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Analysis of nickel- and iron-based
superalloys on smallest
length scales
Dissertation
Robert Lawitzki
Institute for Materials Science
Chair of Materials Physics
2021
Untersuchung von Nickel- und Eisen-basierten
Konstruktionswerkstoffen auf kleinsten
Längenskalen
Von der Fakultät Chemie der Universität Stuttgart
zur Erlangung der Würde eines Doktors der
Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung
Vorgelegt von
Robert Lawitzki
aus Sindelfingen
Hauptberichter: Prof. Dr. Dr. h. c. Guido Schmitz
Mitberichter: Prof. Dr. Dr. h. c. Siegfried Schmauder
Prüfungsvorsitzender: Prof. Dr. Thomas Schleid
Tag der mündlichen Prüfung: 31.05.2021
Institut für Materialwissenschaft der Universität Stuttgart
2021
ii
iii
Zusammenfassung
In dieser Arbeit werden neue Methoden sowohl zur mikroskopischen Charakterisie-
rung der Mikro- und Nanostruktur als auch zur Charakterisierung des mikromechani-
schen Materialverhaltens von Konstruktionswerkstoffen entwickelt. Konstruktions-
werkstoffe sind Hochleistungslegierungen, die z.B. in Flugzeug- und Dampfturbinen
eingesetzt werden und somit sehr hohen Temperaturen und Drücken ausgesetzt sind.
Die gewonnenen Erkenntnisse dieser Arbeit sollen dazu beitragen, verbesserte oder
neue Werkstoffe zu entwickeln, die in Zukunft unter noch extremeren Bedingungen
eingesetzt werden können. Die Kenntnis des mikromechanischen Materialverhaltens
ermöglicht es außerdem, zukünftige Bauteile entsprechend ihrer Belastung besser
auslegen zu können.
Vorarbeiten haben gezeigt, dass die beiden mikrostrukturell sehr ähnlichen Nickelba-
sislegierungen Haynes 282 und Inconel 718 ein unterschiedliches mikromechanisches
Materialverhalten aufzeigen. Um diesen Befund physikalisch erklären zu können, wird
im Rahmen dieser Arbeit die Entwicklung von Makro- und Mikroeigenspannungen in
Zugproben während plastischer Verformung mittels Neutronendiffraktion gemessen
und mit begleitenden kristallplastischen Finite-Elemente Simulationen verglichen. Da-
für wurden in-situ Streuexperimente an der Neutronen-Beamline „Stress-Spec“ am
Forschungsreaktor FRM II am Heinz-Maier Leibnitz Zentrum (MLZ) in Garching durch-
geführt.
Ein weiterer wesentlicher Bestandteil dieser Arbeit ist die Planung und Ausführung von
Experimenten zur hochauflösenden mikroskopischen Untersuchung der Mikrostruktur-
gefüge und deren Korrelation mit Ergebnissen von begleitenden Neutronenbeugungs-
experimenten. Entscheidend dabei ist die Charakterisierung kleinster intermetallischer
Phasen im Nanometerbereich, welche für die herausragenden Kriecheigenschaften
von Konstruktionswerkstoffen verantwortlich sind. Um die Zusammensetzung, die
Größe und den Anteil dieser Phasen zu erfassen, wurden Anlagen zur Elektronenmik-
roskopie (REM der Uni Stuttgart und TEMs der Uni Stuttgart und des KIT), zur Atom-
sondentomographie (APT) der Uni Stuttgart und zur Kleinwinkel-Neutronenstreuung
(SANS) am FRM II in Garching komplementär verwendet. Hierbei werden herkömmli-
che Verfahren und Auswertetechniken angewandt und weiterentwickelt. Insbesondere
iv
wird ein Verfahren entwickelt, um mittels SANS unterschiedliche intermetallische Pha-
sen zu erfassen und zu quantifizieren.
Das an den beiden industriell bewährten Nickelbasislegierungen gewonnene Wissen
und die entwickelten Techniken zur Charakterisierung der Mikrostruktur werden in ei-
nem zweiten Teil der Arbeit an einem fundamental anderen metallischen Konstrukti-
onswerkstoff angewandt. Hierfür wurde die ferritische Legierung FBB-8 gewählt, wel-
che erst 2010 entwickelt wurde und sich ebenfalls durch gute Hochtemperatureigen-
schaften auszeichnet, allerdings mit geringer Duktilität bei Raumtemperatur. Grund-
sätzlich anders ist hier die Kristallstruktur; während Nickel-basierte Konstruktionswerk-
stoffe eine kubisch flächenzentrierte Kristallstruktur aufweisen, ist die Kristallstruktur
der Eisenbasislegierung FBB-8 kubisch raumzentriert. Da die Legierung kommerziell
noch nicht erhältlich ist, musste diese Legierung eigens hergestellt werden. Nach einer
Vorcharakterisierung und verschiedenen Wärmebehandlungen werden zwei Verfah-
ren zur hochauflösenden mikroskopischen Analyse kleinster Ausscheidungen in dieser
Legierung entwickelt (mittels TEM, APT und begleitenden numerischen Simulationen),
welche vor allem für die Sprödigkeit der Legierung bei Raumtemperatur verantwortlich
sind. Die Erkenntnisse dieser Studien lassen sich auf beliebige Legierungen mit klei-
nen Ausscheidungen übertragen.
v
Abstract
In this work, new experimental methods and techniques for the analysis of the micro-
scopic and the micromechanical material behavior of nickel- and iron-based superal-
loys are developed with the emphasis on the material characteristics and processes at
smallest length scales. Superalloys are high performance structural materials which
are for instance used in the turbine sections of aircraft engines and thus, exposed to
very harsh environments. The research in this work will help to design new further
advanced alloys for future applications in even extremer environments.
Previous works have shown that the nickel-based superalloys Haynes 282 and In-
conel 718 display a different micromechanical material behavior despite exhibiting a
similar microstructure. To explain this phenomenon physically, neutron diffraction ex-
periments have been performed at the FRM II research reactor in Garching to analyze
the formation of residual stresses during uniaxial tensile deformation of the specimens.
For interpretation of the experiments, the measurement results are compared to crystal
plasticity based finite element simulations.
Another essential part of this thesis are experiments to investigate the alloys’ micro-
structure by highest resolution microscopy techniques and a correlation of the results
with accompanied neutron diffraction experiments. The focus of this study is the char-
acterization of nano-sized intermetallic phases which are responsible for the outstand-
ing creep performance of superalloys. To analyze the chemical composition, the mor-
phology and the amount of these phases, a complementary study by scanning and
transmission electron microscopy (at the Uni Stuttgart and at the KIT), by atom probe
tomography (at the Uni Stuttgart) and by small angle neutron scattering (at the FRM II
in Garching) is performed. Hereby, techniques for data evaluation are refined with the
aim to develop a new method to differentiate and quantify different intermetallic phases
by small angle neutron scattering.
In the second part of this thesis, the experience and techniques to investigate the mi-
crostructure of nickel-based superalloys are transferred to study a fundamentally dif-
ferent alloy. For this purpose, the ferritic alloy FBB-8 was chosen, which is a recently
suggested iron-based superalloy with a remarkable high temperature performance; but
at the expense of the alloys’ ductility at room temperature. Principally different is the
vi
alloys crystal structure; while nickel-based superalloys exhibit a face-centered cubic
crystal structure, alloy FBB-8 exhibits a body-centered cubic structure. The alloy is
commercially not yet available and therefore fabricated within this study and pre-char-
acterized. After subsequent heat treatments, two new methods for the study of smallest
precipitates that form during cooling and which are responsible for the poor alloy duc-
tility at room temperature are developed. This is achieved by a complementary study
applying transmission electron microscopy, atom probe tomography and assisting nu-
merical simulations. The proposed methods are not only suited to study ferritic alloys
but can also be applied to study smallest precipitates in any other system.
Contents
vii
Contents
Zusammenfassung .................................................................................................. iii
Abstract ..................................................................................................................... v
Contents .................................................................................................................. vii
List of Figures .......................................................................................................... xi
List of Tables ......................................................................................................... xiv
List of Abbreviations .............................................................................................. xv
1 Introduction .................................................................................................... 1
1.1 Scope of this work ...................................................................................... 3
2 Material physical background ....................................................................... 5
2.1 Superalloys ................................................................................................ 5
2.2 Strengthening mechanisms ........................................................................ 9
2.3 Micromechanical material behavior .......................................................... 15
3 Experimental methods ................................................................................. 23
3.1 Electron microscopy ................................................................................. 23
3.2 Atom probe tomography ........................................................................... 35
2.1.1 Alloy design of Inconel 718 and Haynes 282 ..................................... 6
2.1.2 Alloy design of the ferritic alloy FBB-8 ............................................... 7
2.2.1 Strengthening against plastic deformation ......................................... 9
2.2.2 Temperature dependence of strengthening mechanisms ................ 12
2.3.1 Residual stresses ............................................................................ 16
2.3.2 Modelling of the residual stress state .............................................. 17
3.1.1 Scanning electron microscopy ......................................................... 24
3.1.2 Transmission electron microscopy .................................................. 28
3.1.3 Electron dispersive X-ray spectroscopy ........................................... 34
Contents
viii
3.3 Analysis with Neutrons ............................................................................. 43
3.4 Sample preparation .................................................................................. 51
4 Studies on nickel-based superalloys .......................................................... 57
4.1 Differentiation of
-
and
-
precipitates in Inconel 718 ........................... 57
4.2 Micromechanical behavior of Haynes 282 and Inconel 718 ..................... 73
5 Studies on iron-based superalloys ............................................................. 99
5.1 On the formation of nano-sized precipitates after aging ........................... 99
3.2.1 Tomographic Reconstruction and common artifacts ........................ 36
3.2.2 Numeric simulations of APT measurements .................................... 37
3.2.3 Analysis of APT data ....................................................................... 38
3.3.1 Small angle neutron scattering ........................................................ 43
3.3.2 Neutron diffraction ........................................................................... 45
3.3.3 Complementary crystal-plasticity modeling...................................... 49
3.4.1 Alloy fabrication ............................................................................... 51
3.4.2 Round tensile test specimens .......................................................... 51
3.4.3 Standard sample preparation techniques ........................................ 52
3.4.4 Focused ion beam assisted sample preparation ............................. 54
4.1.1 Characterization of the microstructure ............................................. 59
4.1.2 Compositional analysis by APT ....................................................... 66
4.1.3 Evaluation and interpretation of SANS experiments ........................ 67
4.1.4 Discussion ....................................................................................... 70
4.2.1 Microstructure of Inconel 718 and Haynes 282 ............................... 74
4.2.2 Macroscopic mechanical behavior ................................................... 77
4.2.3 Microscopic mechanical behavior .................................................... 80
4.2.4 Discussion ....................................................................................... 90
Contents
ix
5.2 Effect of local magnifications in APT on nano precipitates ..................... 116
6 Conclusions and Outlook .......................................................................... 135
Appendix A ............................................................................................................ 139
Appendix B ............................................................................................................ 142
Appendix C ............................................................................................................ 144
Appendix D ............................................................................................................ 151
Appendix E ............................................................................................................ 151
Bibliography .......................................................................................................... 157
List of publications (2016-2021) .......................................................................... 169
Acknowledgements .............................................................................................. 171
Curriculum vitae ................................................................................................... 173
Declaration of Authorship .................................................................................... 175
5.1.1 Correlation of mechanical alloy behavior and microstructure ........ 101
5.1.2 Introduction and test of a modified cluster search algorithm .......... 105
5.1.3 APT analysis of alloy FBB-8 .......................................................... 109
5.1.4 Discussion ..................................................................................... 113
5.2.1 Set up of the numerical simulations ............................................... 117
5.2.2 Evaluation of the morphology of precipitates ................................. 118
5.2.3 Model to correct the cluster morphology in APT reconstructions ... 120
5.2.4 Test on experimental data ............................................................. 127
5.2.5 Discussion ..................................................................................... 131
x
List of Figures
xi
List of Figures
Figure 2.1: Elementary cells of the most important phases in Ni-based superalloys. . 7
Figure 2.2: Elementary cells of the most important phases in ferritic superalloys....... 8
Figure 2.3: One stack of atoms in a (111) slip plane in a Ni-based alloy .................. 11
Figure 2.4: Schematic representation of different types of residual stresses ............ 16
Figure 2.5: Decomposition scheme of the total deformation gradient ....................... 18
Figure 3.1: Overview on length scales of characterization techniques in this work .. 23
Figure 3.2: Basics of scanning electron microscopy ................................................. 24
Figure 3.3: Scheme of the setup for electron backscatter diffraction (EBSD) ........... 26
Figure 3.4: EBSD and electron channeling contrast imaging ................................... 27
Figure 3.5: Diffraction contrast of B2 ordered precipitates in ferritic steels ............... 30
Figure 3.6: Comparison of SAD and CBED patterns in a Ni-based alloy .................. 31
Figure 3.7: Contrast transfer functions for two different microscopes. ...................... 33
Figure 3.8: Bloch-wave simulation of HRTEM images .............................................. 33
Figure 3.9: Schematic drawing of the setup for atom probe tomography. ................ 36
Figure 3.10: Simulated shape of heterogeneous emitter during field evaporation .... 38
Figure 3.11: ATP analysis techniques demonstrated on experimental data ............. 39
Figure 3.12: Cluster search by maximum separation method................................... 41
Figure 3.13: Experimental setup of a small angle neutron scattering experiment. .... 44
Figure 3.14: Schematic drawing of the neutron diffractometer Stress-Spec ............. 46
Figure 3.15: Setup and evaluation of neutron diffraction experiments ...................... 47
Figure 3.16: Schematic drawing of the measurement of lattice strains ..................... 48
Figure 3.17: Crystal plasticity finite element modelling ............................................. 50
Figure 3.18: Geometry of a round tensile test specimen. ......................................... 51
Figure 3.19: FIB assisted TEM specimen preparation .............................................. 55
Figure 4.1: Performed heat treatments on specimens of alloy Inconel 718. ............. 58
Figure 4.2: Inverse pole figure maps obtained by EBSD .......................................... 60
Figure 4.3: Morphology of
-
phase .......................................................................... 60
List of Figures
xii
Figure 4.4: Quantification of
-
precipitates after different heat treatments. ............. 61
Figure 4.5: Morphology and phase identification by TEM of
-
phase ..................... 62
Figure 4.6: Phase identification of
´
-
phase in state C-870 ..................................... 63
Figure 4.7: Volume-weighted particle size distributions of
-
and
-
precipitates ... 64
Figure 4.8: Compositional analysis by APT .............................................................. 66
Figure 4.9: Experimental ex-situ SANS patterns ...................................................... 67
Figure 4.10: Scattering curve of sample state B-760 fitted with a model .................. 69
Figure 4.11: APT reconstruction of a tip of state C-870 ............................................ 72
Figure 4.12: Microstructure of Inconel 718 (left) and Haynes 282 (right) .................. 76
Figure 4.13: Carbides and nitrides in Haynes 282 .................................................... 77
Figure 4.14: Macroscopic material behavior during uniaxial tensile tests ................. 78
Figure 4.15: Comparison of measured true stress versus lattice strain diagrams .... 81
Figure 4.16: Comparison of measured intergranular strain....................................... 83
Figure 4.17: Comparison of intergranular strains ...................................................... 85
Figure 4.18: Scheme of two specific grain orientations ............................................ 88
Figure 4.19: Comparison of peak widths (FWHM) .................................................... 89
Figure 4.20: Texture analysis on Haynes 282 .......................................................... 90
Figure 4.21: Logarithmic true stress - true strain curves of Haynes 282 ................... 93
Figure 4.22: Simulation of the macroscopic material behavior ................................. 93
Figure 4.23: Comparison of experimental and simulated evolution of lattice strains 95
Figure 4.24: Enlarged view on the evolution of (200)- lattice strains ........................ 96
Figure 4.25: Comparison of the microstructure of Inconel 718 (a) and Haynes 282 . 98
Figure 5.1: Hardness as a function of aging time of alloy FBB-8 ............................ 101
Figure 5.2: Microstructure of the heat-treated specimens ...................................... 103
Figure 5.3: DF TEM images of samples after heat treatment ................................. 104
Figure 5.4: Scheme of the modified cluster search algorithm ................................. 106
Figure 5.5: Test 1 of modified cluster search algorithm .......................................... 107
Figure 5.6: Test 2 of modified cluster search algorithm .......................................... 108
Figure 5.7: Comparison of the cluster analysis on FBB-8 ....................................... 111
List of Figures
xiii
Figure 5.8: Composition of primary (a) and secondary (cooling) (b) precipitates.... 113
Figure 5.9: Method to determine the number of excess atoms ............................... 119
Figure 5.10: Comparison of phase maps (top row) and atomic density .................. 121
Figure 5.11: Comparison of the isotropic radii ........................................................ 122
Figure 5.12: Dependence of the size ratio between the isotropic radii ................... 124
Figure 5.13: Correlation between the ratio in evaporation fields ............................. 124
Figure 5.14: Comparison of the differently calculated radii ..................................... 125
Figure 5.15: Comparison of the radii of the reconstructed precipitates ................... 126
Figure 5.16: Dark field TEM images of ordered NiAl type precipitates ................... 127
Figure 5.17: Method for analysis of single cooling precipitates in alloy FBB-8 ....... 128
Figure 5.18: Reconstruction of the atom probe tip after aging at 700 °C ................ 130
Figure 5.19: (a) Spatial distribution map (SMD) in x- and z- direction .................... 132
Figure 5.20: Reconstructions with varying field compression factor ....................... 133
List of Tables
xiv
List of Tables
Table 2.1: Chemical composition of Inconel 718 and Haynes 282 ............................. 6
Table 2.2: Nominal chemical composition of alloy FBB-8 ........................................... 8
Table 4.1: Parameters of the performed heat treatments on Inconel 718 ................. 59
Table 4.2: Compositions in Inconel 718 measured by APT ...................................... 65
Table 4.3: Comparison of SANS and TEM results .................................................... 70
Table 4.4: Comparison of measured volume fractions of
-
phase ........................... 71
Table 4.5: The theoretical volume fractions of
- phase in Inconel 718 ................... 72
Table 4.6: Comparison of the microstructure of Inconel 718 and Haynes 282 ......... 75
Table 4.7: Microscopic Young’s moduli
Ehkl
of specific reflections (
= 0°). ............. 82
Table 4.8: Texture analysis by neutron diffraction and EBSD................................... 91
Table 4.9: Input parameters of the CPFEM simulations. .......................................... 92
Table 5.1: Evaluated mean radii of cooling and primary precipitates ...................... 102
Table 5.2: Measurement of the composition of different phases in FBB-8 by APT . 109
Table 5.3: Comparison of the evaluated radii and atomic fractions ........................ 129
Table 5.4: Analysis of the APT measurement of the sample aged at 700 °C ......... 134
List of Abbreviations
xv
List of Abbreviations
APT Atom Probe Tomography
BCC Body-Centered Cubic
BSE Backscattered Electrons
BF Bright Field
CBED Convergent Beam Electron Diffraction
CPFE(M) Crystal Plasticity Finite Element (Model)
DF Dark Field
DP Diffraction Pattern
EBSD Electron Backscatter Diffraction
ECCI Electron Channeling Contrast Imaging
EDX Energy Dispersive X-ray Spectroscopy
FCC Face-Centered Cubic
FEG Field Emission Gun
FIB Focused Ion Beam
FWHM Full width at half maximum
KNN Kth Nearest Neighbor
SAD Selected Area Diffraction
(U)SANS (Ultra) Small Angle Neutron Scattering
SDM Spatial Distribution Map
SE Secondary Electrons
SEM Scanning Electron Microscopy
TEM Transmission Electron Microscopy
ToF Time of Flight
VIM Vacuum Induction Melting
xvi
1
1 Introduction
Enhancing the quality and sustainability of our life by improving our technologies is a
perpetual desire of our society. As materials scientists, we contribute to this develop-
ment by improving mechanical, chemical and physical properties of the materials we
are working with and by improving the technologies to study our materials. At the pre-
sent day, one of our biggest challenges is to decrease the emission of greenhouse
gases (especially carbon dioxide CO2) from fossil fuel combustion into the atmosphere.
Common strategies to achieve this aim, are replacing fossil fuels by renewable energy
sources, recycling industrial products rather than producing new ones and increasing
the efficiency of our technologies to reduce the consumption of resources.
The particular motivation of this work is, improving the efficiency of fossil fired energy
producers like steam power plants or turbine engines. According to Carnot’s theorem,
the efficiency of a reversible heat engine is defined by the difference in temperatures
between a hot reservoir (inside the engine) and a cold reservoir (ambient temperature)
divided by the temperature of the hot reservoir. Therefore, the efficiency of heat en-
gines can be generally improved by increasing their working temperature. This requires
specifically designed materials that resist elevated temperatures (typically above
600 °C) and hot gas corrosive environments while maintaining their ductility and tough-
ness over extended service time. In general, an alloy is more heat resistant, the higher
its melting point is. As a rule of thumb, typical metals and alloys start to creep at ap-
proximately 0.3-0.4
Tm
(
Tm
= melting temperature) and fail at temperatures of less than
0.6
Tm
. Alloys for load-bearing applications that can withstand temperatures of more
than 0.6
Tm
(up to 0.9
Tm
)
for an extended period of time are called “superalloys” [1].
Superalloys have been first emerged in 1941 and since that have a huge impact as a
material for aircraft and industrial gas turbines (up to 50 % of the weight), as well as
for nuclear reactors, rocket engines, space vehicles and other high-temperature appli-
cations.
The capability to employ superalloys at temperatures close to their melting point is
achieved by specific alloy design concepts, as will be introduced in more detail in sec-
tion 2.2. In recent superalloys, up to 14 alloying elements are added in minuscule to
major amounts and specific heat treatments are performed for a controlled adjustment
1 Introduction
2
of the microstructure. Thereby, the most important microstructural features are nano-
to micrometer sized ordered intermetallic phases with volume fractions of up to 60 %
that precipitate homogeneously within the alloy and which are responsible for the out-
standing performances at high temperatures. Additional adjustments of the microstruc-
ture are necessary for mechanical and chemical stability of the alloy and for ensuring
excellent processing parameters like the weldability or malleability.
The technological development of high performance superalloys is very complex and
thus, further improving the properties of future alloys further is only possible when un-
derstanding the microscopic origin of any change in the microstructure and its impact
on the macroscopic properties. There exists a wide variety of different materials char-
acterization and simulation techniques that have to be applied complementary to study
the alloys. A technological progress in alloy design requires that these techniques have
to be continuously advanced or that new techniques have to be developed. Improving
the methods and techniques to study superalloys on smallest length scales is the aim
of this research.
1.1 Scope of this work
3
1.1 Scope of this work
The objective of this work is, to get a better qualitative and quantitative understanding
of the nano-sized precipitates and the mechanisms that strengthen superalloys.
In section 2, a brief introduction of the materials physical background is given. It will be
shown, which type of superalloys exist and the specific superalloys that are analyzed
in this work are presented. It will be further demonstrated, how these alloys are
strengthened and how the mechanical performance of such alloys is evaluated.
In the subsequent section 3, the material characterization techniques that have a major
contribution to this work – scanning electron microscopy (SEM), transmission electron
microscopy (TEM), atom probe tomography (APT) and neutron diffraction – are pre-
sented, together with corresponding evaluation and simulation techniques for data in-
terpretation. Furthermore, information on the specimen preparation and material pro-
cessing are given.
In section 4, the results of the research on the two commercial nickel-based superal-
loys Inconel 718 and Haynes 282 are presented. Even though the alloy Inconel 718 is
used and well-studied for around 70 years, the complex microstructure of this alloy is
still not fully understood which is also seen at the large number of new research articles
about this alloy that are published every year. Therefore, in the first part of this section,
the complex microstructure of the alloy is studied by TEM, APT and small angle neu-
tron scattering (SANS) with the emphasis on the identification and quantification of
different precipitates in the alloy Inconel 718. By complementary use of the techniques
TEM, APT and SANS, a new experimental methodology is developed that enables the,
so far, most accurate quantification of the different strengthening phases in the alloy
Inconel 718.
The Ni-based superalloy Haynes 282 is a rather new alloy (first presented in 2006)
with superior creep properties in comparison to Inconel 718 while still featuring a very
good fabricability and weldability. Previous studies [2] have shown that both alloys be-
have, from a micromechanically point of view, differently although they exhibit a similar
microstructure. In order to understand the origin of this behavior, residual stresses in
both alloys are measured by neutron diffraction experiments during uniaxial tensile
deformation. A new experimental setup is applied that enables to measure the lattice
1 Introduction
4
strain of specific grains in arbitrary directions with respect to the load. The data are
interpreted by a continuum micromechanical crystal plasticity model allowing to com-
pare and discuss the obtained results.
With the aim to apply the micromechanical stress analysis and corresponding numeri-
cal simulations not only on face-centered cubic alloy systems, a ferritic superalloy was
chosen and for this reason, analyzed in further studies which are summarized in sec-
tion 5. The recently proposed alloy FBB-8 [3] has been found to be a good candidate
for the research, since its alloy design follows the same concepts as those applied for
Ni-based superalloys. Alloy FBB-8 exhibits good creep properties and oxidation re-
sistance and, due to its comparably lower costs, lower density and higher thermal con-
ductivity, is an attractive alternative to Ni-based alloys. A major drawback of the alloy
is its poor ductility at room temperature, which can be lead back to smallest precipitates
within the alloys. The characterization of precipitates with a size of less than five na-
nometers is experimentally very challenging and is the primary objective of this section.
Two new methodologies that are based on the study of those precipitates with the atom
probe technique are developed and discussed thoroughly.
The work is concluded and an outlook on still ongoing research is given in section 6.
5
2 Material physical background
2.1 Superalloys
Superalloys are materials for long term application at elevated temperatures close to
their melting point. They combine excellent high temperature properties such as re-
sistance to oxidation, corrosion and creep, a good fabricability and high strength. In
general, there are three different classes of superalloys. Depending on the base ele-
ment, one may distinguish Ni-based, Fe-based and Co-based superalloys. Among
these, the Ni-based alloys are most often the material of choice, because they are
tough, ductile and the costs are moderate. Co-based alloys are often too expensive
and show worse oxidation resistance, whereas Fe-based alloys are prone to brittle-
ness. In Co- and Fe-based alloys, one has to further avoid allotropic phase transfor-
mations between the temperatures in and out of operation. Nowadays, a wide range
of different superalloys (e.g. wrought, cast, by powder metallurgy, directionally solidi-
fied and single crystalline) exists varying in their maximum operational temperature
and their specific properties and costs. The highest operational temperatures are
reached in single crystalline alloys, like the 6th generation Ni-based alloy TMS-238,
which can be used at temperatures of up to 1100 °C [4], or even more with additional
coatings. These alloys are used for the hottest parts, such as the turbine blades, in
modern aircraft jet engines.
In this work, superalloys are investigated that aim for applications at intermediate tem-
peratures (600-850 °C) like exhaust and nozzle components in gas turbines or com-
ponents in the transitions sections and other hot-gas-path components in land-based
gas turbines. These alloys have to be creep and corrosion resistant, but furthermore
they should be easily fabricable, weldable and financeable excluding for instance the
use of single crystalline alloys in this sector. The key for the strength of these alloys is
the controlled precipitation of ordered intermetallic phases but with a relatively low vol-
ume fraction of 10-25 % to increase formability; in single crystalline superalloys they
usually exceed volume fractions of 60 %. The two Ni-based superalloys Inconel 718
and Haynes 282 and the ferritic superalloy FBB-8 have been chosen for the studies in
this work and are introduced in the following.
2 Material physical background
6
2.1.1 Alloy design of Inconel 718 and Haynes 282
The alloy Inconel 718 has been introduced in the early 1960s and is one of the most
widely used superalloys for applications in cast and forged components at elevated
temperatures up to 650 °C [5,6]. The chemical composition of Inconel 718 is given in
Table 2.1. The alloy is strengthened by solid solution and precipitation hardening (see
also next section). Cr and Mo are added to strengthen the face-centered cubic
- matrix
(Fm-3m); Cr also increases the corrosion resistance. The high iron content reduces
the costs of the alloy and increases the alloy’s machinability and weldability. Al, Ti and
Nb form precipitates during controlled ageing treatments (see section 3.4.1) [7,8].
Within Inconel 718, three different ordered intermetallic phases can be distinguished
as presented Figure 2.1. A strengthening effect is obtained by coherent
-
precipitates
(L12 structure, Pm-3m) that have a chemical composition of Ni3(Al,Ti) and are spherical
in shape and by semi-coherent tetragonal
-
precipitates (D022 structure, I4/mmm)
with a chemical composition of Ni3(Nb,Ti) that form as disks [9,10]. Since
-
phase is
metastable, it transforms to the stable but incoherent orthorhombic
-
phase (D0a
structure, Pmmn) at temperatures above 850 °C [11].
-
phase has the same chemical
composition as
-
phase but forms plate-shaped precipitates mainly at grain bounda-
ries that contribute only to grain boundary strengthening but also lead to brittleness.
Furthermore, minor amounts of primary MC- type carbides and Laves phases can
form, but these phases are irrelevant for the studies in this work and therefore not
further considered.
Since the use of alloy Inconel 718 is limited to 650 °C, alloys such as Waspaloy,
René 41 or U-720, capable to resist higher temperatures, have been developed; but
these alloys are more expensive and difficult to process. Therefore, the alloy
Haynes 282 was introduced in 2005, which was designed to achieve an optimum bal-
ance in fabricability and high temperature creep strength [12,13]. Haynes 282 shows
Alloy
Ele-
ments
Ni
B
Cr
Co
Fe
Mo
Mn
C
Al
Ti
Nb
Inconel
718
[at.-%]
51.4
0.03
21.2
-
19.7
1.9
-
0.14
1.18
1.15
3.31
[wt.-%]
52.1
0.005
19.0
-
19.0
3.1
-
0.030
0.55
0.95
5.3
Haynes
282
[at.-%]
56.9
0.016
21.9
9.89
0.2
5.1
0.03
0.31
3.2
2.5
-
[wt.-%]
57.9
0.003
19.7
10.1
0.2
8.4
0.03
0.064
1.5
2.1
-
Table 2.1: Chemical composition of Inconel 718 and Haynes 282 in atomic and weight percentage. The
alloys have been provided by Böhler Schmiedetechnik GmbH&CoKG.
2.1 Superalloys
7
excellent creep strength in the temperature range of 650 °C to 900 °C. In comparison
to Inconel 718, the content of Al and Ti is higher (cf. Table 2.1), but no Nb is added;
thus, only coherent
-
precipitates are formed. Instead of Fe, more Mo and Co are
added. Molybdenum atoms are relatively large and immobile and therefore, provide
significant solid-solution strengthening. Co is used to increase the solvus of
-
phase.
As in alloy Inconel 718, minor amounts of primary MC carbides, mainly at grain bound-
aries, are formed, but in Haynes 282, these further decompose to M23C6 and M6C car-
bides between 760 °C and 980 °C [14,15]. The precipitation of M23C6 carbides is in-
tended and controlled during heat treatments, since they provide grain boundary
strengthening.
2.1.2 Alloy design of the ferritic alloy FBB-8
The Fe-based superalloy FBB-8 has been designed by Teng et al. in 2010 [3] and is a
ferritic steel which has a microstructure analogously to
-
strengthened Ni-based sup-
eralloys. The body-centered cubic
- matrix is strengthened by the addition of Al and
Ni (see Table 2.2) that form coherent NiAl- type
- precipitates (B2 structure, Pm3m)
as shown in Figure 2.2. According to Ni-based alloys, Cr and Mo provide solid solution
strengthening and furthermore, control the lattice misfit between
-
matrix and
- pre-
cipitates. The lattice misfit (see also next section) has to be small in order to keep the
precipitate to matrix interface coherent. Cr provides further resistance against oxidation
and Zr is added to slow down diffusion in the matrix.
Figure 2.1: Elementary cells of the most important phases in Ni-based superalloys.
2 Material physical background
8
In comparison to Ni-based alloys and austenitic steels, ferritic alloys have lower ther-
mal expansion coefficients, a higher heat conductivity and are less expensive and
therefore, an attractive candidate material for steam-turbine applications when achiev-
ing comparable creep properties.
Figure 2.2: Elementary cells of the most important phases in ferritic superalloys.
Elements
Fe
Al
Cr
Ni
Mo
Zr
B
[at.-%]
66.07
12.73
10.16
9.00
1.87
0.14
0.024
[wt.-%]
69.85
6.50
10.00
10.00
3.40
0.25
0.005
[wt.-%]
69.0±0.7
6.59±0.07
10.2±0.1
10.1±0.1
3.69±0.13
0.36±0.01
0.0044±0.0003
Table 2.2: Nominal chemical composition of alloy FBB-8 in atomic and weight percentage. The nomi-
nal composition is compared to measured data obtained by ICP-OES (third row).
2.2 Strengthening mechanisms
9
2.2 Strengthening mechanisms
The mechanical properties and thermal stability of superalloys depend strongly on the
alloy’s microstructure and its evolution during deformation and heating. In order to ob-
tain metals of extraordinary high strength, mechanisms that restrict the motion of dis-
locations are necessary. The most important strengthening mechanisms against plas-
tic deformation and creep activity in superalloys are introduced in the following.
2.2.1 Strengthening against plastic deformation
Solid solution strengthening
Solute atoms affect the strength of an alloy because of a difference in size and thereby,
impose lattice strains on surrounding matrix atoms. Lattice strains cause local stress
fields that interact with the stress fields of dislocations. Depending on the size of the
solute atom, it will segregate at the tensile or compressive site of a dislocation and
thus, decrease the entire elastic energy. The required stress to move the dislocation
away from the segregation zone is now increased and therefore, the dislocation immo-
bilized. Small interstitial solutes, such as C and N, move to the tensile side of the dis-
location, whereas larger substitutional solutes, such as here Cr and Mo, move to the
compressive side. Instead of interactions caused by elastic strain fields due to different
atomic radii of the solutes, also chemical, electrostatic and dielastic interactions con-
tribute to solid solution strengthening [16].
Work hardening
The typical observation in metals that increasingly higher stresses are required to con-
tinue plastic deformation is referred to as strain or work hardening. During plastic de-
formation of a material below its recrystallization temperature (and therefore also
sometimes called as “cold working”), the dislocation density increases and therefore,
dislocations start to interact with each other. The higher the dislocation density is, the
higher is the stress required for additional plastic deformation. A simple relationship
between the flow stress
and the dislocation density
is given by
,
[2.1]
2 Material physical background
10
where
0
is the flow stress to move a dislocation in the absence of other dislocations,
a constant between 0.3 and 0.6,
G
the shear modulus and
b
the Burgers vec-
tor [16,17]. In most metals and alloys, the work hardening follows a power law behav-
ior. In the simplest and most used expression as proposed by Hollomon [18], the true
stress
t
is related to the true plastic strain
t
by
,
[2.2]
where
K
is the strength coefficient,
n
the strain-hardening exponent (typically between
0.1 and 0.5 for metals). The exponent measures the ability of a metal to work harden,
where a value of zero represents a perfect plastic solid and one an elastic solid.
Precipitation hardening
Strengthening of an alloy by precipitation hardening is a crucial mechanism in super-
alloys. Fine precipitates, as introduced in the previous section, are formed during con-
trolled heat treatments. Therefore, the alloy is first heated to a one phase region, where
all precipitates are solved in the matrix phase and then quenched to room temperature
or below, so that precipitation is avoided and a supersaturated solid solution is ob-
tained. In a subsequent aging treatment, the precipitates are formed.
Since precipitates present a barrier for the movement of dislocations, a strengthening
effect is obtained. Depending of the nature of the precipitates, dislocations interact
differently with the precipitates. If precipitates are impenetrable to dislocations, they
are bypassed by a dislocation according to the Orowan mechanism. The smaller the
free distance between the precipitates is, the higher is the strengthening effect
,
[2.3]
in which L denotes the center-center distance between two precipitates and
r
the ra-
dius.
Impenetrable precipitates, for instance carbides and Laves phases, are usually hard
and incoherent with the matrix and often do not have a significant strengthening effect
in superalloys due to their very low volume fraction [16].
In superalloys, the interfacial energy between the precipitates (
,
or
) and the
matrix phase (
or
) is relatively low and therefore, the stress to shear a precipitate is
lower than the Orowan stress for dislocation bowing. To ensure a low interfacial energy
2.2 Strengthening mechanisms
11
in the alloys, the lattice misfit between precipitates and matrix has to be minimized.
The lattice misfit is defined by
,
[2.4]
where is the lattice parameter of the precipitate and is the lattice parameter of
the matrix. can be adjusted by considering the elemental partitioning of the alloying
elements into the corresponding phases. The stress required to shear the precipitate
increases with their volume fraction. Since the precipitates in superalloys are ordered
intermetallic phases, typically in cube / cube orientation, a dislocation, e.g. a
dislocation in the
-
phase, cannot shear a
-
precipitate without the formation
of an anti- phase boundary (APB). The formation of an APB requires additional energy
and thus, provides order strengthening, which is a substantial contribution in Ni-based
superalloys [1]. As indicated in Figure 2.3 (a), two
dislocations are
required to restore the order of a
-
precipitate and explain why dislocations always
travel in pairs through the
/ -
structure as so called superdislocations. For the
/ -
structure, this is the same for a
dislocation, whereas even four dislo-
cations in
- and
- direction are required to restore the order as shown in
Figure 2.3b).
Figure 2.3: One stack of atoms in a (111) slip plane in a Ni-based alloy: (a) of the L12 structure corre-
sponding to
-
phase and (b) of the D022 structure corresponding to
-
phase. The arrows indicate
how many dislocations in a specific orientation are required to restore the order.
a) ´- phase b) ´´- phase
Ni
Al, Ti Nb
2 Material physical background
12
For very small precipitates in comparison to the distance of the pair of dislocations, this
hardening effect is proportional to the square root of the radius of the precipitates
(called weak- coupling). The critical resolved shear stress
c
can be expresses by [1]
,
[2.5]
where
APB
is the anti-phase boundary energy,
f
the volume fraction of the precipitates
and
T
the dislocation line tension.
If the precipitate size becomes comparable to the distance of the two dislocations, this
is called strong- coupling and the hardening is inversely proportional to the radius
,
[2.6]
where
w
is a dimensionless constant. It can be shown that a peak strength of approx-
imately is obtained for precipitates with a size of ,
which is at the transition of the two mechanisms as discussed in more detail in the
textbook of R. Reed [1,19]. E.g. for
-
precipitates in typical Ni-based superalloys the
optimum radius for peak strength is in the range of 20 nm to 100 nm.
A further positive effect of the precipitates is that they give the possibility to control the
grain size. Precipitates pin the grain boundaries and therefore, extra energy is needed
to move the grain boundaries (Zener effect) [20]. Phases, such as the M23C6 carbides,
are good grain boundary strengtheners, since they do not dissolve during the super
solvus heat treatment, where the ordered precipitates are fully dissolved.
2.2.2 Temperature dependence of strengthening mechanisms
The design of superalloys aims to maintain a high strength at high temperatures over
the lifetime of the material. Typically, the mechanical performance of a material de-
creases with increasing temperature due to creep, changes in the microstructure or
due to chemical degradation which is elaborated in the following.
Creep
Creep is the thermally (diffusion) activated and time dependent mode of plastic defor-
mation under mechanically applied stresses below the yield strength of the material.
This deformation mode can be compared to a viscous material behavior and becomes
2.2 Strengthening mechanisms
13
typically severe at temperatures above 0.6
Tm
. As a result, structural components such
as turbine blades will expand in the load direction and decrease in cross-sectional area;
eventually the material will failure after a certain time of operation.
Deformation by creep is usually the dominant deformation mechanism at stresses be-
low ~ 4 x 10-3 times the shear modulus of the material. Depending on the stress level
and temperature, different mechanisms can be responsible for creep which is often
summarized in so called Ashby-type deformation maps [21]. At lowest temperatures
and stresses, atoms diffuse mainly along grain boundaries which results in grain
boundary sliding; called Coble creep [22]. For low stresses but high temperatures, bulk
diffusion becomes active and generates a flux of vacancies from regions subjected to
tensile stresses within a grain to regions of lower chemical potential as proposed by
Nabarro and Herring [23,24]. These two mechanisms are therefore very sensitive to
the grain size; larger grains decrease the creep rate. At high temperatures and high
stresses, dislocation (power law) creep is the dominant creep mechanism. Due to the
increased amount of vacancies, pinned dislocation can overcome obstacles by climb
processes and continue to glide [16]. The climb step is hereby rate-controlling and
thus, alloy additions that pin dislocations – e.g. solid solution strengtheners or precipi-
tates – provide creep resistance.
Mechanical stability
For the performance of superalloys it is further crucial that good mechanical perfor-
mance is not only achieved at room temperature, but also at service temperatures.
Usually, the yield strength, the tensile strength and the Young’s modulus decrease with
increasing temperature and thus, plastic deformation is easier at elevated tempera-
tures. A remarkable characteristic of
-
phase strengthened Ni-based superalloys is
the anomalous yielding effect, which is an increase in the yield stress with increasing
temperatures up to about 800 °C. This is due to the anisotropy of the anti-phase bound-
ary (APB) energies which promotes thermally activated cross-slip of segments of a
superpartial dislocation within
-
precipitates from the {111} slip plane (high APB en-
ergy) to the {001} slip plane (low APB energy). These cross-slipped segments, known
as Kear- Wilsdorf locks, cannot glide on the not closed-packed {100} plane and thus,
restrict further deformation [25].
2 Material physical background
14
A further essential factor to preserve the mechanical properties at high temperatures
is to minimize the coarsening of strengthening precipitates through Ostwald ripening.
High temperatures activate diffusion in the alloys and therefore, large precipitates start
to grow at the expense of smaller once, in order to reduce the interfacial area between
precipitates and matrix. According to the theory by Lifshitz and Slyozov [26] and Wag-
ner [27], the cube of the average precipitate radius
r(t)
is proportional to the ageing
time , which is described by
,
[2.7]
where
r0
is the initial average radius of the precipitates and
k
a coarsening rate con-
stant that can be expressed by
,
[2.8]
where
is the interfacial energy,
c
the solute concentration at equilibrium,
2
the molar
volume of the solute,
D
the diffusion coefficient of the solute atoms,
R
the ideal gas
constant and
T
the temperature. Accordingly, low coarsening rates can be achieved
by lowering the interfacial energy and thus, by minimizing the lattice misfit between
precipitates and matrix (Eq. [2.4]).
Chemical degradation
Like for every other material, chemical degradation or corrosion is a further issue to
encounter when operating temperatures are high. Typical damage caused by corrosion
is surface oxidation or stress corrosion cracking which often results in a failure of the
material. Since oxidation of most metals (beside noble metals like gold or platinum)
cannot be avoided, the surface of a superalloy has to be passivated against further
oxidation, either by formation of a stable protective oxide layer, or, by additional coat-
ings (e.g. thermal barrier coatings) which are relevant in application for turbine blades
that have to resist temperatures of more than 1400 °C. For the formation of a continu-
ous, adherent and slowly growing oxide film, it is further important that the volume and
the thermal expansion coefficients of the oxide and the metal are similar to prevent
spallation or cracking of the oxide scale. A protective and thermodynamically stable
Al2O3 oxide layer is formed by the addition of Al and Cr, where Cr promotes a prema-
ture external oxidation of Al as is described in more detail in the article of Giggins and
Pettit [28].
2.3 Micromechanical material behavior
15
2.3 Micromechanical material behavior
In order to tailor a material for specific applications, it is important to understand and
predict material properties and processes – such as the formation of residual stresses,
the deformation of a material, phase transformations or fatigue – because these can
significantly influence the lifetime of a component. Besides theoretical considerations
and experiments it is very useful to perform numerical simulations; especially in com-
plex situations or in case of competing mechanisms, that cannot be accessed by ex-
periments. In practice there are different simulation approaches. For instance, there
are atomistic approaches like molecular dynamics and Monte Carlo methods that focus
on the investigation of lattice and defect dynamics at the atomic scale. On the other
hand, there are continuum models, such as finite element approaches that focus on
rather large-scale mechanical construction problems to determine stress and strain
concentrations under load by averaging constitutive laws.
In contrast to these rather classical simulation methods, recent advances in computa-
tional materials science focus on numerical simulations that bridge the large gap in
size and time-scales between these methods as introduced in greater detail in the text-
books “Computational Materials Science” of D. Raabe [29] and “Multiscale Materials
Modeling” by S. Schmauder and I. Schäfer [30]. Typical approaches to describe micro-
structure- property relationships at the micros- to mesoscopic scale (or lattice defect
scale to grain level scale) are phase field kinetic methods and crystal plasticity simula-
tions. Whereas phase field methods are especially useful to describe kinetic processes
like phase transformations, diffusion or grain growth, crystal plasticity finite element
(CPFE) models are developed to calculate the evolution of the dislocation structure
and residual stresses during plastic deformation which is of interest in this work.
2 Material physical background
16
2.3.1 Residual stresses
Residual stresses are mechanical stresses that remain within a material without exter-
nal forces. They can be both, beneficial or detrimental, for the life of a component,
depending on their sign and the type of stresses during operation. For instance, tensile
residual stresses on the surface of a component can induce crack formation while this
can be avoided when the surface of a component is under compressive residual
stresses. Residual stresses can arise in the production process, during mechanical,
thermal or chemical load or due to internal phase transformations (e.g. precipitation
processes). As shown in Figure 2.4, three different types of residual stresses can be
differentiated depending on their spatial dimension [31]. Type I (
I) residual stresses
are macroscopic stresses and correspond to the averaged stress state across a large
region of the component. In contrast to this, microscopic residual stresses of type II
(
II) or intergranular stresses vary on the grain scale and of type III (
III) or intragran-
ular stresses show variations within single crystallites. Type II stresses arise in poly-
crystalline materials due to anisotropic elastic, plastic and thermal properties of differ-
ently orientated neighboring grains. The type III micro stresses are caused by local
stress fields of lattice defects or secondary phases.
-+
residual stress
residual stress
distance
σI : macroscopic stress
σII : intergranular stress
σIII: intragranular stress
component
grains
Figure 2.4: Schematic representation of different types of residual stresses within a material.
2.3 Micromechanical material behavior
17
2.3.2 Modelling of the residual stress state
For the design of safe and optimized components that do not fail during operation it is
essential to know their residual stress state before and during service. From the stress
state, it is possible to determine for instance which parts of complex constructions are
loaded close to the design limited and therefore, have to be improved by e.g. increas-
ing the cross section or other forms of reinforcement. As will be shown in sections 3.3.2
and 4.2, residual stresses can experimentally be determined non-destructively by neu-
tron diffraction in components of rather simple geometry. For components of complex
geometries or components for future technological applications, where an experi-
mental evaluation of the stress state is not feasible, it is necessary to predict the overall
stress and strain state.
Early approaches to predict the macroscopic behavior of a polycrystal during plastic
deformation described plasticity by models according to simple assumptions; for in-
stance that all grains are subjected to the same stress – lower bound models such as
the Sachs model (1928) [32] – or subjected to the same strain – upper bound models
such as the Taylor model (1938) [33] and later the model by Bishop and Hill (1951)
[34,35]. More realistic considerations by Kröner (1961) [36], Budiansky & Wu (1962)
[37] and Hutchinson (1970) [38] include elastic and elastic-plastic interactions of the
grains and their surroundings in so called self-consistent models. However, none of
these approaches can deal with complex internal and/or external boundary conditions
such as direct grain-to-grain interactions. In real materials, grains that resolve the high-
est shear stress (Schmid factor) will start to deform first and will change their shape.
This change in shape is suppressed by the surrounding grains that do not yet deform
plastically and thus, experience higher elastic stresses, until also these grains deform
plastically. Thereby, the shape change of each grain has to be arranged in a way that
the complete crystal deforms without separation of the individual grains. For reasona-
ble predictions of the plastic behavior of an anisotropic polycrystalline superalloy, many
more aspects such as multiple deformation mechanisms, the interaction between the
grains, crystallographic texture development, but also phase transformations, soften-
ing phenomena or defects have to be considered.
A possibility to deal with these complex boundary conditions are crystal plasticity finite
element (CPFE) methods. Thereby, the complete sample volume is discretized into
2 Material physical background
18
finite elements and finite element approximations are used to find an equilibrium in
forces under compatibility of the displacements of the finite elements. By constitutive
and numerical plasticity formulations any mechanism that renders deformation kine-
matics can be incorporated. A detailed introduction to CPFE modelling is given in
Roters habilitation thesis [39] and his review [40].
In this work, CPFE simulations are performed to model the dynamics of plastic flow in
face centered polycrystalline systems. The simulations are based on a constitutive
plasticity law to calculate plastic shear strain caused by deformation due to dislocation
glide. Therefore, the sample volume is discretized in finite elements, where each of
them is defined by an orientation, a phase (decides which constitutive law is used) and
a homogenization scheme (according to Ref. [41]). A brief introduction of the relevant
constitutive equations used to describe deformation by dislocation glide in a CPFE
framework is given in the following.
Constitutive model describing plastic deformation by dislocation glide
Phenomenological derivation according to Roters et al. [41]:
The motion of a continuum or the local deformation at a material point is described by
the deformation gradient (second order tensor)
,
[2.9]
where is the position function with the reference position vector
and with describing the displacement of the original position of a material point
to its new position. The deformation gradient can be multiplicatively decomposed (cf.
Figure 2.5) as
,
[2.10]
Figure 2.5: Decomposition
scheme of the total deformation
gradient. Taken from Roters et al.
[41]
2.3 Micromechanical material behavior
19
and its time derivative by
,
[2.11]
where describes the elastic (reversible) part of the deformation gradient, the plas-
tic (irreversible) permanent deformation and the time dependency of the finite defor-
mations. In case of dislocation glide, the plastic deformation evolves as
,
[2.12]
where is the plastic velocity gradient which can be further formulated as the sum of
the shear rates of all slips systems
,
[2.13]
where is the tensor (or dyadic) product of the vectors describing the slip
direction and the normal vector to the slip plane of slip system . In phenomeno-
logical models, the shear rate is most often formulated by a power-law behavior as
a function of the resolved shear stress and the current resistance to slip as e.g.
suggested by Hutchinson [38,42]
,
[2.14]
where and are material parameters denoting the rate sensitivity of slip and a
reference shear rate, respectively (cf. Hollomon’s model to predict work hardening in
Eq. [2.2]). The resolved shear stress on slip system depends on the elastic strain
and is defined by the (Frobenius inner) product of 2nd Piola-Kirchhoff stress and the
Schmid factor as
,
[2.15]
where is the stiffness (fourth order) tensor and the Green-Lagrange strain tensor.
After initial yield, strain hardening depends on the influence of the micromechanical
interaction among different slip systems on a fixed slip system and is characterized
by the rate of the increase of the slip resistance
,
[2.16]
2 Material physical background
20
where denotes the increment of slip resistance on slip system due to an incre-
ment of shear on slip system and where is the saturation shear stress, and
are material dependent slip hardening parameters. is the latent hardening param-
eter which accounts for anisotropic hardening and is taken as 1.0 for coplanar slip
systems and (which means that two dislocations are on the same slip plane but
with different Burgers vector) and 1.4, otherwise. For FCC slip systems, the parame-
ters, and are assumed to be independent of the slip system.
It should be mentioned here that it is also possible to describe the material state in
terms of physical-based constitutive models, which rely on internal variables such as
the geometric necessary dislocation density (e.g. as proposed by Ma et al. [43,44]).
These models allow considering local neighborhood information (coupling of neighbor-
ing integration points) and thus can describe mechanical size effects (such as grain
size dependencies) but this is out of the scope of this thesis.
Computational procedure
For the calculation of the mechanical response of a continuum body with specific
boundary conditions, a numerical solver (here ABAQUS) is applied, to establish a me-
chanical equilibrium (a balance of all inertia forces) for each time increment during
deformation. The FEM resolver uses the 1st Piola-Kirchhoff stresses at each integra-
tion point that have to be derived as a response for a given average deformation gra-
dient by applying the constitutive crystal plasticity formulations. The following calcu-
lation procedure is applied:
(i) is partitioned into individual deformation gradients for each crystal depending on
the homogenization scheme (in this work, the isostrain scheme is applied). The
material’s stress response is derived using a predictor-corrector scheme, which is
implemented in the simulation kit DAMASK [45]. (ii) First, a velocity gradient
at the
time is predicted and allows estimating the stress response by a Newton-
Raphson scheme
. The estimated 2nd Piola-Kirch-
hoff stress can now be applied to calculate a new velocity gradient ac-
cording to Eq. [2.13]. These calculations are-times iterated until the residual of the
difference between guessed
and updated converges. (iii) The 1st Piola-
2.3 Micromechanical material behavior
21
Kirchhoff stress , which relates forces in the deformed/current configuration to areas
in the reference configuration, are obtained applying the relation
.
[2.17]
The stresses are finally used by the numerical FEM solver to calculate the static
equilibrium, where zero net forces are acting on the integration points. Based on a
weak form of the principle of virtual work, the FEM approximates the equilibrium in
forces by solving the partial differential equation
,
[2.18]
where is the mass density and the acceleration [45].
22
23
3 Experimental methods
In this section, the analyzed alloy systems and experimental methods are presented.
The main interest in this work is the study of the micro- and nanostructure of the sup-
eralloys. As shown in Figure 3.1, there is no single technique that is capable to image
and quantitatively analyze a material on all length scales. Therefore, a variety of tech-
niques like atom probe tomography (APT), scanning / transmission electron micros-
copy (SEM / TEM) and neutron scattering techniques such as (ultra) small angle neu-
tron scattering (SANS) and neutron diffraction (ND) were applied complementary. A
short introduction of the techniques, their capabilities and on data evaluation and in-
terpretation is given in the following.
3.1 Electron microscopy
Electron microscopy (EM) is an optical microscopy techniques that facilitates the char-
acterization of materials on a micro- to nanometer scale. By irradiating the specimens
with a defined electron beam, numerous signals are produced from the interaction of
the beam with the specimen. These signals can be used to image the specimen as
well as to examine the crystallography or composition of the specimen making EM the
most versatile characterization techniques in materials science. There are two different
types of electron microscopes that have to be distinguished: the scanning electron mi-
croscope (SEM) and the transmission electron microscope (TEM). Whereas in the
SEM, the electrons that are reflected or backscattered create the image, the TEM uses
Figure 3.1: Overview on length scales of characterization techniques in this work.
3 Experimental methods
24
electrons that have transmitted the samples to obtain information about the specimen.
In both types of instruments the electron beam is produced by a so-called electron gun
(thermionic or field emission gun (FEG)) and controlled by electromagnetic and elec-
trostatic lenses and apertures. In this work, instruments with Schottky FEGs have been
utilized, which generate smaller probes with higher brightness and coherency com-
pared to thermionic emitters but require high vacuum conditions in the order of 10-
8 Pa.
3.1.1 Scanning electron microscopy
In the SEM, electrons are accelerated to (0.1 – 30) keV and focused to a probe with a
diameter down to 1 nm (utilizing a FEG), which is deflected onto the specimen. A vari-
ety of signals (secondary, backscattered & Auger electrons, continuum and character-
istic X-rays and fluorescence) are generated when electrons of the primary beam in-
teract elastically or inelastically with the specimen. The region in the specimen where
the signals are generated (interaction volume) increases with the acceleration voltage.
Electrons that escape the sample surface with an energy of less than 50 eV are defined
as secondary electrons (SE) [46] and stem from the first 50 nanometers within the
specimen (see Figure 3.2), since the probability to absorb them increases exponen-
tially with the depth where they are formed. Thus, SE give mainly information on the
specimen topography. Beam electrons that get deflected backwards after numerous of
elastic or inelastic (by emission of SE) scattering events are backscattered electrons
(BSE) with higher energy than 50 eV. Since heavy atoms have a higher probability to
scatter electrons back, BSE give atomic number (compositional) contrast. Due to their
Figure 3.2: Basics of scanning electron microscopy: (a) Schematic drawing of the setup of a SEM: an elec-
tron beam interacts with a specimen and generates secondary electrons (SE), backscattered electrons
(BSE) and X-rays that are collected by dedicated detectors; (b) Schematic spectrum of the energy distribu-
tion of scattered electrons.
Specimen
Primary
e
-
- beam
SE
BSE
X-rays
BSE-
detector
Interaction
volume
SE
BSE
50 eV
Electron energy
Number of electrons
Energy of
primary
electrons
a)
b)
SE-
detector
EDX-
detector
Pole piece
3.1 Electron microscopy
25
higher energy, BSE electrons can penetrate the samples by several micrometers. Fur-
thermore, X-rays are emitted which can be used to obtain quantitative compositional
information (see chapter 3.1.3 for more detailed information). SE and BSE are the
principal signals used to form images in SEM. By scanning the electron beam point by
point over the sample and collecting the SE and BSE by dedicated detectors, a gray-
level image is generated depending on the intensity of the signal detected in each
point. For further general information regarding SEM, the reader is referred to the text-
book of Goldstein et at. [47].
Advanced SEM analysis techniques
By making use of electron diffraction, a SEM can be also applied to obtain crystallo-
graphic information about the specimen. The most widely used technique for the de-
termination of the materials’ microtructure is electron backscatter diffraction (EBSD).
Since electrons are backscattered in every direction within the specimen, they can un-
dergo Bragg diffraction at every set of lattice planes and thereby form two Kossel cones
for each distinct crystallographic lattice plane (see Figure 3.3). By imaging the dif-
fracted signal with a phosphorous screen, each projection of the two Kossel cones on
the screen will create a pair of lines (Kikuchi lines), representing one distinct crystallo-
graphic lattice plane. The distance between the two Kikuchi lines is linked to the Bragg
angle
hkl
which is inversely related to the lattice plane spacing
dhkl
given by the Bragg
equation for a certain wavelength
of the electrons and order of diffraction
n
.
[3.1]
In order to minimize the energy spread and maximize the proportion of BSE, the spec-
imen is usually tilted to an angle of 70° between the specimen surface normal and the
incident electron beam. The indexing (determination of the crystallographic indices) of
the recorded Kikuchi pattern is nowadays a fully automated process. To this end, the
Kikuchi lines are first transformed into points (Hough transformation) and the resultant
point pattern is evaluated by comparison to a look-up table that contains all angle com-
binations possible in the given crystal structure [48]. In modern instruments, the elec-
tron probe is scanned on the specimen surface and EBSD patterns are captured,
stored and indexed with a rate of up to 4500 patterns per second in recent fastest
detectors.
3 Experimental methods
26
By application of a rotation orientation matrix which depends on the orientations of
electron beam and specimen and of specimen and EBSD camera and which depends
on the distance between the specimen and the camera, the macroscopic specimen
coordinate system is linked to the crystal coordinate system. The most common
method for visualization is to represent the crystallographic orientation (hkl) that is par-
allel to a reference direction of the specimen coordinate system [rolling direction RD,
transverse direction TD, normal direction ND] by a color coding according to a colored
standard triangle; known as inverse pole figure map (see Figure 3.4a; the same color
coding is used for all IPF-EBSD maps presented in this work). The inverse pole figure
map provides graphical information about the grain size, orientation, as well as grain
and twin boundaries and local misorientations or lattice strains within the grains. The
texture of a specimen is often represented in a pole figure (PF), which is the stereo-
graphic projection of the specific grain normal vectors. In Figure 3.4 c), the (111)- pole
figure of a grain with normal vector in [101]- direction is shown presenting four inter-
section points (A-D).
Figure 3.3: Scheme of the setup for electron backscatter diffraction (EBSD).
Electron beam
EBSD
camera
Tilted Specimen
Normal
direction
Transverse
direction
Rolling
direction
Kikuchi lines on
phosphorous screen
Kossel cones
20°
2θ
3.1 Electron microscopy
27
Another microscopy technique to visualize crystallographic information is electron
channeling contrast imaging (ECCI) [48–50] (see Figure 3.4 b,d). ECCI makes use of
the dependency of the BSE intensity on the orientation of lattice planes with respect to
the incident electron beam. Therefore, the specimen has to be oriented into Bragg
condition for a specific lattice plane, which is possible by correlation to the orientation
information provided by EBSD. When the lattice planes are parallel to the primary elec-
tron beam, electrons channel deep into the material and the BSE image will appear
dark. In case of local defects, like dislocations or stacking faults, the lattice is locally
distorted and leads to an increased BSE signal. Individual defects will thus appear as
bright lines in a dark matrix (see Figure 3.4 d).
Figure 3.4: EBSD and electron channeling contrast imaging (ECCI) of a Ni-based alloy:(a) Electron
backscatter diffraction (EBSD) inverse pole figure map of a deformed Ni-based superalloy; (b) electron
channeling contrast image of dotted region shown in (a); (c) [111]- pole figure (PF) of a specific grain
oriented in (101) orientation; (d) enlarged view of red region in (b) providing crystallographic information
as a twin boundary, stacking faults and lattice defects.
and
planes appear as sharp lines
because they form almost 90° angles with the specimen surface.
a)
111
101
001
b)
d)
c)
B
C
[111] Pole figure (PF)
b)
d)
PF
Stacking faults
Twin boundary
Lattice defects
C
B
(
)
D
()
A
(
)
(
)
3 Experimental methods
28
SEM analysis in this work
In this work, a focused ion beam (FIB) assisted dual beam SEM (FEI SCIOS FIBSEM)
was utilized to analyze the size, volume fraction and morphology of precipitates in the
super alloys by SE and BSE imaging and to obtain crystallographic information by
ECCI (at 20 kV and beam current of 1.6 nA to 6.4 nA). Furthermore, grain sizes, grain
orientations and the texture of deformed specimens were investigated by EBSD with a
system by EDAX and with the software packages TEAM V4.5 and TSL OIM Analysis 8
for analysis. EBSD was further applied for orientation specific specimen preparation of
TEM and APT samples (see chapter 3.4.4) and for setup of the ECCI conditions for
defect analysis.
3.1.2 Transmission electron microscopy
TEMs are the optical microscopes with the highest resolution capable to obtain atomic
scale structural and compositional information. In contrast to SEMs, electrons that
have transmitted the specimen form the signal. This requires the specimens to be thin
enough (electron transparent). For high resolution analysis and depending on the ma-
terial, specimen thicknesses of usually less than 50 nm are essential. Higher acceler-
ation voltages allow to analyze thicker specimens and increase the theoretical resolu-
tion that is limited by the wavelength of electrons; the wavelength
of electrons is
related to their energy
E
(Louis de Broglie:
~ 1.22
E
-1/2). Since beam damage to the
specimens becomes severe and the improvement in resolution is mainly limited by
chromatic CC and spherical CS aberrations when applying very high voltages, modern
TEMs are operated at intermediate voltages at 120-400 kV and are equipped with extra
systems to minimize or correct the aberrations. TEMs are maybe the most versatile
instruments to characterize materials in the micro- and nanometer range, both spatially
and analytically at the same time. There are almost 40 different image formation
modes, diffraction modes and spectroscopy techniques that provide different infor-
mation about the specimen. For a general introduction into the field of transmission
electron microscopy the reader is recommended to the TEM textbook by Williams &
Carter [51], providing a basic presentation of the TEM techniques. In this work, only a
brief introduction of the techniques that have been applied to analyze ordered precipi-
tates in Ni- and Fe-based superalloys is given.
3.1 Electron microscopy
29
Imaging of ordered precipitates
Before imaging precipitates with a TEM, one has to be aware that the TEM provides
only projected information averaged through the thickness of the specimen and that
imaging a specimen means that image contrast – a difference in intensity of two adja-
cent areas – has to be established. There are two fundamental contrast mechanisms
that have to be distinguished: phase contrast and amplitude contrast, whereas the lat-
ter one is further divided into mass-thickness contrast and diffraction contrast. Phase
contrast is used for the formation of high resolution TEM (HRTEM) images and will be
explained later. Mass-thickness depends on the amount of incoherent elastic scatter-
ing contrast which is rather low for precipitates that do not differ significantly in the
atomic number and are embedded in a surrounding bulk. Therefore, ordered precipi-
tates are better imaged by diffraction (coherent elastic scattering) contrast, which is
controlled by the crystal structure and orientation of the specimen relative to the inci-
dent electron beam. The amplitude of the scattered electron beam
F
(), also known
as structure factor, depends on the type of atoms (the atomic form factor
f
()), the
position of the atom in the unit cell (x,y,z) and the specific atomic planes (hkl), depend-
ing on the crystal structure and is given by
.
[3.2]
Applying Eq. [3.2] e.g. to the bcc structure of iron (A2), the scatter factor is
if is even,
and if is odd.
Replacing Fe by the ordered NiAl- (B2) type precipitates, leads to
if is even,
and if is odd.
Since the atomic form factors of Ni and Al atoms are different, reflections that are for-
bidden in the disordered Fe (A2) structure, become now visible in the presence of or-
dered NiAl-type (B2) precipitates known as superlattice reflections. This is shown in
the selected area diffraction (SAD) pattern of a B2 strengthened ferritic alloy in [001]-
zone axis direction, meaning that the crystallographic direction [001] is parallel to the
electron beam (see Figure 3.5a). In order to maximize the diffraction contrast, the sam-
ple is tilted to a two-beam condition. In this condition (in the ideal case) only one dif-
fracted beam is strong as shown in Figure 3.5b). By the use of an objective aperture
3 Experimental methods
30
selecting only the intensified {100} reflection, a dark field (DF) image with strong con-
trast of B2 precipitates is obtained (see Figure 3.5c).
Convergent beam technique
As shown in Figure 3.5a, by using a parallel incident beam (usually ~1-10 μm in diam-
eter), DPs with sharp maxima can be obtained. The use of a small SAD aperture, al-
lows to decrease the analysis area to ~100 nm in diameter (for instruments with a low
CS
). Since many features (e.g. particles or defects) in nanoscale materials are smaller
than 100 nm, the information of SADs cannot be limited to these features. This problem
can be solved by converging the electron beam on the specimen. Since TEM speci-
mens are thin, convergent beam electron diffraction (CBED) diffraction patterns gen-
erate information from small regions that are not accessible by any other diffraction
techniques. CBED patterns, as introduced by Kossel and Möllenstedt, carry quantita-
tive information on the crystallography, lattice-strains, the specimen thickness and
even line or planar defects.
In this work, CBED was applied to determine the specimen thickness which was nec-
essary for a quantification of precipitates. In contrast to SAD patterns, in CBED pat-
terns the sharp diffraction maxima expand to discs with their diameter depending on
the beam convergence angle and the camera length. A comparison of a SAD and a
CBED pattern on a [
]- zone axis of a Ni-based alloy with L12 precipitates is given
in Figure 3.6. The concentric diffuse fringes in the CBED pattern (in Figure 3.6b) are
known as Kossel-Möllenstedt fringes and carry the thickness information. Each time
Figure 3.5: Diffraction contrast of B2 ordered precipitates in ferritic steels:(a) indexed [001] zone axis
diffraction pattern (DP); (b) DP in two beam excitation; (c) DF image using a {100} superlattice reflection
as indicated by the objective aperture (OA) in figure (b) giving bright contrast of differently sized B2
precipitates.
5 nm-1 5 nm-1 0.5 m
a) b) c)
objective
aperture
3.1 Electron microscopy
31
the specimen thickness increases by one extinction distance
g
, the number of fringes
increases by one. For a more simple interpretation, the sample is tilted to a two-beam
condition with only one strong beam excited (as shown in Figure 3.6c,d). In this case,
the CBED disks contain parallel fringes. The procedure to extract the specimen thick-
ness from the CBED patterns was developed by Kelly et al. [52] and S. Allen [53] and
is based on the calculation of the distances of the intensity minima
i
and the center
of the primary beam
2B
to the center of the diffracted beam (cf. Appendix A for the
specific formulas). Nowadays, the evaluation of the specimen thickness is computer-
ized and was performed with the software CrysTBox diffractGUI 2.21 [54] in this work.
5 nm-1 5 nm-1
5 nm-1 5 nm-1
Zone axis
SAD
Two beam
SAD
Zone axis
CBED
Two beam
CBED
a) b)
c) d)
Kossel-
Möllenstedt
fringes
2B
1
2
Figure 3.6: Comparison of SAD and CBED patterns in a Ni-based alloy:(a) indexed selected area dif-
fraction (SAD) pattern on [
] zone axis; (b) convergent beam electron diffraction (CBED) pattern on
[
] zone axis; (c) two beam SAD pattern with (002) reflection excited; (d) two beam CBED pattern
with (002) reflection excited used calculation of the specimen thickness.
3 Experimental methods
32
A further application of the convergent beam is to scan the fine probe over the speci-
men, like in a SEM but with additional lenses so that the beam is always parallel to the
optical axis. This technique is called scanning transmission electron microscopy
(STEM). BF and DF detectors below the sample are used to form the image from the
direct or diffracted beams, respectively. In this work, STEM is mainly applied to obtain
quantitative X-ray mappings. By
CS
correction of the probe in modern TEMs, a resolu-
tion down to 50 pm is obtained allowing to chemically resolve atomic columns.
High resolution imaging and analysis
Imaging and interpreting the atomic structure of thin specimens is one of the most
important TEM techniques and is called high resolution transmission electron micros-
copy (HRTEM). As already mentioned, the prerequisite to image the crystal structure
is phase contrast, which means a phase shift of diffracted and undiffracted electron
waves after interaction with the specimen. An additional phase shift is induced by
spherical aberration and depends on the amount of defocusing of the objective lens.
Since the phase shift depends further on the spatial frequencies
q
, a correct interpre-
tation of HRTEM images is only possible when it is known how the phase shift was
transferred to an amplitude shift in the image. A simple 1D- approximation of the con-
trast transfer as a function of the spatial frequency
CTF(q)
can be calculated using the
weak phase object approximation (valid for thin objects, where the real part of the wave
function does not show any modulation)
.
[3.3]
The
CTF
describes the contrast transfer by an oscillating term depending on the spher-
ical aberration
CS
, the wavelength
and the amount of defocus
f
, and an exponential
damping term depending on the chromatic aberration
CC
, the energy spread
E
and
the primary energy
E0
of the electrons [55]. The
CTF
of the two TEMs operated in this
work, a Philips FEG CM200 and a
CS
corrected FEI Titan 300, are compared in Figure
3.7 under optimum defocus (Scherzer focus), meaning the first sign change in the CTF
is at the largest spatial frequency. The reciprocal of the spatial frequency where the
CTF changes its sign is the microscopes point resolution and also the frequency to
which an intuitive interpretation of the image is possible. As shown in Figure 3.7, the
3.1 Electron microscopy
33
point resolution is shifted from ~0.2 nm for the CM200 to less than 0.1 nm for the
corrected Titan 300.
The maximum transferred frequency (sometimes defined as depends
on the damping term in Eq. [3.3], and is called information limit. To interpret TEM im-
ages with spatial frequencies in the oscillating region of the
CTF
, a comparison to sim-
ulated HRTEM images is inevitable. There are two principle approaches to simulate
TEM images: the multislice method [56,57] and the Bloch-wave method [58,59]. Figure
3.8 shows simulated Bloch-wave images for a L12 lattice of Ni3Al in [001] orientation
(a) performed with the instrument parameters of the CM200 (b) and the Titan 300 (c).
Figure 3.7: Contrast transfer functions for two different microscopes.
Figure 3.8: Bloch-wave simulation of HRTEM images:(a) input cell: L12 lattice of Ni3Al in [001] orienta-
tion; (b) Bloch-wave image for CM200; (c) Bloch-wave image for the
CS
corrected Titan 300 showing its
capability to resolve all atomic columns in contrast to the CM200.
Thickness [nm]
30
25
20
Defocus [nm-1]
-9 9 27
67 138 209
a) b) c)
L12Ni3Al [001]-oriented
CM200 Titan 300
Ni Al
3 Experimental methods
34
Since the exact amount of defocus and the specimen thickness are often unknown,
the thickness and the amount of defocus are varied in the Bloch-wave simulations to
find the best match between experimental and simulated images. When necessary,
simulations were performed with the JEMS software by P. Stadelmann [60] and com-
pared to the experimental images.
TEM analysis in this work
Most of the TEM analysis in this research work was performed on a Philips FEG
CM200 by standard BF and DF analysis and the study of diffraction patterns. Main
focus was the study of the morphology of ordered precipitates. A quantification of the
precipitates in Ni-based superalloys was achieved by measuring the specimen thick-
nesses by CBED. Smallest precipitates have been analyzed by HRTEM with a
CS-
cor-
rected FEI Titan 300. For HRTEM imaging, a small negative
CS
was adjusted by the
aberration correction system to ensure a more constructive superposition of contrast
contributions which is therefore significantly more robust against noise caused by
amorphous surface layers. Both microscopes have been used also in STEM mode to
obtain EDS mappings.
3.1.3 Electron dispersive X-ray spectroscopy
X-rays are the most important secondary signal emitted when electrons interact with
specimen. There are two kinds of X-rays produced: Bremsstrahlung and Characteristic
X-rays. Bremsstrahlung is emitted when electrons interact with the Coulomb field of a
nucleus and thereby change their momentum. Since bremsstrahlung has a continuous
energy spectrum, it is rather irrelevant for material scientists. Characteristic X-rays on
the other hand, are emitted, when a beam electron ejects an inner-shell (core) electron,
leaving the atom ionized with an electron hole in the inner shell. To reduce its energy,
an electron from an outer shell fills the hole and thereby emits a characteristic X-ray
with an energy that equals the difference in energy between the two electron shells.
Since this energy difference is unique for each element and each shell, an identification
of the element is possible after recording an EDX spectrum with an EDX spectrometer.
Care has to be taken, since there will be a superposition with the continuum brems-
strahlung background, peak overlap of different elements and other detection artifacts
that can cause misleading peaks (escape or sum peaks) in the spectra [47].
3.2 Atom probe tomography
35
Much more challenging is the quantification of the elements in an EDX spectrum. The
probability of X-ray emission depends on the ionic cross sections (increases with in-
creasing atomic number Z), the fluorescence yield (F), which describes the chance to
emit an Auger electron instead of a X-ray (lighter elements emit more Auger electrons)
and the possibility that X-rays are absorbed by the specimen or the detector (A). For
elemental quantification, all these contribution have to be considered, known as ZAF
correction. In case of TEM EDX analysis, the corrections of A and F can be ignored by
the assumption of the specimen as a thin foil. The accuracy of the quantification of
EDX data is nevertheless not more than ± 5% for an analysis using standards and
± 25% for standardless analysis.
3.2 Atom probe tomography
Atom probe tomography (APT) is an analytical microscopy technique with single atom
sensitivity at highest spatial resolution. APT is based on the principles of field evapo-
ration. Needle shaped specimens with a very small apex (< 100 nm) are required (cf.
APT setup in Figure 3.9). By controlled voltage or laser pulsing of the specimen, which
is subjected to a high DC voltage, individual atoms at the surface are field desorbed
and accelerated by a counter electrode towards a position sensitive detector. For each
ion, the time of flight (ToF) and the impact position is recorded. The specimen is placed
in an ultra-high vacuum chamber to reduce the background signal and cooled to cryo-
genic temperatures, to avoid surface diffusion of atoms decreasing the spatial resolu-
tion.
The APT measurements on Ni-based alloys and the majority of the measurements on
the Fe-based alloy in this work were performed in a femtosecond laser- assisted local
electrode tomographic atom probe (TAP) [61] operated with a pulse frequency of
200 kHz using a laser with spot size of 60 μm and a power of 30 mW (for the Ni-based
alloys) and of 33 mW (for the Fe-based alloys) while cooling the specimen to ~50 K. A
laser based system was used because of its higher success rate in avoiding premature
tip fracture which occurs under voltage pulsing more frequently due to high stresses in
the tip region [62].
3 Experimental methods
36
Since Ni- and Fe-based alloys are poor conductors of heat (Inconel 718: 11.4 W/m K,
stainless steel 304: 14.4 W/m K vs. aluminum: 236 W/m K all at 20 °C) the specimens
often do not cool sufficiently between two pulses and thereby rise the base temperature
which leads to thermal tails in the mass spectra. Therefore, additional APT measure-
ments were performed in a wide angle tomographic atom probe (WATAP) [63] by ap-
plying voltage pulses with a repetition rate of 75 kHz and a pulse fraction of 20 %, while
the specimen was cooled to 80 K.
3.2.1 Tomographic Reconstruction and common artifacts
After the APT measurement, the three dimensional volume of the specimen is recon-
structed from the detected events. Unfortunately, the exact ion trajectories cannot be
measured in the experiment and thus, a reconstruction is only possible by assumption
of the shape of the emitter during field evaporation. In the first proposed and until now
(or in slight variations) most applied reconstruction protocol by Bas et al. [64], the re-
construction follows a simple point projection model assuming a stable hemispherical
tip shape. The (x,y)- coordinates of an atom on the tip are calculated from a magnifi-
cation factor
,
[3.4]
Cryo pump:
T< 100 K
+HV DC
(1-15 kV)
Counter electrode
HV pulser
Laser
pulser
Ion trajectories
Radius:
r< 150 nm
Position
sensitive
detector
ToF
Multi
channel
plates
Figure 3.9: Schematic drawing of the setup for atom probe tomography.
3.2 Atom probe tomography
37
where is the emitter-detector distance, an image compression factor since trajec-
tories are bent into the direction of the detector and is the tip’s radius of curvature
with
,
[3.5]
where is the applied potential, the evaporation field strength and the field com-
pression factor depending on the geometry of the emitter. The z- coordinate is calcu-
lated by adding a depth increment for each atom proportional to its atomic volume and
accounting for the detection efficiency and curvature of the surface.
In real experiments, the assumption of a hemispherically shaped emitter apex is often
insufficient, since the real tip shape changes significantly in case of multiphase mate-
rials with different evaporation behavior during the measurement. This is demonstrated
in Figure 3.10 on a simulated APT experiment (see next section for more details) of a
multilayered emitter where the central layer has a higher (b) or lower evaporation field
strength (c). For the layer with a higher evaporation field, this causes the formation of
a protrusion, a region with a locally increased curvature radius during field evaporation,
and results in a local magnification of the layer in the reconstruction; the situation is
vice versa for the layer with lower evaporation field strength. These effects, where the
projection of the detected ions onto a spherical emitter causes distortions in the recon-
struction, are known as local magnification artifacts [65,66] and motivated the reseach
presented in detail in chapter 5.2 of this work.
3.2.2 Numeric simulations of APT measurements
Full scale simulations of APT measurements were performed with the simulation pack-
age TAPSim by C. Oberdorfer [67,68], to improve the understanding of APT recon-
structions that were distorted by local magnification effects caused by the evaporation
of precipitates with a different evaporation behavior. After a model emitter is set up
(see Figure 3.10a), the Poisson equation is solved and ion trajectories of desorbing
atoms and two dimensional detector coordinates are sequentially calculated, compa-
rable to recorded measurement data in experiments, until each atom of the emitter is
3 Experimental methods
38
removed. The model emitters were setup to match the experimental situations, con-
sidering the lattice structure, the composition and size of different phases. The simu-
lated detector data were reconstructed according to the initial radius and shaft angle
of the emitter [69]. More details will be given in chapter 5.2.1.
3.2.3 Analysis of APT data
The reconstruction according to the protocol by Bas et al. [64] and basic analysis of
the APT data was performed with the software Scito [70]. The focus of the analysis
was on the study of the composition of both, matrix and precipitate phases, in the al-
loys. A variety of filters and algorithms for visualization and analysis have been applied
and will be introduced in the following (see examples in Figure 3.11).
Visualization techniques
In order to visualize and identify the presence of different phases, APT reconstructions
are presented by atom maps, local concentration maps, local density maps and iso-
concentration surfaces. For the first three techniques, each ion is visualized by a col-
ored point or sphere. For atom maps, the color is chosen according to a predefined
Initial emitter Emitter during field evaporation
a) b) Local
magnification Local
compression
c)
ΔE= +20 % ΔE= -20 %
Ion
trajectories
Figure 3.10: Simulated shape of heterogeneous emitter during field evapora-
tion: (a) Surface of the input emitter for numerical simulation of an APT meas-
urement with the software TAPSim; (b,c) Shape of the emitter when central
layer has a 20 % higher (b) or lower (c) evaporation field strength during the
field evaporation process.
3.2 Atom probe tomography
39
color palette for each ionic species (Figure 3.11 a). In local concentration (see Figure
4.11 in section 4.1.4) and local density (see Figure 5.7 in section 5.1.3) maps, the
concentration of one species or the atomic density, respectively, around each ion
within a user defined cut off radius is first calculated, and then colored according to a
color specified for each concentration / density. In the isoconcentration analysis, com-
positional fluctuations in the reconstruction are visualized by proving, whether the con-
centration for one species within a voxel (usually sized 1 nm³) is above a certain
threshold value [71]. In a following delocalization step between the grid points, an iso-
concentration surface (short: isosurface) is interpolated allowing to visualize e.g. iso-
lated precipitates by selecting an element with a higher concentration inside the pre-
cipitate than in the surrounding matrix (Figure 3.11 b).
Figure 3.11: ATP analysis techniques demonstrated on experimental data: (a) atom map highlighting
NiAl-type spherical precipitates (in red) and bulk atoms Fe & Cr (in cyan); (b) precipitates visualized by
a 15 at.-% Ni isosurface; (c) typical mass spectrum before and after background correction; (d) and (e)
neighbor distance distribution of Ni atoms of the kth nearest neighbors (KNN) (d) and for only the 10th
nearest neighbor (e) fitted by the sum of two normal distributions (red line) corresponding to
β
’- precip-
itates and
α- matrix phase.
3 Experimental methods
40
Mass spectrum analysis
For accurate compositional analysis, the correct assignment of each mass peak with
a specific ion or molecule is essential. However, in cases of complex alloys, different
ions often have the same mass-to-charge ratio, e.g.
,
and
. Further-
more, mass peaks in alloys with bad thermal conductivity show thermal tails, when
measured by laser pulsing. Thereby, small peaks can be obscured by the tail of large
neighboring peaks. A typical mass spectrum of a Fe-based alloy containing NiAl- type
precipitates and measured by laser pulsing is presented in Figure 3.11c) exhibiting
these aspects.
The following two methods are applied to extract accurate compositional information
despite the before mentioned limitations: Ambiguous peaks due to peak overlap are
split (deconvoluted) with respect to the natural isotopic abundances after examination
of neighboring peaks that could be clearly identified.
For correction of small peaks superimposed by thermal tails of large mass peaks, a
three step fitting procedure is applied. First, the mass peaks are fitted with a Gaussian
function. Then, the largest thermal tail is described by a Pearson VII distribution,
,
[3.6]
with the atomic mass , height , center mean of Gaussian fit, a width , a shape
factor and with the constraint when and. In the third step,
all other thermal tails with a significant contribution are fitted while keeping the widths
and shapes of the distribution constant for all mass peaks. The sum of all (tail)
atoms fitted by a distribution is then added to the counts of the corresponding
elemental mass and subtracted from the background (cf. Figure 3.11c). Thermal tails
can vary significantly when analyzing different materials [72]. The distribution was
found to show the smallest fit residuals for thermal tails of analyzed the alloys.
3.2 Atom probe tomography
41
Cluster analysis
The automatized identification and evaluation of small accumulations of solute atoms
(clusters), is a very intensely discussed topic in the field of APT and will be a central
topic when analyzing cooling precipitates in ferritic alloys in section 5.1 of this work. So
far, isosurfaces have been presented as a technique to localize and identify precipi-
tates; but for precipitates with a size of a few nanometers only, this technique becomes
inaccurate. The coarse graining and delocalization step leads to incorrectly evaluated
sizes and shapes, especially when individual precipitates differ in composition
[71,73,74]. Therefore, a variety of cluster identification algorithms have been devel-
oped [75–82], with the majority of these based on the fact that cluster atoms have a
higher local density or are closer to each other than atoms in the matrix. The most
frequently applied algorithm is the maximum separation method [75,76], which is also
applied in this work and will be introduced in the following.
The maximum separation method relies on two user defined parameters
k
and
dmax
.
Each solute atom is defined as a cluster atom, if its kth nearest neighbor solute atom
Figure 3.12: Cluster search by maximum separation method: Solute atoms are defined as cluster atoms
if they have at least
k = 2 (top row) or k
= 4 (bottom row) nearest neighbors within the critical distance
dmax. (a) and (d) first atom satisfying the requirement; (b) and (e) all connected atoms satisfying the
requirement; (c) and (f) network of all atoms assembled to one cluster.
dmax
dmax
dmax
all cluster atoms
dmax
k 2
dmax
all cluster
atoms
dmax
solute
atom
cluster
atom
a)
b)
c)
d)
e)
k 4
f)
3 Experimental methods
42
is located within a critical distance
dmax
. This step is repeated in the algorithm for all
solute atoms. Atoms within one network of associations are assembled as one cluster
(see Figure 3.12). Often a further parameter, a minimum number of required solute
atoms
Nmin
in each cluster, is defined, to exclude small clusters that are statistically
likely to also occur in randomized systems. As indicated in Figure 3.12, the choice of
the input parameters is not trivial, since it can significantly influence the results of the
analysis. For an objective selection of the parameters, it is common to derive the pa-
rameters
k
and
dmax
from the distance distribution of the kth nearest neighbor (KNN)
[83]. In Figure 3.11 d) the KNN distributions for the 1st, 3rd, 5th, 10th and 15th nearest Ni
neighbors are compared in an iron alloy with NiAl-type precipitates. Each of the distri-
butions (cf. Figure 3.11 d) features two peaks, where the first stems from the nearest
neighbor distances within the clusters and the second one, at larger distances, from
the distances with the matrix. In depth discussion on the choice of optimum parameters
for the cluster search will be provided in section 5.1.2 of this work.
Spatial distribution maps
The extraction of crystallographic information within APT reconstructions is often de-
sired, for example to calibrate the reconstruction parameters or for the study of lattice
site occupancies. A common problem when analyzing superalloys by APT is that the
crystal structure is often only poorly resolved and therefore a statistical-based analysis
is necessary. There have been several methods proposed based on Fourier transform
methods [72,84,85], autocorrelation [85] and, most recently, on spatial distribution
maps (SDM) [86–88]. SDMs have been also calculated in this work, since they have
been shown to show the strongest signal-to-noise ratio. In principle, a SDM is a modi-
fied 3D radial distribution function, and is usually presented by mapping the occur-
rences of inter-atomic distances between each atom and all other atoms in two direc-
tions (xz, yz, or xy) in the dataset (see Figure 5.19 (a) in section 5.2.5). To keep the
calculations as short as possible, small boxes of atoms with an edge length of several
nanometers and inter-atomic distances of less than one or two nanometers are ana-
lyzed. When averaging the 2D SDM in only one direction, a histogram of the atomic
offsets is obtained (see Figure 5.19 (b-d) in section 5.2.5). In order to optimize the
peak-to-noise ratio, the analysis direction is aligned perpendicular to the lattice planes
of interest.
3.3 Analysis with Neutrons
43
3.3 Analysis with Neutrons
The third fundamentally different technique for materials characterization with a sub-
stantial contribution to this work was the analysis with neutrons. Neutrons are electri-
cally neutral subatomic particles and therefore interact only with the atomic nuclei or
magnetic moment of unpaired electrons in matter. As a consequence, they penetrate
deeply into the material and give access to probe the bulk properties. For research
purposes, neutrons are produced in research reactors by nuclear fission or in spallation
sources by accelerating protons (e.g. in a synchrotron) and colliding these with a heavy
tungsten target (e.g. the ISIS neutron source). In this work, the majority of the neutron
experiments have been carried out at the FRM II research reactor Heinz-Maier Leibnitz
(MLZ) in Garching with a thermal power of 20 MW and offering the highest neutron flux
(8 x 1014 neutrons cm-2s-1) in the world. The intensive neutron beam is generated by
nuclear fission of heavy isotopes like Uranium-235 that split into two elements and 2-
3 neutrons that are partly used for the nuclear fission of further Uranium isotopes.
Like electrons, neutrons can behave as particles or as waves and thus, allow to inves-
tigate a variety of material properties. In this work, only the elastic interaction of neu-
trons with matter was utilized. Depending on the length scale of the scattering objects,
it is common to distinguish small angle neutron scattering (SANS) to study large-scale
structures like precipitates and neutron diffraction to analyze materials on the atomic
scale.
3.3.1 Small angle neutron scattering
The typical setup for a SANS experiment is schematically shown in Figure 3.13. A
beam of neutrons is collimated onto a sample and the elastically forward scattered
neutrons are detected on a position sensitive area detector. The scatter vector is
defined as the difference of the wave vectors of scattered
and incident
wave
vectors with a momentum transfer of
,
[3.7]
where is the incident neutron wavelength and the scattering angle. Since the mo-
mentum transfer correlates with the length scale D of the scattering objects by
3 Experimental methods
44
,
[3.8]
SANS is sensitive to inhomogeneities like precipitates, macromolecules or voids with
a dimension up to hundreds of nanometers for typical wavelengths of a few Angstrom.
Experiments in this work have been performed at the SANS-1 beamline [89,90] at the
FRM II in Garching which is a pinhole instrument with a collimation length of up to 20 m
and variable sample to detector distance of 1 m to 20 m. It is further possible to select
a wavelength of the incoming neutron beam in a range of 4.5 Å to 30 Å. The instrument
is equipped with a 1 x 1 m² detector and therefore, dedicated to study objects on length
scales of 1 nm to 300 nm. The measurements in this work were carried out with three
different combinations of collimation length / sample to detector distances (8 m / 2 m,
8 m / 8 m and 20 m / 20 m and with a neutron wavelength of 6 Å.
The SANS experiments were performed to quantify the strengthening phases
´ and
´´ in the nickel-based superalloy Inconel 718 after different aging treatments. An ex-
perimental method has been developed to interpret and distinguish the SANS signal
of the two strengthening phases and will be discussed in section 4.1. The general pro-
cedure for evaluation of the isotropic two dimensional raw data is given in the following.
For interpretation of the SANS experiments, raw data are first corrected for instrument
collimation, background scattering and solid angle, then scaled and finally reduced to
one dimensional curves by azimuthally averaging (see figures in section 4.1.3). For
correction of the measured scattering intensities , the scattering of the electronic
Figure 3.13: Experimental setup of a small angle neutron scattering experiment.
3.3 Analysis with Neutrons
45
background (using a Cd shielding) , the scattering of the empty cell and, for
normalization, the transmission of the sample and the empty cell are measured
and applied to calculate the corrected intensities by
.
[3.9]
From the incoherent scattering of water, an absolute calibration of the intensities to the
absolute macroscopic scattering cross section d
/d
(
q
) is obtained by
,
[3.10]
where and are the scattering and transmission of the water standard,
and are the scattering and transmission of the empty cell used for the water
sample, is the sample thickness, the solid angle that one detector cell covers
and and are the irradiated volumes of water and the sample, respectively [91].
A further scaling factor for the geometry has to be added to Eq. [3.10], if the collimation
length and the sample to detector distance for the measurement of the sample and the
water standard differ. In this work the software BerSANS [92] was used for calibration,
scaling and reduction of the SANS raw data.
3.3.2 Neutron diffraction
The process when elastically scattered neutrons interfere constructively is termed neu-
tron diffraction and defined by Braggs law (Eq. [3.1]). Since the typical neutron wave-
length is in the same order of the magnitude of interatomic distances in crystalline
specimens, neutrons offer the possibility to probe the crystallographic structure in con-
densed matter. In contrast to X-rays, which are scattered by the electrons of an atom
and therefore, only penetrate the surface of a material, neutrons are a valuable tool to
study the arrangement of atoms deeply inside the material. The typical measurement
volume is in the range of mm³ to several cm³. Disadvantages of neutrons in comparison
to X-rays (e.g. produced in a synchrotron) are a higher beam divergence that can be
reduced at the expense of intensity and large scattering angles up to 180°. Therefore,
neutrons have to be detected either by large detector banks that cover a large scatter-
ing angle, but which are very expensive, or by area detectors, that cover only a small
part of the scattering Debye-Scherrer rings.
3 Experimental methods
46
In-situ tensile testing
In this work, neutron diffraction experiments were performed at the materials science
diffractometer Stress-Spec [93] at the FRM II in Garching to study the evolution of
lattice strains during tensile loading in the superalloys Haynes 282 and Inconel 718.
The data presented in this work, were collected during three beam times. The experi-
mental setup of the diffractometer Stress-Spec is shown in Figure 3.14. For the exper-
iments a neutron wavelength of 1.75 Å utilizing the (311) reflection of a Ge-mono-
chromator was chosen. The instrument is equipped with a 25 x 25 cm² ³He-detector
(256 x 256 px) covering a scattering angle of 5-20° depending on the neutron wave-
length and the distance of sample and detector. The detector is moved in a circular
path around the specimen allowing to detect Bragg angles between 25° and 100° or
respectively all Debye-Scherrer rings between the (100) and (221) reflection in the Ni-
based superalloys. A defined gauge volume of 5x5x5 cm³ was chosen by using primary
and secondary slits and a radial collimator with a field of view of 5 mm.
Figure 3.14: Schematic drawing of the neutron diffractometer Stress-Spec; figure taken from [93].
3.3 Analysis with Neutrons
47
For tensile testing, a specimen is fixed within a uniaxial load frame that is mounted on
an Eulerian cradle as shown in Figure 3.15 (a). The macroscopic sample elongation is
measured by clip-on gauge extensometers. The majority of the measurements were
performed at room temperature. One beam time was used to study the behavior of
Haynes 282 at 650 °C. For this measurement, the sample was induction heated while
measuring the sample elongation with a high temperature strain gauge extensometer.
The temperature was monitored by a thermocouple spot welded onto the tensile test
specimen.
In each tensile test run, 35 deformation states are evaluated; 12 in the elastic region
(machine load controlled) and 23 in the plastic region (machine strain controlled). For
each deformation state, 25 measurement orientations are realized. During that time
the strain is hold constant and the detector is moved into five positions to detect the
(111)-, (200)- and (220)- matrix reflections and (100)-, (110), (210)- and (221)- reflec-
tions of ´ precipitates. Additionally, the Eulerian cradle is rotated for each detector
position to measure the diffraction at lattice planes in five different orientations (0°, 30°,
45°, 60°, 90°) to the direction of load as demonstrated schematically in Figure 3.16 and
in Appendix A Figure A.1. So far, no other studies exist, where more than two sample
orientations have been analyzed.
For data evaluation, the radial detector signal (see Figure 3.15 b), is converted into a
1D-intensity curve as a function of the Bragg angle 2θ (see Figure 3.15 c). The intensity
Figure 3.15: Setup and evaluation of neutron diffraction experiments: (a) labelled photo of the experi-
mental setup; (b) Detector signal exhibiting two Debye-Scherrer; (c) radially averaged detector intensi-
ties in dependence of the Bragg angle.
3 Experimental methods
48
of each detector pixel is normalized utilizing a vanadium sample, since V scatters neu-
trons almost only incoherently. Finally all reflections are fitted (by Pseudo-Voigt func-
tions) utilizing the software STeCa2 ver. 2.0.5RC3 and lattice strains are calculated
from the shift of the diffraction peaks during deformation by
- 1 ,
[3.11]
where and are the lattice plane spacings of the reference state in the th defor-
mation state and and are the respective Bragg angles. More detailed infor-
mation on data evaluation and the results of the neutron diffraction experiments are
summarized in section 4.2 of this work.
Due to limited beam time at the FRM II, a few selected experiments with the alloy
Haynes 282 have been performed at the time of flight ISIS Neutron and Muon Source
in Oxfordshire, also utilizing the materials science diffractometer Stress-Spec (see dis-
cussion section 4.2.4).
Figure 3.16: Schematic drawing of the measurement of lattice strains by neutron diffraction during uni-
axial loading: Lattice strains of (a) (200- and (b) (220)- orientated lattice planes (grains) are measured
in five different orientations (0°, 30°, 45°, 60° and 90°) to the direction of load.
3.3 Analysis with Neutrons
49
3.3.3 Complementary crystal-plasticity modeling
For interpretation of the in-situ neutron diffraction experiments, which monitor the evo-
lution of lattice strains during uniaxial loading, the experimental data are compared to
the stress and strain evolution in simulated polycrystalline structures by phenomeno-
logically based CPFE modelling (such as introduced in section 2.3.2). The simulations
are performed on a cubic cell with periodic boundary conditions (see Figure 3.17) that
is subjected to a uniaxial tensile load. To simulate an isotropic polycrystalline material,
the cubic cell is discretized into cube shaped finite elements (C3D8: continuum
stress/displacement, three-dimensional and eight nodes) with the open-source soft-
ware Neper [94]. Depending on the size of the cell, a certain number of neighbored
elements is selected to represent one grain with the same crystallographic orientation
of those elements. The crystallographic orientation of the grains is chosen arbitrarily.
For simulation of the constitutive response, the crystal plasticity model “constitu-
tive_phenopowerlaw.f90” which is implemented in the simulation toolkit DAMASK
[45,95] is utilized. The finite element solver ABAQUS v.6.10 is employed to solve the
resulting system of differential equations [96].
For the hardening model introduced in section 2.3.2 (Eqs. [2.13]-[2.16]), several input
parameters are required. Initial elastic constants C11 = 244.76 GPa, C12 = 149.22 GPa
and C44 = 123.6 GPa were chosen by interpolation of the constants for - Ni and ´-
Ni3Al for a ´- phase volume fraction of 20 % [E2]. A reference shear rate of
and a rate sensitivity of
m
= 20 were chosen. All further parameters
(and) were iteratively adjusted to fit the experimental data on small cubic
cells first (see Figure 3.17 a-c) and finally on large cubic cells (see Figure 3.17 d-f).
The three dimensional stress state in Figure 3.17c,f) is visualized by a contour plot of
the equivalent von Mises stresses at the nodes, which are defined by
.
[3.12]
For the quantitative analysis of the simulations, the longitudinal strains of all crystallo-
graphic directions in the five different orientations (0°, 30°, 45°, 60°, 90°) to the direc-
tion of load are determined for each deformation state and every integration point. To
3 Experimental methods
50
encompass a statistically significant number of specific grains contributing to one anal-
ysis orientation, a misorientation of 7.5° between measurement and crystallographic
direction was allowed and the longitudinal strains averaged by weight (in section 4.2).
Figure 3.17: Crystal plasticity finite element modelling: (a-c) cubic cell discretized into 9x9x9 finite ele-
ments of type C3D8 and including 27 grains and in (d-f) cubic cell discretized into 76x76x76 elements
including 361 grains. (a,b) and (d,e) show the orientation of grains in the initial state (a,d) and after uniaxial
loading to a strain of 10 % (b,e); (c,f) show the distribution of von Mises stresses after the deformation.
3.4 Sample preparation
51
3.4 Sample preparation
3.4.1 Alloy fabrication
Ingots of the Ni-based superalloys Inconel 718 and Haynes 282 were provided by Böh-
ler Schmiedetechnik GmbH&Co Kg. The composition of the alloys is given in Table 2.1
in section 2.1.1.
The ferritic alloy FBB-8 has been prepared in a vacuum arc melting furnace in a work-
shop of the Max-Planck Institute in Stuttgart using raw materials of at least 99.99 %
purity as is given in Table 2.2 in section 2.1.2. For solution annealing, the ingot was
cut into two halves which were sealed in quartz tubes under argon atmosphere and
then solution-treated for 30 min at 1200 °C. To study the influence of different cooling
rates on the formation of precipitates during cooling (section 5.1), one half was cooled
in air and the other half was quenched in a saturated brine.
For a controlled precipitation of ordered precipitates, all alloys were further subjected
to specific one- or two- step aging treatments. The details on these aging treatments
are given in the respective chapters.
3.4.2 Round tensile test specimens
Specimens for uniaxial tensile tests were prepared at the TU Munich. First, rods with
a diameter of 12 mm were cut by spark erosion from large ingots. These rods were
lathed into the geometry shown in Figure 3.18. Finally, threads were tapped into both
ends. During the subsequent aging treatment, the surface of the tensile test specimens
oxidized. The comparison of tensile tests of samples with polished (as shown in Figure
3.18) or unpolished surface has not shown any significant difference.
Figure 3.18: Geometry of a round tensile test specimen.
3 Experimental methods
52
3.4.3 Standard sample preparation techniques
Specimens for Vickers hardness tests and SEM analysis
Slices with a thickness of ~1 mm of the Ni-based tensile test specimens and the ferritic
ingots were sawed with a Struers Accutom-50 while water cooling to prevent changes
of the microstructure. The slices were then mechanically grinded (up to P4000 SiC-
paper). For SEM analysis, the samples were further polished with a 1 μm diamond
suspension.
Specimens for TEM analysis
Standard TEM specimens were prepared by electropolishing. A brief introduction to
the method of electropolishing is given in the reviews of D. Landolt (general proce-
dure) [97] and of B. Kestel (electropolishing of TEM foils) [98].
In this work, thin slices of the alloys were cut with the Struers Accutom-50, mechani-
cally grinded to a thickness of less than 150 μm (finishing with P4000 SiC-paper) and
then punched into disks with a diameter of 3 mm. The discs were finally electropolished
with a Struers Tenupol-5 twin jet electropolishing device. For the Ni-based alloys, the
electropolishing was performed at -30 °C with an electrolyte consisting of 85 % meth-
anol, 12 % perchloric acid and 3 % deionized water. A voltage of 25 V and a pump flow
of 20 was adjusted. The process was stopped when the sample was perforated. The
result is a small hole in the center of the disc with a diameter of 20 μm to 300 μm.
Around the hole, the sample is thin enough (usually less than 100 nm) for analysis by
TEM. The advantage of this method is that a relatively large area can be analyzed
which allows, to analyze grains with different orientations in case of intermediate grain
sizes (as was the case for the Ni-based alloys).
The ferritic alloy was too brittle to apply a mechanical punch; a solution was found by
laser cutting of the discs. Furthermore, the grain size was huge so that only one or two
grains per sample could be analyzed. A site specific sample preparation by FIB (see
section 3.4.4) was found to be more suitable for a quantitative analysis of precipitates.
3.4 Sample preparation
53
Specimens for EBSD analysis
For EBSD analysis, a defect free specimen surface is required. The best results were
obtained by electropolishing of the SEM specimens. Therefore, the specimen was con-
nected as an anode against a flat platinum electrode in a cooled (to ~0 °C) electrolyte
bath. For the Ni-based alloys, the same electrolyte as for the TEM specimen prepara-
tion was applied; for the Fe-based alloy, a mixture of 90 % methanol and 10 % per-
chloric acid was used. While stirring the electrolyte, 3 to 5 short voltage pulses of 20 V
DC were applied.
Specimens for APT analysis
For APT analysis, needle shaped specimen with an apex radius of ~50-100 nm are
required. The standard method to prepare these needles is the preparation by electro-
chemical polishing [99,100] which is the most widely used and fastest method in case
of conductive materials (see textbook of Gault et al. for further methods [72]).
In this work, APT specimen of the Ni-based alloy Inconel 718 were prepared from
blanks with a dimension of 0.3x0.3x15 mm³ that were cut from the tensile test speci-
mens by spark erosion discharge machining. The blanks were then electropolished
against a Pt cathode in a two-step procedure. The first polishing step was performed
in a thin layer of 20 % perlochloric acid in acetic acid on top of an inert liquid (Galden
LS230). The center of the specimen blank was therefore placed inside the electrolyte
layer while applying ~10 V DC pulses until a thin neck was formed. The second polish-
ing step was performed in an electrolyte of 2 % perchloric acid in 2-butoxy-ethanol
applying first 7 V DC pulses and then decreasing the voltage until the lower part of the
specimen broke; both parts could be used for APT analysis.
3 Experimental methods
54
3.4.4 Focused ion beam assisted sample preparation
When the sample preparation by electropolishing was not feasible, a focused ion beam
(FIB) assisted SEM was utilized to prepare APT and TEM specimen. For instance, APT
specimen of the ferritic alloy could not be prepared by electropolishing because the
alloy was too brittle to cut the required blanks by spark erosion discharge machining.
Therefore, APT tips with an apex radius of less than 100 nm were prepared by the
standard FIB lift-out procedure and fixation of a blank onto a tungsten post and subse-
quent annular milling [101]. The tungsten post was prepared by electrochemical pol-
ishing in a NaOH solution (2 mol/l) [102].
Furthermore, the FIB technique facilitates to prepare site specific specimen for TEM
(and APT) investigation. In this work, TEM specimens obtaining certain microstructural
features or aligned in a specific crystallographic orientation have been prepared by a
FIB- liftout procedure as demonstrated in Figure 3.19.
For the preparation of TEM specimen obtaining a specific microstructural feature, the
location for the FIB- liftout is chosen by pre-characterization using for instance
backscattered electrons (BSE) to detect certain particles, EDX analysis to differentiate
various phases or electron channeling contrast imaging (ECCI) to contrast grain / twin
boundaries or lattice defects such as dislocations or stacking faults (see Figure 3.19a).
In order to prepare TEM specimen in a specific crystallographic orientation that is par-
allel to the electron beam in the TEM, the microstructure is first analyzed by EBSD.
From the EBSD map, a grain is then chosen that is already aligned with the orientation
of interest (e.g. the [001]- direction in this work) perpendicular to the specimen surface.
For the FIB- liftout, the stage is finally rotated so that the orientation of interest is ver-
tical in the SEM image and perpendicular to the FIB- liftout lamella (see Figure 3.19 b).
After location or alignment of the area of interest, the TEM specimen is prepared by
the standard FIB- liftout procedure (see review of Giannuzzi et al. [103]) and fixation
on a TEM copper grid provided by PELCO® (see Figure 3.19 c).
3.4 Sample preparation
55
Figure 3.19: FIB assisted TEM specimen preparation: (a) Four alternative ways to find a microstructural
region of interest; (b) procedure to align specific crystallographic orientation by EBSD; (c) standard FIB-
liftout procedure of a TEM lamella.
56
57
4 Studies on nickel-based superalloys
In this section, the experimental results of the studies on nickel-based superalloys are
presented and discussed. In chapter 4.1, a new methodology to distinguish and quan-
tify
-
and
-
precipitates in the nickel-based superalloy Inconel 718 by small angle
neutron scattering is developed. The results of this research (and most figures in this
chapter) are published in the article “Differentiation of
-
and
- precipitates in In-
conel 718 by a complementary study with small-angle neutron scattering and analytical
microscopy” by Lawitzki et al. (2019) [E3]. In chapter 4.2, the results of the microme-
chanical behavior studied by in-situ neutron diffraction during uniaxial tensile testing
and the results of complementary microstructural analysis and crystal plasticity mod-
eling of the alloys Inconel 718 and Haynes 282 are presented. These studies were
performed in a collaboration with the Institute of Materials Science and Mechanics of
Materials and the central scientific institute of the research neutron source Heinz
Maier-Leibnitz (FRM II), both referring to the TU Munich. Parts of the results here are
published in the joint article “Micromechanical behaviour of Ni-based superalloys close
to the yield point: a comparative study between neutron diffraction on different poly-
crystalline microstructures” by Kobylinski and Lawitzki et al. (2019) [E2].
4.1 Differentiation of
-
and
-
precipitates in Inconel 718
Although the alloy Inconel 718 has been proposed more than 70 years ago and is one
of the most frequently produced Ni-based superalloys for gas turbine applications, the
alloy’s complex microstructure is still bringing up new questions. One of the biggest
challenges is, to adjust the alloy’s microstructure depending on the processing tech-
nique (cast, wrought, powder metallurgical or additively manufactured) and the appli-
cation, to achieve a maximum in performance. New technical innovations and im-
proved scientific characterization methods offer the possibility to now obtain a better
understanding of processes at the nano-scale.
In the alloy Inconel 718, it has been found to be experimentally very difficult to precisely
determine the volume fraction of the two strengthening phases
and
in bulk dimen-
sions [104]. Since the unit cells and the chemical composition of both phases is very
4 Studies on nickel-based superalloys
58
similar, a differentiation of the phases by powder neutron diffraction is not possible,
mainly due to the low volume fraction of the precipitating phases [105]. In previous
studies by small angle neutron scattering (SANS), only a joint contribution of
and
precipitates could so far be evaluated [104,106].
On the other hand, it has been shown that a more local phase differentiation and quan-
tification is possible by methods such as electron holography and electron tomography
imaging [107], energy dispersive STEM X-ray mappings [108], FIB assisted scanning
electron tomography [108] or APT [106,109,110], but all of these methods encounter
the problem of poor statistics due to small analysis volumes or are limited in the anal-
ysis of smallest precipitates (such as the SEM based methods).
In this work, a new experimental methodology to evaluate the individual volume frac-
tions of
-
and
-
precipitates is developed that is based on SANS measurements.
For the interpretation of the SANS curves, a structural model is set up according to
complementary information obtained by APT and TEM.
The proposed methodology is demonstrated on specimens of alloy Inconel 718 that
were subjected to four different aging heat treatments to produce contrasting micro-
structures. The heat treatments were chosen according to previous works by Repper
et al. [105,111] as demonstrated in Figure 4.1 and summarized in Table 4.1. The dif-
ferently heat treated specimens are denoted by the letters A, B, C and D followed by
the temperature of the highest aging treatment. Samples A-718 and B-760 were both
subjected to an initial solution treatment to fully dissolve
-
and
-
precipitates, fol-
lowed by a two- step aging treatment. The solution treatment for sample state B-760
was performed above the solvus of
-
phase, which does not precipitate in the following
aging treatment. In the sample C-870, a three- step aging treatment was performed.
Figure 4.1: Performed heat treatments on specimens of alloy Inconel 718.
4.1 Differentiation of γ′- and γ″- precipitates in Inconel 718
59
In the reference sample D-950, only a solution treatment above the solvus of
-
and
-
precipitates was performed followed by quenching in water to suppress the precip-
itation of
-
and
-
phase. By powder neutron diffraction, Repper et al. [105] have
already proven which phases exist in each of the sample states (see Table 4.1) which
is in agreement with the time temperature transformation diagram of Inconel 718 [8].
4.1.1 Characterization of the microstructure
In this section, the results of the characterization of the microstructure by electron mi-
croscopy are summarized. The focus of the analysis was on features (from mirco- to
nano-scale) that can influence the SANS results and are therefore necessary to con-
sider for interpretation of the data.
Grain structure
The grain structure and consequential grain boundaries influence the scattering be-
havior at smallest
q
vectors. Whereas all specimen containing
-
phase featured an
average grain size of (11 ± 5) μm in diameter (see Figure 4.2 a), only state B-760 with
the solution treatment above the
-
solvus temperature at 1050 °C featured a signifi-
cantly larger grain size of (93 ± 38) μm (see Figure 4.2 b).
Sample state
Temperature
[°C]
Time
[h]
Cooling
Phases pre-
sent
A-718
975
718
621
2
8
8
AC
FC to 621 °C (0.81 K min-1)
AC
,
,
and
B-760
1050
760
649
2
10
10
AC
FC to 649 °C (0.85 K min-1)
AC
,
and
C-870
870
815
718
0.5
0.75
2
FC to 815 °C (2.75 K min-1)
FC to 718 °C (3.23 K min-1)
WQ
,
,
and
D-950
950
22
WQ
and
Table 4.1: Parameters of the performed heat treatments on Inconel 718 and the present phases ac-
cording to reference [105]. The samples were cooled in air (AC), in the furnace (FC) or quenched in
water (WQ).
4 Studies on nickel-based superalloys
60
Morphology and quantification of
-
phase
-
phase is found in all sample states except state B-760. Neither the analysis by SEM,
nor the analysis by TEM has indicated the presence of
-
phase in this state. A
backscattered SEM image of the microstructure of state A-718 is shown in Figure 4.3.
-
phase is found to be preferentially located at grain boundaries. From the view of the
specimen surface, the morphology of
-
phase could be plate like or needle shaped,
and in previous works, both possibilities have been stated (e.g. plate like in ref. [112]
or needle shaped in ref. [11]). This discrepancy could be clarified by a cross section
cut with a focused ion beam intersecting a few
-
precipitates (shown in Figure 4.3a).
As demonstrated in Figure 4.3 b), the same precipitates can also be observed in the
image of the cross section verifying a plate like morphology.
Figure 4.2: Inverse pole figure maps obtained by EBSD revealing: a) small grains (diameter: 11 ± 5 μm)
for sample state A-718 and b) large grains (diameter: 93 ± 38 μm) for sample state B-760. (According
to reference [E3])
Figure 4.3: Morphology of
-
phase: a) Backscattered electron image featuring
-
phase in sample A-
718. The image after a FIB cross section cut shown in (b), verifies a plate like morphology of the pre-
cipitates.
4.1 Differentiation of γ′- and γ″- precipitates in Inconel 718
61
After post processing and careful adjustment of a threshold value for binarisation of
the images, the size and area fraction of
-
precipitates was automatically evaluated
with the software ImageJ [113,114]. As shown in Figure 4.4, the equivalent diameter
of
-
precipitates of (0.9 ± 0.4) μm is almost the same in the initial (as delivered)
state (a) and in states A-718 (b) and C-870 (c). The area fractions of
-
phase are
slightly larger after the heat treatments with (2.7 ± 0.1) % in state A-718 and (2.0 ±
0.1) % in state C-870 in comparison to the initial state (1.8 ± 0.2) %. Only in state D-
950, the size and area fraction of
-
phase is significantly larger with (1.2 ± 0.5) μm
and (7.1 ± 0.3) %, respectively.
Quantitative analysis of
-
and
-
precipitates
The analysis of the size and volume fraction of
- and
- precipitates was performed
by TEM. For clear identification of the two different phases, the analysis was performed
Figure 4.4: Quantification of
-
precipitates after different heat treatments.
4 Studies on nickel-based superalloys
62
with the electron beam parallel to grains in the [001] zone axis. A differentiation of
-
and
- precipitates with a diameter of more than 10 nm was possible by diffraction
pattern analysis. The procedure is demonstrated exemplary on sample state C-870 in
Figure 4.5 and in Figure 4.6. In order to record local diffraction patterns with almost
parallel illumination of very small areas (< 50 nm in diameter), the smallest C2 aperture
(30 μm) was chosen and the diffractions lens slightly adjusted. All diffraction patterns
were indexed with respect to the face centered
-
matrix by comparison to simulated
Figure 4.5: Morphology and phase identification by TEM of
-
phase in state C-870 of a grain tilted to
[001]-zone axis orientation: a) darkfield (DF) images showing three variants of
-
precipitates visualized
in the inset figures [A], [B] and [C]; row b) experimental diffraction patterns (DP) according to the areas
highlighted above. [DP Overall] is the DP of the complete area above; the circle highlights the scattered
beams tilted to the optical axis to obtain figure a) [DF]; c) simulated diffraction patterns using the D022
crystal structure in the orientation as shown below in row d); [Sim All] is the superposition of all simulated
diffraction patterns and figure [D022] shows the elementary cell of
-
phase, where no axis is parallel to
the view direction. (According to reference [E3])
4.1 Differentiation of γ′- and γ″- precipitates in Inconel 718
63
diffraction patterns calculated with JEMS [60] using the crystal structure D022 for
-
phase (Figure 4.5 c) and the crystal structure L12 for
´
-
phase (Figure 4.6 d).
From Figure 4.5, it can be concluded that
- phase precipitates in the shape of oblate
ellipsoids, where the shorter rotational axes is parallel to the
c
- axis of the elementary
cell and the longer equatorial axes are parallel to the
a-
axes of the crystal.
- precip-
itates exist in three different orientational variants [A], [B] and [C], where variants [A]
and [B] appear ellipsoidal and variant [C] appears circular in the DF images.
As shown in Figure 4.6,
´
-
phase forms spherical precipitates that are (in state C-870)
much smaller than
- precipitates allowing a simple differentiation of both phases. For
sample state B-760 with similar size of
- and
- precipitates, only the [A] and [B]
variants of
- precipitates are directly distinguished from
´
-
precipitates. The [C] var-
iant can only be distinguished by analysis of local diffraction patterns, since these show
less superlattice reflections than
´
-
precipitates (compare Figure 4.5 c) [001] and Fig-
ure 4.6 d).
In state A-718,
- and
- precipitates were too small (<6 nm in diameter) and the
number density too high, for a clear differentiation of both phases by local diffraction
pattern analysis. A differentiation was facilitated by HRTEM and FFT filtering which is
demonstrated in the Appendix B Figure B.1. The shape of
- precipitates is found to
be oblate ellipsoidal even for the smallest precipitates.
In state D-950, no
- and
- precipitates could be observed and the diffraction patterns
were free of corresponding superlattice reflections.
Figure 4.6: Phase identification of
´
-
phase in state C-870 of a grain tilted to [001]-zone axis: a) DF
image using a (010)-reflection shows large
-
precipitates and small
´
-
precipitates indicated by the
arrows; b) magnified inset figure according to the square in figure a); c) diffraction pattern (DP) of the
area highlighted in figure b); d) simulated DP using the L12 crystal structure in [001] zone axis orientation.
(According to reference [E3])
4 Studies on nickel-based superalloys
64
The volume weighted radii distributions of
- and
- precipitates after their differenti-
ation in sample states A-718, B-760 and C-870 are compared in Figure 4.7. The radii
were evaluated manually with the help of the software ImageJ [113]; for
- precipitates,
by overlaying with circles and for
- precipitates, by fitting only the [A] and [B] variants
of the precipitates with ellipses. Since both equatorial axes have the same length, the
volume of each
- precipitate can be estimated and an equivalent spherical radius
can be calculated by
,
[4.1]
where is the radius of the equatorial axis and is the radius of the rotational axis.
The radii distributions in Figure 4.7 could be approximated by log-normal size distribu-
tions by
,
[4.2]
where
is the median radius, the logarithmic standard deviation and the number
density of the precipitates. The fitted values for and are compared to the results
obtained by SANS later in this work in Table 4.3. Whereas
- precipitates are smallest
in the state A-718 they did not become larger than 11.2 nm after heat treatments at
higher temperatures as in states B-760 and C-870. In contrast to this,
- precipitates
010 20 30 40 50 60
0
5
10
15
20
25
30
'
''
Volume fraction [%]
Radius [nm]
0 5 10 15 20
0
2
4
6
8
10
12
14
16
18
20
22
24
'
''
Volume fraction [%]
Radius [nm]
100 nm
100 nm 100 nm
a) b) c)
0 2 4 6 8 10 12 14
0
5
10
15
20
25
30
''
'
Volume fraction [%]
Radius [nm]
Figure 4.7: Volume-weighted particle size distributions of
-
and
-
precipitates: a) state A-718, b) state
B-760 and c) state C-870. The inset figures show typical darkfield TEM images of
-
precipitates used
for the analysis. (According to reference [E3])
4.1 Differentiation of γ′- and γ″- precipitates in Inconel 718
65
are similar in size in the states A-718 and B-760 (4.6 nm and 7.8 nm, respectively) but
became significantly larger in state C-870 (33.5 nm).
The evaluation of the specimen thickness by CBED (see section 3.1.2) allowed to fur-
ther calculate volume fractions of
- precipitates. Since the contrast between matrix
and
- precipitates in the [C] variant was weak, only the [A] and [B] variants were
considered for the calculation of volume fractions and multiplied by the factor of 3/2
assuming an isotropic distribution of all three variants. The volume fraction of
- pre-
cipitates could not be evaluated, because they were occasionally found in clusters and
showed much less contrast. Furthermore,
- precipitates and the [C] variants of
-
precipitates had similar sizes in states A-718 and B-760 and therefore, could only be
distinguished by DP analysis of single precipitates, which ruled out a large number
quantitative analysis. Considering the bias of precipitate sectioning in the TEM foils
where the particles are smaller than the specimen thickness, the volume fractions of
- precipitates have been corrected applying a Schwartz-Saltykov [115] type correc-
tion (cf. Appendix A) and are presented together with the SANS results in Table 4.2.
Phase
*
A-718
matrix
B-760
matrix
C-870
matrix
D-950
matrix*
Composition
[at.-%]
Ni
70.61 ±
0.11
71.11 ±
0.07
71.72 ±
0.34
50.59 ±
0.06
50.43 ±
0.06
50.93 ±
0.16
51.13 ±
0.12
Nb
6.33 ±
0.05
17.47 ±
0.06
17.10 ±
0.29
2.00 ±
0.02
2.30 ±
0.01
1.86 ±
0.03
2.21 ±
0.21
Al
7.92 ±
0.07
0.96 ±
0.04
0.38 ±
0.04
0.58 ±
0.02
0.55 ±
0.02
0.45 ±
0.05
0.90 ±
0.10
Ti
7.86 ±
0.05
4.29 ±
0.06
3.23 ±
0.06
0.62 ±
0.00
0.63 ±
0.00
0.72 ±
0.02
0.90 ±
0.06
Cr
3.05 ±
0.04
2.44 ±
0.04
2.44 ±
0.11
22.04 ±
0.05
21.97 ±
0.07
21.91 ±
0.14
21.64 ±
0.35
Fe
2.80 ±
0.04
1.85 ±
0.03
3.78 ±
0.10
22.32 ±
0.06
22.27 ±
0.05
22.26 ±
0.12
21.35 ±
0.23
Mo
1.42 ±
0.04
1.89 ±
0.03
1.34 ±
0.05
1.85 ±
0.02
1.84 ±
0.01
1.86 ±
0.08
1.87 ±
0.24
Lattice Pa-
rameters
[nm]
a
3.590
3.600
5.103
3.598
3.598
3.598
3.598
b
4.223
c
7.438
4.516
Scattering
length densi-
ties
[1010 cm-2]
i
7.084
± 0.011
7.328
± 0.008
7.428
± 0.035
7.206
± 0.008
7.202
± 0.007
7.213
± 0.018
7.177
± 0.031
Table 4.2: Compositions in Inconel 718 measured by APT (or TEM EDX in case of the *), LeBail refined
lattice parameters from neutron diffraction as measured by J. Repper [13] and calculated scattering
length densities
of different phases and the precipitate free matrix of the different states. For a partic-
ular phase, the lattice parameters are considered to be the same for all heat-treated states.
4 Studies on nickel-based superalloys
66
4.1.2 Compositional analysis by APT
The composition of the
- matrix and the ordered precipitates was measured by APT.
As shown exemplary in Figure 4.8 for state B-760, 1D-composition profiles along re-
gions containing precipitates were used to analyze the local composition of different
phases. APT reconstructions of all other states are shown in the Appendix B Fig-
ure B.2. Regions with high amounts of Cr (>20 at.-%) were denoted as
- matrix phase,
whereas regions with a high Nb content (> 16 at.-%) were denoted as
- precipitates
and regions increased in Al, Ti and Nb (all more than 5 at.-%) as
- precipitates, re-
spectively. For the final evaluation, only regions that could be clearly identified as one
of the three mentioned phases were considered. The elemental composition of the
different phases in all states is given in Table 4.2. Since no significant variation in the
elemental composition of the precipitate compositions could be determined, an aver-
age value for all states is given. In contrast to this, the elemental composition of the
matrix phase is slightly affected by the heat treatment, which is shown in the differ-
ences in the amount of Ni, Nb, Al, and Ti.
0
15
30
45
60
75
0
5
10
15
20
25
010 20 30 40 50
Ni
Nb
Ti
Al
Concentration of Al, Nb, Ti, Cr [at.-%]
Distance [nm]
Concentration of Ni [at.-%]
230 nm
'
matrix
'' '
Cr
Figure 4.8: Compositional analysis by APT: Left: Reconstruction of a tip of state B-760 where only Cr
(green), Al (red) and Nb (blue) atoms are visualized; right: 1D composition profile along a cylinder placed
inside the tip indicating Cr rich regions (>20 at.-%) belong to the matrix, Nb rich regions (>16 at.-%) to
-
precipitates and Al rich regions (> 5 at.-%) to
´
-
precipitates. The error bars shown for Cr and Nb
consider statistical variations of the compositional analysis. (According to reference [E3])
4.1 Differentiation of γ′- and γ″- precipitates in Inconel 718
67
The measurement of the atomic compositions enables to calculate the scattering
length density
i
of each phase
i
, which is a necessary quantity to interpret the contrast
in the SANS scattering patterns. The scattering length density
i
is defined by [116]
,
[4.3]
where
bj
is the scattering length of each individual atom
j
listed by Sears [117],
xj
is the
atomic concentration and
p
is the atomic volume, which is calculated using the unit
cell parameters that were measured in a previous work by powder neutron diffraction
for the present phases in all sample states by Repper et al. [105]. The unit cell param-
eters and the calculated scattering length densities are listed in Table 4.2. Considering
a statistical error for the average concentrations determined by APT of
± (0.05-0.1) at.-%, the Gaussian error of the scattering length density is ± 108 cm-2.
4.1.3 Evaluation and interpretation of SANS experiments
The azimuthally averaged SANS curves of the differently heat treated sample states
after absolute calibration to the absolute macroscopic scattering cross section
d
/d
(
q
) (Eqs. [3.9] and [3.10]) are compared in Figure 4.9. The scattering data re-
veal clear differences which depend on the specific microstructure of the samples. For
sample A-718 with the lowest heat treatment temperatures, a strong scatter signal is
obtained in the high
q-
range. With increasing temperature of the aging treatment
(states B-760 and C-870), the increase in the scatter signal shifts to lower
q-
values
●●●
■■■■■■■■■
■
■
■
■
■
■
■
■
■■
■
■
■■
■
■
■■
■■
■■■
■■
■
■■■
■■
■
■
■
■
■■■
■■
■■■■■■■■■■■■■■■■■■
■
■
■
■
●●●●●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●●
●●●
●
●
●
●
●
●●
●
●●●
●
●●●●●●●
●
●
●
●●●
●●●●●●●●●●●●●●
●
●
●
▲▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲▲
▲
▲▲
▲
▲▲▲ ▲▲▲
▲▲▲
▲
▲▲
▲▲
▲
▲
▲
◆
◆
◆◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆◆
◆
◆
◆
◆
◆
◆◆
◆
◆
◆
◆
◆◆◆◆◆◆◆
◆◆◆◆◆◆
◆◆
◆◆
◆
◆
◆
◆
◆
◆
A-718
●B-760
■
▲C-870
◆D-950
0.05 0.10 0.50 1
10-3
10-2
0.1
1
10
q[nm-1]
d/d
ΣΩ[cm sr]
-1 -1
Figure 4.9: Experimental ex-situ SANS patterns of the differently heat treated Inconel 718 samples. (Ac-
cording to reference [E3])
4 Studies on nickel-based superalloys
68
which is in accordance to the observation made by TEM (cf. Figure 4.7). For sample
D-950, where no nano-sized precipitates were detected by TEM, correspondingly also
no increase in the SANS curve is observed.
Structural model to describe SANS data
For evaluation of the SANS curves in Figure 4.9, the macroscopic scattering cross
section is formulated by a model that considers the contribution of various scattering
signals by
,
[4.4]
where
B
inc is the signal of incoherent background scattering and the term
Kp q-4
de-
scribes the asymptotic behavior for low
q-
values that is generated by smooth inter-
faces of large scattering structures (such as
-
precipitates and grain boundaries) ac-
cording to Porod’s law [118] with Porod coefficient
Kp
. The scattering of precipitates of
phase
i
(
- and
-
phase) is described by
,
[4.5]
where
i
is the scattering length density difference,
ni(R)
the particle size distribution,
Vi(R)
the volume of the precipitate and
Fi(q,R)
the form factor. The scattering length
density difference
i
between
- matrix and precipitate
i
is calculated by
,
[4.6]
using the scattering length densities listed in Table 4.2. The size distributions
ni(R)
were chosen of log-normal type (Eq. [4.2]) according to the evaluation by TEM. The
form factors for oblate ellipsoidal precipitates are calculated by
[4.7]
with and
[4.8]
where
Re
and
Rrot
are the equatorial axis and the rotational axis, respectively. For the
spherical
- precipitates,
Rrot
equals
Re
.
The SANS model in equation [4.4] was numerically least square fitted to the experi-
mental SANS data which is demonstrated in Figure 4.10 for state B-760 (and in Ap-
pendix B Figure B.3 for fits to data of states A-718 and C-870). As starting parameters
4.1 Differentiation of γ′- and γ″- precipitates in Inconel 718
69
for the least square regression, the parameters of the log-normal size distributions
evaluated by TEM (
and
) were chosen and kept constant. In the first iterations of
the fitting procedure, only the particle number densities were fitted. After this step,
the parameters
and
were fitted. It has to be mentioned, that this was the only way
to obtain converging fits with reasonable results such as realistic precipitate sizes. The
parameters of the log-normal size distributions (
and
) after least square fitting are
very close to the starting parameters, confirming the accuracy of the TEM results (see
Table 4.3). Finally, the volume fraction of each phase is obtained by integration over
all radii
R
of the log-normal size distributed particles
.
[4.9]
Comparing the statistical errors of the TEM and SANS results presented in Table 4.3,
it is obvious, that the accuracy of the SANS data is significantly better. For the median
precipitate sizes
,
the statistical error is practically zero which is due to the large meas-
urement volume of the SANS experiment. Considering a typical precipitate density of
1016 cm-3, roughly a quadrillion times more particles are analyzed by SANS. In contrast
to the analysis by TEM, it was further possible to evaluate a volume fraction for both
kind of nano-precipitates. The statistical errors of the presented volume fractions by
Data state B-760
Fit
γ'
γ''
Porod
0.05 0.10 0.50 1
10-3
10-2
0.1
1
10
q[nm-1]
dd
ΣΩ
/[cm sr]
-1 -1
γ''
γ'
510 15 20
0.00
0.05
0.10
0.15
0.20
0.25
Radius [nm]
Propabilty density
Size distributions
Figure 4.10: Scattering curve of sample state B-760 fitted with a model consisting of three different
contributions: a) log-normal size distributed
-
precipitates, b) log-normal size distributed
-
precipi-
tates and c) a Porod signal with a
q
-4 decay that levels out in a constant incoherent background signal.
The inset figure shows the applied log-normal size distributions in the model. According to refer-
ence [E3])
4 Studies on nickel-based superalloys
70
SANS depend mainly on the accuracy of the compositional analysis and the used size
distribution model, whereas for the TEM analysis they depend on the accuracy of the
measurement of the precipitate radii and the specimen thickness which is further elab-
orated in the article by Lawitzki et al. [E3].
4.1.4 Discussion
Influence of grain boundaries and
-
phase
Since
-
precipitates and especially the grains are too large to give a scattering signal
within the measured
q-
range, only information about the so-called “fractal dimension”
of these scattering objects are obtained at smallest
q-
values. In case of scattering
objects with smooth interfaces, this asymptotic “Porod” regime is described by a
KP
q-
4-
decay (Eq. [4.4]), with Porod coefficient
KP
, which is proportional to the specific sur-
face area
S
of the scattering objects [118,119]. Since the grain structure in the samples
A-718, C-870 and D-950 was comparable, a difference in the Porod decay (cf. Figure
4.9) is most probably generated by differences in the amount of interfaces of
-
precip-
itates. As shown in Table 4.4, this trend is supported when comparing the specific
surface area
S
of
-
phase (calculated from the size and volume fraction of
-
precipi-
tates) with the fitted Porod coefficients. Small differences in the defect structure be-
tween the states and inaccuracies in the estimation of the size and volume fraction
could explain why the ratio of
KP/S
is not fully constant.
Heat Treatment
Phase
Size
Parameter
A-718
B-760
C-870
TEM
SANS
TEM
SANS
TEM
SANS
[nm]
7.31.5
6.9
11.21.0
9.3
9.71.0
3.6
0.30
0.29
0.27
0.25
0.19
0.21
fp
[%]
-
3.10.4
-
4.20.6
-
0.70.1
[nm]
4.60.4
4.0
7.80.5
7.1
33.51.4
28.1
0.30
0.29
0.35
0.26
0.30
0.31
Rrot/Re
0.22
0.22
0.23
0.24
0.19
0.18
fp
[%]
2.00.8
3.80.6
3.71.6
4.10.6
9.14.4
6.00.9
Table 4.3: Comparison of SANS and TEM results: Median sizes
, logarithmic standard deviations
,
volume fractions
fp
and the aspect ratio
Ra
/
Rb
of
- precipitates of the TEM and the SANS analysis are
compared for the differently heat treated states containing
- and
- precipitates.
4.1 Differentiation of γ′- and γ″- precipitates in Inconel 718
71
Check of consistency of the results
In section 4.1.3, it has been shown that the SANS signals of
-
and
-
precipitates
can be distinguished and quantified, when the particle size distributions are known. In
order to prove that the evaluated volume fractions are reasonable, a plausibility test is
performed by checking against the mass balance of Nb and Ti. From the amount of Nb
or Ti that is necessary for the precipitation of
-
phase, it is calculated how much Nb
or Ti, respectively, is left for the precipitation of
-
phase. The calculations are per-
formed by considering the volume fraction of
-
precipitates evaluated either by TEM
or by SANS, as shown in Table 4.5. Details and further discussion of these calculations
are elaborated in the article by Lawitzki et al. [E3]; necessary equations [6.9]-[6.11] are
given in the Appendix B: “Mass balance calculation in section 4.1”.
A good example illustrating the consistency of the SANS results presented in Table
4.5 are the results of sample C-870. The measured volume fraction of
-
precipitates
is 6.0 % (by SANS) or 9.1 % (by TEM) and the volume fraction of
-
precipitates is
0.7 %, which could only be measured by SANS. The measured volume fraction of
-
precipitates has to be compared to the volume fraction of 1.5 %, which is predicted
based on the mass balance calculation with Nb and Ti, determined by SANS. In con-
trast to this and after quantification by TEM, a volume fraction of -6.7 %, based on Nb
mass balance, and -0.1 %, based on the Ti mass balance, is predicted. These even
negative volume fractions for the
-
phase indicate that the volume fraction of
-
phase measured by TEM was too high. Furthermore, the predicted volume fractions
after mass balance with Ti and Nb by SANS show in all states better accordance op-
posed to these, predicted after evaluation by TEM.
Sample state
SEM Volume fraction
of
- phase [%]
Specific surface
area
S
[μm2/μm3]
Porod coefficient
KP
[10-5 sr-1]
Ratio of
KP
/S
C-870
2.00.1
0.14
3.28
239
A-718
2.70.1
0.16
4.38
289
D-950
7.10.3
0.41
5.93
148
Table 4.4: Comparison of measured volume fractions of
-
phase and fitted Porod coefficients.
4 Studies on nickel-based superalloys
72
Precipitation of
-
phase at the
/
-
interface
In general, the evaluated size distributions of
-
and
-
precipitates by TEM and SANS
almost coincide (compare Table 4.3). There is only one significant discrepancy, which
is the average diameter of
-
precipitates that is ~9.7 nm, evaluated by TEM, and
~3.6 nm, evaluated by SANS. The most probable origin of this discrepancy can be
found in the analysis with APT. As indicated in Figure 4.11, the reconstructed tip of
state C-870 reveals enrichments of Al atoms at the interface of
-
and matrix- (
-
)
phase that match well with the composition of
-
phase. The average size of these Al
rich regions is less than 5 nm and thus, coincides with the evaluated size of
-
precip-
itates by SANS. The segregation of
-
phase at the
/
-
interface has been already
observed by Miller et al. [120] and is a consequence of the rejection of Al when
-
precipitates coarsen during aging. The evaluation of the signal of
-
phase by SANS
offered for the first time the chance to even quantify this phase as (0.7 ± 0.1) %.
In contrast to the analysis by APT, the coexistence of
-
precipitates at the
/
-
in-
terface could not be resolved by TEM; only single
-
precipitates that were larger in
size, but obviously also not relevant in their amount could be detected.
Figure 4.11: APT reconstruction of a
tip of state C-870 and the tip after
local concentration analysis using
Nb (c) and Al (d). (According to ref-
erence [E3])
Sample state
A-718
B-760
C-870
Evaluation technique
SANS
TEM
SANS
TEM
SANS
TEM
γ″- phase
Measured
3.8
2.0
4.1
3.7
6.0
9.1
γ′
- phase
Measured
3.1
-
4.2
-
0.7
-
Calculated with Nb balance
3.6
8.5
5.0
6.0
1.5
-6.7
Calculated with Ti balance
3.7
4.6
4.4
4.6
1.5
-0.1
Table 4.5: The theoretical volume fractions of
- phase in Inconel 718 calculated from TEM and SANS
results of
- precipitates are compared to measured values obtained by the SANS evaluation.
4.2 Micromechanical behavior of Haynes 282 and Inconel 718
73
4.2 Micromechanical behavior of Haynes 282 and Inconel 718
Macroscopic residual stresses can significantly affect the service lifetime of superal-
loys, both with favorable or detrimental consequences. Residual stresses arise due to
inhomogenities (such as heterogeneous phases or inclusions), during material pro-
cessing or during material loading (plastic, thermal, chemical) and always exist in a
material. In order to design safe and reliable components, it is therefore of technical
relevance to measure or estimate the macroscopic residual stress state. Generally,
one can distinguish destructive and non-destructive methods for the measurement of
residual stresses. Destructive or mechanical methods (e.g. hole drilling, turning off,
sectioning) measure the strain that is released when the load on the component is
released. These techniques are limited to analyze macrostresses of rather large re-
gions. In contrast, non-destructive methods (here neutron diffraction or X-ray diffrac-
tion, but also ultrasonic or micromagnetic methods exist) are sensitive to microstresses
and allow to measure strains of much smaller areas even within the cross section of
structural parts and even in-situ during external loading of the material [121].
Diffraction methods track changes in the distance of crystallographic lattice planes (lat-
tice strains) and thus, always measure a superposition of macro- and microscopic
strains. For the macroscopic residual stress analysis it is necessary, to cancel out the
influence of microscopic residual (intergranular) stresses; either by choosing crystal
planes that are known to be insensitive to accumulate intergranular strains and feature
a linear lattice strain response far in the plastic regime (e.g. [311]- and [111]- planes in
FCC- metals), or by cutting a reference sample that possesses the same microscopic
stress state as the material of interest [122]. For the latter approach, it is assumed that
the microscopic stress state does not change during extraction of the reference sam-
ple.
In the work of Repper et al. on alloy Inconel 718 [123], it was shown that this assump-
tion is not always fulfilled. In this case, the intergranular strains change during extrac-
tion of the reference sample leading to so called spurious stresses (of up to 25 %) in
the macroscopic residual stress analysis of the uncut specimen. By in-situ neutron dif-
fraction during tensile testing it could be further demonstrated that the microstructure
4 Studies on nickel-based superalloys
74
of Inconel 718 has a significant influence on the amount of intergranular strains that
are accumulated in different crystal orientations [124].
In a more recent work [2], the micromechanical behavior of Inconel 718 during tensile
testing is compared to the behavior of the Ni-based superalloy Haynes 282. This study
has shown that the alloys behave unexpected differently. In contrast to alloy In-
conel 718, the accumulated intergranular strains in alloy Haynes 282 are much lower
and they do not change during stress release. So far, the authors could not explain
these observations but it laid the foundation for more detailed studies of these alloys
in this work.
In the following chapters, the results of a more complex study of the alloys Inconel 718
and Haynes 282 are presented. Neutron diffraction experiments were performed to
track the evolution of residual stresses in form of intergranular strains during tensile
loading of lattice planes not only parallel to the load (0°), but in five different directions
(0°, 30°, 45°, 60° and 90°) as introduced in section 3.3.2. To interpret the measure-
ments on both Ni-based superalloys, the results are compared to complementary
CPFEM simulations (see section 3.3.3) and to microstructural studies by electron mi-
croscopy (EBSD, TEM and APT).
4.2.1 Microstructure of Inconel 718 and Haynes 282
A general overview of the microstructure of the investigated Ni-based superalloys In-
conel 718 and Haynes 282 is given in Table 4.6. The Haynes 282 specimens were
subjected to the standard heat treatment recommended by the manufacturer (solution
annealing at 1010 °C and 8 h aging at 788 °C) which yielded an average grain size of
(13 ± 7) μm in diameter (see also Figure 4.12 d). By dark field TEM imaging, spherical
-
precipitates with an average diameter (13.8 ± 1.9) nm and a volume fraction of
(18.2 ± 0.5) nm could be detected that were homogeneously distributed and coherent
with the matrix (see Figure 4.12 b). Due to the coherency of the precipitates with the
matrix and their small size,
-
precipitates are sheared by dislocations during defor-
mation which could be observed by TEM (see also Appendix C Figure C.1).
4.2 Micromechanical behavior of Haynes 282 and Inconel 718
75
The microstructural analysis of Inconel 718 in section 4.1 has demonstrated that the
microstructure of the alloy strongly depends on the heat treatment. To compare the
two alloy systems, the specimen state A-718 of Inconel 718 (solution annealed at
975 °C and double-step aging at 718 °C for 8 h and 621 °C for 8 h) was selected, be-
cause the grain sizes are comparable and the strengthening phases are rather small
in both alloys (cf. Figure 4.12). Nonetheless, the microstructures do not equal which is
mainly due to the different alloy compositions (cf. Table 2.1). The amount of strength-
ening phases is less in Inconel 718, mainly because the alloy contains less Al and Ti.
Furthermore, the lattice mismatch between
-
precipitates and
-
matrix is larger
(0.2 %) compared to the lattice mismatch in Haynes 282 (0.03 %) [2]. Because of the
lack in Nb, Haynes 282 does not contain any
-
and
-
phases.
-
phase strengthens
the grain boundaries in Inconel 718. In Haynes 282, grain boundary strengthening is
achieved by carbides instead, which is shown in Figure 4.13. By SEM and EDX anal-
ysis, it can be revealed that some of the grain boundaries in Haynes 282 are covered
by M6C- carbides (M = Mo,Cr,Ni), M23C6- carbides (M = Cr,Mo) and non-spherical
-
phases. These phases are products of the decomposition reaction of primary carbides
and
.
[4.10]
Inconel 718
Haynes 282
Heat treatment
975 °C 2 h
718 °C 8 h +
621°C 8 h
1010 °C 2 h
788 °C 8h
Lattice parameters [nm]
- matrix
- precipitates
- precipitates
a
: 3.5972
a
: 3.5900
a
: 3.6004;
c: 7.4382
a
: 3.5917
a
: 3.5909
Grain diameter [μm]
11 ± 5
13 ± 7
- precipitates radius [nm]
7.3 ± 1.3
13.8 ± 1.9
- precipitates equivalent radius [nm]
4.6 ± 0.5
-
- phase volume fraction [%]
3.1 ± 0.4
18.2 ± 0.5
- phase volume fraction [%]
3.8 ± 0.6
-
Table 4.6: Comparison of the microstructure of Inconel 718 and Haynes 282.
Lattice parameters taken from Ref [2].
.
4 Studies on nickel-based superalloys
76
The grain boundary carbides in Haynes 282 are thermodynamically more stable at
higher temperatures than
-
phase, which explains why Haynes 282 can be heated
above 1000 °C without significant grain growth [125]. Beside these secondary car-
bides, there are a few large primary carbides and core-shell nitrides (TiN core with
carbide shell) that are homogeneously distributed within the grains. Due to their low
volume fraction, they do not contribute to the strength of the alloy. Further general
information about the microstructure of Haynes 282 are also summarized in the article
of Matysiak [15]; more details on the microstructure of alloy Inconel 718 are given in
section 4.1.
Figure 4.12: Microstructure of Inconel 718 (left) and Haynes 282 (right): (a) and (b) DF TEM images of
the strengthening precipitates; (c) and (d) inverse pole figure maps obtained by EBSD evaluated for
grain size analysis.
4.2 Micromechanical behavior of Haynes 282 and Inconel 718
77
4.2.2 Macroscopic mechanical behavior
In-situ neutron diffraction experiments during uniaxial tensile loading were performed
on the alloys Inconel 718 and Haynes 282 (see Figure 4.14). In this section, only the
macroscopic mechanical material behavior is discussed. Each data point shown in the
diagrams represents one deformation state where 25 different diffractograms have
been recorded (cf. section 3.3.2); these will be discussed in the next section in detail.
Figure 4.14 (a-c) presents true stress
t
- true strain
t
curves with
and
[4.11]
,
[4.12]
where
Eng
is the applied (engineering) stress and
Eng
the measured strain. In contrast
to prior studies by Wagner et al. [2], the Haynes 282 specimens that were only solution
annealed and subsequently water quenched are significantly softer than the speci-
mens after additional aging at 788 °C for 8 h (see Figure 4.14 a). The reproducibility of
the measurements in this work is confirmed since the tensile curves of three independ-
ent experiments of different specimens are almost identical.
Figure 4.13: Carbides and nitrides in Haynes 282: BSE SEM images (left) and EDX maps (right) of
carbides (a,b) and a Ti-nitride with a primary carbide (MC) shell (c,d).
4 Studies on nickel-based superalloys
78
In Figure 4.14 (b), the stress-strain curves of Haynes 282 and Inconel 718 are com-
pared. Whereas the Young’s Modulus of Inconel 718 of (192.6 ± 0.4) GPa and of
Haynes 282 (204.8 ± 4) GPa (average of six measurements) are quite similar, their
yield stresses are very different. Haynes 282 yields at around 800 MPa, whereas In-
conel 718 starts to yield at stresses above 1000 MPa. Furthermore, the elastic-plastic
transition is smoother for Inconel 718.
At 650 °C (cf. Figure 4.14 c), Haynes 282 starts to yield at around 500 MPa and the
elastic-plastic transition is even sharper, compared to the behavior at room tempera-
ture. At 650 °C, the alloy barely hardens because of dynamic recovery processes.
According to the Hollomon Eq. [2.2] (cf. section 2.2.1), the shape of the true stress -
true strain curves can be simplified by a power-law relationship. A log-log plot of true
stress versus true strain allows to estimate the strain hardening exponent
n
by
Figure 4.14: Macroscopic material behavior during uniaxial tensile tests: (a-c) true stress – true strain
curves of Haynes 282 after solution annealing and after additional aging (a); of Inconel 718 vs Haynes
282 (b) and of Haynes 282 at room temperature and 650 °C; (d) Logarithmic true stress - true strain
curves indicating two different stages (I
and
II) of strain hardening for Haynes 282 and only one stage
for Inconel 718.
4.2 Micromechanical behavior of Haynes 282 and Inconel 718
79
,
[4.13]
as depicted in Figure 4.14 (d) for Inconel 718 at 25 °C and for Haynes 282 at 25 °C
and 650 °C. Both alloys show a different strain hardening behavior. For uniaxial strains
below 7.5 %, the stress-strain relationship for Inconel 718 can be described by the
simple Hollomon equation and a strain-hardening exponent of 0.085 is obtained. In
contrast, the strain-hardening behavior of alloy Haynes 282 cannot be described by
the Holloman equation, since the value of
n
increases during deformation. At least two
Hollomon equations are required, to describe the work hardening of Haynes 282; a
necessity that was initially introduced for steels by W. Morrison [126] and further elab-
orated by M. Atkinson [127] and is considered as the “double-n” method. It should be
mentioned here that also other empirical equations (e.g. by Ludwik, Swift, Voce or
Ludwigson) have been developed to describe work-hardening for non-linear strain-rate
behavior (in log-log coordinates) as discussed by Kleemola and Nieminen [128] and
Praveen et al. [129]. Typically, a change of
n
is addressed to a change in the defor-
mation mechanism. For polycrystalline FCC metals, plastic flow can be divided into
different stages which was demonstrated by G. Zankl already in 1963 [130]. Plastic
deformation starts usually in the largest grains with lowest shear stresses because
resistance induced by dislocation pile-up at grain boundaries is lowest. At a total plastic
strain of around 0.1 %, also the smallest grains start to deform plastically by multiple
slip which is denoted as the start of stage I. In this stage, dislocations glide by multiple
slip to the grain boundaries, where they pile-up until a total sample strain of around
1 % (for Ni). At this point, resistance due to interactions of dislocations within the grains
become so strong that deformation by slip on single slip bands (stage II) becomes
energetically more favorable which causes a inflection of the strain rate; at higher
stresses even shear bands form. The stresses produced by dislocation pile-up at grain
boundaries are eventually so large that they can even activate glide in neighboring
grains. At around 5 % plastic strain, cross slip sets in (stage III).
The intersection of the two linear regressions for Haynes 282 in Figure 4.14 (d) at both,
high and low temperatures, indicates a transition from deformation stage I to II at about
2 % strain. This transition is not observed for Inconel 718. Either stage I does not exist,
or the strain-rates of stages I and II have a similar slope in Inconel 718. Beside the
overall larger strength of Inconel 718, the variation in the strain-hardening behavior
4 Studies on nickel-based superalloys
80
between the two alloys is the most significant difference in the macroscopic material
behavior; to find the origin for this difference, the microscopic mechanical behavior is
investigated in the following.
4.2.3 Microscopic mechanical behavior
The in-situ neutron diffractogramms are evaluated with respect to changes and shifts
of the measured lattice reflections of specific grains in five different orientations (
= 0°,
30°, 45°, 60° and 90°) to the direction of load (see also section 3.3.2 and Appendix A
Figure A.1). In the following sections, the results of the micromechanical characteriza-
tion are presented in four different kind of diagrams (i-iv). i) The measured lattice
strains
hkl
(Eq. [3.11]) are presented in true stress vs lattice strain diagrams (see Fig-
ure 4.15). The elastic regimes are evaluated by linear regression with a slope repre-
senting the hkl dependent microscopic Young’s modulus or diffraction elastic constant
Ehkl
. The deviation of the lattice strains to the linear elastic behavior when the sample
starts to yield, indirectly represent the intergranular
II
stresses (see section 2.3.1) and
is in the following denoted as intergranular strain
hkl,II
and calculated by
,
[4.14]
where
t
is the stress applied to the sample,
0
is a threshold stress in the reference
state which always exists in the data, since the stable mounting of the samples in the
tensile rig required to apply already a small force (200 N) to record the first (reference)
data point. ii) The evolution of intergranular strains during uniaxial deformation is plot-
ted in intergranular strain versus (macroscopic) true plastic strain diagrams (see Figure
4.16). Intergranular strains can be considered (if no slip occurs during unloading) as
the residual strains in the material when no macroscopic stress is applied.
Furthermore, the evolution of (iii) the full width at half maximum (FWHM) (cf. Figure
4.19) and the evolution of (iv) the integrated intensities (cf. Figure 4.20) of the reflec-
tions are cpresented as a function of the macroscopic true strain; the FWHM and the
integrated intensities are closely related to the amount of lattice defects / dislocations
within the grains and the texture development in the specimen, respectively.
4.2 Micromechanical behavior of Haynes 282 and Inconel 718
81
True stress versus lattice strain behavior
The evolution of the lattice strain in (111)-, (200)- and (220)- lattice directions during
uniaxial tensile deformation of Haynes 282 and Inconel 718 are compared in Figure
4.15 for different orientations
with respect to the loading direction. The experimental
errors estimated from the peak fitting procedure are smaller than the symbols drawn.
Figure 4.15: Comparison of measured true stress versus lattice strain diagrams of Haynes 282 (left) and
Inconel 718 (right). Compared are the lattice strains of (111)-, (200)- and (220)- oriented grains with
different orientations (
= 0°, 30°, 45°, 60° and 90°) to the direction of load.
4 Studies on nickel-based superalloys
82
In Appendix C Figures C.2 and C.3, the remaining diagrams of all other measured
reflections and the respective diagrams of the measurements at 650 °C are shown.
The accuracy of the estimation of lattice strains by analysis of the (100)-, (110)-, (210)-
and (221)- reflections of
- precipitates is much less (especially for Inconel 718),
mainly due to their low volume fraction and their small size. The results of the high
temperature experiments are not yet published and will be only briefly discussed here.
In the linear elastic regime, the evolution of lattice strains in Inconel 718 and
Haynes 282 is almost the same, since both materials have a similar elastic anisotropy.
This can be also observed in the comparison of the microscopic Young’s moduli (see
Table 4.7). In absolute values, Haynes 282 exhibits slightly higher moduli (around 10-
14 GPa) than Inconel 718, as was also observed for the macroscopic behavior. At
650 °C, Haynes 282 softens elastically by around 30 GPa. The measured microscopic
Young’s moduli of the alloys are compared to the values for pure nickel calculated with
the Kröner model [2], which are marginally below the values of alloy Inconel 718. The
difference in the microscopic elastic constants is related to the anisotropy of nickel
which can be expressed by 2C44/(C11-C12) and is 2.59 when applying the elastic con-
stants given in section 3.3.3. A value of 1 represents an isotropic material. In aniso-
tropic FCC metals, the (111)- directions are typically the stiffest directions and the
(200)-directions the elastically softest directions. From the measured microscopic elas-
tic constants, an estimate of the specimen anisotropy can be obtained by the ratio
E111/E200
, which is almost the same in Haynes 282 and Inconel 718.
At the elastic - plastic transition however, the micromechanical behavior of Inconel 718
and Haynes 282 can be distinguished. Like in the macroscopic stress-strain curves,
the elastic-plastic transition is smoother in Inconel 718, whereas in Haynes 282, there
is a stronger deviation from the linear elastic behavior directly after the onset of plastic
flow. A significant difference in the yield stress between the different orientations and
reflections for one alloy is not measurable.
Young’s Moduli [GPa]
E111
E200
E220
E111/E200
Haynes 282 (25 °C)
262 ± 2
174 ± 2
231 ± 4
1.51
Inconel 718 (25°C)
248 ± 4
164 ± 3
221 ± 4
1.51
Haynes 282 (650 °C)
230 ± 5
147 ± 2
201 ± 4
1.56
Nickel (Kröner model)
243
150
211
1.62
Table 4.7: Microscopic Young’s moduli
Ehkl
of specific reflections (
= 0°).
4.2 Micromechanical behavior of Haynes 282 and Inconel 718
83
Intergranular strains versus macroscopic plastic strain
The micromechanical behavior in the plastic regime can be better compared when
representing the deviation of the lattice strain from a linear elastic behavior or short
“intergranular strains” (and “intragranular strains” in case of reflections stemming from
precipitates) as a function of the macroscopic plastic strain as shown in Figure 4.16.
Figure 4.16: Comparison of measured intergranular strain versus macroscopic plastic strain diagrams
of Haynes 282 (left) and Inconel 718 (right). Compared are the intergranular strains of reflections stem-
ming from the matrix (111), (200) and (220) in different orientations (
= 0°, 30°, 45°, 60° and 90°) to
the direction of load.
4 Studies on nickel-based superalloys
84
Compared are the intergranular strains within grains with the largest lattice plane spac-
ings (111), (200) and (220) measured in the five different orientations to the direction
of load
during tensile testing. In Appendix C Figure C.4, the respective diagrams are
shown for the intragranular strains stemming from precipitates and in Appendix C Fig-
ure C.5, the intergranular strains of the high temperature measurements of
Haynes 282 are presented. The diagrams demonstrate clearly that the sign and
amount of intergranular strains depends on both, the lattice plane family and its orien-
tation to the direction of load.
Interestingly, the magnitude of intergranular strains is largest in the (elastically) softest
(200)- orientated grains and lowest in the stiffest (111)- grains. In grains with the (200)-
lattice plane normal vectors parallel to the load (0°), the largest positive intergranular
strains are measured. In contrast, if the (220)- direction is parallel to the load, the in-
tergranular strains become negative (or compressive). This behavior has already been
demonstrated in previous (theoretical and experimental) works for polycrystalline ma-
terials and is controlled by the elastic anisotropy of the material [2,131–135].
Clausen et al. [131] have demonstrated by self-consistent modelling that the anisotropy
of a material has a strong influence on the distribution and sign of intergranular strains
in differently oriented grains; for example, in anisotropic polycrystalline metals such as
copper and stainless steels, largest intergranular strains develop in the (200)- planes,
whereas in isotropic aluminum, the (111)- orientated grains accumulate the largest in-
tergranular strains. An explanation for this behavior is given in the article of Wong et
al. [136] and explained by the directional strength-to-stiffness ratio, the ratio of the
Schmid factor and the directional stiffness of a certain crystal orientation with respect
to the direction of load. The authors demonstrate that in anisotropic FCC metals, grains
with the (200)- orientation parallel to the load (highest strength-to-stiffness ratio) re-
quire the largest stress to yield and (220)- oriented grains the lowest stress. This
means, the (220)- oriented grains will in average yield first and as a consequence,
show no further change in the lattice strain evolution. Since grains in different orienta-
tions take up more load before yielding, (220)- oriented grains exhibit negative or com-
pressive intergranular strains in comparison to grains that still deform linear elastic at
the same stress level. Vice versa, (200)- oriented grains require in average the largest
stresses to yield and therefore, take up more load (show larger intergranular strains)
4.2 Micromechanical behavior of Haynes 282 and Inconel 718
85
at stress levels were most of the grains deform already plastically. To conclude this,
the lattice strains of (111)- oriented grains show almost no deviation from the linear
elastic behavior, meaning they yield at intermediate stresses. Once slip is activated in
all grains (end of the elastic-plastic transition), the intergranular strain curves flatten. A
further increase of the intergranular strain accumulation is now only possible by work
hardening (plastic regime).
As shown in Figure 4.16, the sign and magnitude of intergranular strains in Haynes 282
and Inconel 718 are very similar which can now be attributed to the elastic anisotropy
of the alloys which is almost the same (cf. Table 4.7). Nonetheless, the evolution of
intergranular strains in Haynes 282 and Inconel 718 do not equal. This is demonstrated
in Figure 4.17 (b) which compares the intergranular strains after a plastic deformation
to 4.5 %. In general, the (absolute) intergranular strains are larger in Inconel 718. This
Figure 4.17: Comparison of intergranular strains at 4.5 % macroscopic plastic strain versus the meas-
urement orientation for Haynes 282 (a) and in comparison to the intergranular strains in Inconel 718 (b);
in (c) and (d), the intergranular strains of matrix reflections are compared to the intragranular strains
stemming from
- precipitates (or
-
phase) for Haynes 282 (c) and Inconel 718 (d). In (a), the orientation
of the measured lattice plane spacings with respect to the tensile direction is indicated on top. The drawn
bands in the diagrams indicate the experimental uncertainties.
4 Studies on nickel-based superalloys
86
can be related to the macroscopic behavior of alloy Inconel 718, since the alloy yields
at higher stresses. Taking a closer look at Figure 4.17 (b), the curves feature two sig-
nificant differences between Inconel 718 and Haynes 282 that cannot be explained by
measurement errors. In Haynes 282, the (111)- oriented grains accumulate almost no
intergranular strains whereas in Inconel 718, a formation of small compressive inter-
granular strains is observed in all measurement orientations. Furthermore, the (220)-
grains oriented perpendicular (90°) to the direction of load accumulate almost no inter-
granular strains in Inconel 718, in contrast to Haynes 282.
These differences could have been initialized by a texture in the undeformed tensile
test specimens, which will be discussed further in the next section, or by a different
contribution of the precipitates to the micromechanical behavior. Therefore, also the
changes of the lattice plane spacings in the precipitates have been evaluated (see
Figure 4.17 (c,d) and Appendix C Figure C.4). Due to the low intensities of the precip-
itate reflections, these data feature larger uncertainties. For Haynes 282, the (100)-,
(110)- and (210)- reflections of
- precipitates could be measured. Due to the lower
volume fraction of
- precipitates in Inconel 718, only the (110)- reflection could be
measured in this alloy; it should be further mentioned that this reflection could also
stem from
- precipitates, which could not be differentiated with the current experi-
mental setup. In contrast to Haynes 282, the diffractograms of Inconel 718 featured
also reflections stemming from
-
phase. The (020)- reflection of
-
phase, which does
not overlap with any matrix reflection and which is better resolved than the reflections
of
- precipitates, is therefore included in the comparisons in Figure 4.17.
The intergranular strains and the intragranular strains (deviation from linear behavior
of the precipitates) for Haynes 282 are almost identical (see Figure 4.17 c). This
means, the precipitates and the matrix take up the same load and deform jointly since
no type III residual stresses are caused by the precipitates. This behavior was also
demonstrated in a previous study on the nickel-based superalloy RR1000 and is cor-
related to the size of the
- precipitates [133]. As long as the
- precipitates are below
a critical size (in the studied case 90 nm were sufficient), the same slip system is active
in both phases, whereas, if precipitates get larger, a load transfer from
- to
- phase
is observed since dislocations do not penetrate the precipitates any longer. In the
Haynes 282 specimens tested in this work, the
- precipitates have a radius of only
4.2 Micromechanical behavior of Haynes 282 and Inconel 718
87
(13.8 ± 1.9) nm, which is far below the critical size found for alloy RR1000. Surprisingly,
the measurements indicate a small load transfer from
- to
- phase for those grains
where the (220)- plane normal vectors are parallel to the direction of load (0°). While
the (220)- grains, which yield first, develop compressive intergranular strains, the in-
tragranular strains of
- precipitates in these grains show a still linear elastic behavior.
In the measurements performed at 650 °C for Haynes 282 (see Appendix C Figure C.5
and Figure C.6) inter- and intragranular strains show the same characteristics com-
pared to the room temperature data and the phenomenon of the load transfer from
-
to
- phase (only) in (220)- oriented grains is even more pronounced.
In Figure 4.17 (d), intergranular strains and the intragranular strains for Inconel 718
are compared. The comparison of the intergranular strains in (200)- matrix grains to
those formed in
-
phase in (020)- orientation indicates a clear load transfer from
- to
- phase. Delta phase is incoherent to the matrix and very large and thus, dislocations
can only bypass the stiff precipitates according to the Orowan mechanism. On the
other hand
- precipitates, are very small in Inconel 718 (< 10 nm) and therefore
should show a similar behavior as in Haynes 282. The intragranular strains of (110)-
planes in 30°, 45°, 60° and 90° orientation seem to confirm this behavior, but in 0°
orientation, very large compressive residual strains are formed. To this point, it is not
clear if this is due to measurement uncertainties (these data scatter strongly which is
shown in Appendix C Figure C.4e), or if this is due to the overlap of the reflections of
-
and
-
precipitates;
-
precipitates exhibit a longer
c-
axis (cf. Table 4.2).
Concerning Figure 4.16 and Figure 4.17, it should be further mentioned that an “intui-
tive” interpretation of the magnitude and sign of intergranular strains can be achieved
by the directional strength-to-stiffness ratio [136], but only for the grains with their nor-
mal vectors parallel to the direction of load. The diagrams represent clearly that the
intergranular strains depend strongly on the orientation in which they are measured
and can even change their sign (for example compare the intergranular strains in 0°
and 45° orientation). The reason is that only grains with their lattice plane normal in a
direction parallel to the load are in a similar elastic strain state since the deformation is
rotationally symmetric. But when comparing grain subsets with lattice plane normals in
different orientations, such as perpendicular (90°) to the load, the elastic lattice strain
4 Studies on nickel-based superalloys
88
shows great variations, since representatives of these grains, which are arbitrarily ro-
tated around the perpendicular axis, are differently orientated to the tensile axis. Dif-
ferent orientations to the tensile axis mean different strain states. This is illustrated in
Figure 4.18 for a grain with the (002)- plane normal (a) and a (220)- plane normal (b)
parallel to the tensile axis. A rotation of the grains around the tensile axis does not
change the strain state; but for example a rotation of grain (a) around the (200)- plane
normal by 45° establishes the situation shown in figure (b). This means, when meas-
uring the lattice strain of the (200)- reflections perpendicular to the tensile axis, four
sub-set of grains will be in orientation (a), another four in orientation (b) and the re-
maining grains in any another orientation along the (100)-(110) side of the unit triangle.
The measured lattice strain will be an average of the strains in all these grains and can
thus only be interpreted by correlative simulations which was previously shown by a
self-consistent model for intergranular strains measured perpendicular to the
load [131]. A comparison of simulated average strains in all other directions will be
given in the discussion in the next section by a CPFE model.
So far, only the positions of the peaks of the in-situ neutron diffraction data were ana-
lyzed. As shown in Figure 4.19, also the evolution of the peak widths reveals valuable
information about the specimens’ microstructure during deformation. The peak width
is affected by the dislocation density in the material and thus, can be used as a quali-
tative indicator of plastic deformations within the grains. It can be concluded that the
Figure 4.18: Scheme of two specific grain orientations: with the (002)- lattice plane normal (a) and the
(220)- lattice plane normal (b) parallel to the tensile axis. The angles of other specific lattice plane normal
vectors to the tensile axis are indicated.
4.2 Micromechanical behavior of Haynes 282 and Inconel 718
89
amount of dislocations in Inconel 718 is slightly higher. Possibly, the higher amount of
incoherent phases (
– phase) increases the amount of dislocations within the grains
in Inconel 718. The change of the width of the (111)- reflections is less than for the
(200)- reflections. This may not directly be related to the dislocation density but could
be an effect of a larger deviation of the peak positions within the (200)- grains due to
different stress states. This hypothesis is affirmed by the fact that the grains with the
(200)- plane normals parallel to the load (exhibiting all similar stress states as dis-
cussed above) show the smallest change in peak widths.
Only the results for the (111)- and (200)- reflections are presented here, because the
evaluation of the other reflections does not allow further conclusions. The peak broad-
ening in
- and
- phase (not shown here) is the same; this was also demonstrated by
Grant et al. for small
- precipitates which are sheared by dislocations [133].
Figure 4.19: Comparison of peak widths (FWHM) versus macroscopic true strain diagrams of
Haynes 282 (left) and Inconel 718 (right). Compared are the FWHMs of reflections stemming from the
matrix (111) and (200) in different orientations (
= 0°, 30°, 45°, 60° and 90°) to the direction of load.
4 Studies on nickel-based superalloys
90
4.2.4 Discussion
Specimen texture
The change of the integrated intensities of the reflections in the in-situ neutron diffrac-
tion experiments allows a prediction of the development of a specimen texture during
uniaxial deformation (cf. Figure 4.20 (d-f) and Appendix C Figure C.7 and C.8). For a
better comparability of the data, the integrated intensities presented here are normal-
ized to the values obtained at the onset of plastic flow. Values larger than 1 correspond
to an increase in the amount of grains contributing to a reflection in a specific direction
Figure 4.20: Texture analysis on Haynes 282: Comparison of the specimen texture evaluated by EBSD
(see corresponding IPF maps left and generated texture plots in the middle column) and measured by
neutron diffraction (right) in 0° (a,d,g), 45° (b,e,h) and 90° (c,f,i) orientation. Texture plots have been
generated with the software
OIM Analysis
by harmonic series expansion where the color scale repre-
sents the relative fraction of grains (weighted by area) with a certain orientation within the standard
triangle (blue: < 0.9 and red: > 1.3).
4.2 Micromechanical behavior of Haynes 282 and Inconel 718
91
and vice versa. In Table 4.8 (and Figure 4.20 and Appendix C Figure C.7), the normal-
ized integrated intensities of Haynes 282 and Inconel 718 are compared with the tex-
ture that is evaluated microscopically before and after deformation by EBSD. For the
texture analysis by EBSD, areas of 900 x 675 μm² including in average 18100 ± 1400
grains are evaluated. The numbers in Table 4.8 are calculated with TSL OIM Analy-
sis 8 from the orientation distribution function by harmonic series expansion until series
rank 25 (and by averaging over 5° of Gaussian half-width) and represent the relative
fractions of grains (weighted by area) with a given orientation in multiple times of a
random distribution (1 = random). The data obtained by neutron diffraction are statisti-
cally more significant, since a volume of ~5 x 5 x 5 mm³ has been analyzed.
According to Table 4.8, the evolution of the texture of Haynes 282 and Inconel 718 is
almost identical. The largest increase in the number of grains with normal vectors par-
allel to the load is observed in the (111)- and (002)- oriented grains while the amount
of (202)- oriented grains decreases. The data obtained by EBSD and neutron diffrac-
tion match well for Haynes 282. In contrast, the texture analysis on alloy Inconel 718
by EBSD suggests a stronger texture. The reason for this is seen in the comparison
with the data of the undeformed specimens. The specimen of alloy Inconel 718 shows
a certain degree of pre- texture. A texture in the undeformed specimen is possible
since the provided material is a leftover from the company Böhler Schmiedetechnik
and the specimen history is unknown. Apparently, this pre- texture has no influence on
the evolution of the texture but it might be responsible for the differences in the devel-
opment of intergranular strains between Inconel 718 and Haynes 282 as shown in
Figure 4.16 and discussed in the previous section. A specimen texture changes the
0° Orientation
Undeformed
0° Orientation
Deformed
45° Orientation
Deformed
90° Orientation
Deformed
EBSD
ND
EBSD
ND
EBSD
ND
EBSD
Haynes 282
[002]
0.91
1.08
0.99
1.31
0.61
1.51
1.14
0.72
1.41
0.99
1.03
0.92
0.89
1.19
0.71
0.90
1.16
0.90
1.08
1.40
0.84
[202]
[111]
Inconel 718
[002]
1.73
0.58
2.01
1.23
0.48
1.35
1.95
0.45
2.31
0.92
0.95
0.85
-
0.92
1.16
0.86
-
[202]
[111]
Table 4.8: Texture analysis by neutron diffraction and EBSD: Comparison of the normalized integrated
intensities evaluated by neutron diffraction (ND) with the texture in multiples of random orientation eval-
uated by EBSD before and after deformation of Haynes 282 and Inconel 718. For the microscopic anal-
ysis of Haynes 282, specimens were cut also parallel and with an angle of 45° to the loading direction.
4 Studies on nickel-based superalloys
92
local environment of grains and can influence the buildup of intergranular strains in
grains in specific orientations.
Correlation to simulations
A crystal plasticity based finite element model was set-up to describe the macro- and
micromechanical behavior of FCC polycrystalline and anisotropic materials (see sec-
tions 2.3.2 and 3.3.3). In an initial work (see reference [E2]), the input parameters of
the CPFE models were adopted to match the macro- and micro strain response of alloy
Haynes 282. By considering the higher yield point of Inconel 718, the simulated curves
for Haynes 282 could be rescaled by a factor of 1.65 (both, the lattice strain and mac-
roscopic strain), to match the behavior of Inconel 718.
Now, the parameters have been further optimized to match the strain and the stress
response for both alloys, Haynes 282 and Inconel 718 (see Table 4.9). With the opti-
mized parameters, the macroscopic stress - strain and the microscopic stress - lattice
strain curves can be described well by the CPFEM simulations (cells including 50000
elements divided into 500 grains were used for the results presented here).
The macroscopic stress – strain curves are compared in Figure 4.22. For Inconel 718,
the deviation between simulated data and experimental data is within the size of the
symbols for the complete stress - strain curve; and also for Haynes 282, the elastic
and the plastic regime do not exhibit significant deviations. Only in the elastic - plastic
transition, the simulated and experimental data do not coincide. The sharp yield point
of Haynes 282 cannot be described with the current hardening model (cf. Eq. [2.16])
used for the simulations.
Initial parameters
(from reference [E2])
Optimized parameters
for Haynes 282
Optimized parameters
for Inconel 718
c11 [GPa]
244.76
257
249
c12 [GPa]
149.22
165
159
c44 [GPa]
123.6
142
131
h0
[GPa]
1000
1850
1300
n
1.75
1.75
1.75
c,t=0
[MPa]
187
278
408
s
[MPa]
393
590
800
Table 4.9: Input parameters of the CPFEM simulations.
4.2 Micromechanical behavior of Haynes 282 and Inconel 718
93
In order to model a sharper yield point, further simulations have been performed by
manipulation of Eq. [2.16]. Instead of the constant material dependent slip hardening
parameters
h0
and , a strain dependent formulation is used. The value of either
h0
or
n
is changed in dependence of the total strain. For
n
, a larger value (
n
= 18) is chosen
at small strains and then stepwise decreased with increasing strain to a value of 1.75
at around 2 % strain. For
h0
, negative values (
h0
= -1200) are chosen at smallest strains
(implying a softening at the onset of yield) and then stepwise increased with increasing
strain. Since the CPFEM calculation effort increased drastically with the changes in the
hardening behavior, these test simulations could be performed only on a few small test
cells with a maximum of 10000 finite elements. In Figure 4.21, the results of these
variations in the hardening behavior are compared to the experimental data and the
simulated data with constant hardening parameters. Obviously, a sharper yield point
can be achieved, when forcing the increase in slip resistance to be very small (large
n)
or even negative (negative
h0
.), according to Eq. [2.16], at the onset of yield.
Figure 4.22: Simulation of the macroscopic material behavior: (a) True stress- true strain curves and (b)
logarithmic true stress - true strain curves of Haynes 282 and Inconel 718 in comparison to the simulated
curves.
Figure 4.21: Logarithmic true stress - true strain
curves of Haynes 282 measured at the FRM
II
(red
points) and at the ISIS (light green points). Simula-
tions were performed with the presented hardening
model with constant parameters
n
and
h0
(red line)
and when varying the parameter
n
(blue line) or
h0
(dark green line) with increasing sample strain.
4 Studies on nickel-based superalloys
94
A complete fit to the macroscopic stress-strain curve measured at the FRM II could not
be obtained. Even more negative values of
h0
would be necessary. On the other hand,
the simulated curve with variation of the parameter
h0
matches well with the experi-
mental data measured at the ISIS time of flight neutron source. Evidently, the
Haynes 282 specimen measured at the ISIS neutron source exhibited a slightly lower
yield point and more surprisingly, a smoother yield point. Since both specimens stem
from the same alloy ingot, have been subjected to the same aging treatments and have
been measured in the same tensile rig (Stress-Spec) but with different deformation
rates, creep processes at room temperature might be responsible for the experimental
observation of a sharp yield point measured at the FRM II. At the FRM II neutron re-
actor, the recording of the diffractograms in all orientations and of all reflexes took
around 90 minutes for each deformation state. At the ISIS time of flight neutron source,
all reflexes are measured at the same time and therefore, only around 20 minutes were
necessary to record one deformation state. This means the average strain rate is larger
and leaves less time for creep, which would explain a smoother yield point. In conclu-
sion, the CPFEM simulations allow to very reasonably describe the macroscopic ma-
terial behavior and demonstrate that representative volume elements were chosen.
In Figure 4.23, the microscopic mechanical behavior, here in terms of the lattice strains,
of simulated (with the parameters in Table 4.9) and experimental data are compared
in dependence of the macroscopic sample strain. For all grain and measurement ori-
entations, the linear elastic regime is well described by the CPFEM simulation. In the
plastic regime, small deviations between simulated and experimental data are exhib-
ited. Especially for the grains with their lattice plane normal orientated parallel to the
direction of load (and to a lesser degree also 30°), this could be an effect of limited
counting statistics; the rotation of the model structure around the tensile axis enables
to evaluate around ten times more grains in the other measurement orientations. Most
grains are counted in 90° orientation due to the largest solid angle. Further deviations
in the plastic regime may also be affected by a texture or residual stresses in the initial
state. As shown in Table 4.8, the sample of alloy Inconel 718 exhibits a small pre-
texture. This affects the onset of yield which was demonstrated by Fillafer et al. [137]
and could explain the deviations between simulations and experiments observed for
Inconel 718 in the (111)- directions (for 30° and 45°).
4.2 Micromechanical behavior of Haynes 282 and Inconel 718
95
The most significant qualitative differences between experimental and simulated data
are observed at the elastic plastic transition of alloy Haynes 282. Like in the macro-
scopic stress - strain curve, Haynes 282 exhibits pronounced yielding also in the mi-
croscopic strain curves which is in the following considered as “microyielding”. The
Figure 4.23: Comparison of experimental and simulated evolution of lattice strains of Haynes 282 (left)
and Inconel 718 (right). Compared are the lattice strains of reflections stemming from the matrix of (111),
(200) and (220) grains in different orientations (
= 0°, 30°, 45°, 60° and 90°) to the direction of load.
Significant differences between the simulated and experimental data at the elastic-plastic transition in-
dicate microyielding and are indicated by red circles.
4 Studies on nickel-based superalloys
96
strongest effect of microyielding is observed in the (200)- grains oriented 45°, 60° and
90° to the load and in the (220)- grains oriented 0° and 30° to the direction of load; in
these directions, microyielding causes a change in the curvature of the lattice strain
evolution (indicated by red ellipses in Figure 4.23). In the simulated data with hardening
according to Eq. [2.16], changes in the monotony behavior of the lattice strain evolution
at the onset of plastic flow are not observed. However, when applying negative values
for
h0
or large values for
n
at the onset of plastic flow, the CPFEM also indicates a non-
monotonous lattice strain evolution, even though, it may not be as pronounced as in
the experimental data. This is demonstrated Figure 4.24, where the experimentally
measured lattice strain of the (200) grains in 90° orientation are compared to the sim-
ulated data by varying the hardening parameter
n
(cf. Eq. [2.16]) as a function of the
macroscopic strain:
n
= 18 for
≤ 1 %;
n
= 14 for 1 % <
≤ 2 % and
n
= 1.75 for
>
2 % (the corresponding macroscopic stress strain curve is shown Figure 4.21 in blue).
In the article of Kobylinsky and Lawitzki et al. [E2], various mechanisms that could
explain the pronounced microyielding of the alloy Haynes 282 are proposed and dis-
cussed in more detail. Whereas several possible mechanisms (for instance mechanical
twinning, repeated shearing of precipitates, prior thermomechanical deformation, crack
formation or recovery processes) could be ruled out by correlation to the results from
microstructural analysis, two possible mechanisms remain: an unpinning effect of dis-
locations from pinning points (such as segregations or precipitates) or a pronounced
transition in the hardening behavior because of a change in the deformation
mode/stage as was proposed above to explain the macroscopic stress - strain behav-
ior of Haynes 282 (cf. section 4.2.2).
Figure 4.24: Enlarged view on the evolution of
(200)- lattice strains in Haynes 282 measured in
90° orientation to the direction of load. The ex-
perimental data are compared to the simulated
data obtained by varying the hardening parame-
ter
n
in dependence of the strain.
4.2 Micromechanical behavior of Haynes 282 and Inconel 718
97
To test, whether dislocation pinning effects can be made responsible, a further meas-
urement has been performed on a sample that was pre-deformed to 3 % strain and
subsequently heat treated at 788 °C for 4 h (results are not shown here). In analogy to
the formation of Cottrell atmospheres in steels, high temperatures could activate the
diffusion and so induce segregations (carbon or boron in steels) which pin dislocations
and lead to the formation of two yield points and Lüders’ bands. However, in these
experiments, pronounced microyielding did not appear again and thus, pinning effects
can be ruled out as a significant effect.
Therefore, a change in the deformation mechanism is the most plausible explanation
for the pronounced microyielding in alloy Haynes 282. For alloy Inconel 718, the mac-
roscopic stress - strain curve and also the microscopic lattice strain evolutions are well
described, when applying a simple power law relationship in the CPFEM model. For
alloy Haynes 282, it was necessary to modify the power law behavior into a first stage
with almost no hardening (large hardening exponent
n
) and a second stage with normal
hardening (
n =
1.75). With this change in the CPFEM model, it was possible to also
simulate discontinuities in the lattice strain evolution of specific grains (see Figure
4.24).
According to the theory of plastic deformation of polycrystalline metals by G.
Zankl [130], a change in the hardening behavior is observed when dislocations start to
pile-up inside the grains. At the onset of yield, dislocations do not interact with each
other and move on multiple slip systems to the grain boundaries, where they pile-up
(stage I). During ongoing deformation, dislocations start to accumulate also inside the
grains at around 1 % plastic strain, where they now interact with each other and cause
a significant increase in hardening (stage II).
When comparing the microstructure of Haynes 282 and Inconel 718 (see Figure 4.25),
it is obvious that there are much more obstacles for dislocations at the grain boundaries
but also inside the grains in alloy Inconel 718 (mainly
- phases). In contrast, in alloy
Haynes 282, the grains are almost free of any incoherent phases; only a small amount
of carbides is observed. The high density of incoherent phases in Inconel 718 could
be the reason, why no transition from stage I to stage II is observed. Incoherent
phases can only be bypassed by dislocations via the Orowan mechanism and thus,
act as sources for new dislocations and early dislocation interactions. Thereby, the
4 Studies on nickel-based superalloys
98
dislocation density inside the grains increases faster and leads to a one-stage harden-
ing behavior. This is also supported by the larger peak broadening of the reflection in
alloy Inconel 718 during plastic deformation which was shown in Figure 4.19. The same
behavior was already observed for steels [138]. In case of softer steels such as ferrite,
pearlite or upper bainite, a two- stage hardening behavior is observed, whereas harder
microstructures such as lower bainite and martensite, with more hard cementite
phases, exhibited a one-stage hardening behavior. This study motivates to consider
the complex hardening behavior of superalloys in physically (dislocation density) based
crystal plasticity models in future.
Figure 4.25: Comparison of the microstructure of Inconel 718 (a) and Haynes 282 (b): Alloy Inconel 718
exhibits a high density of plate like
- phases at the grain boundaries and also inside the grains; in
contrast, in alloy Haynes 282, only a few of the grain boundaries are decorated by M23C6-, M6C- and
MC- carbides. Further analysis by EBSD has shown that especially low energy grain boundaries (twin
boundaries) are free of carbide phases. These grain boundaries might have formed during recrystalli-
zation which could also explain why some “lines” of carbides exist within the grains.
99
5 Studies on iron-based superalloys
In this section, the results of the studies on the noncommercial Fe-based superalloy
FFB-8 are presented and discussed. The emphasis is on the characterization of hy-
perfine NiAl- type precipitates that form during sample cooling after the aging treat-
ments of the alloy and which have shown to strongly increase the alloys’ hardness at
room temperature. In the first part of this section, the results of mechanical tests of the
alloys’ room temperature performance are correlated to studies of the so called “cool-
ing” precipitates by electron microscopy and APT. For identification of these cooling
precipitates in APT reconstructions, the existing “maximum separation method” for
cluster selection is modified and now uses a Delaunay tessellation which avoids user
defined input parameters and, additionally, directly gives access to the morphology of
the clusters. The results summarized in this section, are published in the article “On
the formation of nano-sized precipitates during cooling of NiAl- strengthened ferritic
alloys” by Lawitzki et al. (2020) [E11]. While it is possible to obtain accurate composi-
tional information of these cooling precipitates by APT, the study has also shown that
the analysis of the exact size and shape of smallest precipitates by APT is not trivial,
since it can be strongly affected by the effect of local magnifications caused by a het-
erogeneous evaporation behavior of different phases. Therefore, numerical simula-
tions of the APT measurements are performed in chapter 5.2, to predict the recon-
structed morphology of precipitates with contrasting evaporation threshold. The simu-
lations are evaluated and a model is developed for the calculation of the original size
of the precipitates which is tested on the experimental APT data sets of alloy FBB-8.
These results are published in the article “Compensating local magnifications in atom
probe tomography for accurate analysis of nano-sized precipitates” by Lawitzki et al.
(2021) [E12].
5.1 On the formation of nano-sized precipitates after aging
The ferritic alloy FBB-8 (see Table 2.2) is a test alloy aiming for future structural appli-
cations in gas turbines capable of operating at 760 °C and steam pressures up to
35 MPa [3,139]. Due to the high thermal conductivity and low thermal expansion coef-
ficient of ferritic alloys, they have attracted wide interest as an economic alternative to
5 Studies on iron-based superalloys
100
Ni-based superalloys or austenitic steels. In comparison to commercial creep resistant
ferritic steels such as T91, P92, P122, X21CrMoV, 12Cr (which are strengthened by
solid-solution hardening and carbide- or nitride-type particle reinforcement) that can
operate up to ~620 °C, the NiAl-type precipitation hardened alloy FBB-8 exhibits su-
perior oxidation resistance and creep resistance at low stress levels up to 700 °C [140–
143]. But there are still two major concerns that restrict the application of the alloy at
these temperatures. On the one hand, the thermal instability of nano-sized precipitates
is still a major issue in ferritic alloys which significantly shortens the service life time at
high temperatures. Later studies by Song and Rawlings et al. [143–145] have shown
that this problem could be bypassed by the addition of Ti and the formation of hierar-
chical Ni2TiAl / NiAl- type precipitates which improve the creep resistance by several
orders of magnitude; but this is out of the scope of this work and will be not further
considered here.
The second major challenge is the alloys’ brittleness at room temperature. To ensure
sufficient workability of the as-cast material, steam turbine materials require a ductility
of more than 10 %, but alloy FBB-8 exhibits a ductility of around 1 % only [3]. Previous
articles have shown that the alloy’s ductility strongly decreases with higher amount of
B2-type precipitates [3,146]. It was further shown, that hyperfine secondary (cooling)
precipitates form during sample cooling following the aging treatment and that these
have a tremendous hardening effect. A huge contribution of the poor room temperature
ductility is attributed to the presence of these cooling precipitates [3,147–149].
Whereas the existence of the cooling precipitates in FBB-8 is proven, there exists al-
most no quantitative information about their chemical nature which was the primary
motivation for the studies presented in the following and which is required, to overcome
the problems caused by cooling precipitates in future alloy design.
The goal of this study is to clarify, at which temperature cooling precipitates form and
if they are of the same crystallographic structure and composition than the primary
precipitates formed during aging. Therefore, the alloy was subjected to isothermal heat
treatments in pre-heated tube furnaces in air at 600 °C, 700 °C, 800 °C and 900 °C for
up to 1000 h (until the volume fraction of the precipitates was in equilibrium in each
sample) and subsequently cooled in air. In the first part, the heat-treated samples are
characterized by electron microscopy and the results are correlated to the hardness of
5.1 On the formation of nano-sized precipitates after aging
101
the alloy. In the succeeding parts, the focus is on the compositional analysis by APT.
The evaluation of the APT data is performed by a modified cluster search algorithm
that is first tested on model datasets and then applied to the heat-treated specimens
of alloy FBB-8.
5.1.1 Correlation of mechanical alloy behavior and microstructure
As a measurement of the alloy’s mechanical behavior at room temperature, the Vickers
hardness of the differently heat-treated specimens is compared as a function of the
aging time in Figure 5.1. The first two data points of 432 HV and 491 HV represent the
initial hardness after the solution treatment at 1200 °C and subsequent cooling in air
(AC) or quenching in brine (WQ), respectively. During the following aging treatments,
the evolution of the hardness shows a strong dependency on the aging temperature.
When aging the WQ specimens at 800 °C and 900 °C, the hardness decreases con-
tinuously and levels out in a plateau; the faster the higher the aging temperature. In
contrast, an increase in hardness of the WQ specimens within the first ~30 min of aging
can be observed, when the aging is performed at 600 °C (up to 525 HV) or 700 °C (up
Figure 5.1: Hardness as a function of aging time of alloy FBB-8 heat-treated at different temperatures.
The hardness of the alloy after solution annealing and following air cooling (AC, red) is much lower in
comparison to the samples that were quenched in salt water (WQ). Lines were drawn to guide the eye
of the reader. (According to reference [E11])
5 Studies on iron-based superalloys
102
to 510 HV). This behavior is well known for e.g.
- precipitate [150],
- carbide [151]
and other (B2- type)
- precipitate strengthened ferritic steels [152,153] and is typically
correlated to the size of the precipitates. For ordered precipitates a maximum in hard-
ness is obtained, when the mechanism of dislocations shearing the precipitates
changes from weakly to strongly coupled, as was already introduced in section 2.2.1
“precipitation hardening”. If precipitates coarsen even more, the alloy is considered
over-aged resulting in a loss in strength. For alloy FBB-8, this behavior was not yet
reported so far; most probably, because the time span before over aging is very short.
An exciting feature when comparing the water quenched specimens after longest aging
times is that the hardness decreases with decreasing aging temperature (~425 HV at
900 °C, ~410 HV at 800 °C, ~390 HV and still decreasing at 700 °C) except for the
specimen aged at 600 °C. This behavior can only be interpreted by complementary
characterization of the microstructure. The results of the analysis by electron micros-
copy are presented in Table 5.1, Figure 5.2 and Figure 5.3.
As demonstrated in Figure 5.2 (a), a precipitation cannot be avoided, even by quench-
ing in iced brine (WQ), precipitates form and already have a radius of (6.6 ± 1.0) nm.
In contrast to this, in the air cooled condition these precipitates have already grown to
a size of (42 ± 13) nm explaining the significantly lower hardness of the AC specimen.
During the subsequent aging, these precipitates grow and are now denoted as primary
- precipitates. With increasing aging temperature, the primary precipitates coarsen
Heat treatment
Cooling precipitates
Primary
- precipitates
Mean radius
r
[nm]
Mean radius
r
[nm]
Volume
fraction [%]
Growth rate
k
[nm/h1/3]
WQ
Initial*
6.6 ± 1.0
-
-
-
600 °C 1000 h
-
25 ± 5
19.2 ± 0.3
2.5 ± 0.2
700 °C 50 h
1.7 ± 0.3
62 ± 17
15.5 ± 0.5
16.8 ± 1.5
800 °C 50 h
6.2 ± 1.2
162 ± 48
10.3 ± 0.3
44.0 ± 4.5
900 °C 50 h
11.7 ± 1.7
298 ± 62
5.9 ± 0.1
80.9 ± 8.0
AC
Initial*
42 ± 13
-
-
-
700 °C 50 h
-
70 ± 13
15.2 ± 0.3
17.4 ± 0.9
Table 5.1: Evaluated mean radii of cooling and primary precipitates in alloy FBB-8 after water quenching
(WQ) or air cooling (AC) from solution annealing treatment (initial) and after subsequent aging treat-
ments. Furthermore, the volume fraction and growth rate of primary precipitates were calculated. It
should be mentioned that the cooling precipitates (marked by *) have formed during cooling after the
solution treatment, whereas the others formed during cooling after the aging treatment.
5.1 On the formation of nano-sized precipitates after aging
103
(cf. size distributions in Figure 5.2g) and their volume fraction decreases (see Figure
5.2h). The growth rate of primary
- precipitates is described by Ostwald ripening (Eq.
[2.7]) and shows an exponential increase with increasing aging temperature. In the
comparison with typical growth rates of e.g.
- precipitates in Ni-based superalloys,
these growth rates are almost one order of magnitude higher [154,155], which explains
why the hardness in the ferritc alloy FBB-8 starts to decrease so early during aging.
The reason for this is the higher diffusivity (due to lower atomic packing) in bcc alloys.
After aging, DF TEM images of the aged specimen reveal additionally small cooling
precipitates (see Figure 5.2 (c,d) and Figure 5.3 (b)). The size and amount of these
600 650 700 750 800 850 900
0
20
40
60
80
Aging temperature [°C]
Growth rate [nm/h1/3]
4
8
12
16
20
Volume fraction
Growth rate
Volume fraction [%]
WQ
AC
800 °C
50 h
900 °C
50 h
Ni
Al
Mo
Cr
Fe
Zr
Ni+Cr+Zr
DF
STEM
EDX- map
800 °C; 50 h
800 °C
50 h
10 100 1000
0
4
8
12
16
20
24 900°C 50h
800°C 50h
700°C 50h
600°C 1000h
Number fraction [%]
Diameter [nm]
WQ
a)
c)
e)
b)
d)
f)
g)
h)
Figure 5.2: Microstructure of the heat-treated specimens: (a-d) dark field (DF) TEM images of the
samples, directly after the solution annealing (a) and (b), and after different subsequent aging
treatments (c) and (d) (DF on a [011] superlattice reflection in two-beam excitation close to a [001]-
zone axis); (e) STEM elemental EDX-map and annular DF image of the sample heat treated at
800 °C for 50 h; (f) larger backscatter SEM micrograph as used for the quantification of primary
NiAl- type precipitates; (g) particle size-distributions for the water-quenched (WQ) and subse-
quently heat treated specimens; (h) exponential growth rate dependency and volume fraction of
the precipitates after WQ as a function of the aging temperature. (According to reference [E11])
5 Studies on iron-based superalloys
104
cooling precipitates is strongly correlated to the temperature of the aging treatment.
Larger and more of these precipitates form when the samples are cooled from higher
temperatures (see also Table 5.1). Since primary precipitates are very coarse in the
aged specimen, mainly the cooling precipitates contribute to alloy hardening and this
is the reason why the hardness of the alloys increases after long aging times at higher
temperatures (see Figure 5.1).
For the samples aged at 600 °C, the situation is different. Here, no cooling precipitates
could be detected by TEM and also the diffraction patterns of areas between primary
precipitates showed no additional reflections beside α- matrix reflections (see Figure
5.3a). Thus, for the samples aged at 600 °C the comparably rather small primary pre-
cipitates still control the hardness.
By nano-diffraction, the crystallographic structure of the cooling precipitates in the
specimens aged at 800 °C and 900 °C could be proven to be the B2 structure. Re-
markably, in the diffraction patterns of the specimen after aging at 700 °C, additional
reflections were observed indicating the presence of a different phase. With the matrix
aligned in [112]- zone axis direction (cf. Figure 5.3 b), this phase could be indexed by
using the software package JEMS [60] as NiAl3 (Pnma with a = 0.648 nm, b = 0.719
nm and c = 0.406 nm) existing in two different orientation relations. Orientation 1: {112}
A2 // {120} Pnma and {1-10} A2 // {002} Pnma; orientation 2: {112} A2 // {012} Pnma
and {1-11} A2 // {200} Pnma.
(020)
(010)
A2 Fe [001]
10 nm
(
10)
(
)
B2 NiAl
[112]
NiAl3 [120] and [012]
(020)
B2 NiAl [001]
a)
b)
Figure 5.3: DF TEM images of samples after heat treatment at 600 °C for 1000 h (a) and at
700 °C for 425 h (b). The quadratic (blue) inset is a magnified image with modified contrast
highlighting the cooling precipitates. Nano-diffraction patterns of a single primary precipitate
and the area in between those are given in the circular (red) inset figures labeled with the
crystal structure and zone-axis. (According to reference [E11])
5.1 On the formation of nano-sized precipitates after aging
105
5.1.2 Introduction and test of a modified cluster search algorithm
Whereas it is possible obtain qualitative information about the chemical nature of pre-
cipitates by analytical electron microscopy such as STEM EDX (cf. Figure 5.2e), these
methods are unsuited to obtain accurate quantitative information about the composi-
tion of small precipitates. If the precipitates are small, the EDX signal of the precipitate
will always be an irrecoverable convolution with the signal of the surrounding
-
matrix.
In this case, APT is the best technique that is capable to obtain accurate compositional
information; but the evaluation of the APT data sets, often containing up to several
hundred million atoms, is not trivial. The challenge is to localize and extract the infor-
mation of smallest precipitates/clusters, which are basically small accumulations of at-
oms of a specific type.
As was introduced in section 3.2.3 cluster analysis, the best techniques to localize
smallest precipitates in large data sets, are cluster identification algorithms such as the
maximum separation method. This method relies at least on the two (but usually on
three) user defined parameters
k
,
dmax
(and
Nmin
) (cf. Figure 3.12), that have to be
selected carefully because they can significantly modify the outcome of the analysis
[156–158]. Various possibilities for an objective selection of these parameters have
benn proposed [80,83,159]. In this work, the critical distance
dmax
and parameter
k
will
be derived from the distribution of nearest neighbor distances (NND) (see Figure
3.11d,e). A disadvantage of the maximum separation method is that the algorithm er-
roneously always includes a thin shell of atoms over the cluster surface that has to be
removed for an accurate evaluation. To remove this shell, usually a further parameter,
the so called erosion distance, is chosen to remove those cluster atoms which are
nearest to the matrix [75,76].
In the following, a new methodology to remove this additional shell around the cluster
is introduced which is demonstrated in Figure 5.4. It avoids the definition of any addi-
tional parameter such as the erosion distance. The example shows a slice through an
APT data set including a cluster (a) that is identified and extracted with the maximum
separation in step I. The extracted cluster, shown in (b), still contains the erroneous
5 Studies on iron-based superalloys
106
thin shell of matrix atoms. To identify those atoms, a Delaunay triangulation is per-
formed to all solute atoms within the cluster (in this work the TetGen library was imple-
mented for this step [160]). The Delaunay tessellation calculates the coordinates of the
solute atoms that form the convex hull of the clusters in step II as shown in (c). In the
last step, all atoms located outside the convex hull are removed. The two most im-
portant advantages of this method are that it does not require a further user defined
input parameter for the erosion step and furthermore, the construct of the convex hull
represents directly the shape of the clusters (as long as the real precipitate is of convex
shape, which is however likely since interface area is avoided). This enables to calcu-
late a proximity histogram (proxigrams) of the elemental distribution in shells of a cho-
sen thickness around the cluster. The performance of this modified algorithm is tested
on two numerically generated atom maps with well-defined cluster distributions. In the
model datasets, atoms are positioned on an arbitrary oriented bcc lattice with the lattice
parameter of FBB-8 of 0.2886 nm. To keep the situation in the model datasets compa-
rable to real APT measurements, a realistic detection efficiency is considered by ran-
domly removing 50 % of the atoms and a limited spatial resolution is modeled by ran-
domly shifting the atoms laterally by ± 0.3 nm and in depth by ± 0.05 nm in half width.
Figure 5.4: Scheme of the modified cluster search algorithm: Clusters in an APT data set are localized
with the maximum separation method in the
I. step. In step II,
a Delaunay tessellation is performed only
to the solute (red) atoms to identify those solute atoms that form the convex hull of the cluster. In step III,
all atoms outside the convex hull are removed and only the cluster atoms are left.
5.1 On the formation of nano-sized precipitates after aging
107
Example 1: Accuracy and optimization of choice of
d
max and
k
For the first example, a model emitter containing 5 at.-% impurity (Ni) atoms (in a Fe-
matrix) was set up (see Figure 5.5c). Then, six clusters with a mean concentration of
40 at.-% Ni and a radius of (5.5 ± 1.5 nm) were randomly distributed. This model emitter
resembles the situation in alloy FBB-8 after cooling from 800 °C. Clusters are searched
with the proposed modified cluster search algorithm. In order to identify the input pa-
rameters for cluster search that yield best results, the parameters
d
max and
k
are varied
and the evaluated fraction of atoms belonging to clusters and the Ni- concentration
inside the clusters is compared with the preset input values of the model structures (cf.
Figure 5.5 a,b).
The critical distances
d
max are chosen based on the distance distributions between the
3rd, 5th, 10th, 15th and 20th nearest neighbors (KNN) of Ni atoms in the generated struc-
ture. As shown in Figure 3.11 (d,e), each of these double-peaked distributions can be
described by a superposition of two Gaussian functions. The upper bound of
d
max (blue
data points) was chosen at the interception of both Gaussians, whereas the lower
bound of
d
max (red data points) at the maximum of the first distribution stemming from
the distances within the clusters.
Figure 5.5: Test 1 of modified cluster search algorithm: Identification of (a) total cluster atoms and (b)
the Ni concentration within the clusters of a numerically generated atom map by the maximum separa-
tion method based on selected nearest neighbor orders
k
. The value of
d
max has been either chosen at
the maximum of the first Gaussian fit (red) or at the intercept between both Gaussian peaks (blue) to
the distribution of nearest neighbor distances. Comparison of these characteristic parameters with the
preset input (solid black) show that clusters are most accurately detected when selecting a
d
max close to
the maximum of the first Gaussian for neighbor orders
k
≥ 10. (c) visualization of numerically generated
model tip. (According to reference [E11])
5 Studies on iron-based superalloys
108
The evaluation of Figure 5.5 shows that the optimum value of
d
max changes with the
number of selected nearest neighbors
k
. For smallest values of
k
, best results are
obtained when localizing
dmax
at the intercept of the two Gaussians, whereas for values
of
k
larger than 10, accurate results are obtained, when choosing
d
max close to the
maximum of the first Gaussian.
Example 2: Ellipsoidal shaped core shell clusters
In the second model tip, 20 ellipsoidal shaped clusters with 50 at.-% Ni, a half axis ratio
of 1:2 and an equivalent spherical mean radius of 1.2 nm are distributed in a matrix
containing 5 at.% Ni (cf. Figure 5.6 a). In this example, each cluster is additionally
surrounded by a 0.4 nm thick shell with a Ni concentration of 75 at.-%.
The maximum separation algorithm was performed with a critical distance of 0.37 nm
for the 10th nearest neighbors. As shown in Figure 5.6 (b), all 20 numerically generated
clusters are successfully detected with this choice of parameters.
Figure 5.6 (c) demonstrates the benefit of triangulating the cluster surface when chem-
ical information of a fine structure have to be determined. In the shown diagram, the
black dotted line is the input Ni concentration in vicinity to the cluster surfaces of the
model dataset. This curve is compared to the proxigrams after cluster search calcu-
lated by two different approaches. The proxigram marked by red color is created by
Figure 5.6: Test 2 of modified cluster search algorithm: Numerically generated dataset with 20 ellipsoidal
shaped core-shell clusters with a core containing 50 at.-% Ni and a 0.4 nm thick shell containing
75 at.-% Ni: (a) visualization of the numerically generated tip containing clusters (red dots); (b) detected
clusters after use of the maximum separation algorithm and visualized by their triangulated convex hulls;
(c) proxigram of all clusters calculated by assumption of a spherical shape (red) and by preserving the
detected shape of the clusters (blue) in comparison to the input of the generated dataset (black dotted
line). (According to reference [E11])
5.1 On the formation of nano-sized precipitates after aging
109
calculating the elemental concentration in concentric spherical shells around the center
of gravity of each cluster. This is the conventional approach, when the shape of the
cluster is not known. In contrast, the proxigram marked by blue color makes use of the
triangulated real shape of the clusters and the concentration is calculated in shells
retaining the shape of the clusters. Obviously, only the second approach is suited to
describe the fine chemical structure of the generated clusters. In the demonstrated
example, the average input cluster size (1.53 ± 0.10) nm almost coincides with the
evaluated size by triangulation of the cluster surfaces (1.54 ± 0.11) nm and the thin
shell is clearly resolved.
5.1.3 APT analysis of alloy FBB-8
The results of the experimental evaluation of local compositions of the differently heat
treated tips by APT (and partly TEM EDX) are summarized in Table 5.2. The analysis
of the cooling precipitates was performed with the modified cluster search algorithm
Heat treatment
Fe
Ni
Al
Cr
600 °C
Matrix
80.9 ± 0.9
1.2 ± 0.1
5.5 ± 0.3
11.6 ± 0.7
(78.5 ± 1.7)
(2.6 ± 0.2)
(5.8 ± 0.2)
(11.9 ± 0.6)
Primary precipitates
5.7 ± 0.4
46.5 ± 2.0
46.0 ± 2.4
0.6 ± 0.1
(5.6 ± 0.4)
(45.1 ± 2.0)
(46.7 ± 2.5)
(0.7 ± 0.1)
Cooling precipitates
-
-
-
-
700 °C
Matrix
78.5 ± 0.4
2.9 ± 0.8
6.6 ± 0.6
11.2 ± 0.3
(75.0 ± 1.4)
(4.9 ± 1.0)
(7.3 ± 0.8)
(11.3 ± 0.8)
Primary precipitates
9.6 ± 0.4
45.3 ± 1.8
44.1 ± 1.0
0.6 ± 0.1
(9.2 ± 0.4)
(46.1 ± 1.7)
(43.6 ± 1.1)
(0.6 ± 0.1)
Cooling precipitates
19.1 ± 1.0
40.2 ± 2.1
39.4 ± 1.8
1.3 ± 0.1
800 °C
Matrix
77.2 ± 0.3
2.9 ± 0.5
7.2 ± 0.7
11.6 ± 0.7
(73.3 ± 1.1)
(5.7 ± 0.7)
(7.8 ± 0.5)
(10.9 ± 0.7)
Primary precipitates*
21.7 ± 1.1
37.8 ± 1.0
39.1 ± 1.8
0.9 ± 0.3
Cooling precipitates
10.0 ± 0.8
44.2 ± 1.4
45.2 ± 0.9
0.7 ± 0.1
900 °C
Matrix
76.1 ± 1.8
3.5 ± 0.8
8.2 ± 0.6
11.1 ± 0.7
(73.2 ± 2.7)
(6.6 ± 0.7)
(7.9 ± 0.7)
(10.7 ± 0.7)
Primary precipitates*
26.8 ± 0.8
35.6 ± 0.4
35.7 ± 0.6
1.7 ± 0.1
Cooling precipitates
13.7 ± 1.6
47.0 ± 1.7
37.2 ± 2.9
1.1 ± 0.2
Table 5.2: Measurement of the composition of different phases in FBB-8 by APT (or TEM EDX in the
case of *). The concentration of Fe, Ni, Al and Cr in primary
β
’- precipitates and in the precipitate free
matrix phase was obtained by evaluation of mass spectra after peak decomposition and background
correction (data before background correction are given in brackets below). The composition of cooling
precipitates (when present) was obtained from proxigrams after cluster analysis. Zr, Mo and B contribute
to other phases and are therefore not presented.
5 Studies on iron-based superalloys
110
which is demonstrated in Figure 5.7 exemplarily for two cases. First, small regions
containing one (a) or several clusters (d) are chosen for the cluster analysis, then the
nearest neighbor distributions are calculated (b,e) and finally the proxigrams are cal-
culated (c,e). For the sample revealing the smallest precipitates (after cooling from
700 °C), the consideration of the 10th nearest neighbors was necessary to unambigu-
ously distinguish the two contributions of precipitates and matrix (e). For consistency,
all samples have been therefore analyzed evaluating the distances of the 10th nearest
Ni neighbor atoms with
dmax
at the maximum of the first Gaussian (according to Figure
5.5) and the additional requirement that at least 30 Ni atoms (
Nmin
) are necessary to
be considered as a cluster.
The compositions of the cooling precipitates presented in Table 5.2 are obtained by
fitting the proxigrams with error functions and evaluating the fitted concentrations at
the cluster cores. Agreeing with the TEM results, no cooling precipitates could be de-
tected after heat treatment at 600 °C. The compositions of the matrix phase is evalu-
ated from the same analysis volumes after extraction of the cluster atoms. For compo-
sitional analysis of large primary precipitates (after heat treatment at 600 °C and
700 °C), local mass spectra of regions within the precipitates are directly analyzed. All
presented compositions in Table 5.2 are corrected for peak overlap of ambiguous
peaks. Additionally, a subtraction of the background was performed for the composi-
tions of matrix and primary precipitates (see also section 3.2.3 mass spectrum analysis
and for further details in reference [E11]).
Since the APT data sets of the specimens after heat treatment at 800 °C and 900 °C
did not contain any primary precipitates (due to a lower number density), data obtained
by TEM EDS are presented in Table 5.2 instead.
Unexpected results of the analysis of cooling precipitates
The comparison of the clusters formed after heat treatment at 700 °C with those formed
at higher temperatures reveals two interesting features.
The first unexpected observation is demonstrated in the local density maps in Figure
5.7 (a,d); the atomic density inside the clusters is lower (~35 atoms/nm³) than the den-
sity inside the matrix (~55 atoms/nm³) when cooling from 800 °C. In contrast, after
5.1 On the formation of nano-sized precipitates after aging
111
cooling from 700 °C, the atomic density inside the precipitates is higher (up to 125 at-
oms/nm³) than in the matrix. Since
- precipitates and
- matrix have a very small
lattice mismatch in alloy FBB-8 [140], a density difference in the APT reconstruction is
first of all not expected and secondly, should be similar for both temperatures. The only
phenomenon that can explain such a behavior are so called local magnification effects
which are caused by phases with heterogeneous evaporation behavior. Local magni-
fication effects and the consequence of these artefacts on APT reconstructions were
Figure 5.7: Comparison of the cluster analysis on FBB-8 of an extracted box containing one cluster of
a sample heat treated at 800 °C (a-c) and an analysis box containing several clusters for a sample
heat treated at 700 °C (d-f). From top to bottom are compared the reconstruction visualized as atom
and density maps (a) and (d), the neighborhood distributions (b) and (e) and the results from cluster
analysis by visualizing the proxigram of Ni, Al (blue and red error function fit) and the density trend
(dashed line) (c) and (f). (According to reference [E11])
5 Studies on iron-based superalloys
112
introduced in section 3.2.1 and are discussed in detail in section 5.2. Typically, phases
with a higher evaporation field form protrusions on the emitter surface during field emis-
sion which results in a lateral expansion of the phase in the APT reconstruction. Vice
versa, a comparably lower evaporation field causes a local compression of this phase
in the reconstruction.
Transferring this to alloy FBB-8, the clusters formed after aging above 800 °C exhibit
a higher evaporation field and these, formed after aging at 700 °C, exhibit a lower
evaporation field compared to the evaporation field of the matrix. Possibly, the clusters
formed by cooling from 700 °C have a different crystal structure/composition which
would influence the chemical bonds and thus, also the evaporation field. The observa-
tion of different superlattice reflections in electron diffraction that could be indexed with
the Al enriched phase NiAl3 in Figure 5.3 (b) supports this hypothesis. A further expla-
nation could be a compositional variation in Al amount, which naturally has a lower
evaporation field (19 V/nm) than Fe (33 V/nm) or Ni (35 V/nm).
The second unexpected result is obtained, when the amount of Ni and Al inside the
clusters is compared. When only the composition inside the cluster cores is consid-
ered, the concentration of Ni and Al is almost the same after cooling from 800 °C and
700 °C; but when all atoms are considered that are located within the proximated sur-
face of the clusters, the situation is different: For the cluster formed after heat treatment
at 800 °C (cf. Figure 5.7 a,c) the amount of Al is roughly 3 at.-% higher (37.2 at.-% Al
vs. 34.1 at.-% Ni). After cooling from 700 °C (cf. Figure 5.7 d,f), the clusters contain
35.7 at.-% Al and 28.5 at.-% Ni and thus, the Al content is even ~ 7 at.-% higher. In
this example, the difference in the Al and Ni content is even more significant, when
reading the local concentrations directly at the proximated convex hull (distance of
0 nm in the proxigrams in Figure 5.7 (f) – 27.6 at.-% Al vs. 17.2 at.-% Ni – which cor-
respond to a ratio of ~3:2 (Al:Ni). It can be concluded that the cooling precipitates ex-
hibit an Al- enrichment in the interphase region to the matrix. The width of this inter-
phase region (~1.0 – 1.4 nm) seems to be independent of the heat treatment temper-
ature. Thus, the composition of smaller precipitates is more influenced when including
the atoms in the shell for the evaluation. The enrichment of Al in the interphase region
might trigger the change in the crystal structure which is indicated in the TEM results
where the Al enriched phase NiAl3 was detected (cf. Figure 5.3b).
5.1 On the formation of nano-sized precipitates after aging
113
5.1.4 Discussion
The measurements of the chemical composition of the precipitates in the alloy FBB-8
by TEM EDX and APT have shown (cf. Table 5.2) that there is a strong correlation
between the aging temperature and the chemical nature of the precipitates. To com-
pare the results, the composition of primary and secondary cooling precipitates are
compared in Figure 5.8 (a,b) in dependence of the aging temperature. The results are
additionally included in the isothermal section of the Fe-Ni-Al ternary phase diagram
at 750 °C reported by Eleno et al. [161]. The composition of the primary precipitates
agrees well with the phase diagram. For higher aging temperatures, the solubility of Fe
increases in the B2 phase and decreases in the A2 phase. This explains why most of
the iron is found in the primary precipitates after aging at 900 °C. It also explains, why
more cooling precipitates can form when cooling from a higher temperature (see Figure
Figure 5.8: Composition of primary (a) and secondary (cooling) (b) precipitates in dependence of the
aging temperature (for secondary core-shell precipitates after heat treatment at 700 °C, the concentra-
tion including the shell is indicated as dashed lines); (c) determination of the volume fraction of precipi-
tates by comparing the difference between the total alloy concentration
c0
and the solute concentrations
in the
- matrix
c
versus the difference between the concentration of the precipitated phase
c
and the
- matrix
c
(lever rule); (d) ternary phase diagram of FeNiAl at 750 °C by Eleno et al. [161] comple-
mented with the measured compositions of the phases: average alloy (star), matrix (crosses) and pre-
cipitates (spheres, triangles) measured by APT after heat treatment at 600 °C (blue), 700 °C (green),
800 °C (brown) and 900 °C (red). (According to reference [E11])
5 Studies on iron-based superalloys
114
5.8 (c), further information and the respective equations are provided in the article
[E11]). Furthermore, the comparison of the Fe content of primary and cooling precipi-
tates allows the conclusion at which temperature cooling precipitates have formed. It
can be followed that most of the cooling precipitates form at ~700 °C when cooling
from 800 °C, and at ~750 °C when cooling the samples from 900 °C and only after
cooling from 700 °C, the Fe content in the cooling precipitates is higher than expected
which might indicated that the precipitation of phases during cooling is not a thermo-
dynamically but kinetically controlled process.
The same conclusion can be drawn when comparing the ratio of Ni to Al atoms inside
the precipitates. For primary precipitates (see Figure 5.8 a) the same amount of Ni and
Al atoms are found within the precipitates independently of the aging temperature. Re-
markably, in the cooling precipitates (see Figure 5.8 b), the ratio between Ni and Al
changes from being enriched in Ni after cooling from 900 °C to being in enriched in Al
(when including the atoms in the shell) after cooling from 700 °C.
Possibilities to explain this behavior can be drawn by the comparison to the partial
diffusivities of Al and Ni in
- iron. These are affected by the magnetic transition tem-
perature which is at 768 °C for pure iron. In the paramagnetic regime, the partial diffu-
sivities of Al and Ni are similar (e.g. at 800°C:
D
Al = 1.810-15 m2/s [162] and
D
Ni =
2.410-15 m2/s [163]). In contrast, in the ferromagnetic regime, e.g. at 700 °C, the diffu-
sivity of Ni (
D
Ni = 3.810-18 m2/s [163]) typically slows down by more than two orders of
magnitude due to ferromagnetic alignments [163,164] in comparison to the diffusivity
of Al (
D
Al = 6.410-16 m2/s [162]). For alloy FBB-8 the Curie temperature is 644 °C, which
could be measured by a quantum design MPMS®3 (see Appendix D Figure D.1). This
means for cooling from 700 °C or lower temperatures, precipitation of cooling precipi-
tates happens mainly in the ferromagnetic regime.
In order to check if the difference in partial diffusivities can influence the precipitation
behavior, the diffusion lengths
of Ni and Al have to be estimated. The samples aged
at
T0
=700 °C cooled down to a temperature
T
of 50 °C within a time
t
of ~45 s, corre-
sponding to an exponential decay time
0
of 17 s, which can be described by
,
[5.1]
Now, the diffusion lengths
are obtained by integration over the cooling time
5.1 On the formation of nano-sized precipitates after aging
115
,
[5.2]
where
D0
is the theoretical maximal diffusion coefficient,
QA
the activation energy for
diffusion and the
R
is the universal gas constant. Applying Eq. [5.2] for the diffusivities
of Al and Ni in the ferromagnetic region, the diffusion lengths of Al and Ni when cooling
from 700 °C are calculated to 27 nm and 2.3 nm, respectively.
The diffusion lengths have to be compared to the distances that matrix atoms have to
migrate to form a cluster of a given size and composition. From the average numbers
of Al and Ni atoms inside a cluster at 700 °C (
NAl
= 513 and
NNi
= 424) and the con-
centrations of Al and Ni in the matrix (7.6 at-% and 3.8 at.-%) before the formation of
cooling precipitates, these required transport distances
i
are estimated as
,
[5.3]
where
a0
= 0.2886 nm is the lattice constant of alloy FBB-8 [140]. For Al, the required
distance is 2.7 nm, which is obviously smaller than the diffusion length (27 nm). In
contrast, for Ni, the required distance is 3.2 nm, but the diffusion length is only 2.3 nm.
This explains, that the diffusion of Ni is kinetically limited and eventually leads to a
depletion of Ni in the shell of the cooling precipitates. This depletion is so strong that it
even enforces the change of the B2 structure to a Pnma (NiAl3) structure, as observed
by electron microscopy Figure 5.3 (b). The kinetic limitation in the Ni diffusion further
explains the observation (shown in Appendix D Figure D.2), why especially small clus-
ters, which nucleated at the lowest temperatures, tend to have the lowest Ni to Al ratio.
When cooling from 800 °C and 900 °C, the formation of most of the cooling precipitates
happens above the Curie temperature where the diffusivities of Al and Ni are similar.
Therefore, the Ni enrichment in the cooling precipitates after cooling from 900 °C can-
not be explained by asymmetric diffusion. Possibly, coherency strains hinder the pre-
cipitation process. Pike et al. [165] have shown, that the lattice parameter of B2-phase
increases with the relative amount of Al and that for an alloy containing 40 at.-% Al,
48 at.-% Ni and 12 at.-% Fe, the lattice parameter of the B2- phase in this alloy
(0.2884 nm) is very close to the matrix lattice parameter of alloy FBB-8 (0.2886
nm) [140]. Thus, an increase in the relative Al content would increase coherency
strains leading to a higher nucleation barrier.
5 Studies on iron-based superalloys
116
5.2 Effect of local magnifications in APT on nano precipitates
In the previous section, it was shown that the APT technique is well suited for the
compositional analysis of nano-sized precipitates in the alloy FBB-8. On the other
hand, it was revealed that the multiphase nature of the material causes distortions or
local magnification artefacts in the APT reconstruction (see Figure 5.7). The main rea-
son is that the APT measurement data were reconstructed following a simple point
projection model assuming a hemispherical apex (see also section 3.2.1); but in reality,
the shape of the apex changes during field evaporation if phases with varying evapo-
ration threshold are present. While the composition of these phases in the reconstruc-
tion may not be severely affected, the size and morphology of precipitates can deviate
significantly [166]. This is also the reason why it is often not recommended to apply
APT to obtain accurate size information.
Attempts to improve the reconstruction protocol to account for local magnification ar-
tefacts during the measurement are rare so far. For example there is the iterative re-
construction protocol by de Geuser et al. [167], which accounts for variations in the
evaporation behavior of different phases after their identification and thereby, homog-
enizes the volume laterally. In 2015, Rolland et al. suggested an analytical model to
describe the field evaporated emitter end-form by assumption of a fixed mean curva-
ture and applied the model for tips of axial symmetry such as bilayers and multilayers
[168,169]. Based on these concepts, they proposed a reconstruction protocol for tips
with stacked multilayer structures two years later [170]. Most recently, Beinke et al.
proposed a concept that can be applied to sample features of almost arbitrary shape
and furthermore, the dimensions of these features, in contrast to the concept by Rol-
land et al., do not need to be known before the reconstruction [171,172]. In this ap-
proach, the real shape of the emitter is directly extracted from the density of the events
on the detector.
Instead of developing improved reconstruction protocols, researchers also started to
predict the emitter shape by simulations of the evaporation process and from these,
identify and understand the artefacts in experimental data caused by local magnifica-
tions [173–177] (see also section 3.2.2). In comparison to the conventional reconstruc-
tion approach, the computation time of these simulations is extensive and thus, it is a
5.2 Effect of local magnifications in APT on nano precipitates
117
very time consuming procedure to derive conclusions from the simulations. Therefore,
it would be more beneficial to still use the point projection protocol and apply valid
quantitative corrections to the geometric parameters that were determined from an er-
roneous reconstruction.
In this section, numerical simulations are performed with the versatile simulation pack-
age TAPSim to study the effect of varying evaporation thresholds on the size, shape
and atomic density of precipitates reconstructed with the conventional point projection
approach. For analysis, a statistical approach is developed and with this, a model is
derived to calculate the original size of precipitates. This model is tested on the exper-
imental APT data obtained from alloy FBB-8. The ferritic alloy is found as an ideal test
material for this study, since the spherical NiAl-type cooling precipitates exhibit a dif-
ferent evaporation behavior depending on the aging temperature (see Figure 5.7), and
thus are differently affected by local magnification effects.
5.2.1 Set up of the numerical simulations
For the APT simulations, needle shaped emitters with a radius of 9 nm, a length of
35 nm and a shaft angle of 5° were set up. The tips were filled with atoms on a bcc
mesh with the lattice parameter of alloy FBB-8 (0.2886 nm). In total, this made approx-
imately 0.8 million grid points (or ~1.5 million grid points when additionally considering
surrounding support points). To model the effect of the limited spatial resolution of APT
and to avoid massive crystalline faceting, all atoms were randomly shifted laterally by
± 0.2 nm and in depth by ± 0.05 nm in half width. To match the microstructure of the
field emitters to the experimental situation in the ferritic alloy, 5.8 at.-% of the atoms
were randomly assigned as solute atoms. Six tips containing one centralized spherical
precipitate but with different radii (1.2, 1.7, 2.2, 2.7, 3.2 and 4.1 nm) were chosen.
Insight the precipitate, 44.0 at.-% of the atoms were assigned as solute atoms. With
the simulation package TAPSim, the emitters were then field evaporated by selecting
varying contrast in the evaporation thresholds between matrix (
E
= 20.0 V/nm) and
precipitate atoms (
E
= (16.0, 17.2, 20.0, 22.8, 24.0, 25.2 or 27.3) V/nm). This made
in total 42 simulations. The simulated two dimensional detector data were finally re-
constructed with a reconstruction protocol by Jeske and Schmitz [69] according to the
initial shape of the emitters (see Appendix E Figures E.1 and E.2).
5 Studies on iron-based superalloys
118
5.2.2 Evaluation of the morphology of precipitates
The analysis of the size of a nano-sized precipitates in an APT reconstruction is not
straight forward, since it can be easily over- or underestimated if too many or too few
atoms are considered as belonging to the precipitate. In most datasets, it is not clear
where exactly the interface between precipitate and matrix is located, even if there are
no distortions by local magnification artifacts. Common techniques to evaluate a pre-
cipitate size are for instance from the volume within an iso-surface or by cluster search
and thus, depend strongly on user defined parameters such as the coarse-graining
process [178] or the chosen critical distance
dmax
, respectively (see also section 5.1.2
Figure 5.5) .
A statistical approach for the size analysis
In order to avoid the requirement of a user defined interface, the precipitate size is
evaluated statistically by calculation of the radius of gyration from the coordinates of
all ( constituent atoms belonging to the precipitate by
,
[5.4]
where is the distance of an individual atom from the center of mass
[179,180]. As long it is clear which atoms belong to the precipitate, Eq. [5.4] can directly
be evaluated. Unfortunately, in most of the experimental cases, matrix and precipitates
consist of the same atomic species only with different compositions and therefore, pre-
cipitates cannot be extracted directly from the APT data. This problem may be solved
by calculation of the solute excess atoms
B
. Considering a volume consisting of
A
and
B
atoms and a matrix with a certain concentration
c0,B
of solute atom
B
, the number of
excess atoms
B
due to an embedded precipitate can be calculated by
,
[5.5]
where
NA
and
NB
are the total number of
A
and
B
atoms, respectively. This means,
Eq. [5.4] can now be generalized so that the radius of gyration
rg
is obtained without
prior identification of the atoms belonging to the precipitates by
.
[5.6]
Accordingly, the centre of mass is given by
5.2 Effect of local magnifications in APT on nano precipitates
119
.
[5.7]
The radius of gyration is transformed into the more intuitively interpreted isotropic
radius of a sphere by
,
[5.8]
which is an approximation by Guinier exactly valid for homogeneous spherical precip-
itates [181]. To solve Eqs. [5.4]-[5.8], the solute concentration
c0,B
in the matrix phase
has to be determined first. An elegant way for this is, to derive
c0,B
from the atoms in
vicinity of the precipitate by expressing Eq. [5.5] as a function of an offset
c0,B
[5.9]
,
[5.10]
where
Ntot
is the total number of atoms in the analysis volume including the precipitate.
As demonstrated in Figure 5.9, the linear relation in Eq. [5.10] may be evaluated for
differently sized volumes around one precipitate (a), and thus different Ntot. For detec-
tion of the precipitate, the modified maximum separation algorithm, as introduced in
section 5.1.2, was utilized. The graph in Figure 5.9 (b) presents the calculated bulk
excess
Bcalc
for differently chosen solute concentrations
c0,B
. Since the excess must
be independent of the chosen volume, the correct excess and solute concentration are
Figure 5.9: Method to determine the number of excess atoms B belonging to a precipitate in an APT-
dataset: (a) Atom probe reconstruction of the model tip showing only
B
atoms containing a precipitate
with an original radius of 2.2 nm. For illustration, precipitate atoms were identified by cluster search and
triangulated to present their convex hull (red). Volumes of three arbitrary shells around the precipitate
(
Ch
+
x
) were cropped and used for generating the diagram in (b): Calculated excess atoms
Bcalc as a
function of the bulk concentration (
c
0,B
+
c
0,B) for three differently sized shells (red solid lines) merge in
one point that represents the correct choice. (According to reference [E12])
5 Studies on iron-based superalloys
120
found at the crossing point of the linear graphs. In principle, any volume fully surround-
ing the precipitate could be chosen, but one has to carefully avoid including parts of
other precipitates in this volume.
Analysis of aspect ratios and density ratios
It was already shown that spherical precipitates with a varying evaporation field com-
pared to the matrix phase appear typically vertically or laterally compressed and with
a different density in the APT reconstruction (cf. Figure 5.7 a,d). To track these
changes in morphology of the reconstructed precipitates, the terms aspect and density
ratio will be used in the following. The aspect ratio is calculated according to
Eq. [5.6], when evaluating the radii of gyration in -, - and - direction separately and
assuming axial symmetry by
.
[5.11]
The density ratios between reconstructed precipitates (β
) and the surrounding
-
matrix phase are obtained by
.
[5.12]
In order to calculate the densities inside the precipitates and inside the matrix in
vicinity of to the precipitates , concentric shells are positioned around the centre of
mass of the precipitate (Eq. [5.7]) and their atomic densities are determined. and
can now be obtained by extrapolating these densities to the centre and far off the
precipitate.
5.2.3 Model to correct the cluster morphology in APT reconstructions
The statistical approach to evaluate the morphology of clusters is now applied on the
numerical simulated APT measurements. Figure 5.10 shows phase and atomic density
maps of the model emitter and the reconstructed data after field evaporation in de-
pendence of the ratio in evaporation thresholds
,
[5.13]
between precipitate (pure ) and matrix phase (pure ). For transparency, the results
shown here, present only the emitter containing the precipitate with an initial radius of
5.2 Effect of local magnifications in APT on nano precipitates
121
2.2 nm. The results of all 42 performed simulations and additionally atom maps are
compared in Appendix E Figures E.1-E.3.
In the first view, the results confirm what other authors have already shown: precipi-
tates with are laterally compressed, whereas precipitates with a ratio of
are laterally expanded (see also [68,166,171,182,183]). In a second view of
Figure 5.10, it becomes clear that the morphology of the reconstructed precipitates is
more complex than simply ellipsoidal. Furthermore, the density distribution around the
precipitates is not homogeneous. In order to get a quantitative understanding of the
evolution of the field emitters after their evaporation, the radii of the reconstructed
precipitates are compared in the dependence of the ratio of evaporation fields in
Figure 5.11 (a). After cluster localization with the maximum separation method (see
section 5.1.2), the radii of the precipitates are calculated with the presented statistical
approach. Dashed lines indicate the initial radius of the precipitate in the model emitter.
In comparison to these, solid lines present the radii evaluated by Eqs. [5.4] and [5.8],
when considering only cluster atoms for evaluation. This evaluation is of course only
possible for the numerically simulated emitters (since precipitate and matrix atoms are
unambiguously assigned), but it can be considered as a best case scenario for the
cluster size analysis. In contrast, dotted lines show the radii calculated by Eqs. [5.6]
and [5.8], when considering matrix and precipitate atoms in a larger volume is the only
Figure 5.10: Comparison of phase maps (top row) and atomic density distribution maps (bottom row) of
APT reconstructions from data sets obtained by simulated field evaporation. The simulations were per-
formed with different ratios between the evaporation thresholds of matrix and precipitate as denoted at
the bottom. The model emitter contained a precipitate sized 2.2 nm in radius. In the left part, the original
input structure of the simulation is shown. (According to reference [E12])
5 Studies on iron-based superalloys
122
reliable way for the evaluation of experimental data. However, satisfyingly both meth-
ods deliver almost identical results (solid and dotted lines) which can be seen as a
confirmation of the statistical size analysis approach. Only for clusters with very large
ratios and radii of less than 2.7 nm, statistical fluctuations cause noticeable devi-
ation.
The validity of the presented statistical size approach is also confirmed, when compar-
ing the measured radii for homogenous evaporation thresholds ( ). In this case,
the evaluated radii equal the original radii. For other field ratios, it is evident that the
isotropic radius , obtained by averaging the radius in x- y- and z- direction
(Eq. [5.6] and [5.8]), is not a suitable direct measure of the original precipitate size
(Figure 5.11a). Interestingly, the radii of precipitates determined only in z- direction
shown in Figure 5.11 (b) perform better and thus, is a much better size esti-
mate of the original precipitate radius (with a relative error of ± 7.3 % considering
all atoms and of only ± 3.0 % when considering cluster atoms only).
For real experimental data, a conclusion from to would be nevertheless
doubtful, since the in-depth scaling of an APT reconstruction (the z- dimension) de-
pends strongly on the reconstruction parameters (e.g. the field compression factor or
the detector efficiency). Without an independent method to prove these parameters,
an accurate in-depth scaling cannot be guaranteed. For this reasons, in the following
Figure 5.11: Comparison of the isotropic radii RExp(x,y,z) (a) and the radii in z- direction RExp(z) (b) of
the precipitates in dependence of the ratio of evaporation fields
FR
. The solid lines are the radii ob-
tained when considering only the precipitate atoms for the evaluation (only possible for simulated emit-
ters, in which the origin of all atoms can be identified) and dotted lines, when matrix atoms are consid-
ered as well in the evaluation. Dashed lines indicate the original precipitate radii
R0.
5.2 Effect of local magnifications in APT on nano precipitates
123
part a concept is developed to calculate the original cluster radius from the isotropic
radius . This model will allow accurate size estimates, even if the in-depth
scaling of the reconstructed tip is not correct (which will be demonstrated in the dis-
cussion in section 5.2.5).
From Figure 5.11 (a), it can be extracted that the isotropic radius depends
linearly, with slope , on the ratio of evaporation fields . Furthermore, at ,
the original radius equals the measured radius. This can be expressed as
[5.14]
.
[5.15]
To estimate the value of slope , the size ratios of the isotropic and original radii
are compared in Figure 5.12 (a). It is shown that the slope is a
function of the original radius and converges to 1 for . As demonstrated in
Figure 5.12 (b), the dependence of slope as a function of the original radius
can be approximated exponentially by,
,
[5.16]
with the parameters
a1
= 2.78 and
b1
= 0.76. This means, if the ratio of evaporation
fields is known, the original precipitate radii could now be evaluated following
Eqs. [5.14] and [5.16] (which is demonstrated in Appendix E Figure E.4).
In most of the experiments, the ratio is not known beforehand and thus, Eqs. [5.14]
and [5.16] can only be solved, if a second quantity, besides the cluster size, is found
that describes the cluster morphology and depends on the ratio of evaporation fields.
As shown in Figure 5.13, both, the aspect ratios (Eq. [5.11]) and the density ratios
(Eq. [5.12]) are suitable candidates for this.
Analogously to the size ratios, the aspect ratios
AR
follow a linear dependence on the
ratio of evaporation fields and can therefore be described (similar to Eq. [5.15]) by
,
[5.17]
where is again approximated exponentially as in Eq. [5.16] with and
which is shown in Appendix E Figure E.5 (a). This means, after measurement
5 Studies on iron-based superalloys
124
of the isotropic radii and the aspect ratios
AR
, the original radii can be calculated
numerically by solving the equation
,
[5.18]
obtained by inserting Eq. [5.17] in Eq. [5.14].
In contrast to the size ratios (Figure 5.12 a) and the aspect ratios (Figure 5.13 a), the
density ratios
DR
do not depend linearly on the ratio of evaporation fields (cf. Figure
5.13 b); instead, a dependency of the cubic root of the density ratios is found. Assuming
that - matrix and - precipitates have the same initial density ( and that the
Figure 5.12: Dependence of the size ratio between the isotropic radii
Rexp
(
x,y,z
) and the original radii
versus the ratio in evaporation fields
FR
(a). The dashed line indicates a 1:1 correlation. (b) The fit
parameter
m1
, describing the dependence of the size ratios shows an exponential decay for increasing
original cluster radii
R0. (According to reference [E12])
Figure 5.13: Correlation between the ratio in evaporation fields
FR
and (a) the aspect ratios
AR
and (b)
the cubic root of the density ratios of the reconstructed. The dashed lines indicate a 1:1 correlation.
(According to reference [E12])
5.2 Effect of local magnifications in APT on nano precipitates
125
matrix density is correctly reproduced in the APT reconstruction, the density ratios
DR
(Eq. [5.12]) can be rewritten as
,
[5.19]
where and are the original volume of the precipitate and the volume after field
evaporation, respectively. Eq. [5.19] suggests that the cubic root of the density ratios
(Figure 5.13 b) should show the same behavior as the size ratios (Figure 5.12 a). Ob-
vious from Figure 5.13 (b), cannot be described solely by a linear function (e.g.
by an analogue to Eq. [5.15]); additionally a quadratic term depending on the ratios of
evaporation fields is necessary to find a well matching description such as
.
[5.20]
With this, can again be approximated exponentially by Eq. [5.16] with
and , which is shown in Appendix E Figure E.5 (b).
The quadratic term in Eq. [5.20] can be indeed justified. Is is necessary,
because the assumption of a spherical cluster shape (and therefore the assumption
) becomes inaccurate for the field evaporated clusters. This is demonstrated
in Figure 5.14, where the isotropic radii (Eq. [5.6]), which could also be
expressed by
,
[5.21]
are compared to the volume equivalent ellipsoidal radii calculated by
Figure 5.14: Comparison of the differently calcu-
lated radii: the isotropic
RExp(x,y,z)
(solid lines),
the
equivalent ellipsoidal radii
REllip
(dotted line) and
the original precipitate radii
R0
are compared in
dependence of the ratio of evaporation fields
FR
(dashed line)
.
(According to reference [E12])
5 Studies on iron-based superalloys
126
.
[5.22]
In order to estimate the field ratio from the measured density ratio, Eq. [5.20] is
rewritten as
.
[5.23]
By inserting Eq. [5.23] into Eq. [5.14], the isotropic radii can now be expressed as
function of the original radii and the density ratios
DR
as
.
[5.24]
Eqs. [5.18] and [5.24] enable to calculate the original precipitate radii directly after
measurement of the isotropic radii and the aspect ratios or density ratios
DR
,
respectively. For derivation of this model with a maximum in accuracy, only the atoms
indexed as precipitate phase were considered in the reconstructed emitters. Now the
model is now tested by considering all atoms in the reconstructed emitters (corre-
sponding to the situation in real experiments). As shown in Figure 5.15, both ap-
proaches (Eqs. [5.18] and [5.24]) are well suited to estimate the original precipitate
size. Especially for precipitates ≥ 2.7 nm and with ratios in evaporation fields of
FR
<
1.3, the total deviation in size is less than 1.5 %. Only in case of high positive field
ratios and for small precipitates, larger deviations to the original radii are observed (±
3.6 % for 2.2 nm sized clusters and ± 5.2 % deviation for 1.7 nm sized clusters), mainly
caused by statistical fluctuations.
Figure 5.15: Comparison of the radii of the recon-
structed precipitates after correction using the
measured isotropic radii
and the density rati-
os
DR
(solid lines) by Eq. (19), after correction us-
ing the measured isotropic radii
and the as-
pect ratios
AR
(dotted lines) by Eq. (14) to the
original radii
in the dependence of the ratio in
evaporation fields
FR
(dashed lines). The results
of the smallest precipitates with a radius of 1.2 nm
(containing only 266 excess atoms) are not shown
in the diagram since their size could not be evalu-
ated to a precision of better than ± 20 %. (Accord-
ing to reference [E12])
5.2 Effect of local magnifications in APT on nano precipitates
127
5.2.4 Test on experimental data
The model to calculate the original size of precipitates that appear local magnified or
compressed in APT reconstructions, as proposed in the previous section, is now ap-
plied to the APT data of the ferritic alloy FBB-8 that were already presented in sec-
tion 5.1.3. The model is tested on the data obtained from the specimens after heat
treatment at 700 °C and 800 °C (see also Figure 5.7). As a reminder, the tip after aging
at 700 °C contains cooling precipitates with a radius of (1.7 ± 0.3) nm (see Figure
5.16a), whereas after aging at 800 °C, a mean radius (6.2 ± 1.2) nm was obtained for
the cooling precipitates (see Figure 5.16 b). Especially for the smallest cooling precip-
itates in the state after heat treatment at 700 °C, the contrast of the precipitates in DF
TEM images fades out at the edges and indicates that conventional TEM characteri-
zation is not well suited for an accurate size estimation. Furthermore, no chemical in-
formation can be obtained (e.g. by EDX or EELS), because the signals of precipitates
and surrounding matrix cannot be distinguished. For this reason, APT measurements
were performed to collect more information about cooling precipitates.
The APT measurements have shown that cooling precipitates appear laterally magni-
fied in the reconstruction and exhibit a lower atomic density than the matrix, when they
form during cooling from 800 °C (see Figure 5.7 a). Surprisingly, this behavior reverses
after aging at 700 °C (see Figure 5.7 d). In this case, the cooling precipitates appear
more stretched in measurement direction and have a higher atomic density.
a)
b)
Figure 5.16: Dark field TEM images of ordered NiAl type precipitates (white contrast) in the alloy FBB-8
after heat treatment at 700 °C (a) and 800 °C (b).
5 Studies on iron-based superalloys
128
Test of the model on precipitates with a high evaporation field
Following the conclusions of the APT simulations, the precipitates formed during cool-
ing from 800 °C have a higher evaporation threshold than the matrix (
FR
> 1), since a
lower density inside the precipitates was observed in the reconstructions, in contrast
to the real material, where the precipitates exhibit almost the same density as the ma-
trix. For the tests, the size and composition of five precipitates were analyzed first by
a “conventional” method (cf. Figure 5.17) and then compared to results obtained by
the new proposed method considering the distortion by local magnification artefacts
(see Table 5.3).
In the conventional method (see Figure 5.17), the precipitates were I. visualized in the
APT reconstruction by isosurfaces. II. Analysis boxes including only one precipitate
were cropped. III. The cluster search with the modified maximum separation method
(cf. section 5.1.2) was performed for the atoms inside the analysis box and the surface
of the cluster was triangulated. IV. The composition and density in vicinity to the trian-
gulated surface were calculated. For the rather large cooling precipitates in this state,
the Ni concentration in the core of the triangulated cluster can be extrapolated
by fitting an error function, revealing ~44 at.-% (see also Table 5.3). In contrast to this,
Figure 5.17: Method for analysis of single cooling precipitates in alloy FBB-8 after heat treatment at
800 °C:
I. Precipitates are visualized by 25 at.-% Ni isosurfaces; II
. The precipitate is selected by an
analysis box;
III
. Cluster search with maximum separation method and Delaunay triangulation is per-
formed; IV. Proxigrams are calculated.
5.2 Effect of local magnifications in APT on nano precipitates
129
the Ni concentration is underestimated, when it is directly calculated from all
atoms within the convex hull (33.5 at.-%), since matrix atoms from the interface region
are included in the triangulated volume; the volume selected by the convex hull corre-
sponds to an equivalent spherical cluster radius of = 4.77 nm in the given example.
In Table 5.3, the results (cluster radii and compositions) from this “conventional” anal-
ysis are compared to the results obtained by the proposed statistical approach applied
on each analysis box without (Eqs. [5.6] and [5.8]) and by additionally considering the
distortions due to local magnification artefacts (Eq. [5.24]). The statistically evaluated
isotropic radii are very similar to the radii calculated directly from the convex hull
. However, when the measured density ratios are considered for calculation of
the evaporation thresholds (here 1.14 ± 0.05), and these are applied to calculate
the original precipitate radii (Eq. [5.24]), the evaluated radii become significantly (in
average by 0.7 nm) smaller.
To cross check which radius ( or ) is more realistic, the composition of the
precipitate is calculated from the number of excess atoms and the radius
of the precipitate by
,
[5.25]
where is the concentration of solute atoms in the matrix and the density of
the precipitate. In alloy FBB-8, the density can be assumed to be the same as the
matrix density . In view of the table, it is obvious that the compositions calculated
from the original radii are much closer to the compositions obtained by extrap-
Table 5.3: Comparison of the evaluated radii and atomic fractions: of Ni atoms of five different precipi-
tates after heat treatment at 800 °C obtained by different analysis methods.
Precipitate
Radius [nm]
Atomic fraction of Ni Atoms [at.-%]
From
convex
hull
Isotropic
radius
Calc.
original ra-
dius
Proxigram
extrapola-
tion
1
5.23
5.36
4.42
33.2
27.6
44.3
43.2
2
5.25
5.53
4.85
38.9
32.4
45.3
47.7
3
4.77
4.93
4.15
33.5
26.6
40.0
44.0
4
3.55
3.73
3.38
36.1
33.5
43.3
48.2
5
4.70
4.83
4.05
34.1
26.5
40.2
42.9
5 Studies on iron-based superalloys
130
olation of the proxigram . In average, they differ by only 2.6 at.-% Ni and con-
sequently, the corrected sizes are convincingly close to the real size of the precipi-
tates.
Test of the model on precipitates with a low evaporation field
After heat treatment at 700 °C, the cooling precipitates showed a higher density in the
APT reconstruction and thus, it can be concluded that they exhibit a lower evaporation
field than the matrix atoms (
FR
< 1).
For the test of the model, one cylinder containing 19 cooling precipitates (as shown in
Figure 5.18) was cropped and further analyzed. Again, the maximum separation algo-
rithm was applied for precipitate localization. Subsequently the sizes of the precipitates
were analyzed by the statistical approach yielding an average isotropic mean radius
of (1.45 0.23) nm (see also Table 5.4 in the discussion). When additionally the
density ratio is considered to estimate the ratio in evaporation thresholds (here
FR
=
0.83 0.03) and applying this to calculate the original radius, a mean radius of (1.80
0.31) nm is obtained (Eq. [5.24]). This corrected radius is very close to the radius
evaluated by TEM, (1.7 ± 0.3) nm (see Table 5.1).
Applying Eq. [5.25] to cross check which radius is more realistic from a compositional
point of view, yields a concentration of (73 16) at.-% Ni using the isotropic
radius but a concentration of (43.7 7.0) at.-% Ni when using the corrected
original radius. In comparison, a concentration of (40.2 2.1) at.-% Ni was ob-
tained by evaluating the proxigrams of all clusters individually (see Table 5.2). Obvi-
ously, only the corrected radius allows a realistic determination of the composition.
The directly evaluated and by local magnification artefacts reduced isotropic size ,
on the other hand, presents only half of the real size of the cooling precipitates in this
example.
Figure 5.18: Reconstruction of the
atom probe tip after aging at 700 °C
visualized by 25 at.-% Ni isosurfaces.
The blue cylinder indicates the analy-
sis volume chosen for cluster analysis.
5.2 Effect of local magnifications in APT on nano precipitates
131
5.2.5 Discussion
The presented tests have demonstrated that the original size of precipitates can be
accurately obtained from an APT reconstruction when correcting their distortion due to
local magnification artefacts. For the presented examples, original radii have been cal-
culated from the measured isotropic radii and applying the ratios of the measured den-
sities of precipitates and matrix for correction (Eq. [5.24]). In section 5.2.3, it was shown
that the original radii could also be calculated by considering the aspect ratios instead
(Eq. [5.18]). Also this approach was tested in this work, but the results have shown that
this approach is less suited to obtain accurate sizes. Especially for the large clusters
after heat treatment at 800 °C (Figure 5.17), the results based on the correction using
the aspect ratios were not satisfactory. This is, because the aspect ratios of features
within the reconstructed APT data depend more strongly on the quality of the recon-
struction than the density. For instance, trajectory aberrations, due to the specimen
crystallography or due to grain boundaries, can cause further deviations of the as-
sumed hemispherical emitter shape [184–186]. Furthermore, the scaling of the APT
reconstruction depends on user defined parameters such as the field compression fac-
tor or the detection efficiency. Whereas the detection efficiency is usually known for
the instrument, the field compression factor or the tip radius has to be derived for each
measurement. This can be achieved if complementary information about the tip exist,
e.g. by matching the distance of reconstructed lattice planes with those in the real ma-
terial [72] or, by matching the geometry or the arrangement of features in the APT
reconstruction until it matches the a-priori determined shape of these features (for ex-
ample determined by correlative electron microscopy) [187–190]. But, if none of these
methods can be applied, a correct in-depth scaling of the APT reconstruction can so
far not be ensured.
For the example shown in Figure 5.17 – a measurement that was performed by laser
pulsing in the METAP instrument – no lattice planes could be observed in the recon-
struction and no complementary information on the tip shape exist. Therefore, the in-
depth scaling might not be accurate. Nonetheless, the derivation of the original precip-
itate sizes from the density ratios yielded still reliable information such as accurate
compositions.
5 Studies on iron-based superalloys
132
The measurement of the tip aged at 700 °C, shown in Figure 5.18, was performed in
a voltage-pulsed instrument (WATAP). In this case, a better in-depth resolution was
obtained so that lattice planes could be resolved in the APT reconstruction. This is
exemplified in the spatial distribution map (SDM) in Figure 5.19 (a) [88]. However, no
clear zone axis or zone lines were revealed in the reconstruction and thus, an easy
identification of the corresponding lattice plane family, visible in the SDM, is not possi-
ble. For this reason, three reconstructions were performed by choosing different values
for the field compression factor . Figure 5.20 demonstrates, how the shape of the
precipitates (and the analysis region) changes by varying from 6.6 (a), to 7.6 (b), to
8.6 (c). The analysis of small regions (4x4x6 nm³) around the zone axis by the SDM
method and subsequent averaging of the intensities in
z
- direction enables, to estimate
the lattice plane distance, as presented in Figure 5.19 (b-d). The lattice plane distance
increases from 0.151 nm (b), to 0.205 nm (c), to 0.264 nm (d) with increasing field
compression factor.
According to Figure 5.20 (a), one could now naively choose a field compression factor
of 6.6, so that the clusters appear more spherical in the reconstruction; but this would
ignore the local magnification effects. So far, there is no method to decide, which of
Figure 5.19: (a) Spatial distribution map (SMD) in x- and z- direction of atoms close to a zone axis in the
sample heat treated at 700 °C; (b-d) radial distribution maps obtained by averaging the SDM in z- direc-
tion and applying different field compression factors
kr
of 6.6 (b) / 7.6 (c) / 8.6 (d) in the APT reconstruc-
tion. The data points where fitted by a sum of Gaussian functions.
5.2 Effect of local magnifications in APT on nano precipitates
133
the reconstructions is best. However, the comparison of the field ratios obtained
from the aspect ratios (Eq. [5.17]) with those obtained by calculation from the den-
sity ratios [5.23], as given in Table 5.4, can give surprisingly an independent an-
swer to this question. Whereas shows a strong dependence on the field compres-
sion factor applied for the reconstruction, is almost constant for all three recon-
structions. Therefore, it can be concluded, that the density ratios are an independent
measure of the field ratio of about 0.83. Only if 7.6 is chosen, the field ratios
estimated from are in correspondence with the values obtained from and in-
dicate that this is the most realistic field compression factor.
A further indication that a value of 7.6 is the best choice is seen, when comparing
the measured lattice plane distances with the theoretical distances in the bcc lattice of
Figure 5.20: Reconstructions with varying field
compression factor : Density maps (top) and
detected precipitates after use of the maximum
separation algorithm and visualized by their tri-
angulated convex hulls (bottom) of one APT
dataset reconstructed by applying three differ-
ent field compression factors
kr
of 6.6 (a), 7.6
(b) and 8.7 (c).
5 Studies on iron-based superalloys
134
alloy FBB-8 (cf. Table 5.4). Only a field compression factor of 7.6 yields a lattice plane
distance (0.2045 nm) that matches to a real distance in the alloy; here to the spacing
of (110)- planes. In contrast, for 8.6, the measured distance of 0.2638 nm is far
off from any theoretical distance (
d
110 = 0.2041 nm is the closest one). The same ar-
gument applies for a value of 6.6, where the measured distance of 0.1511 nm
does not clearly represent a specific lattice plane family in the ferritic alloy (
d
111: 0.1666
nm and
d
200: 0.1443 nm are closest).
Finally, also the comparison of the Ni concentration inside the precipitates, evaluated
from the individual proxigrams (40.2 at.-%), to the concentration evaluated with
Eq. [5.25] by applying the corrected radii estimates, confirms a field compression factor
of 7.6. In this case, the same concentration is obtained, independently whether the
aspect or density ratios are chosen for the estimation of the precipitate size. On the
other hand, for 6.6 and 8.6, the amount in Ni is over- or underestimated, respec-
tively, due to the erroneously estimated radii from the aspect ratios.
This example demonstrates, that the presented model for size evaluation of precipi-
tates in APT reconstruction can be even applied to find the optimum reconstruction
parameters and therefore, adds a new internal calibration to optimize the in-depth scal-
ing of APT measurements.
Table 5.4: Analysis of the APT measurement of the sample aged at 700 °C reconstructed by applying
three different field compression factors
kr
: Compared are the lattice plane distances evaluated by spa-
tial distribution maps (SDM) and the closest theoretical lattice plane distance of the bcc Fe-alloy, the
field ratios
derived from the aspect ratios
(Eq. [5.17]) or the density ratios
(Eq. [5.23]
), the
isotropic radius
and the calculated original radii
(Eq. 14) and
(Eq.
[5.24])
and the elemental concentration of Ni derived from the proxigrams and calculated with Eq. (
[5.25]
) ap-
plying the corrected radii estimates of the precipitates.
Field compression factor
6.6
7.6
8.6
Lattice plane dis-
tance
dhkl [nm]
Measured by SDM
0.1511
0.2045
0.2638
Nearby dhkl in Fe- alloy
d111: 0.1666
d200: 0.1443
d110: 0.2041
d110: 0.2041
Field ratio
Using
0.93 ± 0.10
0.83 ± 0.06
0.74 ± 0.13
Using
0.830 ± 0.028
0.835 ± 0.029
0.828 ± 0.032
Radius [nm]
1.40 ± 0.04
1.45 ± 0.06
1.62 ± 0.05
1.36 ± 0.06
1.79 ± 0.11
2.32 ± 0.12
1.75 ± 0.05
1.83 ± 0.08
2.02 ± 0.08
Ni concentration
[at.-%]
Proxigram
40.2 ± 2.1
Corrected by
84.6 ± 2.4
43.7 ± 2.9
26.6 ± 1.9
Corrected by
43.3 ± 2.1
43.4 ± 1.7
32.5 ± 1.6
135
6 Conclusions and Outlook
The desire to increase the working temperature of high performance structural materi-
als requires to understand the physical processes within the alloys. In this work, new
analysis approaches have been developed to study nickel- and iron-based superalloys
on smallest length scales. In four sub-sections, it was shown how the materials char-
acterization techniques neutron diffraction, electron microscopy and atom probe to-
mography (APT) are complementary applied to improve our understanding of the com-
plex nano-structure and micromechanical relationships within superalloys.
In section 4.1, a new experimental method was presented that allows to distinguish
and quantify the two strengthening
-
and
-
precipitates
in the nickel-based superal-
loy Inconel 718. Before, this could only be achieved by microscopic techniques which
collect information from nano- to micrometer sized volumes. In this work, it was demon-
strated that reliable volume fractions of the precipitates could be obtained by ex-situ
small-angle neutron scattering (SANS); a technique, which probes large sample vol-
umes and thus delivers bulk average values of statistical significance. The challenge
of this work was, to identify the individual scattering signals of the
-
and
-
phases.
This could be achieved by the set-up of a structural model which required complemen-
tary information by transmission electron microscopy (TEM) and APT. By TEM, crys-
tallographic information, size distributions and the morphologies of the precipitates
were obtained, while APT provided compositional information, which was necessary to
calculate the scattering contrast of the phases. The obtained volume fractions of the
precipitates in differently heat-treated Inconel 718 specimens are fully consistent with
associated mass balance calculations.
In section 4.2, the micromechanical behavior of the two nickel-based superalloys In-
conel 718 and Haynes 718 was studied by in-situ neutron diffraction. Thereby, the
lattice strain evolution during uniaxial tensile loading was measured with a new exper-
imental set-up which allowed, to measure the lattice strains of various crystallographic
planes in five different sample orientations with respect to the load. The comparison of
the hardening behavior of Inconel 718 and Haynes 282 revealed significantly different
characteristics in both, the macroscopic and the microscopic material behavior. Alloy
Inconel 718 featured a smooth transition in the evolution of macros- and microscopic
6 Conclusions and Outlook
136
strain in the elastic to plastic regime. In contrast, alloy Haynes 282 featured a sharp
transition in the macroscopic strain response and even local softening and pronounced
yielding in the lattice strain evolution (microyielding) in specific measurement direc-
tions. For interpretation of the results, the experimental data were compared to the
simulated material response modelled by a crystal plasticity finite element model which
utilized a constitutive description (power-law behavior) of the hardening response. The
hardening behavior of Inconel 718 could be accurately described by the proposed
model. On the other hand, the hardening behavior could not be simulated with the
simple power-law based hardening behavior; a more complex hardening behavior that
allows for softening effects at the onset of yield had to be considered instead. This
study motivates to consider phenomena like slip band formation or pronounced mi-
croyielding in more sophisticated physically based crystal plasticity models in future.
The focus of the last two sub-chapters was on the analysis of iron-based superalloys.
The presence of smallest nano-precipitates within the ferritic alloy FBB-8 have shown
to tremendously strengthen the alloy at room temperature. Mainly based on APT meas-
urements, two analysis methods have been developed that allow to better identify and
characterize these precipitates.
In section 5.1, a modified version of the maximum separation algorithm for cluster se-
lection in APT datasets has been proposed. The algorithm was modified by replacing
the erosion step by a Delaunay tessellation which enables to calculate the convex hull
of the precipitates. As a benefit, the Delaunay tessellation does not require any user
defined input parameters and gives directly access to the morphology of the clusters.
The algorithm was first tested on simulated data and then successfully applied to the
ferritic alloy FBB-8 containing hyperfine NiAl- type precipitates. The modified algorithm
is especially useful to detect and analyze clusters with a radius between 1 nm and
10 nm in these alloys. The analysis has further demonstrated that there is a systematic
variation of the Al/Ni- ratio within the NiAl- type precipitates which depends on their
formation temperature. Due to the low diffusivity of Ni in the ferromagnetic iron matrix
at low temperatures (< 700 °C), a kinetically driven enrichment in Al and even a tran-
sition in the crystal structure from cubic B2 NiAl to NiAl3 (Pnma) is favored. However,
at high temperatures, the precipitates exhibited an enrichment in Ni which is assumed
to be caused by a composition-dependent nucleation barrier.
6 Conclusions and Outlook
137
Surprisingly, also the atomic density of the reconstructed precipitates has shown to be
significantly influenced by the formation temperature. This was the motivation for the
further studies presented in the last section 5.2. The observed density differences have
shown to be a consequence of local magnification artefacts caused by a heterogene-
ous evaporation behavior of the different phases in the alloy. Especially for the analysis
of the morphology and the exact size of small precipitates, the APT technique is thus
often considered as “unsuited”. To study the distortion of nano-sized precipitates in the
APT reconstruction in dependence of their size and their evaporation behavior, numer-
ical simulations of APT measurements have been performed with the software
TAPSim. For the analysis, a new statistical approach, based on the calculation of ex-
cess atoms, has been proposed. It revealed a linear relation of the ratio in evaporation
thresholds between matrix and precipitate atoms to the isotropic radius and the aspect
ratios of the precipitates. Furthermore, a quadratic relation of the ratio in evaporation
thresholds to the cubic root of the density ratios between precipitate and matrix densi-
ties was revealed. These relations enabled to derive an empirical model for the extrac-
tion of the original size and exact composition of the precipitates. This method was
successfully tested on the heterogeneous alloy FBB-8 containing precipitates with a
lower (0.83 -times) evaporation field, when formed at temperatures below 700 °C and
a larger (1.14 -times) evaporation field, when formed at higher temperatures. In the
end, it was shown how the method can be further applied as a new concept for the
internal calibration of tomographic reconstructions.
138
Appendix A
139
Appendix A
Complementary figure to section 3.3.2
Quantification of the volume fraction of precipitates by TEM
The volume fraction of precipitates in a specimen can be evaluated by TEM if the shape
of the precipitates and the analysis volume is known. The precipitates in this work are
of spherical or ellipsoidal shape. Therefore, the equivalent volume of the precipitates
can be estimated from the measured area of the precipitates. In case of the ellipsoidal
shaped
-
precipitates, the measurements are performed with the precipitates aligned
with one equatorial axis parallel to the beam and assuming that both equatorial axes
have the same length (see also Eq. [4.1]).
The analysis volume is calculated from the analyzed area and the specimen thickness.
In this work, the specimen thickness
t
is determined by CBED with the specimen
aligned in a two beam condition (cf. Figure 3.6 d). From the distances of the intensity
minima
i
and the center of the primary beam 2
B
to the center of the diffracted beam,
the deviation (or excitation error)
si
of the
i
th fringe can be calculated by
,
[6.1]
Figure A.1: Schematic view of the five different measurement orientations (0°, 30°, 45 °, 60° and 90°)
with respect to the applied load
F
.
Appendix A
140
where
dhkl
is the interplanar spacing of the diffracting plane and
the wavelength of
the electron beam. Depending on the extinction distance
g
of the material in the Bragg
condition, the specimen thickness can be determined by
,
[6.2]
where
nk
is an integer identical to
i
or larger by a constant integer. The number of
fringes increases by one, every time the specimen thickness increases by one extinc-
tion distance. If the extinction distance is unknown, the specimen thickness
t
can be
evaluated graphically by plotting (
si
/
nk
)² versus (
1
/
nk
)² which must be a straight line if
the assignment of value
nk
is correct (
nk
needs not necessarily to be 1 for the first
fringe, since the minimum thickness
can be larger than one extinction distance). From
the plot, the slope corresponds to
g-2
and the intercept corresponds to -2.
Due to the finite thickness of the TEM specimen, the analysis of particle volume frac-
tions from TEM images is biased by the sectioning of large precipitates. Precipitates
at the specimen surface might be sliced and only a smaller projected diameter of the
precipitate is observed in the image. Depending on the sample preparation, it is also
possible that precipitates at the specimen surface are preferentially or protruding pol-
ished. For a correct estimate of the precipitate volume fraction, the sectioning of pre-
cipitates has to be considered.
In this work, precipitates are assumed to be sliced at the specimen surface. This
means, precipitates with their center inside the TEM foil are projected with their full
diameter. If the center of the precipitates is outside the TEM foil, the projected diame-
ters are smaller than the actual diameter of the precipitate. A Schwartz-Saltykov type
correction of the particle size distribution, according to Dorin et al. [115], is applied to
consider the sectioning of these precipitates: The corrected amount
mi
of precipitates
in a size class
i
is the sum of the precipitates that are projected with their full diameter
(center inside the TEM foil) and the amount of precipitates that appear smaller in the
projection. The number of fully projected precipitates is
,
[6.3]
where
Ni
is the measured number of precipitates in this size class and the diameter
of a precipitate in size class
i
(precipitates that are sliced but with their center inside
Appendix A
141
the TEM sample increase the transmitted specimen thickness by a virtual amount of
t
+
di
).
In case of the largest size class (
i = max)
, the probability that a precipitate is sliced by
the specimen surface is
.
[6.4]
The amount of sliced precipitates with radius
rmax
within this size class and with their
center located outside the TEM foil can be calculated by a simple projection model as
.
[6.5]
Therefore, the corrected amount of precipitates in the largest size class is
.
[6.6]
For every other size class, it has to be considered that there is a certain amount of
precipitates from every larger size class that is projected into this size class. For in-
stance, in size class
i =
9, the amount of precipitates projected into size class 8 is
.
[6.7]
Eqs. [6.6] and [6.7] can be expressed more generally by
.
[6.8]
Appendix B
142
Appendix B
Complementary figures to section 4.1
Figure B.1:
-
phase in sample state A-718, which contains the smallest precipitates: a) High resolution
image of the sample tilted to [001] zone axis; b) Fast Fourier Transformation (FFT) images according to
the highlighted squares in figure a); c) Fourier-filtered image to highlight different phases and orienta-
tions. (According to reference [E3])
Figure B.2: APT maps of tips of alloy Inconel 718 after APT reconstruction where only Cr (green),
Al (red) and Nb (blue) atoms are visualized and A-718, B-760, C-870 and D-950 denote the different
heat treatments performed (see section 4.1).
Appendix B
143
Mass balance calculation in section 4.1
The maximum possible phase fraction of each phase by considering only one ele-
ment
i
(here Nb or Ti) and the concentration of the element in the precipitate (given
in Table 4.2) is calculated by
,
[6.9]
where is the total content of element in Inconel 718 (see Table 2.1) and
the concentration of element in the matrix measured by APT (see Table 4.2).
To calculate the maximum possible volume fraction of the precipitated phase,
it is necessary to additionally account for the atomic densities
of each phase by
.
[6.10]
By comparing the experimental (see Table 4.4) and the maximum possible
volume fractions of
-
and
-
precipitates, the volume fraction of
-
precipitates
can be estimated in terms of a mass balance calculation
.
[6.11]
Figure B.3: Fitted SANS curves of sample state (a) A-718 and (b) C-870.
Appendix C
144
Appendix C
Complementary figures to section 4.2
Figure C.1: Visualization of dislocations by TEM in an undeformed Haynes 282 specimen aged for 8 h
at 900 °C: (a) Dark field image of one
-
precipitate sheared by a dislocation and (b) bright field image
of a specimen in two beam orientation close to the (111) zone axis showing a band of dislocations.
Dislocations create a change in the diffraction condition and therefore contrast that can be visualized
as lines (marked as arrows) in the TEM image as long as the scalar product of Burgers vector and the
reflection chosen for imaging is non zero.
Appendix C
145
Figure C.2: Comparison of measured true stress versus lattice strain diagrams of Haynes 282 (left) and
Inconel 718 (right). Compared are the lattice strains of reflections stemming from
- precipitates in dif-
ferent orientations (
= 0°, 30°, 45°, 60° and 90°) to the direction of load. The measurement errors are
significantly larger compared to those obtained for the matrix reflections. The (221)- reflection (not
shown here) was too weak to be evaluated in both alloys. For alloy Inconel 718, also the (210)- and the
(100)- reflections could not be evaluated. The data of the (020)- reflection of
-
phase are presented
instead which were more clear in the diffractogramms. The measurement errors are larger for Inconel
718, since the amount of
- precipitates is much less.
Appendix C
146
Figure C.3: Comparison of measured true stress versus lattice strain diagrams of Haynes 282 measured
at 650 °C. Compared are the lattice strains of reflections stemming from
- precipitates (left) and the
matrix (right) in different orientations (
= 0°, 30°, 45°, 60° and 90°) to the direction of load.
Appendix C
147
Figure C.4: Comparison of measured intergranular strain versus macroscopic plastic strain diagrams of
Haynes 282 (left) and Inconel 718 (right). Compared are the intragranular phase strains of reflections
stemming from
- precipitates (or
-
phase) in different orientations (
= 0°, 30°, 45°, 60° and 90°) to
the direction of load. The measurement errors are significantly larger compared to those obtained for
the matrix reflections.
Appendix C
148
Figure C.5: Comparison of intra- and intergranular strains versus macroscopic plastic strain diagrams
of Haynes 282 measured at 650 °C. Compared are the inter/intra-granular strains of reflections stem-
ming from
- precipitates (left) and the matrix (right) in different orientations (
= 0°, 30°, 45°, 60° and
90°) to the direction of load.
Appendix C
149
Figure C6: Comparison of intergranular strains at 4.5 % macroscopic plastic strain versus measurement
direction for Haynes 282 measured at 25 °C and 650 °C (a) and comparison of the intergranular strains
of reflections stemming from
- precipitates and matrix reflections at 650 °C.
Figure C.7: Texture analysis on Inconel 718: Compared are the results of the texture analysis obtained
by EBSD after deformation (a) with the results obtained by neutron diffraction in 0° (b), 45° (c) and 90°
(d) orientation. Texture plots has been generated with the software
OIM Analysis
by harmonic series
expansion where the color scale represents the relative fraction of grains (weighted by area) with a
certain orientation within the standard triangle (blue: < 0.9 and red: > 1.3).
Appendix C
150
Figure C.8: Comparison of normalized integrated intensity versus macroscopic plastic strain diagrams
of Haynes 282 (left) and Inconel 718 (right). Compared are the normalized integrated intensities of re-
flections stemming from the matrix (111), (200) and (220) in different orientations (
= 0°, 30°, 45°, 60°
and 90°) to the direction of load.
Appendix D
151
Appendix D
Complementary figures to section 5.1
Figure D.1: Measurements with the MPMS3 SQUID-Magnetometer von Quantum Design: a) Measure-
ment of the magnetization as a function of the applied magnetic field at 300 K giving a saturation mag-
netization of 1050 emu/cm³ or a magnetic flux density of 1.3 T; b) Measurement of the magnetization
during sample cooling from 950 K with an applied magnetic field of 100 Oe. From the plot of the recip-
rocal magnetization (shown in the inset) the Curie temperature is determined as 917 K. (According to
reference [E11])
Figure D.2: Dependence of the Ni/Al-ratio (blue squares) and the density (red circles) as a function of
the radius of 87 detected clusters formed by cooling from 700 °C. The regression of linear fits to the
data points has been overlaid by the arrows. (According to reference [E11])
Appendix E
152
Appendix E
Complementary figures to section 5.2
Figure E.1: Phase maps (
´ red,
cyan) of the input APT emitters and the reconstructed emitters after
field evaporation in dependence of the size and the ratio in evaporation thresholds of precipitate and
matrix atoms. (According to reference [E12])
Appendix E
153
Figure E.2: Atom maps (Ni red, Fe
cyan) of the input APT emitters and the reconstructed emitters after
field evaporation in dependence of the size and the ratio in evaporation thresholds of precipitate and
matrix atoms. (According to reference [E12])
Appendix E
154
Figure E.3: Local density maps of the input APT emitters and the reconstructed emitters after field evap-
oration in dependence of the size and the ratio in evaporation thresholds of precipitate and matrix atoms.
(According to reference [E12])
Appendix E
155
Figure E.5: The fit parameters
m2
, describing the dependence of the aspect ratios (a) and
m3
, describing
the dependence of the density ratios (b) on the ratio of evaporation fields, show an exponential decay
for increasing original cluster radii R0. (According to reference [E12])
Figure E.4: Three dimensional graph of the reconstructed isotropic radii
RExp
in the dependence of the
original radii R0 of the clusters and the ratio of evaporation fields
FR
calculated by Eq. [5.14].
156
157
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169
List of publications (2016-2021)
Contributions in peer-reviewed journals
[E1] JOSHI, Y., HADJIXENOPHONTOS, E., NOWAK, S., LAWITZKI, R., GHOSH, P. K. & SCHMITZ,
G. (2018). Modulation of the Optical Properties of Lithium Manganese Oxide via Li-Ion
De/Intercalation. Advanced Optical Materials 6, 1701362.
[E2] VON KOBYLINSKI, J., LAWITZKI, R., HOFMANN, M., KREMPASZKY, C. & WERNER, E. (2019).
Micromechanical behaviour of Ni-based superalloys close to the yield point: a
comparative study between neutron diffraction on different polycrystalline
microstructures and crystal plasticity finite element modelling. Continuum Mechanics
and Thermodynamics 31, 691–702.
[E3] LAWITZKI, R., HASSAN, S., KARGE, L., WAGNER, J., WANG, D., VON KOBYLINSKI, J.,
KREMPASZKY, C., HOFMANN, M., GILLES, R. & SCHMITZ, G. (2019). Differentiation of γ′-
and γ″- precipitates in Inconel 718 by a complementary study with small-angle neutron
scattering and analytical microscopy. Acta Materialia 163, 28–39.
[E4] MARTINE, J., LAWITZKI, R., MA, W., EVERETT, C., SCHMITZ, G. & CSISZÁR, G. (2020).
Beyond linearity: bent crystalline copper nanowires in the small-to-moderate regime.
Nanoscale Advances 2, 3002–3016..
[E5] MENOLD, T., HADJIXENOPHONTOS, E., LAWITZKI, R., SCHMITZ, G. & AMETOWOBLA, M.
(2020). Crystal defects in monocrystalline silicon induced by spot laser melting. Journal
of Applied Physics 127, 093102.
[E6] KRUHLOV, I. O., SHAMIS, O. V., SCHMIDT, N. Y., GULYAS, S., LAWITZKI, R., BURMAK, A. P.,
KONOREV, S. I., KATONA, G. L., SCHMITZ, G., ALBRECHT, M. & VLADYMYRSKYI, I. A. (2020).
Thermally-induced phase transitions in Pt/Tb/Fe trilayers. Thin Solid Films 138134.
[E7] TRÄGER, N., LISIECKI, F., LAWITZKI, R., WEIGAND, M., SCHÜTZ, G., SCHMITZ, G., KUSWIK,
P., KRAWCZYK, M., GRÄFE, J. & GRUSZECKI, P. (2020). Competing spin wave emission
mechanisms revealed by time-resolved x-ray microscopy. Physical Review B 103,
014430.
[E8] BUBECK, C., GOERING, E., LAWITZKI, R., KÜSTER, K., WIDENMEYER, M., STARKE, U.,
RITTER, C., CUELLO, J., NAGEL, P., MERZ, M., SCHUPPLER, S., SCHÜTZ, G., WEIDENKAFF,
A. (2020). High-temperature ferromagnetism in the diluted magnetic semiconductor
system lanthanum tantalum cobalt oxynitride. Submitted to Nature Communications.
[E9] CSISZÁR, G., SOLODENKO, H., LAWITZKI, R., MA, W., EVERETT, C. & CSIZÁR, O. (2020).
Nonlinear elastic aspects of multi-component iron oxide core–shell nanowires by means
of atom probe tomography, analytical microscopy, and nonlinear mechanics. Nanoscale
Advances 2, 5710–5727.
[E10] VON KOBYLINSKI, J., HITZLER, L., LAWITZKI, R., KREMPASZKY, C., ÖCHSNER, A. & WERNER,
E. (2020). Relationship between Phase Fractions and Mechanical Properties in Heat
Treated Laser PowderBed Fused CoBased Dental Alloys. Israel Journal of Chemistry
60, 607–614.
[E11] LAWITZKI, R., BEINKE, D., WANG, D. & SCHMITZ, G. (2021). On the formation of nano-
sized precipitates during cooling of NiAl- strengthened ferritic alloys. Materials
Characterization 171, 110722.
[E12] LAWITZKI, R., STENDER, P. & SCHMITZ, G. (2021). Compensating local magnifications in
atom probe tomography for accurate analysis of nano-sized precipitates. In press in
Microscopy and Microanalysis.
170
[E13] CSISZÁR, G., LAWITZKI, R., & CSIZÁR, O. (2021). Extreme elastic deformable ceramics on
the nanoscale. Submitted to Nano Express.
[E14] SCHULZ, F., LAWITZKI, R., GLOWINSKI, H., LISIECKI, F., Träger, N., KUSWIK, P., GÖRING,
E., SCHÜTZ. G., GRÄFE, J. (2021). Increase of Gilbert Damping in Permalloy Thin Films
due to Heat-Induced Structural Changes. Journal of Applied Physics 129(15), 153903.
[E15] CSISZÁR, G., LAWITZKI, R., EVERETT, C. & SCHMITZ, G. (2021). Elastic behavior of
Nb2O5/Al2O3 core-shell nanowires in terms of short-range order structures. ACS
Applied Materials & Interfaces 13(20), 24238–49.
[E16] SCHWARZ, T., DIETRICH, C., OTT, J., WEIKUM, E., LAWITZKI, R., SOLODENKO, H.,
HADJIXENOPHONTOS, E., GAULT, B., SCHMITZ, G. & STENDER, P. (2021). 3D Sub-
Nanometer Analysis of Glucose in an Aqueous Solution by Cryo-Atom Probe
Tomography. In press in Scientific Reports.
[E17] JOSHI, Y., LAWITZKI, R. & SCHMITZ, G. (2021). Slow-Moving Phase Boundary in
Li4/3+xTi5/3O4. Submitted to Small Methods.
Contributions on conferences and in proceedings
[P1] RASHIDI, M., LAWITZKI, R., ANDRÉN H. & LIU, F. (2016). Tantalum and niobium based Z-
phase in a Z-phase strengthened 12% Cr steel. Advances in Materials Technology for
Fossil Power Plants - Proceedings from the 8th International Conference on Advances
in Materials Technology for Fossil Power Plants. 1058–1066.
[P2] LAWITZKI, R., VON KOBYLINSKI, J., HOFMANN, M., KREMPASZKY, C., WANG, D., SCHMITZ,
G. & WAGNER, J. (2017). Micromechanical behavior in dependence of the microstructure
of nickel-based superalloys. Presentation at DPG conference in Dresden.
[P3] LAWITZKI, R., UL-HASSAN, S., KARGE, L., WAGNER, J., HOFMANN, M., GILLES, R. &
SCHMITZ, G. (2018). Advanced microstructural characterization of nanoprecipitates in
nickel-based superalloys. Presentation at DPG conference in Berlin.
[P4] LAWITZKI, R., GILLES, R., HOFMANN, M., WAGNER, J. & SCHMITZ, G. (2019). Combining
small-angle neutron scattering and analytical microscopy: An advanced method to
characterize nanoprecipitates in Ni-based superalloys. Poster at DPG conference in
Regensburg.
[P5] HITZLER, L., VON KOBYLINSKI, J., LAWITZKI, R., KREMPASZKY, C. & WERNER, E. (2020).
Microstructural Development and Mechanical Properties of Selective Laser Melted Co–
Cr–W Dental Alloy. BT - TMS 2020 149th Annual Meeting & Exhibition Supplemental
Proceedings. 195–202.
171
Acknowledgements
First, I would like to express my deepest gratitude to my academic supervisor Prof. Dr.
Dr. h.c. Guido Schmitz for his guidance and support during my time at the chair for
materials physics. Without his expertise and especially his dedication to discuss and
solve physical problems this work would not have been possible.
I would also like to thank Prof. Dr. Dr. h.c. Siegfried Schmauder for fruitful discussions
on numerical simulations and that he agreed in being the second reviewer of this the-
sis. Further, I thank Prof. Dr. Thomas Schleid for his interest in my work and for being
the main examiner of this dissertation.
This work was funded by the German Research Foundation (DFG). Therefore, I want
to thank the DFG and I want to thank my project partners Dr. Di Wang (INT at the
Karlsruhe Institute of Technology), Jonas Woste & PD Dr. Christian Krempaszky (both
WKM at TU Munich) and Dr. Michael Hofmann (FRM II at TU Munich) for the great
collaboration. Regular project meetings and especially common measurement times
have made this work very particular.
I would also like to thank the Heinz Maier-Leibnitz Zentrum (MLZ) for providing beam-
time and service at the instruments Stress-Spec and SANS-1 at the neutron reactor
FRM II in Garching. Particularly, my gratitude is to the small angle neutron scattering
experts Dr. Ralph Gilles and Dr. André Heinemann for their technical assistance and
advices during and after the SANS measurements.
A special appreciation goes to my colleagues at the chair of materials physics. Espe-
cially, I would like to mention Dr. Patrick Stender for his support and for many discus-
sions on all kind of scientifically related questions and Dr. Daniel Beinke for his assis-
tance in the numerical modelling of atom probe data. My gratitude goes also to our
technicians Peter Engelhardt and Frank Hack, the kind souls who kept the measure-
ment instruments at the institute always running. Further, I want to thank the many
successful collaborations with all postdocs and becoming PhD students Yug Yoshi, Dr.
Efi Hadjixenophontos, Dr. Gábor Csizár, Dr. Sebastian Eich, Samuel Griffiths, Helena
Solodenko and all those, who I did not mention, which made my time as a PhD student
very exciting and special.
172
I am also very thankful to my master and bachelor students Stefan Rotzsche (M.Sc.),
Philip Imhof (M.Sc.), Chang Luo (M.Sc.), Salman-ul-hassan (M.Sc.) and Carsten
Teucher (B.Sc.), who gave me the chance for their supervision, which helped me to
advance as a scientist.
Finally, I would like to gratefully thank my family, my friends and especially my future
wife Carina, who often came too short during late night or weekend experiments, but
who helped me to stay always positive and motivated, even in the most difficult situa-
tions.
173
Curriculum vitae
Personal Information:
Name: Robert Lawitzki
Address: 75391 Gechingen
Email: Robert.Lawitzki@mp.imw.uni-stuttgart.de
Birthday and -place: 02/01/1991 in Sindelfingen, Germany
Nationality: German
Education:
School education: 1997 - 2001: primary school in Weissach - Flacht
2001 - 2010: secondary school in Rutesheim
School qualification: General qualification for university entrance
Undergraduate studies: Bachelor of Materials Science
2011 - 2014: University of Stuttgart
Bachelor thesis: Quantitative analysis of field evaporation sequences
at nanometric tungsten tips
Postgraduate studies: Double Masters Degree Materials Science / Materials Engineering
2014 - 2015: Materials Science at the University of Stuttgart
2015 - 2016: Materials Engineering at Chalmers University of
Technology in Göteborg, Sweden
Master thesis: Microstructure of Z-phase strengthened martensitic
steels: Meeting the 650°C challenge
Doctoral degree: Since 01/09/2016: Ph.D. candidate at the University of Stuttgart
Dissertation: Analysis of nickel- and iron-based superalloys on smallest
length scales
Work experience:
01/07 - 31/12/2010: Military service: Gebirgsaufklärungsbataillon 230 in Füssen
02/03 - 12/08/2011 + Assistant at TRUMPF in the department for Human Resources
05/03 - 05/04/2012: and Social Services
01/12/2012 – 30/09/2013: Working student at TRUMPF in the department for Safety and En-
vironment
15/09/2014 – 30/09/2015: Student research assistant at the Institute of Materials Science at
the Chair of Material Physics in the University of Stuttgart
Topic: Atom probe tomography reconstruction tool
Since 01/09/2016: Student research assistant at the Institute of Materials Science at
the Chair of Material Physics in the University of Stuttgart
Topics: Teaching, supervision and research for PhD thesis
Since 01/11/2019: Additional project: Establishing a facility for a new highest resolu-
tion transmission electron microscope within SFB 1333.
174
175
Erklärung über die Eigenständigkeit der Dissertation
Ich versichere, dass ich die vorliegende Arbeit mit dem Titel
„Untersuchung von Nickel- und Eisen-basierten Konstruktionswerkstoffen auf
kleinsten Längenskalen“
selbständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel
benutzt habe; aus fremden Quellen entnommene Passagen und Gedanken sind als
solche kenntlich gemacht.
Declaration of Authorship
I hereby certify that the dissertation entitled
“Analysis of nickel- and iron-based superalloys on smallest length scales”
is entirely my own work except where otherwise indicated. Passages and ideas from
other sources have been clearly indicated.
________________________________
Robert Lawitzki, Stuttgart, 12.04.2021
