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Title: Metric and topological characterization of the advanced stages of sintering
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Permanent Link: http://ufdc.ufl.edu/UF00089978/00001
 Material Information
Title: Metric and topological characterization of the advanced stages of sintering
Physical Description: Book
Language: English
Creator: Watwe, Arunkumar Shamrao
Publisher: Arunkumar Shamrao Watwe
Publication Date: 1983
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Bibliographic ID: UF00089978
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: alephbibnum - 000506243
oclc - 12203626

Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
        Page iv
    Table of Contents
        Page v
        Page vi
        Page vii
        Page viii
        Page 1
        Page 2
        Page 3
        Page 4
    Evolution of microstructure during sintering
        Page 5
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    Experimental procedure and results
        Page 39
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    Biographical sketch
        Page 163
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Full Text







Copyright 1983


Arunkumar Shamrao Watwe


To My Parents,

Mr. Shamrao Vasudeo Watwe


Mrs. Sharada Shamrao Watwe


I am grateful for the opportunity to conduct my research under

the guidance of Dr. R. T. DeHoff, the chairman of my advisory committee.

An ability to approach any scientific matter with objectivity and logic

has been blissfully passed on by him to all his students.

I thank Drs. R. E. Reed-Hill, J. J. Hren, G. Y. Onoda, Jr.,of the

Department of Materials Science and Engineering and Dr. R. L. Scheaffer

of the Department of Statistics for serving on my advisory committee.

Their helpful advice and encouragement are deeply appreciated.

It is a pleasure to thank my colleagues, Mr. Atul B. Gokhale and

Mr. Shi Shya Chang, for their collaboration in the experimental aspects

of the project.

Mr. Rudy Strohschein, Jr., of the Department of Chemistry assisted

me beyond and above the call of duty in the fabrication of the sintering

apparatus. He saved me a great deal of time and aggravation.

All the credit for the preparation of this dissertation in its final

form must go to Miss Debbie Perrine for her excellent typing.

The financial support of the Center of Excellence of the State of

Florida and the Army Research Office is gratefully acknowledged.



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

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

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



Introduction ........................................... 5
Metric Properties of the Microstructure.................. 5
Fundamentals of Topology................................. 6
Sintering from a Geometric Viewpoint..................... 12
Importance of the Present Research...................... 35


Sample Preparation..............
Topological Measurements........

THREE DISCUSSION.............................................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
Suggestions for Further Study...........................139






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



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.



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


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.



A microstructure is characterized by its metric and topological

properties and therefore the following discussion will be carried out in

terms of variation of these quantities as the sintering proceeds. These

microstructural properties will be defined and the previous investigations

of this type will be discussed in detail; a review of metric studies will

be followed by topological analyses. The principles of quantitative

stereology employed in the estimation of microstructural properties will

be described in the next chapter on experimental procedure and results.

Metric Properties of the Microstructure

These quantities are estimated in terms of geometric properties of
lines, surfaces and regions of space averaged over the whole structure.2

The basic properties are listed in Table 1 and illustrated in Figure 1.

Among the properties listed, VV, SV and MV are used to yield two impor-

tant global averages of the microstructural properties. These are listed

in Table 2.

In a sintered structure, there are two regions of space or phases,

namely, pore and solid, and two surfaces, pore-solid interface and grain

boundaries. Two main linear features of interest are the grain edges and

the lines formed as a result of intersection of pore-solid interface and

grain boundaries. Superscripts are used to identify the properties that

are associated with a particular feature. These notations are listed

in Table 3 and illustrated in Figure 2.

In addition to the metric properties listed above, the microstruc-

ture of a porous body is also characterized by its topological properties.

A brief discussion of the fundamentals of topology will precede the sur-

vey of microstructural studies of sintering.

Fundametals of Topology

The subset of topological geometry of present interest is that of

closed surfaces,24 that is to say, surfaces that may enclose a region of

space. In a sintered body the regions of space are the pore and the

solid phases; the pore-solid interface is a closed surface of interest.

Such a surface may enclose several regions and have multiple connectivity.

A surface is said to be multiply connected if there exist one or more

redundant connections that can be severed without separating the surface

in two. The genus of such a surface is defined as the number of redundant

connections. For complex geometries it becomes difficult to visualize the

topological aspects of surfaces. It has been found very convenient :to

represent surfaces by equivalent networks of nodes and branches. Such an

equivalent network is called the deformation retract of a particular region

of space. It is obtained by shrinking the surface without closing any

openings or creating new openings,27 until it collapses into the said

network that can be represented in the form of a simple line drawing.28

A number of closed surfaces and their equivalent networks are illustrated

in Figure 3. The connectivity, P, of a network is equal to the number of

nearr Features of Length L

L = L

Sectioning Plane of
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

Property Definition

LV Length of a linear feature per unit

S,, Area of a surface per unit volume.

1 1 1-
2 rl r2

MV =f/ HdS

Local mean curvature of a surface
ata point on the surface, where
rl and r2 are the principal radii
of curvature. By convention, a
radius of curvature is positive
if it points into a solid phase.
Thus, a convex solid has a positive
curvature whereas a convex pore has
a negative curvature.

Integral mean curvature of
per unit volume.

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





Table 2

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

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

Table 3



Pore-Solid Interface


Solid Phase

Grain Edges in the

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

Grain Edges Occupied
by the Pore Phase




Area of pore-solid interface
per unit volume

Integral mean curvature of
pore-solid interface per unit

Volume fraction of porosity

Volume fraction of a solid

Length of grain edges or
triple lines per unit volume

Length of intersection lines
of pore-solid interface and
grain boundaries per unit

Length of occupied grain
edges per unit volume


Grain Boundaries of Area SSS


(occ) V V

v L

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-


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

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
decreased linearly with the decrease in V(1218-20) during the second

stage. Surface area may be reduced both by densification and surface
rounding or by surface rounding alone; the linearity between SP and VV

is believed to arise from a balance between surface rounding and densifi-

cation. Support for this hypothesis comes from the observation that sur-

face rounding dominates in pressed and sintered samples until the balance

has been reached,35 as shown schematically in Figure 7. The slope of the

SSP versus VV line is inversely proportional to the initial particle size.20

V (a)

S\ 1I


Figure 6. Two basic topological events that occur in the network
coarsening scenario proposed by Rhines.36 The dotted
lines indicate the occupied grain edges.

There is evidence to suggest that this path of evolution of microstruc-

ture for loose stack sintering is insensitive to temperature.21
Data for hot pressed samples indicate that the SV -Vv relationship
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
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
this minimum occurs at lower VV for finer particle size,3 as illustrated

in Figure 9. According to the convention used, most of the "SP" surface

has positive curvature in the initial stages. Due to decreasing surface

area and increasing negative curvatures there occurs a minimum in MV in

the second stage. As the sintered density approaches the theoretical

density, MV must approach zero and hence the initially high positive

value of MV that becomes negative must go through a minimum. For an

initial stack of irregularly shaped particles, MV varies with VP at a
slower rate and has a minimum earlier in the process, compared to an

initial stack of spherical powders.25 This is illustrated in Figure 10.

In all the cases studied the paths were insensitive to temperature. In

the case of hot pressing, the minimum in MV is much more negative and
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


Figure 7.





cm1 )



Pore-solid interface area versus
conventionally sintered U0222

solid volume fraction for

0 0 KSI
S 1-8 KSI
- A 10-20 KSI
~ \ 30-40 KSI
O 75-90 KSI


I I I I I \





Figure 8.



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


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

these curves become deeper and shift towards lower VV with increasing

pressure,22 as shown in Figures 11 and 12.
The.grain boundary area per unit volume, SV increases until a
network is formed; subsequent grain growth tends to decrease SS.
This was observed for loose stack sintering,8 as shown in Figure 13.

It is evident here that the variation of SS with V is independent of

the initial particle shape in the late second and early third stages.
LSSP increases with decrease in V until the second stage is reached22
when it begins to decrease. In the second stage :L- is significantly

higher than the case for random intersection of "SP" and "SS" surfaces,48

as illustrated in Figure 14.

A new metric property, IA, was discovered in the course of doctoral

research carried out by Gehl22 at the University of Florida; IA is the

measure of inflection points observed on the traces of a surface per unit

area of plane of polish, and is proportional to the integral curvature of

asymptotic lines over saddle surfaces (surfaces that have principal radii

of curvature of opposite signs at all points on the surface). This was
found to decrease smoothly in the second stage22 which means that the

saddle surfaces occupy only a small fraction of the pore-solid interface

at the end of the second stage.

The variation of grain contiguity, grain face contiguity and grain

shapes during conventional sintering and hot pressing were studied in

some detail by Gehl.22 There were two parameters, CS and CSS, defined

for grain contiguity and grain face contiguity, respectively. Four

unitless parameters, Fl, F2, F3 and F4, were used to characterize grain

and pore shapes. These were defined as follows.22


(105 cm-2)



Figure 11.


0.80 0.85 0.90 0.95

Variation of integral mean curvature per unit volume with
the volume fraction of solid during hot pressing of RSR 107
nickel (-170+200) at 1500 psi. Data for spherical copper
(-170+200) loose stack sintered at 10050C included for

(106 cm-2)

Figure 12.

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



60.0 ,



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

[11 t-^

C = S (2)
2SV +S V

Css = 3L~_ss

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

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

LV +3L
F3 SS (6)

SSSP SS)2 (7)
F4 = LVP/2(SV)2 (7)

The fraction of the total area of solid grains shared with other
grains is given by CS. Variation of CS with VP for hot pressed samples
of U2(22) and loose stack sintered spherical and dendritic copper
powder25 is shown in Figure 15. It can be seen from the definition of
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

7V -V



Figure 15.

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

at the same VP when compared to a loose stack sintered sample; SV

was found to vary linearly with CS and the dependence was the same

for widely different precompaction pressures up to very late second

stage. For hot pressed samples, a maximum was observed in CS, believed

to indicate a point where the grain boundary area has increased enough
to form a boundary network that subsequently coarsens. Both LSP and
LSSS exhibited a maximum when plotted versus CS for conventionally

sintered and hot pressed samples.

The grain face contiguity parameter, CSS, indicates the fraction

of edge length of grain faces that is shared with other grains. It

can be shown22 that CS = CSS for the case of random intersection of

grain boundaries and pore-solid interface, and CS > CSS when grain

boundaries intersect pore-solid interface preferentially. For conven-

tionally sintered and hot pressed samples CS was observed to be greater

than CSS which indicated preferential association of grain boundaries

with the pore-solid interface.

The factors FI, F3 and F4 can be used to compare the grain shapes

and F2 the pore shapes; Fl, F2, F3 and F4 were observed to be weakly

linear with VV, whereas a strong correlation was observed between F2,

F3 and F4 and CS for all the samples. It was believed that the above

data indicate a strong influence of the extent of grain contiguity (CS)

on the grain and pore shapes.

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.
During the second stage, S decreased linearly with decrease in VV;

the slope of this line was found to be proportional to GV, genus per unit

volume20 as should be expected from dimensional analysis. Kronsbein et

al.49 carried out serial sectioning of sintered copper samples and found
that even for VV = 0.1, very few pores were isolated. This is in agree-

ment with similar observations made by Barrett and Yust.34

Aigeltinger and DeHoff18 studied loose stacking sintering of copper

powder by measuring metric and topological properties. The genus per

unit mass, Gp, number of isolated pores and number of contacts per parti-

cle, C, were the measured topological quantities. Variation of Gp and

number of isolated pores with V revealed a definite increase in the

former during the first stage and identified the end of the second stage

(Gp = 0). As shown in Figure 16, Gp and number of isolated pores were

inversely proportional to the initial value of mean particle volume.

The same plot for dendritic powder showed that the topological path is

different up to late third stage and that the third stage (Gp 0)
begins at a higher value of VV as compared to spherical powder, Figure

17. The initial decrease in Gp during the first stage for dendritic

Gp or Np
(106 gm )

0 Gp (48 micron)
O Gp (115 micron) x 13.7

* Np (48 micron)

* Np (115 micron) x 13.7

0.6 VV 1 0
I -- II II

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)

0.05 a


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


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.



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



Nickel Powder (Wt.%)

Carbon (typical)





Other Elements



0.1 max

0.15 max

0.001 max

0.01 max



Table 5

Particle Shape Roughly spherical with spiky surface

Average Particle Siz 5.5 microns

Standard Deviation 0.75 microns

Surface Area Per Unit Volume 7.65 x 103 cm/cm

VS of As-Received Powder 0.25

It. was found by trial and error that sintering a loose stack of this

powder at 1250C produced the required series of samples in convenient

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


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)

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

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.


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-
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

Test Feature Quantity Definition

Points Pp Fraction of points of a grid that
fall in a phase of interest

Lines LL Fraction of length of test lines
that lie in a phase of interest

PL Number of intercepts that a test
line of unit length makes with the
trace of a surface on a plane

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

Table 7

Pp = L = VV

PL 2 V

P = LV

TT A = 27TNA = M





dealt with relatively small amounts of porosity (10 percent or lower)

the error in VV introduced by the polishing technique approached that

of the density measurements, namely, about 0.005, as the sintered den-

sity approached the bulk density. Thus, the polishing was accepted for

further characterization if the metallographically determined VP was
within 15 percent of VP obtained from the water immersion method, except

for the samples 97 percent dense and higher for which the limit had to
be relaxed to 30 percent of VV. Since VV values range from 0.15 to 0.02,

the abovementioned limits translate into a few percent of the sample den-

sity as measured metallographically. Typically, the samples 97 percent

dense and lower exhibited a precision of 0.05 of the VV value obtained

from the density measurements. Manual measurements of SP and MSP were

made on the accepted polished surfaces using standard stereological techni-

ques.26 The measurements of V SSP and MSP were made with at least 30

different fields and at magnifications that allowed at least 15 pores to be

viewed in a single field. As a result, the estimates of the properties

were within 5 percent with 95 percent confidence, as illustrated in Figures

20 and 21. Plots of SSP and MSP contained metallographically measured values

of VV to yield the paths of evolution of microstructure in order to partially

compensate for the polishing errors.

Measurement of these metric properties was followed by etching the

specimens to reveal the grain boundaries. Each sample was immersed in a

solution made from equal parts of nitric acid, glacial acetic acid and

acetone for about 30 seconds. The grain boundaries were brought out

clearly with some evidence of facetting of the initially smooth contours

of pore features. Samples in the lower part of the density range exhibited



(102 cm )





0 Loose Stack Sintered

0 Hot Pressed at 2000 psi

SCold Pressed at 60,000 psi


I I I 1 1 1 t I l 1


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.





(105 cm-2)



O Loose Stack Sintered

O Hot Pressed at 2000 psi

- A Cold Pressed at 60,000 psi


I 1.. I I I I I I I I I. I I I I.. ...1 L L- I.


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 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-
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).


* 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).




(cm-1) 300 -

200 -o

100 -

0 90




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.







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.









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







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 '




1 x105


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.










of pores





Figure 41.




1.0 0.90




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.


pressed and

ii Ii I?


. . I ill. I i I ii I

- i

- 0


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


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

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



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-


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

Sample Nickel ring

Epoxy mount

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

genus or the connectivity and the number of isolated pores were

measured by comparing pairs of neighboring sections as follows.

There are listed in Table 9 three possible classes of topologi-

cal events that can be observed when two consecutive sections are

compared, along with the corresponding increments in Gmax, Gmin and

Nis0. The significance of each of such observed events will be

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



Table 9








AGmin ANiso
0 +1

Within a


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


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

)-- -- \--

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
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

Height of Sample, Microns


Counts Gmax

Previous total -

This section

Current total

Unit volume
Current total
Current volume

-- _


until a linearity between AQ and LV was observed. A different criterion

was adapted in the present investigation, and is described presently.

Since the connectivity and the number of separate parts are two

independent quantities, one may begin to exhibit a constant value of

Q after several sections whereas the other may be far from levelling

off. This makes it difficult to establish a criterion for terminating

the serial sectioning for a given sample. If use is made of both the

quantities a standard basis for terminating the analysis can be obtained.

A typical pore subnetwork with convex, concave and saddle elements

of its surface is shown schematically in Figure 48. If a test plane is
swept through a unit volume, a measure of TV, the number of times this

plane is tangential to the pore-solid interface, may be obtained in

principle. This is related to the connectivity and the number of parts

per unit volume26 as follows.

Tet = T + T T = 2(NV GV)


where TV = number of times a concave element (having both the principal

radii of curvature negative by convention) is tangential to

test plane, or the number of "ends" of a feature (see Figure

TV = number of times a convex element (having both the principal

radii of curvature positive by convention) is tangential to

to test plane, very small at this stage of sintering.
T = number of times a saddle element (with principal radii of

curvature of different signs by convention) is tangential

Figure 48. Convention used in the net tangent count (Tnet)
during serial sectioning.



1 x10



Figure 49.

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

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