Mechanical Properties of TransitionMetal Alloys from First-Principles

Theory

Xiaoqing Li

Doctoral Thesis

School of Industrial Engineering and Management, Department of

Materials Science and Engineering, KTH, Sweden, 2015

MaterialvetenskapKTH

SE-100 44 StockholmISBN 978-91-7595-635-0 Sweden

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolanframlägges till offentlig granskning för avläggande av doktorsexamen fredagenden 28 augusti 2015 kl 10:00 i F3, Kungliga Tekniska Högskolan, Lindstedtsvä-gen 26, Stockholm.

© Xiaoqing Li, 2015

Tryck: Universitetsservice US AB

Abstract

The aim of the thesis is to investigate the alloying and temperature effectson the mechanical properties of body-centered cubic (bcc) random alloys. Weemploy the all-electron exact muffin-tin orbitals method in combination with thecoherent-potential approximation. The second-order elastic constants reflect themechanical properties of materials in the small deformation region, where thestress-strain relations are linear. Beyond the small elastic region, the mechanicalproperties of defect-free solids are described by the so called ideal strength.These two sets of physical quantities are the major topic of my investigations.

In part one (papers I and II), the elastic constants and the ideal tensile strengt-hs (ITS) are investigated as a function of Cr and Ti for the bcc V-based randomsolid solution. We find that alloys along the equi-composition region exhibit thelargest shear modulus and Young’s modulus, which is a result of the oppositealloying effects obtained for the two cubic shear elastic constants C′ and C44.The classical Labusch-Nabarro solid-solution hardening (SSH) model extendedto ternary alloys predicts a larger hardening effect in V-Ti than in V-Cr alloy.By considering a phenomenological expression for the ductile-brittle transitiontemperature (DBTT) in terms of Peierls stress and SSH, we show that the pre-sent theoretical results can account for the observed variations of DBTT withcomposition. Under uniaxial [001] tensile loading, the ITS of V is 12.4 GPa andthe lattice fails by shear. Assuming isotropic Poisson contraction, the ITSs are36.4 and 52.0 GPa for V in the [111] and [110] directions, respectively. For theV-based alloys, Cr increases and Ti decreases the ITS in all principal directions.Adding the same concentration of Cr and Ti to V leads to ternary alloys withsimilar ITS values as that of pure V. We show that the ITS correlates with thefcc-bcc structural energy difference and explain the alloying effects on the ITSbased on electronic band structure theory.

In part two (paper III), the alloying effect on the ITS of four bcc refractoryHEAs based on Zr, V, Ti, Nb, and Hf is studied. Starting from ZrNbHf, wefind that the ITS decreases with equimolar Ti addition. On the other hand, ifboth Ti and V are added to ZrNbHf, the ITS is enhanced by about 42 %. Aneven more captivating effect is the ITS increase by about 170 %, if Ti and Vare substituted for Hf. We explain the alloying effect on the ITS based on thed-band filling. We explore an intrinsic brittle-to-ductile transition, which arisesdue to an alloying-driven change of the failure mode under uniaxial tension. Ourresults indicate that intrinsically ductile HEAs with high intrinsic strength canbe achieved by controlling the proportion of group four elements to group fiveelements.

In part three (papers IV and V), the ITS of bcc ferromagnetic Fe-based ran-dom alloys is calculated as a function of compositions. The ITS of Fe is calcu-lated to be 12.6 GPa under [001] direction tension, which is in good agreementwith the available theoretical data. For the Fe-based alloys, we predict that V,Cr, and Co increase the ITS, while Al and Ni decrease it. Manganese yields aweak non-monotonic alloying behavior. We show that the previously establishedideal tensile strengths model based on structural energy differences for the non-magnetic V-based alloys is of limited use in the case of Fe-bases alloys, which is

iv

attributed to the effect of magnetism. We find that upon tension all investigatedsolutes strongly alter the magnetic response of the Fe host from the unsaturatedtowards a stronger ferromagnetic behavior.

In part four (paper VI), the temperature effect on the ITS of bcc Fe andFe0.9Co0.1 alloy is studied. We find that the ITS of Fe is only slightly tempe-rature dependent below ∼ 500 K but exhibits large thermal gradients at highertemperatures. Thermal expansion and electronic excitations have an overall mo-derate effect, but magnetic disorder reduces the ITS with a pronounced 90 %loss in strength in the temperature interval ∼ 500 - 920 K. Such a dramatictemperature effect far below the magnetic transition temperature has not beenobserved for other micro-mechanical properties of Fe. We demonstrate that thestrongly reduced Curie temperature of the distorted Fe lattices compared tothat of bcc Fe is primarily responsible for the onset of the drop of the intrin-sic strength. Alloying additions, which have the capability to partially restorethe magnetic order in the strained Fe lattice, push the critical temperature forthe strength-softening scenario towards the magnetic transition temperature ofthe undeformed lattice. This can result in a surprisingly large alloying-drivenstrengthening effect at high temperature as illustrated in our work in the caseof Fe-Co alloy.

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Preface

List of included publications:

I Elastic properties of Vanadium-based alloys from first-principles theoryXiaoqing Li, Hualei Zhang, Song Lu, Wei Li, Jijun Zhao, Börje Johanssonand Levente Vitos, Phys. Rev. B 86, 014105 (2012).

II Ideal strength of random alloys from first-principles theoryXiaoqing Li, Stephan Schönecker, Jijun Zhao, Börje Johansson and Le-vente Vitos, Phys. Rev. B 87, 214203 (2013).

III Anomalous ideal tensile strength of ferromagnetic FeXiaoqing Li, Stephan Schönecker, Jijun Zhao, Börje Johansson and Le-vente Vitos, Phys. Rev. B 90, 024201 (2014).

IV Ideal tensile strength of ferromagnetic Fe-based alloys from first-principlestheoryXiaoqing Li, Stephan Schönecker, Jijun Zhao, Börje Johansson and Le-vente Vitos, submitted to Phys. Rev. B.

V Ab initio-predicted micro-mechanical performance of refractory high-entropyalloysXiaoqing Li, Fuyang Tian, Stephan Schönecker, Jijun Zhao, and LeventeVitos, accepted by Sci. Rep.

VI Tensile strain-induced softening of iron at high temperatureXiaoqing Li, Stephan Schönecker, Eszter Simon, Lars Bergqvist, HualeiZhang, László Szunyogh, Jijun Zhao, Börje Johansson, and Levente Vitos,submitted to Proc. Natl. Acad. Sci. USA.

Comment on my own contribution:Paper I: project formulated jointly, all calculations, literature survey, data ana-lysis, the manuscript was written jointly.Paper II: project formulated jointly, all calculations and literature survey, dataanalysis jointly (80%), the manuscript was written jointly.Paper III: project formulated jointly, all calculations and literature survey, dataanalysis jointly (80%), the manuscript was written jointly.Paper IV: project formulated jointly, most of calculations (80%), literature sur-vey, data analysis, the manuscript was written jointly.Paper V: I suggested this project, I did the ideal tensile strength calculations,literature survey and data analysis jointly (75%), the manuscript was writtenjointly.Paper VI: project formulated jointly, calculations (90%), literature survey anddata analysis jointly (80%), the manuscript was written jointly.

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Publications not included in the thesis (but most of them very closely relatedto the topic of my thesis):

I Anomalous elastic hardening in Fe-Co alloys at high temperatureHualei Zhang, Xiaoqing Li, Stephan Schönecker, Börje Johansson, andLevente Vitos, Phys. Rev. B 89, 184107 (2014).

II Influence of manganese on the bulk properties of Fe-Cr-Mn alloys: a first-principles studyNoura Al-Zoubi, Xiaoqing Li, Stephan Schönecker, Börje Johansson, andLevente Vitos, Phys. Scr. 89, 125702 (2014).

III Kinetic arrest induced antiferromagnetic order in hexagonal FeMnP0.75Si0.25

alloyGuijiang Li, Wei Li, Stephan Schönecker, Xiaoqing Li, Erna K. Delczeg-Czirjak, Yaroslav O. Kvashnin, Olle Eriksson, Börje Johansson, and Le-vente Vitos, Appl. Phys. Lett. 105, 262405 (2014).

Sammanfattning

För att tillhandahålla teoretiska riktlinjer vid optimering och modellering avmikromekaniska egenskaper hos teknologiskt viktiga övergångsmetallegeringar,så har vi i denna avhandling fokuserat på legerings- och temperatureffekter påmekaniska egenskaper hos kroppscentrerat kubiska (engelska: body-centered cu-bic, bcc) legeringar, genom att använda den exakta muffin-tin orbitalmetoden ikombination med den koherenta potentialapproximationsmetoden. Andra ord-ningens elastiska konstanter beskriver de mekaniska egenskaperna hos materialvid små deformationer, där dragprovskurvan är linjär. Vid större deformationerså beskrivs mekaniska egenskaper hos dislokationsfria solida material av denideala styrkan.

De elastiska konstanterna och ideala dragstyrkorna (ITS) har undersökts somfunktion av Cr och Ti för bcc V-baserade solida lösningar. Legeringar som be-finner sig längs med ekvimolarregionen finns ha de största skjuvningarna ochYoungsmodulerna som ett resultat av de motsatta legeringseffekterna hos de tvåkubiska skjuvelasticitetskonstanterna C′ och C44. Klassiska modeller för solidlösningshärdning (engelska: solid-solution hardening, SSH) förutspår en störrehärdning i V-Ti än i V-Cr legeringar. Genom att betrakta ett fenomenologisktuttryck för den spröda-till-duktila övergångstemperaturen (engelska: ductile-to-brittle transition temperature, DBTT) i termer av Peierlsstress och SSH, så visasdet att de nuvarande teoretiska resulaten kan ta hänsyn till variationerna i DB-TT som funktion av sammansättning. Vid uniaxial [001] dragbelastning är denideala dragstyrkan hos V 12.4 GPa och gittret ger vika genom skjuvning. Genomatt anta isotrop Poissonkontraktion är den ideala dragstyrkan 36.4 respektive52.0 GPa för V i [111]- och [110]-riktningarna. För V-baserade legeringar så ökarCr och minskar Ti den ideala dragstyrkan i alla principalriktningar. Tillägg avCr och Ti i lika koncentration till V leder till terniära legeringar med liknandeideal dragstyrka som hos rent V. Effekten av legering på den ideala dragstyrkanförklaras genom den elektroniska bandstrukturen.

Effekten av legering på ITS hos fyra bcc refraktoriska högentropilegeringar(engelska: high entropy alloys, HEA) baserade på Zr, V, Ti, Nb och Hf harstuderats. Hos ZrNbHf visar det sig att ITS minskar med ekvimolar addition avTi. Å andra sidan, om både Ti och V adderas till ZrNbHf så ökar ITS med runt42%. En ännu mera slående effekt är ökningen av ITS på 170% om Ti och Vbyts ut mot Hf. Legeringseffekten på ITS förklaras med d-bandets fyllnad. Enspröd-till-duktil övergång hittas i termer av felmoder under uniaxial spänning.Dessa undersökningar antyder att duktila HEAs med hög ideal dragstyrka kanuppnås genom kontroll av proportionen av grundämnen från grupp fyra ochgrupp fem i det periodiska systemet.

ITS hos bcc ferromagnetiska Fe-baserade legeringar har beräknats som funk-tion av sammansättning. Den ideala dragstyrkan hos Fe i [001]-riktningen beräk-nas till 12.6 GPa, i överensstämmelse med tillgänglig data. För Fe-baserade lege-ringar så förutspår vi att V, Cr, och Co ökar den ideala dragstyrkan, medan Aloch Ni minskar den. Mangan ger ett svagt icke-monotont beteende vid legering.Vi visar att det begränsade användandet av de tidigare använda ideala dragstyr-kemodellerna, som baseras på strukturella energiskillnader, för Fe-baserade le-

viii

geringar beror på magnetism. Vi finner att för alla undersökta lösningar underspänning så ändras den magnetiska responsen hos Fe-baskomponenten från detomättade till det starkare ferromagnetiska beteendet.

Temperatureffekten på ITS hos bcc Fe och Fe0.9Co0.1 studeras. Vi finner attITS endast visar ett svagt temperaturberodende under ∼ 500 K, men att den vi-sar stora temperaturgradienter vid höga temperaturer. Termisk utvidgning ochelektroniska exciteringar har överlag en måttlig effekt, men magnetisk oordningminskar ITS med en markant förlust i styrka på 90% i temperaturintervallet∼ 500 − 920 K. En sådan dramatisk temperatureffekt har inte observerats iandra mikromekaniska egenskaper hos Fe. Vi visar att den starkt reduceradeCurietemperaturen hos det förvrängda Fe-gittret, jämfört med bcc Fe, är främstansvarigt för påbörjan av förlusten i styrka. Legeringar som har möjligheten attdelvis återställa den magnetiska ordningen i det belastade Fe-gittret ökar denkritiska temperaturen för det upp uppmjukande styrkescenariot mot den magne-tiska övergångstemperaturen hos det odeformerade gittret. Detta kan resultera iöverraskande stora legeringsdrivna styrkeökningar vid höga temperaturer, vilketvisas i vårt arbete på Fe-Co legeringar.

Contents

List of Figures xv

List of Tables xvii

1 Introduction 1

1.1 Vanadium-based Alloys . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Refractory High-entropy Alloys . . . . . . . . . . . . . . . . . . . 2

1.3 Iron and Iron-based Alloys . . . . . . . . . . . . . . . . . . . . . . 2

2 Theory 5

2.1 First-principles Theory of the Electronic Structure . . . . . . . . 5

2.2 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . 6

2.3 Exact Muffin-tin Orbitals Method . . . . . . . . . . . . . . . . . 8

2.4 Coherent-potential Approximation . . . . . . . . . . . . . . . . . 9

2.5 Disordered Local Moment Model . . . . . . . . . . . . . . . . . . 10

3 Mechanical Properties of Materials 11

3.1 Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Solid-solution Hardening . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Ideal Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Methodology for Ideal Strength Calculation . . . . . . . . . . . . 14

4 Magnetism 21

4.1 Stoner Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Finite Temperature Properties of Iron and Iron-based Alloys . . 23

4.2.1 Heisenberg Model and Magnetic Exchange . . . . . . . . . 23

4.3 Curie Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.4 Spin-wave Stiffness for Cubic Symmetry . . . . . . . . . . . . . . 27

4.5 Spin-wave Stiffness for Non-cubic Symmetry . . . . . . . . . . . . 29

x CONTENTS

5 Mechanical Properties of Vanadium and Vanadium-based Al-

loys 31

5.1 Structural Energy Difference . . . . . . . . . . . . . . . . . . . . 31

5.2 Assessing the Accuracy of the Theoretical Predictions for Alloys 32

5.3 Lattice Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.4 Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.4.1 Single Crystal Elastic Constants . . . . . . . . . . . . . . 34

5.4.2 Polycrystalline Elastic Properties . . . . . . . . . . . . . . 37

5.4.3 Ductile/Brittle Properties . . . . . . . . . . . . . . . . . . 38

5.4.4 Solid-solution Hardening . . . . . . . . . . . . . . . . . . . 39

5.5 Ideal Tensile Strength . . . . . . . . . . . . . . . . . . . . . . . . 41

5.5.1 Ideal Tensile Strength of Vanadium . . . . . . . . . . . . . 41

5.5.2 Ideal Tensile Strength of Vanadium-based Alloys . . . . . 44

5.5.3 Structural Energy Difference and Ideal Tensile Strength . 44

6 Ideal Tensile Strength of Refractory High-entropy Alloys 51

6.1 Ideal Tensile Strength of High-entropy Alloys . . . . . . . . . . . 51

6.2 Structural Energy Difference and Average d-band Filling . . . . . 53

7 Ideal Tensile Strength of Iron and Iron-based Alloys 55

7.1 Ideal Tensile Strength of Iron . . . . . . . . . . . . . . . . . . . . 55

7.2 Ideal Tensile Strength of Iron-based Alloys . . . . . . . . . . . . . 56

7.3 Magnetic Effect on the Ideal Tensile Strength . . . . . . . . . . . 59

7.4 Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . 62

7.5 Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

8 Thermal Effect on the Ideal Tensile Strength of Iron 67

8.1 Methodology for Magnetic Contribution . . . . . . . . . . . . . . 67

8.2 Methodology for Phonon Contribution and Electron Excitation . 68

8.2.1 Thermal Expansion . . . . . . . . . . . . . . . . . . . . . 68

8.2.2 Lattice Vibration . . . . . . . . . . . . . . . . . . . . . . . 69

8.2.3 Electron Excitation . . . . . . . . . . . . . . . . . . . . . 70

8.3 Prediction of the Temperature-dependent Ideal Tensile Strength

of Iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

8.4 Comparison of the Ideal Tensile Strength in Ferromagnetic State

and Paramagnetic State . . . . . . . . . . . . . . . . . . . . . . . 72

8.5 Alloying Effect of Cobalt . . . . . . . . . . . . . . . . . . . . . . . 74

CONTENTS xi

9 Concluding Remarks and Outlook 79

Bibliography 85

List of Figures

3.1 Illustration of the lattice distortion of the bcc structure for ten-

sion along the [001] direction . . . . . . . . . . . . . . . . . . . . 16

3.2 The sketches of two failure modes of bcc crystal under [001] tension 17

3.3 Illustration of the lattice distortion of the bcc structure for ten-

sion along the [110] direction . . . . . . . . . . . . . . . . . . . . 18

3.4 Illustration of the lattice distortion of the bcc structure for ten-

sion along the [111] direction . . . . . . . . . . . . . . . . . . . . 19

4.1 The Stoner model for a rectangular density of states. . . . . . . . 21

4.2 The total density of states of nonmagnetic bcc Fe. . . . . . . . . 22

4.3 Exchange interaction J0j for bcc Fe as a function of reduced distance 24

4.4 Binder cumulants of bcc Fe for various sizes of the simulation box 26

4.5 The magnetization curves of Fe from the MC simulation and

Kuz’min’s fit to the experimental data . . . . . . . . . . . . . . . 26

4.6 Behavior of D as a function of reduced distance for various damp-

ing coefficients η for bcc Fe . . . . . . . . . . . . . . . . . . . . . 28

4.7 Shape of the magnetization curve with shape parameters s com-

puted for Fe and Fe90Co10 . . . . . . . . . . . . . . . . . . . . . . 28

4.8 Behavior of Dzz as a function of reduced distance for various

damping coefficients η for bct Fe . . . . . . . . . . . . . . . . . . 29

5.1 Present results and experimental lattice constants, bulk modulus

B, tetragonal shear elastic constant C′ . . . . . . . . . . . . . . . 33

5.2 Theoretical lattice parameters of V1−x−yCrxTiy alloys as a func-

tion of Cr and Ti concentrations . . . . . . . . . . . . . . . . . . 35

5.3 Theoretical single crystal elastic constants of V1−x−yCrxTiy al-

loys as a function of Cr and Ti concentrations . . . . . . . . . . . 36

5.4 Density of states for bcc V, V-7.5Cr, and V-7.5Ti alloys . . . . . 37

xiv LIST OF FIGURES

5.5 Theoretical polycrystalline elastic constants for V1−x−yCrxTiyalloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.6 The ratio of bulk and shear modulus and Cauchy pressure for

V1−x−yCrxTiyalloys . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.7 Solid-solution hardening in ternary V1−x−yCrxTiy alloys . . . . . 40

5.8 A 3D plot of the DBTT for the ternary V1−x−yCrxTiy alloys . . 40

5.9 The stress of bcc V along the [001] direction as a function of the

applied strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.10 The stress of bcc V along [001], [111] and [110] directions as a

function of strain . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.11 The ITS of V-based alloys along [001], [111] and [110] directions

as a function of Cr and Ti. . . . . . . . . . . . . . . . . . . . . . 46

5.12 Density of states for V-based alloys and the fcc-bcc structural

energy difference as a function of d-occupation for 3d transition

metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.13 The uniaxial strain energy of V-based alloys for a stress applied

along the [001] direction . . . . . . . . . . . . . . . . . . . . . . . 48

5.14 The correlation between the real ITS calculation and the one

obtained by SED calculation for V-based alloys along the [001]

direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.15 The correlation between the real ITS calculation and the one

obtained by SED calculation for V-based alloys along the [111]

direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.1 The tensile stress as a function of strain under uniaxial stress

applied along the [001] direction for refractory HEAs. . . . . . . 52

6.2 The correlation between the ITS, the fcc-bcc SED, and the aver-

age valence d-band filling for the HEAs . . . . . . . . . . . . . . 53

7.1 The ITS of ferromagnetic bcc Fe1−xMx alloys as a function of

concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.2 The change of the ITS versus the change of the fcc-bcc SED for

Fe1−xMx alloys assuming a FM fcc state . . . . . . . . . . . . . . 59

7.3 The change of the ITS for constant volume versus the change of

the fcc-bcc SED for Fe1−xMx alloys assuming a FM fcc state . . 59

7.4 The magnetic moments as a function of the tetragonal axial ratio

and the Wigner-Seitz radius for Fe and Fe0.9V0.1 alloy . . . . . . 60

7.5 Correlation between the stress change relative to the ITS of Fe

and the magnetic pressure change . . . . . . . . . . . . . . . . . . 61

LIST OF FIGURES xv

7.6 Band energy cost for an increase of the magnetic moment beyond

its equilibrium value . . . . . . . . . . . . . . . . . . . . . . . . . 62

7.7 The total and partial (t2g and eg) density of states for FM bcc

Fe at equilibrium volume . . . . . . . . . . . . . . . . . . . . . . 63

7.8 The total density of states for Fe and all Fe-based alloys at their

equilibrium volumes . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.9 The position of the t2g↑ shoulder in the majority-spin band rela-

tive to the Fermi energy . . . . . . . . . . . . . . . . . . . . . . . 65

8.1 Experimental lattice parameters of the bcc phases of Fe as a

function of temperature . . . . . . . . . . . . . . . . . . . . . . . 69

8.2 The ITS of bcc Fe in tension along the [001] direction as a function

of temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

8.3 Contour plot of TC of bct Fe as a function of the lattice parame-

ters a and c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.4 The sketch of local spins for the unstrained parent lattice and for

the strained lattice at high temperature . . . . . . . . . . . . . . 73

8.5 Comparison between the full and magnetic ITSs and the ITS

obtained by omitting the structure dependence of "s" . . . . . . . 74

8.6 Total energies for ferromagnetic and paramagnetic Fe as a func-

tion of lattice parameters a and c . . . . . . . . . . . . . . . . . . 75

8.7 Total energy of PM bct Fe for various values of the local magnetic

moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

8.8 The exchange interactions J1 and J2 of bcc Fe and Fe0.9Co0.1 . . 76

8.9 Tensile strength of Fe whiskers under [001] oriented from experi-

mental measurement . . . . . . . . . . . . . . . . . . . . . . . . . 77

List of Tables

3.1 Theoretical ideal strength and experimental maximum strength

for uniaxial and shear loadings . . . . . . . . . . . . . . . . . . . 15

5.1 The ITS and the corresponding strain of V . . . . . . . . . . . . 42

5.2 Comparison between the present and former ideal tensile strength

of V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.3 The ITS and the corresponding strain along [001], [111] and [110]

directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.1 The ITSs of bcc Fe from strain-energy fitting function and stress

tensor as a function of plane wave cut-off energy . . . . . . . . . 56

7.2 The ITS and the corresponding strain in the [001] direction of

ferromagnetic bcc Fe. . . . . . . . . . . . . . . . . . . . . . . . . . 56

7.3 The ITS and the corresponding strain calculated in the [001] di-

rection for Fe1−xMx alloys . . . . . . . . . . . . . . . . . . . . . . 57

8.1 Magnetic quantities used to compute the shape parameter s of

the bcc phases of Fe and Fe-Co . . . . . . . . . . . . . . . . . . . 68

Chapter 1

Introduction

1.1 Vanadium-based Alloys

With the development of technological productivity, the demand for energy in-creases rapidly. Therefore, it is important to exploit new energy resources, forexample fusion energy. Developing fusion energy demands, among many otherfactors, designing materials for fusion reactors. Successful development of struc-tural material is one of the most important tasks for the realization of fusionpower systems. Vanadium-based alloys have been identified as a leading can-didate material for fusion structure applications [1, 2]. This is because certainvanadium alloys exhibit decent thermal creep behavior, high thermal conduc-tivity, good resistance to irradiation-induced swelling and damage, and long op-erating lifetime in the fusion environment [1–6]. Considerable efforts have beenmade to find optimal V-based alloy compositions that can endure the extremeenvironments of fusion reactors. The available experimental data indicate thatreasonable properties can be achieved by introducing a few percent titanium(Ti) and chromium (Cr) into the vanadium (V) matrix. Consequently, the V-Cr-Ti system has attracted a broad interest, and in particular the compositionswith 0-15 at.% Cr and 0-20 at.% Ti have been intensively investigated. Previousstudies of the V-based alloys were mainly focused on the ductile-brittle tran-sition temperature before or after irradiation, swelling properties, and impacttoughness as a function of Cr and Ti concents. As a promising structure materialfor fusion reactors, V-based alloys must not only withstand radiation damagebut should also keep intrinsic mechanical properties and structural strength. Itis well known that the elastic properties and structural strength of materialscan be used to characterize their mechanical deformation and structural stabil-ity under external loading. For example, the bulk modulus (B) is used to unveilthe average bond strength, and the phenomenological ratio of bulk modulusand shear modulus (B/G) is often employed to give a rough measure of theductile/brittle characteristics of materials. Thus it is necessary to study thesefundamental mechanical properties for further optimization of the compositionof V-based alloys.

2 CHAPTER 1. INTRODUCTION

1.2 Refractory High-entropy Alloys

Recently developed high-entropy alloys (HEAs) represent a new field in metal-lurgy and have attracted great attention in the scientific community due to theirappealing properties for a wide range of applications [7, 8]. Different from tradi-tional alloys which contain one or two principal elements, HEAs are composedof several equimolar or near equimolar elements [7]. The solid solution phasesof HEAs often are face-centered cubic (fcc), hexagonal-close packed (hcp) orbody-centered cubic (bcc), which is due to a high mixing (or configurational)entropy. HEAs consisting of refractory metallic elements (Ti, Zr, Hf, V, Nb,Ta, Cr, Mo, and W) are single bcc phase and have been in the focus of sev-eral experimental studies [9–18], which should be attributed to their novel hightemperature properties. High strength and good ductility are essential demandson the mechanical performance of structural materials. Most previous experi-mental works focussed on compressive properties of refractory HEAs. Althoughthe tensile properties are more important for engineering applications, they havehardly been investigated hitherto. On the theoretical side, due to the large num-ber of principal elements (three to thirteen) in the solid solution single phase,it is a big challenge to investigate the properties of the HEAs by conventionalab initio atomistic simulation methods. Here, based on the coherent-potentialapproximation (CPA), we investigate the compositional effect on the intrinsicstrength/ideal strength of four HEAs in bcc phase composed of the refractoryelements Ti, Zr, Hf, V, and Nb. The present study offers a guideline to designintrinsically ductile refractory HEAs having a carefully tailored strength level.

1.3 Iron and Iron-based Alloys

Iron is one of the most abundant elements in the universe and constitutes themajor component of earth’s core [19]. Processed iron is the main ingredient insteels and other ferrous materials, whose technology is a central pillar in today’sindustrial world. Beyond the obvious practical interests, the physical propertiesof Fe have attracted great attention in many fields of sciences. Magnetic orderin solid iron emerges with the formation of a crystalline lattice. The stable bccphase (α-Fe) is characterized by long-range ferromagnetic (FM) order belowCurie temperature (TC). The fcc structure appears in the phase diagram attemperatures above 1189 K in the paramagnetic (PM) state (γ-Fe). Modelingthe temperature effect on the attainable strength of Fe determined by complexmicro-structural properties associated with defects like vacancies, dislocationnetworks, and grain boundaries, is an enormous task for ab initio methods.Fortunately, there are several atomic-level properties which are accessible for ab

initio methods. One relatively new physical characteristic of ideal defect-freecrystals is the intrinsic ideal tensile strength (ITS) which is recognized as keyparameter in fracture theory and defect nucleation. Although the ITS of Fehas been widely studied [20–25], so far all investigations were limited to 0 K.Extending these investigations to high-temperatures will broaden the possibilityfor assessing the impact of ITS in high-temperature mechanical properties ofmicro/nano-scale materials.

Steel is essentially iron and carbon alloyed with certain additional elements.The additional elements play a central role in designing specific materials with

1.3 Iron and Iron-based Alloys 3

desired properties, such as, Cr significantly improves corrosion resistance, andmanganese (Mn) improves hardenability, ductility and wear resistance [26]. Withthe addition of more than 12% Cr, the steel becomes stainless. Stainless steelsare the most widely used engineering materials due to their excellent mechani-cal and chemical properties. For example, Fe-Cr alloys form the basis of ferriticand martensitic stainless steels. These alloys have been considered as the pri-mary structural materials in the first wall and blanket structure of future fusionreactors [27].

Today a detailed mapping of the elastic parameters of Fe as a function ofcomposition, magnetic state and temperature is available. A large number ofworks focused on the effects of various typical solute atoms on the mechanicalproperties of Fe in the small deformation region, where the stress-strain relationsare linear [29–34]. In contrast to that, the ideal tensile strength describes themechanical properties beyond the elastic regime and thus gives informationabout large structural changes upon loading. To further optimize Fe-basedalloys, it is important to investigate the alloying effect on the structure strength.

Chapter 2

Theory

2.1 First-principles Theory of the Electronic Struc-ture

First-principles calculations of the electronic structure are based on the lawsof quantum mechanics and only employ natural constants and some reason-able approximations to investigate the properties of materials. By solving theSchrödinger equation for solids, we can understand electronic, magnetic, me-chanical, and optical phenomena of solid materials.

The time-independent Schrödinger equation is

HΨ = EΨ, (2.1)

where Ψ = Ψ(r1, . . . , rN ,R1, . . . ,RM ) denotes the many-body wave function forN electrons and M nuclei with positions ri, i = 1 . . .N , and Rj , j = 1 . . .M ,respectively.

The Hamiltonian H describing a solid formed by interacting electrons andnuclei is given by [35]

H = − ~2

2me

N∑

i

∇2ri

− ~2

2

M∑

j

∇2Rj

Mj−

N∑

i

M∑

j

e2Zj

|ri − Rj|

+12

N∑

i6=j

e2

|ri − rj | +12

M∑

i6=j

e2ZiZj

|Ri − Rj| , (2.2)

where ~ stands for the reduced Planck constant, e denotes the elementary charge,me is the mass of electrons and Mj are the masses of the nuclei with the atomicnumber Zj . The first term and the second term in the previous equation are thekinetic energy operators for electrons and nuclei, respectively. The third termdescribes the interaction between electrons and nuclei (Coulomb potential). Thelast two terms express the electron-electron and nucleus-nucleus interactions.Due to the huge number of interacting electrons and nuclei in solids (of the orderof the Avogadro number), it is impossible to solve the Schrödinger equation forthe Hamiltonian given in Eq. (2.2). Thus, we have to introduce approximations.

6 CHAPTER 2. THEORY

The first step to overcome this objection is given by the Born-Oppenheimer(BO) approximation [36]. Since the masses of nuclei are much larger than themass of electrons (Mj & 1000me), one can assume that on the timescale ofnuclear motion, the electrons subsystem is always in its stationary state. Thus,one can separate the kinetic energy of the nuclei and consider the electrons tomove in the external potential, Vext, generated by static nuclei. With the BOapproximation, we can simplify Eq. (2.1) to

− ~2

2me

N∑

i

∇2ri

−N

∑

i

M∑

j

e2Zj

|ri − Rj| +12

N∑

i6=j

e2

|ri − rj |

Ψ =

(T + Vext + Vee)Ψ = EΨ, (2.3)

where the operators T , Vext and Vee denote the kinetic energy, electron-nucleusinteraction energy, and electron-electron interaction energy, respectively. Thewave function Ψ in the previous expression depends only parameterically on Rj .Once the Schrödinger equation (2.3) is solved, the nucleus-nucleus interactionenergy is added to the total electronic energy E.

2.2 Density Functional Theory

In this section, we briefly present the basics of non-relativistic density functionaltheory for a non-spin-polarized system following Refs. [35, 37]. Atomic units(~ = me = e = 1) are used throughout.

Even for a system of N interacting electrons moving in a static externalpotential (Eq. (2.3)), it is impossible to solve the Schrödinger equation directly.The very powerful tool to solve Eq. (2.3) for the ground state is by means ofdensity functional theory (DFT). Within DFT, the many-electron problem isreduced to an effective single-electron problem and the electron density n(r)as the main variable is introduced. DFT is based on two important theorems,which were formulated by Hohenberg and Kohn (HK) [38]:

Theorem I: For any system of interacting particles in an externalpotential Vext(r), the potential is determined uniquely, except for aconstant, by the ground state density n0(r).

Theorem II: A universal functional for the energy E(n(r)) in termsof the density n(r) can be defined, valid for any external potentialVext(r). For any particular Vext(r), the exact ground state energy ofthe system is the global minimum value of this functional, and thedensity n(r) that minimizes the functional is the exact ground statedensity n0(r).

With the HK theorems, the energy functional can be defined as

E(n(r)) = F (n(r)) +∫

Vext(r)n(r)dr, (2.4)

where the first term on the right hand side is a universal functional of theelectron density, and the second term describes the interaction with the external

2.2 Density Functional Theory 7

potential. The Hohenberg-Kohn functional F (n(r)) is called universal functionalbecause it does not depend on Vext.

Even though the HK theorems are very powerful, they do not provide a wayof computing the ground-state density of a system in practice. One still needs tofind a good method to carry out DFT calculations. Luckily, in 1965, Kohn andSham (KS) turned DFT into a practical tool for rigorous calculations [39]. Theyassumed that the ground state density of the system of interacting electronsequals the electron density of a fictitious auxiliary system of non-interactingelectrons. Based on KS’s idea, the universal functional is usually formulated as

F (n(r)) = Ts(n(r)) + EH(n(r)) + Exc(n(r)), (2.5)

where the fist term and the second term on the right hand side are the ki-netic energy of a non-interacting system of electrons and the Hartree energy(classical electron-electron interaction), respectively. The third term is theexchange-correlation energy, which contains two parts: the first part is theenergy difference between the real kinetic energy and the kinetic energy of thenon-interacting reference system; the second part is the energy difference be-tween the true electron-electron interaction energy and the classic Coulombenergy.

By applying the variational principle, the famous KS single-particle equationsare obtained [35]

[

−12

∇2 + Veff((n(r)); r)]

ψi(r) = ǫiψi(r). (2.6)

The effective potential is

Veff = Vext(r) +∫

n(r′)|r − r′|dr′ + Vxc(r), (2.7)

where the exchange-correlation potential is defined as Vxc = δExc(n(r))δn(r) . The

electron density can be obtained from the occupied single-electron orbitals

n(r) =occ.∑

i

|ψi(r)|2. (2.8)

The electronic energy (total energy) is given by

E =occ.∑

i

ǫi − EH + Exc −∫

Vxc(r)n(r)dr. (2.9)

The only unknown term in the KS equations is the exchange-correlation en-ergy functional defined in Eq. (2.5). Since the accuracy and the predictive powerof DFT are limited by the employed approximations to describe exchange andcorrelation, modeling Exc represents a very important challenge in the field ofDFT. In general, for a non-spin-polarized system, the local-density approxima-tion (LDA) for the exchange-correlation energy is written as

Exc ≈ ELDAxc (n) =

∫

n(r)ǫxc(n(r))dr, (2.10)

8 CHAPTER 2. THEORY

where ǫxc is the exchange-correlation energy density. The LDA was found toreproduce the ground properties of many systems with high accuracy. Espe-cially, the bulk properties of 4d and 5d transition metals, oxides, and so on,or surface properties of metals are very well described [35, 37]. However, thereare situations where the LDA is inappropriate, the most spectacular failure ofit happens in the case of 3d transition metals. For example, the LDA can notpredict the lowest-energy crystal and magnetic structure for pure Fe.

To overcome the limitations of LDA, the so called generalized-gradient ap-proximation (GGA) has been developed. Within the GGA, the exchange-correlation energy depends not only on the local electron density, but also onits local density gradient ∇n(r)

Exc ≈ EGGAxc (n) =

∫

n(r)ǫxc(n(r),∇n(r))dr. (2.11)

The GGA predicts ground states of solids, including those of 3d metals, whichare in closer agreement with experiments than the corresponding LDA results.The LDA in the parameterization of Perdew and Wang [40] and the GGA inthe parameterization of Perdew, Burke and Ernzerhof [41] are two very popularexchange-correlation functionals.

2.3 Exact Muffin-tin Orbitals Method

Finding an accurate and efficient method to solve the KS equations is a bigchallenge in computational materials science. Nowadays, there are several dif-ferent methods available, such as full-potential methods or muffin-tin orbitalsmethods. In this thesis, mainly the Exact Muffin-Tin (MT) Orbitals Method(EMTO) [42–44] is employed, which is the 3rd generation method in the MTfamily. The EMTO method is an improved screened Korringa-Kohn-Rostoker(KKR) method [42], where the full potential is represented by overlapping MTpotential spheres. Inside these spheres, the potential is spherically symmetricand constant in between.

In the EMTO program, the effective single-electron potential, called MT po-tential VMT, is approximated by spherical potential wells VR(rR) − V0 centeredon lattice sites R with potential radius sR plus a constant potential V0 (V0 iscalled MT zero). We can rewrite Eq. (2.7) as follows

Veff ≈ VMT = V0 +∑

R

[VR(rR) − V0], (2.12)

with the notation rR = r − R, rR = |rR|. For the above MT potential, to solvethe KS equations, we expand the KS orbital ψi(r) in terms of exact MT orbitalsψ

a

RL(ǫi, rR), viz.ψi(r) =

∑

RL

ψa

RL(ǫi, rR)υaRL,i, (2.13)

where υaRL,i are expansion coefficients. These coefficients are determined in

such a way, that the above expansion should be a solution for the KS equations(Eq. (2.6)) in the entire space. L represents a multi-index, L = (l,m), where land m are the orbital and magnetic quantum numbers, respectively.

2.4 Coherent-potential Approximation 9

The exact MT orbitals are constructed using different basis functions at dif-ferent regions of space. Inside the MT potential spheres (rR ≤ sR), the par-tial waves φa

RL(ǫi, rR) are chosen as the basic functions, which are solutionsof the radial scalar-relativistic Schrödinger equation. In the interstitial region,the screened spherical waves ψa

RL(ǫi − υ0, rR) are employed as basic functions,which can be obtained from the solution of the Schrödinger equation at constantpotential υ0. The boundary condition for the Schrödinger equation are givenin conjunction with non-overlapping spheres (hard spheres) centered at latticesites R with radii aR (aR < sR). By means of hard spheres, screened sphericalwaves are localized in space (this is the concept of screening).

The partial waves and screened spherical waves are connected by introducing athird type of basis function: free-electron wave functions with pure lm character.They are matched continuously and differentiable to the partial waves at sR andcontinuously to the screened spherical waves at aR.

The final expression for the exact MT orbitals is

ψa

RL(ǫi, rR) = φaRL(ǫi, rR) + ψa

RL(ǫi − υ0, rR) − ϕaRL(ǫi, rR)YL(rR). (2.14)

The solution of the KS equations in the EMTO method are found by solvingthe so-called kink-cancelation equation [45].

2.4 Coherent-potential Approximation

It is a difficult task to treat correctly random alloys by theoretical means. Onepossible way is that we can build supercells, the distributions of different ele-ments are random in the supercells, however, the above method is very time-consuming for non-trivial case, thus we should find another simple way to dealwith it. Fortunately, the coherent-potential approximation (CPA) [45–48] isassumed to be the most powerful technique to treat systems with random dis-order, i.e., substitutional random alloys. The CPA is based on the assumptionthat the alloy may be replaced by an ordered effective medium, the parametersof which are determined self-consistently. The impurity problem is treated inthe single-site approximation. This means that one single impurity is replacedin an effective medium and no information is provided about the individualpotential and charge density beyond the sphere or polyhedra around this impu-rity. Nowadays, with the CPA method, numerous applications [43, 49–54] haveshown that one can accurately calculate lattice parameters, elastic constants,mixing enthalpies, etc., for random alloys.

Now, we illustrate the principle idea of the CPA within the conventional MTformalism. We consider a substitutional alloy AaBbCc, . . . , where the atoms A,B, C, . . . are randomly distributed on the underling crystal structure. Here, a, b,c . . . are the atomic fractions of the A, B, C, . . . atoms, respectively. We use theGreen function g and the alloy potential Palloy to describe the above system.In a real alloy, due to the environment, the alloy potential shows variationsaround the same type of atoms. In the CPA method, there are two importantapproximations. First, it assumed that the local potentials around a certain typeof atom from the alloy are the same, i.e., the effect of the local environment isneglected. These local potentials are described by the potential functions PA,PB, PC, . . . . Second, the system is replaced by a monatomic set-up described by

10 CHAPTER 2. THEORY

the site independent coherent potential P . In terms of Green functions, the realGreen function g is approximated by the coherent Green function g. For eachalloy component i = A, B, C, . . . a single-site Green function gi is introduced.

In the following, we will show the main steps to build the CPA effectivemedium. First, the coherent Green function is calculated from the coherentpotential with an electronic structure method

g = [S − P ]−1, (2.15)

here S is the structure constant matrix corresponding to the underlying lattice.The Green functions of the alloy components gi are determined by substitutingthe coherent potential of the CPA medium by the atomic potential Pi, which isgiven by

gi = g + g(Pi − P )gi, i = A, B, C, . . . . (2.16)

The previous equation is the Dyson equation in real space. At last, the averageof the individual Green functions should reproduce the single-site part of thecoherent Green functions, i.e.,

g = agA + bgB + cgC. (2.17)

The above three equations are solved iteratively, and the output g and gisare used to determine the electronic structure, charge density and total energyof random alloys. The implementation of the CPA in the EMTO method isdescribed in Ref. [45].

2.5 Disordered Local Moment Model

We use the disordered local moment model (DLM) [55, 56] to deal with the para-magnetic state. Within the DLM picture, each alloy component is described byits spin-up (↑) and spin-down (↓) components, i.e. A −→ A ↑ A ↓. Considerthe temperature effect, the reduction of magnetic long-range order in the FMstate can be described in a similar way by the partially disordered local moment(PDLM) model [57, 58]. In the PDLM model, a ferromagnetically ordered com-ponent of an alloy A can be represented by a random binary alloy A1−x ↑Ax ↓with the concentration, x, varying from 0 to 0.5. This PDLM model allows oneto model the intermediate magnetic states of a system with 0 < x < 0.5, wherex = 0.5 and x = 0 correspond to a completely disordered magnetic state andordered magnetic state, respectively.

Chapter 3

Mechanical Properties ofMaterials

3.1 Elastic Properties

The mechanical behavior of solids is usually defined by constitutive stress-strainrelations. In the region where Hooke’s law is obeyed, the relationship betweenthe stress σij and strain ǫkl can be expressed by

σij =∑

kl

cijklǫkl, (3.1)

where i, j, k and l are indices running from 1 to 3. The cijkl are called elasticconstants and are a fourth-order elasticity tensor, which, in general, has 34 = 81components. Due to the symmetry cijkl = cijlk = cjikl = cklij , the number ofindependent elastic constant reduces to 21 [59, 60]. Sometimes, it is convenient touse matrix notation (Voigt notation), so that the 21 constants can be arrangedin a symmetric 6 × 6 matrix. Employing the Voight notation, we can writeEq. (3.1) as

σα =∑

β

cαβǫβ , (3.2)

where the new α and β indices run from 1 to 6. The number of elastic constantscan be further reduced by crystal symmetry. For cubic crystals, there are threeindependent parameters: C11, C12, and C44.

In this thesis, the C11 and C12 are obtained from the tetragonal shear modulusC′ = (C11 − C12)/2 and the bulk modulus B = (C11 + 2C12)/3. The bulkmodulus is determined from an exponential Morse-type function to computedtotal energys [61]. To obtain the two cubic shear modulus C′ and C44, volume-conserving orthorhombic and monoclinic deformations can be applied on theconventional cubic cell [62]. For the tetragonal shear modulus C′, the following

12 CHAPTER 3. MECHANICAL PROPERTIES OF MATERIALS

orthorhombic deformation is used:

1 + δ0 0 0

0 1 − δ0 0

0 0 1/(1 − δ20)

,

which leads to the energy change

∆E(δ0) = 2V C′δ20 +O(δ4

0). (3.3)

The C44 shear modulus is determined from the monoclinic distortion

1 δm 0

δm 1 0

0 0 1/(1 − δ2m)

,

with energy change∆E(δm) = 2V C44δ

2m +O(δ4

m). (3.4)

Here δ is the strain parameter. We considered six distortions δ=0.00, 0.01, ..., 0.05in the elastic constants calculation.

From the single-crystal elastic constants, the polycrystalline shear modulus(G) can be obtained by the arithmetic Hill average [63], GH = 1/2(GR + GV ),where the Reuss and Voigt bounds [45] are

G−1R = 2/5(C′)−1 + 3/5(C44)−1, (3.5)

andGV = 2/5C′ + 3/5C44. (3.6)

For cubic solids, the polycrystalline bulk modulus is equivalent with the single-crystal one. The polycrystalline Young’s modulus (E) can be expressed asE = 9BG/(3B +G).

3.2 Solid-solution Hardening

Hardness is a very important mechanical property, which can be used to measurethe resistance of solid matter to various kinds of permanent shape change whena force is applied. To improve the hardness of a pure metal, we can use differentways to harden it, such as work hardening, solid-solution hardening (SSH) byinterstitial atoms or by substitutional atoms, and refinement of grain size. Inthis thesis, we only consider the strengthening caused by substitutional soluteatoms and leave out the effect of an interstitial solute.

The SSH is due to dislocation pinning by the randomly distributed soluteatoms, which has been described by different models [64–68]. The Labusch-Nabarro (LN) semiempirical model [66–68] is often used to describe the hard-ening mechanism in alloys. In the LN model, the dislocation pinning is mostlydetermined by the size misfit and elastic misfit parameters. They are calculated

3.3 Ideal Strength 13

from the concentration dependent Burgers vector or lattice parameter, and theshear modulus. According to the LN model, the SSH varies as

∆τ = const.× c2/3 × ε4/3L , (3.7)

where const. is a host-specific constant number (does not depend on c and thesize of the misfit), the Fleischer parameter εL is expressed by

εL = [(ε′

G−LN)2 + (αεb)2]1/2, (3.8)

withε

′

G−LN = εG/(1 + 0.5|εG|), (3.9)

with α being a parameter (usually, the value of α is between 9 and 16, inthis thesis we adopted α = 10), and εb and εG are the volume and modulusmisfit parameters, respectively. These parameters can be obtained from thecomposition-dependent lattice constants and shear modulus of binary alloys,viz.,

εb = [δ(b)/δ(c)]/b(0) and εG = [δ(G)/δ(c)]/G(0), (3.10)

where b is Burger’s vector (lattice parameter), G is the shear modulus, and c isthe atomic fraction of the solute atom.

3.3 Ideal Strength

The elastic constants describe mechanical properties of materials in the smalldeformation region, where the stress-strain relations are linear. Beyond theelastic region, we may use the ideal strength to describe mechanical propertiesof materials. The ideal strength is the possible maximum strength of an idealsingle crystal, which is an intrinsic property of a solid material. This strengthcan offer insight into the correlation between the intrinsic chemical bondingand the crystal symmetry, and has been accepted as an essential mechanicalparameter of single crystal materials [20, 24, 69, 70]. The ideal strength playsan increasingly important role in understanding the mechanical behavior of solidmaterials. Below we give a few relevant examples where the significance of theideal strength in materials science has been recognized.

1, In the theory of fracture, the local inhomogeneous stress distri-bution close to the tip of a crack, and thus the cleavage behavior ofmaterials, is related to the ideal strength. The stress necessary forthe nucleation of a dislocation loop can be identified with the idealshear strength and the local stress for nucleation of a cleavage crackshould overcome the tensile value [71].

2, According to the failure modes, the ideal strength determineswhether a solid will behave in an intrinsically brittle or ductile man-ner [72, 73].

3, The ideal strength is an important factor in defining the point atwhich coherency breakdown occurs at a particle matrix interface [74]and in defining dislocation core radii [75, 76].

14 CHAPTER 3. MECHANICAL PROPERTIES OF MATERIALS

4, In applications where large deformation occurs without disloca-tion movement or for materials with very low defect density, the idealstrength is of particular interest. Examples comprise the deforma-tion behavior of whiskers, the mechanical properties of nanowiresor nanopillars, and nanoindentation experiments on thin films andnanostructures [77–83]. The plastic deformation behavior of a mul-ticomponent solid solutions based on the Ti-Nb binary (called "gummetals") is believed to be governed by the ideal strength rather thanconventional dislocation mobility [84].

5, The values of ideal strength can be used for calibration or checkingof semi-empirical interatomic potential that are used for study ofextended defects [85].

6, Finally, since the ideal strength sets the upper limit to the at-tainable strength, its knowledge enables researchers to assess thegap remaining between the ideal strength and the best achievablestrength of advanced engineering materials.

In some cases the measured maximum strength approximates the theoret-ically predicted one. Table 3.1 shows the calculated ideal strength and theexperimental maximum strength (taken from Refs. [77, 79, 80]) for uniaxial andshear loadings. From Table 3.1, we can see that compared to the theoretical ten-sile strength σms the experimental data are somewhat lower. A possible reasonis that the observed failure initiated at the surface and, therefore, the measuredvalue might not represent the actual bulk strength. For the shear strength Tss,the theoretical and experimental values are reasonably close. The above resultsindicate that a good correspondence between the atomistic ideal strengths andexperimental maximum strengths can be achieved.

Recently, a lot of works about the ideal strength calculations were pub-lished [24, 69, 70, 86–95], but they just focused on pure elements, compounds,or ordered alloys. There is no study on the ITS of random alloys. To predictthe optimized random alloy configuration with excellent mechanical properties,it is necessary to investigate the alloying effect on the ITS of pure elements.

3.4 Methodology for Ideal Strength Calculation

In this section, we present the methodology of ITS calculation with EMTO. Theground state structure of all elements and alloys considered in this thesis is bcc.We apply a uniaxial tensile strain ǫ along a specific crystalline direction whichmimics a certain tensile stress σ. To make sure that no internal forces remainin the crystal in directions perpendicular to the applied stress, for each value ofthe strain, we relaxed the deformed structures. From the above steps, we canobtain energy versus strain and stress versus strain curves. The first maximumon the stress-strain curve defines the ITS σm for the selected strain path. Thestress σ is given by [94]

σ(ǫ) =1 + ǫ

Ω(ǫ)∂E

∂ǫ, (3.11)

3.4

Meth

od

olo

gy

for

Ideal

Str

en

gth

Calc

ula

tion

15

Table 3.1: Theoretical ideal strength and experimental maximum strength for uniaxial (σm) and shear (Ts) loadings in different directions and shearsystems (taken from Refs. [77, 79, 80]).

element σm (GPa) Ts (GPa)theory experiment theory experiment

W 32.4 [110] 28.3 [110] 22.8-24.0 <111>110 22.1-23.3 <111>110Mo 31.9 [110] 19.8 [110] 17.6-18.8 <111>110 15.8-16.7 <111>110Fe 27.7 [111], 12.6 [100] 13.1 [111], 5 [100]Cu 9.3 [100] 6 [100] 2.16 <112>111 1.65 <112>111

16 CHAPTER 3. MECHANICAL PROPERTIES OF MATERIALS

where E is the total energy per atom and Ω(ǫ) is the volume per atom at a giventensile strain. The engineering strain ǫ of the simulation cell in the direction ofthe applied uniaxial force F is defined as

ǫ =l‖ − l0

l0, (3.12)

where l‖ and l0 denote the length of the cell parallel to F in the final state and inthe initial state (without any force), respectively. The initial state correspondsto the equilibrium bcc structure with energy E0. We define the uniaxial strain

energy ∆E(l‖, F) as the total energy change upon deforming the material in thedirection along the applied force, and relaxing it with respect to the dimensionsin the plane perpendicular to F, viz.,

∆E(l‖, F) = mina1,a2

E(a1,a2, l‖) − E0. (3.13)

The minimization is done with respect to the pair of unit cell vectors a1,a2 ⊥F.

bC

bC

bCbC

bCbC

bC

bCbC

abct

cbct

[001]

[100][010]

cbct/abct = 1

(a)

bC

bC

bC

bC

bC

bC

bC

bC

bCbCbCbC

bCbC

bC

bC

bCbC

bCbC

bC

bC

bC

[001]

cfct

afct

cbct/abct > 1

cbct

abct

(b)

Figure 3.1: Illustration of the lattice distortion of the bcc structure due to an applieduniaxial stress the [001] direction. (a) The unit cell in the absence of strain isbcc (in the bct delineation cbct/abct = 1). For finite strain, the lattice sym-metry is distorted to bct (b), and becomes fcc for cbct/abct =

√2. To describe

the bifurcation from the primary tetragonal to the secondary orthorhombicstrain path, the fct reference frame is used.

The symmetry of the bcc structure will reduce during tension in differentways for different directions. First, we chose the [001] axis to be the direction ofthe applied force for 〈001〉 ITS calculations. The symmetry of the bcc lattice isreduced to the body-centered tetragonal (bct) one on the primary deformationpath, see Fig. 3.1. Increasing the axial ratio cbct/abct from 1 to

√2 transforms

the bcc lattice into the face-center cubic (fcc) lattice while the crystal remainsbct during the transformation. This transformation corresponds to the Baintransformation or tetragonal deformation [96]. If we do not constrain the tetrag-onal symmetry under [001] tension, there is a secondary branching away fromthe Bain path before the fcc structure is reached, which is called orthorhom-bic deformation path. The observed orthorhombic branching is triggered by avanishing shear modulus. That is, if the reference frame is bct with x-, y-, andz-axes coinciding with the [100], [010], and [001] directions, respectively, the

3.4 Methodology for Ideal Strength Calculation 17

Figure 3.2: The sketches of two failure modes of bcc crystal under [001] tension. (a):failure by cleavage and (b): failure by shear.

instability condition reads cxyxy ≤ 0, cxyxy being a shear elastic constant of thebct structure. This instability condition seems to lower the bct symmetry tomonoclinic symmetry, but in fact a higher symmetry of the lattice, face-centeredorthorhombic (fco) symmetry, can be found by redefining the orientation andsize of the unit cell. In the fco reference frame, branching originates from face-

18 CHAPTER 3. MECHANICAL PROPERTIES OF MATERIALS

bC

bC

b C

bCbC

bC

bC

bCbC

a

a

[001]

[110]

[110]

bC

bC

bCbC

bC

bC

bC

afco

bfco

cfco

(a)

bC

bC

bC

bCbC

bC

bC

bCbC

[001]

[110]

[110]

bC

bC

bCbC

bC

bC

bC

afco

bfco

cfco

(b)

Figure 3.3: Illustration of the lattice distortion of the bcc structure due to an applieduniaxial stress in the [110] direction. (a) and (b) show the undistorted latticeand the distorted lattice, respectively. Both the fco unit cell used in thecomputation and the (undistorted and distorted) bcc cell are sketched.

centered tetragonal (fct) symmetry, see Fig. 3.1. The instability condition readsC = (c′

xxxx − c′xxyy)/2 ≤ 0 in the fct reference frame (the two instability condi-

tions are identical, merely expressed in the different coordinate systems). C isalso referred to as a shear modulus.

There are two failure modes for a bcc solid under [001] tension, which dependon where the orthorhombic branching happens. If the branching happens beforethe maximum stress is reached along the tetragonal deformation path, we saythat the material fails by shear. If the branching happens after the maximumstress is reached along the tetragonal deformation path, the material fails bycleavage [23]. A sketch of the above mentioned two failure modes is shown inFig. 3.2.

Straining the bcc structure along the [110] axis reduces the lattice symmetryto fco guiding symmetry. In the absence of strain, the fco lattice parametersfulfill afco = bfco =

√2cfco, see Fig. 3.3 for a detailed illustration. Here, strain is

applied to bfco ≡ l‖. Assuming isotropic Poisson contraction, afco and cfco arerelaxed keeping cfco/afco fixed to the initial value of

√2. This assumption was

used to make the computations feasible.

If uniaxial strain is applied along the [111] direction parallel to the body di-agonal of the bcc structure, the symmetry of the lattice is reduced to trigonalsymmetry. The distorted lattice can be equivalently described by a hexago-nal (hex) lattice or a rhombohedral lattice. Here, we chose the hex lattice, assketched in Fig. 3.4. The [0001] axis and the [1010] axis of the hex lattice are ori-ented parallel to the [111] axis and the [110] axis of the bcc lattice, respectively.In the absence of strain, the lattice parameters of the hex unit cell and the bcccell are related by ahex =

√2abcc and chex =

√

3/4abcc, where ahex denotes thein-plane lattice parameter of the hexagonal basal plane (the distance betweentwo atoms of the smallest equal-sided triangle). The out-of-plane lattice param-eter, chex, is oriented parallel to the threefold stacking axis. Bcc (111) planesare stacked along [0001] (ABCABC stacking) and two successive planes are dis-placed by 1/3 of chex (Fig. 3.4). For each applied strain, interatomic distances inthe (0001) planes are relaxed while the in-plane symmetry is preserved. Thus,we assume isotropic Poisson contraction for the [111] type of ITS calculations.

3.4 Methodology for Ideal Strength Calculation 19

bC

bC

bC

bC

bC

bC

bC

bC

bCa

chex ahex

[110],[1010]

[111],[0001]bC

bC

bC z(hex)

bC 0, chexbC 1/3 chexbC 2/3 chex

(a)

bCbC

bCbC

bC

bC

bCbC

bCa

a

ahex

[112]

[110]

[1010]

bC

bC

bC bC

(b)

bC

bC

bCbC

bC

bCbCbC

bC

bC

bCbC

[0001]

[1010]

ahexchex

(c)

Figure 3.4: Illustration of the lattice distortion of the bcc structure due to an applieduniaxial stress in the [111] direction. (a) shows the undistorted bcc lattice withhexagonal delineation (a wedge representing one sixth of the conventional hexunit is drawn), the axial ratio is chex/ahex =

√

3/8. The undistorted unit cellprojected along [111] ([0001]) is depicted in (b) to show more clearly the fullsymmetry of the hex lattice. A finite strain, (c), lowers the symmetry of thebcc lattice to a trigonal one.

According to the assumptions on the lattice geometry, we rewrite Eq. (3.13)for tension along [001], [111] and [110] directions as

∆E prim(cbct, [001]) = minabct

E(abct, cbct) − E0, (3.14)

∆E secon(cfco, [001]) = minafco,bfco

E(afco, bfco, cfco) − E0, (3.15)

∆E(chex, [111]) = minahex

E(ahex, chex) − E0, (3.16)

∆E(bfco, [110]) = minafco

E(afco, bfco) − E0. (3.17)

Based on the above equations in conjunction with Eq. (3.11), we can obtainthe ITS along [001], [111] and [110] directions. The strain energies as well asthe volume of each state of strain were fitted to polynomial fit functions anddifferentiated to obtain the stress as a function of strain.

Chapter 4

Magnetism

A part of our work deals with ferrite and ferritic alloys. Magnetic order insolid iron emerges with the formation of a crystalline lattice. The stable body-centered cubic (bcc) phase (α-Fe) is characterized by long-range ferromagnetic(FM) order below TC = 1044 K. When the temperature is above the Curietemperature and below 1183 K, bcc Fe is paramagnetic (PM). In the following,we introduce some basic notions of intinerant ferromagnets.

4.1 Stoner Model

The Stoner model is a simplified band model for itinerant electron magnetism,which can be used to describe spontaneous ferromagnetic order of a metal inthe ground state [97]. This can be demonstrated by the rectangular d-bandmodel of Fig. 4.1. Starting from the nonmagnetic state, the spin up and downchannels have the same number of electrons, n+ and n−, and identical density

e e

eF

g+(e) g-(e) g+(e) g-(e)

Nonmagnetic

e

g+(e) g-(e)

Ferromagnetic

e+e-

eF

Δe

(a) (b) (c)

n+ n- n+ n- n+ n-

Figure 4.1: The Stoner model for a rectangular density of states. (a) Nonmagneticdensity of states split into the two subbands for the two spin directions. (b)and (c) density of states showing splitting of energy bands.

22 CHAPTER 4. MAGNETISM

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1-30-25-20-15-10-505

1015202530

NM bcc Fe

minority

majority

total

DO

S (s

tate

s/R

y)

Energy (Ry)Figure 4.2: The total density of states of nonmagnetic bcc Fe. The vertical dashed

line indicates the Fermi level.

of states, g+(e) and g−(e), which is shown in Fig. 4.1(a). Imagine we move asmall number of electrons from the spin down channel to the spin up channel 1

as illustrated in Fig. 4.1(b). Since the Fermi energy for both subbands has tobe the same (both spins have the same chemical potential), the above electrontransfer causes a redistribution of electrons, which is displayed in Fig. 4.1(c).Here, e+ − e− = ∆e is called band splitting, and the number of transferredelectrons is g+/−(eF)∆e/2. Based on this result, we can write the total kineticenergy (band energy) change as,

∆Ekin =∫ ∆e/2

0

g+(0)ede−∫ 0

−∆e/2

g−(0)ede =14g+/−(0)(∆e)2, (4.1)

where we set eF = 0. In the Stoner model, the energy difference between aferromagnetic and a nonmagnetic state contains two parts. The first part isthe increase of band energy due to the reoccupation of states near the Fermilevel, which is described above. The second part is the decrease of the exchangeenergy contribution. The total energy difference can be described as,

U =14g+/−(0)(∆e)2 − 1

4Iµ2 =

14µ2/g+/−(0) − 1

4Iµ2 (4.2)

where the first term is the total kinetic energy (band energy) change, and thesecond term is exchange energy. I is the Stoner parameter calculated from the

1For a small number of transferred electrons, one can assume that g+/−(e) = g+/−(eF) +O(e) in the absence of singularities.

4.2 Finite Temperature Properties of Iron and Iron-based Alloys 23

intraatomic exchange interaction and µ is magnetic moment. Due to symmetry,the exchange energy should only have even order terms of µ, and the lowestorder approximation is proportional to µ2. According to the above equation,the nonmagnetic state will be unstable to ferromagnetism if U < 0, which means,

Ig+/−(0) > 1 (4.3)

which is the famous Stoner criterion for the spontaneous onset of ferromag-netism in the ground state. According to this model, if the Stoner criterion isnot fulfilled, then spontaneous ferromagnetism will not occur. A large g+/−(0)or I is favorable for the occurrence of ferromagnetism. However, I is an in-traatomic quantity and rather insensitive to atomic geometry (crystal struc-ture) [97, 98].

Figure 4.2 displays the density of states of nonmagnetic bcc Fe computedfor the theoretical lattice parameter (2.84Å). The lattice parameter was ob-tained within the Perdew-Burke-Ernzerhof (PBE) [41, 99] parameterization ofthe generalized-gradient approximation (GGA) to describe exchange and corre-lation. We can see that there is a big peak at the Fermi level, which correspondsto g+(0) = 23 Ry−1. Using the literature Stoner I = 0.068 Ry [98], we arrive atIg+/−(0) = 1.56, which is larger that 1. Thus, the Stoner criterion is fulfilled.

4.2 Finite Temperature Properties of Iron andIron-based Alloys

In this section, we elaborate on how to compute the Curie temperature (TC)of ferromagnetic Fe and dilute Fe-alloys. To this end, the itinerant electronsystem is mapped on a classical Heisenberg spin model. The parameters of theHeisenberg Hamiltonian (interatomic exchange interaction) are determined bymeans of first principle calculations. This task may be considered as a three-stepprocedure. First, we perform a self-consistent electronic structure calculationfor the ground state collinear spin structure. Second, we employ the magneticforce theorem [100] to compute the magnetic exchange interactions (Jij) ofthe classical Heisenberg model. In the third step, we solve the HeisenbergHamiltonian for TC by means of classical Monte Carlo (MC) simulations.

4.2.1 Heisenberg Model and Magnetic Exchange

The Heisenberg model is spin model for ferromagnetism and other phenomena.The physical motivation for the mapping of an itinerant electron system ona classical Heisenberg Hamiltonian is given by the adiabatic hypothesis. Theadiabatic hypothesis assumes that the time scale of the magnetic moment dy-namics in a solid is much slower than the adaptation of the electronic systemunder the perturbation of temperature or external magnetic field [101, 102]. Thedirections of magnetic moments are treated as classical variables.

The classical Heisenberg Hamiltonian in the quadratic approximation in therelativistic case can be written as [103]

H = −12

∑

i6=j

eiJijej , (4.4)

24 CHAPTER 4. MAGNETISM

0 1 2 3 4 5 6 7 8 9

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

5.0 5.5 6.0 6.5 7.0 7.5 8.0

-0.01

0.00

0.01

0.02

Exc

hang

e in

tera

ctio

n (m

Ry)

Reduced distance (R0j/abcc)

Exc

hang

e in

tera

ctio

n (m

Ry)

Reduced distance (R0j/abcc)

Figure 4.3: Exchange interaction J0j for bcc Fe as a function of reduced distance.R0j denotes the distance between atom j and atom i ≡ 0, and abcc is thetheoretical lattice constants of bcc Fe.

where Jij are 3 × 3 matrices, and the ei’s are unit vectors. Jij is convenientlydecomposed into the following contributions

eiJijej = Jijei · ej + eiJ Sij ej + Dij(ei × ej), (4.5)

where the first and second terms are the isotropic and symmetric relativisticanisotropic exchange interactions, respectively, and the third term is antisym-metric relativistic anisotropic interaction (Dzyaloshinsky-Moriya (DM) interac-tion). The DM interaction is identically zero for crystal lattices with inversionsymmetry like the here considered bcc and bct structures. The anisotropic ex-change is a small relativistic correction to isotropic exchange and neglected inthe following [104]. We have the first term left, which is

Jij =13

Tr(Jij) =13

(J xxij + J yy

ij + J zzij ). (4.6)

So, the classical Heisenberg Hamiltonian in the nonrelativistic case is describedas,

H = −12

∑

i6=j

Jijei · ej. (4.7)

For Fe-based random alloys, Fe1−cMc, M denoting the solute, we employedeffective alloy exchange interactions [100]. That is, Jij =

∑

α,β cαcβJαβij , where

α, β = (Fe, M), Jαβ = Jβα, and cα(β) is the atomic concentration.Several methods have been proposed to compute the magnetic exchange en-

ergy Jij , for example, the frozen magnon approach [101] or the magnetic forcetheorem [100]. In our work, we employed the latter one, which is implementedin EMTO. As an example, the J0js of bcc Fe are shown in Fig. 4.3. From

4.3 Curie Temperature 25

Fig. 4.3, one can see that the dominating interactions are from the first andsecond nearest-neighbors shells. With increasing the distance, the interactionsbecome weaker and can change sign. From the inset figure, we can see thatthe J0j parameters are long range and show a Ruderman-Kittel-Kasuya-Yosida(RKKY) oscillating behavior [105, 106].

4.3 Curie Temperature

There are several approaches to determine TC of the Heisenberg Hamiltonianin Eq. (4.7). The simplest analytic estimate for TC is based on the mean-fieldapproximation (MFA) [97] (here for a Bravais lattice),

kBTMFAC =

13

∑

j 6=0

J0j , (4.8)

where kB is the Boltzmann constant. Generally, the MFA gives larger TC valuescompared to experimental data, because the MFA neglects fluctuations [97]. Toget more reliable TCs, commonly used approaches are the analytic random-phaseapproximation [101] and classic MC simulations. Here, MC simulations with theUppAsd program package [107] were performed to determine TC. Because ofthe stochastic nature of MC methods, good statistical accuracy might requireto simulate several ensembles. Here, to achieve the necessary accuracy, fiveensembles and 30000 MC steps were used, 10000 steps for the initial phase inorder to bring the system into thermal equilibrium, and 20000 steps for themeasurement phase.

To obtain accurate TCs without finite size effect (simulation box), the Binder’smethod was used in our work. With this method, the Curie temperature wasderived from the crossing point of the fourth order Binder cumulant [108],

U4 = 1 −⟨

M4⟩

3 〈M2〉2 , (4.9)

where M is the absolute magnetization per site, and 〈Mn〉 is the ensembleaverage of the n’th moment of M .

From a plot of U4(T ) for different sizes of the simulation box one can obtain TC

by identifying the common intersection point of these plots to which the systemconverges (for too small systems there are correction terms which prevent allcurves from having one common fixed point). As an example, the cumulants U4

of bcc Fe for various sizes of the simulation box are depicted in Fig. 4.4. Thecommon intersection point is at TC = 1066 K.

Classical Boltzmann statistics [107, 110, 111] is employed in classical MC sim-ulations in UppAsd. The computed magnetization curve for Fe derived from thepresent simulation does not reproduce the shape of the experimental magnetiza-tion curve. This can be seen in Fig. 4.5 where the magnetization curves for bccFe from the MC simulation and Kuz’min’s fit to the experimental data [109] aredisplayed. Here, the reduced magnetization and reduced temperature have beendefined as m(T ) = Ms(T )/Ms0, Ms0 = Ms(T = 0 K), and τ = T/TC, respec-tively. Ms denotes the saturation magnetization. In the following section, weshow how we work around this issue, since we require an accurate magnetizationcurve in Chapter 8.

26 CHAPTER 4. MAGNETISM

960 980 1000 1020 1040 1060 1080 1100 1120

0.45

0.50

0.55

0.60

0.65

0.70

10*10*10 20*20*20 30*30*30

Bin

der

cum

ulan

t

T (K)Figure 4.4: Binder cumulants (Eq. (4.9)) as a function of temperature for bcc Fe for

various sizes of the simulation box in terms of the conventional bcc unit cell.The dashed line denotes the crossing point.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Red

uced

mag

netiz

atio

n m

Reduced temperature τ

MC FeKuz’min Fe

Figure 4.5: The magnetization curves of Fe from the MC simulation and Kuz’min’sfit to the experimental data [109]. τ is normalized to the corresponding TC

(T exptC = 1044 K and T theo

C = 1066 K).

4.4 Spin-wave Stiffness for Cubic Symmetry 27

4.4 Spin-wave Stiffness for Cubic Symmetry

An simple analytic expression for m(τ) was suggested by Kuz’min [109]

m(τ) = [1 − sτ3/2 − (1 − s)τ4]1/3, (4.10)

where s is defined as shape parameter, which controls the shape of the magne-tization curve. As τ → 0, m(τ) ≈ [1 − 1

3sτ3/2], which obeys Bloch’s 3/2 power

law at low temperature. The parameter s is related to the spin-wave stiffnessconstant D, the prefactor in the quadratic magnon dispersion relation for smallwave length, viz. [109, 112],

s = 0.1758gµB

M0

(

kBTC

D

)3

2

, (4.11)

where g is the experimental spectroscopic splitting factor [113] and M0 is thevolume saturation magnetization. D is accessible to DFT [114]. The employedmagnetization curve in Chapter 8 was obtained based on Eq. (4.10) with shapeparameter s calculated via Eq. (4.11). Below, we illustrate how to compute theparameter D.

Following Liechtenstein et al. and Turek et al. [100, 105, 114], we may expressD for a cubic lattice by

D =2µB

3µ

∑

j 6=0

J0jR20j , (4.12)

where R0j denotes the distance between atom j and atom i ≡ 0. It turnsout that Eq. (4.12) for D is a numerically not converging sum as a function ofdistance, which is related to the long-range behavior of Jij shown in Fig. 4.3.In practice, Eq. (4.12) is replaced by a formally equivalent expression includinga damping coefficient η, η > 0 [105, 114]

D = limη→0

D(η) (4.13)

D(η) =2µB

3µ

∑

j 6=0

J0jR20j exp(−ηR0j/a), (4.14)

where a is an arbitrary lattice parameter of the underlying lattice. Here, we useabcc.

Figure 4.6 (a) displays the spin-wave stiffness (D(η)) for bcc Fe as a functionof the number of exchange parameters included in the sum on the right handside of Eq. (4.14). The sum is truncated for pairs outside the sphere of radiusR0j/abcc. From Fig. 4.6(a), we can see that D(0) is numerically not converg-ing as a function of reduced distance. For finite η, the long-range oscillatorybehavior of D is damped. D(η) is converged for η ≥ 0.8 at the largest con-sidered distances. From a plot of D(η) extrapolated to η = 0 with a secondorder polynomial function [105, 114], D can be obtained. Figure 4.6(b) showsthe extrapolation of D(η) to η = 0 for bcc Fe and bcc Fe90Co10. From these

extrapolations, the computed Ds are 244 meV Å2

and 285 meV Å2

for bcc Feand Fe90Co10, respectively.

Figure 4.7 displays the MC magnetization curves for bcc Fe and Fe90Co10

with our calculated shape parameters. Clearly seen that the improved curveshows a small deviation from the curve fitted to experiments (s = 0.35) for Fe.

28 CHAPTER 4. MAGNETISM

0 1 2 3 4 5 6 7 8

Reduced distance R0j

/abcc

-200

0

200

400

600

800

1000

Sp

in-w

ave

stif

fnes

s D

(η)

(meV

Å2 )

0.00.20.40.60.70.80.91.01.1

(a)

0 0.2 0.4 0.6 0.8 1

Damping coefficient η

100

125

150

175

200

225

250

275

300

Sp

in-w

ave

stif

fnes

s D

(η)

(meV

Å2 )

FeFe90Co10

(b)

Figure 4.6: (a) Behavior of D as a function of reduced distance for various dampingcoefficients η for bcc Fe. Curves for η ≥ 0.8 are considered converged forR0j/abcc > 8. (b) Extrapolation of D(η) to η = 0 for bcc Fe and bcc Fe90Co10.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Red

uced

mag

netiz

atio

n m

Reduced temperature τ

Kuz’min Fe, s=0.35Fe, s=0.41

Fe90Co10, s=0.43

Figure 4.7: Shape of the magnetization curve (Eq. (4.10)) with shape parameters scomputed for Fe and Fe90Co10 in this work in comparison to the curve fittedto experiments (s = 0.35).

4.5 Spin-wave Stiffness for Non-cubic Symmetry 29

0 1 2 3 4 5 6 7 8

Reduced distance R0j

/abct

-200

0

200

400

600

800

1000

Sp

in-w

ave

stif

fnes

s D

zz(η

) (m

eVÅ

2 )

0.00.40.70.80.91.0

(a)

0 0.2 0.4 0.6 0.8 1

Damping coefficient η0

50

100

150

200

250

300

350

Sp

in-w

ave

stif

fnes

s D

(η)

(meV

Å2 )

Dxx

= Dyy

Dzz

(b)

Figure 4.8: (a) Behavior of Dzz as a function of reduced distance for various dampingcoefficients η for bct Fe (c/a = 1.095). Curves for η > 0.7 are consideredconverged for R0j/abct ≥ 8. (b) Extrapolation of Dxx(η) and Dzz(η) to η = 0for bct Fe.

4.5 Spin-wave Stiffness for Non-cubic Symme-

try

In fact, the spin-wave stiffness introduced in Eq. (4.12) is a tensor [100]

Dαβ =2µB

µ

∑

j 6=0

J0jRα0jR

β0j α, β = x, y, z. (4.15)

Since R20j =

∑

α(Rα0j)2, it is easy to see that for cubic lattices, Dxx = Dyy =

Dzz = 3D with D defined in Eq. (4.12).Assuming in general an anisotropic magnon dispersion relation in the form

ǫq = Dxxq2x + Dyyq2

y + Dzzq2z (principal axes), it can be shown that the D

entering Eq. (4.11) and the Bloch T 3/2 law is the mean geometric [112]

D = (DxxDyyDzz)1/3. (4.16)

For bct geometry, we have Dxx = Dyy, and Eq. (4.14) can be written as

Dαβ(η) =2µB

µ

∑

j 6=0

J0jRα0jR

β0j exp(−ηR0j/abct), (4.17)

here abct is a lattice parameter of the distorted structure, see Chapter 3. As anexample, Fig. 4.8 shows the spin-wave stiffness Dzz(η), Dyy(η), and Dxx(η) ofbct Fe (c/a = 1.095) as a function of reduced distance and damping coefficient.

Chapter 5

Mechanical Properties ofVanadium andVanadium-based Alloys

In this chapter, we discuss the elastic constants and the ideal tensile strength(ITS) of the bcc V1−x−yCrxTiy random alloys as a function of Cr (0 ≤ x ≤ 0.1)and Ti (0 ≤ y ≤ 0.1) concentrations.

5.1 Structural Energy Difference

In the following parts, we will use the structural energy difference (SED) to ex-plain the alloying effect on the tetragonal shear elastic constant C′ and on theITS. Thus, in this part, we give a short introduction to the SED. The SED isthe energy difference between different crystal structures for the same element.Since transition metals crystallize in the fcc, bcc, or hexagonal-close packed(hcp) structure, SEDs for transition metals are typically evaluated for thesethree structures. It is well established [115–117] that the electronic structureand bonding in transition metals is governed by a narrow valence d-band thathybridizes with a broader valence nearly-free-electron sp-band. SEDs for thetransition metal series (disregarding the effect of magnetism in the present dis-cussion) that give rise to the experimentally observed sequence of stable crystalstructures are well understood in terms of band filling within this band pic-ture [118–120]. It is clear that mainly the gradual filling of the d-band not onlydictates crystal structure sequences in all three transition metal series, but alsotrends of the equilibrium volume dependence [117, 121], the cohesive energy [118,121, 122], the bulk modulus [123], and elastic constants [124, 125].

The tetragonal shear elastic constant C′ is often used to describe the struc-tural stability of a cubic solid under tetragonal deformation [126] (small strain),which can be obtained from the curvature of the total energy under a smallvolume-conserving tetragonal deformation (see Eq. (3.3) on Page 12). A cor-relation between C′ and the fcc-bcc SED for the transition metals has beensuggested by previous studies [124, 125, 127, 128]. According to that, large C′

32

CHAPTER 5. MECHANICAL PROPERTIES OF VANADIUM AND

VANADIUM-BASED ALLOYS

for the bcc phase corresponds to a more stable bcc lattice compared to thefcc one. This correlation can be established because the tetragonal deforma-tion which governs C′ corresponds to the constant volume Bain transformationbetween bcc and fcc [125].

The distorted bct lattice of the [001] tetragonal deformation and the dis-torted trigonal lattice of the [111] deformation coincide with the fcc lattice (atcbct/abct =

√2) and with the simple cubic (sc) lattice (at chex/ahex =

√

3/2),respectively. The fcc and the sc structures of V were identified to be the nearestsymmetry-dictated maxima to the bcc phase of the respective energy versusstrain curves [127, 129–131], i.e., the uniaxial strain energy must level off tothe fcc-bcc SED and to the sc-bcc SED for an elongation along [001] and anelongation along [111], respectively, albeit at strains larger than the maximumstrain (ǫm).

5.2 Assessing the Accuracy of the TheoreticalPredictions for Alloys

Before investigating the elastic properties of V ternary alloys, we first comparethe present elastic parameters calculated for binary V-based alloys with theavailable experimental data [132–136], see Fig. 5.1. For binary V-Cr and V-Tialloys, the theoretical lattice constants increase (decrease) with increasing Ti(Cr) concentration. The reason is that bcc antiferromagnetic Cr has smalleratomic radius (1.28 Å) and hcp Ti has the larger one (1.46 Å) as comparedto that of bcc V (1.35 Å) [137]. Furthermore, the bulk modulus B and thetetragonal shear elastic constant C′ increase (decrease) with Cr (Ti) addition.These theoretical predictions agree well with the trends observed in experiments(Fig. 5.1, six upper panels). At the same time, the theoretical C44 shows smallnegative (positive) change with Cr (Ti) addition to V, whereas for V-Cr (V-Ti) the experimental variation of C44 is slightly positive (negative). We willreturn to the theoretical trends obeyed by C44 in Subsection 5.4.1. The discrep-ancies become even more pronounced when comparing the theoretical and theexperimental trends for the shear modulus versus composition (Fig. 5.1, lowerpanels). While theory predicts decreasing G for both V-Cr and V-Ti with in-creasing doping level, experiments reported increasing shear modulus G uponalloying V with either Cr or Ti.

The above deviations between theory and experiment call for a detailed inves-tigation. Numerous previous applications confirm the accuracy of the EMTOapproach for the elastic properties of random solid solutions, and there is no apriori reason why it should perform less accurately for the present V-based bi-nary alloys either. The theoretical results correspond to static conditions (0 K),and temperature might change the compositional trends to some extent. Thisis a question to be investigated in the future. On the other hand, as we willdemonstrate below, the quoted experimental values for G may not correspond torandom solid solutions (as assumed in the present calculations) and are incon-sistent with the single-crystal data. The first problem is related the miscibilitygap in the V-Ti system below 900 K. According to that, the experiments onV-27%Ti and V-47%Ti were performed either on quenched (metastable) or ondecomposed samples. We note that since the present calculations correspond to

5.2 Assessing the Accuracy of the Theoretical Predictions for Alloys 33

2.95

3.00

3.05

3.10

3.15

V-50%CrV-20%CrV

EMTO Expt(a)

a

(A)

V-2.5%CrV-5%CrV-7.5%CrV-10%Cr

V

o

V

V-50%Ti

V-10%Ti

EMTO Expt(a)

V-2.5%TiV-5%Ti

V-7.5%TiV-10%Ti

V

110

130

150

170

190

EMTO Expt(c) Expt(b)

V-2.5%CrV-5%CrV-7.5%Cr

V-10%CrV-59.7%Cr

V-39.6%Cr

V-20.3%Cr

V-17.5%Cr

V-17.5%Cr

V

V

B

(GP

a)

EMTO Expt(b) Expt(d)

V-47%Ti

V-42%TiV-27%Ti

V-19.8%Ti

V-19.8%Ti

V-10%TiV-7.5%Ti

V-5%Ti

V

V V-2.5%Ti

35

45

55

65

75

85

C(G

Pa)

V-39.4%Cr

V-20.3%CrV-17.5%Cr

V EMTO Expt(b) Expt(e)

V-2.5%CrV-5%Cr

V-7.5%CrV-10%Cr

V

V

V-47%Ti

V-27%Ti

EMTO Expt(b)V-2.5%Ti

V-5%TiV-7.5%Ti

V-10%Ti

V

25

35

45

55

V-39.4%CrV-20.3%CrV-17.5%CrV

C44

(GP

a)

EMTO Expt(b) Expt(e)

V-2.5%CrV-5%Cr

V-7.5%CrV-10%Cr

V

V-47%Ti

V

V-27%TiV

EMTO Expt(b)

V-2.5%TiV-5%Ti

V-7.5%TiV-10%Ti

0 10 20 30 40 50

47

49

51

53

55

57

Cr at.%

G (G

Pa)

V-39.4%Cr

V-20.3%Cr

V-17.5%Cr

V

V

EMTO Expt(b) Expt(e)

V-2.5%CrV-5%Cr

V-7.5%CrV-10%Cr

0 10 20 30 40 50

Ti at.%

V-47%Ti

V-27%Ti

V

EMTO Expt(b)

V V-2.5%TiV-5%Ti

V-7.5%TiV-10%Ti

Figure 5.1: Present results and experimental lattice constants, bulk modulus B,tetragonal shear elastic constant C′, C44 and shear modulus G for binary bccV-Cr and V-Ti alloys as a function of composition. Expt(a) from Ref. [134],Expt(b) from Ref. [133], Expt(c) Ref. [136], Expt(d) from Ref. [132] andExpt(e) from Ref. [135].

34

CHAPTER 5. MECHANICAL PROPERTIES OF VANADIUM AND

VANADIUM-BASED ALLOYS

completely disordered phase (modeled by the CPA), future theoretical investi-gations taking into account the local ordering and relaxation effects are neededbe able to answer the above question.

The second concern is the inconsistency between the measured single-crystaland the polycrystalline data. For V-Ti, it was reported that both C44 and C′

decrease and shear modulus G increases with the amount of Ti. However, ac-cording to the Voigt and Reuss models (see Chapter 3), when both C44 and C′

decrease the G should also decrease and vice versa. One might argue that theVoigt and Reuss bounds give upper and lower values, respectively, and the trueshear modulus might still change within these limits. But V and V-27%Ti (as-suming a random solid solution model) are especially isotropic materials (theirexperimental Zener anisotropy ratio C44/C

′ being close to 1) and thus the Voigtand Reuss bounds are close to each other (in fact they differ by less than 1 GPa).This averaging "uncertainty" is definitely below the measured 5 GPa increaseof G when adding 27% Ti to V, meaning that the experimental G, C44 and C′

are not consistent with each other. In order to solve this puzzle, further accu-rate measurements on the single-crystal and polycrystalline elastic parametersof V-based random alloys are needed.

Based on the general agreement between the present theoretical predictionsand the available experimental data from Fig. 5.1, we conclude that our the-oretical tool is able to describe the elastic properties of V-based alloys withsufficiently high accuracy.

5.3 Lattice Parameters

Using the EMTO-CPA method, we first calculated the equilibrium lattice pa-rameters of V as a function of Cr and Ti, which is shown in Fig. 5.2. We can seethat Cr addition shrinks the lattice parameter of pure V, whereas Ti enlargesit. Because of Cr and Ti have opposite alloying effects on the equilibrium vol-ume, the lattice constants of V-Cr-Ti alloys are almost unchanged when equalamounts of Cr and Ti are introduced into V (alloys along the main diagonal inFig. 5.2).

5.4 Elastic Properties

5.4.1 Single Crystal Elastic Constants

Using the above equilibrium lattice constants, we calculated the elastic constantsCij(x, y) for bcc V1−x−yCrxTiy as a function of Cr and Ti contents. Figure 5.3shows the present theoretical single crystal elastic constants as a function ofCr and Ti contents (the calculation method is described in Chapter 3). Fromthese contours, in general, we can see that the effect of alloying on C11(x, y) isgreater than that on C12(x, y) and C44(x, y), which varies from ∼ 258 to ∼ 297GPa. For C12(x, y) and C44(x, y), the maximum changes are about 6 GPa. TheC11(x, y) increases with increasing Cr and decreases with increasing Ti, whichhas an opposite trend to C44(x, y). Different from C11(x, y) and C44(x, y), theC12(x, y) shows a weak dependence on the Cr content.

5.4 Elastic Properties 35

0 2 4 6 8 100

2

4

6

8

10

Cr (at.%)

Ti (

at.%

)

2.9802.9862.9912.9973.0033.0083.0143.0193.025

Figure 5.2: Theoretical lattice parameters (in Å) of V1−x−yCrxTiy (0 ≤ x, y ≤ 0.1)alloys as a function of Cr and Ti concentrations.

We know that C′ is often used to describe the structural stability of a cubicsolid under tetragonal deformation. As seen from Fig. 5.3, C′(x, y) increases(decreases) with increasing Cr concentration (Ti concentration), and as a resultof these opposite effects, C′(x, y) remains almost constant with increasing xand y along the equicomposition diagonal region. According to these trends,the V-Cr-Ti alloys become dynamically less stable with Ti addition and morestable with Cr addition under tetragonal deformation.

Based on the volume effect, we can understand the trends of C′(x, y). Fromthe lattice parameters calculation, we know that Ti expands the average volumeper atom in V-Cr-Ti, which means a decreases of the average bond strength andthus the ability of the alloy to resist the tetragonal shear. A more elaboratedexplanation for the calculated trends of C′(x, y) is possible if we use the cor-relation between C′ and the SEDs (see Section 5.1). Referring to the crystalstructure theory of transition metals, it is known that in nonmagnetic solidswith approximately three d electrons (i.e., close to bcc V) increasing (decreas-ing) d-occupation number stabilizes the bcc (fcc) phase. In the present case, Cr(Ti) addition increases (decreases) the d-occupation number and thus stabilizes(destabilizes) the bcc phase. As a consequence, Cr is expected to increase andTi to decrease the tetragonal shear modulus. Above analysis explains the trendsof C′(x, y).

As seen from Fig. 5.3, the trend of C44(x, y) is opposite to the trend ofC′(x, y),but it has the same trend as the lattice parameters (Fig. 5.2), which decreaseswith Cr and increases with Ti addition to bcc V. The above results mean thatneither the rule of mixing (i.e., linear interpolation between the results obtainedfor V and Cr and between those of V and Ti) nor the volume expansion can

36

CHAPTER 5. MECHANICAL PROPERTIES OF VANADIUM AND

VANADIUM-BASED ALLOYS

0 2 4 6 8 100

2

4

6

8

10

Ti (

at.%

)255.0260.6266.3271.9277.5283.1288.8294.4300.0

C11

0 2 4 6 8 100

2

4

6

8

10

32.0032.8733.7534.6235.5036.3737.2538.1239.00

C44

0 2 4 6 8 100

2

4

6

8

10

Cr (at.%)

Ti (

at.%

)

120.5121.2121.9122.6123.2123.9124.6125.3126.0

C12

0 2 4 6 8 100

2

4

6

8

10

Cr (at.%)

68.0070.2572.5074.7577.0079.2581.5083.7586.00

C

Figure 5.3: Theoretical single crystal elastic constants (in GPa) of bcc V1−x−yCrxTiy(0 ≤ x, y ≤ 0.1) alloys as a function of Cr and Ti concentrations.

account for the predicted trend of C44(x, y). To find a good explanation forthe anomalous behavior, we started from the electronic structure. We knowthat the elastic parameters are computed from the second-order derivative ofthe total energy

E(δ) = E(0) + aδ2 +O(δ4), (5.1)

with respect to the strain parameter δ. Since δ ≤ 0.05, the high-order termsO(δ4) can be neglected and then we can write

C44 ∼ ∆E(δ)/δ2, (5.2)

where ∆E(δ) = E(δ) −E(0) represents the change in total energy upon latticedistortion. According to the force theorem [119], the total energy change withdistortion can be approximated by the change of the one-electron energy ∆Eone,which in turn is determined by the density of states (DOS) calculated as afunction of lattice distortion. Figure 5.4 displays the DOS for pure V, V-7.5Crand V-7.5Ti alloys calculated in the bcc phase (without lattice distortion) andwith 5% distortion used to compute C44. For pure V, there is a peak locatedapproximately at −15 mRy below the Fermi level [marked by a black dashed-dotted line in Fig. 5.4(a)]. Upon monoclinic distortion [Fig. 5.4(b)], this peaksplits and shifts toward the Fermi level, indicating a positive change in the one-electron energy.1 Positive energy change leads to positive C44 in bcc V. In V-Cr

1In bcc V, this peak is dominated by t2g states, which splits into three one-dimensionalrepresentations (b1g, b2g, and b3g) due to the monoclinic deformation (Eq. (3.4)).

5.4 Elastic Properties 37

-0.12 -0.09 -0.06 -0.03 0.00 0.03

10

15

20

25

30

35

10

15

20

25

30

35

V-7.5Cr V V-7.5Ti

(b) 5%C44 distortion

Energy (Ry)

Den

sity

of S

tate

s(st

ates

/Ry)

V-7.5Cr V V-7.5Ti

No distortion(a)

Figure 5.4: Density of states (DOS) for bcc V, V-7.5Cr, and V-7.5Ti alloys (a) and inC44-type monoclinic distorted lattice (b) plotted for energies close to the Fermilevel (marked by dashed line). The dash-dotted lines indicate the position ofthe DOS peak close to the Fermi level. Note that the whole occupied valanceband DOS is shown in Fig. 5.12.

and V-Ti alloys the above scenario is slightly altered by alloying. Namely, thebefore mentioned DOS peak moves toward lower (higher) energy levels uponCr (Ti) doping as compared to that in bcc V. These results indicate that theone-electron energy change of V-7.5Ti (V-7.5Cr) is larger (smaller) than thatof V, suggesting that C44(V-7.5Cr) < C44(V) < C44(V-7.5Ti). Therefore, theelectronic structure of bcc V provides an explanation for the anomalous C44

versus composition.

5.4.2 Polycrystalline Elastic Properties

The polycrystalline elastic parameters of bcc V1−x−yCrxTiy alloys as a functionof composition are shown in Fig. 5.5. The bulk modulus B(x, y) varies betweena minimum value of 166 GPa belonging to V-10Ti, and a maximum value of 183GPa corresponding to V-10Cr. The B(x, y) follows the opposite trend obeyedby the lattice parameter (Fig. 5.2). The bulk modulus measures the resistanceof material to uniform compression. Alloys with larger lattice constant possesslower average bond strength and thus they can be more easily compressed thanthose with smaller lattice constants. From the investigation of the lattice con-stant, we know that Cr decreases it and Ti increases it. These results explainwhy the bulk modulus of V-Cr-Ti alloys decreases with Ti and increases withCr addition.

The variation of the shear modulus is very small within the present composi-

38

CHAPTER 5. MECHANICAL PROPERTIES OF VANADIUM AND

VANADIUM-BASED ALLOYS

0 2 4 6 8 100

2

4

6

8

10

Bulk modulus

Cr(at.%)

Ti (

at.%

)

166.0168.2170.5172.7175.0177.2179.5181.7184.0

0 2 4 6 8 100

2

4

6

8

10

Young's modulus

Cr (at.%)

Ti (

at.%

)

132.5133.0133.5134.0134.5135.0135.5136.0136.5

0 2 4 6 8 100

2

4

6

8

10

Shear modulus

Cr (at.%)

Ti (

at.%

)

48.0048.2348.4548.6848.9049.1349.3549.5849.80

Figure 5.5: Theoretical polycrystalline elastic constants (in GPa) for V1−x−yCrxTiy(0 ≤ x, y ≤ 0.1) alloys as a function of Cr and Ti concentration.

tional map (about 2 GPa). Along the diagonal region, the shear modulusG(x, y)is relatively large compared to the rest of the map, and slightly increases withincreasing total solute concentration when (x+ y) > 0.1 − 0.15. This particularsaddle type of trend of G(x, y) is due to the fact that C44(x, y) and C′(x, y)show opposite variations with alloying (Fig. 5.3) and more specifically due tothe peculiar trend of C44(x, y). The Young’s modulus E(x, y) has similar com-position dependence as that of the shear modulus, with V-10Cr-7.5Ti havingthe largest E value (136.3 GPa).

5.4.3 Ductile/Brittle Properties

The ductile/brittle behavior of V-Cr-Ti alloys is a crucial issue for the perfor-mance of structural materials. Previously, Pugh [138] proposed a phenomeno-logical criterion for the ductile-brittle transition by means of the B/G value: Amaterial is ductile when its B/G ratio is greater than 1.75; otherwise it is in thebrittle regime. The calculated values of B/G for all V-based alloys consideredhere are well above 1.75 (see Fig. 5.6), suggesting that all of these alloys exhibitexcellent ductile property. Alloys with largest B/G ratios correspond to lowTi (<3 at.%) and high Cr (>6 at.%) concentration. The high-Ti-content (>7at.%) alloys possess the lowest B/G ratio (<3.5) with a minimum around V-10Ti. Furthermore, we also find that the V-4Cr-4Ti alloy has marginally betterductility (in terms of B/G) than the V-5Cr-5Ti alloy. This is consistent withthe experimental observation that at temperatures up to 400°C V-4Cr-4Ti ex-hibits a larger uniform elongation (18%-23%) than V-5Cr-5Ti (14%-18%) [3]. Inaddition, the ductile/brittle behavior of a cubic crystal can also be expressed by

5.4 Elastic Properties 39

0 2 4 6 8 100

2

4

6

8

10

Cr (at.%)

Ti (

at.%

)

3.4003.4503.5003.5503.6003.6503.7003.7503.800

B/G

0 2 4 6 8 100

2

4

6

8

10

Cauchy pressure

Ti (

at.%

)

Cr (at.%)

41.1041.6742.2442.8243.3943.9644.5345.1045.6846.2546.82

Figure 5.6: The ratio of bulk and shear modulus and Cauchy pressure (GPa) forV1−x−yCrxTiy (0 ≤ x, y ≤ 0.1) alloys as a function of Cr and Ti concen-tration.

its Cauchy pressure (C12(x, y)-C44(x, y))/2: the Cauchy pressure has a positivevalue if materials possess ductility, whereas for brittle materials, the Cauchypressure is negative. [139] As shown in Fig. 5.6, one can see that all Cauchypressure values are positive, which means all alloys have ductile property. Fur-thermore, the Cauchy pressure as well as the B/G ratio from our calculationsshow similar variation of ductile behaviors with the alloying element concentra-tion.

5.4.4 Solid-solution Hardening

With the Labusch-Nabarro (LN) semiempirical model [66–68] introduced in Sec-tion 3.2, we investigated the SSH of V-Cr-Ti alloys as a function of Cr and Ticoncentration. First, we calculated the SSH of V-Cr and V-Ti binary alloys.We found that both Cr and Ti induce SSH in V, but Ti has a slightly largerstrengthening effect than Cr. These theoretical predictions are in line with theobservations [140, 141]. For the ternary alloys, we simply sum the effects ob-tained for the V-Cr and V-Ti binary alloys. Figure 5.7 shows the SSH for theV-based alloys as a function of composition. One can see that for total soluteconcentrations below ∼ 10%, the SSH has relatively low values (as compared tothe dilute alloys). Obviously, the SSH increases more with Ti content than withCr content as a result of the larger misfit parameters obtained for a Ti-dopedsystem.

The experimental works usually focus on the ductile-brittle transition temper-ature (DBTT), which is a important parameter for structural material design.As a structure material for future fusion reactors, it needs to have good ductilityeven at room temperature. Thus it is necessary to find excellent V-based alloyswith low DBTT. However, it is impossible to investigate the DBTT by theoret-ical tools. Fortunately, it has been found that the DBTT is related to the yieldstrength [142]. In an ideal crystal, the yield strength is decomposed into latticefriction strengthening (Peierls stress) and SSH contributions. The stress neededto move a dislocation across the barriers of the oscillating crystal potential isthe Peierls stress. In metals, this is found to be approximately proportional to

40

CHAPTER 5. MECHANICAL PROPERTIES OF VANADIUM AND

VANADIUM-BASED ALLOYS

3 4 5 6 7 8 9 10

3

4

5

6

7

8

9

10

Cr (at.%)

Ti (

at.%

)

0.10650.12550.14450.16350.18250.20160.22060.23960.25860.2755

Figure 5.7: Solid-solution hardening in ternary V1−x−yCrxTiy (0.025 ≤ x, y ≤ 0.1)alloys as a function of Cr and Ti concentrations.

Figure 5.8: A 3D plot of the DBTT for the ternary V1−x−yCrxTiy alloys as a functionof Cr and Ti concentrations, taken from Ref. [1].

5.5 Ideal Tensile Strength 41

the shear modulus [143]. According to the above correlation, we can study theDBTT as a function of alloying contents. In the following, we combine the shearmodulus and SSH results to investigate the alloying effect on the DBTT.

From the shear modulus shown in Fig. 5.5, we can see that alloys with high(x+y) (above ∼ 0.15) have the highest shear modulus, indicating that these al-loys possess the highest Peierls stress. Alloys outside of this region have slightlylower Peierls stress. Combining this result with the SSH from Fig. 5.7, weconclude that the yield stress should increase with both Cr and Ti addition,and this increase should be more pronounced along the main diagonal (x ≈ y).Hence, our results suggest that the DBTT should show nearly symmetric com-position dependence (as a function of Cr and Ti contents) with a maximum forthe equiconcentration alloys with (x+ y) larger than 0.1 − 0.15. A very similarDBTT map was established from experimental measurements [2], see Fig. 5.8.From Fig. 5.8, we can see that the V-Cr-Ti alloys with total concentration of Crand Ti below 10% show the lowest DBTT, and that the DBTT increases withincreasing the total concentration of Cr and Ti (above 10%), which is consistentwith our prediction.

5.5 Ideal Tensile Strength

5.5.1 Ideal Tensile Strength of Vanadium

To assess reliability of our computational approach for the ITS introduced inSection 3.3, we first calculated the ITS of pure V along the [001], [110] and [111]directions.2 For V, it is found that the deformation starts along the Bain path,but branches away onto an orthorhombic path before the fcc point is reachedin response to tension along [001]. Here, using our computational approach,we investigated this bifurcation for V with the aim to reveal its impact on theattainable ITS. Figure 5.9 shows the stress as a function of strain along the[001] direction. Along the tetragonal deformation path, stress increases withincreasing strain up to a maximum of σm = 18.8 GPa at a strain of ǫm = 16.8%.The significantly lower ideal stress and strain on the orthorhombic deformationpath are clearly different from those corresponding to the tetragonal deformationpath. The stress corresponding to the secondary orthorhombic path reaches amaximum of 12.4 GPa at ǫm = 9.2%. Hence the ITS of V is limited by thebifurcation to the secondary orthorhombic path in the [001] direction, in linewith the previous observations [70, 84].

The ideal tensile strength σm corresponding to the strain ǫm from our andother calculations in the [001] direction and in the other two directions arelisted in Table 5.1 and Table 5.2, respectively. From Table 5.1, it can be seenthat our results are in close agreement with available literature values. FromTable 5.2 we can see that our computed maximum stress for the [110] directionis somewhat larger than the only available theoretical data, Ref. [130]. This

2The results on the ITS shown in this part were published in Paper II [144]. Regrettably, inEquation (1) as published in Ref. [144] a mistake slipped in: the equation for the stress missesthe prefactor (1 + ǫ). The correct equation for the stress is given in Eq. (3.11) on Page 14.As a result, all published numerical data on the ITS are incorrect in Ref. [144], but updatedresults are presented in this section. However, this mistake does neither change the presentedtrends, the given explanations, nor the drawn conclusion in Ref. [144].

42

CHAPTER 5. MECHANICAL PROPERTIES OF VANADIUM AND

VANADIUM-BASED ALLOYS

0 5 10 15 20 2502468

101214161820

Tetragonal Orthorhombic

Stre

ss (G

Pa)

Strain (%)Figure 5.9: The stress of bcc V along the [001] direction as a function of the applied

strain. Open symbols and closed symbols refer to the primary tetragonaldeformation path and to the secondary orthorhombic deformation path, re-spectively.

Table 5.1: The ideal tensile strength σm and the corresponding strain ǫm of V for thetetragonal and orthorhombic deformation paths in the [001] direction. PAWand PP stand for the projector-augmented wave method and pseudo-potentialmethod, respectively.

system methodTetragonal Orthorhombic

σm (GPa) ǫm(%) σm (GPa) ǫm(%)

V

This work 18.8 16.8 12.4 9.2PAW Ref. [130] 19.1 18.0PAW Ref. [70] 17.8 17.0 11.5 10.0PAW Ref. [145] 18.9 18.2

PP Ref. [84] 17 16.0 14 12.0

may be attributed to our constraint relaxation (fixed lattice parameter ratiocfco/afco =

√2). If this constraint is released as done in Ref. [130], the total

energy for each lattice distortion lowers and hence the uniaxial strain-energycurve may be shallower compared to a constraint calculation.

Figure 5.10 displays the stresses as a function of strain along all investigateddirections for V. Compared to the [001] direction, the ideal stresses are muchhigher in the [111] direction and the [110] direction. From these results, weconclude that the [001] direction is the weakest one. In the inset of Fig. 5.10(a), we also show stress as a function of strain in the small deformation region(ǫ ≤ 0.5%). These strain-stress relations are linear and follow Hooke’s law,

5.5 Ideal Tensile Strength 43

Table 5.2: Comparison between the present and former (Ref. [130]) ideal tensilestrength σm and the corresponding strain ǫm of bcc V calculated in the [111]and [110] directions.

element methoddirection/[111] direction/[110]

σm (GPa) ǫm(%) σm (GPa) ǫm(%)

VThis work 36.4 37.0 52.0 38.6

PAW Ref. [130] 31.0 36.0 32.8 42.0

0 5 10 15 20 25 30 35 4005

1015202530354045505560

0.00 0.25 0.500.0

0.5

1.0

(a)

[110] [111] [001]

Stre

ss (G

Pa)

Strain (%)

Figure 5.10: The stress of bcc V along [001] (secondary deformation path), [111] and[110] directions as a function of strain. The insets show the trends of stressversus strain in the smaller strain region.

σ = E · ǫ, where E is Young’s modulus, which depends on the direction of theapplied force [59]. According to the inset figure, we obtained E[001]=200 GPa andE[111]=110 GPa, respectively. Furthermore, we also can use the elastic constants(C11, C12, and C44) to get Young’s modulus along different tension directions,which can be written as [59]

E[001] = 6C′ B

C11 + C12, (5.3)

E[111] = 3C441

1 + C44

3B

. (5.4)

Accordingly, using the theoretical elastic parameters of bcc V [54], we obtainE[001]=205.3 GPa and E[111]=101.4 GPa. These values are in good agreementwith the above data derived directly from Fig. 5.10. The anisotropy between

44

CHAPTER 5. MECHANICAL PROPERTIES OF VANADIUM AND

VANADIUM-BASED ALLOYS

E[001] and E[111] at small strains is distinct from the one for the larger, non-linear deformation region, where [110] ultimately replaces [001] as the strongestdirection.

5.5.2 Ideal Tensile Strength of Vanadium-based Alloys

Based on the same method, we investigated the alloying effect on the ITS forthe V-based alloys. Figure 5.11 shows the composition dependence of the ITSof ternary bcc V-Cr-Ti random alloys along the [001], [111], and [110] directionsand the corresponding numerical data for selected compositions are listed inTable 5.3. From Fig. 5.11, we can see that the ITS increases with increasingCr and decreases with Ti addition for all directions. Due to the opposite effectof alloys Cr and Ti on the ITS, the ITS remains almost unchanged if equalamounts of Cr and Ti are alloyed to V. For example, the ITS of V-5Cr-5Ti is12.6 GPa, which is very close to the value of V. From our results, we can see thatthe [001] direction is the weakest one for all investigated alloys. Table 5.3 showsthe ITS of V-Cr-Ti random alloys along the [001], [111], and [110] directions.From Table 5.3, it can be seen that the orthorhombic deformation significantlyreduce the ITS for all V-based alloys. Li et al. [84] calculated the ITS of Ti-V alloys along the [001] direction for concentrations of Ti ≥ 30 at.% with thevirtual crystal approximation (VCA). Their results confirm our trend, namelythat the more Ti is present in the alloy the more the ITS is decreased. Tofurther assess our results, we calculated the ITS of the V-30Ti alloy along thetetragonal deformation. Our value of 10.7 GPa is close to the 10 GPa obtainedby Li et al. [84] for the same alloy composition.

5.5.3 Structural Energy Difference and Ideal Tensile Strength

In the following, we explain the alloying trend on the ITS for tension alongthe [001] and [111] directions. The deformation along [001] is limited to theprimary deformation path (Bain path, bct lattices) and the deformation in re-sponse to strain along [111] is constrained to trigonal geometries in the discus-sion. Although the branching from the primary tetragonal deformation pathsignificantly reduces the ITS (for [001] uniaxial stress), the alloying effect onthe maximum stress is similar for both deformation paths (Table 5.3). Sinceboth deformation paths are identical before branching, the alloying effect on theITS can be understood based on the primary tetragonal deformation path.

For a quantitative estimation of the ITS, we below assume a state of strainwith constant volume fixed to the theoretical bcc equilibrium volume per atom,Ωbcc. The calculated ITS (σm) is approximated by σSED

m , which is defined by

σm ≈ σSEDm =

1 + ∆ǫΩbcc

∆E∆ǫ

, (5.5)

where ∆E is the energy of the maximum closest to the bcc phase on the re-spective strain-energy curve and ∆ǫ is the strain at constant volume necessaryto transform the bcc lattice into either the fcc lattice or into the sc lattice.Readily one may find, ∆ǫ[001] = (1 − 0.51/3)/0.51/3 ≈ 0.260 and ∆ǫ[111] =(1−0.251/3)/0.251/3 ≈ 0.587 for the [001] distortion and the [111] distortion, re-spectively. Along the [001] and [111] directions, the nearest, symmetry-dictated

5.5

Ideal

Ten

sile

Str

en

gth

45

Table 5.3: The ideal tensile strength σm, corresponding strain ǫm from our calculations in [001], [111] and [110] directions for vanadium-based alloys.

compositionTetragonal/[001] Orthorhombic/[001] direction/[111] direction/[110]σm (GPa) ǫm(%) σm (GPa) ǫm(%) σm (GPa) ǫm(%) σm (GPa) ǫm(%)

V-10Cr 21.7 17.2 14.0 9.5 37.5 36.1 53.5 38.2V-5Cr 20.2 17.1 36.9 36.4 52.8 38.7V 18.8 16.8 12.4 9.2 36.4 37.0 52.0 38.6V-5Cr-5Ti 18.8 16.8 12.6 9.4 36.1 37.2 51.6 38.5V-5Ti 17.3 16.4 35.5 36.6 50.9 39.0V-10Ti 16.0 16.3 11.5 9.3 34.5 36.5 49.8 38.8

46

CHAPTER 5. MECHANICAL PROPERTIES OF VANADIUM AND

VANADIUM-BASED ALLOYS

0 2 4 6 8 100

2

4

6

8

10

[001]

Ti (

at.%

)

Cr (at.%)

11.40

11.74

12.08

12.41

12.75

13.09

13.43

13.77

14.00

0 2 4 6 8 100

2

4

6

8

10

[111]

Ti (

at.%

)

Cr ( at.%)

34.50

34.88

35.25

35.63

36.00

36.38

36.75

37.13

37.50

0 2 4 6 8 100

2

4

6

8

10

[110]

Ti (

at.%

)

Cr (at.%)

49.80

50.26

50.73

51.19

51.65

52.11

52.58

53.04

53.50

Figure 5.11: The ideal tensile strength of V-based alloys along [001], [111] and [110]directions as a function of Cr and Ti. All stress values are in GPa. The [001]stress corresponds to the secondary (orthorhombic) path (i.e. fully relaxedstructure within the plane perpendicular to the strain), whereas the [111]and [110] stresses correspond to isotropic Poisson contraction.

maxima to the bcc phase of pure V are fcc and sc structures, which were iden-tified by the previous works [127, 129–131], i.e., the uniaxial strain energy mustlevel off to the fcc-bcc SED and to the sc-bcc SED for an elongation along the[001] and an elongation along [111], respectively. Using Eq. (5.5), we calcu-lated the ITS along [001] and [111] directions. We obtain σSED

m = 16.1 GPa andσSED

m = 36.3 GPa for the [001] distortion and the [111] distortion, respectively,their ratio being 2.25. We compared these simple estimates to our ab initio data,where we obtained 18.8 GPa and 36.4 GPa for the respective ideal stresses, andtheir ratio is 1.94.

To understand the alloying effect on the basis of band filling arguments, wecalculated the DOS for pure V, vanadium alloyed with 10 % Cr and with 10 %Ti, as well as the DOS of the equi-composition V-5Cr-5Ti alloy (Fig. 5.12).From Fig. 5.12 (left panel), we notice that the curves of V-10Cr and of V-10Ti are almost rigidly shifted with respect to the one of V, while the shapeof their DOSs is merely effected by alloying vanadium with adjacent elementsin the periodic table. This rigid band shift behavior signals an increase of thed-occupation (increase of the number of valence electrons) and a decrease of the

5.5 Ideal Tensile Strength 47

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04Energy (Ry)

0

5

10

15

20

25

30

Den

sity

of

stat

es (

stat

es/R

y)

VV-5Cr-5TiV-10CrV-10Ti

-0.5 -0.25 00

10

20

30

0 2 4 6 8 10 3d-band filling (states/atom)

-30

-20

-10

0

10

20

30

40

EFC

C -

EB

CC (

mR

y/at

om)

Sc Ti V Cr Mn Fe Co Ni ZnCa

EFCC

- EBCC

EBCC

Cu

bcc stable

fcc stable

Figure 5.12: Left panel, the density of states for V-based alloys show that alloyingwith a single element leads mainly to a rigid band shift of the DOS. The insetdisplays the complete occupied valence band of pure V. Energies are givenrelative to the Fermi level which is indicated by a vertical dotted line. Rightpanel, the fcc-bcc structural energy difference as a function of d-occupationfor the elements 20 to 30 in the periodic table (mostly 3d transition metals),taken from Ref. [131]. The data is based on self-consistent non-spin-polarizedLDA calculations. The lines guides the eye.

of d-occupation (decrease of the number of valence electrons) for Cr additionand Ti addition, respectively. Thus, alloying V with Cr and Ti changes the bandfilling. The DOS of V-5Cr-5Ti is virtually not shifted with respect to the one ofV, which indicates a zero net gain in the number of d-electrons in the matrix.It was shown [115–117] that the band filling is the most important parameterdetermining structural stability in transition metals. Thus, it is straightforwardto correlate the d-band filling due to alloying to the ITS via SEDs. Figure 5.12(right panel, taken from Ref. [131]) shows the SEDs as a function of d-occupationfor single elements. From this figure, we can see that Cr (Ti) has bigger (smaller)SED than that of V. Based on the DOSs and the SEDs trend, we expect anincrease (decrease) of the fcc-bcc SED if more Cr (Ti) is added into pure V, anda similar fcc-bcc SED as pure V with the same amount composition of Cr andTi.

Figure 5.13 displays the uniaxial strain energy for V-based alloys in the [001]direction for the tetragonal deformation path. It is apparent that with respectto the curve of V, the energy-strain curve rises more rapidly with increasingCr concentration and decreases more rapidly with increasing Ti concentration.Furthermore, the energy-strain curve of the equi-composition alloy (V-5Cr-5Ti)is similar to that of V. Based on the above trends and according to the behaviorof SEDs with band filling, we expect that the ITS in the [001] direction increases(decreases) with increasing Cr (Ti) concentration.

To confirm our expectation, we calculated the SEDs of V-based alloys, andalso calculated the ITS based on the SEDs data according to Eq. (5.5). Fig-ure 5.14 displays the correlation between the real ITS calculation and the oneobtained by SED calculation along the [001] direction. From Fig 5.14, it isclearly seen that there is a very strong correlation between these two types ofcalculations. From the above results, we infer that the alloying effect on theITS for the V-based alloys can be explained on the basis of SEDs. Note that

48

CHAPTER 5. MECHANICAL PROPERTIES OF VANADIUM AND

VANADIUM-BASED ALLOYS

0 5 10 15 20 2502468

10121416182022

V-10Cr V-5Cr V V-5Cr-5Ti V-5Ti V-10Ti

E (m

Ry/

atom

)

Strain (%)

Figure 5.13: The uniaxial strain energy of V-based alloys for a stress applied in the[001] direction (primary deformation path). The higher the 3d-band occupa-tion the steeper the curve progression.

the influence of the volume change due to alloying (which enters Eq. (5.5)) isabout 2 % and is not primarily responsible for the observed trend of σm sincethe volume effect is much smaller than the effect of ∆E.

According to Eq. (5.5), we also calculated the ITS along the [111] direction.Figure 5.15 shows the correlation between the real ITS calculation and the oneobtained by SED calculation along the [111] direction. From Fig. 5.15, we cansee that these two types of calculations have a strong correlation.

On the basis of our simple estimate of the ITS, we conclude with respect tobulk V an increase of σm for the V-Cr binary alloys and a reduction of theITS in the case of V-Ti binary alloys. We find for the equi-composition alloyV-5Cr-5Ti, that its ITS almost retains the ITS of pure V.

5.5 Ideal Tensile Strength 49

16 17 18 19 20 21 22σ

m (GPa)

13

14

15

16

17

18

19

σSED

m (

GPa

)

V

V-5Cr-5Ti

V-5Cr

V-10Cr

V-5Ti

V-10Ti

Figure 5.14: The correlation between the real ITS calculation and the one obtainedby SED calculation for V-based alloys along the [001] direction.

34.0 34.5 35.0 35.5 36.0 36.5 37.0 37.5 38.034.0

34.5

35.0

35.5

36.0

36.5

37.0

37.5

38.0

38.5

V-5Ti

V-10Ti

V-5Cr-5Ti

V

V-5Cr

V-10Cr

mSE

D (G

Pa)

m (GPa)

Figure 5.15: The correlation between the real ITS calculation and the one obtainedby SED calculation for V-based alloys along the [111] direction.

Chapter 6

Ideal Tensile Strength ofRefractory High-entropyAlloys

6.1 Ideal Tensile Strength of High-entropy Al-

loys

The tensile stress-strain relationships along the [001] direction for bcc ZrNbHf,ZrVTiNb, ZrTiNbHf, and ZrVTiNbHf random alloys are shown in Figure 6.2.The calculated ITS of the ternary ZrNbHf alloy is about 2.64 GPa, and slightlydecreases to 2.28 GPa by adding the fourth element Ti. However, if we add Tiand V to ZrNbHf, the ITS is enhanced by nearly 42 % with respect to the valueof ZrNbHf. An even more pronounced impact of alloying on the ITS occurs ifHf in ZrNbHf is substituted with equiatomic Ti and V, which increases the ITSby about 170 % (to 7.13 GPa).

From our calculations, we found that the ZrNbHf, ZrTiNbHf, and ZrVTiNbHfalloys maintain bct geometry along the [001] direction for strains up to slightlylarger than maximum strain (ǫm). Hence, these alloys fail by cleavage of the(001) planes and are representatives of bcc materials termed intrinsically brittle

owing to their particular deformation and failure behaviors during tension (seeChapter 3). For the ZrVTiNb alloy, a branching from bct to orthorhombicsymmetry occurs before ǫm is reached along the tetragonal deformation path,which is due to the elastic shear instability (see Chapter 3). This branching leadsto a slight decrease of the ITS of ZrVTiNb with respect to the maximum valueon the tetragonal deformation path. The above shear instability can also beunderstood as an activated shear instability of the 〈111〉112 slip system (theresolved shear stress reaches the ideal shear stress under superimposed tensionin [001] direction) [69]. The resolved shear stress for the 〈111〉112 slip systemat the instability is 3.5 GPa for ZrVTiNb, which is a good estimate of its idealshear strength (τm) for the same slip system without superimposed tension [23].The estimated reduced shear strength, given by τm/G〈111〉,

G〈111〉 = 6c44c′/(2c′ + 4c44), (6.1)

52

CHAPTER 6. IDEAL TENSILE STRENGTH OF REFRACTORY

HIGH-ENTROPY ALLOYS

amounts to 0.1 in agreement with other bcc systems [70, 146]. Owing to thisparticular failure behavior, the ZrVTiNb alloy is referred to as being intrinsi-

cally ductile, which represents the preferred failure mode over the intrinsicallybrittle one (see Chapter 3). Thus, ZrVTiNbHf and ZrVTiNb display an intrin-sic brittle-to-ductile transition in terms of their intrinsic failure mode which isaltered upon changing the composition of the alloy, i.e., removal of Hf.

We noticed that the alloy with the highest ITS (ZrVTiNb) contains two el-ements from the fourth group and two group five elements, which is denotedby the ratio 2:2. In contrast, alloy possessing the lowest ITS (ZrTiNbHf), hasa composition dominated by elements from group four (ratio 3:1). The alloywith moderate ITS (ZrVTiNbHf) consists of totally five elements with ratio 3:2.This simple analysis indicates that the ITSs of refractory HEAs are determinedby the ratio of constituent transition metal elements belonging to the fourthgroup and fifth group. Obviously, the smaller is the ratio the higher is the ITS.It further suggests that the ITS of refractory HEAs could be tuned by varyingthe total number of equimolar transition metals elements.

Figure 6.1: The tensile stress as a function of strain under uniaxial stress applied alongthe [001] direction for refractory HEAs. For ZrVTiNb, a branching from thetetragonal (bct) to the orthorhombic (ort) deformation path occurs at a strainof 12.7 % < ǫm shown more clearly in the inset; for larger strains, both thefully-relaxed orthorhombic and the constrained to bct stress-strain curves areplotted. Branching does not occur for the other alloys in the strain intervalshown.

6.2 Structural Energy Difference and Average d-band Filling 53

Figure 6.2: (a) The correlation between the ITS and the fcc-bcc SED for the HEAs.(b) The correlation between the fcc-bcc SED and the average valence d-bandfilling for the HEAs (solid symbols), group 4 and 5, 3d- and 4d transitionmetals and their random binaries. Dashed lines guide the eye.

6.2 Structural Energy Difference and Average

d-band Filling

To understand the above ITS trends, we employed the structural energy dif-ference (SED) model, see Section 5.5.3. Although ZrVTiNb was found to beintrinsically ductile, we also apply the SED model to ZrVTiNb, since its ITS is

54

CHAPTER 6. IDEAL TENSILE STRENGTH OF REFRACTORY

HIGH-ENTROPY ALLOYS

only slightly lowered by branching as discussed above. The correlation betweenthe ITS and the SED is displayed in Figure 6.2(a). One can see that the ITS andthe SED correlate strongly, indicating that this SED model is indeed holds forthe presently investigated HEAs, i.e., higher fcc-bcc SED corresponds to higherITS.

For all three transition metal series, electronic structure theory showed thatthe fcc-bcc SED itself follows a regular pattern as a function of the numberof valence d-electrons [115, 118, 119, 121]. For example, Fig. 5.12 (right panel)displays the fcc-bcc SED as a function of d-electrons for 3d transition metals. Fora d-band filling corresponding to bulk Ti, ∆ESED < 0, but for a d-band fillingcorresponding to bulk V, ∆ESED > 0, which implies that ∆ESED increasesas the number of 3d electrons increases and changes sign. Both properties of∆ESED are confirmed by the present theory and are shown for the 3d (Ti, V)and 4d (Zr, Nb) elements and their equimolar random solid solutions (Ti-V,Zr-Nb) in Fig. 6.2(b). For a more elaborate analysis of the ITSs trend forHEAs alloys, we plotted the determined ∆ESEDs for the present HEAs as afunction of the d-band filling in Fig. 6.2(b). The d-band filling of alloys wasobtained by averaging the number of valence d-electrons of the constituent inthe alloy. Clearly seen from Fig. 6.2(b), the SED of the multicomponent HEAsfollows very closely the trend of the SED inherent to the 3d or 4d transitionmetal series (for the 5d series, see Refs. [119, 147]). The data for the HEAs inFig. 6.2(b) clearly reveals the fact that the SED rises monotonically and nearlylinear as the average d-occupation increases. This strong correlation confirmsthat the SED of the present refractory HEAs is mainly determined by theiraverage d-occupation number. Combining the above discussion and the datain Figs. 6.2(b), our results show that the ITS of the present refractory HEAis correlated with their average d-occupation, i.e., the higher the d-occupation,the larger the ITSs.

It is important to realize that, although Ti, Zr, and Hf are not stable in thebcc structure (at ambient conditions), the HEAs composed to a major part ofelements from the fourth group are, in fact, bcc stable. This in turn suggests,that there is great flexibility to tune the ITS of refractory HEA by varyingcomposition and number of constituent elements as illustrated by the presentstudy. The highest values of the ITS are obtained for HEA containing a largeamount of group five elements due to their high intrinsic ITS (approximately12-14 GPa [70]). The benefit of forming HEA by alloying group four elementsto group five elements is a wide stability range of the bcc phase (in termsof the average d-occupation number, see Fig. 6.2(b)) at the cost of loweringthe ITS. The ZrVTiNb alloy with d-occupation approximately equal to 3.15 isintrinsically ductile as are the elements V, Nb, and Ta with larger d-occupationapproximately equal to 3.6. Given the present finding that the d-occupation isthe controlling variable for the ITS, one might expect, that HEAs with d-bandfillings larger than ∼ 3.2 are also intrinsically ductile and reach ITS values thatapproach those of the group five elements, as the d-occupation approaches ∼ 3.6.

Chapter 7

Ideal Tensile Strength ofIron and Iron-based Alloys

In Chapter 5, we investigated the ITS of nonmagnetic random vanadium-basedalloys. Based on our results, we concluded that the EMTO-CPA approach pro-vides an efficient and accurate theoretical tool to design the ITS of nonmagneticbcc random solid solutions and reveal the composition dependence of this fun-damental physical parameter. In this chapter, using the same method, we willinvestigate the ITS of ferromagnetic Fe-based random alloys. In this work, theselected solute atoms are common in commercial Fe-based steel alloys and theyrepresent simple metal (Al), nonmagnetic (V) and magnetic (Cr, Mn, Co, andNi) transition metals. The concentration of the solutes varied in the range from0 to 10 % except for Mn where the maximum concentration was 5 %. The [001]direction was identified as the weakest direction of bcc crystals [21–23]. Thuswe only focus on the [001] direction in this chapter.

7.1 Ideal Tensile Strength of Iron

In order to establish the accuracy of the EMTO method for the ITS of Fe, wecarried out additional ITS calculations using three different density-functionalcodes. They are the projector-augmented wave (VASP) code [148], an all-electron full-potential localized orbitals (FPLO) code [149], and an all-electronfull-potential linearised augmented-plane wave (FP-LAPW) code (ELK) [150].1

The ITSs were obtained using a strain-energy fitting function for EMTO, ELK,FPLO and FP-LAPW codes. For the VASP calculation, we used two approachesto get the ITS: one is from the strain-energy fitting function, the other one isdirectly from the computed stress tensor [154]. Table 7.1 shows the ITS as afunction of plane wave energy cut-off for both approaches. From Table 7.1, wecan see that the ITS derived from the stress tensor is converged when the planewave energy cut-off is larger than 500 eV. From the fitting function data, we cansee that the ITSs show a small change with increasing energy cut-off (error bardue to fitting). Both approaches give practically similar results.

1The VASP, ELK, and FPLO calculations were done by Stephan Schönecker. For thecomputation details, see Refs. [151–153].

56

CHAPTER 7. IDEAL TENSILE STRENGTH OF IRON AND IRON-BASED

ALLOYS

Table 7.1: The ideal tensile strength (in GPa) of bcc Fe from strain-energy fittingfunction and stress tensor as a function of plane wave cut-off energy (in eV)with VASP code.

Energy cut-off 267 350 500 800 1000

fitting function 11.9 11.7 11.6 11.7 11.6stress tensor 14.5 12.1 11.8 11.8 11.8

The ITS σm of Fe corresponding to the strain ǫm from our and other calcu-lations are listed in Table 7.2. From Table 7.2, we can see that our data fromEMTO, FPLO, and FP-LAPW methods agree well with other results, however,our VASP value is lower than the ones from other calculations. The likely rea-son is that the previous PAW data were obtained with a lower-energy cut-off(below 350 eV) from stress tensor method, however, a 500 eV energy cut-off anda fitting function were employed in this study. Decreasing the energy cut-off to350 eV, we found that the ITS of Fe increases to 12.1 GPa from stress tensor (seeTable 7.1) bringing it close to the ones from others PAW calculations. UsingEMTO method, we found that a bifurcation from tetragonal to orthorhombicsymmetry occurs at ǫorth = 17 %, i.e., well above ǫm = 14.1%. This result is inaccordance with Ref. [20], where the branching was reported to occur at 18%strain.

Table 7.2: The ideal tensile strength σm and the corresponding strain ǫm in the [001]direction of ferromagnetic bcc Fe. The present results (EMTO, FPLO, PAW,FP-LAPW) are compared with previous projector-augmented wave (PAW) [20,21, 155], and full-potential linearized augmented plane wave (FP-LAPW) [22]

.

element methoddirection/[001]

σm (GPa) ǫm(%)

Fe

EMTO (this work) 12.6 14.1FPLO (this work) 13.0 15.2

VASP-PAW (this work) 11.6 14.7FP-LAPW (this work) 12.8 14.3

PAW Ref. [20] 12.6 15PAW Ref. [21] 12.4 16PAW Ref. [155] 12.4 14

FP-LAPW Ref. [22] 12.7 15

7.2 Ideal Tensile Strength of Iron-based Alloys

Figure 7.1a shows the concentration dependence of the ITS of the present binaryalloys along the [001] direction. The corresponding numerical data for selectedcompositions are listed in Table 7.3. The ITS is found to increase with Cr,Co and V and decrease with Ni and Al addition to Fe. For instance, when

7.2 Ideal Tensile Strength of Iron-based Alloys 57

10% Cr, Co, or V is added to bcc Fe, the ITS of Fe increases by 12.7%, 9.5%and 23%, respectively. If however 10% Ni or Al is added to the Fe matrix,the ITS reduces by 13.5%. The ITS of Fe1−xMnx increases by 3% as the Mnconcentration increases from 0 to 2.5%, and then reduces by 6% when up to 5%Mn is added to Fe.

In order to account for the possibility of a branching from the primary bct de-formation path to the secondary orthorhombic deformation path in the presentFe1−xMx alloys, we performed additional structural optimization as a functionof the orthorhombic degrees of freedom. We find that for all binaries consid-ered here and for all x values, the branching occurs at strains larger than ǫm(x)corresponding to the ideal tensile strength of alloy Fe1−xMx. For instance,the branching points are 18 %, 18 % and 17 % for Fe0.95Co0.05, Fe0.95Al0.05 andFe0.95Ni0.05, respectively. These figures should be compared to the correspond-ing ǫm values of 15.4 %, 14.9 % and 14.6 %, respectively (Table 7.3). Althoughfor some alloys, the branching point gets slightly closer to ǫm as compared topure Fe (e.g., in Fe-Co and Fe-Ni), within the present compositional interval(x ≤ 0.1) we can exclude the bifurcation from the primary tetragonal to thesecondary orthorhombic path.

Table 7.3: Theoretical ideal tensile strength (σm, in GPa) and the corresponding strain(ǫm, in %) calculated in the [001] direction for Fe1−xMx alloys. For pure Fe,σm=12.6 GPa and ǫm=14.1 %.

xσm(x) ǫm(x) σm(x) ǫm(x)

xσm(x) ǫm(x)

Fe-Cr Fe-Co Fe-V

0.025 13.6 14.2 12.8 14.8 0.025 13.8 14.40.05 14.1 14.9 13.2 15.4 0.05 14.4 15.20.075 14.2 15.1 13.5 15.4 0.075 15.2 15.70.1 14.2 14.7 13.8 15.4 0.1 15.5 15.8

Fe-Ni Fe-Al Fe-Mn0.025 12.4 14.3 12.0 14.6 0.0125 12.8 14.70.05 12.1 14.6 11.6 14.9 0.025 13.0 14.90.075 11.6 14.3 11.1 15.1 0.0375 12.6 15.00.1 10.9 14.6 10.9 14.8 0.05 12.2 15.1

For the V-based alloys (see Chapter 5), the alloying effect on the ITS wasexplained based on the SEDs model. Hence, for the present Fe-based alloys,we made an attempt to describe and predict the alloying effect on the ITS ofFe-based alloys using the above provided model. We assumed a FM state forfcc Fe and its alloys, since the FM fcc state of Fe was shown to be the near-est maximum of the uniaxial strain energy curve [20, 22, 155, 156]. Figure 7.2displays the correlation between the change of the ITS as a function of con-centration, ∆σm(x) ≡ σm(x) − σm(x0), and the change of the fcc-bcc SED,∆(∆ESED)(x) ≡ ∆ESED(x) − ∆ESED(x0), where x0 is the reference concen-tration. The prefactor 1/Ωbcc is weakly concentration dependent and does notchange the conclusions drawn here. We can see that Al, Mn, and Ni decreasethe ITS and also decrease the SED, however, V, Cr, and Co increase the ITS

58

CHAPTER 7. IDEAL TENSILE STRENGTH OF IRON AND IRON-BASED

ALLOYS

11

12

13

14

15

16

0 1 2 3 4 5 6 7 8 9 10 1111

12

13

14

15

16

17

18

a

Stre

ss (G

Pa)

b

Concentration (at.%)

Fe-Al Fe-V Fe-Cr Fe-Mn Fe-Co Fe-Ni

Stre

ss (G

Pa)

Fe-V

Figure 7.1: The ITS of ferromagnetic bcc Fe1−xMx alloys as a function of concen-tration along the fully-relaxed loading path (σm, panel a). The constrainedITSs at a constant-volume deformation (σΩ

m, filled symbols) and with fixed-magnetic moment along the relaxed loading path (σµ

m, open pentagons) areshown in panel b.

but decrease the SED. We also investigated the correlation between σm and∆ESED increasing the concentration of the solute from 5 % to 10 % (x0 = 0.05and x = 0.10), however the result is qualitatively identical to the one depicted inFig. 7.2. From these results, we infer that the correlation between the SEDs andthe ITSs built in the V-based part is of limited use in the present FM Fe-basedalloys.

7.3 Magnetic Effect on the Ideal Tensile Strength 59

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

FeCrFeV

FeNi FeMn

FeCo

FeAl

Stre

ss c

hang

e (G

Pa)

Energy change (mRy)

Figure 7.2: The change of the ITS, ∆σm, versus the change of the fcc-bcc SED,∆(∆ESED), for Fe1−xMx alloys assuming a FM fcc state and for a concen-tration increase from 0 % to 5 %. Data in unshaded areas affirm a correlationbased on Eq. (5.5).

-3 -2 -1 0-5

-4

-3

-2

-1

0

FeCr

FeV

FeNi

FeCo

FeAl FeMn

a

Stre

ss c

hang

e (G

Pa)

Energy change (mRy)

-1.2 -0.9 -0.6 -0.3 0.0 0.3

-1.2

-0.9

-0.6

-0.3

0.0

0.3

FeAl

FeCo

FeCr

FeNi

FeVb

Energy change (mRy)

Stre

ss c

hang

e (G

Pa)

Figure 7.3: The change of the ITS for constant volume, ∆σΩm, versus the change of the

fcc-bcc SED, ∆(∆ESED), for Fe1−xMx alloys assuming a FM fcc state andfor a concentration increase from 0 % to 5 % and from 5 % to 10 %. Data inunshaded areas affirm a correlation between ∆σΩ

m and ∆(∆ESED).

7.3 Magnetic Effect on the Ideal Tensile Strength

Since the quoted SEDs correspond to the same bcc and fcc volumes, we lookinto the ITSs obtained assuming constant-volume deformations, σΩ

m. In theseadditional studies, the volumes along the deformation paths were fixed to therespective bcc equilibrium volumes. The constrained-ITS values, σΩ

m are dis-played in Fig. 7.1b. According to Fig. 7.1b, the constrained ITS decreases withincreasing concentration for all binaries compared to the value for pure Fe. Alu-minum, nickel, and manganese have a stronger effect than the other elements.Comparing σΩ

m (Fig. 7.1b) with the previously computed ITSs along the loading

60

CHAPTER 7. IDEAL TENSILE STRENGTH OF IRON AND IRON-BASED

ALLOYS

0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.302.60

2.62

2.64

2.66

2.68

2.70

2.72

2.74

2.76

2.78

2.80

w (B

ohr)

c/a

Fe

(a)

2.214

2.276

2.338

2.401

2.463

2.525

2.587

2.650

2.712

0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.302.60

2.62

2.64

2.66

2.68

2.70

2.72

2.74

2.76

2.78

2.80

w (B

ohr)

Fe0.9V0.1

(b)

c/a

1.864

1.898

1.933

1.967

2.001

2.036

2.070

2.105

2.139

Figure 7.4: The magnetic moments (µB) as a function of the tetragonal axial ratio(c/a) and the Wigner-Seitz radius (w) for Fe (panel(a)) and Fe0.9V0.1 alloy(panel(b)).

path with fully relaxed geometry (Fig. 7.1a), we realize that the ITS of pureFe is most significantly affected by the constant-volume constraint, i.e., it in-creases from 12.6 GPa to 18.2 GPa. This is to a much lesser extent the case forFe-alloys, for which the ITS enhancements (σΩ

m − σm) are considerably smallerthan for pure Fe. Figure 7.3 displays the correlation between the change of theconstrained-ITS calculated at the constant volume, ∆σΩ

m(x) ≡ σΩm(x) − σΩ

m(x0),and the change of the FM SED ∆(∆ESED)(x). Alloying effects for an increaseof the concentration of the solute from 0 % to 5 % (Fig. 7.3(a)) and for an in-crease from 5 % to 10 % (Fig. 7.3(b)) are displayed. Interestingly, the change ofσΩ

m correlates well with the change of the SED as illustrated in Fig. 7.3.

The observed correlation may be understood by considering magnetism. Fig-ure 7.4 shows the magnetic moments as a function of the tetragonal axial ratio(c/a) and the Wigner-Seitz radius (w) for Fe and Fe0.9V0.1 alloy as a repre-sentative of all investigated binary alloys. We also displayed the correspondinguniaxial deformation path in the range ǫ:(0 - ǫm). According to the contour plots,the magnetic moment increase ∆µ for Fe along the deformation path from ǫ = 0to ǫm is much larger than the one obtained for Fe0.9V0.1. Namely, ∆µ is 0.3µB

for Fe and 0.08µB for Fe0.9V0.1. These numbers should be contrasted with 0.1µB for Fe and 0.04 µB for Fe0.9V0.1, calculated along the constant-volume defor-mation path (in the range ǫΩ:(0 - ǫΩ

m)). This behavior is universal for all presentFe1−xMx alloys. We should note that ǫΩ

m ≈ ǫm.

The comparatively large increase of the magnetic moment in Fe along theuniaxial deformation path with unconstrained volume and the pronounced dif-ference between σΩ

m and σm (5.6 GPa) are related. We recall that both theincrease of the magnetic moment and the difference σΩ

m − σm for Fe-alloys aresmaller than the values for Fe. To substantiate this correlation, we recalcu-lated the ITSs for the previously determined unconstrained strain paths butwith magnetic moments fixed to their ground state (bcc) values, i.e., the mag-netic moments were not allowed to relax self-consistently. The fixed-momentresults σµ

m are shown in Fig. 7.1b (open pentagons). Fixing the magnetic mo-ment increases the computed strength of Fe by 4.8 GPa. However, this increase

7.3 Magnetic Effect on the Ideal Tensile Strength 61

is strongly diminished at higher solute concentrations: being only 0.2 GPa forFe0.9V0.1. Monitoring Fig. 7.1, we observe that the constrained-volume σΩ

m andthe constrained-moment σµ

m results obey very similar trends. Namely, for bothcases the resulting stresses are essentially much larger than σm of Fe (by 4.8GPa for σµ

m and 5.6 GPa for σΩm) but only slightly to moderately larger than

σm of Fe0.9V0.1 (0.2 GPa for the fixed-moment and 1.7 GPa for σΩm).

The previously discussed test calculations for the ITS of Fe with constrainedvolume and fixed-moment were cross-checked with the FPLO scheme. We ob-tained σΩ

m = 17.1 GPa and σµm = 17.2 GPa meaning an increase of 4.1 GPa and

4.2 GPa with respect to the unconstrained value, respectively. We conclude thatfixing the volume but allowing for a relaxation of the magnetic moments pro-duces essentially the same alloying effect than fixing the magnetic moment buttaking into account structural relaxations (Poisson’s effect).

0.0 0.5 1.0 1.5 2.00

10

20

30

40

50

Fe0.9

V0.1

Fe0.975

V0.025

Fe

Fe1-x Vx , =const.

Fe, =const.

Fe0.975M0.025 , =const.

Fe0.95M0.05 , =const.

Fe0.925M0.075 , =const.

Fe0.9M0.1 , =const.

Pmag

(100 2B/Bohr

3)

Figure 7.5: Correlation between the stress change relative to the ITS of Fe (∆ΣΩ/µ)and the magnetic pressure change (∆Pmag) for the constrained-constant vol-ume deformation (filled symbols) and the fixed-magnetic moment along therelaxed loading path (open symbols). The dashed line guides the eye.

The impact of magnetism on the ITS of Fe and the Fe-alloys can be demon-strated if we realize that the above discussed changes of the ITS are associatedwith the magnetic pressure in itinerant magnets. Within the Stoner model, themagnetic pressure, pmag, is estimated by pmag ∝ kµ2/Ω, where k stands for apositive proportionality factor that depends on potential parameters [157, 158].We consider the excess magnetic pressure

∆Pmag ≡ µ2m/Ωm − µ2

0/Ω0, (7.1)

evaluated at the ITS (index ’m’) with respect to the magnetic pressure inthe ground state (index ’0’). Figure 7.5 shows the variation of the two aux-iliary ITSs relative to the unconstrained ITS and normalized to the ITS of Fe

62

CHAPTER 7. IDEAL TENSILE STRENGTH OF IRON AND IRON-BASED

ALLOYS

(∆ΣΩ/µ ≡ (σΩ/µm − σm)/σm(Fe)) versus the excess magnetic pressure. Accord-

ingly, Fe exhibits the largest excess magnetic pressure, which induces a 38 %stress increase when constraining the magnetic moment. For the fixed-volumeITS, ∆Pmag results in a 45 % increase of the ITS of Fe. The important effectof alloying is that the excess magnetic pressure to the ITS is gradually reducedwith the addition of any investigated soluble. It is evident from Fig. 7.5 thatthis effect occurs for the ITSs derived from the constrained-volume and thefixed-magnetic moment. Obviously, the correlation between ∆ΣΩ/µ and ∆Pmag

is almost perfectly linear. The Fe-alloys with 10 % solute concentration alreadypossess approximately zero excess pressure, meaning that there is only a smallcontribution of ∆Pmag to the their ITSs. ∆Σµ converges to zero as ∆Pmag ap-proaches zero (Fe-V data set) while ∆ΣΩ

m remains finite because the involveddeformation paths are different.

0 0.05 0.1 0.15 0.2∆µ (µB)

0.0

0.5

1.0

1.5

2.0

∆Eba

nd (

mR

y)

FeFe

0.9Al

0.1Fe

0.9V

0.1Fe

0.9Cr

0.1Fe

0.95Mn

0.05Fe

0.9Co

0.1Fe

0.9Ni

0.1

Figure 7.6: Band energy cost for an increase of the magnetic moment beyond its equi-librium value based on a rigid band analysis of the respective bcc total DOSs.

7.4 Electronic Structure

Next we investigate why the magnetic moments of the Fe-alloys become ratherinsensitive to volume change compared to pure Fe. We consider the single-particle band energies, e, of the present binaries in a rigid band model. Em-ploying the force theorem [119, 159], we express the energy change by thechange in the band energy, ∆Eband, when the magnetic moment is increasedby ∆µ = µ− µ0 relative to the equilibrium moment, µ0. Accordingly

∆Eband ≡ Eband↑ + Eband↓

=∫ e↑(µ)

eF

(e′ − eF)N↑(e′)de′ +∫ e↓(µ)

eF

(e′ − eF)N↓(e′)de′, (7.2)

7.4 Electronic Structure 63

-0.5 -0.4 -0.3 -0.2 -0.1 0.0-20

-10

0

10

20

30

minority

majorityFM bcc Fe

total t

2g

eg

DO

S (s

tate

s/R

y)

Energy (Ry)Figure 7.7: The total and partial (t2g and eg) density of states for FM bcc Fe at

equilibrium volume. The vertical dashed line indicates the Fermi level. TheDOSs of the majority (minority) spin channel have positive (negative) sign.

is the band energy change produced by transferring (µ − µ0)/(2µB) electronsfrom the minority-spin band, N↓(e), below the Fermi level, eF, to the majority-spin band, N↑(e), above eF. All required quantities were found from the calcu-lated spin-polarized electronic DOS at the corresponding bcc equilibrium vol-umes of all alloys.

Figure 7.6 illustrates that the band energy cost for an increase of µ is lower forbcc Fe than for the present binaries with x = 0.1 (x = 0.05 in the case of Mn).We notice that the same trend was found for the bct lattice as well. Therefore,an increase of the magnetic moment, e.g., as a result of a lattice distortions,is energetically much more favorable in pure Fe than in Fe alloys. For pureFe, the contribution of N↓ to ∆Eband is approximately twice as large as thecontribution of N↑. That is because the Fermi level in bcc Fe sits at the bottomof the pseudo gap in the minority DOS (where N↓ is small) and at the shoulderof the majority-spin band dominated by t2g↑ states (where N↑ is relatively high),which is shown in Fig. 7.7. In the Al, Co, and Ni containing binaries, alloyingincreases mainly the impact of Eband↑ to ∆Eband exceeding significantly thenearly concentration-independent contribution of Eband↓. With V, Cr, or Mnaddition, alloying still increases Eband↑ much more significantly than Eband↓ butthe main contribution to the increase of the band energy remains the minority-spin band, like in pure Fe.

The common denominator for the alloying-induced increase of Eband↑ is basi-cally due to a downshift of the Fe host states in the majority-spin band, whichcan be seen in Fig. 7.8 for the present binaries with x = 0.1 (x = 0.05 in thecase of Mn). Figure 7.9 illustrates the lowering of the edge of the t2g↑ shoul-

64

CHAPTER 7. IDEAL TENSILE STRENGTH OF IRON AND IRON-BASED

ALLOYS

-1501530

DOS (1/Ry)Fe Fe

Cr

-15

01530

DOS (1/Ry)

FeC

o

-1501530

DOS (1/Ry)

FeV

-15

01530

DOS (1/Ry)

FeA

l

-0.5

-0.4

-0.3

-0.2

-0.1

0en

ergy

(R

y)

-1501530

DOS (1/Ry)

FeN

i

-0.5

-0.4

-0.3

-0.2

-0.1

0en

ergy

(R

y)

-15

01530

DOS (1/Ry)

FeM

n

Figure 7.8: The total density of states for Fe and all Fe-based alloys (10% solute con-centration, 5% in the case of Mn) at their equilibrium volumes. The verticaldashed line indicates the Fermi level. The DOS of Fe is shown in all panels toease the comparison. The DOSs of the majority (minority) spin channel havepositive (negative) sign.

7.5 Importance 65

Al V Cr Mn Fe Co Ni-25

-20

-15

-10

-5

0

5

e t 2g -

eF (

mR

y)

Figure 7.9: The position of the t2g↑ shoulder in the majority-spin band relative to theFermi energy as extracted from the equilibrium DOSs.

der with respect to the Fermi level2. While the edge of the t2g↑ shoulder islocated just above eF in Fe, it is pushed below eF for all binaries, indicating theopening of a small Stoner pseudo gap.3 The associated microscopic mechanismin the case of Al, V, Cr, and Mn doping originates from the hybridization ofthe solute states with the unoccupied s and p bands of Fe [160]. In the caseof Co and Ni, the solutes fill up mainly the majority-spin band of Fe. Bothmechanisms lower N↑(eF), decrease the magnetic susceptibility4 and make theferromagnetism stronger relative to pure Fe.

7.5 Importance

From the above investigations, we highlight some important findings. Under[001] tension, ferromagnetic Fe fails by cleavage of the (001) atomic planes atan attainable strength 12.6 GPa, which is about 55% lower than those of theother bcc metals with the same failure mode (Mo and W) [69, 70]. However,by calculating auxiliary ideal strengths assuming constant volume deformation(σΩ

m) and fixed-spin moment calculations (σµm) along the unconstrained deforma-

tion path, it is found that the ITS increases by about 45% and 38%, respectively.This indicates that the low ITS of Fe originates from its weak ferromagnetic na-ture. If Fe were a strong ferromagnet, it would possess an ideal tensile strengthof ∼ 18 GPa.

2The distance between the inflection point of t2g↑ shoulder and the Fermi level.3The energy difference separating the top of the completely filled d-majority spin channel

and the Fermi energy for strong elemental ferromagnets is called Stoner gap [97]. In analogy,the energy difference separating the top of the Fe d-majority spin channel in the dilute Fe-alloys and the Fermi level may be called pseudo Stoner gap—pseudo since the DOS from thesolute atoms at the Fermi level and above is finite (in both spin channels).

4See discussion in Ref. [97], Chapter 8. The inverse susceptibility 1/χ of a ferromagnet isproportional to (1/N↑(e↑) + 1/N↓(e↓)).

66

CHAPTER 7. IDEAL TENSILE STRENGTH OF IRON AND IRON-BASED

ALLOYS

According to the ITSs of Fe-based alloy, we show that the ITS of Fe canbe sensitively tuned by adding a small amount of simple or transition metals.We predict that vanadium produces the largest strength enhancement amongthe common substitutional dopants used in steels. Based on the analysis ofalloying-induced effects on the peculiar electronic structure of Fe, we find thatthe solutes Al, V, Cr, Mn, Co, and Ni alter the magnetic response of the Fe hostduring tension from the weak towards a stronger ferromagnetic behavior. Themost important finding is that in contrast to the commonly accepted scenariobased on the phenomenological Slater-Pauling curve, our results give an evidencethat a small amount of early 3d metals can also enhance the stability of theferromagnetic order in Fe.

Chapter 8

Thermal Effect on the IdealTensile Strength of Iron

Nowadays, DFT is a very powerful theoretical tool, which leads to an excellentparameter-free descriptions of ground-state properties of bulk metals, orderedand disordered alloys, and thin films in many cases. However, it is a big challengeto investigate the thermal properties of these systems at finite temperature.The ideal strength of elements has been widely studied at 0 K based on DFT.However, it is largely unknown how this intrinsic property of solids behaves atfinite temperature. Extending these investigations to finite temperatures willbroaden the possibility for assessing the impact of the ITS in finite temperaturemechanical properties of micro/nano-scale materials. In this chapter, we predictthe temperature effect on the ITS of bcc Fe from 0 K to the Curie temperature(TC). To model the temperature effect, here, we take into account three basiccontributions arising from phononic, electronic and magnetic degrees of freedom.

8.1 Methodology for Magnetic Contribution

We modeled the effect of thermal magnetic disorder on the total energy bymeans of the partially disordered local moment (PDLM) method (see Chap-ter 2). The PDLM approach describes the energetics underlying the loss ofmagnetic order. Connection to the temperature is provided through the magne-tization curve which maps the computed m to T . It is important to realize thatfor Fe and Fe-based alloys the thermal spin dynamics embodied in the shape ofm(τ) and in the TC value change under the presently applied uniaxial loading.The computational details for TC and m(τ) are given in Chapter 4.

To assess the reliability of our computational approach, we calculated s, TC,and D for bcc Fe and Fe90Co10 alloy. From Table 8.1, we can see that the com-puted D and TC of bcc Fe agree well with the previously published theoreticalresults, although the theoretical values for D are with the exception of Ref. [161]systematically smaller than the measured ones. The presently derived shape pa-rameter for Fe (0.42) is in line with a previous assessment (0.41, Ref. [111]), butboth theoretical assessments are larger than the value fitted to the experimen-tal magnetization curve (0.35(2) [112]). This discrepancy is mainly due to the

68

CHAPTER 8. THERMAL EFFECT ON THE IDEAL TENSILE STRENGTH

OF IRON

Table 8.1: Magnetic quantities used to compute the shape parameter s of the bccphases of Fe and Fe-Co at the theoretical equilibrium volume compared to theavailable experimental and theoretical data.

M0 (kG) g D (meV Å2) TC (K) s

Fe 1.85 2.091 244 1066 0.42220-2872 1015-12703 0.414

1.775 280-3306 10437 0.35(2)8

Fe0.9Co0.1 1.91 2.081 285 1286 0.4311489

1 Experiment; Ref. [113].2 Refs. [112, 161] and references therein.3 Ref. [101] and references therein.4 Ref. [111].5 Experiment; Ref. [162].6 Experiment; Refs. [163–166].7 Experiment; Ref. [167, 168].8 Fit to experimental magnetization curve; Ref. [112].9 Experiment; Ref. [169].

underestimated D. The present TC for Fe90Co10 agrees well with experimentalvalues.

8.2 Methodology for Phonon Contribution and

Electron Excitation

8.2.1 Thermal Expansion

In order to take into account the effect of thermal expansion, the ground-statebcc lattice parameter at temperature T was obtained by rescaling the theoret-ical equilibrium lattice parameter with the experimentally determined thermalexpansion γ(T ) = aexpt(T )/aexpt(T = 0 K). The rescaled bcc lattice parameterat a certain T , a0

T , equals atheo(T = 0 K) · γ(T ) with corresponding volume V 0T .

Figure 8.1 shows the experimental lattice parameters of the bcc phases of Feas a function of temperature, which were used to obtain γ(T ). The expandedvolume can be interpreted as being stabilized by a hydrostatic stress σT . TheσT was derived from the partial derivative of the bcc total energy E with respectto the volume V taken at the expanded lattice parameter corresponding to T ,1

σT =∂E(V )∂V

∣

∣

∣

∣

V 0

T

. (8.1)

1For a hydrostatic pressure, we have (σij ) = diag(−p, −p, −p) and p = −13

Tr (σij). Inanalogy, we refer to a hydrostatic stress as σT = −p and (σij ) = diag(σT , σT , σT ).

8.2 Methodology for Phonon Contribution and Electron Excitation 69

The unstrained state for a given T is defined with respect to V 0T (a0

T =c0T ). For

a chosen strain ǫ, the distorted structure is bct with cT (ǫ) = (1 + ǫ)c0T along the

direction of tension. aT (ǫ) relaxes, so that the associated stress perpendicularto [001], σ⊥, fulfills

σ⊥ =1

aT cT (ǫ)∂E(aT (ǫ), cT (ǫ))

∂aT (ǫ)

∣

∣

∣

∣

aT (ǫ⊥)

≡ σT . (8.2)

0 500 1000 1500 2000Temperature T (K)

2.86

2.88

2.90

2.92

2.94

Lat

tice

par

amet

er a

bcc

(Å

)

TC

Tm

ferromagneticparamagnetic

Figure 8.1: Experimental lattice parameters of the bcc phases of Fe as a function oftemperature. TC and Tm indicate the Curie temperature and the meltingtemperature, respectively. Data were taken from Refs. [167, 168].

The strain energy E(ǫ) for a series of cT (ǫ) was computed and the stress-straincurve is derived from the following equation,

σ(ε) =1 + ǫ

VT (ǫ)∂E(ǫ)∂ǫ

− σT , (8.3)

where VT (ǫ) = cT (ǫ)a2T (ǫ⊥)/2.

8.2.2 Lattice Vibration

To estimate the effect of explicit lattice vibrations on the ITS, we computedthe vibrational free energy as a function of strain in the vicinity of the stressmaximum within the Debye model [59] employing an effective Debye tempera-ture ΘD which was determined from the bulk modulus B and the Wigner-Seitzradius r [61]. The Debye temperature is estimated from the following form,

ΘD = 41.63(

rB

M

)1/2

ΘD in units of K, (8.4)

here M is the atomic mass, B is obtained at evaluated r (the units are kbarand a.u. for B and r, respectively). With the obtained ΘD, the vibrational free

70

CHAPTER 8. THERMAL EFFECT ON THE IDEAL TENSILE STRENGTH

OF IRON

energy per atom in the Debye model can be described as,

Fvibr = Evibr − TSvibr (8.5)

= −kBT

[

3(ΘD/T )3

∫ ΘD/T

0

x3

ex − 1dx− 3 ln

(

1 − e−ΘD/T)

− 98

ΘD

T

]

. (8.6)

Based on the above equation, we can obtain the vibrational free energy changeat strains near the maximum strain. Taking into account this vibrational freeenergy change together with the energy change without vibrational contribution,we can get the relative energy slope change. From the relative energy slopechange, we can estimate the stress change due to lattice vibrations.

8.2.3 Electron Excitation

The temperature-induced contribution of electronic excitations to the ITS wasconsidered by smearing the density of states with the Fermi-Dirac distribu-tion [170].

8.3 Prediction of the Temperature-dependent IdealTensile Strength of Iron

The change in the ITS due to electronic excitations was found to be less than1 % at ∼ 1000 K. The computed explicit phonon contribution decreases theITS by approximately 5% at high temperature (664 K) and leads to about 1 %change at lower temperature (362 K). Both of the above effects are substantiallylower than those due to the magnetic disorder and thermal expansion. Thus,we mainly focus on the magnetic disorder and thermal expansion contributionsin the following part. The temperature effect on the ITS (σm(T )) is shown inFig. 8.2. At 0 K, the ITS of Fe amounts to 12.6 GPa. It remains nearly constantup to 350 K, and decreases then by ∼ 8 % at temperature ∼ 500 K comparedto the 0 K strength. The ITS drops significantly by ∼ 90 % between ∼ 500 Kand ∼ 920 K. At temperatures above ∼ 1000 K, Fe can resist a maximum tensilestrength of ∼ 1.0 GPa. We found that Fe maintains bct symmetry during thetension process and eventually fails by cleavage of the (001) planes [20]. InFig. 8.2, we also shows the ITS of Fe considering merely the thermal magneticdisorder effect (dashed line, open diamonds). Comparing these data to the fullITS (solid line and circles), it can be inferred that the main trend of σm(T ) isgoverned by the magnetic disorder term.

To understand the onset of the significant drop of the ITS from Fig. 8.2,we turn to analyze the Curie temperature of the distorted bct Fe. Figure 8.3displays the Curie temperature as a function of the bct lattice parameters. Onecan see that the TC of bct Fe is strongly reduced compared to that of bcc Fe(black asterisk). From this observation, one may expect that the significance ofthe lattice-distortion-induced reduction of TC compared to bcc Fe is primarily

8.3 Prediction of the Temperature-dependent Ideal Tensile Strength of Iron 71

0 200 400 600 800 10000.0

0.2

0.4

0.6

0.8

1.0

1.2

300 400 500 600 700 800 9000.51.01.52.02.53.03.54.04.5

Max. strength Avg. strength

Ten

sile

Str

engt

h (G

Pa)

Temperature (K)

T0C

Fe Fe (magnetic effect) Fe (constant TC)

Fe0.9Co0.1

Nom

arliz

ed IT

S

Temperature (K)Figure 8.2: The ITS of bcc Fe in tension along the [001] direction as a function of

temperature (σm(T ), solid line and circles); the ITS taking into account onlymagnetic disorder (dashed line, open diamonds); the ITS taking into accountmagnetic disorder and neglecting the change of TC with structural deforma-tion (σc

m, dotted line, open triangles). The Fe0.9Co0.1 alloy (stars) possessesa slightly larger ITS at 0 K, but compared to pure Fe the ITS drastically in-creases at high temperatures. All data are normalized to the ITS of Fe at 0 K(12.6 GPa).

responsible for the onset of the drop of the ITS at ∼ 500 K. To confirm thisexpectation, we investigated the ITS (σc

m(T )) as a function of temperature witha constant TC for bct Fe. We chose without loss of generality the TC of bcc Federived from EMTO (1066 K). The magnetic disorder effect on σc

m(T ) is shownin Fig. 8.2 (dotted line, open triangles). Clearly see from Fig. 8.2, the dropof the ITS is shifted to higher temperatures for σc

m(T ) compared to σm(T ).Hence, this auxiliary ITS calculation indicates that indeed the significant dropof the ITS from ∼ 500 K to ∼ 920 K is due to the strong reduction of TC. Inother words, lattice deformation destabilizes the magnetic order which leads toan unexpected softening of the lattice already at temperatures far below TC ofbcc Fe. Figure 8.4 displays the sketch of local spins for the unstrained parentlattice and for the strained lattice at high temperature, illustrating the magneticdisorder increase with tensile strain.

We have mentioned that for Fe the thermal spin dynamics embodied in theshape of m(τ) [related to the shape parameter s (see Chapter 4)] and in the TC

value change under structural deformation. Above, we have studied the ITS asa function of temperature with and without considering the change in TC withstructural distortion, while in the following we access the effect of s on the ITSof Fe at evaluated temperatures. Figure 8.5 shows the comparison of the ITSwhere the change of s with structural deformation is considered (solid circles and

72

CHAPTER 8. THERMAL EFFECT ON THE IDEAL TENSILE STRENGTH

OF IRON

2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.302.70

2.72

2.74

2.76

2.78

2.80

2.82

2.84

V=const.

TC(K)

9201008

869

664

515 478

362

249

0

Lat

tice

para

met

er a

(Å)

Lattice parameter c (Å)

707

779

851

922

994

1066

Figure 8.3: Contour plot of TC of bct Fe as a function of the lattice parameters a andc. The region where TC is shown is confined by the bcc ground state (blackasterisk) and the failure points associated with σm(T ) (stars, correspondingnumbers denoting T ). The ground state bcc structure possesses the highestcalculated TC in the region shown. The dashed line represents the hyperbolaof constant volume equal to the bcc equilibrium volume.

open diamonds, respectively) and not considered (open circles). Clearly seenthat in contrast to TC, the shape parameter s of bct Fe does not significantlydetermine σm(T ). The reason is that although the individual quantities TC andspin-wave stiffness D depend strongly on tetragonality and volume, both followthe same characteristic trend and their combined effect on s largely cancels out(s ∝ M−1

0 (TC/D)3/2). We found that M0 depends rather weakly on both c/aand volume.

8.4 Comparison of the Ideal Tensile Strength

in Ferromagnetic State and ParamagneticState

Now, one may wonder why the ITS of PM bcc Fe is so much lower than thatof FM bcc Fe, see Fig. 8.2. In the following part, we will answer this questionbased on the electronic structure. It is known that nonmagnetic (NM) bcc Fe isunstable. Bcc iron is susceptible to the formation of sizable magnetic momentsboth in the FM phase (2.2µB) and in the disordered PM phase (2.1µB), butthe mechanisms that stabilize the PM and FM magnetic phases are different.In Chapter 4, we plotted the density of states (DOS) of NM bcc Fe and alsocalculated the Stoner criterion (Ig+/−(0) = 1.56 > 1). We concluded that thepronounced peak in the NM DOS at the Fermi level drives the onset of FM

8.4 Comparison of the Ideal Tensile Strength in Ferromagnetic State and

Paramagnetic State 73

Figure 8.4: The sketch of local spins for the unstrained parent lattice and for thestrained lattice at high temperature, illustrating the magnetic disorder increasewith tensile strain.

order. Accordingly, the majority spin channel and the minority spin channelpopulate and depopulate, respectively, thereby causing an exchange split andmoving the peak away from the Fermi level. The DOS of FM Fe at equilibriumhas been shown in Fig. 7.7, from which one can see that EF is located near thebottom of the pseudo gap at approximately half band filling in the minoritychannel. This explains the pronounced stability of FM bcc Fe reflected by adeep minimum in the energy versus a and c map 8.6(a). The depth of thisenergy minimum in connection to the ITS is best characterized by the totalenergy cost to reach the ideal strain (ǫm) from the unstrained bcc phase (strainenergy), which amounts to 0.426 mRy/atom per % strain.

For the PM state, we plotted the constant-volume Bain path of PM bctFe for various fixed values of the local magnetic moment (µ), which is dis-played in Fig. 8.7(a). We can see that the bcc structure is gradually stabilizedwith increasing µ, and a shallow minimum is formed on the tetragonal energycurve for the self-consistent value of the local magnetic moment (µ0 = 2.1µB).We also calculated the total energy change for an elastic distortion of the bccstructure corresponding to a small volume-preserving tetragonal deformation[∆E(δ) = E(δ)−E(0)] for the corresponding µ values, see the inset of Fig. 8.7(a).One can see that for NM Fe, ∆E(δ) is negative, but increasing µ turns ∆E(δ)positive signaling the mechanical stabilization of PM bcc Fe upon PM momentformation. This result reveals that the mechanical stabilization of bcc PM Fedue to local magnetic moment formation, embodied in the trend of ∆E(δ) as afunction of µ, is in fact determined by the kinetic energy (solid symbols) andthus by the details of the electronic structure. The above analyzes indicate thatthe formation of local magnetic moments in the PM state stabilizes the bccstructure with respect to the NM bcc lattice as shown in Fig. 8.7(a). Differentfrom the FM state, in the PM state, the local moment formation is due to a

74

CHAPTER 8. THERMAL EFFECT ON THE IDEAL TENSILE STRENGTH

OF IRON

0 200 400 600 800 10000.0

0.2

0.4

0.6

0.8

1.0

1.2

T0C

Fe Fe (magnetic effect) Fe (constant s)

Nom

arliz

ed IT

S

Temperature (K)Figure 8.5: Comparison between the full and magnetic ITSs (solid circles and open

diamonds, respectively) and the ITS obtained by omitting the structure de-pendence of "s" (open circles). All data are normalized to the ITS of Fe at 0 K(12.6 GPa)

split-band mechanism for which the average exchange field is zero [56, 171]. Thedisorder-broadened DOSs of PM Fe (Fig. 8.7(b)) reveal that the formation oflocal magnetic moments effectively removes the peak at EF seen for NM Fe.However, this disorder mechanism does not yield a comparably large cohesiveenergy increase as for the FM case 8.6(b). The strain energy cost to reach ǫm

starting from the shallow energy minimum equals 0.024 mRy/atom per % strainin the PM phase, which is approximately 18 times smaller than in the FM case.Hence, the significantly lower ITS of PM Fe compared to that of FM Fe arisesfrom the differences in the magnetism-driven stabilization mechanisms and thecorresponding energy minima within the Bain configurational space.

8.5 Alloying Effect of Cobalt

In Section 7.2, we have investigated the alloying effect on the ITS of Fe at 0 K.We found that the ITS can be sensitively turned by alloying. Here, we study howthe ITS of Fe at evaluated temperatures changes with Co addition. Followingthe same procedure as described for Fe, we computed the ITS of the Fe0.9Co0.1

alloy in the high-temperature interval between 496 K and 1000 K, the data areshown in Fig. 8.5 (stars). At temperatures below 500 K, Fe0.9Co0.1 exhibits a∼ 10 % larger ITS than Fe. However, the strengthening impact of alloying onthe ITS becomes more dramatic in the high-temperature region.

Now, let us explore the physical origin underlying the alloying effect at hightemperatures. The alloying effect of Co on TC is mainly connected to the

8.5 Alloying Effect of Cobalt 75

2.8 2.9 3.0 3.1 3.2 3.32.6

2.7

2.8

2.9

3.0

3.1FM

Lat

tice

para

met

er a

(Å)

Lattice parameter c (Å)

0.0

3.0

5.9

8.9

12

15

2.80 2.82 2.84 2.86 2.88 2.90 2.92 2.94

2.76

2.78

2.80

2.82

2.84

2.86

2.88

2.90PM

Lat

tice

para

met

er a

(Å)

Lattice parameter c (Å)

0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 8.6: Total energies (in mRy) for ferromagnetic (FM) and paramagnetic (PM) Feas a function of lattice parameters a and c. The energies are plotted relative tothe bcc minimum. The gray solid lines show the uniaxial deformation pathsfrom the unstrained state (bcc, marked by red stars) to the correspondingmaximum strain (ǫm) (marked by white stars). Notice the different energyscales for the two panels.

strengthening of the first nearest neighbor exchange interactions (J1) [172, 173].Figure 8.8 shows the influence of Co on the magnetic exchange interaction en-ergy for bcc and bct Fe and Fe0.9Co0.1 (for one representative bct structure,with a = 2.732 Å and c = 3.109 Å). For the bcc phase, we show the influence ofCo on the eight first [J1(8)] and six second [J2(6)] nearest neighbor exchangeinteractions. From Fig. 8.8(a), we can see that J1 between both similar anddissimilar atomic species increases significantly in Fe0.9Co0.1 compared to J1 inpure Fe. At the same time, alloying with Co weakens J2, but this effect is oflesser importance for TC than the strengthening of J1. For bcc Fe and Fe0.9Co0.1

alloy, TCs are 1066K and 1286K, respectively. For the bct phase, we show thealloying effect of Co on J1(8), J2(4), and J3(2) (the two third nearest neighborinteractions), see Fig. 8.8(b). We can see that the influence of Co is a strong in-crease of J1 and a weakening of J2 and J3 similar to the behavior of Co in the bccphase. For the chosen example, the TCs of Fe and Fe0.9Co0.1 amount to 707 Kand 1053 K, respectively, i.e., Co addition results in an increase of TC whichis in magnitude much more pronounced for the bct structure than the impactof Co alloying for the bcc phase. The above results express that Co additionsrestore the magnetic order in the strained lattice (both the magnetic momentand exchange interactions are larger and more stable in the Fe-Co alloy thanin pure Fe upon structural perturbation), push the critical temperature for thestrength-softening scenario in Fe towards the magnetic transition temperatureof the undeformed lattice, which leads to a surprisingly large alloying-drivenstrengthening effect at high temperatures.

Brenner measured systematically the tensile strength of bcc Fe whiskers anddetermined its temperature dependence [82, 83, 174], which is shown in Fig. 8.9.It was found that the maximum tensile strength of [001]-oriented Fe whiskersdrops by about 67 %, and the average tensile strength drops by 60 % as the tem-perature increases from 298 K to 823 K. The failure of whiskers with diameter

76

CHAPTER 8. THERMAL EFFECT ON THE IDEAL TENSILE STRENGTH

OF IRON

0.92 1.00 1.08 1.16 1.24

0

2

4

6

8

10

12

14

16

18

20

22

24

bcc

(a)

(NM)

E-E

0 (m

Ry)

c/a

0.0 0.5 1.0 1.5 2.0-6

-4

-2

0

2

E (

) (ar

b. u

nits

)( )

-0.4 -0.2 0.0 0.20

10

20

30

40

50

60(b)

0 (NM)

1.0 1.5 1.9 2.1

DO

S (s

tate

s/R

y)Energy-EF (Ry)

Figure 8.7: (a) Total energy of PM bct Fe for various values of the local magneticmoment. The energies are plotted with respect to the energy of the PM bccstate, and the volume is fixed to that of PM bcc Fe. The inset displays the totalenergy change (open symbols) and the kinetic energy (solid symbols) changecorresponding to a small volume-preserving tetragonal shear (arbitrary units).(b) Total DOS of PM bcc Fe for different local magnetic moments.

0.0

0.5

1.0

1.5

2.0

2.5

(a)

bcc Fe0.9

Co0.1

bcc Fe

Co-CoFe-CoFe-Fe

Exc

hang

e in

tera

ctio

n (m

Ry)

Fe-Fe

J1 (8)

J2 (6)

0.0

0.5

1.0

1.5

2.0

2.5

(b)

Co-CoFe-Fe Fe-CoFe-Fe

bct Fe0.9

Co0.1bct Fe

Exc

hang

e in

tera

ctio

n (m

Ry) J1 (8)

J2 (4)

J3 (2)

Figure 8.8: (a) and (b) The exchange interactions J1 and J2 of bcc Fe and Fe0.9Co0.1

and J1-J3 of one representative bct structure (a = 2.732 Å, c = 3.109 Å).

< 6µm were reported to occur without appreciable plastic deformation. More-over, the observed temperature gradient of the measured strength is strongerthan the one assuming only the thermally-activated nucleation of dislocations atlocal defects [174]. On the other hand, the present results show a strong temper-ature gradient (mainly due to magnetic disorder) that is nicely reflected in theexperimental data. The qualitative difference in the theoretical and experimen-tal temperature dependence may be ascribed to the low strain-rate inherent inexperiment, which allows for the activation of various mechanisms (dislocation

8.5 Alloying Effect of Cobalt 77

300 400 500 600 700 800 9000.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Max. strength Avg. strength

Ten

sile

Str

engt

h (G

Pa)

Temperature (K)Figure 8.9: Tensile strength of Fe whiskers under [001] oriented from experimental

measurement.

activation, nucleation, etc.) [175–177]. We suggest to measure the ITS of Fe-Cowhiskers at elevated temperatures to investigate the here predicted differencein the ITS between Fe and Fe0.9Co0.1. The output of such experiments couldalso be used to verify the connection between the ITS and the measured tensilestrength of single-crystalline whiskers.

Chapter 9

Concluding Remarks andOutlook

In this thesis using the exact muffin-tin orbitals method (EMTO) in combina-tion with the coherent-potential approximation (CPA), we have investigated theelastic properties of V-based alloys as a function of Ti and Cr concentration. Thetwo cubic shear elastic constants (C′and C44) follow opposite alloying trends,which results in a local maximum in the polycrystalline shear and Young’s mod-uli for the equi-concentration alloys. The anomalous composition dependence ofC44 has been explained using the electronic structure of bcc V. Using the phe-nomenological correlation by Pugh, the present B/G values indicate excellentductility for all V-based alloys considered here. The addition of Cr increases thebulk modulus and the ductility of V-based alloys. Both Cr and Ti enhance thesolid solution hardening (SSH) of V-based alloys but Ti has larger strengthen-ing effect than Cr. The SSH of the ternary alloys increases with increasing thetotal Cr and Ti concentration. We have used our theoretical data to explain thechanges of ductile-brittle transition temperature of V-Cr-Ti alloys as a functionof Cr and Ti concentration.

The ideal tensile strength (ITS) of pure V and Fe has been calculated by theEMTO method. Our results are in good agreement with the available literaturecalculations and confirm that our methodology has the accuracy needed for suchcalculations. In the case of V-based alloys, we have obtained that Cr additionincreases and Ti addition decreases the ITS of bcc V in all three investigatedcrystallographic directions ([001], [110] and [111]). As a consequence, the ITS ofV-Cr-Ti alloys remains virtually unchanged if equal amounts of Cr and Ti areintroduced into the V host. Comparing the ITSs along the [001], [111] and [110]directions, we identify the [001] direction as the weakest one. Along the [001]tension, there is an orthorhombic branching, which significantly decrease theITSs of V and V-based alloys compared to the primary tetragonal deformation.We have shown that the observed alloying effects on the ITS can be understoodon the basis of band filling arguments and structural energy differences.

We have investigated the ITS of four single bcc-phase HEAs: ZrNbHf, ZrVT-iNb, ZrTiNbHf, and ZrVTiNbHf. We have found that the ITS increases withdecreasing the ratio of the fourth group elements to fifth group elements. Thealloying effect on the ITS of HEAs is explained by the d-band filling. In light

80 CHAPTER 9. CONCLUDING REMARKS AND OUTLOOK

of the recent attempts connecting the intrinsic materials properties to the ex-trinsic ductility [178], the present progress offers a guide line to design ductilerefractory HEAs with carefully tailored strength level.

For the Fe-based alloys, we have demonstrated that alloying Fe with fre-quently utilized solutes (Al, V, Cr, Mn, Co, and Ni) with concentrations up to10 % is an effective mean to alter the intrinsic upper bound of the mechanicalstrength in tension along [001]. The ITS increases with increasing concentrationof Cr, Co, and V and decreases with Ni and Al addition to pure Fe. Manganeseshows a small but non-monotonic alloying behavior. Among the investigatedsolutes, V turns out to be one of the most efficient alloying agents producing anenhancement of the ITS by 2.3 % per atomic percent of V. All binary systemsconsidered here fail by cleavage under [001] loading. We have found that thepresent solutes alter the magnetic response of the Fe host during tension fromthe unsaturated towards a more stable ferromagnetic behavior. The underly-ing driving force originates from the alloying induced effects on the peculiarelectronic structure of Fe.

We have discovered a strong magneto-mechanical softening of iron single-crystals upon tensile loading. We have shown that the strength along the [001]direction persists up to ∼ 500 K, however, diminishes most strongly in the tem-perature interval ∼ 500 - 920 K due to the loss of the net magnetization uponuniaxial strain. The strength in the paramagnetic bcc phase is more than 10times lower than that in the ferromagnetic bcc phase. We have found thatFe fails by cleavage at all investigated temperatures. The strength of Fe athigh temperatures is significantly enhanced by alloying with Co. We expectthat the large magnetic effect on the strength exists for all dilute Fe alloys andthat the magneto-mechanical softening temperature can be sensitively tuned byproper selection of the alloying elements. In particular, solutes that strengthen(weaken) the ferromagnetic order in the tetragonally distorted Fe increase (de-crease) the intrinsic strength of defect-free Fe crystals at temperatures above(below) ∼ 600 K.

Based on these achievements, we conclude that the EMTO-CPA approachprovides an efficient and accurate theoretical tool to design the mechanicalstrength of bcc random solid solutions.

The works presented in this thesis give me a solid background to continue myresearch along the ab initio study of the mechanical properties of metals and al-loys. One open question is the connection between extrinsic ductility/brittlenessand intrinsic ductility/brittleness [73, 178] in bcc systems. Extrinsic ductil-ity/brittleness is related to defects, such as the dislocation network and disloca-tion mobility, grain boundaries, etc. In contrast, intrinsic ductility/brittlenessrefers to the failure mode of an ideal infinitely large crystal without any struc-tural imperfections. The B/G ratio employed in our work is a phenomenolog-ical indicator to characterize the extrinsically ductile/brittle fracture behaviorof polycrystalline metals. A more elaborated measure of the extrinsic ductil-ity/brittlness is given by the Rice theory [179], which involves the surface energyand the unstable stacking fault energy. Although extrinsic ductility is the prop-erty we usually experience in our daily life, one may generally argue that bothintrinsic and extrinsic ductility are important. Recently, a connection betweenextrinsic and intrinsic ductility/brittleness for single elements was attempted byPokluda et al. [178]. Their results indicate that the tendency of an engineering

81

material to brittle or ductile behavior is partially predetermined by intrinsicproperties of the crystal lattice of its basic constituents. In my future study, Iwill look for a simple criterion to predict the intrinsic ductility/brittlness in bccalloys.

Acknowledgement

First, I would like to express my sincere gratitude to my supervisor Prof. LeventeVitos for giving me the opportunity to join the AMP group at KTH and hiscontinuous support during my Ph.D. study and research, his enthusiasm, andfor guiding me with his comprehensive understanding of physics.

Grateful acknowledgement also goes to my co-supervisors Prof. Jijun Zhaoand Prof. Börje Johansson for their ongoing encouragement and professionalguidance. I am also grateful to Dr. Stephan Schönecker for his kind, patient, andprofessional guidance in my study. I would like to thank Prof. László Szunyogh,Dr. Eszter Simon, and Dr. Lars Bergqvist to provide new computational toolsand for helpful discussions. Many thanks to our group members: Andreas,Dongyoo, Erna, Fuyang, Guijiang, Guisheng, Hualei, Liyun, Lorand, Narisu,Ruihuan, Shuo, Song, Wei, and Zhihua for help, discussions, and co-working.Especially, many thanks go to Andreas for providing the nice Swedish abstractand Wei for helping in computation issues.

The Swedish Research Council, the Swedish Foundation for Strategic Re-search, the Swedish Steel Producers’ Association, the European Research Coun-cil, the China Scholarship Council, the Knut och Alice Wallenbergs stiftelse, theHungarian Scientific Research Fund, and the National Magnetic ConfinementFusion Program of China (2011GB108007) are acknowledged for financial sup-port. The computations simulations were performed on resources provided bythe Swedish National Infrastructure for Computing (SNIC) at the National Su-percomputer Centre in Linköping.

Last but not the least, I would like to thank my family, relatives and myfriends.

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