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Multi-chip module interconnections at microwave frequencies: electromagnetic simulation and material characterisation

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ABSTRACT
T itle Of Dissertation:
COMPLEX PERMITTIVITY MEASUREMENTS
AND MIXING LAWS OF CERAMIC
MATERIALS AND APPLICATION
TO MICROWAVE PROCESSING
David Louis G ershon, Doctor of Philosophy. 1999
D issertation directed by:
Professor Thom as Antonsen, Jr.
D ep artm en t of Physics
T h e complex perm ittivity of alum ina composites was exam ined w ith respect to
its dependence on th e volume fraction of constituents, m icrostructure, processing
tem p eratu re, and processing m ethod.
In addition, th e effective p erm ittiv ity of
these com posites was q u an titativ ely modeled based on th e perm ittivities, volume
fractions, and m icrostructures of the constituents.
T h e studies focused on th e com plex perm ittivity of alum ina com posites, which
contained th e lossy additives silicon carbide and copper oxide. Tw o com posite
system s were prepared by physically mixing alum ina and one of th e additives. A
th ird com posite system was produced by chemically precipitating copper oxide
onto alum ina.
The two synthesis m ethods produced com posites w ith different
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m icrostructures and complex perm ittivities. T h e im aginary part of the com plex
p erm ittiv ity was generally larger in th e chem ically precipitated composites th a n
in th e physically mixed composites.
T h e dependence of the complex perm ittivities of th e composites on volume frac­
tio n an d m icrostructure were com pared w ith several algebraic mixing laws an d w ith
th ree dim ensional, electrostatic num erical sim ulations. T he algebraic m ixing laws
do n o t tak e into account for the dependence of th e im aginary part of th e com plex
p erm ittiv ity on absorbed w ater and m icrostructure, which is affected by com posite
synthesis. B y incorporating general physical characteristics of the composites, th e
electrostatic sim ulations were able to accurately predict their perm ittivity.
H eating some selected alum ina com posites in conventional and microwave fur­
naces dem o n strate several interesting results. T h e densification and dielectric prop­
erties of th e alu m in a/ copper oxide com posites varied due to processing tem per­
ature. T he changes in these properties depended upon preparation m ethod an d
not on heating m ethod. T he density an d real p art of the complex p erm ittiv ity
of a lu m in a / silicon carbide also varied due to processing tem perature an d not. on
h eating m ethod. Interestingly, th e im aginary p art of th e complex perm ittiv ity of
a lu m in a / silicon carbide did depend
011
h eatin g m ethod. T he electrostatic sim ula­
tions were found to be of lim ited value in predicting th e perm ittivity when th ere is
a lack of d a ta of the volume fraction or p e rm ittiv ity of minor constituents, which
co n trib u te significantly to the overall effective perm ittivity.
Several dielectric measurem ent teclm iques were specifically developed for this
research.
A stainless steel open- ended coaxial probe accurately m easured th e
com plex p erm ittiv ity of solid dielectric m aterials up to 1000C and over a broad
frequency range of 0.3 to
6
GHz. T he p ro b e’s insensitivity to low loss m ateri-
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als constrained accurate dielectric m easurem ents of m aterials w ith a loss tangent
g reater th a n 0.05. A nondestructive resonant cavity was developed to m easure the
dielectric properties of low loss m aterials w ith variable dimensions.
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COMPLEX PERM ITTIVITY MEASUREMENTS
AND MIXING LAWS OF CERAMIC
MATERIALS AND APPLICATION
TO MICROWAVE PROCESSING
by
David Louis Gershon
D issertation subm itted to the F aculty of th e G raduate School of the
University of M aryland. College P ark in partial fulfillment
of the requirem ents for th e degree of
D octor o f Philosophy
1999
A dvisory Committee:
Professor T hom as A ntonsen. Jr.. C hairm an/A dvisor
Associate Professor S teven Anlage
Research Scientist Jeff Calam e
Research Scientist Yuval Carm el
Professor C hristopher J. Lobb
Assistant Professor O tto W ilson. Jr.
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UMI Number: 9942963
Copyright 1999 by
Gershon, David Louis
All rights reserved.
UMI Microform 9942963
Copyright 1999, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized
copying under Title 17, United States Code.
UMI
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Ann Arbor, MI 48103
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© Copyright by
David Louis Gershon
1999
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DEDICATION
To m y parents for their unconditional love and support.
ii
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ACKNOWLEDGEMENTS
I w ould like to gratefully acknowledge the wisdom, guidance, patience,
an d encouragem ent of my advisors Thom as A ntonsen, Jeff Calame,
Yuval Carm el, Isabel Lloyd, and O tto W ilson.
I also would like to
express my appreciation to my other com m ittee m em bers, C hris Lobb
an d Steven Anlage, for their interest in my research and th eir sugges­
tions.
1 like to th an k K en Diller and Victor Yun for th e ir assistance, advice,
and encouragem ent in designing and m achining experim ental equip­
m ent and settin g up th e laboratory. In addition, I really appreciate
th e advise and su pport from Tayo Olorunyolemi, A m ikam Birnboim,
A ndrew Case, Jo h n Curry, K en Hutchenson, V italy Talansky, Shankar
V enkataram ani, and Jim Weaver. I also would like to th a n k th e fol­
lowing IP R personnel: A m non Birman, N olan Ballew, C arol Bellamy,
D orothea F . Brosius, Ed Condon, Alan DeSilva, R ichard Ellis, Ray­
m ond Elton, H ans Griem. M argaret Hess, B art H ogan, J u a n ita Irving,
S ato ru Kobayashi. B aruch Levush, Don M artin, M a tt N aim an, Gregory
Nusinovich, Jo n Orloff, Jay Pyle, John Rodgers, E van P ert, K athleen
Santangelo, Janice Schoonover, M ark W alter and G engfu X u. I also ap­
preciate Jo h n Bimier, Tom Cross, and Neal G reenacre a t th e U niversity
iii
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o f N o ttin g h am (UI<) for hosting m e in their laboratories and introduc­
ing me to dielectric m easurem ents: Ron H utcheon for perform ing high
te m p e ra tu re resonant cavity m easurem ents on ceramic samples; Mike
Janezic a t N IST and Sanjaya R a ja p a trin a a t U M C P for assisting on th e
p robe model; Jam es B ooth for allowing me access to his experim ental
equipm ent an d suggestions: S tan M orrow at O ak Ridge N ational L abo­
ra to ry for his suggestions and encouragem ent; D an Young for assisting
on th e X RD experim ents; Jan et Q uinn for polishing samples and as­
sisting on processing; Jay W allace a t N IST for advice and use of his
equipm ent; an d K im berly Brown for helping w ith F T IR analysis. I ap­
p reciate U.S. D ep artm ent of E nergy and U.S. A ir Force for providing
su p p o rt for th e la st five years and th e Physics D epartm ent for provid­
ing m e w ith a Teaching A ssistantship.
I offer much th an k s to my parents, my sister, an d other family m em bers
for their su p p o rt, encouragem ent, and love. I would also like to th a n k
all m y teachers th ro u g h out m y academ ic career. T hey m ade learning
fun an d exciting.
I would like to conclude w ith an inspirational quote th a t helped m e
o n m y educational p ath .
’’B u t where does th e power come from,
To see th e race to its end?
It comes from w ith in.”
W . J. W eather by
C hariots of F ire
iv
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TABLE OF CO NTENTS
L ist o f T a b le s
v ii
L ist o f F ig u r e s
v iii
1
I n tr o d u c tio n
1
2
B a c k g r o u n d In fo r m a tio n
2.1 S in te r in g .............................................................................................................
2.2 D ielectric P r o p e r t i e s .....................................................................................
2.3 Microwave P ro c e ssin g .....................................................................................
2.4 Differences in Microwave and Conventional P ro cessin g .......................
2.5 Microwave Effect an d Field I n te n s if ic a tio n ............................................
6
7
10
12
14
O p e n -E n d e d C o a x ia l P r o b e for H igh - T e m p e r a tu r e an d B road B a n d D ie le c tr ic M ea su r e m e n ts
3.1 I n tr o d u c tio n ......................................................................................................
3.2 P ro b e D e s i g n ..................................................................................................
3.3 T h eo ry an d C a lib ra tio n .................................................................................
3.4 R esults an d D is c u s s io n .................................................................................
3.5 O ptim izing th e P robe's Sensitivity ..........................................................
3.6 S um m ary an d C o n c lu s io n s ...........................................................................
18
18
26
28
36
56
61
3
4
6
D ie le c tr ic M ix in g L aw s A p p lied to A lu m in a C o m p o site s
62
4.1 I n tr o d u c tio n ...................................................................................................... 62
4.2 D ielectric M ixing L a w s ................................................................................. 65
4.3 Sam ple P rep aratio n and C h aracterizatio n ............................................... 93
4.3.1 Sam ple P re p a ra tio n ........................................................................... 94
4.3.2 M aterial characterization of alu m in a/ copper oxide composites 97
4.4 C om plex P erm ittiv ity M e a s u r e m e n ts .........................................................103
v
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4.5
4.6
Results and D is c u ss io n ......................................................................................109
C o n c lu s io n s .......................................................................................................... 128
5
M icrow ave a n d C o n v en tio n a l P r o c e ss in g o f A lu m in a C o m p o site s 130
5.1 I n tro d u c tio n ..........................................................................................................130
5.2 Experim ental P r o c e d u r e .................................................................................. 139
5.2.1 Conventional and Microwave H e a tin g .............................................139
5.2.2 Dielectric M e a s u re m e n ts .....................................................................140
5.3 Results an d D is c u s s io n ..................................................................................... 141
5.3.1 A lu m in a/ Silicon Carbide C o m p o sites.............................................141
5.3.2 A lu m in a/ Copper Oxide C o m p o s ite s .............................................151
5.4 Sum m ary and C o n c lu sio n s...............................................................................167
6
F u tu re W ork
Complex P erm ittivity
6.2 Complex P erm ittivity
6.3 Chemical P rep aratio n
6.4 Improve T em perature
6 .1
169
M easurem ents at E levated T em peratures . . 169
M easurem ents using a R esonant C avity . . . 173
of A lum ina C o m p o s ite s .........................................174
M easurements: B lackbody R a d i a t o r ................ 175
A
C o m p u ter M o d e lin g o f O p en - E n d e d C o a x ia l P r o b e
178
B
D ie le c tr ic M ix in g Law s
190
C
M a teria ls C h a ra c teriza tio n
194
C .l Transm ission Electron M icroscopy ................................................................ 194
C.2 Therm ogravim etric A n a ly s is ...........................................................................194
G'.3 X-ray D i f f r a c tio n ...............................................................................................200
D
R eso n a n t C a v ity
E M icrow ave S y ste m :
C o n tro l S y s te m
204
F u rn ace, T em p era tu re M e a su r e m e n ts, a n d
212
vi
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LIST OF TABLES
3.1
E rro r Sources for Reflection C o e f f ic ie n t...................................................
4.1
P erm ittiv ity of phases in the following sim ulations...............................
69
4.2
Com position and Synthesis of A lum ina C o m p o s ite s .........................
96
4.3
Infrared absorption peaks of copper carbonate oxalate, which
fore c a lc in a tio n
56
is be­
1 0 0
4.4
X -ray diffraction analysis of chem ically precipitated copper oxide . . 101
4.5
X -ray diffraction analysis of chem ically precipitated copper oxide
onto a l u m i n a ....................................................................................................... 1 0 2
A .l
Convergence of Reflection Coefficient for specified param eters . . . 188
A.2
Convergence of Reflection Coefficient for specified param eters . . . 189
vii
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LIST OF FIG U R ES
3.1
Real p a rt of th e dielectric constant of ZnO sam ples sintered in mi­
crowave an d conventional furnances vs. fre q u e n c y ................................
3.2
Im aginary p art of the dielectric co n stan t of ZnO samples sintered
in microwave and conventional furnances vs. fre q u e n c y ......................
3.3
‘2 0
2 1
Schem atic diagram of a high te m p e ra tu re open- ended coaxial probe.
A spring loaded inner conductor forces th e inner and outer conducters to m aintain intim ate contact
3.4
w ith m aterial under te st..................
C alculation of th e effective a' a t th re e different frequencies of a ZnO
sam ple vs. air gap thickness............................................................................
3.5
24
C alculation of th e effective a" at th re e different frequencies of ZnO
vs. air gap thickness...........................................................................................
3.6
23
25
T h e open- ended coaxial probe contacting a m ultilayer sam ple. Re­
gion 2 represents the air gap layer a n d region 3 represents th e sam ­
ple.
A lthough region 4 represents th e m aterial term ination, th e
sam ple was assum ed to be semi- infinite, L—►
oc.....................................
3.7
30
C ontour plot of th e m agnitude and phase of th e reflection coefficient
over a range in a' and loss tangent a t a frequency of 915 MHz. . . .
viii
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31
3.8
C ontour plot of the calculated m agnitude and phase of th e reflection
coefficient over a range in s ' an d losstangent at a frequency of 2.45
GHz. No air gap and semi- infinite m aterial. L—+ oc.....................
3.9
32
M agnitude of th e error p aram eters for th e high tem perature dielec­
tric probe vs. f re q u e n c y .......................................................................
37
3.10 U ncertainty of the reflection coefficient for calibration param eters
an d ZnO samples vs. f re q u e n c y .........................................................
38
3.11 U ncertainty of the reflection coefficient for calibration param eters
an d ZnO samples vs. f re q u e n c y .........................................................
39
3.12 Percent uncertainty of com plex p erm ittiv ity measured by the probe
at 956 MHz. For a specified com plex perm ittivity, the percentage
uncertainty in z‘ is indicated by solid line contour and in z" is indi­
cated by dashed line contour.................................................................
40
3.13 Percent uncertainty of com plex p erm ittiv ity measured by the probe
at 2.425 GHz. For a specified complex perm ittivity, th e percent­
age uncertainty in s' is indicated by solid line contour and in z" is
indicated by dashed line contour..........................................................
41
3.14 Complex perm ittivity of ZnO (conventionally sintered to p = 53.7%)
m easured by a high tem p eratu re an d H P probes vs. frequency. . . .
43
3.15 Com plex perm ittivity of ZnO (conventionally sintered to p = 72.5%)
m easured by a high tem p eratu re and H P probes vs. frequency. . . .
44
3.16 Com plex perm ittivity of ZnO (microwave sintered to p — 95.0%)
m easured by a high tem p eratu re and H P probes vs. frequency. . . .
ix
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45
3.17 Real p a rt of the complex p erm ittiv ity of Nephelene Syenite mea­
sured vs. tem perature by a resonant cavity (M PN ) and high- tem ­
p eratu re probe at 912 M H z.............................................................................
48
3.18 Im aginary p art of the com plex p erm ittiv ity of Nephelene Syenite
m easured vs. tem perature by a resonant cavity (M PN) an d hightem p eratu re probe at 912 M H z......................................................................
49
3.19 Real p a rt of the complex p erm ittiv ity of Nephelene Syenite mea­
sured vs. tem perature by a resonant cavity (M PN) and high- tem ­
p eratu re probe at 2.46 G H z............................................................................
50
3.20 Im aginary p a rt of the com plex p erm ittivity of Nephelene Syenite
m easured vs. tem perature by a resonant cavity (M PN) an d hightem p eratu re probe at 2.46 G H z.....................................................................
51
3.21 Real p a rt of the complex perm ittiv ity of AI9 O 3 / CaO + fine SiC
grains m easured vs. tem p eratu re by a resonant cavity (M PN ). hightem p eratu re probe, and H P probe (only at room tem perature) at
912 M Hz................................................................................................................
52
3.22 Imaginary- p art of the com plex perm ittivity of AI2 O 3 / CaO + fine
SiC grains m easured vs. tem p eratu re by a resonant cavity (M PN ),
high- tem p eratu re probe, and H P probe (only a t room tem perature)
at 912 M Hz...........................................................................................................
53
3.23 Real p a rt of the complex perm ittiv ity of AI2O3/ CaO + fine SiC
grains m easured vs. tem p eratu re by a resonant cavity (M PN ), hightem p eratu re probe, and H P probe (only a t room tem p eratu re) at
2.46 G H z...............................................................................................................
x
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54
3.24 Im aginary p art of th e complex p e rm ittiv ity o f AI2O3/ CaO + fine
SiC grains m easured vs. tem p eratu re by a resonant cavity (M PN ),
high- te m p eratu re probe, and H P probe (only a t room tem perature)
a t 2.46 G H z...........................................................................................................
55
3.25 C ontour plot o f th e m agnitude and phase of th e reflection coefficient
over range in e7 an d loss tangent a t a frequency of 2.45 GHz. T h e
im pedance of th e probe is 16 fi. T he d iam eter of th e inner conductor
is 4.0 m m an d of th e outer conductor is 5.3 m m ...................................
59
3.26 C ontour plot of th e m agnitude and phase of th e reflection coefficient
over range in e7 an d loss tangent a t a frequency of 2.45 GHz. T h e
im pedance of th e probe is 50 Q. T he diam eter of the inner conductor
is 5.7 m m an d of th e outer conductor is 13.3 m m .................................
4.1
60
C alculated e' for a two phase system vs. th e volume fraction of th e
second phase. T h e perm ittivities of th e phases are e*(ls£ phase) =
1.00055 — 10—°_7* an d e'"(2nd phase) = 2 — 0
4.2
.0 0 2 7
.....................................
71
C alculated s" for a two phase system vs. th e volume fraction of th e
second phase. T h e perm ittivities of th e phases are e*(ls£ phase) =
1.00055 — 10~°j an d e'(2 nd phase) = 2 — 0 .0 0 2 j......................................
4.3
72
C alculated s ' for a two phase system vs. th e volume fraction of th e
second phase. T h e perm ittivities of th e phases are e*(ls£ phase) =
1.00055 — 1 0 ~ ° 7 an d e’ (2nd phase) = 10 — O.OI7 ......................................
4.4
73
C alculated e" for a two phase system vs. th e volume fraction of th e
second phase. T h e perm ittivities of th e phases are e*(ls£ phase) =
1.00055 — 1 0 _
°7
a n d e'(2 nd phase) = 10 — O.OI7 ......................................
xi
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74
4.5
C alculated s" for a two phase system vs. th e volume fraction of the
second phase. The perm ittivities of th e phases are e*(ls£ phase) =
1.00055 - 10~5j and e'(2nd phase) = 10 - 0.02j ........................................
4.6
75
C alculated s" for a two phase system vs. th e volume fraction of the
second phase. The perm ittivities of th e phases are e*(ls£ phase) =
1.00055 — 10~°./ and e*(2nd phase) = 10 — 0 .1 /...........................................
4.7
76
C alculated e! for a two phase system vs. th e volume fraction of the
second phase. The perm ittivities o f th e phases are e*(ls£ phase) =
1.00055 - 10~3j and e.'(2nd phase) = 20 - 0 .0 2 ;......................................
4.8
77
C alculated e" for a two phase system vs. th e volume fraction of the
second phase. The perm ittivities of th e phases are e*(ls£ phase) =
1.00055 - 10~5j and em{2nd phase) = 20 - 0.02j ......................................
4.9
78
C alculated s ' for a two phase system vs. th e volume fraction of the
second phase. The perm ittivities of th e phases are e*(lst phase) =
1.00055 — I0 ~ °j and em(2nd phase) =
50 — 5j ........................................
79
4.10 C alculated s" for a two phase system vs. th e volume fraction of the
second phase. The perm ittivities of th e phases are e*(ls£ phase) =
1.00055 — 10~°j and e*(2nd phase) =
50 — 5j ........................................
80
4.11 C alculated cf for a three phase system vs. th e volume fraction of
th e th ird phase. The volume fraction of phase
a t 40.75c.
1
T he perm ittivities of th e phases are
is held constant
e*(ls£ phase) =
1.00055 — 10-0,). e*(2nd phase) = 10 — O.OI7 . and e’ (3r<i phase) =
50 — 5j ..................................................................................................................
xii
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81
4.12 C alculated s" for a three phase system vs.
th e volume fraction
of th e th ird phase. T h e volume fraction o f phase 1 is held con­
stan t a t 40.7%. The perm ittivities of th e phases are e’ ( l s£ phase) =
1.00055 - 10~sj , e*(2nd phase) = 10 - O.Olj, and e*(3rd phase) =
5 0 - 5 j ...................................................................................................................
82
4.13 Slice through the center of the model space for spheres arranged in
a body- centered- cubic lattice.......................................................................
85
4.14 C alculated s' for a two phase system vs. s' o f th e second phase. The
volume fraction of phase 1 (air) is held co n stan t at 41.1%........................ 8 8
4.15 C om puted s' for a three phase system vs. th e volume fraction of
the th ird phase. T he volume fraction of phase 1 is held constant
at 31.93%. T he perm ittivities of the phases
are e*(ls£ phase) =
1.00055 — 10-0.;, e*(2nd phase) = 10 — O.Oly.
and e*(3rd phase) =
50 — 5j ...................................................................................................................
91
4.16 C om puted s" for a three phase system vs. th e volume fraction of
the th ird phase. T he volume fraction of phase
1
is held constant
at 31.93%. The perm ittivities of the phases
are e*(ls£ phase) =
1.00055 — 10~°j. e*(2nd phase) = 10 — O.Olj,
and e*(3rd phase) =
50 — 5 j ...................................................................................................................
92
4.17 TEM m icrograph showing m orphology o b tain ed by coating copper
oxalate carbonate onto alum ina particles. T h e calcined sam ple is
composed of 83% (mass) AI0O3 + 17% copper oxide..................................98
4.18 Schematic diagram of the resonant cavity w ith a moveable wall. . .
106
4.19 P icture of the vector network analyzer connected to the resonant
cavity w ith a moveable wall...............................................................................107
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4.20 Circular cavity w ith a dielectric sample in region
Region 2 is filled w ith e2 an d
1
w ith eLan d fil .
....................................................................... 107
4.21 Real part of th e com plex perm ittiv ity of a lu m in a / silicon carbide
composites was m easured by th e nondestructive resonant cavity at
2.59 GHz vs. concentration of silicon carbide. For each concentra­
tion of silicon carbide, two sam ples were prepared a n d m easured. . . I l l
4.22 Im aginary p art of th e com plex perm ittivity of a lu m in a / silicon car­
bide composites was m easured by th e nondestructive resonant cavity
a t 2.59 GHz vs. co ncentration of silicon carbide. For each concen­
tratio n of silicon carbide, two sam ples were prepared an d m easured.
1 1 2
4.23 Real p art of th e com plex p erm ittiv ity of a lu m in a / copper oxide
composites was m easured by th e nondestructive resonant cavity at
2.66 GHz vs. concentration of copper oxide. For each concentration
of copper oxide, two sam ples were prepared and m easured..................... 113
4.24 Im aginary part of th e com plex perm ittivity of a lu m in a / copper ox­
ide composites was m easured by th e nondestructive resonant cavity
a t 2.66 GHz vs. concentration of copper oxide. For each concentra­
tion of copper oxide, two sam ples were prepared and m easured.
. . 114
4.25 Volume Fraction of A lum ina an d Air vs. Silicon C arbide w ithin
A lum ina/ Silicon C arbide Com posites. Two sam ples were prepared
for each concentration of silicon carbide.........................................................116
4.26 Volume Fraction of A lum ina and Air vs. C opper O xide w ithin Alu­
m in a/ Copper Oxide Com posites. Two samples were prepared for
each concentration of copper oxide.................................................................. 117
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4.27 M easured an d calculated s ' of a lu m in a / silicon carbide composites
a t 2.59 GHz vs. volume fraction of silicon carbide..................................... 120
4.28 M easured an d calculated s" of a lu m in a / silicon carbide composites
a t 2.59 GHz vs. volume fraction of silicon carbide.....................................121
4.29 M easured an d calculated s' of a lu m in a / copper oxide composites at
2.66 GHz vs. volum e fraction of copper oxide............................................ 123
4.30 M easured an d calculated e" of a lu m in a / copper oxide com posites at
2.66 GHz vs. volume fraction of copper oxide............................................ 124
4.31 C om puted s ' of alum ina particles coated w ith copper oxide vs. vol­
um e fraction of copper oxide. M onolayer encapsulation of alum ina
spheres occurs a t ~10% of copper oxide.........................................................126
4.32 C om puted e" of alum ina particles coated w ith copper oxide vs. vol­
um e fraction of copper oxide. M onolayer encapsulation of alum ina
spheres occurs a t ~10% of copper oxide.........................................................127
5.1
D iagram of th e microwave sam ple crucible. T h e alum ina sample
has a diam eter of 2.84 cm and a height of 1.09 cm or 2.60 cm. The
sam ple is surrounded by alum ina bulk fiber. T h e insulating crucible
is m ade o u t of porous alum ina b oard (6 % theoretical density). Its
has an o u te r diam eter of 7.62 cm. an inner diam eter of 5.08cm, an
outer height of 15.24 cm, and inner height of 10.16 cm............................ 135
5.2
N um erical calculation of th e m axim um core tem perature of an alu­
m ina com posite w ith respect to its im aginary p art of the complex
perm ittivity. T h e applied power is 1500W a t 2.45 GHz. The sam ple
diam eter is 2.84 cm and thickness, h, is indicated ......................................136
xv
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5-3
D ensity of 80% alum ina + 20% silicon carbide com posites vs. max­
im um processing tem perature in a conventional and microwave fur­
naces w ith a nitrogen atm osphere. T h e uncertainty in density mea­
surem ents is ju st less th an
1
%. M easurem ents were perform ed at
room tem p eratu re...........................................................................................
5.4
Real p art of the complex p erm ittiv ity of 80% alum ina -I- 20% silicon
carbide composites was m easured by th e nondestructive resonant
cavity a t 2.76 GHz vs. processing tem p eratu re in conventional and
microwave furnaces w ith a nitrogen atm osphere. T h e average un­
certainty in m easuring e' is ~1%. M easurem ents were performed at
room tem perature...........................................................................................
5.5
Im aginary part of th e complex p erm ittiv ity of 80% alum ina + 20%
silicon carbide composites was m easured by the nondestructive reso­
nant cavity at 2.76 GHz vs. processing tem perature in conventional
and microwave furnaces w ith a nitrogen atm osphere. T he average
uncertainty in m easuring e" is ~ 1 %. M easurem ents were performed
a t room tem perature.....................................................................................
5 .6
M easured Complex P erm ittivity of 80% A lum ina + 20% Silicon Car­
bide by th e high - tem perature open- ended coaxial probe at 2.425
GHz vs. tem perature in a conventional furnace w ith a 95% No +
5% H 2 atm osphere..........................................................................................
xvi
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5.7
Real part of th e complex p erm ittiv ity o f 82.8% alum ina + 17.2%
copper oxide com posites was m easured by th e nondestructive res­
onant cavity an d a commercial open- en d ed coaxial probe at
2 .6 6
GHz vs. m axim um processing te m p e ra tu re in conventional and mi­
crowave furnaces w ith an air atm osphere. C om posites were synthe­
sized by physical mixing and chem ical precipitation. T h e average
uncertainty in m easuring d is
~ 1
%. M easurem ents were performed
a t room tem p eratu re............................................................................................ 152
5.8
Im aginary p art of th e complex p e rm ittiv ity of 82.8% alum ina +
17.2% copper oxide composites was m easured by the nondestruc­
tive resonant cavity and a commercial open- ended coaxial probe at
2.66 GHz vs. m axim um processing te m p e ra tu re in conventional and
microwave furnaces w ith an air atm osphere. Com posites were syn­
thesized by physical mixing and chem ical precipitation. T he average
uncertainty in m easuring e" is ~1 %. M easurem ents were performed
a t room tem p eratu re............................................................................................153
5.9
D ensity of 82.8% alum ina + 17.2% copper oxide composites vs.
m axim um processing tem perature in a conventional and microwave
furnaces with a n air atm osphere. C om posites were synthesized by
physical mixing an d chemical precipitation. T h e uncertainty in den­
sity m easurem ents is ju st less th a n
1
%. M easurem ents were per­
formed at room tem p eratu re..............................................................................154
xvii
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5.10 Real p art of the complex perm ittivity of 85.1% alum ina + 14.9%
copper oxide com posites vs. tem perature by a resonant cavity (MPN)
a t 2.46 GH z. Com posites were synthesized by physical mixing and
chemical precipitation.......................................................................................... 156
5.11 Im aginary p a rt of the complex p e rm ittiv ity of 85.1% alum ina +
14.9% co p p er oxide composites vs. tem p e ra tu re by a resonant cav­
ity (M PN ) a t 2.46 GHz. Composites were synthesized by physical
mixing an d chemical precipitation....................................................................157
5.12 T his figure is expanded scale of fig. 5.11. Im aginary p art of the
complex p erm ittiv ity of 85.1% alum ina + 14.9% copper oxide com­
posites vs.
tem p eratu re (less th an
1 0 0 0
°C) by a resonant cavity
(M PN) a t 2.46 GHz. Composites were synthesized by physical mix­
ing and chem ical precipitation...........................................................................158
5.13 X -ray diffraction of 82.8% alum ina + 17.2% copper oxide composites
heated to 1100°C in conventional and microwave finances w ith an
air atm osphere. Composites were synthesized by physical mixing.
. 162
5.14 X -ray diffraction of 82.8% alum ina + 17.2% copper oxide composites
heated to 1100°C in conventional and microwave finances w ith an air
atm osphere. Com posites were synthesized by chemical precipitation. 163
xviii
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5.15 R eal p art of the complex p erm ittiv ity o f 82.8% alum ina + 17.2%
copper oxide composites was m easured by th e nondestructive res­
onant cavity and a commercial open- ended coaxial probe at
2 .6 6
GHz vs. maximum processing tem p eratu re in conventional and mi­
crowave furnaces w ith an air atm osphere. Composites were synthe­
sized by physical mixing and chemical precipitation. M easurem ents
were perform ed at room tem p eratu re............................................................. 165
5.16 Im aginary p art of th e com plex perm ittiv ity of 82.8% alum ina
17.2% copper oxide composites was m easured by the nondestruc­
tive resonant cavity and a com m ercial open- ended coaxial probe
at
2 .6 6
GHz vs. m axim um processing tem p eratu re in conventional
an d microwave furnaces w ith an air atm osphere. Composites were
synthesized by physical mixing and chemical precipitation. M ea­
surem ents were perform ed at room tem p eratu re.........................................166
6 .1
Side profile of sample in contact w ith a lid sample. T he outline
region in the lid sample indicates cutout region..........................................177
A .l M agnitude of the reflection coefficient for different num ber of modes
w ith respect to number of integration points............................................... 182
A.2 M agnitude of the reflection coefficient for different num ber of inte­
gratio n points with respect to num ber of m odes.........................................183
A.3 P hase of th e reflection coefficient for different num ber of modes w ith
respect to number of integration points......................................................... 184
A.4 P hase of the reflection coefficient for different num ber of integration
points w ith respect to num ber of m odes........................................................185
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A.5 M agnitude of the reflection coefficient for different num ber of modes
w ith respect to upper lim it of in teg ratio n ......................................................186
A.6
Phase of th e reflection coefficient for different num ber of m odes w ith
respect to upper lim it of in teg ratio n ............................................................... 187
C .l T E M m icrograph showing m orphology obtained by coating copper
oxalate carbonate onto alum ina particles. T he calcined sam ple is
composed of 56.896 (mass) AI2 O 3 4- 43.296 copper oxide.......................... 195
C.2 Therm ogravim etric analysis of copper carbonate oxalate. H eating
rate was
1 0
° C / min to 1000°C in flowing air atm osphere.........................196
C.3 Therm ogravim etric analysis of alum ina. H eating ra te was
1 0
°C /
m in to 1000°C in flowing air atm osphere.......................................................197
C.4 Therm ogravim etric analysis of alum ina oxide (94.7%) 4 - copper ox­
ide (5.3%). H eating rate was
1 0
° C / min to 1000°C in flowing air
atm osphere.............................................................................................................. 198
C.5 Therm ogravim etric analysis of alum ina oxide (82.8%) + copper ox­
ide (17.2%). H eating ra te was
1 0
° C / min to 1000°C in flowing air
atm osphere.............................................................................................................. 199
C.6
X-ray diffraction p a tte rn for subm icron alum ina an d chem ically pre­
cipitated copper oxide.......................................................................................... 2 0 2
C.7 X-ray diffraction p a tte rn for 94.7396 alum ina + 5.27% copper oxide
and 82.8% alum ina + 17.2% copper oxide.................................................... 203
E .l
Front Profile of Microwave Processing System w ith C om puter Con­
trol C a r t .................................................................................................................213
E.2 3 kW Microwave Source operating a t: 2.45 G H z ......................................214
xx
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E.3
Ceram ic Sample loaded into Porous A lum ina Crucible sittin g on the
Microwave A pplicator Table...............................................................................215
E.4
Pyrom eter A rray A ttached to Microwave A pplicator Door w ith B ar­
ium Fluoride Lens..................................................................................................218
E.5
Cross Section View of O ptical P a th in th e Microwave Furnace
...
E.6
Look- Table for 80% A lum ina an d 20% Silicon Carbide: R elation
219
between Sample T em perature an d Voltage Reading of th e Single
Color Pyrom eter.................................................................................................... 222
E.7
Sample 80% Alum ina and 20% Silicon Carbide: Tem perature Dif­
ference between M iddle Range P yrom eter and Type C therm ocouple.222
E .8
Sample 80% Alum ina and 20% Silicon C arbide: T em perature Dif­
ference between U pper Range P yrom eter and Type C therm ocouple. 223
xxi
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Chapter 1
Introduction
Microwave processing of ceramic m aterials possess some significant advantages over
conventional th erm al processing. Unlike conventional therm al furnaces, microwave
furnaces can volum etrically deposit energy w ithin a dielectric (or insulating) mate­
rial. Since the processing tim e is almost independent of th e m aterial's therm al con­
ductivity. m aterials can be heated at extrem ely high heating rates (l000°C /m in) in
a microwave furnace. T his rapid heating not only sinters and densities th e material,
b u t it also lim its grain grow th, which can enhance its m echanical an d electrical
properties [1-6].
T he ability of a m aterial to absorb microwave power depends upon the mate­
ria l’s dielectric properties. Unfortunately, several ceram ic m aterials, which have
com m ercial an d m ilitary applications, possess a low loss tangent. T h e loss tangent
is defined as th e ratio of the imaginary p a rt to th e real p art of th e complex per­
m ittivity. Therefore, low- loss m aterials do not absorb sufficient microwave energy'
to h eat to desired tem p eratures in a microwave furnace [7-8].
One technique to increase a m aterial’s loss tan g en t is to combine a lossy material
(additive) w ith a low- loss (host) m aterial [9]. T h e lossy additive absorbs and
1
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converts microwave energy into therm al energy, which raises th e tem p eratu re of
b o th the lossy and low- loss m aterials.
For microwave processing, th e desired
am ount of additive depends upon th e relationship between th e concentration and
dielectric properties of the constituents of th e m aterial.
Such a relationship is
know n as a dielectric m ixing law.
Dielectric mixing laws are helpful in designing composites and modeling mi­
crowave processing. However, current dielectric mixing laws have several problems.
F irst, m ixing laws are lim ited to certain ranges in the constituents’ perm ittivities
or volume fractions. A pplying popular m ixing laws (i.e. Maxwell G arn ett , ef­
fective m edium approxim ation, a n d Landau- Lifshitz- Looyenga form ula [10-12])
outside th eir regimes of validity leads to errors. A nother problem w ith dielectric
m ixing laws is th a t th ey do not take into account details of th e com posite’s mi­
crostructure. In the context of this research, we will be considering composites
consisting of m ixtures of particles of several m aterial types as well as composites
consisting of particles of one m aterial type coated by another. P opular m ixing laws,
which are based on constituent volume fractions and dielectric constants only, do
not distinguish between these two types of composite. We will find, however, th a t
these composites have different effective perm ittivities even for th e sam e volume
fractions of constituent. T he arrangem ent of particles also affects th e interaction
betw een particles. A n appropriate m ixing law m ust account for these interactions
o n th e overall effective perm ittivity. For example, the effective perm ittivities of
layers of m aterial in series or parallel are very different. In addition, th e contact
betw een particles significantly affects the effective perm ittivity. T he electric field is
generally enhanced near th e point of contact of two dielectric particles, and conse­
quently th e effective perm ittivity is sensitive to th e geometry of the contact region.
2
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A gain, com m on mixing laws neglect th e dependence of effective p erm ittiv ity on
p article contact.
T h e effective perm ittivity of a heterogeneous m aterial also can be determ ined
by num erical m ethods [13-16]. C alam e et al. recently used a three- dim ensional
finite- difference electrostatic m odeling to com pute th e effective p erm ittiv ity of
porous ZnO [13]. By m aking a num ber of assum ptions related to the m icrostruc­
tu re o f th e com posite, the electrostatic sim ulations were able to accurately repro­
duce th e m easured results over wide range of fractional volumes. W ith respect
to th e present research, the general physical characteristics of the com posite sys­
tem s un d er consideration, such as arrangem ent of particles and m icrostructure,
have been incorporated into the electrostatic m odel. T he model was th e n used to
accurately calculate th e dependence of th e effective perm ittivity over a range of
param eters.
T h e objective of this research is to m easure a n d quantitatively m odel th e com­
plex p e rm ittiv ity of alum ina com posites w ith lossy sintering aids.
In addition,
physical laws will be developed to describe heterogeneous m ixtures. Specifically,
the com plex p erm ittiv ity is exam ined w ith respect to its dependence on concentra­
tion o f lossy additions, density, an d m icrostructure. In order to accom plish these
objectives, th e research was divided into four different tasks.
F irst, dielectric m easurem ent techniques were developed. Since th ere were no
com m ercial m easurem ent techniques th a t could satisfy our research requirem ents,
two different techniques were developed to m easure th e complex p erm ittiv ity of
solid m aterials.
A high- tem p eratu re open- ended coaxial probe was designed,
built, an d m odeled to m easure th e dielectric properties over a broad frequency
and tem p eratu re range. Also, a nondestructive resonant cavity was developed to
3
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m easure th e dielectric properties of low- loss m aterials.
Second, ceram ic composites were selected, prepared, and characterized. T he
com posite system s were alum ina/ silicon carbide and alu m in a/ copper oxide. Alu­
m in a / silicon carbide composites are high s tre n g th m aterials w ith stru c tu ra l ap­
plications. These com posites were prepared by physically mixing various concen­
tratio n s of ceram ic powder. For th e other com posite system , copper oxide was
selected as th e lossy additive because a lu m in a/ copper oxide com posites could be
synthesized by two different techniques (physically mixed and chem ical precipi­
ta te d ) w hich yielded different m icrostructures. T h e chemical precipitation tech­
nique p artially coated alum ina particles w ith copper oxide. Also, th e chemically
precip itated com posites were more homogeneously m ixed on th e microscopic scale
th a n physically m ixed composites [17]. These differences in m ixing scale and mi­
cro stru ctu re were expected to affect a com posite’s dielectric properties.
P erm ittiv ity an d o th er m aterial properties of these com posites were m easured.
T he arrangem ent of lossy and low- loss constituents significantly affects the effec­
tive p e rm ittiv ity of a composite. For example, alum ina particles coated w ith lossy
additives have a higher imaginary p art of th e complex p erm ittiv ity th a n a ran­
dom m ix tu re of alum ina and lossy additive particles, which has th e sam e volume
fraction of constituents. This is due to the uniform lossy coating shielding m ore
electric field from particles th an the random m ix tu re of particles. Also, th e lossy
coating can provide a continuous conducting p a th through the com posite.
T h ird , some selected composites were processed in conventional and microwave
furnaces. O ur earlier research determ ined th a t th e real and im aginary p arts of the
complex p erm ittiv ity and density of microwave heated samples were significantly
larger th a n conventionally heated samples [18]. Since heating com posite changes
4
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its properties, dielectric and other m aterial properties were exam ined w ith respect
to variations of tem perature, processing m ethod, synthesis technique, and phase
com position. In regard to phase com position, copper oxide reacts w ith alum ina
a t elevated tem peratures and forms a spinel phase. Thus, it was interesting to
exam ine changes in dielectric properties due to a phase change.
Finally, analysis of the com plex p erm ittiv ity data was perform ed. T hree popu­
lar dielectric mixing laws (Maxwell G a r n e t t , effective m edium approxim ation, and
L andau- Lifshitz- Looyenga formula) were used to predict th e dependence of th e
effective perm ittivity of these com posites w ith respect to density an d volume frac­
tio n of constituents. Finite- difference electrostatic sim ulations, which successfully
m odeled two phase system s (ZnO -air), were extended to the case of three- and fourphase system s. The m aterial characterization of composites enabled physically re­
alistic m icrostructures of th e com posites to be incorporated into th e electrostatic
m odel, which provides a more accurate calculation of the effective perm ittivity.
Com parison and contrast between these sim ulations and current dielectric mixing
laws were made.
T his dissertation is organized as follows. C hapter 2 discusses som e background
inform ation on sintering, m aterial processing, and complex perm ittivity. T he de­
velopm ent, modeling, and operation of an open-ended coaxial probe are presented
in ch ap ter 3 [19]. C hapter 4 describes dielectric m easurem ent a n d m odeling of
alum ina composites before firing [20]. T h e design, modeling, and operation of a
nondestructive resonant cavity are also presented in chapter 4. C h ap ter 5 discusses
th e complex perm ittivity m easurem ents and modeling of selected alum ina compos­
ites heated in conventional or microwave furnaces [21]. C hapter 6 outlines open
questions and future paths for research in this field.
5
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Chapter 2
Background Inform ation
T h e following sections provide background on several im p o rtan t areas of sinter­
ing. complex perm ittivity, and microwave processing, which are relevant to this
research.
2.1
Sintering
Sintering or therm al treatm ent of a powder- ceramic sam ple bonds particles to­
gether and improves the sam ple’s stren g th , ductility, conductivity, and corrosion
resistance [22]. Sintering is achieved by heating the sam ple in a microwave or
conventional furnace. Before firing, ceram ic powder is formed (pressed, slip cast,
etc.) into a specific shape, called a "green” sample. W eak chemical bonds (Van
der Waals, hydrogen, etc.) between th e particles m aintain th e green sam ple shape.
Sintering enables atoms to diffuse an d form strong bonds (covalent and ionic)
between particles.
T he m ain mechanisms for mass tra n sp o rt in ceramics are surface, grain bound­
ary, and lattice diffusion.
These mechanism s strongly depend o n tem perature.
6
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O th er tra n sp o rt m echanisms are viscous flow, plastic flow, and evaporation- con­
densation [22.23]. Since the ability of atom s to b reak existing bonds and move to
other sites exponentially increases w ith tem p e ra tu re , significant atomic diffusion
requires raising th e m aterial’s tem perature to g re a te r th a n half its melting tem per­
atu re [22,23]. Diffusion is preferentially directed to m inim ize th e total free energy
of th e system and, in particular, the free energy associated w ith th e surfaces of the
grains [23]. Mobile atom s are energetically favored to reduce surface curvature as
well as surface area. T h is diffusion forms necks betw een grains an d creates strong
interp article bonding [22.23].
D ensification an d grain growth also result from m ass diffusion. Densification
occurs w hen the tra n sp o rt mechanisms decrease th e distance between the centers
of th e grains.
O th er mechanisms lower th e to ta l free energy by grain growth.
However, grain grow th reduces sample stre n g th a n d d uctility which are inversely
proportional to p article size. In summary, heating ceram ics causes the form ation
of stro n g bonds betw een grains, and hence leads to g rain grow th and densification.
2.2
Dielectric Properties
M aterials can be classified by their response to an ex tern ally applied electric field.
In conductors, an applied electric field forces electrons to move freely through
th e m aterial. In insulators, th e electrons are held tig h tly in bound energy bands.
Therefore, an applied electric field does not generate an electric current, b u t po­
larizes th e m aterial. T his polarization creates an opposing secondary internal field
th a t decreases th e to ta l field inside the m aterial.
T h e two types of dipoles are
induced and perm anent dipoles. In the first type, a n applied field forces th e pos-
7
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itive an d negative charges ap art. Atom ic and molecular electrostatic fields limit
this displacem ent because these internal forces increase w ith displacem ent. Perm a­
nent dipoles or p olar molecules result w hen th e center of the positive an d negative
charges do not coincide [24]. A n applied electric field rotates these dipoles or
molecules along th e field fines. T h e polarization, P . is defined as th e dipole mo­
m ent per unit volume. For the m aterials of interest, polarization is assum ed to be
linear an d isotropic. Therefore,
P = N p = e0x eE T
(2.1)
where N is num ber of dipoles per unit volume, p is th e average dipole m om ent, ea
is the p erm ittiv ity of free space, x e 1S the susceptibility, and E T is th e to ta l electric
field. T h e average dipole mom ent is equal to th e to tal polarizability of a m aterial
times th e local electric field.
Different polarization m echanisms contribute to th e to tal dipole m om ent or po­
larization of a m aterial. In ceram ic m aterials, there are basically four polarization
mechanisms, which are th e electronic, ionic, dipole, and interfacial polarization. A
general description of these polarization mechanisms follows.
In electronic polarization, the valence electron cloud surrounding a nucleus is
shifted by an applied electric such th a t its center of charge is displaced very slightly
from th e center of the positive charge. Thus, a neutral ato m in an electric field
acquires an electronic dipole m om ent. In m aterials w ith ionic or p artially ionized
bonds, an electric field creates ionic dipoles by changing the relative distance be­
tween anions and cations. P erm anent dipoles align themselves along th e applied
electric field. T herm al excitation random ly orients these perm anent dipoles and
decreases the net dipole m oment. Finally, some mobile charge carriers do exist
8
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an d can m igrate through insulating m aterial. Interfacial polarization results when
these electrons and ions are stopped a t physical barriers such as grain boundaries,
lattice defects, electrode surfaces, voids, strains, and im purity centers [24].
T h e dielectric properties of a m aterial are th e macroscopic m anifestation of
these various microscopic polarizations. Obviously, the dependence of th e dielec­
tric properties on frequency and tem p eratu re result from the dependence of po­
larization mechanisms on these param eters. T he following explains th e frequency
dependences of these different types of polarizations. A basic model of a dipole
m om ent in a tim e varying electric field is an underdam ped oscillator which is driven
by a n external, sinusoidal force. T h e response of th e dipole depends upon the char­
acteristics of the dipole moment, w hich are its mass, charge, dissipative force, and
n a tu ra l frequency, and of the applied field, which are its am plitude and frequency.
Consequently, the different polarization mechanisms have different characteristics,
such as n atu ral frequency. In general, the n atu ral frequency is about 10loH z for
electronic, 1012 to 1013H z for ionic, 103 to 106H z for dipolar, and less thanlO 2H z
for interfacial polarizabilities. In a tim e varying electric field, th e dipole rotates
along th e direction of the electric field. Due to dissipative forces of th e dipole,
th ere is a phase shift between the tim e dependent polarization and applied electric
field. T h e component of th e polarization out of phase with th e applied electric field
governs th e dissipation of energy from th e applied electric field. Thus, this com po­
nent of th e dipole moment is characterized by the im aginary p art of th e complex
perm ittivity. The component of the dipole moment in phase w ith the applied elec­
tric field contributes to the polarization of the m aterial, which is characterized by
th e real p art of the complex perm ittivity.
T h e polarization mechanisms also depend on tem perature [24,25]. In general,
9
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th erm a l excitation can provide electrons in th e valence b an d sufficient energy to
ju m p across th e band gap an d reach the conduction band. O r, it can enable ions or
holes to ju m p from th e conduction band to th e valence band. T h e num ber of m obile
electrons an d ions exponentially increases w ith tem p eratu re. T hese charge carriers
increase th e m a te ria rs conductivity and thus th e im ag in ary p a rt of the complex
perm ittivity.
Polarizability can also have an inverse relatio n w ith tem p eratu re
[25]. As tem p eratu re increases, th e therm al ex citatio n of th e perm anent dipoles
increases, which decreases th e to tal polarization of th e perm anent dipoles. T hus,
th e te m p eratu re dependences of the different polarization m echanism s varies. O n
th e m acroscopic scale, th e tem perature dependence of th e dielectric properties
is very im p o rtan t in understanding how ceramic m aterials absorb energy during
microwave heating.
2.3
Microwave Processing
Microwave heating of ceram ics is a complex process involving th e interaction of
electrom agnetic fields w ith m aterials. T he m aterial’s dielectric properties affect
th e electrom agnetic fields inside the m aterial and enable conversion of electrom ag­
netic energy into th erm al energy. The microwave field s tre n g th and distribution
depend upon th e furnace’s dimensions and frequency (or frequencies) of operation,
an d sam ple arrangem ent. T h e heating rate and te m p e ra tu re d istrib u tio n through­
o u t th e sam ple also depend upon the m aterial’s specific h e a t, th erm al conductiv­
ity, an d sam ple insulation. H eating ceramic powders also causes mass diffusion,
densification, an d grain grow th, which affect various m ateria l properties. D uring
microwave processing, th e electrom agnetic fields, m aterial properties, and m ass
10
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diffusion processes are dynam ically interrelated. In order to stu d y an d improve
microwave interaction w ith ceram ic com posites, one needs to m easure an d model
th e com plex perm ittivities based on th e constituents’ perm ittivities, volum e frac­
tions, an d various m icrostructural details. Some im portant m icro stru ctu re features
are arrangem ent of constituents, chem ical reactions and phase changes.
Microwave processing of m aterials possesses some significant advantages over
conventional heating.
In conventional therm al furnaces, th e sam ple surface is
heated an d therm al conduction tra n sp o rts th e energy into th e sam ple interior.
Therefore, the processing tim e depends up o n th e therm al conduction of th e sample.
Conventional processing is usually lim ited to slow rates. In co n trast, microwave
rad iatio n can volum etrically deposit energy w ithin a dielectric (or insulating) m a­
terial. Therefore, th is m ethod can rap id ly process m aterials irrespective of their
therm al properties, which can im prove pro d u ct quality an d reduce m anufacturing
costs [7].
C u rren t commercial applications take advantage of selective microwave absorp­
tion w ithin the m aterial. M ost current com m ercial applications involve tem pera­
tures below 500°C. For example, microwaves are used to cook food, d ry solids, and
cure w ood lam inates [7]. Microwave processing of ceramics, which requires heating
above 500°C, has th e potential of producing m aterials w ith im proved properties
an d creating new m aterials which could n o t be produced by other m eans [7]. R apid
processing of ceramics during sintering lim its grain growth, w'hich can enhance me­
chanical and electrical properties.
11
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2.4
Differences in Microwave and Conventional
Processing
T h e m ain difference betw een microwave and conventional h eatin g is the frequency
of th e applied radiation.
T h e p en etratio n depth of electrom agnetic fields into
m aterials is determ ined by th e skin depth, which is th e d istan ce to attenuate th e
am plitude of a plane wave by e - 1 . T h e skin depth is [26]:
( 2 .2 )
w here / is th e frequency, p. is the perm eability of the m aterial. e0 is the perm ittivity
of free space, e is the relative real p art of th e complex perm ittivity, e" is the relative
im aginary p art of the complex perm ittivity, T is tem perature, and p is density. T he
com plex p erm ittiv ity is defined as:
(2.3)
For th e sake of brevity, th e term “relative” will be henceforth om itted. T he real
p a rt relates to the stored electric energy and the im aginary p a rt relates to th e
resistive loss due to th e applied field. T he loss tangent, ta n 8, is th e ratio of the
im aginary to real p art of th e complex perm ittivity. Since m ost ceram ic m a te r ia ls
are poor electrical conductors (e" <C e'), especially at low tem peratures, Eq. 2.2
reduces to:
(2.4)
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M ost im portantly, th e field pen etratio n is inversely proportional to frequency.
T he field pen etratio n stro n g ly affects the tem p e ra tu re d istribution w ithin th e
m aterial which can be d eterm in ed by th e following heat tra n sp o rt equation:
pc?—g
f - = V - (k V T ( r ) ) + P
where p (T, r) is th e density, Cp is the heat capacity,
k
(2.5)
(T , p, r) is th e coefficient of
therm al conductivity, r is th e position, and P (T, p, / , t, r ) is th e power absorbed
p er unit volume of th e ceram ic. Some critical differences betw een th e two pro­
cessing techniques can be d em o n strated from this equation. C onventional therm al
furnaces heat m aterials by em ittin g infrared radiation, which corresponds to a fre­
quency of 1013 to 1014 Hz. Since m aterials possess a n extrem ely sm all skin depth in
th e IR range, this rad iatio n is deposited on the sam ple surface and therm al conduc­
tion tran sp o rts this energy to th e sam ple's interior. (Infrared rad iatio n transport
in low- loss ceramics like alu m in a is not necessarily lim ited by skin depth. The
rad iatio n is sometimes dispersed by scattering an d is th en diffused through the
ceramic.)
The tim e n eeded to equilibrate tem p eratu re differences between the
sam ple surface and core is:
CppLr
c o n v e n tia l
/n
^
( 2. 6)
K
where L is a dimension of th e sample. T his tim e constant depends on th e sample
size and is inversely related to th e m aterial’s th erm al conductivity. For a porous
ZnO sam ple with a thickness of 12.7 m m [27], th e therm al tim e constant
166 s. Since the conventional processing tim e depends
is about
on this tim e constant and
th e heating rate of th e sam p le surface, th e tim e needed to process a sam ple w ith
13
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m inim al tem p eratu re gradients is longer th a n th e tim e co n sta n t, which is presented
by Eq. 2.6. Since porous ceram ic com pacts possess very low th erm a l conductivity,
a un iform heating requires a long processing time, w hich could lead to undesirable
g rain growth.
Experim ental microwave furnaces op erate at a signific a n tly lower frequency (0.9
to 84 GHz). W hen th e skin d e p th of th e microwaves is g re a te r th a n th e sample
dim ensions, samples are volum etrically heated.
T h e tim e required to heat the
sam ple interior is:
* t;
(2.7)
w here ATm is th e change in tem p eratu re during heatin g a n d Pv is the microwave
power absorbed per u n it volum e. A ssum ing an electric field stren g th s of 1000 V /cm
w ithin th e ZnO particles [28], a porous ZnO sam ple can be h eated to 1000°C in
a b o u t 3 s. Since th e h eatin g tim e in microwaves is alm ost independent of the
th erm a l conductivity of th e m aterial and sam ple dim ensions, microwave ovens can
quickly h eat m aterials regardless of their therm al conductivity.
2.5
Microwave Effect and Field Intensification
T h e term s ‘‘atherm al effect” a n d “microwave effect” have been used to describe
nontherm al effects on m ass diffusion in th e presence of a n electrom agnetic field.
Several observations an d theories rep o rt th a t microwave sin terin g of ceramics low­
ers th e sintering te m p eratu re an d activation energy of m ass diffusion [6,29-32].
Specifically, microwave h eatin g densified alum ina and y ttria - stabilized zirconia at
lower tem peratures th a n by conventional heating [31,32]. In o rd er to explain th e
14
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decrease in sintering tem perature, researchers hypothesized th a t microwave heat­
ing somehow enhances mass diffusion by decreasing th e activ atio n energy for grain
boundary diffusion [31,32].
Several experim ental observations of th e “microwave effect” are based on mea­
suring th e sam ple te m p e ra tu re w ith thermocouples. T h e following briefly discusses
some problems of te m p e ra tu re m easurem ents w ith therm ocouples. In spite of these
tem p eratu re m easurem ent problems, our previous experim ental observations and a
few current theories are discussed to explain possible differences in m aterial prop­
erties due to processing m ethod. One of the m otivations for p a rt of th e research
in ch ap ter 5 is to exam ine differences in dielectric properties due to processing
m ethod.
M easuring th e tem p eratu re of ceramics during microwave processing is very
difficult. Therm ocouples, which have been widely used in microwave processing,
alter and intensify th e fields especially a t the therm ocouple tip. These intensified
fields have been found to locally heat th e sample and therm ocouple tip. Further­
more, tem p eratu re gradients are inherent in microwave processing of m aterials
[33]. Even though microwave power can be uniformly and volum etrically absorbed
in th e sample, ra d iatio n from th e sam ple removes energy from th e surface and
lowers th e surface tem p eratu re. T h e low therm al conductivity in porous ceramic
m aterials enables th e m aintaining of an inverted te m p e ra tu re gradient, where the
interior tem p eratu re is higher th a n th e surface 1. Thus, microwave processing heats
different regions w ith in th e m aterial a t different rates [9]. Therefore, a single sur­
1Microwave processing of ceramics w ith two different sources, which generate microwaves at
widely different frequencies, or variable frequency source could decrease temperature gradients
[37][?].
15
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face or core tem p eratu re m easurem ent will not fully characterize the tem perature
distrib u tio n w ithin a sam ple during processing, especially a t high heating rates.
In sum m ary, th e inaccuracies in measuring tem p eratu re w ith thermocouples cast
d o u b t o n several rep o rted observations th a t microwave heating lowers sintering
tem p eratu res and activation energy.
In our previous research, ZnO samples were heated in conventional and mi­
crowave furnaces w ith sim ilar heating profiles [18]. A lthough th e sample tem pera­
tu re was m easured by therm ocouples in b o th furnaces, this experim ent found th a t
th e dielectric properties of ZnO differed greatly depending upon th e processing
m ethod [18]. T h e real and im aginary parts of th e complex perm ittivity of th e
microwave sintered samples were significantly higher th a n conventionally heated
sam ples w ith th e sam e density. Thus, some other factor besides processing tem ­
p eratu re an d density m ust account for th e difference in dielectric properties.
Several theoretical explanations of the “microwave effect” have been based or
enhanced by num erical modeling of the electric fields w ithin porous compacts. J.
C alam e et al. perform ed electrostatic simulations an d found th a t electric fields
significantly intensify near grain contacts [28]. G eom etrical focusing w ithin the ce­
ram ic grains causes this field intensification. A nother physical explanation follows.
Im agine a sphere of dielectric m aterial which is placed inside an originally uniform
electric field. T he field will force charges to the surface of th e sphere. W ith th e
positive and negative charges moving to opposite hem ispheres, this induced field
or polarization will try to cancel out the applied field. T h e to ta l field will be zero
in conductors an d nonzero in dielectrics. Bringing another sphere into the system
also results in inducing surface charges on the new sphere and it becomes polarized.
By decreasing th e distance between the spheres, th e surface charges on each sphere
16
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will induce more surface charges on th e o th er sphere, b u t w ith the opposite sign
[25]. As long as th e applied electric field is n ot perpendicular to the line connecting
th e cen ter of the spheres, the surface charges on each sphere will differ in sign and
increase as th e distance between th e spheres decrease.
Som e “microwave effect” theories could be explained by these electrostatic sim ­
ulations. These electric field stren g th s could be stro n g enough to cause localized
plasm a form ation [28]. Localized ionization phenom ena could provide a new tra n s­
p o rt m echanism betw een grains. Also, it could produce chemical changes a t grain
boundaries th a t alter sintering kinetics [34]. T hese changes could also cause elec­
trically active defects on grain surface an d increase th e effective perm ittivity of th e
m aterial. Field intensification could also su p p o rt an o th er theory of the “microwave
effect.” Ponderm otive theory models m ass tra n sp o rt of charge vacancies due to
high frequency electric fields. T h e field intensification, which is directed along th e
line connecting th e center of spheres, provides for a large net polarization w i t h in
the co n tact region. Thus, the ponderm otive force concentrates charge vacancies
to th e p article surface and contact region [28,35]. Therefore, microwave heating
of a m aterial increases th e density of charged species into regions w ith high field
strengths, which would increase th e m ateria l’s dielectric properties.
17
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Chapter 3
O pen-Ended Coaxial P robe for HighTem perature and Broad- Band
D ielectric M easurem ents
3.1
Introduction
T h e recent grow th in microwave processing of m aterials has been m otivated by
hope of achieving superior electrical, optical, th erm al, and mechanical proper­
ties u n attain ab le by conventional processing m ethods [36]. Microwave processing
o f ceram ics is a dynam ic and complex process encom passing microwave power ab­
so rp tio n an d th erm al and mass tran sp o rt w ith th e m aterial. The average absorbed
m icrowave power p er unit volume is:
Wf = iw e 0£ " c r ,p ,w ) | £ | 2
18
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(3.1)
where u> is th e angular frequency and \E\ is th e average local electric field am ­
plitude. T h e local electric field depends on the input power, applicator design,
an d th e m agnitude and spatial d istrib u tio n of th e complex p e r m it t i v i t y w ithin th e
m aterial.
T h e com plex perm ittivity is a function of tem perature, density, and frequency
of th e applied microwaves. T he dielectric properties of ceramics can also depend
upon processing m ethod [18]. Figures 3.1 and 3.2 show th a t th e com plex perm it­
tiv ity for microwave sintered samples was significantly higher th a n conventionally
sintered sam ples w ith th e sam e heating rate. Since there is a lack of knowledge of
th e dependence of th e complex perm ittiv ity w ith respect to tem p eratu re and den­
sity, it is difficult to model th e m aterial’s power absorption, tem p eratu re distribu­
tion, and densification. Complex p erm ittiv ity m easurem ents over a wide frequency
and tem p eratu re range will improve microwave processing models, increase under­
standing of m aterial’s polarization mechanisms, and optim ize future microwave
processing system s [37.38].
Open- ended coaxial probes nondestructively measure a m aterial’s com plex per­
m ittiv ity over a broad frequency range and up to 1200°C [38,39]. T hese probes
have been thoroughly studied theoretically [40-49] and extensively used in exam­
ining biological m aterials [46,50]. T h e sim ple sample preparation and its nonde­
structive n atu re provide th e open- ended probe w ith a significant advantage over
o ther techniques [51,52]. T he m easurem ent technique basically requires th a t the
sam ple possesses a single flat and sm ooth surface. Its diam eter should b e a t least
two tim es th e probe diam eter. M ost theoretical models assume th a t th e sample
is semi- infinitely thick. In practice, th e sam ple thickness should allow th e electric
field at the far end of th e sam ple to be a t least two orders of m agnitude sm aller
19
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MW sintered to 88% dense: o
MW sintered to 95% dense: +
Conv. sintered to 87% dense: x
Conv. sintered to 95% dense: *
o50
30
20
Frequency (Hz)
Figure 3.1: Real part of the dielectric constant of ZnO sam ples sintered in mi­
crowave and conventional furnances vs. frequency.
20
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45
,
MW sintered to 88% dense: o
MW sintered to 95% dense: +
Conv. sintered to 87% dense: x
Conv. sintered to 95% dense: *
9-30
o25-
O)
o oo
X X
X X X
X X X X X
Frequency (Hz)
Figure 3.2: Im aginary p art of th e dielectric constant of ZnO sam ples sintered in
microwave and conventional fum ances vs. frequency.
21
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th a n a t th e probe/sam ple interface- For th in sam ples, which do not satisfy this
condition, a m ultilayer dielectric m odel w ith an air term ination could be applied
[40].
As show n in figure 3.3, an open- ended coaxial probe is a tru n cated coaxial
transm ission line. A flange is welded to th e ended of th e outer conductor. T his
flange acted as a grounding plane for th e fringing electric fields and provided
a stab le su p p o rt for the m aterial under test, which was brought flush w ith th e
probe. A vector network analyzer (VNA) H P 8520C, which was connected to th e
other en d o f th e probe via a coaxial cable, swept th e frequency of the tra n sm itted
T E M wave. T his incident wave reflected off th e p ro b e / m aterial interface due to
im pedance m ism atch. T he network analyzer m easured th e m agnitude and phase
of the reflected wave relative to th e incident wave. Electrom agnetic field analysis
related th e reflection coefficient to th e m aterial’s complex perm ittivity.
A ccurate probe m easurements require in tim ate contact between probe and m a­
terial [39,40,53,54]. A n air gap or th e space betw een th e probe and surface of th e
sam ple can result from axial m isalignment of th e inner and outer conductors dur­
ing assembly, th e surface roughness of the sam ple [54], and th e differential therm al
expansion of th e inner and outer conductors at elevated tem peratures [39]. T h e re­
flection coefficient sensitivity to extrem ely sm all air gaps has been experim entally
and theoretically proven [40,55].
To em phasize th e probe’s sensitivity to air gaps, theoretical calculations of th e
reflection coefficient were performed on a typical ZnO sample. A t three different
frequencies, th e reflection coefficient was com puted for various thicknesses of air
between th e probe and sample. As th e air gap increased, the m agnitude of th e
reflection coefficient increased to u nity and phase of th e reflection coefficient in-
22
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HBR W a s h e r
0.209
0 . 093 "
Stainless Steel
IT)
03
csi
Spring
r
' Center Contact for
Type N Connector
F igure 3.3: Schematic diagram of a high tem p eratu re open- ended coaxial probe.
A spring loaded inner conductor forces th e inner an d outer conducters to m aintain
in tim ate contact with m aterial under test.
23
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'40
+ : 2 0 0 MHz
* : 2 .4 2 5 GHz
o : 10 G H z
_20
UU10
Lift- O ff (m m )
F igure 3.4: Calculation of th e effective s' at three different frequencies of a ZnO
sam ple vs. air gap thickness.
creased to zero. A n air gap also decreased th e probe’s sensitivity [40]. As shown
in figures 3.4 and 3.5, calculating th e com plex p erm ittiv ity from the reflection co­
efficient, while neglecting th e effect of th e air gap, created significant errors. As
th e air gap or lift- off increased, th e effective complex p e rm ittiv ity for a sam ple of
ZnO a t th re e different frequencies decreased dram atically.
E arlier researchers have exam ined th is air gap problem .
Several capacitive
models include th e air gap into relatin g th e reflection coefficient to th e sam ple
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+ : 200 MHz
* : 2.425 GHz
X10
o.
o : 10 GHz
O 8
u>
•icr
10
Lift- O ff (m m )
Figure 3.5: C alculation of th e effective e" a t th ree different frequencies of ZnO vs.
air gap thickness.
25
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complex p e rm ittiv ity [45.54-56]. D ifferential therm al expansion o f the probe was
m inimized by constructing a m etallized ceramic probe [39]. Since this previous
stu d y found th a t a length difference between the inner and o u ter conductors on
th e order of 10-4 m strongly affected th e m easured reflection coefficient [39], an
open- ended coaxial probe was specifically developed to elim inate air gaps. T his
chapter presents th e design, model, and use of a spring- loaded open- ended coaxial
probe for high tem p eratu re dielectric m easurem ents. A n u n certainty analysis of
th e probe is also presented. In addition, a model is presented to calculate th e
complex p erm ittiv ity of a sample w ith a large surface roughness. Finally, full wave
analysis provides a system atic m ethod for optim izing the probe dimensions for
p articu lar applications.
3.2
Probe Design
There are several im p o rtant design param eters for an open- ended probe. Efficient
transm ission of incident and reflected signals between th e netw ork analyzer and
probe head required th a t th e im pedance of th e probe m atched th e 50 Q im pedance
of th e netw ork analyzer and coaxial cable. Tins condition restricted the probe di­
mensions to fit com m ercially available connectors. T he probe had som e com peting
design constraints. T h e electrom agnetic fields in an axis- sym m etric probe excited
by an incident T E M mode can be represented as a superposition of T M (transverse
magnetic) m odes. If there are asym m etries due to, for example, m isalignm ent th en
TE (transverse electric) modes will be excited as well. In order to minimize th e
effect of these m odes, whose presence is difficult to predict w ithout precise knowl­
edge of th e degree of asymmetry, th e probe operated below the cutoff frequency
26
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of T E modes which lim ited th e m easured frequency range. To increase the cutoff
frequency of T E modes and th e bro ad b an d range of th e probe, th e probe diam eter
would have to decrease. However, th e reflection coefficient strongly depended upon
th e fringing fields interaction w ith th e sam ple. A larger diam eter probe increases
th e interaction of th e fields an d th e p ro b e’s sensitivity [40]. Therefore, a com pro­
mise on the probe dim ensions was m ade betw een probe sensitivity an d frequency
range.
As shown in figure 3.3, this stainless steel coaxial probe w ith an air dielectric
was built to m aintain intim ate contact w ith th e sam ple a t tem peratures up to
1200°C and m easure the reflection coefficient from 0.2 to 10 GHz. T he springloaded inner conductor was centered by a boron n itrid e (HBR) washer at th e
probe head an d by th e center contact of th e type N connector a t th e other end.
T h e inner diam eter of th e w asher was large enough to provide a slip fit for th e inner
conductor at 1200°C. T he spring was positioned inside th e inner conductor, which
was slip fitted into the center contact. T h e center co n tact assisted in electrically
shielding the spring.
T his shielding effectively avoided introducing parasitic reactances w ithin th e
probe and altering the probe’s im pedance. To verify efficient transm ission through
th e probe and connector assem bly (connector, precision adaptor, and coaxial ca­
ble), tim e dom ain analysis using a netw ork analyzer m easured th a t th e reflection
of a sho rt at the probe head was a t least seventeen tim es g reater th a n th e reflection
from the connector. Thus, th e connection of the inner conductor is an effective
high frequency electrical p ath . T h e connector’s dielectric was modified to constrain
th e spring- loaded inner conductor to axial motion.
Intim ate contact between th e probe an d m aterial was assisted by th e pedestal
27
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a t th e probe head. Several theoretical models assum e th a t th e flat g r o u n d in g plane
ex ten d s o u t to infinity. Previous finite element analysis calculations by Blackham
a n d P ollard found an insignificant difference betw een this idealized grounding plane
a n d a finite sized ground plane w ith a pedestal [49].
In order to prevent dam age to th e connector an d other microwave com ponents
a t h ig h tem p eratu re operation, a w ater jacket, w hich is positioned near th e con­
necto r, removes h eat from th e o u ter conductor. In an a tte m p t to remove heat
from th e inner conductor, A rai et al. [38] placed a n alum ina nitrid e (AIN) washer,
w hich has a high therm al conductivity and tra n sp aren t to microwaves, inside a
stainless steel probe near th e type N connector. T herm al tra n sp o rt experim ents
te ste d th e efficiency of a BeO washer to cool th e inner conductor. T he therm al
co n d u ctiv ity of BeO is 62% greater th a n AIN. T h e th erm al tests indicated th at
th e re was insufficient th erm al contact between th e washer a n d conductors to pro­
vide a cooling path. Instead, therm al energy from th e inner conductor appears to
have been transferred prim arily by radiation n ear th e probe head. Then, towards
th e connector end of th e probe, conduction and convection sufficiently cooled the
in n er conductor so th a t it was only a few degrees h o tte r th a n th e outer conductor.
T h u s, a BeO washer was not used w ith the probe. Sim ilarly to A rai et al. [38],
housing th e probe inside a m ullite tu b e w ith a flowing reducing gas of 95% N2 +
5% H 2 m inim ized oxidation of th e probe.
3.3
Theory and Calibration
A full- wave analysis of a T E M wave incident upon a m ultilayer dielectric body
allowed co m putation of th e complex perm ittivity of th e sam ple from the m easured
28
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values of th e reflection coefficient [40]. Figure 3.6 shows a three- layer dielectric
bo d y a t th e end of an open- ended coaxial probe, where region 2 is th e air gap.
region 3 is th e m aterial under test, an d region 4 is th e m edia term ination. T h e m a­
terial in each layer is assum ed to be a linear dielectric, homogeneous, and infinite
in radius. T he electric and m agnetic fields of th e incident TEM wave in teracted
w ith th e m aterial and reflected off it due to th e im pedance mismatch. To relate th e
incident an d reflected T E M an d evanescent T M on waves in the coaxial probe to th e
fields in th e m aterial, th e electric, E p, an d m agnetic, H0, fields were m atched a t
each interface. A lthough this full wave analysis was exact, numerical calculations
required lim iting th e num ber of modes an d integration of the Hankel transform s.
As show n in A ppendix A, extensive com puter modeling determ ined th a t th e re­
flection coefficient converged for th e specified num ber of T M ^ modes and over
the in teg ratio n range. For a specified complex perm ittivity of th e m aterial, th e
reflection coefficient was com puted for b o th 6 and 12 modes. By extrapolating th is
d ata, th e reflection coefficient was approxim ated for an infinite num ber of m odes.
T he estim ated error in calculating th e reflection coefficient is A
|r|
^
3E
— 4 an d
|A0| < 0.03°.
T h e ideal analysis would directly com pute th e sam p les complex p erm ittiv ity for
a specified reflection coefficient, which is m easured by th e VNA. U nfortunately, th e
inverse solution ( r —> e*) can not be solved directly. A n interpolation routine was
devised by m aking a d a ta table of th e reflection coefficient a t a specified frequency
for a range of e7 and ta n S, which is e"/e'. Figure 3.7 and 3.8 show th e contour
plots of th e m agnitude,
|r|,
and phase, 9, of th e reflection coefficient for a sem i­
infinite thick sam ple (L—> oo) w ith no air gap. T he intersection of the m easured
m agnitude and phase contours established the sam ple’s complex perm ittivity. T h e
29
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
connector
z =0
*
€ Rg air gap (2)
*
€ Rs sample (3)
*
€ R{j termination (4)
z=d
z = L+ d
Figure 3.6: T he open- ended coaxial probe contacting a multilayer sample. R egion
2 represents th e air gap layer and region 3 represents th e sample. A lthough region
4 represents th e m aterial term ination, the sam ple was assum ed to be semi- infinite,
L—►oo.
30
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-120
7-90
Line - Magnitude
Dash - P h ase (deg)
60
40
,-60
30
,0.7
-30
20
0.8
0-9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Loss T an g en t
Figure 3.7: C ontour plot of th e m agnitude and phase of th e reflection coefficient
over a range in s ' an d loss tangent at a frequency of 915 MHz.
estim ated error in inverting ( r —»■e*) is A [F| < 10- 3 and |A 0| < 0.004°. The
probe m easurem ent sensitivity can be determ ined from th e contour density. The
higher contour fine density in th e reflection coefficient relates to a sm aller variation
in the complex p erm ittiv ity and a higher m easurem ent resolution. For m aterials
w ith a sm all loss tan g en t, th e contour density of th e m agnitude of th e reflection
coefficient is low, w hich lim its th e probe to accurately m easure m ateria l properties
when ta n 6 > 0.05.
31
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10Q
Line - Magnitude
Dash - Phase (deg)
90
-150
80
CL
.0.9
50
-120
40
,0.8
-90
20
.0.5
-60
0.7
-30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Loss T an g en t
Figure 3.8: Contour plot of th e calculated m agnitude and p hase of th e reflection
coefficient over a range in s ' and loss tangent a t a frequency o f 2.45 GHz. No air
gap an d semi- infinite m aterial, L—■» oo.
32
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C alculating th e complex perm ittivity assum ed th a t th e sam ple is flush w ith
th e probe or at a specified lift- off. A ctual m easurem ents were performed w ithout
intentional lift - off. but th e sample surface roughness restricted perfect contact
w ith th e probe. In order to minimize the air gap error, A rai et al. found th a t
th e sam ple surface roughness should be less th a n 0.5 fim [54].
As mentioned
earlier, th e effect of surface roughness was estim ated b y capacitive models where
one layer is air an d th e other is the M UT [54-56]. To accurately calculate the
com plex p erm ittiv ity of a m aterial w ith a rough surface, th e full wave analysis
m odeled th e surface layer as a separate region. Looking at figures 3.6, Region 2
represented th e surface layer of th e m aterial w ith a thickness two times the average
surface roughness, R a , of the M UT. T he complex p e rm ittiv itj' of the region was
approxim ated by th e Landau, Lifshitz, and Looyenga dielectric m ixture equation
[12]:
e* = {Vair (eal r ) 1 / 3 + VMUT
(3-2)
where v is th e volum e fraction of the particular m aterial. This mixing law has
been found to accurately determ ine e* for a variety of m aterials [12]. In this case,
v air = v m u t = 0.5 an d e\Iu r was the complex p erm ittiv ity of th e M UT in region
3. For sam ples w ith a rough surfaces, d ata tables were calculated for a specified
R a and a t various frequencies.
T he m easured reflection coefficient differed from th e reflection coefficient at the
probe head due to th e im pedance m ism atch betw een th e microwave components
(VNA, coaxial cable, adaptor, connector, and probe) and the phase shift due to the
electrical length of com ponents. This phase shift is also tem p eratu re dependent.
Since th e full wave analysis related this latter reflection coefficient to complex
33
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
p erm ittivity, it was essential to calibrate th e probe to m easure this reflection coef­
ficient. Using an H P 85052D 3.5 m m calib ratio n kit, th e system atic errors of th e
V NA and coaxial cable were accounted for an d th e m easured reflection coefficient
was referenced from th e end of th e cable. A fter connecting the probe assem bly to
th e coaxial cable, another calibration was perform ed to account for th e im pedance
m ism atch betw een th e probe and coaxial cable, electrical length of th e probe, an d
atte n u atio n of the T E M waves. Applying transm ission line theory to this system ,
th e m easured reflection coefficient can be re la te d to th e reflection coefficient a t th e
probe head. The m easured reflection coefficient is:
r>
. . „
m e a su re d — ^ 1 1
e l2r PH
,
i"
1 — e 22L p h
ro
fo .O j
where r meaS-ured is th e m easured reflection coefficient, T PP is the reflection a t th e
probe head, and e^ are the error param eters, which are complex num bers [57].
Referring to figure 3.6, e u is th e directivity error from th e im pedance m ism atch
between VNA and probe, e 1 2 is th e tracking error due to th e attenuation a n d tim e
delay between th e incident and reflected waves, and e22 is th e source m atch error
from th e m ultiple internal reflections betw een th e VNA and probe. In term s of S
param eters, e u =
5
x1 , ex2 = S i2S 2i exp(—2
7
/), and e22 = S 22 exp(—2 y l), w here Sy
are th e S- param eters of the connector, I is th e distance from the connector to th e
probe head, and
7
is th e complex propagation constant. These term s were d eter­
m ined by m easuring th e reflection coefficient of th ree standards (Rexolite 1422, a
short, and a ZnO sam ple) for which T PP is known. Since th e probe h ad a springloaded inner conductor, only solid flat sam ples could force the in n er conductor
into th e sam e plane as the grounding plane.
Rexolite provided a good su b sti­
tu te for an open (air) standard because it possessed low dielectric p e rm ittiv ity
34
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(e* ~ 2.53 —0.00I j) and was easy to polish. T he complex p e rm ittiv ity of th e ZnO
sta n d a rd sam ple was m easured b y th e H P probe, which has a 5% erro r in m easuring
th e com plex perm ittivity of sam ples w ith a ta n S > 0.05 [58].T h e full wave anal­
ysis com puted the expected reflection coefficient, Vp H, for th e R exolite and ZnO
sta n d a rd an d these expected values were th e n used in Eq. 3.3 to determ ine th e val­
ues of Qij. For calibration and, in general, m easurem ents, th e m easured reflection
coefficients were averaged from twelve traces to reduce m easurem ent uncertainty.
C alib ratio n of the VNA and th e probe assem bly thus enabling calculations of the
reflection coefficient a t th e probe head from any m easured reflection coefficient
[42].
D uring high tem perature m easurem ents, therm al expansion of th e probe ex­
ten d ed its physical and electrical length, which added a phase shift to th e reflection
coefficient. Using a sam ple w ith a know n reflection coefficient w ith respect to tem ­
p e ra tu re and frequency, this phase shift was measured. O riginally this technique
used an open or air stan d a rd [59]. However our experim ents utilized a gold plated
sho rt w ith th e known reflection coefficient, V = —1 . A polynom ial fit of this phase
shift w ith respect to tem p eratu re enabled th e phase shift to be su b tra c te d out in
subsequent high tem perature m easurem ents. Since th e m agnitude of th e reflection
coefficient varied m inutely w ith respect to tem perature, no correction in th e mag­
n itu d e was performed. Therm al expansion of th e inner and outer conductor a t the
probe head in the radial direction was probably uniform and resulted in a m inor
error in th e d a ta tables.
35
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.4
Results and Discussion
In this section, com plex p erm ittiv ity m easurem ents conducted with the springloaded probe are com pared to those of a com m ercial open- ended probe a t room
tem perature and a resonant cavity at elevated tem peratures. B ut first, the ac­
curacy and reproducibility of th e probe m easurem ents were examined. R epeated
m easurem ents determ ined th e average and uncertainties of the error param eters,
&ij. Figure 3.9 shows th e average m agnitude of th e error param eters w ith respect
to frequency. Since th e transm ission term through th e probe connector, |e12|, was
about
0
.8 , T E M waves were effectively tra n sm itted betw een the VNA and probe
head. It also proves th e effectiveness of the slip fit connection on the inner con­
ductor. R epetitive reflection m easurem ents of several ZnO samples determ ined
th e variation in th e m easured reflection coefficient as shown in figures 3.10 and
3.11. P ropagating th e uncertainties of the error param eters, calibration standards,
an d various samples resulted in combined uncertainty of <5|r| = ±0.014 and 69
= ±1.09°.
Because th e m easurem ents were averaged over m ultiple traces, this
combined uncertainty in th e m easured reflection coefficient was less th an the un­
certainty of a single tra ce of th e VNA, which is A [TJ < 0.03 and |A0| < 2° [60].
T h e relation between th e combined uncertainty of th e reflection coefficient to the
uncertainty of th e com plex p erm ittiv ity is shown in figures
3 .1 2
and 3.13. These
contour plots illustrate th e percentage uncertainty in the real, e7, and imaginary,
e", parts of the complex p erm ittiv ity and the variability of th e probe’s sensitivity.
For a specific e7 and ta n 6, th e percentage uncertainty in e7 is displayed by the solid
contour line an d in e77 is displayed by the dashed contour line. As expected, the
uncertainty of e77 is quite large for low- loss m aterials. Figure 3.12 and 3.13 have a
low contour density in a low loss tangent region. Therefore, th e uncertainty in e7/
36
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
x
c
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X
x x x x
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1C?
Frequency (Hz)
«
* * J{
>10
101
Figure 3.9: M agnitude of the error param eters for th e hig h tem perature dielectric
probe vs. frequency.
w ith a low loss tangent will be larger th an th e u n certain ty in e" with a higher loss
tan g en t.
In order to dem onstrate th at the spring- loaded p ro b e could accurately m easure
th e com plex perm ittivity over a wide frequency range, this probe and a sta n d a rd
H P probe m easured the dielectric properties of several ZnO samples w ith diffe r in g
dielectric constants a t room tem perature. Since th e average surface roughness of
the ZnO samples was measured to be less th a n 0.3 jum, th e error due to the surface
37
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
O.OE
0.04'□ 0 OO 0 0 0 O
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Frequency (Hz)
F igure 3.10: U ncertainty of th e reflection coefficient for calibration param eters and
ZnO sam ples vs. frequency.
38
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0.35
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Figure 3.11: U ncertainty of th e reflection coefficient for calibration param eters and
ZnO samples vs. frequency.
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
25
Line - ETror(%) in Real Part
Dash - Errar(%) in Imaginary Part
Q.
0.1
0.2
0.3
0.4
0.5
0.6
Loss Tangent
0.7
0.8
0.9
Figure 3.12: P ercent uncertainty of complex perm ittivity measured by th e probe
at 956 MHz. For a specified complex perm ittivity, the percentage uncertainty in
s' is indicated by solid line contour and in e" is indicated by dashed line contour.
40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10Q
Real Part of Complex Permittivity
90
Line - Error(%) in Real Part
Dash - E.rror(%) in Imaginary Part
80
70
,'25
50
40
30-75
>§-
0.2
0.3
0.4
0.5
0.6
Loss Tangent
0.7
0.8
0.9
Figure 3.13: Percent uncertainty of com plex p erm ittiv ity measured b y th e probe
at 2.425 GHz. For a specified com plex perm ittivity, th e percentage u n c e rta in ty in
e' is indicated by solid line contour an d in e" is indicated by dashed line contour.
41
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
roughness can be ignored for the H P probe [54] an d d a ta tables were com puted
w ith no- Lift off. T h e complex perm ittivity m easurem ents of ZnO samples a t room
tem p eratu res are show n in figures 3.14- 3.16. Over th e frequency range 0.3 to
6
GHz, th e average difference betw een the two probes in m easuring d was 2.5% and
th e spring- loaded p ro b e’s average uncertainty is 4%. T hus, th e spring- loaded
pro b e an d HP probe agreed w ithin the expected lim its for th e real p a rt of the
com plex perm ittivity. T h e average percentage difference in e" m easurem ents, which
was 24%, was less th a n th e spring- loaded probe’s average uncertainty, w hich was
51%. A lthough th e im aginary p a rt m easurem ents agree, th e larger percentage
difference and u n certain ty result from m easuring several low- loss samples, which
possessed tan 6 < 0.05. Also, th e H P probe m easurem ents were only valid for ta n
8 > 0.05 [58]. Thus, th e spring- loaded probe accu rately m easured the com plex
p erm ittiv ity over a w ide frequency range for m aterials w ith ta n 5 > 0.05.
A fter dem onstrating th a t th e spring- loaded probe worked well a t room tem p er­
atu re, th e probe was th e n tested a t high tem peratures. Tw o m aterials (porous alu­
m ina composite, w hich is A h O s/C a O + SiC grains, a n d nephelene syenite, which
is 60% SiOo + 24% AI2 O 3 +
1 0
% NaaO + 5% K 2 O) were separately m easured over
a wide tem p eratu re ran g e by th e probe and a resonant cavity. Microwaves P rop­
erties N orth (MPN) perform ed dielectric m easurem ents w ith a resonant cavity at
discrete frequencies (912 MHz and 2.46 GHz) [61]. T h e nephelene syenite sam ple
was h eated in a conventional furnace up to 1000°C. A t specified tem peratures, th e
sam ple was removed from th e furnace and positioned in th e m iddle of a resonant
cavity. T h e sam ple lowered the frequency of th e resonant modes and broadened
th e resonant curves. A vector netw ork analyzer found th e resonant frequency an d
Q- factor of the TMono modes. A computer, which was connected to th e VNA,
42
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5-
I
CD
3
CL
X 2 .5 |
_CD
Q.
E 2
O
O
* : Real Part (HP probe)
o : Real Part (H.T. probe)
* : Imaginary Part (HP probe)
a : Imaginary Part (H.T. probe)
1-51
$ S - *4
Ss s * * s * I
0 .5 -
-
II
°uf
l-
Cf
Frequency (H z)
1
.10
1 0
'
Figure 3.14: Complex p erm ittivity of ZnO (conventionally sintered to p = 53.7%)
m easured by a high tem perature and HP probes vs. frequency.
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
12
10
«4—*
>
1
8
'£L _
CD
Q.
x : Real Part (HP probe)
CL
£
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* : Imaginary Part (HP probe)
□ : Imaginary Part (H.T. probe)
2
ICf
1Cf
1010
F req u en cy (H z)
F igure 3.15: Complex perm ittivity of ZnO (conventionally sintered to p = 72.5%)
m easured by a high tem perature and H P probes vs. frequency.
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
90
x : Real Part (HP probe)
o : Real Part (H.T. probe)
80
* : Imaginary Part (HP probe)
□ : Imaginary Part (H.T. probe)
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Frequency (H z )
F igure 3.16: Com plex p erm ittiv ity of ZnO (microwave sintered to p = 95.0%)
m easured by a high te m p eratu re and H P probes vs. frequency.
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
recorded this inform ation and determ ined th e com plex perm ittivity of the sam ple
based upon this in form ation and sample dim ensions [61]. D uring th e dielectric
m easurem ents, th e sam p le cooled.
E m pirical cooling curves approxim ated th e
sam ple tem p eratu re d u rin g m easurem ents.
Figures 3.17- 3.20 show th e m easured com plex perm ittivity of nephelene syen­
ite a t two different frequencies. Since this glass’ viscosity dram atically decreased
around 875°C, probe m easurem ents were lim ited to tem peratures below 825°C.
T he average difference in e' was 12.7% at 912 MHz and 8.4% a t 2.46 GHz. T h e
average difference in e" was 15.3% at 915 MHz an d 14.4% a t 2.46 GHz. As denoted
in Table 3.1, th erm al cycling of the spring- loaded probe contributed to a 0.005
uncertainty in th e m ag nitude an d 0.018° u n certain ty in phase of th e reflection co­
efficient. This v ariability in th e reflection coefficient due to therm al cycling was
only significant while m easuring low- loss sam ples. T h e activation energy for this
glass was calculated from :
a = a 0 exp (—A / k T ) = uje0e"
(3.4)
where a is the conductivity, A is the activation energy, and k is th e B oltzm ann
constant. Cavity m easurem ents of the activation energy were 0.338 eV a t 912 MHz
and 0.296 eV a t 2.46 G H z which is only 4% different th a n probe m easurem ents.
These were obtained b y fittin g a straight fine to th e logarithm of th e m easured
values of e” versus inverse of th e tem perature in electron volts. Therefore, the probe
has a higher relative accuracy in m easuring changes in e" th a n it does in m easuring
th e absolute value. To verify th a t the th erm al expansion of the inne r and o u ter
conductor at th e probe head in th e radial direction contributed an insignificant
error, th e complex p erm ittiv ity of nephelene syenite a t T = 793° C was calculated
46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
w ith two different d a ta tables. T he first tab le used th e dimension of th e probe at
room te m p e ra tu re an d the other table used th e dimensions of th e probe at 793°C.
T h e calculated complex perm ittivities differed by 0.33%.
Since th e surface roughness of the glass ceram ic was less th a n 0.17^m , calcu­
lating th e com plex perm ittivity utilized d a ta tables w ith no lift- off. However, the
alum ina com posite w ith fine SiC grains was very porous and had a large average
surface roughness of
=
8
fim . C alculations for this sample used d a ta tables
which m odel a 16 fj.m thick surface layer. Figures 3.21- 3.24 show th e measured
com plex p erm ittiv ity of AI0 O 3 / CaO + fine SiC grains at two different frequencies
and up to 1000°C. A lthough th e H P probe does n ot account for surface roughness,
it was used to m easure the alum ina com posite a t room tem perature. T he HP re­
sults differed from th a t of the cavity m easurem ent by 62% for e7 and 83% for e7'.
T he spring- loaded probe w ith surface roughness model differed by only 16% for
e7 and 15% for e". A t elevated tem peratures, th e average error in e7 was 8.7% at
912 MHz and 4.4% a t 2.46 GHz. U nfortunately th e loss tangent of this alumina
com posite was less th a n 0.05 for tem peratures above 100°C. T he average error in
e77 was 358% a t 912 MHz and 252% at 2.46 GHz. Above 700°C, th e general trend
of probe results m atched the cavity m easurem ents.
47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
>24,
g2 0 "
Q.
E18o : Resonant Cavity (MPN)
o : High Temperature Probe
_ T ^ f J_ A
Tem perature (C)
Figure 3.17: Real p a rt of th e complex p erm ittiv ity of N ephelene Syenite m easured
vs. te m p eratu re by a resonant cavity (M PN) and high- tem p eratu re probe a t 912
MHz.
48
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o : Resonant Cavity (M P N )
CL
a : High Tem perature Probe
□
□o
°o
o
1Cf
□0
□ 0
w
1
■S1 o '
CD
0
100
200
300
400
500
Tem perature (C )
600
700
800
Figure 3.18: Im aginary p art of th e com plex p erm ittiv ity of N ephelene Syenite
m easured vs. tem perature by a resonant cavity (M PN) and high- tem perature
probe a t 912 MHz.
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o : Resonant Cavity (MPN)
o : High Temperature Probe
10
-
Tem perature (C )
Figure 3.19: Real p a rt of the com plex perm ittivity of N ephelene Syenite measured
vs. tem p eratu re by a resonant cavity (MPN) and high- tem p eratu re probe a t 2.46
GHz.
50
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
QJ
1
0-10 f
_<d
o : Resonant Cavity (MPN)
□: High Temperature Probe
CL
E
3
□^
ck° 0
•
-s itf
c
p
E
^
O °z
c
f
• 5 ,0 ’ E
C[
»°
D
□
□
1 0 2
0
100
200
300
400
500
Tem perature (C)
600
700
800
Figure 3.20: Im aginary p a rt of the com plex perm ittivity of Nephelene Syenite
m easured vs. tem p eratu re by a resonant cavity (M PN) and high- tem p eratu re
probe at 2.46 GHz.
51
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frO -r1
Q.
o : Resonant Cavity (MPN)
co 5
o : High Temperature Probe
+ : HP probe
200
400
600
800
T em p eratu re (C )
1000
120C
Figure 3.21: R eal p a rt of the com plex p erm ittiv ity of AI2 O 3 / C aO + fine SiC
grains m easured vs. tem perature by a resonant cavity (M PN), high- tem p eratu re
probe, an d H P probe (only at room tem p eratu re) a t 912 MHz.
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
~
.6£
o : Resonant Cavity (MPN)
° : High Temperature Probe
+ : HP Probe
j
? 1 .4 -
T
0 . 605
£ 0 .4
0.2
200
400
600
800
Tem p eratu re (C )
1000
120C
Figure 3.22: Im aginary p art of th e com plex p erm ittiv ity of AI2 O 3 / CaO + fine SiC
grains m easured vs. tem perature by a resonant cavity (M PN ), high- tem perature
probe, an d H P probe (only a t room tem p eratu re) a t 912 MHz.
53
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
>
E
1
_
a>
a.
x
JD
Q.
E
o
O
<4 —
o
■c
CO
Q_
co
a)
DC
o : Resonant Cavity (MPN)
o : High Temperature Probe
+: HP probe
200
400
600
800
Tem perature (C)
1000
1200
F igure 3.23: Real p a rt of th e complex perm ittivity of AI2 O 3 / CaO + fine SiC
grains m easured vs. tem perature by a resonant cavity (M PN ), high- tem perature
probe, and HP probe (only a t room tem perature) a t 2.46 GHz.
54
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.7
Q.
0 . 6-
0-0.4
I 1 J* *
..0
>
□ : High Temperature Probe
o : Resonant Cavity (MPN)
+ : HP Probe
0.1
200
400
600
800
T em p e ratu re (C)
1000
120C
Figure 3.24: Im aginary p a rt of the com plex perm ittivity of AI2 O 3 / C aO T fine SiC
grains m easured vs. tem perature by a resonant cavity (M PN ), high- tem p eratu re
probe, and H P probe (only a t room te m p eratu re) a t 2.46 GHz.
55
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E rro r S o u rc e
A|r|
|A 0|
C o m p u tatio n of T
~3E-4
< 0.03°
<
< 4E -3
Inversion
(r —►e*)
1 0 ~ 3
R ep eatib ity
0.014
1.09°
T h erm al cycling
0.005
0.018
C om bined
0.015
1.23°
Table 3.1: E rror Sources for Reflection Coefficient
3.5
Optimizing the Probe’s Sensitivity
Since th e probe’s sensitivity w ith respect to frequency and com plex perm ittiv­
ity was dem onstrated in figures 3.12 an d 3.13, the following analysis examines
th e p ro b e’s sensitivity w ith respect to probe dimensions. V ariations in th e probe
dim ensions alter th e electrom agnetic fields fringing into th e m ateria l an d conse­
q uently altering the dependence of th e reflection coefficient on th e com plex perm it­
tivity. This analysis enables optim izing a probe’s sensitivity for future dielectric
m easurem ents and illustrates th e basic T vs. e* relation w ith respect to probe
dim ensions and frequency.
T h e probe sensitivity was first exam ined by varying th e size of th e in n e r con­
d u cto r while fixing th e o u ter conductor dimension. T h e inner conductor diam eter
was increased so th a t th e probe h ad a 16 Q im pedance. F igure 3.25 shows the
contour plot of the reflection coefficient over a range in e* and ta n 6 a t 2.45 GHz.
Figure 3.8 shows th e corresponding contour plot for a 50 G p ro b e a t 2.45 GHz.
T h e phase contour density for th e 16 Cl im pedance probe is less th a n th a t for the
50 Cl probe. T he m agnitude contour density is slightly g reater for th e probe with
56
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th e sm all im pedance in th e low loss tangent region. Slight variations of th e in n er
conductor, such th a t th e p ro b e’s im pedance ranged from 40 to 60 Q, generated
a m ore significant change in th e reflection coefficient contours a t 2.45 GHz th an
a t 915 MHz. Therefore, optim izing th e probe’s sensitivity also depends on the
frequency.
In general, the reflection coefficient can be scaled for various probe dimensions
w ith some constraints. T h e reflection coefficient for different p robe sizes is th e same
as long as th e ratio of th e inner an d outer conductors and w avelength tim es size of
th e outer conductor are held constant [41]. However this m eth o d was not used in
calculating th e d a ta tables for a larger 50 Q probe. Figure 3.26 shows the contour
plot of this larger probe a t 2.45 GHz. T his larger probe has a contour density which
is significantly higher for e < 30, b u t very low for e > 30. As th e probe increases
in size, the fields are m ore effectively launched into th e m aterial. This increased
interaction improves the probe sensitivity, especially for low- loss m aterials. As the
dimensions of a 50
probe increases, th e distance between th e conductors increases
an d requires more tim e for fields launched into the m aterial to re tu rn to th e probe.
Thus, th e phase shift in th e reflection coefficient at any specific e increases as the
dimensions of a 50 Q probe increases. In general, th e probe sensitivity could be
optim ized for a specific range in e , ta n 5, and frequency. O ptim izing th e probe
dimensions will invariable resu lt in probes w ith a non- 50 Q im pedance and conflicts
w ith m atching the im pedance of th e probe to the im pedance of th e VNA and
coaxial cable. This situ atio n could be corrected by using a binom ial impedance
m atching section. This section will m atch a non- 50 Q probe to th e VNA over
a lim ited frequency band an d allow efficient transm ission of a microwave signal
[57]. B roadband operation of th e probe would require a set of binom ial impedance
57
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m atching sections to cover th e entire frequency range; however, optim izing the
probe dimension only works over a particular frequency range.
58
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10Q
-150
90
Line - Magnitude
Dash - P hase (deg) Q
80
70
50
- r1 2 0
40
r90
30
,0.9
-60
20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Loss T a n g e n t
Figure 3.25: C ontour plot of th e m agnitude an d phase of th e reflection coefficient
over range in e' and loss tan g en t a t a frequency of 2.45 GHz. T he impedance of
th e probe is 16 Q. T he d iam eter of the inner conductor is 4.0 m m and of the outer
conductor is 5.3 mm.
59
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45-
-180
©30Q.
E
£25-
-120
-90
0.2
0.4
0.6
0.8
1
1.2
Loss Tangent
1.4
1.6
1.8
2
Figure 3.26: C ontour plot of the m agnitude and phase of the reflection coefficient
over range in d and loss tangent a t a frequency of 2.45 GHz. T he im pedance of
th e probe is 50 Q. T h e diam eter of th e inner conductor is 5.7 m m and of th e o u ter
conductor is 13.3 m m .
60
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.6
Summary and Conclusions
A stainless steel open- ended probe accurately m easured th e complex p erm ittiv ity
over a wide frequency range and up to 1000° C. T h e spring loading of inner conduc­
tor forced th e p ro b e to m aintain intim ate contact betw een the m aterial under te st
and probe. T his design elim inated any significant air gaps, which resulted from
differential th erm al expansion of the probe and m isalignment of the probe during
assembly. Full wave analysis of a m ultilayer dielectric body related the reflection
coefficient to th e com plex perm ittivity of th e m aterial under test. A ccurate dielec­
tric m easurem ents of a sam ple with a rough surface utilized this full wave analysis,
which m odeled th e sam ple surface as a separate layer and its properties determ ined
by a dielectric m ixing law. Relating th e m easured reflection coefficient to th e re­
flection coefficient a t th e probe head required calibrating the probe w ith three
known m aterials. Com parisons between th e spring- loaded probe and HP probe
agreed w ithin 5% for e' and 24% for e", which were less th a n the uncertainties in
the complex perm ittivity. Repetitive m easurem ents and error analysis determ ined
the uncertainty of th e complex perm ittivity m easurem ents and dem onstrated th a t
the probe could accurately measure samples a t elevated tem peratures. Dielectric
m easurem ents on nephelene syenite and alum ina com posite samples a t elevated
tem peratures by th e probe and a resonant cavity differed by about
8
% for e' an d
15% for e", w here ta n 8 > 0.05. Large m easurem ent errors occurred for sam ples
w ith very low loss. D uring therm al cycling, sm all errors in the m agnitude of th e
reflection coefficient generated large uncertainties in m easuring e"or ta n 8. Thus,
the lower lim it in dielectric measurements w ith th e spring- loaded probe is ta n 8
equal to 0.05.
61
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Chapter 4
D ielectric M ixing Laws Applied to
Alum ina Composites
4.1
Introduction
T h e interaction o f electrom agnetic waves a n d a homogeneous, dielectric m aterial
is characterized b y its complex perm ittivity. W ith in a heterogeneous m aterial, th e
dielectric constant an d electric field are functions of position. W hen the w avelength
of the incident ra d ia tio n is much larger th a n th e scale size of the variations in th e
m aterial, th e heterogeneous m aterial responds as a homogeneous m aterial w ith an
effective com plex perm ittivity. In this long w avelength lim it, dielectric m ixing laws
are used to estim ate th e effective perm ittiv ity of heterogeneous m aterials based
upo n the perm ittiv ities and volume fractions of th e constituents and th e overall
m icrostructure. D ielectric mixing laws have several other im portant uses. F irst,
th e basic form ulation of dielectric mixing laws is also applicable to other physical
properties, such as perm eability and therm al conductivity [10]. Second, selection
62
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o r form ulation of a m ixing law and com parison w ith actual m easurem ents help
to develop a physical u n d erstan d in g of heterogeneous m aterials. M ixing laws are
used to calculate th e effective p erm ittiv ity by idealizing particle shape, size, and
arrangem ent. T he incorporation of these generalized an d o th er m ateria l properties
into th e mixing law could d em o n strate th eir effect on the effective perm ittivity.
T hus, th e dom inant properties or system param eters could be identified and further
analyzed. T hird, dielectric m ixing laws can be used to design com posites w ith
specified properties for com m ercial applications or dielectric h e a tin g by R F waves
an d microwaves [62-64].
O ne of our research in terests is th e microwave heating of ceram ic m aterials.
Low- loss ceramics, such as AI 2 O 3 , MgO, SiaN4, AIN, and ZrOo, do not absorb
enough microwave power a t low tem p eratu res to raise their te m p e ra tu re s signifi­
can tly [65]. One technique to im prove microwave heating is to com bine microwave
absorbing m aterial w ith a low- loss (host) m aterial [9]. T w o additives, silicon
carbide and copper oxide, were found to increase e" of alum ina. T his approach
to increase a com posite’s loss tan g en t can be of practical as well as theoretical
interest. A lum ina/ silicon carbide com posites are high stre n g th m aterials with
stru c tu ra l applications. T h e a lu m in a / copper oxide com posite form s a copper ox­
ide spinel, which could possibly be used as a chemical sensor or a m odel m aterial
for alumina spinels. A nother reason in selecting copper oxide as a lossy additive
is th a t alum ina/ copper oxide com posites could be synthesized by two different
techniques, which yielded different m icrostructures. T h e m ateria l’s m icrostructure
can significantly affect its effective perm ittivity [13,66-69]. T herefore, predicting
th e effective perm ittivity of heterogeneous m aterials requires a n understan d in g of
its m icrostructure. Dielectric m ixing laws m ust combine this level of inform ation
63
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w ith th e perm ittivities and volum e fractions of constituents if th e mixing laws are
to be useful in predicting behavior in real system s. In this chapter, the com plex
p erm ittiv ity of alum ina com posites is m easured an d quantitatively modeled w ith
respect to th e concentration of lossy additives, m icrostructure, and density. Two
com posite systems, a lu m in a/ silicon carbide and a lu m in a / copper oxide, were pre­
pared by physically m ixing alum ina w ith one of th e lossy additives. A th ird system
was prepared by chemically precipitating copper oxide onto the surface of alu m in a
particles to produce a com pletely different m icrostructure. This system is also
com pared w ith the physically m ixed system w ith m atching composition.
T he effective p erm ittiv ity models developed for these com posite systems are
based on th e finite- difference electrostatic sim ulations done by Calam e et al. [13].
Calam e et al. found th a t com m on dielectric m ixing laws poorly predicted th e
effective perm ittivity of porous ZnO. They used electrostatic sim ulations where
th e two phase system (ZnO a n d air) was m odeled as ZnO spheres arranged in a
simple cubic lattice. W ith a dielectrically inactive, fractal- geom etry boundary
layer between spheres, these sim ulations agreed well w ith the m easured results.
In our work we have used a sim ilar approach to incorporate the general physical
characteristics of th e alum ina composites into an electrostatic model to provide a
m ore accurate prediction of th e ir effective perm ittivity. O ur model is also used to
deduce some m aterial properties. C onstituent p erm ittiv ity and phase changes can
be determ ined from dielectric m easurem ents and m ixing laws.
Background inform ation on common dielectric m ixing laws and a variation
of C alam e’s electrostatic m odel is presented in sect.
4.2.
Sample preparation
and m aterials characterization of alum ina com posites are described in sect.
In sect.
4
.3 .
4.4, the design and m odel of a nondestructive resonant cavity w ith a
64
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moveable wall, w hich m easured the complex p e rm ittiv ity of low- loss m aterials
w ith variable dim ensions, is described. E xperim ental results are presented and
compared w ith dielectric mixing laws and electro static simulations in sect. 4.5.
Section 4.6 provides a brief sum m ary of th e experim ental an d theoretical results.
4.2
Dielectric Mixing Laws
Dielectric m ixing laws are used to estim ate or calculate an average or effective
perm ittivity of heterogenous materials. T he effective perm ittivity depends up o n
the perm ittivities, volume fractions, and m icrostructure (size, shape, and arrange­
ment) of the con stitu en ts. Since exact solutions of th e effective perm ittivity are
only possible for a few cases, dielectric
m ix in g
laws are based on simplified approx­
imations of th e m icro structure of the m aterial [10,68]. For example, the particle
shapes need to b e idealized as spheres, ellipsoids, or confocal geometric shells.
Various arrangem ents of constituents lead to m any different assum ptions an d ap­
proximations. Therefore, there are num erous dielectric mixing laws. E xtensive
theoretical studies have been conducted on heterogeneous materials. Several re­
views and surveys on dielectric mixing laws were w ritten by Landauer [70], Van
Beek [71], Sihvola [10], B ergm an [1 1 ], and others [12,66,72]. In this chapter, th ree
algebraic m ixing laws were applied to predict th e effective perm ittivity of hetero­
geneous m aterials. T h e predicted perm ittivities from th e algebraic m ixing laws
are, in general, accu rate for composites w ith one dom inant constituent. Beyond
this lim ited dom ain of th e constituents, electrostatic sim ulations are needed to
accurately com pute th e effective perm ittivity of heterogeneous materials.
Besides th e aforem entioned applications, m ixing laws have various other appli-
65
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
cations. For example, th e com plex p e rm ittiv ity of homogeneous sam ples m ight be
difficult or impossible to m easure. Some solid m aterials are difficult to machine
to ex act dimensions or to o b ta in in a p u re form [1 2 ]. O ne technique m easures th e
p erm ittiv ity of a com posite m ateria l w hich is a m ixture of th e solid m aterial an d a
know n phase, such as air. D ielectric m ixing laws can th e n be used to ex trap o late
th e p erm ittiv ity of the solid m aterial. As a diagnostic tool, m ining laws have also
been useful in rem ote sensing of ore deposits [62] or w ater, detecting contam ination
in ground w ater [62] or ag ricultural p roducts [73], and detecting m oisture content
[63,74-76].
T h ere are m any different m ixing laws which provide different estim ates. T his
v ariatio n in estim ates highlights som e problem s w ith dielectric m ixing laws and
m akes it difficult to select th e b est m ixing law for a specific heterogeneous m a­
terial. O ne problem is th a t m ixing laws were developed for a certain dom ain of
th e system param eters, which are perm ittivity, volum e fraction, a n d m icrostructu re. A pplying mixing laws o utside th is dom ain leads to errors in th e estim ated
perm ittivity. A nother problem is th e assum ption th a t particles are im m ersed in
a uniform externally applied electric field, which only induces polarization w ithin
th e particles an d distorts fields in th e neighborhood of each particle. T his approx­
im atio n is only valid if th e in teractio n betw een neighboring particles is negligible.
As th e distance between particles decreases, th e induced charges on th e particles
in teract w ith each other and th e resulting changes in th e spatial v ariation of th e
electric fields between particles becom es significant. Therefore, higher m ultipole
interactions m ust be included in th e m ixing laws for close particle spacings. A th ird
problem concerns the contact betw een particles. P article contact strongly affects
th e dielectric properties of a heterogeneous m aterial.
T h e particle contact and
66
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m icrostructure of com pacted ceram ic particles are affected by synthesis and heat
tre a tm e n t. Therefore, the properties of th e particle contact need to be measured
an d incorporated into the m ixing laws.
A general description of th re e algebraic mixing laws and th e electrostatic model
is presented. Also, these dielectric m ixing laws can be used to predict the effec­
tive p erm ittivity for several two and th re e phase systems. T hese simulations help
to illu strate the applicability an d selection of these m ixing laws. T he most com­
m only used dielectric m ixing laws are Maxwell G arn ett th e o ry (MG), the effec­
tive m edium approxim ation (E M A ), an d th e Landau- Lifshitz- Looyenga formula
(LLL). Molecular field m odels, like M G, calculate th e local electric fields inside
inclusions immersed in a host m aterial, which is the m ajor phase [77,78]. W hen
th e inclusions m ake up a sm all volume fraction of th e sam ple, this model assumes
th a t th e individual particles weakly interact w ith each other. For isolated parti­
cles im m ersed in a uniform electric field, th e induced dipole m om ent and electric
field w ithin the particles is determ ined. T h e induced dipole m om ent and electric
field w ithin particles, which have sim ple shapes (like sphere an d ellipsoids), can
be solved exactly. T he higher order m ultipole moments are neglected. As derived
in A ppendix B, th e calculations of th e electric fields in th e inclusions and host
enable predicting th e effective perm ittivity. The predicted effective perm ittivity of
a system with N com ponents is [10,72]:
w here ee/ / is the effective com plex perm ittivity, eh is the com plex perm ittivity of
th e host m aterial, and e, is th e com plex perm ittivity of th e i th m aterial, and
is
th e fractional volume of th e i th m aterial. This model asym m etrically treats the
67
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
various constituents of the heterogenous m aterial. This approximation is accurate
for dilute system s where th e host m aterial is th e constituent w ith the largest volum e
fraction. As th e volume fraction of one of th e constituents is varied from zero to
one, there is som e am biguity in selecting th e host m aterial and inaccuracies in th e
predicted perm ittivity. Figures 4.1 - 4.12 show th e predicted perm ittivity of several
two an d three phase systems. Table 4.1 lists th e perm ittivity of the various phases
in these sim ulated systems. For dense system s, th e predicted perm ittivity could
have m ore th a n one value depending upon th e selected host phase. As th e volume
fraction of constituents vary, th e selected host m aterial could change. T hus, th ere
would b e a discontinuity in th e predicted perm ittivity. Also, these calculations of
th e MG model enable com parison w ith o th er mixing laws.
A nother class of dielectric mixing laws are effective m edium theories. T he ef­
fective medium approxim ation is the m ost widely used m ixing law of this class.
This model is also known as th e Polder- van S anten formula, th e B ottcher formula,
the Bruggem an formula, and th e Coherent P otential Approxim ation [10,79]. T h e
effective m edium approxim ation models th e constituents as surrounded by a host
or m edium w ith th e same (unknown) effective perm ittivity of the heterogeneous
m aterial [80]. T h e dielectric properties of th e host are th e effective complex per­
m ittivity of th e heterogeneous m aterial [72]. Therefore, th e effective p erm ittiv ity
is solved self- consistently, which is derived in A ppendix B [13]. Calculating again
the electric field in th e constituents, EM A predicts th a t th e effective p e rm ittiv ity
can be calculated according to the following equation [72]:
(4.2)
As shown in figures 4.1- 4.12, EMAl was used to predict perm ittivities for various
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure
P erm ittiv ity of
P hase
1
2
4.1
1.00055 - 10~5j
2
-
0
.0 0 2 . 7
e7
4.2
1.00055 - 10~5j
2
-
0
.0 0 2 j
e77
4.3
1.00055
10~sj
1 0
-
j
e7
4.4
1.00055 - 10~5i
1 0
- o.oiy
e"
4.5
1.00055 - 10~5j
1 0
-
0 .0 2
j
e"
4.6
1.00055 - 10~'°j
1 0
-
0 .1
4.7
1.00055 - 10~5i
2 0
-
4.8
1.00055 - 10~5j
2 0
-
4.9
1.00055 - 10~5j
50 - 5j
e7
4.10
1.00055
I0~3j
50 - 5j
e"
4.11
1.00055 - 10~5j
10
-
O .O lj
50 - 5j
e7
4.12
1.00055 - 10"5i
1 0
-
O.OI7
50 - 5j
e77
4.14
1.00055 - 10~5j
e'
4.15
1.00055
-
1 0 '5j
1 0
4.16
1.00055
-
10"5j
e"
0
.0 2 J
e7
0
.0 2 ./
e"
----- ►1
1
-
P erm ittiv ity
j
0 .0 1
j
t—*
:
0 .0 1
o
o
-
3
0
1
-
Effective
e7
0 0
50 - 5j
e7
50 - 57-
e77
Table 4.1: P erm ittiv ity of phases in th e following sim ulations.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
tw o an d th ree phase m aterials. Unlike th e previous model, th is m ixing law tre a ts
th e constituents sym m etrically an d provides a single com puted value. W hen one
co n stitu en t has a large volum e fraction an d th e to ta l volum e fraction of th e o th er
co n stitu en ts is very sm all, b o th equations (Maxwell- G arn ett an d Effective M edium
A pproxim ation) reduce to th e sam e form and calculate th e sam e result.
Power- law approxim ations are an im p o rtan t class of dielectric mixing laws [10].
W ith considerable success, th e L andau- Lifshitz- Looyenga form ula has accurately
p red icted th e perm ittiv ity o f solid m aterials from m easurem ents of the heteroge­
nous m aterial [12,73,81-88]. T h e L andau- Lifshitz- Looyenga form ula assumes t h a t
th e perm ittivities of th e co n stitu en ts are sim ilar an d isotropic. B y averaging over
th e volum e space of these co n stitu en ts w ith sim ilar perm ittivities, the effective
p e rm ittiv ity is [89,90]:
3
. i= l
A m ore thorough derivation o f th is form ula is presented by L andau and Lifshitz
[89]. T his formula tre a ts all co n stitu en ts sym m etrically. T he calculations of LLL
are also shown in figures 4.1 - 4.12. T his formula an d EM A have sim ilar calcula­
tions of perm ittivities over a range in volume fractions and perm ittivities of th e
constituents. For a sm all difference in perm ittivities of th e constituents, th e LLL
form ula is sim ilar to M G an d E M A [89]. T his sim ilarity is shown in figures 4.1
an d 4.2.
T he finite- difference electro static m odel com putes th e p erm ittiv ity of a phys­
ically realistic three- dim ensional representation of th e m aterial [13]. A cubical
volum e space is filled w ith co n stitu en ts of idealized shapes and sizes. T he vol­
um e fraction and p erm ittiv ity o f th e phases in th e m odel space m atch those of
70
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Electrostatic
MG1
MG2
LLL
EMA
1.8
1.2
0.1
0.2
0 .4
0.3
0 .5
0 .6
0.7
Volume Fraction of Second Phase
0.8
0.9
F igure 4.1: C alculated s' for a two phase system vs. the volume fraction of th e
second phase. T he p erm ittivities of th e phases are e* (lst phase) = 1.00055 —10“ 5j
and e*(2 nd phase) =
2
—0 .0 0 2 y.
71
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2.5,
Electrostatic
MG1
MG2
LLL
EMA
O)
0.5
0.1
0.2
0.3
0.4
0 .5
0.6
0.7
Volume Fraction of Second Phase
0.8
0.9
Figure 4.2: C alculated e" for a two phase system vs. th e volume fraction of th e
second phase. T h e perm ittivities of th e phases are e*(lst phase) = 1.00055 —10~5;
and £*(2 ^ phase) =
2
-
0
.0 0 2 ; .
72
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Electrostatic
MG1
MG2
LLL
EMA
CL
0.2
0.3
0.4
0 .5
0.6
0.7
Volume Fraction of Second Phase
0.8
0.9
Figure 4.3: C alculated s' for a two phase system vs. th e volume fraction of the
second phase. T he perm ittivities of th e phases are e*(lst phase) = 1.00055 —10-5j
an d e*(2nd phase) =
1 0
— 0 .0 1 /.
73
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0.012
Electrostatic
MG1
MG2
LLL
EMA
0.01
Q.
oo.ooe-
o .o o :
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Volume Fraction of Second Phase
0.8
0.9
Figure 4.4: Calculated e" for a two phase system vs. the volum e fraction of the
second phase. T he perm ittivities of th e phases are e*(lst phase) = 1.00055 — 1 0 -5 ;'
an d e*(2nd phase) = 10 —O.Olj.
74
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0.025-
'> 0.05 -
Electrostatic
MG1
MG2
LLL
EMA
0.01
O)
0.2
0.3
0.4
0.5
0.6
0.7
Volume Fraction of Second Phase
0.8
0.9
Figure 4.5: Calculated e" for a two phase system vs. th e volu m e fraction of the
second phase. T he perm ittivities of the phases are e*(lst phase) = 1.00055 —10-5,;
and e*(2nd phase) =
1 0
—0 .0 2 j .
75
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0.12
Electrostatic
MG1
MG2
LLL
EMA
0.1
CL
O0.06Q_
0.02-
- J > ___
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Volume Fraction of Second P hase
0.8
0.9
F igure 4.6: C alculated e" for a two phase system vs. th e volum e fraction of th e
second phase. The perm ittivities of th e phases are e*(ls£ phase) = 1.00055 —10_5j
a n d e*(2nd phase) = 10 — O .lj.
76
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E .S .
M G1
MG2
LLL
EMA
0.1
0.3
0.2
0.4
0.5
0.6
0.7
Volume Fraction of Second Phase
0.8
0.9
Figure 4.7: C alculated s ' for a two p hase system vs. th e volume fraction of th e
second phase. T h e perm ittivities of th e phases are e* (lst phase) = 1.00055 —10~5j
and em(2nd phase) =
2 0
-
0
.0 2 j .
77
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0.025
E .S . (regular)
E .S . (random )
MG1
MG2
LLL
EMA
0 . 02 -
0.01
05
0.005n
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Volume Fraction of Second Phase
0.8
0.9
Figure 4.8: C alculated e" for a two phase system vs. th e volume fraction of the
second phase. T he perm ittivities of th e phases are e*(ls£ phase) = 1.00055 —10~5j
an d e*(2nd phase) =
2 0
— 0 .0 2 j .
78
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60
Electrostatic
MG1
MG2
LLL
EM A
50
Q_
40
20
•4 -O -r -
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Volume Fraction of Second P hase
0.8
0.9
F igure 4.9: C alculated e' for a two phase system vs. th e volum e fraction of th e
second phase. T he p erm ittiv ities of th e phases are e’ ( l s£ phase) = 1.00055 —1 0 ~5j
an d e* (2nd phase) = 50 — 5j .
79
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Electrostatic
MG1
MG2
LLL
EMA
X
03
-e
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Volume Fraction of Second Phase
0.8
0.9
Figure 4.10: C alculated e" for a two phase system vs. th e volume fraction of th e
second phase. T h e perm ittivities of the phases are e*(lst phase) = 1.00055 —10-5j
an d e*{2nd phase) = 50 —5j .
80
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•= 25
Electrostatic
MG1
MG2
LLL
EM A
X 20
Q.
20
30
40
60
70
Volume Percentage of Third Phase
Figure 4.11: Calculated s' for a three phase system vs. th e volume fraction of
th e th ird phase. The volume fraction of phase
1
is held constant a t 40.7%. T h e
p erm ittivities of the phases are e*(lst phase) = 1.00055 — 1 0 ~5 j , e‘ (2nd phase) =
10 — 0 .0 1 .7 , an d e*(3rd phase) = 50 —5j .
81
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±1 3
Electrostatic
MG1
MG2
LLL
EMA
9-
2
0 1 .5 -
cn
0.5-
20
30
40
50
60
70
Volume Percentage of Third Phase
Figure 4.12: C alculated e" for a three phase system vs. th e volume fraction of
th e th ird phase. T he volum e fraction of phase
1
is held constant a t 40.7%. T he
perm ittivities of th e phases are e*(ls£ phase) = 1.00055 — 1 0 ~3j , e*(2nd phase) =
10 —0.0l j , an d e*(3rd phase) = 50 —5j .
82
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th e actu al com posite constituents. T his m odel space is th e n divided into n x n x n
cubical cells. In th e following sim ulations, n was set a t 64. T h e discretizing o f th e
model space yields som e individual cubical cells assigned different perm ittivities.
Therefore, th e m a jo r phase in a cubical cell is assigned th e perm ittivity of th e
entire cell. G rid p o in ts are located at th e vertices of th e cells. A potential of one
is assigned to th e to p face of the cubical m odel- space volume and zero is assigned
for th e b o tto m face. T hese boundary conditions produce a ‘capacitor’ out of this
model space [13]. N ext, grid points in th e in terio r of th e m odel space are assigned
initial potentials. N eum ann boundary conditions for potentials and perm ittivities
are applied to th e side faces of the m odel space. T h e conservation of the flux of
electric displacem ent defined a differential eq u atio n a t each grid point:
V • (e V $ ) = 0,
(4.4)
where $ is th e com plex potential. T he differential eq u atio n a t each grid point is
converted to an algebraic equation th a t relates to th e potential a t the grid point
to its neighboring p o tentials by way of th e finite differential approxim ation. T h e
complex potentials w ithin th e model space, w hich are defined by th e ensem ble of
equations, are calculated by an iterative routine. Convergence required th a t th e
changes in th e real and im aginary parts of th e com plex potentials are less th a n
lCn° an d 10- 6 , respectively. A fter convergence, th e electric fields at the b o tto m
face of th e model space were obtained, an d th e effective perm ittiv ity of th e m odel
space was th en calculated using Eq. 4.5.
^
C
J e E -d S
Q
=
K
hot tom surface
=
1------------=
(
Area, \
e‘" {T h ic k n e ss )
83
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( 4 '5 )
Sim ulations for th e alum ina com posites were perform ed by placing spheres
in a body- centered- cubic (bcc) arrangem ent w ithin th e cubical volume, which
is show n in figure 4.13. This is a reasonable approxim ation since alum ina and
lossy additives possess a spherical shape. T o m atch th e volume fraction of air
in th e m odel space, some spheres were ran d o m ly selected to represent air. For
th e physically m ixed composites, random selection designated the p erm ittiv ity of
th e spheres. For th e chemically p recipitated com posites, each sphere in th e m odel
space was labeled as alum ina. C oated phases w ere sim ulated by random ly selecting
th e o u ter layer cells of the alum ina spheres as copper oxide. The physical m odel
a tte m p ts to idealize T E M micrographs of th e co ated particles, which are presented
in sect. 4.3. T h e interface between th e spheres was varied for two different cases:
no con tact a n d contact. A noncontact interface left a space of air betw een th e
spheres. A contacting interface allowed spheres to touch.
As m entioned earlier, previous electrostatic sim ulations performed on two phase
system s accu rately com puted the effective p e rm ittiv ity of porous ZnO an d the
com m on m ixing laws did not [13]. T hree series of electrostatic sim ulations were
perform ed for a variety of volume fractions a n d perm ittivities of the constituents.
T hese predictions are compared w ith calculations from th e three algebraic m ixing
laws. T h e goal is to determ ine agreem ent of th e electrostatic with the algebraic
m ixing laws. T his com parison will enable selecting w hen to use a specific m ixing
law o r require th e electrostatic simulations.
In th e first series, th e electrostatic m odel was used to com pute the effective
p e rm ittiv ity for a two phase system. T he physical system consisted of spheres
arran g ed in a bcc lattice. The radius of th e spheres was varied in order to m odel a
range in volum e fractions of the second phase. A s show n in figures 4.1 - 4.10, th e
84
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Cell Number along Z Axis
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
C ell Num ber along Diagonal o f X Y Plane
Figure 4.13: Slice th ro u g h th e center of th e m odel space for spheres arranged in a
body- centered- cubic lattice.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
real an d im aginary p arts of th e com plex perm ittivities are co m p u ted an d com pared
w ith th e other dielectric m ixing laws. Figures 4.7 and 4.8 also show predictions
o f th e electrostatic model for spheres random ly placed in a b cc lattice.
Since
th is random arrangem ent of spheres is w idely used later in th is chapter, some
sim ulations were perform ed to verify agreem ent between spheres arran g ed placed
regularly or random ly in th e lattice. For sm all concentrations of th e second phase
in figures 4.1 - 4.10, all m odels agree, except for MG 2 where th e second phase is
th e host m aterial. For high concentrations of th e second phase, all m odels agree,
except M G l where the first p h ase is th e h o st m aterial.
E stim ates of th e MG
m odel are not expected to be correct w hen th e assum ed host p h ase is actually a
m inor phase. Since agreem ent betw een th e th re e mixing laws an d th e electrostatic
sim ulations varies w ith respect to in term ed iate volume fractions, n o n e of th e three
analytic mixing laws could be used over th e entire range of th e volum e fraction
o f th e second phase. For a fixed loss tan g en t of the second phase, figures 4.1,
4.3, 4.7, an d 4.9 show th a t th e calculated effective s' becomes m ore nonlinear as
th e difference between s ' of th e phases increases. As shown in figures 4.4, 4.5,
an d 4.6, th e calculated effective e" scales w ith th e e" of th e second phase. It is
also interesting to note th a t th e curves of th e electrostatic sim ulations and LLL
form ula intersect near 60% volum e of th e second phase. Using th e electrostatic
m odel, figures 4.7 and 4.8 show close agreem ent between th e calcu lated real and
im aginary parts of th e complex p erm ittiv ity of th e random and regularly placed
spheres in a bcc lattice.
Since th e actual heterogeneous m aterials prepared in this c h a p te r are about
58% t.d (or volume fraction of second phase), th e second and th ird series of sim u­
lations atte m p t to model m aterials relevant to this research. In th e second series,
86
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th e electrostatic sim ulations were used to com pute th e p erm ittiv ity for two slightly
different physical m odels. In th e first model spheres are random ly placed in a bcc
lattice.
T hese spheres can contact neighboring spheres.
In th e second model,
spheres are regularly placed in a bcc lattice. In order to m atch th e volume fraction
of th e phases in th e two m odels, th e diam eter of th e spheres was decreased in th e
second model. T hus, th e spheres in th e second m odel do n ot contact each other.
W ith th e volume fraction of th e second phase is fixed a t 58.9%, figure 4.14 shows
th e calculated average s? of th e m aterial for a r a n g e in e' of th e second phase.
T he calculated ef of th e electrostatic models are simila r an d diverges as s 7 of th e
second phase increases. T h e first m odel w ith th e contacting spheres possesses a
larger com puted e! th a n th e second model, because electric fields are focused more
through particle contacts.
This increases electric fields an d polarization in th e
second phase. Finally, th e electrostatic model and LLL form ula agree well over
th e entire range. T h is result is surprising since th e LLL form ula is derived for
constituents w ith sim ilar perm ittivities. T his agreem ent betw een models results
from com paring th ese m odels close to where th eir curves intersect. If these cal­
culations were perform ed for 40% t.d . samples, the electrostatic and LLL model
would significantly agree less.
T he th ird series of sim ulations models a th ree phase m aterial which is 59.3%
t.d., w here th e volum e fraction of th e first phase is fixed a t 40.7%. T h e physical
system consisted of spherical particles of two different phases arranged in a bcc
configuration. T he voids betw een sphere are assigned as air. T he spheres were then
random ly selected as second or th ird phases. T he effective real an d im aginary part
of the complex p erm ittiv ity are shown in figures 4.11 an d 4.12. Maxwell G arnett
theory provides for th re e possible results depending up o n th e m ajor phase. The
87
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50
45
40
E .S . (regular)
E .S . (random )
MG1
MG2
LLL
EMA
$3 0
CL
20
30
40
50
60
Real Part of Second Phase
80
90
100
F ig u re 4.14: Calculated s' for a two phase system vs. s' of th e second phase. T h e
volum e fraction of phase 1 (air) is held constant a t 41.1%.
88
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first phase is th e host in M G l an d th e second phase is th e host in M G2. Since
estim ates w ith th e th ird p hase as th e host provides very inaccurate results, th e
estim ates for MG3 are not p lo tted . All th e common m ix in g laws differ from th e
electrostatic m odel by at least 18% for e an d 35% for e". T h e reason for th e large
difference w ith th e electrostatic m odel is th a t th e c o m m o n m ix in g laws do n ot ac­
cu rately account for particle to particle interactions. Since th e perm ittiv ities of th e
various phases are widely different, LLL poorly estim ates th e effective p erm ittiv ity
of this system .
Finally, th e p erm ittivity o f heterogeneous m aterials due to its m icrostructure
is exam ined. W hen th e coating phase has a larger dielectric constant th a n th e core
or host m aterial, th e effective p erm ittiv ity of th e coated com posite is greater th a n
physically m ixed composites. T his has been dem onstrated by experim ental obser­
vations [6 6 ]. T he following electrostatic sim ulations of three phase system s also
prove th e dependence of p e rm ittiv ity on m icrostructure. Spheres were random ly
arranged in a bcc arrangem ent. T h e perm ittivities of th e various constituents are
listed in Table 4.1. For th e ran d o m system , th e perm ittivity of th e spheres axe
random ly selected as a second or th ird phase. For th e coated system , th e spheres
are com posed of th e second phase an d various concentrations of th e th ird p hase
coated these spheres. Then, th e effective real and im aginary p arts of th e com plex
p erm ittiv ity are com puted a n d shown in figures 4.15 and 4.16. B o th e' an d e" of
th e coated system are substantially higher th a n th e random system . B y placing
a higher perm ittiv ity phase onto th e spheres, th e electric fields were shielded for
a larger volume th a n if this high p erm ittiv ity phase was arranged as a random
m ix tu re of spheres. As th e exclusion of electric fields increases, th e effective per­
m ittiv ity increases. In th e co ated system , th e electric field is preferentially directly
89
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th ro u g h th e coated phase which also increases e " of th e m aterial.
90
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Voi. of 1st Phase = 31.93%
Coated System
o
s§
LU
Random System
10
15
20
25
Volume Percentage of Third Phase
Figure 4.15: C om puted ef for a th ree phase system vs. th e volume fraction of
th e th ird phase. T h e volume fraction of phase 1 is held constant a t 31.93%. T he
perm ittivities of th e phases are e*(lst phase) = 1.00055 — 10- 5 j , e’ {2nd phase) =
10 — 0.01/, a n d e*(3rd phase) = 50 — 5j .
91
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0.7
0.6
Vol. of 1st Phase = 31.93%
Coated System
> 0 .3
LLjO.2
0.1
Random System
10
15
20
Volume Percentage of Third Phase
25
Figure 4.16: C om puted si' for a three phase system vs. th e volume fraction of
th e th ird phase. The volum e fraction of phase
1
is held constant a t 31.93%. T h e
perm ittivities of th e phases are e*(lst phase) = 1.0005-5 — 10- 5 j , e*(2nd phase) =
10 — O.Olj, and e*(3rd phase) = 50 — 5j .
92
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4.3
Sample Preparation and Characterization
A lum ina com posites were prepared w ith a range of lossy additions and differ­
ent m icrostructures. T h e additives were selected to increase th e loss ta n g e n t of
th e com posite. Also, th e selected com posites are relevant to future processing of
com m ercially interestin g m aterials in microwave or conventional furnaces. T h e fol­
lowing describes th e selection of lossy additives for alum ina, sam ple p rep aratio n ,
an d m aterial ch aracterization techniques. Several different m aterials (copper(II)
oxide, silicon carbide, carbon [91], m anganese oxide, copper, nickel(II) oxide, iron
oxide, an d m olybdenum oxide) were sep arately com bined w ith th e selected host
m aterial, alum ina. T h e first four additives significantly increased th e loss tan g en t
of alum ina. C a rb o n an d manganese oxide w ere abandoned as additives because
they burned off or th erm ally decomposed below sintering tem peratures. Therefore,
silicon carbide an d copper(II) oxide were selected as additives. These com posites
also have possible relevance in the com mercial sector. T h e alu m in a/ silicon carbide
composites can b e used for structural and microwave absorbing applications. T h e
alu m in a/ copper oxide com posite forms a copper oxide spinel, which has p o ten tial
as a chemical sensor.
P rep aratio n m eth o d can affect the m icrostructure an d macroscopic properties.
A lum ina/ copper oxide composites were synthesized by ball milling and chemi­
cal precipitation. A lu m in a/ silicon carbide com posites were only prepared by ball
milling. Chem ically m ixed alum ina/ silicon carbide was n o t attem p ted due to th e
prohibitively h igh cost o f SiC precursor. Ball or physical mixing combines differ­
ent powders an d random ly intermingles them together. T his m ethod can produce
composites w hich are inhomogeneous on a m icroscopic scale [92]. Chem ical precip­
itation can produce m ore homogeneous com posites on th e microscopic scale th a n
93
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physically m ixed composites [17]. C hem ical precipitation also produces com posites
w ith differen t m icrostracture th a n physically mixing. As discussed in sect. 4.2, the
p e rm ittiv ity of coated spheres has a significantly larger complex p e rm ittiv ity th an
ran d o m m ix tu re of spheres. T hus, th e dependence of th e effective p erm ittivity
o n m icro stru ctu re is examined. T h e m icrostructure also plays a m ajo r role in the
re actio n sintering of alu m in a/ copper oxide, w hich is presented in C hap. 5 and of
a lu m in a / zinc oxide [93].
4.3.1
Sample Preparation
A lu m in a / copper oxide com posites were initially prepared by chem ical precipi­
ta tio n o f copper carboxylate salt onto th e surface of alum ina particles.
The
chem ical p recipitation process was based on W ilson and Roman’s precipitation of
lead carboxylate salts onto subm icron sized polystyrene latex spheres [94]. Ethyl
alcohol (P h arm co P roducts, Brookfield, C T ), deionized water; diethyl oxalate
(A ldrich C hem ical Co., Milwaukee, W I), copper acetate [Cu(O O CCH 3 ) 2 «H 2 0 ]
(Alfa A esar, W ard Hill, MA), sodium hydroxide (Fisher Scientific, Fair Lawn,
N J), polyvinylpyrrolidone (P V P ) (A ldrich Chem ical Co., Milwaukee, W I), and
SMS alum ina (Baikwoski Int. Co., C harlotte, NC) were used in chemical precipi­
ta tio n . V arious am ounts of alum ina particles were ultrasonically dispersed w ithin
a solution o f 800 m l of ethyl alcohol an d 5 g of PVP. T he polym eric dispersant,
P V P , hinders alum ina particles from agglom erating. 40 m l of diethyl oxalate and
20 m l of
1
M sodium hydroxide solution were th e n sim ultaneously dissolved in a
continuously stirred ethanol solution. D iethyl oxalate enabled su b stitu tio n of ox­
alate molecules in the solution and sodium hydroxide raised th e pH by reducing
b icarb o n ate products. Next, 20 g of copper acetate was dissolved into 400 m l deion-
94
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ized water. This 0.05 M solution of copper acetate was poured into th e ethanol
solution. This solution was continuously stirre d for 14 hours at room tem perature.
In this solution of 80% ethanol and 20% w ater, copper acetate was slightly less
soluble th an in water. Therefore, m ixing o f th e aqueous copper acetate solution
w ith the ethanol solution changed th e hydrolysis, stability, and speciation char­
acteristics of copper based solution complexes. Due to stability factors, copper
complexes were driven out of solution an d precipitated on the surface of alum ina
core particles. Also, th e su b stitu tio n of oxalates for carbonates w ithin th e precip­
itates created lattice stra in and reduced th e crystalline order of th e precipitate.
Continuing w ith the sam ple preparation, centrifugation was used to collect and
wash coated particles. T h e precipitates were washed two times in ethanol and
centrifugally separated.
P recipitates were th e n dried for several hours a t 60°C
an d subsequently calcined a t 350°C for 4 hours. H eating copper carbonate ox­
alates to 350° C converts th e m aterial into copper oxide. T his calcination drives off
carbonates and oxalates as CO and C O 2 .
Alum ina / copper oxide com posites were also prepared by ball milling. In order
to m atch the composition, phase, particles size, and im purities for b o th preparation
m ethods, the copper oxide for physical m ixing was prepared by the aforem entioned
precipitation process, excluding th e alum ina powder. Ball m illin g was perform ed
in a 500 ml polypropylene bottle. This b ottle w as half- filled w ith 0.64 cm diam eter
dense alumina cylinders (U.S. Stoneware, E a st Palestine, OH). A fter pouring in
measured amounts of alum ina and copper oxide, th e bottle was sealed and ro ta te d
for
13 hours on a ball mill (U.S. Stoneware). B o th preparation m ethods produced
composites with various concentrations of copper oxide. A lu m i n a / silicon carbide
composites were only prepared by ball milling one micron sized a-A l 2 0
95
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3
and /3-SiC
Com posite
A120 3 / SiC
A120 3 / C uO
A120 3 / CuO
Synthesis
Physical M ixing
Physical M ixing
Chem ical P recip itatio n
M ass Percentage
2.5
2.64
2 .6 6
of Additive
5
5.28
5.27
7.5
10.49
10.55
1 0
14.89
14.87
15
17.21
15.4
16.21
2 0
17.2
Table 4.2: C om position and Synthesis o f A lum ina Composites
(Alfa Aesar). Com posites were prepared w ith various concentration of SiC. T able
4.2 shows the mass percentage of additives in th e th re e alum ina com posite system s.
Com posite samples were form ed by uniaxially cold- pressing w ithout binder in
various m old sizes. T hese cylindrical samples were th e n cold- isostatically pressed
to ~300 MPa.
T h e average sam ple height was a b o u t 1.3cm and the diam eter
depended on the m old size. T h e samples were d ried in an oven at 120° C for 180
m in. T h e samples were stored in a desiccator u n til dielectric m easurem ents.
96
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4.3.2
Material characterization of alum ina/ copper oxide
composites
M aterial ch aracterizatio n was perform ed for th e chem ically precipitated alum ina /
copper oxide com posites to determ ine th eir com position, phase, and particle size.
T h e m icro stru ctu re of th e composites was also exam ined. This m aterial charac­
teriz atio n is also relevant for dielectric m ixing laws. In th e sect. 4.5, th e general­
ized p roperties o f th e m icrostructure of these com posites is incorporated into th e
electro static m odel. Physically realistic arrangem ents of constituents enable ac­
cu rate estim ates o f th e effective complex p erm ittiv ity . T he following sum m arizes
transm ission electron microscopy (TEM ), therm ogravim etric analysis (T G A ), Xray diffraction (X RD ), fourier transform infrared spectroscopy (FT IR ), and particle
size analysis o n th e alu m in a/ copper oxide com posites.
T h e dried p recip itate before calcination was actu ally a complex copper carboxyla te salt (C C O ), which contains oxalate and carb o n ate, on alum ina particles. F irst,
T E M analysis ascertained th a t CCO was co ated onto alum ina particles, as show n
in figure 4.17. T h e copper complexes were am orphous or of low crystalline order.
T his is im p o rtan t for coating particles. C hem ically precipitating copper oxide up
to 17% (mass) did not encapsulate th e alum ina particles. C om plete encapsulation
of th e alum ina particles was observed for com posites w ith 56.8% a lu m i n a. + 43.2%
copper oxide as shown in Appendix C. U pon calcining, T E M analysis determ ined
th a t copper oxide coatings on alum ina rem ained am orphous.
T h e com position o f these precipitates a n d th eir conversion to copper oxide
were fu rth er stu d ied by a Shim adzu TG A -50 (K yoto, Jap an ), which m easured
th e sam ple m ass w ith respect to tem perature. T herm ogravim etric analysis was
97
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Figure 4.17: T E M m icrograph showing m orphology obtained by coating copper
oxalate carb o n ate on to alum ina particles. T h e calcined sam ple is composed of
83% (mass) AI2 O 3 + 17% copper oxide.
98
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perform ed on. CCO and chemically p re c ip ita te d composites. A ppendix C shows
th e m ass loss of CCO and chem ically p recip ita te d composites w ith resp ect to
te m p e ratu re . B y ~235°C, a m ajo rity o f th e carbonates and oxalates are driven
from th ese sam ples. T he lowest te m p e ra tu re to drive off the volatiles or calcine
w ith o u t sintering th e m aterial was 350°C.
D uring chemical precipitation, n o t
all of th e copper from the copper a c e ta te solution coated the particles.
Som e
rem ained in th e discarded solution or w as lost during preparation. T herefore, it
was necessary to determ ine th e co n cen tratio n of copper oxide in th e com posites.
Also show n in A ppendix C, th e m ass ra tio of alum ina to copper oxide in th e
chem ically p recip itated composites was calcu lated an d shown in Table 4.2.
U sing a M agna-IR 550 S pectrom eter Series II (Nicolet), F T IR analysis was
perform ed on th e chemically p recip itated m aterials before and after calcining. T h e
presence of oxalate bands would d e m o n strate th a t oxalate ions were in co rp o rated
in th e p recipitate. T h e incorporation of th e se larger anions would result in la ttic e
stra in defects. T his decreases th e crystalline o rder of th e precipitates and enables
encapsulating o r coating alu m i n a, particles [94]. Also, th e absence of oxalates an d
carb o n ates from th e calcined sam ples w ould verify th a t th e selected calcination
tem p era tu re was sufficient. T he infrared ab so rp tio n peaks are sum m arized in T able
4.3. T h e absorption peaks at 1323, 1431, an d 1720 cm - 1 indicate th e presence of
oxalate bonds in th e uncalcined sam ples.
T h e absorption at these frequencies
increases as th e concentration of copper carb o n a te oxalate increases. Also, th ese
peaks m ostly disappear upon calcining, because carbonates and oxalates convert
into CO an d CO 2 gas. Therefore, h eatin g chem ically precipitated com posites a t
350°C for 4 hours was sufficient to rem ove volatiles. Finally, an absorption b a n d
does form a t 2346 cm - 1 and could result from carb o n bonds.
99
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W a v e n u m b e r ( c m *)
A s s ig n m e n t
1323
C om pound
oxalate
1366
v s {C — O n ) + v (C — Of)
carbonate?
1431
vs ( C - 0 ) + v ( C - C)
oxalate
1720
u a (C = 0 )
ox alate
1938
1976
T ab le 4.3: Infrared absorption peaks of copper carbonate oxalate, which is before
calcin atio n
C arb o n , hydrogen, and nitrogen com bustion analysis (O neidea, Whitesboro,NY)
a tte m p te d to determ ine th e com position of th e calcined p recip ita te d sample. This
analysis o n th ree samples determ ined th a t th e concentrations o f carbon, hydrogen,
an d n itro g en were all below th e precision levels of th e m achine, which was 0 .3 %.
T h e sam ples were a copper oxide p recip itate which was calcined a t 350°C for 4 h,
a copper oxide precipitate which was calcined a t 670°C for 10 h, and a chemical
p re c ip ita te d copper oxide (17.2%)
alum ina oxide (82.8%) com posite. Since the
am o u n t o f carbon in the samples was less th a n 0.3%, this possible contam inant
will insignificantly affect complex p e rm ittiv ity and eventually w ill be driven off as
C O an d CO? upon heating in a microwave or conventional furnace w ith an air
atm osphere.
X -ray diffraction (XRD) experim ents were performed by a R igaku 2 circle Xra y diffractom eter. Four powder sam ples were analyzed: the subm icron a lumina.
pow der, th e chemically p recipitated powder, th e chemically p recip ita ted
100
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9 4
.7 %
L a tt i c e p a r a m e t e r (A )
Com pound
2.5208
CuO
2.4576
C u20
2.3244
CuO
2.1349
CuO
1.8638
CuO
1.5088
CuO, Cu20
Table 4.4: X -ray diffraction analysis o f chem ically precipitated copper oxide
alum ina 4- 5.3% copper oxide com posite, and th e chemically p recip itated 82.8%
alum ina H- 17.2% copper oxide com posite.
X -ray diffraction p atte rn s for these
samples are show n in A ppendix C. T h e lattice constants and corresponding com­
pounds for these samples are sum m arized in tables 4.4 and 4.5. XRD analysis
found th a t th e copper oxide in th e com posites consisted of b o th copper(II) oxide
and copper(I) oxide. Also, it did not find peaks indicative of copper oxalate or
copper carb o n ate com pounds. Thus, calcining a t 350°C for four horns sufficiently
removed oxalates an d carbonates.
The particle sizes for th e subm icron alum ina and chemically p recip itated copper
oxide were m easured to characterize th e com posites and provide relative particle
sizes of constituents for th e electrostatic sim ulations. The particle size analyzer,
which is a M odel 3000 Zetasizer (M alvern, Southborough, MA), uses a fight scat­
tering technique and assumes th a t th e particles were spherical.
T h e m easured
diam eters of th e subm icron alum ina particles was 0.42 fim an d th e copper oxide
diam eter was 0.32 /rm. For com parison, th e m anufacturer’s specification sheet for
101
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L a ttic e p a ra m e te r
(A)
C om pound
7.3155
AI2 O 3
3.4817
AI2 O 3
2.5469
AI2 O 3
2.4576
CU2 O
2.3747
AI2 O 3
2.3244
CuO
2.0817
AI2 O 3
1.7413
AI2 O 3
1.6034
AI2 O 3
1.3739
AI2 O 3
Table 4.5: X -ray diffraction analysis of chemically precipitated copper oxide onto
alum ina
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
th e subm icron alum ina, m easured w ith G ranulom etric Sedigraph 5100, indicates
an average diam eter of 0.28 ,um. From th e T E M pictures, th e particle size of th e
copper oxide, which was p recip itated onto alu m in a particles, was approxim ately
70 nm . F or th e alu m in a/ silicon carbide com posites, a-A l 2 0
1
3
a n d /3-SiC were b o th
fim as rep o rted in th e m anufacturers d a ta sheets.
4.4
Complex Permittivity Measurements
T h e com plex perm ittivities of these com posite sam ples were m easured by, prim ar­
ily, a reso n an t cavity and a com m ercial open- ended coaxial probe. As discussed
in chap. 3, th e probe can m easure th e com plex p erm ittiv ity of m aterials w ith ta n
6 > 0.05. Therefore, it was used to m easure th e com plex p erm ittiv ity of porous sil­
icon carbide an d copper oxide sam ples. R esonant cavities have been widely used to
m easure th e com plex p erm ittiv ity in th e microwave frequency range and especially
for low- loss sam ples [61,74,95-99]. W hen a dielectric m aterial is placed inside a
resonant cavity, th e frequency of th e resonant m odes of th e cavity will shift and
th e resonance curve broadens depending upon th e m aterial’s com plex p erm ittiv ­
ity a n d dimensions. However, th e resonant cavity m ethod is usually considered
d estru ctiv e because th e sam ple has to be m achined to a specific shape and size
[100]. To improve experim ental flexibility and consistency, a nondestructive res­
on an t cavity w ith a moveable wall was constructed to accom m odate cylindrically
sh ap ed sam ples w ith variable dim ensions. A form ula is used to determ ine th e m a­
te ria l’s com plex perm ittivity based on th e sam ple dim ensions, resonant frequency,
an d Q- factor. T he following describes th e design an d m odeling of th e cavity.
T h e nondestructive resonant cavity was built to m easure th e p e r m i t t i v i t y of
103
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sam ples w ith a height between
12.7 a n d 31.8 m m .
8 .8
a n d 19.1 m m and a radius, ideally, between
Figure 4.18 shows th e schem atic diagram of th e resonant
cavity. T h e cavity was m ade of oxygen free high conductivity (O FH C ) copper.
L oading a resonant cavity w ith th ese large ceram ic samples will significantly de­
crease th e frequency of th e resonant m odes. For num erical m odeling of microwave
processing, it is desirable to m easure dielectric properties at or n ear th e operating
frequency o f th e microwave furnace, w hich is 2.45 GHz. Therefore, it was decided
th a t th e resonant frequency of th e T M 0 2 0 m ode in a cavity loaded w ith a typical
alumin a sam ple should be approxim ately equal to 2.45 GHz. To m atch th is design
p aram eter, th e inner diam eter of th e cavity was m achined to 16.400 cm . A nother
design p a ra m e te r required th a t th e m oveable wall be parallel to th e cavity bottom ,
which was square to th e cavity walls. Since th e angle difference betw een the per­
pendiculars of th e top and b o tto m walls is 0.013°, th e cavity walls are considered
parallel a t frequencies less th a n 5 G Hz.
T h e cavity plug allowed loading of th e sam ples into the cavity. T h e OFHC
moveable wall was positioned so th a t th e cavity height equalled th e height of th e
sam ple. G old contact strips (In stru m en t Specialties, Delaware W ater G ap, PA)
were soldered to the edges of th e cavity plug an d moveable wall. T hese strips
provided th e high frequency electrical contact betw een the cavity walls and plug.
T h e sta tio n a ry wall holder was fixed to 1.27 cm diam eter linear shafts. B y turning
a 1”-12 n u t, th e moveable wall moves vertically by sliding on 1.27 cm bore- closed
ball bushings. Since repeated sam ple loading would wear the th read s in th e O FH C
copper, th e blind tapped holes on th e b o tto m of th e cavity were th read ed w ith
1 0
-
32 Heli- Coil stainless steel inserts.
Coupling loops excited and m easured H^ m odes in the cavity. T hese antennas,
104
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w hich were placed 90° apart, were m ade by looping th e inner conductor of a 50 Q
semi- rigid coaxial cable and soldering it to th e outer conductor. Coaxial connectors
(SMA) were soldered on th e other end of these coaxial cables, which were feed
th ro u g h holes in th e cavity walls a n d held into place by modified male C ajon
u ltra - to rr connectors. As shown in figure 4.19, a vector network analyzer (VNA)
H P 8520C, which was connected to th e loop couplers via precision coaxial cables,
sw ept th e frequency and m easured th e tra n sm itte d signal.
T h e resonant frequency, / 0, and Q -factor of th e cavity a t several T M mo modes
were m easured w ith and w ithout th e sam ples in the cavity in order to determ ine
the cavity losses. To reduce m easurem ent uncertainty, the m easured transm itted
signals were averaged from sixteen traces. T he peak and 3 dB points of resonant
m odes betw een
1
and 4 GHz were found an d recorded. T he following exact ana­
lytical solution allowed com putation of complex perm ittivity of th e sam ple based
on these m easurem ents. As shown in figure 4.20, the sample, w hich is centered in
th e cavity an d labeled as region 1, has m aterial properties of e\ and /q . Region 2
has m aterial properties of e2 and fi2- T h e resonant cavity has an inside diam eter
of 2b an d th e sam ple diam eter has a diam eter of 2 a. The height of th e cavity and
sam ple is I.
T h e electric and magnetic fields w ithin th e cavity interact w ith th e m aterial.
To relate th e shift and broadening of th e resonant modes, the electric, fq, fields are
m atched a t each interface. T h e entire solution for th e resonant frequency and Qfactor of a loaded cavity is shown in A ppendix D. Briefly, the general expressions
for an electric field in axial direction an d region i is:
105
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Stationary Wall
Holder with
Threaded Rod
and Nut
16.400 cm
Linear Shaft
Moveable Wall
with Linear
Bearings
3.564 cm
2.54 cm
►i
OFHC C opper
Gold Contact Strips
Profile View
Cavity Plug
Loop Coupling
Ports
Top View
Figure 4.18: Schematic diagram o f th e resonant cavity w ith a m oveable wall.
106
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Figure 4.19: P icture of the vector netw ork analyzer connected to th e resonant
cavity w ith a moveable wall.
M--------- 2b
-------------------------------------------------------- »
1
T
K -2 a
(2)
i
i
-H
:a a s a a ^ 5 g ^ K s a s s a a a s i
|42 £ 2
Figure 4.20: Circular cavity w ith a dielectric sam ple in region 1 w ith eiand fxl .
Region 2 is filled with e2 and fj,2-
107
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Ezi = (AiJm (a iP) + G iN m (otip)) (CtelTn4>+ A e _im0) (E t cos (kz) + Fx sin (kz))
(4.6)
= n fy j1 - k2
(4.7)
Since the diam eter o f th e cavity is greater th a n its height, th e lowest order m odes
in a cylindrical cav ity are T M modes, where B z — 0. B oundary conditions a t
the top an d b o tto m ends of th e cavity require th a t th e electric field parallel to a
conducting surface to be zero. Thus,
kl = pir,p = 0 ,1 ,2 ...
Applying this b o u n d ary condition a t the cavity walls and m atching the tangential
electric an d m agnetic fields a t th e sample interface provides th e following relation­
ship:
ex
fm(aid) _ 62_ Jm (a 2b) N'm (a2a) - J ( a 2a) N m (a 2b)
a: Jm (aid)
a2 . Jm {a2b) N m (a 2a )
Jm (a 2a) N m (a 2b)
(4.8)
The complex p e rm ittiv ity of th e sample can th en be calculated from the m easured
complex angular frequency of th e cavity (Eq. 4.9) an d solving Eq. 4.8.
“
=2 "(/o+j4 )
( 4
' 9 )
where Qs is the Q -factor of th e sample.
Q,
Q,
Q.
108
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4I0)
where Qi is th e Q -factor of a loaded cavity and Qe is th e Q -factor of th e em pty
cavity. A two variable m inim ization routine determ ined th e com plex p erm ittiv ity
of th e m aterial for a given estim ate of th e value. T h e error in calculating e' was
ab o u t 0.005% and e" is a b o u t 0.14%.
4.5
Results and Discussion
In this section, th e com plex perm ittivity m easurem ents of alum ina com posites
are presented.
T h e dependences of th e p erm ittiv ity of alum ina com posites on
concentration of lossy additions, m icrostructure, and density are exam ined. T h e
electrostatic model a n d th ree algebraic m ixing laws are used to predict th e ef­
fective perm ittivities o f th e composites. T he accuracies of these predictions and
comparisons of these m ixing laws are also discussed.
Resonant cavity m easurem ents of th e alum ina com posites were perform ed at
three different T M mo m ode frequencies, n = 1, 2, and 3. D ue to th e different
sam ple sizes and perm ittivities, the resonant frequencies slightly differ. Analysis
of th e resonant frequencies and Q values of th e m odes leads to a set of values for
th e complex p erm ittiv ity a t different frequencies. In order to com pare different
samples, these values w ere used to predict th e p erm ittiv ity a t a designated “av­
erage” frequency. At 2.59 GHz, th e m easured com plex perm ittivity of alu m in a/
silicon carbide com posites are shown in figures 4.21 and 4.22. For each concentra­
tion of silicon carbide, tw o samples were prepared an d measured. Slight variations
in preparation resulted in samples w ith slightly different densities or volum e frac­
tions of air. T he real p a rt of the complex p erm ittiv ity increased linearly w ith th e
concentration of silicon carbide. T he im aginary p a rt of the com plex perm ittiv-
109
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ity has a nonlinear dependence on volum e fraction of additive. In calculating th e
p e rm ittiv ity from th e m easured com plex an g u lar frequency, u n certa in ties in th e
perm ittiv ities were calculated based on uncertainties in th e m easured frequency
an d Q -factor. For A / = 500 kHz and A Q /Q = 0.02, th e u n certain ty in e' was
0.3% an d e" was 2%. T he v ariation in sam ple density is th e m ain reason for
differences in p erm ittiv ity of com posites w ith th e sam e concentration of additives.
T h e m easured complex p erm ittiv ities of a lu m in a / copper oxide com posites are
show n in figures 4.23 and 4.24 a t 2.66 GHz. For each concentration of copper oxide,
two sam ples were prepared an d m easured. Slight variations in p rep aratio n resulted
in sam ples w ith slightly different densities or volum e fractions of air. T h e real p art
o f th e com plex p erm ittivity increased linearly w ith concentration of copper oxide.
T h e im aginary p a rt of the com plex p erm ittiv ity increased w ith th e concentration
of copper oxide an d strongly depended on p rep aratio n m ethod. T h e chemically
p recip itated com posites had a significantly higher e" th an th e physically mixed
com posites. A lthough the m easured perm ittivities of the lossy additives are pre­
sented later, th e following com pares th e perm ittiv ities of th e alum ina com posite
system s. T he real p a rt of th e complex p erm ittiv ity of th e alum ina com posites
increases m ore w ith additions o f silicon carbide, because ef of silicon carbide is
m uch larger th a n copper oxide. Since th e loss tan g en t of copper oxide is larger
th a n silicon carbide, th e im aginary p a rt of th e com plex perm ittivity of th e a l u m i n a
com posites increases more w ith additions of copper oxide.
Dielectric m ixing laws are used to predict th e effective p erm ittivity based upon
th e perm ittivities an d volume fractions of th e constituents and th e overall mi­
crostructure. Therefore, the volume fraction and perm ittivities of th e constituents
in these system s needed to be m easured. T he volume fraction of th e constituents
110
with permission of the copyright owner. Further reproduction prohibited without permission.
Alumina / Silicon Carbide
0 5 .5 -
5
10
15
Volume Percentage of Silicon Carbide
F ig u re 4.21: Real part of th e com plex p erm ittiv ity of a lu m in a/ silicon carbide
com posites was measured by th e nondestructive resonant cavity a t 2.59 GHz vs.
concentration of silicon carbide.
For each concentration of silicon carbide, two
sam ples were prepared and m easured.
ill
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o.os~o.07|-
x
Alumina/ Silicon Carbide
CD
CL
0.00-
X
x
0)
Q.
Eo.oe
o
o
M—
Oo.(M
•c
X
X
CD
CL
_i0.03(O)
CD
E
0 . 02-
0.01L
5
10
15
Volume Percentage of Silicon Carbide
F igure 4.22: Im aginary p a rt of th e complex perm ittivity of a lu m in a./ silicon carbide
composites was m easured by th e nondestructive resonant cavity' at 2.59 GHz vs.
concentration of silicon carbide. For each concentration of silicon carbide, two
sam ples were prepared an d measured.
112
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CD
&
* Alumina/Copper Oxide : Phys. Mixed
+ Alumina/Copper Oxide : Chem. Prec.
3-95-
2
3
4
5
6
7
Volume Percentage of Copper Oxide
Figure 4.23: R eal p a rt of th e complex p e rm ittiv ity of
alu m i n a /
copper oxide com­
posites was m easured by th e nondestructive resonant cavity at 2.66 GHz vs. con­
centration of copper oxide. For each concentration of copper oxide, two sam ples
were prepared an d m easured.
113
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.14>*
£ 0 .12-
=< Alumina/Copper Oxide : Phys. Mixed
+ Alumina/Copper Oxide : Chem. Prec.
L_
<D
CL
%
X 0.1niL
<D
Q_
E
o
O 0 .0 £ «+—
o
■c
□ V o ed)
03
— 0.04-
+
*
-i-
1
2
3
4
5
6
7
Volume Percentage of Copper Oxide
Figure 4.24: Im aginary p a rt of th e complex p e rm ittiv ity of alu m in a/ copper oxide
com posites was m easured by the nondestructive resonant cavity a t 2.66 GHz vs.
concentration of copper oxide. For each concentration of copper oxide, two samples
were prepared an d m easured.
114
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are shown in figures 4.25 and 4.26. A th ird order polynom ial was fitted to the
d a ta so th a t the algebraic m ixing laws could be applied over th e entire range of
volume fraction of lossy additions and not ju s t discrete points. N ext, th e com­
plex p erm ittiv ity of th e constituent m aterials was measured. To o b tain fully dense
sam ples, a micron and subm icron alum ina pure alum ina samples were sintered
in air to 1400° C and m easured in a resonant cavity. T he p erm ittiv ity of micron
alum ina was e*(2.2352GHz, p = 96.8% theoretical density) = 9.22 - 3.7xl0~3/
an d subm icron alum ina was e*(2.2582GHz, p = 97.5% t.d.) = 8.82 - 2 .7 x l0 -4/ .
All four dielectric mixing laws were used to extrapolate th e perm ittivities of solid
subm icron and micron- sized alum ina samples. Due to difficulties in producing
solid copper oxide and silicon carbide sam ples, a commercial open- ended coaxial
probe m easured th a t th e p erm ittiv ity of a porous copper oxide sam ple, which was
e*(2.66GHz, p = 63.0% t.d.) = 4.87-0.37/, an d a porous silicon carbide sample,
which was e*(2.59GHz, p = 62.4% t.d.) = 20.4 - 1.52j . All four dielectric mixing
laws were used to extrapolate th e perm ittivities of solid copper oxide and silicon
carbide. In calculating the effective p erm ittiv ity of a heterogeneous m aterial, the
prediction of a given m ixing law was based on th e extrapolated perm ittivities of
th e solid m aterials, which was derived from th e sam e mixing law.
T he applicability of mixing laws and electrostatic solutions requires th a t the
wavelength of the electrom agnetic rad iatio n m ust be significantly greater th a n the
typical size of th e particles w ithin th e m aterial [10,13]. T he w avelength of the
incident microwaves w ithin silicon carbide, which possess th e largest perm ittivity
of all constituents, is 1.7 cm and th e skin d ep th is 7.0 cm.
Since th e average
particle size is 1 p m for th e a lu m in a/ silicon carbide composites and 0.3 p m for th e
alu m in a/ copper oxide composites, th e com posites can be treated as a homogeneous
115
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0.5E-
Alumina
D)
CD
c 0.5-
0.4E-
Air
0.4
- - 4- _
- t ___
0.3E-
0 4
Volume Percentage of Silicon Carbide
Figure 4.25: Volume Fraction of A lum ina and A ir vs.
Silicon Carbide w i t hin
A lum ina/ Silicon C arbide Com posites. Two sam ples were prepared for each con­
centration of silicon carbide.
116
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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Alumina
0 ) 0 .5 -
0.4E-
Air
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Volume Percentage of Copper Oxide
Figure 4.26: Volume F raction of A lum ina and Air vs. C opper O xide w i t h in Alu­
m in a / Copper Oxide Com posites. Two samples were p rep ared for each concentra­
tio n of copper oxide.
117
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m edia an d th e electrostatic simulations are valid.
A n interesting result is th a t e" value of th e sintered alum ina sam ple was less
th a n th a t of porous alum ina samples. Since th e com plex perm ittivity is expected
to increase w ith density, s o m e t h i n g m ust have decreased e" of these alum ina sam ­
ples d u ring heating. W ithout accounting for th is discrepancy in the im aginary
p a rt o f th e com plex perm ittivity, the predicted e" from th e mixing laws, which use
e" value of th e sintered alum ina sample, would be significantly lower th a n e" of
alum ina com posites. W ater content of th e porous alum ina samples is a distinct
possibility. Several papers have discussed th e im p o rtan ce of w ater content on th e
com plex p e rm ittiv ity of various m aterials [66,95,101-103]. A recent p aper m ea­
su red a “clear correlation between hum idity an d high dielectric losses [103].” Also,
som e qualitative experim ental observations found th a t e" of alum ina composites
significantly increased if exposed to hum id air for as little as one hour. Therm o­
gravim etric analysis was performed on porous alu m in a particles to find how much
m ass is rem oved during heating. T he subm icron sized alum ina had a percentage
m ass loss of 0.19 ± 0 .04% and th e micron sized alu m in a had a percentage mass loss
of 0.06 ± 0.03%. A ssum ing th a t a m ajority of th is removed m atter is w ater, there
was an insufficient am ount of water to form a m onolayer on th e alum ina particles.
T hus, bound w ater m ust be discretely positioned on th e alum ina particle surfaces.
Published p erm ittiv ity m easurements of bound w ater found th a t it is less th a n th e
bulk value [66,74], an d the approxim ate p erm ittiv ity of bound water at 2.66 GHz
is ab o u t 60.8—11.2 j. Incorporating this ex tra phase into th e three different m i x i n g
laws did not provide an accurate prediction of th e p erm ittiv ity of porous alu m i n a
as show n in figures 4.28 and 4.30. E lectrostatic sim ulations attem pted to m odel
a porous alum ina sam ple by spot coating w ater onto alum ina spheres. C ontact-
118
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ing a n d noncontacting sphere arrangem ents were tested. These calculations were
sim ilar to th e o th er mixing laws predictions.
A n o th er physical model postulated th a t bound w ater is located at contact re­
gions. W ater is b o th chemically and physically bonded to a collection of particles.
In itial drying of th e particles to 120°C will drive off a m ajority of the physically
absorbed w ater. T he rem aining water would be preferentially located in th e con­
ta c t are a betw een spheres due to capillary forces. Electrostatic sim ulations were
perform ed again w ith a uniform am ount of w ater in the contact area between
spheres. However, th e am ount or volum e fraction of w ater is less th an th e vol­
um e of cells in th e contact region. Therefore, th e perm ittivity of th e contact area
was approxim ated as proportional to th e am ount of alum ina and bound w ater in
parallel. E lectrostatic simulations w ith w ater in contact regions accurately pre­
dicted th e perm ittivity of porous alum ina as show n in figures 4.28 and 4.30. Due
to th e uncertainty in the am ount of w ater, th e uncertainty in the sim ulations was
A d = 0.9% and A d ' = 15% in the porous subm icron alum ina and A d = 0.6% and
Ae" = 32% in th e porous micron alum ina. It is interesting how a small, but spe­
cific arrangem ent of lossy m aterial can greatly increase the effective perm ittivity
of a system . Previous electrostatic sim ulations on porous m icrostructures found
significant field intensification near grain contacts [28,35]. These intensified fields
coupled w ith th e losses of w ater located in these regions and therefore enhanced
d ' of th e composites.
T h e complex perm ittivities of these four phase system s were calculated by the
four dielectric mixing laws. The p erm ittiv ity m easurem ents and predictions of
th e alu m in a/ silicon carbide samples are presented in figures 4.27 and 4.28. In
th e electrostatic model, equal size spheres were arranged of alum ina and silicon
119
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Alumina / Silicon Carbide
Electrostatic Model
L-L-L Model
EMA Model
MG Model
GU.5-
Volume Percentage of Silicon Carbide
Figure 4.27: M easured and calculated e' of alum ina/ silicon carbide com posites at
2.59 GHz vs. volum e fraction of silicon carbide.
carbide in a bcc lattice and a small am ount of water was incorporated a t th e contact
regions. For th e Maxwell G arnett model, the m ajority phase o f these composites,
alum ina, was selected as th e host phase. T h e electrostatic, LLL, a n d EM A models
predicted e' w ithin 12%. Also, the electrostatic sim ulations p red icted e" w ithin
12%. T h e electrostatic model com puted th e base line value (porous alum ina) and
trend of the p erm ittiv ity b e tte r th a n th e oth er models, especially for th e im aginary
p art of th e complex perm ittivity.
120
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o Electrostatic Model
.... L-L-L Model
EMA Model
-- MG Model
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Volum e Percentage of Silicon Carbide
Figure 4.28: M easured and calculated e" of alu m in a/ silicon carbide com posites at
2.59 GHz vs. volume fraction of silicon carbide.
121
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The p e rm ittiv ity m easurem ents and predictions of th e alum ina/ copper oxide
samples are presented in figures 4.29 and 4.30. Again, alum ina was selected as th e
host phase in th e M axwell G arnett model. L andau- Lifshitz- Looyenga and EM A
models predicted d w ithin
8
% for b o th p rep aratio n m ethods.
A lthough these
algebraic m ixing laws did m atch the tre n d for lossy additives for th e physically
mixed system s, th e y incorrectly predicted e"of th e porous alum ina sam ple or th e
base line value. Also, th e predictions of these m odels do n ot differentiate due to
com posite m icrostructure. For the physically m ixed com posites, the electrostatic
m odel employed equal size spheres of alum ina an d copper oxide in a bcc lattice
an d incorporated a sm all am ount of w ater a t th e contact regions. This m odel
accurately com puted e' and e" of the physically m ixed system s within 8 % and
1 2
%,
respectively. For th e chemically precipitated com posites, th e model placed alum ina
particles in a bcc lattice w ith water a t th e co n tact regions. Various am ounts of
copper oxide were sp o t coated onto the top layer of th e alum ina particles. T h e
predictions of e' were w ithin 7% and e" were only w ithin 25%. Above 6 % (volume)
of copper oxide in figure 4.30, the sharp increase in th e m easured e" was not
predicted by the electrostatic simulations or o th e r mixing laws. T he rem ainder
of this section explains th e basis for the poor predictions of e" of the chemically
precipitated com posites w ith a concentration of copper oxide greater th a n
6
%
(volume).
The electrostatic sim ulations were used to predict th e effective p erm ittiv ity
of physically m ixed and chemically precipitated a lu m in a/ copper oxide compos­
ites. The general dependence of the perm ittivity due to concentration of additives,
density, and synthesis was predicted by th e electrostatic simulations. However,
these simulations did not agree with th e m easured e" of th e chemically precipi-
122
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Figure 4.29: M easured and calculated s ' of alum ina/ copper oxide composites at
2.66 GHz vs. volume fraction of copper oxide.
123
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Figure 4.30: M easured an d calculated s" of alu m in a/ copper oxide composites at
2.66 GHz vs. volume fraction of copper oxide.
124
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ta te d composites w ith large concentrations of copper oxide. Above 6 % (volume) of
copper oxide, th e sh a rp increase in m easured e" could have resulted from percola­
tio n or continuous conducting p a th through th e sam ple. In th e actual com posites,
a monolayer or sufficient coating of copper oxide on th e alum ina particles could
have produced a continuous conducting p a th and cause a large increase in e". T he
following discusses w hy this explanation does not account for th e sharp increase in
m easured e". In m odeling these com posites, copper oxide was random ly d istrib u ted
o nto the surface o f alum ina particles. However, th e coarse m esh size of th e m odel
could have lim ited rep resentation of some m icrostructure features of th e coated
m aterial. Therefore, several sim ulations w ith a finer m esh size were perform ed on
a three- phase system : air, alum ina, and copper oxide. The goal was to determ ine
if random ly coating various am ounts of copper oxide onto alum ina, which are ar­
ranged in a bcc lattice, contributes to percolation a t or below full encapsulation.
Figures 4.31 an d 4.32 show th e com puted perm ittiv ity of alum ina spheres w ith var­
ious am ounts of copper oxide coating. T here is no sharp increase in d a n d e" of th e
sim ulated com posites before or after com plete encapsulation prim arily. T h e lack
o f a sharp increase in p erm ittiv ity is prim arily due to th e relatively sm all e" of th e
chemically p recipitated copper oxide. Therefore, another mechanism m ust account
for th e sharp increase in e" of th e chemically precipitated alum ina com posites.
T h e measured p erm ittiv ity of th e copper oxide was based on th e pow der used
in th e physically m ixed system s. However, th e dielectric properties of th e copper
oxide in the chemically precipitated com posites could differ due to particle size,
phase, and, possibly, contam ination. Even though sam ple preparation a ttem p ted
to control these param eters, T E M analysis found th a t th e approxim ate particle
size of copper oxide in th e chemically precipitated composites was about th ree or
125
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10
12
14
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Volume Concentration of Third Phase
Figure 4.31: C om puted s ' of alum ina particles coated w ith copper oxide vs. volume
fraction of copper oxide. Monolayer encapsulation of alum ina spheres occurs a t
~ 1 0 % of copper oxide.
126
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Volume Concentration of Third Phase
F igure 4.32: C om puted e" of alum ina particles coated w ith copper oxide vs. volume
fraction of copper oxide. Monolayer encapsulation of alum ina spheres occurs a t
~ 1 0 % of copper oxide.
127
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four tim es less th a n in th e physically m ixed com posites. Since several studies have
found th a t e" is inversely proportional to p artic le size [98,104,105], th e perm ittiv­
ity of copper oxide in th e chemically p recip ita te d system s could be larger th a n the
o th er system . M ore su b stan tial evidence was found in X -ray diffraction analysis
of these com posites, which is shown in A p p en d ix C. As shown in figure C.7, th e
relative am ount o f copper(I) to copper(II) oxide is 20% less in th e com posite w ith
5% (mass) copper oxide th a n in the com posite w ith 17%. T hus, th e com posite
w ith 17% copper oxide has a higher percen tag e am ount of conducting m aterial
(copper(II) oxide) th a n th e composite w ith 5% copper oxide. T his relative shift
in th e conducting phase of th e precipitate, w here th e p erm ittiv ity of copper(II)
oxide is 51- 93j , explains th e sharp increase in e" of th e chem ically precipitated
com posites. If ~3% of copper(I) oxide changes into copper(II) oxide, electrostatic
sim ulations calculated th a t th e effective e" o f th e m ixture of copper(II) and copper(I) oxide w ould increase by 75%. W ith such a n increase in e" of th e chemically
precip itated copper oxide, th e electrostatic sim ulations com puted an effective e"
in agreem ent w ith th e m easured e" of th e com posite w ith 17% copper oxide. Ap­
parently, th e m ain error in predicting th e effective p erm ittiv ity of th e chemically
p recip itated com posites resulted from using a n in accu rate value of th e p erm ittiv ity
of copper oxide in th e system . This error can b e corrected by accurately m easuring
th e p erm ittivities o f th e copper oxide, which is coated onto th e alum ina particles.
4.6
Conclusions
Silicon carbide an d copper oxide were, separately, com bined w ith alum ina to in­
crease its effective perm ittivity. The dependence of p erm ittiv ity on m icrostructure
128
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was exam ined by varying th e prep aratio n m ethod of alu m in a/ copper oxide com­
posites. M aterial characterization perform ed on alum ina/ copper oxide composites
found th a t m ixtures of copper(II) and copper(I) oxides coated th e alum ina parti­
cles. T h e complex perm ittivities of com posites w ith various dim ensions were mea­
sured by a nondestructive resonant cavity from 1 to 4 GHz. T he w ater content and
m icrostructure strongly affected th e perm ittiv ity of porous alum ina composites.
Since d of silicon carbide was m uch larger th a n copper oxide, e7 of th e composites
increased more w ith additions of silicon carbide. Since th e loss tan g en t of copper
oxide was larger th a n silicon carbide, e" of th e composites were increased more w ith
additions of copper oxide. T he perform ance of several mixing laws (MG, EMA, and
LLL) in predicting th e effective perm ittiv ity of these heterogeneous samples was
investigated. Although EM A and LLL were able to accurately predict e7. they did
not properly account for th e contribution of bound water, leading to inaccuracies
in th e com putation of e". E lectrostatic sim ulations were extended to these four
phase systems. By idealizing a physically realistic configuration of these phases,
th e effective perm ittivity of th e com posites were accurately predicted except for
e77 of th e chemically precipitated a lu m in a/ copper oxide com posites.
For these
composites, the electrostatic m odel predicted lower values of e". XRD analysis
discovered higher concentration of conducting phase (copper(II) oxide) in some
chemically precipitated com posites th a n physically mixed com posites. Therefore,
e77 of copper oxide in some chemically precipitated composites would be higher
th a n in th e physically m ixed com posites. T hus, th e electrostatic sim ulations and
o th er mixing laws underestim ated e" in some chemically precipitated composites.
129
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Chapter 5
Microwave and C onventional Processing
o f Alum ina C om posites
5.1
Introduction
Microwave processing of m aterials possesses som e significant advantages over con­
ventional heating. In conventional therm al furnaces, th e sam ple surface is heated
an d therm al conduction transports the energy in to its interior. T he processing
tim e prim arily depends upon th e therm al co n d u ctio n of the sample. In order to
minimize therm al gradients w ithin a sample, conventional processing is usually
lim ited to slow heating rates. However, m icrow ave radiation can volum etrically
deposit energy w ithin a dielectric (or insulating) m aterial. Since th e processing
tim e is alm ost independent of th e m aterial’s th e rm a l conductivity, m aterials can
be heated a t extremely high heating rates ( > 1000°C /_min) in a microwave furnace.
Currently, wide commercial applications take ad v an tag e of selective microwave ab­
sorption w ithin the m aterial at tem peratures below 500° C [7]. Microwaves are
130
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used to cook food, dry solids, and cure wood lam inates [7]. Microwave processing
of ceram ics, w hich usually occurs above 500°C, can reduce th e processing tim e
and, possibly, processing tem perature w hich ca n lim it grain grow th and im prove
m aterial stren g th , electrical properties, an d d u ctility [7].
Tw o m ain obstacles hinder wider application of microwave processing. Sev­
eral technologically im p o rtan t ceramic m aterials poorly absorb microwave power,
especially a t low tem peratures. Therefore' th ere is insufficient energy to heat m ate­
rials to sintering tem p eratures [65]. N onuniform te m p e ra tu re d istrib u tio n w ithin
samples is an o th er problem with microwave processing.
T h e volum etric power
absorption d u ring microwave heating does not necessarily ensure a homogeneous
tem p eratu re distrib u tio n. T he radiative losses from th e sam ple surfaces produce
an inverted tem p e ratu re gradient where th e interior is h o tte r th a n th e surface
[2,33,37,106,107]. T he absorbed microwave power, as show n in Eq. 3.1, depends
on e" and th e local electric field. Since th e com plex p e rm ittiv ity is tem p eratu re de­
pendent, tem p eratu re gradients w ithin th e m aterial can lead to nonuniform power
absorption and fu rth er enhance tem perature gradients.
In addition, tem p eratu re gradients can cause nonuniform sintering or ‘th e r­
m al runaw ay.’ Above some critical tem perature, e" of m ost dielectric m aterials
increases exponentially w ith tem perature.
O nce th e h o tte r regions w ithin th e
m aterial reach th e critical tem perature, these regions absorb significantly m ore
microwave power th a n surrounding regions an d rap id ly increase in tem perature.
Since porous ceram ics are poor therm al conductors, th erm al gradients continue to
increase. T his sh arp rise in tem perature in certain regions or ‘therm al runaw ay’
eventually m elts or cracks th e sample.
Various techniques have been used to reduce or elim inate these problems. In
131
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order to microwave h e at low- loss m aterials, th e low- loss sample is surrounded by
a lossy m aterial, such as silicon carbide [108]. In th e form of a ring or rods, th e
lossy m aterial absorbs microwave power a n d tra n sp o rts therm al energy to th e Iowloss sample. This hy b rid heating technique heats th e sam ple until its p erm ittiv ity
increases to a level sufficient to allow th e sam ple to directly absorb th e applied
microwave power.
A nother technique is to combine a microwave absorbing m aterial to a low- loss
host m aterial, which increases th e effective e". W ith sufficient addition of lossy
additives, these com posites can th en be directly heated by microwaves. Microwaves
selectively heat lossy co nstituents w ithin com posite samples. For composites th a t
are homogeneously m ixed on th e microscopic scale, th e therm al energy is effectively
tra n sp o rted from dielectric lossy particles to low- loss particles. Thus,
in s ig n ific a n t
tem p eratu re gradients betw een constituent particles occur [29].
Modeling of th e microwave heating could lead to measures, which would reduce
or avoid processing problem s (i.e. nonuniform tem p eratu re distribution w ithin m a­
terials). Some obvious exam ples of large tem p eratu re gradients are macroscopic
differential densification and cracking of a sam ple, which are discovered after p ro ­
cessing. The presence of sm aller tem p eratu re gradients can be inferred by microstructural analysis. Previous m icrostructural analysis found porosity gradients
w ithin ZnO samples, w hich indicated the existence of tem perature gradients d u r­
ing microwave processing [107]. U nfortunately, this post- processing analysis of
th erm al gradients is tim e consum ing and provides only qualitative inform ation of
th e tem perature gradients. Num erical m odeling can analyze this complex prob­
lem, which involves th e absorption of microwave energy, therm al tra n sp o rt w i t h in
th e m aterial, and mass tra n sp o rt and densification [13]. This analysis can help
132
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im prove insulation design and estim ate optim al heating rate to minimize th erm al
gradients w ithin th e m aterial and avoid therm al runaway. In general, num erical
m odeling significantly reduces th e tim e a n d expense of numerous trial- and- error
experim ents.
Im proving microwave absorption of com posites also requires combining accu­
ra te dielectric measurem ents, num erical m odeling, and electrostatic sim ulations.
F irst, knowledge of th e m aterial properties is crucial. Specifically, dielectric prop­
erties w ith respect to tem perature, density, and frequency are essential in esti­
m ating microwave power absorption, tem p eratu re distribution, and densification
[2,7,108-110]. As discussed in the two previous chapters, the high- tem perature
open- ended coaxial probe and nondestructive resonant cavity were designed and
built to m easure th e complex p erm ittiv ity of solid m aterials and also provide d a ta
for num erical modeling.
N ext, num erical modeling can estim ate the sam ple tem perature during mi­
crowave processing. Previous num erical sim ulations modeled th e power absorp­
tion, therm al tra n sp o rt, and densification of ZnO [37,106]. Using this com puter
program , th e m axim um tem perature of a n alum ina composite was calculated for
constant applied microwave power w ith respect to a range in e" l . A diagram of
th e microwave sintering sample and insulation is shown in figure 5.1. Due to th e
axial sym m etry of th e sample and insulation, a two- dimensional finite difference
code was used to sim ulate th e power absorption and therm al tran sp o rt w ithin a
porous an d sintered alum ina composite. A more thorough description of this twodim ensional cylindrical mesh w ith variable mesh size is provided by A. B im boim
et. al. [106]. Each cell was assigned m aterial properties representative of th e sam •Two- dimensional numerical simulations were performed by A. Birnboim
133
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pie or insulation. M aterial properties (density, specific heat, th erm a l conductivity,
and complex p erm ittiv ity ) were tak en from the literature. T h e alum ina com posite
has d = 5.5 an d e"ranges from 0 to 0.04. For a constant application of 1500W
a t 2.45 GHz, th e local electric field and power absorption were calculated a t each
cell and tim e step [106]. B y exactly solving th e im plicit two- dim ensional therm al
conduction problem , th e te m p e ra tu re of each cell was calculated a t each tim e step.
These sim ulations assum ed th a t th e m aterial properties were co n sta n t w ith respect
to tem p eratu re. Figure 5.2 shows th e calculated
m a x im u m
core tem p e ra tu re of an
alum ina com posite w ith respect to a theoretical im aginary p a rt of th e com plex
perm ittivity.
In order for an alum ina com posite to be microwave h eated above 500°C, this
num erical m odeling predicts th a t e" needs to be greater th a n or equal to 0.04. This
specified dielectric value is th e n related to dielectric m easurem ents an d m odeling in
figures 4.28 a n d 4.30 to determ ine th e required volume fraction o f lossy additives.
Therefore, alum ina com posites w ith a t least 7 (mass)% of silicon carbide or copper
oxide should enable m icrowave heating above 500°C. A lthough th e desired pro­
cessing tem p e ra tu re m ight be significantly higher th a n 500°C, th e initial goal was
to increase th e effective p e rm ittiv ity of alum ina at low tem p eratu res. A t elevated
tem perature, th e dielectric properties are expected to increase w ith tem perature.
T hen, alum ina com posites should absorb m ore microwave pow er a n d continue to
heat to higher tem p eratu res for densification.
These num erical sim ulations highlight th e need to u n d erstan d in g th e contribut­
ing factors to e". For exam ple, e of th e 92.5% alum ina + 7.5% silicon carbide
com posite is 0.035. Microwave heatin g of this com posite w ith a sam ple diam eter
is 2.84 cm an d thickness o f 2.60 cm a t 1500W is initially assisted by th e w ater
134
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1. Sample
2. Porous Alumina Insul ation
3. Setter Powder
Figure 5.1: D iagram of th e microwave sam ple crucible. T h e alum ina sam ple has a
diam eter of 2.84 cm an d a height of 1.09 cm or 2.60 cm. T h e sam ple is surrounded
by alum ina bulk fiber. T h e insulating crucible is m ade o u t of porous alum ina board
(6 % theoretical density). Its has an outer diam eter of 7.62 cm, an inner diam eter
o f 5.08cm, an o uter height o f 15.24 cm, and inner height of 10.16 cm.
135
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soa
... Alumina: k = 3 [W/m*K], h = 2.60 cm
- Sintered Alumina: k = 30 [W/m*K], h = 1.09 cm
— Alumina: k = 3 [W/m*K], h = 1 .OS cm
700-
500-
200
-
100
-
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Imaginary Part of the Complex Permittivity
Figure 5.2: Num erical calculation of th e m axim um core tem perature of an alum ina
com posite w ith respect to its im aginary p a rt of th e complex perm ittivity. T h e ap­
plied power is 1500W a t 2.45 GHz. T he sam ple diam eter is 2.84 cm and thickness,
h, is indicated.
136
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in th e com posite. In chapter 4, electrostatic sim ulations com puted th a t chemi­
cally bonded w ater at grain boundaries increased the effective e" of th e
alu m in a
com posites, which increased th e absorption of microwave power. T h e numerical
m odel calculated th a t the m axim um tem p eratu re of th e microwave heated com­
posite is 650°C. However, heating of an actu al alum ina com posite to 200- 500°C
will drive off chemisorbed w ater and reduce th e effective e to ~0.018. Therefore,
th e asym ptotic core tem p eratu re of th e dehydrated
alu m i n a
com posite is 350°C.
A lthough it is not surprising th a t the m axim um tem p eratu re reduces to half if its
absorbed power decreases by half, such tem p eratu re dependencies of th e dielectric
properties should be accounted for in designing com posite m aterials for microwave
processing.
One m otivation of chapter 4 was preparing alum ina com posites for microwave
heating.
In this chapter, selected com posites were heated in conventional and
microwave furnaces. The m aterial properties and densification of these compos­
ites dep en d upon processing tem perature, synthesis, and processing m ethod. In
general, sintering or heat tre atm e n t of a ceramic causes th e form ation of strong
bonds betw een particles, a reduction in internal surface area, densification, and
g rain grow th. H eating changes th e volume fraction of constituents an d particle
contacts, which significantly affect com posite density and th e perm ittivity. Also,
synthesis affects mixing scale and m icrostructure of th e alum ina composites. At
elevated tem peratures, alum ina chemically reacts w ith copper oxide to produce
a new phase, alu m in a/ copper oxide spinel. Thus, synthesis also affects compos­
ite densification and perm ittivity. Finally, processing m ethod could also affect a
com posite’s perm ittivity. O ur earlier research found th a t th e complex perm ittiv­
ity and density m easurem ents of microwave sintered ZnO sam ples was significantly
137
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higher th an conventionally sin tered sam ples [18]. Additionally, o th e r studies found
microwave processed sam ples required lower sintering tem p eratu res an d improved
m ate ria l properties [6,29-31]. Therefore, this chapter exam ines th e dependence of
physical properties of alum ina com posites on processing te m p eratu re, preparation
m eth o d , and processing m ethod.
Ideally, the complex p e rm ittiv ity and density of these com posites would be
m easured during microwave a n d conventional processing. A Jthough th is can be
co nducted during conventional heating, it can n o t be perform ed d uring microwave
processing.
(As discussed in C hap.
6
and A ppendix E, d en sity an d dielectric
p ro p erties could eventually be m easured during microwave processing.) To pro­
vide com parison in heating m ethod, com posite samples were h e a te d to successively
higher tem peratures in conventional and microwave furnaces. A fter attain in g each
tem p eratu re, the sam ples were cooled to room tem perature. T h e com plex per­
m ittiv ity and density of these m aterials were m easured. M aterial characterization
a n d electrostatic sim ulations w ere perform ed to help u n d erstan d th e behavior of
th ese m aterials. T he dielectric properties of these composites w'ere also m easured
d u rin g conventional heating. T h e n ex t section discusses processing an d m easure­
m ents o f th e alum ina com posites. T hen, th e experim ental results and discussion
are presented. T he last section presents conclusions.
138
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5.2
Experimental Procedure
5.2.1
Conventional and Microwave Heating
A lum ina com posite system s (80%(mass) A12 0 3 + 20% SiC and 82.8% AI2 O 3 -f17.2% copper oxide), which were p repared in chapter 4, were heated to various
tem peratures in conventional therm al as well as microwave furnaces. D ue to th e
extensive sam ple p rep aratio n tim e and try in g to m inim ize sample- to- sam ple vari­
ation, th e sam e sam ple was heated to successively higher tem peratures. Conven­
tional therm al h eatin g of th e alu m in a/ copper oxide com posites was perform ed in
5 cm tu b e furnaces an d an air atm osphere. In order to provide therm al insulation
and a buffer pow der, th e samples were loosely packed w ith powder w ith th e sam e
com position as th e sam ple. T he samples were housed in a porous alum ina enclo­
sure similar to figure 5.1, b u t the outer diam eter of th e insulation was 4.4 cm , th e
inner diam eter was 3.8 cm, and the inner length was 6.4 cm. T he heating ra te was
2°C /m in up to 500° C and th en 5°C /m in to th e final tem perature. T he sam ple
tem p eratu re was m easured w ith a type K therm ocouple in the sample vicinity up
to 1100°C. In a flowing nitrogen atm osphere, the alu m in a / silicon carbide com­
posites were conventionally heating in a sealed
2 .2
cm tu b e furnace to 800 and
1000°C and in a g rap h ite furnace to 1400 and 1600°C. T h e sam ple tem p eratu re
was m easured w ith th e furnace therm ocouple. In order for th e furnace and sam ple
to reach therm odynam ic equilibrium , th e holding tim e was set a t 10 min. A fter
cooling, density was determ ined from m easuring sam ple dimensions and mass.
Microwave h eating was perform ed in th e modified overm oded 2.45 GHz ap­
plicator, which is discussed in A ppendix E. Microwave heated samples were also
housed in an enclosure sim ilar to figure 5.1, b u t the inner height was 6.4 cm an d
139
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outer height was 8.9 cm. T h e heating ra te and atm osphere were the same as con­
ventional m n s. Also, th e sam ple surface tem perature was m easured by a n array
of pyrometers, which is also discussed in Appendix E.
Unfortunately, there were difficulties in processing a lu m in a / copper oxide com­
posites.
Therm al runaw ay occurred when the physically mixed samples were
heated to ~900°C an d th e chemically precipitated sam ples were heated to "1000° C.
In order to hinder therm al runaway, a hybrid heating technique was used. A ring
of a microwave susceptor, SiC, was placed around th e alum ina composite to assist
in uniform heating of th e sam ple. T he susceptor was a SiC ring w ith an inner
diam eter of 4.45 cm, an outer diam eter of 5 cm, and a height of 1.9 cm.
D uring microwave heating, th e alu m in a/ silicon carbide sam ple was too small
to be supported by th e sight tube. In order to support this sample and minimize
energy loss from the sam ple surface, this sample was placed inside a larger cylindrically shaped sam ple of 80(mass)% AI2 O 3 + 20% SiC. B o th samples were th en
heated together in a nitrogen atm osphere.
5.2.2
Dielectric Measurements
Dielectric m easurem ents were perform ed by all the previously mentioned dielectric
m easurem ent techniques (nondestructive resonant cavity, H P open- ended coax­
ial probe, high tem p eratu re dielectric probe, and Microwave Properties N o rth ’s
(M PN) resonant cavity ). R oom tem perature dielectric m easurem ents were per­
formed by the first two techniques. T he nondestructive resonant cavity is espe­
cially useful for m easuring low- loss m aterials and H P probe is especially useful for
m easuring lossier m aterials w ith ta n 8 > 0.05. D uring conventional heating, the
dielectric properties of a 80 (mass)% AI2 O 3 +
2 0
% SiC sam ple were measured by
140
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th e high- te m p e ratu re dielectric probe in a 95% N 2 + 5% H 2 atm osphere, which
is described in ch ap ter 3. Dielectric m easurem ents were perform ed from room
tem p era tu re to 1000°C. Dielectric properties of 85.1% AI2 O 3 -f- 14.9% copper ox­
ide com posites, which were synthesized by ball m illing and chemical precipitation,
were m easured by Microwave P roperties N orth, w hich is discussed in sect. 3.4 [61].
5.3
Results and Discussion
Several physical properties of th e alum ina com posites were exam ined w ith respect
to variations of th e processing tem p eratu re, p rep aratio n m ethod, and processing
m ethod. Specifically, the dielectric properties an d densification of the com posites
were m easured an d th e following presents these experim ental results. Discussion
an d explanation for th e changes of th e physical properties of th e alum ina com pos­
ites are also presented.
5.3.1
Alumina/ Silicon Carbide Composites
Alum ina / silicon carbide composites were heated in conventional and microwave
furnaces up to 1600° C in a nitrogen atm osphere. T h e density of 80% alum ina +
2 0
% silicon carbide composites versus peak processing tem perature is show n in
figure. 5.3. T he average uncertainty in density m easurem ents is about
1
%. Slight
densification occurs during heating from 1400 to 1600° C. To fully density a lu m in a /
silicon carbide com posites, the m aterial needs to be hot pressed a t 1700- 1800°C
[1 1 1 , 1 1 2 ]. However, our microwave furnace can n o t apply pressure to a sam ple an d
th e m axim um processing tem perature of th e conventional furnace was 1600°C.
Therefore, com parative experim ents were lim ited to processing up to 1600° C and
141
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w ith o u t pressure.
T h e p e rm ittiv ity of alum ina/ silicon carbide com posites was m easured by the
nondestructive resonant cavity at room tem p eratu re. Figures 5.4 an d 5.5 show the
real a n d im aginary p a rt of the complex p e rm ittiv ity m easurem ents, respectively,
at 2.76 GHz versus peak processing te m p eratu re.
T he average uncertainty in
m easuring e' is ~1% an d e" is ~2%. T h e high- tem p eratu re probe m easured the
com plex p erm ittiv ity of the composite up to 1000°C and over a broad frequency
range of 0.3 to 6 GHz. Figure 5.6 presents th e m easured complex perm ittiv ity at
2.425 GHz versus tem perature. T h e average u n certain ty in m easuring e' is 10%
an d e" is 220%. T h e peak in e" from 400 - 700° C is a m easurem ent error. This error
ap p eared in a previous probe m easurem ents of a low- loss m aterial, see figure. 3.24.
T h e source of th e error could originate from insufficient contact betw een th e inner
conductor of the probe and the sam ple. In m easuring low- loss m aterials, small
errors or uncertainty in th e reflection coefficient co n trib u te to large uncertainty in
e" m easurem ents.
Figure 5.3 shows th a t minimal densification (~3% ) occurred for alu m in a/ silicon
carbide com posites during heating from room tem p eratu re to 1600°C. Actually,
heating th e com posite up to 1400° C caused no m easurable densification. This ini­
tial stage of sintering enabled m aterial to diffuse to particle contact regions, where
it form ed necks an d grain boundaries betw een particles. Surface d iffu s io n does not
lead to densification. Since th e activation energy o f m ass diffusion of silicon carbide
is higher th a n th e activation energy of alum ina, alum ina has a higher diffusion rate
th a n silicon carbide [113-115]. Therefore, alum ina predom inantly diffused into the
contact region. At tem peratures up to 1400°C, significant diffusion of alum ina was
exhibited earlier. As m entioned in sect. 4.5, m icron- sized alum ina powder was
142
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2.44-
2.42-
2.4-
x Microwave Heated
o Conventionally Heated
co
E2-38-
,o
~3)
o
C/)
§ 2-34o
2.32f-
2.3r
2.28-
200
400
600
800
1000
1200
1400
1600
1800
Peak Processing Temperature (C)
Figure 5.3: Density of 80% alum ina -f- 20% silicon carbide com posites vs. m ax­
im um processing tem p erature in a conventional and microwave furnaces w ith a
nitrogen atm osphere. T he uncertainty in density m easurem ents is ju s t less th a n
1%. M easurements were perform ed at room tem perature.
143
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7.47.27
* Microwave Heated
o Conventionally Heated
05
Q .6 .8 r
05
CD6 . 6 -
a:
6.4
6.2
x
0
6
0
200
400
600
800
1000
1200
1400
1600
Peak Processing Temperature (C)
1800
F igure 5.4: Real p art of th e com plex perm ittivity of 80% alum ina + 20% silicon
carbide composites was m easured by the nondestructive resonant cavity at 2.76
GHz vs. processing tem p eratu re in conventional and microwave furnaces w ith a
nitrogen atmosphere. T he average uncertainty in m easuring e7 is ~1%. M easure­
m ents were performed a t room tem perature.
144
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0 .1 8
i------------------ 1------------------ 1------------------:------------------ 1------------------ 1------------------ !------------------1-----------------x
0.16
x Microwave Heated
o Conventionally Heated
X
^ 0 .1 4
CD
X
CL
>0.1:
CD
C
'o )
CD 0.1
o
o
E
0
0.08
X
0
0.06
o
nu.u^f-----------1
ndr
-----------1-----------1---------- 1-----------'■
----------- '■
----------- 1— --------1--------0
200
400
600
800
1000
1200
1400
1600
1800
Peak Processing Temperature (C)
Figure 5.5: Im aginary p a rt of the complex p e rm ittiv ity of 80% alum ina + 20%
silicon carbide com posites was m easured by th e nondestructive resonant cavity
at 2.76 GHz vs. processing tem perature in conventional and microwave furnaces
with a nitrogen atm osphere.
The average u n certain ty in m easuring e" is ~1%.
M easurem ents were perform ed at room tem p eratu re.
145
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10r
,o '
Ao 0 0 0<><>
•is 7-
Real Part
•*—»
■g
t 6
CP
.
aa
, 0 <>ooooo o 0 0 0 0 0
<W
Oo 0 0 0 0 0 ° ° °
Q_
X 5
©
Q- .
£
o
o 3
A
Imaginary Part
A ° o o 0<>^ o o o o <;>
/naOOOOOO0 0 0
°o^ ^ 0 0 0 0 0 9 0 °
100
200
300
<> oo ooo oooooooo«
..................................................................
i
400
500
600
700
800
900
1000
Temperature (C)
Figure 5.6: M easured Complex P erm ittiv ity of 80% A lum ina + 20% Silicon C ar­
bide by the high - te m p eratu re open- ended coaxial probe at 2.425 GHz vs. tem ­
perature in a conventional furnace w ith a 95% N2 + 5% H2 atm osphere.
146
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densified to 96.8% t.d.(theoretical density) by heating to 1400°C. F u rth er heat­
ing of th e alu m in a/ silicon carbide com posite to 1600°C increased its density by
3%. T h e m ain difference in densities of com posites, which were heated in either
microwave or conventional furnaces, is prim arily due to initial differences in their
green densities. A lthough the difference in densities slightly increases as th e pro­
cessing tem perature increases to 1600° C, this change in differences of density is less
th a n th e uncertainty in density m easurem ents. Therefore, the slight densification
of alu m in a/ silicon carbide com posites does n o t ap p ear to depend on processing
m ethod.
T h e dependence of the perm ittivity on processing tem perature, synthesis, and
processing m ethod is furthered studied. As shown in figure 5.4, the real p a rt of the
com plex p erm ittivity of the alu m in a/ silicon carbide com posite rem ained constant
up to a peak processing tem perature of 1000°C. F u rth er heating of th e composite
to 1400°C increased e' by about 4%. T he form ation of necks and grain bound­
aries increased interparticle contact an d reduced th e air phase between particles.
Thus, surface diffusion probably increased e'. Upon heating to 1600°C, the sam­
ple slightly densified and increased e7 by about 14%. As shown in figure 5.5, the
im aginary p a rt of th e complex p erm ittiv ity of th e com posite decreased by 10%
when heated to 400°C. T he real p art of th e complex perm ittivity also decreased
in this tem p eratu re cycle, but th e decrease was less th a n th e uncertainty in mea­
surem ents. This decrease in e77 is probably due to removal of chemisorbed water
from th e ceramic particles.
In fact, th e dependence of p erm ittiv ity on absorbed w ater was exam ined in
chapter 4. Electrostatic sim ulations com puted th e effective perm ittivity of porous
alum ina composites w ith and w ithout w ater. By rem oving of water from th e par-
147
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tid e contacts, th e electrostatic m odel com puted th a t e" decreased by 15% a n d e'
decreased by 1%. W ith in th e uncertainty of dielectric m easurem ents and sim ula­
tions, the predicted decrease in ^'agrees w ith the dielectric measurements. T h ere­
fore, the decrease in com plex p erm ittiv ity of th e com posite, which was heated to
400°C, is prim arily du e to removal of w ater a t co n tact regions.
F u rth er h eatin g of th e com posites to 1400 an d 1600° C caused significant a n d
unexplained changes in th e p erm ittiv ity of th e com posites. Although no densificatio n resulted from h eating th e sam ple to 1400° C, th e re were substantial increases
in e". T he im aginary p a rt of th e com plex p erm ittiv ity increased by 150% for th e
microwave h eated com posite and 100% for the conventionally heated com posite.
A lthough surface diffusion form ed necks between particles below 1400° C [23], th e
dependence of p erm ittiv ity on surface diffusion is n o t clear. Interestingly, heating
th e composites to 1600°C, which only produced a slight densification of th e com­
posites, resulted in a 19% decreased in e". T he real an d im aginary p arts of th e
complex p erm ittiv ity are expected to increase w ith density.
E lectrostatic sim ulations were used to estim ated th e changes in p e rm ittiv ity
due to neck form ation. A basic physical model of a composite, which is in this
initial stage of sintering, is an arrangem ent of spheres in a bcc lattice w ith e x tra
m aterial in th e contact region between particles. A ssum ing th a t th e p e rm ittiv ity
of th e contact region is th e sam e as in th e bulk m aterial, th e electrostatic m odel
predicted th a t th e ex tra m aterial in th e contact region would increase t" by less
th a n 10%. Therefore, th e changes in th e complex p erm ittiv ity axe the result of
factors besides volum e fraction and arrangem ent of constituents.
In chapter 4 , th e p erm ittiv ity of alum ina com posites was
s i g n if ic a n t l y
increased
by a small am ount of lossy m aterial in th e contact region. The intensified field
148
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near grain co n tacts couples w ith th e lossy m ateria l in these regions a n d increases
e" of th e com posites. T h e electrostatic m odeling of alum ina com posites w ith neck
grow th assum ed th a t th e perm ittivity in th e co n tact region is the sam e as th e bulk.
However, th e dielectric properties of th e g rain boundaries can differ from th e bulk
m aterial [116].
A grain b o u n d ary is an area of lattice m ism atch between two different grains
in a solid m aterial. In order to reduce th e h ig h surface energy of grains, im purities
an d defects preferentially diffuse a t elevated tem peratures to grain boundaries.
Thus, charged species can reside a t th e g ra in boundaries and a space charge can
p e n etrate into th e grains. This Schottky b a rrie r near th e grain surface increases
th e resistivity in this region. W ith th e intensified electric fields in th e contact
region, th e effective im aginary p a rt of th e com plex perm ittivity increases w ith
concentration of defects.
T he concen tratio n of charged species depends u p o n the form ation energy of
cation an d ion vacancies at the grain b o u n d ary and tem perature. T h e defects can
also depend on concentration of im purities, la ttic e stru ctu re, and chem ical affinity
of constituents. For example, silicon carbide has a large chemical affinity b u t it
is lim ited in form ing su bstitutional solid so lu tio n w ith alum ina because th ey have
different lattice stru ctu res. E stim ating th e p e rm ittiv ity of th e grain b o u n d ary is
further com plicated by th e tem perature dependence of th e space charge thickness
and binding energy of defects to various sites [117].
A nother possible explanation for th e increase in e" a t peak processing tem p era­
tures above 800° C is due to unintentional doping of silicon carbide. Silicon carbide
is a wide bandgap semiconductor. T he silicon carbide powder from Alfa- A esar
is 99.8% pure an d possesses m any trace im purities such as N, B, Mo, Al, Fe, Ca,
149
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an d Zr. D uring heating of the a lu m in a / silicon carbide com posites, nitrogen is
also absorbed into silicon carbide [118]. Some of these im purities can be activated
during heating and act as N- or P - ty p e donors [119]. F u rth er research is necessary
to identify dom inant donor species an d their overall effect to th e conductivity of
silicon carbide and com posite a t room tem perature.
F igure 5.5 also shows significant differences in th e m easured e" a t various pro­
cessing tem peratures. Therefore, th e im aginary part of th e com plex perm ittivity of
th e microwave heated composites increased more th an conventional heated com­
posites. T h e microwave field stre n g th in th e contact regions could have caused
localize plasm a form ation or contributed to th e ponderm otive m aterial flow theory
[23,28,120]. Localized ionization phenom ena could provide a new tra n sp o rt mecha­
nism betw een grains. Also, it could produce chemical changes a t g rain boundaries
th a t alter sintering kinetics [34]. T hese changes could also cause electrically active
defects to form on grain surface and increase e" of the com posite. T h e ponderm o­
tive theory models mass tran sp o rt of surface charge vacancies due to high frequency
electric fields. These phenom ena could enhance transport of charged particles into
th e grain boundary and explain w hy th e microwave heated com posite possess a
larger e" th a n conventionally heated com posite [28].
T h e complex perm ittivity of a lu m in a / silicon carbide com posites was also mea­
sured w ith respect to tem p eratu re u p to 1000°C as shown in figure 5.6. T he real
p a rt of th e complex p erm ittivity increases linearly w ith tem p eratu re up to 800°C.
Above 800°C, d increased w ith tem p eratu re. O ther researchers m easured th a t e* of
silicon carbide increased w ith respect to tem perature up to 1000°C [121,122]. T h e
m easurem ents of e* of alum ina only slightly increased w ith respect to tem perature
up to 1000°C. Thus, th e initial increase in e' was due to th e tem p eratu re depen-
150
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dence of silicon carbide w ithin the composites. From 800 to 1000°C, e7 increased
w ith respect to th e tem p eratu re dependence of silicon carbide and th e neck for­
m atio n between particles, which increases contact area between particles. Besides
th e substantial m easurem ent errors from 400 - 700° C, e" increased w ith tem p era­
ture. Thus, th e tem p eratu re dependence of e" is difficult to quantify. Considering
previous dielectric m easurem ents of alum ina an d silicon carbide, th e tem p eratu re
dependence of e" of th e com posite is m ainly due to silicon carbide [1 2 1 , 1 2 2 ].
5.3.2
Alumina/ Copper Oxide Composites
T h e physically mixed a n d chemically precipitated a lu m in a / copper oxide compos­
ites were heated in microwave and conventional furnaces. Figures 5.7 and
5 .8
show
th e m easured com plex perm ittivity of the 82.8% alum ina -f- 17.2% copper oxide
com posites at 2.66 GHz versus peak processing tem p eratu re. The average uncer­
ta in ty in m easuring e7 a n d e77 is
~ 1
% and ~2 %, respectively. The density of th e
processed samples versus peak processing tem p eratu re is presented in figure 5.9.
D uring microwave h eating of the composites, th erm a l runaway occurred w ith
an d w ithout th e silicon carbide ring. H eating these composites to ~T000oC pro­
duced a reaction front o r phase change w ithin th e sam ple.
Regions w ith th e
new phase absorbed significantly more microwave power th an surrounding re­
gions. Thus, nonuniform power absorption resulted in differential densification or
warpage. These processing problems prevented accu rate perm ittivity and density
m easurem ents for com posites, which were heated above 950°C.
T h e dielectric properties of alum ina/ copper oxide composites were also m ea­
sured w ith respect to tem p eratu re w ith a resonant cavity [61]. Figures 5.10,
5
.1 1 ,
and 5.12 show th e m easured complex p erm ittiv ity of 85.1% alum ina + 14.9%
151
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
* Physically Mixed
5.5]- + Chem. Precipitated
^5.5[Q_
15 5
Q
3
<D
a:
"b<
4.f
Conventionally Heated
200
300
400
500
600
700
800
900
1000
1100
Peak Processing Temperature (C)
■t
v
m
CtJ5-5
a
"
—
r
— r
i
*
v
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i
Physically Mixed
Chem. Precipitated
CL
-
A
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<1
t>
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15 5
A
O
0
0
0
oc
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A
V
4.5
A
Microwave Heated
V
to o
200
1
,
300
400
500
V
V
600
700
i
i
800
900
1000
1100
Peak Processing Temperature (C)
Figure 5.7: Real p art of th e com plex p erm ittiv ity of 82.8% alum ina + 17.2% cop­
p er oxide composites was m easured by th e nondestructive resonant cavity and a
com m ercial open- ended coaxial probe a t 2.66 GHz vs. m axim um processing tem ­
p e ra tu re in conventional and microwave furnaces w ith an air atm osphere. Compos­
ites were synthesized by physical m ixing an d chemical precipitation. T h e average
u n certain ty in measuring e' is ~1 %. M easurem ents were perform ed at room tem ­
p erature.
152
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
rFn a- x Physically Mixed
LL *°r + Chem. Precipitated
cd0-6^
c
D D .4 )
03
Mo.
Conventionally Heated
200
300
400
500
600
700
800
900
1000
1100
Peak Processing Temperature (C)
—
1
n 3o 8- v Physically Mixed
ua Chem. Precipitated
cdo-^c
'O D . 4 j -
A
A
03
Mo.2r
A
a
V
V
900
1000
V
Microwave Heated
200
300
400
500
600
700
800
1100
Peak Processing Temperature (C)
F igure 5.8: Im aginary p a rt of th e complex perm ittiv ity of 82.8% alum ina + 17.2%
copper oxide composites was m easured by the nondestructive resonant cavity an d a
com m ercial open- ended coaxial probe a t 2.66 GHz vs. m axim um processing tem ­
p e ra tu re in conventional an d microwave furnaces w ith an air atm osphere. Compos­
ites were synthesized by physical m ixing and chemical precipitation. The average
u ncertainty in m easuring e" is
~ 1
%. M easurements were perform ed a t room tem ­
p eratu re.
153
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2.75- v Microwave Heated, Physically Mixed
a Microwave Heated, Chem. Precipitated
2.7- x Conventionally Heated, Physically Mixed
+ Conventionally Heated, Chem. Precipitated
—2.65co
E 2.(
.O
X
2.5$-
A
f „
CD
□ 2.45-A
+
2.4-
$
2.3E-
200
300
400
500
600
700
800
900
1000
1100
Peak Processing Temperature (C)
Figure 5.9: D ensity of 82.8% alum ina + 17.2% copper oxide composites vs. m ax­
imum processing tem p eratu re in a conventional an d microwave furnaces w ith an
air atm osphere. Com posites were synthesized by physical mixing and chem ical
precipitation. T h e uncertainty in density m easurem ents is ju st less th a n
surem ents were perform ed a t room tem perature.
154
Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission.
1
%. Mea­
copper oxide composites, which w ere synthesized by chemical precip itatio n and
physical m ixing, a t 2.46 GHz. T hese sam ples do not possess th e sam e com position
as th e previous samples because th e re was no more powder w ith 17.2% copper
oxide. A t elevated tem peratures, th e m ateria l’s conductivity and e" increase ex­
p onen tially w ith tem p erature. Since a co n stitu en t m aterial or dom in an t diffusion
m echanism c an be characterized b y its activ atio n energy, its activation energy was
calculated using equation 3.4. T h e effective activation energy for b o th com posites
was ~10 eV.
T h e dependence of th e p erm ittiv ity an d o th e r m aterial properties on process­
ing te m p e ra tu re , synthesis, and processing m eth o d is furthered studied. A n un­
expected p eak in e* of alu m in a/ copper oxide composites occurred a t low tem per­
atu res. As show n in figure 5.12, a p eak in e" of the com posites occurs from 100
to 450°C. Also, figure 5.10 shows a ste p increase in d a t ~250°C. Sim ilar trends
in dielectric properties of copper(II) oxide were m easured by H utcheon and T inga
[61,123]. H utcheon concluded th a t th is peak in dielectric properties was due to
copper(II) oxide acting as a catalyst, which increased its surface energy [61]. In­
teractio n of copper(II) oxide w ith w a te r a t high tem peratures produced a “layer
of O-H groups on th e particle surface [61]” a n d increased surface energy. T his
layer o f O-H groups is resistive in th e microwave frequency range. Therefore, h eat­
ing a lu m in a / copper oxide com posites enabled chemisorbed w ater to react w ith
copper(II) oxide and increase its surface energy an d effective perm ittivity.
T h is reaction also explains th e p eak in e* of th e composites, w hich were heated
to successively higher tem peratures. As show n in figures 5.7 and
5
.8 , a peak in
e* appears from 400 to 600° C. O ne obvious difference between th e two sets of
d a ta (figures 5.7 & 5.8 and 5.10 an d 5.12), is th e tem perature range of the peak
155
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20-
Physically Mixed
+ Chem. Precipitated
"^151q
x
_
03
<D
^10h
-* -x
-k
x
200
400
600
800
1000
1200
Temperature (C)
Figure 5.10: R eal p a rt of th e com plex p erm ittiv ity of 85.1% alum ina + 14.9%
copper oxide com posites vs.
tem p e ra tu re by a resonant cavity (M PN ) a t 2.46
GHz. Com posites were synthesized by physical m ixing and chem ical precipitation.
156
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
35
30-
■e
05
CL
£eo05
c
’a )
0 5 1 5
-
^
x Physically Mixed
+ Chem. Precipitated
101
-
0
0
200
400
600
800
1000
1200
Temperature (C)
Figure 5.11: Im aginary p a rt of th e complex perm ittivity of 85.1% alum ina + 14.9%
copper oxide composites vs. tem p eratu re by a resonant cavity (M PN ) at 2.46 GHz.
Com posites were synthesized by physical mixing and chemical precipitation.
157
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.45-
x Physically Mixed
+ Chem. Precipitated
0.4-
0.35-
tl
CD
Q_ 0.3r
CD
C 0 .2 5 -
'O)
CD
^ 0.2|0.1E-
0.1+
x
0
100
200
300
400
500
600
700
800
900
1000
Temperature (C)
F igure 5.12: This figure is expanded scale of fig.
5.11. Im aginary p a rt of the
com plex p erm ittiv ity o f 85.1% alum ina + 14.9% copper oxide com posites vs. tem ­
p e ra tu re (less th a n 1000°C) by a resonant cavity (M P N ) a t 2.46 GHz. Composites
were synthesized by physical m ixing and chemical precipitation.
158
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in e*. T he peak in figures 5.7 and 5.8 extends to higher tem peratures because th e
alu m in a / copper oxide com posites were eventually cooled from high te m p e ra tu re
to room- te m p e ra tu re for dielectric m easurem ents. W hen these com posites were
cooling from above 450°C, the composites could have subsequently reabsorbed
w ater, which caused a n increase in their surface energy a n d e". This reactivation
behavior could have co n tinued for samples h eated above 600° C. However, heating
com posites above 600° C causes surface diffusion an d densification, which decreases
surface area especially in th e contact region. Therefore, e" of th e contact region is
n o t increased during cooling .
E lectrostatic sim ulations, which were presented in ch ap ter 4, can su p p o rt this
hypothesis. E lec tro sta tic sim ulations calculated th a t th e effective e" su b stan tially
increased w ith a sm all ad d itio n of lossy m aterial in th e contact region. Surface
diffusion an d sin terin g reduce th e contact region betw een particles. T hus, w ater
can not react w ith co p p er(II) oxide in th e contact region and e" of th e contact
region does not increase d uring cooling. W ith o u t an increase in e" of th e contact
region during cooling, th e re was less of an increase in effective e" due to a n increase
in th e surface energy o f th e noncontacting surfaces.
H eating alu m in a / co p p er oxide composites to 800 - 900° C resulted in neck for­
m atio n between particles, which increases interparticle contact, and densification.
T h e complex p e rm ittiv ity would be expected to increase w ith tem p eratu re due to
th e reduction of th e a ir phase an d tem perature dependence of copper(II) oxide.
Above 700°C, th e com plex perm ittivity of pow dered copper(II) oxide was found
to increase w ith te m p e ra tu re [123]. However, th e m easured complex p erm ittiv ity
in figures 5.7 an d 5.8, in general, slightly decreased from 800 to 900°C. Also, th e
m easured com plex p e rm ittiv ity in figures 5.10 an d 5.12 rem ains constant from 500
159
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
to 900°C. In all of these cases, th e com plex p erm ittiv ity is still larger th a n ini­
tial, unfired values. In sp ite of th e decrease in surface energy, the p e rm ittiv ity is
stro n g ly effected by copper(II) oxide a n d not as m uch by density or neck form ation.
F u rth er heating of th e com posites caused changes in perm ittivity, density, and
sam ple color. T h e m ain reason for these changes was th e form ation of C11AI2O4.
X- ray diffraction was perform ed on 82.8% alum ina
17.2% copper oxide com pos­
ites, w hich were h eated to 1100°C in one of th e two furnaces. Figures 5.13 and 5.14
show th e diffraction p a tte rn of alu m in a/ copper oxide composites w ith respect to
lattice spacing. T h e lattice spacing an d relative intensity were com pared w ith ref­
erence d a ta for alum ina, copper(II) oxide, copper(I) oxide, CUAI2 O 4 , and CuAIO-2 Irrespective of th e processing or synthesis m ethod, all composites were com posed
of alum ina, copper (I) oxide, and CUAI2 O 4 . T h e copper oxide spinel formed w hen
copper(II) oxide reacted w ith alum ina. X- ray diffraction also m easured th a t th e
relative am ount of [C11AI2O4] to [AI2O3] was 10% larger for chemically p recipitated
com posites th a n physically mixed com posites. Thus, the chemically p recipitated
com posites form ed m ore CUAI2 O 4 th a n th e physically mixed composites.
Interestingly, th e reaction of copper(II) oxide and alum ina produces this new
phase a t 800°C [124]. T his phase tra n sitio n was observed in earlier experim ents.
D um m y sam ples of alum ina and copper(II) oxide were heated in th e microwave
furnace to calibrate th e pyrom eter arra y (see A ppendix E). Significant increases in
th e sam ple’s complex p erm ittiv ity and form ation of a new phase occurred a t 800825°C. However, CuA 1 2 0
4
did not form in th e alu m in a/ copper oxide com posites
below 950°C.
T h e delay in th e chem ical reaction w ithin alum ina /copper oxide com posites
was due to copper(I) oxide interm ixed w ith copper(II) oxide. The reactivity of
160
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solids depends upon the contact area and stru ctu re of reactan ts, which are copper(II) oxide an d alum ina [125]. Copper(I) oxide reduced th e co n tact area betw een
reactan ts and inhibited th e diffusion of reactan ts. A t higher tem peratures and
longer processing tim e, copper(II) oxide was able to react w ith alumina. Above
950°C, figures 5.10 and 5.11 show th a t th e com plex p e rm ittiv ity increased d ram ati­
cally w ith tem perature. Since the effective activation energy o f th e composites was
m uch g reater th a n the activation energy of alum ina or copper(I) oxide, this signif­
icant increase in e* of th e alum ina/ copper oxide com posites is prim arily due to
CuA12 0 4.
In general, th e effective perm ittivity of a com posite depends on the volume
fraction an d p erm ittivity of its constituents.
However th e volume fraction of
CUAI2 O 4 increased w ith tem perature and processing tim e. Also, the p erm ittiv­
ity of CUAI2 O 4 increases w ith tem perature. U nfortunately, th e volume fraction of
constituents was not m easured w ith respect to tem perature. It is difficult to accu­
rately m easure these phases without destroying th e sample. Therefore, dielectric
m ixing laws could not be used to predict th e perm ittiv ity or activation energy of
CuA 1 2 0
4
or alu m in a/ copper oxide composites.
A lthough this new phase is very conductive a t elevated tem peratures, figures
5.7 an d 5.8 show th a t e" of these composites, which were h eated to 1100°C, is less
a t room - tem p eratu re th a n before the phase transition. Since th ese processed com­
posites are low- loss a t room tem perature and conducting above 950°C, CuA12 0 4
can be classified as a wide bandgap sem iconductor w ith an activation energy less
th a n 10 eV. T he form ation of this new phase was th e problem w ith microwave
heating above 900° C. Once th e new phase forms in a region o f th e sample, more
microwave power is absorbed by C 1 1 A.I2 O 4 th a n th e surrounding regions. W hen
161
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Conv. Heated
10
-
2.5
3.5
25r
820
-
MW Heated
Lattice Size (A)
3.5
Figure 5.13: X-ray diffraction of 82.8% alum ina + 17.2% copper oxide com posites
heated to 1100°C in conventional an d microwave furances w ith an air atm osphere.
C om posites were synthesized by physical mixing.
162
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Conv. Heated
10
-
2.5
3.5
!
2.5
Lattice Size (A)
3.5
1Q
o
o
~
MW Heated
2
Figure 5.14: X-ray diffraction o f 82.8% alum ina + 17.2% copper oxide composites
h eated to 1100°C in conventional an d microwave furances w ith an air atmosphere.
Com posites were synthesized by chem ical precipitation.
163
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th e te m p e ra tu re in regions w ith C11AI2O4 increased significantly more th an o th er
regions, th erm a l runaw ay occurred and samples were nonuniform ly heated, w hich
varied th e dielectric properties, sample shape, a n d color. D uring microwave h eat­
ing, phase transitions, which produce more highly constituents, could produce
nonuniform h eating and, possibly, therm al runaway.
T h e dependence of perm ittivity and densification can also depend upon syn­
thesis an d processing m ethod. As shown in figures 5.10 an d 5.11, the significant
increase in e* o f th e chemically precipitated com posites occurs a t a lower tem ­
p eratu re th a n th e physically mixed com posite. Also, figure 5.9 shows th a t the
chem ically p recip itated composites densified m ore th a n physically mixed compos­
ites irrespective of processing m ethod. T he difference in perm ittivity, densifica­
tion, and form ation of CuAl 2
0 4
results from differences in th e m icrostructure of
th e com posites. T he m icrostructure of th e chem ically p recipitate composites has a
larger contact area an d smaller diffusion length betw een reactants than the phys­
ically m ixed com posites. Thus, chemically p recip itated composites produce more
CUAI2 O 4 an d a t lower tem peratures th a n physically m ixed composites.
Finally, p erm ittiv ity and density of the a lu m in a / copper oxide composite de­
pend m ore on synthesis th a n on processing m ethod. Figures 5.7 and 5.8 show dif­
ferences in e* of com posites, which were heated w ith th e sam e processing m ethod.
As shown in figures 5.15 and 5.16, there are less differences in e* of composites
which were p repared by th e same m ethod and h eated in different furnaces.
164
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Microwave Heated
+ Conv. Heated
a
t 5.5
cc
CL
+
7d 5
+
A
A
+
CD
□C
A
+
+
A
800
900
+
4.5
Chem. Precipitated
J _____________t____________ I____________ I____________ L_
200
300
400
500
600
700
1000
1100
Peak Processing Temperature (C)
t
1
1
1
1
1
I
■1
i
v Microwave Heated
m
03 5'5
x Conv. Heated
X
CL
x
x
"cc 5
X
<D
oc
-
vx
V
4.5
X
Physically Mixed
^
00
V
X
-
V
i
i
»
?
f
i
r
r
200
300
400
500
600
700
800
900
1000
1100
Peak Processing Temperature (C)
Figure 5.15: Real p a rt of th e complex p erm ittiv ity of 82.8% alum ina + 17.2% cop­
per oxide com posites was m easured by th e nondestructive resonant cavity an d a
commercial open- ended coaxial probe a t 2.66 GHz vs. maxim um processing tem ­
p eratu re in conventional and microwave furnaces w ith an air atm osphere. Com pos­
ites were synthesized by physical m ixing an d chem ical precipitation. M easurem ents
were perform ed a t room tem perature.
165
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-e
1
CLO 8- A Microwave Heated
+ Conv. Heated
jgo.e
C
OD.4F
TO
A
A
A^
+
^
&
a
800
900
1000
So.
Chem. Precipitated
200
300
400
500
600
700
1100
Peak Processing Temperature (C)
-E
1|
8
•>,
TO0-6'
r
v Microwave Heated
x Conv. Heated
E
‘OB.4&
C.
v
Vx
T*
800
900
iE 0 .2 -
A
V
Physically Mixed
?00
200
300
400
500
600
700
1000
1100
Peak Processing Temperature (C)
Figure 5.16: Im aginary p art of th e com plex p erm ittiv ity of 82.8% alum ina + 17.2%
copper oxide composites was m easured by th e nondestructive resonant cavity and a
com m ercial open- ended coaxial probe a t 2.66 GHz vs. m axim um processing tem ­
p e ratu re in conventional and microwave furnaces w ith an air atm osphere. Com pos­
ites were synthesized by physical m ixing an d chem ical precipitation. M easurem ents
were perform ed a t room tem p eratu re.
166
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5.4
Summary and Conclusions
A sum m ary of experim ents and conclusions follow:
1) T he lossy additions of silicon carbide an d copper oxide com bined separately
w ith alum ina enables microwave heating to desired tem peratures.
2) T he dielectric properties of alu m in a/ copper oxide com posites had a local
peak from 200 to 600°C. T his peak is due to copper(II) oxide. Copper(II) oxide
reacts w ith w ater a t elevated tem peratures which increases its surface energy.
3) C opper(II) oxide reacts w ith alum ina above 950°C to form CuA 1 2 0
4
- This
new phase is an electrical insulator a t room tem p eratu re and highly conductive
above 900°C. T h e form ation of CUAI2 O 4 was th e cause of th erm a l runaw ay dining
microwave heating.
4) Physical properties of alu m in a/ copper oxide com posites depend on synthesis
an d not on processing m ethod. Chem ically precipitated a lu m in a/ copper oxide
composites have a higher reaction ra te due to larger surface area and reduced
diffusion p a th betw een reactants. Therefore, chemically p recip itated composites
form more C11AI2O4. Also, th e m icrostructure properties o f th e com posites effects
dielectric properties an d densification.
5) A lum ina/ silicon carbide com posites were heated in b o th conventional and
microwave furnaces. T hese was no statistical difference in e' and density at vari­
ous processing tem p eratu res. T he initial green density of com posites had a larger
effect on these m aterial properties. A lthough densification did not occur below
1400° C, surface diffusion enabled neck grow th between particles and, possibly, de­
fect form ation at grain boundaries. T he im aginary part of th e complex perm ittivity
significantly increased w ithout appreciable changes in d ensity or surface area. De­
fects could change th e p erm ittiv ity of th e grain boundary from th e bulk m aterials
167
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and th e effective p erm ittiv ity of the composite. T he processing m ethod also af­
fects e". Localized plasm a form ation or ponderm otive forces could increase the
perm ittivity of th e g rain boundary and increase th e effective e" of th e com posite.
6)
E lectrostatic m odeling was of lim ited value in calculating th e effective per­
m ittivity of these heterogeneous composites or individual constituents, because of
lack of d a ta for th e volum e fraction and e* of constituents. Also, the p erm ittiv ity of
the constituents a t g rain boundaries is unknown and plays a m ajor role in effective
perm ittivity.
168
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Chapter 6
Future W ork
In perform ing th is d issertation work, some im provem ents to th e instru m entation
and topics for fu tu re research becam e app aren t. T h e following sections discuss
m odifications for th e open- ended coaxial probe, resonant cavity, alum ina compos­
ite synthesis, a n d te m p e ra tu re m easurem ent m ethods.
6.1
Complex Permittivity Measurements at El­
evated Temperatures
The stainless stee l open- ended coaxial probe was only tested to 1000 °C in a
reducing atm o sp h ere (N2- H2), even though it w as designed to operate a t higher
tem peratures. Like th e U niversity of N ottingham probe, th is stainless steel probe
should be capable of perform ing complex p erm ittiv ity m easurem ents up to 1200°C
[59]. D uring th e pro b e developm ent and operation, it was necessary to lim it th e
tem perature to lim it to 1000°C in order to lim it oxidation, w arpage of the stainless
steel, and meted diffusion from th e probe into th e m aterial under test (M UT). To
169
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lim it w arpage o f th e m etal an d improve p ro b e stability, future m odifications of the
probe design sh ould reduce th e length of th e probe. This could require m aking
or purchasing a special furnace to heat on ly th e probe head.
Also, th e probe
could also be m ade out of platinum or th e m etal probe could be coated w ith a
nonoxidizing m aterial.
W ith these changes, th e reducing atm osphere would no
longer b e required, allowing high tem p eratu re m easurem ents of oxide m aterials
under equilibrium conditions.
In th e in itial design stages of the probe, it was hoped th a t a probe could op­
erate up to 1600°C. T his design param eter prevented m aking th e probe o u t of
stainless steel, w hich m elts around 1400°C. B u t th e probe could be m ade o ut of
m olybdenum , w hich has a higher electrical conductivity and a lower th erm al ex­
pansion th a n stainless steel. However, it is m ore difficult to m achine th a n stainless
steel. A spring- loaded probe m ade out of m olybdenum could be m achined and
coated w ith m olybdenum disilicide. This coating will protect m olybdenum from
high tem p eratu re oxidation [126]. C oating o n sm all diam eter m olybdenum tubes
has been shown feasible by Bob Rapp from O hio S ta te University.
O ne of our in itial research goals was to perform in situ complex p erm ittiv ity
m easurem ents d u rin g conventional and microwave sintering. However, one problem
does occur at elevated tem peratures. M etal from th e probe diffuses into th e M U T
and alters its dielectric properties. This diffusion can be minim ized by limit i n g
th e m axim um te m p e ra tu re of th e m easurem ents a n d /o r th e contact tim e betw een
probe and M UT. T his la tte r requirem ent is essential for perform ing m easurem ents
during microwave processing.
O perating a n open- ended probe w ithin a microwave furnace could dam age the
network analyzer. T his problem can be avoided by placing the probe outside th e
170
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furnace during sintering. A t specified tem peratures, th e system would tu rn off
microwave power. Then, a probe could be inserted into the furnace. It is crucial
th a t the probe m akes sufficient contact w ith a portion of the sam ple. Dielectric
m easurem ents could th e n be conducted. (To prevent therm al shock to th e sam ­
ple and probe, th e probe would have to be heated to th e sam ple tem perature.)
A lthough th e sam ple would cool tu rn in g th e measurem ent, direct tem p eratu re
m easurem ents o r cooling curves could determ ine th e sample tem p eratu re. A fter
th e m easurem ent, th e probe could th en be w ithdraw n and microwave processing
could th en be continued.
One of th e m ost difficult design param eters is achieving in tim ate contact be­
tw een the sam ple an d probe (or a spring- loaded probe). An open- ended probe is
inherently u n stab le if th e inner an d outer conductors axe not fixed together. T he
slightest deflection of the inner conductor significantly changes th e reflection coef­
ficient during calibration and sam ple m easurem ents. A lthough th e spring- loaded
probe was able to accurately m easure th e complex p erm ittiv ity of m aterials up
to 1000° C, enough em phasis can not be m ade about th e p o ten tial in stab ility of
th e probe an d num erous experim ents and design changes required ob taining re­
producible results.
Considering th a t these m easurem ents were perform ed in a
relatively static probe setup, in situ m easurem ents during microwave processing
w ith an open- ended probe would be extrem ely difficult and require very precise
positioning system s. This researcher believes th a t in situ m easurem ents during
microwave processing w ith a spring- loaded probe are impossible.
A ccurate com plex p erm ittiv ity m easurem ents during microwave h eating could
be achieved by arranging a resonant cavity below th e microwave applicator. T his
experim ental design is sim ilar to H utcheon’s [61] and A rai’s [121] dielectric mea-
171
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surem ents during conventional heating-
D uring microwave heating, th e sam ple
could be housed in a q u artz or alum ina tu b e. At specified tem peratures, this tu b e
could be w ithdraw n from th e microwave furnace and in serted through axial holes in
a reso n an t cavity. Complex p erm ittiv ity m easurem ents could th e n be perform ed.
T h e sam ple tem p eratu re could b e m easured directly o r approxim ated w ith cooling
curves. T h e dielectric properties of th e sam ple holder would have to be factored
ou t.
D uring probe developm ent, we determ ined th a t accu rate dielectric m easure­
m ents of m aterials w ith rough surfaces were possible. A m ore comprehensive stu d y
could m easure th e reflection coefficient off of samples w ith a range of surface rough­
ness an d dielectric properties. In com puting the d a ta tables, which relate T —»• e*,
th e electro static modeling of th e surface layer could provide a m ore accurate pre­
diction of th e perm ittivity an d length of th e surface layer.
A pro b e is lim ited in m easuring th e im aginary p a rt of com plex perm ittivity of
m aterials w ith a ta n 6 > 0.05. It would be interesting to apply the K ram ers- Kronig
relatio n to th e m easurem ents of ef over a broad frequency range and determ ine e"
could be calculated. M easurem ents of e" depend more o n th e m agnitude, |F |, than
th e phase shift of the reflection coefficient. T he contour density of | r | is low for
low- loss m aterials. Therefore, th e uncertainty in e" strongly depends on slight
m easurem ent differences in |T| an d th e accuracy in m easuring e" is less th a n e',
which depends more on phase shift th a n on m agnitude o f th e reflection coefficient.
Irrespective of a m aterial’s loss tangent, a n open- ended probe can accurately
m easure th e real part of th e com plex perm ittivity. A pplying th e K ram er- Kronig
relation to th e real p art of th e complex perm ittivity, th e im aginary p art of the
com plex p erm ittiv ity could be calculated over a subset of this frequency range
172
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[25]. Therefore, th e im aginary p a rt of the com plex p erm ittiv ity over this subset
frequency range depends m ore on m easurem ents of th e phase shift of th e reflection
coefficient. Also, th e u n certain ty in e" over th is subset frequency range decreases.
6.2
Complex Permittivity Measurements using a
Resonant Cavity
T h e resonant cavity w ith a moveable wall accurately m easured th e complex per­
m ittiv ity of sam ples w ith a variety of dim ensions. T h e following are som e possible
im provem ents. F irst, sam ples were visually positioned in th e approxim ate center
o f th e cavity. A m ore consistent, accurate positioning system could be designed
a n d im plem ented to reduce uncertainty in centering th e sam ple an d complex per­
m ittiv ity m easurem ents. Second, a sam ple could be dam aged w hen th e moveable
wall is lowered onto it and its faces are not parallel. T he sam ple faces should be
m illed, sanded, grinded, or sectioned such th a t its faces are parallel. Such sam­
ple p re p a ra tio n could be difficult because th e sam ples have a range in toughness,
h ardness, an d stren g th .
T hird, exposing low- loss m aterials to th e atm osphere
allowed these m aterials to absorb w ater, which affected its com plex perm ittivity.
T h e resonant cavity could b e housed in a low h u m i d i t y controlled environment.
Also, th e w ater content of dielectric samples and laboratory hum idity should be
m onitored. F ourth, electrical connection between th e cavity an d th e m oveable wall
an d b o tto m plug of th e cavity could be improved. U nfortunately, corrosion occurs
betw een th e gold contact strips an d th e copper cavity. In order to reduce corro­
sion an d improve Q -factor of th e cavity, the cavity walls should be polished and
173
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p lated w ith gold. Finally, th e resonant frequency and Q-factor of th e cavity could
b e m ore accurately m easured. T h e resonant frequency and Q-factor of th e cavity
were m easured by finding th e m axim um peak value and 3 dB points on a resonant
curve. J. P etersan an d S. A nlage com pared various techniques in determ ining th e
resonant frequency an d Q -factor of microwave resonator [127]. T hey concluded
th a t th e nonlinear least- squares fit to th e phase vs. frequency and n o n l i n e a r leastsquares fit to a L orentzian curve are th e “m ost accurate and precise” depending on
th e signal to noise ratio [127]. F uture complex p erm ittiv ity measurem ents w ith th e
resonant cavity should incorporate one of these fitting routines to more accurately
determ ine th e resonant frequency and Q -factor of th e cavity.
6.3
Chemical Preparation of Alumina Compos­
ites
Experim ents and theoretical m odeling of com posite system s examined th e depen­
dence of p erm ittiv ity on lossy additives and m icrostructure. Chemically precipi­
ta te d alu m in a/ copper oxide com posites possessed an im aginary p art of th e com­
plex p erm ittivity larger th a n physically mixed composites. However, th e ratios of
CuO to CU2 O phases in th e chem ically precipitated and physically mixed compos­
ites differed. In order to analyze th e dependence of perm ittivity on m icrostructure,
chemical precipitation techniques could be improved to coat alum ina particles w ith
th e sam e ratio of CuO to CU2 O as in th e physically m ixed composites. T he con­
centration of polym eric dispersant and pH of the solution could be varied. Com­
parisons could th en be m ade again between prep aratio n methods.
174
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A lu m in a/ silicon carbide composites are high stren g th and durable m aterials.
T hey m ight have scientific and commercial applications. Chemical precipitation o f
alu m in a/ silicon carbide would produce a m ore homogeneous m icrostructure an d
could improve m aterial properties. Silicon carbide coated onto alum ina particles
could be synthesized by pyrolysis of vinylic polysilane [128]. Various concentrations
of this expensive silicon carbide precursor could be m ixed w ith alumina particles.
T hese com pounds could th en be fired in nitrogen to 1000° C. Since the reactan ts
possess some excess carbon, these com pounds would have to be reheated in air to
600°C. T he complex perm ittivity of a lu m i n a / silicon carbide composites could be
examined w ith respect to volume fraction and preparation m ethod. Conventional
an d microwave processing of these com posites could be perform ed and further
testin g could determ ine differences in the m aterial properties and densification
due to processing m ethod. The chemically p recip itated sintered composites should
densify more and possess b etter mechanical stren g th properties th an physically
m ixed composites.
6.4
Improve Temperature Measurements: Blackbody Radiator
N oncontact tem p eratu re measurements require knowledge of th e m aterial’s emissivity a t a specific frequency or over a frequency range. Since th e m aterial’s emissivity can vary w ith respect to tem perature and density, accurate tem perature
m easurem ents m ight require continuous calibration of th e m easurem ent system .
Two possible techniques can enable accurate tem p eratu re measurem ents.
175
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T he
calibration technique, w hich was discussed in C h ap ter 5, ad ju sts th e em issivity
slope on two- color pyrom eter controllers or creates a look- up calibration curve
to m atch the current te m p eratu re reading w ith th e a c tu a l sam ple tem p eratu re.
A nother technique creates a blackbody radiator o ut of th e ceram ic m aterial. T he
following outlines th e design of a blackbody radiator, which is m ade o ut of th e
ceram ic sam ple.
T h e ideal m aterial for tem p eratu re m easurem ents is a blackbody, which possess
a n em issivity of one. A sam ple in conventional or microwave furnaces could be
fashioned as blackbody rad ia to r. O ne can make an approxim ate blackbody cavity
by drilling a cavity into th e m aterial and focusing th e pyrom eter on th e cavity
opening. T he rad iatio n em itte d from th e cavity depends on th e tem p eratu re of
th e cavity walls w ith an em issivity of one. P yrom eter calibration is sim plified
an d invariant during heating. T he design param eters of th e blackbody rad iato r
require th a t it does not contam inate the sam ple or affect th e firing process. The
blackbody radiator could be designed w ith two pieces of th e sam e m aterial. As
shown in figure 6.1, one piece is th e cylindrically shaped sam ple, which is called the
te st sam ple. T he o ther piece (lid sample) is a cylindrically shaped sam ple w ith a
section carved o u t of it an d a hole th rough the sam ple. D uring processing, th e lid
an d te st sam ples are placed against each other. T he carved o u t section in th e fid
sam ple an d a face of th e te st sam ple forms the blackbody walls. D uring microwave
processing, electrical arcing could weld sim ilar m aterials to g eth er and little energy
would be deposited in th e rest of th e samples. To prevent arcing, a th in boron
n itride washer could be placed betw een th e test and lid sam ples [129]. Also, the
application of boron n itrid e cem ent to th e rim of th e lid sam ple could work. This
setu p should satisfy th e p aram eters provide an accurate technique to m easure the
176
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Figure
6 .1 :
Side profile of sam ple in co n tact w ith a lid sam ple. T he
o u t lin e
region
in th e lid sam ple indicates cu to u t region.
sam ple tem perature. T he lid sam ple should also provide some th erm al insulation
to th e te st sample and decrease co n tam in atio n from sample insulation.
177
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Appendix A
C om puter M odeling o f Open- Ended
Coaxial Probe
C om puter program m ing was conducted in M atlab Basic and F o rtran F77 operat­
ing on th e UNIX system . Since exact calculations would require integrating the
H ankel transform s to infinity an d incorporating an infinite num ber of TM modes,
calculations were perform ed for a finite num ber of m odes and finite integration
range. C om puter sim ulations dem onstrated th a t th ese results converged to the
exact results . Integration and calculation of th e H ankel transform coefficients
required specifying th e probe dimensions, upper lim it of integration, number of
integration points, an d num ber of TM modes.
Some of these param eters were
specified by calculating th e convergence of reflection coefficient w ith respect to the
complex p e rm ittiv ity and frequency. A m ajor problem in com puting th e reflection
coefficient an d m atching th e results to Jarv is’ paper was th e complex perm ittivity
of air. Originally, th e complex perm ittivity of air was 1.0 or, more im portantly, it
was real. A fter receiving N IS T ’s com puter program from M. Janezic, I found th a t
th ey used a complex term , e’ = 1.00055 — O.OOOlj. B y su b stitu tin g this value in
178
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for air, it elim inated th e branch cut problem in th e integration.
In order to estim ate th e system atic erro r in calculating th e reflection coefficient
for a finite param eters (modes, upper lim it of integration, a n d num ber of inte­
g ratio n points), th e reflection coefficient for an infinite num ber o f points, num ber
of m odes and upper lim it of integration was estim ated. T hese calculations would
also dem onstrate if th e reflection coefficient for a finite com p u tin g p aram eters con­
verges. These ex trap o lated results were th e n com pared to th e finite calculations.
Convergence tests were perform ed at frequency of 2.5GHz, no lift- off, a semi- infi­
n ite th ick sample, an d various dielectric properties. T h e reflection coefficient w ith
respect to infinite num ber of modes, num ber of integration p o in ts, an d u p p er limit
of integration was approxim ated by ex trap o latin g from th e following figures. Fig­
ures A .l and A.3 show th e reflection coefficient w ith respect to inverse num ber of
integration points. Figures A.2 and A.4 show the reflection coefficient w ith respect
to inverse num ber of modes. Figures A .5 and A.6 show th e reflection coefficient
w ith respect to inverse upper lim it of integration. For 20,000 a n d 100,000 integra­
tio n points an d an upper lim it of integration of 100,000, th e reflection coefficients
were ex trap o lated for an infinite num ber of modes. Since th e reflection coefficients,
in general, converged w ith respect to integration points > 20,000, th e reflection
coefficient was th en ex trap o lated to an infinite num ber of in teg ra tio n points and
num ber of modes. Finally, th e reflection coefficient w ith 12 m odes an d an inte­
g ration step size of 5 was extrapolated for an infinite u p p er lim it of integration.
A ssum ing th a t th e change in reflection coefficient w ith respect to th e u p p er limit
of integration is th e sam e for 12 modes as for an infinite n um ber of m odes case,
I th en added this change in th e reflection coefficient to th e ex tra p o la te d reflec­
tion coefficient w ith an infinite num ber o f modes and of points. Tables A .l and
179
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A.2 show the reflection coefficients for various dielectric values m ostly a t 2.5GHz.
T h e reflection coefficients for six m odes, integration ste p size of five, and an up­
p er limit of integ ratio n o f 20,000 are in column 2 and th e e x trap o lated reflection
coefficients are in colum n 3. T h e difference between these colum ns is in column 4.
T h e difference betw een th e ex trap o lated and calculated m ag n itu d e of th e reflection
coefficient is m ore th a n a m agnitude less th an th e m easurem ent uncertainty of th e
VNA. T he V N A :s u n c ertain ty in phase, which is 1°, is a b o u t two to three tim es
larger th an th e difference betw een th e extrapolated an d calculated phase values.
Also, this la tte r difference is always positive.
Reflection coefficient converged for modes n > 6 and a s te p size in integration <
4. T h e reflection coefficient also converged with u pper lim it o f integration >25,000
for 6 modes an d w ith an u p p e r lim it of integration > 50,000 for 12 modes. As th e
loss tangent increased, th e num ber of integration points decreased. Also, th e upper
lim it of integration is in dependent of complex perm ittivity. Since th e eigenvalue
of th e sixth m ode is a b o u t 13,000 an d th e eigenvalue of th e tw elfth m ode is about
26,000, it is u n d erstan d ab le th a t th e upper lim it of in teg ra tio n m ust be larger
th a t th e eigenvalue o f th e n th m ode and th a t this limit increased as th e num ber of
modes increased. A n in teresting observation was th a t th e difference between the
extrapolated an d calculated m agnitude of the reflection coefficient increased as the
loss tangent increased. Since th e fields penetration into th e m a teria l decreases as
th e loss tangent increases, th e electrom agnetic field profile has a sh arp er decrease.
Modeling of this decrease in p en etratio n requires m ore m odes as th e loss tangent
increases.
For the special case of no air gap an d semi- infinite long sam ple, th e integrand
te rm which depends on com plex p erm ittiv ity scales w ith resp ect to frequency. B y
180
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including th e observation th a t samples w ith higher complex perm ittivities con­
verge, calculations of reflection coefficients for frequencies > 2.5 GHz w ould also
converge. T he convergence for e* = 4 - O.lj was teste d a t 200 MHz. Its convergence
is also shown in Table A.2. In summary, reflection coefficient calculations for a
finite num ber o f T M ^ modes, num ber of integ ratio n points, and u pper lim it of
integration converged. Also, th e difference betw een the reflection coefficient for
finite co m p u tatio n param eters and an infinite num ber of param eters was, a t least,
a m agnitude less th a n th e accuracy of th e VNA.
181
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0.926&
n = 12
_ 0.9264
0 0.9263
wO.926
Ers= 10 - 1j
-Q
0.926-
n=6
0.925^
1/Number of integration points (10-3)
0.2
Figure A .l: M agnitude of the reflection coefficient for different num ber of m odes
w ith respect to num ber of integration points.
182
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0.9265-!*-r
Absolute Reflection coefficient
0.9264
0.926^
Ers= 10 - 1j
+ = 100K pts
* = 20K pts
0.9263
0.9263
0.9262
I
I
0.9262
0 .9 2 e |
0.926i
°-96 f e r 0.09
0?i
O il
042
013
014
015
Inverse of number of modes
0.16 V l 7
F igure A.2: M agnitude of th e reflection coefficient for different num ber of integra­
tio n points w ith respect to num ber o f modes.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-44.9r
-44.95-45
n = 12
'-45.05o -45.1
-45.15o.
-45.2
Ers= 10-1] -
-45.25-45.3
0.05
0.1
0.15 0.2
0.35 0.4
1/Number of integration points (10-3)
0.45
0.5
F igure A.3: Phase of th e reflection coefficient for different num ber of modes w ith
respect to num ber of in teg ratio n points.
184
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-45
*
-45.05Ers= 10-1]
+ = 100K pts
* = 20 K pts
oT -45.1}
CD
CD
i_
O)
CD
f-4 5 -1 5 -
*
CO
CD
JO
CL
-45.2}-
-45.25-
^ d f c r - a 09
0.1
0.11
0.12
0.13 0.14 0.15
Inverse of number of modes
0.16
0.17
Figure A.4: P hase of th e reflection coefficient for different number of integration
points w ith respect to num ber of modes.
185
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0.93a
0.932-
Ers= 10 - 1j
c
£ 0.9:
0.929<n
<0.92*1-
1/Upper limit (10-3)
1
Figure A.5: M agnitude of th e reflection coefficient for different num ber of modes
w ith respect to u pper lim it of integration.
186
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-42
-42.5
Ers= 1 0 - 1j
-43
-44
CL
-44.5
-45
n=
1 2
n=
6
1/U pper limit (10-3)
0.11
F igure A . 6 : Phase of th e reflection coefficient for different num ber of modes with
respect to upper lim it of integration.
187
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e7
4
tan<5
Ta p p ro x
4
n = oo
step size = 5
step size = oo
upper limit = 20000
upper lim it = oo
freq. = 2.5 GHz
freq. = 2.5 GHz
T = .9914
T = .9916
A r = .0002
9 = -18.87°
9 = -18.75°
A 9 = .12°
T = .9683
A r = .0003
9 = -18.87°
9 = -18.77°
A 9 = .10°
r = .8517
T = .8529
A r = .0012
9 = -18.93°
9 = -18.82°
A 9 = .11°
.025
= .968
.1
.5
r
4
= .7252
T = .7272
A r = .002
9 = -19.05°
A 9 = .12°
T = .9777
A r = .0003
1
9 = -19.17°
r
10
= .9774
.025
9 = -45.11°
6
.1
6
= -45.11°
T = .6944
10
6
A 9 = .37°
= .927
A r = .0007
= -44.70°
A 9 = .41°
r=
.6967
A r = .0023
.5
Q = -46.16°
r
10
= -44.74°
r
T = .9263
10
D iffe r e n c e
n = 6
r
4
r
= .4788
9 = -45.72°
A 9 = .44°
T = .4817
A r = .0029
9 = -49.80°
A 9 = .47°
1
9 = -50.27°
Table A .l: Convergence of Reflection Coefficient for specified param eters
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
e'
tan<5
r
Ta p p r o x
n =
D iffe re n c e
n = oo
6
step size = 5
upper limit =
step size = oo
upper lim it = oo
2 0 0 0 0
freq. = 2.5 GHz
freq. = 2.5 GHz
T = .9352
1 0 0
T = .93-52
9
= -1 5 6.6°
9
T = .9042
1 0 0
T = .9038
= -1 5 6.8°
9
T = .7848
= -1 5 6 .5 °
T = .7833
= -1 61.7°
9
r = .7474
A 9 = .3°
A r = -.0 0 1 5
= -1 6 1 .4 °
T = .7455
A 9 = .3°
A r = -.0 0 1 9
1
9
= -1 6 9.9°
n=
9
= -1 6 9 .7 °
A0
=
.2 °
n=oo
6
step size=5
step size= oo
upper lim it= 2 0 , 0 0 0
upper lim it = oc
freq. = 200 MHz
freq. = 200 MHz
T = .9994
4
A r = -.0 0 0 4
.5
9
1 0 0
A 9 = .3°
= -1 5 6 .3 °
.1
9
1 0 0
A r = - .o o o i
.025
T = .9994
A r = .oooo
.025
9
= -1 .507°
9
= -1 .4 9 8 °
A 9 = .09°
Table A.2: Convergence of Reflection Coefficient for specified param eters
189
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Appendix B
D ielectric M ixing Laws
T he following derives two different m ixing laws. T h e Maxwell- G a rn e tt th eo ry and
th e effective m edium approxim ation are sim ilar dielectric m ixing laws. T h e m ain
difference is th e selected host m aterial. T h e average fields are defined as:
(B .l)
(B-2)
where E is th e average electric field in th e m aterial, fi is th e volum e fraction of
th e ith co n stitu en t, Ei is th e average electric field in th e itk com ponent, D is th e
average displacem ent vector in the m aterial, eef j is th e effective p e rm ittiv ity of th e
m aterial, an d e, is th e perm ittivity of th e it/l constituent, and N is th e num ber of
constituents in th e m aterial. These equations can be su b stitu ted into each o th er
to give:
(B-3)
190
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N
^ 2 f i (e* ff ~ ei) E i = 0i= 1
(B.4)
M axwell- G a rn e tt assumes th a t all constituents or phases except one, which is the
h o st m aterial, co n stitu te a sm all volume fraction of the m aterial. T h e
o f th e host m aterial is the
N £/l
phase or eh. T hese
m in o r
p e rm ittiv ity
phases can then be
assum ed to be well dispersed an d do not in tera ct w ith each other. Therefore, the
electric field inside a spherical particle w ith e,- is [25,26]:
« =
(B.5)
T h is expression of th e electric field in th e m inor constituents is placed into Eq.
B.4.
n
—^
3e
f
\
y ^ /t y —jjT9£hJ
~ ^ ^ h + f h (€eff ~ 6fi) Eh =
(B-6)
T h e volume fraction of the host m aterial can be expressed by:
N- 1
h = i -
(b.7)
1=1
S u b stitu tin g in Eq. B.7 into Eq. B . 6 and pulling out the com m on term , Eh, gives:
N -1
/
Y *
(^e . + 2eh J ^ eff ~ ^ + ^ eff — €h^ ~
o
N -l
\
— eh ) = o
T h e following is sim phfications of Eq. B. 8 :
191
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(B.8 )
N
-
1
eeff - e h + Y j i
1=1
N -l
6e // ~ e/i + €e f f Y j '
i= i
j- .
3 £/!
£i + 2e/t
( e e / / — e i ) — ( e e/ / — e /i )
3e/,
(
e* -F 2 e ,
LV
-
1
+ ^ E
= 0
/ . ( ^ ) = 0
(B.9)
(B.10)
fe //
AT— 1
ee/ / =
i=l
Ci ££ + 2efc
(B-12)
Thus, th e Maxwell- G a rn e tt model estim ates th a t th e effective p erm ittiv ity is:
e* - Cfc
. €i + 26/t
(B-13)
T h e effective m edium approxim ation assumes th a t all constituents are sur­
rounded by a host m aterial, which is ce//- Thus, th e electric field in a spherical
particle w ith e* , which is em bedded in a host m aterial w ith ee/ / , is:
3 €eff
1 ~ €i
* -t- zee/
—f h * ff
(B.14)
S u b stitu tin g Eq. B.14 into Eq. B .l gives:
N
N
(B.15)
192
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B y pulling o u t E ef f of Eq. B.16 and s u b stitu tin g th e following identity:
£ >
= 1,
(B.17)
£=1
gives,
(B-18)
T h e effective m edium approxim ation estim ates th a t th e effective perm ittivity is:
193
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Appendix C
M aterials C haracterization
C .l
Transmission Electron Microscopy
C.2
Thermogravimetric Analysis
F igure C.2 shows th e mass loss of copper carb o n ate oxalate w ith respect to tem ­
peratu re. F ig u re C.4 shows the TG A of chem ically precipitated copper oxide onto
alum ina. T h e m ass percentage of alum ina oxide an d th e coated copper oxide in
figure C.4 was calculated to be 94.7% a n d 5.3%, respectively. Also, T G A o f the
subm icron alu m in a oxide powder is show n in figure C.3 and of the chem ical pre­
cip itated co p p er oxide (17.2%) onto alu m in a oxide (82.8%) is shown in figure C.5.
T h erm ogravim etric analysis was useful in calculating th e mass ratio of alum ina,
oxide to co p p er oxide. T h e change in m ass for an unknow n ratio is:
^TtlCLSS
&unk{TR'AI'zOz
d" 777-cco")
^
AI1 O3
d~ QccoTdcco
( c . 1)
where a unk is th e change in mass for th e sam ple w ith unknown concentration,
194
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Figure C .l: T E M m icrograph showing morphology obtained by coating copper
oxalate carb o n ate onto alum ina particles.
T he calcined sam ple is com posed of
56.8% (mass) AI2 O 3 + 43.2% copper oxide.
195
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-5-1C
8-3C
-3E-4C
-4E-
Temperature (C)
Figure C . 2 : Therm ogravim etric analysis of copper carbonate oxalate. H eating rate
was
1 0
° C f min to 1000°C in flowing air atmosphere.
196
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-
0.
-o.e
-O .Tr
Temperature (C)
Figure C.3: Therm ogravim etric analysis of alumina. H eatin g ra te was 10°C/ m in
to 1000° C in flowing air atm osphere.
197
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A
CD-1
S -3
-5-
Temperature (C)
Figure C.4: T herm ogravim etric analysis of alum ina oxide (94.7%) + copper oxide
(5.3%). H eating ra te was 10°C/ m in to 1000°C in flowing air atm osphere.
198
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-5C/i
CD , r
C0-1C
-3C
-39
Temperature (C)
F igure C.5: Thennogravim etric analysis of alum ina oxide (82.8%) + copper oxide
(17.2%). H eating rate was 10°C / m in to 1000°C in flowing air atm osphere.
199
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0
'Ai2 o3is th e change in m ass for alum inum oxide, a ^ is th e change in m ass for cop­
per carb o n ate oxalate, mAi2 o 3 is m ass of alum inum oxide in th e unknow n sample,
an d rricco is th e m ass of copper carbonate oxalate in th e unknow n sam ple. Solving
for TTlcco'.
_
W-cco
__
(
I
V
a AhQ3 \
'
O tc c o —
C lu n k
J]
0X
C alculating th e m ass of copper oxide a fte r calcining copper carb o n ate oxalate:
racuo = (1 - a-cco)™.^ = (1 - <*«*,)
—
\
&cco
-
^ 2 ° 3 ^ m Ai2 o 3
&unk J
(C.3)
T h e ratio o f copper oxide to alum inum oxide is:
’W'CuO
\ f &unk Q-AlzOz A
---------- = (1 - Oicco) ---------------—
ffl'AlzOz
\ &cco &unk J
(C.4)
T h e m ass p ercentage of copper oxide is then:
m axin
% mc«o = ,
^
C.3
(C.5)
Ol/U203
X-ray Diffraction
X -ray diffraction p attern s for alum ina a n d copper oxide are show n in figure C . 6 .
X-ray diffraction p a tte rn s for alu m in a/ copper oxide com posites are shown in figure
C.7. T h e la ttic e spacing and relative in ten sity was com pared w ith reference d a ta
for alum ina. copper(II) oxide, and copper(I) oxide. T h e lattice space indicative
200
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of copper(I) oxide is 2.4577 a n d copper(II) oxide is 2.3209 . In figure C.7, th e
relative intensities of copper(I) to copper(II) oxide is 20% less in th e com posite w ith
5% (m ass) copper oxide th a n in th e com posite w ith 17%. T hus, th e com posite
w ith 17% copper oxide has a higher percentage am ount o f conducting m aterial
(copper(II) oxide) th a n th e com posite w ith 5% copper oxide.
201
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o
520
—
10
-
Copper Oxide
-
!
2.5
Lattice Size (A)
Figure
C .6 :
X -ray diffraction p a tte rn for subm icron
a lu m in a a n d
cipitated copper oxide.
202
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3.5
chemically pre­
- 5.27% copper oxide
2.25
o
o
2.3
2.35
2.4
Lattice Size (A)
2.45
2.5
2.35
2.4
Lattice Size (A)
2.45
2.5
- 17.2% copper oxide
2.25
2.3
Figure C.7: X -ray diffraction p a tte rn for 94.73% alu m in a -f- 5.27% copper oxide
and 82.8% alum ina + 17.2% copper oxide.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix D
R esonant Cavity
A n exact solution of a loaded cylindrical resonant cavity follows. S tartin g w ith
M axwell’s equations:
v H = P f
(D .l)
V ' B = 0
(D.2)
dB
v * ^ = ~ aT
(D -3)
d~E)
V x H = Jf + —
(D.4)
T$ = €~e!
(D.5)
-F t
For linear m edia:
204
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T his case has no free currents or charges, J f — pj- =
0
.
Applying th e curl to Eq.D.3:
V x V x i? = V
(v •
~ V 2^
=
(v x
H)
v
d 2~S
V ^ = A * e -^ “
9
(D.7)
—
(D-8 )
T h e tim e dependence of th e electric an d m agnetic fields can be w ritten as:
i? (T*, t) =~E? (T*') ex p (—zu;£)
(D.9)
(~r*) exp(—iwt)
(D.10)
1?
where u is th e angular frequency, u) = u>T — i u>i where u r is th e resonant angular
frequency an d Ui =
Substituting Eq. D.9 into Eq. D . 8 gives:
V 2# + p.eu2E = 0
( D .ll)
Since this case has cylindrical symm etry, Eq. D .l l can be expanded in cylindrical
coordinates:
205
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i a ( 8E\
^
[p f y )
1
a2s
+ 7 W
a2E
+ a ? +
„
=
(D.12)
A general solution of this eq u atio n in th e axial direction is:
E z = (A J m (a p ) + GlVm (ap )) (C eim* + D e 'irn*) ( E cos (fcz) + F s i n (Jfcz))
(D.13)
a 2 = pea; 2 — k 2
(D.14)
Following th is analysis, a general solution for th e m agnetic field is in th e axial
direction:
B z = (A J m (ap) + G N m (ap)) (Ce*"* + D e ^ ) ( E cos (kz) + F s in (kz))
(D.15)
Since th e diam eter of the cav ity is less th an its height, th e lowest order modes
in a cylindrical cavity are T M m odes [25].
For T M modes,
Bz = 0
(D.16)
T h e electric field in th e axial d ire c tio n , E2, interact w ith th e m aterial’s dipoles.
In region
E zl =
1
in figure 4.20, Eq.D .13 becomes
A y Jm (a lP)(Cyeim* + Dye-™ *) (Ey cos (kz) + Fy sin (kz))
206
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(D.17)
C?i=0 because Nm(0) asym ptotically diverges to —oo. Also,
2
2
al =
t
2
“ &
In region 2 in figure 4.20,
E ~ 2 = ( A 2 Jm {ct2 p) + G 2 N m ( a 2 p)) ( C 2 eirrui> + D 2 e~tTn4>) (E 2 cos ( kz) + F 2 sin ( kz))
(D.18)
w here
2
2
?
2
<*2 = A^€2o; - rtr
To satisfy th e boundary conditions a t z = 0 and z = I, th e electric field parallel
to a conducting surface, En is zero.
E xpanding eqs. D.9 an d D.10:
= ^'E pp -+- E^tp + E zk j e iu>t
t
=
( B pp + B 4$ ) e"*■*
(D.19)
(D.20)
P lacing these expressions into M axwell’s Eq. D.3 and D.4:
Bp = —
u
1dEz
p d<p
dE,
dz
207
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(D.21)
B* = —
u
Ep =
EP
dz
dBz
uipe p d(f>
1
Erh =
Ez =
8
uipe
dBp
dz
uipep
dEz
dp
(D.22)
dB,
dz
(D.23)
dBz
dp
(D.24)
d<t> ) .
(D.25)
To rela te E# to E z , Eq. D.21 is su b stitu te d into Eq.D.24:
Ea =
dz
/ I d2 E z
u 2pe \pd< pdz
d2 E d
dz2
(D.26)
= ± — E<t, (u)2pe - k 2)
vn
'
(D.27)
1
A t th e cavity plane boundary surfaces,
dEz
12=0,/ = 0
dz
Forcing th is boundary condition on eqs. D .17 and D.18:
dEz
r=o oc Fi cos 0 = 0, Fi = 0
dz
208
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dE
——
—
^ 1 , —f oc E i sin kl =
dz
0
T hus,
k l = p7r, p =
0
, 1 , 2 ...
E quations D.17 and D.18 become:
E zi = AxJ m ( a xp)
2
O f! =
+ Dxe"1'^ ) cos ( ^ )
fXxeiuj 2
-
(D.28)
m
an d
^
2
= (A 2 Jm (a 2 p) + G 2 N m (a 2 p)) (C 2 eim* + Z)2 e - * ^ ) cos
2
2
a 2 = P2e2^ -
(D.29)
J
To satisfy th e boundary condition En (p = b) = 0 ,
£ * 2
(p =
6 ) =
0
209
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(D.30)
A 2
Jin (a 2b)
-f-
G2Nm ( a 2 6 )
=
(D.31)
0
A pplying Eq. D.3 a t th e interface p = a :
E zi = E z2 |P=Q
(D.32)
T his requires C\ = C 2 an d D \ = £>2. M atching equations D.28 an d D.29.
A i J m (ckia) = A 2«/m ( a 2a) + G 2iNrm (o:2a)
(D.33)
A pplying Eq. D .l a t th e b o u n d ary interface p = a :
(D . 34)
= -D2j_ |p=a
d2Ep
dz2
=
(C eim</>+
u 2pe — k 2 v
d2E z
dz d p
(D.35)
G iN ™ (ocip))
7
dp
v
'
A pplying Eq. D.36 to b o u n d ary condition in Eq. D.34:
e\ E pi — e2 E p 2
210
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(D.37)
(a l a) = ~
<*i
(a 2“ ) + ^2-^Ci (<*2^)^
(D.38)
Solving for th e three unknow n constants w ith equations D.31, D.33, and D.38.
Solving for A 2 in term s of G 2. eq u atio n D.31 becomes
a
2
G 2 N m (Q2 6 )
Jm (a 2 b)
(D.39)
S ubstitu tin g th is into eqs. D.38 an d D.33:
S . * / , ( « , « ) _ 2® *
a1
a2
AiJm (^ 1 ®)
— G2
Jm ( 0 :2 6 ) AC (Q2 tt) ~ Jin (<*2 a) N m (a 2 b)
Jm (ct2 b)
Jm (oc2 b) N m ( a 2 a) - Jm (<*2 <i) N m (a 2 b)
(D.40)
(D.41)
Dividing Eq. D.40 by Eq. D.41 gives:
£i J ’m { a i a ) _ _£2_
Jm
0:2
Jm (0 =2 6 ) N.Jm (a 2 a) - f m ( a 2 a) N m (a 2 b)
Jm (oi2 b) N m (a 2 a) - Jm (a 2 a) N m (a 2 b)
211
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(D.42)
Appendix E
Microwave System: Furnace,
Temperature M easurem ents, and
Control System
Microwave processing was performed in th e highly overm oded 2.45 GHz applica­
to r (Model 101 from Microwave M aterials Technology, Inc.) as shown in figure
E .l. This microwave system possesses four key com ponents: microwave source,
applicator, tem p eratu re measurem ent system , a n d control system. The following
describes these im p o rtan t and other supporting com ponents.
As shown in figure E.2, the microwave source is a 2.45 GHz magnetron w ith a
maxim um power of 3 kW . T he microwaves travel along a S-band waveguide which
connected to a circulator, bi- directional coupler, 90° bend, and the applicator. A
quartz window divides th e waveguide and applicator e n v ir o n m e n ts . This a lu m in u m
cylindrical applicator has an inner diam eter of 0.768 m and a length of 1.22 m.
T h e wavelength of microwaves (A = 0.1224 m ) is several times less th a n an y
applicator dimensions. T h e cavity is classified as highly overmoded because th ere
212
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Figure E .l: F ront Profile of Microwave P rocessing System w ith C om puter Control
C art
are m any T E a n d T M modes excited a t th e frequency of operation. A small fan
in the upper p o rtio n of th e applicator mixes th e electrom agnetic modes.
The
interference of th ese modes produces regions in th e cavity w ith uniform microwave
fields [109]. T h e atm osphere in the applicator c an be evacuated w ith a ro ta ry vane
pum p (Model D 30A from Trivac) and backfilled w ith a desired gas (ex. dry air,
nitrogen, an d argon). T he pressure of th e cham ber is slightly overpressurized to
1.1 atm . as a safety measure. As shown in figure E.3, th e sample a n d insulation
are placed on a stainless steel table, w hich is su p p o rted by alum inum rectangular
tubes. D uring microwave processing, th e sam ple, insulation, and applicator heat
up. In order to keep th e chamber an d tab le cool, cold w ater passes along th e
exterior applicator walls and through th e alum inum rectangular tubes. In order to
protect the m agnetron, th e circulator diverts any reflected microwave power from
th e applicator in to a w ater load.
213
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Figure E.2: 3 k W Microwave Source operating at: 2.45 GHz
214
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F igure E.3: Ceramic Sam ple loaded into Porous A lum ina C rucible sittin g o n th e
M icrowave A pplicator Table.
T h e original microwave sy stem m easured the sam ple te m p e ra tu re w ith a type K
or S therm ocouple. In itial research utilized type K therm ocouples to m easure th e
te m p e ra tu re of ZnO sam ples du rin g microwave processing. However,
s ig n ific a n t
problem s lim ited ceramic processing and accurate tem p eratu re m easurem ents. A
b rief description of therm ocouple problem s follows and rep o rted in recent paper
by P e rt et. al. [130]. As m entioned in sect. 2.5, shielded or unshielded therm o­
couples are metallic, th e microwaves can only be perpendicular to th e surface of
th e therm ocouple. Field intensification a t th e tip of th e therm ocouple has resulted
in sp o t h eatin g of the sam ple a n d arcing. This local heatin g of th e sam ple around
th e therm ocouple tip also generates therm al gradients. D u rin g rap id heating or
applying large changes in microwave power, this local h eatin g has lead to therm al
shock a n d destruction of sam ples. E rrors in tem perature m easurem ents also result
from te m p eratu re differences betw een th e sample and therm ocouple tip. For exam-
215
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
pie, th e lack of contact betw een th e therm ocouple a n d sample can be resu lt from
th e sample densifying an d moving away from the therm ocouple. A nother source
of error occurs during rap id heating of samples (>100 °C / m in.). T herm al tra n s­
p o rt from th e m etallic sh eath to th e therm ocouple junction is not instantaneous
a n d results in a tim e delay in m easuring th e sample tem perature. O ther problem s
w ith thermocouples axe th erm al tra n sp o rt of energy away from th e sample, ohm ic
heating of th e therm ocouple tip , and m etal diffusion from the therm ocouple to th e
sam ple at elevated tem p eratures (>1000 °C).
In order to elim inate these problem s and obtain accurate tem p eratu re mea­
surem ents, non- contact or optical tem perature measurem ent techniques were re­
searched and applied to this microwave furnace. Infrared radiation em itted from
a m aterial is indicative of its tem perature. Planck’s formula, Eq. E .l, relates the
em itted radiation w ith respect to sam ple tem perature, emissivity and wavelength.
*(A ) = ^ * e ( X , T , p ) * ( efa/i - _
(E . 1 )
w here R is radiant intensity, A is wavelength, c is speed of light in a vacuum , e
is emissivity, and h is P lanck’s constant. Sample em issivity varies w ith respect to
tem perature, frequency, a n d density. Therefore, optical techniques require knowl­
edge of the sample em issivity o r provide m ethod to compensate. One noted excep­
tio n is the application of m ultiw avelength pyrometry. This technique can m easure
tem p eratu re from 500 to 5000 K w ithout detailed inform ation of th e surface emis­
sivity and in a hostile environm ent [131].
U pon reviewing existing comm ercial and affordable non- contact tem p eratu re
system s, three different pyrom eter system s were selected to measure from room
tem p eratu re to 2500° C [132]. T his array consisted of one single- color (M odel
216
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
67S from M ikron for room tem perature to 500°C) an d two two- color pyrom eters
(Model M77S from M ikron for 400- 1400° C a n d 1000 - 2500° C). T h e single- color
pyrom eter m easures infrared radiation a t a single wavelength.
T h e tw o- color
pyrom eters m easure rad ian t intensity a t tw o different wavelengths. Eq. E .2 shows
th a t th e ratio of rad ian t intensities provides a value independent of th e emissivity,
but dependent on th e ratio of these em issivities.
5
r = R( X 2 ) / R( Xi ) =
£(^LjT , p ) V^ 2 /
'g h c / X jf c T
g h c /\ik T
p
jL
(E.2)
Since th e Ai an d A2 are ab o u t th e same, th is ratio (Ax / A2 ) or “slope” should be
about one an d could depend on tem p eratu re an d density. As shown in figure E.4,
these th ree pyrom eters were attached to a linear stage, w hich is th e n m o un ted to
the front door of th e applicator. An optical p a th directs infrared rad iatio n from
the irradiated sam ple to th e selected pyrom eter. F igure E.5 shows th e sam ple
em itting infrared rad iatio n downward th ro u g h holes in th e alum ina insulation to
the gold m irror. T his m irror reflects IR th ro u g h a 2.54 cm diam eter alum inum tu b e,
which attaches to th e inside of th e microwave furnace. N ext, IR passes th ro u g h
a barium fluoride window or lens and reaches th e desired pyrom eter. A ccu rate
tem perature m easurem ents require careful alignm ent of th is light p a th and focusing
of th e pyrom eters.
In order to accom m odate these changes a n d im prove operating flexibility, a new
control system was designed an d built [132]. As shown in figure E .l, th e co m p u ter
control cart housed a 166 M Hz IBM com patible P C , th ree Mikron te m p eratu re
controllers, an d two therm ocouple controllers. T h e P C interface cards control th e
m agnetron power, linear stage w ith pyrom eter array, and output displays. T h ey
also receive signals from power, tem perature, pressure, and vacuum m easurem ents.
217
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Figure E.4: P y ro m eter A rray A ttached to Microwave A pplicator D oor w ith B arium
Fluoride Lens.
In general, microwave processing can b e perform ed by controlling th e microwave
power, sam ple tem p eratu re, a n d /o r densification ra te of sample. H om e microwave
ovens prim arily control th e microwave power over a specified tim e.
Tem pera­
tu re control system s ad ju st th e applied power in order to m atch a desired sta tic
or dynam ic te m p eratu re profile. Some com m on examples are a household th er­
m ostats, conventional hom e ovens, and furnaces. Microwave furnaces have also
been controlled w ith respect to th e densification rate of th e sam ple in order to
prevent cracking [133,134]. T his new control system enables controlling th e mi­
crowave furnace w ith respect to power, sam ple tem perature, and eventually den­
sification w ith a n optical extensom eter. D uring tem perature control, a dynam ic
PED (P roportional- Integral- Differential) control algorithm ad ju sts th e m agnetron
power so th a t th e sam ple tem perature closely m atches a desired tem p e ra tu re p ro ­
file [132].
T h e com puter also positions th e appropriate pyrom eter into th e IR
218
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.
2.
3.
4.
5.
6.
Sample
Porous Alumina Insulation
Stainless Steel Table
Gold Mrror
Aluminum Tube
Thermocouple
Figure E.5: Cross Section View of O ptical P a th in th e M icrowave Furnace
219
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
light p ath . T he param eters for th e
PED
algorithm are found by
ru n n in g
several
experim ents w ith dum m y sam ples. These param eters are varied until th e sam ple
tem p eratu re closely m atches th e set tem perature.
As m entioned earlier, single- or two- color pyrom eters require knowledge of th e
sam ple em issivity or calibration of the tem p eratu re sensors. T h e pyrometers can
be calibrated against type K or C therm ocouples d u rin g microwave processing. In
sp ite of th e problem s of therm ocouples during microwave processing, a com parison
can be m ade between these different techniques u n d er appropriate conditions. By
slowing heating th e sam ple (~2°C /m in), tem p e ra tu re gradients, field intensifica­
tion, an d arcing can be reduced. T h e single- color pyrom eter (0- 500° C) requires
calibration for each new type of sample and insulation arrangem ent. C alibration
is perform ed by aligning and focusing each pyrom eter. As shown in figure E.5,
a therm ocouple is placed inside a hole in th e sam ple, which is near th e observed
sam ple surface. T he tem p eratu re difference betw een these two points is small. T he
com puter records th e o u tp u t voltage from th e pyrom eters versus th e therm ocouple
m easurem ents. T he single- color pyrom eter produces a nonlinear voltage response
w ith respect to sam ple tem perature. A look- up ta b le is m ade to relate tem per­
atu re an d m easured voltage as shown figure E . 6 . T h e background radiation from
th e barium fluoride lens, gold m irror, and cavity strongly affect the m easurem ent
accuracy of th e low tem p erature single- pyrom eter. Since ceramic sintering does
not or m inim ally occurs below 500 °C, accurate tem p eratu re m easurem ents are
not essential. In this region, th e critical processing p aram eter is measuring th e ap­
proxim ate sam ple tem perature and sm ooth tra n sitio n from th e low to middle range
pyrom eters. T h e switching between these pyrom eters has resulted in tem perature
differences betw een 0 to 100°C. T he general agreem ent between the pyrom eters
220
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
an d th e control system enable for te m p e ra tu re control through tran sitio n regions
a n d has not ham pered microwave processing.
T he two- color pyrom eters o u tp u t a voltage proportional to sam ple te m p e r­
atu re . For th e two- color pyrometers, th e em issivity slope on th e te m p e ra tu re
controllers can be adjusted or look- u p tables could be m ade. Figures E . 7 a n d E . 8
show the tem p eratu re difference betw een th e two- color pyrom eters and th e rm o ­
couple. T he em issivity slope for th is a lu m in a / silicon carbide sam ple and several
o th e r samples was found to be one. Since th e tem p eratu re difference over th e
ran g e in te m p eratu re is relatively small, th e tw o- color pyrom eters can accu rately
m easure th e sam ple surface tem perature above 500° C. During microwave pro­
cessing, this pyrom eter array accurately 'm easures th e sam ple tem p eratu re from
room tem p eratu re to over 1600°C and provides a feedback signal for te m p e ra tu re
control.
221
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450-
400-
02502200 -
10C
5C0.1
(9.05
0.15
0.2
0.25
Voltage of Low Temp Pyrometer (V)
0.3
Figure E . 6 : Look- Table for 80% A lum ina and 20% Silicon Carbide: R elation
betw een Sam ple T em perature an d Voltage R eading of the Single Color Pyrom eter.
X
60 -
£50-
| 30
-2C
^Jijo
500
600
700
800 900 1000 1100 1200 1300 1400
Temperature of Type C (C)
F igure E.7: Sample 80% A ltunina and 20% Silicon Carbide: T em perature Differ­
ence betw een Middle R ange P yrom eter and T ype C thermocouple.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
~^300
1350
1400
1450
1500
1550
Temperature of Type C (C)
1600
1650
Figure E.8: Sample 80% A lum ina and 20% Silicon Carbide: T em perature Differ­
ence betw een Upper R ange P yrom eter and Type C therm ocouple.
223
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