microstructure and embrittlement of leaded copper alloys - Research ...

MICROSTRUCTURE AND EMBRITTLEMENT OF
LEADED COPPER ALLOYS
THÈSE N O 3217 (2005)
PRÉSENTÉE À LA FACULTÉ SCIENCES ET TECHNIQUES DE L'INGÉNIEUR
Institut des matériaux
SECTION DE SCIENCES ET GÉNIE DES MATÉRIAUX
ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE
POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES
PAR
Laurent FELBERBAUM
ingénieur en science des matériaux diplômé EPF
de nationalité suisse et originaire de Lausanne (VD)
acceptée sur proposition du jury:
Prof. A. Mortensen, directeur de thèse
Dr N. Eustathopoulos, rapporteur
Prof. P. Stadelmann, rapporteur
Dr E. Vincent, rapporteur
Lausanne, EPFL
2005

Remerciements
Ce travail de recherche a été mené au Laboratoire de Métallurgie Mécanique de
l’Ecole Polytechnique Fédérale de Lausanne, avec le soutien financier de la
Commisssion pour la technologie et l’innovation
CTI
de l’Office fédéral de la
formation professionnelle et de la technologie (projet 4448.2 KTS), et celui de
Swissmetal®
, partenaire industriel.
Je tiens en premier lieu à remercier mon directeur de thèse, le professeur Andreas
Mortensen pour m’avoir accueilli dans son laboratoire. Je lui suis vivement
reconnaissant pour sa disponibilité sans limite qu’il m’a généreusement offerte, ainsi
que pour sa passion qu’il m’a transmise et pour sa rigueur qu’il m’a enseignée.
Je désire remercier également les membres de mon jury de thèse pour le temps qu’ils
m’ont accordé, Dr. N. Eustathopoulos, Dr. E. Vincent, Prof. P. Stadelmann et Prof.
H.J. Mathieu, président du jury. J’ai tout particulièrement apprécié les commentaires
développés par le Dr. Eustathopoulos au cours de mon examen de thèse.
®
Notre collaboration avec Swissmetal a toujours été positive et source de nombreux
enseignements. Je tiens à remercier particulièrement E. Vincent pour son suivi à la
fois attentif et critique du projet. Je remercie aussi ses collègues Ch. Charbon, P. Isler,
S. Gilléron, M. Roulin, H.-Q. Tran, P.-A. Girard, R. Flahaut et C. Bezençon.
Un grand merci au Dr. Andreas Rossoll qui m’a aidé, guidé, aiguillé et encouragé tout
au long de ce projet. C’est à lui que je dois tous les calculs de simulation. Associé à
Andreas Mortensen, il m’a donné d’innombrables et précieux conseils lors de nos
réunions de travail.
i

REMERCIEMENTS
ii
Je tiens à remercier les étudiants, Jeannette Frei, Patrick Beyeler, Yves Winkler,
Nicolas Barbi, Alban Dubach, Mustapha Laraqui et Olivier Robyr qui ont collaboré
avec moi dans le cadre de projets de semestre ou de diplôme. De par leur curiosité, ils
m’ont beaucoup apporté.
Avec mes complices et cammarades Ali, Jean-François, Rando, Nono, Yves, Aude,
Nicolas, Charpy, Ludger, Chris, Marianna, Doris et Vincent, j’ai passé des moments
de grande amitié: Merci!
Bravo et merci à l’équipe de l’atelier mécanique emmenée par Louis-Henri Masson
puis Pierre-André Despont. Merci Werner, Jean-Marc, Tristan, Yves et Eric pour
votre magnifique travail. J’adresse une mention spéciale à notre ”Père Noël” du
vendredi et à sa fine équipe, santé!
Que Mme Zanetta et Gökhan se voient spécialement remerciés. Eux qui sont toujours
là quand il faut.
Je salue et embrasse ma maman Florence, mon papa Claude, ma soeur Isaline, mon
frère Milan et le chouchou Andrea. Merci de m’avoir lancé et accompagné dans cette
aventure! Grazie Michela du bonheur que tu continues de m’apporter et de l’amour
que tu me donnes chaque jour. Michela et Samuel, je vous aime.

Nunc est bibendum, nunc pede libero
pulsanda tellus, nunc Saliaribus
ornare puluinar deorum
tempus erat dapibus, sodales.
Horace, Odes
I,
xxxvii, 1-4
iii

12
Table of contents
1 Abstract 1
Résumé 3
2 Introduction 5
2.1 Context and goals of the study 5
3 Literature review 9
3.1 Shape of an inclusion (trans- or intergranular) 9
3.1.1 Equilibrium shape: dihedral angle 9
3.1.2 Influence of stress 17
3.2 Cu alloy embrittlement at intermediate temperature 26
3.2.1 Phenomenology 26
3.2.2 GBE 32
3.2.3 LME 39
3.2.4 Copper intermediate temperature embrittlement - Conclusion 49
4 Equilibrium dihedral angle of Pb in Cu 51
4.1 Experimental procedure 52
4.1.1 Material 52
4.1.2 Heat treatments 52
4.1.3 Microstructure characterization 53
4.2 Time for equilibration 54
4.3 Dihedral angle measurement 58
4.3.1 Classical 2D method 58
4.3.2 New 3D method 59
v

vi
4.4 Results 63
4.4.1 Equilibration time 63
4.4.2 γSL
isotropy 64
4.4.3 Measured dihedral angle 65
4.5 Discussion 68
4.5.1 Inclusion equilibration kinetics 68
4.5.2 Solid/liquid interfacial energy 71
4.5.3 Dihedral angle measurement 72
4.5.4 Dihedral angles in Cu-Pb 72
4.6 Conclusion 77
5 Influence of stress on the shape of an embedded liquid inclusion 79
5.1 Theoretical predictions 80
5.1.1 The Eshelby method 80
5.1.2 Inclusion shape 81
5.1.3 Intergranular inclusion 83
5.1.4 Equilibration time 85
5.2 Experimental procedures 87
5.2.1 Material 87
5.2.2 Interrupted creep tests 87
5.3 Results 89
5.3.1 Creep curves 89
5.3.2 Intragranular inclusion 89
5.3.3 Intergranular inclusion 91
5.4 Discussion 97
5.4.1 Theoretical predictions 97
5.4.2 Measured inclusion shape 98
5.4.3 Comparison of theory with experiment 99
5.5 Conclusion 105
6 Cu-embrittlement: the role of Pb 107
6.1 Experimental procedures 108
6.1.1 Materials 108
6.1.2 Mechanical testing 109
6.2 Results 112
6.2.1 Ductility trough 112
6.2.2 Impact energy 118
6.2.3 Damage 119

6.2.4 Fractography 122
6.3 Discussion 124
6.3.1 Embrittlement Mechanisms 124
6.4 Conclusion 139
7 Quench cracking in leaded copper alloys: a case study 141
7.1 experimental procedures 142
7.1.1 Materials 142
7.1.2 Thermal properties 142
7.1.3 Mechanical analysis 144
7.2 Results 146
7.2.1 Mechanical properties of CN8 & NP8 between RT and 800 °C 146
7.2.2 Heat transfer coefficient of different quenching media 147
7.3 Implications 152
7.3.1 Bar diameter 152
7.3.2 Cooling medium 153
7.3.3 Alloy composition 154
7.4 Conclusion 155
8 General conclusions 157
9 Future work 159
10 Appendices 161
10.1 Least square method and dihedral angle measurement 161
10.2 Bulk modulus of liquid lead 170
10.3 Eshelby analysis 172
10.4 Measured values after interrupted creep test 178
11 List of References 179
vii

viii

1
Chapter 1
Abstract
Mechanisms responsible for the embrittlement of leaded "free-machining” copper
alloys at intermediate temperatures ( i.e.
around 400 °C) are examined using a
combination of microstructural and mechanical characterization.
Tensile tests are conducted on pure copper and on industrial high-strength Cu-Ni-Sn
alloys, comparing leaded and unleaded versions, at strain rates ranging from 10-4
to
-1
10 s , and various temperatures between 25 °C and 800 °C. It is found that, in the
leaded samples, both liquid metal embrittlement and grain boundary embrittlement
operate at intermediate temperature,
i.e. , near the melting point of lead. Their
respective importance depends mainly on the strain rate: fracture at high strain rate
-1
(10 s ) displays characteristic signatures of liquid metal embrittlement, while at low
-4
strain rate (10 s-1)
the ductility trough behaviour can be attributed to grain boundary
embrittlement.
In leaded copper and copper alloys, lead is mostly present as discrete inclusions,
frequently located along grain boundaries. The dihedral angle of intergranular lead
inclusions is then the main microstructural parameter besides their size. A new
technique is proposed and demonstrated for the measurement of inclusion dihedral
angle in a high-purity Cu-1 wt.% Pb alloy, based on room temperature
characterization of samples that were held at elevated temperature at times
sufficiently long for shape equilibration of the inclusions and then rapidly quenched.
Quantitative scanning electron microscopic analysis of metallographic surfaces along
1

CHAPTER 1. ABSTRACT
2
which individual inclusions have been dissolved is used to deduce the dihedral angle
using a mathematical fit of their solid/liquid interface. For a specific temperature, we
show that the dihedral angle is not unique; this reflects the fact that high-angle grain
boundary energies are not constant in copper.
Existing micromechanical analyses of shape equilibrium for intergranular inclusions
in the presence of external stress are adapted to produce an improved description at
low stress and to take into account the inclusion bulk compressibility. The derivation
is based on minimization of the global capillary and elastic strain energy associated
with such inclusions. An adimensional parameter that depends on the applied stress,
the inclusion volume, the elastic properties of the solid, and the relevant interfacial
energies emerges from the analysis as the sole factor governing the shape of the
inclusion. If void nucleation occurs easily within the inclusion, such that its apparent
bulk modulus is nil, the inclusion become unstable, degenerating to a crack when this
adimensional parameter exceeds a critical value.
Measurements of the dihedral angle on samples previously subjected to a remote
stress at 400 °C show that applied stress causes a reduction in the apparent dihedral
angle of liquid lead inclusions. Calculated critical stresses for both leaded pure
copper and leaded copper-nickel-tin alloys that fail at high strain rate by liquid metal
embrittlement are calculated for individual inclusions in the metal. These calculated
critical stresses correlate relatively well with the measured fracture stress, suggesting
that shape instability of the inclusions cause fracture of the alloys. This result is
applied towards the suggestion of general strategies for reduction of the susceptibility
of leaded copper alloys to liquid metal embrittlement.
A case study addressing the problem of quench cracking of leaded Cu-Ni-Sn alloys
on the basis of experimentation and finite element thermomechanical simulation,
showing how the problem can be alleviated in production, concludes the thesis.

1
Résumé
Les mécanismes responsables de la fragilisation d’alliages de cuivre au plomb dits de
décolletage sont examinés par le biais de leur caractérisation mécanique et
microstructurale.
Des essais de traction ont été menés à des taux de déformation allant de 10-4
à 10 s-1,
entre la température ambiante et 800 °C, sur des échantillons de cuivre pur et
d’alliage industriel à base de Cu-Ni-Sn, chacun contenant ou non du plomb. Il est
démontré que l’effet fragilisant du plomb aux températures intermédiaires ( i.e. aux
environs de la température de fusion du plomb) est dû au mécanisme de fragilisation
par les métaux liquides ainsi qu’à la ségrégation intergranulaire. L’importance
relative de ces deux mécanismes dépend principalement du taux de déformation: aux
-1
fortes vitesses (10 s ), la rupture se produit selon le premier mécanisme, alors que le
-4 -1
second domine à faible taux de déformation (10 s ).
Le plomb est présent dans le cuivre et ses alliages sous forme de fines inclusions
rassemblées pour la plupart aux joints de grains. L’angle dièdre de ces dernières en
est la caractéristique principale en plus de leur taille. Une nouvelle méthode de
mesure de cet angle a été développée et appliquée à des inclusions individuelles de
plomb liquide à l’équilibre dans un alliage de Cu-1% masse Pb de haute pureté.
Après trempe et préparation métallographique de l’alliage, le contenu des inclusions
est dissout sélectivement. Ceci permet la reconstruction numérique de l’enveloppe
des inclusions. Cette dernière est approximée par deux calottes sphériques dont
3

CHAPTER 1. RÉSUMÉ
4
l’angle d’intersection est l’angle dièdre. Pour une température d’équilibration
spécifique, il est montré que l’angle dièdre n’est pas unique; ceci reflète le fait que
l’énergie des joints de grains de haute énergie du cuivre n’est pas constante.
Des analyses existantes de la forme d’équilibre d’inclusions sous l’influence d’une
contrainte extérieure ont été adaptées afin de donner une meilleure description de la
forme d’équilibre aux contraintes faibles et de prendre en compte la compressibilité
des inclusions. Ces analyses se basent sur la minimisation de la somme des énergies
d’origine capillaire et élastique. Il en ressort qu’un paramètre adimensionnel régit à
lui seul la forme d’équilibre des inclusions; celui-ci dépend de la contrainte
appliquée, de la taille des inclusions, des propriétés élastiques du solide et des
énergies interfaciales mises en jeu. Si un pore germine au sein de l’inclusion, rendant
nul son module de compressibilité apparent, il est montré que l’inclusion devient
instable et dégénère en fissure une fois que ce paramètre adimensionnel atteint une
valeur critique.
Des mesures d’angle dièdre sur des échantillons de Cu-1Pb mis sous contrainte à
400 °C, montrent que l’angle diminue sous l’effet de celle-ci. Pour le cuivre pur au
plomb et l’alliage de Cu-Ni-Sn-Pb sujets à la fragilisation par métal liquide, les
contraintes critiques pour des inclusions spécifiques de ces alliages ont été calculées.
Celles-ci sont en bon accord avec les contraintes de rupture mesurées sur ces
matériaux. Sur la base de ce résultat, des stratégies visant à diminuer le risque de
rupture due à la fragilisation par métal liquide sont proposées.
L’étude de cas d’un alliage industriel présentant des fissures de trempe conclut ce
travail. Sur la base d’expériences et de calculs par éléments finis, une interprétation
quantitative du phénomène est donnée. Des solutions sont proposées afin d’éviter la
fissuration systématique de cet alliage lors de sa production industrielle.

2
Chapter 2
Introduction
2.1 Context and goals of the study
Small additions of lead, on the order of 1 wt. %, are frequently used to improve the
machinability of copper alloys, Fig. 2-1 [1].
Figure 2-1: Chips produced by drilling in a machinability test: Ø 12.7 mm, depth
12.7 mm, load 86 kg, 141 rpm. (a) For a free-machinig brass Cu-34.7Zn-3.4Pb, the
drilling time is 8.4 s, whereas (b) it is 59.6 s for the unleaded brass Cu-37.1Zn; from
[1].
Leaded high-strength copper alloys, however, have a strong tendency to crack at high
temperature during processing, particularly during quenching, Fig. 2-2.
The aim of the present Ph.D. thesis is to further our understanding of the mechanisms
causing the embrittlement of leaded copper alloys.
5

CHAPTER 2. INTRODUCTION
6
Figure 2-2: Fracture surface of a leaded Cu-Ni-Sn alloy cracked during quenching
from 900 °C.
Its starting point was a quench cracking problem encountered in production by the
Swiss copper alloy producer, Swissmetal, with whom this project was initiated. High-
strength ternary copper-nickel-tin alloys, produced at Swissmetal by the Osprey
process and strengthened by spinodal decomposition and precipitation hardening,
could not be produced in a leaded version because of a strong tendency for extruded
bars to crack during quenching from elevated processing temperatures. Somehow, the
presence of lead in these alloys rendered them very weak at elevated temperature,
and it was our initial goal to explain how and why this occurred and propose
remedies.
To this end, an experimental set-up was designed and implemented, allowing tensile
testing of cylindrical copper bars at elevated temperature over a wide range of strain
-4
-1
rates, extending over five orders of magnitude from 10 up to 10 s . Reasons for this
approach were two-fold. First, at the temperature where cracks develop in the
industrial alloys, its homogenized microstructure is metastable and evolves rapidly.
Consequently, a capacity for rapid straining was important. Secondly, it has been
debated in the literature whether the mechanism underlying the elevated temperature
embrittlement of leaded copper alloys is elevated grain boundary cavitation, liquid
metal embrittlement (LME), or an accentuation of grain boundary embrittlement by
segregated lead (GBE). Experimental data, such as the intergranular fracture surface
in Fig. 2-2, show indeed characteristic signatures of each of these mechanisms. The
strain-rate dependence of embrittlement allows to shed some light on which is the
operative mechanism, so for this reason also the approach was deemed interesting.

2.1 CONTEXT
AND GOALS OF THE STUDY
Given the existing controversy on the underlying mechanism of copper
embrittlement by lead, it was also decided to study the more “basic” and
microstructurally simpler system of high-purity copper with and without lead
additions. The goal of this part of the thesis work was to contribute, using our
capacity for tensile testing over a wide range of strain rates coupled with a novel
approach to the characterization of dihedral angles in this system, to a discrimination
of which mechanism operates, and when.
The thesis is structured in a series of chapters that go from the more general to the
more specific. The next chapter, Chapter 3, provides a review of the literature on the
two basic questions that underlie this thesis, namely the shape and evolution under
stress of liquid inclusions, and the general phenomenon of intermediate temperature
embrittlement of copper and its alloys.
Chapter 4 is dedicated to the description of a new method for dihedral angle
measurement. Results are presented for a high-purity Cu-1 wt.% Pb alloy annealed at
temperatures ranging from 400 to 950 °C.
The influence of an applied remote stress on the shape of the liquid inclusions is then
studied in chapter 5. Theoretical predictions are confronted with experimental results
obtained on an industrial Cu-1Pb alloy subjected to interrupted creep tests. Both
measured and predicted dihedral angle values are shown to decrease as a result of the
application of a remote stress.
Chapter 6 is focused on the intermediate temperature tensile behaviour of leaded and
unleaded copper. It is shown that, in fact, several embrittlement mechanisms operate:
GBE dominates at low strain rate, whereas LME is dominant at high strain rates.
Finally, the starting point of this work is presented as an illustration of the
manifestation of LME. This is given in chapter 7, where transient thermal stresses are
shown to be responsible for the systematic cracking of a water-quenched leaded high-
strength Cu-Ni-Sn alloy.
The thesis concludes with an overview of the main points demonstrated by this work,
together with suggestions for future work. Appendices list the Mathcad
7

CHAPTER 2. INTRODUCTION
8
spreadsheet computer codes that have been built for implementation of the analyses
presented in Chapters 4 and 5.

3
Chapter 3
Literature review
3.1Shape of an inclusion (trans- or intergranular)
3.1.1Equilibrium shape: dihedral angle
It was shown by Smith in a classical paper that phase arrangements in equilibrium
microstructures are not fortuitous [2]: surfaces arrange themselves in order to
minimize their total energy. Provided that the interfacial energies are isotropic, an
intragranular inclusion will be spherical, whereas in intergranular inclusion will
adopt a lenticular shape, Fig. 3-1 [2]. Equilibrium holds when:
where
γ 12 γ 23 γ 13
= =
sin φ sin φ sin φ
3
Eq.3-1
γxy
is the interfacial energy between the phases x and y,
and φz
the dihedral
angle corresponding to phase
1
2
z.
Figure 3-1: Interface equilibrium between three immiscible liquids and mechanical
analogy with triangle of forces; from [2].
9

CHAPTER 3. LITERATURE
REVIEW
10
In the specific case of a liquid intergranular inclusion, Fig. 3-2, from Eq. 3-1 and
elementary trigonometry, we have in the absence of torque components that can arise
with very low angle grain boundaries [3]:
where γgb,
and
respectively.
When
γ = 2γ
gb SL
⎛ φ
cos
⎞
⎝ 2⎠
γ
SL
Eq.3-2
are the grain boundary, and solid-liquid interface energies
b
Figure 3-2: Intergranular, lenticular in shape, liquid inclusion.
φ is zero, the liquid wets perfectly the grain boundaries. If φ is lower than 60°,
symmetric grain boundary triple lines are wetted. Finally, if
are spherical and do not interact with grain boundaries, [2].
a
φ/2
φ is 180° the inclusions
Figure 3-3: Influence of the dihedral angle on the shape of an inclusion located at the
intersection of three grains: the cut is perpendicular to the triple line; from [2].
The theoretical shape of secondary phase inclusions located at symmetric four-grains
junctions (for tetrakaidecahedral ideal grains) was calculated by Wray [4]. A uniform
γ SL
γ SL
γ gb

3.1 SHAPE
OF AN INCLUSION ( TRANS-
OR INTERGRANULAR)
dihedral angle was considered. The grain boundary coverage of the secondary phase,
βA,
its volumic concentration,
β
V
and the dihedral angle,
φ,
are shown to be
interdependent. Six different morphologies of the secondary phase inclusions exist
and can be predicted from two out of these three parameters, Fig. 3-4.
corresponds to a perfect tetraedron. It is shown that even if
φ = 70.53°
φ > 60°, the secondary
phase can form an interconnected network provided that its volume fraction is
sufficiently high. Knowing both the evolution of
φ with temperature and the phase
diagram then leads to the easy determination of βV,
and thus to the secondary phase
morphology, for a specific alloy composition at a specific temperature.
Figure 3-4: The shape of secondary phase inclusions at a four-grains junction is given
as a function of its volume fraction , and its equilibrium dihedral angle; from [4].
Given its importance in the microstructures of materials, and given the information it
conveys on interfacial and grain boundary energy values, there has long been high
interest in measuring φ with good precision, e.g. , Ref. [5].
A difficulty encountered in practice when measuring
φ is that many materials are not
transparent to light. Therefore, unlike contact angles for which the sessile drop
technique provides an elegant and geometrically unambiguous three-dimensional
(3D) measurement technique, dihedral angles must often be observed in two
dimensions, along polished metallographic planes that cut randomly through the
sample and its inclusions. In practice, apparent two-dimensional (2D) dihedral angles
are measured either directly at the tip of a surface groove (”external” angle), at the
apex of intergranular inclusions (”internal” angle), or alternatively for regular lens-
11

CHAPTER 3. LITERATURE
REVIEW
12
shaped inclusions such as that depicted in Fig. 3-2 by using a simple equation linking
the apex angle with the width,
2a,
and thickness, 2b,
of a 2D lens [6]:
cos
Eq.3-3
Since metallographic planes cut the inclusions randomly, the problem arises of
φ
2 2
⎛ ⎞
⎝
2 2
2⎠
=
a − b
a + b
converting such 2D angles into their real 3D value
φ defined above. To this end, a
statistical treatment is often used on a sufficiently large number of 2D measurements.
Early measurements of this type were made by Smith [2]. Assuming a single dihedral
angle, the mode (i.e., the most frequently observed value) in a distribution of 2D
angles was considered to be the true dihedral angle. 250 angles were measured,
giving an estimated (statistical) precision of ± 5° [2]. Also assuming a single dihedral
angle, Harker and Parker calculated the theoretical distribution of apparent dihedral
angles measured on a random section [7]. Based on these calculations, Rieger and
Van Vlack reported that the measured apparent angles should follow a normal
distribution. The median of this distribution then corresponds to the value of the true
dihedral angle within 1° [8]. These authors also reported that 25 (or 100) angle
measurements are sufficient to reach a precision of ± 10° (± 5°) within a 96%
confidence interval in the most difficult situation where φ = 90° [8].
Such a precision was not deemed satisfactory by Stickels and Hucke, who calculated
confidence intervals for the population median in samples of size varying between 25
and 1000 [9]. These authors also quantified the effects of (i) a non-unique dihedral
angle, (ii) measuring errors, and (iii) apparent particle size on the distribution of
apparent φ values. It was concluded that the distribution is not necessarily normal but
its median is almost not influenced by these factors (< 1° error). The median thus still
corresponds to the true dihedral angle.
In an extension of this study, the influence of a given distribution of the true 3D
dihedral angle on the distribution of the apparent 2D dihedral angles measured along
a random section was also quantified [10]. A visible effect, namely a relative
broadening of the latter distribution, was shown to result once the true dihedral angle
φ varies by more than 10°. A method for the determination of the true dihedral angle

3.1 SHAPE
OF AN INCLUSION ( TRANS-
OR INTERGRANULAR)
distribution based on distribution discretization and on single angle measurement
along a polished section was also developped [11]. A mathematical treatment was
then used to demonstrate that the mean value of angles measured on a polished
section is the actual average value of the true dihedral angle [12]. In the same article,
De Hoff proposes a new way to measure apparent dihedral angles on a
metallographic section, based on the use of two stereological measurements: tangent
and triple-line counts. Individual edge angles then do not have to be measured on the
microsection, which eases the procedure operationally [12].
Figure 3-5: Distribution of apparent dihedral angles measured on a 2-D cut. N=150
measurements were performed on a polished section of a Cu-10Pb alloy heat treated at
820 °C. The median of the distribution is 57.5°; from [13].
Methods proposed in the literature for the measurement of dihedral angles along
inclusions that intersect a grain boundary are thus often statistical; values reported
are generally gathered using this approach, Fig. 3-5 [13]. Such methods are, however,
cumbersome and tributary to the methods and assumptions used in data analysis.
Another approach is to conduct precise three-dimensional measurements of
individual dihedral angles, as is done in the sessile drop method for wetting angles.
Stereoscopic and statistical mathematical data analysis are then avoided, easing
interpretation and increasing the reliability of the data. Two methods are generally
used to this end.
One is to create and examine the external angle at grain boundary grooves, formed
along the lines of intersection of a grain boundary with a free surface. If the free
13

CHAPTER 3. LITERATURE REVIEW
14
surface of a sample is wetted by the liquid metal, the liquid-filled grain boundary
groove shape is governed by the dihedral angle φ of Eq. 3-2. Three-dimensional
effects are resolved if the orientation of the grain boundary along the free surface is
known (a 2D cut perpendicular to the grain boundary can then be obtained). Mullins’
treatment of the shape of grain boundary grooves [14] can then be used to extrapolate
the liquid/solid interface surface to the bottom of the groove, so as to deduce the
value of φ from the shape of the groove [15].
Figure 3-6: (a) AFM image of a grain boundary groove in Ni-43.6 at.% Al after
annealing in vacuum at 1400 °C for 1h. (b) Typical depth profile of a grain boundary
groove. Angles θ 1 and θ 2 are determined by parabolic interpolation; from [15].
A second technique for the direct measurement of φ is transmission electron
microscopy (TEM) on samples containing inclusions that are sufficiently small to be
embedded within an electron-transparent metallographic sample. With sufficient
contrast between matrix and inclusion, the inclusion can be clearly distinguished.
With adequate tilting of the sample and taking other precautions, the true dihedral
angle of a single inclusion can then be measured, Fig. 3-7.

3.1 SHAPE OF AN INCLUSION (TRANS- OR INTERGRANULAR)
Figure 3-7: Pb nanometric-inclusions within an Al matrix observed by TEM; from
[16].
Gabrisch et al. made in-situ TEM measurements of dihedral angles using this
method, reporting the influence of (i) faceting, (ii) free surfaces, and (iii) projection
errors on the measurement of the dihedral angle on a single inclusion [16]. The
inclusions are typically nanoscopic (100 nm or less), often causing their shape to be
influenced by size effects, such as ”magic size” phenomena observed with inclusions
a few nanometres wide [17, 18]. This approach is also limited to temperatures below
that at which inclusions start migrating along the grain boundaries towards the
sample free surface; with Al/Pb this temperature is around 500 °C [16, 19].
The Cu-Pb system is a good model system for such capillarity studies since lead
melts at a much lower temperature than copper. Moreover the solubility of lead in
solid copper is almost nil, while that of copper in liquid lead is limited. It has been
studied by some authors [20, 21], who were mainly concerned with the growth
kinetics of grain boundary grooves according to the extensive treatment by Mullins,
see e.g. [14].
Two major contributions on the subject are owed to C. S. Smith in the late fourties [2,
22], and N. Eustathopoulos more recently [5, 13, 23-28]. These authors were mainly
concerned with intergranular lead inclusions, and reported specifically the evolution
of the dihedral angle with temperature, Fig. 3-8.
15

CHAPTER 3. LITERATURE REVIEW
16
Figure 3-8: Evolution of the dihedral angle with temperature for the Cu-Pb system;
from [13] and citing [22].
The dihedral angle depends on one hand on the temperature but also on the chemical
composition of both liquid and solid metals, see e.g. [29]. It was deduced from
thermodynamic analysis that the observed evolution of the dihedral angle with
temperature indicates that the temperature coefficients of both γ SL and γ gb are
negative, as schematically shown on Fig. 3-9 (c) [30].
Figure 3-9: Grain boundary groove morphologies (a) below, and (b) above T w , the
wetting transition temperature. (c) Schematic evolution of γ gb and γ SL with temperature;
from [30].
More specifically, the evolution of γ SL with temperature was quantified according to
a simple relationship involving the temperature, solubility, heat of fusion and molar
volume of the two atomic species. From these results, and from the φ(T) data
presented in Fig. 3-8, it was concluded that there is a linear decrease of γ gb with
increasing temperature. This in turn was shown to imply that there is no, or a very
limited, segregation of Pb at both the solid-liquid interface and the Cu grain
boundaries [27].

3.1 SHAPE OF AN INCLUSION (TRANS- OR INTERGRANULAR)
The influence of a third element was also studied [2, 28]. The addition of nickel to
the Cu-Pb system has no influence on its capillary equilibrium since Ni do not
segregate to the solid-liquid interface nor to the grain boundaries, whereas silver or
bismuth additions do influence φ due to strong interfacial segregation Fig. 3-10.
Figure 3-10: Dihedral angle of the liquid phase in Cu-Pb-Bi alloy as a function of the
liquid phase composition. Increasing the Bi content results in a lower dihedral angle;
from [2].
3.1.2Influence of stress
It has been suggested by Smith that the application of a tensile stress may cause the
full penetration of grain boundaries by a liquid metal when in the absence of stress φ
is somewhat greater than zero [2]. This was proposed to result from the formation of
a liquid film (φ = 0°) and its propagation, both being accelerated if a tensile stress is
applied.
Complete grain boundary coverage by a liquid metal can occur even if φ is positive
[4], see the dashed line on Fig. 3-11 (b). Knowing both the evolution of the dihedral
angle with temperature and the phase diagram for a specific system leads to the easy
determination of φ and β V , thus the secondary phase morphology for a particular alloy
composition at a precise temperature. The different domains of each morphology can
accordingly be sketched on the phase diagram.
As an illustration, the Al-Sn phase diagram is presented, Fig. 3-11. Fracture stress
contours are also drawn. These are parallel to the estimated complete coverage line
17

CHAPTER 3. LITERATURE REVIEW
18
(dashed line: β A =1). It has been argued that this suggests that the morphology of a
secondary liquid phase is influenced by stress [4].
(a) (b)
Figure 3-11: (a) Phase diagram of the Al-Sn system [31], and (b) same diagram with
the domains of morphology of the secondary phase, see Fig. 3-4. The dotted line
corresponds to βA =1. Fracture stress contours are also added; from [4].
A quantification of the influence of an applied stress on φ was performed by Stickels
on Ni-2 wt.% Pb samples. Hydrostatic pressure [32], as well as uniaxial stress in both
tension and compression [33] were applied. Dihedral angles were measured on 2-D
polished sections according to the method described in [8]. The influence of stress on
the dihedral angle in leaded copper was also reported [34] and discussed, Fig. 3-12
[35].

3.1 SHAPE OF AN INCLUSION (TRANS- OR INTERGRANULAR)
Figure 3-12: Influence of tensile stress on the distribution of observed dihedral angle.
Material was uniaxially stressed at 650 °C, and subsequently quenched with the load
still on. The median of each distribution is indicated; from [35].
In all instances, the application of stress results in a slight decrease of φ. According to
Stickels et al., this is rationalized by the fact that the dihedral angle can no longer be
expressed in terms of interfacial tensions alone. Citing J.W. Gibbs: “The sum of the
product of the volumes of the masses by their pressures, diminished by the sum of the
products of the area of the surface of discontinuity by their tensions, must be a
maximum“. The equilibrium particle shape, and hence the apparent dihedral angle
are therefore dependent on both the total interfacial energy and the total strain energy
[33].
Cu-1Pb
650 °C
The equilibrium shape of an intragranular void in an elastic solid subjected to a
remote stress has been calculated by considering both interfacial and stored elastic
energies [36]. The latter energy accounts for the stress field perturbation due to the
introduction of a void into a stressed solid. Assuming that the void is ellipsoidal in
shape, and that the solid is linear elastic, this can be calculated analytically using the
Eshelby formalism [37, 38].
12 MPa
φ = 81°
21 MPa
φ = 75°
0 MPa
φ = 93°
7 MPa
φ = 86°
Minimization of the interfacial energy alone leads to a spherical shape, whereas the
elastic interaction strain energy is minimized if the pore collapses to a crack [36].
Once a remote stress is applied, a pore adopts either an ellipsoidal shape or collapses
to a crack. There is a critical parameter, Λ c , above which the pore shape becomes
19

CHAPTER 3. LITERATURE REVIEW
20
unstable. This parameter parallels the Griffith criterion for elastic crack propagation;
however, here Λ c predicts crack nucleation, with:
2
σ ⋅ a0
Λ=
E ⋅γ
Eq.3-4
where σ is the remote stress, a0 , the initial pore diameter, E, the Youngs’ modulus,
and γ, the interfacial energy. The Griffith criterion is:
2
2 2 1 σ a
K1c= Y σ a = Eγ
⇔ =
Y E γ
Eq.3-5
Both Λc and the pore shape are shown to depend on the stress state, Fig. 3-13 (a).
Pores namely remain spherical as Λ is increased to Λ c =1.78 if the solid is subjected
to a hydrostatic stress; however, under uniaxial stress pores are ellipsoidal in shape
provided that Λ < Λ c = 0,91[36].
The same trends are found for the equilibrium shape of a grain boundary void, also
assumed to be ellipsoidal [39]. The equilibrium shape results from the competition
between elastic energy stored in the solid, and both grain boundary and surface
energies. The initial shape of the intergranular void, which depends on the γ gb /γ ratio,
therefore influences Λ c . Under uniaxial loading, and for γ gb /4γ = 1/11 (corresponding
to φ = 159°), Λ c is about 0.7, Fig. 3-13 (b) [39], whereas it is 0.91 in the case of an
intragranular void (φ = 180°) Fig. 3-13 (a) [36].

3.1 SHAPE OF AN INCLUSION (TRANS- OR INTERGRANULAR)
(a) from [36] (b) from [39]
Figure 3-13: Stability conditions of an (a) intragranular, and (b) intergranular pore
within a stressed elastic matrix. m is a shape parameter that varies from 0 to 1 as the
shape varies from a sphere to a crack. Solid lines represent stable equilibrium, whereas
dashed line represent unstable equilibrium. Note that λ=γgb /4γ (i.e. φ = 159°) and
ω=σ1/σ3 in (b), therefore ω=1, and ω=0 correspond to hydrostatic and uniaxial stress,
respectively.
Raj proposed independently a similar model, aiming to calculate the stress to be
applied to induce the formation and unstable spreading of a liquid film from a stable
intergranular liquid inclusion [40]. The volume of the inclusion, V is assumed to
remain constant. The elastic strain energy is also computed with the aid of the
Eshelby formalism [37]. Here, the inclusion is approximated by a penny-shaped
crack. An adimensionnal parameter, W is defined as:
W =
with ν, the Poisson’s ratio, and G, the elastic shear modulus.
Eq.3-6
With an initially spherical inclusion, ν = 0.33, and, for an elastic solid the relation:
we have:
( ) ⎛
2
1−νσ3V⎞ ⎜ ⎟
π G γ ⎝ 2π⎠
E
G =
21+ ν
( )
SL
1/ 3
2
σ ⋅ a0
W(
φ0
= 180° )≈0.
36 = 0. 36 Λ
E ⋅γ
Eq.3-7
Eq.3-8
21

CHAPTER 3. LITERATURE REVIEW
22
It is shown that W describes both stable and unstable equilibrium configurations.
Similarly to Λ c [36], a critical W* is defined, above which stable lenticular inclusions
cannot exist, Fig. 3-14. On this figure, where the case φ 0 = 60° is considered, we read
W* = 0.06. This corresponds to Λ c = 0.57.
Figure 3-14: Stability conditions of an intergranular liquid inclusion imbedded in an
elastic material subjected to a normal remote stress. Here, the dimensionless parameter
W is plotted as a function of the apparent half-dihedral angle,θ 0 .Under stress-free
conditions, the equilibrium dihedral angle φ is here 2θ e =60°. The liquid inclusion
adopts a stable shape for the values of the apparent dihedral angle 2θ 0 ranging from
60° to about 24°. Under this latter value, the inclusion is not stable and evolves to a
crack; from [40].
Following Ref. [36], a further paper refines the calculation of the elastic stored
energy with the aid of a complex variable method [41] applicable to 2-D (cylindrical)
problems. This was motivated by the fact that the stress state at the apex of a pore
evolving to a crack is largely underestimated when assuming an ellipsoidal shape.
However, since this new analysis is 2-dimensional; it is not applicable to real (3-D)
inclusions. Nevertheless the same trends are shown, Λ c values only being lower. In
the case of uniaxial loading, Λ c = 0.27 [41].
The shape of the pores changes via surface diffusion. The shape evolution was also
computed by Wang et al. [41] taking into account this specific path for matter
transport, Fig. 3-15. A characteristic relaxation time is defined:

3.1 SHAPE OF AN INCLUSION (TRANS- OR INTERGRANULAR)
4
a0⋅kT τ =
Ω
⋅γ ⋅Dsδs
Eq.3-9
where Ω is the atomic volume, Ds the surface self-diffusion coefficient, and δs the
thickness of the atomic layer participating in surface diffusion. Above Λ c = 0.27, the
pore collapses very rapidly to a crack, Fig. 3-15 bottom sequence.
Figure 3-15: Two sequences of evolving pores under uniaxial stress (stress axis is
vertical). The stress state of the sequence on the top leads to a stable equilibrium shape:
ΛΛ c ; from [41].
Crack nucleation at the surface of a stressed elastic solid in contact with an agressive
environment was also addressed by Suo et al., Fig. 3-16 (a) [42]. The equilibrium
groove geometry is determined by energy minimization. The contributions of the
surface energy, the grain boundary energy, the energy associated with the evaporation
and condensation of matter and the elastic stored energy are considered. The surface
profile is approximated by a cycloid. This enables the analytical calculation of the
stored elastic energy. The dimensionless parameter Λ is defined similarly to the
previous analysis, save for the fact that the grain size, λ, is considered instead of the
pore radius:
23

CHAPTER 3. LITERATURE REVIEW
24
2
σ ⋅ λ
Λ=
E ⋅γ
Eq.3-10
Below Λ c , grain boundary grooves approach their invariant shape, whereas above Λ c
grooves sharpen into a crack. Λ c is found to decrease linearly from four to zero as the
interfacial energy ratio, γ gb /γ varies from zero to two, corresponding to an apparent
dihedral angle of 180 and 0° respectively. We then have:
( )
2
⎛ σ ⋅ λ⎞
⎜ ⎟ = 2 2γ
−γ
gb
⎝ E ⎠ c
Eq.3-11
Note that this linear relationship has no fundamental physical grounding: it is namely
tributary to the assumed cycloid geometry used to model the surface profile; however
the trend is meaningful [42]. Notice that a groove can form in a stressed single crystal
provided that Λ>π. In that case, crack nucleation is predicted if Λ oversteps the value
of four, Fig. 3-16 (b) [42].
(a) (b)
Figure 3-16: (a) Semi-infinite elastic polycrystal, in contact with an aggressive
environment, subject to a stress parallel to the surface [42]. (b) Stability conditions of a
groove for different γgb /γ ratio (i.e. different initial φ); from [42].
The above analyses [36, 39, 41, 42] derive the global void shape based on global
equilibrium, and assume a chosen regular shape for the void, inclusion or groove.
The dihedral angle, measured at the very apex of an intergranular inclusion or void,
or at the tip of a grain boundary groove, is on the other hand not expected to be
influenced by an applied or residual stress, since very local stresses are always
relieved by diffusion [43], and since capillary phenomena dominate at atomic-

3.1 SHAPE OF AN INCLUSION (TRANS- OR INTERGRANULAR)
dimensions. Namely these scale with r 2 , whereas elastic energy scales with r 3 , where
r is the distance from the triple line. It was moreover shown that at the triple line, the
singularity associated with elastic fields is insufficient for these to overcome
interfacial energy. Equilibrium angles governed by capillarity can therefore not be
modified by elastic effects [44].
A constant dihedral angle, uninfluenced by stress, is accordingly assumed in a study
of the equilibrium shape of intergranular creep cracks [45].
In a more recent numerical analysis of the shape of grain boundary voids, two models
are proposed and compared. The first assumes a constant φ, whereas it is considered
in the second model that the dihedral angle is not always constant, but adopts over
time a stationary value [46]. It is shown that the second model allows much more
accurate predictions of the void shrinkage process. The authors claim that this
suggests that φ is not always constant. This discrepancy with the conclusions of Rice
et al. [43, 45] probably arises from the fact that a numerical approach can not reflect
the actual phenomena at the atomic-scale due to discrete meshing.
As a conclusion, applied stress does influence the overall shape of a grain boundary
groove, intergranular void or inclusion. Therefore the ”macroscopic dihedral angle”
that reflects the energetic optimization of the overall inclusion shape is affected by
stress, even though, as shown by Chuang and Rice, the ”local dihedral angle” defined
at the atomic length scale remains unaffected by stress.
25

CHAPTER 3. LITERATURE REVIEW
26
3.2 Cu alloy embrittlement at intermediate temperature
3.2.1 Phenomenology
”Intermediate Temperature Embrittlement” is frequently observed with copper and
copper-based alloys. Such embrittlement is manifest in a sharp decrease in ductility
(generally measured as tensile elongation, reduction in area, or impact energy
absorption) as the testing temperature of the material is gradually increased by a few
hundred degrees above room temperature. The domain of reduced ductility begins in
the vicinity of 300 ˚C, and frequently extends over a limited temperature range,
above which the ductility increases again. This type of behavior has been termed
”ductility trough behavior” or ”intermediate temperature embrittlement”; an
illustration from the literature is given in Fig. 3-17 [47].
Figure 3-17: Illustration of the ductility trough for a 200 ppm Pb containing Cu. The
elongation, ∆l/l 0 , Reduction of area at fracture, RA, and fracture stress, R m are plotted
as a function of test temperature, T; from [47].
Such a sharp reduction in ductility has, in metal forming, the implication that many
copper alloys are susceptible to cracking, during deformation or afterwards, as a
result either of residual stresses or of transient stresses of thermal origin. This is
generaly termed ”hot shortness”, or more accurately for copper, ”warm shortness”.
An engineering-oriented review of the problem is given in Ref. [48].
Ductility trough behavior is documented even for pure copper; however, the
phenomenon can be far stronger in alloys. The intermediate-temperature
embrittlement of copper and its alloys has been linked with the onset of grain

3.2 CU ALLOY EMBRITTLEMENT AT INTERMEDIATE TEMPERATURE
boundary sliding above about 300 ˚C [49]: this causes voiding at grain boundaries
and shifts the fracture mode towards intergranular fracture. This is observed in OFHC
(> 99.95% pure) copper, and in copper alloyed with known non-embrittling elements
such as Ni or Zn [49-51]. Extensive creep data gathered on CuOFHC at intermediate
temperature, shown on Fig. 3-18, thus report mostly intergranular fracture [52].
Figure 3-18: Fracture-mechanism map for OFHC copper tested in tension. The map
shows four fields corresponding to four modes of failure. Solid symbols mean that the
fracture was identified as intergranular. Shading indicates a mixed mode of fracture;
from [52].
Fracture mechanisms occuring at intermediate temperature, i.e. above 300 °C, are
schematically shown on Fig. 3-19 [52].
Figure 3-19: Schematic view of the main fracture mechanisms occuring at intermediate
temperature; from [52].
27

CHAPTER 3. LITERATURE REVIEW
28
Voids nucleate at strain concentration sites such as intersecting sliding boundary and
slip planes, Fig. 3-20 [53, 54] or at secondary intergranular particles [55]. They are
found to interlink once a critical level of grain boundary sliding is attained [56]. Note
that grain boundary sliding is not a prerequisite for the occurence of such ductility
loss but exacerbates it [57].
Figure 3-20: A mechanism for microcavitation in the grain boundary due to interaction
of primary and secondary slips with a sliding grain boundary; from [54].
In pure copper, the loss in ductility increases as the strain rate is decreased [49, 51],
as one would expect for a time-dependent damage phenomenon such as grain
boundary voiding. This embrittlement ceases at temperatures above that at which
grain boundary mobility, often accompanied by dynamic recrystallization, becomes
operative [49, 51, 58].
General grain boundaries are more prone to grain boundary sliding than special ones.
This is one of the motivations for the grain boundary engineering concept proposed
by Watanabe [54, 59]. The ductility trough was accordingly reported to widen and
deepen as the grain boundary energy increases [57, 60-62].
Ductility trough behavior can be amplified by two main causes (i) embrittlement
caused by the segregation, prior to fracture, of specific atomic species to grain
boundaries, and (ii) metal induced embrittlement (MIE), which includes liquid metal
embrittlement but also vapour and solid metal induced embrittlement, reviewed in
Ref. [63]. The difference between these two classes of mechanisms is that, in MIE,

3.2 CU ALLOY EMBRITTLEMENT AT INTERMEDIATE TEMPERATURE
fracture is caused by the migration of another embrittling metal towards crack tips
via the crack. Grain boundary embrittlement, on the other hand, is caused by
migration, prior to fracture, of impurity atoms from the bulk metal to the grain
boundaries.
Putting together data from the literature, the ductility trough of copper alloys has the
following characteristics:
• The trough widens, both at its lower and at its upper limits (which respectively
decrease and increase) and also deepens (i.e., the measured ductility decreases
further) as the percentage of embrittling impurities named above increases, Fig. 3-21
[47, 50, 53, 64-68].
Figure 3-21: Influence of the lead content in a Cu-Pb dilute alloy on the magnitude of
the ductility trough; from [47].
• The reverse effect has been noted with the addition of certain elements: Zr [69, 70],
and Mg, B, Y, Ce, La, Ca and, but to a lesser extent, P [70-73], and Ti[70]: such
additions have been documented to reduce the trough extent. Foulger also reports that
Mg and Li improve the ductility of cupronickels at 900 ˚C [48]. We note that
elements Li, Mg, B, Y, Ce, La, and Ca all belong to columns II and III of the periodic
table. The role of Zr, Y, Ce and La is explained as being due to sulfur scavenging [69,
70, 72, 73]. A 1995 patent also cites B and Mn as additions decreasing crack
formation (in both casting and deformation) of nickel containing copper alloys [74].
• As the strain rate increases the trough narrows and becomes more shallow; this is
found with both pure and alloyed copper [47, 49, 51, 70, 71, 75-77]. Such behavior is
classical for grain boundary embrittlement [78].
29

CHAPTER 3. LITERATURE REVIEW
30
Figure 3-22: Influence of the strain rate on the magnitude of the ductility trough of
OFHC copper; from [77].
• The recorded ductility trough becomes deeper as the alloy grain size increases [71,
79-81]. This can be rationalized since (i) while the overall grain boundary surface
diminishes, the impurity coverage per area increases, (ii) the dislocation pile-ups are
larger and hence produce higher stresses, (iii) grain boundary sliding is favoured and
(iv) the intergranular voids are larger [82].
• The upper (brittle to ductile transition) limit of the ductility trough is associated
with the onset of grain boundary mobility, often manifest as dynamic
recrystallization, which thus restores the ductility [47, 49, 58, 64, 67, 70]. This
influence of dynamic recrystallization is frequent [52, 83].
Grain boundary mobility is highly dependant on the material purity [84]. The
occurence of dynamic recrystallization is hence shifted to a lower temperature in an
ultra-high purity material. This was documented for copper by Fujiwara in a study of
the hot-ductility of ultra-high purity copper [68]. The ductility trough behaviour is
even found to disappear for 8 nines purity Cu, since dynamic recrystallization occurs
at a temperature as low as 250 °C, Fig. 3-23. It must be emphasized that the strain
rate used by Fujiwara was very low, namely 4.2 10 -5 s -1 , as compared to those on
Fig. 3-22. At such very low strain rate, conventional purity copper exhibits a clear
ductility trough behaviour, see solid circles for 3N-Cu (three-nines copper) on Fig. 3-
23.

3.2 CU ALLOY EMBRITTLEMENT AT INTERMEDIATE TEMPERATURE
Figure 3-23: Ductility trough of copper of different purity. The reduction in area was
measured after tensile testing in vacuum at a strain rate of 4.2 10 -5 s-1 ; from [68].
• There is a time-dependency of the embrittlement phenomenon. This is related to the
need for the embrittling species to migrate to the grain boundary or the site of
fracture [69, 80, 85-90]. The dependence of embrittlement on annealing time and
temperature then shows the classical ”nose” shape on a TTT diagram [80].
31

CHAPTER 3. LITERATURE REVIEW
32
3.2.2 GBE
The ideal cohesive stress of a material can be schematically derived from its curve of
potential energy vs. interatomic distance. Segregation influences the latter, and
therefore can reduce or increase the interatomic cohesion. The strengthening effect of
boron in nickel-base alloys is one classical example of cohesion enhancement.
Ab initio calculations allow to draw potential-distance curves, and interfacial stress
vs. separation curves [91, 92]. This is illustrated in Fig. 3-24 for a NiAl-Mo interface
with and without C, S, or O [91]. The two latter elements are shown to reduce the
cohesion of the interface, whereas carbon is a cohesion enhancer.
(a) (b)
Figure 3-24: (a) Calculated potential-separation, and (b) interfacial stress-separation
curves for an NiAl-Mo interface with and without C, S, or O; from [91].
Charge transfer is often associated with the occurence of GBE. This can be quantified
by ab initio calculations. It is suggested that a localized charge on the impurity
induces directional bonds and weakens the metal-metal bonds as less charge is
available for these. On the other hand, cohesion enhancers form homopolar bonds,
and in turn do not draw charge off the metal atoms, Fig. 3-25 [93]. EELS spectra can
provide experimental evidence for charge transfer associated with GBE [94-96];
however, its detection may fail [97]. Note that minimal charge transfer was also
calculated at Fe grain boundaries embrittled by segregated S or P, in which case

3.2 CU ALLOY EMBRITTLEMENT AT INTERMEDIATE TEMPERATURE
charge transfer cannot account for the well-known embrittlement effect of these
elements in iron [98].
Figure 3-25: Ab initio calculations of the charge density around an interstitial sulphur
or boron atom in a tetrahedral site in nickel. Charge is localized in the former case,
whereas it is more equally distributed in the second case. This suggests that grain
boundary cohesion is dependent on charge transfer, since sulphur embrittles nickel,
whereas boron acts as cohesion enhancer; from [93].
The amount of segregation to a grain boundary has been reported to be related to the
grain boundary energy. Namely twin Σ3 boundaries in copper were never found to
exhibit bismuth segregation [97, 99]. Also, the least embrittled boundaries in a Cu-
0.1Bi alloy are those of lowest energies [100]. This illustrates another motivation for
the concept of grain boundary engineering [101].
Grain boundary embrittlement has been amply documented for copper and its alloys.
Cited embrittling elements are many: Bi (a strong embrittler subject of many studies)
[48, 53, 94, 95, 99, 100, 102, 103], Te, [48, 104, 105], S [48, 69, 106], Sn [48, 85, 86,
107], O [87, 108-110], Sb [48, 107, 110], Pb and Se [48, 64], as well as C and in
some instances (but not always) P [48]. Gavin ranks their potency in the order of
(most to least potent) Bi, Te, Pb, Se, S [64]. More generally, metal grain boundary
embrittling elements tend to belong to columns IV to VI of the periodic table [93,
111]; this is indeed true of all elements mentioned in what precedes. Embrittling
species also tend to be more electronegative than the host metal [111].
33

CHAPTER 3. LITERATURE REVIEW
34
Figure 3-26: Compilation of sublimation enthalpies and atom size for different
elements. Elements below the dashed line reduce the fracture energy of copper; from
[112].
Tabulated values of sublimation enthalpies and atom sizes allow to rank the potency
of embrittlement or cohesion enhancement of segregating elements [112]. This
simple theory, based on pair bonding, is shown to be equivalent to the general
thermodynamic treatment formulated by Rice and Wang [113]. Embrittlement is
predicted if the surface segregation free energy, ∆g(segreg) surf is higher (in absolute
value) than the grain boundary segregation free energy, ∆g(segreg) gb . Consequently,
2γ int , the ”ideal work of reversibly separating an interface against atomic cohesion” is
reduced, since [113]:
( )
2 2 0
γ int = γ int − X ∆g( segreg) −∆g(
segreg)
gb gb surf
Eq.3-12
with 2γ int 0 , the ideal work of separation in absence of segregation, and Xgb , the
concentration of the segregant per unit area of interface.

3.2 CU ALLOY EMBRITTLEMENT AT INTERMEDIATE TEMPERATURE
One necessary condition for intergranular crack propagation to occur instead of crack
tip blunting is that [113]:
2γ int < Gdisl Eq.3-13
with Gdisl , the energy to nucleate a dislocation at the crack tip. Moreover for
intergranular rupture to occur, the stress must reach locally σ m , the cohesive stress, in
order to nucleate a crack. The segregation of impurities is known to reduce both 2γ int
and σ m , and in turn to induce GBE [113].
Note that the term between parentheses in Eq. 3-12 corresponds to the reversible
work of moving a solute atom from an adsorption site along a free surface to a
specific site along a grain boundary. This value is clearly positive in the case of the
system Cu-Bi, indicating its propensity to GBE. Indeed, it is known that bismuth
segregates strongly, both to the surface and to the grain boundaries in copper. After
surface cleaning by sputtering, a Bi film reforms at the surface by spontaneous
diffusion from the general grain boundaries [114]. This experimental observation is
not surprising since grain boundary embrittlement is commonly illustrated by the Bi-
Cu pair [48, 53, 94, 95, 99, 100, 102, 103]. Grain boundary embrittlement can thus be
ascribed to the difference in segregation of embrittling elements at surfaces and grain
boundaries.
Both thermodynamics and kinetics have to be considered in quantitative analyses of
the phenomenon; these are reviewed in [88]. Bi segregation to copper grain
boundaries is a good illustration of these two facets of the phenomenon [115, 116].
The effect of impurity content, temperature, and type of interface on the equilibrium
segregation are presented in Fig. 3-27. These curves are drawn according to the
classical thermodynamical treatment by McLean [88], Eq. 3-14. For grain boundary
segregation in a dilute solution, one has:
Xgb
X − X
max gb gb
θ
= = X
1 − θ
bulk
⎛ −∆g(
segreg)
exp⎜
⎝ RT
gb
⎞
⎟
⎠
Eq.3-14
35

CHAPTER 3. LITERATURE REVIEW
36
with X gb and X bulk , the concentration of the impurity at the grain boundary and in the
bulk respectively. X max gb is the saturation value of X gb , and θ is the grain boundary
coverage.
It follows from Eq. 3-14 that the segregation increases with the alloy impurity
content, as one of course expects; however, only the amount of impurity that is
actually in solution has to be taken into account. Since the solubility changes, and
generally increases with temperature, it cannot be stated for a dilute alloy that
increasing the temperature necessarily results in a decrease of the segregation level.
(a) (b) (c)
Figure 3-27: (a) The equilibrium fractional segregation,θ is plotted as a function of
temperature, T for several impurity contents in solution Xi . (b) The equilibrium
segregation amount is plotted as a function of the impurity contents at several
temperatures. (c) The equilibrium segregation at the surface is compared to that at a
grain boundary (joint de grain) for a specific temperature. In the case where
Xi =10 ppm, the fractional surface coverage, θ S , is much higher than that of the grain
boundary, θ J ; from [117].
A proposed mechanism for GBE is schematically illustrated on Fig. 3-28 [117]. The
term between parentheses in Eq. 3-12 is taken positive; this corresponds to the
situation depicted in Fig. 3-27 (c). A microcrack opens at a grain boundary triple
junction, segregants enrich the free surface, and favour further grain boundary
decohesion [117].

3.2 CU ALLOY EMBRITTLEMENT AT INTERMEDIATE TEMPERATURE
Figure 3-28: Proposed mechansim for GBE. (a) intergranular segregation at a triple
junctions, (b) microcrack formation and surface enrichment, (c) crack propagation, and
further surface enrichment, from [117].
It has been stated above that GBE is caused by the migration, before fracture, of
impurity atoms from the bulk metal to the grain boundaries. This is true in the ideal
case of instantaneous decohesion (constant X gb). Decohesion under fully equilibrium
conditions governed by thermodynamics (constant chemical potential µ) is on the
other hand characterized by a change in X gb , the impurity concentration at the grain
boundary [113]. It was shown that the actual work of separation 2γ int is [118]:
2γ int( const Xgb ) > 2γ int >
2γ
int(
const µ )
Eq.3-15
During interfacial separation, the impurity penetrates into the interface and reduces
its cohesion. The kinetics of impurity migration were analyzed for slow separation
(constant µ), fast separation (constant X gb ) and transient regimes [119]. A high
sensitivity to the strain rate is reported: embrittlement is more severe at lower strain
rate. This parallels the widely documented dynamic embrittlement phenomenon [109,
119-123], defined as resulting from the stress-induced diffusion of an embrittler to
the tip of an intergranular crack. This in turn reduces the strength of the grain
boundary and promotes intergranular fracture. Decohesion occurs when a critical
combination of stress and impurity concentration is reached, Fig. 3-29 [86].
37

CHAPTER 3. LITERATURE REVIEW
38
Figure 3-29: Schematic view of the dynamic embrittlement process; from [86].
An influence of stress on grain boundary segregation was reported in [124]. It was
namely found in a martensitic stainless steel that phosphorus segregates to a higher
extend to grain boundaries lying normal to applied tensile stress, and also to grain
boundaries parallel to a compressive stress [124]. It was also found in one semi-
quantitative study that embrittlement can be obtained as a result of prior exposure
under applied stress to temperatures above the testing temperature [80]. Anisotropic
segregation of sulphur in a low-alloy steel was also found to result from the
application of an applied stress. The enrichment was maximal at grain boundaries
lying perpendicularly to the applied tensile stress. Note that after 25 h, the grain
boundary coverage returned to a homogeneous equilibrium value. This suggests that
the stress-induced redistribution of sulphur on the grain boundaries is a transient
process that may lead to grain boundary decohesion [125]. This redistribution is due
to a diffusional process characterized by an activation energy of 98 kJ/mol (the
activation energy of the grain boundary diffusion of sulphur in iron ranging from 97
to 135 kJ/mol [126]) and a stress exponent close to unity, as in diffusional creep
[127]. The segregation of some known embrittling elements, such as oxygen or
sulfur, to copper grain boundaries can also be enhanced by an applied tensile stress
[89, 121].

3.2.3 LME
3.2 CU ALLOY EMBRITTLEMENT AT INTERMEDIATE TEMPERATURE
A second mechanism that can cause or increase the severity of ductility trough
behaviour in copper and its alloys is liquid metal embrittlement [78, 128-130].
LME differs from grain boundary embrittlement processes in that the embrittling
metal migrates during fracture to the site of crack nucleation and growth. Given its
importance, the phenomenon has been extensively studied; several reviews exist on
the subject [128, 129, 131-139], as well as dedicated conferences [140].
LME refers to the ductility loss in normally ductile metals when stressed while in
contact with a liquid metal [137]. From a phenomenological standpoint, it has the
following characteristics:
• The phenomenon can be revealed and studied using both tensile and fracture
toughness testing. In tensile testing, it leads to the appearance of a ductility trough
[128, 132, 133, 139]. Moreover the stress-strain curves of the embrittled and the non-
embrittled material superpose well until fracture, Fig. 3-30.
Figure 3-30: Polycrystalline pure Al embrittled by various Hg solutions. Note the
apparent correlation between Pauling electronegativity (in brackets) of solute element
and severity of embrittlement, from [131].
In LME, the lower-end (ductile to brittle) ductility transition is generally situated at
the melting point of the embrittling metal; however, many exceptions exist to this
observation. These exceptions occur either because of embrittler undercooling [Roth,
1980 #303] (when it takes the form of small inclusions within the solid metal) or
because solid metal induced embrittlement (SMIE) is also active.
39

CHAPTER 3. LITERATURE REVIEW
40
• LME varies strongly in severity with the material system at hand; this has been
termed the ”specificity” of LME, Fig. 3-31 [141, 142]. Some common features tend
to be exhibited by documented embrittled/embrittler metal pairs: low mutual
solubility, lack of mutual intermetallics, similar electronegativity, dissimilar heat of
fusion [128, 131-133]; however, all rules that have been proposed so far suffer
exceptions [133]. Copper and its alloys are embrittled by liquid Sn, Pb, Bi, Pb-Bi, Sn-
Pb, Pb-Ag, Hg, Ga, Ga-In, Ga-Hg, In, In-Hg, Zn, Li, Na, Cd, Sb, Th, Na [132, 139].
Quoting a Russian study, Nicholas ranks the potency of some these embrittlers in the
order (most to least potent) Bi, Pb, Cd, Sn, Ga [132].
Figure 3-31: Illustration of the LME susceptibility, the LME couples are marked by a
cross. ”P” stands for nominally pure element, ”A” for alloy, ”C” for commercial purity,
and ”L” for laboratory purity, from [141]. We thus read that both pure and alloyed
copper of commercial purity are embrittled by liquid lead.
It is often stated that LME is far more general than published data may imply, to the
point where even the notion of any ”specificity” of the phenomenon has been
disputed, particularly for LME [133, 137, 142]. A main reason why the occurrence of
LME is judged wider than data suggest is that observation of the phenomenon is
highly dependent on testing conditions (temperature, strain rate, presence of a notch

3.2 CU ALLOY EMBRITTLEMENT AT INTERMEDIATE TEMPERATURE
etc), in particular because of the need for good wetting between the embrittling liquid
and the solid metal: this is often difficult to produce in tests [132, 133, 143, 144].
• For LME to appear, the applied stress must generate at least localized plastic
deformation in the solid metal [128, 132, 133, 136, 139]. Crack initiation is
furthermore driven by the presence of stress-concentrating obstacles to slip [139].
• The onset of liquid spreading is often associated with LME failure [34, 65, 145];
this agrees with the fact that liquid metals facilitate LME crack propagation [66].
• LME bears strong similarities with other environmental embrittlement phenomena,
including stress-corrosion cracking or hydrogen-assisted cracking [135, 139]. In
particular, the relation between crack velocity and the crack stress intensity factor is
similar in LME and these other environmental embrittlement phenomena: a threshold
value K th LME below which cracks do not grow and slightly above which crack
growth increases steeply with increasing K until a plateau is reached, the plateau
extending until K reaches the critical stress intensity factor for rapid fracture, Fig. 3-
32 [135].
Figure 3-32: Crack velocity as a function of the stress-intensity factor for aluminium
and titanium alloys tested in liquid metal and aqueous environment; from [135].
In LME, there is though the significant difference that the plateau cracking velocity is
very high (on the order of a meter per second [132, 136, 146]) because supply of
liquid metal to a moving crack tip is comparatively rapid, being limited only by
viscous flow, which is rapid given the low viscosity of liquid metals [147].
Additionally, all these environmental embrittlement phenomena have common
41

CHAPTER 3. LITERATURE REVIEW
42
fractographic signatures [135]. LME is, however, characterized by a lower tendency
for crack branching [139].
• Fracture by LME is generally intergranular in polycrystalline materials [128, 129,
132, 133, 139]; however, this is not always so and even single crystals or amorphous
alloys are sensitive to the phenomenon (e.g., [133, 135]).
Figure 3-33: Transgranular and intergranular fracture surfaces of Cu-1%Pb alloy (a)
room temperature, and (b) 350 °C impact tested respectively; from [130].
A clear correlation exists between the grain boundary energy and the crack extension
force in LME. This was shown for symmetric Al bicrystals in contact with
liquid Hg-Ga: maximum crack extension forces are required with low-energy grain
boundaries, Fig. 3-34 [148].

3.2 CU ALLOY EMBRITTLEMENT AT INTERMEDIATE TEMPERATURE
Figure 3-34: (top) 3.5 µm/s crack extension force as a function of tilt angle for pure Al
bicrystals in contact with liquid Hg-3 at.% Ga [148], and (bottom) corresponding grain
boundary energy; from [149].
It has therefore been argued that, since segregation reduces the grain boundary
energy, prior impurity segregation to grain boundaries should reduce the severity of
LME. Indeed, a beneficial effect of segregated antimony was reported for pure
copper subjected to creep in liquid bismuth [146].
Observations made on the system Cu + liquid Pb-Bi suggest that the resistance to
LME increases with increasing γ SL [150]. The intermediate temperature fracture
stress and ductility are reported to increase with decreasing bismuth content [151],
while γ SL increases accordingly to Morgan quoted in [150].
• By and large, the response of the system to metallurgical and test variables is the
same as that which is observed in low-temperature plasticity and fracture of brittle
metallic materials [129, 133, 139]. Embrittlement is, in other words, promoted by all
factors that tend to increase the intensity of local stress concentration caused by slip,
particularly - but not only - at grain boundaries [133].
• Following this general trend, the severity of embrittlement increases as the strain
rate increases [78, 128, 132, 133, 139]. In this regard, LME differs from grain-
43

CHAPTER 3. LITERATURE REVIEW
44
boundary induced ductility trough behaviour, in particular as observed for copper and
its alloys (see above).
Several models have been proposed for LME. None of these was found to provide a
universally applicable theory. The most relevant models were recently reviewed in
[137-139]. These and most recent models are briefly summarized below. It is in any
case generally accepted that the reduction in surface energy due to the presence of a
wetting liquid is a necessary but not sufficient condition for embrittlement [137].
According to Robertson, an elastic crack propagates in LME through dissolution of
the solid atoms at the crack tip, diffusion of these through the liquid, and redeposition
on the stress-free and flat walls of the crack [152].
The adsorption induced reduction in cohesion model was proposed independently in
1963 by Stoloff and Johnston [153], and Westwood and Kamdar [154]. These authors
assume that the stress to extend a crack is reduced if an embrittler element is
adsorbed at the crack tip Fig. 3-35 (a), the adsorbed atom inducing a decrease of the
cohesive stress of the material, Fig. 3-35 (b). This model resembles thus strongly the
dynamic embrittlement model proposed by McMahon et al., compare Figs. 3-35
and 3-29.
(a) (b)
Figure 3-35: (a) Schematic illustration of an adsorbed B-atom at the crack tip of a
material subjected to tensile stress. (b) potential-separation curves as well as derived
cohesive stress for both cases where a B-atom is adsorbed (U(a) B & σ(a) B ) or not; from
[154].

3.2 CU ALLOY EMBRITTLEMENT AT INTERMEDIATE TEMPERATURE
According to Lynch, adsorption of a third element results rather in a reduction of the
shear strength of atoms at the crack tip, increasing dislocation activity there.
Enhanced dislocation emission then induces a more localized plastic zone, such that
cracking occurs with less energy dissipation, Fig. 3-36 [135]. Arguments for this
theory are largely derived from the presence of ductile fractographic signatures of
metals embrittled by LME.
Figure 3-36: Mechanisms of crack growth by nucleation, growth and coalescence of
voids (a) in absence or (b) in presence of adsorbed species at the crack tip; from [155].
The GALOP model was proposed recently by Glickman [150, 156], GALOP standing
for grooving accelerated by local plasticity. It is proposed that grain boundary crack
propagation is an iterative process where minute Mullins grooves [14] controlled by
diffusion in the liquid are periodically blunted due to the presence of stress at the
groove tips, Fig. 3-37 [150]. The diffusion length thus remains very short. This
induces in turn a much higher crack propagation rate in comparison with classical
Mullins grooving, where the rate-limiting diffusion length equals the total groove
depth [150]. A strong dependence on the equilibrium dihedral angle is then predicted
for the average groove velocity, V ∞ [156]:
V ∞ ∝
tan 3
⎛ φ
⎞
⎝ 2⎠
Eq.3-16
45

CHAPTER 3. LITERATURE REVIEW
46
Figure 3-37: Galop Mechanism: (a) a Mullins groove forms, (b) stress relaxation by
local dislocation activity blunts the groove tip, (c) and (d) crack blunts again after its
extension by a ∆L* increment following Mullins kinetics [150].
Kinetics of grain boundary wetting, or grain boundary penetration by a liquid metal,
was the subject of a round table chaired in 1998 by Glickman [157]. Many
observations and experiences of the participants were put together, but no clear trend
was exposed, nor was any general agreement found. An influence of stress on grain
boundary wetting kinetics is mentioned, but not discussed.
Stress does, however, play a central role in the intergranular penetration of liquid
gallium along pure aluminium grain boudaries. This was demonstrated by
synchrotron radiation X-ray microradiography [158-161]. A link between grain
boundary wetting and liquid metal embrittlement is thus clearly expressed in [161].
Since gallium intergranular penetration is an intrusive process (meaning that the
neighbouring grains are mooved apart [162]), it is no surprise that stress, or more
precisely elastic strain energy, controls the process; see [163]. Accordingly, a net
driving force for Ga penetration along Al grain boundaries was proposed in [164]:
−
Eq.3-17
with G is the Gibbs free energy, U the strain energy, and A the liquid penetration area
∂
∂ =−∂
G U
− 2γ SL + γ
gb
A ∂A in the grain boundary plane.
Intergranular liquid penetration occurs by definition above the wetting transition
[165]. Cahn showed that this transition can result from a change in temperature,
pressure or composition [166]. The grain boundary wetting transition is observable in
alloys within a two-phase region of the phase diagram; however grain boundary
prewetting or premelting can also occur in a single-phase region, Fig. 3-38 [167].

3.2 CU ALLOY EMBRITTLEMENT AT INTERMEDIATE TEMPERATURE
Note that prewetting refers to a new liquid phase, chemically similar to the bulk, that
appears at grain boundaries, whereas premelting characterizes the appearance of a
chemically dissimilar phase that replaces the grain boundary [167].
Figure 3-38: Schematic phase diagram representation with both bulk and grain
boundary solvus lines. T min and T max are the minimum and maximum wetting transition
temperature, corresponding to high-energy and low-energy grain boundaries
respectively [167].
It is expected that the occurrence of perfect grain boundary wetting has a strong
influence on properties such as mobility, diffusivity, energy and composition [168].
Nanometric grain boundary bismuth precursor films (equivalent to premelting films)
were also detected at the grain boundaries of copper or nickel in contact with molten
bismuth [169-171]. Stress-dependent nanometric film propagation was futhermore
shown in [172-174]: under tensile stress, the film propagation rate was estimated to
be increased by two orders of magnitude [172]. This was one motivation for
proposing a new mechanism for liquid metal embrittlement [171].
Liquid
metal
T
A
S
Short
micrometric
film
S+L
Wt. % B
Figure 3-39: Schematic representation of an intergranular liquid film consisting in a
short micrometre-thick and a long nanometric film. According to the authors, LME must
be controlled by phenomena occurring at the tip of the nanometric film.[171]
σ
σ
L
T m
T wmax
T wmin
Very long
nanometric
film (2−4 nm)
LME
47

CHAPTER 3. LITERATURE REVIEW
48
Following the same idea, prewetting or premelting of grain boundaries allows a
sufficiently high supply of embrittler atoms at the crack tip to induce grain boundary
decohesion and high crack growth rates, as observed in LME. Below the perfect
wetting transition temperature, it is suggested that the application of a tensile stress
may induce prewetting or premelting and in turn provoke the rapid rupture [175].
One may note the analogy with precursor films that also form ahead of a solid or
liquid droplet deposited on a free surface [176]. This was also obseved in the Cu-Pb
system [177-179]. High-resolution transmission electron microscope pictures, Fig. 3-
40 clearly demonstrate the phenomenon, showing a Ti-27Ag-5Cu liquid precursor
film on a (0006) TiC surface: the wetting liquid drop is namely found to be preceeded
by a nanometre-thick liquid film [180].
Figure 3-40: High resolution images of the tip of molten Ti-27.4Ag-4.9Cu alloy
spreading on a SiC substrate at 800°C. Arrows denote the tip of the precursor film. (a)
0 s, (b) 3.2 s, (c) 4.4 s, (d) 7.8 s; from [180].
In summary, several theories have been proposed to explain liquid metal
embrittlement. None of these provides a complete and predictive theory of the
phenomenon. By and large, it nonetheless is recognized that, at the heart of LME, is

3.2 CU ALLOY EMBRITTLEMENT AT INTERMEDIATE TEMPERATURE
the fact that the embrittling metal reduces the strength of atomic bonds in the base
metal, just as do solid state embrittling interfacial segregants. This in turn leads to
lowered capillary energy requirements for fracture.
3.2.4 Copper intermediate temperature embrittlement - Conclusion
Focusing back on the Cu-Pb system only, rationalization of the ductility trough is
thus debated in the literature. It is recognized that ductility is recovered with the onset
of dynamic recrystallization; however, there is no general agreement on a single
cause for the lower temperature marking the appearance of the ductility trough.
Some authors claim that the intermediate temperature embrittlement of leaded copper
is simply due to the presence of the finely dispersed molten and thus inherently weak
liquid lead inclusions [47, 50]. While it cannot be disputed that this effect always
exists, added mechanisms of grain boundary embrittlement, or alternatively of liquid
metal embrittlement, have been advanced to explain mechanical or physical
signatures of the observed embrittlement of copper by lead. Since, furthermore, (i)
grain boundary sliding starts to occur at about 300 °C, (ii) thermally activated
processes are known to operate around and above 0.5 of the homologous
temperature, which is about 400 °C for copper, and (iii) the incipient melting
temperature of lead is 327 °C, all three mechanisms start operating at around the
same temperature, which complicates discrimination between these.
It is, in conclusion and in view of what precedes, most likely that all three
mechanisms in fact operate in the Cu-Pb system depending on circumstances under
which the material is deformed. Most probably, it is not the absence of any of these
mechanisms, but rather the boundary that separate it from others, i.e., the parametric
location of dominance of each of these mechanisms over the others that remain to be
determined.
49

CHAPTER 3. LITERATURE REVIEW
50

Chapter 4
4 Equilibrium dihedral angle of Pb in Cu
A new method is presented for the measurement of the dihedral angle of
intergranular, lenticular, lead inclusions. This method is based on the stereoscopic
quantification of the three-dimensional (3D) surface of selected single inclusions.
The dihedral angle is deduced when the shape of the intergranular inclusion is
known, in particular when it consists in two spheres intersecting along the grain
boundary plane.
To this end, inclusions visible along a polished metallographic section are selectively
dissolved. Scanning electron microscopy stereo image pairs are then taken and
processed so as to enable a 3D digital reconstruction of the inclusion/matrix interface
along each inclusion. Spherical caps describing the Cu/Pb interface over non-facetted
orientations are then fitted to the measured digital inclusion envelope
reconstructions. Knowing the center and radius of these spheres, the true dihedral
angle of each specific inclusion is then deduced.
51

CHAPTER 4. EQUILIBRIUM DIHEDRAL ANGLE OF PB IN CU
52
4.1 Experimental procedure
4.1.1 Material
High purity Cu-1wt.% Pb was used in this study. The solubility of Cu in Pb is very
low, and the solubility of Pb in Cu is almost nil, Fig. 4-1. Hence, the microstructure
of the alloy is of pure copper equiaxed grains with lead-rich inclusions. These are (i)
spherical in shape when located within the grains, and (ii) lenticular in shape when at
grain boundaries.
Figure 4-1: Phase diagram of the Cu-Pb system [31].
The material was prepared from 99.999% pure metals by induction melting in a
quartz crucible under a high-purity (≥ 99.999 vol.%) argon atmosphere, and cast in a
Ø 6 mm copper mold.
4.1.2 Heat treatments
Heat treatments were conducted in a primary vacuum (p ≈ 10-2 mbar) for all samples
except for the 930 °C anneal, which was conducted in Formier-gas (95% N 2 , 5% H 2 ,
and p ≈ 10 1 mbar). A two-step treatment was applied:
(i) 1 h at 900 °C to induce microstructural coarsening, causing the inclusions to grow
to a size of roughly 10 µm;

4.1 EXPERIMENTAL PROCEDURE
(ii) a hold at fixed temperature above the melting point of lead, sufficiently long to
reach shape equilibration of the lead inclusions. The specimens were either
encapsulated in a glass ampoule (static vacuum), or positioned in a quartz tube in the
presence of a titanium sponge oxygen scavenger under dynamic primary vacuum.
The duration of the heat treatment was varied as a function of the temperature (it was
60 h at 400 °C and 1 h at 900 °C). Following this hold, the samples were quenched in
water. The quench cools the samples in water in a few seconds, as shown by simple
estimation of the rate of Newtonian cooling with a heat-transfer coefficient typical
for non-agitated water, h ≈ 10 3 W m -2 K -1 [181].
It can be shown by diffusion rate analysis, see Section 4.2, that these time/
temperature combinations are sufficient for shape equilibration of the 10 µm
inclusions during the isothermal hold, while the inclusions do not have time to
change shape during or after the quench.
Additionnal thin (6 mm in diameter, and 1 mm thick) samples were heat-treated at
600 °C for 1 week in flowing Ar - 10 H 2 in order to scavenge and remove oxygen
and sulfur traces. Hydrogen diffuses troughout the sample and forms H 2 O and H 2 S,
both of which evaporate for these long treatment times [182]. A further heat
treatment at 800 °C for 24 h on one sample and at 400 °C for 1 week on a second
sample was conducted in the same atmosphere. Samples were then transferred to the
cold zone of the apparatus. In this way, the sample is brought to room temperature
within less than 10 s. These treatments were kindly performed by Prof. Dominique
Chatain at CRMC2-CNRS in Marseille, France.
4.1.3 Microstructure characterization
Standard metallographical sections were prepared by successive grinding on wet SiC
papers down to 2500 grit, and then by polishing with 6 µm and 1 µm diamond
suspension (Struers ® MD-Dur with Struers ® red lubricant). Final polishing
comprises repeated etching and 1 µm diamond polishing so as to obtain a
deformation-free surface. The etchant is Klemm III (20 g K 2 S 2 O 5 , 100 ml distilled
water, and 11 ml saturated Na 2 S 2 O 5 ”stock solution”). The wet etching-time was
90 s, following [183].
53

CHAPTER 4. EQUILIBRIUM DIHEDRAL ANGLE OF PB IN CU
54
4.2 Time for equilibration
Shape equilibration of a liquid inclusion occurs by the diffusion of atoms of the solid
matrix through the liquid. Diffusion along the solid/liquid interface is usually not
considered, since it is negligible as compared to the above mentionned diffusion path.
It is often concluded that equilibrium shapes are very rapidly reached for nanometric
inclusions based on a simple comparison of the inclusion diameter with diffusion
distances estimated using the coefficient of diffusion in the liquid [184-186]. The
physics of the situation are, however, a little more complicated than this: additional
parameters, such as the (typically very low) solubility of the solid in the liquid, also
intervene in the process. We therefore estimate the equilibration time in what follows.
We assume here that the solubility of Pb in Cu is nil, and that the transport of matter
occurs by diffusion through the liquid. Moreover, no kinetic limitation to the
movement of the solid/liquid interface, nor any strain energy due to misfit, are
considered.
We consider a small portion of the curved interface between the liquid and solid
phases. At local equilibrium, the concentration of Cu atoms in the liquid is X Cu :
X = X + X
⋅V
RT
0 0 γ SL Cu
Cu Cu Cu
X Cu
Eq.4-1
0
where is the equilibrium atomic concentration of Cu in Pb, γSL is the solid/liquid
V Cu
interfacial energy, is the partial molar volume of Cu in Pb, κ is the curvature, and
RT has the usual meaning.
⋅κ
If the shape of the inclusion deviates from that which is dictated by capillary
equilibrium, Cu atoms will diffuse through liquid Pb, bringing the inclusion
geometry nearer that which is dictated by capillary forces, Fig. 4-2.

4.2 TIME FOR EQUILIBRATION
Figure 4-2: Shape changes of a liquid inclusion due to capillary forces. Arrows
describe the atomic flux of Cu atoms through the liquid Pb inclusion (shaded area).
Suppose that, locally, the curvature κ deviates from its equilibrium value, κ e : a flux J
of solute will then cause the interface to move, into the liquid by adjunction of more
Cu atoms along the interface where κ (counted positively towards the solid) exceeds
κ e , and towards the solid where κ < κ e . If we assimilate for simplicity the local
curved inclusion surface to a spherical cap of radius r = 2/κ, a simple mass balance
yields:
2
1 dr 1 r dκ
J = ⋅ = − ⋅ ⋅
Ω dt Ω 2 dt
where Ω Cu is the atomic volume of copper.
Eq.4-2
The flux J depends on the precise geometry of the inclusions and on their capillary
equilibrium shape. As we only aim here for an order-of-magnitude calculation, we
simply write:
Cu Cu
J D
Eq.4-3
where ∆C is the concentration difference in volumetric solute concentration within
C ∆
≈
L
the inclusion between points of maximum curvature difference ∆κ along the
interface, and L is an average diffusion distance between points of maximum
difference in curvature. By analogy with Nabarro-Herring diffusional creep, for
equiaxed inclusions this distance is [187]:
equilibrium shape
J Cu
actual shape
55

CHAPTER 4. EQUILIBRIUM DIHEDRAL ANGLE OF PB IN CU
56
Thus,
r
L = 2
13. 3
Eq.4-4
J D
Eq.4-5
Neglecting variations in molar volume with composition in the liquid lead inclusions,
C ∆
≈ 66 .
r
from Eq. 4-1 we can write:
0 γ SL ⋅ VCu
∆κ
J ≈66 . D⋅CCu ⋅
RT r
Eq.4-6
Writing Eq. 4-2 at the point of maximum curvature κmax , from Eq. 4-6, and
assuming for simplicity that
∆κ = κmax −κmin ≈2κmax −κe
we obtain:
Eq.4-7
0
1 dκ
max 1 d∆κ
13. 3⋅D⋅CCu
⋅γ SL ⋅VCu ⋅ΩCu
≈ ≈ −
3
∆κdt 2 ∆κdt
RT ⋅ r
Eq.4-8
Since derivatives of κ and ∆κ are the same (the equilibrium shape being fixed). We
thus have:
therefore
( )
0
1 d∆κ
d ln( ∆κ) 26. 6⋅D⋅C
⋅γ ⋅V ⋅Ω
= ≈ −
3
∆κ
dt dt
RT ⋅ r
0
0 ⎛ 26. 6 ⋅D⋅C ⋅γ ⋅V ⋅Ω
∆κ = ∆κ
exp⎜−
3
⎝ RT ⋅ r
Cu SL Cu Cu
Cu SL Cu Cu
Eq.4-9
Eq.4-10
where ∆κ ο is the initial maximum difference in curvature along the interface.
Mathematically, the time to full equilibration is infinite; however, after time
⎞
⋅ t⎟
⎠

4.2 TIME FOR EQUILIBRATION
3
4 RT ⋅ r
∆t
=
0
26. 6 ⋅D⋅CCu ⋅γSL ⋅VCu ⋅ΩCu
Eq.4-11
curvature differences have reduced to 2% of their initial value. ∆t given by Eq. 4-11
can, thus, be taken as a simple estimation of the time for shape equilibration of the
liquid inclusion.
57

CHAPTER 4. EQUILIBRIUM DIHEDRAL ANGLE OF PB IN CU
58
4.3 Dihedral angle measurement
The dihedral angle, φ, is dictated by capillary equilibrium at a three-phase junction.
We consider here the case of a liquid lead inclusion located on a planar grain
boundary, Fig. 4-3. The relevant interfacial energies are the grain boundary energy,
γ gb , and the solid/liquid interfacial energy, γ SL . We have:
γ = 2 γ
gb SL
⎛ φ
cos
⎞
⎝ 2⎠
grain boundary plane
Figure 4-3: Schematic view of an equilibrated lenticular inclusion situated on a planar
grain boundary: (a) 3D view, and (b) detail: the cut is perpendicular to both the grain
boundary plane and the triple line.
Eq.4-12
A meaningful dihedral angle can be measured once the microstructure is at
equilibrium. In this case, the grain boundaries are planar, and the solid/liquid
interface displays a constant curvature.
4.3.1 Classical 2D method
φ
detail
Dihedral angles are classically measured along the plane of a polished
metallographic section. Since this plane crosses the inclusions randomly, a
distribution of apparent angles is obtained, Fig. 4-4. It is often assumed that the (real
3D) angle φ takes a unique value across all inclusions, i.e. the interfacial energies γ gb
and γ SL are isotropic. φ is then taken as the median of the distribution [8, 9].
(a)
Pb
φ
γ SL
γ SL
Cu
detail
(b)
γ gb

4.3 DIHEDRAL ANGLE MEASUREMENT
Figure 4-4: Distribution of apparent dihedral angles on a 2D cut. Insets show three
different scenarii where an intergranular inclusion intersects the polished section. The
apparent angle (here: 53°, 60°, and 62°) varies between 0° and 180°.
We determined the median of more than 100 apparent angles visible in the SEM on a
2D longitudinal cut for samples treated at 400, 820, 900 and 930 °C. The angles were
computed from measured width and thickness values of each 2D lens according to
[6]:
cumulative frequency [%]
100
75
50
25
cos
Eq.4-13
with a and b the respective width and thickness of a regular lens-shaped inclusion.
φ
2 2
⎛ ⎞
⎝
2 2
2⎠
=
a − b
a + b
4.3.2 New 3D method
0
gb
53°
0 30 60 90 120 150 180
The inclusions are first dissolved using a procedure designed by superposing the
Pourbaix diagrams for lead and copper, Fig. 4-5. This reveals that, at a pH of 4 and at
zero electrical potential, copper is immune while lead is corroded. Agitated and
desaerated pure acetic acid is thus used to selectively dissolve exposed lead
inclusions along a polished metallographic cut through the samples, leaving the
copper essentially intact. The acid is desaerated by bubbling N 2 in order to prevent
oxidation of the metal surface.
(i)
(i)
(iii)
(ii)
gb 60° gb 62°
apparent dihedral angle φ [°]
(ii) (iii)
59

CHAPTER 4. EQUILIBRIUM DIHEDRAL ANGLE OF PB IN CU
60
Figure 4-5: Pourbaix diagram of pure copper and pure lead [188]. The domains of
corrosion, passivation, and immunity are specified. At zero potential, and at a pH of 4,
copper is immune, whereas lead is dissolved.
Stereo image pairs are then obtained from such samples in scanning electron
microscopy using the “MeX” stereophotogrammetry software [189]. The software
combines images captured after dual tilting of the stage to produce 3D
reconstructions of the sample surfaces along the surface of dissolved inclusions.
An important parameter with samples such as these is the surface roughness that
exists along the metal surface where inclusions were dissolved. The software bases
its reconstruction of the dissolved inclusion surface using the two captured SEM
images on identifying and matching selected points along this surface. This step of
the process is easily conducted on fractographs (for which the software was
originally designed); however, it becomes near-impossible if the copper surface is
perfectly smooth after lead dissolution. A proper compromise must therefore be
reached between (i) leaving, after etching, a surface that is sufficiently smooth so that
the true overall geometry of the inclusion surface is preserved, and (ii) producing a
small degree of surface roughening so as to aid the software in identifying selected
points that can be “matched” along the inclusion surface across the two SEM
micrographs. In this respect, partial solidification along the interface of the (small)
amount of copper initially dissolved into the liquid inclusion is perhaps somewhat
helpful as it can produce small features along the interface. Practically, an adequate
inclusion surface was obtained by fine-tuning the time of the etch (3 hours), and also
by selecting the “best” inclusions in terms of relative surface roughness — in
addition to other criteria such as their relative location with respect to the plane of

4.3 DIHEDRAL ANGLE MEASUREMENT
polish and the degree of planarity of the grain boundary along which they are located,
Fig. 4-6.
Figure 4-6: Typical solid/liquid interface after selective dissolution of a lead
intergranular inclusion. The sample was rotated and tilted in the SEM so that the triple
line appears as a horizontal line on the image (i.e. the grain boundary plane is normal
to the image, and dual tilting allows optimal 3D reconstruction).
From these images, the 3D dataset is calculated and processed so as to enable a three-
dimensional digital reconstruction of the visible copper/lead interface along each
inclusion, Fig. 4-7a. Assuming that the solid-liquid interfacial energy is isotropic (see
below), spherical caps describing the solid-liquid interface over non-facetted
orientations can then be fitted with the aid of the least square method to the measured
digital inclusion envelope reconstructions, Fig. 4-7b. This determines the radius and
center of the two spheres delineating the inclusion.
Figure 4-7: 3D digital reconstruction of the solid-liquid interface of an intergranular
inclusion, and (b) data points from each cap with the two fitting spheres. The Cu-1Pb
sample was heat treated at 930 °C; φ = 78°.
(b)
61

CHAPTER 4. EQUILIBRIUM DIHEDRAL ANGLE OF PB IN CU
62
Provided the inclusion is located along a planar grain boundary, knowing the center
and radius of these two spheres, the true dihedral angle φ along the circular line of
intersection of the two inclusions is then easily deduced, Fig. 4-8. We have:
α = 2( π/ 2−φ)+
φ → φ = π −α
with α calculated from the scalar product:
cos( α ) =
CA 1 ⋅ CA 2
CA ⋅ CA
1 2
A
α
π/2 − φ π/2 − φ
φ
C 1
C 2
Figure 4-8: Determination of φ from the center and radius of the two spheres
Eq.4-14
Eq.4-15

4.4 Results
4.4.1 Equilibration time
4.4 RESULTS
An approximate value of the equilibration time is computed from Eq. 4-11,and is
reported on Table 4.1 and Fig. 4-9. Typical values for r, the radius of curvature of the
inclusion surface, are 2 µm and 5 µm for intragranular and intergranular inclusions
respectively of our specimens. The diffusion coefficient, D, is computed from the
litterature [190]:
−9
D = ⋅
⎛ −4853,
1
419, 35 10 exp
⎞
⎝ T ⎠
2
⎡m
⎤
⎢
⎣ s
⎥
⎦
Eq.4-16
The solid/liquid interfacial energy is computed according to the regular solution
model, where [116]:
liquid solid 2 liquid 2
γ SL = A+ B( CCu −CPb
) ≈ A+ B( CCu
)
Eq.4-17
liquid
solid
where and are respectively the solidus and liquidus atomic
C Cu
concentrations in the Cu-Pb system.
From the phase diagram [31], and literature data [26], we have A = 190 mJ/m 2 and
B = 248 mJ/m 2 .
C Pb
For the molar volume of copper, we use its value at room temperature, thus:
− ⎡ m ⎤
∀ T VCu
= 712⋅10⎢⎥ ⎣mol
⎦
6
3
,
Eq.4-18
63

CHAPTER 4. EQUILIBRIUM DIHEDRAL ANGLE OF PB IN CU
64
Table4.1 : Equilibration time, ∆t as a function of temperature for typical intragranular inclusions
(r = 2 µm) and intergranular inclusions (r = 5 µm) calculated from Eq. 4-11.
T [°C]
It is worthwhile noting that the equilibration time scales with the radius of curvature
of the inclusion to the third power. Thus a large inclusion with r = 15 µm will reach
its equilibrium shape in 30 min at 900 °C or after 10 days at 400 °C.
Figure 4-9: Time to equilibration for lead inclusions of different radii embedded in a
copper matrix as a function of the annealing temperature, according to Eq. 4-11.
4.4.2 γ SL isotropy
D
10 -9 [m 2 /s]
γ SL [J/m 2 ]
C Cu . ΩCu
[at.%]
∆t
r = 2 µm
∆t
r = 5 µm
350 0.18 0.437 0.3 ~ 1 h ~ 16 hr
400 0.31 0.437 0.4 ~ 30 min ~ 8 hr
600 1.63 0.428 2.6 ~ 1 min ~ 20 min
800 4.56 0.385 10.0 ~ 10 s ~ 2 min
900 6.62 0.340 20.7 ~ 3 s ~ 1 min
955 8.08 0.290 34.7 ~ 2 s ~ 30 s
Equilibration time: ∆ t
[s]
10 6
1000
1
0.001
r=100 nm
Cu-Pb
r=0.2 µm
r=2 µm
r=1 µm
r=5 µm
r=10 µm
400 500 600 700 800 900
Temperature [°C]
1 week
1 day
1 min
Observation of intragranular inclusions confirms that the solid-liquid interfacial
energy is isotropic in this system: intragranular inclusions are spherical, save for
isolated facets that form within limited solid angles at low temperature. Faceted
1 hr
1 s

4.4 RESULTS
inclusions were observed in samples heat-treated at 400 °C, the lowest temperature
used, Fig. 4-10.
Figure 4-10: SEM image of a dissolved intragranular inclusion after heat treatment at
400 °C. Three visible facets are indicated by arrows on the figure.
At 930 °C, a mean deviation to sphericity of 3.2% was measured out of 16’000 data
points from a single spherical inclusion envelope, Fig. 4-11.
Figure 4-11: (a) 3D digital reconstruction of the solid-liquid interface of an
intragranular inclusion, and (b) data points with the fitting sphere. The Cu-1Pb sample
was heat treated at 930 °C.
4.4.3 Measured dihedral angle
Using the present dissolution method, the dihedral angle φ of several inclusions was
measured in specimens equilibrated during the second step of heat-treatment at
temperatures ranging from 400 to 970 °C; individual results are plotted in Fig. 4-12.
This plot also gives results for the measurement of φ using many inclusions
according to the ”classical” statistical method, as well as data (also gathered using the
(b)
65

CHAPTER 4. EQUILIBRIUM DIHEDRAL ANGLE OF PB IN CU
66
”classical” statistical method) for this system from two references in the literature,
namely the work of Ikeuye and Smith who performed about 250 measurements with
an optical microscope along a longitudinal cut of a Cu-1Pb alloy annealed in a
reducing atmosphere [22], and Eustathopoulos et al. who made 100 measurements
with similar experimental conditions on a high-purity Cu-10Pb alloy [13].
Dihedral angle [ ]
120
100
80
60
40
20
Ikeuye & Smith 49
Eustathopoulos 74
3-D method, this work
2-D "classical" method, this work
0
300 400 500 600 700 800 900 1000
Temperature [ C]
Cu - 1Pb (5N)
Figure 4-12: Solid/liquid dihedral angle, φ(T), as a function of heat treatment
temperature for the Cu-Pb system. Our measurements, circular symbols, are compared
with literature data [13, 22]. Open dotted circles are measurements on the alloy of this
work using the classical statistical method. Solid circles are measured dihedral angles
of individual single inclusions using the method presented here.
It is seen that the dihedral angle measured on different inclusions with the method
presented here varies significantly from inclusion to inclusion at each temperature.
Adjacent intergranular inclusions located on a single planar grain boundary are, on
the other hand, characterized by the same dihedral angle, Fig. 4-13.

4.4 RESULTS
Figure 4-13: SEM micrographs displaying two sets of neighbouring dissolved
inclusions situated on a planar grain boundary. The dihedral angle measured by 3D
reconstruction of each inclusion is indicated on the figure next to the inclusion.
Fig. 4-14 shows dihedral angle values measured from samples treated in flowing Ar-
10H 2 compared to φ values obtained after our vacuum heat treatments detailed in
Section 4.1.2.
The two samples heat treated under flowing hydrogenated argon display lower φ
values with similar variations from inclusion to inclusion at each temperature.
Average φ values are still larger than those reported in the literature [13, 22].
Dihedral angle φ [°]
120
100
80
60
40
20
2-D classical method 3-D method
.
.
Cu - 1Pb (5N)
literature data
vacuum treatment
vacuum treatment Ar-10H2 treatment .
0
300 400 500 600 700 800 900 1000
Temperature [°C]
Figure 4-14: Solid/liquid dihedral angle, φ(T), as a function of heat treatment
temperature for the Cu-Pb system. Measurements from the two samples heat treated
under flowing Ar-10H 2 , open squares, are compared with those from vacuum treatments
and literature data [13, 22] presented in Fig. 4-12.
.
.
.
67

CHAPTER 4. EQUILIBRIUM DIHEDRAL ANGLE OF PB IN CU
68
4.5 Discussion
4.5.1 Inclusion equilibration kinetics
The kinetics of morphological changes driven by capillarity were extensively studied
by Mullins. Most geometries were treated comprising grain boundary grooves and
spherical cavities [14, 191]. Despite the simplicity of its derivation, Eq. 4-11 is in
complete accordance with relaxation times reported in [191] for spherical inclusions.
The present simple estimation is thus validated.
• For nanometric particles of lead in Al or Cu slightly above their melting point, the
corresponding equilibration times are relatively short: around one milisecond for
particles 10 nm in radius, around one second for particles 100 nm in radius; at higher
temperatures they are shorter still, Fig. 4-15. Shape equilibration of such particles in
a hot-stage electron microscope is thus rapid [19, 185]. We note in passing that this
reinforces the interpretation offered by Gabrisch et al. for particle shape hysteresis
observed with very small lead inclusions in Al upon thermal cycling as being due to
interface kinetics (as opposed to diffusion) [19].
Equilibration time: ∆ t [s]
10 6
1000
1
0.001
r=1 µm
r=100 nm
r=10 nm
r=0.2 µm
r=5 µm
r=2 µm
350 400 450 500 550 600 650
Temperature [°C]
r=10 µm
Al-Pb
1 week
1 day
1 min
Figure 4-15: Time to equilibration for lead inclusions of different radii embedded in an
aluminum matrix as a function of the annealing temperature, according to Eq. 4-11.
Thermodynamical data are from [192, 193].
1 hr
1 s

4.5 DISCUSSION
• In as-cast melt-spun ribbons of alloys such as Al-Pb or Cu-Pb, cooling rates during
solidification are typically on the order of 10 3 to 10 6 K/s, depending on ribbon size,
and also on the moment of ribbon detachment from the wheel, something that is
likely to occur well before final solidification of liquid inclusions in deep monotectic
systems such as Al/Pb. Freezing ranges being at most a few hundred K, the time
available for equilibration of the liquid inclusion shape is then on the order of 10 -5 to
10 -1 s within the ribbons: depending on inclusion size and the precise value of the
cooling rate, inclusions 10 to 100 nm wide may or may not have had time to reach
their equilibrium shape right before they solidify.
• In chill-cast samples, which typically solidify in a matter of several seconds, there is
clearly no time for inclusions larger than 1 µm to equilibrate before they solidify,
Fig. 4-15. This explains why lead particles in chill-cast samples Al/Pb of Ref. [185],
which were above 1 µm in diameter, were only partially faceted in the as-cast alloy,
and became more faceted after heat-treatment for 2’000 s at 350 °C (a time sufficient
to equilibrate particles on the order of one micrometre wide).
A size dependence has been reported for the shape of inclusions [194]. After
prolonged anneals followed by a water quench, namely 100 h at 410 °C or 610 °C,
inclusions below a critical diameter of about 0.4 µm were found to be faceted . On
the other hand, larger inclusions were found to be spherical, as were Pb inclusions
less than 200 nm in size observed in situ at 500 °C in a hot-stage TEM [19]. The
cooling procedure of McCormick may thus not have been rapid enough to freeze the
high temperature equilibrium shape of such micrometric (r = 0,2 µm) inclusions.
Namely, considering the Al diffusion coefficient in liquid Pb, as well as the solid/
liquid interfacial energy from [193], we estimate the time to equilibration for such
inclusions as being in the order of 1 and 0.1 s, at 450 and 550 °C respectively, Fig. 4-
15.
The time of spheroidization, τ, of elongated Pb inclusions in swaged Al-5 wt.% Pb
were found to follow an r 3 law, indicating a process controlled by volume diffusion.
This was observed both by X-ray microphotography on micrometric inclusions
69

CHAPTER 4. EQUILIBRIUM DIHEDRAL ANGLE OF PB IN CU
70
(Ø 25 - 75 µm) after interrupted anneals, Fig. 4-16, and in situ using hot-stage TEM
on smaller inclusions, 0.1 to 1 µm in size [193, 195].
0 min 855 min 1965 min
Figure 4-16: Series of micrographs showing the change in shape of Pb inclusions in Al
as a function of time at 625 °C [193].
The time of spheroidization was found to depend on the equilibrium solubility, the
diffusion coefficient, the solid/liquid interfacial energy, and the temperature. This is
consistent with Eq. 4-11.
3
kBT⋅r τ ∝ 0 2
D⋅CCu ⋅γSL ⋅ΩCu
Eq.4-19
Moreover the proportional constant that fits their experimental data is about 1/3. This
value compares fairly well with the factor 4/26.6 in Eq. 4-11.
The shape of micrometric inclusions is not influenced by misfit strains. This is unlike
very small inclusions where, as shown in the Al/Pb system, “magic size” effects are
observed below a diameter around 35 nm [17]. With inclusions larger than this, the
relative importance of strain energy increases in the purely elastic regime, but so does
the ease of misfit dislocation nucleation in the matrix [196]. Strain energy effects are
thus expected to be minimized with larger inclusions.
As an overall conclusion, the equilibrium shape of submicronic inclusions must be
observed at elevated temperature during in-situ heating in the TEM. Larger micron-
sized inclusions, on the other hand, can be observed at room temperature after rapid
cooling from a higher annealing temperature.

4.5 DISCUSSION
Focusing now on the Pb/Cu system we find that, for particles 5 µm in radius, the time
for equilibration is around 8 hours at 400 °C and around one minute at 900°C,
Table 4.1 and Fig. 4-9. This agrees well with our observations that, for reproducible
shape equilibration of Pb inclusions roughly 10 µm wide in Cu, heat-treatment times
of 60 h at 400 °C or 1 h at 900 °C are appropriate. After such a heat-treatment, if
samples are quenched and reach room-temperature in a few seconds, there is no time
for the shape of such large liquid inclusions to change significantly. As a
consequence, interface geometries and dihedral angles measured using the technique
described above correspond indeed to the capillary equilibrium shape of the inclusion
at the heat-treatment temperature.
4.5.2 Solid/liquid interfacial energy
Apart from facets that appear around a few crystallographic directions at relatively
low temperature (400 °C, Fig. 4-10), the solid Cu-liquid Pb interfacial energy is
found to be isotropic in the temperature range explored. The small measured
deviation of intragranular inclusions from a perfectly spherical shape can easily be
attributed to roughness along the surface of the dissolved inclusions, and/or to error
in the measurement and digital reconstruction of the interface.
This agrees with the literature. It is known that Al/Pb and Cu/Pb systems are
thermodynamically quite similar [185, 186, 197, 198]. In the former, more
extensively characterized system, when the inclusions are liquid, partial faceting
occurs along {111} planes below a temperature that increases with particle diameter
in the range of 30 to 300 nm, approaching 500 °C for larger particles [19, 185].
Above 500 °C, the interfacial energy is also documented to become isotropic [185].
These observations for Al/Pb are fully consistent with present observations on larger
lead inclusions in copper.
71

CHAPTER 4. EQUILIBRIUM DIHEDRAL ANGLE OF PB IN CU
72
4.5.3 Dihedral angle measurement
The method presented here yields measurements of true (3D) dihedral angles for
individual inclusions that are significantly larger (10 µm) than those typically used in
transmission electron microscopy (≤ 300 nm). Size effects, such as a size-dependence
in particle shape or roughening temperature, which are characteristic of submicron
inclusions and become exacerbated at very small sizes, are thus absent here. The
triple line energy, which can intervene in nanometric inclusions, is also safely
neglected.
The dihedral angle value obtained with the present method results from an
extrapolation of overall smooth inclusion interfaces to the triple line, essentially as it
is done in the sessile drop technique for contact angles. As such, the measurement
yields a ”macroscopic” angle φ that reflects overall energetic optimization of the
inclusion shape – as implicit in the derivation of a capillary equilibrium equation
such as Eq. 4-12. This implies for example that the effect of grain boundary facets,
which are important at the atomic length scale and affect the local dihedral angle
where they meet the inclusion apex, are neither important nor captured. To resolve
details of the inclusions’ shape at the atomic scale, the use of scanning tunneling
microscopy, STM, would be needed [199].
The reproducibility of the method is apparent in analysis of several inclusions located
along the same grain boundary, Fig. 4-13: φ is constant to within two degrees across
several inclusions. Compared with ”statistical” methods, the present measurements
are thus quite precise. We note in passing that φ values in Fig. 4-12 are near 90°,
which as mentioned above is deemed the more difficult value for measurement of φ
using ”classical” statistical methods. When φ is very low, on the other hand, the
present method is less convenient because the stereoscopic reconstruction becomes
challenging if the inclusion envelopes feature steep and narrow channel-like walls.
4.5.4 Dihedral angles in Cu-Pb
Variations in the value of φ that we find at each specific temperature using the present
method do not reflect experimental error since adjacent intergranular inclusions

4.5 DISCUSSION
display the same angle. The reason is rather that the grain boundary energy is far
more anisotropic than the solid-liquid interfacial energy, in accordance with Clarke
and Gees’ expectations [163]. If we consider the measurements made on the
specimen annealed at 930 °C (which shows the largest degree of variation in
measured φ), the angles vary between 64 and 103°. From Eq. 4-12, this implies a
ratio of 4/3 between the corresponding extreme values of grain boundary energy, γ gb .
Such a range of variation of high-angle grain boundary energy in copper is consistent
with the literature, including both experimental data and results from atomistic
computer simulations [149, 200].
We note that such variations in φ as a result of high-angle grain boundary energy
anisotropy have been noted before. Protsenko et al. observed a distribution in groove
depths in polycrystalline nickel wetted by silver at 1040 °C and attributed this to the
anisotropy of the grain boundary energy [201]. Disregarding any ”special” low
energy grain boundary, a ~7/5 ratio between the higher and lower grain boundary
energies in nickel (another fcc metal) can be estimated from their work.
The disagreement between dihedral angle values we measure and the data available
in the literature for Cu-Pb [13, 22] does not seem to arise from the method used: our
”2D classical” method measurements yield values that are in good agreement with
the average of our ”3D” measurements, Fig. 4-12, in accordance with De Hoffs’
mathematical treatment [12]. The disagreement therefore must arise from the
presence of grain boundary contaminants in our alloy compared to the higher-purity
samples of Refs. [13, 22].
Results from the heat-treatments performed under flowing hydrogenated argon are in
accordance with this conclusion. Indeed φ values for purified samples lie on average
between data reported in the literature and values measured after vacuum treatment,
Fig. 4-14. Contaminants such as sulphur or oxygen seem therefore to have been
scavenged at least to some extent by hydrogen in the treatment.
Since solute enrichment factors are reported to be more than one order of magnitude
higher at grain boundaries than at solid-liquid interfaces [202], a segregated element
will reduce γ gb more significantly than γ SL , and in turn increase the dihedral angle φ,
Eq. 4-12. Waterhouse and Grubb also provide data documenting an increase in φ due
73

CHAPTER 4. EQUILIBRIUM DIHEDRAL ANGLE OF PB IN CU
74
to grain boundary contamination in Cu-Pb annealed at 650 °C [34]: they report a
value of 93°, which is close to values found here and well above the value of 60°
documented in Refs. [13, 22] for high-purity Cu-Pb. According to Waterhouse, in
reply to Stickels’ comments, the observed increase of φ is due to the presence of
0.08% P added as a desoxidizing agent, the phosphorous segregation thus reducing
the γ gb /2γ SL ratio [35]. We note, however, that oxygen desegregation per se is also
reported to cause a decrease of the dihedral angle of intergranular SiO 2 inclusions in
internally oxidized copper, following a 1000 °C 24 h treatment in vacuum [6, 203].
In one instance, though, it may be that disagreement between our results and
literature data is due to the method used. This is at 400 °C, where inclusions are
facetted. Indeed, facets render the classical method fairly difficult to apply:
”classically” measured values of φ in the literature may therefore be underestimated.
Even in the case of uncontaminated Cu-Pb alloys, lead may segregate at grain
boundaries at such an intermediate temperature, as when it spreads on a free surface
of copper and forms a monolayer ahead of a liquid droplet of lead [177-179]. This
would also produce a decrease of the apparent grain boundary energy, which may
increase the dihedral angle, even in samples of very high purity.
Segregation of dissolved lead at the copper grain boundaries was considered as
negligible by Camel, from an analysis of the evolution of the dihedral angle with
temperature [27]. This is surely true at high temperature. Indeed the evolution of φ
with temperature can be back-calculated, Fig. 4-17, (i) from φ values at specific
(relatively high) temperatures available in the literature [26], (ii) from an assumed
linear decrease of γ gb with temperature, Eq. 4-20 [23], and (iii) from γ SL calculated
according to the regular solution, Eq. 4-17.
γ gb(
T)
γ gb( T) = γ gb(
T ) + T T . ( T C) mJ
T
∂
0 ⋅( − 0)= 620 −04⋅ − 947°
[ ]
∂
Eq.4-20

Dihedral angle φ [°]
4.5 DISCUSSION
Figure 4-17: Evolution of φ back calculated from literature data [26] (squares), and the
estimated evolution of γ SL and γ gb for the Cu-Pb system. Ikeuyes’ data [22] are added
but were not considered for the calculations.
Back-calculated φ values drop to zero at about 940 °C; this reflects the marked
C Cu
120
100
80
60
40
20
back-calculated φ evolution
literature data: Coudurier 77
literature data: Ikeuye 49
0
300 400 500 600 700 800 900 1000
Temperature [°C]
liquid
increase of at high temperature, Fig. 4-1. On the other hand, below 600 °C,
back-calculated φ values are governed by the estimated evolution of γ gb , which
causes φ to decrease. At these temperatures, the predicted evolution is thus in clear
disagreement with both our measured φ values, and those of Ikeuye [22], Fig. 4-18.
It is therefore possible that, below 600°C, Pb segregation to the Cu grain boundaries
induces a γ gb reduction that increases φ above predictions of Eq. 4-20.
75

CHAPTER 4. EQUILIBRIUM DIHEDRAL ANGLE OF PB IN CU
76
Dihedral angle φ [°]
120
100
80
60
40
20
literature data
back-calculated φ
Cu - 1Pb (5N)
0
300 400 500 600 700 800 900 1000
Temperature [°C]
our data
vacuum treatment
Ar-10H 2 treatment
Figure 4-18: Solid/liquid dihedral angle, φ(T), as a function of heat treatment
temperature for the Cu-Pb system. Back-calculated φ values (grain boundary
segregation is not taken into account), solid line, are compared with literature data [13,
22] and our data presented in Fig. 4-14.

4.6 Conclusion
4.6 CONCLUSION
A new method is presented for the determination of the dihedral angle of individual
intergranular liquid inclusions; it is based on quantitative scanning electron
microscopic analysis of metallographic surfaces along which quenched inclusions
have been dissolved. The angle is derived from a mathematical fitting of the solid/
liquid interface geometry around individual inclusions. It reflects the value dictated
by global energy minimization of the inclusion shape under capillary forces, as
expressed by Eq. 4-12. As such, this method parallels the sessile drop method for
contact angle measurement, where the drop shape is used to deduce ”macroscopic”
contact angles.
Compared with classical methods of dihedral angle measurement, no statistical
treatment of the data is needed. The method also overcomes some shortcomings of
other techniques, such as the direct measurement of dihedral angles at grain boundary
grooves or in transmission electron microscopy.
We show that the Cu/Pb solid-liquid interfacial energy is isotropic above 400 °C,
while it exhibits a slight tendency for faceting at 400 °C. Even in the presence of
facets, the method presented here leads to reliable measurements of the true dihedral
angle φ.
For a specific temperature, we show that φ is not unique; this reflects the fact that
high-angle grain boundary energies vary in copper.
77

CHAPTER 4. EQUILIBRIUM DIHEDRAL ANGLE OF PB IN CU
78

Chapter 5
5 Influence of stress on the shape of an embedded liquid inclusion
In this chapter, the combined effect of stress and capillarity on the equilibrium shape
of a liquid inclusion is studied. This shape is determined by global minimization of
the elastic strain energy, quantified by means of the Eshelby method, and interfacial
energies.
Spheroidal intragranular inclusions are first considered. Predicted shapes are
confronted with actual 3-D shapes imaged and quantified following the procedure
described in Chapter 4. The shape of intergranular inclusions is also studied. It is
found that the apparent global dihedral angle decreases under an applied remote
stress. This is in qualitative accordance with our theoretical predictions; quantitative
agreement is, however, more tenuous, which is not unexpected given simplifications
made in analysis.
79

CHAPTER 5. INFLUENCE OF STRESS ON THE SHAPE OF AN EMBEDDED LIQUID INCLUSION
80
5.1 Theoretical predictions
5.1.1 The Eshelby method
The local elastic stress and strain of an inclusion embedded in a linear elastic
stressed matrix can be approximated by the Eshelby calculation, which assumes that
the inclusion is ellipsoidal in shape, Fig. 5-1. It is demonstrated that in such an
inclusion, the stress and elastic strain are homogeneous [37]. The stress field outside
the inclusion can also be rigourously calculated by adding the stress fields of two
simplified situations [38]. The interaction strain energy, ∆W, due to the presence of
the inclusion can also be quantified. The procedure is briefly summarized in what
follows.
Figure 5-1: Oblate spheroidal inclusion with vertical rotation axis, the small and large
semi-axis are b and a respectively. The direction of the applied uniaxial stress is parallel
to the vertical axis.
We use the Eshelby formalism, which is didactically described in [204]. The effect of
dissimilar elastic properties between the inclusion and the matrix can be accounted
for by an equivalent transformation strain, ε* (or eigenstrain). The latter induces (in
the virtual case of similar elastic properties between the ”ghost” homogeneous
inclusion of same shape and the matrix) the same internal stresses in the matrix as ε,
the elastic strain whithin the actual (inhomogeneous) inclusion. The eigenstrain is
deduced from the elastic properties of the matrix and the inclusion, the applied
remote stress, and the Eshelby tensor, S, defined by the inclusion shape and the
Poisson ratio of the matrix:
b
a
σapplied

5.1 THEORETICAL PREDICTIONS
∗
ε =
−1
∗ ( C−C )⋅S−C ∗ ( C
−1
C)⋅ C ⋅σapp
Eq.5-1
where C and C* are the rigidity tensors of the matrix and inclusion, respectively. The
actual elastic stress and strain within the inclusion are respectively ε and σ :
and
Eq.5-2
σ = C ⋅ε
Eq.5-3
It is shown in [38] that, for an ellipsoidal inclusion, the interaction energy ∆W can
*
easily be computed from the eigenstrain, the inclusion volume, V, and the applied
remote stress, σ app :
1
∗
∆W =− V⋅σapp ⋅ε
2
Eq.5-4
This interaction energy is counted as negative since it corresponds to work performed
by an external load.
5.1.2 Inclusion shape
[ ] ⋅ −
−1
* 0 *
ε = C ⋅ σ + S⋅ ε = ε + S⋅ε
app
( )
In order to determine the equilibrium shape of an inclusion, one can consider the
overall energy associated with the inclusion. Energy minimization as a function of
shape then allows to determine the equilibrium situation. Two contributions must be
considered: (i) the strain energy ∆W, and (ii) the overall interfacial energy, Γ .
The latter is only dependent on the interfacial energy, γ, and on the inclusion surface
area, A. In the case of a spheroidal inclusion,
A =
4π
c
⋅ ⎛
3 V 3 V 3 ⎛1+
1−c
⋅⎜⋅ ⋅ 1 − c + ⋅V⋅c⋅ln 2
− c c
⎜
1 ⎝ 2
4 ⎝1−
1−c
3 2
where c is the aspect ratio of the inclusion defined as:
2
2
⎞⎞
⎟⎟
⎠⎠
Eq.5-5
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CHAPTER 5. INFLUENCE OF STRESS ON THE SHAPE OF AN EMBEDDED LIQUID INCLUSION
82
b
c =
a
Eq.5-6
Thus we can express Γ, which is counted positive since an increase of the interfacial
area consumes energy:
Γ= γ ⋅A
Eq.5-7
The overall energy due to the presence of the inclusion is thus a function of its
volume and shape, the applied remote stress, the elastic properties of matrix and
inclusion, and the interfacial energy.
We make in what follows the assumption that the inclusion remains spheroidal, and
optimize its shape by minimizing the total excess energy associated with its presence
in a stressed solid as a function of its aspect ratio at constant volume. This approach
is classical for this type of problem [36, 39, 40] as it eases the problem considerably:
the calculation is reduced to minimization of an analytical scalar function of a single
scalar variable. It must, however, be borne in mind that this assumption, coupled with
the underlying description of the surrounding matrix as a linear elastic solid (which is
obviously not true for copper near an inclusion at elevated temperature), is a
significant simplification of what is, in fact, a fully three-dimensional free boundary
problem of the optimal shape of a liquid inclusion embedded in an elastoviscoplastic
matrix.
At equilibrium, (∆W + Γ) is at a minimum:
with
d( ∆W + Γ)
= 0
dc
Eq.5-8
σ app
∆W = F() c ⋅V⋅ E
Eq.5-9
where F(c) is a function of the elastic properties of both the matrix and the inclusion,
2
and of the shape of the latter. Moreover,

Γ= Gc () ⋅V ⋅
2 3 γ
where G(c) is only a function of the geometry of the inclusion.
5.1 THEORETICAL PREDICTIONS
Thus equilibrium can be described by a single adimensional factor, Λ:
Eq.5-10
dG() c
2 1/ 3
Λ=− dc σ app ⋅V
=
dF() c γ ⋅ E
dc
Eq.5-11
This last parameter describes a global equilibrium at the inclusion length scale. Note
that this adimensional factor is defined here, Eq. 5-11, by taking into account the
volume V of the inclusion to the power one third, whereas Λ defined in Eq. 3-4
considers the initial pore diameter a 0 . Assuming a lenticular shape for the inclusion,
the proportional factor is deduced from Eq. 5-12:
V
a 2
= 2−3cos ⎛ ⎞
cos
2 3 ⎝ 2⎠ 2
+
⎡ π ⎛ φ ⎛ φ⎞⎞⎤
⎢ ⎜
⎝
⎝ ⎠
⎟
⎣
⎠⎥
⎦
1/ 3 0 3
5.1.3 Intergranular inclusion
Eq.5-12
In the case of an intergranular inclusion, the grain boundary energy, γ gb must
additionally be taken into account. Thus the overall interfacial energy due to the
presence of the grain boundary inclusion is as follows:
2
Γ= γ ⋅A−γ gb ⋅π ⋅a
Eq.5-13
Note that the contribution of the grain boundary energy is counted as negative since a
reduction of the grain boundary area restores energy, and
φ
γ gb = 2γ
⋅cos
⎛ ⎞
⎝ 2⎠
Eq.5-14
The actual equilibrium shape of an intergranular inclusion is not ellipsoidal. We
consider two approximate shapes for the calculation of the interfacial area: (i) two
spherical caps intersecting at the grain boundary plane, or (ii) the standard Eshelby
1/ 3
83

CHAPTER 5. INFLUENCE OF STRESS ON THE SHAPE OF AN EMBEDDED LIQUID INCLUSION
84
ellipsoid. In each case the aspect ratio c is defined as the ratio between the small and
large axis, Fig. 5-2.
Figure 5-2: An intergranular inclusion imbedded in a stressed material. The shape of
the inclusion is approximated by two symetric spherical caps (dashed contour) or by an
oblate spheroid.
The first approximation, which implies a constant curvature, matches the actual
shape of the intergranular inclusion under stress-free conditions only. Under stress,
local stresses are not constant along the inclusion/matrix interface; this implies in
turn a non-uniform κ along the liquid/solid interface, Eq. 5-18.
The surface area of the intergranular inclusion is calculated, respectively, as:
or
/
A = V ⋅2π⋅ 1+
c
lenticular
b
( )
2 3 2
Eq.5-15
Aellipsoidal
c
V
c
V
c
c V c
c
c
Eq.5-16
In both cases, (i) & (ii), the equilibrium shape of the intergranular inclusion can be
=
3
4π
⋅
3
2
1 −
⎛ 3
⋅⎜⋅ ⋅
⎝ 2
⎛
2 3 1+ 1 − + ⋅ ⋅ ⋅ln
4
⎜
⎝1−
2
1−
⎞⎞
2 ⎟⎟
1−
⎠⎠
described by the same adimensional factor Λ defined in Eq. 5-11.
a
⎡ 3
⎢
⎣⎢
π ⋅c⋅ 3 + c
2 ( )
applied
grain boundary
The ideal case, where the remote stress direction is normal to one grain boundary
plane along which an inclusion is located, is rarely met, Fig. 5-3 (a). Rotation and tilt
⎤
⎥
⎦⎥
2/ 3
σ

5.1 THEORETICAL PREDICTIONS
angles, respectively ξ and θ illustrated on Fig. 5-3 (b), must be defined in order to
account for actual situations in a material subjected to uniaxial tensile stress.
(a) (b)
Figure 5-3: Schematic representation of an intergranular inclusion imbedded in an
uniaxially stressed material. (a) The grain boundary plane is normal to the remote
stress axis, whereas in (b), its orientation is random.
Accordingly the stress tensor σσ kl considered in the determination of ∆W with the
Eshelby formalism is transformed from the x, y, z system of axes to the x’’, y’’, z’’
axes by successive rotation of ξ about the x axis (to x’, y’, z’ ), and θ about the y’
axis (in the Einstein suffix notation):
σσ = a a σσ
kl ki lj ij
a ij
where is the direction cosine matrix.
Eq.5-17
The shape of the inclusion is again approximated by an oblate spheroid having the
grain boundary plane as symmetric plane.
5.1.4 Equilibration time
σ app
σ app
The equilibrium shape of an inclusion is met once the chemical potential is
homogeneous throughout the whole inclusion. At constant temperature and
σ app
σ app
ξ
θ
85

CHAPTER 5. INFLUENCE OF STRESS ON THE SHAPE OF AN EMBEDDED LIQUID INCLUSION
86
homogeneous chemical composition at the interface, the chemical potential µ [36] is,
quoting [205]:
µκσ ( , )= µ 0 + γΩκ+ Ωw
Eq.5-18
where µ 0 is the equilibrium chemical potential of a planar interface under stress-free
conditions, Ω is the atomic volume, γ the interfacial energy, κ the curvature; w is the
local elastic strain energy. Note that κ is negative, since the copper interface of
interest is concave.
The local driving force for shape equilibration is therefore dependent on both the
local curvature and the local stress. The time for shape equilibration is therefore more
difficult to estimate for a liquid inclusion embedded in a stressed matrix; however,
since the driving force comprises the same capillary term as in the zero-stress
problem, as a first rough estimate, we use Eq. 4-11.

5.2 Experimental procedures
5.2 EXPERIMENTAL PROCEDURES
In order to measure the influence of stress on the shape of an inclusion, interrupted
creep tests were performed on a Cu-Pb alloy at 400 °C (the inclusions are molten at
this temperature). The 3D shape of the resolidified inclusions was observed on
polished microsections and the ”global” dihedral angle was measured with the
method described in Section 4.3.2.
5.2.1 Material
C99 is the industrial alloy used, a leaded copper containing 0.8 to 1.2 wt.% Pb and
0.01 to 0.04 wt.% P added as a desoxidizing agent. This material was provided by our
industrial partner, Swissmetal Boillat, Reconvilier, Switzerland. Processing
comprised (i) billet casting, (ii) hot extrusion, and (iii) room temperature wire
drawing.
A heat treatment at 900 °C for 2 h under primary dynamic vacuum (p ≈ 10 -2 mbar)
was given to the material in order to induce microstructural coarsening. The
temperature was then lowered to 400 °C and kept constant for 60 h in order to allow
the inclusion shape to equilibrate. The material was then water-quenched in order to
keep the shape of the inclusions unaltered.
5.2.2 Interrupted creep tests
Ø 4 mm tensile test specimens, Fig. 5-4, are mounted in a modified hydraulic tensile
test machine (MFL 100 kN), where up to 1000 N dead load (σ ≤ 80 MPa) can be
suddenly applied after a desired hold at temperature under zero load. The heating is
performed with a lamp furnace (Research Inc., Parabolic Clamshell Heater, Model
4068-12-10), allowing rapid heating and temperature stabilization (< 120 s to
400 °C). The temperature is monitored by a Ø 0.2 mm type K thermocouple attached
to the center of the specimen. The deformation is measured by a high temperature
extensometer (MTS 632.53F-30 equiped with Ø 3 mm, 190 mm long Al 2 O 3
extension rods) attached to the specimen gage section.
87

CHAPTER 5. INFLUENCE OF STRESS ON THE SHAPE OF AN EMBEDDED LIQUID INCLUSION
88
M6
Figure 5-4: Tensile specimen geometry. Dimensions are in mm.
The load is applied after a 5 min hold at 400 °C and the temperature is kept constant
for a further 15 min. The furnace is then switched off and rapidly removed, while
forced air is directed to the specimen surface. The specimen is unloaded once its
temperature falls below 50 °C.
Polished longitudinal cuts are then prepared. The visible lead inclusions are
selectively dissolved and the dihedral angle of specific inclusions is measured with
the method described in Section 4.3.2, recording their orientation with respect to the
applied stress axis.
R3
110
30
4
6
15

5.3 Results
5.3.1 Creep curves
5.3 RESULTS
Fig. 5-5 presents creep curves of C99 material subjected to constant loads for 15 min
at 400 °C. Notice the thermal expansion of the specimen during heating and the
successive holding time on the left-hand side of the viewgraph (t

CHAPTER 5. INFLUENCE OF STRESS ON THE SHAPE OF AN EMBEDDED LIQUID INCLUSION
90
modulus of liquid lead, K Pb , was calculated from the tabulated velocity of sound
[206]. Isotropic elastic properties of copper at 400 °C found in [207] were used.
A numerical application for the typical case of a 5 µm diameter lead intragranular
inclusion within uniaxially stressed copper at 400 °C is given in Fig. 5-6 (b). The
interfacial energy is according to Eq. 4-17.
Notice that no equilibrium configuration is predicted by the lower curve above a
specific critical stress. Instability is predicted once the maximum of this curve is
passed.
Λ (c) [-]
5
4
3
2
1
0
2 1/3
σapp
⋅ V
Λ(c) =
γ ⋅ E
0.9
liquid
inclusion
0.7
void
(a) (b)
Figure 5-6: Equilibrium shape of an intragranular inclusion imbedded in a material
subjected to uniaxial remote stress: a) adimensionalized problem, and b) application to
a typical ø5 µm Pb inclusion in our Cu-Pb alloy.
b) Experiments
0.5
K Pb = 27,5 GPa
"K " = 0 GPa
Pb
0.3
aspect ratio: c [-]
0.1
sphere crack
After interrupted creep at 400 °C, dissolved transgranular inclusions were imaged in
the SEM. Extensive faceting was observed. An example is given in Fig. 5-7 with the
corresponding datapoints obtained by the 3D reconstruction. Datapoints are best
fitted by the sphere displayed; black datapoints lie outside the sphere and shaded
σ [MPa]
app
200
150
100
50
0
T = 400 °C
γ SL = 0.432 J/m 2
E Cu = 111 GPa
V = 10 -16
m 3
K Pb = 27,5 GPa
"K " = 0 GPa
Pb
0.9 0.7 0.5 0.3 0.1
sphere crack
aspect ratio: c [-]

5.3 RESULTS
ones lie inside. Note that Fig. 5-7 (b) adjusts to Fig. 5-7 (a) after a 180° rotation about
the horizontal axis.
One can see that the actual shape resembles a prolate spheroid, instead of the
predicted oblate one. Namely the inclusion is elongated in the stress direction, Fig. 5-
7 (a).
(a) (b)
Figure 5-7: Dissolved intragranular inclusion in C99 crept at 400 °C with σ=59 MPa,
the stress axis is vertical: a) SEM image, and b) corresponding datapoints from the
reconstruction of the unfaceted interface. The datapoints are best fitted by the sphere
displayed (stress axis is vertical).
5.3.3 Intergranular inclusion
The predicted shape of an intergranular inclusion is presented in Fig. 5-8. A
comparison is given between a first set of predictions where the inclusion is made of
two spherical caps intersecting at the grain boundary (solid line, the surface area of
the inclusion is calculated according to Eq. 5-15) and a second set of predictions
where the shape of the inclusion is approximated by an oblate ellipsoid (dashed line,
Eq. 5-16).
In both cases, two calculations were conducted:
91

CHAPTER 5. INFLUENCE OF STRESS ON THE SHAPE OF AN EMBEDDED LIQUID INCLUSION
92
(i) one in which the compressibility of the liquid inclusion equalled that of liquid lead
(Bulk modulus K Pb = 27.5 GPa);
(ii) one in which the liquid inclusion bulk modulus is set to zero, i.e., in which the
inclusion behaves, from the mechanical point of view, like a void (K Pb = 0).
Results from the calculation are given in adimensional form in Fig. 5-8 (a), and in
Fig. 5-8 (b) in dimensional form for the case of a 10 µm diameter lead inclusion
located at a grain boundary that is normal to applied uniaxial stress. The horizontal
axis is given in terms of the “apparent macroscopic” dihedral angle of the inclusion,
related to its aspect ratio according to:
2
⎛1
− c ⎞
φ = 2 arccos⎜
2 ⎟
⎝1
+ c
⎠
Eq.5-19
As seen, regardless of the geometrical assumption made in estimating the capillary
energy, there is a clear difference in inclusion behaviour between a liquid inclusion of
finite compressibility and a void:
(i) with zero bulk modulus, there is a maximum in the curve, past which the inclusion
shape is unstable. Past this point, the inclusion is predicted to collapse into a crack;
this result was derived previously [36, 39, 40];
(ii) with a finite bulk modulus, on the other hand, the curve shows an inflexion point
but increases monotonously. The shape instability predicted for a void thus
disappears. It was ascertained by conducting additional calculations that, even with a
very small bulk modulus (K Pb = 1 GPa), the bifurcation point disappears. The
analysis therefore indicates that, under increasing stress, liquid inclusions remain
stable if there is no void nucleation in their midst.

Λ (c) [-]
3
2.5
2
1.5
1
0.5
0
0.6
2 1/3
σapp
⋅ V
Λ(c) =
γ ⋅ E
lenticular
shape
ellipsoidal
shape
0.5
0.4
5.3 RESULTS
(a) (b)
Figure 5-8: Equilibrium shape of an intergranular inclusion imbedded in a material
subjected to an uniaxial remote stress, which is normal to the grain boundary: a)
adimensionalized problem, and b) application to a typical gb inclusion in our Cu-Pb
alloy. Two sets of data are given, (i) the area of the interface is either approximated by
two spherical caps intersecting at the grain boundary (Eq. 5-15), or (ii) it is
approximated by an oblate ellipsoid (Eq. 5-16). For each set of data, the compressibility
of the liquid lead is taken into account (KPb =27.5 GPa) or the inclusion properties are
that of a void (”KPb ”=0 GPa).
a) Experiment
0.3
Apparent (macroscopic) dihedral angles were measured in interrupted (after 15 min)
creep tests at 43, 59, and 75 MPa. An additional creep test performed at 400 °C at a
stress of 83 MPa was interrupted after 10 min since creep had entered its third stage.
After longitudinal cut preparation of this specimen and inclusion dissolution, no
single inclusion displayed a sufficiently regular shape to allow the measurement of
any relevant dihedral angle.
The angle determination was the more difficult to perform, the higher was the remote
stress. This is due to the plastic flow of the material, and to the migration of the grain
boundaries, Fig. 5-9 (a).
K Pb = 27,5 GPa
"K " = 0 GPa
Pb
0.2
aspect ratio: c [-]
0.1
crack
σ [MPa]
app
200
150
100
50
φ 0
0
100
=120°
T = 400 °C
γ SL = 0.432 J/m 2
E Cu = 111 GPa
V = 10 -16 m 3
80
K Pb = 27,5 GPa
60
"K " = 0 GPa
Pb
40
20
apparent dihedral angle: φ [°]
crack
93

CHAPTER 5. INFLUENCE OF STRESS ON THE SHAPE OF AN EMBEDDED LIQUID INCLUSION
94
(a) (b)
Figure 5-9: Intergranular inclusion dissolved after an interrupted creep test
performed at 400 °C under 59 MPa. Material is C99. (a) The grain boundary is
serrated, and the former solid/liquid interface is extensively distorted. This did not
allow the measurement of φ following the method described in Chapter 4. On the other
hand (b) a relevant dihedral angle could be measured (φ = 79°). The dashed contours
correspond to the intersection of the polished plane with the two calculated spheres
allowing the measurement of φ.
Nevertheless some inclusions displayed a sufficiently regular shape to allow the
measurement of a relevant apparent dihedral angle, Fig. 5-9 (b). Resulting data
values show a significant spread, as do data from stress-free samples, Fig. 4-14;
however it nevertheless emerges from the data that φ is sensitive to the applied load.
Decreasing values are generally measured for increasing load, Fig. 5-10 and
Appendix 10.4.
measured dihedral angle: φ [°]
130
120
110
100
90
80
70
60
50
C99
400 °C
0 20 40 60 80 100
applied remote stress: σapp [MPa]
Figure 5-10: Measured φ values after interrupted creep test as a function of the applied
remote stress. Each datapoint corresponds to a measurement on a single inclusion
according to the procedure described in Section 4.3.2.

5.3 RESULTS
Let us first assume that there is a single zero-stress dihedral angle φ o . To account for
the influence of inclusion orientation with respect to stress, the theoretical Λ(φ)
curves are drawn for the corresponding specific grain boundary ξ and θ values,
Fig. 5-3 for each inclusion from Appendix 10.4. An illustration of this is given in
Fig. 5-11 for the specific inclusion marked by an asterisk in Appendix 10.4. As
shown in Fig. 5-11, the predicted φ values are deduced from the two theoretical
curves and Λ, for the case where the inclusion properties are those of a void
(K Pb = 0). Predictions for a finite K Pb show less agreement.
[-]
Λ
3
2.5
2
1.5
1
0.5
φ 0
measured φ
Λ = 0.587
0
100 80 60 40 20
=120°
apparent dihedral angle: φ [°]
crack
Figure 5-11: Determination of the predicted φ values for an inclusion located on a
grain boundary characterized by ξ = 37° and θ = 25° (i.e. changing the stress tensor
accordingly, Eq. 5-17). Here, the experimentally observed inclusion of interest is
characterized by Λ = 0.587 and φ = 69°.
This way, actual measured dihedral angles, Appendix 10.4, can be compared to
predicted φ values for each inclusion, Fig. 5-12.
"K " = 0 GPa
Pb
lenticular
shape
ellipsoidal
shape
95

CHAPTER 5. INFLUENCE OF STRESS ON THE SHAPE OF AN EMBEDDED LIQUID INCLUSION
96
Figure 5-12: Predicted φ values compared to measured values. The inclusion properties
are taken as those of a void (”K Pb ”=0 GPa). Both lenticular and ellipsoidal shape are
considered.
As is clear from Fig. 5-12, experimental data show much greater spread than
predicted values.
predicted dihedral angle: φ [°]
130
120
110
100
90
σ app = 0
80
"K " = 0 GPa
Pb
70
lenticular
shape
60
50
slope = 1
ellipsoidal
shape
50 60 70 80 90 100 110 120 130
measured dihedral angle: φ [°]

5.4 Discussion
5.4.1 Theoretical predictions
5.4 DISCUSSION
Our theoretical predictions can be compared to those of the literature for
intragranular [36], and for intergranular voids [39, 40]. This can be done by
comparing the respective predicted critical adimensional parameter recalculated in
Table 1 according to our definition, Eq. 5-11.
Eq. 5-11
Table 1 : Predicted critical parameter Λ c according to Eq. 5-11 for a void imbedded in a
uniaxially stressed material.
intragranular inclusion intergranular inclusion φ 0 =120°
Sun et al.
[36]
ellipsoidal
shape
2 1/ 3
σ app ⋅
Λ=
γ ⋅
V
E
our analysis
Fig. 5-6 (a)
ellipsoidal
shape
Raj [40]
Wang et al.
[39]
penny shape ellipsoidal
shape
lenticular
shape
our analysis
Fig. 5-8 (a)
ellipsoidal
shape
0.73 1.07 0.62 0.38 0.61 0.44
Our theoretical predictions are in good accordance with the literature. Differences in
Λ c may be due to different assumptions considered in the analysis. As an illustration,
for the Poisson’s ratio, we used ν = 0.355 (tabulated value for pure copper at 400 °C
[207]), whereas ν = 1/3 was taken in [36] for their numerical calculations. It must
also be emphasized that Λ is numerically calculated by taking the ratio between the
first derivative of two complicated functions, Eq. 5-11. It is moreover clear from
Figs. 5-13 and 5-14 below, that the shape approximations defined in Section 5.1.3
induce some degree of error. In particular, it is seen that, when Γ is estimated using
Eq. 5-16, i.e., by assimilating the inclusion to the same spheroid used in calculating
the mechanical energy, the predicted shape at low stress is inconsistent with the
assumed stress-free dihedral angle φ o . This is clearly because Eq. 5-15, (lenticular
shape) appropriate for such an inclusion at zero stress, is required to obtain a global
97

CHAPTER 5. INFLUENCE OF STRESS ON THE SHAPE OF AN EMBEDDED LIQUID INCLUSION
98
inclusion shape that satisfies local capillary equilibrium at the triple line. This
advantage of Eq. 5-15 instead of Eq. 5-16 used by Wang et al. [39] motivated the
present modification of the analysis.
5.4.2 Measured inclusion shape
Theoretical predictions discussed above describe equilibrium situations. Fig. 5-7
suggests, however, that the rate of shape equilibration is lower than the rate of
viscous flow of our creeping material. Indeed, the intragranular inclusion in Fig. 5-7
is found to be slightly elongated in the applied tensile stress direction, whereas it
should be a prolate ellipsoid shape oriented normal to the stress. Since shape
equilibration is governed by diffusion, it is clear that equilibration requires lower
strain rates and smaller inclusions, as observed in [208]. Notice however that
relatively coarse inclusions, more than 5 µm wide, are needed for accurate dihedral
angle measurement with the method presented in Section 4.3.2.
Estimated times for equilibration in the absence of applied stress Eq. 4-11 are, at
400°C, 8 hours for an inclusion 10 µm in diameter, and one hour for an inclusion 5
µm in diameter. Although (as mentioned in Section 5.1.4), equilibration times thus
estimated are not strictly valid in the presence of stress (because stress changes
concentration gradients across the inclusion), the indication that shape equilibration
is probably incomplete in the present samples is consistent with Fig. 5-7. Apparent
dihedral angles measured here may, therefore, be somewhat superior than the
equilibration value, φ(σ), corresponding to the applied stress, for two reasons:
(i) tensile stress lowers φ, Figs. 5-6 and 5-8, thus measured φ values probably lie
between φ o and φ(σ)
(ii) creep will open up pores that lie more or less perpendicular to the stress axis.
Another difficulty is that the frozen microstructure of the creeping C99 material is
not at equilibrium. Indeed grain boundaries are serrated, indicating grain boundary
movement, Fig. 5-9 (a). Our method for dihedral angle measurement is based on the
assumption that the solid/liquid interfaces are spherical. This is not the case in the
presence of stress, since the inclusion equilibrium shape can not display a constant

5.4 DISCUSSION
curvature κ when w is not constant along the whole interface, Eq. 5-18. Nevertheless,
relevant ”global” dihedral angle measurements could be performed on selected
inclusions, Fig. 5-9 (b).
5.4.3 Comparison of theory with experiment
It must first be recalled that our theoretical predictions are based on several
assumptions.
(i) The interaction energy ∆W is calculated with the Eshelby formalism. It is
therefore assumed that the behaviour of both the matrix and the inclusion are linear
elastic. Moreover, a spheroidal shape is assumed for the inclusion. It is clear that our
lenticular lead inclusions are not spheroidal; nor does the creeping copper matrix
behave as a linear elastic solid.
(ii) The shape approximation for the inclusion is also shown to have indeed a
noticable influence on the predicted curves, see e.g. the solid and dashed lines on
Fig. 5-8.
(iii) It was concluded in Chapter 4 that, at a specific temperature, the dihedral angle is
not unique, reflecting mainly the anisotropy of γ gb . In unstressed C99 equilibrated at
400 °C, φ is not unique. It was indeed found to vary between 102 and 125°,
Appendix 10.4. In our theoretical predictions however, we considered a unique value
for the ratio of γ gb /γ SL at 400 °C.
Experimental data of the present study are not the only that document a decrease of
dihedral angle φ of liquid inclusions under applied stress. In stressed leaded nickel,
Stickels et al. reported a similar decrease of φ [32, 33]. In this study, reported
dihedral angles are averages measured as indicated in Section 4.3.1 from two-
dimensional metallographic cuts, taking the average value of 500 measurements for
each data point. The data of Stickels are plotted in Fig. 5-13. Also plotted in that
figure are curves predicted by the present analysis, considering the elastic properties
of nickel at 370 °C [207], the interfacial energy of the Cu-Pb system, and V = 10- 16
m3 corresponding to a 10 µm wide inclusion, similarly as for Fig. 5-8 (b). This
latter value was guessed, since no indication of the lead particle size was given
99

CHAPTER 5. INFLUENCE OF STRESS ON THE SHAPE OF AN EMBEDDED LIQUID INCLUSION
100
(smaller inclusions render the apparent dihedral angle measurement very difficult
under an optical microscope with a 2000 x magnification, as performed in the sixties
by Stickels) [32, 33].
σ [MPa]
app
350
300
250
200
150
100
50
φ 0
0
=51°
40
T = 370 °C
γ SL = 0.432 J/m 2
E Ni = 195 GPa
V = 10 -16 m 3
Figure 5-13: Apparent dihedral angle data from the literature ploted against applied
hydrostatic or uniaxial stress on Ni-2Pb samples at 370 °C; data are from [32, 33].
Lines correspond to our theoretical predictions considering (upper curves) an
intergranular Pb liquid inclusion embedded in hydrostatically stressed Ni, and (lower
curves) a grain boundary void in an uniaxially stressed matrix.
It is seen that the shape of curves traced by the two sets of data points reproduce
relatively well the overall shape of the two curve sets: data for hydrostatic
compression tests reproduce curves derived assuming that the inclusion has the bulk
modulus of liquid lead at that temperature, while uniaxial (tensile and compressive)
test data points delineate a curve that is closer, in shape and value, to the curve
derived under the assumption that the liquid inclusion behaves as a void. All else
being equal, a far higher isostatic compressive stress (e.g., 344 MPa) is needed to
lower the dihedral angle than is the case for uniaxial stress (the corresponding
measured – tensile – stress is 14 MPa, see Fig. 5-13).
30
Ni-Pb
20
K = 27,5 GPa
Pb
σ
hydrostatic app
"K " = 0 GPa
Pb
σ
uniaxial app
10
apparent dihedral angle: φ [°]
crack
theoretical predictions
lenticular shape
ellipsoidal shape
experimental data
hydrostatic stress
compressive
uniaxial stress
tensile & compressive

5.4 DISCUSSION
This makes good sense: hydrostatic compression does not cause global deformation
of the material. Under uniaxial tension or compression, on the other hand, the
material deforms (by creep and/or plasticity), such that the various void nucleation
mechanisms that have been invoked to explain creep fracture can operate to nucleate
a void in the inclusion [209].
Such pore nucleation was indeed also reported in the literature: (i) pores were
namely found to grow to a micrometre size in a T91 steel after prolonged annealing
under stress in liquid lead [210]. (ii) Pore nucleation and growth at liquid Bi
inclusions was also reported for an Al-0.23Ti-1.5Bi alloy strained at elevated
temperature [211].
The influence of an applied tensile stress on φ was also documented in the literature
for a Cu-1Pb-0.08P alloy, close to our C99 material (which contains 0.8 to
1.2 wt.% Pb and 0.01 to 0.04 wt.% P) [34, 35], Fig. 3-12. Again experimental φ
values compare relatively well with our predictions assuming the presence of
voiding in the inclusion (”K Pb ” = 0 GPa), Fig. 5-14. Elastic properties were
extrapolated from [207], γ SL is according to Eq. 4-17, and the inclusion volume V
was estimated from a metallographic plate in [34].
101

CHAPTER 5. INFLUENCE OF STRESS ON THE SHAPE OF AN EMBEDDED LIQUID INCLUSION
102
σ [MPa]
app
100
80
60
40
20
0
φ 0
=93°
Cu-Pb
T = 650 °C
γ SL = 0.422 J/m 2
E Cu = 100 GPa
V = 10 -16 m 3
80
Figure 5-14: Apparent dihedral angle data from the literature ploted against applied
uniaxial stress on Cu-1Pb-0.08P samples at 650 °C, data are from [34, 35]. Lines
correspond to our theoretical predictions considering a grain boundary void in a
stressed matrix.
In both comparisons of theoretical predictions with data from the literature, the
quantitative agreement is far from perfect, but nonetheless acceptable given the many
assumptions made in the derivation (that the matrix remains elastic, for example).
Turning now to experimental data from the present study, comparison is somewhat
more difficult because our data are more specific: each inclusion that was
characterized has a known size, and also a specific (variable but measured)
orientation with respect to the applied stress axis.
This second factor creates the complication that a single parameter (Λ) no longer
describes entirely the mechanical and capillary state of every inclusion: the stress
state assumed in the derivation is no longer a simple tensile stress normal to the grain
boundary; rather, it is multiaxial, given by Eq. 5-17.
70
60
50
To account for this, theoretical Λ(φ) curves were drawn for each inclusion orientation
(given in Appendix 10.4) and compared with the data; an example is given in Fig. 5-
11. As seen the curve reaches higher Λ values than that of Fig. 5-8 (a): this obeys
intuition, since a higher stress will be required to lower the dihedral angle of an
40
theoretical predictions
30
lenticular shape
ellipsoidal shape
experimental data
apparent dihedral angle: φ [°]
"K " = 0 GPa
Pb
uniaxial stress
20
10
crack

5.4 DISCUSSION
inclusion that is oriented along a grain boundary that is inclined with respect to the
stress axis. Again, we stress the key simplification made in the present derivation,
namely that the inclusion shape evolves along a single family of (spheroidal/
lenticular) shapes: under shear, as experienced along grain boundaries inclined with
respect to the stress axis, asymmetric (sheared) inclusion geometries may be closer to
equilibrium.
These theoretical curves were drawn under a second assumption, namely that the
stress-free dihedral angle for the inclusion is known and constant : we have taken
φ ο = 120°. As seen in the previous chapter, dihedral angles in fact vary strongly
depending on the specifics of the grain boundary along which they lie.
Comparison of predicted and measured dihedral angle values is given in Fig. 5-12:
perfect agreement would cause the data points to lie along the diagonal of the plot. As
seen, for both models (which differ by details of the assumed inclusion shape while
both assimilating the inclusion to a void, as exposed above), the predicted influence
of applied stress is less than the observed spread in values: most predicted values (all
but two) lie, for the stress state and inclusion size characteristic of each inclusion,
relatively close to the zero stress equilibrium value (φ ο = 120° for the model
assuming lenticular inclusions).
This discrepancy is also apparent in the data of Waterhouse, Fig. 5-14: the observed
range of variation of measured average dihedral angles is from 93 to 75° as stress
increases from 0 to 21 MPa (note that no account can be taken here of the effect of
inclusion orientation, not reported for data from that study). Theory on the other hand
predicts a range of reduction of apparent dihedral angle of six degrees at most, Fig. 5-
14.
This discrepancy cannot be explained by the following factors:
- lack of time for equilibration would cause the observed dihedral angles to decrease
less than is predicted; experimental variations should therefore be less than is
predicted by theory;
103

CHAPTER 5. INFLUENCE OF STRESS ON THE SHAPE OF AN EMBEDDED LIQUID INCLUSION
104
- similarly, opening of lenticular voids by tensile creep deformation normal to the
grain boundary will increase φ, not lower it below theoretical predictions;
- lack of voiding in the inclusion (implying a finite liquid phase bulk modulus) would
also lower the rate of change of inclusion aspect ratio (or apparent dihedral angle)
with stress Fig. 5-8, whereas experimental data vary more than is predicted.
The somewhat greater apparent dihedral angle reduction seen experimentally to
result from increasing stress may be related to the many assumptions made in the
derivation. In particular, the assumption that the inclusions are smooth spheroids may
underestimate significantly the elastic energy associated with the presence of the
inclusion, and hence with variations of its shape. Alternatively, the following
explanations could also be proposed:
- shear deformation along a sliding grain boundary may elongate the inclusions, in
turn decreasing their aspect ratio c and hence their apparent “macroscopic” dihedral
angle φ,
- grain boundaries that were observed to become mobile in the stressed Cu-Pb
samples of this work, Fig. 5-9 (a), may orient themselves under the action of stress
away from orientations of lowest energy reached during annealing. This, in turn,
would bias the intrinsic, stress-free dihedral angle φ ο of the inclusions towards lower
values, creating an effect that is over and above the effect expected from stress.

5.5 Conclusion
5.5 CONCLUSION
Analysis predicts a change with stress in the shape of a liquid inclusion embedded in
a stressed matrix. Two cases are treated; (i) the inclusion is considered as a void, or
alternatively (ii) the finite compressibility of the inclusion content is taken into
account. The ”apparent dihedral angle”, defined at the inclusion length scale and
influenced by stress is predicted to decrease while the adimensional parameter Λ
defined in Eq. 5-11 increases. This occurs at a far greater rate if the inclusion
compressibility is nil.
In case (i), a critical value of Λ is found in accordance to literature [36, 39-42], above
which the inclusion is unstable and is predicted to collapse to a crack. On the other
hand, no instability is predicted to occur in case (ii).
A finite decrease of the average dihedral angle φ with increasing tensile stress was
measured on C99 samples subjected to interrupted creep uniaxial tension tests at
400 °C, as compared to φ measured on unstressed specimen. Our experimental data
as well as data from the literature follow the same trends as predictions, namely that
φ decreases under increased tensile stress. φ values are best described by predictions
accounting for the shape evolution of a void – case (i) above – within a uniaxially
stressed solid, suggesting that voids nucleate within the liquid inclusions or along the
matrix/inclusion interface.
Experimental data show a greater rate of decrease of the inclusion apparent dihedral
angle with increasing stress. More work would be needed to elucidate whether the
dispcrepancy is due to limited data, assumptions in the derivation, or an effect not
accounted for in the analysis, such as the possible consequences of grain boundary
sliding or mobility.
105

CHAPTER 5. INFLUENCE OF STRESS ON THE SHAPE OF AN EMBEDDED LIQUID INCLUSION
106

Chapter 6
6 Cu-embrittlement: the role of Pb
Small additions of lead, on the order of 1 wt.%, are frequently used to improve the
machinability of copper alloys. At intermediate temperature, namely above
approximately 300 °C, the reduction in ductility that is classically observed in copper
and its alloys is markedly accentuated by the presence of such lead additions.
There has been some debate in the literature regarding whether the underlying
mechanism is liquid metal embrittlement (LME) or an accentuation of grain
boundary embrittlement (GBE) caused in the solid state by segregated lead, see
Section 3.2.4. This chapter is a contribution to the question, which shows that both
mechanisms operate: GBE dominates at low strain rate, whereas LME is dominant at
high strain rates.
Tensile tests were conducted on both pure and leaded copper at different strain rates
below and above 327 °C, the melting point of lead. In unleaded samples, damage was
identified as evolving from ductile cavitation to grain boundary decohesion as the
strain rate is reduced; this is thus a clear symptom of GBE. The addition of 1 wt. %
Pb induces a sharp decrease in the reduction of area at all investigated temperatures.
At a strain rate of 10 -4 s -1 , damage is recognized as being intergranular decohesion
throughout the whole sample in both leaded and unleaded samples. If the strain rate is
increased to 10 s -1 , stress-strain curves of leaded and unleaded copper superimpose
very well until fracture occurs in the former well below the fracture strain of the
latter; this is a signature of LME. We confirm this conclusion with additional
experiments, in which sample surface contact with liquid lead is shown to cause
cracking of the samples.
107

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
108
6.1 Experimental procedures
6.1.1 Materials
Copper, 99.997% pure (hereafter Cu4N) as well as 99,95% pure (hereafter CuOFHC)
were provided by Swissmetal ® . These were in the form of Ø 18 mm cast bars and
Ø 6 mm strain-hardened bars, respectively.
Cu4N material was hot-extruded at 900 °C to Ø 8 mm, and successively cold-drawn
to Ø 7 and 6 mm. Both Cu4N and CuOFHC were heat-treated at 500 °C for 15 min in
air to allow complete recrystallization. The resulting grains are equiaxed, 25 µm in
size.
Cu-1 wt.% Pb alloys were either prepared from high-purity metals in the laboratory,
(hereafter Cu1Pb) or provided by Swissmetal ® (industrial alloy C99, see
Section 5.2.1). The Cu1Pb material was cast into small billets, 20 mm in diameter
and 60 mm in height following the procedure described in Section 4.1.1. Similarly as
for material Cu4N, Ø 6 mm bars of recrystallized Cu1Pb material were prepared by
hot-extrusion, wire drawing, and heat treatment at 500 °C for 15 min.
Lead coating on CuOFHC samples was either obtained
(i) by hot dipping, where samples were immersed for ≥ 15 min at 600 °C in a lead
bath saturated in copper, removed from the bath and cooled with compressed air, or
(ii) by electrodeposition, as follows. 500 ml electrolyte was prepared by the
dissolution in distilled water of 21 g HBF 4 , 15 g H 3 BO 3 , 0.1 g peptone, and 25 g
PbO according to [212]. The cathode current was set to 200 A/m 2 (i.e. 0.1 A for a
Ø 4 mm, 30 mm-long cylinder) according to [213]. Optimal coating was obtained
after 20 min deposition.
Industrial Cu-15 wt.% Ni-8 wt.% Sn alloys with and without a 1 wt.% Pb addition
were also provided by Swissmetal ® . These alloys are respectively denominated NP8
(leaded) and CN8 (lead-free). These alloys were processed on the industrial site by
successive Osprey spray deposition, hot extrusion and room temperature rolling
and drawing. Strain-hardened CN8 material was provided in the form of 6 mm

6.1 EXPERIMENTAL PROCEDURES
diameter bars. NP8 was supplied as air-cooled hot-extruded Ø 20 mm bars (NP8, the
leaded grade of CN8 is not produced industrially). Single-phased material was
obtained afterwards by solutionization at 850 °C for 1 h under flowing Ar (200 L/h),
followed by rapid cooling (within 20 s to 200 °C) in a fluidized bath. For both CN8
and NP8, the microstructure is equiaxed, the average grain size is 100 µm, and lead
inclusions in NP8 are below 1 µm in size. A pseudo-binary phase diagram as well as
a TTT diagram of the Cu-15Ni-8Sn alloy are available in the literature, [214], Fig. 6-
1.
(a) (b)
Figure 6-1: (a) Isopleth at 15 wt.% Ni of the ternary Cu-Ni-Sn phase diagram, and (b)
TTT diagram of the Cu15Ni8Sn alloy obtained by TEM characterizations and electrical
resistivity measurements [214].
6.1.2 Mechanical testing
a) Tensile testing
Elevated temperature tensile testing was performed on a servohydraulic MFL 100 kN
machine (modified for the interrupted creep tests, see Section 5.2.2). The maximum
displacement rate of the piston is 400 mm/s; thus, strain rates up to 10 1 s -1 can be
reached for samples having a 30 mm long gage section.
The lamp furnace described in Section 5.2.2 was used. A Type K, Ø 0.2 mm
thermocouple was spot-welded on the center of the CuNiSn alloys. For low-alloy
109

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
110
copper specimens, this was not possible because of their high conductivity. In this
case, the thermocouple wires were attached mechanically, using Ni wires, Fig. 6-2.
Figure 6-2: CuOFHC (top) and NP8 (bottom) tensile test specimen with attached
thermocouple. The reduced section was grinded before thermocouple attachement.
Tarnishing of the samples is due to the heat treatment, 15 min at 500 °C in air for the
CuOFHC material, and 1 h at 850 °C under flowing Ar for the NP8 material.
In addition to the high-temperature extensometer described in Section 5.2.2, a clip-on
extensometer was used for the room-temperature tests. The compliance of the
machine was thus determined, allowing precise determination of the sample
deformation from the cross-head position.
Threaded-end round tensile specimen were machined according to ASTM Standard
E8.A: a subsize specimen proportional to the standard was chosen, Fig. 5-4. A 4 mm-
diameter sample allows sufficiently rapid heating (about 50 °C/s with a graphite
coating) in the lamp furnace, and contains a sufficient number (more than 20 [215])
of grains across its diameter.
A peripherical notch (semi-elliptical in section; depth: a = 250 µm, width:
2b = 250 µm) was machined on several specimens in order to reach higher local
stresses.
The reduction in area, RA was measured post mortem with the aid of a profilometer.
Four measurement of the diameter (sample is step-rotated about its axis) were done
on each of the two parts of the broken specimen at the fracture surface, Ø f , and away
from it, Ø hom , in order to measure the homogeneous lateral contraction, RA hom .

RA =
RA
hom
ø − ø
2
ø
100
f 0
2 2
=
b) Impact testing
6.1 EXPERIMENTAL PROCEDURES
Eq.6-1
Eq.6-2
Elevated temperature instrumented Charpy impact testing was performed on CN8
and NP8 material with a 100 J apparatus (MFL G.m.b.H. / Model SPH-III / impact
speed: 5000 mm/s).
0
ø − ø
2
ø
[ ] %
100 hom %
0
2 2
[ ]
Rapid heating to the desired temperature was achieved by dipping the samples in a
salt bath. Temperature hold within the latter varied from a few seconds to several
minutes. After that hold, the V-notched 10x10x55 specimen, Fig. 6-3, was rapidly
positioned and fractured. A type K thermocouple spot-welded on the specimen
surface allowed the aquisition of the temperature until specimen rupture.
45°
detail
R0.25
0
detail
55
8
10
Figure 6-3: V-notched specimen for Charpy impact testing. Dimensions are in mm.
10
111

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
112
6.2 Results
6.2.1 Ductility trough
a) Leaded and unleaded pure copper
The measured ductility trough for pure copper is displayed in Fig. 6-4. The reduction
in area is shown to decrease sharply with increasing temperature only at low strain
rate, and particularly for the less pure material. Results obtained at 400 °C are
striking: extensive ductility is obtained for both CuOFHC and Cu4N material at a
strain rate of 10 1 s -1 , whereas at 10 -4 s -1 the ductility loss is more manifest for the
CuOFHC material (open symbols) than for the Cu4N material (crossed symbols).
Reduction in area: RA
[%]
100
80
60
40
20
0
Cu annealed
15 min at 500°C
CuOFHC
Cu4N
CuOFHC
Cu4N
ε = 10 s -1 .
ε = 10 -4 s -1 .
ε = 10 -4 s -1
0 100 200 300 400 500 600
Temperature [°C]
ε = 10 s -1
Figure 6-4: Evolution of the reduction of area measured after elevated temperature
tensile tests at two different strain rates. Square symbols stand for 10-4 s-1 , and round
ones stand for 10 s-1 . Curve fits underline data gathered on the CuOFHC material.
Some data obtained at 400 °C with the Cu4N material are added for comparison
(crossed symbols).
The influence of an addition of 1 wt.% Pb on the intermediate temperature ductility
of OFHC copper is shown on Fig. 6-5. Arrows are added to guide the eye while
comparing the ductility of leaded and unleaded material for the same test conditions.
.
.

Reduction in area: RA [%]
100
80
60
40
20
0
anneal
15 min at 500°C
CuOFHC
Cu1Pb
CuOFHC
Cu1Pb
ε = 10 s -1 .
ε = 10 -4 s -1 .
(i)
ε = 10 -4 s -1
0 100 200 300 400 500 600
Temperature [°C]
ε = 10 s -1
6.2 RESULTS
Figure 6-5: Reduction in area of leaded and unleaded pure OFHC copper after
elevated temperature tensile test for two different strain rates. Square symbols stand for
10-4 s-1 , and round ones stand for 10 s-1 . Curve fits underline data gathered on the
CuOFHC material. Filled symbols are for Cu1Pb material. Arrows are added in order
to help direct comparison between unleaded and leaded material under the same test
conditions.
The flow curves of the CuOFHC, Cu1Pb, and C99 materials strained at 10 s -1 at
400 °C are displayed in Fig. 6-6. The three flow curves superimpose very well, but
differ in the moment of rupture. The Cu1Pb material is characterized by a far lower
ductility (ε r ≈ 10%). The reduction in area at fracture of CuOFHC, Cu1Pb, and C99
are respectively 97, 15, and 33%. Necking of the CuOFHC is thus almost to a point.
.
(ii)
.
(iii)
113

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
114
eng. stress [MPa]
200
150
100
50
0
ε .
= 10 s -1
400°C
CuOFHC
RA = 97%
Cu1Pb
RA = 15%
C99
RA = 33%
0 10 20 30 40 50 60
eng. strain [%]
Figure 6-6: Flow curves of the CuOFHC (symbol is an open circle), Cu1Pb (open
square), and C99 (open diamond) materials at 400 °C, and at a strain rate of 10 s-1 . For
each material, fracture is emphasized by a solid symbol.
The tensile behaviour of these materials at 400 °C, for a strain rate of 10 -4 s -1 , is
shown in Fig. 6-7. The behaviour of the Cu4N material is also shown. Dynamic
recrystallization of this material is clearly operative since waves are observable on
the σ(ε) curve. The yield stress and ultimate tensile strength σ max of these materials
are also much reduced at lower strain rate, compare Fig. 6-6 with Fig. 6-7. Indeed for
CuOFHC: σ max ( = 10 s -1 ) ≈ 150 MPa and σ max ( = 10 -4 s -1 ) ≈ 70 MPa. Moreover,
ε •
the flow curves of the different materials do not superpose at low strain rate. Note
also the high elongation of the C99 industrial material.
ε •

eng. stress [MPa]
100
80
60
40
20
0
ε = 10 -4 s -1
.
400°C
CuOFHC
RA = 28%
Cu1Pb
RA = 16%
C99
RA = 38%
Cu4N
RA
= 51%
0 10 20 30 40 50
eng. strain [%]
6.2 RESULTS
Figure 6-7: Flow curves of the CuOFHC (symbol is an open circle), Cu1Pb (open
square), C99 (open diamond), and Cu4N (open triangle) materials at 400 °C, and at a
strain rate of 10-4 s-1 . For each material, fracture is emphasized by a solid symbol.
The influence of lead as an alloying element was shown above. Liquid lead supplied
from the external surface of copper specimen also influences the tensile behaviour of
pure Cu at 400 °C, as shown on Fig. 6-8. All the curves superimpose well except for
that of the dipped CuOFHC sample, which is somehow lower (probably due to
coarser grains): ductility loss is obvious after such a treatment. Concerning Pb
electroplated samples, only the notched specimen exhibits embrittlement. The
reduction in area of the unnotched specimen is in accordance with the elongation at
fracture. RA is namely 96 and 49% for OFHC copper electroplated with lead, and
dipped in lead, respectively. By comparison, RA=97% for CuOFHC material.
It can also be noted that the copper-embrittlement induced by liquid lead is far
stronger if the lead is pre-alloyed with copper. This is obvious if one compares the
behaviour of the Cu1Pb, Fig. 6-6, and that of Pb-coated CuOFHC, Fig. 6-8.
115

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
116
Figure 6-8: 400 °C flow curves of CuOFHC material at a rapid strain rate of 10 s -1 :
the influence of a Pb coating, and the effect of a notch are shown. Square symbols stand
for electroplated material, and diamond are for dipped material. Grey filled symbols
represent notched specimen. For each material, fracture is emphasized by a solid
symbol.
b) Leaded and unleaded Cu-Ni-Sn alloys
The intermediate temperature tensile behaviour of CN8 and NP8 materials was also
quantified, Fig. 6-9. In the whole temperature range studied, the NP8 material is
more brittle than CN8. For both materials, the ductility decreases above 200 °C. In
addition, above 327 °C (the Pb melting point), the ductility drops sharply for the
lead-containing NP8 material, and then remains roughly constant as the temperature
increases.
eng. stress [MPa]
200
150
100
50
0
= 10 s -1
ε .
400°C
(pn)
(n)
CuOFHC
CuOFHC notched (n)
Cu + Pb plated (p)
Cu + Pb plated & notched (pn)
Cu + Pb dipped (d)
0 10 20 30 40 50 60
eng. strain [%]
(d)
(p)

Reduction in area: RA
[%]
6.2 RESULTS
Figure 6-9: Reduction in area of CN8 and NP8 material strained at 10 -2 s -1 . The
melting temperature of lead, 327 °C is indicated.
The influence of room temperature strain hardening on the intermediate temperature
tensile properties of solutionized and quenched CuNiSn alloys is shown in Fig. 6-10.
Notice the comparatively high flow stress (~200 MPa) of the CN8 alloy in the
annealed monophase state. In the same conditions, the flow curves of CN8 and NP8
superimpose quite well. The ductility of the lead-containing NP8 is about 1% at
400 °C, and drops to essentially zero for specimens strain-hardened at room
temperature.
100
80
60
40
20
0
CN8
NP8
0 100 200 300 327 400 500 600
Temperature [°C]
CN8 & NP8
annealed
60 min at 850°C
& quenched
.
ε = 10 -2 s -1
117

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
118
eng. stress [MPa]
600
500
400
300
200
100
Figure 6-10: 400 °C flow curves of CN8 & NP8 material at a strain rate of 10 -2 s -1 .
Room temperature strain hardened specimen are differenciated with grey filled symbols.
The ductility of NP8 is much lower than that of CN8. For each material, fracture is
emphasized by a solid symbol.
6.2.2 Impact energy
0
400 °C
CN8
CN8 (RT strain hardened)
NP8
NP8 (RT strain hardened)
0 3 6 9 12 15
eng. strain [%]
Charpy impact tests were performed on both CN8, and NP8 material at room- and
intermediate temperatures. Results are summarized on Fig. 6-11. The impact energy
of the unleaded CN8 material is much higher than that of the NP8 material over the
whole temperature range explored. Notice that between 400 and 500 °C, phase
transformation occurs before impact, due to high rate of the phase transformation,
Fig. 6-1 (b), and finite heating time within the salt bath (about 5 s to 400 °C, plus 5 s
to 500 °C, and an additionnal 5 s to position the sample and run the test).
Comparing the Pb-containing NP8 material with CN8, it is clear that lead inclusions
induce a sharp decrease in impact energy at all temperatures.
CN8 & NP8
annealed
60 min at 850°C
& quenched
ε = 10 -2 s -1
.

6.2.3 Damage
Impact energy: CVN [J]
100
80
60
40
20
0
CN8
NP8
6.2 RESULTS
Figure 6-11: Impact energy of monophased CN8 and NP8 material. Grey symbols
underline the occurence of rapid phase transformation just before impact. The melting
temperature of lead, 327 °C is indicated.
Damage in Cu4N, Cu1Pb, CuOFHC, and C99 materials strained at 400 °C at both
10 s -1 , and 10 -4 s -1 was observed on polished longitudinal sections. SEM images are
shown on Fig. 6-12. The lead-containing materials were imaged in the Back
Scattered Electron mode, BSE, in order to reveal the chemical contrast. Left-hand
images were taken from a specimen strained at 10 s -1 , whereas right-hand images are
from samples tested at 10 -4 s -1 .
Intergranular decohesion is most marked in the CuOFHC and Cu1Pb materials
strained at 10 -4 s -1 . Under the same testing conditions, the Cu4N material is less
damaged, and the C99 is almost sound. These observations remain valid if one
considers the damage remote from the fracture surface. Note that for all materials
tested at low strain rate, intergranular damage is more pronounced near the surface of
the specimen than in its interior.
0 100 200 300 327 400 500 600
Temperature [°C]
CN8 & NP8
annealed
60 min at 850°C
At the higher strain rate of 10 s -1 , the CuOFHC, and Cu4N materials fracture in a
ductile manner with almost perfect necking. There is therefore no intergranular
damage near the fracture surface nor in the specimen interior. On the other hand, the
119

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
120
lead-containing materials fracture intergranularly, and damage is only concentrated in
the vicinity of the fracture surface. No damage was detected away from the fracture
surface in the four materials tested. There is also no evidence for the transient
existence of a Pb-rich liquid film in grain boundaries subjected to tensile stress: the
inclusions appear rather as being mostly elongated in the tensile direction, Fig. 6-
12 (g).

121
(a) Cu4N dε/dt=10 s -1 RA=96% (b) Cu4N dε/dt=10 -4 s -1 RA=51%
(c) Cu1Pb dε/dt=10 s -1 RA=15% (d) Cu1Pb dε/dt=10 -4 s -1 RA=16%
(e) CuOFHC dε/dt=10 s -1 RA=97% (f) CuOFHC dε/dt=10 -4 s -1 RA=28%
(g) C99 dε/dt=10 s -1 RA=33% (h) C99 dε/dt=10 -4 s -1 RA=38%
Figure 6-12: Longitudinal cuts at the fracture surface of Cu4N, Cu1Pb, CuOFHC, and
C99 materials after 400 °C tensile test in air. Strain rate is 10 s -1 (left images) and 10 -
4 s -1 (right images). The tensile axis is vertical. Pb-containing materials were imaged in
the BSE mode, and Pb-free materials in the SE mode.

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
122
Liquid lead surface-coatings are on the other hand found to migrate very rapidly with
a propagating crack. This is illustrated in Fig. 6-13 comparing a CuOFHC tensile
specimen dipped in liquid lead for 18 h at 600 °C before (a), and after (b) tensile
testing at 400°C and = 10 s -1 .
(a) (b)
Figure 6-13: SEM images of longitudinal cuts in the BSE mode: lead appears brighter
than copper. (a) Mullins-type groove of a CuOFHC sample dipped in a Pb bath
saturated in Cu for 18 h at 600 °C, the depth of the groove is 12 µm. (b) Similar sample
subsequently strained at 400 °C, and dε/dt = 10 s-1 : a subcritical crack nucleated from
a grain boundary groove, and propagated to a depth d=560 µm, see Fig. 6-8.
6.2.4 Fractography
ε •
It is clear from Fig. 6-14 that the fracture of Cu1Pb strained at intermediate
temperature and elevated strain rate is intergranular. At 400 °C, above the Pb melting
point, the lead-rich inclusions are molten. Inclusions indeed did not melt in the
Cu1Pb material tested at 300 °C; nevertheless, the material fractured intergranularly.

(a) Cu1Pb 300 °C dε/dt=10 s -1 RA=28% (b)
(c) Cu1Pb 400 °C dε/dt=10 s -1 RA=15% (d)
6.2 RESULTS
(e) C99 400 °C dε/dt=10 s-1 RA=33% (f)
Figure 6-14: SEM fractographs in the BSE mode. The Cu1Pb material was tensile
tested in air at dε/dt=10 s-1 at (a) & (b) 300 °C, below the Pb melting point, and (c) &
(d) at 400 °C, where the lead is molten. Fractographs of the C99 material tested at
400 °C are added for comparison (e) & (f).
123

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
124
6.3 Discussion
6.3.1 Embrittlement Mechanisms
The effect of lead in copper may be three-fold: (i) lead inclusions act as stress
concentration sites and pin the grain boundaries; this renders cavitation easier and
reduces the grain boundary mobility retarding dynamic recrystallization, (ii)
segregated lead may decrease the grain boundary cohesion; (iii) once liquid, lead
may promote liquid metal embrittlement.
a) Grain Boundary Embrittlement
GBE is a time-dependent process, which therefore becomes exacerbated with
decreasing strain rate. This results in a widening and deepening of the ductility
trough. A clear illustration of GBE for pure copper is given on Fig. 6-4, where our
CuOFHC and Cu4N materials display a much reduced RA when strained at 400 °C
and = 10 -4 s -1 as compared to the same materials at = 10 s -1 . These results are
ε •
replotted on Fig. 6-15 (a) and are shown to compare very well with literature, Fig. 6-
15 (b) from Ref. [49].
We note that GBE operates in all materials we studied at low strain rate, since none
demonstrates full ductility, Fig. 6-7. On the other hand, at elevated strain rate, there is
no time allowed for grain boundary embrittlement, and pure copper samples exhibit
necking to a point.
ε •

Reduction in area: R [%]
100
80
60
40
20
0
0 200 400 600
6.3 DISCUSSION
(a) (b)
Figure 6-15: (a) Same Figure as Fig. 6-4 displaying the ductility trough of CuOFHC &
Cu4N material tested in air. (b) At the same scale, results from the literature [49]
(tensile test under vacuum: p=10-2 Pa): pure copper is Cu 99,99% polycrystal, and
internally oxydized copper contains dispersed Ø 0.05 µm Al2O3 particles.
Our results are also plotted on the Ashby fracture-mechanism map [52] for CuOFHC,
Fig. 6-16. The ultimate tensile strength is plotted as a function of temperature for our
CuOFHC samples strained at 10 s -1 and 10 -4 s -1 . Solid and open symbols stand
respectively for intergranular and transgranular fracture. The agreement with the
fracture-mechanisms map is excellent. We note that complete necking due to DRX is
predicted for a temperature above 600 °C, in agreement with the results presented on
Fig. 6-15 (b). Ultra-high purity material can, however, display dynamic
recrystallization at a much lower temperature [68]. In this case, the boundary of the
”Rupture” field on Fig. 6-16 is shifted to the left. The occurence of dynamic
recrystallization is manifest on Fig. 6-7 for the Cu4N material. This is recognized by
the wavy evolution of the σ−ε curve. Cu4N is the purest material we tested; at 400 °C
and = 10 -4 s -1 , it demonstrates indeed the highest ductility in term of both strain
ε •
CuOFHC
Cu4N
CuOFHC
Cu4N
e = 10 s -1 .
e = 10 -4 s -1 .
Temperature [ C]
and reduction in area at fracture, Fig. 6-7.
125

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
126
ultimate tensile stress UTS [MPa]
1000
100
10
Figure 6-16: Ashby fracture mechanism map for CuOFHC material, redrawn from
Fig. 3-18, with our datapoints. For these, the y-axis value is the UTS. Circle symbols
stand for a strain rate of 10 s-1 , and squares for 10-4 s-1 . Solid, and open symbols
indicate an intergranular, and intragranular fracture respectively.
Influence of segregated lead
Ductile Fracture
Transgranular
Creep Fracture
Considering Fig. 6-12 (b) & (d), it is clear that the addition of lead to copper
exacerbates grain boundary embrittlement. Indeed Cu1Pb is more damaged than
Cu4N. Moreover the former is much less ductile than the latter, Fig. 6-7. These
findings are in accordance with the literature. It was namely reported that the 400 °C
ductility of Cu OFHC, tested in air, is reduced by a prior treatment at 800 °C under a
Pb vapor atmosphere [216]. It can therefore be concluded that lead, as an impurity in
copper, induces GBE even at very small concentrations.
Influence of segregated oxygen
Dynamic Fracture
Intergranular
Creep Fracture
Rupture
(DRX)
Cu OFHC
ε = 10
1
0 200 400
Temperature [°C]
600 800
-4 s -1
ε = 10 s
.
-1 .
Impurities such as oxygen, segregated to the grain boundaries of copper, are also
expected to induce GBE. Within the materials we tested at = 10 -4 s -1 , intergranular
damage is found throughout the whole specimen and is more pronounced near the
specimen surface than in its interior. This suggests that stress-driven oxygen diffusion
occurs from the surrounding air during the test, and weakens the grain boundaries.
This is in agreement with the literature. Oxygen diffusion along the grain boundaries
of copper was explicitly reported to occur at intermediate temperature only if a
ε •

6.3 DISCUSSION
tensile stress is applied [217]. Moreover, intergranular damage revealed by
synchrotron microradiography in oxygen-saturated copper subjected to creep at
500 °C was also found to be on one hand governed by a diffusion mechanism, and,
on the other hand, to be controlled primarily by the maximum principal stress [109].
Intergranular damage was furthermore documented by Bleakney in pure copper
subjected to creep in air. It is reported that the creep rupture embrittlement of high-
purity copper is caused by a combination of stress-corrosion and internal oxidation of
the grain boundaries [218].
From the above-mentioned considerations, we expect therefore that the Cu4N
material subjected to low strain rate tensile testing at intermediate temperature would
display a higher ductility if tested under vacuum.
Phosphorus is added to the industrial C99 material as a desoxidizing element of the
melt. Therefore the important reduction of intergranular damage of this material as
compared to the others, see Fig. 6-12 (left column), can be attributed to the presence
of phosphorus acting as an oxygen scavenger. This parallels findings on bronze:
oxygen is indeed scavenged by boron or magnesium added to a Cu-8Sn bronze,
suppressing the penetration of oxygen along the grain boundaries, and restoring in
turn the high temperature ductility of the bronze [71].
Note that the embrittling effect of oxygen is reversible. A heat treatment at 950 °C
under a 10 -5 Torr vacuum was namely found to remove the oxygen and restore both
tensile and creep ductility of coarse-grained oxygen saturated 99,995% Cu that was
treated for 2 h at 800 °C under an O 2 atmosphere [108]. Miura et al. accordingly
report that a 950 °C 6 h degassing treatment in vacuum alleviate the GBE of
internally oxidized Cu-SiO 2 polycrystals [57].
This suggests that, for copper alloys of standard purity, intermediate temperature
ductility is enhanced if desegregation occurs or if the embrittling segregated species
are scavenged. On the other hand it is clear from Fig. 6-4 that standard purity copper
is not embrittled at intermediate temperature if rapidly strained, as one would expect
127

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
128
for such a time-dependent damage phenomenon in the absence of other
embrittlement mechanisms.
b) Liquid Metal Embrittlement
Above the lead melting point, the impact energy of pure copper is found to be much
reduced due to the addition of 1 wt.% Pb to Cu, Fig. 6-17 (b). This is a clear
manifestation of LME, since pure copper does not exhibit any embrittlement at such
high strain rates above 327 °C [130]. It was also reported by Roth that mixed
intergranular and transgranular fracture is obtained for leaded copper samples tested
just below 327 °C. This is probably due to remaining undercooled liquid inclusions,
since the sample was impacted while cooling from a temperature higher than 350 °C
[130].
impact energy [J]
40
30
20
10
0
monophase NP8
850 °C annealed
& quenched
(a) (b)
Figure 6-17: Intermediate temperature impact energy of (a) the monophased NP8
material, see Fig. 6-11 and (b) pure copper, and Cu-1Pb alloy [130]. V-notched samples
were tested (a) 10x10 mm 2 in section, and (b) 8 mm in diameter.
The resemblance of these literature results [130] with our results obtained on the
lead-containing NP8 material is clear, Fig. 6-17. This suggests therefore that this
latter material is prone to LME.
327 °C
100 200 300
Temperature [°C]
Comparing our leaded and unleaded materials tensile tested at 400 °C and 10 s-1 leads to the same conclusion when considering the relative reduction in area at
fracture, Fig. 6-6. At this high strain rate, the flow curves of the different materials
tested superpose very well until the sudden rupture of the Pb-containing materials,
Fig. 6-6. This is a classical signature of LME [132].

6.3 DISCUSSION
The onset of a ductility trough due to LME is associated with incipient melting of the
embrittler. A drastic change in fracture surface occurs, from transgranular to fully
intergranular, see e.g. [219]. At 400 °C and = 10 s -1 , fracture is indeed found to be
fully intergranular in Cu1Pb and C99 materials, Fig. 6-14, and damage is
concentrated at the fracture surface. Unleaded copper materials tested in the same
conditions exhibit fully ductile transgranular fracture, Fig. 6-12 (left column).
The sharp ductility loss at 400 °C measured on our Cu1Pb material subjected to high
strain rate tensile testing, Fig. 6-5, also compares well with results obtained on a
phosphorus bronze containing 2 wt.% lead, where it was observed that above 327 °C,
secondary cracks were filled by liquid lead [65]. Intergranular lead inclusion also act
as liquid metal reservoir at grain boundaries: in the present samples: an emptied
lenticular inclusion is illustrated on Fig. 6-18.
(a) (b)
Figure 6-18: SEM images (BSE mode) of a longitudinal cut of a prolonged annealed
(900 °C 1 h and 400 °C 24 h) Cu1Pb material tensile tested at 400 °C and 10 s-1 , the
stress axis is horizontal: (a) intergranular microcrack at low magnification. (b) detailed
view of the tip of the same microcrack.
Pb-coated samples
Turning to Pb-coated samples, we also find a behaviour characteristic of LME. Pb-
coating caused varying effects on the ductility of our CuOFHC specimen. Namely no
influence was noted for the electroplated smooth sample, Fig. 6-8, whereas
spectacular embrittlement was observed in the dipped samples, Fig. 6-13. Such
erratic manifestation of LME is well known: notched specimen can be embrittled in
conditions under which smooth samples are immune [132 , 142]. To rationalize the
ε •
129

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
130
different behaviour of the dipped and the electroplated samples, it must be
emphasized that the former exhibited slight grain boundary grooves, in intimate
contact with lead. In adddition, electroplated notched specimen displayed
embrittlement, Fig. 6-8. This suggests that the migration of liquid lead is associated
with crack propagation; however, crack initiation requires that additional conditions
be met. According to Roth, the presence of a liquid metal wetting the grain boundary
decreases the grain boundary cohesion and favours the nucleation of a brittle crack
once a slip band impinges, on a grain boundary ahead of an intergranular liquid
inclusion [66].
Our results on Pb-coated specimen also agree with results reported on the LME
susceptibility of martensitic steel in contact with liquid Pb-Bi. Namely the presence
of an oxide layer on the T91 steel renders it immune to LME attacks [144].
It is moreover worthwhile noting that Sanchez Medina claimed that Cu-Pb is not
prone to LME since OFHC Cu samples surrounded by a ring of liquid Pb behave like
uncoated specimen at temperature ranging from RT to 1000 °C (similarly to our
electroplated smooth samples, Fig. 6-8) [47].
Summary of observations
We conclude, in summary, that at high strain rates, Cu-Pb alloys fracture by liquid
metal embrittlement. Furthermore, the data provide indications that a critical stress
must be exceeded in order to cause such LME fracture:
• in Fig. 6-10, it is seen that at 400 °C, strain-hardened NP8 fails at 220 MPa before
any macroscopic plastic deformation, whereas with annealed NP8, the initial flow
stress of which is lower, failure occurs after some plastic deformation when roughly
the same stress is reached (200 MPa);
• in lead-coated specimens, failure by LME was only triggered when there was a site
of stress concentration along the surface of the sample, taking the form of a grain
boundary groove or a pre-machined notch, Fig. 6-8.

Is LME gouverned by a critical value of dimensionless parameter Λ ?
6.3 DISCUSSION
The observation that a critical stress must be exceeded for LME to cause fracture of
Cu-Pb specimens above the melting point of lead suggests the possibility that
fracture is dominated by the dimensionless parameter Λ defined in Eq. 5-11 . This
parameter is shown to trigger unstable flattening of liquid inclusions under tensile
stress, i.e., crack formation and catastrophic growth when Λ exceeds a critical value
Λ c , defined at the maximum of curves for K Pb = 0 in Fig. 5-8 (a). This catastrophic
crack propagation is in particular suggested by numerical simulations of Wang,
which indicate much accelerated shape changes for pores (changing shape by surface
and grain boundary diffusion) if Λ>Λ c , Fig. 3-15 [41]. That very rapid crack
propagation can occur at the tip of a crack wetted by the liquid (i.e., by LME) is
shown quantitatively by Robertson [152].
The suggestion that fracture by LME is triggered by attainment of a critical value, Λ c ,
of parameter Λ is supported by the following observations.
1 - As mentioned above, a critical stress must be exceeded;
2 - this critical stress is lowered when the inclusion diameter increases.
This is seen in Fig. 6-19, which compares tensile data for fine-grained copper and
Cu-Pb alloys, and their coarse-grained equivalents obtained by annealings at 900°C
for 1 h prior to testing. During grain growth the lead inclusions coalesce, causing
their average size to increase (see Section 4.1.2). As seen, solid lines drawn on
Fig. 6-19 for the fine-grained materials CuOFHC, C99 and Cu1Pb superimpose quite
well, the leaded samples fracturing prematurely at 212 MPa (C99) and 118 MPa (Cu-
1Pb). This is also true for the dashed lines corresponding to coarse-grained materials
C99 and Cu1Pb; however, in these, the critical stress at which LME causes premature
failure is significantly lower, namely 129 MPa for C99, and 80 MPa for Cu-1Pb.
131

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
132
Figure 6-19: Flow curves of the CuOFHC (symbol is an open circle), Cu1Pb (open
square), and C99 (open diamond) materials at 400 °C, and at a strain rate of 10 s-1 .
Crossed symbol correspond to the fracture. The flow curves of the fine grained material
are evidenced by solid lines. Tensile behaviour of coarse grained materials (no data for
CuOFHC) follow the dashed lines. Lead inclusions size in the fine grained Cu1Pb &
C99 samples is about 1 µm, whereas it is around 10 µm in the coarse grained samples.
3 - The critical fracture stress is decreased by a lowering of the stress-free dihedral
angle φ o .
true stress [MPa]
250
200
150
100
50
Liquid Ga is known to perfectly wet the general grain boundaries of Al (φ o = 0°). In
this case, as is intuitively obvious (and also borne out by the analysis of Chapter 5
with φ o = 0°), the critical value of Λ c , and hence the critical fracture stress σ c , are nil.
Liquid lead, on the other hand, does not wet perfectly the copper grain boundaries at
intermediate temperature, since φ o > 0°, see e.g. Fig. 4-14. In the Cu-Pb system, Λ c is
thus finite positive.
0
ε .
= 10 s -1
400°C
coarse
grained
fine
grained
0 10 20 30 40 50
true strain [%]
CuOFHC Fracture
C99 Fracture
Cu1Pb Fracture
The influence of φ o is also illustrated in Fig. 6-19, where it is seen that, for both
inclusion sizes, Cu1Pb is more brittle than C99. This is probably due to the slight
difference in interfacial chemical composition between the two alloys, reflected by
an average lower φ o in the purer Cu1Pb material. Indeed at 400 °C, the average value
of φ o in Cu-1Pb is 100°, Fig. 4-14; whereas φ is about 120° for the C99 material; this
is shown in Appendix 10.4, where the results of four dihedral angle measurements in
C99 at 400°C are given. Now, Λ c depends on the stress-free initial dihedral angle φ o

6.3 DISCUSSION
(see Chapter 5): a decrease of φ o results in a decrease in Λ c , and hence in σ c , all else
being constant, Fig. 6-20.
We note in passing that such a dependence on the dihedral angle, whereby increasing
the dihedral angle increases the resistance to LME, is in broad agreement with the
grain boundary engineering concept of Watanabe illustrated on Al bicrystals wetted
by Ga [220], and with Kargols’ findings that a lower grain boundary energy (higher
φ) ensures a higher resistance to LME in Al bicrystals in contact with liquid Hg-
3 at.% Ga [148].
Λ [-]
0.8
0.6
0.4
0.2
0
120
Figure 6-20: Predicted Λ(φ) curves for the C99 and Cu1Pb materials at 400 °C. Inputs
for the theoretical calculations are identical save for φ 0 , the equilibrium dihedral angle
under stress-free conditions. Predictions are for a grain boundary void, lenticular in
shape.
Finally, we note that quantitative comparison of the predictions of the analysis in
Chapter 5 and data of Fig. 6-19 is satisfying given the assumptions made in the
model. At 400 °C, we have γ SL = 0.432 J/m 2 , and E = 111 GPa. Inclusions 1 µm and
10 µm in diameter have a volume of about 10 -19 and 10 -16 m 3 respectively. From the
Λ c values shown on Fig. 6-20, the predicted critical stress, σ c is determined from
Eq. 6-3, and given in Table 6-1.
1
φ 0 C99
400 °C
100
φ 0 Cu1Pb
C99
80
Cu1Pb
60
40
apparent dihedral angle: φ [°]
Λ c C99 = 0.61
Λ c Cu1Pb = 0.41
20
crack
133

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
134
σ
Eq.6-3
Table 6-1: Predicted critical stress σ c for the Cu1Pb and C99 materials subjected to uniaxial tension at
400 °C according to Eq. and Fig. 6-20. Experimental values are from Fig. 6-19.
400 °C
As seen, values do not match with precision, which is not surprising given
assumptions of the calculation and the absence of any adjustable parameter; however,
variations caused by changes in alloy composition or grain size, as well as the order
of magnitude of the fracture stress of the C99 and Cu1Pb materials tensile tested at
400 °C, Fig. 6-19, are quite well predicted.
Similarly, a fracture stress can be predicted for the lead-containing NP8 material if
the dimension and geometry of the lead inclusions are known. φ can be measured
from fractographs, Fig. 6-21 (b), using the method described in Section 4.3.2,
slightly modified since we use the 3D reconstruction of the grain boundary plane and
of only one (but whole) former liquid inclusion/solid metal interface, Fig. 6-22.
Optimal measurement is achieved with an emptied inclusion on an intergranular
fracture surface, which has to be unoxidized and free of distortion due to plastic
deformation.
γ
V
SL ⋅ E
c = Λc 1/ 3
theoretical predictions experimental values
σ c Cu1Pb σ c C99 σ max Cu1Pb σ max C99
fine grained 205 MPa 251 MPa 118 MPa 212 MPa
coarse grained 64 MPa 79 MPa 80 MPa 129 MPa

(a) (b)
6.3 DISCUSSION
Figure 6-21: Fracture surface of a NP8 cracked during quenching in oil from 900 °C.
Figure 6-22: Data points (x’s) from the 3D reconstruction of the former solid/liquid
interface of the inclusion illustrated on Fig. 6-21 (b). The sphere fitting these
datapoints, as well as the reconstructed grain boundary plane are shown. φ = 64° and
V = 3.5 10-17 m3 .
The inclusion volume V, and dihedral angle φ are calculated from the determined
equations of the grain boundary plane and fitting sphere. For the inclusion (the only
characterized here) displayed in Fig. 6-21 (b): φ = 64° and V = 3.5 10 -17 m 3 .
A rough estimate of Λ c is calculated taking into account φ o = 64°, the elastic
properties of the NP8 at 800 °C (E = 80 GPa, ν = 0.364), γ SL = 0.387 J/m 2 at 800 °C
estimated for the Cu-Pb system according to the regular solution model, Eq. 4-17,
and taking K Pb = 0 GPa (i.e. considering the inclusion as a void). The calculations
give Λ c = 0.16 and 0.10 for lenticular and ellipsoidal inclusion geometries,
respectively.
Grain Boundary plane
135

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
136
From Eq. 6-3, this gives σ c = 39 and 30 MPa respectively. These values are in good
agreement with the measured resistance of NP8 at 800 °C, namely R m = 44 MPa,
Fig. 7-6, if one remembers that there is no adjustable parameter in the estimation.
Finally, we note that, according to this analysis;
(i) single crystals are also expected to be prone to LME. In that case, a dihedral angle
of 180° (hence γ gb = 0) must be assumed in the calculation of Λ c ;
(ii) embrittlement resulting from an external contact with liquid metal can also be
rationalized according to [42]: an estimation of σ c is given in Eq. 3-11 [42];
(iii) this approach also predicts intuitively an increasing susceptibility to LME with
increasing hardness of the alloy, since higher elastic strain energy can be stored;
however, the present analysis cannot describe this effect, since the mechanical
analysis is limited to linearly elastic materials.
Implications
If, on the basis of what precedes, we now consider Λ c as a parameter quantifying the
specific resistance of a given liquid inclusion-containing alloy to LME, what are the
practical implications ?
Since the critical fracture stress is then given by Eq. 6-3, increasing the resistance of
a given system to LME is tantamount to increasing the right-hand side of Eq. 6-3.
Increasing E is a rather difficult task; however, controling V, the inclusion size (in all
rigor the largest inclusions’ size) can be achieved relatively easily, by deformation
processing for example.
A second possibility is to increase the equilibrium dihedral angle φ o , as this results in
an increase of Λ c . Since:
cos
Eq. 3-2
this can be achieved either by increasing the liquid/solid interfacial energy, or by
φ ⎛ γ o ⎞ gb
⎝ 2 ⎠ 2γSL
=
decreasing the highest grain boundary energy. An increase of γ SL can be obtained by

6.3 DISCUSSION
changing the composition of the liquid metal; however, gross composition changes
would be required since segregants to the solid/liquid interface do not increase γ SL .
On the other hand, a decrease of γ gb (e.g. resulting from grain boundary segregation,
often detrimental to GBE) should enhance resistance to LME. So should ”grain
boundary engineering” if it can be used to decrease all grain boundary energies in the
alloy. Finally, as mentionned above, increasing γ SL should not be envisaged unless
the inclusion composition can be changed radically.
The upper bound of the ductility trough.
In LME, the onset of embrittlement occurs at the melting point of the liquid metal;
this sets the lower bound of the ductility trough. On the other hand, a higher limit of
the ductility trough corresponding to the recovery of ductility may be explained by
the fact that, at this temperature – T R – the resistance of the ”normally ductile metal”
(i.e. R m of pure Cu) becomes lower than the critical stress σ c , Fig. 6-23. In this high-
temperature region, the difference between the liquid inclusion-containing and the
embrittler-free alloy should disappear: in short, signs of LME may disappear at
elevated temperature.
stress [MPa]
250
200
150
100
50
R m
CuOFHC
ε = 10-3 s-1 .
24 h 500°C
anneal
15 min 500°C
anneal
σ c
T E
0
0 100 200 300 400 500 600 700 800
Temperature [°C]
Figure 6-23: Evolution of the tensile strength of CuOFHC material as compared to a
schematic curve of σ c . Both limits of the ductility trough T E and T R are added.
Also, if the strain rate is sufficiently lowered, in the creep-deformation regime the
flow stress and R m will decrease significantly. This could also cause the
T R
137

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
138
disappearance of LME; this was observed in our leaded copper strained at 400 °C and
10 -4 s -1 .
On the other hand, a larger inclusion size would result in a shift of the dashed line to
lower stress values, extending the range of strain rates and temperatures at which
LME is observed.
c) stress concentration sites
It is clear from the above subsections that leaded copper is subjected at intermediate
temperature both to Grain Boundary, and Liquid Metal Embrittlement. The former
mechanism dominates at lower strain rates, whereas the latter operates mainly at
higher strain rate. In some instances, however, embrittlement may be aided
mechanically by the presence of the solid micrometric intergranular lead inclusions.
These act as stress concentration sites, and in turn favour the nucleation of cavities,
as suggested by [47, 50]. The sharp ductility loss at 300 °C measured on the Cu1Pb
material subjected to high strain rate tensile testing, Fig. 6-5, must thus be, at least in
part, due to this purely mechanical influence of the finely dispersed lead inclusions.

6.4 Conclusion
6.4 CONCLUSION
Three different mechanisms have been proposed and opposed in the litterature to
account for the occurence of the ductility trough in leaded copper. We show here that
all three phenomena operate. Their respective importance depend mainly on the
temperature and on the strain rate. This is illustrated on Fig. 6-5, (see arrows):
(i) Below 327 °C and at high strain rate, cavitation at intergranular solid lead
inclusions seems to trigger fracture, predominantly along the grain boundaries.
(ii) Segregation-induced grain boundary embrittlement dominates at low strain rates
in the whole temperature range of the ductility trough.
(iii) Above 327 °C, liquid metal embrittlement is manifest at high strain rates. This
mechanism requires that a critical stress be reached, which decreases as the
inclusions volume increases and the dihedral angle decreases.
This last observation, coupled with the observed dependence of the high strain rate
fracture stress in these alloys, suggests that fracture by LME occurs once the
adimensional parameter Λ, defined in Eq. 5-11, reaches a critical value Λ c , reflecting
the influence of the stress applied, the elastic modulus of the solid, the interfacial
energies, and the volume of the inclusions.
139

CHAPTER 6. CU-EMBRITTLEMENT: THE ROLE OF PB
140

Chapter 7
7 Quench cracking in leaded copper alloys: a case study
Ternary copper-nickel-tin alloys are often called ”spinodal alloys” since they exhibit
spinodal decomposition at intermediate temperature. These alloys are high-strength
copper alloys showing substantial age hardening [221]. A yield stress as high as
1300 MPa with 46% reduction area at fracture can be reached for a Cu12Ni8Sn
alloys after appropriate thermomechanical treatments [222]. This makes them
attractive for applications in electronics requiring high spring strength, good
formability, and the other advantages of a high-copper alloy, such as electrical
conductivity and corrosion resistance. In addition, these alloys are demonstrated to
have excellent wear properties.
The major drawback of these alloys is that they exhibit substantial solute segregation
during conventional solidification. Powder metallurgy allows the production of small
parts, homogeneous in composition. Large fine-grained parts up to 65 cm in
diameter, are also processed with the patented EquaCast process at Brush Wellman
Inc. [223]. Osprey spray deposition is an alternative process that allows the
production of large billets, typically 160 mm in diameter [224]. Rapid solidification
of the sprayed liquid metal droplets ensures chemical homogeneity at the 100 µm-
scale. Billets are subsequently hot-extruded. The extruded bars, typically 15 to
25 mm in diameter, are immediately water quenched to avoid any hardening phase
transformation.
NP8, the leaded version of CN8, cracks systematically during the quench. Our goal
was to identify the underlying embrittlement mechanisms, and devise strategies to
allow the production of this alloy grade.
141

CHAPTER 7. QUENCH CRACKING IN LEADED COPPER ALLOYS: A CASE STUDY
142
7.1 experimental procedures
7.1.1 Materials
The materials used in this case study were leaded and unleaded CuNiSn alloys NP8
and CN8, already described under Section 6.1.1. Annealed monophase material was
studied, since this corresponds to the state of the matter when quenched. The heat-
treatment procedure is also described in Section 6.1.1.
7.1.2 Thermal properties
The thermal properties of CN8 and NP8 were considered to be the same: the thermal
conductivity, λ, as well as the thermal expansion coeficient, α, were determined
experimentally with CN8 material only.
The thermal expansion coefficient was measured between room temperature and
850 °C under argon in a differential dilatometer (Bähr Thermoanalyse, model DIL
802 V).
The evolution of λ with temperature was estimated up to 900 °C from experimental
values between RT and 100 °C by the ”stationary method” (constant heat flux
through a reference brass material and a Ø 18 mm, 50 mm long CN8 cylinder) and
from Refs. [225, 226]. The specific heat, C p , as well as the density, ρ, were also
estimated for the same temperature range from Refs. [225, 227].
The temperature evolution of an instrumented quenched bar was measured in order to
back-calculate the heat transfer coeficient, h(T) of varying quenching media (from
agitated water to ambient air). A CN8 Ø 18 mm bar was instrumented with three
Ø 1 mm Thermocoax ® thermocouples. One was located at the center of the bar, and
the tips ot the two others were located just below the bar surface, Fig. 7-1.

18
7.1 EXPERIMENTAL PROCEDURES
Figure 7-1: Section of the instrumented CN8 bar prolonged by a 2 m long stainless
steel tube. Thermocouple extension wires contained within the tube are not shown.
The CN8 bar attached to a 2 m long stainless steel tube was heated whithin a
resistance furnace to a homogeneous temperature of 900 °C, and subsequently
immersed in the quenching medium. The temperature acquisition rate was 10 Hz.
Such measurements were conducted in the cooling unit of the extrusion press on the
industrial site, as well as in agitated water, oil, fluidized bath or ambient air at the
laboratory.
CN8 bar
From material parameters and with the adequate thermal boundary conditions, the
exact temperature evolution, T(t,r) at any point whithin a quenched cylindrical bar
can be determined by FEM. Since it is assumed that only radial temperature gradients
occur, the thermal problem at hand is one-dimensional, Fig. 7-2.
The boundary conditions are:
Q( 0, l ) = 0
( )
150
TC tips
Eq.7-1
QR ( , l ) = h⋅2πRL⋅ Tinit −T∞
Eq.7-2
with Q, the heat flux, R and L, the bar radius and length, Tinit , the temperature of the
bar just before quenching, and T ∞ the temperature of the quenching medium.
60
1
Steel tube
143

CHAPTER 7. QUENCH CRACKING IN LEADED COPPER ALLOYS: A CASE STUDY
144
Figure 7-2: Parameters involved for the resolution of the thermal problem of a
quenched round bar by FEM. Axial symetry is assumed.
7.1.3 Mechanical analysis
l
r
R = D/2
C p, ρ, λ, T init
h, T
Mechanical properties of both CN8 and NP8 materials were measured between RT
and 800 °C, at 100 °C intervals. Elevated temperature tensile tests were performed
with the aid of the apparatus described in Section 6.1.2. A strain rate of 10 -2 s -1 was
chosen, since it corresponds roughly to the estimated strain rate during the quench.
The deformation of the sample was measured in all tests using a high-temperature
extensometer, in order to gather precise strain data. The dependence of the Poissons’
ratio, ν, on temperature was estimated up to 900 °C from the literature [228].
The elastoplastic mechanical properties of CN8 and NP8 materials were introduced
in a second FEM code to calculate the thermal displacement field from the T(t,r) and
α(T) values, Fig. 7-3. From this and the conditions for mechanical equilibrium, with
the (known) temperature-dependent alloy flow stress, the local constrained σ−ε
conditions were calculated. The temperature distribution and evolution is given by
the above-mentioned thermal problem, whereas the boundary conditions are:
ur( 0,l)= 0
ul ( r,0)=
0
ur ( R,l)= const1
ul ( r, L)= const2
Eq.7-3
Eq.7-4
Eq.7-5
Eq.7-6

7.1 EXPERIMENTAL PROCEDURES
Figure 7-3: Parameters involved for the resolution of the mechanical problem of a
quenched round bar by FEM. Axial symetry is assumed.
Finally, the transient thermal stresses are determined from the finite element
elastoplastic simulation knowing the thermal strain distribution, Fig. 7-4.
Figure 7-4: Schematic representation of a quenched round infinite bar. The
temperature and stress profiles are schematically added.
FEM calculations presented in what follows were conducted by Dr. Andreas Rossoll
at EPFL.
l
r
T(t, r), α, E, ν, σ − ε
R = D/2
145

CHAPTER 7. QUENCH CRACKING IN LEADED COPPER ALLOYS: A CASE STUDY
146
7.2 Results
7.2.1 Mechanical properties of CN8 & NP8 between RT and 800 °C
The evolution of the reduction in area at fracture, as well as the 400 °C tensile
properties of both CN8 and NP8 materials are shown on Fig. 6-9 and Fig. 6-10. At
400 °C, the ductility is significantly reduced by the addition of 1 wt.% lead. Fig. 7-5
shows the resulting fracture surface of CN8 and NP8 materials. Pb-containing NP8
fractures intergranularly, whereas the CN8 fracture is mixed transgranular and
intergranular. On the intergranular fracture surface of the NP8 material, traces of
intergranular lead inclusions are apparent. These hemispherical cups are empty;
suggesting that liquid lead migration occurred.
(a) CN8 dε/dt=10-2 s-1 RA=18% (b) NP8 dε/dt=10-2 s-1 RA=6%
Figure 7-5: Fracture surface after 400 °C tensile testing in air at dε/dt=10-2 s-1 for (a)
CN8, and (b) NP8 material in the monophased state (solutionized and quenched).
The embrittlement is much more marked at 800 °C: NP8 breaks before the onset of
global plasticity, whereas CN8 displays dynamic recrystallization, Fig. 7-6.

eng. stress [MPa]
120
100
80
60
40
20
0
7.2 RESULTS
Figure 7-6: 800 °C flow curves of CN8 & NP8 material at a strain rate of 10 -2 s -1 . For
each material, fracture is emphasized by a solid symbol.
7.2.2 Heat transfer coefficient of different quenching media
The measured temperature evolution at the center of a Ø 18 mm CN8 bar is shown on
Fig. 7-7. ”Noses” of the CN8 TTT-diagram are reproduced from [214]. Strictly
speaking, since the TTT-diagram is developped using small samples isothermally
transformed, the ”nose” of the curves should be moved somewhat to the right for
continuous cooling applications [181]. Hence can be concluded that a Ø 18 mm CN8
bar does not experience any phase transformation while quenched into water. It is
also not surprising that air-cooled material is not single-phased.
Notice that the cooling curves measured with our instrumented bar were very similar
for a water quench in the bath at our laboratory or in the industrial cooling unit of the
extrusion press.
800 °C
CN8
NP8
0 1 2 3 4 5 6
eng. strain [%]
CN8 & NP8
annealed
60 min at 850°C
ε = 10 -2 s -1
& quenched
.
147

CHAPTER 7. QUENCH CRACKING IN LEADED COPPER ALLOYS: A CASE STUDY
148
Temperature [°C]
800
700
600
500
400
300
200
100
water
oil
fluidized bath
Figure 7-7: Cooling curves of a CN8 Ø 18 mm bar immersed into different quenching
media. Temperature was measured at the center of the bar. The TTT-diagram (not CCT)
of the alloy is added.
Transient thermal stresses arise due to temperature gradients whithin a material
characterized by a non-zero expansion coefficient. Newtonian cooling is defined as
the case were temperature gradients are negligible. It is characterized by Bi ≤ 0.1,
where the Biot number is:
spinodal
0
0,1 1 10 100 1000 10 4
air
time [s]
hR
Bi =
λ
Eq.7-7
Charts displaying the temperature response of an infinite cylinder initially at a
uniform temperature, T init , and then subjected to a convective environnement at T ∞
are available in e.g. [181]. Further adimensional parameters, the Fourier number, Fo ,
and the equivalent temperature, T equ , are used:
λ t
Fo = 2
R Cpρ Eq.7-8
T − T∞
Tequ
=
Tinit − T∞
Eq.7-9
Relevant curves from the reference charts [181] are compared to present FEM results
in Fig. 7-8. Adimensionalization of our results was done by considering room-
DO 22
Grain Boundary +
intragranular γ
Discontinuous γ
L1 2 + DO 22

7.2 RESULTS
temperature values for the parameters λ, C p , and ρ. Values taken were 28 W/m K,
380 J/kg K, and 8.94 10 3 kg/m 3 respectively [225].
As seen, for our Ø 18 mm CN8 bar, air cooling can safely be considered as
Newtonian. Therefore negligible transient thermal stresses are expected. Water
quenching induces severe temperature gradients, while the fluidized bath, on the
other hand, seems to be a suitable cooling medium (thermal stresses are negligible)
since Bi is about 0.1.
T equ centre [-]
1
0.8
0.6
0.4
0.2
0
water
oil
fluidized bath
air
Bi = 20
0.1 1
Fo [-]
10
Figure 7-8: Adimensionalized cooling curves from Fig. 7-7 in comparison with
cooling curves from the analytical approach for variing Biot numbers from [181].
The heat transfer coeficient, h(T) of various quenching media was back-calculated by
FEM from experimental cooling data, Fig. 7-9. h describes the efficiency of the
quench. In the case of a water-quench, this parameter varies by more than an order of
magnitude in the temperature range studied. This is mainly due to phase
transformations in the quench-medium: boiling for water, cracking for oil. Due to
differences in chemical composition and molecular weight, each oil is characterized
by a different cooling capability. The latter can also be tailored by small additions of
water. Concerning the fluidized bath, no phase transformation occurs. We
accordingly find a smooth variation of h. Estimation of the heat transfer coeficient
according to the analytical equations describing cooling at constant h [181], Fig. 7-8
lead to values for h ranging between 6 and 60 kW/m 2 K (Bi=2 to 20) for water
quenching. These values agree well with those determined by FEM, Fig. 7-9.
2
4
0.4
0.8
0.2
Bi = 0.1
149

CHAPTER 7. QUENCH CRACKING IN LEADED COPPER ALLOYS: A CASE STUDY
150
[kWm -2 K -1 ]
h
100
10
1
0.1
0.01
0.001
water
0 100 200 300 400 500 600 700 800 900
Temperature [°C]
Figure 7-9: Heat transfer coeficient for different quenching media calculated by FEM
[Andreas Rossoll, EPFL].
FEM simulation-derived transient thermal stresses are plotted as a function of
temperature, and compared to the ultimate tensile strengh of both CN8 and NP8
materials, Fig. 7-10. Transient stresses plotted are the principal stresses acting at the
surface of a Ø 18 mm bar quenched into water or fluidized bath. It is interesting to
note the presence of compressive residual stresses at the surface of the quenched bar.
These are due to the occurence of plastic deformation during the quench.
Fig. 7-10 shows that the mechanical resistance of the CN8 material always exceeds
the transient thermal stresses in the whole temperature range, for both quench
conditions. On the other hand, this is not the case for the lead-containing NP8
material. In conclusion, NP8 Ø 18 mm bars should crack during water-quenching,
but not during a quench in the fluidized bath.
oil
fluidized bath
air

σ surf or R m [MPa]
600
500
400
300
200
100
0
-100
-200
-300
σsurf water quench
σsurf cooling in fluidized bath
0 100 200 300 400 500 600 700 800 900
Temperature [°C]
CN8
NP8
7.2 RESULTS
Figure 7-10: FEM calculated surface transient thermal stresses for a Ø 18 mm bar
water-quenched or cooled in fluidized bath are compared to CN8 and NP8 tensile
strength (each symbol is the result of a single tensile test).
R m
R m
151

CHAPTER 7. QUENCH CRACKING IN LEADED COPPER ALLOYS: A CASE STUDY
152
7.3 Implications
It is clear from Fig. 7-10 that water-quenched NP8 bars, 18 mm in diameter, crack
because of transient thermal stresses. This is a manifestation of LME. It is recalled
that, in production, this material is rapidly cooled from the extrusion temperature to
avoid the occurence of spinodal decomposition; since the material in the monophased
state is most suitable for subsequent cold working. We show on Fig. 7-7 that water is
the only quenching medium that ensures sufficiently rapid cooling for Ø 18 mm bars.
Therefore a trade-off must be found between rapid cooling to avoid phase
transformation, and moderate cooling to ensure low thermal stresses.
We propose three main strategies to overcome this problem: (i) lower the diameter of
the extruded bar, (ii) change the cooling fluid, or (iii) change the alloy composition.
7.3.1 Bar diameter
Cooling curves can be extrapolated for any bar dimension, provided that the bar
geometry and the cooling medium are kept constant. Namely the T equ -Fo curve is
intrinsic for any cooling medium, Fig. 7-8. We show thus on Fig. 7-11 that 6 mm is
the critical radius for a bar to experience no phase transformation during quenching
in a fluidized bath.

Temperature [°C]
800
700
600
500
400
300
200
100
7.3 IMPLICATIONS
Figure 7-11: Calculated cooling curves for a CN8 bar of different diameters immersed
in fluidized bath. TTT-diagram is added [214].
7.3.2 Cooling medium
ø 8
ø 6
ø 4
ø 18
ø 12
0
0,1 1 10 100 1000 10 4
time [s]
CuNiSn alloys are intended to compete with the Cu-Be alloy family. The very high
strength of both of these alloy classes is obtained by strain hardening and subsequent
precipitation hardening. In Cu15Ni8Sn alloys, nucleation of discontinuous γ-
precipitates corresponds to a loss in hardening [224]. The matrix is in any case
depleted in the alloying elements that form ordered precipitates (spinodal, DO 22 or
L1 2 ). Therefore the ”nose” corresponding to the appearance of the discontinuous γ-
precipitates must definitely be avoided during alloy processing. Fig. 7-7 suggests that
air-cooling is not sufficiently rapid to avoid discontinuous precipitation of the γ-
phase; however, this is not the case for a quench in a fluidized bath.
The industrial production of NP8 could therefore become possible with an
appropriate cooling fluid, the severity of which is lower than that of an oil but higher
than that of agitated air. It is recognized in [229] that fluidized-beds are used as
cooling media only to a limited extend. Alternate media according to [229] are slow
DO 22
spinodal
Grain Boundary +
intragranular γ
Discontinuous γ
L1 2 + DO 22
oils, aqueous polymer solutions, forced gases, fogs, sprays or dry dies.
153

CHAPTER 7. QUENCH CRACKING IN LEADED COPPER ALLOYS: A CASE STUDY
154
7.3.3 Alloy composition
Cu15Ni8Sn alloys are characterized by very rapid kinetics of phase transformation,
since spinodal decomposition starts after only 4 s at 400 °C, Fig. 7-12 (a) [214]. It is
also known from the literature that Cu-7.5Ni-5Sn alloys are characterized by much
slower kinetics, Fig. 7-12 (b). Notice that earlier resistivity measurements indicated
the occurence of a phase transformation after a 100 s isothermal stay at 400 °C for a
Cu-9Ni-6Sn alloy [230]. Reducing the alloying element contents renders the water
quenching-step unnecessary; however, this lowers the peak hardness obtainable
[223].
(a) (b)
Figure 7-12: TTT-diagrams (a) of the Cu15Ni8Sn alloy [214], and (b) of the
Cu7,5Ni5Sn alloy [231] obtained by TEM characterizations and electrical resistivity
measurements.

7.4 Conclusion
7.4 CONCLUSION
We have shown that water-quenched NP8 bars, 18 mm in diameter, crack because of
transient thermal stresses during the quench. Failure is under these circumstances can
be attributed to liquid metal embrittlement.
We propose three main strategies to overcome this problem:
(i) lower the diameter of the extruded bar,
(ii) change the cooling fluid, or
(iii) change the alloy composition.
155

CHAPTER 7. QUENCH CRACKING IN LEADED COPPER ALLOYS: A CASE STUDY
156

Chapter 8
8 General conclusions
• A new method is presented for the determination of the dihedral angle of individual
intergranular formerly liquid inclusions. The angle is derived from a mathematical
fitting of the solid/liquid interface 3-D geometry around individual inclusions, and
therefore reflects the dihedral angle dictated by global energy minimization of the
inclusion shape under capillary forces.
• In a high-purity Cu-1 wt.% Pb alloy, and for a specific temperature, we show that φ
is not unique; this reflects the fact that high-angle grain boundary energies vary in the
metal.
• A decrease of φ was measured on Cu-1Pb-0.04P samples subjected to interrupted
uniaxial tensile creep tests at 400 °C, as compared to φ measured on unstressed
specimens. φ values are best described by predictions accounting for the shape
evolution of a zero bulk modulus inclusion within a uniaxially stressed solid,
suggesting that voids nucleate within the liquid inclusions or along the matrix/
inclusion interface.
• We demonstrate that LME as well as GBE operate in leaded-copper at intermediate
temperature. Their respective importance depends mainly on the strain rate.
• According to analysis accounting for inclusion shape stability, we calculate a
critical parameter Λ c , above which inclusions are predicted to collapse to a crack.
157

CHAPTER 8. GENERAL CONCLUSIONS
158
2 1/ 3
σ ⋅ V
Λ =
γ SL ⋅ E
and
Λc = max( Λ) ⇒ σ c =
γ SL ⋅ E
Λc
1/ 3
V
Eq.8-1
The calculated respective critical stress σc is found to be in good agreement with the
respective fracture stress of both leaded pure copper and leaded copper-nickel-tin
alloys tested here, found to fail by LME at high strain-rate.

Chapter 9
9 Future work
In a continuation of this project, it would be of interest to:
(i) Couple the present individual grain boundary inclusion dihedral angle
measurement method with EBSD methods for characterization of grain orientation in
the scanning electron microscope. This would in particular extend the present method
for dihedral angle measurement to provide a new indirect SEM-based method for the
measurement of grain boundary energies.
(ii) Further heat treatments under reducing atmosphere should be performed in order
to cover the whole temperature range between 350 and 1000 °C. Dihedral angle
measurements should be performed in parallel with EBSD analysis in order to check
whether the φ variations observed at a specific temperature reflects indeed the γ gb
anisotropy.
(iii) Make more measurements of φ from fractographs of the quenched NP8, so as to
determine a representative distribution of inclusion sizes.
Applying also that technique to tensile tested coarse-grained C99 and Cu1Pb and
comparing those measurements with our standard procedure (i.e. after metallographic
preparation and selective dissolution) would also be useful to ascertain the validity of
using fracture surfaces.
159

CHAPTER 9. FUTURE WORK
160
(iv) From the known evolution with temperature of both the elastic properties and
interfacial energies, the critical parameter Λ c (T) could be calculated as a function of
temperature. For a given inclusion size, the critical stress associated with LME
fracture σ c (T) could then be estimated. A new line could be accordingly drawn on the
Ashby fracture-mechanisms map, Fig. 3-18.
(v) In order to further verify the influence of the liquid inclusion size on σ c , tensile
tests should be performed at 400 °C, varying both V and , all else being kept
ε •
constant. With variations between 10 -4 to 10 s -1 , the material would therefore be
subjected to maximum stresses ranging from about 70 to 150 MPa for leaded pure
copper, exhibiting LME failure or not, depending on the (largest) inclusion size.
Hardening of the copper matrix (e.g. by oxide dispersion strenghtening) would also
be opportune, since a linear elastic solid is considered in the Eshelby analysis.
(vi) Since an increase of φ is predicted to enhance the resistance to LME, small
additions should be made to leaded alloys in order to decrease γ gb or increase γ SL ,
resulting respectively from beneficial segregation or liquid metal alloying.
(vii) Dihedral angle measurements should also be performed on samples subjected to
elevated temperature hydrostatic compression. This mainly for two reasons: (a) it
could be checked whether the dihedral angle is correctly predicted making the
assumption that the inclusion has a finite positive compressibility (no voiding); and
(b) the microstructure is expected to be at equilibrium, since neither global plasticity,
nor grain boundary motion, should occur to a significant degree.
(viii) Finally, the mechanical analysis should be extended, to address the influence of
plastic and viscous deformation of the surrounding metal, coupled perhaps with grain
boundary sliding, as all are indeed observed in the experimental data. This is a
formidable task, requiring finite-element simulation to be conducted in realistic
fashion. The link of the present analysis with fracture mechanics, which is relatively
obvious given the nature of the critical parameter Λ c , may perhaps provide pathways
easing this exploration.
ε •

Chapter 10
10 Appendices
10.1 Least square method and dihedral angle measurement
The MeX program allows to get the coordinates of points belonging to the former
solid/liquid interface. Assuming that these points belong to a spherical cap, their
coordinates (x i ;y i ;z i ) obey:
( ) + ( − ) + ( − ) − =
f( a, b, c, R)≡ x −a
y b z c R
2 2 2 2
i i i i
where (a;b;c) are the coordinates of the center of the sphere of radius R.
With more than four measuring points:
2
2
2 2
( x1−a) + ( y1−b) + ( z1−c) − R = f1( a, b, c, R)+ r1
2
2
2 2
( x2−a) + ( y2−b) + ( z2−c) − R = f2( a, b, c, R)+ r2
( ) + ( − ) + ( − ) − =
2 2 2 2
xn − a yn b zn c R fn( a, b, c,R)+
rn the system is over-determined, with no solution for a, b, c and R in general.
Eq.10-1
Eq.10-2
a, b, c and R are therefore calculated as the values that give the lowest mean-square
deviation of f from zero over all points of measurement (x i ;y i ;z i ). Since function f is
of second order, it is approximated by its Taylor series about an approximated
solution a o , b o , c o and R o calculated analytically from four measured points:
0
161

CHAPTER 10. APPENDICES
162
with
this yields a system of linear equations:
with
4
∂f
fi( a, b, c, R)≈ fi( a0, b0, c0, R0)+
∑ ( a0, b0, c0, R0)⋅ xk+ ri
∂ x ∆
x
k = 1
⎛ a⎞
⎛ a0
⎞ ⎛ ∆a⎞
⎜ b⎟
0
= ⎜ ⎟ = x + ∆x
⎜ c⎟
⎜ ⎟
⎝ R⎠
⎜ b ⎟ ⎜ b⎟
0 ∆
= ⎜ ⎟ + ⎜ ⎟
⎜ c0
⎟ ⎜ ∆c
⎟
⎜ ⎟ ⎜ ⎟
⎝ R ⎠ ⎝∆
R⎠
k k k
y = A ∆ x + r
[ n× 1] [ n× m] [ m× 1] [ n×
1]
⎡ ∂f
⎢ ∂a
⎢∂f
A=
⎢
⎢ ∂a
⎢
⎢∂f
⎣⎢
∂a
⎛ ∆a⎞
⎜ ∆b⎟
∆ x = ⎜ ⎟
⎜ ∆c
⎟
⎜ ⎟
⎝∆
R⎠
∂f
∂b
∂f
∂b
∂f
∂b
∂f
∂c
∂f
∂c
∂f
∂c
0
∂f
⎤
∂R
⎥
∂f
⎥
⎥
∂R
⎥
⎥
∂f
⎥
∂R
⎦
1 1 1 1
2 2 2 2
n n n n
⎥
( a , b , c , R )
0 0 0 0
( )− ( )
( )− ( )
⎡ f1 a, b, c, R f1 a0, b0, c0, R0
⎤
⎢
⎥
y = ⎢
f2 a, b, c, R f2 a0, b0, c0, R0
⎥
⎢
⎥
⎢
⎥
⎣⎢
fn( a, b, c, R)− fn( a0, b0, c0, R0)
⎦⎥
⎡r1
⎤
⎢r
⎥
2
r = ⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣rn
⎦
k
Eq.10-3
Eq.10-4
Eq.10-5
Eq.10-6
Eq.10-7
Eq.10-8
Eq.10-9

Its solution is ∆ ˆx :
∆ =( ) − T 1 T
ˆx A A A y
10.1 LEAST SQUARE METHOD AND DIHEDRAL ANGLE MEASUREMENT
Eq.10-10
The coordinates of the center of the sphere and its radius, respectively (a;b;c;R) are
finally obtained by additing ∆ ˆx
to the approximate solution (ao ;bo ;co ;Ro ).
The Mathcad spreadsheet developed to this end is given below for the specific case
of a grain boundary inclusion after shape equilibration at 930 °C. Measured φ was
78°, Fig. 4-7.
163

CHAPTER 10. APPENDICES
164
Determination of the equation of spheres knowing >4 points from its
surface with the least square method
1) Equation of the spheres from an approached solution
sphere A:
A1,B1,C1,D1: 4 points from the sphere surface with center S1 and radius r1
A1 :=
Given
�
�
�
�
vec1 := Find S1 , S1 , S1 , r1
0 1 2
vec1 =
−0.67
8.68
6.16
Approximations:
( A1 − S1
0 0)
2
( A1 − S1
1 1)
2
+ ( A1 − S1
2 2)
2
+ r1 2
− = 0
( B1 − S1
0 0)
2
( B1 − S1
1 1)
2
+ ( B1 − S1
2 2)
2
+ r1 2
− = 0
( C1 − S1
0 0)
2
( C1 − S1
1 1)
2
+ ( C1 − S1
2 2)
2
+ r1 2
− = 0
( D1 − S1
0 0)
2
( D1 − S1
1 1)
2
+ ( D1 − S1
2 2)
2
+ r1 2
− = 0
�
�
�
�
�
�
�
�
10.715
−0.13
9.83
14.856
( )
�
�
�
�
B1 :=
�
�
�
�
−0.51
−8.96
5.74
�
�
S1 :=
�
�
�
�
C1 :=
−5
−5
5
�
�
�
�
�
�
Center & radius of the sphere A
centreA :=
−0.26
1.97
0.04
�
�
�
�
�
�
�
�
r1 := 10
vec1 0
vec1 1
vec1 2
�
�
�
�
D1 :=
�
�
�
�
−3.23
−1.59
4.92
�
�
rayonA := vec1
3

centreA =
Sphere A
�
�
�
�
10.715
−0.13
9.83
XA0 := centreA
0
YA0 := centreA
1
ZA0 := centreA
2
RA0 := rayonA
�
�
10.1 LEAST SQUARE METHOD AND DIHEDRAL ANGLE MEASUREMENT
sphere B:
A2,B2,C2,D2: 4 points from the sphere surface with center S2 and radius r2
�
0.12
0.10
0.22
A2 := � 8.62 B2 := � −7.80
C2 := � 1.35 D2 :=
�
�
6.59
6.10
0.11
�
�
�
�
�
�
Approximations:
Given
S2 := 5
�
� 5�
r2 := 10
( A2 − S2
0 0)
2
( A2 − S2
1 1)
2
+ ( A2 − S2
2 2)
2
+ r2 2
− = 0
( B2 − S2
0 0)
2
( B2 − S2
1 1)
2
+ ( B2 − S2
2 2)
2
+ r2 2
− = 0
( C2 − S2
0 0)
2
( C2 − S2
1 1)
2
+ ( C2 − S2
2 2)
2
+ r2 2
− = 0
( D2 − S2
0 0)
2
( D2 − S2
1 1)
2
+ ( D2 − S2
2 2)
2
+ r2 2
− = 0
vec2 := Find S2 , S2 , S2 , r2
0 1 2
vec2 =
�
�
�
�
�
�
−13.049
0.367
8.327
15.639
( )
�
�
�
�
rayonA = 14.856
�
�
�
�
5
�
centreB :=
�
�
�
�
�
�
�
�
Sphere B
centreB =
vec2 0
vec2 1
vec2 2
�
�
�
�
�
�
�
�
�
�
−13.049
0.367
8.327
XB0 := centreB
0
YB0 := centreB
1
ZB0 := centreB
2
RB0 := rayonB
�
�
�
�
�
�
2.38
−1.48
6.57
Center & radius of the sphere B
rayonB := vec2
3
�
�
rayonB = 15.639
165

CHAPTER 10. APPENDICES
166
2) Import the files containing m>4 points x i , y i et z i
XYZA :=
mA := rows( XYZA)
nA := mA − 1
nA 6.462 10 3
= ×
AA
iA, 0
AA
iA, 1
AA
iA, 2
AA
iA, 3
iA := 0.. nA
XA :=
iA
YA :=
iA
ZA :=
iA
3) Determine the matrices A (mx4)
:= 2⋅ XA0 − XA
iA
:= 2⋅ YA0 − YA
iA
:= 2⋅ ZA0 − ZA
iA
:=
\\Imxlmmpc21\..\profiles_exp15_sphere A.xls
Number of measuring points: n
XYZA
iA, 0
XYZA
iA, 1
XYZA
iA, 2
( )
( )
( )
−2RA0 XYZB :=
mB := rows( XYZB)
nB := mB − 1
nB 5.091 10 3
= ×
iB := 0.. nB
XB :=
iB
YB :=
iB
ZB :=
iB
AB
iB, 0
AB
iB, 1
AB
iB, 2
AB
iB, 3
\\Imxlmmpc21\..\profiles_exp15_sphere B.xls
XYZB
iB, 0
XYZB
iB, 1
XYZB
iB, 2
:= 2⋅ XB0 − XB
iB
:= 2⋅ YB0 − YB
iB
:= 2⋅ ZB0 − ZB
iB
:=
( )
( )
( )
−2RB0

Sphere A:
yA
iA, 0
Sphere B:
yB
iB, 0
4) Determine the matrices y (mx1)
∆xA =
( ) 1
∆xA AA T AA
−
:=
�
�
�
�
�
�
centerA :=
−0.664
0.237
−0.688
−0.959
�
�
�
�
�
radiusA := RA0 + ∆xA3 centerA =
�
�
�
�
0
1
2
radiusA = 13.897
10.1 LEAST SQUARE METHOD AND DIHEDRAL ANGLE MEASUREMENT
( ) 2
( ) 2
RB0 2
:= − XB − XB0 − YB − YB0 − ZB − ZB0
iB iB iB
5) We can determine ∆x = (A T A) -1 A T y
AA T yA
6) The precise solution is: (X0+∆x 0 ;Y0+∆x 1 ;Z0+∆x 2 ) for the center
XA0 + ∆xA0
YA0 + ∆xA1
ZA0 + ∆xA2
0
10.051
0.107
9.141
�
�
�
( ) 2
( ) 2
∆xB AB T AB
−
:=
∆xB =
( ) 2
( ) 1
�
�
�
�
�
�
0.538
−0.028
−0.151
−0.503
R0+∆x 3 for the radius
( ) 2
RA0 2
:= − XA − XA0 − YA − YA0 − ZA − ZA0
iA iA iA
centerB :=
�
�
�
�
�
radiusB := RB0 + ∆xB3 centerB =
�
�
�
�
XB0 + ∆xB0
YB0 + ∆xB1
ZB0 + ∆xB2
0
1
2
AB T yB
0
-12.511
0.339
8.176
radiusB = 15.136
�
�
�
167

CHAPTER 10. APPENDICES
168
x:= 6
Given
7) Equations of the spheres:
sphereB( x, y,
z)
:= x − centerB
0
8) Determination of a point at the intersection of the two spheres
sphereA( x, y,
z)
= 0
sphereB( x, y,
z)
= 0
inter1 := Find( x, y,
z)
inter1 =
9) Vectors between the center of the spheres and the intersection point: V1 et V2
V1 := centerA − inter1
V1 =
�
�
�
�
�
�
�
�
( ) 2
sphereA( x, y,
z)
:= x − centerA
0
( ) 2
y := 5
−0.093
6.656
2.262
10.144
−6.549
6.879
�
�
10) Dihedral angle: φ = 180° - angle V1V2
φ := 180 −
�
�
�
�
�
180
π acos ⋅
φ = 77.95
( y − centerA
1)
2
+ ( z − centerA
2)
2
+ radiusA 2
−
( y − centerB
1)
2
+ ( z − centerB
2)
2
+ radiusB 2
−
z:= 3
V2 := centerB − inter1
−12.418
V2 = −6.317
5.914
�
�
�
�
�
�
�
V1⋅V2 V1 ⋅ V2
�
�
��
��

xa
i, j
ya
i, j
za
i, j
10.1 LEAST SQUARE METHOD AND DIHEDRAL ANGLE MEASUREMENT
11) Representation of the spheres and the measuring points
i := 0.. 50 j := 0.. 50
θi i 2π
:= ⋅ φj j
50
π
:= ⋅
50
sphere A sphere B
( )
( )
:= centerA + radiusA⋅sin φ
0
j ⋅cos
θi xb := centerB + radiusB⋅sin φ
i, j 0
j ⋅cos
θi ( )
( )
:= centerA + radiusA⋅sin φ
1
j ⋅sin
θi yb := centerB + radiusB⋅sin φ
i, j 1
j ⋅sin
θi ( )
:= centerA + radiusA⋅cos φ
2
j
zb := centerB +
i, j 2
( xa, ya,
za)
, ( xb, yb,
zb)
, ( XA, YA,
ZA)
,
( XB, YB,
ZB)
( )
( )
radiusB⋅cos( φj) ( )
( )
169

CHAPTER 10. APPENDICES
170
10.2 Bulk modulus of liquid lead
According to Iida and Guthrie [206] p.93,The isentropic compressibility of a liquid
κS is dependent on its density ρ, and on the velocity of sound in this medium U:
1 ∂V
⎞
κ S =− ⎟
V ∂p
⎠
1 1
U = → κ S = 2
ρκ ρ U
S
Moreover the isothermal compressibility κ T is:
1 ∂V
⎞
κ T =− ⎟
V ∂p
⎠
Eq.10-11
Eq.10-12
Eq.10-13
2
Cp
α MT
κT = κS = κS+
CV
ρ Cp
Eq.10-14
where Cp and Cv are the heat capacity at constant pressure and volume respectively,
M is the atomic weight, and α is the isobaric thermal expansivity:
α =
1
V
∂V
⎞
⎟
∂ ⎠
T p
Combining Eq. 10-12 and Eq. 10-14, we have:
κ
T
2
1 α MT
= + 2
ρ U ρ C
The bulk modulus K is therefore:
S
T
p
Eq.10-15
Eq.10-16
1
K =
κ T
1
=
2
1 α MT
+ 2
ρ U ρ
Cp
Eq.10-17
The velocity of sound as a function of temperature is tabulated. A linear decrease is
assumed from the incipient melting temperature T m [206]:

A linear decrease with temperature is assumed for ρ [206]:
10.2 BULK MODULUS OF LIQUID LEAD
Eq.10-18
3 3 kg
ρ( T)= 10. 67 ⋅10 −1. 32( T −Tm)→
ρ(
400° C)=
10. 57 ⋅10
⎡ ⎤
3
⎣⎢ m ⎦⎥
Eq.10-19
The heat capacity Cp is also assumed to decrease linearly with temperature [206]:
With
UT ( )= 1810 − 0. 38( T−Tm)→ U( 400° C
⎡
)= 1782
⎣⎢
2
−3
⎡ kg m
Cp( T)= 32. 43 −3. 10 ⋅ 10 ( T)→ Cp( 400° C)=
30. 34 ⎢
⎣ K mol s
T K 600
m
we have:
= [ ]
− kg
M = 207 5⋅10 ⎡ ⎤
⎣⎢ mol ⎦⎥
3
.
−4 −1
α = 123 . ⋅ 10 [ K ]
K 400 C 27 5 GPa
° ( )= [ ] .
Pb
m
s
⎤
⎦⎥
2
⎤
⎥
⎦
Eq.10-20
Eq.10-21
Eq.10-22
Eq.10-23
Eq.10-24
171

CHAPTER 10. APPENDICES
172
10.3 Eshelby analysis
The Mathcad spreadsheet is given below for the specific case of a grain boundary
liquid inclusion, lenticular in shape. The Λ(c) curve obtained is Fig. 5-8 (a).
φo := 120deg
Stable intergranular lenticular inclusion:
Aspect ratio : b/a = c < 1
Volume of the inclusion: Vol
γSL 0.432 J
m 2
:= ⋅
�
�
γgb 2γSL cos φo
:= ⋅ �
2
�
�
Area of the lenticular inclusion is :
2
3
3 2
Area( Vol, c)
:= Vol ⋅2⋅π⋅1 + c ⋅
( )
�
�
�
3 1
π
c 3 c 2
⋅
⋅ +
( )
Interfacial energy of the inclusion is Γ=ΓSL - Γgb:
ΓSL( Vol, c)
:= γSL⋅Area( Vol, c)
2
3
Γgb( Vol, c)
γSL⋅Vol ⋅2 cos φo
3
:=
⋅ � ⋅π⋅
2
�
�
�
�
Γ( Vol, c)
:=
ΓSL( Vol, c)
− Γgb( Vol, c)
�
�
�
2
�
�
�
3 1
π
c 3 c 2
⋅
⋅ +
( )
�
�
�
2

ECu 111 10 9
:= ⋅ ⋅Pa
ν := 0.355
C :=
C =
C11 :=
C12 :=
Eshelby analysis for an oblate spheroid
Cu elastic modulus at 400°C
Cu Poissons' ratio at 400°C
ECu⋅( 1 − ν)
( 1 + ν)
⋅(
1 − 2⋅ν) ECu⋅ν 1 + ν ⋅ −
ECu
C44 :=
21+ ν
( ) ( 1 2⋅ν) ( )
Elastic constants of Cu, taken as isotropic
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
C11
C12
C12
0
0
0
C12
C11
C12
0
0
0
1.822 10 11
×
1.003 10 11
×
1.003 10 11
×
0
0
0
C12
C12
C11
0
0
0
KPb 27.5 10 9
:= ⋅ ⋅Pa
0
0
0
C44
0
0
0
0
0
0
C44
0
1.003 10 11
×
1.822 10 11
×
1.003 10 11
×
0
0
0
0
0
0
0
0
C44
�
�
�
�
�
�
1.003 10 11
×
1.003 10 11
×
1.822 10 11
×
Pb bulk modulus at 400°C
0
0
0
0
0
0
4.096 10 10
×
Elastic constant of liquid Pb: G=0 (no resistance to shear)
�
KPb KPb KPb 0 0 0
� KPb
�
KPb
Cstar :=
�
0
KPb
KPb
0
KPb
KPb
0
0
0
0
0
0
0
0
0
0
0 0 0 0 0 0
0 0 0 0 0 0
�
�
�
�
�
�
�
�
�
�
0
0
0
0
0
0
4.096 10 10
×
0
10.3 ESHELBY ANALYSIS
0
0
0
0
0
4.096 10 10
×
�
�
�
�
�
�
�
�
�
Pa
173

CHAPTER 10. APPENDICES
174
Sc ( )
aspect ratio of the oblate spheroid: c=a3/a1

x1 :=
x1b ξ
( ) := Q( θ)
⋅x1b(
ξ)
x1c ξ, θ
�
�
�
�
(ii) rotation de θ autour de x2b -> x1c x2c x3c
Transformation matrix: Q
Cosine matrix: Aij
( ) := x1c( ξ, θ)
⋅x1
A11 ξ, θ
( ) := x2c( ξ, θ)
⋅x1
A21 ξ, θ
( ) := x3c( ξ, θ)
⋅x1
A31 ξ, θ
1
0
0
�
�
Transformation matrix : P
( ) := P( ξ)
⋅x1
Stress tensor
-> after successive ξ and θ rotation
A( ξ, θ)
x2 :=
(i) rotation de ξ autour de x1 -> x1b x2b x3b
x2b ξ
:=
x2c ξ, θ
�
�
�
�
�
�
�
�
0
1
0
�
�
P( ξ)
:=
( ) := P( ξ)
⋅x2
Q( θ)
( ) := Q( θ)
⋅x2b(
ξ)
A12 ξ, θ
A22 ξ, θ
A32 ξ, θ
( )
( )
( )
A11 ξ, θ
A21 ξ, θ
A31 ξ, θ
�
�
�
�
:=
1 0
0 cos ξ
0 −sin
ξ
�
�
�
�
cos( θ)
0
sin θ
( )
( )
( )
( ) := x1c( ξ, θ)
⋅x2
( ) := x2c( ξ, θ)
⋅x2
( ) := x3c( ξ, θ)
⋅x2
( )
( )
( )
A12 ξ, θ
A22 ξ, θ
A32 ξ, θ
�
( )
( ) �
0
sin ξ
cos ξ
x3 :=
( ) �
0 −sin
θ
1 0
0 cos( θ)
( ) �
( )
( ) �
A13 ξ, θ
A23 ξ, θ
A33 ξ, θ
x3b ξ
�
�
�
�
�
0
0
1
�
�
x3c ξ, θ
A13 ξ, θ
A23 ξ, θ
A33 ξ, θ
10.3 ESHELBY ANALYSIS
( ) := P( ξ)
⋅x3
( ) := Q( θ)
⋅x3b(
ξ)
( ) := x1c( ξ, θ)
⋅x3
( ) := x2c( ξ, θ)
⋅x3
( ) := x3c( ξ, θ)
⋅x3
175

CHAPTER 10. APPENDICES
176
Stress tensor before transformation: σij
Stress tensor after transformation: σkl
σ1( ξ, θ)
σ2( ξ, θ)
σ3( ξ, θ)
σ4( ξ, θ)
σ5( ξ, θ)
σ6( ξ, θ)
stress:
strain:
eigenstrain ε∗
2 2
:=
A( ξ, θ)
0i , ⋅A( ξ, θ)
0, j⋅σij
� � ( i, j)
i= 0 j = 0
2 2
:=
A( ξ, θ)
1i , ⋅A( ξ, θ)
1, j⋅σij
� � ( i, j)
i= 0 j = 0
2 2
:=
A( ξ, θ)
2i , ⋅A( ξ, θ)
2, j⋅σij
� � ( i, j)
i= 0 j = 0
2 2
:=
A( ξ, θ)
1i , ⋅A( ξ, θ)
2, j⋅σij
� � ( i, j)
i= 0 j = 0
2
:= �
i= 0
2
A( ξ, θ)
0i , ⋅A( ξ, θ)
2, j⋅σij
� ( i, j)
j = 0
2
:= �
i= 0
2
( A( ξ, θ)
0i , ⋅A( ξ, θ)
1, j⋅σij
� i, j)
j = 0
( ) := C S c
εstar c, σ , ξ,
θ
( )
σ° σ, ξ,
θ
ε° σ, ξ,
θ
Interaction strain energy: ∆W=-1/2 Vol σ° ε∗
[ ⋅( ( ) − I)
− Cstar⋅S( c)
] 1 −
⋅ ( Cstar − C)
⋅ε°
σ, ξ,
θ
−1
( )
∆W Vol, c,
σ,
ξ,
θ
( ) C 1
:=
:=
�
�
�
�
�
�
�
�
�
:=
( ) �
(
(
(
(
) �
) �
)
�
�
) �
( ) �
σ1 ξ, θ
σ2 ξ, θ
σ3 ξ, θ
σ4 ξ, θ
σ5 ξ, θ
σ6 ξ, θ
�
0 0 0
σij := � 0 0 0
�
0 0 1
�
10 6
⋅ ⋅σ⋅Pa ( )
− ⋅σ°
σ, ξ,
θ
�� ( ) ��
( )
( )
2 Vol ⋅ σ° σ, ξ θ , ⋅ ⋅εstar
c, σ , ξ,
θ
�
�

Fcξ , , θ
Adimensionalisation:
Gc ( ) 2⋅π 1 c 2
3
3 1
⋅(
+ )
π
c 3 c 2
2
⋅ � ⋅ � 2 cos
� �
⋅ +
φo
3
� �
:=
− ⋅ � ⋅π⋅
� 2 �
�
�
�
−1
�
( ) := �
2
�
�
�
�
Λ( c, 0deg,
0deg)
σ1 ξ, θ
�
(
(
(
(
(
,
,
) � �
�
) � �
) � �
)
�⋅�
� �
) � �
( ) �
� �
σ2 ξ, θ
σ5 ξ, θ
σ6 ξ, θ
( )
�
�
�
�
3 1
π
c 3 c 2
⋅
⋅ +
( )
σ3 ξ θ
⋅ [ C + ( Cstar − C)
⋅S(
c)
]
σ4 ξ θ
1 −
( C − Cstar)
ECu C 1 −
⋅�
⋅�
⋅ ⋅
3
2.5
2
1.5
1
0.5
0
0.6
Λ c, ξ,
θ
Λ=σ 2 V 1/3 /γ SL E
0.53
d
�
�
�
�
�
�
�
�
( )
c ( ) Gc
d
0.47
c Fcξ , θ ,
:= −
d
d
0.4
( )
0.34
c
0.27
�
�
�
�
�
�
�
�
0.21
10.3 ESHELBY ANALYSIS
2
�
��
0.14
�
�
�
�
�
�
�
�
�
( ) ����
���
( ) ����
( ) ����
( )
����
����
( ) ����
( ) ���
����
σ1 ξ, θ
σ2 ξ, θ
σ3 ξ, θ
σ4 ξ, θ
σ5 ξ, θ
σ6 ξ, θ
0.076
0.01
177

CHAPTER 10. APPENDICES
178
10.4 Measured values after interrupted creep test
The measured dihedral angles in 400 °C crept C99 samples at different stresses are
listed. For each single inclusion treated, its specific volume and the orientation of the
grain boundary are indicated. The adimensional parameter Λ is also added:
2 1/ 3
σ app ⋅
Λ=
γ ⋅
V
E
σ app [MPa] φ [°] V [10 -16 m 3 ] ξ [°] θ [°] Λ [-]
0 102
117
125
117
43 125
88
59 79
96
74
69 *
65
84
0.54
0.26
1.06
7.38
2.93
6.32
1.20
1.64
3.32
5.28
14.11
2.08
-
-
-
-
3
64
38
86
54
37
71
10
-
-
-
-
3
8
0
0
7
25
10
0
Eq. 5-11
0
0
0
0
0.256
0.331
0.358
0.398
0.503
0.587
0.814
0.430
75 88 1.93 62 7 0.678

Chapter 11
11 List of References
1.Kuyucak, S. and M. Sahoo, A review of the machinability of copper-base alloys.
Canadian Metallurgical Quarterly, 1996. 35(1): p. 1-15.
2.Smith, C.S., Grains, Phases, and Interfaces — an Interpretation of Microstructure.
Transactions of the American Institute of Mining and Metallurgical Engineers, 1948.
175: p. 15-51.
3.Lupis, C.H.P., Chemical Thermodynamics of Materials. 1983, New York,
Amsterdam: North-Holland, Elsevier. 375-380.
4.Wray, P.J., The geometry of two-phase aggregates in which the shape of the second
phase is determined by its dihedral angle. Acta Metallurgica, 1976. 24(2): p. 125-
135.
5.Eustathopoulos, N., Energetics of solid/liquid interfaces of metals and alloys.
International Metals Reviews, 1983. 28(4): p. 189-210.
6.Mori, T., et al., Determination of the Energies of 001 Twist Boundaries in Cu with
the Shape of Boundary SiO2 Particles. Philosophical Magazine Letters, 1988. 58(1):
p. 11-15.
7.Harker, D. and E.R. Parker, Grain Shape and Grain Growth. Transactions of the
American Society for Metals, 1945. 34: p. 156-201.
8.Riegger, O.K. and L.H. Van Vlack, Dihedral Angle Measurement. Transactions of
the American Institute of Mining and Metallurgical Engineers, 1960. 218(5): p. 933-
935.
9.Stickels, C.A. and E.E. Hucke, Measurement of Dihedral Angles. Transactions of
the Metallurgical Society of AIME, 1964. 230(4): p. 795-801.
10.Jurewicz, S.R. and A.J.G. Jurewicz, Distribution of Apparent Angles on Random
Sections with Emphasis on Dihedral Angle Measurements. Journal of Geophysical
Research-Solid Earth and Planets, 1986. 91(B9): p. 9277-9282.
179

CHAPTER 11. LIST OF REFERENCES
180
11.Fortes, M.A. and A.C. Ferro, Determination of Dihedral Angle Distributions in
Polycrystals - Application to Alpha+Beta-Brass. Metallurgical Transactions A-
Physical Metallurgy and Materials Science, 1988. 19(5): p. 1147-1151.
12.De Hoff, R.T., Estimation of Dihedral Angles from Stereological Counting
Measurements. Metallography, 1986. 19(2): p. 209-217.
13.Eustathopoulos, N., et al., Etude thermodynamique de la tension interfaciale
solide/liquide pour un système métallique binaire. III. Journal de chimie physique,
1974. 71(11-12): p. 1465-1471.
14.Mullins, W.W., Grain Boundary Grooving by Volume Diffusion. Transactions of
the Metallurgical Society of AIME, 1960. 218: p. 354-361.
15.Rabkin, E. and L. Klinger, The fascination of grain boundary grooves. Advanced
Engineering Materials, 2001. 3(5): p. 277-282.
16.Gabrisch, H., U. Dahmen, and E. Johnson, In situ measurement of dihedral angles
at liquid grain boundary inclusions. Microscopy Research and Technique, 1998.
42(4): p. 241-247.
17.Dahmen, U., et al., Magic-size equilibrium shapes of nanoscale Pb inclusions in
Al. Physical Review Letters, 1997. 78(3): p. 471-474.
18.Johnson, E., et al., Shapes and sizes of nanoscale Pb inclusions in Al. Materials
Science and Engineering A, 2001. 304-306: p. 187-193.
19.Gabrisch, H., et al., Equilibrium shape and interface roughening of small liquid
Pb inclusions in solid Al. Acta Materialia, 2001. 49(20): p. 4259-4269.
20.Robertson, W.M., Grain Boundary Grooving and Scratch Decay on Copper in
Liquid Lead. Transactions of the Metallurgical Society of AIME, 1965. 233(July): p.
1232-1236.
21.Ratke, L. and H.J. Vogel, Theory of Grain-Boundary Grooving in the Convective-
Diffusive Regime. Acta Materialia, 1991. 39(5): p. 915-923.
22.Ikeuye, K.K. and C.S. Smith, Studies of Interface Energies in Some Aluminum and
Copper Alloys. Metals Transactions, 1949. 185: p. 762-768.
23.Eustathopoulos, N., Contribution à l'étude de l'interface solide-liquide de métaux
et alliages, in Ecole Nationale Supérieure d'Electrochimie et d'Electrométallurgie.
1974, Université Scientifique et médicale de Grenoble: Grenoble. p. 118.
24.Joud, J.C., et al., Détermination de la Tension Superficielle des Alliages Ag-Pb et
Cu-Pb par la Méthode de la Goutte Posée. Journal de chimie physique, 1973. 70(9):
p. 1290-1294.
25.Eustathopoulos, N., J.C. Joud, and P.J. Desre, Etude thermodynamique de la
tension interfaciale solide/liquide pour un système métallique binaire. II.-
Applications. Journal de chimie physique, 1974. 71(5): p. 777-787.

26.Coudurier, L., et al., Corrosion Intergranullaire du Cuivre par le Plomb Liquide
sous l'effet des Forces Capillaires. Journal de chimie physique, 1977. 74(3): p. 289-
294.
27.Camel, D., N. Eustathopoulos, and P. Desré, Chemical adsorption and
temperature dependence of the solid-liquid interfacial tension of metallic binary
alloys. Acta Metallurgica, 1980. 28(3): p. 239-247.
28.Sanchez Medina, M., L. Coudurier, and N. Eustathopoulos, Microstructure
d'équilibre et tensions interfaciales dans les systèmes biphasés solide-liquide CuPbM
(M : Ag, Ni, Sn, Sb). Mémoires et Etudes Scientifiques Revue de Métallurgie, 1981.
78(11): p. 607-617.
29.Straumal, B., D. Molodov, and W. Gust, Grain boundary wetting phase
transitions in the Al-Sn and Al-Sn-Pb systems. Materials Science Forum, 1996. 207-
209: p. 437-440.
30.Straumal, B., et al., The wetting transition in high and low energy grain
boundaries in the Cu(In) system. Acta Metallurgica et Materialia, 1992. 40(5): p.
939-945.
31.Baker, H., et al., eds. ASM Handbook Volume 03: Alloy Phase Diagrams. 1992,
ASM International, Materials Park, OH.
32.Stickels, C.A. and E.E. Hucke, Some Effects of Temperature + Hydrostatic
Pressure on Interfacial Tensions in Nickel-Lead System. Transactions of the
Metallurgical Society of Aime, 1964. 230(1): p. 20-27.
33.Stickels, C.A. and E.E. Hucke, Effect of Stress on Dihedral Angle in Leaded
Nickel. Journal of the Institute of Metals, 1964. 92(8): p. 234-237.
34.Waterhouse, R.B. and D. Grubb, Embrittlememt of Alpha-Brasses by Liquid
Metals. Journal of the Institute of Metals, 1963. 91(6): p. 216-219.
35.Stickels, C.A., Discussion of "Embrittlememt of Alpha-Brasses by Liquid Metals".
Journal of the Institute of Metals, 1963. 91(12): p. 422-423.
36.Sun, B., Z. Suo, and A.G. Evans, Emergence of cracks by mass transport in elastic
crystals stressed at high temperatures. Journal of the Mechanics and Physics of
Solids, 1994. 42(11): p. 1653-1677.
37.Eshelby, J.D., The Determination of the Elastic Field of an Ellipsoidal Inclusion,
and Related Problems. Proceedings of the Royal Society of London Series A-
Mathematical and Physical Sciences, 1957. 241(1226): p. 376-396.
38.Mura, T., Micromechanics of Defects in Solids. second edition ed. Mechanics of
Elastic and Inelastic Solids, ed. S. Nemat-Nasser and G.A. Oravas. 1987, Dordrecht/
Boston /London: Kluver Academic Publishers. 587.
181

CHAPTER 11. LIST OF REFERENCES
182
39.Wang, H. and Z.H. Li, The three-dimensional analysis for diffusive shrinkage of a
grain-boundary void in stressed solid. Journal of Materials Science, 2004. 39(10): p.
3425-3432.
40.Raj, R., Unstable Spreading of a Fluid Inclusion in a Grain Boundary under
Normal Stress. Journal of the American Ceramic Society, 1986. 69(9): p. 708-712.
41.Wang, W. and Z. Suo, Shape change of a pore in a stressed solid via surface
diffusion motivated by surface and elastic energy variation. Journal of the Mechanics
and Physics of Solids, 1997. 45(5): p. 709-729.
42.Suo, Z. and H. Yu, Crack nucleation on an elastic polycrystal surface in a
corrosive environment: Low dimensional dynamical models. Acta Materialia, 1997.
45(6): p. 2235-2245.
43.Rice, J.R. and T.J. Chuang, Energy Variations in Diffusive Cavity Growth. Journal
of the American Ceramic Society, 1981. 64(1): p. 46-53.
44.Srolovitz, D.J. and S.H. Davis, Do stresses modify wetting angles? Acta
Materialia, 2001. 49(6): p. 1005-1007.
45.Chuang, T.-J. and J.R. Rice, The shape of intergranular creep cracks growing by
surface diffusion. Acta Metallurgica, 1973. 21(12): p. 1625-1628.
46.Takahashi, Y., K. Takahashi, and K. Nishiguchi, A numerical analysis of void
shrinkage processes controlled by coupled surface and interface diffusion. Acta
Metallurgica et Materialia, 1991. 39(12): p. 3199-3216.
47.Sanchez Medina, M., et al., Influence des additions de Pb et de Sn sur la ductilité
du cuivre OFHC entre la température ambiante et 1000 °C. Mémoires et Etudes
Scientifiques Revue de Métallurgie, 1981. 78(2): p. 101-109.
48.Foulger, R.V. and E. Nicholls, Influence of composition and microstructure on
mechanical working properties of copper-base alloys. Metals Technology, 1976. 3: p.
366-369.
49.Davies, P.W., et al., Fracture behaviour of Copper during Hot Deformation.
Journal of the Institute of Metals, 1971. 99: p. 195-197.
50.Sanchez Medina, M., R. Sangiorgi, and N. Eustathopoulos, On the embrittlement
of Cu by Liquid Pb Inclusions. Scripta Materialia, 1981. 15(7): p. 737-738.
51.Reid, b.J. and J.N. Greenwood, Intergranular Cavitation in Stressed Copper-
Nickel Alloys. Transactions of the Metallurgical Society of AIME, 1958. 212: p. 503-
507.
52.Ashby, M.F., C. Gandhi, and D.M.R. Taplin, Fracture-Mechanism Maps and their
Construction for F.C.C. Metals and Alloys. Acta Materialia, 1979. 27(5): p. 699-729.
53.Hondros, E.D. and D. McLean, Cohesion Margin of Copper. Philosophical
Magazine, 1974. 29(4): p. 771-795.

54.Watanabe, T., Grain Boundary Sliding and Stress Concentration during Creep.
Metallurgical Transactions A, 1983. 14A: p. 531-545.
55.Onaka, S., M. Kato, and R. Tanaka, Intermediate Temperature Embrittlement of
Dispersion Hardened Cu-GeO2 Polycrystals. Transactions of the Japan Institute of
Metals, 1987. 28(1): p. 32-40.
56.Ayensu, A. and T.G. Langdon, The Inter-Relationship between Grain Boundary
Sliding and Cavitation during Creep of Polycrystalline Copper. Metallurgical
Transactions A, 1996. 27A: p. 901-907.
57.Miura, H., T. Sakai, and M. Kato, Ductility and Fracture of Copper Bicrystals
with Boundary SiO2 Particles. Materials Science and Technology, 1994. 10(12): p.
1044-1049.
58.Afshar, S. and M. Biscondi, Intergranular cavitation observed during OFHC
copper bicrystal creep. Scripta Metallurgica, 1986. 20(5): p. 707-711.
59.Watanabe, T. and S. Tsurekawa, The control of brittleness and development of
desirable mechanical properties in polycrystalline systems by grain boundary
engineering. Acta Materialia, 1999. 47(15-16): p. 4171-4185.
60.Miura, H., et al., Correlation between Intermediate-Temperature-Embrittlement
and Grain-Boundary Energy. Materials Science Forum, 1996. 207-209: p. 649-652.
61.Monzen, R., et al., Bismuth embrittlement of [011] twist boundaries in copper
bicrystals. Metallurgical and Materials Transactions a-Physical Metallurgy and
Materials Science, 1999. 30(2): p. 483-485.
62.Monzen, R. and T. Okamoto, Grain-boundary diffusion of Bi in Cu bicrystals.
Materials Science Forum, 1999. 294-296: p. 561-564.
63.Kamdar, M.H., Solid Metal Induced Embrittlement, in Corrosion. 2001, ASM
International: Materials Park, OH, USA. p. 184-189.
64.Gavin, S.A., et al., Effect of trace impurities on hot ductility of as-cast cupronickel
alloys. Metals Technology, 1978: p. 397-401.
65.Eborall, R. and P. Gregory, The Mechanism of Embrittlement by a Liquid Phase.
Journal of the Institute of Metals, 1955. 84: p. 88-90.
66.Roth, M.C., G.C. Weatherly, and W.A. Miller, The temperature dependence of the
mechanical properties of aluminum alloys containing low-melting-point inclusions.
Acta Metallurgica, 1980. 28(7): p. 841-853.
67.Wolley, D.J. and A.G. Fox, The embrittlement of leaded and unleaded a+b 60/40
brasses in the temperature range 300 to 500°C. Journal of Materials Science Letters,
1988. 7: p. 763-765.
68.Fujiwara, S. and K. Abiko, Ductility of Ultra High Purity Copper. Journal de
Physique IV, 1995. 5(C7): p. 295-300.
183

CHAPTER 11. LIST OF REFERENCES
184
69.Misra, R.D.K., V.S. Prasad, and P.R. Rao, Dynamic embrittlement in an agehardenable
copper-chromium alloy. Scripta Materialia, 1996. 35(1): p. 129-133.
70.Kanno, M., Effect of a Small Addition of Zirconium on Hot Ductility of a Cu-Cr
Alloy. Zeitschrift für Metallkunde, 1988. 79: p. 684-688.
71.Kanno, M., et al., Effects of a Small Amount of Added Elements on the Ductility of
a Cu-8 mass%Sn Alloy at Elevated Temperature. Transactions of the Japan Institute
of Metals, 1987. 28(1): p. 73-80.
72.Kanno, M. and N. Shimodaira, Improvement in Hot Ductility in a Cu-4.4mol.% Sn
Alloy by Adding a Small Amount of Third Element. Transactions of the Japan Institute
of Metals, 1987. 28(9): p. 742-748.
73.Kanno, M. and N. Shimodaira, Improvement in Ductility at Intermediate
Temperatures by a small Addition of Yttrium in a Cu-30Zn Alloy. Scripta Materialia,
1987. 21: p. 1487-1492.
74.Kubosono, K., et al., Copper-Nickel Based Alloy. 1995, Mitsubishi Denki
Kabushiki Kaisha: US Patent.
75.Ohmori, M., K. Wakasa, and Y. Yoshinaga, Effect of strain Rate on Tensile
Ductility of Polycrystalline Copper and alfa-Brasses at Intermediate Temperatures.
Journal of the Japan Institute of Metals, 1973. 37(11): p. 1188-1194.
76.Ohmori, M., Y. Yoshinaga, and H. Maniwa, The Effect of Strain Rate on
Intermediate Temperature Embrittlement of Copper. Journal of the Japan Institute of
Metals, 1970. 8: p. 791-797.
77.Ohmori, M., et al., High Temperature Tensile Ductility of Copper in Vacuum.
Journal of the Japan Institute of Metals, 1976. 40(8): p. 802-807.
78.Perovic, D.D., G.C. Weatherly, and W.A. Miller, On the strain rate dependence of
ductility troughs observed under liquid metal embrittlement and creep conditions.
Scripta Materialia, 1987. 21: p. 345-347.
79.Nowosielski, R., Ductility Minimum Temperature in Selected Monophase Binary
Brasses. Journal of Materials Processing Technology, 2001. 109: p. 142-153.
80.Sato, S., T. Otsu, and E. Hata, Embrittlement of Copper Alloys during Annealing.
Journal of the Institute of Metals, 1970. 98: p. 135-141.
81.Wang, G.X., Effect of Microstructural Modification on Tensile and Fatigue
Properties of Cu-35%Ni-3.5%Cr Alloy. Acta Metallurgica et Materialia, 1994. 42(7):
p. 2547-2554.
82.Shibayanagi, T., S. Saji, and S. Hori, Improvement in low ductility failure in
Copper. Supplement to Transactions of the Japan Institute of Metals, 1986. 27
Supplement(Supplement): p. 943-950.

83.McQueen, H.J., Dynamic Recovery and Recrystallization, in Encyclopedia of
Materials: Science and Technology, K.H.J. Buschow, et al., Editors. 2001, Elsevier
Science Ltd.: Oxford. p. 2375-2381.
84.Gottstein, G., D.A. Molodov, and L.S. Shvindlerman, Grain Boundary Migration
in Metals: Recent Developments. Interface Science, 1998. 6: p. 7-22.
85.Barrera, E.V., et al., Quasi-Static Intergranular Cracking in a Cu-Sn Alloy; an
Analog of Stress Relief Cracking of Steels. Scripta Metallurgica et Materialia, 1992.
27: p. 205-210.
86.Muthiah, R.C., et al., Tin-Induced Dynamic Embrittlement of Cu-7%Sn Bicrystals
with a Σ5 Boundary. Acta Materialia, 1999. 47(9): p. 2797-2808.
87.Pfaendtner, J.A., et al., Time-dependent Interfacial Failure in Metallic Alloys.
Materials Science and Engineering, 1999. A260: p. 1-11.
88.Hondros, E.D. and M.P. Seah, Segregation to Interfaces. International Metals
Reviews, 1977: p. 262-300.
89.Misra, R.D.K., C.J. McMahon Jr., and A. Guha, Brittle Behavior of a Dilute
Copper-Beryllium Alloy at 200°C in Air. Scripta Materialia, 1994. 31(11): p. 1471-
1474.
90.Hondros, E.D., et al., Interfacial and Surface Microchemistry, in Physical
Metallurgy: Fourth, Revised and Enhanced Edition, R.W. Cahn and P. Haasen,
Editors. 1996, Elsevier Science Publishers, BV: Amsterdam. p. 1201-1289.
91.Raynolds, J.E., et al., Impurity effects on adhesion at an interface between NiAl
and Mo. Acta Materialia, 1999. 47(11): p. 3281-3289.
92.Van der Ven, A. and G. Ceder, The thermodynamics of decohesion. Acta
Materialia, 2004. 52(5): p. 1223-1235.
93.Messmer, R.P. and C.L. Briant, The role of chemical bonding in grain boundary
embrittlement. Acta Metallurgica, 1982. 30(2): p. 457-467.
94.Keast, V.J., et al., Chemistry and bonding changes associated with the segregation
of Bi to grain boundaries in Cu. Acta Materialia, 1998. 46(2): p. 481-490.
95.Bruley, J., V.J. Keast, and D.B. Williams, An EELS study of segregation-induced
grain-boundary embrittlement of copper. Acta Materialia, 1999. 47(15-16): p. 4009-
4017.
96.Pearson, D.H., C.C. Ahn, and B. Fultz, Measurements of 3d Occupancy from Cu
L(2,3) Electron-Energy- Loss Spectra of Rapidly Quenched CuZr, CuTi, CuPd, CuPt,
and CuAu. Physical Review B, 1994. 50(17): p. 12969-12972.
97.Alber, U., H. Mullejans, and M. Ruhle, Bismuth segregation at copper grain
boundaries. Acta Materialia, 1999. 47(15-16): p. 4047-4060.
185

CHAPTER 11. LIST OF REFERENCES
186
98.Rez, P. and J.R. Alvarez, Calculation of cohesion and changes in electronic
structure due to impurity segregation at boundaries in iron. Acta Materialia, 1999.
47(15): p. 4069-4075.
99.Keast, V.J. and D.B. Williams, Quantitative compositional mapping of Bi
segregation to grain boundaries in Cu. Acta Materialia, 1999. 47(15-16): p. 3999-
4008.
100.Roy, A., U. Erb, and H. Gleiter, Grain boundary embrittlement as a function of
boundary structure (energy). Acta Metallurgica, 1982. 30(10): p. 1847-1850.
101.Watanabe, T., Grain boundary Design for the Control of Intergranular Fracture.
Materials Science Forum, 1989. 46: p. 25-48.
102.Joshi, A. and D.F. Stein, An Auger Spectroscopic Analysis of Bismuth Segregated
to Grain Boundaries in Copper. Journal of the Institute of Metals, 1971. 99(JUN): p.
178-181.
103.Powell, B.D. and H. Mykura, The segregation of bismuth to grain boundaries in
copper-bismuth alloys. Acta Metallurgica, 1973. 21(8): p. 1151-1156.
104.Marcus, H.L. and N.E. Paton, Metastable Grain-Boundary Segregation in
Copper Containing Tellurium. Metallurgical Transactions, 1974. 5(10): p. 2135-
2138.
105.Menyhard, M., B. Blum, and C.J. McMahon Jr., Grain Boundary Segregation
and Transformations in Bi-doped polycristalline Copper. Acta Metallurgica
Materialia, 1989. 37(2): p. 549-557.
106.Pineau, A., et al., Grain Boundary Cosegregation and Diffusion in Cu(Fe,S)
Solid Solution. Acta Metallurgica, 1983. 31(7): p. 1047-1052.
107.Westbrook, J.H., Segregation at Interfaces. Metallurgical Reviews, 1964. 9(36):
p. 415-471.
108.Nieh, T.G. and W.D. Nix, Embrittlement of Copper Due to Segregation of
Oxygen to Grain- Boundaries. Metallurgical Transactions a-Physical Metallurgy and
Materials Science, 1981. 12(5): p. 893-901.
109.Benci, J.E., D.P. Pope, and J.L. Bassani, A combined experimental and analytical
investigation of creep damage development in copper. Acta Metallurgica et
Materialia, 1994. 42(1): p. 225-238.
110.Tipler, H.R. and D. McLean, Influence of Interface Energy on Creep Rupture.
Metal Science Journal, 1970. 4: p. 103-107.
111.Briant, C.L. and R.P. Messmer. Chemical Bonding and Grain Boundary
Embrittlement. in Embrittlement by Liquid Metals. 1982. Watervliet NY: TMS
Warrendale, PA, publ. 1983.

112.Seah, M.P., Adsorption-induced interface decohesion. Acta Metallurgica, 1980.
28(7): p. 955-962.
113.Rice, J.R. and J.-C. Wang, Embrittlement of Interfaces by Solute Segregation.
Materials Science and Engineering, 1989. A107: p. 23-40.
114.Chatain, D., P. Wynblatt, and G.S. Rohrer, Anisotropic phenomena at interfaces
in bismuth-saturated copper. Scripta Materialia, 2004. 50(5): p. 565-569.
115.Chang, L.-S., et al., Thermodynamic aspects of the grain boundary segregation
in Cu(Bi) alloys. Acta Materialia, 1999. 47(15-16): p. 4041-4046.
116.Chang, L.-S., et al., Kinetic aspects of the grain boundary segregation in Cu(Bi)
alloys. Acta Materialia, 1999. 47(10): p. 2951-2959.
117.Saindrenan, G., R. Le Gall, and F. Christien, Endommagement interfacial des
métaux. TECHNOSUP. 2002, Paris: Ellipses édition. 256.
118.Hirth, J.P. and J.R. Rice, On the Thermodynamics of Adsorption at Interfaces as
It Influences Decohesion. Metallurgical Transactions a-Physical Metallurgy and
Materials Science, 1980. 11(9): p. 1501-1511.
119.Mishin, Y., P. Sofronis, and J.L. Bassani, Thermodynamic and kinetic aspects of
interfacial decohesion. Acta Materialia, 2002. 50(14): p. 3609-3622.
120.Bika, D. and C.J. McMahon Jr., A Model for Dynamic Embrittlement. Acta
Metallurgica et Materialia, 1995. 43(5): p. 1909-1916.
121.McMahon, C.J., Brittle fracture of grain boundaries. Interface Science, 2004.
12(2-3): p. 141-146.
122.Pfaendtner, J.A., R. Muthiah, and C.J. McMahon Jr. Quasi-Static Brittle Failure.
in Materials Research Society Symposium Procedings. 1999.
123.Xu, Y. and J.L. Bassani, A steady-state model for diffusion-controlled fracture.
Materials Science and Engineering A, 1999. 260(1-2): p. 48-54.
124.Christien, F., R. Le Gall, and G. Saindrenan, Influence of stress on intergranular
brittleness of a martensitic stainless steel, in Diffusion, Segregation and Stresses in
Materials. 2003. p. 275-283.
125.Misra, R.D.K., Issues concerning the effects of applied tensile stress on
intergranular segregation in a low alloy steel. Acta Materialia, 1996. 44(3): p. 885-
890.
126.Kaur, I. and W. Gust, Handbook of Grain and Interphase Boundary Diffusion
Data. Vol. 1. 1989: Ziegler Press, Stuttgart.
127.Misra, R.D.K. and B.P. Kashyap, An analysis of intergranular segregation of
sulphur in a low alloy steel by stress-induced diffusion process. Scripta Materialia,
1996. 35(6): p. 755-760.
187

CHAPTER 11. LIST OF REFERENCES
188
128.Nicholas, M.G. A Survey of Litterature on Liquid Metal Embrittlement of metals
and Alloys. in Embrittlement by Liquid and Solid Metals. 1982. St.-Louis, Missouri:
The Metallurgical Society of AIME.
129.Rostoker, W., Liquid Metal Embrittlement, in Metals Handbook, Desk Edition,
H.E. Boyer and T.L. Gall, Editors. 1985, American Society for Metals: Metals Park,
OH. p. 32.26-32.27.
130.Roth, M.C., G.C. Weatherly, and W.A. Miller, The Role of Liquid Metal
Embrittlement in Free-machining Alloys containing low-melting-point Inclusions.
Canadian Metallurgical Quaterly, 1979. 18: p. 341-348.
131.Westwood, A.R.C., C.M. Preece, and M.H. Kamdar, Adsorption-Induced Brittle
Fracture in Liquid-Metal Environments, in Fracture An Advanced Treatise, H.
Liebowitz, Editor. 1971, Academic Press: New York and London. p. 589-644.
132.Nicholas, M.G. and C.F. Old, Review Liquid Metal Embrittlement. Journal of
Materials Science, 1979. 14: p. 1-18.
133.Stoloff, N.S. Metal Induced Embrittlement - A Historical Perspective. in
Embrittlement by Liquid and Solid Metals. 1982. St.-Louis, Missouri: The
Metallurgical Society of AIME.
134.Lynch, S.P. Mechanisms of Fracture in Liquid Metal Environments. in
Embrittlement by Liquid and Solid Metals. 1982. St.-Louis, Missouri: The
Metallurgical Society of AIME.
135.Lynch, S.P., Environmentally Assisted Cracking: Overview of Evidence for an
Adsorption-Induced Localised-Slip Process. Acta Metallurgica, 1988. 36(10): p.
2639-2661.
136.Fernandes, P.J.L., R.E. Clegg, and D.R.H. Jones, Failure by Liquid Metal
Induced Embrittlement. Engineering Failure Analysis, 1994. 1(1): p. 51-63.
137.Fernandes, P.J.L. and D.R.H. Jones, Mechanisms of liquid metal induced
embrittlement. International Materials Reviews, 1997. 42(6): p. 251-261.
138.Joseph, B., M. Picat, and F. Barbier, Liquid metal embrittlement: A state-of-theart
appraisal. The European Physical Journal Applied physics, 1999. 5: p. 19-31.
139.Kamdar, M.H., Liquid-Metal Embrittlement, in Corrosion. 2001, ASM
International: Materials Park, OH, USA. p. 171-184.
140.Kamdar, M.H., ed. Embrittlement by Liquid and Solid Metals. 1982, The
Metallurgical Society of AIME: St.-Louis, Missouri.
141.Shunk, F.A. and W.R. Warke, Specificity as an aspect of liquid metal
embrittlement. Scripta Metallurgica, 1974. 8(5): p. 519-526.
142.Fernandes, P.J.L. and D.R.H. Jones, Specificity in liquid metal induced
embrittlement. Engineering Failure Analysis, 1996. 3(4): p. 299-302.

143.Fernandes, P.J.L. and D.R.H. Jones, The effects of microstructure on crack
initiation in liquid-metal environments. Engineering Failure Analysis, 1997. 4(3): p.
195-204.
144.Auger, T., et al., Effect of contact conditions on embrittlement of T91 steel by
lead-bismuth. Journal of Nuclear Materials, 2004. 335(2): p. 227-231.
145.Perovic, D.D., G.C. Weatherly, and W.A. Miller, Influence of Temperature and
Strain-rate on Liquid Metal Embrittlement of an Aluminum Alloy containing Bismuth
Inclusions. Acta Materialia, 1988. 36(8): p. 2249-2257.
146.Glickman, E.E. Mechanism of Liquid Metal Embrittlement by Simple
Experiments: from Atomics to Life-Time. in Multiscale Phenomena in Plasticity.
2000. Ouranopolis, Greece: Kluwer Academic Publisher.
147.Gordon, P., Metal-Induced-Embrittlement of Metals - An Evaluation of
Embrittler Transport Mechanisms. Metallurgical Transactions A, 1978. 9A: p. 267-
273.
148.Kargol, J.A. and D.L. Albright, The Effect of Relative Crystal Orientation on The
Liquid Metal Induced Grain Boundary Fracture of Aluminum Bicrystals.
Metallurgical Transactions A, 1977. 8A: p. 27-34.
149.Hasson, G., et al., Theoretical and experimental determinations of grain
boundary structures and energies: Correlation with various experimental results.
Surface Science, 1972. 31: p. 115-137.
150.Glickman, E.E., Grain boundary grooving accelerated by local plasticity as a
possible mechanism of liquid metal embrittlement. Interface Science, 2003. 11(4): p.
451-459.
151.Wolski, K., et al., Liquid metal embrittlement studies on model systems with
respect to the spallation target technology: The importance of nanometre-thick films.
Journal de Physique IV, 2002. 12: p. Pr8-249-261.
152.Robertson, W.M., Propagation of a Crack Filled with Liquid Metal. Transactions
of the Metallurgical Society of AIME, 1966. 236: p. 1478-1482.
153.Stoloff, N.S. and T.L. Johnston, Crack Propagation in a Liquid Metal
Environment. Acta Materialia, 1963. 11(4): p. 251-256.
154.Westwood, A.R. and M.H. Kamdar, Concerning Liquid Metal Embrittlement,
Particularly of Zinc Monocrystals by Mercury. Philosophical Magazine, 1963. 8(89):
p. 787-804.
155.Lynch, S.P., Solid-metal-induced embrittlement of aluminium alloys and other
materials. Materials Science and Engineering A, 1989. 108: p. 203-212.
156.Glickman, E.E., Stress, surface energy and segregation effects in liquid metal
embrittlement: Role of grain boundary grooving accelerated by local plasticity.
Defect and Diffusion Forum, 2003. 216-217: p. 207-216.
189

CHAPTER 11. LIST OF REFERENCES
190
157.Glickman, E.E., et al., Round table discussion II: Grain Boundary Wetting and
Liquid Grooving. Defect and Diffusion Forum, 1998. 156: p. 265-272.
158.Pereiro-Lopez, E., et al., Grain boundary liquid metal wetting: A synchrotron
micro-radiographic investigation. Nuclear Instruments and Methods in Physics
Research Section B: Beam Interactions with Materials and Atoms, 2003. 200: p. 333-
338.
159.Pereiro-Lopez, E., et al., Liquid-metal wetting at grain boundaries: a
synchrotron-micro- radiographic investigation, in Diffusion, Segregation and
Stresses in Materials. 2003. p. 241-248.
160.Pereiro-Lopez, E., W. Ludwig, and D. Bellet, Discontinuous penetration of
liquid Ga into grain boundaries of Al polycrystals. Acta Materialia, 2004. 52(2): p.
321-332.
161.Ludwig, W., E. Pereiro-Lopez, and D. Bellet, In situ investigation of liquid Ga
penetration in Al bicrystal grain boundaries: grain boundary wetting or liquid metal
embrittlement? Acta Materialia, 2005. 53(1): p. 151-162.
162.Ludwig, W. and D. Bellet, Penetration of liquid gallium into the grain
boundaries of aluminium: a synchrotron radiation microtomographic investigation.
Materials Science and Engineering A, 2000. 281(1-2): p. 198-203.
163.Clarke, D.R. and M.L. Gee, Wetting of surfaces and grain boundaries, in
Materials Interfaces Atomic-level structure and properties, D.Y. Wolf, S., Editor.
1992, Chapman & Hall: London. p. 255-272.
164.Hugo, R.C. and R.G. Hoagland, In-Situ TEM Observation of Aluminum
Embrittlement by Liquid Gallium. Scripta Materialia, 1998. 38(3): p. 523-529.
165.Laporte, V. and K. Wolski. Liquid Bismuth Induced Intergranular Embrittlement
of Polycrystalline Copper. in Environmental Degradation of Engineering Materials.
2003. Bordeaux.
166.Cahn, J.W., Critical-Point Wetting. Journal of Chemical Physics, 1977. 66(8): p.
3667-3672.
167.Straumal, B. and B. Baretzky, Grain boundary phase transitions and their
influence on properties of polycrystals. Interface Science, 2004. 12(2-3): p. 147-155.
168.Straumal, B.B., P. Zieba, and W. Gust, Grain boundary phase transitions and
phase diagrams. International Journal of Inorganic Materials, 2001. 3(8): p. 1113-
1115.
169.Marié, N., K. Wolski, and M. Biscondi, Grain boundary penetration of nickel by
liquid bismuth as a film of nanometric thickness. Scripta Materialia, 2000. 43(10): p.
943-949.

170.Marié, N., K. Wolski, and M. Biscondi, Intergranular penetration and
embrittlement of solid nickel through bismuth vapour condensation at 700°C. Journal
of Nuclear Materials, 2001. 296(1-3): p. 282-288.
171.Wolski, K., N. Marié, and M. Biscondi, AES quantification of intergranular film
thickness in the Ni-Bi system with respect to the liquid metal embrittlement
phenomenon. Surface and Interface Analysis, 2001. 31: p. 280-286.
172.Wolski, K., et al., About the Importance of Nanometer-Thick Intergranular
Penetration in the Analysis ol Liquid Metal Embrittlement. Interface Science, 2001.
9: p. 183-189.
173.Joseph, B., et al., Rapid penetration of liquid Bi along Cu grain boundaries.
Scripta Materialia, 1998. 39(6): p. 775-781.
174.Joseph, B., F. Barbier, and M. Aucouturier, Embrittlement of copper by liquid
bismuth. Scripta Materialia, 1999. 40(8): p. 893-897.
175.Rabkin, E. Grain Boundary Embrittlement by Liquid Metals. in Multiscale
Phenomena in Plasticity. 2000. Ouranopolis, Greece: Kluwer Academic Publisher.
176.de Gennes, P.G., Wetting - Statics and Dynamics. Reviews of Modern Physics,
1985. 57(3): p. 827-863.
177.Rao, G., D.B. Zhang, and P. Wynblatt, A determination of interfacial energy and
interfacial composition in Cu-Pb and Cu-Pb-X alloys by solid state wetting
measurements. Acta Metallurgica et Materialia, 1993. 41(11): p. 3331-3340.
178.Moon, J., et al., Effects of concentration dependent diffusivity on the growth of
precursing films of Pb on Cu(111). Surface Science, 2001. 488(1-2): p. 73-82.
179.Prevot, G., et al., Macroscopic and mesoscopic surface diffusion from a deposit
formed by a Stranski-Krastanov type of growth: Pb on Cu(100) at above one layer of
coverage. Physical Review B, 2000. 61(15): p. 10393-10403.
180.Iwamoto, C. and S.-I. Tanaka, Atomic morphology and chemical reactions of the
reactive wetting front. Acta Materialia, 2002. 50(4): p. 749-755.
181.Poirier, D.R. and G.H. Geiger, Transport Phenomena in Materials Processing.
Second Edition ed. 1994, Warrendale, PA: TMS. 267.
182.Chatain, D., V. Ghetta, and P. Wynblatt, Equilibrium shape of copper crystals
grown on sapphire. Interface Science, 2004. 12(1): p. 7-18.
183.Weck, E. and E. Leistner, Metallographic instructions for colour etching by
immersion. Fachbuchserie Schweisstechnick. Vol. 77. 1995, Düsseldorf: Deutscher
Verlag für Schweisstechnick. 60.
184.Moore, K.I., K. Chattopadhyay, and B. Cantor, In situ Transmission Electron-
Microscope Measurements of Solid Al-Solid Pb and Solid Al-Liquid Pb Surface-
Energy Anisotropy in Rapidly Solidified Al-5-Percent-Pb (by Mass). Proceedings of
191

CHAPTER 11. LIST OF REFERENCES
192
the Royal Society of London Series a- Mathematical Physical and Engineering
Sciences, 1987. 414(1847): p. 499-&.
185.Moore, K.I., D.L. Zhang, and B. Cantor, Solidification of Pb particles embedded
in Al. Acta Metallurgica et Materialia, 1990. 38(7): p. 1327-1342.
186.Goswami, R. and K. Chattopadhyay, The superheating and the crystallography
of embedded Pb particles in f.c.c. Al, Cu and Ni matrices. Acta Metallurgica et
Materialia, 1995. 43(7): p. 2837-2847.
187.Gibbs, G.B., Fluage Par Diffusion Dans Les Solides Polycristallins. Memoires
Scientifiques De La Revue De Metallurgie, 1965. 62(10): p. 781-786.
188.Pourbaix, M., Atlas of Electrochemical Equilibria in Aqueous Solutions. 2nd
english edition ed. 1974, Houston, Texas, USA: NACE.
189.MeX software, Alicona Imaging GmbH. 2004: Graz.
190.Gorman, J.W. and G.W. Preckshot, Molecular Diffusion and Interphase Transfer
in the Solid Copper-Molten Lead System. Transactions of the American Institute of
Mining and Metallurgical Engineers, 1958. 212: p. 367-373.
191.Nichols, F.A. and W.W. Mullins, Surface- (Interface-) and Volume-Diffusion
Contributions to Morphological Changes Driven by Capillarity. Transactions of the
Metallurgical Society of AIME, 1965. 233: p. 1840-1847.
192.Predel, B., Al-Pb (Aluminum-Lead), in Landolt-Börnstein - Group IV Physical
Chemistry. 1991, Springer-Verlag: Heidelberg.
193.McLean, M., Kinetics of Spheroidization of Lead Inclusions in Aluminum.
Philosophical Magazine, 1973. 27(6): p. 1253-1266.
194.McCormick, M.A., E.B. Evans, and U. Erb, On the Shapes of Liquid-Lead
Inclusions in Aluminum. Philosophical Magazine a-Physics of Condensed Matter
Structure Defects and Mechanical Properties, 1986. 53(2): p. L27-L30.
195.McLean, M. and M.S. Loveday, In-situ observations of the annealing of liquid
lead inclusions entrained in an aluminium matrix. Journal of Materials Science,
1974. 9: p. 1104-1114.
196.Matthews, J.W., Misfit Dislocations, in Dislocations in Solids, F.R.N. Nabarro,
Editor. 1979, North-Holland Publ.: New York. p. 461-545.
197.Kim, W.T. and B. Cantor, Solidification behaviour of Pb droplets embedded in a
Cu matrix. Acta Metallurgica et Materialia, 1992. 40(12): p. 3339-3347.
198.Goswami, R., K. Chattopadhyay, and P.L. Ryder, Melting of Pb particles
embedded in Cu-Zn matrices. Acta Materialia, 1998. 46(12): p. 4257-4271.
199.Bonzel, H.P., Surface morphologies: Transient and equilibrium shapes. Interface
Science, 2001. 9(1-2): p. 21-34.

200.Wolf, D., Correlation between Structure, Energy, and Ideal Cleavage Fracture
for Symmetrical Grain-Boundaries in Fcc Metals. Journal of Materials Research,
1990. 5(8): p. 1708-1730.
201.Protsenko, P., et al., Misorientation effects on grain boundary grooving of Ni by
liquid Ag, in Diffusion, Segregation and Stresses in Materials. 2003. p. 225-230.
202.Chatain, D., C. Vahlas, and N. Eustathopoulos, Etude des tensions interfaciales
liquide-liquide et solide-liquide dans les systemes à monotectique Zn-Pb et Zn-Pb-
Sn. Acta Metallurgica, 1984. 32(2): p. 227-234.
203.Miura, H., M. Kato, and T. Mori, Temperature-Dependence of the Energy of Cu
[110] Symmetrical Tilt Grain-Boundaries. Journal of Materials Science Letters,
1994. 13(1): p. 46-48.
204.Clyne, T.W. and P.J. Withers, An Introduction to Metal Matrix Composites.
Cambridge Solid State Science Series, ed. E.A. Davis and L.M. Ward. 1993,
Cambridge, U.K.: Cambridge University Press. 509 pp.
205.Herring, C., Surface Tension as a Motivation for Sintering, in The Physics of
Powder Metallurgy, W.E. Kingston, Editor. 1951, McGraw-Hill: New-York.
206.Iida, T. and R.I.L. Guthrie, The Physical Properties of Liquid Metals. 1993,
Oxford, U.K.: Oxford Science Publications, Clarendon Press. 288.
207.Simmons, G. and H. Wang, Single Crystal Elastic Constants and Calculated
Aggregate Properties: A HANDBOOK. 1971, Cambridge, MA: MIT Press.
208.Miura, H. and T. Sakai, Shape change of liquid B2O3 particles dispersed in
copper crystals during high-temperature deformation. Materials Science and
Engineering A, 2001. 319-321: p. 756-759.
209.Courtney, T.H., Mechanical Behavior of Materials. Mc Graw-Hill Series in
Materials Science and Engineering, ed. M.B. Bever and C.A. Wert. 1990, New York,
NY: Mc Graw-Hill. 502-561.
210.Gamaoun, F., et al., Cavity formation and accelerated plastic strain in T91 steel
in contact with liquid lead. Scripta Materialia, 2004. 50(5): p. 619-623.
211.Perovic, D.D., W.A. Miller, and G.C. Weatherly, On the solidification of bismuth
inclusions in stressed aluminum. Scripta Metallurgica, 1987. 21(5): p. 701-703.
212.Agullo, E., et al., Influence of an ultrasonic field on lead electrodeposition on
copper using a fluoroboric bath. New Journal of Chemistry, 1999. 23(1): p. 95-101.
213.Lowenheim, F.A., Electroplating, ed. J. Robison and R. Weine. 1978, New York:
McGraw-Hill. 594.
214.Zhao, J.-C. and M.R. Notis, Spinodal decomposition, ordering transformation,
and discontinuous precipitation in a Cu-15Ni-8Sn alloy. Acta Materialia, 1998.
46(12): p. 4203-4218.
193

CHAPTER 11. LIST OF REFERENCES
194
215.Kocks, U.F., The Relation Between Polycrystal Deformation and Single-Crystal
Deformation. Metallurgical Transactions, 1970. 1(5): p. 1121.
216.Beckman, J.P. and D.A. Woodford, Gas Phase Embrittlement by Metal Vapors.
Metallurgical Transactions A, 1989. 20A: p. 184-187.
217.Lagarde, P. and M. Biscondi, Fluage intergranulaire de bicristaux orientés de
cuivre. Memoires Scientifiques De La Revue De Metallurgie, 1974. 71(2): p. 121-
131.
218.Bleakney, H.H., The Creep-Rupture Embrittlement of Copper. Canadian
Metallurgical Quarterly, 1965. 4(1): p. 13-29.
219.Benson, B.A. and R.G. Hoagland, Crack growth behavior of a high strength
aluminum alloy during LME by gallium. Scripta Metallurgica, 1989. 23(11): p. 1943-
1948.
220.Watanabe, T., M. Tanaka, and S. Karashima. Liquid Metal Gallium-Induced
Intergranular Frcture of Aluminium Bicrystals. in Embrittlement by Liquid and Solid
Metals. 1982. St.-Louis, Missouri: The Metallurgical Society of AIME.
221.Biselli, C. and P. Isler, Neue Kupferlegierungen für Mechanik, Elektrik und
Elektronik. Metall, 2000. 54(4): p. 180-183.
222.Plewes, J.T., High-Strength Cu-Ni-Sn Alloys by Thermomechanical Processing.
Metallurgical Transactions, 1975. A 6(3): p. 537-544.
223.Cribb, W.R. and J.O. Ratka, Copper Spinodal Alloys. Advanced Materials and
Processes, 2002. 160(11): p. 1-4.
224.Hermann, P. and D.G. Morris, Relationship between Microstructure and
Mechanical Properties of a Spinodally Decomposing Cu-15Ni-8Sn Alloy Prepared by
Spray Deposition. Metallurgical Transactions A, 1994. 25: p. 1403-1412.
225.Boillat, S., List of products. 2002, UMS Usines Métallurgiques Suisse SA, Usine
Boillat: Reconvilier.
226.Touloukian, Y.S., et al., Thermal Conductivity: Metallic Elements and Alloys.
Thermophysical Properties of Matter, ed. Y.S. Touloukian. Vol. 1. 1970, New York -
Washington: IFI/Plenum.
227.Zinov'ev, V.E., Handbook of thermophysical properties of metals at high
temperatures. 1996, New York: Nova Science Publishers. 581.
228.Ledbetter, H.M. and E.R. Naimon, Elastic Properties of Metals and Alloys.
II. Copper. Journal of Physical and Chemical Reference Data, 1974. 3(4): p. 897-929.
229.Davis, J.R., et al., eds. ASM Handbook Volume 04: Heat Treating. 1991, ASM
International, Materials Park, OH.

230.Schwartz, L.H., S. Mahajan, and J.T. Plewes, Spinodal decomposition in a Cu-9
wt% Ni-6 wt% Sn alloy. Acta Metallurgica, 1974. 22(5): p. 601-609.
231.Zhao, J.-C. and M.R. Notis, Microstructure and precipitation kinetics in a Cu-
7.5Ni-5Sn alloy. Scripta Materialia, 1998. 39(11): p. 1509-1516.
195

Laurent FELBERBAUM
Av. des Tilleuls 24 né le 5 juin 1975
1203 Genève suisse
Tél. privé 022 344 24 57 marié, un enfant
Tél. prof. 021 693 68 02
laurent.felberbaum@epfl.ch
EXPÉRIENCE
PROFESSIONNELLE
Candidat au Doctorat
2000 à ce jour
Laboratoire de Métallurgie Mécanique,
Ecole Polytechnique Fédérale de Lausanne,
EPFL
Projet en collaboration avec Swissmetal
Mécanismes de fragilisation et microstructures d’alliages de cuivre au plomb
Ingénieur en Matériaux Laboratoires de Métallurgie Physique et Métallurgie
1998-1999 Mécanique, EPFL
Projet en collaboration avec ABB,
Calcom,
EMPA,
SR Technics et Sulzer
Essais mécaniques à haute température sur des superalliages réparés par
soudage laser ou brasage.
Enseignement
Section des Matériaux, EPFL
1998-2004 Animation de nombreux travaux pratiques en Métallurgie.
Conduite de deux projets de diplôme (4 mois).
Direction de séances d’exercices ”Déformation et Rupture” (niveau master).
1998
FORMATION
Diplôme d’Ingénieur en
Département des matériaux, EPFL
Sciences des Matériaux 3ème
année en échange à l’ETH Zurich
1993 Maturité type A Gymnase Cantonal du Bugnon, VD
Baccalauréat Latin-Grec-Mathématiques
LOISIRS
Montagne VTT, randonnées en cabane, ski et snowboard
Sports d’équipe Volleyball et surtout Football