Fig. 5.1).In Table 5.2, we list the calculated (un)stable stacking fault energies for the

random Cu-Al solid solutions. As derived from bond-counting arguments, forelemental metals or homogeneous solid solutions, γisf and γesf are proportionalto the hcp-fcc energy difference though

γisf ' γesf ' 2(Ehcp − Efcc)/A, (5.1)

where Ehcp or Efcc are the total energy per atom of hcp or fcc phase, respectively,and A is the stacking fault area. Equation 5.1 holds up to the second nearest-neighbor interactions, but neglects non-central forces and fails in the case ofinhomogeneous solid solutions [30]. In the present study, the 2(Ehcp − Efcc)/Avalues for different Al contents are also listed in Table 5.2. We can see the valuesfrom Eq. 5.1 are very close to those obtained by employing direct supercellcalculations.Using a supercell approach to account for alloying, Kibey et al. [24] calculated

the γ surface for two Cu-Al alloys. Their results are also listed in Table 5.2.These previous theoretical predictions are in a good agreement with ours. How-ever, they noticed that the results are very sensitive to the alloy configuration.Without experimental data, it would be difficult to decide which configurationcorrectly represents the real case. For instance, Kibey et al. [29] showed thatchanging the Al distribution may lower the calculated γisf from 20 to 18.7 mJ/m2

in Cu-5 at.% alloy. The segregation of the solute atoms to the stacking fault area(Suzuki effect) has a great effect on the GSFE [24, 30]. For instance, in Al-Agalloys, ab initio calculations showed that Ag segregation to the stacking faultsdecreases γisf to a negative value, implying that the stacking faults may serveas a fast nucleation pathway for nanoprecipiation of the Ag-rich particles [30].

Atomic segregation on the γ surface

To address the effect of atomic segregation on the γ surface, here we considertwo particular cases. In the first case (Case I), we calculate the γ surfaces of asystems containing two atomic layers with 4 at.% Al placed at different distancesfrom the slip plane in an otherwise perfect Cu matrix. In this case, only onelayer on one side of the slip plane contains Al, which is similar as in Ref. [24].The corresponding average concentration of the supercell is ∼ 0.57 at.%. Tokeep the symmetry, we put 4 at.% Al in layers 8,14 or 9,13 or 10,12. Within eachof these planes, the system is treated as a random solid solution. Layers 1-7 and2-6 are then displaced for calculating the γ surface from fcc to ISF and fromISF to ESF, respectively. The results are shown in Fig. 5.2 and summarizedin Table 5.3. Note that these energies should not be directly compared to the

26CHAPTER 5. GENERALIZED STACKING FAULT ENERGY OF FCC

METALS AND ALLOYS

Figure 5.2: The γ surface for non-homogenous Cu0.9943Al0.0057 alloy. The 4 at.% Alis located at different planes away from the slip plane in pure Cu matrix.b = 1

6 < 211 > is the Burgers vector of the partial dislocation (in units ofcubic lattice parameter).

Table 5.3: The effect of segregation of 4 at.% Al on the stacking fault energies in Cu.All energies are in units of J/m2. Case I and Case II correspond to distributing4 at.% Al on one side, and two sides of the slip plane in Cu, respectively. LayerNo. refers to the location of the alloying atoms in the supercell.

Case I Case II Case I Case II Case I Case IILayer No. 8,14 1,7,8,14 9,13 2,6,9,13 10,12 3,5,10,12

γisf 37 28 44 39 46 44γusf 169 156 186 186 187 190

fault energies of Cu-4 at.% Al listed in Table 5.2 because of the different latticeconstants. Here the lattice constants are fixed to those of pure Cu. We find thatas long as the alloyed (Al-containing) plane is more than one interlayer distanceaway from the fault plane, the alloying effect is negligibly small. This resultis consistent with the convergence test of the supercell size, showing that theperturbation induced by a stacking fault is short-ranged. Therefore, the localconcentration at/near the slip plane is more crucial compared to the nominal(/average) concentration. In the second case (Case II), the segregation effectson the stacking fault energies are introduced by alloying 4 at.% Al in layers onboth sides of the slip plane. The results are shown in Table 5.3. From the tablewe can see that distributing Al in layers on both sides of the slip plane results inlarger effect on the stacking fault energies than the single layer distribution onlywhen they are close to the slip plane. When they are more than two interlayerdistances away from the slip plane, alloying has minor effect on the stackingfault energies.

5.2 Random Solid Solutions 27

Table 5.4: Theoretical lattice parameters (a), and stable and unstable stacking faultenergies of Cu-Ga alloys. Available experimental results are listed for compar-ison. Unit of lattice parameter is Å and of energies is mJ/m2.

Cu-Ga a γisf γExp.isf γusf γutf γesf

Cu 3.637 47 - 184 217 53Cu-3 at.% Ga 3.645 27 - 173 196 32Cu-6 at.% Ga 3.655 15 30a 161 176 16Cu-9 at.% Ga 3.665 3 10a 156 161 1Cu-12 at.% Ga 3.675 -9 6a 144 144 -12a Values from Reference [97] have been corrected by a factorof 2 by Gallagher [31].

5.2.2 Cu-Ga

The results obtained for Cu-Ga alloys are listed in Table 5.4. The calculatedγisf of Cu-Ga becomes negative at ∼10 at.% Ga at 0K, which agrees well withthe Cu-Ga phase diagram [98]. The phase diagram shows that at room temper-ature the solubility of Ga in Cu is around 15 at.% beyond which a hexagonalphase forms. However, our theoretical γisf values are somewhat smaller than theexperimental values, which may very well be due to the effect of temperatureneglected by the theory.

5.2.3 Cu-Zn

Table 5.5: Theoretical lattice parameters (a), and stable and unstable stacking faultenergies of Cu-Zn alloys. Available experimental results are listed for compar-ison. Unit of lattice parameter is Å and of energies is mJ/m2.

Cu-Zn a γisf γExp.isf γusf γutf γesf

Cu 3.637 47 - 184 217 53Cu-5 at.% Zn 3.645 30 - 175 197 35Cu-10 at.% Zn 3.655 21 35a 161 178 25Cu-14 at.% Zn - - 25a - - -Cu-15 at.% Zn 3.666 13 - 157 169 13Cu-20 at.% Zn 3.676 4 18a 150 157 3a Values from Reference [99] have been corrected by a factorof 2 by Gallagher [31].

Zinc is another common alloying element used to strengthen copper alloysand to achieve low stacking fault energy. The present results for alloying Znin Cu are listed in Table 5.5. It can be seen that Zn decreases the intrinsicstacking fault energy, which agrees well with the experimental observations [31].Unfortunately, there are no previous calculations for the γ surface of Cu-Znalloys to compare.

28CHAPTER 5. GENERALIZED STACKING FAULT ENERGY OF FCC

METALS AND ALLOYS

5.2.4 Cu-Ni

Table 5.6: Theoretical lattice parameters (a), and stable and unstable stacking faultenergies of Cu-Ni alloys. Available theoretical and experimental results arelisted for comparison. Unit of lattice parameter is Å and of energies is mJ/m2.

Ni-Cu a γisf γusf γutf γesf

Cu 3.637 47 184 217 53Cu-20 at.% Ni 3.598 74 217 257 77Cu-40 at.% Ni 3.578 92 244 294 91Cu-60 at.% Ni 3.556 106 269 327 106Cu-80 at.% Ni 3.537 124 297 366 123Ni 3.518 153 330 410 153Nia - 110 273 324 -Ni-2.5 at.% Cud - 86 256 293 -a Reference[24].

The alloying effects (Nb, W, Mn, Fe, Cu) on the GSFE of Ni alloys wasinvestigated by Siegel et al.[22]. They found that all alloying elements decreasethe stacking fault energy of Ni. The ductility and twinnability were discussedbased on the calculated GSFE surfaces. They also suggested that the higherstrain-hardening rate reported for nanocrystalline (nc) Ni-Cu relative to nc-Ni-Fe alloys may be ascribed to the different alloying effects on the unstablestacking fault energy rather than on the stable stacking fault energy [21, 22]. InTable 5.6, we list the present stable and unstable stacking fault energies obtainedfor the Ni-Cu alloys, together with the previous theoretical calculations. The γisfvalues calculated with the present method are in reasonable agreement with theexperimental results, noticing that the reported measured values with respectto composition show very large scattering. For detailed discussions about thecomposition dependence of γisf for Ni-Cu alloys, readers are referred to Lu etal. [33].

For Ni-Cu random solid solutions, increasing Cu content decreases all planarfault energies. The present decreasing ratio averaged over the entire compositionrange (0-100 at.% Cu) is calculated to be around 0.83, 0.77, 1.33 and 1.17 mJ/m2

per at.% Cu for γisf , γesf , γusf , and γutf , respectively. Siegel et al. [22] foundmuch larger ratios. However, their calculations were carried out for segregatedalloys, i.e., the alloying atoms were placed on the fault plane. Therefore, in theirstudy an average concentration of 2.5% Cu means in fact a local concentration of25 at.% Cu on the fault plane. Previous atomic simulations have shown that Cusegregation to the stacking fault area or the partial dislocation cores significantlydecreases the stacking fault energy [100, 101]. Similar to the discussion aboutthe segregation effect in Cu-Al alloys, it is not surprising that Siegel et al. [22]got significantly larger decreasing ratios.

5.2.5 Pd-Ag and Pd-AuPalladium alloys are commonly processed through severe plastic deformationto reach the combination of high strength and high ductility in ultrafine/nano

5.3 The “Universal Scaling Law” in Alloys 29

crystallines [102, 103]. The composition dependence of the plastic deformationmechanisms are usually related to the stacking fault energy. Here, we reportthe stable and unstable stacking fault energies as a function of composition,which may benefit the understanding of the relation between plastic deformationmechanisms and compositions. The results for Pd-Ag and Pd-Au alloys arelisted in Table 5.7. Both Ag and Au decrease the stable and unstable stackingfault energies. Our theoretical γisf for Pd-Ag alloys is in good agreement withthose from experimental measurements [104] and previous calculations [33]. ForPd-Au, we could not find any published experimental γisf data. Jonathan etal. [103] calculated the GSFE of the Pd-Au alloys with an interatomic potentialand showed that γisf and γesf decrease with increasing Au content, whereasγusf and γutf vary only slightly with the Au concentration, which somewhatcontradicts to the present results. Careful experimental work is needed to checktheir results about the dislocation behaviors under plastic deformation fromthe atomic simulations with the same interatomic potential, because dislocationbehavior strongly depends on the γ surface.

Table 5.7: Theoretical lattice parameters(a), and stable and unstable stacking faultenergies of Pd-Ag/Au alloys. Unit of lattice parameter is Å and of energies ismJ/m2.

Alloy a γisf γusf γutf γesf

Ag 4.164 17 120 137 18Pd-90 at.% Ag 4.138 27 134 155 27Pd-80 at.% Ag 4.115 31 146 169 33Pd-70 at.% Ag 4.093 37 158 186 39Pd-60 at.% Ag 4.072 50 174 208 52Pd-45 at.% Ag 4.043 63 194 235 65Pd-40 at.% Ag 4.034 71 202 - -Pd-25 at.% Ag 4.005 99 225 282 101Pd-20 at.% Ag 3.996 111 233 - -Pd-10 at.% Ag 3.976 126 245 - -Pd 3.957 143 260 - -Au 4.176 31 123 150 32Pd-80 at.% Au 4.129 74 167 210 74Pd-60 at.% Au 4.086 96 202 260 101Pd-40 at.% Au 4.045 120 227 295 121Pd-20 at.% Au 4.003 141 247 - 143Pd 3.957 143 260 - -

5.3 The “Universal Scaling Law” in AlloysIn Fig. 5.3, the relation between γutf/γusf and γisf/γusf is plotted for differ-ent alloys, taking the present data for fcc metals, Pd-Ag, Pd-Au and Cu-X(X=Ni,Zn,Ga,Al) alloys together with the available density functional results for

30CHAPTER 5. GENERALIZED STACKING FAULT ENERGY OF FCC

METALS AND ALLOYS

Figure 5.3: Scaling plot of stable and unstable fault energies for different metals andalloys. For Ni-X (X=Ni,W,Mn,Fe,Cu), Cu-Al and Al-X (X=Mg,Ga,Zn,Si,Cu)alloys, data is taken from Reference [22], Reference [24], and Reference [25],respectively (solid symbols). The rest of the data is from the present calcula-tions (open symbols).

Cu-Al [24], Ni-X (X=Ni,W,Mn,Fe,Cu) [22] and Al-X (X=Mg,Ga,Zn,Si,Cu) [25]alloys. We find that except for the Al-X alloys, the relationship assumed byEq. (2.5) is very well obeyed. This leads to a question mark on the results ofAl-X alloys. We notice that using the results from Reference [25], it is foundthat Eq. (2.5) does not hold even for pure Al, so it is not surprising that all theAl-X alloys do not pose themselves on the dashed line in Fig. 5.3. The otherreason may be that the alloying configuration in the supercell can not repre-sent/describe the random solid solution correctly [29]. For pure fcc elementsincluding Al, our results also evident that Eq. (2.5) should be generally obeyed,similarly as shown by Jin et al. [47].

5.4 Alloying Effects on Twinnablity

The twinnability of the Cu-X alloys, rd and T are calculated with Eq. 2.8 and2.6, respectively. Figure. 5.4 shows the alloying effects on rd and T in the presentCu alloys. ∆T and ∆rd are calculated relative to those of pure Cu, respectively.Note that larger T or smaller rd means stronger twinning tendency. We findthat the tendency of twinning in Cu alloys increases with Zn, Al, and Cu alloy-ing but decreases monotonously with Ni alloying. Both twinnability parametersyield the same alloying effects for the tendency of twinning, which is in linewith the experimental observations. Aluminum and gallium have similar effectson the tendency for twinning, while the effect of Zn is weaker. Because of thehigh twinnability due to the high Al concentration (∼16 at.%), multifold defor-mation twins were found in relatively large grains (∼ 100 nm) in Cu-Al alloysprocessed through severe plastic deformation [23]. Based on the alloying effecton twinnability, we may also expect that similar complex twin patterns exist inCu-Ga or Cu-Zn alloys at high alloying concentrations with very high twinning

5.4 Alloying Effects on Twinnablity 31

tendency. However, one should be noted that there are many factors being re-ported affecting twinning tendency, such as loading stress [44], grain size [19,105], crystalline orientation [44], nonequilibrium grain boundaries [106], and soon. In addition to the GSFE curves, other intrinsic material parameters suchas shear modulus and Poisson’s ratio will also affect the final observation oftwins [50]. Temperature and strain rate represent the most important externalfactors that controlling the formation of twins. For example, Cu does not nor-mally deform by twinning due to its relatively low twinning tendency, except atconditions of high strain rate and/or low temperature [105]. Another extremecase is Al, which rarely undergoes twinning due to its very high SFE. Twinning,however, was observed at the crack tip where the stress is concentrated andhigh [107]. Using molecular dynamics simulation, Warner et al. [50] showedthat the temperature dependence of the activation energy barriers for trailingor twinning partials primarily arises from the temperature dependence of the γsurface and elastic constants. However, to date the temperature dependence ofthe γ surface has not been well studied from ab initio theory.

Figure 5.4: Alloying effects on twinnability measures (rd and T ) in Cu-X(X=Zn,Al,Ga,Ni) alloys. ∆T and ∆rd are calculated relative to pure Cu.

Chapter 6

Concluding Remarks andFuture Work

In this thesis, using ab initio alloy theory formulated within the frameworks ofEMTO method in combination with the coherent potential approximation weinvestigated the effect of Al on the formation energy of the α-α′ interfaces inFe-Cr alloys and generalized stacking fault energies of random solid solutionsalloys.The results suggest that the interplay between the relative small critical nu-

clease size and high atomic diffusion rates at 475oC cause the highest rate ofembrittlement which observed experimentally in Fe-Cr alloy at this tempera-ture. In Fe-Cr-Al system (PM2000TM), for the preliminary stage of the phaseseparation we found that Al barely changes the interfacial energy. For the laterstage of phase separation we found that Al increases the interfacial energy, par-ticularly for 0-15 at.% Cr. Aluminum also shifts the local maximum on theinterfacial energy map toward low Cr limit. The effects of Al on the interfacialenergy is related to its particular effects on the formation energies of Fe-Cr-Al alloys. Based on the present findings, we conclude that the suppression ofthe phase separation in Fe-Cr alloys by adding Al can primarily be ascribed tothe effect of Al on the formation energies rather than on the interfacial energy.It is found that Al destabilizes the α′ phase and shifts the α + α′ two phasesboundary toward high Cr content.EMTO-CPA method is capable to account for the composition dependent in-

trinsic energy barriers of random solid solutions. We illustrate the performanceof the EMTO method in the generalized stacking fault energy calculations in thecase of elemental fcc metals. Applications to solid solutions (Cu-Al) where othertheoretical predictions are also available, verify the accuracy of our approach.Starting from the electronic structure of the host and faulted lattices, we pro-vide a plausible explanation for the alloying trends calculated for the stable andunstable stacking fault energies. Using the present tool, we show that alloyingeffects are primarily important within the layers next to the planar faults, a factwhich should be considered when modeling the alloying effects via conventionalsupercell techniques. Based on the present and previous theoretical results onthe γ surface, we find that the universal scaling law for the intrinsic energybarriers holds also in the case of solid solutions. Considering two twinnability

34 CHAPTER 6. CONCLUDING REMARKS AND FUTURE WORK

parameters, we predict that Al, Ga and Zn favor the twin formation in Cu-basedsolid solutions, whereas Ni acts as a full slip agent. The present developmentsshow that modern alloy theory can handle such fundamental question as the γsurface in solid solutions, opening promising opportunities in quantum mechan-ics based computational material modeling.For future work, I will work on the generalized stacking fault energy in steels,

for example, fcc iron and austenitic stainless steels, etc. The effects of magnetismand the composition dependence will be investigated.

Acknowledgement

Thanks to my supervisor Prof. Levente Vitos for giving me the opportunity tojoin the AMP group at KTH and his continuous support and guiding me in myPh.D. study and research.Grateful acknowledgement also goes to my co-supervisors Prof. Qing-Miao

Hu, Prof. Börje Johansson, and Dr. Song Lu for their ongoing encouragementand professional guidance.Many thanks to our group members: Andreas, Dongyoo, Erna, Fuyang, Gui-

jiang, Guisheng, Hualei, Lorand, Narisu, Ruihuan, Song, Stephan and Xiaoqingfor helpful discussions and collaborations.The Swedish Research Council, the Swedish Steel Producers’ Association,

the European Research Council, the China Scholarship Council, the Hungar-ian Scientific Research Fund, and are acknowledged for financial support. Thecomputations simulations were performed on resources provided by the SwedishNational Infrastructure for Computing (SNIC) at the National SupercomputerCentre in Linköping.Last but not the least, I would like to thank my wife, my parents for contin-

uous support and love.

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