• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 Abstract
 Introduction
 Review of literature
 Experimental procedure
 The metric properties of sintered...
 The electrical properties of sintered...
 Discussion
 Summary and conclusions
 Suggestions for future researc...
 Appendix A: The gross tangent...
 Appendix B: The area inflection...
 Appendix C: Notes on the grain...
 Appendix D: Electrical measurements...
 References
 Biographical sketch














Title: Structural evolution and electrical conductivity of sintered uranium dioxide
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Permanent Link: http://ufdc.ufl.edu/UF00097485/00001
 Material Information
Title: Structural evolution and electrical conductivity of sintered uranium dioxide
Physical Description: viii, 246 leaves : ill. ; 28 cm.
Language: English
Creator: Gehl, Stephen Mark, 1947-
Copyright Date: 1977
 Subjects
Subject: Uranium compounds   ( lcsh )
Materials Science and Engineering thesis Ph. D
Dissertations, Academic -- Materials Science and Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 238-245.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Stephen Mark Gehl.
 Record Information
Bibliographic ID: UF00097485
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000206777
oclc - 04042235
notis - AAX3571

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Table of Contents
    Title Page
        Page i
        Page i-a
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
        Page vi
    Abstract
        Page vii
        Page viii
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
    Review of literature
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
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        Page 43
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        Page 46
        Page 47
        Page 48
        Page 49
    Experimental procedure
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
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        Page 61
        Page 62
        Page 63
        Page 64
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        Page 67
    The metric properties of sintered and hot pressed UO2
        Page 68
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        Page 161
        Page 162
    The electrical properties of sintered uranium dioxide
        Page 163
        Page 164
        Page 165
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    Discussion
        Page 173
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    Summary and conclusions
        Page 192
        Page 193
        Page 194
    Suggestions for future research
        Page 195
        Page 196
        Page 197
    Appendix A: The gross tangent count
        Page 198
        Page 199
        Page 200
        Page 201
        Page 202
        Page 203
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        Page 208
        Page 209
        Page 210
        Page 211
    Appendix B: The area inflection point count
        Page 212
        Page 213
        Page 214
        Page 215
        Page 216
        Page 217
        Page 218
        Page 219
        Page 220
        Page 221
    Appendix C: Notes on the grain contiguity and the grain face contiguity
        Page 222
        Page 223
        Page 224
    Appendix D: Electrical measurements for specimens with water vapor adsorbed
        Page 225
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    References
        Page 238
        Page 239
        Page 240
        Page 241
        Page 242
        Page 243
        Page 244
        Page 245
    Biographical sketch
        Page 246
        Page 247
        Page 248
        Page 249
Full Text











STRUCTURAL EVOLUTION AND ELECTRICAL CONDUCTIVITY
OF SINTERED URANIUM DIOXIDE














By

STEPHEN MARK GEHL


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

1977


- 4



































Dedicated to

my wife, Patsy,

and

my parents, Mr. and Mrs. Frank H. Gehl














ACKNOWLEDGEMENTS


The author wishes to acknowledge the encouragement and helpful

discussions in the course of this research provided by the chairman of

his supervisory committee, Dr. R. T. DeHoff, and by Dr. F. N. Rhines.

Thanks are due also to Drs. E. D. Verink, Jr. and C. P. Luehr for

serving on the committee.

The assistance of Drs. P. F. Johnson and L. L. Hench with the

electrical measurements and their interpretation is gratefully acknowl-

edged. In addition, Dr. Johnson's assistance in the art of "disserta-

tioneering" was invaluable.

The encouragement and forbearance of L. A. Neimark, L. R. Kelman,

Dr. R. W. Weeks, and Dr. B. R. T. Frost of the Materials Science Division

of Argonne National Laboratory are also appreciated.

The financial support of the U. S. Atomic Energy Commission and

the National InStitutes of Health is gratefully acknowledged.


iii














TABLE OF CONTENTS


Page
ACKNOWLEDGEMENTS . . . . . . . . ... . . . iii

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

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

CHAPTER I. REVIEW OF LITERATURE . . . . . . . . 6

The Methods and Results of Quantitative Stereology ..... 6

Notation . . . . . . . . ... . . . 7

The Counting Measurements . . . . . . . 7

Total Curvature . . . . . . . .. .. . 12

Contiguity . . . . . . . . .. . . . 14

The Microstructural State and the Path of
Microstructural Change . . . . . . ... 16

Shape Parameters . . . . . . . .... . 20

Sintering . . . . . . . . ... . . . . 23

Sintering Mechanisms . . . . . . . . .. 23

Geometric Changes during Sintering . . . . .. 25

Sintering of UO2 . . . . . . . .... . 31

The Electrical Properties of Heterogeneous Structures . . 32

Experimental Studies of the Electrical Properties of
Heterogeneous Systems . . . . . . . ... 39

The Electrical Properties of Porous Materials ... . 42

The Electrical Properties of UO2 ........... 45







TABLE OF CONTENTS continued



Page

CHAPTER II. EXPERIMENTAL PROCEDURE . . . . . . . 50

Material Characterization . . . . . . . ... 50

Sample Preparation . . . . . . . . ... . . 57

Conventional Sintering . . . . . . . ... 57

Hot Pressing . . . . . . . . ... . . 58

Quantitative Metallography . . . . . . . ... 60

Electrical Measurements . . . . . . . . ... 62

CHAPTER III. THE METRIC PROPERTIES OF SINTERED AND
HOT PRESSED UO ........... .............. 68

The Path of Microstructural Change . . . . . ... 68

Cold Compaction . . . . . . . . .. .... 82

Conventional Sintering . . . . . . . .. 87

Hot Pressing . . . . . . . . .. . . 108

Contiguity . . . . . . . . ... .... .. 129

The Variation of Grain Contiguity during
Sintering and Hot Pressing . . . . . . . 129

Contiguity and Pore-solid Surface Area . . . ... .131

Contiguity and Triple-line Length . ... . . . . 136

Grain-face Contiguity . . . . . . . . . 140

Shape . . . . . . . . . . ..... . 146

The Pore-solid Interface . . . . ... .. ... 146

Grain Shape . . . ... .. . . . . .. 148







TABLE OF CONTENTS continued


Page

CHAPTER IV. THE ELECTRICAL PROPERTIES OF SINTERED URANIUM
DIOXIDE . . . . . . . . . . . . . . . 163

Electrical Conductivity under "Dry" Conditions . . ... .164

Electrical Conductivity and Dielectric Properties
under Humid Conditions . . . . . . . .... . .172

CHAPTER V. DISCUSSION . . . . . . . . . . 173

The Path of Microstructural Change . . . . . ... .173

The Effect of Compaction Pressure . ... . . .... .173

Additional Relationships . . . . . . ... 182

Extensions of the Path of Microstructural Change
to Other Systems . . . . . . . ... .. . 184

Relationships between Electrical Conductivity and the
Quantitative Stereology Parameters .............. 188

CHAPTER VI. SUMMARY AND CONCLUSIONS . . . . . . ... .192

CHAPTER VII. SUGGESTIONS FOR FUTURE RESEARCH . . . . .. 195

APPENDIX A. THE GROSS TANGENT COUNT . . . . . . ... 198

APPENDIX B. THE AREA INFLECTION POINT COUNT ........... 212

APPENDIX C. NOTES ON THE GRAIN CONTIGUITY AND THE
GRAIN FACE CONTIGUITY . . . . . . . . .. . .222

APPENDIX D. ELECTRICAL MEASUREMENTS FOR SPECIMENS WITH
WATER VAPOR ADSORBED . . . . . .. . . . . . 226

LIST OF REFERENCES . . . . . . . . .. . ... . 238

BIOGRAPHICAL SKETCH . . . . . . . . . . . 246







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



STRUCTURAL EVOLUTION AND ELECTRICAL CONDUCTIVITY
OF SINTERED URANIUM DIOXIDE

By

Stephen Mark Gehl

December 1977


Chairman: R. T. DeHoff
Major Department: Materials Science and Engineering


The simple counting measurements of quantitative stereology were

used to determine the sequences of microstructural states attained

during conventional sintering and hot pressing of uranium dioxide

powders. Each such sequence, called a path of microstructural change,

was evaluated by measuring, for a series of specimens, the pore volume

fraction; the area, total curvature, and gross tangent count of the

pore-solid interface; the grain-boundary area; and the lengths of the

triple lines formed by the intersection of the grain boundaries with

each other, and with the pore-solid interface. The paths of microstruc-

tural change were found to be functions of the scale of the system, the

compaction pressure used prior to conventional sintering, and the applied

pressure during hot pressing. Quantitative relationships were developed

between pore volume fraction, pore-solid surface area, grain-boundary

area, and the lengths of both kinds of triple lines. These relation-

ships, with appropriate values for the empirical constants, were valid

for all fabrication conditions explored. The grain contiguity param-

eter was found to provide a useful index of the extent to which the







grain structure had developed during sintering and hot pressing. An

additional parameter, the grain-face contiguity, was defined in this

study and was linearly related, by a factor of 10.6, to the grain con-

tiguity. This relationship, which was valid for both conventional sin-

tering and hot pressing, was found to be an indication that the grain

boundaries preferentially intersected the pore-solid interface, and

avoided intersections with other grain boundaries.

The area density of inflection points formed by the intersection

of a sectioning plane with a saddle surface was demonstrated to be

proportional to the integral of the asymptotic line curvature. The area

inflection point count was evaluated for a series of hot-pressed speci-

mens and was found to decrease by a factor of '50 as pore volume frac-

tion decreased from 0.24 to 0.08.

The gross tangent count was demonstrated to be upper bound for the

total absolute curvature. For saddle surface, the gross tangent count

was found to be approximately equal to the integral of the root mean

square local curvature over the domain of surface area.

Electrical conductivity measurements were performed on a series of

conventionally sintered specimens. The results of these experiments

were compared with the quantitative stereology measurements to determine

the effect of microstructure on electrical conductivity. In the early

stages of sintering electrical conductivity was found to be strongly

correlated with grain-boundary area and grain contiguity, an indication

that the bulk electrical conductivity is controlled by the grain con-

tacts, which are the minimum cross-section area through which current

passes. In the latter stages of sintering, the minimum cross-section

area is no longer controlled by the grain contact area.













INTRODUCTION


All materials, whether by design or accident, contain heteroge-

neities in structure. These heterogeneities range from vacant lattice

sites and dislocations in high-purity single crystals to large slag

entrapments in metal castings. The external surface is a heterogeneity

which is present in all real materials. It is not surprising, there-

fore, that there is much interest in the physical properties of hetero-

geneous systems. The important class of heterogeneity which is the

subject of this dissertation is the microstructure, i.e., the arrange-

ment of volumes and interfaces that define the distribution of the

crystalline and amorphous constituents of a system. It is usual to

restrict the term microstructure to features too small to be resolved

with unaided vision; this convention will be followed here.

Several strategies have been used to explore relations between

structure and properties. These methods may be summarized as follows:

In one type of investigation the heterogeneous system is repre-

sented by a model simple enough to allow a prior calculation of its

physical properties. An important example of this approach is the
1-3
Maxwell-Wagner-Sillars (MWS) theory of heterogeneous dielectrics,

which begins with Maxwell's field equations, and predicts the dielec-

tric properties of a medium in which the electric field is perturbed






by a small amount of a disperse second phase with dielectric constant

different from the matrix. The major limitation of this approach is

the requirement of mathematical tractability. If the set of equations

generated is to be solvable, simplifying assumptions about the geometry

of the heterogeneous system must be made. For the MWS theory, the

second phase is assumed to consist of a small volume fraction of

isolated ellipsoidal particles with an electrical conductivity higher

than the matrix. The danger lies in applying such a theoretical model

to systems in which the model's assumptions are not valid.

In other studies the variation of physical properties with fabri-

cation conditions is explored. Since the processing variables control

the microstructure, which in turn determines the physical properties,

this type of investigation is actually one step removed from a direct

study of the relations between microstructure and properties. While

the use of this indirect method is no longer prevalent, some examples
4-5
can be found in the recent literature.5

A third group consists of studies in which structural characteri-

zation is performed, for which a wide variation exists in the effort

expended toward this end. In some cases, a somewhat limited set of

structural parameters is determined, such as grain size or volume

fraction of a second phase. In other studies, a more detailed struc-

tural investigation is undertaken: the physical properties of hetero-
6
generous structures have been related to mean phase intercept, conti-
7 8 9
guity, genus, and "shape parameters," in addition to volume fraction
10-14
and grain size. The development of an empirical relation, the

form of which may be suggested by theoretical considerations, is the

goal of studies of this type.








Such an empirical .equation will express the physical property of

interest as a function of some set of structural parameters,

P = {P 2' ...PN}. If the relation is general, the physical property

will depend only on the set P, and will be independent of all other

structural parameters. The difficulty in the development of general

relations lies in the determination of the set P. When the makeup of

P is uncertain, the extension of empirical relations to new material or

structure types must be done with caution.

In this study, the problem of generating relations between a struc-

ture and its properties was attacked in the following way. After all of

the accessible parameters of quantitative stereology were determined, an

attempt was made to define, on one or several of these parameters, a

function whose variation from sample to sample would parallel the

behavior of some physical property measured for the same samples. A

long list of stereological properties was measured to increase the

probability of finding a precise relation, with a simple mathematical

form, between the structure and its physical properties. To enable a

large number of samples to be characterized, those parameters which

require time-consuming serial sectioning techniques were not determined.

While it was recognized that the topological properties of number and

genus, which can only be measured by serial sectioning methods, often

directly determine the physical properties of a heterogeneous system,

the necessity of testing a proposed relation against a large number of

widely differing structures was judged to be of primary importance. The

experimental methods actually used in this study provided a critical

test of the ability of the more easily obtained stereological properties

to predict the physical properties of a heterogeneous structure.








The physical properties selected for measurement were the room

temperature electrical conductivity and dielectric constant of sintered

uranium dioxide. The electrical properties were chosen because there

is much interest in the electrical behavior of heterogeneous systems,

and because the electrical properties can be calculated for some simple

structures but not in the general case. Sintered structures were exam-

ined because of their complexity, and because sintering is a common

fabrication method for many ceramic articles. The electrical behavior

of sintered uranium dioxide is of special interest to the nuclear

industry because the present-day laboratory simulation of reactor con-

ditions utilizes the principle of direct electrical heating of UO2 fuel

rods.15

In addition to the development of relationships between the micro-

structure and the electrical properties of UO2, the structural charac-

terization studies performed in the course of this research led to

several advances. A new metric property, the total curvature of the

asymptotic lines on saddle surface, was discovered and methods for its

measurement were developed. A new contiguity parameter, the grain face

contiguity, was defined. For sintering, this parameter was found to be

simply related to an earlier measure of contiguity.6

In this dissertation, three major topics are discussed. These are

quantitative structure analysis, sintering, and the electrical behavior

of heterogeneous structures. In Chapter I the literature on these

topics is reviewed separately. In the results section, Chapter III,

and the discussion section, Chapter V, the analysis of structure is







grouped with the sintering study, since quantitative structure analysis

was the primary tool used to investigate the sintering process. Results

of the electrical measurements and correlations with the stereological

data are presented in Chapter IV. A description of the experimental

procedure is contained in Chapter II.

















CHAPTER I

REVIEW OF LITERATURE



The Methods and Results of
Quantitative Stereology


As used by Underwood,7 stereology includes "not only the quanti-

tative study and characterization of any spatial structure, but also

its qualitative interpretation." Stereology may be differentiated from

other structure-measuring techniques such as gas-adsorption surface

area determination, in which no direct image of the structure is formed.

The basic tools of stereology are a set of counting measurements,

which are performed on a section or series of sections through the

structure. These measurements yield information about the number, size,

distribution, shape, and extent of features in the three-dimensional

structure, when appropriate assumptions about the uniformity of the

structure and the orientation of features with respect to the sectioning

plane are made.

In this section, the counting measurements used in the present

study, and the relations between these measurements and the metric prop-

erties of a three-dimensional structure, are reviewed. Special atten-

tion is given to surface curvature and to measures of contiguity, since

these parameters were determined for many samples in the course of the

experimental work on which this dissertation is based. The concept of

the path of microstructural change is reviewed.









Notation

The notation used in the dissertation for expressing stereology

parameters is derived from the system proposed by Underwood.17 Most of

the parameters are written as upper case letters and are identified in

Table 1-1. The terms usually have obvious meanings, such as "T" for

tangent, "S" for surface area, and "N" for number. The upper-case sub-

scripts "P," "L," "A," and "V" indicate the basis for normalization of

the major term, e.g., number of points, unit line length, unit area, and

unit volume, respectively. The superscripts a, for the crystalline

phase, and P, for the pore phase indicate the volumes that meet to form

the specific feature. For example, aa indicates the grain boundaries

between a crystals, aP the interfaces between the crystalline and pore

phase, aaa the lines formed by the juncture of three a grains, and aaP

the lines formed by the juncture of two a grains with the pore phase.


The Counting Measurements

The counting measurements used in this study are the systematic

point count, the line intercept count, the area feature count, and the

area tangent count.

In the point count, a regular grid of points is placed at random on

the structure. The fraction of the total number of points falling in

the phase of interest, for a sufficiently large number of grid place-

ments, is called the point fraction P The average point fraction has
18
been shown8 to be an unbiased estimator of the volume fraction of a

phase,


p = VV.


(1-1)










Table I-1

Notation Used for the Quantitative Stereology Parameters


I. Root Symbols (indicate the measured quantity).

A area

C contiguity

D caliper diameter

G genus

H mean surface curvature

I inflection point

K Gaussian curvature

L length

M total surface curvature

N number

P point

Q total curvature of lineal features

S surface area

T tangent

V volume

W total torsion of lineal features


II. Subscripts (indicate the normalization base).

A per unit area

L per unit length

P per point

V per unit volume

g per gram









Table I-1 continued


III. Superscripts (indicate the feature being measured by specifying

the bounding phase or phases).

a the solid, i.e., crystalline, phase

P the pore phase

aa the boundaries between a grains

aP the pore-solid interface

aa triple lines at the juncture of 3 a grains

aaP triple lines at the juncture of 2 a grains with the

aP interface

aaa quadruple points at the juncture of 4 a grains

anaP quadruple points at the juncture of 3 a grains with

the aP interface









The line intercept count consists of placing a test line on the

section, and counting the number of intersections of the line with

traces of interface in the three-dimensional structure. The average

value of the line intercept count, PL, is related to the surface area

of the interface producing the trace by19


PL = Sv (1-2)
L 2 V

where SV is the interfacial area in unit volume.

The area feature count is simply the determination of the number,

per unit area, of some feature on the sectioning plane. The feature may

be a point representing the intersection of a linear feature with the
19
sectioning plane. For this case, it can be shown9 that


P = L (1-3)
A 2 V

where PA is the number of points per unit area of sectioning plane and

LV is the length of line in unit volume.

When the feature on the section has some spatial extension, for

example a particle outline, the net number of particles per unit area is

related to the total curvature of particle boundary per unit volume

by20,21


N = M (1-4)
Anet 2ir V

where MV is the total curvature in unit volume and NAnet is the number

of features in unit area summed so that a particle outline with n inter-

nal holes (n > 0) contributes 1-n to the total number.









An alternate method for the determination of total curvature is the
22
area tangent count.2 This measurement consists of sweeping the section

with a test line and counting the number of times the test line makes a

tangent with an arc element of particle outline. The net number of tan-

gents is obtained by choosing the volume on one side of the interface as

the reference phase: tangents formed with elements of arc which are

convex with respect to the reference phase are counted as positive; tan-

gents with concave elements of arc are counted as negative. The net

number of tangents per unit area, summed in this way, is related to the

total curvature by20


TAnet = V (1-5)

The use of the area feature count to determine the length per unit

volume of lineal features has already been mentioned. When the lineal

features are grain edges, they contribute to the total curvature of

interfaces in the structure. The grain edge curvature is given by23



Vedge = LV (1-6)

where 6 is the average dihedral angle between the surfaces that meet to

form the edge.

An additional feature which may be present on polished sections is

an inflection point in the trace of an interface. At inflection points

the curvature of the trace changes sign. The number of inflection

points per unit area, IA, is proportional to the integral of the curva-

ture of asymptotic lines over the domain of saddle surface in unit

volume,


I = ff+- k dS+- (1-7)
A 2 S as V









where k is the curvature of the asymptotic lines, and the superscript
as
+- indicates saddle surface. This relation was discovered as a part of

this research, and is included in this summary of the counting measure-

ments for completeness. A derivation of the relationship is presented

in Appendix B.


Total Curvature

Of the metric properties which can be obtained from the counting

measurements, the volume of a phase, the area of phase or grain bound-

ary, and the length of lineal feature in a unit volume of structure,

are all easily visualized concepts whose intuitive meaning correspond

closely to the geometric definitions. Total curvature is a somewhat

more abstract concept. The local mean curvature of a surface is

defined by


H = + (1-8)
2 R, R2

where R1 and R2 are the principle normal curvatures at a point on the

surface. To illustrate this definition, note that the tip of a needle

has a higher curvature than a pencil point, which in turn has a higher

curvature than the tip of a crayon. Note also that H is a local prop-

erty, e.g., the radii of curvature are not the same on the shank of a

needle as at the point. The total curvature of a surface is simply the

integral of H over the surface, or

M = ffS H(u,v) dudv (1-9a)


where u and v are a set of coordinates on the surface. The total

curvature in unit volume is usually written

= ffS HdSV (1-9b)

where dS = dudv, and the subscript V is added to balance the units.








If the mean curvature is high at a point on the surface, the pres-

sure differential across the interface is also high since24

AP = 2yH (1-10)

where y is the surface energy. Thus, high curvatures are often associ-

ated with large driving forces for microstructural change. Unfortu-

nately, the curvature parameter accessible by quantitative stereology

measurements, MV, is the result of an algebraic summation of the local

values of mean curvature. This means that positive and negative values

of H will cancel when integrated to give MV. Therefore, in many cases

MV will not report the existence of highly curved surface elements.

The total curvature parameter can also give a misleading picture of

the curvature of interfaces in single-phase grain structures. Since all

interfaces except the external surface are shared by two grains, the

curvature of every element of surface of one grain is exactly cancelled

by a contribution of equal magnitude but opposite sign from the adjacent

grain. Thus, while the grain surfaces can be, and almost always are,

curved, their contribution to M is identically zero. (For single-phase

grain structures the only nonzero contribution to Mi is the grain edge

curvature.)

Thus, It is seen to provide somewhat limited information about the

curvature of interfaces in a structure. In order to overcome these

limitations, alternate methods of measuring curvature have been sought.
25
One of these, proposed by Johnson, consists of performing a tangent

count only in selected areas of the structure. For the sintered struc-

tures considered, Johnson counts tangents only near interparticle necks,

since these are the regions whose curvature is assumed to determine the









driving force for densification. The subjective nature of this selec-

tive measurement technique make its results sensitive to the biases of
25
the individual operator, and thus Johnson's method25 does not lie within

the framework of the quantitative techniques used in this research.

Another method for reporting surface curvature is the gross tangent
26
count. If a section through a structure is found to have large

numbers of both positive and negative tangents, the net tangent count

may be very small, but the structure is quite different from one in

which a small number of tangents, all of one sign, are counted. The

gross tangent count,


T = T + T (1-11)
Ag A A

can be used to differentiate two such structures. The gross tangent

count also may be applied to single phase grain structures and to sur-

faces which are not closed. In general, large values of TAg will be

associated with structures with a large amount of highly curved inter-

face. A more precise interpretation of the gross tangent count was

formulated in the course of this research. These results are summa-

rized in Appendix A.


Contiguity

The contiguity of a phase is a concept first formulated by
16
Gurland, who was interested in reporting the results of quantitative

stereological measurements in a manner which might explain observed

physical properties such as the electrical conductivity of two-phase










alloys or the fracture behavior of systems containing both brittle and

ductile constituents. Gurland's contiguity parameter, which will be

called the grain contiguity, is defined by16


2Sa(
Ca V (1-12)
2S + So,
V V

for a two-phase system.

The phase whose contiguity is determined, the a phase, must consist

of a number of particles which form a definite interface when placed in

contact. For most systems, this requirement means that the phase is

crystalline. Gurland's definition was extended to multiple-phase

systems by Dorfler.27 The two-phase definition will be used in the

present discussion.

The definition of contiguity may be combined with the definitions

of mean grain intercept, rg, and mean phase intercept, Ta, to yield28,29


-g
Ca = 1 (1-13)
-0


Other contiguity parameters, such as the lengths of the several

types of triple lines and the total perimeter length of grain contacts,

have been proposed,30 but these are largely dependent on the scale of

the system and on the shape of particles, and thus do not directly indi-

cate the contiguity of a phase. Ratios of surface areas similar to the

grain contiguity Ca have also been used,30 but these are not independent

of the contiguity parameter.









The determination of contiguity parameters for sintered structures

is a relatively new concept. The crystalline phase in a sintering is

always continuous, but its contiguity varies from a value of zero for an

unfired compact (if the initial particles are monocrystalline) to one

for a fully dense body.

A new parameter, similar in form to the grain contiguity but con-

sisting of ratios of triple line lengths, was devised to analyze some

of the stereological data generated in this research. This parameter,

called the grain face contiguity, is discussed in Chapter III and in

Appendix C.


The Microstructural State and the Path of Microstructural Change

The methods of quantitative stereology yield a large amount of

information about a structure. Some method of organizing these data is

useful for describing the results of studies of the stereological

parameters. The method used in this dissertation describes individual

structures in terms of their "quantitative microstructural state," and

the evolution of microstructure by the "path of microstructural change."

The quantitative microstructural state. Each microstructural

feature may be characterized by determining its associated stereological

properties. These properties may be divided into two broad classes:

metric properties, which are sensitive to the size and shape of the

features, and the topological properties of number and genus, which are

independent of the scale of the features and are unchanged by bending

and stretching operations. A determination of the quantitative micro-

structural state consists of measuring all of these stereological prop-

erties, listed in Table 1-2. However, at least at the present stage of






17

















6I


o



H d 6
U 6> > >6a > > > >6>

m 0


0o
6 0






i a P0
6 H







P > 6> e
4C t B p. > a- a
I a> p a a>
w 0


^t O
>
S 0

I 4 6 t






6 0


S 0>

6 0-c H 0
C0 b P

16 p. 0=




1 1-0




r o I s pc a
U











*4 00En En
00 6
CD CO ~ 6 ) 41,
U 6J 0. 6 U 6 6
63 6; 6i 6) 6 6









advancement, all of the topological properties and some of the metric

properties of an opaque material can only be determined by reconstruct-

ing the spatial structure by means of a laborious serial sectioning

procedure. Only those properties which can be determined by measure-

ments on a single polished plane section were determined in this inves-

tigation. These properties are enclosed in a box in Table 1-2.

The path of microstructural change. The quantitative microstruc-

tural state of a specimen may be regarded as a point in an n-dimensional

Euclidean space, where n is the total number of stereological properties

which may be defined for the structure. If the structure is altered by

a physical or chemical process, the point moves through n-space and

traces out a curve. This curve describes the sequence of microstruc-

tural states through which the system passes during the process and is

called the path of microstructural change.32 Since a complete descrip-

tion of the quantitative microstructural state is almost never avail-

able, the path of microstructural change is usually observed as a pro-

jection onto a lower-dimensional space, whose coordinate axes are the

properties actually determined. Figure 1-1 provides an example of the

path in three dimensions for a simple sintering experiment. In practice,

the path of microstructural change is usually presented as a series of

k-1 planar projections, where k is the number of stereological proper-

ties determined.

Most of the graphical presentations of the path of microstructural

change in this dissertation have pore volume fraction as the abscissa.

Pore volume fraction has several advantages as an independent variable

with which to describe the sintering process: 1) volume fraction is

































MV


Figure 1-1. Three-dimensional representation of the path of micro-
structural change for a simple sintering experiment.









commonly used to measure the degree to which a compact has sintered into

a coherent entity, and 2) volume fraction decreases monotonically but

not linearly with time. Use of VP as the independent variable has the

effect of expanding the time scale at short times, when the structural

changes are occurring most rapidly, and compressing the scale at long

times.


Shape Parameters

The problem of specifying the shape of features in complex struc-

tures has received attention in the fields of mathematics, biology, and

materials science.1733-37 The wide range of shapes present in mate-

37
rials systems has been reviewed by Smith, who also summarized the

physical processes (and combinations of processes) that produce the

various classes of shapes.

A complete specification of shape might consist of a description

of the relative numbers, positions and equations of the points, curves,

and surfaces that form the structure. Two broad classes of shape, topo-

logical and geometric, are represented in such a specification. Topo-

logical shape is a description of the number of each type of feature,

the genuses of the surfaces in the structure, and the connectivities of

the lineal networks. Several experimental determinations of one or more

aspects of the topological shape of materials systems are represented in

the literature.38-42 Most of these studies employed serial sectioning

techniques.









Geometric shape is the description of the positions and equations

of the curves and surfaces of a structure. Experimental studies usually

attempt to quantitatively characterize the average geometric shape of a

structure in terms of a "shape parameter," sometimes called a "form

factor." The shape parameter should satisfy the requirement being sen-

sitive to the shape but independent of the scale of structural features.

This requirement means that a shape parameter is a dimensionless quan-

tity. Underwoodl7 has compiled a list of twelve parameters that have

been proposed for the evaluation of shape. Fischmeister has pointed

out that of the parameters on Underwood's list, four pertain only to

oriented structures, four contain quantities that can only be evaluated

by serial sectioning, and one is not a dimensionless quantity.

Fischmeister also showed that the remaining three parameters are all

closely related and may be written as simple products and ratios of the

counting measurements Pp, PL, and NA.

The shape parameters that require serial sectioning analysis for

their determination are calculated from the average dimensions, surface

areas, and volumes of the particles or grains in the structure. For

example, one of Underwood's expressions is


-1/3
f = (1-14)
1 -1/2


where V and S are the average particle volume and surface area

respectively.7








One of the parameters that can be calculated from counting measure-
44
ments on a single plane section was first proposed by DeHoff for

structures consisting of isolated particles of constant size and shape
45
and was later extended by Rhines, DeHoff, and Kronsbein45 to sintered

structures. For the latter case, this parameter may be written

P aP
A A P2 (1-15)
(S )
V

where H = M /S in the average mean curvature of the pore-solid

interface, and P = 4Vv/SP is the mean pore intercept.

It is difficult to ascribe a physical meaning to a given value of

HP-XP in the general case. However, several observations may be made

for particulate structures: First, H A approaches one for prolate

ellipsoids or rod-shaped cylinders and zero for oblate ellipsoids and

plate-shaped cylinders. Second, even for particles of the same class

(e.g., ellipsoids or cylinders), a single value of the shape parameter
44
may be shared by particles with two different aspect ratios. Third,

H A depends on the details of the particle size distribution, even if

all particles have the same shape. Generally IH increases as the dis-

persion of the size distribution increases.

A geometric shape parameter applicable to single-phase grain struc-

tures was proposed by DeHoff and determined for recrystallized aluminum
40 aaa aa 2
structures by Craig. This parameter is LV /(SV ) Several addi-

tional shape parameters, similar in form to the preceding expression but

appropriate to two-phase sintered structures, are developed in Chapter 3.









Sintering


The voluminous nature of the literature on sintering attests to

the controversies which the subject has engendered. These controver-

sies are often kindled by the fact that various researchers in the field

ascribe different meanings to the term sinteringg." A simple definition

which includes all of the various kinds of sintering is that provided

46
by Rhines,4 who defined sintering as "that process by which particles

bond themselves into coherent bodies, usually, although not necessarily,

under the influence of pressure and elevated temperature." This defini-

tion identifies geometrical change as the factor common to all types of

sintering. An investigation of sintering in a particular system might

consist of a description of the geometrical changes which take place and

the identification of the atomic mechanisms which produce the observed

changes. For a variety of reasons, studies of sintering mechanisms and

studies of microstructural change are usually performed separately, so

that a survey of sintering literature may be naturally divided into two

parts. Since the research done for this dissertation was intended to

be a geometric study, more emphasis will be placed on other geometric

studies. A brief review of the mechanistic studies will be presented

first.


Sintering Mechanisms

Early investigators proposed that sintering was due to partial

melting of the powders caused by high pressure or friction at contact

points during pressing, the lower melting point of small par-

ticles, or recrystallization and phase changes.50








51-54
The nature of the sintered bond was explored by Sauerwald.51

He proposed that atomic orbitals extended outward as temperature

increased until atoms, which were initially separated, formed a bond.

These new bonds increased the effective area of contact and thus the

strength of the compact. Other workers established Sauerwald's hypoth-

esis that the bonds which joined particles (or massive bodies) at con-

tact points are the same as the interatomic bonding forces in a single
55,56
crystal.55,56

Surface-tension forces, which were known to smooth the crevices in
57
a rough surface, were identified as the driving force for sintering

by Jones56 and Balke.58 These surface-tension effects were observed to

occur in short times at high temperatures. Since plastic flow and dif-

fusion are promoted by high temperature, these two processes became

popular candidates for the role of the mechanism of sintering. As other

modes of material transport were identified, the list of possible sin-

tering mechanisms grew to include volume diffusion, boundary diffusion,

surface diffusion, evaporation-condensation, plastic flow, and viscous

flow. Shaler and Wulff59 argued that mechanisms could be differentiated

on the basis of their ability to produce densification. Today, it is

generally agreed that surface diffusion and evaporation-condensation

processes can produce only a surface-smoothing effect, while the others

listed above have the potential of producing densification.

Kuczynski60 provided a simple set of rules for determining which

mechanism is active in a particular sintering experiment. In his

method, the growth of interparticle necks was observed for geometrically

simple systems. The time dependence of neck radius was used to deter-

mine the mode of material transport. The appeal of this approach may be










demonstrated by the enthusiasm with which other workers attempted to
61-64
apply Kuczynski's rules to sintering in a wide variety of systems.

The main drawback to this type of mechanism-determining study is the

assumption that a three-dimensional network of particles may be ade-

quately represented by a simple model consisting of a sphere on a

plate,60,61 twisted wires,65 or spools of wires.62 Because of the limi-

tations of the geometrical models, experimentally observed densification

rates on real sintered structures seldom agree with the model's predic-

tions. In order to resolve these discrepancies, more recent sintering

models have included the possibility of the simultaneous action of

several mechanisms, or of a change in the predominant mechanism as

sintering occurs.6


Geometric Changes during Sintering

This discussion will be restricted to the sintering of systems con-

sisting of one solid phase and the vapor phase.

The microstructural evolution which takes place during sintering

has been described by Rhines as a combination of four distinct and

somewhat independent processes: densification, surface rounding, isola-

tion of pores, and coarsening of the pore structure. All of these have

the potential for reducing surface area.

Densification is the most easily observed process, since density

measurements do not rely on microscopy techniques. It is not sur-

prising, therefore, that density was the first geometric parameter to be

carefully monitored for the various steps in a powder metallurgy process

(by Wollaston69 in 1829). During sintering, densification occurs by a

reduction in the average center-to-center distance of neighboring










particles, and not by a decrease in the number of particle centers.

Density, or equivalently volume fraction of either solid or pore phase,

is a convenient parameter for following the progress of a sintering

experiment, and for comparison of the paths of microstructural change

during sintering of different materials, and of different powder types

and sizes. The variations of the other geometrical parameters at a con-

stant volume fraction of porosity will be used for many comparisons of

microstructures in this dissertation.

Surface rounding processes were observed by Desch57 in 1923. In

his experiments, small pieces of gold were melted and allowed to solid-

ify in the shape of droplets. The initial freezing of the outer layer

produced a thin solid skin, which then wrinkled to accommodate the

shrinkage that occurred as the interior of the droplet froze. When the

solid droplet was reheated below the melting point, the wrinkled surface

was smoothed out until it had the same appearance as the surface of a

liquid drop. Two geometric parameters underwent change in the experi-

ment just described: the surface area of the system and gradients in

surface curvature were both reduced. The concurrent reduction of sur-

face area and curvature gradients is a characteristic of surface-

tension-driven processes.*


*Changes in surface area and curvature gradients are not necessarily
tied together for all geometric processes. For example, an initially
flat sheet of paper may be crumpled to introduce curvature and curva-
ture gradients, with no change in surface area.










The most striking change which a powder mass undergoes during

sintering is the conversion of the initially multiply connected pore

phase into a number of isolated, nearly spherical pores. The isolation

of pores is accomplished by breaking the connections which join the pore

phase. If all of the connections or channels leading to a particular

region of porosity are severed, an isolated pore is produced. Rhines

has shown68 that the closing off of a channel usually reduces the genus

of the pore-solid interface by one. Thus, the channel-closing and pore-

isolation process may be followed by measuring the topological proper-

ties of a system as it sinters. Methods for the measurement of topo-

logical properties have been reported by Kronsbein et al.41
42
Aigeltinger4 measured the topological properties of sintered copper,

and showed that most of the channel closure events occurred in a narrow

volume fraction range, from 0.25 to 0.15.

Three sequential stages of the sintering process have been

defined68 in terms of the changes in topological state which the system

experiences. These stages are:

First stage Growth of interparticle welds; genus is
constant or increases slightly as new
contacts form.

Second stage Genus decreases from initial value to
zero; isolation of pores.

Third stage Genus constant at zero; pore volume
continues to decrease.

The final geometric process, coarsening of the pore structure, was

first observed by Rhines et al.70 Coarsening is a redistribution

process in which the larger pores grow at the expense of the smaller

ones. Densification continues to occur, however, so that coarsening of

the porosity in sintered structures is characterized by an increase in










average pore size, and a decrease in pore volume fraction. In the work

of Rhines et al.,70 coarsening of isolated pores was observed during the

latter stages of the sintering of copper powders. Coarsening of multi-

ply connected pore phases has also been reported for the "inhibition

sintering" of antimony powder,71 and for sintering of compacted copper71

and uranium dioxide72 powders.

As a result of the action of the four geometric processes, the

microstructural parameters vary with the volume fraction of porosity in

a characteristic way for sintering. The variation of the topological

properties, genus and number, has already been discussed.

The decrease in surface area which is observed in second stage

sintering is a combined result of the densification, channel closure,

and surface rounding processes.6 As densification occurs, the length

to diameter ratio of channels increases until a tubular channel becomes
68
unstable. As the unstable channel pinches shut, regions of high cur-

vature are momentarily produced at the collapsed end of the channel.

The curvature gradients thus produced are smoothed out and the surface

area reduced by the action of the surface rounding processes.

DeHoff et al.71 found that a straight line which extrapolated to

the origin described the relationship between pore-solid surface area

and volume fraction of porosity for second stage sintering of uncom-

pacted powders. The linear relationship was taken to indicate that a

balance existed between surface rounding and densification processes.

Support for this hypothesis was obtained from experiments in which

powders were compacted prior to sintering. The additional surface area

(per unit volume) produced by precompaction led to a dominance of sur-

face rounding over densification processes. Surface area versus volume









fraction plots for compacted powders were curved lines lying above, and

approaching asymptotically, the straight line characteristic of loose-

stack uncompactedd) sintering. Schematic surface area versus volume

fraction curves for different sintering conditions are shown in

Figure 1-2. For uncompacted powders, the mean pore intercept

(A = 4V/SP ) is a constant; for compacted powders, XP continually

increases during sintering.

While direct measurements of curvature gradients have never been

made, the variation of total curvature during sintering has been

reported by Rhines et al.45 and Rhines and Gregg.73 For uncompacted

spherical powders, MV was found to be greater than zero during first

stage sintering, pass through zero and go through a minimum during
72
second stage, and approach zero during third stage sintering. Tuohig7

reported total curvature values for pressed and sintered UO2 powders,

but made no attempt to determine the effect of compaction pressure on

the total curvature versus volume fraction relationships. One of the

goals of this research was a study of the total curvature of compacted

powder systems.

In addition to changes in the geometry of the pore network and the

pore-solid interface, the grain structure is also subject to rearrange-

ment during sintering. Since the grain structure usually depends on the

geometry of the pore structure, it is convenient to discuss the evolu-

tion of grain structure in terms of the parallel development of the

porosity as the system passes through the three stages of sintering.

For the moment, assume that the unsintered powder consists of single

grain particles. During the first stage of sintering, grain edges of

the type aaP are generated by the formation of weld necks between the


























Sv


Figure 1-2. The effect of fabrication variables on the path of pore-
solid surface area change.










individual particles. Each area of interface is bounded by an aaP

triple line. As neck growth proceeds, the area of aa interface and the

length of oaP line increases. The channel closure events which predom-

inate in second stage sintering will usually result in the joining of

three aa surfaces to form an aaa triple line.

During the third stage of sintering, several distinct geometrical
74-76
changes have been observed. In some studies, the grain boundaries

break away from the pores and coarsen at a much faster rate than the
42
coarsening of the porosity. Both uniform grain growth and exaggerated

grain growth,7 which resembles secondary recrystallization in fully

dense material, have been reported. In other systems, usually oxide

ceramics with additions of aliovalent cations as sinteringg aids," grain

growth does not occur77 in third stage sintering. This stability of the

grain structure has been attributed to the segregation of the impurities
78
to the grain boundaries, which impedes boundary motion.


Sintering of UO2

While UO2 has many characteristics in common with other sinterable

powders, certain of its properties distinguish its sintering behavior

from that of other materials.

The multiple oxidation states of UO2 lead to the existence of at

least four stable oxides. The composition U02, which corresponds to

the stable oxide of lowest valence, crystallizes in the cubic fluorite

structure. The cubic structure is thought to be stable for O/U ratios

ranging from 2.00 to 2.25.79 The stoichiometry of "UO2" affects the

physical properties and the sintering behavior. For example, Williams

et al.80 sintered material with O/U ratios ranging from 2.00 to 2.18 and








found a large increase in densification rate as O/U was increased from

2.00 to 2.02. Further increase in 0/U ratios produced little increase
80
in rate, however. Williams et al. attributed the observed behavior

to increased cation mobility caused by a larger number of lattice

defects. Stoichiometry may affect the path of microstructural change

for sintering of UO2, but no investigation of a possible effect is known

to the author.

The response of a powder to mechanical stress also influences the

sintering behavior. At ambient temperatures, UO2 is very nearly a
81
perfect elastic solid.81 Thus, cold compaction will not plastically

deform the individual particles. At temperatures above about 13000C,
82
U02 becomes capable of plastic flow, so that at usual sintering tem-

peratures, or during hot pressing, plastic flow may be an active densi-

fication mechanism.

Other properties which may affect the sintering of UO2 are its high

vapor pressure (10 Torr at 17000C),83 which promotes surface rounding

processes by the vapor transport mechanism, and the tendency of UO2
83
powder to agglomeration and aggregation, which may cause defects in

either pressed or fired material.



The Electrical Properties of
Heterogeneous Structures


Consider a unit cube of a homogeneous, isotropic substance with

electrical contacts attached to two opposite faces. If a potential is

applied across the electrodes, current will flow. By measuring this

current, the electrical conductivity can be calculated. If a small

volume of material is removed from the interior of the cube and is









replaced by an equal volume of a substance with a different electrical

conductivity, the effective conductivity of the unit cube will be

altered. If the second substance has a higher conductivity than the

first, the conductivity will be increased. If successively larger

amounts of the second material are added, a larger increase in the con-

ductivity of the mixture will be observed. This type of reasoning leads

to the intuitive notion that the conductivity of a heterogeneous system

is a monotonic function of the conductivities and volume fractions of

the phases which make up the structure. That this notion is misleading

is demonstrated in Figure 1-3. Assume that the two phases present are

phase A, a substance with a small but finite conductivity, and phase B,

a high conductivity material. The two structures shown have equal

volumes of B, but the structure with the continuous path of high conduc-

tivity material between the electrodes clearly has the higher effective

conductivity. This demonstration suggests that the conductivity of

heterogeneous systems is a complicated function of the distribution of

phases in a structure. Still, it is possible, at least in principle, to

calculate the conductivity of an arbitrary heterogeneous system in terms

of the conductivities and the spatial distributions of the phases that

make up the structure. The result is obtained by simultaneous solution
84
of the Laplace steady-state heat-flow equation, and the Gauss electric

field equation.85 Because the mathematical methods are entirely analo-

gous, the same treatments can be used to obtain the electrical conduc-

tivity, static dielectric constant, magnetic permeability, thermal

conductivity, and diffusion coefficient of heterogeneous systems.














C3-
O0

\ c -
,01


Figure 1-3. Unit cubes of heterogeneous structures showing
(a) particles dispersed in a matrix, and (b) a single
particle extending the length of the cube.









If an alternating electric field is applied to a heterogeneous

structure, dielectric relaxation may occur. When the observed disper-

sions are due to polarization of the internal interfaces, the relaxation

times may be calculated if the spatial distribution of the interfaces is

known.

The methods outlined above lead to mathematical forms which, for

the general case, require complete information about the spatial distri-

bution of phases in the structure. Even if this information is avail-

able, the mathematical expressions soon become intractable. As a

result, theories of the electrical properties of heterogeneous struc-

tures employ approximate methods. These theories may be conveniently

divided into two broad classes: (1) those theories in which a specific

geometric model of the structure is assumed, and (2) theories in which

no model is assumed. Examples of the first class far outnumber the

second and will be reviewed first.

The initial analysis of the dielectric constant of a heterogeneous

structure was done by Maxwell1 in 1873. He obtained an expression for

the dielectric constant of a system consisting of n parallel slabs of

material, each with a different dielectric constant. A two-phase

layered dielectric is a special case of Maxwell's model. In 1892 Lord

Rayleigh86 extended Maxwell's analysis to additional two-phase geome-

tries. He presented solutions for the dielectric constant of systems of

spheres on a cubic lattice and of parallel arrays of cylinders at right

angles to the field direction. These solutions included the effect of
2
electrostatic interaction between second phase particles. Wagner

neglected electrostatic interaction and obtained the dielectric constant

of a random array of spheres. The assumption of no electrostatic









interaction is valid if the interparticle distance is large compared to

particle radius. In 1925, Fricke87 attacked the problem of electro-

static interaction by adding to the original field a value obtained by

averaging the effect of all the charges on suspended particles over the

sample volume. With this technique, Fricke calculated the conductivity
87
of suspensions of ellipsoids of revolution or spheroids. These solu-

tions were extremely cumbersome but gave good agreement with experi-

mental measurements on several systems. Fricke's experimental results

will be discussed in the next section. Sillars also examined suspen-

sions of spheroids. He followed the same method as Wagner, i.e.,

assumed a dilute suspension, and made the further assumption that the

suspending medium was a perfect dielectric (conductivity = 0). Sillars'

equations treat the dielectric relaxation problem as well as the static

case, but are valid only for low volume fractions of the dispersed phase.
88
Bruggeman8 offered another approach to the problem of large volume

fractions. He differentiated an approximate solution for the conduc-

tivity of suspensions of spheres and assumed that the resulting differ-

ential equation was valid for all volume fractions. By an iterative

integration process, he was able to calculate the dielectric constant

over the full volume fraction range. In another publication, Bruggeman

presented formulae for the dielectric constant of single phase poly-
89
crystals in which the single crystal properties are anisotropic.

Nearly thirty years after his initial publications, Fricke90

extended his average field approach to suspensions of triaxial ellip-

soids. This publication is noteworthy for several reasons. First, with

the exception of multiply connected structures, triaxial ellipsoids may










be used to model most structures in real materials. Second, Fricke

gives complete calculations of static dielectric constants and relaxa-

tion times for the full volume fraction range and for parallel or random

orientation of the ellipsoids. Thus the solutions for a broad range of

practical problems is compiled in one source. Third, the dielectric

dispersion equations predict three relaxation times for randomly oriented

ellipsoids. This result suggests that. more complex geometries will have

very complicated relaxation behavior.

Recently, interest in the microstructure of fiber-reinforced com-

posite materials has led to the use of electrical resistivity measure-

ments as a means of characterizing the composites. Several investiga-

tors91-93 have used an electrical analog technique in which the com-

posite is modeled as a group of resistors. Various series and parallel

combinations of the resistors are used to simulate fiber branching,

nonparallel fibers and fiber termination. In most of these models,91'92

the assumption is made that no field perturbations occur when second

phase particles are embedded in a matrix with a different electrical

conductivity. This simplifying assumption makes the electrical analog

models much less rigorous than any of the solutions derived from the

Maxwell equations. Recently, Watson et al.93 partially overcame this

objection by using Rayleigh's cylinder equation to calculate the resis-

tance of some of the circuit elements in their model.

The results of these geometrical modeling studies indicate that the

electrical properties of heterogeneous structures depend on the amount

of second phase and on particle shape but not on particle size or

details of particle size distribution, as long as shape is independent

of size and the particles are small compared to the specimen size. The










best agreement with experimental results is obtained with the more com-
90
plicated formulae, e.g., those of Fricke.90 The simpler mathematical

expressions, such as those of Wagner, are only valid for a restricted

set of geometries. None of the models treat the case of multiply

connected structures.

The limitations of the geometrical models have led several workers

to devise solutions for the electrical properties (of heterogeneous

mixtures) in which no model is assumed. The results of such studies

should thus apply to a wide range of structures.
94
Brown,9 for example, calculated the dielectric constant of a two-

phase system by writing the polarization vectors as a power series in

the dielectric constants of the two phases, and throwing away higher

95
order terms. Herring95 used a similar Fourier series expansion tech-

nique and applied it to several electrical properties. He restricted

his analysis to small fluctuations in the electrical properties, but

found that the formulae often gave useful approximations for large

fluctuations. For a porous structure, however, in which the conduc-

tivity of one phase is zero, Herring's equation predicts a linear rela-

tion between conductivity and volume fraction, and negative conductivi-

ties at high pore volumes.

96
Hashin and Shtrikman used a variational method to obtain upper

and lower bounds for the magnetic permeability of multiphase materials.

They found that the bounds gave a good estimate of the effective perme-

ability only for relatively small permeability variations between the

phases.










The results of the theories of the electrical properties of hetero-

geneous systems can be summarized as follows:

1. Dispersed phase systems may be effectively described by any
one of several geometrical model solutions based on the
Maxwell Field Equations.

2. Other approximate methods may be used for any system
geometry if the spatial variations in electrical prop-
erties are small enough.

3. All solutions depend on the details of the spatial distri-
bution of the several phases.

4. No solution exists for a general structure, containing
multiply connected and dispersed elements, if large
property fluctuations are present.

The sintered structures studied in this research had multiply

connected pore and solid phases, and a connected grain boundary network.

Fluctuations in electrical conductivity were discontinuous and large.

For a system this complicated, direct experimental observation is the

only means of correlating microstructure with electrical properties.


Experimental Studies of the Electrical Properties
of Heterogeneous Systems

The development of theoretical treatments of the electrical prop-

erties of heterogeneous structures was accompanied by a number of

experimental studies designed to test the validity of the various

theories. Millikan,97 for example, found that Rayleigh's cubical array

equation accurately described the dielectric constant of random dis-

persions of water in benzol-chloroform. Similar experiments were per-

formed by Sillars3 on emulsions of water in paraffin and by Dryden and
Meakins98 on emulsions of water in lanolin.
Meakins on emulsions of water in lanolin.










The use of electrical measurements as a probe with which to measure

structure was first proposed by Fricke87 in 1924. The utility of this

technique was demonstrated in subsequent publications. Fricke and Morse

used conductivity measurements to determine the relative volumes in
99
butterfat-milk mixtures and capacity measurements to estimate the mem-

brane thickness of the red blood cell.100 Fricke's 1925 value of 33 X
101
is remarkably close to recent values determined by electron microscopy.

Other examples of structural determinations by means of electrical

measurements may be found in the literature of the biological sciences,
102
and have been reviewed by Cole.

Electrical measurements have also been used to determine structure

in the materials field. In metals, resistivity measurements have been

used to estimate dislocation density by Blewitt et al.103 and Clarebrough
104
et al., and to monitor the recovery of lattice defects in cold worked

metals by Sharp et al.1

Several studies of the structures produced by varying heat treat-

ments of glass have used dielectric measurements to detect the presence
106
of heterogeneous structures. Owen06 interpreted the loss behavior of

CaO-B203-Al 03 glasses as an effect of a Maxwell-Wagner dispersion.

Charles07 examined a series of lithia-silica glasses and concluded that

the observed dielectric loss behavior was an effect of the heterogeneous

(two-phase) nature of the liquid from which the glass was generated.

Charles corroborated the dielectric behavior with replica electron

micrographs. Kinser and Hench08 examined the effect of thermal history

on structure, also in the lithia-silica system. In contrast to Charles,

they observed no loss peaks in quenched material, but noted relaxation

effects after heat treatment at 5000C. The loss peaks, ascribed to the









presence of a metastable crystalline lithium metasilicate phase, disap-

peared after prolonged heat treatment as the metasilicate dissolved and

was replaced by the equilibrium disilicate phase. Replica and thin
109
film09 electron microscopy and x-ray diffraction studies were used to

support the evidence of the dielectric studies. In a recent review of

the electrical properties of glasses, Hench and Schaake0 conclude that

the use of a heterogeneous dielectric model to explain dielectric losses

in glass must be justified by independent methods of structure analysis

in order to insure that the observed losses are due to a heterogeneous

mechanism and not an atomistic one.

The electrical conductivities of two-phase systems in which the

high-conductivity phase is dispersed when present in low volume frac-

tions, but becomes multiply connected at high volume fractions, have
Whie111 28
been investigated by Blakely and White l and Gurland.28 The electrical

conductivities of sintered silver-alumina bodies showed a marked
111
increase for silver contents in excess of t25 percent, presumably

because the silver phase became interconnected for long distances

through the structure as the volume fraction exceeded a critical value.

A sharp transition in electrical conductivity was observed by Gurland28

for specimens consisting of spheres of silver embedded in a bakelite

matrix. In the latter study, the critical volume fraction was '0.37.

Results of this nature are usually explained in terms of percolation
28,112 113
probability theory,28'1213 which predicts that long-range intercon-

nection of a particulate phase will occur if the average number of con-

tacts between particles of the second phase exceeds a critical value.

The variation in different systems of the volume fraction at which the

second phase becomes interconnected is apparently due to the variation

in particle shape and size distribution.









The Electrical Properties of Porous Materials

Because residual porosity is usually present in articles fabricated

from powders, much attention has been focused on porosity as a special

kind of second phase. The effect of porosity on conductivity and static

dielectric constant can often be described using one of the heteroge-

neous dielectric theories. Because the pore phase is an insulator, it

will not give rise to conduction losses and the attendant dielectric

relaxation phenomena. An important exception occurs when a pore network

is exposed to a gas which will adsorb to the surface of the pores. In

this case, large dielectric losses may result at certain combinations of

frequency and temperature.

Before about 1950, many studies of sintering used electrical con-

ductivity measurements as an indicator of the degree to which a pressed

or sintered compact had been consolidated into a coherent mass.

The early work in this area is typified by the paper of

Streintz.114 The experimental scatter found in his results may have
115
been caused by density gradients in his samples. Later,
116
Kantorowicz reported the curious result that the electrical conduc-

tivity of a tungsten powder pressing decreased if it was pressed

repeatedly to a given pressure. This behavior can be explained if the

continued reapplications of pressure had the effect of fracturing the

powder particles.

It has already been remarked that density or pore volume fraction

is a desirable choice as an independent plotting variable. Since many

of the earlier works present data in the forms of plots of physical

properties against sintering temperature, without giving density values,

it is difficult to construct plots of physical properties against pore









volume fraction. The work of Grube and Schlect17 is an interesting

exception. Figure 1-4, taken from their work, presents electrical con-

ductivity values of sintered nickel powder as a function of volume frac-

tion of porosity. At high pore volumes, data points for three pressures

of cold compaction lie on three separate curves. These curves converge

to form a single trace as densification occurs. Similar behavior has

been observed for the fracture strength of UO2 powder. The latter

effect was explained by noting that green density increased and average

interparticle distance decreased as the powders were pressed. When

sintered to equivalent densities, specimens pressed at lower pressures

have more well-developed necks and larger interparticle contact area.

The effect of contact area is a likely explanation for the data of Grube

and Schlect.1
119
Rhines and Colton used electrical conductivity measurements to

study the homogenization of compacts pressed from mixtures of elemental

copper and nickel powders. They do not report the density changes which

occur during the homogenization treatments but do show that conductivity

changes due to densification are far outweighed by the effects of

homogenization.

Other factors which affect the electrical conductivity of sintered
120
materials include sintering atmosphere and purity of the powder.

Since the observed electrical behavior is the sum of contributions from

several sources, care must be exercised in the interpretations of con-

ductivity data of sintered structures. Independent confirmation of the

mechanisms responsible for electrical conductivity must be sought, just

as in the case of dielectric relaxation effects in heterogeneous systems.














A 800 kg/crri2

0 200 kg/cm2
0 4000 kg/cm2







L








r


5.0 6.0 7.0 8.0

DENSITY (g/cm3


Electrical conductivity versus density for sintered
nickel. This graph was constructed from the data of
Grube and Schlect.117


Figure 1-4.









A large body of literature exists on the dielectric relaxation

effects of gases adsorbed to the surfaces of porous adsorbents.121 In

most cases, the aim of these studies has been to analyze the dielectric

properties of the adsorbate. Often, however, the results contain infor-

mation about the structure of the adsorbent surface, the degree of sur-

face coverage by adsorbate, or the kinetics of the adsorption process.

122 123
For example, Ebert and Langhammer2 and Baldwin and Morrow report

frequency and temperature dependent maxima in plots of loss dielectric

constant against the amount of water vapor adsorbed on the surface of
124
y-alumina. Hench24 reported the change in A.C. conductivity and loss

tangent of reactive MgO powders exposed to air of 85 percent relative

humidity. Equilibration of the water vapor adsorption process had not

occurred after 140 hours of exposure.


The Electrical Properties of UO

Electrical Conductivity. In common with several other oxide

ceramics, the electrical conductivity of UO2 may be due to either ionic

transport or extrinsic or intrinsic semiconduction mechanisms, depending

on the temperature of measurement. At room temperature. conductivity

values ranging from 3 x 10- to 4 x 108 (0 cm)-1 have been reported.125

These values represent extrinsic semiconduction and are markedly influ-

enced by variations in the composition, purity, apparent density, and

crystalline perfection of the samples examined.









126
Ilartmann, by using Hall effect measurements, was the first to
127
identify U02 as a metal deficit or p-type semiconductor. Meyer2

reported both p- and n-type behavior. However, subsequent investiga-

tions of the U-0 phase diagram have failed to confirm the existence of
128
substoichiometric UO2'128 throwing Meyer's claim into doubt. When

n-type semiconduction is observed in "UO ," a close examination will

reveal either a low purity specimen or the existence of a substoichio-
128
metric phase of higher oxidation state, e.g., U409._y

The effect of metallic impurities on the conductivity of UO2 is

consistent with its description as a p-type semiconductor, i.e., lower

valence cationic impurities such as Ca2+ increase the conductivity by

making UO2 more p-type.128 Higher violence cations such as Mo6+ usually

do not decrease the p-type character, however, perhaps because they are

reduced to the +4 state, accompanied by the formation of U5+ cations.128

A somewhat different analysis of the conductivity mechanism in U02
129-131
is that provided by Aronson and coworkers, who measured the elec-

trical conductivity of high density, large grain size, nonstoichiometric

plates of UO2 at temperatures ranging from 5000C to 11500C. These tem-

peratures are below the range where ionic transport would be expected to

contribute to electrical conduction. The observed dependence of con-

ductivity on temperature and O/U ratio is in agreement with a proposed129

description of nonstoichiometry in UO2. The description assumes the

existence of U4 and U5+ ions on lattice sites, and lattice and inter-
2-
stitial 02- ions. Conduction occurs by a "hopping" mechanism, in which

holes are localized at the U5+ sites, but have some freedom to jump to

an adjacent U site, thus transferring charge. In a series of papers
132-134
by Devreese and coworkers, the current carriers are identified









as small polarons, which, in the present case, are holes with an associ-

ated polarization of the nearby ions.135 While experimental evidence

132-134
cited by Devreese and coworkers strongly suggests that small

polarons are responsible for electrical conduction in nearly stoichio-

metric UO2, the evidence does not exclude the possibility that UO2 is a

band-type semiconductor.

The effect of microstructure on the electrical properties of UO2 is

not well known, probably because the effects of stoichiometry and purity

usually dominate the electrical behavior so that the microstructure

plays only a secondary role in determining the electrical properties.

As a consequence, only a few published works concerning the microstruc-

tural effects on the electrical conductivity of UO2 can be found, and

the results of these are often contradictory.

For high density material most workers observe an increase in con-
136
ductivity with increasing grain size, although a decrease in conduc-

137
tivity with increasing grain size and no effect of grain size on
133
conductivity have also been reported.

For studies in which the effects of stoichiometry are investigated,

it is usual to multiply the measured conductivities of porous specimens

by a "correction factor," equal to


VP
1+ V
[1 (VP)2/3
V










which gives the conductivity that would be measured if the specimens
125
were fully dense. The accuracy and range of applicability of this

correction factor has not been established. Willardson et al.125

caution that the correction factor is not appropriate if grain size

variations accompany the density differences among the specimens which

are to be compared.

Dielectric properties. In a series of papers, M. Freymann,

139-141
R. Freymann, and their coworkers, give the results of investiga-

tions into the dielectric properties of several of the oxides of
139
uranium. These studies were summarized by Freymann et al., who

report for sintered samples of U02: 1) the existence of a Debye-type

relaxation peak at low temperatures (1500K at 16 kHz); 2) large losses

at room temperature; and 3) thermal hysteresis of the room temperature

c" value. The low temperature Debye peak was ascribed to the relaxation

of the dipoles associated with crystalline defects, i.e., vacancies or

impurity ions. The high c" values at room temperature correspond to

D.C. conduction losses which become important when the free carriers

become thermally activated. The hysteresis effect was probably due to

the condensation and evaporation of traces of water vapor.
142
Wachtman, working with Th02, found that the presence of 1.5 mole

percent CaO did produce both mechanical and dielectric relaxation at

temperatures in excess of 5000K and frequencies in the kilohertz range.

The relaxation peaks were found to be due to the interchange of oxygen

ions and oxygen vacancies.





49



143
More recently, Tateno and Naito, have measured the dielectric

constant of UO2 as a function of temperature and frequency. The

extremely large values of the dielectric constant (105 at 20 kHz and

room temperature) reported by these workers are probably a result of the

specimen preparation technique or of the presence of adsorbed water.

















CHAPTER II

EXPERIMENTAL PROCEDURE



Material Characterization


The powders used in this investigation were supplied by the U.S.

Atomic Energy Commission. The material originated at the Mallinkrodt

Chemical Works where it was precipitated from solution as ammonium

diruanate, calcined to U03, reduced to U02, arc fused and crushed. A

chemical analysis of the powder is presented in Table II-1.

Preliminary attempts to sinter the as-received powder met with

limited success, even at temperatures in excess of 22000C. Therefore,

a program was undertaken to produce well-characterized powders of

reasonable sinterability. Parts of this program have been previously

described by Tuohig,72 but are included in the following summary for the

sake of completeness.

About 80 percent of the as-received material passed through a

270-mesh seive (U.S. Standard Series) but was retained on a 325-mesh

seive. Because of its abundance, this powder, designated powder lot 3,

was used as a primary source of material for most of the specimens

described in this dissertation. A scanning electron micrograph of this

material, Figure 2-lb, shows the characteristic granular equiaxed appear

ance of the particles. Figure 2-lb also shows a number of particles in

the 20 to 40 pm size range, and many small (1 pm and under) particles


















Table II-i

Analysis of Impurities in




Carbon

Aluminum

Boron

Cadmium

Chromium

Iron

Magnesium

Nickel

Silicon


the UO2 Powders




57 ppm

33

<0.1

<1.0

<5.0

70

<3.0

<3.0

33






























/





1J


Figure 2-1. Scanning electron micrographs of powders used in this study:
(a) lot 6; (b) lot 3; (c) lot 2; and (d) lot 1.


2t -'i









on the surfaces of the larger ones. The presence of these particles,

which are small enough to pass through the 45-im openings in a 325-mesh

sieve, is attributed to the tendency of UO2 to agglomerate. The

adherence of fine particles to larger ones and to each other is a

general feature of all powders used in this research.

Powder lot 3 was found to be easily sinterable to densities in

excess of 80 percent and was used to make some of the specimens described

in the later sections of this dissertation. In addition, this powder

was used as the starting material for a ball-milled and a water-settled

powder.

Ball milling was performed in a rubber-lined U.S. Stoneware mill

with Burundum* cylinders. The powder was wet milled for 16 hours. The

appearance of this powder, called powder lot 2 and shown in Figure 2-lc,

differs from that of the source material, in that the average particle

size is smaller, and the distribution of particle size is somewhat

broader. The estimated average caliper diameter for this material

is %5 pm.

A third powder was obtained by using a sedimentation technique to

remove the larger particles from ball milled powder. The ball milled

powder was first dispersed in water by adding about 20 ppm of a pigment

stabilizer, polyoxyethylene sorbitan monolaurate, to break up the

agglomerates. This slurry was mixed in a blender, poured into a 90-cm


*Carborundum Company









column and allowed to stand for 80 minutes. After this time, the

supernatant was decanted, and the fine powder recovered from this liquid

by evaporation. The settling time and height were calculated from

Stokes' Law to yield a maximum equivalent spherical diameter of 5 pm.

The resulting settled powder, called powder lot 1, is shown in

Figure 2-1d.

In addition to these powders, a number of other powder sizes were

used for specialized purposes in the course of this research. Of these,

the most important were:

(1) a (-270 +325) sieve cut which was a nearly monosized
fraction. This powder is designated as powder lot 5.

(2) powder lot 4, a (-325) fraction seived from the same
starting material as powder lot 4.

These powders were part of a special lot from which the electrical

properties samples were made. In order to minimize the pickup of impuri-

ties, these powders were not milled or settled during preparation.

Powder lot 4 was similar to powder lot 3 in particle size and sintering

characteristics. Powder lot 5 had a somewhat larger particle size than

Powder lot 4. A (-100 +200) seive fraction, powder lot 6, shown in

Figure 2-la, was used for some experiments. This extremely coarse

powder could be successfully hot pressed at temperatures above 15000C

and at a pressure of 5000 psi.

All of these powders were stored in closed containers prior to use.

Since, as discussed in Chapter I, the electrical properties of UO2

are extremely sensitive to the O/U ratio, both the powder and fired

specimens were further characterized by a stoichiometry determination.










To do this, the results of precision lattice parameter measurements were
144
compared with data of Schaner ,who measured the lattice constants of

samples of known O/U ratio.

The x-ray diffraction technique employed either a Norelco dif-

fractometer scanned at 0.5 degrees 20 per minute or a G.E. diffractometer

at 0.2 degrees 20 per minute. The 20 range from 70 to 145 was

examined with filtered copper Ka radiation from a fine-focus tube

operated at 35 KV and 15 mA. Annealed 99.999 percent gold fillings

were used as an internal standard. Mechanical backlash was eliminated

by averaging the peak positions obtained on two runs, increasing 20 on

one run and decreasing it on the other. These data were fitted by
145
Cohen's method of least squares; the analysis was performed with a
146
computer program supplied by Cvildys of Argonne National Laboratory.

Vacuum hot pressing is reported to have a negligible effect on the

O/U ratio, while heating in hydrogen is an effective way of reducing
147
UO to a composition which is usually termed "nearly stoichiometric."
2+x
Lattice constant measurements performed as part of the present study

indicated that the O/U ratio for the as-received powders and for vacuum-

hot-pressed specimens was 2.06. The O/U ratio was reduced to <2.005 by

sintering in H2 at temperatures of 1600 to 17000C for times of one hour

or more. Annealing times at 1600C in H2 of up to 20 hours were required

to obtain an equivalent reduction of vacuum-hot-pressed specimens.
148
Low-temperature gas-adsorption surface-area determinations were

used to further characterize the UO2 powders. Results of these experi-

ments are presented in Table 11-2.


















Table 11-2

Gas-adsorption Surface-area Determinations
of Three U02 Powder Lots




Powder Lot Surface Area


1 1.54 m2/g

2 0.87

3 0.1-0.3









Sample Preparation


Conventional Sintering

Throughout this dissertation the term "conventional sintering" is

used to denote the sintering of loose stacks of powder or of powder com-

pacts. Conventional sintering is distinguished from hot pressing or

pressure sintering in which a load is applied to the specimen at

temperature.

Because of the slow rate of sintering at experimentally accessible

sintering temperatures, loose-stack sintering produced only a limited

range of microstructures. As a result, almost all of the conventionally

sintered specimens were produced by compacting the powder into pellets

prior to sintering.

Powders were mixed with approximately 2 weight percent polyethylene

glycol* as a binder and lubricant prior to pressing into pellets in a

specially designed die mounted in a double acting Haller die table. The

die had a nominal diameter of 0.689 inches and a taper of 8 minutes per

inch. This equipment was capable of producing 0.75 inch tall pellets

which were free of density gradients. The useful operating pressure

range was from 10,000 psi to 80,000 psi. Lower pressures produced

fragile specimens, which usually crumbled when ejected from the die.

Higher pressures often yielded samples with laminar separations, caused

by excessive die wall friction during ejection.


*Union Carbide, Carbowax 4000.









A resistance-heated alumina muffle furnace* was used for sintering.

The pellets were heated to approximately 200C for one hour in a flowing

carbon dioxide atmosphere to volatilize the binder. The muffle was then

vacuum purged and filled with hydrogen before raising the furnace to the

sintering temperature at 1100C/minute. A flowing hydrogen atmosphere

was used for all sintering runs. Sintering temperature was limited to

approximately 18000C by the alumina muffle. The reported sintering

times include only the time at the sintering temperature; the heating

and cooling periods are not included.


Hot Pressing

Hot pressing was used to expand the attainable range of microstruc-

tures for these powders and to evaluate the effect on the path of micro-

structural change of applying pressure at high temperature.

Hot pressing was conducted in an induction heated vacuum hot press.

The powder was loaded into a die assembly shown schematically in

Figure 2-2. The main die cavity served as a susceptor and was made from

graphite. The liner was either boron nitride or graphite lined with

boron-nitride-sprayed molybdenum sheet. The graphite punches were capped

with either boron nitride disks or with boron-nitride-sprayed molybdenum.

Once in the die the powder was compacted to the pressure (1000,

000, or 5000 psi) that would later be applied at the hot pressing tem-

perature. This precompaction step was introduced to minimize the galling

of the die wall caused by large ram travels at high temperature, and to


*Astro Industries, Model 1000B.





























































Figure 2-2. Schematic cross section of the die used for hot pressing.









permit a check of the alignment of the components in the load train.

Since quantitative microstructural analysis was performed on unfired

pellets in the course of this research, the precompacted pellets had

well-characterized microstructures which served as the starting point

for microstructural evolution during hot pressing.

After the preload was released, the chamber was evacuated to a

pressure of approximately 1 x 10-5 Torr, and the die assembly was

heated to the desired temperature, where the load on the sample was

reapplied. Temperatures between 11500C and 20000C were used for times

ranging from 15 minutes to 2 hours. The load was then released and the

assembly allowed to cool under vacuum.

The load on the sample was kept constant to within +10 psi by a

modification of the hydraulic system of the hot press. A fluid-filled

reservoir connected, through an isolation valve, to the main hydraulic

cylinder, was pressurized with a two-stage-regulated gas supply. The

regulated pressure was equal to the pressure on the hydraulic ram, and

could be carefully controlled. The pressure was monitored with a gauge

with scale divisions of 1 psi.



Quantitative Metallography


Specimens for metallographic examination were cut in the planned

plane of observation, vacuum impregnated with Buehler epoxy resin, and

mounted in the same material. The mounts were ground on silicon carbide

metallographic papers through 600 grit. Fine polishing on a rotating

bronze lap was performed with Linde 0.3-and 0.05-pm aluminum-oxide

abrasives. The lap was covered with a nylon cloth for the 0.3-pm









abrasive and with Buehler microcloth for the 0.05-pm abrasive. The

polishing media were suspended in a 2 percent chromic acid solution.

This procedure gave a minimum of edge rounding. Grain "pullout" prob-

lems were minimized by the epoxy impregnation technique. If substan-

tial pullout occurred even with impregnation, the mount was reimpreg-

nated and repolished. Pullout was most commonly observed for very low

density samples.

A solution of 90 percent hydrogen peroxide (stabilized 30 percent)

and 10 percent sulfuric acid was used as a grain boundary etchant. Best

results were obtained by swabbing the specimen for about 45 seconds,

followed by immersion for one to two minutes. In all cases, the quanti-

tative measurements on the pore-solid interface were performed prior to

etching.

The metallographic preparation was checked by performing a point

count on the pore phase with the Quantimet 720 Image Analyzing Computer.

If the density calculated from the point count was within 2.5 percent of

the density as determined by the Archimedes immersion principle, the

polish was accepted and the other stereological measurements were per-

formed. If the density determined by the point count was outside the

2.5 percent limit, the specimen was reimpregnated with epoxy and

repolished.

The Quantimet, interfaced with a metallograph,* provided one of the

two major methods of structure characterization used in this research.

This instrument was used for the point count, the line intercept count

on the pore-solid interface, and the area tangent count.


*Bausch and Lomb, Inc., Research Hetallograph.










Manual counting, the other major characterization method, was used

for the grain boundary intercept, grain triple junction and inflection

point counts. Grain boundary intercepts were measured with a ruled

reticule in the eyepiece of the metallograph; triple junctions and

inflection points were counted on large photographic prints of the

microstructure.



Electrical Measurements


The powder preparation and stoichiometry determination methods used

for the electrical properties samples have already been discussed. After

sintering, these samples were centerless ground and sliced. A disk

about 5 mm thick and 15 mm in diameter taken from the center of the

specimen was used for electrical measurements. An adjacent slice was

mounted and prepared for metallographic examination. Electrodes were

attached by lightly clamping each sample in an aluminum holder which

masked all but a 1-cm area of each sample face. A thin (approximately

500 R) layer of gold was then vapor deposited on the exposed areas. The

samples were stored in a vacuum dessicator until used.

The specimen measuring chamber,* shown in Figure 2-3, held the

samples in place by pressing spring-loaded gold-palladium buttons

against the sample electrodes. The specimen chamber incorporated coax-
14
ially shielded platinum conductors and high resistivity (101 ohm-cm)

insulation. The chamber could be sealed, and evacuated or filled with

various atmospheres. In the present study, dry, high-purity helium

and air saturated with water vapor were used in the specimen chamber.


*Designed by D. Kinser and D. Jenkins.




63



























l


Figure 2-3. Specimen holder used for the electrical measurements.


P6~j










After the specimens had been transferred to the chamber, and before

measurement, helium was introduced and the temperature was raised to

200 C for 30 minutes. The heat served to drive adsorbed water vapor

from the specimen surfaces and to promote adherence of the gold elec-

trodes to the specimen. The extended 3000C heat treatment used by

Kinser149 to anneal vapor deposited electrodes was not used for the UO2

specimens because of the danger of their reacting with trace amounts of

oxygen and thus altering the stoichiometry.

The quality of the gold electrodes was checked by measuring the

electrical response of several samples at three test voltages with peak

amplitudes of 30, 3, and 0.3 volts. Since the measured capacities and

conductances were identical for all three voltage levels, the electrode-

specimen interface was judged to be ohmic.

The measurement circuit, shown schematically in Figure 2-4, con-

sists of two parallel systems, one for the audio frequency range (100 Hz

to 20 KHz) and one for radio frequencies (20 KHz to 1 MHz).

For audio frequencies, a Hewlett-Packard 651A oscillator supplies a

signal to the bridge circuit, which consists of the specimen, a Wayne-

Kerr B221 transformer-ratio-arm bridge, and a General Radio GR1232-A null

detector. At balance, the bridge reports the equivalent parallel con-

ductance and capacity of the sample.

For frequencies between 20 KHz and 100 KHz the measuring circuit is

the same except that a Wayne-Kerr B601 bridge is used. Because the null

detector is limited to a maximum frequency of 100 KHz, additional equip-

ment is necessary for higher frequencies. In the present system, a

General Radio 1232-P1 crystal mixer is used to mix the bridge output

with a signal from a Wayne-Kerr 022D video oscillator. The video


































I I DETECTOR

>20 KHz I
S1 20-I
I '--1- 100 KHz I
L._^ __ -J
RADIO
FREQUENCY
r-----" -- BRIDGE

I i I I
I OSCILLATOR CRYSTAL

1 0 1 MIXER
100 KHz 1 MHz


Figure 2-4. Block diagram of the apparatus used for the electrical
measurements.









oscillator is set for a frequency 100 KHz higher then the test signal.

The output of the crystal mixer contains a 100 KHz beat frequency as one

of its components. The null detector receives the beat component when

set for 100 KHz. When the bridge is balanced, the intermediate fre-

quency will show a null signal strength, which is indicated on the null

detector.

When balanced, the bridge readouts gave the conductance and

capacity of a hypothetical circuit, consisting of a resistor and capac-

itor in parallel, that would have the same response to the applied

sinusoidal signal as did the specimen. The equivalent parallel conduc-

tance and capacity were frequency-independent for tests performed with

high-purity helium in the specimen chamber but were frequency-dependent

if the atmosphere was air saturated with water vapor. The electrical

conductivities and dielectric properties of the specimens were calcu-

lated from the equivalent parallel conductances and capacities with the

following formulae.

Let the conductance, capacity, and measurement frequency be

denoted by G, C, and f, respectively. Assume that the specimen is a

circular disk with diameter d and thickness t. Let A represent the area

of one end of the disk, i.e., A = nd2/4. Then the electrical conduc-

tivity, o, is given by

= G (2-1)
A
-1 -1 -1
The units of o are ohm cm if G is measured in ohm t in cm, and A
2
in cm The real component of the complex dielectric constant is

denoted by e', and is calculated from

C t
S. (2-2)
Co A









The unit of capacity is the farad (F). The constant ,o' the permittiv-

ity of free space, is equal to 8.854 x 10-14 F/cm. The parameter E' is

dimensionless. Dielectric losses are expressed by the tangent of the

angle in the complex plane between the complex voltage vector, V*, and

the complex charge vector, Q*. The loss tangent is denoted by tan ,

and is calculated by


tan6 -= (2-3)
2nfC

The angle between the complex dielectric constant and the real axis is

also equal to 6. Therefore, the imaginary component of the dielectric

constant, e", is given by

E" = E' tan6 (2-4)

For experiments in which large D.C. conductivities are obtained,

the A.C. dielectric-loss parameters are usually calculated by replacing

the total conductance in Equation (2-3) with the alternating current

component of conductance, i.e.,

(G CDC)
tan (G DC (2-5)
AC 2nfC
and

A" = E' tan (2-6)
AC AC


where GDC is the D.C. conductance. For the UO2 specimens examined in

this study, the total conductance was a constant for frequencies between

100 and 1000 Hz. This constant value was used as the direct-current

conductance in Equation (2-5).
















CHAPTER III

THE METRIC PROPERTIES OF SINTERED AND HOT PRESSED UO2


The paths of microstructural change for cold compaction, conven-

tional sintering, and hot pressing are described in this chapter. The

variations of the metric properties, average properties derived from the

metric properties, and shape parameters are reported as functions of the

pore volume fraction and the grain contiguity. New relationships among

several of the metric properties were developed by analyzing the rela-

tionships among the metric properties and contiguity parameters.


The Path of Microstructural Change

The paths of microstructural change for cold compaction, conven-

tional sintering, and hot pressing are described in this section by

P
projecting the n-dimensional paths onto planes containing the VV axis

and the axis of the metric properties of interest. The metric proper-
oP oP an naP sce
ties considered are S M S L and L Results of the
V v' v' V V
inflection-point count for the pore-solid interface of a series of hot-

pressed specimens are reported. The metric properties, average prop-

erties, and the H A-P shape index of the pore-solid interface are listed

in Tables III-1 and III-2 for conventionally sintered and hot-pressed

specimens, respectively. Inflection point measurements for a series of

hot-pressed specimens are listed in Table III-3. The metric properties,

contiguity parameters, and shape indices of the grain structure are

listed in Tables III-4 and III-5.













-:'
O O


0N
C 0


o o
S0










I I
0: <0

C rn


, .-
-CD










o


-V
p0



















o
---I
U







C)



a)
























o


CD -13


O C
0 o
H- H-


0
H -
> C


0 0 0
,.. o o


H m







-m H







m c^
C') C')

I I
















. c4l
C'M C')



r--- '--
0 C'
c') CM
0 0


a)
u

C)






H 0







Q) -l
m o
a)L

CH







4 C
Io






S OH


4-


c l ) -)
H-i CM) C'n 'C a\ .0 I'-


U00 0 0 H~
H_ H


CD 0 '0 '0C C) C') 0 0






0 0




0 0 0 n C 0 0 0 o' Hl l Q)
H H H-1 C m O oo


r o

^ r


Co
a) 0





CC






U 44
H N
* a) --












,--t



o'3

















C I C CI


I( I


I I

I I 0


N)
m 00 rN- 'M ~- m


I rI


NL C) C C0 C)
lo 0 1Lf0 0\
r-i -1 14 14 c


p E U

0




d


-s



e-
N


0

H













e o
iH








0




0 O
0)
r-


C) CM CM
0 \0 C-


0 0 0


m a 0





r- m







-I I oI







r o 0 n
n n -c C

N N) NJ


C CD
O O


0 0 0 0 0 0 0 0


Q N fl >D N- N N N



H- -4 N)l Nr) Nr) Nr) C) Nr) Nf) W) N-) ,- N- -4









N ) M M ) C M ) CM C C0 C Nm CD CM M






NJ N NJ C N N N c N NJ N C NJ N


m N) ) 10 N-I 00 C, CD M CM CM M) M |)
CD CH H 0 CD cn NJ Ni C







-4 -4 -4 4 "


Mn o












o m
-I


'C



C
0r
U


H:
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-4



C


4 C


m N -4
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ES
0-l vY.









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a
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0 N

-4


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p













fa




c
0




to U







C-
0
u



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1-







t=M 0
T-


m 0
CN r-.

H H







M N
H 'N
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H- H-


NH
H0 0

0 o


Sl H-- H H-I N
C11 r\]


0 0 0
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.--4 m a


H- mo
ao


0 0 0


C N l H-


o mmoo0 o
0o 0 r H


0 00 0 D H H H- In 'I
4 H) CI C A m H Hn H H


r-4
,o



0
C


L 0 'l < o0 '0 H co 1 -4 H


I + t 1 I I I


I



SU








I O

0
1-


FP > >


EvU
H ^
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SN






0n
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0 >- ^<













ui

















1<









is




o




C)
(-'4
P u

















-E.
0


















1--
C>
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So





r'




'H
PI <
t= 0









3>
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0


i-l




i)
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CL, u
B
U1


10
')
10 LU) c0
\ H C
H H o






\a
(\1 0
C O 0

CN m o


m m

r- 10


0 CN
) 0-
on r-


o U

C') (-'


Nj 11 0


o o CD 0 C:) C3 C) C) CD
00 00 00 00 0 co 0 00 c)





'H -0 0 ) \ \D)
T- C .I ^ m r


CH C'





0 o
0 0


o o C 0 C)
0- 0 0 0 0D 0
'H o H '


iri ot00
U) U) U) U)


c0 ) rc C) O C) C o) C) O C) ) C o ) oC)
U) C') 'H 1 'H ) C') 00U C) c) 0) U0


N -I






0
C) fCi


-o 'T T -I U) U-) U)T U-1 ) V) U U) U )





co '-H m i) U) In 0- oo (31 o 'H 01 M U)


*0

C


r(.









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I I I I


C.











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(U
















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c N

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m e


Hq 10 U') U') H
'ci C) H H 1) 0
H 0) ci H H 'C
I I I I~u


o -' N
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O c O O q H H H H H H


ci


0





0


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Eco














u'l





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tI-


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S 0










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A) 0


a
moI


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iicS
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i CD 0- c- 0 H' H H 0- 0i 'C-



i in i)n i) in lo lo lo ID 'lo lo lo


0 0 0 0 0 0 CD
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n o
m ca
CN H
















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ci~i


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la





















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ri






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ci





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cn

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CO CO
m C


m cO

a c


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0 rn
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Ci r-I i-H 0
0 Ci 0 Ci


0 0


o co
0 v0
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Ci 0
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rCl r^


-i c I-i rH c- I m cn





c) m ci ci ri m ci ci cn cm Lf)





ci cq cA c "i i ci cq cs c14 c-i cq "





D cc cmi 0 -4 c4 ci -'t Li c co oo
c cc cc i i i i i -


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C C C 0


0 0 0 0 0 0 0 0 0
m m m cn m) mn cn m) f


a, rq ) 'T C o r- co ca o
C- oo co Co co co co oo co co co Ca c


a
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C4 C C NNN NNN N C -I






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HA H H H H H H m


0 0 0





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17 C14 Ir3
)- -





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co o





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0 CO 0


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0 0 0 0 0 0 0







o-co o


0O







cq







N-
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a)

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lO









Cold Compaction

The effect of cold compaction pressure on the pore volume fraction

of unfired compacts made from powder lot 2 is shown in Figure 3-1. A

linear relationship between the logarithm of the compaction pressure and

the pore volume fraction was obtained for samples pressed with and with-

out the addition of a 2 weight percent polyethylene glycol as a binder-

lubricant. The equations of least-squares regression lines through the

data are

VV' = 0.77 0.056 log p (3-1)


for samples pressed without a binder, and


VV' = 0.59 0.067 log p, (3-2)


for samples pressed with a binder, where p is the compaction pressure in

psi, and VV' is the pore volume fraction of the unfired compact. The

linearity of the VV' versus log p curve was first reported by

Bal'shin.150

Polished sections through low-density unfired pellets indicated the

presence of density gradients in the specimens pressed without a binder,

but not in specimens pressed with a binder addition. This result, along

with the high densities of pellets pressed with binder, indicates that

the lubricating quality of the polyethylene glycol helped the particles

to slide past one another, and promoted the breakup of multiple particle

"bridges" or "arches."

Examples of polished sections through two unfired pellets, pressed

without a binder addition, are shown in Figure 3-2. Quantitative

stereology measurements performed on the unfired compacts were used to

determine the path of pore-solid surface area change shown in Figure 3-3.











0.7-



0.6 -



0.5 WITHOUT
BINDER



0.4

0,O4\0 V'ITH BINDER

0.3



0.2 I I I
103 2 5 10 2 5 10

COMPACTION PRESSURE (psi)


Figure 3-1. Pore volume fraction versus log compaction pressure for
cold compaction of powder lot 2.































3ksi 10 VV0.48
10 rm
P---.-I


75 ksi VV =0.34


Figure 3-2. Polished-section micrographs of unfired compacts of powder
lot 2 pressed at 3,000 and 75,000 psi.
















15.0 3.0








0 E

0o 10.0- -2.0 o
0

U) -&--U)






5.0-- 10
0.5 0.4 0.3
P
V






caP cP
Figure 3-3. The variation of SV and S with pore volume fraction
during compaction of powder lot 2 specimens.









aP
The amount of surface area per unit volume of structure, S increased

during cold compaction; however, surface area per unit mass, S ,
g
72
remained nearly constant. This behavior, first observed by Tuohig,72

suggested that the dominant geometrical process during cold compaction

was a rearrangement of the powder particles that moved more particles,

and hence more surface area, into a unit volume of structure. The

latter interpretation of the results was confirmed by the examination of

the microstructures, Figure 3-2, which showed no evidence of particle
aP
fragmentation or cracking, processes that increase S (A third
g
process that can alter surface area during compaction, plastic deforma-

tion, does not occur in U02 at room temperature.) The apparent drop in

SaP at high pressures observed for the present data is the result of

extremely close packing of the smaller particles. These particles are

packed so close together that the gaps between them cannot be resolved

in the optical microscope.
aP aP
If S is a constant, S is a linear function of V, since
g V

(S ) = S 6(1 VP) (3-3)
V g oV
3 aP '
where 6 is the density of UO2, 10.96 g/cm ,and (S ) is the surface

area per unit volume in an unfired compact. The slope of the upper
aP
curve in Figure 3-3 is -6 S ~. The upper line shows the behavior of
og
aP
S for an idealized case in which only particle rearrangement occurs,

and all particles are resolved. The data point for an 80,000 psi com-

paction pressure lies below the line as a result of the inability to

resolve all of the interparticle gaps.









Because of the difficulty in resolving the small particles, no

attempt was made to determine the total curvature of unfired compacts.

However, when the compacts were heated, the fine particles quickly

sintered to the surfaces of the larger ones, as shown in Figure 3-4.

Therefore, resolution problems were only encountered for a few specimens.


Conventional Sintering

The effects of average particle size and compaction pressure of the

unfired specimens on the paths of microstructural change for sintering

are described in this section. Other fabrication parameters that might

have affected the path, such as sintering atmosphere and stoichiometry,

were held constant. Based on the results of an earlier study,4 the

sintering temperature was not expected to affect the paths of micro-

structural change. Although a systematic study of the potential effects

of temperature was not conducted in this investigation, the data in

Tables III-1 and III-4 do not show temperature effects. Note that the

range of temperatures used was limited (1600-17200C, or homologous tem-

peratures of 0.60-0.64) to reduce the probability that the paths of

microstructural change were affected by sintering temperature.

Pore-solid surface area. The paths of pore-solid surface area

change for two UO2 powders are shown in Figures 3-5 and 3-6. The micro-

structural state produced during cold compaction of a pellet is the

starting point for microstructural change during sintering. The cold

compaction curve for powder lot 2 specimens is indicated in Figure 3-5.





















.I)









-mom
(e)












(f)


Figure 3-4. Scanning electron micrographs
hot-pressed specimens.
P
(a) VV = 0.30 (d)

(b) VP = 0.27 (e)
v


(c) =
(c) V v = 0.21


(f)


of fracture surfaces through

P
S= 0.14

V = 0.12

V = 0.05


rri;~
- .~d
~
c 11










14.0

) 0 KSI
/ 1-8 ISI
12.0 A 10-20 KSI

\0 30-50 KSI
O 75-90 KSI
10.0



T 8.0



"L 6.0



4.0



2.0




0.6 0.5 0.4 0.3 0.2 0.1 0

P
V,



Figure 3-5. Pore-solid surface area versus pore volume fraction for
conventionally sintered specimens prepared from powder lot 2.













I
b 1-8 KSI
L 10-20 KSI
0 30-50 KSI
[3 75-90 KSI


0.4 0.3 0.2 0.1 0


Figure 3-6. Pore-solid surface area versus pore volume fraction for
conventionally sintered specimens prepared from powder lot 3.


5.0


4.0 I-


3.0 -


2.0 I-


1.0 -


0
0.5









The initiating points for the sintering curves of specimens pressed with

a binder were calculated from Equation (3-3) using the pore-volume

fraction data in Figure 3-1. Recall that surface area measurements of

unfired compacts, shown in Figures 3-3 and 3-5, were obtained for speci-

mens pressed without a binder. Surface area and volume fraction

decrease from their initial values as the pellet is sintered. The path

of surface area change for the entire densification process was influ-

enced by the cold compaction pressure. Increasing the pressure shifts
aP P
the S -V curves to higher densities and surface areas. As a result,

the paths of surface area change form an envelope of curves in S -V
V V
P aP
space. For a given value of V in the range 0.10-2.25, the SP value of

the sintered specimen can be varied by a factor of 2 by selection of the

compaction pressure.

The envelope of curves collapses at high densities and approaches

the straight line drawn from the origin to the points that represent the

sintering of uncompacted powder. The slopes of the individual curves

decrease and approach the slope of the straight line. Envelopes of

curves with similar characteristics have been used to describe the paths

of surface area change for sintered Cu71 and UO72 in earlier studies.

Total curvature. The evolution of total curvature during conven-

tional sintering is shown for powder lots 2 and 3 in Figures 3-7 and

3-8, respectively. Most of the curves depict only a portion of the path

of curvature change for a particular set of fabrication conditions. How-

ever enough information is present to allow the following description.




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