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Exploring the thermal evolution of nanomaterials: from nanometallic multilayers to nanostructures

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Exploring the thermal evolution of nanomaterials: from nanometallic multilayers to nanostructures

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Exploring the thermal evolution of nanomaterials:
From nanometallic multilayers to nanostructures

J. Sebastian Riano Z.

Dr. Andrea M. Hodge, Advisor

A Dissertation Presented to the Faculty of the USC Graduate School
In Partial Fulfillment of the Requirements for the Degree
Doctorate of Philosophy
(Materials Science)

University of Southern California

May 2019

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Acknowledgments

First, I would like to thank my advisor Professor Andrea Hodge for her unconditional support and patience.
Her insight was fundamental to obtain and analyze the results presented in this work. I cannot thank her
enough for her guidance and mentorship. She has taught how to be a scientist and a better person.

I would also like to express my gratitude to Professors Michael Kassner and Jayakanth Ravichandran for
serving on my dissertation committee. I hope that this works reflects all the lessons of thermodynamics and
diffusion that you taught me, which were important to understand the complex phenomena occurring in
nanomaterials.

I also want to thank all the members of the Hodge group: I-Chung, Mikhail, Leo, Teri, Nathan, Kamia,
Thien, Chelsea, Joel, Alina, Angelica, Karina, and Daniel, who were more than a group, they were family
and friends. Specifically, I am indebted to Leo for all of his help and for teaching me TEM and SEM. I
would like to thank Joel, Chelsea and Alina for their friendship and support during graduate school. All of
you made this experience significantly better.

I am grateful to John Curulli and Dr. Mathew Mecklenburg for their advice and for the opportunity of
working for two years at the Core Center of Excellence in Nano Imaging at USC, where I learned how to
characterize a variety of materials.

I also appreciate the support of the USC staff, who always considered the students a top priority.
Particularly, I am grateful for Kim Klotz, Andy Chen, Cory Reano, Jennifer Gerson, and the Mork Family
Department of Chemical Engineering and Materials Science.

This research would have not been possible without funding from the National Science foundation under
the grant DMR-1709771 and the Office of Naval Research under grant N00014-12-1-0638. Additionally, I
would like to thank the Viterbi School of Engineering for the Andrew and Erna Viterbi Graduate Student
Fellowship, which partially funded my doctoral studies.

I am grateful to my previous mentors, Professors Hugo Ricardo Zea and Luis Carlos Belalcazar at
Universidad Nacional de Colombia and Professor Ilona Kretzschmar at The City College of New York,
who guided me when I started doing research and who motivated me to embark on graduate school. I feel
lucky for the opportunity to learn from them.

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I would like to thank Dr. Deepak Goyal and Dr. Pilin Liu at Intel Corporation for the opportunity to do
research at this company, and for showing me that materials science can be used to design better electronic
products that have an impact in peoples’ life.

I am grateful for my wonderful parents Rosa Maria and Fernando for doing every possible effort to give
me the best possible education, for teaching me not to give up and to strive to pursue my dreams. Without
their unconditional support I would not have pursued graduate studies. I have been fortunate to have them
in my life.

I am also grateful for Pamela and Jeremy for treating me like another son. Last, but not least, I would like
to thank my partner Ryan, you have been my source of strength during the last years, thank you for always
supporting me in every possible way, and making my life better every day.

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Table of Contents
List of Figures ............................................................................................................................................ vii
List of Tables ............................................................................................................................................ xiii
Abstract ........................................................................................................................................................ 1
1 Introduction ......................................................................................................................................... 3
2 Background ......................................................................................................................................... 4
2.1 Nanocrystalline Materials............................................................................................................. 4
2.1.1 Nanometallic Multilayers .......................................................................................................... 5
2.1.2 Synthesis of Nanometallic Multilayers ..................................................................................... 7
2.2 Interfaces in Nanocrystalline Materials ....................................................................................... 9
2.2.1 Grain boundaries ..................................................................................................................... 10
2.2.2 Interface Interphases ............................................................................................................... 15
2.2.3 Complexions ........................................................................................................................... 18
2.3 Thermal Stability of Nanocrystalline Materials .......................................................................... 19
2.3.1 Diffusion in Nanocrystalline Materials ................................................................................... 20
2.3.2 Kinetic Approach .................................................................................................................... 24
2.3.3 Thermodynamic Approach ..................................................................................................... 26
2.3.4 Thermal Stability of NMMs .................................................................................................... 27
2.4 Thermodynamic Models of Segregation ..................................................................................... 30
2.4.1 Gibbs Segregation Isotherm .................................................................................................... 30
2.4.2 Gibbs Adsorption Equation ..................................................................................................... 31
2.4.3 Weissmüller Segregation Model ............................................................................................. 32
2.4.4 Interface Segregation Models Based on Surface Segregation Models.................................... 33
2.4.5 Numerical Calculation of Segregation Properties ................................................................... 38
2.5 Kinetics of Segregation ............................................................................................................... 41
2.5.1 Model of Limited Reaction Rates ........................................................................................... 41
2.5.2 Semi Infinite Solution of Fick Equation at the Grain Boundary ............................................. 41
2.5.3 Layer by Layer Models ........................................................................................................... 43
2.5.4 Phase field models .................................................................................................................. 47

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2.6 Microstructural Transformations in Nanomaterials ................................................................... 48
2.6.1 Recovery ................................................................................................................................. 50
2.6.2 Recrystallization...................................................................................................................... 51
2.6.3 Grain Growth .......................................................................................................................... 51
2.6.4 Precipitation ............................................................................................................................ 52
3 Experimental Methods ..................................................................................................................... 53
3.1 Synthesis of Nanometallic Multilayers ........................................................................................ 53
3.2 Characterization Methods........................................................................................................... 54
3.2.1 Profilometry of Thin Films ..................................................................................................... 55
3.2.2 X-ray Diffraction..................................................................................................................... 55
3.2.3 Scanning Electron Microscopy ............................................................................................... 56
3.2.4 Differential Scanning Calorimetry .......................................................................................... 58
3.2.5 Heat-treatment of NMMs ........................................................................................................ 58
3.2.6 Focus Ion Beam ...................................................................................................................... 59
3.2.7 Transmission Electron Microscopy......................................................................................... 61
3.2.8 Scanning Transmission Electron Microscopy ......................................................................... 63
4 Microstructural Evolution of NMMs: The Hf-Ti system .............................................................. 65
4.1 Synthesis of Hf-Ti NMMs ............................................................................................................ 65
4.2 Microstructural Transitions in HF-Ti NMMs ............................................................................. 66
4.3 Microstructural Evolution of Hf-Ti NMMs ................................................................................. 67
4.4 Recrystallization in Hf-Ti NMMs ................................................................................................ 69
4.5 Grain Boundary Energy of the Hf-Ti System .............................................................................. 69
4.6 Kinetics of Segregation ............................................................................................................... 71
4.7 Crystallization of the Amorphous Ti Precipitates ....................................................................... 72
4.8 Summary ..................................................................................................................................... 74
5 Grain morphology influence on the thermal stability of NMMs: The Ta-Hf system ................. 75
5.1 Synthesis of Ta-Hf NMMs ........................................................................................................... 75
5.2 Microstructure of Ta-Hf NMMs .................................................................................................. 76

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5.3 Interfaces in Ta-Hf NMMs .......................................................................................................... 77
5.4 Thermal evolution of Ta-Hf NMMs ............................................................................................. 78
5.5 Kinetic and thermodynamic stabilization of Ta-Hf NMMs ......................................................... 80
5.6 Summary ..................................................................................................................................... 81
6 Effect of phase transformations on the thermal stability of NMMs: The W-Cr system ............ 83
6.1 Synthesis of W-Cr NMMs ............................................................................................................ 83
6.2 Microstructure of W-Cr NMMs ................................................................................................... 84
6.3 Thermal evolution of W-Cr NMMs ............................................................................................. 85
6.4 W 3Cr precipitation ...................................................................................................................... 87
6.5 Structure of the W 3Cr precipitates .............................................................................................. 89
6.6 Summary ..................................................................................................................................... 90
7 Conclusions and Future Work ......................................................................................................... 91
7.1 Conclusions ................................................................................................................................. 91
7.2 Future Work ................................................................................................................................ 92
References .................................................................................................................................................. 95
Appendix A. Analysis of Characterization Data .................................................................................. 109
Appendix A1. Grain size calculation..................................................................................................... 109
Appendix A2. Diffraction patterns analysis .......................................................................................... 113
Appendix B. Summary of Sputtered Samples ...................................................................................... 117
Appendix B1. Hf-Ti NMMs ................................................................................................................... 117
Appendix B2. Ta-Hf NMMs ................................................................................................................... 120
Appendix B3. W-Cr NMMs ................................................................................................................... 121
Appendix C. Phase Diagrams ................................................................................................................ 123
Appendix C.1 Hf-Ti Phase Diagram ..................................................................................................... 123

vi

Appendix C.2 Ta-Hf Phase Diagram .................................................................................................... 124
Appendix C.3 W-Cr Phase Diagram ..................................................................................................... 125

vii

List of Figures
Figure 1 Yield strength, 𝜏𝑦 , as a function of the inverse square root of the grain size, 𝑑 − 1/2, for
nanocrystalline materials [4]. ....................................................................................................................... 4
Figure 2 Bright field TEM plan-view of (a) as-deposited nanocrystalline Ni and heat-treated for (b) ~1 s,
(c) 20 s, (d) 1 h, (e) 11 h, (f) 120 h at 420 ºC [5]. ......................................................................................... 5
Figure 3. As-sputtered 35 nm Cu/35 nm Nb nanometallic multilayers [8]. .................................................. 6
Figure 4. (a) W-Cr columnar multilayers showing columnar grains with several layers. (b) Nb-Mg equiaxed
(c) brick-like Ta-Hf nanometallic multilayers and showing individual grains between the layers. ............. 6
Figure 5. (a) Schematic representation of the accumulative roll-bonding process. (b) Bright field TEM
image of Cu-Nb nanometallic multilayers prepared by this technique [12]. ................................................ 7
Figure 6. (a) Cross-sectional TEM micrograph of 50 nm Cu / 50 nm α-CuNb (amorphous layer) NMMs.
(b) SAD patter of the multilayers[20]. .......................................................................................................... 8
Figure 7. Thornton diagram showing how columnar grains (Zone I) and equiaxed grains (Zone III) grow
via magnetron sputtering at different pressure and substrate temperatures. The regions in the middle (Zone
T and II) show the transition between the two grain structures [21]............................................................. 9
Figure 8. Dislocation models of (a) asymmetric and (d) symmetric grain boundaries. (c) shows a symmetry
boundary that has a dislocation on every lattice plane [22]. ....................................................................... 10
Figure 9. Low angle grain boundary energy as a function of the misorientation angle [24]. ..................... 11
Figure 10. Energy of symmetric high-angle grain boundaries based on calculations using a dislocation
model, where the dashed line is the contribution due to a uniform array of dislocations [27]. .................. 12
Figure 11. (a) Grain boundary structure displaying the misorientation 𝜃 (one degree), the axis of rotation
𝑀 (two degrees), and the orientation of the grain boundary 𝑁 (two degrees). (b) schematic representation
of the atomic structure at the grain boundary [28]. ..................................................................................... 12
Figure 12. Representation of [100] symmetric tilt grain boundaries (squares) in the (a) {013} (b) {024} and
(d) {037}, where parallel planes are indicated with triangles and circles [30]. .......................................... 13
Figure 13. Representation of 36.87° [100] grain boundaries, where the orientation is symmetric {013} (A
to B), asymmetric (001)/(034) (B to C), and symmetric {012} (C to D). ................................................... 14
Figure 14. Energy of symmetric tilt [110] Cu and Al grain boundaries as a function of the misorientation
angle [33]. ................................................................................................................................................... 15
Figure 15. Schematic representation of different systems that have interphases [22]. ............................... 16
Figure 16. Structure of (a) perfect coherent, (b) tilted coherent interfaces [35]. ........................................ 16
Figure 17. Structure of strained coherent interfaces [35]. ........................................................................... 17
Figure 18. Structure of (a) semi-coherent and (b) incoherent interfaces [35]. ........................................... 17

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Figure 19. Dillon Harmer complexions in Al 2O 3 (a) undoped and (b) doped with 100 ppm of Nd 2O 3, (c)
with 30 ppm of CaO, (d) 200 ppm of SiO 2, (e) 100 ppm of CaO, (f) and 30 ppm of CaO (impinged). (g) to
(l) display similar complexions in other systems [26]. ............................................................................... 18
Figure 20. (a) Diagram showing the different region for complexions as a function of normalized
disorientation and normalized bulk composition, where R 1= 0.6, R 2=1.2, and R 3=3, are reduced
misorientation (b) Grain boundary energy as a function of the bulk composition [39]. ............................. 19
Figure 21. Image showing logarithmic plots of the diffusivities in metals at free surfaces (𝐷𝑠 ) , inter granular
regions (𝐷𝑖𝑔 ) , dislocation-pipes (𝐷𝑑 ) , and inside lattices (𝐷 ) as a function of temperature [45]. ........... 20
Figure 22. Representation of systems with grain boundaries (a) parallel, (b) perpendicular, and (c) random
orientation in comparison to the diffusion direction [45]. .......................................................................... 21
Figure 23. Diffusion maps for (a) FCC and (b) BCC metals where the dominant regions for triple junction
(TJ), grain boundary (GB), and lattice or volume (V) diffusion can be compared [49]. ............................ 23
Figure 24. TEM image of electrodeposited Ni 23 at.% W heat-treated under Ar atmosphere at 700 °C for
24 hours [54]. .............................................................................................................................................. 24
Figure 25. (a) SEM and (b) TEM images of Cu-10 at% Nb after consolidation at 700 °C and heat-treatment
at 900° C for 1 h. (c) shows the grain size for pure Cu, and alloyed with Nb [55]. ................................... 25
Figure 26. (a) grain sizes of W - 20 at.% Ti and unalloyed W showing that the alloyed sample exhibits
minimum grain growth. TEM images of (b) the alloyed as-milled structure, and of (c) the unalloyed and (d)
alloyed samples after heat-treatment at 1100 °C for one week [43]. ......................................................... 26
Figure 27. Cross-sectional TEM images of Hf - 21 at.% Ti showing (a) the as-sputtered multilayers (b) the
equiaxed microstructure after heat-treatment at 800 °C for 96 h [50]. ...................................................... 27
Figure 28. Microstructures observed in an annealing simulation of multilayers A(41 nm, white)/B (5 nm,
black). (a) mesh-like morphology, (b) fiber-like morphology, (c) intermediate morphology, and (d)
dispersed morphology [79]. ........................................................................................................................ 28
Figure 29. Microstructures observed in annealing simulation of immiscible multilayers 6 nm A-B. (a) initial
microstructure, (b) interpenetrating morphology, and (c) coarsened morphology [79]. ............................. 29
Figure 30. Cross-section bright field micrographs of 15 nm Cu-Nb multilayers heat-treated (a) at 600 ºC
for 1 h, (b) at 700 ºC for 30 min, and at 700 ºC for 1h. Spheroidization is noticeable in (b) and (c) [8]. .. 29
Figure 31. Schematic representation of a general interphase limited by the planes AA’ and BB’, with
thickness, Δ𝑡 , and grain boundary energy, 𝛾𝐼 , along the plane SS’ [83]. ................................................... 30
Figure 32. Gibbs energy versus the global solute mole fraction for different gran sizes [89]. ................... 33
Figure 33. McLean segregation isotherm (Equation 24) as function of temperature using equation (27) to
calculate the enthalpy of segregation. The two curves show the behavior of segregation in clustering (Top)
and ordering (Bottom) systems [91]. .......................................................................................................... 35

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Figure 34. Normalized grain boundary energy for the system Fe-Zr as a function of the mole fraction of Zr
at the grain boundary for several grain sizes [95]. ...................................................................................... 36
Figure 35. Nanocrystalline stability maps where stable alloys appear in the top (green), metastable alloys
in the middle (yellow) and unstable alloys in the bottom (red) [101]. ........................................................ 38
Figure 36. Gibbs energy density (denoted by F) at the grain boundary (blue) and in the bulk of the grains
(red) as a function of the global solute composition [105]. ........................................................................ 39
Figure 37. (a) Initial columnar nanocrystalline copper. (b) Sample free of dopants annealed at 1200 K
showing grain growth. (c) Sample with dopants (red) at grain boundaries (green) annealed at 1200 K
showing retention of the grain size. (d) Close view of a junction in the sample with dopants (c) [106]. ... 39
Figure 38. (a) Gibbs energy of the system showing a minimum around a grain size of 5 mm for a system
with a solute concentration of 5 at.% with Δ𝐻𝑚𝑖𝑥 = 20 kJ/mol and Δ𝐻𝑠𝑒𝑔 = 20 kJ/mol. Systems with a
solute concentration of (b) 1 at.% (c) 2 at.% (d) 10 at.% and (e) 15 at.% [107]. ........................................ 40
Figure 39. Grain boundary coverage as a function of normalized time for strongly segregating systems
(𝛽𝑎𝑔𝑏 ≫ 1) [111]. ...................................................................................................................................... 43
Figure 40. Gibbs energy as a function of distance showing the activation energy as grain the layers at the
interphase Δ𝐷𝐷 + Δ𝐺 (Φ) and in the bulk of the grains Δ𝐷𝐷 [23]. ........................................................... 44
Figure 41. Segregation profile of Sn (solute) at a Cu surface as a function of time. Sn segregation occurs as
the surface energy of copper decreases[112]. ............................................................................................. 45
Figure 42. Different subsystems in which the region around the grain boundary is divided. Φ stands for the
grain boundary and B the bulk of the grains[113].................................................................................... 45
Figure 43. Surface segregation profiles calculated as a function of time for a system with Δ𝐺 1 = 60 kJ/mol
[113]. ........................................................................................................................................................... 46
Figure 44. Solute partitioning between the grain boundaries and the bulk of the grains (Υ = 𝑛𝑏𝑔𝑏 /(𝑛𝑏𝑔𝑏 +
𝑛𝑏𝑏 ) ) as a function of time for systems with varying chemical potential difference between the gran
boundaries and the bulk of the grains (𝐺𝑔𝑏𝐵 − 𝐺𝑏𝐵 ) [105]. ................................................................... 47
Figure 45. Classification of transformations in terms of the growth process [115]. ................................... 49
Figure 46. Schematic representation of the magnetron sputtering deposition process. .............................. 53
Figure 47. Sputtering process of nanometallic multilayers, (a) sputtering from source 1 while a shutter
blocks the source 2, (b) sputtering from source 2 while a shutter blocks the source 1. .............................. 54
Figure 48. Image of (a) a Ta-Hf film being scanned by profilometry and (b) of the profilometry scan. .... 55
Figure 49. X-ray diffraction scan of W-Cr nanometallic multilayer showing a highly (110) BCC texture.
.................................................................................................................................................................... 56
Figure 50. (a) Signals produced by the interaction between electrons and the sample and (b) their
corresponding energy [13, 126] . ................................................................................................................ 57

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Figure 51. DSC scans of nanocrystalline Cu (40 nm grain size) stored at 20 °C for 20 min, at 20 °C for 5
days, and at -20 °C for 5 days, showing different recrystallization enthalpies [127]. ............................... 58
Figure 52. Schematic of vacuum furnace setup used in this study. ............................................................ 59
Figure 53. Schematic of dual FIB-SEM system used to perform FIB lift-outs [128]. ................................ 60
Figure 54. Illustration of the lift-out procedure showing (a) the top view of the sample coated with a
protective carbon coating, (b) lamella ready to be extracted from the sample, (c) after attaching it to the
micromanipulator, and (d) a sample thinned to electron transparency. ...................................................... 60
Figure 55. (a) Bright field mode, (b) dark field mode, and (c) selected area diffraction mode [88]. Cross-
sectional TEM micrographs of Hf-Ti nanometallic multilayers showing (d) a bright field image, (e) a dark
field image, and (f) a selected area diffraction pattern. ............................................................................... 62
Figure 56. Cross-sectional (a) bright field STEM image and (b) high angle annular dark field images
showing Hf-rich grains and Ti precipitates at the grain boundaries of Hf-Ti NMMs heat-treated at 800 ºC.
.................................................................................................................................................................... 63
Figure 57. Energy dispersive x-ray spectroscopy maps of Hf-Ti NMMs heat-treated at 800 ºC showing (a)
a composed map, (b) a Hf map, (c) a Ti map, (d) and an O map. ............................................................... 64
Figure 58. (a) Bright field TEM and (b) dark field TEM images and (c) the grain size distribution of the as-
sputtered Hf-Ti multilayers. ........................................................................................................................ 66
Figure 59. Differential scanning calorimetry scan of the Hf-Ti nanometallic multilayers 20 °C to 1000 ºC
showing the multiple transitions occurring between 494 ºC and 990 ºC. ................................................... 67
Figure 60. Cross-sectional bright field TEM images and grain size distributions of samples heat-treated
(b,e) at 500 ºC (breakthroughs in the Ti layers have been circled in yellow), (c,f) at 800 ºC (inset shows a
nanodiffraction pattern of the amorphous Ti precipitates, the diffraction spots evidence short-range order),
and (d,g) at 1000 ºC (inset shows a selected-area diffraction pattern of a crystallized Ti precipitate in the
[221] zone axis)........................................................................................................................................... 68
Figure 61. Normalized grain boundary energy as a function of composition for different temperatures
calculated for a grain size of 50 nm. ........................................................................................................... 71
Figure 62. Time for grain boundary saturation as a function of temperature for a diffusion length of 25 nm,
the dotted red line is the heat treatments time in this study (3.5 x 10
5
s). ................................................... 72
Figure 63. EDS scans of Ti precipitates (bright regions) and Hf grains (dark regions) of samples heat treated
(a) at 800 ºC and (b) at 1000 ºC. The red arrows indicate the respective regions where the scans were taken
from. ............................................................................................................................................................ 73
Figure 64. Bright field TEM images showing (a) the bimodal multilayers, (b) columnar grains at the top,
and (c) brick-like grain at the bottom of the film. The dark-field TEM images display the corresponding

xi

grain morphologies for (d) columnar and (e) brick-like regions. The grain size distributions for the columnar
and the brick-like grains are shown in (f) and (g) respectively. .................................................................. 76
Figure 65. High resolution TEM images of (a) the brick-like grains, showing atomic lattices that extend
over a single layer, and of (b) the columnar grains, displaying atomic layers extending over several layers.
The yellow lines indicate the orientation of atomic layers inside the grains. The FFT patterns (i) to (iv) show
that the brick like grains in consecutive layers have different orientations, while the patterns from (v) to
(viii) indicate that the columnar subgrains have the same orientation. ....................................................... 78
Figure 66. Differential scanning calorimetry scan of the Ta-Hf nanometallic multilayers from 20 °C to 1000
ºC showing that recrystallization and grain growth occur between 482 ºC and 842 ºC. ............................. 79
Figure 67. Bight field, dark field TEM and corresponding grain size distributions of the multilayers
annealed at 550 ⁰C (a,b,c and d), and at 1000 ⁰C (e,f,g and h) after 96 h. The insets in (a) and (e) show the
corresponding SAD patterns with the Hf rings colored green. The dark-field TEM images indicate that
while at both temperature the structure of the columnar grains is preserved, most of the brick-like grains
have recrystallized at 500 ⁰C. Similar grain size distributions are observed at 550 °C (d) and at 1000 °C (g)
suggesting no significant grain growth. ...................................................................................................... 80
Figure 68. (a) Bright field STEM image highlighting the structure of the multilayered samples after
annealing at 1000 °C, where the yellow box indicates an area with both columnar and brick-like grains.
This region was studied in more detail by collecting additional (b) STEM images and (c) EDS maps, which
display Ta rich grains (Turquoise) surrounded by Hf precipitates (Orange). ............................................. 81
Figure 69. (a) Cross-sectional bright field STEM and (b) integrated radial intensity profile of as-sputtered
W-Cr nanometallic multilayer. Only α-Tungsten peaks are observed in the profile, indicating that the β-
Tungsten phase is not present in the sample. .............................................................................................. 84
Figure 70. Differential scanning calorimetry scan of the W-Cr nanometallic multilayers from 20 °C to 1000
ºC showing that recrystallization and grain growth occur between 690 ºC and 990 ºC. ............................. 85
Figure 71. Cross-sectional bright field STEM images and integrated radial intensity profile of W-Cr
nanometallic multilayers heat-treated at (a-d) 550 ºC, (b-e) 800 ºC and (c-f) 1000 ºC. The inset in the STEM
images shows the diffraction patterns and the orientations indexed in the integrated radial intensity profiles.
The images display that while at 500 ºC the as-deposited microstructure is retained, at 800 ºC some
recrystallized grains are observed between the multilayers. At 1000 ºC, when grain growth has occurred the
sample is comprised of W rich grains surrounded by amorphous W 3Cr precipitates. ................................ 86
Figure 72. EDS scans of (a) recrystallized grains at 800 ºC and of (b) W rich grains surrounded by W 3Cr
precipitates rich in Ar. The arrows point to the corresponding regions in the HAADF images. The scans
show that at 800 ºC the recrystallized grains have enriched in Cr. At 1000 ºC, when grain growth has

xii

occurred, the W rich grains (Gray grains) have further enriched to 40 at.% Cr. At this temperature W 3Cr
precipitates (Black grains) are observed at the grain boundaries. ............................................................... 88
Figure 73. High resolution bright field TEM images of (a) a W rich grain surrounded by W3Cr precipitates
and (b) an amorphous W3Cr precipitate at 1000 ºC. The insets show the SAD patterns of the corresponding
regions. (a) displays a crystalline BCC structure, while (b) shows an amorphous W3Cr precipitate
surrounded by a diffusion zone. .................................................................................................................. 90
Figure 74. Delineated (a) columnar and (b) equiaxed grains in Ta-Hf NMMs using the TsView7 software
showing the area and perimeter of each grain. .......................................................................................... 109
Figure 75. (a) Initial dark-field TEM micrograph of the as-sputtered W-Cr NMMs and (b) image processed
using the code to improve contrast. .......................................................................................................... 111
Figure 76. (a) dark-field TEM micrograph of the as-sputtered W-Cr NMMs after the change of the color
threshold, and (b) image showing the grains identified using the methods in ImageJ. ............................ 112
Figure 77. (a) Diffraction pattern of a Ti grain within the Hf-Ti NMMs annealed at 1000 ⁰C, and (b)
symmetry patter in the [221] zone axis used to index the diffraction pattern [130, 213]. ........................ 113
Figure 78. Nanodiffraction pattern of a W 3Cr grain showing a ring corresponding to the amorphous nature
of these precipitates and diffraction spots form the surrounding W rich grains. ...................................... 114
Figure 79. Rings diffraction pattern of a W-Cr sample annealed at 1000 ⁰C analyzed using
ProcessDiffraction [216]. .......................................................................................................................... 115
Figure 80. Average relative intensity profile of the W-Cr NMMs annealed at 1000 ⁰C. ......................... 116
Figure 81. (a) Rings pattern showing the half-circle schematic of a W-Cr sample annealed at 1000 ⁰C, and
(b) the corresponding radial intensity profile showing W BCC and Cr BCC peaks. ................................ 116
Figure 82. Hf-Ti phase diagram showing in red the 20.1 at % Ti isopleth. This is the global composition of
the Hf-Ti NMMs [218]. ............................................................................................................................ 123
Figure 83. Ta-Hf phase diagram showing in red the 22.4 at % Hf isopleth. This is the global composition
of the Ta-Hf NMMs [218]. ....................................................................................................................... 124
Figure 84. W-Cr phase diagram showing in red the 33.1 at % Cr isopleth. This is the global composition of
the W-Cr NMMs [218]. ............................................................................................................................ 125

xiii

List of Tables
Table 1. Nanocrystalline materials stabilized up to T max/T melt via thermodynamic approach [44]. ........... 27
Table 2. Coefficients to evaluate the proclivity for nanocrystalline stability [102] .................................... 37
Table 3. Sputtering conditions used to deposit the Hf-Ti NMMs ............................................................. 117
Table 4. Properties of the as-sputtered Hf-Ti NMMs ............................................................................... 118
Table 5. Heat-treatments of the Hf-Ti NMMs .......................................................................................... 119
Table 6. Sputtering conditions used to deposit the Ta-Hf NMMs ............................................................ 120
Table 7. Properties of the as-sputtered Ta-Hf NMMs............................................................................... 121
Table 8. Heat-treatments of the Ta-Hf NMMs ......................................................................................... 121
Table 9. Sputtering conditions used to deposit the W-Cr NMMs ............................................................. 121
Table 10. Properties of the as-sputtered W-Cr NMMs ............................................................................. 122
Table 11. Heat-treatments of the W-Cr NMMs ........................................................................................ 122

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1

Abstract
Nanocrystalline materials have interesting electrical, magnetic, and mechanical properties, especially when
compared to their coarse-grained counterparts. For example, the Hall-Petch relation shows that
nanomaterials have a greater yield strength than coarse grained systems. However, the application of
nanocrystalline materials has been limited by their low thermal stability, which stems from their high
density of interfaces that act as channels for diffusion. Although several attempts have been made to
improve the thermal stability of nanomaterials via kinetic or thermodynamic mechanisms, there are limited
alternatives to induce the desired stabilization. Thus, there is a need in this field to investigate possible
routes to promote the formation of nanostructures with a decreased propensity for grain growth at elevated
temperatures. In this work, nanometallic multilayers (NMMs) are used to drive microstructural and phase
transformations that result in the formation of stabilized nanostructures after protracted annealing. NMMs
were selected because they allow for control over the local composition, the density of interphases, and the
grain structure of nanocrystalline samples. In the studies presented in this dissertation, the effect of the
initial microstructure on the thermal progression from nanomultilayers to nanostructures was studied by
characterizing Hf-Ti, Ta-Hf, and W-Cr NMMs heat-treated at critical temperatures. Overall, the results
from this investigation highlight a new path to synthesize stable nanomaterials via annealing of NMMs.

2

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3

1 Introduction
Nanocrystalline materials have interesting mechanical, electrical, magnetic and optical properties due to
their high density of grain boundaries. However, the application of these materials has been limited by their
sample dimensions, availability of alloy systems, brittleness, and especially their low thermal stability. In
particular, annealing of nanomaterials at elevated temperatures results in deterioration of the nanograin
structure and consequentially, the degradation of the exceptional properties of nanocrystalline materials.
Although there has been significant research, the low thermal stability of nanocrystalline materials still
hinders future progress.

Extensive efforts have been made to increase the thermal stability of nanocrystalline materials via two main
approaches. First, in a kinetic approach, the grain boundary mobility is decreased through several
mechanisms including Zener pinning, solute drag, and chemical ordering. Second, in a thermodynamic
approach, the driving force for grain growth is reduced by inducing solute segregation to the grain
boundaries. These stabilization approaches can be activated by promoting the formation of special
microstructures via thermally activated transformations, which result in nanostructures with increased
thermal stability.

Therefore, the objective of this project is to improve the thermal stability of nanocrystalline materials by
using nanometallic multilayers (NMMs), which can evolve into stabilized nanostructures after prolonged
annealing. NMMs consist of alternating metallic layers of nanoscale thicknesses that can be synthesized to
control the initial grain size, the local composition, and the density of interfaces. These microstructural
characteristics can be tailored to drive the formation of nanostructures stabilized by the aforementioned
mechanisms, which makes NMMs a suitable system to enhance the thermal stability of nanomaterials.

In the studies presented in this dissertation, NMMs of the Hf-Ti, Ta-Hf and W-Cr systems, which show
proclivity for nanograin stability in thermodynamic models, were synthesized to study the progression from
nano multilayers to stabilized nanostructures. The multilayers were heat-treated at critical temperatures and
characterized by several techniques to explore the microstructure before and after thermally activated
transformations. The analysis of microstructural characteristics at various stages of thermal evolution
helped to understand how the initial structure of the NMMs drives the formation of nanostructures stabilized
by either thermodynamic or kinetic mechanisms. Overall, the processes resulting in the formation of these
nanostructures were correlated to experimental data, as well as to thermodynamic and kinetic calculations,
in order to improve the thermal stability of nanocrystalline materials.

4

2 Background
2.1 Nanocrystalline Materials
Nanocrystalline materials by definition have an average grain size smaller than 100 nm, and usually have
a significant fraction of atoms located at interfaces. For example, when the grain size is 10 nm, around 30
% of the atoms are located at grain boundaries, which increases the energy of the system [1, 2]. Additionally,
nanomaterials have a high density of defects like grain boundaries, triple junctions and quadrupole points,
which induce strain fields that displace the atomic lattices from their equilibrium position. The combined
effect of the grain boundary energy and the strain by defects modifies the properties of nanocrystalline
materials in ways not observed in coarse grain materials.

The exceptional properties of nanocrystalline materials include enhanced diffusivity, active catalytic
properties, increased electrical resistivity, and increased mechanical stability [3]. For example, the strength
of nanograined materials is between one and two orders of magnitude greater than that of coarse grain
materials. This is expressed by the Hall-Petch relation [4]:

𝜏 𝑦 = 𝜏 0
+ 𝑘𝑑
−1/2
(1)
Where 𝜏 𝑦 is the yield strength and 𝑑 is the grain size. Figure 1, a graphical representation of equation (1),
shows that the yield strength of nanocrystalline materials increases as the grain size decreases.

Figure 1 Yield strength, 𝜏 𝑦 , as a function of the inverse square root of the grain size, 𝑑 −1/2
, for
nanocrystalline materials [4].

5

Although nanocrystalline materials exhibit interesting properties, they usually have low thermal stability.
This is due to their high density of interfaces, which increase the energy and cause the system to be more
prone to thermal transformations. Therefore, processes that cause degradation of the nanograined structure
are easily activated, resulting in the deterioration of the material properties. For instance, the grain size of
nanocrystalline Ni increases from 25 nm to 500 nm after heat-treatment for 1 h at 420 ºC (Figure 2) [5].

Figure 2 Bright field TEM plan-view of (a) as-deposited nanocrystalline Ni and heat-treated for (b) ~1 s,
(c) 20 s, (d) 1 h, (e) 11 h, (f) 120 h at 420 ºC [5].

These processes, which are driven by excess energy accumulated at interfaces within the material, occur at
rates that depend on the local temperature and composition [6]. Therefore, a high density of interfaces can
facilitate the activation of such processes. This is especially important for nanometallic multilayers, which
are nanocrystalline systems with two or more alternating layers of different phases in contact.

2.1.1 Nanometallic Multilayers
Nanometallic multilayers (NMMs) are a type of nanostructured materials which consist of metallic layers
of alternating composition and individual thicknesses on the scale of nanometers; an example of Cu/Nb
NMMs is presented in Figure 3 [7]. The multilayered geometry allows for control of the density of

6

interphases, grain size, and local composition. The density of interphases in NMMs, which depends on the
individual layer thicknesses, can be increased or reduced by changing the individual layers thickness.

Figure 3. As-sputtered 35 nm Cu/35 nm Nb nanometallic multilayers [8].

NMMs have either columnar or equiaxed grain structures. Columnar multilayers are comprised of columnar
grains that extend over several layers (Figure 4a). These grains have an internal structure with subgrains
that correspond to the thicker layers, which are bounded by thinner layers forming semi-coherent interfaces
[9]. For example, Figure 4a, shows the structure of columnar W-Cr NMMs. In contrast, equiaxed
multilayers consist of grains that only extend over a single layer and do not have an internal structure
(Figure 4b). A special type of equiaxed system is compressed NMMs, which are comprised of cylindrical
grains with an in-plane size larger than the layer thickness. Due to the compressed multilayers resemblance
of a brick wall in two-dimensional micrographs, such systems are referred to as “brick-like NMMs” (Figure
4c)[10, 11].

Figure 4. (a) W-Cr columnar multilayers showing columnar grains with several layers. (b) Nb-Mg
equiaxed (c) brick-like Ta-Hf nanometallic multilayers and showing individual grains between the layers.

7

The grain size of NMMs depends on the microstructure of the multilayers. The grain size of columnar
NMMs is determined in a two-step process. First, the volume 𝑉 of each grain is estimating assuming a
conical shape. Second, the diameter of a sphere with volume 𝑉 is calculated using the equation 𝑑 = √6𝑉 /𝜋 3
.
This diameter is the size of columnar grains. Although a similar procedure can be followed to find the grain
size of equiaxed NMMs, usually their grain size is close to the thickness of the individual layers. Overall,
the structure of NMMs will depend on the synthesis conditions.

2.1.2 Synthesis of Nanometallic Multilayers
The methods that can be used to synthesize NMMs are categorized into bottom-up and top-down
approaches. In the top-down approach, a macroscopic bulk sample is deformed by techniques such as
rolling until the layers inside the material are of nanometer thickness. In contrast, in bottom-up approaches
the NMMs are prepared layer-by-layer.

2.1.2.1 Top-Down Synthesis of NMMs
The main top-down approach to synthesize NMMs is accumulative roll-bonding (ARB) [12]. In this
technique, a sandwich of multiple layers of a few microns and alternating composition is rolled multiple
times to reduce the thickness of the layers. After a certain number of rolling cycles, the sample is cut in half
and the pieces are staked together. The rolling and cutting cycles continue until the individual layers reach
the desired thickness. In addition, annealing series may be performed after each cycle to enhance the
bonding between the layers. This procedure is schematically represented in Figure 5.

Figure 5. (a) Schematic representation of the accumulative roll-bonding process. (b) Bright field TEM
image of Cu-Nb nanometallic multilayers prepared by this technique [12].

8

2.1.2.2 Bottom-Up Synthesis of NMMs
In bottom-up techniques the NMMs are prepared layer-by-layer via different methods including chemical
vapor deposition (CVD), electroplating, and physical vapor deposition (PVD). In CVD, a volatile
compound of a metal reacts with a gas to produce a non-volatile solid that deposits on a substrate [13]. This
technique allows for control over the stoichiometry of the film, as the composition of each layer can be
tailored using multiple evaporation sources where each precursor is individually heated. A variety of
precursors can be used depending on the metals being deposited [14, 15]. This technique is mostly used in
the semiconductor industry to deposit films with a low density of defects [16].

In electrodeposition, dissolved metallic cations are reduced to form a metallic coating on top of an electrode.
NMMs can be electrodeposited using single-bath (one electrolyte) or dual-bath methods (two electrolytes)
[17]. In single bath deposition, compositional or structural multilayers are deposited by periodically
changing the reduction conditions. In contrast, during deposition using dual baths, the substrate is moved
between two electrolytes such that each alternating layer is deposited from a different solution [18].

During PVD processes, an initial solid or liquid is vaporized, this substance is later transported through a
vacuum environment, and finally deposited onto a substrate [19]. PVD can be used to deposit crystalline or
amorphous layers. Although there are several PVD techniques, including arc vapor evaporation, molecular
beam epitaxy, and ion plating, NMMs are predominantly deposited by magnetron sputtering. An example
of sputtered NMMs showing crystalline and amorphous layers is presented in Figure 6.

Figure 6. (a) Cross-sectional TEM micrograph of 50 nm Cu / 50 nm α-CuNb (amorphous layer) NMMs.
(b) SAD patter of the multilayers[20].

9

Magnetron sputtering offers two main advantages for the deposition of NMMs. First, this technique allows
for precise control of deposition rate by changing the target polarization. Second, by controlling the
deposition pressure and substrate temperature it is possible to tailor either columnar or equiaxed
microstructures as summarized by the Thornton diagram (Figure 7), which is constructed based on the fact
that the formation of columnar grain is driven by shadowing effects and surface diffusion, while the
formation of equiaxed grains is due to recrystallization.

Figure 7. Thornton diagram showing how columnar grains (Zone I) and equiaxed grains (Zone III) grow
via magnetron sputtering at different pressure and substrate temperatures. The regions in the middle
(Zone T and II) show the transition between the two grain structures [21].

Therefore, it is possible to synthesize equiaxed or columnar multilayers via magnetron sputtering by
controlling the deposition conditions. These grain structures have different grain and subgrain interfaces,
which result in distinct rates for microstructural transformations [22]. Thus, details on the structure of
interfaces will be explored in the following section.

2.2 Interfaces in Nanocrystalline Materials
In general, solid-solid interfaces are dividing surfaces between two homogenous regions, which have
atomic structures that depend on the composition, crystallographic structure, and orientation of the crystals
in contact [23]. This structure consists of atomic patterns that are periodically repeated along the surface
[24]. Interfaces can be classified depending on the number of phases in contact at grain boundaries, phase
boundaries, or complexions. Grain boundaries are interfaces between grains of the same composition and

10

crystalline structure that only differ in orientation [25]. In contrast, interphases occur between grains of
different composition or crystalline structure. Finally, complexions are new nanoscale phases between two
grains [26]. Overall, it has been observed that these three types of interfaces extend over a distance of some
atomic diameters with structures that are studied in more detail below [23].

2.2.1 Grain boundaries
Grain boundaries are transition regions where two crystals of different orientations are in periodic
coincident configurations. The structure of the grain boundaries has been explored using different models,
including dislocation arrangements, structural models, and coincident site lattices (CSLs) [22]. These
models are used to understand properties like the energy and the strain fields of grain boundaries. In their
simplest form, grain boundaries can be visualized as an array of dislocations [22]. For example, Figure 8
illustrates asymmetric and symmetric grain boundaries, where θ, the misorientation, is the angle that one
of the crystals needs to be rotated around the axis 𝑀 to make the lattices of both grains coincide. For this
model to be valid the dislocations must be uniformly spaced.

Figure 8. Dislocation models of (a) asymmetric and (d) symmetric grain boundaries. (c) shows a
symmetry boundary that has a dislocation on every lattice plane [22].

When the misorientation is 𝜃 < 15
0
the grain boundary is comprised of dislocations spaced at a distance
𝐷 , such that 𝜃 = 𝑏 ⃗
/𝐷 , were 𝑏 ⃗
is the Burgers vector [24]. Each of these dislocations has an associated elastic

11

strain energy per unit of length 𝐸 ⊥
. Thus, the grain boundary energy (𝛾 𝑔𝑏
) is given by the product of this
energy and the dislocation density 1/𝐷 , that is 𝛾 𝑔𝑏
= 𝐸 ⊥
/𝐷 = 𝐸 ⊥
(𝜃 /𝑏 ) . The same expression per unit of
area is:

𝛾 𝑔𝑏
= 𝐸 0
𝜃 (𝐴 0
− ln𝜃 ) (2)
Where:

𝐸 0
=
𝜇𝑏
4𝜋 (1−𝑣 )
(3) 𝐴 0
= 1 + 𝑙𝑛 (
𝑏 2𝜋 𝑟 0
) (4)
From equations (2) to (4) it is possible to calculate the energy of low angle grain boundaries as a function
of the misorientation:

Figure 9. Low angle grain boundary energy as a function of the misorientation angle [24].

The increase of the grain boundary energy observed in Figure 9 is due to the fact that the density of
dislocations increases proportionally with the misorientation angle 𝜃 . This results in a higher grain
boundary energy for low angle grain boundaries with misorientation close to 𝜃 = 15°. Although the
dislocation model was only developed for low-angle grain boundaries, it can be extrapolated to high angle
grain boundaries. However, as the misorientation increases, the dislocations are not evenly spaced, and
instead it is necessary to represent the grain boundary as a combination of two dislocations arrays, one
uniform and the other non-uniform. Thus, the energy of a symmetric tilt high-angle grain boundary is

12

comprised of contributions from the uniform array of dislocations and from the angular difference between
both distributions (Figure 10).

Figure 10. Energy of symmetric high-angle grain boundaries based on calculations using a dislocation
model, where the dashed line is the contribution due to a uniform array of dislocations [27].

Please note that the structure of the grain boundaries has been described using 3 degrees of freedom, the
misorientation θ and two of the dihedral angles of the rotation axis 𝑀 . However, a better description of the
grain boundary also indicates the orientation of the normal to the grain boundary 𝑁 , see Figure 11 [28].
These dihedral angles add another 2 degrees of freedom, and thus in total 5 degrees are needed to completely
describe a grain boundary.

Figure 11. (a) Grain boundary structure displaying the misorientation 𝜃 (one degree), the axis of rotation
𝑀 (two degrees), and the orientation of the grain boundary 𝑁 (two degrees). (b) schematic representation
of the atomic structure at the grain boundary [28].

13

In general, dislocation models are unable to represent the more complex structure of high-angle grain
boundaries, which have misorientation 𝜃 > 15°, because as 𝜃 increases the dislocation cores enlarge,
deform, and combine to form continuous grain boundaries with more complex structures including ledges,
double ledges, serrations, protrusions, and dislocation networks [22]. Due to the limitations of dislocation
models, structural models have been proposed to describe high-angle grain boundaries [23]. Based on these
models, the position of the atoms at the interface is indicated using a limited number of convex polyhedra,
which include tetrahedrons, pentagonal bipyramids, octahedrons, and dodecahedrons that can be combined
to create other structural units [29]. Thus, a grain boundary with the structural unit |𝐴𝐵 .𝐴𝐵 | is formed by
combining the structural types |𝐴 .𝐴 | and |𝐵 .𝐵 |. These structural units obey the relation |𝐴 .𝐴 | + |𝐵 .𝐵 | →
|𝐴𝐵 .𝐴𝐵 |, which matches the correspondence between Miller indices [23]. For example, Figure 12 shows a
[100] BCC symmetric tilt boundaries with boundary planes {013}, {024} and {037}, which obey the
relation {013} + {024} → {037}.

Figure 12. Representation of [100] symmetric tilt grain boundaries (squares) in the (a) {013} (b) {024}
and (d) {037}, where parallel planes are indicated with triangles and circles [30].

As mentioned previously, the energy of low-angle grain boundaries increases monotonically with the
misorientation. In contrast, the energy of high-angle grain boundaries does not increase monotonically and
instead changes depending on the atomic matching at the interface [31]. When the atomic matching is close
to the structure inside the grains, the interface energy is at a minimum, and the high-angle grain boundary
is denoted as “singular” [23]. On the contrary, when the atomic matching is different from the arrangement
inside the grains, the interface energy increases, and the grain boundary is called “general”. This type of
grain boundary consists of a combination of structural units. An example is shown in Figure 12c.
Additionally, a “vicinal” boundary is a “singular” grain boundary that has been modified by an array of
dislocations. These interfaces combine the properties of both “singular” and “general” configurations.
Finally, the term “special” is used for grain boundaries that maximize the value of properties including
toughness, diffusivity, or corrosion resistance. The properties of high angle grain boundaries can be studied

14

in detail using the coincident site lattice (CSL) approach [31]. This approach is based on the fact that the
grain boundary energy decreases due to increasing atomic bonding as the atomic matching at the interface
increases. Thus, for a given high angle grain boundary the density of coincident sites, where the lattices of
the two crystals at the interface coincide, is computed using the expression [32]:

Σ =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑠𝑖𝑡𝑒𝑠 𝑖𝑛 𝑎 𝑛 𝑒𝑙𝑒𝑚𝑒𝑡𝑎𝑟𝑦 𝑐𝑒𝑙𝑙 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑛𝑒𝑟 𝑜𝑓 𝑠𝑖𝑑𝑒𝑡𝑠 𝑖𝑛 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 𝑐𝑒𝑙𝑙 = 𝛿 (ℎ
2
+ 𝑘 2
+ 𝑙 2
) (5)
Where Σ is the reciprocal of the density of coincident sites, (ℎ,𝑘 ,𝑙 ) are the miller indices of the grain
boundary orientation, and 𝛿 is a constant where 𝛿 = 1 if (ℎ
2
+ 𝑘 2
+ 𝑙 2
) is odd or 𝛿 = 1/2 if
(ℎ
2
+ 𝑘 2
+ 𝑙 2
) is even. For example, Figure 13 shows the atomic pattern for three grain boundaries with a
misorientation relation of 36.87 ⁰ [100]. The first grain boundary, extending from A to B, is symmetric with
a boundary plane {013}. The second one, which extends from B to C is asymmetric with grain boundary
planes (001)/(034). Finally, the third grain boundary, extending from C to D, is symmetric with boundary
plane {012}. Please notice that both symmetric grain boundaries mentioned above have the same Σ
value, Σ = 5.

Figure 13. Representation of 36.87° [100] grain boundaries, where the orientation is symmetric {013} (A
to B), asymmetric (001)/(034) (B to C), and symmetric {012} (C to D).

The Σ value from the CSL approach can be used to understand the grain boundary energy dependence on
the misorientation [33]. For example, the energy of symmetric tilt [11
̅
0] grain boundaries of Cu and Al are
presented in Figure 14. This diagram indicates energy cusps at particular misorientations corresponding to
singular grain boundaries, for example the boundaries in the (113) and (111) planes in Figure 14.

15

Figure 14. Energy of symmetric tilt [11
̅
0] Cu and Al grain boundaries as a function of the misorientation
angle [33].

As the misorientation deviates from the singular values, the number of coincident sites decreases as the
energy of the grain boundaries increases. However, if this deviation is small, the misorientation difference
is accommodated by new individual dislocations, which results in vicinal grain boundaries with similar
properties as special boundaries [31]. Thus, vicinal grain boundaries are usually related to a Σ value by an
angular deviation 𝜈 . The maximum angular deviation from the singular misorientation corresponding to a
specific Σ value is denoted by 𝜈 𝑚 , which is computed using Brandon’s criterion the expression [34]:

𝑣 𝑚 =
𝑣 0
Σ
𝜉 (6)
Where 𝑣 0
= 15 and 𝜉 = 0.5. Equations (5) and (6) are commonly used to characterize high-angle grain
boundaries. In the case of low-angle grain boundaries Σ is taken to be Σ = 1. Overall, the CSL method is
employed to classify grain boundaries and understand their energy in grain boundary engineering studies
[32].

2.2.2 Interface Interphases
In contrast to grain boundaries, interphase interfaces separate grains with different composition and/or
crystalline structures [24]. Interphases are present in any system with two or more phases and usually appear
after diffusional processes such as precipitation or eutectic transformations. For example, Figure 15
summarizes several interphases where two phases are in contact at the boundary SS’ [22].

16

Figure 15. Schematic representation of different systems that have interphases [22].

The structure of interphases is explained in terms of the same microstructural features used to describe grain
boundaries such as dislocations, ledges, and dislocation networks. Although interphases do not have a single
crystallographic orientation normal to an interface plane, it is possible to describe the orientation of the
planes at each side of the interphase by relations of the type {h 1k 1l 1} A || {h 2k 2l 2} B , where A and B are the
phases at the interface [22]. In addition to these relations, the structure of interphases is characterized by
indicating the type of atomic matching at the interface. A useful quantity that can be used to understand this
relation is the lattice misfit, 𝑓 = (𝑎 𝑠 − 𝑎 𝑓 )/𝑎 𝑓 , where 𝑎 𝑠 and 𝑎 𝑓 are the unstrained lattice parameters of the
substrate and the film, respectively [13]. If the two phases have similar interatomic distances and crystalline
structures, the atoms of one crystal can perfectly match the lattices at the opposite site of the interphase and
form a coherent interphase if the grains are in a specific orientation. When the lattice misfit is 𝑓 < 0.09 the
interphase can be perfectly coherent (epitaxial), (Figure 16 and Figure 16b) or strained coherent (Figure
17).

Figure 16. Structure of (a) perfect coherent, (b) tilted coherent interfaces [35].

17

Figure 17. Structure of strained coherent interfaces [35].

Epitaxial interphases form if the lattice mismatch is smaller than approximately 0.015. For example, the
systems CoSi 2/Si, NiSi 2/Si, FeSi 2/Si, Ca/GaAs, Rb/GaAs, Fe/GaAs are epitaxial [13]. On the other hand,
for systems with lattice mismatch 0.015< 𝑓 < 0.09, strain accumulates as the thickness of the phases
around the interfaces increases, which results in strained-coherent boundaries. This strain is released by
dislocation that appear above a critical thickness. For instance, the systems Ge xSi 1-x/Si dislocations appear
at a critical thickness that decreases with increasing concentrations of Ge.

Figure 18. Structure of (a) semi-coherent and (b) incoherent interfaces [35].

For misfits 0.09 < 𝑓 < 0.25 the difference in the lattice parameters is accommodated by dislocations
spaced at a distance 𝐷 ≈
𝑏 ⃗
𝑓 , where 𝑏 ⃗
is the Burgers vector (Figure 18a) [35]. In this case, several types of
defects including threading edge, threading screw, and misfit dislocation extend from the substrate surface
through the thickness of the film, and the interface is semi-coherent [13]. Finally for misfits 0.25 < 𝑓 the
dislocation cores overlap and prevent matching between the atoms, resulting in an incoherent interface
(Figure 18b) [35]. The phase boundaries presented thus far have different atomic structures that result in

18

different energies. While coherent interfaces have relatively low energy (5-200 mJ/m
2
) due to the fact that
the atoms have maximum electronic delocalization in this configuration, semi-coherent boundaries have
higher associated energies (200-800 mJ/m
2
) as a result of several broken bonds at the defect sites. This is
even more pronounced in incoherent interfaces, which are at the end of energy spectrum (800-2500 mJ/m
2
)
[24]. These energy differences result in contrasting atomic mobilities and diffusivities proportional to the
energy of the interphase. Thus, the stability of coherent interfaces is greater than that of semi-coherent
interfaces, which in turn are more stable than incoherent boundaries [36]. On the other hand, the higher
energy of incoherent interfaces facilitates the nucleation of new phases, leading to the formation of
complexions [24].

2.2.3 Complexions
A complexion is an interface where a nanometric phase is in equilibrium with the surrounding grains [37].
This type of interface forms after a new phase forms at a grain boundary via diffusional transformations
like chemical reactions or segregation [26]. The new phase, which usually has a chemical composition or
crystallographic orientation different from that of the surrounding matrix, modifies the interfacial energy
and changes its properties. For example, it has been shown that complexions change the kinetics of grain
boundary diffusion, which results in different rates for grain growth [38]. Complexions either have ordered
or amorphous structures, presented in Figure 19, that can vary from a single layer of atoms to a wetting
layer. The different types of complexions are prepared by changing the composition of the system or the
synthesis conditions [38].

Figure 19. Dillon Harmer complexions in Al 2O 3 (a) undoped and (b) doped with 100 ppm of Nd 2O 3, (c)
with 30 ppm of CaO, (d) 200 ppm of SiO 2, (e) 100 ppm of CaO, (f) and 30 ppm of CaO (impinged). (g) to
(l) display similar complexions in other systems [26].

19

Complexions have been studied using Monte Carlo simulations, which have shown that their structure
depends on the misorientation, composition, and crystallinity of the interface. These simulations, as
summarized in Figure 20a, have found that as the complexion phase grows, the thickness of the complexion
increases and the stability decreases [39]. This is in agreement with Figure 20b, which indicates that the
Gibbs energy increases with the number of atomic layers. Please note that although each of the complexions
in Figure 20 has a different energy, all of them correspond to relative minima.

Figure 20. (a) Diagram showing the different region for complexions as a function of normalized
disorientation and normalized bulk composition, where R 1= 0.6, R 2=1.2, and R 3=3, are reduced
misorientation (b) Grain boundary energy as a function of the bulk composition [39].

Considering that the driving force for complexion formation is the reduction of the interfacial energy,
complexions could be used to thermodynamically stabilize grain boundaries. Specifically, several authors
have studied the possibility of synthesizing solid solutions tailored to induce grain boundary segregation to
drive the formation of complexions, thereby stabilizing the grain boundaries [37, 40]. Complexions can
also interact with dislocations, which could further improve the stability of nanomaterials [40].

2.3 Thermal Stability of Nanocrystalline Materials
The thermal stability of nanocrystalline materials depends on the processes that decrease the density of
grains boundaries. For example, processes like recrystallization and grain growth reduce the number of
interfaces and lower the grain boundary energy. During these processes the grain boundaries migrate with
velocity given by [41]:

𝑣 = 𝑀𝑃 = 𝑀 0
𝑒 −𝑄 /𝑅𝑇
4𝛾 𝑑 (7)

20

Where 𝑀 is the grain boundary mobility, 𝑃 is the driving force, 𝑀 0
is the preexponential factor of the grain
boundary mobility, 𝑄 is the activation energy for grain boundary mobility, 𝛾 is the grain boundary energy
and 𝑑 is the grain size. Equation (7) shows that there are two alternatives to slow grain growth. First, by
inducing the formation of precipitates or inclusions at the grain boundaries, which increases the activation
energy 𝑄 for grain growth, thus decreasing the grain boundary velocity (kinetic mechanism [42]). The
second approach to minimize grain growth is by driving the segregation of solute atoms to the grain
boundary, which then decrease the grain boundary energy 𝛾 , thus lowering the driving force for grain
growth (thermodynamic approach [43]). Both approaches have been applied with some degree of success
and are discussed in further detail at the end of this section [44]. Overall, the use of these interesting
approaches relies on controlling the diffusion properties of nanomaterials to induce the nucleation of new
phases at the interface.

2.3.1 Diffusion in Nanocrystalline Materials
The structure of nanocrystalline materials is comprised of a high density of grains boundaries that influence
diffusive processes in two ways [45]. First, grain boundaries act as sources and sinks that facilitate the
attainment of equilibrium concentration of vacancies, which are responsible for volume or lattice diffusion.
Second, grain boundaries are high-diffusivity paths that drive microstructural transformations [25]. For
example, the formation of precipitates occurs mainly via grain boundary diffusion. Other high diffusivity
paths include dislocations and free surfaces, which have different diffusivities as can be observed in Figure
21.

Figure 21. Image showing logarithmic plots of the diffusivities in metals at free surfaces (𝐷 𝑠 ) , inter
granular regions (𝐷 𝑖𝑔
) , dislocation-pipes (𝐷 𝑑 ) , and inside lattices (𝐷 ) as a function of temperature [45].

21

The diffusivities at free surfaces (𝐷 𝑠 ) , at grain boundaries (𝐷 𝑔𝑏
) , at dislocation-pipes (𝐷 𝑑 ) , and inside
lattices (𝐷 ) follow the trend 𝐷 ≪ 𝐷 𝑑 ≤ 𝐷 𝑔𝑏
≤ 𝐷 𝑠 , which is in agreement with the fact that atomic migration
is more difficult inside perfect lattices where the atoms are constrained [45]. In contrast, at dislocation
cores, where the lattices are distorted, the constrain for atomic motion is decreased, thus facilitating atomic
diffusion. This effect is more pronounced at grain boundaries, which have less packed dislocation structures
and where solute atoms can move faster [25]. Finally, at free surfaces the atoms are the least constrained
and can move the fastest [46].

More specifically, the high density of grain boundaries and triple junctions at the intergranular regions of
nanocrystalline materials induce the formation of solute fringes around the grain boundaries. These fringes
overlap inside the grains and form almost flat diffusion profiles (Kinetics type A) [47]. From a macroscopic
point of view, this indicates that the diffusion process can be represented using an effective diffusion
coefficient. There are two limiting cases for the effective diffusivity that depend on the arrangement of the
grain boundaries, see Figure 22 [45].

Figure 22. Representation of systems with grain boundaries (a) parallel, (b) perpendicular, and (c) random
orientation in comparison to the diffusion direction [45].

First, if the grain boundaries are parallel to the diffusion direction, the effective diffusivity is maximized
by the uninterrupted diffusion through the grain boundaries (Figure 22a), and the value of 𝐷 𝑒𝑓𝑓 is calculated
using the Hart’s equation, which is equivalent to a system of multiple conductance in parallel:

𝐷 𝑒𝑓𝑓 = 𝑓 𝑖𝑔
𝐷 𝑖𝑔
+ 𝑓 𝑙 𝐷 (8)
Where 𝐷 𝑖 𝑔 is the intergranular diffusivity, 𝑓 𝑖𝑔
is the volume fractions of intergranular regions, and 𝑓 𝑙 the
fraction of grain lattices. The intergranular diffusivity 𝐷 𝑖𝑔
is due to the combined diffusion at grain
boundaries and triple junctions. When the grain boundaries are perpendicular to the diffusion direction

22

(Figure 22b) the system is comprised of alternating grains and grain boundaries and is described as a set of
several conductance in series. In that case the effective diffusivity is:

𝐷 𝑒𝑓𝑓 =
𝐷 𝑖𝑔
𝐷 𝑓 𝑖𝑔
𝐷 𝑙 +𝑓 𝑙 𝐷 𝑖𝑔
(9)
Equation (9) is used to calculate a lower limit for the effective diffusivity. For more complex systems such
as equiaxed grains, where the grain boundaries are randomly oriented, the contribution of interfaces to the
effective diffusivity is considered using geometrical approximations like cubic (Figure 22c) or spherical
system or analogies from the theories of heat-transfer or electromagnetism [48]. For example, the effective
diffusivity of nanocrystalline materials can be calculated using the Maxwell-Garnett equation (10), which
is equivalent to an equation originally developed by Maxwell to calculate the effective dc conductivity of
a material comprised of two phases [45]. In this case, the two phases are the grain boundaries, which are
embedded between the grains.

𝐷 𝑒𝑓𝑓 = 𝑓 𝑙 𝐷 + 𝑓 𝑖𝑔
𝐷 𝑖𝑔
+
𝑓 𝑖𝑔
𝑓 𝑙 (𝐷 𝑖𝑔
−𝐷 )
2
𝑓 𝑖𝑔
(𝐷 𝑖𝑔
−𝐷 )−3𝐷 𝑖𝑔
(10)
Where the first two terms of (10) are Hart’s equation, and the last term, which is always negative, captures
the effect of grain boundaries not parallel to the diffusion direction. This equation is a more accurate higher
bound for the effective diffusivity (Equation 10 predicts a diffusivity which ~2 times the experimental
value). A modification to the Maxwell-Garnett equation has been proposed by Chen and Schuh to include
the effect of triple junctions and dislocation networks [49]. In this model, the intergranular region is
comprised of grain boundaries which connect at triple junctions. Thus, a similar analogy to Maxwell’s
equation is used for the intergranular diffusivity:

𝐷 𝑖𝑔
=
1
𝑓 𝑖𝑔
[𝑓 𝑔𝑏
𝐷 𝑔𝑏
+ 𝑓 𝑡𝑗
𝐷 𝑡𝑗
+
𝑓 𝑡𝑗
𝑓 𝑔𝑏
(𝐷 𝑡𝑗
−𝐷 𝑔𝑏
)
2
𝑓 𝑡𝑗
(𝐷 𝑡𝑗
−𝐷 𝑔𝑏
)−2𝑓 𝑖𝑔
𝐷 𝑡𝑗
] (11)
In this equation, it is assumed that the grain boundaries have an average diffusivity 𝐷 𝑔𝑏
, and that the
diffusivity at triple junctions can be represented by 𝐷 𝑡𝑗
. The volume fractions of these microstructural
features are 𝑓 𝑔𝑏
= 𝐻 𝑔𝑏
𝛿 /𝑑 and 𝑓 𝑡𝑗
= 𝐻 𝑡𝑗
𝛿 2
/𝑑 2
, respectively. 𝐻 𝑔𝑏
and 𝐻 𝑡𝑗
are geometric parameters 𝐻 𝑔𝑏
=
2.9105 and 𝐻 𝑡𝑗
= 2.5259 for a log-normal grain size distribution. Substituting (11) in (9):

𝐷 𝑒𝑓𝑓 = 𝑓 𝑙 𝐷 + 𝑓 𝑔𝑏
𝐷 𝑔𝑏
+ 𝑓 𝑡𝑗
𝐷 𝑡𝑗
+ 𝑓 𝑑 𝑡 (𝑓 𝑙 + 𝑓 𝑑 )
1−𝑡 𝐷 𝑑 +
𝑓 𝑡𝑗
𝑓 𝑔𝑏
(𝐷 𝑡𝑗
−𝐷 𝑔𝑏
)
2
𝑓 𝑡𝑗
(𝐷 𝑡𝑗
−𝐷 𝑔𝑏
)−2(𝑓 𝑔𝑏
+𝑓 𝑡𝑗
)𝐷 𝑡𝑗
+
𝑓 𝑔𝑏
𝑓 𝑙 (𝐷 𝑖𝑔
−𝐷 )
2
𝑓 𝑔𝑏
(𝐷 𝑔𝑏
−𝐷 )−3𝐷 𝑔𝑏
(12)

23

Where 𝑓 𝑑 = 1− 𝑓 𝑔𝑏
− 𝑓 𝑡𝑗
− 𝑓 𝑙 is the volume fractions of dislocation cores. The last two terms of (12)
represent corrections due to grain boundaries and triple junctions not parallel to the diffusion direction.
Considering that at most temperatures 𝐷 ≪ 𝐷 𝑔𝑏
≪ 𝐷 𝑡𝑗
, equation (12) can be further simplified:

𝐷 𝑒𝑓𝑓 = 𝐷 + 𝑔 (𝑑 )[
2𝐻 𝑔𝑏
𝛿 𝑑 (𝐷 𝑔𝑏
− 𝐷 )+
𝐻 𝑡𝑗
𝛿 2
𝑑 2
(𝐷 𝑡𝑗
− 𝐷 )] (13)
Where 𝑔 (𝑑 ) is a parameter dependent on the grain size:

𝑔 (𝑑 )=
2𝐻 𝑔𝑏
𝑑 3
+2𝐻 𝑡𝑗
𝛿 𝑑 2
6𝐻 𝑔𝑏
𝑑 3
−(2𝐻 𝑔𝑏
2
−3𝐻 𝑡𝑗
)𝛿 𝑑 2
−3𝐻 𝑔𝑏
𝐻 𝑡𝑗
𝛿 2
𝑑 −𝐻 𝑡𝑗
2
𝛿 3
(14)
Equations (13) and (14) have been used by Polyakov et al. to calculate the effective diffusivity and diffusion
length for Hf-Ti nanometallic multilayers [50]. In addition, Chen and Schuh employed these equations to
create diffusion maps where the different diffusivities can be compare as a function of grain size and
temperature [49]. For instance, Figure 23 shows that as grain size and temperature increase, the effective
diffusivity is dominated by lattice diffusion. On the contrary, when the grain size and the temperature
decrease, the effective diffusivity is predominantly due to the faster mobilities at triple junctions. In the
middle range of grain sizes and temperature, the mass transport process is mostly due to grain boundary
diffusion.

Figure 23. Diffusion maps for (a) FCC and (b) BCC metals where the dominant regions for triple junction
(TJ), grain boundary (GB), and lattice or volume (V) diffusion can be compared [49].

Another approach to calculate effective diffusivities and diffusion lengths is to use the solute mass transport
equations and equilibrium models of vacancies. However, the derivation of these equations is significantly

24

more complex, and the details can be explored in the books by Gusak [51, 52]. The diffusive processes at
the interface have a major influence on the thermal stability of nanocrystalline materials. Specifically,
several authors have demonstrated that the stability is highly dependent on the interface energy, which
controls processes such as recrystallization and grain growth [36]. The various approaches to increase the
thermal stability of nanocrystalline materials are discussed in the following sections.

2.3.2 Kinetic Approach
In a kinetic approach grain boundary motion is inhibited by drag forces that oppose to the displacement of
grain boundaries. For example, in Zener pinning, precipitates at interfaces inhibit grain boundary motion.
These precipitates usually form after annealing above a critical temperature and exert a drag pressure 𝑃 𝑍 at
the grain boundaries given by [53]:

𝑃 𝑍 =
3𝑓𝛾
2𝑅 (15)
Where 𝑃 𝑍 is also known as the Zener pressure, 𝑓 is the volume fraction of precipitates, and 𝑅 is the distance
of interaction between precipitates and grain boundaries. Equation (15) suggests that the greater the volume
fraction of precipitates 𝑓 , the larger the force impeding grain boundary motion 𝑃 𝑍 . An example of Zenner
pinning occurs in the Ni-W system, where WO x precipitates pin NiW grains as depicted in Figure 24, thus
increasing the stability of this system [54].

Figure 24. TEM image of electrodeposited Ni 23 at.% W heat-treated under Ar atmosphere at 700 °C for
24 hours [54].

Grain growth in multiple systems such as Fe-Al, Cu-Zr, Al-Mg, Mg-Cu, Cu-Al, and Cu-Nb (solvent first)
is prevented by inducing Zenner pinning [44]. A remarkable example of this type of stabilization occurs in

25

nanocrystalline Cu. The grain size of pure Cu increases from 20 nm to 60 nm after heat-treatment for 1 h
at 600 °C; however, when alloyed with Nb, which precipitates at the grain boundaries, the grain size only
increases from 10 to 15 nm (Figure 25) after heat-treatment at the same temperature [55].

Figure 25. (a) SEM and (b) TEM images of Cu-10 at% Nb after consolidation at 700 °C and heat-
treatment at 900° C for 1 h. (c) shows the grain size for pure Cu, and alloyed with Nb [55].

In general, kinetic mechanism limit grain growth by decreasing the kinetic coefficient for grain boundary
motion. Other kinetic mechanism include stabilization via porosity drag, chemical ordering, and grain size
stabilization [56, 57]. For instance, grain growth in Fe 3Si with initial grain size of 8 nm is prevented by the
addition of Mn, which induces chemical ordering and stabilizes the system at 12 nm after annealing at 450
°C. An example of grain size stabilization occurs in the Fe-10 wt.% Ni system which reaches a steady grain
size at 20 nm due to the formation of Ni 2O 3 oxides that preferentially move to the grain boundaries at 600
ºC [44, 58].

26

2.3.3 Thermodynamic Approach
In the thermodynamic approach, the excess Gibbs energy of the grain boundaries is reduced by solute
segregation. This process consists in the diffusion of solute atoms to the grain boundary via equilibrium or
non-equilibrium mechanisms [23]. Equilibrium segregation takes place when thermal motion during
annealing induces the movement of solute atoms to the grain boundaries. In contrast, non-equilibrium
segregation occurs when solute atoms that interact with non-equilibrium vacancies are moved to grain
boundaries, this usually occurs during quenching. For example, as can be seen in Figure 26, grain growth
in nanocrystalline W is prevented using a thermodynamic approach. While the grain size of pure W
increases from 22 nm to 604 nm after heat treatment at 1100 ºC for 1 week, the grain size of W 10 at.% Ti
increases to only 24 nm after the same heat-treatment. This absence of grain growth is attributed to
thermodynamic stabilization due to segregation of Ti (white regions) at the grain boundaries [43]. This type
of stabilization has also been observed in other systems, including some indicated in Table 1 [44].

Figure 26. (a) grain sizes of W - 20 at.% Ti and unalloyed W showing that the alloyed sample exhibits
minimum grain growth. TEM images of (b) the alloyed as-milled structure, and of (c) the unalloyed and
(d) alloyed samples after heat-treatment at 1100 °C for one week [43].

27

Table 1. Nanocrystalline materials stabilized up to T max/T melt via thermodynamic approach [44].
Material Solute T max/T melt (°C/°C) Reference
Pd 10-20 at.% Zr 0.80 [59-61]
Ni 6-21 at.% W 0.58 [62]
Co 1.1 at.% P 0.22 [63]
Fe 4.0 at.% Zr 0.76 [64, 65]
Fe- 10-18 at %Cr 1-4 at.% Zr 0.65 [66]
Ni 3.6 at.% P 0.27 [67]
Y 5-20 at.% Fe 0.55 [68]
RuAl 15 at.% Fe 0.48 [69]
TiO 2 0.34 mol Ca 0.57 [70]
W 20 at.% Ti 0.32 [43]
Hf 20 at.% Ti 0.36 [50]

The reduction in grain growth via a thermodynamic approach not only depends on the composition of the
system, but also on the initial microstructure. For example, Polyakov et al. showed that the grain size of a
monolithic Hf-Ti sample increased from 57 nm to 196 nm after heat-treatment at 800 ºC for 96 h [50]. In
contrast, as shown in Figure 27, the grain size of Hf-Ti nanometallic multilayers of the same composition
grow from 25 nm to 50 nm after the same heat treatment [50]. The multilayered geometry can be used to
control microstructural properties, such as the grain size and local composition. This makes nanometallic
multilayers a suitable model system to study thermal phenomena in nanomaterials.

Figure 27. Cross-sectional TEM images of Hf - 21 at.% Ti showing (a) the as-sputtered multilayers (b)
the equiaxed microstructure after heat-treatment at 800 °C for 96 h [50].

2.3.4 Thermal Stability of NMMs
The majority of the research on the thermal stability of NMMs are simulations studies that describe the
degradation of equiaxed multilayers. Although there are some experimental studies, most of them aimed to
show that the structure of the multilayers was conserved at a temperature greater than the melting point of

28

one of the alternating layers, without making an attempt to understand the thermally activated
microstructural transformations or to describe the microstructures observed during annealing [8, 71-76].

The study of Bobeth et al. shows that depending on the initial microstructure and composition of equiaxed
NMMs, several intermediate structures are observed during their heat-treatment [77, 78]. Prolonged
annealing of multilayers at high temperatures results in degradation of the multilayered structure, which is
driven by the interfacial free energy. However, depending on the solubility of the components conforming
the NMMs and the thickness of the alternating layers A and B, which have thicknesses 𝑑 𝐴 and 𝑑 𝐵 , different
mechanisms take place during the degradation process.

Heating of miscible NMMs causes mixing, which flattens the concentration profiles. On the contrary,
annealing of immiscible NMMs, in which 𝑑 𝐴 ≫ 𝑑 𝐵 at temperatures below the roughening temperature
(𝑇 𝑅 = 2𝛾 𝑎 2
/𝜋 𝑘 𝐵 , where 𝑎 is the lattice spacing), results in chemical sharpening [79]. Above this
temperature, thermal fluctuations cause interphase roughening, which develops fast as the interphases act
as short diffusion paths. Later, the roughening fluctuations sharpen at grain boundaries and breakthroughs
develop inside the thin layers. Subsequently, the breakthroughs extend over the thin layers B, and the
multiple microstructures displayed in Figure 28 are observed as time goes on. First, mesh and fiber-like
morphologies are observed. In these structures, the material of the thin B (solute) layers has moved to lateral
regions and forms fibers that are embedded between the thick A (solvent) layers. As the heat-treatment
continues, these fibers break into equiaxed spherical grains surrounded by equiaxed grains resulting from
the fragmentation of the A layers. The rate of transformation of these microstructures is accelerated if the
solute segregates to the grain boundaries.

Figure 28. Microstructures observed in an annealing simulation of multilayers A(41 nm, white)/B (5 nm,
black). (a) mesh-like morphology, (b) fiber-like morphology, (c) intermediate morphology, and (d)
dispersed morphology [79].

29

In the case of heat-treatment of immiscible multilayers with 𝑑 𝐴 ≈ 𝑑 𝐵 , after roughening, breakthroughs
develop in both layers A and B, which results in an interpenetrating morphology that coarsens after
extended annealing, as shown in Figure 29 [79]. As the layer thicknesses increase, the rate of breakthrough
formation decreases, and the processes observed for both cases (𝑑 𝐴 ≫ 𝑑 𝐵 and 𝑑 𝐴 ≈ 𝑑 𝐵 ) take longer [75].
The evolution processes may also take longer depending on the grain morphology [80, 81]. For example,
in columnar NMMs, the development of pinch-offs is more difficult than in non-columnar multilayers [8,
73, 80, 81].

Figure 29. Microstructures observed in annealing simulation of immiscible multilayers 6 nm A-B. (a)
initial microstructure, (b) interpenetrating morphology, and (c) coarsened morphology [79].

The processes described for immiscible NMMs are in agreement with the experimental observations for
multiple systems [82]. For example, for Cu-Nb NMMs, the micrographs displayed in Figure 30 show that
the formation of several breakthroughs which result in the formation of a interpenetrating morphology
comprised of several regions of Cu and Nb and some spherical grains [8].

Figure 30. Cross-section bright field micrographs of 15 nm Cu-Nb multilayers heat-treated (a) at 600 ºC
for 1 h, (b) at 700 ºC for 30 min, and at 700 ºC for 1h. Spheroidization is noticeable in (b) and (c) [8].

The aforementioned processes are driven by the larger diffusion coefficients at high temperatures. The
greater diffusivities can also induce segregation of solute atoms, which activate thermodynamic

30

stabilization mechanisms. To understand this process in more detail, the thermodynamics models of
segregation are explored in the following section.

2.4 Thermodynamic Models of Segregation
The segregation of solute atoms to the grain boundary is driven by the reduction of the grain boundary
energy, which is estimated using models that are based on Gibbs energy balances at grain interfaces. The
interface energy depends on the composition and temperature of the system. Thus, segregation models are
used to construct energy isotherms, which are employed to calculate the composition that minimizes the
grain boundary energy.

2.4.1 Gibbs Segregation Isotherm
The first segregation model, proposed by Gibbs, assumes that between the grains A and B, Figure 31, is a
dividing surface that has no internal structure, which can be understood using common thermodynamics
relations [83].

Figure 31. Schematic representation of a general interphase limited by the planes AA’ and BB’, with
thickness, Δ𝑡 , and grain boundary energy, 𝛾 𝐼 , along the plane SS’ [83].

The Gibbs-Duhem equation for interfaces is:

𝑆𝑑𝑇 − 𝑉𝑑𝑃 + ∑ 𝑛 𝑗 𝑔𝑏
𝑑 𝜇 𝑗 = 0
𝑖 (16)

31

Where 𝑛 𝑗 𝑔𝑏
is the amount of component 𝑗 in the interface. The pressure in the interface is modified by the
interfacial free energy:

𝑉𝑑𝑃 = Δ𝑡𝐴𝑑𝑃 − 𝐴𝑑 𝛾 𝐼 (17)
Where Δ𝑡 is the interface thickness and 𝛾 𝐼 is the interfacial free energy. Substituting (17) in (16) and
dividing by 𝐴 :

𝑑 𝛾 𝐼 = −𝑆 𝐼 𝑑𝑇 + Δ𝑡𝑑𝑃 − ∑ Γ
𝑗 𝑑 𝜇 𝑗 𝑖 (18)
Where 𝑆 𝑗 is the entropy per unit of interface area 𝐴 , and Γ
𝑗 = 𝑛 𝑗 𝑔𝑏
/𝐴 is the amount of component 𝑗 per
unit of interface area 𝐴 . For a binary system with components 1 and 2 at constant temperature and pressure,
equation (18) takes the form of the Gibbs absorption isotherm:

𝜕 𝛾 𝐼 𝜕 𝜇 2
= −(Γ
2
−
𝑥 2
𝑥 1
Γ
1
) = −Γ
21
(19)
Equation (19) relates the interfacial adsorption to changes in the interfacial free energy with composition.
This equation has been used to quantify the segregation occurring in Fe-P, Sn-P, Si-Fe, and S-Fe [84-87].
However, measuring the surface energy, 𝛾 𝐼 as a function of temperature and composition is difficult [88].
Hence, efforts have been made to develop phenomenological relations between the grain boundary energy
and the interfacial and bulk compositions [23].

2.4.2 Gibbs Adsorption Equation
A phenomenological relation for the grain boundary energy can be obtained considering the Gibbs energy
of a system at constant temperature and pressure [29]:

𝑑𝐺 = Σ
𝑖 (𝜇 𝑖 𝑔𝑏
− 𝜇 𝑖 )𝑑 𝑛 𝑖 𝑔𝑏
+ 𝛾 0
𝑑𝐴 (20)
Where 𝜇 𝑖 𝑔𝑏
and 𝜇 𝑖 are the chemical potentials at the grain boundary and at the bulk of the grains, 𝛾 0
is the
interfacial energy of the grain boundary energy at constant composition, 𝐴 is the interfacial area and 𝑑 𝑛 𝑖 𝑔𝑏

represents the amount of component 𝑖 moving from the bulk of the grains to the grain boundary. The
difference in chemical potentials, which is the driving force for segregation, can be written in the form:

𝜇 𝑖 𝑔𝑏
− 𝜇 𝑖 𝑏 = 𝜇 𝑖 𝑔𝑏 ,0
− 𝜇 𝑖 0
+ 𝑅𝑇𝑙𝑛 (
𝑥 𝑖 𝑔𝑏
𝛾 𝑖 𝑔𝑏
𝑥 𝑖 𝛾 𝑖 ) (21)

32

Where 𝛾 𝑖 and 𝛾 𝑖 𝑔𝑏
are the activity coefficients for component 𝑖 in the bulk and the grain boundary,
respectively. From (20), using the thermodynamic definition of the interfacial energy 𝛾 = 𝑑𝐺 𝑑𝐴 ⁄ for a
binary system with components 𝑎 and 𝑏 , which satisfy 𝑑 𝑛 𝑎 𝑔 𝑏 = −𝑑 𝑛 𝑏 𝑔𝑏
, the Gibbs absorption equation is
obtained:

𝛾 = Δ𝐺 𝑠𝑒𝑔 𝜕 𝑛 𝑎 𝜕𝐴
+ 𝛾 0
(22)
Where Δ𝐺 𝑠𝑒𝑔 is:

Δ𝐺 𝑠𝑒𝑔 = 𝜇 𝑎 𝑔𝑏 ,0
− 𝜇 𝑎 0
− (𝜇 𝑏 𝑔𝑏 ,0
− 𝜇 𝑏 0
)+ 𝑅𝑇𝑙𝑛 (
𝑥 𝑎 𝑔𝑏
1−𝑥 𝑎 𝑔𝑏
1−𝑥 𝑎 𝑥 𝑎 𝛾 𝑎 𝑔𝑏
𝛾 𝑏 𝛾 𝑎 𝛾 𝑏 𝑔𝑏
) (23)
Equations (22) and (23) could be used to calculate the grain boundary energy as a function of composition
for a given temperature. Although there are no known models for the activity coefficients 𝛾 𝑎 𝑔𝑏
and 𝛾 𝑏 𝑔𝑏
, for
dilute systems, the Gibbs adsorption isotherm can be expressed in an interesting form proposed by
Weissmüller, which is used to show that the Gibbs energy can be minimized for a given grain size.

2.4.3 Weissmüller Segregation Model
For a dilute system, it is valid to assumed that the activity coefficients in equation (23) are equal to one,
which results in the following expression [89, 90]:

𝛾 = 𝛾 0
− Γ
𝑎 [Δ𝐻 𝑠𝑒𝑔 + 𝑅𝑇𝑙𝑛 (
𝑥 𝑎 𝑔𝑏
𝑥 𝑎 )] (24)
This is the McLean segregation isotherm, where Δ𝐻 𝑠𝑒𝑔 = Δ𝐻 𝑎 𝑠𝑜𝑙
− Δ𝐻 𝑎 𝑠𝑜𝑙 ,𝑔𝑏
. Thus, the total Gibbs energy
of the system is given by:

𝐺 = 𝑛 𝑎 𝜇 𝑎 0
+ 𝑛 𝑏 𝜇 𝑏 0
+ 𝛾 0
𝐴 + 𝑛 𝑎 𝑔𝑏
Δ𝐻 𝑎 𝑠𝑜𝑙 ,𝑔𝑏
+ 𝑛 𝑏 𝑔𝑏
Δ𝐻 𝑏 𝑠𝑜𝑙 ,𝑔𝑏
+ 𝑅𝑇 (𝑛 𝑎 ln(𝑥 𝑎 )+𝑛 𝑏 ln(𝑥 𝑏 )+
𝑛 𝑎 𝑔𝑏
ln(𝑥 𝑎 𝑔𝑏
)+ 𝑛 𝑏 𝑔𝑏
ln(𝑥 𝑏 𝑔𝑏
)) (25)
Where the grain boundary area is calculated using 𝐴 = 3𝑉 /𝑑 , where 𝑑 is the grain size. Using equations
(24) and (25) simultaneously, the Gibbs energy of the system is calculated as a function of composition at
the grain boundary for different grain sizes, as displayed in Figure 32 [89].

33

Figure 32. Gibbs energy versus the global solute mole fraction for different gran sizes [89].

Figure 32 shows that for every grain size, there is a single solute mole fraction that minimizes the energy
of the system and suggests that the system is only thermodynamically stable at this composition. The models
described so far are phenomenological relations between the surface energy and chemical potentials that
can only be used in the dilute limit. Conversely, interface segregation models based on free-surface
segregation models have shown predictive potential to calculate grain boundary energies of concentrated
systems.

2.4.4 Interface Segregation Models Based on Surface Segregation Models
Due to the similarities between interfacial segregation and segregation at free surfaces, models describing
segregation at free-surface have been adapted to calculate the interface energy of systems with grain
boundary segregation [23]. There are two main approaches. First, in the Wynblatt and Ku model the
enthalpy of segregation is calculated adding chemical and elastic contribution, which are due to the different
properties of the solute and solvent atoms [88]. Second, in Miedema’s model, the enthalpy of segregation
is calculated using an equation with tunable constants based on models of the electronic density at the
boundary of the Wigner-Seitz cell of the solid solution.

2.4.4.1 Wynblatt and Ku Model
Wynblatt and Ku proposed the following equation for the Gibbs energy for surface segregation [91]:

Δ𝐺 𝑠𝑒𝑔 = Δ𝐻 𝑠𝑒𝑔 − 𝑇 Δ𝑆 𝑠𝑒𝑔 (26)

34

Where the enthalpy of segregation is given by:

Δ𝐻 𝑠𝑒𝑔 = Δ𝐻 𝑠𝑒𝑔 𝑐 ℎ𝑒𝑚
+ Δ𝐻 𝑠𝑒𝑔 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 (27)
The chemical enthalpy of segregation can be calculated using the Defay and Prigogine equation, which is
derived from the regular solution theory [92]:

Δ𝐻 𝑠𝑒𝑔 𝑐 ℎ𝑒𝑚
= (𝛾 𝑎 − 𝛾 𝑏 )𝜎 −
2ΔH
𝑚 𝑧 𝑥 𝑎 𝑥 𝑏 [𝑧 𝑖𝑛
(𝑥 𝑎 𝑔𝑏
− 𝑥 𝑎 )− 𝑧 𝑜𝑢𝑡 (𝑥 𝑎 − 1/2)] (28)
Where 𝛾 𝑎 and 𝛾 𝑏 are the surface energies of the solute and solvent correspondingly, 𝑧 𝑖𝑛
and 𝑧 𝑜𝑢𝑡 are the in-
plane and out-of-plane coordination numbers, and ΔH
𝑚 is the enthalpy of mixing of 𝑎 and 𝑏 at the
composition of the system. The elastic enthalpy is given Friedel’s equations [93]:

Δ𝐻 𝑠𝑒𝑔 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 = −
2𝐾 𝑏 𝐺 𝑎 (𝑉 𝑏 −𝑉 𝑎 )
2
3𝐾 𝑏 𝑉 𝑎 +4𝐺 𝑎 𝑉 𝑏 (29)
Where 𝑉 𝑎 corresponds to the molar volume, 𝐺 𝑎 and 𝐾 𝑏 are the shear and bulk modulus. The entropy of
segregation is calculated using an expression equivalent to equation (27):

Δ𝑆 𝑠𝑒𝑔 = (𝑆 𝑎 − 𝑆 𝑏 )𝜎 −
2Δ𝑆 𝑚 2𝑥 𝑎 𝑥 𝑏 [𝑧 𝑖𝑛
(𝑥 𝑎 𝑔𝑏
− 𝑥 𝑎 )− 𝑧 𝑜𝑢𝑡 (𝑥 𝑎 −
1
2
)] +
𝑑 𝑑𝑇
[
2𝐾 𝑏 𝐺 𝑎 (𝑉 𝑏 −𝑉 𝑎 )
2
3𝐾 𝑏 𝑉 𝑎 +4𝐺 𝑎 𝑉 𝑏 ] (30)
Where 𝑆 𝑎 and 𝑆 𝑏 are the entropies of the solute and the solvent, correspondingly. Using the Mclean
segregation isotherm (Equation 24), assuming an entropy of segregation equal to zero and calculating the
enthalpy of segregation Δ𝐻 𝑠𝑒𝑔 using equation (27), Wynblatt and Ku confirmed the existence of a transition
region for ordering and clustering in binary systems with negative heats of adsorption as presented in Figure
33 [91]. Figure 33 shows that at low temperatures depending on the enthalpy of mixing, segregation occurs
in different ways for clustering and ordering binary systems. On the other hand, at high temperatures both
clustering and ordering systems follow a similar trend. A useful adaptation of the Wynblatt and Ku model
which allows for comparison of binary system is presented next.

35

Figure 33. McLean segregation isotherm (Equation 24) as function of temperature using equation (27) to
calculate the enthalpy of segregation. The two curves show the behavior of segregation in clustering
(Top) and ordering (Bottom) systems [91].

2.4.4.2 Darling et al. Modification of the Wynblatt and Ku Model
Considering the limited number of experimental correlations of the enthalpy of mixing ΔH
𝑚 with
composition for most binary systems, and the scarcity of literature on surface entropy, Darling et al.
modified the Wynblatt and Ku model to overcome these difficulties and calculate interfacial segregation
properties. In their model, the chemical enthalpy of segregation is calculated using the following equation
[94]:

Δ𝐻 𝑠𝑒𝑔 𝑐 ℎ𝑒𝑚
= (𝛾 𝑎 − 𝛾 𝑏 )(1− 𝛼 )𝜎 −
8Δ𝐻 𝑚 𝑒𝑞
𝑧 [𝑧 𝑖𝑛
(𝑥 𝑎 𝑔𝑏
− 𝑥 𝑎 )− 𝑧 𝑜𝑢𝑡 ((𝑥 𝑎 − 1/2)− 𝛼 (𝑥 𝑎 𝑔𝑏
− 1/2))]
(31)
Where Δ𝐻 𝑚 𝑒𝑞
is the enthalpy of mixing of an equimolar mixture, and 𝛼 is a bond energy interaction
parameter used to differentiate between the bond energy in the bulk of the grains and the grain boundaries.
Values for the equimolar enthalpy of mixing for multiple binary alloys are available and the elastic term of
the enthalpy of segregation is calculated in the same way as in the original model of Wynblatt and Ku [54].
Additionally, the entropy of segregation is calculated assuming an ideal solution at the grain boundary and
the bulk of the grains [95]:

Δ𝑆 𝑠𝑒𝑔 𝑖𝑑𝑒𝑎𝑙 = −𝑅𝑙𝑛 [
𝑥 𝑎 𝑔𝑏
(1−𝑥 𝑎 )
𝑥 𝑎 (1−𝑥 𝑞 𝑔𝑏
)
] (32)

36

The Darling et al. modification of the Wynblatt and Ku model allows for a simplified calculation of the
interfacial segregation properties of binary systems. For instance, Figure 34 shows the grain boundary
energy of the Fe-Zr system as a function of composition calculated using equations (22), (26), (31), and
(32) for different grain sizes [95].

Figure 34. Normalized grain boundary energy for the system Fe-Zr as a function of the mole fraction of
Zr at the grain boundary for several grain sizes [95].

Instead of calculating the chemical enthalpy of segregation Δ𝐻 𝑠𝑒𝑔 𝑐 ℎ𝑒𝑚
using the regular solution theory,
Miedema proposed to calculate Δ𝐻 𝑠𝑒𝑔 𝑐 ℎ𝑒𝑚
using available thermodynamic data, like the vaporization
enthalpies of pure metals and the mixing enthalpies of binary alloys [96-98]. This results in a simplified
calculation of the segregation enthalpy.

2.4.4.3 Miedema Model
Based on previous work on the calculation of the enthalpy of formation using the chemical potential and
the electronic density in a unit cell, Miedema proposed the following equation to calculate the enthalpy of
surface segregation of binary alloys [99]:

𝛥𝐻
𝑠𝑒𝑔 𝑠 𝑢 𝑟𝑓
=
𝑓 3
[𝐻 ̅
𝑎 − 𝛥 𝐻 𝑎 𝑣𝑎𝑝 − 𝛥 𝐻 𝑏 𝑣𝑎𝑝 ] (33)

37

Where 𝐻 ̅
𝑎 is the partial molar enthalpy of A in B. In equation (33) it is assumed that atoms at the surface
have lost one third of their energy of vaporization, this is the reason there is a 1/3 in the expression. In this
equation 𝑓 is a constant equal to 0.71. As proposed by Miedema the enthalpy of vaporization can be
approximated by the equation Δ𝐻 𝑎 𝑣𝑎𝑝 = 𝑐 0
𝛾 𝑎 𝑠 𝑉 𝑎 2/3
where 𝑐 0
is a constant equal to 𝑐 0
= 1.5 × 10
8
, 𝛾 𝑎 𝑠 is the
surface energy and 𝑉 𝑎 is the molar volume [100]. Substituting this formula in equation (33):

Δ𝐻 𝑠𝑒𝑔 =
𝑓 3
[𝐻 ̅
𝑎 − 𝑐 0
𝛾 𝑎 𝑠 𝑉 𝑎 2/3
− 𝑐 0
𝛾 𝑏 𝑠 𝑉 𝑏 2/3
] (34)
Equation (34) in combination with the McLean segregation isotherm (equation 24) were used by Miedema
to calculate free surface enrichment factors of multiple solutes in Cu and Pt [99]. However, in order to
estimate enthalpies of interfacial segregation, it is necessary to modify equation (33).

2.4.4.4 Murdoch and Schuh Modification of Miedema Model
Murdoch and Schuh proposed a modification of equation (33) to calculate the enthalpy of grain boundary
segregation. First, they added a coordination factor 𝑣 = 1/2 which accounts for the increased coordination
at grain boundaries in comparison to free surfaces. Then, an additional term to consider the effect of elastic
strains was added. After the modifications equation (34) takes the form [101]:

Δ𝐻 𝑠𝑒𝑔 =
𝑓𝑣
3
[𝐻 ̅
𝑎 − 𝑐 0
𝛾 𝑎 𝑠 𝑉 𝑎 2/3
− 𝑐 0
𝛾 𝑏 𝑠 𝑉 𝑏 2/3
] + Δ𝐻 𝑒𝑙
(35)
Murdoch and Schuh also suggested that a nanocrystalline alloy is stable with respect to grain growth when
the next expression is valid [102]:

Δ𝐻 𝑠𝑒𝑔 (Δ𝐻 𝑚𝑖𝑥 )
𝑎 > 𝑐 (36)
Where Δ𝐻 𝑚𝑖𝑥 is the enthalpy of mixing, 𝑎 and 𝑐 are empirically calculated constants that depend on the
critical temperature for the alloy, which is calculated as in the regular solution model 𝑇 𝑐𝑟
= Δ𝐻 𝑚𝑖𝑥 /2𝑅 .
Values for these constants are presented in Table 2 [102].

Table 2. Coefficients to evaluate the proclivity for nanocrystalline stability [102]
Temperature 𝒂 (slope) 𝒄 (intercept, metastable) 𝒄 (intercept, stable)
0.35Tcr 0.757 1.7326 2.7680
0.5Tcr 0.661 2.8038 3.7236
0.65Tcr 0.567 4.4250 4.9580

Equation (36) is a limiting condition for stability with respect to grain growth. This inequality is used to
construct stability maps where binary alloys are classified in stable (grain size stable with respect to small

38

and large thermal disturbances), metastable (grain size stable only with respect to small thermal
disturbances) and unstable (grain growth occurs after any thermal disturbance) regions depending on their
enthalpy of segregation Δ𝐻 𝑠𝑒𝑔 and enthalpy of mixing Δ𝐻 𝑚𝑖𝑥 . Figure 35 shows examples of this maps for
the temperatures 0.5𝑇 𝑐𝑟
and 0.65𝑇 𝑐𝑟
[101].

Figure 35. Nanocrystalline stability maps where stable alloys appear in the top (green), metastable alloys
in the middle (yellow) and unstable alloys in the bottom (red) [101].

Stability maps can be used to select potential systems in which grain-growth can be prevented by a
thermodynamic approach. Although these maps do not indicate the global composition that will stabilize
the systems, that information can be determined using the previous segregation models. Additionally,
systems with proclivity for nanograin stability can also be predicted using mesoscale or atomistic
simulations of the segregation process [103].

2.4.5 Numerical Calculation of Segregation Properties
There are two main approaches to solve the diffusion equation at the grain boundary. First, at the mesoscale
level it is possible to solve simultaneously the segregation isotherms and Fick’s first and second laws by
the finite element method [104]. After solving these equations, the solute composition is obtained as a
function of position and time. For example, after equilibration, the Gibbs energy of a segregating system
can be calculated as a function of composition as shown in Figure 36 [105].

39

Figure 36. Gibbs energy density (denoted by F) at the grain boundary (blue) and in the bulk of the grains
(red) as a function of the global solute composition [105].

This approach simultaneously solves thermodynamic and kinetic equations, yielding temporal information
about the segregation process. Additionally, the diffusion problem can be solved at the atomic level using
molecular dynamics or Monte Carlo simulations. Molecular dynamics methods have been used to show
that the addition of dopants to the grain boundaries of polycrystalline Cu decreases the grain boundary
energy and increases the stability of the system, which helps to retain a nanoscale grain size at high
temperatures [106]. For example, Figure 37 shows that grain growth in polycrystalline Cu can be prevented
by doping with Bi [106].

Figure 37. (a) Initial columnar nanocrystalline copper. (b) Sample free of dopants annealed at 1200 K
showing grain growth. (c) Sample with dopants (red) at grain boundaries (green) annealed at 1200 K
showing retention of the grain size. (d) Close view of a junction in the sample with dopants (c) [106].

Otherwise, Monte Carlo simulations have been used to track the different stages of segregation under
different bulk solute concentrations. For instance, the composition that will reduce the grain boundary
energy to a minimum without precipitation at grain boundaries is computed by changing the solute content
of an ideal system (Figure 38) [107].

40

Figure 38. (a) Gibbs energy of the system showing a minimum around a grain size of 5 mm for a system
with a solute concentration of 5 at.% with Δ𝐻 𝑚𝑖𝑥 = 20 kJ/mol and Δ𝐻 𝑠𝑒𝑔 = 20 kJ/mol. Systems with a
solute concentration of (b) 1 at.% (c) 2 at.% (d) 10 at.% and (e) 15 at.% [107].

As shown in Figure 38, the total Gibbs energy of the system depends on the initial composition of the
sample and there are three possible scenarios for the evolution of the system [107]. If the solute
concentration is equal to the equilibrium composition, the solute segregates and effectively decreases the
grain boundary energy, thus preventing grain growth. However, If the solute content is below the
equilibrium value, not enough solute can segregate to fully decrease the grain boundary energy and grain
growth can occur. For concentrations above equilibrium, solute segregation is followed by precipitation,
which depletes the solute at the grain boundary and leads to grain growth. Even if the system has the
required solute concentration for stabilization via a thermodynamic approach, if the rate of segregation is
slower than that of other process like recrystallization, grain growth may still occur. Thus, it is necessary
to consider the effect of the kinetics of segregation on the microstructural evolution of NMMs.

41

2.5 Kinetics of Segregation
During annealing of polycrystalline materials, thermal motion induces atomic diffusion. As discussed in
Section 2.3.1, the diffusive process is significantly faster at grain boundaries, where the higher Gibbs energy
decreases as new metallic bonds with solute atoms are formed. This energy reduction is the driving force
for segregation [23]. The rate of segregation, which depends on the local composition and temperature, is
expressed using kinetic expressions that describe how the system evolves over time while the solute fraction
at the grain boundary tends to equilibrium.

2.5.1 Model of Limited Reaction Rates
Grain boundary segregation is a two step process that depends on the diffusion in the bulk of the grains and
at the grain boundaries. When the bulk diffusion is significantly faster, grain boundary diffusion is the
controlling step. Hence, the rate of solute diffusion from the grains to the interfaces is constant, and the
segregation process is controlled by the difference between the initial and the equilibrium solute
concentrations at the grain boundary:

𝑑𝑥
𝑎 𝑔𝑏
𝑑𝑡
= 𝑘 (𝑥 𝑎 𝑔𝑏 ,𝑒𝑞
− 𝑥 𝑎 𝑔𝑏
) (37)
Where 𝑘 is the rate constant, which is given by the Arrhenius equation. Integrating equation (37) from time
𝑡 to time 𝑡 = ∞:

𝑥 𝑎 𝑔𝑏
= 𝑥 𝑎 𝑔𝑏 ,𝑒𝑞
(1− 𝑒 −𝑘𝑡
) (38)
Equation (38) can be used to calculate the solute concentration at the grain boundary 𝑥 𝑎 𝑔𝑏
as a function of
time 𝑡 . This simple expression is only valid for fast diffusing systems. For example, for diffusion of C in
W [108]. For systems in which the solute and solvent have comparable radius, bulk diffusion needs to be
considered.

2.5.2 Semi Infinite Solution of Fick Equation at the Grain Boundary
When the limiting step is the bulk diffusion, Fick’s second law can be solved near the grain boundary [109].
For a grain boundary between two semi-infinite grains [23]:

𝜕 𝑥 𝑎 𝑔𝑏
𝜕𝑡
= 𝐷 𝑎 ,𝑏 (
𝜕 2
𝑥 𝑎 𝑔𝑏
𝜕 𝑋 2
) (39)

42

Where 𝑥 𝑎 𝑔𝑏
is the fraction of a at the grain boundary, 𝑋 is the position inside the bulk of the grains and
𝐷 𝑎 ,𝑏 is the interdiffusion coefficient of 𝑎 in 𝑏 . For the initial stages of the diffusion process that coefficient
is given by the Nasarov-Gurov expression, equation [110]:

𝐷 𝑎 ,𝑏 =
𝐷 𝑎 𝐷 𝑏 𝑥 𝑎 𝐷 𝑎 +(1−𝑥 𝑎 )𝐷 𝑏 (40)
At conditions near equilibrium the interdiffusion coefficient is given by the Darken equation:

𝐷 𝑎 ,𝑏 = 𝑥 𝑎 𝐷 𝑎 + (1− 𝑥 𝑎 )𝐷 𝑏 (41)
Equation (39) can be solved to obtain the grain boundary composition 𝑥 𝑎 𝑔𝑏
as a function of time 𝑡 . For the
solution Laplace transforms and three boundary conditions are used. The first boundary condition is the
mass balance at the grain boundary:

𝛿 𝑔𝑏
(
𝜕 𝑥 𝑎 𝑔𝑏
𝜕𝑡
) = (
𝜕 𝑥 𝑎 𝜕𝑙
)
𝑋 =0
(42)
The other two boundary conditions are the concentration of 𝑎 at the grain boundary at time 𝑡 = 0, which is
𝑥 𝑎 𝑔𝑏
, and at time 𝑡 = ∞, which is the saturation composition 𝑥 𝑎 ,𝑠𝑎𝑡 𝑔𝑏
. The solution to the partial differential
equation (39) is:

𝑥 𝑎 ,𝑡 𝑔𝑏
−𝑥 𝑎 ,0
𝑔𝑏
𝑥 𝑎 ,𝑠𝑎𝑡 𝑔𝑏
−𝑥 𝑎 ,0
𝑔𝑏
= −𝐸𝑥𝑝 (
𝑋 𝛽 𝑎 𝑔𝑏
𝛿 𝑔𝑏
+
𝐷 𝑎 ,𝑏 𝑡 𝛽 𝑎 𝑔𝑏
𝛿 𝑔𝑏
2
)𝐸𝑟𝑓𝑐 (
𝑋 2
√
𝐷 𝑎 ,𝑏𝑓
𝑡 +
√
𝐷 𝑎 ,𝑏𝑓
𝑡 𝛽 𝑎 𝑔𝑏
𝛿 𝑔𝑏
) (43)
Where 𝛽 𝑎 𝑔𝑏
= 𝑥 𝑎 ,𝑡 𝑔𝑏
𝑥 𝑎 ,0
⁄ is the enrichment parameter at the grain boundary, which relatively indicates the
strength of the segregation process. Equation (43) can be used to calculate the composition at the grain
boundary as a function of time. For instance, Figure 39 shows the surface coverage (left side of equation
43) as a function of normalized time for strongly segregating systems (𝛽 𝑎 𝑔𝑏
≫ 1) [111].

43

Figure 39. Grain boundary coverage as a function of normalized time for strongly segregating systems
(𝛽 𝑎 𝑔𝑏
≫ 1) [111].

This simple model can be used to estimate the diffusion time when the enrichment parameter is known. A
better approach to calculate the segregation time is to solve the diffusion problem at the grain boundary
assuming that the solute concentration inside the grains does not change significantly over time. This is one
of the main assumptions of the layer by layer models.

2.5.3 Layer by Layer Models
In the layer by layer approaches, the composition is calculated in the direction perpendicular to the grain
boundary as a function of time. Therefore, composition profiles are obtained numerically by solving Fick’s
second law for the planes inside and around the grain boundaries. In these models, the atomic fluxes are
calculated using jump probabilities.

2.5.3.1 Model of Hofmann and Erlewein
The model of Hofmann and Erlewein assumes that there are parallel layers of uniform composition in the
direction perpendicular to the grain boundary and that the activation energy for diffusion at the interface is
lower by the factor Δ𝐺 (Φ) . Thus, the fluxes of solute atoms between the layers next to the grain boundaries
are as depicted in Figure 40 [23].

44

Figure 40. Gibbs energy as a function of distance showing the activation energy as grain the layers at the
interphase Δ𝐷 𝐷 + Δ𝐺 (Φ) and in the bulk of the grains Δ𝐷 𝐷 [23].

The fluxes between the layers are a function of the Gibbs energy distribution near the grain boundary:

𝑗 12
=
1
𝑎 2
𝑥 1
𝑤 12
𝑣 1
𝐸𝑥𝑝 [−
(Δ𝐷 𝐷 +Δ𝐺 (Φ))
𝑅𝑇
] (44)
𝑗 𝑖 ,𝑖 ±𝑖 =
1
𝑎 2
𝑥 𝑖 𝑤 𝑖 ,𝑖 ±1
𝑣 1
𝐸𝑥𝑝 [−
Δ𝐷 𝐷 𝑅𝑇
] (45)
Where 𝑗 𝑖 ,𝑖 ±1
is the flux from layer 𝑖 to layer 𝑖 + 1, 𝑎 is the atomic jump distance, 𝑥 𝑖 is the concentration of
solute atoms in the layer 𝑖 , 𝑣 1
is the solute jump frequency, and 𝑤 𝑖 ,𝑖 ±1
is the jump probability factor of the
solute atoms flux from layer 𝑖 to layer 𝑖 + 1. Equation (44) applies to the interface and (45) to the layers
inside the grains. Using these equations Fick’s second law for the layers is rewritten in the form [112]:

𝑑 𝑥 1
𝑑𝑡
= (𝑗 21
− 𝑗 12
)𝑎 2
=
𝐷 𝑎 2
[𝑤 21
𝑥 2
− 𝑤 12
𝐸𝑥𝑝 (Δ𝐺 (Φ)/𝑅𝑇 )] (46)
𝑑 𝑥 𝑖 𝑑𝑡
= (𝑗 𝑖 −1,𝑖 + 𝑗 𝑖 +1,1
− 𝑗 𝑖 ,𝑖 −1
− 𝑗 𝑖 ,𝑖 +1
)𝑎 2
=
𝐷 𝑎 2
[𝑤 𝑖 −1,𝑖 𝑥 𝑖 −1
− 𝑤 𝑖 ,𝑖 −1
𝑥 𝑖 + 𝑤 𝑖 +1,𝑖 𝑥 𝑖 +1
− 𝑤 𝑖 ,𝑖 −1
𝑥 𝑖 ] (47)
Where 𝐷 is calculated using the following equation:

𝐷 = 𝑎 2
𝐸𝑥𝑝 (−Δ𝐺 𝐷 /𝑅𝑇 ) (48)
The solute concentration profile in and around interfaces is obtained by solving numerically equations (46)
and (47) over time. This model was used to estimate the segregation profile of Sn diffusing in Cu (Figure
41), which show that the concentration of Sn increase at the grain boundary as the solute atoms in the
surrounding layers move towards the interface [112].

45

Figure 41. Segregation profile of Sn (solute) at a Cu surface as a function of time. Sn segregation occurs
as the surface energy of copper decreases[112].

Even though the flux of atom between the layers is calculated using jump frequencies, these values are not
readily available for most systems. Therefore, another approach is to calculate these fluxes using chemical
potentials.

2.5.3.2 Model of Du Plessis and Van Wyk
Du Plessis and Van Wyk proposed using chemical potentials to calculate the flux of solute atoms to the
interface, thus approximating the diffusivity using thermodynamic variables. They divided the region
around the grain boundary in multiple subsystems as shown in Figure 42 [113].

Figure 42. Different subsystems in which the region around the grain boundary is divided. Φ stands for
the grain boundary and B the bulk of the grains[113].

The flux of atoms between each subsystem is given by [113]:

𝐽 1
1,𝑔𝑏
= 𝑀 1
𝑐 𝑥 1
1
Δ𝜇 1
1,𝑔𝑏
𝑎 (49) 𝐽 1
𝑖 +1,𝑖 = 𝑀 1
𝑐 𝑥 1
𝑖 +1
Δ𝜇 1
𝑖 +1,𝑔𝑏
𝑎 (50)

46

Where 𝑀 1
is the mobility of component 1, 𝑐 is the concentration of atoms per unit cell, 𝑥 1
𝑖 +1
is the mole
fraction of 1 (the solute) in the subsystem 𝑖 + 1, and Δ𝜇 1
𝑖 +1,𝑖 is the difference in chemical potentials between
the subsystems 𝑖 and 𝑖 + 1:

Δ𝜇 1
1,𝑔𝑏
= Δ𝐺 1
𝑔𝑏
+ 𝑅𝑇𝑙𝑛 (
𝑥 1
1
(1−𝑥 1
𝑔𝑏
)
𝑥 1
𝑔𝑏
(1−𝑥 1
1
)
) = 𝜇 1
0,𝑏 − 𝜇 1
0,𝑏 + 𝜇 2
0,𝑏 − 𝜇 2
0,𝑏 + 𝑅𝑇𝑙𝑛 (
𝑥 1
1
(1−𝑥 1
𝑔𝑏
)
𝑥 1
𝑔𝑏
(1−𝑥 1
1
)
) (51)
Δ𝜇 1
𝑖 +1,𝑖 = 𝑅𝑇𝑙𝑛 (
𝑥 1
𝑖 +1
(1−𝑥 1
𝑖 )
𝑥 1
𝑖 (1−𝑥 1
𝑖 +1
)
) (52)
In (51) and (52) the regular solution model has been used. Using equations (49) and (50), Fick’s second
law is expressed for each subsystem in the form:

𝑥 1
𝑔𝑏
𝜕𝑡
=
𝐽 1
1,𝑔𝑏
𝑎 = [
𝑀 1
𝑥 1
1
𝑎 2
Δ𝜇 1
1,𝑔𝑏
] (53)
𝜕𝑥
1
𝑖 𝜕𝑡
=
𝐽 1
𝑖 +1,1
−𝐽 𝑖 ,𝑖 −1
𝑎 = [
𝑀 1
𝑥 1
𝑖 +1,𝑖 𝑎 2
Δ𝜇 1
𝑖 +1,𝑖 −
𝑀 1
𝑥 1
𝑖 ,𝑖 −1
𝑎 2
Δ𝜇 1
𝑖 ,𝑖 −1
] (54)
Equations (51) to (54) are solved to calculate the solute concentration profile using a similar approach as
in the model of Hofmann and Erlewein. For example, as displayed in Figure 43, using these equations the
concentration profiles in and around the grain boundary are estimated for an ideal positive segregating
system (Δ𝐺 1
𝑔𝑏
> 0).

Figure 43. Surface segregation profiles calculated as a function of time for a system with Δ𝐺 1
=
60 kJ/mol [113].

47

Since the solute profiles around the grain boundary extend over a large number of atomic layers, the layer
by layer models are only capable of predicting approximate concentration profiles near interfaces. A more
realistic approach would resolve solute concentration profiles over multiple grains and interfaces, this is
possible at the mesoscale level using phase field models.

2.5.4 Phase field models
In phase field models, Fick’s first and second laws are solved at the mesoscale by finite element methods
[114]. At every iteration, the local Gibbs energy is calculated using local density functions [104], which are
also used to estimate the potential for segregation. Through this approach it is possible to calculate the
solute partitioning between the grain boundaries and the bulk of the grains (Υ = 𝑛 𝑏 𝑔𝑏
/(𝑛 𝑏 𝑔𝑏
+ 𝑛 𝑏 𝑏 ) ) as a
function of time (Figure 44) [105].

Figure 44. Solute partitioning between the grain boundaries and the bulk of the grains (Υ = 𝑛 𝑏 𝑔𝑏
/(𝑛 𝑏 𝑔𝑏
+
𝑛 𝑏 𝑏 ) ) as a function of time for systems with varying chemical potential difference between the gran
boundaries and the bulk of the grains (𝐺 𝑔𝑏
𝐵 − 𝐺 𝑏 𝐵 ) [105].

When the difference in chemical potential between the grain boundaries and the bulk of the grains
( )
B
b
B
gb
G G − is high (green), the solute initially segregates to grain boundaries. However, at longer times
the strong difference in chemical potentials also drives phase separation and precipitation, which reduces
the solute concentration at the grain boundary and results in grain growth [105]. On the contrary, when this
difference in chemical potentials ( )
B
b
B
gb
G G − is small (blue), after initial segregation, the solute remains

48

dissolved at the grain boundary. These results suggest that to stabilize the system via segregation to the
grain boundaries the difference in chemical potentials should be small, which can only occur when the
chemical potential of the solute is similar to that of the solvent.

Solute segregation usually occurs simultaneously with other thermally activated microstructural
transformations. These processes are driven by a reduction of the excess energy and result in different
microstructures that are observed during annealing of NMMs. For example recovery, recrystallization, grain
growth, and precipitation cause the degradation of the multilayered structure. Therefore, these
transformations are examined in detail in the next section.

2.6 Microstructural Transformations in Nanomaterials
Microstructural transformations in solid materials are understood to be “any extensive rearrangement of the
atomic structure” [115]. The driving force for these transformations is the reduction of excess energy, as
the system moves towards states of lower Gibbs energy. Thus, using thermodynamic calculations, it is
possible to predict if a microstructural transformation can occur (a transformation must be
thermodynamically favorable) or if it is forbidden. However, the description of an ongoing transformation
is only possible using kinetic models, and models of the atomic processes involved in the transformation.

Most of the transformations in solids are heterogeneous and can be classified into two main groups [115].
First, martensitic transformations involve small atomic rearrangements over large volumes. In these
transformations, the new phase develops after the cooperative movements of many atoms at a high velocity,
process which is relatively independent of temperature. It has been observed that the transformation rate
does not change over time. A noticeable change in grain shape is a characteristic of martensitic
transformations. On the other hand, the “nucleation and growth” transformations involve drastic atomic
rearrangements over localized volumes. In this type of transformations, the new phase grows via atom by
atom diffusion across the interphase boundary with a temperature dependent rate. Even if the volume of the
grains changes after these transformations, the initial shape of the grains is conserved.

Considering that a significant number of transformations do not fit this simple scheme, the classification
presented in Figure 45, which takes into account whether growth is athermal (rate almost independent of
temperature) or thermally activated (rate dependent on temperature) was proposed [115]. When growth is
thermally activated it is possible to distinguish between transformations in which nucleation is the limiting
rate (interphase controlled) and transformations in which diffusion to the grain boundary is the controlling
mechanism (long-range transport). The latter transformations can be further categorized between

49

continuous reactions, when the new phase nucleates inside clusters inside the parent material, and
discontinuous reactions, when the new phase forms like lamellae at the grain boundary which extend inside
the parent phase over time.

Figure 45. Classification of transformations in terms of the growth process [115].

50

In heterogeneous transformations, the nucleation rate depends on the interfacial energy. The Gibbs energy
reduction due to the formation of the new phase is opposed by the increasing surface energy of the new
interface. As the surface energy increases, the critical nucleus size increases and the nucleation rate
decreases. Conversely, as the surface energy approaches zero the nucleation rate strongly increases, and the
transformation occurs uniformly through the material. An example of near-homogeneous transformations
are order-disorder transformations. Independent of the type, a thermal activated microstructural
transformation will only take place if there in enough energy available to overcome the activation barrier.
Therefore, due to their low activation energy recovery is the fist process that occurs in nanocrystalline
materials [116].

Recovery occurs due to the high density of dislocation and other defects; it involves the interaction of
mobile dislocations. During this transformation, a fraction of the dislocations annihilates, and the remaining
dislocations arrange in energetically favorable configurations. After recovery, or simultaneously,
recrystallization takes place, and the initial grain structure is replaced by new grains with a lower density
of defects. Recrystallization can be followed by segregation, which may induce precipitation depending on
the global solute content. As the main goal of this research is to understand how NMMs transform into
nanostructures, the dependence of the previous processes on composition, temperature and time is
considered next.

2.6.1 Recovery
At low temperatures, thermal vibrations activate the motion of defects like vacancies and dislocations. This
causes rearrangement and partial annihilation of dislocations without the motion of grain boundaries.
Recovery depends on the amount of strain, the temperature, and the material. This process occurs
homogeneously through the material and can be quantified using either the following empirical kinetic
equations for isothermal recovery [117]:

Kinetics type 1:

𝑋 𝑅 = 𝑐 2
− 𝑐 1
𝑙𝑛 (𝑡 ) (55)
Kinetics type 2:

𝑋 𝑅 −(𝑚 −1)
− 𝑋 0
−(𝑚 −1)
= (𝑚 − 1)𝑐 1
𝑡 for 𝑚 > 1 (56)
ln(𝑋 𝑅 )− ln(𝑋 0
)= 𝑐 1
𝑡 for 𝑚 = 1 (57)

51

Where 𝑋 𝑅 is the recovered fraction, which is measured following the change of a property of the material
(hardness, yield stress, resistivity, heat evolution), 𝑐 1
and 𝑐 2
are rate constants which depend on
temperature, 𝑚 is the order of reaction, and 𝑡 is the time. Equations (55), (56) and (57) show that as time
increases the recovered fraction 𝑋 𝑅 increases with a rate depending on the constants 𝑐 1
and 𝑐 2
. At low
temperatures recovery may partially occur. This is due to the fact that as-sputtered thin films can have a
high density of defects, which make it easier for recrystallization to occur over recovery [118].

2.6.2 Recrystallization
During recrystallization, new strain-free grains consume the recovered microstructure [6]. The two main
steps of this process are nucleation and crystallite growth. In the nucleation stage, crystallites appear
through the sample. Later, the crystallites grow by consuming deformed material. The recrystallization
process can be quantified using the Johnson-Mehl-Avrami generalized equation:

𝑋 𝑉 = 1− 𝐸𝑥𝑝 (−
𝑓 𝑁 ̇ 𝐺 ̇ 3
𝑡 4
4
) (58)
Where 𝑋 𝑉 is the fraction of recrystallized grains at time 𝑡 , 𝑓 is shape factor, 𝑁 ̇ is the nucleation rate, 𝐺 ̇ 3
is
the growth rate given by 𝐺 ̇ = (𝑑 𝑋 𝑉 /𝑑𝑡 )/𝑆 𝑉 ̇ , and 𝑆 𝑉 is the volume fraction of unrecrystallized material
[116]. The temperature of recrystallization, defined as the temperature at which 50% of the grains have
recrystallized, depends on multiple variables including the initial amount of deformation, the annealing
time, and the grain size [116]. In general, as the initial amount of deformation or the annealing time increase,
the crystallization temperature decreases. On the contrary, the recrystallization temperature increases with
the initial grain size. Overall, the formation of new strain-free grains during recrystallization lowers the
energy of the system. The recrystallization process and is then followed by grain growth.

2.6.3 Grain Growth
After recrystallization is complete, the microstructure is not yet in its most stable state and grain growth of
the recrystallized grains can occur [116]. The driving force for this process is the remaining energy stored
in the grain boundaries, which is two orders of magnitude lower than the energy required for
recrystallization (10
-2
MPa). Thus, grain growth can occur at a rate lower than that of recrystallization. The
rate of grain growth is the derivative of the average grain size as a function of time. The average grain size
is described by the Burke and Turnbull equation [116]:

𝑅 ̅
𝑛 − 𝑅 ̅
0
𝑛 = 2𝛼 𝑐 1
𝛾 𝑏 𝑡 (59)

52

Where 𝑅 ̅
is the average grain size, 𝛼 is a geometric factor, 𝑐 1
is a constant, 𝛾 𝑏 is the grain boundary energy,
and 𝑡 is the time. Grain growth is only observed at very high temperatures and the rate of this transformation
increases with temperature. On the contrary, the presence of solutes which pin the grain boundaries or
highly textured grains with a high density of low angle grain boundaries reduce the rate of grain growth
[116].

2.6.4 Precipitation
Precipitation consists in the formation of new phases after annealing, quenching and tempering of a solid
solution composed of a solute with limited solubility [119]. It is possible to distinguish between four types
of precipitation [115]. Continuous precipitation occurs by nucleation of crystals, which grow by depleting
the solute in the surrounding matrix. After continuous precipitation, the orientation and shape of the grain
is conserved. On the other hand, discontinuous precipitation requires nucleation of duplex cells, which grow
from the grain boundary inside the matrix of the untransformed material. A characteristic of discontinuous
precipitation is the abrupt change in composition at the boundaries of the cell. While the kinetics of
continuous precipitation depend on the growth rate of the precipitates (interface or diffusion controlled),
the kinetics of discontinuous precipitation depends on the growth of the duplex cells, which occurs at a
linear rate.

Precipitation may also occur via Guinier-Preston zones or by spinodal decomposition [115]. In both cases,
the initial step of the transformation is the formation of solute rich clusters. After this, in the case of Guinier-
Preston zones, these clusters induce precipitation via small localized fluctuations that result in coherent
precipitates with plate or spherical morphologies. In contrast, in the spinodal decomposition clustering
continues via small fluctuations that extend all over the system until the new phase can be differentiated
from the parent phase.

53

3 Experimental Methods
This section introduces the methods employed in this study to synthesize and characterize NMMs. First,
the deposition of multicomponent thin films via magnetron sputtering is outlined. Next, the equipment used
to heat-treat NMMs under vacuum and obtain their calorimetric profiles are described. Finally,
characterization methods to study the residual stresses, the microstructure, and the local compositions of
as-deposited and heat-treated multilayers are presented.

3.1 Synthesis of Nanometallic Multilayers
In this study, NMMs were prepared via magnetron sputtering. During sputtering deposition, Ar pumped
into a vacuum chamber is ionized using a voltage bias, which forms a plasma [19]. The plasma ions
bombard the source or target (material being deposited), causing ejection of the target atoms, which then
migrate inside the chamber and deposit onto the substrate. A schematic of the deposition process is
presented in Figure 46.

Figure 46. Schematic representation of the magnetron sputtering deposition process.

In magnetron sputtering, the ionization process is facilitated by a magnetic field which acts as a trap for the
electrons. The magnetic field promotes easier Ar ionization and accelerates the resulting ions towards the
target [120]. Magnetron sputtering yields faster deposition rates than conventional sputtering, molecular
beam epitaxy, or electron beam techniques. Multiple targets can also be used, depending on the number of

54

materials being deposited [13]. In our process, two sources which turn on and off intermittently are used to
deposit NMMs, thus controlling the composition of the individual layers as shown in Figure 46.

Figure 47. Sputtering process of nanometallic multilayers, (a) sputtering from source 1 while a shutter
blocks the source 2, (b) sputtering from source 2 while a shutter blocks the source 1.

Additionally, the thickness of the layers inside NMMs is inversely proportional to the density of interphases
in the system. Therefore, by changing the layers thickness it is possible to control the diffusive processes
in the sample and modify the energy available for microstructural transformations. Thus, magnetron
sputtering, which allows for tailoring of thin films composition, grain structure, and density of interfaces
was used to deposit NMMs. After deposition, the morphology and nanoscale features of the films, were
studied using different characterization techniques.

3.2 Characterization Methods
Although NMMs have individual layers with a thickness of some nanometers, the total thickness of these
films is on the order of micrometers. Such dimensions facilitate the use of profilometry methods to measure
the thickness and residual stresses. Additionally, the texture of the sample can be studied using x-ray
diffractometry. However, to the study the morphology of the grains, the composition, and the structure of
the films, scanning and transmission electron microscopy studies are required. Furthermore, before
annealing the samples, differential scanning calorimetry scans are collected to identify thermally activated
transformations of these materials. These and other characterization techniques used in this work are
discussed in the following sections.

55

3.2.1 Profilometry of Thin Films
The thicknesses of as-sputtered films were measured by contact profilometry. In this technique, a 20 nm
diameter diamond stylus of scans the surface of the sample to obtain a Z height profile of the sample (Z
versus distance) [121]. In these profiles, vertical features ranging from 10 nm to 1 mm are detected. An
example of a profilometry scan is presented in Figure 48.

Figure 48. Image of (a) a Ta-Hf film being scanned by profilometry and (b) of the profilometry scan.

From these profiles, the curvature radius of the film is estimated using the maximum deflection of the film.
This value is then used in Stoney’s Equation to determine the amount of residual stresses [122]:

𝜎 =
−𝐸 𝑠 𝑡 𝑠 2
6𝑡 𝑐 (1−𝑣 𝑠 )𝑅 (60) 𝑅 =
𝐿 2
8𝛿 𝑚𝑎𝑥 (61)
Where 𝜎 is the total of residual stresses in the film, 𝑅 is the curvature radius of the film, 𝑡 𝑠 and 𝑡 𝑐 are the
substrate and coating thickness, 𝐸 𝑠 is the elastic modulus, 𝑣 𝑠 is the Poisson’s ratio, 𝐿 is the coating length,
and 𝛿 𝑚𝑎𝑥 is the maximum deflection (obtained from the scans). Equation (60) is only valid when the
thickness ratio (𝑡 𝑐 /𝑡 𝑠 ) is less than ~ 0.05. For the films used in this study, Equation (60) is valid since the
thin films are 2 µm thick and the Si substrate is 250 µm thick, yielding a ratio of 0.008. The calculated
residual stresses are then employed to compute the strain energy stored in the samples. After profilometry,
x-ray diffraction scans of the films were collected.

3.2.2 X-ray Diffraction
X-ray diffractometry (XRD) was used to obtain the orientation and crystallographic information of the
NMMs [123]. In this technique, a diffraction pattern displaying the intensity of diffracted x-rays versus the
diffraction angle 2θ are utilized to identify the crystallographic planes parallel to the surface of film. The

56

texture of the NMMs was determined from XRD scans collected using a Rigaku ultima IV diffractometer.
For instance, the diffraction pattern in Figure 49 shows that the W-Cr NMMs have a strong (110) texture.

Figure 49. X-ray diffraction scan of W-Cr nanometallic multilayer showing a highly (110) BCC texture.

X-ray diffractometry can also be used to estimate the grain size of the films. The orientation and the full
width of half the maximum intensity (FWHM) can be obtained by tilting the sample (rocking curves) around
a single orientation and gathering the reflected x-rays [124]. The in-plane grain size of the films can be
obtained using the FWHM of the main diffraction direction and Scherrer’s Equation:

𝑑 =
0.9𝜆
𝐹𝑊𝐻𝑀 𝐶𝑜𝑠 (𝜃 𝐵 )
(62)
Where 𝑑 is the grain size, 𝜆 is the wavelength of the x-ray source, and 𝜃 𝐵 is the angle at the center of the
peak. Before performing thermal tests, the global composition of the samples was measured by dispersive
x-ray spectroscopy using a scanning electron microscope.

3.2.3 Scanning Electron Microscopy
In scanning electron microscopy (SEM), a beam of electrons, which is produced using a filament or a field
emission gun, is condensed using multiple electromagnetic lenses and apertures into a beam of around 10
nm [125]. This beam of electrons is used to raster the surface of a sample. The different types of interactions
that occur between the electrons and the sample are illustrated in Figure 50. When the interaction is

57

inelastic, the electrons collide with the top atomic layers causing emission of secondary electrons (lower
energy electrons reflected at low angles) and x-rays [13]. If the interaction is elastic, the energy of the
electrons is almost totally conserved, and the colliding electrons are reflected at high angles after advancing
inside the sample (back-scattered electrons). Different information is then obtained depending on the
collected signal:

Figure 50. (a) Signals produced by the interaction between electrons and the sample and (b) their
corresponding energy [13, 126] .

Secondary electrons: This is the basic signal used to form SEM images. Secondary electrons, which are
generated from the top layers of the sample, are used to form images of the surface. The contrast in the
image depends on the depth of the sample. Higher places, which are more exposed, reflect a higher number
of electrons. For example, edges in the sample appear brighter than planes in secondary electron images.

Back-scattered electrons: Back-scattered electrons escape from the sample at near Bragg angles. These
electrons form Kikuchi patterns, a set of bands with varying degrees of intensity, on a phosphorous screen.
In electron backscattered diffractions (EBSD), these patterns are indexed to identify the crystallographic
orientation of each grain on the surface.

X-rays: When the electrons interact inelastically with the surface of the sample, x-rays with different
energies are emitted from the surface. The energy of these x-rays depends on the types of constituent atoms
and their molar fraction. Then, by analyzing the intensity of the x-rays as a function of energy, it is possible
to evaluate the composition of the sample. This technique is called energy dispersive x-ray spectroscopy
(EDS).

Cross-sectional SEM micrographs and EDS scans (obtained using a JSM-7001F-LV with an EDAX EDS
detector) were obtained to measure the thickness and to estimate the global compositions of the samples.

58

Subsequently, differential scanning calorimetry scans were performed to identify microstructural
transitions occurring during heat-treatment of the samples.

3.2.4 Differential Scanning Calorimetry
In differential scanning calorimetry (DSC), the heat flow to the sample is measured as the temperature is
increased at a constant rate. The heat flow changes when thermal transitions occur. For example, during
recrystallization (an exothermic process), the measured heat flow decreases. Furthermore, other processes
such as segregation, precipitation or grain growth, also appear as endothermic or exothermic peaks over a
baseline [127]. For instance, depending on the material history, different recrystallization peaks are
observed in nanocrystalline Cu (Figure 51) [127].

Figure 51. DSC scans of nanocrystalline Cu (40 nm grain size) stored at 20 °C for 20 min, at 20 °C for 5
days, and at -20 °C for 5 days, showing different recrystallization enthalpies [127].

The temperature and enthalpies of recrystallization, precipitation, and grain growth transformations
occurring between 20 °C and 1000 °C were identified through DSC scans (obtained using a Labsys TG-
ATD-DSC and a STA 449 F5 Jupiter thermal analyzers) of the NMMs. These values were used to select
temperatures before and after microstructural transitions to perform heat-treatments under vacuum.

3.2.5 Heat-treatment of NMMs
The NMMs were heat-treated at temperatures ranging between 20 °C and 1000 °C for 96 h in a GSL1100X
tube furnace under vacuum (pressure ≈ 5 x 10
-4
Pa). At the end of the heat-treatments, the samples were

59

quenched in low-pressure vacuum oil (Invoil 705, Inland Vacuum Industries) without breaking vacuum.
Figure 52 shows a schematic of the vacuum furnace used.

Figure 52. Schematic of vacuum furnace setup used in this study.

This furnace was connected to a getter pump to guarantee a vacuum pressure lower than 2 × 10
−5
Pa. After
quenching the samples and waiting for the furnace to cool down, the vacuum was broken, and the samples
were removed. Subsequently, they were cleaned with isopropanol and stored under vacuum. The samples
were then prepared by focus ion beam lift-out for TEM characterization.

3.2.6 Focus Ion Beam
In focus ion beam (FIB) microscopy, Ga ions produced inside of a liquid metal source are extracted by field
emission [128]. The interaction between the ions and the surface of the sample results in the emission of
backscattered ions, sputtered ions, radiation, and ion induced secondary electrons (ISE). Secondary
electrons are then detected by charge electron multipliers to create FIB images, which have a better
channeling contrast than SEM images. This is useful to observe crystals with different orientations.

The sputtering action of the ion beam is used to mill the sample. This feature, combined with
micromanipulators and gas injection systems (GIS) for chemical vapor deposition of carbon or platinum
inside a dual SEM and FIB system (Figure 53), is used to prepare TEM samples via FIB lift-out. In this
technique, a cross-sectional lamella of approximately 20 µm in length and of 2 µm thickness is extracted

60

from the sample and thinned until electron transparency is achieved. An illustration of this technique,
showing how the lamella is milled out of the sample, attached to the micromanipulator, and thinned is
presented in Figure 54.

Figure 53. Schematic of dual FIB-SEM system used to perform FIB lift-outs [128].

Figure 54. Illustration of the lift-out procedure showing (a) the top view of the sample coated with a
protective carbon coating, (b) lamella ready to be extracted from the sample, (c) after attaching it to the
micromanipulator, and (d) a sample thinned to electron transparency.

61

TEM samples of the as-sputtered and heat-treated NMMs were prepared using a JIB-4500 multi-beam FIB.
At the end of the lift-out, the samples were thinned below 60 nm thickness and stored under vacuum before
observation in the TEM.

3.2.7 Transmission Electron Microscopy
In transmission electron microscopy (TEM), a parallel beam of electrons is utilized to observe
microstructures with nanometer-sized features. The electrons, generated in a field emission gun, are
condensed (using condenser lenses) onto a thin specimen [88]. Subsequently, the image formed by the
transmitted electrons is magnified using electromagnetic lenses (objective and projector lenses). After
magnification, an image is formed on a phosphorous screen or a charged couple device (CCD) camera.

The energy of the transmitted electrons depends on how the electrons interact with the sample [13]. When
the interaction is elastic, the transmitted electrons conserve their energy. These electrons are used to form
diffraction patterns. During inelastic scattering, the transmitted electrons lose energy due to absorption and
other scattering processes, which modify the intensity of the beam. This increases the contrast of the
microstructural details. It is possible to collect either elastic or inelastic electrons by changing the position
of the objective aperture (located below the sample and above the objective lenses). This enables the
different TEM operation modes, as illustrated in Figure 55:

Bright field mode (BF): This is the conventional imaging TEM technique. Images of the microstructure are
formed using the rays of the central beam, while the diffracted beams are blocked using different apertures.
Bright field micrographs show the diffraction-contrast of the grains within the sample.

Dark field mode (DF): In this mode, the image of the sample is formed using one of the diffracted beams
and blocking all other beams with the objective aperture. Thus, only the grains in the selected direction will
be visible. Dark field micrographs, which are used to measure the grain size, clearly display the grain
morphology.

Selected area diffraction (SAD): Diffraction patterns provide the orientation and crystallographic
information of crystals, defects and interfaces in the sample. Diffraction patterns can be obtained by several
methods, the most common of which is area diffraction. In this method, the shape and distances of the
patterns are used to identify the crystalline structure and can be used to calculate the lattice parameter.
Furthermore, in convergent beam diffraction, the electron beam is converged to a single spot which is

62

defocused to obtain diffraction patterns, which provide the point group symmetry of the zone axis. These
point groups are then used to determine the crystallographic group.

High-resolution (HR): In high-resolution TEM imaging, the angle of the electron beam is changed to
combine the central and some diffracted beams, preserving their amplitude and phase. This method is used
to capture atomic resolution micrographs where lattices can be distinguished.

Figure 55. (a) Bright field mode, (b) dark field mode, and (c) selected area diffraction mode [88]. Cross-
sectional TEM micrographs of Hf-Ti nanometallic multilayers showing (d) a bright field image, (e) a dark
field image, and (f) a selected area diffraction pattern.

Cross-sectional bright field and high-resolution micrographs of the as-sputtered and heat-treated NMMs
were taken in a JEOL JEM-2100F transmission electron microscope. In addition, multiple dark field images

63

were taken to measure the average grain size of each sample. Later, diffraction patterns were captured to
determine the crystal structure and orientation of areas in the sample. After observing the samples in TEM
mode, compositional EDS maps were obtained in the scanning transmission electron microscopy mode.

3.2.8 Scanning Transmission Electron Microscopy
In scanning transmissions electron microscopy (STEM), the electron beam is focused on a point that moves
in a television like raster pattern to scan the sample. The transmitted electrons form two types of images
[129]. First, the electrons transmitted along the axis of the beam are gathered using a bright field detector
to form bright field STEM images (Figure 56a). Second, high angle incoherent scattered electrons, collected
using a high angle annular dark filed detector, are used to form annular diffraction micrographs (HAADF-
STEM) [130]. HAADF-STEM images have compositional contrast (Figure 56b). The bright field STEM
images, which are complementary to the HAADF-STEM micrographs, are formed via diffraction contrast,
which process images that have a better contrast in comparison to bright-field TEM micrographs [131].

Figure 56. Cross-sectional (a) bright field STEM image and (b) high angle annular dark field images
showing Hf-rich grains and Ti precipitates at the grain boundaries of Hf-Ti NMMs heat-treated at 800 ºC.

As the beam interacts with the sample, x-rays, secondary electrons, scattered electrons, and backscattered
electrons are emitted [88]. The x-rays are used in energy dispersive x-ray spectroscopy (EDS) to obtain
compositional point scans and maps of regions inside the sample with resolution close to the size of the
electron beam (Figure 57). In this technique, the intensity of the x-rays with a particular energy
corresponding to the emission of one element is used to find the relative composition of the sample.

64

EDS maps provide information that can be correlated to compositional changes along with microstructural
transformations. For example, solute segregation and precipitation resulted in decreased solute
concentration in the matrix of the grain, but increased concentrations at the grain boundaries. Additionally,
EDS composition profiles can be used to follow the concentration of impurities in the sample. For example,
Figure 57 shows the EDS maps for annealed Hf-Ti NMMs, which reveal a strong segregation of Ti (Figure
57c) to the grain boundaries.

Figure 57. Energy dispersive x-ray spectroscopy maps of Hf-Ti NMMs heat-treated at 800 ºC showing (a)
a composed map, (b) a Hf map, (c) a Ti map, (d) and an O map.

65

4 Microstructural Evolution of NMMs: The Hf-Ti system
Poor thermal stability has hindered the development of nanocrystalline materials for most applications,
especially at elevated temperatures, where high grain boundary mobility typically results in significant grain
growth and degradation of nanoscale properties [2, 41]. Therefore, extensive efforts have been made to
inhibit grain growth and stabilize grain boundaries via kinetic or thermodynamic mechanisms (see Section
2.3), which have been employed with some success [43, 44, 65, 132].

Although proposed models have shown potential for synthesizing thermally stable nanostructures (see
Section 2.4), there are several open questions concerning the mechanisms controlling the stability; for
instance, to what extent a given process is fully thermodynamically or kinetically driven is still under
debate. Specifically, most experimental studies have been carried out with materials prepared by
mechanical deformation methods, where a high density of dislocations, other defects, and chemical
contamination can modify the energy of the system [133, 134]. As an alternative to those synthesis methods,
a recent publication by Polyakov et al. showed that a sputtered multilayered configuration can be used as a
route to synthesize nanostructures that are stable up to 800 °C based on the aforementioned thermodynamic
models [50]. However, the process by which the multilayers rearranged into equiaxed grains is not fully
understood. Although there have been studies on the thermal stability of multilayered systems, their focus
has been on maintaining the integrity of the layers rather than using nanometallic multilayers (NMMs) as
a route to achieve stable equiaxed nanocrystalline structures [71, 72, 80, 82].

In this investigation, we identified the microstructural transformations occurring in NMMs as function of
temperature. The morphologies of Hf-Ti NMMs heat-treated at temperatures ranging from 500 °C to 1000
°C are presented. Differential scanning calorimetry (DSC) scans and characterization tools were used to
identify critical temperature-induced events and the microstructural changes occurring at each stage. This
study shows the stages of recrystallization, segregation, precipitation and grain growth with respect to the
mechanisms controlling the evolution of heat-treated Hf-Ti NMMs. These findings were published in
Scripta Materialia 142 (2018) 55-60 under the title: “Exploring the microstructural evolution of Hf-Ti: From
nanometallic multilayers to nanostructures” [135].

4.1 Synthesis of Hf-Ti NMMs
Ti/Hf-Ti multilayers were deposited on 25.4 mm round microprobe glass slides by magnetron sputtering at
0.80 Pa Ar using a power of 60 W and 200 W for the Ti and Hf source, respectively [135]. The composition
of the films was tuned by controlling the on-time of the Hf source following procedures listed elsewhere

66

[50]. The structure of the as-sputtered Hf-Ti NMMs is shown in the cross-sectional bright field and dark
field TEM images in Figure 58a and 58b. The overall film thickness is 2 µm, while the layer thicknesses
are 5 nm Ti (99.9 at.% Ti) and 48 nm Hf rich (9.7 at.% Ti), respectively. The sample has a vertical growth
direction with densely packed fibrous columnar grains characteristic of a morphology type zone 1 [136].
XRD scans of the as-sputtered NMMs, obtained using a Rigaku Ultima IV diffractometer, show a highly
(002) HCP texture.

Figure 58. (a) Bright field TEM and (b) dark field TEM images and (c) the grain size distribution of the
as-sputtered Hf-Ti multilayers.

The global composition of the as-sputtered sample (20 at.% Ti) was measured using a JSM-7001 SEM
microscope with an EDS EDAX detector. The grain size distribution (Figure 58c) was obtained by
measuring the in-plane grain size (perpendicular to the growth direction) and the perpendicular grain size
(parallel to the growth direction) of more than 200 grains. In order to present a “grain size” rather than a
“grain width”, the grain size was calculated by approximating every columnar grain as a sphere with equal
volume as the grain [17], where the grain size is the diameter of this sphere. From this method, the average
grain size of the as-sputtered sample was 48 ± 19 nm.

4.2 Microstructural Transitions in HF-Ti NMMs
DSC scans were performed on freestanding Hf-Ti NMMs films to estimate critical temperatures for events
occurring during heat-treatments. Before the scans, the DSC chamber was flushed with the maximum flow
of Ar for 20 min, then the flow was decreased to 40 ml/min and the samples were stabilized for 10 min
before heating began. Figure 59 shows a DSC of the as-sputtered multilayers from 20 °C to 1000 °C
performed at a rate of 10 °C/min under a constant Ar flow of 40 ml/min (Labsys thermal analyzer).

67

Figure 59. Differential scanning calorimetry scan of the Hf-Ti nanometallic multilayers 20 °C to 1000 ºC
showing the multiple transitions occurring between 494 ºC and 990 ºC.

The scan shows that recrystallization (exothermic peak) occurs between 494 ºC and 650 ºC with
Δ𝐻 𝑟𝑒𝑐𝑟𝑦𝑠𝑡𝑎𝑙𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 = -9.7 kJ/mol [127]. During recrystallization Ti segregates to the grain boundaries and
induces Ti precipitation (endothermic peak) around 666 ºC with Δ𝐻 𝑝𝑟𝑒𝑐𝑖𝑝𝑖𝑡𝑎𝑡𝑖𝑜𝑛 = 1.8 kJ/mol. This is
followed by subsequent grain growth (exothermic peak) between 737 ºC and 990 ºC with
Δ𝐻 𝑔𝑟𝑎𝑖𝑛 −𝑔𝑟𝑜𝑤𝑡 ℎ
= -36.7 kJ/mol [137]. Please note that the transitions appeared as endothermic or
exothermic peaks over a baseline, which was removed from the scans [138].

4.3 Microstructural Evolution of Hf-Ti NMMs
To connect the critical events identified by DSC scans to resulting microstructures, freestanding films were
heat-treated at 500 °C, 800 °C and 1000 °C, for 96 h under vacuum (pressure ≈ 5 x 10
-4
Pa) inside a
GSL1100X tube furnace (MTI Corporation). At the end of the heat treatments, the samples were quenched
in a low vapor pressure oil (Invoil 705, Inland Vacuum Industries) without breaking vacuum to avoid the
phase separation that can occur in the Hf-Ti system below 227 ºC (Appendix C.1) [139]. Subsequently,
samples for TEM were prepared by FIB lift out using a FIB-4500 (JEOL). TEM images of the sample
(Figure 60a - 60c) were captured on a JEM-2100F (JEOL) microscope.

68

Figure 60. Cross-sectional bright field TEM images and grain size distributions of samples heat-treated
(b,e) at 500 ºC (breakthroughs in the Ti layers have been circled in yellow), (c,f) at 800 ºC (inset shows a
nanodiffraction pattern of the amorphous Ti precipitates, the diffraction spots evidence short-range order),
and (d,g) at 1000 ºC (inset shows a selected-area diffraction pattern of a crystallized Ti precipitate in the
[221] zone axis).

At 500 ºC, the DSC scan indicates that recrystallization has already started; although, the initial columnar
structure and the multilayered morphology of the as-sputtered sample is mostly preserved (Figure 60a).
However, at this temperature there is some roughening and breakthroughs in the pure Ti layers, highlighted
in yellow in Figure 60a, similar to observations reported in immiscible NMMs [13, 22, 23]. In contrast, the
microstructure at 800 ºC (Figure 60b) is comprised of equiaxed grains surrounded by amorphous
precipitates (Inset Figure 60b), suggesting reorientation of the columnar grains, as well as precipitation at
the grain boundaries. This is in agreement with the results by Polyakov et al., which showed the contrast
between the grains and the precipitates is caused by differences in composition [11]. Figure 60c shows the
micrograph for the sample heat-treated at 1000 ºC, which exhibits Hf-rich grains surrounded by impinged
and crystallized Ti grains arranged in two layers equidistant from the center of the sample (Figure 60c).

The grain growth observed in the TEM images of the heat-treated samples was quantified by calculating
the grain size distribution using the procedures described earlier. Figure 60d shows the grain size
distribution for the sample heat-treated at 500 ºC is similar to that of the as-sputtered sample, but with a
larger average grain size of 68 ± 14 nm. At 800 ºC, after grain growth has started, the distribution spreads

69

and the average grain size increases to 71 ± 37 nm (Figure 60e). As shown in Figure 60f, the grain size
distribution spreads further at 1000 ºC, which is reflected by an average grain size almost three times greater
than that of the as-sputtered sample, 132 ± 62 nm.

4.4 Recrystallization in Hf-Ti NMMs
Although the TEM micrographs clearly point out microstructural changes during heat treatment of the
multilayers, it is necessary to evaluate the intermediate steps between the critical events highlighted in the
DSC scans. First, recrystallization (which starts at 494 ºC as suggested by the DSC scans) is activated by
the strain energy stored in the films, primarily due to the difference in the radius of Hf and Ti. Residual
stresses in the as-sputtered films, calculated using Stoney’s equation and curvature measurements obtained
by profilometry, were used to compute the strain energy (1.3 x 10
9
J/m
3
). Overall, this value is within the
reported range by Thornton et al. for sputtered thin films [140]. As the newly formed crystallites consume
the strained material, the internal stresses in the film relax and the columnar structure transforms into
equiaxed grains [116]. During recrystallization, segregation initiates, and Ti atoms start to diffuse to the
grain boundaries. The segregation process continues until the Ti mole fraction at the grain boundary is close
to the saturation composition, when the grain boundary energy is minimized.

4.5 Grain Boundary Energy of the Hf-Ti System
An estimate of the molar fraction of Ti at grain boundary saturation was obtained by calculating the grain
boundary energy as a function of the grain boundary composition for different temperatures. The grain
boundary energy, 𝛾 , was calculated using the Darling et al. modification of the Wynblatt and Ku model [91,
94], which uses the Gibbs adsorption equation:

𝛾 𝛾 𝑏 = 1+ Γ
𝑇𝑖
Δ𝐺 𝑠𝑒𝑔 𝛾 𝑏 (63)
where 𝛾 𝑏 is the interfacial grain boundary energy (approximately one third of the surface energy of the
solvent [35]), Γ
𝑇𝑖
is the Ti excess at the grain boundary, and Δ𝐺 𝑠𝑒𝑔 is the Gibbs energy of segregation. To
evaluate Γ
𝑇𝑖
, the Ti mole fraction in the bulk, 𝑋 𝑇𝑖
𝑏 , was calculated using the following mass balance
equation:

𝑋 𝑇𝑖
𝑏 =
6(𝑉 𝐻𝑓
)
1/3
𝑑 𝑋 𝑇𝑖
𝑔𝑏
−𝑋 𝑇𝑖 ,0
6(𝑉 𝐻𝑓
)
1/3
𝑑 −1
(64)

70

Here 𝑋 𝑇𝑖
𝑔𝑏
is the Ti mole fraction at the grain boundary and 𝑋 𝑇𝑖 ,0
is the global Ti molar fraction in the
sample. 𝑉 𝐻𝑓
corresponds to the molar volume of hafnium, and 𝑑 is the grain size. By simultaneously
changing 𝑋 𝑇𝑖
𝑔𝑏
and 𝑋 𝑇𝑖
𝑏 , Γ
𝑇𝑖
was computed using the following equation:

Γ
𝑇𝑖
=
2(𝑋 𝑇𝑖
𝑔𝑏
−𝑋 𝑇𝑖
𝑏 )
𝜎 (65)
Where 𝜎 = 𝑁 𝐴𝑣
𝑉 𝐻𝑓
2/3
and 𝑁 𝐴𝑣
is Avogadro’s number. Δ𝐺 𝑠𝑒𝑔 was estimated considering the chemical and
elastic contributions to the enthalpy of segregation, Δ𝐻 𝑠𝑒𝑔 𝑐 ℎ𝑒𝑚
and Δ𝐻 𝑠𝑒𝑔 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 , and the ideal entropy of
segregation using the next equation:

Δ𝐺 𝑠𝑒𝑔 = Δ𝐻 𝑠𝑒𝑔 𝑐 ℎ𝑒𝑚
+ Δ𝐻 𝑠𝑒𝑔 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 − 𝑇 Δ𝑆 𝑠𝑒𝑔 𝑖𝑑𝑒𝑎𝑙 (66)
In this equation, it is assumed that the excess entropy of segregation is equal to zero. The expressions used
to compute the terms in equation (66) are given below [95]:

Δ𝐻 𝑠𝑒𝑔 𝑐 ℎ𝑒𝑚
= (𝛾 𝑇𝑖
− 𝛾 𝐻𝑓
)(1− 𝛼 )𝜎 −
8Δ𝐻 𝑚 𝑍 [𝑧 𝑖𝑛
(𝑋 𝑇𝑖
𝑔𝑏
− 𝑋 𝑇𝑖
𝑏 )− 𝑧 𝑜𝑢𝑡 [(𝑋 𝑇𝑖
𝑏 − 1/2)−
𝛼 (𝑋 𝑇𝑖
𝑔𝑏
− 1/2)]] (67)
Δ𝐻 𝑠𝑒𝑔 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 = −
2𝐾 𝐻𝑓
𝐺 𝑇𝑖
(𝑉 𝐻𝑓
−𝑉 𝑇𝑖
)
2
3𝐾 𝐻𝑓
𝑉 𝑇𝑖
+4𝐺 𝑇𝑖
𝑉 𝐻𝑓
(68) Δ𝑆 𝑠𝑒𝑔 𝑖𝑑𝑒𝑎𝑙 − 𝑅𝑙𝑛 [
𝑋 𝑇𝑖
𝑔𝑏
(1−𝑋 𝑇𝑖
𝑏 )
𝑋 𝑇𝑖
𝑏 (1−𝑋 𝑇𝑖
𝑔𝑏
)
] (69) 𝛽 𝑇𝑖
𝑔 𝑏 =
𝑋 𝑇𝑖
𝑔𝑏
𝑋 𝑇𝑖
𝑏 (70)
where 𝐺 and 𝐾 are the shear and bulk modulus; 𝑧 𝑖𝑛
and 𝑧 𝑜𝑢𝑡 are the in-plane and out-of-plane coordination
numbers, respectively. Surface energies, enthalpy of mixing, bulk modulus, and shear modulus reported in
the literature were used for the calculations [54, 141, 142]. Using equations (63) through (69), the grain
boundary energy (Equation 63) was plotted as a function of the mole fraction of Ti at the grain boundary
for different temperatures (Figure 61). Figure 61 indicates the minimum grain boundary energy occurs
when the molar fraction of Ti at the grain boundary is around 91 at.% Ti for temperatures between 500 ºC
and 1000 ºC. Considering the initial composition of the sample (20 at.% Ti), an enrichment parameter 𝛽 𝑇𝑖
𝑔𝑏

equal to 4.5 is expected (Equation 70) [23]. At this high solute concentration, clusters (small solute
enrichments with the structure of the matrix) form at the grain boundary, where precipitates can nucleate
and grow [119].

71

Figure 61. Normalized grain boundary energy as a function of composition for different temperatures
calculated for a grain size of 50 nm.

4.6 Kinetics of Segregation
To compare the results from Figure 61 to the observed microstructures, it is necessary to estimate the
relationship between time and temperature by computing the diffusion time required to saturate the grain
boundaries at different temperatures. This allows us to identify the time it will take at a given temperature
for precipitation to occur. For this calculation, the semi-infinite solution of Fick’s equation at the grain
boundary was used [23, 45]:

𝑋 𝑇𝑖 ,𝑡 𝑔𝑏
−𝑋 𝑇𝑖 ,0
𝑔𝑏
𝑋 𝑇𝑖 ,𝑠𝑎𝑡 𝑔𝑏
−𝑋 𝑇𝑖 ,0
𝑔𝑏
= 1− 𝐸𝑥𝑝 (
𝑥 𝛽 𝑇𝑖
𝑔𝑏
𝛿 𝑔𝑏
+
𝐷 𝑇𝑖 ,𝐻𝑓
𝑡 𝛽 𝑇𝑖
𝑔𝑏
𝛿 𝑔𝑏
2
)𝐸𝑟𝑓𝑐 (
𝑥 2
√
𝐷 𝑇𝑖 ,𝐻𝑓
𝑡 +
√
𝐷 𝑇𝑖 ,𝐻𝑓
𝑡 𝛽 𝑇𝑖
𝑔𝑏
𝛿 𝑔𝑏
) (71)
where 𝑋 𝑇𝑖 ,0
𝑔𝑏
is the initial Ti mole fraction at the grain boundary, 𝑋 𝑇𝑖 ,𝑠𝑎𝑡 𝑔𝑏
is the Ti saturation mole fraction at
the grain boundary, 𝑡 is the time, and 𝑋 𝑇𝑖 ,𝑡 𝑔𝑏
is the Ti mole fraction at the grain boundary at time 𝑡 . 𝛽 𝑇𝑖
𝑔𝑏
is
the previously defined Ti enrichment parameter, 𝛿 𝑔𝑏
is the thickness of the segregated layer, which was
assumed equal to the thickness of a monoatomic layer at the grain boundary (320 pm) [90, 143], and 𝑥 is

72

the diffusion length, taken as half the initial grain size (25 nm). The inter-diffusion coefficient of Ti in Hf
was calculated using the Nazarov-Gurov equation [45, 144]:

𝐷 𝑇𝑖 ,𝐻𝑓
=
𝐷 𝑇𝑖
𝐷 𝐻𝑓
𝑋 𝑇𝑖
𝑏 𝐷 𝑇𝑖
+(1−𝑋 𝑇𝑖
𝑏 )𝐷 𝐻𝑓
(72)
where 𝐷 𝑇𝑖
and 𝐷 𝐻𝑓
are the self-diffusivities of Hf and Ti. Figure 62 shows the grain boundary saturation
time decreased ten orders of magnitude between 500 ºC and 1000 ºC. This strong dependency on
temperature is expected because segregation is a thermally activated process. Due to the duration of the
heat treatments in this study (3.5 x 10
5
s, indicated by dashed line in Figure 62) precipitation should be
observed at temperatures around 700 ºC and higher.

Figure 62. Time for grain boundary saturation as a function of temperature for a diffusion length of 25
nm, the dotted red line is the heat treatments time in this study (3.5 x 10
5
s).

4.7 Crystallization of the Amorphous Ti Precipitates
From our microstructural studies at 800 ºC, we compared the results of the calculations shown in Figure 62
to the DSC scan. The TEM micrograph at this temperature (Figure 60b) depicts amorphous Ti precipitates

73

surrounding equiaxed grains similar to those observed in the work of Polyakov et al. [50]. The inset
highlights a nanodiffraction pattern of the precipitates which shows no preferential orientations, confirming
the amorphous nature of Ti. However, this nanodiffraction pattern displays some diffraction spots which is
evidence of short-range order [145]. Nucleation via amorphous Ti becomes energetically favorable due to
a lower surface contribution, the energy of an incoherent grain boundary is lower than the interfacial energy
between Hf rich grains [22, 115, 146]. To further understand the precipitation process, the composition of
the precipitates and grains was obtained by EDS scans (EDAX detector) using a JEM-2100F (JEOL)
microscope. Figure 63a displays representative EDS scans, which show the precipitates have a strong Ti
peak and no Hf peaks, indicating they are mostly pure Ti. In contrast, the scans inside the grains show a
strong Hf peak and no Ti peaks, suggesting Ti segregation and precipitation continue until the Ti inside the
grains is depleted. This is in agreement with the result from the grain boundary energy calculations (Figure
61). In both scans, the oxygen peaks are approximately ten times shorter than Hf or Ti peaks, which suggests
that minimal oxidation occurred in the sample.

Figure 63. EDS scans of Ti precipitates (bright regions) and Hf grains (dark regions) of samples heat
treated (a) at 800 ºC and (b) at 1000 ºC. The red arrows indicate the respective regions where the scans
were taken from.

Additionally, as indicated by the DSC scan, grain growth should start at around 796 ºC. This coincides with
the observed average grain size (for both Hf-rich grains and Ti-precipitates) increase from 71 ± 37 nm to
132 ± 62 nm between 800 ºC and 1000 ºC (Figure 60b and c). At 1000 ºC, Ti crystals are observed instead

74

of amorphous Ti precipitates (Figure 63b). As the amorphous Ti precipitates grow and coalesce,
crystallization becomes energetically favorable. At some precipitate size, when the bulk energy dominates
over the interface energy, crystallization takes place since the crystalline phase has lower formation
enthalpy than the amorphous phase [22, 115, 119]. Since Ti crystallization occurs during grain growth (an
exothermic process), the grain growth peak overlays the Ti crystallization peak in the DSC scans.
Subsequently, the Ti precipitates impinge and arrange in two layers equidistant from the center of the
sample. Although the rearrangement causes noticeable microstructural changes between 800 ºC and 1000
ºC, the composition of both Ti precipitates and Hf grains do not change significantly (Figure 63b). This
suggests that the driving force for segregation and precipitation has markedly decreased at 800 ºC, and that
the changes in the microstructure observed between 800 ºC and 1000 ºC are due to the movement and
coalescence of grains of constant composition.

4.8 Summary
The potential usage of NMMs as a route to synthesize stable nanostructures was explored by following the
microstructural evolution of Hf-Ti NMMs. The DSC scan revealed that multiple transformations occur
between 494 ºC and 990 ºC. Furthermore, NMMs samples heat-treated at transition temperatures of interest,
followed by microstructural observations, agree with both the DSC scan and theoretical predictions. After
recovery, recrystallization leads to the transformation of the as-sputtered columnar structure into equiaxed
grains. During recrystallization, Ti segregation becomes kinetically active and continues until reaching
grain boundary saturation around 91 at.% Ti, thus reducing the grain boundary energy. When saturation
occurs, cluster formation and subsequent precipitation of amorphous Ti occurs at the grain boundaries at
about 666 ºC. Higher temperatures lead to grain growth and rearrangement of the crystallized Ti
precipitates, although the microstructure remains at the nanometer scale. Overall, this study presents a
comprehensive guide to microstructural changes in NMMs that can be used for development of new
nanocrystalline systems.

75

5 Grain morphology influence on the thermal stability of NMMs: The Ta-Hf
system
As presented in Section 2.3, several studies have attempted to improve the thermal properties of
nanomaterials via kinetic or thermodynamic approaches [36, 44]. However, another alternative to achieve
thermal stability is to use special architectures that decrease the interface energy such as low angle grain
boundary nanolaminates and NMMs [147]. Specially, NMMs are a potential route towards enhanced
thermal stability, improved strength, and better radiation resistant materials [50, 148-151].

Although NMMs with either columnar or brick-like grains have been studied individually, to the best of
the author’s knowledge, there is no previous research on NMMs with a bimodal grain structure. Previous
studies of bimodal materials have focused on microstructures with two grain size distributions which
combined the high strength of nanocrystalline materials and the increased ductility of ultrafine grain
materials [152-154]. These microstructures have been used to enhance the mechanical properties of
nanocrystalline pure Cu, Ti, Zr, and Ni, as well as Cu-Al, Cu-Ag, Al-Mg, and Ti-Zr alloys [155-164].
Despite the fact that there is limited research on the thermal stability of bimodal materials, a study on the
magnesium alloy AZ91 shows that the combination of two grain sizes promotes the development of low
angle grain boundaries, which exhibit higher thermal stability [116, 165]. Thus, a system with both NMMs
and a bimodal grain structure comprised of columnar and equiaxed brick-like grains, a bimodal NMM,
could provide improved microstructural stability.

The aim of this study was to explore the thermal stability of Ta-Hf bimodal NMMs by performing a detailed
microstructural investigation at different annealing temperatures. The Ta-Hf system, which favors the
formation of columnar grains and thus facilitates the deposition of a bimodal structure, was selected due to
its proclivity for nanocrystalline stability [101]. The results of this project were published in published in
Scripta Materialia 166 (2019) 19-23 under the title “Exploring the thermal stability of a bimodal nanoscale
multilayered system” [166].

5.1 Synthesis of Ta-Hf NMMs
Ta-Hf bimodal multilayers (2.0 µm thick) were deposited on Si (100) substrates by direct current (DC)
magnetron sputtering inside a vacuum chamber evacuated to a base pressure of 1.1 x 10
-4
Pa. During
deposition, the chamber was filled with 6.7 x 10
-1
Pa of Ar to sputter the individual layers using powers of
175.0 W and 50.0 W for Ta (99.95 at. % Ta) and Hf (99.9 at. % Hf) sources, respectively. Under these
conditions, the deposition rates were approximately 30 nm/min for Ta and 9 nm/min for Hf, values which

76

were held constant during the sputtering procedure. The composition of the alternating 2 nm Hf (99.9 at. %
Hf) and 14 nm Ta-Hf (14.6 at. % Hf) layers was adjusted by controlling the sputtering time of the Ta source
as indicated in a previous manuscript [135]. The global composition of the film (22.4 at. % Hf) was
measured using an EDS EDAX detector inside a JSM-7001 SEM microscope. X-ray diffractometry
measurements using a Rigaku Ultima IV diffractometer revealed a highly BCC (110) texture. To study the
morphology of the films, TEM lamellas were prepared by FIB using a FIB-4500 (JEOL). Subsequently,
these TEM samples were characterized using a JEM-2100F (JEOL) microscope.

5.2 Microstructure of Ta-Hf NMMs
The microstructure of the Ta-Hf bimodal NMMs, presented in Figure 1a, is comprised of columnar grains
at the top of the film (Figure 64b) and brick-like grains (Figure 64c) that extend over 600 nm from the
substrate surface; this structure corresponds to a film type Zone 1 in the Thornton diagram [167]. The
selected area diffraction (SAD) pattern from the cross-section of the film (top of Figure 64a) only showed
Ta rings diffracted from the solid solution in the Ta rich layers. This indicates either that the pure Hf layers
are not large enough to diffract sufficient electrons to observe a ring, or that most of the pure Hf layers are
adopting the BCC structure that has been observed in other studies [168, 169].

Figure 64. Bright field TEM images showing (a) the bimodal multilayers, (b) columnar grains at the top,
and (c) brick-like grain at the bottom of the film. The dark-field TEM images display the corresponding
grain morphologies for (d) columnar and (e) brick-like regions. The grain size distributions for the
columnar and the brick-like grains are shown in (f) and (g) respectively.

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The morphology of the columnar and brick-like grains was identified by dark field TEM images (Figures
1d-1e) that were also used to measure grain size distributions for each type of grain (Figure 64f and Figure
64g) following procedures outlined in a previous manuscript [135]. From these distributions, average grain
sizes of 109 ± 43 nm and 31 ± 9 nm (Average grain size ± Standard Error) were calculated for columnar
and brick-like grains, respectively [170].

5.3 Interfaces in Ta-Hf NMMs
The structure of the interphases within these types of grains depends on the surface energy ( 𝛾 ) and the
atomic radius (𝑎 ) of the individual metals 𝐴 and 𝐵 at the interface [171]. When the surface energy mismatch
(Γ
𝐴𝐵
= 2(𝛾 𝐴 − 𝛾 𝐵 )/(𝛾 𝐴 + 𝛾 𝐵 ) ) is Γ
𝐴𝐵
< 0.5 and the ratio of atomic radii (𝑟 = 𝑎 𝐴 /𝑎 𝐵 ) is 𝑟 ≲ 1.00,
relatively stable coherent (5 - 200 mJ m
-2
) or semi-coherent (200 - 800 mJ m
-2
) interphases form between
the subgrains of uninterrupted columnar structures [171, 172]. For example, in studies performed on Hf-Ti
(Γ
𝐻𝑓 ,𝑇𝑖
= −0.06, 𝑟 𝐻𝑓 ,𝑇𝑖
= 1.08) and W-Cr (Γ
𝑊 ,𝐶𝑟
= 0.13 𝑟 𝑊 ,𝐶𝑟
= 0.96) NMMs, the columnar grains are
comprised of subgrains bounded by semi-coherent interfaces [135, 173]. In contrast, when Γ
𝐴𝐵
> 0.5,
highly mobile incoherent interphases (800 – 2500 mJ m
-2
) can interrupt columnar grain structure or drive
the formation of azimuthally misoriented brick-like grains [24, 141, 174]. For instance, it was observed that
in a Mo-Au NMMs study (Γ
𝑀𝑜 ,𝐴𝑢
= 0.92, 𝑟 𝑀𝑜 ,𝐴𝑢
= 1.07), the columnar grains are interrupted by
incoherent interphases between the Mo rich and pure Au layers [175]. Therefore, different elements
promote the formation of certain interphases that can be used to tailor NMMs with specific grain structures.

Particularly, in the high resolution TEM (HRTEM) images of the Ta-Hf NMMs in Figure 65, it is observed
that the interphases between layers in the columnar and brick-like regions have different atomic structures.
For this system (Γ
𝑇𝑎 ,𝐻𝑓
= 0.22, 𝑟 𝑇𝑎 ,𝐻𝑓
= 0.96), the aforementioned variables imply that this system favors
the formation of semi-coherent interphases within uninterrupted columnar grains [141, 174]. However,
under the sputtering conditions used (Zone 1 of the Thornton diagram), shadowing due to substrate
roughness and limited surface diffusion prevent the development of columnar grains at the bottom of the
film [13]. Therefore, close to the substrate the deposition of Hf layers effectively interrupts the growth of
coalesced Ta rich islands that become brick-like grains. Specifically, Figure 65a shows that these grains
have different orientations in consecutive layers (see Fast Fourier Transform or FFT patterns) and are
surrounded by high-angle grain boundaries, which have higher contrast and definition in HRTEM images
[126, 176]. As the film continues to grow, shadowing effects reduce the uniformity of the Hf layers, and in
turn these layers less effectively interrupt the growth of Ta-rich islands, which continue to grow in the form
of columnar grains that extend over several layers. For instance, Figure 65b, highlights a columnar grain at

78

the top of the film comprised of Ta rich subgrains in the (111) zone axis (see FFT patterns) with diffuse
semi-coherent pure Hf layers between the subgrains [13]. Furthermore, the heterogenous distribution of
brick-like and columnar grains makes the density of high-angle grain boundaries greater at the bottom of
the film, which results in a non-uniform distribution of stresses [177]. Accordingly, profilometry
measurements show that the as-sputtered films are in a compressed state with average residual stresses of -
745.6 MPa (Stoney’s equation) [178].

Figure 65. High resolution TEM images of (a) the brick-like grains, showing atomic lattices that extend
over a single layer, and of (b) the columnar grains, displaying atomic layers extending over several layers.
The yellow lines indicate the orientation of atomic layers inside the grains. The FFT patterns (i) to (iv)
show that the brick like grains in consecutive layers have different orientations, while the patterns from
(v) to (viii) indicate that the columnar subgrains have the same orientation.

5.4 Thermal evolution of Ta-Hf NMMs
In order to study the evolution of the bimodal microstructures under thermal loading, the as-sputtered
samples were heat-treated under vacuum at 550 ⁰C and 1000 ⁰C, keeping the temperature constant for 96 h
and quenching the films in a low vapor pressure oil without breaking vacuum (Invoil 705, Inland Vacuum
Industries). These temperatures were selected from DSC scans of the system (Figure 66), where a thermal
event occurring between 482 ⁰C and 842 ⁰C with ∆𝐻 = −9.6 kJ/mol was identified. The scans were
collected using a Labsys thermal analyzer, by heating the films at a constant rate of 10 °C/min from 20 °C
to 1000 °C under constant Ar flow of 40 ml/min.

79

Figure 66. Differential scanning calorimetry scan of the Ta-Hf nanometallic multilayers from 20 °C to
1000 ºC showing that recrystallization and grain growth occur between 482 ºC and 842 ºC.

The microstructure of the samples annealed at 550 °C and 1000 °C, are presented in the bright field TEM
micrographs in Figure 67a and Figure 67e, which are comprised of columnar grains at the top of the film
and equiaxed grains at the bottom. It can be observed that the columnar grains conserve their multilayered
structure (Figure 67b and Figure 67f), although, the pure Hf layers show several breakthroughs, suggesting
that roughening has occurred during annealing [79]. At 550 °C the equiaxed grains at the bottom of the film
(Figure 67c) do not show any multilayers, which indicates that the exothermic transformation identified in
the DSC scans corresponds to recrystallization of the brick-like grains. This process is driven by the strain
energy and the greater mobility of the incoherent high-angle grain boundaries [45, 116, 179]. Furthermore,
the SAD patterns at 550 °C (Inset Figure 67a) show a new Hf (100) ring, which points to the segregation
and clustering of Hf at grain boundaries during recrystallization [180]. Subsequently, at 1000 ⁰C, the
equiaxed grains have spheroidized (Figure 67g) and the Hf clusters have agglomerated to form precipitates
at the grain boundaries, which diffract an additional Hf (102) ring (Inset Figure 67e) [180]. Dark-field TEM
images were used to calculate the grain size distributions between 500 °C and 1000 °C (Figure 67d and
Figure 67h). The measurements show that the average size of the columnar grains moderately increases

80

from 139 ± 52 nm to 157 ± 73 nm, and that of the equiaxed grains grow from 55 ± 32 nm to 60 ± 28 nm,
respectively.

Figure 67. Bight field, dark field TEM and corresponding grain size distributions of the multilayers
annealed at 550 ⁰C (a,b,c and d), and at 1000 ⁰C (e,f,g and h) after 96 h. The insets in (a) and (e) show the
corresponding SAD patterns with the Hf rings colored green. The dark-field TEM images indicate that
while at both temperature the structure of the columnar grains is preserved, most of the brick-like grains
have recrystallized at 500 ⁰C. Similar grain size distributions are observed at 550 °C (d) and at 1000 °C
(g) suggesting no significant grain growth.

5.5 Kinetic and thermodynamic stabilization of Ta-Hf NMMs
The aforementioned results seem to indicate that a bimodal NMM structure can enhance the thermal
stability of the Ta-Hf system via mechanisms that depend on the interface structure. The boundaries of the
columnar subgrains are locked by the semi-coherent pure Hf layers, which have lower mobility [22]. These
layers intersect high-angle grain boundaries at quadrupole points, which prevent the formation of grooves
and decrease the strain at the columnar grains, thus lowering the driving force for recrystallization [80].
This type of stabilization has been studied in detail by comparing the thermal evolution of Cu-Nb NMMs
with either staggered or vertically aligned grains [8, 80, 81, 181]. In addition, the equiaxed grains are
stabilized by the Hf precipitates, which lock the grain boundaries and reduce their energy, resulting in the
simultaneous activation of kinetic and thermodynamic stabilization mechanisms that have been reported
for other systems such as Mo-Au, Cu-Ta, and Cu-Nb [132, 175, 182]. Specifically, at 1000 ⁰C, the Hf
precipitates inside the area highlighted in Figure 68a were studied in more detail. Figure 68b, a TEM

81

micrograph of that area, shows a combination of columnar grains with well-defined layers and equiaxed
grains, which are surrounded by precipitates that display compositional contrast. The corresponding EDS
map (Figure 68c), where Hf is colored orange and Ta turquoise, highlights that while the equiaxed grains
are enriched with Ta, the precipitates along the grain boundaries are Hf rich. These precipitates form a
wetting complexion that reduces the energy and the mobility of the grain interfaces and results in the
combined stabilization of the equiaxed grain boundaries, in agreement with predictions from the Murdoch
and Schuh model for the Ta-Hf system [26, 38, 101]. Furthermore, as the equiaxed grains spheroidize, their
shape tends towards polyhedrons that minimize the grain boundary area and increase the stability of the
system [22, 183]. It should also be noted that an increased temperature threshold for grain coarsening has
been reported for grain sizes below a critical value [184]. Although that particular study focused on pure
Cu and Ni systems with critical grain sizes of 70 nm and 90 nm, respectively, the small grain size of the
brick-like grains could also be increasing the temperature needed for grain growth.

Figure 68. (a) Bright field STEM image highlighting the structure of the multilayered samples after
annealing at 1000 °C, where the yellow box indicates an area with both columnar and brick-like grains.
This region was studied in more detail by collecting additional (b) STEM images and (c) EDS maps,
which display Ta rich grains (Turquoise) surrounded by Hf precipitates (Orange).

5.6 Summary
In this manuscript we explore for the first time, the thermal stability of bimodal Ta-Hf NMMs with both
columnar and equiaxed grains. It was observed that a bimodal microstructure enhances the thermal stability
by providing several stabilizing mechanisms that depend on the interface structure. For example, the
stability of the columnar grains is increased by quadrupole points that lock the grain boundaries and by the
presence of semi-coherent Hf layers, which prevent recrystallization. Although the brick-like grains
recrystallize between 482 °C and 842 °C, the new equiaxed grains are stabilized by Hf rich precipitates at
the grain boundaries, which simultaneously activate both kinetic and thermodynamic stabilization

82

mechanisms. Overall, both columnar and brick-like grains show minimal grain growth, for instance at 1000
°C the grain size of the columnar grain is 157 ± 73 nm and that of the brick-like grains is 60 ± 28 nm. These
results highlight a path to improve the stability of nanocrystalline materials using bimodal NMMs with
engineered interface structures.

83

6 Effect of phase transformations on the thermal stability of NMMs: The W-
Cr system
In order to improve the thermal stability of nanomaterials it is necessary to understand the mechanisms
behind both microstructural and phase transformations during the annealing process. Although, several
authors have successfully studied microstructural transformations at the nanoscale, the study of nanoscale
phase transformations has been limited by the lack of systems that allow for control over the kinetics of the
transformation [5, 116, 185-187]. In addition, previous studies were mainly focused on the effects of
stresses, surface energy, and phase stability for these systems [188-192].

In this study a systematic approach to explore the effect of nanoscale phase transformations and the thermal
evolution of NMMs was developed. Specifically, the phase transformations occurring in W-Cr NMMs with
thin layers (<10 nm) were studied. Such fine layers led to a high density of interphases that increased the
diffusivities of both W and Cr [115]. The greater diffusivities induced mixing and the development of
diffusion zones where new phases were more likely to nucleate [7, 110]. Inside the diffusion zones three
types phases could have formed, the W 3Cr intermetallic, or two BCC solid solutions, one rich in W and the
other in Cr, which coexist at equilibrium below a miscibility gap (Critical temperature of 1677 ºC).

Although, the W 3Cr intermetallic has been observed as a tetragonal phase in diffusion couples held at 1350
°C, as presented in this investigation, this phase could nucleate at lower temperatures inside NMMs with
layers of compositions selected to shift the kinetics of nucleation [45, 193]. The approach used in the current
study is a new alternative to explore phase formation processes in nanoscale systems and their effect on the
thermal stability of NMMs. The results of this investigation have been published in Materialia 2 (2018)
190-195 under the title “Phase transformations in the W–Cr system at the nanoscale” [173].

6.1 Synthesis of W-Cr NMMs
Cr/W-Cr NMMs were deposited on (100) Si substrates by magnetron sputtering at 2.0 Pa using powers of
50 W and 200 W for the Cr and W sources, respectively. The composition of the films was tuned by
controlling the on-time of the W source following procedures previously described [50]. The structure of
the as-sputtered W-Cr NMMs is shown in the cross-sectional bright field TEM image in Figure 69a. The
overall film thickness was 2 microns and the individual layer thicknesses were 1.5 nm Cr (99.9 at.% Cr)
and 6.3 nm W rich (12.4 at.% Cr), respectively. These thin layers increase the density of interphases and
thus the local diffusivity. For example, at 550 ⁰C, calculations made using Darken equation and diffusivity
values obtained from Fisher’s analysis at the interphase, show that the diffusivity of Cr in W is 𝐷 𝐶𝑟 −𝑊 =

84

7.7 × 10
−29
𝑚 2
𝑠 −1
, while the diffusivity of Cr in W at the interphase is 𝐷 𝐶𝑟 −𝑊 𝑖 = 1.7 × 10
−28
𝑚 2
𝑠 −1
,
which is more than a twofold increment [45, 48, 52, 194].

Figure 69. (a) Cross-sectional bright field STEM and (b) integrated radial intensity profile of as-sputtered
W-Cr nanometallic multilayer. Only α-Tungsten peaks are observed in the profile, indicating that the β-
Tungsten phase is not present in the sample.

6.2 Microstructure of W-Cr NMMs
The global composition of the as-sputtered sample, 65.3 at.% W, 34.0 at.% Cr and 0.6 at.% Ar, was
measured using a JSM-7001 SEM microscope with an EDS EDAX detector. The presence of Ar is
attributed to the high sputtering pressure used during the deposition [195]. This sample has a vertical growth
direction with densely packed fibrous columnar grains characteristic of a morphology type zone 1 with
grains interrupted by the pure Cr layers [136]. The average grain size of the as-sputtered sample, 9 nm, was
measured following procedures described in a previous manuscript [135]. In addition, XRD scans of the
as-sputtered sample, acquired using a Rigaku Ultima IV diffractometer, show a highly (110) BCC texture.
That orientation is also observed in the integrated radial intensity profile (Figure 69b), which only shows
α-tungsten peaks; the lack of β-tungsten peaks points to the absence of β-tungsten nucleation reported for
W alloys in another sputtering study [196].

DSC scans of the W-Cr NMMs (Figure 70) mounted on the substrate were collected using a Si (100)
substrate reference from 20 °C to 1000 °C at a rate of 10 °C per minute under a constant Ar flow of 50
ml/min in a STA 449 F5 Jupiter thermal analyzer. The DSC scans showed that recrystallization occurs
between 690 °C and 830 °C with = -1.8 kJ/mol, where recrystallization is then followed by
grain growth which takes place between 830 °C and 990 °C with = -1.8 kJ/mol. These
ization recrystall
H 
growth grain
H
−


85

temperature ranges suggest that above 700 °C, when recrystallization starts and the diffusivities are
significantly large, the grain boundary mobility, which depends linearly on the diffusivity, should be high
enough for the formation of diffusion zones where new phases could nucleate [48, 116]. This is in
agreement with a five order of magnitude increase in the diffusivity of Cr in W at the interphase from
𝐷 𝐶𝑟 −𝑊 𝑖 = 6.3× 10
−25
𝑚 2
𝑠 −1
at 700 ⁰C to 𝐷 𝐶𝑟 −𝑊 𝑖 = 3.1× 10
−20
𝑚 2
𝑠 −1
at 1000 ºC [48, 194].

Figure 70. Differential scanning calorimetry scan of the W-Cr nanometallic multilayers from 20 °C to
1000 ºC showing that recrystallization and grain growth occur between 690 ºC and 990 ºC.

6.3 Thermal evolution of W-Cr NMMs
Guided by the DSC scans, the W-Cr NMMs were annealed at 550 °C, 800 °C and 1000 °C, for 96 h under
vacuum (pressure ≈ 5 x 10
-4
Pa) inside a GSL1100X tube furnace (MTI Corporation). At the end of the heat
treatments, the samples were quenched in a low vapor pressure oil (Invoil 705, Inland Vacuum Industries)
without breaking vacuum. After cleaning the samples with isopropanol, TEM specimens were prepared by
FIB lift out using a FIB-4500 (JEOL). TEM images of the sample and EDS spectra were captured on a
JEM-2100F (JEOL) microscope equipped with an Oxford X-MaxN 100 EDS detector.

86

Figure 71. Cross-sectional bright field STEM images and integrated radial intensity profile of W-Cr
nanometallic multilayers heat-treated at (a-d) 550 ºC, (b-e) 800 ºC and (c-f) 1000 ºC. The inset in the
STEM images shows the diffraction patterns and the orientations indexed in the integrated radial intensity
profiles. The images display that while at 500 ºC the as-deposited microstructure is retained, at 800 ºC
some recrystallized grains are observed between the multilayers. At 1000 ºC, when grain growth has
occurred the sample is comprised of W rich grains surrounded by amorphous W 3Cr precipitates.

87

The DSC scans indicate that at 550 °C no microstructural transitions have occurred. Figure 71a and Figure
71d show that at this temperature the multilayered structure is preserved without apparent changes in the
grain orientation or the average grain size in comparison to the as-sputtered sample. In contrast, at 800 °C
(Figure 71b) the microstructure is comprised of unrecrystallized multilayers and recrystallized grains,
which have an average grain size of 16 nm and show contrast due to change in orientation. Although the
integrated radial intensity profile at 800 ºC (Figure 71e) shows new Cr peaks, no new phases were observed
at this temperature. This indicates that these peaks correspond to Cr clusters forming inside the
recrystallized grains.

Annealing at 1000 °C (Figure 71c) resulted in grain growth and further microstructural changes; the sample
is now comprised of equiaxed grains (average grain size of 78 nm) surrounded by precipitates at the grain
boundaries (average grain size of 9 nm). The radial intensity profile of this sample (Figure 71f) shows that
in comparison to the sample annealed at 800 °C, the intensity of the Cr peaks has increased at 1000 °C,
which suggests that the equiaxed grains have further enriched with Cr.

6.4 W3Cr precipitation
To understand the formation of precipitates observed at 1000 °C, samples heat-treated at 800 °C and 1000
°C were compared by EDS techniques. Figure 72a shows that the recrystallized grains observed at 800 °C
have enriched from the initial composition of the W rich layers (12.4 at.% C) to ≈ 34.5 at.% Cr, which is in
agreement with the observations made from the radial intensity profile (Figure 71e). Furthermore, the EDS
scans show that a low fraction of deposited Ar (0.6 at.% Ar) is uniformly distributed through the sample.
However, at 1000 ºC, the EDS scans (Figure 72b) indicate that the equiaxed grains have further enriched
to 39.5 at.%. Cr, which corresponds to higher intensity Cr peaks in the radial intensity profiles. Inside the
precipitates, the EDS scans show a 3:1 atomic ratio of W and Cr, suggesting that the precipitates could be
the W 3Cr phase previously reported in the literature (Appendix C.3) [193]. From here on, we will refer to
the precipitates at the grain boundaries as W 3Cr precipitates. Moreover, the EDS scans indicate that at 1000
°C, Ar in the equiaxed grains has been completely depleted, while the fraction of Ar in the W 3Cr precipitates
has increased to 11.5 at % Ar. The Ar depletion in the equiaxed grains can be understood by considering
that above 800 °C the diffusivity of Ar increases, which facilitates the segregation of Ar to the grain
boundaries [197]. Subsequently, Ar diffuses along the grain boundaries and dissolves into the W 3Cr
precipitates, minimizing the Gibbs energy, which increases the stability of the system [198, 199].

88

Figure 72. EDS scans of (a) recrystallized grains at 800 ºC and of (b) W rich grains surrounded by W 3Cr
precipitates rich in Ar. The arrows point to the corresponding regions in the HAADF images. The scans
show that at 800 ºC the recrystallized grains have enriched in Cr. At 1000 ºC, when grain growth has
occurred, the W rich grains (Gray grains) have further enriched to 40 at.% Cr. At this temperature W 3Cr
precipitates (Black grains) are observed at the grain boundaries.

To understand the formation of the W 3Cr intermetallic, Miedema’s model was used to calculate the enthalpy
of formation of the W 3Cr intermetallic ∆𝐻 𝑊 3
𝐶𝑟
𝑓 = 1.0 𝐾𝐽 /𝑚𝑜𝑙 , and that of a solid solution of the same
composition ∆𝐻 𝑆𝑜𝑙𝑖𝑑 −𝑠 𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑓 = 8.0 𝐾𝐽 /𝑚𝑜𝑙 . Assuming similar entropies of formation for the W 3Cr and
the solid solution, these values suggest that the formation of the W 3Cr is energetically favorable [200]. In
addition, the fact that the W 3Cr precipitates are localized at the grain boundaries suggests that they formed
via continuous precipitation. During that precipitation process, amorphous Cr clusters appear at the grain
boundaries and continue growing, due to Cr diffusing from the surrounding grains, until their composition
is close to 25 at.% Cr [201]. This is in agreement with the precipitation of amorphous phases (amorphization
reactions) at grain boundaries and interphases observed in other multilayered systems [7, 202]. However,
this result leads to the question of why W 3Cr precipitates were not observed at 800 °C even though the
composition was also favorable and the diffusivities of W and Cr should have been high enough to induce
the development of diffusion zones. The lack of W 3Cr nucleation at 800 °C could be due to two reasons.

89

First, the Gibbs energy of the W 3Cr phase may be higher at 800 °C than at 1000 ⁰C, which would make the
nucleation of this intermetallic unlikely at the former temperature. However, there is lack of accurate
models to compute the mixing entropy of metastable alloys that have a significant contribution from
interphase effects and therefore, the Gibbs energy of the W 3Cr phase cannot be directly calculated. Even
though, as an approach to understand the temperature dependence of the Gibbs energy for the W 3Cr
intermetallic, the Scientific Group Thermodata Europe (SGTE) datasheet was used to calculate the Gibbs
energy isotherms for the W-Cr system [203]. These isotherms showed that the Gibbs energy of this system
decreases with increasing temperature and a similar trend would be expected for W 3Cr [198]. Second,
considering the high melting points of W and Cr, 3422 °C and 1907 °C, respectively, it is expected that the
diffusivities of both species will be slower at temperatures below half the meting point of Cr (around 850
°C). For example, at 800 ⁰C the self-diffusivities of W and Cr are 𝐷 𝑊 ∗
= 1.8 × 10
−31
𝑚 2
𝑠 −1
and 𝐷 𝐶𝑟
∗
=
1.6× 10
−22
𝑚 2
𝑠 −1
respectively, while at 1000 ºC they are 𝐷 𝑊 ∗
= 1.7× 10
−27
𝑚 2
𝑠 −1
and 𝐷 𝐶𝑟
∗
=
1.1× 10
−19
𝑚 2
𝑠 −1
[194]. Thus, the formation of W 3Cr nuclei may have been kinetically unlikely at or
below 800 ⁰C [115].

6.5 Structure of the W3Cr precipitates
To understand the different phases observed at 1000 ºC in more detail, high-resolution TEM images and
SAD patterns of the equiaxed grains and of the W 3Cr precipitates were collected. Figure 73a shows that the
equiaxed grains, between the W 3Cr precipitates, have a crystalline BCC structure with a [100] zone axis
[130]. Due to the small difference between the metallic radius of W and Cr (8.5 % difference), these
elements form a solid solution inside the equiaxed grains, which is in agreement with the Hume–Rothery
rules [204]. On the other hand, Figure 73b reveals that the W 3Cr precipitates are amorphous, which is
confirmed by the concentric diffuse rings in the SAD pattern. In addition, there is a diffusion zone of around
5.1 nm between the precipitates and the equiaxed grains, which is crystalline at the side of the W-rich grains
and amorphous at the side of the W 3Cr precipitates.

The formation of amorphous W 3Cr precipitates at the grain boundaries also indicates that the activation
energy barrier for nucleation is overcome by the excess Gibbs energy at grain interfaces, where the higher
energy induces the nucleation of amorphous aggregates, which continue growing as the diffusivities of W
and Cr increase with temperature. This process should continue above 1000 ºC until the Gibbs energy of
formation of crystalline W 3Cr is lower than that of the amorphous precipitates leading to crystallization.
Although these events, which are highly dependent on temperature, could be affected by the presence of
the Ar impurities in the system, further studies are still required. Thus far, Ar has been shown to decrease

90

grain growth by promoting kinetic stabilization of the grain boundaries [54, 205]. However, this is not
related to the formation of intermetallics.

Figure 73. High resolution bright field TEM images of (a) a W rich grain surrounded by W3Cr
precipitates and (b) an amorphous W3Cr precipitate at 1000 ºC. The insets show the SAD patterns of the
corresponding regions. (a) displays a crystalline BCC structure, while (b) shows an amorphous W3Cr
precipitate surrounded by a diffusion zone.

In addition to the W 3Cr precipitates, the microstructure at 1000 ºC is also comprised of equiaxed grains rich
in W. The composition of these grains suggests that the initial multilayered structure shifts the kinetics of
nucleation towards the formation of solutions rich in W. Although there are two phases below the
miscibility gap of the W-Cr system, we do not observe the Cr rich phase. This is a result of the initial W-
rich layers being in a stable state above 1000 ºC, which lowers the Gibbs energy of W solutions and
increases the energy penalty for nucleation of solutions rich in Cr [193]. Thus, the initial metastable state
of the NMMs should modify the local Gibbs free energy and favor the nucleation of stable states close to
that of the W-rich layers, which suggests that the structure of NMMs can be tuned to favor the formation
of selected stable states.

6.6 Summary
In summary, NMMs were used to control the formation of phases in the W-Cr system during annealing.
Specifically, the high density of interfaces in the nanoscale multilayers increased the rate of phase
transformation, such that thermal processes could be resolved at reduced temperatures. For instance, an
amorphous W 3Cr intermetallic was observed at 1000 ºC, a temperature much lower than that reported in
previous studies. Furthermore, the multilayered structure shifted the kinetics of phase separation towards
the formation of solid solutions rich in W. Overall, we propose a new alternative method to study phase
transformations in the nanoscale by using NMMs to control the thermal evolution of the system.

91

7 Conclusions and Future Work
7.1 Conclusions
The systems explored in Chapters 4 through 6 revealed that NMMs of immiscible systems are an effective
approach to improve the thermal stability of nanocrystalline materials. In these studies, NMMs with thin
layers (< 50 nm), high solute contents, and tailored grain structures were synthesized to drive the formation
of stabilized nanostructures after annealing. These final nanostructures formed after a series of
microstructural and phase transformations that reduce the Gibbs energy of the system, where processes
such as recrystallization, segregation, precipitation, and grain growth were observed during the thermal
evolution of NMMs.

The observed transformations were promoted by NMMs with layers thinner than 50 nm and a solute content
greater than 20 at. %, characteristics which increased the diffusivity of the species in the system. High
diffusivities induced the formation of solute complexions and precipitates that thermodynamically or
kinetically stabilized the grain boundaries. Furthermore, the formation of stabilized nanostructures also
depended on the initial grain structure. For instance, NMMs with columnar grains tended to retard the onset
of thermally activated microstructural transformations and led to an increased thermal stability. In contrast,
brick-like NMMs accelerated the transition to nanostructures. Independent of the initial grain structure, the
NMMs tended to evolve into equiaxed nanostructures after prolonged annealing.

These equiaxed nanostructures were clearly observed after annealing of Hf-Ti NMMs. The thermal
progression of this system started with the development of breakthroughs that signaled the onset of
recrystallization, which resulted in the formation of the equiaxed grains observed at 800 ⁰C. During this
process, Ti segregated to the grain boundaries and induced the formation of clusters, which formed a wetting
complexion that thermodynamically stabilized the recrystallized grains. Further heat-treatment at 1000 °C
drove the movement and coalescence of the equiaxed grains, which resulted in the impingement of the Ti
precipitates and the formation of a heterogeneous microstructure. Despite the fact that at this temperature
grain growth has occurred, the grains remained in the nanoscale (132 nm).

A similar result was observed after heat-treatment of bimodal Ta-Hf NMMs with both columnar and brick-
like grain structures. This system revealed that the grain structure influences the rate of the transformations
during annealing. More specifically, at 550 °C the brick-like grains have undergone recrystallization,
whereas the columnar grains were mostly preserved. These contrasting stabilities indicated that columnar
grains with less mobile semi-coherent interphases between the subgrains can be leveraged to retard or

92

prevent recrystallization and lead to nanocrystalline materials with increased thermal stability. Furthermore,
heat-treatment of the Ta-Hf multilayers at 1000 ⁰C induced the formation of Hf wetting complexions and
precipitates that simultaneously activated thermodynamic and kinetic stabilization mechanisms. This
system showed that it is possible to tailor the grain structure of NMMs to control the formation of stabilized
nanostructures.

Additionally, the high diffusivities of NMMs can lead to stabilization via precipitation of intermetallics,
which was observed in the W-Cr system. These multilayers were synthesized with sufficiently thin layers
(< 10 nm) and a relatively high Cr content (31 at. % Cr) to enhance the diffusive processes and to drive the
precipitation of the W 3Cr intermetallic. Annealing of this system at 800 ⁰C resulted in a partially
recrystallized microstructure without noticeable compositional changes or visible precipitates. However,
after heat-treatment at 1000 ⁰C, a fully recrystallized microstructure with W 3Cr precipitates were observed
to be locking the grain boundaries at triple junctions and quadrupole points. Therefore, the W-Cr multilayers
showed that it is possible to induce the precipitation of intermetallic compounds in order to activate kinetic
stabilization mechanisms. Additionally, this system demonstrated the use of NMMs with sufficiently thin
layers, which facilitate diffusive processes, to explore the formation of nanoscale phases.

In the studies described above, several microstructural or phase transformations occurring during the
thermal evolution of NMMs were identified. These transformations are driven by the excess energy of the
system and occur at rates that depend on both the grain morphology and interphase structure. Furthermore,
the aforementioned processes induce the progression from NMMs to nanostructures with a grain size at
1000 °C ranging between 60 nm and 140 nm due to stabilization by either thermodynamic or kinetic
mechanisms. Overall, NMMs are a promising approach to understand the mechanisms that control the
thermal evolution and to improve the stability of nanocrystalline materials.

7.2 Future Work
The results of the previous studies indicate that NMMs are a path to increase the thermal stability of
nanocrystalline materials. However, there are still several open questions that could be answered in future
studies. For instance, there are no precise relations for the rate of the microstructural processes with
temperature and composition based on atomistic processes. These models could be constructed using
experimental techniques that measure the composition at an atomic scale such as atom probe tomography
(APT), or techniques that display the microstructural and compositional progression of the multilayers over
time, such as in-situ TEM heat-treatments. Additionally, several authors have suggested that NMMs with

93

thicker layers have greater stability, a hypothesis that has yet to be tested. Finally, complete stabilization
using NMMs will only be possible if the mechanical properties of these systems are understood in more
detail. These possible investigations are discussed in the following paragraphs.

The rate at which the processes observed during annealing of the NMMs occur, could be quantified using
compositional profiles inside the grains and at grain boundaries obtained via APT characterization [206].
This technique is suitable to study the local change in solute content due to segregation and to explore the
composition of clusters that induce precipitation at the grain boundaries. This information can be employed
to calculate accurate enrichment parameters, which are required in the different kinetic models of
segregation. Additionally, due to their sensitivity, APT maps can be used to measure the composition of
low molecular weight elements such as nitrogen, carbon, and oxygen which could alter the grain boundary
energy [207]. Therefore, this technique can provide detailed compositional information that could be used
to understand the transformation occurring during annealing of the NMMs.

Although we have observed how the transformations occur after isochronal heat-treatments at specific
temperatures, we have not yet observed how microstructural and phase transformations occur over time
during isothermal annealing. These heat-treatments could be performed in-situ in a TEM, such that the
transformations could be recorded over time [208]. Even though in TEM lamellas (< 60 nm) the rate of
these processes is increased by the larger surface energy, such heat-treatments would allow to resolve the
different morphological and compositional changes occurring during the transformations [209]. This data
would help to identify the mechanisms driving these transformations and develop kinetic models that better
represent the atomistic processes behind them.

On the other hand, several studies on the stability of NMMs suggests that thermally activated
microstructural transformations in NMMs are delayed in systems with thicker layers [80, 210]. The
increased stability stems from pinch-offs being more difficult formation in thicker layers and an overall
reduction of the interface energy. Indeed, it has been shown Cu-Nb NMMs with larger layer size have an
improved thermal stability [79]. Therefore, by increasing the thickness of the individual layers it should be
possible to delay the microstructural evolution of NMMs. For instance, NMMs of the same composition as
the system already studied, with layers 10 times thicker should exhibit a greater thermal stability.

Finally, mechanical characterization of the NMNs will provide insight into the deformation mechanisms
occurring in the multilayers. For example, nanoindentation can be used to measure properties such as the
hardness and Young’s modulus, values which point to the efficacy of interphases as dislocation barriers
[211]. The interaction of dislocations during plastic deformation depends on the structure of the multilayers

94

[212]. For instance, several authors have shown that the flow stress decreases with increasing layer
thickness [74]. Therefore, the size of the individual layers could be tailored to control the plastic
deformation of thin films. Additionally, the correlation between the mechanical properties and the thermal
evolution of the samples would provide more information about the mechanisms controlling the stability
of nanocrystalline materials.

95

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109

Appendix A. Analysis of Characterization Data
Appendix A1. Grain size calculation
The grain size distribution of the samples presented in the previous chapters were calculated using the
equations described by Vander Voort in the book “Metallography Principle and Practices” in Chapter 6-7.3
on arithmetic distributions [170]. To use these equations, it was necessary to count and classify more than
300 grains into size classes that were then used to calculate the average grain size using the equation:

𝑑 ̅
=
1
𝑁 ∑ 𝑁 𝑖 𝑑 𝑖
𝑖 (73)
Where 𝑑 ̅
is the average grain size, 𝑁 𝑖 is the number of grains in the class 𝑖 , 𝑑 𝑖 the grain size of the
class, and 𝑁 = ∑𝑁 𝑖 . The average deviation of the distribution is calculated using the formula:

𝑠 (𝐷 )= [
1
𝑁 ∑ 𝑁 𝑖 (𝑑 𝑖 − 𝑑 ̅
)
𝑖 ]
1/2
(74)
Where 𝑠 (𝐷 ) is the standard deviation. In these formulas, the size of individual grains is computed assuming
a spherical shape using the equation 𝑑 = √4𝐴 /𝜋 , where 𝐴 is the projected area of the grain in two
dimensional micrographs. The projected area is calculated using two approaches depending on the visibility
of the grains in dark-field TEM images. For clearly visible grains, the area is calculated using the TsView7
software by delineating the perimeter of each grain with the cursor as presented in Figure 74 [213].

Figure 74. Delineated (a) columnar and (b) equiaxed grains in Ta-Hf NMMs using the TsView7 software
showing the area and perimeter of each grain.

110

For samples with grains not clearly visible, ImageJ was used to process the images and to obtain the area
of the grains [214]. In order to use the utilities of this software to measure the grains area, they should be
visible against a white background. Therefore, the micrographs were processed in two steps. First, the
contrast of the images was increased to improve the visibility of the grains using the following code [215]:

AUTO_THRESHOLD = 1000000;

getRawStatistics(count, mean, min, max, std);
limit = count/10;
threshold = count/AUTO_THRESHOLD;

nBins = 256;
getHistogram(values, counts, nBins);

i = -1;
condition = 0;
while (condition != 1) {
i = i+1;
count = counts[i];
if (count > limit) { count = 0; }
found = count > threshold;
if (found) { condition = 1; }
else if (i >= 255) { condition = 1; }
}
hmin = i;

i = 256;
condition = 0;
while (condition != 1) {
i = i-1;
count = counts[i];
if (count > limit) { count = 0; }
found = count > threshold;
if (found) { condition = 1; }

111

else if (i < 1) { condition = 1; }
}
hmax = i;

setMinAndMax(hmin, hmax);

getPixelSize(unit, pw, ph);

if(unit == "nm") {
run("Scale Bar...", "width=200 height=25 font=100 color=Black background=White location=[Lower
Right] bold overlay");
} else if(unit == "microns") {
run("Scale Bar...", "width=0.2 height=10 font=40 color=Black background=White location=[Lower
Right] bold overlay");
}

This code, which is an adaptation of the routine created by Kota Miura for Auto Brightness/Contrast in
ImageJ, identifies all non-black pixels and sets their contrast to the maximum value in the initial image.
That code was run to batch process several images using the Macro utility of ImageJ under
Process/Batch/Macro, which saves the processed image in a new file. An example showing how the
previous code increases the visibility of the grains is presented in Figure 75.

Figure 75. (a) Initial dark-field TEM micrograph of the as-sputtered W-Cr NMMs and (b) image
processed using the code to improve contrast.

112

Subsequently, to compute the area of the grains using ImageJ, the color threshold of the processed images
was changed to invert the color of the grains from white to black, and the background from black to white,
which was done using the following two lines of code:

setAutoThreshold("Default dark");
run("Create Mask");

An example of an image with changed color threshold is presented in Figure 76a. Such images were
employed to calculate the area of the visible grains using the routine under Analyze/Analyze Particles
prebuild in ImageJ, which produces a list of the grains found in the image with their corresponding area
and perimeter, and generates a new image where the indexed grains are highlighted in green (see Figure
76b) [214]. This information was used to create the grain size distributions discussed in Chapters 4 to 6.

Figure 76. (a) dark-field TEM micrograph of the as-sputtered W-Cr NMMs after the change of the color
threshold, and (b) image showing the grains identified using the methods in ImageJ.

113

Appendix A2. Diffraction patterns analysis
To study the orientation of grains, different TEM diffraction techniques were employed depending on the
number of grains and their size. For individual grains greater than 100 nm, the orientation was identified
using selected area diffraction patterns (SAED). In these patterns, the distance between spots, which
corresponds to diffracted electrons from specific planes, was measured using the TSview7 software, as seen
in Figure 77 [213]. Then the pattern is indexed using a two-step procedure. First, the patterns were compared
against symmetry tables for different crystalline structures to identify the zone axis (normal orientation) of
the grain. After matching the pattern, it was verified that the planes corresponding to the spots satisfy the
following relation [170]:

𝑟 1
=
𝜆𝐿
𝑎 √ℎ
1
2
+ 𝑘 1
2
+ 𝑙 1
2
(75) 𝑟 2
=
𝜆𝐿
𝑎 √ℎ
2
2
+ 𝑘 2
2
+ 𝑙 2
2
(76)
Where 𝑟 1
and 𝑟 2
are the distances between two spots surrounding the (000) spot, 𝜆𝐿 is the camera
constant, 𝑎 is the lattice constant, and ℎ,𝑘 ,𝑙 are the Miller indices of the planes. After eliminating
the camera length between equations (75) and (76), the following relation results:

𝑟 2
= 𝑟 1
√
ℎ
2
2
+ 𝑘 2
2
+ 𝑙 2
2
√
ℎ
1
2
+ 𝑘 1
2
+ 𝑙 1
2
⁄ (77)
If this relation is satisfied, it is confirmed that the orientation of the grain and that of its symmetry pattern
are the same.

Figure 77. (a) Diffraction pattern of a Ti grain within the Hf-Ti NMMs annealed at 1000 ⁰C, and (b)
symmetry patter in the [221] zone axis used to index the diffraction pattern [130, 213].

114

In the case of grains smaller than 100 nm, nanodiffraction patterns, as seen in Figure 78, were collected to
confirm the presence of either crystalline or amorphous microstructures in high resolution TEM (HRTEM)
images. Although these patterns could be studied by performing quantum mechanical simulations of the
system, which would provide detailed information about the position of the atoms, such simulations were
beyond the extent of the studies presented in this dissertation [145].

Figure 78. Nanodiffraction pattern of a W 3Cr grain showing a ring corresponding to the amorphous nature
of these precipitates and diffraction spots form the surrounding W rich grains.

The analysis of the texture of the films to find the phases present was made using diffraction ring patterns
collected by illuminating most of the cross-sectional area of the sample with a parallel beam. An example
is shown in Figure 79. These patterns, which consist of diffracted spots corresponding to all the grains
intersected by the beam of electrons were analyzed using the ProcessDiffraction software [216].

115

Figure 79. Rings diffraction pattern of a W-Cr sample annealed at 1000 ⁰C analyzed using
ProcessDiffraction [216].

In order to index ring diffraction patterns using ProcessDiffraction, it is necessary to indicate the calibration
constant in (1/nm)/pixel, a value obtained from measurement of the scale bar in the diffraction micrographs
using ImageJ [214, 216]. After setting that parameter, radial intensity profiles in the reciprocal space were
calculated by using the “Calculate distribution” feature. Subsequently, the background of this distribution
was calculated assuming a logarithmic function using more than four fitting points. Later, this background
was subtracted to obtain the net peaks diffracted by the phases present in the sample. These peaks
correspond to scattering vectors of magnitudes that were computed using the utility under Process/Calculate
Net Peaks. An example of a processed radial intensity profile is presented in Figure 80.

116

Figure 80. Average relative intensity profile of the W-Cr NMMs annealed at 1000 ⁰C.

The profile in Figure 80 shows several peaks corresponding to planes that can be identified using Braggs
law [217]:

2𝜋 1
𝑑 (ℎ,𝑘 ,𝑙 )
=
2𝜋 𝜆 2𝑆𝑖𝑛 (𝜃 ) (78)
Where the right side of equation (78) is the magnitude of the scattering vector, which can be used to find
the interplanar distance, and thus the Miller indices corresponding to the planes normal to grains present in
the sample. Another alternative to visualize the peaks identified in the radial intensity profiles is by
combining half of the diffraction pattern and half-circle schematics depicting the position of the peaks with
a circle (Figure 81a). Additionally, the orientation of the peaks can be added to the normalized radial
intensity profiles (Figure 81b).

Figure 81. (a) Rings pattern showing the half-circle schematic of a W-Cr sample annealed at 1000 ⁰C, and
(b) the corresponding radial intensity profile showing W BCC and Cr BCC peaks.

117

Appendix B. Summary of Sputtered Samples
The present appendix consists of tables describing the sputtering conditions for the respective NMMs, their
properties, and the heat-treatments performed.

Appendix B1. Hf-Ti NMMs
Table 3. Sputtering conditions used to deposit the Hf-Ti NMMs

Hf Ti
Sample
Name
Substrate
Material
Pressure
(mTorr)
Power
(W)
Time on /
Time off (s)
Power
(W)
Time on
(s)
Test-3 Si 10.0 200 25.5 on / 25.5 off 60 3720
Test-3 glass Thick glass 10.0 200 25.5 on / 25.5 off 60 3720
Sample A-1 Si 5.0 200 25.5 on / 25.5 off 60 3720
Sample A-1 glass Thick glass 5.0 200 25.5 on / 25.5 off 60 3720
Sample A-2 Si 5.5 200 25.5 on / 25.5 off 60 3720
Sample A-2 glass Thick glass 5.5 200 25.5 on / 25.5 off 60 3720
Sample A-3 Thick glass 5.0 200 25.5 on / 25.5 off 60 3720
Sample B-1 Si 5.0 200 63.6 on / 63.6 off 60 5400
Sample B-1 glass Thick glass 5.0 200 63.6 on / 63.6 off 60 5400
Sample B-2 Si 5.0 200 63.6 on / 63.6 off 60 5400
Sample B-2 glass Thick glass 5.0 200 63.6 on / 63.6 off 60 5400
Sample B-3 Si 2.3 200 63.6 on / 63.6 off 60 5400
Sample B-3 glass Thick glass 2.3 200 63.6 on / 63.6 off 60 5400
Sample B-4 Thick glass 5.0 200 63.6 on / 63.6 off 60 5400
Sample B-5 Thick glass 5.0 200 63.6 on / 63.6 off 60 5400
Sample B-6 Thick glass 5.0 200 63.6 on / 63.6 off 60 5400
Sample B-7 Thick glass 5.0 200 63.6 on / 63.6 off 60 5400
Sample B-8 Thick glass 5.0 200 63.6 on / 63.6 off 60 5400
Sample B-9 Thick glass 5.0 200 63.6 on / 63.6 off 60 5400
Sample B-10 Thick glass 6.0 200 63.6 on / 63.6 off 60 5400
Sample B-11 Thick glass 6.0 200 63.6 on / 63.6 off 60 5400
Sample B-12 Thick glass 6.0 200 63.6 on / 63.6 off 60 5400
Sample B-13 Thick glass 20.0 200 63.6 on / 63.6 off 60 5400
Sample B-14 Thick glass 7.5 200 63.6 on / 63.6 off 60 5400
Sample B-15 Thick glass 7.5 200 63.6 on / 63.6 off 60 5400
Sample B-16 Thick glass 20.0 200 63.6 on / 63.6 off 60 5400
Sample B-17 Thick glass 6.0 200 63.6 on / 63.6 off 60 5400
Sample B-18 Thick glass 6.0 200 63.6 on / 63.6 off 60 5400
Sample B-19 Thick glass 6.0 200 63.6 on / 63.6 off 60 5400
Sample C-1 Si 4.0 200 7200 60 7200
Sample C-1 glass Thick glass 4.0 200 7200 60 7200

118

Sample C-2 Si 4.0 200 7200 60 7200
Sample C-2 glass Thick glass 4.0 200 7200 60 7200
Sample C-3 Thick glass 4.0 200 7200 115 7200
Sample C-4 Thick glass 7.5 200 5400 115 5400
Sample C-5 Thick glass 5.0 200 5400 115 5400
Sample C-6 Thick glass 7.5 200 7200 115 7200
Sample C-7 Thick glass 7.5 200 3600 115 3600
Sample C-8 Thick glass 10.0 200 3600 115 3600
Sample C-9 Thick glass 10.0 200 3600 115 3600
Sample D-1 Si 5.0 200 66 60 127
Sample D-1 glass Thick glass 5.0 200 66 60 127
Sample D-2 Thick glass 5.0 200 66 60 127
Hf-Ti-20 Si 6.0 200 63.5on/63.5off 60 5400
Hf-Ti-21 Si 6.0 200 63.5on/63.5off 60 5400
Hf-Ti-22 Si 6.0 200 635on/635off 60 5715
Hf-Ti-23 Si 6.0 200 635on/635off 60 5715
Hf-Ti-24 Si 6.0 200 635on/635off 60 5715

Table 4. Properties of the as-sputtered Hf-Ti NMMs

Thickness (nm)

Sample
Name
Bilayers
Hf-Ti
layers
Ti
layers
Sample Composition
Crystalline
structure
Test-3 40 15 2 680

P63/mmc
Test-3 glass 40 15 2 680

P63/mmc
Sample A-1 40 15 2 680 23 at % Ti P63/mmc
Sample A-1 glass 40 15 2 680 23 at % Ti P63/mmc
Sample A-2 40 15 2 680 23 at % Ti P63/mmc
Sample A-2 glass 40 15 2 680 23 at % Ti P63/mmc
Sample A-3 40 15 2 680 23 at % Ti P63/mmc
Sample B-1 40 40 5 1800 24 at % Ti P63/mmc
Sample B-1 glass 40 40 5 1800 24 at % Ti P63/mmc
Sample B-2 40 40 5 1800 24 at % Ti P63/mmc
Sample B-2 glass 40 40 5 1800 24 at % Ti P63/mmc
Sample B-3 40 40 5 1800 24 at % Ti P63/mmc
Sample B-3 glass 40 40 5 1800 24 at % Ti P63/mmc
Sample B-4 40 40 5 1800 24 at % Ti P63/mmc
Sample B-5 40 40 5 1800 24 at % Ti P63/mmc
Sample B-6 40 40 5 1800 24 at % Ti P63/mmc
Sample B-7 40 40 5 1800 24 at % Ti P63/mmc
Sample B-8 40 40 5 1800 24 at % Ti P63/mmc
Sample B-9 40 40 5 1800 24 at % Ti P63/mmc
Sample B-10 40 40 5 1800 24 at % Ti P63/mmc
Sample B-11 40 40 5 1800 24 at % Ti P63/mmc

119

Sample B-12 40 40 5 1800 24 at % Ti P63/mmc
Sample B-13 40 40 5 1800 24 at % Ti P63/mmc
Sample B-14 40 40 5 1800 24 at % Ti P63/mmc
Sample B-15 40 40 5 1800 24 at % Ti P63/mmc
Sample B-16 40 40 5 1800 24 at % Ti P63/mmc
Sample B-17 40 40 5 1812 24 at % Ti P63/mmc
Sample B-18 40 40 5 1812 24 at % Ti P63/mmc
Sample B-19 40 40 5 1812 24 at % Ti P63/mmc
Sample C-1 40

23 at % Ti P63/mmc
Sample C-1 glass 40

23 at % Ti P63/mmc
Sample C-2 40

23 at % Ti P63/mmc
Sample C-2 glass 40

23 at % Ti P63/mmc
Sample C-3 40

23 at % Ti P63/mmc
Sample C-4 40

23 at % Ti P63/mmc
Sample C-5 40

23 at % Ti P63/mmc
Sample C-6 40

23 at % Ti P63/mmc
Sample C-7 40

23 at % Ti P63/mmc
Sample C-8 40

23 at % Ti P63/mmc
Sample C-9 40

23 at % Ti P63/mmc
Sample D-1 40 40 10 2000

P63/mmc
Sample D-1 glass 40 40 10 2000

P63/mmc
Sample D-2 40 40 10 2000

P63/mmc
Hf-Ti-20 40 40 5 1800 20 at % Ti P63/mmc
Hf-Ti-21 40 40 5 1800 20 at % Ti P63/mmc
Hf-Ti-22 4 400 50 1800 23 at % Ti P63/mmc
Hf-Ti-23 4 400 50 1800 23 at % Ti P63/mmc
Hf-Ti-24 4 400 50 1800 23 at % Ti P63/mmc

Table 5. Heat-treatments of the Hf-Ti NMMs
Heat-Treatment Sample
Temperature
(°C)
Time
(min)
Quenched Characterization
B-1-HT1 Sample B-1 300 60 Yes TEM
B-1-HT2 Sample B-2 500 60 Yes

B-1-HT3 Sample B-3 600 60 Yes

B-1-HT4 Sample B-4 600 5 Yes

B-1-HT5 Sample B-5 1000 5 Yes

B-1-HT6 Sample B-6 800 5 Yes

B-1-HT7 Sample B-2 750 5 Yes

B-2-HT1 Sample B-2 750 5 Yes

B-2-HT2 Sample B-2 800 5 Yes

B-2-HT3 Sample B-2 800 5 Yes

B-2-HT4 Sample B-2 800 5 Yes

B-2-HT5 Sample B-2 800 60 Yes

120

B-2-HT6 Sample B-2 800 90 Yes

B-2-HT7 Sample B-2 800 85 Yes

B-3-HT1 Sample B-3 800 15 Yes TEM
B-3-HT2 Sample B-3 800 30 Yes

B-3-HT3 Sample B-3 800 30 Yes

B-3-HT4 Sample B-3 800 60 Yes

B-5-glass-TH-1 Sample B-5 800 5760 Yes TEM
B-3-glass-HT-1 Sample B-3 glass 800 5760 Yes

B-14-glass-HT-1-800 Sample B-14 800 5760 Yes TEM
B-11-glass-HT-1-800 Sample B-11 800 5760 Yes TEM
B-11-glass-HT-2-1000 Sample B-11 1000 5760 Yes TEM
B-11-glass-HT-3-500 Sample B-11 500 5760 Yes TEM/EDS
B-17-HT1-800 Sample B-17 800 5760 Yes

B-19-HT-1-800 Sample B-19 800 5760 Yes TEM/EDS
B-19-HT-2-500 Sample B-19 500 5760 Yes TEM/EDS
B-19-HT-3-1000 Sample B-19 1000 5760 Yes TEM/EDS
C-8-glass-HT-1-500 Sample C-8 500 5760 Yes EDS
C-8-glass-HT-2-800 Sample C-8 800 5760 Yes

C-8-glass-HT-3-1000 Sample C-8 1000 5760 Yes TEM

Appendix B2. Ta-Hf NMMs
Table 6. Sputtering conditions used to deposit the Ta-Hf NMMs
Ta Hf
Sample
Name
Substrate
Material
Pressure
(mTorr)
Power
(W)
Time on /
Time off (s)
Power
(W)
Time on
(s)
Ta-Hf-1 Thick glass 5.0 250 19 on/ 11 off 50 3000
Ta-Hf-2 Si 5.0 175 25 on/ 11 off 50 4800
Ta-Hf-3 Si 5.0 175 25 on/ 11 off 50 4800
Ta-Hf-4 Si 5.0 175 25 on/ 11 off 50 4800
Ta-Hf-5 Si 5.0 175 25 on/ 11 off 50 4800
Ta-Hf-6 Si 5.0 175 25 on/ 11 off 50 4800
Ta-Hf-7 Si 5.0 175 25 on/ 11 off 50 4800
Ta-Hf-8 Si 5.0 175 25 on/ 11 off 50 4800
Ta-Hf-9 Si 5.0 175 250 on/ 110 off 50 5040
Ta-Hf-10 Si 5.0 175 250 on/ 110 off 50 5040
Ta-Hf-11 Si 5.0 175 250 on/ 110 off 50 5040

121

Table 7. Properties of the as-sputtered Ta-Hf NMMs

Thickness (nm)

Sample
Name
Bilayers
Ta-Hf
layers
Hf
layers
Sample Composition
Crystalline
structure
Ta-Hf-1 127 20 2 2794

P63/mmc
Ta-Hf-2 127 20 2 2794 22.4 at % Hf P63/mmc
Ta-Hf-3 127 20 2 2794 22.4 at % Hf P63/mmc
Ta-Hf-4 127 14 2 1994 22.4 at % Hf P63/mmc
Ta-Hf-5 127 14 2 1994 22.4 at % Hf P63/mmc
Ta-Hf-6 127 14 2 1994 22.4 at % Hf P63/mmc
Ta-Hf-7 127 14 2 1994 22.4 at % Hf P63/mmc
Ta-Hf-8 127 14 2 1994 22.4 at % Hf P63/mmc
Ta-Hf-9 13 140 17 2041 22.4 at % Hf P63/mmc
Ta-Hf-10 13 140 17 2041 22.4 at % Hf P63/mmc
Ta-Hf-11 13 140 17 2041 22.4 at % Hf P63/mmc

Table 8. Heat-treatments of the Ta-Hf NMMs
Heat-Treatment Sample
Temperature
(°C)
Time
(min)
Quenched Characterization
HT-1-Ta-Hf-1 (Peel-off) Ta-Hf-1 800 60 Not
HT-2-Ta-Hf-1 (Peel-off) Ta-Hf-1 800 60 Not
HT-1-Ta-Hf-2 (Peel-off) Ta-Hf-2 800 60 Not
Ta-Hf-2-HT-1-800 Ta-Hf-2 800 5760 Yes
Ta-Hf-2-HT-2-800 Ta-Hf-2 800 5760 Yes TEM/EDS
HT-1-Ta-Hf-3 (Peel-off) Ta-Hf-3 800 60 Not
HT-1-Ta-Hf-4 (Peel-off) Ta-Hf-4 800 60 Not
Ta-Hf-2-HT-2-550 Ta-Hf-2 550 5760 Yes TEM/EDS
Ta-Hf-2-HT-3-1000 Ta-Hf-2 1000 5760 Yes TEM/EDS

Appendix B3. W-Cr NMMs
Table 9. Sputtering conditions used to deposit the W-Cr NMMs
W Cr
Sample
Name
Substrate
Material
Pressure
(mTorr)
Power
(W)
Time on /
Time off (s)
Power
(W)
Time
on (s)
W-Cr-1 Thick glass 14.6 150 45 on/ 10 off 100 5520
W-Cr-2 Thick glass 15.0 200 16 on/ 6 off 50 5700
W-Cr-3 Si 15.0 200 16 on/ 6 off 50 6600

122

W-Cr-4 Si 15.0 200 16 on/ 6 off 50 14400
W-Cr-5 Si 15.0 200 16 on/ 6 off 50 6600
W-Cr-6 Si 15.0 200 16 on/ 6 off 50 6600
W-Cr-7 Si 15.0 200 16 on/ 6 off 50 6600
W-Cr-8 Si 15.0 200 160 on/ 60 off 50 6600
W-Cr-9 Si 15.0 200 160 on/ 60 off 50 6600
W-Cr-10 Si 15.0 200 160 on/ 60 off 50 6600
W-Cr-11 Si 15.0 200 160 on/ 60 off 50 6600

Table 10. Properties of the as-sputtered W-Cr NMMs
Thickness (nm)
Sample
Name
Bilayers
W-Cr
layers
Cr
layers
Sample Composition
Crystalline
structure
W-Cr-1 247 20 2 5434 28.6 at % Cr Im-3m
W-Cr-2 247 20 2 5434 28.6 at % Cr Im-3m
W-Cr-3 247 6 2 1927 32.7 at % Cr Im-3m
W-Cr-4 247 6 2 1927 32.7 at % Cr Im-3m
W-Cr-5 247 6 2 1927 32.7 at % Cr Im-3m
W-Cr-6 247 6 2 1927 32.7 at % Cr Im-3m
W-Cr-7 247 6 2 1927 32.7 at % Cr Im-3m
W-Cr-8 25 63 15 1950 32.7 at % Cr Im-3m
W-Cr-9 25 63 15 1950 32.7 at % Cr Im-3m
W-Cr-10 25 63 15 1950 32.7 at % Cr Im-3m
W-Cr-11 25 63 15 1950 32.7 at % Cr Im-3m

Table 11. Heat-treatments of the W-Cr NMMs
Heat-Treatment Sample
Temperature
(°C)
Time
(min)
Quenched Characterization
HT-1-W-Cr-1 (Peel-off) W-Cr-1 800 60 Not
HT-2-W-Cr-1 (Peel-off) W-Cr-1 800 60 Not
W-Cr-2-HT-1-800 W-Cr-2 800 5760 Yes TEM/EDS
W-Cr-2-HT-2-550 W-Cr-2 550 5760 Yes TEM/EDS
W-Cr-2-HT-3-1000 W-Cr-2 1000 5760 Yes TEM/EDS

123

Appendix C. Phase Diagrams

Appendix C.1 Hf-Ti Phase Diagram

Figure 82. Hf-Ti phase diagram showing in red the 20.1 at % Ti isopleth. This is the global composition
of the Hf-Ti NMMs [218].

124

Appendix C.2 Ta-Hf Phase Diagram

Figure 83. Ta-Hf phase diagram showing in red the 22.4 at % Hf isopleth. This is the global composition
of the Ta-Hf NMMs [218].

125

Appendix C.3 W-Cr Phase Diagram

Figure 84. W-Cr phase diagram showing in red the 33.1 at % Cr isopleth. This is the global composition
of the W-Cr NMMs [218].

Abstract (if available)

Abstract Nanocrystalline materials have interesting electrical, magnetic, and mechanical properties, especially when compared to their coarse-grained counterparts. For example, the Hall-Petch relation shows that nanomaterials have a greater yield strength than coarse grained systems. However, the application of nanocrystalline materials has been limited by their low thermal stability, which stems from their high density of interfaces that act as channels for diffusion. Although several attempts have been made to improve the thermal stability of nanomaterials via kinetic or thermodynamic mechanisms, there are limited alternatives to induce the desired stabilization. Thus, there is a need in this field to investigate possible routes to promote the formation of nanostructures with a decreased propensity for grain growth at elevated temperatures. In this work, nanometallic multilayers (NMMs) are used to drive microstructural and phase transformations that result in the formation of stabilized nanostructures after protracted annealing. NMMs were selected because they allow for control over the local composition, the density of interphases, and the grain structure of nanocrystalline samples. In the studies presented in this dissertation, the effect of the initial microstructure on the thermal progression from nanomultilayers to nanostructures was studied by characterizing Hf-Ti, Ta-Hf, and W-Cr NMMs heat-treated at critical temperatures. Overall, the results from this investigation highlight a new path to synthesize stable nanomaterials via annealing of NMMs.

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Creator Riano Zambrano, Juan Sebastian (author)

Core Title Exploring the thermal evolution of nanomaterials: from nanometallic multilayers to nanostructures

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Materials Science

Publication Date 04/22/2019

Defense Date 03/20/2019

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag bimodal nanometallic multilayers,brick-like grains,columnar grains,Cr,diffusion,diffusion zones,DSC scans,grain boundary energy,grain growth,HF,interfaces,interphases,nanomaterials,nanometallic multilayers,nanomultilayers,nucleation,OAI-PMH Harvest,precipitation,recrystallization,segregation,Ta,thermal stability,Ti,W,W₃Cr

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Advisor Hodge, Andrea Maria (committee chair), Kassner, Michael (committee member), Ravichandran, Jayakanth (committee member)

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Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-141484

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