Ab initio modelling of mechanical and elastic properties ... · PDF fileDissertation Ab initio...

Dissertation

Ab initio modellingof mechanical and elastic properties of

solids

Petr Lazar

in fulfillment of the academic degree

Doktor Rerum Naturarum

at the Faculty of Physics, University of Vienna

Vienna, January 2006

Abstract

The aim of the thesis is to study mechanical properties of crystalline materialson the basis of density functional theory (DFT) by applying first-principles or ab

initio techniques. Mechanical properties of materials are of crucial importancefor technological applications. How a material breaks, is -however- still not wellunderstood in many aspects. The results of the thesis should demonstrate that ab

initio calculations can provide fundamental insight into the true, namely atom-istic, mechanisms of fracture. For very small loads material behaves in an elasticmanner, and -consequently- the elastic properties of solids are need to be under-stood and calculated. Therefore, after some introductory remarks discussing theab initio concept in chapter 1 the elastic behaviour of solids and results of theactual calculations of elastic constants are discussed in chapter 2. The main partof the thesis focuses on the mechanisms of fracture at the atomic scale, startingwith brittle fracture as discussed in chapter 3. The ab-initio total energy calcu-lations simulating cleavage of material under tensile loading are introduced anddiscussed in the light of classical theories. Consequently, a long standing questionof materials science about the possible connection between critical cleavage stressand elastic properties is addressed in chapter 4. A concept of localisation of theelastic energy is developed, by which a well defined correlation between cleavageand elastic properties is established, at least for some idealized cases of fracture.This concept is applied to a wide range of materials representing different typesof bonding. The calculated and derived cleavage properties are compared to the(rather scarce) experiments and to data of other theoretical concepts, and thebehaviour of the newly introduced materials parameter -the localisation length-is investigated. Interestingly and surprisingly, for brittle cleavage the results sug-gests that by choosing an average, constant value of the localisation length for-almost- all materials critical cleavage stress can be directly estimated from thecleavage energy and the elastic constants within an error of ± 10%. Such a cor-relation, which is also quantitatively useful, was sought for about 80 years inthe scientific community, and finally established in the present work. Chapter 5deals with ductile fracture. For Nial, the criteria of Rice for dislocation emissionfrom a crack tip and the Peierls-Nabarro model are utilised in order to calculateductility and dislocation properties of various slip systems. The 〈111〉 slips in(110) and (211) planes dominate the ductile behaviour. For the first time, thetension-shear coupling in the slip plane is calculated by an ab initio technique.In chapter 6, the application of the previously discussed models is demonstratedfor the simulation of microalloying effects for NiAl, with the aim for finding animprovement of its ductility, which is very important for technological applica-tions. The achieved results suggest that Cr and in particular Mo are promisingcandidates for improving ductility. The ab initio findings are in excellent correla-tion with experimental observations. The short summary of chapter 7 concludesthe thesis.

Abstract

Das Ziel der Dissertation ist die Untersuchung von mechanischen Eigenschaftenfester Materie mit Hilfe von Ab Initio Methoden, die auf der Dichtefunktionalthe-orie beruhen. Mechanische Eigenschaften von Materialien sind von entscheiden-der Bedeutung fur ihre technolgosche Anwendung. Wie eine Material wirklichbricht, ist immer noch nicht gut verstanden. Die Ergebnisse dieser Arbeit zeigen,daß Ab Initio Berechnungen einen tiefen Einblick in die wirklichen, atomistischenVorgange des Materialbruches geben konnen. Fur sehr kleine Belastungen verhaltsich jedes Material elastisch. Die elastischen Eigenschaften mussen daher berech-net werden konnen. Nach einer kurzen Einleitung uber die Ab Initio Methodik inKapitel 1 werden deshalb die elastischen Eigenschaften fester Materie im Kapi-tel 2 diskutiert. Der Hauptteil der Dissertation befaßt sich mit Bruchvorgangenim atomistischen Bereich, wobei Kapitel 3 mit dem ideal bruchigen Verhaltenbeginnt. Die Ab Initio Gesamtenergien der Rechnungen, die das Spalten einesMaterials unter Zugspannung simulieren, werden in Verbindung mit klassischenTheorien diskutiert. Das seit langem offene Problem eines moglichen Zusam-menhangs zwischen der kritischen Spaltspannung und elastischen Eigenschaftenwird im Kapitel 4 angesprochen. Ein Konzept der Lokalisierung der elastischenEnergie wird entwickelt, durch das eine wohldefinierte Beziehung zwischen Spal-tung und elastischen Eigenschaften eingefuhrt werden kann -zumindest fur einigeidealisierte Falle von Bruchtypen. Dieses Konzept wird auf eine große Klasse vonMaterialien mit verschiedenen Typen von chemischer Bindung angewendet. Diedadurch gewonnenen Spalteigenschaften werden mit experimentellen Daten (vondenen es nur wenige gibt) und anderen theoretischen Ergebnissen verglichen. DasVerhalten des neu eingefuhrten Parameters -der Lokalisierungslange- wird unter-sucht. Interessanterweise und uberraschend stellt sich heraus, das fur den idealbruchigen Bruch diese Lange als konstant angenommen werden kann, unabhangigvom Material und der Richtung der Belastung. Damit kann die kritische Span-nung direkt aus den Spaltenergien und den elastischen Konstanten mit einemFehler von ± 10% bestimmt werden. Nach einer solchen Beziehung, die auchquantitive sinnvolle Resultate liefert, wurde mehr als 80 Jahre lange gesucht.In dieser Arbeit ist sie schließlich aufgestellt worden. Kapitel 5 behandelt duk-tiles Bruchverhalten fur NiAl. Die Kriterien von Rice fur Versetzungsemissionendurch eine Rißspitze und das Peierls-Nabarro Modell werden verwendet, um Duk-tilitat und Versetzungseigenschaften von verschiedenen Gleitsystemen zu berech-nen. Die 〈111〉 Gleitungen in den (110) und (211) Ebenen bestimmen das duktileVerhalten. Zum ersten Mal wurde die Kopplung zwischen Zug- und Scherspan-nungen in einer Gleitebene mit einer Ab Initio Methode berechnet. Im Kapi-tel 6 werden die diskutierten Modelle fur NiAl angewendet, um die Effekte desDazulegierens dritter Elemente zu simulieren, um die Duktilitat zu verbessern,was fur technologische Anwendungen sehr wichtig ist. Die Rechnungen deutendarauf hin, daß Cr und Mo erfolgsversprechende Kandidaten sind. Die Ab Initio

Ergebnisse sind in ausgezeichneter Ubereinstimmung mit experimentellen Daten.Eine kurze Zusammenfassung in Kapitel 7 beendet die Dissertation.

Contents

1 Introduction 5

1.1 Fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Electronic structure methods . . . . . . . . . . . . . . . . . . . . . 10

2 Elastic properties of material 13

2.1 Elastic constants and crystal symmetry . . . . . . . . . . . . . . . 14

2.2 DFT calculation of elastic constants . . . . . . . . . . . . . . . . . 16

2.3 Results for selected materials . . . . . . . . . . . . . . . . . . . . 18

2.4 The ideal strength . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Brittle fracture of material 23

3.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Continuum theory . . . . . . . . . . . . . . . . . . . . . . 23

3.1.2 Stress intensity factors . . . . . . . . . . . . . . . . . . . . 24

3.1.3 Griffith’s thermodynamic balance . . . . . . . . . . . . . . 26

3.1.4 Irwin Theory . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.5 Lattice trapping . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 DFT calculations for brittle fracture . . . . . . . . . . . . . . . . 30

3.2.1 Cleavage decohesion . . . . . . . . . . . . . . . . . . . . . 30

3.2.2 Calculation of cleavage decohesion for ideal brittle fracture 32

3.2.3 Advanced applications of the ideal brittle cleavage concept 34

3.2.4 Relaxed cleavage decohesion . . . . . . . . . . . . . . . . . 35

4 Cleavage and elasticity 39

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Orowan-Gilman model . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Ideal brittle cleavage . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4 Localisation length . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.5 Results for ideal brittle cleavage . . . . . . . . . . . . . . . . . . . 45

1

2 CONTENTS

4.5.1 Computational aspects . . . . . . . . . . . . . . . . . . . . 45

4.5.2 Simple metals . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.5.3 Intermetallic compounds . . . . . . . . . . . . . . . . . . . 50

4.5.4 Refractory compounds . . . . . . . . . . . . . . . . . . . . 54

4.5.5 Ionic compounds . . . . . . . . . . . . . . . . . . . . . . . 56

4.5.6 Diamond and silicon . . . . . . . . . . . . . . . . . . . . . 57

4.5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.6 Relaxed cleavage . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.6.1 Correlation between cleavage and elasticity . . . . . . . . . 62

4.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.7 Semirelaxed cleavage . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5 Ductile fracture 75

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 The concept of unstable stacking fault energy . . . . . . . . . . . 77

5.3 Modifications of Rice’s approach . . . . . . . . . . . . . . . . . . 79

5.4 Dislocations properties . . . . . . . . . . . . . . . . . . . . . . . . 80

5.4.1 Continuum model for dislocations . . . . . . . . . . . . . . 81

5.4.2 Peierls-Nabarro model of a dislocation . . . . . . . . . . . 83

5.4.3 Lejcek’s method . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4.4 Peierls stress of a dislocation . . . . . . . . . . . . . . . . . 87

5.5 Calculation of stacking fault energetics . . . . . . . . . . . . . . . 89

5.5.1 Modelling aspects . . . . . . . . . . . . . . . . . . . . . . . 89

5.5.2 Results - slip properties of NiAl . . . . . . . . . . . . . . . 91

5.5.3 Results - dislocation properties of NiAl . . . . . . . . . . . 94

5.6 Tension-shear coupling . . . . . . . . . . . . . . . . . . . . . . . . 98

5.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.6.2 Model for tensile-shear coupling . . . . . . . . . . . . . . . 99

5.6.3 Combined tension-shear relations . . . . . . . . . . . . . . 100

5.6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 Microalloying of NiAl 111

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2 Fracture properties of alloyed NiAl . . . . . . . . . . . . . . . . . 112

CONTENTS 3

6.3 Computational and modelling aspects . . . . . . . . . . . . . . . 113

6.4 Brittle cleavage . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.5 Slips and Ductility . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.5.1 〈111〉(110) and 〈001〉(110) slips . . . . . . . . . . . . . . . 121

6.5.2 〈001〉(100) slip . . . . . . . . . . . . . . . . . . . . . . . . 126

6.5.3 〈111〉(211) slip . . . . . . . . . . . . . . . . . . . . . . . . 126

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7 Summary 131

A Publications 145

B Conference contributions 147

C Acknowledgments 149

D Curriculum Vitae 151

4 CONTENTS

Chapter 1

Introduction

The mechanical properties of materials are of crucial importance for technological

applications. Processing and usage of metals became a central factor of human

civilization, and in particular iron and steel have become indispensable materials

for many purposes. Their applications range from tools, screws, nails etc. to

the objects as large as a ship or a gas transmission line. Another, technologi-

cally extremely important class of materials is based on aluminum and its alloys,

which are used as lightweight materials in particular for aerospace industry. The

crucial role of the mechanical properties of all these materials is obvious. Many of

these objects and materials are subject to large forces and stresses, and their me-

chanical failure can be disastrous. However, until now, efforts in understanding

mechanical properties of materials have been based mainly on phenomenological

and empirical concepts and approaches.

Materials science (at least its scientific version) on the other hand aims to

explain the macroscopic properties of solids on the basis of their microstructure.

In general, this very broad field includes physics and chemistry combined with

metallurgy and mechanical engineering. Following the spirit of materials sci-

ence in combination with fundamental research, the present work tries to link

the interactions between atoms modelled by concepts of quantum physics to the

macroscopic mechanical behaviour of materials.

The most elementary -but very important- mechanical property is elasticity.

It describes the response of a solid material to a (very) small loading which causes

reversible deformations. The fundamental material parameters which character-

ize the elastic behaviour of the solid are the elastic constants. This subject will

be elaborated in chapter 2.

When the stress induced by some external load is increased beyond the elastic

limit, ductile materials undergo a plastic deformation, which is permanent and

irreversible. In crystalline materials, the most important plastic deformation is

5

6 CHAPTER 1. INTRODUCTION

realized by slips of crystallographic planes which might be carried through by

motions of dislocations. The stress for which the elastic limit is exceeded and

plastic deformation begins is called the yield stress. By applying further stress the

material may suffer fracture and break down. Materials characterized as being

ductile can suffer large plastic deformations before they finally break, whereas

brittle materials fail at a much earlier stage. The fracture of brittle materials

is elaborated in chapter 3 and the relation of cleavage and elastic properties of

ideal brittle material is the subject of chapter 4. The categorization of the frac-

ture behaviour of a material is not strict, because many materials (for example,

aluminum) undergo a brittle-to-ductile transition at elevated temperatures. The

mechanisms underlying intrinsically ductile or brittle behaviour of materials are

the subject of chapter 5.

In chapter 6 the application of the models in computational alloy design is

demonstrated for the simulation of microalloying of NiAl, in a survey for the

improvement of its intrinsic ductility. Such a simulation fully exploits the DFT

method, because the change of the electronic structure and bonding of alloyed

interfaces cannot be reasonably described by any of empirical or semiempirical

methods.

1.1 Fracture mechanics

Whereas elastic properties are well studied, both experimentally and theoretically,

the fracture process in solid materials still remains unclear in many aspects.

Fracture is a process by which the material breaks into two or more parts. In

most cases it involves nucleation and a propagation of cracks. Cracks and their

behaviour in the material are not only important for large external loads acting

on the atomistic structure of the material. Crack formation can cause failure

of major structures which are subjected only to relatively moderate loads, e.g.

structures such as a storage tank, a gas transmission line, or an aircraft. In

such cases, cracks usually start in a large time scale at surface defects, and their

slow growth is further aided by chemical effects such as corrosion. When the

external load reaches a certain critical limit, then the crack begins to propagate

on a much faster time scale, and the structure suddenly breaks. A detailed

description of some of catastrophic failures in modern history may be found in

the literature [1, 2].

Therefore, the key questions of fracture mechanics are: when will a crack

nucleate and under what circumstances (i.e. external stresses acting on the ma-

terial) will an already existing crack propagate? The major obstacles are due

to the fact that the length scales relevant for fracture span from macroscopic

1.1. FRACTURE MECHANICS 7

dimensions to the atomistic length scale of chemical bonds between the atoms,

spanning several length scales in between that are associated with, for example,

particles, grains or dislocations. On all these scales the total fracture energy

might be accumulated.

The materials science approach for understanding fracture emphasizes a de-

scription of basic physical processes underlying the fracture of the material. These

processes are material dependent and again spread over several length scales. This

represents the major obstacle for the describing the mechanical response of ma-

terial to external loading. As a consequence, common models focus only at the

physical properties relevant for a specific length scale.

In the macroscopic approach it is usually assumed that the material -

represented usually by a linearly elastic, often isotropic continuum- contains

cracks and the influence of the crack geometry and external load on the process

of fracture is studied. Such models helped to enlighten the discrepancy between

theoretical and observed strength of materials, revealing that due to a stress con-

centration at a crack tip the cohesive strength of the material may be reached

at already moderate loads. Continuum modelling was successfully used in the

engineering approach to fracture mechanics, where the interest lies in the design

of fracture resistant components and structures. However, because the material

is usually treated as an homogeneous continuum the influence of the interaction

between the atoms, comprising the chemistry discreteness and anisotropy of a

solid material cannot be taken into account.

Models at the atomistic level deal with local effects by focusing on the inter-

action of atoms in the immediate vicinity of the crack. Three approaches might

be distinguished: (1) larger mesoscopic-scale methods combining an atomistic

treatment of regions near the crack tip with continuum linear elastic solutions

at larger distances from the tip; (2) atomistic simulations of crack formation

by simplified bond models; (3) accurate ab initio approaches which describe the

bonding between atoms free from any parameters.

The combined models (1) may implement in principle the atomistic structure

into continuum models and the may provide a reasonable description of the crack

behaviour taking into account both, the shape of the crack tip and the character of

the bonding, for which in the last decade the embedded-atom method (EAM) [3]

was frequently applied. Although - in principle - EAM refers to the atomistic

scale, it nevertheless involves several limitations: the model potentials used for

calculations within the atomistic region have to be developed and calibrated for

each material and are known to provide insufficient results for configurations

where atoms are far from the bulk ground state, which is usually used for the

calibration of the atomic potentials. Of course, atoms near the crack tip are


under strong stress and, consequently, and the lattice is strongly distorted from

the bulk equilibrium. Furthermore, these methods lack predictive power, which

is needed to be useful for materials design.

The very powerful and promising approach (2) involves a large number of

atoms (typically 106 or more) interacting as described by model potentials. Be-

cause of the tremendous increase in computing power such many-atoms concepts

are very promising for the future- The success obviously depends -again- on the

development of realistic model of interatomic potentials for which a particularly

large progress was done by the development of bond-order potentials methods [4].

A very recent application for the simulation of brittle cleavage of Ir [5] demon-

strates the power of such large-scale simulations. Nevertheless, these methods

rely on the model potential and, consequently, still some progress has to be made

until predictive power is achieved.

Hence, in order to avoid the ad-hoc choice of atomistic interaction parameters,

in concept (3) ab-initio density functional theory (DFT) [6, 7] methods are

applied. The DFT approaches proved to be of general and predictive nature

for various problems in computational materials science [8]. Truly ab initio DFT

simulations require as an input only positions and atomic numbers of the involved

elements, and they provide accurate descriptions of properties determined from

the electronic structure, which naturally includes all details of the atomic bonds

at the crack tip. However, they are quite demanding for computational resources

and consequently limited to relatively small (in the order of hundreds) number

of atoms. Thus, these methods are used to simulate processes which lie on the

scale of a small number of atoms, or to obtain parameters required for a large-

scale modelling of materials properties. For example, results of brittle cleavage

calculations might provide an input for the cohesive zone models or the γ-surface

energetics may be used to determine dislocation core structures and dislocation

dissociation by means of the Peierls-Nabarro model of dislocation [9].

1.2 Density functional theory

In general, wave-function based ab initio methods approach the atomistic in-

teractions at the fundamental level - quantum physics is utilised by solving

Schrodinger’s equation for the many-body problem of the electronic structure.

The complexity of this approach is obvious - in general the wavefunction of the

many-particle system depends on the coordinates of each particle and, thus, the

treatment of any system larger than a small number of electrons is not feasible.

DFT provides some kind of compromise in the field of ab initio concepts, and

can be applied to the fully interacting system of many electrons. The crucial

1.2. DENSITY FUNCTIONAL THEORY 9

DFT ansatz is based on theorems of Hohenberg and Kohn [6], who demonstrated

that the total ground state energy E of a system of interacting particles is com-

pletely determined by the electron density ρ. Therefore, E can be expressed as a

functional of the electron density and the functional E[ρ] satisfies the variational

principle. Kohn and Sham [7] then rederived the rigorous functional equations in

terms of a simplified wave function concept, separating the contributions to the

total energy as,

E[ρ(r)] = TS[ρ] +∫

V (r)ρ(r)dr +1

2

∫

ρ(r)ρ(r′)

r − r′drdr′ + Exc[ρ(r)], (1.1)

in which TS represents the kinetic energy of a noninteracting electron gas, V the

external potential of the nuclei. The last term, Exc, comprises the many-body

quantum particle interactions, it describes the energy functional connected with

the exchange and correlation interactions of the electrons as fermions.

Introducing the Kohn-Sham orbitals the solution of the variational Euler equa-

tion corresponding to the functional of equation 1.1 results in Schrodinger-like

equations for the orbitals Ψ

(

− h2

2m∇2 + Veff (r)

)

Ψ(r) = εΨ(r). (1.2)

This are the renowned Kohn-Sham equations which are then actually solved (after

introducing the approximations described below). Equation 1.2 transforms the

many-particle problem into a problem of one electron moving in an effective

potential

Veff(r) = V (r) +∫

ρ(r′)

|r − r′|dr′ +

δExc[ρ]

δρ, (1.3)

which describes the effective field induced by the other quantum particles. The

actual role of the auxiliary orbitals is to build up the true ground state density

by summing over all occupied states,

ρ(r) =∑

occ

Ψ∗(r)Ψ(r). (1.4)

In short, the reformulation of Kohn and Sham provides a suitable basis, which

transforms the functional equation into a set of differential equations. The re-

sulting equations can be solved in a self-consistent manner. The crucial point

for actual applications is the functional Exc, which is not known (and therefore

has no analytical expression) and it therefore requires approximations. The his-

torically first and widely used approximation is the local density approximation

(LDA), which is based on the assumption that the exact exchange-correlation


energy can be locally at the point r be replaced by the expression and value for

an homogeneous electron gas,

Exc[ρ] =∫

ρ(r)εxc(r)ρ(r)dr, (1.5)

in which εxc(ρ) is the exchange-correlation energy per particle of the homoge-

neous electron gas. The function εxc(ρ) has to be -partially- approximated as

well, but this can be done accurately by computer simulations. Several methods

have been utilised to parameterize the many-body interactions of a homogeneous

gas of interacting electrons, for instance by many body perturbation theory or by

quantum Monte-Carlo techniques. The differences between the different param-

eterizations are small and, therefore, εxc(ρ) may be considered as a well-defined

quantity.

However, LDA itself is rather crude approximation, although it gives sur-

prisingly reliable results for many cases. Several arguments might be found to

elucidate the success of LDA for a wide range of applications. Nevertheless, due

to its overbinding effects LDA is now considered to be not accurate enough (for

many but not all cases). Various improvements have been proposed by going

beyond the most simple local assumption of LDA taking into account the gradi-

ent of the electron density. Nowadays, this is done by the so-called generalised

gradient approximation (GGA), which counteracts the overbinding of LDA, e.g.

equilibrium volumes are increased whereas cohesive properties are reduced when

compared to standard LDA [10] results. In many applications, GGA provides

a substantially improved description of the ground state properties, in particu-

lar for 3d transition metals, as strikingly demonstrated for the ground state of

iron [11].

Ab initio DFT methods have great capabilities and are widely applied, in par-

ticular since the last two decades. Their usefulness for the scientific community

was demonstrated by the Nobel Prize which was given 1998 to W. Kohn and

J. Pople. DFT proved to be general and predictive tool for calculating various

properties which can be derived from the electronic ground state, such as equilib-

rium crystal structures and lattice parameters, elastic constants, surface energies,

phonon dispersions, etc. [8]

1.3 Electronic structure methods

There is yet a large step from the theoretical considerations outlined in the previ-

ous section to a manageable form of the Kohn-Sham equations, which can be run

on a computer. Because the solution for a solid is desired, a natural condition is

1.3. ELECTRONIC STRUCTURE METHODS 11

to require translational symmetry for the observables, such as the potential,

veff(r + R) = veff(r). (1.6)

There, R is a lattice translation vector. As a consequence of translational sym-

metry, the wave function must fulfill Bloch’s theorem,

ψk(r + R) = eikRψk(r) (1.7)

Then, the variational Kohn-Sham orbital may be expressed as linear combination

of basis functions φ obeying Bloch’s theorem,

ψnk(r) =∑

i

ci,nkφik(r), (1.8)

with band index n and k being a vector of the first Brillouin zone. Building the

energy functional (i.e. the expectation value of the Hamiltonian) and applying

the variational principle, the solution of the Kohn-Sham equations is transformed

into an matrix eigenvalue problem,

∑

i

[〈φjk|H|φik〉 − εnk〈φjk|φik〉]ci,nk = 0. (1.9)

This equation has to be diagonalized for obtaining the eigenvalues ε and eigen-

vectors c, from which the electron density is constructed and -consequently- the

total energy is derived.

At present, the most widely used numerical methods for solving the Kohn-

Sham equations are pseudopotential (and related) methods, the linear muffin-tin

orbitals method and the full-potential linearized augmented plane wave method

(FLAPW). In the present thesis, the Vienna Ab Initio Simulation Package

(VASP) is applied [12, 13], which is the most powerful ab initio DFT package

available at present. VASP is based on the pseudopotential concept. For the

actual calculations a generalization in terms of the so-called projector augmented

waves construction of the potential [12, 13] is applied, which is known to give

very accurate results as tested by comparison to FLAPW benchmarks. VASP

has been already applied to a wide range of problems and materials, to bulk

systems, surfaces, interfaces, e.g. Refs. [14, 15, 16, 17, 18, 19]. VASP provides

framework for the bulk and surface phonon calculations as well [20, 21].

Specific computational and technical aspects, e.g. number of k-points, geom-

etry of the unit cell etc., are discussed later together with the results. The theory

and parameters underlying the VASP code have been addressed in above men-

tioned publications. It should be noted that VASP was applied for materials and

systems which may be considered as ’well-established’ from the computational


point of view. The VASP package served as a tool, which works reliably when

handled with care and knowledge. Convergency aspects were carefully tested

in several cases. Consequently, it can be argued that the results as presented

in following chapters do not depend on inherent technical parameters and are

physically meaningful.

Chapter 2

Elastic properties of material

A solid body which is subject to external forces, or a body in which one part

exerts a force on neighbouring parts, is in a state of stress. If such forces are

proportional to the area of the surface of the given part, the force per unit area

is called the stress. The stress in a crystalline material is a direction dependent

quantity and, therefore, is in general described by the stress tensor σij. If all parts

of the body are in equilibrium and body forces are absent (body forces may be

produced, for instance, by a distribution of electrostatic charges in the presence

of an electric field, but are absent in cases of interest herein), the equation (in

the following Einstein’s convention for the summation is applied)

∂σij

∂xj= 0 (2.1)

must be fulfilled. The symbols xi denote the cartesian axes. The deformations

of the solid caused by the exerted stress are described by the strain tensor. If ui

is the displacement of a point xj in a deformed solid, the strain tensor is then

defined as

εij =1

2

(

∂ui

∂xj+∂uj

∂xi

)

. (2.2)

The diagonal components ε11, ε22 and ε33 are called tensile strains, whereas the

other components are usually denoted as shear strains. Both stress and strain

tensors are symmetrical (in the absence of body torques).

The linear theory of elasticity provides a mathematical description for the

phenomenological fact, that relative elongations and distortions (or strains in

general) are linearly proportional to applied stresses, provided that these stresses

are kept to suitable small magnitudes. Once the stresses are removed, an ideal

linearly elastic body returns to the unstrained state. This theoretical model does

not refer to any model for real matter, and the atomistic nature of matter does

not enter as a prerequisite to this concept. The range of the stress for which the

13

14 CHAPTER 2. ELASTIC PROPERTIES OF MATERIAL

assumption from above applies is called the elastic limit. Beyond the elastic limit

a non-linear effects break the (linear) proportionality between stress and strain,

and for large stresses a plastic dissipation makes the deformation irreversible.

2.1 Elastic constants and crystal symmetry

The most general linear relationship which connects stress to strain is provided

by the generalized version of the well-known Hooke’s law,

σmn = Cmnprεpr, (2.3)

in which σmn denotes the stress tensor, εpr the strain tensor and the elements of

the fourth-order tensor Cmnpr are the so-called elastic constants. Alternatively,

one might express the strains in terms of the stresses by

εmn = Smnprσpr (2.4)

defining the elastic moduli Smnpr. The elastic constants and elastic moduli are

fundamental materials parameters providing a detailed information on the me-

chanical properties of materials. The knowledge of these data may enable predic-

tion of mechanical behaviour in many different situations. Whereas σmn and εpr

are symmetric and have therefore only 6 independent elements, the number of 81

elastic constant is reduced by symmetry arguments to a total of 21. The elastic

energy density U , which is defined as the total energy per volume, is obtained

from the stress tensor (force per unit area) by integration of Hooke’s law

U =E

V=

1

2Cmnprεmnεpr. (2.5)

So far, e.g. the strain tensor has been considered as a tensor of order two of the

form

ε =

1 + exx12exy

12exz

12eyx 1 + eyy

12eyz

12ezx

12ezy 1 + ezz

. (2.6)

Introducing the convenient matrix-vector notation, where the 6 independent el-

ements of stress and strain are represented as vectors (denoted here as Σi and

εj with i, j running from 1 . . . 6 according to the sequence xx, yy, zz, yz, xz, xy),

and furthermore rewriting the fourth order tensor Cmnpr as a 6x6 matrix cij, one

can formulate a more simplified expression,

Σi = cijεj U =E

V=

1

2cijεiεj (2.7)

2.1. ELASTIC CONSTANTS AND CRYSTAL SYMMETRY 15

Table 2.1: The number of independent elastic constants for different lattice sym-metries and point groups (from Ref.[23]).

Lattice (point group) No. of constantsTriclinic 21Monoclinic 13Orthorhombic 9Tetragonal (4, -4, 4/m) 7Tetragonal (422, 4mm, -42/m, 4/mmm) 6Hexagonal and rhombohedral (3, -3) 7Hexagonal and rhombohedral (32, 3m, -32/m) 6Hexagonal (6, -6, 6/m, 622, 6mm, -62m, 6/mmm) 5Cubic 3

Taking into account additional symmetry arguments imposed by the crystal lat-

tice, the number of elastic constants further decreases. In particular, for a cubic

lattice only three independent elastic constants, c11, c12, c44 remain, whereas for

a tetragonal lattice the six elastic constants c11, c12, c13, c33, c44, c66 are sufficient

Since the examples discussed here are cubic and tetragonal crystals, the explicit

form of the tensor is given for these two cases:

ccubic =

c11 c12 c12 0 0 0c12 c11 c12 0 0 0c12 c12 c11 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 c44

(2.8)

ctetragonal =

c11 c12 c13 0 0 0c12 c11 c13 0 0 0c13 c13 c33 0 0 00 0 0 c44 0 00 0 0 0 c44 00 0 0 0 0 c66

(2.9)

Explicit forms for other lattice symmetries may be found for instance in ref-

erence [22]. The total number of independent elastic constants for all crystal

systems is summarized in table 2.1.


2.2 DFT calculation of elastic constants

In principle, there are two ways of computing single crystal elastic constants from

ab initio methods: the energy-strain approach and the stress-strain approach.

The energy-strain approach is based on the computed total energies of properly

selected strained states of the crystal. The crystal is strained in order to extract

the corresponding stiffness values preserving as much symmetry as possible. For

each strain type, several magnitudes of strains are applied and the corresponding

total energies are computed with an ab initio approach. The stiffness is then

derived from the curvature of the energy-strain relation by means of a least-

squares fit making use of equation 2.5. Some of the imposed strains may be related

to a single elastic constant while others are described by a linear combination of

elastic constants, from which the elastic constant tensor is finally evaluated. The

number of necessary distortions is given by the number of independent elastic

constants.

As an example, the deformations commonly used for the calculation of the

elastic constants in a cubic crystal are discussed. Note, that the linear elastic

energy-strain relation of the equation 2.5 is valid for any crystal symmetry, and

by that it is possible to evaluate elastic constants of any crystalline material.

The elastic energy density for a cubic crystal can be expressed as (making use

of equation 2.5):

E

V=

1

2c11(ε

211+ε

222+ε

233)+

1

2c44(ε

223+ε

231+ε

212)+c12(ε11ε22+ε33ε22+ε11ε33). (2.10)

For a tetragonal distortion the shear displacements will be zero and the diagonal

components of the strain tensor are expressed as

ε1 = ε2 =da

a, ε3 =

dc

c, ε4 = ε5 = ε6 = 0. (2.11)

Inserting this relation into expression 2.10, the elastic energy density is given by

E

V= (c11 + c12)ε

21 + 2c12ε1ε3 +

c112ε33. (2.12)

The strains εi can be replaced with a more convenient set of parameters, namely

the c/a ratio (which characterizes the amount of tetragonal deformation) and the

unit cell volume V . Substituting the parameters one arrives at the expression

E =c11 + 2c12

6

(

dV

V

)2

+2c′

3

(

d(c/a)

c/a

)2

, (2.13)

in which dV and d(c/a) denote infinitesimally small change of respective param-

eter. Calculating the total energy along a volume-conserving tetragonal deforma-

tion path close to the equilibrium, the elastic constant c′ = c11−c122

is obtained

2.2. DFT CALCULATION OF ELASTIC CONSTANTS 17

from the curvature of the energy curve at equilibrium. In the same way, the

shear constant c44 is obtained from a trigonal deformation of the cubic lattice.

The total energy expressed in terms of c/a and V for a trigonal deformation is

E =c11 + 2c12

6

(

dV

V

)2

+2c443

(

d(c/a)

c/a

)2

. (2.14)

Finally, hydrostatic isotropic compression may be applied and by that the bulk

modulus B is directly derived from the curvature at the equilibrium volume V0

B =1

3(c11 + 2c12) = V

∂2E

∂V 2. (2.15)

The numerically obtained total energy relation for the isotropic compression may

be fitted by the Birch ansatz [24], or alternatively the Birch-Murnaghan [25]

equation of state. The trigonal and tetragonal paths were selected, because they

preserve as much symmetry as possible and, thus, reduce computational costs and

guarantee a high precision. The choice of distortions is analogous for crystals with

other symmetries.

The stress-strain approach, on the other hand, relies on the feature of VASP

to directly calculate the stress tensor. Once the stress tensor components can be

computed by an ab initio method, the elastic constants matrix can be directly

derived from the generalized Hooke’s law of equation 2.3. For instance, assuming

again cubic symmetry, the elastic constants can be expressed in terms of the

stress tensor by

c44 =1

2

∂σ12

∂ε12(2.16)

c′ =1

2(c11 − c12) = −1

2

∂σ33

∂ε33(2.17)

B =1

3(c11 + 2c12) =

∂σ11

∂ε11. (2.18)

Whereas within the energy-strain approach several magnitudes of strain have

to be evaluated in order to obtain the elastic constant from an analytic fit to

the total energy data, within the stress-strain approach just one evaluation is

in principle sufficient to obtain the same information. However, to ensure high

accuracy three strain values have been applied for all systems calculated here.

Both approaches have been implemented in a symmetry-generalized form [26],

the underlying concepts are discussed in detail in Ref. [23] for the energy-strain

approach and in Ref. [27] for the stress-strain approach.

The ab initio calculation of elastic constants of single crystal has been outlined

so far. By macroscopic averaging, also elastic moduli of polycrystalline materials


can be derived. There are several averaging procedures available to derive the

elastic moduli of a quasi-isotropic polycrystalline material from its single crystal

elastic constants. The averaging encumbers all possible orientations of the crystal,

and there is a well-defined lower and upper limit for the elastic moduli. Based

on the averaging procedures, the ab initio treatment for single crystals can be

extended to polycrystalline samples [28].

2.3 Results for selected materials

In this section the accuracy and reliability of the ab initio concept is demon-

strated by discussing the actual calculation of elastic constants for a range of

materials. The elastic constants are important materials parameters and their

calculation requires a skillful handling of the computer code. The calculation pro-

ceeds as follows: first, the convergency of computational parameters, primarily

k-point grid, is assured. Then, (if the lattice is cubic) the bulk equilibrium value

of the lattice parameter a0 is derived by means of unit cell volume relaxation. If

there are any internal degrees of freedom for the atoms to change their positions,

the equilibrium positions have to be calculated by minimizing the atomic forces.

Number of degrees of freedom (if any) depends on the actual space group sym-

metry of the system. Consequently, the elastic constants are calculated using the

approach outlined above. The calculated values are displayed in table 2.2. For

the intermetallic compound TiAl with tetragonal symmetry, the a and c/a lattice

parameters have to be relaxed for finding the equilibrium shape and volume. For

the six independent elastic constants of the tetragonal symmetry, c11, c33, c12,

c13, c44, c66, the values of 190, 185, 122, 60, 110, and 50 GPa, respectively, are

obtained. Though the tetragonal distortion is rather small because the c/a ratio

of 1.02 is close to 1, the elastic properties display pronounced differences between

related constants (Note, that for a cubic crystal c33 = c11, c13 = c12, c66 = c44).

The overall agreement of calculated results displayed in table 2.2 with ex-

perimentally determined values is excellent, in fact within the error bars of the

experimental methods in most cases. In general the calculated values are slightly

smaller than experimental ones, which is well-known feature of the GGA approxi-

mation (see section 1.2). Several experimental methods are applied for the deter-

mination of the elastic constants of single crystals, the most prominent making

use of ultrasonic waves. However, one has to be careful when comparing exper-

imental elastic constants, which are usually measured at room temperature, to

the ab initio results, which correspond to T=0 K. In general, elastic constants are

reduced with increasing temperature. Nevertheless, the effect of temperature on

elastic properties is small for many materials at room temperature. In principle,

2.3. RESULTS FOR SELECTED MATERIALS 19

Table 2.2: The equilibrium lattice parameters and elastic constants of selectedmaterials calculated by VASP within the GGA PAW approach. The values inbrackets are experimental references (a Ref. [29], b Ref. [30], c Ref. [31], d Ref. [32],e Ref. [33])

a0 c11 c12 c44Al fcc 4.06 110.2 (108.2) 54.8 (61.3) 30.4 (28.5)a

Fe bcc 2.83 302.4 (242) 168.7 (146.5) 102.5 (112)d

W bcc 3.17 540.9 (521) 202.7 (201) 141.1 (160)e

NiAl B2 2.89 202.9 (204.6) 140.3 (135.4) 112.6 (116.8)c

FeAl B2 2.87 278.1 139.9 145.4Ni3Al L12 3.56 225.0 150.6 116.5Al3Sc L12 4.10 185.7 49.4 60.1

VC B1 4.16 646.6 135.6 193.4TiC B1 4.34 514.5 (500) 106.0 (113) 178.8 (175)a

MgO B1 4.22 297.1 (286) 95.4 (87) 156.1 (148)a

NaCl B1 5.01 52.8 (49) 12.3 (12) 12.4 (13)b

C A4 3.6 1050.2 (1076) 125.3 (125) 556.3 (576)a

Si A4 4.04 154.1 (165.7) 57.7 (63.9) 74.7 (79.6)a


high temperature elastic constants may be determined by including the effects of

lattice vibrations and anharmonicity effects. Such a treatment would be possible

with VASP, but it is very time-consuming even for simple cases.

Briefly exploring the results in table 2.2, one realizes the very outstanding

elastic properties of the refractory compounds TiC and in particular VC. Among

the metals, for W the elastic constants are large because of the strong bonding

as is also revealed by the high melting point.

The differences of results for elastic properties between up-to-date DFT meth-

ods are usually rather comparable (if the actual calculations are done with care).

Considerable discrepancies are found only when different exchange-correlation

potentials are used (LDA vs. GGA). Due to the overestimation of bonds LDA

derived elastic constants are always larger than their counterparts obtained with

a GGA potential.

2.4 The ideal strength

Whereas the linear elastic properties outlined above describe the behaviour of a

material with very small strains, studying the ideal strength deals with a ma-

terial’s property for large strains: when an ideal, defect-free crystal is being

loaded until the lattice becomes elastically unstable, the stress at the onset of

elastic instability is called ideal strength [34]. For testing the material for its

ideal strength the loading occurs infinitely slow, and the material does not break

before an elastic instability is reached.

For many materials under real loading conditions it is not possible to load the

material until its ideal strength is reached. Nevertheless, ideal strength resembles

an upper boundary on the strength of a material which can be calculated. The

loading may come close to the ideal strength for some brittle materials like dia-

mond, silicon, and some of the ”super hard” transition metal carbides or nitrides

as well. Recently, a new technologically important class of materials (hard defect

free films [35]) was found to approach the conditions of ideal strength as well.

Since ideal strength is determined by the elastic instability of an ideal crystal,

it can be conveniently calculated within the framework of the ab initio electronic

structure calculations. For instance, the ideal strength in tension is evaluated by

straining the crystal by a series of incremental strains and simultaneously relaxing

the stress components perpendicular to the loading direction. The total energy

as a function of the strain can be derived from DFT calculations, and the stress

may then be directly calculated from the proper derivatives of the calculated total

energy. The maximum of the stress obtained in such a way is the ideal strength

under uniaxial stress conditions. The calculation of the energy and stress without

2.4. THE IDEAL STRENGTH 21

relaxation of the stress components perpendicular to the loading direction would

correspond to uniaxial strain conditions.

During a homogeneous deformation, the ideal stress may be influenced by

a possible existence of higher-symmetry structures along the deformation path.

For instance, if a bcc crystal is sufficiently stretched in the [100] direction (i.e.

the cubic structure is deformed to a tetragonal structure with c/a > 1) it will

eventually be transformed to fcc (for c/a =√

2). Because of symmetry, the stress

vanishes for both, the bcc and fcc structures along this Bain’s path, and the

corresponding deformation energy at the fcc point must reach an extremum (being

a maximum, minimum or a saddle point). Similar deformational paths connect

some other structures as well, for instance a B2 crystal under [111] uniaxial

tension may be transformed to a B1 structure. Another example would be the a

trigonal [111] distortion transforming a bcc lattice to an fcc lattice, via a simple

cubic structure.

The appearance of higher-symmetry structures was used to explain the strong

anisotropy of ideal strength in otherwise elastically isotropic materials [36]. How-

ever, uniaxial tension represents only one kind of possible lattice instability. In

general the crystal may fail by other elastic instabilities (in shear, for example)

prior the ideal strength in tension is reached.

In principle, the ideal strength for arbitrary type of loading can be studied.

However, until recently the calculations were limited to uniaxial tension, simple

shear, or triaxial tension, because the relaxation of the strained solid (in whatever

degrees of freedom it is free to relax) and perquisite precision of calculations were

computationally very demanding. Today’s computational resources and powerful

codes have enabled DFT calculations of the elastic limits for tensile as well as

shear deformations under fully relaxed conditions [37]. An overview on the history

as well as the state-of-the-art of ab initio simulations of ideal strength is given in

a very recent review articles [38, 39].

Clearly, such calculations of ideal strength are substantially different from

the cleavage models as elaborated in the following chapters. The conceptual

difference is that cleavage and fracture involves defects -cracks- in a material

whereas the ideal strength refers to the onset of elastic instability for an ideal

-defect free- crystal. Though the concepts are of different nature and cannot be

directly compared, the results agree well in quite a few cases. A discussion of the

available ideal strength values will be given at the respective place.

The ideal strength of the material may be explored experimentally as well.

The strength is defined for ideal material and, therefore, experiments require

flaw-free crystals. This means specimens with atomically smooth surfaces which

contain no cracks, impurities or dislocations. Tests must be carried without


marring the perfection of specimens. The largest number of the high strength ex-

periments is performed on whisker crystals, despite difficulties arising from their

small size [34]. This is so, because whiskers often grow dislocation free and with

surfaces of required quality. Metals usually permit the gliding of dislocations at

low temperatures and, thus, exhibit high strengths only in whisker form. For

covalent materials, however, relatively large ideal strengths can be reached even

in bulk specimens because of the low dislocation mobility at ambient tempera-

tures. In real materials, however, cracks decrease the strength by several orders of

magnitude. The maximum strength may be reached only locally at the crack tip,

where the stress concentrates. Thus, the strength of common engineering mate-

rials is determined mostly by the properties of cracks and dislocations, which will

be treated in the following chapters.

Chapter 3

Brittle fracture of material

3.1 Fundamentals

3.1.1 Continuum theory

The effect of cracks or dislocations on the properties of a solid is mainly associated

with displacements and the stress fields due to their presence. These processes

are at the macroscopic level described by the elasticity theory and, thus, most

of the continuum theories of mechanical properties of solids make use of linear

elastic solutions when simulating crack- or dislocation-like perturbations in an

infinite solid.

The elasticity theory neglects the atomistic structure and treats materials as

a homogeneous continuum. In order to reduce the mathematic complexity the

continuum is often being assumed to be linearly elastic and isotropic. Although

it may seem as an oversimplification, such a linear elastic treatment provides

basic -but important- analytic solutions and -moreover- helps to find the material

parameters involved. However, even in the simple isotropic linear elastic theory

framework it is difficult to obtain the solutions for general three-dimensional

problems. Thus, most of the problems are further constrained to the state of plane

strains or plane stresses. Under planar strain conditions, which are important in

the theory of straight dislocations, the stress is independent of the displacements

in one direction (for example, z = 0). The equilibrium equation of classical

elasticity (equation 2.1) then yields the form

∂σxx

∂x+∂σxy

∂y= 0 (3.1)

∂σyy

∂y+∂σxy

∂x= 0. (3.2)

These conditions are automatically fulfilled if stresses are expressed in terms of

23

24 CHAPTER 3. BRITTLE FRACTURE OF MATERIAL

a stress function Ψ as

σxx =∂2Ψ

∂y2(3.3)

σyy =∂2Ψ

∂x2(3.4)

τxy = − ∂2Ψ

∂y∂x. (3.5)

Furthermore, the differentiation of equation 2.2 produces for the case of plane

strain the equation∂2εxx

∂y2+∂2εyy

∂x2− 2

∂2εxy

∂y∂x= 0. (3.6)

Combining above relations, the biharmonic equation for the stress function is

obtained

∇2(∇2Ψ) = 0. (3.7)

When a solution of the biharmonic equation is found, the stresses and displace-

ments are obtained from the equation 3.3. The crack problem itself is formulated

by appropriate boundary condition. Near field conditions constrain usually a

surface of the crack (i.e. elliptical crack, straight linear crack) demanding track-

free surface, while far field conditions express the external loading of the crack

(tension σ0, or shear τ0 usually) acting far enough from the crack to make macro-

scopic dimensions of the specimen negligible. The solution of the general plane

strain problem for mode I loading was first provided by Westergaard using the

complex stress function method [40].

3.1.2 Stress intensity factors

There are essentially three basic ways of loading a solid body containing a crack.

These are known as loading modes and represent possible symmetric displace-

ments of the upper crack surface against the lower one. The modes are illustrated

in figure 3.1. The most important one from technological and also scientific point

of view is the so-called mode I (tensile opening), where the crack faces, under

tension, are displaced in a direction normal to the crack plane. The mode I

component is prevalent under common tensile loading of the crack. The mode II

(shear sliding) and mode III (tearing) loadings represent deformations for which

the crack surfaces glide over each other in the same plane, or out of this plane,

respectively. There is a difficulty connected with modes II and III, which ham-

pers their experimental surveys. Because the crack faces are not pulled away

from one another, the contact between the crack faces is unavoidable and results

3.1. FUNDAMENTALS 25

Figure 3.1: Three crack loading modes - tension, shear, and anti-plane shear.

in friction forces along the crack faces which cause difficulties for the experimen-

tal measurements (and also for modelling such situations). Therefore, mode I

loading corresponds most closely to the conditions used in most of experimental

works.

In reality, combinations (called mixed loading) of these displacements occur,

i.e. mixed I/II or I/III loading. The coupling between various components of the

loading becomes important in polycrystalline materials, because of the different

orientations of grains with respect to the external stress direction.

Each of the crack loading modes is associated with a certain stress field in the

neighbourhood of the crack. The stress field can be described using the concept

of the stress intensity factor K, introduced by Irwin [41]. For a crack parallel to

the x axis, the nonzero stress components are

σyy =KI√2πx

fI(θ) σxy =KII√2πx

fII(θ) σyz =KIII√2πx

fIII(θ). (3.8)

The universal functions f(θ) are independent of the crack geometry and describe

the radial and angular variations of the stresses around the crack. Hence, the local


stress field around the crack is fully characterized by the stress intensity factor K.

The factor K proved to be an effective parameter modelling the brittle fracture or

fatigue crack growth. It contains information about the specimen geometry, the

crack length and the applied load. Since the factor K is an outcome of elasticity

theory, it relies on solutions of the crack problems. A range of the methods for

solving elastic crack problems was introduced in the 1970s, and the stress fields

for various crack shapes and loadings were calculated [1, 42, 2].

3.1.3 Griffith’s thermodynamic balance

The first serious theoretical treatment of the fracture was introduced by Griffith,

who considered the problem on the energy level. His simple but general ideas

have stayed as a starting point even for sophisticated modern theories. The

presumptions introduced by Griffith are utilised in our DFT treatment of the

brittle fracture as well, thus his theory is discussed in more detail.

In his work, to account for usually observed discrepancy between calculated

and measured values of the strength of solids, he postulated that a solid contains

cracks, and that the rupture proceeds by spreading of these cracks. He formulated

the condition for propagation of the crack based on a competition between an

excess elastic energy W of the solid due to the presence of cracks and the surface

energy of the crack S. Then, the critical equilibrium state is defined by

∂(S −W )

∂a≡ G = 0. (3.9)

Hence, the crack cannot propagate until the elastic energy from external forces

acting on the solid reaches the surface energy of the newly created crack faces.

Obviously, this is only a necessary condition for crack propagation. The energy

release rate G measures the tendency of the crack to propagate and it is a function

of the specimen geometry and the applied loading. For the propagation, the

energy release rate must exceed some critical value Gc. The parameter Gc is

a material property, called critical energy release rate. In the case of brittle

materials -for which the crack propagation is not accompanied by any energy

dissipation at the crack tip- the analysis above yields the relation

Gc = 2γs. (3.10)

Thus, for a brittle solid, equation 3.9 becomes also a sufficient condition for

crack propagation. The input energy needed to propagate the crack may then be

obtained from the linear elastic solution of the corresponding crack problem.


Thus, a material is defined to be brittle when there is no energy dissipation

(in form of the dislocation emission, for instance) involved in the fracture pro-

cess. For such a solid, the work of fracture approaches the surface energy of

the newly created crack surfaces. Many materials appear to be brittle to a first

approximation, because the energy of plastic flow is very small at low tempera-

tures. Examples are LiF, MgO, CaF2, BaF2, CaCO3, Zn. For these materials,

the surface energies derived from fracture experiments agree quite well with those

theoretically expected [43], proving indirectly Griffith’s prediction. The cleavage

crack propagation is the dominating fracture process in these materials, although

often some dislocation emission occurs. A few materials (Si, Ge, SiC, Al2O3) can

fracture purely in the brittle fashion. Hence, the concept of a brittle material is

not purely an academic model. Furthermore, it provides important information

for the models considering the brittle-ductile transition, which occurs in many of

the technologically important structure materials, e.g. Fe, Al, or various inter-

metallic compounds.

It should be noted, that equation 3.9 is just the energy balance condition

corresponding to the first law of thermodynamics, applied to a solid containing a

crack. It also contains the assumption, that all of the strain energy stored in the

solid body is involved in the fracture process. This is valid in materials where

stress localizes at the crack tips and thus the theory does not apply to highly

deformable materials such as rubber.

3.1.4 Irwin Theory

Griffith’s theory holds in principle only for ideal brittle materials. Experimental

studies, however, showed that there is an evidence of the plastic deformation even

in materials fracturing in a brittle manner. Thus, Irwin [44] and independently

Orowan [45] concluded that the work of the plastic deformation γp at the crack

tip must be considered as well and the surface energy γs in equation 3.10 must be

replaced with the term γs+γp. Following this argument Irwin observed that if the

size of a plastic zone at the crack tip is small with respect to the crack size, the

energy flowing into the tip will come from the bulk and, therefore, will not depend

on subtle details of the stress state in the cohesive zone. As a consequence, this

observation allowed to use linear elastic solutions to calculate the energy release

rate available for fracture. Consequently, the energy release rate became a basis

of most concepts in the theory of fracture.

Utilizing the concept of the stress intensity factor K (see section 3.1.2) Irwin


evaluated the energy release rate G for symmetric loading of a planar crack as

G =K2

E?, (3.11)

in which E? = E reflects the plane stress and E? = E/(1 − ν2) the plane strain

conditions (E denoted Young’s modulus and ν Poisson ratio). The equation 3.11

is a very important relation, because it relates the energy release rate to the stress

intensity factor, which contains all information about the geometry, crack length,

and loading. Furthermore, the proportionality between K2 and G holds also

for anisotropic solids - then, G is related to the appropriate elastic compliance

constant (for plane strain conditions). For a rectilinear anisotropic (orthotropic)

solid under mode I loading, the parameter G is given by [46]

G = K2I

√

√

√

√

s11s22

2

[

(

s22

s11

)1/2

+2s12 + s66

2s11

]

. (3.12)

Assuming that the size and shape of the cohesive zone remain constant as the

crack advances, G can be also used as a characterizing material parameter. Fur-

thermore, the parameter G can be considered as characterizing parameter under

mixed mode loading conditions. Because the work of the plastic deformation γp is

difficult to measure, the critical stress intensity factor Kc is used as characterizing

property of a common engineering material, in which plastic dissipation always

occurs. The parameter Kc can be experimentally determined by introducing a

thin sharp crack into a material and measuring the applied stress necessary to

cause the propagation of this crack. For a straight crack of length a the critical

stress intensity factor is

Kc =σ√πa. (3.13)

In summary, the equations presented above provide links between crucial ma-

terial properties, cracks and external stresses. They represent the basis of the

engineering part of fracture mechanics. The question of the following sections

will be: what is the role of the atomic nature of the solid and how to correlate

macroscopic model parameters with DFT calculations.

3.1.5 Lattice trapping

The models of Griffith and Irwin are based on the continuum representation and

ignore the discrete atomistic true nature of the solid material, in which the crack

tip is formed and develops further. The simplest lattice property of a crack is

its trapping by the lattice analogous to the Peierls trapping of dislocations. For


a given length of a crack there exists a range of loads over which the crack is

mechanically stable. In order for a crack to move by one lattice spacing to the

next stable position a bond at the crack tip must be cut, which represents an

energy barrier for crack growth. Thus, the equilibrium is achieved over much

wider a range of crack lengths than a single length predicted by Griffith theory.

Furthermore, in a periodic lattice the crack moves at a loading larger than the

Griffith critical load. This statement was confirmed by atomistic studies in char-

acteristically brittle materials such as silicon [47] and β-SiC [48] where lattice

trapping raised the critical load over Griffith’s prediction. In contrast, simula-

tion for metallic systems with long-range interatomic potentials reported good

agreement with the Griffith theory [49].

Therefore, the Griffith condition has to be modified. Now, the critical energy

release rate includes a structural term with the periodicity of the lattice.

Gc = 2γs + 2γ1 sin2πa

a0

(3.14)

When such a term is included, the fracture process becomes thermodynamically

irreversible during propagation, because the crack growth becomes unstable with

maxima in the structure term. The lattice trapping explains the experimental

fact that crack healing never occurs at a loading smaller than the critical loading,

although healing is implicitly contained in Griffith’s theory.

The lattice trapping was studied in a number of atomic simulations. However,

the results are difficult to generalize, because discrepancies between simulation

and Griffith’s prediction may be due to another atomic scale features, because

atoms do not form an ideally sharp crack tip. Subsequently, no general relations

which would enable DFT calculations of γ1 have so far been developed.

Though the Griffith criterion may seem oversimplified in the light of the atom-

istic features, it still provides a sound basis for the treatment of brittle fracture. It

provides a lower limit for the energy release rate. Griffith’s theory framework -the

relationship between the critical load for crack propagation, intrinsic crack resis-

tance and the surface energy- was addressed in very recent atomistic simulation

by Mattoni et al. [48]. They considered an elliptical crack in β-SiC, a prototype

of an ideal brittle material. They found that the crack extends in a perfectly brit-

tle way, by preserving atomically smooth (111) cleavage surfaces. On the other

hand, they never observed healing of the crack at values of the loading lower than

the critical one, in agreement with the experimental experience. This is due to

the lattice trapping and the relaxation of the crack surface, which make the crack

propagation process irreversible. Mattoni et al. [48] suggested possible corrections

of the Griffith theory, by modifying assumptions of the linear elastic theory: (1)


the surface energy depends on the state of strain, and (2) the stress-strain curve

needs not be strictly linear over the whole range of the explored loads; in other

words, the Young modulus is not a constant. Nevertheless, in the present treat-

ment of brittle fracture one still considers the Griffith approach, because more

elaborate studies are necessary for providing reliable general framework.

3.2 DFT calculations for brittle fracture

As elaborated in the previous sections, the stress field of a crack falls off with

distance like 1√r, being very long ranged compared to other lattice defects. This

fact seems to make a direct ab initio DFT modelling of a crack impossible, because

a large number of atoms (in the order 106) would be needed, in order to avoid

interactions between cracks which are repeated because of the periodic boundary

conditions. The supercell of such size can be treated only with empirical or

semiempirical atomistic methods. The ab initio methods are rather being used

to determine input parameters for some of advanced larger scale models, e.g.

cohesive zone model. The cohesive zone models use linear elastic solutions at

larger distances from the tip, but involve only a small region around the tip

where cleavage decohesion of atoms is considered. The ab initio calculations of

brittle cleavage decohesion are discussed in following sections.

3.2.1 Cleavage decohesion

Although the macroscopic stress field around a crack constitutes large part of

fracture energy, it is clear that at the atomic level the crack advances by sequent

breaking of individual bonds between atoms. This small portion of total crack

energy can still govern the whole fracture process. If the crack in a brittle solid

is atomically sharp, the critical energy release rate for propagation is governed

by thermodynamic Griffith relation. So the question is, how to incorporate the

process of bond-breaking into the traditional Griffith thermodynamic analysis,

and possibly how to extend such a concept to blunted cracks.

In his analysis, Griffith considered an elliptical crack of propagating across

a uniform plate of unit thickness and showed, that the stress concentration at

the crack tip is responsible for the discrepancy between observed and theoretical

values of the critical strength of materials. However, the stresses near the crack

tip are determined by the interatomic forces and consequently the shape of the

crack close to the tip is given by the materials chemistry and might not be

well represented by an elliptical surface. In a truly brittle material, in which

no dislocation emission or other mechanisms of energy dissipation occur during

3.2. DFT CALCULATIONS FOR BRITTLE FRACTURE 31

Figure 3.2: Sketch of the cohesive zone in front of the tip of an elliptical crack.Vertical lines represent bonds between individual atoms, and their elongation andrupture due to the opening of the crack.

crack propagation, the crack closes smoothly and atomic bonds around the tip

are at different stages of elongation, as sketched in figure 3.2. When the crack

advances by one interatomic distance, each atomic bond ahead of the crack tip

takes the strain held by its predecessor and the sum of all elongations is equal to

complete breaking of one bond. If this process takes place under the conditions of

thermodynamic reversibility, the work required to proceed is the cleavage energy

Gc [34].

Thus, in brittle solids the Griffith condition holds even at the atomic level

(when the lattice trapping is neglected). Of course, very different concepts and

ideas stand behind the Griffith condition in the continuum model and in the dis-

crete crystal lattice theory. The correspondence is reached through the energy,

which connects both processes. Very recently, the validity of Griffith’s approach

was tested utilizing large scale atomistic simulations, which showed very satisfac-

tory agreement with Griffith’s prediction [48].

It should be noted, that the class of intrinsically brittle materials is relatively

small. In particular it includes materials with strong covalent bonding like SiC,

VC as well as Si. Moreover, even in brittle materials the crack propagation is

accompanied by the emission of dislocations, though the emission is being rather

rare. Nevertheless, the cracks propagating in a brittle manner can suddenly break

the material, and then the cleavage energy Gc can be viewed as a lower threshold

for brittle crack propagation. Furthermore, the information about the energy


release rate for brittle cleavage decohesion is important when brittle-ductile com-

petition at the crack is considered . The ductile-brittle transition at ambient

temperatures concerns a large number of materials, including iron or aluminum.

3.2.2 Calculation of cleavage decohesion for ideal brittle

fracture

Within the DFT framework, the cleavage energy Gc can be calculated using the

simple model of ideal brittle cleavage decohesion; between two planes the crystal

is rigidly separated with a distance x into two semi-infinite parts. The change

of total energy E(x) with increasing separation x is obtained from DFT total

energies. The asymptotic value of the interfacial energy E(x) gives then the

ideal cleavage energy Gc, which is identified with Griffith’s critical energy release

rate. Obviously, the relation Gc = 2γs is valid only for the geometrically most

simple cases, in which the two surface planes of the cleaved blocks are equivalent.

Furthermore, from the maximum of the cohesive stress function σ(x) = dE/dx

the critical cleavage strength σc can be obtained.

The cleavage of a single crystal is modelled by a repeated slab scheme of

atomic layers with three-dimensional translational symmetry as utilized in many

surface DFT studies. Of course, in the rigid displacement calculation no atomic

relaxations or surface reconstructions are allowed (although easily possible for

the DFT approach). The remaining interplanar separations within the two sep-

arated blocks are maintained at their bulk equilibrium spacings, except between

the planes at which cleavage occurs. The rigid energy-separation curve pro-

vides information on important limit in the cleavage energetics. Furthermore,

the knowledge of the rigid cleavage energies is needed before the effects of relax-

ations or reconstructions might be evaluated.

The calculation of the cleavage energy and strength within an ab initio DFT

approach was described by Fu [50]. To fit the DFT results so-called univer-

sal binding energy relation (UBER) [51] is used, which conveniently describes

non-linear effects due to changes in the electronic structure during cleavage de-

cohesion. The UBER describes the energy-separation law by

Eb(x) = −Gc

[(

1 +x

l

)

exp(

−xl

)

− 1]

. (3.15)

The critical cleavage stress σc is given by the maximum of the stress dEb(x)/dx,

resulting in

σc =Gc

el. (3.16)


Figure 3.3: Sketch of the brittle cleavage model. Two adjacent planes are rigidlyseparated by a distance x, the spacing of the remaining planes is kept fixed at itsequilibrium bulk value.

The critical stress σc represents the maximum tensile stress perpendicular to

the given cleavage plane, that can be withstood without spontaneous cleavage.

Because no relaxations are allowed the procedure corresponds to uniaxial strain

geometry of tensile loading.

UBER was first proposed for metallic interfaces, and was claimed to be an

universal relation between binding energy and interplanar separation. An exact

derivation of UBER has not been proposed so far. In the original paper [51], its

validity was explained on the basis of a jellium model. Based on the arguments

in [51] it may seem that its universal nature is rooted in metallic screening,

nevertheless it is not limited to metal interfaces or to simple metals. UBER

has provided its validity in transition metals and intermetallic compounds as

well [52, 46]. In fact, UBER has been found to apply accurately to essentially

all classes of materials from stainless steel to chewing gum [53]. Exceptions

are cases in which strong covalent bonds are broken, like for the cleaving of

diamond and silicon between the narrow-spaced (111) planes. Nevertheless, for

an overwhelming number of materials and directions UBER provides reliable

description of ideal fracture and adhesion. Because it is an analytic model with

only three parameters it enables to reduce the number of ab initio total energy

calculations required for a good fit [52]. In the following chapter UBER will be

applied to a wide range of materials with very different types of chemical bonding.

In these cases, UBER provided reliable fits of the energy-separation curves for

with metallic-covalent, strong covalent as well as ionic bonding.


3.2.3 Advanced applications of the ideal brittle cleavage

concept

The critical cleavage properties presented above might as well serve as an input

for other, more intricate models. For instance, following Beltz, Lipkin and Fischer

(BLF) [54], the critical energy release rate Gc for elliptical cracks may be esti-

mated, which, consequently, enables to track down the effect of crack blunting.

BLF represented a blunted crack by an elliptical cut-out in an infinite solid. They

searched for parameters controlling the cleavage in such a configuration. They

utilised the linear elastic solution for the elliptical crack subject to an external

load σ0 [55], which gives the stress σtip at the crack tip in the form

σtip = σ0

(

1 + 2

√

a

r

)

. (3.17)

In this relation, a is the length of a crack and r is the radius of the curvature at

the tip. The crucial -but natural- assumption is that the crack propagates when

the local stress σtip reaches critical cleavage stress σc of the material. Using the

relation 3.11 and solving for the critical energy release rate gives

Gc(r) =πσ2

c

E ′( 1√a

+ 2√r)2

≈ π

4E ′σ2

cr, (3.18)

in which the relation E ′ = E/(1−ν2) holds for plane strain conditions. Therefore,

the energy release rate of the elliptical crack is approximately proportional to the

tip radius r and the slope is given by the square of the critical brittle cleavage

stress. In the limit of a sharp crack (r → 0) the relation 3.18 breaks down

and Griffith’s theory prevails. Thus, as r approaches zero the critical energy

release rate approaches Gc. Accurate shape of Gc(r) at small r may be obtained

utilizing a full solution of the corresponding elasticity problem, which has been

demonstrated recently [56].

Furthermore, using a supercell approach, the influence of substitutional de-

fects or vacancies on the cleavage behaviour of a given interface might be studied,

simulating phenomena like environment-induced embrittlement. DFT calcula-

tions of such systems may provide an insight into the intrinsic influence of defects

on the interface bonding and cohesion. Such simulations are still relatively rare,

mainly due to increased computational demands of large supercell calculations.

Nevertheless, some technologically important problems have been addressed, for

example the effect of boron and sulfur on the cleavage properties of Ni3Al [57],

or hydrogen enhanced local plasticity of Al [58].

In chapter 6the ab initio DFT modelling of the enhancement of ductility

of NiAl by microalloying with various elements is discussed. This simulation

considers the effect of substitute atoms on cleavage interfaces in NiAl as well.


Figure 3.4: A sketch of brittle and relaxed cleavage models: a solid (sketched as astacking of interacting layers with a layer distance a0, panel a) undergoes brittlecleavage (panel b): the crack of size x breaks the material into two rigid blockswithout relaxing the geometry of the layers in the blocks; by the ideal elasticcleavage a process is defined, in which the material reacts perfectly elastic (panelc) up to a critical crack above which it breaks abruptly into two blocks of relaxedatomic geometries (panel d).

3.2.4 Relaxed cleavage decohesion

In contrast to the ideal brittle case now a cleavage process is considered in which

which relaxation is allowed: after cleaving the blocks by a separation x (according

to panel b of figure 3.4) the atomic layers are allowed to relax along the direction

[hkl]: the layer distances vary until the total energy reaches a minimum. If x is

smaller than a critical limit, then the crack will be healed by the elastic response.

If, however, x is too large, the bonds between the cleavage surfaces will break, the

crack remains and the atoms close to the surfaces relax their positions, forming

structurally relaxed surfaces.

The concept of relaxed cleavage was first applied in calculations by Jarvis,

Hayes and Carter [59]. UBER, which was proposed for ideal brittle decohesion be-

tween unrelaxed surfaces, is now not suitable any more. A universal macroscopic

cohesive relation describing the minimum energy path which might be applied to

the relaxed cleavage process, was introduced by Nguyen and Ortiz [60]. Involving


renormalization theory, these authors claimed to have achieved a universal for-

mulation for macroscopic materials consisting of a sufficiently large ensemble of

atomic planes: after opening of a crack-like perturbation of size x there occurs an

elastic expansion until some critical limit at which structurally relaxed surfaces

are formed. The universal law was derived for number of planes N as

E(δ) = min{

C

2Nx2, 2γr

}

=

{

C2Nx2 x ≤ δc

2γr x > δc(3.19)

in which the parameters, e.g. the relaxed surface energy γr and the elastic modu-

lus C, are material and direction dependent. The critical opening δc was expressed

as

δc = 2

√

γrN

C(3.20)

The macroscopic cohesive law adopts the universal form asymptotically in the

limit of large number of planes. As Nguyen and Ortiz pointed out, this behaviour

is universal, i.e. does not depend on the form of the interatomic binding law.

Hayes, Ortiz and Carter [61] applied a repeated slab geometry (as we do) and

demonstrated that the DFT energies for relaxed cleavage follow a universal form

according to Nguyen and Ortiz. However, the results depend on N according

to equation 3.19, i.e. the macroscopic dimension of the material, which is very

unsatisfying because intrinsic properties should be independent of the macro-

scopic dimensions. Furthermore for more complex crystal structures with several

non-equivalent cleavage planes the derivation becomes clumsy.

Such a complication is unnecessary, as described above. One might follow the

spirit of UBER by introducing a critical opening x = lr, at which the materials

should crack abruptly. For smaller x the material should react perfectly elastic

with an energy quadratic in strain. Then, in a rather trivial way the decohesion

relation for x ≤ lr is derived as

G(x) =Gr

l2rx2 (3.21)

with the cleavage energy for relaxed surfaces, Gr. For crack sizes x > lr the

condition G(x) = Gr is required. Clearly, the relation of equation 3.21 fulfills the

required conditions, and does not depend on the number of layers of a macroscopic

material. This is achieved by the materials and direction dependent parameter

lr which has to be validated by fitting the simple law to the DFT data, in the

same way as done for the brittle cleavage. It turned out -rather surprisingly- that

the DFT calculations for realistic materials follow to a large extent the simple

elastic, quadratic relation (as observed also in reference [61] and demonstrated


0

1

2

3

4

5

E (

J/m

2 )

0 1 2 3 4 5x (Å)

0

10

20

30

σ (G

Pa)

0 1 2 3 4 5 6x (Å)lrl

brittle relaxed

NiAl

Figure 3.5: Brittle and relaxed cleavage for [100] direction for NiAl. Full lines:analytic models, symbols: DFT results.

by figure 3.5). Of course, deviations between the simple model and the realistic

cases occur close to the critical crack size, as shown in figure 3.5. Calculating

the first derivative σr(x) = dG(x)dx

, the critical stress is derived as the Hooke-like

relation,σr

A= 2

Gr

A

1

lr. (3.22)

Therefore, the critical stress for relaxed cleavage is independent upon the number

of planes when the critical length lr is introduced. The length lr has a simple

interpretation: it is the crack-like opening which a material can heal under ideal

elastic conditions (the material is able to fully relax). The results obtained from

the relaxed cleavage concept are discussed in the following chapter.

Chapter 4

Cleavage and elasticity

4.1 Introduction

The ability to describe or just to estimate the critical properties of crack for-

mation of a solid material in terms of its elastic properties is an objective both

of scientific as well as technological interest. Having reliable connection between

experimentally accessible macroscopic quantities - lattice parameters, elastic con-

stants, surface energies - and critical fracture properties one could for instance

easily classify new materials after their ideal mechanical properties, making it

attractive for modern materials design.

First attempts to estimate the critical cleavage stress were made by

Polanyi [62] and Orowan [45], later refined by Gilman [63]. The implicit assump-

tions contained in the Orowan-Gilman (OG) model were discussed by Macmillan

and Kelly [64], and further by Smith [65] as well. Compared with more precise

models, the Orowan-Gilman (OG) model overestimates the critical stress sub-

stantially [34]. The same result was obtained, comparing the OG model to the

critical cleavage stress calculated from the modern DFT approach [46]. Never-

theless, it is still used in applications where the simple estimate of the critical

stress of materials is of interest [66], since there is still a lack of more precise

general models. Furthermore, the OG model describes decohesion of solids at

the atomistic level using the Frenkel law with an artificial parameter, the ’range

of interaction’ [67]. This parameter depends strongly on the bond type and is

usually assumed to be approximately of the same value as the lattice parameter

because it cancels out then. Thus, the model does not make an attempt to distin-

guish between different classes of bonding, although the bond type is known to be

important factor in estimating the intrinsic fracture resistance at the atomistic

level.

39

40 CHAPTER 4. CLEAVAGE AND ELASTICITY

4.2 Orowan-Gilman model

The cohesive strength model, as suggested by Orowan and Gilman (OG), is based

on the sequential bond-rupture picture of brittle fracture utilizing an interatomic

cohesive-force function to describe breaking of bonds. The force-separation func-

tion is approximated by a half-sine curve,

σ(x) = σc sin(2πx/a). (4.1)

In order to find an expression for the stress maximum σc involving some macro-

scopic physical quantities, according to Hooke’s law the initial slope of the σ(x)

curve is related to the Young’s modulus E by

E = a0π

aσc. (4.2)

Inserting Young’s modulus a simple estimate for the critical cleavage stress is

obtained

σc =Ea

πa0. (4.3)

However, the unknown parameter a depends strongly on the bond type. Usually,

this parameter is assumed to be of atomic dimension, a ≈ a0, and it follows

σc =E

π, (4.4)

with a0 usually being the bulk-like layer-layer distance. Equation 4.4 gives a direct

relation between the critical stress and Young’s modulus. However, the prefactor

1/π highly overestimates σc. The unknown parameter a might be eliminated in

a more elegant way: as two new surfaces are formed during crack propagation

one can presume that the area under the force-separation curve gives the ideal

brittle cleavage energy Gc (a quantity 2γs was used by OG, where γs is the surface

energy per unit area, however, the relation Gc = 2γs holds only for equivalent

surfaces)∞∫

0

σ(x)dx = Gc. (4.5)

Such a relation is strictly valid only for brittle materials in which no plastic

energy dissipation occur, as discussed in the introduction. The combination of

assumptions represented by equations 4.2 and 4.5 is certainly rather crude since,

as pointed out by Macmillan and Kelly [64], the elastically stressed solid would

separate into a uniformly spaced set of mono-atomic planes. The surface energy

of a single plane, of course, differs from the surface energy of a semi-infinite

4.3. IDEAL BRITTLE CLEAVAGE 41

crystal. Nevertheless, the dependency on a can be eliminated and well known

OG estimate of the critical stress is obtained

σc =

√

EGc

2a0. (4.6)

In case of equivalent surfaces the relation Gc = 2γs is valid and Gc can be sub-

stituted by the surface energy γs. Thus, the OG model predicts that a high

cleavage strength is favored by a large Young’s modulus and large cleavage (sur-

face) energy, together with a short spacing of atomic planes. In applications of

equation 4.6 one has to take into the account the anisotropy of the crystal prop-

erties: Young’s modulus E should be replaced by 1/s11[hkl], in which s11[hkl]

is the elastic modulus for the direction [hkl] considered. The comparison of the

σc values given by the OG model with those computed in a more exact way

showed that the OG model overestimates the theoretical cleavage strength by a

factor of about 2 [34]. The comparison of the OG prediction with the critical

cleavage stress σc calculated by means of a modern DFT method was performed

by Yoo and Fu [46], again demonstrating systematic overestimations. As an ex-

ample, the confrontation of critical stress estimates -obtained utilizing results of

the present DFT calculations in section 4.5- for W, NiAl and VC is displayed in

table 4.1. In addition to the previously mentioned deficiencies, both estimates do

not reproduce even the trends, e.g. the OG model predicts a strong anisotropy of

strength for W, and the E/π estimate gives an opposite direction-dependence of

the critical stress in VC. Clearly, an improvement of the OG estimate is needed.

4.3 Ideal brittle cleavage

The present attempt to improve the OG estimate consists of two crucial modifi-

cations of the OG approach, namely (1) the application of UBER as an analytical

model for the bond breaking energy E(x), and (2) the localisation of the elas-

tic response. In the step (1), let us consider UBER (see equation 3.15) as a

force-separation function. As introduced in the previous chapter, the materials

and direction dependent parameters are the cleavage energy Gc and the criti-

cal length l. The critical cleavage stress is then defined by the two parameters

as σc = Gc/el (see section 3.2.2). As it should be, the UBER energy behaves

quadratically around the equilibrium x = 0, and a Taylor expansion at the equi-

librium yields

Eb(x) =Gc

l2x2 + . . . (4.7)


Table 4.1: Critical stress estimates: simple Orowan’s E/π, Orowan-Gilman esti-mate (equation 4.6) and the critical cleavage stress σc (all in GPa) calculated byVASP for selected compounds and cleavage directions [hkl].

[hkl] E/π σOGc σc

W 100 177 119 47110 171 87 44111 169 148 46

NiAl 100 65 58 26110 90 47 22111 104 89 26211 90 69 24

VC 100 206 70 32110 186 118 46111 180 147 63

The elastic energy density required to open a crack of size x might be expressed

as

Eelast(x)/V =1

2Cδ2, (4.8)

which comprises a dimensionless relative strain δ and appropriate elastic mod-

ulus C. For straining the material along [hkl] for a fixed area A of the planes

perpendicular to [hkl] the modulus C is identified as the uniaxial elastic modulus.

Now, in order to relate cleavage with elastic properties it is assumed, that

for very small cleavage separation there is an unstable equilibrium between the

cleavage decohesion and elastic response. Utilizing this assumption the energy of

elastic elongation (equation 4.8) for small x is set equal to the energy necessary

for cleavage decohesion (equation 4.7)

1

2Gcx2

l2=

1

2ALbC

x2

L2b

. (4.9)

The new unknown parameter Lb is of dimension length and establishes correct

physical dimension in the equation. The cleavage energy is defined as energy

per unit area whereas the elastic energy is contained in the volume of material.

Therefore, the elastic energy has to be rescaled. A physical interpretation of Lb

is given in the following section.

4.3. IDEAL BRITTLE CLEAVAGE 43

Now, by equation 4.9 brittle cleavage and its material parameters Gc and l

are related to the elastic properties described by the uniaxial modulus C and

the length Lb. Equation 4.9 is the basis on which relations between all crucial

parameters might be constructed. For instance, Lb can be expressed as

Lb = ACl2

Gc. (4.10)

All quantities at the right side of equation 4.10 can be calculated by an DFT

approach. For the critical stress one can derive the equation

σc

A=

1

e

√

GcC

Lb

. (4.11)

Obviously, equation 4.11 is very similar to the OG relation. The difference lies

in the constant prefactor 1/e which naturally arises from the use of UBER as

the force-displacement law and the parameter Lb, which substitutes the bulk-like

interplanar distance a0.

Thus, if some general correlations of the Lb with other macroscopic parameters

describing the solid are found, equation 4.10 may well serve as approximate esti-

mate for the critical cleavage stress, because in principle both C and Gc might be

obtained from experiments. The lengths l and Lb, however, are internal materials

parameters which are not directly accessible by experiment. (Of course, they can

be derived from DFT calculations, as demonstrated below). For rigid cleavage

separation, C is identified to be the elastic constant c′11 in a given [hkl] direction,

which can be calculated from measured or calculated elastic constants [22]. For

instance, in a cubic crystal three independent elastic constants c11, c12 and c44are involved and the direction dependent uniaxial modulus is given as

c′11[hkl] = c11 − 2(c11 − c12 − 2c44)(h2k2 + h2l2 + k2l2). (4.12)

Analogous, for a tetragonal lattice the relation

c′11[hkl] = c11(h4 + k4) + c33l

4 + h2k2(2c12 + c66) + l2(1 − l2)(2c13 + c44) (4.13)

is valid (for symmetry classes 4mm, 42m, 422, 4/mmm). The cleavage energy

might be obtained for equivalent surfaces as twice the surface energy γs. For

non-equivalent surfaces this is not possible. It should be noted, that throughout

this section Gc is the ideal cleavage energy obtained for a rigid block separation

and, therefore, the role of surface structural relaxation (or reconstruction) is

neglected. This is done because ideal brittle cleavage should be modelled. The

DFT approach easily allows a full relaxation of any structural degree of freedom,

if wanted. Structural relaxations (i.e. reconstruction of surfaces) usually have a

rather small influence on the cleavage or surface energy (within a couple of per

cents).


4.4 Localisation length

The necessity to rescale the elastic energy appeared also in the OG approach.

There, the work needed to cleave is related to the elastic energy stored between

adjacent atomic planes with bulk-like separations [34]. Using this implicit as-

sumption, the OG relation was derived and that is why the interplanar bulk-like

spacing a0 appears in equation 4.6. Although it may be tempting to accept this

presumption, there is no obvious physical reason why a0 should be the scaling

factor between both energies. As it turns out, this presumption is utterly wrong.

The conceptual problem in correlating cleavage and elastic properties consists

in correlating a non-local property to a local property: the elastic response to a

perturbation is usually described as non-local quantity with its energy distributed

over the macroscopic volume Vmac of the whole material. The cleavage energy,

however, is considered to be localized in some local volume Vloc in the vicinity of

the crack. In a ’gedanken’ experiment, the energy for initializing infinitesimally

small cracks may be consumed by an elastic deformation: then, the elastic re-

sponse and the energy for opening an infinitesimally small cleavage can then be

set equal. Consequently, a correlation between elastic and cleavage properties in

the localized volume Vloc, or -optionally- in the non-local macroscopic volume

Vmac should exist.

In both, the brittle and relaxed cleavage models the solid is cleaved into two

rigid blocks terminated by surfaces of area A. The local volume Vloc can then be

expressed by

Vloc = A L, (4.14)

Therefore, in a general approach some length L, defining the volume Vloc = AL

over which the elastic energy is distributed, has to be introduced. The parameter

L is called localisation length and enters the model as an intrinsic parameter

depending on the material and the direction. Hence, it is assumed that only the

elastic energy Eelast localized within the volume Vloc contributes to the cleavage

process. This is the first time that a rigorous definition of the localisation of

the elastic energy is given. Usually, it is argued that the elastic energy must be

localized somewhere. . . Consequently, the macroscopic volume of the solid body

is defined as

Vmac = A D, (4.15)

where D describes the macroscopic, actual thickness of the material in direction

[hkl]. By equation 4.14 and 4.15 a simple rescaling condition between the two

volumes exists, namely Vmac = Vloc(D/L). The rescaling factor D/L (or its

inverse) transforms local quantities into nonlocal ones (or vice versa for the inverse

4.5. RESULTS FOR IDEAL BRITTLE CLEAVAGE 45

factor). The rescaling between local and nonlocal quantities is symmetric in the

sense that the same relation as the equation 4.11 is obtained when the elastic

energy is distributed in the macroscopic volume, defined by the equation 4.15,

but now the cleavage energy is rescaled by Gc,mac = GcLb/D, and applied in

equation 4.9.

Exploiting equation 4.9, another, very direct relation between the cleavage

stress and the elastic modulus can be formulated

σc

A=

1

e

l

LbC , (4.16)

in which the ratio of the two intrinsic lengths l and Lb enters as a prefactor

for the elastic modulus. The localisation length is independent of the actual

thickness (i.e. the number of layers) due to the rescaling of the elastic energy to

the local volume (or, vice versa, rescaling the cleavage energy to the macroscopic

volume). Because of that, the parameter Lb might be useful in any concepts of

coarse-graining which describes the transition from an atomistic to a macroscopic

form of cohesion [60]. Therefore, the behaviour of Lb and the cleavage properties

in various classes on materials is investigated. The goal is to find a general

correlation of the Lb with other macroscopic parameters describing the solid.

4.5 Results for ideal brittle cleavage

4.5.1 Computational aspects

The exchange-correlation functional was described within the generalized gra-

dient approximation (GGA) according to the parameterization of Perdew and

Wang [68]. Convergency of the total energies with respect to basis size and num-

ber of k points for the Brillouin zone integration was checked. Atomic forces

were relaxed within a conjugate gradient algorithm whenever structural relax-

ations were required.

The elastic constants needed for the derivation of the rigid moduli C[hkl] were

evaluated as described in section 2.2. The cleavage of a single crystal was modelled

by a repeated slab scheme of atomic layers with three-dimensional translational

symmetry. The spacing of the planes inside a slab was fixed at its theoretical

value. This is the standard modelling of surfaces for most of the ab initio DFT

calculations. The consistency of the computational parameters for the elastic

constants and cleavage calculation was secured. By that, all parameters are

obtained with comparable precision and the influence of the computational setup

is minimized.


Table 4.2: The brittle cleavage properties of NiAl vs. slab thickness: cleavageenergy Gc/A (J/m2) and critical cleavage stress σc/A (GPa) with respect to thenumber of layers in the slab. interfaces

no. of planes Gc[100] σc[100] Gc[110] σc[110]2 4.41 24.4 3.23 22.64 4.71 25.2 3.24 21.88 4.79 25.4 3.24 21.812 4.79 25.5 3.24 21.8

Convergency of the cleavage energy as a function of the slab thickness was

tested for NiAl, as demonstrated in table 4.2. The convergency was better for the

[110] direction, for which the slab with 4 atomic planes would be thick enough,

whereas for the mixed-atom (100) interface an 8 atom slab would be needed to

obtain very accurate cleavage energies. Thus, unit cells with 6 atomic layers

separating the (111) and 8 atomic layers separating the (100) and the (110)

cleavage interfaces were employed in the following calculations.

4.5.2 Simple metals

As a first application example, the transition metals Fe and W are chosen. Both

crystallize in the bcc structure but have some distinct properties: Fe is magnetic

whereas the bonding in W is particularly strong (as expressed e.g. by the high

melting point). Both properties strongly influence the elastic and the cleavage

behaviour. In addition, Fe and W are brittle at low temperatures and preferably

crack between (100) planes. Fracture experiments indicated that W primarily

cleaves on (100) planes, but also occasionally prefers (110) planes [69]. In more

recent experiments, (100) and (121) cleavage planes appeared, whereas (110)

planes resisted against crack propagation [70]. In particular, for W the cleavage

plane preference can not be determined by the lowest cleavage energy (see the

values of Gc in table 4.3).

The weak [100] direction in W was explained on the basis of symmetry argu-

ments: if a bcc crystal is sufficiently strained in the [100] direction (i.e. the cubic

structure is deformed to a tetragonal structure with c/a > 1) it will eventually

be transformed to fcc (for c/a =√

2). Such a continuous deformation path is

called Bain’s path. Because of symmetry, the stress vanishes for either the bcc

and fcc structure along the volume-conserving Bain’s path [71]. Therefore, the

corresponding deformation energy at the fcc point must reach an extremum (be-


Table 4.3: Calculated parameters for brittle cleavage of fcc Al and bcc W andbcc Fe in direction [hkl]: uniaxial elastic modulus C (GPa), cleavage energy persurface area Gc/A (J/m2), critical length l (A ), maximum stress per area σc/A(GPa), bulk interlayer distance a0 (A ), and localisation length Lb (A ). Resultsfor Fe derived from spin polarized calculations.

[hkl] C Gc/A l σc/A a0 Lb

Al (fcc) 100 110 1.8 0.57 12 2.03 2.01110 113 2.1 0.64 12 1.43 2.24111 114 1.6 0.54 11 2.34 2.08

Fe (bcc) 100 302 5.3 0.58 34 1.41 1.93110 338 5.0 0.54 35 1.99 1.97111 350 5.8 0.61 35 0.82 2.25

W (bcc) 100 540 8.4 0.66 47 1.59 2.80110 516 6.5 0.55 44 2.24 2.40111 508 7.3 0.64 42 0.92 2.83

ing a maximum, minimum or a saddle point). Similarly, trigonal transformation

(in [111] direction) connects bcc-sc-fcc structures, however, in W the energy dif-

ference between bcc and sc structure was found much larger than bcc-fcc energy

difference [36]. No similar symmetry-dictated extrema exist for the [110] direc-

tion, and therefore the [100] direction seemed to be the direction of the easiest

cleavage [36, 37].

Another explanation of the observed preference of the (100) cleavage in W

was given by Riedle et al. [72]. The brittle cleavage might be anisotropic with

respect to the crack propagation direction within one cleavage plane, and ’easy’

and ’tough’ cleavage systems can be distinguished. Riedle et al. argued that

(100) planes provide two independent easy directions while for a (110) plane

there is only one easy direction but also one tough direction. According to this

observation, a crack with an arbitrary oriented front should generally prefer (100)

cleavage.

Table 4.3 reveals that the critical stress per area of both Fe and W is rather

isotropic but very different in value. There is no obvious relation between stress

and interplanar lattice spacing a0, as it is assumed in the OG model, because the

parameter a0 varies by more than a factor of two. The new length Lb, however,

shows much smaller direction dependence varying within 20% only. All the listed

parameters (in particular C, Gc/A, and σc/A) are significantly larger for W,

which reflects the stronger bonding.


0

2

4

6

Eb (

J/m

2 )

0 2 4 6 8x (Å)

0

1

∆µ (

µ B)

Fe

Figure 4.1: Brittle cleavage for Fe: decohesion energy (upper panel) and thechange of magnetic moment vs. cleavage size x: in [100] (full circles), [110] (righttriangles), and [111] (diamonds) direction. Lines: fit to UBER (upper panel),and guiding the eye (lower panel).

For W, tensile tests were simulated by DFT calculations of Sob et al. resulting

in values of the critical stress of 29 GPa for the [100], 54 GPa for the [110] and

40 GPa for the [111] direction [36], which are significantly different from the

results in table 4.3, because different concepts were applied. We focused on

brittle cleavage for which we utilized the uniaxial (rigid) modulus in order to

find a correlation between cleavage and elastic properties. Sob et al, however,

investigated the elastic response under tensile tests which probe the attainable

stress of ideal crystals without any cracks. It should be noted, that Sob et al

applied the LDA for their DFT application which always yields stronger bonding

in comparison to GGA calculations.

Despite such a tendency to cleave on (100) planes, whether an ideal single

crystal of W fails by fracture or shear depends on the loading direction. Any of

〈111〉 slip systems has the ideal shear strength around 18 GPa [37] and, therefore,


for non-[100] directions the resolved shear stress is always high enough to promote

shear failure. Tensile fracture experiments on microcrystalline W whiskers with

the long axes in [110] direction and different diameters found a maximum strength

of 28 GPa [73], which is significantly smaller than for brittle cleavage for which

a critical stresses larger than 40 GPa (see table 4.3) is obtained. This suggests

that whiskers failed by shear on some of the favorably oriented planes.

Ab initio simulation of tensile tests for Fe [74] indicated that the [100] di-

rection is also the direction with lowest critical stress. However, the symmetry

analysis based on the bcc to fcc Bain’ path is hampered by a following problem:

performing ab initio calculations for fcc Fe yields that it is energetically close

to the bcc ground state and is at least metastable at low temperatures. As a

consequence, the ideal tensile strength in [100] direction would be grossly un-

derestimated. For more information we refer to the papers of Herper et al. [75],

Clatterbuck et al. [76] and Friak et al. [77], who concluded that that the ideal

mechanical strength of Fe is determined by a subtle interplay of crystal structure

and magnetic ordering.

Figure 4.1 shows the change of magnetic moments due to cleavage. For small

separations x the moments increase linearly for all directions, and for separations

x > 2 A saturation is reached because of the formation of free surfaces. The

influence of ferromagnetic ordering is particularly strong for the (100) cleavage.

Finally, Al was chosen as example of an fcc system. The electronic structure

of Al is rather free-electron like and a relatively large k-point grid -in comparison

with W and Fe- was necessary to obtain convergent total energies. According to

table 4.3, for Al the cleavage energy and the critical stress are smaller by at least

a factor of 2 in comparison to the d-d bonded transition metals. The critical

stress is rather isotropic as one would expect for a metal with a free-electron like

electronic structure. The UBER parameter Gc, l and σc are in perfect agreement

to the reported values of Hayes et al. [61]. The localisation length is largest for

the [110] direction, which has shortest plane spacing. In general, the localisation

lengths display relatively moderate values compared to W and Fe, though the

cleavage energies and critical cleavage stresses are much smaller.

Interestingly, in Al are the present cleavage calculations are in excellent agree-

ment with DFT simulation of the tensile test. Li and Wang [78] reported 12.65

GPa and 11.52 GPa for uniaxial deformation (which comprises no relaxations

perpendicular to loading direction) in the [001] and [111] directions respectively.

In case of uniaxial loading, for which lateral relaxations were allowed, Li and

Wang obtained 12.1 GPa and 11.05 GPa, very similar to 12 GPa and 11 GPa as

obtained in the present work.

Furthermore, for Al several experiments studied the maximum tensile


strength. The value 10.9 GPa is reported from the tensile test on whiskers in

[0001] direction [79]. The remarkable agreement of either theoretical estimates

with experiment suggests that Al fails in tension rather than by shear. In may be

noted, that due to the low mobility of dislocations at ambient temperatures the

difference between the strength of whiskers and bulk specimens is relatively low

and, consequently, the strength -as large as 7 GPa- was obtained by rod bending

experiment [80].

4.5.3 Intermetallic compounds

Another interesting class of materials are ordered intermetallic compounds. Their

mechanical properties are to a large extent governed by processes at the atomic

scale, because typical crack mode I propagation or blunting depends on the com-

petition between the cleavage decohesion and the emission of dislocations. At

room temperatures, intermetallic compounds typically fail by brittle fracture,

which has important consequences for the fabrication of such materials for tech-

nological applications.

As table 4.4 reveals, the calculated critical stresses per area are rather

isotropic for all compounds. Only the (100) cleavage of FeAl is exceptional due to

occurrence of magnetic ordering. Again, the interplanar spacings vary strongly

and, consequently, any models for cleavage based on these parameters will clearly

fail. As discussed above for the transition metals, the localisation length Lb varies

in a rather narrow interval of 2.0 A to 2.8 A. In comparison, values for a0 range

from 0.83 A for the (111) stacking of B2 FeAl to 2.37 A for the stacking of L12

Al3Sc in [111] direction. In particular the low values of a0 values for NiAl and

FeAl in the [111] direction lead to a substantial overestimation of σc within the

OG model.

Concerning the strength (i.e. critical stress per area), FeAl, Ni3Al and NiAl

are rather comparable. Note that FeAl in [100] direction is weakened due to

magnetic ordering as shown in table 4.5, otherwise all critical stresses would be

larger than 30 GPa. The compound Al3Sc is the weakest, presumably due to the

weaker d-character of the bonding. No direct measurements of the ideal strength

of the studied intermetallic compounds are available whereas Yoo and Fu [46]

performed pioneering ab initio calculations for the same class of intermetallic

compounds as in the present thesis. However, they did not derive any useful

connection between elastic and cleavage properties because they proposed an

OG-like model involving the interplanar spacings (see equation 2 in their paper)

producing much too high values for σc. Yoo’s and Fu’s results for Gc, C, and σc

are somewhat larger than the present ab initio data, because they applied LDA


Table 4.4: Calculated parameters for brittle cleavage for some selected inter-metallic compounds together with their crystal structures. Further details, seetable 4.3.


NiAl B2 100 203 4.8 0.69 26 1.45 2.01110 284 3.2 0.54 22 2.05 2.59111 311 4.1 0.58 26 0.84 2.68211 284 4.0 0.60 24 1.18 2.56

Ni3Al L12 100 225 4.3 0.66 24 1.78 2.28111 331 3.7 0.52 26 2.06 2.42

FeAl B2 100 278 4.8 0.71 25 1.43 2.92110 354 4.3 0.50 32 2.03 2.06111 380 5.1 0.61 31 0.83 2.77

Al3Sc L12 100 189 2.7 0.61 16 2.05 2.60110 182 2.9 0.65 17 1.45 2.65111 180 2.6 0.61 16 2.37 2.58

TiAl L10 001 185 4.4 0.70 23 2.03 2.06100 190 3.3 0.58 21 2.00 1.98110 240 4.1 0.69 22 1.41 2.82111 268 3.5 0.58 22 2.32 2.57

Table 4.5: Calculated parameters for brittle cleavage for FeAl: comparison ofspin polarized (mag) and non spin polarized calculations. Further details, seetable 4.3.

[hkl] Gc/A l σc/Amag 100 4.8 0.71 25

100 5.7 0.64 33mag 110 4.3 0.50 32

110 4.7 0.52 33mag 111 5.1 0.61 31

111 6.1 0.62 36


[010]

[001]

[110]

[001]

Figure 4.2: The cleavage interfaces in FeAl. The (100) cleavage produces surfacelayers of pure Fe and Al (left panel), whereas (110) planes contain both types ofatoms (right panel).

whereas in the present GGA is used for the inherent approximation to the many

body terms of DFT. It is well-known that in many cases LDA overestimates the

strength of bonding.

Recently, Tianshu et al. simulated by ab-initio calculations tensile tests for

NiAl, FeAl and CoAl [81]. Although tensile test simulations represent different

type of material tests, the reported values of the ideal tensile strength rather agree

with values of σc in the [110] and [111] directions, but differs from the present

results in finding 45 GPa and 19 GPa for the [100] ideal tensile stress NiAl and

FeAl, respectively. The surprisingly very low ideal stress for FeAl was attributed

to a small local maximum of stress at relatively small strains, preceding the global

stress maximum. Consequently, Tianshu et al. suggest that in the [100] direction

FeAl becomes unstable before the global stress maximum is reached. However,

the calculations of Tianshu et al. are non spin-polarised although magnetic mo-

ments might appear in highly strained FeAl. In the ground state FeAl should

be nonmagnetic, standard DFT yields a small magnetic moment. Nevertheless,

the energy difference between the nonmagnetic and ferromagnetic ground state

is very small [82, 83] and has no influence on the present results.

Experimentally, FeAl shows preference for (100) fracture facets, in contrast

to NiAl and CoAl for which such a fracture behaviour is unfavorable [84]. The


0

2

4

6E

b (J/

m2 )

(100) non pol.(110) non pol.(100) spin pol.(110) spin pol.

0 1 2 3 4 5 6 7x (Å)

0

1

2

µ (µ

B)

FeAl

Figure 4.3: Calculated brittle cleavage for FeAl. Decohesion energy (upper panel)and generated surface magnetic moment (lower panel) for the (100) and (110)cleavage. Lines: UBER fit (upper panel), guiding the eye (lower panel).

present calculations for FeAl for brittle cleavage also find that (100) cleavage is fa-

vorable because the magnetic ordering reduces the critical stress for this direction

significantly compared to the (110) case which is preferred in the other transition

metal aluminides. The comparison of the cleavage parameters of FeAl obtained

from non-polarised and spin-polarised calculation is displayed in table 4.5 and

the dependence of cleavage energy and magnetic moment on the opening of the

brittle crack is illustrated in figure 4.3. For the (100) case (in contrast to the

(110) cleavage) the slope (i.e. the stress) of the decohesion energy also changes,

and not only the value of the cleavage energy (i.e. the asymptotic value of the

energy). It should be noted, that the (100) cleavage produces surface layers of

pure Fe and Al, whereas (110) planes contain both types of atoms. The interfaces

of B2 structure are sketched in figure 4.2.


0 1 2 3 4 5 6 7x (Å)

0

1

2

3

4

Eb (

J/m

2 )

0 2 4 6

0

2

4

VC (001)TiC (001)

Figure 4.4: The (100) cleavage of VC and TiC. The circles: VASP results; lines:fit of UBER.

For NiAl, the calculations of Tianshu et al. indicate [111] as a weak direction.

Fracture experiments, however, have found (110) cleavage habit planes (some-

times also the higher-index (511) cleavage planes) [84, 85, 86]. The preference

for (110) planes could be deduced from the calculated data, because Gc and σc

are lowest for the [110] direction. It may be noted, that no magnetic moment

appears during cleavage of NiAl and, therefore, the different cleavage behaviour

of FeAl and NiAl seems to be due to the formation magnetic order in FeAl. This

fact has important consequences in the modelling of the mechanical response of

materials where magnetic ordering may feature, because magnetic properties are

in common neglected in large scale simulations.

4.5.4 Refractory compounds

The properties of the refractory compounds VC and TiC reflect the very strong

covalent-like p-d bonding (e.g. extremely high melting points and hardness) and,

therefore, their strength mainly stems from the properties at the atomic scale.

Their overall materials properties makes the fabrication of well-defined samples

prohibitive (at least as bulk phases). Therefore, ab initio studies of elastic and

mechanical properties are rather valuable.


Table 4.6: Calculated cleavage properties for brittle cleavage for TiC and VC.


VC B1 100 647 3.2 0.37 32 2.08 2.77110 585 7.0 0.55 46 1.47 2.53111 564 9.9 0.58 63 1.20 2.06

TiC B1 100 515 3.5 0.42 31 2.17 2.57110 489 7.7 0.56 51 1.53 1.97111 481 11.6 0.70 61 1.25 2.03

Fracture experiments revealed pronounced preference for (100) cleavage planes

in carbides of cubic crystal structure [87], which is obvious in the calculation as

well: table 4.6 shows a strong variation of the critical stress per area for both, VC

and TiC, which is mainly due to the strong anisotropy of the values for Gc/A.

The energy profile for the (100) cleavage in VC and TiC is displayed in figure 4.4

revealing very steep increase of the cleavage energy with separation compared to

other compounds (see figure 4.3 for example).

It is noticeable, that for the (100) cleavage the critical stress of σc/A=32 GPa

is lowest (and comparable e.g. to FeAl) but C and Lb are the largest when

compared to the other two directions. The low value of Gc for (100) cleavage

might be explained in terms of breaking nearest neighbor bonds when cleaving,

because only one of the six nearest-neighbor p-d bonds is broken when cleaving

(100) planes. Such simple models work only for very strong covalent bonds, they

will fail for intermetallic compounds as discussed for NiAl [88]. Again, the (100)

cleavage is found to be exceptional when the very short critical lengths of l=0.37

A and 0.42 A for VC and TiC are considered (see table 4.6). Presumably, this

feature indicates the brittleness of the carbides.

The much stronger anisotropy of bonding properties of the refractory com-

pounds in comparison to the intermetallic materials is also reflected by the more

expressed direction dependency of the localisation lengths Lb, which is now of a

size comparable to the anisotropy of the bulk interlayer spacings a0.

Since tensile tests of the hard carbides are difficult to perform, no experimen-

tal information is available. Fracture experiments revealed preference for (100)

cleavage planes in carbides of cubic crystal structure [87], in clear agreement to

the present data. Price et al. [89] performed a DFT study for the (100) brit-

tle cleavage of TiC reporting a value of 40 GPa for the critical stress, which is

about 20% larger than the present value in table 4.6. Presumably, this is due to

the application of LDA by Price et al. which results in stronger bond energies.


Also some technical limitations could be influential such as the layer thickness

(in Ref. [89] the slab consisted only of four atomic layers) which certainly is a

rather small number.

4.5.5 Ionic compounds

In order to demonstrate the generality of the discussed concepts, now another

class of materials is studied, namely solids with ionic bonding. MgO is a prototype

for the ionic properties of alkaline-earth oxides, and it is also of technological

importance. It features a dislocation-free zone in front of the crack tip: when a

dislocation is emitted from the crack, it stays at a certain distance from the crack

tip and prevents further emissions. The crack cleaves then in the brittle manner.

The equilibrium distance from the crack is given by the balance between the crack

stress field and the stress of the image dislocation due to the free surface of the

crack [34]. In addition, another classic ionic compound was studied, namely NaCl

because one expects very unusual, very soft cleavage properties (rock salt is easy

to cut, and it is easy to make scratches). Studying the electronic structure of

these large-gap insulators one immediately encounters the usual DFT problem,

namely that standard ab initio calculations result in far too small gaps. This

feature of the excited states spectrum, however, does not influence the ground

state properties needed for the present purpose.

Although UBER was originally proposed for crystals with a covalent (or metal-

lic) bonding character, it also works for the cleavage of ionic compounds (at least,

when non polar surfaces are created, which are considered here). The fit of UBER

in case of NaCl is shown in figure 4.5. Table 4.7 illustrates the strong preference

for the (100) cleavage, similar to the refractory compounds with rock salt struc-

ture (MgO and, of course, NaCl have the same structure). Again, the critical

lengths l are very short for the (100) cleavage. For MgO, the localisation lengths

Lb are comparable to the values e.g. derived for the intermetallic compounds.

NaCl, however, is exceptional in every respect, and in particular for the (100)

cleavage, for which Lb=4.16 A is derived. For no other class of materials it is

found such a large value. Surveying table 4.7 one notices the very weak elastic

modulus, the very low cleavage energy and stress per area.

For NaCl, experimental tests of the maximum strength were performed as

well. The highest strength recorded for a whisker crystal is 1.6 GPa in [100]

tension [90]. This value is in excellent agreement with 2 GPa obtained in the

present study.


0 2 4 6 8x (Å)

0

0.2

0.4

0.6

0.8

Eb (

J/m

2 )

NaCl

Figure 4.5: The brittle cleavage of NaCl. The full circles and diamonds arecleavage energies vs. separation for [100] and [110] derived by VASP; the linesare fit to UBER.

4.5.6 Diamond and silicon

Finally, two elemental prototypes of covalent bonding are discussed: Si and the

very hard material diamond which should be suitable for expecting brittle frac-

ture. For diamond, by transmission electron microscopy cracks have been ob-

served to propagate without the emission of dislocations [91]. The experimentally

claimed preference for (111) cleavage was recently corroborated by ab initio ten-

sile test simulations [92], which derived the significantly lowest tensile strength for

the [111] direction (in comparison to the [110] and [100] directions). The reported

values for the ideal strength of 225 GPa for [100] and 93 GPa for [111] directions

compare well with the present data for σc/A as shown in table 4.8. For Si, the

calculated ideal tensile stress of 22 GPa for the [111] direction by Roundy et.

al. [66] is also in prefect agreement with our value. This leads to the conclusion,

that when the elastic stability of a solid is not governed by the appearance of

some higher-symmetry structures along the corresponding transformation path

the results of ideal tensile test and brittle cleavage are in reasonable agreement,


Table 4.7: Calculated parameters for brittle cleavage for MgO and NaCl.


MgO B1 100 299 1.8 0.37 18 2.11 2.27110 345 4.4 0.54 30 1.53 2.29

NaCl B1 100 52 0.3 0.49 2 2.83 4.16110 45 0.7 0.66 4 2.00 2.84

Table 4.8: The calculated parameters for brittle cleavage for diamond and Si.


C A4 100 1045 18.3 0.35 193 0.89 0.75111 1210 11.5 0.45 93 1.55 2.08

Si A4 100 154 4.3 0.53 30 1.37 1.01111 189 3.1 0.54 21 2.37 1.78

although the elastic response (i.e. the elastic modulus) to the deformation is

different, because for simulating brittle cleavage no lateral relaxation is allowed

in contrast to the ideal strength studies for tensile tests.

Because the diamond lattice consists of two fcc sublattices, for stacking in the

[111] direction two different interplanar spacings occur, a short and a long one

with a ratio of 1 : 3. (It is referred to it as short and long spacing). Obviously,

the weaker bonding between planes separated by the longer spacing makes them

easier to cleave, in comparison to planes separated by the short spacing (see

figure 4.6).

The values in table 4.8 correspond to the data derived from cleaving the long

spacing, for which the number of broken nearest-neighbor bonds per unit area

is three times smaller than for the short spacing. Cleaving the planes with the

short spacing, UBER fails to describe the decohesion energy for larger separations

because of a maximum in the energy curves (see figure 4.6). The usual interpreta-

tion is [93] that strong directional bonds have to be re-oriented when broken, and

this causes the uncommon maximum of cleavage energy at finite separation. This

energy maximum is caused by second-nearest neighbor interactions, as shown by

figure 4.7, in which isolated carbon planes spaced at distances corresponding to

the (111) stacking of diamond are cleaved. When only nearest-neighbor planes

are present, the energy maximum atG(x) curve does not appear at all. It emerges

only when the second pair of planes enters the calculation.

As a demonstration of the difference between GGA and LDA calculations,


0 1 2 3 4 5x (Å)

0

10

20

30

Eb (

J/m

2 )

diamond

Figure 4.6: Brittle cleavage of diamond in [111] direction: cleavage for long in-terlayer spacing (full circles), for short interlayer spacing (diamonds), LDA cal-culation for short spacing (triangles). Lines: UBER fit.

figure 4.6 compares the results for the short spacing cleavage along [111], clearly

showing the enlarged cleavage properties (energy and stress) for the LDA ap-

plications. It also shows that the energy maximum discussed in the preceding

paragraph is not an artificial product of GGA, because it is reproduced by LDA

as well.

The localisation lengths Lb in table 4.8 are rather small, in particular for the

(100) cleavage, indicating a strong localisation of the elastic energy. When releas-

ing this energy (i.e. fully relaxing the structure of the cleavage plane surfaces)

one should note, that reconstruction (i.e. the change of atomic positions in the

layers plays now a -direction dependent- major role, and significantly also influ-

ences the cleavage energy, as illustrated for Si by an abundant number of studies

searching for the stable reconstructed surface (e.g. Ref. [94]).


0 1 2 3 4 5x (Å)

0

5

10

15

20

25

Eb (

J/m

2 ) crack

a)

b)

vacuum

vac.

b)

a)

Figure 4.7: The simulation of diamond (111) cleavage: the isolated carbon planeswith diamond (111)-like spacing are separated. The chosen supercell geometryis sketched inside the figure. The vacuum region surrounds the isolated planesfrom both sides, due to the periodic boundary conditions.

4.5.7 Conclusions

By combining analytic models for the brittle cleavage process with ab initio DFT

simulations well-defined correlations between elastic and cleavage properties were

established. This was made possible by the concept of localizing the energy of

the elastic response and relating the localized energy to the energy of crack-like

perturbation in the spirit of Polanyi [62], Orowan [45] and Gilman [63], the basic

principle was suggested more than 80 years ago. Probably, the main achievement

of the thesis consists in the introduction of a new materials parameter, which

was defined as the localisation length L. By this flexible parameter the bridge

between elastic and cleavage energy (or stress) was built. The actual values of L,

which depend on the material and the direction of cleavage, has to be determined

by fitting to DFT calculations of the decohesive energy as a function of the crack


opening. The concepts were tested for all types of bonding. For brittle cleavage

it turned out, that -at least for metals and intermetallic compounds- an average

value of Lb ≈ 2.4 A would yield reasonably accurate cleavage stresses if one knows

only the uniaxial elastic modulus and the brittle cleavage energy. This means,

that the ”engineer” may estimate the critical mechanical behaviour of a mate-

rial -at least for simpler types of crack formation- purely knowing macroscopic

materials parameters, namely the cleavage energy and the elastic moduli. (Even

if the cleavage energy is not easily accessible experimentally, it could be derived

from a single DFT calculation for each direction, which in many cases is not very

costly.)


4.6 Relaxed cleavage

4.6.1 Correlation between cleavage and elasticity

In contrast to the ideal brittle case a cleavage process is now considered for

which relaxation is allowed. The concept of relaxed cleavage was outlined in

section 3.2.4 in details, here the important results are briefly repeated for the

sake of consistency. In the first step, the cleavage-like preopening x is introduced

into bulk spaced lattice in the same manner as for the brittle cleavage. But then

the atomic layers are allowed to relax along the direction [hkl]. The relaxation

involves the spacings of planes (cleavage plane area A is fixed), thus uniaxial

stress conditions are modelled. If x is smaller than a critical limit then the crack

will be healed by an elastic response. If, however, x is too large, the bonds

between the cleavage surfaces will break, the crack remains and usual structural

relaxation of cleaved surfaces occurs. It should be noted, that surface relaxations

are not involved in this model.

As discussed in the section 3.2.4 for smaller x the material reacts perfectly

elastic with an energy quadratic in strain. In the spirit of UBER a critical opening

x = lr was introduced, at which the materials should crack abruptly. Then, the

decohesion relation for x ≤ lr was derived as

G(x) =Gr

l2rx2, (4.17)

with the cleavage energy for relaxed surfaces, Gr. For crack sizes x > lr the

condition G(x) = Gr is required. Clearly, this relation fulfills the above condi-

tions, and does not depend on the number of layers of a macroscopic material.

Calculating the first derivative σr(x) = dG(x)dx

, the critical stress may be evaluated

σr

A= 2

Gr

A

1

lr. (4.18)

Again, like for the brittle case the correlation between elastic and cleavage prop-

erties is established by setting equal elastic and relaxed cleavage energy for very

small crack size x. It may seem, that the connection can now be done for any

x ≤ lr, because both type of energies are now quadratic in x (For UBER, this was

valid only for x→ 0). But with larger size of the preopening x anharmonic elastic

effects become important and, therefore, the advantage of the simple description

in the terms of volume-independent first-order elastic constants would be lost. In

order to prevent that, the connection has to be established for small crack sizes

analogically to brittle cleavage. Again, a localisation length is introduced as a

4.6. RELAXED CLEAVAGE 63

new materials parameter and by that the key relation is derived,

Grx2

l2r=

1

2AC

x2

Lr

(4.19)

containing only intrinsic materials parameters. Again as for the brittle case, in-

stead of localizing the elastic energy, the cleavage energy can be delocalized by

multiplying with a scaling factor Lr/D. As for the brittle case, the macroscopic

dimension D cancels out from the equations. No unwanted dependency on any

artificial number of layers is necessary, Lr can be determined by fitting the ana-

lytic expressions to a proper set of DFT data. An obvious but elegant relation

can be gained for the critical stress stress

σr

A=

lrLrC. (4.20)

Hereby, the critical stress is directly related to the elastic constant. Obviously,

it is linearly proportional to C with the slope given by ratio of two intrinsic

parameters lr and Lr.

4.6.2 Results

The determination of the parameters is done in a similar way as for the brittle

case, but for the relaxation of atoms after the opening a crack of size x is now

performed. For that, forces acting on the atoms are calculated, and the minimum

of forces is searched for by a conjugate gradient algorithm [95].

For the ideal relaxed cleavage the critical lengths lr and localisation lengths

Lr are much larger than for the brittle case, as shown in table 4.9 and figure 4.8.

This seems to be obvious because for the ideal elastic cleavage the material is

now allowed to relax after the crack initialization and therefore it needs much

larger crack sizes to break it. Also a strong variation of Lr is noticeable, which

is in contrast to the brittle case. Some -but no simple- correlation between the

critical lengths and the localisation lengths exists because, generally, for larger lrthe values of Lr are larger as well. Also the critical strengths σr are significantly

enhanced in comparisons to σc, whereas the cleavage energies Gr -although re-

duced compared to Gc- differ not very strongly from the ideal brittle case for the

metallic cases. For VC and W, in particular, the relaxed critical stresses σr are

drastically increased because of the very large values of the rigid elastic moduli

C[hkl]. The effect is particularly strong for the [100] direction which is also the

nearest-neighbor direction with the largest value for C. Obviously, the strongly

covalent p-d bonding of VC is responsible for these findings.


Table 4.9: Calculated parameters for the relaxed cleavage: energy per surfacearea Gr/A (J/m2), relaxation energy ∆Gr = Gc − Gr (%), critical length lr(A ), maximum stress σr/A (GPa), and localisation length Lr (A ) for selectedcompounds and cleavage directions [hkl].

[hkl] C Gr/A ∆G lr σr/A Lr

Al fcc 100 110 1.8 1 1.9 19 11.0110 113 1.9 8 2.2 17 14.4111 114 1.6 1 2.3 14 18.8

W bcc 100 540 7.8 8 2.0 78 13.9110 516 6.4 2 1.5 85 9.0111 508 6.6 12 1.6 82 9.8

NiAl B2 100 203 4.6 4 2.7 34 16.1110 284 3.1 3 2.0 31 18.3111 311 3.9 5 2.2 35 19.3

Ni3A L12 100 225 4.2 2 2.2 38 13.0111 331 3.6 3 1.6 45 11.8

VC B1 100 647 2.4 25 0.8 60 8.6110 585 6.0 14 1.6 75 12.5111 564 8.4 15 1.6 105 8.8

TiAl L10 001 185 4.2 5 2.0 42 8.8100 190 3.2 3 2.2 29 14.4110 240 3.9 5 2.2 35 14.8


0 1 2 3 4l (Å)

5

10

15

20

L (Å

)

brittlerelaxed

Figure 4.8: Localisation lengths L vs. critical lengths l for ideal brittle andrelaxed cleavage for a variety of materials and directions. Values of L for thesame compound are connected by lines.

Discussing the (111) cleavage of Al, a value for the critical stress of σr=15

GPa was obtained, which is slightly larger than the value for brittle cleavage of

σc=11 GPa. Clearly, the effect of relaxation is very small, because screening of

perturbations (i.e. creation of a surface) is fast due to the free-electron like elec-

tronic structure of Al. In Ref. [61] a layer dependent model for relaxed cleavage

was applied, the critical stress scales according to σr ∝ 1/√N with N being the

number of layers of the macroscopic solid (see equation 5 of Hayes et al. [61]). By

that, an extremely small value for the critical stress of σr=0.16 GPa is derived

for a length of 10µm in the [111] direction. On the other hand, the presented

model and the data for relaxed as well as brittle cleavage are independent of any

macroscopic dimension (as long as the actual slab of material is large enough to

be bulk like). However, for brittle cleavage (see section 4.5.2) the agreement for

UBER parameter of the present calculation and Ref. [61] is perfect.


0 1 2 3 4 5 6 7x (Å)

0

1

2

3

4

5

G (

J/m

2 )

(100) - short axis(110)(001) - long axis

TiAl

Figure 4.9: Relaxed cleavage for TiAl. The gap symbols the formation of relaxedcrack surfaces.

Inspecting the values of σr/A in table 4.9, in NiAl and VC one finds similar

directional anisotropy as found for brittle cleavage. The relaxed critical lengths lrfollow more less the trends of their brittle counterparts l. In contrast, as displayed

in figure 4.9, TiAl breaks first in [001] direction -along longer axis- while in

[100] and [110] directions TiAl can heal larger precrack sizes. One would rather

expect cleavage in [100] direction to precede in forming of the crack, because

elastic moduli in both directions are very close and, therefore, considerably lower

Ge(100) should be reached prior to Ge(001). The possible explanation is that en

route precrack → elastic response some unstable state has to be passed. This

unstable equilibrium is caused by the forces acting on the surface layers and leads

to bad convergency of DFT calculation around x ≈ lr.

As consequence, the unstable state acts like an energy barrier and may even-

tually stabilize the crack prior the energy really reaches Ge, as demonstrated

in figure 4.9. For instance, exploiting (001) relaxed cleavage in TiAl, a stable


opening -one which does not heal- is found at G = 3.32 J/m2, whereas relaxed

cleavage energy Ge is as high as 4.19 J/m2. For (100) and (110) cleavages is this

effect less obvious, nevertheless still apparent. Thus, proposed analytical model

for relaxed cleavage provides reliable description for the energy in cases, where

the energy barrier between preopened state and state with uniformly expanded

planes is low enough.

According to table 4.9, the relative energy differences δG due to surface struc-

tural relaxations are in NiAl, Ni3Al, and TiAl less than 5%, in agreement with

common expectation. The exceptional effect of relaxation is found in VC, where

the cleavage energy is reduced by 25 %, 14 % and 15 % in [100], [110] and [111]

direction, respectively.

It should be noted, that relaxation was only allowed by changing the atomic

layer distances. More complex relaxations in terms of reconstructions (i.e. ge-

ometrical changes also in the planes) which might occur for certain materials

and directions would lead to smaller cleavage energies. However, reconstructions

usually result in a much smaller energy gain than the layerwise relaxations.

4.6.3 Conclusions

An useful and physically sound analytical formulation for the relaxed cleavage

process was found, which utilizes a natural parameter -the critical length for

relaxed cleavage lr- and does not depend on number of layers of the macroscopic

material, as applied in previous approaches [60, 61]. Moreover, the parameter lrgives a measure up to which critical openings an initiated crack is able to heal

under ideal conditions. The connection to elastic properties can be again made

via the localisation of the elastic energy, however the behaviour of Lr for the

relaxed cleavage is less simple to describe and no general trend is observed yet.


4.7 Semirelaxed cleavage

4.7.1 Introduction

Many of macroscopic theories of fracture involve so-called ’cohesive zone’ in front

of the crack tip. The determination of the stress within cohesive zone is based

on the cohesive law, which shape and form is being postulated. In principle,

the cohesive zone might be modelled accurately within the framework of DFT

calculations, however, typical size of engineering models makes such calculation

impossible. Therefore, DFT methods are rather employed to obtain necessary pa-

rameters for chosen cohesive law in a given material under some kind of idealized

conditions, e.g. pure tensile stress acting in the cohesive zone.

The classical and widely used cohesive law is UBER (equation 3.15). The

UBER conveniently catches non-linear effects due to changes in electronic struc-

ture during decohesion and applies accurately to essentially all classes of materials

from the stainless steel to a chewing gum [53]. Furthermore, a recent study has

shown, that the cohesive zone model derived from fully relaxed ab initio calcu-

lations follows UBER curve very closely [96]. However, the application of UBER

within macroscopic crack simulations seems hampered by its inability to capture

shape of cohesive law when structural surface relaxation is involved, as discussed

in previous section.

Thus, the presented modified concept combines preceding brittle and relaxed

cleavage models. The results will demonstrate that the description provided by

UBER may be used even when the surface relaxation is allowed. The procedure

of the calculation is following: the cleavage-like opening x -representing a crack-

is introduced between two bulk-terminated blocks of atomic planes. Then a

plane at each side of the cleaved interface is fixed to conserve initial opening,

while atoms inside separated blocks are allowed to change their positions to their

minimum energy configuration. The unit cell dimension is relaxed in the direction

of cleavage as well, whereas its area A is fixed at a bulk value. The lowest energy

for given separation is consequently used as a data point to fit UBER. This

procedure is called semirelaxed cleavage -in order to distinguish it from relaxed

cleavage- and demonstrate its application in cases of NiAl, W and VC.

4.7.2 Results

The quantities corresponding to the semirelaxed cleavage are denoted by the

subscript s.

As a first example high-strength intermetallic compound NiAl is considered.

According to figure 4.10, which compares rigid and semirelaxed cleavage in NiAl,

4.7. SEMIRELAXED CLEAVAGE 69

0 1 2 3 4 5 6 7 8x (Å)

0

1

2

3

4

5

E (

J/m

2 )

(100) unrelaxed(100) semirelaxed(110) unrelaxed(110) semirelaxed

NiAl

Figure 4.10: The cleavage of NiAl. The cleavage energy as a function of separationalong [100] ( circles), [110] (diamonds) direction. The red lines and symbolscorrespond to semirelaxed cleavage.

Table 4.10: The cleavage parameters obtained from UBER: ideal brittle cleavageenergy Gc (J/m2), brittle critical length lc A, critical stress σc and their semire-laxed counterparts denoted by index s. The values in brackets were obtainedallowing lateral dimension of unit cell to relax, see text.

[hkl] Gc lc σc Gs ls σs

NiAl 100 4.88 0.68 26.4 4.72 0.69 25.2110 3.24 0.49 24.3 3.18 0.49 23.8

W 100 8.53 0.66 47.4 7.99 0.66 44.3110 6.49 0.54 44.2 6.35 0.53 44.1

VC 100 3.15 0.39 29.7 2.97 0.39 27.9


the relaxation acts primarily at larger separations. For x < lc no remarkable

changes of total energy are observed. The shape of energy-separation curve is

essentially unchanged as well and, as consequence, UBER provides reliable fit

for unrelaxed as well as semirelaxed DFT energies. The parameters obtained

from the fit are displayed in table 4.10. The cleavage energy is reduced by 3 %

and 2 % in the [100] and [110] direction, respectively whereas the critical length

l stays unchanged. It should be noted, that the energy reduction was driven

by the relaxation of positions of atoms, while the relaxation of unit cell lateral

dimension brought negligible change of total energy.

VC exhibits strong surface relaxations, as was demonstrated in previous sec-

tion. The cleavage habit planes in cubic carbides are (100) ones, because they

exhibit markedly lower Gc than other high-index planes. Though the fit of UBER

was very satisfactory the calculated value Gs = 2.97 J/m2 is much higher than

fully relaxed cleavage energy Gr = 2.4 J/m2 found in relaxed calculation (see

table 4.10). The fully relaxed cleavage model allows the atoms lying at the crack

surface to relax as well and, consequently, contains additional degree of freedom,

which is responsible for the discrepancy between Gr and Gs. In VC, due to its

strong covalent bonding, this effect is pronounced whereas in the case of NiAl

and W the difference between Gs and Gr lies within the bars of computational

error.

As a last example, W is discussed. Inspecting the results for W in table 4.10

one realizes much larger surface relaxations in [100] direction compared to [110]

direction. Due to the relaxations the critical stress in [100] direction -σs = 44.3

GPa- gets very close to 44.1 GPa found in [110] direction. As discussed above, the

lateral dimension of the unit cell (one in direction of cleavage) is relaxed, but in

cases of NiAl and VC accordant energy changes were within computational noise.

The (100) cleavage of W is the only case displaying considerable effect of cell

relaxation, as is showed in figure 4.11. The cleavage energy is further decreased

by 0.13 J/m2 due to additional cell relaxation, compared to the calculation where

atoms were relaxed but the volume of the cell was fixed at a bulk value. In [110]

direction the cell relaxation caused again negligible change of cleavage energy.

One therefore might conclude, that in general relaxations of atoms prevail and

the unit cell relaxation might be safely neglected to reduce computational costs.

Interestingly, fracture experiments revealed that W cleaves primarily on (100)

planes, which could not be explained on basis of the lowest surface energy (see

section 4.5.2). However, the Griffith thermodynamic treatment -which in brittle

materials relates cleavage plane preference to the surface energy- applies only

for atomically sharp cracks, whereas in blunted crack configurations the critical

energy release rate depend on also the critical stress of material, as discussed

4.7. SEMIRELAXED CLEAVAGE 71

0 1 2 3 4 5 6 7x (Å)

0

2

4

6

8

E (

J/m

2 )

(100) unrelaxed(100) only atoms relaxed(100) semirelaxed(110) unrelaxed(110) semirelaxed

W

Figure 4.11: The semirelaxed cleavage of W. The cleavage energy as a functionof separation along [100] ( circles), [110] (diamonds) direction. The red lines andsymbols correspond to constant volume relaxation of cleavage, the blue symbolsto additional volume relaxations, see text for details.

in section 3.2.1. As shown in table 4.10, when structural relaxations of cleaved

surfaces are considered, the values of critical cleavage stress in both directions

are very similar and, thus, no explicit preference of (110) cleavage planes would

be observed in blunted crack configuration.

The accuracy of the fit provided by UBER in all semirelaxed cases may seem

surprising, because Hayes et al. claimed that UBER cannot describe the cleavage

process involving the surface relaxations. Of course, UBER is relation based

on the decay of the electronic density into vacuum and the cleavage relaxation

proposed by Hayes et al. involves -up to a critical point, where the crack is

really formed- rather an uniform expansion of the atomic planes. In semirelaxed

approach the planes representing the crack boundaries are fixed and the relaxation

concerns only the planes inside a supercell slab and, thus, the presumptions of

UBER are fulfilled.

In summary, a model is presented which incorporates the structural surface

relaxation into the cleavage calculation and demonstrated that UBER provides

reliable description of this process. The procedure of relaxation affected primarily


the cleavage energies, whereas the critical lengths were essentially unchanged.

The relaxed cleavage energy brings better agreement with the experiments, where

the surface energy is always relaxed and deliver more realistic parameters into

the model connecting cleavage and elasticity as well. It turns out, however, that

in this model the surface relaxation causes only a small change of the cleavage

energy, essentially much smaller than the variation of the localisation length.

4.8. SUMMARY 73

4.8 Summary

The correlation between elastic and cleavage properties was established by in-

troducing the concept of the localisation of the elastic energy and relating lo-

calized elastic energy to the crack-like perturbation in the spirit of Polanyi [62],

Orowan [45] and Gilman [63] approach. Consequently, a new materials parameter

is introduced, which is called the localisation length L. By this flexible parameter

the bridge between elastic and cleavage energy (or stress) was built. The actual

values of L, which depend on the material and the direction of cleavage, has to be

determined by fitting to DFT calculations of the decohesive energy as a function

of the crack opening. The concepts were tested for all sorts of bonding.

For the brittle cleavage it turned out, that -at least for metals and intermetal-

lic compounds- an average value of Lb ≈ 2.4 A would yield reasonably accurate

cleavage stresses if one knows only the uniaxial elastic modulus and the brittle

cleavage energy. This means, that the ”engineer” may estimate the critical me-

chanical behaviour of a material -at least for simpler types of crack formation-

purely knowing macroscopic materials parameters, namely the cleavage energy

and the elastic moduli. (Even if the cleavage energy is not easily accessible exper-

imentally, it could be derived from a single DFT calculation for each direction,

which in many cases is not very costly.) It is proposes, that the introduced con-

cept might even hold for the cleavage of more complex solids than single crystals.

Summarizing the results for various materials, interesting interplay of mag-

netism and cleavage in FeAl should be emphasized. It seems to be responsible

for a change of the cleavage habit plane of FeAl compared to NiAl and CoAl.

For both FeAl and Fe it is found that surface magnetic moment generated dur-

ing cleavage lowers the cleavage energy as well as the critical cleavage stress. In

particular the case of FeAl demonstrates the significance of magnetism which is

in common neglected in large-scale or continuum crack simulations.

The relaxed cleavage process involves structural surface relaxations, in con-

trast to the ideal brittle case. A convenient analytical formulation for the relaxed

cleavage process which utilizes a natural parameter -critical length for relaxed

cleavage lr- was found. However, the behaviour of appropriate localisation length

Lr is less simple to describe and no general trend is observed. This issue is

surely the topic for the future calculations. Another possibility how to incor-

porate surface relaxations into the model is the semirelaxed cleavage model, in

which the surface relaxation was introduced into the cleavage calculation in the

spirit of UBER demonstrating that UBER provides sufficient description of the

structurally relaxed surfaces. The connection to the elastic properties may be

then established in the same manner as for the case of the brittle cleavage.

Chapter 5

Ductile fracture

5.1 Introduction

An intrinsic ductile material like copper or aluminum cannot fail in the brittle

fashion, i.e. cannot sustain cleavage crack, but fails by a shear instability or by

a dislocation emission. Certain level of ductility in the material is important

for engineering applications, because it prevents cleavage crack propagation and,

therefore, lower risk of sudden collapse of the macroscopic object. Clearly, the

resolution between brittle and ductile behaviour of given material is of great tech-

nological interest. However, until mid-1950 the engineering materials were said to

be ”ductile” without specific clarification. Several airplane accidents caused by

brittle failure of ”ductile” aluminum brought more attention to the mechanism

underlying brittleness or ductility of materials.

In metals and many other materials as well, a cloud of dislocations screens

the crack from the external stress and, consequently, prevents brittle crack prop-

agation. Such materials are called extrinsic ductile. They may have significant

strength, but at lower temperatures the mobility of dislocations decreases and

dislocations cloud cannot keep up with the propagating crack - the material un-

dergoes transition to brittle behaviour. In fact, many ductile materials -including

important engineering steels or above mentioned Al- turn brittle below certain

critical temperature Tc. The transition from ductile to brittle at ambient tem-

peratures occurs also in modern perspective intermetallic alloys. This kind of be-

havior strongly complicates the engineering usage of extrinsic ductile materials,

because the synthesis involves usually several heat-cold cycles. The microcracks

may appear during heated phase and consequently spread when the material is

cooled down.

The mechanisms behind ductile fracture have begun to be studied in the

beginning of 70s. Kelly, Tyson and Cole made first important contribution by

75

76 CHAPTER 5. DUCTILE FRACTURE

Figure 5.1: The sketch of competing mechanisms -cleavage, or dislocationemission- at the crack tip. The outcoming dislocation in (2) has Burgers vec-tor perpendicular to crack plane and, thus, the crack is blunted by one atomicdistance.

finding that blunting of the crack tip (i.e. ductile response) requires production of

the dislocations. Then, Rice and Thomson [97] proposed the first general model

for emission of the dislocations from the crack tip. In order for a dislocation to

blunt the crack, its Burgers vector must have nonzero component normal to the

crack plane and its glide plane has to intersect the crack plane. The crystals for

which this emission is spontaneous are then expected to behave in ductile manner.

Using condition of equality of the stress field around the crack and stress field

due to a presence of a dislocation, Rice and Thomson arrived to condition for a

material to be ductileµb

γs

> 7.5 − 10, (5.1)

where µ is the shear modulus and the γs surface energy of the material. The

relation first enabled to quantify ductility and make theoretical predictions for

different types of materials. Utilizing equation 5.1 Rice and Thomson predicted

that fcc metals should be ductile while bcc metals, materials with diamond cubic

structure, and ionic materials should be brittle. However, the derivation of equa-

tion 5.1 was based on linear elasticity solutions for fully formed dislocations and

5.2. THE CONCEPT OF UNSTABLE STACKING FAULT ENERGY 77

involves poorly defined parameter - dislocation core cut-off. Therefore, quantita-

tive predictions were still strongly limited.

5.2 The concept of unstable stacking fault en-

ergy

The important breakthrough was brought by Rice [98], who analyzed dislocation

nucleation in the framework of the Peierls concept (see section 5.4.2). Rice pro-

posed that at the atomic scale a material is expected to be ductile when emission

of dislocations is energetically favorable over cleavage at the crack tip. The com-

petition of these processes at the crack tip is sketched in figure 5.1. The crucial

quantity which governs the emission is Gd, the critical energy release rate for

dislocation emission. Because dislocation emission is complex process influenced

by many factors (e.g. the geometry of crack and loading, the type and direction

of nucleated dislocation), the relations between Gd and intrinsic materials param-

eters are to large extent approximate and subject of discussion. In the following

the link between the electronic structure of the material and the prediction of

brittle or ductile behaviour is described. Theoretical considerations will then be

applied to evaluate slip behaviour of NiAl and especially to model ductilization

of NiAl via microalloying.

Rice assumed the periodic relation between shear stress τ and atomic displace-

ment u, and introduced a new material parameter γus, called unstable stacking

fault energy. The γus was defined as a maximum of an energy Φ per unit area

associated with slip discontinuity. The Φ is the energy of block-like shear, along

a slip plane, of one half of a perfect lattice relative to the other.

Rice showed that for an isotropic linear elastic solid in so-called mode II

configuration -pure shear loading of the crack and dislocation emitted on the slip

plane coinciding with the crack plane- Gd is equal to the γus. However, in such

a configuration the crack is not blunted, because Burgers vector of dislocation

has zero component in direction perpendicular to the crack plane. Moreover, the

pure shear loading at the crack tip is rarely to occur.

Thus, much more important is mode I configuration, where tensile loading

acts and, consequently, the most highly stressed slip plane is at nonzero angle θ

with the crack plane. The analytic derivation of Gd is difficult and has not been

given so far. Rice suggested an approximate expression, where γus is scaled with

a geometrical factor

Gd = 8γus1 + (1 − ν) tan2 φ

(1 + cos θ) sin2 θ, (5.2)


Figure 5.2: The geometry of dislocation emission

where θ is an angle between crack plane and slip plane and φ is an angle be-

tween Burgers vector of outcoming dislocation along slip plane and line drawn

perpendicular to the crack tip, as sketched in figure 5.2. The Gd quantity may

be compared with release rate for Griffith cleavage decohesion

G = Gc. (5.3)

Hence, dislocation nucleation is expected to occur when Gc exceeds Gd.

Although the expression for Gd contain several approximations which will be

discussed further, it has brought a bridge between ab-initio calculations and

brittleness-ductility estimates. The energy Φ is identified with the generalised

stacking fault energy γGSF introduced by Vitek [99, 100], which can be cal-

culated by means of quantum mechanical computations. The crucial quantity

γus is simply the maximum of γGSF energy surface along given glide direction

of emitted dislocation. The γus can not be obtained experimentally, however,

it is relatively well accessible by means of atomic potentials or DFT calcula-

tions. Various atomic methods based on pair potentials or atomic potentials like

embedded-atom method (EAM) lead usually to substantially lower estimates of

γus compared to accurate ab-initio DFT methods. The shear displacement of

the lattice can involve considerable charge transfer, which is badly described by

empirical potential based methods.

5.3. MODIFICATIONS OF RICE’S APPROACH 79

5.3 Modifications of Rice’s approach

Rice’s model was found to be rather accurate for mode II loading, where the

emission is predicted in agreement with direct atomic simulation [101]. In mode

I configuration, equation 5.2) seems less reliable [102]. This can be easily ex-

plained: because in mode I configuration the slip plane is at nonzero angle to the

crack plane, a ledge is formed at the crack tip by the emission of an edge dis-

location. Thus, the emission criterion should involve also the energy associated

with formation of the ledge which was neglected in Rice’s analysis. In order to

account for the ledge formation, Zhou, Carlsson and Thomson (ZCT) [102] intro-

duced surface corrections into the misfit energy and found that the crossover from

a ductile to a brittle solid is essentially independent of the intrinsic surface energy

γs when the ledge is present. They suggested new criterion for the prediction of

ductile behaviour,γus

µb< 0.014. (5.4)

There, µ denotes the isotropic shear modulus and b the Burger’s vector of the

emitted dislocation.

Furthermore, Schock [103, 104] treated the problem of dislocation emission on

the energy level. He expressed the total free energy of the system -loaded crack

and incipient dislocation described via associated displacement discontinuity- and

obtained the equilibrium configuration of the incipient dislocation by minimiz-

ing the free energy. Such a treatment enables to account for ledge formation

by including the term describing the ledge energy into variational problem. The

solution of the variational problem with geometrical trail functions then demon-

strated that Rice’s estimate of critical stress intensity KR gives correct order of

magnitude, however the emission occurred at stress intensity smaller than KR.

When the formation of ledges was included into calculations the critical stress

intensity for dislocation emission was increased. In fact it may reach the value of

KR, depending on the ledge energy.

Schock’s findings are in qualitative agreement with results of atomistic simu-

lation by ZTC. In summary, though the brittle-ductile criterion in mode I config-

uration may be well independent on the surface energy and several modifications

have been proposed, all of them share the feature that the γus is crucial physical

quantity which governs the ductility at the atomic level.

Though Rice’s criterion was in the last decade broadened to account for elas-

tic anisotropy [105], realistic slip systems [106] or crack surface tension [107],

the assumption of an atomically sharp crack [98] was not addressed. However,

cracks with a shape near to elliptical are usually observed [34]. An attempt to

extend Rice’s framework to blunted elliptical cracks was made by Fischer and


Beltz [54, 56]. They modelled elliptical cut-out in an infinite medium under

plane strain conditions. They constrained themselves to the cases of crack ad-

vancing directly ahead of the crack tip and emission of edge dislocations with

dislocation lines parallel to the crack tip. To calculate energy release rate for

dislocation nucleation, they applied treatment similar to that of Rice [98] to the

active slip plane. To obtain release rate for crack propagation the cohesive zone

model was used. The distinction between brittle or ductile behaviour was shown

to be dependent on maximum theoretical cleavage stress σc and maximum shear

stress τc. Two new classes of materials were introduced - quasi-brittle materials

which would cleave when their tips are sharp enough, but would tend to nucle-

ate dislocations when their crack tip curvature meets some threshold value. In

contrast, quasi-ductile solid would nucleate dislocations at sharp crack tips and

would cleave at blunted crack tip. The quasi-ductile kind of behaviour is not

likely to be expected in metals, but might occur at metal-ceramic interfaces [56].

5.4 Dislocations properties

The mobility of dislocations is further physical property which may, under cer-

tain conditions, govern ductile behaviour of the material. Rice’s treatment of

dislocation emission from the crack tip assumes that the dislocation can move

easily away from the tip. If it is difficult to move the outcoming dislocation over

the lattice, the dislocation emitted first would block next ones and, consequently

prevent further blunting of the crack. Furthermore, the mobility of dislocations

controls the extrinsic ductility as well. In an extrinsic ductile material disloca-

tions are accompanying the propagating crack, driven by the stress field of the

crack, and form a cloud of dislocations, which screens the crack tip from the

external stress field. However, at lower temperatures it is harder for dislocations

to move over the lattice and the crack -exposed to the external stresses- starts to

propagate in a brittle manner. This mechanism is believed to be responsible for

the ductile-to-brittle transition in many materials including intermetallic com-

pounds. The motion of dislocations, rather than the nucleation, is postulated

to be governing mechanism of ductile-brittle transition in model of Hirsch and

Roberts [108, 109].

The parameter describing the ease of the dislocation movement is the Peierls

stress, which is defined as the stress needed for a dislocation to glide. Direct

DFT calculations of the Peierls stress are impossible, because the unit cell would

be extremely large in order to minimize interactions between dislocations and

their associated stress field in periodic boundary conditions. However, it can be

estimated in the framework of the Peierls-Nabarro model making use of the gen-

5.4. DISLOCATIONS PROPERTIES 81

eralised stacking fault energy γGSF surface, which can be conveniently calculated

by means of an DFT approach.

Peierls and Nabarro [110, 111] provided the first model of dislocations ac-

counting for the lattice periodicity. It combines the dislocation stress field as

determined by the continuum theory with an atomic description of the dislo-

cation core region and, therefore, is capable of taking advantage of the results

of DFT calculations. The model proved to be reliable for determining the core

structures of dislocations, and yields the Peierls stress of a dislocation within

correct order of magnitude [112]. It should be mentioned, that latter theoreti-

cal estimates of Peierls stress were wrong by several orders of magnitude, when

compared to experimental results [113].

5.4.1 Continuum model for dislocations

As discussed in the Introduction, the plastic deformation of material is carried by

the motion of dislocations. Geometrically viewed the dislocations are line defects

in otherwise perfect crystal. The concept of dislocation enabled to answer why

metals deform easily despite high theoretical estimates of the shear stress - a unit

slip associated with dislocation glide along the slip plane requires much lower

stress compared to the shear slip of the whole plane. The behaviour of dislo-

cations underlies phenomena like the work-hardening, melting, or intergranular

brittleness-ductility as well.

The presence of a dislocation in a solid medium involves displacements of

atoms and, as consequence, generates the stress field around the dislocation. The

continuum model provides an analytic solution for the stress field of a straight

dislocation in an infinite linear elastic media. This solutions can be further used

as an input into more advanced models as the Peierls-Nabarro model as discussed

later.

The elementary geometric properties of dislocations as well as displacements

needed to produce dislocation may be found in the literature [114]. The strength

of dislocation is characterized by displacement vector b, called Burgers vector.

Based on orientation of Burgers vector with respect to the dislocation line ξ an

edge and a screw dislocation can be resolved. For the edge dislocation b.ξ = 0,

whereas for the screw dislocation b.ξ = b.

Because displacements associated with given dislocation are known from ge-

ometrical considerations [114], appropriate stress field can be calculated in an

analytical form within linear elasticity theory (see section 3.1.1). The screw dis-

location with a cut surface defined by y = 0 and x > 0 involves only displacements

in the direction of the z axis. The displacement discontinuity fz associated with


the screw dislocation may be then represented by form

fz =b

2πtan−1 x

y, (5.5)

which satisfies the constitutive relations of linear elasticity (equations 2.1 and

2.2). Consequently the stress field associated with screw dislocation may be

determined from basic equations of linear elasticity [114]. Nonzero stress tensor

components are

σxz =µb

2π

y

y2 + x2σyz =

µb

2π

x

y2 + x2. (5.6)

The edge dislocation produces plane strain, so the solution of its stress field is

more difficult. The relations of linear elasticity under plane strain conditions

are outlined in section 3.1.1. Defining plane strain conditions by fz = 0 and

∂fi/∂z = 0, the stress function of equation 3.3 can be expressed as [114]

Ψ = − µby

4π(1 − ν)ln(x2 + y2) (5.7)

and associated stress field can be calculated from the equation 3.3

σxx = − µb

2π(1 − ν)

y(3x2 + y2)

(y2 + x2)2(5.8)

σyy =µb

2π(1 − ν)

y(x2 − y2)

(y2 + x2)2(5.9)

σxy =µb

2π(1 − ν)

x(x2 − y2)

(y2 + x2)2(5.10)

σzz = ν(σxx + σyy). (5.11)

In either case (screw and edge dislocation) the stress fields do not exert any

back force on their source. Thus, in the continuum theory the dislocation can

glide through a medium without any resistance. The Peierls stress (the stress

dislocation experiences when moves through lattice) is purely atomic property,

analogical to the lattice trapping. Therefore, atomistic description has to be used

in order to obtain theoretical estimate of the Peierls stress.

Note that for a finite solid, or a solid containing flaws such as a crack, the

boundary conditions have to be introduced into the problem. Then, an image

dislocation is placed in such a manner that the stress field of a real dislocation

cancels at the surface of the solid or the crack. This allows an extension of

the solutions onto more complex problems. The limit of an infinite solid can be

reintroduced by increasing the dimensions of the solid system. The image stresses

caused by the boundary conditions then decrease, and in the infinity the stress

field is again characteristic to that of a dislocation in an infinite medium.


5.4.2 Peierls-Nabarro model of a dislocation

Peierls and Nabarro [110, 111] provided the first useful model of a dislocation

which reflected the lattice periodicity. Their model has been proved to be reli-

able in determining the core structure and the core energy of a dislocation. It

provides an analytical nonlinear elastic model of a dislocation core, which can

take advantage of the generalised stacking fault energies obtained from atom-

istic or DFT calculations. In the framework of Peierls-Nabarro model also the

misfit energy and the Peierls stress of a dislocation might be estimated, giving

important information about dislocations mobility.

In the Peierls-Nabarro model, the crystal is bisected into two semi-infinite

halves and which are then joined to form a dislocation. Then the upper and the

lower part are subjected to displacements f+ and f−, so as the ideal lattice ar-

rangement is re-established at the infinity. For pure edge dislocations the atoms

are arranged in rows parallel to the dislocation line and, therefore, the problem

can be treated in one-dimension. The shear stress τ(f) resulting from atomic

interactions depends only on the total relative displacement (often called disreg-

istry) f(x) = f+(x) + f−(x) across the glide plane. Thus, the disregistry f(x)

due to a presence of dislocation is to be derived.

Let us imagine straight edge dislocation in a lattice with periodicity b in the x

direction, which is perpendicular to the dislocation line. In the one-dimensional

Peierls-Nabarro model, the displacements in y-direction are negligibly small, and

the problem can be solved across the plane y = 0. Boundary conditions for

disregistry f(−∞) = 0 and f(∞) = b are required. As discussed in the previous

section, a dislocation in an isotropic infinite linear elastic solid generates in plane

y = 0 the stress field (other stress components are zero in this plane)

σxy =K

x, (5.12)

where K is material dependent elastic constant. For linear elastic and isotropic

solid medium characterized by the shear modulus µ and the Poisson ratio ν,

K = µb2π(1−ν)

for the edge dislocation and K = µb2π

for the screw dislocation. The

model is not limited to isotropic solids - anisotropic elastic constant K may be

calculated following the procedure outlined in Ref. [114].

The stress field of equation 5.12 may be interpreted as being generated by a

continuous distribution of infinitesimal edge dislocations with density ρ(x)

ρ(x′) =df(x′)

dx′, (5.13)

where x′ is a distance from the dislocation line. The force at point (x,0) produced


by the distribution of dislocations is

Fdisl = K

∞∫

−∞

1

x− x′ρ(x′)dx′. (5.14)

As the displacement f(x) moves atoms out of their original positions, the atomic

bonds pull them back. The condition of balance between the stress field of a

dislocation and atomic restoring force F [f(x)] forms the Peierls-Nabarro integro-

differential equation

K

∞∫

−∞

1

x− x′ρ(x′)dx′ = F [f(x)]. (5.15)

The question of course is, how to approximate atomic restoring forces. In original

Peierls-Nabarro treatment was used simple sinusoidal Frenkel law (equation 5.50)

with initial slope related to Hooke’s law and, therefore, F [f ] was approximated

by

F [f(x)] = − bµ

2πsin

(

2πf(x)

b

)

(5.16)

In such a case, a simple analytic solution satisfying boundary conditions can by

found

f(x) = − b

2πtan−1 x

ξ, (5.17)

where ξ is the width of a dislocation. The ξ represents a region, where the

disregistry is greater than one-half of its maximum value. Hence, the parameter

ξ gives rough estimate of the core region of a dislocation. Furthermore, because

the continuous distribution of dislocations corresponds to the stress function Ψ,

analytical expressions for the stress field of Peierls-Nabarro dislocation can be

obtained [114]. Nevertheless, the Frenkel model of restoring forces is very crude

approximation, since the structure of dislocation core depends more on the value

of the restoring stress at large displacements than on the value in the elastic limit.

Hence, classical Peierls-Nabarro model provides rather simple analytical solution,

which may serve as a basis for a description of dislocations within more accurate

models.

In order to obtain more reliable results, the restoring forces are approximated

utilizing the generalised stacking fault energy γGSF surface, introduced in previous

section. The restoring force is given by a gradient of γGSF [99]

F [f(x)] =dγGSF (f)

df. (5.18)


The gradient of γGSF catches nonlinear effects associated with the displacement

of atoms. Because it can be obtained accurately by means of the DFT electronic

structure method, it provides important link between atomic level DFT calcula-

tions and mesoscopic scale object such a dislocation is. Using the approximation

of equation 5.18 the Peierls-Nabarro equation becomes

K

∞∫

−∞

1

x− x′ρ(x′)dx′ =

∂γGSF [f(x)]

∂f(x). (5.19)

The solution of such integro-differential equation is difficult. The disregistry f(x)

is usually presumed in general form with several free parameters, which can be

adjusted to given force law. By that, the equation 5.19 is transformed into a set

of nonlinear algebraic equations which can be conveniently solved by means of

numerical iterative methods.

5.4.3 Lejcek’s method

Useful method for the solution of the Peierls-Nabarro equation with general

restoring force law (equation 5.19 was proposed by Lejcek [115]. He showed

that the Peierls-Nabarro equation is an example of a Hilbertian transformation

and represented dislocation density and corresponding force law with Laurent

series

ρ(x) =N∑

k=1

pk∑

n=1

ρnk(x) (5.20)

Kdγ

dx= −

N∑

k=1

pk∑

n=1

gnk(x), (5.21)

and

ρnk =1

2

[

Ank

(x− zk)n+

A?

(x− z?k)

n

]

(5.22)

gnk =−i2

[

Ank

(x− zk)n− A?

(x− z?k)

n

]

, (5.23)

where star denotes complex conjugate and zk = xk + iξk. The number N is

interpreted as the number of partial dislocations and then the parameters xk

determine the positions of the partial dislocations, ξk respective core widths of

partials. It should be noted that because this ansatz determines the force law as a

function of x and not of the disregistry f , one has to integrate f(x) =∫ x−∞ ρ(t)dt

and eliminate variable x from dγ(f(x))dx

in order to determine the dependence on f .

The expressions 5.22 and 5.23 are explicitly listed up to n = 3. Note that

for the sake of simplicity index k is left out, in cases with k > 1 one can simply

substitute an, bn, x and ξ with ank, bnk, x− xk and ξk, respectively.


For the case n = 1

f1(x) =a1

2ln(x2 + ξ2) − b1 arctan

(

x

ξ

)

(5.24)

ρ1(x) =a1x− b1ξ

x2 + ξ2(5.25)

g1(x) = −b1x+ a1ξ

x2 + ξ2(5.26)

Therefore, the Peierls solution of equation 5.17 is essentially obtained, because

logarithm is divergent function of x and, therefore, can be excluded from the

solution as will be discussed later. The parameter ξ measures the width of the

dislocation in analogy with the simple Peierls solution. Now, however, the higher-

order terms in n may be evaluated. They essentially represent the modifications

of the dislocation core structure due to deviations of the stacking fault energy

gradient from the sinusoidal force law.

The case n = 2

f2(x) = −a2x− b2ξ

x2 + ξ2(5.27)

ρ2(x) =a2(x

2 − ξ2) − 2b2ξx

(x2 + ξ2)2(5.28)

g2(x) = −b2(x2 − ξ2) + 2a2ξx

(x2 + ξ2)2(5.29)

The case n = 3

f3(x) =a3(ξ

2 − x2) + 2b3xξ

2(x2 + ξ2)2(5.30)

ρ3(x) =a3x(x

2 − 3ξ2) − b3ξ(3x2 − xi2)

(x2 + ξ2)3(5.31)

g3(x) = −1

2

2a3ξx+ b3(x2 − ξ2)

(x2 + ξ2)3(5.32)

The expressions for the higher-order terms may seem intricate, but they basically

provide the change of the dislocation core structure only because they fall off as 1x2

and 1x3 , respectively. Therefore, higher-order terms in n contribute significantly to

the solution of Peierls-Nabarro equation only in the inner part of the dislocation

core.


Furthermore, the number of independent parameters ank and bnk is reduced

by physical requirement that the disregistry f(x) must be finite for all values

of x. Therefore, either a1k = 0 for all k, or a1i = −a1j for i 6= j, because

ln(x) diverges with increasing x. Furthermore, from the boundary condition for

disregistry (f(−∞) = 0 and f(∞) = b) one derived∑

k b1k = − bπ. It is useful to

define b1k = − bπαk where

∑Nk=1 αk = 1, because the parameters bαk can then be

interpreted as Burgers vectors of partial dislocations. Remaining parameters have

to be estimated to fit given ∂γ∂f

curve. In applications of outlined approach, the

number of partial dislocations i needed for a unique solution can be determined

from the number of inflexion points on the γGSF (f) curve. The higher-order

terms in n provide better description of the dislocation core and are needed in

particular when the partials are strongly coupled, or strong deviations from the

simple sinusoidal shape of the force law occur.

In short, Lejcek’s method provides unified and physically transparent scheme

for solution of the Peierls-Nabarro equation. It can be extended into the gen-

eralised case of the two component displacement field (two-dimensional Peierls-

Nabarro model) [116], which allows to treat dislocations with mixed screw and

edge components as well as dislocation dissociation.

5.4.4 Peierls stress of a dislocation

Although the crystal periodicity and atomic-level description of restoring forces

has been incorporated, Peierls-Nabarro model still treats the solid around the

glide plane as an elastic continuum. As a consequence, in original Peierls-Nabarro

model a dislocation does not experience any stress and can travel through the

lattice without any resistance, because if the function f(x) is a solution of the

equation 5.19, so is f(x − u) (corresponding to a dislocation translated by u)

where u is any constant. Again, periodic nature of the crystal lattice of the

solid has to be incorporated. This can be achieved noting that the displacement

function f(x − u) corresponds to real displacement in the crystal only when an

atomic plane is present [117, 113].

Let a be the spacing of planes in the glide direction. The ma will be then

positions of individual planes. When the dislocation is introduced at the position

u, the planes in the upper half (at positions ma) will be displaced with respect to

the planes in the lower half by f(ma−u). The misfit energy can then be defined

as a sum of misfit energies between pairs of atomic planes [114, 100, 117]

W (u) =∞∑

m=−∞

γGSF (f(ma− u))a. (5.33)


This equation has correct period in a and correct limit for very narrow disloca-

tions [113] as well. It focuses on the energy variation during rigid shift of the

disregistry in glide direction. However, it should be mentioned that the rigid

shift of the disregistry is an approximation. The disregistry itself will change as

the dislocation moves between the atomic positions and, hence, the elastic en-

ergy will be changed as well. Therefore, the misfit energy and stress are slightly

overestimated in the rigid shift approximation.

The Peierls stress is defined as the stress required to overcome the periodic

barrier in W (u)

σp = max σ = max

{

1

b

dW

du

}

. (5.34)

An analytic solution for σ(u) was given by Joos and Duesberry [113]. Assuming

sinusoidal restoring force law and utilizing the Peierls solution of equation 5.17

they derived the stress associated with the misfit energy variation

σ(y) = −Kb2a

sinh 2πΓ sin 2πy

(cosh 2πΓ − cos 2πy)2, (5.35)

where the parameters Γ = ξ/a and y = f/a are the dimensionless width of

the dislocation and the dimensionless disregistry, respectively. The formula 5.35

provided reliable estimate of the σp when it was compared to direct atomistic

calculation of the critical stress [113]. However, the sinusoidal restoring force is

oversimplified in the range of applications and cannot be used for the case of

coupled partial dislocations.

Nevertheless, the assumption of sinusoidal restoring force law is essentially

necessary only for the derivation of the analytic solution of equation 5.35.

Medvedeva et al. proposed alternative treatment [118] which provides accurate

solution for σ(u). First, one uses the Poisson summation rule to simplify the

summation over m in equation 5.33 and obtains an expression

W (u) =a

|a|∞∑

n=−∞

Fγ

(

2πn

a

)

exp(

−2πinu

a

)

, (5.36)

where Fγ is the Fourier transform of γ[f(x)]

Fγ

(

2πn

a

)

=

∞∫

−∞

γ[f(x)] exp(

2πinx

a

)

. (5.37)

The formula 5.37 can be simplified via integration by parts which results in the

relation

Fγ

(

2πn

a

)

=a

2πin

∞∫

−∞

∂γ

∂f

∂f

∂xexp

(−2πinx

a

)

dx. (5.38)

5.5. CALCULATION OF STACKING FAULT ENERGETICS 89

Note that ∂γ∂f

is the restoring force obtained from the DFT calculation and ρ(x) =∂f∂x

is the dislocation density known from the solution of the Peierls-Nabarro

equation. The integral in equation 5.38 converges rapidly with increasing n.

Furthermore, Fγ is an even function of x and, thus, the Poisson sum of the

equation 5.36 may be simplified to a form

W (u) =∞∑

n=0

Fγ

(

2πn

a

)

2 cos(

2πnu

a

)

. (5.39)

Derivative of the equation 5.39 yields the periodic stress which the dislocation

experiences when it glides

σ =4π

ab

∞∑

n=1

nFγ

(

2πn

a

)

cos(

2πnu

a

)

(5.40)

and the stress maximum is the Peierls stress

σp =4π

ab

∞∑

n=1

nFγ

(

2πn

a

)

. (5.41)

In applications of the equation 5.41 the first two terms in n are usually suffi-

cient. Higher-order terms have values at least an order of magnitude lower and,

therefore, might be neglected. Thus, the formula may by considered an accu-

rate solution. The calculated results and their confrontation with Joos formula

(equation 5.35) are discussed in the following section.

It should be noted, that the direct summation in equation 5.33 is possible as

well. As the disregistry f(ma−u) converges to zero when the term ma−u is large,

finite number of m yields reliable estimate of W (u). For example, the number

of terms in the sum may be increased until further summation terms cause only

negligible change of the misfit energy. It is found m ≈ 1000 to be convergent in

this sense and such a calculation can be performed very conveniently on a modern

PC. The stress σ(u) can be then obtained via numerical derivative of W (u).

5.5 Calculation of stacking fault energetics

5.5.1 Modelling aspects

Possible applications of the stacking fault energetics in dislocations modelling

were outlined in previous sections. Now, the DFT calculation of the stacking

fault energy itself is presented.

A stacking fault is formed by an in-plane shift f of one part of the crystal

against the fixed another part. The work needed to generate such a displacement


0 0.1 0.2 0.3 0.4 0.5f/b

0

0.5

1

1.5

γ GSF

(J

/m2 )

atoms relaxedrigid shift

Figure 5.3: The effect of atomic relaxation on γGSF energetics of 〈111〉(110) slipsystem in NiAl. See text for details.

is called the generalised stacking fault energy γGSF (f). As discussed in section 5.2,

the unstable stacking fault energy γus is the maximum of γGSF (f) along given

direction of the slip displacement. This predetermines the method which has to

be used for the calculation.

In the first step, suitable supercell is constructed. It has to be large enough to

minimize interactions between the stacking faults due to the periodic boundary

conditions. Performing series of tests it is found, that at least eight atomic planes

separating the stacking faults are necessary - they provide bulk-like behaviour in

a region between the fault interfaces as well as convergent values of γGSF for NiAl.

Consequently, the whole supercell then contains at least 16 atomic planes in the

direction perpendicular to the stacking fault interface. Hence, in particular for

the stacking faults at higher-index planes relatively high number of atoms per unit

cell might be involved (for example 〈111〉(211) slip in B2 NiAl requires at least

64 atoms per unit cell), making the calculation of γGSF -surface computationally

very demanding.

The calculation proceeds as follows: the upper half of the supercell is shifted

relative to its lower part and the atomic positions are fully relaxed in order

minimize the tensile stress (the problem of the tensile-shear coupling at the slip

plane is discussed in section 5.6). Finally, the stacking fault energy is obtained


0 0.1 0.2 0.3 0.4 0.5f/b

0.5

1

1.5

γ GSF

(J

/m2 )

<111>(110)<001>(100)<001>(110)<111>(211)

Figure 5.4: Generalised stacking fault energy profile γGSF of the most importantslip systems in NiAl.

as the difference of the relaxed total energy of shifted cell with respect to the

unshifted one. Such a calculation is repeated for a series of displacements fi in

order to construct the γGSF (f) profile and determine γus.

The effect of the atomic relaxation is shown in figure 5.3 for the 〈111〉(110) slip

path in NiAl. The result of relaxed γGSF calculation is compared to the unrelaxed

calculation, where only simple rigid shift was applied. Clearly, the relaxation of

atoms lowers the energies γGSF (fi) considerably and changes the shape of γGSF

curve as well. It should be noted that the volume of the supercell was kept

constant during the slip, in order to have well-defined conditions focusing on the

interactions at the interface. The effect of volume relaxation is anyway small

when compared to the effect of atomic relaxation [119].

5.5.2 Results - slip properties of NiAl

The procedure outlined in previous section will be now applied for the calculation

of the stacking fault energetics of various slip systems of NiAl. First, a brief dis-

cussion of slip properties of NiAl are discussed. The compound NiAl crystallizes

in B2 structure and, therefore, one might expect that dislocation properties will

be similar to that of bcc metals. However, the 12〈111〉(110) slip which is typical in


bcc materials because it provides the shortest possible Burgers vector, is unlikely

in NiAl. The reason is simple: by the 12〈111〉 slip in the crystal with B2 struc-

ture an anti-phase boundary is formed. In NiAl is the energy of the anti-phase

boundary relatively high [120] making such a slip improbable.

Therefore, in 〈111〉 direction two possible dislocation configurations exist: a

pair of 12〈111〉 Shockley partial dislocations separated by the anti-phase boundary,

or a 〈111〉 superdislocation formed by slipping full length of the Burgers vector

b. The glide mechanism of the partial dislocations differ from that of the full

dislocations, because, depending on the width of splitting, partials can move

independently or together. If the coupling is strong it is possible to have a

situation where one partial moves up on the energy barrier while the other moves

downwards, hence lowering the total barrier [121]. The splitting of the partials

is mainly determined by the energy of the anti-phase boundary EAPB because

the splitting between dislocations balances the gain in the elastic energy with the

cost for the formation of the anti-phase boundary. The elastic theory gives the

equilibrium separation [114]

d =b2Ksplit

2πEAPB

, (5.42)

where b is the Burgers vector of the partial dislocation and K elastic constant,

which can be obtained from anisotropic elastic constants [114].

The mechanical properties of NiAl gained a lot of attention in the last decade,

which is reflected in number of studies of its stacking fault energetics [120, 122,

118]. The results are compared to other available calculations in the table 5.1.

However, in older calculations the relaxation of atoms was neglected, which led

to substantially higher values of γus energy. For instance, Medvedeva et al. [118]

reported 3.13 J/m2 and 2.28 J/m2 for 〈001〉(100) and 〈001〉(110) slips respectively,

much larger than the results of the relaxed calculation displayed in the table 5.1.

Note that EAM calculation of Ref. [123] obviously underestimated γus of the

(100) plane, which is well-known feature of semiempirical EAM potentials.

Relaxed γGSF (f) profiles are displayed in the figure 5.4 for significant slip

systems. In general, the γGSF -surfaces of NiAl are strongly anisotropic even

within one crystallographic plane. In NiAl the (110) cleavage habit plane is

preferred slip plane as well. Exploring table 5.1, one realizes the dominance of

the 〈111〉 and 〈001〉 slip systems. This is in agreement with the experimental

observations, which report the activity of the 〈001〉 and sometimes the 〈111〉dislocations [124, 125, 126, 127]. The 〈110〉 dislocations activity seems improbable

due to the high stacking fault energy barriers at all planes considered. The 〈110〉slip seems more likely to be formed by the dissociation: 〈110〉 → 〈111〉 + 〈001〉.Such a dissociation seems energetically more favorable.


Table 5.1: Calculated unstable stacking fault energies γus (J/m2) and the ratioGc/Gd, which is evaluated assuming that the crack lies at the (110) cleavageplane and calculating appropriate θ (see section 5.2). The values in brackets areother theoretical results, namely a Ref. [123] Embedded Atom calculation, and b

Ref. [118] FLMTO calculation.

Slip system γus Gc/Gd

〈001〉(100) 1.52 (1.21)a [3.13]b 0.53〈011〉(100) 2.9 (2.00)a 0.28

〈001〉(110) 1.28 [2.28]b 0.63〈110〉(110) 2.09 0.38〈111〉(110) 0.83 [0.97]b 0.96

〈110〉(111) 1.61 0.5

〈110〉(211) 2.84 0.35〈111〉(211) 0.96 0.83

The comparison with dislocations experiments is somewhat difficult, because

dislocation behaviour depends on the loading direction via the resolved shear

stress on various slip systems. Among slip systems of a given 〈h′k′l′〉(hkl) type

will dominate those with the greatest resolved shear stress acting upon them. For

a single crystal under the uniaxial tension σ11 the resolved shear stress on the

glide system is given by [114]

τ12 = cosα cos βσ11, (5.43)

where α is the angle between the tensile axis x1 and the glide direction x′1, and

β is the angle between x1 and the normal vector of the glide plane. Therefore,

for given tensile axis in the crystal, one can directly calculate the resolved shear

stress. It should be noted that the shear stress resolved at given glide plane can

be calculated for shear and torsion loadings as well [114].

The NiAl single crystals generally exhibit two significantly different types of

mechanical behaviour which one distinguishes as the soft and the hard direction.

The soft orientations are non-[001] loading directions and in this case 〈001〉 slips

dominate [126]. The hard orientations are those close to the [001] tensile loading

direction, where 〈001〉 slips experience low resolved shear stress. The deformation

of single crystals with hard orientation of the tensile axis requires considerably


higher stress. In the hard orientation, 〈111〉 slips at the (110), (211) and (123)

planes were reported as preferred slip direction at liquid nitrogen temperatures

(77 K) [127]. Obviously, this findings are in very good agreement with the present

results, which revealed low stacking fault energies for essentially the same slip

systems. Note low γus values for 〈111〉(110) and 〈111〉(211) slips in table 5.1.

5.5.3 Results - dislocation properties of NiAl

Now, the calculated γGSF profiles (figure 5.4) can be utilized, and the dislocation

core structure and the Peierls stress is estimated. The 〈001〉 slips involve single

dislocations and, therefore, their core structure should be relatively easy to de-

scribe. The 〈111〉(110) slip system features two possible configurations, namely

two Shockley partial dislocations separated by the anti-phase boundary, or full

〈111〉 dislocations.

In the first step, one has to evaluate anisotropic values of the elastic con-

stant K (see equation 5.12 and discussion below) for both of directions. Now,

the procedure outlined in Ref. [114] will be applied. Straight dislocations in an

anisotropic media can be conveniently analyzed if one of the reference axes is

oriented parallel to the dislocation line. The 〈111〉 dislocations lie in a direction

other than the cube axes, to which anisotropic constants of NiAl listed in sec-

tion 2.2 refer. Thus, in order to obtain the anisotropic factor K one has to first

transform elastic constants to a system where two axes are in the (110) plane

and one axis is oriented in the [111] direction. Transformed axes can be chosen

in the form (with respect to the Cartesian axes)

i′ =1√3(i − j + k) j′ =

1√2(j + k) k′ =

1√2(i + j). (5.44)

The change to the new coordinates can be expressed in terms of the transforma-

tion matrix Tij

Tij =1√6

√2 −

√2

√2

0√

3√

3√3

√3 0

. (5.45)

Now the tensor transformation rules [22] have to applied because elastic constants

are essentially fourth-rank tensors. In general, the transformation has the form

c′ijkl = QijghcghmnQmnkl, (5.46)

where the 9x9 transformation matrix Qmnkl is obtained as Qmnkl = TkmTln. Per-

forming the matrix multiplication within the program package Maple one obtains


Table 5.2: The elastic constant K for dislocations in NiAl. Isotropic value Kiso isgiven by µ/(1− ν) for an edge dislocation and by µ for a screw dislocation. Theshear modulus µ and the Poisson ration ν are evaluated from the Reuss averageover the elastic constants of NiAl listed in table 2.2, anisotropic values Ke andKs are calculated out of the elastic constants via the procedure outlined in thetext.

Kiso [001] [111]Ke 112.3 85 96Ks 81.5 65 75

the transformed constants, and then the relations of the anisotropic theory of dis-

locations may be applied. The anisotropic elastic theory of straight dislocations

was developed by Eshelby [128] and Stroh [129], and the framework and its appli-

cations are summarized in Ref. [114]. The theory is rather complex and lengthy,

hence the results of concern for us will be only briefly presented.

The general problem of the straight dislocation with mixed edge and screw

components involves the solution of a sixth-order polynomial equation and can

be solved only numerically. Nevertheless, instead of using full sixth-order poly-

nomials one can decompose the problem into a screw and an edge part involving

second order and fourth order polynomials, respectively. For pure edge dislo-

cation the coefficient Ke is then given in terms of transformed elastic constants

by [114]

Ke = (c′11 + c′12) +

[

c′66(c′11 − c′12)

(c′11 + c′12 + 2c′66)c′22

]1/2

, (5.47)

where c′11 =√

c′11c′22. For pure screw dislocation

Ks =√

c′44c′55 − c′245. (5.48)

For the 〈001〉 dislocations, the dislocation line is parallel to cubic axis and the

formula 5.47 can be directly used with cubic elastic constants listed in table 2.2

(substituting c66 with c44, and c22 with c11). The values of K obtained in this

way are summarized in table 5.2 together with isotropic estimate evaluated using

Reuss average over elastic constants.

Calculated γGSF are fitted with the Lejcek’s ansatz as discussed in sec-

tion 5.4.3. The γGSF profiles were calculated within constrained path approx-

imation -the slip energy is calculated only along given direction, whereas the


minimum energy needed to generate given slip displacement might follow some-

what different path- and, therefore, one-dimensional Peierls-Nabarro model will

be utilised. The two-dimension Peierls-Nabarro model can handle dislocations

with mixed edge and screw components (one-dimensional Peierls-Nabarro model

is limited to pure edge, or pure screw dislocations) but requires an order of mag-

nitude larger computational costs because full γ-surface has to be calculated.

Nevertheless, the deformation of NiAl is carried mainly by pure edge disloca-

tions [130, 131], so the description of dislocations within one-dimensional Peierls-

Nabarro model is reasonable.

Two partial edge dislocations with second-order terms in n describing core

structure (equation 5.27) have to be used for 〈111〉(110) system, whereas for

〈001〉 slips third order terms were used to fit single edge dislocation. Using this

parameterization, integro-differential Peierls-Nabarro equation (equation 5.19) is

transformed into a set of nonlinear algebraic equations. In principle, the solution

of a set of nonlinear equations cannot be obtained analytically (upon some special

cases) and some of iterative methods must be utilised. The resulting set of

nonlinear equations was solved by using the Levenberg-Marquardt method, which

represents a kind of Gauss-Newton nonlinear least squares approach.

It may be noted that even for the 1D Peierls-Nabarro model the numerical so-

lution is rather tedious, in particular of the equation corresponding to 〈111〉(110)

slip system where γGSF profile features the anti-phase boundary separating the

Shockley partial dislocations. For instance, one has not obtained a stable solu-

tion using usual Newton’s iterative algorithm. Convergent and stable solutions

were not achieved even by improving Newton’s method with the line search al-

gorithm for finding the next step in the iterative process. Convergent results

were obtained by the Levenberg-Marquardt method. All of these methods are

well described -rather from a theoretical point of view- in Ref. [132], which was

followed in programming of the nonlinear least squares algorithms.

The numerical integrations needed to obtain the Peierls stress from the equa-

tion 5.41 were performed utilizing mathematical program package Maple. Cal-

culated parameters are displayed in table 5.3. Exploring the results, one realizes

that full 〈111〉 dislocations should glide more easily compared to 〈001〉 ones.

Therefore, the mobility of dislocations in not a limiting factor for the activity

of 〈111〉 dislocations. That are probably large structural displacements of the

lattice associated with the nucleation and glide of such a dislocation. Note that

the Burgers vector of the full 〈111〉 dislocation is as long as 5.01 A in NiAl.

The formation of partials is energetically prohibitive because of the energy of the12〈111〉 anti-phase boundary energy, as discussed in the previous section. Explor-

ing the table 5.3 one realizes that the splitting between the 12〈111〉 partials is


Table 5.3: Dislocation core parameters and Peierls stress in NiAl. The dislocationcore width ξ (A), the separation of partials d (A) (the partials are at positions x−dand x+d), the Peierls stress σJ (µ) calculated from Joos formula (equation 5.35)and the Peierls stress σp (µ) calculated from exact formula in equation 5.41. Seetext for more details.

Slip system ξ d σJ σp

〈001〉(100) 1.4 0 0.034 0.036〈001〉(110) 1.6 0 0.024 0.024〈111〉(110) 3.2 7.1 - 0.002

14 A. This value is in reasonable agreement with experimental TEM observation

which reported that partials are about 10 A apart. Higher value of the theoreti-

cal estimate can be explained by constrained path approximation which may not

follow ideal dissociation path. Thus, better agreement may be expected within

two-dimensional Peierls-Nabarro model.

Comparing the Peierls stress values calculated via Joos formula (equa-

tion 5.35) and the accurate formula expressed by equation 5.41, one finds good

agreement for both of 〈001〉 slips. The sinusoidal approximation of restoring

forces works well for these slips with relative simple geometry and the agreement

proves reliability of the approaches for such a slips. The 〈111〉 system cannot be

treated with equation 5.35 because sinusoidal approximation is obviously wrong

in that case.

Of course, the Peierls-Nabarro dislocation model has several limitations. It is

ambiguous in the sense that it uses both continuum and atomistic descriptions.

While a dislocation is represented by a continuous function, the calculation of

the Peierls stress is realized via discrete summation. Nevertheless, when correctly

employed it gives reliable core structure of dislocations and the Peierls stress is

calculated within the correct order of magnitude [112, 119]. Number of modified

approaches (but still more or less based on the classical Peierls-Nabarro formu-

lation) have been recently proposed [116, 133, 134]. The inherent limitations of

the Peierls-Nabarro model are summarized and discussed in the reference [9].


5.6 Tension-shear coupling

5.6.1 Introduction

As discussed in the section 5.2, Rice considered primarily the pure shear loading

in simple geometry with emission plane coplanar with crack plane [98]. Under

tensile loading the most highly stressed slip plane is at nonzero angle θ with the

crack plane. In that case, Rice suggested an approximate criterion (equation 5.2).

However, the extension of the concept onto the tensile state of loading involves

two conceptual problems neglected by Rice: the energy associated with a ledge

formed by the emission of an edge dislocation which Burgers vector has nonzero

component normal to a crack plane and the tensile stress component of a loading

coupled to a shear stress at the emission plane. Whilst the ledge energy con-

tribution has been addressed in several theoretical studies (see section 5.2 and

references therein), the problem of tension-shear (TS) coupling has been studied

just by Sun, Beltz and Rice (SBR) [106] and da Silva [135] so far.

SBR employed embedded atom method (EAM) and found that tensile stress

across a slip plane eases dislocation nucleation at the crack tip. Furthermore, by

comparing the results of atomic calculations to the solution of the exact integral

equation describing dislocation emission from the crack tip, they found that as a

reasonable approximate approach one can use tension-softened γus in the shear-

only model. However, the EAM potential utilised by SBR did not provide reliable

description of the stacking fault energetics. SBR reported an order of magnitude

difference when they compared intrinsic stacking fault energies calculated using

EAM potentials with those obtained using more accurate methods. For instance,

the energies of anti-phase boundaries in Ni and Al reported by SBR are an order

of magnitude lower than experimental values.

The lack of other studies or calculations of the TS coupling seems somewhat

surprising, because -besides the dislocation emission considerations- it constitutes

interesting conceptual problem in the dislocations modelling as well. For instance,

within the models which treat the dislocation glide as the variational problem

for the disregistry f(x) [112, 9] the tensile opening could be treated as another

variational parameter and the effect of the tension on the misfit energy and the

Peierls stress could be elucidated. Such models would require as an input the

tension-modified γGSF -surfaces. Therefore, we performed the simulation of the

TS coupling with an accurate PAW method. It may be mentioned that no ab

initio calculation of tensile-shear coupling has been performed so far, probably

because of considerable computational demands of such a survey.

5.6. TENSION-SHEAR COUPLING 99

x

f

a0

b

Figure 5.5: Block-like slip displacement f and opening separation x of two partsof the supercell

5.6.2 Model for tensile-shear coupling

For the tensile-shear coupling simulation the intermetallic compound NiAl was

chosen, for which slip properties have been calculated in the previous section.

The main slip systems in 〈111〉 and 〈001〉 directions are considered. As was

demonstrated in section 5.5.2, those are preferred slip system in NiAl at low

temperatures.

The methodology of the calculation is illustrated in figure 5.5. The supercell

is bisected into two blocks, which are then subject to the tensile rigid block

opening x− a0. Then the opening separation fixed is kept fixed, the upper block

(slip displacement f) is shifted, and the individual atoms -of course besides the

atoms at the interface- are allowed to fully relax. To prevent any interactions

between the slip interfaces a supercell slab geometry is employed where each of

the two blocks is composed of eight atomic layers in a direction perpendicular

to a slip plane. Finally, the energy of a configuration with combined tensile

opening and slip displacements is calculated taking the difference relative to the

undisplaced supercell.


5.6.3 Combined tension-shear relations

The important issue in TS coupling treatment is the construction of appropriate

constitutive relations, which would describe the stresses associated with the com-

bined displacements (x, f). In an analytic form, the constitutive relations might

be utilised in numerical treatment of the dislocation emission, or to determine

an influence of the TS coupling on the core structure of dislocations within the

Peierls-Nabarro dislocation model [112].

The basic analytic form of constitutive relations was derived by SBR. They

defined a potential Ψ(x, f) generated by the displacements (x, f). The work done

by the tensile stress σ and the shear stress τ may be then expressed as

dΨ(x, f) = σdx+ τdf. (5.49)

In the absence of the tensile stress component, the pure shear stress may be

approximated with the Frenkel sinusoidal formula

τ(f) =πγus

bsin

(

2πf

b

)

(5.50)

and vice-versa, the pure tensile stress of rigid opening may be derived from UBER

(equation 3.15) as

σ(x) =Gc

l2exp

(

−xl

)

. (5.51)

Thus, one naturally requires that general functions τ(x, f) and σ(x, f) should hold

the important characteristics of their predecessors, i.e. periodicity b in shear and

scaling length l in tension. The functions τ(x, f) and σ(x, f) must in limiting

cases x = 0 and τ = 0 agree with the equations 5.50 and 5.51 as well. These

conditions are fulfilled by functions in form

τ(x, f) = A(x) sin

(

2πf

b

)

(5.52)

σ(x, f) =[

B(f)x

l− C(f)

]

exp(

−xl

)

. (5.53)

The functions A(x), B(f), and C(f) are further constrained. The shear stress

must vanish at x→ ∞. Moreover, the existence of the potential Ψ(x, f) requires

that the Maxwell reciprocal relation

∂τ

∂x=∂σ

∂f(5.54)

must be fulfilled. These constraints allowed SBR to obtain functions A,B,C just

with one new parameter introduced. This parameter is the opening displace-

ment x0 corresponding to zero tensile stress at the unstable stacking fault (shear


displacement f = 12b). The analytic form of functions A,B,C may be found in

reference [106]. The resulting potential Ψ(x, f) was derived as

Ψ(x, f) = 2γs

[

1 −(

1 +x

l

)

exp(

−xl

)

+ sin2

(

πf

b

){

q +

(

q − p

1 − p

)

x

l

}

exp(

−xl

)

]

,

(5.55)

where q is defined as ratio γus/2γs and p = x0/l. These dimensionless material

constants quantify the degree of tensile-shear coupling. Xu et al. [136] followed

above treatment and extended it to allow for skewness in the shear resistance

curve utilizing phenomenological non-sinusoidal law for the restoring shear force

by Foreman, Jawson and Wood [137, 138].

However, as pointed out by da Silva et al. [135], the equation 5.55 does not

account for the fact that in real crystals shear stresses develop due to asymmetry

of atomic positions with respect to direction of tension. In order to make Ψ(x, f)

applicable to asymmetric deformations, da Silva et al. proposed to add into

equation 5.55 a term

Ψa = 2γsπxfa exp(−x/l)

bl. (5.56)

This term involves additional fitting parameter a which should represent the

strength of a new coupling mode - the shear stress generated when crystal halves

are pulled apart (x > 0 and f = 0). Nevertheless, when such a term is added

the periodicity in b is lost. Anyway, no additional shear stresses appeared in the

calculations. Therefore, there was no need to include this additional term.

The importance of the potential Ψ(x, f) lies in the fact that its analytic form

might be used in other models. SBR utilised Ψ(x, f) for considerations concerning

the effect of the tension-shear coupling on the dislocation emission within Rice’s

approach. But the TS coupling would influence the parameters associated with

the dislocations glide as well. In principle, the Ψ(x, f) could be utilised in models

which calculate Peierls stress as variational problem of disregistry f [9]. Thanks

to analytic form of Ψ(x, f) the tensile opening could treated as another variational

parameter and the effect of tension on misfit energy and Peierls stress could be

obtained. However, this treatment involves several conceptual obstacles, which

are yet to be solved [139]. This topic remains open and physically very challenging

issue into the future.

5.6.4 Results

First, the effect of the relaxation of individual atomic planes is discussed, which

was neglected in the calculations of SBR. Two approaches are sketched: (1) the

combination of rigid opening and rigid slip displacement and (2) the calculation,


0 0.1 0.2 0.3 0.4 0.5f/b

0.5

1

1.5

E (

J/m

2 )

x = 0.0x = 0.2x = 0.4

Figure 5.6: The effect of relaxation; the stacking fault energy for [111](211) slipsystem calculated for rigid tensile and shear displacements (broken line) and withadditional relaxation of individual atomic planes in direction perpendicular to theslip plane (solid line).

where after opening and slip displacement the atoms are allowed to fully relax.

According to figure 5.6 -where these approaches are compared in the case of

γGSF -profile of the 〈111〉(211) slip system- the relaxation has substantial influ-

ence on the stacking fault energetics. The relaxed stacking fault energy profile

displays weak local energy minimum around f = 0.35 which is not reproduced

when the relaxation of individual atoms is neglected. Furthermore, relaxed cal-

culation identifies the stacking fault energy maximum γus at the position of the

1/2〈111〉(211) anti-phase boundary, while the unrelaxed calculation yields the γus

approximately at f = 0.3. Thus, the relaxation of individual planes may cause

quantitative as well as qualitative changes of the stacking fault energy profile.

Of course, strong changes of topology cannot be expected for simple 〈001〉 slips,

nevertheless the quantitative changes of γus are substantial and cannot be ne-

glected. Therefore, the relaxation of atomic planes was performed in all following

calculations.

The slip systems considered in 〈001〉 direction -(100) and (110)- are displayed

in figure 5.7 and figure 5.8, respectively. Both have simple geometry with γus at

f = 1/2 and display pronounced tension softening of the γGSF surface. The slips


0 0.1 0.2 0.3 0.4 0.5f/b

1

2

3

4E

(J/

m2 )

x = 1.0x = 0.6x = 0.4x = 0.2x = 0.0

Figure 5.7: Tensile-shear coupling for 〈001〉(100) slip system; the energy E as afunction of the slip displacement f with the tensile opening x as a parameter

with such simple geometry are actually only cases, which might be conveniently

fitted with SBR formula (equation 5.55). When the stacking fault energy profile

involves additional extrema along displacement path, Frenkel formula based force

law breaks down. However, even for these slips the fits of the equation 5.55 were

rather rough.

The tension softened γus is less then half of the value obtained in the unrelaxed

calculation at zero tensile opening (simple rigid shift). The effect of the shear

displacement f quickly diminish at larger opening and beyond x ≈ 0.6 A the

energy is dictated only by the tensile separation. In general, the tension has

relatively strong influence on calculated stacking fault energies, in particular on

γus, the quantity which should govern the emission of dislocations into this slip

systems. Thus, the calculations which do not relax tensile stress σ in direction

perpendicular to slip plane might yield highly overestimated values of γus.

It is also worth of notice that softening is certainly stronger when compared

to calculations of SBR utilizing EAM potentials. This fact might indicate that

large charge transfers are involved during such combined crystal displacements,

because significant charge transfer is common reason of the failure of the pair-

potential or the embedded atom based methods.

The slips in 〈111〉 direction have more complex energy profile. The profile of


0 0.1 0.2 0.3 0.4 0.5f/b

0.5

1

1.5

2

2.5

3E

(J/

m2 )

x = 1.0x = 0.6x = 0.4x = 0.2x = 0.0

Figure 5.8: Tensile-shear coupling for 〈001〉(110) slip system

0 0.1 0.2 0.3 0.4 0.5f/b

0.5

1

1.5

2

E (

J/m

2 )

x = 1.0x = 0.6x = 0.4x = 0.2x = 0



0 0.1 0.2 0.3 0.4 0.5f/b

0.5

1

1.5

2

E (

J/m

2 )

d = 1.0d = 0.6d = 0.4d = 0.2d = 0.0


0 0.5 1 1.5γ

GSF (J/m

2)

0.1

0.2

0.3

0.4

0.5

x 0 (Å

)

<111>(110)<001>(110)<001>(100)<111>(211)

Figure 5.11: The zero-stress separation parameter x0 of UBER as a function ofstacking fault energy γ.


0 1 2 3 4 5x (Å)

0

1

2

3

4E

(J/

m2 )

f/b = 0.0f/b = 0.1f/b = 0.21f/b = 0.28f/b = 0.42f/b = 0.5

Figure 5.12: The (110) cleavage of NiAl in the presence of the 〈001〉 stackingfault; the cleavage energy E as a function of opening displacement x with theshear displacement f as a parameter

the (110) system shown in figure 5.9 has the maximum approximately at f = 0.3

and local minimum at f = 0.5 due to formation of the anti-phase boundary. The

(211) system displays local maximum followed by shallow minimum at f = 0.35,

as indicated in figure 5.10. At f = 0.5 the anti-phase boundary is created as well.

It should be noted that γGSF profiles were calculated within the constrained path

approximation, i.e. no deviations from direct slip direction were allowed. In

general, a minimum energy path which generates given stacking fault may be

slightly different from the constrained path.

For the (110) slip systems the effect of the tensile stress is less pronounced

compared to (100) ones. The tension softening of the γGSF is substantially weaker

as well. Moreover, the lowest value of γus is found at much shorter opening

x compared to 〈001〉 slips. Recalling the brittle cleavage properties of NiAl -

Gc = 4.8 J/m2 and l = 0.69 A for the (100) planes, Gc = 3.2 J/m2, l = 0.54 A

for the (110) cleavage planes- one can observe greater cleavage strength of (100)

planes and larger critical length (the length at which cleavage stress reaches its


maximum). This fact might explain the difference in the tension softening of the

(100) and the (110) planes.

Finally, in the case of the 〈111〉(211) slip system, the effect of the tension is

obviously weakest. Herein the relaxation of atoms causes substantial change of

the γGSF profile (see figure 5.6), as was discussed in beginning of this section.

The equation 5.55 did not provide reliable fit to the calculated energy profiles.

The sinusoidal Frenkel formula is too simple force-displacement law and, there-

fore, corresponding γ profiles involved cannot be sufficiently described. In the

absence of tension, a more general expression for stacking fault energy is to be

used [136]. However, a new materials parameters -besides the opening displace-

ment x0 corresponding to the zero tensile stress at the unstable stacking- must

then be introduced.

Exploiting the results, it was found that the zero stress separation x0 scales

linearly with generalised stacking fault energy γGSF for a given displacement.

The correlation between γGSF and x0 is demonstrated in figure 5.11 and seems

valid for all slip systems studied. The physical interpretation of x0 is emphasized

in figure 5.12 which shows the change of the cleavage properties with respect to

the shear displacement f . The parameter x0 represents the equilibrium separa-

tion as given by the UBER [51]. It should be noted that the cleavage energies

were markedly decreased in the presence of the slip displacements. Therefore, in

general the weakening of the cohesive forces at the crack tip might be expected

when also same amount of the shear stress is involved and the crystal might be

more easily cleaved in the presence of the stacking faults.


5.7 Summary

In summary, it was found 〈001〉 and 〈111〉 as the preferred slip directions in

NiAl, in good agreement with fracture experiments. Though calculated values

of the γus are lower in the 〈111〉 direction, the 〈001〉 is dominating slip, because

〈111〉 slips are somewhat hampered by the relatively high anti-phase boundary

energy. The anti-phase boundary prevent formation of the 12〈111〉 dislocations

which occur in metals with bcc structure. Thus, full 〈111〉 dislocations form only

when the resolved shear stress for the preferred 〈001〉 slip is low. The attempts

to improve ductility of NiAl should clearly focus on the lowering high anti-phase

boundary barrier. The splitting of the 12〈111〉 partials was estimated within the

framework of the Peierls-Nabarro model and was found in reasonable agreement

with experimental TEM observation.

The tensile stress acting over the slip plane considerably decreases the unsta-

ble stacking energy and, consequently, lowers the threshold for the dislocation

emission onto that slip plane. The relaxation of planes in the direction of the

tension has to be performed in order to obtain accurate stacking fault energetics.

When the cleavage properties are of concern, similar conclusion can be made -

the cleavage energy is lower in the presence of the stacking faults or shear stress

component. Such a fact is important in the case of the polycrystals, where -due

to various orientations of grains with respect to external stress direction- is some

amount of the resolved shear stress essentially always present. Thus, the resolved

shear stress might weaken a grain interface and make the crack propagation be-

tween grains more favorable over the propagation through crystal bulk. Of course,

more elaborate studies are necessary to elucidate the tensile-shear coupling and

associated processes at the grain boundaries. It should be also noted, that only

the case of NiAl was investigated. Hence, the results for the other crystalline ma-

terials may differ. However, the present calculations show clearly that the tension

acting over the slip plane has essential influence on the γGSF energetics and its

effect on the dislocations properties should be considered in future calculations.

5.7. SUMMARY 109

Chapter 6

Microalloying of NiAl

6.1 Introduction

In the following chapter it is tried to utilize the computational approaches as

described in the previous chapters and to show their technologically oriented ap-

plication. It is attempted to simulate the effect of alloying of NiAl at the atomic

level, endeavoring to find the mechanisms which would improve its room temper-

ature ductility. Ni or Al atoms are substituted with one of selected elements -Cr,

Mo, Ga, Ti- and the change of the cleavage and stacking fault energetics is calcu-

lated and discussed within the framework of latter introduced models. This kind

of simulation fully exploits the DFT method, because the change of the electronic

structure and bonding of the alloyed interface cannot be reasonably described by

any of empirical, or semiempirical methods. Of course, the simulation treats

the effects which span over a few atomic distances and neglects many processes

which play role at the macroscopic level e.g. the solubility and the segregation

of dopants, or the interaction of dopants with dislocations. Nevertheless, the

studies based on the ab initio approach can provide important information on

the influence of the substitutional atoms at the atomic level under well-controlled

conditions. The DFT treatment elucidates the intrinsic effect of the alloying, e.g.

the change of bonding at the cleavage and slip interfaces. However, such calcu-

lations are computationally very demanding and, therefore, only microalloying

in molybdenum disilicide [140] has been treated by means of the DFT method

so far. And last but not least, although the comparison of the results of calcu-

lations with the findings of fracture experiments is always somewhat tricky, the

trends found in calculations nicely correlate with experimental findings, as it is

now demonstrated.

111

112 CHAPTER 6. MICROALLOYING OF NIAL

6.2 Fracture properties of alloyed NiAl

Physical properties of NiAl such as high strength, high melting temperature,

phase stability for a range of varying chemical composition and good corrosion re-

sistance, are of interest for applications in aerospace industry. However, poor duc-

tility at low temperatures and brittle grain-boundary fracture limit its technolog-

ical assignments as well as its synthesis. Therefore, improving ductility has been

tried by many techniques (see [120, 141, 142] and references therein), and amongst

them, microalloying seems to be the most promising approach [124, 125, 143].

Experimental investigations of the mechanical properties, however, are strongly

influenced by rather uncontrollable factors such as impurity content, heat treat-

ment, constitutional defects and surface conditions [144].

The slip properties of pure NiAl were in detail discussed in the previous chap-

ter. In short, the soft or the hard orientation of specimen can be resolved in

NiAl single crystals. The soft orientations are non-[001] loading directions, and

in this case 〈001〉 slips dominate [126]. The hard orientations are those close to

[001] tensile directions, where 〈001〉 slips experience low resolved shear stress.

At liquid nitrogen temperatures (77 K) the preferred slip direction in the hard

orientation of specimens is 〈111〉 at the (110), (211) and (123) planes [127].

In summary, a variety of experimental results indicate that the major defor-

mation mode of NiAl with its cubic B2 structure is the 〈001〉 slip [145, 124, 126].

However, 〈001〉 slip provides only three independent slip systems [145], and -

consequently- the von Mises criterion for a polycrystalline material to be ductile

is not met. von Mises demonstrated that five independent slip systems are re-

quired for a polycrystal to undergo plastic deformation [146]. When the poly-

crystal is deformed a grain within it must deform somehow. If five independent

slip systems are not available, the dislocations are more rarely to nucleate and

grain-boundary sliding, twinning, phase transformation, or brittle grain-boundary

fracture occurs [114].

On the other hand, the 〈111〉 slip -prevalent in other intermetallic compounds

with a B2 structure like CuZn and AgMn- fulfills the von Mises criterion. There-

fore, an improvement of the intergranular ductility of polycrystalline NiAl should

proceed via an activation of systems related to the 〈111〉 slip.

Miracle et al. reported that alloying NiAl with Cr enhances the nucleation and

motion of 〈111〉 dislocations at low temperatures, while at higher temperatures

the 〈001〉 slip is suggested to be prevalent [124]. In contrast, the experimental

results of Darolia et al. indicate that 〈111〉 dislocations are absent in stoichiomet-

ric NiAl single crystals alloyed with Cr [147] and V [148]. Further experiments

revealed that very small additions of ≈ 0.1 - 0.25 at.% of Fe, Ga and Mo enhance

6.3. COMPUTATIONAL AND MODELLING ASPECTS 113

significantly the room temperature ductility of NiAl single crystals loaded in the

[110] direction [125]. A more recent study for the same orientation demonstrated

that high tensile elongations can also occur in pure single crystals [144]. It seems

therefore possible, that the ductility improvement found in the tensile experi-

ments might be an indirect consequence of the process of alloying rather than

an intrinsic property depending only on the chemical composition of the alloying

element.

6.3 Computational and modelling aspects

In the light of amount of experimental results with difficult interpretation, the

studies which would provide reliable data for well-defined, controlled conditions

are inevitably needed. For this reason, an ab initio density functional approach

is applied for a variety of alloying elements modelling mechanical properties of

microalloyed NiAl at low temperatures. Cleavage energies are calculated from

a model for ideal brittle cleavage and the generalised stacking fault energies are

obtained from model studies of active slip systems. Both properties are then

combined for the prediction of brittle fracture behaviour and for the indication of

possible mechanisms of ductility improvement. As alloying elements Cr, Ga and

Mo were chosen, for which experimental data are available. In addition, Ti was

also considered because it was found to improve stress-rupture properties [149]

and creep strength at elevated temperatures [150]. The present investigation

investigation is the first ab initio study for modelling slip processes in a microal-

loyed material. This is done in terms of supercells with the atomic positions fully

relaxed for any finite slip. Approaches based on the simplified concept of inter-

atomic potentials would be much less reliable due to the missing atomic relaxation

and the multi-centered bonding formed by the electronic states of transition metal

elements.

Up to now, ab initio studies were made for deriving the influence of ternary

additions on the anti-phase boundary energies [120], for calculating stacking fault

energies and dislocations properties of pure NiAl [118]. Though these studies do

not address the improvement of ductility of NiAl, they provide a crosscheck on

the accuracy of calculated stacking fault energies.

As outlined in section 3.2.1, brittle cleavage formation is modelled by a re-

peated slab construction with three-dimensional translational symmetry. Con-

vergency of the cleavage energy as a function of the slab thickness and vacuum

spacing was tested. Unit cells with 8 atomic layers separating both the (100) and

and the (110) interfaces were sufficiently thick. In order to minimize artificial

interactions between stacking faults, for the calculations of stacking fault ener-


[010]

[001]

[110]

[001]

Figure 6.1: Interfaces for an AB compound of B2 structure. (100) interface (leftpanel): plane for A (black circles) atoms with a (1x1), (

√2 x

√2) and (2x2)

supercell geometry corresponding to coverage by X of 100, 50, and 25 %; secondinterface plane for B atoms is similar but with white circles. (110) interface (rightpanel): plane for a (1x1) and (1x2) geometry corresponding to coverage of X by50 and 25 %.

gies unit cells of 16 atomic layers had to be used. Because ideal brittleness was

modelled no atomic positions were relaxed during cleaving.

Generalized stacking fault energies as a function of the shear displacement (i.

e. slip) f were calculated by shifting the upper half of a suitable supercell relative

to its lower, fixed part. The atomic positions were always fully relaxed in order

minimize the tensile stress. The problem of tensile-shear coupling was discussed

in section 5.6. The overall volume of the supercell was kept constant also during

the slip, in order to have well-defined conditions focusing on the interactions at

the interface. Effects of volume relaxations are anyway small when compared to

atomic relaxations [119].

Alloying with the elements X=Cr,Mo,Ga,Ti was modelled by substituting X

for Ni or Al in one of the two interface or cleavage planes. Thus, it is implicitly

assumed that cleavage is initiated in a plane containing substitute atoms. This

construction ensures the maximum influence of X on cleavage and slip properties.

In principle, the dopants can replace atoms at both sides of the interface, how-

ever, we consider such a case rather unphysical in the light of the low solubility

(usually 5-10 %, see [151]) of dopants in NiAl. Because of the symmetry of the B2

structure, for (100) planes, only one type of atom fills each layer; consequently,

6.4. BRITTLE CLEAVAGE 115

Table 6.1: For NiAl, UBER parameters as derived from fitting to ab initio cal-culations for cleavage planes of orientation (hkl): cleavage energy per area Gc/Ain J/m2, cleavage energy Gc in eV, critical length l in A , critical stress σc/Ain GPa. N1 and N2 denote the numbers of broken nearest and second nearestneighbor bonds.

(hkl) Gc/A Gc l σc/A N1 N2

(100) 4.79 2.50 0.69 25.5 4 1(110) 3.24 2.40 0.54 22.2 4 2(111) 4.12 3.73 0.58 26.1 4 3

X replaces 100% of the atoms in one of the planes. In the (110) planes, how-

ever, two types of atoms are located. Therefore, X substitutions cover 50% of

this plane. Concentration dependence by reducing the amount of X was studied

via enlarging the supercells. To obtain the dependence on the concentration of

substitutional atoms, both the cleavage and generalized stacking fault energies

were calculated in five supercell geometries, which are displayed in the left panel

of figure 6.1. A representative size of a unit cell for modelling the slips was 64

atoms for both the (2x2) coverage of the (100) interface and the (1x2) geometry

for the (110) interface.

Further justification for placing X in the interface planes is given by a recent

study claiming that Cr substitutions segregate to the cleavage surfaces [152]. The

site preference of ternary additions was recently proposed for X=Ti,Ga preferring

Al sites, and for X=Cr,Mo occupying both sublattice sites, depending on the

concentration x of a Ni1−xAlx compound: for x < 0.5 Ni sites and for x > 0.5

Al sites are preferred [153] by X. Therefore, the alloys for X=Ti,Ga on Al sites

were studied, and for X=Cr,Mo on both sublattice sites. It should be noted that

the site preference reported in [153] is the bulk one and the site preference at the

crack surface may be different. The placement of X on an Al- or Ni-site is denoted

by XAl or XNi, respectively. The alloyed compound is described as NiAl-X.

6.4 Brittle cleavage

Ideal brittle cleavage (i.e. no relaxation of atomic positions during cleavage) is

described in terms of the Griffith energy balance, according to which the crack

under load mode I propagates when the mechanical energy release rate G exceeds

the cleavage energy Gc, defined as the energy needed to separate the solid material

into two blocks. The energy G(x) depends on the cleavage size or separation x of


Table 6.2: For NiAl-X, calculated properties of (110) brittle cleavage. Results ofUBER fit to ab initio data: cleavage energy Gc/A in J/m2, maximum cleavagestress σc/A in GPa, its relative changes ∆σc/a with respect to pure NiAl, andthe length parameter l in A .

Xsite Gc/A σc/A ∆σc/A lCrAl 3.88 26.6 4.3 0.54CrNi 3.74 27.4 5.1 0.50

MoAl 3.47 24.3 2.0 0.53MoNi 3.53 26.4 4.1 0.49

TiAl 3.35 23.7 1.1 0.52

GaAl 2.72 20.7 -1.6 0.49

two blocks of the material. Then, Gc is defined by the limit Gc = limx→∞G(x).

The energy Gc was determined from fits of DFT total energies for a set of given

fixed separations xi. Because the aim is to simulate the ideal brittle behaviour,

no structural relaxations were allowed. The ab initio values for G(xi) are then

fitted to the so-called universal binding energy relation (equation 3.15). The

details concerning the ideal brittle cleavage can be found in section 3.2.1, the

description herein is given for the sake of consistency.

In general, the parametersGc and l depend on the material and the orientation

(hkl) of the actual cleavage plane. Now, they will depend on the kind and position

of substitute atoms at the interfacial plane. The parameters determine the critical

cleavage stress σc = Gc/el as well.

For pure NiAl, the results of UBER fit are given in table 6.1. For (110)

cleavage, the lowest energy Gc = 2.40 eV is obtained, and also the lowest value

Gc/A = 3.24 J/m2 which indicates that the (110) cleavage is preferred, in ac-

cordance to Ref. [123]. For (100) cleavage, Gc = 2.50 eV is very close to the

result for the (110) case, but a substantially larger Gc/A= 4.79 J/m2 is derived

because the area A is smaller by a factor√

2 compared to (110). The rather equal

energies Gc seem to be surprising if the number of broken bonds (see table 6.1)

are inspected because for cleaving (110) twice as many second nearest neighbor

bonds are broken when compared to (110), with the number of broken nearest

neighbor bonds being equal. Analyzing the bond strengths by cleaving the pure

sublattices it turns out that strong Ni-Ni (≈ 0.7 eV) and Al-Al second nearest

neighbor bonds (≈ 0.6 eV) dominate the cleavage properties. The loss in nearest

6.4. BRITTLE CLEAVAGE 117

0 1 2 3 4 5 6x (Å)

0

1

2

3

4G

/A (

J/m

2 )

CrAl

TiAl

NiAl

GaAl

Figure 6.2: For NiAl and NiAl-X, calculated cleavage energy release rate G(x)/Afor (110) cleavage versus cleavage size x for substitutions X=Cr,Ti,Ga at Alsites. The analytic curves are obtained by fitting the ab initio energies (symbols)to UBER.

neighbor Ni-Al bonding, however, varies strongly (≈ 0.15, -0.02, 0.02 eV per bond

for (100), (110), (111), respectively), which consequently makes Gc for the (110)

cleavage the lowest in energy. Obviously, the accommodation of the dangling

bonds arising from cutting Ni-Al bonds depends strongly on the orientation and

size of the cleavage planes.

Inspecting figure 6.2 it is obvious that UBER fits rather well the ab initio data.

The energies G(x) for X=CrNi,MoNi,MoAl are not displayed but they behave very

similar to the shown data. All fitted values for Gc and l are presented in table 6.2.

For the (110) cleavage table 6.3 lists the change in Gc due to alloying for

different coverages of dopants at the interface . The most stabilizing effect is

derived for X=Cr for which the increase of Gc in comparison to pure NiAl is

about 15%, rather independent of the substitution site. Similarly but about half

of the increase of Gc is found for Mo substitutions. However, Ti on Al sites

influences the cleavage properties less significantly because of the rather similar

metallic radii and number of valence electrons of Ti and Al. A very exceptional

case of the present study is Ga, for which Gc/A is reduced by a rather substantial

amount.


Table 6.3: (110) cleavage of NiAl-X for the (1x1) and (1x2) geometries corre-sponding to 50% and 25% coverage by the substitutional atoms X = (Cr, Mo,Ti, Ga) at Al and Ni sites. Cleavage energy change ∆Gc/A in J/m2 with respectto pure NiAl (110).

cover. CrAl CrNi MoAl MoNi TiAl GaAl

50% 0.64 0.50 0.23 0.29 0.11 -0.5225% 0.33 0.21 0.14 0.03 0.15 -0.25

Table 6.4: (100) cleavage of NiAl-X for three different geometries correspondingto 100%, 50% and 25% coverage by the substitutional atoms X = (Cr,Mo,Ti,Ga)at Al and Ni sites. Cleavage energy change ∆Gc/A in J/m2 with respect to pureNiAl (100).

cover. CrAl CrNi MoAl MoNi TiAl GaAl

100% 0.49 0.13 -0.52 -1.45 -0.63 -1.0650% -0.01 0.26 -0.39 -0.18 -0.39 -0.5425% 0.01 0.09 0.07 -0.11 -0.10 -0.29

Cleaving (100) planes, the change of bonding is rather different from the (110)

results. The main difference being that for the (110) cleavage only two nearest

neighbor X-Al or X-Ni bonds are broken (because X replaces only one type of

atom in a 50% coverage) whereas for the (100) plane four of those bonds are

affected (because of the 100% coverage). The stabilisation effects for X=Cr is

still significant but reduced, the reduction being rather substantial for Cr on a Ni

site. The reinforcement of a Ni-terminated (100) interface by Cr was predicted in

an ab-initio study of the interfacial adhesion in NiAl-Cr eutectic composites [154].

For X=Mo the alloy is significantly easier to cleave as compared to pure NiAl,

and similar to Cr, Mo on a Ni site reduces the cleavage energy much more by

about 30%. The elements Ti and in particular Ga on Al sites lower the cleavage

energy by a sizable amount.

Last, the (211) cleavage is calculated. In the table 6.5 are the changes of Gc

compared, of course at the same 25 % coverage. Obviously, the changes caused

by various dopants are similar at different planes, in particular the (211) and

(110) planes display very similar results. Interestingly, CrAl causes pronounced

strengthening of the cleavage planes, whereas, as will be demonstrated in the

6.5. SLIPS AND DUCTILITY 119

Table 6.5: For NiAl-X, calculated brittle cleavage properties for the orientations(hkl). Cleavage energy change ∆Gc/A in J/m2 at 25 % coverage with respect topure NiAl.

(hkl) CrAl CrNi MoAl MoNi TiAl GaAl

(100) 0.01 0.09 0.07 -0.11 -0.10 -0.29(110) 0.33 0.21 0.14 0.03 0.15 -0.25(211) 0.40 0.19 0.29 0.04 0.22 -0.17

next section, its effect on the stacking fault surface is vice-versa.

6.5 Slips and Ductility

On the atomic scale, a material is expected to be ductile when the emission

of a dislocation is energetically favorable over cleavage at the crack tip [97].

The crucial quantity which should govern this process is Gd, the critical energy

release rate for the emission of a dislocation. Because the dislocation emission

is a complex process influenced by many factors (e.g. the geometry of crack and

loading, the type and direction of the emitted dislocation), the relation between

Gd and intrinsic materials parameters are to a large extent approximate and

subject of discussion.

Rice [98] showed that for an isotropic linear elastic solid under mode II loading

(i.e. the dislocation is emitted on the slip plane coinciding with the crack plane)

Gd is equal to the so-called unstable stacking fault energy γus: it is defined as the

maximum of the generalised stacking fault energy by γus = max (γGSF (f)), with

γGSF being an energy per unit area necessary to slip two blocks of the material

against each other in the direction f [99, 100]. For load mode I, the most highly

stressed slip plane is at an angle θ with the crack plane. For that, Rice suggested

the criterion involving the geometrical factor

Gd = γusY (θ);Y (θ) = 8/((1 + cos θ) sin2 θ). (6.1)

The brittle to ductile crossover is given by condition Gd/Gc < 1. For ratios

smaller than 1 the material is considered to be ductile. Rice’s model was found

to be rather accurate for mode II loading [101], whereas for mode I loading

it seems less reliable: in case the dislocation emission plane is at an nonzero

angle to the crack plane, a ledge is formed. Thus the emission involves also the

formation of the surface of the ledge which is not included in Rice’s analysis.

In order to account for the ledge formation, Zhou, Carlsson and Thomson [102]


0 0.1 0.2 0.3 0.4 0.5f/b

0.2

0.4

0.6

0.8γ G

SF (

J/m

2 )NiAlTi

Al

GaAl

MoAl

CrAl

<111>(110)

Figure 6.3: For NiAl and NiAl-X, calculated generalised stacking fault energiesγGSF for a 〈111〉(110) slip with X=Ti,Cr,Mo,Ga on Al sublattice sites. f/b: sliprelative to Burger’s vector.

introduced corrections and found that the crossover from a ductile to a brittle

solid is independent of the intrinsic surface energy when the ledge is present.

They suggested a new criterion for the prediction of ductile behaviour (ZCT),

γus

µb< 0.014. (6.2)

There, µ denotes the isotropic shear modulus and b the Burger’s vector of the

emitted dislocation. A recent study of dislocation emission indicated a similar

effect of the ledge formation [104]. One can roughly estimate the brittle-ductile

crossover by ZCT (see equation 6.2) assuming that the isotropic shear modulus

µ = 80.1 GPa as calculated for pure NiAl remains constant. Then, for a 〈001〉slip ductile behaviour is expected for γus < 0.33 J/m2. In case of a 〈111〉 slip

the ZCT correction cannot be directly applied because the emission of partial

dislocations may occur. Nevertheless, assuming the emission of a full dislocation

ductile behaviour is expected to occur for γus < 0.57 J/m2. Of course, the

anisotropic shear modulus may be calculated, following a procedure outlined

in [114]. The procedure is shortly described in section 5.5.3 and the calculated

values of anisotropic shear modulus of NiAl are displayed table 5.2.

Both criteria -Rice and ZCT- have in common that the ductility is primarily


0 0.1 0.2 0.3 0.4 0.5f/b

0.5

1

1.5γ G

SF (

J/m

2 )

NiAlGa

Al

CrAl

TiAl

MoAl

<001>(110)

Figure 6.4: For NiAl and NiAl-X, calculated generalised stacking fault energiesγGSF for the 〈001〉(110) slip with X=Cr,Ti,Mo on the Al sublattice sites. f/b:slip relative to Burger’s vector.

controlled by the unstable stacking fault energy γus, which is then the key quan-

tity. Therefore, the influence of alloying elements X on γus is studied. The results

obtained from both criteria are demonstrated and discussed in section 6.6.

6.5.1 〈111〉(110) and 〈001〉(110) slips

The (110) cleavage habit plane is preferred slip plane as well. By slip in the 〈001〉direction single dislocations are formed, whereas the 〈111〉 direction features pair

of Shockley partial dislocations separated by the anti-phase boundary formed by12〈111〉 shift displacement.

Observing the γGSF profile of the 〈111〉 slip in figure 6.3, the local minimum at

the displacement f/b = 0.5 corresponds to the geometry of the anti-phase bound-

ary. Consequently, the position of maximum of γGSF is not dictated by symmetry

and lies at f/b ≈ 0.25 for all the studied cases, except Ga. For pure NiAl, an

anti-phase boundary energy of EAPB = 1.00 J/m2 is derived for the geometrically

unrelaxed case, being in excellent agreement to other calculations [50, 118]. The

reported 20% decrease of EAPB due to atomic relaxations [118] is consistent with

our value of EAPB = 0.76 J/m2 for a fully relaxed calculation (see table 6.6).


Table 6.6: For NiAl-X, calculated unstable stacking fault energies γus in J/m2

for 〈001〉 and 〈111〉 slips on the (110) plane, and the energy EAPB of the 12〈111〉

anti-phase boundary.

γus 〈001〉 γus 〈111〉 EAPB

NiAl 1.28 0.83 0.76

CrAl 0.88 0.47 0.07CrNi 1.40 0.79 0.48

MoAl 0.22 0.55 0.06MoNi 1.04 0.70 0.12

TiAl 0.60 0.70 0.30

GaAl 1.05 0.60 0.60

It is noticeable that for X=Cr,Mo at Al sites the profiles look very similar with

very small values EAPB < 0.1 J/m2. For X=Cr,Mo at Ni sites, the maxima of

the profiles are larger by a factor two, and the stacking fault energies are signif-

icantly different as shown in table 6.6. The strong decrease of EAPB for X=Cr

is in agreement with calculations of Hong and Freeman [120]. In the present

work, the reduction effect is even more pronounced, probably due to the neglect

of atomic relaxations in the study of Ref. [120]. The lowering of EAPB due to

alloying might lead to an increased width of splitting between 1/2〈111〉 Shockley

partial dislocations, because due to elasticity theory the equilibrium separation

of partials is inverse proportional to EAPB [114]. Depending on the strength of

their coupling, two partials may move independently or will be coupled and, con-

sequently, their mobility will be substantially influenced. However, the splitting

is also determined by the shape of the γGSF surface. Thus a more elaborate treat-

ment within the dislocation model of Peierls and Nabarro [110, 111] is needed to

elucidate the splitting mechanism. The activation of the 〈111〉 Shockley partial

dislocation is considered to be crucial for improving the intergranular brittleness

of NiAl.

Stacking fault energy profiles for the 〈001〉 slip with X on Al sites are shown

in figure 6.4. In comparison to NiAl, for X=CrAl the energy γus for the 〈111〉 slip

is reduced by 40%, but to a lesser amount for the 〈001〉 slip. Therefore, the nucle-

ation of 〈111〉 dislocations becomes more favorable at the crack tip. Furthermore,

because of the calculated value of γus = 0.47 < 0.57 J/m2 (see table 6.6) duc-


Table 6.7: Unstable stacking fault energy γus in J/m2 for the 〈001〉 [110] slipfor NiAl-X, X=(Cr,Mo,Ti,Ga) substitutions. Results for two concentrations ofdefects. Further details, see text. For NiAl, γus = 1.28 J/m2.

conc. CrAl CrNi MoAl MoNi TiAl GaAl

50% 0.88 1.40 0.22 1.04 0.60 1.0425% 1.12 1.30 0.80 1.07 1.05 1.20

tile behaviour may be expected. These findings agree with the experimentally

observed activity of 〈111〉 dislocations in NiAl-CrAl at low temperatures [124].

There exists, however, contradiction between experimental findings, because in

Ref. [147] no activity of 〈111〉 dislocations is reported for stoichiometric NiAl

single crystals alloyed by Cr. This contradiction may be well explained within

our calculations. Presumably, the Ni-Al composition plays a major role because

-according to our calculations- γus is much larger for X=CrNi than for X=CrAl,

as displayed in figure 6.5. The ’successful’ (in terms of the observed activity of

〈111〉 dislocations) experiments of Ref. [124] alloyed Cr atoms into Al sublattice,

where is their effect obviously stronger than in Ni sublattice due to calculations

herein (see figure 6.3 and figure 6.5). The other experimental group [147] used

stoichiometric NiAl-Cr single crystals and in such an arrangement Cr atoms may

sit at the both sublattice sites [153]. Because the γus for CrNi is relatively large,

the effective reduction of γus due to alloying might be rather moderate and pre-

sumably insufficient to open the 〈111〉 slip system.

The energy profile of γGSF (f) for X=MoAl for the 〈111〉 slip is similar to

X=CrAl, but for the 〈001〉 slip the energy γGSF for X=MoAl is strongly reduced

compared to Cr (see table 6.6). For the 〈001〉 slip a remarkable reduction of γGSF

arises even for a smaller interface coverage by Mo. Thus, for higher coverage

by X=MoAl the NiAl-Mo alloys should display ductile behaviour as predicted

by ZCT. This finding is in excellent agreement with observed enhancement of

ductility for NiAl-Mo single crystals with tensile axis in [110] direction [125].

The [110] orientation of loading provide large resolved shear stress on 〈001〉 slip

system.

Nevertheless, as the observed enhancement of ductility for NiAl-Mo [125] is

probably carried by the activity of the 〈001〉(110) dislocations, the intergranular

ductility of NiAl-Mo polycrystals seems not to be improved. As discussed in

section 6.2, the 〈001〉 slip generates only three independent slip systems and,

thus, does not fulfill von Mises criterion for ductility of a polycrystal.


0 0.1 0.2 0.3 0.4 0.5f/b

0.2

0.4

0.6

0.8γ G

SF (

J/m

2 )

NiAlCr

Ni

MoNi

<111>(110)

Figure 6.5: Generalised stacking fault energy γGSF along the 〈111〉 direction onthe (110) plane for NiAl-X with X=(Cr,Mo) on the Ni sublattice sites. Letter b:respective Burger’s vector.

0 0.1 0.2 0.3 0.4 0.5f/b

0.2

0.4

0.6

0.8

γ GSF

(J/

m2 )

NiAlCr

Al 25%

MoAl

25%Mo

Al 50%

CrAl

50%

<111>(110)

Figure 6.6: Generalised Stacking Fault energy γGSF along 〈111〉 direction onthe (110) plane for NiAl-X with X=(Cr,Mo) on the Al sublattice for two defectconcentrations. Further details, see text.


0 0.1 0.2 0.3 0.4 0.5f/b

0.5

1

1.5

2γ G

SF (

J/m

2 )

NiAlCr

Al

MoAl

TiAl

Figure 6.7: Generalised Stacking Fault energy γGSF along 〈001〉 direction on the(100) plane for NiAl-X with X=(Cr,Mo,Ti) on the Al sublattice. The Ga atomsdid not cause any considerable change of stacking fault energetics. Further details,see text.

In contrast to X=CrAl,MoAl, for X=GaAl no slip direction is significantly fa-

vored and when RC is considered, the lower stacking fault energy barriers are

compensated by the decreased cleavage energies as listed in table 6.5. Never-

theless, the profile for X=GaAl for the 〈111〉 slip (see figure 6.3) indicates that

partials might tend to join into one superdislocation: γus is close to the crossover

value of ZCT. Thus, in case of Ga, the predictions of both criteria differ. The

experiments of Darolia et al. [125] showed high tensile elongations for NiAl-Ga

loaded in the [110] direction. Other experiments indicated that even pure stoi-

chiometric NiAl single crystals are able to undergo high tensile elongations under

certain conditions [144]. Because the results do not strongly indicate an im-

provement of intrinsic ductility of NiAl-Ga, observed larger elongations reached

in NiAl-Ga crystals might also be an indirect product of the process of alloying.

For X=TiAl, only a weak influence on the 〈111〉 slip was derived, when com-

pared to the other studied cases. Also for the 〈001〉 slip one cannot speculate

about an intrinsic ductile alloy. On the other hand, when Ti is used for en-

hancing the creep properties of NiAl at high temperatures [150], no worsening of

low-temperature brittleness is to be expected.


Table 6.8: For NiAl and NiAl-X, the unstable stacking fault energy for 〈001〉(100) slip (in J/m2) calculated in three supercell configurations, see text. PureNiAl has γus = 1.52 J/m2.

supercell cover. % CrAl CrNi MoAl MoNi TiAl GaAl

1x1 100 1.56 2.27 1.46 0.80 2.05 0.91√2x√

2 50 1.66 1.93 1.61 1.44 1.93 1.242x2 25 1.54 1.74 1.54 1.46 1.71 1.32

6.5.2 〈001〉(100) slip

At the (100) plane only s lips along 〈001〉(100) are studied, because the 〈011〉(100)

slip is blocked by a large unstable stacking fault energy (see table 5.1). The

corresponding values are listed in table 6.8 and stacking fault energy profile for 50

% coverage are displayed in figure 6.7. For the 〈001〉(100) slip in pure NiAl a value

of γus = 1.52 J/m2 is calculated, which is about 10% larger than the calculated

value reported by Wu et. al. [122]. This small discrepancy is attributed to the

rather thin slab used in Ref. [122].

Substitutions X=MoAl show some remarkable concentration dependence: for

100% coverage γus is almost half that of NiAl, but at 50% coverage the alloying

effect almost vanishes. This indicates that Mo-Mo bonding is much weaker than

Mo-Al bonding, which is further confirmed by the large γus for the NiAl-MoNi

compound. Hence, macroscopic behaviour of NiAl-MoAl alloys might depend on

diffusion and cluster segregation of Mo on the cleavage plane. In general, because

of the rather large stacking fault energy in the (100) plane for lower concentrations

of X, the emission of 〈001〉 dislocations in this plane seems improbable.

Exploiting table 6.8 one can observe that Ga dopants reduce significantly the

γus of the 〈001〉(100) slip system. According to the experiments of Darolia et

al. [125] NiAl-Ga single crystals loaded in the [110] direction showed high tensile

elongations. This may well be due to the activity of 〈001〉(100) dislocations,

because [110] tensile loading provides large resolved shear stress for this slip

system.

6.5.3 〈111〉(211) slip

The 〈111〉(211) is an active slip system in NiAl as well. As was discussed in the

previous chapter, in the pure NiAl is the stacking fault energy of this slip rela-

tively close to that of 〈111〉(110) slip system. By 12〈111〉(211) shift an anti-phase

6.6. SUMMARY 127

Table 6.9: For NiAl and NiAl-X, calculated anti-phase boundary energies (inJ/m2) for the 〈111〉(211) slip.

NiAl CrAl CrNi MoAl MoNi TiAl GaAl

0.96 0.66 0.82 0.62 0.78 0.84 0.90

boundary is formed. In pure NiAl, the anti-phase boundary energy represents

at the same time the maximum of γGSF (i.e. the maximum of γGSF lies at the

position of the anti-phase boundary, see figure 5.6) and, therefore, the γus is

given directly by the anti-phase boundary energy. The calculated values of the

anti-phase boundary energy for the NiAl-X are listed in table 6.9.

The calculations of the stacking fault energetics of the high-index (211) plane

are costly from computational point of view, because unit cell contains larger

number of atoms. Thus, only one coverage of substitutional atoms was considered,

namely 25% coverage of the (211) interface.

The effect of Cr and Mo is in analogy to the effect on (110) plane properties:

the cleavage energy is elevated, in particular in case of Cr, whereas the unstable

stacking fault energy is considerably reduced. The substitutions into Al sublattice

provided better ductilization on NiAl as in latter cases of (100) and in particular

(110) planes.

It should be noted, that in pure NiAl the stacking fault energy profile features

weak local maximum of γGSF approximately at 0.2b in 〈111〉. If the anti-phase

boundary energy is reduced stronger that the rest of γGSF curve (as found for

〈111〉(110) profile, see figure 6.3), it would be possible that upon alloying this

maximum becomes global. However, the calculation of full γGSF profile would be

very demanding in terms of computational time and the change of the anti-phase

boundary energy reveals well the effect of various substitutes.

6.6 Summary

Table 6.10 summarizes the results about the estimation of a possible ductility im-

provement of microalloyed NiAl. The results indicate, that the most pronounced

improvement of the intrinsic ductility of NiAl-X alloys is expected in particular

for X=Cr,Mo at Al sites. These substitutions decrease substantially the stacking

fault energies of the (110) plane whereas the calculated cleavage properties of the

(110) plane indicate strengthening against brittle fracture. It should be noted,

that Cr and Mo might activate different slip systems (〈111〉(110) for X=MoAl

and 〈001〉(110) for X=CrAl), which might result in significant differences for the


Table 6.10: For NiAl-X, estimation of ductile behaviour. The material is pre-dicted to be ductile according to Rice [98] if listed values of Gd/Gc are smallerthan 1 (Actual values of Gd are derived for θ = 90◦ according to equation 6.1),and according to Zhou et al. [102] (ZCT) if listed values of γus/µb are smallerthan 0.014 (see text for details). Results are derived for 50% concentration ofdopants at the interface. Symbols: + material is ductile; ∼ at the crossover.

Rice ZCTcompound 〈001〉〈111〉〈001〉〈111〉

NiAl 3.13 2.04 0.055 0.021

CrAl 1.82 0.93∼ 0.037 0.011 +CrNi 2.94 1.70 0.060 0.020

MoAl 0.51 + 1.27 0.009 + 0.014 ∼MoNi 2.38 1.59 0.045 0.018

TiAl 1.43 1.67 0.026 0.018GaAl 3.13 1.75 0.045 0.015 ∼

macroscopic behaviour of the corresponding alloys. Because Mo dopants promote

the 〈001〉 slip, the improvement of NiAl-Mo intergranular ductility seems improb-

able, because the 〈001〉 slip does not fulfill von Mises criterion for a ductility of

a polycrystal (see section 6.5.1 for details).

In contrast to NiAl-Cr and -Mo alloys, alloying by X=Ti and Ga has only

a minor effect on the stacking fault energies of the (110) plane. Ti promotes

activity of the 〈001〉(110) slip system, but the reduction of γus is not sufficient for

suspecting ductile behaviour. For X=Ga no ductility improvement at the (110)

plane can be strongly surmised, although -according to ZCT- an NiAl-GaAl alloy

for the 〈001〉(110) slip is close to the limit. In contrast with other elements, Ga

dopants generally decreased cleavage energies.

In general, the effect of dopants was found significantly dependent either on

the slip direction even within one slip plane (compare, for instance, Mo and

Cr effects on the 〈001〉 and 〈111〉 slip at the (110) plane), or the composition

(CrAl with respect to CrNi) which enabled us to interpret discrepancies in the

experimental findings. See section 6.5.1 for details.

It should be noted, that because of the application of standard density func-

tional theory the presented approach neglects temperature dependent effects.

Furthermore, the present investigations are of a model character but nevertheless

6.6. SUMMARY 129

provides reliable data for perfectly known and controlled conditions. The influ-

ence of alloying substitutions X is certainly overemphasized by placing all X in

the cleavage and interface planes, i.e. segregation to the cleavage surfaces and

slip interfaces was assumed. However, the agreement of trends obtained by the

present ab initio approach with experimental findings is remarkable.

Chapter 7

Summary

This thesis was aimed at the role of DFT calculations in the treatment of the

mechanical properties of materials. Though strong development in last decades,

the mechanisms underlying the mechanical response of material still retain much

mystery. Essential processes at the atomic level associated with the mechanical

response of material were discussed and their modelling in the framework of the

DFT method was demonstrated.

Several distinct problems of the materials science were addressed: (1) a con-

ceptual problem of the correlation between cleavage and elasticity, (2) theoretical

approach to the ductility and the dislocation behaviour, and (3) the simulation of

the microalloying of NiAl in a survey for its ductilization. The theme underlying

all these different problems is how to link subtle interactions between the atoms

with the behaviour of the macroscopic piece of material.

The problem (1) involves a conceptual obstacle, because the non-local quan-

tity (elasticity) has to be related to the local quantity (cleavage). It was managed

to establish well-defined correlations between the elastic and cleavage properties

introducing the concept of the localisation of the energy of the elastic response

close to the crack-like perturbation in the spirit of Polanyi [62], Orowan [45] and

Gilman [63]. Probably, the main achievement of this thesis consists in the in-

troduction of a new materials parameter, which is called the localisation length

L. By this flexible parameter the bridge between elastic and cleavage energy (or

stress) was built. The actual values of L, which depend on the material and the

direction of cleavage, were determined by fitting to DFT calculations of the deco-

hesive energy as a function of the crack opening. The concepts were tested for all

types of bonding, and for brittle cleavage it turned out, that -at least for metals

and intermetallic compounds- an average value of Lb ≈ 2.4 A would yield reason-

ably accurate cleavage stresses if one knows only the uniaxial elastic modulus and

the brittle cleavage energy. This means, that the ”engineer” may estimate the

131

132 CHAPTER 7. SUMMARY

critical mechanical behaviour of a material -at least for uniaxial strain loading-

purely knowing macroscopic materials parameters, namely the cleavage energy

and the elastic moduli. Even if the cleavage energy is not easily accessible exper-

imentally, it could be derived from a single DFT calculation for each direction,

which in many cases is not very costly.

Furthermore, it is found convenient analytical formulation for relaxed cleavage

process which utilizes a natural parameter -critical length for relaxed cleavage lr-

and does not depend on number of layers as in previous approaches [60, 61].

Moreover, the parameter lr gives a measure up to which critical openings an

initiated crack is able to heal under ideal conditions. The connection to elastic

properties can be again made via the localisation of the elastic energy, however

the behaviour of Lr for the relaxed cleavage is less simple to describe and no

general trend is observed. In order to take advantage of the reasonable behaviour

of the L for brittle cleavage, a new concept of relaxation -semirelaxed cleavage-

is suggested, which enables structural relaxation of the surface within UBER

framework. This concept may be useful in deriving parameters for cohesive zone

models.

In addition, the cleavage properties and anisotropic elastic constants calcu-

lated for many various technologically significant materials on-the-equal-footing

form rather unique database of basic mechanical properties for number of mate-

rials. The calculated parameters can be used to derive or adjust model potentials

for the large-scale simulations as well.

Treating problem (2), the stacking fault energetics of several slip systems in

NiAl was calculated. It is found 〈001〉 and 〈111〉 as the preferred slip directions

in NiAl, in good agreement with fracture experiments. Though calculated values

of the unstable stacking fault energy are lower in the 〈111〉 direction, the 〈001〉is main slip, because 〈111〉 slips are somewhat hampered by the relatively high

anti-phase boundary energy. The anti-phase boundary prevents the formation of

the 12〈111〉 partial dislocations, which occur in metals with bcc structure. Thus,

the 〈111〉 dislocations form only when the resolved shear stress for the 〈001〉 slip

is low.

In the next step, a problem of the tension coupled to the shear stress at the

slip plane was considered. The calculation presented in this thesis is the first ab

initio simulation of the tension-shear coupling. It revealed that the tensile stress

acting over the slip plane considerably decreases the unstable stacking energies

and, consequently, lowers the threshold for the dislocation emission onto this slip

plane. The relaxation of atoms has to be performed in order to obtain the reliable

stacking fault energetics. When the cleavage properties are of concern, similar

conclusion can be made - the cleavage energy is lower in the presence of the stack-

133

ing fault. Such a fact is important in case of polycrystal, where -due to various

orientations of grains with respect to the external stress direction- some amount

of the resolved shear stress is essentially always present. Thus, the resolved shear

might weaken grain interface and make the crack propagation between grains

more favorable over the propagation through crystal bulk. Of course, more elab-

orate studies are necessary to elucidate tensile-shear coupling and other processes

at grain boundaries. The calculations of this thesis demonstrated clearly that the

tension acting over slip plane has an essential influence on the γGSF energetics and

its effect on dislocations properties should be considered in future calculations.

In the future, it is planned to focus on the derivation of the constitutive rela-

tions for tension-shear coupling based on Lejcek’s solution of the Peierls-Nabarro

equation, because such a model would provide complete, tractable, and physically

transparent -and DFT based- description of dislocations.

The problem (3) is concerned with ductilization of NiAl via the microalloy-

ing. The calculations were focused on the cleavage and stacking fault energetics

in a supercell configuration where Ni or Al atoms at the cleaved or faulted in-

terface were replaced by Cr, Ti, Mo, or Ga atom. The results indicate that

the most pronounced improvement of the intrinsic ductility of NiAl-X alloys is

expected in particular for X=Cr,Mo at Al sites. These substitutions decrease

substantially the stacking fault energies of the (110) plane whereas the calcu-

lated cleavage properties of the (110) plane indicate strengthening against brittle

fracture. It should be noted that Cr and Mo might activate different slip sys-

tems (〈111〉(110) for X=MoAl and 〈001〉(110) for X=CrAl), which might result in

significant differences for the macroscopic behaviour of the corresponding alloys.

The improvement of the ductility of NiAl-Mo was found experimentally, in agree-

ment with the calculations. Somewhat contradictory experimental results have

been reported for Cr dopants, which is discussed in detail in appropriate section.

The strong difference between chromium alloyed into Al or Ni sublattice sites was

found. Based on this finding, the experimental discrepancies were explained by

different stoichiometry of the single crystals used in respective experiments.

In contrast to NiAl-Cr and -Mo alloys, alloying by X=Ti and Ga had only

a minor effect on the stacking fault energies of the (110) plane. Ti promotes

activity of the 〈001〉(110) slip system, but the reduction of γus is not sufficient

for suspecting ductile behaviour. The reported improvement of the ductility of

an NiAl-GaAl (110) oriented single crystal was explained by the activity of the

〈001〉(100) slip, where considerable decrease of γus was observed. In general, the

agreement of the purely theoretical simulation with the experimentally observed

trends suggests that DFT calculations offer an alternative route for modern alloy

design which can be used in synergy with experiments. Such a modelling takes full

134 CHAPTER 7. SUMMARY

advantage of the predictive capability of DFT quantum mechanical calculations.

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Appendix A

Publications

P. Lazar, R. Podloucky, and W. Wolf

Correlating elasticity and cleavage

Applied Physics Letters, 87, 261910 (2005)

P. Lazar, R. Podloucky

Ab initio study of the mechanical properties of NiAl microalloyed by

X=Cr,Mo,Ti,Ga

Physical Review B, accepted for publication


Ab initio study of correlations between elastic and cleavage properties

Physical Review B, submitted


Correlation between elastic and cleavage properties

Progress in Materials Science, Proceedings, Festschrift on the 60th birthday of

D.G. Pettifor, submitted


A new concept of cleavage: an ab-initio study

Modelling and Simulation in Material Science, submitted


Ab initio study of tension-shear coupling at the slip plane

to be submitted to Physical Review B

145

146 APPENDIX A. PUBLICATIONS

Appendix B

Conference contributions


Ab-initio calculation of the influence of Cr- and Ti-microalloying on the me-

chanical properties of NiAl

E-MRS Fall Meeting, Warsaw, Poland (2004)

P. Lazar and R. Podloucky and W. Wolf

Fracture and Elasticity

Meeting of the International Advisory Board at CMS, December (2004)


An ab initio study of the connection between elasticity and crack formation

DPG (Deutsche Physikalische Gesselschaft) year meeting, Berlin, Germany

(2005)

P. Lazar and R. Podloucky and W. Wolf

Correlating Elasticity and Cleavage

Meeting of the International Board at CMS, November (2005)

147

148 APPENDIX B. CONFERENCE CONTRIBUTIONS

Appendix C

Acknowledgments

This thesis would not have been created without help and support of several

people, to whom I am very grateful. At the first place shines Raimund Podloucky,

who suggested that the link between cleavage and elasticity might be of interest,

found sound physical interpretation of results as well as new research direction

and simulated me with many discussions about the topic. But, I am grateful

to him for many more reasons, his good and positive mood, which results in

friendly atmosphere in the office as well as in outdoor drinking sessions. In

addition, he found financial support which enabled me to work on the thesis.

The friendly atmosphere in our group would be unimaginable without other

members of the group, Cesare Franchini, Veronika Bayer and Xing-Xiu Chen. I

would like to mention former member of our group, Doris Vogtenhuber, because

it was pleasure to share office with her.

Further, I would like to thank Walter Wolf, who cooperated on the signif-

icant part of the work, in particular on the calculation of elastic constants. His

also stimulated the work with fruitful discussion and comments.

I thank to Mojmır Sob, who introduced me into the exciting field of the

ab initio DFT calculations of solid state properties. He supported and led me in

my first steps in the role of scientist.

Last, but not least, I thank to family and my girl Zuzana. She deserves

acknowledgment, because she drew several figures and sketches in the thesis and,

thus, saved reader from the boring combination of the text and equations only.

149

150 APPENDIX C. ACKNOWLEDGMENTS

The work was supported by the Austrian Science Fund FWF in terms of the Sci-

ence College Computational Materials Science, project nr. WK04. Calculations

were performed on the Schrodinger-2 PC cluster of the University of Vienna.

Appendix D

Curriculum Vitae

• 21.2.1979 born in Brno, Czech Republic

• 1997 - 2002 graduate study of physics, Masaryk University in Brno

• 1998 - 1999 young research assistant at Plasmochemical laboratory at

Masaryk University; thin films deposition using hollow tube discharge at

atmospheric pressure

• 2000 - 2002 at Solid State Department of M. University

• 2001 - 2002 also at Institute of Physics of Materials, Academy of Sciences

of the Czech Republic

(CZ-61662 BRNO, Zizkova 22)

• 2002 Master Thesis: Martensitic Phase Transformations and Phase Stabil-

ity in group of Prof. Mojmır Sob

• 2002 - 2005 PhD Study of physics at University of Vienna

• 2002 - 2005 also at Center for Computational Materials Science (CMS)

(Gumpendorferstr. 1A, A-1060 Vienna, Austria)

151

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