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Metric and topological characterization of the advanced stages of sintering

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Metric and topological characterization of the advanced stages of sintering
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Watwe, Arunkumar Shamrao
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Curvature ( jstor )
Distance functions ( jstor )
Grain boundaries ( jstor )
Hot pressing ( jstor )
Material concentration ( jstor )
Nickel ( jstor )
Porosity ( jstor )
Property lines ( jstor )
Sintering ( jstor )
Volume ( jstor )

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METRIC AND TOPOLOGICAL
CHARACTERIZATION OF THE ADVANCED
STAGES OF SINTERING










By






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


1983
































Copyright 1983

by

Arunkumar Shamrao Watwe



























Dedicated

To My Parents,


Mr. Shamrao Vasudeo Watwe

and

Mrs. Sharada Shamrao Watwe










ACKNOWLEDGEMENTS


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



PAGE

ACKNOWLEDGEMENTS........................................ ......... .. iv

ABSTRACT....................... ........... ...................vii

INTRODUCTION..................................................... 1

CHAPTER

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


Introduction.....................
Sample Preparation..............
Metallography....................
Topological Measurements........


THREE DISCUSSION.............................................101

Introduction...........................................101
Loose Stack Sintering................................ ..101
Hot Pressing.............................................130
Conventional Sintering...................................134
Comparison of Loose Stack Sintering with Hot Pressing
and Conventional Sintering...............................135

FOUR CONCLUSIONS ............................................137

Introduction............................ .............. 137
Conclusions...........................................137
Suggestions for Further Study...........................139

REFERENCES.........................................................140

APPENDIX THE GEOMETRIC MODEL OF THE PORE PHASE....................143

Introduction.........................................143


..................
..................
..................
..................










PAGE

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


By

Arunkumar Shamrao Watwe-

August, 1983



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

significantly different.

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

isolated pores.

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

isolation events.


viii










INTRODUCTION


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

separate parts.

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

occur,

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

particle size.

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

or lower.2










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.









CHAPTER 1
EVOLUTION OF MICROSTRUCTURE DURING SINTERING


Introduction


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
26
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
V V


Sectioning Plane of
Area A


L _7
A 4


Lines of Intersection
of Length L


Element of Surface




H ( + )
2 r1 r2


Figure 1. Illustration of basic metric properties.


Volume V










Table 1
BASIC METRIC PROPERTIES



Property Definition

LV Length of a linear feature per unit
volume.

S,, Area of a surface per unit volume.


1 1 1-
2 rl r2


MV =f/ HdS
SV


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.


a surface


Length of a trace of surface per
unit area of a plane section.

Volume fraction or volume of a
particular region of space per
unit volume.


Regions of Space


Feature


Lines


Surfaces






9











Table 2
DERIVED METRIC PROPERTIES


Feature Property Definition
MV
Surface H = Average mean curvature of
SV a surface

4V
4V Mean intercept in a particular
Region of Space of space
Sv region of space










Table 3
METRIC PROPERTIES OF A SINTERED STRUCTURE


Property


Description


Pore-Solid Interface


Porosity

Solid Phase


Grain Edges in the
Solid

Lines Formed by the
Intersection of Grain
Boundaries and Pore-Solid
Interface

Grain Edges Occupied
by the Pore Phase


LSSS
V


LSSP
V


LSSS
V(occ)


Area of pore-solid interface
per unit volume

Integral mean curvature of
pore-solid interface per unit
volume

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
volume

Length of occupied grain
edges per unit volume


Feature










Grain Boundaries of Area SSS





SS SS


SSS
LSSS L(occ) LSS
(occ) V V

SSS
v L
V V








SSS edge

SSP lines



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

parts, then

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-

nections.

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
(dotted lines).











A Closed Surface


Deformation Retract


Spurious Node


Spurious Branch


Figure 4. Illustration of a one-to-one correspondence between a
spurious node and a.spurious branch in a deformation
retract.



















































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.


Metric Investigations

It has been observed that in loose stack sintered samples SP
P(12,18-20)
decreased linearly with the decrease in V(1218-20) during the second

stage. Surface area may be reduced both by densification and surface
SP
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
V



















V (a)










S\ 1I










(b)




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
SP
Data for hot pressed samples indicate that the SV -Vv relationship
22 23
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
P
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-
1821 ,22
vex pore has a negative curvature. There is a minimum in'MV;182122
35
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
N V
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
P 23
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


SV















Figure 7.













14.0



12.0



10.0



8.0

-1
cm1 )
6.0


0.6


0.8


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



















1
-\






I I I I I \


sSP
(13
(103


4.0



2.0



0
0


Figure 8.


1.0


).4


























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


Figure 9.































MP (105 cm2)


0 Spherical

- A Dendritic


I I I I f


I I


0.2 0.5 VV 1.0

Spherical ---------- --- I -IIIi-
Dendritic I II III


Figure 10.


Integral mean curvature versus volume fraction of
solid for 48 micron spherical and dendritic copper
powder.25










these curves become deeper and shift towards lower VV with increasing

pressure,22 as shown in Figures 11 and 12.
SS
The.grain boundary area per unit volume, SV increases until a
SS
network is formed; subsequent grain growth tends to decrease SS.
18
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
V V
SSP
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
22
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



M SP
M
(105 cm-2)

-1







-2


Figure 11.


0.75


0.80 0.85 0.90 0.95
VV


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
comparison.23.25





























MS P
V
(106 cm-2)


Figure 12.


The effect of pressure on the path of integral mean
curvature for hot pressed specimens of U02O22


-20.0


-40.0







60.0 ,















300





200


S SP (cm- 1


- I- I


Figure 13.


Grain boundary area per unit volume versus volume fraction
of solid for 48 micron spherical and dendritic copper
powder.2b


1.0
[11 t-^








SS
CS V
C = S (2)
2SV +S V


SSS
Css = 3L~_ss
SSS SSP (3)
3LV +LV


2LSSP SSS
2Lv +3Lv
V V
F1 S= P SS2 (4)
(Sv +2SV )



F2 = 2LSSP/(SSP)2 (5)



SSP SS
LV +3L
F3 SS (6)
2(SVS)2


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)
V (10


Figure 14.


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


SS SP
TrS SSbS
7V -V
4(1-Vv)














1.0


0.4


Figure 15.


The variation of grain contiguity with solid volume
fraction for loose stack sintered copper and hot pressed
U02.22









P SP
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
SSP
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.










Topological Studies

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.
SP P
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
P
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)
P
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


Figure 16.


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
> A



























Gp or Np

(107 gm-1)


A
0
0.05 a


Spherical
Dendritic
Spherical
Dendritic


0 0
Spherical - ---
Dendritic -


Figure 17.


i 1.0

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


0.6










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.
































Figure 18.


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.











CHAPTER 2
EXPERIMENTAL PROCEDURE AND RESULTS


Introduction


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.


Sample Preparation

This section presents the procedure employed to prepare the

sintered samples and the standard for density measurements.


Sintered Samples

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
(1000 X).
















Table 4
CHEMICAL COMPOSITION OF INCO


Element


TYPE 123 NICKEL POWDER



Nickel Powder (Wt.%)


Carbon (typical)

Carbon

Oxygen

Sulphur

Iron

Other Elements

Nickel


0.03-0.08

0.1 max

0.15 max

0.001 max

0.01 max

trace

Balance





















Table 5
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
V










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

sintering times.

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-

tering temperatures.


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

individually.










Density Measurements

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)

Wl
(W2 + weight of pan in air W3) -
(Weight of pan in air W4)

Thus
W (8)
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.


Metallography


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.


Polishing Procedure

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.


quantitative Stereology

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-
26
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
















Table 6
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
section

Area PA Number of points of emergence of
linear feature per unit area of
plane section

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
section















Table 7
STANDARD RELATIONSHIPS OF STEREOLOGY2


Pp = L = VV


PL 2 V


P = LV
TA 2NA MV

TT A = 27TNA = M


(9)


(10)


(11)


(12)










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
V
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
P P
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










23



20






15
SP
SV
(102 cm )



10





5






0


0.8


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


0.9

VV (Metallographic)


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.


1.0


.0


I I I I I I I I I I













0






MP
(105 cm-2)



-50.0










-90.0
0


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.


.8


Figure 21.


0.9

(Metallographic)


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
t t
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.












































Figure 22.


Photomicrograph of INCO 123 nickel powder loose stack
sintered at 12500C to VV = 0.871, etched (approx. 400 X).













qW'


* S


r *
r'-, "












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).













































Figure 33.


Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 1250C to VV = 0.942,
etched (approx. 400 X).












































Figure 34.


Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 12500C to VV = 0.962,
etched (approx. 400 X).











































Figure 35.


Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 12500C to VV = 0.975,
etched (approx. 400 X).












































Figure 36.


Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 12500C to VV = 0.983,
etched (approx. 400 X).








700


600


500


400-
SSS
V
(cm-1) 300 -

200 -o


100 -


0 90
0.90


1.0


0-95

Vv


Figure 37.


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.











3x10"






2x105


Lss

(cm-2)

1x105
lxlO


0.90
0-90


Figure 38.


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.


0.95


1.0










1.5x106






1x106


Lssp
V

(cm-2)

5x10'





1x105
C
C


OLoose Stack Sintered
OHot Pressed at 2000 psi
T ACold Pressed at 60,000 psi


I I I I


).90


I I I


0.95


1.0


Vv


Figure 39.


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 '

















2x105



S(occ)

(cm-2)

1 x105







0.90


Figure 40.


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.


0.95


1.0












1.0

0.9

0.8

0.7

0.6

0.5

0.4


Fraction
of pores


0.3

0.2

0.1

0.
0.


Figure 41.


90


0.95
Vv

(a)


1.0 0.90


0.95


0.90


0.95


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.


1.0


(half-filled)
pressed and


ii Ii I?


II~~i~


. . I ill. I i I ii I


- i


- 0


A










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

described presently.


Topological Measurements


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

analysis.


Serial Sectioning

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.



















Table 8
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
S f


Ah = hi hf


Ah = 0.1428 Ad


Figure 42.


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




















'71


S


Figure 43.


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-

micrographs.

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


Microhardness indentations
used to position the
Specimen







Sample Nickel ring











Epoxy mount




Figure 44. Schematic diagram of a typical specimen used in serial
sectioning.










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

discussed presently.

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

event.







84






Table 9
OBSERVABLE TOPOLOGICAL EVENTS


Appearance


Whole
Subnetworks


Disappearance


Internal

External


AGmax
0

-1


AGmin ANiso
0 +1


Within a
Subnetwork


Between
Subnetworks


Appearance of a branch +1
Disappearance of a branch -1

Connection Different or new 0
Same +1Connection
Same +1



























[jth section]


[(j+l)th section]


Figure 45.


Two typical consecutive sections studied during serial
sectioning that illustrate the topological events listed
in Table 9.
















External Node





J\/


a- End of an Original Subnetwork


Redundant Connection to be Deleted from the
Estimate of Gmax


Figure 46.


Contribution of the end of an original subnetwork towards
the estimate of Gmax.
















External Node












10
)-- -- \--
N






















towards Gmax.
/ I r i /
/ r I / r t








/ Branching within a subnetwork

Redundant connection increasing the connectivity
by one





Figure 47. Illustration of the contribution of a branching event
towards Gmax.










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
iso
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















Table 10
CURRENT ESTIMATES OF Gmax, Gmin and Niso AFTER SECTION #J


Height of Sample, Microns

--


Counts Gmax

Previous total -

This section

Current total

Unit volume
value
Current total
Current volume


Gmin
-- _


Niso
--










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
net
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)


(14)


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

48).
++
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.










5x107



min
Gv
-3




1 x10


0.
O-


Vv
V


Figure 49.


Dependence of connectivity on the volume fraction of solid
during loose stack sintering of INCO 123 nickel powder at
12500C.




Full Text
130
Thus, a spherical pore has more than twice as much Cy as does a
tubular channel. The relative fractions of tubular channels and spheri
cal pores can be computed from the number of branches (by in Table 17)
and the surface-corrected number of isolated, pores (Figure 59). It can
SP
be seen from the variation of f fraction of spheres, Figure 64, that
the curvature would decrease (in magnitude) relatively slowly with the
volume until the maximum in the fraction of spheres; beyond this point
the curvature would decrease sharply. This is supported by the variation
of My, Figure 21. It is suggested that a kinetic model of this type of
geometry should be devised and tested in the future, provided the associ
ation of isolated and connected porosity with the grain boundary network
can be successfully quantified.
The preceding discussion on loose stack sintering will now be
used to describe hot pressing and conventional sintering, in that order.
Hot Pressing
It can be seen from Figure 20 that in the early part of the range
SP
of observation a hot pressed sample has a higher Sy than a loose stack
sintered sample with comparable density. The variation of surface area
with Vy approaches the linear relationship for LS series as the density
increases. Both hot pressed samples and loose stack sintered samples then
continue to exhibit comparable values of surface area. Since the pressure
is applied for the whole duration of hot pressing, the abovementioned
SP
variation of Sy with Vy may have less to do with pressure and thereby
particular mechanisms than with the geometry of the pore structure.


CHAPTER 2
EXPERIMENTAL PROCEDURE AND RESULTS
Introduction
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.
Sample Preparation
This section presents the procedure employed to prepare the
sintered samples and the standard for density measurements.
Sintered Samples
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
c
having densities that are uniformly distributed over the range Vy = 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.
39


123
L(measured) = / / L Cos6Sin0d0d<> = L/2
<¡>=0 0=0
Unit Radius
Figure 61. Illustration of the relation between measured and true
lengths of a branch.


62
Figure 30. Photomicrograph of INCO 123 nickel powder hot pressed
at 1250C to Vy = 0.958, etched (approx. 400 X).


90
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
^ 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
net
swept through a unit volume, a measure of Ty, 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
pc
per unit volume as follows.
T?et = TV~ + TV+ TV~ = 2 where Ty- = 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
48).
Ty+ = 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.
Ty" = number of times a saddle element (with principal radii of
curvature of different signs by convention) is tangential


4
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-
25
cal measurements are time-consuming and since an earlier doctoral research
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.


94
to the test plane per unit volume, or the number of
branching and connection events per unit volume (see
Figure 48).
Cbmputation of is shown;,in Table 11.
not
If the constancy;, of T is used as the basis for terminating
the analysis, both N and G are used and so a more general criterion is
obtained. The serial sectioning was terminated for each sample after
Tye* was found to level off. The values of the parameters G^ax, G^n
and NyS0 are listed in Table 12. Since Gyax depends on the number of
pores intersecting the surface that encloses the volume of analysis, it
is very sensitive to the surface to volume ratio of the "slice" of the
sample used for analysis. Therefore, G^ax does not exhibit a monotonic
decrease with an increase in Vy. On the other hand, Gyin is unaffected
by the surface to volume ratio mentioned above and hence can be used to
draw meaningful inferences about the variation of connectivity. The
variation in Gyin and NyS0 are illustrated in Figures 49 and 50, respec
tively. It can be seen that even at 98 percent of the bulk density the
pore structure has a finite connectivity and thus cannot be regarded as
a collection of simple isolated pores.
Behavior of Connected and Isolated Porosity
A natural reaction to these data would be to wonder what fraction
of porosity is connected and how it varies with the total Vy. It is
possible to distinguish connected porosity from the isolated pores on
any of the series of sections used for topological analysis. Thus, the
volume fraction, surface area and integral mean curvature of the connected


125
Table 17
VALUES OF THE MODEL PARAMETERS USED IN CALCULATIONS
No. Vv
R L
microns microns
cm
-3
1 0.906
2 0.928
3 0.944
4 0.971
2.34
6.63
4.48 x 108
2.1 x 108
2.5 x 108
2.58
8.18
2.42 x 108
6.2 x 107
2.63 x 108
2.79
7.1
8.39 x 107
2.04 x 107
1.02 x 108
2.1
9.56
1.8 x 108
3.0 x 107
2.66 x 108
1.07
25.8
1.29 x 108
3.4 x 107
1.23 x 108
5 0.979


81
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
micrographs.
A set of 4"x5" negatives was obtained by repeating the polishing
and photographic steps. Each was enlarged to a size of 8"xl0" 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 IN. These were regarded as the "internal"
networks and were used to measure the number of separate parts. The


149
which is the same as equation (26). Thus, the number of branches can
be calculated from the number of nodes, and vice versa.
Metric Properties of the Connected Porosity
Three basic properties, namely, volume fraction, area per unit
volume and integral mean curvatre per unit volume are discussed. The
conn cconn Mconn
i S\ cl net Mt;
notations used are listed in Table 20; V
are made
V dna V
up of contributions from branches, one-branch nodes and three-branch
nodes; they will be discussed in terms of these individual contributions,
listed in Table 21.
Three-Branch Node
It can be seen from Figure 67 that r, the radius of a cylindrical
branch, and R, the radius of a three-branch node, are related by the
equation
r = RCos(ir/6) or r = '^-R
(31)
Thus, h, the height of the spherical cap, illustrated in Figure 68, is
given by
h = R/2
(32)
C
The volume of this spherical cap, V is given by
(33)


2r = 3R
Figure 63. Illustration of a typical pore channel in the connected
porosity.


147
The abovementioned parameters of the model are discussed in terms
of the notations listed in Table 19.
Since each one-branch node is associated with a branch and each
three-branch node with three branches, by, the total number of branches
1 h
and Ny and Ny are related by the following equation
+ 3Nyb
2
(26)
since each branch is counted twice. Alternatively, each one-three
branch is associated with one one-branch node and one three-branch
node, whereas each three-three branch is associated with two three-
branch nodes. Thus
,3b
Nr
h13 .33
by + 2by
(27)
since each three-branch node is counted thrice. The number of one-branch
nodes, Nyb, is given by
Njb = bj3 (28)
Equations (26) and (27) give
wlb .33
3b NV + 2bV
V
or
3N?b NJb
(29)
Thus by is given by
K h13 4. h33
bV bv + bV
3|\j3^ N^b
My> + 3_Ni_Ny.
bv =
Nb + 3Nf
or
2
(30)


25
Figure 11. 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 1005C included for
comparison.23.25


112
Figure 57. Illustration of three-branch nodes signified by
branching and connection events.


42
Table 5
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 10^ cm^/cm^
5
Vy of As-Received Powder 0.25


METRIC AND TOPOLOGICAL
CHARACTERIZATION OF THE ADVANCED
STAGES OF SINTERING
By
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
1983


145
Figure 66. Illustration of a) a one-three branch and b) a three-three
branch.


61
Fiaure 29. Photomicrograph of INCO 123 nickel powder hot pressed
at 1250C to Vv = 0.943, etched (approx. 400 X).


22
M
SP
V
Figure 9. Variation of integral mean curvature per unit volume with
the volume fraction of solid for three representative copper
powders sintered in dry hydrogen at 1005C.35


PAGE
Parameters of the Model 143
Metric Properties of the Connected Porosity 149
Surface Corrections 155
BIOGRAPHICAL SKETCH 163
VI


34
Figure 17. 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).2^


INTRODUCTION
Sintering is a coalescence of powder particles into a massive formj
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
2-9
of the pore phase. 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.^'frie traditional
or mechanistic approach involves the study of kinetics and mechanisms of
^Im
material transport;1the 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
separate parts.
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 mechanism(s) of interest.
3) These equations are compared with the experimentally observed
time dependences of the chosen parameter and an attempt is made
to identify the operating mechanism(s).
1


144
Figure 65. Illustration of three-branch and one-branch nodes in a
pore network.


146
2r
Figure 67. Dihedral angle of the edge at the intersection of a
cylindrical branch and a spherical node.


71
Vv
Figure 39. 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 1250C.


APPENDIX
THE GEOMETRIC MODEL OF THE PORE PHASE
Introduction
Main features of the proposed model are stated and followed by the
derivation of equations relating metric properties and the parameters of
the model. The corrections for surface effects are also outlined.
Parameters of the Model
As said in Chapter 3, the pore phase is modeled as a collection of
spherical pores of the same size and a set of networks made up of cylin
drical branches and spherical nodes. Only two kinds of nodes and branches
are assumed to exist in the connected pores; a node is either a one-branch
node or a three-branch node, as illustrated in Figure 65. A branch either
terminates in three-branch nodes at both ends, or in a one-branch node at
. one and a three-branch node at the other, Figure 66. A branch that termi
nates in one-branch nodes at both ends is considered an isolated part and
hence is not included in the connected porosity.
A one-branch node is assumed to be a semi-spherical cap, the radius
of which is the same as that of a cylindrical branch, r, as shown in
Figure 66. A three-branch node is considered as a sphere of radius R
which*is connected to three cylindrical branches of radius r, Figure 67.
An isolated pore is assumed to be a sphere of radius R. All branches
have the same length, "L.
143


21
Figure 8. Pore-solid interface area versus solid volume fraction for
conventionally sintered


43
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
sintering times.
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
tering temperatures.
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
32
change in metal powders.


ACKNOWLEDGEMENTS
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.
iv


136
1) As the grain boundaries are anchored effectively only by
equiaxed pores, a fine grain structure can be obtained if a
sufficient number of equiaxed pores is isolated before grain
growth begins. Grain growth requires a connected grain edge
network and hence a certain minimum grain boundary area,
therefore, an initial powder stack with a relatively coarse
grain structure (low grain boundary area) would have grain
growth beginning at a higher density (after a sufficient number
of pores is isolated); as compared to an initial powder stack
with a finer grain structure. Thus, for a sintered body with
high density and fine grain structure requirements, an initially
coarse grain structure is better than a fine one.
2) If sintering is carried out in such atmosphere that the isolated
pores trap a gas of low diffusivity, these pores are relatively
stable and hence offer effective grain boundary pinning. Care
must be taken to delay the isolation events so that only a few
pores are isolated; otherwise coarsening of these pores would
42
lead to an increase in volume.
3) At least for the conditions of the present investigation, it
can be said that hot pressing leads to both higher density and
a finer grain size in a shorter length of time as compared to
loose stack sintering or conventional sintering at the same
temperature, up to a certain density. Beyond this, a relatively
coarser grain structure is obtained during hot pressing or con
ventional sintering.
The findings of this investigation are summarized and the course of
future research suggested in the next chapter.


124
*1 CQ
Where Ny = number of isolated pores per unit volume and
,iso
= radius of a spherical pore. Thus,
1/3
3V1S0
Ris ( V
4ttN
iso
)
(22)
1 C A
The radius of a spherical -node;:R> was taken to be equal to R The
Al 1 L
values of L, R, by, Ny and Ny are listed in Table 17. It can be seen
that r increases slowly until Vy s 0.97 when it increases significantly.
The reduction in connectivity by way of pinching off of a branch decreases
L whereas elemination of a three-branch node, accomplished when a 1-3
type branch merges into the parent network, leads to an increase in "L,
Figure 62. Since the number of isolation events goes through a maximum,
L should eventually increase significantly once most of the isolation
events or channel closures have taken place. The apparent maximum in
R can be attributed to simultaneous isolation and shrinkage processes.
When a pore shrinks it can be thought of as going from one size class
to the lower one. Since all isolated pores shrink, although at different
rates, there is a "flux" toward the smallest size class in a given size
distribution. Isolation events bring new pores into this collection,
such that there is an influx in all the size classes. Since the number
of isolation events goes through a maximum, the net "flux" in the size
distribution is towards the largest size class when isolation events
dominate over shrinkage and towards the smallest size class when very
few isolation events occur. This leads to a maximum in the estimated
average volume of an isolated pore.
In order to test the model, the metric properties of the connected
porosity were calculated, since those of the isolated fraction were used,


98
w conn
vV
or
V
iso
V
p
Vv (metallographic)
Figure 51. Dependence of the metallographically determined volume
fraction of connected porosity on the volume fraction of
porosity during loose stack sintering of nickel at 1250C.
Data for the volume fraction of isolated porosity included
for comparison.


150
Table 20
CALCULATED METRIC PROPERTIES
Property
Notation
Definition
Volume
..conn
vV
Volume fraction of
connected porosity
Area
-conn
Surface area of connected
porosity per unit volume
Curvature
Mconn
Integral mean curvature of
connected porosity per unit
volume


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9
Table 2
DERIVED METRIC PROPERTIES
Feature
Property
Definition
Surface
- MV
H =-r
Average mean curvature of
sv
a surface
Region of Space
x = 4sVv
Mean intercept in a particular
region of space


7
Figure 1. Illustration of basic metric properties.


3
Consequently, a major school of thought prevails that favors the geometri
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.^ 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
C ip 1 0_0*|
sintering, conventional sintering-cold pressing followed by
sinteringD,<:, and hot pressing*c ~ 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
24
or lower.


83
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, G111111 and
i so
N The significance of each of such observed events will be
discussed presently.
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
event.


26
Figure 12. The effect of pressure on the path of integral mean
curvature for hot pressed specimens of UOg.^


Ad (section thickness) = d^ Ah = h^ Ah = 0.1428 Ad
Figure 42. 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.


no
a mechanistic model and hence derive any mechanistic conclusions from
the time dependences of geometric properties of the pore microstructure.
However, it was possible to construct a variety of geometric models that
describe the evolution of microstructure qualitatively. These models
incorporate connected and isolated porosity of regular geometry so that
the metric properties such as volume, area and integral mean curvature
can be calculated. The models are tested by comparing the calculated
metric values with the experimentally determined quantities. Since '
these models do not have any parameters that characterize the extent
of association with the grain boundary network, properties that depend
SSP SSS
on this association, such as Ly and Ly(occy could not be calculated
and compared with the measured values. These models are described below.
Geometric Models
All of the geometric models mentioned above describe the porosity
as composed of a collection of isolated, spherical pores of the same
size and a set of networks of cylindrical pore channels and spherical
nodes, as illustrated in Figure 56. The parameters of such a model are
listed in Table 13.
Here it is assumed that the connected porosity has only three-branch
nodes and one-branch nodes; these were measured for a unit volume during
the topological characterization in a manner described presently.
Each branching or connection event defines a node formed as a result
of merging of three pore channels, Figure 57. Thus, the number of three-
branch nodes is given by the number of branching and connection events
observed per unit volume (Ty). The number of one-branch nodes is the


74
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
described presently.
Topological Measurements
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
analysis.
Serial Sectioning
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
51 25
to avoid redundancy of measurements. Patterson and Aigeltinger
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.
-P
A reliable measure of the scale of the system is X the mean intercept
_p- p cp
of pore phase. Since x = 4VV/Sy the slope of the straight line in
-P
Figure 20 yields the value of X ~ 4.5 microns. Thus the serial sec
tions for LS Series of samples ought to be roughly one micron apart.


Table 11
COMPUTATION OF T
net
V
Number of Number of Bottom T~~ = Top Ends
Section # Top Ends Ends = A + B + Bottom Ends
A: Disappearance
of internal &
external sub
networks
B: End of a Branch
J+l
J+2
T+ = Branchings __
and Connections TeL = T T
10
cn


86
Figure 46. Contribution of the end of an original subnetwork towards
the estimate of Gmax.


69
Vv
Figure 37. Dependence of grain boundary area on solid volume during
loose stack sintering, hot pressing and conventional sin
tering of INCO 123 nickel powder at 1250C.


128
Table 18
CALCULATED AND ESTIMATED VALUES OF V
conn
V
cconn
bV
and M
conn
V
No.
VS
Vv
..conn
vV
(.conn
-1
, cm
Mconn
Mv ,
_ 2
cm
Calc.
Est.
Calc.
Est.
Calc.
Est.,
1
0.906
0.05
0.09
479
731
-12.4 x 105
-20.3 x 105
2
0.928
0.04
0.053
373
474
-10.3 x 105
-15.2 x 105
3
0.944
0.015
0.017
133
138
-3.6 x 105
-3.1 x 105
4
0.971
0.022
0.017
257
165
-8.9 x 105
-4.0 x 105
5
0.979
0.009
0.02
202
237
-11.2 x 105
-10.7 x 105


105
after about 95 percent density has been reached. Thus, it can be said
that the isolation of pores from an interconnected network continues
until a point beyond which the residual collection of tree-like pores
(those with very low connectivity) does not change appreciably. The
isolated pores, on the other hand, continue to shrink and disappear.
The preceding discussion of evolution of pore and grain boundary struc
tures is expected to present the overall scenario described below.
The Overall Scenario
It is interesting to note that the density range of effective
pinning of the grain boundary network coincides with that of little
changes in the connected porosity; the major change is in the isolated
porosity, Figure 51. If most of the associated porosity that anchors
the grain boundaries is assumed to be a collection of isolated pores,
the removal of isolated porosity in this density range can be explained
by the proximity of grain boundaries that can act as vacancy sinks. If
a balance exists between the number of isolated pores that disappear and
the number of pores that are isolated as a result of channel closures,
there will be sufficient associated porosity maintained to anchor the
grain boundaries. This suggests that the isolated, equiaxed pores anchor
the grain boundaries much more effectively than do the connected pores.
A hypothesis is offered presently that attempts to rationalize the above
contention.
Grain boundary migration takes place during grain growth that
decreases the grain boundary area. If part of the boundary area is
occupied by second phase particles, as illustrated in Figure 54, the


52
Vv (Metal!ographic)
Figure 21. Variation of integral mean curvature with solid volume
during loose stack sintering, hot pressing and conven
tional sintering of INCO 123 nickel powder at 1250C.


72
X/
Figure 40. 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 1250C.


20
Figure 7. 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


33
Gp or Np
(106 gm"1)
Figure 16. 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 1005C. Data for 115 micron spherical copper
included for comparision.25


60
Figure 28. Photomicrograph of INCO 123 nickel powder hot pressed
at 1250C to Vy = 0.93, etched (approx. 400 X).


(a)
(b)
Figure 69. Illustration of a) "IS11 pore branches and b) "SS" pore branches.


117
Table 14
MEASURED VALUES OF THE NETWORK PARAMETERS
No.
vv
b, cm"3
M3b -3
N\/ $ cm
Njb, cm"3
N, cm"3
1
0.906
4.48 x 108
2.1 x 108
2.5 x 108
V
1.07 x 101
2
0.928
2.42 x 108
6.2 x 107
2.63 x 108
3.33 x 101
3
0.944
8.39 x 107
2.04 x 107
1.02 x 108
5.48 x 101
4
0.971
1.8 x 108
3.0 x 107
2.66 x 108
3.95 x 101
5
0.979
1.29 x 108
3.4 x 107
1.23 x 108
1.55 x 101


29
. SSP
LV
Figure 14. Variation of the length of lines of intersection of grain
boundaries and the pore-solid interface (L$SP) with the
corresponding value for the random intersection of the
abovementioned surfaces (LR) for spherical copper powder
loose stack sintered at 10T5C.48


126
Figure 62. Effects of channel closure and surface rounding on L,
the average length of a branch.


CHAPTER 4
CONCLUSIONS
Introduction
The discussion of the results of this investigation is summarized
in a number of conclusions; an outline of suggested research is also
presented.
Conclusions
1) The porosity can be modeled as composed of a set of networks
of cylindrical channels and a collection of monosized isolated spherical
pores during the advanced stages of loose stack sintering, hot pressing
and conventional sintering.
2) During loose stack sintering, a highly interconnected network
of branches and nodes disintegrates into simpler subnetworks which sub
sequently break up to form the isolated pores. The connected or tree-like
pores continue shrinking until Vy r 0.95 when the rate of removal of these
pores becomes significantly slow for the rest of the range of observation,
up to Vy s 0.98.
3) Isolated porosity, on the other hand, goes through a maximum and
diminishes to a very low value when connected porosity is observed to have
been more or less stabilized.
4) The onset of stabilization of connected porosity is coincident
with an arrest of grain growth and rapid reduction in the isolated porosity.
137


51
0.8 0.9 1.0
Vy (Metallographic)
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 1250C.


139
12) It is suggested that this higher boundary area brings the
onset of grain coarsening at a lower density, well before the pores
begin to isolate. The early grain growth in PS leaves only a small
opportunity for subsequently isolated pores to associate with the moving
boundaries. Thus, an absence of an arrest in grain growth is attributed
to the onset of grain growth well before that of isolation events.
Suggestions for Further Study
It should be apparent from the preceding discussions that topologi
cal analysis of the advanced stages of hot pressing and conventional
sintering would resolve the speculations about a higher number of iso
lated pores in the hot pressed and sintered sample. The associated fractions
of isolated and connected porosity during all three processes, when char
acterized, would facilitate the mechanistic study of the advanced stages.
Thus, the course of further research is outlined as follows.
T) An etching procedure should be developed that will facilitate
the determination of associated fractions of isolated and con
nected porosity.
2) These fractions should be measured on sections in the series
studied for topological, characterization.
3) The isolated and connected fractions should be determined on
a section with calibrated polish.
4) The advanced stages of hot pressing and conventional sintering
should be characterized regarding the topological properties
of the pore structure.


80
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.


24
p
these curves become deeper and shift towards lower Vy with increasing
22
pressure, as shown in Figures 11 and 12.
SS
The.grain boundary area per unit volume, Sy > increases until a
SS
network is formed; subsequent grain growth tends to decrease Sy .
18
This was observed for loose stack sintering, as shown in Figure 13.
SS P
It is evident here that the variation of Sy with Vy is independent of
the initial particle shape in the late second and early third stages.
SSP P 22
Ly increases with decrease in Vy until the second stage is reached
SSP
when it begins to decrease. In the second stage Ly is significantly
48
higher than the case for random intersection of "SP" and "SS" surfaces,
as illustrated in Figure 14.
A new metric property, 1^, was discovered in the course of doctoral
22
research carried out by Gehl at the University of Florida; 1^ 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
22
found to decrease smoothly in the second stage 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
22 S SS
some detail by Gehl. There were two parameters, C and C defined
for grain contiguity and grain face contiguity, respectively. Four
unitless parameters, F-j, Fg, Fg and F^, were used to characterize grain
22
and pore shapes. These were defined as follows.


41
Table 4
CHEMICAL COMPOSITION OF
INCO TYPE 123 NICKEL POWDER
Element
Nickel Powder (Wt.%)
Carbon (typical)
0.03-0.08
Carbon
0.1 max
Oxygen
0.15 max
Sulphur
0.001 max
Iron
0.01 max
Other Elements
trace
Nickel
Balance


121:
Type
One-Three
Three-Three
Table 16
TYPES OF BRANCHES
Order of Events Observed During Serial Sectioning
New appearance + Branching
New Appearance -* Connection
Branching -* End of a branch
Connection -* Bottom end
Branching Bottom end
Connection Branching
Branching -* Branching
Connection -* Connection
Branching - Connection


127
in a sense, to estimate L and R. The calculated and estimated (with
surface correction) values of Vy0nn, Sy0nn and My0nn are listed in
Table 18. The estimation of the above values was made by using two
conn P P
experimentally determined values, such as Vy /Vy and Vy; the confidence
intervals therefore were relatively large. High sample surface to volume
ratio led to significant amounts of corrections, both in estimated and
measured values. In light of the difficulties and the geometric simpli
city of the model the agreement between measured and calculated values
seems to indicate that the modeled geometry is qualitatively representa
tive of the real microstructure.
SP
It was said earlier in this section that the linearity between Sy
and Vy probably can be attributed to similar area to volume ratios for
pore channels and isolated pores. From the Appendix, for a tubular
channel with length = T and radius r = (/3/2)R, the area to volume
ratio, Ay, is given by
a = 2tttL = 2 _4_ = 2J1
V irr2L r n/3R R
(23)
as illustrated in Figure 63. For an isolated spherical pore, Ay = 3/R
which is not far from 2.31/R. However, the curvature to volume ratios,
Cy's, are significantly different. For a spherical pore, Cy vis given by
Cy (spherical pore) = ^4 = -4> (24)
-2TrRJ R
and Cy for a tubular channel is given by
C
V
(tubular channel) =
ttL
2r
irr L
1
2
r
4 = 1.33
3R2 R2
(25)


12
branches that can be cut without creating a new isolated part. If
b = number of branches, n = number of nodes, P0 = number of separate
parts, then
P1 = b n + PQ (1)
29
The first Betti number of the network, P-j, 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
30 31
Rhines and Schwarzkopf 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


76
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 136 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 Geotech 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
25
Aigeltinger. The abovementioned investigation dealt with loose stack


141
19. R. A. Gregg and F. N. Rhines, Met. Trans., 4, 1365 (1973).
20. R. T. DeHoff, R. A. Rummel, H. P. LaBuff and F. N. Rhines,
Modern Developments in Powder Metallurgy, p. 310, Plenum Press,
New York (1966).
21. W. D. Tuohig, Doctoral Dissertation, University of Florida (1972).
22. S. M. Gehl, Doctoral Dissertation, University of Florida (1977).
23. A. S. Watwe and R. T. DeHoff, unpublished research.
24. J. S. Adams and D. Glover, Metal Progress, August (1977).
25. E. H. Aigeltinger, Doctoral Dissertation, University of Florida (1969).
26. R. T. DeHoff and F. N. Rhines, eds., Quantitative Microscopy, McGraw
Hill Book Co., New York (1967).
27. L. K. Barrett and C. S. Yust, ORNL Report, No. 4411 (1969).
28. L. K. Barrett and C. S. Yust, Metallography, 3^, 1 (1970).
29. S. S. Cairns, Introductory Topology, The Ronald Press Company,
New York (1961T!
30. F. N. Rhines, Powder Met. Bull., 3^, 28 (1948).
31. P. Schwarzkopf, Powder Met. Bull., 2 74 (1948).
32. F. Thummler and N. Thomma, Met. Review, 12, 69 (1967).
33. R. L. Coble, J. Appl. Physics, 32, 787 (1961).
34. L. K. Barrett and C. S. Yust, Trans. Met. Soc. AIME, 239, 1172 (1967).
35. R. T. DeHoff and F. N. Rhines, Final Report, AEC Contract AT(40-1),
2581 (1969).
36. F. N. Rhines, University of Florida, private communication.
37. G. C. Kuczynski, Acta Met., 4, 58 (1956).
38. P. J. Wray, Acta Met., 24, 125 (1976).
39. W. D. Kingery and B. Francois, Sintering and Related Phenomena,
p. 471, Gordon and Breach Publishers, New York (1965).
40. A. J. Markworth, Met. Trans., £, 2651 (1973).
41. W. Trzebiatowski, Zhurnal Physik Chem., B24, 75 (1934).


8
Table 1
BASIC METRIC PROPERTIES
Feature
Property
Definition
Lines
LV
Length of a linear feature per unit
volume.
Surfaces
sv
Area of a surface per unit volume.
H .
rl r2
Local mean curvature of a surface
ata point on the surface, where
r-i and rz 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.
Mv =// HdS
SV
Integral mean curvature of a surface
per unit volume.
la
Length of a trace of surface per
unit area of a plane section.
Regions of Space
VV
Volume fraction or volume of a
particular region of space per
unit volume.


Figure 31. Photomicrograph of INCO 123 nickel powder hot pressed
at 1250C to Vy = 0.968, etched (approx. 400 X).


Ill
Figure 56. Geometric model of connected and isolated porosity
during the advanced stages of loose stack sintering
of nickel at 1250C.


67
Figure 35. Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 1250C to Vy = 0.975,
etched (approx. 400 X).


161
If by(corr) is defined as the corrected number of branches per unit
volume, T(corr) as the corrected average length of a branch and TT(add) as
the weighted average of the length of additional branches, then
by(corr)L(corr) = by*L + by(add)*L(add)
(63)
where by and T refer to the previously measured values of branches per
unit volume and their average length. The values of by(corr), TT(corr),
Nyb(corr), Ny^ and R1S0 are listed in Table 22; these were used to cal
culate the metric properties of the connected porosity. The surface
corrections used to modify the measured values of the metric properties
of the connected porosity are described presently.
It can be seen from equation (56) that the additional number of
isolated pores per unit volume, Ny (add), is given by
,iso
(64)
N
Miso, 1 S _V DS
Ny (add) 2 Ny 2V
. . 1_ 2V
Since VyS0, SyS0 and MyS0, the metric properties of isolated poro
sity, are directly proportional to the number of isolated pores, the
additions to these properties are given by
.ISO
Vjs0(add) = 4
N
SO
N\/S(add)
or
VwS0(add) = vjso {^4
i DS
1 2V
(65)
Similarly,
sjS0(add) = sjS0
v v 1-DS
W
(66)


119
properties of the connected porosity have also to be corrected; these
corrected values are listed in Table 15. The remaining two parameters
of the model, T and R, were estimated as follows.
Since the model has only one-branch and three-branch nodes, the
branches can be either one-three of three-three type. (One-one type
branches are isolated pores and thus are not included in the connected
porosity.) These branches can be classified further, as shown in Table
16 and illustrated in Figure 60.
Since the procedure described in Chapter 3 involves recording of
all these events, it was possible to measure the number and apparent
lengths of each type of branches. As illustrated in Figure 61 for the
case of randomly oriented branches, the true length of a branch is twice
-IT
the average of lengths measured at different orientations. Thus L and
33
L the true average lengths of 1-3 and 3-3 type branches can be esti
mated from the separation between the pertinent events. The average
lengths of 1-3 and 3-3 branches that cross the boundaries can be taken
as twice the value measured, since for randomly oriented branches the
average length of the part of the branch that is contained in the volume
of analysis is half of its true length. The overall weighted average of
all types of branches yields the required parameter, L.
For a collection of isolated, equiaxed pores, VyS0, the volume
fraction of isolated porosity, is given by
(21)


28
2S
SS
CJ =
2sf+sSp
(2)
,ss
3L
SSS
SSS., SSP
JLv +LV
(3)
F, =
9I SSP.-, SSS
Lv JLv
(sgP+2sgS)2
(4)
F2 = 2Lf P/(sf )2
(5)
F~ =
L^Lg3
2(sgS)2
(6)
F4 = LgSP/2(sgS)2
(7)
The fraction of the total area of solid grains shared with other
c S P
grains is given by C Variation of C with Vy for hot pressed samples
of U02^22^ and loose stack sintered spherical and dendritic copper
pc
powder is shown in Figure 15. It can be seen from the definition of
c c cc
C that high C values indicate high Sy ; this was believed to arise from
polycrystallinity of the particles. It is apparent that as third stage
p
(Vy = 0.1) is approached all data tend to fall on a single curve. Pre
ss S
compaction seems to increase Sy and hence exhibits higher values of C


Copyright 1983
by
Arunkumar Shamrao Watwe


75
Table 8
LOOSE STACK SINTERED SAMPLES USED FOR TOPOLOGICAL CHARACTERIZATION
Number
Sintering Time
Volume Fraction of Solid
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


91
Figure 48. Convention used in the net tangent count (Twet) .
during serial sectioning. v


Figure 18. 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


31
P SP
at the same Vy when compared to a loose stack sintered sample; Sy
c
was found to vary linearly with C and the dependence was the same
for widely different precompaction pressures up to very late second
C
stage. For hot pressed samples, a maximum was observed in C believed
to indicate a point where the grain boundary area has increased enough
SSP
to form a boundary network that subsequently coarsens. Both Ly and
ccc c
Ly exhibited a maximum when plotted versus C for conventionally
sintered and hot pressed samples.
SS
The grain face contiguity parameter, C indicates the fraction
of edge length of grain faces that is shared with other grains. It
22 S SS
can be shown that C = C for the case of random intersection of
grain boundaries and pore-solid interface, and C > C when grain
boundaries intersect pore-solid interface preferentially. For conven
ts
tionally sintered and hot pressed samples C was observed to be greater
SS
than C which indicated preferential association of grain boundaries
with the pore-solid interface.
The factors F-|, Fg and F^ can be used to compare the grain shapes
and ?2 the Pore shapes; F-|, Fg, Fg and F^ were observed to be weakly
P
linear with Vy, whereas a strong correlation was observed between Fg,
C
Fg and F^ and C for all the samples. It was believed that the above
data indicate a strong influence of the extent of grain contiguity (C )
on the grain and pore shapes.


100
Figure 53. Dependence of the integral mean curvature of the connected
porosity on the volume fraction of solid during loose stack
sintering of nickel at 1250C.


2
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
occur,
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 dependences derived in the model studies are influenced
by the geometric detailsJ^~^ 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. As pointed
out by DeHoff et al.,^ 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
particle size.
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
isties, temperature and external pressure would be very helpful in the
control of sintering aimed at desired end properties of the components.


32
Topological Studies
12
It was found that the connectivity or genus, G, stays
nearly constant during the first stage. More precise measurements
18
made by Aigeltinger and DeHoff 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
18
irregularly shaped powders, it was observed that G decreases during
the first stage, due to coalescence of multiple contacts between particles
cp p
During the second stage, Sy decreased linearly with decrease in Vy-,
the slope of this line was found to be proportional to Gy, genus per unit
20
volume as should be expected from dimensional analysis. Kronsbein et
49
al. carried out serial sectioning of sintered copper samples and found
D
that even for Vy = 0.1, very few pores were isolated. This is in agree-
34
ment with similar observations made by Barrett and Yust.
18
Aigeltinger and DeHoff 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
p
number of isolated pores with Vy revealed a definite increase in the
former during the first stage and identified the end of the second stage
(Gp s 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)
p
begins at a higher value of Vy as compared to spherical powder, Figure
17. The initial decrease in Gp during the first stage for dendritic


152
Figure 68. Height of the spherical cap in an extended, spherical,
three-branch node.


45
Density Measurements
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 (W-|), it was coated with wax
and weighed again (Wg). The wax-coated sample was placed in the minia
ture pan in a beaker of distilled water and weighed (Wg). The sample
was then dropped to the bottom of the beaker by gently tilting the pan.
The weight of partially immersed pan was measured (W^). The density of
the sample, p, was calculated as follows:


122
New Appearance
Branching
(b)
Figure 60. Various types of a) one-three and b) three-three branches
and the topological events signifying each type of branch.


78
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, were measured 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


49
Table 7
STANDARD RELATIONSHIPS OF STEREOLOGY
PL = 2 SV
PA = 2 LV
"tA = 2ttNa = MV
(9)
00)
(ID
02)


135
Once most of these are eliminated the pores isolated thereafter have
only a limited opportunity to intersect the moving boundaries. Since
SSP SSS
Ly ancj l_v(occ) quantify the degree of association of porosity with
the grain boundary network, it is apparent that initially a pressed and
sintered sample has a higher amount of associated porosity that decreases
rapidly. Before the onset of isolation processes, most of the porosity
is in the form of an interconnected network mostly associated with grain
edges. If grain coarsening begins well before an appreciable number of
pores are isolated, this initially high associated porosity would decrease
rapidly since the connected pores cannot anchor the boundaries effectively
and are consequently disassociated. The significance of a topological
study of the advanced stages of conventional sintering is stressed here
as these measurements would characterize the associated fractions of iso
lated and connected porosity and thus would test the postulates put for
ward earlier in this section.
Comparison of Loose Stack Sintering with
Hot Pressing and Conventional Sintering
If the paths of evolution of microstructure during these processes
are examined together, a number of general processing parameter-micro-
structure relationships become apparent. Since these relationships have
a potential as possible strategies to control the microstructure and hence
the service properties of a powder-processed component, they are of both
theoretical and practical interest; they are listed below.


92
Figure 49. Dependence of connectivity on the volume fraction of solid
during loose stack sintering of INCO 123 nickel powder at
1250C.


84
Table 9
OBSERVABLE TOPOLOGICAL EVENTS
Appearance AGmax AGmin AN1S0
Whole
Subnetworks
Internal
0
0
+1
I
Disappearance
External
-1
0
0
II
Within a
Appearance of a branch
+1
0
0
Subnetwork
Disappearance of a branch
-1
0
0
III
Between
Subnetworks
Different or
Connection
Same
new 0
+1
0
+1
0
0


115
Figure 58.
Illustration of pores
of analysis and cross
that terminate within the volume
the surface, or "IS" branches.


148
Table 19
PARAMETERS OF THE MODEL
Feature
Notation
Definition
Branches
J3
bV
Number of one-three branches per unit
volume
h33
bV
Number of three-three branches per unit
vol ume
Nodes
Number of one-branch nodes per unit
volume
nf
Number of three-branch nodes per unit
volume
Isolated Pores
nJS0
Number of isolated pores observed to
be wholly contained, per unit volume


157
Figure 70. Caliper diameters for a convex body. D, the mean
caliper diameter, is the average of D's over all
possible orientations of measuring planes.


External Node
Figure 47. Illustration of the contribution of a branching event
towards Gmax.


16
I
Figure 5. Illustration of neck growth and impingement of growing
necks during the first stage of sintering.


Table 10
CURRENT ESTIMATES OF Gmax, Gmin and Niso AFTER SECTION #J
Height of Sample, Microns
Counts
.max
gmin
Previous total --
This section
Current total
Unit volume
value
_ Current total
Current volume


108
into association with fractions of isolated and connected porosity.
Since only simple, isolated pores pin the boundaries effectively, the
grain boundary network is stabilized when it finds itself mostly asso
ciated with isolated pores. By this time, most of the connected pores
are disassociated,as they do not anchor the boundaries. The isolated
porosity that is associated with the boundaries is reduced as the
boundaries provide vacancy sinks for the necessary material transport.
The balance between the number of isolated pores that disappear and the
number of pores isolated by channel closures helps maintain a sufficient
number of isolated pores that is associated with the boundaries that
renders them immobile. Once most of the associated pores have disappeared,
the boundaries become free to migrate and are not pinned by the remaining
connected porosity. This slows the reduction of connected porosity con
siderably.
It is apparent from the preceding discussion that the pore phase
in the advanced stages of loose stack sintering must be modeled as
composed of isolated and connected fractions that vary in a manner
IQ oo OC CO
described above. Thus, the models that involve only connected 5
33 37 54 55
or isolated porosity * are not appropriate for describing the
evolution of microstructure studied in this investigation. mechanis
tic model satisfying the abovementioned geometry requirements may be
devised, at least in principle, if grain boundary-associated fractions
of connected and isolated porosities are measured during the advanced
stages of loose stack sintering. These fractions were not measured
since the sections studied for topological analysis could not be etched
without affecting the pore features. Thus, it was not possible to devise


35
powder is in agreement with higher C = 14 for the initial stack than
_ p
C = 4 at Vy = 0.55. It has been argued that in second stage, on account
of fewer pore channels in the sample sintered from dendritic powder,
p
isolation of pores begins at a higher Vy 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
p
where Vy 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
P
in the range Vy = 0.1 and lower; this requires that the samples
in the series have Vy 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 meeasurements are very tedious in any case.


I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate,
in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
R. T. DeHoff, Chairmar
Professor of Material'Science
and Engineering
I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate,
in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
L n ¡ip
V ..
R. E. Reed-Hill
Professor of Materials Science
and Engineering
I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate,
in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
Professor of Materials Science
atar Engineering
I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate,
in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
Inoda, Jr.
Professor of Materials Science
and Engineering


68
Figure 36. Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 1250C to Vy = 0.983,
etched (approx. 400 X).


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70
X/
Figure 38. 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 1250C.


10
Table 3
METRIC PROPERTIES OF A SINTERED STRUCTURE
Feature
Property
Description
Pore-Solid Interface
CSP
bV
Area of pore-solid interface
per unit volume
MSP
Mv
Integral mean curvature of
pore-solid interface per uni
volume
Porosity
Vp
Vv
Volume fraction of porosity
Solid Phase
VS
Vv
Volume fraction of a solid
Grain Edges in the
Solid
, sss
Lv
Length of grain edges or
triple lines per unit volume
Lines Formed by the
Intersection of Grain
Boundaries and Pore-Solid
Interface
, SSP
Lv
Length of intersection lines
of pore-solid interface and
grain boundaries per unit
volume
Grain Edges Occupied
by the Pore Phase
, sss
LV(occ)
Length of occupied grain
edges per unit volume


METRIC AND TOPOLOGICAL
CHARACTERIZATION OF THE ADVANCED
STAGES OF SINTERING
By
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
1983

Copyright 1983
by
Arunkumar Shamrao Watwe

Dedicated
To My Parents,
Mr. Shamrao Vasudeo Watwe
and
Mrs. Sharada Shamrao Watwe

ACKNOWLEDGEMENTS
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.
iv

TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS iv
ABSTRACT vii
INTRODUCTION 1
CHAPTER
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
Introduction 39
Sample Preparation 39
Metallography 46
Topological Measurements 74
THREE DISCUSSION 101
Introduction 101
Loose Stack Sintering 101
Hot Pressing 130
Conventional Sintering 134
Comparison of Loose Stack Sintering with Hot Pressing
and Conventional Sintering 135
FOUR CONCLUSIONS 137
Introduction 137
Conclusions 137
Suggestions for Further Study 139
REFERENCES 140
APPENDIX THE GEOMETRIC MODEL OF THE PORE PHASE 143
Introduction 143
v

PAGE
Parameters of the Model 143
Metric Properties of the Connected Porosity 149
Surface Corrections 155
BIOGRAPHICAL SKETCH 163
VI

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
By
Arunkumar Shamrao Watwe'
August, 1983
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
significantly different.
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
isolated pores.
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
isolation events.
viii

INTRODUCTION
Sintering is a coalescence of powder particles into a massive formj
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
2-9
of the pore phase. 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.^'frie traditional
or mechanistic approach involves the study of kinetics and mechanisms of
^Im
material transport;1the 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
separate parts.
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 mechanism(s) of interest.
3) These equations are compared with the experimentally observed
time dependences of the chosen parameter and an attempt is made
to identify the operating mechanism(s).
1

2
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
occur,
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 dependences derived in the model studies are influenced
by the geometric detailsJ^~^ 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. As pointed
out by DeHoff et al.,^ 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
particle size.
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
isties, temperature and external pressure would be very helpful in the
control of sintering aimed at desired end properties of the components.

3
Consequently, a major school of thought prevails that favors the geometri
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.^ 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
C ip 1 0_0*|
sintering, conventional sintering-cold pressing followed by
sinteringD,<:, and hot pressing*c ~ 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
24
or lower.

4
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-
25
cal measurements are time-consuming and since an earlier doctoral research
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.

CHAPTER 1
EVOLUTION OF MICROSTRUCTURE DURING SINTERING
Introduction
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
26
lines, surfaces and regions of space averaged over the whole structure.
The basic properties are listed in Table 1 and illustrated in Figure 1.
Among the properties listed, Vy, Sy and My 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
5

6
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
24
closed surfaces, 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
27
openings or creating new openings, until it collapses into the said
28
network that can be represented in the form of a simple line drawing.
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

7
Figure 1. Illustration of basic metric properties.

8
Table 1
BASIC METRIC PROPERTIES
Feature
Property
Definition
Lines
LV
Length of a linear feature per unit
volume.
Surfaces
sv
Area of a surface per unit volume.
H .
rl r2
Local mean curvature of a surface
ata point on the surface, where
r-i and rz 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.
Mv =// HdS
SV
Integral mean curvature of a surface
per unit volume.
la
Length of a trace of surface per
unit area of a plane section.
Regions of Space
VV
Volume fraction or volume of a
particular region of space per
unit volume.

9
Table 2
DERIVED METRIC PROPERTIES
Feature
Property
Definition
Surface
- MV
H =-r
Average mean curvature of
sv
a surface
Region of Space
x = 4sVv
Mean intercept in a particular
region of space

10
Table 3
METRIC PROPERTIES OF A SINTERED STRUCTURE
Feature
Property
Description
Pore-Solid Interface
CSP
bV
Area of pore-solid interface
per unit volume
MSP
Mv
Integral mean curvature of
pore-solid interface per uni
volume
Porosity
Vp
Vv
Volume fraction of porosity
Solid Phase
VS
Vv
Volume fraction of a solid
Grain Edges in the
Solid
, sss
Lv
Length of grain edges or
triple lines per unit volume
Lines Formed by the
Intersection of Grain
Boundaries and Pore-Solid
Interface
, SSP
Lv
Length of intersection lines
of pore-solid interface and
grain boundaries per unit
volume
Grain Edges Occupied
by the Pore Phase
, sss
LV(occ)
Length of occupied grain
edges per unit volume

77

12
branches that can be cut without creating a new isolated part. If
b = number of branches, n = number of nodes, P0 = number of separate
parts, then
P1 = b n + PQ (1)
29
The first Betti number of the network, P-j, 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
30 31
Rhines and Schwarzkopf 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

13
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
32
pore-solid interface decreases with a moderate amount of shrinkage.
Throughout this stage, the pore-solid interface has many redundant con
nections J
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
1 6 *3*3
porosity is in the form of an interconnected network of channels *
that delineate the solid grain edges. The continued reduction in the
volume and the area of porosity is accompanied by a decrease in the
1 35
connectivity of the pore structure. The decrease in the connec
tivity can be explained by either removal of solid branches or closure
36
of pore channels. According to Rhines, 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 stage 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.16,34
30 31
The third stage has begun by the time most of the pores are isolated.
The connectivity of a pore network is now a very small number.1 Coarsening
16 18 35 37-39
proceeds along with the spheroidization of pores * so that the
volume of porosity, the number of pores and pore-solid interface area

14
Figure 3
Some closed surfaces and their deformation retracts
(dotted lines).

15
Figure 4. Illustration of a one-to-one correspondence between a
spurious node and a spurious branch in a deformation
retract.

16
I
Figure 5. Illustration of neck growth and impingement of growing
necks during the first stage of sintering.

17
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.^^0 gas ¡ias en0Ugh pressure to stabilize
16 41 42
the pore-solid interface, the densification rates can be very low.
Since exaggerated or secondary grain growth that results from boundaries
43 44
breaking away from pores has been observed to be accompanied by slow
rates of shrinkage,43,45-47 has been theorized that the grain boundaries
that can act as efficient vacancy sinks are far away from a large number of
43 45-47
pores. 5 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.
Metric Investigations
SP
It has been observed that in loose stack sintered samples Sy
decreased linearly with the decrease in Vy^2*1820) (jur-jng second
stage. Surface area may be reduced both by densification and surface
SP
rounding or by surface rounding alone; the linearity between Sy and Vy
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, as shown schematically in Figure 7. The slope of the
SP 20
Sy versus Vy line is inversely proportional to the initial particle size.

18
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.

19
There is evidence to suggest that this path of evolution of microstruc-
21
ture for loose stack sintering is insensitive to temperature.
SP
Data for hot pressed samples indicate that the Sy -Vy relationship
22 23
is only approximately linear even in the late second stage. The
path of microstructural change was also found to be insensitive to tem-
22 23
perature. The effect of pressure on the path was significant;
increasing pressure delayed the approach to linearity until a lower
p
value of Vy, as shown in Figure 8.
Integral mean curvature per unit volume, My, 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-
18 21 22
vex pore has a negative curvature. There is a miniimum in.*.My; *
35
this minimum occurs at lower Vy for finer particle size, 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 My in
the second stage. As the sintered density approaches the theoretical
density, My must approach zero and hence the initially high positive
value of My that becomes negative must go through a minimum. For an
p
initial stack of irregularly shaped particles, My varies with Vy at a
slower rate and has a minimum earlier in the process, compared to an
25
initial stack of spherical powders. 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 My is much more negative and
P 23
occurs at a lower value of Vy, compared to a loose stack sintered sample;

20
Figure 7. 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

21
Figure 8. Pore-solid interface area versus solid volume fraction for
conventionally sintered

22
M
SP
V
Figure 9. Variation of integral mean curvature per unit volume with
the volume fraction of solid for three representative copper
powders sintered in dry hydrogen at 1005C.35

23
Figure 10. Integral mean curvature versus volume fraction of
solid for 48 micron spherical and dendritic copper
powder.25

24
p
these curves become deeper and shift towards lower Vy with increasing
22
pressure, as shown in Figures 11 and 12.
SS
The.grain boundary area per unit volume, Sy > increases until a
SS
network is formed; subsequent grain growth tends to decrease Sy .
18
This was observed for loose stack sintering, as shown in Figure 13.
SS P
It is evident here that the variation of Sy with Vy is independent of
the initial particle shape in the late second and early third stages.
SSP P 22
Ly increases with decrease in Vy until the second stage is reached
SSP
when it begins to decrease. In the second stage Ly is significantly
48
higher than the case for random intersection of "SP" and "SS" surfaces,
as illustrated in Figure 14.
A new metric property, 1^, was discovered in the course of doctoral
22
research carried out by Gehl at the University of Florida; 1^ 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
22
found to decrease smoothly in the second stage 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
22 S SS
some detail by Gehl. There were two parameters, C and C defined
for grain contiguity and grain face contiguity, respectively. Four
unitless parameters, F-j, Fg, Fg and F^, were used to characterize grain
22
and pore shapes. These were defined as follows.

25
Figure 11. 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 1005C included for
comparison.23.25

26
Figure 12. The effect of pressure on the path of integral mean
curvature for hot pressed specimens of UOg.^

27
SyP (cm-1)
Figure 13. Grain boundary area per unit volume versus volume fraction
of solid_for 48 micron spherical and dendritic copper .
powder.25

28
2S
SS
CJ =
2sf+sSp
(2)
,ss
3L
SSS
SSS., SSP
JLv +LV
(3)
F, =
9I SSP.-, SSS
Lv JLv
(sgP+2sgS)2
(4)
F2 = 2Lf P/(sf )2
(5)
F~ =
L^Lg3
2(sgS)2
(6)
F4 = LgSP/2(sgS)2
(7)
The fraction of the total area of solid grains shared with other
c S P
grains is given by C Variation of C with Vy for hot pressed samples
of U02^22^ and loose stack sintered spherical and dendritic copper
pc
powder is shown in Figure 15. It can be seen from the definition of
c c cc
C that high C values indicate high Sy ; this was believed to arise from
polycrystallinity of the particles. It is apparent that as third stage
p
(Vy = 0.1) is approached all data tend to fall on a single curve. Pre
ss S
compaction seems to increase Sy and hence exhibits higher values of C

29
. SSP
LV
Figure 14. Variation of the length of lines of intersection of grain
boundaries and the pore-solid interface (L$SP) with the
corresponding value for the random intersection of the
abovementioned surfaces (LR) for spherical copper powder
loose stack sintered at 10T5C.48

30
Figure 15.
The variation of grain contiguity with solid volume
fraction for loose stack sintered copper and hot pressed
U02.22

31
P SP
at the same Vy when compared to a loose stack sintered sample; Sy
c
was found to vary linearly with C and the dependence was the same
for widely different precompaction pressures up to very late second
C
stage. For hot pressed samples, a maximum was observed in C believed
to indicate a point where the grain boundary area has increased enough
SSP
to form a boundary network that subsequently coarsens. Both Ly and
ccc c
Ly exhibited a maximum when plotted versus C for conventionally
sintered and hot pressed samples.
SS
The grain face contiguity parameter, C indicates the fraction
of edge length of grain faces that is shared with other grains. It
22 S SS
can be shown that C = C for the case of random intersection of
grain boundaries and pore-solid interface, and C > C when grain
boundaries intersect pore-solid interface preferentially. For conven
ts
tionally sintered and hot pressed samples C was observed to be greater
SS
than C which indicated preferential association of grain boundaries
with the pore-solid interface.
The factors F-|, Fg and F^ can be used to compare the grain shapes
and ?2 the Pore shapes; F-|, Fg, Fg and F^ were observed to be weakly
P
linear with Vy, whereas a strong correlation was observed between Fg,
C
Fg and F^ and C for all the samples. It was believed that the above
data indicate a strong influence of the extent of grain contiguity (C )
on the grain and pore shapes.

32
Topological Studies
12
It was found that the connectivity or genus, G, stays
nearly constant during the first stage. More precise measurements
18
made by Aigeltinger and DeHoff 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
18
irregularly shaped powders, it was observed that G decreases during
the first stage, due to coalescence of multiple contacts between particles
cp p
During the second stage, Sy decreased linearly with decrease in Vy-,
the slope of this line was found to be proportional to Gy, genus per unit
20
volume as should be expected from dimensional analysis. Kronsbein et
49
al. carried out serial sectioning of sintered copper samples and found
D
that even for Vy = 0.1, very few pores were isolated. This is in agree-
34
ment with similar observations made by Barrett and Yust.
18
Aigeltinger and DeHoff 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
p
number of isolated pores with Vy revealed a definite increase in the
former during the first stage and identified the end of the second stage
(Gp s 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)
p
begins at a higher value of Vy as compared to spherical powder, Figure
17. The initial decrease in Gp during the first stage for dendritic

33
Gp or Np
(106 gm"1)
Figure 16. 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 1005C. Data for 115 micron spherical copper
included for comparision.25

34
Figure 17. 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).2^

35
powder is in agreement with higher C = 14 for the initial stack than
_ p
C = 4 at Vy = 0.55. It has been argued that in second stage, on account
of fewer pore channels in the sample sintered from dendritic powder,
p
isolation of pores begins at a higher Vy 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
p
where Vy 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
P
in the range Vy = 0.1 and lower; this requires that the samples
in the series have Vy 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 meeasurements are very tedious in any case.

36
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 theorized161835,37-39 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
34
that the continuity of the solid phase is maintained, as illustrated in
34
Figure 18. According to Barrett and Yust, 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
43 45-47
pores. 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.

Figure 18. 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

38
Mechanical and physical properties of conmercial porous components
are influenced by the geometry of the porosity. Thermal conductivity is
P
influenced by Vy, pore shapes and the relative fractions of connected and
7 8 4
isolated porosity. Permeability to fluids depends on the connectivity,
p cp g 3
Vy and Sy Mechanical strength and thermal shock resistance depend on
2
pore shapes whereas ductility is influenced by pore shapes and spacings.
Thus geometric characterization of porous structures as a function of
adjustible 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.

CHAPTER 2
EXPERIMENTAL PROCEDURE AND RESULTS
Introduction
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.
Sample Preparation
This section presents the procedure employed to prepare the
sintered samples and the standard for density measurements.
Sintered Samples
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
c
having densities that are uniformly distributed over the range Vy = 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.
39

40
Figure 19. INCO 123 nickel powder used in the present investigation
(1000 X).

41
Table 4
CHEMICAL COMPOSITION OF
INCO TYPE 123 NICKEL POWDER
Element
Nickel Powder (Wt.%)
Carbon (typical)
0.03-0.08
Carbon
0.1 max
Oxygen
0.15 max
Sulphur
0.001 max
Iron
0.01 max
Other Elements
trace
Nickel
Balance

42
Table 5
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 10^ cm^/cm^
5
Vy of As-Received Powder 0.25

43
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
sintering times.
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
tering temperatures.
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
32
change in metal powders.

44
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 1250C 10C.
3. Hot Pressed Series
The third series was prepared by hot pressing at 1250C and under
a pressure of 2000 psi in a CENT0RR 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
-5
was placed in the vacuum chamber. After a vacuum of 10 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 5C 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
individually.

45
Density Measurements
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 (W-|), it was coated with wax
and weighed again (Wg). The wax-coated sample was placed in the minia
ture pan in a beaker of distilled water and weighed (Wg). The sample
was then dropped to the bottom of the beaker by gently tilting the pan.
The weight of partially immersed pan was measured (W^). The density of
the sample, p, was calculated as follows:

46
/, _\ Weight of sample
p^9/ I Volume of sample
Weight of sample
. (Volume of sample + immersed part of pan) -
(Volume of immersed part of pan)
M1
(W2 + weight of pan in air Wg) -
(Weight of pan in air W4)
Thus
W1
p (g/cc) = w2 + w4 w3
The densities thus measured were reproducible within 0.2 percent of
the mean of ten values with 95 percent confidence. The density of a
50
piece of pure nickel, known to have a density of 8.902 g/cc, was
measured and found to be within 0.5 percent of the abovementioned value.
Metallography
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.
Polishing Procedure
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

47
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.
Quantitative Stereo!ogy
Metric properties are estimated by making measurements on a
two dimensional plane of polish with the help of standard relations
pc
of stereology. 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-
pc
tion or structure properties provided the structure is sampled uniformly.
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.) are therefore error-prone to some extent. As this investigation

48
Table 6
QUANTITIES MEASURED ON POLISHED SECTION
Test Feature Quantity
Points Pp
Lines Ll
Area
NA
ta
Definition
Fraction of points of a grid that
fall in a phase of interest
Fraction of length of test lines
that lie in a phase of interest
Number of intercepts that a test
line of unit length makes with the
trace of a surface on a plane
section
Number of points of emergence of
linear feature per unit area of
plane section
Number of full features that
appear on a section of unit area
Net number of times a sweeping
test line is tangential to the
convex and concave traces of
surface per unit area of a plane
section

49
Table 7
STANDARD RELATIONSHIPS OF STEREOLOGY
PL = 2 SV
PA = 2 LV
"tA = 2ttNa = MV
(9)
00)
(ID
02)

50
dealt with relatively small amounts of porosity (10 percent or lower)
P
the error in Vy 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
p
further characterization if the metal!ographically determined Vy was
p
within 15 percent of Vy 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 Vy. Since Vy 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
p
dense and lower exhibited a precision of 0.05 of the Vy value obtained
SP SP
from the density measurements. Manual measurements of Sy and My were
made on the accepted polished surfaces using standard stereological techni-
ques. The measurements of Vy, Sy and My 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
SP SP
20 and 21. Plots of Sy and My contained metallographically measured values
of Vy 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

51
0.8 0.9 1.0
Vy (Metallographic)
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 1250C.

52
Vv (Metal!ographic)
Figure 21. Variation of integral mean curvature with solid volume
during loose stack sintering, hot pressing and conven
tional sintering of INCO 123 nickel powder at 1250C.

53
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
quently, SyS, LySS, LySP and Ly^ccj, 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 Sy, the grain boundary area per unit volume, as measured metal!o-
graphically. However, it was found that these errors are small compared
CD CD .
to those in Sy and My ; the trends of grain structure properties remain
unaffected whether plotted versus Archimedes density or the stereo!ogical
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.

54
Figure 22. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to Vy = 0.871, etched (approx. 400 X).

55
Figure 23. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to Vy = 0.906, etched (approx. 400 X).

56
Figure 24. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to Vy = 0.928, etched (approx. 400 X).

57
Figure 25. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to Vy = 0.944, etched (approx. 400 X).

58
Figure 26. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to Vy = 0.971, etched (approx. 400 X).

59
Figure 27. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to Vy = 0.979, etched (approx. 400 X).

60
Figure 28. Photomicrograph of INCO 123 nickel powder hot pressed
at 1250C to Vy = 0.93, etched (approx. 400 X).

61
Fiaure 29. Photomicrograph of INCO 123 nickel powder hot pressed
at 1250C to Vv = 0.943, etched (approx. 400 X).

62
Figure 30. Photomicrograph of INCO 123 nickel powder hot pressed
at 1250C to Vy = 0.958, etched (approx. 400 X).

Figure 31. Photomicrograph of INCO 123 nickel powder hot pressed
at 1250C to Vy = 0.968, etched (approx. 400 X).

64
Figure 32. Photomicrograph of INCO 123 nickel powder hot pressed
at 1250C to Vy = 0.984, etched (approx. 400 X).

65
Figure 33. Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 1250C to Vy = 0.942,
etched (approx. 400 X).

66
Figure 34. Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 1250C to Vy = 0.962,
etched (approx. 400 X). v

67
Figure 35. Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 1250C to Vy = 0.975,
etched (approx. 400 X).

68
Figure 36. Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 1250C to Vy = 0.983,
etched (approx. 400 X).

69
Vv
Figure 37. Dependence of grain boundary area on solid volume during
loose stack sintering, hot pressing and conventional sin
tering of INCO 123 nickel powder at 1250C.

70
X/
Figure 38. 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 1250C.

71
Vv
Figure 39. 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 1250C.

72
X/
Figure 40. 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 1250C.

Fraction
of pores
Figure 41.
Variation of fractions of pores on the triple edges (filled), on the boundaries (half-filled)
and within the grains (open) for a) loose stack sintered, b) hot pressed and c) pressed and
sintered nickel powder at 1250C.

74
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
described presently.
Topological Measurements
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
analysis.
Serial Sectioning
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
51 25
to avoid redundancy of measurements. Patterson and Aigeltinger
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.
-P
A reliable measure of the scale of the system is X the mean intercept
_p- p cp
of pore phase. Since x = 4VV/Sy the slope of the straight line in
-P
Figure 20 yields the value of X ~ 4.5 microns. Thus the serial sec
tions for LS Series of samples ought to be roughly one micron apart.

75
Table 8
LOOSE STACK SINTERED SAMPLES USED FOR TOPOLOGICAL CHARACTERIZATION
Number
Sintering Time
Volume Fraction of Solid
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

76
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 136 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 Geotech 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
25
Aigeltinger. The abovementioned investigation dealt with loose stack

Ad (section thickness) = d^ Ah = h^ Ah = 0.1428 Ad
Figure 42. 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.

78
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, were measured 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

79
Figure 43. Illustration of contributions of subnetworks crossing
the surface towards the estimate of Gmax.

80
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.

81
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
micrographs.
A set of 4"x5" negatives was obtained by repeating the polishing
and photographic steps. Each was enlarged to a size of 8"xl0" 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 IN. These were regarded as the "internal"
networks and were used to measure the number of separate parts. The

82
Figure 44. Schematic diagram of a typical specimen used in serial
sectioning.

83
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, G111111 and
i so
N The significance of each of such observed events will be
discussed presently.
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
event.

84
Table 9
OBSERVABLE TOPOLOGICAL EVENTS
Appearance AGmax AGmin AN1S0
Whole
Subnetworks
Internal
0
0
+1
I
Disappearance
External
-1
0
0
II
Within a
Appearance of a branch
+1
0
0
Subnetwork
Disappearance of a branch
-1
0
0
III
Between
Subnetworks
Different or
Connection
Same
new 0
+1
0
+1
0
0

85
[(j+l)th section]
Figure 45. Two typical consecutive sections studied during serial
sectioning that illustrate the topological events listed
in Table 9.

86
Figure 46. Contribution of the end of an original subnetwork towards
the estimate of Gmax.

External Node
Figure 47. Illustration of the contribution of a branching event
towards Gmax.

88
When a connection is observed between different subnetworks, the
said subnetworks have to be renumbered to keep track of such connections.
ATI 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 N1S0 were updated and tabulated as shown in Table 10.
The values of interest are the unit volume quantities, G!J!ax, Gyin
i so
and Ny If the features that give rise to these quantities are randomly
25
and uniformly distributed in the sample, then it can be expected that
there exists a quantity (Qy) characteristic of the structure and equal to
the unit volume value. Thus AQ (the change in quantity Q) = Qy x AV.
Dividing both sides by aV leads to
"AV =
The slope of AQ versus AV plot therefore should be equal to Qy provided
AV, the volume covered is large enough for a meaningful sampling. In
25 51
the previous investigations of this kind the analyses were continued

Table 10
CURRENT ESTIMATES OF Gmax, Gmin and Niso AFTER SECTION #J
Height of Sample, Microns
Counts
.max
gmin
Previous total --
This section
Current total
Unit volume
value
_ Current total
Current volume

90
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
^ 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
net
swept through a unit volume, a measure of Ty, 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
pc
per unit volume as follows.
T?et = TV~ + TV+ TV~ = 2 where Ty- = 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
48).
Ty+ = 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.
Ty" = number of times a saddle element (with principal radii of
curvature of different signs by convention) is tangential

91
Figure 48. Convention used in the net tangent count (Twet) .
during serial sectioning. v

92
Figure 49. Dependence of connectivity on the volume fraction of solid
during loose stack sintering of INCO 123 nickel powder at
1250C.

93
0-90 0-95 10
Figure 50. Variation of the number of isolated pores per unit volume
(not corrected for surface effects) with the volume fraction
of solid during loose stack sintering of nickel at 1250C.
Data for the connectivity is included for comparison.

94
to the test plane per unit volume, or the number of
branching and connection events per unit volume (see
Figure 48).
Cbmputation of is shown;,in Table 11.
not
If the constancy;, of T is used as the basis for terminating
the analysis, both N and G are used and so a more general criterion is
obtained. The serial sectioning was terminated for each sample after
Tye* was found to level off. The values of the parameters G^ax, G^n
and NyS0 are listed in Table 12. Since Gyax depends on the number of
pores intersecting the surface that encloses the volume of analysis, it
is very sensitive to the surface to volume ratio of the "slice" of the
sample used for analysis. Therefore, G^ax does not exhibit a monotonic
decrease with an increase in Vy. On the other hand, Gyin is unaffected
by the surface to volume ratio mentioned above and hence can be used to
draw meaningful inferences about the variation of connectivity. The
variation in Gyin and NyS0 are illustrated in Figures 49 and 50, respec
tively. It can be seen that even at 98 percent of the bulk density the
pore structure has a finite connectivity and thus cannot be regarded as
a collection of simple isolated pores.
Behavior of Connected and Isolated Porosity
A natural reaction to these data would be to wonder what fraction
of porosity is connected and how it varies with the total Vy. It is
possible to distinguish connected porosity from the isolated pores on
any of the series of sections used for topological analysis. Thus, the
volume fraction, surface area and integral mean curvature of the connected

Table 11
COMPUTATION OF T
net
V
Number of Number of Bottom T~~ = Top Ends
Section # Top Ends Ends = A + B + Bottom Ends
A: Disappearance
of internal &
external sub
networks
B: End of a Branch
J+l
J+2
T+ = Branchings __
and Connections TeL = T T
10
cn

96
Table 12
TOPOLOGICAL PARAMETERS
lin,La ,,S rmax -3 rmin -3 ..iso -3
Number Vy Gy 9 cm Gy 9 cm Ny 9 cm
0.906
1.71 x 108
4.63 x 107
9.55 x 10
0.928
1.39 x 108
1.7 x 107
2.51 x 10
0.944
2.17 x 107
7.23 x 106
4.63 x 10
0.971
3.23 x 107
5.77 x 106
3.33 x 101
0.979
7.6 x 107
3.18 x 106
1.39 x 10(
5

97
porosity, designated respectively as Vy0nn, Sy0nn and My0nn can be
measured unambiguously. In the present context, "connected" porosity
includes all pores that were not wholly contained in the volume of
analysis. The possibility of measuring the properties of the connected
porosity was not foreseen before all the series of parallel sections
were obtained. Thus, the measurements had to be made on sections with
an uncalibrated polish; the ratios Vy0nn/Vy^a^, Sy0nn/Sy0^ and
My0nn/My0tal, were estimated. These dimensionless ratios, when multi
plied respectively by Vy, Sy and My values determined previously on
calibrated polished surfaces, yield the estimated values of Vy0nn,
Sy0nn and My0nn. It is thought that the errors introduced by noncali-
brated polished surfaces are at least partially compensated for by
measuring the abovementioned ratios rather than the absolute values.
These are illustrated in Figures 51, 52 and 53. These quantities were
estimated from the measurements made on uncalibrated sections; also the
amount of isolated porosity in the 98 percent dense sample was too low
to yield a sufficient number of measurements for unbiased estimates of
metric properties of isolated porosity. Therefore, only the trends,
not the actual numbers (at least for the 98 percent dense sample) are
significant. It can be seen that after the porosity is reduced to about
0.06, the connected porosity changes very little, whereas the isolated
porosity continues to decrease.
The results of the present investigation, briefly mentioned in
this chapter, are discussed in detail in the next chapter.

98
w conn
vV
or
V
iso
V
p
Vv (metallographic)
Figure 51. Dependence of the metallographically determined volume
fraction of connected porosity on the volume fraction of
porosity during loose stack sintering of nickel at 1250C.
Data for the volume fraction of isolated porosity included
for comparison.

99
t-conn
(cm-1)
Vy (Metallographic)
Figure 52. Dependence of surface area of the connected porosity
per unit volume on the volume fraction of solid during
loose stack sintering of nickel at 1250C.

100
Figure 53. Dependence of the integral mean curvature of the connected
porosity on the volume fraction of solid during loose stack
sintering of nickel at 1250C.

CHAPTER 3
DISCUSSION
Introduction
Sintering of a loose stack of powder is the simplest consolidation
process and hence forms the basis for investigation of more complex and
involved techniques. Loose stack sintering was therefore studied in the
greatest detail possible in this investigation. It will be discussed at
length in the beginning of this section to establish a framework on which
the descriptions of hot pressing and pressing and sintering (also called
conventional sintering) will be based. This discussion will be concluded
with a number of speculations regarding potential strategies to control
the microstructure of a powder-processed component.
Loose Stack Sintering
The discussion of metric properties will be followed by that of
topological properties. It is expected that this study will indicate
a likely scenario of various geometric events that lead to the observed
paths of evolution of microstructure. Usually such a detailed study
also suggests a variety of plausible geometric models of the structure;
this study is no exception. The model that is in the best agreement
with the data will be described and followed by suggestions for further
research necessary to complete an understanding of the process.
101

102
Metric Properties of the Pore Structure
SP
It is evident from Figure that Sv is linear with V,, during the
v v
entire range of observation. Prior to this investigation, it was antici
pated that this linear relationship is a manifestation of continued removal
of pore sections of constant surface to volume ratio. In this scenario,
a pore section disappears as soon as it attains a certain surface area to
volume ratio, because gradual shrinkage would increase the surface to
volume ratio which is inversely related to the size of the pore. Since
the porosity consists of connected and isolated fractions during the late
second and third stages, this hypothesis must be taken to mean that the
channels that pinch off, and isolated pores that disappear, do so as soon
as they attain certain area to volume ratio. In this "reverse popcorn"
or "instantaeous removal of pore sections" model the continued removal of
identical pore sections would also mean a linearity between My and Vy, a
speculation not supported by the variation of M.,, Figure T-9.
It is suggested here that this apparent contradiction can be explained
if it is assumed that the pore phase consists of tubular channels and spher
ical isolated pores that have comparable area to volume ratios but signifi
cantly different curvature to volume ratios. If the relative fractions of
these channels and isolated pores vary during the sintering process, the
area would still be linear to the volume but the curvature may not be.
This is discussed further in the section on the geometric model of the
porosity.

103
Metric Properties of the Grain Boundary Structure
SS
In addition to the four parameters mentioned earlier, namely, Sy ,
SSS SSS SSP
Lv Ly^occj and Ly ; the association of pores with the grain boundaries
was also characterized. An etched section of the sample was examined to
measure the number fraction of pores observed to reside within the grains,
on the boundaries and on the grain edges. The variations of these frac
tions are illustrated in Figure 41. These numbers indicate the fractions
of porosity associated with and not associated with the grain boundary
network. The twin boundaries do not participate in grain coarsening and
hence were not included in the characterization of the associated porosity.
The grain boundary area per unit volume, Sy illustrated in
Figure 37, can be seen to change only a little in the density range from
Vy = 0.93 to Vy = 0.97. Thus, the grain growth or decrease in grain
boundary area appears to have been appreciably inhibited in this density
range. It is suggested that the pore structure is changing in such a
manner that the associated porosity is able to pin the boundaries during
this phase of the process. This aspect of the grain boundary structure
will be discussed and explored further in the course of the description
of the other grain boundary properties.
Variations of LySS, LySP and -vfoCC) 1",lustrated in Figures 38, 39
SS
and 40, repsectively, also exhibit the arrest observed for Sy in the same
density range. The fraction of porosity associated with the grain boun
dary network decrease in this range of Vy, Figure 41. Thus, the more
or less stable grain boundary network seems to facilitate the
reduction of associated porosity. The following discussion of

104
topological properties of the pore structure will be used to offer
an explanation for the observed interplay between grain boundary
network and the pore phase.
Topological Properties
Samples with Vy in the range from 0.90 to 0.98 were character
ized to obtain the connectivity and the number of isolated pores per
unit volume, illustrated respectively in Figures 49 and 50. The
connectivity (Gyin) monotonically decreases to a very small value,
but remains finite even at 98 percent density. The numbers of iso
lated pores first increases, Figure 50, as the pore network present
at the end of second stage disintegrates. A large part of the porosity
present at this point consists of rather highly interconnected but
isolated separate parts as opposed to isolated, equiaxed parts. Sub
sequent channel closure rapidly reduces these connected pores to smaller
separate parts with increasing density since, in the absence of any
significant amount of redundant connections, each channel closure event
produces a new separate part. As the sintering proceeds, more and more
parts are isolated; at the same time, some of the simple isolated pores
shrink and disappear. The maximum in the number of isolated pores seems
to indicate that beyond a certain point, very few pores are isolated and
hence from there on the number of isolated pores continues to decrease.
It can be seen from Figure 51 that the volume fraction of isolated poro
sity also goes through a maximum and then decreases to a very small value,
whereas the volume fraction of connected porosity changes only a little

105
after about 95 percent density has been reached. Thus, it can be said
that the isolation of pores from an interconnected network continues
until a point beyond which the residual collection of tree-like pores
(those with very low connectivity) does not change appreciably. The
isolated pores, on the other hand, continue to shrink and disappear.
The preceding discussion of evolution of pore and grain boundary struc
tures is expected to present the overall scenario described below.
The Overall Scenario
It is interesting to note that the density range of effective
pinning of the grain boundary network coincides with that of little
changes in the connected porosity; the major change is in the isolated
porosity, Figure 51. If most of the associated porosity that anchors
the grain boundaries is assumed to be a collection of isolated pores,
the removal of isolated porosity in this density range can be explained
by the proximity of grain boundaries that can act as vacancy sinks. If
a balance exists between the number of isolated pores that disappear and
the number of pores that are isolated as a result of channel closures,
there will be sufficient associated porosity maintained to anchor the
grain boundaries. This suggests that the isolated, equiaxed pores anchor
the grain boundaries much more effectively than do the connected pores.
A hypothesis is offered presently that attempts to rationalize the above
contention.
Grain boundary migration takes place during grain growth that
decreases the grain boundary area. If part of the boundary area is
occupied by second phase particles, as illustrated in Figure 54, the

106
Occupying the Grain Boundary
Figure 54. Illustration of an increase in the grain boundary area
that follows the motion of the boundary occupied by a
second phase particle.

107
motion of these boundaries away from the occupying particles requires
that additional boundary area be created. Since this increases the
surface energy of the boundary network, there is a hindrance to the
52
boundary motion by associated, particles. However, if the pore
geometry is such that the motion of the boundary does not increase
the boundary area, the pores then do not anchor these boundaries
effectively. Thus, the boundary area can be pinned by a tree-like
or connected pore only if the branches in the latter intersect the
boundary with their axes at small angles to the plane of the boundary,
as illustrated in Figure 55. At all other orientations of the connected
pore there is no appreciable increase in the boundary area as it migrates.
On the other hand, a simple isolated pore, equiaxed in shape, is an
effective inhibitor to the grain boundary motion at any orientation.
It is therefore suggested that most of the pores anchoring the boundaries
are simple isolated pores. The relatively small changes observed in the
connected porosity can then be attributed to its inability to associate
itself with the grain boundaries. This sequence of microstructural changes
can be summarized as follows.
In the beginning of the second stage the porosity is mostly inter
connected and associated with the grain edges. The decrease in the
connectivity proceeds along with an increase in the grain boundary area
until a grain boundary network is formed. The subsequent grain growth
has decreased the grain boundary area and the migrating boundaries have
disassociated themselves with a part of the pore network by the time
simple, isolated pores begin to appear. Thus, the boundaries are brought

108
into association with fractions of isolated and connected porosity.
Since only simple, isolated pores pin the boundaries effectively, the
grain boundary network is stabilized when it finds itself mostly asso
ciated with isolated pores. By this time, most of the connected pores
are disassociated,as they do not anchor the boundaries. The isolated
porosity that is associated with the boundaries is reduced as the
boundaries provide vacancy sinks for the necessary material transport.
The balance between the number of isolated pores that disappear and the
number of pores isolated by channel closures helps maintain a sufficient
number of isolated pores that is associated with the boundaries that
renders them immobile. Once most of the associated pores have disappeared,
the boundaries become free to migrate and are not pinned by the remaining
connected porosity. This slows the reduction of connected porosity con
siderably.
It is apparent from the preceding discussion that the pore phase
in the advanced stages of loose stack sintering must be modeled as
composed of isolated and connected fractions that vary in a manner
IQ oo OC CO
described above. Thus, the models that involve only connected 5
33 37 54 55
or isolated porosity * are not appropriate for describing the
evolution of microstructure studied in this investigation. mechanis
tic model satisfying the abovementioned geometry requirements may be
devised, at least in principle, if grain boundary-associated fractions
of connected and isolated porosities are measured during the advanced
stages of loose stack sintering. These fractions were not measured
since the sections studied for topological analysis could not be etched
without affecting the pore features. Thus, it was not possible to devise

109
Figure 55. Illustration of a favorable orientation of a
connected pore for effective pinning of a grain
boundary.

no
a mechanistic model and hence derive any mechanistic conclusions from
the time dependences of geometric properties of the pore microstructure.
However, it was possible to construct a variety of geometric models that
describe the evolution of microstructure qualitatively. These models
incorporate connected and isolated porosity of regular geometry so that
the metric properties such as volume, area and integral mean curvature
can be calculated. The models are tested by comparing the calculated
metric values with the experimentally determined quantities. Since '
these models do not have any parameters that characterize the extent
of association with the grain boundary network, properties that depend
SSP SSS
on this association, such as Ly and Ly(occy could not be calculated
and compared with the measured values. These models are described below.
Geometric Models
All of the geometric models mentioned above describe the porosity
as composed of a collection of isolated, spherical pores of the same
size and a set of networks of cylindrical pore channels and spherical
nodes, as illustrated in Figure 56. The parameters of such a model are
listed in Table 13.
Here it is assumed that the connected porosity has only three-branch
nodes and one-branch nodes; these were measured for a unit volume during
the topological characterization in a manner described presently.
Each branching or connection event defines a node formed as a result
of merging of three pore channels, Figure 57. Thus, the number of three-
branch nodes is given by the number of branching and connection events
observed per unit volume (Ty). The number of one-branch nodes is the

Ill
Figure 56. Geometric model of connected and isolated porosity
during the advanced stages of loose stack sintering
of nickel at 1250C.

112
Figure 57. Illustration of three-branch nodes signified by
branching and connection events.

113
Table 13
PARAMETERS OF A TYPICAL GEOMETRIC MODEL
Nature
Parameter
Definition
Topological
N3b
Number of three branch nodes per unit
volume
Nlb
NV
Number of one-branch nodes per unit
volume
bv
Number of branches per unit volume
nJS0
Number of isolated equiaxed pores
per unit volume
Metric
r
Average length of a branch
R
Radius of an isolated spherical
pore, also assumed to be the "radius"
of a spherical three-branch node (see
Figure 56)

114
number of "ends" (Chapter 2) that are not associated with isolated
pores, since an isolated pore has two "ends" denoted by the appearance
and disappearance of a new pore. As illustrated in Figure 58, among
the pores that cross the boundaries of the volume of analysis, there
are some that have an "end" associated with each of these. Thus, there
is an uncertainty as to how many of these pores that cross the boundaries
are isolated and how many are connected. An attempt was made to intro
duce a correction for the sample surface effects. For a collection of
convex particles (pores, in the present context), the number per unit
volume Ny and the number of features observed on a section are related
26
by the following equation.
na = n
(15)
Where D is the average of the mean caliper diameters of the said parti
cles. Thus, for the volume of analysis with surface area S and volume
c
V, the number of pore features that may cross the surface, N is given
by
N
S = Nv
D S
(16)
The number of isolated pores that cross the boundaries per unit volume,
S
Ny, is therefore given by
S s
NV = T = NV D V
07)
Since these Ny pores per unit volume are not wholly contained in
the volume of analysis, they contribute only half as many pores to the

115
Figure 58.
Illustration of pores
of analysis and cross
that terminate within the volume
the surface, or "IS" branches.

116
collection of isolated pores. Thus, the true number of isolated pores,
Ny, is given by
iy = NyS0(wholly contained) + Ny
, 1so V-S
Nv Ny + 2v
nvo #) = nJS0
ISO
NV =
DS
2V
(18)
09)
1 SO
Where Ny is the number of isolated pores observed to be wholly con-
S
tained in the volume of analysis. Ny is then given by
iso
^5 = n =
V V V
n;
i
DS
2V
DS
V
(20)
S
Thus, among the pores that cross the boundaries, Ny gives the number
of those that are isolated; the rest can be taken to be connected pores.
The treatment presented above therefore also facilitates the estimation
of the number of branches that cross the boundaries per unit volume.
Thus, the number of one-branch nodes can be estimated from Ty Ny ,
, S and V. The number, ofbranches per up it volume, by, cam be calcu
lated as shown in the Appendix, where the model is described in detail.
The values of by, Ny Ny and Ny are listed in Table 14. The variation
i so
of Ny, Figure 59, is qualitatively the same as that of Ny Figure 51.
It should be noted that due to the large surface to volume ratio of the
volume of analysis, the surface corrections change the values of by, Nyb
significantly. Thus, the previously measured values of the metric

117
Table 14
MEASURED VALUES OF THE NETWORK PARAMETERS
No.
vv
b, cm"3
M3b -3
N\/ $ cm
Njb, cm"3
N, cm"3
1
0.906
4.48 x 108
2.1 x 108
2.5 x 108
V
1.07 x 101
2
0.928
2.42 x 108
6.2 x 107
2.63 x 108
3.33 x 101
3
0.944
8.39 x 107
2.04 x 107
1.02 x 108
5.48 x 101
4
0.971
1.8 x 108
3.0 x 107
2.66 x 108
3.95 x 101
5
0.979
1.29 x 108
3.4 x 107
1.23 x 108
1.55 x 101

118
Figure 59. Variation of N\j, surface corrected number of isolated
pores per unit volume during loose stack.sintering of
nickel at 1250C. Uncorrected values (N^so) are included
for comparison.

119
properties of the connected porosity have also to be corrected; these
corrected values are listed in Table 15. The remaining two parameters
of the model, T and R, were estimated as follows.
Since the model has only one-branch and three-branch nodes, the
branches can be either one-three of three-three type. (One-one type
branches are isolated pores and thus are not included in the connected
porosity.) These branches can be classified further, as shown in Table
16 and illustrated in Figure 60.
Since the procedure described in Chapter 3 involves recording of
all these events, it was possible to measure the number and apparent
lengths of each type of branches. As illustrated in Figure 61 for the
case of randomly oriented branches, the true length of a branch is twice
-IT
the average of lengths measured at different orientations. Thus L and
33
L the true average lengths of 1-3 and 3-3 type branches can be esti
mated from the separation between the pertinent events. The average
lengths of 1-3 and 3-3 branches that cross the boundaries can be taken
as twice the value measured, since for randomly oriented branches the
average length of the part of the branch that is contained in the volume
of analysis is half of its true length. The overall weighted average of
all types of branches yields the required parameter, L.
For a collection of isolated, equiaxed pores, VyS0, the volume
fraction of isolated porosity, is given by
(21)

120
Table 15
CORRECTED VALUES OF V^onn,
-conn
bV
and M
conn
No.
vf
..conn
vV
cconn -1
Sy cm
Mconn -2
My cm
1
0.906
0.09
731
-20 x 105
2
0.928
0.053
475
-15.2 x 105
3
0.944
0.017
138
-3.1 x 105
4
0.971
0.017
165
-4 x 105
5
0.979
0.02
237
-10.7 x 105

121:
Type
One-Three
Three-Three
Table 16
TYPES OF BRANCHES
Order of Events Observed During Serial Sectioning
New appearance + Branching
New Appearance -* Connection
Branching -* End of a branch
Connection -* Bottom end
Branching Bottom end
Connection Branching
Branching -* Branching
Connection -* Connection
Branching - Connection

122
New Appearance
Branching
(b)
Figure 60. Various types of a) one-three and b) three-three branches
and the topological events signifying each type of branch.

123
L(measured) = / / L Cos6Sin0d0d<> = L/2
<¡>=0 0=0
Unit Radius
Figure 61. Illustration of the relation between measured and true
lengths of a branch.

124
*1 CQ
Where Ny = number of isolated pores per unit volume and
,iso
= radius of a spherical pore. Thus,
1/3
3V1S0
Ris ( V
4ttN
iso
)
(22)
1 C A
The radius of a spherical -node;:R> was taken to be equal to R The
Al 1 L
values of L, R, by, Ny and Ny are listed in Table 17. It can be seen
that r increases slowly until Vy s 0.97 when it increases significantly.
The reduction in connectivity by way of pinching off of a branch decreases
L whereas elemination of a three-branch node, accomplished when a 1-3
type branch merges into the parent network, leads to an increase in "L,
Figure 62. Since the number of isolation events goes through a maximum,
L should eventually increase significantly once most of the isolation
events or channel closures have taken place. The apparent maximum in
R can be attributed to simultaneous isolation and shrinkage processes.
When a pore shrinks it can be thought of as going from one size class
to the lower one. Since all isolated pores shrink, although at different
rates, there is a "flux" toward the smallest size class in a given size
distribution. Isolation events bring new pores into this collection,
such that there is an influx in all the size classes. Since the number
of isolation events goes through a maximum, the net "flux" in the size
distribution is towards the largest size class when isolation events
dominate over shrinkage and towards the smallest size class when very
few isolation events occur. This leads to a maximum in the estimated
average volume of an isolated pore.
In order to test the model, the metric properties of the connected
porosity were calculated, since those of the isolated fraction were used,

125
Table 17
VALUES OF THE MODEL PARAMETERS USED IN CALCULATIONS
No. Vv
R L
microns microns
cm
-3
1 0.906
2 0.928
3 0.944
4 0.971
2.34
6.63
4.48 x 108
2.1 x 108
2.5 x 108
2.58
8.18
2.42 x 108
6.2 x 107
2.63 x 108
2.79
7.1
8.39 x 107
2.04 x 107
1.02 x 108
2.1
9.56
1.8 x 108
3.0 x 107
2.66 x 108
1.07
25.8
1.29 x 108
3.4 x 107
1.23 x 108
5 0.979

126
Figure 62. Effects of channel closure and surface rounding on L,
the average length of a branch.

127
in a sense, to estimate L and R. The calculated and estimated (with
surface correction) values of Vy0nn, Sy0nn and My0nn are listed in
Table 18. The estimation of the above values was made by using two
conn P P
experimentally determined values, such as Vy /Vy and Vy; the confidence
intervals therefore were relatively large. High sample surface to volume
ratio led to significant amounts of corrections, both in estimated and
measured values. In light of the difficulties and the geometric simpli
city of the model the agreement between measured and calculated values
seems to indicate that the modeled geometry is qualitatively representa
tive of the real microstructure.
SP
It was said earlier in this section that the linearity between Sy
and Vy probably can be attributed to similar area to volume ratios for
pore channels and isolated pores. From the Appendix, for a tubular
channel with length = T and radius r = (/3/2)R, the area to volume
ratio, Ay, is given by
a = 2tttL = 2 _4_ = 2J1
V irr2L r n/3R R
(23)
as illustrated in Figure 63. For an isolated spherical pore, Ay = 3/R
which is not far from 2.31/R. However, the curvature to volume ratios,
Cy's, are significantly different. For a spherical pore, Cy vis given by
Cy (spherical pore) = ^4 = -4> (24)
-2TrRJ R
and Cy for a tubular channel is given by
C
V
(tubular channel) =
ttL
2r
irr L
1
2
r
4 = 1.33
3R2 R2
(25)

128
Table 18
CALCULATED AND ESTIMATED VALUES OF V
conn
V
cconn
bV
and M
conn
V
No.
VS
Vv
..conn
vV
(.conn
-1
, cm
Mconn
Mv ,
_ 2
cm
Calc.
Est.
Calc.
Est.
Calc.
Est.,
1
0.906
0.05
0.09
479
731
-12.4 x 105
-20.3 x 105
2
0.928
0.04
0.053
373
474
-10.3 x 105
-15.2 x 105
3
0.944
0.015
0.017
133
138
-3.6 x 105
-3.1 x 105
4
0.971
0.022
0.017
257
165
-8.9 x 105
-4.0 x 105
5
0.979
0.009
0.02
202
237
-11.2 x 105
-10.7 x 105

2r = 3R
Figure 63. Illustration of a typical pore channel in the connected
porosity.

130
Thus, a spherical pore has more than twice as much Cy as does a
tubular channel. The relative fractions of tubular channels and spheri
cal pores can be computed from the number of branches (by in Table 17)
and the surface-corrected number of isolated, pores (Figure 59). It can
SP
be seen from the variation of f fraction of spheres, Figure 64, that
the curvature would decrease (in magnitude) relatively slowly with the
volume until the maximum in the fraction of spheres; beyond this point
the curvature would decrease sharply. This is supported by the variation
of My, Figure 21. It is suggested that a kinetic model of this type of
geometry should be devised and tested in the future, provided the associ
ation of isolated and connected porosity with the grain boundary network
can be successfully quantified.
The preceding discussion on loose stack sintering will now be
used to describe hot pressing and conventional sintering, in that order.
Hot Pressing
It can be seen from Figure 20 that in the early part of the range
SP
of observation a hot pressed sample has a higher Sy than a loose stack
sintered sample with comparable density. The variation of surface area
with Vy approaches the linear relationship for LS series as the density
increases. Both hot pressed samples and loose stack sintered samples then
continue to exhibit comparable values of surface area. Since the pressure
is applied for the whole duration of hot pressing, the abovementioned
SP
variation of Sy with Vy may have less to do with pressure and thereby
particular mechanisms than with the geometry of the pore structure.

131
Figure 64. Variation of the fraction of spherical pores in a
collection of spherical pores and cylindrical channels
with the volume fraction of solid during loose stack
sintering of nickel at 1250C.

132
Thus, it is suggested that the initially dissimilar geometries become
similar so that the two paths of evolution of microstructure begin to
coincide at Vy s 0.90. It should be noted here that the contention is
that of simi.lar, not identical (see below) geometries, based solely on
the surface area-volume relationships. Thus it is speculated that the
hot pressed samples having densities in the range of observation have
porosity in the form of a collection of isolated, equiaxed pores and
a set of networks made up of tubular channels and spherical pores. As
said in the earlier section on loose stack sintering, the model detailed
in the Appendix dictates the pore channels and isolated spherical pores
have comparable area to volume ratios, regardless of their numbers per
unit volume. If the same type of geometry is assumed for porosity in the
hot pressed samples the linearity of area with volume can be explained.
The variation of My with Vy, Figure 21, indicates that hot pressed
samples have more than twice as much integral mean curvature as loose
stack sintered samples with comparable densities until very late in the
process. It is suggested that the geometries are similar in that they
both can be modeled as a collection of spheres and a set of tubular net
works. They are, however, not identical in that a hot pressed sample has
a considerably higher number of isolated spherical pores than a loose
stack sintered sample with the same density. A need for topological
analysis of hot pressed samples becomes apparent, since these analyses
will test the postulate of higher number of isolated pores in hot pressed
samples as compared to loose stack sintered samples.

133
Unlike loose stack sintering, the grain boundary network did not
exhibit an arrest during hot pressing, as illustrated in Figures 37
through 40. The arrest observed in the grain boundary network for LS
series was attributed to isolated, equiaxed pores pinning the boundaries
effectively. If this assertion is to be valid, the presence of a higher
number of isolated pores in HP series postulated earlier in this section
must also imply that the boundaries are more effectively pinned. The
fact that such a pinning effect was not observed at all is taken to indi
cate that the boundaries are able to migrate in spite of the resistance
offered by isolated and associated porosity. It is speculated that the
application of external pressure induces enough amount of grain boundary
sliding for them to overcome the drag offered by the associated pores;
56
Pond et al. have found some evidence for coupling between grain boundary
migration and grain boundary sliding.
ssp SSS
The values of Ly and Ly(occy the quantities that increase with
an increase in the degree of association of porosity with the grain boun
dary network, were observed to be higher than those for LS. Increased
plastic flow, as suggested above, can also account for higher degree of
pinching off of channels and thereby a higher number of isolation events.
The importance of topological study of hot pressing is emphasized
again as it will test the postulate of a higher number of channel closure
events, as compared to loose stack sintering. The characterization of
associated fractions of isolated and connected porosity will shed a great
deal of light on the grain boundary-porosity relationship.

134
Conventional Sintering
Cold pressing that precedes the high temperature consolidation packs
a larger number of powder particles in a unit volume and hence gives rise
to higher Sv in a green compact as compared to a loose stack sample sin
tered to the same density. It can be seen from Figure 20 that this addi
tional surface area is not reduced enough for the two corresponding paths
of microstructural evolution to coincide even at Vy = 0.98, although the
tendency for the paths to converge is evident. The variation of My with
Vy, Figure 21, indicates that a pressed and sintered sample has higher
surface area and integral mean curvature than a loose stack sample sin
tered to the same density. If the model in the Appendix is examined, it
can be seen from a purely geometric point of view that a similar pore
structure with lower "L and R but higher by, Ny and Ny can yield comparable
values of volume but different area and curvature. In other words, if
the porosity in a pressed and sintered body is viewed as finer networks
and larger number of isolated pores, the measured values of Sy and My
can be explained satisfactorily.
The grain boundary network does not exhibit an arrest, as illustrated
in Figures 37 through 40. A considerably higher number of interparticle
contacts and therefore a larger grain boundary area brings the onset of
grain coarsening at a lower density as compared to loose stack sintering.
If the boundaries begin to migrate well before a sufficient number of
pores are isolated to produce pinning points, there may not be enough
hindrance to the boundary migration to arrest grain growth. Thus, only
a small fraction of isolated pores manage to associate with the boundaries

135
Once most of these are eliminated the pores isolated thereafter have
only a limited opportunity to intersect the moving boundaries. Since
SSP SSS
Ly ancj l_v(occ) quantify the degree of association of porosity with
the grain boundary network, it is apparent that initially a pressed and
sintered sample has a higher amount of associated porosity that decreases
rapidly. Before the onset of isolation processes, most of the porosity
is in the form of an interconnected network mostly associated with grain
edges. If grain coarsening begins well before an appreciable number of
pores are isolated, this initially high associated porosity would decrease
rapidly since the connected pores cannot anchor the boundaries effectively
and are consequently disassociated. The significance of a topological
study of the advanced stages of conventional sintering is stressed here
as these measurements would characterize the associated fractions of iso
lated and connected porosity and thus would test the postulates put for
ward earlier in this section.
Comparison of Loose Stack Sintering with
Hot Pressing and Conventional Sintering
If the paths of evolution of microstructure during these processes
are examined together, a number of general processing parameter-micro-
structure relationships become apparent. Since these relationships have
a potential as possible strategies to control the microstructure and hence
the service properties of a powder-processed component, they are of both
theoretical and practical interest; they are listed below.

136
1) As the grain boundaries are anchored effectively only by
equiaxed pores, a fine grain structure can be obtained if a
sufficient number of equiaxed pores is isolated before grain
growth begins. Grain growth requires a connected grain edge
network and hence a certain minimum grain boundary area,
therefore, an initial powder stack with a relatively coarse
grain structure (low grain boundary area) would have grain
growth beginning at a higher density (after a sufficient number
of pores is isolated); as compared to an initial powder stack
with a finer grain structure. Thus, for a sintered body with
high density and fine grain structure requirements, an initially
coarse grain structure is better than a fine one.
2) If sintering is carried out in such atmosphere that the isolated
pores trap a gas of low diffusivity, these pores are relatively
stable and hence offer effective grain boundary pinning. Care
must be taken to delay the isolation events so that only a few
pores are isolated; otherwise coarsening of these pores would
42
lead to an increase in volume.
3) At least for the conditions of the present investigation, it
can be said that hot pressing leads to both higher density and
a finer grain size in a shorter length of time as compared to
loose stack sintering or conventional sintering at the same
temperature, up to a certain density. Beyond this, a relatively
coarser grain structure is obtained during hot pressing or con
ventional sintering.
The findings of this investigation are summarized and the course of
future research suggested in the next chapter.

CHAPTER 4
CONCLUSIONS
Introduction
The discussion of the results of this investigation is summarized
in a number of conclusions; an outline of suggested research is also
presented.
Conclusions
1) The porosity can be modeled as composed of a set of networks
of cylindrical channels and a collection of monosized isolated spherical
pores during the advanced stages of loose stack sintering, hot pressing
and conventional sintering.
2) During loose stack sintering, a highly interconnected network
of branches and nodes disintegrates into simpler subnetworks which sub
sequently break up to form the isolated pores. The connected or tree-like
pores continue shrinking until Vy r 0.95 when the rate of removal of these
pores becomes significantly slow for the rest of the range of observation,
up to Vy s 0.98.
3) Isolated porosity, on the other hand, goes through a maximum and
diminishes to a very low value when connected porosity is observed to have
been more or less stabilized.
4) The onset of stabilization of connected porosity is coincident
with an arrest of grain growth and rapid reduction in the isolated porosity.
137

138
5) It is suggested that because of their equiaxed shape, isolated
pores anchor the grain boundaries effectively whereas the connected pores
do not. Hence, most of the isolated porosity is associated with grain
boundaries.
6) The association of grain boundary network and isolated pores
facilitates rapid reduction of isolated porosity as the associated
boundaries provide immediate sinks of vacancies; this is in contrast
43 45-47
to the traditional viewpoint that once the pores are isolated,
it is very difficult to remove them from the system.
7) The connected porosity finds itself disassociated from the
grain boundary network which slows the reduction of such pores consid
erably.
8) The higher values of curvature, yet comparable values of
area and volume of pore phase for the hot pressed samples as compared
to loose stack sintered samples, are tentatively attributed to similar
geometries but a higher number of isolated, spherical pores.
9) An absence of an arrest of grain growth in hot pressed samples,
in spite of a higher number of equiaxed pores, is believed to be due to
stress-induced grain boundary sliding that promotes grain boundary migra-
ti on.
10) Porosity in pressed and sintered samples is believed to consist
of finer networks and a higher number of isolated pores compared to the
loose stack sintered samples with the same density; this leads to much
higher areas and curvatures in PS than in LS.
11) Due to larger number of interparticle contacts in a green
compact as compared to a loose stack sintered to the same density, a
pressed and sintered sample has much higher grain boundary area.

139
12) It is suggested that this higher boundary area brings the
onset of grain coarsening at a lower density, well before the pores
begin to isolate. The early grain growth in PS leaves only a small
opportunity for subsequently isolated pores to associate with the moving
boundaries. Thus, an absence of an arrest in grain growth is attributed
to the onset of grain growth well before that of isolation events.
Suggestions for Further Study
It should be apparent from the preceding discussions that topologi
cal analysis of the advanced stages of hot pressing and conventional
sintering would resolve the speculations about a higher number of iso
lated pores in the hot pressed and sintered sample. The associated fractions
of isolated and connected porosity during all three processes, when char
acterized, would facilitate the mechanistic study of the advanced stages.
Thus, the course of further research is outlined as follows.
T) An etching procedure should be developed that will facilitate
the determination of associated fractions of isolated and con
nected porosity.
2) These fractions should be measured on sections in the series
studied for topological, characterization.
3) The isolated and connected fractions should be determined on
a section with calibrated polish.
4) The advanced stages of hot pressing and conventional sintering
should be characterized regarding the topological properties
of the pore structure.

REFERENCES
1. F. N. Rhines and R. T. DeHoff, Modern Developments in Powder
Metallurgy, p. 173, Plenum Press, New York (1971).
2. W. Rostoker and S. Y. K. Liu, J. Materials, _5, 605 (1970).
3. R. D. Smith, H. W. Anderson and R. E. Moore, Bull. Amer. Cer. Soc.,
55, 979 (1976).
4. R. T. DeHoff, F. N. Rhines and E. D. Whitney, Final Report, AEC
Contract AT(40-1), 4212 (1974).
5. G. Arthur, J. Inst. Metals, 83, 329 (1954).
6. R. A. Graham, W. R. Tarr and R. T. DeHoff, unpublished research.
7. G. Ondracek, Radex-Rundschau 3/4 (1971)
8. S. Nazare, G. Ondracek and F. Thummler, Modern Developments in Powder
Metallurgy, p. 171, Plenum Press, New York (1971).
9. J. Kozeny, Sitzber. Akad. Wiss. Wien., 136, 271 (1927).
10. M. F. Ashby, Acta Met., 22, 275 (1974).
11. R. T. DeHoff, B. H. Baldwin and F. N. Rhines, Planseeber. Pulvermet.,
JO, 24 (1962).
12. Metals Research Laboratory, Carnegie Institute of Technology, Final
Report, AEC Contract AT(30-1), 1826 (1959).
13. G. C. Kuczynski, Powder Metal1urgy, p. 11, Interscience Publishers,
New York-London (1961).
14. T. L. Wilson and P. C. Shewmon, Trans. Met. Soc. AIME, 236, 48 (1966)
15. G. Matsumara, Acta Met., 19, 851 (1971).
16. F. N. Rhines, C. E. Berchenall and L. A. Hughes, J. Metals, 188,
378 (1950).
17. R. T. DeHoff, Proceedings of the Symposium on Statistical and
Probabilistic Problems in Metallurgy, Special supplement to Advances
in Applied Probability (1972).
18. E. H. Aigeltinger and R. T. DeHoff, Met. Trans., 6A, 1853 (1975).
140

141
19. R. A. Gregg and F. N. Rhines, Met. Trans., 4, 1365 (1973).
20. R. T. DeHoff, R. A. Rummel, H. P. LaBuff and F. N. Rhines,
Modern Developments in Powder Metallurgy, p. 310, Plenum Press,
New York (1966).
21. W. D. Tuohig, Doctoral Dissertation, University of Florida (1972).
22. S. M. Gehl, Doctoral Dissertation, University of Florida (1977).
23. A. S. Watwe and R. T. DeHoff, unpublished research.
24. J. S. Adams and D. Glover, Metal Progress, August (1977).
25. E. H. Aigeltinger, Doctoral Dissertation, University of Florida (1969).
26. R. T. DeHoff and F. N. Rhines, eds., Quantitative Microscopy, McGraw
Hill Book Co., New York (1967).
27. L. K. Barrett and C. S. Yust, ORNL Report, No. 4411 (1969).
28. L. K. Barrett and C. S. Yust, Metallography, 3^, 1 (1970).
29. S. S. Cairns, Introductory Topology, The Ronald Press Company,
New York (1961T!
30. F. N. Rhines, Powder Met. Bull., 3^, 28 (1948).
31. P. Schwarzkopf, Powder Met. Bull., 2 74 (1948).
32. F. Thummler and N. Thomma, Met. Review, 12, 69 (1967).
33. R. L. Coble, J. Appl. Physics, 32, 787 (1961).
34. L. K. Barrett and C. S. Yust, Trans. Met. Soc. AIME, 239, 1172 (1967).
35. R. T. DeHoff and F. N. Rhines, Final Report, AEC Contract AT(40-1),
2581 (1969).
36. F. N. Rhines, University of Florida, private communication.
37. G. C. Kuczynski, Acta Met., 4, 58 (1956).
38. P. J. Wray, Acta Met., 24, 125 (1976).
39. W. D. Kingery and B. Francois, Sintering and Related Phenomena,
p. 471, Gordon and Breach Publishers, New York (1965).
40. A. J. Markworth, Met. Trans., £, 2651 (1973).
41. W. Trzebiatowski, Zhurnal Physik Chem., B24, 75 (1934).

142
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
T. K. Gupta and R. L. Coble, J. Amer. Cer. Soc., 52, 293 (1968).
R. C. Lowrie, Jr. and I. B. Cutler, Sintering and Related Phenomena,
p. 527, Gordon and Breach Publishers, New York (1965).
G. C. Kuczynski, Powder Met., 12, 1 (1963).
M. Paul us, Sintering and Related Phenomena, p. 225, Plenum Press,
New York (1973).
C. S. Morgan and K. H. McCorkle, Sintering and Related Phenomena,
p. 293, Plenum Press, New York (1973).
J. E. Burke, Ceramic Microstructures, p. 681, John Wiley and Sons,
New York (1968^
E. H. Aigeltinger and H. E. Exner, Met. Trans., 8A, 421 (1977).
J. Kronsbein, L. J. Buteau, Jr. and R. T. DeHoff, Trans. Met. Soc.
AIME, 233, 1961 (1965).
Metals Handbook, 8th Edition, Volume 1.
B. R. Patterson, Doctoral Dissertation, University of Florida (1978).
M. Hillert, Acta Met., 12, 227 (1965).
G. C. Kuczynski, Sintering and Related Phenomena, p. 325, Plenum
Press, New York (1976).
A. R. Hingorany and J. S. Hirschhorn, Inti. J. Powder Met., 2, 5 (1966).
A. J. Markworth, Scripta Met., j>, 957 (1972).
R. C. Pond, D. A. Smith and P. W. J. Southerdon, Phil. Mag., A37, 27
(1978).

APPENDIX
THE GEOMETRIC MODEL OF THE PORE PHASE
Introduction
Main features of the proposed model are stated and followed by the
derivation of equations relating metric properties and the parameters of
the model. The corrections for surface effects are also outlined.
Parameters of the Model
As said in Chapter 3, the pore phase is modeled as a collection of
spherical pores of the same size and a set of networks made up of cylin
drical branches and spherical nodes. Only two kinds of nodes and branches
are assumed to exist in the connected pores; a node is either a one-branch
node or a three-branch node, as illustrated in Figure 65. A branch either
terminates in three-branch nodes at both ends, or in a one-branch node at
. one and a three-branch node at the other, Figure 66. A branch that termi
nates in one-branch nodes at both ends is considered an isolated part and
hence is not included in the connected porosity.
A one-branch node is assumed to be a semi-spherical cap, the radius
of which is the same as that of a cylindrical branch, r, as shown in
Figure 66. A three-branch node is considered as a sphere of radius R
which*is connected to three cylindrical branches of radius r, Figure 67.
An isolated pore is assumed to be a sphere of radius R. All branches
have the same length, "L.
143

144
Figure 65. Illustration of three-branch and one-branch nodes in a
pore network.

145
Figure 66. Illustration of a) a one-three branch and b) a three-three
branch.

146
2r
Figure 67. Dihedral angle of the edge at the intersection of a
cylindrical branch and a spherical node.

147
The abovementioned parameters of the model are discussed in terms
of the notations listed in Table 19.
Since each one-branch node is associated with a branch and each
three-branch node with three branches, by, the total number of branches
1 h
and Ny and Ny are related by the following equation
+ 3Nyb
2
(26)
since each branch is counted twice. Alternatively, each one-three
branch is associated with one one-branch node and one three-branch
node, whereas each three-three branch is associated with two three-
branch nodes. Thus
,3b
Nr
h13 .33
by + 2by
(27)
since each three-branch node is counted thrice. The number of one-branch
nodes, Nyb, is given by
Njb = bj3 (28)
Equations (26) and (27) give
wlb .33
3b NV + 2bV
V
or
3N?b NJb
(29)
Thus by is given by
K h13 4. h33
bV bv + bV
3|\j3^ N^b
My> + 3_Ni_Ny.
bv =
Nb + 3Nf
or
2
(30)

148
Table 19
PARAMETERS OF THE MODEL
Feature
Notation
Definition
Branches
J3
bV
Number of one-three branches per unit
volume
h33
bV
Number of three-three branches per unit
vol ume
Nodes
Number of one-branch nodes per unit
volume
nf
Number of three-branch nodes per unit
volume
Isolated Pores
nJS0
Number of isolated pores observed to
be wholly contained, per unit volume

149
which is the same as equation (26). Thus, the number of branches can
be calculated from the number of nodes, and vice versa.
Metric Properties of the Connected Porosity
Three basic properties, namely, volume fraction, area per unit
volume and integral mean curvatre per unit volume are discussed. The
conn cconn Mconn
i S\ cl net Mt;
notations used are listed in Table 20; V
are made
V dna V
up of contributions from branches, one-branch nodes and three-branch
nodes; they will be discussed in terms of these individual contributions,
listed in Table 21.
Three-Branch Node
It can be seen from Figure 67 that r, the radius of a cylindrical
branch, and R, the radius of a three-branch node, are related by the
equation
r = RCos(ir/6) or r = '^-R
(31)
Thus, h, the height of the spherical cap, illustrated in Figure 68, is
given by
h = R/2
(32)
C
The volume of this spherical cap, V is given by
(33)

150
Table 20
CALCULATED METRIC PROPERTIES
Property
Notation
Definition
Volume
..conn
vV
Volume fraction of
connected porosity
Area
-conn
Surface area of connected
porosity per unit volume
Curvature
Mconn
Integral mean curvature of
connected porosity per unit
volume

151
Table 21
METRIC PROPERTIES OF NODES AND BRANCHES
Property
Volume
Area
Notation
Definition
Volume of a three-branch node
Volume of a one-branch node
Volume of a branch
Area of a three-branch node
Area of a one-branch node
Area of a branch
Curvature
Integral mean curvature of a
three-branch node
Me Integral mean curvature of
edges in a three-branch node
(Figure 67)
Mlb Integral mean curvature of a
one-branch node
M
b
Integral mean curvature of a
branch

152
Figure 68. Height of the spherical cap in an extended, spherical,
three-branch node.

153
JL
The volume of a three-branch node, V is then the volume of a sphere
of radius R minus the volumes of three spherical caps. Thus
V = tt R 3( 24 ) or V = -gq ttR (34)
C
The area of a spherical cap, A is given by
AS = 2irRh or AS = ttR2 (35)
Area of a three-branch node, S is thus
S3b = 4ttR2 3(ttR2) or S3b = uR2 (36)
There are three edges formed on a three-branch node as a result of
intersection of three cylindrical surfaces with a spherical surface.
Since integral mean curvature of an edge (26) is given by
M of an edge = -^ X 1 (37)
Where X = dihedral angle or the angle between the surface normals and
1 = the length of an edge. It can be seen from Figure 67 that X = tt/6
and 1 = 2irr or 1 = /3rrR. Thus, Medge, integral mean curvature of edges,
is given by
Medge = 3X|x£x(V3)ttR or Medge = ^ tt2R (38)
Integral mean curvature of the spherical surface of a three-branch node,
c
M is simply
S 1
M = spherical surface area X(-p-) or
MS = (ttR2) (-!) or MS = -ttR
(39)

154
OL
Thus the integral mean curvature of a three-branch node, M is given by
M3b = MS + Medge = -ttR + (^pk2R or
M3b = tt (1 ^L)R (40)
One-Branch Node
Since such a node is a semi-spherical cap of radius r = (*/3/2)R that
is connected to a cylinder of radius r, there are no edges involved. Thus
V1b=|trr3 or Vlb=^R3
(41)
Area of a one-branch node, S^b, is given by
Sb = 2Trr2 or Sb = 2tt(^-) or Sb = ^R2
(42)
Integral mean curvature of a one-branch node, M^b, is given by
Mlb = 27rr2(-l) or Mlb = -2irr = -/3ttR
(43)
Cylindrical Branch
Since all branches have length = T and radius r = C^-)R,
Vb = -nr2!: = : ' (44)
Sb = 2TrrI = ^3ttRI (45)
Mb = -ttL
(46)

155
The Networks
Thus the properties of networks are given by
,1b
conn
V
conn
= b.
Vb + N3b V3b + Nyb
V
conn
= b.
Sb + N3b + S3b + Njb
Mb + Nyb M3b +
Jb
M
lb
(47)
(48)
ly uy II ny n iiy ri (49)
Substitution of the definition of individual contributions leads to
(50)
(51)
(52)
conn u 3irn2r 3b 177Tr,3 Mlb /3 n3
V = bV TR L NV 24"R NV 41tR
Sj0nn = by ^RL + N3b ttR2 + Njb ^R2
M
,conn
t .,3b
//3-it
lb m
-byTrL + Ny TT^-IJR Ny /3ttR
Computation of coefficients of by, N3b and Nyb yield the equations in
the convenient form:
V
S
M
y0nn = 2.36byLR-+2.23N3bR3 + 1.36NjbR3
(53)
50nn = 5.44byLR + 3.14N3bR2 + 4.72NjbR2
(54)
|Cnn = _3.i4byL + 1.13Nyb R 5.44Nyb R
(55)
Surface Corrections
It was assumed during the measurements of metric properties of
connected porosity that all the pores crossing the boundaries of the
volume of analysis are of a network character. A fraction of these
pore features, in fact, belong to the set of isolated pores. A

(a)
(b)
Figure 69. Illustration of a) "IS11 pore branches and b) "SS" pore branches.

157
Figure 70. Caliper diameters for a convex body. D, the mean
caliper diameter, is the average of D's over all
possible orientations of measuring planes.

158
treatment is outlined below that attempts to take the surface effects
into consideration.
1 s
The parameter by is defined as the number of pores per unit volume
that are observed to cross the boundaries and terminate inside the volume
ss
of analysis ("1-S") and by as those that traverse this volume without
terminating inside ("S-S"), illustrated in Figure 69. The fraction of
these "IS" and "SS" pores that belong to the set of isolated pores is
estimated as follows.
For a collection of convex pores, the number per unit volume, Ny,
and the number of pore features observed on a section of unit area, N^,
are related by the equation
NA = Ny D (15)
where 1) is the average of the mean caliper diameters of the pores, as
illustrated in Figure 70. Thus, for a sample of surface area S and
volume V, the number of pores observed to cross the boundaries per unit
c
volume, Ny, is given by
Nv = na f Nv X '17>
Since these pores are not wholly contained in the volume of analysis,
they contribute only half as many pores to the set of isolated pores. If
i so
Ny is the number of wholly contained, isolated pores per unit volume,
the true measure of isolated pores per unit volume, Ny is given by
Nis0
NV = NVS0+fV-| Nv*-V 09)
i DS
"2V
Thus, equations (17) and (19) give

159
NV = Nv "V
DS NV
iso
1-
DS
2V
DS
S
(56)
The number of "IS" and "SS" pores that contribute to the isolated
porosity, Ny, can be estimated from the measurements of NyS0, , S
and V. Estimated ;.D is the average of physical separations between
sections where the isolated pores appear and those where they disappear.
The pores that cross the boundaries but do not contribute to the
set of isolated pores are assumed to terminate in the three-branch nodes.
Thus, a fraction of "SS" pores belong to the class of "3-3" branches,
whereas a subset of "l-S" pores belongs to the class of "1-3" branches.
These fractions are estimated as follows.
Each "SS" pore contributes two pore features observed on the surface
whereas each "IS" pore contributes one pore feature. Thus, the total
number of pore features arising from these pores per unit volume, Ny,
is given by
Nj = bJS + 2b*jS (57)
e
Since the number of pore features that come from the isolated pores, Ny,
is given by equation (55), the number of features that arise from connected
r
pores per unit volume, Ny, is given by
Niso -
mc mt ms v. mc JS ..SS INV DS
Nv Ny Ny or Ny by + 2by y (58)
. DS
2V
r
It is assumed that these Ny features are distributed according to
the relative fractions of "IS" and "SS" type pores. Thus, byS(C), the
number of "IS" pores that belong to the connected porosity per unit

160
SS,
volume, and by (C), the number of "SS" pores that belong to the connected
porosity per unit volume are given by
.IS
JS =
JS,OLSS
by +2by
JS
ISO
or
and
or
bJS(C) =
.IS.,. SS
by +2by
rKlS.9KSS
{by +2by
1-
DS
2V
DS,
V 1
(59)
SS/p\ 1
bV ^ 2 JS
2b
SS
SS NV
by +2bv
bSS Niso
uSS,^ DV ri_ 1S, olSS NV
bV JSjo.SS {bV +2bV ^
by +2by i US^
V V |-2y
DS,
(60)
Since for every two features contributed by "SS" pores there is one "SS"
pore that is counted as an additional "3-3" type branch. Thus, the total
number of additional branches, by(add), is given by
by(add) = \ [bJS(C) + b*S(C)] or
JS,uSS
by +by
ISO
u v ruiS,olSS V
by(add) hlS+9hSSx ibV 2bV
2(by +2by ) DS
DS,
V *
(61)
1-
2 V
since all these branches are considered to contribute half as many to
the volume of analysis.
Each additional 1-3 branch contributes an additional one-branch node,
N^b(add), the additional number of one-branch nodes per unit volume, is
given by
blS Jso
DV r JS.ouSS NV
JS.-SS {bV +2bV c
by +2by ^DS
DS}
V 1
Nyb(add) = byS(C) =
(62)

161
If by(corr) is defined as the corrected number of branches per unit
volume, T(corr) as the corrected average length of a branch and TT(add) as
the weighted average of the length of additional branches, then
by(corr)L(corr) = by*L + by(add)*L(add)
(63)
where by and T refer to the previously measured values of branches per
unit volume and their average length. The values of by(corr), TT(corr),
Nyb(corr), Ny^ and R1S0 are listed in Table 22; these were used to cal
culate the metric properties of the connected porosity. The surface
corrections used to modify the measured values of the metric properties
of the connected porosity are described presently.
It can be seen from equation (56) that the additional number of
isolated pores per unit volume, Ny (add), is given by
,iso
(64)
N
Miso, 1 S _V DS
Ny (add) 2 Ny 2V
. . 1_ 2V
Since VyS0, SyS0 and MyS0, the metric properties of isolated poro
sity, are directly proportional to the number of isolated pores, the
additions to these properties are given by
.ISO
Vjs0(add) = 4
N
SO
N\/S(add)
or
VwS0(add) = vjso {^4
i DS
1 2V
(65)
Similarly,
sjS0(add) = sjS0
v v 1-DS
W
(66)

162
and
Mi50(add) =
Miso {DSm}
V DS
1 2V
(67)
Here it is assumed that the true D of the isolated pores is the same
as that measured for wholly contained isolated pores. The initial
measurements of properties of connected porosity were made assuming
all pores not contained inside are connected; the abovementioned addi
tions to the isolated fraction have therefore to be subtracted from
Vj0nn, Sy0nn and My0nn to correct for surface effects. These corrected
values, termed Vy0nn, Sy0nn and My0nn by the following equations.
Vvconn(C) = v§onn
{DS/2V}
1 -
DS
2V
s5onn(c) + sJonn
ciso
{
}
DS/2V
, DS
1 2V
Mf,nn(C)
= M:
conn
- M
iso
{DS/2V}
1 -
DS
2 V
(68)
(69)
(70)
The geometric model, discussed in detail in the preceding section
of the Appendix, is tested by comparing the corrected values and the
calculated values.

BIOGRAPHICAL SKETCH
Arunkumar Shamrao Watwe was born on June 16, 1952, in Poona, India.
In June, 1967, he finished high school in Bombay, India. In June, 1974,
he received the degree of Bachelor of Technology in Metallurgical
Engineering from the Indian Institute of Technology, Bombay, India.
In August, 1976, he received the degree of Master of Science in Materials
Science and Engineering from Washington State University, Pullman, Washington.
In September, 1976, he came to the University of Florida to pursue the
degree of Doctor of Philosophy in the Department of Materials Science and
Engineering.
He is a fellow member of Alpha Sigma Mu and is a joint student
member of the American Society for Metals and the Metallurgical Society
of AIME.
163

I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate,
in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
R. T. DeHoff, Chairmar
Professor of Material'Science
and Engineering
I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate,
in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
L n ¡ip
V ..
R. E. Reed-Hill
Professor of Materials Science
and Engineering
I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate,
in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
Professor of Materials Science
atar Engineering
I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate,
in scope and quality, as a dissertation for the degree of Doctor of
Philosophy.
Inoda, Jr.
Professor of Materials Science
and Engineering

I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate,
in scope and quality, as a dissertation for the' degree of Doctor of
Philosophy. / ,//
s Rc_L< Scheaffer'
v-^Professor of Statistics
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
August, 1983 Dean, College of Engineering
Dean for Graduate Studies and
Research

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AUTHOR: Watwe, Arunkumar
TITLE: Metric and topological characterization of the advanced stages of
sintering / (record number: 506243'
PUBLICATION DATE: 1983
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55
Figure 23. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to Vy = 0.906, etched (approx. 400 X).


50
dealt with relatively small amounts of porosity (10 percent or lower)
P
the error in Vy 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
p
further characterization if the metal!ographically determined Vy was
p
within 15 percent of Vy 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 Vy. Since Vy 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
p
dense and lower exhibited a precision of 0.05 of the Vy value obtained
SP SP
from the density measurements. Manual measurements of Sy and My were
made on the accepted polished surfaces using standard stereological techni-
ques. The measurements of Vy, Sy and My 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
SP SP
20 and 21. Plots of Sy and My contained metallographically measured values
of Vy 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


134
Conventional Sintering
Cold pressing that precedes the high temperature consolidation packs
a larger number of powder particles in a unit volume and hence gives rise
to higher Sv in a green compact as compared to a loose stack sample sin
tered to the same density. It can be seen from Figure 20 that this addi
tional surface area is not reduced enough for the two corresponding paths
of microstructural evolution to coincide even at Vy = 0.98, although the
tendency for the paths to converge is evident. The variation of My with
Vy, Figure 21, indicates that a pressed and sintered sample has higher
surface area and integral mean curvature than a loose stack sample sin
tered to the same density. If the model in the Appendix is examined, it
can be seen from a purely geometric point of view that a similar pore
structure with lower "L and R but higher by, Ny and Ny can yield comparable
values of volume but different area and curvature. In other words, if
the porosity in a pressed and sintered body is viewed as finer networks
and larger number of isolated pores, the measured values of Sy and My
can be explained satisfactorily.
The grain boundary network does not exhibit an arrest, as illustrated
in Figures 37 through 40. A considerably higher number of interparticle
contacts and therefore a larger grain boundary area brings the onset of
grain coarsening at a lower density as compared to loose stack sintering.
If the boundaries begin to migrate well before a sufficient number of
pores are isolated to produce pinning points, there may not be enough
hindrance to the boundary migration to arrest grain growth. Thus, only
a small fraction of isolated pores manage to associate with the boundaries


132
Thus, it is suggested that the initially dissimilar geometries become
similar so that the two paths of evolution of microstructure begin to
coincide at Vy s 0.90. It should be noted here that the contention is
that of simi.lar, not identical (see below) geometries, based solely on
the surface area-volume relationships. Thus it is speculated that the
hot pressed samples having densities in the range of observation have
porosity in the form of a collection of isolated, equiaxed pores and
a set of networks made up of tubular channels and spherical pores. As
said in the earlier section on loose stack sintering, the model detailed
in the Appendix dictates the pore channels and isolated spherical pores
have comparable area to volume ratios, regardless of their numbers per
unit volume. If the same type of geometry is assumed for porosity in the
hot pressed samples the linearity of area with volume can be explained.
The variation of My with Vy, Figure 21, indicates that hot pressed
samples have more than twice as much integral mean curvature as loose
stack sintered samples with comparable densities until very late in the
process. It is suggested that the geometries are similar in that they
both can be modeled as a collection of spheres and a set of tubular net
works. They are, however, not identical in that a hot pressed sample has
a considerably higher number of isolated spherical pores than a loose
stack sintered sample with the same density. A need for topological
analysis of hot pressed samples becomes apparent, since these analyses
will test the postulate of higher number of isolated pores in hot pressed
samples as compared to loose stack sintered samples.


CHAPTER 1
EVOLUTION OF MICROSTRUCTURE DURING SINTERING
Introduction
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
26
lines, surfaces and regions of space averaged over the whole structure.
The basic properties are listed in Table 1 and illustrated in Figure 1.
Among the properties listed, Vy, Sy and My 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
5


131
Figure 64. Variation of the fraction of spherical pores in a
collection of spherical pores and cylindrical channels
with the volume fraction of solid during loose stack
sintering of nickel at 1250C.


97
porosity, designated respectively as Vy0nn, Sy0nn and My0nn can be
measured unambiguously. In the present context, "connected" porosity
includes all pores that were not wholly contained in the volume of
analysis. The possibility of measuring the properties of the connected
porosity was not foreseen before all the series of parallel sections
were obtained. Thus, the measurements had to be made on sections with
an uncalibrated polish; the ratios Vy0nn/Vy^a^, Sy0nn/Sy0^ and
My0nn/My0tal, were estimated. These dimensionless ratios, when multi
plied respectively by Vy, Sy and My values determined previously on
calibrated polished surfaces, yield the estimated values of Vy0nn,
Sy0nn and My0nn. It is thought that the errors introduced by noncali-
brated polished surfaces are at least partially compensated for by
measuring the abovementioned ratios rather than the absolute values.
These are illustrated in Figures 51, 52 and 53. These quantities were
estimated from the measurements made on uncalibrated sections; also the
amount of isolated porosity in the 98 percent dense sample was too low
to yield a sufficient number of measurements for unbiased estimates of
metric properties of isolated porosity. Therefore, only the trends,
not the actual numbers (at least for the 98 percent dense sample) are
significant. It can be seen that after the porosity is reduced to about
0.06, the connected porosity changes very little, whereas the isolated
porosity continues to decrease.
The results of the present investigation, briefly mentioned in
this chapter, are discussed in detail in the next chapter.


56
Figure 24. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to Vy = 0.928, etched (approx. 400 X).


36
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 theorized161835,37-39 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
34
that the continuity of the solid phase is maintained, as illustrated in
34
Figure 18. According to Barrett and Yust, 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
43 45-47
pores. 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.


151
Table 21
METRIC PROPERTIES OF NODES AND BRANCHES
Property
Volume
Area
Notation
Definition
Volume of a three-branch node
Volume of a one-branch node
Volume of a branch
Area of a three-branch node
Area of a one-branch node
Area of a branch
Curvature
Integral mean curvature of a
three-branch node
Me Integral mean curvature of
edges in a three-branch node
(Figure 67)
Mlb Integral mean curvature of a
one-branch node
M
b
Integral mean curvature of a
branch


114
number of "ends" (Chapter 2) that are not associated with isolated
pores, since an isolated pore has two "ends" denoted by the appearance
and disappearance of a new pore. As illustrated in Figure 58, among
the pores that cross the boundaries of the volume of analysis, there
are some that have an "end" associated with each of these. Thus, there
is an uncertainty as to how many of these pores that cross the boundaries
are isolated and how many are connected. An attempt was made to intro
duce a correction for the sample surface effects. For a collection of
convex particles (pores, in the present context), the number per unit
volume Ny and the number of features observed on a section are related
26
by the following equation.
na = n
(15)
Where D is the average of the mean caliper diameters of the said parti
cles. Thus, for the volume of analysis with surface area S and volume
c
V, the number of pore features that may cross the surface, N is given
by
N
S = Nv
D S
(16)
The number of isolated pores that cross the boundaries per unit volume,
S
Ny, is therefore given by
S s
NV = T = NV D V
07)
Since these Ny pores per unit volume are not wholly contained in
the volume of analysis, they contribute only half as many pores to the


106
Occupying the Grain Boundary
Figure 54. Illustration of an increase in the grain boundary area
that follows the motion of the boundary occupied by a
second phase particle.


85
[(j+l)th section]
Figure 45. Two typical consecutive sections studied during serial
sectioning that illustrate the topological events listed
in Table 9.


153
JL
The volume of a three-branch node, V is then the volume of a sphere
of radius R minus the volumes of three spherical caps. Thus
V = tt R 3( 24 ) or V = -gq ttR (34)
C
The area of a spherical cap, A is given by
AS = 2irRh or AS = ttR2 (35)
Area of a three-branch node, S is thus
S3b = 4ttR2 3(ttR2) or S3b = uR2 (36)
There are three edges formed on a three-branch node as a result of
intersection of three cylindrical surfaces with a spherical surface.
Since integral mean curvature of an edge (26) is given by
M of an edge = -^ X 1 (37)
Where X = dihedral angle or the angle between the surface normals and
1 = the length of an edge. It can be seen from Figure 67 that X = tt/6
and 1 = 2irr or 1 = /3rrR. Thus, Medge, integral mean curvature of edges,
is given by
Medge = 3X|x£x(V3)ttR or Medge = ^ tt2R (38)
Integral mean curvature of the spherical surface of a three-branch node,
c
M is simply
S 1
M = spherical surface area X(-p-) or
MS = (ttR2) (-!) or MS = -ttR
(39)


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
isolated pores.
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
isolation events.
viii


I certify that I have read this study and that in my opinion it conforms
to acceptable standards of scholarly presentation and is fully adequate,
in scope and quality, as a dissertation for the' degree of Doctor of
Philosophy. / ,//
s Rc_L< Scheaffer'
v-^Professor of Statistics
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School, and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
August, 1983 Dean, College of Engineering
Dean for Graduate Studies and
Research


66
Figure 34. Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 1250C to Vy = 0.962,
etched (approx. 400 X). v


58
Figure 26. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to Vy = 0.971, etched (approx. 400 X).


38
Mechanical and physical properties of conmercial porous components
are influenced by the geometry of the porosity. Thermal conductivity is
P
influenced by Vy, pore shapes and the relative fractions of connected and
7 8 4
isolated porosity. Permeability to fluids depends on the connectivity,
p cp g 3
Vy and Sy Mechanical strength and thermal shock resistance depend on
2
pore shapes whereas ductility is influenced by pore shapes and spacings.
Thus geometric characterization of porous structures as a function of
adjustible 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.


155
The Networks
Thus the properties of networks are given by
,1b
conn
V
conn
= b.
Vb + N3b V3b + Nyb
V
conn
= b.
Sb + N3b + S3b + Njb
Mb + Nyb M3b +
Jb
M
lb
(47)
(48)
ly uy II ny n iiy ri (49)
Substitution of the definition of individual contributions leads to
(50)
(51)
(52)
conn u 3irn2r 3b 177Tr,3 Mlb /3 n3
V = bV TR L NV 24"R NV 41tR
Sj0nn = by ^RL + N3b ttR2 + Njb ^R2
M
,conn
t .,3b
//3-it
lb m
-byTrL + Ny TT^-IJR Ny /3ttR
Computation of coefficients of by, N3b and Nyb yield the equations in
the convenient form:
V
S
M
y0nn = 2.36byLR-+2.23N3bR3 + 1.36NjbR3
(53)
50nn = 5.44byLR + 3.14N3bR2 + 4.72NjbR2
(54)
|Cnn = _3.i4byL + 1.13Nyb R 5.44Nyb R
(55)
Surface Corrections
It was assumed during the measurements of metric properties of
connected porosity that all the pores crossing the boundaries of the
volume of analysis are of a network character. A fraction of these
pore features, in fact, belong to the set of isolated pores. A


159
NV = Nv "V
DS NV
iso
1-
DS
2V
DS
S
(56)
The number of "IS" and "SS" pores that contribute to the isolated
porosity, Ny, can be estimated from the measurements of NyS0, , S
and V. Estimated ;.D is the average of physical separations between
sections where the isolated pores appear and those where they disappear.
The pores that cross the boundaries but do not contribute to the
set of isolated pores are assumed to terminate in the three-branch nodes.
Thus, a fraction of "SS" pores belong to the class of "3-3" branches,
whereas a subset of "l-S" pores belongs to the class of "1-3" branches.
These fractions are estimated as follows.
Each "SS" pore contributes two pore features observed on the surface
whereas each "IS" pore contributes one pore feature. Thus, the total
number of pore features arising from these pores per unit volume, Ny,
is given by
Nj = bJS + 2b*jS (57)
e
Since the number of pore features that come from the isolated pores, Ny,
is given by equation (55), the number of features that arise from connected
r
pores per unit volume, Ny, is given by
Niso -
mc mt ms v. mc JS ..SS INV DS
Nv Ny Ny or Ny by + 2by y (58)
. DS
2V
r
It is assumed that these Ny features are distributed according to
the relative fractions of "IS" and "SS" type pores. Thus, byS(C), the
number of "IS" pores that belong to the connected porosity per unit


158
treatment is outlined below that attempts to take the surface effects
into consideration.
1 s
The parameter by is defined as the number of pores per unit volume
that are observed to cross the boundaries and terminate inside the volume
ss
of analysis ("1-S") and by as those that traverse this volume without
terminating inside ("S-S"), illustrated in Figure 69. The fraction of
these "IS" and "SS" pores that belong to the set of isolated pores is
estimated as follows.
For a collection of convex pores, the number per unit volume, Ny,
and the number of pore features observed on a section of unit area, N^,
are related by the equation
NA = Ny D (15)
where 1) is the average of the mean caliper diameters of the pores, as
illustrated in Figure 70. Thus, for a sample of surface area S and
volume V, the number of pores observed to cross the boundaries per unit
c
volume, Ny, is given by
Nv = na f Nv X '17>
Since these pores are not wholly contained in the volume of analysis,
they contribute only half as many pores to the set of isolated pores. If
i so
Ny is the number of wholly contained, isolated pores per unit volume,
the true measure of isolated pores per unit volume, Ny is given by
Nis0
NV = NVS0+fV-| Nv*-V 09)
i DS
"2V
Thus, equations (17) and (19) give


14
Figure 3
Some closed surfaces and their deformation retracts
(dotted lines).


TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS iv
ABSTRACT vii
INTRODUCTION 1
CHAPTER
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
Introduction 39
Sample Preparation 39
Metallography 46
Topological Measurements 74
THREE DISCUSSION 101
Introduction 101
Loose Stack Sintering 101
Hot Pressing 130
Conventional Sintering 134
Comparison of Loose Stack Sintering with Hot Pressing
and Conventional Sintering 135
FOUR CONCLUSIONS 137
Introduction 137
Conclusions 137
Suggestions for Further Study 139
REFERENCES 140
APPENDIX THE GEOMETRIC MODEL OF THE PORE PHASE 143
Introduction 143
v


77


82
Figure 44. Schematic diagram of a typical specimen used in serial
sectioning.


Dedicated
To My Parents,
Mr. Shamrao Vasudeo Watwe
and
Mrs. Sharada Shamrao Watwe


Fraction
of pores
Figure 41.
Variation of fractions of pores on the triple edges (filled), on the boundaries (half-filled)
and within the grains (open) for a) loose stack sintered, b) hot pressed and c) pressed and
sintered nickel powder at 1250C.


40
Figure 19. INCO 123 nickel powder used in the present investigation
(1000 X).


13
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
32
pore-solid interface decreases with a moderate amount of shrinkage.
Throughout this stage, the pore-solid interface has many redundant con
nections J
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
1 6 *3*3
porosity is in the form of an interconnected network of channels *
that delineate the solid grain edges. The continued reduction in the
volume and the area of porosity is accompanied by a decrease in the
1 35
connectivity of the pore structure. The decrease in the connec
tivity can be explained by either removal of solid branches or closure
36
of pore channels. According to Rhines, 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 stage 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.16,34
30 31
The third stage has begun by the time most of the pores are isolated.
The connectivity of a pore network is now a very small number.1 Coarsening
16 18 35 37-39
proceeds along with the spheroidization of pores * so that the
volume of porosity, the number of pores and pore-solid interface area


102
Metric Properties of the Pore Structure
SP
It is evident from Figure that Sv is linear with V,, during the
v v
entire range of observation. Prior to this investigation, it was antici
pated that this linear relationship is a manifestation of continued removal
of pore sections of constant surface to volume ratio. In this scenario,
a pore section disappears as soon as it attains a certain surface area to
volume ratio, because gradual shrinkage would increase the surface to
volume ratio which is inversely related to the size of the pore. Since
the porosity consists of connected and isolated fractions during the late
second and third stages, this hypothesis must be taken to mean that the
channels that pinch off, and isolated pores that disappear, do so as soon
as they attain certain area to volume ratio. In this "reverse popcorn"
or "instantaeous removal of pore sections" model the continued removal of
identical pore sections would also mean a linearity between My and Vy, a
speculation not supported by the variation of M.,, Figure T-9.
It is suggested here that this apparent contradiction can be explained
if it is assumed that the pore phase consists of tubular channels and spher
ical isolated pores that have comparable area to volume ratios but signifi
cantly different curvature to volume ratios. If the relative fractions of
these channels and isolated pores vary during the sintering process, the
area would still be linear to the volume but the curvature may not be.
This is discussed further in the section on the geometric model of the
porosity.


116
collection of isolated pores. Thus, the true number of isolated pores,
Ny, is given by
iy = NyS0(wholly contained) + Ny
, 1so V-S
Nv Ny + 2v
nvo #) = nJS0
ISO
NV =
DS
2V
(18)
09)
1 SO
Where Ny is the number of isolated pores observed to be wholly con-
S
tained in the volume of analysis. Ny is then given by
iso
^5 = n =
V V V
n;
i
DS
2V
DS
V
(20)
S
Thus, among the pores that cross the boundaries, Ny gives the number
of those that are isolated; the rest can be taken to be connected pores.
The treatment presented above therefore also facilitates the estimation
of the number of branches that cross the boundaries per unit volume.
Thus, the number of one-branch nodes can be estimated from Ty Ny ,
, S and V. The number, ofbranches per up it volume, by, cam be calcu
lated as shown in the Appendix, where the model is described in detail.
The values of by, Ny Ny and Ny are listed in Table 14. The variation
i so
of Ny, Figure 59, is qualitatively the same as that of Ny Figure 51.
It should be noted that due to the large surface to volume ratio of the
volume of analysis, the surface corrections change the values of by, Nyb
significantly. Thus, the previously measured values of the metric


113
Table 13
PARAMETERS OF A TYPICAL GEOMETRIC MODEL
Nature
Parameter
Definition
Topological
N3b
Number of three branch nodes per unit
volume
Nlb
NV
Number of one-branch nodes per unit
volume
bv
Number of branches per unit volume
nJS0
Number of isolated equiaxed pores
per unit volume
Metric
r
Average length of a branch
R
Radius of an isolated spherical
pore, also assumed to be the "radius"
of a spherical three-branch node (see
Figure 56)


47
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.
Quantitative Stereo!ogy
Metric properties are estimated by making measurements on a
two dimensional plane of polish with the help of standard relations
pc
of stereology. 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-
pc
tion or structure properties provided the structure is sampled uniformly.
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.) are therefore error-prone to some extent. As this investigation


30
Figure 15.
The variation of grain contiguity with solid volume
fraction for loose stack sintered copper and hot pressed
U02.22


BIOGRAPHICAL SKETCH
Arunkumar Shamrao Watwe was born on June 16, 1952, in Poona, India.
In June, 1967, he finished high school in Bombay, India. In June, 1974,
he received the degree of Bachelor of Technology in Metallurgical
Engineering from the Indian Institute of Technology, Bombay, India.
In August, 1976, he received the degree of Master of Science in Materials
Science and Engineering from Washington State University, Pullman, Washington.
In September, 1976, he came to the University of Florida to pursue the
degree of Doctor of Philosophy in the Department of Materials Science and
Engineering.
He is a fellow member of Alpha Sigma Mu and is a joint student
member of the American Society for Metals and the Metallurgical Society
of AIME.
163


109
Figure 55. Illustration of a favorable orientation of a
connected pore for effective pinning of a grain
boundary.


104
topological properties of the pore structure will be used to offer
an explanation for the observed interplay between grain boundary
network and the pore phase.
Topological Properties
Samples with Vy in the range from 0.90 to 0.98 were character
ized to obtain the connectivity and the number of isolated pores per
unit volume, illustrated respectively in Figures 49 and 50. The
connectivity (Gyin) monotonically decreases to a very small value,
but remains finite even at 98 percent density. The numbers of iso
lated pores first increases, Figure 50, as the pore network present
at the end of second stage disintegrates. A large part of the porosity
present at this point consists of rather highly interconnected but
isolated separate parts as opposed to isolated, equiaxed parts. Sub
sequent channel closure rapidly reduces these connected pores to smaller
separate parts with increasing density since, in the absence of any
significant amount of redundant connections, each channel closure event
produces a new separate part. As the sintering proceeds, more and more
parts are isolated; at the same time, some of the simple isolated pores
shrink and disappear. The maximum in the number of isolated pores seems
to indicate that beyond a certain point, very few pores are isolated and
hence from there on the number of isolated pores continues to decrease.
It can be seen from Figure 51 that the volume fraction of isolated poro
sity also goes through a maximum and then decreases to a very small value,
whereas the volume fraction of connected porosity changes only a little


23
Figure 10. Integral mean curvature versus volume fraction of
solid for 48 micron spherical and dendritic copper
powder.25


160
SS,
volume, and by (C), the number of "SS" pores that belong to the connected
porosity per unit volume are given by
.IS
JS =
JS,OLSS
by +2by
JS
ISO
or
and
or
bJS(C) =
.IS.,. SS
by +2by
rKlS.9KSS
{by +2by
1-
DS
2V
DS,
V 1
(59)
SS/p\ 1
bV ^ 2 JS
2b
SS
SS NV
by +2bv
bSS Niso
uSS,^ DV ri_ 1S, olSS NV
bV JSjo.SS {bV +2bV ^
by +2by i US^
V V |-2y
DS,
(60)
Since for every two features contributed by "SS" pores there is one "SS"
pore that is counted as an additional "3-3" type branch. Thus, the total
number of additional branches, by(add), is given by
by(add) = \ [bJS(C) + b*S(C)] or
JS,uSS
by +by
ISO
u v ruiS,olSS V
by(add) hlS+9hSSx ibV 2bV
2(by +2by ) DS
DS,
V *
(61)
1-
2 V
since all these branches are considered to contribute half as many to
the volume of analysis.
Each additional 1-3 branch contributes an additional one-branch node,
N^b(add), the additional number of one-branch nodes per unit volume, is
given by
blS Jso
DV r JS.ouSS NV
JS.-SS {bV +2bV c
by +2by ^DS
DS}
V 1
Nyb(add) = byS(C) =
(62)


99
t-conn
(cm-1)
Vy (Metallographic)
Figure 52. Dependence of surface area of the connected porosity
per unit volume on the volume fraction of solid during
loose stack sintering of nickel at 1250C.


118
Figure 59. Variation of N\j, surface corrected number of isolated
pores per unit volume during loose stack.sintering of
nickel at 1250C. Uncorrected values (N^so) are included
for comparison.


18
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.


64
Figure 32. Photomicrograph of INCO 123 nickel powder hot pressed
at 1250C to Vy = 0.984, etched (approx. 400 X).


93
0-90 0-95 10
Figure 50. Variation of the number of isolated pores per unit volume
(not corrected for surface effects) with the volume fraction
of solid during loose stack sintering of nickel at 1250C.
Data for the connectivity is included for comparison.


133
Unlike loose stack sintering, the grain boundary network did not
exhibit an arrest during hot pressing, as illustrated in Figures 37
through 40. The arrest observed in the grain boundary network for LS
series was attributed to isolated, equiaxed pores pinning the boundaries
effectively. If this assertion is to be valid, the presence of a higher
number of isolated pores in HP series postulated earlier in this section
must also imply that the boundaries are more effectively pinned. The
fact that such a pinning effect was not observed at all is taken to indi
cate that the boundaries are able to migrate in spite of the resistance
offered by isolated and associated porosity. It is speculated that the
application of external pressure induces enough amount of grain boundary
sliding for them to overcome the drag offered by the associated pores;
56
Pond et al. have found some evidence for coupling between grain boundary
migration and grain boundary sliding.
ssp SSS
The values of Ly and Ly(occy the quantities that increase with
an increase in the degree of association of porosity with the grain boun
dary network, were observed to be higher than those for LS. Increased
plastic flow, as suggested above, can also account for higher degree of
pinching off of channels and thereby a higher number of isolation events.
The importance of topological study of hot pressing is emphasized
again as it will test the postulate of a higher number of channel closure
events, as compared to loose stack sintering. The characterization of
associated fractions of isolated and connected porosity will shed a great
deal of light on the grain boundary-porosity relationship.


27
SyP (cm-1)
Figure 13. Grain boundary area per unit volume versus volume fraction
of solid_for 48 micron spherical and dendritic copper .
powder.25


19
There is evidence to suggest that this path of evolution of microstruc-
21
ture for loose stack sintering is insensitive to temperature.
SP
Data for hot pressed samples indicate that the Sy -Vy relationship
22 23
is only approximately linear even in the late second stage. The
path of microstructural change was also found to be insensitive to tem-
22 23
perature. The effect of pressure on the path was significant;
increasing pressure delayed the approach to linearity until a lower
p
value of Vy, as shown in Figure 8.
Integral mean curvature per unit volume, My, 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-
18 21 22
vex pore has a negative curvature. There is a miniimum in.*.My; *
35
this minimum occurs at lower Vy for finer particle size, 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 My in
the second stage. As the sintered density approaches the theoretical
density, My must approach zero and hence the initially high positive
value of My that becomes negative must go through a minimum. For an
p
initial stack of irregularly shaped particles, My varies with Vy at a
slower rate and has a minimum earlier in the process, compared to an
25
initial stack of spherical powders. 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 My is much more negative and
P 23
occurs at a lower value of Vy, compared to a loose stack sintered sample;


44
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 1250C 10C.
3. Hot Pressed Series
The third series was prepared by hot pressing at 1250C and under
a pressure of 2000 psi in a CENT0RR 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
-5
was placed in the vacuum chamber. After a vacuum of 10 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 5C 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
individually.


107
motion of these boundaries away from the occupying particles requires
that additional boundary area be created. Since this increases the
surface energy of the boundary network, there is a hindrance to the
52
boundary motion by associated, particles. However, if the pore
geometry is such that the motion of the boundary does not increase
the boundary area, the pores then do not anchor these boundaries
effectively. Thus, the boundary area can be pinned by a tree-like
or connected pore only if the branches in the latter intersect the
boundary with their axes at small angles to the plane of the boundary,
as illustrated in Figure 55. At all other orientations of the connected
pore there is no appreciable increase in the boundary area as it migrates.
On the other hand, a simple isolated pore, equiaxed in shape, is an
effective inhibitor to the grain boundary motion at any orientation.
It is therefore suggested that most of the pores anchoring the boundaries
are simple isolated pores. The relatively small changes observed in the
connected porosity can then be attributed to its inability to associate
itself with the grain boundaries. This sequence of microstructural changes
can be summarized as follows.
In the beginning of the second stage the porosity is mostly inter
connected and associated with the grain edges. The decrease in the
connectivity proceeds along with an increase in the grain boundary area
until a grain boundary network is formed. The subsequent grain growth
has decreased the grain boundary area and the migrating boundaries have
disassociated themselves with a part of the pore network by the time
simple, isolated pores begin to appear. Thus, the boundaries are brought


142
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
T. K. Gupta and R. L. Coble, J. Amer. Cer. Soc., 52, 293 (1968).
R. C. Lowrie, Jr. and I. B. Cutler, Sintering and Related Phenomena,
p. 527, Gordon and Breach Publishers, New York (1965).
G. C. Kuczynski, Powder Met., 12, 1 (1963).
M. Paul us, Sintering and Related Phenomena, p. 225, Plenum Press,
New York (1973).
C. S. Morgan and K. H. McCorkle, Sintering and Related Phenomena,
p. 293, Plenum Press, New York (1973).
J. E. Burke, Ceramic Microstructures, p. 681, John Wiley and Sons,
New York (1968^
E. H. Aigeltinger and H. E. Exner, Met. Trans., 8A, 421 (1977).
J. Kronsbein, L. J. Buteau, Jr. and R. T. DeHoff, Trans. Met. Soc.
AIME, 233, 1961 (1965).
Metals Handbook, 8th Edition, Volume 1.
B. R. Patterson, Doctoral Dissertation, University of Florida (1978).
M. Hillert, Acta Met., 12, 227 (1965).
G. C. Kuczynski, Sintering and Related Phenomena, p. 325, Plenum
Press, New York (1976).
A. R. Hingorany and J. S. Hirschhorn, Inti. J. Powder Met., 2, 5 (1966).
A. J. Markworth, Scripta Met., j>, 957 (1972).
R. C. Pond, D. A. Smith and P. W. J. Southerdon, Phil. Mag., A37, 27
(1978).


162
and
Mi50(add) =
Miso {DSm}
V DS
1 2V
(67)
Here it is assumed that the true D of the isolated pores is the same
as that measured for wholly contained isolated pores. The initial
measurements of properties of connected porosity were made assuming
all pores not contained inside are connected; the abovementioned addi
tions to the isolated fraction have therefore to be subtracted from
Vj0nn, Sy0nn and My0nn to correct for surface effects. These corrected
values, termed Vy0nn, Sy0nn and My0nn by the following equations.
Vvconn(C) = v§onn
{DS/2V}
1 -
DS
2V
s5onn(c) + sJonn
ciso
{
}
DS/2V
, DS
1 2V
Mf,nn(C)
= M:
conn
- M
iso
{DS/2V}
1 -
DS
2 V
(68)
(69)
(70)
The geometric model, discussed in detail in the preceding section
of the Appendix, is tested by comparing the corrected values and the
calculated values.


CHAPTER 3
DISCUSSION
Introduction
Sintering of a loose stack of powder is the simplest consolidation
process and hence forms the basis for investigation of more complex and
involved techniques. Loose stack sintering was therefore studied in the
greatest detail possible in this investigation. It will be discussed at
length in the beginning of this section to establish a framework on which
the descriptions of hot pressing and pressing and sintering (also called
conventional sintering) will be based. This discussion will be concluded
with a number of speculations regarding potential strategies to control
the microstructure of a powder-processed component.
Loose Stack Sintering
The discussion of metric properties will be followed by that of
topological properties. It is expected that this study will indicate
a likely scenario of various geometric events that lead to the observed
paths of evolution of microstructure. Usually such a detailed study
also suggests a variety of plausible geometric models of the structure;
this study is no exception. The model that is in the best agreement
with the data will be described and followed by suggestions for further
research necessary to complete an understanding of the process.
101


154
OL
Thus the integral mean curvature of a three-branch node, M is given by
M3b = MS + Medge = -ttR + (^pk2R or
M3b = tt (1 ^L)R (40)
One-Branch Node
Since such a node is a semi-spherical cap of radius r = (*/3/2)R that
is connected to a cylinder of radius r, there are no edges involved. Thus
V1b=|trr3 or Vlb=^R3
(41)
Area of a one-branch node, S^b, is given by
Sb = 2Trr2 or Sb = 2tt(^-) or Sb = ^R2
(42)
Integral mean curvature of a one-branch node, M^b, is given by
Mlb = 27rr2(-l) or Mlb = -2irr = -/3ttR
(43)
Cylindrical Branch
Since all branches have length = T and radius r = C^-)R,
Vb = -nr2!: = : ' (44)
Sb = 2TrrI = ^3ttRI (45)
Mb = -ttL
(46)


103
Metric Properties of the Grain Boundary Structure
SS
In addition to the four parameters mentioned earlier, namely, Sy ,
SSS SSS SSP
Lv Ly^occj and Ly ; the association of pores with the grain boundaries
was also characterized. An etched section of the sample was examined to
measure the number fraction of pores observed to reside within the grains,
on the boundaries and on the grain edges. The variations of these frac
tions are illustrated in Figure 41. These numbers indicate the fractions
of porosity associated with and not associated with the grain boundary
network. The twin boundaries do not participate in grain coarsening and
hence were not included in the characterization of the associated porosity.
The grain boundary area per unit volume, Sy illustrated in
Figure 37, can be seen to change only a little in the density range from
Vy = 0.93 to Vy = 0.97. Thus, the grain growth or decrease in grain
boundary area appears to have been appreciably inhibited in this density
range. It is suggested that the pore structure is changing in such a
manner that the associated porosity is able to pin the boundaries during
this phase of the process. This aspect of the grain boundary structure
will be discussed and explored further in the course of the description
of the other grain boundary properties.
Variations of LySS, LySP and -vfoCC) 1",lustrated in Figures 38, 39
SS
and 40, repsectively, also exhibit the arrest observed for Sy in the same
density range. The fraction of porosity associated with the grain boun
dary network decrease in this range of Vy, Figure 41. Thus, the more
or less stable grain boundary network seems to facilitate the
reduction of associated porosity. The following discussion of


REFERENCES
1. F. N. Rhines and R. T. DeHoff, Modern Developments in Powder
Metallurgy, p. 173, Plenum Press, New York (1971).
2. W. Rostoker and S. Y. K. Liu, J. Materials, _5, 605 (1970).
3. R. D. Smith, H. W. Anderson and R. E. Moore, Bull. Amer. Cer. Soc.,
55, 979 (1976).
4. R. T. DeHoff, F. N. Rhines and E. D. Whitney, Final Report, AEC
Contract AT(40-1), 4212 (1974).
5. G. Arthur, J. Inst. Metals, 83, 329 (1954).
6. R. A. Graham, W. R. Tarr and R. T. DeHoff, unpublished research.
7. G. Ondracek, Radex-Rundschau 3/4 (1971)
8. S. Nazare, G. Ondracek and F. Thummler, Modern Developments in Powder
Metallurgy, p. 171, Plenum Press, New York (1971).
9. J. Kozeny, Sitzber. Akad. Wiss. Wien., 136, 271 (1927).
10. M. F. Ashby, Acta Met., 22, 275 (1974).
11. R. T. DeHoff, B. H. Baldwin and F. N. Rhines, Planseeber. Pulvermet.,
JO, 24 (1962).
12. Metals Research Laboratory, Carnegie Institute of Technology, Final
Report, AEC Contract AT(30-1), 1826 (1959).
13. G. C. Kuczynski, Powder Metal1urgy, p. 11, Interscience Publishers,
New York-London (1961).
14. T. L. Wilson and P. C. Shewmon, Trans. Met. Soc. AIME, 236, 48 (1966)
15. G. Matsumara, Acta Met., 19, 851 (1971).
16. F. N. Rhines, C. E. Berchenall and L. A. Hughes, J. Metals, 188,
378 (1950).
17. R. T. DeHoff, Proceedings of the Symposium on Statistical and
Probabilistic Problems in Metallurgy, Special supplement to Advances
in Applied Probability (1972).
18. E. H. Aigeltinger and R. T. DeHoff, Met. Trans., 6A, 1853 (1975).
140


6
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
24
closed surfaces, 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
27
openings or creating new openings, until it collapses into the said
28
network that can be represented in the form of a simple line drawing.
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


54
Figure 22. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to Vy = 0.871, etched (approx. 400 X).


96
Table 12
TOPOLOGICAL PARAMETERS
lin,La ,,S rmax -3 rmin -3 ..iso -3
Number Vy Gy 9 cm Gy 9 cm Ny 9 cm
0.906
1.71 x 108
4.63 x 107
9.55 x 10
0.928
1.39 x 108
1.7 x 107
2.51 x 10
0.944
2.17 x 107
7.23 x 106
4.63 x 10
0.971
3.23 x 107
5.77 x 106
3.33 x 101
0.979
7.6 x 107
3.18 x 106
1.39 x 10(
5


88
When a connection is observed between different subnetworks, the
said subnetworks have to be renumbered to keep track of such connections.
ATI 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 N1S0 were updated and tabulated as shown in Table 10.
The values of interest are the unit volume quantities, G!J!ax, Gyin
i so
and Ny If the features that give rise to these quantities are randomly
25
and uniformly distributed in the sample, then it can be expected that
there exists a quantity (Qy) characteristic of the structure and equal to
the unit volume value. Thus AQ (the change in quantity Q) = Qy x AV.
Dividing both sides by aV leads to
"AV =
The slope of AQ versus AV plot therefore should be equal to Qy provided
AV, the volume covered is large enough for a meaningful sampling. In
25 51
the previous investigations of this kind the analyses were continued


53
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
quently, SyS, LySS, LySP and Ly^ccj, 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 Sy, the grain boundary area per unit volume, as measured metal!o-
graphically. However, it was found that these errors are small compared
CD CD .
to those in Sy and My ; the trends of grain structure properties remain
unaffected whether plotted versus Archimedes density or the stereo!ogical
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.


48
Table 6
QUANTITIES MEASURED ON POLISHED SECTION
Test Feature Quantity
Points Pp
Lines Ll
Area
NA
ta
Definition
Fraction of points of a grid that
fall in a phase of interest
Fraction of length of test lines
that lie in a phase of interest
Number of intercepts that a test
line of unit length makes with the
trace of a surface on a plane
section
Number of points of emergence of
linear feature per unit area of
plane section
Number of full features that
appear on a section of unit area
Net number of times a sweeping
test line is tangential to the
convex and concave traces of
surface per unit area of a plane
section


79
Figure 43. Illustration of contributions of subnetworks crossing
the surface towards the estimate of Gmax.


65
Figure 33. Photomicrograph of INCO 123 nickel powder cold pressed
at 60,000 psi and sintered at 1250C to Vy = 0.942,
etched (approx. 400 X).


138
5) It is suggested that because of their equiaxed shape, isolated
pores anchor the grain boundaries effectively whereas the connected pores
do not. Hence, most of the isolated porosity is associated with grain
boundaries.
6) The association of grain boundary network and isolated pores
facilitates rapid reduction of isolated porosity as the associated
boundaries provide immediate sinks of vacancies; this is in contrast
43 45-47
to the traditional viewpoint that once the pores are isolated,
it is very difficult to remove them from the system.
7) The connected porosity finds itself disassociated from the
grain boundary network which slows the reduction of such pores consid
erably.
8) The higher values of curvature, yet comparable values of
area and volume of pore phase for the hot pressed samples as compared
to loose stack sintered samples, are tentatively attributed to similar
geometries but a higher number of isolated, spherical pores.
9) An absence of an arrest of grain growth in hot pressed samples,
in spite of a higher number of equiaxed pores, is believed to be due to
stress-induced grain boundary sliding that promotes grain boundary migra-
ti on.
10) Porosity in pressed and sintered samples is believed to consist
of finer networks and a higher number of isolated pores compared to the
loose stack sintered samples with the same density; this leads to much
higher areas and curvatures in PS than in LS.
11) Due to larger number of interparticle contacts in a green
compact as compared to a loose stack sintered to the same density, a
pressed and sintered sample has much higher grain boundary area.


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
By
Arunkumar Shamrao Watwe'
August, 1983
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
significantly different.
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.


46
/, _\ Weight of sample
p^9/ I Volume of sample
Weight of sample
. (Volume of sample + immersed part of pan) -
(Volume of immersed part of pan)
M1
(W2 + weight of pan in air Wg) -
(Weight of pan in air W4)
Thus
W1
p (g/cc) = w2 + w4 w3
The densities thus measured were reproducible within 0.2 percent of
the mean of ten values with 95 percent confidence. The density of a
50
piece of pure nickel, known to have a density of 8.902 g/cc, was
measured and found to be within 0.5 percent of the abovementioned value.
Metallography
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.
Polishing Procedure
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


59
Figure 27. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to Vy = 0.979, etched (approx. 400 X).


17
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.^^0 gas ¡ias en0Ugh pressure to stabilize
16 41 42
the pore-solid interface, the densification rates can be very low.
Since exaggerated or secondary grain growth that results from boundaries
43 44
breaking away from pores has been observed to be accompanied by slow
rates of shrinkage,43,45-47 has been theorized that the grain boundaries
that can act as efficient vacancy sinks are far away from a large number of
43 45-47
pores. 5 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.
Metric Investigations
SP
It has been observed that in loose stack sintered samples Sy
decreased linearly with the decrease in Vy^2*1820) (jur-jng second
stage. Surface area may be reduced both by densification and surface
SP
rounding or by surface rounding alone; the linearity between Sy and Vy
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, as shown schematically in Figure 7. The slope of the
SP 20
Sy versus Vy line is inversely proportional to the initial particle size.


57
Figure 25. Photomicrograph of INCO 123 nickel powder loose stack
sintered at 1250C to Vy = 0.944, etched (approx. 400 X).


15
Figure 4. Illustration of a one-to-one correspondence between a
spurious node and a spurious branch in a deformation
retract.


120
Table 15
CORRECTED VALUES OF V^onn,
-conn
bV
and M
conn
No.
vf
..conn
vV
cconn -1
Sy cm
Mconn -2
My cm
1
0.906
0.09
731
-20 x 105
2
0.928
0.053
475
-15.2 x 105
3
0.944
0.017
138
-3.1 x 105
4
0.971
0.017
165
-4 x 105
5
0.979
0.02
237
-10.7 x 105