Download Ab initio modelling of mechanical and elastic properties of solids

Dissertation
Ab initio modelling
of 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 materials
on the basis of density functional theory (DFT) by applying first-principles or ab
initio techniques. Mechanical properties of materials are of crucial importance
for technological applications. How a material breaks, is -however- still not well
understood in many aspects. The results of the thesis should demonstrate that ab
initio calculations can provide fundamental insight into the true, namely atomistic, mechanisms of fracture. For very small loads material behaves in an elastic
manner, and -consequently- the elastic properties of solids are need to be understood and calculated. Therefore, after some introductory remarks discussing the
ab initio concept in chapter 1 the elastic behaviour of solids and results of the
actual calculations of elastic constants are discussed in chapter 2. The main part
of the thesis focuses on the mechanisms of fracture at the atomic scale, starting
with brittle fracture as discussed in chapter 3. The ab-initio total energy calculations simulating cleavage of material under tensile loading are introduced and
discussed in the light of classical theories. Consequently, a long standing question
of materials science about the possible connection between critical cleavage stress
and elastic properties is addressed in chapter 4. A concept of localisation of the
elastic energy is developed, by which a well defined correlation between cleavage
and elastic properties is established, at least for some idealized cases of fracture.
This concept is applied to a wide range of materials representing different types
of bonding. The calculated and derived cleavage properties are compared to the
(rather scarce) experiments and to data of other theoretical concepts, and the
behaviour of the newly introduced materials parameter -the localisation lengthis investigated. Interestingly and surprisingly, for brittle cleavage the results suggests that by choosing an average, constant value of the localisation length for
-almost- all materials critical cleavage stress can be directly estimated from the
cleavage energy and the elastic constants within an error of ± 10%. Such a correlation, which is also quantitatively useful, was sought for about 80 years in
the scientific community, and finally established in the present work. Chapter 5
deals with ductile fracture. For Nial, the criteria of Rice for dislocation emission
from a crack tip and the Peierls-Nabarro model are utilised in order to calculate
ductility and dislocation properties of various slip systems. The h111i slips in
(110) and (211) planes dominate the ductile behaviour. For the first time, the
tension-shear coupling in the slip plane is calculated by an ab initio technique.
In chapter 6, the application of the previously discussed models is demonstrated
for the simulation of microalloying effects for NiAl, with the aim for finding an
improvement of its ductility, which is very important for technological applications. The achieved results suggest that Cr and in particular Mo are promising
candidates for improving ductility. The ab initio findings are in excellent correlation with experimental observations. The short summary of chapter 7 concludes
the thesis.
Abstract
Das Ziel der Dissertation ist die Untersuchung von mechanischen Eigenschaften
fester Materie mit Hilfe von Ab Initio Methoden, die auf der Dichtefunktionaltheorie beruhen. Mechanische Eigenschaften von Materialien sind von entscheidender Bedeutung für ihre technolgosche Anwendung. Wie eine Material wirklich
bricht, ist immer noch nicht gut verstanden. Die Ergebnisse dieser Arbeit zeigen,
daß Ab Initio Berechnungen einen tiefen Einblick in die wirklichen, atomistischen
Vorgänge des Materialbruches geben können. Für sehr kleine Belastungen verhält
sich jedes Material elastisch. Die elastischen Eigenschaften müssen daher berechnet werden können. Nach einer kurzen Einleitung über die Ab Initio Methodik in
Kapitel 1 werden deshalb die elastischen Eigenschaften fester Materie im Kapitel 2 diskutiert. Der Hauptteil der Dissertation befaßt sich mit Bruchvorgängen
im atomistischen Bereich, wobei Kapitel 3 mit dem ideal brüchigen Verhalten
beginnt. Die Ab Initio Gesamtenergien der Rechnungen, die das Spalten eines
Materials unter Zugspannung simulieren, werden in Verbindung mit klassischen
Theorien diskutiert. Das seit langem offene Problem eines möglichen Zusammenhangs zwischen der kritischen Spaltspannung und elastischen Eigenschaften
wird im Kapitel 4 angesprochen. Ein Konzept der Lokalisierung der elastischen
Energie wird entwickelt, durch das eine wohldefinierte Beziehung zwischen Spaltung und elastischen Eigenschaften eingeführt werden kann -zumindest für einige
idealisierte Fälle von Bruchtypen. Dieses Konzept wird auf eine große Klasse von
Materialien mit verschiedenen Typen von chemischer Bindung angewendet. Die
dadurch gewonnenen Spalteigenschaften werden mit experimentellen Daten (von
denen es nur wenige gibt) und anderen theoretischen Ergebnissen verglichen. Das
Verhalten des neu eingeführten Parameters -der Lokalisierungslänge- wird untersucht. Interessanterweise und überraschend stellt sich heraus, das für den ideal
brüchigen Bruch diese Länge als konstant angenommen werden kann, unabhängig
vom Material und der Richtung der Belastung. Damit kann die kritische Spannung direkt aus den Spaltenergien und den elastischen Konstanten mit einem
Fehler von ± 10% bestimmt werden. Nach einer solchen Beziehung, die auch
quantitive sinnvolle Resultate liefert, wurde mehr als 80 Jahre lange gesucht.
In dieser Arbeit ist sie schließlich aufgestellt worden. Kapitel 5 behandelt duktiles Bruchverhalten für NiAl. Die Kriterien von Rice für Versetzungsemissionen
durch eine Rißspitze und das Peierls-Nabarro Modell werden verwendet, um Duktilität und Versetzungseigenschaften von verschiedenen Gleitsystemen zu berechnen. Die h111i Gleitungen in den (110) und (211) Ebenen bestimmen das duktile
Verhalten. Zum ersten Mal wurde die Kopplung zwischen Zug- und Scherspannungen in einer Gleitebene mit einer Ab Initio Methode berechnet. Im Kapitel 6 werden die diskutierten Modelle für NiAl angewendet, um die Effekte des
Dazulegierens dritter Elemente zu simulieren, um die Duktilität zu verbessern,
was für technologische Anwendungen sehr wichtig ist. Die Rechnungen deuten
darauf hin, daß Cr und Mo erfolgsversprechende Kandidaten sind. Die Ab Initio
Ergebnisse sind in ausgezeichneter Übereinstimmung mit experimentellen Daten.
Eine kurze Zusammenfassung in Kapitel 7 beendet die Dissertation.
Contents
1 Introduction
1.1 Fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Density functional theory . . . . . . . . . . . . . . . . . . . . . . .
1.3 Electronic structure methods . . . . . . . . . . . . . . . . . . . . .
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2 Elastic properties of material
2.1 Elastic constants and crystal symmetry
2.2 DFT calculation of elastic constants . .
2.3 Results for selected materials . . . . .
2.4 The ideal strength . . . . . . . . . . .
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fracture
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3 Brittle fracture of material
3.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Continuum theory . . . . . . . . . . . . . . . . .
3.1.2 Stress intensity factors . . . . . . . . . . . . . . .
3.1.3 Griffith’s thermodynamic balance . . . . . . . . .
3.1.4 Irwin Theory . . . . . . . . . . . . . . . . . . . .
3.1.5 Lattice trapping . . . . . . . . . . . . . . . . . . .
3.2 DFT calculations for brittle fracture . . . . . . . . . . .
3.2.1 Cleavage decohesion . . . . . . . . . . . . . . . .
3.2.2 Calculation of cleavage decohesion for ideal brittle
3.2.3 Advanced applications of the ideal brittle cleavage
3.2.4 Relaxed cleavage decohesion . . . . . . . . . . . .
4 Cleavage and elasticity
4.1 Introduction . . . . . .
4.2 Orowan-Gilman model
4.3 Ideal brittle cleavage .
4.4 Localisation length . .
4.5 Results for ideal brittle
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cleavage
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2
CONTENTS
4.6
4.7
4.8
4.5.1 Computational aspects . . . . . . . . . . .
4.5.2 Simple metals . . . . . . . . . . . . . . . .
4.5.3 Intermetallic compounds . . . . . . . . . .
4.5.4 Refractory compounds . . . . . . . . . . .
4.5.5 Ionic compounds . . . . . . . . . . . . . .
4.5.6 Diamond and silicon . . . . . . . . . . . .
4.5.7 Conclusions . . . . . . . . . . . . . . . . .
Relaxed cleavage . . . . . . . . . . . . . . . . . .
4.6.1 Correlation between cleavage and elasticity
4.6.2 Results . . . . . . . . . . . . . . . . . . . .
4.6.3 Conclusions . . . . . . . . . . . . . . . . .
Semirelaxed cleavage . . . . . . . . . . . . . . . .
4.7.1 Introduction . . . . . . . . . . . . . . . . .
4.7.2 Results . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . .
5 Ductile fracture
5.1 Introduction . . . . . . . . . . . . . . . . . . .
5.2 The concept of unstable stacking fault energy
5.3 Modifications of Rice’s approach . . . . . . .
5.4 Dislocations properties . . . . . . . . . . . . .
5.4.1 Continuum model for dislocations . . .
5.4.2 Peierls-Nabarro model of a dislocation
5.4.3 Lejček’s method . . . . . . . . . . . . .
5.4.4 Peierls stress of a dislocation . . . . . .
5.5 Calculation of stacking fault energetics . . . .
5.5.1 Modelling aspects . . . . . . . . . . . .
5.5.2 Results - slip properties of NiAl . . . .
5.5.3 Results - dislocation properties of NiAl
5.6 Tension-shear coupling . . . . . . . . . . . . .
5.6.1 Introduction . . . . . . . . . . . . . . .
5.6.2 Model for tensile-shear coupling . . . .
5.6.3 Combined tension-shear relations . . .
5.6.4 Results . . . . . . . . . . . . . . . . . .
5.7 Summary . . . . . . . . . . . . . . . . . . . .
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Microalloying of NiAl
111
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2 Fracture properties of alloyed NiAl . . . . . . . . . . . . . . . . . 112
3
CONTENTS
6.3
6.4
6.5
6.6
Computational and modelling aspects
Brittle cleavage . . . . . . . . . . . .
Slips and Ductility . . . . . . . . . . .
6.5.1 h111i(110) and h001i(110) slips
6.5.2 h001i(100) slip . . . . . . . . .
6.5.3 h111i(211) slip . . . . . . . . .
Summary . . . . . . . . . . . . . . . .
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7 Summary
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A Publications
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B Conference contributions
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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, technologically 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 mechanical 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 science 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 characterize 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 fracture 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 material) 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 description 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 material 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 concentration 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 interaction 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
8
CHAPTER 1. INTRODUCTION
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. Because 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] demonstrates 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 largescale 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 interactions at the fundamental level - quantum physics is utilised by solving
Schrödinger’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
9
1.2. DENSITY FUNCTIONAL THEORY
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 completely 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] +
Z
V (r)ρ(r)dr +
1
2
Z
ρ(r)ρ(r0 )
drdr0 + Exc [ρ(r)],
r − r0
(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 equation corresponding to the functional of equation 1.1 results in Schrödinger-like
equations for the orbitals Ψ
h̄2 2
−
∇ + Vef f (r) Ψ(r) = εΨ(r).
2m
!
(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
Z
δExc [ρ]
ρ(r0 )
dr0 +
,
(1.3)
Vef f (r) = V (r) +
0
|r − r |
δρ
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) =
X
Ψ∗ (r)Ψ(r).
(1.4)
occ
In short, the reformulation of Kohn and Sham provides a suitable basis, which
transforms the functional equation into a set of differential equations. The resulting 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 historically first and widely used approximation is the local density approximation
(LDA), which is based on the assumption that the exact exchange-correlation
10
CHAPTER 1. INTRODUCTION
energy can be locally at the point r be replaced by the expression and value for
an homogeneous electron gas,
Exc [ρ] =
Z
ρ(r)εxc (r)ρ(r)dr,
(1.5)
in which εxc (ρ) is the exchange-correlation energy per particle of the homogeneous 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 parameterizations are small and, therefore, εxc (ρ) may be considered as a well-defined
quantity.
However, LDA itself is rather crude approximation, although it gives surprisingly 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 gradient 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 particular 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 particular 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 equilibrium 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 previous 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
11
1.3. ELECTRONIC STRUCTURE METHODS
to require translational symmetry for the observables, such as the potential,
vef f (r + R) = vef f (r).
(1.6)
There, R is a lattice translation vector. As a consequence of translational symmetry, the wave function must fulfill Bloch’s theorem,
ψk (r + R) = eikR ψk (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) =
X
ci,nk φik (r),
(1.8)
i
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,
X
i
[hφjk |H|φiki − εnk hφjk |φik i]ci,nk = 0.
(1.9)
This equation has to be diagonalized for obtaining the eigenvalues ε and eigenvectors 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 KohnSham 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, geometry 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 mentioned publications. It should be noted that VASP was applied for materials and
systems which may be considered as ’well-established’ from the computational
12
CHAPTER 1. INTRODUCTION
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
=0
∂xj
(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
!
1 ∂ui ∂uj
ij =
.
(2.2)
+
2 ∂xj
∂xi
The diagonal components 11 , 22 and 33 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 = Cmnpr pr ,
(2.3)
in which σmn denotes the stress tensor, pr 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
mn = Smnpr σpr
(2.4)
defining the elastic moduli Smnpr . The elastic constants and elastic moduli are
fundamental materials parameters providing a detailed information on the mechanical properties of materials. The knowledge of these data may enable prediction of mechanical behaviour in many different situations. Whereas σmn and pr
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=
1
E
= Cmnpr mn pr .
V
2
So far, e.g. the strain tensor has been considered as a
form

1
1
1 + exx
e
e
2 xy
2 xz
 1
1
e
=  2 eyx 1 + eyy
2 yz
1
1
e
e
1 + ezz
2 zx
2 zy
(2.5)
tensor of order two of the


.
(2.6)
Introducing the convenient matrix-vector notation, where the 6 independent elements 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 = cij εj
U=
1
E
= cij εi εj
V
2
(2.7)
2.1. ELASTIC CONSTANTS AND CRYSTAL SYMMETRY
15
Table 2.1: The number of independent elastic constants for different lattice symmetries and point groups (from Ref.[23]).
Lattice (point group)
Triclinic
Monoclinic
Orthorhombic
Tetragonal (4, -4, 4/m)
Tetragonal (422, 4mm, -42/m, 4/mmm)
Hexagonal and rhombohedral (3, -3)
Hexagonal and rhombohedral (32, 3m, -32/m)
Hexagonal (6, -6, 6/m, 622, 6mm, -62m, 6/mmm)
Cubic
No. of constants
21
13
9
7
6
7
6
5
3
Taking into account additional symmetry arguments imposed by the crystal lattice, 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
0
c12 c11 c12 0
0
0
c12 c12 c11 0
0
0
0
0
0 c44 0
0
0
0
0
0 c44 0
0
0
0
0
0 c44

ctetragonal =



















c11 c12 c13 0
0
0
c12 c11 c13 0
0
0
c13 c13 c33 0
0
0
0
0
0 c44 0
0
0
0
0
0 c44 0
0
0
0
0
0 c66
(2.8)










(2.9)
Explicit forms for other lattice symmetries may be found for instance in reference [22]. The total number of independent elastic constants for all crystal
systems is summarized in table 2.1.
16
2.2
CHAPTER 2. ELASTIC PROPERTIES OF MATERIAL
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 leastsquares 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
1
1
= c11 (ε211 +ε222 +ε233 )+ c44 (ε223 +ε231 +ε212 )+c12 (ε11 ε22 +ε33 ε22 +ε11 ε33 ). (2.10)
V
2
2
For a tetragonal distortion the shear displacements will be zero and the diagonal
components of the strain tensor are expressed as
dc
da
, ε3 = , ε4 = ε5 = ε6 = 0.
(2.11)
a
c
Inserting this relation into expression 2.10, the elastic energy density is given by
ε1 = ε 2 =
E
c11 3
= (c11 + c12 )21 + 2c12 1 3 +
.
(2.12)
V
2 3
The strains i 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
c11 + 2c12
E=
6
dV
V
!2
2c0
+
3
d(c/a)
c/a
!2
,
(2.13)
in which dV and d(c/a) denote infinitesimally small change of respective parameter. Calculating the total energy along a volume-conserving tetragonal deforma12
tion path close to the equilibrium, the elastic constant c0 = c11 −c
is obtained
2
17
2.2. DFT CALCULATION OF ELASTIC CONSTANTS
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
c11 + 2c12
E=
6
dV
V
!2
2c44
+
3
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
1
∂2E
B = (c11 + 2c12 ) = V
.
3
∂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
1 ∂σ12
2 ∂12
1
1 ∂σ33
c0 = (c11 − c12 ) = −
2
2 ∂33
∂σ11
1
B = (c11 + 2c12 ) =
.
3
∂11
c44 =
(2.16)
(2.17)
(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
18
CHAPTER 2. ELASTIC PROPERTIES OF MATERIAL
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 demonstrated 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 proceeds 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 symmetry 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 experimentally 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 approximation (see section 1.2). Several experimental methods are applied for the determination of the elastic constants of single crystals, the most prominent making
use of ultrasonic waves. However, one has to be careful when comparing experimental 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,
19
2.3. RESULTS FOR SELECTED MATERIALS
Table 2.2: The equilibrium lattice parameters and elastic constants of selected
materials calculated by VASP within the GGA PAW approach. The values in
brackets are experimental references (a Ref. [29], b Ref. [30], c Ref. [31], d Ref. [32],
e
Ref. [33])
Al
Fe
W
fcc
bcc
bcc
a0
4.06
2.83
3.17
c11
110.2 (108.2)
302.4 (242)
540.9 (521)
c12
54.8 (61.3)
168.7 (146.5)
202.7 (201)
c44
30.4 (28.5)a
102.5 (112)d
141.1 (160)e
NiAl
FeAl
Ni3 Al
Al3 Sc
B2
B2
L12
L12
2.89
2.87
3.56
4.10
202.9 (204.6)
278.1
225.0
185.7
140.3 (135.4)
139.9
150.6
49.4
112.6 (116.8)c
145.4
116.5
60.1
VC
TiC
B1
B1
4.16
4.34
646.6
514.5 (500)
135.6
106.0 (113)
193.4
178.8 (175)a
MgO
NaCl
B1
B1
4.22
5.01
297.1 (286)
52.8 (49)
95.4 (87)
12.3 (12)
156.1 (148)a
12.4 (13)b
C
Si
A4
A4
3.6
4.04
1050.2 (1076)
154.1 (165.7)
125.3 (125)
57.7 (63.9)
556.3 (576)a
74.7 (79.6)a
20
CHAPTER 2. ELASTIC PROPERTIES OF MATERIAL
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 methods 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 material’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 diamond, 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]. However, 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
22
CHAPTER 2. ELASTIC PROPERTIES OF MATERIAL
marring the perfection of specimens. The largest number of the high strength experiments 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 temperatures. 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 materials 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
3.1.1
Fundamentals
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 ∂σxy
+
=0
∂x
∂y
∂σyy ∂σxy
+
= 0.
∂y
∂x
(3.1)
(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
∂2Ψ
∂y 2
∂2Ψ
σyy =
∂x2
∂2Ψ
τxy = −
.
∂y∂x
σxx =
(3.3)
(3.4)
(3.5)
Furthermore, the differentiation of equation 2.2 produces for the case of plane
strain the equation
∂ 2 xy
∂ 2 xx ∂ 2 yy
+
−
2
= 0.
(3.6)
∂y 2
∂x2
∂y∂x
Combining above relations, the biharmonic equation for the stress function is
obtained
∇2 (∇2 Ψ) = 0.
(3.7)
When a solution of the biharmonic equation is found, the stresses and displacements 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 trackfree 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 macroscopic 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 displacements 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 hampers 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 experimental 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
KI
KII
KIII
σyy = √
fI (θ) σxy = √
fII (θ) σyz = √
fIII (θ).
2πx
2πx
2πx
(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
26
CHAPTER 3. BRITTLE FRACTURE OF MATERIAL
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 )
≡ G = 0.
∂a
(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.
3.1. FUNDAMENTALS
27
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 process. 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 temperatures. 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, Al2 O3 ) 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 intermetallic 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
28
CHAPTER 3. BRITTLE FRACTURE OF MATERIAL
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 − ν 2 ) 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 K 2 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=
v
"
u
u
s22 1/2
2 t s11 s22
K
I
2
s11
#
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. Furthermore, 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
σ
Kc = √ .
(3.13)
πa
In summary, the equations presented above provide links between crucial material 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
29
3.1. FUNDAMENTALS
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 characteristically brittle materials such as silicon [47] and β-SiC [48] where lattice
trapping raised the critical load over Griffith’s prediction. In contrast, simulation 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 sin
2πa
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 atomistic 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 resistance 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 brittle 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)
30
CHAPTER 3. BRITTLE FRACTURE OF MATERIAL
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 treatment 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 √1r , 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 and
rupture 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 discrete 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 satisfactory 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
32
CHAPTER 3. BRITTLE FRACTURE OF MATERIAL
release rate for brittle cleavage decohesion is important when brittle-ductile competition 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 separated blocks are maintained at their bulk equilibrium spacings, except between
the planes at which cleavage occurs. The rigid energy-separation curve provides information on important limit in the cleavage energetics. Furthermore,
the knowledge of the rigid cleavage energies is needed before the effects of relaxations 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 universal binding energy relation (UBER) [51] is used, which conveniently describes
non-linear effects due to changes in the electronic structure during cleavage decohesion. The UBER describes the energy-separation law by
Eb (x) = −Gc
1+
x
x
exp −
−1 .
l
l
(3.15)
The critical cleavage stress σc is given by the maximum of the stress dEb (x)/dx,
resulting in
Gc
.
(3.16)
σc =
el
3.2. DFT CALCULATIONS FOR BRITTLE FRACTURE
33
Figure 3.3: Sketch of the brittle cleavage model. Two adjacent planes are rigidly
separated by a distance x, the spacing of the remaining planes is kept fixed at its
equilibrium 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.
34
3.2.3
CHAPTER 3. BRITTLE FRACTURE OF MATERIAL
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 estimated, 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
r a
σtip = σ0 1 + 2
.
(3.17)
r
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) =
πσc2
π 2
σ r,
≈
1
2
0
2
4E 0 c
E ( √a + √r )
(3.18)
in which the relation E 0 = E/(1−ν 2 ) 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 defects or vacancies on the cleavage behaviour of a given interface might be studied,
simulating phenomena like environment-induced embrittlement. DFT calculations 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 Ni3 Al [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.
3.2. DFT CALCULATIONS FOR BRITTLE FRACTURE
35
Figure 3.4: A sketch of brittle and relaxed cleavage models: a solid (sketched as a
stacking of interacting layers with a layer distance a0 , panel a) undergoes brittle
cleavage (panel b): the crack of size x breaks the material into two rigid blocks
without relaxing the geometry of the layers in the blocks; by the ideal elastic
cleavage a process is defined, in which the material reacts perfectly elastic (panel
c) up to a critical crack above which it breaks abruptly into two blocks of relaxed
atomic 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 between 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
36
CHAPTER 3. BRITTLE FRACTURE OF MATERIAL
renormalization theory, these authors claimed to have achieved a universal formulation 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
C 2
x , 2γr =
E(δ) = min
2N
(
C 2
x
2N
2γr
x ≤ δc
x > δc
(3.19)
in which the parameters, e.g. the relaxed surface energy γr and the elastic modulus C, are material and direction dependent. The critical opening δc was expressed
as
s
γr N
δc = 2
(3.20)
C
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 macroscopic 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 2
x
lr2
(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
37
3.2. DFT CALCULATIONS FOR BRITTLE FRACTURE
σ (GPa)
2
E (J/m )
5
4
3
NiAl
2
1
0
brittle
relaxed
30
20
10
0
0 l1
2
3 4
x (Å)
5
0
1
2 lr 3 4
x (Å)
5
6
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)
, the critical stress is derived as the Hooke-like
dx
relation,
σr
Gr 1
=2
.
(3.22)
A
A lr
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.
38
CHAPTER 3. BRITTLE FRACTURE OF MATERIAL
Chapter 4
Cleavage and elasticity
4.1
Introduction
The ability to describe or just to estimate the critical properties of crack formation 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 constants, 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 assumptions 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 substantially [34]. The same result was obtained, comparing the OG model to the
critical cleavage stress calculated from the modern DFT approach [46]. Nevertheless, 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 distinguish 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
4.2
CHAPTER 4. CLEAVAGE AND ELASTICITY
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 function 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 macroscopic physical quantities, according to Hooke’s law the initial slope of the σ(x)
curve is related to the Young’s modulus E by
π
E = a 0 σc .
a
(4.2)
Inserting Young’s modulus a simple estimate for the critical cleavage stress is
obtained
Ea
.
(4.3)
σc =
πa0
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)
Z∞
σ(x)dx = Gc .
(4.5)
0
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
41
4.3. IDEAL BRITTLE CLEAVAGE
crystal. Nevertheless, the dependency on a can be eliminated and well known
OG estimate of the critical stress is obtained
σc =
s
EGc
.
2a0
(4.6)
In case of equivalent surfaces the relation Gc = 2γs is valid and Gc can be substituted 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 (surface) 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 properties: 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 example, 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 modifications 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 elastic 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 critical 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 equilibrium yields
Gc
Eb (x) = 2 x2 + . . .
(4.7)
l
42
CHAPTER 4. CLEAVAGE AND ELASTICITY
Table 4.1: Critical stress estimates: simple Orowan’s E/π, Orowan-Gilman estimate (equation 4.6) and the critical cleavage stress σc (all in GPa) calculated by
VASP for selected compounds and cleavage directions [hkl].
[hkl]
100
110
111
E/π
177
171
169
σcOG
119
87
148
σc
47
44
46
NiAl
100
110
111
211
65
90
104
90
58
47
89
69
26
22
26
24
VC
100
110
111
206
186
180
70
118
147
32
46
63
W
The elastic energy density required to open a crack of size x might be expressed
as
1
Eelast (x)/V = Cδ 2 ,
(4.8)
2
which comprises a dimensionless relative strain δ and appropriate elastic modulus 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)
x2
1
1 x2
Gc 2 = ALb C 2 .
2 l
2
Lb
(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.
43
4.3. IDEAL BRITTLE CLEAVAGE
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
l2
.
(4.10)
Gc
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
Lb = AC
s
1 Gc C
σc
=
.
(4.11)
A
e Lb
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 estimate 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 c011 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 c44
are involved and the direction dependent uniaxial modulus is given as
c011 [hkl] = c11 − 2(c11 − c12 − 2c44 )(h2 k 2 + h2 l2 + k 2 l2 ).
(4.12)
Analogous, for a tetragonal lattice the relation
c011 [hkl] = c11 (h4 + k 4 ) + c33 l4 + h2 k 2 (2c12 + c66 ) + l2 (1 − l2 )(2c13 + c44 ) (4.13)
is valid (for symmetry classes 4mm, 4̄2m, 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).
44
4.4
CHAPTER 4. CLEAVAGE AND ELASTICITY
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 assumption, 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 response 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 = Gc Lb /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
1 l
=
C,
A
e Lb
(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
4.5.1
Results for ideal brittle cleavage
Computational aspects
The exchange-correlation functional was described within the generalized gradient approximation (GGA) according to the parameterization of Perdew and
Wang [68]. Convergency of the total energies with respect to basis size and number of k points for the Brillouin zone integration was checked. Atomic forces
were relaxed within a conjugate gradient algorithm whenever structural relaxations 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.
46
CHAPTER 4. CLEAVAGE AND ELASTICITY
Table 4.2: The brittle cleavage properties of NiAl vs. slab thickness: cleavage
energy Gc /A (J/m2 ) and critical cleavage stress σc /A (GPa) with respect to the
number of layers in the slab. interfaces
no. of planes
2
4
8
12
Gc [100]
4.41
4.71
4.79
4.79
σc [100]
24.4
25.2
25.4
25.5
Gc [110]
3.23
3.24
3.24
3.24
σc [110]
22.6
21.8
21.8
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 arguments: 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-
47
4.5. RESULTS FOR IDEAL BRITTLE CLEAVAGE
Table 4.3: Calculated parameters for brittle cleavage of fcc Al and bcc W and
bcc Fe in direction [hkl]: uniaxial elastic modulus C (GPa), cleavage energy per
surface area Gc /A (J/m2 ), critical length l (Å ), maximum stress per area σc /A
(GPa), bulk interlayer distance a0 (Å ), and localisation length Lb (Å ). Results
for Fe derived from spin polarized calculations.
[hkl]
100
110
111
C
110
113
114
Gc /A
1.8
2.1
1.6
l
0.57
0.64
0.54
σc /A
12
12
11
a0
2.03
1.43
2.34
Lb
2.01
2.24
2.08
Fe (bcc)
100
110
111
302
338
350
5.3
5.0
5.8
0.58
0.54
0.61
34
35
35
1.41
1.99
0.82
1.93
1.97
2.25
W (bcc)
100
110
111
540
516
508
8.4
6.5
7.3
0.66
0.55
0.64
47
44
42
1.59
2.24
0.92
2.80
2.40
2.83
Al (fcc)
ing a maximum, minimum or a saddle point). Similarly, trigonal transformation
(in [111] direction) connects bcc-sc-fcc structures, however, in W the energy difference between bcc and sc structure was found much larger than bcc-fcc energy
difference [36]. No similar symmetry-dictated extrema exist for the [110] direction, 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.
48
CHAPTER 4. CLEAVAGE AND ELASTICITY
2
Eb (J/m )
6
4
2
Fe
∆µ (µB)
0
1
0
0
2
4
x (Å)
6
8
Figure 4.1: Brittle cleavage for Fe: decohesion energy (upper panel) and the
change of magnetic moment vs. cleavage size x: in [100] (full circles), [110] (right
triangles), 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 Šob 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. Šob 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 Šob 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
h111i slip systems has the ideal shear strength around 18 GPa [37] and, therefore,
4.5. RESULTS FOR IDEAL BRITTLE CLEAVAGE
49
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] direction 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 underestimated. For more information we refer to the papers of Herper et al. [75],
Clatterbuck et al. [76] and Friák 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 Å 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 agreement 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
50
CHAPTER 4. CLEAVAGE AND ELASTICITY
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 competition 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 technological 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 Å to 2.8 Å. In comparison, values for a0 range
from 0.83 Å for the (111) stacking of B2 FeAl to 2.37 Å for the stacking of L12
Al3 Sc 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, Ni3 Al 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 Al3 Sc 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
51
4.5. RESULTS FOR IDEAL BRITTLE CLEAVAGE
Table 4.4: Calculated parameters for brittle cleavage for some selected intermetallic compounds together with their crystal structures. Further details, see
table 4.3.
[hkl]
100
110
111
211
C
203
284
311
284
Gc /A
4.8
3.2
4.1
4.0
l
0.69
0.54
0.58
0.60
σc /A
26
22
26
24
a0
1.45
2.05
0.84
1.18
Lb
2.01
2.59
2.68
2.56
L12
100
111
225
331
4.3
3.7
0.66
0.52
24
26
1.78
2.06
2.28
2.42
FeAl
B2
100
110
111
278
354
380
4.8
4.3
5.1
0.71
0.50
0.61
25
32
31
1.43
2.03
0.83
2.92
2.06
2.77
Al3 Sc
L12
100
110
111
189
182
180
2.7
2.9
2.6
0.61
0.65
0.61
16
17
16
2.05
1.45
2.37
2.60
2.65
2.58
TiAl
L10
001
100
110
111
185
190
240
268
4.4
3.3
4.1
3.5
0.70
0.58
0.69
0.58
23
21
22
22
2.03
2.00
1.41
2.32
2.06
1.98
2.82
2.57
NiAl
B2
Ni3 Al
Table 4.5: Calculated parameters for brittle cleavage for FeAl: comparison of
spin polarized (mag) and non spin polarized calculations. Further details, see
table 4.3.
mag
mag
mag
[hkl]
100
100
110
110
111
111
Gc /A
4.8
5.7
4.3
4.7
5.1
6.1
l
0.71
0.64
0.50
0.52
0.61
0.62
σc /A
25
33
32
33
31
36
52
CHAPTER 4. CLEAVAGE AND ELASTICITY
[001]
[001]
[110]
[010]
Figure 4.2: The cleavage interfaces in FeAl. The (100) cleavage produces surface
layers of pure Fe and Al (left panel), whereas (110) planes contain both types of
atoms (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 moments 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
53
4.5. RESULTS FOR IDEAL BRITTLE CLEAVAGE
µ (µB)
2
Eb (J/m )
6
4
2
FeAl
(100) non pol.
(110) non pol.
(100) spin pol.
(110) spin pol.
0
2
1
0
0
1
2
3
4
x (Å)
5
6
7
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 favorable 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.
54
CHAPTER 4. CLEAVAGE AND ELASTICITY
4
2
Eb (J/m )
3
2
VC (001)
TiC (001)
1
0
0
1
2
3
x (Å)
4
5
6
7
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 (sometimes 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.
55
4.5. RESULTS FOR IDEAL BRITTLE CLEAVAGE
Table 4.6: Calculated cleavage properties for brittle cleavage for TiC and VC.
VC
B1
TiC
B1
[hkl]
100
110
111
C
647
585
564
Gc /A
3.2
7.0
9.9
l
0.37
0.55
0.58
σc /A
32
46
63
a0
2.08
1.47
1.20
Lb
2.77
2.53
2.06
100
110
111
515
489
481
3.5
7.7
11.6
0.42
0.56
0.70
31
51
61
2.17
1.53
1.25
2.57
1.97
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
Å and 0.42 Å 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 compounds 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 experimental 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) brittle 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.
56
CHAPTER 4. CLEAVAGE AND ELASTICITY
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 metallic) 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 structure (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 Å 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.
57
4.5. RESULTS FOR IDEAL BRITTLE CLEAVAGE
0.8
2
Eb (J/m )
0.6
0.4
0.2
NaCl
0
0
2
4
x (Å)
6
8
Figure 4.5: The brittle cleavage of NaCl. The full circles and diamonds are
cleavage energies vs. separation for [100] and [110] derived by VASP; the lines
are 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 fracture. For diamond, by transmission electron microscopy cracks have been observed to propagate without the emission of dislocations [91]. The experimentally
claimed preference for (111) cleavage was recently corroborated by ab initio tensile 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,
58
CHAPTER 4. CLEAVAGE AND ELASTICITY
Table 4.7: Calculated parameters for brittle cleavage for MgO and NaCl.
MgO
B1
NaCl
B1
[hkl]
100
110
C
299
345
Gc /A
1.8
4.4
l
0.37
0.54
σc /A
18
30
a0
2.11
1.53
Lb
2.27
2.29
100
110
52
45
0.3
0.7
0.49
0.66
2
4
2.83
2.00
4.16
2.84
Table 4.8: The calculated parameters for brittle cleavage for diamond and Si.
C
A4
Si
A4
[hkl]
100
111
C
1045
1210
Gc /A
18.3
11.5
l
0.35
0.45
σc /A
193
93
a0
0.89
1.55
Lb
0.75
2.08
100
111
154
189
4.3
3.1
0.53
0.54
30
21
1.37
2.37
1.01
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 interpretation 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 at G(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,
59
4.5. RESULTS FOR IDEAL BRITTLE CLEAVAGE
20
2
Eb (J/m )
30
10
diamond
0
0
1
2
3
4
5
x (Å)
Figure 4.6: Brittle cleavage of diamond in [111] direction: cleavage for long interlayer spacing (full circles), for short interlayer spacing (diamonds), LDA calculation 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 applications. 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 releasing 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 influences the cleavage energy, as illustrated for Si by an abundant number of studies
searching for the stable reconstructed surface (e.g. Ref. [94]).
60
CHAPTER 4. CLEAVAGE AND ELASTICITY
25
a)
2
Eb (J/m )
20
b)
15
crack
a)
vacuum
10
b)
5
vac.
0
0
1
2
x (Å)
3
4
5
Figure 4.7: The simulation of diamond (111) cleavage: the isolated carbon planes
with diamond (111)-like spacing are separated. The chosen supercell geometry
is sketched inside the figure. The vacuum region surrounds the isolated planes
from 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
4.5. RESULTS FOR IDEAL BRITTLE CLEAVAGE
61
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 Å 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 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 experimentally, it could be derived
from a single DFT calculation for each direction, which in many cases is not very
costly.)
62
4.6
4.6.1
CHAPTER 4. CLEAVAGE AND ELASTICITY
Relaxed cleavage
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 2
x ,
lr2
(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 conditions, and does not depend on the number of layers of a macroscopic material.
Calculating the first derivative σr (x) = dG(x)
, the critical stress may be evaluated
dx
σr
Gr 1
=2
.
A
A lr
(4.18)
Again, like for the brittle case the correlation between elastic and cleavage properties 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
63
4.6. RELAXED CLEAVAGE
new materials parameter and by that the key relation is derived,
Gr
x2
x2
1
AC
=
lr2
2
Lr
(4.19)
containing only intrinsic materials parameters. Again as for the brittle case, instead 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 analytic expressions to a proper set of DFT data. An obvious but elegant relation
can be gained for the critical stress stress
σr
lr
=
C.
A
Lr
(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 lr
the 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 reduced 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.
64
CHAPTER 4. CLEAVAGE AND ELASTICITY
Table 4.9: Calculated parameters for the relaxed cleavage: energy per surface
area Gr /A (J/m2 ), relaxation energy ∆Gr = Gc − Gr (%), critical length lr
(Å ), maximum stress σr /A (GPa), and localisation length Lr (Å ) for selected
compounds and cleavage directions [hkl].
[hkl]
100
110
111
C
110
113
114
Gr /A
1.8
1.9
1.6
∆G
1
8
1
lr
1.9
2.2
2.3
σr /A
19
17
14
Lr
11.0
14.4
18.8
Al
fcc
W
bcc
100
110
111
540
516
508
7.8
6.4
6.6
8
2
12
2.0
1.5
1.6
78
85
82
13.9
9.0
9.8
NiAl
B2
100
110
111
203
284
311
4.6
3.1
3.9
4
3
5
2.7
2.0
2.2
34
31
35
16.1
18.3
19.3
Ni3 A
L12
100
111
225
331
4.2
3.6
2
3
2.2
1.6
38
45
13.0
11.8
VC
B1
100
110
111
647
585
564
2.4
6.0
8.4
25
14
15
0.8
1.6
1.6
60
75
105
8.6
12.5
8.8
TiAl
L10
001
100
110
185
190
240
4.2
3.2
3.9
5
3
5
2.0
2.2
2.2
42
29
35
8.8
14.4
14.8
65
4.6. RELAXED CLEAVAGE
20
brittle
relaxed
L (Å)
15
10
5
0
1
2
l (Å)
3
4
Figure 4.8: Localisation lengths L vs. critical lengths l for ideal brittle and
relaxed cleavage for a variety of materials and directions. Values of L for the
same 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 electronic structure of Al. In Ref. [61] a layer dependent model
√ for relaxed cleavage
was applied, the critical stress scales according to σ r ∝ 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 σ r =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.
66
CHAPTER 4. CLEAVAGE AND ELASTICITY
5
2
G (J/m )
4
3
TiAl
2
(100) - short axis
(110)
(001) - long axis
1
0
0
1
2
3
4
5
6
7
x (Å)
Figure 4.9: Relaxed cleavage for TiAl. The gap symbols the formation of relaxed
crack 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 lr
follow 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 eventually 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
4.6. RELAXED CLEAVAGE
67
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 structural relaxations are in NiAl, Ni3 Al, 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. geometrical 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 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 yet.
68
4.7
4.7.1
CHAPTER 4. CLEAVAGE AND ELASTICITY
Semirelaxed cleavage
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 parameters 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 structure 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 calculations 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 crackis 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,
69
4.7. SEMIRELAXED CLEAVAGE
5
2
E (J/m )
4
3
NiAl
2
(100) unrelaxed
(100) semirelaxed
(110) unrelaxed
(110) semirelaxed
1
0
0
1
2
3
4
x (Å)
5
6
7
8
Figure 4.10: The cleavage of NiAl. The cleavage energy as a function of separation
along [100] ( circles), [110] (diamonds) direction. The red lines and symbols
correspond to semirelaxed cleavage.
Table 4.10: The cleavage parameters obtained from UBER: ideal brittle cleavage
energy Gc (J/m2 ), brittle critical length lc Å, critical stress σc and their semirelaxed counterparts denoted by index s. The values in brackets were obtained
allowing lateral dimension of unit cell to relax, see text.
[hkl]
100
110
Gc
4.88
3.24
lc
0.68
0.49
σc
26.4
24.3
Gs
4.72
3.18
ls
0.69
0.49
σs
25.2
23.8
W
100
110
8.53
6.49
0.66
0.54
47.4
44.2
7.99
6.35
0.66
0.53
44.3
44.1
VC
100
3.15
0.39
29.7
2.97
0.39
27.9
NiAl
70
CHAPTER 4. CLEAVAGE AND ELASTICITY
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 section. 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
71
4.7. SEMIRELAXED CLEAVAGE
8
2
E (J/m )
6
W
4
(100) unrelaxed
(100) only atoms relaxed
(100) semirelaxed
(110) unrelaxed
(110) semirelaxed
2
0
0
1
2
3
x (Å)
4
5
6
7
Figure 4.11: The semirelaxed cleavage of W. The cleavage energy as a function
of separation along [100] ( circles), [110] (diamonds) direction. The red lines and
symbols correspond to constant volume relaxation of cleavage, the blue symbols
to 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
72
CHAPTER 4. CLEAVAGE AND ELASTICITY
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
4.8
73
Summary
The correlation between elastic and cleavage properties was established by introducing the concept of the localisation of the elastic energy and relating localized 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 intermetallic compounds- an average value of Lb ≈ 2.4 Å 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 material -at least for simpler types of crack formationpurely 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.) It is proposes, that the introduced concept might even hold for the cleavage of more complex solids than single crystals.
Summarizing the results for various materials, interesting interplay of magnetism 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 during 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 contrast 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 incorporate 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.
74
CHAPTER 4. CLEAVAGE AND ELASTICITY
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 technological 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 propagation. 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 undergoes 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 temperatures occurs also in modern perspective intermetallic alloys. This kind of behavior 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 dislocation
emission- at the crack tip. The outcoming dislocation in (2) has Burgers vector perpendicular to crack plane and, thus, the crack is blunted by one atomic
distance.
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 ductile
µb
> 7.5 − 10,
(5.1)
γs
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 equation 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, quantitative predictions were still strongly limited.
5.2
The concept of unstable stacking fault energy
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 proposed 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 competition 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 parameters 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 displacement 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
1 + (1 − ν) tan2 φ
Gd = 8γus
,
(5.2)
(1 + cos θ) sin2 θ
78
CHAPTER 5. DUCTILE FRACTURE
Figure 5.2: The geometry of dislocation emission
where θ is an angle between crack plane and slip plane and φ is an angle between 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 calculated 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 calculations. 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.
5.3
MODIFICATIONS OF RICE’S APPROACH
79
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 explained: 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 dislocation. 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] introduced 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
< 0.014.
(5.4)
µb
There, µ denotes the isotropic shear modulus and b the Burger’s vector of the
emitted dislocation.
Furthermore, Schöck [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 minimizing 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 demonstrated 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.
Schöck’s findings are in qualitative agreement with results of atomistic simulation by ZTC. In summary, though the brittle-ductile criterion in mode I configuration 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 elastic 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
80
CHAPTER 5. DUCTILE FRACTURE
Beltz [54, 56]. They modelled elliptical cut-out in an infinite medium under
plane strain conditions. They constrained themselves to the cases of crack advancing 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 nucleate 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 certain 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 dislocations 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 compounds. 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 accounting for the lattice periodicity. It combines the dislocation stress field as
determined by the continuum theory with an atomic description of the dislocation 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 theoretical 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 dislocations 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 geometrical considerations [114], appropriate stress field can be calculated in an
analytical form within linear elasticity theory (see section 3.1.1). The screw dislocation 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
82
CHAPTER 5. DUCTILE FRACTURE
the screw dislocation may be then represented by form
fz =
b
x
tan−1 ,
2π
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 =
y
µb
2
2π y + x2
σyz =
µb x
.
2π y 2 + 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
ln(x2 + y 2 )
4π(1 − ν)
(5.7)
and associated stress field can be calculated from the equation 3.3
y(3x2 + y 2 )
µb
2π(1 − ν) (y 2 + x2 )2
µb
y(x2 − y 2 )
=
2π(1 − ν) (y 2 + x2 )2
x(x2 − y 2 )
µb
=
2π(1 − ν) (y 2 + x2 )2
= ν(σxx + σyy ).
σxx = −
(5.8)
σyy
(5.9)
σxy
σzz
(5.10)
(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.
83
5.4. DISLOCATIONS PROPERTIES
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 reliable 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 atomistic 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 arrangement 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 disregistry) 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,
µb
µb
for the edge dislocation and K = 2π
for the screw dislocation. The
K = 2π(1−ν)
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(x0 ) =
df (x0 )
,
dx0
(5.13)
where x0 is a distance from the dislocation line. The force at point (x,0) produced
84
CHAPTER 5. DUCTILE FRACTURE
by the distribution of dislocations is
Fdisl = K
Z∞
−∞
1
ρ(x0 )dx0 .
0
x−x
(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 integrodifferential equation
K
Z∞
−∞
1
ρ(x0 )dx0 = F [f (x)].
0
x−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
!
bµ
2πf (x)
F [f (x)] = − sin
(5.16)
2π
b
In such a case, a simple analytic solution satisfying boundary conditions can by
found
x
b
f (x) = − tan−1 ,
(5.17)
2π
ξ
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)
85
5.4. DISLOCATIONS PROPERTIES
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 calculations and mesoscopic scale object such a dislocation is. Using the approximation
of equation 5.18 the Peierls-Nabarro equation becomes
K
Z∞
−∞
1
∂γGSF [f (x)]
ρ(x0 )dx0 =
.
0
x−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
Lejček’s method
Useful method for the solution of the Peierls-Nabarro equation with general
restoring force law (equation 5.19 was proposed by Lejček [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) =
pk
N X
X
ρnk (x)
(5.20)
k=1 n=1
pk
N X
X
dγ
K
=−
gnk (x),
dx
k=1 n=1
and
ρnk
Ank
A?
1
+
=
2 (x − zk )n (x − zk? )n
"
(5.21)
#
(5.22)
−i
Ank
A?
gnk =
,
(5.23)
−
2 (x − zk )n (x − zk? )n
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
Rx
function of x and not of the disregistry f , one has to integrate f (x) = −∞
ρ(t)dt
dγ(f (x))
and eliminate variable x from 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.
"
#
86
CHAPTER 5. DUCTILE FRACTURE
For the case n = 1
a1
x
f1 (x) =
ln(x2 + ξ 2 ) − b1 arctan
2
ξ
a1 x − b 1 ξ
ρ1 (x) =
x2 + ξ 2
b1 x + a 1 ξ
g1 (x) = − 2
x + ξ2
!
(5.24)
(5.25)
(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 higherorder 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
a2 x − b 2 ξ
x2 + ξ 2
a2 (x2 − ξ 2 ) − 2b2 ξx
ρ2 (x) =
(x2 + ξ 2 )2
b2 (x2 − ξ 2 ) + 2a2 ξx
g2 (x) = −
(x2 + ξ 2 )2
f2 (x) = −
(5.27)
(5.28)
(5.29)
The case n = 3
a3 (ξ 2 − x2 ) + 2b3 xξ
2(x2 + ξ 2 )2
a3 x(x2 − 3ξ 2 ) − b3 ξ(3x2 − xi2 )
ρ3 (x) =
(x2 + ξ 2 )3
1 2a3 ξx + b3 (x2 − ξ 2 )
g3 (x) = −
2
(x2 + ξ 2 )3
f3 (x) =
(5.30)
(5.31)
(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 x12
and x13 , 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.
5.4. DISLOCATIONS PROPERTIES
87
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
P
disregistry (f (−∞) = 0 and f (∞) = b) one derived k b1k = − πb . It is useful to
P
define b1k = − πb αk where N
k=1 αk = 1, because the parameters bαk can then be
interpreted as Burgers vectors of partial dislocations. Remaining parameters have
∂γ
to be estimated to fit given ∂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, Lejček’s method provides unified and physically transparent scheme
for solution of the Peierls-Nabarro equation. It can be extended into the generalised case of the two component displacement field (two-dimensional PeierlsNabarro 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) =
∞
X
m=−∞
γGSF (f (ma − u))a.
(5.33)
88
CHAPTER 5. DUCTILE FRACTURE
This equation has correct period in a and correct limit for very narrow dislocations [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 energy 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)
(
)
1 dW
σp = max σ = max
.
(5.34)
b du
An analytic solution for σ(u) was given by Joós 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) = −
Kb sinh 2πΓ sin 2πy
,
2a (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
∞
2πn
2πinu
a X
Fγ
exp −
,
W (u) =
|a| n=−∞
a
a
(5.36)
where Fγ is the Fourier transform of γ[f (x)]
Fγ
2πn
=
a
Z∞
γ[f (x)] exp
−∞
2πinx
.
a
(5.37)
The formula 5.37 can be simplified via integration by parts which results in the
relation
Z∞
a
2πn
∂γ ∂f
−2πinx
Fγ
=
dx.
(5.38)
exp
a
2πin
∂f ∂x
a
−∞
5.5. CALCULATION OF STACKING FAULT ENERGETICS
89
Note that ∂γ
is the restoring force obtained from the DFT calculation and ρ(x) =
∂f
∂f
is the dislocation density known from the solution of the Peierls-Nabarro
∂x
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) =
∞
X
Fγ
n=0
2πn
2πnu
2 cos
.
a
a
(5.39)
Derivative of the equation 5.39 yields the periodic stress which the dislocation
experiences when it glides
∞
4π X
2πn
2πnu
σ=
nFγ
cos
ab n=1
a
a
(5.40)
and the stress maximum is the Peierls stress
∞
2πn
4π X
σp =
nFγ
.
ab n=1
a
(5.41)
In applications of the equation 5.41 the first two terms in n are usually sufficient. Higher-order terms have values at least an order of magnitude lower and,
therefore, might be neglected. Thus, the formula may by considered an accurate 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
5.5.1
Calculation of stacking fault energetics
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
90
CHAPTER 5. DUCTILE FRACTURE
2
γ GSF (J/m )
1.5
1
atoms relaxed
rigid shift
0.5
0
0
0.1
0.2
0.3
0.4
0.5
f/b
Figure 5.3: The effect of atomic relaxation on γGSF energetics of h111i(110) slip
system 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 h111i(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
91
5.5. CALCULATION OF STACKING FAULT ENERGETICS
2
γ GSF (J/m )
1.5
1
<111>(110)
<001>(100)
<001>(110)
<111>(211)
0.5
0
0.1
0.2
0.3
0.4
0.5
f/b
Figure 5.4: Generalised stacking fault energy profile γGSF of the most important
slip 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 h111i(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 discussion 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 h111i(110) slip which is typical in
92
CHAPTER 5. DUCTILE FRACTURE
bcc materials because it provides the shortest possible Burgers vector, is unlikely
in NiAl. The reason is simple: by the 21 h111i slip in the crystal with B2 structure 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 h111i direction two possible dislocation configurations exist: a
pair of 12 h111i Shockley partial dislocations separated by the anti-phase boundary,
or a h111i 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 EAP B 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]
b2 Ksplit
d=
,
(5.42)
2πEAP B
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 h001i(100) and h001i(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 h111i and h001i slip systems. This is in agreement with the experimental
observations, which report the activity of the h001i and sometimes the h111i
dislocations [124, 125, 126, 127]. The h110i dislocations activity seems improbable
due to the high stacking fault energy barriers at all planes considered. The h110i
slip seems more likely to be formed by the dissociation: h110i → h111i + h001̄i.
Such a dissociation seems energetically more favorable.
5.5. CALCULATION OF STACKING FAULT ENERGETICS
93
Table 5.1: Calculated unstable stacking fault energies γus (J/m2 ) and the ratio
Gc /Gd , which is evaluated assuming that the crack lies at the (110) cleavage
plane and calculating appropriate θ (see section 5.2). The values in brackets are
other theoretical results, namely a Ref. [123] Embedded Atom calculation, and b
Ref. [118] FLMTO calculation.
Slip system
h001i(100)
h011i(100)
γus
1.52 (1.21)a [3.13]b
2.9 (2.00)a
Gc /Gd
0.53
0.28
h001i(110)
h110i(110)
h111i(110)
1.28 [2.28]b
2.09
0.83 [0.97]b
0.63
0.38
0.96
h110i(111)
1.61
0.5
h110i(211)
h111i(211)
2.84
0.96
0.35
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 hh0 k 0 l0 i(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 = cos α cos βσ11 ,
(5.43)
where α is the angle between the tensile axis x1 and the glide direction x01 , 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 h001i slips
dominate [126]. The hard orientations are those close to the [001] tensile loading
direction, where h001i slips experience low resolved shear stress. The deformation
of single crystals with hard orientation of the tensile axis requires considerably
94
CHAPTER 5. DUCTILE FRACTURE
higher stress. In the hard orientation, h111i 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 h111i(110) and h111i(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 h001i slips involve single
dislocations and, therefore, their core structure should be relatively easy to describe. The h111i(110) slip system features two possible configurations, namely
two Shockley partial dislocations separated by the anti-phase boundary, or full
h111i dislocations.
In the first step, one has to evaluate anisotropic values of the elastic constant 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 h111i dislocations lie in a direction
other than the cube axes, to which anisotropic constants of NiAl listed in section 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)
1
i0 = √ (i − j + k)
3
1
j0 = √ (j + k)
2
1
k0 = √ (i + j).
2
(5.44)
The change to the new coordinates can be expressed in terms of the transformation matrix Tij
√ √ 
 √
2 −√ 2 √2
1

(5.45)
Tij = √ 
3 .
 0
√3
6 √
3
3
0
Now the tensor transformation rules [22] have to applied because elastic constants
are essentially fourth-rank tensors. In general, the transformation has the form
c0ijkl = Q̂ijgh cghmn Qmnkl ,
(5.46)
where the 9x9 transformation matrix Qmnkl is obtained as Qmnkl = Tkm Tln . Performing the matrix multiplication within the program package Maple one obtains
5.5. CALCULATION OF STACKING FAULT ENERGETICS
95
Table 5.2: The elastic constant K for dislocations in NiAl. Isotropic value Kiso is
given by µ/(1 − ν) for an edge dislocation and by µ for a screw dislocation. The
shear modulus µ and the Poisson ration ν are evaluated from the Reuss average
over the elastic constants of NiAl listed in table 2.2, anisotropic values Ke and
Ks are calculated out of the elastic constants via the procedure outlined in the
text.
Kiso
112.3
81.5
Ke
Ks
[001]
85
65
[111]
96
75
the transformed constants, and then the relations of the anisotropic theory of dislocations may be applied. The anisotropic elastic theory of straight dislocations
was developed by Eshelby [128] and Stroh [129], and the framework and its applications 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 polynomials one can decompose the problem into a screw and an edge part involving
second order and fourth order polynomials, respectively. For pure edge dislocation the coefficient Ke is then given in terms of transformed elastic constants
by [114]
Ke =
where c̄011 =
q
(c̄011
+
c012 )
"
c066 (c̄011 − c012 )
+
(c̄011 + c012 + 2c066 )c022
#1/2
,
(5.47)
c011 c022 . For pure screw dislocation
Ks =
q
c044 c055 − c02
45 .
(5.48)
For the h001i 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 Lejček’s ansatz as discussed in section 5.4.3. The γGSF profiles were calculated within constrained path approximation -the slip energy is calculated only along given direction, whereas the
96
CHAPTER 5. DUCTILE FRACTURE
minimum energy needed to generate given slip displacement might follow somewhat 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 magnitude larger computational costs because full γ-surface has to be calculated.
Nevertheless, the deformation of NiAl is carried mainly by pure edge dislocations [130, 131], so the description of dislocations within one-dimensional PeierlsNabarro model is reasonable.
Two partial edge dislocations with second-order terms in n describing core
structure (equation 5.27) have to be used for h111i(110) system, whereas for
h001i 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 solution is rather tedious, in particular of the equation corresponding to h111i(110)
slip system where γGSF profile features the anti-phase boundary separating the
Shockley partial dislocations. For instance, one has not obtained a stable solution using usual Newton’s iterative algorithm. Convergent and stable solutions
were not achieved even by improving Newton’s method with the line search algorithm 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 equation 5.41 were performed utilizing mathematical program package Maple. Calculated parameters are displayed in table 5.3. Exploring the results, one realizes
that full h111i dislocations should glide more easily compared to h001i ones.
Therefore, the mobility of dislocations in not a limiting factor for the activity
of h111i 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 h111i dislocation is as long as 5.01 Å in NiAl.
The formation of partials is energetically prohibitive because of the energy of the
1
h111i anti-phase boundary energy, as discussed in the previous section. Explor2
ing the table 5.3 one realizes that the splitting between the 21 h111i partials is
5.5. CALCULATION OF STACKING FAULT ENERGETICS
97
Table 5.3: Dislocation core parameters and Peierls stress in NiAl. The dislocation
core width ξ (Å), the separation of partials d (Å) (the partials are at positions x−d
and x + d), the Peierls stress σJ (µ) calculated from Joós formula (equation 5.35)
and the Peierls stress σp (µ) calculated from exact formula in equation 5.41. See
text for more details.
Slip system
h001i(100)
h001i(110)
h111i(110)
ξ
1.4
1.6
3.2
d
0
0
7.1
σJ
0.034
0.024
-
σp
0.036
0.024
0.002
14 Å. This value is in reasonable agreement with experimental TEM observation
which reported that partials are about 10 Å apart. Higher value of the theoretical 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 Joós formula (equation 5.35) and the accurate formula expressed by equation 5.41, one finds good
agreement for both of h001i 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 h111i 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 formulation) have been recently proposed [116, 133, 134]. The inherent limitations of
the Peierls-Nabarro model are summarized and discussed in the reference [9].
98
CHAPTER 5. DUCTILE FRACTURE
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 contribution 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 shearonly 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.
99
TENSION-SHEAR COUPLING
a0
b
x
f
Figure 5.5: Block-like slip displacement f and opening separation x of two parts
of 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 h111i and h001i 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.
100
5.6.3
CHAPTER 5. DUCTILE FRACTURE
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 combined 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 + τ df.
(5.49)
In the absence of the tensile stress component, the pure shear stress may be
approximated with the Frenkel sinusoidal formula
πγus
2πf
τ (f ) =
sin
b
b
!
(5.50)
and vice-versa, the pure tensile stress of rigid opening may be derived from UBER
(equation 3.15) as
x
Gc
σ(x) = 2 exp −
.
(5.51)
l
l
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
2πf
τ (x, f ) = A(x) sin
b
!
(5.52)
x
x
σ(x, f ) = B(f ) − C(f ) exp −
.
(5.53)
l
l
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
∂σ
=
∂x
∂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 displacement x0 corresponding to zero tensile stress at the unstable stacking fault (shear
5.6.
101
TENSION-SHEAR COUPLING
displacement f = 12 b). The analytic form of functions A,B,C may be found in
reference [106]. The resulting potential Ψ(x, f ) was derived as
"
!(
!
)
#
x
x
Ψ(x, f ) = 2γs
exp −
,
l
l
(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
πxf a exp(−x/l)
Ψa = 2γs
.
(5.56)
bl
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
x
x
πf
1− 1+
exp −
+ sin2
l
l
b
q−p
q+
1−p
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,
102
CHAPTER 5. DUCTILE FRACTURE
1.5
2
E (J/m )
1
x = 0.0
x = 0.2
x = 0.4
0.5
0
0.1
0.2
0.3
f/b
0.4
0.5
Figure 5.6: The effect of relaxation; the stacking fault energy for [111](211) slip
system calculated for rigid tensile and shear displacements (broken line) and with
additional relaxation of individual atomic planes in direction perpendicular to the
slip 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 h111i(211) slip system- the relaxation has substantial influence 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 calculation identifies the stacking fault energy maximum γus at the position of the
1/2h111i(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 h001i slips,
nevertheless the quantitative changes of γus are substantial and cannot be neglected. Therefore, the relaxation of atomic planes was performed in all following
calculations.
The slip systems considered in h001i 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
5.6.
103
TENSION-SHEAR COUPLING
4
2
E (J/m )
3
x = 1.0
x = 0.6
x = 0.4
x = 0.2
x = 0.0
2
1
0
0.1
0.2
0.3
f/b
0.4
0.5
Figure 5.7: Tensile-shear coupling for h001i(100) slip system; the energy E as a
function 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 Å 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 pairpotential or the embedded atom based methods.
The slips in h111i direction have more complex energy profile. The profile of
104
CHAPTER 5. DUCTILE FRACTURE
3
x = 1.0
x = 0.6
x = 0.4
x = 0.2
x = 0.0
2.5
2
E (J/m )
2
1.5
1
0.5
0
0.1
0.2
0.3
0.4
0.5
f/b
Figure 5.8: Tensile-shear coupling for h001i(110) slip system
2
2
E (J/m )
1.5
1
x = 1.0
x = 0.6
x = 0.4
x = 0.2
x=0
0.5
0
0.1
0.2
0.3
0.4
0.5
f/b
Figure 5.9: Tensile-shear coupling for h111i(110) slip system
5.6.
105
TENSION-SHEAR COUPLING
2
2
E (J/m )
1.5
1
d = 1.0
d = 0.6
d = 0.4
d = 0.2
d = 0.0
0.5
0
0.1
0.2
0.3
0.4
0.5
f/b
Figure 5.10: Tensile-shear coupling for h111i(211) slip system
0.5
x0 (Å)
0.4
<111>(110)
<001>(110)
<001>(100)
<111>(211)
0.3
0.2
0.1
0
0.5
2
1
1.5
γGSF (J/m )
Figure 5.11: The zero-stress separation parameter x0 of UBER as a function of
stacking fault energy γ.
106
CHAPTER 5. DUCTILE FRACTURE
4
2
E (J/m )
3
f/b = 0.0
f/b = 0.1
f/b = 0.21
f/b = 0.28
f/b = 0.42
f/b = 0.5
2
1
0
0
1
2
3
4
5
x (Å)
Figure 5.12: The (110) cleavage of NiAl in the presence of the h001i stacking
fault; the cleavage energy E as a function of opening displacement x with the
shear 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 h001i slips. Recalling the brittle cleavage properties of NiAl Gc = 4.8 J/m2 and l = 0.69 Å for the (100) planes, Gc = 3.2 J/m2 , l = 0.54 Å
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
5.6.
TENSION-SHEAR COUPLING
107
maximum). This fact might explain the difference in the tension softening of the
(100) and the (110) planes.
Finally, in the case of the h111i(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, therefore, 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 displacement 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 separation 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.
108
5.7
CHAPTER 5. DUCTILE FRACTURE
Summary
In summary, it was found h001i and h111i as the preferred slip directions in
NiAl, in good agreement with fracture experiments. Though calculated values
of the γus are lower in the h111i direction, the h001i is dominating slip, because
h111i slips are somewhat hampered by the relatively high anti-phase boundary
energy. The anti-phase boundary prevent formation of the 21 h111i dislocations
which occur in metals with bcc structure. Thus, full h111i dislocations form only
when the resolved shear stress for the preferred h001i 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 21 h111i 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 unstable 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 between 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 materials 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
110
CHAPTER 5. DUCTILE FRACTURE
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 application. 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 temperature 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 calculated 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 calculations 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 calculations 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
6.2
CHAPTER 6.
MICROALLOYING OF NIAL
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 resistance, are of interest for applications in aerospace industry. However, poor ductility at low temperatures and brittle grain-boundary fracture limit its technological 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 treatment, constitutional defects and surface conditions [144].
The slip properties of pure NiAl were in detail discussed in the previous chapter. 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 h001i slips dominate [126]. The hard orientations are those close to
[001] tensile directions, where h001i slips experience low resolved shear stress.
At liquid nitrogen temperatures (77 K) the preferred slip direction in the hard
orientation of specimens is h111i at the (110), (211) and (123) planes [127].
In summary, a variety of experimental results indicate that the major deformation mode of NiAl with its cubic B2 structure is the h001i slip [145, 124, 126].
However, h001i 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 required for a polycrystal to undergo plastic deformation [146]. When the polycrystal 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 h111i slip -prevalent in other intermetallic compounds
with a B2 structure like CuZn and AgMn- fulfills the von Mises criterion. Therefore, an improvement of the intergranular ductility of polycrystalline NiAl should
proceed via an activation of systems related to the h111i slip.
Miracle et al. reported that alloying NiAl with Cr enhances the nucleation and
motion of h111i dislocations at low temperatures, while at higher temperatures
the h001i slip is suggested to be prevalent [124]. In contrast, the experimental
results of Darolia et al. indicate that h111i dislocations are absent in stoichiometric 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 experiments 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 microalloyed 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 interatomic 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 repeated slab construction with three-dimensional translational symmetry. Convergency 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-
114
CHAPTER 6.
MICROALLOYING OF NIAL
[001]
[001]
[010]
[110]
Figure 6.1: Interfaces for an AB compound of B2 structure.√(100)√interface (left
panel): 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 %; second
interface plane for B atoms is similar but with white circles. (110) interface (right
panel): plane for a (1x1) and (1x2) geometry corresponding to coverage of X by
50 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, however, 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.
115
BRITTLE CLEAVAGE
Table 6.1: For NiAl, UBER parameters as derived from fitting to ab initio calculations for cleavage planes of orientation (hkl): cleavage energy per area G c /A
in J/m2 , cleavage energy Gc in eV, critical length l in Å , critical stress σc /A
in GPa. N1 and N2 denote the numbers of broken nearest and second nearest
neighbor bonds.
(hkl)
(100)
(110)
(111)
Gc /A
4.79
3.24
4.12
Gc
2.50
2.40
3.73
l
0.69
0.54
0.58
σc /A
25.5
22.2
26.1
N1
4
4
4
N2
1
2
3
X replaces 100% of the atoms in one of the planes. In the (110) planes, however, 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−x Alx 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 XN i , 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
116
CHAPTER 6.
MICROALLOYING OF NIAL
Table 6.2: For NiAl-X, calculated properties of (110) brittle cleavage. Results of
UBER fit to ab initio data: cleavage energy Gc /A in J/m2 , maximum cleavage
stress σc /A in GPa, its relative changes ∆σc /a with respect to pure NiAl, and
the length parameter l in Å .
Xsite
CrAl
CrN i
Gc /A
3.88
3.74
σc /A
26.6
27.4
∆σc /A
4.3
5.1
l
0.54
0.50
MoAl
MoN i
3.47
3.53
24.3
26.4
2.0
4.1
0.53
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 parameters Gc 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 accordance 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.
117
BRITTLE CLEAVAGE
4
2
G/A (J/m )
3
2
Cr
Al
Al
Ti
NiAl
1
Al
Ga
0
0
1
2
3
x (Å)
4
5
6
Figure 6.2: For NiAl and NiAl-X, calculated cleavage energy release rate G(x)/A
for (110) cleavage versus cleavage size x for substitutions X=Cr,Ti,Ga at Al
sites. 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=CrN i ,MoN i ,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.
118
CHAPTER 6.
MICROALLOYING OF NIAL
Table 6.3: (110) cleavage of NiAl-X for the (1x1) and (1x2) geometries corresponding 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 respect
to pure NiAl (110).
cover.
50%
25%
CrAl
0.64
0.33
CrN i
0.50
0.21
MoAl
0.23
0.14
MoN i
0.29
0.03
TiAl
0.11
0.15
GaAl
-0.52
-0.25
Table 6.4: (100) cleavage of NiAl-X for three different geometries corresponding
to 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 pure
NiAl (100).
cover.
100%
50%
25%
CrAl
0.49
-0.01
0.01
CrN i
0.13
0.26
0.09
MoAl
-0.52
-0.39
0.07
MoN i
-1.45
-0.18
-0.11
TiAl
-0.63
-0.39
-0.10
GaAl
-1.06
-0.54
-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.
119
SLIPS AND DUCTILITY
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 to
pure NiAl.
(hkl)
(100)
(110)
(211)
CrAl
0.01
0.33
0.40
CrN i
0.09
0.21
0.19
MoAl
0.07
0.14
0.29
MoN i
-0.11
0.03
0.04
TiAl
-0.10
0.15
0.22
GaAl
-0.29
-0.25
-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 = γus Y (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]
120
CHAPTER 6.
0.6
2
γGSF (J/m )
0.8
MICROALLOYING OF NIAL
NiAl
Al
Ti
Al
Ga
Al
Mo
Al
Cr
0.4
0.2
0
<111>(110)
0.1
0.2
0.3
f/b
0.4
0.5
Figure 6.3: For NiAl and NiAl-X, calculated generalised stacking fault energies
γGSF for a h11̄1i(110) slip with X=Ti,Cr,Mo,Ga on Al sublattice sites. f /b: slip
relative 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
< 0.014.
(6.2)
µb
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 h001i
slip ductile behaviour is expected for γus < 0.33 J/m2 . In case of a h11̄1i 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
6.5.
121
SLIPS AND DUCTILITY
1
2
γGSF (J/m )
1.5
NiAl
Al
Ga
Al
Cr
Al
Ti
Al
Mo
<001>(110)
0.5
0
0.1
0.2
0.3
0.4
0.5
f/b
Figure 6.4: For NiAl and NiAl-X, calculated generalised stacking fault energies
γGSF for the h001i(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 quantity. 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
h111i(110) and h001i(110) slips
The (110) cleavage habit plane is preferred slip plane as well. By slip in the h001i
direction single dislocations are formed, whereas the h111i direction features pair
of Shockley partial dislocations separated by the anti-phase boundary formed by
1
h111i shift displacement.
2
Observing the γGSF profile of the h111i slip in figure 6.3, the local minimum at
the displacement f /b = 0.5 corresponds to the geometry of the anti-phase boundary. 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 EAP B = 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 EAP B due to atomic relaxations [118] is consistent with
our value of EAP B = 0.76 J/m2 for a fully relaxed calculation (see table 6.6).
122
CHAPTER 6.
MICROALLOYING OF NIAL
Table 6.6: For NiAl-X, calculated unstable stacking fault energies γus in J/m2
for h001i and h11̄1i slips on the (110) plane, and the energy EAP B of the 12 h11̄1i
anti-phase boundary.
NiAl
γus h001i
1.28
γus h11̄1i
0.83
EAP B
0.76
CrAl
CrN i
0.88
1.40
0.47
0.79
0.07
0.48
MoAl
MoN i
0.22
1.04
0.55
0.70
0.06
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 EAP B < 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 significantly different as shown in table 6.6. The strong decrease of EAP B 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 EAP B due to
alloying might lead to an increased width of splitting between 1/2h111i Shockley
partial dislocations, because due to elasticity theory the equilibrium separation
of partials is inverse proportional to EAP B [114]. Depending on the strength of
their coupling, two partials may move independently or will be coupled and, consequently, their mobility will be substantially influenced. However, the splitting
is also determined by the shape of the γGSF surface. Thus a more elaborate treatment within the dislocation model of Peierls and Nabarro [110, 111] is needed to
elucidate the splitting mechanism. The activation of the h111i Shockley partial
dislocation is considered to be crucial for improving the intergranular brittleness
of NiAl.
Stacking fault energy profiles for the h001i slip with X on Al sites are shown
in figure 6.4. In comparison to NiAl, for X=CrAl the energy γus for the h111i slip
is reduced by 40%, but to a lesser amount for the h001i slip. Therefore, the nucleation of h111i 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-
6.5.
123
SLIPS AND DUCTILITY
Table 6.7: Unstable stacking fault energy γus in J/m2 for the h001i [110] slip
for NiAl-X, X=(Cr,Mo,Ti,Ga) substitutions. Results for two concentrations of
defects. Further details, see text. For NiAl, γus = 1.28 J/m2 .
conc.
50%
25%
CrAl
0.88
1.12
CrN i
1.40
1.30
MoAl
0.22
0.80
MoN i
1.04
1.07
TiAl
0.60
1.05
GaAl
1.04
1.20
tile behaviour may be expected. These findings agree with the experimentally
observed activity of h111i dislocations in NiAl-CrAl at low temperatures [124].
There exists, however, contradiction between experimental findings, because in
Ref. [147] no activity of h111i 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=CrN i than for X=CrAl ,
as displayed in figure 6.5. The ’successful’ (in terms of the observed activity of
h111i 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 CrN i is relatively large,
the effective reduction of γus due to alloying might be rather moderate and presumably insufficient to open the h111i slip system.
The energy profile of γGSF (f ) for X=MoAl for the h111i slip is similar to
X=CrAl , but for the h001i slip the energy γGSF for X=MoAl is strongly reduced
compared to Cr (see table 6.6). For the h001i 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 h001i slip
system.
Nevertheless, as the observed enhancement of ductility for NiAl-Mo [125] is
probably carried by the activity of the h001i(110) dislocations, the intergranular
ductility of NiAl-Mo polycrystals seems not to be improved. As discussed in
section 6.2, the h001i slip generates only three independent slip systems and,
thus, does not fulfill von Mises criterion for ductility of a polycrystal.
124
CHAPTER 6.
MICROALLOYING OF NIAL
0.6
2
γGSF (J/m )
0.8
0.4
<111>(110)
NiAl
Ni
Cr
Ni
Mo
0.2
0
0.1
0.2
0.3
0.4
0.5
f/b
Figure 6.5: Generalised stacking fault energy γGSF along the h11̄1i direction on
the (110) plane for NiAl-X with X=(Cr,Mo) on the Ni sublattice sites. Letter b:
respective Burger’s vector.
0.6
2
γGSF (J/m )
0.8
NiAl
Al
Cr 25%
Al
Mo 25%
Al
Mo 50%
Al
Cr 50%
0.4
0.2
0
<111>(110)
0.1
0.2
0.3
0.4
0.5
f/b
Figure 6.6: Generalised Stacking Fault energy γGSF along h11̄1i direction on
the (110) plane for NiAl-X with X=(Cr,Mo) on the Al sublattice for two defect
concentrations. Further details, see text.
6.5.
125
SLIPS AND DUCTILITY
2
2
γGSF (J/m )
1.5
1
NiAl
Al
Cr
Al
Mo
Al
Ti
0.5
0
0.1
0.2
0.3
0.4
0.5
f/b
Figure 6.7: Generalised Stacking Fault energy γGSF along h001i direction on the
(100) plane for NiAl-X with X=(Cr,Mo,Ti) on the Al sublattice. The Ga atoms
did 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 favored and when RC is considered, the lower stacking fault energy barriers are
compensated by the decreased cleavage energies as listed in table 6.5. Nevertheless, the profile for X=GaAl for the h11̄1i 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 stoichiometric NiAl single crystals are able to undergo high tensile elongations under
certain conditions [144]. Because the results do not strongly indicate an improvement 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 h111i slip was derived, when compared to the other studied cases. Also for the h001i slip one cannot speculate
about an intrinsic ductile alloy. On the other hand, when Ti is used for enhancing the creep properties of NiAl at high temperatures [150], no worsening of
low-temperature brittleness is to be expected.
126
CHAPTER 6.
MICROALLOYING OF NIAL
Table 6.8: For NiAl and NiAl-X, the unstable stacking fault energy for h001i
(100) slip (in J/m2 ) calculated in three supercell configurations, see text. Pure
NiAl has γus = 1.52 J/m2 .
supercell
√1x1√
2x 2
2x2
6.5.2
cover. %
100
50
25
CrAl
1.56
1.66
1.54
CrN i
2.27
1.93
1.74
MoAl
1.46
1.61
1.54
MoN i
0.80
1.44
1.46
TiAl
2.05
1.93
1.71
GaAl
0.91
1.24
1.32
h001i(100) slip
At the (100) plane only s lips along h001i(100) are studied, because the h011i(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 h001i(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-MoN i
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 h001i dislocations in this plane seems improbable.
Exploiting table 6.8 one can observe that Ga dopants reduce significantly the
γus of the h001i(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 h001i(100) dislocations,
because [110] tensile loading provides large resolved shear stress for this slip
system.
6.5.3
h111i(211) slip
The h111i(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 relatively close to that of h111i(110) slip system. By 21 h111i(211) shift an anti-phase
6.6.
127
SUMMARY
Table 6.9: For NiAl and NiAl-X, calculated anti-phase boundary energies (in
J/m2 ) for the h111i(211) slip.
NiAl
0.96
CrAl
0.66
CrN i
0.82
MoAl
0.62
MoN i
0.78
TiAl
0.84
GaAl
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 h111i. If the anti-phase
boundary energy is reduced stronger that the rest of γGSF curve (as found for
h111i(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 improvement 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 (h111i(110) for X=MoAl
and h001i(110) for X=CrAl ), which might result in significant differences for the
128
CHAPTER 6.
MICROALLOYING OF NIAL
Table 6.10: For NiAl-X, estimation of ductile behaviour. The material is predicted to be ductile according to Rice [98] if listed values of Gd /Gc are smaller
than 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 smaller
than 0.014 (see text for details). Results are derived for 50% concentration of
dopants at the interface. Symbols: + material is ductile; ∼ at the crossover.
Rice
ZCT
h11̄1i
0.021
compound
NiAl
h001i
3.13
h11̄1i
2.04
h001i
0.055
CrAl
CrN i
1.82
2.94
0.93∼
1.70
0.037
0.060
0.011 +
0.020
MoAl
MoN i
0.51 +
2.38
1.27
1.59
0.009 +
0.045
0.014 ∼
0.018
TiAl
GaAl
1.43
3.13
1.67
1.75
0.026
0.045
0.018
0.015 ∼
macroscopic behaviour of the corresponding alloys. Because Mo dopants promote
the h001i slip, the improvement of NiAl-Mo intergranular ductility seems improbable, because the h001i 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 h001i(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 h001i(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 h001i and h111i slip at the (110) plane), or the composition
(CrAl with respect to CrN i ) 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 functional 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 influence 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.
130
CHAPTER 6.
MICROALLOYING OF NIAL
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 conceptual 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 quantity (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 introduction 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 decohesive 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 Å 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
131
132
CHAPTER 7. SUMMARY
critical mechanical behaviour of a material -at least for uniaxial strain loadingpurely 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.
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 cleavageis 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 calculated for many various technologically significant materials on-the-equal-footing
form rather unique database of basic mechanical properties for number of materials. 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 h001i and h111i 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 h111i direction, the h001i
is main slip, because h111i slips are somewhat hampered by the relatively high
anti-phase boundary energy. The anti-phase boundary prevents the formation of
the 21 h111i partial dislocations, which occur in metals with bcc structure. Thus,
the h111i dislocations form only when the resolved shear stress for the h001i 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 elaborate 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 relations for tension-shear coupling based on Lejček’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 microalloying. 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 interface 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 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 (h11̄1i(110) for X=MoAl and h001i(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 agreement 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 h001i(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
h001i(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
P. Lazar, R. Podloucky, and W. Wolf
Ab initio study of correlations between elastic and cleavage properties
Physical Review B, submitted
P. Lazar, R. Podloucky, and W. Wolf
Correlation between elastic and cleavage properties
Progress in Materials Science, Proceedings, Festschrift on the 60th birthday of
D.G. Pettifor, submitted
P. Lazar, R. Podloucky
A new concept of cleavage: an ab-initio study
Modelling and Simulation in Material Science, submitted
P. Lazar, R. Podloucky
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
P. Lazar, R. Podloucky
Ab-initio calculation of the influence of Cr- and Ti-microalloying on the mechanical 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)
P. Lazar, R. Podloucky, and W. Wolf
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 significant 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 Mojmı́r Šob, 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 Science College Computational Materials Science, project nr. WK04. Calculations
were performed on the Schrödinger-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 Stability in group of Prof. Mojmı́r Šob
• 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