microstructural analysis of cemented tungsten carbide ......sintering and grain growth of fine-grain...
TRANSCRIPT
MICROSTRUCTURAL ANALYSIS OF CEMENTED TUNGSTEN
CARBIDE USING ORIENTATION IMAGING
MICROSCOPY (OIM)
by
Vineet Kumar
A thesis submitted to the faculty of The University of Utah
in partial fulfillment of the requirements for the degree of
Master of Science
Department of Metallurgical Engineering
The University of Utah
May 2008
Copyright © Vineet Kumar 2008
All Rights Reserved
THE UNIVERSITY OF UTAH GRADUATE SCHOOL
SUPERVISORY COMMITTEE APPROVAL
of a thesis submitted by
Vineet Kumar
This thesis has been read by each member of the following supervisory committee and by majority vote has been found to be satisfactory.
-
Chair: Zhigang Zak Fan! -
Ravi Chandran
Dinesh K. Shetty
THE UNIVERSITY OF UTAH GRADUATE SCHOOL
FINAL READING APPROVAL
To the Graduate Council of the University of Utah:
I have read the thesis of Vineet Kumar in its final form and have found that (1) its fonnat, citations, and bibliographic style are consistent and acceptable; (2) its illustrative materials including figures, tables, and charts are in place; and (3) the fmal manuscript is satisfactory to the supervisory committee and is ready for
submission to The Graduate School.
>/-v6-lo�-DatI I Zhigang Zak Fang
Chair: Supervisory Committee
Approved for the Major Department
{J 1. D. Mille;'C
Chair
Approved for the Graduate Council
David S. Chapl\1an Dean of The Graduate School
{
ABSTRACT
Cemented tungsten carbide is one of the most widely produced powder
metallurgy products. For the past 75 years cemented tungsten carbide tools have been
performing at an increasingly popular rate. Fine-grain, especially nano-grain, cemented
tungsten carbides make it possible to achieve a new range of properties that are improved
from their present counterparts. Developments in powder processing enable us to produce
true nano-size tungsten carbide powder « 30 nm). Producing true nano-grain cemented
carbide compacts remains a challenge. The conventional grain growth inhibitors are not
able to inhibit grain growth of nano-grain carbides, and liquid phase sintering is not
suitable for nano-grain carbides due to rapid grain growth. The only possible way to
sinter of nano-grain size cemented tungsten carbide is solid state sintering. Several
studies have focused on the sintering behavior of nanocrystalline WC-Co. The majority
of densification and grain growth in the specimen occurs in solid state. Hence in order to
achieve fully dense nano-grain size WC-Co, it is necessary to understand the underlying
mechanism of densification and grain growth. In this study, a comprehensive
microstructural analysis was carried out during solid state and liquid phase sintering on
micron grade samples using Orientation Imaging Microscopy (OIM). Several
microstructural parameters were analyzed to investigate the grain growth mechanisms. A
comprehensive grain boundary analysis was also done to investigate grain boundary
evolution during sintering, especially for the preferred misorientation. Since cemented
carbide showed preferred prism shape in the microstructure, a faceting analysis was also
carried out. OIM software does not provide tools for all of the analysis that were carried
out in this study, so an algorithm for faceting analysis was generated and implemented.
v
TABLE OF CONTENTS
ABSTRACT . .... . .... ....... .. ............ . . . . . _ .... . ... .. .. . . . . .. . ......... . . . .............. iv
ACKNOWLEDGMENTS . . .. . ..... . .............. . .. ............... .. . . . . ........... . .... ix
1. INTRODUCTION .. .. .. . .... . ......... . . . . .... ............ ... ........... .. ........ . .... .. I
1. 1 Scope ............. . ... . .. ... ... . .. ......... . ... . . . .. . .... . ...... ................... 2
2. BACKGROUND ............. . .... .. ... . ................... . ................... . . . ... . ..... 5
2.1 Sintering of Fine-Grain Cemented Carbides . .. .. ............ ..... ........... 5 2.1. 1 Submicron-Grain Cemented Carbides . . . .. .............. . . . .. . .. . ..... 6 2.1.2 Ultrafine-Grain Cemented Carbides ....... . .. . .. .. .... ... ....... ..... . . 7 2.1.3 Nanocrystalline Cemented Carbides ................. .. .... . . . ... . .... 7 2.1.4 Grain Growth ........ . .............. .... .. .. ...... ... ... ..... .... ....... .......... . 8
2. 1.4.1 Grain Size Distribution . . . . .... ... ..................... .. ....... 10 2.2 Electron Backscattered Diffraction ........................... . .. ... . . .. . . .. ... 13
2.2.1 Electron Diffraction .... . ............ . . . . . .. ...... . . .... ................. . 13 2.2 .2 Formation of Kikuchi Patterns .. ... .............. . ..... ... .. .... . ... .... 14 2.2.3 Identification and Indexing of Kikuchi Patterns ... . ........ . .. ..... ... 15 2.2.4 Indexing the Patterns ... .... . . . .. ......... . .......... .. ...... ..... ..... ........ ..... 17
2.2.4.1 Confidence Index . . ........ .. ...... ...... .. .... .. .. .............. 19 2.2.4.2 Fit. ....................... . .. .. .... . .. .. . .. .. . . . . ... . .. . . .. . . .. . ... .. 19 2.2.4.3 d-spacing Fit. . .. .. . .. .. . . . .. . ... . . . ........ . ........ . ............. 19 2.2.4.4 Image Quality . . .... . ..... .. .. ... .. .... . ........ . .. . ..... .. ....... . 19
2.2.5 Phase Identification ....... . ..... .... .. .. .. ..... . .......................... 20 2.2.6 Orientation Determination .. . ................... . .. . .. . .. .. . . . . ..... .. ... 20 2.2.7 Data Collection . .. ....... ............... ... .. ... . .......................... 20 2.2.8 OIM Analysis . ... . ........ . ... .. .... .................... . . . ......... . ... . .. 21
2.2.8.1 Grain Size Analysis . .. . ............... .. .. ... ... . .... . .. .. ....... 21 2.2.8.2 Orientation Analysis and Representation . ...... ..... . ... ... . . 21
2.2.8.2.1 Crystal direction map ........... ... . ... .. ........... . .. 22 2.2.8.2 .2 Texture index . . ........ . .. .. .. .. ... ... .. ...... ....... .. .. 22
2.2.9 Misorientation Analysis .. .. . .......... . .. ... .......... . . . . ... .. . ... .. . .. . . 22 2.2.9.1 Misorientation Angle Chart .... ............... . .... . . .. ......... 23 2.2.9.2 Misorientation Distribution Function (MODF) ........ ...... . 23 2.2.9.3 Misorientation Texture Index . ......... . .. . . . ............ .. .. ... 24 2.2.9.4 Faceting Analysis .. ... . . ... . . . ........... . ..... . ... . ............. . 24
3. EXPERIMENTAL PROCEDURES .. .. ...... .. ........ .. ..... .................. .. .............. ... . 25
3.1 Sample Preparation ............... . ..... .. .. .. ............ .. . ....... . . . .. .... .... 25 3.1 .1 Powder Preparation .. . . .. ............ . ........... ... . . ........ . ......... ... 25 3.1.2 Compaction and Dewaxing . ....... .. . . ............ . . ........... .... .. . ... 25
3.2 Sintering ........ .. ... . ... .. ..... . .... . .. . .. .. .... .... . . ..... . ... .. .. . . ... . ......... 26 3.2. 1 Cutting, Mounting, and Polishing . ............. . ..... ... . ... ... .. . ....... 26 3.2 .2 SEM and OIM Data Collection .. .. . . .................... ... . . ..... . . .. .. 26
3.3 OIM Analysis . .......... . .. .. ....... . ........ .. , . . .. . .. . .. . . .... .................. 28 3.3.1 Faceting Analysis . .............. . . ...... . ..... .... ... ...... . ..... . . . ... .... 28
3.3.1.1 Data Cleaning .. ... . . . ................. . ......... .. . .... . . . .. . . .. .. 29 3.3.1.2 Reconstructed Boundaries . . .................... . . . ..... ... . .. .. 30 3.3.1.3 Angles with Low Energy Planes ........ ........ , ....... . ,. " .. 31 3.3.1.4 Quantitative Analysis of Faceted Boundaries .... .... .. .... .. ..... 31
4. RESULTS AND DiSCUSSION ... ... .... ........... .. .. . .... . ..... .. .. .. .... ... . ..... . 32
4.1 Microstructural Analysis ........................... . . ............ . ........... ... 32 4.1.1 Morphology ... . . .. . .. .... ....... ..... . .... . .... , . ... ... . ............ .. ...... 33 4.1.2 Orientation .. . . . . ... . . .. . .... .. .. .......... ..... .. .. . ...... . ... . ... . ......... 58 4.1.3 Qualitative Faceting Analysis .... . .... ..... .... .. .. . ........ .. .. ... . . .. .. 59
4.2 Faceting Analysis . ................... ........................... .. .. . .. . ......... 59 4.3 Misorientation of WC-WC Boundaries .. .. ................... ... ..... . . . .. ... 65 4.4 Area Fraction of Tungsten Carbide in Microstructure ..................... . 73 4.5 Grain Size and Grain Size Distribution . . .... . .. . ................... ... . . .. .. . 73
4.5.1 Comparison of Grain Growth with Existing Models ........ .. . . . .. ... 76 4.6 Proposed Grain Growth Mechanism . ........... .. ... .... . . ............. . ..... 80
5. CONCLUSIONS ......... ......... ................. .. ..... ...... .. .. . .. ........... . . .. .. .. 82
APPENDIX: FACETING ANALYSIS CODES .. ... ... .. ..... ... .. ..... .. .. . . .. ...... 83
REFERENCES ..... .. .. . . , . .. . ..... .... .. .... .... ....... . . ... ....... ....... ........ . . ..... ... 91
VlIl
ACKNOWLEDGMENTS
This is a sincere expression of thanks to all those who made it possible for me to
successfully complete the experimental work of this study. I thank Dr. Zhigang Zak Fang
for the opportunity to work on this study and benefit from his expert guidance and
instructions. I would also like to thank Dr S. I. Wright for time to time discussions that
helped me learn the OIM technique. I thank Dr. Dinesh Shetty and Dr. Ravi Chandran for
supervision on my thesis.
I thank my colleague Praveen Maheshwari for preparing specimens for my study.
I would also like to thank him for a good get together to discuss the experimental results.
My thanks are also extended to Ms. Karen Haynes and Ms. Kay Argyle, who
provided excellent help for my study and research in the Department of Metallurgical
Engineering.
CHAPTER 1
INTRODUCTION
The potential of fine-grain cemented carbides was realized a century ago [I] , but
due to lack of proper powder preparation methods, it was difficult to produce fine-grain
materials. Recently, there have been many developments in powder making [2] that have
opened the way to achieve nano-grain size tungsten carbide powders. Fine-grain
cemented tungsten carbide materials have the potential to improve mechanical properties.
As a result they are expected to replace the conventional coarse-grain tungsten carbides.
Based upon grain sizes, the fine-grain carbides can be classified into three
categories:
1. submicron-grain (0.5-1 ~m)
2. ultrafine-grain«0.5 ~m)
3. nano-grain «100 nm)
The conventional way to prepare tungsten carbide tools is liquid phase sintering.
Typically, grain growth inhibitors, such as carbides of vanadium, tantalum, and
chromium, are used to hinder the grain growth. The grain growth inhibitors are not
effective for very small grain sizes. Solid state sintering is one of the possible ways to
produce bulk nano-crystalline cemented carbides.
Solid state sintering has been of interest due to grain coarsening in liquid phase
sintering. Some solid state sintering studies [3 , 4] show that more than 50% densification
2
occurs III solid state. The densification rate is higher during heating up and reduces
during the isothermal hold. It is realized that a comprehensive study of sintering behavior
in solid state is required.
The microstructural evolution reflects the progress of sintering. The
microstructural changes show a considerable agglomeration of we grains. The grain size
increases with sintering temperature and sintering time. The analysis of grain size and
grain size distribution during sintering can be helpful in determining sintering mechanism
and kinetics. The we grains evolve as faceted grains with their preferred prim shape
during sintering, as shown in Figure 1.1. This is due to the highly anisotropic character of
the we interface with we or cobalt [5]. There is a possibility that the faceting nature of
grains will evolve during sintering. Several crystallographic/textural parameters can be
analyzed in microstructural evolution during solid state sintering.
The above reasons constitute the motivations for a microstructural analysis during
solid state sintering. Specifically, some unresolved issues in general materials science,
such as how grains grow and what the coalescence mechanisms are, indicate that a study
of this kind can be helpful.
1.1 Scope
The objective of this study is to understand the grain growth mechanisms during
sintering. The main research tasks of this study are summarized below:
1. Grain size analysis - The change in grain size of carbide particle is analyzed as a
function of sintering temperature.
3
Figure 1.1. A microstructure showing faceted tungsten carbide grains
4
2. Misorientation analysis - The misorientation relationship evolution during
sintering IS analyzed. Cemented carbides show a strong misorientation
relationship. Analysis of misorientation evolution IS also an objective of this
study.
3. Shape analysis of carbide particles - Carbide particles in microstructure show a
tendency for faceting. The quantitative measurement of faceting is also one of the
objectives of this study.
CHAPTER 2
BACKGROUND
This chapter is divided into two sections. The first half (section 2.1) describes
sintering and grain growth of fine-grain cemented carbides and the second half explains
the OIM technique that is extensively used in this study.
2.1 Sintering of Fine-Grain Cemented Carbides
The cemented tungsten carbide composite was discovered during the First World
War (1923) as a substitute for diamond in wire drawing applications. Very soon its
applications were found in cutting tools, wear-resistant parts, and various other machine
parts. Several other carbide systems and their combinations have been studied in order to
achieve better properties than cemented tungsten carbides. However, cemented tungsten
carbide proved to be the best in terms of properties and versatility. Therefore, the WC-Co
system has drawn attention, and now the focus is on improving its properties. To date,
WC-Co is the most widely used material in cutting tools and various other wear and
abrasion resistant machine parts. This is because of its outstanding mechanical properties
and wear resistance.
In the past decade, it was realized that fine-grain cemented tungsten carbide has
the potential to show the superior mechanical properties as compared to the coarse-grain
cemented carbides. The current focus is now shifted to produce bulk cemented tungsten
6
carbides with nano-size grains. The fine-grain WC-Co hard metals can be categorized
into three categories based upon the grain size.
2.1.1 Submicron-Grain Cemented Carbides
The submicron-grain cemented tungsten carbide refers to 0.5 to I micrometer
gram size materials. This group of hard metals was introduced in 1927 [1] when the
production of nano-grain carbide powder was a difficult task. The submicron-size WC
powder was produced by a milling operation. The surface energy for small grain-size
powders is high due to the high surface-to-volume ratio. The high surface energy acts as a
driving force for rapid grain growth in fine-grain powders. It is difficult to inhibit grain
growth by sintering at low temperatures because rapid grain growth occurs during the
early stages of sintering. It has been found that certain carbide forming additions, called
grain growth inhibitors, are helpful in preventing grain growth. VC and Cr}C2 were found
to be among the most effective grain growth inhibitors. The inhibitors are known to
dissolve in binder phase cobalt and reduce the interfacial energy anisotropy at carbide
binder interface, which leads to grain growth inhibition [6]. Some of the studies showed a
low sintering temperature for submicron sized WC-Co grains. The sintered specimens
showed properties that were superior to coarse-grain compacts [7, 8, 9] particularly
hardness and wear resistance. A small reduction in transverse rupture strength was also
reported. The submicron-grain cemented tungsten carbide tools exhibit higher toughness
[10]. The tool life was reported to improve by 1.3 - 1.5 times [9].
7
2.1.2 U1trafine-Grain Cemented Carbides
The improved properties of submicron-grain cemented carbides have motivated
researchers to further reduce the grain size. This resulted in an increased amount of
research in powder production as well as consolidation methods. The ultrafine cemented
carbides possess grain size in the range of 200 to 500 nanometers. Schubert et al. [II]
studied consolidation of ultrafine-grain cemented carbides and found that rapid
densification takes place during the early stages of sintering before the formation of
liquid phase. Grain growth also occurred at the same time because densification and grain
growth followed similar trends. Grains grew large before the onset of liquid phase
sintering. The production of ultrafine-grain powders necessitates the inhibition of early
stage grain growth. Vanadium carbide (0.65 wt%) is a suitable grain growth inhibitor
[12]. This inhibition of grains indicates a close relationship between grain growth and
densification processes. Schubert et al. [13] proposed that face specific adsorption, face
orientation deposition and blocking of active growth centers are some of the possible
grain growth inhibition mechanisms. The ultrafine-grain materials proved to be better
than submicron-grain materials, especially for small-sized tools. However, it was found
that the above precautions and inhibitors did not result in reduction of grain size below
200-300 nanometers during liquid-phase sintering [I].
2.1.3 N anocrystalline Cemented Carbide
One of the hurdles in producing bulk nano-crystalline cemented tungsten carbides
is the production of nano-crystalline « 30 run) powders. The first production of WC
powder with smallest grain size in the range of 20-50 nm and an average particle size of
75 flm was reported by McCandlish et al. [2]. Many routes for nano-powder production
8
were found [I] thereafter. Some studies on nano-grain size we powder ranging in size
from ISO to 500 nm have shown significantly improved properties. Fang et al. [13]
produced sintered compacts from 50 nm powder and compared them with the compacts
produced from conventional micron size powder. It was found that nano-powders started
densifying at lower temperatures than micron sized specimens and at 1200e, the nano
powder specimens achieved almost 90% densification. It was also observed that grains
grow rapidly during the intermediate stage of densification. The nano-grain size materials
offer better crack resistance for the same hardness level. Overall, nano-grain size
cemented carbide promises a new class of properties that are superior to conventional
cemented tungsten carbide cobalt materials.
2.1.4 Grain Growth
The grain growth and densification mechanisms follow a similar course. The
driving force for grain growth is reduction in interfacial energy. Large grains grow at the
expense of smaller grains, thus reducing the interfacial energy. The increase in average
grain size reduces the total chemical potential of the system.
There have been efforts [14-18] to quantify the kinetics of grain growth. The
mean grain size variation follows Equation 2.1 for isothermal conditions.
(2.1 )
where G is average grain size, t is isothermal time, and n is grain growth exponent. The
grain growth exponent varies from 2 to 4 for most materials. Most of the cemented
tungsten carbide studies are done during liquid phase sintering. In general, the exponent
value for dissolution and precipitation controlled grain growth is close to 3. The model
9
for diffusion-controlled grain growth is known as the LSW model, named after Lifshitz
and Slyozov [15] and Wagner [16]. It is also known as the Ostwald ripening mechanism
and is given by Equation 2.2.
(2.2)
where Go is initial grain size, k is rate constant, and t is time.
The grain growth can also be controlled by interfacial reaction. The reaction takes
place at interfaces between a growing grain and a shrinking grain. Grain growth is
interfacial reaction controlled when diffusion is faster than the interface reaction. In this
case, the interfacial reaction is the rate-determining step. Under this condition, the grain
size follows Equation 2.3 [19].
G 2 = G 2 + 256YsL CQk,.l , 81kt
(2.3)
where k, is the interfacial reaction rate constant. The interfacial reaction-controlled grain
growth is more sensitive to temperature owing to higher activation energy.
The models discussed above have key assumptions that are different from WC-Co
sintering. The models assume spherical particle shape whereas tungsten carbide particles
have preferred prismatic shapes. The other assumptions, e.g., contact between the grains,
isotropic surface energy, no contact between the grains, and a mean concentration of
solid in liquid, do not apply to cemented tungsten carbide sintering. The models also
assume widely dispersed particles whereas in compacts, carbide particles form a
connected skeleton. Coalescence, which is one of the important mechanisms, is ignored
10
by most of the models. For cemented carbides, the rearrangement of particles and closing
of porosities play an important role in grain growth, but these were not taken into account
in grain growth models.
When the two grains that have low energy boundary between them come into
contact, coalescence occurs by fusion of the grains to one large grain. Low angle
boundaries and coincidence lattice site (CSL) boundaries possess low energy. The
contacts made by these boundaries have high chances of coalescence. Due to preferred
prismatic shape formed by low energy planes of tungsten carbide particles, there is a
higher probability of finding a low energy boundary contact. Also, the rearrangement of
particles increases the favorab le contacts. It is therefore expected that coalescence also
contributes to grain growth of cemented tungsten carbides.
2.1.4.1 Grain Size Distribution
Several mass transport mechanisms contribute to gram growth. The mass
transport mechanisms determine the rate of grain growth and densification. The final
grain size distribution can give some insight into the applicable mechanisms.
The LSW theory, or the Ostwald ripening theory, is based upon dissolution of
smaller grains and deposition over the large grains. The dissolution and reprecipitation
rate is controlled by curvatures of grain. The grain size distribution for the LSW model
can be expressed as follows (20):
P -3p ((p,t) - (2 _ p) 5 exp 2 _ P when 0< p < 2
((p,t) = 0 when p > 2
II
where p is defined as p = 8 r / 9r and rand r are particle size and mean particle size
respectively.
The coalescence mechanisms are not well known. Haslam et al. [21) simulated
grain growth using coalescence on a mesoscale. Grain rotation was assumed to be a
major coalescence mechanism. The mesolevel and the two-dimensional (20) nature of
simulation are the major drawbacks of this study. Kaysser et al. [21) studied grain
coalescence of iron-copper liquid phase sintering. A statistical approach was applied in
modeling the coalescence phenomena. It was assumed that coalescence occurs between
grains having low energy boundaries (low angle and CSL boundaries) between them.
Coalescence produces a different kind of grain size distribution because the two grains
fuse and form a large coalesced grain. Therefore, the final grain can be a combination of
two or more smaller grains. The finer grains contributing to coalescence vanish, thus
leading to a decrease in the number of finer grains. The coalesced grains belong to an
asymptotic tail in grain size distribution curves. The grain size distribution for
coalescence is expressed as follows [17):
f(u) = 2.13u ' exp(-O.712Iu l) (2.4)
where u is the ratio of the individual-to-average particle size. The model proposed by
Haslam et al. [21) produces similar grain size distribution.
A comparison of grain size distribution produced by the LSW mechanism,
coalescence, and their combinations, is given in Figure 2.1. In Figure 2.1, u" refers to the
contribution of the LSW mechanism towards grain growth, u" = 1 refers to the LSW
being the only grain growth mechanism, whereas u" = 0 refers to coalescence. It can be
Figure 2. 1. The nonnalized particle size distribution for concurrent coalescence and LSW
model. (a). particle volume fraction qJ- 0, (b). particle volume fraction qJ = 0.8. Reprinted
with pennission from [22]
13
inferred from Figure 2.1 that a small contribution of coalescence to grain growth brings
grain growth distribution very close to coalescence.
Fine-grain cemented tungsten carbides promises improved properties as compared
to their coarse-grain counterparts. Solid state sintering is one way to achieve fine-grain
cemented tungsten carbides. TIle grain growth and densification behavior is not
understood during solid state sintering. Therefore, understanding the grain growth and
densification can give insight to achieve full densification with minimal grain growth.
The classic grain growth models make some assumptions that are not valid for cemented
tungsten carbides. Some models consider coalescence as grain growth mechanism. A
comparison with a classical grain growth and coalescence mechanism can give insight
into the grain growth mechanism in cemented tungsten carbides.
2.2 Electron Backscattered Diffraction
This study uses the Orientation Imaging Microscopy (orM), also known as the
Electron Backscattered Diffraction (EBSD) technique, to study grain growth mechanism
in tungsten carbide-cobalt. This technique utilizes electron backscattered diffraction
patterns to extract crystallographic information, which is used in orientation and
misorientation analysis.
2.2.1 Electron Diffraction
Electrons show both particle and wave nature. The relationship between its energy
and wavelength is given as
A. = h .J2meU
(2.5)
14
where A is the wavelength of electrons, U is the applied accelerating voltage, m is the
mass of electron, and e is the charge of electron. In SEM, the applied voltage is in the
range of 5-40 kY, which leads to an electron wavelength in the range of 0.0061 - 0.0172
nm. The electron wavelength is of the same order as atomic spacing in the crystals, which
leads to electron diffraction by the atomic planes in crystal. Diffraction occurs, according
to Bragg's law, as:
2dsinB = nA (2.6)
where d is interplanar spacing, B is Bragg's angle, n is the order of diffraction, and A is
the wavelength of radiation. In practice, the Bragg's angle is very small (less than 2
degrees). This indicates that the atomic planes are nearly parallel to the primary electron
beam and reflecting electron beam.
The electron beam interacts with bulk specimen III interaction volume. The
interaction of the electron with specimen atoms leads to several scattering events. When
an electron beam interacts with a bulk specimen (as in SEM), signals such as
backscattered electrons, secondary electrons, auger electrons, etc. are generated. The
backscattered electrons are generated by elastic scattering of electrons with atoms in
specimen. When backscattered electrons come out of the specimen, it gets diffracted by
lattice planes close to the surface. These diffracted electrons are used as a signal in OIM
technique.
2.2.2 Formation of Kikuchi Patterns
The electron beam has less penetration depth, which leads to a small interaction
volume. The interaction volume in an SEM is of the order of the electron beam diameter,
15
which is around 2 nrn. The interaction volume is smaller than the grain size. The signals
come from a single grain at a time unless grain size is very small or the electron beam is
focused on the grain boundary. The back scattered electrons, while coming out of the
sample, get diffracted by atomic planes near the surface. These diffracted electrons make
shallow cones because of the small Bragg's angle. These cones are projected as bands on
the camera screen. In this way, a pattern of bands is obtained. This pattern is called the
Kikuchi pattern or the electron backscattered diffraction pattern. The Kikuchi patterns are
recorded on a camera. An area of a sample for OIM scan is to be selected. This area is
divide into grids, the width of which determines the resolution of the scan. The electron
beam scans the specific area of the sample. At the same time the camera records the
pattern from each point.
2.2.3 Identification and Indexing of Kikuchi Patterns
It is easier to detect some bright spots on a 20 figure than lines on a 20 plane.
Following this concept, the Kikuchi pattern is transformed into a Hough transform where
every line is represented by a point. Every line on Kikuchi pattern can be written as:
p=xcosB+ y sinB (2.7)
where p is the perpendicular distance from origin and B describes the angle of line as
shown in Figure 2.2. Equation 2.7 converts the Kikuchi pattern(x. y) into the Hough
transformation (p ,B). Each point (p,B) in the Hough transformation similarly has a
corresponding line in the Kikuchi pattern. Conversely, a point in the Kikuchi pattern
transforms into a sinusoidal line in the Hough transformation. Intensity corresponding to
each pixel in a Kikuchi pattern is binned corresponding to each pixel in sinusoidal line.
40
'" 30
,. >- 20 ,.
10
5
0
0
x
Figure 2.2. Hough transformation of Kikuchi patterns. Reprinted with permission from [23]
16
17
After binning, the Hough transformation is plotted on gray scales. The corresponding
bands come out as peaks in the Hough transformation. It is now easier to detect these
peaks rather than bands in Kikuchi patterns. Finally, the peaks are recorded as the
location of bands in the Kikuchi pattern.
2.2.4 Indexing the Patterns
The indexing is done by comparisons of possible Kikuchi patterns with detected
Kikuchi patterns. The materials files containing crystallographic information and
theoretically calculated Kikuchi patterns for possible phases in specimen are used from
the database. Sometimes, electron dispersive spectroscopy (EDS) is used to find the
possible phase in the specimen. For indexing, the bands or respective planes are
compared with the planes that would show diffraction bands. For example, the FCC
phase will diffract {III}, {200}, {220}, and {311} planes and their symmetric
equivalents. Three bands are needed to index a Kikuchi pattern. A look-up table is made
using the combination of major diffracting planes for a possible phase. A set of three
bands is taken at a time and compared for the angle between the bands in the look-up
table as shown in Figure 2.3. A vote is given for a possible combination to be the right
indexing solution. After all the combinations of triplets are compared, the solution having
the highest votes is taken as the solution for the Kikuchi pattern. A reconstructed Kikuchi
pattern is generated for the accepted solution. Some parameters, associated with each
index point that represents the accuracy of indexing, are used in analysis. Some of them
are described in following sections.
Indexing Solutions
, , 3 • , , 1 • , '0 " -- x x x
-- x --- x -- -.. - x x x x x Q. -' ': - x x x x I- -'0 - X X X C -II -III x -- x --- x x -- x x x x -
1: , • ,. , , , , , , , ,
Figure 2.3. Look-up table used for indexing the Kikuchi patterns. Reprinted with permission from [23]
18
19
2.2.4.1 Confidence Index
The difference of votes between the solution with the highest and second highest
vote divided by the number of possible combinations of triplets gives the confidence
index for that point. For the solutions having confidence index > 0.1, the probability of it
being correct is more than 95%.
2.2.4.2 Fit
Fit is defined as the difference in the position of bands in the detected pattern with
the position of bands in reconstructed (or solution) pattern. This parameter indicates the
angular difference between detected and recalculated patterns.
2.2.4.3 d-spacing Fit
The width of bands depends upon d-spacing of reflecting planes, specimen-to
screen distance, and the applied voltage. The d-spacing fit refers to the average difference
between widths of the bands in a detected Kikuchi pattern as calculated from the Hough
transform and widths of bands in the reconstructed Kikuchi pattern.
2.2.4.4 Image Quality
Image quality is the parameter that refers to the quality of a detected Kikuchi
pattern. Quantitatively image quality is the sum of peaks in Hough patterns. The image
quality is also a parameter that can be used to separate good data points from bad ones.
The image quality depends upon materials conditions as well as the electron beam
configuration. If a sample is mechanically deformed or not very flat, the image quality
goes down. The image quality also goes down when an electron beam hits a grain
boundary. If an image quality map is plotted on gray scale, it resembles a microstructure.
20
2.2.5 Phase Identification
Different phases have different Kikuchi patterns depending upon their lattice
parameters and space group. The materials files containing this information can generate
theoretically calculated Kikuchi patterns. The possible materials files are loaded into a
program and a detected Kikuchi pattern is compared with the calculated Kikuchi patterns
of all possible phases. Generally, a comparison with a wrong phase leads a to low
confidence index and fit. The possible phase having highest confidence index is accepted
as a phase present in the specimen.
2.2.6 Orientation Determination
The orientation of a point III the OIM scan refers to the difference between
specimen system and crystal axis system. The specimen axis system is assigned during
microscope calibration and is known to OIM program. The crystal axis system is found
from the location of bands in the reconstructed pattern. There are several ways to
represent the orientation. The most popular one is Euler angles representation. The Euler
angles are a set three angles (<PI , <P, <P2). If the sample axis system is rotated by Euler
angles in a sequence, it should coincide with the crystal axis system.
2.2.7 Data Collection
The electron beam raster scans across a specified area and generates the Kikuchi
pattern at each point. The pattern is indexed for phase identification and orientation. Each
point/pixel is recorded in an output file . The output file contains information about the
properties of phases and scan along with information associated with each point. The
21
information related to the point is its coordinate, phase, image quality, fit , confidences
index, and Euler angles. This information is used for analysis.
2.2.8 0 IM Analysis
OIM analysis 4, a product ofTSL-EDAX, is used for representation ofOIM data.
This program provides options in almost all conventional texture representation ways.
The ways in which the present study is done are discussed below.
2.2.8 .1 Grain Size Analysis
The grains in the OIM are defined as a set of neighboring points having
orientation within a given range, termed tolerance. The grain size is defined as the
diameter of a circle having the same area as a grain in the microstructure. The grain size
distribution can be represented as number fraction or area fraction.
2.2.8.2 Orientation Analysis and Representation
The orientation of each point is assigned as Euler angles associated with it. The
most accurate way of describing the orientations is in terms of orientation distribution
functions. The orientation distribution functions are difficult to interpret. Also this space
is not linear, which further makes interpretation difficult . There are some other
conventional ways that are easy to interpret, for example, inverse pole figures and pole
figures. There are some pictorial ways to represent the orientations. Pictorial
representation relates the orientation information with microstructure. There are also
some quantitative methods, such as texture index to represent the orientation, that do not
indicate the distribution but intensity. Some of the representation methods are discussed
below.
22
2.2.8.2.1 Crystal direction map. Crystal direction map shows each point colored
by an automatic grayscale for a particular direction using a unit triangle of crystal
direction. For example, <0 0 0 1> crystal direction maps are used in this work. The
orientation of crystals is shown by <0 0 0 I> crystal direction. The grains having <0 0 0
I> direction parallel to the sample's normal direction (NO) are marked as white grain. As
the angle between NO of the sample and <0 0 0 1> increases, the grain shades becomes
grayer, black when NO and <0 0 0 I> are perpendicular to each other.
2.2.8.2.2 Texture index. The degree of the texture can be represented by the
texture index. The texture can be expressed in terms of the coefficient of harmonic series
expansion in generalized spherical harmonics. This parameter does not consider the
orientation distribution. The value of texture index varies from I for random orientation
to infinity for single crystals. The texture index is given by following Equation 2.8 .
. _" I r p' /- L...~'
p.,., 21 + I (2.8)
where j is the texture index, C's are coefficients in harmonic series and t, f.l and v are
the parameters used for texture calculations. The values of these parameters can be
obtained from texture export files.
2.2.9 Misorientation Analysis
The misorientation between two points is defined as the difference in the crystal
axis system. As the orientation of each point is known, the difference in the orientation
can be calculated. The misorientation can be represented by a matrix when operated on
23
one crystal system, coinciding with a crystal system belonging to other systems.
Misorientation matrix is given by Equation 2.9.
(2 .9)
where g A and g B are the orientation matrix for two points. The misorientation between
two crystals can be visualized in the following way. There can be a common direction in
both of the crystals that are parallel. If one of the crystals is rotated around that axis with
certain angle, the axes systems coincide. This combination of angle and axis defines the
misorientation, and this is called angle/axis misorientation. There are some other ways to
represent misorientations such as Rodrigues space, but they are difficult to interpret.
2.2.9.1 Misorientation Angle Chart
A misorientation angle chart is a plot of a misorientation angle as described above
vs. frequency. There may be a number of angles for the same misorientation. In this case,
the minimum angle is considered as the misorientation angle. Crystal symmetry does not
allow the misorientation angle to be larger than a certain value, which defines the range
of misorientation angle.
2.2.9.2 Misorientation Distribution Function (MODF)
The misorientation distribution function represents boundary information in terms
of axis and angles . For a range of misorientation angles, the axis is plotted in the pole
figures. This way of representation is very helpful 111 interpreting the misorientation.
There are some other ways of representation, such as Rodrigues vector, but they are
difficult to interpret.
24
2.2.9.3 Misorientation Texture Index
Misorientation texture index indicates the overall misorientation intensity
irrespective of distribution. The calculations are similar to the orientation texture
calculations.
2.2.9.4 Faceting Analysis
The carbide particles in matrix grow and take the shape of a triangular prism. The
prism faces and base have low energy {O 0 0 I} or {lOT O} planes forming the facets.
In this method, the facets forming the boundaries are determined and shown in
microstructure. The quantitative analysis is done using length fraction or area fraction.
This method is discussed in the experimental section.
CHAPTER 3
EXPERIMENTAL PROCEDURES
The experimental section is divided into two parts: The first part describes the
specimen preparation, and the second part describes the shape-analysis algorithms.
3.1 Sample Preparation
The samples were prepared by compaction of the powders and sintering the
compacts at various temperatures. The specimen preparation method is described briefly
below, the complete details of which can be found elsewhere [24].
3.1.1 Powder Preparation
The commercially available tungsten carbide powder of grain size I 11m and
commercially available cobalt was used in specimen preparation. WC, 10% Co, and 2%
wax were milled in an attrition mill in heptane. The particle size was determined by a
Fisher particle size analyzer.
3.1.2 Compaction and Dewaxing
Specimens of 1 11m average particle size were prepared by pressing the powder at
200 MPa in a cylindrical die. The specimens were dewaxed in the presence ofH2/Ar (1 :5)
gas mixture.
26
3.2 Sintering
The dew axed specimens were sintered in a high-temperature vacuum furnace at
sintering temperatures. Time periods for different specimens are given in Table 3.1. For
sintering, the specimens were heated from room temperature to sintering temperature at a
heating rate of looe per minute. The specimens were held for the desired hold time and
then furnace cooled to room temperature.
3.2.1 Cutting, Mounting, and Polishing
The specimens were cut using ISOMET'"M 1000 precision saw into two halves
and cleaned in an ultrasonic bath. Low viscosity epoxy (EPO-THINTM) was used to
mount the samples in the vacuum chamber. The specimens were polished using diamond
suspensions. Sequential polishing was perfonned employing diamond slurry with
decreasing diamond particle sizes of 9, 6, 3, I , 0.5, and 0.1 fun. The final polishing was
done using colloidal silica suspension of particle size 0.05 [lm. Silica colloidal suspension
was slightly basic and had a pH of 10. Silica colloidal suspension removes mechanically
deformed regions from the surface, making the specimen more suitable for OIM. It also
etches cobalt from the specimen surface, which helps in reducing magnetic interference
of cobalt with electron beam.
3.2.2 SEM and OIM Data Collection
SEM and OIM data were collected using Phillips 30XL FEGTM scanning electron
microscope. The scans for specimens sintered below 13000 e were taken at EDAX-TSL,
Draper, whereas the rest of the scans were taken at Brigham Young University, Provo.
27
Table 3.1. The sintering temperature and time for specimens prepared
s. No Temperature COc) Time (min)
I 800 I
2 1000 I
3 1200 I
4 1300 I
5 1400 I
6 1400 10
7 1400 30
8 1400 60
28
An accelerating voltage of 20 kV and spot size 5 were used for SEM imaging and OIM
data collection. The OIM data were collected using OIM Data Collection 4™ software. A
scan step size of 0. 1 ~m was used for specimens that were sintered below l300°C,
whereas a scan step size of 0.25 ~m was used for the rest of the specimens.
3.3 OIM Analysis
The OIM analysis was done using OIM Data Collection 4™ software. The
algorithms for analysis, except faceting analysis, can be found in software help files [23].
The faceting analysis algorithm is described below.
3.3.1 Faceting Analysis
Cemented tungsten carbide grains have a tendency to acquire prismatic shape,
formed by low energy planes ({O 0 0 I} and {lOT O}) of a hexagonal crystal system.
After sintering, these grains can be seen as rectangles and triangles in a 20
microstructure. The rectangular and triangular shape comes from truncation prisms
particles. These prism-shaped grains are called faceted grains.
A grain boundary between two tungsten carbide particles can have faceted or
unfaceted grains on either side. Grain boundaries can be categorized into three types
based on the type of grains on either side of the boundary. The three types of boundaries
are as follows:
I. Unfaceted boundary that is defined as a boundary between two unfaceted grains.
2. One-faceted boundary that is defined as a boundary between one faceted and one
unfaceted grain.
3. Two-faceted boundary that is defined as a boundary between two faceted grains.
29
The faceting analysis was crystallographic infonnation to detennine the faceting nature
of a grain. The grain boundaries can be characterized quantitatively based upon the nature
of adjacent grains. The algorithm for faceting analysis can be described as follows:
I . The data are cleaned for analysis. A pseudo-orientation is assigned to low quality
points such as holes and porosity.
2. Reconstructed boundaries [23] are calculated along with the orientation on the left
and right side of the boundaries.
3. A vector parallel to the reconstructed boundary is found in the crystal axis system.
The angles between the vector and four possible low energy planes are calculated.
4. Based upon the angle with low energy planes within a tolerance limit (chosen 10°
for this study), the boundaries are characterized as faceted or nonfaceted.
5. The number fraction and the length fraction of faceted and unfaceted boundaries
are calculated.
The steps are described in detail in the following sections:
3.3.1.1 Data Cleaning
The data were cleaned to eliminate low quality data points. The cleaning was
done in the following three steps:
I. Grain CI standardization - The confidence indices of all the points were matched
with the highest CI among all the data points in a grain.
2. Neighbor orientation correlation The data sets were cleaned using neighbors
orientation correlations at level 0, 1,2, and 3 successively (for CI < 0.1) followed
by grain CI standardization. Neighbor orientation correlation cleaning operates on
points having CI less than a user-defined value and a finite nwnber of neighbors
30
with orientations different from the rest of the neighbors depending on the
cleaning level. For example, level 2 cleaning changes the orientation of a pixel,
which has four out of six neighbors with the same orientation and two with
different ones, to orientation of four pixels.
3. Replacement of low quality data with pseudo orientation - When an electron
beam is focused on a hole or pore, the Kikuchi pattern quality is poor. The OIM
program is unable to index such patterns, so zero CI with no phase is assigned to
those data points . The data are exported from OIM analysis program to analysis
program and data points having confidence index less than 0.1 , which were
replaced by some pseudo-orientation. The data are again put back into analysis
software.
3.3.1 .2 Reconstructed Boundaries
The OIM data collection program uses hexagonal pixel; hence a boundary can
have an angle of 60, 120, or 2400 angle. Reconstructed boundaries are drawn to make
boundary close to the boundaries in the sample. The grains are defined as a group of
pixels having orientation in a given range. The grain boundary triple points are found.
Based upon the tolerance chosen, the grain boundaries are constructed between suitable
grain boundary triple points. The grain boundary data are then exported for faceting
analysis. The exported data contain the following information about a boundary:
1. Coordinates of ends of boundary segments
2. Average orientations of grains on the left and right of the boundary
3. The grain boundary trace
4. Grain boundary length
31
The reconstructed boundaries are then used in faceting calculation.
3.3.1.3 Angles with Low Energy Planes
The orientation matrix is calculated using the average orientation of the grains.
(cos ¢, cos ¢, - sin ¢, sin ¢, cos ¢) (s in ¢, cos ¢, - cos ¢, sin 1/>, cos 1/» (s in 1/>, sin t/J) g(t/J, , ¢, t/J , ) = (cos t/J, sin ¢, - sin ¢, cos 1/>, cos t/J) (- sin t/J, sin 1/>, - cos ¢, cos 1/>, cos t/J) (cos 1/>, sin ¢)
sin ¢, sin I/> - cos 1/>, sin ¢ cos ¢
where 91' ¢ , and ¢, are the Euler angles for orientation of grain.
The orientation matrix on multiplication with the grain-boundary vector gives a
vector parallel to grain boundary in the crystal axis system as Equation 3.1.
Vector = g *(X2-XI .YrYI .O) (3.1 )
where g is orientation matrix, and (Xl , Yl) and (X2, Y2) are the coordinates of the end
points of boundary segment.
The angle between the planes is calculated with (0 0 0 I), (I 0 I 0), (0 I 0 0) and
(I 0 I 0) low-energy planes, which can be represented in a cubic system as (0 0 I), (1 .5
0.8660), (01.7320), and (1.5 0.8660), respectively.
The angle between the plane and vector is calculated using Equation 3.2.
, - I g * vector angle = 90 - cos I I I I g * vector
(3.2)
3.3.1.4 Quantitative Analysis of Faceted Boundaries
The angle tolerance for a boundary to be faceted or un faceted is chosen to be 10°
in this study. The boundaries making angle less than or equal to 10° with low energy
32
planes are considered to be faceted boundaries. The quantitative analysis is done using
length fraction as well as number fraction offaceted and non-faceted boundaries.
ERRATUM
Page 32 was assigned twice in this manuscript.
CHAPTER 4
RESULTS AND DISCUSSION
The SEM technique was used to collect the SEM micrographs, and the OIM
technique was used to collect the crystallographic infonnation. The OIM data are
represented in several ways to extract the relevant infonnation as described in Chapter 2.
In order to find the evidence of the grain growth mechanism, the following
microstructural analyses were carried out:
I. Microstructural analysis (morphology, orientation, and qualitative faceting)
2. Quantitative faceting analysis
3. Misorientation analysis
4. Area fraction of tungsten carbide
5. Grain size and grain size distribution analysis, including comparison with existing
models
In the following sections, the above microstructural characteristics are described. The
grain size and grain size distributions are compared with existing models. A grain growth
mechanism is proposed based upon the observations.
4.1 Microstructural Analysis
The SEM micrographs were taken with FEG SEM with a tilt angle of 70°. The
micrographs were high-resolution images that look like three-dimensional (3D) images
33
due to tilt correction. The SEM micrograph, crystal direction map, and faceting map for
sample sintered at 800°C are shown in Figures 4.1 , 4.2 and 4.3, respectively. Similarly,
Figures 4.6 to 4.24 show SEM micrographs, crystal direction maps, and faceting maps for
rest of the specimens.
4.1.1 Morphology
The specimen sintered at 800°C is treated as if it was not sintered and therefore
represents the conditions of the original powder. This is reasonable because there was
little grain growth and densification at this temperature. Figure 4.1 shows the
microstructure of specimen sintered at 800°C for I minute. It shows large multigrain
clusters and regions having small grains. The topology of clusters shows that they are
multigrain clusters. It is believed that neck formation between particles takess place at
low sintering temperatures. The particles in compact were bonded by neck formation at
800°C. The bonding makes polishing possible and eventually the microscopy. It can be
noted that the large multigrain clusters are far away from each other. This could be due to
unbonded particles that fell apart during polishing. The same morphology is shown by
specimens sintered at 1000°C and 1200°C for I minute as shown in Figures 4.4 and 4.7,
respectively.
The specimen sintered at 1300°C for I minute shows a transition from unsintered
to sintered condition (Figure 4.10). There are two main features to be noted. First, the
porosity is not continuous, and, second, some of the grains show truncated prism shapes.
The small grains seemed to be rearranged, and the large clusters seemed to have
truncated prism shape made by low energy planes of the hexagonal crystal system.
34
Figure 4. I. SEM image for specimen sintered at 800°C for 1 minute
35
Figure 4.2. Crystal direction map for a specimen sintered at 800°C for I minute. The
grains having <0 0 0 I> parallel to ND are shown black. The color scheme becomes
brighter on increasing the angle between ND and <0 0 0 1 >
36
Figure 4.3. Faceting map for specimen sintered at 800°C for 1 minute. The unfaceted,
one-faceted and two-faceted boundaries are marked as dotted, thin-solid, and thick-solid
lines
37
Figure 4.4. SEM image for specimen sintered at lOOO°C for 1 minute
I ' ~ k " t
38
Figure 4.5. Crystal direction map for a specimen sintered at 1000°C for 1 minute. The
grains having <0 0 0 1> parallel to NO are shown black. The color scheme become
brighter on increasing the angle between NO and <0 0 0 1 >
39
Figure 4.6. Faceting map for specimen sintered at lOOO°C for 1 minute. The unfaceted,
one-faceted and two-faceted boundaries are marked as dotted, thin-solid, and thick-solid
lines
40
Figure 4.7 . SEM image for specimen sintered at 1200°C for I minute
41
Figure 4.8. Crystal direction map for a specimen sintered at l200°C for 1 minute. The
grains having <0 0 0 1> parallel to ND are shown black. The color scheme becomes
brighter on increasing angle between ND and <0 0 0 1 >
42
Figure 4.9. Faceting map for specimen sintered at 1200°C for I minute. The unfaceted,
one-faceted and two-faceted boundaries are marked as dotted, thin-solid, and thick-solid
lines
43
Figure 4.10. SEM image for specimen sintered at 1300°C for 1 minute
44
Figure 4.11. Crystal direction map for a specimen sintered at l300D C for I minute. The
grains having <0 0 0 I> parallel to NO are shown black. The color scheme become
brighter on increasing the angle between NO and <0 0 0 I>
45
Figure 4.12. Faceting map for specimen sintered at 1300°C for 1 minute. The unfaceted,
one-faceted and two-faceted boundaries are marked as dotted, thin-solid, and thick-solid
lines
46
Figure 4.13. SEM image for specimen sintered at 1400°C for 1 minute
47
Figure 4.14. Crystal direction map for a specimen sintered at 1400°C for 1 minute. The
grains having <0 0 0 I> parallel to NO are shown black. The color scheme becomes
brighter on increasing the angle between NO and <0 0 0 1 >
48
o 10 20 30 40 50 60
Figure 4.15. Faceting map for specimen sintered at 1400°C for 1 minute. The unfaceted,
one-faceted and two-faceted boundaries are marked as dotted, thin-solid, and thick-solid
lines
49
Figure 4.16. SEM image for specimen sintered at 1400°C for 10 minutes
50
Figure 4.17. Crystal direction map for specimen sintered at 1400°C for 10 minutes. The
grains having <0 0 0 1> parallel to NO are shown black. The color scheme becomes
brighter on increasing the angle between NO and <0 0 0 I>
51
o 10 20 30 40 50 60
Figure 4.18. Faceting map for specimen sintered at 1400°C for 10 minutes. The
unfaceted, one-faceted and two-faceted boundaries are marked as dotted, thin-solid, and
thick-solid lines
52
Figure 4.19. SEM image for specimen sintered at 1400°C for 30 minutes
53
Figure 4.20. Crystal direction map for a specimen sintered at 1400°C for 30 minutes. The
grains having <0 0 0 1> parallel to NO are shown black. The color schemes become
brighter on increasing the angle between NO and <0 0 0 1 >
54
f ,/
o 10 20 30 40 50 60
Figure 4.21. Faceting map for specimen sintered at 1400°C for 30 minutes. The
unfaceted, one-faceted and two-faceted boundaries are marked as dotted , thin-solid, and
thick-solid lines
Figure 4.22. SEM image for specimen sintered at 1400°C for 60 minutes
56
Figure 4.23. Crystal direction map for a specimen sintered at l400°C for 60 minutes. TIle
grains having <0 0 0 1> parallel to ND are shown black. The color scheme becomes
brighter on increasing the angle between ND and <0 0 0 1 >
57
Figure 4.24. Faceting map for specimen sintered at l400°C for 60 minutes. The
unfaceted, one-faceted and two-faceted boundaries are marked as dotted, thin-solid, and
thick-solid lines
58
The microstructure of the specimen sintered at 1400°C for 1 minute is shown in Figure
4.13. This micrograph shows that the majority of the carbide grains had acquired their
equi librium shapes of truncated prisms. Regions having small grains (- I ~m) in a group
could be seen in this specimen too , but they also had faceted shapes. Large grains had
grown to 4-5 microns in size. The cobalt pools were somewhat visible at this stage, but
were almost invisible before. The morphology remained the same for the specimens
isothermally held at 1400°C for 10, 30, and 60 minutes (Figure 4.16, 4.19, and 4.22).
4.1.2 Orientation
The particles have a random distribution in a green compact. The multigrain
particles in the compact possess grains having preferred misorientation, but this does not
reflect in texture of compact because multi grain particles are distributed without any
preference of orientation. Figure 4.2 shows the crystal direction map as described in
section 2.2.8.2.1 for specimen sintered at 800°C for 1 minute. The orientation of each
point in the map is represented by an automatic gray scale. In this study, the <0 0 0 I>
crystal direction maps are used. The orientation of crystals is described by showing <0 0
o 1> crystal direction. The grains having <0 0 0 I> direction parallel to the specimen
normal direction (ND) are marked in white. As the angle between the ND and <0 0 0 1 >
direction increases, the grain shade becomes grayer and is black when ND and <0 0 0 I>
direction are perpendicular to each other. It can be seen that grains of specimen sintered
at 800°C for 1 minute do not possess any preferred orientation as there is no color
preference in the crystal direction map.
59
The same behavior was observed in all the other specimens analyzed in this study
(Figure 4.5, 4.8, 4.14, 4.17, 4.20, and 4.23). All the specimens showed a random
orientation distribution of grains.
4.1.3 Qualitative Faceting Analysis
The microstructure can give some insight into faceted grains because they possess
straight boundaries and a triangular or rectangular shape. Orientation calculations as
described in section 3.3.1 make faceting analysis more definitive. The faceting map for
specimen sintered at 800°C for I minute is shown in Figure 4.3. Unfaceted, one-faceted,
and two-faceted boundaries are marked with dotted, thin, and thick lines, respectively. It
can be noted that most of the boundaries are un faceted or one-faceted. The faceting maps
remain similar for specimens sintered at 1000°C and 1200°C for 1 minute (Figure 4.6,
4.9). Figure 4.12 shows a faceting map for specimen sintered at 1300°C for I minute. A
sharp change in faceting can be noted. Quantitative analysis of change in faceting is
discussed in section 4.2. For specimen sintered at a higher temperature (specimens
sintered at I 400°C), the faceting maps show that most all of the grains are faceted
(Figures 4.15, 4.18, 4.18 and 4.24).
4.2 Faceting Analysis
A faceted carbide particle is prismatic in shape. In a 2D cross section, the
truncated prismatic carbide particles have the shape of triangles or rectangles. These
shapes can easily be identified visually in a microstructure. For quantitative analysis, the
faceting information is extracted using crystallographic calculation as described in
section 3.3.1. The calculations are based on the angles between the grain boundary trace
60
and low energy planes within a given tolerance. A tolerance of 10° was used in this
analysis. The total-facted boundaries consist of one-faceted and two-faceted boundaries.
In faceting maps, un faceted, one-faceted, and two-faceted boundaries are marked by
dotted, thin-solid and thick-solid lines, respectively. Inspection of faceting maps indicates
that the microstructure becomes faceted as sintering progresses. Quantitatively, the
faceted boundaries can be analyzed using number/length fractions as well as
number/length per unit area.
The variation in number fraction of various faceted boundaries with sintering
conditions is shown in Figure 4.25 . [n this figure, the data for specimens sintered at
800°C to 1400°C for I minute are shown by the corresponding temperatures. For
isothermally sintered specimens, the time axis is drawn on the top of the chart. The time
axis describes the hold time at 1400°C. The number fraction of un faceted and one-faceted
boundaries decreases on increasing the sintering temperature. The majority change takes
place between specimen sintered at 1200°C for I minute and specimen sintered at
1400°C for 1 minute. The equilibrium phase diagram shows that the liquid starts forming
at 1280°C. The decrease of un faceted and one-faceted boundaries reflects as increment in
two-faceted boundaries. The reduction in one-faceted boundaries is due to the conversion
of some of the one-faceted boundaries to two-faceted boundaries. These trends indicate
that grains take their equilibrium prismatic shapes as sintering progress. The same trend
can be observed in length fractions (Figure 4.26).
Figure 4.27 and 4.28 show the number per unit area and length per unit area of
faceted boundaries respectively. The per unit area quantities are corrected for partition
c: o
0.9
0.8
0.7
0.6
~ 0.5 co ol:: <> 0.4 Z
0.3
0.2
0.1
o 700
±
Faceted no. fraction Time (min)
1 10 30
• +./ ---------<. + +
• •
•
B
900 1100
b
<>--....... • ,
< : • : 1300
Temperature (' C)
I
•
:
50
Figure 4.25. Number fraction of faceted/unfaceted boundaries
70
•
:
61
• unfaceted no .
b one-faceted no.
o two-faceted no.
+ total faceted no.
0.9
0.8
0.7
§ 0.6 ., u ~ 0.5 -:; ",0.4
" .. ..J 0.3
0.2
0.1
0 700
I !
f
900 1100
Faceted length fraction Time (min)
1 10 30 50 70
1---....... --.~------~I~------~--
1300
Temperature (' C)
Figure 4.26. Length fraction offaceted/unfaceted boundaries
62
-- unfaceted length
one-faceted length
~ two-faceted length
- total faceted length
C 14 e
.~ 12 E ,;-en 10 :;.
'" ~ 8 '" "" c::: :::l ... ., Q.
.; Z
6
4
2
0 700
1 l 900
Faceted boundary no. per unit area Time (min)
1 10 30 50 70 , ,
, I ; t
1100 1300 Temperature (. C)
Figure 4 .27. Number of faceted/unfaceted boundaries per unit area
63
- unfaceted no.
- one..faceted no.
__ two-faceted no.
~ lotal faceted no.
4.5
4.0 .
C 3.5 -0 ~
<.> ·e 3.0 -::.. co 2.5 -.. ~ co - 2.0 ~ ·2
" ~ 1.5 -'" Co s: - 1.0 -'" c: .. ..J 0.5 -
0.0 700
.-
900
Faceted boundary length per unit area Time (min)
1100
1 10
•
1300 Temperature ( ' C)
30 50
o
Figure 4.28. Length of faceted/unfaceted boundaries per unit area
70
64
-+- unfaceted length
one-faceted length
.... two-faceted length
-<>- total faceted
65
fraction, which takes care of the difference in porosity at different sintering temperatures.
The length per unit area and number per unit area calculations show simi lar trend. As
grain growth and densification progresses, the number and length of grain boundary
reduce because the reduction in grain boundary area is the driving force for grain growth.
This phenomenon is reflected in all the per unit area parameters.
The relative changes in different faceted boundaries give information on the type
of boundaries playing an important role in grain growth. The length and number per unit
area of faceted boundaries reduce sharply from 1200°C to 1400°C. The drop in total
faceted boundary length per unit area occurs mainly from the decrease of one-faceted
boundaries. The one-faceted boundaries show a drop for two reasons:
I. There is a reduction of boundaries during grain growth and densification.
2. The reduction of two-faceted boundaries is attributed to grain growth. However,
due to the conversion of unfaceted and one-faceted boundaries simultaneously,
the rate of decrease of two-faceted boundaries is delayed.
From the above discussions, it can be concluded that the faceting follows the same trends
as grain growth and densification.
4.3 Misorientation of WC-WC Boundaries
The misorientation between two grains can be described as a combination of axis
and angle. If one of the grains is rotated about a misorientation axis by an angle equal to
the misorientation angle, both of the grains will have the same orientation. There can be
different combinations of angle and axis for same misorientation depending upon the
crystal symmetry. The misorientation is defined by an axis and an angle with the smallest
66
misorientation angle. Figure 4.29 and 4.30 show misorientation angle chart and
67
Misorientation angle chart
0.12 -- Observed distribution
0.10
c: 0.8 0
.... . . . Random distribution
<:: (.)
0.6 «:I ol:: 0 0.4 z .- - _ .. _--- -". - ~ .. - .. - -
0.2
0.0 . - ..
0 10 20 30 40 50 60 70 80 90 100 Misorientation angle (degree)
Figure 4.29. Misorientation angle chart for WC-WC boundaries. The observed
misorientation distribution is compared with random misorientation distribution. Two
peaks can be discerned, at 30° and 90°
0 0 10· 2Qo 3D·
1010 1010 1010 1010
~~ o oeoo I ;:110 QliO I :::110
1010 1010 lai D loio
~~L1Lj 0001 .::1100001 ;:1100001 .::i10 0001 :::110
80· 90·
1010 1010
~~. : 'I,
, "' .i:"-:- , > .. . _ 000 I ::1"10 0001 ::ilo
max= 11.011
7.382
4.949
3.318
2.225
1.492
1.000
0.670
min = -0 .130
68
Figure 4.30. Misorientation distribution function (MODF) for sample sintered at 1400°C
for I minute. The MODF shows a high intensity at 90° @ <I 0 -I 0>
69
misorientation distribution function (MOD F), respectively, for specImen sintered at
1400°C for I minute. All of the specimens showed a similar misorientation angle chart
and MODF. The low angle boundaries indicated from Figure 4.30 might be an artifact
originating from mechanical deformation at surface and other polishing defects [25].
These low-angle boundaries show high intensity in MODF due to nonlinear nature of
misorientation space [26]. For this reason, the low misorientation angle boundaries are
not included in this analysis. The misorientation angle chart has a lower limit of 10° at
the misorientation axis. The grain boundaries belonging to 30° are also excluded from the
analysis because of the small difference in peak value and the corresponding random
misorientation intensity value as discussed in section 2.2.9
The 90° @ <loT 0> boundary is a special boundary. This boundary is similar to
L 2 boundaries [27]. The grain boundary energy shows a rapid decrease for special
boundaries due to its low energy. The grain boundary energy should have a sharp cusp at
90°. Grain boundary energy increases steeply with grain boundary angle deviation from
90°. For this reason, a tight tolerance of 2.5° was chosen in the calculation of volume
fraction of 90° @ <loT 0> boundary. The length fraction and length per unit area for
90° @ < loT 0> boundaries are shown in Figure 4.31 and 4.32. The number fraction and
number per unit area parameters also show the same trend as the length fraction and
length per unit area parameters. The length fraction of 90° @ <loT 0> type boundaries
decreases on increasing the sintering temperature. A gradual decrease in length fraction
can be noted between the specimen sintered at 1200°C for I minute and the specimen
sintered at 1300°C for I minute. Above 1300°C sintering temperature, the length
fractions variation is relatively small. It is difficult to a make conclusion about the
c:: 0 ;:: u '" .... -.c: -'" c:: .. ..J
Length fraction of 90· @ < 1 010> type boundaries Time (min)
70
4.5 1 10 30 50 70
r----------r-...:,:.--..:.;:.--~--...:.; -+- length fraction of
4.0
I 3.5
3.0
t 2.5
2.0
1.5
1.0
0.5
0.0
700 900
I I
I !
1100 1300
Temperature ( ' C)
[ I
90 ·@ <1010>
type boundaries
Figure 4.3 1. Length fraction 0[90° @ <I 0 To> type boundaries
71
Length per unit area 01 90'@ <1 010> type boundaries Timetm1n)
1 10 30 50 70 2.5 - length per unit area
90 ' @<1010> C 0 2.0
i 1
type boundaries ~
" E ::;. • .. 1.5 .. t ~ .. '" I: :::J ~ 1.0 .. • c. f .I: -'" I: 0.5 .!
0.0 700 900 1100 1300
Temperature t ' C)
Figure 4.32. Length per unit area of 90° @ <1 0 0> type boundaries
72
changes in the specimens isothermally held at 1400°C for 10, 30, and 60 minutes. It
seems that the length fraction for isothermal-held specimens remains the same or first
increases and then decreases. The conclusion that it remains the same is consistent with
other parameter variations such as grain growth and densification. The grain boundary
number per unit area and length per unit area decreases during sintering because grain
growth and densification are driven by the reduction in grain boundary area. The drop in
the length fraction of 90° @ <loT 0> type boundaries indicates its preferential
reduction compared to other types of boundaries as the sintering temperature was
increased from 800°C to 1400°C. The length per unit area measurements are shown in
Figure 4.32. A rapid drop in length per unit area can be observed for this type of
boundary between 1200°C and 1300°C, which is because most of the sintering (or
densification) occurs in this temperature range. Here again, it is difficult to make a
conclusion for specimens held isothermally at I 400°C. This is consistent with length
fraction variation.
To summarize, the 90° @ <I 0 1 0> type boundaries decreased in length and
number per unit area as the sintering temperature was increased from 800°C to 1400°C.
The decrease was sharp when the sintering temperature increased from 1200°C to
1400°C. For specimens held isothermally at 1400°C, the length per unit area for 90° @
<loT 0> type boundaries remains almost constant. The drop in 90° @ <I 0 1 0> type
boundaries is more than that of any other type of boundaries at least for sintering
temperatures below 1200°C.
73
4.4 Area Fraction of Tungsten Carbide in Microstructure
The quality of Kikuchi patterns was not good when the electron beam was
focused at a pore since the signals get hindered by pore walls. The OIM program was
unable to index these patterns and did not assign any phase to these data points. The
regions where phases are assigned by the program can be used to calculate the volume
fraction of cemented tungsten carbide. The area fraction of the WC phase in the
microstructure is also called the partition fraction of tungsten carbide. The partition
fraction or area fraction of WC is a term similar to relative density. The area /Taction and
relative density would follow the same trend as pores are eliminated on sintering. Figure
4.33 shows the variation of area fraction with sintering temperature.
4.5 Grain Size and Grain Size Distribution
The OIM technique was also used for determining gram size and gram Slze
distribution of different specImens. Grain size calculations, based on OIM, are more
accurate than conventional microscopy smce orientation information is used in this
technique. The average grain size variation of various specimens is shown in Figure 4.34.
It can be observed from the figure that the average grain size remains almost constant as
the sintering temperature is increased from 800°C to 1200°C. For the specimens sintered
below 1300°C, the average grain size depended upon the area scanned because of the non
uniform nature of the microstructure, whereas for specimens sintered at temperatures
higher than 1300°C, the microstructure became uniform and average grain size variation
was small for different OIM scans of the same specimen. As can be seen /Tom Figure
4.34, there was not much change in the grain growth when the specimens were sintered
1.1
1.0
c 0.9 0 ',"
" .. ~
0.8 -.. .. ~
c(
0.7
0.6
f 0.5
700 900
1
Area fraction of WC·Co in microstructure
Time(min)
1 10 30 50 70
• •
r 1100 1300
Temperature (OC)
74
__ Area fraction
Figure 4 .33. Area fraction of cemented tungsten carbide in microstructure
75
Average grain size Time (min)
1 10 30 50 70 2.5
~ Grain size (avg. area)
C 2.0 0 -- Grain size (avg. no.)
~
.!:! g 1.5 ..
N ·in
" .;6 • I ~ 1.0 '" .. '" to ~ .. 0.5 > «
0.0 700 900 1100 1300
Temperature ('C)
Figure 4.34. Average grain sizes at various sintering temperatures
76
below 1200°C. There was a rapid grain growth when the specimens were sintered in the
temperature range 1200°C to 1400°C for 1 minute. It was found that there was no effect
on grain size of isothermally holding the specimen at 1400°C.
Figure 4.35 shows grain size distribution for all the specimens. The grain size
distribution curves are overlapping for specimens sintered at 800, 1000, and 1200°C. The
smallest grain size in these specimens was found to be 0.2 micron with the majority of
grains having size around 1 micron. The minimum grain size was 0.2 micron because the
step size of aIM scan was 0.1 micron. The minimum grain size is defined as two
consecutive data points having the same orientation. The area fraction for the smallest
grain size is very low indicating fewer of these small grains. The maximum value of grain
size distribution was around 3.5 microns. The grain size distribution curve of specimens
sintered at 1400°C for I, 10, 30, and 60 minutes also overlap, showing size distribution
ranging from 0.5 micron to 6 micron with a peak around 1.2 microns. The specimen
sintered at 1300°C for I minute showed a grain size distribution that was between the
specimens sintered below 1200°C and the specimens sintered at 1400°C for different
times. It is evident from Figure 4.34 that the specimen sintered at 1300°C for 1 minute
showed an average grain size that was between that of specimens sintered 1200°C and
1400°C for I minute.
4.5.1 Comparison of Grain Growth with Existing Models
There are two types of grain growth models reported in the literature. One is
based on the average grain size, whereas the other is on the grain size distribution. The
models based upon the average grain size use the variation of grain size with time at a
constant temperature. Most of the models based on the average grain size follow
0.16
0.14
0.12
a 0.1 '5 .m 0.08
~ 0.06
0.04
0.02
0
0 1
Gai n size distri bLtion
2 3
--<>-- 800°C - 1 !li n -<>- 1000 - 1 !lin -6-1200°C-1 !lin -<>-1300°C-1 !lin -ll- 1400°C - 1 !lin -+- 1400°C -10!lin __ 1400°C-30!lin
-1400°C-60 !lin
4 5
Gain size (!liaon)
Figure 4.35 . Grain size distributions
77
6
78
Equation 2.1 and describe the grain growth with a power law [18], whereas some of the
models [28] predict an exponential grain growth in the presence of coalescence. All
models that are based on the average grain size correlate the grain growth with the time
for isothermal heating conditions. In this study, most of the grain size changes are
observed during the heating stage of sintering (between sintering temperatures of 1200°C
to I 300°C). The grain size data can be fitted to the grain growth equation by changing the
rate constants. This does not make much sense because conditions for experiments are
different from the models.
The models based on the grain size distribution also deal with the isothermal grain
growth. As these models utilize the normalized grain size, the time and temperature
factors do not come into the picture. The comparison of the shape of the grain size
distribution curve with the different models can provide insight into possible grain
growth mechanism or mechanisms.
Figure 4.36 shows the variation of normalized gram sIze distribution with
sintering temperatures. The normalized grain size distribution is around the same for all
of the specimens sintered under different conditions. The comparison with the
coalescence and LSW model is also shown in Figure 4.36. It can be seen from the figure
that the main peak of the grain size distribution curve does not match either of the
models. It could possibly be because of the low resolution of the OIM scans chosen for
this study. A grain is defined in an OIM scan by consecutive points with the same
orientation, so the grain size is an integral multiple of the step size, which is not the case
in real life. This factor is more important for smaller grains. This makes grain sized
distribution a little skewed. If a small enough step size is chosen, the grain size
0.2
0.18
0.16
§ 0.14
0.12
E 0.1
~ 0.08
0.06
0.04
0.02
0
0
fIbTnaIized grain size distributions
1
o 800°C - 1 nin o 1000°C-1 nin is 1200°C-1 nin )( 1300°C-1 nin )I( 1400°C 1 niIL
o 1400°C 10 niIL
I 1400°C 30 niIL -- 1400°C 60 niIL --Coalescen::e rrodel
- LSWrrodei
2
Norrmlized grain size 3
79
4
Figure 4.36. Normalized grain size distributions for all of the specimens. The comparison
with LSW model and coalescence model is shown with bold lines
80
distribution should closely match the models. For larger gram size, the gram size
distribution shows a tail. There is no tail at larger grain size for the LSW model, whereas
an asymptotic tail is present in the coalescence model. Figure 2.1 shows that a
combination of Ostwald ripening and coalescence results in a tail at large grain size. This
indicates that the grain growth is a combination of Ostwald ripening and coalescence
mechanism.
4.6 Proposed Grain Growth Mechanism
The microstructure for unsintered cemented carbides shows that the tungsten
carbide - cobalt powder consists of fine particles as well as multigrain clusters (Figure
4.1). The particles are loosely bonded in a compact. The compact can be considered as
unsintered until the sintering temperature reaches 1200°C. Some necking takes place by
this stage because specimens are easy to polish, though there is not much difference in
microstructure. There is also a possibility of some coalescence inside the clusters. The
length fraction of 90° @ <loT 0> decreases during this stage, i.e. , between 800°C to
1200°C (Figure 4.31). The 90° @ < loT 0> type of boundary is a low energy boundary
as it is similar to the CSL boundary in structure and coherency. This type of boundary is
believed to play an important role in grain growth by coalescence. Most of the grain
growth and densification occur during 1200°C and l300°C sintering temperatures. The
liquid is supposed to form between 1280°C and 1320°C. At these temperatures (1200°C
l300°C), the diffusivity of cobalt is high. Cobalt plays a crucial role in grain growth by
the Ostwald ripening mechanism. At these temperatures, the capillary forces are also
high. Due to grain growth by Ostwald ripening and thermal/mechanical vibrations,
carbide particles rearrange themselves. The capillary forces help in bringing particles
81
together. As the particles have surfaces fonned by low energy planes with preferred
prismatic shapes, the volume fraction of grains having low energy boundary between
them is high. In this way rearrangement supports the grain growth by coalescence. The
area fraction of carbide particles in microstructure becomes 90% by the time the sintering
temperature reaches 1300°C. This indicates that 90% densification occurs by this
temperature (Figure 4.33). On heating untill 1400°C, a similar trend continues with
relatively less grain growth. There is not much grain growth for isothennal holding at
1400°C. For isothennal hold, the grain growth is expected to occur by a diffusion-based
mechanism rather than coalescence because of less rearrangement of particles. For this
reason, the grain growth is relatively low for this stage. After 1400°C sintering
temperature, the specimen is fully dense with faceted grains.
CHAPTERS
CONCLUSIONS
The microstructural evolution based on crystallographic details is described in this
study. The OIM technique brings certainties in analysis as it utilities orientation
infonnation in contrast to SEM. The comparison between grain size distribution with
calculated ones indicates that the grain growth is a combination of Ostwald ripening and
coalescence. By comparison it is difficult to calculate the contribution of each mechanism
to overall grain growth. This can be achieved in calculating rate constant for both of the
mechanisms. A high resolution OIM is also needed to get grain size accurately, especially
at smaller grain size.
APPENDIX
FACETING ANALYSIS CODES
The Matlab™ code for faceting analysis is given below. The subroutines are
given in subsections A. I, A.2 and A.3 .
fid _ w = fopen('C:\Documents and Settings\vineet\My Documents\project\OIM
fi les\micro\800\TSL Scans\800C scan 1 \facetingl O.txt','w');
fid = fopen('C :\Documents and Settings\vineet\My Documents\project\OIM
files\micro\800\TSL Scans\800C scan 1 \reconstructtxt','r') ;
tolerance = 10; % here u change the tolerance
%reading the scan values
for i=1 :7;
tline=fgets(fid);
% count = fprintf(fid_w,'%s',tline);
end
11=0; ml=O; nl =O; count_the_lines = 0; check_flag=O; count! = 0;
unfacetedno = 0; onefacetedno = 0; twofacetedno = 0; unfacetedlength=O;
onefacetedlength=O; twofacetedlength = 0;
count = fprintf(fid w,'the description of the columns \n 1- the boundary no. same as in
reconstrust output file \n 2 and 3 - boundary length and boundary trace \n 4 to 7- the
angle between basal planes and prism planes for grain one grain \n 8-flag for faceting for
ERRATUM
Page 83 was assigned twice in this manuscript.
83
one grain \n 9-12 angle with basal and prism planes for other grain \n 13-flag for faceting
for other grain\n');
count = tprintf(fid _ w,'\nThe flag description: \n O-unfaceted or fake unfaceted \n I or 10-
faceted \n');
count = tprintf(fid_w,'\nThe color coding:\n black- both of the grains show un faceting \n
blue- one of the grain is faceted \n green- both of the grain are faceted\n\n');
n=O' ,
while feof(fid) == 0
n=n+I;
[A, count) = fscanf(fid,'%f%f%f%f%f %f%f%f%f%f%f %f\n',[1,12)); % I-
phil 2- phi 3-phi2 4-phil' 5-phi' 6phi2' 7-length 8-trace 9-xl IO-yl II-x2 12-y2
if count the lines =27
check flag = I
cJc
end
BL=A(1,7);
BT=A(I,8);
%here are the calcuations for the grain on left side of the boundary
%listl I is a function (Iistll.m) that ca1cuates the angle between vector parallel to
boundary and habit planes
C=listll(A(1,I),A(l,2),A( I,3),A(I ,9),A(l,lO),A(1,II) ,A(J,12)) ; % phil phi
phi2 xl yl x2 y2
baseylane = C(I,I); %angle between vector and base plane
sideylane_1 = C(1,2); %angle between vector and prism plane 1
sideylane_2 = C(I ,3); %angle between vector and prism plane 2
sideylane_3 = C(1 ,4); %angle between vector and prism plane 3
planel = 0; plane2 = 0; %setting flag to zero, this flag keep track of faceted or
unfaceted boundary
if((A(l,1 )=0.300)&(A(l ,0.620)==0)&(A(1 ,3)=5.656))
planel = 0;% fake unfaceted
else
if((sideylane_ 1 < tolerance) l(sideylane_2 < tolerance)l(sideylane_3 <
tolerance) l(baseylane <tolerance)) % the range is set to 5 degrees
planel =1; % faceted
else
planel = 0; % unfaceted
end
end
%writing output to the file output.dat
84
%count 1 =fjJrintf( fid _ w, '%f\t%f\t%f\t%f\t%f\t%f\t%f\t%f\t\t' ,count_the Jines,BL,
BT,C,planel);
%Here are the calcuations for grain in the right of the boundary
D= listll (A(l ,4),A(l ,S),A(l ,6),A(l ,9),A(l , I O),A(l , II ),A(l ,12));
% phil' phi' phi2' xl yl x2 y2
base ylane = D(l, 1);
sideylane_ 1 = D(l,2);
sideylane_2 = D(I,3);
side ylane _3 = D(l ,4);
if «A(I ,4)==0.300)&(A(I ,5)==0.620)&(A(l ,6)=5.656))
plane2=0; %fake unfaceted
else
if«side -'plane _I < tolerance) l(sideylane_2 < tolerance)l(side ylane_3 <
tolerance)l(baseylane < tolerance))
plane2 = 10; %faceted
else
plane2 = 0; %unfaceted
end
end
% count= fprintf(fid_w,'%f\t%f\t%f\t%f\t%t\n',D,plane2) ; %writing output to a file
flag_value = planel + plane2;
%plotting the boundaries
xl = [A(1,9), A(l,II)] ;
yl = [100-A(I,IO),100-A(l ,12)];
if«flag_ value ==1 O) I(flag_ value =1)) % one faced BLUE
plot(x I ,yl ,'b','LineWidth', 1.5)
onefacetedno = onefacetedno + I;
onefacetedlength = onefacetedlength + A(l , 7);
elseif«flag_ value == 11)) % both faceted GREEN
85
86
plot(x 1 ,yl ,'g-','LineWidth', 1.5)
twofacetedno = twofacetedno + 1 . ,
twofacetedlength = twofacetedlength + A(l, 7);
elseif(flag_ value ==O)%both of them are unfaceted BLACk
plot(x 1 ,yl ,'k','LineWidth', 1.5)
unfacetedno = unfacetedno + I;
unfacetedlength = unfacetedlength + A(l , 7);
end
hold on
% pausing the program at some particular stage, This put arrows on a boundary you want
to point
%if check_flag == 1
%text(xl ,y 1 ,'\leftarrow')
%A*180/pi
%pause(2)
%break %stitch it on when u wish to stop at the boundary calculation and keep off
when u want all data set to be calculated
%end
%check_flag =0;
end
axis image %setting square axis
hold off
87
totalno = unfacetedno + onefacetedno + twofacetedno' ,
totallength = unfacetedlength + onefacetedlength + twofacetedlength;
unfacetedl engthfraction=unfacetedlengthltotall ength * I 00;
facetedlengthfraction = 100 - unfacetedlengthfraction;
unfacetednofraction=unfacetedno/totalno* 1 00;
facetednofraction = 100 - unfacetednofraction;
count=fjJrintf(fid w,'\n\n\n\nunfaceted no = \t%f\n unfaceted length = \t%f\n onefaceted
no = \t%f\n onefaceted length = \t%f\n twofaceted no = \t%f\n two faceted length =
\t%f\n total no = \t%f\n total length = \t%f\n U have a nice
day\n\n',unfacetedno,unfacetedlength,onefacetedno,onefacetedlength,twofacetedno,twofa
cetedlength,totalno,totallength);
count=fjJrintf(fid_w,'\n\n\nI tell u the fractions \n unfaceted length = \t%f\t no.- %r,
unfacetedlengthfraction, unfacetednofraction);
count=fjJrintf(fid_w,'\n faceted length = \t%f\t no.- \t%r, facetedlengthfraction,
facetednofraction);
status=fclose(fid); %close the file for reading
status=fclose(fid w); % close the file for writing
A.I Code for Replacing Low Quality
Data With Fake Orientation
fid_ w = fopen('C:\Documents and Settings\vineet\My Documents\project\OIM
files\micro\800\TSL Scans\800C scan I \scan 1 imp.ang','w');
fid _r = fopen(,C:\Documents and Settings\vineet\My Documents\project\OIM
files\micro\800\TSL Scans\800C scan 1 \export.ang','r');
for i=1 :53;
tline=fgets(fid J);
count = fprintf(fid _ w,'%s',tline);
end
n=O;
while feof(fidJ) = 0
%n=n+1
[data yoint, count] = fscanf(fid _T,'%f\t%f\t%f\t%f\t%f\t%f\t%f\t%f\t%f\t%f
\n',[ 1,10]);
data yoint(1 ,7)
count
if data yoint(1, 7) <0. 1
count=fprintf(fid w,' %f %f %f %f %f %f %f %f %f
88
%f\n',0.3,0.62,.42,datayoint(1 ,4),data yoint(1 ,S),data yoint(! ,6),0, 1 ,data yoint(1,9),dat
a yoint(1, 1 0»;
else
count=fprintf(fid_w,' %f %f %f %f %f %f %f %f %f
%f\n',data yoint(1 , 1 ),data yoint( 1 ,2),data yoint(! ,3 ),data yoint(1 ,4),data yoint(! ,5),data
yoint(1 ,6),data yoint(1, 7),data yoint(1 ,8),data yoint(l ,9),data yoint(l , 1 0»;
end
end
status=fclose(fid r) ; %close the file for reading
status=fclose(fid _ w); % close the file for writing
A.2 Function for Calculating Angle
Between Two Vectors
function [dangleres) = anglell(dxl , dyl, dzl , dx2, dy2, dz2)
% this function calculates the angle between two directions.
if((dxl *dxl +dyl *dyl +dzl *dz l )*(dx2*dx2 +dy2*dy2 + dz2*dz2)-=0)
dangleres = acos((dxl *dx2 + dyl *dy2 + dzl *dz2)/sqrt((dx l *dxl +dyl *dyl
+dz l *dzl)*(dx2*dx2 +dy2*dy2 + dz2*dz2)));
end
A.3 Function for Calculating Angle Between
Grain Vector and Low Energy Planes
function [ solution_matrix) = li stll (Phi I , phi, phi2, x I ,yl ,x2,y2)
% this function calculate the angle between the vector and all habit planes.
% g = orientation matrix
g= [cos(Phi I )*cos(phi2)-sin(phi I )*sin(phi2)*cos(phi),
sin(phi 1 )*cos(phi2)+cos(phi I )*sin(phi2)*cos(phi) , sin(phi2)*sin(phi); -
cos(phi I )*sin(phi2)-sin(phi I )*cos(phi2)*cos(phi), -
sin(phi I )*sin(phi2)+cos(phi I) *cos(phi2)*cos(phi ), cos(Phi2)* sin(phi);
sin(phi I )*sin(phi), -cos(phi l)*sin(phi), cos(Phi»);
gphi I =[ cos(phi I ),sin(phi I ),O;-sin(phi 1 ),cos(Phi I ),0;0,0, I);
gphi = [I ,0,0;0,cos(Phi),sin(phi);0,-sin(phi),cos(phi»);
gphi2 = [cos(Phi2),sin(phi2),0;-sin(phi2) ,cos(phi2) ,0;0,0, I) ;
g2=gphi2 * gphi * gphi I ;
%n = [x2-xl; y2-yl ;0);
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n = [y2-yl ; x2-xl ; 0]; %sample direction parallel to boundary
% the x and y values are interchanged to get a match with software calculations
vector = g2*n; % crystal direction parallel to the boundary
% n,g2, vector %display of calculation
%anglell is a function that calculates the angle between two vectors
pi =abs(180/pi*(pi/2 - anglell(vector(l , I),vector(2,1),vector(3,1), 0, 0, I))); %angle
between base plane and vector in degrees
p2 =abs(180/pi*(pil2 - anglell(vector(l,I),vector(2,1),vector(3,1), 1.5, 0.866025, 0)));
%angle between prism plane I and vector in degrees
p3= abs(l80/pi*(pil2 - angle II (vector(l, I) , vector(2, I ),vector(3, I), 0, 1.732051 , 0)));
%angle between prism plane 2 and vector in degrees
p4 = abs(l80/pi*(pi/2 - anglell(vector(l,I) ,vector(2,1),vector(3,1), -1.5,0.866025,0)));
%angle between prism plane 3 and vector in degrees
solution matrix = [pl ,p2,p3 ,p4] ; %retum the values
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