Title Page
 Table of Contents
 List of Tables
 List of Figures
 Research objectives and accomp...
 Materials and experimental...
 Summary and conclusions
 Suggestions for future work
 Biographical sketch

Title: Removal of polymethyl methacrylate from alumina compacts using microwave energy
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Permanent Link: http://ufdc.ufl.edu/UF00090916/00001
 Material Information
Title: Removal of polymethyl methacrylate from alumina compacts using microwave energy
Series Title: Removal of polymethyl methacrylate from alumina compacts using microwave energy
Physical Description: Book
Creator: Moore, Edmund Harvey,
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Bibliographic ID: UF00090916
Volume ID: VID00001
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Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
        Page iv
        Page v
    Table of Contents
        Page vi
        Page vii
        Page viii
    List of Tables
        Page ix
        Page x
    List of Figures
        Page xi
        Page xii
        Page xiii
        Page xiv
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    Research objectives and accomplishments
        Page 83
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        Page 87
    Materials and experimental procedure
        Page 88
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    Summary and conclusions
        Page 231
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    Suggestions for future work
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    Biographical sketch
        Page 255
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Full Text







Copyright 1993


Edmund Harvey Moore

I would like to dedicate this dissertation to my loving
and caring family: with all of my love, thanks.


I would like to thank a number of people for their

support, suggestions and contributions towards this study as

well as to the overall quality of my existence. First, I

would like to express the utmost and most sincere thanks to my

committee chair, research director and acquaintance, Dr. D.E.

Clark. I thank him for his support, guidance, constructive

criticism and other intangibles over the past couple of years,

which made graduate life fruitful.

I would also like to thank Drs. R.T. DeHoff, M.D. Sacks,

C.D. Batich and J.H. Adair, all of the Department of Materials

Science and Engineering, and Dr. K.B. Wagener of the

Department of Chemistry, for useful comments, encouragement,

advice and assistance.

I would like to thank all of my present and former

colleagues in Dr. Clark's research group, the Department of

Materials Science and Engineering, the Department of

Chemistry, my colleagues at Wright-Patterson Air Force Base,

the University of Florida Advanced Materials Research Center

and the Atomic Energy of Canada Limited Research Company.

A special thanks goes out to Salwan Al-Assafi, Zakaryae

Fathi, Dr. Iftikhar Ahmad, Rebecca Schulz, Diane Folz, Richard

Linert and Alex Cozzi for research assistance. A special


thanks also goes out to Dr. Ron Hutcheon, Guy Latorre, Dr.

M.D. Sacks, Amy Bagwell, Gary Scheiffele, Richard Crockett,

John West, Dr. W.S. Brey, Dr. Lee Smithson, Dr. R. Kerans,

Rudy Strohschein, Dr. R.S. Duran, J. Patton and J. Linert for

various analyses, research assistance and materials

fabrication. A special thanks goes out to Richard Napier,

Sr., for reviewing this manuscript, for being there for

support, and for his many helpful suggestions.

Finally, I would like to thank my parents, brother,

grandmother, niece, aunts, uncles, the Warren Temple United

Methodist Church family, The Mount Carmel Baptist Church

family, Omega Psi Phi Fraternity members and friends for their

constant support, encouragement and belief in me. I would

like to thank God without whom this work would not be






ABSTRACT . . . . .



General Introduction .
Outline of Study . .

2 BACKGROUND . . . .

Microwaves ... . . . . . . .
Microwave Generation . . . . .
History of Microwave Heating . . .
Industrial Applications of Microwave
Heating . . . . . . . .
Ceramic Processing . . . .
Ceramic, Polymer and Ceramic/Polymer
Processing Applications . . .
Single Mode and Multimode Microwaves .
Theoretical Aspects of Microwave Interactions
and Heating . . . . . . . .
Polarization Mechanisms . . . . .
Electronic polarization . . .
Atomic (ionic) polarization . .
Dipole (orientational) polarization
Interfacial polarization . . .
Polarization Theory . . . .
Complex Form of Dielectric Constant . .
Rate of Increase in Temperature . . .
Penetration Depth of Microwaves . . .
Critical Temperature and Thermal Runaway
Tailoring Systems to Reach a
Material's T . . . .......




S 32


. . ..Pa. iv
. . . . . . .iv

. . . . . . . xi

. . . . . . . xxi

Dielectric Mixture Models . . . .. 58
Microwave Heating . . . . . ... 62
Conventional Heating . . . . ... 64
Polymeric Binders . . . . . . .. 66
Binder Properties and Characteristics . 66
Classification of Binders . . . ... 69
Binder Tacticity . . . . .... .. 72
Thermal Degradation of Polymethyl
Methacrylate (PMMA) and Diffusion . . 72
Equations for Addition of Binder to
Ceramic Materials . . . . ... 79
PMMA Binder Removal . . . . . . . 80
Conventional Binder Removal . . ... 81
Microwave Binder Removal Work ..... .81


Research Objectives . . . . .
Accomplishments . . . . . .


S . . 83
S . . 85

. . 88

Materials . . . . . . . . . 88
Materials Characterization . . . . ... 90
Gel Permeation Chromatography ...... .90
Nuclear Magnetic Resonance . . . .. 91
X-ray Sedimentation. . . . . . 92
Nitrogen Gas Adsorption Method . . .. 92
Experimental Procedure . . . . . .. 93
Analytical Techniques . . . . . ... 97
Thermal Analysis . . . . . ... 98
Thermal gravimetric analysis (TGA) . 98
Differential scanning calorimetry
(DSC) . . . . . . . 99
Microwave thermogravimetric
analyzer (MTGA) . . . . .. 99
Composition and Microstructural
Analysis . . . . . . . .. 100
Energy dispersive spectroscopy
(EDS) . . . . . . .. 100
Scanning electron microscopy (SEM) . 100
Dielectric Analysis . . . . . .. 101
Residue Analysis . . . . . .. 104
Fourier transform infrared
spectroscopy (FTIR) . . . .. 104
Inert gas fusion instrument ... . 105
Binder Burnout . . . . . ... 106
Conventional furnace . . . . 106
Microwave thermogravimetric analyzer
(MTGA) . . . . . . . 108
Raytheon microwave . . . ... 112

5 RESULTS . . . . . . . . . .

Introduction . . ...........
Thermal Analysis . ....
Dielectric Analysis . . . . .
Binder Burnout . . . . . .
Microstructure . . . . . .
Dielectric Mixture Model . . . .

6 DISCUSSION . . . . . . . . .

S. 116


S. 212

Introduction . . ........ ... .212
Differences or Advantages of Microwave
Binder Removal . . . . .... . 213
Effect of Tacticity on Microwave Binder
Removal . . . . . .... .. 215
Effect of Sample Mass on Binder Removal 217
Effect of Binder Molecular Weight on
Binder Removal . . .... . . 218
Effect of Binder Concentration on Binder
Removal .... . . . . . 219
Effect of Alumina Particle Diameter on
Binder Removal . . . ... ..... 220
Effect of Carbon Powder Additives on
Binder Removal . . . . . .. 222
Microwave Interactions With the
Composite System . . . . ... 223

7 SUMMARY AND CONCLUSIONS . . . . . ... 231


REFERENCE LIST . . . . . . . . ... 242

BIOGRAPHICAL SKETCH . . . . . . . ... .255







Table 4.3

Table 4.4





Table 5.3

Table 5.4

Table 5.5


Classification Of Binder Materials . .

Physical Properties Of The Alumina Powders

Physical And Chemical Properties Of The
PMMA Binders . . . . . . . .

Calculated Dyad Tacticities Of
PMMA Binders . . . . . . . .

Materials Used To Prepare 156 Grams Of
4 w/o PMMA/Alumina . . . . . .

FTIR Peak Assignments For PMMA Resins . .

Compositional Analysis For 12 And 24
Gram 2010 PMMA/Alumina Samples Heated In
The MTGA And Unheated Alumina . . . .

Compositional Analysis For 12 And 24
Gram 2010 PMMA/Alumina Samples Heated
In A Furnace . . . . . . . .

Compositional Analysis For 8, 12 And 16
Gram 2010 PMMA/Alumina Samples Heated
In A Furnace And The MTGA . . . . .

Compositional Analysis For 12 And 24
Gram 2010 PMMA/Alumina Samples Heated
In A Furnace And The MTGA . . . . .











Table 5.6 Room Temperature Measured Dielectric Data
At 2450 MHz . . . . . . . . 209

Table 5.7 Room Temperature Measured Dielectric Data
At 2450 MHz . . . . . . ... .210

Table 5.8 Room Temperature Calculated Bulk
Dielectric Data At 2450 MHz . . ... 211

Table 6.1 The Surface Area To Volume (SA/V) Ratio For
Cylindrical Compact Samples (Pressed At 4000
psi) . . . . . . . . . 229


Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Interaction of microwaves with
materials, adapted from Sutton
[Sut89] . . . . . . . .

The electromagnetic spectrum, adapted
from Sisler [Sis80] . . . . .

Typical examples of critical temperatures
in ceramic materials . . . . .

An illustration of the field patterns
(standing wave or mode) inside a single
mode microwave waveguide, adapted
from Liao [Lia88] . . . . .

Schematics of mechanisms of polarization
in ceramics: (a) electronic, (b)
atomic or ionic, (c) high-frequency
oscillatory dipoles, (d) low frequency
cation dipole, (e) interfacial space
charge polarization at the electrodes
and (f) interfacial polarization at
heterogeneities. Adapted from Hench
[Hen90] . . . . . . .

Frequency dependence of polarization
mechanisms in dielectrics: (a)
contributions to the charging constant
with representative values of K' and
(b) contributions to the loss angle
with representative values of tanS.
Adapted from Hench [Hen90] . . .



Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.12

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

The effect of temperature on the
maximum tanS over a wide range of
frequencies, adapted from Kingery
[Kin76] . . . . . . .

Model for the calculation of an internal
field [Von59] . . . . . .

Charging density (Jc) and loss current
density (J.) of the polarization
current [Kin76] . . . .

Loss tangent versus temperature for a
(a) low loss material containing
microwave absorbing additives, and
(b) the low loss material without
the microwave absorbing additives .

Monomer unit of PMMA . . . . .

Depolymerization of PMMA . . .

Flow chart of sample preparation . .

Experimental design flow chart . .

Schematic of the dielectric analysis
instrument [Hut92a] . . .

Schematic of the Lindberg box furnace

The l/e penetration depth for P-SiC at
2450 MHz and from room temperature
to 700C . . . . . . .

MTGA schematic . . . . ..

A schematic of the insulation cavity














Figure 4.8

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7

A schematic of the Raytheon
microwave oven . . . . .

Heat flow versus temperature for
PMMA from 60"C to 240*C using a
flowing helium atmosphere
(DSC analysis) . . . . .

Heat flow versus temperature for
PMMA from 60C to 2400C using a
flowing helium atmosphere
(DSC analysis) . . . . .

Heat flow versus temperature for
PMMA from 600C to 2400C using a
flowing helium atmosphere
(DSC analysis) . . . . .




Heat flow versus temperature for
isotactic PMMA from 60C to 240C
using a flowing helium atmosphere
(DSC analysis) . . . . . .

The weight percent versus temperature
for 4 w/o PMMA/AKP-30 alumina composed
of (a) 2008, (b) 2010 and (c)
2041 PMMA from 50'C to 1000C
using a flowing air atmosphere
(TGA analysis) . . . . . .

The weight percent versus temperature
for 4 w/o PMMA/AKP-30 alumina composed
of (a) 2010 and (b) isotactic PMMA
from 50C to 1000C using a flowing
air atmosphere (TGA analysis) . .

The weight percent versus temperature
for 4 w/o 2010 PMMA/alumina composed
of (a) AKP-30 and (b) AKP-15 alumina
from 50C to 1000C using a flowing
air atmosphere (TGA Analysis) . .









Figure 5.8

The weight percent versus temperature
for 4 w/o PMMA/AKP-30 alumina
containing 1.0 w/o carbon from
500C to 10000C using a flowing air
atmosphere (TGA analysis) . . .

Figure 5.9

Figure 5.10

Figure 5.11

Heat flow versus temperature for the
4 w/o 2010 PMMA/AKP-15 alumina from
600C to 2400C using a flowing helium
atmosphere (DSC analysis) . . .

Heat flow versus temperature for the
4 w/o 2010 PMMA/AKP-30 alumina from
60*C to 2400C using a flowing helium
atmosphere (DSC analysis) . . .

Heat flow versus temperature for the
8 w/o 2010 PMMA/AKP-30 alumina from
600C to 2400C using a flowing helium
atmosphere (DSC analysis) . . .


S 131


Figure 5.12

Figure 5.13

Figure 5.14

Figure 5.15

Temperature and change in weight percent
of sample versus time for 4 gm 4 w/o
PMMA/alumina samples composed of
different particle diameter alumina
(MTGA data) . . . . . . .

Temperature and change in weight percent
of sample versus time for 2 gm and 4 gm
4 w/o PMMA/alumina samples composed of
AKP-30 alumina (MTGA data) . . .

Temperature and change in weight percent
of sample versus time for 4 w/o PMMA/
AKP-30 alumina sample composed of
2008, 2010, 2041 or isotactic PMMA
(MTGA data) . . . . . . .

The typical heating schedule used to
measure the dielectric properties
data (e' and e") for the materials
used in this study . . . . .






Figure 5.16

Figure 5.17

Figure 5.18

Figure 5.19

Figure 5.20

Figure 5.21

Figure 5.22

Figure 5.23

The loss tangent versus temperature,
from room temperature to 1000*C,
for AKP-15 and AKP-30 alumina
at both 915 MHz and 2450 MHz . . .

The loss tangent versus temperature,
from 200*C to 11000C, for a 94% purity
alumina, adapted from [A1A92], and a
99.99% purity alumina at 2450 MHz .

The loss tangent versus temperature for
2008 and isotactic PMMA are shown at
915 MHz, from ambient to 500C . .

The loss tangent versus temperature for
2008 and isotactic PMMA are shown at
2450 MHz, from ambient to 5000C . .

The loss tangent versus temperature of
4 w/o PMMA/AKP-30 alumina samples
prepared with 2008, 2010, 2041 or
isotactic PMMA is given up to 1000C
at 2450 MHz . . . . . . .

The loss tangent versus temperature of
4 w/o PMMA/AKP-30 alumina samples
prepared with 2008, 2010, 2041 or
isotactic PMMA is given up to 1000C
at 915 MHz . . . . . . .

The loss tangent versus temperature of
4 w/o PMMA/AKP-15 alumina samples
prepared with 2008, 2010, 2041 or
isotactic PMMA is given up to 1000C
at 2450 MHz . . . . . . .

The loss tangent versus temperature of
4 w/o PMMA/AKP-15 alumina samples
prepared with 2008, 2010, 2041 or
isotactic PMMA is given up to 1000C
at 915 MHz . . . . . . .









Figure 5.24

Figure 5.25

Figure 5.26

Figure 5.27

Figure 5.28

Figure 5.29

Figure 5.30

Figure 5.31

Figure 5.32

The loss tangent versus temperature of
8 w/o PMMA/AKP-15 alumina samples
prepared with 2008, 2010, 2041 or
isotactic PMMA is given up to 1000C
at 2450 MHz . . . . . . .

The loss tangent versus temperature of
8 w/o PMMA/AKP-15 alumina samples
prepared with 2008, 2010, 2041 or
isotactic PMMA is given up to 1000*C
at 915 MHz . . . . . . .

The loss tangent versus temperature of
8 w/o PMMA/AKP-30 alumina samples
prepared with 2008, 2010 or 2041 PMMA
is given up to 10000C at 2450 MHz .

The loss tangent versus temperature of
8 w/o PMMA/AKP-30 alumina samples
prepared with 2008, 2010 or 2041 PMMA
is given up to 10000C at 915 MHz . .

Temperature versus time for 8 gm, 16 gm
and 24 gm 2008 PMMA heated in a 700 watt
2450 MHz microwave oven . . . .

Temperature versus time for 8 gm, 16 gm
and 24 gm 2010 PMMA heated in a 700
watt 2450 MHz microwave oven . . .

Temperature versus time for 8 gm, 16 gm
and 24 gm 2041 PMMA heated in a 700
watt 2450 MHz microwave oven . . .

The critical temperature and time
versus temperature of 24 gm 2008,
2010 and 2041 samples heated in a 700
watt 2450 MHz microwave oven . . .

Heating schedule for compact samples
heated in the MTGA and analyzed
by FTIR . . . . . . . .










Figure 5.33

Figure 5.34

Figure 5.35

Figure 5.36

Figure 5.37

Figure 5.38

Figure 5.39

Heating schedule for compact samples
heated in a conventional furnace
and analyzed by FTIR . . . . .

FTIR spectra for 4 w/o 2010 PMMA/
AKP-15 alumina compact samples
unheated and heated to 400C in
a conventional furnace and in
the MTGA . . . . . . . .

Temperature and change in weight percent
of the sample versus time for 12
gm 4 w/o 2010 PMMA/alumina samples
composed of either AKP-15 or AKP-30
alumina (MTGA data) . . . . .

Temperature and change in weight percent
of the sample versus time for 24
gm 4 w/o 2010 PMMA/alumina samples
composed of AKP-15 or AKP-30
alumina (MTGA data) . . . . .

An illustration of the compact sample
cut in half and the locations
(Xl, X2, X3 and X4) from which
the powdered samples were collected
for carbon residue analysis . . .

Temperature versus time for the 12 gm
4 w/o 2010 PMMA/alumina compacts
composed of AKP-15 or AKP-30
alumina, heated in a furnace using
two different heating schedules . .

Temperature versus time for the 24 gm
4 w/o 2010 PMMA/alumina compacts
composed of AKP-15 or AKP-30
alumina, heated in a furnace using
two different heating schedules . .









Figure 5.40

Figure 5.41

Figure 5.42

Figure 5.43

Figure 5.44

Figure 5.45

Figure 5.46

Figure 5.47

Temperature and change in weight percent
of the sample versus time for a
buoyancy run, 8 gm, 12 gm and 16
gm compacts of 4 w/o 2010 PMMA/
AKP-30 alumina (MTGA data) . . .

Temperature and change in weight percent
of the sample versus time of 8 gm
compact samples composed of 4 w/o
2010 PMMA/AKP-30 alumina heated
in a furnace and in the MTGA . . .

Temperature and change in weight percent
of the sample versus time of 16 gm
compact samples composed of 4 w/o
2010 PMMA/AKP-30 alumina heated
in a furnace and in the MTGA . . .

Temperature and change in weight percent
of the sample versus time of 12 gm
compact samples composed of 8 w/o
2010 PMMA/AKP-30 alumina heated
in a furnace and in the MTGA . . .

Temperature and change in weight percent
of the sample versus time of 24
gm compact samples composed of 8
w/o 2010 PMMA/AKP-30 alumina heated
in a furnace and in the MTGA . . .

Compact 12 gm PMMA/alumina samples heated
at 2400 watts to 400C for a total
of 90 minutes to remove the binder
using pure microwave heating . . .

Compact 24 gm PMMA/alumina samples heated
at 2400 watts to 400C for a total
of 90 minutes to remove the binder
using pure microwave heating . . .

Samples heated to the highest
temperatures possible using pure
microwave heating in a 3200 watt
microwave oven . . . . . .










Figure 5.48

Figure 5.49

Figure 5.50

Figure 5.51

Figure 5.52

Figure 5.53

Temperature and change in weight percent
of sample versus time for 12 gm
4 w/o 2010 PMMA/AKP-30 alumina
compact samples containing either
0.1 w/o carbon, 0.5 w/o carbon or
1.0 w/o carbon (MTGA data) . . .

Temperature and change in weight percent
of sample versus time for 24 gm 4
w/o 2010 PMMA/AKP-30 alumina compact
samples containing either 0.1 w/o
carbon, 0.5 w/o carbon or 1.0 w/o
carbon (MTGA data) . . . . .

X-ray maps at 20x for the 4 w/o PMMA/
AKP-30 alumina sample pressed at
10,000 psi (a) carbon map, (b)
oxygen map, (c) aluminum map, (d)
composite map, and (e) elemental
analysis . . . . . . . .

X-ray maps at 20x for the 8 w/o PMMA/
AKP-30 alumina sample pressed at
10,000 psi (a) carbon map, (b)
oxygen map, (c) aluminum map,
(d) fluorine map, (e) composite
map, and (f) elemental analysis . .

SEM micrographs of unheated samples
pressed at 4000 psi (a) 4 w/o
PMMA/AKP-30 alumina at 10,000x,
(b) 4 w/o PMMA/AKP-30 alumina
at 20,000x, (c) 4 w/o PMMA/AKP-30
alumina, heated to 400*C, at 10,000x,
and (d) 4 w/o PMMA/AKP-30 alumina,
heated to 4000C, at 20,000x . . .

SEM micrographs of unheated samples
pressed at 4000 psi (a) 4 w/o
PMMA/AKP-15 alumina at 10,000x
and (b) 4 w/o PMMA/AKP-15 alumina
at 20,000x . . . . . . .







Figure 5.54

Figure 5.55

SEM micrographs of unheated samples
pressed at 4000 psi (a) 8 w/o
PMMA/AKP-30 alumina at 10,000x and
(b) 8 w/o PMMA/AKP-30 alumina at
20,000x . . . . . . . .

Schematics of particles of alumina
alone, alumina containing 4 w/o
binder and alumina containing
8 w/o binder . . . . . . .



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




May 1993

Chairperson: Dr. David E. Clark
Major Department: Materials Science and Engineering

Microwave energy is becoming more important in the

processing and fabrication of ceramic materials because of the

potential advantages of uniform, volumetric, ultra-rapid and

selective heating of materials. Economic benefits due to

reduced processing times and enhanced properties produced via

microwave energy are also potential advantages. Organics are

added to ceramic powders to act as binders, lubricants,

plasticizers and dispersants during forming processes.

Forming processes include extrusion, slip casting, pressing

and injection molding. The removal of organic additives, such

as binders, can be time-consuming and can lead to damage of

the ceramic (i.e., bloating or cracking).

The removal of polymethyl methacrylate (PMMA) binder from

pressed compacts of alumina is investigated using both

microwave and conventional heating means at temperatures


:400C. Binder removal studies are carried out using several

variables: sample mass, binder tacticity, binder molecular

weight, binder concentration, carbon additives, heating type

and alumina particle diameter.

Information related to binder removal is obtained using

scanning electron microscopy (SEM), Fourier transform infrared

spectroscopy (FTIR), thermal gravimetric analysis (TGA), an

inert gas fusion instrument (for carbon residue analysis) and

differential scanning calorimetry (DSC). A microwave

thermogravimetric analyzer (MTGA) is developed to monitor the

weight change of a sample, heated by microwave energy, as a

function of time and temperature. Dielectric measurements

over a wide range of temperatures and frequencies were made

for alumina, PMMA/alumina mixtures, and for other materials.

Energy dispersive spectroscopy (EDS) and SEM are used to

observe the binder distribution within the compacts.

Results show that for compacts composed of PMMA and

alumina, heated by microwave hybrid heating, the binder is

removed in similar times regardless of sample size.

Identically prepared samples heated by conventional means show

a delay in binder removal as sample mass increases. Binder is

removed from microwave heated samples more efficiently as

sample mass and binder concentration increase.

Degradation mechanisms are investigated on a limited

basis; however, it is assumed that PMMA primarily degrades by

a depolymerization mechanism. The effects of variables used


in this study on binder removal are discussed and are related

to the materials' physical, chemical and dielectric




General Introduction

Organic binders are important additives in ceramic

processing. Ceramic processing operations include slip

casting, injection molding, extrusion and pressing. Organic

polymers are used as binders, lubricants, plasticizers and

dispersants. Inorganic additives are also used as binders,

lubricants and for other purposes.

Before a ceramic piece is sintered, the polymeric binder

must be removed completely to avoid damage to the ceramic.

Damage to the ceramic may include cracking, bloating, lower

densification yields and detrimental physical properties.

There are several reasons for investigating the

polymethyl methacrylate (PMMA) and alumina system. Alumina is

an important technical ceramic, is a low loss material

(insulator) and has a large penetration depth for microwaves.

The room temperature penetration depth of alumina is

calculated, using appropriate dielectric properties data, to

be on the order of hundreds to thousands of centimeters at

2450 MHz. PMMA is used as a binder in the production of

ceramic substrates in the electronics industry, is a clean


burning binder (leaves very small amounts of carbon residuals)

and has a smaller penetration depth (1/e attenuation distance)

than alumina. The room temperature penetration depth of PMMA

is calculated, using appropriate dielectric properties data,

to be on the order of tens to hundreds of centimeters at 2450

MHz. Due to the greater penetration depth of alumina, large

samples may be heated more uniformly and volumetrically with

microwaves than by conventional means. Also, because PMMA has

a smaller penetration depth than alumina, selective heating of

the PMMA to effect binder removal may be possible.

Therefore, the PMMA/alumina system is investigated to

determine if microwave energy can be used to effectively

remove the binder and, in the future, sinter the ceramic piece

in one processing step. Sintering of the alumina, however,

will not be discussed in this study.

A mixture of binders, carbon and other additives may be

used to assist in heating the PMMA/alumina system to its

critical temperature (Tc). The Tc as it relates to microwave

processing will be discussed in Chapter 2. This may allow

pure microwave heating to be used to remove the additives

completely and to sinter the part in one step, without any

detrimental effects to the part. Dielectric properties data

(as a function of frequency, temperature, composition, and

density) may allow one to better understand the processing of

this system and other composite systems.


One advantage of using microwave energy to heat

materials, when compared to conventional heating, is that heat

is generated uniformly and volumetrically throughout the

material. The material itself is being heated and not the

atmosphere surrounding the material. Thus, both large and

small samples may be heated rapidly, and uniformly with a

reduction in thermal gradients. This is important because

severe thermal gradients may lead to cracking in the piece

being heated. Microwave heating may result in lower energy

cost, shorter processing times, enhanced properties and

improved product uniformity.

Conventional heating of a material is accomplished via

the material being heated by an external source of heat. The

atmosphere surrounding the material is heated first, the

surface of the material is heated second by convection and

radiation, and heat is then conducted from the materials

surface into its interior by conduction. In some processing

methods, a vacuum is used and convection is not a method of

heat transfer. Ceramic materials, in general, are not very

good thermal conductors, as they have a low thermal

conductivity. It may take some time before the samples

surface and interior temperatures equilibrate. Therefore,

slow heating rates are generally used to avoid thermal

shocking the material. This may lead to nonuniform heating in

large ceramic pieces, which results in undesirable physical

and microstructural materials properties.

Outline of Study

In Chapter 1 a general introduction is given regarding

the importance of removing binders, organic additives and

inorganic additives from ceramic pieces. Several reasons are

given for selecting the PMMA and alumina system. Potential

advantages of processing materials by microwave heating versus

conventional heating are also discussed. Chapter 2 contains

background information on microwaves (theory, properties,

heating, applications, generation and other areas) and

conventional heating. Polymeric binders (usage, properties,

characteristics, classification, thermal degradation and

mixing with ceramic powders) and a literature review of binder

removal by both microwave and conventional heating are

discussed. Research objectives and accomplishments are

presented in Chapter 3. A discussion of the materials,

experimental procedure and analytical techniques used in this

study is provided in Chapter 4. Experimental results are

given in Chapter 5. Information on samples heated by

microwave hybrid heating (a combination of microwave and

conventional heating), pure microwave heating and conventional

heating are provided. Dielectric data and various analytical

techniques conducted on samples used in this study are

presented. Results of this study are discussed in Chapter 6

with respect to dielectric properties data, thermal analysis

and processing variables used. Chapter 7 contains a summary


of the major findings of this study and conclusions. Finally,

Chapter 8 contains numerous suggestions for future work.



Microwaves are being studied and utilized in materials

processing because of potential opportunities to (a) reduce

manufacturing cost (time and energy) [Das87, Swa88], (b)

improve product uniformity [De90, De91a, De91b] and yield

[Sut89] and (c) synthesize new and improved materials

[Sut89]. Both microwaves and infrared radiation fall under

the electromagnetic (EM) spectrum. The radiation involved in

microwave processing ranges from 0.3 GHz to 300 GHz, which

corresponds to a wavelength of 1 m to 1 mm. Since microwaves

are a part of the EM spectrum (e.g., ultraviolet, infrared,

visible light and radio waves), they may be reflected,

transmitted or absorbed, and obey the laws of optics by

traveling in straight lines at the speed of light. A general

rule is that microwaves are reflected by opaque metal or

metallic objects, are absorbed by some dielectrics, and are

transmitted without appreciable absorption by many insulators.

Figure 2.1 gives an illustration of the interaction of

microwaves with materials [Sut89]. In Figure 2.2, the entire

EM spectrum is shown [Sis80].


(Low Loss Insulator)
Total Penetration

No Penetration

(Lossy Insulator)
Partial to Total

Matrix is Low Loss

Additives are Absorbing Material
Partial to Total Penetration

Figure 2.1 Interaction of microwaves with materials, adapted
from Sutton [Sut89].


CD M iia ss



0.3GHz 300GHz


I m 1mm

v, cycles sec-1 3x1019 3x1015 3x1011 3x107 3x105
I I I I --
WAVELENGTH 10-12 10-8 10-4 1 104 108
I (cm) I I 1 I I I I I
(a) y rays
SUltraviolet Rays
Visible Light
SInfrared Rays
Short Radio Waves
L- Broadcast Band
----- --------- Microwaves h I Radio
Long Radio Wave

WAVELENGTH 4000 5000 6000 7000 8000
S(A) I I I I I
(b) I Blue I IYellow I Red
Violet Green Orange

v (cycles sec-1) 75x1014 5.0x1014 3.75x1014

Figure 2.2 The electromagnetic spectrum, adapted from Sisler

Microwave Generation

Microwave power may be generated by klystrons,

magnetrons, vacuum tubes and solid state devices [Met83]. The

most widely used method for generating continuous wave (CW)

microwave power is with magnetron tubes. Magnetron tubes

allow high microwave power output to be produced at low

capital cost, and efficiently with excellent frequency

stability. Therefore magnetron tubes are commonly used on a

commercial basis. Vacuum tubes may be applied to produce

high-power microwaves but are too expensive and complex for

industrial microwave processing. The klystron, an amplifier

of microwave signals, has been used to generate low power

microwaves. The klystron is a low-cost device that provides

excellent frequency stability, but the disadvantages of this

device are that it usually requires liquid cooling. Solid-

state devices have been used to generate low-power microwaves

(e.g., radar). The power levels produced by the solid-state

devices (even in large arrays), however, are too low for

industrial usage [Dec86, Met83, Rod86, Vel87].

Once microwaves are generated, they are usually

transmitted through hollow metallic tubes called waveguides.

Metallic tubes are usually used because they reflect


History of Microwave Heating

Before the advent of the home microwave oven, the most

frequent applications for microwaves were limited to the


communications area. Other areas of usage included heating,

medical and biological applications. But the most wide spread

application for microwaves, by far, has been the microwave

oven for heating and cooking food.

Very little work was reported on microwave heating before

World War II. A comprehensive history of microwaves, given by

Osepchuk [Ose84], includes work by Kassner, who attempted to

change the chemistry and molecular state of materials without

heating. During World War II, many efforts were made to

measure the dielectric properties of materials. This work led

to the development of radar, telephone and other forms of

communication. Work done by von Hippel [Von59] and colleagues

at the Massachusetts Institute of Technology's (MIT)

Laboratory for Insulation Research has led to many important

contributions towards understanding and utilizing radio

frequency (RF) and microwave heating.

Osepchuk [Ose84] discussed the origins of the usage of

microwave energy for commercial heating, which will now be

briefly summarized. After World War II, several companies

expressed an interest in the microwave tube industry for

heating applications. These companies included General

Electric (GE), Raytheon, Radio Corporation of America (RCA)

and Westinghouse. The Federal Communications Commission (FCC)

established a frequency allocation procedure. General

Electric and Raytheon petitioned the FCC for a microwave oven

frequency. Raytheon favored the 2450 MHz frequency due to the


advantages of more uniform heating (because of more modes per

cavity), and smaller loads could be heated. General Electric

favored the 915 MHz frequency because of the advantages of

deeper penetration and better control of thermal runaway. The

FCC allowed both of these frequencies for commercial use, and

commercial ovens at both frequencies were made available.

Soon the 915 MHz oven was phased out for domestic use because

of its large size and was only used for higher power

industrial applications. The 2450 MHz units are still widely

used for domestic purposes, because of their compact size.

Industrial Applications of Microwave Heating

The major applications of microwave heating still remain

in the area of food processing [Dec86]. Microwave processing,

however, is expanding rapidly to include many ceramic,

polymeric and metallic materials. An overview of many of the

important applications utilizing microwave heating was

presented by Sutton [Sut89] and Metaxas [Met90]. A few

applications of microwave heating currently being used are

discussed below.

In the food industry, microwave energy is used to dry

pasta, process potatoes, bake foods, defrost frozen foods,

process poultry, roast cocoa beans and sterilize food


In the lumber industry, microwaves are used to

continuously dry lumber [Bar76]. This process allows for

savings in both time and energy costs.


In the polymer industry, the main usage of microwave

energy involves curing rubber tires, bonding composites,

thermoplastics and thermosets in the automobile industry

[Rai88]. DuPont has been using microwave energy to dry nylon

fibers [Ose84] and Litton has used microwaves to heat urethane

foam [Ose84]. Microwave heating has also been applied in the

aircraft industry, to manufacture carbon fiber thermoplastic

parts [Lin91].

A few medical applications for microwave energy include

treatment, sterilization and diagnostic techniques. In one

technique, the body temperature is deliberately elevated to

kill cancerous tumors [Leh78]. Microwaves also may be used to

measure the blood flow, for imaging purposes, and to measure

changes in the content of lung water [Isk88]. The United

States Bureau of Mines is examining microwaves to reduce the

energy (by 50 to 70%) needed to grind minerals in the

benefication process. This bureau has shown that many

minerals absorb microwaves and are rapidly heated and that

thermal stresses are induced within them to cause the minerals

to crack. This improves grinding efficiency and reduces the

total energy required to extract minerals [Wal88].

In the waste treatment field, microwave energy is being

applied to treat and dispose of waste materials (regular

garbage and radioactive waste). Radioactive waste treatment

and disposal by microwaves are being investigated by Schulz et


al. [Sch91], who has researched the usage of microwave heating

to process simulated nuclear waste glass.

Microwave heating has also been used to produce nuclear

fuels and other nuclear alloys in reduced processing times


Ceramic Processing

Microwaves are used in the processing of ceramic

materials because advantages in processing, microstructure and

properties may be achieved. In order for these processing

advantages to be realized, special knowledge is needed to

understand and efficiently use microwave energy. The proper

microwave heating equipment should be applied to optimize

microwave processing of ceramic materials. The effect of the

ceramic systems composition, moisture content, surface

chemistry, rheology, forming procedure, microstructure,

electric properties, and magnetic properties needs to be

better understood. In addition, dielectric data at elevated

temperatures and at several microwave frequencies need to be

obtained to correlate with ceramic processing data.

During the processing of ceramics, in general, they must

be heated to high temperatures (>700C). Therefore the

ceramic must efficiently couple with microwave energy and


In general, uniform and volumetric heating is believed to

occur in large ceramic pieces over a short time period.

Uniform and volumetric heating, however, may not always occur


due to the existence of hot spots within a microwave oven.

Mode stirrers, turntables, conveyor belts, or a combination of

these are usually used to eliminate this problem. Thermal

stresses are usually reduced when processing large ceramic

pieces with microwaves because of an inverse temperature

gradient and uniform and volumetric heating.

Many dielectric materials are transparent to microwaves

at ambient temperatures, but upon heating above a critical

temperature (Tc), these materials will absorb and efficiently

couple with microwave energy. In order to selectively heat a

microwave transparent material (e.g., alumina) at ambient

temperatures, conductive or magnetic materials may be added to

the material. The conductive or magnetic materials will

couple with and absorb microwaves at temperatures below the Tc

of the matrix or microwave transparent material. Selective

heating of these conductive or magnetic materials will allow

for the transfer of energy (in the form of heat) to the

microwave transparent material, until its Tc is reached.

Typical examples of the Tc of ceramic materials are shown

in Figure 2.3, where the microwave absorption (in arbitrary

units) versus temperature is plotted. The Tc provides an

approximate temperature at which the absorption of microwave

energy by the material increases rapidly.

Another method of heating low loss ceramic materials is

by preheating it to a temperature above its Tc, where it will

couple efficiently with microwave energy. The application of

I I I I -

' Tc (Silicon Carbide)


0 200 400 600 800 1000 1200 1400 1600
Temperature (C)

Figure 2.3 Typical examples of critical temperatures in
ceramic materials.




an insulation cavity used in conjunction with a microwave

suscepting material (e.g., SiC), may be utilized to preheat a

sample above its TC.

In some instances, the power of the microwave energy

applied to the load may be increased to more efficiently heat

the low loss material [McG88, She88]. This option, however,

is not available in all cases or applications.

Ceramic, Polymer and Ceramic/Polymer
Processing Applications

A summary of applications for microwave processing of

ceramics may be found in an overview of microwave processing

by Sutton [Sut89] and by Metaxas [Met90]. A few of these

applications will be discussed in this section.

Low temperature applications include process control,

liquid state processing, ceramic drying, binder removal and

surface modification.

One area to apply microwave energy is in process control.

Microwave energy may be used to control processing by

measuring the moisture content, flaws and gauging thickness

(without physical contact) [Cam73].

Another area of ceramic processing utilizing microwave

energy is liquid state processing. Microwaves have been used

in several processing steps for slip casting [Cha85, Oda87]

and to reduce the time for wet-chemical analysis of minerals,

ceramics and alloys, by greatly enhancing the dissolution

rates in acids. Microwaves can be used to process liquids in


the form of solutions or suspensions to analyze or synthesize


Microwaves have also been used to dry ceramics [Bea88,

Cha85] to efficiently remove the low water content from thick

ceramic bodies. A combination of microwave and conventional

heating seems to be the best system for achieving drying of

materials with a water content in excess of 10%.

Microwave energy may be applied to remove binders.

Binders have been removed from polycrystalline lead zirconate

titanate (PZT) and lead-based lanthanum doped zirconate

titanate (PLZT) during the early stages of sintering of these

materials [Har88]. Clark and Folz [Cla91] have reported

research on using microwave energy to remove polymeric binder

from ceramic compacts.

Microwaves have also been used to cure polymers and

composites and to dry polymers and other materials. Microwave

energy has been used for drying plastics, polymer blending,

drying wood flour, and the biomass conversion process [Geo88].

Thermoplastic modified epoxy resins have been cured with

microwaves by Lewis et al. [Lew88]. Giammara et al. [Gia91]

have applied microwave energy to polymerize epoxy and other

commercial resins, which are used for biological and

biomedical applications. Microwave processing of

thermosetting systems have been shown to lead to curing with

a 10 to 20% reduction in processing time [Hed91]. Microwaves

have been used to process graphite/epoxy composites [Jam88],

polyester and polyester/glass composites [Hot91], composites

[Lee84] and epoxy-matrix/glass-fibre composites [Boe92].

Microwaves have been used to modify the surfaces of

materials. Fathi et al. [Fat91, Fat92] have employed

microwave heating to modify the surface of sodium

aluminosilicate glasses. The extent and the rate of ion

exchange was enhanced when compared to a conventional heating


Higher temperature applications of microwave energy for

ceramic processing include joining of ceramics, fabrication of

optical fibers, processing of superconductors, combustion

synthesis, and sintering of ceramics and composites.

Ceramic and composite materials have been sintered at

2450 MHz and at higher frequencies. Ceramic materials

sintered include ferrites [Kra81], lead-based perovskite

ceramics [Ali87], boron carbide [Hol91, Kat88a], titanium

diboride [Hol91, Kat89], alumina [De90, De91a, De91b, Jan88,

Pat91, Swe91, Tia88a], A120/TiC [Tia88b], A1203/SiC [Kat88b,

Kat91a, Mee87b], composites [Bla86] and partially stabilized

ZrO2 [Hol88, Jan91, Wil88]. Tian [Tia91] has discussed the

problems of using a single mode microwave applicator to sinter

various ceramic materials and solutions to these problems have

been discussed.

Ceramics have been joined by microwave energy. An

interlayer of A100H gel was applied to join alumina [A1A91,

AlA92a, AlA92b]. Silberglitt et al. [Sil91] used an


interlayer material to join SiC and the Quest Research

Corporation have joined ceramic materials via microwave

heating [Pal88, Pal89]. Ceramics have also been joined with

microwave heating at the Toyota Research and Development Labs

[Fuk88, Fuk90]. Ahmad et al. [Ahm92] have applied microwave

heating to join reaction bonded SiC without the aid of an

interlayer material. Ceramic-ceramic seals have been formed

using microwave energy [Mee86]. Joints produced by microwave

heating exhibited greater strength than as received materials,

did not fail near the joint and were not detected on a

microscopic scale.

A Modified Chemical Vapor Deposition (MCVD) station was

used to fabricate optical fibers for the telecommunications

industry [Baj87]. These optical fibers have been manufactured

with microwave plasmas at atmospheric pressure.

Microwave energy has been used to dry, remove binder, and

calcine and sinter PZT and PLZT to densities as high as 99.6%

of theoretical values [Har88]. In some cases, better

properties were obtained with microwave heating than by

conventional heating.

Superconductors have been processed using microwave

energy [Ahm88, Coz91a, Coz91b, Gie91]. The sintering and

annealing of the superconductors were superior to those

processed with conventional heating, but more work needs to be

done in this area.


Combustion synthesis of ceramics and composites was

achieved using microwave energy [Cla91]. This process was

called Microwave Ignition and Combustion (MICOM).

Research on mass transport and solid state reactions

between zinc oxide and alumina oxide, using microwave heating,

have been reported at the University of Florida [Ahm91a,

Ahm91b]. Katz et al. [Kat91b] investigated sintering of

ceramics with microwave heating to determine if enhanced

diffusion occurred. The nitridation of pure Si in a nitrogen

atmosphere to form Si3N4 has been reported [Kig91].

Single Mode and Multimode Microwaves

The most commonly used applicator for microwave heating

is with the multimode cavity, an example of which is the home

microwave oven. Multimode microwave cavities are most often

utilized in industrial and commercial applications. Single

mode microwave applicators or cavities are mostly used in


Before discussing the advantages and the disadvantages of

single mode and multimode microwave applicators, the concept

of the mode will be defined. A mode is essentially a

particular pattern of electromagnetic energy which has been

distributed within a confining structure and is caused by the

interaction of two or more travelling waves. In an empty

single mode cavity, one standing wave or mode is established

and maintained. In a multimode cavity, the cavity is

typically large (much larger than a home microwave oven) in


relation to the free wavelength, which allows a number of

different standing waves or modes to be established.

A single mode microwave allows for more precise and

predictable control over fields (electric and magnetic) being

produced within a cavity and the material being heated. This

type of applicator is ideal for localized heating, as a

material may be placed into the location of maximum field

(electric or magnetic) strength and this will allow the

material to be heated rapidly and uniformly.

A single mode cavity may have the shape of a rectangle,

a cylinder, or any other shape, such that the electromagnetic

field inside the cavity satisfies Maxwell's equations when

subjected to certain boundary conditions. The boundary

conditions to be satisfied are that the electric field

tangential to and the magnetic field normal to the metal walls

must vanish [Lia88]. These boundary conditions, along with

Maxwell's equations, have been solved by Liao [Lia88] for the

circular, rectangular and semicircular single-mode cavities.

Only the rectangular cavity will be discussed, as that is the

type of cavity used in our research. The rectangular cavity

used in our research is a multimode microwave type.

A rectangular cavity with dimensions of width, a, height,

b, and length, d, along the x, y and z directions,

respectively, is considered. The wave equations for this

rectangular resonator should satisfy the boundary conditions

of zero tangential electric field strength at four of the


walls. By choosing the appropriate harmonic functions in z to

satisfy this boundary condition at the remaining two walls,

the Hz and Ez may be solved [Lia88]. The separation equation

for both the transverse electric TE and the transverse

magnetic TM modes is given by

k2 = (mr/a)2 + (nr/b)2 + (p7r/d)2 (2.1)

where k is the wavenumber, 7 (= 3.14), m (= 0,1,2,...) denotes

the number of half-waves of electric or magnetic intensity in

the x-direction, n (= 0,1,2,...) denotes the number of half-

waves of electric or magnetic intensity in the y-direction, p

(= 1,2,3,...) denotes the number of half-waves of electric or

magnetic intensity in the z-direction. For a lossless


k2 = -2ep (2.2)

where e is the relative permittivity, is the relative

permeability and o (= 2rf) is the angular frequency. Thus the

resonant frequency fr for the TEp and TM.P may be given as

fr = (1/2(E/ )o5')[(m/a)2 + (n/b)2 + (p/d)2]-0.5 (2.3)

For a>b
amplitude of the standing wave occurs when the frequency of

the transmitted signal is equal to the resonant frequency.

Waveguides follow the same principle as for the cavities, with

the exception that the length of the waveguide is considered

to be infinite. Therefore the transverse electric and the

transverse magnetic modes of wave guides are designated as TEn

- --

-" _--

/ilk\ i 1

Top view


Side view

(a) TE1o mode

Side view

- Electric field lines

- -- Magnetic field lines


End view

End view

(b) TMn, mode

Figure 2.4 An illustration of the field patterns (standing
wave or mode) inside a single mode microwave waveguide,
adapted from Liao [Lia88].


and TMm, respectively. In Figure 2.4 an illustration of the

field patterns inside of a single mode waveguide is shown.

A multimode cavity is essentially a closed box that can

handle many sized loads. The electrical field distribution

inside of the multimode cavity, however, is not uniform, which

may lead to hot spots and nonuniform heating of the load.

Therefore, mode stirrers, turntables, conveyor belts or a

combination of these methods are used to eliminate or control

the problems of hot spots and uneven heating. A mode stirrer

is a device which alters the modes (i.e., continuously disturb

the field distribution) in a cavity to give a more uniform

heating effect. Turntables and conveyor belts alter the

position of the load in the multimode cavity to ensure that

more uniform heating is occurring. All three methods expose

the load to a more uniform electric field.

The multimode cavity, in principle, should be several

wavelengths long in at least two directions. Such a cavity

will support a large number of resonant modes (Elm and Him)

over a narrow frequency range [Met83]. In an empty cavity,

each of these modes is distinct and characterized by a sharp

resonance response at a given frequency. In the case of a

rectangular cavity of dimensions a, b and d, each mode must

satisfy the equation

,Lm/c = (17/a)2 + (mr/b)2 + (nr/d)2 (2.4)
where 1, m and n are integers corresponding to the number of

half-wavelengths of the field along the principal coordinate


axis, w.m is the angular frequency of the 1, m and n mode and

c is the speed of light. But o = 2rf = 27c/l. Note that, in

both the single mode cavity and the multimode cavity, the

presence of a sample (load) will alter the field (electric or

magnetic) strength or energy distribution within the cavity.

In a multimode cavity, in general, the uniformity of the

field is improved as the size of the cavity is increased. The

distribution of the modes in a multimode cavity is complex.

However, Metaxas and Meredith [Met83] have discussed methods

of calculating and measuring the field strength inside of a

multimode cavity (these methods make many assumptions and are

not trivial).

Theoretical Aspects of Microwave Interactions
and Heating

Within this section, the following topics will be

discussed: polarization mechanisms, microwave power

absorption, rate of increase in temperature, critical

temperature, thermal runaway, penetration depth, tailoring

systems to reach critical temperature and dielectric mixture

rules. This will be followed by a discussion of microwave

hybrid heating, pure microwave heating and conventional

heating. The discussion of polarization mechanisms,

polarization theory, complex forms of the dielectric constant,

rate of increase in temperature, and the penetration depth of

microwaves has been adapted from Kingery [Kin76], Debye


[Deb29], Metaxas [Met83, Met90], Hench [Hen90], and von Hippel


Polarization Mechanisms

Insulation materials may be heated with microwave energy

via polarization mechanisms. The origin of the heating is due

to the ability of the electric field to polarize charges in

the material and the inability of the polarization to follow

rapid reversals of the electric field. This lag in the

polarization being able to follow the applied electric field

(at lower frequencies) results in the dissipation of power

into the material. At extremely high frequencies, i.e., far

above the resonant frequency of the material, the inability of

the load to follow the field may not lead to losses and the

dissipation of power into the material. Dielectric materials

can be heated by polarization effects and through direct

conduction. An example is a redistribution of charged

particles under the influence of an externally applied

electric field, which forms conducting paths in the

heterogeneous material.

Debye [Deb29] first discussed the phenomenon of

dielectric dispersions that occur in materials containing

polar molecules. A dielectric may have permanent dipoles or

induced dipoles which arise due to the application of an

applied field. A dielectric material in the presence of an

electric field becomes polarized because of the relative

displacement of positive and negative charges in the material.


The magnitude of the dipole moment on a molecule is directly

dependent on its size and symmetry. Molecules having no

center of symmetry are termed polar, and molecules having a

center of symmetry are termed nonpolar (i.e., have no dipole

moment). The permittivity dielectricc constant) of a material

is influenced by the dipole moment of the material [Hil69].

The relative permittivity of a material is the ratio of the

permittivity of the material to the permittivity of free space

[Mea61]. The greater the polarizability of the molecules, the

higher the permittivity of the material.

The four primary mechanisms of polarization in ceramics

and polymers are electronic polarization (Pe), atomic (ionic)

polarization (Pa), dipole polarization (Pd) and interfacial

polarization (P) Dipole polarization is the primary

polarization mechanism of polymers. These mechanisms involve

a short-range motion of charge carriers influenced by an

electric field and contribute to the materials' total

polarization. Nonpolar molecules placed in an applied field

give rise to induced dipoles as a result of the displacement

of charged particles from their equilibrium positions.

Induced polarization mechanisms involve electronic and atomic

polarization. Polar molecules placed in an applied field give

rise to dipole polarization as a result of rotations of polar

molecules in the applied field. Figure 2.5 shows the four

primary mechanisms of polarization [Hen90]. In Figure 2.6,

the resonance absorption peaks due to various forms of

<-- E
+ Shift in
-- 8 j--- Electron

atom M atom N

----- E


Figure 2.5 Schematics of mechanisms of polarization in
ceramics: (a) electronic, (b) atomic or ionic, (c)
high-frequency oscillatory dipoles, (d) low frequency
cation dipole, (e) interfacial space charge polarization
at the electrodes and (f) interfacial polarization at
heterogeneities. Adapted from Hench [Hen90].

(d) +




Atomic (or Ionic)

Si Pd2Fa TL (a)
103 104 107 109 1011 1013 1015
Log Frequency

10-3 -1
105 1- 1
10-3 10-1 10 103 104 107

I WM (b)
109 1011 1013 1015

Log Frequency

Figure 2.6 Frequency dependence of polarization mechanisms in
dielectrics: (a) contributions to the charging constant
with representative values of K' and (b) contributions to
the loss angle with representative values of tanS.
Adapted from Hench [Hen90].

Interfacial Polarization
SDipole Polarization
, (low freq.) (high freq.)






polarization are given over a wide range of frequencies

[Hen90]. In Figure 2.7, the effect of temperature on tanS

(loss tangent) versus frequency over a range of frequencies is

shown for Georgia kaolin. The tan6 will be defined and

discussed on pages 45 through 49 of this chapter. In Figure

2.7 the maximum tan5 shifts to the right or to higher

frequencies with increasing temperature over several decades

of frequency. If a material has tan6 properties which are

very dependent on frequency, then the tanS will peak over a

very narrow frequency range with increasing temperature. This

is the case reported for atomic or ionic polarization, where

a resonant frequency is obtained for a particular loss

mechanism [Kin76]. If the maximum tanS shifts to higher

frequencies with increasing temperature, then this is an

indication of dipolar and/or interfacial loss mechanisms

[Kin76]. This shift to higher frequencies may result from a

combination of loss mechanisms or due to a change in loss

mechanisms. One may also correlate the Tc of a material (at

a distinct frequency) with the maximum tan6 value at a

particular temperature. Concepts of Tc and thermal runaway,

as related to tanS, will be discussed later in this chapter.

Electronic polarization

Electronic polarization is due to a shift of the valence

electron cloud of ions in a material with respect to the

positive nucleus (see Figure 2.5a). These polarization

mechanisms occur at very high frequencies (1015 Hz) located in

10 103 10 1
102 10o 104 105


Frequency (cycles/sec)

Figure 2.7 The effect of temperature on the maximum tanS over
a wide range of frequencies, adapted from Kingery


the ultraviolet (UV) optical range. Moreover, this mechanism

of polarization gives rise to a resonance absorption peak

located in the optical range [Hen90] (see Figure 2.6). A

material's index of refraction is dependent on the electronic


Atomic (ionic) polarization

Atomic polarization occurs due to the displacement of

positive and negative ions in a material with respect to each

other (see Figure 2.5b) and occurs in the infrared range (1012

to 1013 Hz), as shown in Figure 2.6. A resonance absorption

occurs at frequencies characteristic of the bond strength

between ions. The existence of several types of ions in an

insulator or a broad distribution in bond strengths (e.g.,

glasses) will result in a broad infrared absorption.

Dipole (orientational) polarization

Dipole polarization occurs in some dielectric materials

that contain permanent dipoles, due to an asymmetric charge

distribution of unlike charge in the molecule (e.g.,

polymers). In those cases, ionic or molecular dipoles may

reorient under the influence of a charging electric field to

give rise to orientational (dipolar) polarization (see Figures

2.5c and 2.5d). Dipolar polarization occurs in the

subinfrared range of frequencies.

Dipolar polarization may be broken down into two

mechanisms. First, molecules (liquids, gases and polar

solids) containing a permanent dipole moment may be rotated


against an elastic restoring force about an equilibrium

position. The frequency of this relaxation is high, (1011 Hz)

at room temperature, as shown in Figure 2.6. Under an applied

ac field, bonds may oscillate about an equilibrium position

(see Figure 2.5c). A second mechanism is important at room

temperature for ceramics and glasses. This mechanism involves

the rotation of dipoles between two equivalent positions, as

shown in Figure 2.5d. This polarization results from a motion

of charged ions between the interstitial positions within the

ionic structure of the material. This polarization occurs

over a frequency range of 103 to 106 Hz at room temperature.

Due to the involvement of mobile cations which contribute to

dc conductivity, this mechanism is also referred to as

migration losses.

Interfacial polarization

Interfacial polarization is also called space charge or

Maxwell-Wagner polarization. This type of polarization occurs

in heterogeneous dielectrics containing a phase or component

with a higher electrical conductivity than the other.

Interfacial polarization occurs as a result of mobile charge

carriers piling up at the interfaces between components in the

heterogeneous system. This physical barrier inhibits charge

migration and charges pile up at the barrier to produce

localized polarization within the material. It is usually

associated with the presence of impurities which form separate

and conducting phases. (Refer to Figure 2.5e and 2.5f for the


two cases of interfacial polarization.) Usually the frequency

range of the interfacial polarization is quite low (10-3 Hz),

but if the density of charges contributing to this

polarization is quite large, frequencies may be extended into

the kilocycle range (103 Hz). (Refer to Figure 2.6 for the

frequency range associated with orientation polarization.)

Polarization Theory

The polarization theory in this section may be found in

works by Debye [Deb29], von Hippel [Von54], Kingery [Kin76]

and Metaxas [Met83]. A capacitor will allow an electrical

charge Q (1 Coulomb = Coul = Amp-s) to be stored. A capacitor

is simply an electrical condenser that allows an electrical

charge to be stored. The charge on a capacitor is given by

Q = CV (2.5)

where V is the applied voltage (volts) and C (1 Farad = F =

Coul/V) is the capacitance. The capacitance contains a

geometrical factor and a materials factor. For a large plate

capacitor with an area A (m2) and thickness 1 (m) the

geometrical capacitance in a vacuum is

Co = (A/l)eo (2.6)

where Eo (= 8.854 x 10-12 F/m) is the permittivity dielectricc

constant) of free space. If one inserts a ceramic or

dielectric material of permittivity E' between two capacitor

plates, one obtains

C = Co(e'/eo) (2.7)

C = CoK' (2.8)


where K' is the relative permittivity or the dielectric

constant of the material.

Similarly, the induction coil may be considered as an

energy storing device. The inductance L (1 Henry = H = V-

s/Amp) is the creation of an electromotive force by means of

varying current in an electric circuit. The capacity of a

coil to store current results from a moving charge in the coil

setting up a magnetic field parallel to the coil axis.

Current charges in an induction coil creates opposing voltages

which must be sustained by counter-voltages

V = L(dI/dt) (2.9)

where t (s) is the time and I (Amp) is the current. The

inductance contains a geometrical factor and a material factor

and can be written as

L = Lo,('/Ao) (2.10)

L = LK ', (2.11)

where Lo is the geometrical inductance of a long coil of N

turns, A' is the permeability of the medium embedding the

coil, jg (= 1.257 x 10-6 H/m) is the permeability of a vacuum,

K', is the relative permeability, A (m2) is the area of the

coils, and 1 (m) is the length of the coil. The value of the

geometrical inductance is

L, = N2(A/l) o (2.12)
Combining Lo with the previous equation 2.10 yields

L = N2(A/l) /' (2.13)


Dielectric materials react to an electric field in a

different fashion than with free space. This is due to the

presence of charge carriers within a dielectric material,

which can be displaced, and charge displacements, which can

neutralize a portion of the applied field. Equations 2.5 and

2.8 may be combined and rewritten as

V = Q/C = Q/CoK' (2.14)

When a dielectric material reacts to an applied electric field

(Eap = E), only a fraction of the total charge sets up an

electric field and voltage towards the outside. This is known

as the free charge Q/K'. The remaining fraction of the total

charge is neutralized by a polarization of the dielectric

material via countercharges. This is known as the bound

charges and is given by Q(1 1/K'). The total electric flux

density D (Coul/m2) that can transverse the dielectric

material can be represented as a sum of the densities of the

electric field lines e E and the dipole chains P as

D = Eapp + P (2.15)

D = e'Ea8 (2.16)

The polarization P can represent either the surface charge

density of the bound charge or the dipole moment per unit

volume of the material

P = NA, (2.17)

where N is the number of dipoles per unit volume and Aa is the

average dipole moment induced in each molecule by a

polarization P. The elementary electric dipole moment can be


represented by two electric charges of opposite polarity and

separated by a distance d on a microscopic scale.

I = Qd (2.18)

The polarization P is related to the permittivity E' by

rewriting equation 2.15 in terms of P and substituting it into

equation 2.16 for D to obtain

P = D oEapp (2.19)

P = E'Eap oEapp (2.20)

Dividing both sides by 1/E, and using K' = E'/E0, one obtains

(1/eo) [P = E'Epp EoEapp] (2.21)

P/o = Eapp(E'/o E0/0) (2.22)

P = Eo(K' l)Eapp (2.23)

where Eo is the permittivity of free space, Eapp is the applied

electric field and K' is the relative permittivity. Another

measure of the ratio of polarization to the applied field is

a dimensionless term called the electric susceptibility X

(used to describe the capacitive behavior on a macroscopic

basis). The electric susceptibility X is the ratio of the

bound charge density to the free charge density.

X = Q(l 1/K')/(Q/K') (2.24)

X = K'(1 1/K') (2.25)

X = K' 1 (2.26)

X = P/oEapp (2.27)
The average dipole moment 1a of an elementary particle is

proportional to the local electric field E' acting on the


particle (used to describe the polarization on a microscopic


ua = aE' (2.28)
where the polarizability a (a proportionality factor) is a

measure of the average dipole moment per unit of local

electric field strength E'. Polarization may be represented


P = (K' 1) Epp (2.29)

P = NaE' (2.30)

This links the macroscopic measured permittivity K' to three

molecular parameters (N, a and E'). But E' is the locally

acting electric field or the internal field E,.i, a (m3) is the

polarizability and N is the number of contributing elementary

particles per unit volume.

Note that relations for the magnetic moment (which will

not be derived here) may be treated in a similar fashion to

yield the following pertinent results,

B = AoH + AoM (2.31)

B = AI'H (2.32)

M = nPm (2.33)

M = naH (2.34)

X, = K' 1 (2.35)

X, = M/H (2.36)

Xm = Nam (2.37)
where B (V-s/m2 = Wb/m2) is the total magnetic flux density or

magnetic induction, o is the permeability of free space, A'


is the effective permeability of the material, H (Coul/m-s =

Amp-turn/m) is the magnetic field strength, M (Coul/m-s = Amp-

turn/m) is the magnetization of the material, n is the number

of elementary magnetic dipoles per unit volume, Pm is the

magnetic moment of the elementary magnetic dipoles, am is the

magnetizability of the elementary constituents, Xm

dimensionlesss) is the magnetic susceptibility and x"' (A'/Io)

is the relative magnetic permeability dimensionlesss). Since

the magnetic properties do not play a role in the

electromagnetic waves interacting with the PMMA/alumina

system, these magnetic property relations equations 2.31

through 2.37 will not be further deduced or discussed.

The total polarizability of a molecule is given by aT and


aT = ae + aa + ao + ai (2.38)

where ae, aa, a, and a. are the electronic, atomic, orientation

and interfacial contributions to the polarizability,


Using equations 2.29 and 2.30 yields

Eo(K' 1)Eap = NaTEint (2.39)

where (K' = E'/Eo) is the relative dielectric constant or

relative permittivity of the material. This equation links

the macroscopic quantity K' to the molecular parameters (N, a

and E') in the dielectric. Since there are three molecular

parameters, one would like to eliminate two of these to obtain

a relation that will provide the most useful information.

Therefore the relation will be written in terms of a, which

contains the primary information on electric charge carriers

and their polarizing actions. The Mosotti equation for the

two fields

Eint = Eap + P/3Eo (2.40)

Eint = (K' + 2)Epp/3 (2.41)
will allow N and Eap to be eliminated.

To digress, the derivation of Mosotti's equation for two

fields along with the assumptions used will follow. A result

of Mosotti's equation, equation 2.41 is that the internal or

local field (E' = Eint) is always greater than the applied

field (E = Eapp) when K' >1. When K' = 1 (the case for air),

then Eint = Eap. This is also the case for an externally

applied field EaP to gases at low pressures where the

interactions between molecules can be neglected. For solids

or condensed phases and liquids at higher pressures, however,

the field acting on a reference molecule A may be noticeably

altered by the polarization of the surroundings. Mosotti's

equation was derived using several assumptions and these will

be revealed in the following discussion. To take these

concerns into consideration, the following model shown in

Figure 2.8 was developed by Mosotti [Deb29, Kin76, Met83,

Von54, Von59].

In the reference model, a molecule A is assumed to be

surrounded by an imaginary sphere of such an extent that

beyond it the dielectric may be treated as a continuum. It

Figure 2.8 Model for the calculation of an internal field


was assumed that the molecules inside the sphere were removed

and that the polarization on the outside of the sphere remains

frozen (static). Thus the two fields acting on molecule A

arises from three sources: (1) from the free charge at the

electrodes of the capacitor plate, El; (2) from the free ends

of the dipole chains that line the cavity walls, E2; and (3)

from molecules inside the sphere close enough to molecule A

such that their shapes and individual positions should be

considered, E3. Thus the locally acting field E' (= Eit) may

be written as

Eint = E1 + E2 + E3 (2.42)

The contributions from the free charges at the electrodes are

equal to the field intensity.

E, = Eapp (2.43)

The contributions of E2 may be calculated from the bound

charges lining the cavity and have been calculated by Debye

[Deb29] to be

E2 = P/3E0 (2.44)

E2 = Eapp (i 1)/3 (2.45)

In order to calculate E3, one needs to understand the

individual actions of molecules and the shapes of these

molecules within the sphere. Since this is difficult, Mosotti

assumed that this information was already recognized in the

calculation of E2 by assuming that the cavity was scooped out

without disturbing the state of the polarization of the

remaining dielectric. Thus, Mosotti assumed that any


additional individual field effects of the surrounding

molecules on molecule A would mutually cancel

E3 = 0 (2.46)

This is a good assumption when elementary particles are

neutral and without permanent dipole moments or when they are

arranged in highly symmetrical arrays or arranged in complete

disorder. Thus, this relation allows one to substitute for

the unknown molecular parameter E' (= Ein) using

Eint = El + E2 (2.47)

Eint = El + P/3Eo (2.48)

Substituting equation 2.29 into equation 2.48 yields

Eint = Eapp + (K' 1) oEapp/3 (2.49)

Eint = Eap + (K' 1)Ea/3 (2.50)

Eint = Epp + 'EapP3 Eap3 (2.51)

Ent = Eap(l + K'/3 1/3) (2.52)

Eint = Eapp(2/3 + K'/3) (2.53)

Eint =Eapp(2 + K')/3 (2.54)

The following assumptions are used to obtain the Mosotti

equation for two fields:

(1) Based on gases (and nonpolar liquids) where the

interactions between molecules can be neglected,

(2) the locally acting field E' (= Eint) is the same as

the externally applied field E (= Ea.P) for air,

(3) assumes the local field can be calculated where a

particular molecule is surrounded by an imaginary sphere,


sufficiently large for the dielectric beyond it to be treated

as a continuum,

(4) assumes the sphere is cut out of the solid and the

polarization remains unchanged, and

(5) assumes contributions to the electric field (arising

from individual molecular interactions is zero if arranged in

either complete disorder or in a symmetric array). For

solids, however, the polarization of the surrounding medium

can have a marked effect on the local field acting on a

particular molecule.

Upon inserting the local Mosotti field, equation 2.54

into equation 2.39, one obtains the relationship between

polarizability per unit volume (Na) and K'.

Na/3Eo = (K' 1)/(K' + 2) (2.55)

Nevertheless, for gases at low pressures, (K' 1) << 1, which

means that (K' + 2) ; 3. This is the same as replacing the

local E' (= Ein) with the applied field Ea8. Then equation

2.55, using (K' + 2) = 3, simplifies to

Na/3Eo = (K' 1)/3 (2.56)

Na/Eo = (K' 1) (2.57)

Na/E, = X (2.58)

where X is the susceptibility of a gas.

The number of molecules per unit volume N (molecules/m3)

is related to the number of molecules per mole by Avogadro's

number No (= 6.023 x 1023 molecules/mole) by

N = Nop/M (2.59)


where M (gm/mole) is the molecular weight and p (gm/m3) is the

density. Now equation 2.59 may be substituted into equation

2.55 and solved for aT (the total polarizability) to obtain

aT = 3MEo(K' 1)/Nop(K' + 2) (2.60)

"T = 3eoPfNo (2.61)

Now E' (= E t) and N have been eliminated, and the

polarizability a is the only unknown molecular parameter.

In the case where molecules themselves are the dipole

carriers, the dependence of the polarization on the density of

the material may be eliminated by referring to it as the

polarizability per mole P,. Substituting equation 2.59 into

equation 2.55 yields

aNp/3ME, = (K' 1)/(K' + 2) (2.62)

aN/3e, = M(K' l)/p(K' + 2) (2.63)

P, = aN/3 (2.64)

Pm = (M/p)[(K' 1)/(K' + 2)] (2.65)
The units of P, are m3/mole and is called the molecular

polarizability or the polarizability per mole and is directly

proportional to the atomic polarizability.

Complex Form of Dielectric Constant

Kingery [Kin76] has discussed the origin of the complex

form of the dielectric constant. When a dielectric is placed

into an alternating field, the time required for polarization

appears as a phase retardation of the charging current. A

time lag occurs between creating field and inducing a moment

which causes a phase shift between the voltage and the


current, as shown in Figure 2.9. This results in reducing the

phase angle of the polarization and magnetization currents

below 900. Instead of being 90* advanced, the charging

current is only advanced by an angle of 900 6. This loss

angle 6 is called the loss tangent. Loss currents appear that

are not due to charge-carrier migration. The storage of

electric and magnetic energy is usually accompanied by the

dissipation of energy. For a simple plate capacitor, a

sinusoidal current may be applied which results in a charge

current Jc given by

Jc = io'Eapp (2.66)

and a loss current J, given by

Jl = E"Eap (2.67)

J = Eapp (2.68)

where a is the dielectric conductivity (including both ohmic

conductivity due to migrating charge carriers as well as

losses associated with the formation or orientation of

electric dipoles). Similarly, the charging current may also

indicate dipole polarization or result from field distortion

of space charge formation. The loss factor (tanS) is given by

the ratio of the loss current to the charging current.

tanS = Ji/Jc (2.69)

tanS = w"Eapp/oe'Eapp (2.70)

tanS = e'/E' (2.71)

tanS = Eapp/o'Eapp (2.72)

tanS = a/oe' (2.73)

------ -

J/= cO"E
J/ Jc tan 5

Figure 2.9 Charging density (Jc) and loss current density (J )
of the polarization current [Kin76].


Kingery [Kin76] represents the electric field E and the

displacement flux density D in a complex notation as

E = Eoei't (2.74)

D = Doei(t 6) (2.75)

and making use of the relation D = K*E, yields

D = K*E = Doei(wt -6) (2.76)

K*Eoei't = Doei("t ) (2.77)

K* = (D0/Eo) e-i (2.78)

Now xs = D/Eo is the static dielectric constant which gives

K* = Kxse-i (2.79)

and using e-'i = cos6 isin6 yields

K' = Ks(cos6 isin6) (2.80)

For equation 2.80, the K' and the K" are

K' = KsCOSS (2.81)

K" = K5sinS (2.82)

One obtains for K* (the complex form of the dielectric

constant) using equations 2.81 and 2.82 the following

K* = K' iK" (2.83)

K* = E*/E0 (2.84)

K* = (1/E,) (E' ie") (2.85)

Using equations 2.83 and 2.85, the loss tangent is given by

tanS = K"/K' = E'/E' (2.86)

Losses arising from various forms of polarization are

better understood by considering the complex form of the

dielectric constant, given as

K* = E*/Eo = K' iK" (2.87)


6* = 6' iE" (2.88)

where E' is the dielectric permittivity (real or storage part)

and e" is the imaginary part (loss factor). In an electric

field there are some currents flowing (conductivity) and

losses will occur due to current flow. With the majority of

measuring techniques, it is difficult to separate the losses

due to conduction from those due to polarization, unless

variable frequencies are used. Therefore, all forms of losses

(conductivity and polarization) are grouped together and

described as an effective loss factor e"eff

6"eff = (E"e + ea + no + E"i) + e"c (2.89)
where e"c = a/O6o is the loss due to conduction. The losses

due to polarization are e"e (electronic), E"a (atomic), E"

(orientational) and e6" (ionic). Thus the complex dielectric

constant may be defined as

e* = 6o(6'r ier,eff) (2.90)

K* = K' iK"eff (2.91)

where 6o is the permittivity of free space (eo = 8.86 x 10-12

F/m) E (K') is the relative permittivity and i is an

imaginary number (i = [-1]0o5). The ratio of the effective

loss factor to relative permittivity is commonly called the

effective loss tangent and used to describe losses in a

material by

tan6eff = "r,eff/ r = K"eff/K' (2.92)

The effective loss tangent may also be given as

tan&eff = 6"r,eff/E'r (2.93)


tan6eff = a/2rfeo (2.94)

taneff = /oE (2.95)
where f is frequency and a (= oe" r,eff) is the total effective

conductivity caused by conduction and displacement currents.

The e"reff factor includes contributions from polarization as

well as conductivity (E"c = a/27rfeo)

Rate of Increase in Temperature

A material placed inside of a microwave cavity absorbs

microwave energy and its temperature increases. The rate of

the increase in the temperature of the material is dependent

on the power P (W) absorbed by the material, the specific heat

capacity cp (W-s/gm-"C) of the material at constant pressure

and the mass M (gm) of the heated material. From the relation

Q = McpAT, the heat equation, the power required to increase

the temperature of a mass of material from To ("C) to Tf (C)

in a time t (s) is

P = Q/t = (McpAT)/t (2.96)

where Q (W-s)is the heat generated and AT (= Tf To) is the

change in temperature. The power dissipated per unit volume

V (m3) may be defined as

P/V = 0 Ee"r,effl El 2 + (4o/ r,effl HI 2 (2.97)

P/V = alEl2 = oEE r,efflEl2 (2.98)

where E (V/m) is the magnitude of the electric field inside of

the material being heated (= Er) H (A/m) is the magnitude of

the magnetic field, w (= 2rf) is the angular frequency in Hz,

Ao (H/m) is the permeability of free space, V is the volume of


the material and p"r,eff is the relative effective magnetic loss

[Met90]. The relationship between the maximum electric field

Eo and the root mean square electric field Erm strengths are

(2Erm)0.5 = E0. In equation 2.98, it is assumed that the

materials magnetic contribution to the losses are negligible.

The rise in temperature as a result of microwave heating

(assuming no appreciable magnetic field) is found by

substituting equation 2.98 into 2.96 and solving for AT/t

AT/t = (e~or,eff El 2V/MCp (2.99)

AT/t = oeoer,eff El 2/pCp (2.100)

The density p (gm/m3) of a material is related to the

materials mass M (gm) and volume V (m3) by (p = M/V). This is

the bulk density of a sample pressed from a mixture of ceramic

powders or other phases (e.g., a polymeric binder) containing

some porosity. If no porosity is contained within the sample,

however, then this is the theoretical density.

In a multiphased system, the cp, E, e" (= E"reff) and the

p of each phase is different. Therefore, AT/t for a two

component system may be represented by the additions of both

of these terms

AT/t = EOE"11 E11 2/PCp,1 + E06"21 E21 2/P2Cp,2 (2.101)

AT/t = oEoE C"il Eil 2/Pic,i (2.102)

The E used in these equations is the internal field. These

relations, equations 2.101 and 2.102, may not be additive, but

all phases will contribute to AT/t.


Equation 2.97 shows that the power per unit volume

dissipated into a material is directly proportional to f and

tanS (or "reff) but varies with the square of the electric

field. These equations assume that power (P) is absorbed

uniformly throughout the heated material, which is not usually

the case. It should also be noted that f, E', E" rff tan

and E are all interdependent. Moreover, the electric field is

dependent on the size, geometry and location of the material

within the microwave cavity, as well as on the design and

volume of the cavity. Equation 2.97, however, gives a good

approximation of the power absorbed by a material. Moreover,

when mixtures of different phases are involved, interfacial

polarization may occur.

Varadan [Var88] has shown that as the bulk density of

materials heated by microwaves decreases (or porosity

increases), the heating rates are increased. For a material

system investigated by Varadan, this observation was true up

to about 50% porosity. However, as the material's porosity

increases significantly above 50%, the heating rate decreases.

Changes in the heating rate, due to changes in porosity, may

be related to changes in a materials dielectric properties.

Penetration Depth of Microwaves

Microwaves interact with a material and are reflected,

absorbed and/or transmitted. As electromagnetic energy

penetrates into the interior of a material, it is attenuated

to an extent dependent on the e"r,eff. Microwaves penetrating


into a material are dissipated, absorbed and attenuated. The

distance from the surface of the material at which the power

decays to 1/e of the value at the surface is called the

penetration depth (D) The power P at the penetration depth

d (= Dp) is related to an attenuation constant a by

P = Poe'2ad (2.103)

where Po is the incident power and e is equal to 2.718.

Rearranging the equation, setting P/Po equal to e-1, and

taking the natural logarithmic ln(x) yield

Dp = 1/2a (2.104)

Metaxas [Met83, Met90] has given the attenuation constant a as

a = O( ol/'e'reO/2)0.5 [(1 + (er,eff/E'r)2)0.5 1]0.5 (2.105)

There are two cases to apply a [Met90], and, in the first

case, one may consider a low loss medium, where 6"r,eff/Er <<

1. Using a Binomial Series for the term [(1 + ("reff/'.r)2)0.5

- 1]0.5 yields

[(1 + (E"r,eff/l'r)2)0.5 1]0.5 = r,eff/ 'r(2) 0.5 (2.106)

After substitution of o = 2rf = 2rc/Ao, p' = 1 and c = (oEo) -0.5

yields for Dp = 1/2a

Dp(l/e) = lo(E') )05/27E "reff (2.107)

where 10 is the wavelength in free space. From equation 2.107

one observes that the depth of penetration increases with

increasing wavelength or decreasing frequency. In general,

lower frequencies (or longer wavelengths) result in greater

penetration depths. But heating does not necessarily increase


since the internal field E may be lower, as it depends on the

materials' dielectric properties [Sut89].

There are no typical ceramics, but the room temperature

1/e penetration depth of a ceramic may range from microns for

superconductors, to centimeters (for SiC) to hundreds of

centimeters (for alumina).

In the second case [Met90], one may consider a highly

lossy medium, where E r,eff/ r >> 1 and the a for equation

2.105 reduces to

a = (p)(o'E'rEO2)0.5 (E"r,eff/E'r)0"5 (2.108)

a = (aojl/2) 05 (2.109)

The inverse of the attenuation constant a is defined as the

skin depth 6s, i.e.,

6s = 1/a (2.110)
which is the depth at which the magnitude of the energy decays

to 1/e of its value at the surface [Met83]. This is the case

for a conducting material (e.g., metal) and assuming t' = 1

gives for the skin depth

6,(1/e) = (2/soa)0.5 (2.111)

Skin depth decreases with increases in conductivities and the

frequency of the applied microwave field. The skin depth of

most metals, due to their high conductivity, is on the order

of a few microns at 2450 MHz. Most of the power is reflected

from the metallic surface with some power being dissipated to

induce a flow of currents.

Critical Temperature and Thermal Runaway

The loss tangent (tan6eff = "reff/Er') or effective loss

factor (e"r,eff) is used as a description of losses in a

material dissipated in the form of heat. Both the tan6eff and

e"reff are interrelated and affected by temperature. In low
loss materials, initially the tanSeff (=tan6) rises very slowly

with increasing temperature until a Tc is reached. Beyond the

Tc, the tanS increases rapidly. The Tc of a material, as shown

in Figure 2.3, is the point at which a material absorbs

microwave energy efficiently and the temperature of the

material increases rapidly [She90]. The TC is dependent on

dielectric properties, composition, porosity, microstructure,

density and other properties of the material system. The TC

is the point at which thermal runaway begins.

Metaxas [Met83] defined thermal runaway as the

uncontrolled temperature rise in a material heated by high

frequency (microwave) energy due to a positive rate of change

of the effective loss factor with temperature (dE"reff/dT).

Thermal runaway is a condition in a microwave heated material,

where the tan6 rises very rapidly once a Tc is reached. As

the tanS begins to rapidly increase, the material absorbs

microwaves more efficiently and the temperature rises rapidly

[Sut89]. The temperature rise and TC of materials vary

widely. Thermal runaway can be undesirable (cause hot spots

and uneven heating of a material) or be beneficial (heat

materials rapidly). In order to control thermal runaway, one

may pulse microwave levels or design the microwave applicator

to deposit microwave energy in a controlled fashion [Sut89].

Tailoring Systems to Reach a Material's T.

Additives or other microwave absorbing phases may be

added to a single component or multicomponent material to aid

in heating the low loss (high Tc) material above its Tc. Once

a low loss material is heated above its Tc, the material may

then couple with and absorb microwave energy on its own and be

heated to higher temperatures. In Figure 2.10, the loss

tangent versus temperature is shown for a low loss material

containing microwave absorbing additives and the same low loss

material without microwave absorbing additives. The Tc is

reached in shorter processing times and shifted to lower

temperatures for the low loss material, under identical

processing conditions.

A >99% high purity alumina is more difficult to heat than

a lower purity (<94%) alumina. This may be due to glassy

phases at the grain boundaries that interact with microwave

energy to heat the lower purity (<94%) alumina. In the higher

purity (>99%) alumina, these glassy phases are absent.

The addition of microwave absorbing (conductive or

magnetic) phases enables these low Tc materials to absorb and

couple with microwave energy. This energy is transferred (in

the form of heat) to the low loss (high Tc) matrix or

material. A variety of additives (sodium nitrate and

glycerol) have been added to alumina-silicon carbide whisker

10 I I I I------

9 /
U- (a)
) 7
0 /(b)




g 2

1 -
T / T
0 C / CI
0 200 400 600 800 1000 1200 1400 1600
Temperature (OC)

Figure 2.10 Loss tangent versus temperature for a (a) low
loss material containing microwave absorbing additives,
and (b) the low loss material without the microwave
absorbing additives.


and alumina-silicon nitride whisker composites and pyrex-

silicon carbide whisker composites [Mee86, Mee87b]. Other

additives used in materials include carbon, zirconium

oxynitrate solution (ZON), niobium, tantalum carbide, silicon

carbide, molybdenum disilicide, iron and copper [Mee87b].

Dielectric Mixture Models

Most ceramic materials of importance consist of mixtures

of particles of various shapes, sizes and compositions. In

order to understand polycrystalline or multiphased materials,

the effects of grain boundaries, porosity and phase mixture

should be considered. Even green ceramics consisting of

alumina, porosity and polymer are of importance in ceramic

processing applications. Thus, the dielectric properties of

the mixture and space charge polarization (which can result

from mixtures of components having different resistivity

characteristics) must be considered.

Electromagnetic waves interact with a variety of

microscopic boundary conditions, which affect local field

values. Several models have been developed to calculate the

localized microwave field strengths and resultant bulk

dielectric constants. Dielectric mixture rules or theory may

be used to determine these values.

Dielectric mixture models have been proposed by Tinga

[Tin88], Varadan et al. [Var88] and Meek [Mee87a]:


(1) Tinga proposed a multiphased dielectric mixture

theory to predict the internal fields in

multiphase ceramics,

(2) Varadan discussed a multiple scattering theory to

better understand the microwave sintering process,


(3) Meek used Maxwell's expression for mixtures

[Kin76] to apply to the sintering of a ceramic

system in a microwave field.

The most practical model appears to be Maxwell's

dielectric mixture model, as reviewed by Kingery [Kin76] and

Meek [Mee87a]. Several assumptions, however, appeared to have

been made by Kingery [Kin76] and Meek [Mee87a] for this model:

(1) The model assumes a two phased system, with one

phase being a ceramic material and the second phase

porosity [Mee87a] or another phase (e.g., ceramic,

binder) [Kin76],

(2) No new phases or materials are being formed,

(3) The applied electric field is uniform, and

(4) The material is a porous monolithic ceramic that is

homogeneous [Mee87a] or a multiphased ceramic

[Kin76] that is homogeneous.

Kingery [Kin76] considered a mixtures rule using two

ideal cases. In both cases it was assumed that layers of

materials were placed between two capacitor plates.


In the first case, when layers of material are placed

parallel to the capacitor plates, the structure corresponds to

capacitive elements in series, and the inverse capacitances

are additive, as is true for the inverse conductivities (a =

WK"eff = WoCE r,eff). Then

1/K" = v1/K", + v2/K"2 (2.112)

1/K' = V1/K'1 + V2/K'2 (2.113)

where v, and v2 are volume fractions of each phase and equal

to the relative plate thickness.

In the second case it is assumed that layers of a

material are arranged normal to the capacitor plates. When

the plate elements are arranged normal to the capacitor

plates, the applied field is similar for each of the elements

such that the capacitances are additive. Then

K" = V1K + v2K2 (2.114)

K' = V1K'1 + V2K'2 (2.115)

General equations yield

K" n= SVi(Ki)n (2.116)

where n = 1 for the normal case and n = -1 for the parallel

case. As n approaches zero, Kn equals 1 + nlogK and one

obtains equation 2.117, which is the so-called logarithmic

mixture rule [Kin76]. This equation, which may be used for K'

or K", gives values which are intermediate between the

extremes of equations 2.112 through 2.115.

logK = EivilogKi (2.117)


Considering a dispersion of spherical particles of

dielectric constant K'd in a matrix of dielectric constant K'm,

Maxwell derived a relationship for the mixtures:

K' = [VmK'm(2/3 + K'/3c'm) +

VdK'd]/[V,(2/3 + K'd/3K',) + Vd] (2.118)
With the volume fractions and the bulk dielectric constants of

both components, the bulk dielectric constant of the composite

material may be calculated. Maxwell's relation for the

mixtures may also be used for K". Using the four assumptions

mentioned earlier and Maxwell's dielectric mixture model,

these calculated values may be correlated with measured bulk

dielectric properties data. This assumes that all conditions

of the model are satisfied. Kingery [Kin76] has pointed out

that Maxwell's dielectric mixture model, equation 2.118, comes

very close to the logarithmic expression, equation 2.117, when

the dispersed phase has a higher dielectric constant than the

matrix material.

Meek [Mee87a] used this model for a sintering study,

i.e., for a material composed of a ceramic (alumina) and

porosity. Kingery [Kin76] discussed a high dielectric matrix

material (TiO2) mixed with another phase, being either

polystyrene, clay or ZrO2. Kingery discussed the following


(1) The geometrical configuration of the TiO2 particles

in the matrix is apparently controlling,


(2) For small differences in the dielectric constant of

both materials, equations 2.117 and 2.118 give similar


(3) For small amounts of porosity (10 to 15%), equations

2.114, 2.115 and 2.118 are appropriate for most circumstances,

(4) If there are great differences between the

dielectric constants of the phases, use equation 2.117.

Space charge or interfacial polarization may arise from

differences between the conductivity of the phases present in

the multiphased material. The resulting polarization due to

heterogeneity at high temperatures will appear in the form of

a high dielectric constant and cause a peak in the loss

tangent (resulting from a buildup of charges at the

interface). For the compositions, frequencies and

temperatures used in this study, interfacial and dipolar

polarization are expected to be important.

Microwave Heating

Microwave heating may be divided into two areas: pure

microwave heating (called microwave heating) and microwave

hybrid heating (MHH).

Pure microwave heating is accomplished by placing a

material into a microwave oven. The material is either placed

into the microwave oven alone or a microwave transparent

insulation is placed around the material. Microwave energy is

directly absorbed by the material, which results in energy

being dissipated and heating. A major problem of heating some


materials to high temperatures is that they absorb microwave

energy and are heated, but the heat is dissipated from the

surface of the material, which results in inefficient heating.

The purpose of a low loss insulation cavity is to allow heat

generated within the sample and dissipated from the surface of

the sample to be retained in close proximity to it and this

will allow the sample to be heated more uniformly to higher


The MHH is a combination of microwave and conventional

heating. A low loss insulation cavity containing a suscepting

material, such as silicon carbide, is used to enclose a sample

to be heated within a microwave oven. The purpose of the

insulation cavity (containing the suscepting material) is to

both retain heat produced by the sample and to aid in heating

the low loss material to its Tc. Once the Tc of the material

is reached, the material itself absorbs microwave energy

efficiently and heats of its own accord.

The insulation cavity may also be completely composed of

a suscepting material, such as graphite or SiC [Stu91b].

Several different insulation cavity configurations (generally

called susceptors) used for MHH have been used by Krage

[Kra81], Janney and Kimrey [Jan88], De [De90] and Holcombe


In order to increase the microwave heating efficiency,

the applied microwave power may be increased, magnetic or

conductive phases may be added to the material to


preferentially absorb microwave energy, or the microstructure

of the material may be altered (if possible).

Conventional Heating

Conventional heating is important when compared to

microwave hybrid and pure microwave heating. Heat transfer

may be broken down into three mechanisms: convection,

conduction and radiation. The following conventional heating

theory, with regard to ceramics and other materials, has been

discussed by Kingery [Kin76] and Incropera [Inc81].

Conventional heating in a dielectric ceramic is dependent

upon the material's thermal conductivity (k). The k (W/m-C)

is a material's constant or property that relates the rate of

heat flow to a resultant temperature gradient. A one-

dimensional definition of heat flux (q) or heat transfer by

conduction may be defined as

q = -k(dT/dx) (2.119)

where q (W/mZ) is the heat transfer in the x-direction per

unit area normal to the direction of transfer and proportional

to the temperature gradient dT/dx in this direction [Inc81].

The minus sign is due to the fact that heat is transferred in

the direction of decreasing temperature. This is also known

as Fourier's Law and it is a phenomenological law, i.e., it is

an observed phenomena.

The heat rate by conduction (Q) is the product of q and

area (A) and given by

Q = -kA(dT/dx)


where Q (W) is the amount of heat dq (W/m2) flowing normal to

an area A (m2) in time dt (sec). Both the heat flux and the

heat rate are proportional to a temperature gradient (dT/dx)

and to the thermal conductivity (k). The thermal

conductivity, at a given temperature, is only dependent on the

chemical and physical composition of the material being heated


Under steady-state conditions, equation 2.120 may be

integrated for a particular shape of interest to determine Q.

Under steady-state conditions, the Q and T at each point are

independent of time.

Generally, the heating of materials occurs under non-

ideal or unsteady-state conditions. Therefore small volume

elements in rectangular coordinates, dx, dy and dz, are

considered. Cylindrical or spherical coordinates may also be

used. The difference between the heat entering and leaving

along each axis must be equal to that stored. Neglecting

variations of k with T, and assuming that the material is

homogeneous and isotropic, one obtains for the one-dimensional

case at constant pressure

(k/pcp) (a2T/ax2) + q'/pCp = aT/at (2.121)

where q' (W/m3) is the rate at which energy is generated per

unit volume of the medium, p (gm/m3) is the density of the

material, and cp (W-sec/gm-C) is the specific heat capacity

of the material at constant pressure. The materials constant

(k/pcp) is called the thermal diffusivity (a) and has units of


m2/sec. The three-dimensional case for heat flow or heat

rate, at constant pressure, yields

aT T k ) T T T q
(cT ( ~ ) + (2.122)
at pcP ax2 ay2 az2 pep

The heat rate increases for a material as the k increases or

as pcp term decreases, which is comparable to a increasing.

The q' for a material heated with microwaves is the heat

generated within the sample, i.e., q' = co6EE",ff EitJ2

(assuming no contributions from a magnetic field). This

equation is difficult to solve and it is usually solved by

finite element analysis [Isk91].

Polymeric Binders

Within this section, binder properties, binder

characteristics, the classification of binders, binder

tacticity, and the thermal degradation of PMMA and diffusion

are discussed. The mixture of polymers and ceramic materials

will also be discussed as it relates to mass fraction (weight

percent) and volume fraction (volume percent) additions of

polymeric binder to a ceramic system. The effect of density

will be discussed to a small extent and a literature review of

binder removal using conventional and microwave heating (for

the PMMA/alumina system) is presented.

Binder Properties and Characteristics

Binders are used in industry to process a variety of

ceramics. The major functions of the binder are to

(1) provide plasticity to shape a ceramic,


(2) bond ceramic particles together before further

processing (e.g., sintering) for green strength, and

(3) be easily removed during firing.

Many binder systems, consisting of a single or a combination

of several binders, have been developed.

A good binder has the following properties [Ger91]

(1) allows formability of the ceramic,

(2) gives adequate strength to the ceramic,

(3) is inexpensive,

(4) is not toxic,

(5) leaves no carbon residue after firing,

(6) is stable during storage, and

(7) is soluble.

The green strength imparted to a ceramic body by a binder

is dependent on the amount and the distribution of the binder

within the green body. Ceramics need plasticity to be formed

into shapes and green strength to maintain the formed shapes.

The binder bonds ceramic particles to attain that purpose.

Binders may be removed by a thermal process at low

temperatures (<600C) and the shaped ceramic may be fired at

elevated temperatures (>7000C).

A theoretical study of some characteristics of organic-

ceramic interactions, used to strengthen ceramic bodies, was

presented by Onoda [Ono76]. This study indicated that three

classes of binder distribution exist in green compacts

(1) a nonwetting state,

(2) a wetting (pendular) state, and

(3) a coated state.

The pendular state produces the highest green strength.

Results show that important properties needed to produce the

highest green strength are binder amount, binder strength and

the particles packing density.

It has been found that the amount of binder used for most

ceramic forming applications has a ratio of binder volume to

particle volume of 0.08 to 0.15 [Ono78]. This ratio applies

to most ceramic forming processes, but excludes injection and

compression molding. This is the maximum amount of binder

that can be added without closing pore spaces in the body,

which could be detrimental to binder removal [Ono78].

Onoda [Ono78] also looked at the effect of theological

characteristics of the binder solution (including viscosity,

pseudoplasticity and gelation). Onoda reported that the

binder viscosity is one of the primary considerations needed

to select a binder for a specific process.

In general, removing binders from the green body is the

most important step in the processing of ceramics. Several

methods (other than binder burnout) have been developed and

used to remove organic binders. These binder removal methods

include capillary action, evaporation and solvent extraction.

However, these methods are costly and time-consuming and are

only used for low-molecular weight binders (such as oils or

paraffin waxes). Binder removed by thermal degradation or


binder burnout is the most commonly used method. Binder

removal by thermal degradation can require up to a hundred

hours, but is a relatively inexpensive method for binder


Classification of Binders

Organic polymers are important additives in ceramic

processing operations. Ceramic processing operations include

casting (slip or tape), injection molding, extrusion and

pressing (dry and wet). Organic polymers are used as binders,

lubricants, dispersants, plasticizers and for a variety of

other reasons. Nonpolymeric (inorganic) additives may also be

used for a few of these purposes. In general, organic

additives have a temporary or fugitive role in the processing

of ceramics. Before processing at higher temperatures or

sintering, the organic additives must be completely removed.

Incomplete removal of organic additives may have an adverse

effect on the densification behavior and final use properties

of the ceramic. Ceramics may even bloat or crack in some

severe cases.

Binders may fall under two categories: clay based and

nonclay based. Table 2.1 gives the classification of binder

materials, with examples of binders listed under each category


Clay binders are used to make pottery, mud dams, bricks

and other items. Clay may be mixed with water to impart

plasticity to form virtually any desired shape. This shape is

Table 2.1
Classification Of Binder Materials.


Organic Inorganic

Microcrystalline Cellulose Kaolin
Ball Clay


Organic Examples Inorganic Example

Natural Gums







Xanthan Gum,
Gum Arabic

Refined Starch

Paper Waste,

Na, NH4,

Soluble Silicates

Organic Silicates

Soluble Phosphates

Soluble Aluminates





Methyl Cellulose,
Hydroxyethyl Cellulose,
Sodium Carboxymethyl

Polyvinyl Alcohol

Polyvinyl Butyral

Acrylic Resins Polymethyl Methacrylate

Polyethylene Glycol

Paraffin, Wax Emulsions,
Microcrystalline Wax

(Adapted From [Ree88])




retained after an elapsed time which allows the formed part to

attain strength and rigidity. The strength of dried clay is

due to the bonding between platelets of clay with a high

surface area [Wil71]. Impure clays may not be used for all

purposes; therefore, other binders have been developed from

natural or synthetic materials.

Nonclay binders may fall into two categories, i.e.,

permanent and temporary binders.

A permanent binder may be composed of inorganic materials

which develop bonds due to physical structural changes. These

binders (e.g., polycarbosilanes) retain excellent bonding

properties, even after being further processed and fired.

These are often used in the refractory industry where nonclay

ceramics, such as alumina and silicon carbides, are used.

Temporary binders are organic compounds composed of

natural or synthetic materials. Natural materials include

rubbers and cellulose. Synthetic materials include plastics

and glues. Synthetic binders may be further divided into two

groups: thermoplastics and thermosets. A thermoplastic may

be heated, reshaped, and used again, whereas a thermoset can

not be reused in the same fashion.

The binding properties of temporary binders, which are

organic materials, may be attributed to their high molecular

weights, the interactions between their polar groups with

active sites on the ceramic surface, or a combination of

these. Examples of a thermoplastic binder include polymethyl


methacrylate or polyethylene. Examples of a thermoset plastic

include an epoxy or a phenolic resin.

Binder Tacticity

A vinyl polymer (CH2=CXY) may be stretched out such that

one or more carbon atoms in the backbone of each monomer is

joined to four different groups. This is a tertiary carbon

and it constitutes a center of asymmetry. The tacticity of

this vinyl polymer is defined by observing the configuration

at each pseudo-carbon site. If the backbone of PMMA is

stretched out to form a zig-zag, with all carbon atoms in the

same plane, then each COOCH3 (= X) may lie above or below the

plane (the Y group is CH3). When the COOCH3 groups all lie on

the same side of the plane, then the polymer is defined as

isotactic. When the COOCH3 groups are randomly located above

and below the plane, the polymer is defined as atactic. When

the COOCH3 groups alternate above and below the plane, the

polymer is defined as syndiotactic.

Thermal Decradation of Polymethyl Methacrylate (PMMA) and

The degradation of PMMA alone and the degradation of PMMA

binder in a ceramic (e.g., alumina) have been studied in great

detail [Sun88a, Sun88b] using conventional heating. When a

binder such as PMMA is heated, the binder changes or

transforms to lower molecular weight species, liquids and

gases. In a ceramic body these gases and liquids form paths

and are transported out of the body. The burnout or binder

degradation is affected by several variables which include the


polymer distribution, powder distribution, porosity,

permeability and packing density of the green body [Ree88].

During the degradation of PMMA from a ceramic body, a

counter-diffusion of gases exist. Oxygen is diffused inward

and the degradation gas products are diffused outward. In

order for the carbon residue products to be completely

removed, a chemical reaction must occur between the residue

and oxygen.

Important variables in the binder removal process include

the atmosphere of decomposition, temperature and heating

schedule. Catalytic reactions occurring on the surface of the

ceramic should also receive consideration, as reactions

between the binder decomposition products and the surface of

the ceramic may impede complete binder removal at lower

temperatures [Edi91].

A study on the diffusion of carbon residue from ceramic

compacts has been performed by Strijbos [Str73], and by Weisz

and Goodwin [Wei63]. An important result is that gas

transport plays an important role in the burnout process and

that an increase in the two-way gas diffusion may increase

binder burnout kinetics. However, the degradation and removal

of a polymeric binder from a ceramic/polymer system is

complicated due to many processing variables. These include

the type of heating (microwave and/or conventional) and

composite material properties (porosity, homogeneity,

permeability and packing density of the green body). Many of


these variables are changing during polymer degradation and

removal from the ceramic/polymer compact during the binder

burnout process. There has not been many written works or

articles published concerning the area of binder degradation

with regard to diffusion.

Results from Strijbos [Str73] were in agreement with an

earlier study performed by Weisz and Goodwin [Wei63]. Results

for carbon residue burnout in an air or oxygen environment


(1) at low temperatures the combustion process proceeds

uniformly and chemical reactions between carbon residue and

oxygen is the rate limiting factor,

(2) at sufficiently high temperatures, the process

proceeds from the outside toward the inside with a sharp

interface between the burnout and nonburnout zones, and the

rate of combustion is primarily controlled by oxygen diffusion

through the reaction zone,

(3) at an intermediate temperature, a transition state

exist where the chemical reaction competes with diffusion, and

(4) the characteristic time for total burnout is

determined by the initial carbon content, body thickness, gas

diffusivity and oxygen partial pressure.

Equations for burnout were derived by Strijbos [Str73]

for the burnout of carbon residue from a porous body and these

equations were based upon the following mass transport steps:


(1) chemical reactions occurring between carbon residue

and oxygen,

(2) transport of oxygen from the bulk gas to the

exterior surface of the body and reverse transport of product

gases, and

(3) diffusion of oxygen into a porous body and reverse

diffusion of the reaction products.

The degradation and diffusion of PMMA from a compact of

PMMA/alumina is a heterogeneous reaction because more than one

phase and several steps are involved, i.e., new phases are

formed, diffusion of reactants to or from a reaction site

occur, reactions at phase boundaries occur and diffusion of

products of reaction away from phase boundaries occur. The

rate of the reaction is limited by the slowest of these steps.

For the PMMA/alumina system, one may suspect that

microwave heating may lead to enhanced binder degradation and

removal due to potential advantages of uniform and volumetric

heating. Nevertheless, uniform and volumetric heating depend

on sample size, sample properties, microwave frequency and

other processing variables.

Diffusion occurs in solids, liquids and gases. In order

for diffusion to occur an energy barrier must be overcome

(i.e., an activation energy and/or a concentration gradient

must exist). The activation energy is usually supplied by a

thermal source (microwave and/or conventional heating) and

atoms or diffusing species move from regions of high


concentration to low concentration. The flux (J) of atom

movements along a concentration gradient (dC/dx) is given by

J (mol/cm2-s) = -D(dC/dx), where C (mol/cm3) is the

concentration, x (cm) is the distance and D (cm2/s) is the

diffusion coefficient. The dependence of D on temperature may

be given by D = Ae-Q/kT, where k is Boltzmann's constant (=

1.381 x 10-23 J/K), A (cm2/s) is a materials constant and T (K)

is the temperature. The D increases with increasing

temperature, as species become smaller and as the melting

temperature of compounds decrease. In general,

Dsolid Binder burnout may proceed in several stages, i.e., vapor

flow, liquid flow and diffusion. Strijbos [Str73] developed

a model for burning out carbon residue from a porous

cylindrical body and a simple equation was developed for the

length of the process. At temperatures below 660C, the

chemical reaction competes with diffusion throughout the

porous body. For temperatures $480C, the rate of the process

is controlled by the reaction rate. An equation used to

explain diffusion is De = K62/2Z(tt tL) and the diffusion

coefficient is represented as D = Do(T/To)n, where De (cm2/s) is

the effective diffusivity in residue free material, K

(mol/cm3) is the concentration of carbon at t=0, Z (mol/cm3)

is the oxygen concentration at x=0, tt (s) is the length of

the total process, tt (s) is the length of the first period of

the burning out process, 6 (cm) is the thickness of the porous

body, D (cm2/s) is the diffusion coefficient, Do (cm2/s) is the
diffusion coefficient at standard conditions, T ("C) is the
absolute temperature, To (C) is the standard temperature and
n (ranges from 1.75 to 2) is a constant for the pair of gases
The PMMA is formed from carbon, oxygen and hydrogen.
This organic binder is composed of many units which are
connected to each other by chemical bonds, and these units are
called monomers. The monomer unit of PMMA is shown in Figure

/ J "
N CH (s, 250C)

Figure 2.11 Monomer unit of PMMA.
Organic polymers degrade at much lower temperatures than
most metallic or ceramic materials. At higher temperatures,
the chemical bonds in the backbone of the polymer chain and
side group constituents are ruptured to form lower molecular
weight polymer species. These low molecular weight species
may be volatilized as is or ruptured into smaller species
before volatilization.

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