METRIC AND TOPOLOGICAL
CHARACTERIZATION OF THE ADVANCED
STAGES OF SINTERING
ARUNKUMAR SHAMRAO WATWE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
Arunkumar Shamrao Watwe
To My Parents,
Mr. Shamrao Vasudeo Watwe
Mrs. Sharada Shamrao Watwe
I am grateful for the opportunity to conduct my research under
the guidance of Dr. R. T. DeHoff, the chairman of my advisory committee.
An ability to approach any scientific matter with objectivity and logic
has been blissfully passed on by him to all his students.
I thank Drs. R. E. Reed-Hill, J. J. Hren, G. Y. Onoda, Jr.,of the
Department of Materials Science and Engineering and Dr. R. L. Scheaffer
of the Department of Statistics for serving on my advisory committee.
Their helpful advice and encouragement are deeply appreciated.
It is a pleasure to thank my colleagues, Mr. Atul B. Gokhale and
Mr. Shi Shya Chang, for their collaboration in the experimental aspects
of the project.
Mr. Rudy Strohschein, Jr., of the Department of Chemistry assisted
me beyond and above the call of duty in the fabrication of the sintering
apparatus. He saved me a great deal of time and aggravation.
All the credit for the preparation of this dissertation in its final
form must go to Miss Debbie Perrine for her excellent typing.
The financial support of the Center of Excellence of the State of
Florida and the Army Research Office is gratefully acknowledged.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS........................................ ......... .. iv
ABSTRACT....................... ........... ...................vii
ONE EVOLUTION OF MICROSTRUCTURE DURING SINTERING............. 5
Introduction ........................................... 5
Metric Properties of the Microstructure.................. 5
Fundamentals of Topology................................. 6
Sintering from a Geometric Viewpoint..................... 12
Importance of the Present Research...................... 35
TWO EXPERIMENTAL PROCEDURE AND RESULTS...................... 39
Loose Stack Sintering................................ ..101
Comparison of Loose Stack Sintering with Hot Pressing
and Conventional Sintering...............................135
FOUR CONCLUSIONS ............................................137
Introduction............................ .............. 137
Suggestions for Further Study...........................139
APPENDIX THE GEOMETRIC MODEL OF THE PORE PHASE....................143
Parameters of the Model .................................143
Metric Properties of the Connected Porosity..............149
Surface Corrections................................ 155
BIOGRAPHICAL SKETCH.............. ............... ............. .. 163
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
METRIC AND TOPOLOGICAL
CHARACTERIZATION OF THE ADVANCED
STAGES OF SINTERING
Arunkumar Shamrao Watwe-
Chairman: Dr. R. T. DeHoff
Major Department: Materials Science and Engineering
Measurements of the metric properties of porosity and the grain
boundary network during the advanced stages of loose stack sintering,
conventional sintering and hot pressing of spherical nickel powder
(average size 5.5 microns) were followed by topological analysis of
the loose stack sintered samples.
Linearity between area and volume of the pore phase for the loose
stack sintered series was approached by the conventionally sintered and
hot pressed series whereas the curvature values for these series remained
An arrest in grain growth during loose stack sintering was concurrent
with the removal of most of the isolated porosity. Subsequent resumption
of grain growth coincided with the stabilization of connected porosity.
It is suggested that isolated, equiaxed pores pin the boundaries more
effectively than do the connected pores. Increase in the boundary area
accompanies the boundary migration for all orientations of an equiaxed
pore whereas this is true only for a limited number of orientations of
a connected pore. Consequently, isolated pores are removed via transport
of vacancies to the occupied boundaries; subsequent resumption of grain
growth slows the reduction of residual connected porosity. Porosity in
loose stack sintered samples is modeled as a set of tubular networks and
a collection of monodispersed spheres. Comparison of metric properties
of loose stack sintered samples with those of conventionally sintered
and hot pressed samples led to the speculations that a higher number of
isolated pores exist during hot pressing and that the porosity in con-
ventionally sintered samples is composed of finer networks and smaller
Absence of an arrest in grain growth during hot pressing is believed
to be due to boundary migration that is induced by grain boundary sliding.
A similar absence of an arrest in grain growth during conventional sin-
tering is attributed to the onset of grain growth well before that of
Sintering is a coalescence of powder particles into a massive form,1
wherein the densification is accompanied by a variety of profound geo-
metrical changes in the pore-solid composite. The mechanical and physical
properties of a powder-processed compact are influenced by the geometry
of the pore phase.2-9 Thus, the manner in which the reduction of porosity
takes place is of great practical and theoretical interest.
There are two approaches to the study of sintering.1 The traditional
or mechanistic approach involves the study of kinetics and mechanisms of
material transport; the geometric viewpoint focuses on the geometry of the
pore phase as it evolves during sintering. The latter approach involves
estimation of size- and shape-dependent quantities (volume, area, etc.)
and topological properties such as the connectivity and the number of
Mechanistic studies essentially consist of three steps:
1) A laboratory model of a particulate system is selected that is
amenable to mathematical treatment of desired sophistication.
Assumptions are made regarding the geometrical changes during
sintering and the identities of source and sink of matter.
2) Kinetic equations are derived that describe the variation of a
measurable parameter (width of an interparticle contact or "neck,"
density, etc.) for the particular mechanisms) of interest.
3) These equations are compared with the experimentally observed
time dependence of the chosen parameter and an attempt is made
to identify the operating mechanismss.
The details of the geometry determine
1) the initial and boundary conditions for the flow equations,
2) the areas through which the material fluxes are assumed to
3) the separation between sources and sinks,
4) the relationship between the variation of the chosen parameter
and densification, and
5) the state of stress (important in plastic and viscous flow).
Hence the time dependence derived in the model studies are influenced
by the geometric details. Different mechanisms exhibit different
variations with temperature; thus, the relative importance of various
mechanisms should depend on temperature and the chemical composition
of the powder, as observed in several investigations.1316 As pointed
out by DeHoff et al.,11 sintering requires that densification, surface
rounding, channel closure and removal of pores proceed in cooperation.
Since all these involve different geometric events, the time exponent
n in the relation x, the monitored parameter = (t)n varies with the
Any mechanistic arguments must ultimately explain the observed
geometrical changes taking place during sintering. It is thus evident
that study of the changing geometry or microstructure should precede
mechanistic investigations. Knowledge of the dependence of this micro-
structure on various process parameters such as initial powder character-
istics, temperature and external pressure would be very helpful in the
control of sintering aimed at desired end properties of the components.
Consequently, a major school of thought prevails that favors the geometric
approach. The present investigation was undertaken to study the advanced -
stages of sintering (wherein the porosity values vary from ten to a few
percent of the total volume) from this point of view.
A main feature of this approach is the concept of path of evolution
of microstructure. A given microstructure is characterized by its geo-
metric properties such as volume, area, curvature, connectivity, etc. A
microstructural state is defined as a point in a n-dimensional space where
each dimension denotes a particular microstructural property. As the
microstructure evolves during a process, the resultant locus of such
points represents the sequence of microstructural states that is obtained
during the process. This sequence is termed the path of evolution of
microstructure.17 It is convenient to represent two-dimensional projec-
tions of this path (two geometric properties at a time); usually one of
them is the relative density or the volume fraction of solid.
Previous studies of microstructural evolution during loose stack
sintering,6,12,18-21 conventional sintering-cold pressing followed by
sintering6,2223 and hot pressing6,21-23 have provided a coherent pic-
ture of these processes for all but the last ten percent of the porosity.
A detailed study of the late stages (porosity ten percent or lower) of
loose stack sintering, conventional sintering and hot pressing would
complete the picture of evolution of microstructure during these pro-
cesses. The practical interest in the behavior of porosity at these
stages stems from the fact that a variety of commercial products made
by powder technology are required to have porosities in the range 0.1
The objectives of this research were to determine the paths of
evolution of microstructure during the advanced stages of loose stack
sintering, conventional sintering and hot pressing. Since the topologi-
cal measurements are time-consuming and since an earlier doctoral research25
dealt with topological characterization of loose stack sintering in the
porosity range 0.1 and higher, it was planned to estimate the topological
parameters for loose stack sintered series only. Metric properties of
the pore structure and grain boundary network were estimated for all three
series of samples.
Previous investigations of this type are discussed in Chapter 1,
followed by experimental procedure and results in Chapter 2. These
results are discussed in Chapter 3 and the important findings and con-
clusions summarized in Chapter 4.
EVOLUTION OF MICROSTRUCTURE DURING SINTERING
A microstructure is characterized by its metric and topological
properties and therefore the following discussion will be carried out in
terms of variation of these quantities as the sintering proceeds. These
microstructural properties will be defined and the previous investigations
of this type will be discussed in detail; a review of metric studies will
be followed by topological analyses. The principles of quantitative
stereology employed in the estimation of microstructural properties will
be described in the next chapter on experimental procedure and results.
Metric Properties of the Microstructure
These quantities are estimated in terms of geometric properties of
lines, surfaces and regions of space averaged over the whole structure.2
The basic properties are listed in Table 1 and illustrated in Figure 1.
Among the properties listed, VV, SV and MV are used to yield two impor-
tant global averages of the microstructural properties. These are listed
in Table 2.
In a sintered structure, there are two regions of space or phases,
namely, pore and solid, and two surfaces, pore-solid interface and grain
boundaries. Two main linear features of interest are the grain edges and
the lines formed as a result of intersection of pore-solid interface and
grain boundaries. Superscripts are used to identify the properties that
are associated with a particular feature. These notations are listed
in Table 3 and illustrated in Figure 2.
In addition to the metric properties listed above, the microstruc-
ture of a porous body is also characterized by its topological properties.
A brief discussion of the fundamentals of topology will precede the sur-
vey of microstructural studies of sintering.
Fundametals of Topology
The subset of topological geometry of present interest is that of
closed surfaces,24 that is to say, surfaces that may enclose a region of
space. In a sintered body the regions of space are the pore and the
solid phases; the pore-solid interface is a closed surface of interest.
Such a surface may enclose several regions and have multiple connectivity.
A surface is said to be multiply connected if there exist one or more
redundant connections that can be severed without separating the surface
in two. The genus of such a surface is defined as the number of redundant
connections. For complex geometries it becomes difficult to visualize the
topological aspects of surfaces. It has been found very convenient :to
represent surfaces by equivalent networks of nodes and branches. Such an
equivalent network is called the deformation retract of a particular region
of space. It is obtained by shrinking the surface without closing any
openings or creating new openings,27 until it collapses into the said
network that can be represented in the form of a simple line drawing.28
A number of closed surfaces and their equivalent networks are illustrated
in Figure 3. The connectivity, P, of a network is equal to the number of
nearr Features of Length L
L = L
Sectioning Plane of
Lines of Intersection
of Length L
Element of Surface
H ( + )
2 r1 r2
Figure 1. Illustration of basic metric properties.
BASIC METRIC PROPERTIES
LV Length of a linear feature per unit
S,, Area of a surface per unit volume.
1 1 1-
2 rl r2
MV =f/ HdS
Local mean curvature of a surface
ata point on the surface, where
rl and r2 are the principal radii
of curvature. By convention, a
radius of curvature is positive
if it points into a solid phase.
Thus, a convex solid has a positive
curvature whereas a convex pore has
a negative curvature.
Integral mean curvature of
per unit volume.
Length of a trace of surface per
unit area of a plane section.
Volume fraction or volume of a
particular region of space per
Regions of Space
DERIVED METRIC PROPERTIES
Feature Property Definition
Surface H = Average mean curvature of
SV a surface
4V Mean intercept in a particular
Region of Space of space
Sv region of space
METRIC PROPERTIES OF A SINTERED STRUCTURE
Grain Edges in the
Lines Formed by the
Intersection of Grain
Boundaries and Pore-Solid
Grain Edges Occupied
by the Pore Phase
Area of pore-solid interface
per unit volume
Integral mean curvature of
pore-solid interface per unit
Volume fraction of porosity
Volume fraction of a solid
Length of grain edges or
triple lines per unit volume
Length of intersection lines
of pore-solid interface and
grain boundaries per unit
Length of occupied grain
edges per unit volume
Grain Boundaries of Area SSS
LSSS L(occ) LSS
(occ) V V
Figure 2. Illustration of metric properties characterizing grain
boundaries, pore-solid interface and their association.
branches that can be cut without creating a new isolated part. If
b = number of branches, n = number of nodes, Po = number of separate
P1 = b n + P (1)
The first Betti number of the network, P1'29 is equal to the genus
of the surface it represents.
It may be apparent from Figure 3 that there exists some ambiguity
as to the number of nodes and branches in a deformation retract. As
illustrated in Figure 4, a number of additional nodes and branches
can be used to represent the same region of space. Such spurious
branches and nodes do not change the value of the connectivity because
each spurious node introduces one and only on spurious branch.
Quantities such as connectivity and number of separate parts or
subnetworks are estimated by examining a series of parallel polished
sections that cover a finite volume of sample, as described in Chapter
2. The investigations dealing with the study of sintering from the
geometric viewpoint will be discussed presently.
Sintering from a Geometric Viewpoint
Three Stages of Sintering
Rhines30 and Schwarzkopf31 were among the first investigators to
point out three more or less geometrically distinct stages that a sin-
tering structure traverses.
The first stage is characterized by formation of initial inter-
particle contacts and their growth until these contact regions or necks
begin to impinge each other, as illustrated in Figure 5. Due to differ-
ent crystallographic: orientations of adjacent particles, grain boundaries
form in the interparticle contact regions. In this stage, the area of
pore-solid interface decreases with a moderate amount of shrinkage.32
Throughout this stage, the pore-solid interface has many redundant con-
During the second stage, the distinguishing features are not
the interparticle contacts or "necks" but the pore channels formed as
a result of the impingement of neighboring necks. Virtually all of the
porosity is in the form of an interconnected network of channels16'33
that delineate the solid grain edges. The continued reduction in the
volume and the area of porosity is accompanied by a decrease in the
connectivity of the pore structure.1,35 The decrease in the connec-
tivity can be explained by either removal of solid branches or closure
of pore channels. According to Rhines,36 the connected pore network
coarsens, analogous to a grain edge network in a single phase polycrystal
(driven by excess surface energy) as illustrated in Figure 6. In this
scenario, a fraction of solid branches (necks) are pinched off and no new
pores are isolated. Although a finite number of isolated pores observed
during the late second stage35 can be explained only by channel closure
events, a closer scrutiny is needed to resolve this issue. The isolated
pores may be irregular in shape.1634
The third stage has begun by the time most of the pores are isolated.30'31
The connectivity of a pore network is now a very small number. Coarsening
proceeds along with the spheroidization of pores16'18'3537-39 so that the
volume of porosity, the number of pores and pore-solid interface area
Figure 3. Some closed surfaces and their deformation retracts
A Closed Surface
Figure 4. Illustration of a one-to-one correspondence between a
spurious node and a.spurious branch in a deformation
Figure 5. Illustration of neck growth and impingement of growing
necks during the first stage of sintering.
continue to decrease. If the pores are filled with a gas of low solu-
bility or very slow diffusivity, then coarsening leads to an increase
in volume of porosity.16,40 If this gas has enough pressure to stabilize
the pore-solid interface, the densification rates can be very low.16,41,42
Since exaggerated or secondary grain growth that results from boundaries
breaking away from pores43,44 has been observed to be accompanied by slow
rates of shrinkage,43'45-47 it has been theorized that the grain boundaries
that can act as efficient vacancy sinks are far away from a large number of
pores.43'45-47 The end of the third stage is of course the disappearance
of all pores, although that is rarely accomplished in practice.
The three stages described above provide a common framework for the
discussion of microstructural studies that are reviewed presently. This
review is expected to demonstrate the potential that the present research
has for providing a perspective of sintering that is more profound than
the current one.
It has been observed that in loose stack sintered samples SP
decreased linearly with the decrease in V(1218-20) during the second
stage. Surface area may be reduced both by densification and surface
rounding or by surface rounding alone; the linearity between SP and VV
is believed to arise from a balance between surface rounding and densifi-
cation. Support for this hypothesis comes from the observation that sur-
face rounding dominates in pressed and sintered samples until the balance
has been reached,35 as shown schematically in Figure 7. The slope of the
SSP versus VV line is inversely proportional to the initial particle size.20
Figure 6. Two basic topological events that occur in the network
coarsening scenario proposed by Rhines.36 The dotted
lines indicate the occupied grain edges.
There is evidence to suggest that this path of evolution of microstruc-
ture for loose stack sintering is insensitive to temperature.21
Data for hot pressed samples indicate that the SV -Vv relationship
is only approximately linear even in the late second stage.2223 The
path of microstructural change was also found to be insensitive to tem-
perature.22'23 The effect of pressure on the path was significant;
increasing pressure delayed the approach to linearity until a lower
value of VV, as shown in Figure 8.
Integral mean curvature per unit volume, MV, has been measured for
loose stack sintering, conventional sintering (cold pressing followed by
sintering) and hot pressing in the density range characteristic of late
second stage. A convex particle has a positive curvature whereas a con-
vex pore has a negative curvature. There is a minimum in'MV;182122
this minimum occurs at lower VV for finer particle size,3 as illustrated
in Figure 9. According to the convention used, most of the "SP" surface
has positive curvature in the initial stages. Due to decreasing surface
area and increasing negative curvatures there occurs a minimum in MV in
the second stage. As the sintered density approaches the theoretical
density, MV must approach zero and hence the initially high positive
value of MV that becomes negative must go through a minimum. For an
initial stack of irregularly shaped particles, MV varies with VP at a
slower rate and has a minimum earlier in the process, compared to an
initial stack of spherical powders.25 This is illustrated in Figure 10.
In all the cases studied the paths were insensitive to temperature. In
the case of hot pressing, the minimum in MV is much more negative and
occurs at a lower value of VV, compared to a loose stack sintered sample;
Schematic representation of the variation of surface area
with solid volume for loose stack sintering and conventional
sintering. The approach to the linear relation from a range
of initial conditions is emphasized.35
Pore-solid interface area versus
conventionally sintered U0222
solid volume fraction for
0 0 KSI
S 1-8 KSI
- A 10-20 KSI
~ \ 30-40 KSI
O 75-90 KSI
I I I I I \
MSP (105 cm2)
MV (1 cm )
A (-170+200) Spherical Cu
O (-200+230) Spherical Cu
D (-270+325) Spherical Cu
0.8 0.9 1.0
Variation of integral mean curvature per unit volume with
the volume fraction of solid for three representative copper
powders sintered in dry hydrogen at 10050C.35
MP (105 cm2)
- A Dendritic
I I I I f
0.2 0.5 VV 1.0
Spherical ---------- --- I -IIIi-
Dendritic I II III
Integral mean curvature versus volume fraction of
solid for 48 micron spherical and dendritic copper
these curves become deeper and shift towards lower VV with increasing
pressure,22 as shown in Figures 11 and 12.
The.grain boundary area per unit volume, SV increases until a
network is formed; subsequent grain growth tends to decrease SS.
This was observed for loose stack sintering,8 as shown in Figure 13.
It is evident here that the variation of SS with V is independent of
the initial particle shape in the late second and early third stages.
LSSP increases with decrease in V until the second stage is reached22
when it begins to decrease. In the second stage :L- is significantly
higher than the case for random intersection of "SP" and "SS" surfaces,48
as illustrated in Figure 14.
A new metric property, IA, was discovered in the course of doctoral
research carried out by Gehl22 at the University of Florida; IA is the
measure of inflection points observed on the traces of a surface per unit
area of plane of polish, and is proportional to the integral curvature of
asymptotic lines over saddle surfaces (surfaces that have principal radii
of curvature of opposite signs at all points on the surface). This was
found to decrease smoothly in the second stage22 which means that the
saddle surfaces occupy only a small fraction of the pore-solid interface
at the end of the second stage.
The variation of grain contiguity, grain face contiguity and grain
shapes during conventional sintering and hot pressing were studied in
some detail by Gehl.22 There were two parameters, CS and CSS, defined
for grain contiguity and grain face contiguity, respectively. Four
unitless parameters, Fl, F2, F3 and F4, were used to characterize grain
and pore shapes. These were defined as follows.22
0.80 0.85 0.90 0.95
Variation of integral mean curvature per unit volume with
the volume fraction of solid during hot pressing of RSR 107
nickel (-170+200) at 1500 psi. Data for spherical copper
(-170+200) loose stack sintered at 10050C included for
The effect of pressure on the path of integral mean
curvature for hot pressed specimens of U02O22
S SP (cm- 1
- I- I
Grain boundary area per unit volume versus volume fraction
of solid for 48 micron spherical and dendritic copper
C = S (2)
2SV +S V
Css = 3L~_ss
SSS SSP (3)
F1 S= P SS2 (4)
(Sv +2SV )
F2 = 2LSSP/(SSP)2 (5)
F3 SS (6)
SSSP SS)2 (7)
F4 = LVP/2(SV)2 (7)
The fraction of the total area of solid grains shared with other
grains is given by CS. Variation of CS with VP for hot pressed samples
of U2(22) and loose stack sintered spherical and dendritic copper
powder25 is shown in Figure 15. It can be seen from the definition of
S S SS
C that high C values indicate high SV ; this was believed to arise from
polycrystallinity of the particles. It is apparent that as third stage
(VVP 0.1) is approached all data tend to fall on a single curve. Pre-
compaction seems to increase SS and hence exhibits higher values of C
LSP (104 cm-2)
Variation of the length of lines of intersection of grain
boundaries and the pore-solid interface (LSSP) with the
corresponding value for the random intersection of the
abovementioned surfaces (L ) for spherical copper powder
loose stack sintered at 10l5C.48
The variation of grain contiguity with solid volume
fraction for loose stack sintered copper and hot pressed
at the same VP when compared to a loose stack sintered sample; SV
was found to vary linearly with CS and the dependence was the same
for widely different precompaction pressures up to very late second
stage. For hot pressed samples, a maximum was observed in CS, believed
to indicate a point where the grain boundary area has increased enough
to form a boundary network that subsequently coarsens. Both LSP and
LSSS exhibited a maximum when plotted versus CS for conventionally
sintered and hot pressed samples.
The grain face contiguity parameter, CSS, indicates the fraction
of edge length of grain faces that is shared with other grains. It
can be shown22 that CS = CSS for the case of random intersection of
grain boundaries and pore-solid interface, and CS > CSS when grain
boundaries intersect pore-solid interface preferentially. For conven-
tionally sintered and hot pressed samples CS was observed to be greater
than CSS which indicated preferential association of grain boundaries
with the pore-solid interface.
The factors FI, F3 and F4 can be used to compare the grain shapes
and F2 the pore shapes; Fl, F2, F3 and F4 were observed to be weakly
linear with VV, whereas a strong correlation was observed between F2,
F3 and F4 and CS for all the samples. It was believed that the above
data indicate a strong influence of the extent of grain contiguity (CS)
on the grain and pore shapes.
It was found12 that the connectivity or genus, G, stays
nearly constant during the first stage. More precise measurements
made by Aigeltinger and DeHoff18 indicated a definite increase in G
during the first stage. This can be viewed as formation of additional
interparticle contacts as particles come closer by densification. For
irregularly shaped powders, it was observed18 that G decreases during
the first stage, due to coalescence of multiple contacts between particles.
During the second stage, S decreased linearly with decrease in VV;
the slope of this line was found to be proportional to GV, genus per unit
volume20 as should be expected from dimensional analysis. Kronsbein et
al.49 carried out serial sectioning of sintered copper samples and found
that even for VV = 0.1, very few pores were isolated. This is in agree-
ment with similar observations made by Barrett and Yust.34
Aigeltinger and DeHoff18 studied loose stacking sintering of copper
powder by measuring metric and topological properties. The genus per
unit mass, Gp, number of isolated pores and number of contacts per parti-
cle, C, were the measured topological quantities. Variation of Gp and
number of isolated pores with V revealed a definite increase in the
former during the first stage and identified the end of the second stage
(Gp = 0). As shown in Figure 16, Gp and number of isolated pores were
inversely proportional to the initial value of mean particle volume.
The same plot for dendritic powder showed that the topological path is
different up to late third stage and that the third stage (Gp 0)
begins at a higher value of VV as compared to spherical powder, Figure
17. The initial decrease in Gp during the first stage for dendritic
Gp or Np
(106 gm )
0 Gp (48 micron)
O Gp (115 micron) x 13.7
* Np (48 micron)
* Np (115 micron) x 13.7
0.6 VV 1 0
I -- II II
Variation of genus per unit mass (Gp) and the number of
isolated pores per unit mass (Np) with the volume fraction
of solid for 48 micron spherical copper powder loose stack
sintered at 10050C. Data for 115 micron spherical copper
included for comparisonn.5
Gp or Np
Spherical - ---
II "---* II
a) Genus per gram (Gp) the number of isolated pores per
gram (Gp) versus volume fraction of solid for 48 micron
dendritic copper powder. Data for 48 micron spherical
powder included for comparison. b) Enlarged part of
lower right corner of (a).25
powder is in agreement with higher C = 14 for the initial stack than
C = 4 at VV = 0.55. It has been argued that in second stage, on account
of fewer pore channels in the sample sintered from dendritic powder,
isolation of pores begins at a higher VV value than for the sample made
from spherical powder. The maximum in the number of separate parts
observed during the third stage was attributed to simultaneous shrinkage
and coarsening. Initiation of rapid grain growth coincided with the
approach of connectivity towards zero.
Importance of the Present Research
Microstructural characterizations of the last stages of sintering
where VV goes from about 0.1 to nearly zero have been sketchy. The
reasons for such a lack of data are evidently
1) For an aggregate of coarse powder particles that is convenient
for serial sectioning, very long sintering times are required
to obtain samples with such low values of porosity.
2) For a given range of densities, the paths of evolution of
microstructure can be determined with a higher degree of
confidence if a larger number of distinct microstructural
states can be obtained and examined. Thus, it is desirable
to have a sufficient number of samples that have the densities
in the range VV = 0.1 and lower; this requires that the samples
in the series have VP values that are only a percent or so apart
from each other. Due to this requirement and that of long
sintering times, much preliminary experimental work is necessary
to establish the required sintering schedules.
3) The topological measurements are very tedious in any case.
The present investigation that dealt with the microstructural
characterization of the advanced stages of sintering has a potential
for enhancing and quantifying the existing sketchy picture of the late
stages of sintering. The theoretical and practical importance of this
work can be appreciated from the following discussion.
It has been theorized16'18'35'37-39 that the spheroidization of
pores proceeds along with coarsening during the advanced stages. It
is necessary to couple topological analysis with the metric measurements
to study the spheroidization and coarsening of isolated pores. To date,
there has been no such direct observation of the behavior of isolated
porosity. If a pore of higher than average size is surrounded by a shell
of higher than average density with finer pore channels, then early clo-
sure of these channels pulls the solid shell away from the large pore so
that the continuity of the solid phase is maintained,34 as illustrated in
Figure 18. According to Barrett and Yust,34 most of the reports of
coarsening are in fact the observed removal of smaller channels before
the larger ones. Another disputed contention is that of deceleration of
densification due to separation of grain boundaries from isolated
pores.43,45-47 A pore that is observed to be isolated on a two dimen-
sional section may or may not be so in the third dimension, whether
associated with the grain boundaries or not. The topological analysis
of grain boundary-porosity association alone can determine the true
extent of association of isolated porosity with the boundaries. A
detailed geometric study of porosity in the advanced stages will clarify
some aspects of microstructural evolution mentioned above.
Illustration of coarsening of a relatively large pore channel that results
from an early closure of surrounding finer channels so that the solid conti-
nuity is maintained.34
Mechanical and physical properties of commercial porous components
are influenced by the geometry of the porosity. Thermal conductivity is
influenced by VV, pore shapes and the relative fractions of connected and
isolated porosity.7'8 Permeability to fluids depends on the connectivity,
VV and S P.9 Mechanical strength and thermal shock resistance3 depend on
pore shapes whereas ductility is influenced by pore shapes and spacings.2
Thus geometric characterization of porous structures as a function of
adjustable process parameters would suggest a number of potential strate-
gies to control the final service properties.
It is apparent from the review of previous microstructural studies
of sintering that the present investigation is expected to offer a much
needed general and quantitative picture of the advanced stages of sin-
tering. The experimental procedure employed in the present research is
described in detail in the next chapter.
EXPERIMENTAL PROCEDURE AND RESULTS
Microstructural characterization involved sample preparation,
metallography and in the case of loose stack sintering, also serial
sectioning. These are described in detail in this chapter, followed
by results of this investigation.
This section presents the procedure employed to prepare the
sintered samples and the standard for density measurements.
Three series of samples of sintered nickel powder were prepared:
1) loose stack sintered (LS), 2) pressed and sintered (PS) and 3) hot
pressed (HP). In order to study the path of evolution of microstructure
during the late stages of sintering, it is desirable to obtain samples
having densities that are uniformly distributed over the range VV = 0.85
to 1.0. Accordingly, preliminary experiments were designed to determine
the processing parameters, such as temperature, pressure and time, that
yield the desired series of samples made from the selected metal powder.
INCO type 123 nickel powder, illustrated in Figure 19, supplied by the
International Nickel Company, Inc., with the chemical and physical prop-
erties listed in Tables 4 and 5, was used in the present investigation.
Figure 19. INCO 123 nickel powder used in the present investigation
CHEMICAL COMPOSITION OF INCO
TYPE 123 NICKEL POWDER
Nickel Powder (Wt.%)
PHYSICAL PROPERTIES OF INCO TYPE 123 NICKEL POWDER
Particle Shape Roughly spherical with spiky surface
Average Particle Siz 5.5 microns
Standard Deviation 0.75 microns
Surface Area Per Unit Volume 7.65 x 103 cm/cm
VS of As-Received Powder 0.25
It. was found by trial and error that sintering a loose stack of this
powder at 1250C produced the required series of samples in convenient
This sintering temperature was also used for PS and HP series, in
order to ensure that the differences among the paths of evolution of
microstructure for LS, PS and HP series were not due to different sin-
1. Loose Stack Sintered Series
In order to have the same initial microstructure for all the
samples in a series, they were prepared from the same initial loose
stack of powder. The first sample of the series was prepared by
heating a loose stack of powder (tapped to yield a level top surface)
in an alumina boat (6 x 12 x 75 mm) under a flowing dry hydrogen
atmosphere for the specified sintering time. A small piece (about
5 mm thick) was cut off and stored for subsequent characterization;
the rest of the sintered body was used to yield the remaining samples
in the series by the repetition of the procedure described above for
an appropriate sequence of accumulated sintering times. It required
11 minutes for the sample transferred from the cold zone to the hot
zone to reach the sintering temperature. Although this time was not
negligible compared to the time spent at the sintering temperature and
although this procedure takes the samples through an increasing number
of heating and cooling cycles with longer sintering times, it has been
shown that these cycles do not influence the path of microstructural
change in metal powders.32
2. Pressed and Sintered Series
A CARVER hydraulic hand press was used to prepare cylindrical
pellets about 15 mm in diameter and typically 3 mm in height. Cold
pressing at 60,000 psi followed by sintering at 1250C yielded the
desired series of samples. Due to the small size of these pellets
and their patterns of potential inhomogeneity it was not feasible
to prepare the series of samples from a single initial compact, as
in the loose stack case. Instead, samples in this series were pre-
pared individually by sintering the green compacts in an alumina boat
under a flowing dry hydrogen atmosphere for preselected sintering times
at 12500C 100C.
3. Hot Pressed Series
The third series was prepared by hot pressing at 12500C and under
a pressure of 2000 psi in a CENTORR high vacuum hot press. A loose
stack of powder was placed in a cylindrical boron nitride die 2.54 cm
in diameter and tapped; the die with the top punch resting on the powder
was placed in the vacuum chamber. After a vacuum of 10-5 Torr was reached
the induction coil was switched on. The attainment of sintering temper-
ature which nominally required one hour was followed by the application
of a pressure of 2000 psi. The pressure was maintained and the tempera-
ture controlled to 50C for the specified sintering times; the pressure
was then released and the induction coil turned off. After the sample
was allowed to cool overnight, air was admitted and the die assembly
removed. As in the case of PS series, samples in this series were made
The most common procedure for measuring density of a specimen is
the liquid displacement method, wherein the volume of a specimen is
estimated by measuring the volume of water displaced when it is immersed
in water. Since in the cgs system of units the density of water is unity,
this volume is numerically equal to the weight of water displaced, which
is equal to the decrease in weight of the sample when immersed, according
to the Archimedes principle. The major source of error in the case of
this method lies in measuring the weight of the sample in water. A thin
coating of paraffin wax, typically weighing a few tenths of a percent of
the weight of the sample, was used to seal the surface pores during water
immersion. The samples were suspended by placing them in a miniature
rigid metal pan, thus eliminating the need to tie odd-shaped samples with
a wire. Further, the use of this pan made it easy to correct for the
volume of water displaced by the immersed part of the pan, whereas a
similar correction in the case of a wire is not made easily. An elec-
tronic balance accurate to 0.1 mg was used to achieve the required
high degree of accuracy.
After the sample was weighed in air (W1), it was coated with wax
and weighed again (W2). The wax-coated sample was placed in the minia-
ture pan in a beaker of distilled water and weighed (W3). The sample
was then dropped to the bottom of the beaker by gently tilting the pan.
The weight of partially immersed pan was measured (W4). The density of
the sample, p, was calculated as follows:
Weight of sample
p g/c = Volume of sample
Weight of sample
(Volume of sample + immersed part of pan) -
(Volume of immersed part of pan)
(W2 + weight of pan in air W3) -
(Weight of pan in air W4)
p (g/cc) = + W 3 (8)
The densities thus measured were reproducible within 0.2 percent of
the mean of ten values with 95 percent confidence. The density of a
piece of pure nickel, known to have a density of 8.902 g/cc,5 was
measured and found to be within 0.5 percent of the abovementioned value.
The polishing procedure will be described and followed by a
brief discussion of principles of quantitative stereology involved
in the estimation of metric properties. The estimated microstructural
properties will be presented thereafter.
The wax coating on the samples was dissolved in hexane and the
samples were sectioned; a vacuum impregnation method was used to
mount the samples, surrounded by a nickel ring, in epoxy. The purpose
of the ring will be discussed later in this section. Rough polishing
was done on wet silicon carbide papers of increasing fineness from
180 grit through 600 grit. Fine polishing was done by using 6 micron
diamond paste, followed by 1 micron diamond paste, 0.3 micron alumina
and finally 0.05 micron alumina.
Metric properties are estimated by making measurements on a
two dimensional plane of polish with the help of standard relations
of stereology.26 A set of test lines, arranged in a grid pattern,
also provide a set of test points and a test area to characterize
the plane section; these are usually used to make the measurements
listed in Table 6. The relationships between these measurements and
the globally averaged properties of the three dimensional microstruc-
ture are listed in Table 7. The relations yield estimates of popula-
tion or structure properties provided the structure is sampled uniformly.2
Stereological counting procedure and the estimated properties
will be discussed presently.
Each metallographically prepared surface was calibrated by measuring
the volume fraction of porosity by quantitative stereology and comparing
the result with the value obtained from density measurements. A definite
amount of plastic deformation by the polishing abrasive media leads to a
smearing effect that introduces some error in quantifying the information
on a polished section. This effect can be viewed as local movements of
traces of the pore-solid interface; all the counted events (number, inter-
cept, etc.) aretherefore error-prone to some extent. As this investigation
QUANTITIES MEASURED ON POLISHED SECTION
Test Feature Quantity Definition
Points Pp Fraction of points of a grid that
fall in a phase of interest
Lines LL Fraction of length of test lines
that lie in a phase of interest
PL Number of intercepts that a test
line of unit length makes with the
trace of a surface on a plane
Area PA Number of points of emergence of
linear feature per unit area of
NA Number of full features that
appear on a section of unit area
TA Net number of times a sweeping
test line is tangential to the
convex and concave traces of
surface per unit area of a plane
STANDARD RELATIONSHIPS OF STEREOLOGY2
Pp = L = VV
PL 2 V
P = LV
TA 2NA MV
TT A = 27TNA = M
dealt with relatively small amounts of porosity (10 percent or lower)
the error in VV introduced by the polishing technique approached that
of the density measurements, namely, about 0.005, as the sintered den-
sity approached the bulk density. Thus, the polishing was accepted for
further characterization if the metallographically determined VP was
within 15 percent of VP obtained from the water immersion method, except
for the samples 97 percent dense and higher for which the limit had to
be relaxed to 30 percent of VV. Since VV values range from 0.15 to 0.02,
the abovementioned limits translate into a few percent of the sample den-
sity as measured metallographically. Typically, the samples 97 percent
dense and lower exhibited a precision of 0.05 of the VV value obtained
from the density measurements. Manual measurements of SP and MSP were
made on the accepted polished surfaces using standard stereological techni-
ques.26 The measurements of V SSP and MSP were made with at least 30
different fields and at magnifications that allowed at least 15 pores to be
viewed in a single field. As a result, the estimates of the properties
were within 5 percent with 95 percent confidence, as illustrated in Figures
20 and 21. Plots of SSP and MSP contained metallographically measured values
of VV to yield the paths of evolution of microstructure in order to partially
compensate for the polishing errors.
Measurement of these metric properties was followed by etching the
specimens to reveal the grain boundaries. Each sample was immersed in a
solution made from equal parts of nitric acid, glacial acetic acid and
acetone for about 30 seconds. The grain boundaries were brought out
clearly with some evidence of facetting of the initially smooth contours
of pore features. Samples in the lower part of the density range exhibited
(102 cm )
0 Loose Stack Sintered
0 Hot Pressed at 2000 psi
SCold Pressed at 60,000 psi
I I I I IA II
I I I 1 1 1 t I l 1
Figure 20. Variation of surface area of the pore-solid interface
with solid volume during loose stack sintering, hot
pressing and conventional sintering of INCO 123 nickel
powder at 12500C.
I I I I I I I I I I
O Loose Stack Sintered
O Hot Pressed at 2000 psi
- A Cold Pressed at 60,000 psi
I I I I I I I I I
I 1.. I I I I I I I I I. I I I I.. ...1 L L- I.
Variation of integral mean curvature with solid volume
during loose stack sintering, hot pressing and conven-
tional sintering of INCO 123 nickel powder at 12500C.
grain structures that were too fine to be studied optically; these
samples were not included in the measurement of grain structure prop-
erties. A number of etched microstructures are illustrated in Figures
22 through 36. Typically, the scale of the grain structure was such
that the information contained in a single plane section was not enough
to yield estimates with the desired precision of 10 percent. Conse-
SS SSS SSP SSS
quently, SV LV LV and LVcc) defined earlier in this report,
were measured by repeating the polishing, etching and counting steps a
number of times to obtain at least 100 different fields of view. The
grain structure properties are illustrated in Figures 37 through 40.
The apparent local movements of the traces of the pore-solid interface,
mentioned earlier in this section, are likely to introduce some errors
in the estimation of grain structure properties whenever the pores are
associated with the boundary network. For example, an enlargement-of
pore features residing on grain boundaries would underestimate the value
of SS the grain boundary area per unit volume, as measured metallo-
graphically. However, it was found that these errors are small compared
to those in SSP and MSP; the trends of grain structure properties remain
unaffected whether plotted versus Archimedes density or the stereological
density. The quantities in Figures 37 through 40 are thus plotted versus
the Archimedes density.
The pores observed on a polished and etched surface can be classi-
fied as to their association with the grain boundaries, that is, according
to whether they appear to be inside a grain, on the grain boundary or on
a grain edge. The relative fractions of pore features regarding their
association with the boundary network were measured. These are illustrated
in Figure 41.
Photomicrograph of INCO 123 nickel powder loose stack
sintered at 12500C to VV = 0.871, etched (approx. 400 X).
Figure 23. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to Vv = 0.906, etched (approx. 400 X).
Figure 24. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to VV = 0.928, etched (approx. 400 X).
Figure 25. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 12500C to VV = 0.944, etched (approx. 400 X).
Figure 26. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to VV = 0.971, etched (approx. 400 X).
Figure 27. Photomicrograph of INCO 123.nickel powder loose stack
sintered at 1250% to V = 0.979, etched (approx. 400 X).
Figure 28. Photomicrograph of INCO 123 nickel powder hot pressed
at 12500C to VV = 0.93, etched (approx. 400 X).
Figure 29. Photomicrograph of INCO 123 nickel powder hot pressed
at 12500C to VV = 0.943, etched (approx. 400 X).
Figure 30. Photomicrograph of INCO 123 nickel powder hot pressed
at 12500C to VV = 0.958, etched (approx. 400 X).
Figure 31. Photomicrograph of INCO 123 nickel powder hot pressed
at 12500 to VV = 0.968, etched (approx. 400 X).
Figure 32. Photomicrograph of INCO 123 nickel powder hot pressed
at 1250C to VV = 0.984, etched (approx. 400 X).
Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 1250C to VV = 0.942,
etched (approx. 400 X).
Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 12500C to VV = 0.962,
etched (approx. 400 X).
Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 12500C to VV = 0.975,
etched (approx. 400 X).
Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 12500C to VV = 0.983,
etched (approx. 400 X).
(cm-1) 300 -
Dependence of grain boundary area on solid volume during
loose stack sintering, hot pressing and conventional sin-
tering of INCO 123 nickel powder at 12500C.
Dependence of the length of grain edges per unit volume
on the volume fraction of solid during loose stack sin-
tering, hot pressing and conventional sintering of INCO
123 nickel powder at 12500C.
OLoose Stack Sintered
OHot Pressed at 2000 psi
T ACold Pressed at 60,000 psi
I I I I
I I I
Variation of the length of lines of intersection of grain
boundary and pore-solid surfaces per unit volume with the
volume fraction of solid during loose stack sintering,
hot pressing and conventional sintering of INCO 123 nickel
powder at 12500C.
2 1 I '
Variation of the length of occupied grain edges per unit
volume with the volume fraction of solid during loose
stack sintering, hot pressing and conventional sintering
of INCO 123 nickel powder at 12500C.
Variation of fractions of pores on the triple edges (filled), on the boundaries
and within the grains (open) for a) loose stack sintered, b) hot pressed and c)
sintered nickel powder at 1250C.
ii Ii I?
. . I ill. I i I ii I
The metric measurements of pore structure and grain structure
properties were followed by topological characterization of loose stack
sintered samples. The experimental procedure employed in the latter is
As mentioned earlier in Chapter 1 only the loose stack sintered
samples having densities that are typical of late stages were analyzed
to yield the topological parameters. These samples and their densities
are listed in Table 8. The procedure for serial sectioning is described
below, followed by the algorithm used and the results of the topological
The first step in the technique of serial sectioning is to develop
and standardize the procedure for removing a layer of desired thickness.
This optimum thickness is such that it is small enough to encounter the
smallest structural feature for a number of sections; yet large enough
to avoid redundancy of measurements. Patterson51 and Aigeltinger25
tackled this problem very systematically and found that the optimum
thickness is of the order of one-fifth of that of scale of the structure.
A reliable measure of the scale of the system is XP, the mean intercept
of pore phase. Since = 4VV/SV the slope of the straight line in
Figure 20 yields the value of 3 : 4.5 microns. Thus the serial sec-
tions for LS Series of samples ought to be roughly one micron apart.
LOOSE STACK SINTERED SAMPLES USED FOR TOPOLOGICAL CHARACTERIZATION
Number Sintering Time Volume Fraction of Solid, V
1 128.7 min. 0.906
2 190.0 min. 0.928
3 221.0 min. 0.944
4 352.5 min. 0.971
5 400.0 min. 0.979
This fine scale ruled out the possibility of employing an established
procedure for measuring thickness such as using a micrometer. A new,
simple procedure was developed to measure the section thicknesses
and is presently outlined.
A microhardness tester was used to make square-based, pyramid-
shaped indentations on the polished surface of a sample. It is known
that the apex angle of the diamond indentor is 1360 and hence the ratio
of the depth of an indentation to the diagonal of the impression is
equal to 0.1428. As illustrated in Figure 42, the decrease in the depth
of an indentation is 0.1428 times the decrease in the legnth of the
diagonal. The hardness tester has a capability of a wide variety of
loads and magnifications, so indentations of a wide variety of sizes
can be made and measured with desired accuracy. Thus, the section thick-
ness can be easily measured by measuring the decrease in the length of
the diagonals of an indentation.
A GeotechTM automatic polisher was used to achieve a reproducible
combination of polishing speed, load on the sample and polishing time
that would yield the desired magnitude of material removal. A trial
sample of sintered nickel was polished, indented with 30 indentations
and the section thickness was measured several times by repeatedly
polishing and measuring the diagonals until the polishing technique
and measurement of section thickness were established with a high
degree of confidence.
An elaborate and rigorous procedure for topological analysis of
porous bodies was developed in the course of doctoral research by
Aigeltinger.25 The abovementioned investigation dealt with loose stack
Ad (section thickness) = d. df
Ah = hi hf
Ah = 0.1428 Ad
Illustration of the relation between the decrease in the length of diagonals of a
microhardness indentation and the decrease in the depth of the indentation.
sintered samples having densities in the range from 50 percent to 90
percent of the bulk value and hence exhibited pore structures of a large
variety of scales and complexities. Since the samples used in the present
investigation had densities higher than those used in this research, their
pore structures were typically relatively simple. This made it possible
to streamline and simplify the topological analysis to a great extent.
The revised algorithm is presently described in detail.
Algorithm for Topological Analysis
Two topological parameters, namely, the connectivity and the number
of separate pores, weremeasured in this investigation. Since the connec-
tivity is a measure of the number of redundant connections, there is an
inevitable uncertainty regarding the connections between pores that inter-
sect the boundaries of the volume of analysis (which is a very small
fraction of the sample volume). It is not possible to determine whether
such pore sections intersect each other or meet with themselves outside
the volume covered by the series of parallel areas of observation. This
has led to the necessity of putting maximum and minimum limits on the
estimate of connectivity. As illustrated in Figure 43, an upper limit
on connectivity is obtained when all the pores meeting the boundaries
of the volume of analysis are regarded as meeting at a common node, and
is called Gmax. A lower limit is derived by considering all such pores
to be terminating or "capped" at the boundaries, and is called Gmin. The
quantity Gmin then consists solely of redundant connections or "loops"
observed within the volume of analysis. The number of separate parts is
Illustration of contributions of subnetworks crossing
the surface towards the estimate of Gmax,
obtained by counting the separate pores that appear and disappear
within the volume observed and do not intersect the boundaries. The
actual algorithm is as follows.
The surface of a loose stack sintered sample, one from the series
designated for topological analysis, was conditioned by polishing it on
a microcloth with 1 micron diamond paste abrasive for about half an hour.
This effectively removed all plastically deformed material, the result of
an earlier alumina polish. The sample was cleaned in an ultrasonic cleaner,
dried and viewed under a microscope to check for polishing artifacts. If
the nickel ring, mentioned earlier in this section, was polished uniformly
all around, the sample was examined for undue number of scratches that
would hinder the analysis. If the polishing was uniform and had only a
small number of scratches, it was deemed ready for further analysis;
otherwise it was returned to the polishing step.
Since the contours of the hardness indentation are mixed with those
of the pore sections when observed for the measurement of diagonals, the
thickness measurements become more difficult the more the sample is
polished or the smaller the square-shaped impression. The trial sample
of sintered nickel mentioned previously was polsihed, ten indentations
were made on the nickel ring and the specimen and the sample repolished.
This was followed by measuring the diagonals of impressions on both ring
and specimen. The repetition of this procedure demonstrated that the
extents of polishing (removal of material or layer thickness) of the
ring and the sample were not statistically different. Indentations in
the ring were therefore used to measure the section thickness.
Nine indentations were made on the nickel rings of each sample.
Three indentations were made on the specimen so that the same area
could be located and photographed after each polishing step. The
pattern of indentations is schematically illustrated in Figure 44.
The first photomicrograph of the serial sectioning series was taken
by positioning the three indentations on the sample in a manner that
can be easily reproduced. The magnification was selected so that at
least 70 pore features could be observed in a single field of view.
A Bausch and Lomb Research Metallograph II was used for all the photo-
A set of 4"x5" negatives was obtained by repeating the polishing
and photographic steps. Each was enlarged to a size of 8"x10" so that
even the smallest pores were easily seen. A smaller rectange of
6"x8" was marked on print #1; this identified the area of observation.
This manner of delineating the area was adapted to help minimize the
misregistry error. A similar rectangle was marked on successive prints
such that the pores observed on the consecutive sections were in the
same position relative to the boundaries of the rectangle. Xerox copies
of these prints were used for further analysis, which involves marking
each pore on the area of observation for easy identification.
The pores seen on Section #1 were numbered beginning with 1.
These are all connected to the "external" networks and thus were not
included in the count of separate parts. Pores that first appeared
thereafter on successive sections were numbered with a number and a
letter N, beginning with 1N. These were regarded as the "internal"
networks and were used to measure the number of separate parts. The
Microhardness indentations used to
measure section thickness
used to position the
Sample Nickel ring
Figure 44. Schematic diagram of a typical specimen used in serial
genus or the connectivity and the number of isolated pores were
measured by comparing pairs of neighboring sections as follows.
There are listed in Table 9 three possible classes of topologi-
cal events that can be observed when two consecutive sections are
compared, along with the corresponding increments in Gmax, Gmin and
Nis0. The significance of each of such observed events will be
Two typical consecutive sections are shown schematically in
Figure 45, wherein the types of events mentioned above are also
illustrated. The simplest of these events is the appearance and
disappearance of whole pores or subnetworks. When an external sub-
network disappears, the number of possible "loops" or redundant
connections that are assumed to exist outside the volume of analysis
is reduced by one, as illustrated in Figure 46. When an internal
subnetwork appears, it cannot be determined whether the said subnet-
work is wholly contained in the volume of analysis or is connected
to the external pores. Thus, this event does not change any of the
parameters. However, the disappearance of an internal subnetwork
signifies a whole separate part and thus the number of separate parts
is increased by one.
Within a subnetwork, a branch may appear. When that happens,
the number of possible loops, terminating in a single external node,
is increased by one, as shown in Figure 47. When such a branch is
observed to disappear, the abovementioned number is decreased by one,
to account for the increase assumed prior to an observation of this
OBSERVABLE TOPOLOGICAL EVENTS
Appearance of a branch +1
Disappearance of a branch -1
Connection Different or new 0
Two typical consecutive sections studied during serial
sectioning that illustrate the topological events listed
in Table 9.
a- End of an Original Subnetwork
Redundant Connection to be Deleted from the
Estimate of Gmax
Contribution of the end of an original subnetwork towards
the estimate of Gmax.
)-- -- \--
/ I r i /
/ r I / r t
/ Branching within a subnetwork
Redundant connection increasing the connectivity
Figure 47. Illustration of the contribution of a branching event
When a connection is observed between different subnetworks, the
said subnetworks have to be renumbered to keep track of such connections.
All the subnetworks involved in such connections are marked with the
lowest of the numbers of these connecting subnetworks. If an internal
subnetwork is observed to be connected to an external one, the said
internal subnetwork is marked with the number designating the external
subnetwork. A connection between previously unconnected subnetworks,
those with different numbers, does not change any of the three parameters.
A connection between two or more subnetworks with the same number signi-
fies a complete loop observed entirely inside the sample volume, and thus
increases the count of Gmin by one. Since Gmax includes such internal
loops, it is also increased by one.
After each comparison of consecutive sections, the counts of Gmax,
Gmin and N1so were updated and tabulated as shown in Table 10.
The values of interest are the unit volume quantities, Gax, Gmin
and NV If the features that give rise to these quantities are randomly
and uniformly distributed in the sample,25 then it can be expected that
there exists a quantity (QV) characteristic of the structure and equal to
the unit volume value. Thus AQ (the change in quantity Q) = QV x AV.
Dividing both sides by AV leads to
S= QV (13)
The slope of AQ versus AV plot therefore should be equal to Qv provided
AV, the volume covered is large enough for a meaningful sampling. In
the previous investigations of this kind25,51 the analyses were continued
CURRENT ESTIMATES OF Gmax, Gmin and Niso AFTER SECTION #J
Height of Sample, Microns
Previous total -
until a linearity between AQ and LV was observed. A different criterion
was adapted in the present investigation, and is described presently.
Since the connectivity and the number of separate parts are two
independent quantities, one may begin to exhibit a constant value of
Q after several sections whereas the other may be far from levelling
off. This makes it difficult to establish a criterion for terminating
the serial sectioning for a given sample. If use is made of both the
quantities a standard basis for terminating the analysis can be obtained.
A typical pore subnetwork with convex, concave and saddle elements
of its surface is shown schematically in Figure 48. If a test plane is
swept through a unit volume, a measure of TV, the number of times this
plane is tangential to the pore-solid interface, may be obtained in
principle. This is related to the connectivity and the number of parts
per unit volume26 as follows.
Tet = T + T T = 2(NV GV)
where TV = number of times a concave element (having both the principal
radii of curvature negative by convention) is tangential to
test plane, or the number of "ends" of a feature (see Figure
TV = number of times a convex element (having both the principal
radii of curvature positive by convention) is tangential to
to test plane, very small at this stage of sintering.
T = number of times a saddle element (with principal radii of
curvature of different signs by convention) is tangential
Figure 48. Convention used in the net tangent count (Tnet)
during serial sectioning.
Dependence of connectivity on the volume fraction of solid
during loose stack sintering of INCO 123 nickel powder at