Solid-State Diffusion of Bismuth in Tin-Rich, Lead-Free Solder Alloys · iii temperatures, the...
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Solid-State Diffusion of Bismuth in Tin-Rich, Lead-Free Solder Alloys by André Delhaise A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Materials Science & Engineering University of Toronto © Copyright by André Delhaise 2018
Transcript of Solid-State Diffusion of Bismuth in Tin-Rich, Lead-Free Solder Alloys · iii temperatures, the...
Solid-State Diffusion of Bismuth in Tin-Rich, Lead-Free Solder Alloys
by
André Delhaise
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy
Department of Materials Science & Engineering University of Toronto
© Copyright by André Delhaise 2018
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Solid-State Diffusion of Bismuth in Tin-Rich, Lead-Free Solder
Alloys
André Delhaise
Doctor of Philosophy
Materials Science & Engineering
University of Toronto
2018
Abstract
A significant reliability concern with traditional lead-free solder alloys is the degradation of
mechanical properties after aging, due to microstructure coarsening. The inclusion of bismuth (Bi)
in these alloys stabilizes properties over time and improves reliability. An examination of the
microstructure reveals that homogenization of Bi in the tin (Sn) matrix occurs via solid-state
diffusion. The goal of this thesis is twofold – to ascertain whether these trends hold over a wide
range of aging conditions, and to characterize the previously unknown diffusional mechanisms
behind these effects.
Several Bi-containing alloys were aged at one of several temperatures/times. The microstructures
were evaluated using Scanning Electron Microscopy (SEM), and Rockwell Superficial hardness
(15X) was used to measure the mechanical response. Bismuth precipitates became more uniformly
distributed over time. Alloy hardness did not change, except for that of Sn-5Bi, which was slightly
reduced after 252 days of aging at 25°C.
Sn-Bi diffusion couples were prepared and annealed at several temperatures. After a
microstructure examination using SEM, diffusion profiles were collected using Electron Probe
Microanalysis and analyzed using traditional methods or a forward simulation technique. At all
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temperatures, the diffusivity of Bi in Sn demonstrated a low anisotropy ratio - samples oriented
perpendicular to the Sn ‘c’ axis yielded slightly higher diffusivities than those parallel.
Subsequently, low angle grain boundaries in Sn can convolute the orientation relationship. The
Arrhenius parameters for lattice diffusivity are greater parallel to the ‘c’ axis than perpendicular.
An analysis of concentration-dependent diffusivity was then conducted, and it was found the ratio
between lattice and effective impurity diffusivity decreases with temperature, due to grain growth
and the active nature of Sn-based systems. Harrison ‘Type B’ kinetics were lastly considered to
study the effects of grain boundaries on diffusivity in polycrystalline Sn; further work is necessary
to determine grain boundary diffusivity.
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Acknowledgments
I would first like to thank my supervisor, Professor Doug D. Perovic, for his guidance throughout
this project, as well as keeping me on track and ensuring I kept my scope reasonable.
Special thanks to my other PhD committee members, Professors Zhirui Wang and Uwe Erb, for
the fruitful discussions and input during our yearly meetings, and to Professor Tom Coyle and
Dr. Paul Vianco for examining my final external defense.
A huge thank you to Dr. Polina Snugovsky for her mentorship, guidance and encouragement
throughout this project. Several additional individuals I had the pleasure of working with over
the course of this project as well as in the various Refined Manufacturing Acceleration Process
(ReMAP) projects I owe many thanks for the wonderful discussions, feedback, and suggestions,
in particular David Hillman (Rockwell-Collins), Jeffrey Kennedy (Celestica), and Dr. Stephan
Meschter (BAE Systems).
Numerous people aided with the technical aspects of this project; without any of them this thesis
would not be possible: Dr. Yanan Liu from the Department of Earth Sciences at the University of
Toronto for assistance with EPMA, Zhangqi Chen and Qiaofu Zhang from Ohio State University
for their efforts in developing and optimizing the diffusion simulation code, Harlan Kuntz from
the Toronto Nanofabrication Facility for support with sputter deposition, Sal Boccia for great
discussions and help with electron microscopy, Dr. Dan Grozea for Rockwell hardness testing,
and Diana Vucevic for assistance with WinWulff software.
Two additional people at UofT were also a great help: Dr. Leonid Snugovsky for his valuable
metallurgical expertise and assistance in the early stages of my project, and Ivan Matijevic for
the enjoyable discussions and suggestions about the scope of this thesis.
I would finally like to thank my family for their unwavering support and love as I pursued this
degree, in particular my parents Paul and Susan.
Funding from ReMAP, the Department of Materials Science & Engineering, Surface Mount
Technology Association (SMTA), Celestica, and Ontario Graduate Scholarship (OGS) are
greatly appreciated.
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For my grandmother, Catherine Madeleine Delhaise. Tu vas me manquer, Bamo.
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Table of Contents
Acknowledgments.......................................................................................................................... iv
Table of Contents ........................................................................................................................... vi
Published References ..................................................................................................................... xi
List of Figures ............................................................................................................................... xii
List of Tables ............................................................................................................................... xxi
List of Symbols, Acronyms, and Variables ............................................................................... xxiv
Chapter 1 Introduction .....................................................................................................................1
1.1 Overview ..............................................................................................................................1
1.2 Goals and Scope of Research ...............................................................................................2
Chapter 2 Background .....................................................................................................................3
2.1 History of Soldering .............................................................................................................3
2.1.1 Tin Lead (Sn-Pb) Alloys ..........................................................................................3
2.1.2 Pb-Free Alloys – First Generation ...........................................................................5
2.1.3 Pb-Free Alloys - Second Generation .......................................................................9
2.1.4 Pb-Free Alloys – Third Generation........................................................................10
2.1.4.1 History .....................................................................................................11
2.1.4.2 Recent Results .........................................................................................13
2.2 Strengthening Mechanisms in Metals and Alloys .............................................................17
2.2.1 Solid-Solution Strengthening .................................................................................18
2.2.2 Dispersion Strengthening .......................................................................................20
2.2.3 Grain Boundary Strengthening ..............................................................................22
2.2.3.1 Recovery, Recrystallization & Grain Growth .........................................24
2.2.3.2 Particle Stimulated Nucleation (PSN) .....................................................24
2.3 Solid-State Diffusion .........................................................................................................26
2.3.1 Fick’s (Continuum) Laws of Diffusion..................................................................27
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2.3.1.1 Classic Solutions .....................................................................................28
2.3.1.2 Inverse and First Principles Methods ......................................................29
2.3.2 Atomic Mechanisms ..............................................................................................32
2.3.3 Factors which Influence Diffusivity ......................................................................32
2.3.3.1 Temperature .............................................................................................32
2.3.3.2 High Diffusivity Paths .............................................................................33
2.3.3.3 Anisotropy in Non-Cubic Materials ........................................................39
2.3.4 Diffusion Study Methods .......................................................................................41
2.3.5 Interdiffusion and Impurity Diffusion ...................................................................42
2.3.5.1 Impurity Diffusion ...................................................................................43
2.3.5.2 Interdiffusion ...........................................................................................43
2.3.6 Diffusion in Sn-based Systems ..............................................................................45
2.3.6.1 Self and Impurity Diffusion .....................................................................45
2.3.6.2 Fast Impurity Diffusion ...........................................................................46
2.4 Analytical Techniques .......................................................................................................49
2.4.1 Imaging Techniques ...............................................................................................50
2.4.2 X-Ray Microanalysis .............................................................................................51
2.4.3 Electron Backscatter Diffraction (EBSD) ..............................................................52
Chapter 3 The Effects of Aging on Microstructure and Properties of Bismuth-Containing,
Lead-Free Solder Alloys ...........................................................................................................56
3.1 Introduction ........................................................................................................................56
3.2 Experimental Methodology ...............................................................................................57
3.2.1 Test Matrix .............................................................................................................57
3.2.2 Sample Preparation ................................................................................................59
3.2.3 Microstructure Evaluation .....................................................................................60
3.2.4 Mechanical Property Testing .................................................................................60
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3.3 Results & Discussion .........................................................................................................62
3.3.1 As-Cast ...................................................................................................................62
3.3.2 Elevated Temperature Aging .................................................................................65
3.3.3 Room Temperature Aging .....................................................................................72
3.4 Concluding Remarks ..........................................................................................................77
Chapter 4 Methodology for Study of Solid-State Diffusion of Bismuth in Tin ............................79
4.1 Introduction ........................................................................................................................79
4.2 Test Matrix .........................................................................................................................81
4.3 Sample Preparation ............................................................................................................83
4.3.1 Casting ...................................................................................................................83
4.3.2 Metallography ........................................................................................................85
4.3.3 Diffusion Couple Preparation ................................................................................87
4.3.3.1 Coarse-Grained and Monocrystalline Sn .................................................87
4.3.3.2 Polycrystalline Sn ....................................................................................88
4.3.4 Cross-Sectioning ....................................................................................................89
4.4 Characterization Techniques ..............................................................................................91
4.4.1 Scanning Electron Microscopy (SEM) ..................................................................91
4.4.2 Electron Backscatter Diffraction............................................................................92
4.4.3 Electron Probe Microanalysis ................................................................................94
4.5 Diffusivity Analysis ...........................................................................................................95
4.5.1 Slab Source Model .................................................................................................95
4.5.2 Inverse Methods .....................................................................................................96
4.5.3 First Principles Simulation .....................................................................................96
Chapter 5 Solid-State Diffusion of Bismuth in Polycrystalline Tin ............................................100
5.1 Overview ..........................................................................................................................100
5.2 Electron Microscopy Results ...........................................................................................100
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5.2.1 As-Solidified Microstructure ...............................................................................100
5.2.2 Diffusion Microstructure .....................................................................................103
5.2.3 Diffusion Profiles .................................................................................................106
5.3 Diffusivity Analysis .........................................................................................................108
5.3.1 Inverse Methods ...................................................................................................109
5.3.2 Forward Simulation .............................................................................................111
5.3.3 Arrhenius Analysis...............................................................................................113
5.3.4 Comparison between Models ...............................................................................115
5.4 Concluding Remarks ........................................................................................................117
Chapter 6 Solid-State Diffusion of Bismuth in Coarse-Grained and Monocrystalline Tin .........120
6.1 Overview ..........................................................................................................................120
6.2 Electron Microscopy Results ...........................................................................................121
6.2.1 As-Solidified Microstructure (Coarse-Grained Sn) .............................................121
6.2.2 As-Sputtered Microstructure ................................................................................121
6.2.3 As-Annealed Microstructure (Coarse-Grained Sn) .............................................124
6.2.4 As-Annealed Microstructure (Monocrystalline Sn) .............................................128
6.2.5 Orientation Analysis & Maps ..............................................................................129
6.2.6 Diffusion Profiles .................................................................................................131
6.3 Diffusivity Analysis .........................................................................................................133
6.3.1 Comparison between Models ...............................................................................134
6.3.2 Phase Diagram Analysis ......................................................................................136
6.3.3 Effects of Anisotropy ...........................................................................................138
6.3.4 Effects of High Diffusivity Pathways ..................................................................143
6.3.5 Effects of Temperature ........................................................................................147
6.3.6 Effects of Solute Atomic Radius and Molar Volume ..........................................149
6.3.7 Effects of Bi Concentration .................................................................................151
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6.4 Correlations between Diffusivity in Coarse-Grained and Polycrystalline Sn .................154
6.5 Concluding Remarks ........................................................................................................158
Chapter 7 Conclusions and Recommendations ............................................................................162
References ....................................................................................................................................166
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Published References
This thesis is based on the following refereed publications:
A. Delhaise, D. Perovic, and P. Snugovsky, “The Effects of Aging on the Microstructure and
Mechanical Properties of Bi-containing Sn-rich Alloys,” Journal of Surface Mount Technology,
30, 2, 2017
Chapter 3
A. Delhaise and D. Perovic, “Study of the Solid-State Diffusion of Bi in Polycrystalline Sn
Using Electron Probe Microanalysis,” Journal of Electronic Materials, 47, 3 (2018)
Chapter 2, Chapter 4 & Chapter 5
A. Delhaise, Z. Chen, D. Perovic, “Solid-State Diffusion of Bi in Sn – Effects of β-Sn Grain
Orientation,” Journal of Electronic Materials (2018) https://doi.org/10.1007/s11664-018-6621-
y
Chapter 2, Chapter 4 & Chapter 6
A. Delhaise, Z. Chen, D. Perovic, “Solid-State Diffusion of Bi in Sn – Effects of Anisotropy,
Temperature, and High Diffusivity Pathways,” JOM (in press, to be published January 2019)
Chapter 2 & Chapter 6
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List of Figures
Chapter 2
Figure 2.1: Sn-Pb binary phase diagram [11]. The range of near-eutectic compositions (38-40
wt% Pb) is highlighted. ................................................................................................................... 4
Figure 2.2: Optical micrographs of eutectic (a) and hypereutectic Sn-Pb alloy with proeutectic Pb
phase [12]. The dark phase is Pb; the light phase is Sn. ................................................................. 5
Figure 2.3: Body-centered tetragonal unit cell and Sn lattice parameters [15] .............................. 6
Figure 2.4: Pb-free solder phase diagrams. Sn-Ag [19] (a); Sn-Cu [20] (b); Sn-Ag-Cu [21] (c). .. 7
Figure 2.5: Some typical microstructures observed in SAC solder joints [23]. Ball grid array
(BGA) joint showing dendritic Sn structure and primary Ag3Sn intermetallics (a), interdendritic
region featuring two binary eutectics (b). ....................................................................................... 8
Figure 2.6: Sn-Bi binary phase diagrams: NIST [37], based on Lee et al. [25] (a); Okamoto [38],
based on Vizdal [26] (b); Braga [27] (c), indicating variance of reported phase equilibria in Sn-Bi
system. .......................................................................................................................................... 12
Figure 2.7: Evolution of microstructure and hardness of Bi-containing alloys after aging. Violet
(Sn-2.25Ag-0.5Cu-6Bi), as-cast (a) and aged at 125°C for 24 hours (b) microstructure [7, 40];
hardness [6] of the seven selected Bi-containing alloys in the Celestica-UofT low melt project
after aging at 100°C (c). ................................................................................................................ 14
Figure 2.8: Further effects of aging on the microstructure of Violet. Electron Backscatter
Diffraction (EBSD) maps [7] (Euler coloring) of the as-cast alloy (a) and alloy aged at 125°C for
50h (b). Microstructure of Violet after aging at 70°C for 300 hours (c), showing Ostwald
ripening [7, 40].............................................................................................................................. 15
Figure 2.9: Effects of heat treatment on the mechanical properties and reliability of Violet.
Nanoindentation results [7, 40] showing improvement in creep resistance after heat treating the
alloy at 120°C (a). ATC data [43] showing improvement in the Weibull characteristic life θ after
heat treatment at 125°C (b). .......................................................................................................... 16
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Figure 2.10: Proposed flow chart [32] illustrating the practical implementation of the heat
treatment during the life cycle of an electronic device. ................................................................ 17
Figure 2.11: A larger substitutional solute will exert a compressive stress on the surrounding
lattice. Sn (white) and Bi (orange) atoms are to scale. Schematic is not representative of atom
positions in Sn lattice. Adapted from Callister [12]. .................................................................... 18
Figure 2.12: Hypothetical location of Bi atoms at the tensile strain portion of a dislocation in the
Sn matrix. Sn (white) and Bi (orange) atoms are to scale. Schematic is not representative of atom
positions in Sn lattice. Adapted from Callister [12]. .................................................................... 20
Figure 2.13: Effect of particle size on the cutting and bypassing stresses, showing some optimal
particle size R* will yield the highest required stress for successful proliferation of dislocations
through the particle-strengthened alloy [48]. ................................................................................ 21
Figure 2.14: Ostwald ripening of Bi precipitates, showing large particles surrounded by a region
devoid of Bi [28]. .......................................................................................................................... 22
Figure 2.15: Effect of misorientation between two grains on the grain boundary energy γ and on
the transition from a LAGB to a HAGB [50]. .............................................................................. 23
Figure 2.16: EBSD maps showing grain structure evolution of Violet after aging. As cast (a);
after aging at 120°C for 24h (b), 50h (c), and 300h (d). ............................................................... 26
Figure 2.17: Schematic of thin and thick layer geometry [56]. .................................................... 29
Figure 2.18: Schematic of a material featuring several high diffusivity paths, showing diffusion
fringes [56]. ................................................................................................................................... 34
Figure 2.19: Diagram showing the three diffusion regimes in a polycrystalline sample [56]. ..... 35
Figure 2.20: HREM image [56] of GB region in Au, indicating δ is on the order of interatomic
distance (0.5nm). ........................................................................................................................... 36
Figure 2.24: Schematic of sputtering process [76] ....................................................................... 42
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Figure 2.21: Kirkendall voiding in Sn-Pb (a) and Pb-free (b) solder joints during the formation of
Cu3Sn IMC [68]. ........................................................................................................................... 44
Figure 2.23: Plot of diffusivity (log scale) against solute atomic radius for impurity diffusivity in
Sn at 125°C, for orientations parallel and perpendicular to the ‘c’ axis, indicating species with a
higher atomic radius have lower diffusivities. Sb appears to be an outlier. Assuming the
diffusivity of Bi follows this trend with respect to atomic radius (156pm), an rough estimate of
the range of diffusivities of Bi in Sn is annotated on the plot using trendlines. ........................... 46
Figure 2.25: Effect of beam voltage and atomic number on the dimensions of the interaction
volume [79]. .................................................................................................................................. 50
Figure 2.26: Wavelength Dispersive X-Ray Spectrometry schematic [79] (a) and superior peak
resolution compared with EDS [81] (b). ....................................................................................... 52
Figure 2.27: Formation of Kikuchi lines [82] (a); Geometrical relationships between Kikuchi
lines [83] (b); Example EBSP image of cadmium. The zone axis with the three intersecting
Kikuchi bands is the <001> zone axis, which contains sixfold symmetry (hexagonal crystal
structure) [82] (c). ......................................................................................................................... 53
Figure 2.28: The stereographic projection of Sn with the <0 0 1> primary axis. The symmetry of
the tetragonal crystal structure allows the stereographic projection to be collapsed into a
spherical triangle bounded by the <0 0 1>, <0 1 0> and <1 1 0> poles. ...................................... 54
Figure 2.29: Inverse pole figures showing the texture of a β-Sn sample in each of the X (a), Y
(b), and Z (c) directions. Coloring is based on the RGB scale in (d). X, Y, and Z directions are
shown on a cylindrical sample with the circular face corresponding with the sample surface (e).
....................................................................................................................................................... 55
Figure 2.30: EBSD maps showing the texture of a β-Sn sample. IPF maps in the X (a), Y (b), and
Z (c) directions; Euler map (d)...................................................................................................... 55
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Chapter 3
Figure 3.1: Sn-10Bi aged at 100°C for 3 days, showing ‘decoration’ of GBs by Bi .................... 58
Figure 3.2: Rockwell Hardness test method [87].......................................................................... 61
Figure 3.3: As cast microstructure of SAC 305 (a), Sn-0.7Cu-1Bi (b), Sn-5Bi (c), and Sn-0.7Cu-
5Bi, with all pertinent phases labelled (d). ................................................................................... 62
Figure 3.4: Hardness Data for as-cast alloys ................................................................................ 64
Figure 3.5: Evolution of microstructure of Sn-5Bi after aging at 125°C. As-cast (a); 1 day (b); 7
days (c); 14 days (d). ..................................................................................................................... 66
Figure 3.6 (previous page): Microstructure of samples aged at elevated temperature. Sn-0.7Cu-
5Bi aged at 100°C for 3d (a), 100°C for 14d (b), 125°C for 14d (c); SAC 305 aged at 125°C for
3d (d) and 14d (e); Sn-0.7Cu-1Bi aged at 100°C for 1d (f) and 7d (g). ....................................... 70
Figure 3.7: Evolution of alloy hardness after elevated temperature aging at 70°C. ..................... 70
Figure 3.8: Evolution of alloy hardness after elevated temperature aging at 100°C. ................... 70
Figure 3.9: Evolution of alloy hardness after elevated temperature aging at 125°C. The 14 day
timepoint for Sn-3Bi is missing due to testing equipment malfunction. ...................................... 71
Figure 3.10: Evolution of microstructure of Sn-5Bi after RT aging. As cast (a); 28 days (b); 63
days (c); 168 days (d). ................................................................................................................... 74
Figure 3.11: Microstructure of samples aged at room temperature. Sn-0.7Cu-5Bi as-cast (a), aged
after 10d (b) and 168d (c); SAC 305 as-cast (d), aged after 10d (e) and 168d (f); Sn-0.7Cu-1Bi
as-cast (g), aged after 10d (h) and 168d (i). .................................................................................. 75
Figure 3.12: Evolution of alloy hardness after room temperature aging. ..................................... 76
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Chapter 4
Figure 4.1: Failure of mixed metallurgy solder joint (Sn-58Bi paste and SAC 305 BGA ball)
likely caused by segregation of Bi to the package-ball interface [90] .......................................... 80
Figure 4.2: Sample preparation of coarse-grained Sn-Bi diffusion couples. Casting crucible/mold
(a); furnace door propped open (b); cast samples inserted in epoxy mount with through-holes (c);
mount and samples after Bi sputtering (d). ................................................................................... 86
Figure 4.3: Post-sputtering of coarse-grained and monocrystalline diffusion samples. Samples
placed in graphite crucible for annealing (a); crucible with samples loaded in hollowed fire brick
(b). ................................................................................................................................................. 88
Figure 4.4: Polycrystalline diffusion triple preparation. Schematic of Sn-Bi-Sn diffusion triple
(a); fixture with hex bolts, along with graphite collar, preheating on hot plate (b); side view of
diffusion triple, with Bi pieces on either side of Sn piece (c). ...................................................... 89
Figure 4.5: Cryo-storage of coarse-grained samples in custom wire mesh basket ....................... 90
Figure 4.6: Schematic of how the 6Dt criterion for exclusion of sample extremities is applied in
SEM analysis of diffusion samples. .............................................................................................. 92
Figure 4.7: Example of superposition of Sn IPF on stereographic projection to determine grain
orientation. IPF Y map of sample slow-cooled using method described above (a); superposition,
indicating sample is oriented close to <1 1 1> (specifically <7 7 6>) in the ‘Y’ direction
(labelled) (b), IPF legend (c). ........................................................................................................ 93
Figure 4.8: Example of diffusion sample showing EPMA line scan (50µm long) locations. ...... 94
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Chapter 5
Figure 5.1: Electron Backscatter Diffraction of pure Sn. Quenched (a-b), air-cooled (c-d), aged
(120°C for 7 days) after quenching (e-f)..................................................................................... 102
Figure 5.2: Representative microstructures (800x magnification) of diffusion zone at different
temperatures. Bi precipitates are spherical at 85°C and 100°C (a,b); mixture of spherical and
lamellar at 115°C (c), lamellar at 125°C (c). .............................................................................. 105
Figure 5.3: Sample EBSD data from diffusion samples, demonstrating increase in average grain
size with temperature. 85oC Inverse Pole Figure (IPF) Y coloring map (a); 100oC IPF Y coloring
map (b); 125oC IPF Y coloring map (c); IPF Y for 85°C map (d); IPF Y for 100°C map (e); IPF
Y for 125°C map (f); IPF color coding (g). ................................................................................ 106
Figure 5.4: Sample raw Bi concentration profiles from 100oC samples..................................... 107
Figure 5.5: Averaged concentration profiles for 100oC. Entire, unsmoothed profile, with
additional Sn-rich and Bi-rich points indicated with red ovals (a). Sn-rich side, unsmoothed
profile (b). Entire, smoothed profile (c). Sn-rich side, smoothed profile (d). ............................. 110
Figure 5.6: MatLab-generated plots used to determine interdiffusivity for the 100°C data.
Interdiffusivity plot, showing Sauer-Friese and Hall results (a); Hall plot of u vs. λ, showing the
linear fit used to determine Hall result (b). ................................................................................. 111
Figure 5.7: Comparison between Experimental Data and Simulation, for 125°C Polycrystalline
Diffusion Data ............................................................................................................................. 113
Figure 5.8: Arrhenius plots for impurity diffusivity of Bi in polycrystalline Sn. Best fit lines from
which intrinsic parameters were derived is shown. .................................................................... 114
Figure 5.9: Interdiffusivity plot for 100°C. The black dotted line represents the diffusivity
calculated by the forward simulation. ......................................................................................... 116
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Chapter 6
Figure 6.1: Example EBSD maps of coarse-grained Sn (IPF Y coloring), along with
corresponding IPFs. A wide range of structures are observed, featuring LAGBs and interlaced
grains. .......................................................................................................................................... 122
Figure 6.2: As-sputtered microstructure of Sn-Bi diffusion couples. Bi film sputtered onto a Si
wafer, with thickness oriented parallel to the electron beam (a); and cross-sectioned couple,
revealing moderately uniform columnar structure (b). Abnormal ‘whiskerlike’ growths on
surface of film (c) were sometimes captured in cross-section (d). EBSD IPF Y coloring map (e)
and IPF (f) of as-sputtered Sn sample indicates sputtering did not alter the Sn grain structure. 123
Figure 6.3: Diffusion microstructures at 125°C, indicating the presence of Bi induces
recrystallization of Sn. It is also indicated that the <7 7 6> orientation (a-b) may yield higher
diffusivity than the <1 3 4> orientation (c-d). Corresponding IPFs for the <7 7 6> (e) and <1 3 4>
(f) orientations, as well as the color legend (g). All EBSD data shown is in the ‘Y’ direction. . 125
Figure 6.4: SEM images indicating growth of h after annealing at 115°C. As-sputtered film,
average h = 0.70µm (a). On average, h increased (b-d), 1.06µm (c), 1.14µm (d). Yellow arrows
indicate h; red arrows indicate Kirkendall voids. ....................................................................... 126
Figure 6.5: SEM analysis comparing diffusion microstructures from different temperatures.
115°C (a), 100°C (b), 85°C (c), and 25°C (d). Lower magnification image at 25°C (e) with
corresponding EBSD ‘Y’ map, revealing recrystallization of Sn in the absence of Bi precipitates
(f)................................................................................................................................................. 127
Figure 6.6: Example EBSD ‘Y’ maps of annealed ‘monocrystalline’ Sn samples. The maps are
oriented with the Bi deposition on the top, similar to Figure 6.3, Figure 6.4, and Figure 6.5. ... 129
Figure 6.7 (previous page): Orientation maps for coarse-grained diffusion samples, based on the
<0 0 1> stereographic projection for β-Sn. Poles corresponding to sample orientations are
indicated in red. 125°C (a), 115°C (b), 100°C (c), 85°C (d), 25°C (e). ...................................... 131
Figure 6.8: Example of concentration profile (a) of Bi diffusion in coarse-grained Sn. SEM
imaging (b) indicates that the ‘spikes’ in the profile data (for example at 5 and 7.5 µm) may be
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the result of the electron beam impinging an area of the sample rich in Bi, such as a precipitate.
..................................................................................................................................................... 132
Figure 6.9: EBSD ‘Y’ map of 25°C diffusion couple, showing recrystallization of Sn close to the
diffusion interface. Portions of each EPMA line scan location are superimposed, indicating the
maximum depth of recrystallization. .......................................................................................... 132
Figure 6.10: Relationship between diffusivities calculated using the slab source (x axis) and
forward simulation (y axis) models ............................................................................................ 134
Figure 6.11: Examples of diffusivity calculation using slab source model (a&b) and simulation
model (c&d). Figures a & c show all data in the profile; figures b & d show only the data below
the solid solubility of Bi in Sn at 25°C (1.1 at%) ....................................................................... 135
Figure 6.12: Example of calculation of remaining Bi-rich phase in deposited layer. Original BSE
image (a); cropped BSE image of sputtered layer, with all precipitates within the substrate
removed (b); precipitate isolation prior to area calculation (9.9µm2, or 23.1% of initial layer
composition) (c). ......................................................................................................................... 137
Figure 6.13: Deviation of the calculated remaining Bi in the film from the experimental results.
The Lee diagram appears to be the best representation of the experimental diffusivity data. .... 137
Figure 6.14: Scatter plots showing relationship between sample orientation <u v w> relative to
the ‘c’ axis and the diffusion coefficient (data from all samples plotted). 115°C (a); 100°C (b);
85°C (c); 25°C (d). ...................................................................................................................... 139
Figure 6.15: Scatter plot containing all diffusivity data from 125°C, showing a large degree in
scattering (a). EBSD ‘Y’ maps (b&c) are matched with their corresponding diffusivities via color
coding on the scatter plot. ........................................................................................................... 140
Figure 6.16: Orientation data with substantial LAGBs removed, curve fitted to the anisotropy
equation, for 125°C (a), 115°C (b), 100°C (c), 85°C (d) and 25°C (e). For the 125°C plot, the
two data points at 90° seen in Figure 6.14a were from <1 1 0> and <1 2 0> orientations and are
considered as one orientation here. ............................................................................................. 141
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Figure 6.17: High diffusivity pathways in Sn-Bi diffusion samples. SEM images of corner (a&b)
indicating edge effects, and of region in the bulk (c). EBSD ‘Y’ mapping (d) and IPF/WinWulff
analysis (e) suggests the fringe in (c) may be caused by a grain boundary. ............................... 144
Figure 6.18: Example of the effects of grain boundaries on diffusion of Bi in Sn. Composite
SEM image (a) of a location featuring several ‘fringes’ of Bi precipitation and Sn
recrystallization; EBSD ‘Y’ map (b) of the same location indicating the fringes were likely
caused by grain boundaries. ........................................................................................................ 145
Figure 6.19: SEM image (a) and EBSD ‘Y’ map (b) of location with grain boundary diffusion,
indicating the locations of EPMA line scans .............................................................................. 146
Figure 6.20: The effects of high diffusivity pathways on the concentration profile. Profile 1 in a
region with predominantly bulk diffusion (a) looks considerably different from Profile 4, which
was influenced by the HAGB between the two grains (b).......................................................... 147
Figure 6.21: Arrhenius plot for the estimated diffusivities of Bi in Sn for the orientations parallel
and perpendicular to the ‘c’ axis. ................................................................................................ 148
Figure 6.22: Plot of solute atomic radius against impurity diffusivity in Sn, for both the ‘a’ / ‘b’
(D ⊥) and ‘c’ (D||) axes, identical to Figure 2.23, with the addition of Bi from this study. The
diffusivities for Bi in Sn show good agreement with the literature data, which suggests that as r
increases, the diffusivity decreases (other than Sb). ................................................................... 150
Figure 6.24: Plot of impurity diffusivity data perpendicular to the ‘c’ axis (normalized to Sn self-
diffusivity) with respect to corrected molar volume (normalized to the Sn molar volume)
calculated using the Miedema-Niessen model. Most of the data falls on a straight line, however
the data from Bi from this study is a noticeable outlier. ............................................................. 151
Figure 6.25: Forward Simulation results assuming a concentration-dependent diffusion
coefficient. Arrhenius plot of DC, 1 values (for Bi at% → 0) (a); D(C) plot (b). The blue data
points are from the Sauer-Friese plot. ......................................................................................... 153
Figure 6.26: Arrhenius plot comparing lattice impurity diffusivity (DC, 1), effective or
polycrystalline impurity diffusivity (Deff) and triple product (P) for Bi in Sn. ......................... 157
xxi
List of Tables
Chapter 2
Table 2.1: First and Second Generation Pb-free Solder Alloys .................................................... 10
Table 2.2: Select physical properties of Pb-free solder alloying elements pertaining to solid
solubility ....................................................................................................................................... 19
Table 2.3: Density and Lattice Data for β-Sn and Bi phases ........................................................ 25
Table 2.4: Impurity Diffusivity data in Sn-X systems (125°C diffusivity estimated using
Arrhenius parameters Do and QA) ................................................................................................. 45
Table 2.5: Miedema-Niessen Analysis of Several Diffusing Species in Sn, with V0 =
16.30 cm3
mol⁄ ............................................................................................................................ 48
Chapter 3
Table 3.1: Metallographic Sample Preparation Procedure for Aging Samples ............................ 60
Table 3.2: Metallurgical Explanation for Aging Performance of Alloys ..................................... 71
Chapter 4
Table 4.1: Calculated Sputter Yields for Select Metals (500 eV Ar ions) [92] ............................ 81
Table 4.2: Number of profiles (samples) for each condition ........................................................ 82
Table 4.3: Estimated Ratio of Diffusion Temperatures to Solidus [25] Temperature (at solubility
limit) .............................................................................................................................................. 83
Table 4.4: Metallographic Sample Preparation Procedure for Diffusion Interfaces .................... 85
xxii
Table 4.5: Metallographic Sample Preparation Procedure for Cross-Sectioning of Diffusion
Couples ......................................................................................................................................... 90
Table 4.6: First Principles Simulation Parameters ........................................................................ 97
Table 4.7: Bi Self-Diffusivity Parameters [97] ............................................................................. 98
Table 4.8: Solubility of Bi in Sn in wt% (at%) based on the Lee [25], Vizdal [26], and Braga [27]
Sn-Bi binary phase diagrams ........................................................................................................ 98
Chapter 5
Table 5.1: Grain Size Statistics (diameters) for Sn samples ....................................................... 101
Table 5.2: Summary of Grain Size Data from Polycrystalline Diffusion Samples .................... 106
Table 5.3: Calculated Interaction Volume Depth (Kanaya-Okayama Range) for Sn and Bi ..... 108
Table 5.4: Experimentally Calculated Impurity Diffusivities of Bi in Polycrystalline Sn using the
Hall Inverse Method ................................................................................................................... 111
Table 5.5: Experimentally Calculated Diffusivities of Bi in Polycrystalline Sn using Forward
Simulation ................................................................................................................................... 112
Table 5.6: Arrhenius parameters for impurity diffusivity of Bi in polycrystalline Sn ............... 115
Chapter 6
Table 6.1: Example Orientation Data from Purchased Monocrystalline Sn samples ................. 129
Table 6.2: Sample EPMA Data of 25°C Diffusion Couple ........................................................ 133
Table 6.3: Wellness of fit of experimental data to anisotropy equation ..................................... 139
xxiii
Table 6.4: Calculated diffusivities of Bi in Sn parallel and perpendicular to the ‘c’ axis, using
experimental diffusion data......................................................................................................... 142
Table 6.5: Summary of Diffusivities in High Angle Grain Boundary Region ........................... 146
Table 6.6: Arrhenius data for the diffusivity of Bi in Sn parallel and perpendicular to the ‘c’ axis
..................................................................................................................................................... 147
Table 6.7: Calculated average D(C)l values for Bi in Sn, compared to Dl values ...................... 154
Table 6.8: Arrhenius data for DC,1 and Dl for Bi in Sn ............................................................... 154
Table 6.9: Comparison of lattice and effective impurity diffusivities of Bi in Sn ...................... 155
Table 6.10: Harrison Kinetics Parameters .................................................................................. 155
Table 6.11: Arrhenius data for P, compared with DC,1 and Deff for Bi in Sn ............................. 157
xxiv
List of Symbols, Acronyms, and Variables
Ag – silver
Al – aluminum
Ar – argon
Au – gold
Bi – bismuth
Cu – copper
Ge – germanium
In – indium
Ni – nickel
Pb – lead
Sb – antimony
Si – silicon
Sn – tin
Zn – zinc
AES – Auger Electron Spectroscopy
ATC – accelerated thermal cycling
A&D – aerospace & defense
BCT – body-centered tetragonal
BSE – backscattered electrons
CSL – coincident site lattice
xxv
CTE – coefficient of thermal expansion
DSC – differential scanning calorimetry
EBSD – Electron Backscatter Diffraction
EBSP – Electron Backscatter Pattern
EDM – Electric Discharge Machining
EDS – Energy Dispersive X-Ray Spectrometry
EPMA – Electron Probe Microanalyser / Microanalysis
GB – grain boundary
HAGB – high angle grain boundary
HREM – high resolution electron microscopy
IMC – Intermetallic compound
IPF – Inverse Pole Figure
JCAA/JG-PP – Joint Council on Aging Aircraft / Joint Group on Pollution Prevention
LAGB – low angle grain boundary
LD – linear density
NCMS – National Center for Manufacturing Services
PCB – printed circuit board
PE – primary electrons
PSN – Particle Stimulated Nucleation
ReMAP – Refined Manufacturing Acceleration Process
xxvi
RoHS – Restriction of Hazardous Substances
RSS – residual sum of squares
SAC – Sn-Ag-Cu or Tin-Silver-Copper
SE – secondary electrons
SEM – Scanning Electron Microscopy
SIMS – Secondary Ion Mass Spectrometry
SSS – solid solution strengthening
TEM – Transmission Electron Microscopy
Tg – glass transition temperature
Th – Homologous temperature
WDS – Wavelength Dispersive X-Ray Spectrometry
WEEE – Waste Electronic and Electrical Equipment
A – atomic mass
c – solubility limit
C – concentration
𝐶𝑜 – initial concentration
𝐶𝐿 – concentration at left hand side of diffusion couple
𝐶𝑅 – concentration at right hand side of diffusion couple
d – grain size
𝑑𝑎 – interplanar spacing
xxvii
D – diffusion coefficient / diffusivity
𝐷𝐶,1 – concentration dependent diffusivity, where solute content approaches zero
𝐷𝐶,1 – concentration dependent diffusivity, where solute content approaches the solubility limit
𝐷𝑎 – diffusivity oriented with ‘a’ axis
𝐷𝑏 – diffusivity oriented with ‘b’ axis
𝐷𝑐 – diffusivity oriented with ‘c’ axis
𝐷𝑑 – dislocation diffusivity
𝐷𝑒𝑓𝑓 – effective (polycrystalline) diffusivity
𝐷𝑔𝑏 – grain boundary diffusivity
𝐷𝑔𝑏,𝑜 – Arrhenius pre-exponential / frequency factor for grain boundary diffusivity
𝐷𝑙 – lattice diffusivity
𝐷𝑠 – surface diffusivity
𝐷𝑜 – Arrhenius pre-exponential / frequency factor for diffusivity
𝐷𝜃 – diffusivity at angle θ
𝐷|| – diffusivity parallel to unique axis
𝐷⊥ – diffusivity perpendicular to unique axis
𝐷|| 𝐷⊥
⁄ – anisotropy ratio for diffusivity
�̃� – interdiffusion coefficient
𝐸𝑜 – electron microscope beam energy / voltage
erf – error function
xxviii
f – volume fraction of grain boundaries in polycrystalline material
h – deposited / thin film thickness
j – slope of grain boundary ‘tail’ in Harrison ‘Type B’ analysis
J – diffusion flux
K – proportionality constant relating solubility to segregation factor (Seah)
m – mass
M – number of diffusing particles per unit area
𝑁𝑖 – molar / atomic concentration of species i
𝑛𝑊𝑆 – electron density at the boundary of the Wigner-Seitz cell
P – triple product for diffusivity in a polycrystalline material
𝑃𝑜 – Arrhenius pre-exponential / frequency factor for triple product
𝑃0 – constant in Miedema-Niessen model, equivalent to 1.5
q – constant for geometry of grain boundary (q=1 for parallel GBs; q=3 for cubic grains)
𝑄𝐴,𝑖 – Arrhenius activation energy (‘i' may represent processes such as grain boundary
diffusivity, segregation, etc)
r – atomic radius
R – precipitate/particle radius
𝑅𝑖 – ideal gas constant
𝑅𝐾𝑂 – Kanaya-Okayama range
𝑅∗ – critical precipitate/particle radius
s – segregation factor
xxix
𝑠𝑚𝑎𝑥 – maximum segregation factor
𝑠𝑜 – Arrhenius pre-exponential / frequency factor for segregation factor
t - time
T - temperature
𝑇𝑔 - glass transition temperature
𝑇ℎ - homologous temperature
𝑇𝑚 - melting temperature
𝑇𝑠𝑜𝑙 – solvus temperature
𝑇𝑠𝑜𝑙𝑖𝑑 – solidus temperature
u – Hall (inverse method) parameter
v – moving velocity of diffusion boundary / interface
𝑉𝑖𝑚𝑝 – corrected molar volume of solute after alloying (Miedema-Niessen model)
𝑉𝑖𝑚𝑝0 – molar volume of pure solute prior to alloying (Miedema-Niessen model)
∆𝑉𝑖𝑚𝑝 – change in molar volume of solute after alloying (Miedema-Niessen model)
𝑉0 – molar volume of solvent (Miedema-Niessen model)
x – (diffusion) distance
𝑥𝑀 – Matano plane
�⃗� – one of two vectors in cosine similarity equation
�⃗� – one of two vectors in cosine similarity equation
𝑌𝑆𝐹 – concentration ratio (Sauer-Friese method)
xxx
Z – atomic number
15HRX – Rockwell Superfical Hardness 15X scale
α – primary phase in binary eutectic system
𝛼𝐻 – Harrison ‘Type B’ kinetics parameter
𝛼𝐻,𝑚𝑎𝑥 – maximum Harrison ‘Type B’ kinetics parameter
α+β – multiphase region composed of α and β
β – secondary phase in binary eutectic system
𝛽𝐻 – Harrison ‘Type B’ kinetics parameter
γ – interfacial energy
δ – grain boundary width
𝛿𝑜 – Arrhenius pre-exponential / frequency factor for grain boundary width
θ – misorientation angle between grains
𝜃𝐵 – Bragg angle
ξ – Boltzmann’s similarity variable
𝜆𝐻 – Hall (inverse method) parameter
ρ – density
𝜙 – electronegativity
[a b c] – crystallographic direction vector for β-Sn unit cell with indices a=b=0.583nm;
c=0.318nm
[u v w] – crystallographic direction vector with indices u, v, and w
1
Chapter 1 Introduction
1.1 Overview
For many years, tin-lead (Sn-Pb) solders such 63Sn-37Pb (wt%) were widely employed as
joining materials in electronic devices. These alloys have historically performed very well in
service – demonstrating excellent mechanical and wetting properties as well as good
performance in reliability tests such as thermal cycling, vibration, and drop testing. However, in
the early 2000s, due to ongoing concerns with the environmentally hazardous production and
end-of-life disposal processes of electronic devices, lead –containing solders were eliminated
from use. In particular, in the European Union, legislation such as the Restriction of Hazardous
Substances (RoHS) and Waste Electronic and Electrical Equipment (WEEE) were enacted which
stipulated that lead be removed from solder materials. This brought about a need for the use of
Pb-free solders as replacement alloys. One major family of alloys that is commonly used today is
tin-silver-copper (Sn-Ag-Cu, or SAC), in particular SAC 305 (Sn-3.0Ag-0.5Cu).
Unfortunately, SAC has shown to be an ill-suited replacement alloy to Sn-Pb when employed in
the same applications. One problem which is the main focus of this research project is the
accelerated degradation of physical properties over time. In numerous studies, SAC alloys aged
at both room and elevated temperatures demonstrate reduced mechanical properties such as yield
strength and creep resistance [1, 2]. Typical operating conditions represent high homologous
temperatures 𝑇ℎ, where 25°C corresponds to a 𝑇ℎ of 0.61 and 100°C to a 𝑇ℎ of 0.76, based on the
alloy’s melting point 𝑇𝑚 of 217°C and where diffusion tends to be very active [3]. As a result,
the microstructure changes drastically where interphase spacing and secondary phase (e.g.
intermetallic compound IMC coarsening will increase. This leads to an increased number of
failures and consequently, replacement of parts. While no ideal drop-in replacement for current
lead-free solders has been found, bismuth (Bi) has shown some interesting and attractive
properties when added in small amounts to lead-free solders. Bismuth-containing alloys have
demonstrated superior performance in a wide range of accelerated reliability tests such as
thermal cycling and vibration, and the mechanical properties are highly stable over long
durations of time, most notably at harsh temperatures [4, 5, 6, 7].
2
1.2 Goals and Scope of Research
While it has been shown that Bi provides an improvement to the properties and performance of
Pb-free solder alloys, it is not known precisely why and how. It is suspected, from examination
of the Sn-Bi phase diagram, that strengthening mechanisms such as solid solution strengthening
and precipitation/age hardening are operative with the inclusion of Bi. Examination of the
microstructures before and after aging gives a strong indication that these mechanisms are driven
by the solid-state diffusion of Bi in the Sn matrix, in which a homogenized microstructure is
produced. It is therefore important to study the diffusion properties in the Sn-Bi system, to fully
understand and characterize the strengthening mechanisms in Bi-containing Pb-free solder
alloys. The goal of this thesis, therefore, is to characterize the effects of Sn grain orientation and
diffusion temperature on the solid-state diffusion of Bi in Sn, to allow for more accurate
predictions of the homogenization of Bi-containing solder alloys after an aging treatment.
After the compilation of background information from the literature (Chapter 2), an extensive
aging study is presented (Chapter 3), in which a wide range of Bi-containing alloys are aged at
one of many aging conditions. The evolution of the microstructure and resulting change in
mechanical properties is evaluated. The study of solid-state diffusion of Bi in Sn is investigated.
Experimental methods are given in Chapter 4, followed by the results from diffusion in
polycrystalline Sn (Chapter 5) and coarse-grained / monocrystalline Sn (Chapter 6). This
includes the characterization of the microstructure, diffusion profile analysis, and calculation of
diffusivity. In addition, the effects of temperature, high-diffusivity pathways such as grain
boundaries, and anisotropy are considered. In the latter chapter, correlations between the two
crystallinities are explored. Finally, a comprehensive set of conclusions and recommendations
for future studies is provided (Chapter 7).
This project is funded by and is in close collaboration with the Refined Manufacturing
Acceleration Process (ReMAP), a Canadian-based Business-led National Centre of Exellence
(BL-NCE) which strives to bring manufacturing innovations towards commercialization [8]. The
secondary goal of this research is to support ReMAP and other industry partners in the complete
characterization of the properties of Bi-containing solder alloys. Such a vast body of work would
accelerate the adoption of Bi-containing alloys in electronic devices, to improve reliability and
performance, particularly in high reliability applications.
3
Chapter 2 Background
2.1 History of Soldering
Soldering is defined as the connection of two or more materials using a lower melting material;
these materials are usually metals or alloys. Historically, the most commonly used alloy was the
eutectic or near-eutectic tin-lead (Sn-Pb) alloy. In the late 1990s, the toxicity of Pb emerged as a
significant environmental risk, and the Restriction of Hazardous Substances (RoHS) directive in
2002 sought to eliminate Pb from electrical and electronic devices. Tin-silver-copper (Sn-Ag-Cu,
or SAC) alloys such as SAC 305 (Sn-3.0Ag-0.5Cu) became widely used, however the high Ag
content was a detriment mainly to alloy reliability. Second generation Pb-free alloys such as
SAC 105 (Sn-1.0Ag-0.5Cu) reduced Ag content but presented new challenges. Currently, the
industry is investigating third generation Pb-free alloys containing additional alloying elements,
which may improve the microstructure, properties and reliability.
2.1.1 Tin Lead (Sn-Pb) Alloys
Lead-containing solders have been used for thousands of years in a range of applications, dating
back to the Bronze Age [9]. For soldering of interconnects, however, the eutectic Sn-37Pb alloy,
or near-eutectic compositions (with 38-40 wt% Pb), emerged as the most ideal materials [10]
This is highlighted in the schematic Sn-Pb binary phase diagram shown in Figure 2.1 [11]. Tin-
lead alloys are inexpensive (due to high Pb content), can wet substrates such as copper (Cu)
easily with only mildly active flux materials, form no brittle intermetallic (IMC) compounds
(other than with the substrate), and have a relatively low melting temperature (𝑇𝑚), resulting in
processing temperatures which many typical electronic components can withstand during
assembly [10].
4
Figure 2.1: Sn-Pb binary phase diagram [11]. The range of near-eutectic compositions (38-40
wt% Pb) is highlighted.
Tin-lead alloys solidify to form a eutectic or near-eutectic microstructure (Figure 2.2), with an
intermixing of Sn-rich and Pb-rich phases, each with limited solubility of the other element [10].
At the exact eutectic composition, solidification occurs at a single temperature, and redistribution
of the Sn and Pb components of the alloy occurs via atomic diffusion, producing a layered
lamellar ‘eutectic’ structure [12]. In an off-eutectic composition e.g. 60Sn-40Pb, solidification
occurs over a range of temperatures, termed the melting or pasty range. As this alloy is cooled
across the liquidus temperature, primary (i.e. proeutectic) β-phase, or Pb-rich solid, starts to
form, and co-exists with the liquid phase. As cooling proceeds, the β-phase grows, and upon
crossing the eutectic isotherm, the remaining liquid, possessing a eutectic composition, solidifies
into the eutectic structure [12]. The resulting microstructure is composed of primary Pb-rich β-
phase, and a binary eutectic phase composed of Sn-rich α-phase and Pb-rich β-phase. Tin-lead
solders are known to undergo drastic changes over time, with coarsening of the microstructure.
This is accompanied by a reduction in mechanical properties such as the elastic modulus and
yield strength [10].
While Sn-Pb alloys demonstrate several desirable qualities, Pb is a highly toxic metal. Of
significant concern is the effects of Pb on the environment, particularly as it pertains to end-of-
life disposal of electronic devices; this was addressed by the RoHS and Waste from Electrical
and Electronic Equipment (WEEE) environmental directives in 2002 [13]. The latter is focused
on sustainable disposal of electronic products, while the former targets the reduction of
5
hazardous materials in the manufacturing of electronic products. The original 2002 RoHS
directive implored all applicable industries to comply by July 1st, 2006; several products were
initially exempt. The updated RoHS 2 directive was published in 2011 and is more expansive,
covering all electrical/electronic devices, and goes into effect in July 2019. Under this directive,
the usage of Pb is limited to 1000 ppm or less, far below what is used in Sn-Pb solder and Pb-
containing surface finishes [13]. This has led the industry to seek out reliable Pb-free
replacement alloys.
Figure 2.2: Optical micrographs of eutectic (a) and hypereutectic Sn-Pb alloy with proeutectic
Pb phase [12]. The dark phase is Pb; the light phase is Sn.
2.1.2 Pb-Free Alloys – First Generation
With the removal of Pb, solder alloys are now predominantly Sn-based, with several additional
alloying elements such as Ag and Cu. It is therefore expected that Pb-free, Sn-rich alloys should,
at least loosely, behave comparably to Sn.
Beta (β)-Sn, the most prevalent allotrope present in solder joints, takes on a body-centered
tetragonal (BCT) crystal structure, with lattice parameters a=b=0.581nm and c=0.318nm [14]
(Figure 2.3). Tin is highly anisotropic with regards to several properties; notably, the orientation
with the highest stiffness, the ‘c’ axis or the <001> direction, also has the highest coefficient of
thermal expansion (CTE). Accordingly, large differences in microstress are possible between
adjacent grains, leading to localized plastic deformation [9]. The anisotropy of Sn is
accommodated by Pb (a softer, more isotropic metal) in Sn-Pb alloys. In addition, the self-
(a) (b)
6
diffusivity of Sn is also anisotropic, with the diffusivity along the ‘c’ axis slightly higher than
that along the ‘a’ axis [14]. Diffusivity is discussed in further detail in Section 2.3.
As observed in Sn-Pb alloys, ideal Pb-free alloys should be eutectic or near-eutectic to ensure
optimal manufacturability [3]. Early efforts to investigate Pb-free options led the industry to
adopt SAC alloys. While the melting point of these alloys (SAC 305 melts at 217°C) are
significantly higher than that for Sn-Pb, they were lower than several other Pb-free options such
as binary Sn-Ag and Sn-Cu alloys. Incorporation of both Ag and Cu is advantageous for the
following reasons:
• Ag: Lowers 𝑇𝑚 and improves wetting of the solder
• Cu: Reduces Cu dissolution from substrates during assembly processes
Figure 2.3: Body-centered tetragonal unit cell and Sn lattice parameters [15]
Zinc (Zn) and Bi were also considered as they lower the melting point. Zinc presents severe
concerns with corrosion, which results in poor wetting properties [3]. Inclusion of Bi presented
concern surrounding the possible formation of low-melting phases with Pb, which originates
from Sn-Pb surface finishes. For example, the ternary Sn-Pb-Bi eutectic melts at around 95°C [3,
16, 17]. This phase may result in solder joint opens and reduced reliability, particularly in high
temperature applications [18].
The binary phase diagrams for Sn-Ag and Sn-Cu, as well as the ternary Sn-Ag-Cu phase diagram
(Sn-rich end) are shown in Figure 2.4 [19, 20, 21]. The solubility of Ag and Cu in Sn is limited,
and intermetallic (IMC) phases will form (Ag3Sn and Cu6Sn5). Both SAC 305 (Sn-3.0Ag-0.5Cu)
and SAC 405 (Sn-4.0Ag-0.5Cu) were first considered as ‘1st generation’ Pb-free solder alloys.
7
While both showed comparable reliability, SAC 305 was preferred due to slightly lower Ag
content, which lowered the cost of the alloy and produced fewer primary Ag3Sn precipitates [22].
Figure 2.4: Pb-free solder phase diagrams. Sn-Ag [19] (a); Sn-Cu [20] (b); Sn-Ag-Cu [21] (c).
The microstructure of a SAC solder joint can demonstrate considerable variation, depending on
factors such as, but not limited to, the peak temperature, cooling rate, and proportions of alloying
elements. In general, the microstructure will consist of a globular (in cross-section) dendritic Sn
matrix along with secondary Ag3Sn and Cu6Sn5 IMCs [3]. The cooling rates present in typical
solder joints are significantly faster than ideal equilibrium cooling that can be predicted using the
phase diagram [9]. Nonequilibrium cooling results in undercooling – this results in the liquid
cooling below its liquidus temperature to overcome the free energy required to produce a solid-
liquid interface. Typical degrees of undercooling in SAC alloys range from 20-30°C [9].
Constitutional undercooling is prevalent in Pb-free solder joints; as a phase nucleates from the
melt, the melt is depleted of the constituents of that phase. As a result, the liquidus of the
resulting melt composition is shifted above the temperature of the melt, promoting the nucleation
of subsequent phases [23]. For example, the liquidus of Ag3Sn is often the highest of the three
possible solid phases (including β-Sn and Cu6Sn5) – this results in the initial nucleation of
primary, needle or plate-like Ag3Sn precipitates (Figure 2.5a). The depletion of Ag from the
melt results in constitutional undercooling of the β-Sn phase, which is subsequently formed [23].
The finely dispersed binary eutectics Ag3Sn-Sn and Cu6Sn5-Sn will then form last in the
interdendritic spaces (Figure 2.5b); the ternary eutectic may also form [3]. Primary Ag3Sn
typically forms if the Ag content in the alloy is high, if the cooling rate is slow (approaching
(a) (b) (c)
8
equilibrium cooling), and if the peak temperature is high [23]. If the Ag content is low, cooling
rate is fast or the amount of heating is minimal, β-Sn can form as a primary phase, and Ag3Sn
only forms as a binary or ternary eutectic [23]. In cases where the Cu content is high (e.g. Sn-
4.7Ag-1.7Cu, or via Cu dissolution from the substrate), Cu6Sn5 can form as the primary phase
[9].
At the interface between solder and a Cu substrate, a Cu6Sn5 IMC layer is present; this is the
result of a diffusion reaction between Sn and Cu prior to the solidification of the joint. Different
metallizations of the substrates (e.g. Ni) can alter the chemistry and morphology of the interfacial
IMCs.
Figure 2.5: Some typical microstructures observed in SAC solder joints [23]. Ball grid array
(BGA) joint showing dendritic Sn structure and primary Ag3Sn intermetallics (a), interdendritic
region featuring two binary eutectics (b).
Tin-silver-copper is not an ideal family of solder alloys to replace Sn-Pb. Several disadvantages
of this alloy are listed below:
• Higher melting temperature - This causes processing and assembly temperatures to
increase, which drives up energy expenditure and subsequently cost of manufacturing.
Higher temperatures necessitate the usage of higher 𝑇𝑔 (glass transition temperature)
board materials in assembly. These are more susceptible to failure modes such as pad
cratering. Temperature-sensitive parts are also more prone to thermal damage.
• High material cost – The inclusion of Ag increases the cost of the alloy.
(a) (b)
9
• Inferior manufacturability – Tin-silver-copper alloys have shown to wet substrates
such as Cu less effectively than Sn-Pb. For example, in a study comparing Sn-Pb and
various SAC solders, the wetting time was roughly twice as high for SAC 305 than
eutectic Sn-Pb [24]. Wetting angles have also consistently been reported to be larger for
SAC compared to Sn-Pb, indicating poorer wetting.
• Reduced toughness – IMCs such as Ag3Sn and Cu6Sn5 contribute to higher strength and
elastic modulus [3]. While smaller IMCs are effective strengtheners, primary IMC phases
are very large and may serve as stress concentrators in the alloy. For example, Ag3Sn
needles may have a deleterious effect on the drop/shock reliability [3].
• Longer Sn whiskers – Whiskers are highly conductive, thin, single crystal filaments
which can form in many metallic systems. In Sn and Sn-rich alloys, they often grow
because of stress relaxation, and can form bridges between adjacent solder joints, causing
short circuit failures. Whiskers were present in Sn-Pb, however Pb limited their growth,
so they were not a significant reliability concern.
• Microstructure and Property Degradation – Tin-silver-copper alloys, like Sn-Pb,
demonstrate reduced strength after aging – this infers that the reliability of the alloy will
also be degraded over time. This is caused by coarsening of the microstructure – Sn grain
size increases resulting in less grain boundary (GB) strengthening, and IMCs also
increase in size, rendering them less effective as dispersion strengtheners (see Section
2.2).
2.1.3 Pb-Free Alloys - Second Generation
Reducing the Ag content in the alloy became the primary focus during the development of ‘2nd
generation’ Pb-free solder alloys. Less Ag would lower the cost of the alloy, but more
importantly reduce the amount of primary Ag3Sn, which contributes to poor drop/shock
performance, increases the flow stress, and makes the alloy less compliant [18, 22].
Unfortunately, reducing Ag increases the melting temperature (one of the reasons binary Sn-Cu
alloys were quickly ruled out); accordingly high glass transition temperature (𝑇𝑔) boards are a
continued necessity for assembly, and there is increased risk for thermal damage to the printed
circuit board (PCB) and components. It has also been demonstrated in numerous reliability
studies that the accelerated thermal cycling (ATC) performance of SAC 105 (Sn-1.0Ag-0.5Cu) is
inferior to that of SAC 305 [18, 22]. Finally, reducing Ag does not affect the propensity for tin
10
whisker formation, nor does it prevent the degradation of microstructure and properties after
aging.
Table 2.1 contains a summary of several 1st and 2nd generation Pb-free alloys, including
constituents, compositions, and melting range.
Table 2.1: First and Second Generation Pb-free Solder Alloys
Alloy
Constituents
(wt%)
Melting
Range
(°C) Sn Ag Cu
SAC 305 96.5 3.0 0.5 217-220
SAC 405 95.5 4.0 0.5 217-225
SAC 105 98.5 1.0 0.5 215-227
SAC 387 95.5 3.8 0.7 217-220
Sn-3.5Ag 96.5 3.5 0 221
Sn-0.7Cu 99.3 0 0.7 227
Certain industries were given a deadline to adhere to the RoHS standards by 2006, and, at the
time of the writing of this thesis, several high-reliability industries such as aerospace and defense
(A&D), medical, and automotive, are temporarily exempt under RoHS-2 from ceasing to use Sn-
Pb until July 2019 [13]. This is primarily because 1st and 2nd Pb-free solder alloys such as SAC
305 and SAC 105 do not currently satisfy industry performance requirements. Since these
exemptions will eventually come to an end, there has been significant interest in the development
of new, 3rd generation Pb-free solder alloys.
2.1.4 Pb-Free Alloys – Third Generation
Based on the performance of 1st and 2nd generation Pb-free solder alloys, it is clear there is no
‘drop-in’ replacement for Sn-Pb alloys. Various researchers and consortia began investigating
‘3rd generation’ Pb-free solder alloys, which deviate from SAC compositions and make use of
additional alloying elements.
Bismuth currently has shown the most promise – the Sn-Bi system is a simple binary eutectic
(Figure 2.6), with Bi having moderate solubility and not forming IMCs with Sn. Additions of Bi
lower the melting point of the alloy, and Bi is inexpensive. While the inclusion of Bi in the alloy
was initially a concern due to the possible formation of the low-melting ternary Sn-Pb-Bi phase,
11
RoHS legislation has effectively eliminated Sn-Pb finishes from the supply chain, so there is no
longer any risk of encountering this deleterious ternary phase.
Figure 2.6 shows three separate iterations of the binary eutectic Sn-Bi phase diagram [25, 26,
27]. It is noted that each diagram predicts different phase boundaries (e.g. liquidus, solidus,
solvus lines) and maximum solubilities, which may present difficulty in the design of Bi-
containing solder alloys. Belyakov [28] used Differential Scanning Calorimetry (DSC) to
experimentally verify the solidus and liquidus lines in the Sn-Bi system. To ensure samples
(ranging from 1.5 wt% to 25 wt% Bi) were fully homogenous with no concentration gradients or
non-equilibrium phases, extensive equilibration (up to 10 months) slightly below the reported
solidus temperature was conducted. It was concluded that the most accurate phase equilibria
were those reported by, or similar to Lee [25]. The inaccuracies in other reported diagrams were
likely the result of insufficient equilibration of samples; Braga, for example, reported
equilibration of 120°C for one hour for all compositions [27]. This is believed to be insufficient
time for complete sample homogenization.
2.1.4.1 History
Several industry consortia projects in the late 1990s / early 2000s have evaluated Bi-containing
alloys as possible alternatives to SAC and determined that these alloys demonstrate excellent
mechanical and thermomechanical properties [18]. These include the National Center for
Manufacturing Sciences (NCMS) Lead-Free Solder Project and Joint Council on Aging Aircraft /
Joint Group on Pollution Prevention (JCAA/JG-PP) Lead-Free Solder project, completed in 2001
and 2006, respectively [29, 30, 31].
An extensive study by Vianco indicated that increasing Bi content may improve wetting
properties, and inclusion of Bi greater than 4.8 wt% does not provide any significant
improvements to alloy strength [32, 33]. Furthermore, Bi was not observed to segregate to the
IMC-solder interface after joint solidification. Witkin determined that the creep properties of Bi-
containing alloys are not significantly affected by aging, contrary to what is observed in SAC
305; this is likely caused by the retardation of dislocation motion by solute Bi atoms [34]. In
separate studies, Witkin observed a similar trend concerning the yield and ultimate tensile
strengths of these alloys [35, 36].
12
Figure 2.6: Sn-Bi binary phase diagrams: NIST [37], based on Lee et al. [25] (a); Okamoto [38],
based on Vizdal [26] (b); Braga [27] (c), indicating variance of reported phase equilibria in Sn-Bi
system.
In 2009, Celestica began a collaborative project with the University of Toronto, focused on the
development of new 3rd generation Pb-free solder alloys [18]. The initial motivation for this
project stemmed from the desire to eliminate the pad cratering failure mode, improve drop/shock
performance, and reduce the cost of the alloy. Based on the results from the NCMS and
JCAA/JGPP projects, as well as the Vianco and Witkin studies, Bi was selected as the alloying
element of focus. Bismuth lowers the melting point of the alloy, allowing for standard board
materials to be used in assembly, reduces pad cratering, and improves reliability. Moreover,
preliminary studies have shown that Bi may possibly mitigate tin whisker formation [39]. Three
industry available solder alloys were selected as baselines, and after analysis of the ternary Sn-
(a)
(c)
(b)
13
Ag-Bi and Sn-Cu-Bi phase diagrams, 23 new compositions (Sn-Ag-Bi, Sn-Cu-Bi, or Sn-Ag-Cu-
Bi) were selected for further study; these were downselected to 7 alloys, all of which met the
required criteria established by the project.
2.1.4.2 Recent Results
In 2014, the Refined Manufacturing Acceleration Process (ReMAP) was established [8]. This is
a Canadian-led initiative aimed at advancing commercialization of novel manufacturing
technologies, materials, and processes. Three projects were set up focused on Bi-containing, Pb-
free solder alloy development, effectively a continuation of the low-melt collaborative project
undertaken by Celestica and the University of Toronto. One of these projects, ‘M3: Aging,’ is
focused on the study of the long-term properties of Bi-containing alloys.
One early M3 project study looked at the effects of aging on the microstructure and hardness of
seven Bi-bearing alloys, five of which contain low or no Ag [6]. All alloys were aged at 100°C
for 25 or 100 hours. The microstructure of all alloys changed significantly after aging – Bi
migrated from the interdendritic spaces into the Sn dendrites to produce a more homogenous
distribution (Figure 2.7a&b). The hardness of all alloys was observed to increase (Figure 2.7c);
this differs strongly from SAC alloys, whose strength is degraded after aging.
Based on the results from the aging study, it was believed that the homogenization of Bi in the
solder may be related to the increasing solubility of Bi in Sn at higher temperatures, as well as
the solid-state diffusion of Bi in Sn after dissolution at higher temperatures to produce a more
homogenous distribution of Bi in the alloy. Upon cooling, as the solid solubility is reduced, Bi is
forced out of solution to form an even dispersion of Bi precipitates. A homogeneous
microstructure is considered to be more robust than a non-homogeneous (as-solidified)
microstructure in typical alloy systems. To that end, it was postulated that Bi-containing alloys
could be purposefully heat treated to resolutionize the Bi in the Sn matrix and prolong the life of
the solder joint. The treatment temperature is important to consider – it should effectively
dissolve all Bi in the Sn matrix. The minimum temperature at which all second phase particles
dissolves is termed the solvus temperature 𝑇𝑠𝑜𝑙.
14
Figure 2.7: Evolution of microstructure and hardness of Bi-containing alloys after aging. Violet
(Sn-2.25Ag-0.5Cu-6Bi), as-cast (a) and aged at 125°C for 24 hours (b) microstructure [7, 40];
hardness [6] of the seven selected Bi-containing alloys in the Celestica-UofT low melt project
after aging at 100°C (c).
Based on Matijevic’s experimental determination of 𝑇𝑠𝑜𝑙 of these practical alloys using a
combination of DSC and metallography, the ReMAP team decided on an aging treatment
temperature of 125°C, to be subjected for no greater than 48 hours [40]. Matijevic and
Snugovsky examined the evolution of the grain structure of Violet compared with SAC 305 after
aging at 125°C (Figure 2.8a&b) [7, 40]. It was found that in the as-cast conditions, the grain
(a) (b)
(c)
15
size of Violet is quite large, with localized recrystallization only within interdendritic spaces. As
aging proceeds, the grain size is reduced, and a polycrystalline grain structure is formed. This is
the result of the Bi entering solid solution and distributing evenly throughout the Sn matrix.
When Bi is forced out of solution as the alloy is cooled from 125°C, Bi precipitates form, which
exert stress on the surrounding Sn matrix. Recrystallization occurs as the alloy attempts to
relieve the stress; this is known as Particle-Stimulated Nucleation (PSN) [41, 42]. It was also
found that the solvus of the alloys with higher Bi content (Violet and Sunflower) lies above the
typical operating temperature of solder joints (~70°C) and Ostwald ripening occurs after aging at
this lower temperature (Figure 2.8c).
Figure 2.8: Further effects of aging on the microstructure of Violet. Electron Backscatter
Diffraction (EBSD) maps [7] (Euler coloring) of the as-cast alloy (a) and alloy aged at 125°C for
50h (b). Microstructure of Violet after aging at 70°C for 300 hours (c), showing Ostwald
ripening [7, 40].
(a) (b)
(c)
16
Matijevic performed a nanoindentation creep study to evaluate the effects of the heat treatment
on the microstructure and mechanical response of the alloy under a prolonged load [7, 40]. It was
found that aging Violet above solvus results in a significant increase in creep resistance. While
polycrystalline materials tend to demonstrate lower creep resistance than their monocrystalline
counterparts due to increased grain boundary sliding (creep mechanism), the aged Violet alloy
also contains an even dispersion of Bi precipitates, which likely pin the grain boundaries and
retard creep deformation (Figure 2.9a). Further work was done by Hillman to evaluate the
efficacy of this heat treatment as it relates to solder joint reliability [4]. Accelerated Thermal
Cycling was performed on a series of assemblies; some boards were ‘preconditioned’ prior to
cycling. It was found that for the -40°C to 70°C cycling range, preconditioned Violet solder
joints outperformed their ‘as-assembled’ counterparts by roughly 15%, with respect to Weibull
characteristic life (Figure 2.9b). These results further confirmed that a combination of Bi in the
alloy as well as engineering the alloy microstructure via heat treatments can produce more
reliable solder joints.
Figure 2.9: Effects of heat treatment on the mechanical properties and reliability of Violet.
Nanoindentation results [7, 40] showing improvement in creep resistance after heat treating the
alloy at 120°C (a). ATC data [43] showing improvement in the Weibull characteristic life θ after
heat treatment at 125°C (b).
(a) (b)
17
These results prompted the ReMAP team to file a conditional patent in Canada and the United
States for this heat treatment, designed to ‘precondition’ the alloy microstructure after joint
solidification, or ‘restore’ the alloy after some time in service (when Ostwald ripening may occur
– coarse Bi particles may be a detriment to solder joint reliability) [44, 45]. Figure 2.10 shows a
flowchart illustrating the execution of this heat treatment during the life cycle of a solder joint
[43]. Further work is currently underway to test the restoration properties of the heat treatment
after some time in service (during which time Bi precipitates would undergo Ostwald ripening)
and the process to file a full patent in several jurisdictions is ongoing.
Figure 2.10: Proposed flow chart [43] illustrating the practical implementation of the heat
treatment during the life cycle of an electronic device.
2.2 Strengthening Mechanisms in Metals and Alloys
Alloys are known to have superior physical and mechanical properties over pure metals. The
addition of secondary (alloying) elements may provide a strengthening effect to the material,
depending on the base material, selected alloying elements, and the quantity of these elements. In
many alloy systems, solid solutions exist at low concentrations of the alloying element; further
additions result in a secondary dispersive phase, either a precipitate or IMC. Both solid solutions
and dispersive particles can strengthen materials via alloying. Reducing the grain size (and
increasing GB content) is another method of strengthening materials, which may or may not be
related to secondary alloying elements. For example, Particle Stimulated Nucleation (PSN) is a
18
well-known phenomenon in which the presence of secondary phases induces recrystallization in
an alloy [41, 42].
2.2.1 Solid-Solution Strengthening
A solid solution is formed when impurity atoms (solute) are added to a pure metal (solvent),
without changing the structure of the solvent [12]. Solid solutions are typically formed at lower
impurity levels; in many systems further additions of solute result in the formation of a
secondary phase such as a precipitate. The concentration at which the second phase forms is
typically referred to as the solubility limit and is determined by the solvus phase boundary.
There are two types of solid solution – substitutional and interstitial. In a substitutional solid
solution, the solute atoms replace the host (solvent) atoms – a solute with a smaller or larger
atomic radius r than the solvent will exert a tensile or compressive stress, respectively, on the
surrounding matrix (Figure 2.11). In an interstitial solid solution, the solute atoms occupy
interstitial sites in the lattice. Hume-Rothery developed a set of rules (for both substitutional and
interstitial solid solutions) which dictate the conditions necessary for a solid solution to form,
given a specific solvent-solute system. The conditions which improve the range of substitutional
solid solution between solute and solvent are given below; not all conditions are required [46]. If
any conditions are not met, it is possible an IMC phase may form instead:
Figure 2.11: A larger substitutional solute will exert a compressive stress on the surrounding
lattice. Sn (white) and Bi (orange) atoms are to scale. Schematic is not representative of atom
positions in Sn lattice. Adapted from Callister [12].
1. Proximity of r of solute relative to solvent
2. Similarity of crystal structures between the solute and solvent
3. Higher valency of the solute than the solvent
19
4. Similar electronegativities between solute and solvent
Table 2.2 contains data on these four attributes for Sn and several lead-free solder alloying
elements, as well as their approximate solubilities at room temperature and at 125°C.
Table 2.2: Select physical properties of Pb-free solder alloying elements pertaining to solid
solubility
Species r (nm) Crystal
Structure
Valency
(most
common)
Electro-
negativity
Solubility
at RT
(wt%)
Solubility
at 125°C
(wt%)
Sn 0.140 Body-Centered
Tetragonal
+2, +4 1.96 N/A N/A
Ag 0.144 Face-Centered
Cubic
+1 1.93 ~0 [19] ~0 [19]
Cu 0.128 Face-Centered
Cubic
+2 1.90 ~0 [20] ~0 [20]
Bi 0.156 Rhombohedral +3 2.02 1.97 [37] 17.39 [37]
Sb 0.140 Rhombohedral +3, +5 2.05 ~0 [47] 0.77 [47]
Notably, the crystal structures of these solute species are largely different than the β-Sn body-
centered tetragonal structure, and generally the solubility of these species in Sn is quite limited.
Bismuth and antimony (Sb) have higher solubility than Ag and Cu; this may be related to the
valencies of these species, which are higher than that of Sn. Silver, copper, and antimony all tend
to form intermetallics with Sn, due to their low solubility; Bi forms a binary eutectic system with
Sn and no intermetallics are formed under equilibrium conditions.
Solute atoms tend to diffuse towards dislocations [12]. This diffusion lowers the overall
dislocation strain energy; for example, a Bi atom, having a larger atomic radius and inducing a
compressive strain on the Sn lattice, should tend to partially cancel out the dislocation’s tensile
strain (Figure 2.12). It will then require more stress to move dislocations past impurity atoms,
resulting in an increase in the material’s strength.
20
Figure 2.12: Hypothetical location of Bi atoms at the tensile strain portion of a dislocation in the
Sn matrix. Sn (white) and Bi (orange) atoms are to scale. Schematic is not representative of atom
positions in Sn lattice. Adapted from Callister [12].
2.2.2 Dispersion Strengthening
Second phase precipitates can have a very strong influence on the mechanical properties of an
alloy. These particles, often harder than the matrix, serve as obstacles to dislocation motion. For
plastic deformation to occur, the dislocation must be able to bypass the second phase particles;
this requires additional energy. The size of a particle, i.e. radius R and particle spacing have a
very strong influence on its effectiveness as an obstacle to dislocation motion. There are two
types of stress which describe the interaction between a dislocation and an array of obstacles –
the cutting stress and the bypassing, or Orowan stress [48]. The cutting stress scales with √𝑅,
while the Orowan stress scales with 1 𝑅⁄ . A dislocation will tend to adopt the mechanism that
requires the least energy - at some critical particle radius 𝑅∗ the behavior of the dislocation will
transition between cutting and bowing (Figure 2.13); this represents the optimal aged conditions
[48, 49].
In a system with limited solubility and at equilibrium, a precipitate phase will form when the
solid solubility is exceeded, usually when the temperature is lowered. For the nucleation of a β
phase in an α matrix, solute atoms first diffuse and collect in a localized region to attain the
composition of the β phase; this is followed by rearrangement into the β crystal structure [50].
Not all nuclei will result in the formation of a β phase particle; the most successful nuclei will
tend to have the smallest nucleation barrier energy. The smallest barriers to nucleation typically
occur in a system with heterogeneous nucleation, in which a non-equilibrium defect such as a
GB serves as a nucleation site [50]. Homogeneous nucleation requires equilibrium lattice sites to
21
serve as nucleation centers; homogenous nucleation is therefore quite rare. In a material, the type
and concentration of nucleation site dictate the rate of precipitation of a second phase [50].
Certain nucleation sites such as free surfaces may have higher free energies, which is lowered by
the precipitation reaction.
Figure 2.13: Effect of particle size on the cutting and bypassing stresses, showing some optimal
particle size R* will yield the highest required stress for successful proliferation of dislocations
through the particle-strengthened alloy [48].
Ostwald ripening is a coarsening of second phase particles. In an alloy consisting of many small
second phase precipitates, the area of phase boundaries and thus the interfacial energy γ is high.
The total energy of the system can be reduced via particle coarsening which lowers the total
phase boundary area [51, 52]. Ostwald ripening usually occurs during aging below 𝑇𝑠𝑜𝑙. Below
𝑇𝑠𝑜𝑙, larger precipitates remain out of solution, while smaller precipitates will dissolve in the
matrix. Solute atoms then migrate towards the larger precipitates, which coarsen with time.
Ostwald-ripened Bi precipitates (Figure 2.14) are surmised to have a detrimental effect on alloy
reliability – as they are larger and more widely spaced, they are less effective obstacles to
dislocation motion than an array of smaller, more closely spaced precipitates [48].
*
22
Figure 2.14: Ostwald ripening of Bi precipitates, showing large particles surrounded by a region
devoid of Bi [40].
Intermetallic compounds, typically containing two or more species arranged in a specific
stoichiometric ratio, also provide dispersion strengthening to the matrix [12]. They are also quite
brittle; IMC coarsening can result in the degradation of mechanical properties of Pb-free solder
alloys after aging. After alloy solidification, IMCs in eutectic regions are finely dispersed and are
situated in the interdendritic spaces, as discussed in Section 2.1.2. These fine particles are quite
effective at pinning dislocations and retarding plastic deformation. After aging, IMCs tend to
coarsen, rendering them less effective at pinning dislocations.
The Sn-Bi system is a simple binary eutectic; no IMC phases form. This is one of the attributes
which makes Bi such an attractive alloying element. The Brinell hardness of Bi is approximately
94, compared to 51 for Sn [53]. It can therefore be inferred that Bi may serve as an excellent
dispersion strengthener in a Sn-rich solder alloy, provided the Bi precipitates are not allowed to
coarsen.
2.2.3 Grain Boundary Strengthening
Crystalline materials such as metals and alloys are composed of grains, each possessing a
different orientation. The interface between two grains is known as a grain boundary (GB); in
these locations there is mismatch between the atomic planes in each grain and subsequently
irregular bonding between atoms [12, 50]. Dislocations are needed to accommodate the
23
irregularity of the bonding between atoms along the GB; this results in interfacial grain boundary
energy γ. At low degrees of misorientation, γ is lower as fewer dislocations are needed to
accommodate the mismatch; these are known as low-angle grain boundaries (LAGBs) [50].
As misorientation increases, γ increases at a decreasing rate before levelling off at around 10-15°
(Figure 2.15). At this point, dislocation strain fields overlap, forming high angle grain boundary
(HAGB) structures [50]. These boundaries generally demonstrate a very open structure and poor
fit between the adjacent grains. Some exceptions exist at discrete misorientations (material
dependent); these are known as special HAGBs. These boundaries allow significantly better fit
between the two adjacent grains, lowering the interfacial energy [50]. Some examples of these
boundaries include twins and coincident site lattice (CSL) boundaries. Twins, common in
materials such as Sn, may be either coherent or incoherent boundaries; the energy of a coherent
twin boundary is typically far less than that of an incoherent twin boundary.
Figure 2.15: Effect of misorientation between two grains on the grain boundary energy γ and on
the transition from a LAGB to a HAGB [50].
Grain boundaries are known to be effective strengtheners in metals and alloys. During plastic
deformation, a dislocation will migrate along a certain atomic plane in one grain but will need to
change its direction of travel when it reaches the GB to pass into the second grain. For HAGBs,
dislocation pile-ups may occur at the boundary, which may induce the nucleation of additional
dislocations in neighboring grains [12, 48]. The Hall-Petch effect suggests that a material with
higher GB content (smaller grains) will be stronger than one with fewer or no GBs [12].
However, solder alloys typically operate in the time-dependent deformation regime and undergo
24
creep. As discussed above, increased grain boundary content usually results in poorer creep
resistance; however the uniform Bi precipitation likely contributes to the improved creep
performance in the aged Violet alloy, compared with the as-solidified alloy.
2.2.3.1 Recovery, Recrystallization & Grain Growth
The plastic deformation of a material often results in stored strain energy, with strain fields
situated around dislocations. Heating the material may revert it to its pre-deformed state and
release this strain energy; the processes involved are known as recovery, recrystallization, and
grain growth (coarsening) [12, 42].
Recovery involves the rearrangement of dislocations to reduce the overall strain energy [12].
Recrystallization follows, as the grains are still under high internal strain, and involves the
formation of a new arrangement of strain free-equiaxed grains via the formation and migration of
HAGBs [42]. These grains form initially as small nuclei and grow from the pre-existing strained
material until it is consumed. The difference in energy between the original and recrystallized
material serves as the driving force for this microstructural change. Recrystallization can occur
during deformation (dynamic), or after deformation (static), and is thermally activated [42]. The
recrystallization temperature is that at which the entire material recrystallizes within one hour; it
is usually between one third and half the absolute melting temperature of the material [12]. There
also exists a critical deformation to initiate recrystallization – increased deformation will lower
the recrystallization temperature. These together will affect the nucleation rate of recrystallized
grains and the final grain size.
Grain growth may occur after recrystallization; the driving force is the reduction of interfacial
energy at GBs [12, 42]. This phenomenon occurs at extended holding times at elevated
temperature.
2.2.3.2 Particle Stimulated Nucleation (PSN)
Very often in multiphase materials, the presence and nucleation of second phase particles
induces recrystallization to occur. As the second phase is often of a different density than the
matrix, a volumetric mismatch is formed between the two phases. In addition, crystal structure
and/or lattice mismatch may also exist if the lattice/unit cell parameters differ greatly between
matrix and precipitate. This exerts strain on the surrounding lattice, forming a particle
25
deformation zone; recrystallization may originate anywhere within this zone [41]. This is known
as Particle Stimulated Nucleation (PSN) of recrystallization, and is analogous to heterogeneous
nucleation of second phase particles from a solid solution, as discussed above [41, 42]. Particle
Stimulated Nucleation typically occurs when the second phase precipitate particles are on the
microscale and larger particles may result in the nucleation of several grains.
The recrystallization observed in Bi-containing solder alloys is likely caused by PSN. The
densities ρ, crystal structures, and lattice parameters of Sn and Bi are given in Table 2.3. It
appears as the alloy is cooled from the elevated treatment temperature, Bi is forced out of solid
solution, and as Bi is denser than Sn and has a different crystal structure, there is mismatch
between the Sn lattice and the newly formed Bi precipitates. This exerts stress on the
neighboring Sn matrix, which is relieved via recrystallization. The recrystallization is possibly
caused by PSN induced by the precipitation of Bi, as no recrystallization is observed within Sn
dendrites in the as-solidified microstructures, and the final grain size is equiaxed with weak
texture [41].
Table 2.3: Density and Lattice Data for β-Sn and Bi phases
Material ρ
(g/cm3)
Crystal
Structure
Lattice Parameters
a b c α β γ
Sn 7.27 Body-Centered
Tetragonal 0.583 nm
0.318
nm 90°
Bi 9.78 Rhombohedral 0.454 nm 1.185
nm 90° 120°
The extent of recrystallization and grain size is influenced by the spacing of second phase
particles and subsequently nucleation sites; this can be controlled by alloy processing [41, 42].
Matijevic and Snugovsky found that the extent of recrystallization in Bi-containing alloys
increases with aging time. The as cast structure is fairly coarse, and grain size is reduced as the
aging time is increased, up to 50 hours. Further aging past 50 hours did not change grain size
appreciably, indicating the Bi precipitate distribution did not change (Figure 2.16) [7, 40].
It is unknown whether the PSN-influenced recrystallization in Bi-containing alloys is an example
of static (occurring after precipitation is complete) or dynamic (occurring during precipitation)
recrystallization. This would need to be verified using in situ electron microscopy. Further, while
it appears no apparent tempering or heat treatment is necessary to induce recrystallization as in
26
other systems [54], it is noted that room temperature (at which microscopy was performed) is
high relative to the absolute melting temperature of the alloy. Thus it is possible that the critical
recrystallization temperature is achieved simply by holding the alloy at room temperature
between cooling and electron microscopy.
Figure 2.16: EBSD maps showing grain structure evolution of Violet after aging [7]. As cast (a);
after aging at 120°C for 24h (b), 50h (c), and 300h (d).
2.3 Solid-State Diffusion
Many changes in the structure of metals occur via solid-state diffusion [55]. For example, in Bi-
containing solder alloys, the homogenization of Bi precipitates after aging, accompanied by the
Particle Stimulated Nucleation of recrystallized Sn, can be explained by the diffusion of Bi in the
Sn matrix. Diffusion can be modelled using several methods, including the differential equations
(a) (b)
(c) (d)
27
developed by Adolf Fick in 1855, as well as with atomistic approaches. The former method is
generally applicable to a wide range of systems as few assumptions are made and is the main
focus in this thesis [55]. Several factors which affect diffusivity that are relevant to Bi-containing
solder alloys such as temperature, high diffusivity paths, and the anisotropy of the β-Sn lattice
are discussed, followed by a brief overview of impurity diffusion and interdiffusion. Finally,
several methods used to study diffusion are highlighted.
2.3.1 Fick’s (Continuum) Laws of Diffusion
In 1855, Adolf Fick developed a set of basic differential equations to describe diffusion. These
equations ignore the atomic structure and assume that the material is a continuum [55, 56].
Fick’s first law (Eqn 2.1), also known as the flux equation, describes steady-state mass transport,
i.e. the change in concentration at a point does not change with time [55, 56, 57]. It assumes that
the net flow of matter will decrease the concentration gradient. If matter flows along some
direction x, parallel to the concentration (C) gradient, the flux J is given as follows:
𝐽 = −𝐷𝜕𝐶
𝜕𝑥
(Eqn 2.1)
D is a proportionality constant known as the diffusivity or diffusion coefficient [56]. In most
practical cases, steady state does not exist, and the concentration gradient will change with time
t. After some derivation via combining Fick’s first law and the continuity equation [56, 58], we
arrive at Fick’s second law (Eqn 2.2), also known as the diffusion equation [55, 56, 57, 58]. This
is a second order partial differential equation:
𝜕𝐶
𝜕𝑡=
𝜕
𝜕𝑥𝐷
𝜕𝐶
𝜕𝑥
(Eqn 2.2)
The various solutions to Fick’s laws are divided into two groups – those in which the diffusivity
is assumed to be a constant, and those where the diffusivity is assumed to be a function of
concentration. Experimental diffusivity data can also be modelled based on first principles
simulations.
28
2.3.1.1 Classic Solutions
The solutions to the flux equation are not discussed here as they are not typically relevant to
practical diffusion systems. Fick’s second law has been extensively utilized as the underlying
model for most diffusion calculations in a variety of sample geometries [57]. Several solutions
exist for Fick’s second law. They can be divided into two types, whether D is constant for all
concentrations C, or whether it is a function of composition D(C). Several examples of each
follow.
When D is considered to be a constant, Fick’s second law transforms into (Eqn 2.3) [55]:
𝜕𝐶
𝜕𝑡= 𝐷
𝜕2𝐶
𝜕𝑥2
(Eqn 2.3)
The instantaneous source solution of Fick’s second law for non-steady state diffusion, assuming
some amount M of diffusant per unit area is deposited on a substrate at distance x=0 and allowed
to spread inwards for some time t, is given by the following equation (Eqn 2.4) [55, 56, 57, 58]:
𝐶(𝑥, 𝑡) =𝑀
√𝜋𝐷𝑡𝑒𝑥𝑝 (−
𝑥2
4𝐷𝑡)
(Eqn 2.4)
where C is the concentration at x and t, and D is the diffusivity. M is denoted as the mass of
diffusant per unit area. This solution is only valid if the thickness of deposited material is
significantly less than the diffusion distance 2√𝐷𝑡, [57]. A schematic of the geometry of this
solution is given in Figure 2.17.
A solution assuming thick layer geometry (Eqn 2.5) is valid when the thickness of the deposited
material h is comparable to the diffusion distance 2√𝐷𝑡 [56, 57, 58]:
𝐶(𝑥, 𝑡) =𝐶𝑜
2[𝑒𝑟𝑓 (
𝑥 + ℎ
2√𝐷𝑡) − 𝑒𝑟𝑓 (
𝑥 − ℎ
2√𝐷𝑡)]
(Eqn 2.5)
29
where once again C is the concentration at x and t, and D is the diffusivity. Co is the initial
concentration when t=0, and erf denotes the error function. This solution is relevant in diffusion
couples where a thin film of solute is deposited on a surface.
Figure 2.17: Schematic of thin and thick layer geometry [56].
2.3.1.2 Inverse and First Principles Methods
It is typically assumed that the diffusion coefficient D is independent of composition. However,
this often is not the case. In situations where D is a function of composition, Fick’s second law
cannot be solved conventionally. Inverse methods are commonly used to circumvent this
problem – given the concentration field data collected experimentally, one would solve for the
diffusivity as a function of composition and Fick’s second law transforms into (Eqn 2.6) [58]:
𝛿𝐶
𝛿𝑡=
𝛿
𝛿𝑥(𝐷(𝐶)
𝛿𝐶
𝛿𝑥)
(Eqn 2.6)
The Boltzmann-Matano method requires the non-linear Fick’s second law to be transformed into
an ordinary differential equation by removing all instances of space (x) and time (t) and by
introducing ξ, Boltzmann’s similarity variable [55, 56, 58]. ξ is defined as the following (Eqn
2.7):
𝜉 =𝑥 − 𝑥𝑀
2√𝑡
(Eqn 2.7)
30
Where xM is the Matano plane, which is defined as the imaginary interface across which equal
amounts of matter have diffused to the left and right [55, 56, 58]. When the transformed equation
is solved and reverted back from ξ to space-time coordinates, this yields (Eqn 2.8):
𝐷(𝐶′) = [−1
2𝑡
𝑑𝑥
𝑑𝐶]
𝑐′∫ (𝑥 − 𝑥𝑀)𝑑𝐶
𝐶′
𝐶𝑅
(Eqn 2.8)
The Boltzmann-Matano method is not an optimal approach to solve for concentration-dependent
diffusivity, as errors may be introduced when solving for the location of the Matano plane, and
solving at the extremities of the diffusion zone is unreliable [58, 59]. Several alternative methods
can be used which reduce or eliminate these sources of error; two of these are the Sauer-Friese
and Hall methods [60, 61].
With the Sauer-Friese method, there is no need to evaluate the position of the Matano plane [59]
[60]. In this method, a concentration ratio 𝑌𝑆𝐹 is defined (Eqn 2.9), resulting in an equation for
the diffusivity, which is a function of 𝑌𝑆𝐹 (Eqn 2.10). The Sauer-Friese method must be used
instead of the Boltzmann-Matano method if the volume of the solvent changes appreciably as a
function of the solute during diffusion [56].
𝑌𝑆𝐹(𝑥) =𝐶(𝑥) − 𝐶𝐿
𝐶𝑅 − 𝐶𝐿
(Eqn 2.9)
𝐷(𝐶′) =1
2𝑡 (𝑑𝑌𝑆𝐹
𝑑𝑥⁄ )
𝑥′
{−[1 − 𝑌𝑆𝐹(𝑥′)] ∫ 𝑌𝑆𝐹(𝑥)𝑑𝑥𝑥′
−∞
+ 𝑌𝑆𝐹(𝑥′) ∫ [1 − 𝑌𝑆𝐹(𝑥)]𝑑𝑥+∞
𝑥′
}
(Eqn 2.10)
Using the Hall method, it is possible to reliably solve for the diffusivity at the two extremities of
the concentration profile (where the composition approaches the initial concentration of the two
original constituent metals) [59, 61]. A profile of u vs 𝜆𝐻 is analyzed which has a linear fit u=
h𝜆𝐻 + k, with u (Eqn 2.11) and 𝜆𝐻 (Eqn 2.12) defined as:
31
𝑌𝐻 =1
2[1 + 𝑒𝑟𝑓𝑐 (𝑢)]
(Eqn 2.11)
𝜆𝐻 = 𝑥√𝑡⁄
(Eqn 2.12)
These yield the following function for the diffusivity (Eqn 2.13):
𝐷(𝐶) =1
4ℎ12 {1 +
2𝑘1
√𝜋𝑒𝑥𝑝 (𝑢2)[𝑌𝐻(𝐶)]}
(Eqn 2.13)
Armed with these analytical approaches, it becomes possible to solve for an unknown diffusivity
as a function of concentration using any set of experimental diffusion data.
As the classic solutions to Fick’s laws are often too simplistic for practical cases, and the inverse
methods can prove unwieldly or present small amounts of error, simulations based on first
principles can be used to model experimental diffusion data. Researchers at Ohio State
University have developed a ‘forward simulation’ approach, which showed good agreement with
diffusion data in the literature [59, 62, 63]. This simulation makes use of Fick’s laws, the moving
boundary condition (Eqn 2.14), and several other conditions to solve for diffusivities, either
constants or functions of concentration. Further details about the technique are given in Section
4.5.3.
𝑣 =𝐽𝐵𝑖 − 𝐽𝑆𝑛
𝑐𝐵𝑖 − 𝑐𝑆𝑛
(Eqn 2.14)
Where 𝑣 is the velocity of the moving boundary between Sn and Bi, 𝐽𝑆𝑛 and 𝐽𝐵𝑖 are the mass
fluxes of Sn and Bi and 𝑐𝑆𝑛 and 𝑐𝐵𝑖 are the corresponding solubility limits.
32
2.3.2 Atomic Mechanisms
The atomistic description of diffusion in solids is referred to as the ‘random walk’ model [55, 56,
58]. The time required for an atomic jump is extremely short compared with the average time an
atom is situated at a new lattice site. It is therefore postulated that within a lattice for some time
an atom undergoes many discrete jumps in random directions. The random walk model uses the
volume of jumps combined with the atomic spacing and lattice coordination number to
determine the average distance at atom moves from its initial location. This in turn is used to
determine the diffusion coefficient. Lattice defects can mediate atomic jumping; the vacancy
mechanism involves a solute atom migrating through the lattice via periodic switches with
vacancies in the lattice and explains the diffusion of substitutional solutes [55, 56, 58]. The
process of jumping requires an atom to pass through a region (e.g. directly between two lattice
atoms) which often requires additional (activation) energy. Both the jump rate and the site
fraction of vacancies of a crystal are thermally activated [55, 56, 57]. Higher temperatures will
both facilitate a higher frequency of successful atomic jumps and introduce a higher proportion
of vacancies with which to facilitate atomic jumps - subsequently a higher mass flow of atoms is
achieved.
2.3.3 Factors which Influence Diffusivity
The diffusivity in metals and alloys can be influenced by several factors, both external
(surrounding environment) as well as internal (specific to the material). A few of these pertinent
to Pb-free solders are discussed below; these include temperature, high diffusivity paths such as
GBs, and anisotropy of the crystal structure of the solvent species (in this case, Sn).
2.3.3.1 Temperature
Increasing temperature results in a larger diffusion coefficient. The Arrhenius equation is used to
calculate D at some absolute temperature T. This equation and variables within were derived
using thermodynamic principles and is given as [55, 56, 57, 58] (Eqn 2.15):
𝐷 = 𝐷𝑜𝑒𝑥𝑝 (−𝑄𝐴
𝑅𝑖𝑇)
(Eqn 2.15)
33
Where 𝐷𝑜 is the pre-exponential, or frequency factor, 𝑄𝐴 is the apparent activation energy for
diffusion, and 𝑅𝑖 is the ideal gas constant [56, 57, 58]. To determine these values from some
experimental values of D, the natural logarithm is taken on both sides of the equation, and 1 𝑇⁄
and 𝑙𝑛(𝐷) are plotted on the x and y axes respectively [55, 56]. The data should fall on a straight
line if Arhennius behavior is observed. The slope of the line is equal to −𝑄𝐴
𝑅𝑖⁄ and the y-
intercept of the line is equal to 𝑙𝑛(𝐷𝑜), from which 𝑄𝐴 and 𝐷𝑜 respectively can be calculated.
Departures from strict Arrhenius behavior may occur for several reasons, including additional
atomic migration mechanisms interactions with microstructural features such as lattice defects,
and high temperatures (often greater than 0.7𝑇𝑚) [56, 58]. The total diffusivity in a system with
n diffusion mechanisms is simply the summation of the individual diffusivities 𝐷1, 𝐷2, . . . 𝐷𝑛
from each mechanism (Eqn 2.16):
𝐷 = 𝐷1 + 𝐷2+. . . +𝐷𝑛
(Eqn 2.16)
The Arrhenius diagram will depart from a straight line into a curve. The mechanism with the
highest activation energy will be more dominant at the higher temperature [56]. If multiple
diffusion mechanism are acting in parallel (independent of one another), the mechanism with the
smaller activation energy will dominate the overall process. Such behavior occurs in highly
polycrystalline films (which include lattice and grain boundary diffusion, discussed in the next
section).
2.3.3.2 High Diffusivity Paths
In previous sections, non-equilibrium material defects were not assumed to take part in diffusion.
However, practical materials contain high-energy defects such as GBs, dislocations and free
surfaces; the atomic jump rate and subsequently the diffusivity will be higher in these regions
[55]. In general, the hierarchy between lattice diffusion 𝐷𝑙 and high-diffusivity pathways (grain
boundaries 𝐷𝑔𝑏, free surfaces 𝐷𝑠, and dislocations 𝐷𝑑) is as follows (Eqn 2.17). It is expected
that as special GBs such as CSLs and twins possess a lower interfacial energy, the diffusivity
along these pathways will be lower than 𝐷𝑔𝑏. The opposite trend is observed for the activation
34
energy [56]. These diffusivities follow a similar Arrhenius relationship to lattice diffusivity with
respect to temperature.
𝐷𝑙 ≪ 𝐷𝑑 < 𝐷𝑔𝑏 < 𝐷𝑠
(Eqn 2.17)
Alloying elements are prone to segregate at these defects when the solubility is lower [55].
Segregation lowers the interfacial energy and may stabilize the defect, or create a new structure,
filling in the open spaces and inhibiting further solute from diffusing along the defect [55]. High-
diffusivity paths can be viewed in diffusion samples as ‘deep penetrating diffusion fringes,’
departing from lattice diffusion which is determined by the diffusion length √𝐷𝑡. A schematic
showing examples of high-diffusivity paths and corresponding diffusion fringes is shown in
Figure 2.18 [56]. As solute atoms build in these paths, lattice sites in adjacent lattice sites may
readily receive solute, resulting in diffusion fringes [50]. To avoid accentuation of diffusion by
lateral free surfaces, one would remove or neglect the region of material measuring 6√𝐷𝑡 from
the sample edges in a diffusion experiment.
Figure 2.18: Schematic of a material featuring several high diffusivity paths, showing diffusion
fringes [56].
35
It has been experimentally determined in several diffusion systems that GB diffusion in metallic
materials is between 4 and 8 orders of magnitude faster than bulk diffusion [56, 64]. This is the
result of lower 𝑄𝐴 because the atomic packing is lower within the GB and there exists more free
volume for atomic jumps to occur [56]. The activation energy for grain boundary diffusivity is
typically between 40% and 60% that of bulk diffusivity [64]. 𝐷𝑜, on the other hand, are
approximately the same in the lattice and in GBs [55, 56]. It has been demonstrated in several
studies that at high temperature (which is likely dependent on the solvent), diffusivities in both
polycrystalline and monocrystalline sample converge and become equivalent, likely due to grain
growth in the polycrystalline material [55]. Harrison defined a classification of kinetics regimes
for diffusion in polycrystalline samples [56, 64]. These are known as Type A, Type B, and Type
C; a schematic of the geometries of each type is shown in Figure 2.19.
Figure 2.19: Diagram showing the three diffusion regimes in a polycrystalline sample [56].
For ‘Type A,’ diffusion fringes overlap and a near-planar diffusion front is observed. Hart’s
equation for diffusion in a polycrystalline material defines an effective diffusion coefficient 𝐷𝑒𝑓𝑓
which is a mixture of the GB diffusivity 𝐷𝑔𝑏 and the lattice diffusivity 𝐷𝑙 (Eqn 2.18) [56, 64]:
36
𝐷𝑒𝑓𝑓 = 𝑓𝐷𝑔𝑏 + (1 − 𝑓)𝐷𝑙
(Eqn 2.18)
where f is the volume fraction of GBs in the polycrystalline material. It is defined based on the
grain size d, and the GB width δ (Eqn 2.19):
𝑓 =𝑞𝛿
𝑑
(Eqn 2.19)
where q is a constant, approximately equal to 1 for parallel GBs to the diffusion direction, and 3
for cubic grains. δ is on the order of interatomic distance based on High Resolution Electron
Microscopy (HREM) imaging (Figure 2.20) and is commonly referred to be equal to 0.5nm
[56].
Figure 2.20: HREM image of GB region in Au, indicating δ is on the order of interatomic
distance (0.5nm) [56].
‘Type A’ kinetics are valid if the diffusion distance (𝐷𝑡)1/2 is greater than the grain size [56]. If
(𝐷𝑡)1/2 < 𝑑, ‘Type B’ kinetics become valid. Very often this is encountered with large grain
size and low annealing temperature and time and is observed with non-overlapping diffusion
fringes.
37
For solute diffusion, such as Bi in Sn, f must be multiplied by the segregation factor s. s is
defined as the ratio of solute concentration within the GB to that within the lattice and is
inversely proportional to temperature [65]. As solubility c often increases with temperature, s is
also inversely proportional to solubility. s can be estimated directly from solubility data (Eqn
2.20) [66].
𝑠 =𝐾
𝑐
(Eqn 2.20)
where K is a constant and 1 < 𝐾 < 20.
If 𝛿 < √𝐷𝑙𝑡 < 𝑑, ‘Type B’ kinetics become valid, in which diffusion fringes do not overlap.
Diffusion profiles in this regime typically consist of two regions – the first, close to the diffusion
interface, is dominated by lattice diffusion, and the second, penetrating deep into the sample, is a
‘tail’ dominated by GB diffusion [56].
Analysis of these profiles requires plotting 𝑥6
5⁄ on the x axis and ln(𝐶) on the y axis [56, 64].
The beginning of the ‘tail,’ where lattice diffusion becomes negligible, is assumed to begin at a
diffusion distance between 3√𝐷𝑙𝑡 and 4√𝐷𝑙𝑡 [56]. Beyond this depth, the GB tail can be linearly
fitted, with a slope j.
It follows that the triple product P for a semi-infinite source can be calculated (Eqn 2.21) using j,
𝐷𝑙 and diffusion time t, as [56, 64]:
𝑃 = 1.322√𝐷𝑙
𝑡(−𝑗)−5
3⁄
(Eqn 2.21)
For an instantaneous source, the constant 1.322 is replaced with 1.308. P is defined as (Eqn
2.22):
𝑃 = 𝐷𝑔𝑏𝑠𝛿
38
(Eqn 2.22)
Eqn 2.21 is only valid if two conditions (Eqn 2.23) and (Eqn 2.24) are met [56, 64]:
𝛽𝐻 =𝑃
2𝐷𝑙√𝐷𝑙𝑡> 10
(Eqn 2.23)
𝛼𝐻 =𝑠𝛿
2√𝐷𝑙𝑡> 0.1
(Eqn 2.24)
The value of 𝛽𝐻 indicates how pronounced the diffusion fringe is along the GB [56]. Similar to
diffusivity, the thermal activation of P can be calculated using an Arrhenius relationship (Eqn
2.25).
𝑃 = 𝑃𝑜𝑒−
𝑄𝑃𝑅𝑖𝑇⁄
(Eqn 2.25)
The three constituents of the triple product (𝐷𝑔𝑏, s, and δ) also follow an Arrhenius relationship.
Therefore, using some mathematical derivation using rules of exponents, the apparent pre-
exponential 𝑃𝑜 is equal to the product of the pre-exponentials of each constituent (Eqn 2.26), and
the apparent activation energy 𝑄𝐴,𝑃 is equal to the sum of the activation energies of each
constituent (Eqn 2.27). 𝐷𝑔𝑏 and δ increase with temperature, however s decreases with
temperature i.e. a solute is less inclined to segregate at a grain boundary at a higher temperature.
It follows that the slope of the Arrhenius plot for s is positive rather than negative, and 𝑄𝐴,𝑠 has
the opposite sign of 𝑄𝐴,𝑔𝑏 and 𝑄𝐴,𝛿 [56]. In Eqn 2.27, this distinction is shown with 𝑄𝐴,𝑠 being
denoted as negative with respect to 𝑄𝐴,𝑔𝑏 and 𝑄𝐴,𝛿. In most cases, δ is assumed to be constant
and equal to 0.5 nm and thus the terms 𝛿𝑜 and 𝑄𝐴,𝛿 can be omitted from Eqn 2.26 or Eqn 2.27,
respectively.
𝑃𝑜 = 𝐷𝑔𝑏,𝑜𝑠𝑜𝛿𝑜
39
(Eqn 2.26)
𝑄𝑃 = 𝑄𝐴,𝑔𝑏 − 𝑄𝐴,𝑠 + 𝑄𝐴.𝛿
(Eqn 2.27)
To accurately determine all three constituents of P (and their respective Arrhenius parameters),
one would need to perform additional experiments to study ‘Type C’ kinetics, in which the mass
flow is contained within GBs with no penetration into adjacent grains [11][19]. Typically, very
low annealing temperatures and times are required. 𝐷𝑔𝑏 can be determined directly; subsequently
s can be calculated using (Eqn 2.22.
It is recommended that to properly study lattice diffusion, monocrystals should be used to
minimize or eliminate high diffusivity paths. In lieu of monocrystals (due to cost and/or
availability), coarse-grained samples can be used to study lattice diffusion, however it is
important to note that this only minimizes high-diffusivity pathways and care should be taken to
avoid these regions [56].
2.3.3.3 Anisotropy in Non-Cubic Materials
For materials with non-cubic lattices, multiple principal diffusion coefficients are necessary to
determine diffusion properties; these are based on the symmetry of the lattice structure [55, 56,
58]. In general, the treatment of diffusion in such a material involves the use of second-order
tensors, and the various components of the vector equations can be eliminated based on
symmetry. In a cubic crystal, the ‘a’, ‘b’ and ‘c’ axes are indistinguishable from one another,
thus D is the same in all directions (Eqn 2.28) [55, 56]:
𝐷𝑎 = 𝐷𝑏 = 𝐷𝑐 = 𝐷
(Eqn 2.28)
In the tetragonal crystal structure (such as β-Sn), the ‘a’ and ‘b’ axes are indistinguishable from
one another but different from the ‘c’ axis, and thus the diffusivity is anisotropic (Eqn 2.29) [55,
56]:
𝐷𝑎 = 𝐷𝑏 ≠ 𝐷𝑐
40
(Eqn 2.29)
In this thesis, the diffusivity parallel to the unique ‘c’ axis <001> is referred to as 𝐷|| and that
perpendicular to the ‘c’ axis (such as the ‘a’ axis) is referred to as 𝐷⊥. Often, it may be desired to
calculate the diffusivity at some intermediate orientation 𝐷𝜃, subtending an angle θ from the
unique axis with direction vector <uvw>. The equation for 𝐷𝜃 is determined using 𝐷|| and 𝐷⊥, as
well as the angle θ (Eqn 2.30) [55].
𝐷𝜃 = 𝐷|| cos2 𝜃 + 𝐷⊥ sin2 𝜃
(Eqn 2.30)
To determine θ, a modified version of the cosine similarity equation was used (Eqn 2.31) [67].
This calculates the angle θ between two vectors �⃑� and �⃑�.
cos 𝜃 =(�⃑� ∙ �⃑�)
||�⃑�|| ∙ ||�⃑�||
(Eqn 2.31)
The vectors x⃑⃑ (Eqn 2.32) and y⃑⃑ (Eqn 2.33) are related to the <uvw> and <001> directions via
the unit cell parameters a, b, and c of the β-Sn BCT crystal structure.
�⃑� = [𝑢𝑣𝑤
] ∗ [𝑎𝑏𝑐
] = [𝑢𝑎𝑣𝑏𝑤𝑐
] = [0.583𝑢0.583𝑣0.318𝑤
]
(Eqn 2.32)
�⃑� = [001
] ∗ [𝑎𝑏𝑐
] = [𝑢𝑎𝑣𝑏𝑤𝑐
] = [00
0.318]
(Eqn 2.33)
41
2.3.4 Diffusion Study Methods
Several experimental methods for studying diffusion exist; these are classified into two main
groups. Direct methods are based on Fick’s laws, while indirect methods based on atomic
mechanisms. Only the former group is discussed here.
The diffusion couple technique involves two dissimilar metals or alloys being polished, mated
together, and annealed at high temperature to promote diffusion [55, 56, 58, 59]. After annealing,
the couple is cross-sectioned and elemental analysis is performed to collect the composition
profile as a function of diffusion distance. This is accomplished using one of several methods:
Secondary Ion Mass Spectrometry (SIMS), Auger Electron Spectroscopy (AES), or Electron
Probe Microanalysis (EPMA). The latter method is discussed in depth in Section 2.4.2. Very
often, diffusion multiples, with three or more metal/alloy pieces, are created to maximize the
amount of data collected [59].
In cases where the diffusion is more rapid in one direction compared to the other, if one or both
species are susceptible to deformation, or if radiotracers are being used to study self-diffusivity, a
thin film approach (Figure 2.17) may be used, in which the solute species is deposited on a
substrate (solvent). The radiotracer method can be used for both self- and impurity diffusivity
studies; this is highly sensitive but requires radioisotopes with reasonable half lives. Sputter
deposition, evaporation, or electrodeposition are candidate methods to apply a thin film of solute
to a substrate.
It is in most cases desirable to not alter the substrate chemically during the deposition of the
diffusing species, so a non-reactive approach is ideal. Non-reactive thin films can be deposited in
several ways, including evaporation, electrodeposition and sputtering [68]. Evaporation typically
generates immense heat as it is required to vaporize the source material. Electrodeposition has a
much higher throughput than sputtering, however the process may require toxic chemicals, may
generate excess heat, and may also generate contaminants in the deposited layer which may
retard diffusion. Sputtering is typically performed under high vacuum, sputtering targets are
affordable and accessible, and deposition doesn’t generate many artifacts. Sputtering yield is
defined as the average number of emitted atoms per incident particle (typically, Ar (argon) ions
are used as Ar is a noble gas and will not interact with most materials) [69]. Sputtering yield is
42
dependent on a number of parameters, including those of the target material (structure,
composition), as well as on those of the incident particles (energy / power).
The sputter deposition setup consists of a cathode (target) and substrate (anode) with Ar gas
introduced into the chamber, which serves as the medium for electrical discharge [70]. The Ar
gas is ionized, which forms a plasma, visible as a glow in the chamber. A plasma is defined as a
“quasineutral gas which exhibits collective behavior in the presence of an applied
electromechanical load [70].” During sputtering, kinetic energy from ionized Ar+ ions in plasma
is transferred to target atoms at the cathode. Target atoms are ejected once they have acquired
sufficient energy to overcome binding energy of the material (Figure 2.21) [69]. These atoms are
transported through the plasma from cathode to anode and deposit on the substrate, producing a
thin film.
Figure 2.21: Schematic of sputtering process [69]
Depth profiling is often used if the diffusion distance is short, which may involve mechanical or
sputter sectioning. Cross-sectioning is an alternative method if the diffusion distance is long,
followed by SIMS, AES, or EPMA for profiling.
2.3.5 Interdiffusion and Impurity Diffusion
In diffusion systems in which two dissimilar materials A and B are joined together and annealed,
there will be a simultaneous interchange of matter across the interface, known as interdiffusion,
in which A diffuses into B and vice versa. In almost all cases, the intrinsic diffusivities of A in B
43
and B in A are different from one another, which yields interesting phenomena such as the
movement of the diffusion interface (in the direction of the highest diffusivity) as well as the
Kirkendall effect. These are briefly discussed in this section. Finally, some empirical diffusion
data in Sn is compared, including self-diffusivity and impurity diffusivities of some common
alloying elements employed in Pb-free solder alloys.
2.3.5.1 Impurity Diffusion
In substitutional or interstitial diffusion, when the solute has a nominal concentration of less than
1 at%, it is often defined as impurity diffusion [56]. While several abnormal cases exist, normal
impurity diffusion typically features the following characteristics [56]:
• At temperatures between 2 3⁄ 𝑇𝑚 and 𝑇𝑚, impurity diffusivities are within two orders of
magnitude of the self-diffusivity.
• 𝐷𝑜 for impurity diffusion is within one order of magnitude of that for self-diffusion.
• 𝑄𝐴 for impurity diffusion is within 25% of that for self-diffusion.
There are not any encompassing trends which can be used to predict impurity diffusivities across
a range of solutes and solvents. However it has been noted that substitutional impurities behave
similarly in solvents with similar lattice structure. It is agreed that the difference in diffusivities
between various solute atoms can be explained by variance in atom-vacancy exchange rates.
2.3.5.2 Interdiffusion
The diffusion coefficient described in the previous discussion on inverse methods is known as
the interdiffusion coefficient �̃�, which describes the intermixing of a two-part diffusion system
[55, 56]. The driving force is the chemical potential across the entire diffusion couple [57]. As
interdiffusion constitutes the simultaneous diffusion of two species in the same system, there
exist intrinsic diffusion coefficients 𝐷𝑖 of each species i, often different from one another. This
results in a net mass flow, known the Kirkendall effect, in which the location of the diffusion
interface is shifted with respect to the ends of the diffusion couple. The evidence of the
Kirkendall effect has also helped support the vacancy-mediated mechanism of diffusion. When
extensive interdiffusion occurs in a system where the intrinsic diffusivities vary greatly, or in
non-equilibrium conditions, a buildup of vacancies may occur, which develop into porosity at the
diffusion interface, known as Kirkendall voiding [55]. These features are sometimes observed in
44
the IMCs of solder joints during the formation of a Cu3Sn IMC situated between the Cu substrate
and the Cu6Sn5 IMC (Figure 2.22) after either extensive holding times above liquidus or after
long-term high temperature isothermal aging [71].
Figure 2.22: Kirkendall voiding in Sn-Pb (a) and Pb-free (b) solder joints during the formation
of Cu3Sn IMC [71].
Darken performed separate analysis of the Kirkendall effect, assuming the diffusion couple
consisting of parts A and B is sufficiently thick such that the concentration gradients never reach
the ends of the couple, and that the molar volume of the couple remains constant [58]. Darken’s
first law (Eqn 2.34) considers the marker speed 𝑢𝑀, or speed of the diffusion interface is a
function of the molar concentrations 𝑁𝑖 and intrinsic diffusivities 𝐷𝑖 of the two species [58]:
𝑢𝑀 = 𝐷𝐴
𝜕𝑁𝐴
𝜕𝑥+ 𝐷𝐵
𝜕𝑁𝐵
𝜕𝑥
(Eqn 2.34)
After combination of Darken’s first law with the convective form of the diffusion equation, we
arrive at Darken’s second equation (Eqn 2.35), in which the interdiffusion coefficient �̃� can be
described as a function of 𝑁𝑖 and 𝐷𝑖 [58]:
�̃� = 𝑁𝐵𝐷𝐴 + 𝑁𝐴𝐷𝐵
(Eqn 2.35)
(a) (b)
45
Darken analysis is commonly used in conjunction with inverse methods such as the Boltzmann-
Matano, Sauer-Friese and Hall methods described in Section 2.3.1.2.
2.3.6 Diffusion in Sn-based Systems
Diffusion data from several systems (in which Sn is the solvent) exists in the literature and is
necessary to consider prior to evaluating diffusivity in a previously uncharacterized system (such
as Sn-Bi). The self-diffusion properties in β-Sn are first considered, followed by several
impurities. Finally, further discussion of several unique impurity diffusion systems, in which
diffusivity is anomalously low and activation energy is high, is warranted.
2.3.6.1 Self and Impurity Diffusion
Significant work has been done to characterize the diffusion behavior of various solutes in Sn.
Table 2.4 contains diffusion data from the literature - 𝐷𝑜 and 𝑄𝐴, as well as the estimated
diffusion coefficients at 125°C in each of the ‘a’ (𝐷⊥) and ‘c’ (𝐷||) axes of the Sn lattice,
computed using these Arrhenius parameters). In addition, the atomic radii r of these solutes are
included – generally, the diffusion coefficient of a solute will decrease with increasing r as larger
lattice distortions are necessary to accommodate such solute atoms [58].
Table 2.4: Impurity Diffusivity data in Sn-X systems (125°C diffusivity estimated using
Arrhenius parameters Do and QA)
Solute r (nm)
Do (cm2/s) QA (kJ/mol) D at 125°C (cm2/s) Anisotropy
ratio 𝑫||/
𝑫⊥
Ref. || to ‘c’
axis
⊥ to
‘c’ axis
|| to
‘c’
axis
⊥ to
‘c’
axis
|| to ‘c’
axis
⊥ to ‘c’
axis
Sn 0.140 12.8 21.0 108.9 108.5 6.52 x 10-
14
1.21 x 10-
13 0.54 [72]
Ag 0.144 0.0071 0.18 51.5 77.0 1.24 x 10-9 1.41 x 10-
11 90.91 [73]
Cu 0.128 0.001 0.0024 16.7 33.1 6.43 x 10-6 1.09 x 10-
7 58.82 [74]
Ni 0.124 0.020 0.019 18.1 54.2 8.38 x 10-5 1.44 x 10-
9 5.81 x 105 [75]
Sb 0.140 79.1 76.6 121.8 123.1 8.17 x 10-
15
5.34 x 10-
15 1.53 [72]
In 0.167 12.2 34.1 107.2 108.0 1.04 x 10-
13
2.28 x 10-
13 0.46 [76]
Au 0.144 0.0058 0.16 46.1 74.1 5.16 x 10-9 3.01 x 10-
11 171.42 [73]
46
The vast majority of these studies employed a tracer method. The diffusion coefficients, D, at
125°C (log scale) are plotted against r and there is relatively good agreement with this trend
(Figure 2.23). Sb appears to be an outlier. Assuming the diffusivity of Bi follows this trend with
respect to atomic radius (156pm), an rough estimate of the range of diffusivities of Bi in Sn is
annotated on the plot using trendlines. The data in Table 2.4 and Figure 2.23 also suggests that
the anisotropy ratio (defined as the ratio between D|| and D⊥) for diffusivity in Sn can vary
depending on the solute.
Figure 2.23: Plot of diffusivity (log scale) against solute atomic radius for impurity diffusivity in
Sn at 125°C, for orientations parallel and perpendicular to the ‘c’ axis.
2.3.6.2 Fast Impurity Diffusion
It is evident that several species, such as Cu, Ni, Zn, Ag, and Au, demonstrate very high
diffusivities and low activation energies, particularly in the direction parallel to the ‘c’ axis. In
addition, the anisotropy ratios of these species are also very high. It was noted in Section 2.3.5.1
Ni
Cu
Bi?
Sb
In
Zn Au Ag
47
that typical impurity diffusivities are within two orders of magnitude of the self-diffusivity at
23⁄ 𝑇𝑚 < 𝑇 < 𝑇𝑚 and typical activation energies are within 25% of that of the self-diffusivity.
The diffusion properties of the aforementioned species fall outside both these ranges, suggesting
a mechanism other than substitutional (vacancy-mediated) diffusion, is dominant. In contrast,
several species such as Sb and In demonstrate properties that suggest vacancy diffusion.
Several studies have suggested an interstitial mechanism may be responsible for this anomalous
behavior. Indeed, the interstitial spacings running parallel to the ‘c’ axis are greater than those
along the ‘a’ axis in b-Sn, forming ‘large open chimneys’ [72, 74]. Hägg’s rule, based on the
hard sphere model, suggests that a solute atom may be able to diffuse interstitially if the diameter
ratio between solute and solvent is less than 0.59, or the corresponding volume ratio is less than
0.21 [77, 78]. However, in the case of all the species in Table 2.4, this condition is violated.
Several mechanisms have been proposed to explain this abnormality; most revolve around the
concept that the ionic radius, not the atomic radius, determines whether an interstitial solid
solution will form [73]. In addition, the valence of the diffusing species may also be a factor.
This has been suggested to occur for several of the above species in β-Sn. For example, Huang
notes that while Cd and Zn both form divalent ions, the smaller ionic radius of Zn2+ (0.074nm)
compared with Cd2+ (0.097nm) explains the tendency for the former to diffuse more rapidly in β-
Sn [72].
Miedema and Niessen developed a semiempirical model (Eqn 2.36) based on similar concepts in
which the change in the molar volume of an impurity A Δ𝑉𝑖𝑚𝑝 with initial molar volume 𝑉𝑖𝑚𝑝0
when it is added to a solvent B with molar volume 𝑉0 is considered [78]. The charge transfer of
electrons, which is dependent on the difference in electronegativity 𝜙 and electron density at the
boundary of the Wigner-Seitz cell 𝑛𝑊𝑆 between the solute and solvent, also dictates whether a
volumetric change will occur:
Δ𝑉𝑖𝑚𝑝(𝐴) =𝑃0(𝑉𝑖𝑚𝑝
0)2
3⁄(𝜙𝐴 − 𝜙𝐵)
(𝑛𝑊𝑆𝐴)−1
3⁄ + (𝑛𝑊𝑆𝐵)−1
3⁄[(𝑛𝑊𝑆
𝐴)−1 − (𝑛𝑊𝑆𝐵)−1]
(Eqn 2.36)
48
Where 𝑃0 is an empirical constant equal to 1.5. The ratio of the corrected molar volume of the
solute 𝑉𝑖𝑚𝑝 = 𝑉𝑖𝑚𝑝0 + Δ𝑉𝑖𝑚𝑝 to that of the solvent 𝑉0 dictates whether an interstitial solid
solution, and thus fast diffusion, will occur. Fast diffusion typically occurs when this ratio
𝑉𝑖𝑚𝑝
𝑉0⁄ is less than 0.65 to 0.70, with the transition from slow (or normal) to fast diffusion
taking place when the ratio equals approximately 0.55. Several corrected molar volumes using
the Miedema-Niessen model are given in Table 2.5. Several of these data (black font) were
taken directly from Bakker’s paper [78]. Others (in red font) were calculated using the 𝑉0, 𝜙,
and 𝑛𝑊𝑆 parameters experimentally derived and tabulated by Miedema and Niessen for over
seventy elemental species [79]. In addition, each solute is classified as either a ‘slow’ or ‘fast’
diffuser in Sn, based on the volume ratio.
Table 2.5: Miedema-Niessen Analysis of Several Diffusing Species in Sn, with 𝑽𝟎 =
𝟏𝟔. 𝟑𝟎 𝒄𝒎𝟑
𝒎𝒐𝒍⁄
Solute 𝑽𝒊𝒎𝒑𝟎 𝚫𝑽𝒊𝒎𝒑 𝑽𝒊𝒎𝒑
𝑽𝒊𝒎𝒑
𝑽𝟎⁄ Notes
Cu 7.12 -0.23 6.89 0.42 Fast
Au 10.19 -1.30 8.89 0.55 Fast
Ag 10.25 -0.12 10.13 0.62 Fast
Cd 13.00 0 13.00 0.80 Slow
Hg 14.08 0 14.08 0.86 Slow
In 15.73 -0.14 15.59 0.96 Slow
Sb 16.96 0 16.96 1.04 Slow
Ni 6.60 -1.36 5.25 0.322 Fast
Zn 9.16 0.019 9.18 0.563 Fast
Bi 21.37 0 21.37 1.31 Slow?
The data in Table 2.5 agrees with that in Table 2.4 and the conclusions given in the literature. It
is noted that some Δ𝑉𝑖𝑚𝑝 values are equal to zero, implying no volumetric change; this is
because either the 𝜙 or 𝑛𝑊𝑆 values for the solute are equivalent to those of Sn and as such,
corresponding term(s) in (Eqn 2.36 are equal to zero. This is the case for the estimated result for
Bi in Sn, which also suggests given the large ratio (1.31), that Bi is not expected to undergo
interstitial or fast diffusion in Sn, and should demonstrate behavior suggesting a vacancy-
mediated diffusion mechanism.
49
As observed in earlier work, the solid-state diffusion of Bi in Sn is largely responsible for the
microstructural evolution of the alloy. Bismuth-containing alloys have demonstrated interesting
microstructural changes after aging, at both elevated and room temperatures. Bulk, or matrix
diffusion, is more likely to be dominant in the early stages of the life of a solder joint – as-
manufactured joints typically consist of only a few large grains and any diffusion of Bi in these
joints would likely occur entirely within the grains [9]. The anisotropic nature of the β-Sn
tetragonal crystal structure indicates that diffusion in a Bi-bearing solder joint might vary
significantly depending on the orientation of the joint. It is possible that over the lifetime of the
solder joint Bi could segregate in undesirable locations. It has been shown that the segregation
(which may be caused by anisotropic diffusion) of Bi at already brittle IMC interfaces may
degrade reliability [80]. Over time, as Bi becomes more evenly distributed throughout the alloy,
Bi-containing alloys undergo extensive β-Sn recrystallization [7]. Grain boundary diffusion will
therefore become more prevalent. It is therefore important to study diffusion of Bi in
monocrystalline (or coarse-grained) Sn (representing as-solidified solder joints) as well as in
polycrystalline Sn (representing aged solder joints). The quantification of these properties will
therefore aid in the understanding of the behavior of Bi and its effects on properties and
reliability. Further, this data may aid in the development of predictive models of alloy
performance.
2.4 Analytical Techniques
Scanning Electron Microscopy (SEM) is a useful tool for the characterization of materials such
as metals and alloys. In the SEM, a focused beam of primary electrons (PE) is used to produce
images of a sample. When the beam interacts with the sample, a wide range of interactions
occur, each of which can provide the analyst with different information about the sample,
including phases, grain structure, and elemental composition. The electron beam may undergo
scattering events (either elastic or inelastic), which produce a range of signals. In an elastic
scattering event, the kinetic energy of the electrons is largely unchanged, while their direction of
travel can vary. In an inelastic scattering event, there is a loss of energy to the PE; this loss is
transferred to the sample and produces different types of signal.
50
2.4.1 Imaging Techniques
Two of the more commonly employed signals in SEM are backscattered (BSE) and secondary
(SE) electrons. Backscattered electrons arise from elastic collisions and possess very high
energies (the majority with more than 60% of the initial beam energy). Secondary electrons are
produced via inelastic collisions, ejecting conduction electrons from the sample. These electrons
typically possess energies of a far lesser magnitude than BSE (<50eV), and are detected from
near the sample surface.
The overall interaction behavior of the beam electrons with the sample can be described using
the interaction volume [81]. The interaction volume (Figure 2.24) is strongly affected by atomic
number (Z), beam energy 𝐸𝑜 (voltage), and sample tilt. A sample with a higher Z will have larger
atomic size and a greater likelihood to scatter beam electrons – this leads to a smaller interaction
volume. Using a higher 𝐸𝑜, the electrons are given sufficiently more energy to penetrate deeper
into the sample, thus the interaction volume is larger. When the sample is tilted, the interaction
volume becomes smaller, as a larger proportion of electrons propagate nearer the surface of the
sample and can thus escape more readily [81]. This feature is used to great effect in Electron
Backscatter Diffraction (EBSD) analysis (Section 2.4.3).
Figure 2.24: Effect of beam voltage and atomic number on the dimensions of the interaction
volume [82].
In SEM imaging, contrast is used to differentiate between the features in a sample, and the
various types of contrast are more observable in different imaging modes. Backscattered electron
signal has a very strong dependence on atomic number – a sample with higher Z will undergo
51
more elastic scattering events, and therefore a larger yield of BSE than one with low Z. In a two-
phase material, for example a Sn-Bi alloy, Z contrast is used to differentiate between Bi (Z = 83)
and Sn (Z = 50). Channeling contrast is caused by orientation differences of grains in a
polycrystalline material and is easily observed in BSE imaging [83].
2.4.2 X-Ray Microanalysis
Scanning electron microscopy is often used to perform elemental analysis using the techniques
of energy-dispersive x-ray spectrometry (EDS) or wavelength-dispersive x-ray spectrometry
(WDS). When incident electrons interact with the sample, a lower-level electron may be ejected,
and the atom enters an excited state. An electron from an outer, higher-energy shell is transferred
to fill this recently vacated site and de-excite the atom. The surplus of energy is emitted from the
sample as a characteristic x-ray [81, 82, 83]. The energy of the x-ray is dictated by the difference
in binding energies between the outer and inner shells (unique to each shell) and is also
dependent on the atomic number of the atom (unique to each element).
In EDS, characteristic x-rays are differentiated from one another based on x-ray energy. All x-
rays with a wide range of energies are collected and analyzed simultaneously via the production
of electron-hole pairs in the detector, which is composed of a semiconductor such as Si or Ge
[83]. These pairs form a charge pulse, which is converted into a voltage pulse and amplified [81].
The main disadvantage of EDS is that the energy resolution is quite poor, with peak widths
greater than 100eV [82]. This peak broadening is largely caused by a statistical (Gaussian)
distribution in the number of charge carriers that represent a specific photon energy, and may
prove problematic when differentiating between very similar elements, or quantifying very low
(ppm) concentrations [81]. The accuracy of EDS as a quantitative tool is limited; its main utility
is to quickly identify species present in a sample.
A WDS system is found most often in a specialized type of electron microscope known as an
Electron Probe Microanalyzer (EPMA). The main difference between WDS and EDS is that
characteristic x-rays are differentiated based on their wavelength λ rather than energy [82]. The
technique makes use of the Bragg condition (Eqn 2.37), which suggests a strong reflection of
some incident radiation is only possible if a specific angle 𝜃𝐵 between the incident beam and
atomic planes with spacing 𝑑𝑎 [81, 82, 83].
52
𝜆 = 2𝑑𝑎 sin 𝜃𝐵
(Eqn 2.37)
In the EPMA, a WDS system consists of a detector combined with several analyzing crystals
(allowing for a large range of λ to be collected) each with different 𝑑𝑎-spacings (Figure 2.25a).
Only x-rays with specific λ will diffract and arrive at the detector. Each diffracted x-ray produces
a current pulse in the detector, which are summed to produce a count rate [82]. Higher
processing times are required compared with EDS, as only one λ can be measured at any given
time [82]. However, because the Bragg condition is very selective, x-ray peaks collected in WDS
are extremely narrow, over an order of magnitude less than those for EDS. This allows for
similar species to be distinguished from one another, and very low concentrations can be reliably
and accurately measured (Figure 2.25b).
Figure 2.25: Wavelength Dispersive X-Ray Spectrometry schematic [82] (a) and superior peak
resolution compared with EDS [84] (b).
2.4.3 Electron Backscatter Diffraction (EBSD)
Electron Backscatter Diffraction (EBSD) resembles WDS in that the signal is dependent on
electron diffraction; the main difference is that the diffracted electrons are collected directly from
the sample. Because the Bragg and structure factor conditions are very specific, only a small
fraction of incident beam electrons will diffract [83]. The sample is tilted at a high angle, usually
70° to the horizontal, to minimize the interaction volume and obtain the greatest possible
(a) (b)
53
emission from the near surface of the sample. These electrons form beams of characteristic
energy and diverge outwards in a conical pattern (Figure 2.26a). These beams are collected by a
phosphor detector; the edges of these cones (which form a 2D projection on the detector screen)
are known as Kikuchi lines and constitute an electron backscatter pattern (EBSP). The
geometrical relationships between Kikuchi lines represent the geometry of the actual crystal
lattice (Figure 2.26b&c) in reciprocal space and can be used to determine properties of the
sample such as grain orientation, GB types, and local misorientation of the grains.
Figure 2.26: Formation of Kikuchi lines [85] (a); Geometrical relationships between Kikuchi
lines [86] (b); Example EBSP image of cadmium. The zone axis with the three intersecting
Kikuchi bands is the <001> zone axis, which contains sixfold symmetry (hexagonal crystal
structure) [85] (c).
The stereographic projection is a projection of all the possible EBSPs in a material onto a plane.
The pole figure, based on the stereographic projection, indicates which sample directions are
aligned with which crystallographic poles [85]. Depending on the symmetry of the crystal lattice,
the stereographic projection can be collapsed to a spherical triangle (Figure 2.27); the size of the
triangle increases with decreasing symmetry [83]. This spherical triangle is used as the basis for
the inverse pole figure (IPF), which shows which poles are aligned with the directions in the
sample - X (horizontal direction), Y (vertical direction), and Z (orthogonal direction) (Figure
2.28), with the X and Y directions in the sample surface plane [85]. Using the IPF, the texture of
a sample can be defined by the orientations within. IPFs may be used in conjunction with the
(a) (b) (c)
54
stereographic projection to determine the precise orientation (indices) of a site of interest in a
sample with respect to the sample directions X, Y, and/or Z.
Figure 2.27: The stereographic projection of Sn with the <0 0 1> primary axis. The symmetry of
the tetragonal crystal structure allows the stereographic projection to be collapsed into a
spherical triangle bounded by the <0 0 1>, <0 1 0> and <1 1 0> poles.
Electron Backscatter Diffraction is often used for mapping. In an EBSD map, the coloring
represents a specific crystal orientation; the most common and intuitive coloring scheme is based
on the IPF and uses a red-green-blue (RGB) scale. There are three possible IPF colorings,
corresponding to the three sample directions. At least two maps are required for analysis; one
map alone may suggest the sample has a very strong texture, while another may reveal the
sample to have a weaker texture. Euler coloring is another method to portray EBSD map data;
this is based on the Euler angles [85]. Examples of EBSD maps of a β-Sn sample are given in
Figure 2.29, the first three corresponding to the IPFs in Figure 2.28. The ‘X’ map shows a
stronger texture than the ‘Y’ and 'Z’ maps, and a weaker texture than the Euler map. Additional
map elements that are possible to include (often in post-processing software) include GBs,
special boundaries, and strain analysis.
55
Figure 2.28: Inverse pole figures showing the texture of a β-Sn sample in each of the X (a), Y
(b), and Z (c) directions. Coloring is based on the RGB scale in (d). X, Y, and Z directions are
shown on a cylindrical sample with the circular face corresponding with the sample surface (e).
Figure 2.29: EBSD maps showing the texture of a β-Sn sample. IPF maps in the X (a), Y (b),
and Z (c) directions; Euler map (d).
(a) (b)
(d)
(c)
(e)
(a) (b) (c)
(d)
56
Chapter 3 The Effects of Aging on Microstructure and Properties of Bismuth-
Containing, Lead-Free Solder Alloys
3.1 Introduction
In 2009, Celestica began a low-melt program, which aimed to develop new Pb-free alloy
compositions to meet the stringent requirements of high reliability industries (such as aerospace
and defense), while satisfying the manufacturability standards in the electronics manufacturing
industry, along with the environmental requirements imposed by the Restriction of Hazardous
Substances (RoHS) [6]. This was in response to the need by the industry to find a reliable Pb-free
alloy to replace SnPb solders; a number of researchers have determined traditional Pb-free alloys
(such as SAC 305) demonstrate poor reliability, and, over time, a loss in mechanical properties
[1, 2, 87]. After a series of experiments, a cell of 23 alloys was narrowed down to seven,
including ‘Violet’ (Sn-2.25Ag-0.5Cu-6.0Bi), an alloy co-invented by Celestica and the
University of Toronto [6, 88].
In one preliminary study, several Bi-containing alloys were subjected to an aging treatment at
100°C for either 25 hours or 100 hours [6]. The microstructure was observed to become more
homogenous, with Bi precipitating equally throughout the Sn matrix, with time. In addition, the
hardness of the alloy was observed to increase with aging, with a significant improvement seen
after short aging times. The results from this work inspired two graduate projects at the
University of Toronto, including this thesis, focused on more in-depth microstructural
characterization.
Matijevic focused on determining 𝑇𝑠𝑜𝑙 of the practical alloy compositions, in order to determine
an ideal ‘aging’ temperature (heat treatment) which could be used, intentionally, to improve
microstructure and properties of the alloy [40]. It was found that at sub-solvus temperatures,
second phase Bi precipitates may coarsen and embrittle the alloy as a result of Ostwald ripening,
while very uniform particle size can be achieved by aging above solvus (Figure 2.8c).
A heat treatment of 120°C was proposed, and the effects of this heat treatment on the creep
performance of the alloy were evaluated [7, 40]. Nanoindentation creep was performed on the
alloy after solidification, following exposure at 70°C (below 𝑇𝑠𝑜𝑙, simulating device operating
57
temperatures), and after a sequential exposure, firstly at 70°C, followed by the 120°C heat
treatment of 120°C. It was found that creep resistance could be markedly improved after this heat
treatment (Figure 2.9a).
This chapter contains a more extensive study of aging alloys and conditions, to supplement the
earlier work. While not entirely novel, the results in this aging study serve to confirm the
findings in the literature that the strength of Bi-containing alloys can be preserved or even
improved over time, as compared to traditional lead-free alloys, whose properties degrade over
time. These results, along with those in related projects under the Refined Manufacturing
Acceleration Process (ReMAP) umbrella, helped support the development of a patented thermal
restoration treatment by the ReMAP Pb-free team (Figure 2.10) [4, 6, 7, 40].
3.2 Experimental Methodology
The methodology used in this study strongly resembled that used by Snugovsky [6]. The alloys
selected were not chosen based on manufacturability (pasty range, melting temperature,
wettability); the focus was both to have varying amounts of Bi in the alloy and to have as few
additional alloying elements as possible (i.e. Ag, Cu). This decision was made so focus could be
placed on the effects of Bi on the properties of the alloy, without complicating the system by
including additional alloying elements. Several alloys are binary Sn-Bi alloys; the others are
ternary alloys containing a small amount (0.7 wt%) of Cu, which are more practical
compositions of interest in the industry. Alloy microstructure was examined in the Scanning
Electron Microscope (SEM) followed by Rockwell superficial hardness testing.
3.2.1 Test Matrix
Six alloys were considered for this study. Five contain Bi and were selected based on the Sn-Bi
binary phase diagram and are different from the earlier low-melt study (Figure 2.7). The sixth is
a Pb-free baseline alloy, SAC 305.
• Sn-1Bi – expected to be a single-phase material, with all Bi present in solid solution with
Sn;
• Sn-3Bi – expected to have a small amount of Bi precipitation in addition to the solid-
solution strengthened Sn matrix. 3wt% is the amount of Bi that is included in one of the
alloys being studied by Celestica/ReMAP (“Senju” – Sn-2.0Ag-0.5Cu-3.0Bi) [88].
58
• Sn-5Bi – expected to have a significant amount of Bi precipitation in addition to the
solid-solution strengthened Sn matrix. The 5wt% level of Bi has been observed to be
approximately the ideal amount of Bi in lead-free alloys, at which further additions do
not contribute significant improvements to mechanical properties. It has been shown that
in the Sn-10Bi alloy, Bi precipitates encapsulate the Sn grains, producing a brittle GB
region (Figure 3.1) [89].
• Sn-0.7Cu-1Bi – expected to have two phases, with Cu6Sn5 intermetallics present in a Sn
matrix solid-solution strengthened by Bi.
• Sn-0.7Cu-5Bi – expected to have three phases, with Bi precipitates and Cu6Sn5
intermetallics present in Sn matrix solid-solution strengthened by Bi.
• Sn-3.5Ag-0.5Cu – also known as SAC305, this is one of the lead-free alloys of choice in
industry today, and was the baseline alloy of this portion of the study.
Figure 3.1: Sn-10Bi aged at 100°C for 3 days, showing ‘decoration’ of GBs by Bi.
The two Sn-Cu-Bi alloys follow a similar composition trend as two alloys that were considered
in an earlier joint Celestica / UofT study – Sn-0.7Cu-7Bi (Sunflower) and Sn-0.7Cu-10Bi
(Cornflower) – but contain less Bi, as it has been shown that at very high Bi content, the
encapsulation of Sn grains by Bi precipitates occurs, which may severely weaken GBs and result
in reduced strength [6, 89]. The corresponding binary Sn-Bi alloys (with the same Bi content)
were also included, to analyze the effect of Bi on the alloy without any potentially influential
59
effects of intermetallic phases such as Ag3Sn or Cu6Sn5, and to directly compare the effects of
including 0.7wt% Cu.
Samples were left either in as-cast conditions, or subjected to aging at room temperature (in the
laboratory, roughly 25°C) or elevated temperature (in an oven at 70°C, 100°C or 125°C). These
temperatures are typically selected in studies focusing on aging of solder alloys [1, 2, 5, 6, 87].
For all alloys, room temperature aging was performed for 10, 28, 63, 112, 168, 252, and 365
days. Elevated temperature aging was performed for 1, 3, 7, or 14 days for all alloys at all three
temperatures.
3.2.2 Sample Preparation
Samples were 10g in size to allow for a uniform temperature gradient during the aging process.
The Bi-containing samples were prepared using large 50g ingots of two master alloys purchased
from ACI Alloys – one containing 7 wt% Bi (either Sn-0.7Cu-7Bi or Sn-7Bi), the other containing
no Bi (either Sn-0.7Cu or pure Sn). These ingots were melted, cut in half, and remelted into smaller
pieces, Appropriate amounts of each were weighed to yield the desired alloy compositions. The
SAC 305 samples were prepared from larger 50g ingots which were similarly melted and cut in
half repeatedly to ensure near-identical composition between samples. All samples were melted in
an alumina crucible on a laboratory hot plate. Upon melting, these were then removed from the
hot plate and allowed to cool on an iron slab at room temperature. Using a flatbed chart recorder
equipped to a thermocouple, the cooling rate across the alloy liquidus was approximately 3°C/s,
similar to that experienced by a solder joint in a typical assembly process. It is noted that since the
sample volume is several orders of magnitude greater than that in a solder joint, it is difficult to
precisely replicate assembly conditions.
For each alloy / aging condition, two samples were prepared. Samples underwent a standard
metallographic sample preparation procedure to allow the microstructure to be observed in SEM
as well as create a flat, planar surface for Rockwell hardness testing. This procedure is shown in
Table 3.1.
60
Table 3.1: Metallographic Sample Preparation Procedure for Aging Samples
# Step Comments
1 Wet grind at 250
rpm
SiC papers (FEPA scale) used were 120, 320, and 500
Grinding for 1-2 minutes per paper
2 Wet grind at 125
rpm
SiC papers (FEPA scale) used were 800, 1200, 2400, and 4000
Grinding for 2-3 minutes per paper
3 Cleaning Rinse with water between grinding steps; water and ethanol rinse
after 4000 paper
4 Polish at 125 rpm Diamond slurries of 3µm and 1µm
5 Cleaning Rinse with water and ethanol after polishing steps
6 Final polishing Colloidal silica + distilled water slurry; hand polishing
7 Final cleaning Rinse with water, ultrasonic cleaning with ethanol for 40s, rinse with
ethanol
8 Drying Aspirator vacuum system (using running tap water) used to remove
residual moisture prior to SEM analysis
3.2.3 Microstructure Evaluation
Microstructure of both samples of each alloy / aging condition were first examined using a
Hitachi SU-3500 SEM, operated at 20kV. To ensure the observed microstructure was truly
representative of the bulk alloy microstructure and that of the aging condition, sample
preparation was done immediately preceding SEM inspection. Imaging was conducted at 300x,
500x, 1000x, and 2000x magnification; both secondary electron (SE) and backscattered electron
(BSE) images were collected. Backscattered electron images are shown in this thesis as the
various phases (β-Sn, Cu6Sn5 intermetallics, and Bi precipitates) are easier to distinguish via Z-
contrast [81].
3.2.4 Mechanical Property Testing
After SEM analysis, samples underwent hardness testing to evaluate the absolute strength of the
alloy via time-dependent deformation. Testing was performed using a Rockwell hardness tester
equipped with a ¼” diameter tungsten carbide ball indenter. Rockwell superficial hardness testing
was conducted using the 15X scale. This scale was determined as the most appropriate for
measuring the hardness for all six alloys; Rockwell hardness numbers fall in the range of 0-100.
The 15T scale (using a 1/8” diameter ball) was initially used to test Celestica’s Bi-containing
alloys, however SAC 305 was too soft to be measured using this scale, necessitating the acquisition
of a ¼” diameter indenter [6].
61
The sample surface allowed for up to nine measurements to be collected per sample, with indent
spacing adhering to the American Section of the International Association for Testing Materials
(ASTM) standard [90]. To test hardness (Figure 3.2), the indenter head is brought into contact
with the sample and a primary load of 3 kgf is applied (this load is applied for all Rockwell
superficial hardness scales) [90]. A secondary load is then applied and held for some (dwell)
time. The secondary load is dependent on the scale; in this study, the 15X scale indicates the
total (sum of primary and secondary) load is 15 kgf. Therefore, the secondary load is 12 kgf [90].
The secondary load is removed, returning to the primary load, which is then held for some
(dwell) time. The tester measures the final depth of the indentation, and the Rockwell hardness is
determined via the difference between the indentation depths under the primary load (before and
after the secondary load is applied) [90]. For each condition, two samples were tested, and
eighteen hardness measurements were taken. Of these, the two highest and two lowest
measurements were discarded, and the remaining measurements were averaged. Data is given via
the average hardness accompanied by standard error.
Figure 3.2: Rockwell Hardness test method [90]
62
3.3 Results & Discussion
Microstructure and hardness results are divided into the three aging conditions studied – as-cast,
elevated temperature aged (70°C, 100°C, and 125°C), and room temperature aged.
3.3.1 As-Cast
SAC 305 solidified into a dendritic microstructure, with fine Ag3Sn and Cu6Sn5 nucleating in the
interdendritic spaces. This microstructure appears to suggest that β-Sn nucleated first, followed by
binary eutectic consisting of β-Sn and Ag3Sn, followed by ternary eutectic consisting of β-Sn,
Ag3Sn, and Cu6Sn5 (Figure 3.3a) [91]. No primary Ag3Sn was present, which is periodically
observed in SAC 305 solder joints – this is likely related to cooling rate and sample size
differences. It is still likely that some undercooling was present during solidification.
Figure 3.3: As cast microstructure of SAC 305 (a), Sn-0.7Cu-1Bi (b), Sn-5Bi (c), and Sn-0.7Cu-
5Bi, with all pertinent phases labelled (d).
β-Sn dendrites
Bi precipitates
Cu6Sn5
(a) (b)
(d)
(c) (a) (b) (c)
(d)
63
All Bi-containing alloys also take on a dendritic structure. As the solid solubility of Bi in Sn at
room temperature is roughly 2wt%, Bi is not visible in Sn-1Bi or Sn-0.7Cu-1Bi; in the latter alloy
Cu6Sn5 is present in the interdendritic spaces as in SAC 305, likely present in a binary eutectic
along with β-Sn. In the case of the two alloys with 5 wt% Bi, the solid solubility of Bi in Sn is
exceeded, and Bi is present in both solid-solution and as a second phase precipitate (Figure
3.3b&c). More Bi precipitates are observable in Sn-0.7Cu-5Bi than in Sn-5Bi – there is limited
solubility of Cu in the β-Sn matrix, which likely reduces the solid solubility of Bi in Sn. While this
limits the solid solution strengthening effect of Bi in the alloy, Cu additions to the alloy are
beneficial for reducing dissolution from the substrate, as indicated in Section 2.1.2. In these alloys,
Bi tends to nucleate in large clusters, decorating the interdendritic spaces, and the microstructure
in general is quite non-uniform. This Bi distribution seems to indicate that these precipitates form
late in the solidification process. As the alloy cools, the solubility of Bi in Sn decreases, and Bi is
rejected from the β-Sn phase into the interdendritic spaces. Sufficient accumulation of Bi atoms
results in the nucleation and growth of a secondary Bi precipitate phase. The Cu6Sn5 IMCs in Sn-
0.7Cu-5Bi form in a similar morphology to the other alloys containing these phases. The phases
in Sn-0.7Cu-5Bi are labelled (Figure 3.3d).
Hardness testing of the as-cast alloys indicates that alloys with a higher bismuth content show
greater hardness (Figure 3.4), for both the binary and ternary alloys. Solid-solution strengthening
and precipitation hardening are likely the primary mechanisms behind these changes. The hardness
values of SAC and Sn-3Bi in the as-cast conditions lie somewhere between those of the alloys
containing 1 wt% Bi and those with 5 wt% Bi.
In addition, the inclusion of Cu to the alloy has a larger effect on hardness when the alloy contains
1wt% Bi than when it contains 5wt% Bi. (i.e. the difference in hardness is larger between Sn-
0.7Cu-1Bi and Sn-1Bi than between Sn-0.7Cu-5Bi and Sn-5Bi). This is likely the result of
competing mechanisms and some ‘saturation’ in strengthening in the alloy: Bi solid-solution
strengthening (SSS), Bi precipitation hardening, and Cu6Sn5 IMC dispersion strengthening. At low
Bi concentration, insufficient Bi is in the alloy to precipitate a second phase. Thus, the Sn-1Bi
binary alloy is only strengthened by Bi in solid solution. Adding Cu to the alloy forms a Cu6Sn5
intermetallic secondary phase, which imparts some dispersion strengthening to the alloy which
was not present in the binary, single phase alloy. In the case of the alloys containing 0.7 wt% of
Cu, assuming all Cu forms Cu6Sn5 IMCs and no Cu is dissolved in the Sn matrix, this would result
64
in approximately 1.8 wt% of the alloy composed of this secondary phase, as shown in the following
calculations. The first (Eqn 3.1) converts the Cu:Sn ratio in the stoichiometric formula Cu6Sn5 to
weight percent.
𝐶𝑢6𝑆𝑛5 → 54.55 𝑎𝑡% 𝐶𝑢 = 39.1 𝑤𝑡% 𝐶𝑢
(Eqn 3.1)
Figure 3.4: Hardness Data for as-cast alloys
The second equation (Eqn 3.2) assumes all Cu is consumed to form Cu6Sn5, and calculates the
required amount of Sn required to form enough IMC to use up all 0.7 wt% of Cu in the alloy (1.1
wt%). The sum of the Cu and Sn gives the total weight percent of Cu6Sn5 IMC in the alloy.
39.1 𝑤𝑡% 𝐶𝑢
61.9 𝑤𝑡% 𝑆𝑛(𝑤𝑒𝑖𝑔ℎ𝑡 𝑟𝑎𝑡𝑖𝑜 𝑖𝑛 𝐼𝑀𝐶)
=0.7 𝑤𝑡% 𝐶𝑢
1.1 𝑤𝑡% 𝑆𝑛(𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑛 𝑎𝑛𝑑 𝐶𝑢 𝑖𝑛 𝐼𝑀𝐶𝑠 𝑖𝑛 𝑎𝑙𝑙𝑜𝑦)
→ 1.8 𝑤𝑡% (𝑜𝑓 𝐶𝑢6𝑆𝑛5 𝑖𝑛 𝑎𝑙𝑙𝑜𝑦)
(Eqn 3.2)
65
For the binary Sn-5Bi alloy, the Sn matrix is still enriched in dissolved Bi atoms, imparting a
solution strengthening effect, however the excess Bi atoms have been forced out of solution
when the alloy is cooled, forming a second phase and strengthening the alloy. Based on the solid
solubility of Bi in Sn at room temperature (approximately 2 wt%), about 3 wt% of Bi in the alloy
has formed a precipitate phase. Therefore, since there are already many second phase
strengthening particles (Bi precipitates) in the Sn-5Bi alloy, it is evident that the inclusion of Cu
(only 0.7 wt%, forming about 1.8 wt% of Cu6Sn5 IMCs) does not add any appreciable additional
strengthening benefit to the alloy, as indicated in the as-cast hardness data. It is possible that the
alloy is optimally strengthened by Bi precipitates, and the addition of further Cu6Sn5 precipitates
does not lend any further strengthening benefits to the alloy. It is unknown precisely how much
of a strengthening effect each of these three attributes impart to the alloy on a quantitative basis –
this would need to be verified using Transmission Electron Microscopy (TEM).
3.3.2 Elevated Temperature Aging
The microstructure of Sn-5Bi (Figure 3.5) and Sn-0.7Cu-5Bi (Figure 3.6a-c) changed
significantly after aging at 100°C and 125°C – Bi particle spacing and size becomes more uniform,
and the average size of precipitates decreases. Non-uniform spacing in the as-cast alloy results in
local regions (Sn dendrites) in which dislocation passage is made easier as precipitates exist solely
within interdendritic spaces, which results in lower strength. Further, dislocations tend to cut
through, rather than bow around finer precipitates during deformation, which requires more energy
[48]. 𝑇𝑠𝑜𝑙 of Sn-5Bi is approximately 60°C, and that of Sn-0.7Cu-5Bi is likely slightly higher due
to the presence of Cu in the alloy [40]. Therefore, heating the sample to 100°C or 125oC allows
for all Bi to enter solid solution. As diffusion of Bi in the Sn matrix occurs more rapidly at higher
temperatures, Bi was able to distribute itself more equally at 125°C than at 100°C for the same
aging time (Figure 3.6c). The IMC size and distribution were not affected by aging temperature
or time for the Cu-containing alloy. Microstructure evaluation after aging at 70°C was not
performed, however Ostwald ripening may occur at this temperature for Sn-0.7Cu-5Bi, as the
solvus of this alloy may be higher than 70°C [40].
SAC 305 (Figure 3.6d&e) coarsened after elevated temperature aging, with the IMCs enlarging
slightly. Sn-0.7Cu-1Bi (Figure 3.6f&g) and Sn-1Bi (not shown) showed slight losses in
dendritic structure but, similar to Sn-0.7Cu-5Bi, the IMCs in Sn-0.7Cu-1Bi did not change
66
significantly. Sn-3Bi (not shown) showed similar microstructural changes as Sn-5Bi and Sn-
0.7Cu-5Bi. In agreement with earlier studies, the hardness of SAC 305 decreases as time
progresses after elevated temperature aging at all temperatures (Figure 3.7, Figure 3.8, &
Figure 3.9) [1, 2, 87]. The difference in hardness evolution between the three aging temperatures
is negligible. The hardness of the Bi-containing alloys does not change appreciably after aging at
any of the three temperatures, in agreement with earlier studies [5, 6, 88, 89]. No observable
changes in hardness relating to Ostwald ripening of Bi precipitates were seen after aging at 70°C
– this is either because all alloy solvus temperatures were lower than the aging temperature, or
the aging time was not long enough to produce sufficient Ostwald ripening to have a detrimental
effect on the hardness.
Figure 3.5: Evolution of microstructure of Sn-5Bi after aging at 125°C. As-cast (a); 1 day (b); 7
days (c); 14 days (d).
(a) (b)
(c) (d)
(a) (b)
(c) (d)
67
Unlike typical age-hardened materials such as aluminum alloys, in which the strength increases
during aging (eventually decreasing during overaging), hardness is relatively consistent over the
full aging treatment for these Sn-Bi and Sn-Cu-Bi alloys. The hardness of SAC 305, on the other
hand, is reduced. There are several metallurgical factors that need to be considered to understand
the evolution of hardness in all six alloys:
1. Bismuth in saturated solid solution strengthens the Sn matrix. Bismuth atoms (r=0.156 nm) are
slightly larger than Sn atoms (r=0.140 nm), which introduces a compressive stress on the Sn
lattice. The resulting elastic strain field impedes dislocation motion [12].
2. When the solubility of Bi in Sn is exceeded, Bi forms a secondary precipitate phase. These
precipitates are harder than the matrix, and the introduction of a new phase boundary serves as
an impediment to dislocation motion via either the introduction of misfit strains (caused by
differences in lattice parameters) or volume misfit (caused by the volume change during the
phase transformation of precipitation) [50]. This will strengthen the alloy. As a rough estimate,
the proportional volume change when a mass of Bi dissolved in Sn at a higher temperature is
transformed into Bi precipitates is given as follows (Eqn 3.3):
𝑉𝐵𝑖,𝑝𝑝𝑡
𝑉𝐵𝑖,𝑠𝑠=
𝑚𝐵𝑖,𝑝𝑝𝑡
𝑚𝐵𝑖,𝑠𝑠
𝜌𝑆𝑛
𝜌𝐵𝑖
(Eqn 3.3)
Note it is assumed the density of the Bi in solid solution is equivalent to the density of β-Sn. The
masses of the Bi in both states are equivalent, thus (Eqn 3.4):
𝑉𝐵𝑖,𝑝𝑝𝑡
𝑉𝐵𝑖,𝑠𝑠=
𝜌𝑆𝑛
𝜌𝐵𝑖=
7.27𝑔
𝑐𝑚3⁄
9.78𝑔
𝑐𝑚3⁄= 0.743
(Eqn 3.4)
Thus, it is estimated that the precipitation of Bi from solid solution results in a reduction in the
localized volume by about 25.7%; this places a tensile stress on the surrounding β-Sn matrix.
68
3. Grain boundaries are barriers to dislocation motion due to change in orientation between
grains. A material with fewer, larger grains will therefore be weaker than a material with many,
smaller grains.
4. There is some optimal particle size (Figure 2.13) which will produce the greatest strengthening
effect; this is related to the crossover between the two mechanisms for dislocation interaction
with second phase particles (cutting versus bowing). Both very small particles and very large
particles are not as effective at strengthening materials as those with this ‘optimal’ size [48].
5. No signs of overaging were observed. In Al alloys, strength is reduced with extensive aging
times. It is likely that the overaging time for Sn-Bi alloys is longer the time frames studied in
these experiments (>14 days). While the properties of the binary Sn-Cu alloy are known to
degrade over time, addition of Bi appears to stabilize Sn-Cu IMCs and delay or inhibit this
aging effect.
Some observations from typical Pb-free alloys, as well as from Bi-bearing alloys are as follows:
• In Pb-free solder materials, IMC particles are known to coarsen over time. In addition, the
Sn grain size is observed to coarsen [1, 87].
• The precipitation of Bi has been shown to cause the surrounding Sn grains to recrystallize,
likely caused by Particle Stimulated Nucleation (PSN) [41]. This arises via the relief of the
stress in the Sn lattice that develops due to lattice parameter and volumetric mismatch
between β-Sn and Bi when the latter precipitates out of solid solution.
• It has been established that aging Bi-containing Pb-free solder alloys results in
homogenization of the Bi in the alloy [6, 7, 40]. Both Bi precipitate size and spacing
become more uniform. This also produces a more equiaxed grain structure due to uniform
PSN of recrystallization [7].
69
(a) (b)
(d)
(f)
(c)
(e)
(g)
(a) (b)
(c) (d)
(e) (f)
(g)
70
Figure 3.6 (previous page): Microstructure of samples aged at elevated temperature. Sn-0.7Cu-
5Bi aged at 100°C for 3d (a), 100°C for 14d (b), 125°C for 14d (c); SAC 305 aged at 125°C for
3d (d) and 14d (e); Sn-0.7Cu-1Bi aged at 100°C for 1d (f) and 7d (g).
Figure 3.7: Evolution of alloy hardness after elevated temperature aging at 70°C.
Figure 3.8: Evolution of alloy hardness after elevated temperature aging at 100°C.
71
Figure 3.9: Evolution of alloy hardness after elevated temperature aging at 125°C. The 14 day
timepoint for Sn-3Bi is missing due to testing equipment malfunction.
It is hypothesized that these factors affect the hardness of the alloys after aging as follows (Table
3.2):
Table 3.2: Metallurgical Explanation for Aging Performance of Alloys
Alloy Factor(s) Result Explanation
SAC 305 IMC coarsening
GBs (-) Degradation
IMC particles coarsen and required stress for
circumventing particles decreases. In addition
(not observed in this study), grain size likely
increases and reduces the number of GBs in
the material.
Sn-1Bi Solid solution
strengthening
Stabilization
SSS roughly the same over time; Bi remains
dissolved uniformly during solidification.
Because no precipitation of Bi occurs, no
changes to GB content are expected.
Sn-3Bi Solid solution
strengthening
Bi precipitation
GBs (+)
Bi precipitates strengthen the Sn matrix,
induce PSN. Bi enriches and likely embrittles
GBs slightly. These effects are both more
pronounced over time, and possibly cancel
each other out to result in a net stabilization
of strength.
Sn-5Bi
Sn-0.7Cu-
1Bi
Solid solution
strengthening
Same mechanism as for Sn-1Bi. No
coarsening of IMCs observed (possible
stabilization by Bi?). Grain structure also
expected to not change during aging.
72
Alloy Factor(s) Result Explanation
Sn-0.7Cu-
5Bi
Solid solution
strengthening
Bi precipitation
GBs (+)
Same mechanism as for Sn-3Bi and Sn-5Bi.
As with Sn-0.7Cu-1Bi, no coarsening of
IMCs is observed, so these do not affect
hardness over time.
3.3.3 Room Temperature Aging
Figure 3.10 shows the evolution of Sn-5Bi after room temperature aging for all time points
between 10 days and 168 days, inclusive. Bi precipitates remained clustered after 28 days, however
the clusters grew larger. After 168 days, Ostwald ripening was observed – some precipitates grew
larger than the others, whereas several other precipitates became smaller.
Figure 3.11 shows the evolution of the microstructure of Sn-0.7Cu-5Bi, SAC 305, and Sn-0.7Cu-
1Bi, after 10 days and 168 days. In Sn-0.7Cu-5Bi, Bi precipitate morphology underwent similar
changes as the binary Sn-5Bi alloy, and the IMC size / distribution did not differ appreciably, even
after 168 days. SAC 305 demonstrated significant loss in dendritic structure and slight coarsening
(a) (b)
(c) (d)
(e) (f)
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
73
of the IMCs after room temperature aging. Sn-0.7Cu-1Bi showed very little change to its dendritic
structure or IMC size / distribution, similar to Sn-0.7Cu-5Bi. The microstructure of Sn-1Bi (not
shown) remained nearly identical owing to the single phase present, and Sn-3Bi underwent similar
changes to Sn-5Bi.
The hardness of SAC 305, as anticipated, dropped off immediately after casting (Figure 3.12),
and stayed relatively consistent after 28 days. The Bi-containing alloys again demonstrate a
general maintenance of hardness from casting through to 365 days of aging at room temperature.
This stabilization was much more consistent for the Cu-containing alloys (Sn-0.7Cu-1Bi and Sn-
0.7Cu-5Bi); the binary alloys showed some deviation after 112 days (this appears to be
statistically insignificant due to the spread in the data – error bars).
The mechanisms which contribute to the preservation of alloy strength after aging at elevated
temperatures (Table 3.2) are likely also operating during room temperature aging, albeit at a
significantly reduced rate. This includes Ostwald ripening, which has been shown to occur
during aging below solvus. Matijevic found that under these conditions, while the Bi will
generally become more evenly dispersed throughout the alloy over time, some particles (those
which remained out of solution) will coarsen and become surrounded by a particle-denuded zone
[40]. It is hypothesized that as these particles become enlarged, their effectiveness at
strengthening the alloy will diminish and eventually become a detriment to alloy strength [7].
This hypothesis would need to be tested using more advanced techniques such as TEM and
nanoindentation (to evaluate the strength at the denuded zones compared with regions devoid of
coarsened precipitates.
74
Figure 3.10: Evolution of microstructure of Sn-5Bi after RT aging. As cast (a); 28 days (b); 63
days (c); 168 days (d).
(a) (b)
(c) (d)
(a) (b)
(c) (d)
75
Figure 3.11: Microstructure of samples aged at room temperature. Sn-0.7Cu-5Bi as-cast (a),
aged after 10d (b) and 168d (c); SAC 305 as-cast (d), aged after 10d (e) and 168d (f); Sn-0.7Cu-
1Bi as-cast (g), aged after 10d (h) and 168d (i).
The following observations can be made about the trends in alloy hardness after room
temperature aging (Figure 3.12):
• SAC 305: Alloy hardness dropped off immediately after 10 days. Hardness levelled off
thereafter (no statistically significant change in hardness), similar to previous studies and
after elevated temperature aging.
• Sn-1Bi: The hardness of the alloy increases slightly over time; this may be explained by
homogenization of the solid solution of Bi. There is a large, statistically significant drop
in hardness after 112 days; this may be explained by sample variation.
(a) (b)
(c) (d)
(e) (f)
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
76
• Sn-3Bi: Hardness was relatively stable over time, due to the counteracting effects of
homogenization of Bi / increased GB content and segregation of Bi at recrystallized GBs.
Any small deviations may be explained by sample variation. It is possible, due to the low
Bi content, that insufficient Ostwald ripening occurred to have a detrimental effect on
hardness.
• Sn-5Bi: In contrast to Sn-3Bi, hardness decreased over time. This may be explained by
Ostwald ripening – some Bi particles may have coarsened sufficiently to not only lose
effectiveness in inhibiting dislocation motion, but also embrittle the microstructure. Some
small deviations exist; these again may be explained by sample variation.
• Sn-0.7Cu-1Bi: Similar behavior to the binary Sn-1Bi alloy was observed; the increase in
hardness was slighter. This may be because the diffusivity of Bi in Sn-Cu is lower than
that in pure Sn.
• Sn-0.7Cu-5Bi: The hardness of this alloy behaved considerably different than the binary
Sn-5Bi alloy. Hardness was stable (statistically) across all aging times. This result may be
due to the likely reduced diffusivity of Bi in Sn-Cu compared to that in pure Sn –
insufficient Ostwald ripening occurred to have a detrimental effect on the hardness.
Figure 3.12: Evolution of alloy hardness after room temperature aging.
77
Tin-based alloys are noteworthy in that room temperature represents a large percentage of the
melting temperature – the microstructure is susceptible to evolve even at this relatively low
temperature. Therefore, room temperature aging is very important to study and is considered in
the solid-state diffusion study in this thesis (Chapters 4 and 6).
3.4 Concluding Remarks
These aging results have confirmed the results of prior studies that the inclusion of Bi as an
alloying element is beneficial in Pb-free solder alloys. A wide range of alloy compositions were
tested under many aging conditions (temperatures, times) and hardness was, for the most part,
stable over all testing conditions.
Bismuth can serve to solution strengthen the Sn matrix (at low concentrations) as well as via
precipitation hardening at higher concentrations. After aging, Bi precipitates tend to transition
from tightly-bunched clusters with large particle size within the interdendritic spaces to becoming
finer and more homogeneously distributed throughout the Sn matrix. These microstructural
changes, along with several other mechanisms including grain boundary strengthening and solid
solution strengthening, cause the hardness of the alloy to remain relatively constant throughout the
entire aging treatment. One exception was observed for the binary Sn-5Bi alloy aged at room
temperature, whose hardness decreased by roughly 5% after 252 days of aging. This may be
explained by Ostwald ripening of the Bi precipitates.
The results from this study indicate that for up to 14 days at elevated temperature and 365 days at
room temperature, Bi-bearing alloys are (other than Sn-5Bi aged at RT for ≥ 252 days) extremely
robust. It is possible that the aging times selected in this study were largely insufficient to degrade
the microstructure and properties of the alloys, and further experiments with extended aging times
are recommended to determine when overaging (similar to what is observed in Al alloys) occurs.
Currently, the Center for Advanced Vehicle and Extreme Environment Electronics (CAVE3) group
at Auburn University is planning an aging study featuring extremely long, high temperature
storage conditions, with a test matrix including many 3rd generation solder alloys (including
‘Violet’). It will also be beneficial to perform room temperature aging for longer durations;
Ostwald ripening appears to be largely responsible for the degradation of strength of Sn-5Bi and
this mechanism is not expected to occur at elevated temperatures for the compositions that were
tested in this study.
78
A saturation effect was also observed with respect to Cu and Bi strengthening in the alloy. For
alloys with 1 wt% Bi, the inclusion of small amounts of Cu improves hardness from approximately
42.8 to 59.9 HR15X (40% increase), whereas there is no statistical change in strength at higher Bi
concentrations (Sn-5Bi and Sn-0.7Cu-5Bi demonstrated hardness values of 75.3 and 75.5 HR15X,
respectively). This is an indication that there is some optimal total dispersion (precipitate + IMC)
content which may differ depending on the types and amounts of each particle present (including
Ag3Sn – Ag was not included in any of the Bi-bearing alloys). The inclusion of Ag and Cu provides
additional benefits to the alloy, including wetting and reduction of Cu dissolution from the
substrate. All of these factors are necessary to consider in the development of new alloy
compositions, and likely influenced the outcomes of the Celestica / University of Toronto low melt
project.
The inclusion of Bi in solder joints may improve long-term reliability, compared with traditional
SAC alloys. It is evident that the distribution of Bi throughout the Sn matrix over time, at all
temperatures, is driven by solid-state diffusion. One of the main challenges in the
commercialization of Bi-containing alloys is the development of a complete model which serves
to predict the microstructural evolution and performance of the alloy under a variety of
environmental and mechanical conditions. Understanding and being able to predict the solid-
state diffusion of Bi in Sn over time will be very useful data to supplement the development of
this model. The following three chapters in this thesis are focused on the test methods (Chapter
4) and results (Chapters 5 & 6) from the solid-state diffusion study.
79
Chapter 4 Methodology for Study of Solid-State Diffusion of Bismuth in Tin
4.1 Introduction
It was established in related work under ReMAP that Bi-containing alloys undergo extensive β-
Sn recrystallization (Figure 2.8 and Figure 2.16) over the course of aging. As discussed in
Chapter 2, this is stimulated by the precipitation of homogenously distributed Bi precipitates,
which likely are distributed via solid state diffusion [7, 40]. This process produces many small,
equiaxed grains, and greatly increases the grain boundary content of the material. This finding,
along with the fact that the solidification structure of solder joints typically consists of one or two
large grains, indicates that the diffusion of Bi in the alloy may change significantly during the
lifetime of the solder joint. For example, recrystallization is known to occur not only after
isothermal aging, but also after mechanical and thermomechanical fatigue [92]. It is expected
that over time, diffusion of Bi will be amplified as grain boundary content increases [55, 56]. It is
therefore important to study diffusion of Bi in coarse-grained or monocrystalline Sn
(representative of early life), as well as in polycrystalline Sn (representative of later life).
Furthermore, it is known that during aging below the solvus of the alloy (such as during typical
operating conditions of a commercial electronic device), that Ostwald ripening of Bi precipitates
may occur [40]. While small precipitates of Bi are known to strengthen the alloy, Bi is inherently
brittle and larger precipitates may degrade the reliability of the alloy. It was shown by Coyle et
al. that the segregation of Bi towards the package-joint or PCB-joint interfaces (which may
already be brittle due to the presence of IMCs) may lead to early failures in the joint (Figure 4.1)
[93]. While the testing in Coyle’s study was done post-assembly (with no aging), it is possible
that if diffusion were favored in certain orientations, that such segregation may occur over the
product’s lifetime, degrading joint integrity. For this reason, the dependence of orientation on the
diffusion of Bi in Sn is very important to study.
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Figure 4.1: Failure of mixed metallurgy solder joint (Sn-58Bi paste and SAC 305 BGA ball)
likely caused by segregation of Bi to the package-ball interface [93]
Bulk/matrix diffusion can be studied either by using specially-produced monocrystals, or by
preparing coarse-grained samples. While reproducible, specific orientations (low-index
orientations such as <001> are typically of interest and are cited when listing orientation-
dependent properties) can be achieved with monocrystals, they are quite expensive, making
studies with large test matrices impractical [94]. Solder joints are known to solidify in random
orientations, which can be emulated by producing coarse grains in the lab [92]. This method is
significantly more cost-effective, allowing for more comprehensive studies to be completed. The
major drawbacks of producing coarse grains in the lab are that it is difficult or impossible to
control the exact orientation, and that orientations are not easily reproducible. In addition, care
must be taken to avoid conducting analysis near grain boundaries, which are known to
accentuate diffusion [55, 56].
There are several techniques to produce diffusion couples or multiples. The most common
method is to metallographically prepare samples of each species and press them together to
promote diffusion. In cases where the diffusivity is largely one-sided, i.e. diffusion occurs
significantly more rapidly in one direction compared to the other, or if one or both species are
81
susceptible to deformation, using a thin film approach may be better, in which one species is
passively deposited onto the other. It is important to note that the geometry varies between cases
(semi-infinite vs finite layer), and thus the modelling (mathematics) may be different.
In this work, both the classic diffusion couple technique as well as the thin film method are
employed to evaluate the diffusion of Bi in Sn. As mentioned previously, the diffusion of Bi in
the as-cast alloy can be studied using coarse-grained or monocrystalline Sn. To ensure the Sn
remains as unchanged as possible microstructurally during couple preparation, the thin film
method is ideal. Diffusion of Bi in the aged alloy, where Sn is largely polycrystalline, can be
studied using either approach; in this study the classic approach was used. Even though Sn is a
relatively soft metal, the classic approach introduces additional deformation to the alloy, which
may promote grain recrystallization. In addition, it is expected, given the great extent of
recrystallization expected in the aged Bi-bearing alloy, that the dislocation density is fairly high
and this will also be the case for couples prepared using the classic approach.
To ensure that the coarse-grained or monocrystalline Sn substrates remain intact after deposition,
sputtering was selected to produce thin film diffusion couples. This method is particularly
attractive as Bi is predicted to have a high sputtering yield compared with other metallic species
(Table 4.1) [95]. Sputter yield is defined as the number of ejected target atoms per incident ion
(in this case Ar+).
Table 4.1: Calculated Sputter Yields for Select Metals (500 eV Ar ions) [95]
Material Sputter
Yield
Al 1.37
Ni 1.51
Cu 1.91
Au 2.47
Bi 4.35
4.2 Test Matrix
Table 4.2 shows the entire test matrix for the solid-state diffusion portion of this thesis. It is
noted that although the polycrystalline methodology and results are described after the coarse-
grained and monocrystalline methodology and results, the polycrystalline work was completed
first. This is because the approach used for the polycrystalline samples was easier and more
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established, so it was sensical to use that method first as a ‘proof-of-concept.’ In addition, and as
discussed below in Section 4.3.1, several monocrystals of Sn with known orientations were
purchased to verify the results obtained using coarse-grained Sn. There are thus three types of Sn
samples that are used in this study: coarse-grained, monocrystalline, and polycrystalline.
Table 4.2: Number of profiles (samples) for each condition
Diffusion Conditions Sample Type
Temperature Time Coarse-
Grained
Mono-
crystalline*
Poly-
crystalline**
25°C 99, 115, or 144 days 51 (17)
50°C 28 days 54 (18)
70°C 14 days 54 (18)
85°C 7 days 54 (18) 16 (2) 12 (6)
100°C 2 days 48 (15) 16 (2) 12 (6)
115°C 30 hours 54 (18) 16 (2) 12 (6)
125°C 1 day 61 (20) 16 (2) 12 (6) *For the monocrystalline samples, one sample of each orientation (two total orientations) are analyzed, each with 8
profiles.
**For the polycrystalline samples, diffusion triples are prepared, and thus 3 triples are equivalent to 6 samples.
Several annealing temperatures were selected (25°C, 100°C and 125°C) as they are commonly
used in thermomechanical reliability and long-term aging studies of solder alloys in industry, and
to be consistent with the approach in the aging study covered in Chapter 3. In addition, 70°C was
selected as this is a typical temperature experienced by a solder joint during device operation.
115°C and 85°C were selected as intermediates between 100°C and 125°C, and 70°C and 100°C,
respectively.
It is typical that diffusivity departs from strict Arrhenius behavior at around 0.7𝑇𝑚 [58]. For Sn-
Bi, in which melting occurs over a pasty range, 𝑇𝑚 can be replaced with the alloy solidus
temperature 𝑇𝑠𝑜𝑙𝑖𝑑 (the temperature at which liquid begins to form upon heating). In Table 4.3,
the solubility limits (assumed maximum possible Bi concentration) at each diffusion temperature
are listed (assuming the phase equilibria predicted by Lee [25]). Also included are the estimated
solidus temperatures at the solubility limit and the ratio between the diffusion temperature and
solidus. While it is evident that the concentration will decrease with diffusion distance and thus
so will 𝑇 𝑇𝑠𝑜𝑙𝑖𝑑⁄ (the homologous temperature), part of the diffusion zone at most temperatures
will be above 70% of the local absolute solidus temperature. Thus, it may be difficult to fit the
diffusivity data to the Arrhenius relationship. For this reason, an additional low annealing
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temperature of 50°C is included in the test matrix – all compositions in diffusion couples
annealed at both 50°C and 25°C will be below 0.7𝑇𝑠𝑜𝑙𝑖𝑑.
Table 4.3: Estimated Ratio of Diffusion Temperatures to Solidus [25] Temperature (at
solubility limit)
Diffusion
Temperature Solubility
Limit (at%)
𝑻𝒔𝒐𝒍𝒊𝒅 at
solubility limit 𝑻
𝑻𝒔𝒐𝒍𝒊𝒅⁄
°C K
25 298 1.1 522K 0.60
50 323 1.9 492K 0.66
70 343 3.2 483K 0.71
85 358 4.9 471K 0.76
100 373 6.7 458K 0.81
115 388 9.0 442K 0.88
125 398 10.7 430K 0.93
4.3 Sample Preparation
The methodology for sample preparation in this study was relatively straightforward but
consisted of many steps:
• Tin samples were cast into their desired shape.
• The surface(s) which form the interface between Sn and Bi were prepared using standard
metallography techniques.
• The diffusion couples were assembled using either a sputtering technique or mating of Sn
and Bi via compression.
• The couples were then annealed, mounted in epoxy, and subsequently cross-sectioned
metallographically.
4.3.1 Casting
Both coarse-grained and polycrystalline Sn samples were produced in the lab. A mold consisting
of a graphite trough, graphite block (with several through-holes 4mm in diameter), and glass
slide (coarse-grained Sn only) was used (Figure 4.2a). This mold was first heated on a hot plate
to roughly 260°C (far exceeding the melting point of Sn). Pieces of Sn were cut from large 50g
ingots, and melted in the through holes of the mold to produce cylindrical samples. The average
length and mass of these samples were 8mm and 1 gram, respectively. Blemishes in the mold
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such as cracks and pores are known to be heterogenous nucleation sites [96]. Placing the smooth
glass slide at the bottom of the trough helped promote the solidification of coarse Sn grains.
For the coarse-grained samples, the mold and samples were subsequently placed into a
Lindberg/Blue M Mechanical Oven, also set to a temperature of roughly 260°C. The temperature
of the furnace was set to cool to 40°C, and the door was propped open (roughly an inch between
the end of the door and the frame of the furnace) to allow for some air flow into the chamber and
speed up the cooling process slightly (Figure 4.2b). This relatively slow cooling process resulted
in a low degree of undercooling, which results in formation of fewer critical nuclei for nucleation
of Sn grains [96]. Fewer nuclei allow for the heterogenous nucleation of Sn at the interface
between the melt and the mold surface to dominate the solidification process, resulting in the
formation of coarse grains.
It required about 55 minutes for the chamber to reach 40°C, upon which the mold was removed
from the furnace and allowed to cool to room temperature on an iron slab. The average cooling
rate was determined to be approximately 4°C/min over the whole cooling cycle. As the samples
solidified at 232°C (𝑇𝑚 of Sn), the cooling rate was determined to be roughly 8°C/min.
In the case of the polycrystalline Sn samples, after being melted on the hot plate, the mold was
quenched in an ice bath consisting of 1.6L of water and 400 mL of ice. The rapid cooling effect
caused the Sn samples to solidify with a highly polycrystalline grain structure. Quenching
resulted in a greater degree of undercooling, which is known to decrease the critical radius for
nucleation [96]. The vacancy concentration in crystalline materials increases with temperature;
when the sample is cooled, this concentration must be lowered to achieve equilibrium.
Quenching does not provide sufficient time for this equilibrium concentration to be achieved,
and the vacancy concentration becomes supersaturated. The excess vacancies will form clusters
and subsequently dislocation loops, which serve as heterogenous nucleation sites. Therefore, as
the cooling rate is increased, more heterogenous nucleation sites are formed for Sn grain
nucleation and the critical radius for successful nucleation site is reduced, resulting in a
decreased average grain size [50].
In addition, similarly sized samples of Bi were also cast in a similar fashion, except these were
not quenched in an ice bath; they were cooled on an iron slab. Since diffusion triples (Sn
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sandwiched between two pieces of Bi) were utilized in this case, it was necessary to cast twice as
many Bi samples as Sn.
In addition, several monocrystals of Sn were purchased to verify the results of the coarse-grained
study. Two crystals each of <100> and <001> orientations, with dimensions of 4mm diameter
and 12mm length were obtained. Each crystal was cut into two smaller pieces, roughly 5.5mm in
length, using electric discharge machining (EDM), yielding four crystals of each orientation.
4.3.2 Metallography
For all Sn samples produced in this study, after casting the samples were removed from the
graphite mold and inserted into epoxy mounts with through holes, such that the samples snugly
fit and did not require additional adhesion (Figure 4.2c). This temporary mount was beneficial
for efficiently preparing multiple samples at once and allowing for the samples to be easily
removable from the mount. The latter point is exceptionally important as typical two-part epoxy
materials for metallography are not designed for use at high temperatures (otherwise the samples
would be mounted then annealed). The samples were thinned to be flush with the surface of the
epoxy using a standard metallographic procedure (Table 4.4):
Table 4.4: Metallographic Sample Preparation Procedure for Diffusion Interfaces
# Step Comments
1 Wet grind at 250
rpm
SiC papers (FEPA scale) used were 120, 320, and 500
Done until sample surfaces close (within 1mm) of surface of epoxy
mount
2 Wet grind at 125
rpm
SiC papers (FEPA scale) used were 800, 1200, 2400, and 4000
Done until sample surfaces are flush with surface of epoxy mount
3 Cleaning Ultrasonic cleaning with ethanol for 45 seconds
4 Polish Diamond slurries of 3µm and 1µm
5 Cleaning Ultrasonic cleaning with ethanol, between each polish step
6 UV cleaning Used for 30 minutes, performed under vacuum
Ultrasonic cleaning was used to ensure no cross-contamination between preparation steps (more
susceptible in this case, as samples were not truly mounted in epoxy). UV cleaning, performed
using a ZONE-SEM cleaning system developed by Hitachi High-Technologies Canada, was used
to ensure optimal adhesion of the thin layer of Bi by eliminating any residual contamination after
the sample preparation process [97]. Mounts were then placed into a desiccator to ensure no
86
moisture or particulates in the air could contact the sample and compromise the quality of the
polished surface.
Since diffusion triples were used rather than couples for polycrystalline Sn, it was necessary to
polish the Sn samples on both sides. After the first side was finished, Kaptan tape was used to
cover the surface to avoid damage and/or contamination while the second side was prepared. In
addition, metallography was also performed on the Bi samples. Different polishing cloths from
those used for Sn were used, to avoid contamination.
Figure 4.2: Sample preparation of coarse-grained Sn-Bi diffusion couples. Casting
crucible/mold (a); furnace door propped open (b); cast samples inserted in epoxy mount with
through-holes (c); mount and samples after Bi sputtering (d).
(a) (b)
(c) (d)
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4.3.3 Diffusion Couple Preparation
The method for joining Sn and Bi into a diffusion couple varied between the coarse-
grained/monocrystalline and polycrystalline samples. For the former (Section 4.3.3.1), a sputter
deposition technique was used to lay down a thin film of Bi on the freshly polished Sn surface.
For the latter (Section 4.3.3.2), the polished Sn and Bi pieces were mechanically compressed
together to form a more traditional diffusion couple.
4.3.3.1 Coarse-Grained and Monocrystalline Sn
Sputter deposition was conducted using an AJA International ATC Orion 5 system equipped
with a Bi target (4N purity, purchased from ACI Alloys). The epoxy disc with embedded
samples was attached to the sample holder. Sputtering was performed under high vacuum (<
3x10-5 Torr) and samples were maintained at room temperature throughout the entire process. A
uniform layer approximately 0.67µm (based on a deposition rate of 3.7 �̇� 𝑠⁄ ) in thickness of Bi
was deposited on the disc and Sn samples (Figure 4.2d) at a power of 75W – this required about
30 minutes. The sputtering time was in almost all cases not continuous, as arcing in the chamber
caused the plasma to become unstable. On average, it required three exposures of 10 minutes
each to produce the desired thin film thickness. Sputtering was performed on all samples for each
given temperature simultaneously i.e. all samples that underwent annealing at 85°C were
sputtered at the same time. All eight monocrystalline Sn samples underwent sputter deposition at
the same time, despite the fact several temperatures were investigated for these samples. This
was done to ensure that all metallography was performed in the same, optimal manner (the cost
of these samples was substantial, and it was desired to not have to waste material or possibly
replace the crystals).
The samples were ejected from the epoxy disc after sputtering. Several were set aside (for room
temperature diffusion); the remainder were placed in a graphite crucible for elevated temperature
diffusion (Figure 4.3a). This crucible was placed within a pre-heated fire brick (Figure 4.3b) in
a Lindberg/Blue M box furnace, set to the desired diffusion temperature. A thermocouple was
inserted into the fire brick through the roof of the furnace and placed near the samples. This
allowed for careful monitoring and periodic adjustments as needed such that the target
temperature was maintained as the diffusion temperature for the entire desired time of diffusion.
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For monocrystalline Sn, one sample of each orientation was placed in the oven using a similar
arrangement.
4.3.3.2 Polycrystalline Sn
The polycrystalline Sn samples were first ejected from the temporary epoxy mount. An
aluminum fixture with three hex bolts, along with a graphite collar, were used to assemble
diffusion triples. One triple consists of one Sn sample sandwiched between two Bi samples
(Figure 4.4a); this allowed for three triples (six Sn-Bi interfaces), to be prepared per run. The
fixture and collar were preheated on a laboratory hot plate (Figure 4.4b) for approximately 30
min prior to diffusion triple sample assembly. The hot plate was set to approximately (+/- 10°C)
the diffusion temperature; this was verified by a thermocouple. The metal pieces were stacked in
the collar (Figure 4.4c) and compressed with the hex bolts (2.5 rotations after finger tightening).
The polycrystalline samples were left secured in the hex bolt fixture during annealing. The entire
assembled fixture was inserted into the fire brick and monitored in a similar fashion to the
coarse-grained/monocrystalline samples.
Figure 4.3: Post-sputtering of coarse-grained and monocrystalline diffusion samples. Samples
placed in graphite crucible for annealing (a); crucible with samples loaded in hollowed fire brick
(b).
(a) (b)
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The reason for preheating in this case is because the aluminum fixture is relatively large
compared to the graphite crucible the coarse-grained samples were placed in, and the
temperature ramp rate would be a lot slower. Graphite has a very high thermal conductivity, and
thus the coarse-grained samples would reach the desired diffusion temperature substantially
faster when preheating the Al fixture.
After annealing, samples were removed from the oven. Diffusion couples are typically quenched
after annealing to ‘lock’ the solute atoms in the solvent matrix and avoid two-phase (e.g. α+β)
regions [62, 63]. In this study, the polycrystalline samples were quenched after annealing; the
coarse-grained and monocrystalline samples were not quenched as it was shown in preliminary
work that quenching Sn results in a reduction of grain size due to supersaturated vacancy
concentration, as explained in Section 4.3.1. This would make it difficult to identify original
grain orientations. This is not a concern with the polycrystalline Sn, as specific grain orientation
and the anisotropy of diffusion are not considerations for these samples. Coarse-grained and
monocrystalline Sn samples were cooled in air.
Figure 4.4: Polycrystalline diffusion triple preparation. Schematic of Sn-Bi-Sn diffusion triple
(a); fixture with hex bolts, along with graphite collar, preheating on hot plate (b); side view of
diffusion triple, with Bi pieces on either side of Sn piece (c).
4.3.4 Cross-Sectioning
After cooling, sets of 3-6 samples were mounted and cross-sectioned. The sample preparation
procedure is outlined in Table 4.5.
(a) (b) (c)
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Table 4.5: Metallographic Sample Preparation Procedure for Cross-Sectioning of Diffusion
Couples
# Step Comments
1 Embed samples in
epoxy
Used EpoFix 2-part, slow curing (12 hours) epoxy. Mounts roughly 1
inch wide. Between 3 and 6 samples embedded in same mount
2 Wet grind at 250
rpm
SiC papers (FEPA scale) used were 120, 320, and 500
Grinding to roughly 1/3 through sample
3 Wet grind at 125
rpm
SiC papers (FEPA scale) used were 800, 1200, 2400, and 4000
Grinding to roughly center of samples
4 Cleaning Rinse with water between grinding steps; water and ethanol rinse
after 4000 paper
5 Polish at 125 rpm Diamond slurries of 3µm and 1µm
6 Cleaning Rinse with water and ethanol after polishing steps
7 Final polishing Colloidal silica + distilled water slurry; hand polishing
8 Final cleaning Rinse with water, ultrasonic cleaning with ethanol for 40s, rinse with
ethanol
9 Drying Aspirator vacuum system (using running tap water) used to remove
residual moisture prior to SEM analysis
As only a limited number of samples could be analyzed at any given time (on average, one set of
samples could be analyzed each week), and 15-18 samples were prepared for each temperature,
any remaining annealed samples were stored in liquid nitrogen to halt any further, low
temperature diffusion. Custom-made wire baskets (Figure 4.5) which can be easily lowered into
and lifted out of a dewar were used to hold the samples. Each week, the basket was lifted from
the dewar and allowed to warm to room temperature for roughly 20 minutes prior to sample
preparation.
Figure 4.5: Cryo-storage of coarse-grained samples in custom wire mesh basket
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As only three polycrystalline diffusion triples could be assembled at any given time, no storage
in liquid nitrogen was required for these samples. After annealing the diffusion triples with
polycrystalline Sn, Bi pieces were removed mechanically from the Sn. Initially they were left
attached but smearing of Bi onto the Sn during subsequent cross-sectioning tampered with the
elemental analysis.
4.4 Characterization Techniques
Electron microscopy was used to characterize diffusion samples. Scanning Electron Microscopy
imaging was used to observe the diffusion microstructure, and EBSD was utilized to
determine/verify Sn grain orientation as needed. Finally, compositional data from diffusion
profiles was collected using EPMA.
4.4.1 Scanning Electron Microscopy (SEM)
The cross-sectioned samples were first examined in a Hitachi SU-3500 SEM, operated at 15kV.
It was necessary to operate the SEM under a partial pressure of 50 Pa to eliminate charging of
the epoxy mount. A conductive coating was not used as this degrades pattern quality for EBSD
[98]. Imaging was performed at 1000x and 4000x using BSE imaging mode, to identify potential
locations for further analysis. These locations would demonstrate appreciable diffusion, as
evidenced by depletion of the sputtered Bi layer and possible visible proliferation of Bi into the
Sn substrate (as a second phase precipitate). The annealing temperatures are greater than the
analysis (room) temperature and, based on the Sn-Bi phase diagrams, the solubility of Bi in Sn
increases with temperature. This implies that a greater concentration of Bi will be present in Sn
than is soluble at room temperature during annealing, and upon cooling the excess Bi is forced
out of solid solution to form an easily observable second phase using BSE imaging. For the 25°C
diffusion couples, where the ‘annealing’ temperature is the same as the analysis temperature,
precipitation is not expected and EDX was used to verify the presence of Bi in solid solution
with Sn. On each sample, 2-4 locations were selected for further analysis using EBSD.
The extremities of the sample were neglected due to lateral diffusion effects – it has been
observed that near free surfaces (sample sides), diffusion is amplified, and this study focused on
diffusion in the bulk. It is standard practice to remove (or ignore) the region from the sample
edge to about 6√𝐷𝑡 (Eqn 4.1) from the sample sides inward [56, 58]. In this case, a diffusivity of
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1.10 x 10-11 cm2/s (which was determined using the results from the diffusion of Bi in
polycrystalline Sn at 125°C – these conditions would be expected to produce the largest
diffusivity) was assumed:
𝟔√𝑫𝒕 = 𝟔√(𝟏. 𝟏𝟎𝒙𝟏𝟎−𝟏𝟏 𝒄𝒎𝟐𝒔⁄ ) ∗ 𝟖𝟔𝟒𝟎𝟎𝒔 = 𝟎. 𝟎𝟓𝟖𝒎𝒎 = 𝟓𝟖µ𝒎
(Eqn 4.1)
This distance was increased conservatively to 100µm. A schematic showing the sample
extremities not considered in diffusion analysis is shown in Figure 4.6.
For the coarse-grained and monocrystalline samples, the higher magnification SEM images at
the selected locations were analyzed using ImageJ software to quantify the depletion of the
sputtered Bi layer after annealing. The area of Bi remaining in the film was calculated and
divided by the cross-sectional area of the thin film (based on the initial thickness of 0.67µm and
the field of view in the SEM images) to obtain a percentage of film remaining.
Figure 4.6: Schematic of how the 𝟔√𝑫𝒕 criterion for exclusion of sample extremities is applied
in SEM analysis of diffusion samples.
4.4.2 Electron Backscatter Diffraction
The two selected sites on each sample then underwent EBSD at 20kV. The spot intensity, which
dictates the amount of signal that can be generated from a given location, was increased as
electron diffraction generally generates little signal compared with typical electron beam-sample
interactions. Samples were mounted on a pre-tilted sample stub at a 70° angle to the horizontal to
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allow for optimum pattern quality. The magnification was set to 1000x, and each map required
about 25 minutes of beam time.
For the coarse-grained Sn samples, the step size in EBSD was set to approximately 0.7µm; this
ensured the entire field of view could be mapped at 1000x. Maps, along with accompanying
inverse pole figures (IPFs), were examined to identify specific locations (with minimal grain
boundary content) for elemental analysis. Orientations were determined based on the IPF ‘Y’
map; an example is shown in Figure 4.7a. The IPFs were superimposed on the stereographic
projection of β-Sn, obtained from WinWulff software, to identify sample texture (Figure 4.7b-
c). As shown, the orientation of a typical sample is clustered around several similar orientations,
denoting subgrains bound by low-angle grain boundaries (LAGBs). The orientation of each map
was selected based on the most centralized pole in the IPF orientation cluster. Further analysis
was conducted within specific subgrains to ensure no amplification of diffusion was included via
the LAGBs.
Figure 4.7: Example of superposition of Sn IPF on stereographic projection to determine grain
orientation. IPF Y map of sample slow-cooled using method described above (a); superposition,
indicating sample is oriented close to <1 1 1> (specifically <7 7 6>) in the ‘Y’ direction
(labelled) (b), IPF legend (c).
To compare unique orientations for monocrystalline and coarse-grained Sn samples, (Eqn 2.32
was first used to calculate the direction vector �⃗� which relates the <uvw> indices to the β-Sn
lattice parameters. Then, the cosine similarity equation (Eqn 2.33) was used to calculate the
angle θ between the ‘c’ axis and the diffusion direction in the sample. These angles were then
plotted against diffusivities using Eqn 2.30.
(a) (b)
(c)
94
Electron Backscatter Diffraction was only performed on the polycrystalline samples to verify the
samples consisted of many small grains. The step size was reduced to about 0.4µm as the
average grain size was substantially smaller than for the coarse-grained samples. To ensure that
each map could be collected within the same timeframe of 25 minutes, EBSD maps of
polycrystalline samples covered a smaller field of view (about 25µm wide).
4.4.3 Electron Probe Microanalysis
Elemental analysis was performed using a JEOL JXA8230 Electron Probe Microanalyser. One
of two locations that underwent EBSD was selected; within this region three sets of linescan data
were collected in the EPMA, as shown in Figure 4.8.
Figure 4.8: Example of diffusion sample showing EPMA line scan (50µm long) locations.
For the coarse-grained Sn samples, these were within large grains with single orientations where
possible; the path bisecting the most grains was selected for the polycrystalline samples. The
EPMA detection limit, which defines the minimum amount of Bi that can be reliably detected,
was set to be roughly 350 ppm. The detection limit is inversely proportional to analysis time;
while more reliable analyses of low concentrations can be conducted by decreasing the detection
limit, the analysis time will be increased. A detection limit of 350ppm corresponds to an analysis
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time of approximately five minutes per spot; due to EPMA booking limitations this was deemed
optimal for obtaining sufficient composition data. The length of each line scan varied depending
on the crystallinity of the Sn as well as the diffusion temperature/time:
• Polycrystalline Sn: length 70µm, point spacing 3µm
• Coarse-grained and monocrystalline Sn (all temperatures except 25°C): length 50µm,
point spacing 2.5µm
• Coarse-grained Sn (25°C only): length between 36µm and 45µm, point spacing between
2µm and 2.5µm
4.5 Diffusivity Analysis
The diffusivity was calculated using two methods – using the slab source model, as well as a
simulation technique. The latter utilizes both traditional inverse methods such as the Sauer-Friese
and Hall methods as well as a forward simulation method to determine diffusivity [59, 62, 63].
Prior to diffusivity calculations/simulation, data sets were modified slightly:
• Any data in which the combined wt% of Sn and Bi was less than 95 wt% was removed.
• Any data below the EPMA detection limit was removed.
• Since some of the total wt% data were not precisely 100%, these data were normalized
such that the total wt% equalled 100%, while maintaining the ratio of Sn and Bi from the
original data
• The data was then converted from wt% to at% (required for the simulation)
Average concentrations were then calculated for each condition. For the coarse-grained and
monocrystalline samples, averages were determined from three profiles for each sample/Sn grain
orientation. For the polycrystalline samples, 12 profiles for each temperature were averaged.
4.5.1 Slab Source Model
Using the slab source model, D was determined using trial and error. Different values of D were
inputted into the model to produce a ‘simulation’ of the concentration profile. For all values of D
that were tried, the error between the model and the actual experimental data was calculated at
all values of x where the Bi concentration was lower than its solubility in Sn at room
96
temperature. These error values were averaged, and the diffusivity was determined based on the
inputted value that led to the lowest average error. The additional variables required by the
model were the diffusion distance x and concentrations C from the EPMA data, as well as the
diffusion time t and sputtered layer thickness h, assumed to be 700nm. This model was only used
for the coarse-grained data to verify the results from the simulation (Section 4.5.3).
4.5.2 Inverse Methods
Zhang and Zhao developed a comprehensive MATLAB program which, given some
experimental composition data from a diffusion couple, will smooth the data and perform a
simulation of the diffusion conditions using either the Sauer-Friese or Hall inverse methods [59].
This technique was only used for the polycrystalline data, since the deposited Bi layer was very
thin for the coarse-grained and monocrystalline samples, and inverse methods assume semi-
infinite media on both sides of the diffusion interface [58].
In the EPMA, only data on the Sn-rich end of the diffusion couple was collected, as no diffusion
of Sn in Bi appeared evident. To create a ‘true’ profile representative of the entire diffusion
couple, extra points were added at the Sn-rich (Bi wt% = 0) and Bi rich (Bi wt% = 100) ends of
the profile. The solubility of Sn in Bi is zero to account for the latter set of points [25]. Moving
average smoothing (range of 3-5, applied 5-7 times) and smoothing splines (increased number of
data points from ~30 to ~5000) were used to smooth the data. Care was taken to preserve the
overall shape of the concentration profile as much as possible and avoid introducing artifacts to
the data. The smoothed data was then inputted into the program to extract the interdiffusivity as a
function of Bi content using the Sauer-Friese and Hall methods.
4.5.3 First Principles Simulation
Diffusivities in both coarse-grained and polycrystalline Sn were calculated using a forward
simulation technique, conducted using Python software, and based on the Python library
pyDiffusion [59, 62, 63, 99]. The model, initially designed for classic diffusion couple
geometries (with semi-infinite thicknesses of each species), was modified to account for the thin
(finite) layer of deposited Bi on semi-infinite, coarse-grained Sn. The diffusion simulation is
based on Fick’s first and second laws, and considers the moving boundary condition (Eqn 2.14):
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It is not necessary to append additional points to either extreme of the diffusion profiles (largely
because the sputtered layer was finite in thickness), nor is it necessary to smooth the data.
Several parameters were required for the simulation; these are described in Table 4.6. The only
difference between the simulations in coarse-grained in polycrystalline Sn is the assumed
deposited film thickness – this was set at 300µm for polycrystalline Sn (effectively semi-infinite
with respect to diffusion distance) and 0.67µm for coarse-grained Sn.
Table 4.6: First Principles Simulation Parameters
Parameter Value / Notes
Diffusivity in Bi thin film (cm2/s) Assumed using Bi self-diffusivity data, see Table
4.7 [100]
Diffusion time (s) Depending on temperature (see Table 4.2)
Solubility in thin film (atomic fraction) Dependant on phase diagram iteration, see Figure
2.6 and Table 4.8. Solubility in substrate (atomic fraction)
Thin film composition (atomic fraction) Pure Bi (5N)
Deposited film thickness (µm) 0.67µm for coarse-grained samples; 300µm for
polycrystalline samples
To account for interdiffusion between the Sn and Bi phases, the diffusivity of Sn in Bi is
required. As this data is not currently available, the self-diffusivity of Bi was utilized [100]. The
specific values used are taken as an average of the diffusivities calculated using the diffusivities
along the ‘a’ and ‘c’ axes of the Bi rhombohedral lattice (Table 4.7). Future work will seek to
evaluate the true diffusivity of Sn in Bi.
As indicated in Section 2.1.4, Belyakov determined that the discrepancies between the phase
equilibria in the Sn-Bi binary phase diagrams (Figure 2.6) reported in the literature are the likely
result of variance in sample homogenization, and that the diagram proposed by Lee is most
accurate [28]. The Lee, Vizdal, and Braga diagrams (featuring alternatively reported solvus
lines) were considered in the forward simulation – the objective was to use the simulation as an
alternative method to verify which phase diagram is most accurate [25, 26, 27]. The solubilities
predicted by each diagram, for both Sn and Bi solid solutions, at each diffusion temperature,
were extracted using GetGraph Data Digitizer software (Table 4.8). Using the software, each
phase diagram was digitized by first setting the scale (based on the x and y axes of the phase
diagram, at% and temperature, respectively). Points corresponding to the solubilities of interest
were then manually selected.
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Table 4.7: Bi Self-Diffusivity Parameters [100]
‘a’ axis ‘c’ axis ‘Average’
𝑫𝒐 6.60 x 1053 cm2/s 3.40 x 10-2 cm2/s -
𝑸𝑨 602.87 kJ/mol 134.89 kJ/mol -
D
25°C 1.39 x 10-48 cm2/s 7.70 x 10-18 cm2/s 3.27x 10-33 cm2/s
85°C 7.13 x 10-31 cm2/s 7.07 x 10-14 cm2/s 2.24 x 10-22 cm2/s
100°C 2.46 x 10-27 cm2/s 4.37 x 10-13 cm2/s 3.28 x 10-20 cm2/s
115°C 4.52 x 10-24 cm2/s 2.35 x 10-12 cm2/s 3.26 x 10-18 cm2/s
120°C 4.87 x 10-23 cm2/s 4.00 x 10-12 cm2/s 1.40 x 10-17 cm2/s
125°C 4.94 x 10-22 cm2/s 6.72 x 10-12 cm2/s 5.76 x 10-17 cm2/s
The solubility limit is used to set the starting composition (at x=0) of the simulated diffusion
profiles; this assumes that all Bi will remain in solid solution during annealing. This is only true
if some Bi remains in the film after the simulated annealing time. If the Bi film is depleted prior
to the end of the annealing time, the Bi present in the Sn substrate will begin to homogenize,
which will lower the composition at x=0.
Table 4.8: Solubility of Bi in Sn in wt% (at%) based on the Lee [25], Vizdal [26], and Braga
[27] Sn-Bi binary phase diagrams
Temperature
Phase Diagram
Solubility of Bi in Sn in wt% (at%) Solubility of Sn in Bi in wt% (at%)
Lee Vizdal Braga Lee Vizdal Braga
25°C 2.0 (1.1) 3.2 (1.9) 1.9 (1.1) 0.0 (0.0) 0.6 (0.4) 1.9 (1.1)
50°C 3.6 (1.9) 3.5 (2.1) 1.9 (1.1) 0.0 (0.0) 1.0 (0.7) 2.4 (1.4)
70°C 5.9 (3.2) 3.8 (2.2) 2.4 (1.4) 0.0 (0.0) 1.6 (1.1) 2.7 (1.6)
85°C 8.4 (4.9) 4.0 (2.3) 3.2 (1.9) 0.0 (0.0) 2.1 (1.2) 3.1 (1.8)
100°C 11.2 (6.7) 5.1 (3.0) 4.0 (2.3) 0.1 (0.2) 2.7 (1.6) 3.3 (1.9)
115°C 14.8 (9.0) 6.1 (3.6) 5.1 (3.0) 0.1 (0.2) 3.6 (2.1) 3.5 (2.0)
125°C 17.4 (10.7) 9.2 (5.5) 6.1 (3.6) 0.1 (0.2) 4.2 (2.4) 3.7 (2.2)
The initial state before the simulation is a step profile between the thin layer of deposited Bi and
the semi-infinite Sn. Since there is limited solubility of Bi in Sn and thus the concentration range
within the Sn solid solution is small, a constant interdiffusion coefficient with respect to
concentration was assumed. In the forward simulation, the diffusivity of Bi in Sn was estimated
and then compared to the experimental data via an error value (sum of least squares) with each
repeated simulation. The diffusivity was subsequently adjusted until the minimum error between
the simulation and experimental data was achieved [59, 62, 63]. To calculate the error between
the experimental data and simulated profile, the following steps are taken:
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1. For each value of diffusion distance, the difference between the corresponding
experimental concentration and simulated concentration (in atomic % Bi) is taken.
2. The absolute value of each difference is taken.
3. The absolute difference values are averaged to calculate the error (also in Bi atomic %).
In addition to the diffusivity, the amount of sputtered Bi remaining (in µm) is also calculated by
the simulation. It is expected that the calculated amount of Bi remaining will vary along with the
diffusivity, depending on which phase diagram is considered. The calculated Bi remaining (as a
percent of the initial thickness of the film) for all three diagrams can therefore be compared to
the measured percentage of Bi remaining (from the SEM images) to verify which diagram best
represents the experimental data. Because the coarse-grained Sn samples had a finite layer of Bi
deposited, this difference is significantly easier to observe with these samples compared with the
polycrystalline Sn samples (where the Bi source was ‘infinite’).
For the coarse-grained samples, the diffusivities simulated using the best-fitting diagram were
compared to Sn grain orientation to deduce the effect of Sn grain anisotropy. These experimental
diffusivities were fitted to the diffusion anisotropy relation (Eqn 2.30) [55].
100
Chapter 5 Solid-State Diffusion of Bismuth in Polycrystalline Tin
5.1 Overview
This chapter covers the results from the study of solid state diffusion of bismuth in
polycrystalline tin. As mentioned in Chapter 4, the grain structure of a Bi-containing solder alloy
will undergo recrystallization and subsequent reduction in grain size over the course of solder
joint lifetime [7]. It would therefore be logical to focus on the diffusion in monocrystalline Sn
first, however in this thesis, polycrystalline diffusion was covered first. This was because the
polycrystalline diffusion analysis is better served as a ‘proof of concept’ and is far easier to
analyze compared with monocrystalline diffusion.
In the polycrystalline diffusion study, as indicated in Section 4.2, four temperatures were
considered – 85°C, 100°C, 115°C, and 125°C. Six diffusion triples, each producing two
interfaces for a total of twelve diffusion interfaces, were prepared for each temperature. The
sample microstructure was characterized using SEM and EBSD. Compositional data was
collected using EPMA. The diffusivities were calculated using MatLAB and Python software,
using inverse methods (Sauer-Friese and Hall) and a forward simulation technique, respectively.
Finally, Arrhenius parameters were calculated using the data from both techniques. These results
will be compared with those from the coarse-grained and monocrystalline diffusion results to
evaluate the effects of grain boundaries on the diffusivity of Bi in Sn, in the solid state.
5.2 Electron Microscopy Results
Electron Backscatter Diffraction was the primary tool used to characterize the polycrystalline Sn
samples. Characterization was performed before and after assembly of the diffusion triples, to
understand the changes in grain structure during the entire experimental procedure. Some select
EPMA data is also shown in this section, although it is emphasized that diffusivity was derived
from the average of several data sets.
5.2.1 As-Solidified Microstructure
Prior to assembling the diffusion couples, EBSD was performed on as-quenched Sn samples to
verify the crystallinity. These are compared to air cooled Sn, with cooling performed at standard
101
ambient conditions. After both quenching (Figure 5.1a-b) and air cooling (Figure 5.1c-d), the
texture is relatively weak, with a wide range of grain orientations and many HAGBs present. The
quenched Sn samples produced smaller average grain size than the air-cooled Sn samples with
less variance, and the overall texture was slightly weaker.
Electron Backscatter Diffraction was performed on Sn samples that were aged at 120°C for 7
days after quenching (to simulate a heat treatment during diffusion) to observe the changes in the
grain structure of Sn over time. Aging the Sn results in a much stronger texture than both as-
solidified cases (quenched and air cooled), with fewer HAGBs and a more preferred orientation
(in this case, close to <001>). In addition, the aged Sn samples generated comparable grain size
and distribution to the air-cooled samples (Figure 5.1e-f). It is noted that the aging treatment
used here (120°C for 7 days) is harsher than any of the diffusion conditions planned in this work,
however it is evident that the metallurgy of Sn is very temperature sensitive and it can be
expected that some changes to the grain structure and orientation could occur during diffusion.
For all EBSD results, the inverse pole figure (IPF) ‘Z’ direction is shown. Grain size (d) statistics
are shown in Table 5.1. The critical misorientation, which delineates one grain from another,
was set to be 10°; this corresponding to the LAGB misorientation limit.
Table 5.1: Grain Size Statistics (diameters) for Sn samples
Statistic Quenched Air
Cooled
Quenched
+ Aged
Average (µm) 12.24 25.39 21.13
Std Deviation (µm) 6.77 15.87 14.24
Minimum (µm) 4.79 6.56 6.70
Maximum (µm) 42.88 103.8 86.16
102
Figure 5.1: Electron Backscatter Diffraction of pure Sn. Quenched (a-b), air-cooled (c-d), aged
(120°C for 7 days) after quenching (e-f).
(a)
(b)
(c)
(d)
(e)
(f)
103
5.2.2 Diffusion Microstructure
Based on the various Sn-Bi binary phase diagrams, it is known that the solid solubility of Bi in
Sn is low at room temperature (the conditions at which the diffusion microstructure was
characterized) [25, 26, 27]. At concentrations greater than the solid solubility, it is expected that
a two-phase region consisting of a β-Sn matrix and Bi precipitates would exist. At concentrations
lower than the solid solubility, Bi remains completely dissolved in the β-Sn matrix. These
characteristics were observed in the polycrystalline diffusion samples, despite being quenched
after annealing. This is contrary to what is observed in other diffusion systems, where the second
phase remains in a supersaturated solid solution after quenching [62, 63]. It is likely that Bi
nevertheless precipitated out of solid solution because room temperature is very close to the
annealing temperatures and the melting point of Sn. Thus, the Sn-Bi system is quite active at
‘low’ temperatures such as room temperature [58]. The presence of Bi precipitates close to the
interface, therefore, was used as an indicator to identify whether diffusion of Bi in Sn took place.
These precipitates formed in a ‘layer’ close to the diffusion interface.
Example diffusion microstructures are shown in Figure 5.2. The precipitate layer was not
uniform and varied in thickness, not only between samples, but also within samples. The surface
area of the compressing hex bolt was smaller than the surface area of the Sn and Bi pieces – this
may have resulted in variance in the adherence of the two species. In addition, localized
oxidation of the Sn or Bi samples may have created a barrier against atomic movement across the
interface. No PSN-induced recrystallization was observed; and as the samples were already
highly polycrystalline, it was difficult to distinguish the as-quenched grain structure from
mechanically-induced recrystallization. These locations were not selected for EPMA analysis;
only regions representing the most consistent extent of precipitation were chosen.
Better adhesion between Sn and Bi may promote diffusion of Bi for the following reasons:
• More ‘intimate’ contact between the two species allows for easier atomic movement
across the diffusion interface.
• As the samples were mechanically compressed together, the stress in the system will
promote recrystallization of the Sn; grain boundaries serve as high-energy pathways
which accentuate diffusion [55, 56, 58].
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• The mechanical stress in the system will increase the dislocation density; networks of
dislocations serve as high-energy pathways which accentuate diffusion [56, 58].
Bismuth precipitate morphology also varied between test temperatures. At lower temperatures,
for example at 85°C (Figure 5.2a), Bi precipitates were smaller and more spherical, while at
higher temperatures (such as 125°C, Figure 5.2d), precipitates were more lamellar. It is evident
that the polycrystalline diffusion couples contain significantly high dislocation density and
recrystallization because of quenching and mechanical compression. There are therefore ample
high diffusivity pathways in these samples (such as grain boundaries and dislocation pipes) in
which Bi atoms will tend to accumulate during annealing. It is possible that the difference in
morphology can be explained by the difference in the amount of cooling from the annealing
temperature to room temperature. At the higher temperatures, cooling occurs over a longer time
period, promoting coarsening of precipitates. These precipitates coarsen along the grain
boundaries / dislocation pipes, forming lamellar precipitates. At lower temperatures, therefore,
less coarsening occurs, forming more spherical precipitates.
Some sample EBSD maps are shown in Figure 5.3, along with accompanying inverse pole
figures (IPFs). It is noted that the grain size increases with depth from the interface (at the top of
each map). This is likely the result of the deformation that was introduced near the surface of the
Sn via the compression with the hex bolts in addition to the recrystallization of Sn caused by Bi
precipitation. It is also evident that the average grain size increases with temperature. This can be
seen in the IPFs – the IPF of the sample annealed at 85°C (Figure 5.3a&d) shows a wide range
of orientations and a weaker texture, while the IPF of the sample annealed at 125°C (Figure
5.3c&f) shows only a few orientations are present and a stronger texture. This may be a result of
a greater driving force for grain growth at higher temperatures – larger grains will relieve the
internal stress from the Bi precipitation and compression of samples. Such changes to grain size
reduce the grain boundary content in the Sn samples and sufficient grain growth at higher
temperature would produce a diffusivity result more closely resembling that of lattice diffusion.
As will be shown in Chapter 6, this is indeed the case. It is unknown at which point in the
annealing treatment (and diffusion) the grain growth occurred, however it is likely that for all
temperatures, diffusion was initially significantly faster due to the higher grain boundary content,
and slowed as annealing time increased due to the reduction of grain boundary content in the
sample.
105
It is noted that the grain size statistics in Table 5.2 were determined by considering only grains
larger than 5µm (the minimum grain size seen in the as-quenched Sn samples was 4.7µm). This
is a rough approximation of the grain structure prior to cooling, and filters out the very small Sn
grains resulting from Bi precipitation. The critical misorientation was once again set at 10°.
Figure 5.2: Representative microstructures (800x magnification) of diffusion zone at different
temperatures. Bi precipitates are spherical at 85°C and 100°C (a,b); mixture of spherical and
lamellar at 115°C (c), lamellar at 125°C (c).
(b)
(c)
(a)
(d)
106
Figure 5.3: Sample EBSD data from diffusion samples, demonstrating increase in average grain
size with temperature. 85oC Inverse Pole Figure (IPF) Y coloring map (a); 100oC IPF Y coloring
map (b); 125oC IPF Y coloring map (c); IPF Y for 85°C map (d); IPF Y for 100°C map (e); IPF
Y for 125°C map (f); IPF color coding (g).
Table 5.2: Summary of Grain Size Data from Polycrystalline Diffusion Samples
Temperature Average d
(µm)
Standard
Deviation
(µm)
85°C 10.26 8.17
100°C 10.54 7.94
115°C 12.66 12.31
125°C 19.40 23.77
5.2.3 Diffusion Profiles
For each sample, the composition profile varied greatly between sites; as shown in Figure 5.4, in
which three raw profiles are superimposed on the same plot. This is largely due to a difference in
grain orientation and variance in the depth of grain orientation changes. It is expected, as grain
boundaries are known to accentuate diffusion, localized ‘jumps’ in Bi concentration will occur at
or near grain boundaries (depending on the exact location of the EPMA analysis point). In
addition, localized jumps in orientation may be explained by the presence of Bi precipitates.
(a) (b) (c)
(d)
(e)
(f)
(g)
(a) (b) (c)
(d) (e)
(f) (g)
107
Figure 5.4: Sample raw Bi concentration profiles from 100oC samples.
This can be explained by knowledge of the physics of the interaction volume in electron
microscopy. Although the electron beam impinges on a very small site on the surface,
characteristic x-rays originate from an interaction volume wider than the spot size of the electron
beam [81]. As observed in the various Sn-Bi binary phase diagrams, the solubility of Sn in Bi is
limited [25, 26, 27]. This implies that the Bi precipitates in the α+β phase are composed largely
of Bi. According to precipitation theory, when the diffusion couples are cooled to room
temperature, Bi atoms within the Sn matrix undergo long-range diffusion to nucleate a small Bi-
rich volume, which then rearranges itself into the rhombohedral Bi crystal structure. The
interface between the Sn matrix and Bi gives rise to a free energy, which is reduced as further Bi
migrates to the interface, resulting in precipitate growth [50]. As the precipitates consist almost
entirely of Bi and contain little Sn, the elemental makeup of α+β phase region is, on the
microscale, quite heterogeneous.
The depth of the interaction volume for both Sn and Bi can be approximated using the Kanaya-
Okayama Range formula (Eqn 5.1) and these values are given in Table 5.3 [81].
𝑅𝐾𝑂 =0.0276𝐴𝐸𝑜
1.67
𝑍0.889𝜌
(Eqn 5.1)
108
Where A is the atomic mass in g/mol (118.71g/mol for Sn; 208.98g/mol for Bi), 𝐸𝑜 is the
accelerating voltage of the EPMA (15kV), Z is the atomic number (50 for Sn; 83 for Bi), and ρ is
the density in g/cm3 (7.27g/cm3 for Sn; 9.78g/cm3 for Bi).
Table 5.3: Calculated Interaction Volume Depth (Kanaya-Okayama Range) for Sn and Bi
Element Kanaya-Okayama Range (µm)
Sn 1.28
Bi 1.07
In a two-phase region composed of a β-Sn matrix and non-uniformly distributed Bi precipitates,
where the EPMA points are separated by 2.5µm, it is reasonable to expect that the reported Bi
concentration at one point may vary drastically from that at another point, because of the
following:
• The EPMA collected elemental data derived from x-rays from interaction volumes with
𝑅𝐾𝑂 between 1.07 and 1.28µm.
• These interaction volumes were spaced 2.5µm apart (EPMA point spacing).
• The observed interparticle spacing is comparable to the interaction volume and particle
size.
In addition, x-rays may originate from Bi precipitates residing in the sample subsurface, which
are not visible using BSE imaging.
These features can be observed in Figure 5.4, which contains data from one diffusion profile.
Here one can see large jumps in Bi concentration within 20µm of the interface (indicative of the
two-phase region). As Bi concentration decreases, any ‘jumps’ in concentration may be
explained by the change in grain orientation, and within each grain the gradient is quite smooth.
In the polycrystalline diffusion samples, 12 profiles were averaged, which resulted in moderately
smooth diffusion profiles prior to moving average smoothing. In coarse-grained and
monocrystalline diffusion samples (Chapter 6), these jumps/spikes are more relevant as fewer
profiles were averaged.
5.3 Diffusivity Analysis
For the diffusion data from polycrystalline Sn, two of the methods described in Chapter 4 were
used to analyze the data and determine diffusion coefficients – inverse methods as well as the
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forward simulation [59, 60, 61, 62, 63]. It is important to note that of these, only the inverse
methods consider the diffusivity to be a function of concentration. The forward simulation used
in this study assumes the diffusion coefficient is independent of composition, however it can be
modified to depend on concentration. Using the diffusion coefficients calculated by each method,
the Arrhenius parameters (𝐷𝑜and 𝑄𝐴) were then determined.
5.3.1 Inverse Methods
As mentioned in Chapter 4, 12 profiles per temperature were averaged. Example profiles after
averaging, addition of extra points, and smoothing are shown in Figure 5.5. The resulting
average profile was relatively smooth, so only a small amount of moving average smoothing was
required.
Figure 5.6 shows the interdiffusivity plot (with Sauer-Friese [60] and Hall [61] data) and Hall
plot for the 100°C data. In the interdiffusivity plot, the blue data points show how the simulation
calculated the interdiffusivity in Sn-Bi as a function of mole fraction, using the Sauer-Friese
method (Figure 5.6a). Using this method, the interdiffusivity is very unreliable at low Bi
concentrations, owing to the drastic difference in order of magnitude that can be observed. This
is likely the result of limitations with the EPMA data acquisition – as the detection limit is set at
350ppm, any values of mole fraction at or below this level will not be precise. In the Hall plot
(Figure 5.6b), λ<0 represents the Sn-rich side of the diffusion couple; this is where the linear fit
was selected using the Hall methodology to approximate the interdiffusivity at low Bi
concentrations. As indicated in Section 2.3.1.2, the Hall method addresses the downfall of the
Sauer-Friese method yielding unreliable results below the EPMA detection limit, as seen in
Figure 5.6a [61]. This approximation was superimposed on Figure 5.6a. From the
interdiffusivity plot, the impurity diffusivity of Bi in Sn (𝐷𝐵𝑖) at each temperature can be
extracted. This is accomplished by utilizing a simplified version of Darken’s second law [58]
(Eqn 2.35) [58]. If 𝑁𝑆𝑛 → 1, 𝑁𝐵𝑖 → 0, and Darken’s law simplifies to (Eqn 5.2:
�̃�(𝐶) = 𝐷𝐵𝑖
(Eqn 5.2)
110
Figure 5.5: Averaged concentration profiles for 100oC. Entire, unsmoothed profile, with
additional Sn-rich and Bi-rich points indicated with red ovals (a). Sn-rich side, unsmoothed
profile (b). Entire, smoothed profile (c). Sn-rich side, smoothed profile (d).
As previously mentioned, since there is a lot of uncertainty in the interdiffusivity data collected
using the Sauer-Friese method [60] at low Bi concentration, the Hall method data was utilized to
evaluate the impurity diffusivities at each temperature. This was done by linearly extrapolating
the Hall data to the y-axis of the interdiffusivity plot, where the Bi mol fraction = 0. These
impurity diffusivities are given in Table 5.4.
(a)
(c)
(b)
(d)
111
Figure 5.6: MatLab-generated plots used to determine interdiffusivity for the 100°C data.
Interdiffusivity plot, showing Sauer-Friese and Hall results (a); Hall plot of u vs. λ, showing the
linear fit used to determine Hall result (b).
Table 5.4: Experimentally Calculated Impurity Diffusivities of Bi in Polycrystalline Sn
using the Hall Inverse Method
Diffusion
Temperature
Bi Impurity
Diffusivity
(cm2/s)
85°C 6.53 x 10-12
100°C 5.69 x 10-11
115°C 4.39 x 10-10
125°C 6.52 x 10-10
5.3.2 Forward Simulation
The results from the simulation of the polycrystalline diffusion data is given in Table 5.5. The
data was simulated assuming solid solubilities from the three reported Sn-Bi phase diagrams [25,
26, 27]. The error (in atomic percent) is also given.
As indicated in Section 4.5.3, the diffusivity of Sn in Bi was approximated using the self-
diffusivity data for Bi from the literature [100]. While there is inconclusive evidence to suggest
that these values are a good representation of the diffusion of Sn in Bi, the simulation results are
nearly identical if the assumed diffusivity is adjusted by several orders of magnitude. The
‘extreme’ values of 6.72 x 10-12 cm2/s (parallel to the ‘c’ axis at 125°C) and 7.13 x 10-31 cm2/s
(parallel to the ‘a’ axis at 85°C), for example, both yield diffusivities of 1.11 x 10-11 cm2/s for the
diffusivity of Bi in polycrystalline Sn at 125°C (assuming the Lee diagram), compared with the
(a) (b)
112
value of 1.10 x 10-11 cm2/s (shown in Table 5.5) if the average of the ‘a’ and ‘c’ axis self-
diffusivities at 125°C is assumed. The error values are also nearly identical.
Table 5.5: Experimentally Calculated Diffusivities of Bi in Polycrystalline Sn using
Forward Simulation
Diffusion
Temp.
Lee [25] Vizdal [26] Braga [27]
D (cm2/s) Error D (cm2/s) Error D (cm2/s) Error
85°C 4.17 x 10-12 8.03 x 10-4 5.45 x 10-12 4.63 x 10-4 8.77 x 10-12 4.25 x 10-4
100°C 7.39 x 10-12 1.19 x 10-3 1.11 x 10-11 6.77 x 10-4 1.93 x 10-11 3.91 x 10-4
115°C 8.22 x 10-12 4.89 x 10-4 1.45 x 10-11 1.86 x 10-4 1.70 x 10-11 2.00 x 10-4
125°C 1.10 x 10-11 2.79 x 10-4 1.42 x 10-11 2.89 x 10-4 1.82 x 10-11 3.24 x 10-4
Two results in Table 5.5 are not thermodynamically valid (in bold font):
• For simulations in which the Vizdal solubility was considered, the calculated diffusivity
is lower at 125°C than 115°C.
• For simulations in which the Braga solubility was considered, the calculated diffusivity is
higher at 100°C than 115°C and 125°C.
Based on these results, neither the Vizdal nor Braga diagrams are representative of the diffusion
observed in this study. In Figure 5.7, the simulated results from each phase diagram are
compared to each other and the experimental data. Based on this figure, the Lee diagram is more
than likely the best fit to the experimental data, considering that it best matches the entire
experimental profile. The polycrystalline diffusion data sets (12 per temperature) were averaged,
yielding a moderately smooth profile across all diffusion distances. Every experimental data
point from 0µm < x < 24µm exceeds the corresponding simulated Vizdal and Braga data points,
while the data points lie approximately equally above and below the Lee simulated profile. The
latter case would be expected for the very locally heterogenous microstructures observed in this
study, with Bi-rich precipitates embedded in a Sn-rich matrix. These results indicate that the Lee
diagram may be the more appropriate fit to the experimental data, in agreement with the results
reported by Belyakov [28].
There exists another output from the simulation which can confirm the most appropriate phase
diagram for this analysis – the thickness, or amount of Bi remaining in the Bi layer. For the
polycrystalline diffusion samples, this value cannot be determined, because the Bi pieces were
removed from the Sn prior to metallography. The thin film remaining term will be considered in
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the coarse-grained and monocrystalline diffusion analysis (Chapter 6), as the film is not
removed, and it is easier to measure since the layer is very thin.
Figure 5.7: Comparison between Experimental Data and Simulation, for 125°C Polycrystalline
Diffusion Data
5.3.3 Arrhenius Analysis
From the impurity diffusivity values determined using both the inverse methods and forward
simulation, it is possible to calculate the intrinsic diffusion parameters using the Arrhenius
relationship for temperature activated processes. Figure 5.8 shows the plotting of the data using
this relationship and two sets of diffusivity data (inverse Hall method, as well as the simulations
assuming solubility from the Lee diagram), along with linear lines of best fit. Moderate linear
correlation is evident for the forward simulation results; a poor fit is seen for the Hall method
results. From each of these sets of data, the instrinsic parameters were predicted. These are
shown in Table 5.6, along with 95% confidence bounds; these suggest the true value of the
parameter has a 95% probability to lie within this interval enclosed by these bounds [101]. In
addition, several parameters are included which describe how well the anisotropy model fits the
entire data set [101, 102]:
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• R2 indicates the proportion of variation in the experimental data that can be explained by
the model. R2 values range from 0 (no variability can be explained) to 1 (all variability
can be explained).
• The residual sum of squares (RSS) measures the discrepancy between the model and
empirical data, summing the square of deviations between each empirical data point and
the predicted value from the model. A smaller value of RSS indicates a better fit of the
model to the data.
It is noted that because 𝐷𝑜 and 𝑄𝐴 are based on experimental data, the values given should
considered as empirical approximations of these parameters [58].
Figure 5.8: Arrhenius plots for impurity diffusivity of Bi in polycrystalline Sn. Best fit lines
from which intrinsic parameters were derived is shown.
It can be seen from the results in Table 5.6 that confidence bounds are extremely wide – this is
likely indicative of the lack of data points (only four temperatures were considered). The inverse
Hall method yields distinctly different results to the forward simulation method and is not fitted
as strongly to the Arrhenius relationship. 𝐷𝑜 is twelve orders of magnitude lower in the simulated
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data, and 𝑄𝐴 is nearly four times smaller. In studies comparing inverse methods and the forward
simulation, it was found that there is a good match between both techniques, contrary to what is
observed here [103]. The suggested reasons behind the discrepancies between the methods are
given in the following section.
Table 5.6: Arrhenius parameters for impurity diffusivity of Bi in polycrystalline Sn
Hall Method [61] Simulation (Lee diagram [25])
Estimate 95% confidence
bounds Estimate
95% confidence
bounds
𝑸𝑨
(kJ/mol) 102.9
-200.3
406.1 26.59
5.32
47.86
𝑫𝒐 (cm2/s) 1.61 x 104 2.18 x 10-38
1.20 x 1046 3.39 x 10-8
3.93 x 10-11
2.92 x 10-5
R2 0.516 0.935
RSS 6.517 0.032
5.3.4 Comparison between Models
The main difference between the two methods is that the inverse methods assume a
concentration-dependant interdiffusion coefficient, and the impurity diffusivities reported are
those where the Bi concentration approaches zero [61], while the simulation assumes the
diffusivity is constant across all compositions, and does not truly represent an impurity
diffusivity. It is likely the discrepancy between the two methods is related to these
characteristics. In addition, the results from the Hall method showed relatively poor fit to an
Arrhenius relationship compared with those from the forward simulation.
Because the Hall method/Darken equation was used to estimate diffusivity at the Sn-rich (𝑁𝐵𝑖 →
0) end of the concentration profile, the impurity diffusion coefficients determined using this
approach are most representative of the data at the Sn-rich end of the profile, rather than the
entire profile. That said, the Hall method is fairly subjective in determining the linear fit to the
Hall plot (Figure 5.6b) and results may not be accurate. This characteristic prompted the
development of the forward simulation technique [63]. The Sauer-Friese data in the
interdiffusivity plots (Figure 5.9) suggests that the diffusion coefficient varies greatly (by an
order of magnitude, from 10-15 m2/s to 10-16 m2/s across the concentration range from 0.005 to
0.03 mole fraction). The diffusion coefficient calculated by the forward simulation may almost
be thought of as an ‘average’ of the concentration-dependent diffusivities on the Sauer-Friese
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plot, because the simulation minimized the average error between all experimental data points
and the simulated profile.
For all temperatures, the impurity diffusivity calculated using inverse methods was higher than
that calculated using the forward simulation. This can be observed in the comparison between
the experimental data and the simulated diffusion profile (Figure 5.7). Where 𝑁𝐵𝑖 → 0, the
experimental data is generally higher than the simulated data. Recalling that the Hall method
calculates the diffusivity based on the data where 𝑁𝐵𝑖 → 0, it can be assumed that the Hall result
superimposed on this plot would correlate more strongly with the experimental data. This would
indicate a higher calculated diffusivity from the Hall method than from the simulation.
It is noted that the forward simulation technique can be used to solve for concentration-
dependent diffusion coefficients, however in this study it was initially decided to use a constant
diffusion coefficient with respect to concentration, due to the relatively narrow solubility range
of Bi in β-Sn. Using this approach would likely yield the most accurate diffusivity data and
preliminary results are given in the following chapter, after consideration of lattice diffusion
using a constant diffusion coefficient.
Figure 5.9: Interdiffusivity plot for 100°C. The black dotted line represents the diffusivity
calculated by the forward simulation.
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5.4 Concluding Remarks
For the first time, solid-state diffusion couples were assembled using Sn and Bi. Quenching of
Sn was conducted to produce a highly polycrystalline grain structure. Several methods were used
to estimate diffusivity of Bi in Sn, including traditional inverse methods such as the Sauer-Friese
and Hall methods (which assume diffusivity is dependent on concentration) and a novel forward
simulation technique which has demonstrated success in other alloy systems such as Fe-Ni.
Several experimental challenges were encountered in this study:
• Grain growth occurred during annealing; the average grain size of samples annealed at
125°C (19.40µm) was nearly twice as large as those annealed at 85°C (10.26µm). This is
indicative of a greater driving force for grain growth at elevated temperatures. It is
evident that the polycrystalline Sn structure was highly metastable and prone to evolution
even with short annealing times.
• The mechanical compression of the Sn and Bi in this study was a likely contributor to the
inconsistent diffusion front observed across the sample cross-sections, and likely induced
more deformation in the Sn than would be expected during the typical solder joint life
cycle.
• Quenching of the diffusion couples did not prevent the Bi from precipitating from solid
solution, likely because room temperature is relatively active compared to the melting
point of Sn and thus the Bi is prone to precipitate from solid solution without any
substantial driving force. The second phase precipitates, being rich in Bi, caused ‘spikes’
in the diffusion profiles; these were largely smoothed out as each resultant profile at each
temperature was averaged from a dozen experimental profiles. Precipitation would be a
significant issue if fewer profiles were to be averaged, as will be seen in Chapter 6.
The forward simulation was successfully used to independently verify the results of Belyakov
and determine the Lee diagram most accurately represents the experimental Sn-Bi diffusion
results from this study. Simulations using the Vizdal and Braga diagrams yielded results which
are not thermodynamically possible; for example, using the Braga diagram the diffusivity at
100°C was roughly 14% larger than that at 115°C. Furthermore, the simulated diffusion profile
using the Lee diagram fits the experimental data better than does the simulated Vizdal or Braga
profile. Results from the Lee diagram were subsequently used for Arrhenius analysis.
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The two methods for determining diffusivity in this study, the inverse (Hall) method, and the
forward simulation, yielded very different diffusivities for all temperatures, as well as Arrhenius
parameters. There was significantly better fit to Arrhenius model for forward simulation than the
inverse Hall method. The RSS was roughly 60 times smaller and 94% of the variability can be
explained by the model, compared with 52% for the Hall method. Furthermore, the confidence
bounds were substantially wider (for example, for activation energy, an order of magnitude) for
the inverse Hall method than the forward simulation. From the forward simulation, the activation
energy for diffusion of Bi in polycrystalline Sn was 26.59 kJ/mol, with a pre-exponential of 3.39
x 10-8 cm2/s. This value of activation energy is significantly lower than that of Sn self-diffusivity
in the lattice reported in the literature [72] and shown in Table 2.4. This value is still lower than
what would be expected for short-circuit diffusivity; the activation energy for grain boundary
diffusivity is typically between 40% and 60% that of lattice diffusivity. This result suggests that
multiple mechanisms may be operating which accentuate the diffusivity, such as dislocation
pipes caused by mechanical compression of the diffusion triples. In Section 2.3.6.2, fast diffusion
in Sn was discussed and it was estimated that fast diffusion of Bi is not likely to occur.
The Hall method and simulation assume different dependence of composition on diffusivity,
which is the most likely reason for the difference in results. Most practical diffusion systems
demonstrate some dependence of impurity concentration on diffusivity, thus it is likely that the
forward simulation (assuming a constant diffusivity) yields an inaccurate result and results in an
abnormally low activation energy. The Hall method likely too yields an inaccurate result as the
activation energy (102.9 kJ/mol) appears large considering the high grain boundary and (likely)
dislocation pipe content, and that this value is nearly equivalent to the lattice self-diffusivity of
Sn (average 108.7 kJ/mol). This may be caused by the subjective nature of the Hall fitting.
Further analysis on this is carried out in Chapter 6, including some preliminary work with a
concentration dependent diffusivity using the forward simulation.
Some recommendations for future work are as follows:
• Assemble diffusion couples using sputter deposition rather than mechanical clamping.
This will produce more consistent adhesion between Sn and Bi and eliminate the
extraneous dislocation density likely present in this work.
• Adapt the forward simulation model to treat diffusivity as a function of concentration.
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• Perform diffusion experiments using polycrystalline Sn at several additional temperatures
to produce a more conclusive fit to the Arrhenius relationship.
Despite these issues, the data shown in this chapter provides a rough understanding of the
diffusion properties in aged Bi-containing solder alloys. As the alloy progresses through its life
cycle, it may experience many stimuli including isothermal aging, thermal cycling, and vibration,
all of which may cause the Sn-rich phase to recrystallize. Such a material with increased grain
boundary content would allow for increased diffusion of Bi, which may affect the deformation
and failure mechanism of the alloy.
In the next chapter, the influence of grain boundaries will be initially removed from the analysis
by using coarse-grained and monocrystalline Sn. This will allow for the understanding of the
more fundamental bulk diffusion characteristics of the Sn-Bi system, representative of a typical,
as-reflowed solder joint. These results will then be compared to the results from polycrystalline
Sn to examine the effects of grain boundaries on diffusivity.
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Chapter 6 Solid-State Diffusion of Bismuth in Coarse-Grained and
Monocrystalline Tin
6.1 Overview
In the polycrystalline diffusion study, diffusion was likely highly accentuated by the presence of
grain boundaries – these are well-known high-energy pathways that are favorable for diffusion
[55, 56, 64]. It was thus not possible to determine diffusivity in the lattice. The mechanical
clamping of the Sn and Bi to achieve good adhesion also likely introduced immense strain in the
sample, leading to further recrystallization and induced large quantities of dislocations, which are
known to form networks which enhance diffusion in other alloy systems [55, 56]. The clamping
force was likely not uniform across the whole sample, meaning that adhesion was poorer in some
locations, and subsequently diffusion was non-uniform. This non-uniformity may also be caused
by oxidization of the Sn and/or Bi samples. It was thus difficult to determine representative
locations for further analysis. Sputter deposition was performed (in vacuum) immediately after
sample preparation, which would significantly minimize any oxidization of Sn.
To determine the true lattice diffusion properties in materials, it is desirable to use either
specially prepared monocrystals, or samples with very coarse grains within which lattice
diffusion can be reliably measured without the effects of grain boundaries. Monocrystals are
expensive to purchase, however very specific orientations can be produced. For Sn, much of the
diffusion data in the literature [104] is given in terms of two orientations, parallel to the ‘c’ axis,
or <0 0 1>, and parallel to the ‘a’ axis, or <1 0 0> / <0 1 0> [15]. The latter orientations are
perpendicular to the ‘c’ axis and theoretically should display the largest deviation in diffusivity
from <0 0 1>. Coarse-grained samples are an inexpensive alternative to study lattice diffusion,
however precautions must be taken to avoid regions in close proximity to grain boundaries.
In this study, several true monocrystalline Sn samples were purchased from Accumet Materials,
Co., however it was desired to examine several intermediate orientations. This was accomplished
by producing coarse-grained Sn samples in the lab using standard equipment, as discussed in
Chapter 4. The goal was to evaluate the fit of the various diffusivities (and their corresponding
orientations) to the anisotropy model for diffusivity (Eqn 2.30) [55, 56, 58]. Seven temperatures
(25°C, 50°C, 70°C, 85°C, 100°C, 115°C and 125°C) were selected for diffusivity analysis using
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coarse-grained Sn samples; four of these (85°C, 100°C, 115°C and 125°C) were chosen for
analysis using specially prepared Sn monocrystals with orientations parallel to the ‘a’ and ‘c’
axes. These four temperatures were also studied in polycrystalline Sn; it is also of interest to
compare the results from the two studies and examine the effects of grain boundaries (which are
observed to occur in greater quantities in aged Bi-containing Pb-free solder alloys) on the
diffusivity of Bi in Sn.
6.2 Electron Microscopy Results
Prior to preparing samples for sputtering of Bi, the microstructure of the coarse-grained samples
was characterized using EBSD mapping. After sputtering, SEM imaging and EBSD were used to
determine what, if any, effects the sputtering process had on the microstructure. These same
techniques were used to characterize the microstructure of the diffusion samples after annealing
and determine sample orientations. Finally, EPMA was used to collect the diffusion profile data.
6.2.1 As-Solidified Microstructure (Coarse-Grained Sn)
As observed in Figure 4.7, the texture of the as-solidified Sn samples was very coarse – very
large grains were formed, containing several subgrains bound by LAGBs. Several additional
EBSD maps of coarse-grained Sn are shown in Figure 6.1; it can be seen that the structure
varied between samples, however the average grain size was very large, with the same texture
persisting across the entire sample diameter.
6.2.2 As-Sputtered Microstructure
Some images of sputtered Bi are shown in Figure 6.2. As expected for sputtered films, a
columnar structure was formed, with slight variance in film thickness h (Figure 6.2a-b). Several
whiskerlike structures were also observed, however they were sparsely distributed (Figure 6.2c-
d). No further characterization was done on the film, however it is believed that the film was
highly polycrystalline – the ‘whiskers’ were likely the result of abnormal grain growth in certain
orientations of Bi. Good adhesion between the sputtered Bi and Sn substrate was observed, with
no gaps or voiding. Finally, sputter deposition did not alter the grain structure of the Sn substrate
(Figure 6.2e-f).
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Figure 6.1: Example EBSD maps of coarse-grained Sn (IPF Y coloring), along with
corresponding IPFs. A wide range of structures are observed, featuring LAGBs and interlaced
grains.
(a) (b)
(c) (d)
(e) (f)
(g) (h)
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Figure 6.2: As-sputtered microstructure of Sn-Bi diffusion couples. Bi film sputtered onto a Si
wafer, with thickness oriented parallel to the electron beam (a); and cross-sectioned couple,
revealing moderately uniform columnar structure (b). Abnormal ‘whiskerlike’ growths on
surface of film (c) were sometimes captured in cross-section (d). EBSD IPF Y coloring map (e)
and IPF (f) of as-sputtered Sn sample indicates sputtering did not alter the Sn grain structure.
(a) (b)
(c) (d)
(e)
(f)
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6.2.3 As-Annealed Microstructure (Coarse-Grained Sn)
Figure 6.3 shows representative diffusion microstructures from 125°C. After annealing of the
diffusion couples, the Bi diffused into the Sn and produced a uniform precipitate layer close to
the interface (where the solid solubility was exceeded). The Bi in the sputtered layer was
replaced by Sn. The EBSD mapping indicated that the Sn grain structure was mostly preserved
after annealing. It is highly likely the Bi precipitating out of solid solution upon cooling caused
the Sn to recrystallize via Particle Stimulated Nucleation (PSN), a consequence of relieving
stress brought upon by the density and crystal structure mismatch between the Sn and Bi phases
[41].
Within each temperature, some samples produced drastically different SEM microstructures – for
example in Figure 6.3b, the sputtered layer was consumed to a lesser extent than in Figure 6.3a
and fewer Bi precipitates were present. Examining the corresponding EBSD maps, less
recrystallization of Sn occurred in Figure 6.3d than in Figure 6.3c. It was determined after
comparing IPFs (Figure 6.3e&f) that these samples were closer to the ‘c’ axis in the Sn
tetragonal unit cell. This suggests that Sn oriented closer to the ‘c’ axis may demonstrate a lower
diffusivity, as more Bi tends to remain close to the interface and less migrates into the Sn.
Two additional noteworthy features are also visible, shown in Figure 6.4. Firstly, h increased
from 0.7µm (Figure 6.4a) to around 1.12µm. This increase (60%) is not completely accounted
for by ρ difference of 34% between Sn (7.27 g/cm3) and Bi (9.78 g/cm3), suggesting an
inequality in mass flow of the two species and that the diffusivity of Bi in Sn exceeds that of Sn
in Bi. Due to the uneven sputtered surface, h was calculated using an objective method. A grid
was superimposed on the BSE image; the gridlines were used as unbiased locations to take
measurements of h (Figure 6.4b). For each site, two images were taken, for a total of ten
measurements of h. Secondly, small gaps are visible at certain locations in the diffusion interface
(Figure 6.4c&d). Along with the apparent inequality in mass flows and based on the vacancy
mechanism for substitutional diffusion, this gap suggests that a surplus of vacancies is formed at
the interface [56]. These precipitate into Kirkendall voids and subsequently coalesce over time
[55].
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Figure 6.3: Diffusion microstructures at 125°C, indicating the presence of Bi induces
recrystallization of Sn. It is also indicated that the <7 7 6> orientation (a-b) may yield higher
diffusivity than the <1 3 4> orientation (c-d). Corresponding IPFs for the <7 7 6> (e) and <1 3 4>
(f) orientations, as well as the color legend (g). All EBSD data shown is in the ‘Y’ direction.
(a) (b)
(c) (d)
(e) (f)
126
As expected, diffusion microstructures also varied between temperatures (Figure 6.5). It is
observed that as the annealing temperature is decreased, the α+β region (consisting of Bi
precipitates in a Sn matrix) becomes narrower. This is accompanied by reduced depletion of Bi
(or substitution of Bi with Sn) in the original sputtered layer. These observations are an
indication of the reduction in solubility of Bi in Sn as temperature is decreased, justifying the
assumption made in the diffusion simulation that the maximum Bi that can be present in the Sn
substrate is equivalent to the solubility limit. The results from room temperature diffusion
couples (Figure 6.5d&e) further confirm that all diffusion occurs within solid solution as no Bi-
rich phases exist on the Sn-rich side of the diffusion interface.
Figure 6.4: SEM images indicating growth of h after annealing at 115°C. As-sputtered film,
average h = 0.70µm (a). On average, h increased (b-d). Yellow arrows indicate h; red arrows
indicate Kirkendall voids.
(a) (b)
(c) (d)
127
Figure 6.5: SEM analysis comparing diffusion microstructures from different temperatures.
115°C (a), 100°C (b), 85°C (c), and 25°C (d). Lower magnification image at 25°C (e) with
corresponding EBSD ‘Y’ map, revealing recrystallization of Sn in the absence of Bi precipitates
(f).
(a) (b)
(c) (d)
(e) (f)
128
Recrystallization was observed in regions beyond the α+β region (Figure 6.3c&d), and also in
25°C diffusion couples, where no α+β region is possible given the annealing and analysis
temperatures are identical (Figure 6.5f). This is contrary to what was predicted in the aging
study in Chapter 3 and suggests Bi in saturated solid solutions may cause Sn to recrystallize, or
recrystallization may precede precipitation. Further analysis of this phenomenon using EPMA
data is discussed in Section 6.2.6.
As mentioned in Section 4.3.4, to control the duration of room-temperature storage across the
entire set of samples, samples were temporarily stored in liquid nitrogen. It is known that Sn
undergoes an allotropic transformation from BCT β-Sn to diamond α-Sn below approximately
13°C. No α-Sn was observed in any region of the coarse-grained samples. Only five of the seven
desired annealing temperatures were studied; 50°C and 70°C were dropped due to extensive
equipment delays.
6.2.4 As-Annealed Microstructure (Monocrystalline Sn)
It was expected that the monocrystalline Sn samples would, similarly to the coarse-grained Sn
samples, retain their initial orientations except for recrystallization near the diffusion interface.
Unfortunately, the sample structure generally consisted of a multitude of low angle grain
boundaries, and aside from some small regions, the orientations were not identified as <0 0 1> or
<0 1 0>. Electron Probe Microanalysis was still performed on these samples, however they were
treated as coarse-grained Sn and only 3-4 line scans were performed on each.
It is likely that the incorrect orientations were produced by Accumet, and possible that sample
preparation induced the formation of LAGBs. It is also possible that the samples may not have
been sectioned exactly through the midline and thus the observed orientation in EBSD is slightly
misaligned from the crystal orientation parallel to the sample ‘Z’ axis (Figure 2.28e). The angles
between the observed and desired orientations of a few of the purchased monocrystalline Sn
samples were calculated and provided in Table 6.1. This data indicates that the most likely
reason for the incorrect orientation is supplier-related. Some sample EBSD maps, corresponding
to the data in Table 6.1, are shown in Figure 6.6.
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Figure 6.6: Example EBSD ‘Y’ maps of annealed ‘monocrystalline’ Sn samples. The maps are
oriented with the Bi deposition on the top, similar to Figure 6.3, Figure 6.4, and Figure 6.5.
Table 6.1: Example Orientation Data from Purchased Monocrystalline Sn samples
Sample Orientation Angular Deviation
a <2 11 7> 71.1° from <0 0 1>
b <2 15 3> 9.8° from <0 1 0>
c <0 4 10> 36.3° from <0 0 1>
d <1 6 5> 65.9° from <0 0 1>
6.2.5 Orientation Analysis & Maps
From the inverse pole figures (IPFs), approximate orientations of each sample were determined
and plotted on the Sn stereographic projection, one for each temperature (Figure 6.7). Again,
due to equipment issues, 50°C and 70°C were not considered. A wide range of orientations were
(a) (b)
(c) (d)
130
collected; the <1 1 0> orientation was the most reproducible. Angles between sample
orientations and <0 0 1> were then plotted against diffusivity; results are shown in Section 6.3.
(a) (b)
(c) (d)
(e)
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Figure 6.7 (previous page): Orientation maps for coarse-grained diffusion samples, based on
the <0 0 1> stereographic projection for β-Sn. Poles corresponding to sample orientations are
indicated in red. 125°C (a), 115°C (b), 100°C (c), 85°C (d), 25°C (e).
6.2.6 Diffusion Profiles
Referring to Section 5.2.3 and examining some sample data from this study, the challenges with
modelling diffusivities in systems such as Sn-Bi, where quenching the diffusion couples after
annealing is not ideal, are evident. As shown in Figure 6.8a, even after averaging the three
profiles per orientation, the first few data points in the profile produce large anomalous spikes of
high Bi content; this is indicative of Bi precipitation that occurs when the Bi content exceeds the
solid solubility at room temperature (Figure 6.8b). For each sample, only 3-4 profiles were
collected and averaged – this does not allow for natural smoothing via averaging (as in the
polycrystalline diffusion profiles) of the first few data points as was observed in Chapter 5 using
polycrystalline Sn, in which twelve profiles were averaged.
Therefore, depending on where the electron beam interacts with the sample surface, the reported
Bi content can vary in the α+β region for the same diffusion depth, and within the region overall.
Therefore, for this study, any data collected in the α+β region is not representative of the true
concentration at the respective depth. This is why quenching is often desirable to ‘freeze’ the
solute atoms (Bi) in their high-temperature configuration (solid solution) [59] [63]. Since the
error between simulated and experimental data would be very high within the α+β region, it was
decided to only consider data below the solid solubility at room temperature (25°C, the
approximate analysis temperature). For example in Figure 6.8, all data from diffusion depths
less than 20µm exceeds the solid solubility and was thus not compared to the simulated profile
via error and not used to evaluate the diffusivity.
As discussed in Section 6.2.3, it is apparent that recrystallization may also be caused by
dissolved Bi in the Sn matrix. Figure 6.9 is the same EBSD map as in Figure 6.5; the
superimposed lines represent the portion of the EPMA line scans traversing the recrystallized
region. The recrystallization depth was measured, and the Bi content (at%) was approximated by
linear interpolation between EPMA data points. The data is provided in Table 6.2.
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Figure 6.8: Example of concentration profile (a) of Bi diffusion in coarse-grained Sn. SEM
imaging (b) indicates that the ‘spikes’ in the profile data (for example at 5 and 7.5 µm) may be
the result of the electron beam impinging an area of the sample rich in Bi, such as a precipitate.
Figure 6.9: EBSD ‘Y’ map of 25°C diffusion couple, showing recrystallization of Sn close to
the diffusion interface. Portions of each EPMA line scan location are superimposed, indicating
the maximum depth of recrystallization.
(a) (b)
1 2 3
133
Table 6.2: Sample EPMA Data of 25°C Diffusion Couple
Line Recrystallization
depth (µm)
Approx
Bi at%
1 15.4 0.09
2 13.5 0.12
3 8.4 0.30
It is difficult to determine exactly what composition of Bi is necessary to induce recrystallization
from these results, since the recrystallization of new grains is initiated with nucleation and radial
growth, and it is not possible to observe the change in composition and microstructural evolution
over time [12, 41, 42]. At the location of the third line scan, recrystallization terminates at around
8.4µm with 0.3 at% Bi; the EPMA detected 0.264 at% of Bi at 10µm, where no recrystallization
is present. It can therefore be speculated that the minimum required Bi composition to induce
recrystallization is about 0.3 at%. A possible reason for the deeper recrystallized regions at scans
1 and 2 is that the growth of one grain may induce further recrystallization nearby.
It is therefore possible that at all annealing temperatures, the evolution of grain structure is
dynamic and occurs as Bi migrates through the Sn matrix, rather than solely after cooling (upon
precipitation) as was previously surmised. This implies that it may be impossible to avoid
recrystallized grains accentuating diffusion, although it is likely given that small amounts (< 0.3
at%) of Bi are present in regions that have not undergone recrystallization, that the diffusion
front is unaffected by recrystallization and occurs entirely in the bulk. These hypotheses would
need to be verified using in situ scanning or transmission electron microscopy.
6.3 Diffusivity Analysis
The EPMA data were first analyzed using both the slab source model (Eqn 2.5) and the forward
simulation initially developed by Zhang and Zhao for the 125°C diffusion data only [59]. After
analyzing the results, further analysis on the remaining temperatures was conducted using the
forward simulation method. Curve fitting of the diffusivity data to the anisotropy equation was
conducted to estimate the diffusivities parallel to the ‘c’ axis (𝐷||) and perpendicular to the ‘c’
axis (𝐷⊥). Arrhenius analysis was then conducted using these estimated diffusivity values to
obtain 𝐷𝑜 and 𝑄𝐴 for each extreme orientation. Finally, preliminary analysis was conducted
assuming a concentration-dependent diffusion coefficient; future work will involve optimization
of the model.
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6.3.1 Comparison between Models
Figure 6.10 shows how the diffusivities from the slab source model and forward simulation
compare. The dotted line represents equality between the two models; all data points lie above
the line, indicating all diffusivities reported higher for the forward simulation. Figure 6.11 shows
examples of how both models (slab source and simulation) compare with the experimental data
from 125°C. It can be seen that the slab source model (Figure 6.11a-b) fits the experimental
fairly poorly. The simulation model, on the other hand, fits the experimental data significantly
better (Figure 6.11c-d). These observations were observed for all data sets.
Figure 6.10: Relationship between diffusivities calculated using the slab source (x axis) and
forward simulation (y axis) models
As mentioned earlier, the boundary condition for the slab source model requires the h be on the
same order as the diffusion distance 2√𝐷𝑡 [56, 57, 58]. Using the diffusivity values determined
using both models, the validity of the slab source model was back-checked by calculating this
diffusion distance and comparing to the actual thickness of the sputtered Bi layer. As an
example, using the diffusivity derived from Figure 6.11a&b (D = 1.87 x 10-12 cm2/s), the
diffusion distance was calculated (Eqn 6.1):
135
2√𝐷𝑡 = 2√(6.93 𝑥 10−12 𝑐𝑚2𝑠⁄ ) ∗ (86400𝑠) = 15.5 µ𝑚 ≫ 0.67µ𝑚 = ℎ
(Eqn 6.1)
Figure 6.11: Examples of diffusivity calculation using slab source model (a&b) and simulation
model (c&d). Figures a & c show all data in the profile; figures b & d show only the data below
the solid solubility of Bi in Sn at 25°C (1.1 at%)
This calculation indicates the thickness of the Bi layer should be 2 orders of magnitude larger
than was actually sputtered. This confirms that the slab source model is inappropriate for
determining diffusivities of Bi in Sn in the sample geometries used in this study. Results from
this model were therefore discarded and only the results from the simulation were considered
further.
(a) (b)
(c) (d)
136
6.3.2 Phase Diagram Analysis
In Section 5.3.2, it was determined that for polycrystalline Sn, the Lee diagram [25] was a better
fit to the diffusivity data. This shows good agreement with the previously reported findings by
Belyakov et al [28]. For the coarse-grained data, the diffusion data from one temperature only
(125°C) was initially simulated using all three phase diagrams. This was done to verify the
polycrystalline diffusion results of this study, as well as those of Belyakov. Considering one
temperature only for this purpose also reduces total simulation computation time.
The simulation calculates the thickness of the Bi film remaining after annealing; for the coarse-
grained Sn diffusion samples, this term was calculated using each phase diagram and compared
to the experimental data to determine which phase diagram is the best approximation. The Bi
layer is finite in thickness, and as observed in Section 6.2, any changes to the layer after
diffusion are easily quantifiable. For each coarse-grained (125°C) diffusion sample, four BSE
images were taken at 4000x (each with a field of view of 64µm). Using ImageJ software, color
thresholding was used to isolate the Bi precipitates within the sputtered layer visible in each BSE
image, and statistical analysis was performed to calculate the total area of the Bi-rich phase in
the layer (Figure 6.12). This was divided by the initial area of the sputtered layer, assumed to be
0.67µm (h) x 64µm / image x 4 images. This calculation assumes negligible dissolution of Sn in
the Bi-rich phase. In a similar fashion, the amount of film remaining in the simulation was
divided by the initial layer thickness (0.67µm).
Both proportion values were then converted to percentages. In Figure 6.13, the difference
between the observed and calculated percentages of film remaining are shown for each phase
diagram. The calculated result from the Lee diagram is closest to that measured using SEM (a
difference of zero indicates a perfect match between experimental and simulated results) [25].
Based on this finding, it can be confirmed that the Lee diagram best represents the experimental
results.
137
Figure 6.13: Deviation of the calculated remaining Bi in the film from the experimental results.
The Lee diagram appears to be the best representation of the experimental diffusivity data.
Figure 6.12: Example of calculation
of remaining Bi-rich phase in
deposited layer. Original BSE image
(a); cropped BSE image of sputtered
layer, with all precipitates within the
substrate removed (b); precipitate
isolation prior to area calculation
(9.9µm2, or 23.1% of initial layer
composition) (c).
138
Despite the best fit shown with the Lee diagram, the experimental result and simulation are not
identical. There are four possible reasons for this. The first may be the result of the crystal
structure of the sputtered Bi layer and the diffusivity of Sn in Bi. The sputtered layer consists of
columnar grains, each likely with a different orientation, and the diffusivity in the film may
differ depending on orientation. The second possible reason may relate to the fact that the phase
diagram assumes equilibrium conditions, with a perfectly homogenous distribution of Bi in Sn.
Similar to Belyakov’s analysis, in which the diagrams of Vizdal and Braga were deemed
inaccurate due to concentration gradients, it is obvious that a concentration gradient exists in Sn-
Bi diffusion couples [28]. Because the initial concentration is always assumed to be equal to the
solubility limit and no greater, this is a limiting factor affecting mass flow into the Sn substrate.
If no solubility limit existed, there would be no limiting factor inhibiting Bi atoms from diffusing
across the interface and the concentration gradient would decrease, approaching equilibrium
conditions. This is a likely reason why the simulation, on average, reports less Bi diffusing into
the Sn substrate and more remaining in the film. The third reason may be because the simulation
assumed a constant diffusion coefficient and a better fit to the experimental data may be
achieved by considering a concentration dependent diffusivity. The fourth reason may simply be
due to experimental error.
6.3.3 Effects of Anisotropy
Figure 6.14 and Figure 6.15 show the relationship between the β-Sn grain orientation relative to
the ‘c’ axis and Bi diffusivity, using the diffusivities obtained via the simulation model, based on
the phase equilibria from the Lee diagram, for all five temperatures. The wellness of fit
parameters to the anisotropy equation (Eqn 2.30) R2 and RSS are provided in Table 6.3.
139
Figure 6.14: Scatter plots showing relationship between sample orientation <u v w> relative to
the ‘c’ axis and the diffusion coefficient (data from all samples plotted). 115°C (a); 100°C (b);
85°C (c); 25°C (d).
Table 6.3: Wellness of fit of experimental data to anisotropy equation
Temperature 125°C 115°C 100°C 85°C 25°C
R2 0.283 0.005 0.359 0.048 0.269
RSS 0.481 0.944 0.334 0.346 0.951
Examination of the data in Figure 6.14, Figure 6.15a, and Table 6.3 indicates the following:
• The data is highly scattered, with low R2 and high RSS values, and there is no clear trend
between orientation and diffusivity.
• All diffusivities are within about one order of magnitude, suggesting that the diffusivity
of Bi in Sn is not strongly anisotropic.
The EBSD data were re-examined to investigate whether the microstructure might be the cause
of the scatter. In the case of the 125°C data (Figure 6.15a), 11 of 18 samples contained LAGBs;
examples of sample texture are shown in Figure 6.15b-c. Both samples yielded nearly identical
diffusivities, however the <5 9 2> orientation (Figure 6.15c) contains a LAGB close to
(a) (b)
(c) (d)
140
perpendicular with the diffusion direction – this feature may have caused the diffusivity to be
under-reported. Removing the eleven data points containing LAGBs resulted in a much stronger
orientation relationship and fit to the anisotropy relation (Figure 6.16a) [55, 56, 58].
Figure 6.15: Scatter plot containing all diffusivity data from 125°C, showing a large degree in
scattering (a). EBSD ‘Y’ maps (b&c) are matched with their corresponding diffusivities via color
coding on the scatter plot.
Grain boundaries have a more open structure and thus there is a greater propensity for atomic
movement than in the lattice [56, 64]. Because diffusion is so favorable in grain boundaries, the
solute atoms often discharge into neighboring grains [50]. This implies grain boundaries will
always amplify the diffusivity, however many of the excluded data were ‘under-reporting’
diffusivity. This may be because the segregation factor of Bi in Sn LAGBs is very high, and
atoms are not likely to be discharged into adjacent grains. Thus, the low diffusivities observed
may be explained by the accumulation and net lateral (LAGBs close to perpendicular to the
original diffusion direction) transport of Bi atoms. Some excluded data over-reported diffusivity;
this was likely due to LAGBs running parallel to the original diffusion direction. This result
indicates that LAGBs can have a strong short-circuiting effect on the bulk diffusivity when the
anisotropy ratio is low, such as in the Sn-Bi system.
(a) (b)
(c)
141
Similar results were observed for the remaining four diffusion temperatures (Figure 6.16b-e),
although the fit was not nearly as good as with the 125°C data. The calculated 𝐷|| and 𝐷⊥ values
at each temperature determined via curve fitting of the experimental data (with all GBs removed)
to the anisotropy equation, are shown in Table 6.4. 95% confidence bounds are also included, as
well as updated R2 and RSS values.
Figure 6.16: Orientation data with substantial LAGBs removed, curve fitted to the anisotropy
equation, for 125°C (a), 115°C (b), 100°C (c), 85°C (d) and 25°C (e). For the 125°C plot, the
two data points at 90° seen in Figure 6.15a were from <1 1 0> and <1 2 0> orientations and are
considered as one orientation here.
(a) (b)
(c) (d)
(e)
142
Table 6.4: Calculated diffusivities of Bi in Sn parallel and perpendicular to the ‘c’ axis,
using experimental diffusion data
Temperature 125°C 115°C 100°C 85°C 25°C
𝐷||
(cm2/s)
Estimate 2.51 x 10-12 7.08 x 10-13 6.92 x 10-13 3.72 x 10-13 2.63 x 10-15
95%
confidence
bounds
1.35 x 10-12
4.57 x 10-12
6.03 x 10-14
8.32 x 10-12
2.69 x 10-13
2.00 x 10-12
1.74 x 10-13
7.94 x 10-13
2.45 x 10-16
2.88 x 10-14
𝐷⊥ (cm2/s)
Estimate 7.24 x 10-12 3.63 x 10-12 2.82 x 10-12 7.94 x 10-13 5.75 x 10-14
95%
confidence
bounds
5.89 x 10-12
9.12 x 10-12
1.95 x 10-12
5.01 x 10-12
2.00 x 10-12
4.07 x 10-12
6.92 x 10-13
8.91 x 10-13
3.16 x 10-14
1.07 x 10-13
R2 0.809 0.191 0.676 0.464 0.633
RSS 0.016 0.315 0.071 0.019 0.244
The following observations can be made about the results from Table 6.4:
• Compared to Table 6.3, the fit of the data after removing data corresponding to LAGBs
is substantially stronger, with higher R2 and lower RSS values for all temperatures.
• For all temperatures, the estimated 𝐷|| values demonstrated wider confidence bounds than
the 𝐷⊥ values due to the lack of data close to or at the <0 0 1> orientation.
• The confidence intervals in general are very wide for all temperatures; this is largely the
result of fitting to a small population of data points.
• The data from 115°C showed the poorest fit to the model, with the lowest R2 and highest
RSS values. The data from 125°C demonstrated the best fit.
• The data from 85°C showed a low RSS despite a low R2 value. This is likely the result of
the tightly bunched data with θ between 83° and 90° showing low deviation from the
model. This result indicates the importance of considering both residuals and R2 to
evaluate wellness of fit.
While it was necessary to exclude many diffusivities from the plots in Figure 6.14 and Figure
6.15 due to the presence of LAGBs, there simply were not enough data points representative of
bulk diffusion to obtain a good fit to the anisotropy equation and serve as an adequate predictor
of 𝐷|| and 𝐷⊥, in particular the former. Inclusion of more diffusivity data on each plot
representative of bulk diffusion would likely narrow the confidence intervals and improve the
wellness of fit parameters [101]. Modelling the data using the anisotropy model was also made
more challenging because of the inability to acquire the correct orientations of monocrystalline
143
Sn from the supplier. Having these specific orientations would have certainly allowed for more
confident predictions of 𝐷|| and 𝐷⊥.
6.3.4 Effects of High Diffusivity Pathways
Several examples of high diffusivity paths were observed in a few samples, including free
surfaces and high angle grain boundaries (Figure 6.17 and Figure 6.18). Diffusion fringes can
be visually observed via Bi precipitates and Sn grain recrystallization at lower Bi concentrations.
The microstructures seen at the sample corners (Figure 6.17a-b) appeared on all samples
studied; this justified the decision to only analyze locations distant from the corners – a
minimum distance of approximately 6√𝐷𝑡 = 50µ𝑚 (rounded up) was chosen, with D and t
taken as 1.10 x 10-11 cm2/s and 24 hours respectively (from the polycrystalline diffusion results
using the forward simulation). Fringes were also seen in locations in the bulk (Figure 6.17c and
Figure 6.18a); EBSD confirmed these are caused by grain boundaries (Figure 6.17d and Figure
6.18b). For the sample with the diffusion fringe at a grain boundary shown in Figure 6.17c&d,
the orientations of each grain were determined using the stereographic projection (Figure 6.17e)
to be <9 12 4> and <9 13 14>.
The angle between these orientations was calculated to be 17.6°. This result suggests that the
grain boundary was high angle (based on Figure 2.15) compared with the subgrain boundaries
typically seen in coarse-grained Sn samples [50] and is thus highly favorable for diffusion.
Further characterization was performed using EPMA. Six line scans were taken (Figure 6.19)
and diffusivities were calculated using the forward simulation technique at each location,
numbered from left to right (Table 6.5). Slightly higher error was observed compared with
conventional simulations as no averaging of data was performed prior to simulation. Sample
diffusion profiles (from Profiles 1 and 4) are shown in Figure 6.20.
144
Figure 6.17: High diffusivity pathways in Sn-Bi diffusion samples. SEM images of corner (a&b)
indicating edge effects, and of region in the bulk (c). EBSD ‘Y’ mapping (d) and IPF/WinWulff
analysis (e) suggests the fringe in (c) may be caused by a grain boundary.
(a) (b)
(c) (d)
<9 12 4>
<9 13 14>
(e)
145
Figure 6.18: Example of the effects of grain boundaries on diffusion of Bi in Sn. Composite SEM image (a) of a location featuring
several ‘fringes’ of Bi precipitation and Sn recrystallization; EBSD ‘Y’ map (b) of the same location indicating the fringes were likely
caused by grain boundaries.
(b)
(a)
146
The diffusivity increased by an order of magnitude from Profile 1 to 4, indicating a strong short-
circuiting effect. Interestingly, Profile 5 (which was taken almost directly on top of the estimated
location of the HAGB) showed a very low diffusivity, comparable to the near-bulk diffusivities
calculated using Profiles 1 and 6. A possible reason for this is significant diffusion into the
adjacent lattices took place, depleting the grain boundary of solute. Another reason may be that
the grain boundary was located slightly to the left, between Profiles 4 and 5, and Profile 5 was
situated within the <9 13 14> grain. The diffusivity at Profile 5 was then likely higher than
Profile 6 because of its proximity to the grain boundary.
Figure 6.19: SEM image (a) and EBSD ‘Y’ map (b) of location with grain boundary diffusion,
indicating the locations of EPMA line scans
Table 6.5: Summary of Diffusivities in High Angle Grain Boundary Region
Profile
#
Line
Length
(µm)
Point
Spacing
(µm)
Recrystalliz-
ation Depth
(µm)
D (x 10-12
cm2/s) Error Notes
1 50 2.5 20.5 3.1 9.02 x 10-4 Near bulk <9 12 4>
2
69 3
34.7 16.4 5.59 x 10-4 Within recrystallized
region on <9 12 4>
side of ‘interface’
between grains
3 51.6 28.8 1.41 x 10-3
4 63.6 31.1 9.57 x 10-4
5 > 69 3.5 7.38 x 10-4
Directly on
‘interface’ between
grains
6 50 2.5 15.8 1.2 6.33 x 10-4 Near bulk <9 13 14>
(a) (b)
147
Figure 6.20: The effects of high diffusivity pathways on the concentration profile. Profile 1 in a
region with predominantly bulk diffusion (a) looks considerably different from Profile 4, which
was influenced by the HAGB between the two grains (b).
6.3.5 Effects of Temperature
Despite the generally poor fit of the diffusivity data to the anisotropy equation and subsequent
low confidence in the estimated 𝐷|| and 𝐷⊥ values, the Arrhenius parameters 𝐷𝑜 and 𝑄𝐴 for 𝐷||
and 𝐷⊥ values were calculated [55, 56, 58]. Table 6.6 contains a summary of these values as
well as the confidence bounds and goodness of fit parameters; Figure 6.21 is an Arrhenius plot
of the data. The following observations can be made about the data:
• 𝐷𝑜 is higher for 𝐷|| than 𝐷⊥.
• 𝑄𝐴 is higher for 𝐷|| than 𝐷⊥.
• 𝐷|| is not fitted as well to the Arrhenius relationship as 𝐷⊥, with a higher RSS and lower
R2, and wider confidence bounds.
Table 6.6: Arrhenius data for the diffusivity of Bi in Sn parallel and perpendicular to the
‘c’ axis
Parallel to ‘c’ axis Perpendicular to ‘c’ axis
Estimate 95% confidence
bounds
Estimate 95% confidence
bounds
𝑸𝑨 (kJ/mol) 64.95 46.48, 82.93 46.62 34.32, 58.91
𝑫𝒐 (cm2/s) 7.27 x 10-4 1.71 x 10-6, 3.09 x 10-1 7.66 x 10-6 1.22 x 10-7, 4.83 x 10-4
RSS 0.6284 0.2943
R2 0.9778 0.9798
(a) (b)
148
Figure 6.21: Arrhenius plot for the estimated diffusivities of Bi in Sn for the orientations parallel
and perpendicular to the ‘c’ axis.
Similar to the results from polycrystalline Sn, the calculated activation energies for diffusion in
coarse-grained Sn are also lower than expected for a substitutional solute (within 25% of the self-
diffusivity) [56]. As these activation energies were calculated based on samples with no visible
grain boundaries, this result suggests a possible fast diffusion mechanism. However, as stated in
Section 2.3.6.2 and Section 5.4, fast diffusion of Bi in Sn is not expected according to the
Miedema-Niessen model [78]. This is further supported by the result that 𝐷⊥ > 𝐷||, which is also
not expected for fast diffusion in Sn. Further discussion is provided in the following two
sections.
It is stressed that since the anisotropy equation (from which 𝐷|| and 𝐷⊥ were derived) generally
did not fit the experimental data strongly, the Arrhenius data displayed here is not necessarily a
good predictor of the true quantitative effects of temperature on the diffusivity of Bi in Sn.
149
6.3.6 Effects of Solute Atomic Radius and Molar Volume
Overall (given the uncertainty in the data in Table 6.4), these orientation results are comparable
to those determined for the Sn-Sb and Sn-In systems in which the diffusivity in the <0 1 0>
direction is within the same order of magnitude as the diffusivity in the <0 0 1> direction [72,
76]. The 125°C data set yielded the soundest results, so these are used for comparisons to the
other solutes discussed in Section 2.4.3.2.
As shown in Figure 2.23, solutes with a higher atomic radius r generally demonstrate lower
diffusivity in Sn at 125°C. In vacancy-mediated substitutional diffusion which is expected for the
diffusion of Bi in Sn (despite the analogous activation energies shown in Section 6.3.5), the
inclusion of solute atoms in the solvent matrix imparts strains to the lattice [58]. These strains
would increase as the solute radius increases, which lowers the diffusivity. In Figure 6.22, the
𝐷|| and 𝐷⊥ values estimated in Section 6.3.3 are added to the plot of r versus diffusivity, and it is
noted that the values determined in this study fit quite well to the trend. In Figure 6.22, it is also
observed that the anisotropy ratio also decreases as the atomic radius increases. The similar
anisotropy ratios of Bi, Sb, and In are notable as all three species are situated closest to Sn in the
periodic table (by group). This may be related to the valence of these atomic species, or the
filling of atomic orbitals, as suggested in the case of Sb relative to Zn by Huang et al [72].
Some deviations exist in the data displayed in Figure 6.22, particularly concerning Sb, but
possibly affecting other species. These might be explained by the bonding characteristics of the
solute paired with the solvent possibly affecting the diffusion energy barriers [105]. Based on the
Sn-Sb phase diagram, for example, Sb has very low solubility in Sn at 125°C; it may be more
favorable to form SbSn (β) IMC phases [47]. This would raise the diffusion energy barrier and
might explain the abnormally low diffusivity [72]. In addition, the diffusion anisotropy can be
affected by bond angle distortion [105]. These topics can be further explored using techniques
such as density functional theory and are beyond the scope of this thesis.
150
Figure 6.22: Plot of solute atomic radius against impurity diffusivity in Sn, for both the ‘a’ / ‘b’
(𝑫⊥) and ‘c’ (𝑫||) axes, identical to Figure 2.23, with the addition of Bi from this study. The
diffusivities for Bi in Sn show good agreement with the literature data, which suggests that as r
increases, the diffusivity decreases (other than Sb).
The Miedema-Niessen model suggests that as the ratio between the corrected molar volume of an
impurity to that of the solvent increases, the ratio between the impurity diffusivity to the self-
diffusivity at the melting temperature decreases [78]. The data for several solutes in Sn, using the
corrected molar volumes from Table 2.4 and the calculated impurity diffusivities at the melting
point of Sn (505K) are plotted in Figure 6.23, showing a moderately strong linear relationship
between these two ratios, both parallel and perpendicular to the ‘c’ axis. The data for Bi is added
(with diffusivity calculated using the activation energies from this study) which falls outside this
linear relationship, indicating that the diffusivities of Bi in Sn calculated using the forward
simulation are greater than what is expected for Bi based on its molar volume relative to that of
the Sn matrix. This result is further evidence suggesting the method of assuming constant
diffusivity independent of Bi concentration may be responsible for both higher diffusivity and
151
lower activation energy than expected. Simulations using a concentration-dependent diffusivity
are required to test this hypothesis and preliminary results are provided in the following section.
Figure 6.23: Plot of impurity diffusivity data perpendicular to the ‘c’ axis (normalized to Sn
self-diffusivity) with respect to corrected molar volume (normalized to the Sn molar volume)
calculated using the Miedema-Niessen model.
6.3.7 Effects of Bi Concentration
The forward simulation code was modified to account for a concentration dependence on the
diffusivity of Bi in Sn. For the first preliminary calculations, a linear function of D(C) was
assumed; future work will further optimize this to higher order functions. Rather than one
diffusivity being assumed at the start of each simulation, a range of diffusivities is assumed for
two diffusion coefficients:
• 𝐷𝐶,1 – diffusion coefficient at 0 at% Bi, and
• 𝐷𝐶,2 – diffusion coefficient at the solubility limit (for example, 10.7 at% at 125°C)
152
For the initial simulations, these ranges are very large, spanning several orders of magnitude. A
partition number is also required for each of 𝐷𝐶,1 and 𝐷𝐶,2; this number dictates how many
values within each of these ranges will be simulated for each run. For example, if each of 𝐷𝐶,1
and 𝐷𝐶,2 are given partition numbers of 8, a total of 64 simulations will be conducted per run.
After each run of simulations, the Python software provides a best guess to the values of 𝐷𝐶,1 and
𝐷𝐶,2, as well as a recommended (narrower) set of new ranges of 𝐷𝐶,1 and 𝐷𝐶,2 to test for the
following run. This process is repeated until the range of diffusivities suggested by the software
is effectively (within three significant figures) a single value. The 𝐷𝐶,1 values are of interest as
they represent the impurity diffusivity and are most comparable to data in the literature. For the
preliminary simulation of the concentration dependent diffusivity of Bi in Sn, all concentration
profiles obtained from samples devoid of grain boundaries were averaged to yield one
concentration profile for each diffusion temperature. This was not done for the 25°C data as a
wide range of diffusion times were studied and thus the profiles are not directly comparable to
one another. Similar to the constant diffusivity, only concentrations lower than the solubility
limit at room temperature are compared to evaluate error. An updated Arrhenius plot using the
𝐷𝐶,1 values is shown in Figure 6.24a, along with an example of a D(C) plot from the 125°C
simulations (Figure 6.24b). It is observed that there is a strong linear relationship between 1 𝑇⁄
and ln (𝐷𝐶,1), and that the diffusivity increases with Bi concentration. This agrees with what is
expected for Bi in Sn. As additions of Bi lower the melting point, the diffusivity of Bi in Sn
increases with concentration as the diffusion temperature approaches the melting point.
The values for 𝐷𝐶,1 and 𝐷𝐶,2 (with error), along with the Arrhenius parameters, are provided in
Table 6.7 and Table 6.8 respectively; and are compared to those assuming a constant diffusivity.
It is observed that the 𝐷𝐶,1 values are slightly lower than the constant D values and the error is
lower, indicating a better fit of the simulated concentration profile to the experimental profile.
The corresponding activation energy is slightly higher when a concentration dependence is
assumed. It is noted that the updated activation energy value remains outside the expected range
of that for impurity diffusivity compared with self-diffusivity (+/- 25%). This may be because in
this simulation, a linear relationship of D(C) is too simplistic of an assumption and a higher order
function may be more appropriate.
153
Figure 6.24: Forward Simulation results assuming a concentration-dependent diffusion
coefficient. Arrhenius plot of 𝑫𝑪,𝟏 values (for Bi at% → 0) (a); D(C) plot (b). The blue data
points are from the Sauer-Friese plot.
(a)
𝑫𝑪,𝟏
𝑫𝑪,𝟐 (b)
154
Table 6.7: Calculated average 𝑫(𝑪)𝒍 values for Bi in Sn, compared to 𝑫𝒍 values
Temperature 125°C 115°C 100°C 85°C
D(C)
𝑫𝑪,𝟏 (cm2/s) 4.54 x 10-12 3.02 x 10-12 1.93 x 10-12 5.34 x 10-13
𝑫𝑪,𝟐 (cm2/s) 1.43 x 10-10 8.01 x 10-12 4.59 x 10-12 1.35 x 10-12
Error 3.04 x 10-4 1.30 x 10-4 2.68 x 10-4 1.55 x 10-4
D 𝑫𝒍 (cm2/s) 8.23 x 10-12 3.32 x 10-12 2.54 x 10-12 7.89 x 10-13
Error 4.95 x 10-4 1.67 x 10-4 3.23 x 10-4 1.77 x 10-4
Table 6.8: Arrhenius data for 𝑫𝑪,𝟏 and 𝑫𝒍 for Bi in Sn
𝑫𝑪,𝟏 𝑫𝒍
Estimate 95% confidence
bounds
Estimate 95% confidence
bounds
𝑸𝑨 (kJ/mol) 61.25 19.79, 102.68 49.26 37.82, 60.71
𝑫𝒐 (cm2/s) 5.47 x 10-4 1.03 x 10-9, 2.90 x 102 1.77 x 10-5 3.74 x 10-7, 8.36 x 10-4
RSS 0.1218 0.2943
R2 0.9528 0.9798
6.4 Correlations between Diffusivity in Coarse-Grained and Polycrystalline Sn
For an adequate comparison to the results from Chapter 5, the 𝐷𝐶,1 results from Table 6.7 are
compared with the effective diffusivities 𝐷𝑒𝑓𝑓 from the polycrystalline Sn samples (determined
using the forward simulation using the Lee diagram), in Table 6.9. For all four temperatures,
polycrystalline Sn yielded higher impurity diffusivities than coarse-grained Sn. Furthermore, the
difference between the two diffusivities decreases as the annealing temperature is increased. This
can be explained by the increasing grain size of the polycrystalline Sn samples with temperature,
combined with the short-circuiting effect of grain boundaries in these samples. As the
temperature is increased, the grain size increases, reducing the number of grain boundaries, and
subsequently the contribution of grain boundaries to the diffusivity is reduced. As a result, the
diffusivity in polycrystalline Sn approaches that of lattice diffusion.
Armed with the lattice impurity diffusivities 𝐷𝐶,1 from the coarse-grained Sn samples and the
effective impurity diffusivities 𝐷𝑒𝑓𝑓 from the polycrystalline Sn samples, it is possible to
consider Harrison’s three kinetics regimes for analyzing polycrystalline diffusivity and
characterizing grain boundary diffusion. Since no specific ‘Type C’ experiments were conducted
155
(possible future work) on Sn-Bi diffusion couples, only ‘Type A’ and ‘Type B’ regimes are
possible to analyze within this thesis. The various parameters used for the Harrison analysis are
given in Table 6.10. In this section, 𝐷𝐶,1 is analogous to 𝐷𝑙 (should the reader wish to refer back
to the Harrison ‘Type B’ empirical derivations in Section 2.3.3.2).
Table 6.9: Comparison of lattice and effective impurity diffusivities of Bi in Sn
Diffusion Temp. 𝑫𝒆𝒇𝒇
𝑫𝑪,𝟏
125°C 2.42
115°C 2.69
100°C 3.83
85°C 7.81
The grain boundary diffusivity 𝐷𝑔𝑏 can be estimated directly using Hart’s equation provided the
grain boundary diffusion occurs in the ‘Type A’ regime [56, 64]. To check if this approach is
valid, the diffusion distance √𝐷𝐶,1𝑡 is calculated and compared to the average grain size of the
polycrystalline Sn samples, at all temperatures.
Table 6.10: Harrison Kinetics Parameters
Parameter 125°C 115°C 100°C 85°C
Average grain size d
(µm) 19.40 12.66 10.54 10.26
√𝑫𝑪,𝟏𝒕 (µm) 6.26 5.71 5.77 5.68
𝟒√𝑫𝑪,𝟏𝒕 (µm) 25.04 22.84 23.08 22.73
P (cm3/s) 1.79 x 10-13 1.26 x 10-13 3.38 x 10-14 8.39 x 10-15
𝜷𝑯 31.48 36.53 15.16 13.82
𝒔𝒎𝒂𝒙 (estimated,
assuming K = 20) 187.27 222.97 298.95 406.50
𝒂𝑯,𝒎𝒂𝒙 (using
estimated 𝒔𝒎𝒂𝒙) 7.48 x 10-3 9.76 x 10-3 1.29 x 10-2 1.79 x 10-2
The data in Table 6.10 suggests that the ‘Type A’ regime is not valid for the polycrystalline Sn
samples in this study as √𝐷𝐶,1𝑡 < 𝑑 at all temperatures; and ‘Type B’ kinetics must next be
considered. Extending the diffusion time, increasing the annealing temperature (likely not an
option for this system as the Sn-Bi eutectic temperature is 138°C), or decreasing Sn grain size
(likely not possible due to the highly active nature of polycrystalline Sn, as observed in Chapter
5) would be sufficient for this diffusion experiment to be considered as ‘Type A’ [56, 64].
156
For ‘Type B’ analysis, 4√𝐷𝐶,1𝑡 was first calculated at each temperature (Table 6.10) as a
conservative estimate of the beginning of the grain boundary ‘tail’ in the modified
polycrystalline diffusion profile of 𝑥6
5⁄ vs ln(𝐶). The slope of the linear fit to this region for
4√𝐷𝐶,1𝑡 < 𝑥 < 72.5µ𝑚 was determined and a value of the triple product P (defined in (Eqn
2.22) was calculated for each temperature using (Eqn 2.21 (Table 6.10). After checking the
corresponding values of 𝛽𝐻 using (Eqn 2.23, it was determined that 𝛽𝐻 > 10 (Table 6.10) and
suggests that ‘Type B’ kinetics are valid for all temperatures.
(Eqn 2.24 was then used to check if 𝛼𝐻 < 0.1 for all temperatures. s was estimated using
assuming K = 20; this yields the largest theoretically possible segregation factor 𝑠𝑚𝑎𝑥, which will
in turn result in the largest theoretically possible 𝛼𝐻, 𝛼𝐻,𝑚𝑎𝑥 (Table 6.10). Based on these
assumptions, for all temperatures, 𝑎𝐻,𝑚𝑎𝑥 ≪ 0.1, therefore it can be confidently stated that for all
values of 1 < 𝐾 < 20, the 𝛼𝐻 < 0.1 requirement is satisfied.
Based on these calculations, ‘Type B’ assumptions and methods are appropriate for analyzing the
kinetics of diffusion of Bi in polycrystalline Sn. The two diffusivities (𝐷𝐶,1 and 𝐷𝑒𝑓𝑓), as well as
P, are compared on an Arrhenius plot (Figure 6.25) and the Arrhenius parameters are tabulated
(Table 6.11).
The effective impurity diffusivity has a lower activation energy (26.59 kJ/mol) than the average
coarse-grained impurity diffusivity; this is because the polycrystalline Sn samples featured many
HAGBs, so the required energy for a Bi atom to jump to an adjacent lattice site is lower as grain
boundaries have a more open structure and tend to attract diffusing atoms [56]. The pre-
exponential for the effective diffusivity (3.39 x 10-8 cm2/s) was also lower than that for coarse-
grained diffusivity by several orders of magnitude; this is contradictory to what is typically
observed in other systems, in which the two are approximately equivalent [55, 56]. This may be
caused by the changing grain size; it was observed that polycrystalline Sn is metastable,
particularly at high temperatures, with average grain size increasing as the annealing temperature
is increased.
157
Figure 6.25: Arrhenius plot comparing lattice impurity diffusivity (𝑫𝑪,𝟏), effective or
polycrystalline impurity diffusivity (𝑫𝒆𝒇𝒇) and triple product (P) for Bi in Sn.
Table 6.11: Arrhenius data for P, compared with 𝑫𝑪,𝟏 and 𝑫𝒆𝒇𝒇 for Bi in Sn
𝑫𝑪,𝟏 𝑫𝒆𝒇𝒇 P
Estimate
95%
confidence
bounds
Estimate
95%
confidence
bounds
Estimate
95%
confidence
bounds
𝑸𝑨 (kJ/mol)
61.25 19.79,
102.68 26.59 5.32, 47.86 93.53
127.20,
59.88
𝑫𝒐
(cm2/s) 5.47 x 10-4
1.03 x 10-9,
2.90 x 102 3.39 x 10-8
3.93 x 10-11,
2.92 x 10-5 4.04 x 10-1
9.07 x 10-6,
1.80 x 104
R2 0.953 0.935 0.986
RSS 0.122 0.032 0.080
From Table 6.11, 𝑄𝐴,𝑃 > 𝑄𝐴,𝐶,1 > 𝑄𝐴,𝑒𝑓𝑓. If δ is assumed to be constant, then (Eqn 2.27
becomes the following:
𝑄𝐴,𝑃 = 𝑄𝐴,𝑔𝑏 − 𝑄𝐴,𝑠
158
If this equality holds, 𝑄𝐴,𝑔𝑏 > 𝑄𝐴,𝑃. Since 𝑄𝐴,𝑃 > 𝑄𝐴,𝐶,1, this condition is not thermodynamically
possible as this requires 𝑄𝐴,𝑔𝑏 > 𝑄𝐴,𝐶,1; 𝑄𝐴,𝑔𝑏 is typically between 40 and 60% that of 𝑄𝐴,𝑙 [64].
Therefore, δ cannot be assumed to be constant, and it must be considered to follow an Arrhenius
relationship and possess an activation energy 𝑄𝐴,𝛿.
To accurately determine 𝐷𝑔𝑏 and a precise value for s, supplemental experiments considering
‘Type C’ kinetics are necessary.
6.5 Concluding Remarks
To study lattice diffusion of Bi in Sn using coarse-grained Sn, a novel sample preparation
method was utilized to produce a wide range of Sn grain orientations. The use of slow oven
cooling and a glass substrate to initiate solidification promoted the nucleation of very large Sn
grains, often encompassing the entire sample volume. To metallographically prepare and sputter
many samples at once and consistently, a temporary epoxy mount was used to insert samples,
which allowed for easy removal prior to annealing. This process was not perfectly ideal as many
samples contained LAGBs, which serve as high-diffusivity pathways and do not truly reflect
lattice diffusion.
Diffusion microstructures indicated that the diffusivity of Bi in Sn is likely higher than that of Sn
in Bi: the initial sputtered layer appeared to enlarge (for 115°C, by roughly 60%), and Kirkendall
voids were present. The depth of the α+β (Sn matrix + Bi precipitate) layer increased with
diffusion temperature; this is indicative of higher solid solubility of Bi in Sn. Contrary to what
was previously believed, recrystallization appears to occur without the influence of Bi
precipitation. This suggests that some degree of Bi in solid solution (lattice strain) is sufficient to
nucleate new grains. Despite moderately long storage times in liquid nitrogen (up to one month),
the allotropic transformation from β-Sn to α-Sn was not observed in any samples that were stored
cryogenically after annealing. This suggests that the transformation either does not occur or is
itself retarded at low temperatures.
Using the film remaining calculation in the forward simulation, it was confirmed that the Sn-Bi
phase diagram proposed by Lee is the best fit to the experimental data; this agrees with the result
in Chapter 5, as well as the result determined by Belyakov using DSC. This indicates the
simulation is a powerful, versatile tool for not only determining diffusivity but verifying phase
159
equilibria. The fact that diffusion couples are not homogeneous is a likely reason why slight
deviation (-5% average) between the experimental data and the simulation exists, even with the
Lee diagram. This deviation may also be explained by the assumption of a constant diffusion
coefficient.
Analysis of the EBSD results alongside the diffusivity data (which demonstrates very low
anisotropy), it is apparent that LAGBs have a strong influence on the perceived diffusivity of Bi
in Sn, resulting in a high degree of scatter in the anisotropy plot and weaker fit to the anisotropy
equation. For example, for 125°C, R2 decreases from 0.81 to 0.28 and RSS increases from 0.02
to 0.48 when samples featuring LAGBs are included in the fitting. Due to supplier issues with
the monocrystals oriented parallel to the ‘c’ and ‘a’ axes, it was challenging to obtain a
conclusive fit to the anisotropy equation due to the lack of diffusivity data parallel to the ‘c’ axis.
The data at 125°C demonstrated the best fit to the anisotropy equation and was compared to the
data from the literature for diffusion of other solutes in Sn. The anisotropy ratio for diffusivity of
Bi is close to unity (𝐷|| = 2.51 x 10-12 cm2/s and 𝐷⊥ = 7.24 x 10-12 cm2/s); this is more
representative of ‘slow’ diffusing solutes in Sn which are believed to undergo vacancy-mediated
diffusion rather than ‘fast’ diffusion. The anisotropy result appeared to directly contradict the
Arrhenius results. 𝐷𝑜 was two orders of magnitude larger parallel to the ‘c’ axis than
perpendicular. Directions with higher packing generally demonstrate higher diffusivity as the
atomic jump distance is shortened. 𝑄𝐴 was larger parallel (64.95 kJ/mol) to the ‘c’ axis than
perpendicular (46.62 kJ/mol); this is likely the result of the large Bi atom size combined with the
shorter ‘c’ axis, generating greater lattice strain fields. It requires more energy for Bi atoms to
overcome these strains and perform successful atomic jumps.
The forward simulation was modified to consider the influence of Bi concentration on
diffusivity. Such results would be more comparable to the data in the literature, derived from
tracer experiments. As observed in Chapters 5 and 6, constant diffusion coefficients appeared
high and corresponding activation energies low compared to those from self-diffusivity
experiments, and to those from comparable solutes such as Sb or In. A linear function of D(C)
was assumed, with 𝐷𝐶,1 values representing impurity diffusivity. A strong fit to an Arrhenius
relationship was observed, and D was determined to increase with Bi concentration at all
temperatures. 𝐷𝑜,𝐶,1 was nearly identical to 𝐷𝑜,𝑙, and the activation energy increased from 49.26
160
to 61.25 kJ/mol when Bi concentration was considered. While this result remains low compared
to what is expected for substitutional solutes such as Bi, this is a useful preliminary result
suggesting concentration dependence is important to consider in diffusivity calculation and is
relevant in the Sn-Bi system. Further optimizations to the simulation model are needed.
The 𝐷𝐶,1 from the coarse-grained Sn and 𝐷𝑒𝑓𝑓 from the polycrystalline Sn were compared to
examine the effects of HAGBs on the diffusivity of Bi in Sn. The effective diffusivity at 85°C
was 7.81 times greater than the lattice diffusivity at 85°C, whereas at 125°C, 𝐷𝑒𝑓𝑓 was only 2.42
times greater than 𝐷𝐶,1. This is the result of increased grain growth of the polycrystalline Sn at
higher temperatures (average grain size at 125°C was approximately twice as large as that at
85°C). It is evident that as the temperature increases, the polycrystalline Sn trends towards
resembling coarse-grained Sn and thus the diffusivities become comparable.
Harrison diffusion kinetics were used to attempt to determine 𝐷𝑔𝑏. ‘Type A’ kinetics were not
valid as the grain size of polycrystalline Sn at all temperatures was greater than the diffusion
distance √𝐷𝐶,1𝑡 (3.1 times greater at 125°C). ‘Type B’ kinetics were then considered and the
triple product P was calculated for each temperature. As no ‘Type C’ studies were conducted in
this thesis, the exact value of 𝐷𝑔𝑏 could not be determined, however the Arrhenius parameters
were calculated for P. The activation energy for P was determined to be 93.53 kJ/mol; this is
greater than that for average lattice impurity diffusion (61.25 kJ/mol). This result suggests that if
the grain boundary width cannot be treated as a constant, as 𝑄𝐴,𝑔𝑏 ≯ 𝑄𝐴,𝐶,1.
Overall, despite the various challenges, these results for the diffusivity of Bi in coarse-grained Sn
are very valuable and will help to further understanding of the long-term properties of Bi-
containing solder alloys, specifically relating to the evolution of microstructure during aging.
Some possible future work opportunities to supplement the findings from this research include:
• Continuing to optimize the forward simulation model with respect to concentration
dependent diffusivity. In this thesis, some preliminary calculations were conducted using
a linear function of D(C); higher order functions may yield more precise values of 𝐷𝐶,1
and 𝐷𝐶,2.
161
• Study of the solid-state diffusion of Sn in Bi by performing sputter deposition of Sn onto
a Bi substrate. This would verify the hypothesis that the diffusivity of Bi in Sn is greater
than that of Sn in Bi.
• Acquisition of correctly oriented Sn monocrystals to conclusively determine the
anisotropy ratio for diffusion.
• Delays with lab equipment prevented the characterization of diffusivity of Bi in Sn at
very low temperatures of 50°C and 70°C; these temperatures are of interest as they are
below 0.7𝑇𝑚 and diffusivity may behave differently with respect to temperature.
• Using TEM, strain field analysis of various substitutional solutes in Sn and in situ
diffusion studies may be used to verify hypotheses concerning the relationship between
the anisotropy of the Sn unit cell and solute atom radius.
• ‘Type C’ Harrison kinetics study of diffusion of Bi in Sn grain boundaries. This would
conclusively determine 𝐷𝑔𝑏 and the segregation factor. Transmission Electron
Microscopy analysis of grain boundary width (including effects of temperature) would
also be important to consider.
162
Chapter 7 Conclusions and Recommendations
This thesis characterized the long-term mass transport mechanisms of Bi in Sn as it pertains to
the improved properties seen after aging of Bi-containing solder alloys. The primary focus was to
determine the previously unknown diffusivity parameters of Bi in Sn in the solid state; an
extensive aging study of binary Sn-Bi and ternary Sn-Cu-Bi alloys was first considered to
provide impetus for the diffusion analysis and understand how Bi-bearing alloys evolve
microstructurally and mechanically over time. Two crystallinities (coarse-
grained/monocrystalline and polycrystalline) of Sn were considered in this study based on results
from earlier ReMAP work, which determined the grain structure of Bi-containing alloys
undergoes extensive recrystallization due to dissolution, diffusion, and precipitation of Bi in the
β-Sn matrix over time. The following conclusions can be drawn from this study:
1. The effect of Cu on the strength of a Bi-containing alloy is greater when Bi content is low
(e.g. 1 wt%) compared when Bi content is high (e.g. 5 wt% Bi). In this study, adding 0.7
wt% of Cu to the Sn-1Bi alloy resulted in an average increase of 17 HR15X, whereas an
increase (not statistically significant) of only 2 HR15X was observed when the same
amount of Cu was added to the Sn-5Bi alloy. This is in agreement with the literature,
which suggests the optimal secondary element content in Bi-bearing alloys is between 5
wt% and 7 wt%.
2. The hardness of Bi-containing alloys was preserved after up to 14 days of aging at either
70°C, 100°C, or 125°C, or up to 365 days at room temperature, except for Sn-5Bi, in
which hardness decreased by around 5% between 168 days and 252 days of aging at
room temperature. This was likely the result of Ostwald ripening of Bi precipitates. It is
believed that further aging would cause all alloys to overage and experience a similar
reduction in hardness.
3. After annealing, interdiffusion between the sputtered Bi layer and Sn substrate suggests
that the diffusivity of Bi in Sn is greater than that of Sn in Bi. The diffusion interface
migrated towards the Sn-rich end of the couple, the deposited layer increased in thickness
by approximately 60%, and Kirkendall voids appear to be present at the interface.
4. Recrystallization was not only present near Bi precipitates (suggesting PSN took place),
but also in regions with Bi in solid solution (including room temperature diffusion
163
couples). This suggests that recrystallization may precede precipitation and the lattice
strain from Bi atoms in substitutional solid solution may be sufficient to nucleate
recrystallized Sn grains.
5. The forward simulation technique was not only a powerful tool for determining
diffusivities from experimental data, but also was versatile in determining that the binary
Sn-Bi phase diagram proposed by Lee was more accurate than the diagrams proposed by
Vizdal and Braga. For example, using the diffusion data from coarse-grained Sn, the
deviation between the experimental data and the simulation varied by only 8% on average
for the Lee diagram, compared with 23% and 42% for the Vizdal and Braga diagrams,
respectively.
6. The diffusivities determined using the forward simulation were a better fit to the
experimental data than the inverse Hall method (polycrystalline Sn) or the slab source
model (coarse-grained Sn).
7. The lattice diffusivity of Bi in Sn demonstrates very low anisotropy, with all calculated
diffusivities within one order of magnitude at all temperatures. As a result, low angle
grain boundaries (LAGBs) can strongly convolute the effect of orientation on diffusivity
– R2 values increase from 0.193 to 0.555 and RSS values decrease from 0.611 to 0.133,
on average, when diffusivities corresponding to samples with high LAGB content are
excluded. However, the diffusivities exceed what is expected for Bi in Sn based on the
Miedema-Niessen model, which considers the influence of molar volume of a solute (Bi)
with respect to a given solvent (Sn).
8. The pre-exponential for lattice diffusivity is greater parallel to the ‘c’ axis (7.27 x 10-4
cm2/s) than perpendicular to the ‘c’ axis (7.66 x 10-6 cm2/s), as is the activation energy
(64.95 kJ/mol || to the ‘c’ axis; 46.62 kJ/mol ⊥ to the ‘c’ axis). For the latter, this is
believed to be the result of the large atom size of Bi in conjunction with the shorter ‘c’
axis in the β-Sn lattice – there is likely greater lattice strain parallel to the ‘c’ axis and
thus it requires more energy for a Bi atom to jump between adjacent lattice sites. These
activation energies appear low for substitutional solutes and the most likely explanation
for this result is the assumption of a constant diffusivity with respect to Bi concentration.
9. A linear function of diffusivity with respect to concentration was assumed and the
average concentration profiles from coarse-grained Sn at each temperature were
simulated to determine the impurity diffusivity 𝐷𝐶,1. In comparison with the constant
164
diffusivity results, 𝐷𝑜 was unchanged and 𝑄𝐴 increased from 49.26 kJ/mol to 61.25
kJ/mol. The recalculated activation energy remains outside the expected range for
substitutional solute in Sn (self-diffusivity of 108.1 kJ/mol), however this result suggests
a concentration dependence is more realistic for modelling the diffusivity of Bi in Sn.
10. Comparing effective (polycrystalline) diffusivities 𝐷𝑒𝑓𝑓 and lattice diffusivities 𝐷𝐶,1 for
impurity diffusion at the same temperature, it was found that the ratio 𝐷𝑒𝑓𝑓
𝐷𝐶,1⁄
decreases from 7.81 to 2.42 as the temperature increases from 85°C to 125°C. This is
accompanied by an increase in the grain size and indicates the highly metastable nature of
polycrystalline Sn at high (𝑇 > 0.7𝑇𝑚) temperatures.
11. Harrison kinetics were considered to correlate the diffusivity results from polycrystalline
Sn with those from coarse-grained Sn. The triple product P for polycrystalline diffusivity
of Bi in Sn was calculated using ‘Type B’ analysis and the activation energy and pre-
exponential were determined. It was found that 𝑄𝐴,𝑃 = 93.53 kJ/mol, which is greater
than the activation energy for lattice diffusivity (𝑄𝐴,𝐶,1 = 61.25 kJ/mol) and effective
(polycrystalline) diffusivity (𝑄𝑒𝑓𝑓 = 26.59 kJ/mol). This suggests that the grain boundary
width δ cannot be treated as a constant (typically, δ = 0.5nm) and it must be considered to
follow an Arrhenius relationship, to solve for grain boundary diffusivity 𝐷𝑔𝑏 and
segregation factor s.
Several opportunities for future work have been identified and are listed as follows:
1. As almost all alloys/aging conditions did not demonstrate overaging, it is recommended
to perform aging experiments for extended durations, to determine when overaging
occurs. It was noted that Sn-5Bi begins to degrade between 168 and 252 days of aging at
room temperature; decreasing the time interval between analyses may be of interest to
determine a more accurate time necessary for overaging.
2. The sample preparation for coarse-grained Sn produced inconsistent LAGB content
which made analysis with respect to Sn grain orientation challenging. Fine-tuning the
process to reduce or eliminate LAGBs would ensure more ‘useful’ data sets for future
diffusion experiments.
3. For better fit to an Arrhenius relationship, it is preferable to consider as many
temperatures as possible. This is especially true for this study of diffusion in
165
polycrystalline Sn. For consistency and better comparison with the coarse-grained
samples, sputter deposition of Bi on polycrystalline Sn is also recommended.
4. In this study, a constant diffusivity with respect to concentration was assumed in the
forward simulation for most calculations, however better fit to the experimental data (and
more realistic activation energies (with respect to Sn self-diffusivity, for a substitutional
solute) were determined when a linear function of D(C) was assumed. It is of interest to
consider rerunning the simulation assuming a higher order function of D(C).
5. To determine a more conclusive fit to the anisotropy equation, it would be beneficial to
acquire correctly oriented monocrystals of Sn.
6. As the melting point of Sn and Sn-Bi alloys is very low compared to the diffusion
temperatures in this study, it is of interest to consider several additional ‘low’
temperatures such as 50°C and 70°C to examine whether there is any deviation from an
Arrhenius relationship at ‘high’ temperatures.
7. Transmission Electron Microscopy analysis (including in situ) may be beneficial to
examine the microstructure and strain fields arising from the inclusion of Bi atoms in the
Sn lattice, and directly observe the diffusion mechanisms in these alloys.
8. In this study, the triple product for diffusivity of Bi in polycrystalline Sn was estimated at
several temperatures, however the grain boundary diffusivity and segregation factor could
not be determined. To do so, conducting a ‘Type C’ Harrison kinetics study would be
necessary. It would be of interest to examine a wide range of boundaries, including
LAGBs, HAGBs, twins and CSL boundaries, as each possess different atomic
arrangements and energies and would thus demonstrate different diffusivities and
segregation factors.
In conclusion, not only does this project set up numerous potential future research initiatives
focused on the metallurgy and diffusion properties in Bi-containing solder alloys, it also
supplements the current work in industry (for example, reliability and environmental testing)
aimed at the commercialization of these alloys. One major obstacle for the industry in adopting
these alloys into mainstream products is the lack of knowledge of the metallurgy of Sn-Bi, and
related ternary (Sn-Ag-Cu and Sn-Cu-Bi) and quaternary (Sn-Ag-Cu-Bi) systems. The analysis
of the solid-state diffusion of Bi in Sn undertaken in this study should help fill these gaps and
contribute to the acceleration of industry implementation.
166
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