Citation
Analysis of the sintering force in copper

Material Information

Title:
Analysis of the sintering force in copper
Creator:
Gregg, Roger Allen, 1938- ( Dissertant )
Rhines, F. N. ( Thesis advisor )
Reed-Hill, R. E. ( Reviewer )
Bailey, T. L. ( Reviewer )
Blake, R. G. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1968
Language:
English
Physical Description:
xiii, 126 leaves. : illus. ; 28 cm.

Subjects

Subjects / Keywords:
Curvature ( jstor )
Density ( jstor )
Geometry ( jstor )
Interfacial tension ( jstor )
Particle density ( jstor )
Particle size classes ( jstor )
Porosity ( jstor )
Sintering ( jstor )
Specimens ( jstor )
Surface areas ( jstor )
Copper ( lcsh )
Dissertations, Academic -- Metallurgical and Materials Engineering -- UF
Metallurgical and Materials Engineering thesis Ph. D
Sintering ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
The sintering force is defined as the external load which will balance the contractile tendency of a sinter body in the direction of its application. Experimental measurements of the sintering force were made upon uneompacted, sintered copper specimens as a function of apparent density, sintering temperature and particle size. All measurements were obtained from specimens sintering in dry, deoxidized hydrogen. Measurements of void volume and of surface area and average mean curvature of the void-solid interface were made through the procedures of quantitative metallography. The sintering force was found to increase from zero at the onset of sintering, pass through a maximum at approximately 90 per cent of theoretical density and decrease toward zero at bulk density. This general behavior was observed for each particle size. The magnitude of the sintering force at any density increased with a decrease in particle size. There was no effect of temperature on the sintering force in copper over the range 950 to 1050 C. The surface area per unit volume-density relationship was observed to be linear for each particle size. Variation of the average mean surface curvature with density was found to be qualitatively identical to that observed for the sintering force. The results are analyzed on the basis of a balance between the externally applied force and the surface tension forces promoting shrinkage. A quantitative expression of the sintering force in terms of the surface geometry and surface tension of the sinter body is derived from basic concepts of capillarity. A comparison is made of the sintering forces predicted by this relation and experimental value; Sensitivity of the rate of shrinkage to the geometry of the microstructure of the sinter body will be demonstrated through simultaneous application of the concept of the sintering force, an empirical ^treatment of the mechanical behavior of the sinter body and a stress-strain rate relation based on deformation by creep.
Thesis:
Thesis - University of Florida.
Bibliography:
Bibliography: leaves 122-125.
Additional Physical Form:
Also available on World Wide Web
General Note:
Manuscript copy.
General Note:
Vita.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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ANALYSIS OF THE SINTERING FORCE
IN COPPER













By
ROGER ALLEN GREGG


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA
1968

































Dedicated to

my wife

Susan

















ACKNOWLEDGMENTS


The author wishes to express his sincere gratitude to

Dr. F. N. Rhines, chairman of his supervisory committee, for suggest-

ing the subject of this research and for his invaluable assistance in

establishing the conceptual framework of the problem.

The author is indebted to Dr. R. T. DeHoff for lending his

many talents to the discussion of the problem and especially for his

timely contributions to the field of quantitative metallography.

The author wishes to thank Dr. R. E. Reed-Hill, Dr. T. L.

Bailey and Dr. R. G. Blake for serving on his supervisory committee.

The financial support of this research by the Atomic Energy

Commission was appreciated, and is hereby acknowledged.














TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS . . . . . . . . . i

LIST OF TABLFS . . . . . . . . . vii

LIST OF FIGURES . . . . . . . . . . viii

ABSTRACT . . . . . . . .. . . . . . xii

Chapter

I. INTRODUCTION . . . . . . . .. . . 1

1.1 General Characteristics of the Sintering
Process . . . . . . . . . 1

1.2 Previous Investigations of the Sintering
Process . . . . . . . . . 2

1.3 Purpose and Scope of this Research . . . 7

II. EXPERIMENTAL PROCEDURE . . . . . . . .. 13

2.1 Material Specifications . . . . . .. 13

2.2 Particle Size Classification . . . . .. 14

2.3 Specimen Preparation . . . . . . .. 14

2.4 Experimental Determination of the
Sintering Force . . . . . . .. 17

2.41 General features of the apparatus . . 17

2.42 Strain measurement . . . . . 21

2.43 Calibration of beam force and
deflection versus indicated strain . .. 21

2.44 Test temperature and environment
control . . . . . . . .. 23

2.45 Test procedure . . . . . ... 23








TABLE OF CONTENTS (Continued)


Chapter

II. (Continued)

2.5 Post-test Inspection of the Specimens . .

2.51 Density determination . . . . .

2.52 Metallographic preparation . . . .

2.53 Metallographic examination . . . .

III. EXPERIMENTAL RESULTS . . . . . . . . .

3.1 Sintering Force . . . . . . . .

3.2 Quantitative Matallography . . . . .

3.21 Surface area of the void-solid interface

3.22 Area tangent count . . . . . .

3.23 Average mean surface curvature . . .

3.24 Effect of uniaxial constraint on
sintering behavior . . . . . .

IV. DISCUSSION . . . . . . . . . . .

4.1 Surface Energy and Surface Tension of Solids

4.11 Pressure difference across a curved
surface . . . . . . . .

4.12 Measurement of surface tension effects
in solids . . . . . . . .

4.2 The Geometry of Sintering . . . . . .

4.21 The evolution of curvature of the
void-solid interface . . . . .

4.3 Analysis of the Sintering Force . . . .

4.31 Comparison of predicted and experi-
mentally measured sintering forces . .


Page


26

27

27

28

30

30

38

45

45

50


50

58

59


61


63

69


73

78


83








TABLE OF CONTENTS (Continued)


Chapter Page

IV. (Continued)

4.4 Comparison of the Results of This Research
with Previous Investigations . . . .. 89

4.5 Application of the Sintering Force to the
Measurement of Surface Tension of Solids . 90

4.6 Kinetics of the Sintering Process ..... 92

4.61 Estimation of stresses created
by external load . . . . . . 92

4.62 Correlation of the sintering force
with shrinkage rates . . . . . 97

4.63 Comments on the mechanism of shrinkage 105

V. CONCLUSIONS . . . . . . . .. . 107

APPENDICES . . . . . . . .. . . . . 109

I. APPLICATION OF THE PRINCIPLES OF TOPOLOGICAL
GEOMETRY TO THE DESCRIPTION OF THE SINTERING
PROCESS (after F. N. Rhines [l]) . . . . .. 110

II. THE QUANTITATIVE ESTIMATION OF AVERAGE MEAN
SURFACE CURVATURE (after R. T. DeHoff [41]) .... 116

REFERENCES. . . . . . . . .. .. . . . 122

BIOGRAPHICAL- SKETCH . . . . . . . . .. .. . 126













LIST OF TABLES


Table Page

1. Experimental values of sintering force for the
48 micron particle size as a function of
density and temperature . . . . . ... .31

2. Experimental values of sintering force for the
30 micron particle size as a function of
density and temperature . . . . . ... .33

3. Experimental values of sintering force for the
12 micron particle size as a function of
density and temperature . . . . . ... .35

4. Quantitative metallography data for the 48 micron
particle size as a function of density and
temperature . . . . . . . .... . 39

5. Quantitative metallography data for the 30 micron
particle size as a function of density and
temperature . . . . . . . .... . 41

6. Quantitative metallography data for the 12 micron
particle size as a function of density and
temperature . . . . . . . . .. . 43













LIST OF FIGURES


Figure Page

1. Split graphite mold used for presintering the
sintering force specimens . . . . . . .. 15

2. Schematic presentation of experimental setup for
determination of the sintering force . . . ... 18

3. Apparatus used for measurement of the sintering force . 19

4. Cantilever beam assembly of sintering force apparatus . 20

5. Calibration curves of indicated strain and change in
specimen length, AL, versus cantilever beam force . 22

6. Typical specimen behavior during experimental
measurement of the sintering force . . . ... 25

7. Experimental values of the sintering force as a
function of density for the 4S micron particle
size spherical copper powder . . . . . ... 32

8. Experimental values of the sintering force as a
function of density for the 30 micron particle
size spherical copper powder . . . . . . 34

9. Experimental values of the sintering force as a
function of density for the 12 micron particle
size spherical copper powder . . . . . ... 36

10. Variation of surface area per unit volume with
density for the 48 micron particle size
spherical copper powder . . . . . . ... 40

11. Variation of surface area per unit volume with
density for the 30 micron particle size
spherical copper powder . . . . . . ... 42

12. Variation of surface area per unit volume with
density for the 12 micron particle size
spherical copper powder . . . . . . ... 44


viii







LIST OF FIGURES (Continued)


Figure Page

13. Comparison of the linear relationships between
density and surface area per unit volume for
the 48, 30 and 12 micron particle sizes . . . .. 46

14. Variation of the net area tangent count with
density for the 48 micron particle size
spherical copper powder . . . . . .... ... 47

15. Variation of the net area tangent count with
density for the 30 micron particle size
spherical copper powder . . . . ... .. .. . 48

16. Variation of the net area tangent count with
density for the 12 micron particle size
spherical copper powder . . . . . . . .. 49

17. Variation of the average mean curvature of the
void-solid interface with density for the
48 micron particle size spherical copper powder . . 51

18. Variation of the average mean curvature of the
void-solid interface with density for the
30 micron particle size spherical copper powder . . 52

19. Variation of the average mean curvature of the
void-solid interface with density for the
12 micron particle size spherical copper powder . . 53

20. Correlation of the slope of the surface area-
density relationship with initial particle size
for spherical copper powder . . . . . . .. 55

21. Cross-sectional areas of sintering force specimens
after testing . . . . . . .... . . 56

22. Schematic of surface stresses in a solid . . ... 60

23. Resolution of surface tension into a pressure acting
normal to each point of a curved surface ...... 62

24. Two arrangements for balancing the shrinkage of
metal foils by inducing creep with an external force 65

25. Schematic of surface tension forces and grain boundary
array in wire geometry used for determination of
surface tension in metals . . . . . ... 68









LIST OF FIGURES (Continued)


Figure Page

26. Variation of genus (connectivity)and number of
separate parts of the void-sclid interface with
density for the 48 micron special copper powder [43] 72

27. Comparison of the average surface curvature as a
function of density for the 48, 30 and 12 micron
particle sizes of spherical copper powders . . . 74

28. Variation of average surface curvature with density
for a mixture of sizes of an irregular, dendritic,
electrolytic copper powder [62] . . . . .... .79

29. Resolution of surface tension forces emanating from
the cut surface of a spherical pore in a direction
perpendicular to the sectioning plane . . . ... 81

30. Comparison of the experimental values of the sinter-
ing force with predicted values for the 48 micron
particle size powder . . . . . . .... .84

31. Comparison of the experimental values of the sinter-
ing force with predicted values for the 30 micron
particle size powder . . . . . . . ... 35

32. Comparison of the experimental values of the sinter-
ing force with predicted values for the 12 micron
particle size powder . . . . . . . ... 86

33. Room temperature yield stress at 2 per cent elonga-
tion as a function of density for a range of
particle sizes of the spherical copper powders
used in this investigation [67] . . . . .... .94

34. Variation of the average solid area and the effec-
tive solid area estimated from experimentally
observed mechanical behavior with density for
spherical copper powders . . . . . . ... 96

35. Estimated operating stresses created by externally
applied force on sintering force specimens . . .. 98

36. The relation between strain rate, e, and stress, C,
as affected by the scale of the microstructure
for isothermal high temperature deformation of
pure metals . . . . . . . . ... . . 100








LIST OF FIGURES (Continued)


Figure Page

37. Variation of experimentally determined linear shrinkage
rates with density for the 43 [70] and 30 [71] micron
particle sizes of spherical copper powder . . .. 101

33. Relationship between linear shrinkage rates and
estimated effective stresses for the h8 and 30 micron
particle sizes of spherical copper powder ...... 102

39. Correlation of linear shrinkage rate--estimated
operating stress relationship with deformation rate
law based on stress-directed diffusion . . ... 104

40(a). Examples of bodies bounded by surface of genus zero,
one and two . . . . . . . .... ... 112

40(b). Variation of genus with number of contacts
per particle . . . . . . . .... .. 112

41(a). Separation of sintering into three stages on the
basis of changes in genus of the void-solid
interface . . . . . . . . ... .. . 114

41(b). Effect of genus of the void-solid interface on its
minimal surface area configuration . . . ... 114

42. Orientation relationships that exist between an
arbitrary element of surface (dudv) and a section-
ing plane; n is the surface normal; u and v are
the directions of principal curvature; I is normal
to the sectioning plane . . . . . . .. 120








Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy


ANALYSIS OF THE SINTERING FORCE IN COPPER

By

Roger Allen Gregg

June, 1968


Chairman: Dr. F. N. Rhines
Major Department: Metallurgical and
Materials Engineering


The sintering force is defined as the external load which will

balance the contractile tendency of a sinter body in the direction of

its application. Experimental measurements of the sintering force were

made upon uncompacted, sintered copper specimens as a function of

apparent density, sintering temperature and particle size. All meas-

urements were obtained from specimens sintering in dry, deoxidized

hydrogen.

Measurements of void volume and of surface area and average

mean curvature of the void-solid interface were made through the proce-

dures of quantitative metallography.

The sintering force was found to increase from zero at the

onset of sintering, pass through a maximum at approximately 90 per cent

of theoretical density and decrease toward zero at bulk density.

This general behavior was observed for each particle size. The magni-

tude of the sintering force at any density increased with a decrease

in particle size. There was no effect of temperature on the sinter-

ing force in copper over tha range 950 to 10500C.

xii









The surface area per unit volume-density relationship was

observed to be linear for each particle size. Variation of the aver-

age mean surface curvature with density was found to be qualitatively

identical to that observed for the sintering force.

The results are analyzed on the basis of a balance between the

externally applied force and the surface tension forces promoting

shrinkage. A quantitative expression of the sintering force in terms

of the surface geometry and surface tension of the sinter body is

derived from basic concepts of capillarity. A comparison is made of

the sintering forces predicted by this relation and experimental values.

Sensitivity of the rate of shrinkage to the geometry of the

microstructure of the sinter body will be demonstrated through simul-

taneous application of the concept of the sintering force, an empir-

ical treatment of the mechanical behavior of the sinter body and a

stress-strain rate relation based on deformation by creep.


xiii












CHAPTER I


INTRODUCTION


The atoms or molecules on the free surface of a solid possess

higher energy than those within the interior. Therefore, a solid sys-

tem comprised of finely divided powders has higher total energy than

a single large particle of the same material and equal miss. By re-

arranging material so as to reduce its surface area, the system of

powder particles can lower its energy and become thermodynamically

more stable. When powder particles are held in contact at a temper-

ature close to, but below, their melting point, material rearrangement

driven by the excess surface energy can produce permanent adherence

between the particles and result in a single, solid framework. This

process is known as sintering. Most powdered materials will exhibit

sintering under the proper conditions of temperature and environment.

This includes crystalline, vitreous and organic materials.


1.1 General Characteristics of the Sintering Process

When powdered materials are sintered the following general

features are observed. The particles form permanent connections at

their points of contact, which increase in size with time. The sur-

faces of the pores become smooth and the total volume of porosity

within the powder aggregate decreases with time, increasing the appar-

ent density of the system. Every gecmerric change driven by surface








energy is accompanied by a decrease in the total surface area in, the

system. Under the proper conditions, the apparent density of the

sinter body may approach the absolute density of the solid material.

The evolution of the internal structure of a sinter body is

separable into stages. The classification of the stages of sintering

as proposed by Rhines [1] will be applied here. In the first stage,

particle contacts broaden into weld necks and surface contours are

smoothed. In the second stage, the principal geometric change is a

decrease in connectivity of the pore network through closure of the

connecting links, or channels, of the network. The completion of this

process signifies the end of the second stage. The third and final

stage is characterized by the elimination of the isolated porosity.


1.2 Previous Investigations of the Sintering Process

Considerable effort has been expended in fundamental research

of sintering, primarily directed toward determination of the controlling

mechanism. There exist several mass transport processes which can act

either individually or in concert in their participation in the mater-

ial rearrangement observed during sintering. These are: evaporation-

condensation, surface diffusion, volume diffusion, viscous flow and

bulk deformation by shear processes. Consequently, sintering is a com-

plex phenomenon, a fact evident in the diversity of the theories of

sintering which have been proposed.

Saurwald, in a series of papers, was the first to attempt to

formulate a general theory of sintering. He concluded [2] that adhesive


Numbers in brackets refer to te references.
Numbers in brackets refer to the references.









forces between the powder particles are responsible for their con-

solidation, which in his opinion, had striking similarities to the

recrystallization processes observed in metals. Balshin [3,4,5]

attempted a refinement of this approach suggesting that densification

would result from recrystallization initiated at particle contacts,

while a decrease in density would be produced by recrystallization

within the particles. Jones[6] agreed that the forces of atomic cohe-

sion determine the sintering process and concluded that since these

forces decrease with increasing temperature, the increase in sintering

rate with temperature is due to a rapid decrease in the resistance to

plastic flow. Baike [7], Wretblad and Wulff [8], and also Rhines [9],

were among the first to suggest that surface tension played a major

role in promoting sintering. Balke visualized a "zipper action" in

closure of the void space which was initiated at the contact points

between the particles. Wretblad and Wulff suggested the application of

the equations of capillarity in calculating the stresses at the contact

points as a function of the shape of the solid surfaces and their

energy. They further suggested that the stresses estimated in this

manner could exceed the elastic strength of the solid and result in

plastic deformation. Rhines concurred that the magnitude of the forces

derived from surface tension would be greatest in the regions of highly

curved surfaces. Interest in this viewpoint was spurred by theoretical

[10,11] and experimental analyses [12] of the viscous flow of metals

under the action of surface tension. It is now generally agreed that

surface tension is the primary force in achieving the geometric changes

observed during sintering. However, agreement has not been reached









concerning the manner in which the action of surface tension acccm-

plishes these changes.

Pines [13] and also Shaler and Wulff [14] suggested that the

mechanisms) responsible for smoothing the particle surfaces and round-

ing the pores could be distinct from the transport mode producing

densification. Shaler and Wulff pointed out that densification requires

changes in shape and size of the particle network or "skeleton." In

order to achieve this, material must be removed from within the particles

along the line joining their centers. It is now generally acknowledged

[15,16,17] that transport mechanisms which are restricted to the surface,

such as evaporation-condensation and surface diffusion, can produce

only surface-rounding, not densification. Rhines [18] made the impor-

tant suggestion that a transfer of void "space" from the pores to the

external surface, which would produce densification, could occur by a

lattice vacancy diffusion process.

To avoid the complications of a multiconnected, three-

dimensional stack of particles, Kuczynski [19] used simple models of

a single sphere on a flat surface and two spheres in contact to eval-

uate the controlling mechanism in the first stage of sintering. He

invoked the equations of capillarity to calculate the vapor pressure,

vacancy concentration and internal pressure in terms of surface geo-

metry. For these simple geometries he was able to derive equations

relating the rate of growth of the weld neck to the transport mechanism

assumed. lis experimental evidence on metals [19] indicated that neck

growth was controlled by volume diffusion in coarse particle systems,

and by surface diffusion for small particles. In glass [20] he reported








a viscous flow mechanism. Mackenzie and Shuttleworth [21] pointed out

that the removal of vacancies by volume diffusion to the exterior of

the sinter body is too slow to explain the rates of densification

observed. In addition, densification by this process would progress

inward from the external surface, and the rate of densification would

be a function of the size and shape of the system; a situation which

does not develop in sintering.

Using the technique of wrapping wires on a spool, Geach and

Jones [23] observed that the grain boundaries formed at contact points

were stable for long periods of time at sintering temperatures. Using

the same experimental procedure, Alexander and Balluffi [24] claimed

that pores in the wire compact continued to shrink only if they re-

mained connected to a grain.boundary. Correa da Silva and Mehl [25]

and Pranatis et al. [26] suggested that vacancy removal, required for

densirfication by volume diffusion, could occur at grain boundaries

through collapse of the vacancies collected there; Herring [11] had

previously suggested this mechanism for creep by volume diffusion.

Kuczynski [27] adopted the grain boundary effect into his diffusion

model and found reasonable agreement between the rates of pore closure

predicted by volume diffusion and those observed in photographs of

wire-compacts published by Alexander and Balluffi [24,28].

Herring [29] showed that in two-particle systems with similar

geometries but differing by a scale factor X, the times required to

reach identical stages in sintering should be related through this


This has been reported by Rhines et al. [22] to occur only
in the pores immediately adjacent to the external surface.









factor. If, for example, (powder), is X times larger than (powder)2,

then the times required to reach similar stages of sintering are

given by


At
At2


where n = 1, for evaporation-condensation;

n = 2, for viscous Flow;

n = 3, for volume diffusion; and

n = 4, for surface diffusion.

Alexander and Balluffi tested this relation for their wire compacts

and found reasonable support for volume diffusion.

However, DeHoff et al. [30] applied Herring's analysis to the

rate of weld neck growth between twisted wires and found the exponent, n,

to vary with the size of the wires. Small wires moved together during

sintering while large wires did not, thereby resulting in dissimilar

geometries and making Herring's analysis inapplicable.

Shewmon and Wilson [31] measured the shrinkage and weld neck

size during sintering of chains of particles. They reported that less

than 10 per cent of the volume of the weld neck resulted from shrink-

age, an observation which casts doubt on the ability of Kuczynski's

model to reveal the mechanism of densification. They reported that

Herring's analysis supported surface diffusion as the mechanism for

weld neck growth.

Mackenzie and Shuttleworth [21] chose to support the point of

view that surface tension, acting through the curvature of the pore

surface, creates stresses sufficient to result in macroscopic plastic









deformation of the solid framework. They derived a relation for rate

of densification based on the interaction of the capillary pressure of

the pores in raising the stress level in the solid between the pores

past that required for yielding. Clark and White [32] and Clark,

Cannon and White [33] developed a model based on viscous flow, within

a surface layer, under the forces of surface tension. They applied

their model, and also the equation of Mackenzie and Shuttleworth, to

experimental densification rates and achieved some degree of fit.

Rhines and Cannon [34] found that the application of small

compressive loads to a sintering system had the same effect on densi-

fication rates as an increase in stress level has on creep rates. In

other words, no new mechanism was introduced through the application

of small compressive loads to the sinter body. Williams and Westnacott

[35] showed that one of the accepted rate equations for transient

creep of metals could be used to develop a relation for rate of weld

neck growth which is so similar to Kuczynski's equation supporting

volume diffusion, that the available experimental data could not dis-

tinguish between them. Lenel and co-workers [17,36] have observed that

the effect on shrinkage rate produced by varying the level of small

compressive stresses is predictable from creep equations based on

deformation by dislocation processes.


1.3 Purpose and Scope of this Research

In spite of the obvious differences in the conclusions drawn in

the theoretical and experimental analyses of sintering, there exist

some important consistencies:









(a) surface tension is the force through which
densification is achieved, and

(b) the magnitude of the effect of surface tension
is determined by the geometry of the void-solid
interface.

Recognition of the importance of statement (b) provoked the

use of the simple spherical models of Kuczynski, the wire compacts of

Alexander and Balluffi and the particle chains of Wilson and Shewmon.

The complex theories of Mackenzie and Shuttleworth, and of Clark and

co-workers are also based on geometric effects consistent with state-

ments (a) and (b). However, direct experimental evidence that surface

tension is the force which produces densification has never been sat-

isfactorily obtained, nor has the relationship between surface tension

and the geometry of the void-solid interface in real sintering systems

been defined.

The first detailed study of the geometrical evolution of the

pore structure during sintering was published by Rhines et al. [22].

They reported that although the total volume and number of pores

decreased with increasing density, the average pore size increased.

Arthur [37] measured the relative amounts of interconnected and closed

porosity and found the porosity to be mainly interconnected; only

after reaching a density of approximately 95 per cent of theoretical

density does the porosity become completely closed to the external

surface. Both of these results were unexpected and clearly revealed

the inadequacies of simple models in explaining the behavior of real

complex systems.

Rhines [1] has shown that application of the principles of

descriptive topology to sintered structures provides a simplified









method for describing the complex geometric changes observed. In

developing this approach (see Appendix I), Rhines pointed out that the

minimal surface area possible is a function of the degree of connec-

tivity of the void-solid surface; i.e., the topological state. As the

connectivity decreases. the minimal area decreases; thus, once minimal

configuration has been reached, further decrease in surface area

requires a decrease in connectivity of the surface. DeHoff et al. [38]

have produced experimental evidence that, in the second stage of sinter-

ing, a linear correspondence exists between the density of the sintered

system and the surface area it contains in unit volume. This behavior

has been found to hold for all shapes, sizes, and compositions of pow-

dered materials investigated, demonstrating the existence of a unique

geometrical evolution for all sintering systems. In addition, they

also reported that the slope of the linear relationship was predeter-

mined by the initial connectivity of the particle stack. Recently,

Barrett and Yust [39] have demonstrated by serial sectioning technique

that the isolated parts of the porosity network which are developed in

the second stage of sintering "occur first as widespread volumes of

interconnected porosity, definitely nonspherical in shape." They con-

cluded that these sections of the void volume become separated from

the rest of the porous network, and from the external surface of the

sinter body, by the closing of parts of the void; a similar process

has previously been described by DeHoff et al. [38] as a closure of

channels in the porous network.

Techniques for quantitative determination of many of the geo-

metric variables which characterize sintering (e.g., volume fraction









of porosity, surface area and mean free path in the void or solid

phases) have been available for some time [40]. Recently, DeHoff

[41] and Cahn [42], in simultaneous and independent efforts, developed

a new fundamental relationship of quantitative metallography which

permits the experimental determination of the average value of the

mean surface curvature over the total void-solid interface. The details

of the derivation of this important relation are presented in

Appendix II.

A technique for experimental determination of the topological

properties of sintered structures has been presented by Aigeltinger

[43]. In his procedure a series of closely spaced microsections,

obtained by the use of a microtome, is used to synthesize the struc-

ture. Thus, the degree of connectivity of the void-solid surface and

the number of isolated parts of the porosity can be determined as a

function of the density of the sinter body.

The geometric and topological information now available cer-

tainly constitutes a significant basis for understanding the sintering

behavior of real systems. However, in order to explain the kinetics of

sintering it is necessary to discover the relations which exist between

the kinetics and the geometry of the sintered structure. Rhines [44]

has suggested that insight into the latter problem might be gained by

an analysis of the force of contraction generated by a sintering powder

system. Since the mechanical equivalent to the sintering force exists

regardless of the mechanism through which the solid responds to the

thermodynamic driving forces, no a priori assumptions as to operative

mechanism need be made in order to use this approach.









Except for the minor effects of gravity [45], sinter bodies

shrink isotropically. In order to stop total shrinkage, and in effect

balance the force of sintering isotropically, a uniform hydrostatic

tension would have to be imposed on the sinter body. Since it is not

practical to achieve this condition experimentally, the sintering force

has been defined by Rhines [44] as the force necessary to stop the

contraction of the sinter body in the direction of an uniaxially applied

load.

An attempt to measure such forces was made by Young [46]. He

used the method which was employed by Udin [12] to measure surface ten-

sion effects in fine metal wires. By suspending various weights from

a series of identically prepared, sinrered specimens, Young was able

to determine the load which just balanced the tendency to contract in

the direction of the applied load. He reported that, in irregularly

shaped -325 mesh copper powder, the sintering force increased with

increasing density over the range of densities he investigated; from

45 to 60 per cent of the theoretical density. However, he was unable

to discern any apparent relation between this unexpected functional

dependence on density of these external forces, and the internal forces

which are expected to be determined by the geometric evolution accom-

panying densification.

It was the purpose of this research to develop more completely

the concept of the sintering force, experimentally measure it and

*
Intuitively, the force of sintering would be expected to
decrease with increasing density since sintering rates decrease sharply
(often several orders of magnitude) with increasing density.









determine phenomenologically its relation to the physical, geometric

and mechanical properties of the sintered structure. If these goals

can be achieved, a strong link between geometry and kinetics will have

been forged.

The sintering behavior of a system of powder particles is

largely determined by the characteristics of the powder. Therefore,

the choice of the material used in this exploratory research effort

was predicated by the necessity for easily controlled experimental

variables and a high degree of reproducibility in the qualities of the

sintered specimens. For these reasons, copper was chosen and has the

following desirable features:

(1) reliable experimental values of surface tension,

(2) unique compatibility with hydrogen as the sintering
environment;

(a) all oxides of copper are reduced by hydrogen,

(b) the high diffusivity of hydrogen in copper
prevents any pressure buildup within closed
porosity,

(3) a melting point (1083 C) which creates no serious
experimental difficulties, and

(4) mechanical properties as a function of stress and
temperature which are documented as completely as
for any metal.














CHAPTER II


EXPERIMENTAL PROCEDURE


2.1 Material Specifications

The work of DeHoff et al. [38] made quite clear the importance

of the topological state of the initial particle stack in determining

the specific path of geometrical evolution observed during densifica-

tion. They pointed out that the most efficient stacking for any given

particle shape is never achieved, and it is generally possible to

increase the number of contacts per particle by vibratory packing or

mechanical compaction. The latter was not a part of the experimental

procedure in this research. However, it was necessary to handle the

mold after filling it with powder; therefore a powder which would pro-

duce a stable and reproducible stack was highly desirable. In this

respect particle shape is of primary importance, for it has been known

[47,48] for some time that, in general, spherical powders produce the

most efficient, and consequently the most stable, loose particle stack-

ing. The copper powders employed in this investigation were prepared

by the Linde Company by atomization of liquid copper in an inert .gas

atmosphere. The powders possessed a high degree of sphericity and

were of high purity. Of particular importance to the geometric analy-

sis of the sintered structure, the amount of copper oxide within the

interior of the particles was negligible and the formation of gas









porosity, which results from the reduction of internal copper oxide by

the hydrogen sintering atmosphere, was also negligible.


2.2 Particle Size Classification

Standard ASTM sieve analysis procedures were used for particle

size classification. In this method, the number which is used to

designate a sieve size corresponds either to the average number of

apertures per square inch in the wire mesh screen or to the average

size of the apertures. If the particles will pass through a sieve,

a minus (-) sign precedes the sieve size number; a plus (+) sign pre-

cedes the number for a sieve through which the particles will not pass.

The powder used in this investigation was initially separated

into two size classes: -270 + 325 mesh (-52 + 44 microns) and -325 mesh

(-44 microns) on a standard Ro-Tap mechanical shaker. Later in the

course of the work the -325 mesh powder was used as the source of a -20

micron classification performed on an Allen-Bradley Sonic Sifter. The

particle size distribution, as reported by the powder manufacturer, was

used to calculate the average particle diameters within each size classi-

fication. The average diameters for the -52 + 44. -44 and -20 micron

sizes were estimated to be 48, 30 and 12 microns, respectively.


2.3 Specimen Preparation

Specimens used for measurement of the sintering force were pre-

pared by presintering to the desired shape in the split graphite mold

shown in Figure 1. The basic shape of the specimen was similar to a

cylindrical tensile test specimen. However, the shoulders ateachendof

the reduced section were gently sloped to reduce constraint at these
























































Figure 1. Split graphite mold used for presintering
the sintering force specimens.










positions while the specimen was shrinking during presintering or cool-

ing from the presintering temperature. Connection between the specimen

and the load column was provided by presintering 0.9375 inch diameter

tungsten rods into each end of the specimen. The tips of the rods

embedded in the specimen were notched to provide interlocking between

the tungsten and the copper and increase the mechanical strength of the

connection.

All specimens were presintered at 7000C in dry, deoxidized

hydrogen. Presintering times were 15, 20 and 30 minutes for the 48, 30

and 12 micron particle sizes, respectively. The density of the loose

particle stack prior to presintering was calculated by determining the

volume of the mold (by filling with water) and weighing the volume of

the powder which filled the mold. Several determinations of the loose-

stack density revealed it to be very reproducible and always between
3
5.1 and 5.3 gms per cm for each particle size. The densities of the

presintered specimens were found to lie in the range 5.4 to 5.6 gms
3
per cm

As a result of storage and handling in hunid atmospheres the

powder particles were covered with thin oxide films. No attempt was

made to remove these films before presintering; however, close inspec-

tion after presintering revealed all surfaces, both internal and exter-

nal to the specimen, were clean and bright. Internal surfaces were

inspected in specimens fractured immediately after presintering.









2.4 Experimental Determination of the
Sintering Force

2.41 General features of the apparatus

The apparatus used for measurement of the sintering force,

shown schematically in Figure 2, was designed to provide the following:

(1) A continuously variable and manually adjustable force which
would permit the application of the instantaneous force
required to stop contraction of the sintering specimen at
any given density.

(2) Maximum sensitivity to change in specimen length; required
to measure accurately the sintering force at high densities
where shrinkage rates are extremely slow.

(3) Easy entry and removal of the specimen from the test rig;
required due to the weakness of the presintered specimens
and in order to preserve the density which existed at the
moment of final force measurement.

(4) Minimum deleterious effects from changes in ambient and
furnace temperature so that the control possible for these
variables would be adequate.

Simultaneous application and measurement of the externally

applied load as well as direct measurement of length changes in the

specimen were accomplished through the use of the cantilever beam

assembly in Figure 3. Length changes were revealed by the change in

beam deflection: beam force was monitored by measurement of the strain

created in the beam by its deflection.

The design of the beam is shown in Figure 4. The reduced section

of the beam provided a location of concentrated stress in the region of

the fillet where measurable strain was maximized for a given deflection.

The beam was fabricated from an iron-base material, Iso-elastic alloy,

so named for its low temperature coefficient of the elastic modulus.

As reported by the alloy supplier, J. Chatillon and Sons, the value of




18






Cantilever beam





Strain
indicator







0 0






0 s
0o o

Resistance ---













Figure 2. Schematic presentation of experimental setup
for determination of the sintering force.




19







I


Figure 3. Apparatus used for measurement
of the sintering force.
























































Figure 4. Cantilever beam assembly of sintering force
apparatus.









this coefficient for this alloy was 20 x 10- per F as compared to

a value of -190 x 106 per F for spring steel. In addition, errors

due to anelastic and hysteresis effects in this material were also

small; .02 and .05 per cent of the total deflection, respectively.


2.42 Strain measurement

The output of the four strain gages employed on the beam was

measured by a Baldwin-Lima-Hamilton model 120 A strain indicator. Two

gages each were placed on the top and bottom sides of the beam near

the fillet farthest from the free end of the beam. The arrangement of

the gages into the Wheatstone Bridge of the strain indicator served

two purposes:

(1) Measurable strain was increased by a factor of four over
the true strain at this position of the beam.

(2) Strains induced in the gages by thermal expansion or
contraction of the beam were essentially cancelled.


2.43 Calibration of beam force and deflection
versus indicated strain

In order to know accurately the beam force as a function of

total indicated strain, standard weights were applied to the beam and

the resultant strain recorded. The deflection of the beam, as a func-

tion of beam force, was determined by direct measurement with an
-4
optical system accurate to 0.5 x 10-4 inches. The calibration curves

for these quantities are presented in Figure 5. They were found to be

reproducible to within 1 per cent of the value of the beam force.

It was assumed that all changes in beam deflection could be attributed

to changes in specimen length. Based on the sensitivity of the strain


































Strain


1500


Figure 5. Calibration curves of indicated strain and change in
specimen length, AL, versus cantilever beam force.


250




200




150




100


500 1000

AL (x 105 inches); Strain (in/in x 106)









indicator to a change in measurable strain of 2 x 10-6, the apparatus

was sensitive to a change of 3.5 x 10-6 inches in specimen length which

corresponded to a change of 0.3 gms in beam force.


2.44 Test temperature and environment control

All sintering force tests were made in an atmosphere of dry,

deoxidized hydrogen. The gas pressure within the furnace tube was

maintained just above atmospheric pressure by constant gas flow to

prevent the entry of oxygen into the system. Test temperatures were

provided by a nichrome element resistance furnace which had a temper-

ature variation of 2.5 OC over 4 inches of the hot zone. Test tem-

peratures were measured with platinum, platinum-10 per cent rhodium

thermocouples and controlled by a Leeds and Northrup Precision Set-Point

Control System, which maintained the temperature of the hot zone to

within 0.10C.


2.45 Test procedure

The presintered specimens were integrated into the load column

by inserting the L-shaped end of the upper connecting rod into the end

of the pull-red. With the furnace at the desired test temperature, the

load column was lowered into the hydrogen-filled furnace tube. The

bottom connecting rod was inserted into a slot at the bottom of the

apparatus, lowered below the restraining ledge and then rotated such

that elevation of the L-shaped tip of the bottom connecting rod, either

by manual adjustment of the beam position or by shrinkage of the speci-

men, would place the specimen under tensile constraint.









In practice, it was found that the connecting rods were often

presintered into the specimen at slight angles to the specimen axis,

a situation which produced bending stresses upon initial loading.

Therefore, each specimen was either initially brought into creep by

manual adjustment of the beam force, or was allowed to shrink against

the constraint of the beam for some time. This procedure helped to

straighten the load column and produced approximately uniaxial loading

during the remainder of the test.

Typical behavior of a specimen during the test is shown in

Figure 6. Notice that the use of a cantilever beam to supply the exter-

nal force results in an inverse relationship between specimen length

and the magnitude of the beam force on the specimen. If the specimen

can shrink against the initial load, its decrease in length will

increase the force which the beam exerts upon it. If, by manual adjust-

ment of the beam position, the force is increased, then the specimen

may be made to creep. The elongation of the specimen in creep reduces

the beam force on the specimen until the sintering force within the

specimen is sufficient to balance the external beam force. This balance

of forces is only temporary since radial shrinkage of the specimen is

unopposed, and the density of the specimen is constantly increasing.

In practice, the time interval over which the forces appeared to be

balanced ranged from a few seconds to several hours depending on the

shrinkage rate of the specimen, as affected by temperature, particle

size and density. The behavior indicated in Figure 6 is that of a

specimen whose density would lie in the range where the sintering force

was observed to increase with increasing density. Consequently, after















Beam ------
force A














Specimen
length




Time


Figure 6. Typical specimen behavior during experimental measurement of the
sintering force. The dashed line indicates the desired balance
of internal and external forces.









the initial force balance, the external force against which the speci-

men can shrink increases as the specimen density increases. For speci-

mens in this range of density, the test was aborted at the moment the

specimen first revealed that it had progressed from creep to shrinkage.

At this point (indicated by dashed lines in Figure 6) the external force

was removed by manually lowering the beam support. The specimen was

then immediately raised to the cool end of the furnace tube in order to

preserve the density which corresponded to the final force measurement.

The bottom of the load column was essentially suspended from

the rest of the column, therefore the minimum load on the specimen

throughout the test was the beam force plus the weight of the bottom

connecting rod. However, since each point along the length of the

specimen had to support the weight below it, the average load on the

specimen was taken as the sum of the beam force, the weight of the

bottom connecting rod and 0.5 times the weight of the specimen.


2.5 Post-test Inspection of the Specimens

After testing, each specimen was inspected for excessive bend-

ing, necking or obvious failures due to cracks. Any specimen which

possessed any of these defects after testing was discarded. As a fur-

ther check on uniformity the diameter of the cylindrical specimen was

measured at several locations along the gage length. A Jones and

Lamson optical comparator with a reported accuracy of 0.0001 inch was

used for this purpose.









2.51 Density determination

Apparent densities of tested specimens were determined through

Archimedes principle using the standard ASTM procedure for porous

bodies. Samples for density determination were taken from the middle

of the gage length. After obtaining the weight of the sample in air,

the pore openings in its surface were sealed by impregnation with

liquid wax; the excess wax was wiped from the surface. The weight of

the impregnated sample was then obtained in air and in water, the dif-

ference in the two being equal to the volume of the sample. The appar-

ent density of the sample was calculated by dividing the weight (with-

out wax) by the sample volume. Reproducibility of this method was

estimated to be 1 per cent of the value of the density.


2.52 Metallographic preparation

Samples taken from the specimen for metallographic inspection

were mounted either in bakelite or in epoxy mixed with 0.3 micron

alumina. Samples were mounted so that polishing e:cposed a section

parallel to the cylindrical axis of the specimen. Mounting in bakelite

was accomplished by the standard hot-pressing procedure. Mounting in

alumina-dispersioned epoxy was carried out in the following way. After

blending the epoxy, 0.3 alumina powder was added in a 1 : 1 ratio by

volume. The sample was pl-aced in a hollow plastic cylinder sitting on

its end on a glass plate. The cylinder was then filled to the desired

level with the epoxy-alumina mixture and the entire assembly placed in

a vacuum dessicator attached to a mechanical vacuum pump. The pressure

on the system was reduced until the air within the porous sample










appeared to be removed. Pressure was then reapplied by opening the

system to the atmosphere, thus forcing the liquid into the porous net-

work of the sample. The alumina particles, due to agglomeration with-

in the epoxy, did not make their way into the sample. However, their

presence in the epoxy improved the abrasive qualities of the mount and

helped to preserve the edges of the sample during polishing. The

epoxy within the sample helped in preserving the true nature of the

void-solid interface. For this reason, very low density specimens

were usually mounted in the epoxy-alumina mixture; mounting in bakelite

proved to be more efficient and adequate for intermediate and high

density samples.

Mounted specimens were rough polished by hand on wet silicon

carbide abrasive papers. Fine polishing was performed, in the sequen-

tial steps indicated below, on standard rotating-type polishing wheels

covered with microcloths saturated with the following materials:

(1) 600 grit silicon carbide particles and water,

(2) 0.3 micron alumina and water, and

(3) 0.25 micron diamond and lapping oil.


2.53 Metallographic examination

All specimens were metallographically examined without etching.

Bausch and Lomb bench microscopes and metallograph were employed,

depending on the magnification required, for all microscopy. The fol-

lowing quantitative metallography parameters were experimentally

determined:









(1) Pp; the fraction of points of a grid, superimposed on the
microstructure, which fall within the phase of interest,
void or solid.

(2) N ; the number of times a test line, superimposed on the
microstructure, intercepts the void-solid interface, per
unit length of test line.

(3) TAnet= TA TA_; TA+ and TA_ are the number of tangents

which occur between a test line swept across the micro-
structure and the convex and concave segments, respec-
tively, of the void-solid interface, per unit area
traversed by the sweeping test line.

The total length of test line employed in experimental measure-

ment of NL was determined by observation of the effect of accumulated

length of test line on the accumulated average value of N When the

average value of NL fluctuated no more than 5 per cent with further

increase in total length of test line, this value was accepted. Experi-

mental values of TAnet were accepted when increase in total measured

area caused no fluctuations greater than 10 per cent in the accumulated

average value.





















Throughout this work, convex (positive) segments of void-solid
interface are those which are convex with respect to the solid phase.















CHAPTER III


EXPERIMENTAL RESULTS


3.1 Sintering Force

Experimental values of the sintering force were determined as

a function of density at three temperatures for copper powders with

average particle diameters of 48, 30 and 12 microns. These data are

compiled in Tables 1, 2 and 3 and presented in Figures 7, 8 and 9.

For a given particle size the behavior of the sintering force with

increasing density may be described as follows: in the first and

second stages of densification the sintering force increases at a

decreasing rate; this is followed by a sharp rise to a maximum value

from which the sintering force decreases rapidly toward the zero value

expected at the theoretical density of copper. This general functional

dependence on density was obtained for each of the particle sizes;

however, the maximum value appears to shift toward higher density with

decreasing particle size.

Existence of the maximum value of the sintering force was

verified by continuous measurement of the beam force which the specimen

imposed upon itself (through shrinkage or creep in the direction of

the applied load) as its density was increased primarily through radial

shrinkage. In this way the force against which the specimen could

shrink was clearly observed to pass through a maximum. Although the














Table 1. Experimental values of sintering force
for the 48 micron particle size as a
function of density and temperature


Final Sintering Test
Density Force Temperature

(gis/cm ) (gms) ( C)


5.75

5.78

5.80

5.90

5.98

6.57

6.90

7.02

7.16

7.26

7.45

7.55

7.60

7.60

7.73

8.18


8.20


25

23

24

28

31

38

40

35

30

48

44

38

52

48

52

75

65*
56


950

950

950

950

950

1000

1000

1000

1050

1000

1000

1050

1050

1050

1050

950


1050


Maximum (peak) sintering force.
Maximum (peak) sintering force.









0 950C
O 10000C
O 10500C


o B;
0 0
9_ -- [
,---^s- O


20L


7.0


8.0


8.5


Density (gms/cm3 )

Figure 7. Experimental values of the sintering force as a function of density for the
48 micron particle size spherical copper powder. The solid lines indicate
that the specimens corresponding to the data points passed through a max-
imum in sintering force.


_ I I I IJ


o-O









Table 2. Experimental values of sintering force
for the 30 micron particle size as a
function of density and temperature


Sintering Test
Force Temperature
)(gms) (C)


Final
Dens ity

(gms/cm3'

6.35
6.41

6.48
6.70

6.78

6.94
6.95

6.97

7.19
7.36
7.46
7.48

7.53
7.54
7.86

7.88
7.93

8.00

8.10
8.20

8.23

8.26

8.37


8.40


65
73
79
64

91

90
100

99

95
100

110
100

95
103
120

114
90

103
108
112
138*
129
104
147*
130

145*
125


Maximum (peak) sintering force.


950
1000
1000
1000

1000

1000
1000

1000
1000

1000
1000

1000
1000
1000
1000

1000

1000
1000

950
1000

1050

1000

1050


1050















13 0
[] 11 E]E]


I \

D/' C
0


6.0


6.5


7.0


7.5 8.0


8.5


Density (gms/cm3 )


Figure 8. Experimental values of the sintering force as a function of density for the
30 micron particle size spherical copper powder. The solid lines indicate
that the specimen corresponding to the data point passed through a max-
imum in sintering force.


9.0


1.40


120 L


0 9500C
O 10000C
A 1050C


100 L


80 L


El//


/
















Table 3. Experimental values of sintering force
for the 12 micron particle size as a
function of density and temperature



Final Sintering rest
Density Force Temperature

(gms/cm ) (gms) (C)


6.16

6.50

6.59

7.13

7.26

7.68

7.71

7.82

8.17

8.22


8.52


8.59


8.68


89

120

126

145

150

158

155

154

176

183

198*
182

214*
S203

199*
192


1000

1000

1000

1000

1000

1000

950

950

1000

950


1000


1050


1000


Maximum (peak) sintering force.








200-
I

0 95000
180- O Iooo00

r o10500C
/
160 /


0O

140
a /
o i
4
0


120-
/
/
/
/
100 /
I



/
I I I I I I I
6.0 6.5 7.0 7.5 8.0 8.5 9.0

Dens ity (gms/cm3)

Figure 9. Experimental values of the sintering force as a function of density for the 12 micron
particle size spherical copper powder. The solid lines indicate that the specimens
corresponding to the data points passed through a maximum in sintering force.









evolution of the sintering force could be accurately measured by this

procedure, only the final density of the specimen could be determined.

Therefore, the lines drawn in Figures 7, 8 and 9 to indicate the behav-

ior of the specimens in the region of the maximum sintering force are

accurate with respect to force but estimated with respect to density.

After the specimen had clearly passed through a maximum in sintering

force, it was removed and its density determined. The data points at

the ends of the lines drawn correspond to the final force and density.

Measurements of the sintering force were made at 950, 1000 and

1050 C. However, a complete documentation of the sintering force at

each of these temperatures was not made since initial measurements

revealed that the effect over this range of temperatures was negligible

compared to the normal scatter of the experimental values. This knowl-

edge was put to use in determination of the sintering force at high

densities; the higher temperatures provided faster shrinkage rates and

therefore greater sensitivity in the measurement of change in specimen

length.

The lower limit in density at which measurement of the sinter-

ing force could be made was determined by the increase in density which

the specimen achieved during the time (approximately 5 minutes) allowed

for thermal equilibration of the load column after its insertion into

the furnace. Consequently, the lowest density for which sintering

force data were obtained increased with decreasing particle size.









3.2 Quantitative Metallography

The point count was used to calibrate the volume fraction of

solid in the polished microstructure against the volume fraction pre-

dicted from the apparent density of the specimen. Polished specimens

whose point count density was within 2 per cent of their apparent

density were accepted for quantitative evaluation; those outside this

limit were repolished.

Experimental values of NL and TAnet obtained from unetched

microsections were used to calculate the surface area per unit volume,

SV, and the average value of mean curvature, H, of the void-solid

interface according to the relations


NL = S (1)

and

TA
-- TAnet
H = rr (2)
NL


SV, H and TAnet data for the 48, 30 and 12 micron particle sizes are

compiled in Tables 4, 5 and 6, respectively. They are plotted as a

function of density in Figures 10, 11 and 12, and 14 through 19.

Since NL is simply related to S it has been omitted from the tables

and figures.

1 1
Mean curvature is defined by H = (- + -), where r and r
r1 r2 1 2
are the principal normal radii of curvature. Its average value over
Sf HdS
the surface is defined [41] by H- ; where the integration is
i dS
performed over the surface, S.
















Table 4. Quantitative metallography data for the 48 micron particle
size as a function of density and temperature


Density Temperature SV TAnet H

(gms/cm3) (OC) (cm-1) (cm-2x0-2) (cm-


5.90 950 458 340 466

5.98 950 520 225 272

6.57 1000 460 470 642

6.90 1000 362 545 946

7.02 1000 356 572 1010

7.16 1050 330 521 1011

7.55 1050 250 473 1189

7.60 1050 226 395 1098

7.60 1050 235 460 1230

7.73 1050 218 450 1297

8.18 950 137 320 1468





9.0


0 9500C
O 10000C

8.0 / 10500C







S7.0







6.0
0



0 200 400 600 800
-1-1







SV (cm- )

Figure 10. Variation of surface area per unit volume with density for the 48 micron
particle size spherical copper powder.















Table 5. Quantitative metallography data for the 30 micron particle
size as a function of density and temperature




Density Temperature S, TAnet

(gms/cm ) (OC) (nc- ) (cm-2x102) (cm-)


6.48

6.70

6.94

6.97

7.36

7.93

8.00

8.10

8.20

8.23

8.26

8.37

8.40


1000

1000

1000

1000

1000

1000

1000

950

1000

1050

1000

1050

1050


940

860

664

830

614

366

322

314

230

136

248

164

126


2520

2232

2793

2280

2710

1382

1604

1206

1511

1192

896

1196

685


1671

1631

2643

1726

2763

2373

3130

2413

4128

5507

2207

4582

3416














8.







' 7.0
S [Q 10000C [ ]

A 10500C




6.0




0 200 400 600 800 1000
SV (cm1)

Figure 11. Variation of surface area per unit volume with density for the
30 micron particle size spherical copper powder.
















Table 6. Quantitative metallography data for the 12 micron particle
size as a function of density and temperature



Density Temperature S TAnet H

(gmis/cm3) ( C) (cm-1) (cm-2x 10-) (m- )


6.16 1000 1778 2560 905

6.59 1000 1624 6570 2542

7.13 1000 1120 6080 3473

7.26 1000 1210 6190 3241

7.68 1000 744 3830 3234

7.82 950 740 4720 4008

.8.17 1000 470 2960 3957

8.22 950 430 2550 3726

8.52 1000 214 2560 7576

8.59 1050 210 1420 4249

8.68 1000 122 930 4790














8.0







7.0







6.0


600 1200

SV (cm-1)


Figure 12. Variation of surface area per unit volume with density for the
12 micron particle size spherical copper powder.


1800















8.0






S12 p
-A-
7.0




30 p

48 P
6.0





600 1200 1800
S (cm- )


Figure 13. Comparison of the linear relationships between density and surface
area per unit volume for the 48, 30 and 12 micron particle sizes.









3.21 Surface area of the void-solid interface

It is evident from Figures 10, 11 and 12 that the relationship

between SV and density is linear for each particle size. The lines

drawn through the data for each particle size are presented for compar-

ison in Figure 13. Their extrapolation to zero surface area produced

reasonable agreement with the theoretical density of copper; maximum

deviation was found for the 30 micron particle size for which extrap-
3
olation yielded a value of 8.8 gms per cm as compared to the theoret-

ical density of 8.94 gms per cm .

The S data was collected from specimens sintered at each of

the three test temperatures, 950, 1000 and 10500C. The variation of

temperature over this range had no effect on the surface area-density

relationship.


3.22 Area tangent count

The algebraic sum, TAnet, of the number of tangents per unit

area with convex (TA+) and concave (TA_) segments of the void-solid

interface, as determined as a function of density for the three particle

sizes, is presented in Figures 14, 15 and 16. It is apparent that

TAnet was also insensitive to changes in test temperature.

For the loose stack of particles, all surfaces are convex (as

previously defined) and TAnet must therefore be positive. As contacts

form between particles, concave elements are created and with the con-

tinued evolution of the geometry accompanying densification TAnet passes

through zero and becomes negative. As evidenced by Figures 14, 15 and

16, TAnet was found to be negative over the entire range of densities

for which the sintering force was determined. The functional dependence











6.5 7.0


Density (gms/cm3 )

7.5 8.0


-l- -


O 950C
] 1000C
A 1050C
X 10000c*


Figure 14. Variation of the net area tangent count with density for the
48 micron particle size spherical copper powder. (*Obtained
from specimen sintered without external constraint.)


6.0


0
x

a
u



H


9.0


-200




-400




-600


-800








Density (gms/c3 )


v. .u 8.U 9.0






-1000 M H o

OeA
CN Q .
-, -tfX l

S -2000
oo
-


S- 0 9500C

a ] o ooo00c
-3000
D 10500C




-4000



Figure 15. Variation of the net area tangent count with density for the
30 micron particle size spherical copper powder.


r 0


7 n


- -










Density (gms/cm3 )


7.0 7.5 8.0
I i i


9.0
l


0 O

/
El/
O'EI
WY


-4000 L


-10


0 950C

0 1000C

A 1050C


Figure 16. Variation of the net area tangent count with density for the
12 micron particle size spherical copper powder.


+2000


-2000


6.0
I


6.5
1


N
<-1

0

CN



C4


-6000 L


-8000


_ _I 1


I


a-6









of TAnet on density was similar for each particle size; increasing to

a maximum (negative) value in the region of 75 SO per cent of theo-

retical density and then decreasing toward the zero value which must

exist when densification is complete.


3.23 Average mean surface curvature

Average values of the mean surface curvature, H, are presented

in Figures 17, 18 and 19 for the 48, 30 and 12 micron particle sizes,

respectively. Since H derives its sign from TAnet, the average values

of the mean surface curvature were also negative for all sintering force

specimens. In addition, since NL and TAnet were used to calculate H,

H was also insensitive to changes in test temperature. The general

shape of the curve depicting the mean surface curvature-density rela-

tionship indicates that H progresses from positive to negative values in

the region of 65 70 per cent of theoretical density (approximately 6.0

gms/cm ). The similarities between the functional dependence of i on

density and that exhibited by the sintering force (in Figures 7, 8 and 9)

are striking and a most significant discovery in terms of the relation-

ship between the microstructure of a sinter body and its sintering behav-

ior. Further elaboration of this point may be found in Chapter IV.


3.24 Effect of uniaxial constraint on
sintering behavior

It was reasonably expected that the effect of uniaxial loading

on a sinter body, attempting to shrink isotropically, would distort the

microstructure of the specimen and in so doing produce artificial behav-

ior in the development of the geometric properties. Indeed, it was






Density (gms/cm3 )


6.0 7.0 8.0 9.0




O 9500C
", 1000oC
S -500 \ 10500'






-1000 CD






-1500



Figure 17. Variation of the average mean curvature of the void-solid interface
with density for the 48 micron particle size spherical copper powder.





Density (gms/cm3 )


7.0


8.0
1


-% El
El- -
El


I A

Figure 18. Variation of the average mean curvature of the void-solid interface
with density for the 30 micron particle size spherical copper powder.


+1000_


6.0


-1000


9.0


-2000




-3000


O 950C

O 1000 C

Z 10500C


-4000




-5000


_I _ 1 __ _








Density (gms/cm3 )


8.0


EL.


o~- fE


O 9500C
0 10000C
L iooo00
A 10500C


Figure 19. Variation of the average mean curvature of the void-solid interface
with density for the 12 micron particle size spherical copper powder.


+2000


6.0


-2000L


9.0


-4000




-6000 .


-8000


YI I I I










observed, in the very beginning of this research, that extreme over-

loading above the force against which the specimen could shrink produced

internal necking and severe distortions of the void-solid interface.

However, as long as the overload was maintained close to the minimum

needed to produce creep of the specimen, it was impossible to detect any

effect of the external load on the geometric development of the micro-

structure. The veracity of this statement has been experimentally sub-

stantiated. For example, NL measurements which were made parallel with,

perpendicular to and at a 450 angle to the direction of external load

revealed no statistically significant differences from the values meas-

ured without respect to specimen orientation. In addition, the slopes

of the S -density plots (see Figure 13) were found to correlate with

the average particle diameter in the same way as previously reported by

DeHoff et al. [38] for unconstrained sinterings made from the same

powder material; this correlation is presented in Figure 20.

In Figure 21, the cross-sectional areas calculated from the

diameters of the specimen after testing are presented as a function

of density. The upper boundary in the figure represents the cross-

sectional area which would develop during unconstrained isotropicc)

shrinkage. The lower boundary is that predicted by assuming that

external constraint is applied in a fashion which limits all shrinkage

to the radial directions of the cylindrical specimen. Essentially all

specimens fall within these boundaries, indicating that the overload-

ing of the specimens used either initially in straightening the speci-

men in the load column or in determination of the balance point between











S Obtained from sintering
force specimens

O Obtained from uncon-
strained sinterings [38]


20 40 60

D (microns)
o


80 100 120


Figure 20. Correlation of the slope of the surface area-density
relationship with initial particle size for spherical
copper powder.


.000


.002L


.004





.006


,008L


CM4
N
U)
E


0
0
c-4
M


-.010


-.012L


_ ___ I I 1 I ?









AA 2 AV
_ sotropic shrinkage; A 3 V

AA AV
= -V; radial shrinkage only

Average


A A


\QN


NQ

N.
N


7.0 7.5 8.0
Density (gms/cm3)


A


A A


8.5


9.0


Cross-sectional areas of sintering force specimens after testing. The areas predicted
by isotropic shrinkage lie on the upper boundary; those which should result from
radial shrinkage only lie on the lower boundary.


.1600




.1500


A


1400O


] 48
A 30
0 12


microns
microns
microns


.1300




.1200




.1100


\ (


Figure 21.


I __ _ __ L _ _





57




elongation and shrinkage, did not unduly distort the specimen macro-

scopically. The scatter in the cross-sectional areas was produced by

variations in the fraction of the total time at temperature during

which the specimens were under external constraint.


















CHAPTER IV


DISCUSSION


The results of this investigation have revealed the sintering

force to be an experimentally measurable and reproducible property

whose functional dependence on density, particle size and temperature

are consistent with and relatable to the geometric evolution of the

sintered structure. To analyze these results, it is necessary to

consider the general relationships between surface geometry and sur-

face tension forces and the details of the geometric evolution of the

void-solid interface during densification. A method for estimating

the magnitude of the surface tension forces in a sinter body will be

formulated from these considerations and compared with experimental

values of the sintering force.

The ultimate objective of this research was to establish,

through the concept of the sintering force, a relationship between

the microstructure of the sinter body and its sintering kinetics.

Sensitivity of the rate of sintering to the geometry of the micro-

structure will be demonstrated through the conjoint application of an

empirical treatment of the mechanical behavior of the sinter body and

a stress-strain rate relation based on deformation by creep.










4.1 Surface Energy and Surface Tension of Solids

Specific surface energy, y, is defined as the energy necessary

to increase the surface by unit area, and has units of energy per unit

area. Surface tension, C, is the force which resists stretching of the

surface; it has units of force per unit length and may be considered to

act in the surface, perpendicular to any line drawn in the surface.

The equality of surface energy and surface tension in liquids has been

documented by many experiments in liquid capillarity. Gibbs [49], in

his rigorous formulation of the thermodynamics of interfaces, pointed

out that this equality does not always exist in solid surfaces.

Shuttleworth [50] has developed a general relationship between surface

energy and surface tension which reveals the conditions under which

these two surface properties have the same magnitude. This relation-

ship may be developed, following Shuttleworth's arguments, in the fol-

lowing way. Imagine a crystal surface cut by a plane perpendicular

to it and penetrating only a short depth below the surface. To maintain

the surface on both sides of the cut in equilibrium, it is necessary to

apply, in the plane of the surface, equal and opposite forces to each

side of the cut. For anisotropic surfaces, the total force per unit

length of the cut is defined as a surface stress which may vary with

the orientation of the cut with respect to the atomic arrangement of the

crystal surface. Consider the deformations dA1 and dA2 of the segment

A of the solid surface in Figure 22, by forces working against the

surface stresses C0 and 02. If these deformations are performed

reversibly and isothermally, then the work done against the surface










stresses will be equal to the increase in the total (Helmholtz) free

energy,

l dA1 + G2dA2 = d(Ay) (3)


1




S dA




A dA i-- 2







Figure 22. Schematic of surface stresses
in a solid.


In the absence of surface anisotropy, as in a liquid, the

surface stresses, aC and C2, are equivalent and may be considered as

a single value, a surface tension, which characterizes the surface.

Under these conditions equation (1) reduces to


a = y + A() (4)





For a unary system, at constant temperature and volume, in
which y is independent of changes in area, y = dF/dA, where F is the
Helmholtz free energy. For systems in which Y is a function of area,
the total change in free energy will be d(yA).









In the deformation of any surface, liquid or solid, where the mobility

of the atoms or molecules is sufficiently high to maintain the original

surface density of atoms or molecules throughout the deformation, ()
dA
will be zero, and surface tension and surface energy will be equivalent

in magnitude. In liquids, this mobility is easily obtained; in solids,

it may be reasonably assumed that the conditions required exist dur-

ing slow deformation of surfaces at temperatures close to the melting

point where atomic mobility is high. Since these conditions were

satisfied by the testing procedure in this research, it has been assumed

that surface energy and surface tension are equivalent in magnitude.

In keeping with this assumption, the symbol for surface energy, y, has

been used interchangeably for surface tension and surface energy. In

addition, it has been assumed that the random, polycrystalline nature

of the sintered structure assures a statistical distribution of all

orientations of the crystal surfaces at the void-solid interface which

obviates any corrections for anisotropy of surface tension.


4.11 Pressure difference across a curved surface

It has been demonstrated that for a surface of zero curvature,

the surface tension can be balanced by applying external forces par-

allel to the surface at its periphery. The experimental application

of this principle will be discussed in the next section. However,

for curved surfaces, surface tension forces result in pressures normal

to each point of the surface. The magnitude of this normal pressure

may be related to the local curvature of the surface in the following

way. Consider the square segment of surface shown in Figure 23, with





62





n






























S
2
e2

























I r


an


Figure 23. Resolution of surface tension into a pressure acting
normal to each point of a curved surface.









sides of arc length s and normal radii of curvature r1 and r2. From

the edge view it is apparent that the components of surface tension

parallel to the surface normal are equal on opposite sides of the
9
element. If the element is extremely small, then sin is approx-

imately equal to -, and the forces have total components (parallel
2 2
to the surface normal) of Y- and Y-- The difference in pressure,
rl r2
AP, between the volumes separated by the surface is the normal pressure

at each point of the surface. The normal component of force resulting
2
from this pressure is Aps Therefore, the condition for mechanical

equilibrium of the surface is


AP = y( + ) (5)
rl r2

Thus, the effect of surface tension on an arbitrarily curved segment

of surface can be analyzed in terms of the geometry of the surface.


L.12 Measurement of surface tension effects
in solids

Surface forces are small in magnitude and are negligible when

compared to the loads required to overcome the elastic strength of

solids at normal temperatures. However, as discussed above, their

effects become noticeable at high temperatures. The effect of surface

tension in solids was first reported by Faraday [51] who observed

that thin metal sheets would shrink when heated. However, it remained

for Chapman and Porter [52] to attribute this effect to the presence

of forces in the metal surface. Schottky [53], Sawai and Nishida

[54] and Tammann and Boehme [55] were among the first to use the

shrinkage of metal foils to determine values of surface tension.











They employed the method, originally suggested by Gibbs [49], of oppos-

ing the shrinkage of the foil with known external forces. More

recently, Udin et al. [12] measured the surface tension of copper by

determining the external load required to stop the shrinkage of fine

wires. Since the sintering force, as defined in this research,

embodies the same principle of a balance of forces it is appropriate

to consider the analyses of the foil and wire experiments.

The mechanical conditions for balancing the shrinkage of foils

with external loads have been reviewed by Fisher and Dunn [56]. The

simplest geometrical arrangement for balancing the surface tension of

a foil is presented in Figure 24(a). The top and vertical sides of

the foil are constrained while the remaining edge is free. However,

the two vertical sides of the foil are allowed to slide up and down

without friction. If the thickness, t, of the foil is neglected, the

external load which just balances the surface tension forces may be

calculated from the relation


F = 2yw (6)


where y is the specific surface energy (tension) and w is the foil

width. However, these experimental conditions cannot be achieved.

Figure 24(b) shows the more accessible situation in which the vertical

sides are also free. In this condition the horizontal shrinkage of

the foil will contribute to the length changes in the vertical direc-

tion and requires a different analysis. If it is assumed that a con-

dition of plane stress exists, then the stresses in the foil are











































(b) I
F

Figure 24. Two arrangements for balancing the shrinkage
of metal foils by inducing creep with an
external force.


~i~










C_ F 2yw
V wt

and (7)


a = 2
h t


where av and oh refer to the vertical and horizontal directions,

respectively. Plastic strain in the vertical direction may be esti-

mated by


S- [v 2(h)] (8)


where E is the plastic modulus and Poisson's ratio has been assumed

to be 1/2. If it also assumed that the strain rate, E in the vertical

direction is proportional to the stresses producing strain in this

direction, then the stress state for e = 0 under external load is
v

1
v =- T h (9)


which, on substitution, reduces to


F = yw (10)


For the circular cross section wire used by Udin et al. [12]

the same principles apply; however, the stress state is now three-

dimensional. For zero longitudinal strain rate in the wire, the assump-

tion of constant volume results in zero radial strain rate. These con-

ditions are consistent with the existence of a hydrostatic stress state.

Therefore, the conditions for balance under the external force are


a F 2n ry = = (11)
2 r r
rrr










or


F = nry (12)


where r is the radius of the wire cross section, and L and U are
Sr
the longitudinal and radial stresses, respectively. Udin [57] cor-

rected equation (12) by deriving a new relationship based on the ener-

getics of the process. Included in this analysis was the effect of

grain boundaries within the wire. As shown in Figure 25, the grain

boundaries were aligned perpendicular to the cylindrical axis of the

wire in a configuration referred to as "bamboo" structure. For a

reversible process, the change in potential energy of the external

load must be balanced by the total energy change in the wire. If the

latter is composed only of changes in grain boundary and external sur-

face area, then


Fdt = ydA + ydA (13)


where the prime notation refers to the grain boundaries. If it is

assumed that the changes in length and radius of the wire are related

by a Poisson's ratio of 1/2, equation (13) becomes


F = rrry nn r y' (14)


where n is the number of grain boundaries per unit length of the wire.

The grain boundary effect -in Udin's data changed the experimental

value of the surface tension of copper at 1000C from 1430 to 1670

dynes per cm; an increase of approximately 17 per cent. The use of

foil geometry has been critized due to its complicated grain boundary

configuration and the assumptions required to analyze its state of




















2 2r






Grain
Boundaries





















External Force




Figure 25. Schematic of surface tension forces and grain
boundary array in wire geometry used for
determination of surface tension in metals.










stress. The few values of the surface tension of metals which have

received general acceptance have been determined from the wire geo-

metry.

It has been demonstrated that the effect of surface tension

forces in solids can be experimentally analyzed through knowledge of

the geometry of the surface. The shape of the void-solid interface

in a sinter body is extremely complex; consequently, it is necessary

to develop some understanding of its geometry before attempting to

analyze the surface tension forces which exist within it. The analy-

sis which will then be developed will not include an effect of grain

boundaries. It has been assumed that the effect of grain boundaries

on the experimental values of the sintering force is subordinate to

the effect of the void-solid interface and within the scatter of the

experimental results. This assumption is based on preliminary meas-

urement of the ratio of grain boundary to void-solid surface area in

the specimens used in this investigation [58], and also consideration

of the fact that the energy of grain boundaries in copper is approximately

1/3 of the energy of a free copper surface [59].


4.2 The Geometry of Sintering

The degree to which one may describe the complex structural

changes in a three-dimensional sinter body, which occur during its

progression frcm a loose particle stack toward complete densification,

has been summarized by DeHoff and Rhines [60]. They pointed out that

it is now possible to determine experimentally the following geometric

parameters of a sintered structure:









(1) total void volume,

(2) total area of the void-solid interface,

(3) average mean curvature of the total void-solid
interface [41] (see Appendix II),

(4) connectivity of the pore network (expressed as the
genus of the void-solid interface, (see Appendix I), and

(5) the number of separate parts of the void volume.

The first three of these parameters are sensitive to the dimensions of

the system; the last two are topological parameters, insensitive to

dimension but necessary to characterize the topological shape of the

system. Experimental evaluation of these parameters has enabled Rhines

and DeHoff to deduce a reasonably complete and comprehensive descrip-

tion of the structural evolution of a sintering system.

The application of descriptive topology, as developed by Rhines

[1], separates the structural evolution into three stages. These sepa-

rations are based on the variation in connectivity of the void phase

with increasing density. Experimental evaluation of this parameter for

the 48 micron copper powder [60] is presented in Figure 26. The forma-

tion and growth of interparticle contacts constitutes the first stage

of sintering. In this stage the independent processes of surface round-

ing (primarily by surface or vapor transport) and shrinkage should pro-

ceed without affecting the connectivity of the pore network. However,

as evident in Figure 26, the connectivity increases during the first

stage. This behavior is the result of the formation of "bridges" in

the loose particle stack. Particles in these regions, originally not

in contact, are brought together by the shrinkage of the particle









framework, thereby increasing the total connectivity of the void phase.

The contributions to the decrease in total surface area by surface and

vapor transport, and by shrinkage, continue to be independent until

the minimal surface area is achieved for the existing values of volume

and connectivity of the void phase; this signals the end of the first

stage. Further decrease in surface area requires a new set of values

of connectivity and total pore volume. Rhines (see Appendix I) has

demonstrated that the system can progress to lower values of the min-

imal surface area through continual decreases in the connectivity of

the void volume. There exist two experimental findings which support

the minimal area concept in the second stage. First, the relation

between the density of the system and the surface area it contains in

unit volume is linear throughout the second stage. Second, the onset

of this linear relation coincides with the beginning of a strictly

monotonic decrease in the connectivity of the pore network with in-

creasing density; as shown in Figure 26. This decrease in connectivity

is produced by closure of the channels in the multiply-connected void

volume. The criterion for closure of a channel, in terms of local

instability of the void-solid interface, is revealed to a degree by

the linear relationship between the density and surface area per unit

volume; DeHoff and Rhines [60] have concluded that this behavior is

indicative of channel closure events which remove elements of the void

phase having a fixed ratio of volume to associated surface area. This

conclusion is consistent with their previous finding [38] of a constant














First Second Third
stage stage stage

5xl06




4x1O06
SGenus


6
3x10
E l



S2x106

Separate parts

1x106
IxlO6 P C,





6.0 7.0 8.0 8.94
Dens ity (gns/cm3)



Figure 26. Variation of genus (connectivity) and number
of separate parts of the void-solid inter-
face with density for the 48 micron special
copper powder [43].










value of the mean lineal intercept of porosity in the second stage of

sintering. Channel closure continues to be the dominant structural

change throughout the second stage. Eventually the pore network is no

longer multiply connected and the genus is then equal to zero. In the

third stage, the number of separate parts of the void volume is in-

creased through further division of the complex but simply-connected

"tree-like" [60] cavities by channel closure. The number of separate

parts decreases from the maximum value primarily through a reduction

in number resulting from conglobation with other portions of the void

volume.

4.21 The evolution of curvature of the
void-solid interface

To determine the relationship between the surface tension

forces and the microstructure of the sinter body, it is necessary to

consider the information available on the evolution of the curvature

of the void-solid interface. Constant reference by the reader to Fig-

ure 26, which depicts the three stages of sintering and to Figure 27,

a compilation of the average mean curvature values, H, for the three

particle sizes used in this investigation will serve to clarify.the

following discussion.

In the initial loose stack of particles, the curvature of all

void-solid interfaces is positive, by definition. The average value of


Mean lineal intercept, is the average length of all lines
traversing the phase of interest and is calculable from Fullman's [51]
relation, pertain to the phase of interest.
relation, \ = 4 where V and S pertain to the phase of interest.
V









+3000
12 microns

S---- 30 microns

+2000 -- 48 microns




+1000 _
\ Density (gms/cm3)

5.0 6.0 7.0 8.0 8.94






S -1000




-2000




-3000




-4000




-5000





Figure 27. Comparison of the average surface curvature as a
function of density for the 48, 30 and 12 micron
particle sizes of spherical copper powders.









mean surface curvature of the loose stack may be estimated by


= ( + -) 2 4 (15)
rl r2 r d


where d is the average particle diameter for each powder size. These

calculated values were used to extend the curves in Figure 27 from

the experimentally determined region (approximately 6.0 to 8.3 gms/cm3)

to the estimated loose stack density, 5.3 gms/cm3. The formation of

permanent contacts, or weld necks, between the particles introduces

negative curvature into the system and creates curvature gradients

between the nack region and the positively curved surface of the nearly

spherical particles. The magnitudes of these gradients control the

rates of vapor and surface transport which smooth the surface and

reduce the surface area in the system. The amount of negatively curved

surface in the system increases with further densification and the

average value of mean curvature drops rapidly through zero and becomes
3
negative in the density range of 5.8 to 6.0 gms/cm Average mean

curvature becomes an increasingly large negative value throughout the

remainder of the first stage. The slope of the average mean curvature-

density curves in the first stage, in Figure 27, is related to the

change in the magnitude of the curvature gradients; as the gradients

dissipate with increasing density, the slope decreases. This effect is

evident also in the change in the slope produced by a change in the

particle size. Curvature gradients are steeper in a fine particle

size and the slope is correspondingly greater.










The end of the first stage is revealed in the average mean

curvature-density relationship by an almost abrupt decrease in the

slope of the curves in Figure 27. This implies that the effective-

ness of the remnants of the original curvature gradients has been

considerably reduced and is consistent with the attainment of a min-

imal surface area at the end of the first stage. The absolute minimum

value of surface area possible for a given volume of porosity would

exist in a system containing a single sphere. However, Rhines [1] has

demonstrated that the shape changes in the first stage are localized

and prevent such a structural development. This is due to the fact

that the curvature gradients change the direction of material transport

by vapor and surface mechanisms within distances approximating half

a particle diameter. Under this limitation, the system can minimize

its surface area, for a given void volume, by separating the void phase

into separate spherical pores. In order to achieve this configura-

tion the channels in the pore network must be closed. Therefore, the

coincidence of a decrease in connectivity, a linear relationship between

density and surface area per unit volume, and a decrease in the rate

of change of surface curvature is evidence of a decrease in the abil-

ity of curvature gradients to promote shape changes in the pore net-

work. If this were not true, tnan further decrease in surface area

per unit volume could occur without requiring a decrease in connectiv-

ity, and the rate of change of curvature with increasing density would

not be so diminished by the end of the first stage.









The evolution of average surface curvature is also sensitive

to the variation in connectivity of the pore network during the second

stage. As shown in Figures 17, 18 and 19, a rise in curvature occurs

during the last half of the second stage. The rate of decrease in con-

nectivity may be seen to diminish in the same range of density in

Figure 26. According to DeHoff and Rhines [60], a channel can decrease

in size until it attains the set of critical dimensions which render it

unstable and make it collapse. The surface rounding processes of the

first stage produce some channels which can close early in the second

stage. The remaining channels must decrease in size through shrinkage

of the solid framework. This increases the average curvature of the

remaining channels and therefore of the system. The separate parts

developed by the channel closure process reveal their inability to

shrink by decreasing their surface area through coarsening [63] in the

third stage. This process is considered to be responsible for the

occurrence of a maximum value of average curvature. Thus, the evolu-

tion of average mean surface curvature, in agreement with the other

geometric parameters which have been measured, strongly indicates that

after an initial period (the first stage) of adjustment in the shape

of the void-solid interface, the system proceeds along a path of geo-

metric evolution consistent with tne concept of a series of minimal

surface area configurations. The instantaneous values of the minimal

surface area are conditioned on the values of the remaining volume and

connectivity of the void phase. Furthermore, and more important to

this research effort, is the evidence that the average value of mean









surface curvature should be indicative of the average value of the

shrinkage forces in the system. The veracity of the latter state-

ment is substantiated in the obvious similarities in the variation

with density of the average mean curvature (Figures 17, 18 and 19)

and of the sintering force (Figures 7, 8 and 9). The scatter in H

data and the limited number of values do not clearly reveal the max-

imum in average mean curvature implied by the sintering force results.

However, a maximum in average curvature has been reported by DeHoff

and Slean [62] to occur at a density of 7.9 gms per cm in an electro-

lytic copper powder; this is shown in Figure 28.


4.3 Analysis of the Sintering Force

On the basis of the geometric arguments presented in the pre-

vious section it will be assumed in this analysis that the average

surface tension forces in the system may be calculated in terms of its

average value of mean surface curvature. Although it is true that the

external force applied may not balance the internal forces at every

location in the system, it is assumed that it does so on the average.

Using these assumptions, the average pressure, P, acting to

compress the void phase can be calculated from the basic equation

P = yHi (16)

If it is further assumed that this pressure is distributed throughout

the system without dilution by the solid phase, then the force com-

ponent of this pressure which acts perpendicular to an area A of the

sinter body is equal to yHA. The average force, F, which counteracts

the effect of the external load is given by


















Density (gms/cm )


6.0


7.0


-I000L


-2000


-3000


Figure 28. Variation of average surface curvature with density
for a mixture of sizes of an irregular, dendritic,
electrolytic copper powder [62].


+1000


8.0


V ---- ~- -r`l--------T-










F = HA (17)


where A is the average cross-sectional area perpendicular to the direc-

tion of loading. This relation would be directly applicable to a sys-

tem composed of porosity separated by very thin interfaces; e.g., a

soap froth.

An alternative, and more realistic approach, is to consider a

simple balance of forces between the external load and the sun of all

force components emanating from the void-solid interface in a direc-

tion parallel to the applied load. For example, if the desired compo-

nent of force per unit length is F. and is single-valued over the

length, 1., of the line of intersection between the void-solid inter-

face and the plane perpendicular to the applied load, then the total

resolved force would be equal to the sun E F.I. taken over the total
i L i

length of all such intersections. This sun cannot be rigorously calcu-

lated from the geometric description of the void-solid interface pres-

ently obtainable. However, the general principles of this approach may

be demonstrated through the application of a simplified geometry. Con-

sider a system containing a homogeneous spatial distribution of spher-

ical pores of radius R. On any cross-sectional plane, the circular

intersections with these pores will vary in size with the largest having

a radius r = R. The force contribution from any single intersection can

2
be calculated in terms of the curvature, R, of the system. As shown in
R'
Figure 29, the desired force component, F., is constant around the total

length of the perimeter of the intersection. Therefore, the total force

from this pore intersection, perpendicular to the cross-sectional





















F.
1i



Figure 29. Resolution of surface tension forces emanating
from the cut surface of a spherical pore in a
direction perpendicular to the sectioning plane.


F.
plane, is 27 r F. Since sin = r this becomes y()r2 or

yHA, where A is the area of the circular intersection. The total

force in the system acting across the intersecting plane is equal to

the s-um


F = yHE A. (18)
J J

th
where A is the area of the j circular intersection. The total area

of pore intersections on a cross-sectional plane of area A is equal to

AAVA where AAV is the area fraction (equivalent to the volume fraction)

of porosity in the system. Therefore, the working form of this rela-

tion becomes


F = v HAAVA (19)


The effect of increase in curvature and decrease in area fraction of

porosity (i.e., increase in density) on the sintering force predicted









by this relation may be shown as follows. The maximum component of the

surface tension occurs when the radius of the circular intersection in

Figure 29 is equal to the radius of the sphere; i.e., when the inter-

secting plane cuts the diameter of the sphere. Under these conditions

e is 900 and the force component is equal to the surface tension, y.

Allowing the sphere to shrink is equivalent to increasing the density

of the system. However, this decreases the total line length (perimeter

of the circle) over which the force, y, exists and therefore predicts

a decrease in the sintering force with increasing density. This is

contrary to the experimental results which show an increase in the

sintering force over the entire second stage of sintering. This reveals

the inadequacy of the sphere model in representing the geometrical evo-

lution of the multiply-connected pore network in real systems. However,

the principle of a balance of forces expressed by the relation derived

rigorously for the spherical model is correct. Examination of equation

(19) will show that if the increase in curvature, H, is greater than the

decrease in the area fraction of porosity, AAV, then an increase in the

sintering force, F, is predicted. This is not possible in a system of

spheres since their curvature, intersected area and volume relationships

are fixed by their shape. However, in real sintering systems, the

shape of the pore network is constantly changing and H and AAV can vary

somewhat independently. The change in curvature for a system of shrink-

ing spheres would be smooth and continuous, but the change in average

mean surface curvature for a real sintering system is sensitive to the

changes in connectivity of the void network, as is evident by a compar-

ison of Figure 26 and Figure 27. Therefore, equation (19) should be










applicable to real sintering systems for which the average mean surface

curvature, H, and the average area fraction of porosity, AAV on an aver-

age cross-sectional area, A, are known.


4.31 Comparison of predicted and experimentally
measured sintering forces

The sintering forces predicted by equations (17) and (19) are

presented as a function of density in Figures 30, 31 and 32, for com-

parison with experimentally obtained values. The value of surface ten-

sion, y, used for the calculations was 1650 dynes per cm (1.68 gms per

cm), the accepted value for copper at 10000C. The temperature coeffi-

cient of the surface tension for copper as reported by Udin et al. [12]

is approximately 0.50 dynes per cm per C. A variation in temperature

of 1000C, corresponding to the range of temperatures (950 to 10500C)

investigated in this research, produces a total change in y of only

50 dynes per cm, or approximately 3 per cent. Thus, the effect of tem-

perature is well within the normal scatter of the experimental values

of the sintering force and was not included in the calculation of the

predicted values.

The force values predicted by equation (17) are designated as

maximum values since they correspond to equation (19) when the area

(volume) fraction of porosity is unity; i.e., when the volume of the

solid phase is essentially negligible. Although the predictions of

equation (17) are, as expected, greater than the experimental values,

the qualitative behavior of the two curves as a function of density is

strikingly similar. Although qualitative in nature, this observation














F = y H AA A

F = yHA
S- max

F
exp


6.0


7.0

Density (gms/cm )


Figure 30. Comparison of the experimental values of the sintering
force with predicted values for the 48 micron particle
size powder.


300[


^ 200


o
a,


IooL


I





i I_ I


8.0









800
F= y AA A

max mA

700 ____ F
exp




600




500




400
0



300




200




100 1






6.0 7.0 8.0
Density (gms/cm3)


Figure 31. Comparison of the experimental values of the sintering
force with predicted values for the 30 micron particle
size powder.






1000



900


/


/




/


Z


6.0


7.0
Density (gms/cm )


8.0


Figure 32. Comparison of the experimental values of the sintering
force with predicted values for the 12 micron particle
size powder.


F = v H AA A
F Y HA
max A
Fex
exp


/


800L


700L


600L


500L


400L


200L


100[


I_ _










definitely relates the sintering force, as defined in this research,

to the average value of the mean surface curvature of the sintered

structure, and strengthens the assumptions made in regard to this

relation.

The forces predicted by equation (19) are in better agreement

with the experimental values; however, the two curves do not possess

the same shape. Although the predicted curve rises rapidly from zero

at zero average curvature to the proper level of the sintering force,

it begins to decrease early in the second stage of densification.

This discrepancy can be qualitatively reconciled in the following way.

In the experimental measurement of the sintering force the external

force is applied to balance the shrinkage of the system. Therefore,

elements of the void phase which contribute the most to the overall

shrinkage are primarily responsible for the level of external load

required to balance the shrinkage. The values of average surface

curvature and area fraction should be more heavily weighted in favor

of these elements of the void phase. The recognition of these elements

on a two-dimensional microsection cannot be unambiguously performed.

It may be assumed that the smaller pores on a two-dimensional section

are those which will continue to shrink, or if they are intersections

with channels, to collapse, but a quantitative delineation on this

basis must be considered arbitrary. It is obvious that division of

the void volume into portions based on their relative contribution to

shrinkage will reduce the selected area of porosity below the average

value, AAVA. However, the portions which do produce volume changes in